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HYDRAULICS 


BY 

F.   C.   LEA, 

B.Sa  (London  Engineering), 

8SNI0B  WmrWOBTH   8CH0LAB;    ASSOC.  B.  COL.  80.;     A.  M.INST.  0.  £. ; 

TELVORD   PBIZBMAM;    LBCTUBEB  IK  APPLIED   MECHANICS   AND   ENOINEEBINO   DESIGN 

m    THE  CITT  AND   GUILDS   OF  LONDON   CBNTBAL  TECHNICAL  COLLEGE. 


LONDON 
EDWARD   ARNOLD 
41   &  43,   MADDOX  STREET,   BOND  STREET,   W. 
1908 

[All  Rights  rexerrad] 


THE  NEW  YORK 

ASTOR,  LffNOX  AMD 
TIL^EN  FOuS'OATlONa. 


J 


XJSTHEN  the  author  undertook  fioma  time  ago  to  writ^  thi 
IT  worlE,  it  was  under  the  iiopressionj  which  irapression  wa 
shared  by  many  teachers,  that  a  book  was  required  by  Engineering 
etndents  dealing  with  the  subject  of  Hydraulics  in  a  wider  sense 
th^D  that  covered  by  existing  tejtt  books.  In  addition  tho  author 
'  '  '  ^  thoagh  several  excellent  text  books  were  in  existence, 
,r  amount  of  experimental  re&earch  carried  out  during  the 
last  10  or  lt5  years,  very  little  of  which  has  been  done  in  this 
eotantry,  on  the  subject  of  the  flow  of  water,  had  not  received  the 
atteBtiun  it  deserved.  The  great  developments  in  turbines  and 
ceotrifagal  pumps  also  mt^'ited  some  notice* 

An  attempt  has  been  made  to  embody  the  results  of  the  latest 
r€>9earcha^  in  the  book,  and  to  give  sufficient  details  to  indicate 
the  methods  used  in  obtaining  these  resuhs,  especially  in  those 
cables  where  such  information  and  the  references  thereto,  are 
hkely  to  prove  of  value  to  those  desirous  of  carrying  out  ex- 
perimeTits  on  the  flow  of  water. 

Perfiapa  in  no  branch  of  Applied  Science  is  it  more  difficult 
to  co-ordinate  results  and  express  them  by  general  formulae  than 
in  Hydraulics.  Practical  Engineers  engaged  in  the  design  of 
WEter  channels  frequently  complain  of  the  large  differences  they 
obtain  in  the  calculated  dimensions  of  such  channels  by  using 
thp   ff»rmnlae  put  forward  by  different  authorities.     Before  any 

can  be  used  with  assurance  it  is  necessary  to  have  son 
Mi-.i...4ge  of  the  data  used  hi  determining  the  empirical  con^ 
wmniM  in  the  formula.  For  this  reason  a  little  attention  Jms  been 
giv^n  to  the  historical  development  of  the  formulae  for  determining 
the  flow  in  pipes  and  channels,  and  some  particulars  of  the  data 
from  which  the  constants  were  determined  are  given.  In  thia 
respect  the  loganthmic  analysis  of  experimental  data,  especially 
in  Chapter  YI,  tngetlier  with  the  plottings  of  Fig.  114  and  the 
references  to  experiments,  will  it  is  hoped  be  of  assistance  to 


IV  PREFACE 

cngineerB  in  enabling  them  to  choose  the  coefficients  saitable  to 
given  circujiiatances,  and  it  is  further  hoped  that  the  methods  of 
analysis  given  will  be  educational  and  useful  to  studentSj  and 
h^jlpful  ill  the  interpretation  of  experiments. 

The  chapter  on  the  flow  of  w^ater  in  pipes  is  an*anged  so  that  a. 
student  who  reads  as  far  as  section  93  should  be  able  to  solve 
a  large  number  of  problems  on  flow  of  water  in  pipes,  without 
further  readings  At  the  end  of  the  chapter  the  formulae  derived 
in  the  chapter  are  Bummarisedj  and  various  kinds  of  practical 
problems  solved,  and  arithmetical  examples  worked  out.  In  the 
chapter  on  flow  in  channels  the  student  who  reads  to  section  119, 
and  then  sections  124  and  129  should  be  able  to  foHow  the 
problems  at  the  end  of  the  chapter,  and  to  work  the  examples. 
Chapter  YIII  enables  the  student  who  is  desirous  of  studying 
the  elementary  theory  of  the  impact  of  water  on  vanes,  and  of 
turbines,  to  do  so  apart  from  the  details  of  turbines,  and  the  more 
practical  problems  that  arise  in  connection  with  their  design. 

The  principles  of  construction  of  the  various  types  of  turbines 
are  illuBti-ated  in  Chapter  IX  by  diagrams  of  the  simpler  and 
older  types,  as  well  as  by  drawings  of  the  more  complicated 
modern  turbines.  The  dravdngs  have  been  made  to  scale,  and 
in  particular  cases  sufficient  dimensions  are  given  to  enable  the 
student  acquainted  with  the  principles  of  machine  design  to 
design  a  turbine.  The  author  believes  the  analysis  given  of  the 
form  of  the  vanes  for  mixed  flow  turbines  and  also  for  parallel 
flow  turbines  is  new. 

The  subject  of  centrifugal  pumps  is  treated  somewhat  fully, 
because  of  the  complaint  the  author  has  often  heard  of  the 
difficulty  engineers  and  students  have  in  determining  what  the 
performance  of  a  centrifugal  pump  is  likely  to  be  under  varying 
conditions*  The  method  of  analysis  of  the  losses  at  entrance  and 
exit  as  given  in  the  text,  the  author  believes,  is  due  to  Professor 
Unwin,  and  he  willingly  acknowledges  his  obligation  to  him, 
Tlie  general  formula  given  in  article  237  is  believed  to  be  new, 
and  the  examples  given  of  its  application  in  sections  235,  etc,, 
shuw  that  by  such  an  equation,  w^hich  may  be  called  the  character- 
istic equation  for  the  pump,  the  performance  of  the  pump  under 
varying  conditions  can  be  approximately  determined* 

The  effects  of  inertia  forces  in  plunger  pumps  and  the  effect  of 
air  vessels  in  diminishing  these  forces  are  only  imperfectly  treated, 
as  no  attempt  is  made  to  deal  with  the  variations  of  pressure  in 
the  air  vessel  Sufficient  attention  is  however  given  to  the 
snbject  to  emphasise  the  importance  of  it,  and  it  is  probably 
tr^ted  as   fully  as    is    desirable,   considered  from    a  practical 


PREFACE  V 

engineeriiig  standpoint.  The  analysis  of  section  260,  although 
too  refined  for  practical  purposes,  is  of  vahitj  to  the  student  in 
ihat,  neglecting  losses  which  cannot  very  well  be  determined,  it 
enables  htm  to  realise  how  the  energy  given  as  velocity  head  to 
the  vrat*;r  both  in  the  cylinder  and  in  the  suction  pipe  is  recovered 
before  the  end  of  the  stroke  is  reached.  The  examples  given  of 
**  Hydraulic  Machines  "  have  been  chosen  as  types*  and  no  attempt 
himB  been  made  to  introduce  veiy  special  kinds  of  machines*  Tlie 
antbor  has  had  a  wide  experience  of  this  cla^  of  machinery^  and 
lie  think«  the  examples  illustrate  sufficiently  the  principles  and 
prBCtice  of  the  design  of  such  machines. 

The  last  two  chapters  have  been  introduced  in  the  hope  that 
ibrnf  will  be  of  assistance  to  Umversity  studentii^  and  to  candidates 
fofT  the  Institution  of  Civil  Engineers  examinations* 

if r  Fronde's  experiments,  on  the  frictional  resistance  of  boards 

\mg  through  water,  are  considered  in  Chapter  XII  simply  in 
I    ir  relationship  to  the  resistance  of  ships^  and  no  attempt  has 

n  made,  as  is  frequently  done,  to  use  them  to  determine  so- 
ciiilJed  taw*s  of  fluid  friction  for  water  flowing  in  pipes  and 
cbanneb. 

The  author  harfUy  dares  to  hope  that  in  the  large  amount  of 
arithmetical  work  involved  in  the  exercises  given,  mistakes  will 
liMt  have  crept  in^  and  he  ^vill  be  grateful  if  those  discovering 
miMtake<t  will  kindly  point  them  out. 

The  author  wishes  to  express  his  sincereat  thanks  to  his 
friend,  Mr  W.  A.  Taylor,  Wh,Sc.,  A.K.CS.,  for  his  kindness  in 
reading  proofs,  and  for  many  valuable  suggestions,  and  also  to 
Mr  W,  Hewsunj  B.8c.,  who  has  kindly  read  through  some  of  the 
proofs. 

Ty  the  following  firms  the  author  is  ander  great  obligation  for 
the  ready  way  in  which  they  acceded  to  Ins  request  for  information; 

Messrs  Kscher,  Wj^ss  and  Co,  of  Zurich  for  drawings  of 
torbtnes  and  for  loan  of  block  of  turbine  filteri 

Messrs  Piccai*d,  IHctet  and  Co.  of  Geneva  for  di^wings  of 
turbinas* 

Messrs  Worthington  and  Co*  for  drawings  of  centrifugal 
pctm^j^^  and  fur  loan  of  block. 

Meagre  fielding  and  Piatt  of  Gloucester  for  drawings  of 
BceixmvlBtoT. 

Messrs  Tangye  of  Birmingham  for  drawings  of  pumps. 

&li*^rs  Glen  Held  and  Kennedy  of  Kilmarnock  for  drawijigs  of 
tneter  and  for  loan  of  blocks. 

Messrs  G,  W.  Kent  of  London  for  description  and  loan  of 
blockjt  of  Vt*niuri  meter  recording  gear* 


1 


VI  PREFACE 

Messrs  W.  and  L.  E.  Gurley  of  Troy,  N.Y.,  U.S.A.  for  loan 
of  block  of  current  meter. 

Messrs  Holden  and  Brooke  of  Manchester  for  drawing  of 
Leinert  meter. 

Messrs  W.  H.  Bailey  and  Co.  of  Manchester  for  drawing  of 
hydraulic  ram. 

Messrs  Armstrong,  Whitworth  and  Co.  for  drawings  of  crane 
valves. 

Messrs  Davy  of  Sheffield  for  loan  of  block  of  forging  press. 


F.  C.  LEA. 


Gbntral  Tbchnical  GOLLSaE, 
November,  1907. 


CONTENTS. 
CHAPTER  I. 

FLUIDS   AT   REST. 

Introdnction.  Fluids  and  their  properties.  Compressible  and  incom- 
pressible fluids.  Density  and  specific  gravity.  Hydrostatics.  Intensity 
of  pressure.  The  pressure  at  a  point  in  a  fluid  is  the  same  in  aU  directions. 
The  pressure  on  any  horizontal  plane  in  a  fluid  must  be  constant.  Fluids 
at  rest  with  free  surface  horizontal.  Pressure  measured  in  feet  of  water. 
Pressure  head.  Piezometer  tubes.  The  barometer.  The  differential  gauge. 
Transmission  of  fluid  pressure.  Total  or  whole  pressure.  Centre  of 
pressure.    Diagram  of  pressure  on  a  plane  area.    Examples     .    Page  1 


CHAPTER   II. 

FLOATING   BODIES. 

Conditions  of  equilibrium.  Principle  of  Archimedes.  Centre  of 
buoyancy.  Condition  of  stability  of  equilibrium.  Small  displacements. 
Metacentre.  Stability  of  rectangular  pontoon.  Stability  of  floating  vessel 
containing  water.  Stability  of  floating  body  wholly  immersed  in  water. 
Floating  docks.     Stability  of  floating  dock.    Examples        .        .    Page  21 


CHAPTER  III. 

FLUIDS  IN   MOTION. 

i^teady  motion.  Stream  line  motion.  Definitions  relating  to  flow  of 
^*ter.  Energy  per  pound  of  water  passing  any  section  in  a  stream  line. 
^niottilli's  theorem.  Venturi  meter.  Steering  of  canal  boats.  Extension 
^  Bernouilli^s  theorem.    Examples Page  37 


Vlll 


CONTENTS 


CHAPTER  IV. 

FLOW   OF   WATER  THROUGH    OEIFICES  AND  OVER   WEIRS. 

Velocity  of  clischargc?  frgni  aii  oriUce.  Coefficient  of  contraction  for 
sharp -edged  orifice.  Coefficient  of  Telocity  for  ehaj^- edged  orifice.  Bazin'st 
experiments  on  a  sharp -edged  orifice.  Distribution  of  Telocity  in  tlie  plane 
of  the  orifice.  Pressure  in  the  plajie  of  tlie  orifice.  Coefficient  of  discharge 
Efi^ect  of  fjiipprcBBed  con  traction  on  the  coefficient  of  discharge.  The  form 
oi  the  jet  from  sharp-edged  orifices.  Large  orifices.  Drowned  orifices. 
Partially  drowned  orifice.  Velocity  of  approadx.  Coefficient  of  resistance, 
Sndden  enlargement  of  a  current  of  water.  Sudden  contraction  of  a 
current  of  water.  Loss  of  head  due  to  sharp* edged  entrance  into  a  pipe  or 
mouthpiece.  MouUipieceB.  Borda*8  mouthpiece.  Conical  mouthpieces 
and  nozzleu.  Flow  througli  orifices  and  mouth  pieces  under  constant 
pressure.  Time  of  emptying  a  tank  or  reservoir.  Notches  and  weirs. 
Bectangulai*  sharp -edged  weir.  Derivation  of  the  weir  formula  from  that 
of  a  large  orifice*  Thomson'^  principle  of  similarity*  Discliarge  tlirongh 
a  trianglar  notch  by  the  principle  of  similarity.  Diaeharge  through  a 
rectangular  weir  by  the  principle  of  similarity.  Rectangular  w^eir  with 
end  contractions,  Bazin's  formula  for  the  discharge  of  a  wx*ir.  Bazin*s 
and  tlio  CorneO  experiments  on  weirs.  Velocity  of  approach.  Influence  ei 
the  height  of  the  weir  sill  above  the  beil  of  tlie  stream  on  the  contractioii, 
Disdiarge  of  a  weir  when  ttie  air  is  not  fi*eely  admitted  beneath  the  nappe. 
Form  of  the  nappe.  Depressed  nappe.  Adhering  nappes.  Drovnied  or 
wetted  nappes.  Instability  of  the  form  of  the  nappe.  Drowned  weirs  with 
sharp  crests*  Yei'tical  weirs  of  small  thickness.  Depressed  and  wetted 
nappes  for  flat-crested  weirs.  Drowned  nappes  for  flat-crested  weirs.  Wide 
flat-created  weirs.  Flow  over  dams.  Form  of  weir  for  accurate  gau 
Boussinesq'a  theory  of  the  discharge  over  a  weir*  Determining  by 
proximation  the  discliarge  of  a  weir,  when  the  velocity  of  approach  1?" 
unknown.  Time  required  to  lower  the  water  in  a  reservoir  a  given  dbtanoe 
by  means  of  a  weir.     Examples     .......    Page  60 


CHAPTER  V. 


FLOW    THROUGH    PIPEK. 


Resistances  to  the  motion  of  a  fluid  in  a  pii>e.  Loss  of  head  by  friction* 
Head  lost  at  the  entrance  to  tlie  pipe.  Hydrauhc  gradient  and  virtual 
slope*  Determination  of  the  loss  of  head  due  to  friction.  Reynold'a 
apparatus.  Equation  of  flow  in  a  pipe  of  uniform  diameter  and  determi- 
nation of  the  head  lost  due  to  friction.     Hydrauhc  mean  dex>th.     Empiriod 


CONTENTS 


IX 


LtUae  for  lo^  at  head  due  to  friction,  Formola  of  Darcy,  Variatioa 
ol  C  m  the  formula  v  —  Cs^mi  with  service,  Gangtdllet  and  Kutter*s 
foo^itilA.  BeynoUrsi  experiroents  and  the  logarithmic  formula.  Critical 
f«locit>%  Critical  yelocity  by  the  method  of  colour  baade.  Law  of 
frictional  resistance  for  velocities  above  tlie  critical  velocity*  The  de* 
terminAlidii  ol  the  values  of  C  given  in  Table  XII.     Variation  of  k,  in  the 

lulft  i  =  It*,  with  Hm  diameter,  Oriiieifiin  of  experiments.  Piezometer 
fittings.  Effect  of  temperature  on  the  velocity  of  flow,  Lo«s  of  head  due 
to  bands  and  elbows.  Variations  of  the  velocity  at  the  cross  section  of  a 
liadrical  pipe.  Head  necessary  to  give  the  meaa  velocity  v^  to  the 
in  Uie  pipe.  Practical  problcniB.  Velocity  of  flow  in  pipeB,  Trans- 
snteioil  of  power  along  pipes  by  hydraulic  pressure.  The  hmiting  diameter 
«l  cwai  iitm  pipes.  Pressures  on  pii>e  bends.  Pressure  on  a  plate  in  a  pipe 
filled  With  flowing  water.     Pressure  on  a  cylinder.     Examples  ,    Page  112 


^Bfosmi 

^■fittJl! 


CHAPTER  VI. 


FLOW   IN   OPEN   CHANNELS. 

Vamly  of  the  forma  of  channels.    Steady  motion  in  uniform  channeli* 

^  for  the  flow  when  the  motion  is  uniform  in  a  channel  of  uniform 

land  slope.     Formula  of  Chezy,     Fonnulae  of  Prony  and  Eytelwein. 

i1a  €d  Darcy  and  Bajdn,     GangniUet  and  K  utter *8  formula.     Bazin's 

bi.    Variations  of  the  coefficient  C.     Logarithmic  formula  for  flow  In 

Approximate  formula  for  the  flow  in  earth  channels.     Diwtribu- 

L  of  velocity  in  tiie  cross  section  of  open  ohannek,      Fomi  of  the  curve 

I  velocitiets  on  a  vertical  section.    The  dopes  of  channels  and  the  velocities 

rallown]  in  them.     Sections  of  a(|uo<liictft  and  sewerH.     Siphons  forming 

pan  of  a^edncte.    The  best  form  of  channel.     Depth  of  flow  in  a  circular 

diasne]  for  maximum  velocity  and  maximum  chscharge.     Curves  of  velocity 

ttud  discharge  for  a  channel.     Apphcations  of  the  formulae.     Problems. 

Dpleci ,...-,   Page  178 


CHAPTER   VIL 

QAtrOiNG   THE   FLOW   OF  WATER 

[  the  flow  of  water  by  weighing.  Meters.  Measuring  the  flow 
of  an  orifice.  JleaJHiiring  the  flow  in  oi^en  channels.  Surface 
Doable  floats.  Bod  floats).  The  current  meter,  Pitot  tube.  Cali^ 
bvmtiim  of  Pitot  tubes.  Gauging  by  a  weir.  The  hook  gauge.  Oaujring 
|h#  (low  in  pipes ;  Venturi  meter.  Deacon 'u  waste -water  meter,  Kennedy's 
Oanging  the  flow  of  streams  by  chemical  means.     Examples 


Page  2H 


CONTENTS 


CHAPTER  YllL 

IMPACT   OF   WATER   OK   VAlflTS. 

Deflnitioii  of  ?ector.  Sum  of  two  vectors.  Resultant  of  two  Telocitiee, 
Difference  of  two  sectors.  Impulse  of  water  on  vanes*  Relative  velocity* 
Befinition  of  relative  velocity  as  a  vector.  To  find  the  presstire  on  a 
moving  vane,  and  the  rate  of  doing  work.  Impact  of  water  on  a  vane 
when  the  directions  of  motion  of  the  vane  and  jet  are  not  paralleL 
Conditions  which  the  vanes  of  hydra viltc  machines  should  fiatisfy* 
De&nition  of  angular  m  omen  torn.  Change  of  angnl&r  moment  nm.  Two 
important  principles.  Work  tlone  on  a  series  of  vanea  filled  to  a  wheel 
expressed  in  terms  of  the  velocities  of  whirl  of  the  water  entering  and 
leaving  the  wheel.  Carved  vanes.  Pel  ton  whecL  Force  tending  to  move 
a  vessel  from  which  water  is  issuing  through  an  orifice.  The  propulsion 
of  ships  by  water  jets.     Examples    .         ,        «        .        >        .        Pa^fe  261 


CHAPTEE   IX- 


WATER    WHEELS  AND  TURBINES. 

Overehot  water  wheeb,     Brea4^t  wheel     Sagehien  wheels.     Impulse 

wheelB.    Poncelet  wheel.    Turbines.    Reaction  turbines.    Outward  flow 

turbines.    Losses  of  head  due  to  frictional  and  other  resistances  in  outward 

flow  turbines.     Some  actual  outward  flow  turbines.     Inward  flow  turbiueo. 

Some   actual  inwarrl   flow   turbines.      The  best  peripheral  velocity  for 

inward  and  outward  flow  ttirbines,     Exi^erimental  determination  of  the 

beat  peripheral  vekjcity  for  inward  and  outward  flow  turbines.     Value  of  e 

Yv 
to  be  used  in  the  formula  —  =^H.    The  ratio  of  the  velocity  of  wliirl  Y  to 

the  velocity  of  the  inlet  periphery  t\  The  velocity  with  which  water 
leaves  a  turbine.  Bernouilli's  equations  for  inward  and  outward  flow 
turbinei  neglecting  friction.  Bemouilli*s  eixuations  for  the  inward  and 
outward  flow  turbinea  including  friction.  Turbine  to  develope  a  given 
horse -power.  Parallel  or  axial  flow  turbinea.  Regulation  of  the  flow  to 
parallei  flow  turbines,  Bernouilli's  equations  for  axial  flow  turbii 
Mixed  flow  turbinea.  Cone  turbine.  Effect  of  changing  the  direction 
the  guide  blade >  when  altering  the  flow  of  inward  flow  and  mixed  flow 
turbines.  Effect  of  diminishing  the  flow  thi^ough  turbinea  on  the  velocity 
of  exit.  Kegulatiou  of  the  flow  by  means  ^  cylindrical  gates.  The  Bwain 
ptte.  The  form  of  the  wheel  vanes  between  the  inlet  and  outlet  of 
turbines*  The  limiting  head  for  a  single  stage  reaction  tnrbine.  Series 
or  multiple  stage  reaction  turbines.  Impulse  turbines.  The  form  of  tlie 
vanes  for  impulse  turbines,  neglecting  ifriction.  Triangles  of  velocity  for 
an  axial  flow  impulse  turbine  considering  friction.  Impulse  turbine  lor 
Mgh  head,  Pelton  wheel.  Oil  pressure  governor  or  regulator.  Water 
pressure  regulators  for  impulse  turbines.  Hammer  blow  in  a  long  turbine 
supply  pipe.     Examples     *,,.,,.<        Page  283 


J 


CONTENTS 


CHAPTER  X, 


PUMPS. 


Cesitrifiig&l  &nd  tor&me  pimips.  Starting  centrifugal  or  torbme  pnmps. 
I  of  the  v^nes  of  centrifugal  pumps.  Work  done  on  the  water  by  the 
^ifbeel*  lUMo  ol  velocity  of  whirl  to  peripheral  velocity.  The  kinetic  energy 
of  ihm  water  &t  exit  from  the  wbeeL  Gross  lift  of  a  centrifugal  punip^ 
£fficsietiides  of  a  centrifugEl  pump.  Experimental  deteimlnatioti  of  the 
dficieocy  of  a  centrifiigal  pump.  Desi^  of  pump  to  give  a  discharge  Q, 
Tlie  centiiftigal  head  impressed  on  the  water  by  the  wheel.  Head -velocity 
citiiFe  ef  &  centrifugal  pump  at  zero  discbarge.  Variation  of  the  discbarge 
of  &  oenfcrifngal  pomp  with  tlio  head  when  the  speed  ia  kept  constant* 
BemomUi's  equations  applied  to  centrifugal  pumps.  Ltoeses  in  centrifugal 
Variation  of  the  bead  with  diHcbarge  and  with  the  speed  of  a 
I  pump.  The  effect  of  the  variation  of  the  centrifugal  head  and 
the  low  hf  friction  on  the  discharge  of  a  pump.  The  c^ect  of  the  diminu- 
ol  ibe  oentiifugal  head  and  tlie  increase  of  the  friction  head  as  the 
ft,  (m  the  velocity.    Di^targe  curve  at  constant  head»    Special 

ftnmiigeniOBta  for  converting  tbe  velocity  bead  —  ,  with  which  the  water 

laftves  ttie  wheel.,  into  pressure  bead.  Tm-bino  pumps.  Losses  in  tbe 
flpltml  aiam^  of  centrifugal  pumps*  General  equation  for  a  oentrifngal 
|»r^|i-  The  Tjpiit.ing  height  to  which  a  single  wheel  centrifugal  pump  caa  ] 
lie  laed  to  raiae  water.  The  suctioa  of  a  centrifugal  pump.  Series  or 
mulsi-ti&age  ttirfaine  pumps^  Advanta^^es  of  ceotrifngal  pumps.  Pump 
«Urfwiig  into  a  long  pipe  line.  Parallel  flow  turbine  pump.  Inward  flow 
tBrMoe  pnmp*  Reciprocating  pumps-  Coeflicient  of  discharge  of  the 
{mmp.  Slip^  Diagram  of  work  done  by  tbe  pump*  The  accelerations 
ol  the  pomp  plunger  and  tbe  water  in  tlie  suction  pipe.  The  effect  of  ' 
oioeelonition  of  tbe  plonger  on  tbe  pressure  in  Uie  cylinder  during  the 
sncttoii  steoike.  Accelerating  forces  in  the  delivery  pipe.  Variation  of 
IHiiiiiirr  in  tbe  eylinder  due  to  friction.  Air  vessel  on  the  suction  pipe. 
Air  TCBBcl  on  the  delivery  pipe.  Separation  daring  the  suction  stroke. 
K^sliTe  slip,  Heimration  in  tbe  delivery  pipe,  Diagram  of  work  done  i 
ooiyddering  tbe  variable  qnantity  of  water  in  the  cylinder.  Head  lost  at 
tbe  ww^&OMk  talve.  Variation  of  tbe  pressure  in  bydrauhc  motors  due  to 
iaciriia  incces.  Worked  examples.  High  pressure  plunger  pump,  Tangye 
Oeplex  fiiisi|i.  Tbe  hydraulic  ram.  Lifting  water  by  comprised  air. 
Kxaifitiipa Page  392 


CHAPTER  XI, 

HYDRAULIC    MACHI!?ES. 

Jomta  tmd  packings  used  in  hydraulic  work.  The  acciimulator.  Dif- 
feffpiiftl  oocamnk.tor.  Air  accumulator.  Intenaifiers.  Steam  intensiiiera., 
HydeMiUo  forging  preis.  Hydraulic  cranes.  Double  power  crane 
Bjdrsnlie  crane  valveSi  Hydraubc  press.  Hydraulic  riveter.  Brother* 
hood  Odd  Bigg  hydraulic  engines.    Examples     .        ,        .        .    Page  4B^ 


Xll  CONTENTS 

CHAPTER  XII. 

RESISTANCE  TO  THE  MOTION  OF  BODIES  IN   WATER. 

Froude*8  experiments  on  the  resistance  of  thin  boards.  Stream  line 
theory  of  the  resistance  offered  to  motion  of  bodies  in  water.  Determination 
of  the  resistance  of  a  ship  from  that  of  the  model.    Examples  .    Page  507 

CHAPTER  XIII. 

STREAM  LINE  MOTION. 

Hele  Shaw's  experiments.  Curved  stream  line  motion.  Scouring  of 
river  banks  at  bends    ...•••...    Page  517 

Answers  to  Examples •        .    Page  521 

Index  Page  525 


HYDEAULICS. 


CHAPTER   I. 


FLUIDS  AT   REST. 


1.    IntrodnctioiL 

The  science  of  Hydraulics  in  its  limited  sense  as  originally 
understood^  had  for  its  object  the  consideration  of  the  laws 
regulating  the  flow  of  water  in  channels,  but  it  has  come  to 
have  a  wider  significance,  and  it  now  embraces,  in  addition,  the 
study  of  the  principles  involved  in  the  pumping  of  water  and  other 
fluids  and  their  application  to  the  working  of  different  kinds  of 
machines. 

The  practice  of  conveying  water  along  artificially  constructed 
channels  for  irrigation  and  domestic  purposes  dates  back  into 
it  antiquity.     The  Egyptians  constructed  transit  canals  for 
purposes,  as  early  as  3000  B.C.,  and  works  for  the  better 
of  the  waters  of  the  Nile  were  carried  out  at  an  even 
According  to  Josephus,  the   gardens  of   Solomon 
|utiful  by  fountains  and  other  water  works.     Tlie 
some  of  which  were  constructed  more  than 
among  the  "  wonders  of  the  world,"  and 
\ Athens  is  partially  supplied  with  water  by 
bt  constructed  probably  some  centuries  before 


ydraulics,  however,  may  be  said  to  have  oi 

ce  at  the  end  of  the  seventeenth  century  wh 

osophers  was  drawn  to  the  problems  involv 

of  the  fountains,  which  came  into  considerable 

idscape   gardens,  and  which,  according  t 

great  beauty  and  refreshment."     The  fou' 

Torricelli  and  Marriott  from  the  exper' 

from  flie  theoretical,  side.    TThe  experip 

of  Marriott  to  determine  the  discharge  -^  ^  ™* 

fees  in  the  sides  of  tanks  and  through  shr    ...    , 

°         /  cubic  inches. 


L.  B. 


The  Aqueducts  of  Rome.    FroDtinus,  translajf  85*9  cubic  feet. 

1—2 


2  HYDRAUUGS 

mark  the  first  attempts  to  determine  the  laws  regulating  the 
flow  of  water,  and  Torricelli's  famous  theorem  may  be  said  to 
be  the  foundation  of  modem  Hydraulics.  But,  as  shown  in  the 
chapter  on  the  flow  of  water  in  pipes,  it  was  not  until  a  century 
later  that  any  serious  attempt  was  made  to  give  expression  to  the 
laws  regulating  the  flow  in  long  pipes  and  channels,  and  firacti- 
cally  the  whole  of  the  knowledge  we  now  possess  has  been 
acquired  during  the  last  century.  Simple  machines  for  the 
utilisation  of  the  power  of  natural  streams  have  been  made  for 
many  centuries,  examples  of  which  are  to  be  found  in  an  interest- 
ing work  Hydrostatika  and  Hydrauliks  written  in  English  by 
Stephen  Swetzer  in  1729,  but  it  has  been  reserved  to  the  workers 
of  the  nineteenth  century  to  develop(»  all  kinds  of  hydraulic 
machinery,  and  to  discover  the  principles  involved  in  their  correct 
design.  Poncelet's  enunciation  of  the  correct  principles  which 
should  regulate  the  design  of  the  "floats"  or  buckets  of  wafer 
wheels,  and  Foumeyron's  application  of  the  triangle  of  velocities 
to  the  design  of  turbines,  marked  a  distinct  advance,  but  it  must 
be  admitted  that  the  enormous  development  of  this  class  of 
macliinery,  and  the  very  high  standard  of  efficiency  obtained,  is 
the  outcome,  not  of  theoretical  deductions,  but  of  experience, 
and  the  careful,  scientific  interpretation  of  the  results  of 
experiments. 

2.    Fluids  and  their  properties. 

The  name  fluid  is  given,  in  general,  to  a  body  which  offers 
very  small  resistance  to  deformation,  and  which  takes  the  shape 
of  the  body  with  which  it  is  in  contact. 

If  a  solid  body  rests  upon  a  horizontal  plane,  a  force  is  required 
to  move  the  body  over  the  plane,  or  to  overcome  the  friction 
between  the  body  and  the  plane.  If  the  plane  is  very  smooth 
the  force  may  be  veiy  small,  and  if  we  conceive  the  plane  to  be 
perfectly  smooth  the  smallest  imaginable  force  would  move  the 
body. 

If  in  a  fluid,  a  horizontal  plane  be  imagined  separating  the 

'^lid  into  two  parts,   the  force  necessary  to  cause  the  upper 

'   slide  over  the  lower  will   be  very  small  indeed,  and 

however  small,  applied  to  the  fluid  above  the  plane 

'  to  it,  Avill  cause  motion,  or  in  other  words  will  cause 

of  the  fluid. 

a  very  thin  plate  be  immersed  in  the  fluid  in  any 

te  can  be  made  to  separate  the  fluid  into  two 

'^tion  to  the  plate  of  an  infinitesimal  force, 

!t  fluid  this  force  would  be  zero. 


FLUIM  AT  REST  3 

luide  found  in  nature  are  not  perfect  and  are 

cosity-  but  when  they  are  at  rest  the  conditions 

equilibritiin  can  be  obtained,  with   sufficient   accuracy,  on 

aasumption  that  they  are  perfect  fluids,  and  that  therefore 

^tangential  stresses  can   exist  along  any   plane  in  a    fluid. 

Its  branch  of  the  study  of  fluids  is  called  Hydrostatics;  when 

laws  of  movement  of  fluids  are  consideredj  as  in  Hydraulics, 

5)8    tangential,   or    firictional    forces    have    to  be  taken    into 

ideration. 


1 


Compressible  and  inooDELpresslble  fluidB. 

There  are  two  kinds  of  fluids,  gases  and  liquids,  or  those  which 
are  easily  compressed,  and  those  which  are  compressed  with 
*  ""     *  V,    The  amount  by  which  the  volumes  of  the  latter  are 
ftir  a  very  large  variation  in  the  pressure  is  so  small  that 

practical  problems  this  variation  is  entirely  neglected,  and 
bey  are  therefore  considered  as  incompressible  fluids. 

In  this  volume  only  incorapreasible  fluids  are  considered,  and       . 
Attention  is  eonflned,  almost  entirely,  to  the  one  fluid,  water.  ^H 

4*    Density  and  specific  gravity. 

The  deusity  uf  any  substance  is  the  weight  of  unit  volume  at 
he  standard  temperature  and  pressure. 

f  specific  gravity  of  any  substance  at  any  temperature  and 
Bnpe  id  the  ratio  of  the  weight  of  unit  volume  to  the  weight 
tmit  volume  of  pure  water  at  the  standard  temperature  and 
ttxre. 

The  variation  of  the  volume  of  liquid  fluids,  with  the  pressure, 
as  eluted  above,  is  negligible,  and  the  variation  due  to  changes  of 
tempfn^lnrc,  such  as  are  ordinarily  met  with,  is  so  small,  that  in 
practical  problems  it  is  unnecessary  to  take  it  into  account. 

In  tht?  case  of  water,  the  presence  of  salta  in  solution  is  of 
,ter  importance  in  determining  the  density  than  variation 
temperature,  as  will  be  seen  by  comparing  the  densities  of  sea 
Vator  and  pure  water  given  in  the  foUowing  table. 


■f   t€ 


TABLE  I. 
Useful  data. 

One  cnbio  foot  of  water  at  391'  F.  weighs  62-425  lbs. 

60*  F.  „       62-86     „ 

One  oablo  foot  of  average  sea  water  at  60°  F.  weighs  64  Ibe* 
0De  gallon  of  pure  water  at  60*  F,  weighs  10  lbs. 
One  gallon  of  ptue  water  bas  a  volome  of  277*25  cubic  mches* 
One  kai  ol  pure  water  at  60"  F.  has  a  volume  of  85*9  cubic  feet. 

1—2 


HTDRA.ULICB 
Table  of  dendtiea  of  pure  water. 


Temperatnn 

^rees  Fahxenheii 

Demdty 

82 

-09967 

89*1 

1-000000 

M) 

OiW978 

60 

0-99906 

80 

0-99664 

104 

0il9288 

From  the  above  it  will  be  seen  that  in  practical  {irobleiiis  it 
will  be  KufHciently  near  to  take  the  weight  of  one  cubic  foot  of 
fresh  water  as  62*4  lbs.,  one  gallon  as  10  ponnds,  6124  gallons  in  a 
cubic  foot,  and  one  cubic  foot  of  sea  water  as  64  pounds. 

5.  Hydrostatics. 

A  knowledge  of  the  principles  of  hydrostatics  is  very  helpful 
in  approaching  the  subject  of  hydraulics,  and  in  the  wider  sense 
in  which  the  latter  word  is  now  used  it  may  be  said  to  include  the 
former.  It  is,  therefore,  advisable  to  consider  the  laws  of  fluids 
at  rest. 

There  are  two  cases  to  consider.  First,  fluids  at  rest  under  the 
action  of  gravity,  and  second,  those  cases  in  which  the  fluids  are 
at  rest,  or  are  moving  very  slowly,  and  are  contained  in  closed 
vessels  in  which  pressures  of  any  magnitude  act  upon  the  fluid, 
as,  for  instance,  in  hydraulic  lifts  and  presses. 

6.  Intensity  of  pressure. 

The  intensity  of  pressure  at  any  point  in  a  fluid  is  the  pressure 
exerted  upon  unit  area,  if  the  pressure  on  the  unit  area  is  uniform 
and  is  exerted  at  the  same  rate  as  at  the  point. 

Consider  any  little  element  of  area  a,  about  a  point  in  the  fluid, 
and  upon  which  the  pressure  is  uniform. 

If  P  is  the  total  pressure  on  a,  the  Intensity  of  Pressure  p,  is  then 

P 

or  when  P  and  a  are  indefinitely  diminished, 

^ap 

^     da' 

7.  The  pressure  at  any  point  in  a  fluid  is  the  same  in  all 
directions. 

It  has  been  stated  above  that  when  a  fluid  is  at  rest  its  resist- 
ance to  lateral  deformation  is  practically  zero  and  that  on  any 
plane  in  the  fluid  tangential  stresses  cannot  exist.  From  this 
experimental  fact  it  follows  that  the  pressure  at  any  point  in  the 
fluid  is  the  same  in  all  directions. 


FLUIDS  AT   BEST 


Consider  a  smaU  wedge  ABC,  Pig,  1^  floating  immersed  in  a 
II aid  at  rest. 

Sine©  there  caimot  be  a  tangential 
reag  on  any  of  the  planes  AB,  BC,  or  AC, 
tf  prussurad  on  them  must  be  normaL 
Lei  p,  pi  and  ps  be  the  intenaitiea  of 

tires  on  these  planes  respectively, 
H©  weight  of  the  wedge  will  be  very 
fftnal]  and  may  he  neglected* 

An  thtt  wedge  m  in  equihbriiim  under  the  forces  acting  on 
three  faces,  the  resolved  components  of  the  force  acting  on 
;C  in  the  directions  of  p  and  pi  mnst  balance  the  forces  acting 
AB  and  BC  respectively* 
Therefore  p, .  AC  cos  0  -  p ,  AB, 

d  psACsin^^PaBC, 

But  AB  =  ACcose, 

And  BC^ACsin^* 

Thert*fore  p  =  pj  =  p, , 

8.    The  pressure  on  any  horizontal  plane  in  a  fluid  must 
eonfilant. 

Confiider  a  small  cylinder  of  a  fluid  joining  any  two  points  A 
nd  B  OQ  the  same  horissontal  plane  in  the  fluid. 

Since  there  can  be  no  tangential  forces  acting  on  the  cylinder 

purallel  to  the  axiB^  the  cylinder  must  be  in  equilibrium  under  the 

pressures  on  the  ends  A  and  B  of  the  cylinder,  and  since  these 

re  of  equal  area,  the  pressure  must  be  the  same  at  each  end  of 

he  cylinder* 


9.    Fluids  at  rest,  with  the  free  surface  horizontal, 

'Hie  pres.sure  per  unit  area  at  any  depth  h  below  the  free 
of  a  finid  due  to  the  weight  of  the  fluid  is  equal  to  the 

|b.t  of  a  column  of  fluid  of  height  k  and  of  unit  sectional  area. 

Let  the  pressure  per  unit  area  acting  on  the  surface  of  the 
Sold  be  p  lbs.     If  the  fluid  is  in  a  closed  vessel,  the  pressure  p  may^ 
bave  any  assigned  value,  but  if  the  free  surface  is  exposed  to  tin 
aCiQO&phere,  p  will  be  the  atmospheric  pressure. 

klf  a  small  open  tube  AB,  of  length  h^  and  cross  sectional  area  a, 
placed  in  the  fluid,  the  weight  per  unit  volume  of  which  is 
Ib&,  with  its  axis  vertical,  and  its  uppor  end  A  coincident  with 
9  0tif£ftC0  of  the  fluid,  the  weight  of  fluid  in  the  cylinder  uiUBt  be 
ip.apA  Ibfls.  The,  pressure  acting  on  the  end  A  of  the  colunm 
is  pa  Ib^. 


a 


HTDRAUUCS 


Since  there  cannot  be  any  force  acting  on  the  colnmn  parallel 
to  the  eddes  of  the  tube,  the  force  of  imiA  lbs.  +  pa  lbs.  most  be 
kept  in  equilibrium  by  the  pressure  of  the  external  fluid  acting  on 
the  fluid  in  the  cylinder  at  the  end  B. 

The  pressure  per  unit  area  at  B,  therefore, 


= ^  =  (wh  +  p)  lbs. 


The  pressure  per  unit  area,  therefore,  due  to  the  weight  of  the 
fluid  only  is  tch  lbs. 

In  the  case  of  water,  w  may  be  taken  as  62*40  lbs.  per  cubic 
foot  and  the  pressure  per  sq.  foot  at  a  depth  of  h  feet  is,  therefore, 
62*40%  lbs.,  and  per  sq.  inch  '433b  lbs. 

It  should  be  noted  that  the  pressure  is  independent  of  the  form 
of  the  vessel,  and  simply  depends  upon  the  vertical  depth  of  the 
point  considered  below  the  surface  of  the  fluid.  This  can  be 
illustrated  by  the  different  vessels  shown  in  Fig.  2.  If  these 
were  all  connected  together  by  means  of  a  pipe,  the  fluid  when 
at  rest  would  stand  at  the  same  level  in  all  of  them,  and  on  any 
horizontal  plane  AB  the  pressure  would  be  the  same. 


-T-UJ- 


.^/ 


Pressure  an  the  Pixuie  A^^w-h  lbs  persq Foot, 
Fig.  2. 


If  now  the  various  vessels  were  sealed  from  each  other 
by  closing  suitable  valves,  and  the  pipe  taken  away  without 
disturbing  the  level  CD  in  any  case,  the  intensity  of  pressure  on 
AB  would  remain  unaltered,  and  would  be,  in  all  cases,  equal 
to  wh. 

Example,  In  a  condenser  containing  air  and  water,  the  pressure  of  the  air  is 
2  lbs.  per  sq.  inch  absolute.  Find  the  pressure  per  sq.  foot  at  a  point  8  feet  below 
the  free  surface  of  the  water. 

I>  =  2xl44  +  dx62*4 
=475*2  lbs.  per  sq.  foot. 


FLUIDS  AT  REST 


10.    PreiBiires  meaBiired  in  feet  of  water.    Fressure  head. 

It  is  oonvenient  in  hydrostatics  and  hydraulics  to  express  the 
inreesore  at  any  point  in  a  fluid  in  feet  of  the  fluid  instead  of  pounds 
per  sq.  foot  or  sq.  inch.  It  follows  from  the  previous  section  that 
if  the  pressure  per  sq.  foot  is  p  lbs.  the  equivalent  pressure  in  feet 

of  water,  or  the  pressure  head,  is  A  =  ^  ft.  and  for  any  other  fluid 

w 

having  a  specific  gravity  p,  the  pressure  per  sq.  foot  for  a  head 

h  of  the  fluid  is  p  =  w.p.hy  or  A  =  — . 

top 

IL    Piesometer  tubes. 

The  pressure  in  a  pipe  or  other  vessel  can  conveniently  be 
measured  by  fixing  a  tube  in  the  pipe  and  noting  the  height  to 
which  the  water  rises  in  the  tube. 

Such  a  tube  is  called  a  pressure,  or  piezometer,  tube. 

The  tube  need  not  be  made  straight  but  may  be  bent  into  any 
form  and  carried,  within  reasonable  limits,  any  distance  horizon- 
tally. 

The  vertical  rise  h  of  the  water  will  be  always 

where  p  is  the  pressure  per  sq.  foot  in  the  pipe. 

If  instead  of  water,  a  liquid  of  specific  gravity  p  is  used  the 
height  h  to  which  the  liquid  will  rise  in  the  tube  is 

w  .p 

Example.  A  tube  having  one  end  open  to  the  atmosphere  is  fitted  into  a  pipe 
containing  water  at  a  pressure  of  10  lbs.  per  sq.  inch  above  the  atmosphere.  Find 
the  height  to  which  the  water  will  rise  in  the  tube. 

The  ^*ater  will  rise  to  such  a  height  that  the  pressure  at  the  end  of  the  tabc  in 
the  pipe  due  to  the  column  of  water  will  be  10  lbs.  per  sq.  inch. 


Therefore 


^     10x144     ^„^^  ^    , 
h= =23-08  feet. 


12.    The  barometer. 

The  method  of  determining  the  atmospheric 
pressure  by  means  of  the  barometer  can  now  be 
understood. 

If  a  tube  about  3  feet  long  closed  at  one  end  be 
completely  filled  with  mercury,  Fig.  3,  and  then 
turned  into  a  vertical  position  with  its  open  end 
in  a  vessel  containing  mercury,  the  liquid  in  the 
tube  falls  until  the  length  h  of  the  column  is  about 
30  inches  above  the  surface  of  the  mercury  in  the 
vessel. 


Fig.  3. 


HYDRAULICS 


Since  the  pressure  p  on  the  top  of  the  mercury  is  now  zero,  iYm-^ 
pressure  per  unit  area  acting  on  the  section  of  the  tube,  level  witX^ 
the  surface  of  the  mercury  in  the  vessel,  must  be  equal  to  ih^ 
weight  of  a  column  of  mercury  of  height  h. 

The  specific  gravity  of  the  mercury  is  13*596  at  the  standard 
temperature  and  pressure,  and  therefore  the  atmospheric  pressure 
per  sq.  inch,  pa,  is, 

80"  X  13-596  X  62-4     -,.^„  .    , 

Pa= ^2  X  144 ~       '        •  P®^  ^*  "^ 

Expressed  in  feet  of  water, 

14*7  X  141 
A  =  -i>  I— =  33-92  feet. 
62  4 

This  is  so  near  to  34  feet  that  for  the  standard  atmospheric 
pressure  this  value  will  be  taken  throughout  this  book. 

A  similar  tube  can  be  conveniently  used  for  measoring  low 
pressures,  lighter  liquids  being  used  when  a  more  sensitive  gauge 
is  required. 

13.    The  differential  gauge. 

A  more  convenient  arrangement  for  measuring  pressures^  and 
one  of  considerable  utility  in  many  hydraulic  exx>eriment8,  iB 
known  as  the  differential  gauge. 

Let  ABCD,  Fig.  4,  be  a  simple  U  tube 
containing  in  the  lower  part  some  fluid  of 
known  density. 

If  the  two  limbs  of  the  tube  are  open  to 
the  atmosphere  the  two  surfaces  of  the  fluid 
will  bo  in  the  same  horizontal  plane. 

If,  however,  into  the  limbs  of  the  tube  a 
lighter  fluid,  which  does  not  mix  with  the 
lower  fluid,  be  poured  until  it  rises  to  C  in 
one  tube  and  to  D  in  the  other,  the  two 
surfaces  of  the  lower  fluid  ^vill  now  be  at 
different  levels. 

Let  B  and  E  be  the  common  surfaces  of 
the  two  fluids,  h  being  their  difference  of 
level,  and  hi  and  th  the  heights  of  the  free 
surfaces  of  the  lighter  fluid  above  E  and  B  respectively. 

Let  p  be  the  pressure  of  the  atmosphere  per  unit  area,  and  d 
and  di  the  densities  of  the  lower  and  upper  fluids  respectively. 
Then,  since  upon  the  horizontal  plane  AB  the  fluid  pressure  must 
be  constant, 

p  +  rfjfes  =  p  +  djii  +  dh, 

d\  (Jh  —  hi)  =  dh. 


Fig.  4. 


FLUIDS  AT  REST 


9 


li  now^  instead  of  the  two  limbs  of  the  U  tube  being  open  to 
»tmcMipha?e,  they  are  ooimected  by  tubes  to  closed  vessels  in 
ich  the  pressures  are  pi  and  p^  pounds  per  sq*  foot  respectively^ 

d  ^1  and  fh  are  the  vertical  lengths  of  the  colnnms  of  fluid  above 

ind  B  respectively^  then 

Pt^di^kt  =Pi  +  di^hi  +  d.hf 
p^-  Pi^d,k-di{fh-hi). 

kik  application  of  snch  a  tube  to  determine  the  difference  of 
fmuTe  at  two  pointa  in  a  pipe  containing  flowing  water  is  shown 
I  %  88,  page  116. 

Fluidk  generally  used  in  such  U  ttihee.    In  hydraulic  experiments 

f  Bpper  part  of  the  tube  is  filled  with  water,  and  therefore  the 
lid  b  the  lower  part  must  have  a  greater  density  than  water. 
fWthe  difference  of  pressure  is  fairly  large,  mercury  is  generally 
tli8$peci£c  gravity  of  which  ia  13'596.  When  the  difference 
'  pemife  is  small,  the  height  k  is  difficult  to  measure  iv^th 
m  that|  if  this  form  of  gauge  is  to  be  used,  it  is  desirable 

i»pli«e  the  mercury  by  a  lighter  liquid.     Carbon  bisulphide 

s  Ijeen  used  but  its  action  is  sluggish  and  the  meniscus  between 
and  the  water  is  not  always  well  defined, 
jth)-bi*imne  ^v^  good  results^  its  prin- 
'1*1  fault  being  that  the  falling  meniscus 
^  DOt  very  quickly  assume  a  definite 
ape. 

Hf  inverted  air  gauge,  A  mure  sen- 
^*^  gauge  can  be  made  by  inverting  a 

^  and  enclosing  in  the  upper  part 
**rt&iii  quantity  of  air  as  in  the  tube 
Be,  Fig.  5. 

I^  the  pressure  at  D  in  the  limb  DF 

Pi  poimds  per  square  foot,  equivalent 
;*liead  hi  of  the  fluid  in  the  lower  part 
^  gauge,  and  at  A  in  the  limb  AK  let 
*  Iffmure  be  p%^  equivalent  to  a  head  h%. 
W  A  bt*  the  difference  of  level  of  G  and  C* 

Tfien  if  CGH   contains  air^  and   the   weight  of  the  air  be 

[beted,  being  very  small,  the  pressure  at  C  muBt  equal  the 

at  G I  and  since  in  a  fluid  the  pressure  on  any  horizontal 

constant  the  pressure  at  C  is  equal  to  the  pressure  at  D, 

the  pressure  at  A  equal  to  the  pressure  at  B.    Again  the 

at  G  is  equal  to  the  pressure  at  K. 

Th^tiefore  ht-h^ht^ 

Pf-pt-pw.h, 


H 


».|^ 


iBlCJ 


^  F 


u 


Fig.  &. 


v.i  uy  using,  in  tli« 

upper  part  of  tlie  tube,  an  oil  liglitei 
than  water  instead  of  air,  as  sliowii 
in  Fig.  6. 

Let  pi  and  pj  be  the  pressures  in 
the  two  limbs  of  the  tube  on  a  given 
horizontal  plane  AB,  hi  and  }h  being 
the  equivalent  heads  of  water.  The 
oil  in  the  bent  tube  will  then  take  up 
some  such  position  as  shown,  the 
plane  AD  being  supposed  to  coincide 
with  the  lower  surface  C. 

Then^  since  upon  any  horizontal 
plane  in  a  homogeneous  fluid  the 
pressure  must  be  constant,  the  pres- 

■es  at  G  and  H  are  equal  and  also 

>se  at  D  and  C. 

Let  pi  be  the  specific  gravity  of 

)  water,  and  p  of  the  oil. 

m  Pihi-ph  =  Pi(hi'-h), 

i    irefore    fc  (pi-p)  =Pi  (^-fei) 

(Pi  -  p) 
)stituting  for  hi  and  h^  the  values 


FLUIDS  AT  EEST 


thi^t  either  kerosene,  gasoline,  or  sperm  oil  gave  excellent  results, 
h\x%  flperm  oil  waa  too  sluggish  in  its  action  for  rapid  work. 
Kei?«fieiia  gave  the  be^t  results. 

nkmperaiure  coeffictent  of  the  inverhd  oil  gauge,  Dnhke  the 
inverted  far  gaog^  the  oil  gauge  has  a  considerable  temperature 
oc^^rfficietit,  aa  will  be  seen  from  the  table  of  specific  gravities  at 
viLriotis  t^'Uipera tares  of  water  and  the  kerosene  and  gasoline  used 
br    >Villiams,  HubbeU  and  Fenkell 

In  this  Cable  the  specific  gravity  of  water  is  taken  as  unity 


Wat^ 


T 


40       !      60 


1-00092 


100 


EefQseiie 


I'OOOO  *9041 


40        60       100 
■7955  *7879  '7725 


Gft«oliii6 


40         60  80 

*72147  71587  '70547 


jf^^  caltbrafwn  of  ike  hiverted  oil  gauge.  Messrs  Williams, 
i^  >-il>beU  and  Fenkell  have  adopted  an  ingenious  method  of 
<^^lihniting  the  oil  gauge.  This  will  readily  be  understood  on 
'**f»PBiioe  to  Fig.  6- 

*The  difference  of  level  of  E  and  F  clearly  gives  the  difference 
^     liand  acting  on  the  plane  AD  in  feet  of  water,  and  this  from 

<^^^tion  (1)  equals  M?l:ip)  , 

Pi 
"VFatar  is  put  into  AE  and  FD  so  that  the  surfaces  E  and  F 
*''^*^  <m  the  same  levels  the  common  surfaces  of  the  oil  and  the 
*^^ter  abo  being  on  the 'same  level,  this  level  being  aiero  for  the 
^^^  Water  is  then  run  out  of  FD  until  the  surface  F  is 
f^^djy  I  inch  below  E  and  a  reading  for  h  taken.  The  surface  F 
^  ^^in  lowered  1  inch  and  a  reading  of  h  taken*  This  process 
*  csontinued  tmtfl  F  is  lowered  as  far  as  convenient,  and  then 
irt^  wati*r  in  EA  is  drawn  out  in  a  similar  manner.  When  E 
^^itrl  p  are  again  level  the  oil  in  the  gauge  should  read  zero. 

14.    Tranamission  of  fluid  pressure. 

If  an  external  pressure  be  applied  at  any  point  in  a  fluid,  it  ia 
^^nsmifcted  equally  in  all  direc- 
^u^B  through  the  whole  mass. 
T^UJ  is  proved  experimentally 
h  means  of  a  simple  apparatus 
^ch  aa  shown  in  Fig.  7, 

I^  lire  P  is  exerted  upon 

>^'         ,      on  Q  of  a  sq-  inches  Fig.  7- 


12  HYDRAULICS  j 

P 

area,  the  pressure  per  unit  area  p  =  — ,  and  the  piston  at  B  on  the 

same  level  as  Q,  which  has  an  area  A,  can  be  made  to  lift  a  load  W 

P 

equal  to  A  — ;  or  the  pressure  per  sq.  inch  at  R  is  equal  to  the 

pressure  at  Q.  The  piston  at  R  is  assumed  to  be  on  the  same  level 
as  Q  so  as  to  eliminate  the  consideration  of  the  small  differences  of 
pressure  due  to  the  weight  of  the  fluid. 

If  a  pressure  gauge  is  fitted  on  the  connecting  pipe  at  any 
point,  and  p  is  so  large  that  the  pressure  due  to  the  weight  of  the 
fluid  may  be  neglected,  it  will  be  found  that  the  intensity  of 
pressure  is  p.  This  result  could  have  been  anticipated  from  that 
of  section  8. 

Upon  this  simple  principle  depends  the  fact  that  enormous 
forces  can  be  exerted  by  means  of  hydraulic  pressure. 

If  the  piston  at  Q  is  of  small  area,  while  that  at  R  is  large, 
then,  since  the  pressure  per  sq.  inch  is  constant  throughout  the 
fluid, 

W__A 
P  "a' 
or  a  very  large  force  W  can  be  overcome  by  the  application  of 
a  small  force  P.    A  very  large  mechanical  advantage  is  thus 
obtained. 

It  should  be  clearly  understood  that  the  rate  of  doing  work 
at  W,  neglecting  any  losses,  is  equal  to  that  at  P,  the  distance 
moved  through  by  W  being  to  that  moved  through  by  P  in 
the  ratio  of  P  to  W,  or  in  the  ratio  of  a  to  A. 

Example,  A  pomp  ram  has  a  stroke  of  3  inches  and  a  diameter  of  1  inch.  The 
pump  supplies  water  to  a  lift  which  has  a  ram  of  5  inches  diameter.  The  force 
driving  the  pomp  ram  is  1500  lbs.  Neglecting  all  losses  dne  to  friction  etc., 
determine  the  weight  lifted,  the  work  done  in  raising  it  5  feet,  and  the  nnmber 
of  strokes  made  by  the  pomp  while  raising  the  weight. 

Area  of  the  pump  ram ='7854  sq.  inch. 

Area  of  the  lift  ram  =  19*6  sq.  inches. 

Therefore  W = 1?1|J^= 37,500  lbs. 

Work  done  =  37,600  x  5  =  187,500  ft.  lbs. 

Let  N  equal  the  number  of  strokes  of  the  pump  ram. 
Then  N  x  A  x  1500  lbs.  =  187,600  ft.  lbs. 

N= 600  strokes. 

or  ^  or  whole  pressure. 

From  (2)  it  leessure  acting  on  a  surface  is  the  sum  of  all  the 
different  from  thacting  on  the  surface.    K  the  surface  is  plane  all 
differences  of  presiLel,  and  the  whole  pressure  is  the  sum  of  these 
•Frou 


PLUIJDS   AT  REST 


13 


Ijet  any  sarfece,  which  need  not  be  a  planej  be  immerBed 
a  finid.  Let  A  be  the  area  of  the  wetted  aiirface,  and  k  the 
head  at  the  centre  of  gravity  of  the  area.  If  the  area 
is  inLHiersed  in  a  fluid  the  presisure  on  the  surface  of  which  m  zero, 
tiie  free  snrface  of  the  fluid  will  be  at  a  height  h  above  the  centre 
of  gravity  of  the  area*  In  the  case  of  the  area  being  immeraed  in 
a  flnidf  the  surface  of  which  is  exposed  to  a  pressure  p,  and  below 
vliick  the  depth  of  the  centre  of  gravity  of  the  area  is  ih^  then 

w 

If  the  area  exposed  to  the  fluid  pressure  is  one  face  of  a  body^ 

the  opposite  faee  of  which  m  tsxposed  to  the  atmospheric  pT*e&aure, 

in  the  case  of  the  side  of  a  tank  containing  water,  or  the 

dam  of  Fig,  14,  or  a  valve  closing  the  end  of  a  pipe  as 

Fig.  8|  the   pressure  due    to   the 

itmosphere  is  the   same  on  the   two 

■fec^ts  and  therefore  may  be  neglected. 

Let   IP  be  the  weight  of  a  cubic 

foot  of  the   fluid.     Then,   the  whole 

preesure  on  the  area  is 

If  the  surface  is  in  a  horizontal 
plane  the  theorem  is  obviously  true, 
fliDce  the  intensity  of  pressure  is  eon- 
slant  and  equals  w  ,  K 

In  general,  imagine  the  surface. 
Fig-  %  divided  into  a  large  number  of  small  areas  a>  ai,  o^  ...  * 

Let  «  be  the  depth  below  the  free  surface  FS^  of  any  element 
of  area  a ;  the  pressure  on  this  element  =  tc  .  a; ,  a. 

Tlie  whole  pressure  P  ^  Sw,  ac  *  a. 

But  tr  is  constant,  and  the  sum  of  the  moments  of  the  elements 
[of  the  area  about  any  axis  equals  the  moment  of  the  whole  area* 
fal>ont  the  same  axis,  therefore 

5iE ,  fi  =  A .  ftj 
juid  P  =  W7 .  A .  A. 

16*     Centre  of  pressure. 

The  centre  of  pressurt?  of  any  plane 
snrfo^re  acted  upon  by  a  fluid  is  the 
point  of  action  of  the  resultant  pressure 
acting  upon  the  surface. 

J^^  of  ihs  centre  of  presswre.  Let 
DBC,  Fig.  9,  be  any  plane  surface 
exposed  to  fluid  pressure. 

*  Em  %eit-book8  on  Me^haoiei. 


Fig.  8. 


B( 


ffi 


C 

Fig.  9. 


14  HYDRAULICS 

Let  A  be  the  area,  and  h  the  pressure  head  at  the  centre  of 
gravity  of  the  surface,  or  if  FS  is  the  free  surface  of  the  fluid,  h  is 
the  depth  below  FS  of  the  centre  of  gravity. 
Then,  the  whole  pressure 

F  =  w.A.h. 
Let  X  be  the  depth  of  the  centre  of  pressure. 
Imagine  the  surface,  as  before,  divided  into  a  number  of  small 
areas  a,  ai,  Os,  ...  etc. 

The  pressure  on  any  element  a 

=  w  .a.x, 
and  P  =  ^wax. 

Taking  moments  about  FS, 

P.X=  (wax*  +  waia>i*+  ...) 
=  Siooa;', 


or  X  = 


wAh 


Ah  ' 

When  the  area  is  in  a  vertical  plane,  which  intersects  the 
surface  of  the  water  in  FS,  Soa?*  is  the  "second  moment"  of  the 
area  about  the  axis  FS,  or  what  is  sometimes  called  the  moment 
of  inertia  of  the  area  about  this  axis. 

Therefore,  the  depth  of  the  centre  of  pressure  of  a  vertical 
area  below  the  free  surface  of  the  fluid 

moment  of  inertia  of  the  area  about  an  axis  in  its  own  plane 

_         ' and  in  the  free  surface 

~"  area  x  the  depth  of  the  centre  of  gravity  ' 

or,  if  I  is  the  moment  of  inertia, 

Areao  A.h 

^^  t  of  Inertia  about  any  axis.    Calling  I©  the  Moment 
Thereforebout  an  axis  through  the  centre  of  gravity,  and  I  the 

_   ,  -      nertia  about  any  axis  parallel  to  the  axis  through  the 
Work  done   ..  ,     .       ,.  .  i.  jl.        -.l 

»     ^,        ,  xty  and  at  a  distance  h  from  it. 
Let  N  equal  ti*^  t     t        a  7  • 

Then  I  =  Io^AA«. 

*ea  is  a  rectangle  breadth  6  and  depth  d. 
P=ir.6.d./i, 

or                   '  or  6^ 

From(2)itieess.  "12+^^'*' 

different  from  thacti.  ^+wfc« 

differences  of  presdel,  X=- 


Proc. 


hdh 


FLUIDS  AT  REST 


16 


If  the  free  sorfiuie  of  the  water  is  level  with  the  upper  edge  of  the  rectangle, 
,=  |,  mod  X  =  |.^ 


2' 


is  a  drole  of  radios  B. 


X= 


^+xR.*. 


TSFT 


If  the  top  of  the  circle  is  jast  in  the  free  sorfaoe  or  A=B, 

X  =  1R. 


TABLE  n. 
Table  of  Moments  of  Inertia  of  areas. 


Form  of  area 

Moment  of  inertia  abont 
an  axle  AB  through  the 
C.  of  G.  of  the  section 

Rectangle 

JlJ 

A^' 

Triangle 

k'''' 

Circle 

1©° 

64 

Semicircle 

rj^ 

About  the  axis  AB 

8 

Paarabola 

i 

> 

16 


HYDRAUUCS 


17.    Diagram  of  pressure  on  a  plane  area. 

If  a  diagram  be  drawn  showing  the  intensity  of  pressure  on 
a  plane  area  at  any  depth,  the  whole  pressure  is  equal  to  the  volume 
of  the  solid  thus  formed,  and  the  centre  of  pressure  of  the  area  is 
found  by  drawing  a  line  through  the  centre 
of  gravity  of  this  solid  perpendicular  to  the 
area. 

For  a  rectangular  area  ABCD,  having  the 
side  AB  in  the  surface  of  the  water,  the 
diagram  of  pressure  is  AEFCB,  Fig.  10.  The 
volume  of  AEFCB  is  the  whole  pressure  and 
equals  ^bcPw,  b  being  the  width  and  d  the 
depth  of  the  area. 

Since  the  rectangle  is  of  constant  width, 
the  diagram  of  pressure  may  be  represented 
by   the   triangle   BCF,  Fig.   11,  the   resultant  pressure   acting 
through  its  centre  of  gravity,  and  therefore  at  id  from  the  surface. 


^'''  ""-s.  12  and  13. 


h  g  f^ 

ow  cucL. 

Fig.  12. 

For  a  vertical  circle  the  diagram  of  pressure  is  as  shown  in 

The  intensity  of  pressure  ah  on  any  strip  at  a 

o»  oovii  }i^  ig  ^h^^  The  whole  pressure  is  the  volume  of  the  truncated 

^Sk'er  efkh  and  the  centre  of  pressure  is  found  by  drawing  a 

'T)endicular  to  the  circle,  through  the  centre  of  gravity 

^^^<uncated  cylinder. 

Work  done 

Let  N  eqaal-i 

Then 


or  '  Oi 

From  (2)  it  lee^o) 
different  from  thaci 
differences  of  presde. 

♦  IVoc  Pig.  18. 


^, 


FLUIDS   AT    REST 


It 


Another,  and  fretiuently  a  very  convenient  method  of  dpfcer- 
tniniiig^  the  depth  of  the  centre  of  pressure,  when  the  whole  of  the 
«x*ea   is  itt  some  dietanco  below  the  surface  of  the  water,  is  to 
eoiiisider  the  pressure  on  the  area  as  made  up  of  a  uniform  pressure  1 
over  the  whole  surface,  and  ii  pressure  of  variable  intensity. 

Take  again,  as  an  example^  the  vertical  circle  the  diagrams  of 
pressure  for  which  are  shown  in  Figs.  12  and  13. 

At  any  depth  h  the  intensity  of  pressure  on  the  strip  ad  is 
wh„  -  wHa  ^  y^fh ' 

The   pressure  on  any  strip  ad  is^  therefor*?,  made  up  of  a 
4'  '  presssure  per  unit  area  wtiA  and  a  variable  pressure  wJh ; 

ii :  ^vhole  pressure  is  equal  to  tlie  volume  of  the  cylinder  efghj 

tSg.  12,  together  with  the  circular  wedge  fkg. 

The  wedge  fkg  is  equal  to  the  whole  pressure  on  a  vertical 
circle,  the  tangent  k>  which  is  in  the  free  surface  of  the  water  and 

equals  *r  *  A .  ^  ,  and  the  centre  of  gravity  of  this  wedge  will  he  at 

the  same  vertical  distance  from  the  centre  of  the  circle  as  the 
centre  of  pressure  when  the  circle  touches  the  surface*  The  whole 
pressure  P  umy  be  supposed  therefore  to  be  the  resultant  of  two 
pttndlel  forces  Pi  and  Pa  acting  through  the  centres  of  gravity  of 
the  cylinder  efght  and  of  the  circular  wedge  fkg  respectively,  the 
ma^tudes  of  P^  and  Pj  being  the  volumes  of  the  cylinder  and 

the  wedge  respectively* 

Tu  find  the  centre  of  pressure  on  the  circle  AB  it  is  only 

iieoeeeary  to  find  the  resultant  of  two  parallel  forces 

Pi^A.whj,  and  P^^w.Al^jt 

<jt  irhicli  P,  act**  at  the  centra  c,  and  P,  at  a  point  c-t  which  is  at 
a  distance  from  A  of  |  r. 

£iai^.  A  tsJMonry  dam,  Fig*  1-1, 
jM«i hetirtit  of  80  fe*t  tmm  the  founda- 
Ml  aa4  tbe  »&ter  tmi^  is  ineliJaed  At 
*^^K^tm  10  the  vcTiical ;  find  the  whole 
f^*o»*  m  the  fkce  tlue  to  the  water  per 
•*^»iiilii  ijf  the  dftm^  «nd  the  oehtr^  of 
y*^*Oft,  «h«i3  th*  Wfttcr  Harf*o«  t«*  levpl 
^*4ie  tii|)  of  lh«  imm,  Thu  atmo- 
*9^  pnmam  m^  be  »etlecicd< 
^^  vJlole  pnsmme  will  be  the  force 
J^J^  to  flff •rtura  tii«}  iUtti,  Biace  Uie 
"•**w»Ul  oQiufMJiii^iii  *yf  I  111  prefisute 
«  A»  4«f  to  ^    wiU  be 

IJ«wit*  of  .c  LiieBfittre  on 

^  ^*^  "  !.■  the  prsMure 

•&*!»«  a  ijuimiii.  jiiiii  the  miaufiiiy 

U  proportional  to  tUa  ^eplli, 


RUtAereSfttLajit  tAtitst 
on  tlie-  /jasi^^  IIBoftd  arfs 
oJt  tJie  poirU^  E, 

Fig,  14. 


i-n. 


^■ii 


18 


HYDRAULICS 


the  diagram  of  pressure  on  the  face  AB  will  be  the  triangle  ABC,  BC  being  equal 
to  wd  and  perpendicular  to  AB. 

The  centre  of  pressure  is  at  the  centre  of  gravity  of  the  pressure  diagram  and  is, 
therefore,  at  |  the  height  of  the  dam  from  the  base. 

The  whole  pressure  acts  perpendicular  to  AB,  and  is  equal  to  the  area  ABC 

=  i  iccP  X  sec  10°  per  foot  width 

=  i  .  62-4  X  6400  X  1-054  =r  20540  lbs. 

Combining  P  with  W,  the  weight  of  the  dam,  the  resultant  thrust  R  on  the  base 
and  its  point  of  intersection  £  with  the  base  is  determined. 

Example.  A  vertical  flap  valve  closes  the  end  of  a  pipe  2  feet  diameter ;  the 
pressure  at  the  centre  of  the  pipe  is  equal  to  a  head  of  8  feet  of  water.  To  determine 
the  whole  pressure  on  the  valve  and  the  centre  of  pressure.  The  atmospherie 
pressure  may  be  neglected. 

The  whole  pressure  P  =  irirR^ .  8' 

=  62-4.x.  8  =  1670  lbs. 

Depth  of  the  centre  of  pressure. 

The  moment  of  inertia  about  the  free  surface,  which  is  8  feet  above  the  centre 
of  the  valve,  is 


82 


Therefore 


1=    — +irR*''. 

X  =  ^**^=8'(^" 

IT  .  O 


That  is,  }  inch  below  the  centre  of  the  valve. 

The  diagram  of  pressure  is  a  truncated  cylinder  efkh.  Figs.  12  and  13,  ef  and  hk 
being  the  intensities  of  pressure  at  the  top  and  bottom  of  the  valve  respectively. 

Example.  The  end  of  a  pontoon  which  floats  in  sea  water  is  as  shown  in  Fig.  15. 
The  level  WL  of  the  water  is  also  shown.  Find  the  whole  pressure  on  the  end  of 
the  pontoon  and  the  centre  of  pressure. 


Fig.  15. 

The  whole  pressure  on  BE 

=  64  lbs.  X  lO'  X  4-5'  x  2-25'  =  6480  lbs. 
The  depth  of  the  centre  of  pressure  of  BE  is 

1.4-6=8'. 
The  whole  pressure  on  each  of  the  rectangles  above  the  quadrants 
=ii>.  6  =  320  lbs., 

and  the  depth  of  the  centre  of  pressure  is  |  feet. 

Tlie  two  quadrants,  since  they  are  symmetrically  placed  about  the  vertieal 
centre  line,  may  be  taken  together  to  form  a  semicircle.  Let  d  be  the  distance 
below  the  centre  of  the  semicircle  of  any  element  of  area,  the  distance  of  the 
element  below  the  surface  being  h^ . 


FLUIDS   AT   REST 


19 


i  the  mteBsitT-  of  preisare  Jil  deplh  h^ 
wlM>le  prcsAiire  on  the  lemicirde  Le  P  = 


I 

Hp  lemlciyeto  «1i«d  the  diameter  \&  la  ^be  eurfnee  of  tb^  waler. 

^TW  diateiioe  of  the  t^niu  of  gmvitj  of  &  s^toitiirde  from  the  oentre  of  tho 


2 


2'  -*-  thi  whole  presstire 


Ik 

'^^  =S01B*  +  42  66R>^  1256 4- 666  lbs. 

I      Tb#  ^etAb  of  the  oentni  of  pteaBUre  c>f  tb@  semieir^^Ie  when  t!i«  sorfiaoe  of  the 


P=wfB*+ 


virB>  4B 


li  ml  the  oenire  at  the  cir^ 


ptefiBU 


X^= 


8 


,R 


75^ 
s 


16 


-M7'> 


^^■■^^MiM^the  whrrk  pressure  ort  th«  9emi«!ircle  is  the  tium  of  two  Forces, 
^^^HHHH|HH6  Ibi^.^  acU  at  th^  ceutre  of  gr&vity,  ar  At  &  diKtance  of  3  06'  from 
■iPBiCBBff  0(  r>66  Ibi^.  acta  ut  u  diati^tice  of  3^47'  from  AD. 

T^tfi  taklti^  momenta  al»out  AD  thi^  product  of  the  preBSiire  on  the  whole  area 
isiCc*  the  (t«.'nth  of  lii&  ceiitr«*  of  pressure  is  equal  to  the  moiuetittt  of  i^ll  th«  forced 
OB  the  afv»,  a  boat  AI),    The  depth  of  this  centre  of  priMiiF&  in,  therefore, 

6%m  Ibi.  X  ^'  +  320  Ibi.  X  2  X 1'  + 1266  lbs.  x  306  +  S66  Ibw,  x  3^47' 


^a*9a  feet. 


64ti€  +  ti40'f  1355  +  666 


EXAMPLES. 

l\    A  f«AMfiBl*^  ^&o^  1^  ^^^  l<^^g*  ^  ^^^  wide,  a^d  5  leet  deep  is 

with  ni^ 
F'ind  ih«  iota)  prassure  on  an  end  and  side  of  the  tank. 

Finil  the  total  presj^ure  and  the  centre  of  preaaure,  on  a  vertical 
e,  eLrctdiir  in  fortn,  2  feet  in  diameter,  the  centre  of  which  is  4  feet 
Ibe  sorface  of  tlje  water.     [M.  S.  T.  Cambridge,  1901,] 

A  oiaaonry  dam  vertical   on  the  water  side  supports  water  of 

I  feet-  depth.    Find  the  pressure  per  square  foot  at  depths  of  20  feet  and 

.  iitiBi  the  Miirface ;  also  the  total  pressni^  on  1  foot  length  of  Uie  dam, 

0)    A  dodc  gate  is  hinged  horiEontallj  at  the  hottoni  and  Bttpported  in 
Im  ^viiicai  pcMttion  hj  horiacontal  chains  at  the  top. 

i  of  ^te  45  feet,  width  80  ft.     Depth  of  water  at  ope  siJe  of  the 
[  And  20  feet  on  the  other  side.     Find  the  tension  in  the  diains. 
cr  weigbfl  ©4  poitada  per  cubic  foot, 

II  ui<erctir>'  ia  ISJ  times  as  heavy  as  water,  find  the  height  of  a 
com^sponduig  to  a  pressure  of  100  lbs.  per  square  inch, 

,  sirai^t  pipe  6  inches  diameter  has  a  right-angled  bend  connected 
^  the  end  of  ilie  bend  being  closed  by  a  flange. 
\  oaDtaifis  water  at  a  pressure  of  700  !bs,  per  sq.  inch.   Determine 
I  In  ttie  bults  at  both  ends  of  tlie  elbow, 

2—2 


^^ 


20 


HYDRAULICS 


(7)    The  end  of  a  dock  caisson  is  as  shown  in  Fig.  16  and  the  water 
level  is  AB. 

Determine  the  whole  pressure  and  the  centre  of  pressure. 


Ut —  44)  0 -H 

Fig.  16. 

(8)  An  U  tube  contains  oil  having  a  specific  gravity  of  1*1  in  the  lower 
part  of  the  tube.  Above  the  oil  in  one  limb  is  one  foot  of  water,  and  above 
the  other  2  feet.    Find  the  difference  of  level  of  the  oil  in  the  two  limbs. 

(9)  A  pressure  gauge,  for  use  in  a  stokehold,  is  made  of  a  glass  U  tabs 
with  enlarged  ends,  one  of  which  is  exposed  to  the  pressure  in  the  stokehold 
and  the  other  connected  to  the  outside  air.  The  gauge  is  filled  with  water 
on  one  side,  and  oil  having  a  specific  gravity  of  0*95  on  the  other — ^the 
surface  of  separation  being  in  the  tube  below  the  enlarged  ends.  If  the 
area  of  the  enlarged  end  is  fifty  times  that  of  the  tube,  how  many  inches  of 
water  pressure  in  the  stokehold  correspond  to  a  displacement  of  one  inch 
in  the  surface  of  separation  ?    [Lond.  Un.  1906.] 

(10)  An  inverted  oil  gauge  has  its  upper  U  filled  with  oil  having  a 
specific  gravity  of  0*7955  and  the  lower  part  of  the  gauge  is  filled  with 
water.  The  two  limbs  are  then  connected  to  two  different  points  on  a  pipe 
in  which  there  is  flowing  water. 

Find  the  difference  of  the  pressure  at  the  two  points  in  the  pipe  when 
the  difference  of  level  of  the  oil  surfaces  in  the  limbs  of  the  U  is 
15  inches. 

(11)  An  opening  in  a  reservoir  dam  is  closed  by  a  plate  8  feet  sqnare^ 
which  is  hinged  at  the  upper  horizontal  edge ;  the  plate  is  inclined  at  an 
angle  of  60"*  to  the  horizontal,  and  its  top  edge  is  12  feet  below  the  suifaoe 
of  the  water.  If  this  plate  is  opened  by  means  of  a  chain  attached  to  the 
centre  of  the  lower  edge,  find  the  necessary  pull  in  the  chain  if  its  line  of 
action  makes  an  angle  of  45°  with  the  plate.  The  weight  of  the  plate  ia 
400  pounds.    [Lond.  Un.  1905.] 

(12)  The  width  of  a  lock  is  20  feet  and  it  is  closed  by  two  gates  at  each 
end,  each  gate  being  12'  long. 

If  the  gates  are  closed  and  the  water  stands  16'  above  the  bottom  on  one 
side  and  4'  on  the  other  side,  find  the  magnitude  and  position  of  the  resultant 
pressure  on  each  gate,  and  the  pressure  between  the  gates.  Show  also  that 
the  reaction  at  the  hinges  is  equal  to  the  pressure  between  the  gates.  One 
cubic  foot  of  water=62-5  lbs.    [Lond.  Un.  1905.] 


CHAPTER  11. 

FLOATING  BODIES. 

IB.      ConditionB  of  equilibxium. 

Wlieii    a  body  floats  in  a  fluid  the  surface  of  the  body  in 
contact  -^th   the  fluid  is  subject  to  hydrostatic  pressures,  the 
intensity    of    pressure  on  any  element  of  the  surface  depend- 
ing upon    its   depth  below  the  surface.     The  resultant  of  the 
vertical    components  of  these  hydrostatic    forces  is  called  the 
buoyancy,  and  its  magnitude  must  be  exactly  equal  to  the  weight 
ol  the  body,  for  if  not  the  body  will  either  rise  or  sink.    Again 
the    horizontal    components    of    these    hydrostatic   forces    must 
be  in  equilibrium  amongst  themselves,  otherwise  the  body  will 
have  a  lateral  movement. 

The  position  of  equilibrium  for  a  floating  body  is  obtained 
when  (a)  the  buoyancy  is  exactly  equal  to  the  weight  of  the 
body,  and  (b)  the  vertical  forces — the  weight  and  the  buoyancy — 
act  in  the  same  vertical  line,  or  in  other  words,  in  such  a  way  as 
to  produce  no  couple  tending  to  make  the  body  rotate. 

Let  G,  Fig.  17,  be  the  centre  of  gravity  of  a  floating  ship  and 
BK,  which  does  not  pass  through  G,  the  line  of  action  of  the 
resultant  of  the  vertical  buoyancy  forces.     Since  the  buoyancy 


Fig.  17. 


Fig.  18. 


must  equal  the  weight  of  the  ship,  there  are  two  parallel  forces 
each  equal  to  W  acting  through  G  and  along  BK  respectively, 
and  these  form  a  couple  of  magnitude  Wx,  which  tends  to  bring 
the  ship  into  the  position  shown  in  Fig.  18,  that  is,  so  that  BK 


22 


HYDRAULICS 


passes  through  G.  The  above  condition  (6)  can  therefore  only  be 
realised,  when  the  resultant  of  the  buoyancy  forces  passes  through 
the  centre  of  gravity  of  the  body.  K,  however,  the  body  is 
displaced  from  this  position  of  equilibrium,  as  for  example  a  ship 
at  sea  would  be  when  made  to  roll  by  wave  motions,  there  will 
generally  be  a  couple,  as  in  Fig.  17,  acting  upon  the  body,  which 
should  in  all  cases  tend  to  restore  the  body  to  its  position  of 
equilibrium.  Consequently  the  floating  body  will  oscillate  about 
its  equilibrium  position  and  it  is  then  said  to  be  in  stable  eqm- 
librium.  On  the  other  hand,  if  when  the  body  is  given  a  small 
displacement  from  the  position  of  equilibrium,  the  vertical  force* 
act  in  such  a  way  as  to  cause  a  couple  tending  to  increase  the 
displacement,  the  equilibrium  is  said  to  be  unstable. 

The  problems  connected  with  floating  bodies  acted  upon  by 
forces  due  to  gravity  and  the  hydrostatic  pressures  011I7, 
resolve  themselves  therefore  into  two, 

(a)  To  find  the  position  of  equilibrium  of  the  body. 

(b)  To  find  whether  the  equilibrium  is  stable. 

19.    Principle  of  ArchimedeB. 

When  a  body  floats  freely  in  a  fluid  the  weight  of  the  body  i» 
equal  to  the  weight  of  the  fluid  displaced. 

Since  the  weight  of  the  body  is  equal  to  the  resultant  of  the 
vertical  hydrostatic  pressures,  or  to  the  buoyancy,  this  principle 
will  be  proved,  if  the  weight  of  the  water  displaced  is  shown  to  ^ 
equal  to  the  buoyancy. 

Let  ABC,  Fig.  19,  be  a  body  floating  in  equilibrium,  AC  bei^^ 
n  the  surface  of  the  fluid. 

thv 
16  L 

(11 
which  « 
angle  of  i 
of  the  wal 
centre  of  th. 
action  make(» 
400  pounds.    [ 

(12)    Thewi. 
end,  each  gate  be. 

sideand^d'ontheoa^^*^'  element  ab  of  the  surface,  of  area  a  and 
pressoie  on  each  gattPf  the  element  being  inclined  at  any  angle  6  to 
the  reaction  at  the  hinen,  if  to  is  the  weight  of  unit  volume  of  the 
cubic  foot  of  water«62  sure  on  the  area  a  is  wha,  and  the  vertical 
sure  is  seen  to  be  wha  cos  0, 


Fig.  19. 


FLOATIJfO   BODIES 


23 


IniRgine  now  a  vertical  cylinder  standing  on  this  area^  the  top 
^  which  is  in  the  snrface  AC. 

le  horizontal  sectional  area  of  this  cylinder  m  a  cos^,  the 

is  ha  cos  0  and  the  weight  of  the  water  filling  this  volume 

fshaccmB^  and  is,   therefore,  equal  to   the   buoyancy  on   the 

stmilar  cylinders  be  imagined  on  all  the  little  elements 

I  area  which  make  up  the  whole  immersed  surface,  the  total 

iome  of  these  cylinders  is  the  volume  of  the  water  displaced, 

L  ihe  total  buoyancy  is,  therefore,  the  weight  of  this  displaced 

li   the   body   is   wholly  immersed   as    in 

iy  is  supposed  to  he  mmle  up  of  small 

ffrticttl  cylinders  intersecting  the  surface  of 

j\\e  hAj  in  the  elements  of  area  ah  and  ab\ 

ch  are  inclined  to  the  horizontal  at  angles 

^  and  having  areas  a  and  aj  resi^ctively, 

w  vertical  component  of  the  pressure  on  ah 

111]  be  itha  cos  0  and  on  a'^'  wnll  be  whiaj  cos  ^. 

« C(M§  0  must  equal  aj  cos  ^,  each  being  ^^^^  ^*'' 

I  to  the  horizontal  section  of  the  small  cylinder.     The  whole 
is  therefore 

^wha  cos  0  -  Swh^at  cos  <^j 

fiBafam  equal  to  the  weight  of  the  water  displaced* 

h  this  case  if  the  fluid  be  assumed  to  be  of  conBtant  density 

^i  th  weight  of  the  body  as  equal  to  the  weight  of  the  Huid 

r^  tie  same  volume,  the  body  will  float  at  any  depth.    The 

F«ligtt*^  increase  in  the  weight  of  the  body  would  cause  it  to 

(«iik  njitil  it  reached  the  bottom  of  the  vessel  containing  the  Huid, 

^litfe  a  very  small  diminution  of  its  weight  or  increase  in  ita 

^*^iiiii(?  would  cause  it  to  rise  immediately  to  the  surface.     It 

wcruld  cle?fcrly  be  practically  impossible  to  maintain  such  a  body 

I  *n  t'qiiilibrium,  by  endeavouring  to  adjust  the  weight  of  the  body, 

|%piimping  out*  or  letting  in»  water,  as  has  been  attempted  in  a 

|ni  type  of  submarine  boat.   In  recent  submarines  the  lowering 

'^liringof  the  boat  are  controlled  by  vertical  screw  propellers, 

20.    Centre  of  buoyancy. 

^irict*  the  bunyancy  on  any  element  of  area  is  the  weight  of 

Tierrical  cylinder  of  the  fluid  above  this  area,  and  that  the 

\  ancy  is  the  sum  of  the  weights  of  all  these  cylinders,  it 

■llowH,  that  the  resultant  of  the  buoyancy  forces  rami 

1  through  the  eentre  of  gravity  of  the  water  displaced,  and  this 

Bt  is,  therefore,  called  the  Centre  of  Buoyancy. 


24 


HYDRAUUCS 


21.    Condition  of  stability  of  equilibrium. 

Let  AND,  Fig.  21,  be  the  section  made  by  a  vertical  plane 
containing  G  the  centre  of  gravity  and  B  the  centre  of  buoytincy 
of  a  floating  vessel,  AD  being  the  surface  of  the  fluid  when  the 
centre  of  gravity  and  centre  of  buoyancy  are  in  the  same  vertical 
line. 


Fig.  21. 


Fig.  92. 


Let  the  vessel  be  heeled  over  about  a  horizonal  axis,  PE  being    J 
now  the  fluid  surface,  and  let  Bi  be  the  new  centre  of  buoyancy,    J 
the  above  vertical  sectional  plane  being  taken  to  contain  G,  B, 
and  Bi.    Draw  BiM,  the  vertical  through  Bi,  intersecting  the  Kne 
GB  in  M.    Then,  if  M  is  above  G  the  couple  W .  x  will  tend  to 
restore  the  ship  to  its  original  position  of  equilibrium,  but  if  M  i&^ 
below  G,  as  in  Fig.  22,  the  couple  will  tend  to  cause  a  furthe^^ 
displacement,  and  the  ship  will  either  topple  over,  or  will  heel  ova^-^ 
into  a  new  position  of  equilibrium. 

In  designing  ships  it  is  necessary  that,  for  even  large  displac^^. 
ments  such  as  may  be  caused  by  the  rolling  of  the  vessel,  tkr^^ 
point  M  shall  be  above  G.  To  determine  M,  it  is  necessary  "fc^ 
determine  G  and  the  centres  of  buoyancy  for  the  two  positiox:ii 
of  the  floating  body.  This  in  many  cases  is  a  long  and  somewH^ 
tedious  operation. 

22.    Small  displacements.    Metacentre. 

When  the  angular  displacement  is  small  the  point  M  is  call^^ 
the  Metacentre,  and  the  distance  of  M  from  G  can  be  calculated. 

Assume  the  angular  displacement  in  Fig.  21  to  be  small  an(^ 
equal  to  0. 

Then,  since  theVolume  displacement  is  constant  the  volume  of 
the  wedge  CDE  muk  equal  CAF,  or  in  Fig.  23,  CiCaDE  must  equal 
CiCAF. 


25 


Let  Gi  and  Gj  be  the  centres  of  gravity  of  the  wedges  C1C2AF 
and  CiCsDE  reflectively. 


df     D 

Fig.  23. 

The  heeling  of  the  ship  has  the  effect  of  moving  a  mass  of 
witer  equal  to  either  of  these  wedges  from  Gi  to  G2,  and  this 
movement  causes  the  centre  of  gravity  of  the  whole  water 
di^Iaced  to  move  from  B  to  Bj. 

Let  Z  be  the  horizontal  distance  between  Gi  and  Gj,  when  FE 
is  horizontal,  and  S  the  perpendicular  distance  from  B  to  BiM. 

Let  V  be  the  total  volume  displacement,  v  the  volume  of  the 
^wredge  and  tc  the  weight  of  unit  volume  of  the  fluid. 
Then  tr.v.Z  =  'M7.  V.  S 

=  w  .Y .  BM .  sin  0, 

Or, ance ^ is  small,  ='m?.V.BM.^  (1). 

The  restoring  couple  is 

tr.V.HG  =  ir.V.GM.^ 

=  ii?.V.(BM-BG)^ 
=^w.v,Z-w.Y.BG,e  (2). 

Bat  If .  v .  Z  =  twice  the  sum  of  the  moments  about  the  axis 
CiCijOf  all  the  elements  such  as  acdh  which  make  up  the  wedge 
CCiDE. 

Taking  ah  as  ar,  hf  is  x6,  and  if  ac  is  SZ,  the  volume  of  the 
element  is  ia^^.aZ. 

The  centre  of  gravity  of  the  element  is  at  fa;  from  C1C2. 

3      


Therefi 


:ore 


tr  .  r  .  Z  =  2w 


7?dl 


p 


.(3). 


But,  -^  is  the  Second  Moment  or  Moment  of  Inertia  of  the 

element  of  area  aceb  about  C2C1,  and  2  /     .5  -  is,  therefore,  the 

.'0    'J 

Moment  of  Inertia  I  of  the  water-plane  area  AC1DC2  about  C1C2- 
Therefore  w  ,v  ,Z  =  w  ,1.0 (4). 


26  HYDRAULICS 

The  restoring  couple  is  then 

wI0-w.Y.BG.e. 
If  this  18  positive,  the  equilibrium  is  stable,  but  if  negative  it  is 
unstable. 

Again  since  from  (1) 

wv.Z  =  w.Y.BU.e, 
therefore  w.Y.BM.0  =  wI0y 

and  BM  =  ;5    (5), 

If  BM  is  greater  than  BG  the  equilibrium  is  stable,  if  less  than 
BG  it  is  unstable,  and  the  body  will  heel  over  until  a  new  position 
of  equilibrium  is  reached.  If  BG  is  equal  to  BM  the  equilibrium 
is  said  to  be  neutral. 

The  distance  GM  is  called  the  Metacentric  Height,  and  varies 
in  various  classes  of  ships  from  a  small  negative  value  to  a  positive 
value  of  4  or  5  feet. 

When  the  metacentric  height  is  negative  the  ship  heels  until 
it  finds  a  position  of  stable  equilibrium.  This  heeling  can  be 
corrected  by  ballasting. 

ExampU,  A  ship  has  a  displacement  of  15,400  toDS,  and  a  dranght  of  27*5  feet. 
The  height  of  the  centre  of  buoyancy  from  the  bottom  of  the  keel  is  15  feet. 

The  moment  of  inertia  of  the  horizontal  section  of  the  ship  at  the  water  line 
is  9,400,000  feet«  units. 

Determine  the  position  of  the  centre  of  gravity  that  the  metacentric  height  shall 
not  be  less  than  4  feet  in  sea  water. 

M00^000x64 
16,400x2240 
=  171  feet. 
Height  of  metacentre  from  the  bottom  of  the  keel  is,  therefore,  32*1  feet. 
As  long  as  the  centre  of  gravity  is  not  higher  than  0*6  feet  above  the  anrfiace  of 
the  water,  the  metacentric  height  is  more  than  4  feet. 

23.    Stability  of  a  rectangular  pontoon. 

Let  RFJS,  Fig.  24,  be  the  section  of  the  pontoon  and  G  its 
centre  of  gravity. 

Let  VE  be  the  surface  of  the  water  when  the  sides  of  the 
pontoon  are  vertical,  and  AL  the  surface  of  the  water  when  the 
pontoon  is  given  an  angle  of  heel  0, 

Then,  since  the  weight  of  water  displaced  equals  the  weight  of 
the  pontoon,  the  area  AFJL  is  equal  to  the  area  VFJE. 

Let  B  be  the  centre  of  buoyancy  for  the  vertical  position, 
B  being  the  centre  of  area  of  VFJE,  and  Bi  the  centre  of  buoyancy 
for  the  new  position,  Bi*  being  the  centre  of  area  of  AFJL.  Then 
the  line  joining  BGr  must  be  perpendicular  to  the  surface  VE  and 

*  In  the  Fig.,  B,  is  not  the  centre  of  area  of  AFJL,  as,  for  the  sake  of  cleamesi^ 
it  is  farther  removed  from  B  than  it  actually  should  be. 


FLOATING  BODIES 


27 


s  the  direction  in  which  the  buoyancy  force  acts  when  the  sides 
){  the  pontoon  are  vertical,  and  BiM  perpendicular  to  AL  is  the 
direction  in  iwrliich  the  buoyancy  force  acts  when  the  pontoon  is 
heeled  over  ttrough  the  angle  0.    M  is  the  metacentre. 


Fig.  24. 

The  forces  acting  on  the  pontoon  in  its  new  position  are,  W  the 
weight  of  the  pontoon  acting  vertically  through  G  and  an  equal  and 
parallel  buoyancy  force  W  through  Bj . 

There  is,  therefore,  a  couple,  W  •  HG,  tending  to  restore  the 
pontoon  to  its  vertical  position. 

If  the  line  BiH  were  to  the  right  of  the  vertical  through  G,  or 
in  other  words  the  point  M  was  below  G,  the  pontoon  would  be  in 
unstable  equilibrium. 

The  new  centre  of  buoyancy  Bi  can  be  found  in  several  ways. 
The  following  is  probably  the  simplest. 

The  figure  AFJL  is  formed  by  moving  the  triangle,  or  really 
the  wedge-shaped  piece  CEL  to  CVA,  and  therefore  it  may  be 
imagined  that  a  volume  of  water  equal  to  the  volume  of  this  wedge 
is  moved  from  G2  to  Gi .  This  will  cause  the  centre  of  buoyancy 
to  move  parallel  to  GiGj  to  a  new  position  Bi,  such  that 

BBi  X  weight  of  pontoon  =  G1G2  x  weight  of  water  in  CEL. 

Let  b  be  half  the  breadth  of  the  pontoon, 
I  the  length, 

D  the  depth  of  displacement  for  the  upright  position, 
d  the  length  LE,  or  AV, 
and       w  the  weight  of  a  cubic  foot  of  water. 

Then,  the  weight  of  the  pontoon 

W  =  2b.B.l.w 

and  the  weight  of  the  wedge  CLE  =  -^  ^  I  .w. 


28  H7DRAUUCS 


Therefore  BBi.26.D  =  - 


2 


and  BBi  =  -tjz  GriGg. 

Besolving  BBi  and  GiGj,  which  are  parallel  to  each  other,  along 
and  perpendicular  to  BM  respectively, 

d  r.  ^       d  /2^A      bd     6»tan^ 


B.Q  =  ^G.K=^(|26)  = 


3D        3D    ' 

A  TiT>-Tin  G^  _hd  d  _  iT  _  ^'tan'tf 

and  ^*^-^»'^G.K"3D26~6D"     6D     ' 

To  find  the  distance  of  the  point  Mfrom  G  and  the  value  of  the 
restoring  couple.  Since  BiM  is  perpendicular  to  AL  and  BM  to 
YE,  the  angle  BMBi  equals  6. 

Therefore  QM  =  B,Q  cot«  =  ^  cottf  =  ^. 

Let  z  be  the  distance  of  the  centre  of  gravity  G  from  C. 
Then  QG  =  QC -z  =  BC-BQ -a 

_D     5'tan*g 
2         6D        '• 
Therefore 

rnur    rnur     nr      ^*      D^fc'tan'tf^ 

And  since  HG  =  GM  sin  0, 

the  righting  couple, 

w  Tin    w  •    af^'      D  .  5'tan*g  .    \ 

The  distance  of  the  metacentre  from  the  point  B,  is 
QM  +  QB  =  B,Q  cot«  +  ^^^ 

3D  "^      6D     • 

When  0  is  small,  the  term  containing  tan*tf  is  negligible,  and 

This  result  can  be  obtained  from  formula  (4)  given  in 
section  22. 

I  for  the  rectangle  is  j\l  (25)»  =  ilb\  and  V  =  2bl>l. 

Therefore  BM  =  ^. 

If  BG  is  known,  the  metacentric  height  can  now  be  found. 


FLOATING  BODIES 


29 


Example,  A  pontoon  hM  a  displaoemeDt  of  200  tons.  Its  length  is  50  feet. 
The  centre  of  gravity  is  1  foot  above  the  centre  of  area  of  the  oross  section.  Find 
tbe  fareadth  and  depth  of  the  pontoon  so  that  for  an  angular  displacement  of  10  degrees 
the  metaoentre  shall  not  be  less  than  8  feet  from  the  centre  of  gravity,  and  the  free- 
board shall  not  be  less  than  2  feet. 

BeCerring  to  Fig.  24,  G  is  the  centre  of  gravity  of  the  pontoon  and  O  is  the 
eeotre  of  the  eroes  section  BJ. 

Then,  GO =1  foot, 

Fo=2  feet, 

GM=3  feet. 
Ijet  D  be  the  depth  of  displacement.    Then 

D  X  26  X  62-4  x  50  lbs. =200  tons  x  2240  lbs. 

Therefore  D6=71-5 (1). 

The  height  of  the  centre  of  bnoyancy  B  above  the  bottom  of  pontoon  is 

BT=JD. 
Since  the  free-board  is  to  be  2  feet, 

OT=4(D-|-2). 
Then  B0  =  1'  and  BG=2'. 

Therefore  BM=5'. 

Bnt  BM  =  QM  +  BQ 


__^      6»tan«g 
■"8D"*"      6D     •' 


(2). 


Moltiplying  namerator  and  denominator  by  6,  and  substitating  from  equation  (1) 

6»        &»tan«^    ^, 
-=6, 


from  which 

therefore 

and 


214-6  ^      429 

6»(2  +  (-176)«)=5x429, 

6=101  ft.. 

D  =  71  ft., 

The  breadth  B= 20-2  ft.) 

,,    depth 


3  =  20-2  ft.) 
=  7-lft.| 


Ans, 


24.    Stability  of  a  floating  vessel  containing  water. 

If  a  vessel  contains  water  with  a  free  surface,  as  for  instance 
the  compartments  of  a  floating  dock,  such  as  is  described  on  page 
31,  the  surface  of  the  water  in  these  compartments  will  remain 
horizontal  as  the  vessel  heels  over,  and  the  centre  of  gravity  of 
the  water  in  any  compartment  will  change  its  position  in  such 
a  way  as  to  increase  the  angular  displacement  of  the  vessel. 

In  considering  the  stability 
of  such  vessels,  therefore,  the 

taming  moments   due   to    the      jj  ^^^'""""'    Q 

water   in   the  vessel   must  be 
taken  into  account.  -^ 

As   a  simple  case  consider       E 
the  rectangular  vessel.  Fig.  2o,       H  ^    \      jj^ 
which,  when  its  axis  is  vertical, 
floats  with  the  plane  AB  in  the  ^^8-  ^5. 


■€^ 


K 
D 


30 


HYDRAULICS 


surface  of  the  fluid,  DE  being  the  surface  of  the  fluid  in  the 
vessel. 

When  the  vessel  is  heeled  through  an  angle  ^,  the  surface  of 
fluid  in  the  vessel  is  KH. 

The  effect  has  been,  therefore,  to  move  the  wedge  of  fluid  OEH 
to  ODK,  and  the  turning  couple  due  to  this  movement  is  tu .  t?  .Z, 
V  being  the  volume  of  either  wedge  and  Z  the  distance  between 
the  centre  of  gravity  of  the  wedges. 

If  2b  is  the  width  of  the  vessel  and  I  its  length,  v  is-^l  tan  6, 

Z  is  |6  tan  ^,  and  the  turning  couple  is  w  ft'  I  tan*  0. 

If  0  is  small  wvZ  is  equal  to  wI0y  I  being  the  moment  of  inertia 
of  the  water  surface  KH  about  an  axis  through  0,  as  shown  in 
section  22. 

For  the  same  width  and  length  of  water  surface  in  the 
compartment,  the  turning  couple  is  the  same  wherever  the 
compartment  is  situated,  for  the  centre  of  gravity  of  the  wedge 
OHE,  Fig.  26,  is  moved  by  the  same  amount  in  all  cases. 

If,  therefore,  there  are  free  fluid  surfaces  in  the  floating  vessel, 
for  any  small  angle  of  heel  ^,  the  tippling-moment  due  to  these 
surfaces  is  'S.wIOy  I  being  in  all  cases  the  moment  of  inertia  of  the 
fluid  surface  about  its  own  axis  of  oscillation,  or  the  axis  through 
the  centre  of  gravity  of  the  surface. 


E 
H 


0-4 


pH 


D 


Fig.  26. 


Fig.  27. 


25.    Stability  of  a  floating  body  wholly  immersed. 

It  has  already  been  shown  that  a  floating  body  wholly  im- 
mersed in  a  fluid,  as  far  as  vertical  motions  are  concerned,  can 
only  with  great  difficulty  be  maintained  in  equilibrium. 

If  further  the  body  is  made  to  roll  through  a  small  angle,  the 
equilibrium  will  be  unstable  unless  the  centre  of  gravity  of  the 
body  is  below  the  centre  of  buoyancy.  This  will  be  seen  at  once 
on  reference  to  Fig.  27. .  Since  the  body  is  wholly  immersed  the 
centre  of  buoyancy  cannot  change  its  position  on  the  body  itself, 
as  however  it  rolls  the  centre  of  buoyancy  must  be  the  centre  of 
gravity  of  the  displaced  water,  and  this  is  not  altered  in  form  by 


FLOATING  BODIES 


31 


any  movement. of  the  body.  K,  therefore,  G  is  above  B  and  the 
body  be  given  a  small  angular  displacement  to  the  right  say,  G 
will  move  to  the  right  relative  to  B  and  the  couple  will  not  restore 
the  body  to  its  position  of  equilibrium. 

On  the  other  hand,  if  G  is  below  B,  the  couple  will  act  so  as  to 
brin^  the  body  to  its  position  of  equilibrium. 

26.     Floatiiig  docks. 

Figs.  28  and  29  show  a  diagrammatic  outline  of  the  pontoons 
forming  a  floating  dock,  and  in  the  section  is  shown  the  outline  of 
a  ship  on  the  dock. 


Fig.  29. 

To  dock  a  ship,  the  dock  is  sunk  to  a  sufficient  depth  by 
admitting  water  into  compartments  formed  in  the  pontoons,  and  the 
ship  is  brought  into  position  over  the  centre  of  the  dock. 

Water  is  then  pumped  from  the  pontoon  cliarubers,  and  the 
dock  in  consequence  rises  until  the  ship  just  rests  on  the  keel 
blocks  of  the  dock.  As  more  water  is  pumped  from  the  pontoons 
the  dock  rises  with  the  ship,  which  may  thus  be  lifted  clear  of 
the  water. 

Let  Gi  be  the  centre  of  gravity  of  the  ship,  Ga  of  tlie  dock  and  its 
water  ballast  and  G  the  centre  of  gra\4ty  of  the  dock  and  the 
ship. 

The  position  of  the  centre  of  gi-avity  of  the  dock  >W11  vary 


32  HTDRAULICS 

relative  to  the  bottom  of  the  dock,  as  water  is  pumped  from  the 
pontoons. 

As  the  dock  is  raised  care  must  be  taken  that  the  metacentre 
is  above  G  or  the  dock  will  "  list." 

Suppose  the  ship  and  dock  are  rising  and  that  WL  is  the 
water  line. 

Let  Ba  be  the  centre  of  buoyancy  of  the  dock  and  Bi  of  the 
portion  of  the  ship  still  below  the  water  line. 

Then  if  Vi  and  V2  are  the  volume  displacements  below 
the  water  line  of  the  ship  and  dock  respectively,  the  centre  of 
buoyancy  B  of  the  whole  water  displaced  divides  BsBj,  so  that 

BBi_Va 

BBa"Vx- 

The  centre  of  gravity  G  of  the  dock  and  the  ship  divides  GiGt 
in  the  inverse  ratios  of  their  weights. 

As  the  dock  rises  the  centre  of  gravity  G  of  the  dock  and  the 
ship  must  be  on  the  vertical  through  B,  and  water  must  be 
pumped  from  the  pontoons  so  as  to  fulfil  this  condition  and  as 
nearly  as  possible  to  keep  the  deck  of  the  dock  horizontal. 

The  centre  of  gravity  Gi  of  the  ship  is  fixed,  while  the  centre  of 
buoyancy  of  the  ship  Bi  changes  its  position  as  the  ship  is  raised. 

The  centre  of  buoyancy  Ba  of  the  dock  will  also  be  changing, 
but  as  the  submerged  part  of  the  dock  is  symmetrical  about  its 
centre  lines,  Ba  will  only  move  vertically.  As  stated  above,  B 
must  always  lie  on  the  line  joining  Bi  and  Ba,  and  as  G  is  to  be 
vertically  above  B,  the  centre  of  gravity  Ga  and  the  weight  of 
the  pontoon  must  be  altered  by  taking  water  from  the  various 
compartments  in  such  a  way  as  to  fulfil  this  condition. 

Quantity  of  water  to  he  'pumped  from  the  pontoons  in  raising  the 
doch.  Let  V  be  the  volume  displacement  of  the  dock  in  its  lowest 
position,  Vo  the  volume  displacement  in  its  highest  position.  To 
raise  the  dock  mthout  a  ship  in  it  the  volume  of  the  water  to  be 
pumped  from  the  pontoons  is  V  -  Vo . 

If,  when  the  dock  is  in  its  highest  position,  a  weight  W  is  put 
on  to  the  dock,  the  dock  will  sink,  and  a  further  volume  of  water 

W         .  . 

—    cubic  feet  will  be  required  to  be  taken  from  the  pontoons  to 

raise  the  dock  again  to  its  highest  position. 

To  raise  the  dock,  therefore,  and  the  ship,  a  total  quantity  of 
water 

w 
cubic  feet  will  have  to  be  taken  from  the  pontoons. 


FLOATING   BODIES 


33 


Ifft^vlf,     K  lUmtmg  dock  ms  fibotrtt  di mentioned  m  Fir.  2B  is  made  np  of  s 
not!  540  feet  long  x  OG  fc«l  wide  x  14-75  feet  deep,  two  aide  pontooDt 
gx  13  ft»et  wide  y  4^8  feet  de«p,  the  bottom  of  these  poutoonB  being  ^ 
"<.*vf-  Ihc  bcittoin  of  the  dock^  aud  two  side  chambers  on  the  top  of  thm  I 
poatooti  447  feel  long  by  8  feet  deep  imd  2  feet  wide  at  the  top  and  8  fe«l  ftt  | 
.        itoiD.     The  kf^l  blocks  may  be  taken  as  4  feet  deep. 
I  h«  dock  is  Ui  lift  »  sbip  of  15,400  ttiris  dij^plncetneul  and  27'  6"  dratlght. 
l)ttj^nriiine  the  amouDt  of  water  that  mu^t  be  pumped  from  the  dock,  to  raJBe 
ifi  tm  that  the  deck  of  the  lowest  pontoon  is  m  the  water  surface. 
»:ji  the  xbip  jnst  t^kes  to  the  keel  bloeks  on  the  dock^  the  bottom  of  the 
atw  *  im  27*5' ^  14 '75' +  4' =  46 '25  feet  below  the  water  line, 

Tht  voloiik^  displaovment  of  the  dock  is  then  I 

14-76  *  MO  ^  96  ^'  2  ^  44-25  X  X3  X  aSO  +  447  X  8  X  5'  =  1,3K7,6(H)  cabic  feet,  I 

Tb«  foltime  of  dock  diaplacemeat  when  ih©  deck  is  just  awash  m  I 

540  X  96  M  14-75  +  2  k  3B0  x  13'  >^  ( l4*7o  -  2)  =  H90,OC)O  mhm  feel.  I 

The  TolTim«  djsplacemeat  of  the  ahip  ii  I 

15.400  >c 2240     e.rt^^      t-    ,    , 
^7 = 640, 000  ctibio  feet , 

ASid  tlik  ttfOftlt  Ihe  weight  of  the  ship  in  cubic  feet. 

of  tb«  ^1,000  citbic  feel  di»placemeni  wheu  the  ship  is  olear  of  the  water^ 
&S1  /Wll  eotiic  feel  J!i  therefore  reiiTtired  to  support  Ihe  dock  aloDe. 

'I  riiply  lei  tninm  the  dock  through  31*5  feet  the  amount  of  wat^r  to  l^e  pumped  ia 

Teretkcs  i>f  the  diaplacemenlH^  and  is,  tlierefon*,  347i*K]0  cubic  feet. 

"'^   lh#  ehip  with  the  dock  an   additiooal  ^40,000  cubits  feet  must  be 

':7aD  the  pontoous* 

:%)  q^tAolitjt  Iherefore,  to  be  taken  from  the  pontoons  from  the  time  the 

kkem  to  iht  ked  blocks  lo  when  the  pontoon  deck  is  in  the  HurCaoe  of  the 

^  887.600  cubic  feet  ^  25,380  tons,  , 

27.     Stability  of  the  floatiBg  dock. 
As  some  (jI  the  compartments  of  tlie  dock  are  partially  filled 
w^iiierT  it  in  necesHaiy,  in  considering  the  stability,  to  take 
•    'f  the  tipplirig-moments  eauaed  by  the  movement  of  the 
^  r.ce  of  the  water  in  these  compartments. 

kL         Ii  G  i»  the  centre  of  gra^nty  of  the  dock  and  ship  on  the 
"  ih^^c^  B  the  centre  of  bnoyancyj  I  the  moment  of  inertia  of  the 
n  of  the  ship  and  dock  by  the  water-plane  abont  the  axis  of  ] 
ti,  and  Ii,  Is  etc.  the  moments  of  inertia  of  the  water 
in  the  compartments  about  their  axes  of  oscillation^  the 
n^    moment  when    the   dock    receives   a   amaU    angle    of 

ifW- ti? (Yi  +  Va) BO^- t£vi9  (I,  + 1,  +...).  I 

llir   moment  of    inertia    of    the  water-plane   section  YarieAl 
eoiu»iclerahly  ub    the    dock   ib   raised,  and   the  stability  varies 

When  tht;  i*hip  is  immersed  in  the  water,  I  is  equal  to  the 
jnamenT  of  inertia  of  the  horizontal  section  of  the  ship  at  the 
ter    fiarfAC«»,   together   with    the   moment   of    inertia   of    the 
■   '    -ection    of    the   side    pontoons,    about    the    axis   of 


t-  a- 


li^Mlktflki 


34  HTDRAUUCS 

When  the  tops  of  the  keel  blocks  are  just  above  the  SDrEaea: 
of  the  water,  the  water-plane  is  only  that  of  the  side  pontoon^, 
and  I  has  its  minimum  value.  If  the  dock  is  L-shaped  as 
Fig.  30,  which  is  a  very  convenient  form 
for  some  purposes,  the  stability  when 
the  tops  of  the  keel  blocks  come  to  the 
surface  simply  depends  upon  the  moment 
of  inertia  of  the  area  AB  about  an  axis 
through  the  centre  of  AB.  This  critical 
point  can,  however,  be  eliminated  by 


ii 


fitting  an  air  box,  shown  dotted,  on  the  p.     ^ 

outer  end  of  the  bottom  pontoon,  the 

top  of  which  is  slightly  higher  than  the  top  of  the  keel  blockB* 

Example.    To  find  the  height  of  the  metacentre  above  the  centre  of  baoymoflj  if " 
the  dock  of  Fig.  28  when 


{a\    the  ship  just  takes  to  the  keel  blocks, 

{h)    the  keel  is  just  clear  of  the  water, 

\c)    the  pontoon  deck  is  just  above  the  water. 


Take  the  moment  of  inertia  of  the  horizontal  section  of  the  ship  at  tht 
water  line  as  9,400,000  ft>  units,  and  assume  that  the  ship  is  symmetricallj 
placed  on  the  dock,  and  that  the  dock  deck  is  horizontal.  The  horizontal  distanee 
between  the  centres  of  the  side  tanks  is  111  ft. 

(a)     Total  moment  of  inertia  of  the  horizontal  section  is 

9, 400,000  4-  2  (380  x  1 3'  x  66  S'  4-  tV  x  380  x  13») = 9,400,000  +  30,430,000  + 139,000. 
The  volume  of  displacement 

=640,000  +  1,237,600  cubic  feet. 
The  height  of  the  metacentre  above  the  centre  of  buoyancy  is  therefore 
39,968,000    «...    . 
^^=1,932,000=^^^^^*- 

(h)  When  the  keel  is  just  clear  of  the  water  the  moment  of  inertia  is 
30,569.000. 

The  volume  displacement  is 

540  X  96  X  14-75  +  380  x  2  x  13  x  (14-75  +  4-2) 

=  930,000  cubic  feet. 

Therefore  BM  =  32*8  feet. 

(c)    When  the  pontoon  deck  is  just  above  the  surface  of  the  water, 

I  =  30,569,000  +  iV  x  5  40'  X  96» 

y..  =70,269,000. 

W   "^he  volume  displacement  is  890,000  cubic  feet. 

—  CUL    . 

W  '^fow  BM= 79-8  feet. 

raise  the  (X^^  of  ^^^  centre  of  buoyancy  above  the  bottom  of  the  dock  can  be 
rn    -^ -^p  '  finding  the  centre'  of  buoyancy  of  each  of  the  parts  of  the  dock,  and 
lo  raise   »  ^  jjj  jjj^  water,  and  then  taking  moments  about  any  axis, 
water  To  find  the  height  h  of  the  centre  of  buoyancy  of  the  dock  and 

^  ship  just  comes  on  the  keel  blocks, 
oyancy  for  the  ship  is  at  15  feet  above  the  bottom  of  the  keel, 
vancy  of  the  bottom  pontoon  is  at    7*375'  from  the  bottom. 
I.-     r     i.      ^^^\.      ^  »»       side  pontoons        „     24-125'       ,, 

cubic  feet  Will  have        ,,       „    chambers       .,    i7-94'       ,; 


IXOATUIQ  BODIES 


35 


\  tnometits  &boat  the  bottom  of  the  dock 

h  (540,000  +  437,000  +  765,0004  35,760) 

=  540.000  X  33*75  +  765,000  x  7'375 

+  437,00O  X  *i4'i25  i- 3S,760  X  17  "95, 

A -19 -7  feet, 

{^)  ihe  metiio^iitre  is^  tberefi^re,  40*3'  above  the  bottom  of  the  dock.    If 

«si]iT«  of  griL^i^  of  Ibe  dcHsk  &ad  sUip  u  kuowa  the  metacentrio  height 


EXAMPLES. 

A  «titp  ^bcn  fully  loaded  has  a  total  burden  of  10,000  tons.    Find 
i  displacemeDt  in  aea  water. 

The  aides  d  a  ship  are  vertical  near  the  water  lino  and  the  area  of 
sittl  section  at  the  water  line  ia  22,000  aq.  feet.    The  total  weight 
r  sJiip  IB  10,000  tons  when  it  leaves  the  river  dock, 

tbe  diffei^iice  in  draught  in  the  dock  and  at  sea  after  the  weight 
F  ship  ii&a  been  rednoed  by  consiimption  of  ooaU  etc.,  by  1500  totis, 
>  8  be  Ibe  difference  in  draught, 
I  c  K  22.000= the  difference  in  volume  displacement 
10,000  ic  2340  _  6500x2240 
"        62-48  64 

=61S0  cnbic  feet. 
^■i*2'rSf6et 
sS  34  inches. 

Tlie  moment  of  inertia  of  the  section  at  the  water  line  of  a  boat 
foot*  tmita;   the  weight  of  the  boat  is  11 '5  tons. 

I  the  height  of  the  metacentre  above  tlie  centre  of  buoyancy. 

{%}     k  ship  has  a  total  displacement  of  15.000  tons  and  a  draught  of 


1  the  ship  is  lifted  by  a  floating  dock  so  that  the  depth  of  the  bottom 
\  ked  is  16*5  feet,  tlie  centre  of  buoyancy  is  10  feet  from  the  bottom  of 
^kael  aad  tlie  displacement  is  9000  tons. 

f  nwiiDeait  of  inertia  of  Uie  water-plane  is  7t600,000  foot*  units. 
Iiorueoiital  section  ol  the  dock*  at  tlie  plane  16*5  feet  above  the 
of  ttw  keel^  consists  of  two  rectangles  380  feet  k  11  feet,  the  distance 
i  td  the  «enlre  lines  of  the  rectangles  b^ing  114  feot- 
The  TobtBie  displacement  of  tlie  dock  at  this  level  is  1,244,000  cubic  feet. 
Tbeciotra  of  buoyancy  for  the  dock  alone  is  24*75  feet  below  the  surface 
f  water. 

line  ia]  The  centre  of  buoyancy  for  the  whole  ship  and  the  dock. 
Tlw  height  of  tlie  metacentre  above  the  centre  of  buoyancy. 


h 


fV 


A  rectangulai'  pontoon  60  feet  long  is  to  have  a  displacement  of 

%  fre^^  board  of  not  leas  than  3  feet,  and  the  metacentre  in  not  to 

than  S  feet  above  the  centre  of  gravity  when  the  angle  of  heel 

The  centre  of  gravity  coincides  with  the  centre  ol  tigure. 
tbe  vddUi  and  depth  ol  the  i>ontooii. 

3—2 


36  HYDRAULICS 

(6)  A  rectangular  pontoon  24  feet  wide,  50  feet  long  and  14  feet  deep» 
has  a  displacement  of  180  tons. 

A  vertical  diaphragm  divides  the  pontoon  longitudinally  into  two 
compartments  eadi  12  feet  wide  and  50  feet  long.  In  the  lower  part 
of  each  of  these  compartments  there  is  water  ballast,  the  8iir&<»  of  the 
water  being  free  to  move. 

Determine  the  position  of  the  centre  of  gravity  of  the  pontoon  that  it 
may  be  stable  for  small  displacements. 

(7)  Define  "metacentric  height**  and  show  how  to  obtain  it  graphicallj 
or  otherwise.  A  ship  of  16,000  tons  displacement  is  600  feet  long,  60  leel 
beam,  and  26  feet  draught.  A  coefficient  of  ^  may  be  taken  in  the  moment 
of  inertia  term  instead  of  ^  to  allow  for  the  water-line  section  not  beii^ 
a  rectangle.  The  depth  of  the  centre  of  buoyancy  from  the  water  line  » 
10  feet.  Find  the  height  of  the  metacentre  above  the  water  line  and 
determine  the  position  of  the  centre  of  gravity  to  give  a  metacentric  hei^ 
of  18  inches.    [Lond.  Un.  1906.] 

(8)  The  total  weight  of  a  fully  loaded  ship  is  5000  tons,  the  water  line 
encloses  an  area  of  9000  square  feet,  and  the  sides  of  the  ship  are  vertical 
at  the  water  line.  The  ship  was  loaded  in  fresh  water.  Find  the  changB 
in  the  depth  of  immersion  after  the  ship  has  been  sufficiently  long  at  sea  to 
bum  500  tons  of  coal. 

Weight  of  1  cubic  foot  of  fresh  water  62}  lbs. 
Weight  of  1  cubic  foot  of  salt  water  64  lbs. 


CHAPTER    III. 

FLUIDS  IN  MOTION. 

28.     Steady  motion. 

The  xnotioii  of  a  fluid  is  said  to  be  steady  or  permanent,  when 
the  particles  -which  succeed  each  other  at  any  point  whatever 
have  the  same  density  and  velocity,  and  are  subjected  to  the  same 
pressure. 

In  practice  it  is  probably  very  seldom  that  such  a  condition  of 
flow  is  absolutely  realised,  as  even  in  the  case  of  the  water  flowing 
steadily  along  a  pipe  or  channel,  except  at  very  low  velocities,  the 
velocities  of  succeeding  particles  of  water  which  arrive  at  any 
point  in  the  channel,  are,  as  will  be  shown  later,  not  the  same 
either  in  magnitude  or  direction. 

For  practical  purposes,  however,  it  is  convenient  to  assume 
that  if  the  rate  at  which  a  fluid  is  passing  through  any  finite  area 
is  constant,  then  at  all  points  in  the  area  the  motion  is  steady. 

For  example,  if  a  section  of  a  stream  be  taken  at  right  angles 
to  the  direction  of  flow  of  the  stream,  and  the  mean  rate  at  which 
water  flows  through  this  section  is  constant,  it  is  convenient 
to  assume  that  at  any  point  in  the  section,  the  velocity  always 
remains  constant  both  in  magnitude  and  direction,  although  the 
velocity  at  different  points  may  not  be  the  same. 

Mean  velocity.  The  mean  velocity  through  the  section,  or  the 
mean  velocity  of  the  stream,  is  equal  to  the  quantity  of  flow  per 
unit  time  divided  by  the  area  of  the  section. 

29.     Stream  line  motion. 

The  particles  of  a  fluid  are  generally  regarded  as  flowing  along 
definite  paths,  or,  in  other  words,  the  fluid  may  be  supposed  to 
flow  in  thread-like  filaments,  and  when  the  motion  is  steady  these 
filaments  may  be  supposed  to  be  fixed  in  position. 

In  a  pipe  or  channel  of  constant  section,  the  filaments  are 
generally  supposed  to  be  parallel  to  the  sides  of  the  channel. 


38 


HYDRAULICS 


An 


DaUuiLlLe^ 


Fig.  81. 


30.  Definitions  relating  to  flow  of  water. 

Pressure  head.  The  pressure  head  at  a  point  in  a  fluid  at  rest 
has  been  defined  as  the  vertical  distance  of  the  point  from  the  free 

surface  of  the  fluid,  and  is  equal  to  — ,  where  p  is  the  pressure  per 

sq.  foot  and  w  is  weight  per  cubic  foot  of 
the  fluid.  Similarly,  the  pressure  head  at 
any  point  in  a  moving  fluid  at  which  the 

pressure  is  p  lbs.  per  sq.  foot,  is  -  feet, 

w 

and  if  a  vertical  tube,  called  a  piezometer 
tube.  Fig.  31,  be  inserted  in  the  fluid,  it 
will  rise  in  the  tube  to  a  height  h,  which 
equals  the  pressure  head  above  the  atmo- 
spheric pressure.  K  p  is  the  pressure  per 
sq.  foot,  above  the  atmospheric  pressure, 

h=  —,  but  if  p  is  the  absolute  pressure  per  — 

sq.  foot,  and  px  the  atmospheric  pressure, 

W        W 

Velocity  head.  If  through  a  small  area  around  the  point  B, 
the  velocity  of  the  fluid  is  v  feet  per  second,  the  velocity  head  is 

rt- ,  g  being  the  acceleration  due  to  gravity  in  feet  iter  second  per 

second. 

Position  head.  If  the  point  B  is  at  a  height  z  feet  above  any 
convenient  datum  level,  the  position  head  of  the  fluid  at  B  above 
the  given  datum  is  said  to  be  z  feet. 

31.  Energy  per  pound  of  water  passing  any  section  in 
a  stream  line. 

The  total  amount  of  work  that  can  be  obtained  from  every 
pound  of  water  passing  the  point  B,  Fig.  31,  assuming  it  can  fall  to 
the  datum  level  and  that  no  energy  is  lost,  is 

2  +  ^' +  2  ft.  lbs. 
w     2g 

Proof.  Work  available  due  to  pressure  head.  That  the  work 
which  can  be  done  by  the  pressure  head  per  pound  is  ~  foot 

pounds  can  be  sho'WTi  as  follows. 

Imagine  a  piston  fitting  into  the  end  of  a  small  tube  of  cross 
sectional  area  a,  in  which  the  pressure  is  h  feet  of  wator  as  in 


I 


FLUIDS   IN   MOTION  39 

g.  32,  And  let  a  small  quantity  ^Q  cubic  feet  of  water  enter  the 
ibe  and  move  the  piston  through  a  small  dis- 
knce  6as. 

Then  dQ^a.dx. 

The   iwork   done  on  the  piston  as  it  enters 

Nrill  he 

w.h.a.dx  =  w  .hdQA 

But  the  ^weight  of  5Q  cubic  feet  iaw.dQ  pounds,  ^'8.  82. 

and  the  ^wrork  done  per  pound  is,  therefore,  A,  or  —  foot  pounds. 

A.  pressure  head  h  is  therefore  equivalent  to  h  foot  pounds  of 
energy  ijer  ix>und  of  water. 

Work  available  due  to  velocity.  When  a  body  falls  through 
a  height  h  feet,  the  work  done  on  the  body  by  gravity  is  h  foot 
pounds  i)er  pound.  It  is  shown  in  books  on  mechanics  that  if  the 
body  is  allowed  to  fall  freely,  that  is  without  resistance,  the 
velocity  the  body  acquires  in  feet  per  second  is 

V  =  J2ghy 

And  since  no  resistance  is  offered  to  the  motion,  the  whole  of 
the  work  done  on  the  body  has  been  utilised  in  giving  kinetic 

energy  to  it,  and  therefore  the  kinetic  energy  per  pound  is  ^  ~  • 

In  the  case  of  the  fluid  moving  with  velocity  v,  an  amount  of 

energy  equal  to  ^y  foot  pounds  per  pound  is  therefore  available 

before  the  velocity  is  destroyed. 

Work  available  due  to  position.  If  a  weight  of  one  pound 
falls  through  the  height  z  the  work  done  on  it  by  gravity  will  be 
z  foot  pounds,  and,  therefore,  if  the  fluid  is  at  a  height  z  feet  above 
any  datum,  as  for  example,  water  at  a  given  height  above  the 
sea  level,  the  available  energy  on  allowing  the  fluid  to  fall  to 
the  datum  level  is  z  foot  pounds  per  pound. 

32.    Bemouilli's  theorem. 

In  a  steady  moving  stream  of  an  incompressible  fluid  in  which 
the  particles  of  fluid  are  moving  in  stream  lines,  and  there  is  no 
loss  by  friction  or  other  causes 

p      v^ 

w     2g 
is  constant  for  all  sections  of  the  stream.    This  is  a  most  important 
theorem  and  should  be  carefully  studied  by  the  reader. 


40 


HYDRAULICS 


It  has  been  shown  in  the  last  paragraph  that  this  expression 
represents  the  total  amount  of  energy  per  pound  of  water  flowing 
through  any  section  of  a  stream,  and  since,  between  any  two 
points  in  the  stream  no  energy  is  lost,  by  the  principle  of  the 
conservation  of  energy  it  can  at  once  be  inferred  that  this 
expression  must  be  constant  for  all  sections  of  a  steady  flowing 
stream.    A  more  general  proof  is  as  follows. 

Let  DE,  Fig.  33,  be  the  path  of  a  particle  of  the  fluid. 


Fig.  33. 

Imagine  a  small  tube  to  be  surrounding  DE,  and  let  the  flow 
in  this  be  steady,  and  let  the  sectional  area  of  the  tube  be  so  small 
that  the  velocity  through  any  section  normal  to  DE  is  uniform. 

Then  the  amount  of  fluid  that  flows  in  at  D  through  the  area 
AB  equals  the  amount  that  flows  out  at  E  through  the  area  CF. 

Let  Pj)  and  Vd,  and  p^  and  ve  be  the  pressures  and  velocities  at 
D  and  E  respectively,  and  A  and  a  the  corresponding  areas  of  the 
tube. 

Let  z  be  the  height  of  D  above  some  datimi  and  Zi  the  height 
of  E. 

Then,  if  a  quantity  of  fluid  ABAiBi  equal  to  3Q  enters  *at  D, 
and  a  similar  quantity  CFCiFi  leaves  at  E,  in  a  time  dtj  the 
velocity  at  D  is 


Vj,= 


AuV 


and  the  velocity  at  E  is 


Ve  = 


aot' 


The  kinetic  energy  of  the  quantity  of  fluid  uQ  entering  at  D 


FLUIDS  IN  MOTION  41 

and  tJiat  of  tlie  liquid  leaving  at  E 

Since  tlie  flow  in  the  tabe  is  steady,  the  kinetic  energy  of  the 
portion  ABGF  does  not  alter,  and  therefore  the  increase  of  the 
kinetic  energy  of  the  quantity  dQ 

The  work  done  by  gravity  is  the  same  as  if  ABBiAi  fell  to 
CFFiCi  and  therefore  equals 

The  total  pressure  on  the  area  AB  is  po  •  A,  and  the  work  done 
at  X>  in  time  ot 

=  PdAi?d9^=Pd9Q, 
and  the  work  done  by  the  pressure  at  E  in  time  t 

But  the  gain  of  kinetic  energy  must  equal  the  work  done,  and 
therefore 

^^ .  (ve'- V)  =  wdq  (z - 2,)  +  Pd ^Q-PeC'Q. 


From  which 


2g      2g  w      w^ 


or  ^«%2?^^^  =  V+e5  +  ;,  =  constant. 

2g     w  2g      w 

From  this  theorem  it  is  seen  that,  if  at  points  in  a  steady 
moving  stream,  a  vertical  ordinate  equal  to  the  velocity  head  plus 
the  pressure  head  is  erected,  the  upper  extremities  of  these 
ordinates  will  be  in  the  same  horizontal  plane,  at  a  height  H 

equal  to  —  +  s~  "•"  ^  above  the  datum  level. 
^  ttj     2g 

Mr  Froude*  has  given  some  very  beautiful  experimental  illus- 
trations of  this  theorem. 

In  Fig.  34  water  is  taken  from  a  tank  or  reservoir  in  which 
the  water  is  maintained  at  a  constant  level  by  an  inflowing 
stream,  through  a  pipe  of  variable  diameter  fitted  >vith  tubes 
at  various  points.  Since  the  pipe  is  short  it  may  be  supposed  to 
be  frictionless.  If  the  end  of  the  pipe  is  closed  the  water  will  rise 
in  all  the  tubes  to  the  same  level  as  the  water  in  the  reservoir,  but 
if  the  end  C  is  opened,  water  will  flow  through  the  pipe  and  the 
water  surfaces  in  the  tubes  will  be  found  to  be  at  different  levels. 
*  British  Assoc.  Report  1875. 


42 


HYDRAULICS 


The  quantity  of  water  flowing  per  second  through  the  pipe  can  be 
measured,  and  the  velocities  at  A,  B,  and  C  can  be  found  by 
dividing  this  quantity  by  the  cross-sectional  areas  of  the  pipe  at 
these  points. 


Fig.  34. 

If  to  the  head  of  water  in  the  tubes  at  A  and  B  the  ordinates 

S  2 

1^  and  g^  be  added  respectively,  the  upper  extremities  of  these 

ordinates  will  be  practically  on  the  same  level  and  nearly  level 
with  the  surface  of  the  water  in  the  reservoir,  the  small  difference 
being  due  to  frictional  and  other  losses  of  energy. 

At  C  the  pressure  is  equal  to  the  atmospheric  pressure,  and 
neglecting  friction  in  the  pipe,  the  whole  of  the  work  done  by 
gravity  on  any  water  leaving  the  pipe  while  it  falls  from  the 
surface  of  the  water  in  the  reservoir  through  the  height  H,  which 
is  H  ft.  lbs.  per  pound,  is  utilised  in  giving  velocity  of  motion  to 
the  water,  and,  as  will  be  seen  later,  in  setting  up  internal  motions. 

Neglecting  these  resistances. 

Due  to  the  neglected  losses,  the  actual  velocity  measured  will  be 
less  than  vq  as  calculated  from  this  ^nation. 

If  at  any  point  D  in  the  pipe,  the  sectional  area  is  less  thaai  the 
area  at  C,  the  velocity  will  be  greater  than  Vc,  and  the  pressure 
will  be  less  than  the  atmospheric  pressure. 

If  V  is  the  velocity  at  any  section  of  the  pipe,  which  is  supposed 
to  be  horizontal,  the  absolute  pressure  head  at  that  section  is 

-"     -"     "-     ^2g     w^'2g     2g' 


w     w 


Pa  being  the  atmospheric  pressure  at  the  surface  of  the  water  in 
the  reservoir. 

At  D  the  velocity  Vd  is  greater  than  Vo  and  therefore  p©  is  less 


FLUIDS   IN  MOTION 


43 


than  p..     If  coloured  water  be  put  into  the  vessel  B,  it  will  rise  in 
the  tube  DE  to  a  height 


w 


w 


If  the  area  at  the  section  is  so  small,  that  p  becomes  negative,  the 
fluid  will  be  in  tension,  and  discontinuity  of  flow  will  take  place. 

If  the  fluid  is  water  which  has  been  exposed  to  the  atmosphere 
and  which  consequently  contains  gases  in  solution,  these  gases 
will  escape  from  the  water  if  the  pressure  becomes  less  than  the 
tension  of  the  dissolved  gases,  and  there  will  be  discontinuity  even 
before  the  pressure  becomes  zero. 

Figs.  35  and  36  show  two  of  Froude's  illustrations  of  the 
theorem. 


Fig.  86. 


Fig.  36. 


At  the  section  B,  Fig.  36,  the  pressure  head  is  h^  and  the 
velocity  head  is 


=  /l-/lB  =  H. 


If  a  is  the  section  of  the  pipe  at  A,  and  a^  at  B,  since  there 
is  continuity  of  flow. 


and 


If  now  a  is  made  so  that 


2g 


K 


the  pressure  head  /ia  becomes  equal  to  the  atmospheric  pressure, 
and  the  pii)e  can  be  divided  at  A,  as  shown  in  the  figure. 

Professor  Osborne  Reynolds  devised  an  interesting  experiment, 
to  show  that  when  the  velocity  is  high,  the  pressure  is  small. 

He  allowed  water  to  flow  through  a  tube  |  inch  diameter 
under  a  high  pressure,  the  tube  being  diminished  at  one  section  to 
01)5  inch  diameter. 


£4 

W 

2ff 

=  2? 

w 

2ff 

w     2*7' 

H  = 

Pi" 

2ff 

If  'Ua  is  equal  to  Vi,  pa  is  theoretically  equal  to  pi,  but  there  is 
vays  in  practice  a  slight  loss  of  head  in  the  meter,  the  difference 
-  Pa  being  equal  to  this  loss  of  head. 

•  Tramactiom  Am,S.C,E,,  1887. 


"1 


HYDRAULICS 

At  this  diminished  section,  the  velocity  was  very  high  and  the 
assure  fell  so  low  that  the  water  boiled  and  made  a  hissing 
jse. 

33.    Venturi  meter. 

An  application  of  Bemouilli's  theorem  is  found  in  the  Venturi 
ter,  as  invented  by  Mr  Clemens  Herschel*.  The  meter  takes 
name  from  an  Italian  philosopher  who  in  the  last  decade  of  the 
h.  century  made  experiments  upon  the  flow  of  water  through 
lical  pipes.  In  its  usual  form  the  Venturi  meter  consists  of  two 
mcated  conical  pipes  connected  together  by  a  short  cylindrical 
►e  called  the  throat,  as  shown  in  Figs.  37  and  38.  The  meter  is 
erted  horizontally  in  a  line  of  piping,  the  diameter  of  the  large 
is  of  the  frustra  being  equal  to  that  of  the  pipe. 
Piezometer  tubes  or  other  pressure  gauges  are  connected  to 
)  throat  and  to  one  or  both  of  the  large  ends  of  the  cones. 
Let  a  be  the  area  of  the  throat.  |  ^ 

Let  ai  be  the  area  of  the  pipe  or  the  large  end  of  the  cone 
A. 

Let  Oa  be  the  area  of  the  pipe  or  the  large  end  of  the  cone 
C. 

Let  p  be  the  pressure  head  at  the  throat. 
Let  pi  be  the  pressure  head  at  the  up-stream  gauge  A. 
Let  Pa  be  the  pressure  head  at  the  down-stream  gauge  C. 
Let  H  and  Hi  be  the  differences  of  pressure  head  at  the  throafe 
i  large  ends  A  and  C  of  the  cone  respectively,  or 

w     w^  .  ^ 

d  H.  =  ^-2. 

w     w 

Let  Q  be  the  flow  through  the  meter  in  cubic  feet  per  sec. 
Let  V  be  the  velocity  through  the  throat. 
Let  Vi  be  the  velocity  at  the  up-stream  large  end  of  cone  A. 
Let  Va  be  the  velocity  at  the  down-stream  large  end  of  cone  0. 
Then,  assuming  Bemouilli's  theorem,  and  neglecting  friction, 


PiOTDS  IN  MOTION 


46 


HYDRAULICS 


The  velocity  t?  is  — ,  and  i?i  is  — 


Oi 


Therefore  Q^  (^,  -  ^,)  =  2flf .  H, 


and 


Q  = 


aai 


Joa  -  d^ 


V2^TH. 


Due  to  friction,  and  eddy  motions  that  may  be  set  up  in  the 
meter,  the  discharge  is  slightly  less  than  this  theoretical  value,  or 

4  being  a  coefficient  which  has  to  be  determined  by  experiment. 

For  a  meter  having  a  diameter  of  25*5  inches  at  the  throat  and 
54  inches  at  the  large  end  of  the  cone,  Herschel  found  the 
following  values  for  fc,  given  in  Table  III,  so  that  the  coefficient 
varies  but  little  for  a  large  variation  of  H. 

TABLE  m. 


Herschel 

Coker 

Hfeet 

h 

in  cu.  ft. 

k 

1 

2 

6 

12 

18 

28 

•995 

•992 

•985 

•9785 

-977 

•970 

-0418 
-0319 
•0254 
•0185 
•0096 
•0084 

•9494 
•9587 
•9572 
•9920 
1-2021 
1-8588 

Professor  Coker*,  from  careful  experiments  on  an  exceedingly 
well  designed  small  Venturi  meter.  Fig.  38,  the  area  of  the  throat 
of  which  was  '014411  sq.  feet,  found  that  for  small  flows  the 
coefficient  was  very  variable  as  shown  in  Table  III. 

These  results  show,  as  pointed  out  by  Professor  Coker  from  an 
analysis  of  his  own  and  Herschel's  experiments  on  meters  of 
various  sizes,  that  in  large  Venturi  meters,  the  discharge  is  very 
approximately  proportional  to  the  square  root  of  the  head,  but  for 
small  meters  it  only  follows  this  law  for  high  heads,  and  for  low 
heads  they  require  special  calibration. 

Example.  A  Venturi  meter  having  a  diameter  at  the  throat  of  86  inohes  is 
inserted  in  a  9  foot  diameter  pipe. 

The  pressure  head  at  the  throat  gauge  is  20  feet  of  water  and  at  the  pipe  gauge 
is  26  feet. 

*  Canadian  Society  of  Civil  Engineers,  1902. 


FLUIDS  IN  MOTION 


47 


?xiid  the  diaeharge,  and  the  Telocity  of  flow  through  the  throat. 
The  area  of  the  pipe  is  63*5  sq.  feet. 

throat     7-06 
The  differenee  in  prewore  bead  at  the  two  gauges  is  6  feet. 

-^ ,  ^       63-6  X  7-06       , 

Therefore  Q=    =====:  ^2x32*2x6 

=  137  c.  ft.  per  second. 
The  Tdoeity  of  flow  in  the  pipe  is  2*15  ft.  per  sec. 

„  „  through  the  throat  is  19*4  ft.  per  sec. 

34.    Steering  of  canal  boats. 

An  interesting  application  of  Bemouilli*8  theorem  is  to  show 
the  effect  of  speed  and  position  on  the  steering  of  a  canal  boat. 

l\Tien  a  boat  is  moved  at  a  high  velocity  along  a  narrow 
and  shallow  canal,  the  boat  tends  to  leave  behind  it  a  hollow 
which  is  filled  by  the  water  rushing  past  the  boat  as  shown 
in  Figs.  39  and  40,  while  immediately  in  front  of  the  boat  the 
impact  of  the  bow  on  the  still  water  causes  an  increase  in  the 
pressure  and  the  water  is  "  piled  up ''  or  is  at  a  higher  level  than 
the  still  water,  and  what  is  called  a  bow  wave  is  formed. 


Fig.  39. 


Fig.  41. 


A 

Fij?.  40. 

Let  it  be  assumed  that  the  water  moves  past  the  boat  in 
stream  lines. 

If  vertical  sections  are  taken  at  E  and  F,  and  the  points  E  and 
F  are  on  the  same  horizontal  line,  by  Bemouilli's  theorem 

w      2g     w      2g' 
At  E   the   water  is  practically  at  rest,  and  therefore  Vs  is 
zero,  and 

w      w      2g' 
The  surface  at  E  will  therefore  be  higher  than  at  F. 


48  HTDRAUUCS 

Wlien  the  boat  is  at  tlie  centre  of  the  canal  the  stream  lines  on 
both  sides  of  the  boat  will  have  the  same  velocity,  but  if  the  boat 
is  nearer  to  one  bank  than  the  other,  as  shown  in  the  figures,  the 
velocity  vr  of  the  stream  lines  between  the  boat  and  the  neaarer 
bank,  Fig.  41,  will  be  higher  than  the  velocity  Vr  on  the  other 
side;    But  for  each  side  of  the  boat 

w      w      2g      w       2g  ' 

And  since  vr  is  greater  than  vv,  the  pressure  head  pf  is 
greater  than  pn  or  in  other  words  the  surface  of  the  water  at 
the  right  side  D  of  the  boat  will  be  higher  than  on  the  left  side  R 

The  greater  pressure  on  the  right  side  D  tends  to  push  the 
boat  towards  the  left  bank  A,  and  at  high  speeds  considerably 
increases  the  diflSculty  of  steering. 

This  difficulty  is  diminished  if  the  canal  is  made  sufficiently 
deep,  so  that  flow  can  readily  take  place  underneath  the  boat. 

35.    Extension  of  Bemooilli's  theorem. 

In  deducing  this  theorem  it  has  been  assumed  that  the  fluid 
is  a  perfect  fluid  moving  with  steady  motion  and  that  there  are  no 
losses  of  energy,  by  friction  of  the  surfaces  with  which  the  fluid 
may  be  in  contact,  or  by  the  relative  motion  of  consecutive  ele- 
ments of  the  fluid,  or  due  to  internal  motions  of  the  fluid. 

In  actual  cases  the  value  of 

p      v^ 

w     2g 

diminishes  as  the  motion  proceeds. 

If  hf  is  the  loss  of  head,  or  loss  of  energy  per  pound  of  fluid, 

between  any  two  given  points  A  and  B  in  the  stream,  then  more 

generally 

£^  +  |i.%,,  =  2B^|LV,3.;,, (1). 

w      2g      "^     w      2g 

Tht  EXAMPLES. 

.  ^  "^  The  diameter  of  the  throat  of  a  Venturi  meter  is  |  inch,  and  of 
various  S.^  which  it  is  connected  IJ  inches.  The  discharge  through  the 
approxima-)  minutes  was  found  to  be  814  gaUons. 

small  metenerence   in   pressure  head  at  the  two  gauges  was  49  feet, 
heads  they  ree  coefficient  of  discharge. 

Example.    A  V^i  meter  has  a  diameter  of  4  ft.  in  the  large  part  and 
inserted  in  a  9  foot  oat.    With  water  flowing  through  it,  the  pressure  head  is 

is  26  feer^^^^"^  ^^^^  P*^  ^^^  ^"^  ^'  **  ^®  ^i^oai.    Find  the  velocity  in  the 
1  discharge  through  the  meter.    Coefficient  of 
Cat. 


FLUIDS   IN   MOTION 


49 


A  pipe  AB,  100  ft  loiig^  has  an  inclinatton  ot  1  in  5.  The  haad  dae 
0  iliie  pre^xne  at  A  is  45  tt^  the  velocity  Is  3  ft.  per  aecondf  ajid  the  section 
kf  the  pipe  is  3  sq,  ft.     Find  the  head  due  ta  the  preasure  at  B,  where  the 

is  1|  sq.  ft.    Take  A  as  the  lower  end  of  the  pipe. 


(4^  The  section  pipe  d  a  pump  h  laid  at  an  inclinatioii  of  1  in  5,  and 
is  pnnaped  ihrongb  it  at  6  ft.  per  fiecond.  Suppose  the  air  in  the 
>ter  is  disengaged  if  the  pressure  ialls  to  more  than  10  lbs.  below 
^eric  pceesare.  Then  deduce  the  greatest  practicable  length  of 
ptpa    Friction  neglected. 

Wsler  is  delivered  to  an  Inward-flow  turbine  under  a  head  of  100  feet 
IX j.     The  pressure  just  outside  the  wheel  ia  25  lbs.  per 
jliy  gan^e. 

Fihe  velocity  with  which  the  water  approaches  the  wheel*   Friction 


(0^     A  sh£»rt  conical  pipe  varying  in  diameter  from  4^  6''  at  the  large  end 
feel  at  tiie  small  end  forms  part  of  ei  horizontal  water  main.     Tlie 
!  head  at  the  large  end  is  found  to  be  1 00  feet,  and  at  the  small  end 

[  the  disohaige  through  the  pipe.     Coefficient  of  discharge  nnity^ 

Three  cubic  l^t  of  water  per  second  flow  along  a  pipe  which  as  it 
in  diameter  from  6  inches  to  12  inches.     In  50  feet  the  pipe 

ieei.     Pne  to  rarions  causes  there  is  a  leas  of  head  of  4  feet* 
(aj  the  lo6S  ol  energy  in  fcxjt  ponndfi  par  minute^  and  in  horse- 
id  the  difference  in  pressure  head  at  the  two  points  50  feet  apart. 

eqnmtioii  1,  section  35*) 

|9i     A  boriso&tal  pipe  in  whidi  the  sections  vary  gradually  has  sections 
1 10  9go«r«  feet*  I  square  foot,  and  10  square  feet  at  sections  A,  B,  and  G* 
haad  at  A  i%  100  feet,  and  the  yelocity  S  feet  per  second., 
head  and  velocity  at  B. 
tliftt  in  another  case  the  difference  of  the  pressure  heads  at  A 
,  B   It*  2  f«?et.     Find  the  velocity  at  A, 

A  Ventnri  meter  in  a  water  main  consists  of  a  pipe  converging  to 

fcod  enlarging  again  gradually.     The  section  of  main  is  9  sq.  ft. 

of  UuToat  1  sq.  ft.     The  difference  of  presstire  in  the  main  and 

ibroAi  is  12  feet  of  water*     Find  the  discharge  of  the  main  per  hour* 

If      If  tbe  inlet  area  of  a  Venturi  meter  is  n  times  the  throat  area,  and 
Ijp  mm  the  ^lelodty  and  preB«!.ure  at  the  throat,  and  the  inlet  pressure 

thmt  Up  Asd  mp  are  observed,  t^  can  be  found. 


CHAPTER   IV. 

FLOW  OF  WATER  THROUGH  ORIFICES  AND 
OVER  WEIRS. 

36.    Flow  of  fluids  through  orifices. 

The  general  theory  of  the  discharge  of  fluids  through  orifices, 
as  for  example  the  flow  of  steam  and  air,  presents  considerable 
difficulties,  and  is  somewhat  outside  the  scope  of  this  treatise. 
Attention  is,  therefore,  confined  to  the  problem  of  detennining  the 
quantity  of  water  which  flows  through  a  given  orifice  in  a  given 
time,  and  some  of  the  phenomena  connected  therewith. 

In  what  follows,  it  is  assumed  that  the  density  of  the  fluid  k 
constant,  the  effect  of  small  changes  of  temperature  and  pressure  j 
in  altering  the  density  being  thus  neglected.  ! 

Consider  a  vessel.  Fig.  42,  filled  with  water,  the  free  surEace  of 
which  is  maintained  at  a  constant  level ;  in  the  lower  part  of  tbe 
vessel  there  is  an  orifice  AB. 


Fig.  42. 


Let  it  be  assumed  that  although  water  flows  into  the  vessel  ^ 
as  to  maintain  a  constant  head,  the  vessel  is  so  large  that  at  soi^^ 
surface  CD,  the  velocity  of  flow  is  zero. 

Imagine  the  water  ir^  the  vessel  to  be  divided  into  a  number  ^ 
stream  lines,  and  consider  any  stream  line  EF. 

Let  the  velocities  at  £3  and  F  be  Ve  and  t^p,  the  pressure  heads 
h^  and  h^  and  the  positio^  heads  above  some  datum,  z^  and  9ft  j 
respectively. 


FLOW  THBOUGH  ORIFICES 


61 


Then,  applying  Bemouilli's  theorem  to  the  stream  line  EF, 


If  tf  is  zero,  then 


1  t^E*  T  V9 


217 


=  fcp~fcB  +  2ji— 2£, 


But  from  the  figure  it  is  seen  that 
is  equal  to  A,  and  therefore 


Ve 


=fc, 


or 


2g 

Since  Ae  is  the  pressure  head  at  E,  the  water  would  rise  in 
a  tabe  having  its  end  open  at  E,  a  height  /^e>  and  h  may  thus 
be  called — ^following  Thomson — the  fall  of  "free  level  for  the 
point  B." 

,  At  some  section  GK  near  to  the  orifice  the  stream  lines  are  all 
practically  normal  to  the  section,  and  the  pressure  head  will  be 
equal  to  the  atmospheric  pressure ;  and  if  the  orifice  is  small  the  fall 
of  free  level  for  all  the  stream  lines  is  H,  the  distance  of  the  centre 
of  the  section  GK  below  the  free  surface  of  the  water.  If  the 
orifice  is  circular  and  sharp-edged,  as  in  Figs.  44  and  45,  the  section 
GK  is  at  a  distance,  from  the  plane  of  the  orifice,  about  equal  to 
its  radius.  For  vertical  orifices,  and  small  horizontal  orifices, 
H  may  be  taken  as  equal  to  the  distance  of  the  centre  of  the 
orifice  below  the  free  surface. 

The  theoretical  velocity  of  flow  through  the  small  section  GK 
is,  therefore,  the  same  for  all  the  stream  lines,  and  equal  to  the 
velocity  which  a  body  will  acquire,  in  falling,  in  a  vacuum, 
through  a  height,  equal  to  the  depth  of  the  centre  of  the  orifice 
below  the  free  surface  of  the  water  in  the  vessel. 

The  above  is  Thomson's  proof  of  Torricelli's  theorem,  which 
was  discovered  experimentally,  by  him,  about 
the  middle  of  the  17th  century. 

The  theorem  is  proved  experimentally  as 
follows. 

If  the  aperture  is  turned  upwards,  as  in 
Fig.  43,  it  is  found  that  the  water  rises 
nearly  to  the  level  of  the  water  in  the  vessel, 
and  it  is  inferred,  that  if  the  resistance  of  the 
air  and  of  the  orifice  could  be  eliminated,  the 
jet  would  rise  exactly  to  the  level  of  the 
surface  of  the  water  in  the  vessel. 


1! 
1^ 









' 

Fig.  43. 


4—2 


52 


HTDBAUU08 


Other  experimentB  deflcribed  on  pages  54— S6,  also  bIiow  thal^ 
with  carefully  constracted  orifices,  the  mean  velocity  throogh  the 
orifice  differs  from  ^2gB,  by  a  very  small  quantify.  ; 

37.    Coeffloient  of  contraotion  for  aharp-edged  orifice. 

If  an  orifice  is  cut  In  the  flat  side,  or  in  the  bottom  of  a  veeBe^. 
and  has  a  sharp  edge,  as  shown  in  Figs.  41  and  45,  the  stream  Knfli; 
set  up  in  the  water  approach  the  orifice  in  all  directions,  as  shovi 
in  the  figure,  and  the  directions  of  flow  of  the  particles  of  wato^  [ 
except  very  near  the  centre,  are  not  normal  to  the  plane  of  Am  ^ 
orifice,  but  they  converge,  producing  a  contraction  of  the  jet    \ 


Fig.  44. 


Fig.  46. 


At  a  small  distance  from  the  orifice  the  stream  lines  become 
practically  parallel,  but  the  cross  sectional  area  of  the  jet  is 
considerably  less  than  the  area  of  the  orifice. 

If  w  is  the  area  of  the  jet  at  this  section  and  a  the  area  of  the 

orifice  the  ratio  -  is  called  the  coefficient  of  contraction  and  may 
a 

be  denoted  by  c.    Weisbach  states,  that  for  a  circular  orifice,  the 

jet  has  a  minimum  area  at  a  distance  from  the  orifice  slightly  less 

than  the  radius  of   the   orifice,  and  defines  the  coefficient  of 

contraction  as  this  area  divided  by  the  area  of  the  orifice.    For  a 

circular  orifice  he  gives  to  c  the  value  0*64.    Recent  careful 

measurements  of  the  sections  of  jets  from  horizontal  and  vertical 

sharp-edged    circular   and    rectangular   orifices,  by  Bazin,   the 

results  of  some  of  which  are  shown  in  Table  IV,  show,  however, 

that  the  section  of  the  jet  diminishes  continuously  and  in  fact  hss 

no  minimum  value.    Whether  a  minimum  occurs  for  square  orifice^ 

is  doubtful. 

The  diminution  in  section  for  a  greater  distance  than  tha-* 

given  by  Weisbach  is  to  be  expected,  for,  as  the  jet  moves  aw»y 

from  the  orifice  the  centre  of  the  jet  falls,  and  the  theoretical 

velocity  becomes  J2g  (R  +  y),y  being  the  vertical  distance  betweeO- 

the  centre  of  the  orifice  and  the  centre  of  the  jet. 


FLOW  THROUGH   ORIFICES 


5a 


M  a  Muall  distance  away  from  the  orifice,  however,  the  stream 
iiueaitpe  prscdcally  parallel,  and  very  h'ttb  error  is  introduced  in 
tile  coeieieut  of  contraction  by  measuring  the  stream  near  the 

Poncelet  and  Lesbros  in  1828  fotmdj  for  an  orifice  '20  m.  square, 
I  a  minimom  section  of  the  jet  at  a  distance  of  *3  uu  from  the  orifice 
ianti  Bt  lim  section  c  was  '.56^3.  M.  Bazin^  in  discussing  these 
IiwdI^  re-marks  tlmt  at  distances  greater  than  0*3  m,  the  section 
j  becoinf»  very  difficult  to  measure^  and  althongh  the  vein  appears 
jto  expand,  the  sides  become  hollow,  and  it  is  uncertain  whether 
[tlieur^  is  really  diminiBhed, 

Compkii*  eontractimi.  The  maxinnim  contraction  of  the  jet 
aitef  place  when  the  orifice  is  sharp  edged  and  is  well  removed 
Mm  Ike  Hides  and  bottom  of  the  vesseL  In  this  case  the  contrac- 
on  i»  said  to  be  complete*  Experiments  show,  that  for  complete 
»&tfaeti(>n  the  dista^nce  from  the  orifice  to  the  sides  or  bottom  of 
p  reesel  should  not  be  lea®  than  one  and  a  half  to  twice  the  least 
ijter  of  the  orifice. 

fipMe  or  supprmmd  contractimi.     An  example  of  incom- 

tmtraction   is  shown  in  Fig.  46,  the  lower  edge  of  the 

Jar  orifice  being  made  level  with  the  bottom  of  the  vessel. 

time  effect   is  produced  by  pla^ring  a  horizontal   plate  in 

VB»el  level   with   tlie  bottom   of  the  orifice.     The  stream 

ii  rlie  lower  part  of  the  orifice  are  normal  to  its  plane 

'the  contraction  at  the  lower  edge  is  consequently  suppressedp 


Fig,  a. 


Similarly,  if  the  width  of  a  rectangular  orifice  is  made  equal 
^?6selj  or  the  orifice  ahcd  m  pro\'ided  with  side  walls 
J  he  side  or  lateral  contraction  is  suppressed.    In  any 

ttippressed  contraction  the  discharge  is  increased,  but,  as 
later,  the  discharge  coeificient  umy  vary  more  than 
n traction  is  complete.     To  suppress  the  contraction 
ly»  the  orifice  must  be  made  of  such  a  form  that  the 
liiie&  biM^otne  parallel  at  the  orifice  and  normal  to  its  plane. 


54 


HYDRAULICS 


Experimental  deiermination  of  c.  The  section  of  the  stream 
from  a  circular  orifice  can  be  obtained  with  considerable  accu- 
racy by  the  apparatus  shoT^Ti  in  Fig,  40,  which  consists  of  a 
ring  having  four  radial  set 
screws  of  fine  pitch,  Tlie 
screws  are  adjusted  until  the 
points  thereof  touch  the  jet. 
M.  Bazin  has  recently  lised  an 
octagonal  frame  v^ith  twenty- 
four  set  screws^  all  radiating 
to  a  common  centre,  to  deter- 
mine the  form  of  the  section 
of  jets  from  various  kinds  of 
orifices. 

The  screws  were  adjusted 
until  they  just  touched  the  jet.  The  frame  was  then  placed  upcm 
a  sheet  of  paper  and  the  positions  of  the  ends  of  the  screwi 
marked  upon  the  paper.  The  forms  of  the  8^:stions  could  theo 
be  obtained,  and  the  areaa  measured  with  considerable  accuracy. 
Some  of  the  results  obtained  are  shown  in  Table  IV  and  also  in 
the  section  on  the  form  of  the  liquid  vein, 

38.    Coefflcient  of  velocity  for  sliarp-edged  orifice. 

The  theoretifai  velocity  through  the  contracted  section  is,  m 
shown  in  section  36,  equal  to  V2^H,  but  the  actual  velocity 
Vi  is  slightly  less  than  this  due  to  friction  at  the  orifice.    The 

ratio  —  =  4  is  called  the  coeflicient  of  velocity.  ^ 

Eayperimental  determination  of  k,  Tliere  are  two  methods 
adopted  for  determining  k  experimentally. 

First  method.  The  velocity  is  determined  by  measuring  the 
discharge  in  a  given  time  under  a  given  head,  and  the  cross 
sectional  area  w  of  the  jet*  as  explained  in  the  last  paragraph,  is 
also  obtained.  Then,  if  Vi  is  the  actual  velocity,  and  Q  the 
discharge  per  second, 


and 


Jc^-^ 


Second  method.    An  orifice,  Fig,  50,  is  formed  in  the  aide  of  a 
vessel  and  water  alloweil  to  fiow  from  it.    The  water  after  leaving 
the  orifice  flows  in  a  parabolic  curve.    Above  the  orific  e  is  fixed 
a  horizontal  scale  on  which  is  a  slider  carrying  a  vertical  scale,  i 
to  the  bottom  of  which  is  clamped  a  bent  piece  of  wire,  w  ^tli  a  sharp  1 


i 


FLOW  THROUGH   ORIFICES 


55 


point  The  vertical  scale  can  be  adjusted  so  that  the  point  touches 
the  upper  or  lower  surface  of  the  jet,  and  the  horizontal  and  vertical 
distances  of  any  point  in  the  axis  of  the  jet  from  the  centre  of  the 
orifice  can  thus  be  obtained. 


Fig.  50. 
-Assume  the  orifice  is  vertical,  and  let  Vi  be  the  horizontal 
velocity  of  flow.    At  a  time  t  seconds  after  a  particle  has  passed 
the  orifice,  the  distance  it  has  moved  horizontally  is 

X  =  Vit (1). 

The  vertical  distance  is 

y  =  hgt' (2). 

Therefore  y  =  \g  —% 

and  -.  =  V|- 

The  theoretical  velocity  of  flow  is 

Therefore  h  =  -J=  =  ^  ?-—  . 

j2gK     2'JyB. 

h  is  better  to  take  two  values  of  x  and  y  so  as  to  make 
allowance  for  the  plane  of  the  orifice  not  being  exactly  perpen- 
dicular. 

B  the  orifice  has  its  plane  inclined  at  an  angle  0  to  the 
^^cal,  the  horizontal  component  of  the  velocity  is  Vi  cos  0  and 
^e  verrical  component  Vi  sin  9. 

At  a  time  t  seconds  after  a  particle  has  passed  the  orifice,  the 
horizontal  movement  from  the  orifice  is, 

X  =  Vi  COS  Ot (1), 

and  the  vertical  movement  is, 

y  =  v,smet  +  yt^   (2). 

After  a  time  ii  seconds        ah  =  ViCos^^ (3), 

yi  =  ViamOti  +  igt{' (4). 


56  HYDRAULICS 

Substituting  the  value  of  t  from  (1)  in  (2)  and  U  from  (3) 
in  (4), 

y=''*^^*^2i#^    ®' 

and,  y,  =  «,tan«  +  2^,^ (6). 

From  (5),  2V^.^W1     

Substituting  for  Vi'  in  (6), 

i^^e^yi^^^^ (8). 

Having  calculated  tan  6^  sec  0  can  be  found  from  mathematics^ 
tables,  and  from  (7)  Vi  can  be  calculated.    Then 

^     sl2gR' 

39.    Bazin's  experiments  on  a  sharp-edged  orifice. 

In  Table  IV  are  given  values  of  A;  as  obtained  by  Bazin  from 
experiments  on  vertical  and  horizontal  sharp-edged  orifices,  for 
various  values  of  the  head. 

The  section  of  the  jet  at  various  distances  from  the  orifice  waa 
carefully  measured  by  the  apparatus  described  above,  and  the 
actual  discharge  per  second  was  determined  by  noting  the  time 
taken  to  fill  a  vessel  of  known  capacity. 

The  mean  velocity  through  any  section  was  then 

Q  being  the  discharge  per  second  and  A  the  area  of  the  section. 

The  fall  of  free  level  for  the  various  sections  was  different,  and 
allowance  is  made  for  this  in  calculating  the  coefficient  h  in  the 
fourth  column. 

Let  y  be  the  vertical  distance  of  the  centre  of  any  section 
below  the  centre  of  the  orifice ;  then  the  fall  of  free  level  for  that 
section  is  H  +  y  and  the  theoretical  velocity  is 

V2sr(H  +  y). 

The  coefficients  given  in  column  3  were  determined  by  dividing 
the  actual  mean  velocity  through  different  sections  of  the  jet  by 
J2gT3.y  the  theoretical  velocity  at  the  centre  of  the  orifice. 

Those  in  column  4  were  found  by  dividing  the  actual  mean 
velocity  through  the  section  by  \/2g  (H  +  y),  the  theoretical 
velocity  at  any  section  of  the  jet. 

The  coefficient  of  column  3  increases  as  the  section  is  taken 
further  from  the  jet,  and  in  nearly  all  cases  is  greater  than  unity. 


\ 


FLOW  THBOUGH  ORIFICES 


57 


TABLE  IV. 
Sharp-^dged  Orifices  Contraction  Complete. 

Table  showing  the  ratio  of  the  area  of  the  jet  to  the  area  of 
the  orifice  at  definite  distances  from  the  orifice,  and  the  ratio  of 
the  mean  velocity  in  the  section  to  >/2grH  and  to  J2g.(R  +  y)f 
H  being  the  head  at  the  centre  of  the  orifice  and  y  the  vertical 
distance  of  the  centre  of  the  section  of  the  jet  from  the  centre  of 
the  orifice. 

Vertical  circular  orifice  0*20  m.  ('656  feet)  diameter,  H  =  '990  m. 
(3-248  feet). 

Coefficient  of  discharge  m,  by  actual  measurement  of  the  flow  is 

m  =  -5977*. 

Mean  Velocity 

>/2</(H  +  y) 
=  k 


I^tanoe  of  the  Beetion 

^ni  the  plane  of  the 

orifice  in  metres 

(K)8 

013 

017 

0-235 

0-335 

0-516 


Area  of  Jet 

Area  of  Orifice 

=c 

•6079 
•6971 
•5951 
•6904 
•5830 
•5690 


Mean  Velocity 

0-983 
1001 
1-004 
1-012 
10^5 
1-050 


•998 

•999 

1003 

1^007 

1^010 


Horizontal    circular    orifice    0*20  m.    ('656    feet)    diameter, 
fl  =  -975m.  (3-198  feet). 


0-075 
0-093 
0-110 
0128 
0-145 
0163 


m  =  0*6035. 

0-6003 
0-5939 
0-5824 
0-5734 
0-5658 
0-5597 


1-005 
1016 
1036 
1053 
1-067 
1-078 


0-968 
0-971 
0-982 
0-990 
0-996 
0-998 


Vertical  orifice  '20  m.  (-656  feet)  square,  H  =  '953  m.  (3126  feet). 
m  =  0'6066. 


0-151 

0-6052 

1-002 

•997 

0175 

0-6029 

1-006 

1-000 

0-210 

0-5970 

1-016 

1-007 

0-248 

0-5930 

1-023 

1-010 

0-302 

0-5798 

1046 

1-027 

0-350 

0-5788 

1-049 

1024 

The  real  value  of  the  coefficient  for  the  various  sections  is 
lowever  that  given  in  column  4. 

For  the  horizontal  orifice,  for  every  section,  it  is  less  than 
inity,  but  for  the  vertical  orifice  it  is  greater  than  unity. 

Bazin's  results  confirm  those  of  Lesbros  and  Poncelet,  who  in 


See  section  42. 


58  HTDRAULICS 

1828  found  that  the  actual  velocity  through  the  contracted  section 
of  the  jet,  even  when  account  was  taken  of  the  centre  of  the 
section  of  the  jet  being  below  the  centre  of  the  orifice,  waa 
■^  greater  than  the  theoretical  value. 

This  result  appears  at  first  to  contradict  the  principle  of  the 
conservation  of  energy,  and  Bemouilli's  theorem.  1 

It  should  however  be  noted  that  the  vertical  dimensions  of  the    ! 
orifice  are  not  small  compared  with  the  head,  and  the  explanation 
of  the  apparent  anomaly  is  no  doubt  principally  to  be  found  in  the 
fact  that  the  initial  velocities  in  the  different  horiuzontal  filaments 
of  the  jet  are  different. 

Theoretically  the  velocity  in  the  lower  -ps^rt  of  the  jet  is  greater 
than  J2g  (R  +  y),  and  in  the  upper  part  less  than  J2g  (H  +  y). 

Suppose  for  instance  a  section  of  a  jet,  the  centre  of  which  is 
1  metre  below  the  free  surface,  and  assume  that  all  the  filaments 
have  a  velocity  corresponding  to  the  depth  below  the  free  surface, 
and  normal  to  the  section.  This  is  equivalent  to  assuming  that 
the  pressure  in  the  section  of  the  jet  is  constant,  which  is  probably 
not  true. 

Let  the  jet  be  issuing  from  a  square  orifice  of  '2  m.  ("656  feet) 
side,  and  assume  the  coefficient  of  contraction  is  '6,  and  for 
simplicity  that  the  section  of  the  jet  is  square. 

Then  the  side  of  the  jet  is  '1549  metres. 

The  theoretical  velocity  at  the  centre  is  \/2gr,  and  the  discharge 
assuming  this  velocity  for  the  whole  section  is 

'6  X  '04  X  J2g  =  '024  J2g  cubic  metres. 

The  actual  discharge,  on  the  above  assumption,  through  any 
horizontal  filament  of  thickness  dh,  and  depth  fe,  is 

oQ  =  0-1549xdAxN/2^, 
and  the  total  discharge  is 

/-l-OTTS 

Q  =  0-1549^2^  h^dh 

^  ^  y-9225 

se^..  =  '0241  n/2^. 

'e  theoretical  discharge,  taking  account  of  the  varying  heads 
The  co?^®'  r004  times  the  discharge  calculated  on  the  assumption 

jn^    ,      -head  is  increased  this  diiierence  dimmisnes,  and  when 

^    J      ®.      CTeater  than  5  times  the  depth  of  the  orifice,  is  very 
Those  m  cc 

velocity  througi^  ^jg^^  agrees  very  approximately  with  that  given 
velocity  at  any  se.  gquare  orifice,  where  the  value  of  k  is  given  bs 

The  coefficient  . 
further  from  the  jet, 


rum  THROUGH  ORIFICES 


59 


Uris  partly  then,  ejq^lams  the  anomalous  values  of  k^  but  it 

UPO<  be  Ifxiked  upon  as  a  complete  explanation* 

B^  <  tual  jet  are  not.  exactly  those  assumed, 

H)»  fry  normal  to  the  plane  of  the  section  is 

EtttUly  much  more  complicated  than  here  Essumed. 

iAm  Baein  further  points  out,  it  m  probable  that,  in  jets  like 

|pn^m  the  square  urifiee,  which,  as  will  be  seen  later  when  the 

jbof  the  jet  i«  considered,  are  sabject  to  considerable  deformation, 

^diverfent^-e  of  m^me  of  the  iilaments  gives  rise  to  pressures  less 

BQ  that  of  the  atmosphere. 

^k  litteinpted  to  demonstrate  thist  experimen tally ,  and 

Hti>  ',  Fig,  150,  registered  pressures  less  than  that  of  the 

Imosphen*;    but  he  doubts  the  reliability  of  the  results,  and 

Mte  out  the  extreme  difficulty  of  satisfactorily  determining  the 

^pix*  in  the  jet, 

^^imt  Uie  inequality  of  the  velocity  of   the  filaments  is  the 

tmmrf  CHUt^e,    receives   support   from    the    fact   that   for    the 

^vnoQlal  orifice,  discharging  downwards,   the  coefficient  k  is 

luili  difhtly  less  than  unity.     In  this  case,  in  any  horizontal 

lolimj  below  tlie  orifice,  the  head  is  the  same  for  all  the  stream 

1166,  fti>d  the  velocity  of  the  filaments  is  practically  constant. 

np:  '*^"       ^  of  velocity  is  never  less  than  '96,  bo  that  the  loss 

^P''  rnal  friction  of  the  liquid  is  very  small, 

40.  Distribtitioii  of  velocity  in  the  plane  of  the  orifice. 

Btttn  has  examined  the  distribution  of  the  velocity  in  the 
ioois  sections  of  the  jet  by  means  of  a  fine  Pi  tot  tube  (see 
^  215).  In  the  plane  of  the  orifice  a  minimum  velocity 
teow,  which  for  vertical  orifices  is  just  aliove  the  centre,  but  at  a 
|||4kiaiice  from  the  orifice  the  minimum  velocity  is  at  the  top 

F()r  orifices  having  complete  contraction  Bazin  found  the 
toimaiii  velocity  to  be  '62  to  '64  n/2^H,  and  for  the  rectaTigular 
ttifioft,  frith  lateml  contraction  suppressed,  0'69  v^2^H. 

Ab  the  ilii^tance  from  the  plane  of  the  orifice  increases,  the 
riucities  in  the  transverse  section  of  the  jets  from  horizontal 
rtfices,  rapidly  become  uniform  throughout  the  transverse  section* 

For  vertical  orifices,  the  velocities  below  the  centre  of  the  jet 
1*^  greater  than  those  in  the  upper  part. 

41.  Presstire  in  the  plane  of  the  orifice. 
M*  Lagerjelm  stated  in  1826  that  if  a  vertical  tnbe  open  at 

was  placed  with  its  lower  end  near  the  centre,  and  not 
uiy  below  the  plane  of  the  inner  edge  of  a  horizontal 


I 

I 
\ 


60  HYDRAULICS 

orifice  made  in  the  bottom  of  a  large  reservoir,  the  water  rose  in 
the  tube  to  a  height  equal  to  that  of  the  water  in  the  reservoir, 
that  is  the  pressure  at  the  centre  of  the  orifice  is  equal  to  the  head 
over  the  orifice  even  when  flow  is  taking  place. 

M.  Bazin  has  recently  repeated  this  exx)eriment  and  founds 
that  the  water  in  the  tube  did  not  rise  to  the  level  of  the  water  in 
the  reservoir. 

If  Lagerjelm's  statement  were  correct  it  would  follow  that  the 
velocity  at  the  centre  of  the  orifice  must  be  zero,  which  again  does 
not  agree  with  the  results  of  Bazin's  experiments  quoted  above. 

42.    Coefficient  of  discharge. 

The  discharge  per  second  from  an  orifice,  is  clearly  the  area 
of  the  jet  at  the  contracted  section  GK  multiplied  by  the  mean 
velocity  through  this  section,  and  is  therefore, 

Q=c.fc.aN^H. 
Or,  calling  m  the  coefficient  of  discharge, 

This  coeflScient  m  is  equal  to  the  product  c .  A;.  It  is  the  only 
coefficient  required  in  practical  problems  and  fortunately  it  can 
be  more  easily  determined  than  the  other  two  coefficients  c  and  h. 

Experimental  determination  of  the  coefficient  of  discharge. 
The  most  satisfactory  method  of  determining  the  coefficient  of 
discharge  of  orifices  is  to  measure  the  volume,  or  the  weight  of 
water,  discharged  under  a  given  head  in  a  known  time. 

The  coefficients  quoted  in  the  Tables  from  M.  Bazin*,  were 
determined  by  finding  accurately  the  time  required  to  fill  a  vessel 
of  known  capacity. 

The  coefficient  of  discharge  m,  has  been  determined  with 
a  great  degree  of  accuracy  for  sharp-edged  orifices,  by  Poncelet 
and  Lesbrost,  WeisbachJ,  Bazin  and  others  §.  In  Table  IV 
Bazin's  values  for  m  are  given. 

The  values  as  given  in  Tables  V  and  VI  may  be  taken  as 
representative  of  the  best  experiments. 

For  vertical,  circular  and  square  orifices,  and  for  a  head  of 
about  3  feet  above  the  centre  of  the  orifice,  Mr  Hamilton  Smith| 
junr.  II,  deduces  the  values  of  m  given  in  Table  VI. 

♦  Annales  dfs  Pont$  et  ChausBies^  October,  1888. 

t  Flow  through  Vertical  Orijices, 

X  Mechanics  of  Engineering. 

§  Experiments  upon  the  Contraction  of  the  Liquid  Vein.  Bazin  translated  bj 
Trautwine. 

II  Tfw.  Flow  of  Water  through  Orijices  and  over  Weirs  and  through  open  CondmUi  ' 
and  Pipes,  Hamilton  Smith,  junr.,  1886. 


FLOW  THBOUOH  ORIFICES 


61 


TABLE  V. 


Si|MriiiieDtar 

Partioulara  of  orifice 

Coefficient  of 
discharge  m 

Baxm 

Pbooelet  and 

Leabitn 

Bam        ' 

n 
n 

"        1 

Vertioal  aquaie  orifice  side  of  square  0-6662  ft. 

»                                        »»                                        t«                                        9, 

Vertical  Rectangular  orifice  -666  ft.  high  x  2-624 

ft  wide  with  side  contraction  suppressed 
Vertical  circular  orifice  0*6662  ft.  diameter 
Horizontal 

0-8281 

0-606 
0-606 

0-627 

0-698 

0-6086 

0-6068 

TABLE  VI. 
Cvrcvlar  orifices. 


DiuKterof 
infti 


0O197 
I 
0-627 


0O296 
0^17 


0-089 
0-611 


0O492 
0-606 


0-0984 
0-608 


I 


0-164 
0-600 


0-328 
0-699 


0-6662 
0-698 


0-9848 
0-597 


Square  orifices. 


IT 


Side^of^uare      ^^^   I  ^y^^^   \  ^^^ 


0-681 


0-612 


0-607 


0-197   !   0-6906   I   0-9843 

0-605    I   0-604      I   0-603 

I  ! 


TABLE  VIL 

'       Table  showing  coefficients  of  discharge  for  square  and  rect- 
angular orifices  as  determined  by  Poncelet  and  Lesbros. 


1 

HeftJ  of  water ' 

Width  of  orifice  -6502  feet 

Width  (»f  orifice 
1  -908  feet       i 

ilwre  the  top  , 

of  the  orifice  : 

Depth  of  orifice  in  feel 

• 

in  feet 

0328 

-0656 

•0984 

•1640 
•607 

•3287 

6562 

•0656 

6562 

•0328 

701 

•660 

•630 

•0656 

694 

-659 

-634 

•615 

-596 

572 

•643 

-1312 

683 

•658 

•640 

•623 

•603 

582 

•642     ] 

595 

•26-24 

670 

-656 

•638 

•629 

•610 

589 

•640     1 

601 

•3937 

663 

•653 

•636 

•630 

•612 

593 

•638     1 

603 

•6562 

655 

•648 

•633 

•630 

•615 

598 

•635     1 

605 

1-640          1 

642 

•638 

•630 

•627 

•617 

604 

•630     1 

607 

3-281 

632 

-633 

-628 

•626 

•615 

605 

•626     1 

605 

4-921 

616 

-619 

-620 

•620 

•611 

602 

•623     1 

602 

6-562 

611 

-612 

-612 

•613 

•607 

601 

•620     , 

602 

9-^*48 

609 

•610 

-608 

•606 

•603 

601 

•615     1 

601 

62  HTDRAUUCT 

The  heads  for  which  Bazin  determined  the  coefficientB 
Tables  IV  and  V  varied  only  from  2"6  to  3"3  feet,  but,  as  wilJ 
seen  from  Table  YII,  deduced  from  reaults  given  by  Poiicelet 
Lesbros*  in  their  cla^cal  work,  when  the  variation  of  head  is 
small,  the  coefficienta  for  rectangular  and  square  orifices  vary 
considerably  with  the  head. 

43.  Effect  of  suppressed  contraction  on  the  coeffideni 
of  discharge* 

Sharp^ged  orifice.  When  &ome  part  of  the  contraction  of  & 
transverse  section  of  a  jet  issuing  from  an  orifice  is  suppressed^ 
the  cross  sectional  area  of  the  jet  can  only  be  obtained  witk 
diflSculty. 

The  coefficient  of  ditwharge  can,  however^  be  easily  obtained, 
as  before,  by  determining  the  discharge  in  a  given  time.  Ttfr^ 
most  complete  and  accurate  experimenta  on  the  effect  of  contrac- 
tion are  those  of  Lesbros,  some  of  the  results  of  which  are  quoted 
in  Table  VIII.  The  coefficient  is  most  constant  for  square  or 
rectangular  orificee  when  the  lateral  contraction  is  suppressed.  The- 
reason  being,  that  whatever  the  head,  the  %'ariation  in  the  section 
of  the  jet  i^  confined  to  the  top  and  bottom  of  the  orifice,  the 
width  of  the  stream  remaining  constant,  and  therefore  in  a  greater 
part  of  the  transverse  section  the  stream  lines  are  normal  to  the 
plane  of  the  orifice* 

According  to  Bid  one,  if  iP  is  the  fraction  of  the  periphery  of  a 
sharp-edged  orifice  upon  which  the  contraction  is  suppressed,  and 
m  the  coefficient  of  discharge  when  the  contraction  is  complete, 
then  the  coefficient  for  incomplete  contraction  is, 

Wi  =  m  (1  +  "IStc), 
for  rectangular  orifices,  and 

for  circular  orifices, 

Bidone's  formulae  give  result*  agreeing  fairly  well  with 
Lesbros'  experiments. 

His  formulae  are,  howeverj  unsatisfactory  when  x  approaches 
unity,  as  in  that  case  mi  should  be  nearly  unity. 

If  the  form  of  the  formula  is  preserved,  and  m  taken  ae  '606^ 
for  mi  to  be  unity  it  would  require  to  have  the  valne,  i 

jrii  -  m  (1  +  '65ir),  ^ 

For  accurate  measurements,  either  orifices  with  perfect  a 
traction  or,  if  possible,  rectangular  or  square  orifices  with 
lateral  contraction  completely  suppressed,  should  be  used.    It  ^ 

*  Experiences  hydrauliquee  tur  Us  lois  de  Vicoulement  de  Veau  h 
^eSf  etc.,  1882.    Ponoelet  and  Lesbros. 


FLOW  THK017OH   ORIFICES 


6S 


Qf  be  neceeeaiy  ki  caljbmte  the  orifice  for  variotis  heads, 
W  as  shown  above  the  coefficient  for  the  latter  kind  is  more 
likely  b]  be  constant. 

TABLE  Vni, 

Table  showing  the  effect  of  soppressing  the  contraction  on  the 
efiidetit  of  discharge.     Lesbroa  *, 

Square  vertical  orifica  0"65d  feet  square. 


1  R«v4  ,j  walCT 

Side  con- 

Contrfttitiot] 

Contraction 

Ppet 

Sh&rp^^dged 

tr&otjon 

Buppres^d  al 

HnppFeased  at 

the  hmet  tmd 
Bide  edgen 

t  ■  ■            -mfiee 

Bappreseed 

the  lower  edge 

1 

mm 

0572 

0-599 

0-1640 

i>585 

0*6S1 

0^608 

09281 

0-592 

0'631 

0-615 

0*6562 

0-596 

0*632 

0-621 

0*708 

,    imo 

0-6(^ 

0^631 

0-623 

0-680 

W81 

0-606 

(>-628 

0-624 

0676 

4^931 

0-602 

0*627 

0*624 

0'672 

fl^62 

0-601 

0^626 

0-619 

0-668 

^^§& 

0^1 

0-624 

0*614 

0^665           ' 

Fig*  51,     Section  of  Jet  from 
cireokr  orifice. 


W,    Tlie  fonn  of  tlie  jet  from  sharp-edged  orifices. 
From  a  circular  orifice  the  Jet  emerges  like  a  cylindrical  rod 
iirf  retaimt  a  form  nearly  cylindrical  for  some  distance  from  the 

hg.  51  3how8  three  sections  of  a  jet  from  a  vertical  circular 
[(►nfice  u  varying  distances  from  the 
e,  m  given  by  M,  Bazin, 
The  flow  from  square  orifices  m 
Dpanied  by  an  int-eresting  and 
|c«riou«  phenomenon  called  the  in- 
of  the  jet. 
At  a  vitry  small  distance  from 
llhH  tjrifice  the  section  becomes  as 
Y^ifmi  in  Fi|f*  52,    The  aides  of  the 
t  ant  concave  and  the  comers*  are 
ve  sections.     The 
lies  octagonal  as  in 
\  and  afterwards  takes  the  form  of  a  square  with  concave 
[and  rounded  comers,  the  diagonals  of  the  square  being 
cuhir  to  the  sides  of  the  orifice,  Fig.  54» 


DOO 

Figs.  62 — 54.     Section  of  jet  froni 
aqtiare  orifice* 


*  Kjp^fimefiU  h§drmUiqueM  tur  Ut  loi$  d^  ricmdent^nt  de  Vtau^ 


^?':riare  orifice 
^«  tlxe  coe 

.j^T-    GO-  ^"^ 


c^^^ 


64 


HTDRAUUCS 


45.    Large  orifioea 


Table  VII  shows  very  clearly  that  if  the  depth  of  a  vertical  orifice 
is  not  small  compared  with  the  head,  the  coefficient  of  discharge 
varies  very  considerably  with  the  head,  and  in  the  discussion  of 
the  coeflScient  of  velocity  fc,  it  has  already  been  shown  that  the 
distribution  of  velocity  in  jets  issuing  from  such  orifices  is  not 
uniform.  As  the  jet  moves  through  a  large  orifice  the  stream 
lines  are  not  normal  to  its  plane,  but  at  some  section  of  the  stream 
very  near  to  the  orifice  they  are  practically  normal. 

If  now  it  is  assumed  that  the  pressure  is  constant  and  equal  to 
the  atmospheric  pressure  and  that  the  shape  of  this  section  is 
known,  the  discharge  through  it  can  be  calculated. 

Rectangular  orifice.  Let  efgh,  Fig.  55,  be  the  section  by  a 
vertical  plane  EF  of  the  stream  issuing  from  a  vertical  rectangular 
orifice.  Let  the  crest  E  of  the  stream  be  at  a  depth  h  below 
the  free  surface  of  the  water  in  the  vessel  and  the  under  edge 
F  at  a  depth  h^. 


Fig.  66. 

At  any  depth  h,  since  the  pressure  is  assumed  constant  in  the 
section,  the  fall  of  free  level  is  h,  and  the  velocity  of  flow  through 
the  strip  of  width  dh  is  therefore,  kJ2gh,  and  the  discharge  is 
kbJ2ghdh. 

If  &  be  assumed  constant  for  all  the  filaments  the  total  discharge 
in  cubic  feet  per  second  is 

Q  =  kJ2^j\hidh  =  f  J2gkb  (hi*- ho*). 

Here  at  once  a  difficulty  is  met  with.  The  dimensions  ^,  hi 
and  b  cannot  easily  be  determined,  and  experiment  shows  tJial 
they  vary  with  the  head  of  water  over  the  orifice,  and  that  they 
cannot  therefore  be  written  as  fractions  of  Ho,  Hi,  and  B. 


FLOW  THBOUQH  ORIFICES 


65 


By  replacing  %«,  &i  and  b  by  Ho,  Hi  and  B  an  empirical 
formnla  of  the  same  form  is  obtained  wbicli,  by  introducing  a 
coefficient  c,  can  be  made  to  agree  with  experiments.    Then 

Q  =  5c^^.B(Hlt-Ho*), 
or  replacing  |c  by  n, 

Q  =  nV^.B(Hi»-Ho») (1). 

The  coefficient  n  varies  with  the  head  Ho,  and  for  any  orifice 
the  simpler  formnla 

Q=m.a.^J2gR (2), 

a  being  the  area  of  the  orifice  and  H  the  head  at  the  centre, 
can  be  used  with  equal  confidence,  for  if  n  is  known  for  the 
particular  orifice  for  yarious  values  of  Ho,  m  will  also  be  known. 

From  Table  YII  probable  values  of  m  for  any  large  sharp- 
edged  rectangular  orifices  can  be  interpolated. 

Rectangular  slvices.  If  the  lower  edge  of  a  sluice  opening  is 
some  distance  above  the  bottom  of  the  channel  the  discharge 
through  it  will  be  {vractically  the  same  as  through  a  sharp-edged 
orifice,  but  if  it  is  flush  with  the  bottom  of  the  channel,  the 
contraction  at  this  edge  is  suppressed  and  the  coefficient  of 
discharge  will  be  slightly  greater  as  shown  in  Table  VIII. 

46.    Drowned  orifices. 

"WTien  an  orifice  is  submerged  as  in  Fig.  56  and  the  water  in 
the  up-stream  tank  or  reservoir  is  moving  so  slowly  that  its  velocity 
may  be  neglected,  the  head  causing  velocity  of  flow  through  any 
filament  is  equal  to  the  difference  of  the  up-  and  down-stream 
levels.  Let  H  be  the  difference  of  level  of  the  water  on  the  two 
sides  of  the  orifice. 


Fig.  56. 


L.  H. 


66 


HYDRAULICS 


■h^^hf 


Consider  any  stream  line  FE  which  passes  through  the  orifice 
at  B.  The  pressure  head  at  E  is  equal  to  Aj,  the  depth  of  E  below 
the  down-stream  level.    If  then  at  F  the  velocity  is  zero, 

29[ 

or  Ve  =  ^2g  (h  -  A,) 

or  taking  a  coefficient  of  velocity  k 

VE  =  kJ2g.B., 
which,  since  H  is  constant,  is  the  same  for  all  filaments  of  the 
orifice. 

If  the  coefficient  of  contraction  is  c  the  whole  discharge  through 
the  orifice  is  then 

Q  =  ckas/2gR 

=  m.a,  J2gK. 

47.  Partially  drowned  orifice. 

If  the  orifice  is  partially  drowned,  as  in 
Fig.  57,  the  discharge  may  be  considered  in 
two  parts.  Through  the  upper  part  AC  the 
discharge,  using  (2)  section  45,  is 

and  through  the  lower  part  BC 
Q2=mi.ai.N/2sr.Hi. 

48.  Velocity  of  approach. 


BS^= 


I 

Fig.  57. 


It  is  of  interest  to  consider  the  efEect  of  the 
water  approaching  an  orifice  having  what  is 
called  a  velocity  of  approach,  which  will  be  equal  to  the  velocity 
of  the  water  in  the  stream  above  the  orifice. 

In  Fig.  56  let  the  water  at  F  approaching  the  drowned  orifice 
have  a  velocity  Vi. 

Bemouilh's  equation  for  the  stream  line  drawn  is  then 


2g 


+  h,: 


J.^'"' 

■'^^2^' 


and  Ve=>/2^(r^^)  , 

which  is  again  constant  for  all  filaments  of  the  orifice. 
Then  Q  =  m.a.N^.(H+gy. 


SUDDEN  ENLABOEMEirr  OF  A  STREAM  67 

49.  Effecit  of  velocity  of  approach  on  the  discharge 
throng  a  large  reetangnlar  orifice. 

If  the  water  approaching  the  large  orifice,  Fig.  55,  has 
a  velocity  of  approach  t?i,  Bemonilli's  equation  for  the  stream  hne 
passing  throug^h  the  strip  at  depth  hy  will  be 

w     2g     w            2gr' 
Pm  being  the  atmospheric  pressure,  or  putting  in  a  coefficient  of 
velocity,  

The  discharge  through  the  orifice  is  now, 

50.     CoefELcient  of  resistance. 

In  connection  with  the  flow  through  orifices,  and  hydraulic 
plant  generally,  the  term  "  coefficient  of  resistance  "  is  frequently 
nsed.  Two  meanings  have  been  attached  to  the  term.  Some- 
times it  is  defined  as  the  ratio  of  the  head  lost  in  a  hydraulic 
system  to  the  effective  head,  and  sometimes  as  the  ratio  of  the 
head  lost  to  the  total  head  available.  According  to  the  latter 
method,  if  H  is  the  total  head  available  and  h/  the  head  lost, 
the  coefficient  of  resistance  is 

Cr       jj. 

51.     Sudden  enlargement  of  a  current  of  water. 
It  seems  reasonable  to  proceed  from  the  consideration  of  flow 
throagh   orifices  to  that  of  the  flow  through  mouthpieces,  but 
before    doing  so  it  is   desirable   that  the    effect  of    a  sudden 
enlargement  of  a  stream  should  be  considered. 

Suppose  for  simplicity  that  a  pipe  as 

in  Fig.  58  is  suddenly  enlarged,  and  that 

there  is  a  continuous  sinuous  flow  along 

the  pipe.    (See  section  284.) 

I  At  the  enlargement  of  the  pipe,  the 

J       stream  suddenly  enlarges,  and,  as  shown 

L       in  the  figure,  in  the  comers  of  the  large 

pipe  it  may  be  assumed  that  eddy  motions  p.     gg 

are  ^t  up  which  cause  a  loss  of  energy. 

5—2 


68 


HYDRA  ULICS 


Ck>nsider  two  sections  oa  and  dd  at  each  a  distance  from  bh 
that  the  flow  is  steady. 

Then,  the  total  head  at  dd  equab  the  total  head  at  oa  minaii 
the  loss  of  head  between  oa  and  dd^  orH  hm  the  loss  of  head  doe 
to  shock,  then 

Va        Vj       Pd        Vd         1 

w     2g      w     2g 
Let  A«  and  A^  be  the  area  at  oa  and  dd  respectively* 
Since  the  flow  past  oa  a      '     ^       ^^^^  ^^ 

Then,  assuming  that  each 
velocity  Vo,  and  r^  at  dd^  tlie  m' 

which  passes  aa  in  unit  time  i^  < 

of  the  water  that  passes  dd  is 

t 


nt  of  fluid  at  aa  has 

iin  of  the  quantity  of  waterl 


w 


-  Afli'rt'i  and  the  tDOmentmii 
9  ' 


the  momentum  of  a  mass  of  M  ide  moving  with  a  velo 

V  feet  per  second  being  Mt*  pounds  feet. 
The  change  of  tnonientum  is  therefore, 

The  forces  acting  on  the  water  between  tm  and  dd  to  produce 
this  change  of  moment um,  are 

p<tAa  acting  on  aa,  p^A^  acting  on  dd^ 
and,  if  p  is  the  mean  pressure  per  unit  area  on  the  annular  riu^ 
hhy  an  additional  force  p{Ad-  A^)^ 

There  is  considerable  doubt  as  to  what  is  the  magnitude  of  the 
pressure  p,  but  it  is  generally  assiuwed  that  it  is  equal  to  pa,  for 
the  following  reason. 

The  water  in  the  enlarged  portion  of  the  pipe  may  be  looked 
upon  as  divided  into  two  parts,  the  one  part  having  a  motion  of 
translation,  while  the  other  part,  which  is  in  contact  with  the 
annular  ring,  is  practically  at  rest*     (See  section  284 J 

K  this  assumption  is  correct,  then  it  is  to  be  expected  that  the 
pressure  throughout  this  atill  water  will  be  practically  eqtial  at  all 
points  and  in  all  directions,  and  must  be  equal  to  the  pressure  ia 
the  stream  at  the  section  65,  or  the  pressure  p  is  equal  to  p«. 

Therefore 

PcjA<|  -  Pa  (Ad  -  A«)  -paAa  =  'M?-— -(V«-Vd), 


KkVa 


9 


from  which        (pd  -  po)  A4  =  to (t?o  —  t?d) ; 


SUDDEN  ENLARGEMENT  OF  A  STREAM  69 

and  aince  A«t7«  =  A^rVd, 


9 


Adding  ^   to  both  sides  of    the  equation  and   separating 


2Sr 
~  into  two  parts. 


vr 


or  &  the  loss  of  head  dae  to  shock  is  eqaal  to 
According  to  St  Venant  this  quantity  should  be  increased  by 

1  Vd^ 

wi  amount  equal  to  5  nZ  >  ^^^  this  correction  is  so  small  that  as 
&  nile  it  can  be  neglected. 

52.    Sudden  contraotion  of  a  current  of  water. 

Suppose  a  pipe  partially  closed  by  means  of  a  diaphragm  as  in 
Fig.  59. 

Afi  the  stream  approaches  the  diaphragm 
""■which  is  supposed  to  be  sharp-edged — 
it  contracts  in  a  similar  way  to  the  stream 
passing  through  an  orifice  on  the  side  of 
a  vessel,  so  that  the  minimum  cross  sec- 
tional area  of  the  flow  will  be  less  than  the  Fig.  59. 
area  of  the  orifice. 

The  loss  of  head  due  to  this  contraction,  or  due  to  passing 
through  the  orifice  is  small,  as  seen  in  section  39,  but  due  to 
the  sodden  enlargement  of  the  stream  to  fill  the  pipe  again,  there 
is  a  considerable  loss  of  head. 

Let  A  be  the  area  of  the  pipe  and  a  of  the  orifice,  and  let  c  be 
the  coefficient  of  contraction  at  the  orifice. 

Then  the  area  of  the  stream  at  the  contracted  section  is  ca,  and, 
therefore,  the  loss  of  head  due  to  shock 


2g 


70  HTDRAULICS 

If  the  pipe  simply  diminishes  in  diameter  as  in  Fig.  58,  the 
section  of  the  stream  enlarges  from  the  contracted  area  oa  to  fill  j 
the  pipe  of  area  a,  therefore  the  loss  of  head  in  this  case  is 

"-'iil-^)' «• 

Or  making  St  Venant  correction 

^-m-')'^i} «)• 

Valtte  of  thecoefficient  c.  The  mean  valneof  cfor  aaharp-edged 
circular  orifice  is,  as  seen  in  Table  IV,  about  0*6,  and  this  maj  be 
taken  as  the  coefiicient  of  contraction  in  this  formula. 

Substituting  this  value  in  equation  (1)  the  loss  of  head  k 

found  to  be  -^— ,  and  in  equation  (2),  -g—  ,v  being  the  velodUy iB  J 

0*5o^ 
the  small  pipe.     It  may  be  taken  therefore  as  i^a''    ^^^'*'^^] 

experiments  are  required  before  a  correct  value  can  be  assigned. 

53.  Loss  of  head  due  to  sharp-edged  entrance  Into  a  pipi^ 
or  mouthpiece. 

When  water  enters  a  pipe  or  mouthpiece  from  a  vessel  throogk 
a  sharp-edged  entrance,  as  in  Fig.  61,  there  is  first  a  contraction,  and 
then  an  enlargement,  as  in  the  second  case  considered  in  section  52. 

The  loss  of  head  may  be,  therefore,  taken  as  approximately -5-- 

and  this  agrees  with  the  experimental  value  of  — ^ given  by 

Weisbach. 

This  value  is  probably  too  high  for  small  pipes  and  too  low  far 
large  pipes*. 

54.  Mouthpieces. 

If  an  orifice  is  provided  with  a  short  pipe  or  mouthpiece,  througfc 
which  the  liquid  can  flow,  the  discharge  may  be  very  differem.'^ 
from  that  of  a  sharp-edged  orifice,  the  difference  depending  npo* 
the  length  and  form  of  the  mouthpiece.  If  the  orifice  is  cylindric^i^ 
as  shown  in  Fig.  60,  being  sharp  at  the  inner  edge,  and  so  shor^ 
that  the  stream  after  converging  at  the  inner  edge  clears  th-^ 
outer  edge,  it  behaves  as  a  sharp-edged  orifice. 

Short  external  cylindrical  mouthpieces.  If  the  mouthpiece  1J| 
cylindrical  as  ABFE,  Fig.  61,  having  a  sharp  edge  at  AB  anC3 
a  length  of  from  one  and  a  half  to  twice  its  diameter,  the  je^*^ 

*  See  M.  Bazin,  Exp€rience»  nouvelUs  sur  la  distribution  des  vite$se»  dam^ 
le$  tuyaux. 


FLOW  THROUGH  MOUTHPIECES 


71 


contracts  to  CD,  and  then  expands  .to  fill  the  pipe,  so  that  at  EF 
it  discharges  foil  bore,  and  the  coefficient  of  contraction  is  then 
onitF.  Experiment  shows,  that  the  coefficient  of  discharge  is 


— B 


Fig.  60. 


Fig.  61. 


from  080  to  0"85,  the  coefiicient  diminishing  with  the  diameter 
of  the  tube.  The  coefficient  of  contraction  being  unity,  the 
coefficients  of  velocity  and  discharge  are  equal.  Good  mean 
^iies,  according  to  Weisbach,  are  0*815  for  cylindrical  tubes, 
w»d  0*819  for  tubes  of  prismatic  form. 

These  coefficients  agree  with  those  determined  on  the  assump- 
^on  that  the  only  head  lost  in  the  mouthpiece  is  that  due  to 
sndden  enlargement,  and  is 

0-5i;" 

^  being  the  velocity  of  discharge  at  EF. 
Applying  Bernouilli's  theorem  to  the  sections  CD  and  EF,  and 

taking  into  account  the  loss  of  head  of  -p— ,  and  pa  as  the  atmo- 
spheric pressure, 

w       2g      w      2g      2g  w^ 

or  -7^^ —  =  H. 


Therefore 


2g 
v^  =  '66  X  2gR 


md  i?  =  -812N/2^H. 

The  area  of  the  jet  at  EF  is  a,  and  therefore,  the  discharge 
er  second  is 


a.v  =  '8V2as/2gR. 

Or  m,  the  coefficient  of  discharge,  is  0*812. 
The  pressure  head  at  the  section  CD.     Taking  the  area  at  CD 
s  0-606  the  area  at  EF, 

rcD  =  l'66t7. 


72  HYDRAULICS 

Therefore  S^  =  S^  .  ^' -  2|2l^  Pa  _  lf^\ 

or  the  pressure  at  C  is  less  than  the  atmospheric  pressure. 

K  a  pipe  be  attached  to  the  mouthpiece,  as  in  Fig.  61,  and  tt^ 
lower  end  dipped  in  water,  the  water  should  rise  to  a  height  of  abo^c- 

— s —  feet  above  the  water  in  the  vessel. 

55.    Borda's  mouthpiece. 

A  short  cylindrical  mouthpiece  projecting  into  the  vessel,  as  ^B 
Fig.  62,  is  called  a  Borda's  mouthpiece,  arid  is  of  interest,  as  tfcr^ 
coefficient  of  discharge  upon  certain  assumptions  can  be  readiK- 
calculated.  Let  the  mouthpiece  be  so  short 
that  the  jet  issuing  at  EF  falls  clear  of  GH. 
The  orifice  projecting  into  the  liquid  has 
the  effect  of  keeping  the  liquid  in  contact 
with  the  face  AD  practically  at  rest,  and 
at  all  points  on  it  except  the  area  EF  the 
hydrostatic  pressure  will,  therefore,  simply  " 

depend   upon   the   depth  below    the   free  ^^' 

surface  AB.     Imagine  the  mouthpiece  produced   to   meet  tli0 
face  EC  in  the  area  IK.    Then  the  hydrostatic  pressure  on  AID, 
neglecting  EF,  will  be  equal  to  the  hydrostatic  pressure  on  BC, 
neglecting  IK. 

Again,  BC  is  far  enough  away  from  EF  to  assume  that  the 
pressure  upon  it  follows  the  hydrostatic  law. 

The  hydrostatic  pressure  on  IK,  therefore,  is  the  force  whicl 
gives  momentum  to  the  water  escaping  through  the  orifice,  ove^ 
comes  the  pressure  on  EF,  and  the  resistance  of  the  mouthpiece. 

Let  H  be  the  depth  of  the  centre  of  the  orifice  below  the  free 
surface  and  p  the  atmospheric  pressure.  Neglecting  frictional 
resistances,  the  velocity  of  flow  v,  through  the  orifice,  is  j2glEL. 

Let  a  be  the  area  of  the  orifice  and  cu  the  area  of  the  transverse 
section  of  the  jet.    The  discharge  per  second  will  be  it; .  w  J2gK  lbs. 

The  hydrostatic  pressure  on  IK  is 

pa  +  wdK  lbs. 

The  hydrostatic  pressure  on  EF  is  pa  lbs. 

The  momentum  given  to  the  issuing  water  per  second,  is 

M  =  -.o,.2^H. 

Therefore  pa  +  —  o>  2gK  =  pa  +  wdR, 

and  (I)  =  ia. 


FLOW  THROUGH  MOUTHPIECES  73 

The  coefficient  of  contraction  is  then,  in  this  case,  equal  to 

one  half. 

Experiments  by  Borda  and  others,  show  that  this  result  is 
justified,  the  experimental  coefiicient  being  slightly  greater 
than  \. 

56.  Conical  monthpieoes  and  nozzles. 

These  are  either  convergent  as  in  Fig.  63,  or  divergent  as  in 
Fig.  64. 


j=^ 


Fig.  63.  Fig.  64. 

Calling  the  diameter  of  the  mouthpiece  the  diameter  at  the 
ontlet,  a  divergent  tube  gives  a  less,  and  a  convergent 
tabe  a  greater  discharge  than  a  cylindrical  tube  of  the 
®^  diameter. 

Experiments  show  that  the  maximum  discharge  for  a 
convergent  tube  is  obtained  when  the  angle  of  the  cone 
is  from  12  to  13^  degrees,  and  it  is  then  0*94 .  a .  J2gh, 
K  instead  of  making  the  convergent  mouthpiece  conical, 
te  sides  are  curved  as  in  Fig.  65,  so  that  it  follows  as 
ear  as  possible  the  natural  form  of  the  stream  lines,  the 
^efficient  of  discharge  may,  with  high  heads,  approxi- 
ate  very  nearly  to  unity. 
Weisbach*,  using  the  method  described  on  page  55 
determine  the  velocity  of  flow,  obtained,  for  this 
>nthpiece,  the  following  values  of  k.  Since  the  mouth-  p.  gg 
?ce  discharges  full  the  coefficients  of  velocity  k  and 
icharge  m  are  practically  equal. 


Head  in  feet 

0-66 

1-64 

11-48 

55-8 

888 

k  and  m 

•959 

•967 

•975 

•994 

•994 

According  to  Freeman  t,  the  fire-hose  nozzle  shown  in  Fig.  66 
3  a  coefficient  of  velocity  of  *977. 

•  Mechanics  of  Engineering. 

t  TramaetionM  Am,  Soc.  C.E.,  Vol.  xxi. 


74  HYDRAULICS 

If  the  mouthpiece  is  first  made  convergent^  and  then  divergent, 


Fig.  66. 

as  in  Fig.  67,  the  divergence  being  sufficiently  gradual  for  the 
stream  lines  to  remain  in  contact  with  the  tube,  the  coefficient  of 
contraction  is  unity  and  there  is  but  a 
small  loss  of  head.  The  velocity  of  efflux 
from  EP  is  then  nearly  equal  to  >/2gfH 
and  the  discharge  is  m .  a .  N/2grH,  a  being 
the  area  of  EP,  dnd  the  coefficient  m 
approximates  to  unity. 

It  would  appear,  that  the  discharge 
could  be  increased  indefinitely  by  length- 
ening the  divergent  part  of  the  tube  and 
thus  increasing  a,  but  as  the  length  increases,  the  velocity 
decreases  due  to  the  friction  of  the  sides  of  the  tube,  and  farther, 
as  the  discharge  increases,  the  velocity  through  the  contracted 
section  CD  increases,  and  the  pressure  head  at  CD  consequently 
falls. 

Calhng  Pa  the  atmospheric  pressure,  pi  the  pressure  at  CSD> 
and  Vi  the  velocity  at  CD,  then 

w 


and 


2g  w 

w  w     2g 


If  s^  is  greater  than  H  +  —  ,  2^  becomes  negative. 

As  pointed  out,  however,  in  connection  with  Proude's  apparatoBi 
page  43,  if  continuity  is  to  be  maintained,  the  pressure  cannot  be 
negative,  and  in  reaUty,  if  water  is  the  fluid,  it  cannot  be  lew 
than  7  the  atmospheric  pressure,  due  to  the  separation  of  the  air 
from  the  water.  The  velocity  Vi  cannot,  therefore,  be  incr 
indefinitely. 


FLOW  THROUGH    MOUTHPIECES 


75 


Assmnin^  th©  pressure  can  J  list  become  zero,  a,nd  taking  the 
eric  preamre  as  equivalent  to  a  head  of  34  tL  of  water,  the 
mm  possible  Yelocitjr,  is 

Ur=^%/2ff  CH  +  34ft.) 
id  the  tnaiciTnnm  ratio  of  the  area  of  EF  to  CD  is 


^A^ 


S4ft 
H 


Praclically^  tlie  maximtim  value  of  vi  may  be  taken  as 
and  the  maximnin  ratio  of  EF  to  CD  as 


v/-^. 


The  maxim  HID  discharge  is 


y- 


^ 


Tlie  ratio  gWen  of  EF  to  CD  may  be  taken  as  the  masdnaum 
beiween  the  area  of  a  pipe  and  the  throat  of  a  Venturi  meter 
[lo  be  used  in  the  pipe. 

57.    Flow  tliroiigh  orifices  and  mouthpieces  under  constant 
freasure. 

The  head  of  water  causing  flow  through  an  orifice  may  be 
produced  \yf  a  pomp  or  other  mechanical  means,  and  the  discharge 
iB^  tdce  place  int-o  a  vessel,  such  as  the  condenser  of  a  steam 
,  in  which  the  pressure  is  less  than  that  of  the  atmosphere. 
example,  suppose  water  m  be  discharged  from  a  cylinder 
I  a  vemel  B,  Fig,  6H^  through 
It  orifice  or  moathpiece  by  means 
U  iMtoEi  loaded  with  P  lbs.,  and 
fft^preettre  per  sq.  foot  in  B 

Lpt  the  area  of  the  piston  be 
A  ijuare  feet.  Let  h  be  the  height 
of  the  water  in  the  cylinder  above 
*fc«  ORiti«  of  the  orifice  and  fh  of 
^  water  in  the  vesael  B.  The 
tlworetical  effective  head  forcing  water  through  the  orifice  may 
be  writlm 


Fig,  68. 


Aw 


w 


76  HTDRAITLICS 

If  P  is  large  K  and  h  will  generally  bo  negligible. 

At  the  orifice  the  pressure  head  la  K-^^t  a^id  therefore  for 

w 

any  stream  line  through  the  orifice,  it  there  is  no  friction^ 
2g  w      Aw 

The  actual  velocity  will  be  less  than  u,  due  to  frictioG,  and  Kl] 
IB  a  coefiicient  of  velocity,  the  velocity  is  then 

and  the  discharge  i^i  Q-m.a*J2gH. 

In  practical  examples  the  cylinder  and  the  vessel  will  geneiaDy 
be  connected  by  a  short  pipe,  for  which  the  coefficient  of  velocity 
will  depend  upon  the  length. 

If  it  is  only  a  few  feet  long  the  principal  loss  of  head  will  be 
at  the  entrance  to  the  pipe,  and  the  coefficient  of  discharge  will 
probably  vary  between  0*65  and  0'85, 

The  effect  of  lengthening  the  pipe  mil  be  understood  aft'er  tbd 
chapter  on  flow  through  pipes  has  been  read. 

Example.    Water  is  di^ehai-t^ed  from  a  pamp  inta  &  condenaer  in  ^hioh  Hift 
pressure  is  3  lbs.  per  aq.  inch  Ibroui^li  a  f^ort  pipe  S^  inchea  diameter. 
The  pressure  in  the  puiup  i&  2Q  iba.  per  kij.  tneh. 

Find  the  dischargi^  into  the  condenser,  takinR  the  coefficieut  of  discharge  0'7S* 
The  effective  head  is 


^^20x144     3jcU4 


as  k^ 

62-4 

section  v^. 

=39  a  feet. 

falls. 

f     Q=-?i>v '7M54 

^^-  V  .  /- 


'  i**  ' 


jjij4:^4i  y.  3li*2  cubic  feet  per  sec 


i 


Calling  J       =1'84  cubic  ft.  per  sec. 

and  Vi  the  Vi 

e  of  emptying  a  tank  or  reservoir. 

reservoir  to  have  a  sharp-edged  horizontal  orifice 
It  is  required  to  find  the  time  taken  to  empty 
and 

-UjS  vf  the  horizontal  section  of  the  reservoir  at  any 

If  ^  is  greater  t^  ^^^^  ^^  ^  gq   f^^^  ^^^  ^1^^  ^^^  ^j  ^^ 

As  pointed  out,ho>^  let  the  ratio  ~  be  sufficiently  large  that  the 
page  43,  if  continuity  .  ?  ,  ,        -. 

negative,  and  in  reaUt?  the  reservoir  may  be  neglected, 
than  i  the  atmospheric  J*  *he  water  is  at  any  height  h  above  the 
from  the  water.    The  veV  Ao^s  through  the  orifice  m  any  time  dt 
indefinitely. 


FLOW  THROUGH  MOUTHPIECES  77 

The  amoimt  dh  by  which  the  surface  of  water  in  the  reservoir 
falls  in  tKe  time  dt  is 

j_maJ2ghdt 

A. 

;..         Aoh 
or  or  =  -  -  . —  -. . 

ma  v2gA* 

The  time  for  the  water  to  fall  from  a  height  H  to  Hi  is 

^^  f^    Adh_  ^    i_  r«  Adh 

J  ^tma  ^gh     a  J2g  J  h,  mh^  ' 
If  A  is  constant,  and  m  is  assumed  constant,  the  time  required 
for  the  surface  to  fall  from  a  height  H  to  Hi  above  the  orifice  is 

_  .     Adh 
ma  ^2g  • 


^Vc-^H-n/Hi), 


ma  J2g 
and  the  time  to  empty  the  vessel  is 

^  ^  2 A  n/H 
mu  \/2g* 
or   is  equal   to  twice  the  time  required  for  the  same  volume  of 
wwter  to  leave  the  vessel  under  a  constant  head  H. 

Time  of  emptying  a  lock  with  vertical  drowned  sluice.  Let  the 
'water  in  the  lock  when  the  sluice  is  closed  be  at  a  height  H, 
Fig^.   56,  above  the  down-stream  level. 

Then  the  time  required  is  that  necessary  to  reduce  the  level  in 
the  lock  by  an  amount  H. 

"When  the  flow  is  taking  place,  let  x  be  the  height  of  the  water 
Ij^  the  lock  at  any  instant  above  the  down-stream  water. 

\jet  A  be  the  sectional  area  of  the  lock,  at  the  level  of  the 
^vrater  in  the  lock,  a  the  area  of  the  sluice,  and  m  its  coefficient  of 
discharge. 

The  discharge  through  the  sluice  in  time  dt  is 

cQ  =  m .  a  >/2gx .  ot. 

If  da?  is  the  distance  the  surface  falls  in  the  lock  in  time  cty  then 

Adx  =  m^  J2gxdt, 

Aox 

or  ^^  ="    "     /?r-  1  • 

m^  y/2gx^ 


\ 


To  redace  the  level  by  an  amount  H, 

f^     Adx 
0  ma  *J2gxi 


H 


.J  > :  ni:  of 

-:    -.■■t±;:t-n:  cf 
"-:-  iTri  A  is 


r 


•  i  :•:&.;  ^f  40  a    To 
iischai^e  10  eft 


FLOW  THROUGH   ORIFICES  AND  MOUTHPIECES  79 

(5)     A  jet  3  in.  diameter  at  the  orifice  rises  vertically  50  ft.    Find  its 
at  25  ft.  above  the  orifice. 


(6^)  An  orifice  1  sq.  ft.  in  area  discharges  18  c.  ft.  per  second  under  a 
head  of  9  ft.  Assuming  coefficient  of  velocity =0*98,  find  coefficient  of 
ooatraction. 

{7)  The  pressure  in  the  pump  cylinder  of  a  fire-engine  is  14,400  lbs. 
per  sq.  ft.;  assuming  the  resistance  of  the  valves,  hose,  and  nozzle  is  such 
that  the  coefficient  of  resistance  is  0*5,  find  the  velocity  of  discharge,  and 
the  height  to  which  the  jet  will  rise. 

<8)  The  pressure  in  the  hose  of  a  fire-engine  is  100  lbs.  per  sq.  inch; 
the  jet  rises  to  a  height  of  150  ft.    Find  tlie  coefficient  of  velocity. 

<9)  A  horizontal  jet  issues  under  a  head  of  9  ft.  At  6  ft.  from  the 
orifice  it  lias  faUen  vertically  15  ins.    Find  the  coefficient  of  velocity. 

i  10)  Required  the  coefficient  of  resistance  corresponding  to  a  coefficient 
of  velocity =0-97. 

Ill)  A  fluid  of  one  quarter  the  density  of  water  is  discharged  from  a 
Tessel  in  which  the  pressure  is  50  lbs.  per  sq.  in.  (absolute)  into  the 
fttmosphere  where  the  pressure  is  15  lbs.  per  sq.  in.  Find  the  velocity  of 
d&Mharge. 

(12)  Find  the  diameter  of  a  circular  orifice  to  discharge  2000  c.  ft.  per 
hour,  under  a  head  of  6  ft.     Coefficient  of  discharge  0*60. 

(18 1  A  cylindrical  cistern  contains  water  16  ft.  deep,  and  is  1  sq.  ft.  in 
cross  section.  On  opening  an  orifice  of  1  sq.  in.  in  the  bottom,  the  water 
lervel  fell  7  ft.  in  one  minute.     Find  the  cocfficicut  of  discliarge. 

(14)  A  miner*s  inch  is  defined  to  be  the  discharge  through  an  orifice  in 
a  vertical  plane  of  1  sq.  in.  area,  under  an  average  head  of  6J  ins.  Find 
the  supply  of  water  per  hour  in  gallons.     Coefficient  of  discharge  0*62. 

(15 1  A  veJisel  fitted  with  a  piston  of  12  sq.  ft.  area  discharges  water 
imder  a  head  of  10  ft.  What  weight  placed  on  the  piston  would  double  the 
jmte  of  discharge? 

(16)  An  orifice  2  inches  square  discharges  under  a  bead  of  ICX)  feet 
1*888  cubic  feet  per  second.  Taking  the  coefficient  of  velocity  at  0*97,  find 
the  coefficient  of  contraction. 

(17)  Find  the  discharge  per  minute  from  a  circular  orifice  1  inch 
diaiiieter,  under  a  constant  pressure  of  84  lbs.  per  sq.  inch,  taking  0*60  as 
the  coefficient  of  discharge. 

(18)  The  plunger  of  a  fire-engine  pump  of  one  quarter  of  a  sq.  ft.  in 
ana  is  driven  by  a  force  of  9542  lbs.  and  the  jet  is  observed  to  rise  to  a 
hdigjit  of  150  feet.     Find  the  coefficient  of  resistance  of  the  apparatus. 

(19)  An  orifice  8  feet  wide  and  2  feet  deep  has  12  feet  head  of  water 
■bofe  Hb  centre  on  the  up-stream  side,  and  the  backwater  on  the  other 

ie  is  at  the  level  of  the  centre  of  the  orifice.    Find  the  discharge  if 
fli.»i«CH$2. 


80 


HYDRA  tTLlCS 


(20)  Tan  e.  ft,  of  watat  per  second  flow  thfQu^  a  pipe  of  1  aq.  ft.  Bxe^ 
which  suddenly  enlarges  to  4  sq.  ft.  area.    Taking  the  pressure  at  100  lbs; 
par  sq.  ft.  in  the  mmaUar  part  of  the  pipe,  find  (1)  the  head  lost  in  aho 
(*2j  the  preaaure  in  the  larger  part,  (3)  the  work  expended  in  forcing 
water  through  the  enlargement. 

(21)  A  pipe  of  3"  diameter  ia  suddenly  enlarged  to  5'^  diameter*    A  \ 
tuba  containing  marcnry  is  connected  to  two  points,  one  on  eaeh  side  of 
enlargement,  at  points  where  the  flow  ia  steady.     Find  the  differenee  : 
level  in  the  two  Linabs  of  the  U  when  water  flows  at  the  rate  of  2  c,  ft» 
second  from  tlie  small  to  the  large  section  and  i^ice  versd.     The  sp 
gravity  of  mercury  is  IB'6,     Lond»  Un, 

(22)  A  pipe  is  suddenly  enlarged  from  2|  indies  in  diameter  to 
inches  in  diameter.    Water  flows  tlirough  these  two  pipes  from  the  i 
to  the  larger,  and  the  discharge  from  the  end  of  the  higger  pipe  is  twd 
gaDons  per  second.     Find:— 

(a)  The  loss  of  head,  and  gain  of  pressure  head,  at  the 
ment. 

(b)  The  ratio  of  head  lost  to  velocity  bead  In  small  pipe. 
(28  J     The  head  and  tail  water  of  a  vertical -sided  lock  differ  in  levdl 

12  ft  The  area  of  the  lock  bai^in  is  TOO  sq.  ft.  Find  the  time  of  emp^flQf I 
the  lock,  through  a  sluice  of  5  sq.  ft.  area,  with  a  coefficient  0"6,  Tli»1 
sluice  discharges  helow  tail  water  level. 

(24 1  A  tank  X200  sq.  ft.  in  area  discharges  through  an  orifice  1  a^ilt  J 
in  area.  Calculate  the  time  retj^aired  to  lower  the  level  in  the  tank  frodi  j 
60  ft.  to  26  ft.  ahove  the  orifice.    Coefficient  of  discharge  0"6. 

(26)  A  vertical-sided  lock  is  65  ft.  long  and  18  ft,  wide.  Lift  li5  H  ] 
Find  the  area  of  a  sluice  helow  tail  water  to  empty  tlie  lock  in  5  minute!.  | 
Coefficient  0'6* 

(26)  A  reservoir  has  a  bottom  width  of  100  feet  and  a  length  of  3S0  J 
feet. 

The  sides  of  the  reservoir  are  vertical. 

The  reservoir  ia  connected  to  a  second  reservoir  of  the  same  dimensioM  I 
by  means  of  a  pipe  2  feet  diameter.     The  surface  of  the  water  in  tlie  &til| 
reservoir  is  17  feet  above  tliat  in  the  other.     The  pii>6  is  Iwlow  the  su 
of  the  water  in  both  reservoirs.     Find  the  time  taken  for  the  water  in  1 
two  reservoirs  to  become  level*     Coefficient  of  discharge  0*8. 


59*    Notches  and  Weirs. 

Wben  the  sides  of  an  orifice  are 
prod  need  J  so  that  they  extend  be- 
yond the  free  surface  of  the  water, 
as  in  Figs.  69  and  70,  it  is  called  a 
notch, 

Notches  are  generally  made  tri- 
angular or  rectangular  as  shown 
in  the  fignres  and  are  largely  used 
for  gauging  the  flow  of  water* 


Fig.  09*    Triangukt  KoKfh, 


FLOW   OYER   WEIRS 


81 


For  example,  if  the  flow  of  a  small  stream  is  required,  a  dam  is 
ooQstrticted  scroti  t^e  stream  and  the  wate^r  allowed  to  pasg 
tliTotigli  a  noteli  cut  in  a  board  or  metal  plate  * 


Fig.  70,     B£eta.»gulftr  Notch. 

They  can  conveniently  be  used  for  measuring  the  compensation 
[irater  to  be  supplied  from  collecting  reservoirSj  and  also  to  gauge 
f  fiupplj  of  water  to  water  wheels  and  turbines. 
The  tenn  weir  is  a  name  givon  to  a  structure  used  to  dam  np 
I  a  strnun  and  over  which  the  water  flows. 

The  conditiomi  of  flow  are  practically  the  same  as  through  ■ 
a  rectangtilar  notchj  and  hence  such  notches  are  generally  caUedl 
ifreiri,  and  in  wlmt  folio wft  ihe  latter  term  only  is  used.     The  top 
of  the  weir  corresponds  to  the  horisfiontal  edge  of  the  notch  and  is 
called  the  sill  of  the  weir. 

r  ijf  water  flowing  over  a  weir  or  through  a  notch  is 
g*  V  1  lied  the  vein,  sheetj  or  nappe* 

The  shape  of  the  nappe  depends  upon  the  form  of  the  sill  and 
sid«6  of  the  weir,  tlie  height  of  the  sill  above  the  bottom  of  the 
oi>€ti^«ra  channel  J  the  width  of  the  up-stream  channel,  and  the 
c^iDitrocd/m  of  the  channel  into  which  the  nappe  falls. 

The  effect  of  the  form  of  the  Bill  and  of  the  down-stream, 
lehaimiel  will  be  considered  later,  butj  for  the  present,  attentioi 
J  will  lie  confined  to  weirs  with  sharp  edges,  and  to  those  iu  which 
[the  air  has  free  access  under  the  nappe  so  that  it  detaches  itself 
ply  from  the  weir  as  shown  iu  Fig,  70, 

60,    Rectangular  sharp-edged  weir. 

If  the  crest  and  sides  of  the  weir  are  made  sharp-edged^  aa 
^wn  in  Fig.  70,  and  the  weir  i^  narrower  than  the  approaching 
",and  the  sill  some  distance  above  the  bed  of  the  stream, 
at  the  sill  and  at  the  ddes,  contraction  similar  to  that  at 
» abarp^ged  orifice, 
Tbr  imrface  of  the  water  m  it  approaches  the  weir  falls,  taking  J 
eitmid  form,  so  that  the  thickness  K,  Fig.  70,  of  the  vein  over^ 
weir,  id  teea  than  H,  the  height,  above  the  sill,  of  the  water  at 

t.a.  6 


&ni&iULIC8 


some  distance  from  the  weir.  The  height  H,  whicjh  is  ealledl|fl 
head  over  the  weir,  should  be  carefully  measured  at  such  a  distaoM 
from  itj  that  the  water  surface  has  not  commenced  to  cumaj 
Fteley  and  Steams  state,  that  this  distance  should  be  equal  tol 
2^  times  the  height  of  the  weir  above  the  bed  of  the  stream^  1 

For  the  present,  let  it  be  assumed  that  at  the  point  where  H  if  J 
measured  the  water  is  at  rest.  In  actual  cases  the  water  will  I 
always  have  some  velocity,  and  the  effect  of  thi^  velocity  will  haFel 
to  be  considered  later,  H  may  be  called  the  still  water  head  oyer  I 
the  weir,  and  in  aU  the  fonnulae  following  it  has  this  meaning,      I 

Side  cofUraclwi,  According  to  Fteley  and  Stearns  the  amount  I 
by  which  the  stream  is  contracted  when  the  weir  is  sharp-edg^  I 
is  from  0*06  to  0'12H  at  each  side,  and  Francis  obtained  a  mean  of  I 
O'lH.  A  wide  weir  may  be  divided  into  several  baj^B  by  parti- 1 
tions,  and  thert*  may  then  be  mt>re  than  two  contractions,  at  each  1 
of  which  the  effective  width  of  the  weir  will  be  diminished,  if  1 
Francis'  value  be  taken,  by  0*1  H.  I 

If  L  is  the  total  width  of  a  rectangLilar  weir  and  N  the  mimber  I 
of  contractions,  the  effective  ^^'idth  l^  Fig.  70,  is  then,  I 

(L-O'lN).  I 

When  L  is  very  long  the  lateral  contraction  may  be  neglect^*  I 

Suppresdo7i  of  the  cmitractton.  The  side  contraction  can  be  I 
completrely  suppressed  by  mating  the  approaching  channel  with  I 
vertical  sides  and  of  the  same  width  as  the  weir,  as  was  done  fof  I 
the  orifice  shown  in  Fig.  47.  The  width  of  the  stream  ia  then  I 
equal  t-o  the  \\^dth  of  the  sill.  I 

61.  Derivation  of  the  v^eir  formula  f^om  that  of  a  liidtJ 
orifice.  I 

If  in  the  formula  for  large  orifices,  p,  64,  /i&  is  made  equal  te| 
zero  and  for  the  effective  \\idth  of  the  stream  the  length  I  ill 
substituted  for  b^  and  k  is  unity,  the  formula  becomes  I 

Q  =  t^2gJ.h^    ....a),   j 

If  instead  of  hi  the  head  H,  Fig,  70,  is  substituted^  ondl 
a  coefficient  H  introduced,  I 

The  actual  width  I  is  retained  instead  of  L,  to  make  allowBan 
for  the  end  contraction  which  as  explained  above  is  equal  to  0*13 
for  each  contraction,  I 

If  the  width  of  the  approaching  channel  is  made  equal  to  thi 
width  of  the  weir  I  is  equal  to  L,  J 

With  N  contractions    I  =  (L  -  OIN),  I 

and  Q-|C^.(L-01N)H*.  I 

If  C  is  given  &  mean  value  of  0*625,  I 

Q  =  333CL-0aN)H*  ..,-,,..(2).  I 


FLOW  OVER   WEXBS 


83 


well-known  formula  deduced  by  Prancb*  from^ 
of  €jcperiments  on  sharp-edged  w^eirs, 
itila,  as  an  empirical  one,  is  approximately  cMjrrect  and'j 
Me  \*alne^  for  the  discharge, 

Ki^thcid  of  obtaining  it  from  that  for  large  orifices  is, 
Lipen  to  very  serious  objection,  as  the  velocity  at  F  on 
EP,  Fig*  70,  is  clearly  not  equal  to  zero,  neither  is  the 
flaw  at  the  surface  perpendicnlar  t^  the  section  EF, 
[»re«i&ttre  on  EF,  as  will  be  underst*:jod  later  (section  83) 
(Tkely  to  be  constant, 

Ilhc*  directions  and  the  velocities  of  the  stream  lines  are 
irorn  those  through  a  section  taken  near  a  sharp-edged 
[seen  by  comparing  tho  thickness  of  the  jet  in  the  two  | 
li  the  Ci:»effictent  of  discharge, 
%e  tthiirp-edged  oritic^  ^vith  mde  contractions  suppressed, 
of  the  thicknees  of  the  jet  Uj  the  depth  of  the  orifice  is  not 
afferent  fr*itn  tlie  coefficient  of  discharge,  being  about  0  625, 
llicknetos  EF  of  the  nappe  of  the  weir  is  very  nearly  0'78Hj 
lie  coefficient  of  discharge  is  practically  0'625,  and  the 
[is  therefore  V2A  times  the  coefficient  of  discharge, 
pars  therefore,  that  although  the  assumptions  made  in 
the  t!ow  through  an  orifice  may  be  justifiable,  pro\nding 
[above  the  top  of  the  orifice  is  not  very  small ^  yet  when 
riies  xero,  the  assumptions  are  not  approximately  true. 
Igles  which  the  stream  lines  make  with  the  plane  of  EF 
I  very  different  from  90  degrees,  so  that  it  would  appear, 
?rror  principally  arises  fa^om  the  asaamption  that  the 
bronghout  the  section  is  uniform. 

f«ir  special  cases  has  carefully  measured  the  fall  of  the 
i*  and  the  tliickness  EF,  and  if  the  assumptions  of  constant 
•e  and  stream  lines  perpendicular  to  EF  are  madei  the 
•ge  through  EF  can  Vie  calculated. 

r,  example,  the  height  of  the  point  E  above  the  sill  of  the 
ae  of  Baziu'S  experiments  was  0'n2H  and  the  tliickness 
HBU.     The  fall  of   the  point  F  is,  therefore,  0*1 08H. 
constant  pressure  in  the  section,  the  discharge  per  foot 
He  weir  is,  then, 

Mmm 
^f  V2^,Hi  {C888}*^C108)^l 


•  Lowall,  BudrnMe  Mi^erimtm,  New  York,  186S. 


6—2 


84 


HYDRAULICS 


The  actual  discharge  per  foot  width,  by  experiment,  was 
g  =  0-433  n/2^.H*, 

so  that  the  calculation  gives  the  discharge  1*228  greater  than  the 
actual,  which  is  approximately  the  ratio  of  the  thickness  EF  to 
the  thickness  of  the  stream  from  a  sharp-edged  orifice  having 
a  depth  H.  The  assumption  of  constant  pressure  is,  therefore, 
quite  erroneous. 

62.    Thomson's  principle  of  similarity. 

"  When  a  frictionless  liquid  flows  out  of  similar  and  similarly 
placed  orifices  in  similar  vessels  in  which  the  same  kind  of  liquid 
is  at  similar  heights,  the  stream  lines  in  the  different  flows  are 
similar  in  form,  the  velocities  at  similar  points  are  proportional  to 
the  square  roots  of  the  linear  dimensions,  and  since  the  areas  of 
the  stream  lines  are  proportional  to  the  squares  of  the  linear 
dimensions,  the  discharges  are  proportional  to  the  linear  dimensions 
raised  to  the  power  of  4*-" 

Let  A  and  B,  Figs.  71  and  72,  he  exactly  similar  vessels  with 
similar  orifices,  and  let  all  the  dimensions  of  A  be  n  times  those 
of  B.  Let  c  and  Ci  be  similarly  situated  areas  on  similar  stream 
lines. 


Fig.  71. 


Fig.  72. 


Then,  since  the  dimensions  of  A  are  n  times  those  of  B,  the 
fall  of  free  level  at  c  is  n  times  that  at  Ci.  Let  v  be  the  velocity 
at  c  and  Vi  at  Ci. 

Then,  since  it  has  been  shown  (page  36)  that  the  velocity  in 
any  stream  line  is  proportional  to  the  square  root  of  the  &I1  of 
free  level, 

.'.    V  :  Vi  ::  vn  :  1. 

Again  the  area  at  c  is  n^  times  the  area  at  Ci  and,  therefore, 
the  discharge  through  c        ^    .-      . 
the  discharge  through  Ci  ~  ^  s/n^n  , 

which  proves  the  principle. 

*  British  Association  Reports  1858  and  1876. 


FLOW   OVER   WEIRS 


S5 


63.    Discharge    throngh     a    triangular    notch     by    the 
principle  of  similarity. 

Lies  AI>C\  Figs,  73  and  74,  b©  a  triangular  notch. 


^gr 

rr^ 

^=^y^ — 

4  '*-'• 

yV^  — 

D 

Fig.  74. 


the  depth  of  the  flow  through  the  notch  at  one  time  be  H 
and  at  another  n ,  H. 

Stippfj^^  the  area  of  the  stream  in  the  two  cases  to  be  divided 
into  the  same  number  of  horissontal  elements,  such  as  ab  and  aibi , 

Then  clearly  the  thickness  of  ab  will  be  n  times  the  thickness 

of  Oi^w 

Let  di&i  be  at  a  distance  m  from  the  apex  B,  and  ah  at  a 
distance  m^;  then  the  width  of  ab  is  clearly  n  times  the  width 

The  area  of  ab  will  therefore  ben*  times  the  area  of  ^i&i , 
Again,  the  head  above  ab  is  n  tiniea  the  head  above  ai6i  and 
therefore  the  velocity  thraugh  ah  will  be  \/n  times  the  velocity 
through  a^bj  and   the  discharge   through  ab   will  be  n'   times 
that  through  ai&i. 

More  generally  Thomson  expresses  this  as  follows : 
**  If  tm'o  triangular  notches,  similar  in  form,  have  water  flowing 
through  them  at  different  depths,  but  with  similar  passages  of 
apfiroach,  the  cross  sections  of  the  jets  at  the  notches  may  be 
limitarly  divided  into  the  same  number  of  elements  of  area,  and 
the  area^  of  corresponding  elements  will  be  proportional  to  the 
aquaree  of  the  Hnea!  dimensions  of  the  cross  sections,  or  pro- 
portionai  to  the  squares  of  the  heads.*^ 

Ab  the  depth  h  of  each  element  can  be  expressed  as  a  fraction 
€f  the  head  H,  the  velocities  through  these  elements  are  propor- 
tional  to  the  stjuare  root  of  the  head,  and,  therefore,  the  discharge 
ift  propiirtional  to  H*. 
~    Therefore  Q  a«o  H^ 

being  a  coefficient  which  has  to  be  determined  by  experiment. 
Prom  experiments  with  a  sharp-edged  notch  having  an  angle 
the  rertex  of  90  degrees,  he  found  C  to  be  practically  constant 
'jM  baads  and  equal  to  2*635.    Tlien 

Q-2  635.H* .,,,..(3), 


Mmta 


86  HTDRAULICS 

64.    Flow  tbrongh  a  triangular  notch. 

The  flow  through  a  triangalar  notch  is  frequently  given  as 

Q=Aw>/27.BHt. 

in  which  B  is  the  top  width  of  the  notch  and  n  an  experimental  coefficient. 

It  is  deduced  as  follows: 

Let  ADC,  Fig.  74,  be  the  triangular  notch,  H  being  the  still  water  bead  over 
the  apex,  and  B  the  width  at  a  height  H  above  the  apex.     At  any  depth  h  the 

width  b  of  the  strip  a, 6,  is r^ — -  • 

If  the  velocity  through  this  strip  is  assumed  to  be  v=kj2gh^  the  width  of  the 
stream  through  a^fii ,  — — ,  and  the  thickness  dh,  the  discharge  tbrongh  it  is 

The  section  of  the  jet  just  outside  the  orifice  is  really  less  than  the  ares  EFD. 
The  width  of  the  stream  through  any  strip  a,&j  is  less  than  a^bj,  the  surface  is  lovff 
than  EF,  and  the  apex  of  the  jet  is  some  distance  above  B. 

The  diminution  of  the  width  of  a^b^  has  been  allowed  for  by  the  coefficient  c,  vA 
the  diminution  of  depth  might  approximately  be  allowed  for  by  integrating  between 
^=0  and  ^=:H,  and  introducing  a  third  coefficient  c^. 

Then  Q  =  kecy^  /'^B(H-fe)  ^  —  ^^ 

=TVcCifcN/2^-B-H*, 

Replacing  ee-Jt  by  n 

Q=T^.nV27.BHt (4). 

Calling  the  angle  ADC,  6, 

B=2Htan|, 

and  Q=^^tn^/2^.tan|.H*. 

When  6  is  90  degrees,  B  is  equal  to  2H,  and 

Taking  a  mean  value  for  n  of  0*617 
Q= 2-636.  H*, 

which  agrees  with  Thomson's  formula  for  a  right-angled  notch. 

The  result  is  the  same  as  obtained  by  the  method  of  similarity,  but  the  method 
of  reasoning  is  open  to  very  serious  objection,  as  at  no  section  of  the  jet  are  all  th0 
stream  lines  normal  to  the  section,  and  A;  cannot  therefore  be  constant.  The 
assumption  that  the  velocity  through  any  strip  is  proportional  to  Jh  is  also  open 
to  objection,  as  the  pressure  throughout  the  section  can  hardly  be  uniform. 

65.    Discharge    through     a    rectangoiar    weir    by    the 
principle  of  similarity. 

The  discharge  through  a  rectangular  weir  can  also  be  obtained 
by  the  principle  of  similarity. 


FLOW  OVSB  WEIRS 


87 


Consider  two  rectangular  weirs  each  of  length  L,  Figs.  75 
and  76,  and  let  the  head  over  the  sill  be  H  in  the  one  case  and 
Hi,  or  nH,  in  the  other.  Assume  the  approaching  channel  to  be 
of  such  a  form  that  it  does  not  materially  alter  the  flow  in  either 
case. 

K- L ^ 


^ 


Fig.  75. 


B 

Fig.  76. 


To  simplify  the  problem  let  the  weirs  be  fitted  with  sides 
projecting  up  stream  so  that  there  is  no  side  contraction. 

Then,  if  each  of  the  weirs  be  divided  into  any  number  of  equal 
XiartB  the  flow  through  each  of  these  parts  in  any  one  of  the  weirs 
will  be  the  same. 

Supxx)se  the  first  weir  to  be  divided  into  N  equal  parts.    K 

N  H 
then,  the  second  weir  is  divided  into    J^     equal  parts,  the  parts 

±±1 

in  the  second  weir  will  be  exactly  similar  to  those  of  the  first. 

By  the  principle  of  similarity,  the  discharge  through  each  of 

the  parts  in  the  first  weir  will  be  to  the  discharge  in  the  second 

H* 

as  — ; ,  and  the  total  discharge  through  the  first  weir  is  to  the 

Hi* 

discharge  through  the  second  as 

N.H^         H^_  1 
N.H.Hi^     Hi^     n^' 

Hi 

Instead  of  two  separate  weirs  the  two  cases  may  refer  to  the 
same  weir,  and  the  discharge  for  any  head  H  is,  therefore,  pro- 
pjrtional  to  H^ ;  and  since  the  flow  is  proportional  to  L 

Q  =  C.L.H^ 
in  which  C  is  a  coefficient  which  should  be  constant. 

66.    Rectangular  weir  with  end  contractions. 

If  the  width  of  the  channel  as  it  approaches  the  weir  is  greater 
than  the  width  of  the  weir,  contraction  takes  place  at  each  side, 
and  the  effectual  width  of  the  stream  or  nappe  is  diminished ;  the 
amount  by  which  the  stream  is  contracted  is  practically  inde- 
pendent of  the  width  and  is  a  constant  fraction  of  H,  as  explained 
above,  or  is  equal  to  &H,  k  being  about  0*1 


88  HTDRAULTCS  I 

Let  the  total  width  of  each  weir  be  now  divided  into  three  I 
parts,  the  width  of  ea4?h  end  part  being  equal  to  » .  fc ,  H,  The  I 
width  of  the  end  parts  of  the  transTerae  section  of  the  stream  will  ] 
each  be  i7i  —  l)k  .  H,  and  the  width  of  central  part  L-  2«tH.  I 

The  flow  through  the  central  part  of  the  weir  will  be  equal  to    I 

Qi  =  C(L-2tim)Hi  1 

Now,  whatever  the  head  on  the  weir,  the  end  pieces  of  the  I 

stream,  since  the  width  is  (n  — 1)  JcK  and  fe  is  a  conBtsjit,  will  he  I 

similar  figures,  and,  therefore,  the  flow  through  them  can  be  I 

expressed  as  I 

Qa  =  2C,(n-l)fcHHl  I 

The  total  flow  is,  therefore,  I 

Q  =  C  (L  -  2rifcH)  H^  +  2C»  (n  - 1)  fcHH^-  I 

If  now  Ci  is  assumed  equal  to  C  I 

Q  =  C(L-2^H)Hl  1 

If  instead  of  two  there  are  N  contractions,  due  to  the  weir] 

being  divided  into  several  bays  by  posts  or  partitions,  the  formula  ] 

becomes  I 

Q  =  C(L-NOa.H)Hi  I 

This  is  Francis*  formula,  and  by  Thomson's  theory  it  is  thual 

shown  to  be  rational*  I 

67-    Bazin's*  formula  for  the  discharge  of  a  weir,  ] 

The  discharge  through  a  weir  with  no  side  contraction  may  be  I 
written  _  J 

or  Q  =  mL%/2^.H,  ^J 

the  coefficient  m  being  equal  to  -7=. .  ^H 

v2^  ^1 

Taking  Francis*  value  for  C  as  3'33,  m  is  then  0'415. 

From  experiments  on  sharp-crested  weirs  with  no  side 
traction  Bazin  deduced  for  mt  the  value 

m  ^  0'405  ^     ■□■     . 

In  Table  IX,  and  Fig.  77,  are  shoviTi  Bazin'ti  values  for  m  foi 
different  heads,  and  also  those  obtained  by  Rafter  at  Cornell  upon 
a  weir  similar  to  that  used  by  Bazin^  the  maximum  head  in  thi 
Cornell  experiments  being  much  greater  than  that  in  Baxin': 
experiments.  In  Fig*  77  are  also  shown  several  values  of  m,  lu 
calculated  by  the  anthorj  from  Francis'  experimental  data* 

t  "  Eiperimeots  oa  flow  over  Weira^"  Am.8.C.E^  Td.  xx^n. 


FLOW  OVEK  WEIRS 


TABLE  IX. 


89 


Valueti  of  the  coeflScient  m  in  the  f onmda  Q  =  mL  ^2g  H^* 
W^ir*  aharp-crestedj  6 '56  feet  wide  with  free  oyerfall  and  lateraX 
Dotitrmctioti  suppressed,  H  being  the  atill  water  head  over  the  weir, 
>r  ihe  nuiafliirted  head  A*  corrected  for  irelocity  of  approach. 


Bazin, 

bileet 

a-164 

0^328 

0*656 

0'984 

1*312 

1-64 

1968 

fit 

0448 

0483 

0421 

0-417 

0414 

0412 

0-409 

m 

-tHOS^"-^. 

Rafter. 

^^H 

He&d  in 

f«et 

m 

C 

^^^^ 

0^1 

0-4286 

3437 

^^^H 

0*5 

0-4230 

8-392 

^^^H 

1*0 

0*4174 

3348 

^^^^H 

l'& 

0-4136 

3-317 

^^^H 

M 

0-4106 

8-293 

^^^H 

S^ 

0'4094 

3283 

^^^^1 

8^ 

0*4094 

s-m^ 

^^^H 

S'5 

0*4099 

8-288 

^^^H 

4-0 

0*4112 

8^298 

^^^H 

4'fi 

04125 

8-808 

^^^f 

m 

0-4188 

8-815 

^^^^ 

§'& 

04135 

8-816 

H 

e-0 

0-4136 

3317 

W.    Bazta's  and  the  Cornell  experiments  on  weirs* 

Bazin's  experinient«  were  made  on  a  weirt  6'56  feet  long 
faATiiig  the  approaching  channel  the  same  width  as  the  weir,  so 
lluit  the  lateral  contractions  were  suppressed,  and  the  discharge 
was  meftsiired  by  noting  the  time  taken  to  fill  a  concrete  trench  of 
Iehovth  capacity. 

The  bead  over  the  weir  was  measured  by  means  of  the  hook 
lESOge,  page  249.  Side  chambers  wei^e  constructed  and  connected 
to  the  channel  by  means  of  circular  pipes  0*1  nu  diameter. 

The  water  in  the  chambers  was  very  steady,  and  its  level 
cotaUd  therefore  be  accurately  ganged.  The  gauges  were  placed 
h  metree  from  the  weir.  Tht^  maximum  head  over  the  weir  in 
Bazin%  experiments  was  however  only  2  feet. 

Tlic  experiments  for  higher  heads  at  Cornell  University  were 
tuaile  on  a  weir  of  practically  the  same  width  as  fiassin's,  6'53  feet, 
fher  conditioDii  being  made  as  nearly  the  aame  as  possible; 
Liiaximum  head  on  tlie  weir  was  6  feet. 

*  Sed  page  90. 

i  Mnnalm  4e$  Font*  tl  Vkau9s^f*,  p.  445,  Yq],  ii.  Ig91. 


J 


00 


HYDRAULICS 


The  results  of  these  experiments,  Fig,  77,  show  that  the 
cciefficient  m  diminishes  and  then  increases,  having  a  minimmn 
value  when  H  is  between  2'5  feet  and  3  feet. 


"I  3      3 ^ 

JfiKzO'  oitffxajmt  atryes  ^r  Sharp -^d^mt  H'oLrs 

A  FnMfuxa*  "       (Deduced  by  ths  ajuUh^r} 

Fig,  77. 

It  is  doubtful,  however,  although  the  experiments  were  made 
with  great  care  and  skill,  whether  at  high  heads  the  deduced 
coefficients  are  absolutely  reliable* 

To  measure  the  iiead  over  the  weir  a  1  inch  galvanised  pipe 
with  holes  Jinch  diameter  and  opening  downwards^  6  inchea 
apart,  was  laid  acrosus  the  channel  To  this  pipe  were  connected 
\  inch  pipes  pfi-ssing  through  the  weir  to  a  convenient  point  beloir 
the  weir  where  they  couki  be  connected  to  the  gauges  by  rubber 
tubing.  The  gauges  were  glas^  tubes  J  inch  diameter  mounted 
on  a  frame,  the  height  of  the  water  being  read  on  a  seal* 
graduated  to  zmm*  spaces. 


69,    Velocity  of  approach. 

It  should  be  clearly  understood  that  in  the  formula  given,  it^ 
has  been  assumed,  in  giving  values  to  the  coefficient  m  that  H  is 
the  height,  above  the  sill  of  the  weir,  of  the  still  water  surface. 


A 


FLOW  OVER  WEIRS  91 

In  actual  cases  the  water  where  the  head  is  measured  will  have 
ome  velocity,  and  due  to  this,  the  discharge  over  the  weir  will  be 
acreased. 

If  Q  is  the  actual  discharge  over  a  weir,  and  A  is  the  area  of 
he  up-stream  channel  approaching  the  weir,  the  mean  velocity  in 

he  channel  is  t?  =  ? . 
A 

There  have  been  a  number  of  methods  suggested  to  take  into 
iccouit  this  velocity  of  approach,  the  best  perhaps  being  that 
idopted  by  Hamilton  Smith,  and  Bazin, 

This  consists  in  considering  the  equivalent  still  water  head  H, 
Dfver  the  weir,  as  equal  to 

a  being  a  coefficient   determined   by   experiment,    and   h   the 
measured  head* 

The  discharge  is  then 

<i.mj2ih(h*^' (5), 

<3...L(<..f)y2,(A.g). 

CLV 

Expanding  (5),  and  remembering  that  ^-7  is  generally  a  small 
quantity, 

The  velocity  v  depends  upon  the  discharge  Q  to  be  determined 
and  is  equal  to  ^  • 

Therefore         Q  =  ^LA>/2^  ( 1  +  |  2£0   ^^^• 

From  five  sets  of  experiments,  the  height  of  the  weir  above  the 
bottom  of  the  channel  being  different  for  each  set,  Bazin  found 
the  mean  value  of  a  to  be  1*66. 

This  form  of  the  formula,  however,  is  not  convenient  for  use, 
since  the  unknown  Q  appears  upon  both  sides  of  the  equation. 

If,  however,  the  discharge  Q  is  expressed  as 

Q  =  nL  ^2gh,  h, 
the  coefficient  n  for  any  weir  can  be  found  by  measuring  Q  and  h. 

It  will  clearly  be  different  from  the  coefficient  m,  since  for  m 
to  be  used  h  has  to  be  corrected. 

From  his  experimental  results  Bazin  calculated  n  for  various 
heads,  some  of  which  are  shown  in  Table  X. 


..,u). 


:iC .: 


.:t'  :o  the 
f-r:  above 


FLOW  OVER  WEIRS  93 

coefficient   to    determine  Q,  h   most   first   be  corrected,  or  Q 
calculated  from  formula  9. 

Baher  in  determining  the  values  of  m  from  the  Cornell  ex- 

periments,  increased  the  observed  head  ^  by  o;-  only,  instead  of 

by  1-66^. 
2g 

Fteley  and  Steams*,  from  their  researches  on  the  flow  over 

weirs,  f onnd  the  correction  necessary  for  velocity  of  approach  to 

be  from 

1-45  to  1-5^. 

Hamilton    Smith  t    adopts  for  weirs  with    end    contractions 
sappressed  the  valaes 

1-33  to  1-40^, 
and  for  a  weir  with  two  end  contractions, 
11  to  1-251^. 

TABLE  X. 
Coefficients  n  and  m  as  calculated  by  Bazin  from  the  formulae 
Q=  nL'J2gh^ 
and  Q  =  mLN/^H^ 

h  being  the  head  actually  measured  and  H  the  head  corrected  for 
velocity  of  approach. 


Head 
h  in  feet 

Height  of  sill 
p  in  feet 

Coefficient 
n 

Coefficient 
m 

0-164 

0-656 
6*560 

0-458 
0-448 

0-448 

0-984 

0-656 
6-560 

0-500 
0-421 

0-417 

1-640 

0-656 
6-560 

0-500 
0-421 

0-4118 

An  example  is  now  taken  illustrating  the  method  of  deducing 
ilie  coefficients  n  and  m  from  the  result  of  an  experiment,  and  the 
difference  between  them  for  a  special  case. 

EaoM^U.  In  one  of  Bazin*8  experiments  the  width  of  the  weir  and  the 
■ffMiMiiliiiHj,  channel  were  hoth  6-56  feet.  The  depth  of  the  channel  approaching 
mm  wmr  mminrnd  at  a  point  2  metres  up  stream  from  the  weir  was  7-544  feet  and 

hflad  menBiired  over  the  weir,  which  may  be  denoted  by  A,  was  0-984  feet.  The 
~  diaehazge  was  21*8  cubic  ft.  per  second. 

♦  TranMoetiom  Am,S,C.E,,  Vol.  xn. 
t  Hydrauliet, 


94 


HTBRAULICS 


The  Telooitj  at  the  sectiou  whsre  h  was  me&Bnrfld»  and  which  may  be  called  Ihe  I 
Tilooitj  of  approach  was,  therefore, 

Q  _        2ia 

^  ~  7'54i  X  B  56' '  7  544  jc  eW 

=  0^44  feet  jier  aeccmd. 
If  now  the  formala  for  discharge  be  written 

and  n  ia  qaleuLated  from   this  formula   by   Bubstftuimg  the   known   valaea  of] 
Q,  L  and  h 

n  =  0*421. 
Correeting  h  for  veiooitj  of  approacb. 


2ff 


Then 

from  whtoh 


=  •9888, 
918 


'emj2g.^dms 


=  0-415. 


It  will  sieem  froin  Table  X  that  when  the  h«ip;hi  p  of  the  dll  of  the  weir  abore 
the  stream  bed  is  amall  compared  with  tba  head,  Ihe  dilFerenoe  majr  be  much  ' 
larger  than  for  thia  eiample. 

When  the  head  is  1-64  feet  and  larger  than  p,  the  coeffiolent  n  ia  eighteen] 
per  oent.  greater  than  m.  In  auoh  eases  failnTe  to  eoireet  the  eoefficient  wUJ  letil 
to  considerable  inaccDracy, 

70.  Influence  of  the  height  of  the  weir  slU  above  the  bed 
of  the  stream  on  the  contraction. 

The  nearer  the  sill  is  to  the  bottom  of  the  streamj  the  less  the 
contraction  at  the  sill,  and  if  the  depth  is  small  compared  with  H,  ] 
the  diminution  on  the  contraction  may  conBiderably  affect  the] 
flow. 

When  the  sill  was  Vlb  feet  above  the  bottom  of  a  channel,  I 

of  the  same  width  as  the  weir,  Bazin  found  the  ratio  ^   (Fig.  85) 
to  be  0^097,  and  %vhen  it  was  S70  feet,  to  be  0'112.     For  greater  ] 
heights  than  these  the  mean  value  of  ^  was  OlS. 

71.  Discharge  of  a  weir  when  the  air  is  not  fireelj ' 
admitted  beneath  the  nappe.    Form  of  the  nappe. 

Francis  in  the  Lowell  experiments,  found  that,  by  making  the 
width  of  the  channel  below  the  weir  equal  to  the  width  of  the 
weir,  and  thus  preventing  free  access  of  air  to  the  underside  of  the 
nappe  J  the  discharge  was  increased.  Bazin*,  in  the  experiments  | 
already  referred  to,  has  investigated  very  fully  the  effect  upon 
the  discharge  and  upon  the  form  of  the  nappe,  of  restricting  the 
free  passage  of  the  air  below  the  nappe.  He  finds,  that  when  the 
flow  is  sufficient  to  prevent  the  air  getting  under  the  nappe,  it  niay 
assume  one  of  three  distinct  forms,  and  that  the  discharge  for 
*  Annalev  d^t  FmU  ti  Chau4s€ea,  1891  and  1896. 


L=> 


ite^ 


FLOW  OVXB  WEIBS 


95 


I  them  may  be  28  per  cent,  greater  tlian  when  the  air  is 
admitted,  or  the  nappe  is  ''free.''  Which  of  these  three 
the  nappe  assomes  and  the  amount  by  which  the  discharge 
ater  than  for  the  ''free  nappe/'  depends  largely  upon  the 
over  the  weir,  and  also  upon  the  height  of  the  weir  above 
ater  in  the  down-stream  channel. 

18  phenomenon  is,  however,  very  complex,  the  form  of  the 
'  for  any  head  depending  to  a  very  large  extent  upon 
ler  the  head  has  been  decreasing,  or  increasing,  and  for  a 
head  may  possibly  have  any  one  of  the  three  forms,  so  that 
jscharge  is  very  uncertain.  M.  Bazin  distinguishes  the  forms 
^pe  as  follows : 

)  Free  nappe.  Air  under  nappe  at  atmospheric  pressure, 
70  and  78. 

)     Depressed  nappe  enclosing  a  limited  volume  of  air  at  a 
ire  less  than  that  of  the  atmosphere.  Fig.  79. 
)    Adhering  nappe.    No  air  enclosed  and  the  nappe  adher- 
the  down-stream  &ce  of  the  weir.  Fig.  80.    The  nappe  in  this 
oay  take  any  one  of  several  forms. 


Tcpafauam£l\  ! 


Fig.  79.  Fig.  80. 

Drowned  or  wetted  nappe,  Fig.  81.  No  air  enclosed  but 
ppe  encloses  a  mass  of  turbulent  water  which  does  not  move 
le  nappe,  and  which  is  said  to  wet  the  nappe. 


Fig.  81. 


96 


ETBEA0LICS 


72*    Depressed  nappe. 

The  air  below  the  nappe  being  at  less  than  the  atmofipherie" 

pressure  the  excess  pressure  on  the  top  of  the  nappe  causes  it  to^ 
be  depressed*    There  is  also  a  rise  of  water  in  the  down-st 
channel  under  the  nappe. 

The  discharge  is  slightly  greater  than  for  a  free  nappe.     On  i 
weir  2*46  feet  above  the  bottom  of  the  up-stream  channel ^  thej 
nappe  was  depressed  for  heads  below  0'77  feet,  and  at  this  head 
the  coefficient  of  discharge  was  1*08  wii,  mi  being  the  absolute 
coefficient  for  the  free  nappe, 

73*    Adhering  nappes. 

As  the  head  for  tliis  weir  approached  0*77  feet  the  air 
rapidly  expelledj  and  the  nappe  became  vertical  as  in  Fig,  80,  ita 
surface  having  a  corrugated  appearance*     The  coefficient  of  di 
charge  changed  from  r08  Wi  to  r28TOi-     This  large  change  ia| 
the  coefficient  of  discharge  caused  the  head  over  the  weir  to  fall 
to  0'69  feet,  but  the  nappe  still  adhered  to  the  weir, 

74.    Browned  or  wetted  nappes. 

As  the  head  was  further  increased,  and  approached  0'97  fe 
the  nappe  came  away  from  tlie  weir  face,  assuming  the  drowned| 
form,  and  the  coefficient  suddenly  fell  to  119  mi.     As  the  he 
was  further  increased  the  coefficient  diminished^  becoming  V12\ 
when  the  head  was  above  IS  feet. 

The  drowned  nappes  are  more  stable  than  the  other  two,  bu 
whereas  for  the  depressed  and  adliering  nappes  the  discharge 
not  affected  by  the  depth  of  wat^r  in  the  down-stream  channd, 
the  height  of  the  water  may  influence  the  flow  of  the  dro^ 
nappe.    If  when  the  drowned  nappe  falls  into  the  down  streanl 
the  rise  of  the  water  takes  place  at  a  distance  from  the  foot  of  thai 
nappe^  Fig,  81  j  the  height  of  the  down-stream  water  does  not  affe 
the  flow.    On  the  other  hand  if  the  rise  encloses  the  foot  of  tia 
nappe.  Fig.  82,  the  discharge  is  affected.    Let  K^  be  the  differenc 


Fig.  as. 


FLOW  OVER  WEIRS  97 

erf  level  of  the  siJI  of  the  weir  and  the  water  below  tho  weir.  The 
coefficieiit  of  discharge  in  the  first  ease  iB  independent  of  h^  bat  is 
dependent  upon  p  the  height  of  the  sill  above  the  head  of  the  up- 
stream channel,  and  is 

m^  =  m,  ^0878  +  0128  |Y.. ..(11). 

Bazin  funnd  that  the  drowned  nappe  could  not  be  formed  if  h 
is  lees  than  0*4  p  and,  therefore,  r  cannot  be  greater  than  2'5, 

Snbatitnting  for  f»i  its  value  ■ 

£rom  (10)  page  92 

w»,  =  0"470  +  0TO75^'  .,....„... (12). 

In  the  second  case  the  coefficient  depends  upon  ^,  and  ia, 

m^mi(im  +  0'my(^-om)^  „,(13), 

for  iprhich,  with  a  safficient  degree  of  approximation,  may  be 
nbptitated  the  simpler  formula, 

m,  =  mj(r05+M5^) ...,....(14). 

The  limiting  value  of  ?%  is  1*2  m^  for  if  hi  becomes  greater 
ths^  h  the  nappe  is  no  longer  drowned. 

Further,  the  rise  can  only  enclose  the  foot  of  the  nappe  when 
h^  i»  lesa  than  Q  p-  k).  Ay  h^  passes  this  value  the  rise  is  pushed 
duwn  arream  away  from  the  foot  of  the  nappe  and  the  coefficient 
chAHgr^e  to  that  of  the  preceding  case. 

fB*    Instability  of  the  form  of  the  nappe, 

'  ul  at  wliich  the  form  of  nappe  changes  depends  upon 

^rh*-  i*'  head  is  increasing  or  diminishing,  and  the  depressed 

ajid  adiiering  nappes  are  very  nnstable^  an  accidental  admission 

'jr  other  mterference  causing  rapid  change  in  their  form, 

r,  the  adhering  nappe  is  only  formed  under  special  circum- 

,  and  as  the  air  is  expelled  the  depressed  nappe  generally 

,1.1  --^.  r^  directly  to  the  drowned  form, 

tf,  therefore,  the  air  is  not  freely  admitted  below  the  nappe 

iven  head  is  very  uncertain  and  the  discharge 

L  with  any  great  degree  of  assurance. 

With  the  weir  2  46  feet  above  the  bed  of  the  channel  and  6  56 

'  '*       i  obtained  for  the  same  head  of  0*656  feet,  the  four 

^  ; :  .    ,  the  coefficients  of  discharge  being  as  follows ; 


98  fiTDEAULlCS 

Free  nappe,  0*433 

Depre^ed  nappe,  0*460 
Drowned   nappe,   level   of  water  down  stream 

O'^l  feet  below  the  crest  of  the  weir,  0"4OT 

Nai>pe  aflhering  to  down-stream  ince,  0'554 
The  dii?charge  for  tliis  weir  w^hile  the  head  was  kept  oonstant, 
thus  vari<.Hi  26  per  cent. 

76.    Drowned  weirs  with  sharp  crests*. 

When  the  surface  of  the  water         n  stream  is  higher  than  t^ 
sill  of  the  weir,  as  in  Fig.  83,  the  w        s  said  to  be  drowned- 


T 


r 


:^^ — 


Fig,  83. 


* 


Bazin  gives  a  formula  for  deducing  the  coefficients  for  such  a 
weir  from  those  for  the  sharp-edged  weirs  with  a  free  nappe,  w^bicli 
in  its  simplest  form  is, 


ITIe 


--bH'^'^^/'n^] "^'. 


^ 


Aa  being  t!ie  height  of  the  down-stream  water  above  the  sill  of 
the  weir,  k  the  head  actually  measured  abov^e  the  weir,  p  the 
height  of  the  sill  above  the  up-stream  channel,  and  mi  the 
coefficient  ((10),  p,  02)  for  a  sharp-edged  weir.  This  oxpre^isioti 
gives  the  same  value  within  1  or  2  per  cent,  as  the  formulae  (13) 
and  (14). 

Example.    The  heiid  ovex  a  mnr  is  I  foot,  and  tbe  height  of  the  sill  abova 
np-stream  channel  Ia  fj  feet.     The  leugth  is  ti  Seet  &iid  the  eurface  of   the 
in  the  down -stream  channel  is  6  inches  alM>vi)  the  Rill,     Find  the  diachajge. 

From  formnla  (10)i  pa^e  f^2,  the  eoeffici«nt  fitj  for  a  b harp-edged  weir  with  tim 
nappe  is 

*  AttempU  have  been  made  to  esprega  the  diiieharge  over  a  drowned  wdx 
equivalent  to  thnt  throagh  a  drowned  orifice  of  an  area  equal  to  LA^,  under  a  hi 
h-ha*  together  with  a  disobarge  over  a  weir  of  length  L  when  the  head  ia  h^hm. 

I^e  dUscharge  is  then 

n^2ghh^{h-h^)i  +  m^lj{h-h^)^, 
n  and  m  being  coefficients. 


^m  FLOW  O^EB   WEIRS  99 

^^^K»  m,== '4215  [1^(1 +-031}  0-761] 

^^^^^  ^3440, 

I     Thai  Q  =  *a44>^V2^.1* 

I  =22i)S  eubio  ft*  }ser  second. 

I      77.    Vertical  weirs  of  small  thickness. 

I  likhtt^ail  of  makiiig  the  sill  of  a   weir  sharp-edged,   it   niaj^ 

I  bave  a  flat  sill  of  thickness  c.  This  will  frequently  be  the  ease  in 
nbetict^  the  weir  being  constructed  of  timbers  of  uniform  width 
placed  mvi*  upon  the  other.     The  conditions  of  flow  for  these  weirs 

I  My  be  v^ry  dilferent  from  those  of  a  sharp-edged  weir, 

I  I'he  Ttappes  of  such  weirs  present  two  distinct  forms,  according 

I  as  the  WHter  is  in  contact  with  th€+  crent  of  tlie  weir,  or  becomes 

I  tWtat'liwl  at  the  up-stream  edge  and  leaps  over  the  crest  without 

I  *<itjcliiTig  the  down-stream  edge.    In  the  second  case  the  discharge 

I  » tile  hHtne  as  if  the  weir  were  sharp-edged.     When  the  head  k 

I  'jrt'er  the  weir  is  more  than  2c  this  condition  is  realised,  and  may 

I  *»^tftiii  when  h  pa?+ses  |c.     Between  these  two  value**  the  nappe  is 

I  t«»coTuHtian  of  unstable  equilibrium;  when  k  is  less  than  fc  the 

I  oapjx'  adheres  to  the  sill,  and  the  coefficient  of  discharge  is 

I  m^  =  mi  (070  + 0*185^), 

V  *3^  oxtemal  perturbation  such  as  tlie  entrance  of  air  or  the 
F    ^'^'^^-r  1,1  a  floating  body  causing  the  detachment. 
\  li^ppL*  adheres  between  ic  and  2c  the  coefBcient  m^  varies 

I  ^lik  Mnii  to  r07iwi,  bat  if  it  is  free  the  coefficient  w^^mi. 
I  "  n«D  B  =  Jc,  m^  is  79mi .  If  therefore  the  coefficients  for  a 
I  *wtfp-etiin?d  weir  are  used  it  is  clear  the  error  may  be  con- 
HfnabliC 

HP^  formula  for  iWis  gives  approximately  correct  results  when 
*fc^  width  of  the  sill  m  great,  from  3  to  7  feet  for  example. 

If  the  up-{^tream  edge  of  tlie  weir  is  rounded  the  discharge  is 

iDCi^.Hed.     Ilic  di&charge*  for  a  weir  having  a  crest  6'bG  feet 

^di\  when  the  up-stream  edge  was  rountk?d  to  a  radius  of  4  inches, 

•^  mcreaeed  by  14  per  cent.,  and  that  of  a  weir  2'624  feet  wide 

ky  12  per  cent. 

The  roofiding  of  the  comers^  due  to  wear,  of  timber  weirs  of 
dmienmonsi  to  a  radius  o£  1  inch  or  less,  will,  therefore, 
IpcI  the  flow  eonsiderahly. 

78.    Depressed  and  wetted  nappes  for  flat-crested  weirs. 

The  nappes  of  weirs  having  tiat  sills  may  be  depre^ed,  and 
ijf  become  drowned  as  for  sharp-edged  weirs, 

•  dnmla  du  Pmtt  et  Ch&u»»iti,  Val  u.  18^. 

7—2 

416V02 


100  HYDRAULICS 

The  coefficient  of  discharge  for  the  depressed  nappes,  whether 
the  nappe  leaps  over  the  crest  or  adheres  to  it,  is  practically  the 
same  as  for  the  free  nappes,  being  slightly  less  for  low  heads  and 
becomes  greater  as  the  head  increases.  In  this  respect  they  differ 
from  the  sharp-crested  weirs,  the  coefficients  for  which  are  always 
greater  for  the  depressed  nappes  than  for  the  free  nappes. 

79.  Drowned  nappes  for  flat-crested  weirs. 

As  long  as  the  nappe  adheres  to  the  sill  the  coefficient  m  may 
be  taken  the  same  as  when  the  nappe  is  free,  or 

When  the  nappe  is  free  from  the  sill  and  becomes  drowned, 
the  same  formula 

77io  =  mi(o-878+0-128|), 

as  for  sharp-crested  weirs  with  drowned  nappes,  may  be  used. 
For  a  given  limiting  value  of  the  head  h  these  two  formulae  give 
the  same  value  of  m©.  When  the  head  is  less  than  this  limiting 
value,  the  former  formula  should  be  used.  It  gives  values  of  m 
slightly  too  small,  but  the  error  is  never  more  than  3  to  4  per  cent 
When  the  head  is  greater  than  the  limiting  value,  the  second 
formula  should  be  used.  The  error  in  this  case  may  be  as 
great  as  8  per  cent. 

80.  Wide  flat-crested  weirs. 

When  the  sill  is  very  wide  the  surface  of  the  water  fali^ 
towards  the  weir,  but  the  stream  lines,  as  they  pass  over  the  weimr^ 
are  practically  parallel  to  the  top  of  the  weir. 

Let  H  be  the  height  of  the  still  water  surface,  and  h  the  depbii 
of  the  water  over  the  weir,  Fig.  84. 


T 


L' 


Fig.  84. 

Then,  assuming  that  the  pressure  throughout  the  section  of  tb^ 
nappe  is  atmospheric,  the  velocity  of  any  stream  line  is 

v=J2g{B."h), 
and  if  L  is  the  length  of  the  weir,  the  discharge  is 

Q  =  ^Lhs/W^) (16). 


FLOW  OVER  WBIBS  101 

For  the  flow  to  be  pennanent  (see  page  106)  Q  must  be  a 
naximom  for  a  given  value  of  h,  or  -^  must  equal  zero. 
Therefore 

From  which  2(H-fe)-fe  =  0, 

ind  h  =  f  H. 

Substituting  for  h  in  (16) 

=  0-385L  n/^ . H  =  308L  VH . H. 
The  actual  discharge  will  be  a  little  less  than  this  due  to 
friction  on  the  sill,  etc. 

Bazin  found  for  a  flat-crested  weir  6*56  feet  wide  the  coefficient 
m  was  0-373,  or  C  =  2-991. 

Lesbros'  exx>eriment8  on  weirs  sufficiently  wide  to  approximate 
to  the  conditions  assumed,  gave  '35  for  the  value  of  the  co- 
efficient w. 

In  Table  XI  the  coefficient  C  for  such  weirs  varies  from  2'66 
to  310. 

81.   Flow  over  dams. 

Weirs  of  various  forms,  M.  Bazin  has  experimentally  investi- 
?^^rf  the  flow  over  weirs  having  (a)  sharp  crests  and  (6)  flat 
crests,  the  up-  and  down-stream  faces,  instead  of  both  being  vertical, 


(1)  vertical  on  the  down-stream  face  and  inclined  on  the 
"P-stream  face, 

(2)  vertical  on  the  up-stream  face  and  inclined  on  the  down- 
^ream  face, 

(3)  inclined  on  both  the  up-  and  down-stream  faces, 
^^d  (c)  weirs  of  special  sections. 

The  coefficients  vary  very  considerably  from  those  for  sharp- 
c^ted  vertical  weirs,  and  also  for  the  various  kinds  of  weirs. 
^^fficients  are  given  in  Table  XI  for  a  few  cases,  to  show  the 
^^essity  of  the  care  to  be  exercised  in  choosing  the  coefficient  for 
^^y  weir,  and  the  errors  that  may  ensue  by  careless  evaluation  of 
the  coefficient  of  discharge. 

For  a  full  account  of  these  experiments  and  the  coefficients 
^otained,  the  reader  is  referred  to  Bazin's*  original  papers,  or  to 
Rafter's  t  paper,  in  which  also  will  be  found  the  results  of  experi- 

•  Afmaie*  de$  Fonts  et  Chau*s4ett,  1898. 

t  Traruactions  of  the  Am.S.C.E.,  Vol.  XLiv.,  1900. 


102 


HYDRAULICS 


TABLE  XI.  ^^ 

Values  of  the  coefficient  C  in  the  formula  Q  =  CL  .  fc%  for  weirs 
of  the  sections  shown,  for  various  values  of  the  observed  head  A. 

Bazva. 


Section  of 

Head  iu  feet 

weir 

0-3 

0-6 

10 

1-3 

20 

30 

40 

6-0 

6-0 

V31S 

2*66 

2-66 

2*90 

8*10 

1 

1 

-i 

8-61 

8*80 

401 

8*91 

y^ 

^ 
s 

402 

416 

418 

416 

< 

\s 

8-46 

8-67 
8-49 

8*86 

8*80 

*< 

^^^^C 

8*46 

8*59 

8*68 

gc:^^ 

808 

808 

819 

8*22 

FLOW  OVER  WEIBS 


103 


TABLE  XI   (continued). 
Baain. 


Section  of 

Head  in  feet 

weir 

0-8 

0-6 

10 

1-3 

20 

30 

40 

60 

60 

— ^    '€S 

810 

8-27 
8-05 

8-78 

8-90 
8-78 

i^-Y^ 

T^ 

2-76 

8-62 

Section  of 
weir 


— -J3^ 


Rafter. 


Head  in  feet 


0-3 


0-6        10        1-3        20       30        40        50       60 


8-85 


814 


2-95 


8-68 


8-42 


816 


8-88 


8-52 


8-27 


8-77 


8-61 


8-45 


8-68 


8-66 


8-56 


8-70 


8-66 


8-71 


8-64 


8-7] 


8-6£ 


8-61 


8-66 


8-6^ 


104 


ByDRACTLlCS 


inents  made  at  Cornell  University  on  the  discharge  of  weirs,  sitaiJar 
to  those  used  by  Bazin  and  for  heads  higher  than  lie  used,  and 
also  weirs  of  sections  approximating  more  closely  to  tho&e  oi 
existing  masonry  dams,  nsed  as  weirs*  From  Bazin's  and  Rafter's 
experimentSj  curves  of  discharge  for  varying  heads  for  some  of 
these  actual  weirs  have  been  dra\vn  up, 

82.  Form  of  weir  for  accurate  gauging. 

The  uncertainty  attaching  itself  to  the  correction  to  be  applied 
to  the  measured  head  for  velocity  of  approach,  and  the  difficulty 
of  making  proper  allowance  for  the  imperfect  contraction  at  the 
sides  and  at  the  sill,  when  the  sill  is  near  the  bed  of  tlie  channel 
and  is  not  sharp-edged,  and  the  inetabihty  of  the  nappe  and 
uncertainty  of  the  form  for  any  given  head  when  the  admission  of 
air  below  the  nappe  is  imperfect,  make  it  desirable  that  as  far  as 
possible,  when  accurate  gaugings  are  reqniredj  the  %veir  should 
comply  with  the  following  four  conditions,  aa  laid  down  by 
Bas&in, 

(1)  The  sill  of  the  weir  must  ba  made  as  high  as  possible 
above  the  bed  of  tlie  stream, 

(2)  Unless  the  weir  is  long  compared  with  the  head,  the 
lateral  contraction  should  be  suppressed  by  making  the  channel 
approaching  the  weir  with  vertical  sides  and  of  the  same  width  as 
the  weir. 

(3)  The  sill  of  the  weir  must  be  made  sharp-crested. 

(4)  Free  access  of  air  to  the  sides  and  under  the  imppe  of 
the  weir  must  be  ensured. 

83.  Boussinesq's*  theory  of  the  discharge  over  a  weir. 

As  stated  above,  if  air  is  freely  admitted  below  the  nappe  of 
a  weir  there  is  a  contraction  of  the  stream  at  the  sharp  edge  of  the 
sillj  and  also  due  to  the  falling  curved  surface. 

If  the  top  of  the  sill  is  well  removed  from  the  bottom  of  thd 
channel,  the  amount  by  which  the  arched  under  side  of  the  nappe 
is  raised  above  the  sill  of  the  weir  is  assumed  by  Boussinesq— And 
this  assumption  has  been  verified  by  Bazin's  expeinments—to  be 
some  fraction  of  the  head  H  on  the  weir. 

Let  CDj  Fig,  85,  be  the  section  of  the  vein  at  wliich  the 
maximum  nBe  of  the  bottom  of  the  vein  occurs  above  the  sill,  and 
let  e  be  the  height  of  D  above  S. 

Let  it  be  assumed  that  through  the  section  CD  the  stream 
lines  are  moving  in  curved  paths  normal  to  the  section^  and  that 
they  have  a  common  centre  of  cur^^ature  0. 

*  Comptf*  Eendm,  XSSI  ftjad  1889. 


FLOW  OYER  WEIRS 


105 


Let  H  be  the  height  of  the  snr&ce  of  the  water  up  stream 
ove  the  silL  Ijet  R  be  the  radius  of  the  stream  line  at  any 
bt  E  in  CD  at  a  height  x  above  S,  and  Bi  and  Bs  the  radii  of 
rvatore  at  D  and  C  respectively.  Let  V,  Vi  and  Vj  be  the 
locities  at  E,  D,  and  C  respectively. 


Fig.  85. 

Consider  the  equilibrium  of  any  element  of  fluid  at  the  point 
E,  the  thickness  of  which  is  8R  and  the  horizontal  area  is  a.  If  w 
is  the  weight  of  unit  volume,  the  weight  of  the  element  is  w .  a3R. 

Since  the  element  is  moving  in  a  circle  of  radius  R  the  centri- 

V8R 
fngal  force  acting  on  the  element  is  wa  — ^~  lbs. 

gix 

The  force  acting  on  the  element  due  to  gravity  is  iraSR  lbs. 

Let  p  be  the  pressure  per  unit  area  on  the  lower  face  of  the 

element  and  p  +  Sp  on  the  upper  face. 

Then,  equating  the  upward  and  downward  forces, 

/       ^  \  ST>  .  'M?aV'SR 

(p  +  op)  a  +  tcaoK  =  pa+       -^ — . 

gti 

From  which  ^dR=-l^^  ^^^- 

Assuming  now  that  Bemouilli's  theorem  is  applicable  to  the 
ream  line  at  EF, 

w     2g 
Differentisitingy  and  remembering  H  is  constant, 
,       dp     YdV 


w 


=  0, 


ldp_     ^      YdY 
■w  dx  g.dx 


IM  ffTTntjfcrxics 


fa 
xR- 

vrv 

tSt.  ' 

iW*  ■ 

-V,iK  =  0- 

trakos  of  die 

the 

pressme 

is 

=  *-ig  H--U 


"^n  =  T  3i,  JBBt.  3,  i^m  die  ignre  is  (B,  +  *  —  «),  therefore, 


Kt 


--'^^-''^ZJZ,  (2). 


'-^^^^'(s;^.)'^ 


=  ^ix  3-*  Rt  I 


Bi^jr-e 


n.-K 


=-2?  ^-^  B.Ioe^=g-^    (3). 

^'^   1  -at  TOW  .»v«r  -ii?  WOT  is  permanent,  the  thickness  K  <rf 
:t»   :»|f|K»   auBC  iiUit&gf  :t&ik£.  iu  due  for  the  given  head  H  the 

'*Sr  ^HiuatttflBt  ^w  iuwi*Ter  cui  only  take  place  if  each 
'flSff  ^'v^'xvtt  i>F  Jft^  tai?  maximnm  velocity  possible  to 
:at«  iliuneiLGs  will  be  accelerated;  and 
^•i-  -  .;^v«t  iteiciMciQi^  -n*  tan:icii>tst^  «•  k  therefore  a  minimnm,  or 
vc  -*r  .^r*««>   ^^^  "C  ^  ^iK  xbscfurae  is  a  maximnm.    That  is,  when 

^  '::feto<^<ktK^  A  ^litt  J*  wmtMi  J6  a  fimction  of  fc,,  the  valae  of 
^    ^NiUv.u   "iiat^    4  ^  -^^^^ni^iin^  can  be  determined  by  differ- 

A. 


Therefore,  A«  =  (H  -  e)  (1  -  »«), 

Ri  =  w(l  +  n)(H~e). 
Sabstitatm£^  this  value  of  Bi  in  the  expression  for  Q, 

Q=V2^.(H~6)*(n  +  n»)logi 


FLOW  OVER  WEIRS  107 


n' 


lAich,  smce  Q  is  a  Tnaximnin  when  ^  =  0,  and  A  is  a  function 
%  is  a  maxiiniiin  when  --p  =  0. 


Bifferentiatincr  &i^d  equating  to  zero^ 

(l+2n; 
D  solution  of  which  gives 


(l+2n)log--(l  +  n)  =  0, 

TV 


n  =  0-4685, 
and  tlierefore,  Q  =  05216  ■J2g(a-  e)* 

=  0-5216>/^(l-g)*H* 
=  0-5216  (l-g)*-s^.H* 

•  Ae  coefficient  m  being  equal  to 
^  0-5216  (l  -  g)*. 

M.  Bazin  has  found  by  actual  measurement,  that  the  mean 
i^^  iralne  for  ^^  when  the  height  of  the  weir  is  at  considerable 
<   disfcanoe  from  the  bottom  of  the  channel,  is  0*13. 

Then,  (l-g)*  =  0-812, 

and  m  =  0'423. 

It  will  be  seen  on  reference  to  Fig.  77,  that  this  value  is  very 
near  to  the  mean  value  of  m  as  given  by  Francis  and  Bazin,  and 
the  Cornell  experiments.    Giving  to  g  the  value  32*2, 

Q  =  3'39  H*  per  foot  length  of  the  weir. 

If  the  length  of  the  weir  is  L  feet  and  there  are  no  end  con- 
tractions the  total  discharge  is 

Q  =  3-39L.Ht, 
and  if  there  are  N  contractions 

Q  =  3-39(L-N01H)Hi 


108  HYDRAULICS 

The  coefficient  3*39  agrees  remarkably  well  with  the  mean 
value  of  C  obtained  from  experiment. 

The  value  of  a  theory  must  be  measured  by  the  closeness  of 
the  results  of  experience  with  those  given  by  the  theory,  and  in 
this  respect  Boussinesq's  theory  is  the  most  satisfactory,  as  it  not 
only,  in  common  with  the  other  theories,  shows  that  the  flow  is 
proportional  to  H*,  but  also  determines  the  value  of  the 
constant  C. 

84.  Solving  for  Q,  by  approximation,  when  the  veloeitj 
of  approach  is  nnknown. 

A  simple  method  of  determining  the  discharge  over  a  wdr 
when  the  velocity  of  approach  is  unknown,  is,  by  approximation, 
as  follows. 

Let  A  be  the  cross-sectional  area  of  the  channel. 

First  find  an  approximation  to  Q,  without  correcting  for 
velocity  of  approach,  from  the  formula 

Q  =  mLh  J2gh, 
The  approximate  velocity  of  approach  is,  then. 


and  H  is  approximately 


«=!. 


A  nearer  approximation  to  Q  can  then  be  obtained  by  sub- 
stituting H  for  fe,  and  if  necessary  a  second  value  for  v  can  be 
found  and  a  still  nearer  approximation  to  H. 

In  practical  problems  this  is,  however,  hardly  necessary. 

Example,  A  weir  without  end  contractions  has  a  length  of  16  feet  The  beal 
as  measured  on  the  weir  is  2  feet  and  the  depth  of  the  channel  of  approaeh  belov 
the  sill  of  the  weir  is  10  feet.    Find  the  discharge. 

„»=0-405  +  ^5^=.4099. 
Therefore  C=3«28. 

Approximately,  Q=3-28  2^.16 

=r  148  cubic  feet  per  second. 

The  velocity  v  =  — — -^ = '77  ft.  per  sec. , 

and  1^=. 0147  feet. 

A  second  approximation  to  Q  is,  therefore, 

Q  =  3-28  (20147)^16 
=  150  cubic  feet  per  second. 
A  third  value  for  Q  can  be  obtained,  but  the  approximation  is  snffioieiiily  Mtf 
for  all  practical  purposes. 

In  thin  case  the  error  in  neglecting  the  velocity  of  approaoh  altogether,  ii 
probably  less  than  the  error  involved  in  taking  m  as  0*4099. 


PLOW  OVER  WEIRS  109 

85.  Time  required  to  lower  the  water  in  a  reservoir  a 
given  distance  hy  means  of  a  weir. 

A  reservoir  has  a  weir  of  length  L  feet  made  in  one  of  its  sides, 
and  kavin^  its  siU  H  feet  below  the  original  level  of  the  water  in 
tihe  reservoir. 

It  is  reqnired  to  find  the  time  necessary  for  the  water  to  fall  to 
m  level  H«  feet  above  the  sill  of  the  weir.  It  is  assumed  that  the 
area  of  the  reservoir  is  so  large  that  the  velocity  of  the  water  as 
it  approaches  the  weir  may  be  neglected. 

When  the  snr&ce  of  the  water  is  at  any  height  h  above  the  sill 
the  flow  in  a  time  dt  ia 

Let  A  be  the  area  of  the  water  surface  at  this  level  and  dh  the 
distance  the  surface  falls  in  time  dt. 

Then,  CUfidt  =  Adh, 

The  time  required  for  the  surface  to  fall  (H-Ho)  feet  is, 
therefore, 

^^1  (^  Adh 

The  coefficient  C  may  be  supposed  constant  and  equal  to  3'34. 
If  then  A  is  constant 

^2A/_1 1_\ 

CLWilo      n/H/ 

To  lower  the  level  to  the  sill  of  the  weir,  Ho  must  be  made 
equal  to  0  and  t  is  then  infinite. 

That  is,  on  the  assumptions  made,  the  surface  of  the  water 

never  could  be  reduced  to  the  level  of  the  sill  of  the  weir.    The 

time  taken  is  not  actually  infinite  as  the  water  in  the  reservoir  is 

not  really  at  rest,  but  has  a  small  velocity  in  the  direction  of  the 

^Bveir,  which  causes  the  time  of  emptying  to  be  less  than  that 

^ven  by  the  above  formula.    But  although  the  actual  time  is 

not  infinite,  it  is  nevertheless  very  great. 

9.x 
When  Ho  is  iH,  t  = 


WTien  EU  is  ^H,  t  = 


CLn/H' 
6A 


clVh' 

So  that  it  takes  three  times  as  long  for  the  water  to  fall  from 
\n  to  iVH  as  from  H  to  iH. 


110  HYDRAULICS 

Example  1.  A  reservoir  has  an  area  of  60,000  sq.  yards.  A  weir  10  feet  long 
has  its  sill  2  feet  below  the  surface.  Find  the  time  required  to  redaoe  the  level  of 
the  water  1'  11". 

H,=A'.        H  =  2'. 

Therefore  t=^-^^^^  (Sid -0109), 

o'o4 .  lU 

2.640.000  ^ 

3-34.10         '^ 

=  89,000  sees. 

=  24-7  hours. 

So  that,  neglecting  velocity  of  approach,  there  will  be  only  one  inoh  of 

the  weir  after  24  hours. 


Example  2.    To  find  in  the  last  example  the  dischaige  from  the  reeerfoir  in 
15  hours. 

2.  A 


Therefore  54  000=p^^-^  ( -l=-  -  -^)  . 

C.LV^Ho      V2/ 

From  which  jEQ  =  0'i21, 

Ho=0176  feet. 

The  discharge  is,  therefore, 

(2-0-176)  540,000  cubic  feet 

=  984,960  cubic  feet. 


EXAMPLES. 

(1)  A  vreir  is  100  feet  long  and  the  head  is  9  inches.  Find  the  disdiaigs 
in  c.  ft.  per  minute.     C  =  3'34. 

(2)  The  discharge  through  a  sharp-edged  rectangular  weir  is  500 
gallons  per  minute,  and  the  still  vrater  head  is  2^  inches.  Find  the  effective 
length  of  the  vreir.    m  =  -43. 

(3)  A  vreir  is  15  feet  long  and  the  head  over  the  crest  is  15  incheii 
Find  the  discharge.  If  the  velocity  of  approach  to  this  weir  were  5  feel 
per  second,  what  would  be  the  discharge  ? 

(4)  Deduce  an  expression  for  the  discharge  through  a  right-ao^fid 
triangular  notch.  If  the  head  over  apex  of  notch  is  12  ins.,  find  the 
discharge  in  c.  ft.  per  sec. 

(5)  A  rectangular  weir  is  to  discharge  10,000,000  gallons  per  di^ 
(1  gallon =10  lbs.),  with  a  normal  head  of  15  ins.    Find  the  length  at  tilt ! 
weir.    Choose  a  coefficient,  stating  for  what  kind  of  weir  it  is  applioaUai 
or  take  the  coefficient  C  as  8*38. 

(6)  What  is  the  advantage  in  gauging,  of  using  a  weir  without  end 
contractions? 

(7)  Deduce  Francis'  formula  by  means  of  the  Thomson  principle  d 
similarity. 

Apply  the  formula  to  calculate  the  discharge  over  a  weir  10  feet  widr 
under  a  head  of  1*2  feet,  assuming  one  end  contraction,  and  neglecting  thi 
effect  of  the  velocity  of  approach. 


I 


FLOW    OVER    WEIRS 


111 


W 


(S>  A  ratnfAtl  of  fy  inch  per  hoar  i»  discharged  from  a  catohment  area 
ai  5  aqua^re  mtlea.  Find  tbe  still  water  head  when  this  Tolutue  flowB  ot^^er 
a  weir  mill  free  oTerfftll  30  feet  m  longib^  (x»nstract^  in  m%  bays,  each 
^  iee4  iride,  tulriiyg  0*415  as  Basin's  coefficdent. 

A  dteSrici  of  6500  acxes  (1  acreB43,560  aq.  ft.)  drains  into  a  large 
I  neserroir.  The  ma^iuum  late  at  which  rain  falls  in  the  dii^trict  ia 
jl  ins^  is  24  hoars.  Whan  r&in  talk  after  tbe  reservoir  is  full^  the  water 
x^aii^es  lo  be  discharged  over  a  weir  or  bye-wanh  which  has  its  crest  at 
tbe  ordinary  top- water  level  of  the  reservoir.  Find  tVie  length  of  such  a 
m^mtMT  fof  tlie  abore  reaervoir^  under  the  condition  that  the  water  in  the 
seMSrroir  aball  never  rin^  more  than  18  ins.  above  it^  top- water  level 

Tlie  top  of  tlie  weif  may  be  Bopposed  flat  and  abotit  18  inches  wide 

(10>  Compare  rectangular  and  V  notches  in  regard  to  aocnraey  and 
^amTenieDae  when  there  is  considerable  variation  in  tl)e  flow. 

In  ft  rectangnl&r  notch  50^'  wide  the  still  water  surface  level  is  15''  above 
iba  BilL 

If  the  same  quantity  of  water  Bowed  over  a  right-angled  V  notch,  what 
^vroiiltl  be  the  height  of  the  still  water  surface  above  the  apex  ? 

Lf  the  channels  are  narrow  how  would  you  correct  for  velocity  of 
jkpproAch  tti  each  case?    Lton.  Ua.  1906. 

ll  1  \     Tbe  heaviest  daily  record  of  rainfall  for  a  catchment  area  was 

if!Hi.iid  to  be  4*2^  mlUion  gallonB.     Assuming  two-thirds  of  the  rain  to  reach 

^^iotage  retnervoir  and  to  ftass  over  the  waste  weir,  find  the  length  of 

•mi  erf  the  wa&te  weir*  so  that  the  water  shall  never  riae  more  than  two 

f[0^  above  the  fiOL 

(12)  A  weir  is  300  yards  long.  What  Ib  the  disebarge  when  the  liead 
hk  4  feoi  t    Take  Basdn's  coeMcieut 

m="40SH ,  —  , 

(iSf  Sa|ipoae  tbe  water  approaches  the  weir  m  the  last  question  in  a 
gbattPf*'  f$*  6''  ^e#f>  and  nOO  yarda  wide.  Find  by  approximation  the  dis- 
cbaqfe.  t*kiitg  into  acoouut  tlie  velocity  of  approach. 

(14^  The  area  of  the  water  surface  of  a  re^rvoir  is  20,000  square 
ymt^m-  Ffa>d  Uie  ttme  rei|u]ru(l  for  tbe  surface  to  fall  one  foot,  when  the 
wmier  diadiuiges  over  a  sbarp^edged  weir  5  feet  long  and  the  original  head 
crrrr  tbe  weir  is  2  feet, 

1 15>  Find,  from  the  following  data,  the  horse-power  available  is  a  given 
wsterCaU:- 

Available  height  of  fall  120  feet. 

A  tTH-tangitlar  notch  above  the  fall,  10  feet  long,  is  used  to  measure 
%km  qoantity  of  water,  and  the  mean  head  over  the  notch  la  found  to  be 

tDicbefet  when  tlie  velocity  of  approach  at  the  point  where  ttie  head 
i  m  100  feet  per  miniita     Lon.  Un.  1905. 


CHAPTER  V. 


FLOW  THROUGH  PIPES. 


86.    ResiBtances  to  the  motion  of  a  fluid  in  a  pipe. 

When  a  fluid  is  made  to  flow  through  a  pipe,  certain  resistances 
are  set  up  which  oppose  the  motion,  and  energy  is  consequently 
dissipated.  Energy  is  lost,  by  friction,  due  to  the  relative  motion 
of  the  water  and  the  pipe,  by  sudden  enlargements  or  contractions 
of  the  pipe,  by  sudden  changes  of  direction,  as  at  bends,  and  by 
obstacles,  such  as  valves  which  interfere  with  the  free  flow  of  the 
fluid. 

It  will  be  necessary  to  consider  these  causes  of  the  loss  of 
energy  in  detail. 

L088  of  head.  Before  proceeding  to  do  so,  however,  the  student 
should  be  reminded  that  instead  of  loss  of  energy  it  is  convenient 
to  speak  of  the  loss  of  head. 

It  has  been  shown  on  page  39  that  the  work  that  can  be 
obtained  from  a  pound  of  water,  at  a  height  z  above  datum, 
moving  with  a  velocity  v  feet  per  second,  and  at  a  pressure  head 

*  is  —  +  s-  +  «  foot  pounds. 
w^       w     2g  ^ 

If  now  water  flows  along  a  pipe  and,  due  to  any  cause,  K  foot 
pounds  of  work  are  lost  per  pound,  the  available  head  is  clearly 
diminished  by  an  amount  h. 

In  Fig.  86  water  is  supposed  to  be  flowing  from  a  tank  throngli 
a  pipe  of  uniform  diameter  and  of  considerable  length,  the  end  B 
being  open  to  the  atmosphere. 


Fig.  86.    Lobs  of  head  by  friction  in  a  pipe. 


FLOW  THROUGH  PIPES  113 

Let  *^  l>e  the  head  due  to  the  atmospheric  pressure. 

Then  if  there  were  no  resistances  and  assuming  stream  line 
flow,  Bemonilli's  equation  for  the  point  B  is 

w     2g  w* 

from  which  st  =  Zp-Zb  =  H, 

or  rB=  V2grH. 

The  whole  head  H  above  the  point  B  has  therefore  been 
utilised  to  give  the  kinetic  energy  to  the  water  leaving  the  pipe  at 
B.  Experiment  would  show,  however,  that  the  mean  velocity  of 
the  water  would  have  some  value  v  less  than  Vb,  and  the  kinetic 

energy  would  be  oZ« 

Ahead  /,  =  ___  =  H-2^ 

ha3  therefore  been  lost  in  the  pipe. 

By  carefully  measuring  H,  the  diameter  of  the  pipe  d,  and  the 
discharge  Q  in  a  given  time,  the  loss  of  head  h  can  be  determined. 

For                                  "-7^' 
and  therefore  A  =  H ^*--; 

The  head  h  clearly  includes  all  causes  of  loss  of  head,  which, 
in  this  case,  are  loss  at  the  entrance  of  the  pipe  and  loss  by 
friction. 

87.    Loss  of  head  by  friction. 

Suppose  tubes  1,  2,  3  are  fitted  into  the  pipe  AB,  Fig.  86,  at 
equal  distance  apart,  and  with  their  lower  ends  flush  with  the  inside 
of  the  pipe,  and  the  direction  of  the  tube  perpendicular  to  the 
direction  of  flow.  If  flow  is  prevented  by  closing  the  end  B  of  the 
pipe,  the  water  would  rise  in  all  the  tubes  to  the  level  of  the  water 
in  the  reservoir. 

Further,  if  the  flow  is  regulated  at  B  by  a  valve  so  that  the 
mean  velocity  through  the  pipe  is  v  feet  per  second,  a  permanent- 
regime  being  established,  and  the  pipe  is  entirely  full,  the  mean 
velocity  at  all  points  along  the  pipe  will  be  the  same ;  and  there- 
fore, if  between  the  tank  and  the  point  B  there  were  no  resistances 
offered  to  the  motion*  and  it  be  assumed  that  all  the  particles 

L.H.  8 


114  HYDRAULICS 

have  a  velocity  equal  to  the  mean  velocity,  the  water  would  again 
rise  in  all  the  tubes  to  the  same  height,  but  now  lower  than  the 

surface  of  the  water  in  the  tank  by  an  amount  equal  to  sr- 

It  is  found  by  experiment,  however,  that  the  water  does  nofc  i 
rise  to  the  same  height  in  the  three  tubes,  but  is  lower  in  2  thaa  j 
in  1  and  in  3  than  in  2  as  shown  in  the  figure.  As  the  fluid  movai  \ 
along  the  pipe  there  is,  therefore,  a  loss  of  head. 

The  difference  of  level  fh  of  the  water  in  the  tubes  1  and  2  iB 
called  the  head  lost  by  friction  in  the  length  of  pipe  12.    In  any    . 
length  I  of  the  pipe  the  loss  of  head  is  h. 

This  head  is  not  wholly  lost  simply  by  the  relative  movement 
of  the  water  and  the  surface  of  the  pipe,  as  if  the  water  wotb 
a  solid  body  sliding  along  the  pipe,  but  is  really  the  sum  of  tiia 
losses  of  energy,  by  friction  along  the  surface,  and  due  to  relatifft 
motions  in  the  mass  of  water. 

It  vrill  be  shown  later  that,  as  the  water  flows  along  the  pipe^  ; 
there  is  relative  motion  between  consecutive  filaments  in  the  jripe^ 
and  that,  when  the  velocity  is  above  a  certain  amount,  the  waler 
has  a  sinuous  motion  along  the  pipe.  Some  portion  of  this  headk 
is  therefore  lost,  by  the  relative  motion  of  the  filaments  of  water, 
and  by  the  eddy  motions  which  take  place  in  the  mass  of  fliB 
water. 

When  the  pipe  is  uniform  the  loss  of  head  is  proportional 
to  the  length  of  the  pipe,  and  the  line  CB,  drawn  through  the  tops 
of  the  columns  of  water  in  the  tubes  and  called  the  hydraulic 
gradient,  is  a  straight  line. 

It  should  be  noted  that  along  CB  the  pressure  is  equal  to  that 
of  the  atmosphere. 

88.    Head  lost  at  the  entrance  to  the  pipe. 

For  a  point  B  just  inside  the  pipe,  Bernouilli's  equation  is 

—  +  j^  +  head  lost  at  entrance  to  the  pipe  =  Aa  +  ~ , 

2?  being  the  absolute  pressure  head  at  E. 

The  head  lost  at  entrance  has  been  shown  on  page  70  to  be 

about  -P5 — ,  and  therefore, 
2g 

w      w  2g 

That  is,  the  point  C  on  the  hydraulic  gradient  vertically  above 

l'5v' 
E,  is  -^ —  below  the  surface  FD. 
'       2g 


FLOW  THKOUQH  PIPES 


115 


If  the  pipe  is  bell-moathed,  there  will  be  no  head  lost  at  entrance, 
id  the  point  C  is  a  distance  equal  to  ^  below  the  surface. 

89.    Hydraulic  gradient  and  virtual  slope. 

The  line  CB  joining  the  tops  of  the  columns  of  water  in  the 
nhe,  is  called  the  hydraulic  gradient,  and  the  angle  i  which  it 
Doakes  with  the  horizontal  is  called  the  slope  of  the  hydraulic 
gradient,  or  the  virtual  slope.    The  angle  i  is  generally  small,  and 

mi  may  be  taken  therefore  equal  to  i,  so  that  j  =  ^« 

In  what  follows  the  virtual  slope  -j  is  denoted  by  i. 

More  generally  the  hydraulic  gradient  may  be  defined  as  the 
line,  the  vertical  distance  between  which  and  the  centre  of  the 
pipe  gives  the  pressure  head  at  that  point  in  the  pipe.  This  line 
will  only  be  a  straight  line  between  any  two  points  of  the  pipe, 
wben  the  head  is  lost  uniformly  along  the  pipe. 

If  the  pressure  head  is  measured  above  the  atmospheric 
fffessure,  the  hydraulic  gradient  in  Fig.  87  is  AD,  but  if  above 
lero,  AiDi  is  the  hydraulic  gradient,  the  vertical  distance  between 


LD  and  AiDi  being  equal  to 
ressuru  per  sq.  inch. 


w 


-,  Pa  being  the  atmospheric 


Fig.  87.     Pipe  rising  above  the  Hydraulic  Gradient. 

If  the  pipe  rises  above  the  hydraulic  gradient  AD,  as  in  Fig.  87, 
c*  pressure  in  the  pipe  at  C  will  be  less  than  that  of  the  atmosphere 
r  a  head  equal  to  CE.  If  the  pipe  is  perfectly  air-tight  it  will 
:t  a>  a  siphon  and  the  discharge  for  a  given  length  of  pipe  will 
jt  }je  altered.     But  if  a  tube  open  to  the  atmosphere  be  fitted  at 

8—2 


116 


HYDRAULICS 


the  highest  point,  the  pressure  at  C  is  equal  to  the  atmospheric 
pressure,  and  the  hydraulic  gradient  will  be  now  AC,  and  the  flow 
will  be  diminished,  as  the  available  head  to  overcome  the  resSst* 
ances  between  B  and  C,  and  to  give  velocity  to  the  water,  will  only 
be  CF,  and  the  part  of  the  pipe  CD  will  not  be  kept  full. 

In  practice,  although  the  pipe  is  closed  to  the  atmosphere,  yeA 
air  will  tend  to  accumulate  and  spoil  the  siphon  action. 

As  long  as  the  point  C  is  below  the  level  of  the  water  in  tihs 
reservoir,  water  will  flow  along  the  pipe,  but  any  accumulation  ol 
air  at  C  tends  to  diminish  the  flow.  In  an  ordinary  pipe  line  it  it 
desirable,  therefore,  that  no  point  in  the  pipe  should  be  allowed  to 
rise  above  the  hydraulic  gradient. 

90.  Determination  of  the  loss  of  head  due  to  frietton. 
Reynolds'  apparatus. 

Fig.  88  shows  the  apparatus  as  used  by  Professor  Reynolds*  for 
determining  the  loss  of  head  by  friction  in  a  pipe. 


Fig.  88.    BeyuoldB'  apparatus  for  determining  loss  of  head  by  friction  in  a  pipe^ 

A  horizontal  pipe  AB,  16  feet  long,  was  connected  to  the  water 
main,  a  suitable  regulating  device  being  inserted  between  tlie 
main  and  the  pipe. 

At  two  points  5  feet  apart  near  the  end  B,  and  thus  at  a  distance 
sufiiciently  removed  from  the  point  at  which  the  water  entered 
the  pipe,  that  any  initial  eddy  motions  might  be  destroyed  and  i 
steady  regime  established,  two  holes  of  about  1  mm.  diameter  wero 
pierced  into  the  pipe  for  the  purpose  of  gauging  the  pressoie,  it 
these  points  of  the  pipe. 

Short  tubes  were  soldered    to  the  pipe,  so  that  the  hofc 
communicated  with  these  tubes,  and  these  were  connected  1 

*  PhiU  TratiB.  1883,  or  Vol.  n.  Scienti/ic  Papen^  Bejmolds. 


rnSianil 


FLOW  THROUGH   PIPES 


111 


ibber  pipes  ki  the  limbs  of  a  siphon  gauge  G^  made  of  glass] 
and  which  contained  mercuiy  or  bisulphide  of  carbon.     Sealetl 
wi?re  fixed  behiiKl  tbt*  tubes  ^y  that  the  height  of  the  columns 
in  each  limb  of  the  gauge  could  be  n?ad. 

For  i'ery  small  differences  of  level  a  cathetometer  was  used*, 
'""'  <  made  to  flow  through  the  pipe,  the  diffeiience  in 

'  columns  in  the  two  limbs  of  the  siphon  measured 
presfure  at  the  two  points  A  and  B  of  tlie  pipeJ 
ouM  iiiM-  fiM.i'Mied  the  lo&s  of  head  due  to  friction.  ■ 

If  *  is  the  specific  gravity  of  the  liquid,  and  H  the  difference 
in  ht'tght  of  the  columns,  the  loss  of  head  due  to  friction  in  feet  oil 
waitir  is  A^H  (^-1).  I 

The  quantity  of  water  fl(»wing  in  a  time  t  was  obtained  b|il 
actual  measttTemeEit  in  a  graduated  flask . 

Calling  r  th©  mean  velocity  in  the  pipe  in  feet  per  eecondi  Q 
the  dischafg^  in  cubic  feet  per  secondi  and  d  the  diameter  of  thai 
{ripe  m  f00l,  J 

The  lo»  of  head  at  different  velocities  was  carefully  measured^ 
tlia  law  connecting  head  lost  in  a  given  length  of  pipe,  with 
T«lodty»  determined* 

Hie  resolU  obtained  by  Reynolds,  and  others^  using  tins 
of  expt^rimeniing,  will  be  refen-ed  to  later. 

Bqmation    of   flow  in  a  pipe  of  uniform   diameter 
detenaination  of  the  head  lost  due  to  friction. 
Lei  ci  he  the  length  of  a  ginall  element  of  pipe  of  uniform 
r.  Fig.  m 

A 


Fig.  81). 

I,,  rea  of  the  transverse  section  be  uj,  P  the  length  of 

lu        .     jutact  of  the  water  and  the  surface  on  this  section,  or 
wetu-tl  perimeter,  n  the  inclination  of  the  pipe,  p  the  pressm-e 

'  unit  area  on  AB,  iind  p-dp  the  pressure  on  CD* 
•  p.  268,  Vol.  I,  ScieniifU  Papif*^  Bejnoldf. 


118  HYDRAULICS 

Let  V  be  the  mean  velocity  of  the  fluid,  Q  the  flow  in  cubic 
feet  per  second,  and  w  the  weight  of  one  cubic  foot  of  the  fluid. 
The  work  done  by  gravity  as  the  fluid  flows  from  AB  to  CD 

=  Qw  .dz-io,v  .w  .dz. 
The  work  done  on  ABCD  by  the  pressure  acting  upon  the  area 
AB 

=  p.io.v  it.  lbs.  per  sec. 

The  work  done  by  the  pressure  acting  upon  CD  against  the 

flow 

=  (p  —  dp)  .  ci> .  17  f t.  lbs.  per  sec. 

The  frictional  force  opposing  the  motion  is  proportional  to  the 
area  of  the  wetted  surface  and  is  equal  to  F .  P .  oZ,  where  P  is  some 
coefficient  which  must  be  determined  by  experiment  and  is  the 
frictional  force  per  unit  area.    The  work  done  by  friction  per  aecX 
is,  therefore,  F .  P .  5Z .  v. 

The  velocity  being  constant,  the  velocity  head  is  the  same  ttt 
both  sections,  and  therefore,  applying  the  principle  of  the  con- 
servation of  energy, 

p.tt},v  +  <o,v.w.dz=  (p- dp)  CO . i;  +  F . P . 3Z . «. 

Therefore  w .  w .  92?  =  -82) .  w  +  F .  P .  3Z, 

,         dp     F.P.dZ 
or  dz  =  — -  +  . 

Integrating  this  equation  between  the  limits  of  z  and  Zi,  p  and 
Pi  being  the  corresponding  pressures,  and  I  the  length  of  the  pipe^ 


z- 

-Zi 

w 

w 

F.PZ 

W      ci» 

w 

+  z 

=£1 

w 

+  Zl  + 

FP  I 

W      CD* 

Therefore, 

FPZ 
The  quantity  —  is  equal  to  A  of  equation  (1),  page  52,  and  it 

the  loss  of  head  due  to  friction.  The  head  lost  by  friction  it 
therefore  proportional  to  the  area  of  the  wetted  surface  of  the  pipe 
PZ,  and  inversely  proportional  to  the  cross  sectional  area  of  the 
pipe  and  to  the  density  of  the  fluid. 

92.    Hydraulic  mean  depth. 

The  quantity  p  is  called  the  hydraulic  radius,  or  the  hydranlio 

mean  depth. 

If  then  this  quantity  is  denoted  by  m,  the  head  h  lost  hf 
friction,  is 

w  .m 


FLOW  THBOtJOH   PIPES 


119 


Thio  quantity  F,  which  hns  been  oalled  above  the  friction  per 
imit  mr^A,  is  found  by  experiments  to  vaiy  with  the  density^ 
^imoom^^  and  velocity  c»f  the  flaid,  and  v%nth  the  diameter  and 
lOO^hneas  cif  the  internal  mirface  of  the  pipe. 

In  Hydraulics,  the  fluid  considered  is  water,  and  any  variationB 
in  dtrnnity  or  viscosity,  doe  to  changes  of  temperature^  are  generally 
nie-Gfliiribl^,  F^  therefore,  may  be  taken  as  proportional  to  the 
«^  *  <r  to  the  weight  w  per  cubic  foot,  to  the  roughness  of  the 

;  ,  .  -.  i  as  some  function,  /{v)  of  the  mean  velocity,  and  f{d)  of 
rh**  «liaaEM»ter  of  the  pipe, 


Then, 


j^^pfM/m 


m 


in  which  expresion  ^  may  be  called  the  coefficient  of  friction. 

li  will  be  seen  later^  that  the  mean  velocity  v  is  different  from 
the  Ttdathe  Telocity  u  of  the  water  and  the  surface  of  the  pipe, 
arMi  it  probably  would  be  better  to  express  F  as  a  function  of  «, ' 
1  itftelf  probably  varies?  with  the  roughness  of  the  pipe  and] 

^\__    -:uer  circumstances,  and  cannot  directly  be  deteruiined,  it, 
mnplifictd  matters  to  express  F,  and  thus  A,  as  a  function  of  i?. 

93.    Empirical  formulae  for  loss  of  head  due  to  friction, 

Tha  difficulty  of  con^ctly  determining  the  exact  value  of 
/(t)  f{d)^  has  led  to  the  use  of  empirical  formulae,  which  have 
poporcd  of  great  practical  service,  to  express  the  head  h  m  terms  of 
the  trelodty  and  the  dimenMons  of  the  pipe. 

Tbe  fiiniplest  formula  assumes  that  the  friction  simply  varies  as 

tlif*  nqaare  of  the  velocity,  and  is  independent  of  the  diameter  of 

ape,  or  f{v)  fid)  =  at^. 


Then, 


I 


or  writiiig  gi  for  a, 


m 


i-^*^ 

'^"C^ 


•(1), 


.(2), 


frcmi  which  is  deduced  the  well-known  Chezy  formula, 

v  =  C  ^/m.j^ 

or  1?  -  C  ^Jmt. 

Another  form  in  which  formnla  (1)  ia  often  found  is 


120 


HYDRA  ITLICS 


or  since  wi  =  7  for  a  circular  pipe  full  of  water, 

t     4,f.vH 
^~2g.d    


.(3), 


in  which  for  a  of  (1)  is  substituted 


L 
29  • 


The  quantity  2g  was  introduced  by  Weisbach  so  that  h  is 
expressed  in  terms  of  the  velocity  head. 

Adopting  either  of  these  forms,  the  values  of  the  coefficients  C 
and  /  are  determined  from  experiments  on  various  classes  of  pipes. 

It  should  be  noticed  that  C  =  a/  -? . 

Values  of  these  constants  are  shown  in  Tables  XTT  to  XTV  for 
different  kinds  and  diameters  of  pipes  and  different  velocities. 

TABLE  Xn. 

Values  of  C  in  the  formula  v  =  G  Jrni  for  new  and  old  cast-iron 
pipes. 


New  oast- 

iron  pipes 

Old  oaafc. 

iron  pipes 

Velocities  in  ft.  per  second 

1 

3 

6 

10 

1 

3 

6 

10 

Diameter  of  pipe 

8" 

95 

98 

100 

102 

68 

68 

71 

78 

6" 

96 

101 

104 

106 

69 

74 

77 

79 

9" 

98 

105 

109 

112 

78 

78 

80 

84 

12" 

100 

108 

112 

117 

77 

82 

85 

88 

16" 

102 

110 

117 

122 

81 

86 

89 

91 

18" 

105 

112 

119 

125 

86 

91 

94 

97 

24" 

111 

120 

126 

131 

92 

98 

101 

104 

30" 

118 

126 

131 

186 

98 

108 

106 

109 

86" 

124 

181 

136 

140 

108 

108 

111 

114 

42" 

180 

186 

140 

144 

106 

111 

114 

117 

48" 

185 

141 

145 

148- 

106 

112 

115 

118 

60" 

142 

147 

150 

162 

For  method  of  determining  the  values  of  C  given  in  the  tables, 
see  page  102. 

On  reference  to  these  tables,  it  will  be  seen,  that  C  and  /  are 
by  no  means  constant,  but  vary  very  considerably  for  different 
kinds  of  pipes,  and  for  different  values  of  the  velocity  in  any 
given  pipe. 


FLOW  THROUGH  PIPES 


121 


The  fact  that  C  varies  with  the  velocity,  and  the  diameter  of 
the  pipe,  suggests  that  the  coefficient  C  is  itself  some  function  of 
the  velocity  of  flow,  and  of  the  diameter  of  the  pipe,  and  that 
/i/(r)  f{d)  does  not,  therefore,  equal  at;'. 

TABLE  Xni. 


Values  of  /  in  the  formula 


,  _4/t;»J 


1 

New  east-iron  pipes 

Old  cast-iron  pipes 

VeloeitiMm 
ft.p«rieeoDd 

1 

3 

6 

10 

1 

8 

6 

10 

DiAiiLofpipe 

8" 

•0071 

-0067 

•0064 

•0062 

•0152 

•0189 

•0128 

•0122 

6" 

•007 

•0068 

•006 

•0057 

•0135 

•0117 

•0108 

•0103 

9^ 

•0067 

•0058 

•0055 

•0051 

•0122 

•0105 

•010 

•0092 

IT 

-0064 

•0056 

•0051 

•0048 

•0108 

•0096 

•0089 

•0084 

15" 

-0062 

•0058 

•0048 

•0048 

•0099 

•0087 

•0081 

•0078 

1            18" 

•0058 

•0051 

•0045 

•0041 

•0087 

•0078 

•0073 

•0069 

1           24" 

•0058 

•0045 

•0040 

•0037 

•0076 

•0067 

•0063 

•0060 

1         W 

•0046 

•0040 

•0087 

•0035 

•0067 

•0061 

•0067 

•0066 

86" 

•0042 

•0087 

•0085 

•0038 

•0061 

•0056 

•0062 

•0060 

1           42" 

-0068 

•0085 

•0038 

•0081 

•0058 

•0052 

•005 

•0048 

1           48" 

-0086 

•0082 

•0081 

•0029 

•0067 

•0061 

•0049 

•0046 

60" 

-0032 

•0030 

•0029 

•0028 

TABLE  XIV. 
Valaes  of  C  in  the  formula  v-C  -Jmi  for  steel  riveted  pipes. 


Velocities  in  ft  per  second 

1 

8 

5 

10 

Diameter  of  pipe 

8" 

81 

86 

89 

92 

11" 

92 

102 

107 

115 

111" 
15^ 

93 

99 

102 

105 

109 

112 

114 

117 

88" 

113 

113 

113 

118 

42" 

102 

106 

108 

111 

48" 

105 

105 

105 

105 

72"* 

110 

110 

111 

111 

72" 

93 

101 

105 

110 

108" 

114 

109 

106 

104 

See  pages  124  and  137. 


122  HYDRAULICS 

94.    Formula  of  Daroy. 

In  1857  Darcy*  published  an  account  of  a  series  of  experiments 
on  flow  of  water  in  pipes,  previous  to  the  publication  of  which,  it 
had  been  assumed  by  most  writers  that  the  friction  and  consequently 
the  constant  C  was  independent  of  the  nature  of  the  wetted  surface 
of  the  pipe  (see  page  232).    He,  however,  showed  by  experimenis 
upon  pipes    of    various    diameters    and   of    different  materiah^ 
including  wrought  iron,  sheet  iron  covered  with  bitumen,  lead, 
glass,  and  new  and  old  cast-iron,  that  the  condition  of  the  internal 
surface  was  Of  considerable  importance  and  that  the  resistance  was 
by  no  means  independent  of  it. 

He  also  investigated  the  influence  of  the  diameter  of  the  pipe 
upon  the  resistance.  The  results  of  his  experiments  he  expressed 
by  assuming  the  coefficient  a  in  the  formula 

T     al     2 
m 

Q 

was  of  the  form  a  =  o  +  - 

r ' 

r  being  the  radius  of  the  pipe. 

For  new  cast-iron,  and  wrought-iron  pipes  of  the  same 
roughness,  Darcy's  values  of  a  and  P  when  transferred  to  English 
units  are, 

a  =  0-000077, 
/J  =  0-000003235. 
For  old  cast-iron  pipes  Darcy  proposed  to  double  these  values. 
Substituting  the  diameter  d  for  the  radius  r,  and  doubling  ft  for 
new  pipes, 

;.=(o-oooo77.«:«o^)^ 

=  0-00000647  (1-2^1)  ^^ 
or  "^^^^V  i2dTl^^ (*) 

^^^s/mri-^' (5)- 

Substituting  for  m  its  value  2  >  ^^^  multiplying  and  dividing 

by  29, 

\-'M'^mW^  w- 

For  old  cast-iron  pipes, 

;.=o-ooooi294(i-^)^ 

\     a     /  m 


-«'"('^is)S-s o- 


2g 

*  Reeherehea  ExpirimentaUt. 


^^^^1  FLOW  THROUGH   PIPES  123 

To!^  v  =  27S^^^^^i   (8) 

I  -^^\^l^l^^ t»>' 

■  As  the  atudant  cannot  possibly  retain,  witliout  unnecessary 
Biboiff,  Tallies  of  /  sod  C  for  different  diameters  it  is  convenient 
■to  Temember  the  rim  pie  fonnSj 

I  for  Dew  pipes,  and 

■  for  old  pipes. 

H     According  to  Darcy,  therefore,  the  coefficient  C  in  the  Chezy 

■  fontiala  varies  only  with  the  diameter  and  roughness  of  the  pipe. 
I     The  %*mlae8  of  C  afi  calcolated  from  his  experimental  results,  for 

■  t^m?  of  the  pipes,  were  practically  constant  for  all  velocitieBj  and 
■Jiotubty  for  those  pipes  which  had  a  comparatively  rough  internal 
^■picef  hut  for  smooth  pipes,  the  value  of  C  varied  from  10  to 
^Hper  ct*nL  for  the  siame  pipe  as  the  velocity  changed.  The 
Experiments  of  other  workers  show  the  same  results, 

V  aption  that  ^f{v)f(d)  =  av^  in  which  a  is  made  to 

Tii^  :  h  the  diameter  and  roughness,  or  in  other  words,  the 

aoiunptian  that  h  is  proportional  to  ij"  is  therefore  not  in  general 
stilted  by  experiments. 

As  stated  above,  the  formulae  given  must  be  taken  as 

^npiriea],   and  though  by  the  introduction  of  suitable 

they  can  be  made  to  agree  with  any  particular  experi- 

even  set  of  experiroents,  yet  none  of  them  probably 

rprt!!Si^s  truly  the  laws  of  fluid  friction. 

The  formula  of  Chezy  by  its  simplicity  has  found  favour,  and 
I  is  likely,  that  for  aome  time  to  come^  it  i^411  continue  to  be  used, 
in  the  form  t?  =  C  vmi,  or  in  its  modified  f onu 

^      2gd  ' 
In  making  calculations,  values  of  C  orf,  which  most  nearly  suit 
ly  given  case,  can  be  taken  from  the  tables, 

00.    Yariatioii  of  C  in  the  formula  v  =  C  -/mi  with  aervioe. 
It  should  be  clearly  borne  in  mind,  however,   that  the   dis- 

diaf^ging  capacity  of  a  pipe  may  be  considerably  diminished  after 

m  few  yeanf*  service. 

Dwtjy'B  re«u]t«  show  that  the  loss  of  head  in  an  old  pipe  may 
doable  that  in  a  new  one,  or  since  the  velocity  v  is  taken  as 


^hm  doi 


124 


HYDRAULICS 


proportional  to  the  square  root  of  A,  the  discharge  of  the  old  pif 
for  the  Bame  head  will  be  -^  time^  that  of  the  new  pipe,  or  aboafj 

30  per  cent.  less. 

An  experiment  by  Sherman*  on  a  3d-inch  cast-iron  main  sh( 
that  after  one  year's  gervice  the  discharge  was  dmiinishei 
23  iiev  cent.,  but  a  second  year's  service  did  not  make  any  furthi?r 
altc^ration. 

Experiments  by  Kuichlingt  on  a  36-inch  cast-iron  main  showed 
that  the  discharge  during  four  years  diminished  36  per  cent*|  while 
experiments  by  Fitzgerald  t  on  a  cast-iron  main,  coated  with  tar, 
which  had  been  in  use  for  16  years,  showed  that  cleaning  inci^eased 
the  discharge  by  nearly  40  per  cent,  Fitzgerald  also  found  tlat 
the  discharge  of  the  Sudbury  aqueduct  diminished  10  per  ceni.  in 
one  year  due  to  accumulation  of  slime. 

The  experiments  of  Marx,  Wing,  and  Hoskins|  on  a  72-inc)i  stt'el 
main,  when  new,  and  after  two  years'  service,  showed  that  ther# 
had  been  a  change  in  the  condition  of  the  internal  surface  of  tfiB 
pipe,  and  that  the  disscharge  had  diminished  by  10  per  cent,  at  lo^ 
velocities  and  about  5  per  cent,  at  the  higher  velocities. 

If,  therefore,  in  calculations  for  pipes,  values  of  C  or /are  used 
for  new  pipes,  it  will  in  most  cases  be  advisable  to  make  the  pipe 
of  such  a  size  that  it  will  discharge  under  the  given  bead  at  least 
from  10  to  30  per  cent,  more  than  the  calculated  value. 

97*    Ganguiliet  and  Kiitter*s  formula. 

Granguillet  and  Katter  endeavoured  to  determine  a  form  foi 
the  coefficient  C  in  the  Chezy  formula  v  =  G  ^mi,  applicable 
to  all  forms  of  channels,  and  in  which  C  is  made  a  function  of  the 
virtual  slope  i,  and  also  of  the  diameter  of  the  pipe. 

They  gave  C  the  value, 

1-811     0-00281 


41-6 


C  =  ^ 


1,(41-0^2:29^)  « 

This  formula  is  very  cumbersome  to  use^  and  the  value  of  th 

coefficient  of  roughness  n  for  different  cases  is  uncertain.     Tabk 
have  however  been  prepai*ed  which  considerably  facilitate  the 
of  the  formula. 


*  Tram.  Am^SX.E.  Vol  xjurv,  p,  85. 
t  Tram.  Am.S.CE,  Vol.  eliv,  p.  56. 
t  Tram.  Am.S.C.E.  ToL  xw¥,  p*  87. 
g  Sea  Table  No.  KIV, 


FLOW  THROirOH   PIPES  Iti 

Folties  of  H  in  Ganguilht  ami  Kutler*s  formula. 
Wood  pip€*s  ^  '01,  may  be  as  high  as  "015. 

Cast-iron  and  sttnA  pipes  =  *011,  „  „  02* 

Glaj^  eartheuwar©         =  *013* 

&8*    Keynolds'  experiments  and  the  logarithmic  formtila. 

The  formulae  for  \om  of  head  due  to  friction  previously  givea 
iiave  all  teen  founded  upon  a  probable  law  of  %*ariation  of  h 
"Wiik  V,  bat  oo  rational  basis  for  the  assumptions  has  been  adduced. 

It  has  been  ^tjited  in  Bection  93,  that  on  the  assumption  that  h 
vam  with  17^,  the  ci:^etticient  C  in  the  formula 


-Cy/ 


h 


ilitolf  a  ftinction  of  the  \'elocity. 

Tilt?  experiments  and  deductions  of  Reynolds,  and  of  lati^r 
workers,  throw  considerable  light  upon  this  subject,  and  show  that 
i  i»  proportional  to  t'**,  where  n  is  an  index  which  for  very  small 
^locities*— as  previously  shown  by  Poiseuille  by  experiments  on 
CBfiillary  tubes — is  equal  to  unity,  and  for  higher  velocities  may 
We  jk  variable  vahte,  which  in  many  cases  approximates  to  2. 

}  '^  experLments  marked  a  decifled  advaneej  in  showing 

^1"  -  -ily  that  the  roughness  of  the  wetted  surface  has  an 
fiffisct  upon  the  loss  due  to  friction^  so  Heynolds'  work  marked 
her  step  in  showing  that  the  index  w  depends  upon  the  state 
intt'mal  surface,  being  generally  greater  the  rougher  the 


I 


student  will  be  better  able  to  follow  Reynolds,  by  a  brief  | 
tion  of  one  of  his  experiments, 
n  Table  XV  are  shown  the  results  of  an  experiment  made  | 
f  Reynolds  with  apparatus  as  illustrated  in  Fig.  88. 
In  calumiis   1   and  5  are  shown  the  experimental  values 

=  J,  and  «  peKpjectively, 

The  curves.  Fig.  90,  were  obtained  by  plotting  v  as  abscissae 
kd  t  88  ordinates. 
For  velocities  up  to  1*347  feet  per  second,  the  points  lie  very  close 
A  straight  line  and  i  m  simply  proijortional  to  the  velocity,  or 

i  =  hv  „, (IIX 

bein^  a  coefficient  for  this  particular  pipe. 
Above  2  feet  per  second j  the  points  he  very  near  to  a  continuous^ 
I  the  equation  to  which  is 

i-itu** ,.........(12). 

•  Phil  Tmns,  1863, 


126 


HYDRAULICS 


Taking  logarithms, 


log  i  =  log  h  +  nlog  t;. 


The  curve,  Fig.  90  a,  was  determined  by  plotting  log  i  as 
ordinate  and  logv  as  abscissae.  Reynolds  calls  the  lines  of  this 
figure  .the  logarithmic  homologues. 

Calling  logi,  y,  and  log  v,  a?,  the  equation  has  the  form 

y  =  k-i-  7WJ, 

which  is  an  equation  to  a  straight  line,  the  inclination  of  which  to 
the  axis  of  x  is 

^  =  tan""^7i, 

or  n  =  tan  0, 

Further,  when  »  =  0,  y  =  fc,  so  that  the  value  of  Jc  can  reardily  be 
found  as  the  ordinate  of  the  line  when  x  or  logv  =  0,  that  is, 
when  v  =  1. 

Up  to  a  velocity  of  1*37  feet  per  second,  the  points  lie  near  to 
a  line  inclined  at  45  degrees  to  the  axis  of  v,  and  therefore,  n  is 
unity,  or  as  stated  above,  i  =  kv. 

The  ordinate  when  v  is  6qual  to  unity  is  0*038,  so  that  for  the 
first  part  of  the  curve  k  =  '038,  and  i  =  '0381?. 


FLOW  THROUGH  PIPES 


127 


Above  the  velocity  of  2  feet  i)er  second  the  points  lie  about 
ft  second  strai^Ht  line,  the  inclination  of  which  to  the  axis  of  t;  is 

»  =  tan-U-70. 
Therefore  log  i  =  1*70  log  v  +  k. 

The  ordiixate  -when  v  equals  1  is  0*042,  so  that 

fc  =  0-042, 
and  t  =  0-042t;i^. 


Fig.  90  a.    Logarithmic  plottings  of  t  and  v  to  determine  the  index  n  in 
the  formula  for  pipes,  t  =  A;i;*^. 

In  the  table  are  given  values  of  i  as  determined  experimentally 
and  as  calculated  from  the  equation  i  =  k,  v^. 

The  quantities  in  the  two  columns  agree  within  3  per  cent. 


128 


HTDRAUUCS 


Lead  Pipe. 


TABLE  XV. 
Experiment  on  Resistance  in  Pipes. 
Diameter  0*242".    Water  from  Mancliester  Main. 


Slope 

"1 

k 

n 

Velocity  ft  per 
second 

Experimental  value 

Calculated  from 

•0086 

•0092 

•038 

-289 

•0172      • 

•0172 

•038 

•451 

•0258 

•0261 

•038 

•690 

•0346 

•0347 

•038 

•914 

•0480 

•0421 

•038 

1109 

•0516 

•0512 

-038 

1-849 

•0602 

•  •> 

1482 

•0682 

... 

1573 

•0861 

... 

1-671 

•1033 

... 

1775 

•1206 

... 

1-857 

•1378 

•1362 

•042 

1-70 

1-987 

•1714 

•1610 

•042 

1-70 

2-208 

•3014 

•2944 

•042 

170 

8141 

•4306 

•4207 

•042 

1^70 

8-98 

•8185 

•8017 

•042 

1-70 

5-66 

1021 

1033 

•042 

170 

6-57 

1-438 

1-476 

•042 

170 

811 

2-455 

2^404 

•042 

170 

10-79 

8^274 

3-206 

•042 

1^70 

12-79 

3^878 

3-899 

•042 

1-70 

14-29 

Note.    To  make  the  columns  shorter,  only  part  of  Beynolds'  results  are  ghcs. 

99.    Critical  velocity. 

^t  appears,  from  Reynolds'  experiment,  that  up  to  a  certun 

-  which  is  called  the  Critical  Velocity,  the  loss  of  head  li 

which  is  a)nal  to  v,  but  above  this  velocity  there  is  a  definite  change 

the  axis  ow  connecting  i  and  t;. 

'cperiments  upon  pipes  of  different  diameters  and  the 

Qj,  variable  temperatures,  Reynolds  found  that  the  critical 

hich  was  taken  as  the  point  of  intersection  of  the  two 
Further,  ^^^^ag 

found  as  the  _ '0388? 

when  t?  =  1.  ^* "     D      ' 

Up  to  a  velooeing 

a  line  inclined  at     p_. 1 ,^qx 

unity,  or  as  stated  a        1  +0-0336T  +  •0000221P  ^^^^' 

The  ordinate  wheature  in  degrees  centigrade  and  D  the  diameter 
first  part  of  the  curve. 


^ 


¥upw  through  pipes 


129 


Critical  velocity  by  the  method  of  colour  bands. 

exisftnice   of   the   rrirical   velocity  has   been   beautiftill}' 
by   Reynolds,  by  tho   metliod  of  colour  baTid^^  and  his 
-  r^  slIs*j  explain  why  there  is  a  sudden  change  in  the  law 
^  I  and  r. 

Water  was  drawn  through  tubes  (Figs.  91  and  92),  ont  of 
^rigp  gla?^  tank  in  which  the  tnbcg  were  immersedj  and  in 
th  ihi*  water  had  been  allowe<l  to  come  to  rest,  arrangements 
made  %B  shown  in  tlie  figure  so  that  a  streak  or  streaks  of 
liy  coloured  water  entered  the  tubes  with  the  clear  water," 


^ 


3 


Fig.  91. 


^ 


Fig.  92, 

'  Tfsults  were  a^  follows  :— 

"  (I)     When  the  velocities  were  sufficiently  low,  the  streak 

<iliiur  extended  in  a  beautiful  straight  line  through  the  tube*' 

**{2>     A«  the  velocity  was   increa^^d  by  small  stages,  at 
r  pc»int  in  the  rube,  always  at  a  considerable  distance  from  the 
jimjjrt-«ha|)ed  intake,  the  colour  band  would  all  at  once  mix  up 
the  Huri'MnTnlitig  water,  and  fill  the  rest  of  the  tube  with 

»if  coltHired  water"  (Fig.  92), 
ij*  Httdden  change  takes  place  at  the  cntical  velocity. 

such  a  change  takes  place  is  also  shown  by  the  appai-atus 
Bjstrated  in  Fig.  ^S'f  when  the  critical  velocity  is  reached  there  is 
liiknt  disturbance  of  the  meruury  in  the  U  tube. 

iJij  therefore,  a  definite  and  sudden  change  in  the  con- 

of  fl€>w.     For  vehjcities  below  the  critical  velocity,  the  flow 

to  the  tubes,  or  is  **  Stream  Line"  flow,  but  after  the 

Jocity  ha.s  been  passed,  the  motion  parallel  to  the  tube  is 

Qinpaniecl  liy  eddy  motions,  which  cause  a  definite  change  to 

plut-e  iu  the  law  of  resistance. 
Banie^i  and  Coker*  have  determined  the  critical  velocity  by 
l^<iting  the  sudden  change  of  temperature  of  the  water  when  its 
ttduii  changes.    They  have  also  found  that  the  critical  velocity, 
determined  by  noting  the  velocity  at  which  stream-line  flowr 

PrxK4t4imji>  *>/  iht    Hmjat   Socuqf,  Vol.    lxMV.    1904;    Phil,    TranMctioni, 


Tliat 


There 


130 


HYDRAULICS 


breaks  up  into  eddies,  is  a  much  more  variable  quantity  tlian 
that  determined  from  the  points  of  interBection  of  the  two  \mm 
as  in  Fig*  90*  In  the  former  case  the  critical  velocity  depends 
upon  the  oondition  of  the  water  in  the  tank,  and  when  it  k 
perfectly  at  rest  the  stream  lines  may  be  maintained  at  much 
higher  velocities  than  those  given  by  the  formula  of  Reynolds. 
If  the  water  is  not  perfectly  at  rest,  the  results  obtained  by  both 
methods  agree  with  the  formula. 

Barnes  and  Cokor  have  called  the  critical  velocity  obtained  by 
the  method  of  colour  bands  the  upper  limit,  and  that  obtained  by 
the  interiiiection  of  the  logarithmic  homologues  the  lower  critical 
velocity*  The  first  gives  the  veloc^ity  at  which  water  flowing  from 
rest  in  stream-line  motion  break.s  up  into  eddy  motion,  while  the 
second  gives  the  velocity  at  which  water  that  is  initially  dii^turbed 
persists  in  flowing  with  eddy  motions  throughout  a  long  pipe,  or 
in  other  words  the  velocity  ib  too  high  to  allow  stream  line^  to  be 
formed. 

That  the  motion  of  the  water  in  large  conduits  is  in  a  similar 
condition  of  motion  is  shown  by  the  experiment  of  Mr  G*  H, 
Benzenberg*  on  the  discharge  through  a  sew-er  12  feet  in  diameter* 
2534  ft,,  long. 

In  order  to  measure  the  velocity  of  water  in  the  sewer,  red 
eosine  dissolved  in  water  was  suddenly  injected  into  the  sewer, 
and  the  time  for  the  coloured  water  to  reach  the  outlet  half  a 
mile  away  was  noted.  The  colour  was  readily  perceived  and  it 
was  found  that  it  was  never  distributed  over  a  length  of  more  than 
Q  feet.  As  >vill  be  seen  by  reference  to  section  130,  the  velocities 
of  translation  of  the  particles  on  any  cross  section  at  any  instant 
are  very  difFerentj  and  if  the  motion  were  streatn  line  the  colour 
must  have  been  spread  gut  over  a  much  greater  length. 


101.    Law  of  factional  resistance  for  velocities  above 
critical  velocity* 

As  seen  from  Reynolds'  formula,  the  critical  velocity  ea 
for  very  small  pipes  is  so  very  low  that  it  is  only  nece&sary  S^ 
practical  hydraulics  to  consider  the  law  of  frictional  resistance  for 
velocities  above  the  critical  velocity* 

For  any  particular  pipe, 

and  it  remains  to  determine  k  and  n» 

From  the  plottings  of  the  results  of  his  own  and    Dmr 

*  Tramaeiiom  Am.SM,E,  18t3|  and  alto  Fn>eeeMttfft  Am,S.€.E.^  VoL  m.rm* 


FLOW  THROUGH  PIPES 


131 


experiments,  Beynolds  found  that  the  law  of  resistance  "  for  all 
pipes  and  all  velocities"  could  be  expressed  as 

-t  - 


Transposing, 


,     /BD    \« 
B'D'.u'.P' 


t  = 


and 


4= 


AP-.D* 
B"P-" 


.(14). 
.(15), 


A  D"-* 

D  is  diameter  of  pipe,  A  and  B  are  constants,  and  P  is  obtained 
from  formula  (13). 

Taking'  the  temperature  in  degrees  centigrade  and  the  metre 
as  nnit  length, 

A  =  67,700,000, 

B  =  396, 

1 


P  = 


or 


in  w 


t  = 


y=. 


1  + -00361 +  -000221T" 
B'.t)'.F-»        y.v* 

67,700,000  D»-»"  D»-» 
B»P^" 


.(16), 


^^^  '    67,700,000- 

Values  of  y  tchen  the  temperature  is  10'  C. 


n 

7 

1-76 
1-86 
1-96 
2-00 

0-000266 
0000888 
0-000687 
0-000704 

\ 


The  values  for  A  and  B,  as  given  by  Reynolds,  are,  however, 
only  applicable  to  clean  pipes,  and  later  experiments  show  that 
alt^oagli 

it  is  doubtful  whether 

p  =  3  -  w, 

as  given  by  Reynolds,  is  correct. 

Value  of  n.  For  smooth  pipes  n  appears  to  be  nearly  I'lo. 
Reynolds  found  the  mean  value  of  n  for  lead  pipes  was  1*723. 

Saph  and  Schoder*,  in  an  elaborate  series  of  experiments 
carried  out  at  Cornell  University,  have  determined  for  smooth 

*  TranBoeticnt  of  the  American  Society  of  Civil  Engineers,  May,  1903.  See 
czerdte  31,  psge  172. 

9—2 


132  HYDRAULICS 

brass  pipes  a  mean  value  for  n  of  1*75.  Coker  and  Clements 
found  that  n  for  a  brass  pipe  *3779  inches  diameter  was  1*731.  In 
column  5  of  Table  XVI  are  given  values  of  n,  some  taken  from 
Saph  and  Schoder's  paper,  and  others  as  determined  by  the 
author  by  logarithmic  plotting  of  a  large  number  of  experiments. 

It  will  be  seen  that  n  varies  very  considerably  for  pipes  of 
different  materials,  and  depends  upon  the  condition  of  the  surface 
of  a  given  material,  as  is  seen  very  clearly  from  Nos.  3  and  4. 
The  value  for  n  in  No.  3  is  1*72,  while  for  No.  4,  which  is  the 
same  pipe  after  two  years'  service,  the  value  of  n  is  1*93.  The 
internal  surface  had  no  doubt  become  coated  with  a  deposit  of 
some  kind. 

Even  very  small  differences  in  the  condition  of  the  surfafCe, 
such  as  cannot  be  seen  by  the  unaided  eye,  make  a  considerable 
difference  in  the  value  of  7i,  as  is  seen  by  reference  to  the  xdines 
for  galvanised  pipes,  as  given  by  Saph  and  Schoder.  For  large 
pipes  of  riveted  steel,  riveted  wrought  iron,  and  cast  iron,  the 
value  of  n  approximates  to  2. 

The  method,  of  plotting  the  logarithms  of  i  and  v  determined 
by  experiment,  allows  of  experimental  errors  being  corrected 
without  difficulty  and  with  considerable  assurance. 

102.  The  determination  of  the  values  of  C  given  in 
Table  XH. 

The  method  of  logarithmic  plotting  has  been  employed  for 
determining  the  values  of  C  given  in  Table  XII. 

If  values  of  C  are  calculated  by  the  substitution  of  the 
experimental  values  of  v  and  i  in  the  formula 

s/mi 
many  of  the  results  are  apparently  inconsistent  with  each  other 
due  to  experimental  errors. 

The  values  of  C  in  the  table  were,  therefore,  determined  as 
follows. 

Since  i  =  kv^ 

and  in  the  Chezy  formula 

v  =  C  J  mi, 

.     mC 
or  1  =  — 3-, 

therefore  —pa  =  fry* 

and  2  log  C  =  2  log  v  -  (log  m  +  log  A;  +  n  log  t?)    (17). 

The  index  n  and  the  coefficient  k  were  determined  for  a 
number  of  cast-iron  pipes. 


FLOW  THROUGH    PIPES 


Valoes  of  C  for  i.'tJodties  fnim  1  to  10  were  calculated.    Carvi 
w<-T\^  tlieTi  plf»tt^^  for  diffei^nt  velocities,  having  C  as  oMinai 
atid  dianieters  as  abscisisae,  and  the  values  given  in  the  table  we; 
Ai?<iaced  from  the  curves. 

The  vBlneB  of  C  so  interpolated  diifer  very  conaiderably, 
fOiue   cases,    from    the    es(>t*riinental    values*      The    diffieulties 
&Umding  the  accurate  deterii*ination  of  i  and  v  are  very  great, 
and  tbe  %*alties  of  C,  for  any  given  pipe,  as  calculated  by  substi 
titing  in  the  Chezy  formula  the  losses  of  head  in  friction  and  th 
vt^liiciries    as    determined   in    the   experiments,   were  frequent!; 
inamsiMeiit  wHth  each  father. 

As,  for  example,  in  the  pipe  of  3*22  ins.  diameter  given  i 

Tible  XTI  which  was  one  of  Darcy's  pipes,  the  variation  of  C  a 

calculated  from  h  and  r  given  by  Darcy  m  from  78'8  to  100. 

Chi  plotting  log  ft  and  logr  and  correcting  the  readings  a 

tiw  they  all  lie  on  one  line  and  recalculating  C  the  variation  wa 

^ '  r  >  be  only  from  ^5*9  to  101. 

■tar  correotioni^  liave  been  made  in  other  cases. 

Thv  author  thinks  this  procedure  is  justified  by  the  fact  that 

mauy  of  the  best  experiments  do  not  show  any  such  inconsistencies. 

An  attempt  to  draw  up  an  interpolated  table  for  riveted  pipes 

mt-i  not  8atisfjictor>%     The  author  has  therefore   in  Table  XI Y 

givtm  the  values  of  C  as  calculated  by  formula  (17),  for  %^arioua 

Vidoctties,  and  the  iliameters  of  the  pipes  actually  experimentec 

ap»kn.     If  curves  are   plotted   from   the  values  of  C   given   ii 

Tntde  XIV,  It  will  be  seen  that,  except  for  low  velocities,  th< 

-es  are  not  continuous,  and,  until  further  experimental  evidenc< 

i^  i^irthcoming  for  riveted  pipes,  the  engineer  must  be  content 

with  ch*X)sing  values  of  C,  which  most  nearly  coincide,  as  far 

be  can  judge,  with  th©  case  he  is  considering. 

103.    Vwiation    of  k,  in  the  formula  i  =  kv",  with   the 
diameter. 

It  has  beeii  shown  in  section  98  how  the  value  of  fc,  for 
iveo  pipe,  can  be  obtiiined  by  the  logarithmic  plotting  of  t  and  t?; 
In  Ttihle  XVI,  are  given  values  of  A\  as  determined  by  the 
authcirt  by  plotting  the  results  of  different  experiments,     Saph 
-   hoder    found    that    for   smooth   hard-drawn   In-ass    pipes 
us  siases  n  varied  between  173  and  1*77,  the  mean  value 
being  1'75.  ■ 

By  plotting  logff  tm  abscissae  and  log  ft  as  ordinates,  as  in' 
Fig*  I'ti,  for  these  bniss  pi|X'S  the  points  He  nearly  in  a  straight  line 
which  hoA  aJi  inelination  &  with  the  axis  of  d,  such  that 

tan^--P25 


134 


HYDRAULICS 


and  the  equation  to  the  line  is,  therefore, 

\ogh  =  logy-p\ogd, 
where  p  =  1*26, 

and  log  y  =  log  Jc 

when  d  =  1. 

From  the  figure 

y  =  0-000296  per  foot  length  of  pipe. 


•05 
-03. 


(m 


EqucUwntu  liive 
Zo€f.  'k^Log  m  -  1-2SLog  d  ' 


dz     03  Of    -06 -08  ho         W^    '3 

Log  di 


Fig.  93.    Logarithmio  plottings  of  fc  and  d,  to  determine  the  index  p  in  the  fonnul 

.    7 .  r* 

On  the  same  figure  are  plotted  logd  and  log  A;,  as  deduce 
from  experiments  on  lead  and  glass  pipes  by  various  workers.  I 
will  be  seen  that  all  the  points  lie  very  close  to  the  same  line. 

For  smooth  pipes,  therefore,  and  for  velocities  above  tb 
critical  velocity,  the  loss  of  head  due  to  friction  is  given  by 


%  = 


_yv 
~  d^ 


the  mean  value  for  y  being  0*000296,  for  n,  1*75,  and  for  p  l'2o. 

From  which,  v  =  104i""d'^', 

or  log  V  =  2-017  +  0-572  log  i  +  0*715  log  d. 


FLOW  THROUGH  PIPES  135 

The  value  of  p  in  this  formula  agrees  with  that  given  by 
Reynolds  in  his  formnla 

Pnrfessor  Unwin*  in  1886,  by  an  examination  of  experiments 
on  cast-iron  pipes,  deduced  the  formula,  for  smooth  cast-iron 
pipes, 

•0004t?^* 


*  = 


cf 


and  for  rough  pipes,  %  =     ^^     • 

M.   Flamantt  in   1892  examined  carefully  the  exi)eriment8 
available  on  flow  in  pipes  and  proposed  the  formula, 

for  all  classes  of  pipes,  and  suggested  for  y  the  following  values : 
Lead  pipes  ^ 

Glass    „  [  •000236  to -00028, 

Wrought-iron  (smooth)  J 
Cast-iron  new  '000336, 

„      „    in  service  •000417. 

If  the  student  plots  from  Table  XVI,  log  d  as  ordinates,  and 
log"  A-  as  abscissae,  it  will  be  found,  that  the  points  all  lie  bqtween 
two  straight  lines  the  equations  to  which  are 

log  k  =  log  -00069  - 1-25  log  d, 
and  log  &  =  log  '00028  - 1-25  log  d. 

Further,  the  points  for  any  class  of  pipes  not  only  He  between 
these    two  lines,  but  also  lie  about  some  line  nearly  parallel  to 
these  lines.     So  that  p  is  not  very  different  from  r25. 
From  the  table,  n  is  seen  to  vary  from  1*70  to  208. 
A  general  formula  is  thus  obtained, 

,     -00028  to  •00069t;^'^^^^-^/ 

The  variations  in  y,  n,  and  p  are,  however,  too  great  to  admit 
of  the  formula  being  useful  for  practical  purposes. 
For  new  cast-iron  pipes, 

,     -000296  to  -0004181?^^ '«^*^Z 
fe  = ^ • 

If  the  pipes  are  lined  with  bitumen  the  smaller  values  of  y  and 
T<  may  be  taken. 

*  Industries,  1SS6. 

t  AimaUs  des  Pouts  et  Chauss^et,  1892,  Vol.  ii. 


136 


HYDRAULICS 


For  new,  steel,  riveted  pipes, 

^_'0004to'00054i;^"^'«'Z 
^-  d}^ 

Fig.  94  shows  tlie  result  of  plotting  logfe  and  logd  for  all 
the  pipes  in  Table  XVI  having  a  value  of  n  between  1*92  and  1*94. 
They  are  seen  to  lie  very  close  to  a  line  having  a  slope  of  1^, 
and  the  ordinate  of  which,  when  d  is  1  foot,  is  '000364. 
,     •000364r^"*'Z 


Therefore 


d}'' 


or  r  =  59i-««d"' 


very  approximately  expresses  the  law  of  resistance  for  particular 
pipes  of  wood,  new  cast  iron,  cleaned  cast  iron,  and  galvanised 
iron. 


Fig.  94. 

Taking  a  pipe  1  foot  diameter  and  the  velocity  as  3  feet  per 
second,  the  value  of  i  obtained  by  this  formula  agrees  with  that 
from  Darcy's  formula  for  clear  cast-iron  pipes  within  1  per  cent. 

Use  of  the  logarithmic  formula  for  'practical  calculations.  A 
very  serious  difficulty  arises  in  the  use  of  the  logarithmic 
formula,  as  to  what  value  to  give  to  n  for  any  given  case,  and 
consequently  it  has  for  practical  purposes  very  little  advantage 
over  the  older  and  simpler  formula  of  Chezy. 


TABLE  XVI. 


1 

Diameter 
(in  int.) 

Velocity  in 

Value  of  n        ^ 

/alue  of  k 

enter       1 

Kind  of  pipe 

fk  per  seo. 
from         to 

in  formula        1 

n  formula 

le           ^ 

Wood 

44 

8-46  —  4-416 

1-78 

0001254 

«» 

54 

2-28  —  4-68 

1-75 

000088 

skissV 

w 

72-5 

1       ~  4 

172 

000061 

M 

72-6 

1       —  6-6 

1-98 

000048 

witcham 

Riveted 

8 

1^88 

00245 

oith 

Wrought 

11 

1-81 

000516 

iron  or  steel 

11} 

1^90 

000470 

ff 

16 

194 

000270 

ding 

»» 

88 

•506—  1-254 

2^0 

000099 

^ 

»» 

42 

210  -  499 

198 

00011 

»» 

48 

2       -  6  (?) 

2-0 

000090 

^ipg) 

»» 

72 

1—4 

199 

000055 

99 

72 

1        —  66 

1^85 

000077 

2bel 

»9 

108 

1        —  45 

2-08 

000036 

cy 

Cast  iron 

8*22 

•289— 1071 

1-97 

00156 

new 

6-89 

•48  — 158 

1-97 

00079 

tf 

7-44 

•678—1617 

1^956 

00062 

ff 

12 

1779 

000823 

iZHfl 

}f 

16-25 

1^858 

000214 

pe 

»» 

16-5 

2-48  —  8-09 

1^80 

000267 

«f 

19-68 

1-88  —  87 

1^84 

00022 

nan 

»i 

36 

4       —  7 

2* 

000062 

-ns 

»» 

48 

1-248-  8-28 

1^92 

0000567 

FenkeU 

30 

2 

00003 

ZJ 

Cast  iron 

1-4136 

•167—  2077 

1-99 

0098 

old  and 

31296 

•403-  3-747 

1-94 

0085 

tuberculated 

9-575 

1-007—12-58 

1-98 

0009 

aan 

^1 

20 

2-71  —  5-11 

?i 

36 

11     —  4-5 

2 

000105 

raid 

48 

1.176—  3-533 

2-04 

0(KX)83 

♦> 

48 

1-135—  3-412 

2-00 

00(X)85 

:y 

Cast-iron 

1-4328 

-371—  3-69 

1-85 

(K)41 

old  pipes 

3-1536 

-633—  5-0 

1-97 

(K)lHo 

cleaned 

11-68 

-8     —10-368 

20 

(KK);J75 

raid 

yj 

48 

3-67  —  5-6 

202 

0(K)082 

" 

48 

•895—  7-245 

1-94 

()(KX)59 

:y 

Sheet- iron 

1055 

-098—  8-225 

1-76 

(K)74 

,^ 

3-24 

•328-12-78 

1-81 

00154 

,*,' 

7-72 

-591-19-72 

1-78 

(X)059 

11-2 

1-296— 10-52 

1-81 

00039 

Gas 

-48 

-113—  3-92 

1-83 

0278 

1-55 

•205—  8-521 

1-86 
1-91 

00418 
0072 

Sclioder 

Galvanised 

-364 

1-96 

0352 

•494 

1-91 

0181 

,, 

-623 

1-86 

0182 

•824 

1-80          1 

0095 

,^ 

1-048 

1-93 

•(K)82 

Hard -drawn 

15  pipes 

1-75 

•00025  to 

brass 

up  to  1-84 

-0(K)85 

.Ids 

/ 

Lead 

1-782 

•55      1 
1-61       1 

1-761        \ 

-0\^e> 

1^783        \ 

•0^^^ 

138 


HTDBAUUCS 


TABLE  XVn. 

Showing  reasonable  values  of  y,  and  n,  for  pipes  of  TarioDf 
kinds,  in  the  formula, 


Takes  for     j 

1                                 1 

i^ 

« 

y 

i 

Clean  caBt-iron  pipes 
Old  cast  iron  pipes 
Riveted  pipes 
Galvanieod  pipes 
Sheet-iron  pipes  cover- 
ed with  bitumen 
Clean  wood  pipes 
BrasR  and  lead  pipes 

-00029  to  -000418 
'00O47  to  '00069 
'00040  to  *00054 
-00035  to  -00045 

'00030  to  *00088 

'00056  to  '00063 

1*80  to  1*97 
1'94  to  2*04 
1-93  to  2'08 
1*80  to  1-96 

l'76tol'81  ; 

1-72  to  1-75  ' 

■00086 

-00060 
•00050 
'00040 

-00034 

■O0060 
*00080 

2 
t 

vm 
m 

When  further  experiments  have  been  performed  on  pipes,  rf 
which  the  state  of  the  internal  surfaces  is  accurately  known,  vA 
special  care  taken  to  ensure  that  all  the  loss  of  head  in  a  git® 
length  of  pipe  is  due  to  friction  only,  more  definiteness  maybe 
given  to  the  values  of  y,  n,  and  p. 

Until  such  evidence  is  forthcoming  the  simple  Chezy  formnl* 
may  be  used  with  almost  as  much  confidence  as  the  mow 
complicated  logarithmic  formula,  the  values  of  C  or/  being  takea 
from  Tables  XII — XIV.  Or  the  formula  h  =  fcr*  may  be  used, 
values  of  k  and  n  being  taken  from  Table  XVI,  which  most  nearly 
fits  the  case  for  which  the  calculations  are  to  be  made. 


I 


104.    Criticism  of  experiments. 

The  difficulty  of  differentiating  the  loss  of  head  due  to  friction  . 
from  other  sources  of  loss,  such  as  loss  due  to  changes  in  direction, 
change  in  the  diameter  of  the  pipe  and  other  causes,  as  well  as  the 
possibilities  of  error  in  experiments  on  long  pipes  of  large  diameter, 
makes  many  experiments  that  have  been  performed  of  very  little 
value,  and  considerably  increases  the  difficulty  of  arriving  at 
correct  formulae. 

The  author  has  found  in  many  cases,  when  log  i  and  log  d  were 
plott<3d,  from  the  records  of  experiments,  that,  although  the  results 
seemed  consistent  amongst  themselves,  yet  compared  with  other 
e-Yperiments,  they  seemed  of  little  value. 


\  for  one  ol  Couplot^s*  e3:periments  on  a  lead  and 
ipe  being  &b  low  as  1*56,  while  the  results  of  an 
iSunpecint  cm  a  cust-iroii  pipe  gave  n  as  2*5.  In  the 
fe  were  a  number  uf  bends  in  the  pipe* 
experiments  for  loss  of  head  due  to  friction,  it  is 
I  the  pipe  should  be  of  uniform  diameter  and  aa 
Uible  between  the  points  at  which  the  pressure  head 
Further,  special  care  Rhould  be  taken  to  ensure  the^ 
I  air,  and  that  a  perfectly  steady  flow  m  established 
Fhere  the  pressure  is  taken* 

Eometer  fittings, 

h-eme  iuiixirtance  that  the 

ktmections  shall  be  made 

JBference  in  the  pressures 

any  two  pcjints  shall  be 

fiction^  and  friction  only, 

points, 

Hiiptes  that  there  shall 

^P^B  to  interfere  with  the 

Se  w*ater,  and  it  h,  there- 

bntial  that  all  burrs  sliall 

hom  the  inside  of  the  pipe, 

bients  un  small   pipits  in 

JT  the  best  results  are  no 

bd  by  cutting  the  pipe 

irough  at  the  connection 

t'ig.  95,  which  illustrates 

I  connection   use<l   by  Dv  p|-  gg 

(u  experiments   cited   on 

lie  two  ends  of  the  pipe  ai*e  not  more  than  u^nr 

^B  the  method  adopted  by  Marxj  Wing  and  Hoskins 
riments  on  a  72^inch  wooden  pipe  to  ensure  a  correct 
B  pressure. 

I?  X  was  connected  to  the  top  of  the  pipe  only  while 
jted  at  four  points  as  shown. 

(erences  were  observed  in  the  readings  of  the  t%vo 
ji  they  thought  were  due  to  some  accidental  circum- 
big  the  gauge  X  only,  as  no  change  was  obsoiTcd 
of  Y  when  the  points  of  communication  to  Y  were 
of  the  cocki*, 

utici,  HamUton  Smith.  Jnnr. 

iinfft  fif  the  Ttntitufe  of  Vint  Euffineertf  1855, 


^Ita 


140 


HYDRAULICS 


106.    Effect  of  temperature  on  the  velocity  of  flow. 

Poiseuille  found  that  by  raising  the  temperature  of  the  water 
from  50**  C.  to  100'  C.  the  discharge  of  capillary  tubes  was 
doubled. 


Fig.  96.    Piezometer  connections  to  a  wooden  pipe. 

Reynolds*  showed  that  for  pipes  of  larger  diameter,  the  effect 
of  changes  of  the  temperature  was  very  marked  for  velocities 
below  the  critical  velocity,  but  for  velocities  above  the  critical 
velocity  the  eifect  is  comparatively  small. 

The  reason  for  this  is  seen,  at  once,  from  an  examination  of 
Reynolds'*  formula.  Above  the  critical  velocity  n  does  not  differ 
very  much  from  2,  so  that  P"*  is  a  small  quantity  compared  >vith 
its  value  when  n  is  1. 

Saph  and  Schodert,  for  velocities  above  the  critical  velocity, 
found  that,  as  the  temperature  rises,  the  loss  of  head  due  to 
friction  decreases,  but  only  in  a  small  degree.  For  brass  pipes  of 
small  diameter,  the  correction  at  60°  F.  was  about  4  per  cent,  per 

•  Scientific  Papers,  Vol.  ii. 

+  See  also  Barnes  and  Coker,  Proceeding*  of  the  Royal  Society ^  Vol.  utx.  1904 ; 
Coker  and  Clements,  Transactions  of  the  Royal  Society,  Vol.  cci.  Proceeding* 
Am.S.C.E.  Yol  xxix. 


FLOW  THROUOH  PIPES  141 

0  degrees  F.     With  galvanised  pipes  the  correction  appears  to 
le  from  1  per  cent,  to  5  per  cent,  per  10  degrees  F. 

Since  the  head  lost  increases,  as  the  temperature  falls,  the 
lischarge  for  any  given  head  diminishes  with  the  temperature, 
mt  for  practical  purposes  the  correction  is  generally  negligible. 

107.    Ij088  of  head  due  to  bends  and  elbows. 

The  loss  of  head  due  to  bends  and  elbows  in  a  long  pipe  is 
generally  so  small  compared  with  the  loss  of  head  due  to  friction 
Q  the  straight  part  of  the  pipe,  that  it  can  be  neglected,  and 
onsequently  the  experimental  determination  of  this  quantity  has 
lot  received  much  attention. 

Weisbach*,  from  experiments  on  a  pipe  1^  inches  diameter, 
rith  bends  of  various  radii,  expressed  the  loss  of  head  as 


*-(«^*T)g. 


being  the  radius  of  the  pipe,  B  the  radius  of  the  bend  on  the 
pntre  line  of  the  pipe  and  v  the  velocity  of  the  water  in  feet  per 
?<rond.     If  the  formula  be  written  in  the  form 

7       at?* 

le  table  shows  the  values  of  a  for  different  values  of  ^  . 

r 

R 

•1  -167 

•2  -250 

•6  -626 

St  Venantt  has  given  as  the  loss  of  head  h^  at  a  bend, 

Ab  =  -001524  y^^=Ol|  4  yi  nearly. 

being*  the  length  of  the  bend  measured  on  the  centre  line  of  the 
?nd  and  d  the  diameter  of  the  pipe. 
ANTien  the  bend  is  a  right  angle 


RVR     2VR' 


When 


d 

=  1, 

•5, 

% 

d 

=  1-57, 

Ml, 

•702 

iB  = 

■  -'"g' 

•111 

29' 

< 

.  •  Ueehatue$  of  Engineering, 
i  Compte*  Rendtii,  1862. 

143 


HTTDRiUXICS 


Kt^ent  experiments  by  Williams,  Hubbell  and  Fenkell^onc 
iron  inpes  asphalted,  hy  Saph  and  &boder  on  brajss  pipe.**, 
oth*  r*  by  Alexander t  on  wooden  pipes,  show  that  the  loss  of  ] 
in  lK>nds,  as  in  a  straight  pipe,  can  be  expressed  as 

n  Wing  a  variable  for  different  kinds  of  pipes,  while 


Ar- 


y'Hr 


y  being  a  constant  coeffieii  lipe. 

For  the  cast-ircin  pipes  or  and  Penkell,  y^  n,  m,  and  f  ♦ 

have  approximately  the  {ollowm^         les. 


Dia&Mter  of  pipe 


ir 
ir 
mr 


<€0« 


1-TB 

i-m 


i-m 


^^^len  r  is  3  feet  per  second  and  p"  is  i,  the  bend  being  a  righl 

angle^  tlie  lo^  of  head  as  calculated  by  thia  formula  for  difi'j 

i2-ineh  pipe  is      \^ ,  and  for  ta©  dU-mch  pipe    -^-  - . 

For  the  brass  pipes  of  Saph  and  Schoder,  2  inches  diametO'^ 
Alexander  found, 

and  for  vanitshecl  wood  pipes  when  ^  is  less  than  0'2, 

and  wlieii  u  is  betn^een  0"2  and  O'o, 
11 

He  further  found  for  \*amished  wood  pipes  that^  a  bend  of 
radias  equal  to  5  times  the  radius  of  the  pipe  gives  the  mini  mum 
loss  +  ^^-'rf}  '»*^'i  f^->*  ^*^  !-<»si?^tance  is  equal  to  a  straight  pipe  3"3S 
times  the  length  of  the  bend. 

Messrs  Williams,  Hubbell  and  Fenkell  also  state  at  the  end 
their  elaborate  paper,  that  a  bend  having  a  radius  equal  to  : 

•  Proe.  Amer.  Soc,  Civil  EmgiHeen,  Y6L  xxm. 
t  Proc,  Intt.  Civil  Emgineen,  Vol.  clix. 


rLOW  THROUGH   TlfES 


143 


loffers  hm  remstanoe  to  the  flow  of  water  than  those  of 
aditm.  It  should  not  be  overlooked,  howeverj  that  although 
nf  liead  in  a  bend  of  radius  equal  to  2A  diameterB  of  the 
ess  than  for  any  other,  it  does  not  follow  that  the  loss  of 
JT  unit  length  of  the  pipe  measured  along  its  centre  line 
cniiiit&um  value  for  bends  of  this  radiuB. 


Variations  of  the  velocity  at  the  cross  section  of  a 
rieal  pipe* 

(eriinents  show  that  when  water  flows  through  conduits  of 

rm,  the  velocities  are  not  the  same  at  all  points  of  any 

turn   aeetion,  but  decrease  from  the  centre  towards  the 

terence. 

f  first  experimentsi  to  determine  the  law  of  the  variation  of 

[ocity  in  cylindrical  pipes  were  those  of  Darcy,  the  pipes 

If  in  diameter  frfjm  7"8  inches  to  19  incites.     A  complete 

tf  of  the   exi>eriraents   is   to   be   found   in   his  Recherchm 

tnmiiales  dan^  les  iuyauit^ 

•  Telocity  was  me^isured  by  means  of  a  Pitot  tube  at  five 

on  a  vertical  diamfetjer,  and 

suits    plotted   as   iihown  in 

r. 

ling  V  the  velocity  at  the 
of  a  pipe  of  radins  R^  u  the 
Y  at  the  circumference,  tv 
san  velocity,  y  the  vt^wity 
distance  r  from  the  centre, 
losB  of  head  per  unit 
[the  pipe,  Darcy  deduced  the  formulae 

\  the  unit  is  the  metre  the  value  of  k  is  \\%  and  20'4  when 

ft  IB  the  English  foot. 

'jBT  earperiments  commenced  by  Darcy  and  continued  by 

on  the  distribution  of  velocity  in  a  semicircular  channel, 
rface  of  the  water  being  maintained  at  the  horizon tal 
g"  and  in  which  it  was  assumed  tho  conditions  were  similar 
^Ki  a  cylindrical  pipe,  showed  that  the  velocity  near  the 
^OT  the  pipe  diminished  much  more  rapidly  than  indicated 

iormula  of  Darcy, 


144  HYDRAUUCS 

Bazin  substituted  therefore  a  new  formula, 

Y-v  =  38y/Ri(^J  (1), 

or  since  t7»  «  C  Jmi  =  —j^  JTU 

-C'^ii)' ■■■ <^>- 

It  was  open  to  question,  however,  whether  the  conditions  of  flow 
in  a  semicircular  pipe  are  similar  to  those  in  a  pipe  discharging 
full  bore,  and  Bazin  consequently  carried  out  at  Dijon*,  experi- 
ments on  the  distribution  of  velocity  in  a  cement  pipe,  2'73  feel 
diameter,  the  discharge  through  which  was  measured  by  means 
of  a  weir,  and  the  velocities  at  different  points  in  the  transverse 
section  by  means  of  a  Pitot  tubet. 

From  these  experiments  Bazin  concluded  that  both  formulae  (1) 
and  (2)  were  incorrect  and  deduced  the  three  formulae 

V-..38^RS((0-(iy*(g}    (3), 

V-.^VBi{38(^)\49(g(l-rigy    (4), 

V-r  =  N/Rt53-5{l-yi--95(j)'} (5), 

the  constants  in  these  formulae  being  obtained  from  Bazin's  by 
changing  the  unit  from  1  metre  to  the  English  foot. 

Equation  (5)  is  the  equation  to  an  ellipse  to  which  the  sides  of 
the  pipes  are  not  tangents  but  are  nearly  so,  and  this  formula 
gives  values  of  v  near  to  the  surface  of  the  pipe,  which  agree  much 
more  nearly  with  the  experimental  values,  than  those  given  by 
any  of  the  other  formulae. 

Experiments  of  WilliamSy  Hvhhell  and  FenkellX*  An  elaborate 
series  of  experiments  by  these  three  workers  have  been  carried  out 
to  determine  the  distribution  of  velocity  in  pipes  of  various 
diameters,  Pitot  tubes  being  used  to  determine  the  velocities. 

The  pipes  at  Detroit  were  of  cast  iron  and  had  diameters  of  12, 
16,  30  and  42  inches  respectively. 

The  Pitot  tubes  §  were  calibrated  by  preliminary  experimentB 
on  the  flow  through  brass  tubes  2  inches  diameter,  the  total 

*  '  *  Memoire  de  TAcad^mie  des  Soiences  de  Paris,  Becueil  des  SavantsEtraxigdiv," 
Vol.  XXXII.  1897.    Proc,  Am,S.C,E.  Vol.  xxvii.  p.  1042. 

t  See  page  241. 

X  **  Experiments  at  Detroit,  Mich.,  on  the  effect  of  carvatare  on  the  flow  of 
water  in  pipes,*'  Proc.  Am.S.C.E.  Vol.  xxvu.  p.  318. 

§  See  page  246. 


FLOW  THROUGH  PIPES  145 

discliarge  being  determined  by  weighing,  and  the  mean  velocity 

thas   determined.     From  the  results  of  their  experiments  they 

came  to  the  conclusion  that  the  curve  of  velocities  should  be  an 

ellix>8e  to  winch  the  sides  of  the  pipe  are  tangents,  and  that  the 

velocity  at  the  centre  of  the  pipe  V  is  TlOvm,  t?»  being  the  mean 

velocity. 

These  results  are  consistent  with  those  of  Bazin.    His  experi- 

V 
mental  value  for  —  for  the  cement  pipe  was  1*1675,  and  if  the 

constant  "95,  in  formula  (5),  be  made  equal  to  1,  the  velocity  curve 

be<x>me8  an  ellipse  to  which  the  walls  of  the  pipe  are  tangents. 

V 
The  ratio  —  can  be  determined  from  any  of  Bazin's  formulae. 

Substituting  —^  for  >/Ri  in  (1),  (3),  (4)  or  (5),  the  value  of 
V  at  radius  r  can  be  expressed  by  any  one  of  them  as 

'-^-#--/(S)- 

Then,  since  the  flow  past  any  section  in  unit  time  is  VmyfR\  and 
that  the  flow  is  also  equal  to 

2'7rrdr .  v, 
f  0 

therefore  v^^W  =  2t  J^|v  -  5^"/(^)jrdr (6). 

(t\  ^lA*** 

-g^j ,  its  value  -^  from  equation  (1),  and 

integrating. 

;;  =  ^"-c-  (7), 

and  by  substitution  oifi^j  from  equation  (4), 

l-'*v <«. 

V 
so  that  the  ratio  —  is  not  very  different  when  deduced  from  the 

simple  formula  (2)  or  the  more  complicated  formula  (4). 
Wlien  C  has  the  values 

C  =  80,      100,  120, 

from  (8)  ~  =  1-287,  123,  ri9. 

The  value  of  C,  in  the  30-inch  pipe  referred  to  above,  varied 
between   109"6  and  123*4  for  different  lengths  of  the  pipe,  and 
i^  H.  10 


/: 


146 


HYDRAUUCS 


the  mean  value  was  116,  so  that  there  is  a  remarkable  agreement 
between  the  results  of  Bazin,  and  Williams,  Hubbell  and  Fenkell. 

The  velocity  at  the  8v/rface  of  a  pipe.  Assuming  that  the 
velocity  curve  is  an  ellipse  to  which 
the  sides  of  the  pipe  are  tangents,  as 
in  Fig.  98,  and  that  V=  ri9t?«,  the 
velocity  at  the  surface  of  the  pipe 
can  readily  be  determined. 

Let  u  =  the  velocity  at  the  surface 
of  the  pipe  and  v  the  velocity  at  any 
radius  r. 

Let  the  equation  to  the  ellipse  be 

in  which  a?  =  v  -  u, 

and  b  =  y  —  u. 

Then,  if  the  semi-ellipse  be  revolved  about  its  horizontal  axis, 
the  volume  swept  out  by  it  will  be  |^rR''6,  and  the  volume  of 
discharge  per  second  will  be 

r^ 

7rR'»t7^  =       27rrdr .  i;  =  ttR'  .  u  +  §7rR«6, 

J  0 

itt  +  f  X  ri9r«, 
and 


u  =  '621v„,. 


u 


Using  Bazin's  elliptical  formula,  the  values  of  —  for 


C  =  80, 


are 


u 


-=•552,  -642, 

Vm 


100,    120, 
702. 


The  velocities,  as  above  determined,  give  the  velocity  of 
translation  in  a  direction  parallel  to  the  pipe,  but  as  shown  by 
Reynolds'  experiments  the  particles  of  water  may  have  a  much 
more  complicated  motion  than  here  assumed. 

109.  Head  necesisaky  to  give  the  mean  velocity  Vm  to 
the  water  in  the  pipe. 

It  is  generally  assumed  that  the  head  necessary  to  give  a  mean 

V  ^ 
velocity  Vm  to  the  water  flowing  in  a  pipe  is  ^,  which  would  be 

correct  if  all  the  particles  of  water  had  a  common  velocity  Vm. 

If,  however,  the  form  of  the  velocity  curve  is  known,  and  on  the 
assumption  that  the  water  is  moving  in  stream  lines  with  definite 
velocities  parallel  to  the  axis  of  the  pipe,  the  actual  head  can 
be  determined  by  calculating  the  mean  kinetic  energy  per  lb.  of 

water  flowing  in  the  pipe,  and  this  is  slightly  greater  than  ^ . 


FLOW  THBOUOH  PIPES  147 

As  bef  ore,  let  v  be  the  velocity  at  radius  r. 
The  kinetic  energy  of  the  quantity  of  water  which  flows  past 
any  aectioii  per  second 

w .  2vrdr .  t; .  ^r- , 
0  2g' 

ir  being  tlie  ^vreight  of  1  c.  ft  of  water. 

The  kinetic  energy  per  lb.,  therefore, 

f^w.2irrdrv* 


«io         2g 

^^^ 

1    w .  2xr  dn? 

2g}oV 

V2r.  ^/r\l»   , 

(9). 

The 

simplest  value  for  / 

(^  is  that  of  Bazin's 

formula  (1) 

above. 

from 

which 

■(-^-F) 

and 

/G 

D=^i" 

Substituting  these  values  and  integrating,  the  kinetic  energy 
per  1^-  is  |— ,  and  when 

C  is  80,    100, 
a  is  112,  1-076. 

On  the  assumption  that  the  velocity  curve  is  an  ellipse  to  which 
the  ijvalls  of  the  pipe  are  tangents  the  integration  is  easy,  and  the 
value  of  a  is  1*047. 

Using  the  other  formulae  of  Bazin  the  calculations  are  tedious 
and  the  values  obtained  differ  but  slightly  from  those  given. 

The  head  necessary  to  give  a  mean  velocity  Vm  to  the  water  in 

the  pipe  may  therefore  be  taken  to  be  o"  >  t;he  value  of  a  being 

ftbout  1'12.  This  value*  agrees  with  the  value  of  1*12  for  a, 
obtained  by  M.  Boussinesq,  and  with  that  of  M.  J.  Delemer  who 
finds  for  a  the  value  11346. 

IIO.    Practical  problems. 

Before  proceeding  to  show  how  the  formulae  relating  to  the 
loss  of  head  in  pipes  may  be  used  for  the  solution  of  various 
problems,  it  will  be  convenient  to  tabulate  them. 

*  Flamant's  Hydrauliqw. 

10—2 


148  hydraulics 

Notation. 
fc  =  los8  of  head  due  to  friction  in  a  length  Z  of  a  straight  pipe. 

.    i  =  the  virtual  slope  =  y . 

t?  =  the  mean  velocity  of  flow  in  the  pipe, 

d  =  the  diameter. 

m  =  the  hydraulic  mean  depth 
A  i*ftfli  A      fl 

Fcyrmuh,!.  h  =  ^=^. 

This  may  be  written    y  =  7^5— , 

or  1;  =  C  J  mi. 

The  values  of  C  for  cast-iron  and  steel  pipes  are  shown  in 
Tables  XII  and  XIV. 

Formula  2.  h  =  rf-^ , 

f  ,        .  .  1 

^  in  this  formula  being  equal  to  7^  of  formula  (1). 

Values  of /are  shown  in  Table  XIII. 

Either  of  these  formulae  can  conveniently  be  used  for 
calculating  fc,  t?,  or  d  when  /,  and  Z,  and  any  two  of  three 
quantities  fe,  v,  and  d,  are  known. 

Formula  3.  As  values  of  C  and  /  cannot  be  remembered  for 
variable  velocities  and  diameters,  the  formulae  of  Darcy  are 
convenient  as  giving  results,  in  many  cases,  with  sufficient 
accuracy.    For  smooth  clean  cast-iron  pipes 

A  =  -005(l^|^^, 
or  ^=m^j^M 

For  rough  and  dirty  pipes 
or  «  =  i39y_|_V5f 


=27«yi2^^- 


FLOW  THROUGH  PIPES  149 

II  d  is  the  unknown,  Darcy^s  formulae  can  only  be  used  to  solve 
for  d  by  approximation.   The  coefficient  ( 1  +  t^  j  is  first  neglected 

and  an  approximate  value  of  d  determined.  The  coefficient  can 
then  be  obtained  from  this  approximate  value  of  d  with  a  greater 
degree  of  accuracy,  and  a  new  value  of  d  can  then  be  found,  and 
so  on.     (See  examples.) 

Formula  4.     Known  as  the  logarithmic  formula. 

h     .    y .  r* 

Values  of  y,  n,  and  p  are  given  on  page  138. 

By  taking  logarithms 

log  h  =  log  y  +  n  log  v  +  log  I  —  p  log  d, 

from  which  h  can  be  found  if  i,  v,  and  d  are  known. 

If  A,  Z,  and  d  are  known,  by  writing  the  formula  as 

n  log  V  =  log  h  -  log  I  -  log  y  +  p  log  d, 

T  can  be  found. 

If  A,  /,  and  v  are  known,  d  can  be  obtained  from 

p  log  d  =  log  y  +  n  log  v  +  log  I  -  log  h. 

This  formula  is  a  little  more  cumbersome  to  use  than  either  (1)  or 

(2)  but  it  has  the  advantage  that  y  is  constant  for  all  velocities. 

Formula  5.    The  head  necessary  to  give  a  mean  velocity  v  to 

ri2i7' 
the  iJvater  flowing  along  the  pipe  is  about  — ^ — ,  but  it  is  generally 

convenient  and  sufficiently  accurate  to  take  this  head  as  ^ ,  as 

was  done  in  Fig.  87.  Unless  the  pipe  is  short  this  quantity  is 
negligible  compared  with  the  friction  head. 

Formula  6.^  The  loss  of  head  at  the  sharp-edged  entrance  to  a 
pij>e  is  about  -^    and  is  generally  negligible. 

Formula  7.    The  loss  of  head  due  to  a  sudden  enlargement  in 

a  pipe  where  the  velocity  changes  from  Vi  to  rj  is       ^        . 

^9 
Formula  8.    The  loss  of  head  at  bends  and  elbows  is  a  very 

variable  quantity.    It  can  be  expressed  as  equal  to  -y-  in  which 

9 
a  varies  from  a  very  small  quantity  to  unity. 

ProbUm  1.  The  difference  in  level  of  the  water  in  two  reservoirs  is  h  feet, 
FUr.  99,  And  thej  are  connected  by  means  of  a  straight  pipe  of  length  I  and 
diameter  d ;  to  find  the  discharge  through  the  pipe. 


150 


HYDRAULICS 


Let  Q  be  the  number  of  eabic  feet  discharged  per  second.  The  head  h  is  utilised 
in  giving  velocity  to  the  water  and  in  overcoming  resistance  at  the  entrance  to  the 
pipe  and  the  frictional  resistances. 


Fig.  99.    Pipe  connecting  two  reservoirs. 

Let  V  be  the  mean  velocity  of  the  water.    The  head  necessary  to  give  the  water 
this  mean  velocity  may  be  taken  as  —= — ,  and  to  overcome  the  resistance  at  the 


entrances 
Then 


•6t;« 


^'-    2g     "^   2g'^2g,d' 


Or  using  in  the  expression  for  friction,  the  coefficient  0, 


;i =-0174v3+ -0078179+ 


=  •025r2  + 


4fa« 


C^d 


I  . 


If  -  is  greater  than  300  the  head  lost  dae  to  friction  is  generally  great  compared 


with  the  other  quantities,  and  these  may  be  neglected. 
iflv^  _  4Zt7« 

~C^,d* 
C      /dh 


Then 


h  =  - 


and 


"2 


As  the  velocity  is  not  known,  the  coefficient  C  cannot  be  obtained  from  the 
table,  but  an  approximate  value  can  be  assumed,  or  Darcy*s  value 


C  =  394 


y 


12d  +  l 


for  clean  pipes, 


C=278  a/toT-i  *^  *^®  P^P®  "  ^*y» 


and 

can  be  talcen. 

An  approximation  to  v — which  in  many  cases  will  be  sufficiently  near  or  will  be 
as  near  probably  as  the  coefficient  can  be  known — ^is  thus  obtained.  From  the 
table  a  value  of  C  for  this  velocity  can  be  taken  and  a  nearer  approximation  to 
V  determined. 


Then 


Q=^dM;. 


The  velocity  can  be  deduced  directly  from  the  logarithmic  formula  A=^^, 
provided  y  and  n  are  known  for  the  pipe. 


FLOW  THROUGH  PIPES  151 

The  bydxmnlic  gradient  is  EF. 

At  any  point  C  distant  x  from  A  the  pressore  head  -  is  eqaal  to  the  distance 
rtween  the  centre  of  the  pipe  and  the  hydranlio  gradient.    The  pressure  head 

Lftt  inside  the  end  A  of  the  pipe  is  Aa kz—  «  ^^^  ^^  ^^®  ^^^  ^  ^^^  pressure  head 

last  be  eqaal  to  Ab.  The  hesd  lost  due  to  friction  is  h,  which,  neglecting  the 
oiall  qnantity  — ^ —  ,  is  eqoal  to  the  difference  of  level  of  the  water  in  the  two 

inks. 

Example  1.  A  pipe  3  inehes  diameter  200  ft.  long  connects  two  tanks,  the 
ifferenoe  of  lerel  of  the  water  in  which  is  10  feet,  and  the  pressure  is  atmospheric. 
find  the  discharge  assuming  the  pipe  dirty. 

Using  Darey'8  coefficient 

17  =  278  ^y^  ^/SA^A"=69•5^/,i^ 

=3'8dft.  per  sec. 

For  a  pipe  3  inches  diameter,  and  this  velocity,  C  from  the  table  is  about  69,  so 

iiat  the  approximation  is  sufficiently  near. 

^  ^                                            ,.     -00064^1^/ 
Taking  h= -^,^ , 

r=3-88  ft.  per  sec, 

.    oooei?*! 

pves  v=d'85  ft.  per  sec. 

Example  2.  A  pipA  18  inches  diameter  brings  water  from  a  reservoir  100  feet 
&bove  datum.  The  total  length  of  the  pipe  is  15,000  feet  and  the  last  5000  feet 
^re  at  the  datum  level.  For  tbis  5000  feet  the  water  is  drawn  off  by  service  pipes  at 
me  Quifonn  rate  of  20  cubic  feet  per  minute,  per  500  feet  length.  Find  the  pressure 
&t  the  end  of  the  pipe. 

The  total  quantity  of  flow  per  minute  is 

^     6000x20    ^^      u  .r    . 

0  = =200  cubic  feet  per  minute. 

Area  of  the  pipe  is  1*767  sq.  feet. 
The  velocity  in  the  first  10,000  feet  iw,  therefore, 
200 

The  head  lost  due  to  friction  in  this  length,  is 

4./^0^.2;888a 

2i/.l-5 

In  the  last  5000  feet  of  the  pipe  the  velocity  varies  uniformly.    At  a  distance 

1*888^ 
X  feet  from  the  end  of  the  pipe  the  velocity  is    ^^  . 

In  a  length  dx  the  head  lost  due  to  friction  is 

4./.  l'888».j'dg 
2^.1-5.5000»    ' 
and  the  total  loss  by  friction  is 

.^/ll^??.      /"^   .^_4/.  (l-888)«  6000 

^~2^.1'5.6000»jo  2^.1-5     '    3     * 

The  total  head  lost  due  to  friction  in  the  whole  pipe  is,  therefore, 

H=;r-^.  1-8882  (10,000 +  A«yui). 


152  HYDRAULICS 

Taking  /  as  *0082,  H  =  14*3  feet. 

Neglecting  the  velocity  head  and  the  loss  of  head  at  entrance,  the  pressure  head 
at  the  end  of  the  pipe  is  (100  -  H)  feet =85-7  feet 

Problem  2.    Diameter  of  pipe  to  give  a  given  discharge. 

Beqoired  the  diameter  of  a  pipe  of  length  I  feet  which  will  discharge  Q  onbie  feet 
per  second  between  the  two  reservoirs  of  the  last  problem. 
Let  V  be  the  mean  velocity  and  d  the  diameter  of  the  pipe. 

Then  v=^-^    (1), 

and  ft=-025t>»+^. 


Therefore, 

^/W     A-  /  .nos_i 

Squaring  and  transposing, 

^g    0-0406.  (yd 

If  Hs  long  compared  with  d, 


h 


.(2). 


A- vs. 


(8). 


Since  v  and  d  are  unknown  G  is  unknown,  and  a  value  for  C  must  be  pro- 
visionally assumed. 

Assume  G  is  100  for  a  new  pipe  and  80  for  an  old  pipe,  and  solve  equation  (3) 
for  d. 

From  (1)  find  v,  and  from  the  tables  find  the  value  of  G  corresponding  to  the 
values  of  d  and  v  thus  determined. 

If  G  differs  much  from  the  assumed  value,  reoaloultfte  d  and  v  using  this  aeoond 
value  of  C,  and  from  the  tables  find  a  third  value  for  G.  This  will  generally  be 
found  to  be  sufficiently  near  to  the  second  value  to  make  it  unnecessary  to  oaloolate 
d  and  v  a  third  time. 

The  approximation,  assuming  the  values  of  G  in  the  tables  are  correct,  can  be 
taken  to  any  degree  of  accuracy,  but  as  the  values  of  G  are  uncertain  it  will  not  as 
a  rule  be  necessary  to  calculate  more  than  two  values  of  d. 

Logarithmic  formula.    If  the  formula  h  =  ^^^  be  used,  d  can  be  found  direct, 

from 

jp  log  d=n  log  V +log7+log  £ -log  ^. 

Example  3.  Find  the  diameter  of  a  steel  riveted  pipe,  which  will  discharge 
14  cubic  feet  per  second,  the  loss  of  head  by  friction  being  2  feet  per  mile.  It  is 
assumed  that  the  pipe  has  become  dirty  and  that  provisionally  G  =  110. 

From  equation  (3) 


^    2-55.14        /6280 

^=^[Io-V^-' 


or  ^^  log  <i= log  16-63, 

therefore  d = 3-08  feet. 

For  a  thirty-eight  inch  pipe  Euichling  found  G  to  be  113. 

The  assumption  that  G  is  110  is  nearly  correct  and  the  diameter  may  be  taken 
as  37  inches. 

Using  the  logarithmic  formula 

^    •00045ri-»Z 


FLOW  THROUGH  PIPES 


153 


od  sabetitating  for  v  the  yalne  --£- 

Tom  whieh 

5-15  log  d  ==  log -000i5  - 1-95  log  0-7854  + 1*95  log  14  +  log  2640, 
ind  <i=d-07  feet. 

Short  pipe.     If  the  pipe  Ib  short  so  that  the  velooity  head  and  the  head  lost  at 
Atruftee  are  not  negligible  oompared  with  the  loss  due  to  friction,  the  equation 

•0406Q«d  _  6-5/Q* 
^  h        ^    C*h    ' 

rhen  a  Talne  is  given  to  C,  can  be  soWed  graphically  by  plotting  two  carves 


LZld 


yi= 


_-0406<y 


.d  + 


6-5«Q« 


h  '      Ch    ' 

The  point  of  intersection  of  the  two  corves  will  give  the 
HaxaetesT  d. 

It  is  however  easier  to  solve  by  approximation  in  the 
Dlloving  manner. 

Neglect  the  term  in  d  and  soNe  as  for  a  long  pipe. 

Chooee  a  new  value  for  C  corresponding  to  this  ap- 
roximate  diameter,  and  the  velocity  corresponding  to  it, 
nd  then  plot  three  points  on  the  cnrve  y=d^t  choosing 
mines  of  d  which  are  nearly  eqnal  to  the  caloolated  value 
f  d,  and  two  points  of  the  straight  line 


•0406QSd 
yi= r^  + 


6-6/Q* 


Fig.  100. 


•6 


The  enrve  y=(^  between  the  three  points  can  easily 
e  drawn,  as  in  Fig.  100,  and  where  the  straight  line  cuts 
be  cnr-re,  gives  the  required  diameter. 

KxamepU  4.  One  hundred  and  twenty  cubic  feet  of  water  are  to  be  taken 
«T  minate  from  a  tank  through  a  cast-iron  pipe  100  feet  long,  having  a  square- 
diged  entrance.    The  total  head  is  10  feet.     Find  the  diameter  of  the  pipe. 

Neglecting  the  term  in  d  and  assuming  G  to  be  100, 

^=6M?4.  =  -026. 


nd 


Therefore 


100.100. 
(i=:  4819  feet. 
2 

j(-4819)« 


10" 


=  10-9  ft.  per  sec. 


From  Table  XIl,  the  value  of  C  is  seen  to  be  about  106  for  these  values  of 
and  r. 
A  second  value  for  d"  is 

^,^6-5.100.4 


.10 


=  •0233, 


106* 
rom  which  d= -476'. 

The  schedule  shows  the  values  of  d'  and  y  for  values  of  d  not  very  different 
rom  the  calculated  value,  and  taking  C  as  i06. 

d  -4  -5  -6 

d»  -01024  08125  0776 

yi  -0297  0329 

The  line  and  curve  plotted  in  Fig.  100,  from  this  schedule,  intersect  at|>  for  which 

d= -498  feet. 


154  HYDRAULICS 

It  is  seen  therefore  that  taking  106  as  the  Talae  of  C,  oeglectiDg  the  term  in  i, 
makes  an  error  of  -022'  or  -264". 

This  problem  shows  that  when  the  ratio  -z  is  about  200,  and  the  virtual  slope  k 

even  as  great  as  ^,  for  all  practical  purposes,  the  friction  head  only  need  be  oon- 

sidered.    For  smaller  values  of  the  ratio  -r  the  quantity  *025t^  may  become  in- 

portant,  but  to  what  extent  will  depend  upon  the  slope  of  the  hydraulic  gnuiieol. 

The  logarithmic  formula  may  be  used  for  short  pipes  but  it  is  a  little  man 
cumbersome. 

Using  the  logarithmic  formula  to  express  the  loss  of  head  for  short  pipes  witt 
square-edged  entrance, 

•025Q»  7.Q».t 

or  d*»-»^i«  -  •0406Q2d«*-«^=  V^t'^  • 

When  suitable  values  are  given  to  y  and  n,  this  can  be  solved  by  plotting  tin 
two  curves 

and  j,^=.0406(yd*»-aT5+l:^^  ^ 

the  intersection  of  the  two  curves  giving  the  required  value  of  d. 

Problem  3.    To  find  what  the  discharge  between  the  reservoirs  of  problem  (1) 
would  be,  if  for  a  given  distance  l^  the  pipe         , 
of  diameter  d  is  divided  into  two  branches         I  i  • 

laid  side  by  side  having  diameters  d,  and  dj,         k—    J,^  — >W L    — H 

Fig.  101.  I  ^        ii_         t  ! 

Assume  all  the  head  is  lost  in  friction.  A^  ^       fi^       cL         \C. 

Let  Qi  be  the  discharge  in  cubic  feet.  ^ *;         "^  /^        * 

Then,  since  both  the  branches  BC  and  BD         , j ^  V        t 

are  connected  at  B  and  to  the  same  reservoir,         j  ^^       ^         |  D 

the  head  lost  in  friction  must  be  the  same  in         |  j 

BC  as  in  BD,  and  if  there  were  any  number         I*  ^  *i 

of  branches  connected  at  B  the  head  lost  in  Yia,  101 

them  all  would  be  the  same. 

The  case  is  analogous  to  that  of  a  conductor  joining  two  points  between  iHikh 
a  definite  difference  of  potential  is  maintained,  the  conductor  being  divided  between 
the  points  into  several  circuits  in  parallel. 

The  total  head  lost  between  the  reservoirs  is,  therefore,  the  head  lost  in  AB 
together  with  the  head  lost  in  any  one  of  the  branches. 

Let  17  be  the  velocity  in  AB,  v^  in  BC  and  Vj  in  BD. 

Then  vd^^v^d^^  +  v^^ (1), 

and  the  difference  of  level  between  the  reservoirs 


4?3r2     4/,V 
C*d  "^Ci^di  ' 

I  the  same  as  i 
iqual  to  Cg 


C*d  ^  Ci*di <^'- 

And  since  the  head  lost  in  BC  is  the  same  as  in  BD,  therefore, 

\  C,»d,    C,«d, (')• 

If  proTirionally  Ok  be  taken  as  equal  to  C,, 


FLOW  THROUGH  PIPES 


155 


ThaeCore, 


d,*+ 


'-'■A 


.(4). 


Fitni  (2),  V  ean  be  found  by  sabstitatiiig  for  v^  Arom  (4),  and  thus  Q  can 
It  dBiarmiiied. 

If  AB,  BC,  mnd  CD  are  of  the  same  diameter  and  2^  is  eqaal  to  2,,  then 

ProfrinR  4.  Pipes  eonneeting  three  reservoin.  As  in  Fig.  102,  let  three  pipes 
AB,  EC,  and  BD,  connect  three  reservoirs  A,  C,  D,  the  level  of  the  water  in  each 
•f  which  remains  constant. 

Lei  V,,  V,,  and  v,  be  the  velocities  in  AB,  BG,  and  BD  respectively,  Q,,  Qs, 
md  Q,  Uie  qoantities  flowing  along  these  pipes  in  cubic  feet  per  sec.,  Z] ,  l^,  and  ^ 
|h0  Vngtha  of  the  pipes,  and  d^ ,  d,  and  d,  their  diameters. 


Fig.  105^. 

Jjgt  x^,  r,,  and  £,  be  the  heights  of  the  surfaces  of  the  water  in  the  reservoirs, 
lad  X,  the  height  of  the  junction  B  above  some  datum. 
Liet  h^  be  the  pressure  head  at  B. 

/^ffgyiTTM*  all  losses,  other  than  those  due  to  friction  in  the  pipes,  to  be  negligible. 
Xhe  head  lost  due  to  friction  for  the  pipe  AB  is 


tud  for  tlie  pipe  BC, 


.(1), 

.(2), 


ihe  upper  or  lower  signs  being  taken,  according  as  to  whether  the  flow  is  from,  or 
Uywards,  the  reservoir  G. 

For  the  pipe  BD  the  head  lost  is 


(3). 


Cs'di"^'"*"   ""''' 

Sinee  the  flow  from  A  and  G  must  equal  the  flow  into  D,  or  else  the  flow 
from  A  most  equal  the  quantity  entering  G  and  D,  therefore, 

or  Vi'±r2^'»=rA' (4). 

There  are  foor  equations,  from  which  four  unknowns  may  be  found,  if  it  is 
farther  known  which  sign  to  take  in  equations  (2)  and  (4).  There  are  two  cases  to 
eooaider. 


156  HYDRAULICS 

Ccue  (a).  Given  the  levels  of  the  surfaces  of  the  water  in  the  reflerfoin  aai 
of  the  junction  B,  and  the  lengths  and  diameters  of  the  pipes,  to  find  the  quaali^ 
flowing  along  each  of  the  pipes. 

To  solve  this  problem,  it  is  first  necessary  to  obtain  by  trial,  whether  water  floH 
to,  or  from,  the  reservoir  C. 

First  assume  there  is  no  flow  along  the  pipe  BC,  that  is,  the  pressure  bead  \  al 
B  is  equal  Xo  z^-Zq. 

Q 

Then  from  (1),  substituting  for  v^  its  value  -^^f 

<J.=t/^ » 

from  which  an  approximate  value  for  Qj  can  be  found.    By  solving  (3)  in  the  Mi 
way,  an  approximate  value  for  Q,,  is. 


Q, 


=T\/'^^   (^ 


If  Q,  is  found  to  be  equal  to  Qj,  the  problem  is  solved ;  but  if  Q.  is  greater  te; 
Qi ,  the  assumed  value  for  \  is  too  large,  and  if  less,  h^  is  too  smaU,  for  a  diBur~ 
tion  in  the  pressure  head  at  B  will  clearly  diminish  Q,  and  increase  Qj,  and  ^ 
also  cause  flow  to  take  place  from  the  reservoir  C  along  GB.     Increasbig  te 
pressure  head  at  B  will  decrease  Q|,  increase  Q,,  and  cause  flow  from  B  to  C. 

This  preliminary  trial  will  settle  the  question  of  sign  in  equations  (2)  snd  (A 
and  the  four  equations  may  be  solved  for  the  four  unknowns,  v^,  v^,  v^  and  \.  A 
is  better,  however,  to  proceed  by  "trial  and  error." 

The  first  trial  shows  whether  it  is  necessary  to  increase  or  diminish  h^  and : 
values  are,  therefore,  given  to  h^  until  the  calculated  values  of  v, ,  v^  and  r|  sati^^ 
equation  (4). 

Case  (&).  Given  Q^,  Q^,  Q,,  and  the  levels  of  the  surfaces  of  the  water  i& 
the  reservoirs  and  of  the  junction  B,  to  find  the  diameters  of  the  pipes. 

In  this  case,  equation  (4)  must  be  satisfied  by  the  given  data,  and,  therelbn^ 
only  three  equations  are  given  from  which  to  calculate  the  four  unknowns  6y 
d],  d^  and  Iiq.  For  a  definite  solution  a  fourth  equation  must  consequently  n 
fonnd,  from  some  other  condition.  The  further  condition  that  may  be  taken  ii 
that  the  cost  of  the  pipe  lines  shall  be  a  minimum. 

The  cost  of  pipes  is  very  nearly  proportional  to  the  product  of  the  length  tai 
diameter,  and  if,  therefore,  Udi  +  l^+l^^  is  made  a  minimum,  the  cost  dL  tbi 
pipes  will  be  as  small  as  possible. 

Differentiating,  with  respect  to  h^ ,  the  condition  for  a  minimnm  is,  that 

^dh^^^dK^^dh,       ^ 

Substituting  in  (1),  (2)  and  (3)  the  values  for  17^,  r,  and  v,, 


V 


=_2l, 
■■^' 

-dJ 
4^ 

Q, 
-d^ 


differentiating  and  substituting  in  (7) 


1 


FXOW  THBOUOH  PIPES 


157 


Pottiiic  the  ^mlaes  of  Qi,  Qa,  and  Q,  in  (1),  (2),  (8),  and  (8),  there  are  four 
mitaoM  as  before  for  four  anlmown  quantities. 

U  wUI  be  beUer  however  to  solve  by  approximation. 

Qiie  some  arbitrary  valoe  to  say  d,,  and  oalcolate  Aq  from  equation  (2). 

Then  calenlate  cL   and  dm  by  putting  h^  in  (1)  and  (8),  and  substitute  in 

t|Mli0ll(8). 

If  this  equation  ie  satisfied  the  problem  is  solved,  but  if  not,  assume  a  seoond 
^ihw  for  dL  and  try  again,  and  so  on  until  such  values  of  d,,  dL,  d.  are  obtained 
«al  (8)  is  satisfied. 

In  this,  as  in  simpler  ^stems,  the  pressure  at  any  point  in  the  pipes  ought  not 
to  hJl  below  the  atmoepherie  pressure. 

flow  through  a  pipe  of  constant  diameter  when  the  flow  i$  diminishing  at  a 
ms\f$ru  rate.    Let  I  be  the  length  of  the  pipe  and  d  its  diameter. 

Let  h  be  the  total  loss  of  head  in  the  pipe,  the  whole  loss  being  assumed  to  be 
^frietion. 

Let  Q  be  the  number  of  oobio  feet  per  second  that  enters  the  pipe  at  a  section  A, 
mi.  Q|  the  number  of  cnbie  feet  that  passes  the  section  B,  I  feet  from  A,  the 
eaanti^  Q-Qi  being  taken  from  the  pipe,  by  branches,  at  a  uniform  rate  of 


Q-Q, 

■    .   *  eoUe  feet  per  foot. 


I 

Then,  if  the  pipe  is  assumed  to  be  continued  on,  it  is  seen  from  Fig.  103,  that 
M  the  rate  of  discharge  per  foot  length  of  tlie 
I  is  kepi  oonstant,  the  whole  of  Q  will  be 
1  in  a  length  of  pipe. 

The  diaeharge  past  any  seetion,  x  feet  from 
CwiUbe 


!       lO.^ 

H 

t-^--^B 

M-                      j^.      . 

The  vekxity  at  the  seetion  is 


Fig.  103. 


4(Q-Q,)x 


Aseoming  that  in  an  element  of  length  dx  the  loss  of  head  due  to  friction  is 


nd  eabetituting  for  v^^  its  value 


-d« 


L^d» 
4 


le  lose  of  head  due  to  friction  in  the  length  I  is 

*dx 


[^       /  4Q  \»x»< 
JL-i'^ULd'y    d' 
_    7    /  4Q  Y 

If  Qj  is  zero,  /  is  equal  to  L,  and 


»{L«*^-'-(L-0*+*} 
di« 


_7_/4Q\»J_ 
n  +  l\rd^J   di-»' 
The  result  is  simplified  by  taking  for  dh  the  value 

4v^dx 


dh= 


od  asBTiming  C  constant. 
Then 


C«d 


3T«C«d» 


|M, 


r 


FLOW  THROUGH  PIPES  159 

IjbA  Y  be  the  ^eloci^  of  the  water  in  the  pipe. 

Then,  nnoe  there  is  oontinaitT  of  flow,  v  the  velooity  .with  which  the  water 

V.D" 
BMB  the  noszle  ie     ^    . 

The  bead  lost  by  friotion  in  the  pipe  is 

2g.D~     2gD^     ' 


r» 


i    Tbm  kinetie  energy  of  the  jet  per  lb.  of  flow  as  it  leaves  the  nozzle  is  ~  . 

»«*"  ^^=25  V^  V  ) W' 

fan  vfaieh  by  transposing  and  taking  the  square  root, 

/    2gT>^h    \4 

^-[D^hm)  ^'>- 

The  weight  of  water  which  flows  per  second szjtP.v.w  where  10 = the  weight  of 


.(3). 


%  oobie  foot  of  water. 

Therefore,  the  kinetie  energy  of  the  jet,  is 


t 


Thia  is  a  maximnm  when  j^=0> 
neiefore 


irir      /     2gWi    \* 


~  d (I>»+4/W*)* (2^/iD»)*  -ll^^ (2pftD»)*  (16/W»)  (D»  +  4/W*)*=0...(4), 


vlikh  D»  +  4/W*=12/W*, 

D»=8/W\ 


t-  ^=^/87! (^)- 

If  the  nozzle  id  not  circular  but  has  an  area  a,  then  since  in  the  circular  nozzle 
pf  tiie  same  area 

-d'=a, 

!      ^y  anbstituting  the  value  of  D'  from  (5)  in  (1)  it  is  at  once  seen  that,  for 
iMBmam  kinetic  energy,  the  head  lost  in  friction  is 

[  f "  •'■ 

I     UnHem  7.    Taking  the  same  data  as  in  problem  6,  to  find  the  area  of  the 

^%tm^  thai  the  momentum  of  the  issuing  jet  is  a  maximum. 
I    Dm  momentum  of  the  quantity  of  water  Q  which  flows  per  second,  as  it  leaves 
lMi«Uif  !_iE  lbs.  feet.    The  momentum  M  is,  therefore, 


9 


9    4 


lor  v*  from  equation  (1),  problem  6, 


160 


HYDRAULICS 


and 


Differentiating,  and  equating  to  zero, 

D»-4/W*=0, 
D» 

4/r 

If  the  nozzle  has  an  area  a. 


d= 


D»  =  5/^», 


and 


.=•392^: 


Ti' 


Sabstitating  for  D^  in  eqoation  (1)  it  is  seen  that  when  the  momentom  ii  a 
maximum  half  the  head  h  is  lost  in  friction. 

Problem  6  has  an  important  application,  in  determining  the  ratio  of  the  mm 
of  the  supply  pipe  to  the  orifice  supplying  water  to  a  Pelton  Wheel,  while  problem  7 
gives  the  ratio,  in  order  that  the  pressure  exerted  by  the  jet  on  a  fixed  plaa* 
perpendicular  to  the  jet  should  be  a  maximum. 

Problem  8.  Lom  of  head  due  to  friction  in  a  pipe,  the  diameter  of  wkieh  vmUi 
uniformly.  Let  the  pipe  be  of  length  I  and  its  diameter  vary  anifonnly  from  4^ 
to  d,. 

Suppose  the  sides  of  the  pipe  produced  until  they  meet  in  P,  Fig.  104. 

8.  =  ^.«.  «=^ 


Then 


r  =  ^  and  S= 


..av 


8  +  i     do do-di 

The  diameter  of  the  pipe  at  any  distance  x  from  the  small  end  ia 

~~   s     • 

The  loss  of  head  in  a  small  element  of  length  dx  is  ^^     ,  v  being  the  veloeity 
when  the  diameter  is  d. 


Fig.  104. 


If  Q  is  the  flow  in  cubic  ft  per  second 

t;  =  -^  =  i  Q 

The  total  loss  of  head  %  in  a  length  I  is 
^64Q».  dx 
ir»C»d» 
64.Q«8»dj? 
oir«C«di»(S+j;)» 
16Q«.S» 


(s*    (S+OV' 


Substituting  the  value  of  S  from  equation  (1)  the  loss  of  head  doe  to 
can  be  determined. 

Problem  9.     Pipe  line  comitting  of  a  number  ofpipei  ojf  different  dtOMetefi.    tk- 
practice  only  short  conical  pipes  are  used,  as  for  mstance  in  the  limbs  of  a  VaBtn 
meter.  | 

If  it  is  desirable  to  diminish  the  diameter  of  a  long  pipe  line,  instead  of  ndq 
a  pipe  the  diameter  of  which  varies  uniformly  with  the  length,  the  line  is  mads  ^ 
of  a  number  of  parallel  pipes  of  different  diameters  and  lengths. 


I 


FLOW  THBOUQH   PIPES 


161 


tiel  L«  l^*  t^ .««  he  tiie  leogth*  and  d^,  d^^d^.,,  the  diiuneters  retpeotiv^elj,  of 
Ibi  w^timmi  df  ibid  mpe, 

1!be  loial  loss  of  B«a^  due  lo  bielioti^  if  G  b^  aaaiimdd  oonatanti  is 

(?\d,^  d,^  d,  '-} 

Tlie  iTIiitifrtfr  4  of  the  pip«,  which,  for  ihe  mme  total  length ,  would  ghe  the 
iMne  diTJiTgn  for  the  i&me  lou  of  head  due  (o  friotiou,  can  be  found  from  the 


Tlic  leDftb  Js 
for  the 


oX  ft  pp^T  c^f  4^ostant  diameter  D,  which  wiU  give  the  lame 


ae  loe«  of  head  by  friction,  is 


■'"ii- 


+  — ^  +  — 


^PrtMam  10.  Pipe  aetinff  at  a  tiptore.  It  U  iome times  i:ieoesBAry  to  t^ke  a 
pipe  tiae  otst  aome  obetraclion^  snch  ae  a  hill,  which  neoe&Eitatea  the  pipe  rising » 
ttol  onl^  »tiOTe  the  hydraolic  gradient  as  in  Fig.  87,  but  even  above  the  origlEial 
Ifffivl  o(f  tiM  water  in  the  reservoir  from  which  the  supply  [fl  derived » 

Ii«t  It  be  BQippoaed*  as  in  Fig.  105,  that  water  ie  to  be  delivered  from  the  resefvoir 
3  to  lb»  fieaervoir  C  through  the  pipe  BAG,  which  at  the  poiat  A  rificfi  h^  feet  above 
Ihe  iev^  af  the  lorlace  of  the  water  in  the  npper  reeer voir. 


Fig.  105. 
X4»t   atm  iifftfenee  is  level  of  the  s^rfaoes  of  the  water  in  the  xeservoiFB 

I«et  *.  b«  the  prea^ni^  bead  equivBlent  to  the  atmospbcrio  prcaatire* 

To  ttlan  the  flow  in  the  pipe,  it  will  he  ncoeasary  lo  fill  it  by  a  pump  or  other 

\j^  tl  be  a^eumed  that  the  flow  ia  allowed  to  take  place  and  is  regulated  ao  that 
^  i»  ixyfitiiioo^  and  the  velocity  i?  is  aa  large  as  p<»seible. 

aei^ecting  the  velocity  heatl  and  reBistaoeee  othef  than  that  due  to  fVictionf 


(1). 


Xft  asul  il  beilif  the  length  and  diameter  of  the  pipe  reepoctively^ 

Tkm  h7draalie  gradient  is  practleaUy  the  straight  line  DE. 

Tb^cMVtieally  II  AF  i^  made  greater  than  k^^  which  is  about  34  feet^  the  pfefeure 
■t  A  \mtif^mf'^  nr^^ative  and  the  flow  will  oeaa^« 

Fcsrel>'  nnot  l»e  made  much  greater  than  25  feet. 

X«jl«''  '''»*Tn  vettKit^  poa*ibU  in  the  ruinfj  limb  AB,  to  that  tfu  preuure 

<^^§d  mi  A  t^ini'ljiiit  b^  UFO. 

Lei  e^m  ^  ^^  telomty.    Lei  the  datum  level  be  the  snr&ce  of  the  water  in  C^ 


£^  B. 


II 


162 


HYDRAUUC3 


Then 
fiai 


Therefore 


h^-^hu-t^n  — 


20.  d 


+  H  +  ^. 


H  =  ftB  +  -»B  +  Ai* 


.-s/' 


2g{h„-.h,).d 


1/«AB 


.(»). 


If  the  pre&sare  head  is  not  to  be  lees  thaa  10  f«et  of  w^ter^ 

If  i^m  la  loBB  Ihan  i>i  the  dischafge  of  tha  siphon  will  be  determined  lij  i 
Umititig  velociij,  hhi]  it  will  be  nuce^arv  to  throttle  tlie  (jipe  &t  C  by  meiuis  of  I 
valve,  Ao  tkB  to  keop  the  limb  AC  full  and  to  keep  the  *'  aiphoa  *'  from  bf^mg  bfbk«D.  1 

In  designing  Htidh  a  siphon  it  lA,  therefore,  ned&ss&iy  to  determine  whether  ihi 
flow  through  the  pipe  as  a  whole  nnder  a  bead  h^  is  greater,  or  le^a  Ibftn^  the  flow 
in  the  rising  limb  under  a  head  h^-  hi. 

If  AB  is  flhort,  or  A^  io  imall  that  v^  is  greater  than  t>,  the  head  absorbed  bf  | 
friction  in  A0  will  be 

2pd  ^ 

If  the  end  Oof  the  pipe  U  open  to  the  atmosphere  instead  of  heing  ootmeotad  I 
a  reserrotr,  the  total  head  available  will  be  h^  tustead  of  h^. 

111.  Velocity  of  flow  in  pipes. 
The  mean  velocity  of  flow  in  pipes  is  generally  about  3  f© 

per  second,  but  in  pipes  supplying  water  to  hydraulic  machme 
and  in  short  pipes,  it  may  be  a^^  high  as  10  feet  per  second. 

If  the  velocity  is  high,  the  loss  of  head  due  to  friction  in  lo 
pipes  becomes  excessive,  and  the  risk  of  broken  pipes  and  valve 
through  att-empts  to  rapidly  cheek  the  flow,  by  the  sudden  clodi 
of  valves,  or  other  causes,  is  considerahly  increased. 

On  the  other  hand,  if  the  velocity  is  too  small,  unless  the  wnti 
is  very  free  from  suspended  matter,  sediment*  tends  to  collt?ct  i 
the  lower  parts  of  the  pipe,  and  farther,  at  low  velocities  it 
probable  that  fresh  water  sponges  and  polyzoa  will  make  thdd 
abode  on  the  surface  of  the  pipe,  and  thus  diminish  it%  \ 
capacity. 

112,  Transmission   of  power   along  pipes   by  hydraoHa 
pressure. 

Power  can  be  transmitted  hydraulically  through  a  constderabll 
distance,  >vith  very  great  efficiency,  as  at  high  pressures  the 
centage  loss  due  to  friction  is  small* 

Let  water  be  delivered  into  a  pipe  of  diameter  d  feet  under  i 

head  of  H  feet,  or  pressure  of  p  lbs,  per  aq,  foot,  for  which 

n 
equivalent  head  is  H  =  -  feet. 


*  An   interesting   example   of   this   is    quoted    on   p. 
Vol.  XUT. 


82    Trnw,  JflUS^ai 


FLOW  THBOUOH  PIPES  163 

Let  the  velocity  of  flow  be  v  feet  per  second,  and  the  length  of 
the  pipe  Li  feet. 

The  head  lost  due  to  friction  is 

^       2g.d      ^^^' 

and  the  energy  per  ix)und  available  at  the  end  of  the  pipe  is, 
therefore, 

w      2gd 
The  efficiency  is 

B.-h_.     h 
H    "^    H 

2gdB.' 
The  fraction  of  the  giyen  energy  lost  is 

h 

For  a  given  pipe  the  eflSciency  increases  as  the  velocity 
diiniiiishes. 

If  /  and  L  are  supposed  to  remain  constant,  the  efficiency  is 

constant  if  jj^  is  constant,  and  since  v  is  generally  fixed  from 

other  conditions  it  may  be  supposed  constant,  and  the  efficiency 
then  increases  as  the  product  dR  increases. 

If  W  is  the  weight  of  water  per  second  passing  through  the 
jape^  the  work  put  into  the  pipe  is  W .  H  foot  lbs.  per  second,  the 
available  work  per  second  at  the  end  of  the  pipe  is  W  (H  -  fe),  and 
the  horse-iX)wer  transmitted  is 


XiX 

-        560        -  550  ^^     '"'• 

Since 

W  =  62%^d^v, 

the  horse-power 

4     550     \^     2gd  J 

=  -089»d'H(l-m). 

From  (1) 

"*°-  2gd  ' 

therefore, 

..-1      VdmB. 

and  the  horse-power 

I 

=  0-357  ^^d*H*(l-m). 

11—2 


164  HYDRAULICS 

If  p  is  the  pressure  per  sq.  incli 

TT_pl44 
^^  62-4  ' 

and  the  horse-power     =1*24  a/  jj  d*p*  (1  -  m). 

From  this  equation  if  m  is  given  and  L  is  known  the  diameter  d 
to  transmit  a  given  horse-power  can  be  found,  and  if  c2  is  known  the 
longest  length  L  that  the  loss  shall  not  be  greater  than  the  given 
fraction  m  can  be  found. 

The  cost  of  the  pipe  line  before  laying  is  proportional  to  its 
weight,  and  the  cost  of  laying  approximately  proportional  to  its 
diameter. 

If  t  is  the  thickness  of  the  pipe  in  inches  the  weight  per  foot 
length  is  3T5irdt  lbs.,  approximately. 

Assuming  the  thickness  of  the  pipe  to  be  proportional  to  the 
pressure,  i.e.  to  the  head  H, 

t  =  Jcp=JcK, 
and  the  weight  per  foot  may  therefore  be  written 

w  =  kid .  H. 

The  initial  cost  of  the  pipe  per  foot  will  then  be 
C=fefeidH  =  K.d.H, 

and  since  the  cost  of  lajdng  is  approximately  proportional  to  d, 
the  total  cost  per  foot  is 

P  =  K.d.H  +  Kid. 
And  since  the  horse-power  transmitted  is 

HP  =  -357  ^^ d*H»  (1  -m), 

for  a  given  horse-power  and  efficiency,  the  initial  cost  per  horse- 
power including  laying  will  be  a  minimum  when 


0-357  ,y^  d*H*  (1  -  m) 


Jk 

K.d.H  +  Kid 
is  a  maximum. 

In  large  works,  docks,  and  goods  yards,  the  hydraulic  trans- 
mission of  power  to  cranes,  capstans,  riveters  and  other  machines 
is  largely  used. 

A  common  pressure  at  which  water  is  supplied  from  the  pumps 
is  700  to  750  lbs.  per  sq.  inch,  but  for  special  purposes,  it  is 
sometimes  as  high  as  3000  lbs.  per  sq.  inch.  These  high  preesnres 
are,  however,  frequently  obtained  by  using  an  intensifier  (Ch,  XI) 
to  raise  the  ordinary  pressure  of  700  lbs.  to  the  pressure  required. 


FLOW  THROUGH   PIPES 


165 


The  demand  for  hydraulic  power  for  the  working  of  liftB,  etc. 

1  led  to  the  laying  down  of  a  network  of  mains  in  several  of  the 

cities  of  Grieat  Britain.    In  London  a  mean  velocity  of  4  feet 

second  is  allowed  in  the  mains  and  the  presaur^  is  750  Ihs* 

sq^.  inch,     In  later  installationSj  pressures  of   1100  lbs.  per 

inch  are  used. 

113.    The  limiting  diameter  of  cast-iron  pipes. 

The  diameter  d  for  a  cast-iron  pipe  cannot  be  made  very  large 
[  if  the  presscre  is  high. 

If  f>  is  the  safe  internal  pressure  per  sq.  inch,  and  s  the  safe 
i^3^ees  per  sq.  inch  of  the  metal^  and  r^  and  r^  the  internal  and 
external  radii  of  the  pipe^ 


p=- 


r,*  +  ri* 


For  a  pressnre  p  =  1000  lbs.  per  aq.  inch,  and  a  stress  a  of 
lbs.  per  sq.  inch,  Tt  is  5' 65  inches  when  n  is  4  inches,  or  the 
ipe  req aires  to  be  1*65  inches  thick. 

If,  therefore,  the  internal  diameter  is  greater  than  8  inches,  the 
jpe  becomefi  Teiy  thick  indeed. 

The  largest  cast-iron  pipe  used  for  this  pressure  is  between 
and  ST  internal  diameter. 

tTstng  a  nsanmam  velocity  of  5  feet  per  second,  and  a  pipe 
inches  diameter,  the  ma^mum  horse-power,  neglecting  friction, 
it  can  be  transmitted  at  1000  lbs.  per  sq,  inch  by  one  pipe  is 
„^    4418x1000x5 
^*^^'^   550^ 
-400, 

The  following  example  shows  that,  if  the  pipe  is  13,300  feet 
15  per  c^nt.  of  the  power  is  lost  and  the  maximum  power 
can  be  transmitted  with  this  length  of  pipe  is,  therefore, 
320  torse-power. 

Steel  mains  are  much  more  suitable  for  high  pressures,  as  the 

rkin^  stress  may  be  as  high  as  7  tons  per  sq,  inch.    The  greater 

ity  of  the  metal  enables  them  to  resist  shock  more  readily 

cast-iron  pipes  and  slightly  higher  velocities  can  be  used, 

A  pipe  15  inches  diameter  and  |  inch  thick  in  which  the 

is  1000  lbs.  per  sq.  inch,  and  the  velocity  5  ft,  per  second, 

to  transmit  1600  horse-power, 

E^x^tmpU.     Power  !■  InutBmitted  ulong  ^  c&st^iron  m&in  7^  mcliea  diameter  at 
A  pwsmmrt  of  1000  Iba.  p«r  ^,  meli.     The  velooity  of  the  wmUa  u  5  feel  per  aeoond. 
Fioil  ib»  tnAiiniiim  4istMio©  tbe  p&w^r  can  be  tranBmitted  so  tldat  the  effleienQy 

*  Swing's  Strength  of  Materktli, 


4 


166 


therefore 

Then 
from  which 


HTDRAULICS 

A  =  0*l5  3ca3O0 

4  X  0^0104  x_26^ 

2^  X  0-626         • 
345K&|-4x0^e25 


S4fi'  = 


L  = 


0-0104  X  100 
:^  13,900  f^t 


rest,  the  intensity  of  preasare 

13^ 


£ 


t^ 


Fig,  106, 


Pig^m 


114.    Pre&aures  on  pipe  her 

If  a  bent  pipe  contain  a  fluic 
being  the  same  in  all  directiot 
the  resultant  force  tending  to  mr 
the  pipe  in  any  direction  will 
the  pressure  pernnitarea  moltipl 
by  the  projected  area  of  the 
on  a  plane  perpendicular  to 
direction. 

If  one  end  of  a  right-angled 
elbow,  as  in  Fig.  106,  be  bolted  to 
a  pipe  full  of  wat-er  at  a  pressure  p 
pounds  per  sq.  inch  by  gauge,  and  on  the  other  end  of  the  elboir 
is  bolted  a  flat  cover,  the  tension  in  the  bolts  at  A  will  be  tie 
same  as  in  the  bolts  at  B.  The  presi^ure  on  the  cover  B  is  clearly 
'7854pcP,  d  being  the  diameter  of  the  pipe  in  inches.  If  the  elbow 
be  projected  on  to  a  vertical  plane  the  projection  of  ACB  is  dmfc^ 
the  projection  of  DEF  is  ahcfe.  The  resultant  pressure  on  the 
elbow  in  the  direction  of  the  arrow  is,  therefore,  p .  ahcd  =  '7S54jxf*. 

If  the  cover  H  is  removed,  and  water  flows  through  the  pipe 
with  a  velocity  v  feet  per  second,  the  horizontal  momentum  ol  the 
water  is  destroyed  and  there  is  an  additional  force  in  the  direction 
of  the  arrow  equal  to  '78'54irrfV, 

When  flow  is  taking  place  the  vertical  force  tending  to  lift  tbe 
elbow  or  to  shear  the  bolts  at  A  is 

If  the  elbow  is  less  than  a  right 
angle,   as  in  Fig.   108,  the    total 
tension  in  the  bolts  at  A  is 
T  =  p  (daehgc  -  aefgc) 

+  -7854ii;dVcos^, 

and  since  the  area  aehgcb  is  common  to  the  two  projected  areas, 
T  =  '1854<P(p'-pcoQe'\-wv'co&e). 


PLOW  THROUGH   PIPES 


167 


[^Consider  now  a  pipe  bent  m  shown  in  Fig.  109,  the  limbs  AA 
and  the  water  being  supposed  at  rest» 
direction  AA  ia 


FF  being  fmrallel, 

acting 


total  f  Mrce 


the  i 


P^p  {degkea - asfgcb ^ dcgKea  - aef*ge¥\ 
dh  dearly  is  equal  to  0. 


V   m^tead  of  the  fluid  being  at  rest  it  has  a  uniform 
'  ht*  pressure  must  remain  constantj  and  since  there  is  no 
of  velocity  tljere  is  no  change  of  momentum^  and  the  re- 
tant  pK?asttre  in  the  direction  paraUel  to  AA  is  still  zero. 
There  is  however  a  couple  acting  upon  the  bend  tending  to 

it  in  a  clockwise  direction. 
Let  p  and  q  be  the  centres  of  gravity  of  the  two  areas  daekgc 
1  a^fgch  respectively,  and  m  and  «  the  centres  of  gravity  of 
dWeiig'c  and  aef*gch\ 

Through  these  points  there  are  parallel  forces  acting  BM  shown 

the  arrows,  and  the  couple  iB 

M  —  K  -  m7i  -  P » pq. 

Tli^  '  •  P*pg  is  also  equal  to  the  pressure  on  the  semicircle 

Ic  II'.  i  by  the  distance  between  the  centres  of  gravity  of 

k and  efg^  and  the  couple  P' .  7nn  is  equal  to  the pmssure  on  ad*c* 
iltiplic^d  by  the  distance  between  the  centres  of  gra-^nty  of  a'd't^ 

Ufg. 

Tlu*n  the  resiUtant  couple  is  the  pressure  on  the  semicircle  efg 
Inpbed  by  the  distance  between  the  centres  of  gravity  of  efg 

u'r9. 

of  FF  atid  AA  are  on  the  same  straight  line  the 

as  the  force  J  becomes  aero, 

%  can  also  be  shown,  by  similar  reasoningj  that,  as  long  as  the 

it4?r«  at  F  and  A  are  equal,  the  velocities  at  these  aections 

therefore  equal,  and  the  two  ends  A  and  F  are  in  the  same 

ight  line,  the  force  and  the  couple  are  both  zero,  whatever  the 

of  the  pipe.    If,  therefore,  as  stated  by  Mr  Fronde,  "  the 


J 


168  HYDRAULICS 

two  ends  of  a  tortuous  pipe  are  in  the  same  straight  line,  there  is 
no  tendency  for  the  pipe  to  move." 

115.    Pressure  on  a  plate  in  a  pipe  filled  with  flowing  water. 

The  pressure  on  a  plate  in  a  pipe  filled  with  fiowing  water,  with 
its  plane  perpendicular  to  the  direction  of  fiow,  on  certain  assump- 
tions, can  be  determined. 

Let  PQ,  Fig.  110,  be  a  thin  plate  of  area  a  and  let  the  sectional 
area  of  the  pipe  be  A. 

The  stream  as  it  passes  the  edge  of 
the  plate  will  be  contracted,  and  the 
section  of  the  stream  on  a  plane  gd  will 
be  c(A-a),  c  being  some  coeflScient  of 
contraction. 

It  has  been  shown  on  page  52  that 
for  a  sharp-edged  orifice  the  coeflBcient  Fig.  no. 

of  contraction  is  about  0*625,  and  when 

part  of  the  orifice  is  fitted  with  sides  so  that  the  contraction  is 
incomplete  and  the  stream  lines  are  in  part  directed  perpendi- 
cular to  the  orifice,  the  coeflBcient  of  contraction  is  larger. 

If  a  coeflBcient  in  this  case  of  0*66  is  assumed,  it  will  probably 
be  not  far  from  the  truth. 

Let  Vi  be  the  velocity  through  the  section  gd  and  V  the  mean 
velocity  in  the  pipe. 

The  loss  of  head  due  to  sudden  enlargement  from  gd  to  ef  is 

2g      ' 
Let  the  pressures  at  the  sections  db,  gd,  ef  be  p,  p,  and  p,  pounds 
per  square  foot  respectively. 

Bemouilli's  equations  for  the  three  sections  are  then, 

w     2g    w      2g    ^^^, 

and  £..|l^E,,^,(I^* (2). 

Adding  (1)  and  (2) 

(V.-V)' 

The  whole  pressure  on  the  plate  in  the  direction  of  motion  is  then 

(V.-Y)' 


\w     wJ 


F=(p-pi).a  =  w.a.       2 


FLOW  THBOUQH    PIPES 


P  ^  4^a  5-  nearly. 


P  = 


116,    Pre&aitre  om  a  cylinder, 

1£  inBtBfid  of  a  thiii  plate  a  cylinder  be  placed  in  the  pif 
Iwitli  U«  aads  coincident  with  the  axis  of  the  pipe,  Pig<  111,  there 
fare  two  anlargements  of  the  section  of  the  water. 

As  the  stream  passes  the  up-stream  edge  of  the  cylinder,  it 

contracts  to  the  section  at  ccl,  and  then  enlarges  to  the  section 

ef.     It  again  eniargi?s  at  the  down-stream  end  of   the  cylindc 

I  ftom  the  section  a/  to  the  section  gh. 

«'  |C  .^  J9 


Fig.  111- 

l^t  ^i,  1%,  v»^  v^  be  the  velocities  at  ai,  cd,  ef  and 
lepectively,  €4  and  Vt  being  eqnah 

Betw^eti  cd  and  ^/ there  is  a  loss  of  head 

between  e/and  gh  there  is  a  loss  of 

2g      • 
The  BemouiUi's  eqnations  for  the  sections  are 

w     2g     w     2g'^' ' " 


gk  re- 


w     2g     w  2g  w     2g  ^       ' 

w  2g  w     2g         2g      ' 
Miing  (2>  and  (3), 

«?  2g  2g      ' 


(1), 

.(2), 
.(3). 


ii^Mik 


170  HTDRAtTLlCS 

If  the  coefficient  of  oontractioti  at  cd  is-e,  the  area  at  cd 


-c 


A 


Then  v^= — > v'     "v  ^-nd  i'>=^^^. 

Therefore 

atid  the  pressure  on  the  cylinder  is 


EXAMPLES, 

(1)  A  new  caBt'iroB  pipe  is  2000  ft.  loDg  and  3  ins.  diameter.  Itisbo 
discharge  50  c.  ft,  of  water  per  miniite.  Find  the  lofis  of  bead  in  tndma 
and  the  virtual  elope. 

(2)  What  is  the  head  loet  per  mile  in  a  pipe  2  ft.  diameter*  diflchitf|iiiC 
6,000,000  gallons  in  24  hours  ?    /=  -007. 

(3)  A  pipe  tig  to  r^upply  40,000  gallons  in  24  hours.  Head  of  witcf 
above  point  of  discharge  =  86  ft.  Length  of  pipe=2J  miles*  Find  iti 
diameter.    Take  C  from  Table  XII. 

(4)  A  pipe  is  12  ms.  in  diameter  and  3  rnile^  in  length.  It  oohmscH 
two  FeaerroirB  with  a  difference  of  level  of  20  ft.  Find  the  disehazga  pflf 
minute  in  a  ft.     Use  Darcy's  coefficient  for  corroded  pipes. 

{5)  A  water  main  has  a  Tirtual  slope  of  1  in  900  and  di£chaf-ge636C*(L 
per  second.     Find  the  diameter  of  the  main.     Coefficient  /  is  0*007. 

(0)  A  pipe  12  inB.  diameter  is  suddenly  enlarged  to  IB  ins.»  and  tlian  to 
24  ins.  diameter.  Each  section  of  pipe  is  100  foet  long*  Find  the  h&k  d 
head  in  friction  in  each  length,  and  the  loss  due  to  shock  at  eaeli  ealMge- 
ment.  The  discharge  is  10  c.  ft.  per  second^  and  the  coefficient  of  fricti» 
/=^'O106,     Draw,  to  scale,  tlie  hydraulic  gradient  of  the  pipe. 

(7)  Find  an  oxpresaion  for  the  relative  discharge  of  a  square^  aad  * 
circular  pipe  of  the  same  section  and  slope. 

(8)  A  pipe  is  6  ins.  diameter,  and  is  laid  for  a  quarter  mile  at  a  ^ops 
of  1  in  60:  for  another  quarter  mile  at  a  slope  of  1  in  100;  and  for  ftUM 
quarter  mile  is  level.    The  level  of  the  water  is  20  ft.  above  the  inlet  efli 
and  9  ft.  above  the  outlet  end.     Find  the  diacharge  (neglecting  all 
except  skin  friction)  and  draw  the  hydrauMc  gradient.    Mark  Uie 
in  the  pipe  at  each  quarter  mile. 

(9)  A  pipe  2000  ft.  long  discharges  Q  c.  ft.  per  second.    Find  bybo* 
much  the  discharge  would  be  increased  if  to  the  last  1000  ft*  a  second  ppl 
of  the  same  siase  were  laid  alongside  the  first  and  the  water  allowed  to 
equally  well  along  either  pipe. 


FLOW  THBOUOH  PIPES 


171 


.Ibei 


I) 


►)     A  naacrffoir,  the  level  of  wkicl)  is  50  ft,  above  datum,  diBcharges 

neeood  rmmvm  80  ft.  ^bove  datum,  through  a  12  in.  pipe,  5000  ft. 

find  the  disc^hiurge.    AIbd,  taking  the  levelB  of  tlie  pipe  at  the 

;  Rud  at  each  successive  1000  ft.,  to  be  40,  25,  12,  12, 10,  15, 

d»liiin,  wriie  down  Uie  pressure  at  each  of  tbeie  poiats,  and 

posltieii  of  the  line  of  hydraulic  gradient, 

1%  m  reqained  to  draw  off  the  water  of  a  reservoir  Uirough  a 
lOed  harixoatAllj.  Diameter  of  pipe  6  ins.  Length  40  ft*  Ef- 
lead  20  ft    Find  the  disch^^e  per  geoond. 

Given  the  data  of  Ex.  11  find  the  diBcharge^  taking  into  atccount 
oi  h#ad  if  the  pipe  ifi  not  bell -mouthed  at  either  end* 

A  pipe  4  ins.  diameter  and  100  ft.  long  discharges  ^  c.  ft*  per 
Find  the  head  expended  in  giving  velocity  of  entry^  in  overcoming 
reatstasce,  and  in  Mctdon, 

14)     BeqQired  the  diameter  of  a  pipe  having  a  fall  of  10  ft.  per  mUe, 
Mm  ol  delivering  water  at  a  velocity  of  3  ft  per  second  when  dirty. 

TaMsg  the  coefficient  /  as  0*01  (l  +  f^^)i  ^^  ^ow  much  water 

be  diaehatged  through  a  12^inch  pipe  a  mile  long,  connecting  two 
i  with  a  differeoce  ol  level  of  20  feet, 

Watetr  flows  through  a  12 -inch  pipe  liaving  a  virtual  slope  of  8  feet 
I  feel  at  a  velocity  of  8  feet  per  second. 
I  tlie  Action  per  sq.  ft.  of  surface  of  pipe  iu  lbs, 
iihc  Talne  of  /  in  the  ordinary  formula  for  flow  in  pipes. 

Find  the  relative  discharge  of  a  6 -inch  main  witli   a  slope  of 
40Ov  and  a  4  Inch  main  with  a  slope  of  1  in  50. 

A  6'inch  main  7  mUes  in  length  vrith  a  virtual  slope  of  1  in  100 
hj  4  miles  of  Sinch  main,  and  S  miles  of  4-iiich  main.    Wowld 
►  be  altered,  and,  if  so*  by  how  much  ? 

[  (i^)  Find  the  velocity  of  flow  in  a  water  main  10  miles  longi  con- 
[  Iwo  reaervoirs  with  a  dificTence  of  level  of  200  feet.  Diameter  of 
>  ifldiea.    Goef&cient  /=U^009. 

'  (10)  Ftod  the  discharge,  if  the  pipe  of  the  last  question  is  replaced  for 
I  flxnt  5  BuLas  by  a  pipe  20  inches  diameter  and  the  remainder  by  a  pipe 
idkmeter. 

Ctl)    Calculate  the  loss  of  head  per  mile  in  a  10- inch  pipe  (area  of  cross 
1 0^54  fl(|.  ftj  when  the  discliarge  is  2^  c,  ft  per  second. 

A  pipe  canaiata  6f  ^  a  mile  of  10  inch,  and  4  a  mile  of  5 -inch  pipe, 
I  oomr^ya  $|  e*  It  per  second.     State  from  the  answer  to  the  previous 
k  the  losB  of  head  in  each  section  and  sketch  a  hydraulic  gradient* 
» head  al  the  ontlet  is  5  ft. 

What  is  the  head  lost  in   friction   in  a  pipe  8  feet  diameter 
6,000,000  galioDS  in  12  hours? 

'  {Uj    A  pipe  2000  feet  long  and  8  inches  diameter  is  to  discharge  65  o.  ft. 
f  ndnnte*    What  mufit  be  the  head  of  water  f 


ITi 


1(19) 


^^whargel 


J^^b 


172 


HTDRAULICS 


(25)  A  pipe  6  mcbes  dlanaeter,  50  feet  long,  is  connected  to  the  bcitkrnti 
of  a  tank  50  feet  long  by  40  feet  wide.  The  original  head  over  the  openl 
end  of  the  pipe  is  15  feet.  Find  the  time  of  emptying  the  tank^  assmning] 
the  entrance  to  the  pipe  in  sharp -edged. 

If  /i^the  head  ovet  the  exit  of  the  pipe  at  any  moment, 
t?*     -St?"     4/t^5Q^ 
°2g^  %^2£rx05' 


from  'which,  v 

In  time  dt^  the  discharge  is 


*  1*5 +400/' 


144  l'5  +  400/ 

In  time  ct  the  Borface  falls  an  amoimt  dh. 

Therefore  0_m^^^^^^^^h 

1-5  +  400/  ^4 

Integratmg, 

^_2QQ0  (1-5  +  400/)  2  ^j^^  79QQ0(l'5  +  4OO/)  ^^ 
0-196^%  V2g 

(26)  The  internal  diameter  of  the  tubea  of  a  condenser  is  5*654  Inc 
The  tubes  are  7  feet  long  and  the  n amber  of  tubes  is  400*     The  number  < 
gallons  per  minute  flowing  through  the  condenaer  is  400.     Find  the  lo»  ot| 
head  due  to  friction  as  the  water  flowB  through  the  tubes*    /^  0*006. 

(27)  Aasundng  fluid  friction  to  vary  as  the  square  of  the  velodty,  J 
an  expression  for  the  work  done  in  rotating  a  disc  of  diameter  d  at  i 
angular  velocity  a  in  water. 

(26)     What  horse  power  can  be  convej'ed  through  a  6- in,  main  if  ihs  I 
working  presBure  of  the  water  supplied  from  the  hydraulic  power  station  tt 
700  lbs,  per  sq,  in;?    Assume  that  the  velocity  of  the  water  Is  limited  l9 
8  ft,  per  second, 

(29)  Ten  horae-power  is  to  be  transmitted  by  hydranlic  presanie  i 
distance  of  a  mile.  Find  the  diameter  of  pipe  and  pfdssm-e  required  for  li 
efficiency  of  ^  when  the  velocity  is  5  ft.  per  sec 

The  frictional  loss  is  given  by  equation 

2g      d 

(BO)    Find  the  inclination  necessary  to  produce  a  velocity  of  4|  leei  p*  I 
second  in  a  steel  water  main  31  inches  diameter,  when  roniiing  full  ifl4 
digchargiug  with  free  outlet,  using  the  formula 

■    ■0005  tJ^^ 
*"      di-»      ' 

(Bl)    The  following  values  of  the  slope  i  and  the  velocity  v  w«re 
determined  from  an  experiment  on  flow  in  a  pipe  '1296  ft  diam* 
i     -00022     '00182      '00650       ^02889       '04348    -12815       '22409 
V     206        *606        1*252        2*585        8'693      6310         8'521 


di^ii 


FLOW  THROOGH   PIPES 


173 


Delermme  k  and  n  m  the  formola 

i—ki)'^. 


AIbo  determine  T&Itiee  of  O  for  this  pipe  for  velocitieB  of  *5,  1,  3^  5  and 
1  feel  per  sec 

(S2^    The  total  length  of  the  Coolgsrdie  steel  aqueduct  is  SOTf  miles 
ftttd  the  di&nieter  30  inches.    The  discharge  per  daj  Z£iay  be  5f600|000 
The  water  is  lifted  a  total  height  of  1499  feet, 
(a)    the  head  lo^t  dtie  to  friction, 
{b}    the  total  work  done  per  minote  In  raising  the  water. 

A  pipe  2  feet  diameter  and  500  feet  long  without  bende  furnishee 

wmlier  to  a  tnrhine*    The  tnrbine  works  under  a  head  of  25  feet  and  uses 

ID  e.  fL  ol  water  per  second.    What  |)ercentage  of  work  of  the  fall  is  lost 

^A  MctioD  in  the  pipe  ? 


Find 


im 


Coiifficiaitt 


/-'007 


{'""m)^ 


C84>  Sight  thousand  gallons  an  hour  ha^e  to  be  disoharged  through 
^^ch  of  mx  nozzles,  and  the  jet  haM  to  reach  a  height  of  80  ft. 

If  the  water  supply  i&  1^  tuilee  away,  at  what  elevation  above  the 
ooj^lea  wcpnld  you  place  the  required  reseryoir,  and  what  would  you 
make  the  diameter  of  the  supply  main  ? 

QtTB  the  dimensions  of  the  refiervoir  you  would  provide  to  keep  a 
OODStant  supply  for  six  hours.     Loud.  Uu.  1903. 

(85  ^  The  pipes  laid  to  connect  tlie  Vymwy  dam  with  Liverpool  are 
4S  inebea  diameter.  How  much  water  will  snch  a  pipe  snpply  in  gallons 
^m  diiBBi  if  the  slope  of  the  pipe  is  4^  feet  per  mile  9 

At  0Q«  point  on  the  line  of  pipes  the  gradient  m  6|  feet  per  mile«  and  the 
fipe  diam^er  ia  reduced  to  89  inches;  is  thJB  a  reasonable  reduction  in  the 
^BieBsioii  of  the  croas  section  ?    Loud.  Un.  1905. 

"*'ater  under  a  head  of  60  feet  is   diachajged  through   a  pipe 

I  meter  and  150  fe^t  long,  and  then  through  a  nozzle  the  area  of 

-  -tenth  tlie  area  of  the  pipe.    Neglecting  all  losaes  except  friction, 

locity  witli  which  tbe  water  leaves  the  nozzle* 

^    Two  rectangular  tanks  each  50  feet  long  and  50  feet  broad  are 

ftmnected  by  a  horiiEuntal  pipe  4  inches  diameter,  1000  feet  long.     The 

Wd  over  the  centre  of  tlie  pipe  at  one  tank  is  12  feet,  and  over  the  other 

i  ke%  when  flow  commences. 

Determine  tlie  time  taken  for  the  water  in  the  two  tanks  to  come  to  the 
^am  level.    Aianme  the  coefficient  €  to  be  constant  and  equal  to  90. 

(885  Two  reaervoirs  are  cotinectcd  by  a  pipe  1  mile  long  and  10'' 
Kuieter;    Ih^e    difference    in    the    water    surface    levels    being    25    ft. 

B^terminic  the  flow  through  the  pipe  in  gallons  per  hour  and  find  by 
modi  tlia  discharge  would  be  increased  if  for  the  last  2000  ft.  a  second 
ol  lOT  diameter  is  laid  alongside  the  first.     Loud.  Un.  1905. 

(99^     A  pipe  18^  diameter  leads  from  a  reservoir,  SflO  ft.  above  the  ; 
od  ia  continued  for  a  lengtli  of  5000  ft.  at  the  datum,  the  length' 
iSjOOO  ft    For  the  Last  5000  ft.  of  its  length  water  is  drawn  off  by 


174 


HYDRAULICS 


fiendoe  pipes  at  the  rate  of  10  c.  it,  per  nun*  per  500  fL  tmiformly,    FmiM 
the  pressure  at  the  end  of  the  pipe.    Loud.  Un.  1906.  I 

(40)  B50  horse -power  ia  to  be  transmitted  by  hydraulic  preasoxe  m 
distance  of  1^  miles.  I 

Find  the  nmnber  of  6  ins.  diameter  pipes  and  the  preadoi^e  required  fDcl 
an  efficiency  of  92  per  cent,    /—Ol*    Take  t*  as  3  ft,  per  see  I 

(41)  Find  the  loaa  of  head  due  to  friction  in  a  water  maiD  L  feet  Iohh 
whicJi   receives  Q  cubic  feet  per  second  at  the  inlet  end  and  dahTMl 

P  cubic  feet  to  branch  mains  for  each  foot  of  its  length*  I 

What  is  the  form  of  the  hydraulic  gradient  ?  I 

(42)  A  reservoir  A  aupphes  water  to  two  other  reservoirs  B  and  CLl 
The  difference  of  level  between  the  surfaces  of  A  and  B  is  75  feet,  anil 
between  A  and  C  97-5  feet,  A  common  8 -inch  cast- iron  main  supplies  foci 
the  hrst  850  feet  to  a  point  T>*  A  6'inch  main  of  length  1400  feet  m  tbeal 
earned  on  in  the  same  straight  line  to  B^  ajid  a  5 -inch  main  of  lengtlil 
630  feet  goes  to  C*  The  entrance  to  the  8«iiich  main  is  bell-tnouthed«  and! 
losses  at  pipe  exits  to  the  reservoirs  and  at  the  junction  may  be  neglaoleU 
Find  tlie  quantity  discharged  per  minnte  into  the  reservoirs  B  and  GJ 
Take  the  coefficient  of  friction  (/)  as  *01.     Lond,  0n.  1907.  1 

(43)  Describe  a  method  of  finding  tlie  '*  loss  of  head  "  in  a  pipe  due  tol 
the  hydraulic  resistances^  and  sta,te  how  you  would  proceed  to  find  IbJ 
loss  as  a  function  of  the  velocity.  I 

(44)  A  pipe*  I  feet  long  and  D  feet  in  diameter,  leads  water  from  ■ 
tank  to  a  nozzle  who^e  diameter  is  d,  and  whose  centre  is  h  feet  bekM 
the  level  of  water  in  the  tank*  The  jet  impinges  on  a  fixed  plaal 
surface.  Assuming  that  the  loss  of  head  due  to  hydraulic  resistanoe  m 
given  by  M 

show  that  the  preasure  os  the  Boriace  in  a  maxiiiiniii  wheD  I 

^  m-  1 

(45)  Find  the  flow  through  a  sewer  consisting  of  a  cast-iron  ptpJ 
12  inches  diameter,  and  having  a  fall  of  8  feet  per  mllet  when  dLschafgufl 
full  bore,    c  =  100.  1 

(4d)  A  pipe  9  inches  diameter  and  one  mile  long  alop^  for  the  Snfl 
half  mile  at  1  in  200  and  for  the  aeoond  half  mite  at  1  in  100.  The  pr€ii^ 
head  at  the  higher  end  is  found  to  be  40  feet  of  water  and  at  the  Iohm 
20  feet.  I 

Find  the  velocity  and  flow  through  the  pipe.  I 

Draw  the  hjrdrauhc  gradient  and  find  the  pressure  in  feet  at  500  ktm 
and  lOOO  feet  from  the  higher  end.  I 

(47)  A  town  of  250,000  inhabitants  is  to  be  suppUed  with  water.  EmM 
the  daily  supply  of  32  gallons  per  head  is  to  be  delivered  in  8  hours*  I 

The  service  reservoir  is  two  miles  from  the  town^  and  a  fall  of  10  §&m 
per  mile  can  be  allowed  in  the  pipe.  M 

What  must  be  the  size  of  the  pipe?     C  =  90,  I 


PLOW  THROtJOH   PIPES 


175 


(4$)  A  w&iar  pipe  is  lo  be  laid  in  a  street  800  jardfl  long  with  houseB 
Mbolli  Biidea  of  ibe  street  of  24  feet  frontage.  The  average  number  ol 
iiilml)itsiits  ai  aacb  honse  is  6,  and  the  aTemge  eoneumption  of  water  far 
eacli  peCBcm  ia  80  gallon  b  in  8  brs.  On  the  assumption  that  the  pipe  is  laid 
In  loar  equal  lengths  of  200  yards  and  has  a  uniform  dlope  of  j^,  and  that 
Hie  wbo)e  of  tha  water  flows  through  the  first  lengthy  three-fourtlis  through 
Ibe  Mooml*  one  half  through  the  third  and  ona  quarter  through  the  fourth, 
lad  Uwl  Ibe  Talne  of  G  is  90  lor  the  whole  pipe^  lind  the  diameters  of  the 
ter  parte  of  the  pipe. 

(49)  A  pipe  3  miles  long  has  a  nniform  slope  of  20  feet  per  mile,  and  is 
19  inclMM  diameter  for  the  first  mile,  30  inches  for  the  second  and  21 
faicbes  for  the  third.  The  pressure  heads  at  the  liigher  and  lower  ends  of 
fbe  pipe  are  100  feet  and  40  feet  respectively.  Find  the  discharge  through 
th«^  [Hprn  and  determine  the  pressure  heads  at  the  comman cement  of  the 
3ij  inchem  diameter  pipe,  and  abo  of  the  21  inches  diameter  pipe, 

(SOi  The  difference  of  lerel  of  two  i^eserroirs  ten  miJes  apart  in  60  feet, 
A  pq^  Is  Tnquired  to  connect  tliem  and  to  convey  45,000  gaUona  of  water 
pm  boor  fr«>m  the  higher  to  the  lower  reservoir. 

Find  the  necefisary  diameter  of  the  pipe,  and  sketch  the  hydraulic 
gtafdienU  aMsanusg/^OOl. 

Tlw  middle  part  of  the  pii>e  is  120  feet  below  the  surface  of  the  upper 
reaerrotr.  Osiculate  the  pressure  head  in  the  pipe  at  a  point  midway 
betw^iSl  tiie  two  reservoirs. 


f51)  Sonne  hydraulic  machines  are  served  with  water  under  pressure 
hf  a  pipe  1000  feet  long,  the  pressure  at  the  machines  being  600  lbs.  per 
ai^oara  indi.  The  horBe^power  developed  by  the  machine  is  300  and  ihe 
fricdoa  iione-powes-  in  the  pipes  120.    Find  the  necessary  diameter  of  the 

ptp&^  taking  the  loss  of  head  in  feet  as  0*03  li^^  and   *i3  lb.  per  square 

laeli  aa  ac^aiTiilefit  lo  1  foot  head.  Also  determine  the  pressure  at  which 
Aa  w«t6r  ia  deliirered  by  the  pump. 

Wliat  ts  the  ma^mum  horse ^power  at  which  it  would  be  possible  to 
work  the  machines,  the  pump  pressure  remaining  the  same  ?  Lond.  Un. 
1906* 

(53}  Discnsa  Beynolds'  work  on  the  critical  velocity  and  on  a  general 
law  oi  resistance,  describing  tJie  experiments  appai-atus,  and  showing  the 
ooB&edioii  with  the  experiments  of  Poisemlle  and  D'Arcy.    Load.  Un, 

(58)    In  a  condenser,  the  water  entars  through  a  pipe  (section  A)  at  the 

ol  the  lower  water  head,  passes  through  the  lower  neat  of  tubes, 

thzerii^  the  uppernestof  tubes  into  the  upper  water  head  (section  B). 

I  '9m  teetional  areas  at  sections  A  and  B  are  0-196  and  0^95  sq.  ft.  respec- 

#i«ly;  the  total  sectional  area  ol  flow  of  the  tubes  forming  the  lower  nest 

Ji  C^S14  81]*  ft^  and  of  the  npper  nest  0  75  sq.  ft.,  the  number  of  tnbea  being 

feoipecltrely  35a  and  826.    The  length  of  all  the  tubes  is  6  feet  2  incheSp 

IVIifiii  the  Tolume  of  the  circulating  water  was  1^21  c,  ft.  per  sec.,  the 

Ered  dtfferance  ol  preasure  head  (by  gauges)  at  A  and  B  was  6  5  feet. 
Ihe  total  actual  head  necessary  to  overcome  frictional  resistancei  and 


176 


BYDaATJLICa 


the  coelScieiit  of  hydraulic  reciBtanoe  referred  to  A.    If  i^3^  oodEdest  d 
Motion  (4/)  for  the  tubed  is  t&ken  to  be  *015t  find  the  coefficient  of  bj^ilttiiifi . 
reflifitanee  for  the  tubes  alone,  and  compare  with  the  actual  l  niiiiTipcit  ] 
Lend.  Un.  1906.     iCr  =  head  lost  dimded  bp  veL  he^  at  A.) 

(54)  An  open  stream,  which  19  discharging  20  c.  ft.  of  water  M^ 
BeGood  ifl  paeied  under  a  road  by  bl  siphon  of  smooth  stoneware  pipev  |H 
eectioii  of  the  siphon  being  cylindrical,  and  2  feet  in  diameter.  When  toVi 
stream  enters  ttiis  siphon,  the  siphon  descends  vertioally  12  feet,  H 
then  has  a  horizontal  length  of  100  feet«  and  again  rises  12  feet^  If  all  Hm  I 
bends  are  sharp  right-angled  bends,  what  is  the  total  Iohb  of  head  in  ibi  I 
tunnel  due  to  the  bends  and  to  the  friction  ?     G  =  117.     Loud.  Un.  190?.       I 

(50)  It  has  been  shown  on  page  159  that  when  the  kinetic  ene^y  of  jfl 
jet  issuing  from  a  nozzle  on  a  long  pipe  line  is  a  maximum^  H 

^      8/L'  ^ 

Hence  find  the  minimum  diameter  of  a  pipe  that  will  supply  a  ¥^im  I 

Wheel  of  70  per  cent,  efficiency  and  500  brake  horse-power,  the  airaikUJ 

head  being  600  feet  and  tlie  length  of  pipe  3  miles.  ^M 

(56)  A  fire  engine  supphes  water  at  a  pressure  of  40  lbs,  per  sqo^H 
inch  by  gauge,  and  at  a  velocity  of  6  feet  per  second  into  a  pipe  3  indi^l 
diameter.  The  pipe  is  led  a  distance  of  100  feet  to  a  nozzle  25  feet  aba^| 
the  pump.  If  the  coeJScient  /  (of  friction)  in  the  pipe  be  '01,  and  the  ad^l 
lift  of  the  jet  is  |  of  that  due  to  the  velocity  of  effiux,  find  the  actnal  tiei^| 
to  which  the  jet  will  rise,  and  the  diameter  of  the  nozzle  to  eati^  ^| 
conditions  of  the  problem.  ^H 

(57)  Obtain  expressions  (a)  for  the  head  lost  by  friction  (e^kpre^ej^l 
feet  of  gas)  in  a  main  of  given  diameter,  when  the  main  is  horizontaL  i^| 
when  the  variations  of  pressure  are  not  great  enough  to  cause  any  importtiH 
change  of  volume,  and  (b)  for  the  discharge  in  cubic  feet  per  saoond.        ^ 

Apply  year  resnlte  to  the  following  example:—  I 

The  main  is  16  Inches  diameter,  the  length  of  the  main  is  300  jrwfc  j 
the  density  of  the  gas  is  0'56  (that  of  air^  l),and  the  difference  of  presMiii] 
at  the  tw^o  ends  of  the  pipe  is  |  inch  of  water  j  find:^  I 

(a)     The  head  lost  in  feet  of  gas,  " 

(h)     The  discharge  of  gas  per  hour  in  cubic  feet. 
Weight  of  1  cubic  foot  of  air=0'08  lb,;  weight  of  1  cubic  foot  of  waier 
J^ba.;  coefficient  /  (of  friction)  for  the  gas  against  the  waUs  of  the  pipe 
J*^"^  Loud,  Un.  1905.  J 

on  f    t   '^^  ^^^ '  substitute  for  w  the  weight  of  cubic  foot  of  gas,)  I 

Find  thetee  reservoirs  A,  B  and  C  are  connected  by  a  pipe  le«di4g 
Draw  the  a  junction  box  P  situated  450'  above  datum, 
and  1000  feet  ^  of  the  pipes  are  respectively  10^000' ♦  5000' and  6000"  and  ilm 
(47)     A  tow*^^  water  sorface  in  A,  B  and  C  are  SOff,  600'  and  200'  &bot9 

^j^^^^^^^magnitude  and  indicate  the  direction  of  mean  Telocity  itt  ^ 
permilecanbeaD€^^=>10*>*^'*^*'  ^^  pip^s  being  aU  the  same 
WliatmustbethP"'^^' 


FLOW  THROUGH  PIPES  177 

(00)  A  pipe  8*  fiT  diameter  bends  through  45  degrees  on  a  radius  of 
7  IS  leek  Deiennine  the  dieplaGing  foroe  in  the  direction  of  the  radial  line 
\  MinnHnc  the  aii|^  between  the  two  limbs  of  the  pipe,  when  the  head  of 
P    wiler  in  the  pipe  is  260  feet 

^         Show  alao  that,  if  a  nniformly  distributed  pressure  be  applied  in  the 
/  plftBe  of  the  oentre  lines  of  the  pipe,  normally  to  the  pipe  on  its  outer 
'^    ^w**^,  and  of  intensity 
f  _  49M«   ,^ 

par  unit  length,  the  bend  is  in  equilibrium* 
B»Tadins  of  bend  in  feet 
d— diameter  of  pipe. 
h»head  of  water  in  the  pipe. 


12 


CHAPTER  VL 


FLOW  IK  OPEN  CHANNELS, 


117,  Variety  of  the  forms  of  channels. 
The  study  of  the  flow  of  water  in  open  cliannelB  is  much  more 

complicated  than  in  the  case   of  closed   pipeSj   because  of  the 
infinite  variety  of  the  forms  of  the  channels  and  of  tlie  dii&reit 
degrees  of  ro ugliness  of  the  wetted  BurfaceSj  varying^  as  they  do, 
from  channels  lined  with  smooth  boarda  or  cementj  to  the  irregnkr  , 
beds  of  rivers  and  the  rough,  pebble  or  rock  strewn,  moiEidiift| 
stream. 

Attempts  have  been  made  to  find  formulae  wliich  are  applicalJft  I 
to  any  one  of  these  very  variable  conditions,  bat  as  in  the  case  d  I 
pipes,  the  logarithmic  formulae  vary  with  the  roughness  of  ttel 
pipe,  so  in  this  case  the  formulae  for  smooth  regular  shaped  chanBeU  I 
cannot  with  any  degree  of  assurance  be  applied  to  the  CBlculahon  I 
of  the  flow  in  the  irregular  natural  streams. 

118.  Steady  motion  in  imifomi  channels. 

The  experimental  study  of  the  distribution  of  velocitiefi 
water  flowing  in  open  channels  reveals  the  fact  that,  as  in  tlil'l 
case  of  pipesj  the  particles  of  water  at  different  points  in  a  crowl 
section  of  the  stream  may  have  very  difFerent  velocities,  and  the] 
direction  of  flow  is  not  always  actually  in  the  direction  of  the  flow  [ 
of  the  stream. 

The  particles  of  water  have  a  sinuous  motion,  and  at  any  jkxuh 
it  is  probable  that  the  condition  of  flow  is  continually  changinfJ 
In  a  channel  of  uniform  section  and  slope,  and  in  which  the  totJ 
flow  remains  constant  for  an  appreciable  time,  since  the  lyiii 
quantity  of  water  passes  each  section,  the  mean  velocity  i?  in 
direction  of  the  stream  is  constant,  and  is  the  same  for  all 
sections,  and  is  simply  equal  to  the  discharge  divided  by  the  ami 
of  the  cross  section.  This  mean  velocity  is  purely  an  artificial] 
quantity,  and  does  not  represe^nt,  either  in  direction  or  magnitit<}«^| 
the  velocity  of  the  particles  of  water  as  they  pass  the  section. 


FLOW   IK  OPEN   CHANNELS 


179 


?fits  with  current  meters,  to  determine  the  distribution 

iii  diannels,  show,  however,  that  &t  any  point  in  the 

tion,  the  corapOBent  of  velocity  in  a  direction  parallel  to 

fefction  of  flow  renmins  practically  constant.    The  considers- 

pf  the  motion   m  consequently  simplified  by  assuming  that 

iter  moves  in  paralh*]  fillets  or  stream  lines,  the  velocities  in 

•ne  different,  but  the  velocity  in  each  sti-eam  line  remains 

This  is  the  aesutnption  that  is  made  in  investigating 

rational  formulae  for  the  velocity  of  flo^v  in  channels, 

[tt  should  not  be  overWked  that  the  actual  motion  may  be 

more  complicated. 

Formula  for  the  flow  when  the  motion  is  uniform 
Otiftiuael  of  onifonn  section  and  slope. 

I  this  sasumptionj  the  conditions  of  flow  at  similarly  situated 

i  C  and  D  in  any  two  cross  sections  AA  and  BB^  Figs.  112 

[118^  of  a  channel  of  uniform  slop©  and  section  are  exactly  the 

I;  the  velocities  are  equal,  and  since  C  and  D  are  at  the  same 

lice  below  the  free  surface  the  pressures  are  also  equal     For 

it  CD,  therefore, 

«?     2^      w      2g ' 
B,  HiBce  the  same  is  true  for  any  other  filament, 


for  the  two  sectii 

:»us. 

C 

i-""~"       — '—^ 

mm 

fi».m. 


Fig,  113. 


k-H  be  the  mean  velocity  of  the  stream,  i  the  fall  per  foot 
J  ^  the  surface  of  the  water,  or  the  slope,  al  the  length 
tn  AA  and  BB,  w  the  cross  sectional  area  EF&H  of  the 

fttiit  P  the  wetted  perimetor,  i.e,  the  length  EF  +  FG-  +  GH, 

1  *e  the  weight  of  a  cubic  foot  of  water. 

Let  ^  s  HI  be  called  the  hydraulic  mean  depth* 

I  bet  iff  be  the  fall  of  the  surface  between  AA  and  BB* 
11  ds-i.  cL 


Since 


12—1 


180 


HTDBAUUCS 


If  Q  cubic  feet  per  second  fall  from  AA  to  BB,  the  wort  dow 
upon  it  by  gravity  will  be : 

Then,  since  ^  £  "^  2^) 

is  constant  for  the  two  sections,  the  work  done  by  gravity 
be  equal  to  the  work  done  by  the  frictional  and  other  resistaiios 
opposing  the  motion  of  the  wa***** 

As  remarked  above,  all  the 
so  that  there  is  relative  motion 
and  since  water  is  not  a  perfect 
done  by  gravity  is  utilised  in  ova 
relative  motion.     Energy  is  also 
eddy  motions,  which  are  neglect 
and  some  resistance  is  ako  offer 
surface  of  the  water. 

The  principal  cause  of  loss 
of  the  sides  of  the  channel^  a;.» 
work  done  by  gravity  is  utilised  in  overcoming  this  resistance 

Let  F .  v  be  the  work  done  per  unit  area  of  the  sides  of  tlie 
channel,  v  beiug  the  mean  velocity  of  flow,  F  is  often  called  tk 
frictional  resistance  per  unit  area,  but  this  assumes  that  tlie  relatire 
velocity  of  the  water  and  the  sides  of  the  channel  is  equal  to  the 
mean  velocity,  which  is  not  correct. 

The  area  of  the  surface  of  the  channel  between  AA  and  BB 
isP.dZ. 

Then,  wi^idl-FvPdl^ 


its  have  not  the  same  velodtf^ 
bween  consecutive  likmeDti, 
id  some  portion  of  the  work 
Euing  the  friction  daetotliifi 
due  to  the  cross  currents  or 
1  assuming  stream  line  flow, 
o  the  flow  by  the  air  on  ih 

ever^  the  frictional  reskt&a* 
s  assumed  that  the  whole  A 


therefore 


or 


«  ,_  F 

.     F 
w 


F  is  found  by  experiment  to  be  a  function  of  the  velocity  ani 
also  of  the  hydraulic  mean  depth,  and  may  be  written  J 

b  being  a  numerical  coefficient. 

Since  for  water  to  is  constant  -  may  be  replaced  by  k  and 

therefore,  mi^k  ./{v)  f(m).  ] 

The  form  of /(i?)  /(?n)  must  be  determined  by  experiment- 
120.    Formula  of  Chezy. 
The  first  attempts  to  determine  the  flow  of  water  in  channels 


FLOW  IN  OPEN  CHANNELS  181 

ith  precision  -wrere  probably  those  of  Chezy  made  on  an  earthen 
Jialy  at  Coax^alet  in  1775,  from  which  he  concluded  that 
/(t;)  =  t;»and/(m)  =  l, 

id  therefore  m%  =  kv' (1). 

Writing  C  for  -^ 

v  =  C  vmi, 

'hich  is  known  as  the  Chezy  formula^  and  has  already  been  given 
1  the  chapter  on  pipes. 

lai.     JPormulae  of  Prony  and  E3rtelwem. 
Prony  adopted  the  same  formula  for  channels  and  for  pipes^  and 
Bsmned  that  F  was  a  function  of  v  and  also  of  t?',  and  therefore, 

mi  =  av  +  W. 
By  an  examination  of  the  experiments  of  Chezy  and  those  of 
)a  Buat*  made  in  1782  on  wooden  channels,  20  inches  wide  and 
988  than  1  foot  deep,  and  others  on  the  Jard  canal  and  the  river 
layne,  Prony  gave  to  a  and  h  the  values 

a  =  -000044, 
5  =  -000094. 
This  formula  may  be  written 


mx 


■(-:*')•■■ 


>r  ^  =       / J  mi. 


s/~v 


+  b 


The  coefficient  C  of  the  Chezy  formula  is  then,  according  to  Prony, 
a  function  of  the  velocity  v. 

If  the  first  term  containing  v  be  neglected,  the  formula  is  the 
same  as  that  of  Chezy,  or  

V  =  103  Jmi. 

Eytelwein  by  a  re-examination  of  the  same  experiments 
together  with  others  on  the  flow  in  the  rivers  Rhine  t  and  Weser  J, 
gave  values  to  a  and  b  of 

a  =  -000024, 

6  =  -0001114. 
Xeglecting  the  term  containing  a, 

V  =  95  y/mi, 

•  Prifieipei  d'hydraulique, 

t  ExperimeDts  by  Funk,  180S-6. 

X  Experimenis  by  Braoings,  1790-92 


182 


HTDRAULICB 


Afi  in   the  ease  of  pipes,  Prony  and  Eytelwein   mcorrectlf 
assumed  that  the  constants  a  and  h  were  independent  of  the  j 

nature  of  the  bed  of  the  channel. 

122.    Formula  of  Barcy  and  Bazin, 

After  completing  his  classical  experiments  on  flow    in  pipes  I 
M,  Darcy  commenced  a  series  of  experiments  upon  open  channels 
— afterwards  completed   by  M.   Bazin — to   determine,   how  the  ] 
frictional  resistances  varied  with   the  material  mth  which  the 
channels  were  lined  and  also  with  the  form  of  the  channel. 

Experimental  channels  of  semicircular  and  rectang-ular  secdon  I 
were  construct^  at  Dijon,  and  lined  mth  different  materiak. 
Experiments  were  also  made  upon  the  flow  in  small  earthen 
channels  (branches  of  the  Burgoyne  canal),  earthen  channels  hned 
with  stonesj  and  similar  channels  the  beds  of  which  were  cove^^ 
with  mud  and  aquatic  herbs*  The  results  of  these  experimental 
published  in  1858  in  the  monumental  work,  Rechsi^he^  Hydrtnc* 
liquss^  very  clearly  demonstrated  the  inaccuracy  of  the  assutop- 
tions  of  the  old  writers,  that  the  frictional  resistances  were 
independent  of  the  nature  of  the  wetted  surface. 

From  the  results  of  these  experiments  M.  Bazin  proposed  for  J 
the  coefficient  k^  section  120,  the  form  used  by  Darcy  for  pipes^ 


-i'^i). 


a  and  0  bein^  coefficients  both  of  which  depend  upon  the  nat 
of  the  lining  of  the  channel. 

Thus,  mi  =  {a-r  -Av^ 


or 


V- 


1 


y 


a  + 


mt- 


The  coefficient  C  in  the  Chezy  formula  is  thus  made  to 
with  the  hydraulic  mean  depth  m^  as  well  as  with  the  roughne 
of  the  surface* 

It  is  convenient  to  write  the  coefficient  k  as 


\       am/ 


Taking  the  unit  m  1  foot,  Bazin^s  values  for  a  and  ft  mi 
values  of  k  are  shown  in  Table  XVIII. 

It  will  be  seen  that  the  influence  of  the  second  term  inc 
very  considerably  with  the  roughness  of  the  surface, 

123.    Ganguillet  and  Kutter,  from  an  elimination  of  Baim*j 


FLOW  IN  OPEN  CHANNEI^ 


183 


periments,  together  with  some  of  their  own,  found  that  the 
efficient  C  in  the  Chezy  formula  could  be  written  in  the  form 

I  which  a  is  a  constant  for  all  channels,  and  b  is  a  coefficient  of 


TABLE  XVin. 

Showing  the  values  of  a,  fiy  and  k  in  Bazin's  formula  for 
channels. 


1 

a 

P 

k 

Planed  boards  and  smooth 

cement 
Bon^  boards,  bricks  and 

concrete 

Aiihl^r  masonry 

;Earth 

Gravel      (Ganguillet      and 
Katter) 

•0000467 
•0000580 
•0000780 
•0000864 
•0001219 

•0000046 

•0000188 

•00006 

•00085 

•00070 

•0000457  (l+^*^«) 

•000078   (l  +  ^^) 
•0000854(1+^-1) 
•0001219(1  +  ^"^^) 

The  results  of  experiments  by  Humphreys  and  Abbott  upon 
the  flow  in  the  Mississippi*  were,  however,  found  to  give  results 
inconsistent  with  this  formula  and  also  that  of  Bazin. 

They  then  proposed  to  make  the  coefficient  depend  upon  the 
slope  of  the  channel  as  well  as  upon  the  hydraulic  mean  depth. 

From  experiments  which  they  conducted  in  Switzerland,  upon 
the  flow  in  rough  channels  of  considerable  slope,  and  from  an 
examination  of  the  experiments  of  Humphreys  and  Abbott  on  the 
flow  in  the  Mississippi,  in  which  the  slope  is  very  small,  and 
a  large  number  of  experiments  on  channels  of  intermediate  slopes, 
they  gave  to  the  coefficient  C,  the  unit  being  1  foot,  the  value 


41-6  + 


0-00281     1-811 


C  = 


n 


l.(41-6.-«2281)    »' 
\  *     /  Vm 


^0  which  n  is  a  coefficient  of  roughness  of  the  channel  and  has  the 
values  given  in  Tables  XIX  and  XIX  A. 

.  *  Report  on  the  Hydraulics  of  the  Missunippl  River,  1861 ;  Flow  of  water  in 
^^  and  canaU,  Tnutwine  mnd  HeriDg,  1893. 


184  HYDRAULICS 

TABLE  XIX. 
Showing  values  of  n  m  the  formala  of  Gaugtiillet  and  Kuttjer. 

Chumd  n 

Very  smooth,  cement  and  planed  boards        »„  -009  to  -01 

Smooth,  boards,  brickHi,  concrete ^latoW 

Smooth,  covered  wiUi  slime  or  tnbercula-ted •        .,,  -015 

Roagh  ashlar  or  rubble  masonry  .,         «^«  Ditto  019 

Very  firm  gravel  or  pitched  witli  stone  ,..        *.,         ...  D2 

Earth,  in  ordinary  conditioii  free  fron^  m  and  weeds  *..  D25 

Earth,  not  free  from  atones  and  weed  *. .,  "030 

Gravel  in  bad  condition  , ...        H)65  to  ^ 

Torrential  streamH  with  rough  stony  »  ,  *0S 

TABLE  Ca. 

Showing  values  of  n  in  the  fa  i  of  GangiiiUet  and  Kutter^ 

determined  from  recent  experimc 

K 

Rectangular  wooden  flume,  very  mno  ,»,        ..»        „.        .^,  *0M 

Wood  pipe  6  ft.  diameter  ...        ...        -OlSI 

Brick,  washed  with  cementf  basket  sharped  sewer^  6'?c6'8*\  nearly 

new -Olto 

Brick,  washed  witli  cement,  basket  shaped  newer,  6'x6'B",  one 

year  old         ,..         ,.,  '0148 

Brick,  washed  with  cement,  basket  shaped  sewer,  Wx&&\  font 

years  old       .,,  •0153 

Brick,  washed  with  cement,  circular  sewer,  9  ft-  diameter,  nearly 

new „         „,  ^116 

Brick,  washed  with  cement,  circular  sewer,  9  ft.  diameter,  four 

years  old -Om 

Old  Croton  aqueduct,  lined  with  brick  *0i5 

New  Croton  aqueduct*  .-.         ,        ...         D12 

Sudbury  aqueduct  ^ Dl 

Glasgow  aqueduct,  lined  with  cement ...  D124 

Steel  pipe,  wetted,  clean,  1897  (mean)  "0144 

Steel  pipe,  1899  (mean) DUB 

This  formula  has  found  favour  with  English,  American  and 
German  engineers,  but  French  writers  favour  the  simpler  formula 
of  Bazin. 

It  is  a  peculiarity  of  the  formulaj  that  when  m  equals  unity 

then  C  =  -  and  is  independent  of  the  slope;  and  also  when  m^B 
n 

large,  C  increases  as  the  slope  decreases. 

It  is  also  of  iitiportance  to  notice  that  later  experiments  apou 
the  Mississippi  by  a  special  commissionj  and  others  on  the  flow  of 
the  Irrawaddi  and  various  European  rivers,  are  inconsistent  with 

*  Meport  New  York  Aqueduct  Committion^  ^ 


FLOW  IK   OPEN  CHANNELS 


185 


early  experiments  of  Humplireya  and  Abbott,  to  which 
G&Dg^illet  and  Katter  attached  very  considerable  import^ice  in 
fraiiittig^  their  formula,  and  the  later  experiments  show,  as  described 
later^  that  the  experimental  determination  of  the  How  in,  and  the 
dope  of,  large  natural  etreams  is  beset  with  anch  great  difBcidtieB,  ■ 
'  '  any  formula  deduced  for  channels  of  uniform  section  and 
'*.*  cannot  with  confidence  be  applied  to  natural  streams,  and 
trier  v^rsS.^ 

The  application  of  this  formula  to  the  calculation  of  uniform 
«5hannels  gri%'^Sj  however^  excellent  results,  and  providing  the  value 
of  n  in  known,  it  can  be  used  with  confidence* 

It  is,  however,  very^  cumbersome,  and  does  not  appear  to  give 
results  mare  accurate  than  a  new  and  simpler  formula  suggested 
I  lecsently  hy  Bazin  and  which  is  given  in  the  next  section. 

124-      BI.  Spin's  later  formula  for  the  flow  in  chaxmela. 
M.  Baxin    had  recently  (AiirJiali^  des  Fonts  ei  ChausseeB^  1897, 
VbL  IV-  p,  20),  made  a  careful  examination  of  practically  all  the 
ble   experinient'S  upon  channels,  and  has  propa^ed  for  the 
icieut  C  in  the  Chezy  formula  a  form  originally  proposed  by 
Gan^uillet  and  Kufcter,  which  he  writes 

1 


c=- 


«t  + 


c= 


when  the  unit  is  one  foot, 


1+4= 

m 


4: 


c  = 


157-5 
m 


.(2), 


s/i 


in  which  a  is  ccmstant  for  all  channels  and  ^9  is  a  coefficient  of 
roo^hnass  of  the  channel- 
Taking  1  metre  as  the  unit  a  -  '0115,  and  writing  y  for  -, 

c  =  -2I- (1). 


tike  ralne  of  y  in  (2)  being  I'Slly,  in  formula  (1), 

The  values  of  y  as  found  by  Bazin  for  various  kinds  of  channels 
ar^  shown  in  Table  XX,  and  in  Table  XXI  are  shown  values  of 


186 


HYDRAULICS 


C,  to    the    nearest    whole    number,   as    deduced    from    Bazin's 
coefficients  for  values  of  m  from  "2  to  50. 

For  the  channels  in  the  first  four  columns  only  a  very  feir 
experimental  values  for  C  have  been  obtained  for  values  of  m 
greater  than  3,  and  none  for  m  greater  than  7*3.  For  the  earth 
channels,  experimental  values  for  C  are  wanting  for  small  valuei 
of  m,  so  that  the  values  as  given  in  the  table  when  m  is  greater 
than  7*3  for  the  first  four  columns,  and  those  for  the  first  ihiee 
columns  for  m  less  than  1,  are  obtained  on  the  assumption,  tlurff 
Bazin's  formula  is  true  for  all  values  of  m  within  the  limits  dt  tiifr  i 
table. 

TABLE  XX. 


Values  of  y  in  the  formula, 


C  = 


157.5 


1  + 


-s/m 


Very  smooth  surfaces  of  cement  and  planed  boards  ... 

Smooth  surfaces  of  boards,  bricks,  concrete 

Ashlar  or  rubble  masonry  

Earthen  channels,  very  regular  or  pitched  with  stones, 
tunnels  and  canals  in  rock 

Earthen  channels  in  ordinary  condition  

Earthem  channels  presenting  an  exceptional  resistance, 
the  wetted  surface  being  covered  with  detritus, 
stones  or  weed,  or  very  irregular  rocky  surface 


unit  metre 

nnit  fool 

•06 
•16 
•46 

-106S 
•88 

•86 
1-80 

1-54 
2-86 

1-7 


817 


125.    Glazed  earthenware  pipes. 

Vellut*  from  experiments  on  the  flow  in  earthenware  pii>e8  has 


given  to  C  the  value 


in  which 


or 


C  = 


41-7  +  i 
n 


1  + 


75^' 


71  = -0072, 
181 


C  = 


,     -54  ' 


This  gives  values  of  C,  not  very  different  from  those  given  by 
Bazin's  formula  when  y  is  0*29. 

In  Table  XXI,  column  2,  glazed  earthenware  pipes  have  been 
included  with  the  linings  given  by  Bazin. 

♦  Proe,  I,  C.  E,,  Vol.  cli.  p.  482. 


FX^OIV     IM^     OPEX  CHANNELS 


187 


TAJBLiE  XXI. 

Values  of  C  in  the   f  ormiila  v  =  C^/m^  calculated  from  Bazin's 
{ormiila,  the  unit  of  len^i^lx  l>eiii^  1  foot, 

157-5 


C  = 


1  + 


y/m 


ChannelR 


126     Basin.*«  xnetliod  of  determining  a  and  JS. 


The  meftiod  used  by  Bazin  to  determine  the  values  of  a  and  /? 
'    of  sufficieiit  interest  and  importance  tx)  be  considered  in  detail. 


1 


TTe  first  calculated  values  of  -t=  and 
■^  vm 


>/: 


mi 


from  experimental 


^jita,  and  plotted  these  values  as  shown  in  Fig. 

-  Jim 
abBciBme,and  -^ 


114,  -^ 
vm 


as 


as  ordinates. 


188 


HTDBADLICS 


As  will  be  seen  on  reference  to  the  figure,  points  Iiave  bea 
plotted  for  four  classee  of  channels^  and  the  points  lie  close  to  km 
straight  lines  passing  through  a  common  point  P  on  the  aja 

of  y. 


-•i 


The  equation  to  each  of  these  lines  is 

y  =  a  +  )8aj, 


5 


\\ 


I 

■ 

s 


i  \ 


B  FI^OW  m  OPEN  CHANNELS  189 

a  bein^  the  intercept  on  the  axis  of  ^,  or  the  ordinate  when  -j^  is 
ieto,  a^d  /3,  ^irluch  is  variable,  is  the  inclination  of  any  one  of 
Hhcfwi  linae  to  tKe  ajds  of  oj  ;  for  when  -i=  is  ^ero,  - —  =  a,  and 
traBspoamg  ihB  ©qaation, 

whicli  IB  clearly  the  tangient  of  the  angle  of  inclination  of  the  line 
to  the  aiUB  of  ae. 

It   should  he  notedj  that  since  - — -  =  n  j  ^^^  ordinatea  give 

acteikl  eiqperimental  values  of  ^  ,  or  by  inverting  the  Bcale,  values 

of  C.      Two  scales  for  ordinates  are  thua  shown. 

In    addition  to  the  points  shown  on  the  diagram,  Fig,  114, 

Bazin  plotted  the  results  of  some  hundreds  of  experiments  for  all 

londs  of  channels^  and  found  that  the  points  lay  abont  a  series  of 

lines,  all  passing  through  the  point  P,  Fig.  1 14,  for  which  a  is  '00635, 

B 
and  the  values  of  - ,  i.e.  y,  are  as  shown  in  Table  XX. 
c 

BsLsin  therefore  concluded,  that  for  all  channels 


^*  = '00636  +  -^. 

V  Sim 


I 


the  iralae  of  p  depending  upon  the  roughness  of  the  channel, 

For  very  smooth  channels  in  cement  and  planed  boards,  Bazdn 
plotted  a  large  number  of  points,  not  shown  in  Fig.  114,  and  the 
luie  for  which  y  =  '109  passes  very  nearly  through  the  centre  of 
the   »cirie  occupied  by  these  points. 

TTie  line  for  which  y  is  0"29  coincides  well  with  the  mean  of 
the  plotted  points  for  smooth  channels,  but  for  some  of  the  points 
y  msky  he  as  high  as  0'4* 

It  is  further  of  interest  to  noticej  that  where  the  surfaces  and 
0ectioiiH  of  the  channels  are  as  nearly  as  possible  of  the  same 
character,  as  for  instance  in  the  Boston  and  New  York  aqueductS| 
the  values  of  the  coefficient  C  differ  by  about  6  per  cent,,  the 
differenix*  being  probably  due  to  the  pointing  of  the  sides  and 
^rch  of  the  New  York  aqueduct  not  being  so  carefully  executed 
aa  for  the  Boston  atiueduct.  By  simply  washing  the  walls  of  the 
latter  with  cement^  E*teley  found  that  its  discharge  was  increased 
20  per  cant. 


190 


HYDRAULICS 


y  is  also  greater  for  rectangular-shaped  channalB,  or  those 
which  approximate  to  the  rectangular  form,  than  for  those  rf 
circular  form^  as  is  seen  by  comparing  the  two  channels  in  wood 
W  and  P,  and  also  the  circular  and  basket-shaped  sewers, 

M*  Bazin  also  found  that  y  was  slightly  greater  for  a  veiy 
smooth  rectangular  channel  lined  with  cement  than  for  one  of 
semicircular  section. 

In  the  figure  the  author  has  also  plotted  the  results  of  some 
recent  experimentSj  which  show  clearly  the  effect  of  slime  aid 
tuberculatiousj  in  increasing  the  resistance  of  very  smooth  channels. 
The  %"alue  of  y  for  tlie  basket-shaped  sewer  lined  with  brici, 
washed  with  cement,  rising  from  %  to  "642  during  4  years*  service. 

12 7«    Variations  in  the  coefficient  G. 

For  channels  lined  ^dth  rubble,  or  similar  materials,  some  iif 
the  experimental  points  give  values  of  C  differing  very  consideTjlily 
from  those  given  by  points  on  the  line  for  which  y  is  0'B3,  Fig.  1 
but  the  values  of  C  deduced  from  experiments  on  parti 
channels  show  similar  discrepancies  among  themselves. 

On  reference  to  Bazin^s  original  paper  it  will  be  seen  that, 
channels  in  earthy  there  is  a  atilj  greater  variation  between 
experimental  values  of  C,  and  those  given  by  the  formula,  but 
experimental  results  in  these  cases,  for  any  given  channeli 
even  more  inconsistent  amongst  themselves. 

An  apparently  more  serious  difficulty  arises  with  respect 
Bazin's  formula  in  that  C  cannot  be  greater  than  157'5. 
maximum  value  of  the  hydraulic  mean  depth  m  recorded  ii 
any  series  of  experiments  is  74"3,  obtained  by  Humphreys 
Abbott  from  measurements  of  the  Mississippi  at  Carrollton  in  l 
Taking  y  as  2*35  the  maximum  value  for  C  would  then  be  121 
Humphreys  and  Abbott  deduced  from  their  experiments  valoeft 
of  C  as  large  as  254.  If,  therefore,  the  experiments  are  reliable 
the  formula  of  Basin  evidently  gives  inaccurate  results  for  axoep* 
tional  values  of  m. 

Tlie  values  of  C  obtained  at  Carrollton  are,  however,  incon- 
sistent %vith  those  obtained  by  the  same  workers  at  Yicksbuig, 
and  they  are  not  confirmed  by  later  experiments  carried  out  at 
CatTollton  by  the  Mississippi  commission.  Further  the  Telocitiil 
at  Carrollton  were  obtained  by  double  floats,  and,  according  to 
Grordon*,  the  apparent  velocities  determined  by  such  floats  should 
be  at  least  increased,  when  the  depth  of  the  water  is  large,  by  ten 
per  cent, 

Bazin  has  applied  this  correction  to  the  velocities  obtained  by 
*  Gordon,  Proeeedinff§  Intt  Civil  Eng,,  189S. 


FLOW  IN   OPEH   CHAKNELS 


iphrejB  and  Abbott  at  Yickabm^  and  also  to  those  obtained 
be  MiflstHsippi  Commissioti  at  Catrollton,  and  shows,  that  the 
immB  vala€>  for  C  is  theHj  probably,  only  122. 
%st  the  t-alties  of  C  as  deduced  from  the  early  experiments  on 
Mwiesippi  are  unreliable,  is  more  than  probable,  since  the 
iUefi(  slope,  as  measured,  was  only  '00000*34,  which  is  less  than 
kch  per  mile*  It  is  almost  impossible  to  believe  that  such  small 
ervnces  of  level  could  be  measured  with  certainty,  as  the 
^Il<?«t  ripple  would  mean  a  very  large  percentage  error,  and 
■^rther  probable  that  the  local  variations  in  level  would  be 
HR  than  this  measured  difference  for  a  mile  length.  Further, 
itoiug  the  slope  is  cx)rrectj  it  seems  probable  that  the  velocity 
gijch  a  fait  won  Id  be  less  than  some  critical  velocity  similar 
that  obtained  in  pipesp  and  that  the  velocity  instead  of  being 
>portional  t4>  the  square  root  of  the  slope  t,  is  proportional 
i.  That  either  the  measured  slope  was  unreliable,  or  that  the 
ocity  was  less  than  the  critical  velocity,  seema  certain  from  the 
:,  that  experiments  at  other  part8  of  the  Mississippi,  upon  the 
ftwaddi  by  Gordon,  and  upon  the  large  rivers  of  Europe,  in  no 
gtva  iralaas  of  C  greater  than  124. 
The  experimental  endence  for  these  natural  streams  tends, 
er,  clearly  to  show,  that  the  formulae,  which  can  with 
ifidtrnce  be  applied  to  the  calculation  of  flow  in  channels  of 
form,  cannot  with  assurance  be  used  to  determine  the 
of  riirerH*  The  reason  for  tins  is  not  far  to  seek,  as 
lit  ions  obt-aining  in  a  river  bed  are  generally  very  far 
ed  from  those  assumed,  in  obtaining  the  formula.  The 
pttoti  that  the  motion  is  uniform  over  a  length  sufficiently 
lo  be  able  to  measure  with  precision  the  fall  of  the  surface, 
be  ixLT  from  the  truth  in  the  case  of  riv^ers,  as  the  irregu- 
ia  the  crosa  section  must  cause  a  corresponding  variation 
n  velocities  in  those  sections, 

deri%'ation  of  the  formula,  frictional  resistances  only 
n  into  account,  whereas  a  considerable  amount  of  the 
tie  on  the  falling  water  by  gravity  is  probably  dissipated 
eddy  motions,  set  up  as  the  stream  encounters  obstructions  in 
bed  of  the  river,     Tliese  eddy   motions   must  depend  very 
li  OB  local  circunuitance-s  and  will  be  much  more  serious  in 
liar  channels  and  those  strewn  with  weeds,  stones  or  other 
tioiis,  than  in  the  regTilar  channels.    Another  and  probably 
Otis  difficulty  is  the  assumption  that  the  slope  is  uniform 
oui  the  whole  length  over  which  it  is  measured,  whereas 
between    two    cross  sections  may   vary  considerably 
bank  and  bank.    It  is  also  doubtful  whether  locally 


192 


HYDRAULICS 


there  is  always  eqailibriam  between  the  resisting  and  accele 
forces.    In  those  eases,  therefore,  in  which  the  beds  are  rock 
covered  Mrith  weeds,  or  in  which  the  stream  has  a  very  ir 
shape,  the  hypotheses  of  uniform  motion,  slope,  and  sectmi}  i 
not  even  be  approximately  realised* 

128.    Logarithmic  formula  for  the  flow  In  channels. 

In  the  formulae  discussed,  it  has  been  assumed  that  the  fricti 
resistance  of  the  channel  varies  as  the  square  of  the  velocity, 
in  order  to  make  the  formulae  fit  the    qp^eriments,  the  coefficient 
has  been  made  to  vary  with  the  velo 

As  early  as  1816,  Du  Buat*  p 
increased  at  a  less  rate  than  the 
half  a  century  later  St  Venant  pr 

mi  =  W040' 
To  determine  the  discharge  of  1 
Crimp  has  suggested  the  formula 

t^  =  124m*      • 
and  experiments  show  that  for  sewers  that  have  been  in  use 
time  it  gives  good  results.    The  formula  may  be  written 

0*00006 1>* 


ited  out,  that  the  slope 
[uare  of  the  velocity, 
>i»0d  the  formula 

ck-lined  sew^ers,  Mr 


i- 


m 


An  examiTiatton  of  the  results  of  experiments,  by  logarit 
plotting,  shows  that  in  any  uniform  channel  the  slope 

^  '  m^  ' 

Ic  being  a  numerical  c inefficient  which  depends  upon  the  rouglme«] 
of  the  surface  of  the  channel,  and  n  and  p  also  vary  with 
nature  of  tht.^  surface. 

Therefore,  in  the  formula. 


fMfirn)^ 


.P't' 


From  what  follows  it  will  be  seen  that  n  varies  between  IIS  ] 
and  2'1,  while  p  varies  between  1  and  I'S, 

Since  m  is  constant,  the  formula  t  -  — ^  may  be  written  i  -  friT, 


m" 


b  being  equal  to 
Therefore 


log  i  =  log  b  +  n  log  17. 


♦  Principcs  d'Hydraulique,  Vol.  i.  p.  29,  1816. 


FXOTV    IN    OPEN   CHANNELS 


193 


Pig.  115  are  shoiwii  plotted  the  logarithms  of  i  and  v 
A  from  an  eaqperiment  by  Bazia  on  the  flow  in  a  semi- 
r  cem^nt-lixied  pipe.  The  x)oint8  lie  about  a  straight  line, 
Qgent  of  the  inclination  of  which  to  the  axis  of  v  is  1*96 
le  intercept  on  the  axis  of  %  through  v  =  ly  or  log  v  =  0,  is 
08. 


Fig.  115. 


liOgarithmio  plottings  of  t  and  v  to  determine  the  index  n  in 
the  formola  for  channels,  %=—=, 


For  this  experimental  channel,  therefore, 

i  =  -00008085t;^*«. 
In  the  same  figure  are  shown  the  plottings  of  log  i  and  log  v  for 
18  siphon-aqueduct*  of  St  Elvo  lined  with  brick  and  for  which 
I  is  2*78  feet.    In  this  case  n  is  2  and  b  is  -000283.    Therefore 

t=-000283i?^ 
If,  therefore,  values  of  v  and  i  are  determined  for  a  channel, 
rhile  m  is  kept  constant,  n  can  be  found. 


•  AnnaUi  d£$  Fonts  et  Chatusiei,  Vol.  tv.  1897. 


U  B. 


\z 


194  HYDRAULICS 

To  determine  the  ratio  - .    The  formula, 
may  be  written  in  the  form, 

or  log  m  =  log  (-J  +  -  log  v. 

By  determining  experimentally  m  and  r,  while  the  slope  i  is 
kept  constant,  and  plotting  log  m  as  ordinates  and  log  v  as 
abscissae,  the  plottings  lie  about  a  straight  line,  the  tangent  of  the 

inclination  of  which  to  the  axis  of  v  is  equal  to   -,  and  the 

P 
intercept  on  the  axis  of  m  is  equal  to 

JcV 


(;) 


In  Fig.  116  are  shown  the  logarithmic  plottings  of  m  and  v  for 
a  number  of  channels,  of  varying  degrees  of  roughness. 

The  ratio  -  varies  considerably,  and  for  very  regular  channels 

increases  with  the  roughness  of  the  channel,  being  about  1"40  tat 
very  smooth  channels,  lined  with  pure  cement,  planed  wood  or 
cement  mixed  with  very  fine  sand,  1*54  for  channels  in  unplaned 
wood,  and  r635  for  channels  lined  with  hard  brick,  smooth 
concrete,  or  brick  washed  with  cement.    For  channels  of  greater 

roughness,  -  is  very  variable  and  appears  to  become  nearly  equal 

to  or  even  less  than  its  value  for  smooth  channels.     Only  in  one 

case  does  the  ratio  -  become  equal  to  2,  and  the  values  of  m  and 

P 
V  for  that  case  are  of  very  doubtful  accuracy. 

As  shown  above,  from  experiments  in  which  m  is  kept  constant^ 

n  can  be  determined,  and  since  by  keeping  i  constant  ~  can  be 

found,  n  and  p  can  be  deduced  from  two  sets  of  experiments. 

Unfortunately,  there  are  wanting  experiments  in  which  m  is 
kept  constant,  so  that,  except  for  a  very  few  cases,  n  cannot 
directly  be  determined. 

There  is,  however^  a  considerable  amount  of  experimental  data 
for  ciannels  similarly  lined,  and  of  different  slopes,  but  here 


FLOW  IN  OPBN  CHANNELS 


195 


Fig.  116.    Logarithmie  plottings  of  m  and  v  to  determine  the 
ratio  -  in  the  formula  t= — - . 


TABLE  XXII. 
Particulars  of  channels,  plottings  for  which  are  shown  in  Fig.  116. 


1.  Semieirealar  ehannel,  very  smooth,  lined  with  wood  

3.                  •>             »t           tf          ft          ft        }»    cement    mixed   with 
Tery  fine  sand  

3.  Bectangnlar  channel,  very  smooth,  lined  with  cement 

4.  ,.  »,  „  „  „        „    wood,  1'  7"  wide 

5.  f,  tf  smooth  „        „        „      slope  '00208 
6-                    t»               «f                 fi  »»         »»         M         ti      '0043 

#.  »i  »f  »t  ♦»       »»       »»       It     '004y 

8.  „  „  M  „         M         „         „      -00824 

9.  New  Croton  aqnednet,  smooth,  lined  with  bricks  (Report  New  York 

Water  Sapply) 

10.  Glasgow  aqueduct,  smooth,  lined  with  concrete  IProc.  I,  C.  E.  1896) 

11.  Sodbary        „  „  ,,        ,,     brick  well  pointed  (Tr,  Am. 

S.C.E.  1883) 

12.  Boston  sewer,  circular,  smooth,  lined  with  brick  washed  with  cement 

{Tr.  Am.S,  C.  E,  1901)  

13.  Hectangular  channel,  smooth,  lined  with  brick  

14.  9f  tf  •>  tf        tt     ^o<xi  

15.  t,  t,  „  „        „     small  pebbles        

15^1.  Bectangnlar  slniee  channel  lined  with  hammered  ashlar       

l«>w-  tt                tt          tf          ff        »f            ff            If 

16.  tt  ehannel  lined  with  large  pebbles 

17.  Torlofiia  ionnel,  rock,  partly  lined         

18.  Ordinary  ehannel  lined  with  stones  covered  with  mud  and  weeds   ... 

19.  tf  9»  tt  ff 

50.  BiTerWeeer  

51.  «.        ff  

S3.  «t         tt 

23.  Eitrth  dianneL    Ores  bois 

S4«  Catoot  canal  

S5.  Bhrer  Seiiifl 


n 

P 

1-45 

1-36 
1-44 
1-38 
1-64 
1-64 
1-54 
1-64 

1-74 
1-636 

1-635 

1-635 
1-635 
1-655 
1-49 
1-36 
1-36 
1-29 
1-49 
1-18 
-94 
1-615 
1-65 
2-1 
1-49 
1-5 


XZ—'l 


196 


HYDRAULICS 


again,  as  will  appear  in  the  context,  a  difficulty  is  encountered, 
even  with  similarly  lined  channels,  the  roughness  is  in  no  t' 
cases  exactly  the  same,  and  as  shown  by  the  plottings  in  Fig.  1. 
no  two  channels  of   any  class   give  exactly  the  same   yali 

for  - ,  but  for  certain  classes  the  ratio  is  fairly  constant. 

Taking,  for  example,  the  wooden  channels  of  the  group  (Nos 

to  8),  the  values  of  -  are  all  nearly  equal  to  1*54. 

The  plottings  for  these  channels  are  again  shown  in  Fig.  11 
The  intercepts  on  the  axis  of  m  vary  from  0*043  to  0'14. 


Fig.  117.    Logarithmio  plottings  to  determine  the  ratio  -  for  smooth  ch&nne 
Let  the  intercepts  on  the  axis  of  m  be  denoted  by  y,  then 

JcV 


-(!). 


FLOW  IN  OPEN  CHANNELS 


197 


and 


logy=-log*--logi. 


If  fc  and  'p  are  constant  for  these  channels,  and  logt  and 
logy  are  plotted  as  abscissae  and  ordinates,  the  plottings  should  lie 
about  a  straig^lit  line,  the  tangent  of  the  inclination  of  which  to  the 

axis  of  i  is  —  •  and  when  logy  =  0,  or  y  is  unity,  the  abscissa  %-lc. 

V 
we.  the  intercept  on  the  axis  of  %  is  k. 

In  Y\%.  118  are  shown  the  plottings  of  log  %  and  log  y  for  these 
channels,   froni    which   p=l'14   approximately,  and   %  = '00023. 

Therefore,  n  is  approximately  1*76,  and  taking  -  as  1*54 

._'00023t?^^  ^ 


' 

N 

~ 

' 

N. 

\   ^ 

- 

i 

\ 

1 

1 

\ 

\ 

< 

^ 

s 

s 

-S^-i 

s 

\ 

\ 

w 

4 

\ 

trr 

T 

\ 

«     - 

J 

N 

*t-— 

_ 



.__ 

— 

'"" 

~" 

-J 

" ^^ '         '" 

""  ~'^' '" 

:^ 

- 

^*- 

— 

~ 

_. ^-    , 

■'^—^—'—' 

^ 

«IUi 

1  , 

toLn.d}-%i 

i 

\ 

U- 

COOXt. 

S 

b^ 

■oc 

Jff 

'^ 

br 

oc 

f                     -Qi 

n 

m 

IS 

^i 

Log,  iy 

Fi^.  118.    Logmrithmio  plottings  to  determine  the  value  of  p  for  smooth 
channels,  in  the  formula  t=— ;:• 


Since  the  ratio  -  is  not  exactly  1*54  for  all  these  channels,  the 

valaes  of  n  and  p  cannot  be  exactly  correct  for  the  four  channels, 
but,  as  will  be  seen  on  reference  to  Table  XXIII,  in  which  are 
shown  yaloes  of  t?  as  observed  and  as  calculated  by  the  formula, 
the  calculated  and  observed  values  of  v  agree  very  nearly. 


198 


HYDEAUtlCS 

TABLE  XXm. 


Values  of  t?,  for  rectangular  channels  lined   with  wi 
determined  experimentally,  and  as  calculated  from  the 


formn 


Slope  -O020e 

Slope  -0019 

Slope  -00824 

e  ob- 

V ©ftJcu- 

V  ob* 

t;  ealea- 

V  ob- 

r^ftleu- 

m  m 

served 

lated 

m  in 

served 

Jated 

m  m 

served 

UUd 

metres 

metres 

metres 

metres 

metres 

meirea 

metres 

metres 

metres 

per  B«o. 

per  iee. 

per  sec. 

per  Bee. 

pel- sec. 

per  »a«. 

0*1881 

0*962 

0W2 

0^042 

1-825 

1814 

-0882 

1-594 

1-580 

'1609 

1'076 

1-07 

-1224 

1-479 

1-459 

'1041 

1-776 

1-764 

'1833 

1162 

1*165 

-1882 

1-612 

1-58 

-1197 

1-902 

liSSS 

1076 

1-259 

1'228 

-1585 

1-711 

1-690 

-1313 

2-053 

2-051 

'2146 

1-324 

1-290 

-1668 

1-818 

1-782 

-1420 

2-186 

2158 

'2318 

1-874 

1*854 

-1789 

1-898 

1-S5S 

^548 

2-268 

2-276 

^2441 

1-440 

1402 

a9l3 

1-967 

1-947 

■1649 

2-S57 

2-87T 

*2578 

1-487 

1*452 

*2018 

2-045 

2014 

*1744 

2-447 

2-460 

^2681 

1-562 

1-49 

-2129 

2-102 

2-089 

-1842 

2-518 

2*j^ 

■2809 

1'687 

1-552 

-2215 

2-179 

2143 

-1919 

2-612 

2*618 

As  a  further  example,  which  also  shows  how  n  and  p  increase 
with  the  roughness  of  the  channel,  consider  two  channels  built  ia 
hammered  ashlar^  for  which  the  logarithmic  plottings  of  m  and 

are  shown  in  Fig*  116j  Nos.  15  a  and  15  6^  and  -  is  1"36. 

The  slopes  of  these  channels  are  "101  and  *037,  By  plotting 
log't  and  logi^,  p  is  found  to  be  r43  and  k  '000149.  So  thai  for 
these  two  channels 

The  calculated  and  observed  velocities  are  shown  in  Table  XXXI 

and  agree  remarkably  well. 


Very  smooth  channels. 


The  ratio  -  for  the  four  very  smooth 
p 


channels,  shown  in  Fig.  116,  varies  between  V36  and  1'45,  tha 
average  value  being  about  1'4.    On  plotting  log^  and  logt  the 
points  did  not  appear  to  lie  about  any  particular  line,  so  that 
conld  not  be  determined,  and  indicates  that  k  is  different  for  tbo 
four  channels.    Trial  values  of  n  ^  175  and  p  ^  1*25  were  teken,  oe 

._^k.v'^ 

and  values  of  ft  calculated  for  each  channeL 


FUm   IN    OPEN   CHAKKELS 


Velocities  b&  determined  experimentally  and  as  calculated  for 
'    ^      ehaimels  are  shown  in  Tablo  XXIII  from  which  it  will 
it  fe  vanes  from  '00006516  for  the  channel  lined  with 
eetnent,  to  "0001072  for  tlie  rectangular  shaped  section  lined 
th  cai^fuHy  planed  boards. 
It  will  be  seen,  that  although  the  range  of  velocities  is  con- 
iderabie,  there  is  a  remarkable  agreement  between  the  calculated 
obfierved  values  of  t%  so  that  for  very  smooth  channels  the 
Ities  of  n  and  p  taken,  can  be  tised  with  considerable  confidence, 
Channeb  moderattly  imwoik.     Ite  plottings  of  logm  and  logtj 
chatHiels  Uned  with  brick,  concrete,  and  brick  washed  with 
^izLetit  are  shown  in  Fig.  116,  Mos.  9  to  13. 

It  will  be  seen  that  the  value  of  -  is  not  so  constant  as  for  the 

P 
ro  classes  prevjouBly  considered,  but  the  mean  value  is  about 

"635,  which  is  exactly  the  value  of  -  for  the  Sudbury  aqueduct, 

mi 

Tut  the  New  Croton  aqueduct  -  is  as  Mgh  as  r74,  and,  as  shown 
^m  Fi^*  114j  this  aqueduct  is  a  little  rougher  than  the  Sudbuiy. 

The  variable  values  of  -   show  that  for  any  two  of  these 

P 

iel«  either  n,  or  p,  or  both,  are  different »    On  plotting  log* 

r  a*  was  done  in  Fig*  115,  the  points,  as  in  the  last  casei 

lid  not  W  said  to  lie  about  any  particular  straight  line,  and  the 

la^  of  p  IB  therefore  uncertain*     It  was  assumed  to  be  115,  and 

n 


pre,  teJdng  -  as  1*635,  n  is  I'88. 


n 


Since  no  two  channels  have  the  same  value  for  -,  it  is  to  be 

P 
ted  that  the  coefficient  k  will  not  be  constant. 

lo  the  Tables  XXIV  to  XXXHI  the  values  of  v  as  observed 

as  Gslciilated  from  the  formula 

aW  the  value  of  k  are  given. 
It  will  be  seen  that  k  varies  very  considerably,  but,  for  the 
large  aquedncts  which  were  built  with  care,  it  is  fairly 
It. 
The  effect  of  the  sides  of  the  channel  becoming  dirty  with 
time^  i«  revy  well  seen  in  the  case  of  the  circular  and  basket- 
sewers.  In  the  one  case  the  value  of  k^  during  four  years' 
varied  from  00006124  to  00007998  and  in  the  other  from 
'•00006405  to  0001096-     It  is  further  of  interest  to  note,  that  when 


200 


HYDRAULICS 


m  and  v  are  both  unity  and  k  is  equal  to  ^000067,  the  value  of  i  a 

the  same  as  given  by  Bazin'a  formula,  when  y  is  *29,  and  when  k  is  ] 
'0001096,  as  in  the  case  of  the  dirty  basket-shaped  sewer,  th©  value 
of  y  is  '642^  which  agrees  with  that  shown  for  this  sewer  on 
Fig,  114  ' 

Channels  in  Tnasonry^  HamTnered  ashlar  and  rfihble.  AttentiaQ 
has  already  been  called^  page  198,  to  the  result-s  given  in 
Table  XX XT  for  the  two  channels  lined  with  hammered  ashlar* 

The  values  of  n  and  p  for  these  two  chaonels  were  determined 
directly  from  the  logarithmic  plottings,  but  the  data  is  tnsuiBeietit 
to  give  definite  values,  in  general,  to  n,  p,  and  k. 

In  addition  to  these  two  channels,  the  results  for  one  id 
Bazin's  channels  lined  with  small  pebbles,  and  for  other  channels 
lined  with  rabble  masonry  and  large  pebbles  are  given.  The 
n 


ratio  -  is  quoted  at  the  head  of  the  tables  where  possible. 


In  the, 


other  cases  n  and  p  were  determined  by  trial. 

The  value  of  n,  for  these  rough  channels,  approximates  to  2,] 
and  appears  to  have  a  mean  value  of  about  r96,  while  p  vari$lj 
from  1*36  to  1"5, 

Earthen  channels,    A  very  large  number  of  experiments  h*f 
been  made  on  the  fiow  in  canals  and  rivers,  but  as  it  is  generally  J 

impracticable  to  keep  either  i  or  m  constant,  the  ratio  -  c&n  only] 

be  determined  in  a  very  few  cases,  and  in  these,  as  wiJl  be  seen! 

from  the  plottings  in  Fig,  116,  the  results  are  not  satisfactory,  and] 


n 


appear  to  be  unreliable,  as  -  varies  between  '94  and  2*18, 

P 


It 


probable  that  p  is  between  1  and  1*5  and  n  from  1*96  to  2'15< 
Logarithmic  formulae  for  tmrimts  classes  of  channels. 
Very  smooth  channels,  lined  with  cement,  or  planed  boards^ 


„17S 


1- ("000065  to '00011)-^, 


m* 


Smooth  channels,  lined  with  brick  well  pointed,  or  concrete, 

*  =  '000065  to  *00011^, 

Channels  lined  with  ashlar  masonry,  or  small  pebbles, 

i  = '00015^., 
m}* 

Channels  lined  with  rubble  masonry,  large  pebbles,  rock^  and] 
exceptionally  smooth  earth  channels  free  from  deposits, 

i  =  *00023    ^,,^,;,. 


\ 


FLOW  IN  OPEN  CHANNELS  201 

Earth  channels, 

h  varies  {rem  '00033  to  '00050  for  channels  in  ordinary  condition 
and  from  '00050  to  *00085  for  channels  of  exceptional  resistance. 

120.  Approximate  formula  for  the  flow  in  earth 
nhamialii 

The  author  has  by  trial  found  n  and  p  for  a  number  of 
channels,  and  except  for  very  rough  channels,  n  is  not  very 
different  from  2,  and  p  is  nearly  1*5.    The  approximate  formula 

t?  =  C  vm^i, 

may,  therefore,  be  taken  for  earth  channels,  in  which  C  is  about 

50  for  channels  in  ordinary  condition. 

In  Table  XXXTTT  are  shown  values  of  t?  as  observed  and 

calculated  from  this  formula. 

The  hydraulic  mean  depth  varies  from  '958  to  14*1  and  for  all 

valaes  between  these  external  limits,  the  calculated  velocities 

Wbgree  with  the  observed,  within  10  per  cent.,  whereas  the  variation 
of  C  in  the  ordinary  Chezy  formula  is  from  40  to  103,  and 
according  to  Bazin's  formula,  C  would  vary  from  about  60  to  115. 
With  this  formula  velocities  can  be  readily  calculated  with  the 
ordinary  slide  rule. 

TABLE  XXIV. 
Very  smooth  channels. 
Planed  wood,  rectangular,  1*575  wide. 


i 

=  •0001072^, 
log*  =  4-0300. 

mfeet 

V  ft.  per  sec    v 
observed         ( 

ft.  per 
»loalai 

•2872 

8-65 

8-57 

•2811 

4-00 

408 

•8044 

4-20 

4-26 

•8468 

4-67 

4-68 

•8717 

4-94 

4-94 

•8980 

511 

512 

•4124 

5-26 

5-80 

•4811 

5*49 

5-47 

202 


HTDRAULICS 


TABLE  XXIV  (continued). 
Pure  cement,  semicircular. 


*=m-> 

•00006516^, 

log*  =  5-8141. 

m 

V  observed      v 

oalonlated 

•508 

8-72 

8-66 

•682 

4-69 

4-55 

•760 

4-87 

4^87 

•916 

557 

5-62 

1-084 

614 

614 

y  fine 

sand,  semicircular. 

^I'n 

1  = -0000759^, 

log*  =  6-8802. 

t;  ft.  per  seo.    v 

ft  per  see. 

mfeet 

observed          ( 

Bsloalated 

•879 

2-87 

2-74 

•629 

8-44 

8*49 

•686 

8-87 

8-98 

•706 

4-80 

4-80 

•787 

4-51 

4*59 

•889 

4-80 

4-84 

•900 

4-94 

610 

•941 

5-20 

5-26 

•988 

5^88 

5-48 

1^006 

5-48 

5-58 

1^02 

555 

5-58 

1-04 

5-66 

TABLE  XXV. 

5-66 

Boston  circular  sewer,  9  ft.  diameter. 

Brick,  washed  with  cement,  t  =  7xnnr  (Horton). 


i  = -00006124- 


m' 


log  V  =  -6118  log  m  +  -5319  log  t  +  2-2401. 


V  ft.  per  sec. 

V  ft.  per  seo. 

mfeet 

observed 

caloalated 

•928 

2-21 

2*84 

1208 

2*70 

276 

1^408 

808 

8-08 

1-880 

8*48 

8^56 

1*999 

878 

8-75 

2809 

4^18 

410 

FLOW  IN  OPEN  CHANNELS  208 

TABLE  XXV  (continued). 
The  same  sewer  after  4  years'  service. 

t  =  -00007998^, 

log  t;  = -6118  log  m  + -SSIO  log  i  +  2-1795. 
m  V  ohsenred    v  oaloulated 

1120  2*88  2*29 

1-606  2*82  2-76 

1-952  8-16  8-22 

2-180  8-80  8-89 

TABLE  XXVI. 
New  Croton  aqueduct.    Lined  with  concrete. 

i  =  -000073^, 

log  V  =  -6118  log  m  +  -5319  log  i  +  2'200. 

V  ft.  per  860.    V  ft.  per  seo. 
iifeet  obsenred    .      ealcolated 

1-000  1-87  1-87 

1-260  1-59  1-67 

1-499  1-79  1-76 

1-748  1-96  1-98 

2-001  211  210 

2-260  2-27  2-26 

2-600  2-41  2-40 

2-749  2-52  2-65 

2-998  2-66  2*68 

8-261  2-78  2-82 

8-508  2-89  2-96 

8-750  8-00  8-08 

8-888  8-02  8-12 

TABLE  XXVII. 
Sudbury  aqueduct.    Lined  with  well  pointed  brick. 


i  = 

•00006427^., 

081ogt?  = 

•6118  log  m  + 

•5319  log  1 

V  ft.  per  860. 

t;  ft.  per  seo. 

iifeet 

obBenred 

oaloulated 

•4987 

1-186 

1-142 

-6004 

1-269 

1-279 

-8005 

1-616 

1-526 

1-000 

1-765 

1-762 

1-200 

1-948 

1-964 

1-400 

2149 

2147 

1-601 

2-882 

2-881 

1-801 

2-518 

2-511 

2-001 

2-651 

2-672 

2-201 

2-844 

2-882 

2-886 

2-929 

2-987 

204  HYDRAULICS 

TABLE  XXVm. 
Rectangular  channel  lined  with  brick  (Baziii). 

i=-000107^. 

V  ft.  per  860.    V  ft.  per  seo. 
m  feet           observed         calealated 

•1922  2-76  2-90 

-2888  8-67  8*68 

-8654  4-18  4-80 

•4285  4-72  4-71 

•4812  5^10  5-09 

•540  5-84  5-46 

•5828  5-68  577 

•6197  6-01  5-94 

-6682  615  6-22 

-6968  6-47  689 

-7888  6-60  6-62 

•7788  6-72  688 

Glasgow  aqueduct.    lined  with  concrete. 
i  =  *0000696^„ 

log  V  =  -6118  log  m  +  -5319  log  i  +  2-2118. 

V  ft.  per  seo.    t;  ft.  per  eeo. 
m  feet  observed         calculated 

1-227 
1-478 
1-478 
1^489 
1-499 
1-499 
1-548 
1-597 
1-607 
1-610 
1-620 
1-627 
1-788 
1-811 


Charlestown  basket-shaped  sewer  6'  x  6'  8". 
Brick,  washed  with  cement,  i  =  ^Tnnr  (Horton), 

t  = -00008405  ^5, 

log  V  =  -6118  log  m  +  -5319  log  i  +  2-1678. 

V  ft.  per  sec.    t;  ft.  per  see. 
m  feet  observed  calculated 

•688  1^99  2^05 

•958  2-46  2-52 

1187  2-82  2-87 

1-589  8-44  8*86 


1-87 

1-89 

2-07 

2-11 

2-106 

2-11 

2-214 

218 

2-18 

214 

215 

214 

218 

2-22 

2-21 

2-28 

228 

2-28 

2-22 

2-24 

2-24 

2-24 

2-25 

2-27 

2-26 

2-88 

2-47 

2-40 

3  XXTX, 

FLOW   IN  OPEN  CHANNELS 


205 


TABLE  XXIX  (continued). 
rhe  same  se^irer  after  4  years'  service, 


i=-0001096- 


m' 


log  V  =  -6118  log  m  +  -5319  log  i  +  2*  1065. 


mfeet 
1-842 
1-506 
1-645 


V  ft.  per  aeo.    v  ft.  per  see. 
obserred  oalcnlated 


2-66 
2-86 
804 


2-68 
2-88 
8-04 


TABLE  XXX, 

Left  aqnednct  of  the  Solani  canal,  rectangular  in  section,  lined 
ill  rabble  masonry  (Cunningham), 


i  =  -00026^. 

i 

mfeet 

V  ft.  per  sec. 
observed 

V  ft.  per  seo. 
calculated 

-000225 
•000206 
•000222 
•000207 
•000189? 

6-48 

6-81 

7-21 

7-648 

7-94 

8-46 
3-49 
8-70 
3-87 
4-06 

3-50 
3-47 
3-84 
3-83 
3-88 

aeduct, 

i  = -0002213  ^,. 

t 

m 

V  observed 

V  calculated 

•000195 
-000225 
-000205 
-000198 
-000198 
-000190 

3-42 
5-86 
6-76 
7-43 
7-77 
7-96 

2-43 
3-61 
3-73 
3-87 
3-93 
406 

2-26 
3-58 
3-76 
3-89 
4-04 
4-06 

"orlonia   tunnel,  partly  in  hammered  ashlar,  partly  in  solid 

i=  00104, 


•00022- 


m' 


m 

V  observed 

V  calculated 

1-932 

3-382 

3-45 

2172 

3-625 

8-73 

2-552 

4-232 

4-16 

2-696 

4-324 

4-32 

8-251 

5046 

4-90 

8-488 

4-965 

5-08 

8-581 

4-908 

5-18 

8-718 

5-858 

5-37 

HYDRAUUCS 


TABLE  XXXI. 


Channel  lined  with  hammered  ashlar, 

-  =  1-36, 


m  feet 

•824 
•467 
•580 
•662 


t  =  -000149 
log  *  =  4-1740. 


m* 


{=•101 


V  ft.  per  seo. 
observed 

12-80 
16-18 
18-68 
21-09 


t;  ft.  per  sec. 
calculated 

12-80 
16-18 
18-97 
20-8 


t  =  -087 


m  feet 

-424 
-620 
-746 
-862 


V  ft.  per  see. 
observed 

9-04 
11-46 
18-66 
16-08 


V  ft.  per  MflL 
oal(»lattd 

902 
11-86 
18-52 
14-98 


Channel  lined  with  small  pebbles,  i  =  '0049  (w=l'96,  p  =  l'32 
will  give  equally  good  results). 

^  =  1-49, 
■     P 


i  =  -000152  ■ 
log  ifc  =  41913. 


m' 


V  ft.  per  sec. 

V  ft.  per  sec. 

nfeet 

observed 

calculated 

-250 

216 

2-84 

-867 

2-95 

2-97 

-450 

8-40 

8-47 

•520 

8-84 

8-82 

•588 

414 

4-16 

-644 

4-48 

4-48 

•700 

4-64 

4-66 

-746 

4-88 

4^88 

-786 

612 

5-05 

-882 

5-26 

5-25 

-871 

5-48 

648 

-910 

5-67 

6-68 

FU>W  IN  OPSN  CHANNELS 


207 


TABLE  XXXn. 
Cluumel  lined  with  large  pebbles  (Bazin), 
t  =  -000229^, 

log  i  =  4-3605. 

V  ft.  per  seo.    v  ft.  per  see. 
m  feet  observed       oalcalated 

1-79  1-84 

2-48  2-44 

2-90  2-90 

8-27  818 

8-56  8-45 

8-86  8-67 

4^08  8-91 

4-28  4-88 

4-48  4-58 

4-60  4-69 

4-78  4-84 

4-90  6-00 


TABLE  XXXm. 
Velocities  as  obsery^and  as  calculated  by  the  formula 


c^X' 


t;=Cvm* 

i.     C  =  50. 

f'-"" 

GaTiges 

Canal. 

i 

mfeet 

V  ft.  per  sec. 
observed 

V  ft.  per  sec. 
calcolated 

•000155 
•000229 
•000174 
•000227 
•000291 

5-40 
8-69 
7-82 
9-84 
4-50 

2-4 

8-71 

2-96 

402 

2-82 

2-84 
8-80 
8-08 
4-00 
2d8 

River 

Weser. 

t 

m 

V  obserred 

t;  calcolated 

•0006608 
•0005608 
•0002494 
•0002494 

8-98 
18-86 
141 
10-6 

6-29 
7-90 
6-69 
4-75 

60 
8-18 
6-70 
4-78 

Missouri. 

i 

m 

V  observed 

V  calculated 

•0001188 
•0001782 
•0001714 
•0002180 

10-7 
12-8 
15-4 
17-7 

8-6 
4-88 
6-03 
619 

8-28 
4-87 
4-80 
6*26 

208 


HYDRAULICS 

Cavon/r  Caned. 

i 

m 

V  obsenred 

t;  calculated 

•00029 
•00029 
•00088 
•00088 

782 
5-16 
6-63 
4-74 

870 
8^10 
840 
8^04 

8*80 
2-92 
8-14 
2-91 

Earth  channel  (branch  ofBurgoyne  canal). 
Some  stones  and  a  few  herbs  upon  the  swrfcLce. 

C  =  48. 

17  ft.  per  sec.    v  ft.  per  sec 
t  m  feet  observed       calculated 

•000957  -958  1^248  1^80 


•000929 

1-181 

1-702 

1-66 

•000998 

1-405 

1-797 

1^94 

•000986 

1588 

1-958 

2^06 

•000792 

•958 

1-288 

1-26 

•000808 

1-210 

1-666 

1-66 

•000858 

1-486 

1-814 

1-79 

•000842 

1-558 

1^998 

2-08 

130.  Distribution  of  the  velocity  in  the  oroas  aectioii 
of  open  channels. 

The  mean  velocity  of  flow  in  channels  and  pipes  of  small  ctosb 
sectional  area  can  be  determined  by  actually  measuring  the  weight 
or  the  volume  of  the  water  discharged,  as  shown  in  Chapter  Vll, 
and  dividing  the  volume  discharged  per  second  by  the  ckw 
section  of  the  pipe.  For  large  channels  this  is  impossible,  and 
the  mean  velocity  has  to  be  determined  by  other  means,  usually 
by  observing  the  velocity  at  a  large  number  of  points  in  the  same 
transverse  section  by  means  of  floats,  current  meters*,  or  Htot 
tubes  t.  If  the  bed  of  the  stream  is  carefully  sounded,  the  cross 
section  can  be  plotted  and  divided  into  small  areas,  at  tiie  centres 
of  which  the  velocities  have  been  observed.  K  then,  the  observed 
velocity  be  assumed  equal  to  the  mean  velocity  over  the  smaD 
area,  the  discharge  is  found  by  adding  the  products  of  the  areas 
and  velocities. 

Or  Q  =  Sa .  t;. 

M.  Bazint,  with  a  thoroughness  that  has  characterised  lus 
exi)eriments  in  other  branches  of  hydraulics,  has  investigated  ths 
distribution  of  velocities  in  experimental  channels  and  also  in 
natural  streams. 

In  Figs.  119  and  120  respectively  are  shown  the  cross  section 
of  an  open  and  closed  rectangular  channel  with  curves  of  eaas 

*  See  page  288.  t  See  page  241. 

X  Bazin,  Recherches  HydraiUique^ 


FLOW  IN  OPEN   CHANNELS 


209 


velocity  drawn  on  the  section.  Curves  showing  the  distribution 
of  velocities  at  different  depths  on  vertical  and  horizontal  sections 
are  alao  shown. 


^aj^j^sar  ofequat  VeU^cify 
ion  Rectcuiffular  Chaitttd/, 

Fig.  119. 


VertLcaLS^cUjons. 


Eori^ciUal  S  fictions. 


1 


Fig.  120. 


L  Xt  will  be  seen  that  the  maximum  velocity  does  not  occur  in 
L^  free  surface  of  the  water,  but  on  the  central  vertical  section 
Sit  some  distance  from  the  surface,  and  that  the  surface  velocity 
\  may  be  very  different  from  the  mean  velocity.  As  the  maximum 
m  velocity  does  not  occur  at  the  surface,  it  would  appear  t\iat  \n 

I  U  H.  \4 


210 


MTDRAULIC^ 


assuming  the  wetted  perimeter  to  be  only  the  wetted  sarbeej 
the  chaniii'l^  some  error  is  iritroducefL  That  the  air  hsi  aOll 
same  influcfna;  tm  if  the  water  wer*?  in  ccmtact  with  a  sur 
similar  to  that  of  the  sides  of  the  channel,  is  very  cle 
shown  by  comparing  the  carves  of  equal  velocity  for  the  ch 
rectangular  channel  as  showii  in  Fig.  119  with  thase  of  Fig.  1211 
The  air  resistancej  no  doubt,  accounts  in  some  measure  for  tltl 
surface  velocity  not  being  the  nmxinmm  velocity,  but  that  it  dosij 
not  wholly  acc<mnt  for  it^  in  shinm  by  the  fact  that^  whether  to] 


wind  is  blowing  up  oi 
below  the  surface.    J 
why  the  maximum  veio 
the  water  is  \em  eonstraini 
movements  of    all    k: 
utilised  in  giving  mouio 
translation. 

Depth  rm  a7ty  vsrtkal  at 
velocity.  Later  is  discussed, 
on  the  verticals  of  any  crosa 


the  maximum  velocity  i&«t3t I 
ggest^  as  the  principal  reason  [ 
ot  ocrcur  at  the  surface, 
e  surface,  and  that  ir 
op,  and  energy  is  thepel»| 
rater  not  in  the  direction  dfl 

i$mheity  u  equal  to  ihi  mmX 
3^  tiie  dktrtbntion  of  reloci^ 
,  and  it  will  be  seen^  thst  it  \ 
is  the  mean  veUx*ity  on  any  vertical  st^ctioti  of  the  channt'I,  tte] 
depth  at  which  the  velocity  is  equal  to  the  mean  velocity  is  about 
0'6  of  tht'  total  depth.  This  depth  varies  with  the  roughne«!i  of 
the  stream,  and  u  deeper  the  greater  the  ratio  of  the  depth  to 
the  width  of  the  stream.  It  varies  between  '5  and  *55  of  the  depth 
for  rivers  of  small  depth,  ha\ing  beds  of  fine  sand,  and  from  '5S 
to  'Q^  in  large  rivers  from  1  to  3f  feet  deep  and  ha\ing  stitffig 
bedst. 

As  the  banks  of  the  stream  are  approached,  the  point  at  wliidi 
the  mean  veh^city  occurs  falls  nearer  still  to  the  bed  of  the  stream, 
but  if  it  falls  v^ery  low  there  m  generally  a  second  point  near  tk 
surface  at  which  the  velocity  is  also  equal  to  the  mean  velocity. 

When  the  river  is  covered  with  ice  the  maxim  ima  velocity  of 
the  current  \^  at  a  depth  of  '3-5  to  '45  of  the  total  depth,  and  tl« 
mean  velocity  at  two  points  at  depths  of  '08  to  '13  and  '68  to'JV 
of  the  total  depth  J.  ■ 

If,  therefore,  on  various  verticals  of  the  cross  section  of  a  strfsS^ 
the  velocity  is  determined,  by  means  of  a  current  meter,  or  I*ik>t 
tube,  at  a  depth  of  about  '6  of  the  total  depth  from  the  surface, 
the  velocity  obtained  may  be  taken  as  the  mean  velocity  upon  ibe 
vertical. 


*  Hydraulique. 

f  I,f  GSnie  Civil,  April,  1906,  ''Analysis  of  a  communication  by  Marphj  to 
the  Hydrological  section  of  the  Institute  of  Geology  of  the  United  States." 
"^  Cunningham,  Experiments  on  the  Ganges  Canal. 


FLOW  IN  OPEN  CHANNELS 


211 


The  total  discliarge  can  then  be  found,  approximately,  by 
ividing  the  cross  section  into  a  number  of  rectangles,  such  as 
M,  Fig.  120a,  and  multiplying  the  area  of  the  rectangle  by  the 
"velocity  measured  on  the  median  line  at  0*6  of  its  depth. 


Fig.  120  a. 

The  flow  of  the  Upper  Nile  has  recently  been  determined  in 
Akway. 

Gaptain  Cunningham  has  given  several  formulae,  for  the  mean 
idocity  u  upon  a  vertical  section,  of  which  two  are  here  quoted. 

u=\(y-^^{) (1), 

tt  =  i(2t?j-t;j  +  2yj)  (2), 

V  being  the  velocity  at  the  surface,  v^  the  velocity  at  i  of  the  depth, 
V|  at  one  quarter  of  the  depth,  and  so  on. 

Form    of  the    curve    of  velocities    on   a   vertical 


13L 
notion. 

M.  Bazin*  and  Cunningham  have  both  taken  the  curve  of 
telocities  upon  a  vertical  section  as  a  parabola,  the  maximum 
velocity  being  at  some  distance  hm  below  the  free  surface  of  the 
water. 

Let  V  be  the  velocity  measured  at  the  centre  of  a  current  and 
as  near  the  surface  as  possible.  This  point  will  really  be  at  1  inch 
iff  more  below  the  surface,  but  it  is  supposed  to  be  at  the  surface. 

Let  V  be  the  velocity  on  the  same  vertical  section  at  any  depth 
A,  and  H  the  depth  of  the  stream. 

Bazin  found  that,  if  the  stream  is  wide  compared  to  its  depth, 
the  relationship  between  v,  V,  h,  and  i  the  slope,  is  expressed  by 
the  formula, 


V-t; 


»r  i;  =  V-i(g)VHi    (1), 

;  being  a  numerical  coefficient,  whicli  has  a  nearly  constant  value 
»f  36'2  when  the  unit  of  length  is  one  foot. 


ns. 


Recherches  Hydrauliquef  p.  228 ;  Annales  des  Fonts  et  Chansn^es,  2nd  Vol., 

\4— 'i 


tV 


tV 


;,\ocw; 


•  \-  ...uve  ^^V  »:,   v\vU«*V'    ,  ,..t  '>^'^;".„   and  t^*'  \w,  " 


\\p 


^^^•'l  >rg>'  "a\iuc^ 


\V 


ateY 


lAOt 


\n 


.•^'",f;; 


«--^:^s:^??t;>cs;--■:f> 


3<' 


ivu" 


to 


tlieP^;", 


..euv<:::t'^*"«^* 


to*?' 


opoi 


litits 


atvV'^^^faeptVv:- 
t\tet«t^^...^.otv 


ot  t\^« 


,Tio*! 


\cal» 


oi*«»T 


vevt^'^,.  rf^ 


a>y 


^^''*;e\octty 


vottv 


,V.eBS^^!!lt:Ui>^ 


Cttii^ 


.tiit^»^*' 


212 


HYBRAULICS 


lb  detetmtne  ihe  depth  tyn  aiiy  vertical  at  wM^M  the  velocity  i 
eqtml  to  the  mean  veI>ocity,  Let  u  be  the  mean  velocity  on 
vertical  section,  and  hu  the  depth  at  which  the  velocity  is  equal  I 
the  mean  velocity. 

The  discharge  through  a  vertical  Btrip  of  width*  dl  is 

rH 

uEdl^dlj    v.dh. 


Therefore  uR  =  j^  (y  -  ^s/Ri  )dA, 


and 


u^y-^Jm 


-(2). 


Substitnting  u  and  A,  in  (1)  and  equating  to  (2), 


and 


A«  =  '577H. 


This  depth,  at  which  the  velocity  is  equal  to  the  mean  velocityJ 
is  determined  on  the  assumption  that  Jc  is  constant,  which  is 
true  for  sections  very  near  to  the  centre  of  strea^ms  which  si«| 
wide  compared  with  their  depth. 

It  will  be  seen  from  the  curves  of  Fig.  120  that  the  deptli  ai  1 
which  the  maximum  velocity  occurs  becomes  greater  as  the  sides  J 
of  the  channel  are  approached,  and  the  law  of  variation  of  \'elocitf  J 
also  becomes  more  complicated ♦  M,  Baxin  ako  found  that , 
depth  at  the  centre  of  the  stream,  at  winch  the  maximum  vek 
occurSj  depends  upon  the  ratio  of  the  width  to  the  depth,  j 
reason  apparently  being  that,  in  a  stream  which  is  wide  comf 
to  its  depth  J  the  flow  at  the  centre  is  but  slightly  affected  byl 
resistance  of  the  sides,  but  if  the  depth  m  large  compared  with  1 
width,  the  effect  of  the  sides  is  felt  even  at  the  centre  of 
stream*  The  farther  the  vertical  section  considered  is  remO 
from  the  centre,  the  effect  of  the  resistance  of  the  aides* 
increased,  and  the  distribution  of  velocity  is  influenced  to 
greater  degree.  This  effect  of  the  sides,  Bazin  expressed 
making  the  coefficient  k  to  vary  with  the  depth  h^  at  wl 
the  maximum  velocity  occurs. 

The  coefficient  is  then, 

36"2 


Jfc- 


('-fer 


Further,  the  equation  to  the  parabola  can  be  written  in 
of  v„^  the  maximum  velocity,  instead  of  V. 


FLOW  IN  OPEN  CHANNELS  213 

Th»,  ,.,,.88-2Vffift-t.)- (3, 

The  mean  velocity  t*,  upon  the  vertical  section,  is  then, 
1  f^ 
riJo 

Therefore 

'-"*(r:|)-(5-H-H.)-;:(^<'-w- 

When  t7  =  tA,    fe  =  /i«, 

«  1        fcm       /l«'       2huhm 

and  therefore,  3  "  H  "^  H* W~ ' 

The  depth  ^  at  which  the  velocity  is  a  maximum  is  generally 
len  than  *2H,  except  very  near  the  sides,  and  hu  is,  therefore,  not 
-very  different  from  '6H,  as  stated  above. 

Ratio  of  maximum  velocity  to  the  mean  velocity.  From 
equation  (4), 


.,  - 36'2>/Hi71     K^K\ 


In  a  wide  stream  in  which  the  depth  of  a  cross  section  is  fairly 
constant  the  hydraulic  mean  depth  m  does  not  differ  very  much 
rfrom  H,  and  since  the  mean  velocity  of  flow  through  the  section  is 
( C  ^mi  and  is  approximately  equal  to  u,  therefore, 
v^    .  36-2        /I     K  ,  hj\ 

^  Cfl-M^^^     H      hV- 

Assuming  A.  to  vary  from  0  to  '2  and  C  to  be  100,  —  varies 

from    1*12  to   r09.    The  ratio  of    maximum  velocity  to  mean 
"Vidocity  is,  therefore,  probably  not  very  different  from  Tl. 

132.    The  slopes  of  channels  and  the  velocities  allowed 
in  them. 

The  discharge  of  a  channel  being  the  product  of  the  area  and 
ihe  velocity,  a  given  discharge  can  be  obtained  by  making  the 
rea  small  and  the  velocity  great,  or  vice  versa.    And  since  the 
riocity  is  equal  to  Cvmi,  a  given  velocity  can  be  obtam^id  \>y 


214 


HYDRAULICS 


varying'  either  vt  or  i.     Since  m  will  in  general  increase  with  the 

area>  the  area  will  be  a  niininuiuj  when  i  is  as  lar^e  as  possible. 

^Butf  as  the  coat  of  a  channel,  including  land,  excavation  and 

Icxmstructionj  will^  in  many  cases,  be  almost  proportional  to  ite 

tcroes  eectiona!  area,  for  the  first  cost  to  be  small  it  is  desirable 

rthat  i  should  be  large.     It  should  be  noted,  however,  that  the 

discharge  ia  generally  increased  in  a  greater  proportion,  by  an 

increase  in  A,  than  for  the  same  proportional  increase  in  L 

Assume,  for  instance,  the  channel  to  be  semicirctilar* 

The  area  is  proportional  to  ^,  and  the  velocity  t*  to  vd ,  i. 

Therefore  Q  oc  cp  Jdl 

If  d  is  kept  constant  and  i  donfeled,  the  discharge  i^  increased 
to  %^2Q,  but  if  d  18  doubled,  i  being  kept  constant^  the  diacbargi? 
will  be  increased  to  5"6Q,  The  maximum  slope  that  can  bt*  given 
will  in  many  cases  be  determined  by  the  difference  in  level  of  til 
two  points  connected  by  the  channel. 

When  water  is  to  be  conveyed  long  distance-s,  it  is  often 
necessary  tci  have  several  pumping  stations  en  rout^^  as  gutficient 
fall  cannot  be  obtained  to  admit  of  the  aqueduct  or  pipe  line  being 
laid  in  one  continuous  lengtb. 

The  mean  velocity  in  large  aqueducts  is  about  3  feet  per 
second,  while  the  slopes  vary  from  1  in  2000  to  1  in  10,000,  ITje 
slope  may  be  as  high  as  1  in  1000^  but  should  not^  only  in  eioep" 
tional  circumatancesj  be  less  than  1  in  1 0,000, 

In  Table  XXXIV  are  given  the  slopes  and  the  maximoiB 
velocities  in  them,  of  a  number  of  brick  and  masonry  liuai 
aqueducts  and  earthen  channels,  from  which  it  will  be  seen  thai 
the  maximum  velocities  are  between  2  and  5J  feet  per  second, 
and  the  slopes  vary  from  1  in  2000  to  1  in  7700  for  the  brick  aad 
masonry  lined  aciueducts,  and  from  1  in  300  to  1  in  20,000  for  the- 
earth  channels.  The  slopes  of  large  natural  streams  are  in  iomt 
Leases  even  less  than  1  in  100^000,  If  the  velocity  is  too  small 
P suspended  matter  is  deposited  and  slimy  growths  adhere  to  the  sidefi 

It  is  desirable  that  the  smallest  velocity  in  the  channel  aliall 
such,  that  the  channel  is  "  self -cleansing,"  and  as  £ar  as  poiiibi 
the  growth  of  low  forms  of  plant  life  prevented. 

In   sewers,   or   ctiannels    conveying    unfiltered    wTaters^  it 
especially  desii-able  that  the  velocity  shall  not  be  too  stnall, 
ehouldj  if  possible,  not  be  less  than  2  ft,  per  second. 

^  TABLE  XXXIV. 

Showing  the  slopes  of,  and  maximum  velocities^  as  deterauoe 

experimental  lyj  in  some  existing  channels. 


FLOW  IN  OPEN  CHANNELS 


215 


/ 

Smooth  aqueducts. 

Slope 

Maximum  velocity 

New  Croton  aqueduct         -0001326 

3       ft.  pel 

•  second 

Sudbory  aqoednct 

•000189 

2-94        „ 

)) 

Glasgow  aqueduct 

•000182 

2-25        „ 

»» 

Paris  Dhuis 

•000180 

Avre,  Istpart 

-0004 

„      2nd  part 

-00038 

Manchester  Thirlmere       -000815 

Naples 

-00050 

4^08 

»> 

Boston  Sewer 

-0005 

844        „ 

>» 

»»            >f 

•000388 
Earth  chamiels. 

4^18        „ 

*» 

Slope            Maximum  velocity 

Lining 

ages  canal 

•000806        4-16  ft.  per  second 

earth 

±er      „ 

-008              4-08 

>»        »» 

»» 

ith        „ 

•00037          5-63 

»        »» 

gravel  and 
some. stones 

FOOT         „ 

•00083          342 

))         fi 

amen    „ 

•0070            874 

»        »♦ 

earth 

azilly  cut 

•00085          1^70 

t                   )9 

1  earth,  stony, 
\  few  weeds 

•00048          1^70 

»                  »» 

the  bottom  of  the  canal) 

•00005          3 

»                  >» 

»»                  M 

TABLE  XXXV. 

Showing  for  varying  values  of  the  hydraulic  mean  depth  m,  the 
limum  slopes,  which  brick  channels  and  glazed  earthenware 
es  should  have,  that  the  velocity  may  not  be  less  than  2  ft. 
second. 


m  feet 

slope 

•1                   ] 

L    in    93 

•2                  3 

L     „     275 

•3                  J 

I     „     510 

•4                  ] 

L     „     775 

•5                  ] 

L     „     1058 

•6                  ] 

L     „     1880 

•8                  ^ 

L     „     2040 

i-o             ] 

L     „     2760 

m  feet 

slope 

125                1 

in    8700 

15                 1 

„     4700 

175                1 

„     5710 

20                 1 

„     6675 

25                  1 

„     9000 

30                  1 

„      11200 

40                  1 

„      15850 

lie  slopes  are  calculated  from  the  formula 

157-5     ,--. 


1  + 


n/w 


m 


he  value  of  y  is  taken  as  0*5  to  allow  for  the  channel  becoming 
For  the  minimum  slope  for  any  other  velocity  v,  multiply 

(2\* 
-  j  .    For  example,  the  minimum  slope 

velocity  of  3  feet  per  second  when  m  is  1,  is  1  in  1227. 


216  HYDRAULICS 

Velocity  offimv  in,  and  $lope  of  earth  channels*  If  th©  relodtf 
is  high  in  earth  channels,  the  sides  and  bed  of  the  ehaji&el  sii  | 
eroded,  while  on  the  other  hand  if  it  is  too  sroall,  the  c:i|>;u  ify  of 
the  channel  will  be  rapidly  diminished  by  the  depositi.u  *►!  ;<n4 
and  other  suspend  til  nmfcter,  and  the  growth  of  aijuittic  plant*. 
Da  Buat  gives  '*">  f(H>t  per  second  as  the  minimum  velocity  that 
mad  shall  not  be  dei)o«ite<l,  while  Belgrand  ttlluws  a  miminma 
of  '8  foot  per  second. 

TABLE  XXX\T 

Showing  the  velocities  above  ih^  according  to  Du  Bttiit, ' 
and  as  quoted  by  Rankine,  erofiioi  hanuebof  vanous  matemli  j 
takes  place. 

Soft  clay  Om  tL  per  sec^iid 

Fine  sand  0^^  „  „ 

Coarse  sand  and  gravel  m  large  a^  p  OW  „  n 

Gravel  1  inch  diameter  2*25  „  ^ 

Pebbles  1^  inchea  diameteT  8*33  „  ,, 

Heavy  sbmgle  4'00  „  «. 

Soft  rock,  brick,  eartbeuware  4'50  „  |, 

Rock,  various  kinds  600  „  ,,      and  upwurii 

133.  Sections  of  aqueducts  and  sewers. 

The  forms  ^jf  sections  given  to  scjme  aqueducts  and  sewers  are 
shown  in  Figs.  121  to  131,  In  dt^signing  snch  aqueduct-s  ^d 
sewers,  con  side  nit  ion  has  to  be  given  to  problems  other  than  tlie 
comparatively  simple  one  of  determining  the  size  and  slope  to 
be  given  to  the  channel  to  convey  a  certain  quantity  of  water. 
The  nature  of  the  strata  through  which  the  aqueduct  is  to  be 
cut,  and  whether  the  excavation  can  best  be  accomplished  by 
tunnelling,  or  by  cut  and  cover,  and  also^  whether  the  aqnediict 
is  to  be  lined,  or  cut  in  eolid  rock,  must  be  considered.  In  many 
cases  it  is  desinible  that  the  aqueduct  or  sewer  should  have  9QcI 
a  form  that  a  man  can  conveniently  walk  along  it,  although  its 
sectional  area  is  not  required  to  be  exceptionally  large.  la 
such  cases  the  section  of  the  channel  is  made  deep  and  narrow. 
For  sewers,  the  oval  section.  Figs.  126  and  127,  is  largely 
adopted  because  of  the  facilities  it  gives  in  this  respect,  and  it  has  ' 
the  further  advantage  that,  as  the  flow  diminishes,  the  cross 
section  also  diniinishes,  and  the  velocity  remains  nearly  constant 
for  all,  except  very  small,  discharges.  ITiis  is  im]>ortant,  as  at 
small  velocities  sediment  tends  to  collect  at  the  bottom  of  the 
sewer. 

134.  Siphons  forming  part  of  aqueducts. 

It  is  frequently  necessary  for  some  part  of  an  aqueduct  to  be 
constructed  as  a  siphon,  as  when  a  valley  has  to  be  crossed  or  the 


FLOW  IN  OPEN  CHANNELS 


217 


aciaedact  taken  nnder  a  stream  or  other  obstruction,  and  the 
aqaednct  must,  therefore,  be  made  capable  of  resisting  con- 
siderable pressure.  As  an  example  the  New  Croton*  aqueduct 
from  Croton  Lake  to  Jerome  Park  reservoir,  which  is  33*1  miles 


rrs 


Fig.  121. 


Fig.  122. 


<— 7.^— > 


I 
_- IT- 


Fig.  123. 


i<- is:s^ ->i 

Fig.  126. 


rr\ 


« 


Fig.  127. 


Fig.  128. 


Fig.  129. 


Fig.  ISO. 


Fig.  131. 


218  HYDRAULICS 

long,  is  made  up  of  two  pnrt^a.  The  first  m  a  tnasonry  conduil 
the  section  shown  in  Fig.  121,  23'9  miles  long  and  haipHng  a 
of  '0001326,  the  second  consists  almo«t  pntirely  of  a  brick  lio^ 
siphon  6*83  miles  long,  12'  3'  diameter,  the  maximum  head  in 
which  is  126  feet,  and  the  differenci*  in  level  of  the  two  ends  is 
6*19  feet.  In  such  casen,  however,  the  siphon  is  fretiuentljr  made 
of  steel,  or  cast-iron  pij^es,  as  in  the  case  of  the  new  Edinburgh 
aqueduct  (see  Fig,  131)  which j  where  it  crosses  the  vaUeja,  m 
made  of  cast-iron  pipes  S3  inches  diameter. 

135.    The  best  form  of  cham 

The  best  form  of  channel,  or  nnel  of  least  resistance,  is 
that  which,  for  a  gii'-en  slope  and  ?a,  will  give  the  majdmonL 
discharge. 

Since  the  mean  velocity  in  a  cl        el  of  given  slope  is  propor- 

A 

tional  to  p ,  and  the  discharge  is  A        the  best  form  of  channel  for 

a  given  area,  is  that  for  which  P  linimum. 

The  form  of  the  cliannel  whicj        i  the  minimura  wetted 
meter  for  a  given  area  is  a  semicir        for  which  p  if  r  is  the  nuiina, 

the  hydraulic  mean  depth  is  ^. 

More  convenient  forms,  for  channels  to  be  excavated  in  roct 
or  eartli,  are  those  of  the  rectangular  or  trapezoidal  section, 
Fig.  133.  For  a  given  dis^charge,  the  best  forms  for  these  I 
channels,  will  be  those  for  which  both  A  and  P  are  a  minimmD;  I 
that  is,  when  the  ilifTerentialw  (^A  and  oV  are  respectively  equal  to  ^ 
zero. 

Rectangular  chanwL     Lot  L  be  the  >\ndth  and  h  the  depttj 
Fig.  132,  of  a  rectangular  channel ;  it  is  required  to  find  the  ratio 

^  that  the  area  A  and  the  wetted  perimeter  P  may  both  be  a 

minimum,  for  a  given  discharge. 
A  =  L/i, 

therefore  dA  =  h.dh+hdh  =  0    (1), 

P  =  L  +  2/i, 

therefore  dP  =  dh  +  2Jh  =  0 (2>  - 

Substituting  the  value  of  dL  from  (2)  in  (1), 

L  =  2A. 

Therefore  m  =  -rj-  =  ^ . 

4/1      2 

Since  L  =  2h,  the  sides  and  bottom  of  the  channel  touch  a  cir^^^ 

having  h  as  radius  and  the  centre  of  which  is  in  the  free  surf^*^^ 

of  the  water. 


FLOW  IN  OPEN  CHANNELS 


219 


Earth  ch4innels  of  trapezoidal  form.    In  Fig.  133  let 

I  be  tlie  bottom  width, 

h  tbe  depth, 

A.  the  cross  sectional  area  FBCD, 

P  tbe  length  of  FBCD  or  the  wetted  perimeter, 

i  tbe  slope, 
and  let  tbe  slopes  of  the  sides  be  t  horizontal  to  one  vertical ;  CG 
is  then  equal  to  th  and  tan  CDG  =  t. 


th-H 


and 


Fig.  132.  Fig.  133. 

Let  Q  be  the  discharge  in  cubic  feet  per  second. 

Then  K^hl-^th? 

P=Z  +  2/iV^^"+~l  

h(l±th)_ 

"^'U^hjwvi 


.(3), 
.(4), 

.(5). 


For  tbe  channel  to  be  of  the  best  form  ^P  and  dA  both  equal 

zero- 

From  (3)  K  =  hl-^th\ 

and  tberefore  dA  =  hdl  +  ldh  +  2thdh  =  0 

From  (4)                P  =  Z  +  2hs/FTl 
^^a  dP  =  dl'i-2s/¥^ldh=^0  

Substituting  the  value  of  dl  from  (7)  in  (6) 


Therefore, 


l=2hy/F^-2th  . 


.(6). 
.(7). 
.(8). 


m  = 


4h^/¥Tl  -  2ht 
h 
T 
L«et  O  be  the  centre  of  the  water  surface  AD,  then  since  from  (8) 

I 


therefore,  in  Fig.  133,       CD  =  EG  =  OD. 


^SO  STDBATUCS  1 

Draw  'j¥  mii  •'•£  geryniifnifar  ta  CD  and  BC  respectirely. 

T^usi.  'i«£aaae  -zsb  ansfe  «>FD  s  a  TiaAt  an^le,  the  mn^ks  CD6 
inii  JOD  M*  OTiaL:  ami  mce  OF  =  ODcc«FOD,  auid  DG  =  OE,   J 
Ami  I«5-  =  »:D  ^:«CIfe-  -iu»5t. »,  OE=OF;  mod  ance  OEC  and 
•jFC  ir»  T:jnir  ing^i**.  a  orcie  wim  tj  as  centre  will  touch  the  sides 
if  The  'hannt*!,  a6  in  die  v:at«e  of  die  rectangular  channeL 


in  a  ebaimd  of  given  fbim  thai, 
a  TBaiimiim,  (b;  tiie  diacharge  majr 


and  tracispiDfin^. 


For  a  grven  slope  and  ron^faness  of  the  channel  v  is,  therefore, 
propc-rti'^cal  to  the  h3fiiraiiUc  mean  depth  and  will  be  a  mRYimnm 
wl^n  H  is  a  maxiniiim. 

That  isj  when  the  differential  of  p  is  zero,  or 

PJA-ArfP  =  0 (1). 

For  nummom  discharge,  Ar  is  a  maximum,  and  therefore, 


.(p)'iB: 


f  SL  maxm[iQm. 
Differentiating  and  equating  to  zero, 

^?^PdA-»A^P=0 (2). 

Affixing  values  to  n  and  p  this  differential  equation  can  be 
solved  for  special  cases.  It  will  generally  be  sufficiently  accurate 
to  assume  w  is  2  and  p  =  1,  as  in  the  Chezy  formula,  then 

n-^'p _  3 
n    ""2' 
and  the  equation  becomes 

3PdA-AdP  =  0 (3). 

137.  Depth  of  flow  in  a  circnlar  channel  of  given 
radius  and  slope,  when  the  velocity  is  a  maxinoLum. 

Lot  r  bo  tlio  radius  of  the  channel,  and  2<^  the  angle  subtended 
by  tho  Hiirfaco  of  the  water  at  the  centre  of  the  channel,  Pig.  184, 


FLOW  IN  OPEN   CHANNELS 


221 


Then  the  i^etted  perimeter 

The  area      A  =  r»*-r»8in*co8*  =  r^(^-5^V 
;  and  dA  =  r»<i^ -7^(508  2^(2^. 

Snbstitating  these  values  of  dP  and  dA  in   equation    (3), 
■ectaon  136, 

tan2<^  =  2<^. 

The  solution  in  this  case  is  obtained 
directly  as  follows, 

A     r  /^      sin  2<^N 


m 


P~2V  24  J' 


This  will  be  a  TnaTrimuTn  when  sin2<^ 
18  n^;ative,  and 

sin2<» 
2* 
18  a  maximnni,  or  when 


d  /sin2^\_^ 
d^\   24  )^   ' 


Fig.  184. 


.'.  2*cos2<^-8in2<^  =  0, 
and  tan2^  =  2^. 

The  solution  to  this  equation,  for  which  24  is  less  than  360^  is 
2^  =  257'' 27'. 
Then  A  =  2-738r», 

P  =  4-494r, 
m  =  -608r, 
and  the  depth  of  flow  d  =  l-626r. 

138.  Depth  of  flow  hi  a  ch'cular  channel  for  maximum 
diflcharge. 

Substituting  for  dP  and  dA  in  equation  (3),  section  136, 
6r*^<^  -  6r»«^  co8  2iM«^  -  2r»^«^  +  r»sin  2«^d</»  =  0, 
from  which  4^-66  cos  2<^  +  sin  2<^  =  0, 

and  therefore  <^  =  154'. 

Then  A  =  3044r», 

P  =  5-30r, 
m  =  •573r, 
and  the  depth  of  flow  d  =  l*899r. 

Similar  solutions  can  be  obtained  for  other  forms  of  channels, 
and  may  be  taken  hj  the  student  as  useful  mathematical  exercises 
bat  thejr  |ure  not  of  much  practical  utility. 


222 


HYDRA  trues 


139.    Curves    of    velocity    and    (iiscliarge    for    a   giif 
channel. 

The  depth  of  How  for  maximum  vrfocity,  or  discharge,  aai  U* 
determined  very  readily  hy  drawing  curres  of  velocity  atid  dis- 
charge for  different  depths  of  flow  in  the  channeL  This  metliud 
is  useful  and  instructive,  especially  to  those  students  who  are  not 
familiar  with  the  differential  calculus. 

As  an  exiumple,  velocities  and  discharge,  for  different  depths 
of  flow,  have  l>een  calculated  for  a  ^--ge  aqueductt  the  profUi^  of 
which  is  shown  in  Fig.  135,  and  the  pe  i  of  which  is  000013^31 
The  velocities  and  discharges  are  m  by  the  cnrves  drawn  in 

the  figure. 


Fig.  135. 

Values  of  A  and  P  for  different  depths  of  flow  were  first  deter- 
mined and  m  calculated  from  them. 

Tlie  velocities  were  calculated  by  the  formula 

using  values  of  C  from  column  Z,  Table  XXI. 

It  will  be  seen  that  the  velocity  does  not  vary  very  much  for 
all  depths  of  rt<jvv  greater  than  3  feet^  and  that  neither  the  velocity 
nor  the  discharge  is  a  niaxinnini  when  the  aqueduct  is  full;  the 
reason  being  that,  as  in  the  circular  channel,  as  the  surface  of  the 
water  approaches  the  top  of  the  aqueduct  the  wetted  perimeter  \ 
increases  muuh  muic!  i apidly  than  the  area. 

The  maximum  velocity  is  obtained  when  m  is  a  maximum 
and  equal  to  3*87,  but  the  maximum  discharge  is  given,  when  the 
depth  of    flow  is  greater  than   that  which   gives  the   greatest 


FLOW  IX  OPEN  CHANNELS  223 

Telocity.      A.  circle  is  shown  on  the  figure  which  gives  the  same 
maxinmm  discliarg^e. 

The  student   should  draw  similar  curves  for  the  egg-shaped 
aewer  or  otKer  form  of  channel. 


140.     AjfpiiBmtiflM  of  the  finrmiiU. 

Problewi  1.     To  find  the  flow  in  a  channel  of  given  section  and  slope. 

This  is  the  simplest  problem  and  can  be  solved  by  the  application  of  either  the 
logftnllmiie  fonnuls  or  by  Bazin's  formula. 

The  only  difficoltj  that  presents  itself,  is  to  affix  values  to  k,  n,  and  p  in  the 
logsnthmie  fonnohi  or  to  7  in  Bazin's  formula. 


(1)     By  the  logdritkmie  formula. 
Wnwt 


agn  some  value  to  k^  n,  and  p  by  comparing  the  lining  of  the  channel 

mfA  those  given  in  Tables  XXIV  to  XXXin.    Let  w  be  the  cross  sectional  area  of 
the  water. 

Then  since  i  = -^^  , 

log  V  =  -  log  t  +  —  log  ffl  -  -  log  Ae, 

n  n  ti 

mnd  Q=«.r, 

^^  logQ  =  log«  +  -logi+^logm- -log*. 

n  11  n 

(3)      By  the  Chezy  formula,  using  BazirC$  coefficient. 

The  coefficient  for  a  given  value  of  m  must  be  first  calculated  from  the  formula 

c=.  "^-^ 


or  taken  from  Table  XXI. 
Then 


157-5 
1  +  ^^ 


Example'.     Determine  the  flow  in  a  circular  culvert  9  ft.  diameter,  lined  with 
sooth  brick,  the  slope  being  1  in  2000,  and  the  channel  half  full. 


^^  =i=2-25'. 


Wetted  perimeter     4 
(1)      By  the  logarithmic  formula 


t=  00073^ 
m 


116  • 


Therefore,   log ,=:^  log  0005+  J;^ log  2-25  -  ^ log  00007. 

17=4*55  ft.  per  sec., 
«='-2— =31-8  sq.ft., 
Q=145  cubic  feet  per  sec. 

i2)     By  the  Chery  formula,  unng  Baxin's  coefficient, 

^2-25 
r  =  182  V2~25T^^= 4-43  ft.  per  sec. 
Q  =  31-8  X  3-35  =  141  cubic  ft.  per  sec. 


224  HYDRAULICS 

Problem  2.  To  find  the  diameter  of  a  eiroular  ohaxm^  of  giTen  slope,  for  whidi 
the  maximum  discharge  is  Q  cubic  feet  pear  second. 

The  hydraulic  mean  depUi  m  for  maximum  discharge  is  *678r  (oeetion  138)  tad 
A  =  3044r». 


Then  the  velocity  is 

t;=•767C^/^^ 

and 

Q=2•37Cr^^/^. 

therefore 

*'=Ri3V  OT' 

and  the  diameter 

»— v^- 

The  coefficient  0  is  unknown,  but  by  assuming  a  value  for  it,  an  approzunatioa 
to  D  can  be  obtained ;  a  new  value  for  0  can  then  be  taken  and  a  nearer  appraxi- 
mation  to  D  determined ;  a  third  value  for  G  will  give  a  still  nearer  approximatiiii 
to  D. 

Example.  A  circular  aqueduct  lined  with  concrete  has  a  diameter  of  6'  9"  and 
a  slope  of  1  foot  per  mile. 

To  find  the  diameter  of  two  oast-iron  siphon  pipes  5  miles  long,  to  be  put  m 
series  with  the  aqueduct,  and  which  shall  have  the  same  disd^arge ;  the  diffenoot 
of  level  between  the  two  ends  of  the  siphon  being  12*6  feet. 

The  value  of  m  for  the  brick  lined  aqueduct  of  cirenlar  section  when  tilt 
discharge  is  a  maximum  is  *573r=-64  feet. 

The  area  A=:8'044rS=25  sq.  ft. 

Taking  C  as  130  from  Table  XXI  for  the  brick  culvert  and  110  for  the  cait-inn 
pipe  from  Table  XII,  then 


64 
5280 


Therefore  dl=__^__   ^_ 


2-26' 
d=400  feet. 

Problem  3.  Having  given  the  bottom  width  {,  the  slope  t,  and  the  side  slopes  I 
of  a  trapezoidal  earth  channel,  to  calculate  the  discharge  for  a  given  depth.     | 

First  calculate  m  from  equation  (5),  section  135. 

From  Table  XXI  determine  the  corresponding  value  of  C,  or  calculate  C  from 
Bazin's  formula, 


then 

and 

A  convenient  formula  to  remember  is  the  approximate  formula  for  oidinHj 
earth  channels 

t;  =  50vm*i 

For  values  of  m  greater  than  2,  v  as  calculated  from  this  formula  ia  yen 
equal  to  v  obtained  by  using  Bazin's  formula. 

rrv    r         ,                                 .    •CH)037t;«'» 
The  formula  i= ^^ — 

may  also  be  used. 


FLOW  IN  OPEN  CHANNELS 


225 


Examiple.     An  og6iDMXj  euth  duumel  has  a  width  1=  10  feet,  a  depth,  d=4  feet, 
adailope<  =  v^T-     Side-alopes  1  to  1.    To  find  Q 

A=46  aq.  ft., 
P=21-212  ft., 
»=2*16ft, 
157-6 


C=- 


1  + 


Ta 


-=60-6, 


From  tbe  formnla 


^/?16 
vs  1-625  ft  per  sec., 
Q=r74*7  oabio  fk.  per  see. 


v=l*68  ft.  per  see., 
Q=75  cubic  ft.  per  sec., 
From  the  logarithmio  fbrmnla 

.    •00087t?«-^ 

• W^^' 

vs  1-649  ft.  per  sec., 
Q=75*8  cubic  feet  per  sec. 

Prtiblew^  4.     Haying  given  the  flow  in  a  canal,  the  slope,  and  the  side  slopes,  to 
(hwl  the  dimensions  of  the  profile  and  the  mean  velocity  of  flow, 
(a)    When  the  canal  is  of  the  best  form. 
(5)    When  the  depth  is  given. 

In  the  first  ease  m=^  *  ^'^^  ^^  equations  (8)  and  (4)  respectively,  section  18i> 

T=l  +  2h  ,JF+T. 
A 


Therefore 


Sabetitoting  ^  for  si 


m= r=r 


UHl 


Bet 

Therefore 


uid 


4hjt'»+l-2th 


2>j€^  +  l-t 
A«=M(2N/«*+i-t)2. 

2    hH2y/wn-t)^ 


(1). 


c«.t(2^A>+r-t)« 

A  ▼alne  for  C  should  be  chosen,  say  C=70,  and  h  calculated,  from  which  a  mean 
vmine  for  m=^aaihe  obtained. 

A  neeJ'er  approximation  to  h  can  then  be  determined  by  choosing  a  new  valae  of  C, 
fgffgo  T»ble  XXI  corresponding  to  this  approximate  valae  of  m,  and  recalcalatiD^ 
h  from  equation  (1). 

jr^ampU,  An  earthen  channel  to  be  kept  in  very  good  condition,  having  a  slope 
ai  1  in  10,000,  and  side  slopes  2  to  1,  is  required  to  discharge  100  cubic  feet 
«MMad  :  to  find  the  dimensions  of  the  channel;  take  C=70. 


troni 

the  ad»  iloMt  C»  »e>  an4  Uie  M 


TV  »Mui  T^omKj 


c^  m  whkh  is  nni 
tytr  nf  dJK  Ke  sol 


ja»i 


:^  r  aoififlMnt  C  i*  unknowfi,  ainm  k  ^peads  upon  ih»  ^ 
.  ml  «vm  if  1  tmlm:  fnvChv^  uemned  the  equAUaa  cusot  I 
'  *  !<  dcwftlkk,  Uieretore,  u>  toltt  bj  ftppn)^iiuLtioQ> 
-ini.  fisid  from  mlamn  4«  T^e  XXI,  the  con^sponjuf 


ate 


vhkli  viU  probfthlT  not  btttl^ 


Fv  wmtm  iDt  ^  &oin  the  fbnnala 


TW  ycv^tMi  cift:^  hw  fodtwl  In  m 


TW  wiJ^M^M  « loii^  I 


A  Moe^  Tilac  for  L  1 

Igr  ■■hillT^hmt  ^titdm  fonnok  (2),  it  ^  in  ^paai!^ 
.   if  eo.  ^  »FF^  , 
d  as  4iiiMimil  bj  luing  thi  i vtiM <£ ' 
mmiij  Avail  )ir«eii«»l  ptir  poMp 


'  ««3r  by  ^  kifaritbi&k  fom  lik 


t«a-l^miI'$i^^EBttf^,  mdllM-flOflg 


• 


FLOW  TS  OPEN  CHANNELS 


227 


£xmmtpir.     The  depth  of  &a  ardinary  earth  ch&nn&t  Ib  4  feet,  the  uide  slopes 
1  ta  I4  tbe  Alope  1  ID  €0€0  and  the  diiohaife  is  to  be  7000  oublc  feet  per  minute. 
Find  the  bottom  width  of  the  fhnnn^K 
Ammaaxe  m  "emi&t  for  m,  iaj  2  feet. 
Frc»iii  the  lof^tilhinic  formula 

tl  log  r=log  1  +  1*6  log  TO -4-6682 , (3), 

V  - 1*122  feet  per  aeo* 
7000 
'^^^  ^"1-022:00"^^ '^'l*^*^*' 

,    104 -IS    „„,^ 

..  1= — 3 ^32  feet. 

4 

SulvtitiiliiiK  thi^  mine  for  I  in  eqtuitioo  (2) 

4x23  +  16    „  ,« 

22  +  8^2 
Bcwlcnlatinir  p  &om  fonucLla  (S^ 

Tlnflii  A  =  75  feet, 

1=14-75  feet, 
id  in  =2*88  feet. 

Tbe  ^iTst  ralue  of  i  ia,  therefore,  too  large,  and  this  second  valati  U  too  small 
Tbird  irmloes  were  foiuijd  to  be      17  =£  1*455, 

A=eo-2, 

f=  16*05. 
in  =  2-9B5* 
X^i*  ^vmlna  cf  1 19  again  too  lafge. 

▲  tatrtti  ^coiatioD  gave  17- 1*475, 

A  =  79  2, 
i=15  8, 

afifirDiimatioii  has  been  carried  auffidentlj  far^  and  ereQ  further  tlian  ii 
_  tor  met  chattnele  the  coeffiqient  of  roughness  k  eaiiuot  b»  trusted  to 

mm  Acenimcjr  correvpondlng  to  the  stDall  differeDoe  between  tli^  third  and  four^i 
wm^lnes  of  ^ 

PfiM^tm  6.    HftTiDg  giTen  the  boltom  width  £,  the  slope  t  and  the  tilde  alnpes  of 
tjmpe«Qidal  channel,  to  hud  the  depth  d  for  a  given  diBcharge* 
Thlm  problem  is  aolTsd  exaetlj  as  the  last,  by  first  afisumlng  a  value  for  m,  and 
J  an  approiima(«  value  for  p  from  the  formula  v  =  C  Vm7. 
bj  BQbetitntion  in  ei^Qation  (I)  of  the  last  i^oblem^  and  solving  the 


Tbm^ 


/q     I     I 


atmg  this  valae  for  d  In  equation  (2),  a  new  valne  for  m  can  be  found » 
beoeev  a  Moond  approiimation  to  fi,  and  bo  on. 
Um«g    t^  logadthjnio  formula  the   procedure  is  exaotlj  the   same  as  for 

^Pro^Uml*.  HaTing a  natural  stieam  BC,  Fig.  13dri,  of  ^ven  slope,  it  in  required 
iluiafTiiIpe  the  point  C,  at  whiqh  a  canal,  of  trapezoidal  s^ciioti,  whkh  is  to 
jf«r  m  ^finite  qoantity  of  water  to  a  given  point  A  at  a  given  level,  shall  be 
I  to  jptn  ibe  stream  so  that  the  cost  of  the  canal  is  a  minimum. 

Tbe  aolnlion  here  given  10  praotioally  the  same  as  that  given  bj  M.  Flamant 
t  fiim  «3L0Qlktit  treatise  Hydmuli^ue, 


and  tbKt^fore 

from  which  wfi= 


228 

Let  I  be  the  ilope  of  the  ntrmtn,  i  of  the  msud,  k  Ihfi  height  ftboTt  ioa»  j 
of  the  Burfooe  of  the  watef  at  A,  and  h^  of  ib« 
water  in  the  tttr^tu  at  B^  at  «om€  diatanoe  X4 
from  C. 

Let  L  be  alfto  the  length  and  A  the 
MCtioiial  area  of  the  canal,  aod  t«t  it  ba 
aaaamed  that  the  ««ctLOD  of  the  canal !»  of  the 

moat  eoonomieo]  form,  or  m  =  - . 


Th«  nde  elopes  of  Ihe  canal  wil)  be  tw^ 
aooording  to  the  tiatai^  of  the  atrata  through  which  the  canal  u  eat,  uid  1 
8uppof»«<i  to  be  known. 

Then  the  level  of  the  water  at  C  ia 

Therefore 

Let  f  be  the  bottom  width  of  the 
■ectio&  la  then  dl^t^^  and 

i 

SobttitDtiug  f  m  for  d,       


4 


The  ooefficient  C  in  the  formula  v=C  iJnU  may  be  assmned  constant. 
Then  r«=C*iiM, 

and  r*=C*m«i«. 

Q 

For  V  snbetitnting  ^ ,  and  for  m'  the  above  value, 

<y  C^At< 

A*' 


and  ^'•'= ^  <2  >/?+l  - 1). 


Therefore 


The  cost  of  the  canal  will  be  approximately  proportional  to  the  product  of  the 
length  L  and  the  eroes  sectional  area,  or  to  the  cubical  content  of  the  exctvitioii. 
Let  £ik  be  the  price  per  cubic  yard  including  buying  of  land,  excavation  etc  Let  Ir 
be  the  total  cost. 

Then  £x=£k,h.A 


This  will  be  a  minimum  when  jt  =0. 

Differentiating  therefore,  and  equating  to  zero, 

|i*=tl.-i, 
and  t  =  fL 

The  most  economical  slope  is  therefore  |  of  the  slope  of  the  natural  streun. 

If  instead  of  taking  the  channel  of  the  best  form  the  depth  is  find,  tli^ 
slope  tss^.I. 


FliOW    IN   OPEN   CHANNELS 


229 


There  hate  heea  l#o  lusiiiDptions  mftde  m  the  calcuJAlioti,  neitliei  of  whifih  is 
ri^dljr  trite^  tho  &i^t  beiog  that  the  coefficieQt  C  ia  cod  slant,  And  the  second  th&X 
the  price  of  the  isftnaJ  ia  proportional  to  ita  cross  sectiontU  aTCft. 

It  win  not  &]vvjs  be  po««ible  to  adopt  the  elope  tbuH  founds  as  the  mean 
^reloctt^  mafit  be  maiDlained  within  thif  limits  Kiven  on  page  216,  and  It  ta  not 
adviaable  llial  the  slope  shonld  be  loss  than  1  in  10,000. 


EXAMPLES. 

(1)  The  areft  of  flow  in  a  sewer  was  found  to  be  (h2d  sq.  feet;  the 
peruneter  1*60  feet;  the  mcliaation  1  in  38' 7.    The  mean  Telocity 

llo^r  was  S'12  ^t  per  second*     Find  the  value  of  G  in  the  formula 

(2)  Tbe  dramage  area  of  a  certain  difitrict  was  19'32  aerefli  the  whole 
being  impermeahle  to  rain  water.     The  maximum  intensity  of  the 

was  0*300  ins^  per  hour  and  the  maximum  rate  of  discharge  regis - 
in  the  sewer  was  96%  of  the  total  rainfall. 
Find  the  size  of  a  circular  glazed  earthenware  culvert  having  a  slope  of 
is  50  aoi table  for  carrying  the  storm  water. 

(0)  Draw  a  curve  of  moan  velocities  and  a  curve  of  discharge  for  an 
«i|g^Hili&ped  brick  iewer,  using  Bazin's  coefficient.  Sewer,  6  feet  high  by 
^  ieei  groatesi  width;  dope  1  in  1200. 

(4>  The  sewer  of  the  previous  question  is  required  to  join  into  a  main 
Odt^ftll  «ewer.  To  cheapen  the  junction  with  the  main  outfall  it  is  thought 
ad^satile  to  make  the  last  100  feet  of  the  sewer  of  a  circular  steel  pipe 
feet  diameter,  the  junction  between  tbe  oval  sewer  and  the  pipe  being 
Jy  shaped  so  that  tliere  is  no  impediment  to  the  flow* 
Find  what  fall  the  circular  pipe  should  have  no  that  its  maximum 
^  shall  be  equal  to  the  maxlmmn  discharge  of  the  sewer.  Having 
ihe  slope*  draw  out  a  curve  of  velocity  and  discharge. 

i$)    A  oftnal  in  earth  has  a  slope  of  1  foot  in  20,000,  side  slopes  of 
faodflcmtal  to  1  vertical  >  a  depth  of  22  feet,  and  a   bottom  width  of 
find  the  volume  of  discharge. 
Baxin^  coefficient  y^2'35. 

^fi)  Oire  ihe  diameter  of  a  circular  brick  sewer  to  rnn  half -full  for  a 
peipQlation  of  80,000,  the  dinmal  volume  of  sewage  being  75  gallons  per 
head,  the  period  of  maximum  flow  6  hours^  and  tlie  available  fall  1  in  lOOO. 

Inst  C.  E.  1906, 

(7)    A  channel  is  to  be  cat  with  aide  slopes  of  1|  to  1 ;  depth  of  water, 
ak^iet  9  inches  per  mile:  discharge,  6,000  cubic  feet  per  minute, 
f  approximation  dimensions  of  clianneL 

0f    An  area  of  irrigated  land  requires  2  cubic  yards  of  water  per  hoar 
f  acre.     Find  dimensions  of  a  channel  3  feet  deep  and  with  a  side  slope 
^1  lo  L     Fall.  1^  feet  per  mile.    Area  to  be  irrigated,  6000  acres,     (Solve 
*F  apfiroadmation^)    y = 2*35* 

(9)  A  trapezoidal  channel  in  earth  of  the  most  economical  form  has  a 
i  0f  10  feet  uad  side  slopes  of  1  to  1.  Find  the  discharge  when  the 
\  li  IS  inches  per  mile.    y=2*35. 


S30 


HYDRAULtCS 


(10)    A  river  has  tlie  loUowiiig  section :— top  width,  800  feet ;  depth 
water^  20  feet ;  mde  slopes  1  to  1 ;  fail,  1  foot  per  mile*    Find  the  disdiAig^ 
iismg  Bazin'B  coefficient  for  earth  channels* 

(U)  A  channel  ib  bo  be  ooiwtnicted  for  a,  discharge  of  2000  cubic  feet 
per  second ;  the  fall  ii  1|  feet  per  mile ;  side  ^opes,  1  to  I ;  bottom  widlhr 
10  times  the  depth.    Find  dimensions  of  diannel.     Use  the  approximate' 

formula,  v=50^ftrL 

(12)  Find  the  dimenstoim  of  a  trapezoidal  earth  channel,  of  the 
economical  form,  to  convey  800  cubic  feet  per  second,  with  a  fall  of 
per  mile,  and  side  slopes,  1|  to  h     (Approxiniate  formula,) 

(13)  An  irrigation  channel,  with  side  slopes  of  H  to  1,  receiver  600 
cubic  feet  per  second.  Design  a  suitable  channel  of  3  feet  depth  lad 
determine  its  dimensions  and  slope*  The  mean  velocity  is  not  to  exceed 
2^  feet  per  second,    y  =  2  ■  3  5 . 

(14)  A  eanal^  excavated  in  rock,  has  vertical  sides,  a  bottom  widtli  d 
160  feet,  a  depth  of  22  feet,  and  the  slope  is  1  foot  in  20^000  feet.  Find  tlm 
discharge,    y  =  1  '64, 

(15)  A  length  of  the  canal  referred  to  in  question  (14)  is  in  earth.  It 
has  side  slopes  of  2  horizontal  to  1  vertical ;  its  width  at  the  water  ]kt 
is  290  feet  and  its  depth  22  feet. 

Fmd  the  slope  this  |x>rtion  of  the  canal  should  have,  taJdng  y  aa  S"!^ 

(16)  An  aqueduct  95  iJ  miles  long  is  made  op  of  a  cud  vert  &0J  miJn 
long  and  two  steel  pipes  8  feet  diameter  and  45  miles  long  laid  side  byst^ 
The  gradient  of  the  culvert  is  20  inches  to  tlie  mile,  and  of  the  pipes  2 1«e* 
to  the  mile.  Find  the  dimensions  of  a  rectangnlar  culvert  lined  with  ^ 
pointed  bricks  so  that  the  deptli  of  flow  shall  be  equal  to  the  wid^  crf  lb* 
culvert^  when  the  pipes  are  giving  their  maximum  dischaxge. 

Take  for  the  culvert  the  formula 


and  for  the  pipes  the  formula 


.    -000061  tJ^*88 


i^ 


■00050.  t?" 


(17)  The  Ganges  canal  at  Taoli  waa  found  to  ha^e  a  slope  el 
and  its  hydraulic  mean  deptli  m  was  7^0  feet ;  tlie  velocity  8 
by  vertical  floats  was  2*80  feet  per  second;  find  the  value  of  C  io3"' 
value  of  y  in  Bazin's  equation, 

(18)  The  following  data  were  obtainad  from  an  aqueduct  lined  fi^ 
brick  carefully  pointed : 


mmeties 

ID  me  Ires  per  S4». 

■229 

0-0001326 

'336 

■S81 

tt 

484 

•5S8 

t* 

■596 

'680 

It 

'691 

'888 

II 

*769 

991 

fi 

'848 

1148 

If 

'918 

M70 

n 

'922 

PLOW  IN   OPEN  CHANNELS  231. 

^^=  is  oirdin&tes,  — -  ba  absciesae;  Imd  T&liieaof  a  and  ^  in  Barb's 

1^  and  thos  dedaee  a  valne  of  y  for  this  aqueduct. 

Aa  aqtiedoct  l(fl\  miles  long  eonsbts  of  13|  miles  of  siphon,  and 
niodsr  of  a  masonry  cnivert  6  feet  10|  inches  diameter  with  a  gradient 
I  iOPCl  The  niphone  coniiit  of  two  lines  of  caet-irQii  pipes  43  inches 
Air  tiaruig  a  slope  of  1  in  500,     Determine  the  maximum  cUaaharge. 

An  sqaednet  ocmststs  partly  of  the  section  shown  in  Fig,  131, 
iT»  ft&d  fMutly  (ie,  when  crossing  valleys)  of  38  inches  diameter  cast- 


tbe  minimnm  dope  of  the  siphons^  so  that  the  aqueduct 
liacisafge  15,000,000  giillons  per  day^  aiid  the  slope  of  the  masonry 
so  that  the  water  shall  not  be  more  than  4  feet  6  inches  deep  in 


)  Calealaie  the  qtiantity  delivered  by  the  water  main  in  question  (30), 
|Lper  day  of  34  hours, 
^Hpiooiit,  representing  the  water  supply  of  a  city,  is  discharged  into 
Hn  at  tha  rate  of  one -half  the  total  daily  volume  in  6  hours,  and  m 
lebled  by  rainfalL  Find  the  diameter  of  the  circular  brick  outfall 
which  wUi  carry  off  the  combined  flow  when  running  half  full,  the 
hie  Ml  being  1  in  1500,     Use  Bamn's  eoefiicient  for  brick  channels. 

\)     Determine    for    a   smooth   eyhndrical   cast-iron  pipe  the  angle 
at  the  centre  by  the  wetted  perimeter,  when  tlie  velocity  of  flow 
Determine  the  hydraulic  mean  depth  of  the  pipe  nnder 
Land,  Un,  1905. 

%  A  8*fliQh  drain  pipe  is  Laid  at  a  slope  of  1  in  150,  and  the  valne  of 
17  {v^e^mi).  Find  a  general  expression  for  the  angle  subtended  at 
mtre  by  the  water  hue.  and  the  velocity  of  flow ;  and  indicate  how  Uie 
d  e^uatioB^  may  be  solved  when  the  discharge  is  given.    Loud.  Un. 

Ir    Sifter^  Qit^e^unt  ef  tf%«  hUiarical  ihi^^hpm^nt  of  the  pip^  and  ehannel  formulae. 

kui  reniJirki^bte  Uint,  although  tti^  pmctiise  of  conduGting  water  ulnng  piped 

bapek   for  doniaitio  and   other  purpnieg   haa   been  cfLrried  on   for  many 

^Hhio  aerioiii  ittempt   bo  discover   the   la^'fi   regulating   th^   fiow  fteems 

Wmmtk  att^topted  nntiJ  Ih*^  eigblc^nth  centurj.     It  soema  diMcult  to  realise 

It  giffmnlie  «i?b ernes  of  wnti^r  ihaii-ibiiiion  of  the  ancstetit  Gitiea  cnuld  hftve  be^n 

~  viibout  E^ycb  kouw ledge,  but  ci'rtain  it  i«,  tb&t  whatever  infonnnticin  they 

I.  it  WM  lost  diiriuf;  ibe  middle  ages, 

■i  if  pMoiiar  interest  to  note  tbe  trouble  taken  by  the  Boman  engineers  in 

of  their  aqueducts.      In  order  to  keep  the  elope  oonBtftnl  they 

thnjugb  biUs  and  a&rried  their  fujuedocts  on  mnf^iSeant  arehea.     The^  j 

a^nedtiet  wai  38  milei  loug  and  dad  a  coiii^tAnt  dope  of  five  feel  per  mile, 

mtiy  they  were  unaware  of  ihe  »tmple  fact  that  it  is  not  necessary  for  a  pipe 

idaet  aonneoting  two  r^aervoira  to  be  laid  perfectly  straight t  or  sine  they 

the  water  tt  all  parla  of  the  aqoedticts  to  b^  at  atmospheric  pressure, 

Seliwet^er  in  his  interesting  treatise  on  bydroHtatiiiB  and  hydraulics 
in  1729  quotes  experiments  by  Marriott  Ehowitig  that,  a  pipe  1400  yards 
i|  loobaA  diameter,  only  gave  |  of  the  diiM^harge  which  a  bole  1|  inches  diameter 
idt  of  m  tiitik  wonld  giv«  under  the  saoie  heudp  and  also  eiplama  that  the 
of  the  Hquid  in  the  pipes  is  diminished  by  friction,  but  he  is  entirely 
biws  regulatinjj  the  flow  of  Huida  through  pipes.     Even  as  late  as 


2^2  HYDRAULICS  1 

17B6  Dq  Bta^t*  wrote,  "We  mm  jet  m  absolute  ignoranoe  of  the  laws  lo  wblek  ih«] 
moTement  of  wat^r  is  enbjected.*^  I 

Thei  <»adii^»t  recorded  experimetils  of  any  valu<}  on  long  pipes  lyne  tboM  oil 
Couplet,  in  whiish  he  measoced  the  flow  through  the  pipes  wm^  aupplicHl  tbal 
Ibmoiia  fountains  of  Versailles  in  1732,  In  1771  Abb^  Bossni  made  experimefite  nal 
iow  in  pipes  and  ohannals,  these  bein^  followed  by  tba  expedments  of  Do  BuaU  wbal 
eironeotiflly  argued  that  the  loss  of  head  due  |p  f notion  iu  a  pip«  was  indsiiendeall 
of  the  internal  »urfa43e  of  the  pipe,  and  gave  a  oomplieated  formula  for  the  Tdoei^l 
of  ^ow  when  the  head  and  the  leo^th  of  the  pipe  were  known.  I 

In  1T75  M.  Chezj  from  expenmeuts  upon  the  flow  in  an  open  eatial,  caine  IdH 
the  conclusion  that  the  6uiJ  friction  was  propoTtioual  to  the  velooitj  squared,  and' 
that  the  nlope  of  Ihe  channel  multiplied  by  the  croB»  sectional  area  of  the  stt«*in, 
wag  equal  to  the  prodoct  of  the  length  of  the  wetted  surface  measured  on  tbe  enw 
aeetiont  the  relooitj  iquared^  and  aotue  eotiatant*  or 

iA^Fav* :,, .,.41}, 

I  being  the  slope  of  the  bed  of  the  channel,  A  the  croe^  sectional  area  of  the  atieam, 
P  Ihe  wetted  perimeter^  and  d  a  coefficient. 

From  thi^  ie  dedtieed  the  well-known  Ghezj  formula 


^-^^  ^i-ojmu 


Pronyf ,  applying  to  the  iow  of  water  in  pipes  the  results  of  the  cl&aaical  esp«ii* 
ments  of  Coulotoli  on  fluid  frietion,  from  whieh  Coulomb  had  dednoed  the  lav  thst 
fluid  frieiioB  was  proportionai  to  av-k-bty^t  arriyed  at  the  formula 

«i(= at?  +  ^«  =  [-  +  p\  r/^. 

This  is  elmilar  to  the  Chesj  formola*  (  -  +  /9  j  being  equal  io  ^ , 

By  an  eiatnination  of  the  experimentii  of  Couplet,  Boseut,  atid  Du  Bnat,  Proii|| 
gave  values  to  a  and  ^  whiah  when  trane formed  into  British  units  are, 

0^-00001733, 
/3  =  '00010614, 

For  TclooitieSi  above  S  feet  per  seeond,  Pronj  negleotod  the  term  containing  tbej 
first  power  of  the  velocity  and  deduced  the  formula 

Ue  continued  the  mistake  uf  Du  Bunt  and  assumed  that  the  Criotion  wai  in< 
dependent  of  the  eoodition  of  the  internal  ^orfaoe  of  the  pipe  and  gave  the  foUowioir 
eiplanation :  *' When  the  fluid  flow?  in  a  pipe  or  upon  a  wette^l  surface  a  Slnx'' 
fluid  adhere;;  to  the  surface,  and  tltis  film  may  be  regarded  an  enoloiing  the  tail 
of  fluid  in  raotiont-*'    That  such  a  film  encloses  the  moving  water  reoeiyes  soppenl 
from  the  eiperiments  of  ProfesHor  Hele  Shswf.     The  expei-iment!?  were  made  npAl 
sueh  K  KmaD  seals  that  it  is  dilBcult  to  say  how  far  the  results  obtained  arft  indie**' 
tivti  of  the  conditiona  of  flow  in  large  pipes,  and  if  the  ilm  exists  it  do^  not  sera 
to  m\  in  the  way  argued  by  Prony. 

H 

The  value  of  t  in  Prony's  formula  was  equal  tn  y ,  H  inaluding,  not  onlj  tt» 

loaa  of  head  doe  to  friction  bat*  as  measured  by  Couplet,  Bossut  and  Da  I' 
it  ako  included  the  head  neoensary  to  give  velocity  to  the  water  and  to  overc. 
leflistance^  at  the  entrance  to  the  pipe. 

Eytelwein  and  also  Aubisson,  both  niade  aUowaoces  for  these  loisea^  hs  ^ 

tracting  from  H  a  quantity  x— ,  and  then  determined  new  yalnes  for  a  and  I  ia  tb« 

formnla 

*  Zf  XhVcowTf  priliminaire  de  m*  PHnciptt  d'k^draulique, 

f  See  also  Girard'a  Movement  dc9  fluids  dans  let  tub^t  eapittairew,  1817, 

^  Traiti  d*hydrauliqut,  f  Engineer,  Aug,  1897  and  Maj  ISUB. 


FLOW  m  OPEN   CHANNELS 


233 


Thgy  iAv«  lo  a  and  b  the  rollowing  TaJaes. 

Eytelwem    (r  -  ^000023584, 

h=  ^000085434. 

Aubiaaon'     a  ==  'OOOO 1 8837, 

6=1)00104392. 

By  iMg)«ctui^  the  term  oonlaiiiiiig  v  to  the  first  power,  and  transform ing  the 
I  Atibisflon'B  formolA  reduces  to 


•'=«\^JT 


Hd 


3S-6er  ■ 
ToiiJDfc*  tQ  tbd  Efuryc^cpQ^dta  Bnl^nf^t^u,  gave  a  complicated  formula  for  ?  when 
»cul  ^  were  known,  bnt  gave  the  fiimfilified  formula,  for  veJocitiee  unoh  oa 
Are  geiQ^imllj  met  with  m  practice, 

8t  Teoant  made  ft  decided  depftrtEue  by  making  -  prdportlonal  to  v'r  imtead  of 

I  r*  M  to  lite  Ch«^  IbimiilA. 
Wbioi  eipiMKjd  IP  EngHflh  feei  aa  unitSi  hia  formula  becomes 


v  =  206(>?fi)T^^ 


I  by  an  examination  of  the  early  experiments  together  with  ten  others  bv 

avH 
and  on*  by  M.  GneyttArd  gave  to  the  ooeflScient  a  in  the  formula  ft  =  — - 

rftlne 

,  |b«  be  made  it  to  vaiy  with  the  Telocity* 

valiiia  of  a  and  ^  being  ol^QOI  U  , 

Wram  lins  formnlA  iablas  were  drawn  up  by  Weiabach,  and  in  England  by 
*  J,  which  were  considerably  need  for  calciilations  relating  to  flow  of 
in  pipes. 

l^iflcy,  as  explained  in  Chapter  \\  made  the  coefticient  a  to  vary  with  the 
r,  and  Qag«n  proposed  to  make  it  v&ry  with  both  the  velocity  aod  the 


M^  formula  then  beoame 


«■=&.-")'"• 


The  fbrtnclae  of  Qangmllet  and  Etitter  and  of  Biusin  hare  b#eii  given  in 
I  V  and  VI. 
lAmpe  &om  axperimentB  on  the  Dantzig  maina  and  other  pipes  proposed 
ElonBUlft 

*-"ffi*  < 

modifying  St  Tenant's  formula  «nd  antioipattng  the  formulae  of  Beynoldif 
a^d  Unwrnn,  Id  which, 

i  ftfid  p  being  variable  coeMcienta, 

*  Traiti  d'htfdfaulique. 


CHAPTER  VII. 

GAUGING  THE  FLOW  OF  WATER, 

142.  MeaBuidng  the  ^ow  of  water  by  weigMng. 
In  the  laboratory  or  workshop  a  flow  of  water  can  geiws 

be  tueRsiired  by  collecting  the  water  in   tanks ^  and  either 
direct  weigh  iiigi  or  by  measuring  the  voltime  from  the  \m 
capacity  of  the  tank,  the  discharge  in   a  given  time  cau 
determined,     lliis  is  the  most  accurate  method   of  measu 
water  and   should   be  adopt-ed   where  possible  in  ex  per 
work. 

In  pump  trials  or  in  measuring  the  supply  of  water  to  boilfl 
determining  the  quantity  by  direct  weighing  has  the  di^ct 
advantage  that  the  results  are  not  materially  afTtctetl  by 
changes  of  temperature*  It  is  generally  necessiirj"  tu  have  W 
tanks,  one  of  which  is  filling  while  the  other  is  being  weifbeJ 
and  emptied.  For  facility  in  weighing  the  tanks  should  etJ 
on  the  tables  of  weighing  machines. 

143,  Meters. 

Lirwri  meter.     An  ingenious  direct  weighing  meter 
gauging  practically  any  kind  of  liquid*  is  constrnckHl  . 
Figs.  136  and  137. 

It  consists  of  two  tanks  A"  and  A*,  each  of  which  can 
on  knife  edges  BB.    The  liquid  is  allowed  to  fall  into  a  she 
wliich  sl\^vels  about  the  centre  J,  and  fi'om  which  it  falls 
either  A*  or  A'  according  to  the  position  of  the  shoot.    The  1 
have  weights  D  at  one  end,  which  are  so  adjusted  that  wi 
**  -ta-in  weight  of  water  has  run  into  a  tank*  it  swings  ovt*r  i 
formu.-j^l^^  position,  Fig.   136j  and   flow  commences  thrtvagt 
H  "oe  C.    When  the  level  of  the  liquid  in  tlie  tank 

^^H         ,  j^  Dhc^^^^'  ^^^  i^t^igbts  D  cause  the  tank  to  come  back] 
^^H        t  See  also  csitioUj  but  the  siphon  continues  in  action  until  I 


OAUOiya  THE  FLOW  OF  WATEB 


235 


lenly  tilts  over  tlie  shoot  F^  and  the  liquid  is  discharged 

other  tank.     An  indicator  H   registers  th©  nuTiiber  of 

le  tanks  are  tUledjatid  as  at  each  tippling  a  detinite  weight 

is  etnptied  from  the  tank,  the  indicator  can  be  marked 

>unds  or  in  any  other  unit. 


Fig.  im. 


Fig.  VAl. 


Liner t  direct  weighing  meier. 

Meaatirmg  the  flow  by  means  of  an  orifice 
c<:»t?fficient  of  discharge  of  sharp-edged  orifices  can  be 
ith  cousiderablo  precisioUj  from  the  tables  of  Chapter  IVj 
coefficient  for  any  given  orifice  can  be   deterniined  for 
hm  he^ids  by  direct  measurement  of  the  How  in  a  given  time^ 
tdeicnbed  above*     Tlien,  knowing  the  coefficient  of  discharge  at 
■tidii  a  curve  of  rate  of  discharge  for  the  orifice,  as  in 
may  be  drawn,  and  the   orifice  can   then  b©  uised  to 
*  a  continuous  flow  of  water. 

!  orilice  should  be  made  in  the  side  or  bottom  of  a  tank.     If 

I  the  side  i>f  the  tank  the  lower  edge  sh<juld  be  at  k^ast  one  and 

If  to  twice  its  depth  above  the  bottom  of  the  tank,  and  the 

of  the  orifice  whether  hori?.ontal  or  vertical  should  be  at 

|0tie  and  a  half  to  twice  the  width  from  the  sides  of  the  tank, 

ink  i»hould  be  provided  with  baffle  plates,  or  some  other 

pnient,  fur  destroying  the  velocity  of  the  incoming  water 

t  cnFuring  quiet  water  in  the  neighbourhood  of  th©  orifice.    The 

pent  of  dischargu  i^  otherwise  indefinite.    Th©  head  over  the 

jnhould  be  observed  at  stated  intervals.     A  head-time  curve 

\  h^ftd  aas  ordinate^  and  time  as  abscissae  can  then  be  plotted 

,  189. 

From  the  head-discharge  curve  of  Fig.  138  the  rate  of  discharge 

found  for  any  head  h^  and  the  curve  of  Fig.  139  plotted 

of  this  curve  between  any  two  ordinates  AB  and  CD, 


I 


I 
I 
I 

I 
] 


236 


HYDRAULICS 


which  is  the  mean  ordinate  between  AS  and  CD  multiplied  bytbej 
time  tj  gives  the  discharge  from  the  orifice  in  tinae  t. 

The  head  h  can  be  measured  by  fixing  a  scale,  hBrlng  '•    " 
coinciding  ^vith  the  centre  of  the  orifice,  behind  a  tube  on  : 
of  the  tank. 


Fig,  im. 


145.    Meaauring  tlie  flow  in  open  channels. 

Large  open  channels :  floats.    The  oldest  and  simplest  me 
of  determining  approximately  the  discharge  in  an  open  cha 
by  means  of  floats. 

A  part  of  the  channel  as  straight  as  possible  is  selected,  i 
which  i\m  flow  may  be  considered  as  uniform. 

The  readings  should  be  taken  on  a  calm  day  as  a  down^st 
wind  will  accelei*ate  the  floats  and  an  up-stream  wind  retard  1 

Two  cords  are  stretched  across  the  channel,  as  near  to  J 
surface  as  possible,  and  perpendicular  to  the  direction  of  flow, 
distance  apart  of  the  cords  should  be  as  great  as  possible  coi:  - 
with  uniform  fiow^  and  should  not  be  less  than  150  feet,  i 
boat,  anchored  at  a  point  not  less  than  50  to  70  feet  aboTe  alreafflt  j 
so  that  the  float  shall  acquire  before  reaching  the  first  Hneaj 
imiform  velocity,  the  float  is  allowed  to  fall  into  the  streiim  audi 


GAUOIKG  THE   FLOW  OF  WATEH 


237 


m  time  carefuDy^  noted  hy  means  of  a  clironometer  at  wliich  it 
KMoa  l>ot]i  the  first  aod  aecond  line.  If  the  velocity  ig  slow,  the 
Issenrei*  ma^y  walk  along  the  bank  while  the  float  is  moving. from 
me  cortl  to  the  other,  but  if  it  is  greater  than  200  feet  per  minute 
M  observers  will  generally  be  reqaired,  one  at  each  line, 
^kA  better  method,  and  one  which  enables  any  deviation  of  the 
Hi  fmni  a  path  perpendicular  to  the  lines  to  be  determined,  is, 
Sr  two  observers  provided  with  box  sextan tSj  or  theodoHtes,  to  be 
ioned  at  the  points  A  and  B,  which  are  in  the  planes  of  the 
hneo.  As  the  float  passes  the  line  AA  at  D,  the  observer 
L  fign&Is^  and  the  observer  at  B  measures  the  angle  ABD 
if  both  ane  ppivided  with  watches,  each  notes  the  time. 
en  the  float  passes  the  line  BB  at  E,  the  observer  at  B  signals, 
the  obsen^er  at  A  measures  the  angle  BAE,  and  both 
again  note  the  time.  The  distance  DE  can  then  be 
arately  determined  by  calculation  of  by  a  sc^le  drawing,  and 
\  mean  velocity  of  the  float  obtained,  by  dividing  by  the  time. 
I  To  ensure  the  mean  velocities  of  the  floats  being  nearly  equal 
be  mean  velocity  of  the  particles  of  water  in  contact  with 
,  their  horizontal  dimensions  should  be  as  small  as  possible, 
to  reduce  friction,  and  the  portion  of  the  float  above  the 
of  the  water  should  be  very  small  to  diminish  the  effect  of 

^.pointed  out  in  section  130,  the  distribution  of  velocity  in 

averse  section  is  not  by  any  means   uniform   and   it  is 

(T,  therefore^  to  obtain  the  mean  velocity  on  a  number  of 

planes,  by  finding  not  only  the  surface  velocity,  but  also 

JvekxHty  at  various  depths  on  each  vertical. 

1 146.    Surface  floats, 

5m^€e  floats  may  consist  of  washers  of  cork,  or  wood,  or 

small  fioating  bodies,  weighted  so  as  to  Just  project  above 

surface-    The  surface  velocity  is,  however,  so  likely  to 

It?cted  by  wind,  that  it  is  better  to  obtain  the  velocity  a 

distance  below  the  surface* 

147.    I>ouble  floats. 

To  uiesi^ur*?  the  velocity  at  points  below  the  aur&ce  double 
are  employed*    They  consist  of  two  bodies  connected  by 
of  a  fine  wire  or  cord,  the  upper  one  being  made  as  small 
lible  so  as  to  reduce  its  resistance, 

1%  on  the  Irrawaddi,  used  two  wooden  floats  connected 
\  fine  fittkmg  linej  the  lower  float  being  a  cylinder  I  foot  longi 

•  Proe.lmL  V.  E„  IS98. 


riib 


238 


HYDRAULICS 


and  6  iBches  diameter,  hollow  undemeatli  and  loaded  with 

gink  it  to  any  required  depth }  the  upper  floaty  which  swam  od 
surface,  waa  of  light  wcKMi  1  inch  tliiuk,  and  carried  a  small  Am 
The  surface  velocity  was  obtained  by  sinking  the  lower  I 
to  a  depth  of  3J  feet,  the  velocity  at  this  depth  being  not  i 
different  from  the  surface  velocity  and  the  motion  of  the  float  i 
independent  of  the  effect  of  the  wind* 


Kg.  141.     Gurby'f  ourrent  m©t«r< 

Subsurface  velocities  were  measured  by  increasing  the 
of  the  lower  float  by  lengths  of  Si  feet  until  the  bof 
reached. 


OAnGINe   THE   FLOW  OF  WATER 


H  Gordon  has  cM>mpared  the  results  ob tamed  hy  floats  with  those 
^Btftizied  hf  mestis  of  a  cuiTeiit  meter  (see  section  149).  For 
HpaU  depths  and  low  velocities  the  results  obtained  by  double 
^■KtB  are  ^rly  accurate^  but  at  high  velocities  and  great  depths, 
^Ee  Talocities  obtained  are  too  high.  The  error  is  from  0  to  10 
per  cent. 

Double  floats  are  soinetiinea  made  with  two  eimilar  floats,  of 
the  same  dimensioiiS|  one  of  which  is  ballasted  so  as  to  float  at  any 
reiiuired  depth  and  the  other  floats  just  below  the  surface.  The 
wwiodtf^  of  the  float  is  then  the  mean  of  the  surface  velocity 
mad  the  velocity  at  the  depth  of  the  lower  float. 

148.  Bod  floats. 

The  mean  velocity,  on  any  verticalj  may  be  obtained  ap- 
pro jdmately  by  means  of  a  rod  float,  which  consists  of  a  long  rod 
baring  at  the  lower  end  a  small  hollow  cylinder,  which  may  be 
fillad  with  lead  or  other  ballast  so  a^  to  keep  the  rod  nearly 
vertical. 

The  rc»d  is  made  sufficiently  long,  and  the  ballast  adjusted,  so 
thBt  ita  lower  end  is  near  to  the  bed  of  the  stream,  and  its  upper 
end  project*  slightly  above  the  water.    Its  velocity  is  approximately 

tn^&n  velocity  in  the  vertical  plane  in  wliich  it  floats. 

149.  The  onrrent  nieteT. 
The  discharge  of  large  channels  or  rivers  can  be  obtained  most 

|ccitiv'oni*?ntly  and  accurately  by  determining  the  velocity  of  flow 
number  of  pi>ints  in  a  transverse  section  by  means  of  a  current 

The  arrangement  shown  in  Fig.  141  is  a  meter  of  the  anemo- 
ineter  type,  A  wheel  is  mounted  on  a  vertical  spindle  and  has 
fi%-e  conical  buckets.  The  spindle  revolves  in  bearings,  from 
^rkicfa  all  water  is  excluded,  and  which  are  carefully  made  so 
tbat  the  fricrion  shall  remain  constant.  The  upper  end  of  the 
«tpixidle  extends  above  its  bearingj  into  an  air-tight  chamber,  and 
im  shaped  to  form  an  eccentric.  A  light  spring  presses  against 
eecentric,  and  successively  makes  and  breaks  an  electric 
r  ■  -  the  wheel  revolves.  The  number  of  revolutions  of  the 
r  njcorded  by  an  electric  register,  which  can  be  arranged 

any  convenient  tlistance  from  the  wheeL  When  the  circuit  is 
taade,  an  electro^magnet  in  the  register  moves  a  lever,  at  the  end 
of  which  is  a  pawl  carrying  forward  a  mtchet  wheel  one  tooth 
I  for  each  revolution  of  the  spindle.  The  fi'ame  of  the  meter,  which 
im  made  of  bronze^  is  pivoted  to  a  hollow  cyUnder  which  can  be 
clamped  in  any  desired  position  to  a  vertical  rod.    At  the  rightr 


240 


HTDBAtTttCS 


hand  side  is  a  rudder  having  four  lig^ht  metal  wing^  which 
balances  the  wheel  and  its  frame.  Wlien  the  meter  is  being  miei 
in  deep  waters  it  m  suspended  by  means  of  a  tine  cable,  and 
the  lower  end  of  the  rod  is  fijced  a  lead  weight.  The  el 
circuit  wires  are  passed  through  the  trunnion  and  so  haw 
tendency  to  pull  the  meter  out  of  the  line  of  current, 
placed  in  a  current  the  meter  is  free  to  move  about  the  hoi 
axis,  and  also  about  a  vertical  axis,  ao  that  it  adjosta  il 
the  direction  of  the  current. 

The  meters  are  rated  by  experiment  and  the  makers 
the  following  method.    The  meter  should  be  attached  to 
al  a  boati  as  shown  in  Fig.  1^,  and  immersed  in  still  wmter 
less  til  an  two  feet  deep*     A  thin  rope  should  be  attached 
boatj  and  pa^ed  round  a  pulley  in  lint*  %vith  the  course  in 
the  boat  is  to  move.    Two  parallel  lines  about  200  feet 
should  be  staked  on  shore  and  at  right  angles  to  the  course 
boat.     The  boat  should  be  without  a  rudder,  but  in  the  huai 
the  observer  should  be  a  boatman  to  keep  the  boat  from 


Fig,  142, 

into  the  shore.    The  boat  should  then  be  liauled  between  i 
ranging  lines  at  varying  speeds,  which  during  each  passage  i 
be  as  unifonn  as  possible.     With  each  meter  a  reduction 
supplied  from  which  the  velocity  of  the  stream  in  feet  jm^^t  b 
can  be  at  once  determined  from  the  niunber  of  revolutions  j 
per  second  of  the  wheel. 

The  Haftkell  meter  has  a  wheel  of  the  screw  ] 
revolving  upon  a  horizontal  axis.     Its  mode  of   a. 
eimilar  to  the  one  described. 

Comparative  tests  of  the  discharges  along  a  rectangoiarj 
as  measured  by  these  two  meters  and  by  a  sharp-edged  ' 
had  been  carefully  calibrated,  in  no  case  differed  by  moit^ 
5  per  cent,  and  the  agreement  was  generally  much  closer*. 


*  Mnrphj  on   current  Meter  and  Wdr  lU^abargesv  Pruceedinp  i» 
Vol  TZYiu  p.  779. 


GAT7GINO  THE  FLOW  OF  WATEB 


241 


150.    ntot  tube. 

Another  apparatus  which  can  be  used  for  determining  the 
docity  at  a  point  in  a  flowing  stream,  even  when  the  stream  is  of 
ull  dimensions,  as  for  example  a  small  pipe,  is  called  a  Pitot 
die. 

In  its  simplest  form,  as  originally  proposed  by  Pitot  in  1732, 


=*=, 


Jll 


E 


Fig.  143.    Pitot  tube. 


3k  ooDBisEtB  of  a  glass  tube,  with  a 
BmII  orifice  at  one  end  which  may 
be  tomed  to  receive  the  impact  of 
Ik  stream  as  shown  in  Fig.  143. 
Be  water  in  the  tube  rises  to  a 
iMght  h  above  the  free  surface  of 
Me  water,  the  value  of  h  depending 
i  the  velocity  v  at  the  orifice  of 

Btnbe.    If  a  second  tube  is  placed 

I  the  first  with  an  orifice  0  parallel  to  the  direction  of  flow, 
tte  water  will  rise  in  this  tube  nearly  to  the  level  of  the  free 
■Bzface,  the  &11  h  being  due  to  a  slight  diminution  in  pressure 
9X  the  mouth  of  the  tube,  caused  probably  by  the  stream  lines 
laving  their  directions  changed  at  the  mouth  of  the  tube.  A 
&rther  depression  of  the  free  surface  in  the  tube  takes  place> 
if  the  tube,  as  EF,  is  turned  so  that  the  orifice  faces  down  stream. 

Theory  of  the  Pitot  tube.  Let  v  be  the  velocity  of  the  stream 
at  the  orifice  of  the  tube  in  ft.  per  sec.  and  a  the  area  of  the 
orifice  in  sq.  ft. 

The  quantity  of  water  striking  the  orifice  per  second  is  wav 

pound;s. 

to 
The  momentum  is  therefore  -  .  a .  v'  pounds  feet. 

If  the  momentum  of  this  water  is  entirely  destroyed,  the 
pressure  on  the  orifice  which,  according  to  Newton's  second  law  of 
motion  is  equal  to  the  rate  of  change  of  momentum,  is 


P  = 


wav 


find  the  pressure  per  unit  area  is 

9 
The  equivalent  head 

h  -  V!^  =  - 

According  to  this  theory,  the  head  of  water  in  the  tube,  due  to 
le  impact,  is  therefore  twice  |- ,  the  head  due  to  the  velocity  v,  and 


242 


the  water  shotild  rise  in  the  tube  bo  a  height  above  th©( 
equal  to  h. 

Experimoiit  allows  that  the  actual  height  the  water  ] 

tube  is  more  nearljr  equal  to  the  velocity  head 
and  the  head  b  ig  thum  generally  taken  as 


c  being  a  coefficient  for  any 
is  fairly  conKtantv 

Similarly  for  given  tubes 


tube,  which  escperunent  i 


and 

The  coefRcienta  are  detei 
the  velocities  of  which  are 
body  which  moves  through 
carefully  meitj^uring  h  for  diUereu^  velocities 


t 
7 

by  placing  the  tubes  in  i 
I,  or  by  attaching  them  \ 
iter  with  a  known  vclocit 


.i. 


Fig.   144. 

Darcy*  was  the  first  to  use  the  Pitot  tube  as  an  instrumei 
precision.  His  improved  apparatus  as  used  in  open  channels 
sisted  of  two  tubes  placed  side  by  side  as  in  Fig.  144,  the  or 
in  the  tubes  facing  up-stream  and  down-stream  respectively. 


Reeherchef  Hydrauliques,  etc.,  1857. 


QAI7GIKO  THE   FLOW  OF   WATER 


2m 


bobes  were  connected  at  the  top^  a  cock  C^  being  placed  in  the 
tnon  tube  to  allow  the  tubes  to  be  opened  or  closed  to  the 
At  the  lower  end  both  tubes  could  be  closed  at  the 
e  time  by  uieans  of  cock  C.  Wlien  the  apparatus  is  put  into 
in^  water,  the  cc»cks  C  and  C*  being  open,  the  free  surface 
is  the  tube  B  a  height  hi  and  is  depressed  in  D  an  amount 
Tbfi  cock  C^  i&>  then  closed,  and  the  apparatus  can  be  taken 
i  the  wster  and  the  difference  in  the  leYel  of  the  two  columns, 
h-hi^  fhj 

with  cotusiderable  accuracy* 

deeiredi  air  can  be  aspirated  from  the  tubes  and  the  eolunms 

to  rise  to  convenient  levels  for  observation,  without  moving 

The  difference  of  level  will  be  the  same,  whatever 

in  the  upper  part  of  the  tubeB, 

i  145  shows  one  of  the  forms  of  Pitot  tubes,  as  experimented 

by  Pnifeasor  Gardner  Williams*,  and  used  to  determine 

ribution  of  velocities  of  the  water  flowing  in  circular  pipes, 

arrangement  shown  in  Fig.  146,  ia  a  modified  form  of  the 

tias  used  by  Freeman  t  to  determine  the  distribution  of 

in  a  jet  of  water  issuing  from  a  fire  hose  under  con- 

ile  pressure.    As  shown  in  the  sketch,  the  small  orifice  0 

the  impact  of  the  stream  and  two  small  holes  Q  are  drilled 

be  T  in  a  direction  perpendicular  to  the  Bow,    The  lower 

^paratns  OV,  as  shown  in  the  sectional  plan,  is  made 

so  as  to   prevent  the   formation   of   eddies  in  the 

hood  of  the  orifices.    The  pressure  at  the  orifice  0  is 

ed  through  the  tube  OS,  and  the  pressure  at  Q  through 

QR.    To  measure  the  difference  of  pressure,  or  head, 

two  tubeS)  OS  and  QR  were  connected  to  a  differential 

SLttiilar  to  that  described  in  section  13  and  very  small 

of  head  could  thus  be  obtained  with  great  accuracy, 

lobe   shown  iii   Pig*  145  has  a  cigar-shaped  bulb,  the 

orifice  O  being  at  one  end  and  communicating  with  the 

OS.     There  are  four  small  openings  in  the  side  of  the  bulb, 

M  any  Tariations  of  pressure  outside  are  equalised  in  the 

The  pres^upes  are  transmitted  through  the  tubes  OS  and 

>  m  differential  gauge  as  in  the  case  above, 

r  I%»  147  is  shown  a  special  atuffing-bojc  used  by  IVofessor 

mOBBtto  allow  the  tube  to  be  moved  to  the  various  positions  in 


'•the  I 
ipri 


of  Fit0t  tiibea  api  atetl  by  ProfeSBor  WiHiflma,  E.  S<  Oele  tLnd 

of  the  Jm^\C,E,,  VpL  uli. 

16^^ 


1 


I 
I 


244 


HYDRAULICS 


the  cross  section  of  a  pipe,  at  which  it  was  desired  to  detenmne 
the  velocity  of  translation  of  the  water*. 

Mr  E.  S.  Colet  has  nsed  the  Pitot  tube  as  a  continaoas  meteri 
the  arrangement  being  shown  in  Fig.  148.  The  tabes  were  con- 
nected to  a  U  tube  containing  a  mixture  of  carbon  tetrachloride 
and  gasoline  of  specific  gravity  1'25.  The  difference  of  level  ct 
the  two  colunms  was  registered  continuously  by  photography. 


^s^szsi^^ssm 


\^A^WA^JJ?MJJ^JM'fJJ^J>>JJJi^M>}^>.  ■^■-gry?g 


Fig.  147. 


Fig.  146. 


Fig.  148. 


The  tubes  shown  in  Figs.  149 — 150,  were  used  by  Bazin  to 
determine  the  distribution  of  velocity  in  the  interior  of  jets  issoiiig 

*  See  page  144. 

t  Proc,  A.M.S.C.E.,  Vol.  xztu.    See  also  experiments  by  Murphy  and  TomuMi 
in  B&me  voiume. 


OAUGIKG  THE  FLOW  OF  WATER 


245 


{rem  orifices,  and  in  the  interior  of  tlie  nappes  of  weirs.  Each 
tobe  consisted  of  a  copper  plate  1*89  inches  wide,  by  '1181  inch 
thick,  sharpened  on  the  upjier  edge  and  having  two  brass  tubes 
"0787  inch  diameter,  soldered  along  the  other  edge,  and  having 
orifices  '059  inch  diameter,  0'394  inch  apart.  The  opening  in  tube 
A  was  arranged  perpendicular  to  the  stream,  and  in  B  on  the  face 
of  the  plate  parallel  to  the  stream. 


cai — 


I£ 


2 


Kg.  149. 


Fig.  160. 


15L    Calibration  of  Pilot  tubes. 

Whatever  the  form  of  the  Ktot  tube,  the  head  h  can  be 
expressed  as 


h  = 


cv' 


or 


2g' 


=  ksf2gh, 
Ic  being  called  the  coefficient  of  the  tube. 

This  coefficient  h  must  be  determined  by  exi)eriment  under 
conditions  as  near  as  possible  like  those  under  which  the  tube  ^vill 
be  used  to  determine  velocities. 

To  calibrate  the  tubes  used  in  the  determination  of  the  distri- 
bution of  velocities  in  open  channels,  Darcy  *  and  Bazin  used  three 
distinct  methods. 

(a)  The  tube  was  placed  in  front  of  a  boat  which  was  drawn 
through  still  water  at  different  velocities.  The  coefficient  was 
1"034.  This  was  considered  too  large  as  the  bow  of  the  boat 
probably  tilted  a  little,  as  it  moved  through  the  water,  thus  tilting 
the  tube  so  that  the  orifice  was  not  exactly  vertical. 

(h)  The  tube  was  placed  in  a  stream,  the  velocity  of  which 
was  determined  by  floats.    The  coefficient  was  TOOG. 

(c)  Readings  were  taken  at  different  points  in  the  cross 
section  of  a  channel,  the  total  flow  Q  through  which  was  carefully 
measured  by  means  of  a  weir.    The  water  section  was  divided 

*  J^eherehes  Hydrauliquet, 


246 


HTDIUUUC3 


[  two  metikoAB  of  caUbit 

^nbes  throQgli  still  wfttlflr; 
a   cmmmlerential 
i  8  inches  de^p  wa&  buflt 
sntre  line,  which  W8§  tbs 
hes.     The  tube  to  b^  rafe 
i  a  central  shaft  wkicli  m 
and  oeilitigrf  and  whicb 
observer.     The  gang^  ^as 


into  areas,  and  about  the  cemm  of  mdt  a  raaiiitto'  of  tW 
waa  taken.  CaDtng  m  the  araa  of  oa»of  theaa Hctio&9,  and ^ 
reading  of  the  tube,  the  coefficient 

i  =  — 9= 

and  was  foimd  to  be  *fl93- 

Darcy*  and  Ba^o  also  fotmd  that  by  changing  the  positioti 
the  orifice  in  the  pressure  tube  the  coefficients  changed 
giderably. 

Williama,  Hnbbell  and  Fenkel 
which  gave  very  diffei^nt  re^nlts 

The  first  methi^d  was  to  mo'v 
known  velocities.  For  this  pc 
rectangolar  in  section,  9  inches  i 
galvanised  iron.  The  diameter  t 
the  path  of  the  tube,  wag  1 1  feet 
WB8  supported  npon  an  arm  atta^ 
free  to  revolve  in  bearings  on  i 
supported  the  gauge  and  a  sear  n. 
connected  with  the  tabe  by  rubber  hose.  The  arm  carrying  the 
tube  was  revolved  by  a  man  walking  behind  it,  at  as  unifonaa 
rate  as  possible,  the  time  of  the  revolution  being  taken  by  means 
of  a  watch  reading  to  i  of  a  second.  The  velocity  was  mail* 
tained  as  nearly  constant  as  possible  for  at  least  a  period  of 
5  minutes.  The  \'a!ue  of  A:  as  determined  by  this  method  was  ■9'2( 
for  the  tube  sho^^-n  in  Fig*  145, 

In  the  second  method  adopted  by  these  workers,  the  tabe  iras 

inserted  into  a  brass  pipe  2  inches  in  diameter,  the  discharge 

through  which  was  obtained  by  weighing.     Readings  were  tak^a 

at  various  positions  on  a  diameter  of  the  pipe,  while  the  6ow  in  th? 

pipe  was  kept  constant.    The  values  of  J2gk^  which  may  be  called 

the  tube  velocities,  could  then  be  calculated,  and  the  mean  value  « 

Vfl,  of  them  obtained.    It  was  found  that,  in  the  caaes  in  which  the  I 

form  of  the  tube  was  such  that  the  volume  occupied  by  it  in  the  pipe 

was  not  sufficient  to  modify  the  fiow,  the  velocity  was  a  maxim  am 

at,  or  near,  the  centre  of  the  pipe*    Calling  this  maximum  velocity 

V 
Vc,  the  ratio  ^  for  a  given  set  of  readings  was  found  to  be  '8L  J 

Previous  experiments  on  a  cast-iron  pipe  line  at  Detroit  having 

shown  that  the  ratio  ^  was  practically  constant  for  all  velocities, 

a  similar  condition  was  assumed  to  obtain  in  the  case  of  the  brass 


*  Reeherchet  Hydraulique$, 


ailFOmG  THE  FLOW  OF  WATER 


ii 


ipe.     The  fcube  wae  then  fixed  at  the  centre  of  the  pipe,  anfl 

lugs  taken  for  varinaB  rates  of  discharge,  the  mean  velocity 

determined  by  weight,  varying  from  ]-  to  6  feet  per  second. 

For   the  valnes  of  h  thna  determined,  it  was  found  that     /— 

sf2gh 
i  practically  cgnstanti    This  ratio  was  *729  for  the  tube  shown 
Fig-  145. 
Then  since  for  any  reading  h  of  the  tuhe^  the  velocity  v  is 

u 

^_  U  V. 


I  acttml  m^n  velocity 


fe-. 


Bui 

Thi^rcffore 


&- 


ratio  of  U  to  V. 


814'  ^^' 


ratio  of  Ym  to  V* 
For  the  tube  ahowTi  in  Fig.  146,  some  of  the  values  of  h 
'by  the  two  methods  differed  very  considerably* 
i^'fm  fff  the  values  of  k  by  the  two  methods.     It  will 
ihat  the  value  of  k  as  determined  by  moving  the  tube  throug] 
iQ    water    differs    very  considerably  from    that    obtained    i 
l^rotining  water.     In  the  latter  case  the  pressure  was  considerably 
higher  than  in  the  former,  and  it  appears  therefore,  that  k  depends 
only  upon  the  form  of  the  tube  but  upon  the  pressure  under 
ch  it  is  workings     It  is,  clearly,  of  considerable  importance 
the  value  of  k  shall  be  determined  for  conditions  similar 
those    under    which    the    tube   is    to    be   finally   used.     This 
aty  of  the  value  of  the  coefficient  under  varying  con- 
of  prensure,  and  the  difficulty  in  any  caae  of  accurately 
silling  it,  and  the  danger  of  its  alteration  by  objects  floating 
Lflw  ftream,  makes  the   use  of  the  Pitot  tube   as  a  velocity 
somewhat  uncertain,  and  it  should  be  used  with  con* 
able  care.     In  the  handg  of  Darcy  and  Bazin  it  proved  an 
client  instrument  in  the  measurement  of  small  velocities  in 
canakj  but   for  the  determination   of  velocities  in   closed 
^Is  in  which  the  pressure  is  greater^  it  does  not  seem  so 
Hit. 

U2.   Ganglxig  by  a  weir. 
( When  a  stream  is  so  small  that  a  barrier  or  dam  can  be  easily 
ett?d  acrc^s  it,  or  when  a  large  quantity  of  water  is  required 
^  be  gauged  in  the  laboratory,  the  flow  can  be  determined  bjr 
^*^*"  of  a  notch  or  weir. 


it- 


248 


HYDRAtTHOS 


The  chaimel  as  it  approachBe  the  weir  should  be  as  far  aa 

poBflible  uniform  in  sectioTij  aud  it  is  desirable  for  accurate 
gauging*,  that  the  sides  of  the  channel  be  made  verticalj  and  fhe 
width  equal  to  the  mdth  of  the  weir.  The  sill  should  be  sharp- 
edged,  and  perfectly  horizontalj  and  as  high  as  possible  abore  th& 
bed  of  the  stream,  and  the  dowii-sti^ani  channel 
should  be  vender  than  the  weir  to  ensure  atmospheric 
pressure  under  the  nappe*  The  difference  in  level 
of  the  sill  and  the  surface  of  the  water^  before  it 
begins  to  slope  towards  the  weir,  should  be  ac- 
curately Tneasured.  This  is  best  done  by  a  Boyden 
hook  gauge. 

153.    Tlie  hook  gauge. 

A  simple  form  of  hook  gauge  as  made  by  Gurley 
is  shown  in  Fig>  15L  In  a  rectangular  groove  foruied 
in  a  frame  of  wood,  three  or  four  feet  long,  slides 
another  piece  of  wood  S  to  which  is  attached  a  scale 
graduated  in  feet  and  hundredths,  similar  to  a  leTel 
staff*  To  the  lower  end  of  the  scale  is  connected  a 
hook  Hj  which  has  a  sharp  point.  At  the  upper  end 
of  the  scale  is  a  screw  T  which  passes  through  a  lug, 
connected  to  a  second  sliding  piece  L.  This  sliding 
piece  can  be  clamped  to  the  frame  in  any  position 
by  means  of  a  nut,  not  shown.  The  scale  can  then 
be  moved  J  either  up  or  down,  by  means  of  the  milled 
nut.  A  vernier  V  is  fixed  to  the  frame  by  two  small 
screw^s  passing  through  slot  holes,  which  allow  for  a 
slight  adjustment  of  the  zero.  At  some  point  a  few 
feet  up-stream  from  the  weir*,  the  frame  can  be 
fixed  t-o  a  post,  or  better  still  to  the  side  of  a  box 
from  which  a  pipe  runs  into  the  stream.  The  level 
of  the  water  in  the  box  will  thus  be  the  same  as  the 
level  in  the  stream.  The  exact  level  of  the  crest  of 
the  weir  must  be  obtained  by  means  of  a  level  and  a 
line  marked  on  the  box  at  the  same  height  as  the 
crest.  The  slider  L  can  be  moved,  so  that  the  hook 
point  is  nearly  coincident  with  the  mark,  and  the 
final  adjustment  made  by  means  of  the  screw  T. 
The  vernier  can  be  adjusted  so  that  its  zero  is 
coincident  with  the  i^ero  of  the  scale,  and  the  slider 
again  raised  until  the  hook  approaches  the  surface  of 
By  means  of  the  screw,  the  hook  is  raised  slowly^  until, 


*  See  Hootton  82. 


GA^UaiNQ    THE   FLOW  OF  WATER 


249 


J^  15S.    SMgJn'B  Hook  Gaage. 


250 


HYDRAULICS 


fche  surface  of  the  water,  it  causes  a  distortion  of  the  light  refleete 
from  the  surface.  On  moving  the  hook  downwards  again  vm 
slightly,  the  exact  surface  will  be  indicated  when  the  distortio 
disappears^ 

A  more  elaborate  hook  gauge,  as  used  by  Bazin  for  his  expef 
mental  work,  is  shown  in  Fig.  152. 

For  rough  gauging®  a  po«t  can  be  driven  into  the  bed  of  tl 
channel  J  a  few  feet  above  the  weir,  until  the  top  of  the  post 
level  with  the  sill  of  the  weir.    Tlie  height  of  the  water  sutCm 


Fig.  154.    Eecording  App«ratms  Kent  Teaturi  Meier. 


GAUGING  THE  FLOW  OF  WATER  251 

the  top  of  the  post  can  then  be  measured  by  any  convenient 

154.    Ga^gSjig  the  flow  In  plpea;  Venturi  meter* 
Sucb  methods  as  already  dei^ribed  are  inapplicable  to  the 
?meiit  of  the  flow  in  pipes,  in  which  it  is  necessary  that 
ah&II  be  no  discontinuity  in  the  flow,  and  special  meters  have  i 
ingiy  been  devised* 
For  large  pipes,  the  Ventnri  meter,  Fig.  153^  is  largely  used  in 

f  and  is  coming  into  favonr  in  this  country. 
The  theory  of  the  meter  hae  already  been  discussed  (p,  4i), 
it  waa  shown  that  the  discharge  is  proportional  to  the  square 
*  of  the  difference  H  of  the  head  at  the  throat  and  the  bend  in 
^pipCi  or 

^*  being  a  coefficient. 

For  measuring  the  pressure  heads  at  the  two  ends  of  the  conei 
W*  G.  Kent  nses  the  arrangement  shown  in  Fig.  154. 


Fig>  154.    B«oording  drum  of  the  Kent  TeDtmi  Meter. 
•  Seepage  4a 


i^^li 


252 


HYDBAULICS 


The  two  pressure  tubes  from  the  meter  are  connected  to  a  U  tube 
consisting  of  two  iron  cylinders  containing  mercury.  Upon  the 
surface  of  the  mercury  in  each  cylinder  is  a  float  made  of  iron  and 
vulcanite;  these  floats  rise  or  fall  with  the  surfaces  of  the  mercmy. 


Fig.  156.    Integrating  dram  of  the  Kent  Yentori  Meter. 

When  no  water  is  passing  through  the  meter,  the  mercury  in  tibl 
two  cylinders  stands  at  the  same  level.  When  flow  takes  platt 
the  mercury  in  the  left  cylinder  rises,  and  that  in  the  xi^ 
cylinder  is  depressed  until  the  difference  of  level  of  the  sox&oh 


OAUGINO  THE  FLOW  OF  WATER 


253 


of  the  mercoiy  is  equal  to  — ,  «  being  the  specific  gravity  of  the 

mercury  and  H  the  difference  of  pressure  head  in  the  two 
cylinders.  The  two  tabes  are  equal  in  diameter,  so  that  the  rise 
in  the  one  is  exactly  equal  to  the  fall  in  the  other,  and  the  move- 
ment of  either  rack  is  proportional  to  H.  The  discharge  is 
proportional  to  VH,  and  arrangements  are  made  in  the  recording 
apparatus  to  make  the  revolutions  of  the  counter  proportional  to 
^/H.  To  the  floats,  inside  the  cylinders,  are  connected  racks,  as 
shown  in  Fig.  154,  gearing  with  small  pinions.  Outside  the 
mercury  cylinders  are  two  other  racks,  to  each  of  which  vertical 
motion  is  given  by  a  pinion  fixed  to  the  same  spindle  as  the  pinion 
gearing  with  the  rack  in  the  cylinder.  The  rack  outside  the  left 
cylinder  has  connected  to  it  a  light  pen  carriage,  the  pen  of  which 


Fig.  157.    Kent  Vestori  Meter.    Development  of  Integrating  drum. 

makes  a  continuous  record  on  the  diagram  drum  shown  in 
Fig-  155.  This  drum  is  rotated  at  a  uniform  rate  by  clockwork, 
and  on  suitably  prepared  paper  a  curve  showing  the  rate  of 
discharge  at  any  instant  is  thus  recorded.  The  rack  outside  the 
right  cylinder  is  connected  to  a  carriage,  the  function  of  which  is 
to  regulate  the  rotations  of  the  counter  which  records  tlie  total 
flow.  Concentric  with  the  diagram  drum  shown  in  Fig.  155,  and 
within  it,  is  a  second  drum,  shown  in  Fig.  156,  wliich  also  rotates 
at  a  uniform  rate.  Fig.  157  shows  this  internal  drum  developed. 
Hie  snrface  of  the  drum  below  the  parabolic  curve  FEG  is  recessed. 
If  the  right-band  carriage  is  touching  the  drum  on  ttie  xece'g.^eA. 


HTDRAUUGS 

,  the  coonter  gearmg  is  in  sc^on^  but  is  pat  out  of  a^tioB 
th*  c&rmge  tottches  the  cylinderr  on  the  nuaad  portm 
*boTe  FB.  Sn^pomB  ihie  mercury  in  the  right  cylindE^r  to  fill  a 
beight  i»oportioa»l  to  H,  then  the  carriage  will  be  m  KXi-wm 
with  the  dram^  aa  the  dmm  rofeales,  along  the  line  CD,  bm  i» 
wiU  imly  be  in  operBtion  while  the  CArriage  is  ia 
along  the  kfngth  CE,  Since  FG  ts  a  parabolic  carve  ik 
frmctkm  of  the  drnunlerence  CE  =  m ,  ^fW^  vi  being  &  constant; 
r  for  M*w  «iiftn]st^«*>«n^«ti  Tf  cif  the  Hoats  the  ct juater  k 
I  action  far  a  poricxl  pToptff' 
ae  top  of  the  right  cytind^frt 
im^  and  in  contact  wiili  i^ 
ferolntion  and  no  flow  i 
s  in  itB  lowest  position  tht 
*inn»  and  flow  is  registW 
111'  recording  apparati^cii 
«  ism  than  1000  feet  fnM 
'jp^  larger  as  the  diHtaon 


of 

boulto^.    Wk 
tfae  carnage  m  at 
nised  portifii  for  u 
Whm  tl 
is  al  the  b 
tianng  the  whole  of  a  .«, 
bo  placed  ai  any  oonTOi- 
tlie  meter,  tbe  conaecaog 
B  tQcreaeed. 


155.    Deacon's  waste-water  meter. 

An  ingenious  and  very  diuple  meter  designed  by  Mr  G.  K 
Deacon  prmcipally  for  detecting  the  leakage  of  water  from  pipes 
is  a^  shc'Wn  in  Fig,  lad. 


The  body  of  the  meter  which  is  made  of  cast*iron,  has  fittefl  | 
into  it  a  hollow  cone  C  made  of  brass.  A  disc  D  of  the  same  diameter  | 
3fc^  the  upper  end  of  the  cone  is  suspended  in  this  cone  bymeao^cl  { 
a  fine  winv  -—       er  a  pulley  not  shown ;  the  other  end 

of  the  >Wre  carries  a  balance  weight. 


GAUOtKG  THK   FLOW  OF  WATER 


255 


no  water  paaaes  throu^li  the  meter  the  disc  is  drawn  to 

{top  of  the  cone,  bat  when  water  is  drawn  through,  the  disc  is 

downwards  to  a  position  depending  upon  the  quantity  of 

pasmiig*    A  pencil  is  attached  to  the  wire,  and  the  motion 

imc  can  then  be  recorded  upon  a  drum  mad©  to  revolve  by 

rork-    The  posirion  of  the  pencil  indicates  the  rate  of  flow 

through  the  meter  at  any  instant. 

used  as  a  waste-water  meter,  it  is  placed  in  a  by-pass 
^  from  the  main,  as  shown  diagrammatically  in  Fig,  159, 


(^ 


s.v: 


Bi  sy 


3 


s.\: 


D 


Fig,  16a 

\ulv»-ii  A  and  B  are  closed  and  the  valve  C  opened.  The 
€v*ii!naniption  in  the  pipe  AD  at  those  hours  of  the  night 
the  actual  consumption  is  very  amall,  can  thus  be  detei^ 
i^  and  an  estimate  made  as  to  the  probable  amount  wasted. 
*  If  wm^te  is  taking  place,  a  cai-efiil  inspection  of  the  district 
Hiliod  by  the  main  AD  may  then  be  made  to  detect  where  the 
Hte  is  occurring. 

^i&B.    Kennedy's  meter. 
This  is  a  positive  meter  in  which  the  volume  of  water  passing 
ragh  the  meter  is  measured  by  the  displacement  of  a  piston 
ing  in  the  measuring  cylinder. 

long  hollow  piston  P^  Fig.  157,  fits  loosely  in  the  cylinder 

hot  m  made  water-tight  bj*  means  of  a  cylindrical  ring  of 

iter  which  rijUs  between    the   piston  and   the  inside  of  the 

ider,  the  friction  being  thus  reduced  to  a  minimum.     At  each 

of  the  cylinder  is  a  rubber  ring,  which  makes  a  water-tight 

ii  when  the  pisrton  is  forced  to  either  end  of  the  cylinder,  so 

it  the  rtibber  roller  has  only  to  make  a  joint  while  the  piston  is 

to  move, 

water  ©titers  the  meter  at  A,  Fig.  161  i,  and  for  the 
ition  shown  of  the  regulating  cock,  it  Hows  down  the  passage 
id  under  the  piston*  ^Flie  piston  rises,  and  as  it  does  so  the 
R  turns  the  pinion  8,  and  thug  the  pinion  p  which  is  keyed 
he  .same  spindle  as  S*  This  spindle  also  carries  loosely 
ighted  !ever  W,  which  is  moved  as  the  spindle  revolves  by 
r  of  two  projecting  fingers.  As  the  piston  continues  to 
id,  the  weighted  lever  is  moved  by  one  of  the  fingers  until  its 


-^  -^  --— ^- 


250 


HYDRAULICS 


centre  of  gravity  passes  the  vertical  positioHi  wlien  it  suddei 
falls  on  to  a  buffer j  and  in  its  motion  moves  the  lever  L^  whi 
tuma  the  cock.  Fig.  161  fe,  into  a  position  at  right  angles  to  tl 


r~7   rn  ^ 


R^ibb^SeaHiiff 


Fig.  160. 


GArOlKG  THE   rLOW  OF  WATER 


257 


lown.      The  water  now  passes  from  A  throngh  the  paasage  C, 

syUnder,  and  as  the  piston  descends, 


Fig.  161  6, 


k 


jMilk 


HYBIUtrUCS 


Ao  wmter  that  is  below  it  pas(s<^  to  the  outlet  B-    The 
the  pinkm  H  is  now  re^ersedj  and  the  weight  W  lilted 
mgam  x^eecbee  the  vertical  podttion,  wheti  it  faik,  hxm% 
opek  C  into  Ibo  position  shown  in  the  fignre^  and  another  i 


Fig.  161  c, 

oke  i«  commenced.    The  rise  illations  of  the  pinion  p  bt^\ 
to  the  counter  niecVmnisui  through  the  pinions  p, 
161  a^  in  each  of  which  i^  a  ratchet  and  pawl.    The  eofl 
thtta  rotated  in  the  same  direction  whichever  way  the  p 

IBl.  dauging  the  flow  of  streams  by  chemical  meauL 
ilr  Stromeyer*  has  very  sacceesfully  ganged  the  quantity 
water  supplied  to  boilers,  and  also 
the  flow  of  stretams  by  mixing 
with  the  stream  doring  a  deliuite 
ime  and  at  a  uniform  rate,  a 
auwn  quantity  of  a  concentrated 
llation  of  some  chemical,  the 
91100  of  which  in  watar^  even 
in  very  small  quantities,  can  be 
]y  detected  by  some  sensitive 
&nt.  Suppose  for  instance 
iraler  ia  flowing  along  a  small 
Two  stations  at  a  known 
i  apart  are  taken^  and  the 
determined  which  it  takes 
the  water  to  traverse  the  dis- 
"HDce  between  them.  At  a  stated 
*e,  by  means  of  a  special  ap- 
"ns— Mr  Stromeyer  uses  the  gLL 

'^ment  shown  in  Fig.  162  Fig,  169. 

nc  acid,  say,  of  known 

run  into  the  stream  at  a  known  rate»  at  the  up. 

f  Navai  AnMifCti,  1896  ;  Proctidingf  inwL  C,B*,  Vol,  CLI. 


^ 


-^^i"^ 


K 


J 


GAUaiKG  THE   FLOW   OF  WATER 


259 


While  the  acid  is  being  pat  into  the  stream,  a  small 
k»ce  op-stream  from  where  the  acid  is  introduced  samples  of 
rr    are   taken   at   definit€r  ititervals*    At    the    lower    station 

tg  is  commenced,  at  a  time,  after  the  insertion  of  the 
the  apper  station  is  started,  equal  to  that  required  by  the 
W  to  trarerse  the  distance  between  the  stations,  and  samples 
then  taken,  at  the  same  intervalsj  as  at  the  upper  station, 
quantity  of  acid  in  a  known  volume  of  the  samples  taken 
he  tipper  and  lower  station  is  then  determined  by  analyaiB* 
I  Ttjlnme  V^  of  the  samples,  let  the  difference  in  the  amount  of 
hfiric  acid  be  equivalent  to  a  volume  tv  of  pure  sulphuric 

k If  in  a  time  ^,  a  volume  V  of  water j  has  flowed  down  the 
^  and  there  has  been  mixed  with  this  a  volume  v  of  pure 
ric  acid,  then,  H  the  acid  has  mixed  uniformly  Anth  the 
^  the  ratio  of  the  quantity  of  water  flowing  down  the  stream 
BqBanlity  of  acid  put  into  the  stream,  is  the  same  as  the 
^fe  the  volnme  of  the  sample  tested  to  the  difference  of  the 
me  of  the  acid  in  the  samples  at  the  two  stations,  or 

Pf  Stromeyer  considers  that  the  flow  in  the  largest  rivers  can 
iterxnined  by  this  method  within  one  per  cent,  of  its  true  value. 
d  large  streams  special  precautions  have  to  be  taken  in 
the  chemical  solution  into  the  water,  to  ensure  a  uniform 
and  also  special  precautions  must  be  adopted  in  taking 


[okliar  important  information  upon  this  interesting  method 
lorin^  the  flow  of  water  the  reader  is  referred  to  the  two 
uted.  above. 

iratos  for  accurately  gauging  the  flow  of  the  solution 
in    Fig,   162*    The  chemical  solution  is  delivered  into 
lindrieal  tank  by  means  of  a  pipe  L    On  the  surface  of  the 
1  floats  a  cork  which  carries  a  siphon  pipe  SS,  and  a  balance 
keep  the  cork  horizontal.     After  the  flow  has  been 
the  head  h  above  the  orifice  is  clearly  maintained 
irhaterer  the  level  of  the  surface  of  the  solution  in  the 


\1—1 


260 


HYDBAtJLlCS 


EXAMPLES, 


(1)    Same  obaerrfttions  aro  made  by  towing  a  coireot  meter,  witli  ifa4 
following  resnlta:— 


Speed  in  ft  per  lec. 

1 
5 
Find  an  eqii^tion  far  the  m^ter^ 

(2)    Deecribe  two  methods  of  gi 
in  vertical  and  horizontal  planes;  , 
obtained. 

If  the  croBs  section  of  a  met  is 
discharge  may  bo  oaUmated  by  ob 
alone. 

(8)     The  following  observations  dl 
were  made  ill  connection  with  a  weir 

Head  m  feet       ...  ...    01     01 

Discharge  in  cubic  feet  per 
sec  per  foot  width        ,.,    017 


Be?i.  of  meter  per  min, 
80 

560 


bulge  river,  from  observatioiii 
^te  the  nature   of  the  resultB 

explain  how  the  apptosiiii«be 
n  of  the  mid-anzface  f^pi^ 

nd  the  oorresponding  disdiai^H 
et  wide. 

8-0    !   3-5  I  4i) 


■1 19-82 


25 
18*08 


17-03  21-54 126^4 


1-2 1», 
Assuming  the  law  connecting  the  hoad  h  with  the  discharge  Q  as 

Q  =  mh  .  A", 
find  m  and  n.     ^Plot  log&ritlimH  of  Q  and  h,} 

(4)  The  following  values  of  Q  and  h  were  obtained  for  a  shacp-ed^ 
weir  6'58  feet  longT  without  lateral  contraction.  Find  the  coefl&djent  d 
discharge  at  veltious  heads. 


lead  A  ... 

•1    j-4       -6 

-8 

1*0 

1*5 

2*0   1   25       30 

3-5 

4-0 

€-5 

W 

H 

1  per  foot- 

1 

tength  ... 

•17   -87  j  1-56 

237 

sas 

ei 

9-32  15-08  17-03 

3154 

26-4 

31*62 

37-09 

43^81 

(5)  The  following  values  of  the  head  over  a  weir  10  feet  long  were 
obtained  at  5  minutes  Intervals, 

Head  m  feet  35    -36     37     37     38     80     40    *41    *42    -40     *8&    "41 

Taking  tlie  coefficient  of  discharge  C  as  8  36,  find  the  discharge  m 
one  hour. 

(6)  A  Pitot  tube  was  calibrated  by  moving  it  through  still  water  in  & 
tank,  the  tube  being  tixed  to  an  arm  wliidi  was  made  to  revolve  at 
constant  speed  about  a  fixed  centre.  The  following  were  the  velocities  of 
the  tube  and  the  heads  measured  in  inches  of  water. 


Velocities  ft.  per  sec.   1*432 
Head  in  inches 
of  water  -448 


1-788 
•663 


2-275  I  2-718 
1-02    1 1-69 


8-286 
2-07 


8-878 
2-88 


4-988  I  5*584 

5-40     6-97 


6143 
8-51 


Determine  the  coefficient  of  the  tube. 

For  examples  on  Venturi  meters  see  Chapter  U. 


CHAPTER  VIIL 

IMPACT  OF  WATER  ON  VANES. 

168.  Definition  of  a  vector.  A  right  line  AS,  considered  as 
haying  not  only  length,  bat  also  direction,  and  sense,  is  said  to  be 
a  vector*.    The  initial  point  A  is  said  to  be  the  origin. 

It  is  important  that  the  difference  between  sense  and  direction 
should  be  clearly  recognised. 

Suppose  for  example,  from  any  point  A,  a  line  AB  of 
definite  length  is  drawn  in  a  northerly  direction,  then  the 
direction  of  the  line  is  either  from  south  to  north  or  north  to 
soath,  but  the  sense  of  the  vector  is  definite,  and  is  from  A  to  B, 
that  is  from  sooth  to  north. 

The  vector  AB  is  equal  in  magnitude  to  the  vector  BA,  but 
they  are  of  opposite  sign  or, 

AB  =  -BA. 

The  sense  of  the  vector  is  indicated  by  an  arrow,  as  on  AB, 
Fig.  163. 

Any  quantity  which  has  magnitude,  direction,  and  sense,  may 
be  represented  by  a  vector. 


D 


For  example,  a  body  is  moving  with  a  given  velocity  in  a 
given  direction,  sense  being  now  implied.  Then  a  line  AB  drawn 
parallel  to  the  direction  of  motion,  and  on  some  scale  equal  in 

*  Sir  W.  Hamilton,  QuaUmioru, 


262 


HYDRAULICS 


length  to  the  velocity  of  the  body  is  the  velocity  vector;  the  seise 
is  from  A  to  B< 

159.    ♦  Stun  of  two  vectora. 

If  a  and  ^,  Fig.  163,  are  two  vectors  the  euoi  of  these  vecton 
is  found,  by  drawing  the  vectors,  so  that  the  beginniiig  of  j8  ii  il  j 
the  end  of  a,  and  joining  the  beginning  of  a  to  the  end  of  ftj 
Thus  y  is  the  vector  sum  of  a  and  fi. 


160.     ReanltaJEit  of  two 

When   a   body   has 
velocities^  the  resultant 
direction  is  the  vector  si 
may  be  stated  in  a  way 
problems  to  be  hereai 
moving  with  a  given  i 
velocity  is  inipressed  op 
vector  sum  6f  the  initial 


Teiocitias* 

)n  it  at  any  instant  two 
16  body  in  magnitude  nni 
impressed  velociti**^,  TIlei 
iefiaitely  applicable  to  the 
as  follows.  If  a  body  ii 
en  direction,  and  a  ieoomd 
he  resnltant  Telocity  is  the 
velocities. 


Example*     Bfippos^  ^  pAri 
with  a  velocitj  V^,  rolative  U 

If  the  Tune  is  at  rest,  the  |»i  ^u*^  ^"  at  A  with  this  velocity. 

If  the  ^ane  iti  timd«  to  move  m  the  ^±44^iiiOEL  EF  with  a  velocity  r,  and  l3ot 
particle  haa  still  a  Vf^locitj  V^  r«lalive  to  the  vane,  and  remams  in  contact  with  tbs 
vane  until  th«  point  A  i»  reach^,  the  vdocity  of  the  water  as  it  learea  the  ^n«  it 
A,  will  be  the  ^^otor  (lum  7  of  a  and  ^,  i.e^  of  V,.  and  V,  or  h  equal  to  i^, 

161.    DtfiTerencQ  of  two  vectors. 

The  difference  of  two  vectors  a  and  0  is  f onnd  by  drawing  bo^ 
vectors  from  a  common  origin  A,  and  joining  the  end  of  ^  to  thei 
end  of  a.  Thus,  CB,  Fig,  165,  is  the  difference  of  the  two  vectora 
a  and  A  or  y  =  a-ft  and  BC  is  equal  to  i^-of,  or^-a^-7. 


Fig.  166. 

162.    Absolute  yelooity. 

By  the  terms  "  absolute  velocity  "  or  "  velocity "  without  the 
adjective,  as  used  in  this  chapter,  it  should  be  clearly  understood, 
is  meant  the  velocity  of  the  moving  water  relative  to  the  earth,  or 
to  the  fixed  part  of  any  machine  in  which  the  water  is  moving. 


Henrioi  and  Tamer,  Veeton  tmd  Baton, 


IMPACT  OF  WATER  ON  VANES  263 

To  avoid  repetition  of  the  word  absolute,  the  adjective  is 
■eqaently  dropped  and  "  velocity  "  only  is  used. 

163.  When  a  body  is  moving  with  a  velocity  U,  Fig.  166,  in 
ny  direction,  and  has  its  velocity  changed  to  U'  in  any  other 
iirection,  by  an  impressed  force,  the  change  in  velocity,  or  the 
'elocity  that  is  impressed  on  the  body,  is  the  vector  difference  of 
he  final  and  the  initial  velocities.  It  AB  is  U,  and  AC,  U',  the 
mpressed  velocity  is  BC. 

By  Newton's  second  law  of  motion,  the  resultant  impressed 
Sorce  is  in  the  direction  of  the  change  of  velocity,  and  if  W  is  the 
wmght  of  the  body  in  pounds  and  t  is  the  time  taken  to  change 
ihe  velocity,  the  magnitude  of  the  impressed  force  is 

W 

P  =  -T  (change  of  velocity)  lbs. 
gt 

This  may  be  stated  more  generally  as  follows. 
The  rate  of  change  of  momentum,  in  any  direction,  is  equal  to 
ihe  impressed  force  in  that  direction,  or 

P=  — .;^lbs. 

g    at 

In  hydranlic  machine  problems,  it  is  generally  only  necessary 
o  consider  the  change  of  momentum  of  the  mass  of  water  that 
fccts  open  the  machine  per  second.  W  in  the  above  equation  then 
lecomes  the  weight  of  water  per  second,  and  t  being  one  second, 

W 

P  =  —  (change  of  velocity). 

164.  Impulse  of  water  on  vanes. 

It  follows  that  when  water  strikes  a  vane  which  is  either 
noving  or  at  rest,  and  has  its  velocity  changed,  either  in  magni- 
nde  or  direction,  pressure  is  exerted  on  the  vane. 

As  an  example,  suppose  in  one  second  a  mass  of  water,  weighing 
iV  lbs,  and  moving  with  a  velocity  U  feet  per  second,  strikes  a 
ixed  vane  AD,  and  let  it  glide  upon  the  vane  at  A,  Fig.  167,  and 
eave  at  D  in  a  direction  at  right  angles  to  its  original  direction 
>f  motion.  The  velocity  of  the  water  is  altered  in  direction  but 
lot  in  magnitude,  the  original  velocity  being  changed  to  a  velocity 
kt  right  angles  to  it  by  the  impressed  force  the  vane  exerts  upon 
he  water. 

The  change  of  velocity  in  the  direction  AC  is,  therefore, 

W 
qual  to  U,  and  the  change  of  momentum  per  second  is  — .U 

oot  lbs. 


264 


HYDRAULICS 


Since  W  lbs.  of  water  strike  the  vane  per  second,  the  pressi 
P,  acting  in  the  direction  C  A,  required  to  hold  the  vane  in  positi 
is,  therefore, 

W 


Pig.  167. 

Again,  the  vane  has  impressed  upon  the  water  a  velocity  U 
the  direction  DF  which  it  originally  did  not  possess. 
The  pressure  Pi  in  the  direction  DF  is,  therefore, 

W 
Pi  =  P  =  — .U. 
9 

The  resultant  reaction  of  the  vane  in  magnitude  and  directi 
is,  therefore,  R  the  resultant  of  P  and  Pi. 

This  resultant  force  could  have  been 
found  at  once  by  finding  the  resultant 
change  in  velocity.  Set  out  clc^  Fig.  168, 
equal  to  the  initial  velocity  in  magnitude 
and  direction,  and  ad  equal  to  the  final 
velocity.  The  change  in  velocity  is  the 
vector  difference  cd,  or  cd  is  the  velocity 
that  must  be  impressed  on  a  particle  of 
water  to  change  its  velocity  from  ac  to 
od.  ^«-  ^^ 

The    impressed  velocity  cd    is  V  =  VU*  +  TP,   and    the  to) 
impressed  force  is 


W- 


W 


9  9 


n/2W 
9 


IMPACT  OF  WATEB  ON   VANES 


S0S 


It  at  once  folIow*s,  that  if  a  jnt  of  water  strikes  a  fixed  plan€3 
[perpendicularly^  with  a  velocitjr  U,  and  glides  along  the  plane,  the 

[normal  proas ure  on  the  plane  is^  ^  U. 


A  ttrtuD  of  water  1  eq,  fcN?t  in  seetion  ft&d  having  a  Telocity  of 
^inft  par  Mooad  glMea  on  lo  a  fixed  vane  ia  a  dUreolion  makiDg  an  angle  of 
iifMft  wUh  a  gifen  direction  AB. 
Tbm  vaae  tatna  tbe  jet  ihroogh  an  angle  ot  §C  degrees, 

Whid  Iha  Pfcwmm  cm  Ihe  Tane  io  the  direction  parsiLlsl  lo  AB  and  the  reittltant 
wmom  tibe  taaa. 

Fle»  167.  A€  ia  the  oiiginal  direction  of  the  jet  and  DF  the  tnal  direction. 
lae  ctntply  ehangea  the  directioti  of  the  water*  the  ^nal  velocity  being  oqnal 

triaagle  it  ar4,  Fif?^  ISS^  ae  and  ad  being  equal, 
of  vel<>£i^  In  magnitude  and  direction  is  cd^  the  vector  difference  of 
;  roaolTing  cd  parallel  to,  and  perp^ndienlar  to  AB^  ce  Is  the  ohangie  of 
pafsOlel  to  aB. 
eallog  off  C9  and  oalling  it  v^ »  the  for^e  to  be  applied  along  BA  to  keep  the 
•I  rmt  ia, 

Bat  c4=j2,l0 

C4  =  cd  oo»  1 5^ 


Tli0  premsum  aomml  to  AB  ii^ 


=  2fi4lbfl. 


■      9 


Tbe  rvsoltaat  ia 


B= 


m,m4    ,      1007^2.62  4 


32*2 


3SS 


274  Ibe. 


IBS.     RelaUye  yeloclty. 

Betfore  going  on  to  the  consideration  of  moving  vanes  it 
^^Bfi^tit  that  the  student  should  have  clear  ideaa  as  to  what  iS 
f/fmk'  by  relative  i^elocity. 

I      A  rrain  is  said  to  have  a  velocity  of  dxty  miles  an  honr  when| 
iti   .  '    Tied  in  a  Btraight  line  at  a  constant  velocity  for  one 

^^j-^  Ad  travel  sixty  miles.     What  is  meant  is  that  the  train 

^  fir^ying  «^  sixty  miles  an  hour  relative  to  the  earth. 

^^   t^wo  trains  run  on  parallel  lines  in  the  same  direction,  oi 
.ty    And    the  other  at  forty  miles  an   hour,   they  have  a 
!•  Other  of  20  miles  an  hour.     If  they  move 
^^^^^^,_  [hey  havo  a  relative  velocity  of  100  miles 

boor.  If  one  of  the  trains  T  is  travelling  in  the  direction  AB, 
^  169^  and  th^  other  T,  in  the  direction  AC,  and  it  be  supposed 
fnt  the  line©  on  which  they  are  travelling  cross  each  ottei  at  A^ 


Lin     ^ 
n(^^ 


266 


HYDRAULICS 


and  the  trains  are  at  any  instant  over  each  other  at  A,  at  the  end 

of  one  minute  the  two  trains  will  be  at  B  and  C  respectively,  at 

distances  of  one  mile  and  two-thirds  of  a 

mile  from  A.    Relatively  to  the  train  T 

moving  along  AB,  the  train  Ti  moving 

along  AC  has,  therefore,  a  velocity  equal 

to  BC,  in  magnitude  and  direction,  and 

relatively  to  the  train  Ti  the  train  T  has 

a  velocity  equal  to  CB.    But  AB  and  AC 

may  be  taken  as  the  vectors  of  the  two 

velocities,  and  BC  is  the  vector  difference 


% 


Fig.  169. 

of  AC  and  AB,  that  is,  the  velocity  of  Ti  relative  to  T  is  the 
vector  difference  of  AC  and  AB. 

166.  Definition  of  relative  velocity  as  a  vector. 

If  two  bodies  A  and  B  are  moving  with  given  velocities  v  and 
t?i  in  given  directions,  the  relative  velocity  of  A  to  B  is  the  vector 
difference  of  the  velocities  v  and  Vi. 

Thus  when  a  stream  of  water  strikes  a  moving  vane  the 
magnitude  and  direction  of  the  relative  velocity  of  the  water  and 
the  vane  is  the  vector  difference  of  the  velocity  of  the  water  and 
the  edge  of  the  vane  where  the  water  meets  it. 

167.  To  find  the  pressure  on  a  moving  vane,  and  the 
rate  of  doing  work. 

A  jet  of  water  having  a  velocity  U  strikes  a  flat  vane,  tie 
plane  of  which  is  perpendicular  to  the  direction  of  the  jet,  and 
which  is  moving  in  the  same  direction  as  the  jet  with  a  velocitv  ft 


I 


Fig.  170. 


U^ 


Fig.  171. 


The  relative  velocity  of  the  water  and  the  vane  is  U— «,  the 
vector  difference  of  U  and  v.  Fig.  170.  If  the  water  as  it  strikei 
the  vane  is  supposed  to  glide  along  it  as  in  Fig.  171,  it  will  do 


IMPACT  OF  WATER  ON   VANES  267 

80  with  a  velocity  equal  to  (U— v),  and  as  it  moves  with  the  vane 

it  will  still  have  a  velocity  v  in  the  direction  of  motion  of  the 

vazie.     Instead  of  the  water  gliding  along  the  vane,  the  velocity 

U  —  V  may  be  destroyed  by  eddy  motions,  but  the  water  will  still 

liave  a  velocity  v  in  the  direction  of  the  vane.    The  change  in 

nelocity   in   the  direction  of   motion  is,  therefore,  the  relative 

relocity  U-r,  Fig.  170. 

For  every  pound  of  water  striking  the  vane,  the  horizontal 

XJ  — t? 
ihan^e  in  momentum  is ,  and  this  equals  the  norm^/l  pressure 

^  on  the  vane,  per  pound  of  water  striking  the  vane. 
The  work  done  per  second  per  pound  is 

Pt?  = .  V  foot  lbs. 

9 

The   original  kinetic  energy  of  the  jet  per  pound  of  water 

.   XP 
triking  the  vane  is  s—  >  and  the  efficiency  of  the  vane  is,  therefore, 

"liicli  is  a  maximum  when  v  is  ^U,  and  e  =  J.  An  application  of 
icH  vanes  is  illustrated  in  Fig.  185,  page  292. 

Nozzle  and  single  vane.  Let  the  water  striking  a  vane  issue 
XMn  a  nozzle  of  area  a,  and  suppose  that  there  is  only  one  vane. 

"Let  the  vane  at  a  given  instant  be  supposed  at  A,  Fig.  172.  At 
le  end  of  one  second  the  front  of  the  jet,  if  perfectly  free  to 
love^  would  have  arrived  at  B  and  the  vane  at  C.  Of  the  water 
lat  has  issued  from  the  jet,  therefore,  only  the  quantity  BC  will 
ave  hit  the  vane. 


Fig.  172. 
The  discbarge  from  the  nozzle  is 

W  =  62-4.a.U, 
id  the  'Weight  that  hits  the  vane  per  second  is 

W.(U-t>) 

u 

The  chftn^r^  of  momentum  per  second  is 

W(U-t))' 
g      U     ' 


HTDKAUUCa 


froiQ  ilie  posxkil 


ttitd   hsm   no 

^  wmmm^    If  them  wre  m 
€tiket,  tlie  wholB  of  tlie  w^r 
mnti  Urn  work  done  is 

IP       • 

^  w»ter  wheel,  witb  imdial  bbdoi^l 
m  P%-  19§^  cuiBoC  IxftTe  an  effici^^cj^  of  more  than  50  per  cent 

IG6-    Impact  of  wftter  on  &  T&ne  when  the  dlxectioni  € 
wrf^"^  of  the  Ymne  and  jet  are  not  parallel. 

L««  r  be  tke  ^ckiciix  erf  »  j^  ol  wuier  and  AB  its  directkitf 

J> 


ne 


fif.  tm 


Ltfi  «bt»  f<48«  A,<ii  Ae  na^  AC  be  morbg  witli  a  Tielodtrn 
tbe  ittiiiTT^  velocity  Vr  of  tbe  water  and  the  Taoe  at  A  is  DB 
tfe  tnu^le  DAB  it  is  aeeti  tbAt.^  tbe  rector  sunt  of  tht 
of  the  Tane  and  tb«^  r^latire  f^kMriljr  of  the  jet  ai>d  d4 
«^W  ii  tqasl  to  tlie  T^k^^it^  ol  tbe  jec;  for  clearl|^  IT  is  tbe  t&^ 
aaB  vi  r  at»l  Tr. 

If  tbe  dbvetiiJii  vi  tbe  tip  ol  tbe  Tmne  at  A  is  made  panUel  td 
DB  tbe  water  will  gbde  on  to  tbe  T%ne  in  exaetlj^  the  same  vsf 


J 


IMPACT  OF  WATER  ON   VANIS 


2G9 


it  were  mt  rest,  and  the  water  were  moving  in  tha  direction 
L     This  h  the  condition  that  no  eDerg7  shall  be  lost  by  shock, 
"Wli^n  the  water  leaves  the  vane,  the  relative  velocity  of  the 
and  the  vBne  must  be  parallel  to   the  diroction  of  the 
^mit  to  the  vane  at  the  point  where  it  leaves,  and  it  is  equal  to 
vector  difference  of  the  absolute  velocity  of  the  water,  and 
Tane.     Or  the  absolute  velocity  with  which  the  water  leaves 
mape  is  the  vect-or  sum  of  the  velocity  of  the  tip  of  the  vane 
I  the  relative  velocity  of  the  water  to  the  vane* 
l^t  CO  be  the  direction  of  the  tangent  to  the  vane  at  C.    Let 
i  be  ri  p  the  velocity  of  C  in  magnitude  and  direction,  and  let  CF 
ihe  absoliite  velocity  Ui  with  which  the  water  leaves  the  vane. 
Draw  EF  pai-aUel  to  C(t  to  meet  the  direction  CF  in  Fj  then 
\  relative  velocity  of  the  water  and  the  vane  is  EF,  and  the 
ocity  with  which  the  water  leaves  the  vane  is  equal  to  CF. 
If  Vi  and  the  direction  CG  are  given,  and  the  direction  in  which 
I  water  leaves  the  vane  is  given^  the  triangle  CEF  can  be 
IWB,  and  CF  determined. 

If  on  the  other  hand  Vj  is  given,  and  the  relative  velocity  Vr  is 
in  magnitude  and  direction,  CF  can  be  found  by  measuring 
along  EF  the  known  relative  velocity  iv  and  joining  CF. 
If  t?j  and  Ui  are  given,  the  direction  of  the  tangent  to  the  vane 
then,  a«  at  inlet,  the  vector  difference  of  Ui  and  t^i. 

It  will  be  seen  that  when  the  water  either  strikes  or  leaves  the 
ne,  the  relative  velocity  of  the  water  and  the  vane  m  the  vector 
erence  of  the  velocity  of  the  water  and  the  vane,  and  the  actual 
Oi3ty  of  the  water  as  it  leaves  the  vane  is  the  vector  sum  of  the 
ocity  of  the  van©  and  the  relative  velocity  of  the  water  and 
rane. 


I 
I 
I 
■ 
I 

I 


I 
I 


,    The  difectiott  of  tbe  tip  of  the  vivne  at  the  out^r  circumferrence  of  a 
witli  T&fiai,  m*ke«  aa  Angle  of  165  degiee:^  with  the  direction  of  motioa 
_      kip  of  the  TADe. 
7li«  Tclodty  of  the  tip  at  the  outer  circtimf«rf  tjc«;  in  82  feet  p&r  F^cnzid^ 

«mier  leaves  the  ^h&e\  m  auch  a  direct ii>n  and  with  such  a  Velocity  that  the  ' 
■enl  i^  13  fmx  per  second- 

wbmAute  velocLty  of  tbe  water  in  direetion  and  magnitude  and  the 
of  the  wat«r  and  the  wheeL  ^ 

m  tbe  tri*iigl«  of  velocities,  set  out  A 15  uqual  to  B2  feet,  and  make  tha^ 
^  eQiiaJI  to  IS  degneefi.    BQ  Ih  then  parallei  to  the  tip  of  the  vane.  V 

irallel  lo  ABp  and  at  a  diittatjce  from  It  eqn&l  to  1^  fe«t  &aA 
in  C- 
,\C  t*  tha  vector  Aum  of  AB  and  UC,  and  ia  the  absolate  velocitj  of  the 
directiaa  and  magnttTide, 
Ihgoiiometrically 

AC^  -  (82  -  13  oot  isy  + 133 

=  33^5*HhlB*  and  AC  =  air?  ft  per  seo. 


i 


270 


HYDRAULICS 


169.  Conditions  wMch  the  vanes  of  bydraalio  naadiini 
should  satiflfy* 

In  all  properly  deBigned  hydraulic  machines,  sneh  as  tnrbti 
veater  wheels,  and  centrifugal  pumps,  in  which  water  flowing  id 
a  definite  direction  impinges  on  mo^^ng  vanes,  the  relative  ^ 
of  the  water  and  the  vanes  should  be  parallel  to  the  din- 
the  vaoes  at  the  point  of  contact.     If  not,  the  water  breaks  into 
eddies  aa  it  moves  on  to  the  vanes  and  energy  is  lost. 

Again,  if  in  such  machines  the  water  is  required  to  leave  the 
Tanes  with  a  given  velocity  in  magnitude  and  direction,  it  is  only 
neceaaary  to  make  the  tip  of  the  vane  parallel  to  the  vect<jr 
difference  of  the  given  velocity  with  which  the  water  is  to  leav^? 
the  vane  and  the  velocity  of  the  tip  of  the  vane, 

ExampU  (1|,  A  jet  of  water,  Fig.  174,  movea  in  a  direction  AB  making  as  as  .  • 
of  30  def^ees  with  the  direct! od  of  motion  AC  of  a  vane  moving  in  tlie  atmosi/t 
The  jet  has  a  velocity  of  30  ft.  per  second  and  the  vaue  of  lo  ft.  per  si^ood.  Tt»  umiI 
(ii}  the  direction  ^f  the  vnoe  at  A  eo  tbal  tlie  water  may  cnttir  wiChoQl  aliook;  {h}  tk* 
direction  of  the  taugeni  to  tiie  vane  where  the  water  leaves  it,  so  that  the  abtolulf 
velocity  of  th€  water  when  it  teavei^  the  vane  is  in  a  direction  perpendiotiJAr  to  ^Cfi 
(r)  the  pressure  on  the  vane  and  the  work  done  par  second  per  poand  of  ' 
striking  the  vane,    Frictioii  U  neglected. 


Fig,  174. 


The  relative  Telocity  T^  of  Ihe  water  and  the  vatje  %i  A  ie  CB,  and  for  no  i 
the  vane  at  A  must  be  parallel  to  CB. 

Sinee  there  is  no  friction,  the  relative  velocity  V,,  of  the  watar  ajid  ibe  hm] 
cannot  alter,  and  tharefore,  the  triangte  of  velocities  at  exit  is  ACD  or  FA*C*. 

The  point  D  is  fouodt  by  taking  G  as  centre  and  CB  as  radiua  uid  sitwoff  \ 
are  ED  to  cut  the  known  dlreotion  AD  in  D. 

The  total  change  of  velocity  of  the  jet  i«  the  vector  difference  DB  of  the  L^ltkl  ] 
and  final  velocities,  and  the  change  of  velocity  in  the  direction  of  motion  u  E£^  1 
Calling  this  velocity  Y^  the  presaare  exerted  npon  ihe  vane  in  the  direolkc  <i 
motion  In 

V 

—  lbs,  per  lb.  of  water  striking  the  vane^ 

Tlie  work  done  p^  lb.  is,  therefore^  —  ft.  lbs,  asd  the  efEdencj,  sinc«  Ibtl*  ii  | 

no  loss  by  friction,  or  shock,  h 

\v    _ %Vv 


•'25 


IMPACT  OP  WATER   ON   VANES  271 

The  change  in  the  kinetic  energy  of  the  Jet  i$  equal  to  the  work  done  by  the  jet. 
The  kinetio  energy  per  lb.  of  the  original  jet  is  ^  and  the  final  kinetic  energy  is 

El! 

The  work  done  is,  therefore,  ^ —  o^  ^^-  ^^^*  ^^^  ^^®  efficiency  is 

'2g 
It  can  at  once  be  seen  from  the  geometry  of  the  figure  that 

Vr  __  Ua     Ui« 


For 

AB«=AC«+CB«+2AC.CG, 

knd  aince 

CD=CB  and  CD^=AC«  +  AD», 

lierefore. 

AB«-AD*=2AC(AC  +  CG) 

=  2i;V. 

Bat 

AB«-AD«=U«-Uj«, 

herefore, 

2g           g  ' 

If  the  water  instead  of  leaving  the  vane  in  a  direction  perpendicular  to  t\  leaves 
t  with  a  velocity  Uj  having  a  component  V^  parallel  to  v,  the  work  done  ou  the 
-ane  per  pound  of  water  is 

9 

If  Ui  be  drawn  on  the  figure  it  will  be  seen  that  the  change  of  velocity  in  the 

V  -V 
Lirection  of  motion  is  now  (V  -  V^),  the  impressed  force  per  pound  is ^ ,  and 

/  V  — V  \ 
be  work  done  is,  therefore,  (  j  ^'i  '^'  ^^^'  P^^  pound. 

A!»  before,  the  work  done  on  the  vane  is  the  loss  of  kinetic  energy  of  the  jet,  and 
.herefore, 

(V-V,)v,^U»-Ui« 

9  'k 

The  work  done  on  the  vane  per  pound  of  water  for  any  given  value  of  Uj ,  is, 
Jierefore,  independent  of  the  direction  of  U^ . 

Example  (2).  A  series  of  vanes  such  as  AB,  Fig.  175,  are  fixed  to  a  (turbine) 
whmel  which  revolves  about  a  fixed  centre  C,  with  an  angular  velocity  u. 

The  radius  of  B  is  B  and  of  A,  r.  Within  the  wheel  are  a  number  of  guide 
na«rr(p"i  through  which  water  is  directed  with  a  velocity  U,  at  a  definite  inchnatiou 
f  with  the  tangent  to  the  wheel.  The  air  is  supposed  to  have  free  accesn  to  the 
wbe«L 

To  draw  the  triangles  of  velocity,  at  iulet  and  outlet,  and  to  find  the  directions 
yi  tlM  tips  of  the  vanes,  so  that  the  water  moves  on  to  the  vanes  without  shock  and 
Learee  the  wheel  with  a  given  velocity  Uj.    Friction  neglected. 

Aa  in  the  last  example  the  velocity  relative  to  the  vane  must  remain  constant, 
^a%A  therefore,  V^  and  v^  are  equal,  but  v  and  v^  are  unequal. 

The  tangent  AH  to  the  vane  at  A  makes  an  angle  0  with  the  tangent  AD  to  the 
irheel,  so  (hat  CD  makes  an  angle  0  with  AD.  The  triangle  of  velocities  ACD  at 
nlgt  18,  therefore,  as  shown  in  the  figure  and  does  not  need  explanation. 

To  draw  the  triangle  of  velocities  at  exit,  set  out  BQ  equal  to  t'j  and  perpen- 
lieolar  to  the  radius  BC,  and  with  B  and  G  as  centres,  describe  circles  with  U,  and 

.  which  is  equal  to  V^— as  radii  respectively,  intersecting  in  E.    Then  G£  is 

l^rmliel  to  the  tangent  to  the  vane  at  B. 


272 


HYDRAULICS 


If  there  is  a  loss  of  head,  /t/,  by  friction,  as  the  water  moves  over  the  vane  tl 
Vf  IB  less  than  V^,  if  h/  is  known,  it  oan  be  found  from 

29      2g     "^^ 

(See  Impulse  turbines.) 

Work  done  on  the  wheel.  Neglecting  Motion  etc.  the  work  done  per  poond 
water  passing  through  the  wheel,  since  the  pressure  is  constant,  bong  equal  to ' 
atmospheric  pressure,  is  the  loss  of  kinetic  energy  of  the  water,  and  is 

2g 

The  work  done  on  the  wheel  can  also  be  found  from  the  consideration  of  1 
change  of  the  angular  momentum  of  the  water  passing  through  the  whed.  Befi 
going  on  however  to  determine  the  work  per  pound  by  this  method,  tiie  notati 
that  has  been  used  is  summarised  and  sevoral  important  principles  oonsideredL 


^  -  ^  ft.  lbs. 


Notation  used  in  connection  with  vaneSy  tturbines  and  centrifug 
pwm/pa.    Let  U  be  the  velocity  with  which  the  water  approach 
the  vane,  Fig.  175,  and  v  the  velocity,  perpendicular  to  the  radii 
AC,  of  the  edge  A  of  the  vane  at  which  water  enters  the  wheel. 
Let  V  be  the  component  of  U  in  the  direction  of  t?, 
u  the  component  of  U  perpendicular  to  t?, 
Vr  the  relative  velocity  of  the  water  and  vane  at  A, 
t?i  the  velocity,  perpendicular  to  BC,  of  the  edge  B  of  the  vw 
at  which  water  leaves  the  wheel,  ^ 

Ui  the  velocity  with  which  the  water  leaves  the  wheel, 
Vi  the  component  of  Ui  in  the  direction  of  t^i, 


IMPACT  OF  WATER  OV   VANES 


273 


Ui  the  component  of  Ui  perpendicular  to  r,,  or  along  BC, 
tv  the  relative  velocity  of  the  water  and  the  vane  at  B. 
ITelociiies  of  whirl.    The  component  velocities  V  and  Vi  are 

called   the  velocities  of  whirl  at  inlet  and  outlet  respectively. 

This  temi  will  frequently  be  used  in  the  following  chapters. 

170.  Definition  of  angular  momentum. 

If  a  weight  of  W  pounds  is  moving  with  a  velocity  U,  Figs.  175 
and  176,  in  a  given  direction,  the  perpendicular  distance  of  which 
is  S  feet  from  a  fixed  centre  C,  the  angular  momentum  of  W  is 

W 

—  .  U .  S  pounds  feet. 

171.  Change  of  angular  momentiun. 

If  after  a  small  time  t  the  mass  is  moving  with  a  velocity  Ui  in 

a  direction,  which  is  at  a  perpendicular  distance  Si  from  C,  the 

W 
angular    momentum   is   now   —  UiSi;    the    change  of   angular 

momentum  in  time  t  is 

-J  (US -U,  SO; 
and  the  rate  of  change  of  angular  momentum  is 

^(US-UaSO. 


Fig.  176.  Fig.  177. 

172.     Two  important  principles. 

(1)  Work  done  by  a  coti/pUy  or  turning  moment.  When  a 
body  is  turned  through  an  angle  a  measured  in  radians,  under  the 
action  of  a  constant  turning  moment,  or  couple,  of  T  pounds  feet, 
the  -work  done  is  Ta  foot  pounds. 

If  the  body  is  rotating  with  an  angular  velocity  w  radians 
per  second,  the  rate  of  doing  work  is  Tw  foot  pounds  per  second, 

and  the  horse-power  is  g^. 


I^   H. 


\^ 


274  HYDRAULICS 

Suppose  a  body  rotates  about  a  fixed  centre  C,  Fig.  177,  and 
a  force  P  lbs.  acts  on  the*  body,  the  perpendicular  distance  from 
C  to  the  direction  of  P  being  S. 

The  moment  of  P  about  C  is 

T  =  P.S. 

If  the  body  turns  through  an  angle  ca  in  one  second,  the 
distance  moved  through  by  the  force  P  is  « •  S,  and  the  work 
done  by  P  in  foot  pounds  is 

Pa>S=T«. 

And  since  one  horse-power  is  equivalent  to  33,000  foot  pounds 
per  minute  or  550  foot  pounds  per  second  the  horse-power  is 

(2)  The  rate  of  change  of  angular  momentv/m  of  a  bodf 
rotating  about  a  fixed  centre  is  equal  to  the  couple  acting  wpm 
the  body.  Suppose  a  weight  of  W  pounds  is  moving  at  any  instant 
with  a  velocity  U,  Fig.  176,  the  perpendicular  distance  of  which 
from  a  fixed  centre  C  is  S,  and  that  a  couple  is  exerted  upon  W 
so  as  to  change  its  velocity  from  U  to  Ui  in  magnitude  and 
direction. 

The  reader  may  be  helped  by  assuming  the  velocity  U  is 
changed  to  Ui  by  a  wheel  such  as  that  shown  in  Fig.  175. 

Suppose  now  at  the  point  A  the  velocity  Ui  is  destroyed  in  a 
time  dt^  then  a  force  will  be  exerted  at  the  point  A  equal  to 

P-W  U 
g  'cV 
and  the  moment  of  this  force  about  C  is  P .  S. 

At  the  end  of  the  time  d^,  let  the  weight  W  leave  the  wheel 
with  a  velocity  Ui.  During  this  time  dt  the  velocity  Ui  might 
have  been  given  to  the  moving  body  by  a  force 

p_WU, 

acting  at  the  radius  Si. 

The  moment  of  Pi  is  Pi  Si ;  and  therefore  if  the  body  has  been 
acting  on  a  wheel.  Fig.  175,  the  reaction  of  the  wheel  thus  exerting 
the  couple  upon  the  body,  or  a  couple  has  been  exerted  upon  it  in 
any  other  way,  the  couple  required  to  change  the  velocity  of  W 
from  U  to  Ui  is 

T  =  J^(US-U.S,) (1). 

Let  the  wheel  of  Fig.  175,  or  the  couple  which  is  acting  npoA 
the  body,  have  an  angular  velocity  w. 


DCPAGT  OF  WATER  ON   VANES  275 

In  a  time  dt  the  angle  moved  through  by  the  couple  is  cud^, 
and  therefore  the  work  done  in  time  dt  is 

T.a)3^  =  — co(US-UiSO  (2). 

Suppose  now  W  is  the  weight  of  water  in  pounds  per  second 
iprhich  strikes  the  vanes  of  a  moving  wheel  of  any  form,  and  this 
-water  has  its  velocity  changed  from  U  to  Ui,  then  by  making  dt 
in  either  equation  (1)  or  (2)  equal  to  unity,  the  work  done  per 
second  is 

Ta>=  — co(US-U,S,), 

and  the  work  done  per  second  per  pound  of  water  entering  the 
wheel  is 

^(US-UiSx). 

This  result,  as  will  be  seen  later  (page  337),  is  entirely  inde- 
pendent of  the  change  of  pressure  as  the  water  passes  through  the 
wheely  or  of  the  direction  in  which  the  water  passes. 

173.  Work  done  on  a  aeries  of  vanes  fixed  to  a  wheel 
expreased  in  terms  of  the  velocities  of  whirl  of  the  water 
entering  and  leaving  the  wheel. 

Outvard  flaw  ta/rbine.  If  water  enters  a  whgelat  the  inner 
circumference,  as  in  Pig.  175,  the  flow  is  said  to  be  outward. 
On  reference  to  the  figure  it  is  seen  that  since  r  is  perpendicular 
to  V,  and  S  to  U,  therefore 

r_U 

And  for  a  similar  reason 

R_Ui 

A^ain  the  angular  velocity  of  the  wheel 

"  =  r  =  R' 
therefore  the  work  done  per  second  is 

And  the  work  done  per  pound  of  flow  is 

g       g 

Inward  fl^ow  twrhine.  If  the  water  enters  at  the  outer  cir- 
cunrference  of  a  wheel  with  a  velocity  of  whirl  V,  and  leaves  at 
the  inner  circumference  with  a  velocity  of  whirl  Vi,  tYve  veVoaVdfe^ 


276 


HYDRAULICS 


of  the  inlet  and  outlet  tips  of  the  vanes  being  v  and  ri  respectively 
the  work  done  on  the  wheel  is  still 

Vr     ViTi 
V        9 
The  flow  in  this  case  is  said  to  be  inward. 

Parallel  flour  or  axial  fl^vc  turbine.  If  vanes,  such  as  those 
shown  in  Fig.  174,  are  fixed  to  a  wheel,  the  flow  is  parallel  to  tiie 
axis  of  the  wheel,  and  is  said  to  be  axial. 

For  any  given  radius  of  the  wheel,  Vi  is  equal  to  Vy  and  the 
work  done  per  pound  is 

which  agrees  with  the  result  already  found  on  page  271. 

174.    Curved  vanes.    Pelton  wheel. 

Let  a  series  of  cups,  similar  to  Figs.  178  and  179,  be  moving 
with  a  velocity  t?,  and  a  stream  with  a  greater  velocity  U  in  the 
same  direction. 

The  relative  velocity  is 

Vr=(U-r). 

Neglecting  friction,  the  relative  velocity  Vr  will  remain  con- 
stant, and  the  water  will,  therefore,  leave  the  cup  at  the  point  B 
with  a  velocity,  Vr,  relative  to  the  cup. 


If  the  tip  of  the  cup  at  B,  Fig.  178,  makes  an  angle  *  with  the 
direction  of  v,  the  absolute  velocity  with  which  the  water  leaves 
the  cup  will  be  the  vector  sum  of  v  and  Vr,  jftnd  is  therefore  Ui. 
The  work  done  on  the  cups  is  then 

IP     U^« 
2g      2g 


IMPACT  OF  WATER  ON  VANES  277 

per  lb.  of  water,  and  the  efficiency  is 

For  TJi,  the  value 

Ui  =  'J{v  -  (U  -  v)  cos  e\'  +  (U  -  vy  sin  ^r-* 
can  be  substituted,  and  the  efficiency  thus  determined  in  terms  of 
r,  U  and  0. 

JPelton  wheel  (mps.  If  0  is  zero,  as  in  Fig.  178,  and  U-v  is 
equal  to  t?,  or  XJ  is  twice  v,  Ui  clearly  becomes  zero,  and  the  water 
drops  away  from  the  cup,  under  the  action  of  gravity,  without 
possessing  velocity  in  the  direction  of  motion. 

The  whole  of  the  kinetic  energy  of  the  jet  is  thus  absorbed 
and  the  theoretical  efficiency  of  the  cups  is  unity. 

TTie  work  done  determined  from  consideration  of  the  cltange  of 
mofnenium.  The  component"  of  Ui,  Fig.  178,  in  the  direction  of 
motion,  is 

17  —  (U  -  V)  COS  d, 

and  the  change  of  momentum  per  pound  of  water  striking  the 
vanes  is,  therefore, 

U~t?-f  (U -  v)jDOS 0 

9 
The  work  done  per  lb.  is 

t?  {U  - 1?  +  (U  - 1?)  coa^} 

9 
and  the  eflSciency  is 

^  2v{U-v  +  (U-t7)cos  0} 
W 
When  0  is  0,  cosO  is  unity,  and 

e=-   u— > 
which  is  a  maximum,  and  equal  to  unity,  when  v  is  -^^ . 

175.  Force  tending  to  move  a  vessel  firom  which  water 
is  iflsnintf  through  an  orifice. 

When  water  issues  from  a  vertical  orifice  of  area  a  sq.  feet, 
in  the  side  of  a  vessel  at  rest,  in  which  the  surface  of  the  water  is 
maintained  at  a  height  h  feet  above  the  centre  of  the  OT\&ce^t\v^ 


278  HYDRAULICS 

pressure  on  the  orifice,  or  the  force  tending  to  move  the  vessel 
in  the  opposite  direction  to  the  movement  of  the  water,  is 

F=2w.a,h\hs., 
w  being  the  weight  of  a  cubic  foot  of  water  in  pounds. 

The  vessel  being  at  rest,  the  velocity  with  which  the  water 
leaves  the  orifice,  neglecting  friction,  is 

t7=  J2ghy 
and  the  quantity  discharged  per  second  in  cubic  feet  is 

The  momentum  given  to  the  water  per  second  is 

9 
=  2w  ,a.h. 
But  the  momentum  given  to  the  water  per  second  is  equal  to 
the  impressed  force,  and  therefore  the  force  tending  to  move  the 
vessel  is 

F  =  2w.a.h, 

or  is  equal  to  twice  the  pressure  that  would  be  exerted  upon  a 
plate  covering  the  orifice.  When  a  fireman  holds  the  nozzle  of  a 
hose-pipe  through  which  water  is  issuing  with  a  velocity  r,  there 
is,  therefore,  a  pressure  on  his  hand  equal  to 

2wav^  _  wav^ 

"W"     9    • 
If  the  vessel  has  a  velocity  V  backwards,  the  velocity  U  of  the 
water  relative  to  the  earth  is 

and  the  pressure  exerted  upon  the  vessel  is 

F=^--^'^lbs. 
9 
The  work  done  per  second  is 

F .  V  = ^^ foot  lbs., 

9 

or  =  — ^ foot  lbs. 

9 
per  lb.  of  flow  from  the  nozzle. 

The  efficiency  is  e  =  — ^—r — - 

_2V(t?-V) 

which  is  a  maximum,  when 

.i?  =  2V 

and  e  =  J. 


IMPACT  OF  WATER  ON  VANES  279 

176.    The  proptdflion  of  Bhips  by  water  jets. 

A  method  of  proi)eUing  ships  by  means  of  jets  of  water  issuing 
from  orifices  at  the  back  of  the  ship,  has  been  used  with  some 
sxiccess,  and  is  still  employed  to  a  very  limited  extent,  for  the 
propalsion  of  lifeboats. 

Wat^r  is  taken  by  pumjw  carried  by  the  ship  from  that 
surrounding  the  vessel,  and  is  forced  through  the  orifices.  Let 
tr   be  the  velocity  of  the  water  issuing  from  the  orifice  relative 

to  the  ship,  and  V  the  velocity  of  the  ship.    Then  ^  is  the 

hea.d  h  forcing  water  from  the  ship,  and  the  available  energy 
per  pound  of  water  leaving  the  ship  is  h  foot  pounds. 

The  whole  of  this  energy  need  not,  however,  be  given  to  the 
water  by  the  pumps. 

Ima^ne  the  ship  to  be  moving  through  the  water  and  having 
a  pip^e  with  an  open  end  at  the  front  of  the  ship.  The  water  in 
front  of  the  ship  being  at  rest,  water  will  enter  the  pipe  with  a 

velocity  V  relative  to  the  ship,  and  having  a  kinetic  energy  ^y 

per  pound.  K  friction  and  other  losses  are  neglected,  the  work 
that  the  pomps  will  have  to  do  upon  each  pound  of  water  to  eject 
it  at  the  back  with  a  velocity  v  is,  clearly, 

2g     2g' 

As  in  the  previous  example,  the  velocity  of  the  water  issuing 

from  the  nozzles  relative  to  the  water  behind  the  ship  is  v  -  Y, 

v-Y 
and  the  change  of  momentum  per  pound  is,  therefore,  -         .    If  a 

is  the  area  of  the  nozzles  the  propelling  force  on  the  ship  is 
and  the  work  done  is 


9 


9 
The  efficiency  is  the  work  done  on  the  ship  divided  by  the 

work   done  by  the  engines,  which  equals  wav  (5 —  n " )  ^"*^' 

therefore, 

.2V(i;-V) 


e  = 


2V 

■«  +  ¥' 


280 


HYDBAULICS 


which  can  be  made  ae  near  unity  as  is  desired  hy  makiiig  r  and 

V  approxiraate  to  equality. 

Bat  for  a  given  area  a  of  the  orifices^  and  velocity  t\  the  nearer 

V  approximates  to  V  the  less  the  propelling  fore©  F  becomes,  and 
the  size  of  ship  that  can  be  driven  at  a  g:iven  velocity  V  for  ihn 
given  area  a  of  the  orifices  diminishes. 

If  ris2Y,  e-|. 


EXAMPLES. 

(1)  Ten  cubic  feet  of  water  per  second  ore  diiicharged  ttom  &  ^tationtfj 
jet*  llie  eectional  area  of  which  is  1  square  foot.  The  water  irapiagea  not* 
mally  on  a  fiat  surface,  moring  in  the  directiDii  of  the  jet  w^tb  a  velodiy 
of  %  feet  per  second.  Find  the  pressure  on  the  plane  in  lbB,«  and  the  ^mak 
done  on  tlie  plane  in  horse -iK>wer. 

(2)  A  jet  of  water  deUyering  100  gallons  per  Becond  with  a  Telocity  d 
20  feet  per  second  iiQpiuge&  perpendicularly  on  a  wall.  Find  the  pre^itre 
on  the  w^all* 

(8)  A  jet  delivers  160  cubic  feet  of  water  per  minute  at  a  velocity  of 
I  feet  per  Heeond  and  strikes  a  plane  perpendicularly.    Find  the  preestiM 

tlie  plane^(l)  when  it  is  at  rest;  {%}  when  it  is  moviug  at  5  feet  per 
second  In  tlie  direction  of  the  jet.  En  the  latter  case  find  the  work  c]oo« 
per  second  in  driving  the  plane. 

(4)  A  fire*engine  hose^  8  inches  bore,  discharges  water  at  a  velocity  of 
100  feet  per  seccnd.  Supposing  the  jet  directed  normally  to  tJie  side  d  a 
binding,  find  the  presaure. 

(6)  Water  iBsues  horizontally  from  a  fixed  thin -edged  orifice^  0  indbai 
sqnare,  under  a  head  of  25  feet.  Tlie  jet  impinges  normally  on  a  pbii^ 
moving  in  tlie  same  direction  at  10  feet  per  second.  Find  tlie  pres^suns  <A 
the  plane  in  lbs.,  and  the  work  dome  in  horse-power.  Take  the  coefficicBl 
of  discharge  as  *64  and  the  coefficient  of  velocity  as  '97. 

(6 1  A  jet  and  a  plane  surface  move  in  directions  inclined  at  30%  witb 
velocities  of  30  feet  and  10  feet  per  second  resjiectively.  \Mm%  i»  tlw 
relative  velocity  of  the  jet  and  surface  ? 

(7)  Let  AB  and  BC  be  two  lines  inclined  at  80%  A  jet  of  water  moffli 
in  the  direction  AB,  w^th  a  velocity  of  25  feet  per  second,  and  a  »eric»  o( 
vanes  move  in  the  direction  CB  with  a  velocity  of  15  feet  per  second.  Fm^ 
the  form  of  the  vane  so  that  the  w^ater  may  come  on  to  it  tangentially,  afiJ 
leave  it  in  the  direction  ED,  perpendicular  to  CB. 

Supposing  that  the  jet  is  1  foot  wide  and  1  inch  thick  before  impinging 
find  the  effort  of  the  jet  on  the  vanes. 

(8)  A  enrved  plate  is  mounted  on  a  slide  so  that  tlie  plate  i 
move  along  tlie  shde.  It  receives  a  jet  of  water  at  an  angle  of  3i  • 
normal  to  the  direction  of  sliding,  and  the  jet  leaves  tlie  plate  at  m  iagi» 


IMPACT  OF  WATEE  ON  VANES 


281 


"With  the  «eiiia  noimial.    Find  the  force  which  must  be  applied  to 

be  to  the  direction  of  slidiag  to  hold  it  at  rest^  and  also  tii©  nt^moL 
!e  on  the  slide.  Quantity  of  water  flawing  is  500  Iha.  per  minute 
■vekxaty  of  B5  feet  per  second. 

A  fixed  Fane  receives  a  jet  of  water  at  an  angle  of  120^  with  a 
a  AB-     Find  what  angle  the  jet  must  be  turned  tlirough  in  order 
I  pffcasore  on  the  vane  in  the  direction  AB  may  be  40  Ibs.^  when  the 
\  water  m  45  Ibs^.  per  second  at  a  velocity  of  30  feet  per  second. 

Water  under  a  head  of  60  feet  m  discharged  through  a  pipe  6  indies 
t  and  150  feet  long,  and  then  tlirough  a  nois^le,  the  area  of  which 
nth  the  area  of  the  pipe, 

all  losses  but  the  friciioti  of  the  piper  determine  the  preaaure 
ed  plate  placed  in  front  of  the  nozzle. 

A  jet  of  watier  4  tnehee  diameter  impinges  on  a  fixed  cone,  the 
Adding  with  that  of  the  jet,  and  the  apex  angle  being  BO  degEeeSi 
llocity  of  10  feet  per  second.  Pind  the  pressure  tending  to  move  the 
tJie  direction  of  its  asds. 

A  veeael  containing  water  and  having  in  one  of  it»  vertical  sides 
orMoe  1  inch  diameter^  which  at  first  is  phigged  up,  in 
ided  in  such  a  way  tliat  any  diiiplacing  force  can  be  accurately 
^ed.  On  the  removal  of  the  plug,  the  horii^ontai  force  rec}uired  to 
&  venal  in  place,  applied  opposite  to  the  orifice^  is  3*6  lbs.  By  the 
measimng  tank  the  discharge  m  found  to  be  31  gallons  (>er  minute^ 
tol  of  the  water  in  the  vessel  being  maintained  at  a  constant  height 
M  wihove  the  orifica  Determine  the  coefficients  of  velc>cityi  con^ 
X  and  di»cliaige. 

A  train  carrying  a  Eamsbottom's  scoop  for  taking  water  into  the 
18  nmiiiiig  at  24  miles  an  hour.  What  is  the  greatest  height  at 
Ibe  8000|k  will  deliver  tlie  water  ? 

A  locomotive  going  at  40  miles  an  hour  scoops  up  water  from  a 
The  tank  is  8  feet  above  the  mouth  of  the  scoop*  and  tlie  delivery 
M  ftJi  area  of  50  square  inches.  If  half  the  available  head  is  wasted 
msce,  find  tlie  velocity  at  wliich  the  water  is  delivered  into  the  tank, 
number  of  tons  Ufted  in  a  trench  500  yards  long.  What,  utider 
ions*  is  the  increased  resistance;  and  what  is  the  minimum 
at  which  the  tank  can  be  filled  ?     Lond.  Un.  1906. 

A  stream  deliyeiing  3000  gallons  of  water  per  minute  with  a 
<d  40  feet  per  second,  by  impinging  on  vanes  is  caused  freely  to 
Ihroogh  an  angle  of  10'",  the  velc^city  being  dimmialied  to  M  feet 
ond.  Delejmine  the  pressure  on  tlie  vanes  due  to  impact.  If  the 
m  moving  in  the  direction  of  that  pressure,  find  their  velocity  and 
the  luefni  hor»e-power*    Lond.  Un.  1906, 

Wsttif  flows  from  a  2-inch  pipe,  without  contraction,  at  45  feet  per 

imtne  the  maadmuni  work  done  on  a  machine  carrying  moving 
I^AjoUowiDg  emm  and  tlie  respective  efficiencieti;^ 


282  HYDRAULICS 

(a)  When  the  water  impinges  on  a  single  flat  plate  at  right  angles  and 
leaves  tangentiaUy. 

(b)  Similar  to  (a)  but  a  large  number  of  equidistant  flat  plates  an 
interposed  in  the  path  of  the  jet. 

(c)  When  the  water  glides  on  and  off  a  single  semi-cylindrical  cup. 
{(i)    When  a  large  number  of  cups  are  used  as  in  a  Pelton  wheeL 

(17)  In  hydraulic  mining,  a  jet  6  inches  in  diameter,  discharged  unte 
a  hesid  of  400  feet,  is  delivered  horizontally  against  a  vertical  cliff  ho$B 
Find  the  pressure  on  the  face.  What  is  the  horse-power  delivered  by  tin 
jet? 

(18)  If  the  action  on  a  Pelton  wheel  is  equivalent  to  that  ol  a  jet  on  a 
series  of  hemispherical  cups,  find  the  efficiency  when  the  speed  of  the 
wheel  is  five-eighths  of  the  speed  of  the  jet. 

(19)  If  in  the  last  question  the  jet  velocity  is  50  feet  per  second, 
and  the  jet  area  0*15  square  foot,  find  the  horse-power  of  the  wheeL 

(20)  A  ship  has  jet  orifices  8  square  feet  in  aggregate  area,  and  dii- 
charges  through  the  jets  100  cubic  feet  of  water  per  second.  The  speed «( 
the  ship  is  15  feet  per  second.  Find  the  propelling  force  of  the  jets,  te 
efficiency  of  the  propeller,  and,  neglecting  friction,  the  horse-power  of  fli 

engines. 


CHAPTER   IX. 

WATER  WHEELS  AND   TURBINES. 

Water  ^wKeels  can  be  divided  into  two  classes  as  follows. 

(a)  Wheels  upon  which  the  water  does  work  partly  by 
mpolae  but  almost  entirely  by  weight,  the  velocity  of  the  water 
rfien  it  strikes  the  wheel  being  small.  There  are  two  types  of 
bis  class  of  wheel,  Overshot  Wheels,  Figs.  180  and  181,  and 
Ireast  Wheels,  Figs.  182  and  184. 

(6)  Wheels  on  which  the  water  acts  by  impulse  as  when 
he  wheel  utilises  the  kinetic  energy  of  a  stream,  or  if  a  head  h  is 
available  the  whole  of  the  head  is  converted  into  velocity  before 
lie  water  comes  in  contact  with  the  wheel.  In  most  impulse 
rheels  the  water  is  made  to  flow  under  the  wheel  and  hence 
hey  are  called  Undershot  Wheels. 

It  will  be  seen  that  in  principle,  there  is  no  line  of  demarcation 
letween  impulse  water  wheels  and  impulse  turbines,  the  latter 
mly  differing  from  the  former  in  constructional  detail. 

177.     Overshot  water  wheels. 

This  type  of  wheel  is  not  suitable  for  very  low  or  very  high 
leads  as  the  diameter  of  the  wheel  cannot  be  made  greater  than 
he  head,  neither  can  it  conveniently  be  made  much  less. 

Figs,  180  and  181  show  two  arrangements  of  the  wheel,  the 
mly  difference  in  the  two  cases  being  that  in  Fig.  181,  the  top  of 
he  wheel  is  some  distance  below  the  surface  of  the  water  in  the 
ip-0tream  channel  or  penstock,  so  that  the  velocity  v  with  which 
he  water  reaches  the  wheel  is  larger  than  in  Fig.  180.  Tliis  has 
he  advantage  of  allowing  the  periphery  of  the  wheel  to  have  a 
i^her  velocity,  and  the  size  and  weight  of  the  wheel  is  conse- 
uently  diminished. 

The  buckets,  which  are  generally  of  the  form  shown  in  the 
gures,  or  are  curved  similar  to  those  of  Fig.  182,  are  coll- 
ected to  a  rim  M  coupled  to  the  central  hub  of  the  >N\\ee\  b^ 


284 


HYDRAULICS 


suitable  spokes  or  framework.  This  class  of  wheel  has 
considerably  used  for  heads  varying  from  6  to  70  feet,  but  u 
becoming  obsolete,  being  replaced  by  the  modem  turbine,  y 
for  the  same  head  and  power  can  be  made  much  more  com 
and  can  be  run  at  a  much  greater  number  of  revolutions  pei 
time. 


Fig.  180.    Overshot  Water  Wheel. 


Fig.  181.    Overshot  Water  Wheel. 

The  direction  of  the  tangent  to  the  blade  at  inlet  for  no  si 
can  be  found  by  drawing  the  triangle  of  velocities  as  in  Figs, 
and  181.  The  velocity  of  the  periphery  of  the  wheel  is  t?  and 
velocity  of  the  water  U.  The  tip  of  the  blade  should  be  par 
to  Vr.    The  mean  velocity  U,  of  the  water,  as  it  enters  the  wl 


WATER  WHEELS  285 

in  Fig.  181,  will  be  Vo  •^kyj2gB,,  v^  being  the  velocity  of  approach 
t|{  the  water  in  the  channel,  H  the  fall  of  the  free  surface  and  k 
a  ooeflScient  of  velocity.  The  water  is  generally  brought  to  the 
wlieel  along  a  wooden  flume,  and  thus  the  velocity  U  and  the 
•opply  to  the  wheel  can  be  maintained  fairly  constant  by  a  simple 

dmce  placed  in  the  flume. 

The  best  velocity  v  for  the  periphery  is,  as  shown  below, 

equal  to  iU  cos  tf,  but  in  practice  the  velocity  v  is  frequently 

much  greater  than  this. 

In  order  that  XJ  may  be  about  2v  the  water  must  enter  the 

wheel  at  a  depth  not  less  than 

below  the  water  in  the  penstock.    When 

r  =  4-5  feet,  H  =  0-63  feet, 

mnd  when  r  =  8  feet,     H  =  1  foot. 

If  the  total  fall  to  the  level  of  the  water  in  the  tail  race  is  h, 
the  diameter  of  the  wheel  may,  therefore,  be  between  h  and 

9 

Since  IT  is  equal  to  >/2flrH,  for  given  values  of  U  and  of  /i,  the 
larger  the  wheel  is  made  the  greater  must  be  the  angular  distance 
from  the  top  of  the  wheel  at  which  the  water  enters. 

With  the  type  of  wheel  and  penstock  shown  in  Fig.  181,  the 
bead  H  is  likely  to  vary  and  the  velocity  U  will  not,  therefore,  be 
constant. 

If,  however,  the  wheel  is  designed  for  the  required  power  at 
minimnm  flow,  when  the  head  increases,  and  there  is  a  greater 
quantity  of  water  available,  a  loss  in  efficiency  \vill  not  be 
important. 

The  horse-^fxncer  of  the  wheel.  Let  D  be  the  diameter  of  the 
wheel  in  feet  which  in  actual  wheels  is  from  10  to  70  feet. 

Let  N  be  the  number  of  buckets,  which  in  actual  wheels  is 
generally  from  2|  to  3D. 

Let  Q  be  the  volume  of  water  in  cubic  feet  of  water  supplied 
per  second. 

Let  ••  be  the  angular  velocity  of  the  wheel  in  radians,  and  n 
the  number  of  revolutions  per  sec. 
Let  b  be  the  width  of  the  wheel. 

Let  d,  which  equals  rj  — n,  be  the  depth  of  the  shroud,  which 
en  actual  wheels  is  from  10"  to  20". 


i 
286 

Wbmfii       the  form  of  tbe  backete  the  capacity  of  each  bucket  j 

S 

Tfce  BTiiiiber  of  bocketa  whieh  pa^  tbe  str^m  pet  m:miii 


U  «  fnctbii  i  of  each  bucket:  is  filled  mth  water 


or 

Tlie  Iractidii  1  %  from  |  to  |. 

If  k  »  die  &L  flie  level  of  die  tail  race  ^i 

the  efficiefic7  of  ti  l-power  is 

.  50       ' 
and  the  widtli  b  for  a  given  horse-power,  HP,  18 
llOOHP     ^j^,^     HP 


^<p*ci  o/ reiUn/wi^of /ot^^.  As  the  wteel  revolves,  the  siir&« 
of  the  water  in  the  buckets,  due  to  centrifugal  forces,  takes  up  i 
parabolic  form. 

It  h  shown  on  page  33-S  that  when  a  mass  of  water  having  tt 
inner  radios  r^  and  outer  radius  ri  revolves  about  a  fijced  cenlie 
^^-ith  angular  velocity  «,  the  pressure  headj  due  to  centrifogi 
forces,  at  any  radius  r,  is 

ir"        2g 

To  balance  this  prepare  head  the  surface  of  the  water  in  asj 
bucket,  at  the  point  Cj  of  radius  r,  must  be  raised  above  tb 
hori^>iitaI  through  A  a  distance 

This  is  the  equation  to  a  parabola,  and  the  surface  of  the  water, 
therefore,  assumes  the  form  of  a  parabolic  curve. 

Let  To  be  the  radius  at  the  centre  of  the  surface  of  the  watoin 
any  cup  and  ^  the  inclination  of  the  radius  r©  to  the  horisontaL 

Then  since  n  is  nearly  equal  to  ri,     ^    =  n  nearly. 


WATER  WHEELS  287 

Then  y=2^(n+r)  (r-rO 

=  ^r.(r-n)  nearly. 

Therefore,  y  is  approximately  proportional  to  r— n,  and  the 
irface  AB  is  approximately  a  straight  line  inclined  at  an  angle 
,  the  tangent  of  which  is 

tan  B  = cos  ^. 

g 

Losses  of  energy  in  overshot  wheels. 

V  * . 
(a)     The  whole  of  the  velocity  head  ^  is  lost  in  eddies  in  the 

nekets. 

In  addition,  as  the  water  falls  in  the  bucket  through  the 
ertical  distance  EM,  its  velocity  will  be  increased  by  gravity, 
nd  the  velocity  thus  given  will  be  practically  all  lost  by  eddies. 

Again,  if  the  direction  of  the  tip  of  the  bucket  is  not  parallel  to 
Tr  the  water  will  enter  with  shock,  and  a  further  head  will  be 
Dst.  The  total  loss  by  eddies  and  shock  may,  therefore,  be 
rritten 

'^^^^^ 

[)r  Ai  +  *i2^, 

fc  and  Tci  being  coefficients  and  hi  the  vertical  distance  EM. 

(6)  The  water  begins  to  leave  the  buckets  before  the  level  of 
the  tail  race  is  reached.  This  is  increased  by  the  centrifugal 
forces,  as  clearly,  due  to  these  forces,  the  water  will  leave  the 
buckets  earlier  than  it  otherwise  would  do.  If  h^  is  the  mean 
Iieight  above  the  tail  level  at  which  the  water  leaves  the  buckets, 
»  h«id  equal  to  fc»  is  lost.  By  fitting  an  apron  GH  in  front  of  the 
wheel  the  water  can  be  prevented  from  leaving  the  wheel  until  it 
i  very  near  the  tail  race. 

(c)  The  water  leaves  the  buckets  with  a  velocity  of  whirl 
equal  to  the  velocity  of  the  periphery  of  the  wheel  and  a  further 

l^ead  JT  is  lost. 

(d)  If  the  level  of  the  tail  water  rises  above  the  bottom  of 
Awheel  there  will  be  a  further  loss  due  to,  (1)  the  head  K  equal  to 
•Ke  height  of  the  water  above  the  bottom  of  the  wheel,  (2)  the 
^pact  of  the  tail  water  stream  on  the  buckets,  and  (3)  the 
■C^ndency  for  the  buckets  to  lift  the  water  on  the  ascending  side  of 
^  wheel. 


288  HTDRAUUCS 

In  times  of  flood  there  may  be  a  considerable  rise  of  Urn 
down-stream,  and  h^  may  then  be  a  large  fraction  of  A.  If  on' 
the  other  hand  the  wheel  is  raised  to  such  a  height  above  the  td 
water  that  the  bottom  of  the  wheel  may  be  always  clear^  Urn 
head  km  will  be  considerable  during  dry  weather  flow,  and  d» 
greatest  possible  amount  of  energy  will  not  be  obtained  from  tfa 
water,  just  when  it  is  desirable  that  no  energy  shall  be  wasted. 

If  h  is  the  difF erence  in  level  between  the  up  and  down-stresa 
surfaces,  the  maximum  hydraulic  efficiency  possible  is 

J-^'^t^^ ,. 

and  the  actual  hydraulic  efficiency  will  be 

^^  h 

k,  ki  and  h  being  coefficients. 

The  efficiency  as  calculated  from  equation  (1),  for  any  giToi 
value  of  hm,  is  a  maximum  when 

2g^2g 

is  a  minimum. 

From  the  triangles  EKF  and  KDF,  Fig.  180, 

(U  cos  e-vy^  (U  sin  ey = va 

Therefore,  adding  v^  to  both  sides  of  the  equation, 

Vr"  +  i;'  =  U*cos''^-2Ut;cos^  +  2v»  +  U>sin*tf, 

which  is  a  minimum  for  a  given  value  of  U,  when  2Uv  cos^-2f* 
is  a  maximum.  Differentiating  and  equating  to  zero  this,  and 
therefore  the  efficiency,  is  seen  to  be  a  maximum,  when 

U       /J 
V  =  -^^  cos  u. 

The  actual  efficiencies  obtained  from  overshot  wheels  vary 
from  60  to  80  per  cent. 

178.    Breast  wheel. 

This  type  of  wheel,  like  the  overshot  wheel,  is  becoming 
obsolete.  Fig.  182  shows  the  form  of  the  wheel,  as  designed  by 
Fairbairn. 

The  water  is  admitted  to  the  wheel  through  a  number  of 
passages,  which  may  be  opened  or  closed  by  a  sluice  as  shown  in 
the  figure.  The  directions  of  these  passages  may  be  made  so  thai 
the  water  enters  the  wheel  without  shock.    The  water  is  retained 


WATER   WHEELS 


*  backet)  by  the  breast,  until  tb©  bucket  reaches  the  tail  race, 
greater  fraction  of  tlie  head  m  therefore  utilised  than  in 
verahot  wheel.  In  order  that  the  air  may  enter  and  leave 
i^keta  freely,  they  are  partly  open  at  the  inner  vim.  Since 
iter  in  the  tail  iBce  runs  in  the  direction  of  the  motion  of 
9ttom  of  the  wheel  there  is  no  serious  objection  to  the  tail 
level  being  6  inches  above  the  bottom  of  the  wheel. 
The  losses  of  head  will  be  the  same  as  for  the  overshot  wheel 
S^t  that  k^  will  be  practically  ^ero,  and  in  addition,  there  will 
I08B  by  friction  in  the  guide  passages,  by  friction  of  the  water 

t moves  over  the  breast,  and  further  loss  due  to  leakage 
&n  the  breast  and  the  wheel* 


ling  to  Rankine  the  velocity  of  the  rim  for  ovei^hot  and 
wbeeb,  shoald  Le  from  4^  to  8  feet  per  second,  and  the 
j/dty  XJ  fihoaltl  be  about  2i^ 

depth  o!  the  shroud  which  is  equal  to  n-tt  is  from  1  to 

liet  it  be  denoted  by  d.    Let  H  be  the  total  fall  and  let 

aed  that  the  efficiency  of  the  wheel  is  65  per  eeiit.    Then, 


mmM 


mi 


290 


HYDRAUUGS 


the  quantity  of  water  required  per  second  in  cubic  feet  for  a 
given  horse-power  N  is 

^_        N.550 
^"■62-4xHxO-65 
_  13'5N 
H    • 
From  ^  to  f  of  the  volume  of  each  bucket,  or  from  ^  to  |  of  the 
total  volume  of  the  buckets  on  the 
loaded  part  of  the  wheel  is  filled  with 
water. 

Let  6  be  the  breadth  of  the  buckets. 
If  now  V  is  the  velocity  of  the  rim,  and 
an  arc  AB,  Fig.  183,  is  set  off  on  the 
outer  rim  equal  to  v,  and  each  bucket 
is  half  full,  the  quantity  of  water 
carried  down  per  second  is 

iABCD.6. 
Therefore 


«-»(^') 


vdb. 


Equating  this  value  of  Q  to  the  above  value,  the  width  b  is 

^^      27KD 

(ti  +  Ti)  vdR ' 

D  being  the  outer  diameter  of  the  wheel. 

Breast  wheels  are  used  for  falls  of  from  5  to  15  feet  and  the 

diameter  should  be  from  12  to  25  feet.    The  width  may  be  as 

great  as  10  feet. 

Example,    A  breaat  wheel  20  feet  diameter  and  6  feet  wide,  working  on  a  fifl 

of  14  feet  and  having  a  depth  of  shroud  of  1'  S",  has  its  backets  f  falL     The  meta 

velocity  of  the  backets  is  5  feet  per  second.    Find  the  horse-power  of  the  wh«d» 

assaming  the  efficiency  70  per  cent. 

xjxy     K     ^oK     a     ^      62-4  X  0-70x14' 

HP  =  5x  1-25x6x5  X =^77 

o  ooU 

=  26-1. 

The  dimensions  of  this  wheel  should  be  compared  with  those  calonlated  for  M 

inward  flow  turbine  working  under  the  same  h^  and  developing  the  aame  hofM* 

power.    See  page  339. 

179.    Sagebien  wheels. 

These  wheels,  Fig.  184,  have  straight  buckets  inclined  to  the 
radius  at  an  angle  of  from  30  to  45  degrees. 

The  velocity  of  the  periphery  of  the  wheel  is  very  small,  nerer 
exceeding  2i  to  3  feet  per  second,  so  that  the  loss  due  to  the  witer 
leaving  the  wheel  with  this  velocity  and  due  to  leakage  betweoi 
the  wheel  and  breast  is  small. 


WATER    WHEELS 


291 


An  efficiency  of  over  80  per  cent,  has  been  obtained  wi^ 
wheels* 

The  water  enters  the  wheel  in  a  horizontal  direction  wit 
velocity  V  equal  to  that  in  the  penstock,  and  the  triangle  of 
reloeities  is  therefore  ABC* 

If  the  bucket  is  made  parallel  to  Vr  the  water  entei-s  without 

ck,  while  at  the  same  time  there  is  no  lofts  of  bead  due  to 

icnon  of  guide  passages,  or  to  contraction  as  the  water  enters  or 

iv€3  them ;  moreover  the  direction  of  the  stream  has  not  to  be 

aoged. 


Fig.  1S4.    Sftgfbien  WUeel. 

Tlie  iucUned  srraight  bucket  has  one  disadvantage ;  when  the 
[lower  part  of  the  whee!  is  drowaedj  the  buckets  as  they  ascend  are 
{more  nearly  perpendicular  to  the  surface  of  the  tail  water  than 
rheti  the  blades  are  radial,  but  as  the  peripheral  speed  is  very 
flow  the  renistance  due  to  this  cause  is  not  considerable, 

ISO.    Impulse  wheels. 

In  OvL-rshot  ami  Breast  wheels  the  work  is  done  principally 
Iby  the  weight  of  the  water.     In  the  wheels  now  to  be  considered 
ie  whole  of  the  head  available  is  converted  into  velocity  before 
le  wAter  strikes  the  wheel,  and  the  work  is  done  on  the  wheel 
"rig  the  niomeutum  of  the  mass  of  moving  water,  or  in 
i  -,  by  changing  the  kinetic  energy  o£  tlae  w&^«nc. 


J 


292 


HYDRAULICS 


Undershot  wheel  with  fiat  blades.  The  simplest  case  is  wlien 
a  wheel  with  radial  blades,  similar  to  that  shown  in  Fig*.  185,  is 
put  into  a  nmning  stream. 

If  6  is  the  width  of  the  wheel,  d  the  depth  of  the  stream  under 
the  wheel,  and  U  the  velocity  in  feet  per  second,  the  weight  of 
water  that  will  strike  the  wheel  per  second  isb.d.w.TJ  lbs,,  and 
the  energy  available  per  second  is 

b.d.WcT foot  lbs. 

Let  V  be  the  mean  velocity  of  the  blades. 

The  radios  of  the  wheel  being  large  the  blades  are  similar  to 
a  series  of  flat  blades  moving  parallel  to  the  stream  and  the  water 
leaves  them  with  a  velocity  v  in  the  direction  of  motion. 

As  shown  on  page  268,  the  best  theoretical  value  for  the 
velocity  v  of  such  blades  is  iXJ  and  the  maximum  possible 
efficiency  of  the  wheel  is  0'5. 


Fig.  185.    Impalse  Wheel. 


By  placing  a  gate  across  the  channel  and  making  the  bed  near 
the  wheel  circular  as  in  Fig.  185,  and  the  width  of  the  wheel 
equal  to  that  of  the  channel,  the  supply  is  more  under  control,  and 
loss  by  leakage  is  reduced  to  a  minimum. 

The  conditions  are  now  somewhat  different  to  those  assmned 
for  the  large  number  of  flat  vanes,  and  the  maximum  possiUe 
efficiency  is  determined  as  follows. 

Let  Q  be  the  number  of  cubic  feet  of  water  passing  through 
the  wheel  per  second.  The  mean  velocity  with  which  the  wrtir 
leaves  the  penstock  at  ah  is  \5  =  hJ2gh.    Let  the  defyth  of  tht 


WATER  WHEELS  293 

stream  at  a&  be  ^.  The  velocity  with  which  the  water  leaves  the 
wheel  at  the  section  cd  is  Vy  the  velocity  of  the  blades.  K  the 
width  of  the  stream  at  c(2  is  the  same  as  at  a&  and  the  depth 
is  A«,  then, 

^  X  t?  =  ^  X  XJ, 

or  ^  =  — . 

V 

Since  TT  is  greater  than  v,  h^  is  greater  than  ty  as  shown  in 
the  figure. 

The  hydrostatic  pressure  on  the  section  cd  is  ^ho^bw  and  on 
the  section  a6  it  is  ifbw. 

The  change  in  momentum  per  second  is 

and  this  must  be  equal  to  the  impressed  forces  acting  on  the  mass 
of  water  flowing  per  second  through  ab  or  cd. 

These  impressed  forces  are  P  the  driving  pressure  on  the  wheel 
blades,  and  the  difference  between  the  hydrostatic  pressures  acting 
on  cd  and  ab. 

If,  therefore,  the  driving  force  acting  on  the  wheel  is  P  lbs., 
then, 

P  +  iWfcw;  -  ^^"fctr  =  Q^  (U  -  tj). 
Substituting  for  hoy  — ,  the  work  done  per  second  is 

Or,  since  Q  =  6 .  ^ .  U, 

w-9=(n-.)-|«j(2-i). 

The  efficiency  is  then, 

f  (U-v)     t  /U     v\ 
g  2\v     U/ 

*= — — w ' 

2g 
which  is  a  maximum  when 

2v^  -  4u»U  +  gtU''  +  gtv^  =  0. 

The  best  velocity,  v,  for  the  mean  velocity  of  the  blades,  has 
been  found  in  practice  to  be  about  04U,  the  actual  efficiency  is 
from  30  to  35  per  cent.,  and  the  diameters  of  the  wheel  are 
generally  from  10  to  23  feet. 

Floating  wheels.  To  adapt  the  wheel  to  the  rising  and 
lowering  of  the  waters  of  a  stream,  the  wheel  may  be  mounted  ou 


294 


HYDRAULICS 


a  frame  which  may  be  raised  or  lowered  as  the  stream  rises,  Gr  thd 
axle  carried  upon  pontoons  so  that  the  wheel  rises  automaticalljr 

r'th  the  stream. 
181,    Poncelet  wheel. 
^rhe  efficiency  of  the  straight   hlade  impulse  wheels  is 
small,  due  to  the  large  amount  of  energj^  lost  by  shock,  and  to 
velocity  with  which  the  water  leaves  the  w^heel  in  the  direction 
motion, 

Tlie  efficiency  of  the  wheel  is  doubled,  if  the  blades  are  of  such 
a  form,  that  the  direction  of  the  blade  at  enti-ance  is  parallel  to 
■the  relative  velocity  of  the  water  and  the  blade,  as  fiist  suggestad 
l)y  Poncelet,  and  the  water  is  made  to  leave  the  wheel  with  do 
component  in  the  direction  of  mution  of  the  periphery  of  tha 
wheel. 

Fig,  186  shows  a  Poncelet  wheel. 


I 


Fig.  186*    Undershot  WlieeL 

Suppose  the  water  to  approach  the  edge  A  of  a  blade  mil 
velocity  U  making  an  angle  ^  mth  the  tangent  to  the  wheel  at  t 

Then  if  the  direction  of  motion  of  the  water  is  in  the  directifl 
AC  the  triangle  of  velocities  for  entrance  is  ABC* 

The  relative  velocity  of  the  water  and  the  wheel  is  V^  and  if 
the  blade  is  made  sufficiently  deep  that  the  water  does  not  overflow 
the  upper  edge  and  there  is  no  loss  by  shock  and  by  fiictiun,  i 
particle  of  water  will  rise  up  the  blade  a  vertical  height 


WATER   WHEELS  295 

a  begins  to  fall  and  arrives  at  the  tip  of  the  blade  with  the 
ty  Vr  relative  to  the  blade  in  the  inverse  direction  BE. 
he  triangle  of  velocities  for  exit  is,  therefore,  ABE,  BE  being 
1  to  BC. 
.'he  velocity  with  which  the  water  leaves  the  wheel  is  then 

AE  =  U,. 
It  has  been  assomed  that  no  energy  is  lost  by  friction  or  by 
»ck,  and  therefore  the  work  done  on  the  wheel  is 

ad  the  theoretical  hydraulic  efficiency*  is 

=  l-§* (1). 

This  will  be  a  maximum  when  Ui  is  a  minimum. 

Now  since  BE  =  BC,  the  perpendiculars  EF  and  CD,  on  to 
AB  and  AB  produced,  from  the  points  E  and  C  respectively,  are 
equaL  And  since  AC  and  the  angle  0  are  constant,  CD  is  constant 
for  all  values  of  Vy  and  therefore  FE  is  constant.  But  AE,  that  is 
Ui,  is  always  greater  than  FE  except  when  AE  is  perpendicular 
to  AD.  The  velocity  Ui  will  have  its  minimum  value,  therefore, 
when  AE  is  equal  to  FE  or  Ui  is  perpendicular  to  v. 

The  triangles  of  velocities  are  then  as  in  Fig.  187,  the  point  B 
bisects  AD,  and 

V  =  JU  cos  6, 

For  maximum  efficiency,  therefore, 

V  =  fU  cos  0. 

*  In  what  follows,  the  terms  theoretical  hydraulic  efficiency  and  hydniulio 
•flSeiency  will  be  freqaently  osed.  The  niazimom  work  per  lb.  that  can  be  utilised 
bj  ftoy  hydxmolio  maohine  supplied  with  water  under  a  head  H,  and  from  which 

the  water  ezluuuts  with  a  velocity  u  is  H  -  ^t-  .    The  ratio 

H    Hi 

11  the  theoratieel  hydraulic  effidenoy.    If  there  are  other  hydraulic  losses  in  th 
mnifhiw^i  eqairalent  to  a  head  h/  per  lb.  of  flow,  the  hydraulic  efficiency  is 

— H — • 

The  aetoal  effioieiiey  of  the  maohine  is  the  ratio  of  the  external  work  done  per 
of  water  hj  the  machine  to  H. 


BTDftlDIiCS 
'  cmn  »bo  be  found  t>K  oaoitdeiiiig  Qm  dtangf  oT 

TW  totfti  cb»oge  of  Telocity  jmprewed  <m  the  water  is  CE^  aoJ 
ife  cittiig^  in  th^  directioii  of  tsMmx  is 


fwm^  FD,  1%.  im  AIL^C 

A«d»aiBEi.ttloltoBC,FBiii     L^^^^'h'^^ 

FD  =  2(Uco6i-r).  Kg.  m. 


uid  tke 


r 

9-e) 


4 


DtteRaDtiMtmg  wttli  i  ind  eqoatiii^  to  zero, 

or  r  =  yU  cos  d. 

The  velocity  Ui  with  which  the  water  leaves  the  wheel,  is  then 
perpendicular  to  r  and  is 

Ui=Usind. 

Substitnting  for  r  its  value  JU  cos  B  in  (2),  the  maximum  efficient 
is  cos^  0, 

The  same  result  is  obtained  from  equation  (1),  by  substitutini 
for  Ui,  Usin^. 

The  maximum  efficiency  is  then 

^     ,     IPsin'^  ,. 

E  =  1 ™ —  =  cos'^. 

A  common  ^'alue  for  0  is  15  degrees,  and  the  theoretia 
hydraulic  efficiency  is  then  0"933. 

This  increases  as  0  diminishes,  and  would  become  unity  if 
could  be  made  zero. 

If,  however,  6  is  zero,  U  and  r  are  parallel  and  the  tip  of  tl 
blade  will  be  perpendicular  to  the  radius  of  the  wheel. 

This  is  clearly  the  limiting  case,  which  practically  is  n 
realisable,  Avithout  modifying  the  construction  of  the  wheel.  Tl 
necessarj'  modification  is  shown  in  the  Pelton  wheel  described  < 
page  377. 

The  actual  efficiency  of  Poncelet  wheels  is  from  55  to  65  p 
nt. 


B'ATER  WHEELS 


tirfthe  bed.    Water  enters  i\w  wheel  at  all  points  between 
untl  R,  and  for  no  shock  the  bed  of  the  channel  PQ  should  be 
it  of  such  a  fomi  tlmt  the  diroction  of  the  stream,  where  it, 
aliTs  fhe  wheel  at  any  point  A  between  E  and  Q,  should  make 
constant  angle  &  ^vith  the  radius  of  the  wheel  at  A, 
Willi  0  as  centre,  draw  a  circle  touching  the  line  AS  which, 
liriikef^  the  given  angle  ^  with   the  radius  AO.    Take  several 
point*  on  the  circumference  of  the  wheel  between  R  and 
and  draw  tangents  t4>  the  circle   STV*      If   then    a   curve 
'  m  drawn  normal  to  thcfle  several  tangents,  and  the  stream 
are  parallel  to   PQ,  the  ivater  entering  any  part  of  the 
fl  between  R  and  Qj  will  make  a  constant  angle  0  \rith  the 
iiijss  and  if  it  enters  without  shock  at  A,  it  will  do  so  at  all 
rinti*.    Tlje  actual  velocity  of  the  water  U,  as  it  moves  along  the 
PQj   will   be   less  than    V2gH,  due   to    friction,  etc.    The 
efficient  of  velocity  iv  in  most  cases  will  probably  be  between 
I  and  0*95,  so  that  taking  a  mean  value  for  fey  of  0*92*1, 

U  =  0-925  s/%H. 
The  best  value  for  tfte  velocity  v  tahhig  frictum  mio  ctcc&wni, 
\  determining  the  best  velocity  for  the  periphery  of  the  wheel  no 
Qowance  has  been  made  for  the  loss  of  energy  due  to  friction  in 
*  wheel. 

If  V,  is  the  relative  velocity  of  the  water  and  wheel  at  entrance^ 
is  Uy  be  expected  that  the  velocity  relative  to  the  wheel  at  exit 
!  be  le^  than  Vr,  due  to  friction  and  interference  of  the  rising 
3d  filing  particles  of  water. 

Tlie  case  is  somewhat  analogous  to  that  of  a  stone  thrown 
ertically  up  in  the  atmosphere  with  a  velocity  v.  If  there  were 
Mnesistance  to  its  motion,  it  would  rise  to  a  certain  height, 


: 


A.= 


2ff' 


,  di^ii  descend,  and  when  it  again  reached  the  earth  it  would  j 
a  velocity  equal  to  its  initial  velocity  t*.     Due  to  resistances,^ 
height  to  which  it  rises  will  be  less  than  h^  j  and  the  velocity 
ith  which  it  reaches  the  gi^ound  will  be  even  less  than  that  due 
foiling  freely  through  this  diminished  height. 
Let  the  velocity  relative  to  the  wheel  at  exit  be  nVr,  «  being 
,  fraction  less  than  unity. 

Tlie  triangle  of  velocities  at  exit  will  then  be  ABE,  Fig,  188.  j 
he  change  of  velocity  in  the  direction  of  motion  is  GH,  which 
Itials 

BHH-GB  =  BH(l  +  n) 

=  {1 +n)(U  cos  e-r). 


298  HTDRAUUCS 

If  the  velocity  at  exit  relative  to  the  wheel  is  only  nVn  thai 
must  have  been  lost  by  friction  etc.,  a  head  equal  to 

The  work  done  on  the  wheel  per  lb.  of  water  is,  therefore, 
{(l+n)(Ucose-v)}v     Yr\.     „r, 

-g  -^(l-"^- 


Fig.  188. 

Let  (1  -  TJ?)  be  denoted  by  /,  then  since 

V/  =  BH"  +  CH»  =  (U  cos  «-«)'  +  U»  sin'tf, 
the  efficiency 

{(l  +  n)(Ucos«-«)}--^{(Ucostf-t>)»  +  U»8in»tf} 
e  = a-^ . 

Differentiating  with  respect  to  v  and  equating  to  zero, 
2  (1  +  w)  Ucos^-4  (1  +  n)  t;  +  2U/cos  6  -  2t7/=0, 
from  which 

_{(l+n)+/}Ucos^ 
""  /+2(l  +  n) 

^(2  +  n~?i')Ucosg 
3-n*4.2n 
If /is  now  supposed  to  be  0*5,  i.e.  the  head  lost  by  friction,  et 

is  — ^.^ —  ,  n  IS  0  71  and 

V  =  -5617  cos  ^. 
If /is  taken  as  075, 

v  =  0'6Ucos^. 

Dimemsiotis  of  Ponceht  wheels.  The  diameter  of  the  wheel 
should  not  be  less  than  10  feet  when  the  bed  is  curved,  and  noi 
less  than  15  feet  for  a  straight  bed,  otherwise  there  will  be  con- 
siderable loss  by  shock  at  entrance,  due  to  the  variation  of  the 
angle  0  which  the  stream  lines  make  with  the  blades  between  B 
and  Q,  Fig.  186.    The  water  will  rise  on  the  buckets  to  a  heigte 


WATER  WHEELS 


%m 


Bttrlj  equal  to  ^ ,  and  since  the  water  first  entera  at  a  point  R, 

be  blade  depth  d  must,  therefore,  be  greater  than  this,  or  the 
f»t^*  will  overflow  at  the  upper  edge.  The  clearance  between 
lie  bed  and  the  bottom  of  the  wheel  should  not  be  less  than  f\ 
Ibe  peripheral  distance  between  the  consecutive  blades  is  taken 
torn  S  inches  to  18  inches, 

^  Morse^p&wer  of  Ponctlet  wheels.  If  H  is  the  height  of  the 
Ir&oe  of  water  in  the  penstock  above  the  bottom  of  the  wheel, 
be  velocity  U  will  be  about  ^ 

0*92  v^^,  , 

tid  V  m&Y  be  taken  aa 

0*55  X  0*92  n/%H  =  0'5  V%H. 

Let  D  be  the  diametei*  of  the  wheel,  and  &  the  breadth,  and  let 
Ibe  the  depth  of  the  orifice  EP*    Then  the  number  of  revolutions 

BT  minute  is  

I  0-5  V2ffH 

I  IT,  D 

The  coeflicient  of  contraction  c  for  the  orifice  may  be  from  0*6, 
I  it  is  shaTp-edged,  to  1  if  it  is  carefully  rounded^  and  may  be 
iketi  as  0^8  if  the  orifice  is  formed  by  a  flat-edged  sluice. 

The  quantity  of  water  striking  the  wheel  per  second  is,  then^ 

Q  =  0'92d6%/2^. 

1  If  the  efficiency  is  taken  as  60  per  cent.,  the  work  done  per 
bond  is  0-6  ^  624QH  ft.  lbs, 
I  The  horse-power  N  is  then 

^     ■  5,50  •  I 

182«    Turbines. 

Although  the  water  wheel  has  been  developed  to  a  considemble 
kgre^  of  perfection,  efficiencies  of  over  80  per  cent  having  been 
plained,  it  is  being  ahuost  entirely  superseded  by  the  turbine. 
I  The  old  water  wheels  were  required  to  drive  slow  moving 
hichiiiery,  and  the  great  disadvantage  attaching  to  them  ot 
iring  a  small  angular  velocity  was  not  felt.  Such  slow  moving 
WMth  are  however  entirely  unsuited  to  the  driving  of  modern 
lacKinery,  and  especially  for  the  d^i^^Ilg  of  dynamos,  and  they 
1^  further  quite  unsuited  for  the  high  heads  which  are  now 
Billed  for  the  generation  of  power. 

I  Turbine  wheels  on  the  other  hand  can  be  made  to  run  at  either 
pr  or  veiy  high  speeds,  and  to  work  under  any  head  vaTrjm^ 


^1^ 


^dM 


800 


BYBRAOLICS 


fn^jiii  1  foot  to  2000  feet,  and  the  speed  can  be  regulated 
much  greater  precision. 

Due  to  the  slow  B|}eeds,  the  old  water  wheels  ecmld  not  deirelop 
large  power,  the  niaximmn  being  about  100  horse-power,  wherea*! 
at  Niagara  Falls,  turbines  of  10,000  horse-pjower  have  recently | 
been  installed. 

Types  of  Tm'hifM&. 

Turbines  are  generally  divided  into  two  cl&saes^  impulse,^ 
free  deviation  turbiues,  and  reaction  or  pressure  turbines. 

In  both  kinds  of  turbines  an  attempt  is  made  to  shape 
vanes  so  that  the  water  enters  the  wheel  without  shock ;  tliat  is 
the  direction  of  the  relative  velocity  of  the  water  and  the  vi 
parallel  to  the  tip  of  the  vane^  and  the  direction  of  the  leai 
edge  of  the  vane  is  made  so  that  the  water  leaves  in  a  speciie^ 
direetion. 

In  the  first  class,  the  whole  of  the  available  he-ad  is  com 
into  velocity  before  the  water  strikes  the  turbine  wheel,  am 
pressure  in  the  driving  fluid  as  it  moves  over  the  vanes  rei 
constant^  and  equal  to  the  atmospheric  pressure.    The  wheel 
vanes,  therefore,  must  be  so  formed  that  the  air  has  free 
between  the  vanes,  and  the  space  between  two  consecutive  vaneij 
must  not  be  full  of  water.    Work  is  done  upon  the  vanes,  or  i 
other  wordsj  upon  the  turbine  wheel  to  which  they  are  fixedti! 
virtue  of  the   change  of  momentum  or  kinetic   energy  of 
moving  water,  as  in  examples  on  pages  270 — 2* 

Suppose  water  supplied  to  a  turbine,  as  in  Fig.  258,  under  ic^ 
effective  head  H,  which  may  be  suppoiied  equal  to  the  total  Ht^J 
miiius  losses  of  head  in  the  supply  pipe  and  at  the  noEsle*  Tin 
water  issues  from  the  nozzle  with  a  velocity  U  =  j2gKj  and  ihi 
available  energy  per  pound  is 

IP 


H  = 


%^ 


Work  is  done  on  the  wheel  by  the  absorption  of  the  who?e,  c^r 
part,  of  this  kinetic  energy. 

If  Uj  is  the  velocity  with  which  the  water  leaves  the  wb«r:u 
the  energy  lost  by  the  water  per  pound  is 

2g      2g  * 

and  this  is  equal  to  the  work  done  on  the  wheel  together  w'4 
energy  lost  by  friction  etc*  in  the  wheel. 

In  the  second  class,  only  part  of  the  available  head  ib 
verted  into  velocity  before  the  water  enters  the  wheel,  and 


Blocity  and  preesore  both  vary  as  the  water  pasaefi  through  the 
rbeel-  It  is  therelore  essential,  that  the  wlieel  shall  always  be 
pe|»t  full  of  water*  Work  is  done  upon  the  wlieel,  ae  will  be  seen 
B  the  sequence,  partly  by  changing  the  kinetic  energy  the  water 
when  it  enters  the  wheel,  and  partly  by  changing  itB 
or  potential  energy, 
Suppose  water  is  supplied  to  the  turbine  of  Fig.  191,  under 
[  effective  hi?ad  H ;  the  velocity  U  with  which  the  water  enters 
wheel,  is  only  Borae  fraction  of  J2gilj  and  the  pressure  head 
the  inlet  fco  the  wheel  mil  depend  upon  the  magnitude  of  U 
And  upon  the  position  of  the  wheel  relative  to  the  head  and  tail 
■vmter  siirface^i*  The  turbine  wheel  always  being  full  of  water, 
jUiere  is  continuity  of  flow  through  the  wheel^  and  if  the  !iead 
faBpreeaed  upon  the  water  by  centrifugal  action  is  determint^d,  as 
pa  pAge  335,  the  equations  of  Bernouilli  *  can  be  used  to  determine 
Ed  aajr  gi^en  case  the  difference  of  pressure  head  at  the  inlet  and 
of  the  wheeL 

tlie  preasure  head  at  inlet  is  —  and  at  outlet  — ,  and  the 

ity  with  which  the  water  leavea  the  wheel  is  Ui,  the  v^ork 
on  the  wheel  (see  page  338)  is 

^  -  —  +  ,ir-  -  TT-  per  pound  of  water, 

ir  work  is  done  on  the  wheel,  partly  by  changing  the  velocity 
lead  and  partly  by  changing  the  presBure  head.     Such  a  turbine 
i  called  a  reaction  turbine,  and  the  amount  of  reaction  is  measured 
^  the  ratio 
r  p     Pi 

riy,  if  p  is  made  equal  to  p^  the  limiting  case  is  reached, 
f  tho  turbine  becomes  an  impntsej  or  free-de\^ation  turbine. 

Id  be  clearly  understood  that  in  a  reaction  turbine  no 
ue  on  the  wheel  merely  by  hydrostatic  pressure,  in  the 
III  which  work  m  done  by  the  pressure  on  the  piston  of  a 
engine  or  the  ram  of  a  hydraulic  Hft. 

13,    Eeactioii  turbiBes. 

tie  oldest  furm  of  turbine  is  the  simple  reaction,  or  Scotch 
ine,  which  in  its  simplest  form  is  illustrated  in  Fig.  189. 
%*ertieal  cnbe  T  has  two  horizontal  tubes  connected  to  it,  the 
ends  of  which  are  bent  round  at  right  angles  to  the  direction 


*  See  page  SSC 


302 


HTDRAUUCS 


of  length  of  the  tube,  or  two  holes  O  and  Oi  are  drilled  as  in  tiie 
figure. 

Water  is  supplied  to  the  central  tnbe  at  such  a  rate  as  to  keep 
the  level  of  the  water  in  the  tube 
constant,  and  at  a  height  h  above 
the  horizontal  tubes.  Water  escapes 
through  the  orifices  O  and  Oi  and 
the  wheel  rotates  in  a  direction 
opposite  to  the  direction  of  flow  of 
the  water  from  the  orifices.  Tur- 
bines of  this  class  are  frequently- 
used  to  act  as  sprinklers  for  distri- 
buting liquids,  as  for  example  for 
distributing  sewage  on  to  bacteria 
beds. 

A  better  practical  form,  known  as  the  Whitelaw  turbine,  is 
shown  in  Fig.  190. 


K-R-s 


Fig.  189.    Sootoh  TuImim. 


1 


1 


Fig.  190.    Whitelaw  Turbine. 

To  understand  the  action  of  the  turbine  it  is  first  necessary  t< 
consider  the  effect  of  the  whirling  of  the  water  in  the  arm  upot 


TURBINES  303 

4ie  discharge  from  the  wheel.    Let  v  be  the  velocity  of  rotation 
^  the  orifices,  and  h  the  head  of  water  above  the  orifices. 

Imagine  the  wheel  to  be  held  at  rest  and  the  orifices  opened ; 
the  head  causing  velocity  of  flow  relative  to  the  arm  is 
iply  A,  and  neglecting  friction  the  water  will  leave  the  nozzle 
with  a  velocity 

Vo  =  'J2gh. 

Now  suppose  the  wheel  is  filled  with  water  and  made  to  rotate 
ttt  an  angular  velocity  f»,  the  orifices  being  closed.  There  will 
aofw  be  a  xnressure  head  at  the  orifice  equal  to  h  plus  the  head 
iminesBed  on  the  water  due  to  the  whirling  of  each  particle  of 
water  in  the  arm. 

Assume  the  arm  to  be  a  straight  tube,  Fig.  189,  having  a  cross 
sectional  area  a.    At  any  radius  r  take  an  element  of  thickness  dr. 

The  centrifugal  force  due  to  this  element  is 

a/= — - — . 

The  pressure  per  unit  area  at  the  outer  periphery  is,  therefore, 

9 

~    2g   ' 
and  the  head  impressed  on  the  water  is 

P  __  Ctf  V 

Let  V  be  the  velocity  of  the  orifice,  then  t?  =  cor,  and  therefore 

w''2g' 

If  now  the  wheel  be  assumed  frictionless  and  the  orifices  are 
opened,  and  the  wheel  rotates  with  the  angular  velocity  co,  the 
head  causing  velocity  of  flow  relative  to  the  wheel  is 

K=.h  +  ^  =  h+^     (1). 

w  2g 

Let  Vr  be  the  velocity  relative  to  the  wheel  >vith  which  the 
water  leaves  the  orifice. 

-^-^  ^  =  ^-^1   (2). 

The  velocity  relative  to  the  ground,  with  which  the  water 
leaves  the  wheel,  is  Vr-t?,  the  vector  sum  of  Vr  and  v. 


_1  f^  wau^rdr 
^"ajo        g 


304 

The  water  leaves  the  wheel,  therefore,  with  a  velocity  relttnf | 
to  the  ground  of  M= Vr-  »,  »nd  the  kinetic  energy  lost  ia 

The  theoretical  hydraulic  efficiency  is  theni 


Since  from  (2)|  Tr  becomes 
increases,  the  energy  lost  per  | 
and  the  eflicieacy  E,  therefor 

The  efficifmty  of  the  reuctiou  . 
As  before, 


6  nearly  equal  to  t^  a&  f 

diminishes  b&  v  im 

a^a  with  !?• 

C  whenfnetion  is  cofwidertd. 


^9 


-(3), 


kY/ 


Assuming  the  head  lost  by  friction  to  be  ^h^,  the  total  head 


must  be  equal  to 


2ff 


-(4). 


The  work  done  on  the  wheel,  per  pound,  is  now 


h- 


and  the  hydraulic  efficiency  is 

A- 


2ff       2ff' 


Substituting  for  h  from  (4)  and  for  ^j  Y^-v^ 

2v(Yr-v) 


Let 
then 


e  = 


{l+k)Yr'-v'' 

Yr  =  nv, 

2(71-1) 


(l+/L-)w»-r 
Differentiating  and  equating  to  zero, 

n2(l  +  A:)-2n(l4.fr)  +  l  =  0. 


TURBINES 


305 


the  efficiency  is  a  maxiYnnni  when 

T 


1  +  k' 


Fig.  191.    Outward  Flow  Torbine. 


U.  B. 


"l^ 


806 


HTDRAUUC8 


184.    Outward  flow  turbines. 

The  outward  ilow  tarhine  was  invented  in   1838  hy 
neyron.    A  cylindrical  wheel  W,  Figs.  191,  192,  and  201,  har 
a  number  of  suitably  shaped  vanes,  is  &xed  to  a  vertical  ami 
The  water  enters  a  cylindrical   chamber  at  the  centre  of  d»J 
turbine,  and  is  directed  to  the   wheel  by  auitable  tiated 
blades  G,  and   flows  through  the  wheel   in  a  radial  dir 
outwards.     Between  the  guide  blades  and  the  wheel  is  a  ryhndiij 
cal  sluice  R  which  is  used  Uj        '    >1  the  flow  of  water  th 
the  wheel. 


*v 


f 


Fjg.   191  <t. 

This  method  of  regnlating  the  flow  is  very  imperfect,  as  wbm 
the  gate  partially  closes  the  pas^sageSj  tbere  must  be  a  suddii^ 
enlargement  as  the  water  enters  the  wheel,  and  a  loss  of  h 
ensues,  "llie  efficiency  at  "  part  gate "  is  consequently  reif 
much  less  than  when  the  flow  in  unchecked.  This  difficaltf 
partly  o%'ercome  by  dividing  the  wheel  into  sevei^l  distinct 
compartments  by  horizontal  diaphragmsj  as  shown  in  Pig.  19^ 
so  that  when  working  at  part  load,  only  the  efficiency  of  oa* 
compartment  is  affected* 

The  wheels  of  outward  flow  turbines  may  have  their  asse^ 
either  horizuntal  or  vertical,  and  raay  be  put  either  above,  * 
below,  the  tail  water  level. 

The  '' atuclion  hihty  If  plact.Mi  above  the  tail  water,  ti* 
exhaust  must  take  place  down  a  '*  suction  pipe,"  as  in  Fig.  2!0l 
page  317,  the  end  of  which  must  be  kept  drowned,  and  the  pip* 
air-tight,  so  that  at  the  outlet  of  the  wheel  a  pressure  less  tbii 
the  atmospheric  pressure  raay  be  maintained*  If  A  i  is  the  he^ 
of  the  centre  of  the  discharge  periphery  of  the  wheel  al 
tail  water  level,  and  pa  is  the  atmospheric  pressure  in  pow 
square  foot,  the  pressure  head  at  the  discharge  circumferoiui? 

w 


TURBINES 


SOT 


I  wiieel  catmot  be  more  than  34  feet  above  the  level  of  the  tail 
iter,  or  the  preasore  at  the  outlet  of  the  wheel  will  be  negativei 
\d  practTcally,  it  cannot  be  greater  than  25  feet- 
It   is  Ehowii  later  that  the  effective  head,  under  which  the 
rbine  works,  whether  it  h  drowned,  or  placed  in  a  suction  tnbe, 
the  total  fall  of  the  water  to  the  level  of  the  tail  race. 


Fif,  19a,    FoQTfieyron  Out  word  Flow  Turbine. 

ttm  af  the  suction  tube  has  the  advantage  of  allowing  the 
'     I  be  placed  at  some  distance  above  the  tail  water  J 
r.j  bearings  can  be  readily  got  atj  and  repairB  caili 
r  emmiy  executed. 

mokiDg  the  suction  tube  to  enlarge  as  it  deseendBt  the 
of  exit  can  be  dimimghed  very  gradually,  aiii  '\^  %mX. 


308  HYDRAULICS 

value  kept  small.    If  the  exhaust  takes  place  direct  froi 
wheel,  as  in  Fig.  192,  into  the  air,  the  mean  head  available 
head  of  water  above  the  centre  of  the  wheel. 

Triangles  of  velocities  at  inlet  and  outlet  For  the  wai 
enter  the  wheel  without  shock,  the  relative  velocity  of  the 
and  the  wheel  at  inlet  must  be  parallel  to  the  inner  tips  < 
vanes.  The  triangles  of  velocities  at  inlet  and  outlet  are  i 
in  Figs.  193  and  194. 


Fig.  193. 


Let  AC,  Fig.  193,  be  the  velocity  XJ  in  direction  and  magr 
of  the  water  as  it  flows  out  of  the  guide  passages,  and  let  A 
the  velocity  v  of  the  receiving  edge  of  the  wheel.  Then  DC 
the  relative  velocity  of  the  water  and  vane,  and  the  recc 
edge  of  the  vane  must  be  parallel  to  DC.  The  radial  comp 
GC,  of  AC,  determines  the  quantity  of  water  entering  the  ^ 
per  unit  area  of  the  inlet  circumference.  Let  this  radial  vel 
be  denoted  by  u.  Then  if  A  is  the  peripheral  area  of  the 
face  of  the  wheel,  the  number  of  cubic  feet  Q  per  #3cond  ent 
the  wheel  is 

Q  =  A.i^, 

or,  if  (2  is  the  diameter  and  b  the  depth  of  the  wheel  at  inlet 
t  is  the  thickness  of  the  vanes,  and  n  the  number  of  vanes, 
Q  =  (vd -n.t)  .b.u. 

Let  D  be  the  diameter,  and  Ai  the  area  of  the  discharge 
phery  of  the  wheel. 

The  peripheral  velocity  Vi  at  the  outlet  circumference  is 

v.D 


TtJRBINES 


309 


t  Ui  he  tlie  radial  component  of  velocity  of  exit,  then  what- 
per   the  direction  with  which  the  water  leayes  the  wheel  the 
tciiAl   component  of  velocity  for  a  given  discharge  is  coustant- 
The  trmngle  of  velocity  can  now  be  drawn  as  follows : 
Set  off  BE  equal  to  t^i.  Fig.  194,  and  BX  radial  and  equal 

Let  it  BOW  be  supposed  that  the  direction  EF  of  the  tip  of  the 
Itne  at  discharge  is  knoiJ^Ti,  Draw  EF  parallel  to  the  tip  of  the 
^ne  at  D,  and  tlirotigh  K  draw  KF  parallel  to  BE  to  meet  EF 

F, 

Then  BF  is  the  velocity  in  direction  and  magnitude  with  which 
lie  ir^ter  leaves  the  wheel j  relative  to  the  gi'ound,  or  to  the  fixed 
of  the  turbine.  Let  this  velocity  be  denoted  by  Uj.  If, 
istead  of  the  direction  EF  being  given,  the  velocity  Uj  is  given 
I  direction  and  magnitude,  the  triangle  of  velocity  at  exit  can  be 
ri&im  by  setting  out  BE  and  BF  equal  to  Vi  and  Ui  respectively, 
ttd  joining  EF,    Then  the  tip  of  the  blade  must  be  made  parallel 

EF, 

Fof  »uy  given  value  of  Ui  the  quantity  of  water  flowing 
iroQgh  the  wheel  is 

Q  =  AiUiCotj^=A,w,. 

Work  drjtw  tm  the  wheel  neglecting  friction,  etc.  The  kinetic 
a©rgy  of  the  wmter  as  it  leaves  the  turbine  wheel  is 

^  per  pound, 

if  the  discharge  is  into  the  air  or  into  the  tail  water  this 
is  of  necessity  lost.    Neglecting  friction  and  other  losses, 
'  available  energy  per  pound  of  water  is  then 

H-5i!  foot  lbs., 

.  the  theoretical  hydraulic  efficiency  is 


E  = 


H      ' 


"ant  for  any  given  value  of  Ui,  and  independent  of  the 
.„.,  f  Uj-  This  efficiency  must  not  be  confused  with  the 
1  efficiency,  which  is  much  less  than  E» 

be  smaller  Uj ,  the  greater  the  theoretical  hydraulic  efficiency, 

^nnce  for  a  given  flow  through  the  wheel,  Uj  will  be  least 

it  18  radial  and  equal  to  ttj,  the  greatest  amount  of  work 

be  obtained  for  the  given  flow,  or  the  efficiency  will  be  a 

imtuti,  when  the  water  leaves  the  wheel  radially.    M  tToss 


ML 


310  HTDRAULICS 

water  leaves  with  a  velocity  Ui  in  any  other  direction,  the 
efficiency  will  be  the  same,  but  the  power  of  the  wheel  wfll  be 
diminished.  If  the  discharge  takes  place  down  a  snction  tnbe^ 
and  there  is  no  loss  between  the  wheel  and  the  outlet  from  the 
tube,  the  velocity  head  lost  then  depends  upon  the  velocity  TTi 
with  which  the  water  leaves  the  tube,  and  is  independent  of  the 
velocity  or  direction  with  which  the  water  leaves  the  wheel. 

The  velocity  of  whirl  at  inlet  and  outlet  The  component  of 
XJ,  Fig.  193,  in  the  direction  of  v  is  the  velocity  of  whirl  at  inlel^ 
and  the  component  of  Ui,  Fig.  194,  in  the  direction  of  ih,  is  the 
velocity  of  whirl  at  exit. 

Let  y  and  Yi  be  the  velocities  of  whirl  at  inlet  and  oatlek 
respectively,  then 

V  =  Ucos^ 

and  Vi  =  Uisini8  =  t^itani8. 

Work  done  on  the  wheel.  It  has  already  been  shown, 
section  173,  page  275,  that  when  water  enters  a  wheel,  rotating 
about  a  fixed  centre,  with  a  velocity  XJ,  and  leaves  it  with  velodi^ 
Ui,  the  component  Yi  of  which  is  in  the  same  direction  as  Vi,  the 
work  done  on  the  wheel  is 

Yv     Yii?i  , 
per  pound, 

and  therefore,  neglecting  friction, 

y-ir-^-^  w- 

This  is  a  general  formula  for  all  classes  of  turbines  and  shaoU 
be  carefully  considered  by  the  student. 
Expressed  trigonometrically, 

t;U  cos  S  __  ViUitoxiP  _  TT  _  Hl  fc%\ 

9  9        -^     2g   <2). 

If  F  is  to  the  left  of  BK,  Yi  is  negative. 

Again,  since  the  radial  flow  at  inlet  must  equal  the  radial  floir| 
at  outlet,  therefore 

AUsintf  =  AiXJiCos/9  ,,..,, ^...(g). 

When  Ui  is  radial,  Yi  is  zero,  and  th  equals  t^i  tan  a. 

'^^'^  T=°-|* • ^^ 

from  which  ?l£2i«  =  H-?^^ ® 

g  2g 

and  from  (3)  ATTsin^^  AiVitana  (f^ 


TURBINES 


311 


and 


If  the  tip  of  the  Tana  is  radial  at  inlet,  i.e.  Vr  is  radial, 

V  =  r 

2sr 

V*  tan'  a 


9      9  ' 


=  H-- 


29 


..(7) 
.(8). 


In  actual  turbines  P-  is  from  '02H  to  '07H. 

Exaw^U.  An  oatwmrd  flow  turbine  wheel.  Fig.  195,  has  an  internal  diameter  of 
6'349  feet,  and  an  external  diameter  of  6*25  feet,  and  it  makes  250  revolutions  per 
minnte.  The  wheel  has  32  yanes,  which  may  be  taken  as  |  inch  thick  at  inlet  and 
1}  inches  thick  at  outlet  The  head  is  141*5  feet  above  the  centre  of  the  wheel  and 
the  ezhaoat  takes  place  into  the  atmosphere.  The  effective  width  of  the  wheel  face 
mt  inlet  and  outlet  is  10  inches.  The  quantity  of  water  supplied  per  second  is 
915  cubic  feet. 

Neglecting  all  frictional  losses,  determine  the  angles  of  the  tips  of  the  vanes  at 
inlet  and  oaUiet  to  that  the  water  shall  leave  radially. 

The  peripheral  velocity  at  inlet  is 

v=«-  X  6-249  X  W=69  ft-  P«r  8«o-. 
ftodatootlet  Vi=«'x6-25x  V/=:82ft.      „     „ 


Fig.  195. 

The  i»dial  Tcloeity  of  flow  at  inlet  is 

215 


w  X  5-249  x  li  -  f  I X  J 
=  18*35  ft.  per  sec. 
The  zadiAl  Tekwty  of  flow  at  exit  is 

^ 215 

*^"»x6*25xH-Hxf' 
=  16*5  ft.  per  sec. 

^=4-23  ft. 


312 


Then 


and 


^  =  14^5 

=  137-27  ft, 
137'«7>cS2-2 


=  64  ft.  p^  Mc» 


To  dr&w  the  triangle  of  Tdocitks  at  mtel  Bet  out  p  and  u  at  Hglit  ai]ig]e& 

Then  mnee  V  is  64^  and  ia  |b«  tan^eotia]  coEDpooeat  of  U>  k\A  u  in  tkniM 
compon<?tit  of  U,  tb«  ^ir^tioD  and  magnitude  of  U  is  determined. 

By  joinmg  B  and  C  the  relative  velocity'  \%  is  ohtained,  and  BC  i«  p&Ealkl  lo  Ik 
tip  of  thi'  \'une. 

The  triangle  of  v«lodtkfl  at  exit  ii  DBF,  and  the  ti^  of  the  t-ane  mmt  be  ptfiOK 
toEF. 


-    ~-V^*»" 


-^ 


tf,-S^ 


!    B 

t/"'' 

"c" 

E 

Fig,  m 


Fig.  197, 


-aitW  — 


The  angiefi  i?»  ^,  and  a  can  be  caloulated;  for 

tanS==  ^-1^-0-2867, 
b'4 

tan^=  -  — !^=  -3-670 


and 

and,  therefore, 


tanB--^^0vt994, 


^  =  105°  ir, 
a=iri7'. 
It  will  be  seen  later  how  these  aiiKles  are  modt^ed  when  friction  ia  oonsiderei 
Fig.   1^8  rHows  the    form   the   guide  blades  and   ?anea   of  the   wheel  woold 
probably  lake. 

7*^  path  of  the  water  thrQugh  thi  wheel.     The  average  l^adial  vetoc^itj  chroogfa 
the  wheel  may  be  taken  aa  17*35  feet. 

The  time  taken  for  a  i>article  of  water  to  get  through  the  wheel  i«t  thereibrc^ 
R  ^  r       0'5        ^  ^ 

The  an^le  turned  through  bj  the  wheel  in  thia  time  is  0^S9  radians. 
Set  off  the  arc  AB,  Fig.  198,  ec|ual  to  -39  radian,  and  divide  it  into  four  eqntl 
parts,  and  draw  the  radii  fUffb,  pc  and  Bit. 

Divide  AD  atiM  into  four  equal  parts,  Aud  draw  cireles  through  A^^  A,,  &nd  A,' 

Suppose  a  pai-ticlo  of  ^ater  to  enk^r  the  vrhed  &i  A  in  contact  wiib  a  vooe  aM 

Buppoef  it  to  remam  id  contact  with  the  vau<*  during  itB  poBsage  through  the  vhts^    ' 

Then,  a&snmiug  the  radial  velocity  is  coo&tant,  while  the  wheel  turns  throTJcb  tbe 

ft  the  water  will  mote  radially  a  dietanoe  AA^  and  a  particle  that  came  o^  ^ 


TURBINES 


313 


B  TA&e  ftt  A  will,  therefore,  be  in  oontaot  with  the  Tane  on  the  arc  throngh  A^ . 
i«  Tane  initially  passing  throngh  A  will  be  now  in  the  position  el,  al  being 
Qml  to  hJ  and  the  partiele  will  therefore  be  at  1.  When  the  particle  arrives  on 
e  are  throng  A,  the  vane  will  pass  throngh/,  and  the  particle  will  consequently 
at  S,  63  bdng  eqoal  to  mn.  The  curve  A4  drawn  through  Al  2  etc.  gives  the 
^tli  of  the  water  relative  to  the  fixed  casing. 


Fig.  198. 

185.    Losses  of  head  due  to  Motional  and  other  resistances 
i  outward  flow  turbines. 
The  losses  of  head  may  be  enumerated  as  follows : 

(a)  Loss  by  friction  at  the  sluice  and  in  the  penstock  or 
ipply  pipe. 

If  Vo  is  the  velocity,  and  ha  the  head  lost  by  friction  in 
18  pipe, 

(b)  As  the  water  enters  and  moves  through  the  guide 
assages  there  will  be  a  loss  due  to  friction  and  by  sudden  changes 
1  the  velocity  of  flow. 

This  head  may  be  expressed  as 


being  a  coefficient. 


*  See  page  119. 


814 


HYDRAULICS 


(c)  There  is  a  loss  of  head  at  entrance  due  to  shock  as 
the  direction  of  the  vane  at  entrance  cannot  be  determined 
with  precision. 

This  may  be  written 

V« 

2a  ' 

that  is,  it  is  made  to  depend  upon  Yr  the  relative  velocity  of  the 
water,  and  the  tip  of  the  vane. 

(d)  In  the  wheel  there  is  a  loss  of  head  hd,  due  to  fricticniy 
which  depends  upon  the  relative  velocity  of  the  water  and  the 
wheel.  This  relative  velocity  may  be  changing,  and  on  any  small 
element  of  surface  of  the  wheel  the  head  lost  will  diminish,  as  the 
relative  velocity  diminishes. 

It  will  be  seen  on  reference  to  Figs.  193  and  194,  that  as  the 
velocity  of  whirl  Vi  is  diminished  the  relative  velocity  of  flow  tv  at 
exit  increases,  but  the  relative  velocity  Vr  at  inlet  passes  through 
a  minimum  when  V  is  equal  to  r,  or  the  tip  of  the  vane  is  radial 
If  Vo  is  the  relative  velocity  of  the  water  and  the  vane  at  any 
radius,  and  b  is  the  width  of  the  vane,  and  dl  an  element  of 
length,  then, 

Jci  being  a  third  coefficient. 

If  there  is  any  sudden  change  of  velocity  as  the  water  poonrw 
through  the  wheel  there  will  be  a  further  loss,  and  if  the  turbine 
has  a  suction  tube  there  may  be  also  a  small  loss  as  the  water 
enters  the  tube  from  the  wheel. 

The  whole  loss  of  head  in  the  penstock  and  guide  passages  may 
be  called  H/  and  the  loss  in  the  wheel  h/.      Then  if  U*  is  the 


Fig.  199. 


TUJEIBINES 


315 


city  with  which  the  water  leaves  the  tufbme  the  effective 
is 

In  well  designed  inward  and  outward  flow  turbines 


2ff 


+  A/+H/ 


^-^'  =  eH 


ies  from  O'lOH  to  *22H  and  the  hydraulic  efficiency  is,  therefore^ 
[jm  90  to  78  per  cent 
The  efficiency  of  inward  and  outward  flow  tnrbines  including 
Dhanical  losses  is  fruin  75  to  88  per  cent. 

Calling  the  hydraulic  efficiency  e,  the  general  formula  (1), 
tion  184,  may  now  be  written 

&         9 

=  78to-9H 

Outward  flow  turbines  were  made  by  Boyden*  about  1848  for 

rhich  he  claimed  an  efficiency  of  88  per  cent.    The  workmanship 

of  the  highest  quality  and  great  care  was  taken  to  reduce 

losses  by  friction  and  shock.    The  section  of  the  crowns  of  the 

rhoel  of  the  Boyden  turbine  is  shown  in  Pig,  199,     Outside  of 

turbine  wheel  was  fitted  a  "diffuser"  through  which,  after 

Irving  the  wheel,  the  water  moved  radially  with  a  continuously 

imlni^iung  velocity,  and   finally  entered   the  tail   race  with  a 

!ity  much  less,  than  if  it  had  done  so  direct  from  the  wheeL 

lo«s  by  velocity  head   was  thus  diminished,  and   Boyden 

claimed  that  the  diffuser  increased  the  efficiency  by  3  per  cent. 

186.    Some  actual  outward  flow  turbines. 

l^'uhh'  oufward  flaw  turhin&f.    The  genera!  arrangement  of  an 
ontward  flow  turbine  as  installed  at  Ch^vres  is  shown  in  Fig.  200, 
bere  are  four  wheels  fixed  to  a  vertical  shaft,  two  of  which 
ire  the  water  from  below,  and   two  from  above.    The  fall 
i.ries  from  27  feet  in  dry  weather  to  14  feet  in  time  of  flood. 

Tlje  uppt.*r  wheels  only  work  in  time  of  flood,  while  at  other 
imes  the  full  power  is  developed  by  the  lower  wheels  alone,  the 
'cylindrical  sluices  which  surround  the  upi>er  wheels  being  set  in 
^«uch  a  position  as  to  cover  completely  the  exit  to  the  wheel. 

The  water  after  leaving  the  wheels,  diminishes  gradually  in 

locity,  in  the  concrete  passages  leading  to  the  tail  race,  and  the 

of  head  due  to  the  velocity  with  which  the  water  enters  the 


4 


Lamdl  HffdrauUc  E^eHmettttt  J.  B.  Franoisi  1S56. 


i 


316 


tail  race  is  consequently  small.  These  passagee  eemre  _,  ™^, 
purpose  as  Bo^den^s  diifuser,  and  as  tlie  enlarging  saction  tabej 
in  that  they  allow  the  velocity  of  exit  to  diminish  gradiiaUj^. 


High 


Fig.  200.     Double  Outward  Flow  Turbine,     (E  so  her  W3*sa  and  Co.) 

Outward  fioie  furhine  with  horizontal  tu^m  Fig,  201  shows  b 
section  through  the  wheel,  and  the  supply  and  exhaust  pipes,  of  Rn 
outward  flow  turbine,  having  a  horizontal  axis  and  exhausting 
dowm  a  "  suction  pipe."  The  water  after  leaving  the  wheel  enters 
a  large  chamber,  and  then  passes  down  the  exhaust  pipe,  the 
lower  end  of  which  is  below  the  tail  race. 

The  supply  of  water  to  the  wheel  is  regulated  by  a  horizontal 
cylindrical  gate  S,  between  the  guide  blades  G  and  the  wheel.  The 
gate  is  connected  to  the  ring  R,  which  slides  on  guides,  outside 
the  supply  pipe  P,  and  is  under  the  control  of  the  governor. 

The  pressure  of  the  water  in  the  supply  pipe  is  prevented  from 
causing  end  thrust  on  the  shaft  by  the  partition  T,  and  between 
T  and  the  wheel  the  exhaust  water  has  free  access. 

Outward  flow  turbines  at  Niagara  Falls.  The  first  turbines 
installed  at  Niagara  Falls  for  the  generation  of  electric  power, 


TURBINES 


817 


^rere  outward  flow  turbinee  of  the  type  shown  in  Figs*  202  and 
203. 

TTier©  are  two  wheels  on  the  same  vertical  ahaft-^  the  water 
being  brought  to  the  chamber  between  the  wheels  by  a  Tei-ticai 
penBtock  7'  &*  diameter.  The  water  passes  upwards  to  one  wheel 
And  downwards  to  the  other. 


Fig,  SOI.    Ontw&rd  Flow  TnTbine  with  Hoetion  Tube. 

Am  shown  in  Fig.  202  the  water  preBBiire  in  the  chamber   is 
vented  from  acting  on  the  lower  wheel  by  the  partition  MN, 
fa  allowed  to  act  on  the  lower  side  of  the  upper  wlieel,  the 
{MrtitioD  HK  having  holes  in  it  to  allow  the  watar  free  access 
iderneath  the  wheoL    The  weight  of  the  vertical  shaft;  and  of 
wheels,  is  thus  balanced,  by  the  water  pressure  itself. 
The  lower  wheel  is  fixed  to  a  solid  shaft,  which  passes  through 
Iho  centre  of  the  upper  wheel,  and  is  connected  to  the  hollow 
of  the  upper  wheel  as  shown  diagram matically  in  Fig.  202, 
i¥e  this  connection^  the  vertical  shaft  is  formed  of  a  hollow 


thD 


318 


tube  38  inclieB  diameter,  except  where  it  passes  through  tb 
bearii)gB,  where  it  is  solid,  and  11  inches  diameter. 

A  thrust  block  is  also  provided  to  carry  the  tmbaiftnced  I 
weight* 

The  regulatiug  sluice  is  external  to  the  wheel.  To  maintaias  I 
high  efficiency  at  part  gate,  the  wheel  is  divided  into  three  sepaniti  ] 
compartments  as  in  Fourneyron's  wheeL 


Fig.  20*2.     Diagrammatic  section  of  OQt\rard  Flow  Turbine  it  Niagtftft  Fftlk 

A  vertical  section  through  the  lower  wheel  is  shown  in  Fig. 
203,  ami  a  part  sectional  plan  of  the  wheel  and  guide  blades  in 
Fig.  1115. 

(Further  particulars  of  these  turbines  and  a  description  of  the 
governor  will  be  found  in  Cassier's  Magazine,  Vol.  III.,  and  in 
Turbines  Actuelle)  Buchetti,  Paris  1901. 

187.    Inward  flow  turbines. 

In  an  inward  flow  turbine  the  water  is  directed  to  the  wheel 
through  guide  passages  external  to  the  wheel,  and  after  flowing 
radially  finally  leaves  the  wheel  in  a  direction  parallel  to  the  axis. 

Like  the  outward  flow  turbine  it  may  work  drowned  or  with  a 
suction  tube. 

The  water  only  acts  upon  the  blades  during  the  radial 
movement. 


819 


improvud  by  Francis •,  in  1840,  the  wheel  was  of  the  form 
in  Pig.  204  and  was  called  by  its  inventor  a  "central  rent 


I 


ted 
O 

1 


§ 

M 

g 


1^ 

1^ 


I'.'fi  m  t  ill  runi  uii  a  vertical  shaft.,  resting  cm  a  footstep, 
rted  by  a  collar  bearing  placed  above  the  gtaging  S. 

•  Ltwetl  Hjfdmtilii  E^peHmenUf  F,  B.  Ffioeii,  1856. 


320 


HTURAULICS 


LboT0  tie   wheel  is  a   heavy  castm^  C,  supported   by  l>d| 
from  the  staging  S,  which  acts  as  a  guide  for  the  cylindric 
,,alaice  F,  and  carries   the  bearing  B  for  the  shaft.     There 
\  wmnm  in  the  wheel  shown,  and  40  fixed  guide  blades,  the  f  ormf 
aing  made  of  iron  one  quarter  of  an  inch  thick  and  the  lat 
three-eixteenthB  of  an  inch. 


Fig.  204.    Francis"  Inward  flow  or  Centrml  v^ni  TiubiB«. 

The  triangles  of  velocities  at  inlet  and  outlet,  Fig.  ^'' 
drawn,  exactly  as  for  the  outward  flow  turbine,  the  only  d/> 
being  that  the  velocities  v,  U,  V,  Vr  and  u  refer  to  the  owlet 


TURBINES 


321 


eriplieryy  and  th,  Ui,  Vi,  Vr  and  lii  to  the  inner  periphery  of  the 
rheel. 

The  work  done  on  the  wheel  is 

YH_Yl5lft.ib8.perlb., 
9        9  *~       ' 

iiid  neglecting  friction, 

9         9  ^ 

For  maximum  efficiency,  for  a  given  flow  through  the  wheel, 
Ji  should  be  radial  exactly  as  for  the  outward  flow  turbine. 


Fig.  205. 

The  student  should  work  the  following  example. 

,  The  oater  diameter  of  an  inward  flow  tortfine  wheel  is  7*70  feet,  and  the  inner 

^bmeler  6-8  feet,  the  wheel  makes  55  revolntions  per  minute.     The  head  is 

'^'8  feet,  the  Telocity  at  inlet  is  25  feet  per  sec.,  and  the  radiid  velocity  may  be 

— iiimJ  constant  and  equal  to  7*5  feet.     Neglecting  friction,  draw  the  triangles  of 

2^oeities  at  inlet  and  outlet,  and  find  the  directions  of  the  tips  of  the  vanes  at 

^Jet  and  oatlet  so  that  there  may  be  no  shock  and  the  water  may  leave  radially. 

Lass  of  head  by  friction.  The  losses  of  head  by  friction  are 
•*«u3ar  to  those  for  an  outward  flow  turbine  (see  page  313)  and 
^e  general  formula  becomes 

9         9 
■^^en  the  flow  is  radial  at  exit, 

9 
The  value  of  e  varying  as  before  between  0*78  and  0*90. 

^^uExaMjȣf  (1).  An  inward  flow  turbine  working  under  a  head  of  80  feet  lias 
2^ial  blades  at  inlet,  and  discharges  radially.  The  angle  the  tip  of  the  guide 
rJJ^le  makes  with  the  tangent  at  the  inlet  is  30  degrees  and  the  radial  velocity  is 
^^^tant.  The  ratio  of  the  radii  at  inlet  and  outlet  is  1*75.  Find  the  velocity  of 
^  inlet  drenmference  of  the  wheel.    Neglect  friction. 

L.  H.  ^\ 


^-^^  =  6H. 


322 


HYDRAULICS 


Sinoe  tlM  diach&rge  it  radial,  tli«  Tvlodtj'  ftt  wt  h 

Ui=:=Pjtan3(r 

V 


1-75 


taoSr. 


Then 


7"         1T5«""^^* 


and  sinoe  the  Uiid^  A^m  radial  «i  inlet  V  ia  eqnaX  Ui  i^, 
therefore 


from  whieh 


/82  r  an 


E^  r^ 


Fig.  206. 

Example  (2).  The  outer  diameter  of  the  ^he«l  of  an  inward  flow  tcrbiiii* 
200  horse-power  U  2*41^  fret,  the  inner  diameter  is  l^^tfiS  feet.  The  eDfeenT*iwhb 
of  the  wheel  at  inlets  ILS  feet.  The  head  b  39'5  feet  and  59  eobic  feet  of 
water  per  second  ^iv  supplied.  The  radial  velocity  with  which  the  wat^r  \^^^ 
the  wheel  may  he  taken  us  10  feet  per  ftecond. 

Determine  the  theori^tical  hydraulic  eMciencj  £  aod  the  aeitial  effioien^'i^ 
the  turbine,  and  de^-vign  suitable  vane>^. 

300x550 
^i"a9-5x5yxti3-5"       '"* 


Theoretical  hydranUo  efficiency 


S9'5" 


10* 


m-b 


=  96% 


The  radial  velocity  of  flow  at  in  let, 
59 


2  46xrxM5 


=  fi*7  feet  per  sec* 


TURBINES 


323 


ripbenJ  yekmij 

r=2-46.  XX  W=88-6  feet 
loctty  0/  wkiH  V.     Aasamiiig  a  hydraulic  effioienqj  of  B6%,  from 

^_S9'5x32-ax'85 
88-6 
=:S8*0  feet  per  lee. 
gU  e.    Sinoe  if=6-7  ft.  per  sec.  and  y=38*0  ft.  per  sec. 

tan  ^=^=0-289, 


igle  ^.    Since  V  is  ] 


^=180  27'. 
\  than  V,  0  is  greater  than  90°. 


0=162<>. 
le  water  to  discharge  radially  with  a  velocity  of  10  feet  per  sec. 
^  10x60  ^^^^ 

*"'=l-«68xxx800=°"^' 
a =18°  nearly, 
leoretical  vanes  are  shown  in  Fig.  206. 
9U  (3).    Find  the  valoes  of  ^  and  a  on  the  assumption  that  e  is  0*80. 

nsorCa  inward  flow  twrhvne.  In  1851  Professor  James 
n  invented  an  inward  flow  turbine,  the  wheel  of  which 
rounded  by  a  large  chamber  set  eccentrically  to  the  wheel, 
n  in  Figs.  207  to  210. 

reen  the  wheel  and  the  chamber  is  a  parallel  passage,  in 
ire  four  guide  blades  Gr,  pivoted  on  fixed  centres  C  and 
:jan  be  moved  about  the  centres  C  by  bell  crank  levers, 
1  to  the  casing,  and  connected  together  by  levers  as  shown 
207.  The  water  is  distributed  to  the  wheel  by  these  guide 
and  by  turning  the  worm  quadrant  Q  by  means  of  the 
the  supply  of  water  to  the  wheel,  and  thus  the  power  of 
bine,  can  be  varied.  The  advantage  of  this  method  of 
ing  the  flow,  is  that  there  is  no  sudden  enlargement  from 
ide  passages  to  the  wheel,  and  the  efficiency  at  part  load 
much  less  than  at  full  load. 

i,  209  and  210  show  an  enlarged  section  and  part  sectional 
)n  of  the  turbine  wheel,  and  one  of  the  guide  blades  G. 
tails  of  the  wheel  and  casing  are  made  slightly  different 
bose  shown  in  Figs.  207  and  208  to  illustrate  alternative 
is. 

t  sides  or  crowns  of  the  wheel  are  tapered,  so  that  the 
jral  area  of  the  wheel  at  the  discharge  is  equal  to  the 
3ral  area  at  inlet.     The  radial  velocities  of  flow  at  inlet 

itlet  are,  therefore,  equal. 

41—2 


TURBINES 


325 


iriea  of  velocities  for  the  inlet  and  outlet  are  shown  in 

water  leaving  the  wheel  radially, 
^of  the  water  through  the  wheel,  relative  to  the  fixed 
"^  shown  and  was  obtained  by  the  method  described 

•w  turbines  with  adjustable  guide  blades,  as  made  by 
!ral  makers,  have  a  much  greater  number  of  guide 
I'ig.  238,  page  352). 


tion  through  wheel  and  casing  of  Thomson  Inward  Flow  Turbine. 

»ine  actual  inward  flow  turbines. 
jnn  of  the  Francis  inward  flow  turbine  as  designed  by 
/O.,  and  having  a  horizontal  shaft,  is  shown  in  Fig.  212. 
id  is  doable  and  is  surrounded  by  a  large  chamber 
water  flows  through  the  guides  G-  to  the  wheel  W. 
g  tihe  wheel,  exhaust  takes  place  down  the  two  suction 
38  allowing  the  turbine  to  be  placed  well  above  the 
hile  ntilifliiig  the  full  head. 

nhting  dnioe  F  consists  of  a  steel  cylinder,  which 
parallel  to  the  axis  between  the  wheel  and 


3S6 


HYDEAULICS 


Fig.  209.  Fig.  210* 

Detail  of  wbeel  and  galde  blade  of  Thomson  Inward  Flow  Turbine. 


Fig.  211, 


m-hl-Wi 


TURBINES 


327 


Hie  wheel  is  divided  into  five  separata  compartments,  so  that 
iny  time  only  one  can  be  partially  closedj  and  loss  of  head  by 
ittTaction  and  sudden  enlargement  of  the  streanij  only  takes 
tee  in  this  one  compartment* 


TUBBINES 


329 


ita^e  or 
mjh  the 
^■he  tti 
Ha  h 
Bessar 


Ripened  or  clewed  as  required  by  the  steel  cylindrical  sluice  CC 
^VDUtidtng  the  distributor* 

^"WThen  one  of  the  stages  is  only  partially  closed  by  the  eluice, 
h  loss  of  efficiency  must  take  place,  but  the  efficiency  of  this  one 
ita^  only  is  diminisbedj  the  stages  that  are  still  open  working 
their  full  efficiency.  With  this  construction  a  high  efficiency 
turbine  is  maintained  for  partial  How,  With  normal  flowSj 
head  of  about  62-5  feetj  the  three  lower  stages  only  are 
iry  t-o  give  full  power,  and  the  efficiency  is  then  a 
mnximum.     In  times  of  H<x>d  there  is  a  large  volume  of  %vater 

S reliable,  but  the  tail  watar  rises  so  that  the  head  is  only  about 
\9  te0tf  the  two  upper  stages  can  then  be  brought  int-o  operation 
to  accommodate  a  larger  floWj  and  thus  the  same  power  may  be 
obtained  under  a  less  head.  The  efficiency  is  less  than  when  the 
Oiree  stages  only  are  working,  but  as  there  is  plenty  of  wat-er 
availablCj  the  loss  of  efficiency  is  not  serious* 

I  The  cylinder  C  is  carried  by  four  vertical  spindles  S,  having 
ks  R  fixed  to  their  upper  ends.  Gearing  with  these  racks,  are 
ions  Pf  Fig.  213,  all  of  which  are  worked  simultaneously  by  the 
^lator,  or  by  hand,  A  bevel  wheel  fixed  Uj  the  vertical  sliaft 
f9  with  a  second  bevel  wheel  on  a  horizontal  shaft,  the  velocity 
b  being  3  to  L 
^89*  The  best  peripheral  velocity  for  inward  and  outward 
tow  turbines. 

IjWheii  the  discharge  is  radial,  the  general  formula,  as  showTi  on 
b  315,  is 
I                            —  =  eH  =  0-78toO-90H 
L                              ^ 
K  the  blades  are  radial  at  inlet,  for  no  shock,  t^  should  be  equal 
r , and                                                       
I                         ^^Yi^039    toO'45V2gH, 
f                          t^  -V-0-624to0  67v^2^, 
This  is  aametimes  called  the  best  velocity  for  t;,  but  it  should  be 
rly  understood  that  it  is  only  so  when  the  blades  are  radial  at 


.(1). 


W 


190.     Experimental  determination  of  the  best  peripheral 
Telocity  for  inward  and  outward  flow  turbines. 

fw  an  outward  flow  turbine,  working  under  a  head  of  14  feet, 
mdial  at  inlet*  Francis*  found  that  when  v  was 

0  626  V2^, 

•  Lowell,  Hydmuth  Exfifrlmentst 


' 


390 


HYDRAULICS 


From  the  formula  — -  ■ 
9 


the  efficiency  was  a  maximum  and  equal  to  79 "37  per  cent.  Tlie 
efficiency  however  was  over  78  per  cent,  for  all  \^lues  of  tf 
between  0'545  -J^E  aud  *671  J2gR.  If  3  per  cent,  be  allowt'd 
for  the  mechanical  losses  the  hydraulic  efficiency  may  be  taken 
^B  82"4  per  cent, 

VtJ 

-  '824H,  and  taking  V  equal  to  i\ 

V  ^  '64  %^yH, 

so  that  the  result  of  the  experiment  agrees  well  with  the  formnla* 

For  an  inward  flow  turbine  having  vanes  as  shown  in  Fig,  205, 

I  the  ttjtal  efficiency  was  over  79  per  cent,  for  values  of  r  between 

PO'624  V%B   and  0'708  J2^,  the  greatest  efficiency  being  W! 

per   cent*    when    v    was    0*708  v2gH    and    again    when   t?  WM 

It  will  be  seen  from  Fig.  205  that  although  the  tip  of  the  irwa^ 
at  the  convex  side  is  nearly  radial,  the  general  direction  of  tha 
vane  at  inlet  is  inclined  at  an  angle  greater  than  90  degree  \f> 
the  directioii  of  motion,  and  therefore  for  no  shock  Y  ^should  hi 
less  than  v,  

Wben  V  was  708  V2gH,  V,  Fig.  205^  was  less  than  i\  Tbfi 
value  of  V  was  deduced  from  the  following  data»  which  is  abo 
useful  as  being  taken  from  a  turbine  of  very  high  efficiency. 

Diameter  of  wheel  9'338  feet. 

Width  between  the  crowns  at  inlet  0'999  foot. 

There  were  40  vanes  in  the  wheel  and  an  equal  number  of 
fixed  guides  external  to  the  wheel. 

The  minimum  width  of  each  guide  passage  was  0"1467  foot  ini 
the  depth  r0066  feet. 

The  quantity  of  water  supplied  to  the  wheel  per  second  *i* 
ni2'526  cubic  feet,  and  the  total  fall  of  the  water  was  134  fe<*- 
Kthe  radial  velocity  of  flow  u  was^  therefore,  3*86  feet  per  secoii 

The  velocity  through  the  minimtmi  section  of  the  guide 
was  19  feet  per  second. 

When  the  efficiency  was  a  maximum,  t'  was  20'8  feet  per 
Then  the  radial   velocity  of   flow  at   inlet  to  the  wheel 
3'S6  feet,  and  U  being  taken  as  19  feet  per  second,  the 
of  velocities  at  inlet  is  ABC,  Pig.  205,  and  V  is  18'4  feet  per 

K  it  is  assumed  that  the  water  leaves  the  wheel  radially,  thfi^ 

L  eH=— =  1185feet 

r  ^ 

11*85 
The  efficiency  e  should  be  j^rx  ==88'5  per  cent.,  which  is  9| 

cent  higher  than  the  actual  efficiency. 


TURBINES 


331 


The  actual  efficiency  howeirer  mcladea  not  only  the  fluid  losses 
it  also  the  mechanical  losses,  and  these  would  probahly  be  from 
to  8  per  c^nt,,  and  the  actual  work  done  by  the  turbine  on  the 
laft  IB  probably  between  80  and  S&5  per  cent,  of  the  work  done 
r  the  water.  j 

Vv  I 

19L    Value  of  e  to  be  used  in  the  formula  — -  ^  eH*  1 

i 

In  general,  it  may  be  said  that,  in  using  the  formula  —  =  eH, 

e  value  of  e  to  be  used  in  any  given  case  is  doubtful,  as  even 
oQ^b  the  efficiency  of  the  class  of  turbiues  may  be  known,  it  is 
fficult  to  say  exactly  how  much  of  the  energy  is  lost  mechanically 
id  how  moch  hydrauUcally, 

A  trial  of  a  turbine  without  load,  would  be  useless  to  deter- 
ine  the  mechanical  efficiency,  as  the  hydraulic  losses  in  such  a 
tal  would  be  very  much  larger  than  when  the  turbine  is  working 
full  load.  By  revolving  the  turbine  without  load  by  means  of 
i  electric  motor,  or  through  the  meditnn  of  a  dynamometer,  the 
art  to  overcome  friction  of  bearings  and  other  mechanical  losses 
uld  be  found.  At  all  loads,  from  no  load  to  full  load,  the 
iotional  resistances  of  machines  are  fairly  constant,  and  the 
schanical  losses  for  a  given  class  of  turbineSj  at  the  normal  load 
r  which  the  vane  angles  are  calculated,  could  thus  approximately 
\  obtained.  If,  however,  in  making  calculations  the  difference 
tween  the  actual  and  the  hydraulic  efficiency  be  taken  as,  say, 
per  cent.,  the  error  cannot  be  very  great,  as  a  variation  of  5  per 
ntp  in  the  value  assumed  for  the  hydraulic  efficiency  e,  will  only 
kke  a  difference  of  a  few  degrees  in  the  calculated  value  of 
I  angle  <^. 

The  beat  value  for  0,  for  inward  flow  turbines,  is  probably  0*80, 
1  ejcperience  shows  tliat  this  value  may  be  used  with  confidence, 

ffiwilpfr.     Tftkiag  thedjiU  ikM  u^^en  in  the  exacnple  of  ae^tiou  184,  and  aasaming 

MMtWQT  t^'  ^^  tutbioe  of  75  per  eeat.^  the  horfie-power  k 

iKJK  215  y  62-4  >£Ul%5x  75x60  ■ 

==2600  horse -power 
If  the  hjdr&iUie  efficiencj  ij   anppoaed   to  be  SO  per  centt^   the   velocitj  of 
d  V  sbonld  be 

^      eff.H^  08,32 -Ul-6 

r      "  60  I 

=53  feet  per  a«e*  I 

riten  tmm^       18  35         -18  85  | 

Cbcti  *~5a-6S~      17      ' 

^-182°  47', 
Kippoae  the  torbiise  to  be  itill  ^nersting  2600  hor»e*power,  and  to  h&ye 
\j  of  80  pet  oenL,  ^nd  »  hydrauito  effioie^cy  of  65  per  eent. 


332 


HYDRAUUCS 


Then  the  qaantity  of  water  reqaired  per  second,  is 

^    216x0-76    o^      u-    #    * 

Q  =  — zr^ —  =  200  cubic  feet  per  sec. 

and  the  radial  velocity  of  flow  at  inlet  will  be 
18-36x200 


215 


=  17*1  ft,  per  sec. 


-,    •86.32.141-6     __.-^ 

V  = nn =66*4  ft.  per  sec. 


Then 


tan0=: 


69 
171 


66-4-69 
=  128°.  24'. 


-171 
13-6 


192.    The  ratio  of  the  velocity  of  whirl  V  to  the  velodtj 
of  the  inlet  periphery  v. 

Experience  shows  that,  consistent  with  Yv  satisfying  the  gfenenl 

formula,  the  ratio  ^y  may  vary  between  very  wide  limits  without 
considerably  altering  the  efficiency  of  the  turbine. 

Table  XXXVII  shows  actual  values  of  the  ratio  • ,         taken 

from  a  number  of  existing  turbines,  and  also  corresponding  valnai 


Fig.  214. 


TURBINES 


333 


=^ ,  V  beinff  calculated  from  — : 


'  0'8H.    The  corresponding 


.tion  in  the  angle  <t>y  Fig.  214,  is  from  20  to  150  degrees, 
or  a  given  head,  v  may  therefore  vary  within  wide  limits, 
h  allows  a  very  large  variation  in  the  angular  velocity  of  the 
1  to  suit  particular  circumstances. 

TABLE  XXXVII. 

howing  the  heads,  and  the  velocity  of  the  receiving  circum- 
ice  V  for  some  existing  inward  and  outward,  and  mixed  flow 
nes. 


Hfeet 

vfeet 
per  see. 

J29H 

Ratio 

V 

sl2gR 

H.P. 

V 
Ratio  -7=- 
>/2pH 
V  being  calculated 

from— =  -8H 
9 

.rdflaw: 

gara  Falls* 

146 

70 

96-8 

0-72 

5000 

0-655 

infelden 

14-8 

22 

80-7 

0-71 

840 

0-666 

Theodor) 
and  Co.  j 

28-4 

89 

42-6 

0-91 

0-44 

60-4 

82-2 

62-8 

0-52 

0-77 

et  and  Co. 

188-7 

611 

76-8 

0-47 

800 

0-85 

»» 

184-6 

46-6 

65-6 

0-505 

800 

0-79 

6-25 

16-6 

20 

0-88 

0-48 

ji 

80 

25-75 

44 

0-58 

700 

0-69 

♦» 

88-5 

50-8 

0-77 

200 

0-52 

Lz  and  Co. 

112 

64-8 

84-6 

0-54 

0-74 

»9 

226 

64-7 

120 

0-64 

682 

0-58 

ber  and  Co. 

10-66 

15-2 

26 

0-685 

80 

0-69 

^rdflow : 

gara  Falls 

141-6 

69 

95-2 

0-725) 

5000 

0-55 

;et  and  Co. 

180-6 

69 

91-6 

0-750) 

0-58 

LZ  and  Co. 

96-1 

88-7 

780 

0-495 

290 

0-81 

»» 

228 

65-6 

1200 

0-46 

1200 

0-87 

♦  Escher  Wyss  and  Co. 

'or  example,  if  a  turbine  is  required  to  drive  alternators 
rt,  the  number  of  revolutions  will  probably  be  fixed  by  the 
nators,  while,  as  shown  later,  the  diameter  of  the  wheel  is 
tically  fixed  by  the  quantity  of  water,  which  it  is  required  to 
through  the  wheel,  consistent  with  the  peripheral  velocity  of 
v^heel,  not  being  greater  than  100  feet  per  second,  unless,  as 
ie  turbine  described  on  page  373,  special  precautions  are 
1.  This  latter  condition  may  necessitate  the  placing  of  two 
3re  wheels  on  one  shaft. 


334 


BYDBAmJCa 


Suppose  then,  the  number  of  reT-olutions  of  the  wheel  to 
given  and  d  is  fixed,  then  v  has  a  definite  value,  and  V  muii  1 
made  to  satisfy  the  equation 

s 

Fig.  214  is  drawn  to  illustrate  three  cas€«s  for  which  Yri] 
constant,    Tht?  angles  of  the  vanes  at  outlet  are  the  same  fori 
three,  but  the  guide  angle  ^  and  the  vane  angle  ^  at  inlet 
considerably^. 


193.    The  velocity  witli 
In  a  well-deaigned  turbine 
leaves  tht*  turbine  should  be  as 
keeping  the  turbine  whrel  an 
dimensions. 

In  actual  turbines  the  1 
varies  from  2  to  8  per  cei 
suction  pipe  the  water  may  t 
with  a  fairly  high  velocity  a 
conical  sv»  ai^  to  allow  the  actual 


water  leaves  a  turbme. 
locity  with  which  the  ' 
as  possible,  condsteal  i 
>ftTi-take  within 

h 

^due  to  this  velocity 
I  turbine   is   fitted  with 
jd  to  learc  the  wheel 
iacharge  pipe  can  be  \ 
irge  velocity  to  be  as  i 


as  desired.  It  should  however  be  noted  that  if  the  w^ater  leavtf* 
the  wheel  with  a  high  velocity  it  is  more  than  probable  that  there 
^^^ll  be  Jiome  loss  of  head  due  to  shock,  as  it  is  difficult  to  ensure 
that  wattT  ?4*  di^chargtKi  shall  have  its  velocity  changed  gradiiallT. 

194.  BemoniUi's  equations  applied  to  inward  and  out- 
ward flow  tnrbmes  neglecting  firiction. 

Ctntrifn^al  head  imprejtsed  on  the  water  by  the  wheei  Tk 
theory  of  rhe  reaction  turbines  is  best  considered  from  the  point 
of  N-iew  of  Bemouilli's  equations ;  but  before  proceeding  to  discuss 
them  in  detail,  it  is  necessary  to  consider  the  "  centrifugal  head" 
impivssed  un  the  water  by  the  w^heeh 

This  head  has  already  been  considered  in  connection  with  tk 
Scotch  turbim*,  page  303, 

Let  r^  Kig.  216j  be  the  internal  radius  of  a  wheel,  and  B  tb 
external  radius. 

At  the  internal  circumference  let  the  wheel  be  covered  with  i 
cylinder  e  S4>  that  there  can  be  no  flow  through  the  wheel,  and  1h 
it  W  suppixsed  that  the  wheel  is  made  to  revolve  at  the  angular 
vehxnty  f^  which  it  has  as  a  turbine,  rhe  wheel  being  full  of  wat€f 
and  surrounded  by  water  at  rest,  the  pressure  outside  the  wbed 
being  sufficient  to  prevent  the  water  being  whirled  out  of  the 
whet^l.  Lei  d  be  the  depth  of  the  wheel  between  the  crowia 
Consider  any  element  of  a  ring  of  radius  n  and  thickness  dr,  and 
subtending  a  smaU  angle  6*  at  the  centre  C,  Fig.  210* 


TURBINES 


335 


The  weight  of  the  element  is 

UDToO  .dr,d, 

and  the  centrifugal  force  acting  on  the  element  is 
uTo^  ,dr,d,  «Vo 


9 


lbs. 


Iiet  p  be  the  pressure  per  unit  area  on  the  inner  force  of  the 
element  and  p-^dp  on  the  outer. 

fjPToB ,dr  .d  . «Vo 
op  = 


Then 


g.nO.d 


Fig.  215. 


Fig.  216. 


The  increase  in  the  pressure,  due  to  centrifugal  forces,  between 
-r  and  R  is,  therefore, 


-r 


T'-*"  I  •'<"'"'■' 


i^nd 


For  equilibrium,  therefore,  the  pressure  in  the  water  surround- 
ing the  wheel  must  be  pe. 

If  now  the  cylinder  c  be  removed  and  water  is  allowed  to  flow 
tlnroiigh  the  wheel,  either  inwards  or  outwards,  this  centrifugal 
lieftd  will  always  be  impressed  upon  the  water,  whether  the  wheel 
u  driven  by  the  water  as  a  turbine,  or  by  some  external  agency, 
mnA  acts  as  a  pump. 

BernouilWa  equations.  The  student  on  first  reading  these 
equations  will  do  well  to  confine  his  attention  to  the  inward  flow 
terbine,  Fig.  217,  and  then  read  them  through  again,  confining  his 
attention  to  the  outward  flow  turbine,  Fig.  191. 


336 


HYDRAULICS 


Let  p  be  the  pressure  at  A,  the  inlet  to  the  wheel,  or  in  the 
clearance  between  the  wheel  and  the  guides,  pi  the  pressure  at 
the  outlet  B,  Fig.  217,  and  pa  the  atmospheric  pressure,  in  pounds 
per  square  foot.  Let  H  be  the  total  head,  and  H«  the  statical 
head  at  the  centre  of  the  wheel.  The  triangles  of  velocities  are 
as  shown  in  Figs.  218  and  219. 

Then  at  A 


Pa 


TP 


w  w     2g 


.(1). 


Between  B  and  A  the  wheel  impresses  upon  the  water  the 
centrifugal  head 

2g     2g' 
V  being  greater  than  Vi  for  an  inward  flow  turbine  and  less  for  the 
outward  flow. 


Fig.  217. 

Consider  now  the  total  head  relative  to  the  wheel  at  A  and  BL 
The  velocity  head  at  A  is  -^  and  the  pressure  head  is  ^i>il||H 

at  B  the  velocity  and  pressure  heads  are  ^  and  ^  respectivdy. 

If  no  head  were  impressed  on  the  water  as  it  flows  thtongli 
the  wheel,  the  pressure  head  plus  the  velocity  head  at  A  and  B 
would  be  equal  to  each  other.  But  between  A  and  B  thero  i« 
impressed  on  the  water  the  centrifugal  head,  and  therefore, 


Pi     Vr^     v^     ^^'  =  P  I  ^^* 
w     2^     2g     2g     w     2g 


.(2). 


TURBINES 


.337 


This  equation  can  be  used  to  deduce  the  fundamental  equation, 

^-^  =  fc (3). 

9         9 

From  the  triangles  CDE  and  ADE,  Fig.  218, 
Vr»=(V-t7)«  +  ti'andV»  +  tt«  =  U', 
ind  from  the  triangle  BFG,  ¥ig  219, 

vr'  =  (vi  -  YiY  +  u,'  and  Va»  +  V  =  Vi\ 
Therefore  by  substitution  in  (2), 

i?""       2^  2g     2g^2g     w^      2g      ^  2g    "'^^^' 

From  which 

ti7        g        2g      w     2g       g  ' 

ad 


g         g       w     w     2g      2g 


Substituting  f or  *^  +  ^5-  from  (1) 


w 


VV       VlVl  ^  ^    ,    Pa       Pi        Ul' 


2sr 


.(5). 


(6). 


Pig.  218. 


Wheel  in  suction  tube.    K  the  centre  of  the  wheel  is  ho  feet 
the  snr&ce  of  the  tail  water,  and  Uo  is  the  velocity  witli 
e  water  leaves  the  down-pipe,  then 


^^m  th 


'-      2g      ^    w      2g 


w 


Snbstitntingr  for  ^  +  §^'  in  (6), 


w      2g 
9  "    9 


-         -  WW  2g 


=  H-U'' 


2ff- 


I.     B. 


^•1 


338  HYDRAULICS 

IfVisO,  —^R-^---h. 

9  2flr 

The  wheel  can  therefore  take  full  advantag^e  of  the  head  H 
even  though  it  is  placed  at  some  distance  above  the  level  of  the 
tail  water. 

Drowned  wheel.  If  the  level  of  the  tail  water  is  CD,  Fig.  217, 
or  the  wheel  is  drowned,  and  hi  is  the  depth  of  the  centre  of  the 
wheel  below  the  tail  race  level, 

W  W 

and  the  work  done  on  the  wheel  per  pound  of  water  is  again 
vV     Yiv,    „    W     , 
9         9  ^9 

vY 
IfViisO,  —  =  h. 

From  equation  (5), 

vV  _  ViYj  ^  P  _  Pi  ^  U! _  W 
9         9        w     w     2g      2g  ^ 
so  that  the  work  done  on  the  wheel  per  pound  is  the  difference 
between  the  pressure  head  plus  the  velocity  head  at  entrance  and 
the  pressure  head  plus  velocity  head  at  exit. 

In  an  impulse  turbine  p  and  pi  are  equal,  and  the  work  doi« 
is  then  the  change  in  the  kinetic  energy  of  the  jet  when  it  strikes 
and  when  it  leaves  the  wheel. 

A  special  case  arises  when  pi  is  equal  to  p.  In  this  case  a 
considerable  clearance  may  be  allowed  between  the  wheel  and  the 
fixed  guide  without  danger  of  leakage. 

Equation  (2),  for  this  case,  becomes 

2g      2g     2g     2g' 

and  if  at  exit  Vr  is  made  equal  to  Vi,  or  the   triangle  BFG, 
Fig.  219,  is  isosceles, 

2g      2g^ 

and  the  triangle  of  velocities  at  entrance  is  also  isosceles. 
The  pressure  head  at  entrance  is 

w  2g' 

and  at  exit  is  either        —  +  fei,  or  —  -  ^. 
w       ^        w 


TURBINES  339 

lieref  ore,  since  the  pressures  at  entrance  and  exit  are  equal, 

<r  else  Ho  +  /iD  =  H. 

The  water  then  enters  the  wheel  with  a  velocity  equal  to  that 
Lue  to  the  total  head  H,  and  the  turbine  becomes  a  free-deviation 
NT  impulse  turbine. 

195.  Bemonilli's  equations  for  the  inward  and  outward 
low  turbines  including  friction. 

If  H/  is  the  loss  of  head  in  the  penstock  and  guide  passages, 
^  the  loss  of  head  in  the  wheel,  he  the  loss  at  exit  from  the  wheel 
ttid  in  the  suction  pipe,  and  Ui  the  velocity  of  exhaust, 

£  +  U!=H.  +  £e-H, (1), 

w^2g^2g-2g-w^2-g-^^  ^^^' 


w     w 


w     w 


rom  which  —  =  H-f -^^  +  A/+H/+fee)  (4). 

f  the  losses  can  be  expressed  as  a  fraction  of  H,  or  equal  to  KH, 

hen 

Vt? 

—  =  (l-K)H  =  eH 

^  =0-78H  to  0-90H*. 

196.    Turbine  to  develop  a  given  horse-power. 

Let  H  be  the  total  head  in  feet  under  which  the  turbine  works. 

Let  n  be  the  number  of  revolutions  of  the  wheel  per  minute. 

Let  Q  be  the  number  of  cubic  feet  of  water  per  second  required 
>y  the  turbine. 

Let  E  be  the  theoretical  hydraulic  efficiency. 

Let  e  be  the  hydraulic  efficiency. 

Let  Sm  be  the  mechanical  efficiency. 

Let  fii  be  the  actual  efficiency  including  mechanical  losses. 

Let  Ui  be  the  radial  velocity  with  which  the  water  leaves  the 
irheel. 

Let  D  be  the  diameter  of  the  wheel  in  feet  at  the  inlet  circum- 
erence  and  d  the  diameter  at  the  outlet  circumference. 

Let  B  be  the  width  of  the  wheel  in  feet  between  the  crowns 
it  the  inlet  circumference,  and  b  be  the  width  between  the  crowns 
it  the  outlet  circumference. 

Let  N  be  the  horse-power  of  the  turbine. 
*  See  page  815. 


340  HYDRAULICS 

The  number  of  cubic  feet  per  second  required  is 

^    e,H. 62-4.60  ^^'• 

A  reasonable  value  for  ei  is  75  per  cent. 

The  velocity  Uo  with  which  the  water  leaves  the  turbine,  since 

is  Uo=>/25f(l-E)Hft.per8ec (2). 

If  it  be  assumed  that  this  is  equal  to  thy  which  would  d 
necessity  be  the  case   when    the    turbine    works    drowned,  or 
exhausts  into  the  air,  then,  if  Hs  the  peripheral  thickness  of  the 
vanes  at  outlet  and  m  the  number  of  vanes, 
(ird-mOUo6=Q. 

If  Uo  is  not  equal  to  Ui,  then 

(^d-mt)uib  =  Q  (3). 

The  number  of  vanes  m  and  the  thickness  t  are  somewhat 
arbitrary,  but  in  well-designed  turbines  t  is  made  as  small  as 


As  a  first  approximation  mt  may  be  taken  as  zero  and  (3) 
becomes 

7rdbUi=Q  (4). 

For  an  inward  flow  turbine  the  diameter  d  is  fixed  from 
consideration  of  the  velocity  with  which  the  water  leaves  the 
wheel  in  an  axial  direction. 

If  the  water  leaves  at  both  sides  of  the  wheel  as  in  Pig.  208, 
and  the  diameter  of  the  shaft  is  do,  the  axial  velocity  is 

t^=  y ft.  per  sec. 

2j(d'-d.') 

The  diameter  do  can  generally  be  given  an  arbitrary  value,  or 
for  a  first  approximation  to  d  it  may  be  neglected,  and  u%  may  be 
taken  as  equal  to  i^.    Then 

^-^"^ <«• 

From  (4)  and  (5)  b  and  d  can  now  be  determined. 

A  ratio  for  -v  having  been  decided  upon,  D  can  be  calcvlatedy 

and  if  the  radial  velocity  at  inlet  is  to  be  the  same  as  at  oatletk 
and  to  is  the  thickness  of  the  vanes  at  inlet, 

(^D-mMB  =  S  =  (ird-m06  (6). 

til 


TURBINES  341 

For  rolled  brass  or  wrought  steel  blades,  to  may  be  very  small, 
and  for  blades  cast  with  the  wheel,  by  shaping  them  as  in  Fig.  227, 
to  is  practically  zero.    Then 

If    now   the  number  of  revolutions  is  fixed  by  any  special 

condition,  such  as  having  to  drive  an  alternator  direct,  at  some 

definite  speed,  the  peripheral  velocity  is 

frDn  -.  ,^,^. 

t7  =  -^  ft.  per  sec (/). 

Yv 
Then  V"^^' 

and  if  e  is  given  a  value,  say  80  per  cent., 

V  =  '^  ft.  per  sec (8). 

V 

Since  u,  V,  and  v  are  known,  the  triangle  of  velocities  at  inlet 
can  l>e  drawn  and  the  direction  of  flow  and  of  the  tip  of  vanes 
at  inlet  determined.    Or  B  and  <^,  Fig.  214,  can  be  calculated  from 

tan^  =  ^- (9) 

and  '         tan<^  =  ;^^    (10). 

Then  U,  the  velocity  of  flow  at  inlet,  is 

U  =  V  sec  0, 

Trdn  «, 
At  exit  fi  =  -^  ft.  per  sec, 

and  taking  Ux  as  radial  and  equal  to  Uy  the  triangle  of  velocities 

can  be  drawn,  or  a  calculated  from 

u 
tan  a  =  -  . 

If  BU  is  the  nead  of  water  at  the  centre  of  the  wheel  and  11/  the 
head  lost  by  friction  in  the  supply  pipe  and  guide  passages,  the 
pressure  head  at  the  inlet  is 

Example-     An  inward  flow  turbine  is  required  to  develop  300  horse-power  under 
A  bead  60  feet,  and  to  run  at  250  revolutions  per  minute. 
To  determine  the  leading  dimensions  of  the  turbine. 
Aaeuming  ^i  to  be  75  per  cent., 

Q_      300x33,000 
^~ -76 x60  x~02Tx^60 
=  58*7  cubic  feet  per  sec. 


342  HTDRAUUCS 

Assmmiig  E  is  95  per  cent.,  or  five  per  cent,  of  the  head  iq  lost  by  Telo^ 
of  exit  and  u^^u, 

=^=•05.60 
2g 

and  tt=  13*8  feet  per  sec. 

Then  from  (5),  page  340, 


^^-^W^-J^-^^ 


=  1-65  feet, 
say  20  inches  to  make  allowance  for  shaft  and  to  keep  eyen  dimenBion. 

Then  from  (4),  * = fH "=  '^  ^^* 

=9}  inches  say. 

Taking  -  as  1*8,  D=3*0  feet,  and 
a 

V  =rir .  8 .  W  =  39*3  feet  per  sec., 

and  B=5}  inches  say. 

Assuming  «  to  be  80  per  cent., 

„    -80x60x32    ^^^^ 


"  "~ 

39-3       ~*>''""*P«^"«- 

" 

*     n    13*8 
tan^=  39  ' 

and 

^=19^80'. 

*«'*=^'.-  -  ^^ 

and 

0=91'^  15'. 

13*8  X  1*8 
**"*=-39*3-' 

and 

a  =  32°  18'. 

The  velocity  U  at  inlet  is 

U=>/39*0»  +  (13*8)« 

=41*3  ft.  per  sec. 

The  absolute  pressure  head  at  the  inlet  to  the  wheel  is 

n  n  41  'V 

-  =  Ho  +  -^  — hf,  the  head  lost  by  friction  in  the  down  pipe 

=Ho+34-26*6-/i/. 

The  pressure  head  at  the  outlet  of  the  wheel  will  depend  upon  the  height  of  t 
wheel  above  or  below  the  tail  water. 

197.    Parallel  or  axial  flow  turbines. 

Fig.  220  shows  a  double  compartment  axial  flow  turbine,  tl 
guide  blades  bein  g  placed  above  the  wheel  and  the  flow  throuj 
the  wheel  being  parallel  to  the  axis.  The  circumferential  secti 
of  the  vanes  at  any  radius  when  turned  into  the  plane  of  t 
paper  is  as  shown  in  Fig.  221.    A  plan  of  the  wheel  is  also  shov 

The  triangles  of  velocities  at  inlet  and  outlet  for  any  radi 
are  similar  to  those  for  inward  and  outward  flow  turbines,  t 
velocities  v  and  t^i,  Figs.  222  and  223,  being  equal. 


TUUBINES 


343 


The  general  formula  now  becomes 

For  maximum  efficiency,  for  a  given  flow,  the  water  should 
leave  the  wheel  in  a  direction  parallel  to  the  axis,  so  that  it  has 
no  momentum  in  the  direction  of  v. 


Fig.  220.    Doable  Compartment  Parallel  Flow  Turbine. 


Figs.  221,  222,  223. 


Then    takii^  friction  and  other  losses  into  account, 

9 


344 


HrDRACILICS 


The  ineloeit^  r  will  be  proportional  to  the  radius,  mi  th^t  if  1 
w^ter  is  to  enter  and  lettTO  the  wheel  withoat  shock,  the  anfltsl^l 
^  and  m  tous^  faiy  with  the  radius. 

TW  rariMion  m  the  form  of  the  r&ne  with  the  imdios  is  skmr^] 
by  an  exaai{iki^ 

A  Jcfowml  wheel  has  an  internal  diameter  of  5  feet  mud 
extremal  diameter  of  B"^.     The  depth  of  the  wheel  is  7  iiicli 
TTie  h^ttd  is  15  feet  and  the  wheel  makes   55  tevolationf 
minnre,    The  flow  b  300  cubic  feet  per  second. 

Ti>  find  the  h^iriMes- power  of  th< — ^  id,  and  to  design  the  wh 

Let  Ti  be  the  immn  radins,  a: 
ai  the  inner  and  oater  ciirmnfer 


r  -  i'o  feet      attd  r  =  2a 
fi  =  ^^*>  feet  and  rj  =  S 
ff^= 4l25  ket    and  r^  -  & 

The  mean  axial  Telocity  is 


ind  Ft  the  radii  of  the  wh 
e^iectiTely.    Then 

=  14"4  feet  per  sec,^ 

=  21*5  feet  per  sec,| 

=24^5  fe^  per  sec. 


.   1     jv  =8'la  !i,  per  sec* 


*4-*^^  — 


^:,_H%^;AML  ~I^^M,j  , Jt  C 


F^.  aa*. 


TriAa^J^s  f»l  Tdodlies  ai  mki  md  aailet  %t  tbree  dif ei«nt 
imlit  of  m  Pkimllel  Fl^v  Tarbine. 


Taking  ^  as  0^80  at  each  radius, 

i4'4  ^  ^^  ®^'' 


14  4 


385 
Vt  -  ^YjT^  ==17^  ft-  per  sec,, 

\  ,  =  ^ ,  i  =  la  f  ft,  per  sec 

ImcJimatiim  of  ik^  ran^s  at  inlei.  The  triangles  of  velocities 
for  the  ihrt*^^  T*dii  n  rj,  r-  are  siniwn  in  Fig.  224.  For  example, 
at  nftditt^  r,  ADC  i^  the  triangle  of  Yelocities  at  inlet  and  ABC  tie 


TURBINES  345 

ian^le  of  velocities  at  oatlet.    The  inclinations  of  the  vanes  at 
let  are  found  from 

8*15 
tan  <^  =  2&f:^UA '  ^^^  ^*^^^  *  "  ^^  ^ ' 

^^"^^  179 -^21-5  ^"^^  *!  =  113^50',     . 

8'15 
tan  4>i  =  157^24^5 »  ^^^  which  «^  =  137**  6'. 

2^he  inclination  of  the  guide  blade  at  each  of  the  three  radii. 

tan^  =  267' 
om  'which  ^  =  17', 

tan^i  =  j^  and  ^,=24^30', 

tan^5  =  ?~  and  ^2  =  27'*  30'. 
lo7 

The  inclination  of  the  vanes  at  exit, 

*''°"  =  14-4  =  29'36', 

tan  a,  =  5'^  =  20°  48', 

Zl  o 
tan  03=  1^^=18^  22'. 

If  now  the  lower  tips  of  the  guide  blades  and  the  upper  tips 

[)f  the  wheel  vanes  are  made  radial  as  in  the  plan,  Fig.  221,  the 

inclination  of   the  guide   blade  will   have   to   vary  from    17   to 

27^  degrees  or  else  there  will  be  loss  by  shock.     To  get  over  this 

diflBculty  the  upper  edge  only  of  each  guide  blade  may  be  made 

radial,  the  lower  edge  of  the  guide  blade  and  the  upper  edge  of 

each  vane,  instead  of  being  radial,  being  made  parallel  to  the 

npper  edge  of  the  guide.     In  Fig.  225  let  r  and  R  be  the  radii 

of  the  inner  and  outer  crowns  of  the  wheel  and  also  of  the  guide 

hlades.     Let  MN  be  the  plan  of  the  upper  edge  of  a  guide  blade 

and  let  DG  be  the  plan  of  the  lower  edge,  DG  being  parallel  to 

HX.    Then  as  the  water  runs  along  the  guide  at  D,  it  will  leave 

"the  guide  in  a  direction  perpendicular  to  OD.    At  G  it  will  leave 

in  a  direction  HG  perpendicular  to  OG.     Now  suppose  the  guide 

«M;  the  edge  DG  to  have  an  inclination  fi  to  the  plane  of  the  paper. 

H  then  a  section  of  the  guide  is  taken  by  a  vertical  plane  XX 

XKrpendicnlar  to  DG,  the  elevation  of  the  tip  of  the  vane  on  this 

X^e  will  be  AL,  inclined  at  P  to  the  horizontal  line  AB,  awd  AG 


»46 


HYDRAULICS 


will  be  the  intersection  of  the  plane  XX  with  the  plane  tangei 
to  the  tip  of  the  vane. 

Now  suppose  DE  and  GH  to  be  the  projections  on  the  plai 
of  the  paper  of  two  lines  lying  on  the  tangent  plane  AC  ao 
perpendicular  to  OD  and  OG  respectively.  Draw  EF  and  HI 
perpendicular  to  DE  and  GH  respectively,  and  make  each  < 
them  equal  to  BC.  Then  the  angle  EDF  is  the  inclination  of  tl 
stream  line  at  D  to  the  plane  of  the  paper,  and  the  angle  HGE : 
the  inclination  of  the  stream  line  at  G  to  the  plane  of  the  pape 
These  should  be  equal  to  0  and  0^, 


-JC-Z/fl; 


btcut&l.  of  upper  edg&  ofiwrvt 
Fig.  225.    Plan  of  guide  blades  and  vanes  of  Parallel  Flow  Tarbines. 

Let  y  be  the  perpendicular  distance  between  MN  and  DG 
Let  the  angles  GOD  and  GrOH  be  denoted  by  <^  and  a  respectively 

Since  EF,  BC  and  HK  are  equal, 

ED  tan^  =  y  tan)3 (1), 

and  GH  tan  ^2  =  y  tan  ^ (2). 


But 

and 

Therefore 
and 

Again, 


^  =  cos  (a  +  «^), 


JL 


=  cos  a. 


GH 

tan  B  -  cos  (a  +  <^)  tan  fi. 
tan  ^3  =  cos  a  tan  j3  

V 


sina  = 


R 


.(3), 
.(4). 

.(5). 


There  are  thus  three  equations  from  which  a,  ^  and  fi  can  ta 
determined. 

Let  X  and  y  be  the  coordinates  of  the  point  D,  O  being  tta 
intersection  of  the  axes. 


TURBINES 


347 


len 


cos  (a  +  ^)  = 


•om  (5) 


cosa 


bstitating  for  cos  (a  +  ^)  and  cos  a  and  the  known  values  of 
»id  tan  ^s  in  the  three  equations  (3 — 5),  three  equations  are 
led  with  X,  y^  and  fi  as  the  unknowns. 
Iving  simultaneously 

X  =  ri4  feet, 

y  =  2-23  feet, 
tan /3  =  0-67, 
«rhich  i8  =  34\ 


Fip.  226. 


Fig.  227. 


Fig.  228. 


e  length  of  the  guide  blade  is  thus  found,  and  the  constant 
at  the  edge  DG  so  that  the  stream  lines  at  D  and  G  shall 
he  correct  inclination. 

now  the  upper  edge  of  the  vane  is  just  below  DG,  and  the 
:  the  vane  at  D  and  G  are  made  as  in  Figs.  226 — 228,  ^  and 


:U^ 


HTDKJLUIJCS 


^  bein^  '^  '¥/  ami  I-TT  ^  respectively,  the  wmter  wiD  more  on  to 
the  vTkEie  wTtho'iit  «hf xrk. 

The  plane  •:•£  the  lower  edge  of  the  Tame  may  now  be  taken  ii 
VG\  riz,  22^^  and  che  cErcnlar  aectbos  DIX,  FQ,  and  GG'  at  th 
three  nuifi.  r,  r,  and  Tj  are  th^i  as  in  FigSL  226 — ^228. 

198.    BegoIationofthellowtopanaidlloirtiiiliiniaL 

To  regulate  the  dow  through  a  paralld  flow  tarbinesy  FontuM 
placed  Juices  in  the  guide  passagesy  as  in  Fig^.  229,  connected  to 
a  ring  which  could  be  raised  or  lowered  by  three  Tertical  lodi 
having  nuts  at  the  upper  ends  fixed  to  toothed  pinions.    What 


Fig.  32!^.     Fontaine's  Slniee*. 


Fig.  230.     Adjustable  guide  blades  for  Parallel  Flow  Turbine. 

the  sluices  required  adjustment,  the  nuts  were  revolved  together 
by  a  central  toothed  wheel  gearing  with  the  toothed  piniotf 
canying  the  nuts.  Fontaine  fixed  the  turbine  wheel  to  a  hoUoir 
shaft  which  was  carried  on  a  footstep  above  the  turbine.  In  son* 
modem  parallel  flow  turbines  the  guide  blades  are  pivoted,  as  in 
Fig.  230,  so  that  the  flow  can  be  regulated.  The  wheel  may  ^ 
made  v^-ith  the  crowns  opening  outwards,  in  section,  similar  to 
the  Girard  turbine  shown  in  Fig.  254,  so  that  the  axial  vdodtf 
with  which  the  water  leaves  the  wheel  may  be  small. 

The  axial  flow  turbine  is  well  adapted  to  low  falls  with  variable 
head,  and  may  be  made  in  several  compartments  as  in  Fig.  220. 
In  this  example,  only  the  inner  ring  is  provided  with  gates.  In 
dry  weather  flow  the  head  is  about  3  feet  and  the  gates  of  the 
inner  ring  can  be  almost  closed  as  the  outer  ring  will  give  the  full 


TURBINES  349 

»r.  During  times  of  flood,  and  when  there  is  plenty  of  water, 
head  faJls  to  2  feet,  and  the  sluices  of  the  inner  ring  are 
ed.  A  larger  supply  of  water  at  less  head  can  thus  be 
red  to  pass  through  the  wheel,  and  although,  due  to  the  shock 
le  guide  passages  of  the  inner  ring,  the  wheel  is  not  so  efficient, 
ibundance  of  water  renders  this  unimportant. 

cample,    A  doable  compartment  Jonval  turbine  has  an  outer  diameter  of 

and  an  inner  diameter  of  6  feet. 
le  radial  width  of  the  inner  compartment  is  1'  9"  and  of  the  outer  compart- 

1'  6".  Allowing  a  velocity  of  flow  of  8*25  ft.  per  second  and  supposing  the 
Dum  Call  is  V  Q^,  and  the  number  of  revolutions  per  minute  14,  find  the  hone- 
:  of  the  wheel  when  all  the  guide  passages  are  open,  and  find  what  portion  of 
iner  compartment  must  be  shut  off  so  that  the  horse-power  shall  be  the  same 
>  a  head  of  3  feet.  Efficiency  70  per  cent. 
>gleoting  the  thickness  of  the  blades, 

the  area  of  the  outer  compartments  j  (12-5'-9'5^  =  52*6  sq.  feet. 

„  „      inner  „  =^  (9'6»-6«)  =  42-8  sq.  feet. 

>tal  area=95-4  sq.  feet 

16  weight  of  water  passing  through  the  wheel  is 

W=95'4  X  62*4  X  3*25  lbs.  per  sec. 
=  19,800  lbs.  per  sec. 
he  horse-power  is 

19,800  X 1-66  X  0-7     .^.^ 

°^= 660 =  *^®- 

(saming  the  velocity  of  flow  constant  the  area  required  when  the  head 
eet  is 

40-8x83,000 

60x62-6x8x-7 
=55-6  sq.  feet, 
•  outer  wheel  will  nearly  develop  the  horse-power  required. 

99.  Bemouilli's  equations  for  axial  flow  turbines. 
Tie  Bemouilli's  equations  for  an  axial  flow  turbine  can  be 
ten  down  in  exactly  the  same  way  as  for  the  inward  and 
^rd  flow  turbines,  page  335,  except  that  for  the  axial  flow 
ine  there  is  no  centrifugal  head  impressed  on  the  water 
een  inlet  and  outlet. 

Tien,  P^l^^.P^^'f^h,, 

which,  since  v  is  equal  to  Vi, 

p     V*-2Vt?  +  t?'     u*     pi     r'~2ViV  +  V,^     t^»    , 

p     V     Vi;     t^'^p.     V.'^t^'     Y,v 
tore     — +0::  — Z"^  oZ^  ~'^  lyZ'^  oZ  — 1~  +  %> 


tc     2g       g      2g     w      2g      2g       g 
g        g       w     2g      2g      w      ^' 


330 


BrDRAin.lC3 


If  U«  ie  udal  and  equal  to  u,  as  in  Fig.  228, 

200.    Mix^d  flow  tturbinas* 

By  a  taodtiicatioti  of  tht*  sIuijh*  uf  the  viineei  of  an  jnwarl 
turbiiu*,  tht*  mixed  flow  turbine  in  obtained.  In  tie  inwwt 
outward  flow  turbine  the  water  only  acti*  upon  the  whfiel  wl 
is  raaving  in  a  radial  direction,  but  in  tho  mix<?d  flow  tarbi; 
viuies  are  so  formed  that  the  water  aotfi  u|)Oii  tliem  aiais 
flowing  axially. 


modern 
Fig.  230, 
made  with 
the  (iirard  tu 
w^th  which  tbt 
The  axial  fl 
head,  and  may  \ 
In  this  example, 
dry  weather  flo>^^  ^  diagrammatic  section  through  the  whee 
inner  rinff  can  b^^^^^^>  ^^®  ^^®  ^^  which  is  vertical.     Tlie  w 


Fig.  231.     Mixed  Flow  Turbine. 


k 


TU^KIS 


351 


tlie  wheel  in  a  horizontal  direction  and  leaves  it  ¥@rtically^|] 

leaTes    the   di^hargiiig  edge   of   the   vanes  in   different 

ions.      At  the   upper   part   B   it   leaves    the  vanes  nearly 

lly,  and  at  the  lower  part  A,  axially;     The  vanes  are  spoon- 

1,  BM  shown  in  Fig.  2*32,  and  should  be  so  formed,  or  in  other 

fc,  the  inclination  of  the  discharging  edge  gihould  so  vary, 

i^herever  the  water  leaves  the  vanes  it  should  do  so  with  no 

ment  in  a  direction  perpendicular  to  the  axis  of  the  turbine, 

ith  no  velocity  of  whirl.     The  regulation  of  the  supply  to 

b]  in  the  turbine  of  Fig,  231  is  effected  hy  a  cylindrical  ] 

'  speed  gate  between  tlie  fixed  guide  blades  and  the  wheel. 


Fig.  232.     Wht^l  i^f  Mixed  Flow  Turbme. 

Sg-  2*i*^  shows  a  section  through  the  wheel  and  casing  of  a 

le  tnixrd   flow   turbine  having  adjustable  guide  blades  to 

ate  the  How,     Fig,  234  shows  a  half  longitndinal  section  of 

rbine,  and  Fig.  23o  an  outside  elevation  of  the  guide  blade 

idng  gean    The  guide  blades  are  surrounded  by  a  lar^e 


852 


HYDRAULICS 


vortex  chamber,  and  the  outer  tips  of  the  guide  blades  are  of 
variable  shapes,  Fig.  283,  so  as  to  diminish  shock  at  the  entrance 
to  the  guide  passages.  Each  guide  blade  is  really  made  in  two 
parts,  one  of  which  is  made  to  revolve  about  the  centre  C,  while 
the  outer  tip  is  fixed.  The  moveable  parts  are  made  so  that  the 
flow  can  be  varied  from  zero  to  its  maximum  value.     It  will  be 


Fig.  233.     Section  through  wheel  and  guide  blades  of  Mixed  Flow  Turbine. 

noticed  that  the  mechanism  for  moving  the  guide  blades  is 
entirely  external  to  the  turbine,  and  is  consequently  out  of  the 
water.  A  further  special  feature  is  that  between  the  ring  B 
and  each  of  the  guide  blade  cranks  is  interposed  a  spiral  spring. 
In  the  event  of  a  solid  body  becoming  wedged  between  two  of 
the  guide  blades,  and  thus  locking  one  of  them,  the  adjustment  ol 
the  other  guide  blades  is  not  interfered  with,  as  the  spring  con- 
nected to  the  locked  blade  by  its  elongation  will  allow  the  linf 
to  rotate. 

As  with  the  inward  and  outward  flow  turbine,  the  mixec 
flow  turbine  wheel  may  either  work  drowned,  or  exhaust  into  I 
"suction  tube." 


^ 


TURBINES 


353 


For  a  given  flow,  and  width  of  wheel,  the  axial  velocity 
vitli  which  the  water  finally  flows  away  from  the  wheel  being  the 
tame  for  the  two  cases,  the  diameter  of  a  mixed  flow  turbine  can 
)e  made  less  than  an  inward  flow  turbine.  As  shown  on  page  340, 
;lie  diameter  of  the  inward  flow  turbine  is  in  large  measure  flxed 


{. 


JL 


=L 


ZFegt 


Fig.  234.    Half-loDgitudinal  section  of  Mixed  Flow  Tarbine. 

\gy  the  diameter  of  the  exhaust  openings  of  the  wheel.  For  the 
■ame  axial  velocity,  and  the  same  total  flow,  whether  the  turbine 
is  an  inward  or  mixed  flow  turbine,  the  diameter  d  of  the  exhaust 
<^>eniiig8  must  be  about  equal.  The  external  diameter,  therefore, 
of  the  latter  wiQ  be  much  smaller  than  for  the  former,  aivd  ticva 

L.   H.  'i^ 


354 


HTDRAUUGS 


general  dimensions  of  the  turbine  will  be  also  diminished, 
a  given  head  H,  the  velocity  v  of  the  inlet  edge  being  the  san 
the  two  cases,  the  mixed  flow  turbine  can  be  run  at  a  hi 
angular  velocity,  which  is  sometimes  an  advantage  in  dri 
dynamos. 


o 


a 


!2 
'a 


TURBINES 


355 


J'Wm  of  ths  varies.    At  the  receiving  edge,  the  direction  of  the 

is  found  in  the  same  way  as  for  an  inward  flow  turbine* 
ABC,  Fig,  236,  is  the  triangle  of  velocities^  and  BC  is  parallel 
the  tip  of  the  blade*    This  triangle  has  been  drawn  for  the  data 
he  turbine  shown  in  Figs.  23S— 235 ;  v  is  46*5  feet  per  secondj 
from 

9 

Y  =  33*5  feet  per  second* 
The  angle  ^  is  189  degrees. 


Triangles  of  V^ctHjeG 
at  raomrmff  fdge^ 

Fig.   236, 


The   best  form  for  the   vane  at  the  discharge  is  somewhat 

icult  to  determine,  as  the  exact  direction  of  flow  at  any  point 

the  discharging  edge  of  the  vane  is  not  easily  found.    The 

aditioii  to  be  satisfied  is  that  the  water  must  leave  the  wheel 

;  any  component  in  the  direction  of  motion* 
The  following  constmction  gives  approximately  the  form  of 
vane. 

Hake  a  section  through  the  wheel  as  in  Fig.  237.  The  outline 
the  discharge  edge  FG-H  is  shown.  This  edge  of  the  vane  is 
til  be  on  a  radial  plane,  and  the  plan  of  it  is,  therefore, 
f  Tsdins  of  the  wheels  and  upon  this  radius  the  section  is  taken* 
It  ifi  now  neoeasary  to  draw  the  form  of  the  stream  lines,  as 
would  be  approximatelyj  if  tlie  water  entered  the  wheel 
ly  and  flowed  out  axially,  the  vanes  being  removed* 
Ihvide  04,  Fig.  237,  at  the  inlet,  into  any  number  of  equal 

say  four,  and  subdivide  by  the  points  a,  i,  /i,  e. 
Tike  any  point  A,  not  far  from  Cj  as  centre,  and  describe 
circle  MMi  touching  the  crowns  of  the  wheel  at  M  and  M,. 
AM  and  AMi, 

]>i^w  a  flat  curve  Mi  Mi  touching  the  hues  AM  and  AM,  in  M 
Mt  respectivety,  and  k%  near  as  can  be  estimated,  perpendicular  j 


856 


HYDRAULICS 


to  the  probable  stream  lines  through  a,  6,  d,  e,  which  can 
sketched  in  approximately  for  a  short  distance  from  04. 

Taking  this  curve  MMi  as  approximately  perpendicular  to 
stream  lines,  two  points/  and  g  near  the  centres  of  AM  and  A 
are  taken. 


Fig.  287. 
Let  the  radius  of  the  points  g  and  /  be  r  and  n  respectively- 
If    any  point   Ci   on  MMi  is  now  taken    not  far  from  A,  tfe 
peripheral  area  of  Mci  is  nearly  2irrMci,  and  the  peripheral  art* 
of  MiCi  is  nearly  2w-riMiCi. 

On  the  assumption  that  the  mean  velocity  through  MiM  ^ 
constant,  the  flow  through  Mci  will  be  equal  to  that  throng* 
MiCi,  when, 

Mci.r  =  MiCi.ri. 


TURBINES 


357 


If,  therefore,  MM|  ib  divided  at  the  point  Cj  ao  that 

Mj_Ci  _  r 

thB  point  rj  will  approximateljr  be  on  the  stream  line  through  c. 

If  now  when  the  stream  line  cci  is  carefully  drawn  in^  it  is 
perpendicular  to  MMi,  the  point  c,  cannot  be  much  in  error. 

A  near€*r  approximation  to  d  can  be  found  by  taking  new  values 
for  r  and  n,  obtained  by  moving  the  points  /  and  g  bo  that  they 
more  tiearly  coincide  with  the  centres  of  CjM  and  cjMi.  If  the 
two  corves  are  not  perpend icularj  the  cui've  MM^  and  the  point  Ci 
ajre  not  quite  correct,  and  new  vsiluea  of  r  and  Ti  wiU  have  to  be 
obtained  by  moving  the  points  /  and  g.  By  approximation  Cj  can 
"be  thus  found  ^vith  considerable  accuracy. 

By  diawing  other  circles  to  touch  the  crown  of  the  wheelSj  the 
curv«i  MrM«,  M^Ma  etc.  normal  to  the  stream  lines,  and  the  points 
45i,  Cs,  etc.  on  the  centre  stream  line,  can  be  obtained. 

The  curve  22,  thereforej  divides  the  stream  lines  into  equal 
parf^ 

Proceeding  in  a  similar  manner,  the  curves  11  and  33  can  be 
'I'htaitied^  dividing  the  stream  lines  into  four  equal  parts,  and 
iii^-^  again  subdivided  by  the  curves  oa,  hh,  rfd,  and  ee,  which 
inttfrst»ct  the  outlet  edge  of  the  vane  at  the  points  F,  G,  H  and  e 
Tie*pectively. 

To  dei ermine  the  direction  of  the  tip  of  the  varw  at  points  mi  the 
<^i^chirg{ng  edge.  At  the  pcdiUs  F,  Gr,  H,  the  directions  of  the 
<to?am  lines  are  known,  and  the  velocities  tip,  t%,  uu  can  be  found, 
«mce  the  flows  through  01,  12,  etc,  are  equal,  and  therefore 


uvR^qt  =  Uii'RiTtui  =  tiuRiW't' 


Q 


8tr- 


Draw  a  tangent  FK  to  the  stream  line  at  F,  This  is  the  inter- 
*^*cti«ui,  with  the  plane  of  the  paper,  of  a  plane  perpendicular  to 
^  ^U'  p*»|>er  and  tangent  to  the  stream  line  at  F. 

The  piiint  F  in  the  plane  of  FK  m  moving  perpendiculai*  to  the 
Wane  of  the  paper  with  a  velocity  equal  to  *w,Rm,  «»*  being  the 
^n^kr  velocity  of  the  wheel,  and  H,  the  radius  of  the  point  F. 

If  a  circle  be  struck  on  this  plane  with  K  as  centre,  this  circle 
^ay  be  taken  as  an  imaginary  discharge  circumference  of  an 
*^^ward  Row  rnrbinej  the  velocity  v  of  which  is  mR^^  and  the  tip  of 
^^  =  '*  Made  i&  w  have  such  an  inclination,  that  the  water  shall 
'**>chiirgo  radially*  i.*;.  along  FK,  with  a  velocity  nr .  Turning  this 
'ff'cle  >nto  the  plane  of  the  paper  and  drawing  the  triangle  of 
^>--l^K.nties  FST,  the  inclination  a^  of  the  tip  of  the  blade  at  F  in 
^hm  pliiTie  FK  is  obtained, 


358 


HYDRAULICS 


At  G  the  stream  line  is  nearly  vertical,  but  «R»  can  be  set  ont 
in  the  plane  of  the  paper,  as  before,  perpendicular  to  uq  and  the 
inclination  oeq,  on  this  plane,  is  found. 

At  H,  an  is  found  in  the  same  way,  and  the  direction  of  the 
vane,  in  definite  planes,  at  other  points  on  its  outlet  edge*  can  be 
similarly  found. 


Fig.  288. 


Fig.  239. 

Sections  of  the  vane  by  planes  OGby  and  OiHd.  These  are 
shown  in  Figs.  238  and  239,  and  are  determined  as  follows. 

Imagine  a  vertical  plane  tangent  to  the  tip  of  the  vane  at 
inlet.  The  angle  this  plane  makes  with  the  tangent  to  the  whed 
at  b  is  the  angle  <^,  Fig.  236.    Let  BC  of  the  same  figure  be  the 


TURBINES 


359 


plan  of  a  liomontal  line  lying  in  this  plane,  and  BD  the  plan  of 
tbe  radius  of  the  wheel  at  6.     The  angle  between  these  lines  is  y. 

Liet  0be  the  inclination  of  the  plane  Oiyh  to  the  horizontal. 

From  D,  Fig,  236,  set  ont  DE,  inclined  to  BD  at  an  angle  ft 
aiul  intersecting  AB  produced  in  E,  and  draw  BF  perpendicular 
to  CB. 

Make  BF  equal  i<.\  BE  and  join  CF. 

Now  set  out  a  triangle  BGDj  ha^nng  BG  equal  to  CF,  DiG 
eqtial  to  DE,  and  the  angle  BGDj  a  right  angle*  In  the  figure 
Di  and  D  happen  to  coincide. 

The  angle  BGD  is  the  angle  yi,  which  the  line  of  intersection  of 
tte  plane  OG^,  Fig,  237,  witli  the  plane  tangent  to  the  inlet  tip  of 
tlie  vane,  makeB  irith  the  radius  O^. 

Ill  Fig-  238  the  inclination  of  the  inlet  tip  of  the  blade  is  yi  as 

To  detamiine  the  angle  a  at  the  outlet  edge,  resolve  tt©,  Fig, 
*2S7\  along  and  perpendicular  to  OS,  ti^  being  the  component 
along  (Xt, 

Draw  the  triangle  of  velocities  DEF,  Fig.  238, 
Tlte  tangent  to  the  vane  at  D  is  parallel  to  FE, 
In  the  same  way^  the  section  on  the  plane  Hti,  Fig.  237,  may  be 
letentiinetl ;  the  inclination  at  the  inlet  is  y^.  Fig,  239. 

MU*d  Jl^tt  hirhim  worki^ig  m  open  stream,  A  double  turbine 
rnrking  in  open  stream  and  discharging  tlirough  a  suction  tube 
shown  in  Fig.  2-1^^.  This  is  a  convenient  arrangement  for 
^^oderately  low  falls.  Turbines,  of  this  class,  of  1500  horne- 
p(n«rer,  having  four  wheels  on  the  same  shaft  and  working  under 
m  head  of  25  feet,  and  making  150  revolutions  per  minute,  have 
rt^'einly  tieen  installed  by  Messrs  Escher  Wyss  at  Wangen  an  der 
A  a  re  in  SwitxcTland, 


201.    Cone  ttirbltie. 

Another  type  of  inward  flow  turbine,  which  is  partly  axial  and 

ly  radial,  \b  shown  in  Fig.  241,  and  is  known  aa  the  cone 

Mne,     It  has  been  designed  by  Messrs  Escher  Wyss  to  meet 

\ie  demand  for  a  turbine  that  can  be  atlapt-ed  t«  variable  flow^s. 

l*he  example  shown  has  been  erected  at  Cusset  near  Lyons  and 
ikes  120  revolutions  jx?r  minute. 

Tlie  wheel  is  divided  into   three  distinct  compartments,  the 

applj'  of  Waaler  being  regulated  by  three  cylindrical  sluices  S,  Si 

S*.     The  sluices  S  and  8i  are  each  moved  by  three  vertical 

?^uch  as  A  and  Aj  which  carry  racks  at  their  upper  ends. 

_-    -wo  aluic^es  move  in  opposite  directions  and  thus  balance 

ch  oilier*    The  sluice  Sa  is  normally  out  of  actioni  the  upper 


360 


HYDRAULICS 


compartment  being  closed.  At  low  heads  this  upper  compartrxne 
is  allowed  to  come  into  operation.  The  sluice  Sj  carries  a  "ma 
which  engages  with  a  pinion  P,  connected  to  the  vertical  sha.  ^ 


Fig.  240. 

The  shaft  T  is  turned  by  hand  by  means  of  a  worm  and 
wheel  W.     When  it  is  desired  to  raise  the  sluice  Sg,  it  is  revolved 
by  means  of  the  pinion  P  until  the  arms  F  come  between  collar* 
D  and  E  on  the  spindles  carrying  the  sluice  Si,  and  the  sluice S» 
then  rises  and  falls  with  Si .     The  pinion,  gearing  with  racks  on  A 
and  Ai ,  is  fixed  to  the  shaft  M,  which  is  rotated  by  the  rack  B 
gearing  with  the  bevel  pinion  Q.     The  rack  R  is  rotated  by  two 
connecting  rods,  one  of  which  C  is  shown,  and  which  are  under 
the  control  of  the  hydraulic  governor  as  described  on  page  378. 
The  wheel  shaft  can  be  adjusted  by  nuts  working  on  the 
square-threaded  screw  shown,  and  is  carried  on  a  special  collar 
bearing  supported  by  the  bracket  B.    The  weight  of  the  shaft  is 
partly  balanced  by  the  water-pressure  piston  which  has  acting 
underneath  it  a  pressure  per  unit  area  equal  to  that  in  the  supply 
chamber.     The  dimensions  shown  are  in  millimetres. 


TDRBINES 


36t 


362 


HTDEAPLICS 


202.    Effect  of  changing  the  direction  of  the  guide  bladi|| 
when   altering   the  flow  of  inward   flow   and   mixed  floi 
turbine  3« 

As  long  as  the  velocity  of  a  wheel  remains  constant,  fhj 
backward  head  impreased  on  the  water  by  the  wheel  is  the  sam^ 
and  the  pressure  head,  at  the  inlet  to  the  wheel,  will  remm 
practically  constant  as  the  guides  are  moved^  The  velocity  d 
flow  U»  through  the  guides,  will,  thereforej  remain  constajit 
but  as  the  angle  ^,  which  the  guide  makes  with  the  tangent  to  the 
wheel,  diminishes  the  radial  component  u^  of  U,  diminishes^ 


Fig.  242. 

Let  ABC,  Fig.  242,  he  the  triangle  of  velocities  for  full  opariafi . 
and  suppose  the  inclination  of  the  tip  of  the  blade  i.^  made  paraM 
to  BC.  On  turning  the  guides  into  the  dotted  position*  tlie  incti* 
nation  being  «^'),  the  triangle  of  %^elocities  is  ABCi,  and  the  relati« 
velocity  of  the  water  and  the  periphery  of  the  wheel  is  now  BC^ 
which  is  inclined  to  the  vane,  and  there  is,  consequently,  loss  <te 
to  shock. 

It  vnW  be  seen  that  in  the  dotted  position  the  tips  of  the  gtt>«i* 
blades  are  some  distance  from  the  periphery  of  the  wheel  and  it  ii 
probable  that  the  stream  Hues  on  leaving  the  guide  blades  folio* 
the  dotted  curves  SS,  and  if  so,  the  inclination  of  these  ^tre^B 
lines  to  the  tangent  to  the  wheel  will  be  actually  gre-aler  thiiD^']» 
and  BCi  will  then  be  more  nearly  parallel  to  BC.  The  lt»ss  owtf 
be  approximated  to  as  follows : 

As  the  water  enters  the  wheel  its  radial  component  will  remwfi 
nnaltereilj  but  its  direction  will  be  suddenly  changed  from  flCita 
BCj  and  it-a  magnitude  to  BCj;   CiCt  is  drawn  parallel  to  Ai 
A  velocity  equal  to  dCs  has  therefore  to  be  suddenly  impr 
the  water* 

On  page  68  it  has  been  shown  that  on  oertain  assumptiuui  ^ 


TURBINES  863 

t  ^when  the  velocity  of  a  stream  is  suddenly  changed 
to  179  is 

2g      ' 
it  is  equal  to  the  head  due  to  the  relative  velocity  of 
?«. 

CiC  is  the  relative  velocity  of  BCi  and  BCs,  and  therefore 
1  lost  at  inlet  may  be  taken  as 

a  coefficient  which  may  be  taken  as  approximately  unity. 

.  Effect  of  diTniniiihiTig  the  flow  through  turbines  on 
Locity  of  exit. 

p-ater  leaves  a  wheel  radially  when  the  flow  is  a  maximum, 
lot  do  so  for  any  other  flow. 

angle  of  the  tip  of  the  blade  at  exit  is  unalterable,  and  if 
Uo  are  the  radial  velocities  of  flow,  at  full  and  part  load 
ively,  the  triangles  of  velocity  are  DBF  and  DBFi,  Fig.  243. 

part  flow,  the  velocity  with  which  the  water  leaves  the 
s  tt,.  K  this  is  greater  than  u,  and  the  wheel  is  drowned, 
3xhaust  takes  place  into  the  air,  the  theoretical  hydraulic 
3y  is  less  than  for  full  load,  but  if  the  discharge  is  down  a 

tube  the  velocity  with  which  the  water  leaves  the  tube  is 
iu  for  full  flow  and  the  theoretical  hydraulic  efficiency  is 

for  the  part  flow.  The  loss  of  head,  by  friction  in  the 
iue  to  the  relative  velocity  of  the  water  and  the  vane, 
8  less  than  at  full  load,  should  also  be  diminished,  as  also, 
3  of  head  by  friction  in  the  supply  and  exhaust  pipes, 
chanical  losses  remain  practically  constant  at  all  loads. 

Br^T ! i^A 


Fig.  243.  Fig.  244. 

fact  that  the  efficiency  of  turbines  diminishes  at  part  loads 
lerefore,  in  large  measure  be  due  to  the  losses  by  shock 
icreased  more  than  the  friction  losses  are  diminished, 
suitably  designing  the  vanes,  the  greatest  efficiency  of 

flow  and  mixed  flow  turbines  can  be  obtained  at  some 

of  full  load. 


304 


HTDBAULU^ 


204.    Begulaiion  of  the  £ow  by  cylindriGal  gata&. 

\\ lieu  tlie  &]>eed  of  the  turbine  is  adjusted  by  a  gate  betw«i 
the  guidefi  and  the  wheel,  and  the  fio\r  m  less  than  the  nomtftJ,  tk 
velfjtcity  XJ  with  which  the  water  leav^es  the  gmde  is  alteral  ic 
mBgnitade  bat  not  in  direction* 

Let  ABC  be  the  triangle  of  velocitiaSj  Fig*  244>  whem  the  iiowii 
nonimL 

IM  tht*  How  bo  diminished  until  the  velocity  with  wMchlit 
water  leRves  the  guides  is  Uoj  equal  to  AD, 


llien  BD  is  the 
rsdial  velocity  of  flfi 

Draw  DK  pan 
the  VMIB  a  sodden  reiociiyi 

the  water,  and  there  i»  a  ] 

To  keep  the  velocity 
intrmluoed  the  gate  shown 
connected  to  the  guide  bla 
bladee  as  well  as  tlie  gat«  & 


^r  of  Uo  and  r,  an*i  «s  ii  lil 

for  the  water  to  mmv  aii 
o  KD  must  be  im 
k(KDY 


equal  to 


% 


early  constant  Mr  Swain  ha 
245.  The  gate  g  i*  npif 
to  adjust  the  flow  tW  p\k 
L  The  effe<;tive  width  of  i| 
guides  is  thereby  inarle  approximately  pmpirtional  to  thetpiantitf 
of  flow,  and  the  velocity  D  remains  more  nearly  constant.  If  the 
gate  is  raised,  tlie  width  h  of  the  wheel  opening  will  be  greater 
than  t,  the  widtli  of  the  gate  opening,  and  the  radial  velocif)'*! 


Fig.  2iS.     SwalQ  Ga«. 


Fig,  246, 


TURBINES 


365 


wheel  will  consequently  be  less  than  the  radial  velocity  u 

guides.    If  n  is  assumed  constant  the  relative  velocity  of 

)r  and  the  vane  will  suddenly  change  from  BC  to  BCi, 

Or  it  may  be  supposed  that  in  the  space  between  the 

id  the  wheel  the  velocity  U  changes  from  AC  to  ACi. 

k  (CC,)^ 


OS8  of  head  will  now  be 


2flr 


The  form  of  the  wheel  vanes  between  the  inlet  and 
f  turbines. 

form  of  the  vanes  between  inlet  and  outlet  of  turbines 
»e  such,  that  there  is  no  sudden  change  in  the  relative 
of  the  water  and  the  wheel. 

ider  the  case  of  an  inward  flow  turbine.    Having  given 
3  the  vane  and  fixed  the  width  between  the  crowns  of  the 
le  velocity  relative  to  the  wheel  at  any  radius  r  can  be 
follows. 

any  circumferential  section  ef  at  radius  r,  Pig.  247.  Let 
?  effective  width  between  the  crowns,  and  d  the  effective 
"  between  the  vanes,  and  let  q  be  the  flow  in  cubic  feet 
id  between  the  vanes  Ae  and  B/. 


RelatiTe  Telocity  of  the  water  and  the  Tanes. 


Fig.  348. 


366  HTDRAUUCS 

The  radial  velocity  through  e/is 

Find  by  trial  a  point  O  near  the  centre  of  ef  such  that  a  circle 
drawn  with  0  as  centre  touches  the  vanes  at  M  and  Mi. 

Suppose  the  vanes  near  6  and  /  to  be  struck  with  arcs  of  cirdcB. 
Join  0  to  the  centres  of  these  circles  and  draw  a  curve  MCMi 
touching  the  radii  OM  and  OMi  at  M  and  Mi  respectively. 

Then  MCMi  will  be  practically  normal  to  the  stream  lines 
through  the  wheel.  The  centre  of  MCMi  may  not  exactly 
coincide  with  the  centre  of  e/,  but  a  second  trial  will  probably 
make  it  do  so. 

If  then,  h  is  the  effective  width  between  the  crowns  at  C, 
h  .  MMi .  ^v  =  q, 
MMi  can  be  scaled  off  the  drawing  and  Vr  calculated. 

The  curve  of  relative  velocities  for  varying  radii  can  then  be 
plotted  as  shown  in  the  figure. 


Fig.  249. 

It  will  be  seen  that  in  this  case  the  curve  of  relative  velocities 
changes  fairly  suddenly  between  c  and  A.  By  trial,  the  vanes 
should  be  made  so  that  the  variation  of  velocity  is  as  unifonn 
as  possible. 

If  the  vanes  could  be  made  involutes  of  a  circle  of  radius  &» 


( 


TURBINES  367 

in  Fig.  240,  and  tlie  crowns  of  the  wheel  parallel,  the  relative 
locity  of  the  wheel  and  the  water  would  remain  constant, 
lis  form  of  vane  is  however  entirely  unsuitable  for  inward 
>w  turbines  and  could  only  be  used  in  very  special  cases  for 
itward  flow  turbines,  as  the  angles  ^  and  0  which  the  involute 
akes  with  the  circumferences  at  A  and  B  are  not  independent, 
IT  from  the  figure  it  is  seen  that, 

8ind  =  ?5 
r 

nd  sin  ^  =  g^ , 

ainO     R 
r  -T— -T  =  -  . 

8m9      r 

The  angle  0  must  clearly  always  be  greater  than  <^. 

206.  The  llmitiTig  head  for  a  single  stage  reaction 
arbine. 

Reaction  turbines  have  not  yet  been  made  to  work  under  heads 
ligher  than  430  feet,  impulse  turbines  of  the  types  to  be  presently 
Lescribed  being  used  for  heads  greater  than  this  value. 

Prom  the  triangle  of  velocities  at  inlet  of  a  reaction  turbine, 
.g.  Fig.  226,  it  IB  seen  that  the  whirling  velocity  V  cannot  be 
greater  than 

V  +  u  cot  ^. 

Assuming  the  smallest  value  for  ^  to  be  30  degrees,  and  the 
oaximum  value  for  u  to  be  0*25  v  2grH,  the  general  formula 

9 
becomes,  for  the  limiting  case, 

viv-^2j3s/R)=e.g.R. 
K  t?  is  assumed  to  have  a  limiting  value  of  100  feet  per  second, 
rhich  is  higher  than  generally  allowed  in  practice,  and  e  to 
ye  0*8,  then  the  maximum  head  H  which  can  be  utilised  in  a  one 
(tage  reaction  turbine,  is  given  by  the  equation 

25-6H- 346  n/H  =  10,000, 
from  which  H  =  530  feet. 

207.  Series  or  multiple  stage  reaction  turbines. 
Professor  Osborne   Reynolds  has  suggested  the  use  of  two 

or  more  turbines  in  series,  the  same  water  passing  through  them 

successively,  and  a  portion  of  the  head  being  utilised  in  each. 

For  parallel  flow  turbines,  Reynolds  proposed  that  the  wheels 


S6S 


HYDRAUUCS 


and  fixed  blades  be  arranged  alternately  as  shown  in  Fig-  S 
This  arrangeinentj  although  not  used  in  water  turbines,  is 
largely  caed  in  3*eactioii  steam  turhinos. 


^^^^^^^ 

^ 

-^^*==^w 

J    J  E^'t^  ^  ^ 

J\ 

.^^^si<^^^ 

Fig.  260. 


ImihjtdL 

Figfl,  251,  252.     AiIaI  Flow  Impulse  TorlJim. 
*  Taken  from  Prof,  R^yuolda*  Scimtijis  Papyri,  VoL  i. 


TURBINES 


369 


Fig,  353. 


L     impulse  turbines. 

md  turbine.    To  overcome  the  difficulty  of  diimnution  of 

hf  with  diroinatioii  of  flow, 

(intToduced,  about  1850,  the 

krrmtion  or  partial  admission 

flead   of   the   water  beitig 

to  the  wheel  throughout 

jle  circumference  as  iu  the 

turbines,  in  the  Girard 

it  18  only  allowed  to  enter 

^1  through  guide  passages 

I  diametrically     oppcis^ite 

( as  shown  in  Figs,  2*52— 

tn  the  first  two,  the  flow  is 

knd  in  the  last  radiah 

[Fig,  252  abo%'e  the  guide  crown  are  two  quadrant-shaped 

l^nted  2  and  4^  which  are  made  to  rotate  about  a  vertical 

^Qmns  of  a  toothed  wheel.     When  the  gates  are  over  the 

tJB  2  and  4,  all  the  guide  passages  are  open,  and  by  turning 

in  the  direction  of  the  arrow,  any  desired  number  of  the 

can  be  clost^d.      In  Fig.  2*S4  the  variation  of  flow  i^ 

by  means  of  a  cylindrical  quadrant-shaped  sluice,  which, 

c  previous  case^  can  be  made  to  close  any  desired  number 

gnide  paseages*    Several  other  types  of  regulators  for 

turbines  were  introduced  by  Girard  and  others, 

.  25^^  shows  a  regulator  employed  by  Fontaine.     Above  the 

l>lade6,  and  fixed  at  the  opposite  ends  of  a  diameter  DD, 

ro  indianibber  bands,  the  other  ends  of  the  bands  being 

Sled  to  two  conical  rollers.    The  conical  rollers  can  rotate 

artialB,  formed  on  the  end  of  the  arms  which  are  connected 

coocbed  wheel  TVV,     A  pinion  P  gears  with  TW,  and  by 

Ig  the  spindle  carrying  the  pinion  P,  the  rollers  can  be  made 

irrap,  or  wT-ap  up,  the  indiarubber  band,  thus  opening  or 

m  the  guide  passages. 

I  the  Girard  turbine  is  not  kept  full  of  water,  the  whole  of 
reliable  head  is  converted  into  velocity  before  the  water 
the  wheel,  and  the  turbine  is  a  pure  impulse  turbine, 
prevent  loss  of  head  by  broken  water  in  the  wheel,  the  air 
.  be  freely  admitted  to  the  buckets  as  shown  in  Figs»  252 
A. 

small  heads  the  wheel  must  be  horizontal  but  for  large 
it  may  be  verticaL 
b  clase  of  turbine  has  the  disadvantage  that  it.  c^^^'&.c^'^ 


1 


370 


HYDBAUUCa 


rati  drowned,  and  tence  must  alwajB  b©  plaeod  aboTe  the 
water.    For  low  aod  variable  heads  the  full  head  cannot  thef©^ 
bii  utilisod,  for  if  the  wheel  is  to  be  clear  of  the  tail  water, 
amount  of  hmd  equal  to  half  the  width  of  the  wheel  mast 
necessity  be  lost. 


Fig»  26-t.     Girard  Hftdial  flow  Impulse  TurWoe, 


To  overGome  this  difficalty  Girard  placed  tJie  wheel 
tight  tube,  Fig.  254,  the  lower  end  of  which  is  below  ihc 
level,  and  into  which  air  is  pumped  hy  a  small  auxiliary 
the  pressure  being  maintained  at  the  necessary  value  t* 
surface  of  the  water  in  the  tube  below  the  wheel. 


TURBINES 


an 


t  H  be  the  total  head  above  the  tail  watar  level  of  the  snppl^ 

~  the  pressure  head  dae  to  the  atmospheric  preasare^  H, 

lee  of  the  centre  of  the  wheel  below  the  sarface  of  the 
i?atery  and  h^  the  distance  of  the  surface  of  the  water  in 
Inbe  below  the  tail  water  level.  Then  the  air-pressure  in 
tttbe  must  be 

W  ' 

the  head  causing  velocity  of  flow  into  the  wheel  is^  therefore, 

w  Xw       7 

b  that  wherever  the  wheel  is  placed  in  the  tube  below  the  tail 

the  full  fall  H  is  utilised. 
Ihis  system,  however,  has  not  foand  favour  in  practice,  owing 

difficulty  of  preserving  the  pressure  in  the  tube* 


The  form  of  the  vanee  for  impulfle  turbines,  neg- 
ng  fkictloiL 

Jhe  receiving  tip  of  the  vane  should  be  parallel  to  the  relative 
Atf  Vr  of  the  water  and  the  edge  of  the  vane,  Fig<  255. 
t  exit  the  relative  velocity  iv,  Fig.  256,  neglecting  friction, 
be  equal  to  the  relative  velocity  V,.  at  inlet, 
'  the  angle  a  which  the  tip  of  the  vane  at  exit  makes  with 
lirectiou  of  Vi  is  known  the  triangle  of  velocities  can  be  drawn, 
fffeting  out  DE  equal  to  Vj  and  EF  at  an  angle  a  with  it  and 
1  to  Vr.  Then  DF  is  the  velocity  with  which  the  water  leaves 
irbeeh 

W  the  aidal  flow  turbine  Vi  equals  t*,  and  the  triangle  of 
ntiee  at  e3dt  is  AGB,  Fig.  255. 

t  the  velocity  with  which  the  water  leaves  the  wheel  is  Ui, 
iieorotjcal  hydraulic  eflSciency  is 


E- 


H      "^      IP 


idependent  of  the  direction  of  Uj . 

»iild  be  observed,  however,  that  in  the  radial  flow  turbine 

of  the  section  of  the  stream  by  the  circuniference  of  the 

for  a  given  flow,  will  depend  upon  the  radial  component  of 

\xd  in  the  axial  flow  turbine  the  area  of  the  section  of  the 

by  a  plane  perpendicular  to  the  axis  will  depend  upon  the 

I  component  of  Uj,    That  is,  in  each  case  the  area  ^vill  depend 

the  component  of  Ui  perpendicular  to  Vi. 


372 


HYDRAULICS 


Now  the  section  of  tlie  stream  most  not  fill  the  outlet  area''o{ 
the  wheels  and  the  minininTn  area  of  this  outlet  so  that  it  is  just 
not  filled  will  clearly  be  obtained  for  a  given  value  of  XJi  when  Ui 
is  perpendicular  to  Vi*,  or  is  radial  in  the  outward  flow  and  axial  in 
the  parallel  flow  turbine. 

For  the  parallel  flow  turbine  since  BC  and  B6,  Fig.  255,  are 
equal,  Ui  is  clearly  perpendicular  to  Vi  when 

v  =  ^  =  ^'J2gRcosO, 

and  the  inclinations  a  and  ^  of  the  tips  of  the  vanes  are  equaL 


Figs.  255,  256. 


Fig.  257. 


If  R  and  r  are  the  outer  and  inner  radii  of  the  radial  flow 
turbine  respectively, 

R 
r 

*  It  is  often  stated  that  this  is  the  condition  for  mazimam  effieieiiOT  bat  it  ctif 
is  80,  as  stated  above,  for  mazimam  flow  for  the  given  maohine.    Tat 
only  depends  upon  the  magnitude  of  Ui  and  not  apon  its  direotion. 


Far  Ui  ta  be  radial 


TURBINES 

Yr~Vi  sec  a 


373 


sec  a. 


Y  Y 

and  if  u  is  made  equal  to ^,  Yr  from  Fig,  255  is  equal  to  ^  sec  <^a 

and  therefore, 

sec  ot  =  ^  «©c  9. 

210.  Txian^es  of  velocity  for  an  axial  flow  Impulsa  tur- 
bine confiidering  friction* 

The*  velocity  witb  which  the  water  leave©  the  guide  passages 
may  be  takeu  as  from  0'94r  to  0'97  \^2^H,  and  the  hydraulic  losses 
ia  tlie  wheel  are  from  5  to  10  per  ceut* 

If  the  angle  between  the  jet  and  the  direction  of  motion  of  the 
vane  is  taken  as  30  degrees,  and  U  ia  assumed  as  0"95  n/2^H,  and  v 
ms  0*45  v^2gH,  the  triangle  of  veloeitiL^s  is  ABC,  Fig.  257. 

lUdng  10  per  cent,  of  the  head  as  being  lost  in  the  wheel^  the 
relative  velocity  tv  at  ©sit  can  be  obtained  from  the  expression 

%      ^  

K  now  the  velocity  of  exit  Ui  be  taken  as  0'22s/2^H,  and 
circles  with  A  and  B  as  centres,  and  Ui  and  iv  as  radii  be 
described,  intersecting  in  D,  ABD  the  triangle  of  velocities  at  exit 
is  obtaijiedj  and  U]  is  practically  axial  as  shown  in  the  figure. 
On  these  assumptions  the  best  velocity  for  the  rim  of  the  wheel  is 
iber<?fore  "45  •J2gB.  instead  of  *5  *J2gR. 

The  head  lost  due  to  the  water  leaving  the  wheel  with  velocity 
u  i»  *M8H^  and  the  theoretical  hydraulic  efficiency  is  therefore 
^"2  per  cent. 

The  velocity  head  at  entrance  is  0*9025H  and,  therefore,  *097H 
lm&  been  lost  when  the  water  enters  the  wheel. 

The  efficiency^  neglecting  axle  friction,  will  be 

H  -  O'lH  - 0O48H  - O'OQTH 
e  = g 

=  76  per  centt  nearly. 

21L    Imptdse  turbine  for  bigli  beads. 

For  high  heads  Girard  introduced  a  form  of  impulse  turbine, 
of  which  the  turbine  shown  in  Figs.  258  and  259,  is  the  modem 
deirelopment. 

J        The  water  instead  of  being  delivered  through  guides  over  an 
mrc  of  a  circle,  is  delivered  through  one  or  more  adjustable  nozzles. 


^ib 


TURBINES 


876 


pie  shown,  the  wheel  has  a  mean  diameter  of  6*9  feet 
-revolations  per  minute;  it  develops  1600  horse- 

lead  of  1935  feet. 

pipe  is  of  steel  and  is  1'312  feet  diameter. 

the  orifices  has  been  developed  hy  experience,  and 

)re  is  no  sadden  change  in  the  form  of  the  liquid 

juently  no  loss  due  to  shock. 

of  water  to  the  wheel  is  regulated  by  the  sluices 

258,  which,  as  also  the  axles  carrying  the  same, 
the  orifices,  and  can  consequently  be  lubricated 

ae  is  at  work.    The  sluices  are  under  the  control 

►vemor  and  special  form  of  regulator. 

d  of  the  turbine  tends  to  increase  the  regulator 

11  crank  lever  and  partially  closes  both  the  orifices. 

I  speed  of  the  turbine  causes  the  reverse  action  to 

igh  peripheral  speed  of  the  wheel,  205  feet  per 
?s  a  high  stress  in  the  wheel  due  to  centrifugal 
ng  the  weight  of  a  bar  of  the  metal  of  which  the 
le  square  inch  in  section  and  one  foot  long  as 
tress  per  sq.  inch  in  the  hoop  surrounding  the 


/= 


3-36.  t;^ 
9 


=  4400  lbs.  per  sq.  inch. 

iger  of  fracture,  steel  laminated  hoops  are  shrunk 

ery  of  the  wheel. 

arrying  the  blades  is  made  independent  of  the  disc 

that  it  may  be  replaced  when  the  blades  become 
ri  entirely  new  wheel  being  provided. 

of  the  vanes  at  the  inner  periphery  is  171  feet  per 
herefore,  0*484  v2gH. 

ity  U  wth  which  the  water  leaves  the  orifice  is 
5(7 H,  and  the  angle  the  jet  makes  with  the  tangent 
K)  degrees,  the  triangle  of  velocities  at  entrance  is 
and  the  angle  <f>  is  53'5  degrees. 

1*1  of  the  outer  edges  of  the  vanes  is  205  feet  per 
iming  there  is  a  loss  of  head  in  the  wheel,  equal  to 


'^9 


2^ 


-0-06H, 


=  123*5  ft.  per  second. 


876 


HTDRAULIGS 


If  then  the  angle  a  is  30  degrees  the  triangle  of  velocities  ! 
exit  is  DBF,  Fig.  261.   . 

The  velocity  with  which  the  water  leaves  the  wheel  is  th< 
Ui  =  95  feet  per  sec.,  and  the  head  lost  hy  this  velocity  is  140  fe 
or  -OTSH. 


Fig.  260. 


Fig.  261. 


The  head  lost  in  the  pipe  and  nozzle  is,  on  the  assumpti( 
made  above, 

H-(0-97)»H=0-06H, 

and  the  total  percentage  loss  of  head  is,  therefore, 

6  +  7-3  +  6  =  19-3, 
and  the  hydraulic  efficiency  is  80*7  per  cent. 


Fig.  262.     Pelton  Wheel. 


TURBINES 


377 


The  actual  efficiency  of  a  similar  turbine  at  full  load  was  found 
by  experiment  to  be  78  per  cent.,  which  allows  a  mechanical  loss 
of  2*7  per  cent. 

212.    Pelton  wheel. 

A  form  of  impulse  turbine  now  very  largely  used  for  high  heads 
is  known  as  the  Pelton  wheel. 

A  number  of  cups,  as  shown  in  Figs.  262  and  266,  is  fixed  to  a 
wheel  which  is  generally  mounted  on  a  horizontal  axis.  The 
water  is  delivered  to  the  wheel  through  a  rectangular  shaped 
nozzle,  the  opening  of  which  is  generally  made  adjustable,  either 
by  means  of  a  hand  wheel  as  in  Fig.  262,  or  automatically  by  a 
regulator  as  in  Fig.  266. 

As  shown  on  page  276,  the  theoretical  efficiency  of  the  wheel  is 
unity  and  the  best  velocity  for  the  cups  is  one-half  the  velocity  of 
the  jet.  This  is  also  the  velocity  generally  given  to  the  cups 
in  actual  examples.  The  width  of  the  cups  is  from  2J  to 
4  times  the  thickness  of  the  jet,  and  the  width  of  the  jet  is  about 
twice  its  thickness. 

The  actual  efficiency  is  between  70  and  82  per  cent. 

Table  XXXVIII  gives  the  numbers  of  revolutions  per  minute, 
the  diameters  of  the  wheels  and  the  nett  head  at  the  nozzle  in 
a  number  of  examples. 

TABLE  XXXVIII. 
Particulars  of  some  actual  Pelton  wheels. 


Head 

infect 

Diameter 

of  wheel 

(two  wheels) 

Bevolations 
per  minute 

r 

U 

H.  p. 

262 

89-4" 

875 

64-5 

129 

500 

*233 

7" 

2100 

64 

125 

5 

*197 

20" 

650 

66-5 

112 

10 

722 

89" 

650 

111 

215 

167 

882 

60" 

800 

79 

156 

144 

♦289 

54" 

810 

73 

186 

400 

508 

90" 

200 

79 

180 

300 

•  Pieard  Pictet  and  Co.,  the  remainder  by  Eschar  Wyss  and  Co. 

213.    Oil  pressure  governor  or  regulator. 

The  modem  applications  of  turbines  to  the  driving  of  electrical 
^Xiachinery,  has  made  it  necessary  for  particular  attention  to  be 
X^^d  to  the  regulation  of  the  speed  of  the  turbines. 

The  methods  of  regulating  the  flow  by  cylindrical  speed  gates 
^»^:id  moveable  guide  blades  have  been  described  in  connection  with 


378 


HTBRAULICS 


various  turbines  but  the  means  adopted  for  moving  the  gatae  i 
gtiidt?8  have  not  been  discussed,  ^M 

Until  recent  years  some  form  of  differential  goven^H 
almoat  entirely  used,  but  these  have  been  almost  corople 
superseded  by  hydranlic  and  oil  governors. 

Figs.  26^3  and  264  show  an  oil  governor,  aa  construct 
Messrs  Escher  Wym  of  Zurich.  ~ 


FigB.  263.  Mi.     Oil  PresBure  Begalator  for  TarbineB. 


A  piston  P  having  a  larger  diameter  at  one  end  than  \ 
other^  and  fitted  with  leathers  I  and  ^i,  fits  into  a  double  cylbS 
Ci.  Oil  under  pressure  m  continuously  supplied  through  app* 
into  the  annulus  A  between  the  pistons,  while  at  the  back  of  \ 
iarge  piston  the  pressure  of  the  oil  is  determined  by  the  refuU 


TURBINES 


Fig,  265. 


^ppo^  the  regulator  to  be  in  a  definite  poBitioiij  the  spacej 
»Iiinti  the  krge  piston  being  full  of  oil,  and  the 
rbine  rtinning  at  its  normal  speed.  The  valve  Y 
enlarged  diagrammatic  section  is  ehowii  in 
1 265)  will  be  in  such  a  position  that  oil  cannot 
or  e^icape  from  the  large  cylinder,  and  the 
sure  in  the  annular  ring  betfcveen  the  pistons 
ill  keep  the  regulator  mechanism  locked. 
If  the  wheel  increases  in  speed,  due  to  a 
inntioD  of  load,  the  balls  of  the  spring  loaded 
remor  G  move  outwards  and  the  sleeve  M 
For  the  moment,  the  point  D  on  the  lever 
is  fixed,  and  the  lever  turns  about  D  as  a 
ftilerum,  and  tljus  raises  the  valve  rod  KV,  This 
Howe  oil  under  pressure  to  enter  the  large 
blinder  and  the  piston  in  consequence  moves  to 
ft  '  - 1  moves  the  turbine  gates  in  the  manner  descnbed  later, 

Hi  moves  to  the  right,  the  rod  R,  which  rests  on  the 
^edge  W  connected  to  the  piston,  falls,  and  the  point  D  of  the 
Bver  MD  consequently  falls  and  brings  the  valve  V  back  to  its 
iginal  ptjsition.  The  piston  P  thus  takes  np  a  new  position 
:irresponding  to  the  required  gate  opening.  The  speed  of  the 
irbine  and  of  the  governor  is  a  little  higher  than  before,  the 
>  in  speed  depending  upon  the  sonsitiverjess  of  the  governor* 
the  other  hand,  if  the  speed  of  the  wheel  diminialies,  the 
M  and  also  the  valve  V  falls  and  the  oil  from  behind  the 
irge  piston  escapes  through  the  exhaust  E,  the  piston  moving 
the  left.  The  wedge  W  then  lifts  the  fulcrum  D,  the  valve  V 
I  automatically  brought  to  its  central  position^  and  the  piston  P 
ikes  up  a  new  position,  consistent  with  the  gate  opening  being 
ifficient  to  supply  the  necessary  water  required  by  the  wheeh 
A  hand  wheel  and  screw.  Fig*  264,  are  also  pro\^ded,  so  that 
gates  can  be  moved  by  hand  when  necessary. 
Tlie  piston  P  is  connected  by  the  connecting  rod  BE  to  a  crank 
^f  which  rotates  the  vertical  shaft  T.  A  double  crank  KK  is 
OBnected  by  the  two  coupling  rods  shown  to  a  rotating  toothed 
wheel  R,  Fig.  241^  turning  about  the  vertical  shaft  of  the  turbine, 
^and  the  movement,  as  described  on  page  360,  causes  the  adjusts 
&nt  of  the  speed  gates. 


214.    Water  pressure  regulators  for  impulse  turbines. 
Fig.  266  shows  a  water  pressure  regulator  as  applied  to  regulate 
fche  flow  to  a  Pelton  wheel, 

Tlie  ar^a  of  the  supply  noazle  is  adjusted  by  a  beak  B  which 


TURBINSS 


381 


:  the  centre  O.  The  pressare  of  the  water  m  the 
icting  on  this  beak  tends  to  lift  it  and  thos  to  <ipei] 
The  piston  P,  working  in  a  cylinder  C,  is  also  acted 
mder  side,  by  the  pressare  of  the  water  in  the  supply 
connected  to  the  beak  by  the  connecting  rod  D£L 
:he  piston  is  made  sofficiently  large  so  that  when  the 
{ton  is  relieved  of  pressare  the  pmll  on  the  coDnecting 
nt  to  close  the  orifice. 

p  conveys  water  nnder  the  same  presBore.  Uj  the 
;h  maybe  similar  to  that  described  in  ocmnectKiD  with 
re  governor,  Fig.  265. 

rod  passes  throngh  the  top  of  the  cylinder.and  carries 
screws  on  to  the  square  thread  cut  on  the  r>d.  A 
268,  which  is  carried  on  the  fixed  f  alcnuii  <:,  iut  made 
1  the  piston.  A  link  /A  oonnect£  ef  with  the  lerer 
M  of  which  moves  with  the  governor  d*jei'e  and  the 
is  connected  to  the  valve  rod  XV.  Tlje  vaJie  V  m 
neutral  position. 


■o; 


"Sleteve 


(a 


now  the  speed 


of    til*:    turf^;jj*r    Vj    iu*:r*ii»*i^.      TLe 

the  lever  MS  tumfc  i^'yjvi  ti**:  f ulcmin 

The  va;v*:  V  laii*s  aiid  o^iens  the 

Til*:  ;;r<i*«ur»r  Oli  th*-  pist^.iij 

the   U'/ZZ^l^:    \VU^    di»iM;iKnTr,p 

he  puRogu  nw^  :t  L-ft^^  a^aiii  tht- 

A/f  aod   C:io«*  th*:  va}Tf  T.      J^ 

vveacbefd.    If  the  igteed  cf  the 


382 


HTBHAULICS 


governor  decreases  the  governor  sleeve  falls,  th©  valve  T 

and  wfLter  pressure  ia  admitted  to  the  top  of  the  piaton,  which  i 
then  in  equihbriuin,  and  the  pressure  on  the  beak  B  causes  it  i 
move  upwards  and  thus  open  the  noaszle* 

Hydraulic  ^^alve  for  water  regulatfjr*  Instead  of  the  simple 
piston  valve  controlled  mechanically,  Messi*s  Escher  Wyss  use,  fo 
hi^h  heads,  a  hydraulic  double-piston  valve  Pp,  Fig*  260. 

This  piston  valve  has  a  small  bore  through  its  centre  by  meansi 
of  which  high  pressure  water  which  is  admitted  below^  the  valve  j 
can  pass  to  the  top  of  the  large  piston  P,    Above  the  piston  is  \ 
small  plug  valve  Y  which  is  opened  and  closed  by  the  governor. 


Fig.  S69*     Hjdranlio  valve  for  autornKtle  regalfttbn. 

If  the  speed  of  the  governor  decreases,  the  valve  V  is  open^ 
thus  allowing  water  to  escape  from  above  the  piston  valve,  and  tif 
pressure  on  the  lower  piston  p  raises  the  valve.  Pressure  water  if 
thus  admitted  above  the  regulator  piston^  and  the  prefisiire  on  tk 
beak  opens  the  nozzle*  Aa  the  governor  falls  the  vslre  V  c]n(S«» 
the  exhaust  is  throttledj  and  the  pressure  above  the  piston  P  riitf^ 
"WTien  the  exhaust  through  V  is  throttled  to  such  a  degree  dm 
the  pressure  on  P  balances  the  pressure  on  the  under  face  of  tkt 
piston  pf  the  valve  is  in  equilibrium  and  the  regulator  pfetoft  tf 
Joeked* 


rittHy 


TUHBINKS 


383 


the  speed  of  the  ^Fernor  increases,  the  valve  V  is  closed^ 
and  tlie  excess  pressure  on  th^  tipper  face  of  the  pieton  valve 
eaojM^s  it  to  descend,  thus  connecting  tlie  regulator  cylinder  to 
ejEhatist.  The  pressure  on  tlie  under  face  of  the  regulator  piston 
tbem  closes  the  uozzle. 

Filter.  Between  the  conduit  pipe  and  the  governor  valve  V, 
is  placed  a  filter,  Figs.  270  and  271,  to  remove  any  sand  or  grit 
contained  in  the  water. 

Wit  111  n  the  cylinder,  on  a  hexagonal  fi-anje,  is  stretched  a 
pn'eee  of  canvas.  Tlie  water  enters  the  cylinder  by  the  pipe  E,  and 
after  passing  through  the  canvas^  enters  the  central  perforated 
pipe  and  leaves  by  the  pipe  S. 


FiffK  370,  27L    Water  Filter  for  Impuke  Turbine  Reguktor. 

To  clean  the  filter  while  at  work,  the  canvas  frame  is  revolved 
by  me^ns  nf  the  handle  shown,  and  the  cock  R  is  opened.  Each 
mde  of  tlie  hexagonal  frame  is  brought  in  turn  opposite  the 
chamber  A,  and  water  tlowB  outwards  through  the  canvas  and 
ibroa^h  the  cock  R^  carrying  away  any  dirt  that  may  have 
collected  outside  the  canvas. 

Af^iliary  valve  to  prei^eni  hammer  action.  When  the  pipe  line 
^  long  an  auxiliary  valve  is  frequently  fitted  on  the  pipe  near  to 

nozzlet  which  is  anfcomatically  opened  by  means  of  a  catai^ct 
motion*  as  the  nozzle  closes,  and  when  the  movement  of  the  nozxle 
beak  is  finished^  the  valve  slow^ly  closes  again. 

If  no  such  provision  is  made  a  rapid  closing  of  the  nozzle 
ixieans  that  a  large  mass  of  water  must  have  its  momentum 

'^ly  changed  and  very  large  pressures  may  be  set  up,  or  in 
:  words  hammer  action  is  produced,  which  may  cause  fracture 
of  the  pipe. 

WTien  there  is  an  abundant  supply  of  water,  the  auxiliary 
valve  is  connected  to  the  piston  rod  of  the  regulator  and  opened 
and  elo3*4*d  as  the  piston  rod  moves,  the  valve  being  adjusted  so 
tliat  the  opening  increases  by  the  same  amount  that  the  area  of 
tbe  orifice  dimiiiishes* 

*  See  Enffin^ttt  VoU  ic,  p.  265, 


384 


HYDRAULICS 


If  the  load  an  the  wheel  does  not  vmiy  through  a  large  m 
the  quantity  of  water  wasted  is  not  large. 

215.    Hammer  blow  in  a  long  turbine  supply  pipe. 
Let  L  be  the  length  of  the  pipe  and  d  its  diameter. 
The  weight  of  water  in  the  pipe  is 


4 


Let  the  velocity  change  by  an 

rate  of  change  of  momentum  is 

the  lower  end  of  the  column  of  w 
be  applied  equal  to  this. 

Therefore 


"-}• 


^nnt  dv  in  time  cL  Then  &9 
-J  and  on  a  cross  sectioii  rf 
'  in  the  pipe  a  force  P  tmat 

ft' 


lepth  of  the  orifice  and  di  it« 
1  the  centre  about  which  UitS 


Referring  to  Fig.  266,  let  b  be 
width. 

Then,  if  r  is  the  distance  of  1 
beak  turns,  and  r,  h  the  distance  ot  the  elf  ising  edge  oi  the  ^►ejik 
from  this  centre,  and  if  at  any  moment  the  velocity  of  the  piston 
is  r»  feet  per  sei^ond,  the  velocity  of  closing  of  the  beak  ^nll  be 

r 

In  any  small  element  of  time  ct  the  amount  by  which  the 
nozzle  will  close  is 


BJ- 


r^Fi 


Bt 


Let  it  1^  ass^umed  that  U*  the  velocity  of  flow  through  tis 
nozzle,  remains  constant.  It  will  actually  vary,  due  to  tk 
ivs^istances  varying  with  the  vel^xnty,  hut  unless  the  pipe  is  very 
long  the  errv^r  is  n*>t  gr^^t  in  neglecting  the  variation.  If  then  r 
is  the  velocity  in  the  pipe  at  the  commencement  of  this  element  of 
time  and  r  -  c  r  at  the  end  of  it,  and  A  the  area  of  the  pipe, 

r-A=fr.d,.U ......a) 


and  (r-cr>A-(t-^d(yd^.U 

Subtracting  \2i  tnmi  (I), 

r 


{il 


ct  '' 


r     A 


.(3). 


TURBINES  385 

If  W  is  the  weight  of  water  in  the  pipe,  the  force  P  in  pounds 
it  will  have  to  be  applied  to  change  the  velocity  of  this  water 
cv  in  time  dt  is 

g   of 
Therefore  p^Wr.^o 

id  the  pressure  per  sq.  inch  produced  in  the  pipe  near  the 
ozzle  is 

W  r,  diUi^o 
^  =  7r"A»    • 
Suppose  the  nozzle  to  be  completely  closed  in  a  time  t  seconds, 
nd  during  tlie  closing  the  piston  P  moves  with  simple  harmonic 
notion. 

Then  the  distance  moved  by  the  piston  to  close  the  nozzle  is 

br 

md  the  time  taken  to  move  this  distance  is  t  seconds. 
The  maximum  velocity  of  the  piston  is  then 

vbr 

ad  substituting  in  (3),  the  maximum  value  of  -r  is,  therefore, 

dv_  vbrridiU 
ot"    2trirA  ' 
ad  the  maximum  pressure  per  square  inch  is 

^*         2gtA'  2g.t.A'     2t' gA' 

here  Q  is  the  flow  in  cubic  feet  per  second  before  the  orifice 
e^an  to  close,  and  v  is  the  velocity  in  the  pipe. 

ExamvU,  A  500  horse-power  Pel  ton  Wheel  of  75  per  cent,  efficiency,  and  worki  ng 
Bder  a  head  of  260  feet,  is  sapplied  with  water  by  a  pipe  1000  feet  long  and 
8"  diameter.  The  load  is  suddenly  taken  off,  and  the  time  taken  by  the 
igtilator  to  close  the  nozzle  completely  is  6  seconds. 

On  the  assumption  that  the  nozzle  is  completely  cloned  (1)  at  a  uniform  rate, 
id  (2)  with  simple  harmonic  motion,  and  that  no  relief  valve  is  provided, 
dtermine  the  pressure  produced  at  the  nozzle. 

The  quantity  of  water  delivered  to  the  wheel  per  second  when  working  at  fall 
Dwer  is 

^  500x33,000  „  _      , .    ,    , 

Q=260x62-4x'76x60=^^-^  <="'»*  '*«'• 

The  weight  of  water  in  the  pipe  is 

W = 62-4 X  J.  (2-25)5x1000 
=  250,000  lbs. 
L.  H.  ^"^ 


386 


The  Telodty  is  ^^  =  5-^6  ft.  per  seo. 


In  case  (1)  the  total  pT«sffii]:«  acting  on  the  hyvet  en^  of  thm  4»lixmii  nf  «il»| 

the  pipe  is 

=  8200  lbs. 
The  presson  per  sq.  inch  ia 

y  =  -^14  5  Iba.  per  sq,  iiioh^ 

w   W   I? 
In  case  (2)  p^=^  ^  -^^  =22-1       u  per  wq. 


exam:      s. 


(I)  Find  tteHMMWiic^l  horne^pow* 
diameter,  usingt  9XCXXM)00  gailons  of  m 
of  25  feet. 


!  an  oversliot  watet-wbed  S3  &et  | 
•  per  24  hours  under  a  IxM  had 


(2)     An  over^ot  water-wheel  has  i        imetez^  of  24  Ceet,  and  ma^aif^ 

revolutions  per  minute.     The  vekx'ity  oi  trie  water  a.^  it  enters  the  bucketi 
is  to  be  twice  that  of  the  wheel" a  periphery. 

If  the  angle  wliicli  the  water  make«  with  the  periphery  is  to  be  U 
degrees,  find  the  fhrection  of  the  tip  of  the  bucket,  and  the  relkfeive  velodty 
of  the  water  and  the  bucket. 

(8)  The  sluiee  of  an  overshot  water-wheel  12  feet  diameter  \&  vertioBj 
above  the  centre  of  the  wheel.  The  surface  of  the  water  in  the  shm* 
channel  is  2  feet  6  ineheH  above  the  top  of  tlje  wheel  and  the  centre  of  Uae 
sluice  o{)ening  is  B  iuclies  above  the  top  of  thc^  wheel.  The  velocity  of  ^ 
wheel  periphery  is  to  be  one-half  that  of  the  water  aa  it  enters  the  backets 
Determine  the  nuuibt^r  of  rotationw  of  the  whoeit  the  point  at  wludi  the 
water  enters  the  buckets,  and  tlie  direction  of  the  edge  of  tiie  bucket. 

(4)  An  overt^iu^t  wheel  25  feet  diaujeter  having  a  widtJi  of  5  feeUMui 
depth  of  crowns  12  inches,  receiveit  450  cubic  feet  of  water  per  minute^  ao(i 
makes  6  revolutions  jier  minute.     There  are  64  buckets. 

The  water  enters  the  wheel  at  15  degreca  from  the  cro^Ti  of  the  wlieel 
witli  a  velocity  e^^iial  to  twice  that  of  the  periphery,  and  at  an  angle  of  D 
degrees  with  the  tan  gent  to  the  wheel. 

Assuming  the  buckets  to  be  of  the  form  shown  in  Fig,  180,  the  lengtb 
of  the  radial  i>ortiou  being  one -half  tlie  length  of  the  outer  face  of  the 
bucket,  tind  how  much  water  enters  each  bucket,  and,  allowing  for  oeatn 
fugal  forces,  the  point  at  wliich  the  water  l)egins  to  leave  the  buckets^ 

(5)  An  overi^hot  wheel  S2  feet  diameter  has  Rhrouds  14  inches  deep^ 
and  is  required  to  give  9  horse ■i>ower  when  making  5  revolutions  per  micafe- 

Assuming  tlie  buckets  to  be  one- third  filled  with  water  and  of  the  same 
form  as  in  the  last  question,  find  the  width  of  the  wheel,  when  the  tot^l 
fall  is  32  feet  and  Uie  efficiency  60  per  cent 


TUBBINIS 


387 


I  Assodmiig  the  Telocity  of  tlie  water  la  the  penstock  to  Ve  1}  times  that 
K  tl]«  wliael's  periphciry,  and  the  bottotD  of  the  penstock  level  with  the  top 
m.  tba  ^wbci&l,  find  the  pomt  at  which  the  water  entem  the  wheel.  Fmd  alao 
vhere  water  begins  to  discharge  frona  the  buckets. 

■  (S>  A  ra^dial  blade  impnlie  wheel  of  the  name  width  as  the  channel  in 
vliicb  it  mud,  is  15  feet  diameter.  The  depth  of  tlie  sluice  opening  i^ 
■^  focbaa  and  the  head  above  the  centre  of  the  sluice  is  H  feet.  Assuming 
K  coefficient  ol  vielocity  of  0*8  and  ttiAt  tlie  edge  ef  the  sluice  is  rounded  so 
iliAt  there  m  no  contraction,  and  tlie  velocity  of  the  rim  of  the  wlicel  is  0'4 
pile  velocity  of  flow  through  the  aluicet  iind  the  theoreticid  efficiency  of 
lllie  'vrbeeL 

I       (7 1     An  overshot  wheel  has  a  supply  of  80  cubic  feet  per  second  on  a  fall 

Lf  Mfeet 

I       DetienDine  tlie  probable  horse-power  of  the  wheel,  and   a  suitable 

l^riillii  for  the  wheeL 

I        (8)     The  water  impinges  on  a  Poncelet  float  at  15°  with  the  tangent  to 

m^UB  iHideL,  and  the  velocity  of  the  water  is  double  that  of  the  wheel.    Find, 

Ihj'  oorastmciioD,  the  proper  toclination  of  the  tip  of  the  float. 

I  <]d)  In  a  Ponoelet  wheel,  the  direction  of  tlie  jet  impinging  on  tlie  floats 
kwkflfi  an  angle  of  W^  with  the  tangent  to  the  circumference  and  the  tip  of 
nllB  floats  inakies  an  angle  of  W>°  with  the  aame  tangent.  Supposing  tlte 
ftr^locitj  of  the  jet  to  be  20  feet  per  second.  And,  graphically  or  otherwise» 

■  (1)  the  proper  velocity  of  ttie  edge  of  the  wheel,  (2)  the  height  to  which  the 
nrater  will  rise  on  tlie  float  above  the  point  of  admission,  (8)  the  velocity 
B  ttod  direction  of  motion  of  the  water  leaving  the  float. 

I       00)     Show  that  the  efficiency  of  a  simple  reaction  wheel  increases 

■  niyi  the  speed  when  frictional  resistances  are  neglected^  but  is  greatest 
BMI  a  fiaiite  speed  when  they  are  taken  into  account. 

m  U  the  speed  of  the  orifices  be  that  due  to  the  head  (1)  And  the  efficiency, 
I  0egl€<:ting  friction ;  f'2)  assuming  it  to  be  the  speed  at  maximum  efficiency, 
m  ibow  that  j  of  the  head  is  lost  by  friction,  and  }  by  final  velocity  of  water, 

I  (ll.>  Explain  why^  in  a  vortex  turbine,  the  inner  ends  of  the  vanes  are 
I  lodined  backwards  instead  of  being  radial. 

I  ii2<  An  inward  flow  turbine  wheel  has  radial  blades  at  tlie  outer 
I  F^iphtfry.  and  at  the  inner  periphery  tlje  blade  makes  an  angle  of  30""  with 
K  Ft 

m  iSbb  tangeBt.    The  total  bead  is  TO  feet  and  ^"^  *    Find  the  velocity  of  the 

■  tUb  of  (lie  wheel  if  the  water  discharges  radially.    Friction  neglected. 

■  III  I  The  inner  and  outer  diameters  of  an  inward  flow  turbine  wheel 
■*&  I  loot  and  %  feet  resfjectively.  The  water  enters  the  outer  circumference 
^P^  It  with  the  tangent,  and  leaves  the  inner  circumference  radially.  The 
Br^^  velocity  of  flow  is  6  feet  at  both  circumferences.  The  wheel  makes 
^pJ  rerqhitions  per  seconds  Determine  the  angles  of  the  vanes  at  both 
V^tnsoferenceaf  and  the  theoretical  hydrauhc  efficiency  of  the  turbine. 

V_  (14)  Water  ia  supplied  to  an  inward  flow  turbine  at  44  feet  per  second, 
r^  al  10  d^rees  to  the  tangent  to  the  wheel    The  wheel  makes  200 


88B 


HYDRAULICS 


nmsltitiQtts  per  miniite^    The  inlet  radius  is  1  foot  &iid  the  ontei  nism  I 

%  feet.    The  radiaJ  velocity  of  flow  Uirough  the  wheel  is  oom&tttut        I 

Find  the  iudlnation  of  the  vanes  at  iniet  and  outlet  ol  the  wheel  I 

Determine  the  ratio  of  the  kinetic  Gmstg^  of  the  water  eiitanflg  tbd  I 

wheel  per  pound  to  the  work  done  on  the  whed  per  pound.  I 

(15)  The  suppler  of  water  far  an  inward  flow  reaction  turbine  s  S(K^  1 
cable  feet  per  minnte  and  the  available  head  is  40  feet.  The  vASea  sn  1 
radial  at  the  inlet,  Qxe  outer  radius  is  twice  the  inner,  tlie  couitisl  I 
velocity  of  fiow  is  4  feet  per  aecond,  and  the  revolutions  are  3S0  pif  I 
minute.  Find  the  velocity  of  the  wheeU  the  guide  and  vane  angles  1 
tlie  inner  and  outer  diameters^  and  the  width  of  the  bucket  at  inlet  ill| 
outlet.    Lomd.  Un.  1906.  I 

(16)  An  inward  tiow  turbine  on  15  feet  fall  has  aji  inlet  radius  of  1  Ml 
and  an  outlet  radius  of  6  inches.  Water  enters  at  15*  with  the  tangent  to  j 
the  circumference  and  is  discharged  radially  with  a  velocity  ol  S  feet  per  J 
Be<»ncL  The  actual  velocity  of  water  at  inlet  is  22  feet  per  secofid  fhem 
G^^^nnifereutial  velocity  of  the  Inlet  surface  of  the  wheel  m  ld|  feet  pttl 
second.  I 

Construct  the  inlet  and  outlet  angles  of  the  turbine  vanes*  I 

Determine  the  theoretical  hydrauhc  efficiency  of  the  turbine.  I 

If  the  hydraulic  efficiency  of  the  tm-bine  is  assumed  80  per  cent  ind  tli«  I 

vane  angles.  I 

(17)  A  quantity  of  water  Q  cubic  feet  per  second  flows  throi^ti  A  j 
tnrbin&t  a^^  ^^^  initial  and  final  directions  and  velocities  are  kuavSiJ 
Apply  the  principle  ol  equality  of  angular  impulse  and  momani  di 
momentum  to  find  tlie  couple  exerted  on  the  turbine,  I 

(18)  The  wheel  of  an  inward  flow  turbine  has  a  peripheral  vebcttftf  1 
50  feet  par  second.  The  velocity^  of  whirl  of  the  incoming  water  ig  4«)  tet  I 
per  second^  and  the  radial  velocity  of  flow  5  feet  per  second.  DetenniBB  I 
the  vane  angle  at  inlet.  I 

Taking  the  flow  as  20  cubic  feet  per  second  and  the  total  losees  •*  I 
20  per  cent,  ol  the  available  energy,  determine  the  horse *iKjwer  of  tJnJ 
turbinoi  and  the  head  H.  ^^1 

If  5  per  cent,  of  the  head  is  lost  in  friction  in  the  supply  pipe,  an^H 
centare  of  the  turbine  is  15  feet  above  the  tail  race  level,  find  the  preaia]i| 
head  at  the  inlet  circumference  of  the  wheel.  I 

(10)  An  inward  flow  turbine  is  required  to  give  200  harse*pewer  is^y 
a  head  of  100  feet  when  running  at  500  revolutions  per  minnte.  WM 
velocity  with  which  the  water  leaves  tlie  wheel  axiaUy  ruay  be  tall^H 
10  feet  per  secoud^  aod  tho  wheel  is  to  have  a  double  outlet  The  diftd^H 
of  the  outer  circumference  may  be  taken  as  1|  times  tlie  inner.  Defeei^^l 
tlio  dimensionB  of  the  tiu-bine  and  the  angles  of  the  guide  bMay^| 
vanes  of  the  turbine  wheel.  The  actual  efficiency  is  to  be  taken  as  I^H 
cent,  and  the  hydraulic  efficiency  as  80  per  cent.  ^H 

(20)  An  outward  flow  turbine  wheel  has  an  internal  diameter  of  fi^Sl 
feet  and  an  external  diameter  of  0*25  feet.  The  head  above  the  turlitflUU 
14V5  feet    The  width  of  the  wheel  at  inlet  is  10  inches,  and  the  iifii|^| 


I 


TURBIKES 


389 


plied  per  HeconcT  m  215  cnbic  feei.    Asanmmg  the  hydraulic 
«?«  nrc  211  |ior  oent.,  determine  tlie  angles  of  tips  o!  the  Tanes  ^a  that 
^Ixg  w«i4^r  shall  leave  the  wheel  radially,     Determiiie  the  horse -power  o£ 
tiirbme  and  verify  the  work  done  per  poand  from  the  triangleB  of 


121)  The  total  head  arailable  for  an  inward -flow  ttu-bitie  is  100  feet 
The  tnrbine  wheel  is  pl&oed  15  feet  above  the  tail  water  ley  eh 
When  the  flow  is  somud,  tliere  is  a  loss  of  head  in  the  supply  pipe  of 
•  jmi  t  i'ut  of  the  head  I  in  tlie  goide  passages  a  loss  of  5  per  cent. ;  in  the 
'^'"lieel  9  pi?r  cent ;  in  the  down  pipe  I  per  cent,  i  and  the  velocity  of  flow 
^«^m  the  wheel  and  in  the  supply  pipe,  and  also  from  the  down  pipe  ib 
^    feet  iMjr  decond- 

Tljr  diameter  of  the  inner  circnmference  of  the  wheel  is  9^  inches  and 
tht»  outer  19  inches,  and  the  water  leaves  the  wheel  vanes   radially* 
TtTie  wheel  haa  nnUal  vanes  at  inlet 

f  >et«^nnine  tlie  tiamber  of  revolutiona  of  the  wheeL  the  pretssure  head  in 
\  #ye  of  the  wheel*  the  pressure  head  at  the  circumference  to  the  wheel, 
^  |3n%amire  head  at  the  eu  trance  tti  the  guide  chamber,  and  the  velocity 
vliicb  thr  water  Ijaa  when  it  enters  tlie  wheeL    From  the  data  given 


j^ 


2)     A  horizontal  inward  flow  tnrbine  has  an  internal  dXameter  of 

4  iuchef*  and  an  external  diameter  of  7  feet    The  crowns  of  the 

I  aro  parallel  and  are  8  inches  apart     The  diHerence  in  level  of  the 

i  and  tail  water  in  6  feet,  and  the  upper  crown  of  the  wheel  is  jost  below 

I  latt  water  level.    Find  the  angle  the  guide  blade  niakea  witti  the  tangent 

wbnel,  when  the  wheel  makea  32  revolutious  per  minute,  and  the 

■15  cnbic  foet  per  second.     Neglecting  friction,  determine  the  vane 

,  lite  horse -power  of  the  wheel  and  the  theoretical  hydranhc  efficiency. 

9)     A  parallel  flow  tnrbine  has  a  mean  diameter  of  11  feet. 
The  number  of  revolutions  per  minute  is  15,  and  the  a^dal  velocity  of 
is  3'5  feet  per  second.    The  velocity  of  the  water  along  the  tips  of  the 
.  is  15  feet  per  second. 
LN^termine  tlie  inclination  of  tlie  gnide  blades  and  the  vane  angles  that 
li«  water  shall  enter  withont  shock  and  leave  the  wheel  axially, 

Dc?ieriiiine  the  work  done  per  pound  of  water  passing  thi'ough  the  wheeL 

i*M)  The  diameter  of  Uie  inner  crown  of  a  parallel  flow  pressure  turbine 
f  5  feet  atid  the  diameter  of  tlie  outer  crown  is  B  feet.  The  head  over  the 
whie^l  iit  11*  feett  The  number  of  revolutions  per  minute  is  52*  The  radial 
Jocitv  of  flow  tlirough  the  wheel  is  4  feet  per  second. 

Aaaiiining  a  hydraulic  efficiency  of  0*8,  determine  the  guide  blade  angles 

1  rani^  angles  at  inlet  for  the  three  radij  2  feet  6  inches^  8  feet  8  inches 

i4Ceet 

Assrunlng  the  depth  of  the  wheel  is  S  inches,  draw  am  table  sections  of 

»  Tanes  at  the  three  radii* 

Find  al»o  ihe  width  of  the  guide  blaile  in  plan,  if  the  up])er  and  lower 
are  paraUel*  and  the  lower  edge  makes  a  constant  angle  with  the 


HTDEArXICS 


Huee  mt  the  miidr  And  the 

head  of  64  iBtLl 
Uml  speed  ol  ibe  whed^l 


tmliiike  has  an  inner  diajneter  of  5  leiti] 
d  iftdiBA,  ftud  makes  4S0  reTolatioiiA^  ] 

at  H  lewes  tlie  nozzles  is  dotibla  the  Telodt|*l 
I  Hm  wlifiel,  and  tlie  dtreetion  of  the  water  iDaJi««  m\ 
wUli  Hie  csmttnference  ol  the  wheel. 

at  miei,  and  the  angle  of  tlie  Tane  at  outlet  i 
mier  AmXt  learo  Ibe  idieel  nidiaUy. 

llie  tbeocHml  bydr&idie  efficiency.    U  8  per  cent,  of  ^e  head! 

at  the  mamM^  ia  lost  in  the  wheel,  find  tJie  vane  angle  at  exit  thoft] 

Uft  leanie  imdiaUj. 

■0V  the  l^dnnfo  didbncy  of  the  turbine  ? 


P 


In  aA  asdal  flow  Oinrd  turbine,  kt  V  be  the  velocity  doe  to  Die  I 
head.  Siqppose  the  water  issiies  from  the  guide  blades  with  Qm  1 
O^T,  and  It  dbehazged  tnillj  with  a  velocity  ^12  V*  Lei  ^  | 
iJotily  ctf  the  teoeinigMddiafahaigiiig  edges  be  0-55  V, 

^1^  of  the  guide  liladeB,  receiving  and  dischajrging  angles  d 

■d  Iry^snbe  effdepey  cl  the  torbine. 


m^ 


imi 


Waier  k  sn^fiied  to  an  axial  flow  impulse  turbine,  having  a  mma 
r  ef  t  fibit^  and  niakmg  144  revolntioiis  per  minate,  tmder  a  hmd  d  \ 
.  Tin  aag^  of  t^  guide  blade  at  entrance  is  30%  and  the  angja  tbe 
le  direction  of  motion  at  exit  is  80"".  Eight  per  eeoi  of 
I  the  supply  pipe  and  guide.  Determine  ihe  ralAtlvt 
wtkicHy  «i  mier  end  wtieil  at  entrance,  and  on  the  assumption  Utat  10  f«f 
eesi.  ci  fiie  kpt^  hesd  is  lost  in  frictfon  and  shock  in  tlie  wheel,  detemunt 
Hie  veitoeily  wHh  which  the  water  leav^  the  wheel.  Find  the  Uydnolk 
ofthei 


|W  Hke  golde  bladea  ef  an  inward  dow  turbine  are  inclined  M  SD 
iliyeui»  «Bd  the  Telocity  U  tlong  tl je  tip  of  the  blade  is  60  feet  pet  scooni 
The  i^odtyoC  Hie  wheel  periphery  is  55  feet  per  second.  The  guide  blidei 
wm  trailed  an  thnt  tbey  ate  inclined  at  an  angle  of  15  degrees,  the  ve 
C  T^mftimng  eoasluit.    Find  the  loss  of  head  due  to  »hock  at  entranc 

If  the  nidtQt  of  the  iimer  peripheiy  ie  one-half  that  of  the  outer  aad^ 
tidkU  velocity  thrangb  the  wheel  ie  constant  for  any  flow^  and  the  inl^ 
kfl  Ibe  wheel  twlbliy  in  the  first  case,  &id  the  direction  in  which  it  leaf«« 
JM  liie  eeeond  om&    The  inlet  radius  is  twice  the  outlet  radius. 


03f)     Tile  suppler  of  water  to  a  turbbie  is  controlled  by  a  speed  \ 
between  tlie  guides  and  the  wheel.     If  when  tlie  gate  ia  fully  opea  ^ 
i^ocity  witli  which  the  wmter  approaches  the  wheel  is  70  foot  pe;  seooA^ 


^y^ 


I 

t 
f, 

!  TURBINES  391 

m  and  it  TiMkkes  an  angle  of  15  degrees  with  the  tangent  to  the  wheel,  find 
the  loss  of  Ikead  by  shock  when  the  gate  is  half  closed.  The  velocity  of 
the  inlet  periphery  of  the  wheel  is  75  feet  per  second. 

(81)  A  Pelton  wheel,  which  may  he  assumed  to  have  semi-cylindricaJ 
buckets,  is  2  feet  diameter.  The  available  pressure  at  the  nozzle  when  it 
is  doeed  is  200  lbs.  per  sqnare  inch,  and  the  supply  when  the  nozzle  is 
open  is  lOO  cuIhc  feet  per  minute.  If  the  revolutions  are  600  per  minute, 
estJTnate  the  horse-power  of  the  wheel  and  its  efficiency. 

(82)  Show  that  the  efficiency  of  a  Pelton  wheel  is  a  maximum — 
neig^ecting  fric!tional  and  other  losses — when  the  velocity  of  the  cups  equals 
half  the  velocity  of  the  jet. 

25  cnbic  feet  of  water  are  supplied  per  second  to  a  Pelton  wheel  through 
m  nozzle,  the  area  of  which  is  44  square  inches.  The  velocity  of  the  cups 
is  41  feet  per  second.  Determine  the  horse-power  of  the  wheel  assuming 
an  efficiency  of  75  per  cent. 


CHAPTER  X. 

PUMPS. 

Pumps  are  machines  driven  by  some  prime  mover,  and  used 
for  raising  fluids  from  a  lower  to  a  higher  level,  or  for  impartmg 
energy  to  fluids.  For  example,  when  a  mine  has  to  be  drained 
the  water  may  be  simply  raised  from  the  mine  to  the  surface,  and 
work  done  upon  it  against  gravity.  Instead  of  simply  raising  the 
water  through  a  height  h,  the  same  pumps  might  be  used  to 
deliver  water  into  pipes,  the  pressure  in  which  is  wh  pounds  per 
square  foot. 

A  pump  can  either  be  a  suction  pump,  a  pressure  pump,  or 
both.  If  the  pump  is  placed  above  the  surface  of  the  water  in 
the  well  or  sump,  the  water  has  to  be  first  raised  by  suction; 
the  maximum  height  through  which  a  pump  can  draw  water, 
or  in  other  words  the  maximum  vertical  distance  the  pump  can 
be  placed  above  the  water  in  the  well,  is  theoretically  34  feet,  but 
practically  the  maximum  is  from  25  to  30  feet.  If  the  pump 
delivers  the  water  to  a  height  h  above  the  pump,  or  against  a 
pressure-head  A,  it  is  called  a  force  pump. 

216.    Centriftigal  and  turbine  pumps. 

Theoretically  any  reaction  turbine  could  be  made  to  work  as 
a  pump  by  rotating  the  wheel  in  the  opposite  direction  to  that  in 
which  it  rotates  as  a  turbine,  and  supplying  it  with  water  at  the 
circumference,  with  the  same  velocity,  but  in  the  inverse  direction 
to  that  at  which  it  was  discharged  when  acting  as  a  turbine.  Up 
to  the  present,  only  outward  flow  pumps  have  been  constructed, 
and,  as  will  be  showTi  later,  difficulty  would  be  experienced  in 
starting  parallel  flow  or  inward  flow  pumps. 

Several  types  of  centrifugal  pumps  (outward  flow)  are  shown 
in  Figs.  272  to  276. 

The  principal  difference  between  the  several  types  is  in  the 
form  of  the  casing  surrounding  the  wheel,  and  this  form  has  con- 
siderable influence  upon  the  efficiency  of  the  pump.    The  reason 


PtJMPS 


393 


this  can  be  easily  seen  in  a  general  way  from  the  following 

cnnsideration.  The  water  approaches  a  turbine  wheel  with  a 
lii^li  velocity  and  in  a  direction  making  a  small  angle  with  the 
directioti  of  motion  of  the  inlet  circumference  of  the  wheel,  and 


Fig,  1^3.     Diagrum  of  Centrifui^al  Pump. 

[thus  it  has  a  large  velocity  of  whirl  When  the  water  leaves  the 
]  wheel  its  velocity  is  small  and  the  velocity  of  whirl  should  be  zero. 
jIn  the  centrifugal  pump  these  conditions  are  entirely  reversed; 
[the  water  enters  the  wheel  with   a  small  velocity,   and  leaves 


394 


HYDBAUUCS 


it  with  a  high  velocity.     If    the  case  surronnding  the  whfid 
admits  of  this  velocity  being  diminished  gradually,  the  Mneiac 
energy  of  the  water  is  converted  into  useful  work,  but  if  not,  itia 
destroyed  by  eddy  motions  in  the  casing,  and  the  efficiency  of  tlid 
pump  is  accordingly  low. 

In  Fig.  272  a  circular  casing  surrounds  the  wheel,  and  prac- 
tically the  whole  of  the  kinetic  energy  of  the  water  when  it  leaves 
the  wheel  is  destroyed ;  the  efficiency  of  such  pumps  is  generally 
much  less  than  50  per  cent. 


Fig.  273.     Centrifugal  Pump  with  spiral  casing. 

The  casing  of  Pig.  273  is  made  of  spiral  form,  the  sectional 
area  increasing  uniformly  towards  the  discharge  pipe,  and  thus 
being  proportional  to  the  quantity  of  water  flowing  through  the 
section.  It  may  therefore  be  supposed  that  the  mean  velocity  of 
flow  through  any  section  is  nearly  constant,  and  that  the  stream 
lines  are  continuous. 

The  wheel  of  Fig.  274  is  surrounded  by  a  large  whirlpool 
chamber  in  which,  as  shown  later,  the  velocity  with  which  the 
water  rotates  round  the  wheel  gradually  diminishes,  and  tie 
velocity  head  with  which  the  water  leaves  the  wheel  is  partly 
converted  into  pressure  head. 

The  same  result  is  achieved  in  the  pump  of  Figs.  275  and  276 


PUMPS 


395 


'>;  allowing  the  water  as  it  leaves  the  wheel  to  enter  giiide 

p8«sag^,  similar  to  those  used  in  a  turbine  to  direct  the  water 

to  the  wheeL    The  area  of  these   passages  gi-adually  increaaeB 

md  a  eoiieiderahle  portion  of  the  velocity  head  is  thus  converted 

into  preeenre  head  and  is  available  for  lifting  water. 

Tliis  cliiaB  of  centrifugal  punip  is  known  as  the  tarbine  pump. 


Pig,  274.     EHa^ram  of  Centrifagftl  Pump  with  Whirlpool  Chamber. 

21f .    Starting  centrifugal  or  turbine  pmnps, 

A  centrifngal  pump  cannot  commence  delivery  unless  the  wheel, 
casings  and  suction  pipe  are  full  of  water. 

K  the  pump  is  below  the  water  in  the  well  there  is  no  difficulty 
in  starting  as  the  casing  will  be  maintained  full  of  water » 

WTien  the  pump  is  above  the  water  in  the  well,  as  in  Fig,  272, 
Ison-retum  valve  V  must  be  fitted  in  the  suction  pipe,  to  prevent 

kj^tup  when  stopped  from  being  drained.  If  the  pump  becomes 
ff  or  when  the  pump  is  first  set  to  work,  special  meuna  have 
to  Ije  provided  for  filling  the  pump  case.  In  large  pumps  the  air 
may  be  expelled  by  means  of  steam,  which  becomes  condensed  and 
ihe  water  rises  from  the  well,  or  they  should  be  provided  with 


HTBRAUI,ICS 


i  ftir-piimp  or  ejector  as  an  auxiliaTy  to  the  patdp*    StiiaU  pmnps 
geiie?mll>r  be  easily  filled  by  hand  throug^h  a  pipe  sach  m  \ 
Aomn  at  P,  Fig,  276. 

With  some  classes  of  pomps,  if  the  pump  ha&  to  commence 
delivery  against  full   head,  a  stop  valv^e   on   the   rising  maiii, , 
Fig.  296,  is  closed  until  the  painp  has  attained  the  speed  necessary 
to  commence  delivery*,  after  which  the  stop  valve  is  dowlir  | 
opened. 


It  will  be  seen  later  that,  under  special  circumstances,  other 
pronsions  will  have  to  be  made  to  enable  the  piimp  to  commeDOJ 
dt^liwry. 

818.  Form  of  the  Tanea  of  centrlftigal  pumps, 
Tii^^^  conditions  to  be  satisfied  by  the  vanesi  of  a  centrifuia! 
pump  are  exactly  the  aame  as  for  a  turbine.  At  inlet  the  directioa 
of  the  vane  shoukl  be  parallel  to  the  direction  of  the  relativE 
n^loeity  ivf  tlie  water  and  the  tip  of  the  vane,  and  the  velocity 
witli  which  the  M-ater  leaves  the  wheel,  relative  to  the  pump  o»ft 
\H  tht^  vtvtor  ffiini  of  the  velocity  of  the  tip  of  the  vane  and  tl# 
veUK*ity  n*lative  to  the  vane, 

*  Set  ptge  4m, 


PDlfPS 


397 


Suppoae  the  wheel  and  casing  of  Fig.  272  is  full  of  water^  and 
the  wheel  is  rotated  in  the  dii'ection  of  the  arrow  mth  such  a 
velocity  that  water  enters  the  wheel  in  a  known  direction  with  a 
Telocity  XT,  Fig.  277 j  not  of  necessity  radial. 

t^t  r  he  the  velocity  of  the  receiving  edge  of  the  vane  or  inlet 
eirciimferenee  of  the  wheel;  Vi  the  velocity  of  the  discharging 
circnmfereiice  of  the  wheel ;  Ui  the  absolute  velocity  of  the  water 
&6  it  leaves  the  wheel ;  V  and  Vj  the  velocities  of  wlurl  at  inlet 
and  ontlet  respectively;  Vr  and  iv  the  relative  velocities  of  the 
water  and  the  vane  at  inlet  and  outlet  respectively;  u  and  iti  the 
indial  velocities  at  inlet  and  outlet  respectively. 

The  triangle  of  velocities  at  inlet  is  ACD,  Fig.  277,  and  if  the 
vsae  at  A,  Fig*  272,  ia  made  parallel  to  CD  the  water  will  enter 
the  wheel  without  shock. 


wangle  of  vdifoiies 


Fig.  277. 


at  e*ixt . 
Fig.  278. 


The  wheel  being  full  of  watePj  there  is  continuity  of  How,  and 
if  A  and  Aj  arc5  the  circumferential  areas  of  the  inner  and  outer 
cipDumferences,  the  radial  component  of  the  velocity  of  exit  at  the 
Doter  circumference  ia 

Au 


«i  = 


A," 


If  the  direction  of  the  tip  of  the  vane  at  the  outer  circnm- 
ference  is  known  the  triangle  of  velocities  at  exit,  Fig.  278,  can  be 
drawn  as  follows. 

Set  out  BG  radially  and  equally  to  %,  and  BE  equal  to  v^ 
Draw  GF  parallel  to  BE  at  a  distance  from  BE  equal  to  ft,, 
ad  EF  parallel  to  the  tip  of  the  vane  to  meet  GF  in  F. 

Then  BF  is  the  vector  sum  of  BE  atid  EF  and  is  the  velocity 
ith  which  the  water  leaves  the  wheel  relative  to  the  fixed  casing, 

219.    Work  done  on  the  water  by  the  wheel, 

Lt*t  R  and  r  be  the  radii  of   the  discharging  and  receiving 

enmferences  respectively. 

The  change  in  angular  momentum  of  the  water  as  it  passes 
Ihrgugh  the  wheel  is  V,R  +  Yr/g  per  pound  of  flow,  the  plus  sign 
becing  used  when  V  is  in  the  opposite  direction  to  Vj,  as  in 
Kgs,  277  and  278, 


398  HYDRAULICS 

Neglecting  frictional  and  other  losses,  the  work  done  by  the 
wheel  on  the  water  i)er  pound  (see  page  275)  is 

g  "  g  ' 

If  n  is  radial,  as  in  Fig.  272,  Y  is  zero,  and  the  work  done  on 
the  water  by  the  wheel  is 

-^  foot  lbs.  per  lb.  flow. 

If  then  H«,  Fig.  272,  is  the  total  height  through  which  the  water 
is  lifted  from  the  sump  or  well,  and  Ud  is  the  velocity  with  which 
the  water  is  delivered  from  the  delivery  pipe,  the  work  done  on 
each  pound  of  water  is 

and  therefore, 

g       ^     2g 

Let  (180*  -  <^)  be  the  angle  which  the  direction  of  the  vane  at 
exit  makes  with  the  direction  of  motion,  and  (180"  -  &)  the  angle 
which  the  vane  makes  with  the  direction  of  motion  at  inlet.  Then 
ACD  is  e  and  BEF  is  *. 

In  the  triangle  HEF,  HE  =  HF  cot  <^,  and  therefore, 

Vi "  t'l  —  tti  cot  ^. 
The  theoretical  lift,  therefore,  is 

2g  g 

If  Q  is  the  discharge  and  Ai  the  peripheral  area  of  the  dis- 
charging circumference, 

Q 

Q     .^ 
Vi  -  Vi  -^  cot  4> 

and  H  = ^ (1). 

g 

If,  therefore,  the  water  enters  the  wheel  without  shock  and  all 

R 

resistances  are  neglected,  the  lift  is  independent  of  the  ratio  — ,  and 

dex^ends  only  on  the  velocity  and  inclination  of  the  vane  at  the 
discharging  circumference. 

220.    5B»tioofVitoVi. 

As  in  the  ca^  of  the  turbine,  for  any  given  head  H,  Vi  and  I'l 
can  theoretically'  have  any  values  consistent  with  the  product 


\ 


PUMPS 


399 


Vit?i  heing  equal  to  grH,  the  ratio  of  Vi  to  Vi  simply  depending  upon 
the  magnitude  of  the  angle  ^. 

The  greater  the  angle  ^  is  made  the  less  the  velocity  Vi  of  the 
periphery  must  be  for  a  given  lift. 


Fig.  279. 

This  is  shown  at  once  by  equation  (1),  section  219,  and  is 
illustrated  in  Fig.  279.  The  angle  4^  is  given  three  values, 
30  degrees,  90  degrees  and  150  degrees,  and  the  product  Yv  and 
also  the  radial  velocity  of  flow  tti  are  kept  constant.  The  theo- 
retical head  and  also  the  discharge  for  the  three  cases  are  there- 
fore the  same.  The  diagrams  are  drawn  to  a  common  scale,  and  it 
can  therefore  be  seen  that  as  ^  increases  Vi  diminishes,  and  Ui 
the  velocity  with  which  the  water  leaves  the  wheel  increases. 

221.  The  kinetic  energy  of  the  water  at  exit  from  the 
wheeL 

Part  of  the  head  H  impressed  upon  the  water  by  the  wheel 
increases  the  pressure  head  between  the  inlet  and  outlet,  and  the 
remainder  appears  as  the  kinetic  energy  of  the  water  as  it  leaves 


400  HTD&AULICS 

the  wheel.    This  kinetic  energy  is  equal  to  -^ ,  and  can  only  be 

utilised  to  lift  the  water  if  the  velocity  can  be  gradually  diminished 
so  as  to  convert  velocity  head  into  pressure  head.  This  however 
is  not  very  easily  accomplished,  without  h^ng  accomi)anied  by  a 
considerable  loss  by  eddy  motions.    If  it  be  assumed  that  the  same 

proportion  of  the  head  ^  in  all  cases  is  converted  into  useful 

work,  it  is  clear  that  the  greater  Ui,  the  greater  the  loss  by  eddy 
motions,  and  the  less  efficient  will  be  the  pump.  It  is  to  be  ex- 
pected, therefore,  that  the  less  the  angle  <^,  the  greater  will  be 
the  efficiency,  and  experiment  shows  that  for  a  given  form  of 
casing,  the  efficiency  does  increase  as  <^  is  diminished. 

222.  Gross  lift  of  a  centriftigal  pomp. 

Let  ha  be  the  actual  height  through  which  water  is  lifted; 
hg  the  head  lost  in  the  suction  pipe ;  ha  the  head  lost  in  the  delivery 
pipe ;  and  it^  the  velocity  of  flow  along  the  delivery  pipe. 

Any  other  losses  of  head  in  the  wheel  and  casing  are  incident 

to  the  pump,  but  fe„  ha,  and  the  head  5-  should  be  considered  as 

external  losses. 

The  gross  lift  of  a  pump  is  then 

h=^ha  +  A,  +  fcd  +  o~  > 

and  this  is  always  less  than  H. 

223.  Efficiencies  of  a  centrifugal  pump. 

Manometric  efficiency.    The  ratio  =^ ,  or 

e= ^ 


0 

vi  -  vi  -?  cot  4> 

Ai 


is  the  manometric  efficiency  of  the  pump  at  normal  discharge. 

The  reason  for  specifically  defining  e  as  the  manometric 
efficiency  at  normal  discharge  is  simply  that  the  theoretical  lift  H 
has  been  deduced  from  consideration  of  a  definite  discharge  Q, 
and  only  for  this  one  discharge  can  the  conditions  at  the  inlet  edge 
be  as  assumed. 

A  more  general  definition  is,  however,  generally  given  to  e,  and 
for  any  discharge  Q,  therefore,  the  manometric  efficiency  maj 
be  taken  as  the  ratio  of  the  gross  lift  at  that  discharge  to  the 
theoretical  head 

Vi'  -  Vi  A   cot  4^ 
Ai 


CENTRIFUGAL   PUMPS  401 

This  manometric  efficiency  of  the  pump  must  not  be  confused 
with  the  efficiency  obtained  by  dividing  the  work  done  by  the 
pump,  by  the  energy  required  to  do  that  work,  as  the  latter  in 
many  pumps  is  zero,  when  the  former  has  its  maximum  value. 

Hydraulic  efficienSy,  The  hydraulic  efficiency  of  a  pump  is 
the  ratio  of  the  work  done  on  the  pump  wheel  to  the  gross  work 
done  by  the  pump. 

Let  W  =  the  weight  of  water  lifted  per  second. 

Let  h  =  the  gross  head 

Let  E  =  the  work  done  on  the  pump  wheel  in  foot  pounds 
per  second. 

Let  ek  =  the  hydraulic  efficiency.    Then 

W.h 

The  work  done  on  the  pump  wheel  is  less  than  the  work  done 
on  the  pump  shaft  by  the  belt  or  motor  which  drives  the  pump, 
by  an  amount  equal  to  the  energy  lost  by  friction  at  the  bearings 
of  the  machine.  This  generally,  in  actual  machines,  can  be 
approximately  determined  by  running  the  machine  without  load. 

jkchuil  efficiency.  From  a  commercial  point  of  view,  what  is 
g-enerally  required  is  the  ratio  of  the  useful  work  done  by  the 
pump,  taking  it  as  a  whole,  to  the  work  done  on  the  pump  shaft. 

Let  E,  be  the  energy  given  to  the  pump  shaft  per  sec.  and 
em  the  mechanical  efficiency  of  the  pump,  then 

E  =  E«.em, 
and  the  actual  efficiency 

_W.fea 

ea-    ^    . 

Gross  efficiency  of  the  pump.  The  gross  efficiency  of  the  pump 
itself,  including  mechanical  as  well  as  fluid  losses,  is 

W.h 


ea  = 


K 


224.  Experimental  determination  of  the  efficiency  of  a 
eentriftigal  pomp. 

The  actual  and  gross  efficiencies  of  a  pump  can  be  determined 
directly  by  experiment,  but  the  hydraulic  efficiency  can  only  be 
determined  when  at  all  loads  the  mechanical  efficiency  of  the 
pump  is  known. 

To  find  the  actual  efficiency,  it  is  only  necessary  to  measure 
the  height  through  which  water  is  lifted,  the  quantity  of  water 

L.  H.  ^^ 


40S  HfDRAITLICS 

dischiirged,  and  tlie  energy  E,  given  to  the  pump  shaft  in  oj 
tim©, 

A.  very  convenient  method  of  determiniiig  £«  with  a  hk 
degree  of  accuracy  is  to  drive  the  pump  shaft  direct  by  an  electrie 
motor,  the  efficiency  cun""e*  for  which  at  varying  loads  is  kn 
A  better  method  ia  to  use  some  form  of  transmission  dynjuui^ 
meter  t. 

225.    Design  of  pump  to  give  a  discbarge  Q. 

If  a  pump  is  required  to  give  a  discliarge  Q  under  a  grom 
lift  hf  and  from  pre\^ous  experience  the  probable  manom^tnc 
efliciency  e  at  this  discharge  is  known,  the  problem  of  determinTtig 
suitable  dimensions  for  the  wheel  of  the  pump  is  not  diificulL 
The  difficulty  really  arises  in  gi\nng  a  correct  value  to  e  and  ia 
making  proper  allowance  for  leakage. 

Tins  difficulty  w^tl  be  better  appreciated  after  the  losses  m 
various  kinds  of  pumps  have  been  considered.  It  will  then  be 
seen  that  e  depends  upon  the  angle  ^,  the  velocity  of  the  wbeeJ* 
the  dimensions  of  the  wheeU  the  form  of  the  vanes  of  the  wheet 
the  discharge  through  the  wheel,  and  upon  the  form  of  the  casmg 
surrounding  the  wheel ;  the  form  of  the  casing  being  Just  as 
important,  or  more  important,  than  the  form  of  the  wheel  in 
determining  the  probable  value  of  e. 

Design  of  the  tvheel  of  a  ptiffip  fftr  a  given  discharge  und^fr  o 
given  Jiead.  If  a  pump  is  required  to  give  a  discharge  Q  under  an 
effective  head  h^^  the  gross  head  h  can  only  be  determined  if  A»i 

hit  and  ,j^  f  are  known. 

Any  suitable  value  can  be  given  to  the  velocity  tu^  If  the 
pipes  are  long  it  should  not  be  nmch  greater  tlian  5  feet  per  sec<mi 
for  reasons  explained  in  the  chapter  on  pipes,  and  the  velocity'  % 
in  the  suction  pipe  should  be  equal  to  or  less  tlian  lij.  Tlie 
velocities  ta,  and  n^t  having  been  settled,  the  losses  h*  and  h^  can  k 
approximated  to  and  the  gross  head  h  found.  In  the  suction  pip^^ 
as  explained  on  page  395,  a  foot  valve  is  generally  fitted,  at  ubicK 
at  high  velocities,  a  loss  of  head  of  several  feet  may  itcenr. 
The  angle  <^  is  generally  made  from  10  to  90  degrees.  Theoreti- 
cally, as  already  stated^  it  can  be  made  much  greater  tim 
90  degrees,  but  the  efficiency  of  ordinary  centrifugal  pumps  Mlglii 
be  very  considerably  diminished  as  <f>  is  increased* 

The  manometric  eificiency  e  varies  very  considerably ;  with 
radial  blades  and  a  circular  casing,  the  efficiency  is  not  generally 

'  See  Etfciriml  Engineerinff,  Thomilen-Howe,  p<  195* 
t  Bee  |>apef  by  Stanlon,  Pmc.  ItnL  Meek  Eufft^^  1909, 


CENTRIFUGAL  PUMPS  403 

more  than  0*3  to  0*4.  With  a  vortex  chamber,  or  a  spiral  casing, 
and  the  vanes  at  inlet  inclined  so  that  the  tip  is  parallel  to  the 
relative  velocity  of  the  water  and  the  vane,  and  i>  not  greater  than 
90  degrees,  the  manometric  efficiency  e  is  from  0*5  to  0*75,  being 
greater  the  less  the  angle  <^,  and  with  properly  designed  guide 
blades  external  to  the  wheel,  e  is  from  0*6  to  '85. 

The  ratio  of  the  diameter  of  the  discharging  circumference  to 
the  inlet  circumference  is  somewhat  arbitrary  and  is  generally 
made  from  2  to  3.  Except  for  the  difficulty  of  starting  (see 
section  226),  the  ratio  might  with  advantage  be  made  much 
smidler,  as  by  so  doing  the  frictional  losses  might  be  considerably 
reduced.  The  radial  velocity  t^  may  be  taken  from  2  to  10  feet 
per  second. 

Having  given  suitable  values  to  u,  and  to  any  two  of  the  three 
quantities,  e,  v,  and  <^,  the  third  can  be  found  from  the  equation 

7     e  W  ~  Vitti  cot  0) 

n  = . 

9 

The  internal  diameter  d  of  the  wheel  will  generally  be  settled  from 

consideration  of  the  velocity  of  flow  tt,  into  the  wheel.    This  may 

be  taken  as  equal  to  or  about  equal  to  u,  but  in  special  cases 

it  may  be  larger  than  u. 

Then  if  the  water  is  admitted  to  the  wheel  at  both  sides,  as  in 

Fig.  273, 

from  which  d  can  be  calculated  when  u^  and  Q  are  known. 

Let  b  be  the  width  of  the  vane  at  inlet  and  B  at  outlet,  and  D 
the  diameter  of  the  outlet  circumference. 

and  B  = 


Then  *  =  ^    , 


If  the  water  moves  toward  the  vanes  at  inlet  radially,  the 
inclination  0  of  the  vane  that  there  shall  be  no  shock  is  sucli  that 

tan  ^  =  - , 

and  if  guide  blades  are  to  be  provided  external  to  the  wheel,  as  in 
Fig.  275,  the  inclination  a  of  the  tip  of  the  guide  blade  with  the 
direction  of  Vi  is  found  from 

tan a=  -- , 

The  guide  passages  should  be  so  proportioned  that  the  velocity 
Ui  is  gradually  diminished  to  the  velocity  in  the  delivery  pipe. 


404 


HYDRAULICS 


Limitifig  velocity  of  the  rim  of  the  whe&L  Quite  apart  from 
lead  lost  by  friction  in  tlie  wheel  due  to  the  relative  motion  of 
he  water  and  the  wheel,  there  is  also  conBiderable  loss  of  energy 
external  to  the  wheel  due  to  the  relative  motdon  ol  the  water  mi 
the  wheel.  Betiyeen  the  wheel  and  the  casing  there  is  in  most 
pumps  a  film  of  water,  and  between  this  film  and  the  wheel, 
frictional  forces  are  set  up  which  are  practically  proportional  to 
tlie  square  of  the  velocity  of  the  wheel  periphery  and  to  the  ane* 
of  the  wheel  crowns.  An  attempt  is  frequently  made  to  dimitiisli 
tlris  loss  by  fixing  tlit*  vanes  to  a  central  diaphragm  only,  the 
wheel  thus  being  without  crowns,  the  outer  casing  beiog  so 
fonned  that  there  is  but  a  small  clearance  between  it  and  ih 
outer  edges  of  the  vanes.  At  high  velocities  these  frictional  rdd/ft- 
anoes  may  be  considerable.  To  keep  them  small  the  surface  d 
the  wheel  crowiis  and  vanes  must  be  made  smooth^  and  to  this 
eiid  many  high  speed  wheels  are  carefully  finished. 

Until  a  few  years  ago  the  periphei'al  velocity  of  pump  wheals 
twas  generally  less  than  50  feet  per  second,  and  the  best  velocity 
ms  supposed  to  be  about  30  feet  per  second.  They  are  now,  how- 
ever, run  at  much  higher  speeds,  and  the  limiting  velocities  ar© 
fixed  from  consideration  of  the  stresses  in  the  wheel  due  to  centri- 
fugal forces.  Peripheral  velocities  of  nearly  200  feet  per  second 
are  now  frequently  used,  and  Bateau  has  constructed  small  pump* 
with  a  peripheral  velocity  of  250  feet  per  second*. 

Exatn^U.  To  hnd  the  proportions)  of  a  pump  with  radial  bl&d^  at  oullH 
(i.e«  ^=±&0°)  lo  Uft  10  cubio  fe«l  uf  water  per  »e<]oti(l  figainat  n  head  of  50  feet 

Ai»mnie  there  are  two  euotion  pipes  and  that  the  water  ei^tera  the  vheelfrto 
both  itde«,  a^  ti)  Fig,  S7S,  ako  that  Uie  velocity  in  the  auction  arid  delii^^  ftip!« 
ftnd  the  radial  velocitj  through  the  wheel  are  6  feet  per  eecondt  s^d  the  manooKtik 
efficiei^f^y  ia  75  per  cent. 

Fir&t  to  find  lu.    . 


■7fi'-^  =  50, 


Sinoe  ihe  bladei  aire  mdi&l, 

from  which  t*!  :=  46  feet  per  l^ee, 

To  find  Che  diameter  of  the  suction  pipes. 
The  diseharse  is  10  oubk  feet  per  second,  therefore 

4 

from  which  ii  =  l*03'^12|". 

If  the  radiiiB  R  of  the  ^xteroal  circ^mferenoe  be  taken  as  2r  and  r  is  taken  < 
to  the  radiua  of  the  auction  pipes,  then  R  =  12|",  and  the  namher  of 
p«T  second  win  be 

Th«  felooit;  of  the  inner  edge  of  the  vane  ia 
p=2S  feel  per  see. 


Engineer,  1903* 


CENTRIFUGAL   PUMPS  405 

The  inolination  of  the  vane  at  inlet  that  the  water  may  move  on  to  the  vane 
^tlioQt  abock  U 

tan<?=A, 
lad  the  water  when  it  leaves  the  wheel  makes  an  angle  a  with  v^  such  that 

tana=^. 
If  there  are  guide  Uadea  sarronnding  the  wheel,  a  gives  the  inclination  of  these 


=  •268' 


The  width  of  the  wheel  at  discharge  is 

Q  10 


».D.6'"». 206x6 
=ft|  inches  about. 
The  width  of  the  n^ieel  at  inlet =6J  inches. 

226.  The  centrifligal  head  impressed  on  the  water  by 
thm  wheeL 

Head  against  which  a  pump  vrill  commence  to  discharge.  As 
shown  on  page  335,  the  centrifugal  head  impressed  on  the  water  as 
it  passes  through  the  wheel  is 

^^-2g  2g^ 
1>at  this  is  not  the  lift  of  the  pump.  Theoretically  it  is  the  head 
which  will  be  impressed  on  the  water  when  there  is  no  flow 
through  the  wheel,  and  is  accordingly  the  difference  between  the 
prosonre  at  inlet  and  outlet  when  the  pump  is  first  set  in  motion ; 
or  it  is  the  statical  head  which  the  pump  will  maintain  when 

ronning  at  its  normal  speed.    If  this  is  less  than  — ^-^ ,  the  pump 

theoretically  cannot  start  lifting  against  its  full  head  without 
being  speeded  up  above  its  normal  velocity. 

The  centrifugal  head  is,  however,  always  greater  than 

2g     2g' 
as  the  water  in  the  eye  of  the  wheel  and  in  the  casing  surrounding 
the  wheel  is  made  to  rotate  by  friction. 

For  a  pump  having  a  wheel  seven  inches  diameter  surrounded 
by  a  circular  casing  20  inches  diameter,  Stanton*  found  that,  when 
the  discharge  was  zero  and  the  vanes  were  radial  at  exit,  he  was 

— s— ,  and  with  curved  vanes,  i>  being  30  degrees,  he  was  —^ — . 

For  a  pump  with  a  spiral  case  surrounding  the  wheel,  the 
centrifugal  head  he  when  there  is  no  discharge,  cannot  be  much 

greater  than  ^ ,  as  the  water  surrounding  the  wheel  is  prevented 

from  rotating  by  the  casing  being  brought  near  to  the  wheel  at 
one  point. 

♦  Proceedings  Inst,  M.  £.,  1903. 


406 


HY|>RAriJCS 


Parsons  foimd  for  a  pomp  having  a  wheel  14  inches  diameter 
with  radial  vanes  at  outlet,  and  ninning  at  300  reTolatians  per 

minute,  that  the  head  maintained  without  discharge  was    ^    , 

and  with  an  Appold*  wheel  running  at  320  revolutions  per  minitte 

the  statical  head  was  -^ — - .     For  a  pump,  with  spiral  casing, 

having  a  rotor  1*54  feet  diameter,  the  least  velocity  at  which 
it  commeTiced  to  discharge  against  a  head  of   14"67  feet  wn& 

392  revolutions  per  minute^  and  thus  he  was    ^  "^  >  ^^^  ^^^  1^*^ 

velocity  against  a  head  of  17'4  feet  was  424  revolutions  per 

minute  or  K  was  again     ^^—^  .    For  a  pump  with  circular  casing 

1  '05t'  * 
larger  than  the  wheel,  he  was     ^   — .    For  a  pump  having  guide 

passages  surrounding  the  wheel,  and  outside  the  guide  passages 
a  circular  chamber  as  in  Fig.  275j  the  centrifugal  head  may  alaa 


he 


larger  than  ^;  the  mean  actual  value  for  tJiis  pump 


found  to  be  1067^. 

Stanton  found,  when  the  seven  inches  diameter  wheels  mentioned 
above  discharged  int-o  guide  passages  surrounded    by  a  circuljix 

chamber  *20  inches  diameter,  that  he  was     ^-       when  the  vanes  ot 

the  wheel  were  radial,  and     ,-^    ^  when  <^  was  30  degrees. 

That  the  centrifugal  head  when  the  wheel  has  radial  vanes  is 
likely  to  be  greater  than  when  the  vanes  of  the  wheel  are  set  hnck 
is  t-o  be  seen  by  a  consideration  of  the  manner  in  which  the  water 
in  the  chamber  outside  the  guide  passages  is  probably  set  m 
motion.  Fig,  2B0.  Since  there  is  no  discharge,  this  rotation  cauaot 
be  caused  by  the  water  passing  through  the  pump,  but  mu^  he 
due  to  internal  motions  set  up  in  the  wheel  and  casing,  Th^ 
water  in  the  guide  chamber  cannot  obviously  n.*tate  about  die 
axis  0,  but  there  is  a  tendency  for  it  to  do  so,  and  consequeuily 
stream  line  motions,  as  shown  in  the  figure,  are  prt)bably  «i 
up.  The  layer  of  water  nearest  the  outer  circumference  of  the 
wheel  will  no  doubt  be  dragged  along  by  friction  in  the  directs 
shown  by  the  arrow,  and  water  will  flow  from  the  outer  casing  to 
take  its  place ;  the  stream  lines  will  give  motion  to  the  wai/er  in  ^ 
the  outer  casing. 

*  See  pftge  4l€. 


CTNTEIFUOAL   PUMPS 


407 


Wlien  the  vanes  m  the  wheel  are  radial  and  as  long  as  a  vane  is 
'mo^'iTig  between  any  two  guide  vanes,  the  straight  vane  prevents 
the  frictioii  between  the  water  outside  the  wheel  and  that  insidsj 
from  dragging  the  wat^r  backward:^  along  the  vanej  but  when  the 
vane  is  set  back  and  the  angle  <^  is  greater  than  90  degrees,  there  j 
will  bt^  a  tendency  for  the  water  in  the  wheel  to  tnove  backwardi 
while  that  in  the  guide  chamber  moves  forward,  and  conseciuently 
the  velocity  of  the  stream  liTiea  in  the  casing  will  be  less  in  the 
latter  caee  than  in  the  former.  In  either  case,  the  general 
direction  of  fiqw  of  the  stream  liiieSj  in  the  guide  chamber,  will 
b©  in  the  direction  of  rotation  of  the  wheelj  but  due  to  frtction 
and  eddy  raotions,  even  w4th  radial  vaneSj  the  velocity  of  the  stream 


Fig.  2m. 

^  will  he  less  than  the  velocity  I'l  of  the  periphery  of  the  wheel. 

outride  the  guide  chambers  the  velocity  of  rotation  will  b© 

than  i\.     In  the  outer  chamber  it  is  to  he  expected  that  the 

rater  will  rotate  as  in  a  free  vortex,  or  itB  velocity  of  whirl  will 

be  inversely  proportional  to  the  distance  from  the  centre  of  the 

TOtor,  or  will  rotate  in  some  manner  approximating  to  this. 

The  liead  which  a  pumpj  with  a  vortes^  chambeTy  tcill  th^oreti^ 
llif  rnaifUain  when  the  discharge  is  zero.    In   this  case  it  is 
[)bable  that  as  the  discharge  approaches  zero,  in  addition  to  the 
ktcfT  in  the  wheel  rotating,  the  water  in  the  vortei  chamber  will 
also  rotate  because  of  friction. 


408  HYDRAUUCS 

The  centrifagal  head  dne  to  the  water  in  the  wheel  is 

If  R= 2r,  this  becomes  -j  ^ . 

The  centrifugal  head  due  to  the  water  in   the   chamber  is, 
Fig.  281, 

f^wv^dr 

Jr^    gn  ' 
To  and  Vo  being  the  radius  and  tangential  velocity  respectively  of 
any  ring  of  water  of  thickness  dr. 


Fig.  281. 

If  it  be  assumed  that  t;oro  is  a  constant,  the  centrifugal  head 
due  to  the  vortex  chamber  is 


g    k   n'     2g\Tj     njJ' 
The  total  centrifugal  head  is  then 

^'-2g     2g^2g\rJ     R«,V ' 
If  rto  is  2r  and  R^  is  2r«„ 


2g 
The  conditions  here  assumed,  however,  give  K  too  high.    In 

Stanton's  experiments  he  was  only  — ^ — -  .     Decouer  from  experi- 

^g 


CENTRIFUGAL  PUMPS  409 

ments  on  a  small  pomp  with  a  vortex  chamber,  the  diameter  being 

l'3t'  * 
aboat  twice  the  diameter  of  the  wheel,  found  he  to  be  -rr-^  . 

Let  it  be  assomed  that  K  is  -^  in  any  pump,  and  that  the  lift 

of  the  pump  when  working  normally  is 

7     e Vi  Vi     e  W  -  Vit^  cot  4^) 
/i,  =  —     = . 

Then  if  fe  is  greater  than  v^-  \  the  pump  will  not  commence  to 

discharge  unless  speeded  up  to  some  velocity  Va  such  that 
mvj      efa*--t;ittiCot<^) 

^g  ^  g 

After  the  discharge  has  been  commenced,  however,  the  speed 
may  be  diminished,  and  the  pump  will  continue  to  deliver  against 
the  given  head*. 

For  any  given  values  of  m  and  e  the  velocity  fa  at  which  delivery 
commences  decreases  with  the  angle  ^.  If  ^  is  90  or  greater  than 
©0  degrees,  and  m  is  unity,  the  pump  ynW  only  commence  to 
discharge  against  the  normal  head  when  the  velocity  is  t'l,  if  the 
manometric  efficiency  e  is  less  than  0*5.  If  <^  is  30  degrees  and  m 
is  unity,  v^  is  equal  to  Vi  when  e  is  0*6,  but  if  <^  is  150  degrees  rj 
is  equal  to  t?i  when  e  is  0*428. 

Nearly  all  actual  pumps  are  run  at  such  a  speed  that  the 
centrifugal  head  at  that  speed  is  greater  than  the  actual  head 
against  which  the  pump  works,  so  that  there  is  never  any 
difficulty  in  starting  the  pump.  This  is  accounted  for  (1)  by  the 
low  manometric  efficiencies  of  actual  pumps,  (2)  by  the  angle  <t> 
never  being  greater  than  90  degrees,  and  (3)  by  the  wheels  being 
surrounded  by  casings  which  allow  the  centrifugal  head  to  be 

greater  than  ^. 

It  should  be  observed  that  it  does  not  follow,  because  in  many 
cases  the  manometric  efficiency  is  small,  the  actual  efficiency  of 
the  pump  is  of  necessity  low.     (See  Fig.  286.) 

227.  Head-velocity  curve  of  a  centrifugal  pump  at  zero 
discharge. 

For  any  centrifugal  pump  a  curve  showing  the  head  against 
which  it  will  start  pumping  at  any  given  speed  can  easily  be 
determined  as  follows. 

On  the  delivery  pipe  fit  a  pressure  gauge,  and  at  the   top 

*  See  pages  411  aud  410. 


410 


HYDRAULICS 


of  the  suction  pipe  a  vacuum  gauge.  Start  the  pomp  with 
the  delivery  valve  closed,  and  observe  the  pressure  on  the  two 
gauges  for  various  speeds  of  the  pump.  Let  p  be  the  absolute 
pressure  per  sq.  foot  in  the  delivery  pipe  and  pi  the  absolute 

pressure  per  sq.  foot  at  the  top  of  the  suction  pii)ey  then  ^  -  ^ 
is  the  total  centrifugal  head  K- 


teoo       leoo       200Q       2200 

RevoUttums  per  Minute. 

Fig.  282. 


240? 


A  curve  may  now  be  plotted  similar  to  that  shown  in  Fig.{282 
which  has  been  drawn  from  data  obtained  from  the  pump  shown 
in  Fig.  275. 

When  the  head  is  44  feet,  the  speed  at  which  delivery  would 
just  start  is  2000  revolutions  per  minute. 

On  reference  to  Fig.  293,  which  shows  the  discharge  under 
different  heads  at  various  speeds,  the  discharge  at  2000  revolutions 
per  minute  when  the  head  is  44  feet  is  seen  to  be  12  cubic  feet 
per  minute.  This  means,  that  if  the  pump  is  to  discharge  againrt 
this  head  at  this  speed  it  cannot  deliver  less  than  12  cubic  to 
per  minute. 

228.  Variation  of  the  discharge  of  a  oentriftigal  puiiV 
with  the  head  when  the  speed  is  kept  constant*. 

Head-discharge  curve  at  constant  velocity.  If  the  speed  of* 
centrifugal  pump  is  kept  constant  and  the  head  varied,  the  dis- 
charge  varies  as  shown  in  Figs.  283,  285,  289,  and  292. 

*  See  also  page  418. 


CENTRIFUGAL   PUMPS 


411 


The  curve  No.  2,  of  Fig.  283,  shows  the  variation  of  the  head 
with  discharge  for  the  pump  shown  in  Fig.  275  when  running  at 
1950  revolutions  per  minute;  and  that  of  Fig.  285  was  plotted 
from  experimental  data  obtained  by  M.  Bateau  on  a  pump  ha\nng 
a  wheel  ll'S  inches  diameter. 

The  data  for  plotting  the  curve  shown  in  Fig.  289*  was 
obtained  from  a  large  centrifugal  pump  having  a  spiral  chamber. 
In  the  case  of  the  dotted  curve  the  head  is  always  less  than  the 
centrifugal  head  when  the  flow  is  zero,  and  the  discharge  against 
a  given  head  has  only  one  value. 


RadilMod^  of  How  fronhWheA. 
Fig.  283.     Head-discharge  curve  for  Centrifugal  Pump.    Velocity  Constant. 


Fig.  284.     Velocity-discharge  curve  for  Centrifugal  Pump.     Head  Constant. 

In  Fig.  285  the  discharge  when  the  head  is  80  feet  may  be 
either  '9  or  3*5  cubic  feet  per  minute.  The  work  required  to  drive 
Che  pamp  will  be  however  very  different  at  the  two  discharges, 
teid,  as  shown  by  the  curves  of  efficiency,  the  actual  efficiencies 
for  the  two  discharges  are  very  different.  At  the  given  velocity 
tlierefore  and  at  80  feet  head,  the  flow  is  ambiguous  and  is 
Unstable,  and  may  suddenly  change  from  one  value  to  the  other, 
^>r  it  may  actually  cease,  in  which  case  the  pump  would  not  start 
^gain  without  the  velocity  Vi  being  increased  to  70*7  feet  per 
^lecond.     This  value  is  calculated  from  the  equation 

•  Proeeedingi  InsL  Mech.  Engs.,  1903. 


412 


HYDRAULICS 


the  coefficient  m  for  this  pump  being  1"02.  For  the  flow  to  b 
stable  when  delivering  against  a  head  of  80  feet,  the  pnmp  shool 
be  run  with  a  rim  velocity  greater  than  70*7  feet  x)er  second,  i 
which  case  the  discharge  cannot  be  less  than  4^  cubic  feet  pe 
minate,  as  shown  by  the  velocity-discharge  curve  of  Fig.  2K 
The  method  of  determining  this  curve  is  discussed  later. 

Pump  Wheel  flScUanv. 
Rew,  per  minute  1290. 


Fig.  2S5. 


1  Jl  3 

DisduMT^  in,  cfL  per  mJav. 


Fig.  286. 


Fig.  287. 


Example,  A  oentrifngal  pump,  when  disoharging  normally,  has  a  peripher 
velocity  of  50  feet  per  second. 

Tbe  angle  ^  at  exit  is  30  degrees  and  the  manometric  effioienoy  is  60  per  ceo 
The  radial  velocity  of  flow  at  exit  is  '^Jh. 

Determine  the  lift  h  and  the  velocity  of  the  wheel  at  which  it  will  start  delivei 
nnder  fall  head. 

S      _ 
V=60-(2VA)coBl30 


-50-1-nJh. 


CENTRIFUGAL  PUMPS  413 

9 
from  whieh  A =87  feet. 

Ijei  Oj  be  the  Telocity  of  the  rim  of  the  wheel  at  which  pumping  commences. 
Then  Mraming  the  centrifogal  head,  when  there  is  no  discharge,  is 

r,=48'6  ft.  per  sec. 

229.    Bemonilli's  equations  applied  to  centrifagal  pumps. 

Consider  the  motion  of  the  water  in  any  passage  between  two 
consecutive  vanes  of  a  wheel.  Let  p  be  the  pressure  head  at 
inlet,  pi  at  outlet  and  pa  the  atmospheric  pressure  per  sq.  foot. 

If  the  wheel  is  at  rest  and  the  water  passes  through  it  in 
the  same  way  as  it  does  when  the  wheel  is  in  motion,  and  all 
losses  are  neglected,  and  the  wheel  is  supposed  to  be  horizontal,  by 
Bemouilli's  equations  (see  Figs.  277  and  278), 

w     2g     w     2g     ^^^• 

But  since,  due  to  the  rotation,  a  centrifugal  head 

'•=1:1 <^) 

is  impressed  on  the  water  between  inlet  and  outlet,  therefore, 

w     2g     w      2g      2g     2g  ^'*''' 

**'  w     w     2g     2g*  2g     2g  ^*^- 

From  (3)  by  substitution  as  on  page  337, 

w^2g      w^2g*    g    -  g      •••^^^. 

and  when  U  is  radial  and  therefore  equal  to  Uy 

E.  +  U!«=£^^+Y..^i    (6). 

w      2g     w     2g       g  ^^ 

If  now  the  velocity  Ui  is  diminished  gradually  and  without 

shock,  so  that  the  water  leaves  the  delivery  pipe  with  a  velocity 

U4,  and  if  frictional  losses  be  neglected,  the  height  to  which  the 

water  can  be  lifted  above  the  centre  of  the  pump  is,  by  Bernouilli's 

equation, 

w     2g      w      2g    ^'''• 

If  the  centre  of  the  wheel  is  K  feet  above  the  level  of  the  water 
in  the  sump  or  well,  and  the  water  in  the  well  is  at  rest, 

P^  =  K^P^f (8). 

w  w     2g 


414  HTDRAUUCS 

Substituting  from  (7)  and  (8)  in  (6) 

9  ^^ 

=  H.+  |  =  H   (9). 

This  result  verifies  the  fundamental  equation  given  on  page  888. 
^      Further  from  equation  (6) 

Example.  The  centre  of  a  centrifngal  pmnp  is  15  feei  above  Um  level  of  tlit 
water  in  the  sump.  The  total  lift  U  60  feet  and  the  velocity  of  dieehafge  from  flu 
delivenr  pipe  is  5  feet  per  second.  The  angle  0  at  diachaige  is  135  degrees,  and 
the  radial  velocity  of  flow  through  the  wheel  is  6  feet  per  second.  AMnffiipg  tfien 
are  no  losses,  find  the  pressure  head  at  the  inlet  and  oatlet  cironmfiereiioes. 

At  inlet  ^=34'-16'-^ 

tr  64 

=  18*6  feet. 
The  radial  velocity  at  outlet  is 

iii=5  feet  per  second, 
and  y^^.,'..u,.,cot  450^3^^ 

9  V  64'         . 

and  therefore,  ©i' +  5i7j  =  1940 (1), 

from  which  17^  =  41*6  feet  per  second, 

and  V,=46-r,       „  „ 

Then  |l'  =  LLtii>^34feet. 

The  pressure  head  at  outlet  is  then 

tr      IT  2g 

=  45  feet. 
To  find  the  velocity  v^  when  <p  is  made  80  degrees. 

cot  ip=»JSt 
therefore  (1)  becomes  rj'  -  6  /^S .  Vj  =  1940, 

from  which  V]=48*6  ft.  per  sec. 

and  V,=:40 

Then  5l  =  25-4  feet,  and  ?^=63-6  feet. 

2*7  w 

230.    Losses  in  oentrifogal  pumps. 

The  losses  of  head  in  a  centrifugal  pump  are  due  to  the  same 
causes  as  the  losses  in  a  turbine. 

L088  of  head  at  exit.  The  velocity  Ui  with  which  the  water 
leaves  the  wheel  is,  however,  usually  much  larger  than  in  the 
case  of  the  turbine,  and  as  it  is  not  an  easy  matter  to  diminisli 
this  velocity  gradually,  there  is  generally  a  much  larger  loss  of 
velocity  head  at  exit  from  the  wheel  in  the  pump  than  in  the 
turbine. 


CENTRIFUGAL   PUMPS  415 

In  many  of  the  earlier  pumps,  which  had  radial  vanes  at  exit, 

the  whole  of  the  velocity  head  ^  was  lost,  no  special  precautions 

being  taken  to  diminish  it  gradually  and  the  efficiency  was 
constantly  very  low,  being  less  than  40  per  cent. 

The  effect  of  the  angle  4^  on  the  efficiency  of  the  pump.  To 
increase  the  efficiency  Appold  suggested  that  the  blade  should  be 
set  back,  the  angle  4^  being  thus  less  than  90  degrees.  Fig.  272. 

Theoretically,  the  effect  on  the  efficiency  can  be  seen  by 
considering  the  t^iree  cases  considered  in  section  220  and  illustrated 

in  Fig.  279.    When  <^  is  90  degrees  -^  is  '543!,  and  when  <^  is 

30  degrees  -^  is  'SdH.     K,  therefore,  in  these  two  cases  this  head 

is  lost,  while  the  other  losses  remain  constant,  the  efficiency  in 
the  second  case  is  18  per  cent,  greater  than  in  the  first,  and  the 
efficiencies  cannot  be  greater  than  46  per  cent,  and  64  per  cent. 
re8i)ectively. 

In  general  when  there  is  no  precaution  taken  to  utilise  the 
energy  of  motion  at  the  outlet  of  the  wheel,  the  theoretical  lift  is 

^'"■7""2^ ^^^' 

and  the  maximum  possible  manometric  efficiency  is 

Sabstituting  for  Vi,  i^i  -  tti  cot  <^,  and  for  Uj',  V,'  +  tt,', 

TT        V        "l"  »J. 

^'  =  2^-2^^^'^^*' 
,  ^ .      (vi-  u,  cot  <^)'  +  th^ 

2  W  —  ^\th  cot  <^) 
_  Vi^  -  Vg^  cosec'  4> 

2Vi  (Vi  -  Va  cot  <^)  ' 

When  r,  is  30  feet  per  second,  Ux  5  feet  per  second  and  <t> 
150  degrees,  e  is  56  per  cent,  and  when  <^  is  90  degrees  e  is 
48*5  per  cent. 

Experiments  also  show  that  in  ordinary  pumps  for  a  given  lift 
and  discharge  the  efficiency  is  greater  the  smaller  the  angle  <t>. 

Parsons*  found  that  when  i>  was  90  degrees  the  efficiency  of  a 
pamp  in  which  the  wheel  was  surrounded  by  a  circular  casing 
was  nearly  10  per  cent,  less  than  when  the  angle  <t>  was  made 
about  165  degrees. 

•  Proceedings  Inst,  C,  E,,  Vol.  xLvn.  p.  272. 


416 


HYBRA0LICS 


Stanton  found  that  a  pump  7  inches  diameter  having  niiial 
vanes  at  discharge  had  an  efficiency  of  S  per  cent,  less  than  wto 
the  iiTigle  ^  at  delivery  was  150  degrees.  In  the  first  case  tht 
maximum  actual  efficiency  was  only  39"6  per  cent,,  and  in  thi 
second  vnse  50  i>er  cent* 

It.  han  been  suggested  by  Dr  Stanton  that  a  second  reason  fo 

the  greater  efficiency  of  the  pump  having  vanee  curved  hack  ^ 

outlet  is  to  be  found  in  the  fact  that  with  these  vanes  the  variatitm 

of  the  relative  velocity  of  the  water  and  the  wheel  is  less  than 

L  when  the  vanes  are  radial  at  outlet.     It  has  been  shown  erperi- 

l  mentally  that  when  the  section  of  a  stream  is  diverging,  that  is 

the  velocity  is  diminishing  and  the  pressure  increasing,  there  is 

a  tendency  for  the  stream  lines  to  flow  backwards  towards  die 

sections  of  least  pressure.     Tliese  return  stream  lines  cause  a  losi 

of  eiierg^'^  by  eddy  motions.     Now  in  a  pump,  when  the  vanes  am 

radial,  there  is  a  greater  difference  between  the  relative  velcunty 

of  the  water  and  the  vane  at  inlet  and  outlet  than  when  the  angb 

L  ^  is  \em  than  fK)  degrees  (see  Fig.  279),  and  it  is  prt^bahle  tliere- 

i  fore  that  there  is  more  loss  by  eddy  motions  in  the  wheel  in  the 

former  case. 

Liiss  of  head  at  entry.  To  avoid  loss  of  head  at  entry  the  vanu 
must  bo  pamnel  to  the  relative  velocity  of  the  water  and  the 
vane, 
I  Unless  guide  blades  are  provided  the  exact  direction  in  whicl 
pthe  water  approaches  the  edge  of  the  vane  is  not  knoivn.  If  there 
were  no  friction  between  the  water  and  the  eye  of  the  wheel  it 
would  be  expected  that  the  stream  lines,  which  in  the  suction  pi|K 
are  parallel  to  tlie  sides  of  the  pipe,  would  be  simply  turned  to 
approach  the  vanes  radially. 

It  has  already  been  seen  that  when  there  is  no  flow  tbe  water 
in  the  eye  of  the  wheel  is  made  to  rotate  by  friction,  and  il  ii 
probable  that  at  all  flows  the  water  has  some  rotation  in  the  ep 
of  the  wlieel,  but  as  the  delivery  increases  the  velocity  of  rc»tati«i 
probably  diminishes.  If  the  water  has  rotation  in  the  swb© 
direction  as  the  wheel,  the  angle  of  the  vane  at  inlet  will  c\mx}f 
have  to  be  larger  for  no  shock  than  if  the  flow  is  radial.  Thi* 
the  water  has  rotation  before  it  strikes  the  vanes  seems  te  he 
indicated  by  the  experiments  of  Mr  Livens  on  a  pumpj  the  Tanes 
of  which  were  nearly  radial  at  the  inlet  edge.  (See  section  33^.) 
Tlie  efficiencies  claimed  for  this  pnmp  are  so  high,  that  thm 
conld  have  been  very  little  loss  at  inlet* 

If  the  pump  has  to  work  under  variable  conditions  and  the 
water  be  assumed  to  enter  the  wheel  at  all  discharges  in  the  same 
direction,  the  relative  velocity  of  the  water  and  the  edge  of  the 


CENTRIFUGAL  PUMPS  417 

»n  only  be  parallel  to  the  tip  of  the  vane  for  one  discharge, 
;  other  discharges  in  order  to  make  the  water  move  along 
me  a  sadden  velocity  must  be  impressed  upon  it,  which 

a  loss  of  energy. 
b  t«s,  Pig.  288,  be  the  velocity  with  which  the  water  enters  a 

and  0  and  v  the  inclination 
jlocity  of  the  tip  of  the  vane        i*  -  u^  ->l 
ft  respectively. 

e  relative  velocity  of  th  and  v 
,  the  vector  difference  of  u% 

e  radial  component  of  flow 

fh  the  opening  of  the  wheel 

be  equal  to  the  radial  com-        '  j,.     ^  • 

t   of    Uij  and   therefore  the 

e  velocity  of  the  water  along  the  tip  of  the  vane  is  Vr. 

Ui  is  assumed  to  be  radial,  a  sudden  velocity 

u,  -  t?  -  tta  cot  0 
us  to  be  given  to  the  water. 

ih  has  a  component  in  the  direction  of  rotation  u»  will  be 
Lshed. 

has  been  shown  (page  67),  on  certain  assumptions,  that  if 
Y  of  water  changes  its  velocity  from  Va  to  Vd  suddenly,  the 

ost  is  ^-^^ —     >  or  is  the  head  due  to  the  change  of  velocity. 

this  case  the  change  of  velocity  is  u«,  and  the  head  lost  may 

lably  be  taken  as  -^.    K  A;  is  assumed  to  be  unity,  the 

ve  work  done  on  the  water  by  the  wheel  is  diminished  by 
u^_  (r  -  tta  cot  Oy 

now  this  loss  takes  place  in  addition  to  the  velocity  head 
lost  outside  the  wheel,  and  friction  losses  are  neglected, 

jL_Vi^i     tTi'     (t?-^,cot^)' 

=  |L',|L'eo8ec'»-^^"^"^^^)' 
2g     2g  ^  2g 


=  —  -    ^  a  cosec'  4^  - 


(.-gcot^y 


2g     2^^^^^^^  ^  2g 

v^     u*     u,'  -  .     2t?«,  c< 

=  o —  oT  "  o~  cosec'  4>  +  — s~" 

2g     2g     2g  2g 

H.  Vk 


r,'     u*      u,'  ,..2t?«,cot^        a      .a>,  ,.v 


418 


HTDRJl0LtCS 


Es^Mmpte.  The  rftdi&l  Tdo«itj  of  flow  thfoogh  b  pump  w  S  f««|  per 
The  u^a  ^  ii  SO  degriec  and  the  ADgle  i  if  15  de^ltem..  Tbe  reiodt? 
oqut  cireiunfiertoae  U  50  f««t  per  »ee.  mnd  the  mdliu  is  twic^  ihas  of  tae 

Find  Ikw  tbtonticy  lift  on  the  u«&m|>tiofi  ih&l  the  wtM»l«  of  Ihe  kjnme  mag 
U  lo«t  &t  ezii* 

=  S7*5  feet 


The  theorvtiefti  lid   &«glecticg  aJl  Io« 
^f&cienej  U  Iberefore  68  |ier  cent. 

231.     Variation  of  the  head 
sp«ed  of  a  centriftigal  pump. 

It  is  of  intereet  to  study  by  » 
the  variation  of  the  dischargic 
wbeo  fc  ta  constant,  and  th( 
discharge   when   the   relocitj 
coiui>are    the    results    with 
experiment. 

The  full  curve  of  Fig.  2 
with  the  discharge  when  the 
Tim  data  for  which  the  cm. 
the  figure. 


Ml   ia  64*9  foet,  u^  the  naammm 

with  discharge  and  witb  iM 

^  of  equation  (])^  sed^j 
th  the  velocity  of  the 
ition  of  the  head  wiUi 
e  \mmp  {»  conetant,  hid  ^ 
tofll    results    obtained  kam 

i  the  ranatioBs  of  the  hai 
of  a  wheel  is  kept  fionstot 
been  plotted  is  indicated  to 


Normal  rajdtuML^tlacify^fSlaiviS' 

\t        \2      _  3        ^         \5  ^'^%       b 


Fi^.  289,     Head'diflcharge  carve  at  constant  Telocitj. 


WHion  the  discharge  is  zero 


2g     2g 


10^5  feet. 


Tlie  velocity  of  flow  -f-  at  outlet  haa  been  assumed  equal  h^ 

2  at  inlet. 

Values  of  1,  2,  etc.  were  given  to  ^  and  the  correspondiji^ 
valuiiis  of  /*  found  from  equation  (1). 


CENTRIFUGAL  PUMPS 


419 


WTieii  the  discharge  is  normal,  that  is,  the  water  enters  the 
thoat  shock,  ^  is  4  feet  and  ^  is  14  feet.    The  theoretical 
ing  no  losses  is  then  28  feet  and  the  manometrio 
cy  is  thus  SO  per  cent.     For  leas  or  greater  values  of  ^ 

^e^atl   diminishes  and  also  the  efficiency* 

riie   ciir%*e  of   Fig.  290  shows  how  the  flow  varies  with  the 

city  for  a  constant  value  of  h^  which  is  taken  as  12  feet. 


if-  2W1. 


Rjadinl  Vttod^  thrmtph  WfieeL 
VdocitjHiiicharg©  curve  at  cans  taut  head  for  Ceatrifugal  Pump. 


It  wU  be  seen  that  when  the  velocity  Vi  is  31*9  feet  per  second 
&  velocity  of  discharge  may  be  either  zero  or  8*2  feet  per  second, 
1^  meatm  that  if  the  head  is  12  feet,  the  pnmp^  theoretically, 
ill  only  start  when  the  velocity  is  Zl'Q  feet  per  second  and  the 
Aocity  of  discharge  will  suddenly  become  8"2  feet  per  second, 
'  iMJw  the  velocity  v^  is  diminished  the  pump  still  continues  to 
bdmrge^  and  mil  do  so  as  long  as  i?i  is  great-er  than  26*4  feet  per 
Nx»tid.  The  flow  is  however  tmstable,  as  at  any  velocity  Vc  it  may 
llddenly  change  from  CE  to  CD,  or  it  may  suddenly  cease,  and  it 
BU  not  start  again  until  Vj  is  increased  to  Sl'O  feet  per  second. 

232,  The  effect  of  the  variation  of  the  centrifugal  besul 
Hd  the  loss  by  firiction  on  the  discharge  of  a  piunp* 

If  then  the  lo.s.ses  at.  inlet  and  outlet  were  as  above  and  were 
be  only  lusses^  and  the  centrifugal  head  in  an  actual  pump  was 
^ual  to  the  theoretical  centrifugal  head,  the  pump  could  not  be 
tede  to  deliver  water  against  the  normal  head  at  a  small  velocity 
I  discharge.  In  the  case  of  the  pump  considered  in  section  281, 
^■btd  not  safely  be  nm  with  a  rim  velocity  less  than  31*9  ft, 
PPKec.,  and  at  any  greater  velocity  the  radial  velocity  of  flow 
[>iild  not  be  less  than  8  feet  per  second. 


490  HTDRAULIGS 

In  mctoal  pumpfi,  however,  it  lias  been  seen  that  the  oentrifagal 
head  at  oommenoement  is  greater  than 

2^     2sr- 

lliere  is  also  loss  of  head,  which  at  high  velocities  and  in  smsH 
pumps  is  considerable,  dne  to  friction.  These  two  causes  consider- 
ably modify  the  head-discharge  curve  at  constant  velocity  and  the 
Tidocity-discharge  curve  at  constant  head,  and  the  centrifugal 
head  at  the  normal  speed  of  the  pump  when  the  discharge  is  zn% 
is  generally  greater  than  any  head  under  which  the  pump  works, 
and  many  actual  pumps  can  deliver  variable  quantities  of  wator 
against  the  head  for  which  they  are  designed. 

The  centrifugal  head  when  the  flow  is  zero  is 

m  being  generally  equal  to,  or  greater  than  unity.  As  the  flow 
increases,  the  velocity  of  whirl  in  the  eye  of  the  wheel  and  in 
the  casing  will  climinish  and  the  centrifugal  head  will  there&nre 

Let  it  be  assumed  that  when  the  velocity  of  flow  is  u  (supposed 
constant)  the  centrifugal  head  is 

k  and  n  being  constants  which  must  be  determined  by  experiment 
When  II  is  zero 

and  if  «•  is  known  Iq  can  at  once  be  found. 

Let  it  further  be  assumed  that  the  loss  by  friction*  and  ed 

C*tt* 

melons,  apart  from  the  loss  at  inlet  and  outlet  is  -^  . 


*  The  loss  of  besd  by  frieiioii  will  no  doubt  depend  not  only  upon  «  bat  slM 
apon  the  Tek>eity  r^  of  the  wheel,  and  should  be  written  as 

^  +  ^+etc.. 

or,  as  ^"^^'*'  ••®*^- 

If  it  be  snppoeed  it  ean  be  expressed  by  the  latter,  then  the  oonvetion 

if  proper  Tallies  are  giTen  to  &,  n^  and  Jk|,  takes  into  aoooont  the  Yariatioo  of  tki 
OsntrifUgal  head  and  also  the  firietion  head  «,. 


Lz: 


\ 


CENTRIFUGAL   PUMPS 


421 


The  gross  bead  h  is  then, 


Vi 


v'      t^ 


''  =  2^-2^-2^°°^'*^ 


2vu  cot  S 


-w'cof^ 


2ff 
(kvi  —  wit)*     (^V? 

2ff  23 


.(2). 


-^(«*-'> 


If  BOW  the  head  Ai  and  flow  Q  be  detepmined  experimentally, 
the  difference  between  h  as  determined  from  equation  (1),  page  4 J  7, 
atiid  the  experimental  value  of  A,  muat  be  equal  to 

2g  2^~W^    'W 

~   2g  2g  2^  ' 

Iti  bein^  equal  to  (c*  — n*). 

The  coefficient  fe  being  known  from  an  experiment  when  u  is 
sero,  two  other  exx)eriment9  giving  corresponding  values  of  k  and 
i»  will  determine  the  coefficients  n  and  fcj* 

The  head-discharge  curve  at  constant  velocity,  for  a  pump  such 
as  the  one  already  considered,  would  approximate  to  the  dotted 
Corve  of  Fig,  289.  This  curve  has  been  plotted  from  equation  (2), 
by  taking  k  as  0*5,  n  as  7*64  and  h%  as  —  38. 

Substituting  \m)ue8  for  fc,  n,  fti,  cosec  ^  and  cot  ^j  equation  (2) 
beoomea 

.  *  =  ^*-%H^H.C«' (3), 


2ff    ■     2(7 
■  and  Ct  being  new  coefficients ;  or  it  may  be  written 


».5^-.^.C.Q. W, 

Q  being  the  flow  in  any  desired  units,  the  coefficients  C^  and  Ct 
'varying  with  the  units.  If  equation  (4)  is  of  the  correct  form, 
three  experiments  will  determine  the  constants  m,  Ca  and  Ci 
din^ctly,  and  having  given  values  to  any  two  of  the  three 
^variables  h,  \\  and  Q  the  third  can  be  found, 

233.  The  effect  of  the  diminution  of  the  centrifugal  head 
and  the  inoreaa^  of  the  friction  head  as  the  flow  increaaes,  on 
%he  velocity -discharge  curve  at  constant  head* 

Using  the  corrected  equation  (2),  section  232,  and  the  given 
*ir&laes  of  ^5  W|  and  h  the  dotted  curve  of  Fig.  200  has  been  plotted. 

Frtjra  the  dotted  curve  of  Fig>  289  it  is  seen  that  u  cannot 
Im.*  greati^r  than  5  feet  when  the  head  is  12  feet,  and  therefor©  the 
Hew  curve  of  Fig.  290  is  only  drawn  to  the  point  where  tt  is  5. 

ITie  panip  starts  delivering  when  v  is  27'7  feet  per  second  and 
the  discharge  increases  gradually  as  the  velocity  increases. 


MUM 


422  HTDRAUUGS 

The  pump  wfll  deliver,  therefore,  water  under  a  head  of 
12  feet  at  any  velocity  of  flow  from  zero  to  5  feet  per  secoBd. 

In  sach  a  pomp  the  manometric  efficiency  must  have  its 
mayiTnnTn  valae  when  the  discharge  is  zero  and  it  cannot  be 
greater  than 

w 

fi*  —  txUx  cot  0 ' 

9 
This  is  the  case  with  many  existing  pomps  and  it  explains  wliy, 
when  running  at  constant  speed,  they  can  be  made  to  give  any 
discharge  varying  from   zero   to  a  maximum,   as   the    head  is 
diminished. 

234.    Special  arrangements  for  converting  the  velocity 

TP 
head  ^  with  which  the  water  leaves  the  wheel  into  preasoro 

head. 

The  methods  for  converting  the  velocity  head  with  which  tlie 
water  leaves  the  wheel  into  pressure  head  have  been  indicated  on 
page  3W.    They  are  now  discussed  in  greater  detail. 

TfcowwowV  vortex  or  tthirlpool  chamber.  Professor  James 
Thomson  first  suggested  that  the  wheel  should  be  surroimded  by 
a  chamber  in  which  the  velocity  of  the  water  should  gradoftlly 
change  from  Ui  to  u^  the  velocity  of  flow  in  the  pipe.  Such  a 
chamber  is  shown  in  Fig.  274.  In  this  chamber  the  water  forms 
a  frve  vortex,  so  called  because  no  impulse  is  given  to  the  water 
while  moving  in  the  chamber. 

Any  fluid  particle  ai,  Fig.  281,  may  be  considered  as  moving 
in  a  circle  of  radius  r#  with  a  velocity  ro  and  to  have  also  a 
radial  velocity  u  outwards. 

Let  it  be  supposed  the  chamber  is  horizontal. 

If  W  is  the  weight  of  the  element  in  pounds,  its  momentum 

Wr 
perpendicular  to  the   radius  is  ^  and  the  moment  of   mo- 

Wr  r 
meutum  or  angular  momentum  about  the  centre  C  is  — —, 

9 

For  the  momentum  of  a  body  to  change,  a  force  must  act  upon 

it>  and  for  the  moment  of  momentum  to  change,  a  couple  must  act 

up^^n  the  body. 

But  since  no  turning  effort,  or  couple,  acts  upon  the  element 

after  leaWng  the  wheel   its  moment    of   momentum    must   be 

constant. 


Therefore, 


CENTRIFUGAL  PUMPS  423 

WVqTo 


9 
is  constant  or  v^Vo  =  constant. 

If  the  sides  of  the  chamber  are  parallel  the  peripheral  area  of 
the  concentric  rings  is  proportional  to  r©,  and  the  radial  velocity  of 
flow  t*  for  any  ring  will  be  inversely  proportional  to  Tq,  and  there- 
fore, the  ratio  —  is  constant,  or  the  direction  of  motion  of  any 

Vq 

element  with  its  radias  Vq  is  constant,  and  the  stream  lines  are 
eqaiAngnlar  spirals. 

If  no  energy  is  lost,  by  friction  and  eddies,  Bernouilli's  theorem 
iwill  hold,  and,  therefore,  when  the  chamber  is  horizontal 

2g     2g     w 

is  constant  for  the  stream  lines. 

This  is  a  general  property  of  the  free  vortex. 
If  1^  is  constant 

PT"  +      =  constant. 
2g     w 

Let  the  outer  radius  of  the  whirlpool  chamber  be  Ru,  and 
the  inner  radius  r^.  Let  Vr^  and  Vr^  be  the  whirling  velocities 
at  the  inner  and  outer  radii  respectively. 

Then  since  VqTo  is  a  constant, 

and  ^'  "^  5!;  =  constant. 


w     2g 
w       w       2g       2g 

w       2g  V      njJ' 
WTien  R«,  =  2r«., 

w       w      4'  2<7  * 
If  the  velocity  head  which  the  water  possesses  when  it  leaves 
the  vortex  chamber  is  supposed  to  be  lost,  and  i^i  is  the  head  of 
crater  ahove   the  pump  and  pa  the  atmospheric  pressure,  then 
neglecting  friction 


or 


w 

--h,+ 

Va 

W  ' 

hr- 

w 

23 

w 

If  duoL  i,  »  die  be^s  of  die  psnp  above  the  well,  the  total 

Tlttrcfore 

'^     w     w     2g      2gV     B.V     2g  • 
fiat  from  eqamdaa  (8)  paige  419, 

Tberrfore 

***2g       g        2g"2i^V       K.V- 

This  mi^t  hare  been  wntten  down  at  once  from  equation  (I), 
section  230.     For  clearly  if  there  is  a  gain  of  pressore  head 

in  the  Tortex  chamber  of  ^  ( 1  -  ^"i),  the    velocity  head  to 

be  lost  win  be  leas  by  this  amount  than  when  there  is  no  Yortex 
chamber. 

Snbetitnting  for  Vi  and  Hi  the  theoretical  lift  h  is  now 

,      r,*-r,ttiCot*     tfi'     (ri-t*,cot*)'  R*  r^^.       m 

*=  9  ^ ^— R^'    ^^^-    J 

When  the  discharge  or  rim  velocity  is  not  normal,  there  is » 
farther  loss  of  head  at  entrance  equal  to 

9 
and 


g  2gA^*  2g  R.« 

When  there  is  no  discharge  Vr»  is  equal  to  Vi  and 
"'  g      2gR«»     2g- 


.(2). 


\ 


CENTRIFUGAL   PUMPS  425 


If  R  =  iR«,  and  i?  =  ivi, 

Correcting  equation  (1)  in  order  to  allow  for  the  variation  of 
centrifugal  head  with  the  discharge,  and  the  friction  losses, 
»i'  -  ViUi  cot  ^     tti'     (t'l  -  Ml  cot  ^)*R* 


h  = 


2 


g  2g  2gBJ 

(v  —  u  cot  Oy     Vv^  ^  2nkwvi  _  kiu^ 
2^  ^  2g         2g         2g  ' 

^hich  reduces  to        h=^^'%^^^. 

2g        2g        2g 

The  experimental  data  on  the  value  of  the  vortex  chamber 
^09r  «c,  in  increasing  the  efficiency  is  very  limited. 

Stanton*  showed  that  for  a  pump  having  a  rotor  7  inches 
^diameter  surrounded  by  a  parallel  sided  vortex  chamber  18  inches 
^Uameter,  the  efficiency  of  the  chamber  in  converting  velocity  head 
th>  pressure  head  was  about  40  per  cent.  It  is  however  questionable 
"whetiher  the  design  of  the  pump  was  such  as  to  give  the  best  results 
IKWsible. 

So  far  as  the  author  is  aware,  centrifugal  pumps  with  vortex 
chambers  are  not  now  being  manufactured,  but  it  seems  very 
probable  that  by  the  addition  of  a  well-designed  chamber  small 
oentrifngal  pumps  might  have  their  efficiencies  considerably  in- 
creased. 

235.    Turbine  pomps. 

Another  method,  first  suggested  by  Professor  Reynolds,  and 
now  largely  used,  for  diminishing  the  velocity  of  discharge  Ui 
gradually,  is  to  discharge  the  water  from  the  wheel  into  guide 
passages  the  sectional  area  of  which  should  gradually  increase 
from  the  wheel  outwards.  Figs.  275  and  276,  and  the  tangents  to  the 
tips  of  the  guide  blades  should  be  made  parallel  to  the  direction 
ofU,. 

The  number  of  guide  passages  in  small  pumps  is  generally  four 
or  five. 

If  the  guide  blades  are  fixed  as  in  Fig.  275,  the  direction  of 
the  tips  can  only  be  correct  for  one  discharge  of  the  pump, 
but  except  for  large  pumps,  the  very  large  increase  in  initial  cost 
of  the  pump,  if  adjustable  guide  blades  were  used,  as  well  as 
the  mechaniical  difficulties,  would  militate  against  their  adoption. 

Single  wheel  pumps  of  this  type  can  be  used  up  to  a  head  of 
100  feet  wit  i  excellent  results,  efficiencies  as  high  as  85  per  cent. 
*  Proceedingi  Irut.  C.E.,  1903. 


/ 


4dl  HnMULULiCS 


mvaiff  l«e&  du^cd.  They  mre  now  bem^  used  to  deliver  water 
MKoiiS  kaBd§  cf  o'VBr  dSO  feet,  mud  M.  Bateaa  lias  used  a  single 
^ntf^  ^1^  mdtoK  diameter  Tmming  at  18,000  revelations  per 
snuffle  i£>  detivvr  against  a  Iiead  of  936  feet. 

iMm  ^  Mai  mi  tke  emiramee  to  ike  guide  passages.  If  the 
i»  aie  fixed,  tiie  directicHi  of  the  tips  can  only  be  correct 
c£  xhe  pomp.  For  any  other  discharge  than  the 
^.^rsBhL  i^  dxnKCiGQ  cf  the  water  as  it  leaves  the  wheel  is  not 
piKraljei  <o  ik^  fixed  grnde  and  there  is  a  loss  of  head  due  to 

Ijb%  a  lie  tk^  indiBatioci  of  the  gnide  blade  and  ^  the  vane 
aaiic^  as  ^xii. 

tiS  Ik  fce  Ae  radsal  velocity  of  ^ — 3P^r j 

*:w.  T^KL  RE.  Fig.  291,  is  the  ^^^^^^^^^'"^  ^N.  '^ 
Twxisy  wA  which  the  watm^  leaves    o^ ^ ^  ^^  ^ 

Xii^  radial  i^efcciiy  with  which 
t^  wastnr  e&ters^  the  guide  passages  most  be  «i  and  the  velocity 
adco^  ^le  jirside  isv  therefore,  BF. 

TVre  is  a  swiden  change  of  velocity  from  BE  to  BF,  and  on 
die  iBSSit^spdoii  that  the  Ices  of  head  is  eqnal  to  the  head  due  to  the 
TwaitTv  Tvicvity  FE.  the  head  lost  is 

t  T:  -  Ml  cot4~  Hi  cot  g)^ 

A:  ^rijec  tfee  kxsjs^  of  head  is 

ir-ncot^)* 


*r5d  rise  rbev^r^^x-al  Hft  is 


^9 


r^*  -  r\»t  c\>t  ^  _  (^r  ~  «  cot  BY  _  (ri  -  Ui  cot  ^  ~  tii  cot  a)' 

r^*      r*     ir^nicota     2nicot^ 
^~2g 
Ux^  (cot  *  -^  cot  o)'     tt*  cot*  0 


ij?  2g  2g 


2g  2^         (1)- 

Tv>  ov^r^^t  for  the  dimination  of  the  centrifugal  head  and  to 
allow  fi.>r  friciion^ 

iV_2tr,H.ti,      ,   u^ 
2g  2g  ^^  2g' 

mu$t  be  addeiU  and  the  lift  is  then 
*  ^  «\*  _  r*  ^  2tM*»  cot  a     2ri«  cotf  _  V  (cot  ^  ■^  cot  a)* 
ig '  2g^        2g        ^       2g  2g 

t^cot^^     tV     2fatriUi     fc«i' 
2g      '^  2g  2g  2g   ' 


CENTRIFUGAL  PUMPS 


427 


which,  since  u  can  always  be  written  as  a  multiple  of  Ui,  reduces 
to  the  form 

2gh  =  mvi^  -^  CuiVi-i^  CiUi'' (2). 

Equations  for  the  turbine  pum/p  shown  in  Fig.  275.    Character- 
istic cwrvea.    Taking  the  data 

tf  =  5  degrees,  cot  tf  =  11-43 
<^  =  30       „        cot*=   1-732 
o»  3       „        cot  a  =19-6 
D  =  2-5d 
equation  (2)  above  becomes 

2^A  =  '84»i»  +  48-3i^t7i-587t^* (3). 


piat^UMrge^  igv  Cubic  F^  perlHnu^ 


O  12  8  4.^ 

Sndialf  VAoeUy  aJbBadJb  ffxnv  Ovt  WheA.  Te9t  par  SmcorvdU 

Fig.  292.     Head-disoharge  oarres  at  constant  speed  for  Turbine  Pump. 

From  equation  (3)  taking  r,  as  50  feet  per  second,  the  head- 
discharge  curve  No.  1,  of  Fig.  283,  has  been  drawn,  and  taking  h 
as  35  feet,  the  velocity-discharge  curve  No.  1,  of  Fig.  284,  has  been 
plotted. 

In  Figs.  292 — 4  are  shown  a  as^'ies  of  head-discharge  curves  at 


428 


HTDRAUUGS 


constant  speed,  velocity-discharge  curves  at  constant  head,  and 
head-velocity  curves  at  constant  discharge,  respectively. 

The  points  shown  near  to  the  curves  were  determined  experi- 
mentally, and  the  curves,  it  will  be  seen,  are  practically  the  mean 
curves  drawn  through  the  experimental  points.  They  were  how- 
ever plotted  in  all  cases  from  the  equation 

2gh  =  r087t?i»  +  2'26uiVi  -  62-lu,*, 
obtained  by  substituting  for  m,  C  and  Ci  in  equation  (2)  the  values 
r087,  2*26  and  -62*1  respectively.  The  value  of  m  was  obtuned 
by  determining  the  head  A,  when  the  stop  valve  was  closed,  for 
speeds  between  1500  and  2500  revolutions  per  minute.  Fig.  282. 
The  values  of  C  and  Ci  were  first  obtained,  approximately,  by 
taking  two  values  of  v^  and  Vi  respectively  from  one  of  the 
actual  velocity-discharge  curves  near  the  middle  of  the  series,  for 
which  h  was  known,  and  from  the  two  quadratic  equations  thus 
obtained  C  and  Ci  were  calculated.  By  trial  C  and  Ci  were  then 
corrected  to  make  the  equation  more  nearly  fit  the  remaining 
curves. 


SpmecL  —KeyolBUions  per  liUnuU^. 

Fig.  293.    Velocity-Disoharge  corYes  at  Constant  Head. 

No  attempt  has  been  made  to  draw  the  actual  mean  curves  in 
the  figures,  as  in  most  cases  the  difPerence  between  them  and  the 
calculated  curves  drawn,  could  hardly  be  distinguished.  The 
reader  can  observe  for  himself  what  discrepancies  there  are  between 
the  mean  curves  through  the  points  and  the  calculated  curves.    It 


CENTRIFUGAL  PUMPS 


429 


will  be  seen  that  for  a  very  wide  range  of  speed,  head,  and 
discharge,  the  agreement  between  the  curves  and  the  observed 
points  is  very  close,  and  the  equation  can  therefore  be  used  with 
confidence  for  this  particular  pump  to  determine  its  performance 
under  stated  conditions. 

It  is  interesting  to  note,  that  the  experiments  clearly  indicated 
the  unstable  condition  of  the  discharge  when  the  head  was  kept 
constant  and  the  velocity  was  diminished  below  that  at  which  the 
discharge  commenced. 


Fig.  294.    Head-velocity  carves  at  Constant  Discharge. 


236.    Losses  in  the  spiral  casings  of  centrifugal  pumps. 

The  spiral  case  allows  the  mean  velocity  of  flow  toward  the 
discharge  pipe  to  be  fairly  constant  and  the  results  of  experiment 
seem  to  show  that  a  large  percentage  of  the  velocity  of  the  water 
at  the  outlet  of  the  wheel  is  converted  into  pressure  head. 
Mr  Livens*  obtained,  for  a  pump  having  a  wheel  19^  inches 
diameter  running  at  550  revolutions  per  minute,  an  efficiency  of 
71  per  cent,  when  delivering  1600  gallons  per  minute  against  a 
head  of  25  feet.  The  angle  ^  was  about  13  degrees  and  the  mean 
of  the  angle  0  for  the  two  sides  of  the  vane  81  degrees. 

For  a  similar  pump  21f  inches  diameter  an  efficiency  of  82  per 
cent,  was  claimed. 


*  Proeeedingi  Inst.  Mech,  Engs.^  1903. 


480 


HTDRAULICS 


Tho  author  finds  th©  equation  to  the  head-discharge  curreiar 
the  19i  inches  diameter  purop  from  Mr  Livens'  data  to  be 

and  for  thiB  2U  inckm  diameter  pump 

l'lSvi*-iMiVi  =  2gh  _. ,_®. 

The  velocitf  of  rotation  of  the  water  round  the  wheel  will  h 
less  than  the  velocity  with  wliich  the  i^-ater  leaves  the  wheel  sni 
there  will  be  a  losa  of  head  due  to  the  j§udden  change  in  velodty* 


/!=' 


Let  this  losa  of  head  be  v* 

radial  velocity  of  flow  at  eri 
wheel  nidially,  is  then 

trj'-t^iUiCot^ 
9 

Taking  friction  and  the 
account, 

n  5= ~ ^ —  - 

9  2g 

which  again  may  hr  written 


I.  TTt 


^.     Tlie  head,  when  u,  Utb 
aasmniBg  the  water  eaten  tti^ 

tjoo  of  oentrifagal  bed  its 


2g       2^' 


h  = 


^g 


Cuin     Cifii' 


%         ^9 

The  vHluei>  of  ;«,  C  and  Ci  are  given  for  two  pumps  in  eqva&di 
(1)  uml  VI). 

237.     General  equation  for  a  centrifugal  pump. 
The  CH^ nations  for  the  gross  head  ft  at  discharge  Q  as  detennmfti 
for  the  seveiTil  classes  of  pumps  have  been  shown  to  beof  thefonn 

'^      2ff  ^    2g    ^   2g  ' 

or,  if  a  is  the  velocity  of  How  from  the  wheel, 
,      mv^     Quv     Ci?i* 


'^9 


2^ 


2g 


in  which  m  varies  t jet  ween  1  and  To.     The  coefficients  Ciandu 
for  any  pump  will  depend  upon  the  unit  of  discharge. 

As  a  further  example  and  illuj^trating  the  case  in  whieli  ^ 
certain  speeds  the  How  may  lie  unstable,  the  curves  of  Flf^ 
285 — 2J^7  may  be  now  considered.  When  Vi  is  60  feet  per  saw 
the  equation  to  the  head  discharge  curve  is 

Q  being  in  cubic  feet  per  minute. 


CENTRIFUGAL  PUMPS 


431 


The  velocity-discharge  cnrve  for  a  constant  head  of  80  feet  as 
colated  fmiu  tluM  equation  is  shown  in  Fig*  287. 

Tu  stiirt  the  pump  against  a  head  of  80  feet  the  periphf?ral 
jdciiy  has  to  be  707  feet  per  second,  at  which  velocity  the 
^har^*  Q  suddenly  rises  to  4 "3  cubic  feet  per  minute. 

The  curves  of  actual  and  nianonietric  efficiency  are  shown  in 
b.  286,  the  maximum  for  the  two  cases  occurring  at  different 
leharges. 

338-     Tbe  Limiting  height  to  which  a  single  wheel  centri- 
Bil  pump  can  he  used  to  raise  water. 
The  maximum  height  to  which  a  centrifugal  pump  can  raise 
Iter,  depends  theoretically  upon  the  niaxiinam  %'elocity  at  which 
*  lim  of  thc^  wheel  can  be  run. 

It  has  already  been  stated  that  rim  velocities  up  to  250  feet 
•rsetoiiil  IjavL*  been  used.  Assuming  radial  vanes  and  a  mano- 
Btric  efficiency  of  50  per  cent.,  a  pump  running  at  this  velocity 
lift  against  a  head  of  9HQ  feet, 
t  t}w^'  very  high  velocities,  however,  the  wheel  must  be  of 
mnterial  such  as  brauKe  or  cast  steel,  having  considerable 
nee  t^  tensile  Rtresses,  and  ftpecial  precautions  must  be 
to  balance  the  wheeL  The  hydraulic  losses  are  also 
derable,  and  manometric  efficiencies  greater  than  50  per 
hardly  to  be  expected. 

ding  to  M.  Rateai!*,  the  limiting  head  against  which  it  is 
We  to  raise  water  by  means   of   a  single  wheel  is  about 
feet,  and  the  maximum  demrable  velocity  of  the  rim  of  the 
i  is  about  100  feet  per  second. 

ngle  wheel  pumps  to  lift  up  to  3oO  feet  are  however  being 
At  tliis  velocity  the  stress  in  a  hoop  due  to  centrifugal  forces 
mt  7250  lbs.  per  sq.  incht. 

139.    The  sxictioii  of  a  centrifugal  pump. 

The  greatest  height  through  which  a  centrifugal  or  other  class 

amp  will  draw  water  is  about  27  feet.    Special  precaution  has 

I  taken  to  ensure  that  all  joints  on  the  suction  pipe  are  perfectly 

^htj  and  especially  is  tliis  so  when  the  suction  head  is  greater 

15  feet;  iJTify  under  hipecial  circumstances  is  it  therefore  de* 

for  the  suction  head  to  be  greater  than  this  amount^  and  it 

^  advisable  to  keep  the  suction  head  as  small  as  possible^ 


Centriftiaes,*'  ©tc.,  Itnttetin  d*  la  SociiU  de  I'Tjidtittrie  minfmtfj 
r.  p.  aafi,  Mnrch,  1902. 
witijis  Sirrnfrih  of  SJatermlt;  WooJ's  Strength  o/  Strttffural  Membert; 
Turbine^  Stodola. 


CENTBIFUOAI.  PUMPS 


433 


lO.      Series  or  multi-Btage  turbine  pomps, 
has  l>eeii  stated  that  the  limiting  economical  head  for  a  single 
L  purap   is  about  100  feet,  and  for  high  heads  series  pumps 
low  generally  used. 


^f^^nss^ 


^i3i»  296.     General  Arrangement  of  Worthington  Multi-stage  Turbine  Pump. 

By  putting  several  wheels  or  rotors  in  series  on  one  shaft,  each 
>tor  giving  a  head  varying  from  100  to  200  feet,  water  can  be 
^ed    to   practically  any  height,  and   such  pumps  have  been 

U  H.  "1^ 


HTOEAt?t.ICS 


coDJ^tTuct^  to  work 


head  of  2O0O  feet.    The 


I 


to  work  against  a  beail  or  W^M3  feet.  iHe  ntiiDD 
of  rocorsi  oo  one  eimft*  m^y  be  &QfEi  one  to  twelve  acconliisi 
lo  II10  total  liead*  For  a  given  hefid,  tlia  greater  the  number  1 
rtilors  used,  the  leas  the  peripheral  velocity^,  aod  within 
hitiitB  the  greater  the  e&CMmej^ 

FlgB^  295  and  296  show  a  longitndixial  section  and  i 

regpecti%'ely,  of  a  seried,  or  muld-stagt^  pump, 
hf  the  Worthkigtoii  Pnnip  Conipaiij--     On  the  mi* 
shaft  are  fixed  three  pboephor^brcmze  rotors^  alternating  with  I 
giudeSi  which  are  Tigidl|-  oonnected  to  the  outer  casings  iind  I 
the  bearings.    The  water  is  drawn  in  thrL>ugb  the  pipe  at  tb ) 
of  the  pisp  and  enters  the  fir«t  wheel  axialty.    The  watar  lea^ 
the  first  wheel  at  the  outer  circmnlerence  and  paj^ises  aloti^  1 
expanding  pa^ssagB  in  which  the  velocttj'  is  gradually  dtndtiisb 
at^  enters  the  second  wheel  axially.    The  vane«  in  the 
are  of  hard  phosphor-bronze  made  very  smooth  to  rtnluce  fricti^ 
losses  to  a  minimum.    The  water  parses  through  the  remaintl 
rotors  and  guides  in  a  ^miJar  manner  and  is  finally  discharf 
into  the  caaiiig  and  thence  into  the  delivery  pipe. 


Fig.  f97.    Siilx«r  Malti-sUge  Turbine  Ptiiiip. 

The  difference  in  pressure  head  at  the  entrances  to  any  tw*> 

consecutive  wheels  is  the  head  impressed  on  the  water  by  *J»«? 

wheel.     If  the  head  is  h  feetj  and  there  are  n  wheels  tl^ 

lift  is  nearly  nh  feet.    The  vanes  of  each  wheel  and  the  dirt 

the  guide  vanes  are   detennine<l  as  ejc  plained  for  the  mg\s 

*Sf>  that  losses  by  shock   are  redxiced  10  a  miniuiiitiif  mi 

's  and  gnide  passages  are  made  smooth  so  as  to  reduce 


the  back  of  each  wheel,  just  above  the  boes^  *^ 

'  holes  which  allow  water  to  get  behind  part  of  tlw 

*he  pressure  at  which  it  enters  the  wheel,  to  balauct? 

which  would  otherwise  be  set  up. 


CENTRIFUGAL  PUMPS  435 

The  pomps  can  be  arranged  to  work  either  vertically  or 
rizontally,  and  to  be  driven  by  belt,  or  directly  by  any  form 
motor. 

Fig.  297  shows  a  multi-stage  pump  as  made  by  Messrs  Sulzer. 
e  rotors  are  arranged  so  that  the  water  enters  alternately 
m  the  left  and  right  and  the  end  thrust  is  thus  balanced. 
Sciencies  as  high  as  84  per  cent,  have  been  claimed  for  multi- 
•ge  pumps  lifting  against  heads  of  1200  feet  and  upwards. 
The  Worthington  Pump  Company  state  that  the  efficiency 
DiBishes  as  the  ratio  of  the  head  to  the  quantity  increases,  the 
it  results  being  obtained  when  the  number  of  gallons  raised 
r  minute  is  about  equal  to  the  total  head. 

ExampU.  A  pamp  is  to  be  driven  by  a  motor  at  1450  revolutions  per  minute,  and 

•quired  to  lift  45  oubio  feet  of  water  per  minute  against  a  head  of  320  feet. 

i&iad  the  diameter  of  the  suction,  and  delivery  pipes,  and  the  diameter  and 
r  of  the  rotors,  assuming  a  velocity  of  5*5  feet  per  second  in  the  suction  and 
^  I^pes,  and  a  manometrio  efficiency  at  the  given  delivery  of  50  per  cent. 
one  provisionally  that  tbe  diameter  of  the  boss  of  the  wheel  is  8  inches. 

Lei  d  be  the  external  diameter  of  the  annular  opening.  Fig.  295. 


144      ""  60  X  5-5  * 
B  which  d=:6  inches  nearly. 
Fakiog  the  external  diameter  D  of  the  wheel  as  2d,  D  is  1  foot. 

^  1450  _^-   , 

Phen,  Vi = -g^  X  r = 76  feet  per  sec. 

(kflBoming  radial  blades  at  outlet  the  head  lifted  by  each  wheel  is 

76^ 
A=0.5.i|feet 

=  90  feet. 
Poor  wheels  would  therefore  be  required. 

24L    Advantages  of  centrifugal  pumps. 
There  are  several  advantages  possessed  by  centrifugal  pumps. 
In  the  first  place,  as  there  are  no  sliding  parts,  such  as  occur  in 
iprocating  pumps,  dirty  water  and  even  water  containing  coni- 
•atively  large  floating  bodies  can  be  pumped  without  greatlj- 
iangering  the  pump. 

Another  advantage  is  that  as  delivery  from  the  wheel  is 
istant,  there  is  no  fluctuation  of  speed  of  the  water  in  tlu» 
tkni  or  delivery  pipes,  and  consequently  there  is  no  necessity 
air  vessels  such  as  are  required  on  the  suction  and  delivery 
es  of  reciprocating  pumps.  There  is  also  considerably  less 
iger  of  large  stress  being  engendered  in  the  pipe  lines  by 
ater  hammer*." 

Another  advantage  is  the  impossibility  of  the  pressure  in  tlu^ 
*  See  page  384. 


436 


HYDRAUMCS 


in  the  pipe  line  it  will  gfatt 
[n  th€  second  case  a  simikr 
^heii  the  water  falls  below  a 
hough  convenient  15  unooo- 
Oiing  no  effective  work,  ihs 
%y  be  more  than  50  per  ceol 
ing  masimttm  discharge. 
ip  may  be  made  to  deUwr 
which  water  may  be  takai 


pump  casing  riBing  above  that  of  the  majdmnm  head  wiiici  xh 
rotor  is  capable  of  impressing  npon  the  -w^ter.  If  the  dt'liirry 
is  closed  the  wheel  vn\l  rotate  witliout  any  danger  of  the  pfeasuf 
in  the  casing  becoraing  greater  than  the  centrifugal  hmd  (pa^ 
335).  This  may  be  of  use  in  those  cases  where  a  pump  is  de- 
livering into  a  reservoir  or  pumping  from  a  reaervoin  hi  tie  iit 
case  a  float  valve  may  be  fitted,  which,  when  the  Tnnter  ri^  to 
a  particular  height  in  the  reservoir,  closei^  the  dehverr.  Tk 
pump  wheel  will  continue  to  rotate  but  without  delivering  wat^r, 
and  if  the  wheel  is  running  at  s  ^  a  velocity  that  the  ct'iitn- 
fugal  head  is  greater  than  the  hei 
delivery  when  the  valve  is  opened 
valve  may  be  used  to  stop  the  flo 
certain  level.  This  arrangement 
nomical,  as  although  the  pump  j 
power  required  to  drive  the  pumj. 
of  that  required  when  the  pump  is 

It  follows  that  a  centrifugal 
water  into  a  closed  pipe  system  fr 
regularly,  or  at  intervals,  while  the  punip  continues  to  rotate  at  a 
constant  velocity. 

Pump  delivering  into  a  long  pipe  lin€.  When  a  centrifagiil 
pump  or  air  fun  is  delivering  into  a  lung  pipe  line  the  resistances 
will  vary  approximately  as  the  square  of  the  quantity  of  water 
delivered  by  the  pump* 

Let  p2  be  the  absolute  pressure  per  square  inch  which  h^ 
to  be  maintaimni  at  the  end  of  the  pipe  line,  and  let  the 
resistances  vary  ns  the  square  of  the  velocity  r  along  the  pip?. 
Tlien    if   the  re?iij*tances   are   equivalent   to   a    head   h/^  frr",  tk 

pressure  head     ]  at  the  pump  end  tif  the  delivery  pipe  must  be 

w     w 

A  being  the  sectional  area  of  the  pipe. 

Let  -  be  the  pressure  head  at  the  top  of  the  suction  pipe,  then 
the  gross  lift  of  the  pump  is 

u*      w     w      A       w 
If,  therefore,  a  curve,  Fig,  298,  be  plotted  having 


CENTRIFUGAL  PUMPS 


437 


a.s  ordinates,  and  Q  as  abscissae,  it  will  be  a  parabola.  If  on 
"fclie  same  figure  a  cnrve  having  h  as  ordinates  and  Q  as  abscissae 
'be  drawn  for  any  given  speed,  the  intersection  of  these  two 
enures  at  the  point  P  will  give  the  maximum  discharge  the  pump 
*will  deliver  along  the  pipe  at  the  given  speed. 


JKdfeharffe  uv  C.Ft,per  SecondU 
Fig.  29S. 

242.    Parallel  flow  turbine  pump. 

By  reversing  the  parallel  flow  turbine  a  pump  is  obtained 
which  is  similar  in  some  respects  to  the  centrifugal  pump,  but 
differs  from  it  in  an  essential  feature,  that  no  head  is  impressed  on 
the  water  by  centrifugal  forces  between  inlet  and  outlet.  It 
therefore  cannot  be  called  a  centrifugal  pump. 

The  vanes  of  such  a  pump  might  be  arranged  as  in  Fig.  299, 
the  triangles  of  velocities  for  inlet  and  outlet  being  as  shown. 

The  discharge  may  be  allowed  to  take  place  into  guide 
passages  above  or  below  the  wheel,  where  the  velocity  can  be 
gradually  reduced. 

Since  there  is  no  centrifugal  head  impressed  on  the  water 
between  inlet  and  outlet,  Bemouilli's  equation  is 

tr     2gr     K?      2gr  * 
From  which,  as  in  the  centrifugal  pump, 
XT     Vifi     pi      J)  .  U*     u^ 
g       w     w     2g     2g 
If  the  wheel  has  parallel  sides  as  in  Fig.  299,  the  axial  velocity 
of  flow  will  be  constant  and  if  the  angles  <^  and  B  are  properly 
chosen,  Vr  and  Vr  may  be  equal,  in  which  case  the  pressure  at 
inlet  and  outlet  of  the  wheel  will  be  equal.    This  would  have 
the  advantage  of  stopping  the  tendency  for  leakage  through  the 
clearance  between  the  wheel  and  casing. 


438 


UYDRAULICS 


Such  a  pump  is  airnilar  to  a  reversed  impulBe  tarbine,  tk 
guide  passages  tif  which  are  kept  full.  The  velocity  witii  wM 
tlie  water  leavt-B  the  wheel  wuuld  however  be  great  and  theB 
above  the  pamp  would  depend  upon  the  percentage  of  the  Vt4tjciiy 
liead  that  could  be  converted  into  preissure  head. 


Since  there  is  no  centrifugal  d  impresgsed  upon  tie  watw, 
the  parallel-flow  pump  cannot  ci  vence  discharging  unless  ik 
water  in  the  pinup  is  first  set  in  tion  by  some  external  ra^i^s 
but  as  soon  as  the  flow  ia  coroine«*.,d  through  the  wheel,  tbeM 
discliarge  under  full  head  can  be  obtained. 


Fij?.  300. 


To  commence  the  discharge,  the  pump  would  generally  have  to 
be  placed  below  the  level  of  the  water  to  be  lifted,  an  auxilian* 
discliarge  pipe  being  fitted  with  a  discharging  valve,  and  a  non- 
return valve  in  the  discliarge  pipe,  arranged  as  in  Fig.  300. 


CENTRIFUGAL  PUMPS  439 

3  pamp  could  be  started  when  placed  at  a  height  ho  above 
,ter  in  the  sump,  by  using  an  ejector  or  air  pump  to  exhaust 
r  from  the  discharge  chamber,  and  thus  start  the  flow 
:h  the  wheel. 

3.    Inward  flow  turbine  pump. 

:e  the  parallel  flow  pump,  an  inward  flow  pump  if  constructed 
not  start  pumping  unless  the  water  in  the  wheel  were  first 
motion.  If  the  wheel  is  started  with  the  water  at  rest 
ntrifugal  head  will  tend  to  cause  the  flow  to  take  place 
•ds,  but  if  flow  can  be  commenced  and  the  vanes  are 
ly  designed,  the  wheel  can  be  made  to  deliver  water  at  its 
periphery.  As  in  the  centrifugal  and  parallel  flow  pumps, 
water  enters  the  wheel  radially,  the  total  lift  is 

H  =  Yi^'=2>_£+U'_^ (1). 

g       w     w     2g     2g  ^  ^ 

>m  the  equation 

£.     Xl  -'Pi     Vl.     a^  ^Vl 
w'^  2g~w  '^2g'^2g     2g' 

be  seen  that  unless  Vr*  is  greater  than 

2g     2sr     2g^ 
less  than  p,  and  ^  will  then  be  greater  than  the  total 

ry  special  precautions  must  therefore  be  made  to  diminish 
locity  U  gradually,  or  otherwise  the  efficiency  of  the  pump 
)  very  low. 

3  centrifugal  head  can  be  made  small  by  making  the 
nee  of  the  inner  and  outer  radii  small. 

2g^2g     2g 

le  equal  to  -^ ,  the  pressure  at  inlet  and  outlet  will  be  the 

and  if   the   wheel    passages  are  carefully  designed,  the 

re  throughout  the  wheel  may  be  kept  constant,  and  the 

becomes  practically  an  impulse  pump. 

3re  seems  no  advantage  to  be  obtained  by  using  either 

llel  flow  pump  or  inward  flow  pump  in  place  of  the  centri- 

pump,  and  as  already  suggested  there  are  distinct  dis- 

.ages. 

I.    Reoiprooating  pumps. 

simple  form  of  reciprocating  force  pump  is  shown  dia- 
latically  in  Fig.  301.    It  consists  of  a  plunger  P  working  in 


440 


Fig.  sola.     Vertical  Single-acting  Reciprocating  Pump. 


RECIPROCATING   PUMPS  441 

I  cylinder  C  and  has  two  valves  Vs  and  Vd,  known  as  the  suction 
ind  delivery  valves  respectively.  A  section  of  an  actual  pump 
8  shown  in  Pig.  301a. 

Assume  for  simplicity  the  pump  to  be  horizontal,  with  the 
jentre  of  the  barrel  at  a  distance  h  from  the  level  of  the  water 
n  the  well;  h  may  be  negative  or  positive  according  as  the 
>ump  is  above  or  below  the  surface  of  the  water  in  the  well. 

Let  B  be  the  height  of  the  barometer  in  inches  of  mercury, 
rhe  equivalent  head  H,  in  feet  of  water,  is 

H.l?:f-^  =  1133B, 

which  may  be  called  the  barometric  height  in  feet  of  water. 

When  B  is  30  inches  H  is  34  feet. 

When  the  plunger  is  at  rest,  the  valve  V©  is  closed  by  the  head 
)f  water  above  it,  and  the  water  in  the  suction  pipe  is  sustained  by 
.he  atmospheric  pressure. 

Let  ha  be  the  pressure  head  in  the  cylinder,  then 

At)  =  H  —  fe, 
)r  the  pressure  in  pounds  per  square  inch  in  the  cylinder  is 

p  =  -43(H-fe), 
0  cannot  become  less  than  the  vapour  tension  of  the  water.    At 
)rdinary  temperatures  this  is  nearly  zero,  and  hn  cannot  be  greater 
han  34  feet. 

If  now  the  plunger  is  moved  outwards,  very  slowly,  and  there 
s  no  air  leakage  the  valve  Vg  opens,  and  the  atmospheric  pressure 
^uses  water  to  rise  up  the  suction  pipe  and  into  the  cylinder, 
lo  remaining  practically  constant. 

On  the  motion  of  the  plunger  being  reversed,  the  valve  Vs 
rloses,  and  the  water  is  forced  through  Vd  into  the  delivery 
ripe. 

In  actual  pumps  if  ho  is  less  than  from  4  to  9  feet  the 
lissolved  gases  that  are  in  the  water  are  liberated,  and  it  is  there- 
ore  practically  impossible  to  raise  water  more  than  from  25  to 
W)  feet. 

Let  A  be  the  area  of  the  plunger  in  square  inches  and  L  the 
troke  in  feet.  The  pressure  on  the  end  of  the  plunger  outside  the 
cylinder  is  equal  to  the  atmospheric  pressure,  and  neglecting 
he  friction  between  the  plunger  and  the  cylinder,  the  force  neces- 
sary to  move  the  plunger  is 

P  =  -43  {H  -  (H  -  A) }  A  =  -43^ .  A  lbs., 

ind  the  work  done  by  the  plunger  per  stroke  is 

E  =  -43^  .  A .  L  ft.  lbs. 


pet  ffmte  ti  Ik  )fain 


•  "ttAL  Hm^  ud  tb 


■ft  i  —  ,^^" ,    i.^cAiiL  i  fci^ 

^  vliefc  dip  waMir  larii  tk  d^mf 
K  bf  ilui  pliiflftr  dnna^tk 

■  Ai  ■  ■»  «iiiv«  tki«fc«t  *  *  JS *  ^  fool  poooflipfrfKmid. 

iv  «  ^firi  bkmm  m  Urn  pn^  md  Am  bmiitd  Md 

triwi.  mi  ikr  vehae  ^  vuer  UM  per  ai^tioii  ftn^b  rH 
WmmbM  It  ligWr  ha  tin  ^nshuw  moml  tbi^  b;  ^ 

1^      '     tr  tk  viifk  U  wuer  lifted  per  minate,  and  K  tiF 
[  viki  tkr  wilpr  is  lifted, 

^bjtlirpQiiipk  W.^  foot  poondipff 


HP= 


siooo* 


i 


j^T     -  _^.*  -t  -  "icbufe  of  thfi  pomp.    Slip. 
^  ^-p  -i  &  p)tingier  pump  is  the  rite  I 

'  ^u.  -4  It  ^  l^^i^  P^  £tn)ke  multiplied  by  the  nmahsifi  | 

^    *^m  «f  ^  fcibigr  pir  stroke  to  ihe  rolnme  digpli<^lffl 

:,xS'jct  hiS^Km  d^^  qvBiinei  ti  called  the  SKf. 
^^i^'  tM  ««»I  &cfcirgr  H  l»  tbi  tie  tb^retical  th0  ( 

^  la  a  ««dr  ^^^rkiiig  pm 
4fai;«]a  be  le»  than  five  r 

Tbe  c»a«*  «>f  »««*»"^ 
^^kg^pbw  will  be  disc 


Tnltai  luii 


RECIPROCATING   PUMPS 


443 


246.    Diagram  of  work  done  by  the  pomp. 

Theoretical  Diagram.  Let  a  diagram  be  drawn,  Fig.  302,  the 
"dinates  representing  the  pressure  in  the  cylinder  and  the  abscissae 
le  corresponding  volume  displacements  of  the  plunger.  The 
flumes  will  clearly  be  proportional  to  the  displacement  of  the 
lunger  from  the  end  of  its  stroke.  During  the  suction  stroke, 
n  the  assumption  made  above  that  the  plunger  moves  very 
owly  and  that  therefore  all  frictional  resistances,  and  also  the 
lertia  forces,  may  be  neglected,  the  absolute  pressure  behind  the 
lunger  is  constant  and  efqnsA  to  H  -  fe  feet  of  water,  or  62*4  (H  -  h) 
onnds  per  square  foot,  and  on  the  delivery  stroke  the  pressure  is 

62'4  f  Z  +  H  +  o-  )  pounds  per  square  foot. 

he  effective  work  done  per  suction  stroke  is  ABCD  which  equals 
I'i.h,  V,  and  during  the  delivery  stroke  is  EADF  which  equals 


62-4  (Z.^), 


nd  EBCP  is  the  work  done  per  cycle,  that  is,  during  one  suction 
ad  one  delivery  stroke. 


Pressure^ 


Fig.  302.    Theoretical  diagram  of  pressare  in  a  Reciprocating  Pump. 


l/V^ 


S$ 


Strokes  per  nunju/yo 


-^ 


Fig.  303. 

Actual  diagram.  Fig.  303  shows  an  actual  diagram  taken  by 
eans  of  an  indicator  from  a  single  acting  pump,  when  running 
'j  a  slow  speed. 

The  diagram  approximates  to  the  rectangular  form  and  only 


444 


HYDRAULICS 


differs  from  the  above  in  that  at  any  point  p  in  the  soction  strda^ 
pq  in  feet  of  water  is  equal  to  h  plus  the  losses  in  the  suction 
pipe,  including  loss  at  the  valve,  pins  th©  head  required  to 
accelerate  the  wat^r  in  the  suction  pipe,  and  qr  is  the  lied 
required  to  lift  the  water  and  overcome  all  losses,  and  to  accelerate 
the  water  in  the  deliveiy  pipe*  The  velocity  of  the  plunger  beii^ 
small,  these  correcting  quantities  are  practically  inappreciable. 

The  area  of  this  diagrani  represents  the  actual  work  done  cm 
the  water  per  cyck^  and  is  equal  to  W  (Z  ^  h}^  together  with  ^ 
head  due  to  velocity  of  discharge  i  ^,  all  losses  of  energy  in  tiie 
suction  and  delivery  pipes. 

It  will  be  seen  later  that  althot  at  any  instant  the  pressnn 
in  the  cylinder  i«  effected  by  the  ^rtia  forces,  the  total  work 
done  in  accelerating  the  water  is       *o. 

247.  The  accelerationB  of  t  pump  piunger  and  of  ^ 
water  in  the  suction  pipe. 

The  theoretical  diagramj  Fig.  2,  has  h^u  dra^^Ti  on  die 
assumption  that  the  velc»city  of  '  plunger  is  very  small  and 
without  reference  to  the  variatioTi     f  the  velocity  and  of  the 

acceleration  of  the  plunger,  but  it  is  now  necessary  to  consider 
this  variation  and  its  effect  on  the  motion  of  the  water  in  the  suction 
and  (lch"vci-y  pipes.  To  realise  how  the  velocity  and  acceleration 
of  tlic  ])hnigcr  varj",  suppose  it  to  be  driven  by  a  crank  and 
connecting  rod,  as  in  Fig.  304,  and  suppose  the  crank  rotates  with 
a  uniform  angular  velocitj'  of  ^  radians  per  second. 


Fig.  304. 


If  r  is  tlie  radius  of  the  crank  in  feet,  the  velocity  of  the  crank 

pin  is  V  =  o>r  feet  per  second.     For  any  crank  position  OC,  it  is 

proved  in  l)ooks  on  mechanism,  that  the  velocity  of  the  point  B  is 

V    ()f\ 

()\)     •     ^y  luaking   BD   equal  to  CK  a  diagram  of   velocities 

EDF  is  found. 

When  CB  is  very  long   compared  with  CO,  OK  is  equal  to 
OC  sin  6^,  and  the  velocity  v  of  the  plunger  is  then  Vsin^,  and 


RECIPROCATING   PUMPS 


445 


EDF  is  a  semicircle.  The  plunger  then  moves  with  simple 
harmonic  motion. 

If  now  the  suction  pipe  is  as  in  Fig.  300,  and  there  is  to  be 
eontiniiity  in  the  column  of  water  in  the  pipe  and  cylinder,  the 
velocity  of  the  water  in  the  pipe  must  vary  with  the  velocity  of 
the  plunger. 

Let  V  bo  the  velocity  of  the  plunger  at  any  instant,  A  and 

a  the  cross-sectional  areas  of  the  plunger  and  of  the  pipe  respect- 

V  A 
ively.     Then  the  velocity  in  the  pipe  must  be  — —  . 


Fig.  305. 

As  the  velocity  of  the  plunger  is  continuously  changing,  it  is 
continuously  being  accelerated,  either  positively  or  negatively. 

Let  I  be  the  length  of  the  connecting  rod  in  feet.  The 
acceleration*  F  of  the  point  B  in  Fig.  305,  for  any  crank  angle 
0,  is  approximately 

F  =  caV  (cos  ^  +  ^  cos  2^  V 

Plotting  F  as  BG,  Fig.  305,  a  curve  of  accelerations  MNQ  is 
obtained. 

When  the  connecting  rod  is  very  long  compared  with  the 
length  of  the  crank,  the  motion  is  simple  harmonic,  and  the 
acceleration  becomes 

F  =  wV  cos  ^, 

and  the  diagram  of  accelerations  is  then  a  straight  line. 

Velocity  and  acceleration  of  the  water  in  the  suction  pipe.  The 
velocity  and  acceleration  of  the  plunger  being  v  and  F  respectively, 
for  continuity,  the  velocity  of  the  water  in  the  pipe  must  be 
A 


V  —  and  the  acceleration 
a 


/.= 


F.A 


*  8«e  BaUmeing  of  Enginet,  W.  E.  D«lby. 


44*; 


HYDRAULICS 


248.    TIM  e£Eect  of  acccleratlofzi  of  the  plnxi^er  on  Xt% 
pressore  in  tiie  cylinder  during  the  suction  stroke. 

^^'!;en  the  Trfix  iiy  of  ibe  plmiig<?r  is  increaairig,  F  is  poeitiT^, 
ani  :.  mcoeleTmlki'  Hie  wmter  in  tte  suction  pipe  m  foroe  P  ii 
T>e«::i:rv-i.  Tfce  •tTi3->?pheric  preasisiie  has,  thereforej  not  only  U) 
.'jr,  thr-  v:»ter  an^i  oi-en^ouie  the  remtance  in  the  snetion  pipe, 
V-u:  ::  h.As  mbo  to  provide  the  necee^iry  force  to  accelerate  tk 
'w-^ter,  ind  the  prr--^-'*  in  tW  cylinder  is  ccinseqnently  dimimshed 
.  &5  the  Telocity  of  the  plunger  decreases, 
piston  has  **  exert  a  reaction  upon  the 
relocttyj  or         pre6sure  on  the  plangvr  is 


:rie  ocher 
rr«tive,  an 


F 

w-^Tc^r  :  >  diminisl. 

L«r:  L  be  the  *      :'h  of  the  m 
5^n  -„;il  arm  in  t*  feet,  /»  tl 

:'::-  I  ::>e  at  any  in^iant  in  feet  per 
^c^-xti:    f  a  cohic  fuiot  c4  water, 

Tr-vn  the  masfr  of  water  in  the  i 

;•  -^T.  1-,  and  since  by  Newton's  sec 

acce  ermdng  force  =  ai 

:.  -  aCvv'eratinsr  f  -vv  rt^juired  is 

p     fr  .  a,  L 

g 

T:.-   :-\><urv  ivr  unit  arva  is 

:i'   1  :. .^  -;u:vLi\:::  head  .f  water  is 

:7  ■ 
.     F .  A 


tl  pipe  in  feet,  a  its  cross- 
Dceleration  of  the  i^^ater  iu 
and  per  second,  and  u-  tke 

to  he  accelerated  is  ii?.a«X 
law  of  morion 

« acceleration, 

K  lbs. 


./.lbs., 


^. 


(1 


^i.= 


LA 

(7.a  ' 


F 


arce  if  any  one  of  the  three  quantities,  L,  — ,  or 


and    other    losses    the    pressure    in   the 


v'v'..:..;r  r  :>  ::•  w 

a:;d  :V.v  h^^ad  r>:>:sring  the  UK^rion  of  the  piston  is  h  +  ha, 

249.     Pressure  in  the  cylinder  during  the  suction  stroke 
when  the  plunger  moves  with  simple  harmonic  motion. 

It  ::\o  p'ur.c^^r  Iv  supp^^sed  driven  by  a  crank  and  very  long 


RECIPROCATING  PUMPS 


447 


innecting  rod,  the  crank  rotating  uniformly  with  angular  velocity 

radians  per  second,  for  any  crank  displacement  ^, 

F  =  <»i"rcos^, 

J  ,      L.A.wV        /, 

nd  ha  = .  cos  (f. 

g.a 

The  pressure  in  the  cylinder  is 

LAui'rcos^ 


H-A-- 


ga 


When  B  is  zero,  cos  B  is  unity,  and  when  ^  is  90  degrees,  cos  B 
\  zero.  For  values  of  B  between  90  and  180  degrees,  cos^  is 
e|B^tive. 

The  variation  of  the  pressure  in  the  cylinder  is  seen  in 
'ig.  306,  which  has  been  drawn  for  the  following  data. 


Fig.  306. 

Diameter  of  suction  pipe  3|  inches,  length  12  feet  6  inches. 
Oiameter  of  plunger  4  inches,  length  of  stroke  7^  inches. 

Number  of  strokes  per  minute  136.  Height  of  the  centre  of 
:he  pump  above  the  water  in  the  sump,  8  feet.  The  plunger  is 
issumed  to  have  simple  harmonic  motion. 

The  plunger,  since  its  motion  is  simple  harmonic,  may  be 
supposed  to  be  driven  by  a  crank  3J  inches  long,  making  68  revo- 
lutions per  minute,  and  a  very  long  connecting  rod. 

The  angular  velocity  of  the  crank  is 

u>=      '      =  7*1  radians  per  second. 

The  acceleration  at  the  ends  of  the  stroke  is 
F  =  a,>.r  =  7-rx  0-312 

=  15'7  feet  per  sec.  per  sec, 
A     /    4    V     . 


a     \3- 


125/ 


and 


K  = 


12-5. 15-7. 1-63 
32 


1-63, 

=  10  feet. 


448 


HY0EAULICS 


The  preesnre  in  the  cylinder  neglecting  the  wn^tev  m  xh 
cylinder  at  the  beginning  of  the  stroke  is,  therefore, 

and  at  the  end  it  is  34-8  +  10  =  36  feet-    That  is,  it  is  gnatrt 
than  the  atmospheric  pressure. 

WTien  ^  is  90  degrees,  cos  0  is  aero,  and  A^  is  therefore  jscro, 
and  when  0  is  greater  than  4K)  degrees,  eos  ^  is  negative. 

The  area  AEDF  is  clearly  equal  to  GADH,  and  the  work  iam 
per  suction  stroke  is,  therefore,  not  altjered  by  the  accielemtiiig 
forces;  but  the  rate  at  which  thr  ^^tnger  ia  working  at  varii'«i 
points  in  the  stroke  is  affected  by  m,  and  the  force  requiivi]  t^> 
move  the  plunger  may  be  very  m^ 

In  the  above  example,  for  t 
move  the  piston  at  the  commei 
more  than  doubled  by  the  ace 
remaining  constant  and  equal  t4 
varies  from 

P=  4iJ(8 
to  P  =  '43(8 

Air  i^esseh,  Tu  quick  running  t^timpe,  or  when  the  leitgtk 
of  the  pipe  is  long,  the  effects  of  these  accelerating  forces  tend  to 
become  serious,  not  only  in  causing  a  very  large  increase  in  the 
stresses  in  the  parts  of  the  pump,  but  as  will  be  shown  later,  under 
certain  circumstances  they  may  cause  separation  of  the  water  in 
the  pipe,  and  violi^nt  hammer  actions  may  be  set  up.  To  reduce 
the  effects  of  the  accelerating  forces,  air  vessels  are  put  on  the 
suction  ami  delivery  pipes.  Figs.  310  and  311. 

250.  Accelerating  forces  in  the  delivery  pipe  of  a  plunger 
pump  when  there  is  no  air  vessel. 

When  the  plunger  commences  its  return  stroke  it  has  not  only 
to  lift  the  water  against  the  head  in  the  delivery  pii^e,  but,  if  no 
air  vessel  is  i)rovided,  it  has  also  to  accelerate  the  water  in  the 
cylinder  and  the  delivery  pipe.  Let  D  be  the  diameter,  a^  the  area, 
and  Li  the  length  of  the  pipe.  Neglecting  the  water  in  the 
cylinder,  the  acceleration  head  when  the  acceleration  of  the  piston 
is  F,  is 

L,.A.F 


nereaiied. 

ice,   the  force  necessary  ki 
ent  of  the  stroke  has  be«n 
ting  force,  and  instead  di 
,8.  A  during  the  stroke,  it] 

)A 


ha  = 


gcii 


and  neglecting  head  lost  by  friction  etc.,  and  the  water  in  the 
cylinder,  the  head  resisting  motion  is 


7j  +  ha  + 


If  F  is  negative,  ha  is  also  negative. 


'^9' 


RECIPBOCATINO   PUBCPS 


449 


When  the  plunger  moves  with  simple  harmonic  motion  the 
iagram  is  as  shown  in  Fig.  307,  which  is  drawn  for  the  same 
ats  as  for  Fig.  306,  taking  Z  as  20  feet,  In  as  30  feet,  and  the 
iameter  D  as  S^  inches. 


Fig.  307. 


The  total  work  done  on  the  water  in  the  cylinder  is  NJKM, 
rhich  is  clearly  equal  to  HJKL.  If  the  atmospheric  pressure  is 
cting  on  the  outer  end  of  the  plunger,  as  in  Fig.  301,  the  nett 
rork  done  on  the  plunger  will  be  SNRMT,  which  equals  HSTL. 

251.  Variation  of  pressure  in  the  cylinder  due  to  firiction 
irhen  there  is  no  air  vessel. 

Head  lost  by  friction  in  the  auction  and  delivery  pipes.    If  t?  is 

he  velocity  of  the  plunger  at  any  instant  during  the  suction 

troke,  d  the  diameter,  and  a  the  area  of  the  suction  pipe,  the 

elocity  of  the  water  in  the  pipe,  when  there  is  no  air  vessel,  is 

A 

—  ,  and  the  head  lost  by  friction  at  that  velocity  is 

,  _4fifJf^L 

^^~   2gda'    ' 
Similarly,  if  ai,  D,  and  Li  are  the  area,  diameter  and  length 
espectively  of  the  delivery  pipe,  the  head  lost  by  friction,  when 
he  plunger  is  making  the  delivery  stroke  and  has  a  velocity  v,  is 

^^      2gDa,'  ' 
When  the  plunger  moves  with  simple  harmonic  motion, 
V  =  <i>r  sin  ^, 


nd 


,  _4/A'<oVsin'gL 
^•^■"         2gda' 


L.  H. 


29 


430 


HVDRAUTJCS 


If  the  pump  mftkeA  n  strokes  per  aeoond,  or  tli@  nitmber  d 
revoloticiiis  el  tha  cmnk  is  ^  per  second,  and  4  is  tlia  lengtK  d 
the  strokei 

and  /,  =  2r. 

Snbvtttatiiig  for  i>»  and  r, 

Plotting  values  of  A/  at  vai '         points  along  the  stntke, 
parabolic  cnrv©  E>tP,  Fig.  808, 
When  ^  i»  90  degree®,  ain^ 
The  mean  ordinate  of  the  pan 
head,  is  then 

2^ 
3 


.amed 

QJty^  and  ^  is  a  maTiinmru 

which  ia  the  mean  fricti<nial 


and  since  the  mean  frictional  head  is  equal  to  the  energy  lost  per 
pound  of  water,  the  work  done  per  stroke  by  friction  is 

2gda' 
all  dimensions  being  in  feet. 


foot  lbs., 


r;^ 

G 

H 

'^"^  i 

N 

A 

Sf 

if 

n 

i      ! 

•J    I 

R 

£ 
B 

1 [ 

N^-^^ 

F    i 

S  B 

T 

i 

<3-i- 

Fig.  309. 

Let  Do  be  the  diameter  of  the  plunger  in  feet.     Then 


and 


RECIPROCATING   PUMPS  451 

Therefore,  work  done  by  friction  per  suction  stroke,  when 
here  is  no  air  vessel  on  the  suction  pipe,  is 

The  pressure  in  the  cylinder  for  any  position  of  the  plunger 
.uring  the  suction  stroke  is  now.  Fig.  309, 

Ao  =  H  —  fe  —  fea~  fe/. 

At  the  ends  of  the  stroke  h/  is  zero,  and  for  simple  harmonic 
lotion  hm,  is  zero  at  the  middle  of  the  stroke. 

The  work  done  per  suction  stroke  is  equal  to  the  area 
LEMFD,  which  equals 

ARSD  +  EMF  =  62-4feV  +  ^^'P/^^'* . 

Similarly,  during  the  delivery  stroke  the  work  done  is 

62-4ZV  +  §^^^g^^*. 

The  friction  diagram  is  HKG,  Fig.  309,  and  the  resultant 
iagram  of  total  work  done  during  the  two  strokes  is  EMFGrKH. 

252.    Air  vessel  on  the  suction  pipe. 

Afi  remarked  above,  in  quick  running  pumps,  or  when  the 
sngths  of  the  pipes  are  long,  the  effects  of  the  accelerating  forces 
leeome  serious,  and  air  vessels  are  put  on  the  suction  and  delivery 
apes,  as  shown  in  Figs.  310  and  311.  By  this  means  the  velocity 
Q  the  part  of  the  suction  pipe  between  the  well  and  the  air 
essel  is  practically  kept  constant,  the  water,  which  has  its 
elocity  continually  changing  as  the  velocity  of  the  piston 
hanges,  being  practically  confined  to  the  water  in  the  pipe 
between  the  air  vessel  and  the  cylinder.  The  head  required  to 
accelerate  the  water  at  any  instant  is  consequently  diminished, 
>nd  the  friction  head  also  remains  nearly  constant. 

Let  l^  be  the  length  of  the  pipe  between  the  air  vessel  and 
he  cylinder,  I  the  length  from  the  well  to  the  air  vessel,  a  the 
rosa-sectional  area  of  each  of  the  pipes  and  d  the  diameter  of  the 
ripe. 

Let  h^  be  the  pressure  head  in  the  air  vessel  and  let  the  air 
'essel  be  of  such  a  size  that  the  variation  of  the  pressure  may  for 
implicity  be  assumed  negligible. 

Suppose  now  that  water  flows  from  the  well  up  the  pipe  AB 
ontinuously  and  at  a  uniform  velocity.  The  pump  being  single 
icting,  while  the  crank  makes  one  revolution,  the  quantity  of 
rater  which  flows  along  AB  must  be  equal  to  the  volume  the 
)lunger  displaces  per  stroke. 

29—2 


*:t 


>^J^\ 


F^-  31  '- 

::.:i-//.:'>  :he;rv:i:,  including  friction  and  the  velocity 
c>  r^rir.a:  r.e^'.rvtcd.  the  atmospheric  head 
-         A'  «V     4f'A-<uV/ 


.(1). 


.1  r.uTth  qaannnes  of  the  right-hand  part  of  the 
:-:-^l'y  be  very  small  and  hr  is  practically  equal 

u::  ::  >rr>:ke  is  raking  place,  the  water  in  the  pip^ 

:l  J  ':v  :r:r  ;rvs5j^;irv  head  at  the  point  B,  when  the  velocity 
u'^jTt  r  i>  r  :\v:  ivr  sev.vnd,  and  the  acceleration  F  feet  per 


><x\  •  .1  ;vr  ><Ov 


RECIPROCATING  PUMPS  453 

Let  hf  be  the  loss  of  head  by  friction  in  AB,  and  h/  the  loss  in 

iC.    The  velocity  of  flow  along  BC  is  — ,  and  the  velocity  of 

ow  from  the  air  vessel  is,  therefore, 

t?.A     Aft>r 
a        ira 

Then  considering  the  pipe  AB, 

.nd  from  consideratioii  of  the  pressures  above  B, 

<vA    A«i»r\' 


ls=K*h-{-2—^^ 


2g      /• 

Neglecting  losses  at  the  valve,  the  pressure  in  the  cylinder  is 

hen  approximately 

,,    AkF 

Tx    J.    A*o»V    ,      ,,    Ali¥ 
2gnra'      ^      ^        ag 

A*a)V 

Neglecting  the  small  quantity  ^    9  « , 

For  a  plunger  moving  with  simple  harmonic  motion 

I.      XT    7,     4/wVAVZ^,    .  ,.\     AZiiuVcosd 

ho  =  lB.-h-  -^ — rj-(3  +  *i8m*^) . 

2ga^d    \ir  Jag 

By  putting  the  air  vessel  near  to  the  cylinder,  thus  making 
I  small,  the  acceleration  head  becomes  very  small  and 

Ao  =  H  -  fe  -  fe/  nearly, 

nd  for  simple  harmonic  motion 

The  mean  velocity  in  the  suction  pipe  can  very  readily  be 
etermined  as  follows. 
Let  Q  be  the  quantity  of  water  lifted  per  second  in  cubic  feet. 
Then  since  the  velocity  along  the  suction  pipe  is  practically 

onstant  Vm  =  —  and  the  friction  head  is 


BECIPROOATIKG  PUMPS  455 

Let  k.  Fig.  311,  be  the  length  of  the  pipe  between  the  pump  and 
the  air.  vessel,  Id  be  the  length  of  the  whole  pipe,  and  ai  and  D  the 
area  and  diameter  re8i)ectively  of  the  pipe. 

Let  At  be  the  height  of  the  surface  of  the  water  in  the  air  vessel 
above  the  centre  of  the  pipe  at  B,  and  let  Ho  be  the  pressure  head 
in  the  air  vessel.  On  the  assumption  that  Ho  remains  constant, 
the  velocity  in  the  part  BC  of  the  pipe  is  practically  constant. 

Let  Q  be  the  quantity  of  water  delivered  per  second. 

The  mean  velocity  in  the  part  BC  of  the  delivery  pipe  will  be 

Q 

:=■ 

The  friction  head  in  this  part  of  the  pipe  is  constant  and  equal  to 

Considering  then  the  part  BC  of  the  delivery  pipe,  the  total 
head  at  B  required  to  force  the  water  along  the  pipe  will  be 

But  the  head  at  B  must  be  equal  to  H»  +  /la  nearly,  therefore, 
H.  +  A,  =  Z  +  ^-^ga.-W  +  H    (1). 

In  the  part  AB  of  the  pipe  the  velocity  of  the  water  will  vary 
with  the  velocity  of  the  plunger. 

Let  V  and  F  be  the  velocity  and  acceleration  of  the  plunger 
respectively. 

Neglecting  the  water  in  the  cylinder,  the  head  Hr  resisting  the 
motion  of  the  plunger  will  be  the  head  at  B,  plus  the  head 
necessary  to  overcome  friction  in  AB,  and  to  accelerate  the  water 
in  AB. 

Therefore,     Hr  =  Ht,  +  fea  + il  V^    a  + -- 

For  the  same  total  length  of  the  delivery  pipe  the  acceleration 
head  is  clearly  much  smaller  than  when  there  is  no  air  vessel. 
Substituting  for  H»  +  /la  from  (1), 

If  the  pump  is  single  acting  and  the  plunger  moves  with  simple 
harmonic  motion  and  makes  n  strokes  per  second, 

Q  =  A2rJ, 

J                                              Arn 
and  tt= . 


be  la.  .4^  FA  rr^' 


RECIPBOCATINQ  PUMPS 


467 


X)me  below  the  zero  line  BC  as  in  Pig.  312  •.  The  pressure  in  the 
blinder  then  becomes  negative;  but  it  is  impossible  for  a  fluid 
k>  be  in  tension  and  therefore  discontinuity  in  the  flow  must 
CKxmrt, 

In  actual  pumps  the  discontinuity  will  occur,  if  the  curve  EFG 
aJls  below  the  pressure  at  which  the  dissolved  gases  are  liberated, 
>r  the  pressure  head  becomes  less  than  from  4  to  10  feet. 


Fig.  812. 

At  the  dead  centre  the  pressure  in  the  cylinder  just  becomes 
sero  when  A  +  A^  =  H,  and  will  become  negative  when  fe  +  fea  >  H. 
Theoretically  for  no  separation  at  the  dead  centre,  therefore, 

ha^R-h  or  ^^^R-h. 
ga 

If  separation  takes  place  when  the  pressure  head  is  less  than 
iome  head  h^y  for  no  separation, 

/ia  ^  H  —  hm ""  fe, 


md 


a    *^  I 


Neglecting  the  water  in  the  cylinder,  at  any  other  point  in  the 
itroke,  the  pressure  is  negative  when 

v^  A  * 
^    2g  a' 


h  + +  fe/+  ^ — a  >  H. 

a    g      ^     2g  a: 


That  is,  when 

And  the  condition  for  no  separation,  therefore,  is 


FA 


,(H-/^-/.--;^.fe,) 


.(2). 


*  See  also  Fig.  816,  pafi^e  459. 

t  Sor&oe  tension  of  flaids  at  rest  is  not  alladed  to. 


458 


HYDRAULICS 


255.     Separation   during   the   suction    etroke   when  tlu 
plunger  moves  with  simple  harmonie  motion. 

When  the  plungt^r  is  driven  by  a  crank  and  very  long  om- 
necting  r<>d,  the  acceleration  for  any  crank  angle  0  is 

F  =  w'rcos^, 
or  if  the  pnmp  makea  n  amgle  strokes  per  second, 

•1=  wn. 


and 


I,  being  the  length  of  the  stroke, 

F  is  k\  maximiim  when  ^  is  ze 
place  at  the  end  of  the  stroke  if 

9  a 


a 


nd  separab'on  will  not  tata 


and  ^^^ll  just  not  take  place  whei 

A   ,  ^    A 

—  <tt'r  or  ^  .—n 
a  2    a 

The  minimam  area  of  the  si 
therefore, 

^         Aoi^rL 

""-giU-h^-h)  

«ind  the  maximum  number  of  single  strokes  per  second  is 


I  pipe  for  no  separation  tSj 
(3) 


^_1    /2g{R-}i^-h)j 
''"ttV    "     A./,.L 


.(4). 


Separation  actually  takes  place  at  the  dead  centre  at  a  less 
number  of  strokes  than  given  by  formula  (4),  due  to  causes 
which  could  not  very  well  be  considered  in  deducing  the  fornmla. 

Example.  A  single  acting  pump  has  a  8troke  of  Ih  inches  and  the  plnngeris 
4  inches  diameter.  The  diameter  of  the  suction  pipe  is  3^^  inches,  the  Ieni?th 
12-5  feet,  and  the  height  of  the  centre  of  the  pump  above  the  water  in  the  well  is 
8  feet. 

To  find  the  number  of  strokes  per  second  at  wliich  separation  ^^ill  take  plact, 
assuming  it  to  do  so  when  the  pressure  head  falls  below  10  feet. 

H-/i  =  26  feet, 

-:-l-63. 


and,  therefore. 


_1        /     64x26 
""ir  V   1-63  X  7-5 


xl2 
xl2-5 


=  11  =  36 


=  210  strokes  per  minute. 

Nearly  all  actual  diagrams  taken  from  pumps,  Figs.  313—315, 
have  the   corner  at  the   commencement  of    the   suction   stroke 


RECIPROCATING  PUMPS 


459 


Toonded  off,  so  that  even  at  very  slow  speeds  slight  separation 
occurs.  The  two  principal  causes  of  this  are  probably  to  be  found 
first,  in  the  failure  of  the  valves  to  open  instantaneously,  and 
second,  in  the  elastic  jrielding  of  the  air  compressed  in  the  water 
at  the  end  of  the  delivery  stroke. 


DMA/'cry 


Zero  JPre^suLre/ 


Fig.  314. 


.Atnu 


Line 


Fig.  815. 

The  diagrams  Figs.  303  and  313 — 315,  taken  from  a  single-acting 
pomp,  having  a  stroke  of  7^  inches,  and  a  ram  4  inches  diameter, 
illustrate  the  effect  of  the  rounding  of  the  comer  in  producing 
separation  at  a  less  speed  than  that  given  by  equation  (4). 

Even  at  69  strokes  per  minute,  Fig.  303,  at  the  dead  centre  a 
momentary  separation  appears  to  have  taken  place,  and  the  water 
has  then  overtaken  the  plunger,  the  hammer  action  producing 
vibration  of  the  indicator.  In  Figs.  313 — 315,  the  ordinates  to  the 
line  rs  give  the  theoretical  pressures  during  the  suction  stroke. 
The  actual  pressures  are  shown  by  the  diagram.    At  136  strokes 


..   .11  If  resting  to  endeavour  t 
sh«Hild  overtake  the  plunger  ; 

\NTiile  the  piston  moves  fi 
70  degrees,  in  yts  •  Tr9  second 
points  the  preesure  in  the  c 
therefore  the  head  available 
resistances  and  to  accelerate  tl 

The  height  of  the  centre  o 
in  the  samp.  The  total  len, 
12*5  feet,  and  its  diameter  is 

Assuming  the  loss  of  head  a 
to  have  a  mean  valne  of  2*5  f et 
rating  the  water  in  the  pipe  is 
is,  therefore, 

20-5x82 
12-5 

When  the  piston  is  at  g  th 
behind  the  piston.  Let  this  c 
velocity  of  the  water  be  u  f e< 
takes  the  crank  to  torn  throng 
through  a  distance 

=  0101tt  +  i5 

=  l-2tt  +  3-2i 

The  horizontal  distance  a&  ii 

should  be  equal  tn  i*^* 


RECIPROGATINO  PUMPS  461 

The  frictional  resistances,  which  vary  with  the  velocity,  will  be 

small.    Assmning  the  mean  frictional  head  to  be  '25  foot,  the  head 

canning  acceleration  is  12*55  feet  and  the  mean  acceleration  of  the 

water  in  the  pipe  while  the  piston  moves  from  rest  to  g  is, 

therefore, 

.     12-55x32     ^.    ^ 
fm  =  — Toic —  =  ^^  *6®*  P©r  86C.  per  sec. 

The  velocity  in  the  pipe  at  the  end  of  0*058  second,  should 
therefore  be 

t?  =  32  X  -058  =  1*86  feet  per  sec. 

and  the  velocity  in  the  cylinder 

u  =  y:^  =  ri2  feet  per  sec. 

Since  the  water  in  the  pipe  starts  from  rest  the  distance  it 

should  move  in  0'058  second  is 

12.i32.C058)«  =  -65in., 

and  the  distance  it  should  advance  in  the  cylinder  is 

0-65  .         . ,  . 
j;g3ms.=  4m.; 

so  that  z  is  0*4  in. 

Then  z  +  4*2  ins.  =  4*6, 

and  V2u  +  3'2  ins.  =  4'57  ins. 

The  agreement  is,  therefore,  very  close,  and  the  assumptions 
made  are  apparently  justified. 

256.  Negative  slip  in  a  plunger  pump. 

Fig.  315  shows  very  clearly  the  momentary  increase  in  the 
pressure  due  to  the  blow,  when  the  water  overtakes  the  plunger, 
the  pressure  rising  above  the  delivery  pressure,  and  causing 
discharge  before  the  end  of  the  stroke  is  reached.  If  no  separa- 
tion had  taken  place,  the  suction  pressure  diagram  would  have 
approximated  to  the  line  rs  and  the  delivery  valve  would  still 
have  opened  before  the  end  of  the  stroke  was  reached. 

The  coeflBcient  of  discharge  is  1*025,  whereas  at  59  strokes 
per  minute  it  is  only  0*975. 

257.  Separation  at  points  in  the  suction  stroke  other  than 
at  the  end  of  the  stroke. 

The  acceleration  of  the  plunger  for  a  crank  displacement  B 

mV     A 

is   — '- —  COS  ^,  and  therefore  for  no  separation  at  any  crank 
angle  B 

-^costf^^H-fc^^fc--^^^, h,) .(1). 


•L=A.(l*4^)c«». 


f  1  -^  ^~)  »  eqiml  to 


of  tJbe  cylisdM^ 
take  i^ftae  as  tbe  ikiad 


.  ~:a-    :*i_zi^>w    rL   :^^  -   *   -lie.T    to   be   less  than  Ar,  and 
- :-  '    *  >    c  7  ^•T*jv-<^skry  Tv   onsi-der  the  condition  for  nv 

.^«5.     S«c&rs=Dcir  vt:^  &  -^r^c  ^^r  Tessel  on  the  snctioii  pipe. 

T_:i    vir-^itrr  ?trcitr^rj:c:  ^«^H   lAke  plaoe  with   a  large  air 

^'->t^       .:    "^♦r    sijT'.c   Ti^r^-  -'   ->   -'ii-y  neo^ssarr  to  substitute  in 

r»Mi*~    :-    -     r^.'o  c  .v^  jkz>£    o  .    4  ,  section  2o6,  A»  of  Fig.  310 

^  r  r.       ~  r  _.  i.-:«i  i    r  r  >,     Iz.  FjC-  olO,  k_  is  negative. 

y  r  :«  ?^oij~ic:»  c  ^iie^  i^je  ri.:Lz^>rr  :>  ai  the  end  of  the  strc'ke 
::it    T.-.i^m  Lzi   4r*f>i    .  z  Z2sT  ZLztz   r^er^-errr.  ihe  air  vessel  and  the 


:^-r  fr  ^=lf^^- 
.^  .  A  -  : 


*.  1 ',  section  253,  and 


.^A'      4-"L*VA-    * 


:•!   ^>    :t:t 


frticiD.c   ':>^fcas^  in   the   denominator,  W 
.^:c:irikrv\i  w-.ii^  <  H  -  n  L  then. 


RECIPROCATING  PUMPS 


463 


The  maxmmm  number  of  strokes  is 


^1     /2g(R-h-K)a 


A  pump  can  therefore  be  run  at  a  much  greater  speed,  without 
of  separation,  with  an  air  vessel  on  the  suction  pipe,  than 
*ithoat  one. 

259.    Separation  in  the  delivery  pipe. 

Consider  a  pipe  as  shown  in  Fig.  316,  the  centre  of  CD  being  at 
height  Z  above  the  centre  of  AB. 

Let  the  pressure  head  at  D  be  Ho,  which,  when  the  pipe 
ischarges  into  the  atmosphere,  becomes  H. 

Let  ly  1%  and  k  be  the  lengths  of  AB,  EC  and  CD  respectively, 
f^  hf,  and  A^,  the  losses  of  head  by  friction  in  these  pipes  when  the 
lunger  has  a  velocity  v,  and  hm  the  pressure  at  which  separation 
ctoally  takes  place. 

' — i/O 


rw^ 


^3•    N 


( 


>-   I  -M 


fij^y. 


;    I 

/ 


Fig.  316. 

Suppose  now  the  velocity  of  the  plunger  is  diminishing,  and  its 

"etardation  is  F  feet  per  second  per  second.    If  there  is  to  be 

F  A 
jontinuity,  the  water  in  the  pipe  must  be  also  retarded  by  — '- — 

:eet  per  second  per  second,  and  the  pressure  must  always  be 
positive  and  greater  than  fe.,. 

Let  Ho  be  the  pressure  at  C ;  then  the  head  due  to  acceleration 
in  the  pipe  DC  is 

FAZ. 

9 
and  if  the  pipe  CD  is  full  of  water 

H«  =  Ho :: —  A/, 


which  becomes  negative  when 

FAZ, 
9 


g 

>  Ho  ~  h/. 


alid 


ar 


tor 


for 


Qo  $^t 


andf, 


or  n. 


b*^  deihvrv 
P'aiv  at  B. 


RECIPROCATING   PUMPS  405 

260.  Diagram  of  pressure  in  the  cylinder  and  work  done 
curing  the  suction  stroke,  considering  the  variable  quantity  of 
rater  in  the  cylinder. 

It  is  instructive  to  consider  the  suction  stroke  a  little  more  in 
etail. 

Let  V  and  F  be  the  velocity  and  acceleration  respectively  of 
:xe  piston  at  any  point  in  the  stroke. 

As  the  piston  moves  forward,  water  will  enter  the  pipe  from  the 
""dl  and  its  velocity  will  therefore  be  increased  from  zero  to 

•  — ;  the  head  required  to  give  this  velocity  is 

^'-2^ (1)- 

On  the  other  hand  water  that  enters  the  cylinder  from  the  pipe 

3  diminished  in  velocity  from  —  to  v,  and  neglecting  any  loss  due 

o  shock  or  due  to  contraction  at  the  valve  there  is  a  gain  of 
iiressure  head  in  the  cylinder  equal  to 

^-2^-^-2^    ^2). 

The  friction  head  in  the  pipe  is 

,      4/Lv'A' 

^^=2^^  (3). 

The  head  required  to  accelerate  the  water  in  the  pipe  is 

^'  =  -^     • W- 

The  mass  of    water  to  be  accelerated  in  the  cyUnder  is  a 
variable  quantity  and  will  depend  upon  the  plunger  displacement. 
Let  the  displacement  be  x  feet  from  the  end  of  the  stroke. 

The  mass  of  water  in  the  cylinder  is lbs.  and  the  force 

required  to  accelerate  it  is 

and  the  equivalent  head  is 

P  ^x.F 
wA       g 

The  total  acceleration  head  is  therefore 


9  \         a  / 


9 

L.  H.  30 


I 


466  I  I  BTDRACLICS 

Now  hit  J     be  tli£  pressure  kead  in  thte  cylinder^  thm 

„    ,     «»     4/LAV    F/       LAV 

When  the  plunger  moves  with  simple  harroonic  motioii^  and  a 
driven  by  a  crank  of  radium  r  rotating  uniformly  with  angnlir 
velocity  %  the  displacement  of  the  plmnger  from  the  end  of  tta 
stroke  ife  r(l-  cos  ^),  the  reiocitv  vr  ein  0  and  ita  aceelemtioa  ie 
w'r  cos  0. 

Therefor©  i 

w-r  —  c  ^-     -  + .,M% 

9       ^  g  9 

Worh  dime  during  iks  swc>  ok^*     Assuming  atmospli^c 

pressure  on  the  face  of  the  pi  the  pressnr©  per  square  fool 

resisting  itii  motiDn  ia 

(E  p. 

For  any  small  plunger  displacement  tj,  the  work  done  is, 
therefore, 

A  (H-Ho)w'.aa', 

and  the  total  work  done  during  the  stroke  is 
E=  r  A(H-Ho)7r.caj. 

The  displacement  from  the  end  of  the  stroke  is 
a*  =  r  (1  -  cos  ^), 
and  therefore  d.r  =  r  sin  OdO, 

and  E  -  I'u' .  A  (H  -  Ho)  r  sin  OdO. 

Jo 

Substituting  for  Ho  its  value  from  equation  (6) 
,     f'       4/LAVr\sin'^     a>Vsin-^ 


2gda'  2g 


a>VcOS^       (oVcOS*^       LA     2  A      -     n.^ 

+ + tuVcos  0}  sm  OdS. 

9  g  9  (^  i 

Tlie  sum  of  the  integration  of  the  last  four  quantities  of  this 

expression    is    equal    to   zero,   so    that   the   work    done   by   the 

accelerating  forces  is  zero,  and 


E-«'Ar  r(h+hf)amOde 
Jo 

i/LAV 

29 .  da 


RECIPROCATING   PUMPS 


467 


Or  the  work  done  is  that  required  to  lift  the  water  through 
%  height  h  together  with  the  work  done  in  overcoming  the 
resistance  in  the  pipe. 

Diagrams  of  pressn/re  in  the  cylinder  and  of  work  done  per 
iCroJke.  The  resultant  pressure  in  the  cylinder,  and  the  head 
ransting  the  motion  of  the  piston  can  be  represented  diagram- 
Btttically,  by  plotting  curves  the  ordinates  of  which  are  equal  to 
B^  and  H~Ho  as  calculated  from  equations  (2)  and  (3).  For 
jdeamess  the  diagrams  corresponding  to  each  of  the  parts  of 
eqiiation  (2)  are  drawn  in  Figs.  318 — 321  and  in  Fig.  317  is  shown 
Ihe  combined  diagram,  any  ordinate  of  which  equals 

H  -  A  -  (feZ  +  cd  +  ef-  gh). 


t^^^—^-rdS* 


$h. 


^ 

Figs.  318,  819,  820. 


w 


Figs.  821,  822. 


In  Fig.  318  the  ordinate  cd  is  equal  to 

and  the  carve  H  JK  is  a  parabola,  the  area  of  which  is 

2   4/LA»  ,  ,, 
-3-  W"'^^- 


30—2 


468 


In  Fig.  819,  the  ordinate  efis 


-f"-. 


and  the  ordinate  gk  of  Fig.  320  is 


+ COS*  B, 

9 


The  areas  of  the  curves  are  respectively 

1  wV 

and  are  therefore  equal ;  and  nin      the  orditiatea  are  atwa|v  d 
opposite  sign  the  sum  of  the  twc       ^eas  m  K0ro, 
In  Fig.  822,  km  is  equal  to 

and  A-Z  to 

—  COSE'l  JC-         J. 

g  \         a   / 

Since  cos  B  is  negative  betweei  )*  and  180*  the  area  ^TT  is 
equal  to  YZU. 

Fig.  821  has  for  its  ordinate  at  any  p^int  of  the  stroke,  tke 
head  H  -  H,,  resisting  the  niution  of  the  piston. 

This  equals  /*  +  kl  i-  cd  +  ef-ghf 

and  the  curve  NPS  is  clearly  the  cnrve  GFE,  inverted. 

The  area  \'XST  measured  on  the  proper  scale,  is  the  work  done 
per  stroke,  and  is  equal  to  VMHT  +  HJK. 

Tlie  scale  of  the  diagram  can  be  determined  as  follows. 

Since  h  feet  of  water  =  62"4/i  lbs.  per  square  foot,  the  pressure 
in  pounds  resisting  the  motion  of  the  piston  at  any  point  in  the 
stroke  is 

62-4  .  A  .  /i  lbs. 

If  therefiu-e,  VXST  be  measured  in  square  feet  the  work  done 
per  stroke  in  ft. -lbs. 

=  62-4  A  .  YNST. 

261.     Head  lost  at  the  suction  valve. 

In  determining  the  pressure  head  Ho  in  the  cylinder,  no  account 

has  been  taken  of  the  head  lost  due  to  the  sudden  enlargement 

from  the  pij)e  into  the  cylinder,  or  of  the  more  serious  loss  of  head 

due  to  the  water  passing  through  the  valve.    It  is  probable  that  the 

r*A^ 
whole  of  the  velocity  head,  ^^  — g,  of  the  water  entering  the  cylinder 

from  the  pipe  is  lost  at  the  valve,  in  which  case  the  available  head 
H  will  not  only  have  to  give  this  velocity  to  the  water,  but  ynW 


RECIPROCATING   PUMPS  469 

dso  have  to  give  a  velocity  head  ^  to  any  water  entering  the 

sylinder  from  the  pipe. 

The  pressure  head  H©  in  the  cylinder  then  becomes 

„     „    ,     i;«   A"     t;*     4/Lt;'A'     F/       IA\ 

262.  yariation  of  the  pressure  in  hydraulic  motors  due 
lo  inertia  forces. 

The  description  of  hydraulic  motors  is  reserved  for  the  next 
chapter,  but  as  these  motors  are  similar  to  reversed  reciprocating 
pumps,  it  is  convenient  here  to  refer  to  the  effect  of  the  inertia 
forces  in  varying  the  effective  pressure  on  the  motor  piston. 

If  L  is  the  length  of  the  supply  pipe  of  a  hydraulic  motor,  a 

the  cross-sectional  area  of  the  supply,  A  the  cross-sectional  area 

of  the  piston  of  the  motor,  and  F  the  acceleration,  the  acceleration 

.FA 
of  the  water  in  the  pipe  is — - —  and  the  head  required  to  accelerate 

the  water  in  the  pipe  is 

,      FAL 
ga 

If  p  is  the  pressure  per  square  foot  at  the  inlet  end  of  the 
supply  pipe,  and  hf  is  equal  to  the  losses  of  head  by  friction  in  the 
pipe,  and  at  the  valve  etc.,  when  the  velocity  of  the  piston  is  v,  the 
pressure  on  the  piston  per  square  foot  is 

p«  =  p  -  wha  -  whf. 

When  the  velocity  of  the  piston  is  diminishing,  F  is  negative, 
ind  the  inertia  of  the  water  in  the  pipe  increases  the  pressure  on 
ihe  piston. 

Example  (1).  The  stroke  of  a  double  acting  pump  is  15  inches  and  the  number  of 
strokes  per  minute  is  80.  The  diameter  of  the  plunger  is  12  inches  and  it  moves 
vith  simple  harmonic  motion.  The  centre  of  the  pump  is  13  feet  above  the  water 
n  the  well  and  the  length  of  the  suction  pipe  is  25  feet. 

To  find  the  diameter  of  the  suction  pipe  that  no  separation  shall  take  place, 
Lssuming  it  to  take  place  when  the  pressure  head  becomes  less  than  7  feet. 

As  the  plunger  moves  with  simple  harmonic  motion,  it  may  be  supposed  driven 
)y  a  crank  of  7^  inches  radius  and  a  very  long  connecting  rod,  the  angular 
relooity  of  the  crank  being  2ir40  radians  per  minute. 

The  acceleration  at  the  end  of  the  stroke  is  then 

4ir»  .  40a  .  r 
60 

Therefore,  ||Jx40»x|^=84' -20', 

rom  which  -  =  1*64. 

a 


470 


Therefore 

and  d^^r. 

Ar  is  clearly  lees  th&n  fjT^  tberefora  a^pamtion  o&iuiol  take  pkci  mi  u^olluo 
point  in  the  stroke. 

Example  (2).     The  pump  of  example  (I)   deli^rs  water  into  a  nsiAg  mm 
1225  feet  long  and  5  incbefi  diameter,  which  U  fitted  with  on  aii-  T^asBi 

The  water  is  lifte^i  thr^anh  a  total  height  of  920  feet. 

Neglecting  all  lonieft  escept  frictic*a  in  the  deliTery  pipe*  deleimine  the  h«ifr 
power  required  to  ^ork  the  pomp.    /=0l0i5. 

Since  there  is  an  air  vesHel  in  the  delivezj  pipe  the  Teloei^  of  flcrw  «  viD  W 
practically  aniform. 

Let  A  and  a  be  tlie  cross -»eeUo&al  areas 
ively. 

_ip    10    d 
"  25  '  a  '  6i 

The  head  h  lost  due  to  friction  is 

=  1764  fei 

The  total  lift  is  therefore 

220  +  l76--4=l 
The  weight  of  water  lift^  per  minale  is 


A,2r,80 


he  pnmp  (^Under  and  pipe  respot- 
2r.  80 

■6  ft,  per  WG, 
feafc,. 


Therefore, 


.    —  .  80  X  62-5  lbs.  =  4900  lbs. 
12 

«^=      33,000      ="«■«• 


Exmuple  (3).  If  in  example  (2)  the  air  vessel  is  near  the  pump  and  the  mean 
level  of  the  water  in  the  vessel  is  to  be  kept  at  2  feet  above  the  centre  of  tlie 
pump,  find  the  pressure  per  sq.   inch  in  the  air  vessel. 

The   head  at  the  junction  of  the  air  vessel  and  the  supply  pipe  is  the  head 
necessary  to  lift  the  water  207  feet  and  overcome  the  friction  of  the  pipe. 
Th.refore,  H,  + 2' =  207  +  176-4, 

H„=  381-4  feet, 
381-4  X  62-5 

^=^—144 

=  165  lbs.  per  sq.  inch. 

Kxdtnple  (4).  A  single  acting  hydraulic  motor  making  50  strokes  per  minute 
has  a  cylinder  8  inches  diameter  and  the  length  of  the  stroke  is  12  inches.  The 
diameter  of  the  supply  pipe  is  3  inches  and  it  is  500  feet  long.  The  motor  i- 
supplied  with  wat^r  from  an  accumulator,  see  Fig.  339,  at  a  constant  pressure  of 
300  lbs.  per  sq.  inch. 

Neglecting  the  mass  of  water  in  the  cylinder,  and  assuming  the  piston  moves 
with  simple  harmonic  motion,  find  the  pressure  on  the  piston  at  the  beginning  and 
the  centre  of  its  stroke.  The  student  should  draw  a  diagram  of  pressure  for  one 
stroke. 

There  are  25  useful  strokes  per  minute  and  the  volume  of  water  supplied 
per  minute  is,  therefore. 


2,-).  -rf'^  =  8-725  cubic  feet. 
4 


,502 


At  the  commencement  of  the  stroke  the  acceleration  is  ir^ -— r,  and  the  velocitT 

oO^ 
in  the  supply  pipe  is  zero. 


RECIPROCATING   PUMPS  471 

The  head  required  to  accelerate  the  water  in  the  pipe  is,  therefore, 

ir».50a.l.8<.500 
«~      60». 2.3^32 
^  tiou  xeeiif 
which  is  equivalent  to  165  lbs.  per  sq.  inch. 

The  effective  pressure  on  the  piston  is  therefore  135  lbs.  per  sq.  inch. 
At  the  end  of  the  stroke  the  effective  pressure  on  the  piston  is  465  lbs. 
per  sq.  inch. 

At  the  middle  of  the  stroke  the  acceleration  is  zero  and  the  velocity  of  the 
piston  is 

II  «T=  1  '31  feet  per  second. 

The  friction  head  is  then 

•04^1^12^8^  600' 

=  15-2  feet. 
The  pressure  on  the  plunger  at  the  middle  of  the  stroke  is 

300  lbs.  - 1^^^^^= 293-4  lbs.  per  sq.  inch. 
144 

The  mean  friction  head  during  the  stroke  is  1. 15*2=10*1  feet,  and  the  mean 
loss  of  pressure  is  4*4  lbs.  per  sq.  inch. 

The  work  lost  by  friction  in  the  supply  pipe  per  stroke  is  4*4 .  j  .  8^ .  2^ 

=222  ft.  lbs. 
The  work  lost  per  minute  =  5500  ft.  lbs. 
The  net  work  done  per  minute  neglecting  other  losses  is 

(3001b8.-4*4).^.Z..83.25 

=  370,317  ft.  lbs., 
and  therefore  the  work  lost  by  friction  is  comparatively  small,  being  less  than 
2  per  cent. 

Other  causes  of  loss  in  this  case,  are  the  loss  of  head  due  to  shock  where  the 
water  enters  the  cylinder,  and  losses  due  to  bends  and  contraction  at  the  valves. 

It  can  safely  be  asserted,  that  at  any  instant  a  head  equal  to  the  velocity  head, 
of  the  water  in  the  pipe,  will  be  lost  by  shock  at  the  valves,  and  a  similar  quantity 
at  the  entrance  to  the  cylinder.  These  quantities  are  however  always  small,  and 
even  if  there  are  bends  along  the  pipe,  which  cause  a  further  loss  of  head  equal  to 
the  velocity  head,  or  even  some  multiple  of  it,  the  percentage  loss  of  head  will  still 
be  small,  and  the  total  hydraulic  efficiency  will  be  high. 

This  example  shows  clearly  that  power  can  be  transmitted  hydraulically  very 
efficiently  over  comparatively  long  distances. 

263.    High  pressure  plunger  pump. 

Fig.  323  shows  a  section  through  a  high  pressure  pump 
suitable  for  pressures  of  700  or  800  lbs.  per  sq.  inch. 

Suction  takes  place  on  the  outward  stroke  of  the  plunger,  and 
delivery  on  both  strokes. 

A  brass  liner  is  fitted  in  the  cylinder  and  the  plunger  which, 
as  shown,  is  larger  in  diameter  at  the  right  end  than  at  the  left, 
is  also  made  of  brass ;  the  piston  rod  is  of  steel.  Hemp  packing 
is  used  to  prevent  leakage  past  the  piston  and  also  in  the  gland 
box. 

The  plunger  may  have  leather  packing  as  in  Fig.  324. 

On  the  outward  stroke  neglecting  slip  the  volume  of  water 


RECIPEOCATINO    PUMP8 


473 


into  th©  oylinder  is  j  D/ » L  cubic  feet,  D^  being  the  dia- 

leter  of  the  piston  and  L  the  length  of  the  stroke.    Tlie  quantity 
t  water  forced  into  the  delivery  pipe  through  the  valve  V^,>  is 


■         J  (Do'  -  <f )  L  cubic  feet, 

being  the  diameter  of  the  small  part  of  the 
"lunger. 

On  the  in-stroke,  the  suction  valve  is 
loeed  and  water  is  forced  through  the 
elivery  valve;  part  of  this  water  enters 
l»  delivery  pipe  and  part  flows  behind  the 
^^1  tJirough  the  port  P> 
^TThe  amount  that  flowa  into  the  delivery  pipe  is 


Fig.  'i2A. 


^T^  therefore*  {D/^ef)  is  made  equal  to  (^^  or  D^  is  V2(i,  the 
Lelivsry^  during  each  stroke,  is  ^  Da%  cubic  f eetj  and  if  there  are 
ces  per  minute,  the  delivery  is  4245D„%k  gallons  per 


J^M^Sl^riUM^ii 


^b  Fig.  33S.     TftDgye  Duplex  Pump. 

^^■L    Duplex  feed  pump. 

^^P^  32->  show^s  a  section  through  one  pump  and  steam  cylinder 

rf  a  Taiigye  double-acting  pump. 


474 


HYDRAITUCS 


There  are  mro  stmm  cylinders  side  hy  aide^  one  of 
is  afaowD,  aad  two  pump  cylinderB  in  Kne  with  the  srteam  cjHm 

In  iho  pump  the  two  lower  valves  an*  sucticm  valveF  and  lb 
two  upper  delivery  valvea.  As  the  pnmp  piston  P  moves  to  i^ 
right,  the  lefl-haad  lower  valve  opens  and  wat^^r  is  drawn  iski^ 
pump  from  the  auction  chamber  C.  Daring  this  stroke  the  r^ 
upper  valve  is  open,  and  vvater  is  delivered  into  the  delivi^  Ci- 
^^^len  the  piston  moves  to  the  left,  the  water  m  drawn  in  thrmigii 
the  lower  right  valve  and  delivered  through  the  upper  left  ^*alTe, 

The  steun  engine  has  douhle  j— *-  at  each  end.     As  the  piston 
appn  ka«he«  the  end  at  ita  stroke  t^        earn  valve,  Fig.  82d,  m  at  tm 
and  e*?veTs  the  steam  port  1  whi 
to  eschauHl.    VVTien  the  piston  p 
enck*!iied  in  the  c>*linder  acta  as 
and  plunger  gradually  to  rest. 


inner  stej^m  port  2  m  vspm 
dia  steatn  ]:M>rt  2,  rlie  ^mm 
ihkm  and  brings  the  friisou 


F\^.   'A'lM. 


Fig,  S27- 


lji*r  tilt*  tnw  t'ligino  anil  pump  F^hown  in  ,'^'^ctiuTi  be  called  A  aini 
the  otlipr  engine  nud  pmup,  not  j?huwn,  Le  called  B, 

As  the  piston  of  A  moves  from  right  to  left,  the  lever  L,  Figs. 
32o  luiil  827,  nitates  a  spindle  to  the  othur  end  of  whicli  i^  fixed  a 
cnink  ^!,  wbii-h  moves  tlie  valve  of  the  cylinder  B  from  left  to 
right  :ind  ojx^ns  the  left  j>irt  of  the  cylinder  B.  Just  befure  tlie 
piston  of  A  reaches  the  left  end  of  its  strcfko,  the  piston  of  B, 
then*fore,  commences  its  stroke  from  left  t<i  rights  and  bv  a  lever 
Li  and  cniiik  Mj  lnnvl^s  the  valve  of  cyli ruler  A  also  from  left  lo 
right,  an  si  the  piston  of  A  can  then  commence  its  !^*turn  i^troke. 
It  should  be  noted  that  while  the  piston  of  A  is  moving,  that  c^f 
B  is  practically  at  rest,  and  ricr  verm. 

265.    The  hydraulic  ram. 

Tilt*  bycininlic  ratn  is  a  machine  which  utilises  the  momentum 
of  a  stream  of  water  falling  a  small  height  to  raise  a  part  of  the 
wat-t^r  tti  a  grt^att^r  hciglit. 

In  the  iirrruigcment  sho\\ni  in  Fig.  328  water  is  supplied  from  a 
tank,  or  stream,  througli  a  pipe  A  into  a  chamber  B,  which  ha^  rw^o 


PUMPS 


+75 


V  and  Yi-  Wlien  no  flow  is  taking  place  the  valve  Y  falls 
seating'  and  the  valve  Vj  rests  on  its  seating.  If  water  is 
fed  tci  flow  along  the  pipe  B  it  vnl\  escape  through  the  open 
V.  The  contraction  of  the  jet  through  the  valve  opening, 
3r  as  in  the  case  of  the  plate  obstructing  the  flow  in  a  pipe, 
168,  causes  the  pressure  to  b©  ^eater  on  the  under  face  of 
ftlve,  and  when  the  pressure  is  sufficiently  large  the  valve 
x^mmence  to  close.  As  it  closes  the  pressure  will  increase 
le  rate  of  closing  will  be  continually  accelerated.  Tlie  rapid 
\g  of  the  valve  arrests  the  motion  of  the  water  in  the  pipe, 
bere  m  a  sudden  riaa  in  pressare  in  B,  which  causes  the 

fVt  to  open,  and  a  portion  of  the  water  passes  into  the  air 
C.  Tlie  water  in  the  supply  pipe  and  in  the  vessel  B,  after 
\  brought  to  rest,  recoils,  like  a  ball  thrown  against  a  wall, 
ikB  pres.snre  in  the  vessel  is  again  diminished,  allowing  the 
to  once  more  escape  through  the  valve  V.  The  cycle  of 
l&cmB  is  then  repeated,  more  water  being  forced  into  the  air 
ber  C,  in  wliich  the  air  is  compressed,  and  water  is  forced  up 
rfivery  pipe  to  any  desired  height. 


Fig,  328, 

it  A  be  the  height  the  water  falls  to  the  ram,  H  the  height  to 
the  water  is  lifted, 
f  W  lbs.  of  water  descend  the   pipe  per  second,  the  work 
ble  per  second  is  Wh  foot  lbs.,  and  if  e  is  the  efficiency  of  the 
e  weight  of  water  lifted  through  a  height  H  will  be 

H  ■ 
iciency  &  diminishes  as  H  increases  and  may  be  taken  as 
^  eent*  at  high  heads. 

j'ig.  329  shows  a  section  through  the   De  Cours  hydraulic 
[the  valves  of  which  Brre  controlled  by  sprhigs.     The  springs 


w  = 


nOetSt* 


PUBfPS 


471 


the  fitphon  action  of  the  water  escaping  from  the  discharge^ 
;  sir  to  be  drawn  in  past  the  spindle  of  the  valve.    A  cushion 
air  is  thus  formed  in  the  box  B  every  stroke,  and  some  of  thil^H 
is  carritKl  into  C  when  the  ^^Ive  Vj  opens.  ^M 

The  extreme  sirapHcity  of  the  hydraulic  ram,  together  with 
with  which  it  can   be  adjusted  to  work  mth  vaiying 
im  of  water,  render  it  particularly  suitable  for  pumping 
out-of-the-way  places,  and  for  supplying  water,  for  fountains 
nd  domestic  purposes j  to  country  houses  situated  near  a  stream.* 


260.    Lifting  water  by  compressed  air. 
A  very  simple  method  of  raising  water  from  deep  wells  is  b; 
fis  of  compressed  air.     A  delivery  pipe  is  sunk  into  a  well^ 
[the  open  end  of  the  pip©  being  placed  at  a  considerable  distance 
[low  the  surface  of  the  water  in  the  well. 


J 


/O — Iw^ 


WaJLet* 


f   AirTuj^e 


Fig.  030. 


Fig.  mi. 


In  the  arrangement  shown  in  Fig,  330,  there  is  surrounding  ths 
delivery  tube  a  pipe  of  larger  diameter  into  which  air  is  pumped 
by  a  compressor. 

The  air  rises  up  the  delivery  pipe  carrying  with  it  a  quantity  of 
water.    An  alternative  arrangement  is  shown  in  Fig.  dSl. 

Wbether  the  air  acts  as  a  piston  and  pushes  the  water  in  front 

it,  or  forms  a  mixture  with  the  water,  according  to  Kelly**, 
lepends  very  largely  upon  the  rate  at  which  air  is  supplied  to  the 
[pump. 

In  the  pump  experimented  upon  by  Kelly,  at  certain  rates  of 


I 


478 


HYDEAUUCS 


working  the  discharge  was  continuoue,  the  air  and  the  water  being 
mixed  together,  while  at  low  discharges  the  action  was  intermittent 
and  the  pump  worked  in  a  definite?  cycle;  the  discharge  commeneed 
slowly;  the  velocity  then  gradually  increased  imtil  the  pipi 
di.sdiarged  full  bore;  this  was  followed  by  a  rush  of  air,  after 
whicli  the  flow  gradually  diminished  and  finally  stopped ;  after  a 
period  of  no  flow  the  cycle  commenced  again.  When  the  rate  at 
which  air  was  supplied  w^as  further  diminished,  the  water  tme 
up  the  delivery  tube,  but  not  sufficiently  high  to  overflow,  and  tbe 
air  escaped  without  doing  useful  work. 

Tlie  efficiency  of  these  pumps  is  very  low  and  only  in  exeeptioial 
cnsm  does  it  reach  50  per  cent.  The  volume  v  of  air,  in  cubic  feet^ 
at  atuiospheric  pressure,  required  to  lift  one  cubic  foot  of  ifnt^r 
through  a  height  k  depends  upon  the  efficiency.    With  an  ef* 

ticiency  of  30  per  cent,  it  is  approximately  ^'  =  95' 


and  with  an 


efficiency  of  40  per  cent.  *«  =  He  approximately. 

It  is  necessary  that  the  lower  end  of  the  delivery  be  at  a  gresier 
distance  below  the  surface  of  the  water  in  the  well,  than  the  hriglit 
of  the  lift  above  the  free  surface,  and  the  well  has  consequently  ^ 
be  made  very  deep. 

On  the  other  hand  the  well  is  much  smaller  in  diameter  than 
would  be  required  for  reciprocating  or  centrifugal  pumps,  aiid  the 
initial  cost  of  constructing  the  well  per  foot  length  is  considerayj' 


EXAMPLES, 

(1)  Pind  tlie  horse-power  required  to  raise  100  cable  feet  of  water  p^f 
minute  to  s.  height  of  125  fciet,  by  a  pump  whose  efficiency  is  h 

(2)  A  centrifugal  pump  has  an  inner  radius  of  4  Laches  and  an  outer 
radius  of  12  inches*  The  angle  the  blade  makes  with  the  direction  ^ 
motion  at  exit  is  153  degrees.   The  wheel  naakes  545  revolutions  per  miDnt^* 

The  discharge  of  the  pump  is  3  cubic  feet  per  second.  The  sidej  d  tlio 
wheel  are  jmrallel  and  2  inches  apart. 

Determine  the  inch  nation  of  the  tip  of  the  blades  at  inlet  so  that  tb«^ 
shall  be  no  shock,  the  velocity  with  which  the  water  leayes  the  wheel  ^ 
the  theoretical  lift.  If  the  head  due  to  tbe  velocity  with  which  the  w»t0 
leaves  tlie  wheel  ia  lost,  find  the  theoretical  lift. 

(3)  A  centrifugal  pump  wheel  has  a  diameter  of  7  inches  and  ma^^ 
1358  revolutions  jier  minute. 

The  blades  are  formed  so  that  the  water  enters  and  leaves  the  wheel 
witliont  shock  and  t!ie  blades  are  ra^lial  at  exit  The  water  is  lifted  bj  tlw 
pump  29 "4  feet*    Find  the  manometric  efficiency  of  the  pnmp. 


PUMPS 


479 


{41  A  cefltrifng&l  pump  wheel  11  mchee  diameter  whicli  runs  at  1308 
roToldtiotts  per  minute  ib  suiroimded  by  a  Yortex  chambet  22  iuchofi 
di&tiieUT,  autl  has  radi&l  blades  at  exit  The  prensure  head  at  the  circum- 
lonuice  of  thti  wheel  is  23  feot.  The  water  is  hfted  to  a  height  of  43'5 
ImI  abov€  the  centre  of  the  pomp.  Find  the  efficiency  of  the  whirlpool 
duutiber, 

(5>     The  radial  velocity  of  flow  through  a  pump  is  5  feet  per  second,  i 
Uie  felocity  of  the  outer  periphery  is  60  feet  per  aeeond. 

The  angle  tlie  tangent  to  the  blade  at  outlet  makes  with  the  direction 
of  Diotion  in  120  degrees.  Determine  the  pressure  bead  and  velocity  head 
wbi?re  the  water  leaves  the  wheel*  assuming  the  pressure  head  in  tie  eye 
of  tlie  wheel  in  atmospheric,  and  thus  determine  the  theoretical  lift. 

(6)  A  ctrutrifugal  pump  with  vanes  curved  back  has  an  outer  radius  of 
10  indieH  and  an  inlet  rsdius  of  4  inches,  the  tangents  to  the  vanes  at  outlet 
1>eing  Lnelined  at  40^  to  the  tangent  at  tlie  outer  periphery.  The  section  of 
the  wheel  is  ^^ach  that  the  radial  velocity  of  0ow  is  constant,  6  feet  per 
second ;  and  it  runs  at  700  revolutions  per  minute. 

Deteruiipc  :— 

^1)     the  angle  of  the  vane  at  inlet  so  that  tliere  shall  be  no  shock, 
^S|     tlie  theoretical  lift  of  tlie  pump, 

(3)    Die  Yelocity  bead  of  the  water  as  it  leaves  the  wheel*    Loud. 
L1906. 

(7)  A  centrifugal  pump  4  feet  diameter  running  at  200  revolotions  per 
pumps  5O0O  tons  of  water  from  a  dock  in  45  minutes,  the  meaii| 

tliellig  00  feet.    The  area  through  the  wheel  periphery  is  1200  sqoaroJ 
\  and  the  angle  of  the  vanes  at  outlet  is  26^.   Determine  the  hydraulic f 
Dcy  and  estimate  the  average  horse -power.     B'ind  also  the  lowest  i 
I  to  start  pumping  against  the  head  of  20  feet,  the  inner  radius  being 
[  thti  outer.    Loud.  Un.  1906. 

(8)  A  centrifugal  pump^  delivery  1500  gallons  per  minute  with  a  lift  of 
I  fckot^  has  an  outer  diameter  of  16  inches,  and  the  vane  angle  i»  80^    All 

\  kinetic  energy  at  dischai*ge  is  tost,  and  is  equivalent  to  50  per  cent,  of 
actual  Uft^     Find  the  revolutions  per  minute  and  the  breadth  at  the 
Jet,  tlie  velocity  of  whirl  being  half  the  velocity  of  the  wheel,     Lond. 
1906. 

(9)  A  centrifugal  pump  has  a  rotor  19|  inches  diameter ;  the  width  of 
the  outer  periphery  is  3^^^  inches.  Using  formula  {1),  section  236,  deter- 
tnine  the  discbarge  of  the  putnp  when  the  hea<l  is  30  feet  and  t^i  is  50. 

(10)  The  angle  tp  at  the  outlet  of  the  pump  of  question  (9)  m  IX. 
Find  the  velocity  witli  which  the  water  leaves  the  wheel,  and  the 

;muin  proportion  of  the  velocity  head  that  must  be  converted  into  work, 
tlie  other  losses  are  15  per  cent,  and  the  total  efficiency  70  per  cent* 

fill    The  inner  diameter  of  a  centrifugal  pump  is  12 j^  inches^  the  outer 
diameter  21 1  inches.   The  width  of  the  wheel  at  outlet  is  8g  i^chen.    Using 
uation  |2k  section  236,  find  the  discharge  of  the  pump  when  the  head  is 
*5  f^ot^  and  ^e  number  of  revolutions  per  minute  is  440. 


tofOw 


BT1ML4ITUCS 


ntmiiiig  at  550  leTola^ 

msi^B  Ihe  tip  of  ih^  vane  makes 

wdgp  of  tbe  vane  is  99  de^tes. 

m&  ti^  ^TOCtiOn  of  motion  of  the 

of  iiie  wheel  ui  11|  tnehtft  and  tbe 

idien  a  leaves  the  wheel. 

is  losi  by  frictioii,  aitd  that  ooe- 
A&  exit  s  Vsl^ Ibad  tlie  bead  lost  at  inlet  trhd^ 
I  Ibe  pvobabfe  veldei;^  impcefieod  on  tbe  ^^bsu 


(U)  Deaeoba  a  foeed  TottaL,  aad  sketch  tiia  footn  of  tlici  &^  sashcb 
wbea  the  aaffolar  w^locaty  ia  conatapt, 

In  a  oiBlrifiiiial  pmi^  A«fwuliiug  bonsoBtaDj  imder  water,  tbe  dLimiet^r 
tie  CBBda  frf  Ibe  paMka  ^  1  tac^  aad  of  tbe  oatiide  2  feet,  ajid  tbe 
9  tuiuliua  at  400  ivivilotiaiES  per  aiinute.  Find  approximittclj  bow 
L  tfie  «m«ar  won^  lie  liftad  abotv  the  tail  waler  level. 

(14)  KTp^aiii  llie  ac^oB  of  a  oantrifogal  pump,  and  deduce  an  exprmon 
r  iia  ellcieBej.  If  sacii  a  paai|i  were  requiicd  to  deliver  ICXK^  gailosB  iB 
■r  lo  a  liei^% cl ^  feet,  liow  wtwld  jua.  design  it?    Loud.  Un.  190S. 

(li)    Fuid  fte  i^etd  of  rotasjoa  of  a  wheel  of  a  centrifu^  pomp  wMxt 

[  lo  tin  30O  toBB  of  water  5  feet  high  in  one  mmtite ;  haviiig  ^fiii 

'  w  Ml    The  velocity  of  flow  through  ibe  wheel  is  4*5  feet  pv 

,  aad  Ihie  vanes  are  carved  backward  so  that  the  angle  between  ilm 

and  a  tangent  to  the  cirerunferenoe  ts  20  degtees*    Loml  V^ 

(16)  A  oentrifogal  ptunp  is  leqaired  to  lift  2000  galloiia  ef  water  pf^ 
minnte  ^iroogh  90  feetv    The  velocity  ot  liow  throogh  the  wheel  hi 
per  second  and  the  efficiency  0-6,     The  angle  the  iip-of  tbe  vane  At  ootU* 
makes  witli  the  direction  of  motion  ib  150  degrees.    The  outer  radius  d  tho 
wheel  is  twioe  tbe  inner.     Determine  tbe  dimensions  of  the  wbeeL 

(17)  A  donble-acting  phinger  pump  hss  a  piston  6  inches  cliametor 
and  tbe  length  of  the  strokes  Is  12  inches.  The  gross  heaxl  i&  500  ttsei^ 
and  the  pomp  makes  BO  strokes  per  minute.  Assuming  no  alip,  Md  tbts 
discharge  and  horse-power  of  the  pump.  Find  also  the  necessary'  diiuaeter 
for  the  steam  cylinder  of  an  engine  driving  the  pomp  diret^  ajssumiiig 
steam  pressure  is  100  lbs.  per  square  inch,  and  the  mechanical  dficklK| 
of  tbe  combination  is  85  per  cent. 

(18)  A  plunger  pump  is  plaoed  above  a  tank  containing  water  tt  & 
temperature  of  200'  F,  Tbe  weight  of  the  sncticni  valve  is  2  lbs.  aud  iUi 
diameter  1  j  inches.  Find  the  maxim  am  height  abc^ve  the  ^i^V  at 
the  pump  may  be  placed  so  that  it  will  draw  water,  tJie  barometer  standlog 
at  30  inches  and  the  pump  being  assumed  perfect  and  ^^itliout  clearMCtt 
(The  vapour  tension  of  water  at  200"  F.  is  about  11  "6  lbs,  per  aq.  inch/ 

(19)  A  pump  cylinder  is  8  inches  diameter  and  the  stroke  of  the  plung* 
is  one  foot.    Calculate  ^e  maximum  velocity,  and  the  acceleration  of 


1. 


PUMPS 


481 


mtet  in  the  eucUon  and  delivery  pipes,  aBBximing  tlieir  respective  diametera 
>  be  7  inchee  and  5  inches,  the  motion  of  the  piston  to  be  simple  harroonic, 
ad  the  pistcm  to  make  86  strokes  per  minute. 

(20 1  Taking  the  data  of  question  (19)  ealculate  the  work  done  on  the 
iciion  stroke  of  the  pnmpi 

(1)    neglecting  the  friction  in  the  suction  pipet 

•  (2)    indndiog  the  friction  in  the  suctiou  pipe  and  assuming  that  the 
suction  pipe  is  25  feet  long  and  that/— 0 '01. 
The  height  of  the  centre  of  the  pump  above  the  water  in  the  aump  is 

(21)  If  the  pump  in  queBtion  (20)  delivers  into  a  rising  main  against 
head   of  120  feet,   and   if  the   length    of    the  main  itself   is  250  feet, 

nd  the  total  mork  done  per  revolution,  Assuming  the  pnmp  to  be  douhl© 
efeing,  find  the  i.h.  p,  required  to  drive  the  pumpj  the  efl&ciency  being  '72 
Ii4  no  slip  in  the  pump.  Find  the  delivery  of  the  pumpt  assuming  a  slip 
f  5  per  cent 

(22)  The  piston  of  a  pump  moves  with  simple  harmonic  motion,  and  it 
^fives  at  40  itrokes  per  minute.  The  stroke  is  one  foot.  The  suction 
Wk  ia  25  feet  long,  and  the  suction  valve  is  19  feet  above  the  surface  of  the 
mmt  in  the  sump.  Find  the  ratio  between  the  diameter  of  the  suction 
lape  and  the  pump  cylinder,  so  that  no  separation  may  take  place  at  the 
bad  pointe.     Water  barometer  84  feet. 

(23)  Two  douhle-actiug  pumps  deliver  water  into  a  main  without  an 
lir  TeflseL  Each  is  driven  by  an  engine  with  a  fly-wheel  heavy  enough  to 
^ecp  tbe  speed  of  rotation  uidf orm,  and  the  connecting  rods  are  very  long. 

Let  Q  be  the  mean  dehvery  of  tlie  pumps  per  second,  Qi  the  quantity  of 
Witeif  in  the  main.  Find  the  pressure  due  to  acceleration  (a)  at  the  begin - 
aing  of  a  stroke  when  one  pump  h  delivering  water ^  (&)  at  the  beginning 
fi  the  stroke  of  one  of  two  double-acting  pumps  driven  by  cranks  at  right 
n^m  when  both  are  delivering*     When  is  the  acceleration  ^ero  ? 

(S4)  A  double*acting  horizontal  pump  has  a  piston  6  inches  diameter 
<tlie  diameter  of  the  piston  rod  is  neglected^  and  tlie  stroke  is  one  foot. 
The  water  is  pumped  to  a  height  of  250  feet  along  a  delivery  pipe  450  feet 
kiiig  and  4|  incheH  diameter.  An  air  vessel  is  put  on  the  dehvery  pipe 
10  feet  from  tlie  delivery  valve. 

Find  the  press  tire  on  the  pump  piston  at  the  two  ends  of  the  stroke 
when  the  pomp  is  making  40  strokes  per  minute^  assuming  the  piston 
movea  with  simple  harmonic  motion  and  compare  these  l^esBiires  with  the 
1  when  there  is  no  air  vessel.    /=  *0075. 


(36)  A  single  acting  hydraulic  motor  makes  160  strokes  per  minute  and 
E  vrith  simple  harmonic  motion. 

The  motor  is  supplied  witli  water  from  an  accumulator  iu  which  the 
presanre  is  maintained  at  WO  lbs.  per  square  inch. 

The  cylinder  is  6  inches  diameter  and  12  inches  stroke.  The  delivery 
|iipG  IS  200  feet  long,  and  the  coefficient,  which  includes  bss  at  bends,  etc, 
may  be  taken  as/^0'2. 


^    UH. 


^\ 


d  ifae  wiabfe 


met  p^ap  m  SSiBoi  long  and  4  infihes 
r  B  6  Bciw  mud  the  stroke  1  foot 
mmtaEK.  90lBet  long,  and  the  Iwad  ai 
t  m  mo  wit  wmml  en  the  pomp.   The 
m  above  the  levdd  the  water  in  the 

2i  Bflipie  liamionic  tnotJon  and  niskBi 
cctical  dfagrain  far  the  pnnupi 
qaaattjcl  water  in  the  cylinder  and 


'iT  •oL  «nzasKiL  laks  CKiee  anywhere  in  the  deliyery  pipe  of  the 
janm*  itas  saSA  si  wsndc.  s  f^vcn  in  qaestkm  i26),  if  the  pipe  first  rose 
3iiE:»2icaJ.7  ±T  yiil  ieec  aad  •hea  TcrticaDy  for  40t»  or  rises  40  feet  im- 
Tiwffnapij  ^-nxL  2Kf  TozzLp  aal  ^mb  mns  horiaontally  for  50  feet,  and 
KpacssaaL  TaLsat  ^mx  wben  ihe  piummi  head  falls  below  5  feet? 

:3Si  A  jiimn  bns  ihree  smgjLe-acting  plungers  29^  inches  dianwter 
.srirniL  W  ,39aks  13  IJl>  decEees  wish  e«ch  o^^  The  stroke  is  5  feet  and 
7St>  x^mbinr  :f  scrokes  per  minase  40.  The  soctkui  is  16  feet  and  theloigtfa 
':t  ^re  sxkrcara  pipe  b  ±2  &«ec  The  dehTery  pipe  is  8  feet  diametor  and 
^j  »ii  Irc^.    Tbe  hMa  as  the  dellTerr  Talve  is  214  feet. 

Fzzf  i  z  ibe  ziizi2.:im  dsamecer  of  the  suction  pipe  so  that  there  is  no 
sep&ruoD:!!.  '--Qg— ^^^  do  air  ressei  and  that  separation  takes  place  when 
she  rr^=sasir^  b^ccoes  zenx 

V  The  bcrse-power  of  the  pump  when  there  is  an  air  Yeesel  on  the 
delirerr  vtry  zear  to  the  pomp.    /^-OOT. 

^The  ssciient  should  draw  out  three  cosine  curves  differing  in  phase  by 

12i3  degrees.      Then  remembering  that  the  pump  is  single  acting,  the 

resultant  curve  of  accelerations  will  be  found  to  have  Tn«.TimniTi  poeitiTe 

•V.A 
and  also  negative  values  of  ——-^    every  60  degrees.     The  maximiun 

acceleration  head  is  then  h^=  ± — r-— —  • 

For  no  separation,  therefore,  a »       '^  . 

(29)  The  piston  of  a  double-acting  pump  is  5  inches  in  diameter  and 
the  stroke  is  1  foot.  The  delivery  pipe  is  4  inches  diameter  and  400  feei 
long  and  it  is  fitted  with  an  air  vessel  8  feet  from  the  pump  cylinder.  Tbe 
water  is  pumped  to  a  height  of  150  feet.  Assuming  that  the  motion  of  the 
piston  is  simple  harmonic,  find  the  pressure  per  square  inch  on  the  pision 
at  the  beginning  and  middle  of  its  stroke  and  the  horse-power  of  the  pomp 
when  it  makes  80  strokes  per  minute.  Neglect  the  effect  of  the  variaUe 
quantity  of  water  in  the  cylinder.    Lond.  Un.  1906. 


PUMPS 


483 


(30)  The  plunger  of  a  p«mp  moves  with  simple  harmonic  motioo* 
Eod  the  condition  that  Bepai^ation  shall  not  take  place  on  tlie  suction 
roke  and  show  why  the  speed  of  the  pump  may  he  increased  if  an  air 
Nisel  Is  put  in  the  suction  pipe.  Sketch  an  indic^itoT  diagram  showing 
jpuc&tioD.     Explain  ^^  negative  slip."    Loud.  Un.  I9()6. 

I)  In  &  Bingle-acting  force  pump,  the  diameter  of  the  plunger  ia 
ties,  stroke  6  inches,  length  of  suction  pipe  63  feet,  diameter  of  suction 
pe  !2|  inches,  suction  head  0'07  ft.  When  going  at  10  revolutions  per 
innte,  it  is  found  that  the  average  loss  of  head  per  ntroko  between  the 
Letion  tank  and  plunger  cy Under  is  0*28  ft.  Assuming  that  the  friction al 
WSmB  vary  as  the  square  of  the  speed,  find  the  absolute  head  on  the  suction 
da  of  thJe  plunger  at  the  two  ends  and  at  the  middle  of  the  stroke,  the 
ETolntions  being  50  per  minute,  and  the  barometric  head  M  feet.  Draw  a 
a^Eftm  of  pressures  on  the  plunger — aimplo  hannonic  motion  being 
Lond.  Un-  1906. 


(S2)  A  single-actittg  pump  without  an  air  vessel  has  a  stroke  of 
\  inches.  The  diameter  of  the  plunger  is  4  inches  and  of  the  suction 
tpe  a|  inches.  The  lengtli  of  the  suction  pipe  is  12  faet,  and  the  centre 
I  ahe  pmnp  is  ^  feet  above  the  level  in  tlie  sum^p. 

Determine  the  number  of  single  strokes  per  second  at  which  theoretl- 
liiy  separation  will  take  place,  and  explain  why  separation  will  actually 
ike  place  when  the  number  of  strokes  is  less  tlian  the  calculated  value. 

(38)  Hxpiain  carefully  the  use  of  an  air  vessel  in  tlie  delivery  pipe  of  a 
omp.  The  pump  of  question  (32)  makea  100  single  strokes  per  minnte, 
nd  dehvers  water  to  a  lieight  of  100  feet  above  the  water  in  the  well 
brough  a  dehvery  pipe  1000  feet  long  and  2  inches  diameter.  Large  air 
edeels  being  put  on  tlie  suction  and  dehvery  pipes  near  to  the  pamp. 

On  the  assumption  that  aU  losses  of  head  other  than  by  friction  m 
he  delivery  pip©  are  neglected,  determine  the  horse-power  of  the  pump, 
Thefpe  is  no  slip. 

(34)  A  pump  plunger  has  an  acceleration  of  8  feet  per  second  per 
leoond  when  at  the  end  of  the  stroke,  and  Hie  sectional  area  of  the  plunger 
I  twice  the  sectional  area  of  the  delivery  pipe.  The  dehvery  pipe  is  152 
bet  bug.  It  runs  from  the  pump  horizontally  for  a  lengtli  of  45  feet,  then 
rertically  for  40  feet^  then  rises  5  feet,  on  a  slope  of  1  vertical  to  8  hori* 
kttstal,  and  finally  nms  in  a  horizontal  direction. 

Find  whether  separation  will  take  place,  and  if  so  at  whiuh  section 
of  ihe  pipe,  if  it  be  assumed  tliat  separation  takes  place  when  the  prassnre 
bead  m  the  pipe  becomes  7  feet. 

|S5)  A  pump  of  the  duplet  kind,  Fig,  325,  in  which  the  steam  piston  is 
Kunected  directly  to  the  pump  piston,  works  against  a  head  of  h  feet  of 
rtter,  the  head  being  supphed  by  a  column  of  watec'  in  the  dehvery  pipe« 
rhe  piston  area  is  Aot  the  pUmger  area  A,  the  delivery  pipe  area  a,  the 
angth  of  the  delivery  pipe  t  and  the  constant  steam  pressure  on  the  piston 
^Jba,  per  square  foot.     The  hydranho  resistance  may  be  represented  by 


r-i  V  being  the  Telocity  of  the  plunger  and  F  a  ooe^cieut. 


^V— ^ 


484  HTDRAUUGS' 

Show  thai  when  the  plunger  has  moTed  a  diatanoe  x  from  the  beguming 
ol  the  stroke 


r»«^(^-*)(l-e-^).  Land.  Un.  1906. 


(96)  A  pomp  Talye  of  Ivass  has  a  specific  gravity  of  8|  with  a  lift  of 
^  foot,  the  stroke  ol  the  piston  being  4  feet,  the  head  of  water  40  feet  and 
the  ratio  of  the  fall  valye  area  to  the  pistcm  area  one-fifth. 

If  the  Talve  is  neither  assisted  nor  meets  with  any  reeistanoe  to  closing, 

find  the  time  it  will  take  to  dose  and  the  ''slip"  due  to  this  gradnal  closing. 

7*6 
Timetodoseisgrvod  byfennnla,  Ss^.    /«g;gX82-2.    Lond.  Un.  1906. 


CHAPTER  XI. 

HYDRAULIC  MACHINES. 

267.    Joints  and  packings  used  in  hydraulic  work. 

The  high  pressures  used  in  hydraulic  machinery  make  it 
necessary  to  use  special  precautions  in  making  joints. 

Figs.  332  and  333  show  methods  of  connecting  two  lengths  of 
pipe.     The  arrangement  shown  in  Fig.  332  is  used  for  small 


Fig.  832. 


Fig.  888. 


Fig.  834. 


486 


HYDEATJLICS 


WTOugbt^iroti  pipes,  no  pacldtig  being  required.    In  Fig,  333  tho 
packJDg   material   is  a  gutta-percha  ring,      Pig<  336  siioT^rs  an 

ordinary  siK*ket  joint  for  a  cast-iron  hydraulic  main.  To  maka 
the  joint,  a  few  cords  of  hemp  or  tarred  rope  are  driven  into* 
the  socket.  Clay  is  then  put  round  the  outside  of  the  socket  and 
molten  lead  run  in  it*  The  lead  is  then  jammed  into  the  sockel 
with  a  caulking  tool.  Fig.  33*5  showg  various  forms  of  packiag 
l^rthersj  the  applications  of  which  will  be  seen  in  the  esampli 
given  of  hydraulic  machines. 


Nffk  letUher 


Ring  Utith^r 


Cup  leather 
Fig.  835. 


Fig.  aa§. 


Hemp  twine,  carefully  plaited,  and  dipped  in  hot  taUow, 
makes  a  good  packing,  when  used  in  suitably  designed  glanda 
(see  Fig.  339)  and  is  also  very  suitable  for  pump  bucketSi 
Fig.  323.  Metallic  packings  are  also  used  as  shown  in  Figs*  337 
and  338, 


^>^. 


Fig.  337. 


iig.  a-js. 


268,    The  accumulator. 

The  accumulator  is  a  deWce  used  in  connection  with  hydruulic 
machinery  for  storing  energy* 

In  the  form  generally  adopted  in  practice  it  consists  of  a  long 
cylinder  C^  Fig.  339,  in  which  elides  a  ram  R  and  into  wliich  water 
is  delivered  from  pumps.  At  the  top  of  the  ram  is  fixed  a  rigid 
cross  bead  which  carries,  by  means  of  the  bolts,  a  large  cylin*ler 
which  can  be  filled  with  slag  or  other  heavy  material,  or  it  imj 
be  loaded  with  caet-iron  weights  as  in  Fig.  340.    The  water  h 


HTDRAULIC  MACHINES 


487 


tig.  Sd'j.    lljdrauUc  Acctimulmtor, 


=1 


488 


HTDRAtFLICS 


admitted  to  the  cylinder  at  any  desired  pressures  through  a  pipe 
connected  to  the  cylinder  by  the  flange  shown  dotted^  and  tlie 
weight  is  so  adjusted  that  when  the  pressure  per  sq.  inch  in 
the  cylinder  is  a  given  amount  the  ram  rises. 

If  d  is  the  diameter  of  the  ram  in  inches,  p  the  pressure 
in  lbs.  per  sq.  inch,  and  h  the  height  in  feet  through  which  the 
ram  can  be  lifted,  the  weight  of  the  ram  and  its  load  is 

W  =  p/j<flbs., 

and  the  energy  that  can  be  stored  in  the  accumulator  is 

E  =  p.|d'./i  foot  lbs, 

Tlie  principal  object  of  the  accumulator  is  to  allow  hydraulic 
machines,  or  lifta,  which  are  being  supplied  \rith  hydraulic  power 
from  the  pumps,  to  work  for  a  short  time  at  a  much  greater  rata 
than  the  pumps  can  supply  energy.     If  the  pumps  are  connect<Mi 
directly  to  the  machines  the  rate  at  which  the  pumps  can  supply 
energy  must  be  equal  to  the  rate  at  which  the  machines  are 
working,  together  with  the  rate  at  which  energy  is  being  lost  hf 
friction,  etc.,  and  the  pump  must  be  of  such  a  capacity  as  to  supply 
energy  at  the  greatest  rate  required  by  the  machineB,  and  tJm 
frietional  resistances.     If  the  pump  supplies  wat^er  to  an  acciuDU- 
lator,  it  can  be  kept  working  at  a  steady  rate,  and  during  the  rinit; 
when  the  demand  is  less  than  the  pump  supply,  energy  can  he 
stored  in  the  accumulator* 

In  addition  to  acting  as  a  storer  of  energy,  the  accumnlator 
acts  as  a  pressure  regulator  and  as  an  automatic  arrangement  for 
starting  and  stopping  the  pumps. 

When  the  pumps  are  delivering  into  a  long  main,  the  demand 
upon  which  is  varjnng,  the  sudden  cutting  off  of  the  whole  or 
a  part  of  the  demand  may  cause  such  a  sudden  rise  in  the  pressure 
as  to  cause  breakage  of  the  pipe  line,  or  damage  to  the  poinp. 
With  an  accumulator  on  the  pipe  line,  unless  the  ram  » 
descending  and  is  suddenly  brought  to  rest,  the  pressure  camioi 
rise  very  much  higher  than  the  pressure  p  which  will  lift  the  ram* 

To  start  and  stop  the  pump  automaticallyj  the  ram  as  il 
approaches  the  top  of  its  stroke  moves  a  lever  connected  1*^ 
a  chain  wliich  is  led  to  a  throttle  valve  on  the  steam  pipe  of  tlie 
pumping  engine,  and  thuB  shuts  off  steam.  On  the  ram  agtin 
falling  below  a  certain  level,  it  again  moves  the  lever  and  opem 
the  throttle  valve*  The  engine  is  set  in  motion,  pumping  n- 
commences,  and  the  accumulator  rises. 


HYDRAtTLlC   MACHINES 


489 


^^am^te.     A.  bydjanlio  er«tie  working  at  &  pf«Bimre  of  100  Iba.  per  iq.  ineh  b&e 
30  cwts.  &t  a  rate  of  SOO  feet  per  minute  through  a  height  of  50  feet,  onoe 

J  1|  mliiute^.     The  effioieoaj  of  the  crane  ifl  IQpet  cent,  aad  aji  a<K!i]  mala  tor 
is  piorided. 

Fiiiii  Ihf  volame  of  the  cjlinder  af  the  cranes  the  minimum  horBe^power  for  the 
pomp,  mud  the  minimum  capacity  of  the  accamulatar, 

£^  A  be  the  sectional  are*  of  the  mm  of  the  orane  cylinder  m  sq^  feet  and  L 
\he  length  of  the  stroke  in  feet. 

Then,  I>a44.  A.  Lx 070- 30x112x60', 

!  _    30x112x50 

H  *^~        0  70x144x700 

^P  =2-33  eubio  feet. 

Ihe  rate  of  doing  work  in  the  lift  e^lindtr  k 
112  :<  30x200 


I 


07 


=  960,000  ft^  Iba.  per  minute. 


lli«  ve^rk  done  in  lifting  50  feet  ia  210,000  ft,  lbs.     Since  thia  has  to  be  done 
•rery  one  aud  hiilf  miuuteH,  the  work  the  pump  munt  supply  in  one  and  half 
atee  is  at  ka^st  240,000  ft.  lbs.,  imd  the  minimym  bort^e-power  is 

240,000     ^ 


1*5 


=  40,000  ft,  lbs. 


The  work  done  by  the  pnmp  while  the  crane  h  lifting  is 

B  240,000x0-25 


efoergy  stored  in  the  aeoumuktor  mn&t  be,  therefore^  at  least  200,000  ft.  Ibfk 
!rhererore^  if  T^  it  its  minimum  capacity  in  cubic  feet, 
V^xTOOx  144  =  20O.0O0» 
V^,  =  2  cubic  feet  iieaily* 


269.    DifTeretitial  accumulator  •• 

TweddelFa  differential  accamalator,  shown  in  Fig.  340,  has  a 
fixed  ram,  the  lower  part,  of  which  is  made  slightly  larger  than 
the  upper  by  forcing  a  brass  liner  upon  it.  A  cylinder  loaded 
ifith  heavy  caet-iron  weights  elides  upon  the  i*am,  water-tight 
jomtB  hein^  made  by  means  of  the  cup  leathers  shown.  Water 
is  pumped  into  the  cylinder  through  a  pipe,  and  a  passage  drilled 
ajdally  along  the  lower  part  of  the  ram. 

Let  p  be  the  pressure  in  lbs,  per  sq,  inch,  d  and  di  the  dia- 
meters of  the  upper  and  lower  parts  of  the  ram  respectively, 
^ba  weight  lifted  (neglecting  friction)  is  then 


r 


if  /t  is  the  lift  in  feet,  the  energy  stored  is 
E-p.^W-e?)feJootlbs, 


The  difference  of  the  diameters  dt  and  d  being  small,  the  pres- 
sure p  can  be  very  great  for  a  comparatively  small  weight  W. 

The  capacity  of  the  accumulator  is,  however,  very  smaU. 
33us   is    of    advantage    when    being    used    in    connection    with 


*  Procetdingt  Imt.  Meeh.  Engf,,  1874, 


490 


n" 


J 


^^■d 


HYDRAULICS 


V 


< 


\ 


Fig,  340. 


1. 


11 


JJif 


;:? 


-3- 


X 


Fig,  841.     Hydraulic  Inteiisiflflr. 


HYDRAULIC    MACHINES  461 

hydraolie  rivetera,  aB  when  a  demand  is  made  upon  the  ac- 
cttmalatorj  the  ram  falls  qaickly,  hut  is  suddenly  arrested  when 
the  ram  of  the  riveter  comes  to  rest,  and  there  m  a  consequent 
increase  in  the  pressure  in  the  cylinder  of  the  riveter  which 
clinches  the  rivet.  Mr  Tweddell  estimatea  that  when  the  ac- 
cumulator is  allowed  to  fall  suddenly  through  a  distance  of  from 
IS  to  24  inches,  the  pressure  is  increased  by  50  per  cent. 

270*    Air  accumolatoir* 

The  air  accumulator  is  simply  a  vessel  partly  filled  with  air  and 
into  which  the  pumps^  which  are  supplying  power  to  machinery, 
deliver  water  while  the  machinery  is  not  at  work* 

Such  an  atr  vessel  has  already  been  considered  in  connection 
with  reciprocating  pumps  and  an  application  is  shown  in  connection 
with  a  forging  press.  Fig,  343, 

If  V  is  the  volume  of  air  in  the  vessel  when  the  pressure  is 
p  pounds  per  sq.  inch  and  a  volume  v  of  water  ia  pumped  into 
the  vessel  J  the  volume  of  air  is  (V  —  v), 

AsBuming  the  temperature  remains  constant^  the  pressure  pi  in 
the  vessel  will  now  be 

p.V 

If  V  is  the  volume  of  air,  and  a  volume  of  water  v  is  taken  out 
of  the  vessel, 

271,    Intensiflers. 

It  is  frequently  di-ginible  that  special  machines  shall  work  at 
a  higher  pressure  than  k  available  from  the  hydraulic  mains.  To 
increase  the  pressure  to  the  desired  amount  the  intensifier  is  used. 

One  form  is  shown  in  Fig.  34L  A  large  hollow  ram  works  in 
a  fixed  cylinder  C,  the  ram  being  made  water-tight  by  means  of  a 
§tafling-box.  Connected  to  the  cylinder  by  strong  bolts  is  a  cross 
head  which  has  a  smaller  hollow  ram  projecting  fi*om  it,  and 
Altering  the  larger  ram^  in  the  upper  part  of  which  is  made  a 
Btoffing-box.  Water  from  the  mains  is  admitted  into  the  large 
cylinder  and  also  into  the  hollow  i*am  through  the  pipe  and 
the  lower  valve  respectively  shown  in  Fig.  342, 

If  p  Iba.  per  aq.  inch  is  the  pressure  in  the  main,  then  on 
the  underside   of  the  large  ram   there  is  a  total  force  acting 

of  p  7  D*  pounds^  and  the  pressure  inside  the  hollow  ram  rises  to 

D* 

p  ^  pounda  per  sq,  inch,  D  and  d  being  the  external  diameters 

of  the  large  ram  and  the  small  ram  respectively. 


492 


HYDRAULICS 


The  form  of  iiiteiiBifier  kere  ahovm  is  used  in  connection  with 
a  large  flanging  press.  The  cylinder  of  the  press  and  the  tipper 
part  of  the  intensitier  are  fiUed  with  water  at  700  Ihs.  per  sq.  inch 
and  the  die  brought  to  the  work.  Water  at  the  same  pressore  i 
admitted  below  the  large  ram  of  the  intensifier  and  the  preasti 
in  the  upper  part  of  the  intenaifierj  and  thus  in  the  press  cylinder^ 
rises  to  2000  lbs,  per  sq*  inch,  at  which  preesure  the  flanging 
is  finished* 


JbSmaU 


ptrsq.  indL. 


^ 


Tp^ 


iM 


1 


Tb  Large  ^Undjet  at  IntenrnJUF 


Nofh  Return  Vcdvee  for 

Intensi/Ur 

Fig.  342. 


^    1 


aiioon 


272,  Steam  intensifiers. 

The  large  cylinder  of  an   intensifier  may  be  supplied  wit 
steam,  instead  of  water,  as  in  Fig*  343^  which  shows  a  steam 
tenaifier  used  in  ^con junction  with  a  hydraulic  forging  press,   Thes©| 
intensitiers  have  also  been  used  on  board  ship*  in  connection  wit 
hydraulic  steering  gears. 

273,  Hydraulic  forging  preaa,  with  steam  intensifier  and] 
air  accumulator. 

The  application  of  hydraulic  power  to  forging  presses  is  illu^  I 
trated  in  Fig.  343.  This  press  is  worked  in  conjunction  with  ft] 
steam  intensifier  and  air  accumulator  to  allow  of  rapid  world^iij 
The  whole  is  controlled  by  a  single  lever  K,  and  the  pre« 
capable  of  making  80  working  strokes  per  minute. 

When  the  lever  K  is  in  the  mid  position  everything  is  at  rest  j 

on  moving  the  lever  partly  to  the  rights  steam  is  admitted  into  th^ 

cylinders  D  of  the  press  through  a  valve.     On  mo\nng  the  lever  w 

its  extreme  position,  a  finger  moves  the  valve  M  and  admits  w»ler 

*  Prootedinffi  IntU  MecK  Enff».^  187i^ 


HTDRAITLIC  MACHIKES 


493 


relay  piston  shown  at  the  top  of  the  figure,  which  opens 
E  at  the  top  of  the  air  vessel*     In  small  presses  the  valve 
by  levers.    The  ram  B  now  ascends  at  the  rate  of 


foot  per  secondj  the  water  in  the  cylinder  c  being  forced 
accnmiiiatorf  On  moving  the  lever  K  to  the  left^  a«  soon 
M  passed  the  central  position  the  valve  L  is  opened  to 


494 


WTDEATJLICS 


exhauBt,  and  water  from  the  sir  veeael,  assisted  by  gravity,  forces! 
down  the  ram  B,  the  velocity  acqtiired  being  about  2  feet  perl 
second,  until  the  press  head  A  touches  the  work.  The  movementj 
of  the  lever  K  being  continuedj  a  valve  situated  above  the  vah 
J  is  opened,  and  steam  is  admitted  to  the  inteneifier  cylinder  H;' 
the  valve  E  closes  automatically,  and  a  large  pressure  is  exerted 
on  the  work  under  the  press  head* 

K  only  a  very  short  stroke  is  requiredj  the  bye-pass  valve  L  i 
tamporarily  discoimectedj  so  that  steam  is  supplied  eontiiitiousl]^ 
to  the  Hfting  cylinders  D*    The  lever  K  is  then  sim^ply  used 
admit  and  exhaust  steam  from  the  inteneifier  H,  and  no  water 
enters  or  leaves  the  accamulator.     An  automatic  controlling  ge 
is  also  fitted,  which  opens  the  valve  J  sufficiently  early  to  preventj 
the  intensifier  from  overrunning  its  proper  stroke. 


BTDRAITLIC  MACHINES 


495 


On©  end  of  a  wire  rope,  or  chaiiij  is  fixed  to  a  lug  L  on  the 
cylijiderj  and  the  rope  ia  then  passed  alternately  round  the  upper 
and  lower  pulleys,  and  finally  over  the  puJley  on  the  Jib  of  the 
crane.  Fig.  346,  In  the  crane  shown  there  are  three  pulleys  on 
the  ram,  and  neglecting  friction,  the  pressure  on  the  ram  is  equally 
divided  among  the  iix  ropes.  The  weight  lifted  is  therefore  one- 
aixth  of  the  pressure  on  the  ram,  but  the  weight  is  lifted  a  distance 
equal  to  six  times  the  movement  of  the  ram. 

Let  the  number  of  pulleys  on  the  end  of  the  ram  of  any  crane 

he  J ,  arranged  as  in  Fig.  347- 

The  movement  of  the  weight  wiH  then  he  n  times  that  of 
tlie  ram. 

Let  p  be  the  pressure  in  lbs.  per  sq.  inch  in  the  cylinder  and 
d  the  diameter  of  the  ram  in  inches. 

The  pressure  on  the  i*ara  is 

and  the  energy  supplied  to  the  crane  per  foot  travel  of  the  ram  is 
therefore  P  foot  pounds, 

kThe  energy  supplied  per  unit  volume  displacement  is  144  *  p. 
The  actnal  weight  lifted  is 


W  =  6^^pd»lbs., 


0  being  the  efficiency. 

^\'hen  full  load  is  being  lifted  e  is  between  07  and  0^8. 

For  a  given  lift  of  the  weight,  the  number  of  cubic  feet  of  water 
aaed,  and  consequently  the  energy  supplied,  is  the  same  whatever 
th©  load  lifted,  and  at  light  loads  the  efficiency  is  very  small. 

275.    Double  power  cranes. 

To  enable  a  crane  designed  for  Iieavy  work  to  lift  light  loads 
iHth  reasonable  efficiency,  two  lifting  rams  of  different  diameters 
art*  employed,  the  smaller  of  which  can  be  used  at  light  loads. 

A  convenient  arrangement  is  as  shown  in  Fig^.  348  and  349, 
the  smaller  ram  R'  working  inside  the  large  ram  B* 

^lien  Ught  loads  are  to  be  lifted j  the  large  ram  is  prevented 
from  momkg  by  strong  catches  0,  and  the  volume  of  water  used 
IS  only  eciual  to  the  diameter  of  the  small  mm  into  the  length  of 
the  stroke.  For  large  loadsj  the  catches  are  released  and  the 
two  rama  move  together. 

Another  arrangement  is  shown  in  Fig,  350,  water  being  ad- 
mitted to  both  faces  of  the  piston  when  light  loads  are  to  be 
lift^,  and  to  the  face  A  only  when  heavy  loads  are  to  be  raised. 


HTDRAtrtlC  MACHINES 


49? 


For  a  given  stroke  B  of  the  mm,  the  energy  supplied  in  the 
&l«t  ease  is 


■ted  in  the  second  case 


«r|)-{D^-^)ft.  lbs., 


Fi|«  350.     AitnitroGg  Double -pow«r  Hjdrftulio  Crftne  Cjlindar. 

276,    Hydraulic  crane  valves. 
Figs.  361   and  352  show  two  forms  of  lifting  and  lowering 
talves  tLsed  by   Armstrong,  Whitworth   and  Co.  for  hydraulic 

In  the  arrangement  shown  in  Fig,  351  there  are  two  inde- 
pendent valves,  the  one  on  the  left  being  the  pressure,  and  that 
cm  the  right  the  exhaust  valve. 


|,  851,    Ann  stTQQg'Wli  It  worth 
Hydraalie  Cr&ne  ValTe. 


I      u  a. 


Fig.  SS2.     Armstroog- Whitworth 
MjriirftuUe  Cr&ae  SMd«  Valve. 


4» 


HTDBAtlUCS 


tbowu  in  Fi|f .  S52  a  single  D  ^de  vake  i 

llv  Talvp  chest  duon^  the  prefi&ure  puaga 

in  the  neiitrml  podtion^     If  llie  Wve 

enters  the  cylinder,   but  if  it  is  rights 

dbe  ejriiixder  tkr^u^h  die  port  of  Uie  slids 


faydmUlc  pnn.    Fig.  353  is  a  section  tbroagh 

kydmnlie  preas^  naed  for  teotui^  springs. 

is  fitted  widi  m  brsas  linear,  and  axiaUf 

a  rod  Fr   ^*  a  platan  P  at  thjb  irsie  end^ 

il  ram  is  hoUow^  the  umsr 

id  fmm  the  largt?  cy lindur 
m  top  of   the   cylinder  ADif 
1  cylmder  ini^de  the  tsm  ii 
0 1]^  a  hole  drilled  along  tk 
the  water   ts  cotitinuooBlf 
I  the  annnl^r  ring  RR. 
I  prerenled  hf  means  d  i  ' 
G  ^nd  1  rrae  If^ther  r.  and  leakage  past  the 
--i-    i-  i  ri>c  n  P  by  cup  leathers  L  and  Lq. 
«^      -_  :1-   vilve  >p:r.dle  is  moved  to  the  right,  the  port  p  is 
-tvcvi    ^  ->.    :":ir  rxhaust,  and  the  ram  is  forced  up  by  the 
<->^      :  "1-  wjkTcr     n  the  annular  ring  RR.     On  moving  the 
-  -:   -  V-       r-  :     :he  left,  pressure  water  is  admitted  into  the 

-  -  '  >"  i  :r.r  r-i::i  :>  f  «n.>e^  down.     Immediately  the  pressure 

>,-*:•;.    ':t  -i:::  .    ni^s  baok  a^rain. 

-  r  "    -*     n    "ir    i::i meter  of   the  ram,  ti   the  diameter  of  the 
:V   -    -        i:^:v.r:cr  of  the  piston  P,  and  p  the  water  pressure 


rve  acting  on  the  ram  is 

-jT  :"r.e  ram  when  pivjsurf  is  released  from  the 


..f-»:  =  p^(iy-rf.')  lbs., 


F 


r  -  I •1;-  -  'Pi  lbs. 


.^"  vA.ve  spindle  S,  has  a  chamber  C  cast  in  it, 

:    >:x   holes    m   each  ring  are   drilled   through 

:  the  chamber.     These  rings  of  holes  are  at 

.^wX^rt  that,  when  the  spindle  is  moved  to  the 

i     I  ivssite  to  the  exhaust  and  the  other  opposite 

I  Nvhti:  the  spindle  is  moved  to  the  left,  the  holes 


L  Fig,  a53.    HydratUio  Pr^aa  with  Lutlie  Valve. 

Qrdratilic  riveter. 

n  through  the  cylinder  and  ram  of  a  hydraulic  riveter 
Fig,  354. 


600 


HYDRAULICS 


h  III  1^1  I  ■-    1 


^i   Inlft  Va/ie 


Fig.  355.    V&lTes  for  Hydraalio  Biveter. 


HYDRAULIC  MACHtJ^ES 


^01 


e  mode  of  working  is  exactly  the  same  as  that  of  the  small 
described  in  section  277. 

^  enlarged  section  of  the  valves  is  shown  in  Fig*  355*  On 
g  the  lever  L  to  the  right,  the  inlet  valve  V  18  opened,  and 
\re  water  is  admitted  to  the  large  cylinder^  forcing  out 
to-  Wlien  the  lever  is  in  mid  position,  both  valves  are 
,  hy  the  springs  S,  and  on  moving  the  lever  to  the  left,  the 
ftst  valve  V,  is  opened,  allowing  the  water  to  escape  from  the 
ier*  The  pressure  acting  on  the  annular  ring  inside  the 
ram  then  brings  back  the  ram*  Tlie  methods  of  preventing 
[©  are  clearly  shown  in  the  figures, 

B,    Hydraulic  engiBea, 

rdraulic  power  is  admirably  adapted  for  machines  having  a 
CK^ting  motion  only,  especially  in  those  cases  where  the  load 
Really  constant. 


Fig,  356.     Hydraulic  Capstati. 


501 


HTDRAUUCS 


II  hmB  masearer  been  successfully  applied  to  the  drivini 
mdk  WB  capstaiLS  and  winches  in  which  a  reciprocal 
i  ocMif^ned  into  a  rotarj  motion. 
Tlid  kjn&miLtie-eEigine  shown  in  Figs,  356  and  357,  haa  H 
fl^liiideri  in  one  castixigi  the  axes  of  which  meet  on  the  axis  of 
on&k  eliaft  &  Hie  motion  of  the  piston  P  is  transmitted  to 
crmnk  pin  hf  short  eoDneetin^  rods  R.  Water  is  admitted 
ejchausted  tluxmgh  m  Tmlve  V,  and  ports  p. 


The  face  of  the  valve  is  as  shown  in  Fig.  358,  E 
exhansi  port  oonnected  through  the  centre  of  the  valve  i^ 
exliaiisl  pipe,  luid  KM  the  pressure  port,  connect^  to  the  msft 
chamber  H  hy  a  small  port  through  the  side  of  the  valve.  1 
valve  seating  is  generally  made  of  lignnm-vitae,  and  has  A 
circnlar  ports  as  shown  dotted  in  Fig.  358»  The  valve  receiv^a 
motion  from  a  small  auxiliary  crank  T,  revolved  by  a  project 
from  the  cranX  pin  G.  When  the  piston  1  is  at  the  end  of 
stroke,  Fig*  359^  the  port  p,  should  be  just  opening  to  the  preA 
port,  and  just  closing  to  the  exhaust  port  E*  The  port  pi  shoi 
be  fnDy  open  to  pressure  and  port  pa  fully  open  to  exhai 
When  the  crank  has  turned   through  60  degrees,  piston  3  i 


A 


HYDRAULIC  MACHINES 


503 


be  at  tlie  inner  end  of  its  stroke,  and  the  edge  M  of  the  pressure 
port  ahoold  be  just  closing  to  the  port  p^.  At  the  same  instant  the 
edg^  1^  of  the  exhaust  port  should  be  coincident  with  the  lower 
ed^e  of  the  port  pt.  The  angles  QOM,  and  LON,  therefore, 
ahoald  each  be  60  degrees.  A  little  lead  may  be  given  to  the 
valve  i>orts,  i.e.  they  may  be  made  a  little  longer  than  shown  in 
the  Fig"-  358,  so  as  to  ensure  full  pressure  on  the  piston  when 
commencing  its  stroke.  There  is  no  dead  centre,  as  in  whatever 
position  the  crank  stops  one  or  more  of  the  pistons  can  exert  a 
turning'  moment  on  the  shaft,  and  the  engine  will,  therefore,  start 
in  any  position. 


Fig.  358. 


Fig.  359. 


The  crank*  effort,  or  turning  moment  diagram,  is  shown  in 
Fig.  359,  the  turning  moment  for  any  crank  position  OK  being 
OM.  The  turning  moment  can  never  be  less  than  ON,  which  is 
the  magnitude  of  the  moment  when  any  one  of  the  pistons  is  at 
the  end  of  its  stroke. 

This  tyi)e  of  hydraulic  engine  has  been  largely  used  for  the 
driving  of  hauling  capstans,  and  other  machinery  which  works 
intermittently.  It  has  the  disadvantage,  already  pointed  out  in 
connection  with  hydraulic  lifts  and  cranes,  that  the  amount  of 
^water  supplied  is  independent  of  the  effective  work  done  by  the 
machine,  and  at  light  loads  it  is  consequently  very  inefRcient. 
There  have  been  many  attempts  to  overcome  this  diflSculty, 
notably  as  in  the  Hastie  engine  t,  and  Bigg  engine. 


*  See  text  book  on  Steam  Engine. 

t  Proceedings  Inst.  Mech,  Engs,,  1874. 


lfTDRAlJt.lCS 


i      280«    Bigg  hydranUc  engine. 

To  adapt  the  quajitity  of  wat^sr  nsed  to  the  work  done*,  1 
hafi  modified  the  three  cylinder  engine  by  fbdng  the  crank  pin,  and 
aUawmg  the  cylinders  to  revolve  about  it  aa  centre. 

The  three  pistons  Pi,  Pa  and  P*  are  connected  to  a  disc, 
Fig.  360,  by  three  pins.  Thii  disc  revolves  about  a  fixed  ceutre  A. 
The  three  cylinders  rotate  about  a  centre  6,  which  is  capahle  d 

'  being  moved  nearer  or  furtlier  away  from  the  point  A  as  desired* 
The  stroke  of  the  pietons  is  twice  AG>  whether  the  crank  or  the 
cylinders  revolve,  and  since  the  cylinders,  for  each  stroke*  have  to 
be  filled  with  high  pressure  water,  the  quantity  of  water  supplied 
per  revolution  is  clearly  proportional  to  the  length  AGr. 


Fig.  360.    Higg  Hydrnnlio  Engine. 

The  alteration  of  the  length  of  the  stroke  is  effected  by 
of  the  subsidiary  hydraulic  engine,  shown  in  Fig.  36L  There 
two  cylinders  C  and  d,  in  which  slide  a  hollow  doable  ended 
ram  PPi  which  carries  the  pin  G,  Fig.  S60.  Cast  in  one  piece  iMth 
the  ram  is  a  valve  box  B.  R  is  a  fixed  ram,  and  through  it  wat^r 
enters  the  cylinder  Ci,  in  which  the  pressure  is  continuously 
maintained.  The  difference  between  the  etfectivB  areas  of  P  and 
Pv  when  water  is  in  the  two  cylinders,  is  clearly  equal  to  the  at«» 
of  the  ram  head  Ri . 


See  lilso  Engimgrt  Tgl.  LXxxv»  189S. 


HrDBAUUC   MACHINES 

From  the  cylinder  Ci  the  water  is  led  along  the  passages 

^OTfn   to  the   valve   V»     On   opening    this   valve    high-pressure 

^ter  19  admitted  to  the  cylinder  C.     A  second  valve  similar  to 

V,  bat  not  shown,   is   used   to  regulate  the  exhaust  from   the 

cylinder  C.     A?VTien  this  valve  is  opened,  the  ram  PPi  moves  to 

th  left  and  carries  with  it  the  pin  (t.  Fig*  360.     On  the  exhaust 

hehig  closed  and  the  valve  V  opened,  the  full  pressure  acts  upon 

l»ctth  ends  of  the  ranij  and  since  the  effective  area  of  P  is  great 

th^n  Pi  it  is  moved  to  the  right  carrying  the  pin  G,     If  botkj 

ralves  are  closed,  water  cannot  escape  fron>  the  cylinder  C  and 

the  ram  is  locked  in  position  by  the  pressure  on  the  two  ends* 


Water 


Fig.  361. 


EXAMPLES. 


ill  The  ram  of  &  hydraulic  crane  is  7  inches  diameter.  Water  is 
flopplied  to  the  crane  at  700  Iba.  per  square  inch.  By  suitable  gearing  the 
load  is  lifted  0  times  ob  quickly  as  the  raw.  Assuming  the  total  efficiency 
of  tb©  crane  is  70  per  eent.,  find  the  weight  lifted. 

(*2,t  An  accumulator  has  a  stroke  of  23  feet ;  the  diameter  of  the  ram  is 
23  iJM!he<i;  the  working  pressure  is  700  lbs.  per  squaje  inch.  Find  the 
capacity  ol  the  a^ocumalatoi*  in  horse -power  hour». 

id)     The  total  weight  on  the  cage  of  an  ammonition  hoist  is  8250  lbs. 

The  velocity  ratio  between  the  cage  and  the  ram  is  eix,  and  the  extra  load 

QOI  ilie  cage  due  to  friction  may  be  taken  as  30  per  cent,  of  the  load  on  the 

ea^.    The  steady  spaed  of  the  ram  is  6  inches  per  second  and  the  available  i 

re  at  the  working  valre  is  700  lbs,  per  square  inch. 

B^ttimate  the  loss  of  head  at  the  entrance  to  the  ram  cylinder^  and 

ling  this  was  to  be  doe  to  a  smJden  enlargement  in  passing  through 

^port  to  the  cyhnder^  estimate*  on  the  usual  asaumption,  tlie  area  of  the 

-yorl.  the  ram  cylinder  being  9|  inclies  diameter.     Lond,  Un,  1906, 

The  eflective  preaanre     p- * 

4 


506 


HTTDRAITLICS 


Lobs  of  head 


Area  of  port 


t?= velocity  through  the  ir&lv6i. 
4 


(4)  DeBcribe,  with  sketche^^  some  form  of  hydraulic  Rccumiilatot  so 
able  for  use  in  connection  with  riTstin^.     Explain  by  tl>e  aid  of  dia 
if  pOHsible,  the  general  nature  of  the  curve  of  pre^gxire  on  Uie  riveter  i 
during  the  stroke ;  and  point  out  tlte  re^aaona  of  the  Tariation^.     Lond  Uij 
1905,     (See  sections  262  and  269,) 

(5)  Describe  with  sketche*j  a  hydraulic  intensifier. 
An  intejosifler  is  required  to  increase  the  pressure  of  700  IbR,  per  squa 

inch  on  the  mains  to  SOOO  lbs.  per  square  inch.    The  stroke  of  the  int 
fier  is  to  be  4  feet  and  ita  capacity  Uiree  gallons.     Detennine  the  ( 
of  the  rama.    Inat.  C,  E.  1905. 

(6)  Sketch  in  good  proporttou  a  section  through  a  differential  hydra 
accumulator.    What  load  would  be  necessary  to  produce  a  pressuFe  of  1 1 
|ier  square  inch,  if  tlie  diameters  of  the  two  rams  are  4  incliea  and  4|  i 
respectively  ?    Neglect  the  Mction  of  the  packing.    Give  an  instance  of  tlw' 
use  of  auch  a  machine  and  state  why  accumulators  are  used. 

(7)  A  Tweddeirs  differential  accumulator  is  supplying  water  to  riTetung 
machines.  The  diameters  of  the  two  rams  are  4  inches  and  4|  icchef 
respectively,  and  tlie  pressure  in  the  accumulator  b  1  ton  per  square  mrJi. 
Suppoae  when  the  valve  is  closed  Uie  accumulator  is  falling  at  a  vekxity 
of  5  feet  per  second,  and  the  time  taken  to  bring  it  to  rest  is  2  seconds.  tioJ 
the  increase  in  pressure  in  the  pipe. 

(8)  A  hit  weighing  12  tons  is  worked  by  water  pressure,  the  pn 
in  tlie  maiu  at  the  accumulator  being  1200  lbs.  per  square  inch ;  the  lengtiil 
of  tlie  supply  pipe  which  is  Sj  inches  in  diameter  is  flOO  yards,  Wh^i  >r| 
the  approximate  speed  of  ascent  of  this  Ijftt  on  the  assumption  that  lb*  I 
friction  of  the  Btufiing-box,  guides,  etc.  is  equal  to  6  per  cent,  of  the  j 
load  lifted  and  the  ram  is  8  inches  diameter  ? 

(9)  Explain  what  is  meant  by  the  "  coefficient  of  hydraulic  resiHtaEoe" 
as  applied  to  a  whole  system,  and  what  assumption  is  usually  made  rt<gsnl*| 
ing  it?  -A  direct  acting  lift  having  a  ram  10  inches  diameter  i«f  '^nppll 
from  an  accumulator  working  under  a  pressiu^e  of  750  lbs.  i>er  si] 
When  carrying  no  load  the  ram  moves  through  a  distance  of  t^^ ' 
uniform  speed,  in  one  minute,  the  valves  being  fully  open.     Estiroatt  i 
coefficient  of  hydraulic  resistance  referred  to  the  velocity  of  the  ram,  I 
als^  how  long  it' would  take  to  move  the  same  distance  when  tli«  i 
carries  a  load  of  20  tons.     Loud.  Vn.  1905. 

^''^^ead  1  ost  =  -;^:;^^      .   Assumption  is  made  that  resistance  varies  «a  •'J 


64 


62-4 


CHAPTER  XII. 


EESISTAJS'CE  TO  THE  MOTION   OF  BODIES  IK  WATER. 


281,     Froude^s*   expeiimenta    to   determine   Motional  re- 
istances  of  tliin  boards  when  propelled  in  water. 

It  has  been  shown  that  the  frictional  resiatance  to  the  flow  of 
gr  along  pipes  is  proportional  to  the  velocity  raised  to  some 
BF  n,  which  approximates  to  two,  and  Mr  Froude's  classical 
ents,  in  connection  with  the  resistance  of  ahips,  show  that 
ince  to  motion  of  plane  vertical  boards  when  propelled 
II  w^ftter,  follows  a  similar  law. 


^ft  Fig.  303, 

'^The  experiments  were  carried  out  near  Torquay  in  a  parallel 
lidted  tank  278  feet  longj  36  feet  broad  and  10  feet  deep.  A  light 
teilway  on  '*  which  ran  a  stout  framed  truck,  suspended  from  the 
\xlem  of  two  pairs  of  wheels,"  traversed  the  whole  length  of  the 
^ok,  about  20  inches  above  the  water  level*  Tlie  truck  was  pro- 
lelled  by  an  endless  wire  rope  wound  on  to  a  barrel,  which  could 
le  made  to  revolve  at  varjang  speeds,  so  that  the  truck  could 
llftvirne  the  length  of  the  tank  at  any  desired  velocity  between 
DO  and  1000  feet  per  minute, 

*  Bril.  Amu.  Rep&tU,  li^72-4* 


508 


HTDRAULIOS 


Planes  of  wood,  about  ^  incli  thick,  the  Bui^ces  of  widch  were 
covered  with  variona  materials  as  set  out  in  Table  XXXIX,  were 
made  of  a  uniform  depth  of  19  inches^  and  when  under  experi- 
ment were  placed  on  edge  in  the  water,  the  uppefr  edge  hmg 
ibont  \j  inches  below  the  surface.  The  lengths  were  varied  from 
;  to  50  feet. 

T\\e  ap^mratus  as  used  hj  Fronde  is  illustrafced  and  described 
in  the  British  A^sociution  Reports  for  1872. 

A  later  adaptation  of  the  apparatus  as  uaed  at  Haslar  for 
determining  the  resistance  of  ships*  models  is  shown  in  Fig.  361 
An  arm  L  is  connected  to  the  model  and  to  a  frame  beam,  which 
is  carried  on  a  double  knife  edge  at  H,  A  spring  S  is  attached  to 
a  knife  edge  on  the  beam  and  to  a  fisted  knife  edge  N  on  the 
frame  of  the  truck*  A  link  J  connects  the  upper  end  of  the  beam 
to  a  multiplying  lever  which  moves  a  pen  D  over  a  recordiBg 
cylinder.  This  cylinder  is  made  to  revolve  by  means  of  a  worm 
and  wheel,  the  worm  being  driven  by  an  endless  belt  from  the  ajcle 
of  the  truck.  The  extension  of  the  spring  S  and  thus  the  move* 
ment  of  the  pen  D  is  proportional  to  the  resistance  of  the  model, 
and  the  rotation  of  the  drum  is  proportional  to  the  distance  mOTedi 
A  pen  A  actuated  by  clockwork  registers  time  on  the  cylinder. 
The  time  taken  by  the  truck  to  move  through  a  given  distance 
can  thus  be  determined. 

To  calibrate  the  spring  Sj  weights  W  are  hung  from  a  knife 
edge,  which  is  exactly  at  the  same  distance  from  H  as  the  pointi 
of  attachment  of  L  and  the  spring  S. 

From  tlie  results  of  these  experiments,  Mr  Frond©  mad©  the 
following  deductions, 

(1)  The  frictional  resistance  varies  very  nearly  with  tlie 
square  of  the  velocity. 

(2)  Tlie  mean  resistance  per  square  foot  of  surface  for  leugdia 
up  to  50  feet  diminishes  as  the  length  is  increased,  but  is  pmc* 
tically  constant  for  lengths  gr^^at^r  than  50  feet, 

(3)  The  frictional  resistance  varies  very  couidderably  wil 
the  roughness  of  the  surface. 

Expressed  algebraically  the  frictional  resistance  to  the  motii 
of  a  plane  surface  of  area  A  wheii  moving  with  a  velocity  o 
per  second  is 


/  being  equal  to 


A 

10- • 


KESJSTAiCCE  TO  THE  MOTION   OF   BODIES   IN   WATER 


509 


TABLE  XXXIX, 

Showing  the  result  of  Mr  Froude'a  experiments  on  the  ftnctional  ^ 
(resiivtatice  to  the  motion  of  thin  vertical  boards  towed  through 
in  a  direction  parallel  to  its  plane. 
Width  of  boards  19  inches,  thickness  fV  inch, 

power  or  index  of  speed  to  which  resistance  is  approxi* 
^tely  proportionah 

f^^  tlie  mean  resistance  in  pounds  per  square  foot  of  a  surface, 
length  of  which  is  that  specified  in  the  heading,  when  the 
relocity  is  10  feet  per  second. 

/i  =  the  resistance  per  square  foot,  at  a  distance  from  the 
ing  edge  of  the  boards  equal  to  that  specified  in  the  heading^ 
a  velocity  of  10  feet  per  second. 

As  an  example,  the  resistance  of  the  tinfoil  surface  per  square 
at  8  feet  from  the  leading  edge  of  the  board,  at  10  feet  per 
cond,  is  esttinated  at  0^263  pound  per  square  foot;  the  mean 
resistance  ia  0"278  pound  per  square  foot. 


Soffaoe 
J^-^irea  with 


'^ttmiah 


Letigth  of  plameB 


2  feet 


2^ 

2^16 

1-93 

20 

2-0 

20 


0-41 

0*90 
0-87 
061 
0-90 
110 


0890 

0*295 
0-725 
0-690 
0-780 
0880 


8  feel 


20  ie%i 


1-99 

I '92 
2-0 
2^0 
2*0 


0-325 
0-278 
0-626 
0-583 
0625 
0-714 


0-264 
0*263 
0*504 
0-400 
0*488 
0-520 


n        f,         /j 


1-85 

1-90 

1-89 

2-0 

20 

20 


0-278  0-240 
0*262  '  0-244 
0*581  0-447 
0*480  0-384 
0-584  0-465 
0-588  0^490 


50  feet 


1*83 
1*88 
X-87 
2-06 
2*00 


0-250 
0-246 
0-474 
0*405 
0-486 


0-226 
0-282 
0-423 
0*887 
0-456 


The  dirainution  of  the  resist^-nce  per  unit  area,  with  the  length, 
is  principally  due  to  the  relative  velocity  of  the  water  and  the 
b<mrd  not  being  constant  throughout  the  whole  length, 

A&  the  board  moves  through  the  water  the  frictional  resistance 
of  the  first  foot  length,  say,  of  the  board,  imparts  momentum  to 
tlia  water  in  contact  with  it,  and  the  water  is  given  a  velocity  in 
til©  direction  of  motion  of  the  board.  The  second  foot  length  will 
tlierefore  be  rubbing  against  wat^r  having  a  velocity  in  its  own 
direction,  and  the  frictional  resistance  w411  be  less  than  for  the 
fir&t  foot*  The  momentum  imparted  to  the  water  up  to  a  certain 
point,  is  accumulative,  and  the  total  resistance  does  not  therefore 
increase  proportionally  with  the  length  of  the  board. 


510 


FTBIlAtJLfCS 


282.    Stream  line  theory  of  the  resistance  offered  to  the 
motion  of  bodies  in  water. 

Rem^tance  of  ships*    In  con^deriBg  the  motion  of  water  along 
ptpee  and  channels  of  uniform  section,  the  water  has  been  aesumed 
to  move  in  **  stream  lines/'  which  have  a  relative  motion  to  the 
sides  of  the  pipe  or  channel  and  to  each  other^  and  the  readstaiw 
to  the  motion  of  the  water  has  been  considered  as  due  t-o  the 
friction  between  the  consecutive  stream  lines,  and  between  the    j 
water  and  the  surface  of  the  channel,  these  frictional  resistance^ 
above  certain  speeds  being  such  as  to  cause  rotational  motions 
the  mass  of  the  water. 


theT 


Fig.  363. 


Fig.  8S4. 

It  has  also  been  shown  that  at  any  sudden  enlargement  olj 
stream,  energy  is  loat  due  to  eddy  motions,  and  if  bodies, 
as  are  shown  in  Figs-  110  and  111,  be  placed  in  the  pipe,  there  \ 
a  pressure  acting  on  the  body  in  the  direction  of  motion  of  the 
water.  The  origin  of  the  remetance  of  ships  is  best  realise*!  by 
the  "stream  line^'  theory,  in  which  it  i^  assumed  that  relative  to 
the  ship  the  water  is  moving  in  stream  lines  as  shown  in  Fig*. 
363,  364,  consecutive  stream  lines  also  having  relative  motion. 


lEISTANCE  TO  THE   MOTION  OF  B0DI^1N^^^^»511 

Lecordmg  to  this  theory  the  resistance  is  divided  into  three 

L)  Frictional  resistance  due  to  the  relative  motions  of  con- 
scutive  stream  Unesj  and  of  the  stream  lines  and  the  surface 
f  the  ship. 

(2)  Eddy  motion  resistances  dae  to  the  dissipation  of  the 
aergy  of  the  stream  Hues,  all  of  which  are  not  gradually  brought 

St. 

J)     Wave  making  resistances  due  to  wave  motions  set  up  at 
i§tirface  of  the  wat^er  by  the  ship,  the  energy  of  the  waves 
leing'  dissipated  in  the  surrounding  water. 

According  to  the  late  Mr  Froude,  the  greater  proportion  of 
be  resistance  is  due  to  frictionj  and  especially  is  this  so  in  long 
h]p<s^  Tftith  ftue  lines,  that  is  the  cross  section  varies  very  gradually 
rtiTu  the  bow  towards  midsliipSj  and  again  from  the  midships 
DWards  the  stern.  At  speeds  less  than  8  knots,  Mr  Fronde  has 
huim  that  the  frictional  resistance  of  ships,  the  full  sx>eed  of 
rhich  is  about  13  knots,  is  nearly  90  pt^r  cent,  of  the  whole 
^eBistance,  and  at  full  speed  it  is  not  much  less  than  60  per  cent, 
3e  has  further  shown  that  it  is  practically  the  same  as  that 
■existing  the  motion  of  a  thin  rectangle,  the  length  and  area  of 
ihe  two  sides  of  which  are  equal  to  the  length  and  immersed 
urea  respectively  of  the  ship,  and  the  surface  of  which  has  the 
lame  degree  of  roughness  as  that  of  the  ship. 

If  A  is  the  area  of  the  immersed  surface,  /  the  coefficient  of 
friction,  which  depends  not  only  upon  the  roughness  but  also 
Upon  the  length,  V  the  velocity  of  the  ship  in  feet  per  second,  the  _ 

resistance  due  to  friction  is  ^| 

^5  value  of  the  index  n  approximating  to  2» 

The  eddy  resistatice  depends  upon  the  bluntness  of  the  stem  of 
Ihe  boatu,  and  can  be  reduced  to  a  minimum  by  diminishing  the 
iMtion  of  the  ship  gradually,  as  the  stem  is  approached,  and  by 
^piding  a  thick  stern  and  stern  post. 

As  an  extreme  case  consider  a  ship  of  the  section  shown  in 
JPig,  364j  and  suppose  the  stream  lines  to  be  as  shown  in  the 
figare.  At  the  st«rn  of  the  boat  a  sudden  enlargement  of  the 
Btn^m  lines  takes  place,  and  the  kinetic  energy,  which  has  been 
g]¥en  to  the  stream  lines  by  the  ship,  is  dissipated.  The  case  is 
analogous  to  that  of  the  cylinder,  Fig.  lU,  p*  169,  Ehie  to  the 
Io6s  of  energy,  or  head,  there  is  a  resultant  pressure  acting  up<jn 
the  ahip  in  the  direction  of  flow  of  the  stream  lines,  and  con- 
leqoently  opposing  ite  motion* 


512 


HTtlRA0LlCS 


If  the  ship  baa  fine  lines  towards  the  stem,  as  in  Fi^,  363, 
the  velocitiei  of  tlie  stream  lines  are  diminished  gradually  and  tk 
logg  of  energy  by  eddy  motions  becomes  very  smalL  In  actiml 
ships  it  is  probably  not  more  than  8  per  cent,  of  the  whde 
resistance. 

The  wave  making  resistance  depends  upon  the  length  and  tli6 
form  of  the  ship,  and  especially  upon  the  length  of  the  '^entraiic*'* 
and  ^*  run/'  By  the  "  entrance  *'  is  meant  the  front  pan  of  the 
sliip,  which  gradually  increases  in  section*  until  the  middle  body, 
which  is  of  uniform  section,  is  reached,  and  by  the  *^  nin,"  the 
hinder  part  of  the  ship,  which  diminishes  in  section  from  tie 
middle  hody  to  tlie  stern  post. 

Beyond  a  certain  speed,  called  the  critical  speed,  the  rate  of 
increase  in  wave  making  resistance  is  very  much  greater  than 
the  rate  of  increase  of  speed.  Mr  Fronde  found  that  for  tb 
S*S,  "  Merkara "  the  wave  making  resistance  at  13  kmM,  tlie 
normal  speed  of  the  ship,  was  17  per  cent,  of  the  whole,  but  at  19 
knots  it  was  60  per  cent.  The  critical  speed  was  about  18  kiiDt** 
An  appro3ciinate  formula  for  the  critical  speed  V  in  knot^  is 

L  being  the  length  of  entrance,  and  Li  the  length  of  the  rim  b 
feet. 

The  mode  of  the  formation  by  the  ship  of  waves  can  be  partly 
realised  aa  follows. 

Suppose  the  ghip  to  be  moving  in  smooth  water,  and  the  stT^affl 
lines  to  be  passing  the  ship  as  in  Fig.  363.  As  the  bow  of  tb 
boat  strikes  the  dead  water  in  front  there  is  an  increase  in 
pressure,  and  in  the  horizontal  plane  SS  the  pressure  mB  k 
greater  at  the  bow  than  at  some  distance  in  front  of  it,  and 
consequently  the  water  at  the  bow  is  elevated  above  the  nofiBii 
surface. 

Now  let  AA,  BE,  and  CC  be  three  sections  of  the  ship  and  ^ 
stream  lines. 

Near  the  midship  section  CC  the  stream  lines  will  be  m(0 
closely  pa<iked  together,  and  the  velocity  of  flow  will 
therefore,  than  at  AA  or  BB*    Assuming  there  is  no  1(» 
in  a  stream  Hue  between  AA  and  BB  and  applying  Hemotiilli'i 
theorem  to  any  stream  line^ 

w      2g     w     2g     w      2g* 


*  See  Sir  W.  While's  Kami  ATcMieeture,  Tramaetmm  ttf  Nm^  Anhint^ 

1877  aad  1881, 


RBSISTAKCE  TO  THE  MOTtOK  OF  BODIES  IN  WATEB 
id  since  TjL  and  t^  are  less  than  i'^^ 

^  and  —  are  greater  than  — 


w 


w 


w 


513 


4 


*rhe  Burface  of  the  water  at  AA  and  BB  is  therefore  higher 
%n  at  CC  and  it  takes  the  form  shown  in  Fig.  363, 
jTwo  sets  of  waves  are  thus  fomiedj  one  by  the  advance  of  the 
and  the  other  by  the  stream  hnes  at  the  st^m,  and  tbesal 
motions  are  transmitted  to  the  surroanding  water,  where 

energy  is  dissipated.     This  energy,  as  well  as  that  lost  in 

idy  motions,  must  of  necessity  have  been  given  to  the  water  by^| 
■"  ship,  and  a  corresponding  amount  of  work  has  to  be  done  by 
:ie  i^hip's  propeller.    The  propelling  force  required  to  do  work 
|t&al  to  the  loss  of  energy  by  eddy  motions  is  the  eddy  resist^^f 
ice,  and  the  force  required  to  do  work  equal  to  the  energy  of™ 
waves  set  up  by  the  ship  is  the  wave  resistance. 
To  reduce  the  wave  resistance  to  a  ndnimuin  the  sliip  should 
Qiade  very  long,  and  should  have  no   parallel  body,  or  the 
sfcire  length  of  the  ship  should  be  devoted  to  the  entrance  and 
JH-    On  the  other  hand  for  the  frictional  resistance  to  be  small, 
ne  area  of  immersion   must  be  small,  so  that  in  any  attempt 
design  a  ship  the  resistance  of  which  shall  be   as  small  as 
Dsgible,  two  conflicting  conditions  have  to  be  met,  and,  neglecting^ 
eddy  resistances,  the  problem  resolves  itself  into  making  th^| 
of  the  frictional  and  M^ave  resistances  a  minimum. 
Total  rensiance.    K  R  is  the  total  resistance  in  pounds,  T/  the_ 
ictiotml  resistance,  n  the  eddy  resistance,  and  r„  the  wave 
stance, 

^^Tf  +  rt  +  r^. 

be  frictional  resistance  r/  can  easily  be  determined  when  tl; 
latitre  of  the  surface  is  known.     For  painted   steel  ships  /  is' 
[iractieally  the  same  as  for  the  varnished  boards,  and  at  10  feet 
Br  second  the  frictional  resistance  is  therefore  about  \  lb.  per 
|tiare  foot,  and  at  20  feet  per  second  1  lb,  per  square  foot,    The^ 
If  satisfactory  way  to  determine  r^  and  r„  for  any  ship  is  tijH 
lake  experiments  upon  a  model,  from  which,  by  the  principle  of 
ailarity,  the  corresponding  resistances  of  the  ship  are  deducec 
be  horse-power  required  to  drive  the  ship  at  a  velocity  of  V  fe 
sr  second  is 

EV 


the 
a^v^l 

th6| 


HP  = 


550' 


To  determine  the  total  resistance  of  the  model  the  apparatus 
bown  in  Fig,  362  is  used  in  the  same  way  as  in  determining  the 
ictional  resistance  of  thin  boards. 


L.  a. 


33 


283.  BetemiixLation  of  the  resistance  of  a  Bhlp  firotn  the 
resistance  of  a  model  of  the  ship. 

Tu  obtain  the  resistance  of  the  ship  from  tiie  experimental 
i^eaifltaiicQ  of  the  model  the  priociple  of  sim^Uantyj  as  stated  bf 
Mr  Froudej  is  used.  Let  the  linear  dimensions  of  the  ship  be  D 
tunes  those  of  the  modeL 

Cmre^pofuUfig  speeds.  According  to  Mr  Fronde's  theor>%  for 
any  speed  V„  of  the  models  the  speed  of  the  ship  at  which  m 
resistanoa  mnst  be  compared  with  that  of  the  model,  or  th 
ocMn^ponding  speed  V*  of  the  ship,  is 

CorrBsponJUng  remstances.    If  lU  is  the  resistance  of  the  model 
^  the  velocity  Vmt  and  it  be  assmmed  that  the  coefficients  d 
'friction  for  the  ship  and  the  model  are  the  same,  the  redstamoe  & 
of  the  ship  at  the  corresponding  speed  V*  is 

Ab  an  example,  snppose  a  model  oue-dxteentb  of  the  siis 
of  the  ship;  the  corresponding'  speed  of  the  ship  will  be  fonr  timei 
the  speed  of  the  model,  and  the  resistance  of  the  ship  at  eon^ 
spondiBg  speeds  wOl  be  1^  or  4006  times  the  resistance  of  tJie 

Comadtbii  for  the  difffsrente  of  the  coeffim^nts  of  frietimi  for  HiM 
mocbl  OffM?  0hip.  Tlie  material  of  which  the  immersed  surface 
tlie  model  is  made  is  not  generally  the  same  as  that  of  the  ship^ 
and  conseqeentiy  R,  mnst  be  corrected  to  make  allowance  for  the 
difference  of  ronghnees  of  the  surfaces.  In  addition  the  ship  i 
very  much  longer  than  the  model,  and  the  coefficient  of  frictiaii» 
0f«ii  if  the  surfaces  were  of  the  same  degree  of  roughness,  would 
tbetBfore  be  less  than  for  the  model. 

let  Am  he  the  immersed  surfoce  of  the  model  and  A. 
the  shipw 

LeC  fm  be  the  coefficient  of  friction  for  the  model  and  /,  for 
ahipi  the  values  being  made  to  depend  not  only  upon  the  rougJiD* 
bat  also  upon  the  length.    If  the  resistance  is  assumed  to  vaiy 
V*t  Ihe  frietioinal  resistance  of  the  model  at  the  velocity  Y^  is 

•ai  for  Iks  ship  at  the  corresponding  speed  Y«  the  frietioQii 

r,=/,A.V/. 

A,  =  A»I>* 


But 


SBBISTANCS  TO  THE  MOnOH  OF  BODIES  IN  WATER 


515 


and,  therefore,  r,=yiA«V«*D* 

Then  the  Tesistance  of  the  ship  is 

R.=  (R,-r.)D'  +  r. 


=  {R.^r.(^-l)}D.. 


Determination  of  the  cwrve  of  resistance  of  the  ship  from  the 
curve  of  resistance  of  the  model.  From  the  experiments  on  the 
model  a  curve  having  resistances  as  ordinates  and  velocities  as 
abscissae  is  drawn  as  in  Fig.  365.  If  now  the  coefficients  of 
friction  for  the  ship  and  the  model  are  the  same,  this  curve,  by 
an  alteration  of  the  scales,  becomes  a  curve  of  resistance  for  the 
ship. 

For  example,  in  the  figure  the  dimensions  of  the  ship  are 
supposed  to  be  sixteen  times  those  of  the  model.  The  scale  of 
Telocities  for  the  ship  is  shown  on  CD,  corresponding  velocities 
being  four  times  as  great  as  the  velocity  of  the  model,  and  the 
scale  of  resistances  for  the  ship  is  shown  at  EH,  corresponding 
resistances  being  4096  times  the  resistance  of  the  model. 


H 


Fig.  365. 


^r  Froude^s  method  of  correcting  the  curve  for  the  difference  of 
the  €X>€fficienfs  of  friction  for  the  ship  and  the  model.  From  the 
formula 

rm  —  Jm-A-m  V  m  , 


516 


HTBRAUI.ICS 


the  frictional  resistance  of  the  model  for  several  values  of  V, 
is  calculated,  and  the  curve  FF  plotted  on  the  same  scale  as  aad 
for  the  curve  RR.  The  wave  and  eddy  making  resistance  at  an; 
velcw::ity  is  the  ordinate  between  FF  and  RR.  At  velocities  c 
200  feet  per  second  for  the  model  and  800  feet  per  second  fi 
the  shipj  for  example^  the  wave  and  eddy  making  resistance  is  bt 
measured  on  the  scale  BG  for  the  model  and  on  the  iK^ale  EH  t 
the  ship. 

The  frictional  reaiBtance  of  the  ship  is  now  calculat-ed  from 
form  u la  r,  =/jA«W,  and  ordinate^  are  set  down  from  the  c 
FFj  equal  to  r«j  to  the  scale  for  ship  rpsistance.     A  third  curve 
thus  obtained  J  and  at  any  velocity  the  ordinate  between  this  cum 
and  RR  is  the  resistance  of  the  ship  at  that  velocity.   For  exampl 
when  tlie  ship  has  a  velocity  of  800  feet  per  second  the  resistanc 
is  aCj  measured  on  the  scale  EH. 


EXAMPLES. 

(1)  Taking  skin  Miction  to  be  D'4  lb.  per  sq^uare  foot  at  10  faet  per 
aecondt  find  the  skill  resietaace  of  a  sMp  of  12^000  square  feet  immeraed 
surface  at  15  kxtots  (1  kuot = I'd9  leet  per  second).  Also  tind  the  horse-pQwt*r 
to  drive  the  ship  against  tliis  resistance. 

(2)  If  the  skin  friction  of  a  ship  is  05  of  a  pound  per  square  foot  otf 
immersed  surface  at  a  speed  of  6  knota»  what  horse-power  will  prob*blj 
be  required  to  obtain  a  speed  of  14  knota*  if  the  immersed  surface  is  18,0110 
square  feet  ?  You  may  assume  the  maximmn  speed  for  which  the  ship  Is 
designed  is  17  knots. 

(8)  The  resistance  of  a  vessel  is  deduced  from  that  of  a  model  |V^  ^ 
Hnear  size.  The  wetted  surface  of  the  model  is  20*4  square  feet,  the  ekii 
friation  per  square  foot^  in  fresh  water,  at  10  feet  per  Beoond  is  0*3  lb.,  asd 
the  index  of  velocity  is  1'94.  Tlie  skin  friction  of  tlie  vessel  in  salt  i 
is  60  iba.  per  100  square  feet  at  10  knots,  and  the  index  of  velocity  is ! 
The  total  resistance  of  the  model  in  frei^h  water  at  200  feet  per  jimt 
1-46  lbs.  Estimate  the  total  resistance  of  the  vessel  in  salt  water  4 
speed  corresponding  to  200  feet  per  minute  in  ihe  model.     Lond.  Uo.  M 

(4)  How  from  model  experiments  may  the  resistance  of  a 
inferred  ?  Point  out  what  corrections  have  to  be  made.  At  a 
800  feet  per  minute  in  fresh  water,  a  model  10  feet  in  length  with  *  ' 
akin  of  24  square  feet  has  a  total  resistance  of  2*39  lbs*»  2  lbs.  being  dws  ^ 
skin  resistance^  and  -39  lb,  to  wave-making.  What  will  be  the  total  i 
ance  at  the  corresponding  speed  in  salt  water  of  a  diip  25  tijnes  tbe  I 
dimensions  of  the  model,  having  given  that  the  surface  friction  per  tfp^ 
foot  of  the  ship  at  that  speed  is  1-3  lbs,  7    Lond,  Un.  1906* 


CHAPTEK  XIIL 


STEEAM  LINE  MOTION. 


284.     Hele    Shaw's    experiments    on   the   fLow   of   thin 
slieetB  of  water, 

l-Vofessor  Hele  Shaw*  has  very  beautifully  shown,  on  a  small 

e|^le,  the  fomi  of  the  stream  lines  in  moving  masses  of  water 

imder  varying  circumstances^  and  has  exhibited  the  change  from 

stream  line  to  sinuous,  or  rotational  flowj  by  experiments  on  the 

flow  of  water  at  varying  velocities  between  two  parallel  glass 

plnUft^.     In  some  of  the  exi:)eriinents  obstacles  of  various  forma 

were  placed  between  the  plates,  past  which  the  water  had  to  flow, 

and  in  others,  channels  of  various  sections  were  formed  through 

which  tlie  water  was  made  to  flow.    The  condition  of  the  water 

as  it  Howed  between  the  plates  was  made  visible  hy  mixing  with 

it  a  certain  quantity  of  air,  or  else  by  allowing  thin  streams  of 

coloured  water  to  flow  between  the  plates  along  with  the  other 

water.     When   the   velocity   of   flow   was   kept   sufficiently   low, 

whatever  the  form  of  the  obstacle  in  the  path  of  the  water,  or 

the  form  of  the  channel  along  which  it  HoweJj  the  water  persisted 

ia  stream  line  flow*    When  the  channel  between  the  plates  was 

tnade  to  enlarge  suddenly,  as  in  Fig.  58,  or  to  pass  through  an 

orifice,  as  in  Fig.  59,  and  as  long  as  the  flow  was  in  stream  lines, 

•no  eddy  motions  were  produced  and  there  were  no  indications 

Lof  \ij8^  of  head.     Wien  the  velocity  was  sufficiently  high  for  the 

I  flow  to  become  sinuous,  the   eddy  motions   were   very   marked. 

^len  the  motion  was  sinuous  and  the  water  was  made  to  flow 

l^st,  ohstacles  similar  to  those  indicated  in  Figs.  110  and  III,  the 

^ikter  immediately  in  contact  with  the  down-stream  face   waa 

*h<mn  to  be  at  rest.     Similarly  the  water  in  contact  with  the 

tttiiiulor  ring  surrounding  a  sadden  enlargement  appeared  to  be 

'^t  TL^i  and  the  assumption  made  in  section  51  was  thus  justified. 

*  Proc€tdinff»  of  Naval  ArchiUcU,  1S9T  &iid  IS9B.     Engineer,  Aug.  IS^T  and 


518 


HTDRAUUCS 


Wlien  the  Arm  was  dan^  channels  and  eiQUOCUS,  ilie  stnoonglf 
movm^  iratr-r  appeared  to  be  separated  from  the  sides  c(  llie 
ckannel  bj  a  thm  filiii  of  water,  wbich  Profeaoor  Hele  Sbaw 
5a.«g^?>ted  was  mornig  in  stream  lines,  tke  velocity  of  which  ia 
the  ilm  dii  ":  as  tbe  surface  of  the  channel  is  approached, 

Thie   experi  -  miao  indicaSfMi  that  a  stmilar  film  surroimded 

obc^t;kcIe^  of  fiu{i4tke  wad  other  forms  placed  in  flowing  wata't 
And  :r  waa  ^  -reA  by  Pr^jfe^tsr  Hele  Shaw  thatj  snrrT^unding 
a  ioip  ^s  it  ^  tkrom^  sdH  waler,  there  ia  a  thin  film  moving 

m.  5t7*?ikiiL  b  ely  to  the  shi     ''i©  shearing  forces  between 

wiiicii  Ami  .. :  wOodiBS  waiei        ^  ap  eddy  motions  which 

.accv:cL:i:  fan  ^km  faidici  of  th       lip. 

285.    C^rrvd  abeam  Una  m<  i* 

Ijtt  A  &ju9  oC  feiil  be  moriii^  i  trred  ^ream  lineSf  and  kl 

JlKFx  tr-    — --^^Mieoftheato  fines. 

A~  Aaj  -t  tW  rai&a  Kfiaiun*  of  the  etiieaiti  iiBfi 

O  ci^iie     \^.   .^ r-iitB«faiB         net&i  o&dd  finrrcrandnig  ^B 

Ic'-  '^  :tr  -ji-f  T^r :x:i:    f  this  element. 

.:  ':»r  -^T  Tr^s-cLT^  per  z.ziiz  -^reik  on  the  face  hd. 

-  -    -    >  "-!•:  7rf>«?cLre  per  uziit  area  on  the  face  ae, 

^    t:  -.:-    -      -jitivn  -t  ine  :;ing>rnt  to  the  stream  line  at  c 


i-'^i    *  fiw:i    :*  :~e  fiice^s  >f  and  a^. 
:    »   ~7    f  :i:»e  <cr^-az:  line  at  •:, 

-c^^il:::!  jjitf  -5  .z  i  Tem:-^.  plane  the  forces  acting 


W- 


-^^     ^rix^  "^  r->f   -^-  icrji^  alzng"  the  radius  away 


:r^ss^^    r  r   ictinx  aI-  n^  :'~^    ra^iins    towards  the 


>v>^  --^      i. 


..-■^""-w... 


=  '~"-r:-:.r  •  D. 


STREAM  LINE  MOTION 


519 


TOtind  the  bend  of  a  river,  Oc  is  horizontal  and  the  component  of 
"W  along  Oc  is  zero. 

Then  ^=^^  (2). 

dr     g  r  ^  ^ 

Int^prating  between  the  limits  R  and  Bi  the  difference  of 
piOflompe  on  any  horizontal  plane  at  the  radii  R  and  Ri  is 

^"■^"I/r  r^ ^^^' 

"wliich  can  be  integrated  when  v  can  be  written  as  a  function  of  r. 
Now  for  any  horizontal  stream   line,  applying   Bemouilli's 
equation, 


or 


Differentiating 


—  +  TT-  18  constant, 
to     2g 

w     2g 
Idp,  vdv  _  dH 
w  dr     gdr  ~~  dr 


.(4). 


Then 


.(5). 


Fig.  367. 

Free  vortex.  An  important  case  arises  when  H  is  constant  for 
all  the  stream  lines,  as  when  water  flows  round  a  river  bend,  or  as 
in  Thomson's  vortex  chamber. 

1  dp_  -vdv 
w  dr~^    gdr 

Substituting  the  value  of  ^  from  (5)  in  (2) 

—  yyo  dv_wt^ 
g    dr~  g  '  r' 
from  which  rdv  +  vdr  =  0, 

and  therefore  by  integration 

vr  =  constant  =  C 


620  HTDBAULIC8 

Equation  (3)  now  becomes 

Pi - p     C  f^dr 
w         g  JR  f* 


=  C!/i_j_\ 

2g\B?    RiV' 


2sr 

Forced  vortex.  If ,  849  in  the  turbine  wheel  and  centrifugal 
pump,  the  angular  velocities  of  all  the  stream  lines  are  the  same, 
then  in  equation  (3) 

t?  =  ciir 


w        g  jR 


and  tL^^-       ^dr 

g  jR 

Scou/ring  of  the  banks  of  a  river  at  the  bends.  When  water 
runs  round  a  bend  in  a  river  the  stream  lines  are  practically 
concentric  circles,  and  since  at  a  little  distance  from  the  bend  the 
surface  of  the  water  is  horizontal,  the  head  H  on  any  horizontal 
in  the  bend  must  be  constant,  and  the  stream  lines  form  a  free 
vortex.  The  velocity  of  the  outer  stream  lines  is  therefore  less 
than  the  inner,  while  the  pressure  head  increases  as  the  outer 
bank  is  approached,  and  the  water  is  consequently  heaped  up 
towards  the  outer  bank.  The  velocity  being  greater  at  the  inner 
bank  it  might  be  expected  that  it  will  be  scoured  to  a  greater 
extent  than  the  outer.  Experience  shows  that  the  opi)osite  effect 
takes  place.  Near  the  bed  of  the  river  the  stream  lines  have  a 
less  velocity  (see  page  209)  than  in  the  mass  of  the  fluid,  and,  as 
Lord  Kelvin  has  pointed  out,  the  rate  of  increase  of  pressure  near 
the  bed  of  the  stream,  due  to  the  centrifugal  forces,  will  be  less 
than  near  the  surface.  The  pressure  head  near  the  bed  of  the 
stream,  due  to  the  centrifugal  forces,  is  thus  less  than  near  the 
surface,  and  this  pressure  head  is  consequently  unable  to  balance 
the  pressure  head  due  to  the  heaping  of  the  surface  water,  and 
cross-currents  are  set  up,  as  indicated  in  Fig.  367,  which  cause 
scouring  of  the  outer  bank  and  deposition  at  the  inner  bank. 


ANSWERS  TO  EXAMPLES. 


Chapter  I. 

<1)  8900  lbs.    9872  lbs.  (2)    784  lbs.  (8)    78*6  tons. 

(4)  6880  lbs.  (5)     17*1  feet.  (6)     19800  lbs. 

(7)  P=865,e00  lbs.    X=12-6ft.  (8)     -91  foot.  (9)     -089  in. 

<10)  15-95  lbs.  per  sq.  ft.  (11)    5400  lbs.  (12)    87040  lbs.      ^ 

Chapter  II. 

(1)     85,000  eft  (8)    2-98  ft. 

(4)     Depth  of  C. of  B. =21-95  ft.    BM= 14*48  ft.  (5)     19-1  ft.    69  ft. 

(6)  Less  than  18*8  ft.  from  the  bottom.  (7)    1*57  ft.  (8)    2*8  ins. 

Chapter  m. 

(1)  -945.  (2)    14-6  ft.  per  sec.     18-3  c.  ft.  per  sec.  (8)    26-01  ft. 

(4)  115  ft  (5)    58*8  ft.  per  sec.  (6)    63  c.  ft.  per  sec. 

(7)  44928  ft  lbs.     1-36  h.  p.    8*84  ft.  (8)    86-2  ft.     11-4  ft.  per  sec. 
(9)  1048  gallons. 

Chapter  IV,  page  78. 

(1)  80-25.  (2)    3906.  (8)     37-636.  (4)    5  ins.  diam. 

(6)  8-567  ins.  (6)     -763.  (7)    86  ft.  per  sec.    115  ft. 

(8)  -806.  (9)     -895.  (10)     -058.  (11)     144-8  ft.  per  sec. 
(12)  2-94  ins.            (18)     *60.            (14)     572  gaUons.  (15)     22464  lbs. 
<16)  -6206.            (17)     5-58  eft.            (18)     -755.            (19)     102  c.  ft. 
(90)  -875  ft    186  lbs.  per  sq.  foot    545  ft.  lbs. 

(21)     10-5  ins.    29-85  ins.  (22)     •688  ft.  (23)    4-52  minutes. 

(24)     17*25  minates.  (25)     -629  sq.  ft.  (26)     1-42  hours. 

Chapter  IV,  page  110. 

(1)     18,170  eft.  (2)    415  ft 

(8)  69-9  c  ft  per  sec.    129*8  c.  ft.  per  sec.  (4)     2-685. 
(6)     18-28.                (7)    48-3  c.  ft.  per  sec.                (8)     1-676  ft. 

(9)  89-2  ft  (10)    2-22  ft.  (11)     5-52  ft.  (12)     23,500  c.  ft. 
(18)     24,250 eft                (14)     105  minutes.                (15)    284  h. p. 


522  AN8WEB8  TO  SXAJfPLES 


Chapter  V. 

(1)  27*8  ft.  (2)    142  ft.  (4)    -65.  (5)    2-888  ft 

(6)  10-76.    1-4  ft.     -88  ft.    -782  ft.    "0961  ft 

(8)  -61  eft.    28-54  ft.    25-8  ft.    9  ft.  (9)    28  per  oent 

(10)  1-97.    21ft.    80  ft.    26  ft    24  ft     15  ft  (11)    8-64  eft 

(12)  8-08  eft.  (18)    -674  ft    •267  ft    7-72  ft  (14)    2-1  ft 

(16)  1-86  c.  ft  per  sec.  (16)    F=-08181bB.    /-•006868. 

(17)  1-028.  (18)    -704.  (19)    2-9  ft  per  sec 
(20)  4*4  c.  ft.  per  sec.                       (21)    If  pipe  is  clean  46  ft 
(22)  28  ft.    786  ft.               (28)    Diri^  cast-iron  6'1  feet  per  mile. 
(24)  8-18  feet                              (26)    1  f oot 

(27)  —--A(\  —  »  ^ = friction  per  unit  area  at  unit  velocity. 

(28)  108  H.  p.  (29)    1480  lbs.    1*08  ins.  (80)    -002825. 
(81)  A:=-004286.  n=l-84.       (82)    (a)  940  ft.    (b)  2871  h.  p.       (88)  -045 
(84)  If  d»9^  v^5  ft.  per  sec.,  and/=-0066,  ;i=102  and  Hsl82. 

(86)  1487x10*.    Yes.  (86)    68-16  ft  (87)     lhour48nmL 

(88)  46,260  gallons.    Increase  17  per  cent  (89)    296*7  feet 

(40)  6  pipes.    480  lbs.  per  sq.  inch. 

(42)  Velocities  6*18,  6-08,  8-16  ft.  per  sec.    Quantity  toB=60c.ftperim: 
Quantity  to  C —66*6  c.  ft.  per  min.  (46)    *468  c.  ft  per  sec 

(46)  Using  formula  for  old  cast-iron  pipes  from  page  188,  t? =8*62  ft  per  se 

(47)  2-91  ft.  (48)    d=8-8ins.    d,=8-4in8.    d2=2-9  ins.    d3-2'2ini 
(49)  Taking  G  as  120,  first  approximation  to  Q  is  14-4  c.  ft.  per  sec. 
(61)  d=4'18  ins.    v=20*66  ft.  per  sec.    p=840  lbs.  per  sq.  inch. 

(68)  7-069  ft.    801ft.    C,.=ll-9.     C^ for  tubes « 6-06. 

(64)  Loss  of  head  by  friction ='78  ft. 

A  head  equal  to  ^  will  probably  be  lost  at  each  bend. 

(66)  43*9  ft.     -936  in. 

(67)  7i=68'.    Taking  -006  to  be/in  formula  ^=y^  .  v=16-6  ft.  persec. 

(68)  Vi=8-8  ft.  per  sec.  from  A  to  P.    V8=4-96  ft.  per  sec.  from  B  to  P 

1^3= 13-75  ft.  per  sec.  from  P  to  C. 


Chapter  VI. 

(1)    88-6.  (2)     1-lft.  diam. 

(8)  Value  of  m  when  discharge  is  a  maximum  is  1-857.  «■=  17*62.  C=121 

Q  =  76  c.  ft.  per  sec. 
(4)     -0136.  (6)     16,250  c.  ft.  per  sec.  (6)     8  ft. 

(7)    Bottom  width  15  ft.  nearly.  (8)    Bottom  width  10  ft.  nearly. 

(9)  630  c.  ft.  per  sec.  (10)    96,000  c.  ft.  per  sec. 
(11)    Depth  7-85  ft.                          (12)    Depth  10*7  ft. 

(18)    Bottom  width  76  ft.    Slope -00052.  (17)    C  =  87-5. 


ANSWERS  TO   EXAMPLES  523 

Chapter  Vm. 

124*8  lbs.    -456  H.  p.  (2)    628  lbs. 

104  lbs.    58*7  lbs.    294  ft.  lbs.  (4)    960  lbs. 

261  lbs.    4-7  H.  p.  (6)    21-8.  (7)    57  lbs.  (8)^   194  lbs. 

Impressed  velocity=28-5  ft.  per  sec.    Angle =57'.  (10)*   181  lbs. 

•98.    -678.    -68.  (18)    19-2. 

YeL  into  tank —84*8  ft.  per  sec.     Vel.  through  the  orifice =41*6  ft.  per 

sec.    Wt.  lifted  » 10-8  tons.    Increased  resistance »  2880  lbs. 
125  lbs.    8*4  ft.  per  sec.    1*91  h.  p. 

Work  done,  575,  970, 1150, 1940  ft.  lbs.    Efficiencies  ^,  '50,  |f ,  1. 
1420  H.  p.  (18)     -9875.  (19)    82  H.  p. 

8666  lbs.    161  H.  p.    62  per  cent. 

Chapter  IX. 

105H.P.  (2)    <^=29*.    V^=7ft.persec 

14*8  per  min.    ll"*  from  the  top  of  wheel,    ^s70^ 
1*17  eft  (5)    4*14  ft.  (8)    82^2'. 

10-25  ft.  per  sec.    1*7  ft.    5*8  ft.  per  sec.    11°  to  radios. 
r= 24-7  ft.  per  sec.  (18)    </)=47''80'.    a=27"20'. 

79' 16'.     19**  26'.     58. 

85-6  ft  per  sec    6'24'.    28^  ms.    llfins.    12^89'.    16§ms.    82jins. 
99  per  cent.    <^=78%  a=18'.    <^=120',  a=18°. 
</)=158'28'.    H= 77*64  ft.    h.  p. =14116.    Pressure  head =48*58  ft. 
d=l*22ft.    D=2*14ft.    Angles  12'' 45',  125*22',  16°  4'. 
<^=184'58',  ^=16'25',  a=9n0'.    H.p.=2760. 
616.    Heads  by  gauge,  - 14,  85*6,  81.    U=51*5  ft.  per  sec. 
<^  =  158"68',a  =  25'.    H.  P. =29*8.    Eff.  =  *957. 
Blade  angle  18°  80'.    Vane  angle  80°  25'.    8*92  ft.  lbs.  per  lb. 
At  2'  6"  radius,  6^  10°,  <^ = 28°  45',  a  =  16°  24'.  At  8'  8"  radius,  ^=12°  11' 
(f>  =  78°  47',  a  =  12°  46'.   At  4'  radius,  $  =  15°  46',  (f> = 152°  11',  a = 10°  21 . 
79°  80'.    21°  40'.    41°  30'. 

58°  40'.    86°.    24°.    86*8  per  cent.    87  per  cent. 
12°  45'.    62°  15'.     81°  45'. 

r=45*85.    U=77.    V,.=44.    v^=86.    Ui=28.    €=78-75  per  cent. 
'86  ft.    40°  to  radius.  (80)     About  22  ft. 

H.  p.  =80*8.    Eff . = 92*5  per  cent. 


Chapter  X. 

47*4  H.  p.     (2)  26°.  68-1  ft.  per  sec.  94  ft.  60  ft. 
55  per  cent.  (4)  52*5  per  cent. 

^^  =  106  ft.  ^'=61  ft.  ^-1=56  ft. 
g  2g  w 

11°  86'.  106  ft.  47*4  ft. 

60  per  cent.    161  h.  p.    197  revs,  per  min. 

700  revs,  per  min.    -81  in.    Radial  velocity  14*2  ft.  per  sec. 

15*6  ft.  lbs.  per  lb.    806  ft.     14  ft.  per  sec. 


624  ANSWERS  TO  EXAMPLES 

(16)    v=28-64  ft  per  sec    V=ll-8. 

(16)  d=a9im8.    D  SB  19  ins.    Revs,  per  min.  472  or  higher. 

(17)  15  H.  p.    9-6  ins.  diam.  (18)    6*6  ft 

(19)  Vels.  1-28  and  2-41  ft  per  sec    Max.  accel.  2*82  and  4*55  ft  per  see. 

per  sec 

(20)  898  ft  lbs.    Mean  friction  head=*0268,  therefore  work  doe  to  frictiaa 

is  very  small. 

(21)  4*61  H.  p.    11-91  c  ft  per  min.  (22)    -888. 

4ii^QQi  IT 

(28)  p^  — ^^'    Acceleration  is  zero  when  ^=j(iii+2),  m  being  any 

integer. 
(27)    Separation  in  second  case. 

(29)  67-6  and  66*1  lbs.  per  sq.  inch  respectively,    h.  p.  sd'14. 
(81)    7-98  ft    25*8  ft    41*98  ft  (82)     -648.  (88)    *6. 
(84)    Separation  in  the  sloping  pipe. 


Chapter  XI. 

(1)    8150  lbs.  (2)    8*88  H.P.  hours.  (5)    4*7  ins.  and  9*7  ins. 

(6)    8*888  tons.  (7)    175  lbs.  per  sq.  inch. 

(8)    2*8  ft.  per  sec.  (9)    4*2  minutes. 


Chapter  Xn. 

(1)  80,890  lbs.  1425  H.  p.  (2)  8500  H.  p. 

(8)  4575  lbs.  (4)  25,650  lbs. 


^                              [All  numb^i  refer  to  pages.]                             ^^^^^^^^H 

Abw>kte  Tdoeilj  262 

B«ods,  loaa  of  bead  due  to  140                ^^M 

Acoeleratiot]    io    pumps*  effect  of   (ue 

BemouilU'a  theorem   39                                ^^^ 

Eaciprocalmg  piuup) 

appUed    to    centrifugal    pomps    4 IS,             1 

Aecumxilatars 

423,  437*  439                                                   J 

Air  4S1 

applied  to  turbineB  334,  349                    ^M 

difleratiluU  482 

examples  on   46                                           ^^| 

bydnulic  486 

eipen mental  illtiBtrations  of  41             ^^| 

Air  gauge,  mTeited  0 

ext^n^ion  of  4B                                          ^^| 

Air  vesflels  on  pumps  451 ,  455 

Borda'a  mouthpiece   72                                 ^^B 

Angtil&r  momoutum   S73 

Boui^diucBq'B  theory  for  discharge  of  a           ■ 

Angaidir  moment uirii  rate  of  change  of 

weir   104                                                  ^J 

eqoAl  to  a  conply    274 

Boyden  diffuser  314                                    ^^B 

Appold  eentrlfu^al  pump   415 

Brotherhood  hyilraulic  engine  501             ^^H 

Aiimedueifl   1,  189,  195 

Buoyancy  of  Hoating  bodies  21                  ^^M 

eeclioDft  of  216 

centre  of  23                                                 ^^M 

Archimedes*  principle  of  22 

^^H 

Canal  boats,  steering  of  47                        ^^H 

497 

Capstan,  hydraulic   aOl                                ^^U 

AliDo«phede  piretttire  8 

Centre  of  buojanory   23                                 ^^H 

Centre  of  prestjore   13                                   ^^| 

Bfteon  1 

Centrifugal  force,  effect  of  in  discbarge      ^^^ 

:B*me«  ftnd  Goker   129 

from  water  wheel   286                                   1 

Centrifugal  head                                                  1 

Sftsin's  eiperitnents  on 

in  oentrifugal  pumps  405,  408,  409,            1 

eftlih^tion  of  Piboi  tnbe   245 

419t  421                                                     ^J 

diiitribtilioQ  of  preflsiur^  in  Ihe  plane 

m  reaction  turbtnea  303,  334                ^^M 

of  an  ori^ee  59 

Ceutrifugul  pumpa,  see  Pumpe                    ^^H 

'        distiibatioD  of  velod^  in  the  orosa 

CbanjislB                                                             V 

leoUon  of  a  ahaimel  206 

circular,  de£^th  of  flow  for  maitimum           1 

diatributbii  of  vdoeity  in   the  crous 

discharge   221                                               ■ 

eectiOD  of  a  pipe   144 

circular,  depth  of  flow  for  maiimum            1 

iiatribntion  of  Telocity  in  the  plane  of 

velocity  22U                                            ^J 

mn  orifice   69,  244 

coefficienta  for,  in  formnlaa  of              ^H 

1        flow  uj  chAnoeU   182 

Bazin    166,  137                                      ^H 

flow  OTcr  dama   102 

Darcy  and  Basin    183                             ^H 

1        flow  over  weirs   89 

QanguiUet  and  Kuttt^r   184                  ^H 

flow  through  ori^ces  56 

coeffioientB    for,    in  logarithmic    for-      ^f 

form  of  the  jet  from  orifioea  63 

muloe  300-203                                                1 

3«&iii'fl  formulae  for 

ooefficienta,  variation  of  190                   ^J 

cbAoneLfl    182,   185 

curves  of  veloc i  ty  and  di  seharj^  for  222     ^H 

ori^ccfs  aharp' edged   57,  51 

dimensions  of,   for  given  flow  deter*     ^^H 

¥«loeily   al  any  depth  in   a  Tertical 

miued  by  approximation  225-227             V 

section  of  a  channel  212 

diameter  of,  for  given  maiimum  die*           M 

▼docity   at  any  point  in    the  orosa 

charge   224                                                ^H 

teetion  of  a  pipe  144 

distribution  of  velocily  in  croaa  i€^    ^^H 

weir,  Hat  created  9*J 

tion  of  208                                             ^H 

1         ueir,  iharp*creeted  97-99 

earthy  of  trapezoidal  form  2111             ^^M 

L^  weir,  sill  of  amaU  thickneas  99 

eroAion  of  earth  216                               ^^M 

t»DEX 


in 


of 
US 

of  I7e 

lor  flow  in 
of  SaS 

lor  mx^  SOI,  SOT 
i  S3S 

1*1  ohetbod  c^f  determining  tlie 
in   187 

im 

^  '  bller«   183,  184 

It  of  231 
19t~lSe-9CMI 

of  119 


iJBt,  101.  lae.  197.  flOQ,  i06 

183»  ISA,  187,  195,  901 
l€3,  184,  107.  ISS.  195,  197, 

1&4,  186,    187,  19a, 

1^,  1^.  im,  187,  901.  207 
18a.  1§4 
1^  lift.  ISI,  906 

18i,  186,  1S7.  2^ 
r  m-196 
of,  te  prcD  vdoaty 


GuiMivifaiar  m 


Gummt  meleia  tSS 

«alil»r»tioii  of  940 

aurlej  938 

BiAkdl  S40 
Curred  ttf«uii  linia  mottcm  51§ 
CyUndncal  mouthpiece  79 

Dftms,  flow  oyer   101 

ej^i^zimeDtit  on  flow  in  ehonneb  M 
experun«Dt«  on  flow  in  pipei  12S 

fortDQlA    for    flow   in    «*haJTLwak    1^ 

formolA  for  flow  ld  plpei  111 

Deacon's  w&Bte-waler  meter  ^ 

Denftitj  3 
of  gmooline  11 
of  keroiltie  11 
of  m«t^ii]j  8 
of  pare  water  4  11 

Depth  of  e^Gtre  of  pr««»nie  13 

Di&griim  of  prefi£me  on  a  pkne  uM 
lU 

Diagram  of  pressure  on  i  vcrSioi]  out^ 
16 

Diagram  of  work  done  in  a  recipiooiUiig 
pump  413,  459.  467 

Diflbrenti&i  aooDmalaior  489 

IHJfiscBDtial  ga.nee  8 

Diftdivge 
cocffii^ient    o^    for    oriflcei  60  if 

Orifieefl) 
ooeflkient  of,  for  Vdntohweicr  IS 
of  ft  channel  ITS  {»et  Olmnneii} 
ov«T  weirs  83  (<ee  Weirs) 
thrODgh  notches  S5  i*ct  Noichet) 
throngb  oriflceB   50  {Met  Orifioei) 
thiongh  pip^   112  {$€e  ¥ipmj 

DiBtrihotion  of  ?eloeity  on  gtom  MClka 
of  a  channel  206 

Dusribataon  of  veloeitj  on  crote  aeite 
of  a  pipe  148 

DiTecgeol  monthpitoes  7S 

Dock  cuBson   181,  192,  926 

Doelis,  floftling  31 

Diowned  nappes  of  weiis  9fi,  100 

Drowned  orificee  65 

Drowned  weirs  9$ 

appcoKunate  fixmink  for  301, 
eoeffictenla  for   in   Beam's    fonei^ 

187 
eoeffieienla  for  in  Duej  vid  Ba^'i 

f<»^nJa  183 
noeflkientis    for    in    Qanguillel 

£utter*fi  formula  184 
smaoa  of  216 
Elbows,  ktea  of  head  due  to    140 
Ra^itee,  ludxanlic  mi 
BnH^eriiood  501 
BMlie  S03 
mm  £04 

of  eextli  channels  316 


mDEX 


527 


Ximiapliii,  $oltitiotta  to  wMah  iire  given 
in  the  text — 

P  Boiler,  time  of  omptjing  Uirough  a 
moQthpieoe  78 
Centrifugal  pumpa^  determlaniioa  of 
preftmre  heftd  at  Inl^  and   outlet 

^^^■Omtril^igal  pampB,  dimensiooa  for  a 
^M    given  diaoharge  404 
^FCetitrifugal   pumpat    seriea,    number 
f      of  whtida  for  a  givua  lift  435 
CenCfifogal  pi;mp»,  veiocitj  at  which 

deliT«ij  start?   412 
Clu»iiitle»   droukr  diameter,    for  a 

glvoi  majrimiini  discharge  224 
^CMsnela,  diameter  of  siphon  pipes 
to    given   same    dboharge    m    an 
ac|iitidaot  224 

"  aun«)a»  dimensions  of  a  canal  for 
i  gtven  flow  and  slope  2*25,  22t»»  227 
dinchargo    of    an    earth 
225 

flow  ini  for  giren  section 
•nd  slope  223 
12,  489 
P^lotttiog  docket  height  of  metacentre 

of  34 
^  floating  doeks,  water  to  be  pumped 
ftom   S3 
Bad  of  wa£«r  7 

Hjdraolic    ma<!hitiery,    capacity    of 
aoecunQlator    for    working    a    hy- 
draulic  crane  489 
Hjdraulic    motors    variation    of    the 

pressure  on  the  plunger  470 
IsifMcl  on   vanesj  form   of  vane  for 
w«t«T  to  enter  wit  bout  ^hmk  and 
leave  in  a  given  direction  271 
ItDpat^t  on  vanest  presanrc  on  a  vane 
when  a  jet  tn  contact  with  ia  tamed 
Ihnmgh  a  given  angle  267 
Impaot    on    vaneci,    turbine   wheel, 

form  of  vanes  on   272 
Impact    o»    vanes,    turbine    wheel, 

water  leaving  the  vancf;  of  269 
trnpaot  on   vanes,  work  done  on   a 

vane   271 
Metaoentrei  height  of,  for  a  floating 
■  34 

Eitre,  height  of,  for  a  ship  36 
bpiece,  dteeharg^  through^  into 
[  eondenssr  76 

rfehpieoe,    time    of    emptying    a 
boiler  by  m^ui«  of  76 
oQlhpleoe^    time    of    emptying    a 
ttmxww  hj  means  of  78 

"       ftar  of,  for  a  given  die- 
charge   152.  153 
I  Pipes,  discharge  along  pipe  connecting 

two  T titter voir^  Idl,  154 
KpipeB  m  parallel   154 
>1^pet|  preavore  at  end  of  a  service 
pipe  1§1 


dock 


^tamplea  {cunt.) 
Pontoon^   dimifntionB   for  given  da^ 

plaeement  29 

Pressure  on  a  flap  valve  13 

PreBsure  on  a  masonry  dam   13 

Pressure  on  the  end  of  a  pontoon  18 

Iteoiprocating  pump  htted  with  an 
air  vessel  470 

Reciprocating  pump,  horsepower  of, 
with  long  delivery  pipe  470 

Eeciprocating  pump,  presBure  iu  an 
air  vessel  470 

Eeciprocating  pump.  Beparation  io^ 
diameter  of  suction  pipe  for  no  469 

Eeciprocating  pump,  separation  in 
the  delivery  pipe  464 

Eeeiproeating  pomp,  aeparation  in, 
number  of  strokes  at  which  sepa- 
ration takes  place   458 

Reciprocating  pnmp,  variation  of 
pleasure  in,  due  to  inertia  fonies 
470 

Eeaerroira,  time  of  emptying  by  weir 
108 

Beservoirs^  time  of  emptying  through 
orifice  78 

Ship,  height  of  metacentre  of  26 

Transmisaion  of  fiuid  preasofe  12 

Turbine,  design  of  vanes  and  de- 
termiDAtion  of  effitfiency  of,  con- 
sidering friction  331 

Torbine^  design  of  vaues  and  de- 
termination of  eMciency  of,  frio- 
tion  neglected   322 

TurbiDt;,  tlimeDMiQEis  and  form  of 
varies  for  given  horse -power   341 

Turbine,  double  eompartment  parallel 
!iow   349 

Turbine,  form  of  vanes  for  an  out- 
ward flow   311 

Turbine,  hammer  blow  in  a  supply 
pipe   385 

Turbine,  velocity  of  the  wheel  for  a 
given  he^d  321 

Venturi  meter   46 

Water  wheel,  diameter  of  breast 
wheel  for  given  horae-power  290 

Weir,  CO rr fiction  of  cooflloient  for 
velocity  of  approach   1)4 

Wtjir,  dischargt^  of  94 

Weir,  disoharge  of  by  approsimation 

loa 

Weir,  time  of  emptying  reservoir  by 
of  110 


4 


Fall  of  free  level  51 
Fire  hose  nozzle  73 
Flap  valve,  prefigure  on  18 

centre  of  IS 
Floatliig  bodiofl 

Archimedes,  principle  of  22 

booyanoy  of  21 

oentre  of  buoyancj  of  23 


coaditions  of  eq.iiilibnTim  of  21 

cottt&inimg  ^at«i-,  stubilitj  of  29 

exomplefl  on  34*  516 

inetaoentTe  of  24 

resifiiftnoe  to  the  motion  of  507 

email  di^plaoements  of  24 

HtAbility  of  eqoilibrium,  ooodltion  of 
24 

stability  of  flo&iing  dock  33 

eiabililj  of  reot&Dgular  pontoon  26 

stability  of  veeael  i^oQt&miDg  wat^r  29 

stability  of  Teaad    wholJy  imtnerBed 
30 

weight  of  fitiid  displaced  23 
FloAtiBR  docks   31 

stability  of   33 
TioKis,  doable  237 

rod   239 

iriirfac«  237 
Flow  of  water 

deinitions  reladng  to  36 

enetrgy  per  pound  of  flowing  vater  88 

m  QpetL  obdnnek  176  (k^i*  ChannelB) 

over  damB  IQl  (*«  DaniB) 

over  weirs   81  [tite  Weirs) 

tbmugh  notches  80  (#««  NotcheF) 

through  orificea  50  {s^e  Orifices) 

through  pipes   112  {*ee  Pipes) 
FlnldB  (iiquidj^) 

U  ttBt   3-19 

exam  pies  on   19 

ooin^resaible  3 

deohity  of  3 

flow  of,  tkrough  orifices  50 

iijconipressible  3 

in  motion   37 

presijnre  in^  la  the  same  in  all  direo- 
tionii  4 

pi^sanre  on  an  area  in  12 

pressnra  on  a  horizontal  plane  in,  is 
constant  5 

Bpecifio  gravity  of  3 

steady  motion  of  37 

stream  line  motion  in  37,  517 

trflDfimisaioii  of  pressure  by   11 

used  in  U  tubea  9 

Tlaeosi^  of  2 
Forging  press,  bydrsnlic   492 
Foumoyron  turbine  307 
Friction 

coefficients  of,  for  ships*  surfaces  509, 
615 

effect  of,  on  discharge  of  centrifagal 
pump   421 

e^ect  ofp  on  velocity  of  exit  from  Im- 
pulse Turbine  373 

effect  of,  on    velocity   of  eiit   from 
Poncebt  Wheel  297 

Froude^a  experiments  on  fimd  507 

in  centrifugal  pumps  400 

in  cbanuela  180 

in  pipes  113,  113 


Friction  (fiwit.) 

in  reciprof!Ating  pumps  449 
m  tnibinea  313,  321,  339,  37ft 


Ganguillet  and  Kutter 

coefficients  in  formula  of  1%,  lU 

experiment*  of  183  

formula  for  channels  184 

formula  for  pipe«  124 
Oasoline,  ffpeciSc  granty  of  11 
Gauges,  preeaure 

differential   H 

inverted  air  9 

Inverted  oil   10 
Gauging  the  flow  of  water  234 

by  an  orifice  235 

by  a  weir  247 

by  chemical  means  2B8 

by  floats  239  {*«  Floats) 

by  meters  234,  2^1  {tee  Met< 

by  Pitot  tnbea  241 

by  weighing  2M 

examples  on  260 

in  open  ehannela  236 

in  pipes  251 
Glaz^  earthenware  pipes  136 
Gurley*B  current  meter  238 

Hammer  blow  m  a  long  pip€  3S4 
Haskell's  cuirent  meter  240        _ 
Hastie'fi  engine  503 
Head 

position   39 

preasure  7*  89 

velocity  39 
High  pressure  pump  471 
Historical    devetopment    of    pip 

channel  formulae  231 
Hook  gauge  248 
Hydraulic  aocumulator    4S6 
Hydraulic  capstan   501 
Hydranh'o  crane   494 

double  power  495 

valves  497 
Hydraulic  differential  accumulftti 
Hydraulic  engines  501 

orank  effort  diagram  for  509 
Hydraulic  forging  press  402 
Hydraulic  gradient   115 
Hydraulic  inteusifler  491 
Hydraulic  machines   485 

conditions    which    vanea   of, 
flatisfy  270 

examples  on   489,  505 

joints  for  484 

ma^cimum  efhclency  of  29S 

packings  for   485 
Hydraulic  mean  depth   119 
Hydraulic  motors,  vwiationR  of  j 

in,  due  to  inertia  forces  469 
Hydraulic  ram  474 
Hydraulic  riveter  499 
Hydraulics,  definition  of  1 


INDEX' 


629 


4-19 

»r  on  ▼anes  261  {see  Vanes) 
.  in  hydranlio  motors  469 
}  in  reciprocating  pumps 

ent  of  14 
f^oge  9 
;auge  9 
lydranlic  491 
valves  for  492 
5team  493 

turbines    275,    818   {$ee 
) 

n  hydraulic  work  485 

er  255 
meter  253 
!cific  gravity  of  11 

hydraulic  packiugs  486 

formulae  for  flow 

i   192 

5 

plottings 

iB  195 

27,  133 

499 

I   17 

,vity  of  8 
barometer  7 
U  tubes  8 
leight  of  24 


aste  water  254 

55 

i 

,  75,  251 

lertia  14 

ane  of  floating  body  25 

id  law  of  263 
54 

of  discharge  for 
73 
3 

il  71,  76 
le  78 
of  velocity  for  71,  73 

73 
73 
3 

n   78 

h,  under  constant  pressure 

d  at  entrance  to  70 
ptying  boiler  through  78 


MonthpieoM  (cofit.) 
time  of  emptying  reservoir  throogh 
78 

Nappe  of  a  weir  81 

adhering  95 

depressed  95 

drowned  or  wetted  95 

free  95 

instability  of  the  form  of  97 
Newton's  second  law  of  motion  268 
Notation  used  iu  connection  wiUi  vanes, 
turbines  and  centrifugal  pumps  278 
Notches 

coefficients  for  rectangular  {ae  Weizs) 

coefficients  for  triangular  85 

rectangular  80  ($ee  Weirs) 

triangular  80 
Nozzle  at  end  of  a  pipe  159 
Nozzle,  fire  74 

Oil  pressure  gauge,  inverted  10 

calibration  of  11 
Oil  pressure  regulator  for  turbines  877 
Orlflcea 
Bazin's  coefficients  for  57,  61 
Bazin's  experiments  on  56 
coefficients  of  contraction  62,  56 
coefficients  of  discharge  57,  60,  61» 

63 
coefficients  of  velocity  54,  57 
contraction  complete  58,  67 
contraction  incomplete  or  aappraiiecl 

53,  63 
distribution  of  pressure  in  plane  of 

59 
distribution  of  velocity  in  plane  of  59 
drowned  65 
drowned  partially  66 
examples  on  78 
flow  of  fluids  through  50 
flow  of  fluids  through,^  under  constant 

pressure  75 
force  acting  on  a  vessel  when  water 

issues  frcon  277 
form  of  jet  from  63 
large  rectangular  64 
partially  drowned  66 
pressure  in  the  plane  of  59 
sharp-edged  52 
time  of  emptying  a  lock  or  tank  by 

76,  77 
Torricelli's  theorem  51 
velocity  of  approach  to  66 
velocity  of  approach  to,  effect  on  dis- 
charge from  67 

Packings  for  hydraulic  machines  485 
Parallel  flow  turbine  276,  342,  868 
Parallel  flow  turbiue  pump  487 
Pelton  wheel  276,  377,  880 
Piezometer  fittings  139 
Piezometer  tubes  7 


1H0KX 


lomvi 


C  Itt 


/liii= 


la 


'^ 
for  «Mt  koa^  ne*  cni  «M  190^ 

Itl,  121,  IS,  ttl 
lor  mml  tiraied  121 

133,  isa 

B«r  ftad  old  ISS,  137« 
Il§ 

i)M»  14s 

riveted  nm,  IM 
VDod  l3S«  ISB 
wtohqJu  ir^Mi   l^'i,  las.  13$ 
n  In  GingniUfi  aud  Ku tier's  fi^nstUa 
«tt«t  iitm,  new  atid  old  125 
for  elftsped  caithentiriu-e  125 
for  stod  riveted  184 
for  wood  {ripet  125,  164 
T&miioii  of,  vitfa  8erdoe  123 
OOiUuciiiig  llirie  TCKTrmtv   I06 
oounectiag  two  r«eeir?oirs   149 
oOimfifftJiig  two  teaer¥oirs,  difljueter  of 

for  ^ma  ditcbarge  152 
ootksl  velocity  in  128 
DftrcT'a  foimula  fo(r  122 
deilettmimtioD   of   the   coeffioient   C, 
m  giTcn  in  tables  hj  logarilhmio 

pbtiing  las 

dtaiaptcr    of,    for     git«a    discharge 

dianieler  for  miniitiiiin  eont  158 

diameter  varjing   160 

diTided  into  ivo  brmnches  154 

elbow4  for   111 

empirical  fortniila  for  head  lost  in  119 

empirieal  formula  fot  relocitj  gf  Sow 

in    lia 
equation  of  flow  ia   117 
examples  on  flow  in  149-162,  170 
expert  uieoiai  detertnmation  of  loss  of 

head  by  frictiot]  m   lid 
experimentfi  on  djstributioii  of  velocity 

m  U4 
expetimeny  on  flow  in,  Gjitici&ni  of 

136 
fl^tednieiita  on  loes  of  head  at  bends 

142 
«ipfrimi>nts  on  loss  of  head  in    122, 

i2d,  m,  131.  laa,  136 

experimentB    on    loss    of    head    in, 

criticism  of  138 
flow  through  112 
flow  diminishuig  at  imifonn  rate  id 

157 
formula  for 

Ch^^y   119 

Darcy   1^2 


Fipia  {oiut,) 
fortnnia  fqr  (^oni.) 

logarithmic  125,  xBi^  las,  m4 

Beynolds   ISl 
ma&m%ry  <4  146 
f^docltj'&t  anj  poiat  in  1  oroM 
section  of   143 

Mctioo  In,  loss  of  bead  bf  113 

determination  of  116 
Gangoillet  and  Enttet*«  tmmak  kt 

124 
gmugmg  the  flow  in  Ml 
hAmmer  blow  in  ^1 
bead  lost  St  entrance  of  70,  lU 
head  loet  by  friclioD  m  US 
head  loat   by   friction    in,  emiukit 

formula  for  119 
head  loel  by  Mction  Iil,  exaaiplii  «a 

150^163,  170 
bea^i  loflt  by  friction  in,  logimUmiiO 

formula  for   ItB^  133 
head    rcquinjd    lo    ^ive   tieioaity  Vi 

wai^F  m  the  pip«  146 
bead  rvij  aired  to  give  velodtj  to  v^^ 

in  the  pipe,  «{]rproximate  ^lu  iU 
hydratilio  gfadient  lor  113 
hydratilii;  mean  depth  of  lid 
joints  for  485 
law  of  frictional  resistaiioe  foi|  n^D^ 

the  crjtical  velocity  130 
law  ol  5iotion&l  xesistanoe  fat,  Wjjv 

the  critical  velocity    125 
Htniting  diameteir  of  165 
logarithmic  formula  for  135 
logarithm  it;  fomiola  for,  coeffideiiti 

m  138 
logarithmic  formula,  use  of,  foi  pii^ 

tical  ealculftiionB   136 
logarithinic  plotting^  for  136 
nozzle  at  diichafige  end  of,  ares  of 

when  energy  of  jet  ia  a  madmoiD 

when  momentum  of  jet  is  a  matl* 
mum  159 
pie7,ometer  fillings  for  in^ 
pre&snre  on  bt^nds  of   160 
ptesaure  on  a  cyhnder  in   169 
preafiure  on  a  plate  in    1S8 
problems   147  {*^^  Problema| 
pumping   watc^r   thjougb    long  p$»^ 

diameter  of  for  m  inimajn  ooet  lii 
reg  [stance  to  motion  of  Haid  In  Uf 
TiBing  iibove  hydtHulle  gndieat  111 
short   153 
siphon    161 
temperBtore,  elfeet  of,  ua 

flow  in   131,  140 
tranBrnieeion  of  power  along,  h;  l^ 

draulic  pressure  162 

Taluee  of  C  in  the  formula  r=0%^ 
for  120,  121 

variation  of  C  in  the  formola r=C^'i 
for    123 


IlfDEX 


T^rimuon  of  the  diftch«rge  of»  with 

strviee   133 
veltK^itj  of  £lo^  ullowable  in   102 
velocity,  heftd  riiqiiired  to  give  Telocity 

to  water  in    I  ■IB 
velo^itj,  varifition  of,  in  a  crosaMetion 

of  n  pipe  f43 
virtual  siope  of  115 
Pitot  tube   241 

CAlib  ration  of  245 
Poacelet  water  wh^l   294 
P'oDtooii,  pfert«Tire  on  end  of  18 
P^tiou  hei^  29 
hctsa,  forging   493 
PfttS,  bydraulid  49S,  49B 
pMinre 
&t  anj  point  in  a  6uid  4 
atmonphenc,  In  feet  of  w&ter   B 
ginges   8 
head   7 

mettinred  in  feet  of  water   7 
on  a  horizontal  piftue  in  a  fluid  5 
on  A  plate  in  a  pipe   IBS 
on  pipe  bend!4   IHti 
Principle  of  ArcliimedeB   19 
Principle  of  siiniliirity   64 
Problems,  ^olutiooB  of  which  ait  given 
in  the  t*ixt — 
chiLnnelH 
difljueter  of,  for  a  given  maximnm 

discharge  ^i24 
dimennionB   of,    for   a    given   flow 

««rth  ditfchmge  along,  of  given  di* 
mensioms  and  elope  22  \ 
_    flow  in,  of  given  section  and  Blope 

■     n^ 

^L  filope  of^  for  tninimttm  coat   227 

^m  fioltitaons    of,     b)-     approxiination 

^     225^227 

,  ptpe^ 

acting  as  a  siphon   161 
connecting  three  Psaervoira  155 
connecting  two  reflen^lri   149 
diameter  of,  for  a  given  diMchar^ 

152 
divided  info  two  branches   154 
head  lost  in^  wlien  flow  diminiahea 

ftt  nniform  rate   157 
lam  of  hyad  in,  of  varying  diameter, 

leo,  lei 

ptumplng  water  along,  diameter  of, 
for  minininro  oo»t   1S8 

with  nozzle  at  the  end    158,  159 
E^pnlajon  of  shipa  by  water  jeta    279 
Pmmping  water  through  long  pipea  158 

Pomp* 

G«>ntdfngai  392 
advantoges  of   43y 
Appold    415 

Bemonilli's    equation    applied    to 
4U 


Fiimpi  (conL) 
centrifugal  {conL] 

centrifugal  head,  effect  of  variation 

of  on  discharge  421 
tientrifugal  head,  impressed  on  the 

water  by  the  wheel   405 
design  of,  for  given  diucharge  402 
discharge,   effect  of    the   variation 

of  ttiu  cen trifilgal  head  and  loaa 

by  friction  on   419 
diflcharget    head  ^velocity    curve    at 

^ero   409 
diiicharge,    varmtion    of    with   the 

h£ad  at  oo octant  speed   410 
discharge,  variation  of  with  speed 

at  constant  heod  410 
effideucies  of  401, 
efEoieneieft    of;    (experimental    de> 

tennination  of  401 
examples   od    404»   412,   411,    418, 

43o,  478 
form  of  vanes  39B 
friction,  effect  of  on  dbchorgt  419, 

421 
general  equation  for  421,  425,  428, 

4B0 
gross  lift  of  400 
head- discharge   cMirve   at   constant 

velocity   410,  412,  427 
head  loet  in  414 
head,  variation  of  with  discharge 

and  flpeed   418 
bead- velocity    curve    at    constant 

discharge  429 
head -velocity    curve    at   zero    dis- 
charge 409 
kinetic    energy    of   water    at   exit 

a99 
limiting    height    to    which    single 

wheel  pump  will  raiae  water  4B1 
liuiiting  velocity  of  wheel   404 
lo9He£t  of  head  in   414 
multi'fltafie    433 
»eriea   433 

spiral  casing  for  394,  429 
starting  of  395 
suction  at  431 
Bulzer  series  434 
Thomson's  vortei  chamber  397,  407. 

422 
triangles  of  velocities  at  Inlet  and 

exit   397 
vane  angle  at  exit,  effect  of  variation 

of  on  the  efficiency  415 
velocity- discharge  curve  at  conatant 

heiul  411,412,421.428 
velocity,  heail-dischargf  curve  fof  at 

couBtant  410 
velocity  head,  sp^^iil  amngement 

for  con%'erting  into  pressure  head 

422 
velocity,  limiting,  of  rim  of  wheel 

404 


532 


tKDKX 


Telocity  of  whirl,  ratio  of,  to  fdocitj 

of  outlet  e4g«  of  vmoe  S98 
TOftex  chfijuber  of  397,  407,  423 
with  whirlpool  or  vortex  ch«imber 

B97,  ^37,  in 
work  done  on  wmler  by  9t7 
oompTesHed  air  477 
duplfx    473 
exAmpW  tm  458,  464,  460»  47S 

kigh  pramtire  472 
hjdrftnlic  mm  476 
{iiacking«  for  plangera  of  47S|  436 
reciproeating  439 
ftOOGle ration,  elfeot   of  on  preasura 

in  oylinder  of  a  446,  448 
acce)er&tioD  of  the  plaoger  of  444 
acceleration  of  the  water  id  delivery 

pipe  of  448 
acceleration  of  the  water  in  inotiOD 

pipe  of  445 
ur  Teaael  on  dellverj  pipe  of  454 
air  veaiel  on  eactioD  pipe  of  451 
air  vessel  on  taction  pipci  effect  of 

on  a©p«kration  462 
eodfflcient  of  diaaharge  of  442 
diagram  of  work  done  by  443,  4S0, 

U%  467 
diaebftigtt  coefficient  of  443 
di3pl«i  473 
examples    on    458,   464,  469i  470| 

480 
frictino,  variation  of  ptesattre  in  the 

cylinder  due  to  449 
head  lost  at  auction,  valve  of   468 
head  lONt  bj  Mction  in  the  auotion 

and  deUvery  pipes  449 
high  pre»Bur6  plunger   471 
preasufe   in  cylinder  of  when  the 

plnngoj  move&  with  simple  har- 
monic motion   446 
preaisurt^  in  the  cylinder,  variation 

of  dtie  to  friction   449 
separation  in  delirerv  pipe  463 
separation    dnring   suction    atroke 

456 
separation    daring    suction    stroke 

when  pi  anger  moves  with  fiimple 

harmonic  motion  4o8*  461 
alip  of  442,  451 
snetion  stroke  of   441 
suction  stroke,  separation  in  456, 

4fll,  463 
Tangye  duplex   HH 
Tertlcal  single  acting  440 
work  done  by  441 
work  done  by,  diagmm  of  445, 459, 

467 
turbiue  396,  42€ 
heafl'disoharge  currea  at  oonatant 

speed  427 


lor  bine  {ei>nt.\ 

head'Tdocity    ctLrv@« 
discharge  429 

inward  6ow  439 

multi-i^lage  433 

parallel  6ow   437 

veloei  ly  diBcharge  carvei  si  eciif biat 
head  42B 

Worthington    432  J 

work  done  by  44B  H 

work  done  by,  diagram  of  (KiS 

oiprocating  pompfi) 
work  done  by,  series  43$ 

Heaotion  turbines  301 

limiting  bead  for  367 

series  367 
Beaction  wheels  301 

efficieney  of  304 
Beciprocating  pumps   4S9  Ue* 
Bectangular  pontoon,  etabihtv  uf 
Beotangalar  sharp- edged  weir  §1 
Beclangular  sluices  65 
Rectangular    wetr    with   end   oostiM 

tiona  88 
Begnlation  of  turbines   306,  317,  tU 

328,  343 
Regulators 

oil  pressure,  for  impulse  tnrbine  R71 

water  proisure,  for  impuife  tur^i 
379 
Relative  velocity  265 

as  a  vector  2&% 
Beservoirs,  time  of  emptying 

orifioo   76 
ReservoirB,  time  of  emptying  over  « 

100 
Reaiatance  of  ^hip  510 
Bigg  hvdrAulic  engine   503 
Rivera,' flow  of   191,  207*  211 
Rivera,  acouring  biinkB  of  *%2(> 
Riveter,  hydraulic  500 

Scotch  turbine  301 

Second  law  of  motion   2G3 

Sepamtion  (*i*f  Pumpe) 

Sbaq>-edKed  orificea 

Bazm'a  experiments  on  66 
di  Kiribati  on  ofveiocity  in  the  ptanj 
pressure  in  the  plane  «f  59  m 
table  of  coeffioltmta  for,  «bfll 
traction  is  complete  57,  6^ 
table  of  coeffloi«nta  for,  whtn 
traction  it  iuppreased   63 

Sharp- edged  weir  81  {»et  Weirs) 

Shipa 
ppopalsion  of  by  water  J  eta 
rcBistanoe  of  510 
resistance  of,  from  model  oXJ 
streani  line  theory  of  the 
of  510 

Similarity,  principle  of  M 


J! 


1 


ITTOEI 


S33 


forming  part  of  ^ueduct   31G 

lei 

pomiM  U2t  461 
65 
gnl&ttng  tuirbmes  {ite  Tarbinefl) 

»lme   11 
!Vien«  11 

I,  Tftriatton  of^  with  temperatiire 

re  water  4 

iatioi]  of,  with  tempemture  11 

r  ol 

%  hoij  ^  35 

ig  dock  SI 

ig  ve&st^  QOQtuDiEig  water  29 

tgttlar  pontoon  26 

paodoD  of  fluids  37 

nien&ifier  493 

line  motion  37,  1^3,  517 

I  SIS 

Shawns  eicperiments  oti  384 

line    thaotj   of   resistaiioe    of 
m  510 

oentrifagal  piunp  431 
redprooating  pnmp  441 

Hbe  of  tttrbine   S06 

sontnu^tiQn    of   a    cuirent    of 

m 

lar^ment  of  a   current  of 

67 

inlti-st&ge  pamp  434 
1  contracttiQH   53 
K  on  dlsdmrge  from  ori6de 

\  on  diicharge  of  a  w«ir  @3 


,  i«wers  and  aitnedu^li,  par- 
I  of,    and  values   of   -   in 

ftU  4=*^;    195 


Baofai 


maxlmtim  velooitiea  of 
^    15 

a  and  ^  in  Bazln's  formnla 


EM  of  V  atid  i  as  deC^^rmmed 

Ep£rimenlalij  and  >^  c&leuliit«d 

oa  lopintbniie  tarmaia^  lUS, 

»l-208 

tttita  for  duna  103 

for    shju^-ed^ed    oHfioe, 

ocmmlete  57.  61 
|lbr   «narp<  edged    orifiee, 
duppre^sed  63 
ahftrp^adged  wetri  89, 

for  Venturi  meters  46 
intieli;,  v^loeities  aboTe  which 
takes  place  216 


minimum  dopei  for  varjing  valuea 
of  the  hjdraultc  mean  depth  of 
brick  £)lLivnneb  that  the  veloci^ 
tnajf  not  be  less  than  2  ft,  per 
eeeond  215 

momenta  of  Inertia  15 

PeltOD   wheels «  particutt^m  of  377 

pipes 
lead,  slope  of  Rnd  velocity  of  6ow 

in  128 
reasonable  valaes  of  y  and   n  in 

the  formula  ft  =  ^^   188 

Taloes    of^   C     in     the     formula 

r^Qs^mi   120,  131 
valiaea     of     f    in     the      formula 

2ffd 
vaiiaefl    of    n    in    Oangaitlet    and 

Kutt^*s  formula  IMl  iM 
valuea  of  n  and  k  in  the  formula 
(=itr»  137 
resistance  to  motion  of   boards   in 

Anids  509 
turbines,    peripheral    Teloeiiiea    and 
heads  of  Inward  and  ouc^ard  f  ow 
333 
uBefnl  data  3 
Thomson  f    centrifagal     pump,     vortei 
chamber  for  397,  407,  423 
principle  of  simihurity   63 
turbine  323 
Time  of  emptying  tank  or  reservoir  by 

an  orificp    76 
Time  of  emptyinj;?  a  tank  or  reaervoir 

by  a  weir   10<J 
Torricelli^s  theorem   1 

proof  of  51 
Total  pressure   12 
Triaiiji^ular  notches  80 

dificlifljge  through   85 
TnrMnes 
axial  flow  276i  343 
axial  flowt  impuUe  368 
axial  flow,  prGHaure  or  reaction  343 
axial  fiow,  section  of  tJie  vane  with 

the  variation  of  the  rftdius  344 
Bernuuilli*a  equation  a  far   334 
best  peripheral  velocit>   for  329 
central  vent  320 
oentrifuffal  head  impressed  on  watar 

by  wheel  of  334 
eone  'd&\t 

design  of  vanes  for  346 
efficiency  of  315,  331 
examples  on   311.  321,  333,  331,  341, 

340,  385,  387 
flow  through,  eflect   of  diminishing, 
by  means  of  moreable  guide  blades 
363 
flow  through,   effect  of  dimiuiabing 
by  means  of  aluices  BM 


si*                                                    INDEX                                 ^^^^^^H 

Turliiuti  {€f>ni.) 

Turbines  {cotit,^                                ^^^| 

Jlow  ibitttigbf  e£f60t  of  dttniniBhuig 

ImB  of  head  in  313,  3*21               ^^M 

on  velooHy  O'f  exit  H63 

imixed  ^ow  350                            ^^^1 

FoDlaine,  rcigiilAting  slaioes   343 

form  of  vaiipi  of  355               ^^^| 

form  of  Timefl  for  308,  347,  Bti5 

guide    blade    regulating    gesaF  for 

l^oomeyroD  B06 

352-354 

general  formula  for  31 

in  open  atneam  Bi\0 

geoeral    forToalik,    including   frietion 

Swain  gate  for    374 

315 

triangle:^     of    velocltjea    for     3S5— 

guid^  btadefl  for  320,  3S6,  048,  35S, 

dm 

362 

wheel  of  351 

fuidd  blades,  «fleet  c^f  ehonging  the 

Hiagaju  falln  313 

direotton  of  362 

oil  preiiiore  regulator  for  377 

gotde  blades,  vaHatioti  of  the   angle 

outward  fiow,  275,  306 

of,  for  parallel  flow  turbil)«ii  344 

Beroooilh^B     equatini^B     for     334« 

hors«    power,    to    develop    a    given 

339 

33y 

b«st  peripheral  volooity  far,  at  inlet 

impnltte  3O0,  369-384 

329 

AtM  flow   363 

Boy  den   314 
diffuser  for  314 

«Kampleft  387 

for  high  beads  373 

doable  316 

form  of  Tanea  for  371 

examples  on  311^  3B7 

Girard  369,  37D,  379 

eiperimental  deteniiinatioa  of  iha 
best  velocity   for  329 

bjdmiilic  efRcieiicy  of  37l»  378 

in  airtight  chamber   370 

Foiirneyrt>n  307 

oil  preBRure  regoUtor  for  377 

lo{>&6B  of  head  in  313 

radiiLl  flow  370 

Niagara  falls  318 

tmngles  of  vetocitiei!    for    37*2 

Buctjoa  tube  of    308,  317 

triaogleH  of  velocities  tor  conaideriog 

triangles  of  velocities  for   308 

friction  373,  376 

work  done  on  Iho  wheel  per  lb,  d 

water  prii&gnra  r^^lator  for  379 

water  310,  315 

w&ter  pressure  reijulator,  hjdraiili« 

pamllel  flow  !^76,  34:^ 

valve  for  3#2 

adjustable  guide  bludes  for    MS 

water  presBiire  regulator,  water  filter 

Bernouilli'fl  rquationa  for   S4S 

for  383 

deaipi  of  vanes   for  344 

work  doue  on  wheel  per  Ibu  of  water 

double  compattmtut  MB 

272,  277»  323 

example!  on  349,  3B7 

incliuation  of  vanes  at  inUt  of  wheel 

regulation  of  the  flow  to   341* 

308,  321,  344 

triangle  of  velooitieai  for  344 

inclinatioo  of  vanea  at  outlet  of  whe^l 

reaction  301 

ms,  321,  345 

aiial  flow  276-342 

in  open  atream   360 

cone  359                                        ^^H 

inward  iow   275,  318 

inward  flow  275,  318                 ^^M 

Btroooilli's  eqiiatmiii  for  334,  339 

miied  flow  350 

be«t    peripheral    vdooity    for,     at 

outward  flow   306 

inlet  329 

paraUel  flow  276-342 

central  vent   320 

Scotch   302 

examples  on   321,  931,  341,  mi 

aeries  368 

Batpeiimeotal  determination  of  the 

regulation  of  306,  317.  318,  323,  $18, 

best  velocity  for  329 

S50.  352,  360,  362.  BU 

for  low  and  variable  fulls  328 

Scotch  301 

Franeia  320 

Bluicea  for  305,  307,    316.    317,  31», 

borieontal  ajcis  337 

327,  328,  348,  350,  S61,   Mi 

lom&B  in  S21 

auction  tube  of   306 

Themaon  334 

Bwain  gale  for  364 

to    develop    a    given    horse*  power 

ThomBon's  inward  flow   323 

339 

to  develop  given  borae-power  338 

tnanglea  of  veboitiea  for  322,  326, 

trianglea  of    velocities    at   inlet  an! 

332 

outlet  of  impulae  372,  376 

work  done  on  the  wheel  per  lb.  of 

triangles   of  velocities   at    inlet  and 

water  321 

outlet  of  inward  flow   3U« 

limiting   head    for    reaction    turbine 

triangles   of  velocitieu    at    inlet  mbi 

867 

outlet  of  nii:xed  flow  356 

INDEX 


535 


Tiiztiiii08  (ecmt.) 

triangles  of  velocities  at  inlet  and 

outlet  of  oatward  flow  844 
triangles  of  velocities  at  inlet  and 

outlet  of  parallel  flow  844 
types  of  800 
vanes,  form  of 

between  inlet  and  outlet  866 

for  inward  flow  321 

for  mixed  flow  351,  356 

for  outward  flow  311 

for  parallel  flow  344 
Velocity  of  whirl  278,  310 

ratio  of,  to  velocity  of  inlet  edge 
of  vane  332 
velocity  with  which  water  leaves  384 
wheels,  path  of  water  through  312 
wheels,  peripheral  velocity  of  333 
Whitelaw  302 
work  done  on  per  lb.  of  flow,   275, 

304,  815 
Turning  moment,  work  done  by  273 
Tweddell's  differential  accumulator  489 

XJ  tubes,  fluids  used  in  9 
Undershot  water  wheels  292 

Valves 

crane  497 

hydraulic  ram  476 

intensifier  492 

Luthe  499 

pump  470-472 
Vanes 

conditions  which  vanes  of  hydraulic 
machines  should  satisfy  270 

examples  on  impact  on  269,  272,  280 

impulse  of  water  on  263 

notation    used   in   connection    with 
272 

Pelton  wheel  276 

pressure  on  moving  266 

work  done  266,  271,  272,  275 
Vectors 

definition  of  261 

difference  of  two  262 

relative   velocity   defined   as    vector 
266 

sum  of  two  262 
Velocities,  resultant  of  two  26 
Velocity 

coefficient  of,  for  orifices   54 

bead  89 

of  approach  to  orifices  66 

of  approach  to  weirs  90 

relative  265 
Venturi  meter  44,  75,  251 
Virtual  slope   115 
Viscosity  2 

Water 

definitions  relating  to  flow  of  38 


Water  (cont,) 
density  of  3 
sjpecific  gravity  of  3 
viscosity  of  2 
Water  wheels 
Breast  288 
effect  of  centrifugal  forces  on  water 

286 
examples  on  290,  886 
Impulse  291 
Overshot  288 
Poncelet  294 
Sagebien  290 

Undershot,  with  flat  blades  292 
Welra 
Bazin's  experiments  on  89 
Boussinesq's  theory  of  104 
coefficients 

Bazin's  formula  for 
adhering  nappe  98 
depressed  nappe  98 
drowned  nappe  97 
flat  crested  99,  100 
free  nappe  88,  98 

Bazin's  tables  of  89,  93 

for  flat-crested  99,  100 

for  sharp-crested  88,  89,  93,  97, 
98 

for  sharp-crested,  curve  of  90 

Rafter's  table  of  89 
Cornell  experiments  on  89 
dams  acting  as,  flow  over  101 
discharge  of,  by  principle   of   simi- 
larity 86 
discharge  of,   when   air  is    not  ad- 
mitted below  the  nappe  94 
drowned,  with  sharp  crests  98 
examples  on  93,  98,  108,  110 
experiments  at  Cornell  89 
experiments  of  Bazin  89 
flat-crested  100 

form  of,  for  accurate  gauging  104 
formula  for,  derived  from  that  of  a 

large  orifice  82 
Francis*  formula  for   83 
gaugiug  flow  of  water  by  247 
nappe  of 

adhering  93,  96 

depressed   95,  98,  99 

drowned  95,  96,  98 

free  88,  95,  98 

instability  of  97 

wetted  95,  96,  99 
of  various  forms  101 
principle  of  similarity  applied  to   86 
rectangular  sharp-edged  81 
rectangular,    with    end    contractions 

82 
side  contraction,  suppression  of  82 
sill,  influence  of  the  height  of,  on 

discharge  94 
sill  of  small  thickness  99 


S36  on>KX 


(eoMf.) 

rvqvired    to    lomr   wter  in  ^doeitj  of   ftpproaAh,   effect   of  on 

mail  oil  tj  meini  of  109  dudbaige  90 

wi:>si  forss  of  101  wide  llftt-crested   100 

w^txXT    c<  ftnxoeefa,   eoneeuoo  ol      Wbitelaw  tozbine  302 

eot&eaemt  for  9f  Whole  prenore    12 

viuac^T   of   effTCMdL,  eonectiofi  of      Worthington  multi-stage  pomp  433 
■pies  on  M 


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Electrical  Traction. 

Bv  ERNEST  WILSON,  Whit.  Sch.  M.LE^E.p 

F^tjfcssor  or  t^tectricat  Engineering  in  the  Siecncns  LAbarAtot^v  Ring's.  CoUcf^Ci  Lciiiciant 

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In  dealing  with  ihis  ever-incrcaaingly  important  subject  ihe  authors  have 
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and  other  reasons,  most  convenient^  namel}',  the  utilization  of  direct  and 
alternating  currents  respectively.  Direct  current  traction  taking  ihe 
first  place,  the  first  volume  is  devoted  to  electric  tramways  and  direct- 
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A  Text- Book  of  Electrical  Engineering. 

BY  Dr*  ADOLF  THOMALEN< 

Translated  by  GEORGE  W.  O.  HOWE,  M*Sc.,  Whit.  Sch*, 

A*MJ*E*E*, 

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Alternating  Currents. 

A  Tesct-Book  for  Students  of  En^neenng* 
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Clue  Colkg«i,  CAinbridgc  ;  A«a^ocUte  Meiitber  cf  the  In&iituUon  ef  Klectrical  Ensineen; 
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Logarithmic    and    Trigonometric    Tables  (To 

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Technical  and  Scigntific  PuUimtions 


Organic  Chemistry  for  Advanced  Students* 


I 


Bv  JULIUS  B.  COHEN;  Ph.U.,  B.SCm 

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The  Chemical  Synthesis  of  Vita!  Products  and 

^the  Inter-relations  between  Organic 
Compounds. 
By  RAPHAEL  MELDOLA,  F.R,S.,  V.P.C.S.,  FJ.C,  etc., 
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TTie  great  achievements  of  modem  Organic  Chemistry  in  the  domain  of 
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•  have  not  of  late  years  been  systematically  recorded.  The  object  of 
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reference  which  will  be  found  indispensable  for  teachers j  students, 
and  investigators,  whether  in  the  fields  of  pure  Chemistry,  of  Chemical 
physiology,  or  of  Chemical  Technology. 

L Lectures  on  Theoretical  and  Physical 
Chemistry. 
BY  DR.  J,  H,  VAN  T  HOFF, 
■  Profcsjbor  ot  Cbembtry  ai  the  University  af  Berltts. 

\  Translated  by  R*  A.  LEHFELDT,  D*Sa, 

Profteiior  of  Physics  fit  the  "I  miisvaal  Technical  Initimtc,  Jo hjntTjesburj. 

I  three  volumes,  demy  Svo.^  cloth,  iSs.  net,  or  separately  as  follows  : 
[  PAJtT  L  CHEMICAL  DYNAMICS.    254  P^ges,  with  63  Illustra^ 
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Part  U.  CHEMICAL  STATICS.   156  pages,  with  33  llluslratiom. 
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Part    HL    RELATIONS    BETWEEN    PROPERTIES    AND 
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and  coi7rn'no^° 


Exercises  fo 


'**«'essorofO 

DemoBMiworand' 

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The  book  contains , 

electrolysis  to  i 

book  literature  s 

methods  of  prec 
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"Ppreciate  witho. 

Introduction 


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Mi +  252 


^«l*P®"nient 

i?  ^  Students. 

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Tecknkai  and  Sciefitifie  Pudliiaikfis 


The   Becquerel   Rays   and   the    Properties 
of  Radium. 

By  the  Hon.  R*  J*  STRUTT,  F.R.S,, 

Fellow  of  Trinity  College,  Cam  bridge 

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Astroaomical  Discovery. 

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An  Introduction  to  the  Theory  of  Optics. 

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Recently  Professor  of  Pbytlcs  at  the  Univertity  of  Manchester- 

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EW  Edition*     Revised  anu  Enlargkd.      Demy  8vo.,  12s,  6d.  net. 

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Manual  of  Alcoholic  Fermentation  and  the 
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ir,  Edwmrd  Atwc/d's  Tec  Anita/  <5^  ^tenfifi^  Books 


Tlie  Eiroliitk»ii  Tbeory.     By  Dk.  Aogust  Weis* 

«tf  HtttaAl  Ifirtixj  in  tbe  Unitmity  of  Abctdeca  ;  and 
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Behaviour.       By     Professor    C.     Llovd 

'■&4-j%«  F^e^*  ^^l>  26  DliiitntinBi      Lu^  crovii  Svow,  dotb,  iqk.  6d. 
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BY  THE  SAME  AiTTBOR,  •^ 

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^am  m  aiibsaBAil  fo^^JSMB  of  cuvidly  dhacf^ed  Iftcts  sBd  IqekHl  Indociioti  fiooi 

BY  THE  SAMB  AUTHOR. 

The  Sprini^s  of  Conduct.      Cheaper    Edition, 

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dy»  nd  Mml,  Cbokt,  Fecluig  a»d  < 


,  Sody»  nd  MlBfl,  Cbmot]  Fecluig  a»d  Candoci, 
BY  THE  SAME  AUTHOR. 

Fsycbology  for  Teachers,    New  Edition,  entirely 


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E 


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