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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 T jpii t.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 - 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) -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 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 and on a'^' wnll be whiaj cos ^. 

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

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

^wha cos - 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 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 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 _ 6500x 2240 
" 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 
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 to the 
^^cal, the horizontal component of the velocity is Vi cos 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 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_ maJ2ghd t 

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 
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 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 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 
g b att Pf*' 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 

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(^)\4 9(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 

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 

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- , 
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 . 

= =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. 

T h erefore, 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 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 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>/Hi7 1 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 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 = (1), 

P = L + 2/i, 

therefore dP = dh + 2Jh = (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 = 

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 TBaiimi i m , (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 = (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 = (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 is unknown, but by assuming a value for it, an approzunatioa 
to D can be obtained ; a new value for 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 
tjm p e« 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 



Th gy 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 
KM oa 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 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 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 

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 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 » o port io a»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 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^ 



Li n ^ 
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 i tti i iTT ^ 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 with the tangent AD to the 
irheel, so (hat CD makes an angle 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 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 

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 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^ 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 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 
m nifhiw^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 is 15 degrees, and the theoretia 
hydraulic efficiency is then 0"933. 

This increases as 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 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 \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 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 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, 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 

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 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 as centre touches the vanes at M and Mi. 

Suppose the vanes near 6 and / to be struck with arcs of cirdcB. 
Join 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 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 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= ^ 





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 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 
pr oso nre 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 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 <^) 
_ V i^ - 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 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 
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 ' 

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 2nico t^ 
^~2g 
Ux^ (cot * -^ cot o)' tt* cot* 



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.^ 

Sndial f 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 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 

re gpe ct i%'e ly, 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), 
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 is aero, and A^ is therefore jscro, 
and when 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 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 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/— '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 tno tJon 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 pi u mm i 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 *^~ 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., 



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 
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 th e 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 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 faid i ci 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 Kfi aiun* 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 
p i Oflom pe 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 


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 

mai l 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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