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Full text of "Hydraulics for engineers and engineering students"

V 

OF THE 

UNIVERSITY 
OF 




u 



HYDEAULICS 

FOE 
ENGINEERS AND ENGINEERING STUDENTS 



HYDRAULICS 

FOR 

ENGINEERS AND ENGINEERING STUDENTS 



BY 

F. 0. LEA, 

D,Sc. (ENGINEERING, LONDON) 

SENIOR WHITWOBTH SCHOLAR ; ASSOC. R. COL. SC. ; M. INST. C. E. J 

TELFORD PRIZEMAN ; PROFESSOR OF CIVIL ENGINEERING 

IN THE UNIVERSITY OF BIRMINGHAM. 



THIRD EDITION 

SECOND IMPRESSION 



LONDON 

EDWARD ARNOLD 

41 & 43, MADDOX STREET, BOND STREET, W. 

1919 
{All Rights reserved] 



Engineering 
Library 



J^ 



PREFACE TO THE THIRD EDITION 

OINCE the publication of the first edition of this book there 
has been published a number of interesting and valuable 
papers describing researches of an important character which add 
materially to our knowledge of experimental hydraulics. While 
the results of these supplement the matter in the original text 
they have not made it necessary to modify the contents to any 
considerable extent and instead, therefore, of attempting to 
incorporate them in the original relevant chapters summaries of 
the researches together with a few critical notes have been added 
in the Appendix. This arrangement has the advantage that, from 
the point of the student, he is able to obtain a grasp of a subject, 
which is essentially an experimental one, without being over- 
burdened. At a later reading he will find the Appendix useful 
and of interest not only as an attempt to give some account of the 
most recent researches but also as a reference to the original papers. 
As proved to be the case when the original book was written, 
so in the present volume the difficulty of selection, without going 
far beyond the original purpose of the book and keeping the 
volume within reasonable dimensions, has not been easy. 



F. C. LEA. 



BIRMINGHAM, 
June 1916. 



424806 

a 3 



CONTENTS. 
CHAPTER I. 

FLUIDS AT REST. 

Introduction. 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 all 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. 

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



viii CONTENTS 

CHAPTER IV. 

FLOW OF WATER THROUGH ORIFICES AND OVER WEIRS. 

Velocity of discharge from an orifice. Coefficient of contraction for 
sharp-edged orifice. Coefficient of velocity for sharp-edged orifice. Bazin's 
experiments on a sharp-edged orifice. Distribution of velocity in the plane 
of the orifice. Pressure in the plane of the orifice. Coefficient of discharge. 
Effect of suppressed contraction on the coefficient of discharge. The form 
of the jet from sharp-edged orifices. Large orifices. Drowned orifices. 
Partially drowned orifice. Velocity of approach. Coefficient of resistance. 
Sudden 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. Mouthpieces. Borda's mouthpiece. Conical mouthpieces 
and nozzles. Flow through orifices and mouthpieces under constant 
pressure. Time of emptying a tank or reservoir. Notches and weirs. 
Rectangular sharp-edged weir. Derivation of the weir formula from that 
of a large orifice. Thomson's principle of similarity. Discharge through 
a trianglar notch by the principle of similarity. Discharge through a 
rectangular weir by the principle of similarity. Rectangular weir with 
end contractions. Bazin's formula for the discharge of a weir. Bazin's 
and the Cornell experiments on weirs. Velocity of approach. Influence of 
the height of the weir sill above the bed of the stream on the contraction. 
Discharge of a weir when the air is not freely admitted beneath the nappe. 
Form of the nappe. Depressed nappe. Adhering nappes. Drowned or 
wetted nappes. Instability of the form of the nappe. Drowned weirs with 
sharp crests. Vertical weirs of small thickness. Depressed and wetted 
nappes for flat-crested weirs. Drowned nappes for flat-crested weirs. Wide 
flat- crested weirs. Flow over dams. Form of weir for accurate gauging. 
Boussinesq's theory of the discharge over a weir. Determining by ap- 
proximation the discharge of a weir, when the velocity of approach is 
unknown. Time required to lower the water in a reservoir a given distance 
by means of a weir. Examples . Page 50 



CHAPTER V. 

FLOW THROUGH PIPES. 

Resistances to the motion of a fluid in a pipe. Loss of head by friction. 
Head lost at the entrance to the pipe. Hydraulic gradient and virtual 
slope. Determination of the loss of head due to friction. Reynold's 
apparatus. Equation of flow in a pipe of uniform diameter and determi- 
nation of the head lost due to friction. Hydraulic mean depth. Empirical 



CONTENTS IX 

formulae for loss of head due to friction. Formula of Darcy. Variation 
of C in the formula v=G\^mi with service. Ganguillet and Kutter's 
formula. Reynold's experiments and the logarithmic formula. Critical 
velocity. Critical velocity by the method of colour bands. Law of 
frictional resistance for velocities above the critical velocity. The de- 
termination of the values of C given in Table XII. Variation of fc, in the 
formula i=kv n , with the diameter. Criticism of experiments. Piezometer 
fittings. Effect of temperature on the velocity of flow. Loss of head due 
to bends and elbows. Variations of the velocity at the cross section of a 
cylindrical pipe. Head necessary to give the mean velocity v m to the 
water in the pipe. Practical problems. Velocity of flow in pipes. Trans- 
mission of power along pipes by hydraulic pressure. The limiting diameter 
of cast iron pipes. Pressures on pipe bends. Pressure on a plate in a pipe 
filled with flowing water. Pressure on a cylinder. Examples . Page 112 



CHAPTER VI. 

FLOW IN OPEN CHANNELS. 

Variety of the forms of channels. Steady motion in uniform channels. 
Formula for the flow when the motion is uniform in a channel of uniform 
section and slope. Formula of Chezy. Formulae of Prony and Eytelwein. 
Formula of Darcy and Bazin. Ganguillet and Kutter's formula. Bazin's 
formula. Variations of the coefficient 0. Logarithmic formula for flow in 
channels. Approximate formula for the flow in earth channels. Distribu- 
tion of velocity in the cross section of open channels. Form of the curve 
of velocities on a vertical section. The slopes of channels and the velocities 
allowed in them. Sections of aqueducts and sewers. Siphons forming 
part of aqueducts. The best form of channel. Depth of flow in a circular 
channel for maximum velocity and maximum discharge. Curves of velocity 
and discharge for a channel. Applications of the formulae. Problems. 
Examples . , V_ . Page 178 



CHAPTER VII. 

GAUGING THE FLOW OF WATER. 

Measuring the flow of water by weighing. Meters. Measuring the flow 
by means of an orifice. Measuring the flow in open channels. Surface 
floats. Double floats. Rod floats. The current meter. Pitot tube. Cali- 
bration of Pitot tubes. Gauging by a weir. The hook gauge. Gauging 
the flow in pipes ; Venturi meter. Deacon's waste-water meter. Kennedy's 
meter. Gauging the flow of streams by chemical means. Examples 

Page 224 



X CONTENTS 

CHAPTER VIII. 

. IMPACT OF WATER ON VANES. 

Definition of vector. Sum of two vectors. Resultant of two velocities. 
Difference of two vectors. Impulse of water on vanes. Relative velocity. 
Definition of relative velocity as a vector. To find the pressure 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 hydraulic machines should satisfy. 
Definition of angular momentum. Change of angular momentum. Two 
important principles. Work done on a series of vanes fixed to a wheel 
expressed in terms of the velocities of whirl of the water entering and 
leaving the wheel. Curved vanes. Pelton wheel. Force tending to move 
a vessel from which water is issuing through an orifice. The propulsion 
of ships by water jets. Examples ...... Page 261 

CHAPTER IX. 

WATER WHEELS AND TURBINES. 

Overshot water wheels. Breast wheel. Sagebien wheels. Impulse 
wheels. 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 turbines. 
Some actual inward flow turbines. The best peripheral velocity for 
inward and outward flow turbines. Experimental determination of the 
best peripheral velocity for inward and outward flow turbines. Value of e 

to be used in the formula = eH. The ratio of the velocity of whirl V to 

y 

the velocity of the inlet periphery v. The velocity with which water 
leaves a turbine. Bernoulli's equations for inward and outward flow 
turbines neglecting friction. Bernoulli's equations for the inward and 
outward flow turbines including friction. Turbine to develope a given 
horse-power. Parallel or axial flow turbines. Regulation of the flow to 
parallel flow turbines. Bernoulli's equations for axial flow turbines. 
Mixed flow turbines. Cone turbine. Effect of changing the direction of 
the guide blade, when altering the flow of inward flow and mixed flow 
turbines. Effect of diminishing the flow through turbines on the velocity 
of exit. Regulation of the flow by means of cylindrical gates. The Swain 
gate. The form of the wheel vanes between the inlet and outlet cf 
turbines. The limiting head for a single stage reaction turbine. Series 
or multiple stage reaction turbines. Impulse turbines. The form of the 
vanes for impulse turbines, neglecting friction. Triangles of velocity for 
an axial flow impulse turbine considering friction. Impulse turbine for 
high 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 



CONTENTS XI 

CHAPTER X. 

PUMPS. 

Centrifugal and turbine pumps. Starting centrifugal or turbine pumps. 
Form of the vanes of centrifugal pumps. Work done on the water by the 
wheel. Katio of velocity of whirl to peripheral velocity. The kinetic energy 
of the water at exit from the wheel. Gross lift of a centrifugal pump. 
Efficiencies of a centrifugal pump. Experimental determination of the 
efficiency of a centrifugal pump. Design of pump to give a discharge Q. 
The centrifugal head impressed on the water by the wheel. Head-velocity 
curve of a centrifugal pump at zero discharge. Variation of the discharge 
of a centrifugal pump with the head when the speed is kept constant. 
Bernouilli's equations applied to centrifugal pumps. Losses in centrifugal 
pumps. Variation of the head with discharge and with the speed of a 
centrifugal pump. The effect of the variation of the centrifugal head and 
the loss by friction on the discharge of a pump. The effect of the diminu- 
tion of the centrifugal head and the increase of the friction head as the 
flow increases, on the velocity. Discharge curve at constant head. Special 

U 2 
arrangements for converting the velocity head ^ , with which the water 

leaves the wheel, into pressure head. Turbine pumps. Losses in the 
spiral casings of centrifugal pumps. General equation for a centrifugal 
pump. The limiting height to which a single wheel centrifugal pump can 
be used to raise water. The suction of a centrifugal pump. Series or 
multi-stage turbine pumps. Advantages of centrifugal pumps. Pump 
delivering into a long pipe line. Parallel flow turbine pump. Inward flow 
turbine pump. Reciprocating pumps. Coefficient of discharge of the 
pump. Slip. Diagram of work done by the pump. The accelerations 
of the pump plunger and the water in the suction pipe. The effect of 
acceleration of the plunger on the pressure in the cylinder during the 
suction stroke. Accelerating forces in the delivery pipe. Variation of 
pressure in the cylinder due to friction. Air vessel on the suction pipe. 
Air vessel on the delivery pipe. Separation during the suction stroke. 
Negative slip. Separation in the delivery pipe. Diagram of work done 
considering the variable quantity of water in the cylinder. Head lost at 
the suction valve. Variation of the pressure in hydraulic motors due to 
inertia forces. Worked examples. High pressure plunger pump. Tangye 
Duplex pump. The hydraulic ram. Lifting water by compressed air. 
Examples Page 392 

CHAPTER XL 

HYDRAULIC MACHINES. 

Joints and packings used in hydraulic work. The accumulator. Dif- 
ferential accumulator. Air accumulator. Intensifiers. Steam intensifiers. 
Hydraulic forging press. Hydraulic cranes. Double power cranes. 
Hydraulic crane valves. Hydraulic press. Hydraulic riveter. Brother- 
hood and Rigg hydraulic engines. Examples . . . Page 485 



Xll CONTENTS 

CHAPTER XII. 

RESISTANCE TO THE MOTION OF BODIES IN WATER. 

Froude's 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 

APPENDIX 

1. Coefficients of discharge ...,,. Page 521 

2. The critical velocity in pipes. Effect of temperature Page 522 

3. Losses of head in pipe bends Page 525 

4. The Pitot tube * Page 526 

5. The Herschel fall increaser . . . . . Page 529 

6. The Humphrey internal combustion pump . . Page 531 

7. The hydraulic ram . . , . , . Page 537 

8. Circular Weirs Page 537 

9. General formula for friction in smooth pipes . Page 539 

10. The moving diaphragm method of measuring the 

flow of water in open channels .... Page 540 

11. The Centrifugal Pump . . . . \ . Page 542 

ANSWERS TO EXAMPLES 6 Page 553 

INDEX Page 557 



HYDEAULICS. 



CHAPTER I. 

FLUIDS AT BEST. 

1. Introduction. 

The science of Hydraulics in its limited sense and 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 
great antiquity. The Egyptians constructed transit canals for 
warlike purposes, as early as 3000 B.C., and works for the better 
utilisation of the waters of the Nile were carried out at an even 
earlier date. According to Josephus, the gardens of Solomon 
were made beautiful by fountains and other water works. The 
aqueducts of Borne*, some of which were constructed more than 
2000 years ago, were among the "wonders of the world," and 
to-day the city of Athens is partially supplied with water by 
means of an aqueduct constructed probably some centuries before 
the Christian era. 

The science of Hydraulics, however, may be said to have only 
come into existence at the end of the seventeenth century when 
the attention of philosophers was drawn to the problems involved 
in the design of the fountains, which came into considerable use 
in Italian landscape gardens, and which, according to Bacon, 
were of "great beauty and refreshment." The founders were 
principally Torricelli and Mariotte from the experimental, and 
Bernoulli from the theoretical, side. The experiments of Torri- 
celli and of Mariotte to determine the discharge of water through 
orifices in the sides of tanks and through short pipes, probably 
* The Aqueducts of Rome. Frontinus, translated by Herschel. 

L.H. 1 



mark the v fitit*fejfcf<eiilpis':fcb.^etet*isime the laws regulating the 
flow of water, and Torricelli's famous theorem may be said to 
be the foundation of modern Hydraulics. But, as shown at the 
end of the chapter on flow in channels, 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 practi- 
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 Hydrostatiks and Hydrauliks written in English by 
Stephen Swetzer in 1729, but it has been reserved to the workers 
of the nineteenth century to develope 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 water 
wheels, and Fourneyron'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 
machinery, 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 very 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 
fluid into two parts, the force necessary to cause the upper 
part to slide over the lower will be very small indeed, and 
any force, however small, applied to the fluid above the plane 
and parallel to it, will cause motion, or in other words will cause 
a deformation of the fluid. 

Similarly, if a very thin plate be immersed in the fluid in any 
direction, the plate can be made to separate the fluid into two 
parts by the application to the plate of an infinitesimal force, 
and in the imaginary perfect fluid this force would be zero. 



FLUIDS AT REST 3 

Viscosity. Fluids found in nature are not perfect and are 
said to have viscosity; but when they are at rest the conditions 
of equilibrium can be obtained, with sufficient accuracy, on 
the assumption that they are perfect fluids, and that therefore 
no tangential stresses can exist along any plane in a fluid. 
This branch of the study of fluids is called Hydrostatics; when 
the laws of movement of fluids are considered, as in Hydraulics, 
these tangential, or frictional forces have to be taken into 
consideration. 

3. Compressible and incompressible fluids. 

There are two kinds of fluids, gases and liquids, or those which 
are easily compressed, and those which are compressed with 
difficulty. The amount by which the volumes of the latter are 
altered for a very large variation in the pressure is so small that 
in practical problems this variation is entirely neglected, and 
they are therefore considered as incompressible fluids. 

In this volume only incompressible fluids are considered, and 
attention is confined, almost entirely, to the one fluid, water. 

4. Density and specific gravity. 

The density of any substance is the weight of unit volume at 
the standard temperature and pressure. 

The specific gravity of any substance at any temperature and 
pressure is the ratio of the weight of unit volume to the weight 
of unit volume of pure water at the standard temperature and 
pressure. 

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

In the case of water, the presence of salts in solution is of 
greater importance in determining the density than variations 
of temperature, as will be seen by comparing the densities of sea 
water and pure water given in the following table. 

TABLE I. 

Useful data. 

One cubic foot of water at 391 F. weighs 62-425 Ibs. 

60 F. 62-36 

One cubic foot of average sea water at 60 F. weighs 64 Iba. 
One gallon of pure water at 60 F. weighs 10 Ibs. 
One gallon of pure water has a volume of 277-25 cubic inches. 
One ton of pure water at 60 F. has a volume of 35-9 cubic feet. 

12 



4 HYDRAULICS 

Table of densities of pure water. 

Temperature 

Degrees Fahrenheit Density 

32 -99987 

391 1-000000 

50 0-99973 

60 0-99905 

80 0-99664 

104 0-99233 

From the above it will be seen that in practical problems it 
will be sufficiently near to take the weight of one cubic foot of 
fresh water as 62*4 Ibs., one gallon as 10 pounds, 6*24 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 

'--.. 

or when P and a are indefinitely diminished, 

8P 
pa3 5- 

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 REST 5 

Consider a small wedge ABC, Fig. 1, floating immersed in a 
fluid at rest. 

Since there cannot be a tangential 
stress on any of the planes AB, BC, or AC, 
the pressures on them must be normal. 

Let p, pi and p a be the intensities of 
pressures on these planes respectively. 

The weight of the wedge will be very Fig. 1. 

small compared with the pressures on its 
faces and may be neglected. 

As the wedge is in equilibrium under the forces acting on 
its three faces, the resolved components of the force acting on 
AC in the directions of p and pi must balance the forces acting 
on AB and BC respectively. 

Therefore p 2 . AC cos = p . AB, 

and p 2 AC sin = p^ BO. 

But AB = AC cos 0, and BC = AC sin 0. 

Therefore p = PI = p 2 . 

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

Consider a small cylinder of a fluid joining any two points A 
and B on the same horizontal plane in the fluid. 

Since there can be no tangential forces acting on the cylinder 
parallel to the axis, the cylinder must be in equilibrium under the 
pressures on the ends A and B of the cylinder, and since these 
are of equal area, the pressure must be the same at each end of 
the cylinder. 

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

The pressure per unit area at any depth h below the free 
surface of a fluid due to the weight of the fluid is equal to the 
weight of a column of fluid of height h and of unit sectional area. 

Let the pressure per unit area acting on the surface of the 
fluid be p Ibs. If the fluid is in a closed vessel, the pressure p may 
have any assigned value, but if the free surface is exposed to the 
atmosphere, p will be the atmospheric pressure. 

If a small open tube AB, of length h, and cross sectional area a, 
be placed in the fluid, the weight per unit volume of which is 
w Ibs., with its axis vertical, and its upper end A coincident with 
the surface of the fluid, the weight of fluid in the cylinder must be 
w.a.h Ibs. The pressure acting on the end A of the column 
is pa Ibs. 



6 



HYDRAULICS 



Since there cannot be any force acting on the column parallel 
to the sides of the tube, the force of wah Ibs. + pa Ibs. must 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, 



wah 



f , ,, 

= (wh + p) Ibs. 



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

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

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. 




D 




Pr&sure an the Plane AB~w-& Ws per sq 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 Ibs. per sq. inch absolute. Find the pressure per sq. foot at a point 3 feet below 
the free surface of the water. 

j> = 2x 144 + 3x62-4 
= 475 -2 Ibs. per sq. foot. 



FLUIDS AT REST 



Y 



10. Pressures measured in feet of water. Pressure head. 

It is convenient in hydrostatics and hydraulics to express the 
pressure 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 Ibs. the equivalent pressure in feet 

of water, or the pressure head, is h = ft. and for any other fluid 
having a specific gravity /o, the pressure per sq. foot for a head 
h of the fluid is p = w.p.h, or h = 

11. Piezometer 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 



w 



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 Ibs. per sq. inch above the atmosphere. Find 
the height to which the water will rise in the tube. 

The water will rise to such a height that the pressure at the end of the tube in 
the pipe due to the column of water will be 10 Ibs. per sq. inch. 

Therefore h 



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, the 
pressure per unit area acting on the section of the tube, level with 
the surface of the mercury in the vessel, must be equal to the 
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, p ay is, 

30" x 13-596 x 62-4 ,. 
Pa = 10 IXM = 14 7 Ibs. per sq. inch. 



12 x 144 
Expressed in feet of water, 
,147x144 
62-4 



= 33'92 feet. 



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 measuring 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 experiments, is 
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 be 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 will 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 h z 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 + dih^ = p + djii + dh 9 
or di (Tia hi) = dh. 



D 

*" 

f 
S 



iJB 



Fig. 4. 



FLUIDS AT REST 9 

If now, instead of the two limbs of the U tube being open to 
the atmosphere, they are connected by tubes to closed vessels in 
which the pressures are pi and p 2 pounds per sq. foot respectively, 
and hi and h are the vertical lengths of the columns of fluid above 
B and B respectively, then 



= P! + d l . ^ + d . h, 



or 



An application of such a tube to determine the difference of 
pressure at two points in a pipe containing flowing water is shown 
in Fig. 88, page 116. 

Fluids generally used in such U tubes. In hydraulic experiments 
the upper part of the tube is filled with water, and therefore the 
fluid in the lower part must have a greater density than water. 
When the difference of pressure is fairly large, mercury is generally 
used, the specific gravity of which is 13'596. When the difference 
of pressure is small, the height h is difficult to measure with 
precision, so that, if this form of gauge is to be used, it is desirable 
to replace the mercury by a lighter liquid. Carbon bisulphide 
has been used but its action is sluggish and the meniscus between 
it and the water is not always well defined. 
Nitro-benzine gives good results, its prin- 
cipal fault being that the falling meniscus 
does not very quickly assume a definite 
shape. 

The inverted air gauge. A more sen- 
sitive gauge, than the mercury gauge, 
can be made by inverting a U tube and 
enclosing in the upper part a certain 
quantity of air as in the tube BHC, Fig. 5. 

Let the pressure at D in the limb DF 
be PI pounds per square foot, equivalent 
to a head hi of the fluid in the lower part 
of the gauge, and at A in the limb AE let 
the pressure be p 2 , equivalent to a head h 2 . 
Let h be the difference of level of Gr and C. 




Fig. 5. 



Then if CHG contains air, and the weight of the air be 
neglected, being very small, the pressure at C must equal the 
pressure at Gr ; and since in a fluid the pressure on any horizontal 
plane is constant the pressure at C is equal to the pressure at D, 
and the pressure at A equal to the pressure at B. Again the 
pressure at Gr is equal to the pressure at K. 

Therefore h*-h = h 1 



or 



10 



HYDRAULICS 




If the fluid is water p may then be taken as unity ; for a given 
difference of pressure the value of h will clearly be much greater 
than for the mercury gauge, and it has the further advantage that 
h gives directly the difference of pressure in feet of water. The 
temperature of the air in the tube does not affect the readings, as 
any rise in temperature will simply depress the two columns 
without affecting the value of h. 

The inverted oil gauge. A still more sensitive gauge can 
however be obtained by using, in the 
upper part of the tube, an oil lighter 
than water instead of air, as shown 
in Fig. 6. 

Let pi and p 2 be the pressures in 
the two limbs of the tube on a given 
horizontal plane AB, hi and h 2 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- 
sures at G- and H are equal and also 
those at D and C. 

Let PI be the specific gravity of 
the water, and p of the oil. 

Then pi hi-ph = pi (h^-h). 

Therefore h (pi - p) = Pi Oa - hi) 



/ 

-f 


- 




1 

7 


1 




f*N 


i 


ft] 






1 


1 


G 




H * 


1 




T 




i 




Kt 




ll 




B JrC 


Q 


=5= 


fcw<4 




^<S 



Fig. 6. 



and -fe^r 

Substituting for hi and h* the values 



(1). 



h = 



PL 



and h<> = 



-Pi 



. (PI-P) 



or 



(2), 
.(3). 



From (2) it is evident that, if the density of the oil is not very 
different from that of the water, h may be large for very small 
differences of pressure. Williams, Hubbell and Fenkell* found 
that either kerosene, gasoline, or sperm oil gave excellent results, 
but sperm oil was too sluggish in its action for rapid work. 
* Proceedings Am.S.C.E., Vol. xxvn. p. 384. 



FLUIDS AT REST 



11 



Kerosene gave the best results. The author has used mineral oils 
lighter than water of specific gravities varying from 0'78 to 0'96 
and heavier than water of specific gravities from 1*1 to 1'2. 

Temperature coefficient of the inverted oil gauge. Unlike the 
inverted air gauge the oil gauge has a considerable temperature 
coefficient, as will be seen from the table of specific gravities at 
various temperatures of water and the kerosene and gasoline used 
by Williams, Hubbell and Fenkell. 

In this table the specific gravity of water is taken as unity 
at 60 F. 



Temperature F. 
Specific gravity 



Water 


Kerosene 


Gasoline 


40 
1-00092 


60 100 
1-0000 -9941 


40 60 100 
7955 -7879 '7725 


40 60 80 
72147 -71587 '70547 



The calibration of the inverted oil gauge. An arrangement 
similar to that shown in Fig. 6 can conveniently be used for 
calibrating these gauges. 

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

equation (1) equals . 

Pi 

Water is put into AB and FD so that the surfaces B and F 
are on the same level, the common surfaces of the oil and the 
water also being on the same level, this level being zero for the 
oil. Water is then run out of FD until the surface F is 
exactly 1 inch below E and a reading for h taken. The surface F 
is again lowered 1 inch and a reading of h taken. This process 
is continued until F is lowered as far as convenient, and then 
the water in EA is drawn out in a similar manner. When E 
and F are again level the oil in the gauge should read zero. 

14. Transmission of fluid pressure. 

If an external pressure be applied at any point in a fluid, it is 
transmitted equally in all direc- 
tions through the whole mass. 
This is proved experimentally 
by means of a simple apparatus 
such as shown in Fig. 7. 

If a pressure P is exerted upon 
a small piston Q of a sq. inches Fig. 7. 



R 




12 HYDRAULICS 

p 
area, the pressure per unit area p = , arid 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 

Cb 

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 E, is large, 
then, since the pressure per sq. inch is constant throughout the 
fluid, 

W_A 
P ~a f 

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 pump 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 pump ram is 1500 Ibs. Neglecting all losses due to friction etc., 
determine the weight lifted, the work done in raising it 5 feet, and the number 
of strokes made by the pump while raising the weight. 

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

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

Therefore W = 1 - 

Work done = 37,500 x 5 = 187,500 ft. Ibs. 

Let N equal the number of strokes of the pump ram. 
Then N x T 3 5 x 1500 Ibs. = 187,500 ft. Ibs. 

and N = 500 strokes. 

15. Total or whole pressure. 

The whole pressure acting on a surface is the sum of all the 
normal pressures acting on the surface. If the surface is plane all 
the forces are parallel, and the whole pressure is the sum of these 
parallel forces. 



FLUIDS AT REST 13 

Let any surface, which need not be a plane, be immersed 
in a liuid. Let A be the area of the wetted surface, and h the 
pressure head at the centre of gravity of the area. If the area 
is immersed in a fluid the pressure on the surface of which is zero, 
the free surface of the fluid will be at a height h above the centre 
of gravity of the area. In the case of the area being immersed in 
a fluid, the surface of which is exposed to a pressure p, and below 
which the depth of the centre of gravity of the area is h , then 



w 

If the area exposed to the fluid pressure is one face of a body, 
the opposite face of which is exposed to the atmospheric pressure, 
as in the case of the side of a tank containing water, or the 
masonry dam of Fig. 14, or a valve closing the end of a pipe as 
in Fig. 8, the pressure due to the 
atmosphere is the same on the two 
faces and therefore may be neglected. 

Let w be the weight of a cubic 
foot of the fluid. Then, the whole 
pressure on the area is 



T" 
i 

i 




If the surface is in a horizontal 
plane the theorem is obviously true, 
since the intensity of pressure is con- 
stant and equals w . h. 

In general, imagine the surface, Flg * 8t 

Fig. 9, divided into a large number of small areas a, Ch, Oa ... . 

Let 05 be the depth below the free surface FS, of any element 
of area a ; the pressure on this element = w . x . a. 

The whole pressure P = ^w .x.a. 

But w is constant, and the sum of the moments of the elements 
of the area about any axis equals the moment of the whole area* 
about the same axis, therefore 

2# . a = A . h, 
and P = w . A . h. 



16. Centre of pressure. 

The centre of pressure of any plane 
surface acted upon by a fluid is the 
point of action of the resultant pressure 
acting upon the surface. 

Depth of the centre of pressure. Let 
DEC, Fig. 9, be any plane surface 
exposed to fluid pressure. 

* See text-books on Mechanics. 



s 




14 HYDKAULICS 

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 

P = 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, Oi, 03, ... etc. 

The pressure on any element a 

= w . a . x t 

and P = 2wax. 

Taking moments about FS, 

P . X = (way? + wciiX? + ...) 



or 



wAh 



~ A/i ' 

When the area is in a vertical plane, which intersects the 
surface of the water in FS, 2a# 2 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, 



Moment of Inertia about any axis. Calling I the Moment 
of Inertia about an axis through the centre of gravity, and I the 
Moment of Inertia about any axis parallel to the axis through the 
centre of gravity and at a distance h from it, 

I-Io + ATz, 2 . 

Examples. (1) Area is a rectangle breadth 6 and depth d. 

P=w.b.d.h, 




FLUIDS AT REST 15 

If the free surface of the water is level with the upper edge of the rectangle, 

(2) Area is a circle of radius B. 



X= 



E 2 , 

~Th + h - 
tk 

If the top of the circle is just in the free surface or 7&=B, 

X=B. 

TABLE II. 

Table of Moments of Inertia of areas. 





Form of area 


Moment of inertia about 
an axis AB through the 
C. of G. of the section 


Rectangle 


rf*l 

jtr 


Y^ 


Triangle 


E^- 

H-6--H 


k 


Circle 


ifcz 


Trd 4 
64 


Semicircle 


J73&T 
A B 


About the axis AB 


rr* 
8 


Parabola 


H-.fr-^ 

l !*f^B 

iW 


| 



1G 



HYDRAULICS 



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 %bd?w, b being the width and d the 
depth of the area. 

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





L 




a, b -Intensity of pressure, 
ojv a/ci. 



Fig. 11. Fig. 12. 

For a vertical circle the diagram of pressure is as shown in 
Figs. 12 and 13. The intensity of pressure ab on any strip at a 
depth \ is wh . The whole pressure is the volume of the truncated 
cylinder efJch and the centre of pressure is found by drawing a 
line perpendicular to the circle, through the centre of gravity 
of this truncated cylinder. 




Fig. 13. 



FLUIDS AT REST 



17 



Another, and frequently a very convenient method of deter- 
mining the depth of the centre of pressure, when the whole of the 
area is at some distance below the surface of the water, is to 
consider the pressure on the area as made up of a uniform pressure 
over the whole surface, and a 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 



The pressure on any strip ad is, therefore, made up of a 
constant pressure per unit area wh\ and a variable pressure whi ; 
and the whole pressure is equal to the volume of the cylinder efgh, 
Fig. 12, together with the circular wedge fkg. 

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

equals w . A . , and the centre of gravity of this wedge will be 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 may be supposed therefore to be the resultant of two 
parallel forces PI and P 2 acting through the centres of gravity of 
the cylinder efgh, and of the circular wedge fkg respectively, the 
magnitudes of PI and P 2 being the volumes of the cylinder and 
the wedge respectively. 

To find the centre of pressure on the circle AB it is only 
necessary to find the resultant of two parallel forces 

Pi = A.wh A and P 2 = i0.^ 

of which Pi acts at the centre c, and P 2 at a point Ci which is at 
a distance from A of r. 



Example. A masonry dam, Fig. 14, 
has a height of 80 feet from the founda- 
tions and the water face is inclined at 
10 degrees to the vertical ; find the whole 
pressure on the face due to the water per 
foot width of the dam, and the centre of 
pressure, when the water surface is level 
with the top of the dam. The atmo- 
spheric pressure may be neglected. 

The whole pressure will be the force 
tending to overturn the dam, since the 
horizontal component of the pressure 
on AB due to the atmosphere will be 
counterbalanced by the horizontal com- 
ponents of the atmospheric pressure on 
the back of the dam. Since the pressure 
on the face is normal, and the intensity 
of pressure is proportional to the depth, 

L. H. 




D 

R is the> reswLtcLTLt thrust 
OIL the base, DB and, defy 
E. 
Fig. 14. 



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 

= %wd? x sec 10 per foot width 

= \ . 62-4 x 6400 x 1-0154 = 202750 Ibs. 

Combining P with W, the weight of the dam, the resultant thrust R on the base 
and its point of intersection E 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 atmospheric 
pressure may be neglected. 

The whole pressure P =wirW . 8' 

= 62-4. TT. 8 = 1570 Ibs. 

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 



Therefore 



x =1-f= 8 ' *"- 



That is, f inch below the centre of the naive. 

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. 



W 


A 1 


3 


] 


D 


L 




f 




4 

JL 


f 

y 




V 





Fig. 15. 


K 



The whole pressure on BE 

= 64 Ibs. x 1CK x 4-5' x 2-25'= 6480 Ibs. 
The depth of the centre of pressure of BE is 

$. 4-5 = 3'. 
The whole pressure on each of the rectangles above the quadrants 

= w. 5 = 320 Ibs., 
and the depth of the centre of pressure is feet. 

The two quadrants, since they are symmetrically placed about the vertical 
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 g . 



FLUIDS AT REST 19 

Then the intensity of pressure at depth 7? 

= to . 2 + to . d. 
And the whole pressure on the semicircle is P = w 2* + the whole pressure 

on the semicircle when the diameter is in the surface of the water. 

The distance of the centre of gravity of a semicircle from the centre of the 
circle is 



Therefore, 

= 201R 2 + 42 -66 E 3 = 1256 + 666 Ibs. 

The depth of the centre of pressure of the semicircle when the surface of the 
water is at tho centre of the circle, is 




2 ' '6ir 

And, therefore, the whole pressure on the semicircle is the sum of two forces, 
one of which, 1256 Ibs., acts at the centre of gravity, or at a distance of 3'06' from 
AD, and the other of 666 Ibs. acts at a distance of 3 : 47' from AD. 

Then taking moments about AD the product of the pressure on the whole area 
into the depth of the centre of pressure is equal to the moments of all the forces 
acting on the area, about AD. The depth of the centre of pressure is, therefore, 

_ 6480 Ibs. x 3' + 320 Ibs. x 2 x f' + 1256 Ibs. x 3-06 + 666 Ibs. x 3-47' 
= 2-93 feet. 



EXAMPLES. 

(1) A rectangular tank 12 feet long, 5 feet wide, and 5 feet deep is 
filled with water. 

Find the total pressure on an end and side of the tank. 

(2) Find the total pressure and the centre of pressure, on a vertical/ 
sluice, circular in form, 2 feet in diameter, the centre of which is 4 feet 
below the surface of the water. [M. S. T. Cambridge, 1901.] 

(3) A masonry dam vertical on the water side supports water of 
120 feet depth. Find the pressure per square foot at depths of 20 feet and 
70 feet from the surface; also the total pressure on 1 foot length of the dam. 

(4) A dock gate is hinged horizontally at the bottom and supported in 
a vertical position by horizontal chains at the top. 

Height of gate 45 feet, width 30 ft. Depth of water at one side of the 
gate 32 feet and 20 feet on the other side. Find the tension in the chains. 
Sea- water weighs 64 pounds per cubic foot. 

(5) If mercury is 13| times as heavy as water, find the height of a 
column corresponding to a pressure of 100 Ibs. per square inch. 

(6) A straight pipe 6 inches diameter has a right-angled bend connected 
to it by bolts, the end of the bend being closed by a flange. 

The pipe contains water at a pressure of 700 Ibs. per sq. inch. Determine 
the total pull in the bolts at both ends of the elbow. 

22 



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. 



43 



A\ 



*A 

5 



L.'M 



B 



k 40.0- *! 
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 tube 
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 3 feet square, 
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 surface 
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 is 
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 Ibs. [Lond. Un. 1905.] 



CHAPTER II. 



FLOATING BODIES. 

18. Conditions of equilibrium. 

When a body floats in a fluid the surface of the body in 
contact with 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 
of 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 (6) 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 Gr, 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 "Wo?, which tends to bring 
the ship into the position shown in Fig. 18, that is, so that BK 



22 



HYDRAULICS 



passes through Gr. The above condition (&) can therefore only be 
realised, when the resultant of the buoyancy forces passes through 
the centre of gravity of the body. If, 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 equi- 
librium. On the other hand, if when the body is given a small 
displacement from the position of equilibrium, the vertical forces 
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 only, 
resolve themselves therefore into two, 

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

(6) To find whether the equilibrium is stable. 

19. Principle of Archimedes. 

When a body floats freely in a fluid the weight of the body is 
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 be 
equal to the buoyancy. 

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




Fig. 19. 

Consider any small element ab of the surface, of area a and 
depth h, the plane of the element being inclined at any angle to 
the horizontal. Then, if w is the weight of unit volume of the 
fluid, the whole pressure on the area a is wha, and the vertical 
component of this pressure is seen to be wha cos 0. 




FLOATING BODIES 23 

Imagine now a vertical cylinder standing on this area, the top 
of which is in the surface AC. 

The horizontal sectional area of this cylinder is a cos 0, the 
volume is ha cos and the weight of the water filling this volume 
is wha cos 0, and is, therefore, equal to the buoyancy on the 
area ab. 

If similar cylinders be imagined on all the little elements 
of area which make up the whole immersed surface, the total 
volume of these cylinders is the volume of the water displaced, 
and the total buoyancy is, therefore, the weight of this displaced 
water. 

If the body is wholly immersed as in 
body is supposed to be made up of small 
vertical cylinders intersecting the surface of 
the body in the elements of area ab and ab', 
which are inclined to the horizontal at angles 
and 4> and having areas a and ai respectively, 
the vertical component of the pressure on ab 
will be wha cos and on ab' will be wh^a\ cos <. 
But a cos must equal i cos <, each being Fi 8- 

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

2>wha cos ^whitti cos <, 
and is again equal to the weight of the water displaced. 

In this case if the fluid be assumed to be of constant density 
and the weight of the body as equal to the weight of the fluid 
of the same volume, the body will float at any depth. The 
slightest increase in the weight of the body would cause it to 
sink until it reached the bottom of the vessel containing the fluid, 
while a very small diminution of its weight or increase in its 
volume would cause it to rise immediately to the surface. It 
would clearly be practically impossible to maintain such a body 
in equilibrium, by endeavouring to adjust the weight of the body, 
by pumping out, or letting in, water, as has been attempted in a 
certain type of submarine boat. In recent submarines the lowering 
and raising of the boat are controlled by vertical screw propellers. 

20. Centre of buoyancy. 

Since the buoyancy on any element of area is the weight of 
the vertical cylinder of the fluid above this area, and that the 
whole buoyancy is the sum of the weights of all these cylinders, it 
at once follows, that the resultant of the buoyancy forces must 
pass through the centre of gravity of the water displaced, and this 
point is, therefore, called the Centre of Buoyancy. 



HYDRAULICS 



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 buoyancy 
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. 







M 



B 



71 



Fig. 21, 



Fig. 22. 



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

In designing ships it is necessary that, for even large displace- 
ments such as may be caused by the rolling of the vessel, the 
point M shall be above G. To determine M, it is necessary to 
determine G and the centres of buoyancy for the two positions 
of the floating body. This in many cases is a long arid somewhat 
tedious operation. 

22. Small displacements. Metacentre. 

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

Assume the angular displacement in Fig. 21 to be small and 
equal to 0. 

Then, since the volume displacement is constant the volume of 
the wedge ODE must equal CAF, or in Fig. 23 ; dC 3 DE must equal 



FLOATING BODIES 



25 



Let G-i and G- 2 be the centres of gravity of the wedges 
and CiC 2 DE respectively. 




df B 

Fig. 23. 

The heeling of the ship has the effect of moving a mass of 
water equal to either of these wedges from GK to Gr 2 , and this 
movement causes the centre of gravity of the whole water 
displaced to move from B to BI . 

Let Z be the horizontal distance between GK and Gr 2 , 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 
wedge and w the weight of unit volume of the fluid. 

Then w.v.Z = w. V. S 

= .V.BM.sin0. 

Or, since is small, =w.V.BM.0 (1). 

The restoring couple is 



_ T7 . "V7" ~D/^ /Q / O\ 

But w . v . Z = twice the sum of the moments about the axis 
C 2 Ci, of all the elements such as acdb which make up the wedge 



Taking ab as x, bf is o?0, and if ac is 9Z, the volume of the 
element is J# 2 # . 3Z. 

The centre of gravity of the element is at \x from CiC a . 









w s r/J 

~ 

o 



(3). 



But, -5- is the Second Moment or Moment of Inertia of the 
o 

element of area aceb about C 2 Ci, and 2 / -=* is, therefore, the 

Jo o 

Moment of Inertia I of the water-plane area ACiDC 2 about dCa. 
Therefore w .v .Z = w .1.0 ........................ (4). 



26 HYDRAULICS 

The restoring couple is then 



If this is positive, the equilibrium is stable, but if negative it is 
unstable. 

Again since from (1) 

wv.Z = w.Y 
therefore w . Y . BM . 6 = wlO, 

and 



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

The distance GrM 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. 

Example. A ship has a displacement of 15,400 tons, and a draught 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 4 units. 

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

9,400,000x64 
~ 15,400x2240 
= 17-1 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 surface 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 Gr its 
centre of gravity. 

Let YE 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 YFJE. 

Let B be the centre of buoyancy for the vertical position, 
B being the centre of area of YFJE, 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 YE and 

* In the Fig.,Bj is not the centre of area of AFJL, as, for the sake of clearness, 
it is further removed from B than it actually should be, 



FLOATING BODIES 



27 



is the direction in which the buoyancy force acts when the sides 
of the pontoon are vertical, and BiM perpendicular to AL is the 
direction in which the buoyancy force acts when the pontoon is 
heeled over through 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 BI . 

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 Or, 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 GEL to CYA, and therefore it may be 
imagined that a volume of water equal to the volume of this wedge 
is moved from G 2 to Gi . This will cause the centre of buoyancy 
to move parallel to GiG 2 to a new position BI, such that 

BBi x weight of pontoon = GiG 2 x weight of water in GEL. 

Let 6 be half the breadth of the pontoon, 
I the length, 

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

Then, the weight of the pontoon 

W = 2b.D.l.w 

and the weight of the wedge CLE = -~- x I . w. 



28 HYDRAULICS 

Therefore HB, . 26 . p^^M, 

and BR = ^GA. 

Besolving BB> and GriGr 2 , which are parallel to each other, along 
and perpendicular to BM respectively, 

TiO d rK- d f 2 W\ ld &nan ^ 
Bl Q = 4D GlK ~ 4D V3 2 M = 3D = ^D~' 

^ -R -D _ -o n &2K - M <L - d * - Vt&rfQ 

J ^'a 1 K~3D26~6D~ 6D ' 

To find the distance of the point M from G- and the value of the 
restoring couple. Since B X M is perpendicular to AL and BM to 
VE, the angle BMBi equals 0. 

Therefore QM = B X Q cot B = J^ cot = Jg . 

Let z be the distance of the centre of gravity G from 0. 
Then QG = QC -3 = BC-BQ -z 

P 6 2 tan 2 
2 6D 
Therefore 



And since HGr = GrM sin ^, 

the righting couple, 

D 6 2 tan 2 



The distance of the metacentre from the point B, 13 
QM + QB = B,Q cot + ^~~ 



_ 

" 3D 6D 
Wlien is small, the term containing tan 2 is negligible, and 



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

I for the rectangle is T y (26) 3 = %W, and V = 2bDL 

Therefore 



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



FLOATING BODIES 



29 



Example. A pontoon has a displacement of 200 tons. Its length is 50 feet. 
The centre of gravity is 1 loot above the centre of area of the cross section. Find 
the breadth and depth of the pontoon so that for an angular displacement of 10 degrees 
the metacentre shall not be less than 3 feet from the centre of gravity, and the free- 
board shall not be less than 2 feet. 

Referring to Fig. 24, G is the centre of gravity of the pontoon and is the 
centre of the cross section KJ. 

Then, GO = 1 foot, 

F =2 feet, 

GM = 3feet. 

Let D be the depth of displacement. Then 

D x 26 x 62-4 x 50 Ibs. =200 tons x 2240 Ibs. 
Therefore D6 = 71'5 .......................................... (1). 

The height of the centre of buoyancy B above the bottom of pontoon ia 

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



Then 

Therefore 

But 



B0 = l' and BG = 2*. 
BM=5'. 



6D 



Multiplying numerator and denominator by 6, and substituting from equation (1) 

6 s & 3 tan 2 , 
= 5, 



from which 

therefore 

and 



214-5 ' 429 
6(2-K-176) 2 ): 

6 =10- 1ft., 

The breadth B = 20-2 ft. 
depth =7-1 ft. 



An*. 



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 
turning moments due to the 
water in the vessel must be 
taken into account. 

As a simple case consider 
the rectangular vessel, Fig. 25, 
which, when its axis is vertical, 
floats with the plane AB in the Fi - 25 




so 



HYDRAULICS 



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

When the vessel is heeled through an angle 0, 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 w . v . Z, 
v being the volume of either wedge and Z the distance between 
the centre of gravity of the wedges. 

If 26 is the width of the vessel and I its length, v is -^ I tan 0, 

Z is |5 tan 0, and the turning couple is w |6 3 1 tan 2 0. 

If is small wvZ is equal to wI0, 1 being the moment of inertia 
of the water surface KH about an axis through O, 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 0, the tippling-moment due to these 
surfaces is 2i0I0, 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. 




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. If, therefore, Gr is above B and the 
body be given a small angular displacement to the right say, Gr 
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 Gr is below B, the couple will act so as to 
bring the body to its position of equilibrium. 

26. Floating 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 chambers, 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 Gri be the centre of gravity of the ship, G 2 of the dock and its 
water ballast and G the centre of gravity of the dock and the 
ship. 

The position of the centre of gravity of the dock will vary 



32 HYDRAULICS 

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 Gr or the dock will " list." 

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

Let B 2 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 Y 2 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 B 2 Bi, so that 



r 

The centre of gravity G- of the dock and the ship divides GiGr 2 
in the inverse ratios of their weights. 

As the dock rises the centre of gravity Gr 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 G^ 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 B 2 of the dock will also be changing, 
but as the submerged part of the dock is symmetrical about its 
centre lines, B 2 will only move vertically. As stated above, B 
must always lie on the line joining BI and B 2 , and as Gr is to be 
vertically above B, the centre of gravity Gr 2 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 be pumped from the pontoons in raising the 
dock. Let V be the volume displacement of the dock in its lowest 
position, YO the volume displacement in its highest position. To 
raise the dock without a ship in it the volume of the water to be 
pumped from the pontoons is Y Y . 

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 
w 

raise the dock again to its highest position. 

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

water 

W 

+ Y-YO 

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



FLOATING BODIES S3 

Example. A floating dock as shown dimensioned in Fig. 28 is made up of a 
bottom pontoon 540 feet long x 96 feet wide x 14-75 feet deep, two side pontoons 
380 feet long x 13 feet wide x 48 feet deep, the bottom of these pontoons being 
2 feet above the bottom of the dock, and two side chambers on the top of the 
bottom pontoon 447 feet long by 8 feet deep and 2 feet wide at the top and 8 feet at 
the bottom. The keel blocks may be taken as 4 feet deep. 

The dock is to lift a ship of 15,400 tons displacement and 27' 6" draught. 

Determine the amount of water that must be pumped from the dock, to raise 
the ship so that the deck of the lowest pontoon is in the water surface. 

When the ship just takes to the keel blocks on the dock, the bottom of the 
dock is 27-5' + 14-75' + 4' =46 -25 feet below the water line. 

The volume displacement of the dock is then 

14-75 x 540 x 96 + 2 x 44-25 x 13 x 380 + 447 x 8 x 5'= 1,219, 700 cubic feet. 
The volume of dock displacement when the deck is just awash is 

540 x 96 x 14-75 + 2 x 380 x 13' x (14-75 - 2) = 890,600 cubic feet. 
The volume displacement of the ship is 

15,400 x 2240 . 

- =539,000 cubic feet, 

and this equals the weight of the ship in cubic feet of water. 

Of the 890,600 cubic feet displacement when the ship is clear of the water, 
351,600 cubic feet is therefore required to support the dock alone. 

Simply to raise the dock through 31'5 feet the amount of water to be pumped is 
the difference of the displacements, and is, therefore, 329, 100 cubic feet. 

To raise the ship with the dock an additional 539,000 cubic feet must be 
extracted from the pontoons. 

The total quantity, therefore, to be taken from the pontoons from the time the 
ship takes to the keel blocks to when the pontoon deck is in the surface of the 
water is 

868,100 cubic feet =24,824 tons. 

27. Stability of the floating dock. 

As some of the compartments of the dock are partially filled 
with water, it is necessary, in considering the stability, to take 
account of the tippling-moments caused by the movement of the 
free surface of the water in these compartments. 

If Gr is the centre of gravity of the dock and ship on the 
dock, B the centre of buoyancy, I the moment of inertia of the 
section of the ship and dock by the water-plane about the axis of 
oscillation, and Ii, I a etc. the moments of inertia of the water 
surfaces in the compartments about their axes of oscillation, the 
righting moment when the dock receives a small angle of 
heel 0, is 



The moment of inertia of the water-plane section varies 
considerably as the dock is raised, and the stability varies 
accordingly. 

When the ship is immersed in the water, I is equal to the 
moment of inertia of the horizontal section of the ship at the 
water surface, together with the moment of inertia of the 
horizontal section of the side pontoons, about the axis of 
oscillation 0. 

L. H. 3 



34- HYDRAULICS 

When the tops of the keel blocks are just above the surface 
of the water, the water-plane is only that of the side pontoons, 
and I has its minimum value. If the dock is L-shaped as in 
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 AR about an axis 
through the centre of AB. This critical 
point can, however, be eliminated by 




fitting an air box, shown dotted, on the Fig 30 

outer end of the bottom pontoon, the 

top of which is slightly higher than the top of the keel blocks. 



Example. To find the height of the metacentre above the centre of buoyancy of 
the dock of Fig. 28 when 

(a) the ship just takes to the keel blocks, 

(b) 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 the 
water line as 9,400,000 ft. 4 units, and assume that the ship is symmetrically 
placed on the dock, and that the dock deck is horizontal. The horizontal distance 
between the centres of the side tanks is 111 ft. 

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

9, 400,000 + 2 (380 x 13' x 55 -5 a + T^ x 380 x 13 3 ) = 9,400,000 + 30,430,000 + 139, 000. 
The volume of displacement 

= 539,000 + 1,219,700 cubic feet. 
The height of the metacentre above the centre of buoyancy is therefore 



(6) 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 -I- 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 + & x 5 40' x 96 

= 70,269,000. 

The volume displacement is 890,600 cubic feet. 
Therefore BM = 79'8 feet. 

The height, of the centre of buoyancy above the bottom of the dock can be 
determined by finding the centre of buoyancy of each of the parts of the dock, and 
of the ship if it is in the water, and then taking moments about any axis. 

For example. To find the height h of the centre of buoyancy of the dock and 
the ship, when the ship just comes on the keel blocks. 

The centre of buoyancy for the ship is at 15 feet above the bottom of the keel. 
The centre of buoyancy of the bottom pontoon is at 7 '375' from the bottom. 
side pontoons 24-125' 

,, ,, chambers 17'94' 



FLOATING BODIES 35 

Taking moments about the bottom of the dock 

h (510,000 + 437,000 + 76,5,000 + 35,760) 
= 540,000 x 33-75 + 765,000 x 7'375 
+ 437,000 x 24-125 + 35,760 x 17 '95, 
therefore ft =19 '7 feet. 

For case (a) the metacentre is, therefore, 40*3' above the bottom of the dock. If 
now the centre of gravity of the dock and ship is known the metacentrio height 
can be found. 

EXAMPLES. 

(1) A ship when fully loaded has a total burden of 10,000 tons. Find 
the volume displacement in sea water. 

(2) The sides of a ship are vertical near the water line and the area of 
the horizontal section at the water line is 22,000 sq. feet. The total weight 
of the ship is 10,000 tons when it leaves the river dock. 

Find the difference in draught in the dock and at sea after the weight 
of the ship has been reduced by consumption of coal, etc., by 1500 tons. 
Let 9 be the difference in draught. 
Then 9 x 22,000= the difference in volume displacement 
_ 10,000 x 2240 8500 x 2240 

62-43 64 

=6130 cubic feet. 
Therefore 8 = -278 feet 

=3*34 inches. 

(3) The moment of inertia of the section at the water line of a boat 
is 1200 foot 4 units; the weight of the boat is 11'5 tons. 

Determine the height of the metacentre above the centre of buoyancy. 

(4) A ship has a total displacement of 15,000 tons and a draught of 
27 feet. 

When the ship is lifted by a floating dock so that the depth of the bottom 
of the keel is 16'5 feet, the centre of buoyancy is 10 feet from the bottom of 
the keel and the displacement is 9000 tons. 

The moment of inertia of the water-plane is 7,600,000 foot 4 units. 

The horizontal section of the dock, at the plane 16*5 feet above the 
bottom of the keel, consists of two rectangles 380 feet x 11 feet, the distance 
apart of the centre lines of the rectangles being 114 feet. 

The volume displacement of the dock at this level is 1,244,000 cubic feet. 

The centre of buoyancy for the dock alone is 24-75 feet below the surface 
of the water. 

Determine (a) The centre of buoyancy for the whole ship and the dock. 

(6) The height of the metacentre above the centre of buoyancy. 

(5) A rectangular pontoon 60 feet long is to have a displacement of 
220 tons, a free-board of not less than 3 feet, and the metacentre is not to 
be less than 3 feet above the centre of gravity when the angle of heel 
is 15 degrees. The centre of gravity coincides with the centre of figure. 

Find the width and depth of the pontoon. 

32 



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 each 12 feet wide and 50 feet long. In the lower part 
of each of these compartments there is water ballast, the surface 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 graphically 
or otherwise. A ship of 16,000 tons displacement is 600 feet long, 60 feet 
beam, and 26 feet draught. A coefficient of ^ may be taken in the moment 
of inertia term instead of fo to allow for the water-line section not being 
a rectangle. The depth of the centre of buoyancy from the water line is 
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 height 
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 change 
in the depth of immersion after the ship has been sufficiently long at sea to 
burn 500 tons of coal. 

Weight of 1 cubic foot of fresh water 62 Ibs. 
"Weight of 1 cubic foot of salt water 64 Ibs. 



CHAPTER III. 

FLUIDS IN MOTION. 

28. Steady motion. 

The motion 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 in motion are frequently regarded as 
flowing along definite paths, or in thread-like filaments, and when 
the motion is steady these filaments are 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. 
It will be seen later that such an ideal condition of flow is only 
realised in very special cases, but an assumption of such flow if 
not abused is helpful in connection with hydraulic problems. 



38 



HYDRAULICS 



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 Ibs. 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 t which 
equals the pressure head above the atmo- 
spheric pressure. If p is the pressure per 
sq. foot, above the atmospheric pressure, 

h = , but if p is the absolute pressure per 
sq. foot, and p A the atmospheric pressure, 




\L*A 



Fig. 31. 



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 

5- , g being the acceleration due to gravity in feet per 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 



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 shown 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 water as in 



FLUIDS IN MOTION 39 

Fig. 32, and let a small quantity 3Q cubic feet of water enter the 
tube and move the piston through a small dis- 
tance dx. 

Then dQ,=a.dx. 

The work done on the piston as it enters 
will be 

w . h . a . dx = u 



But the weight of dQ cubic feet is w . 9Q pounds, Fl 8- 32 - 

and the work done per pound is, therefore, h, or foot pounds. 

A pressure head h is therefore equivalent to h foot pounds of 
energy per pound 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 per 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 = \i2ghj 

* -L 

2-g = h ' 

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

v 2 
energy to it, and therefore the kinetic energy per pound is ^- . 

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

Q 

energy equal to -j 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. Bernoulli'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 

f) V* 

+ cT + z 
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 OF. 

Let p D and VDJ and p E and V E 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 datum and z^ 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 tit, the 
velocity at D is 

_3Q_ 
VD ~Ad*' 

and the velocity at E is VE = ^ 

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



FLUIDS IN MOTION 41 

and that of the liquid leaving at E 



Since the flow in the tube is steady, the kinetic energy of the 
portion ABCF 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 
i and therefore equals 

w . 8Q (z - Zi). 

The total pressure on the area AB is p D . A, and the work done 
at D in time dt 



and the work done by the pressure at B in time t 

= pE #UE dt = PE dQ. 

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

-nj (t>B 2 - ^D 2 ) = wdQ l (z- Zi) + p D 3Q - PB ^Q. 
From which 



^ + PE + + PD + tant> 

2gr 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 

/VJ /* 

equal to + ?r- + z above the datum level. 
w 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 with 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. Keport 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 
5^- and ^ 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 fractional 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. Ibs. 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 v c as calculated from this equation. 

If at any point D in the pipe, the sectional area is less than the 
area at C, the velocity will be greater than V G , 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 



w w 2g w 2<7 2g' 

p a being the atmospheric pressure at the surface of the water in 
the reservoir. 

At D the velocity -UD is greater than v and therefore p^ is less 



FLUIDS IN MOTION 



43 



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



w 



w 



2g' 



If the area at the section is so small, that p becomes negative, the 
!luid 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. 35. 



Fig. 36. 



&t the section B, Fig. 36, the pressure head is hs and the 
velocity head is 



H. 



v 

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



and 



If now a is made so that 



the pressure head h A becomes equal to the atmospheric pressure, 
and the pipe 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 f inch diameter 
under a high pressure, the tube being diminished at one section to 
0'05 inch diameter. 



44 HYDRAULICS 

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

33. Venturi meter. 

An application of Bernoulli's theorem is found in the Venturi 
meter, as invented by Mr Clemens Herschel*. The meter takes 
its name from an Italian philosopher who in the last decade of the 
18th century made experiments upon the flow of water through 
conical pipes. In its usual form the Venturi meter consists of two 
truncated conical pipes connected together by a short cylindrical 
pipe called the throat, as shown in Figs. 37 and 38. The meter is 
inserted horizontally in a line of piping, the diameter of the large 
ends of the frustra being equal to that of the pipe. 

Piezometer tubes or other pressure gauges are connected to 
the throat and to one or both of the large ends of the cones. 

Let a be the area of the throat. 

Let 0,1 be the area of the pipe or the large end of the cone 
at A. 

Let a 2 be the area of the pipe or the large end of the cone 
atC. 

Let p be the pressure head at the throat. 

Let pi be the pressure head at the up-stream gauge A. 

Let p 2 be the pressure head at the dcrwn-stream. gauge C. 

Let H and H a be the differences of pressure head at the throat 
and large ends A and C of the cone respectively, or 

H =P>-, 

w w 9 
and H, = -E. 

W W 

Let Q be the flow through the meter in cubic feet per sec. 
Let v be the velocity through the throat. 
Let v l be the velocity at the up-stream large end of cone A. 
Let v 2 be the velocity at the down-stream large end of cone C. 
Then, assuming Bernouilli's theorem, and neglecting friction, 

+ + Sfc+SL 

w 2g w 2g w 2g* 

and H = ^. 

20 

If v 2 is equal to Vi, p 2 is theoretically equal to pi, but there is 
always in practice a slight loss of head in the meter, the difference 
pi ~ Pa being equal to this loss of head. 

* Transactions Am.S.C.E., 1887. 



FLUIDS IN MOTION 



The velocity v is , and v l is - . 
a a\ 

Therefore Q* (^ - ^] = %q . H, 
\a efc / 



and 



-a 



45 




46 



HYDRAULICS 



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



v ch 2 a 2 



(1) 



*Jc being a coefficient which has to be determined by experiment. 
For meters having a throat diameter not less than 2 inches and for 
pipe line velocities not less than 1 foot per second a value of 0'985 
for h will probably give discharges within an error of from 2 to 2*5 
per cent. For smaller meters and lower velocities the error may 
be considerable and special calibrations are desirable. 

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 III. 



Herschel 


Coker 


Hfeet 


k 


Discharge 
in cu. ft. 


i 


1 


995 


0418 


9494 


2 


992 


0319 


9587 


6 


985 


0254 


9572 


12 


9785 


0185 


9920 


18 


977 


0096 


1-2021 


23 


970 


0084 


1-3583 



Professor Coker t, from careful experiments on an exceedingly 
well designed small Yenturi 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. 

Example. A Venturi meter having a diameter at the throat of 3G inches 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. 

* See paper by Gibson, Proc. Inst. C.E. Vol. cxcix. 
t Canadian Society of Civil Engineers, 1902. 



FLUIDS IN MOTION 

Find the discharge, and the velocity of flow through the throat. 
The area of the pipe is 63'5 sq. feet. 
throat 7-05 

The difference in pressure head at the two gauges is 6 feet. 



47 



Therefore 



x 32-2x6 



= _*4o_ ^/sse 
= 137 c. ft. per second. 
The velocity 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 Bernoulli's theorem is to show 
the effect of speed and position on the steering of a canal boat. 

When 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 

Fig. 40. 

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

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

PE + V = Pv + V 
w 2g n 2g ' 

At B the water is practically at rest, and therefore v s is 
zero, and 

Pv _ PI + v 
w w 2g' 

The surface at E will therefore be higher than at F. 



4S HYDRAULICS 

When the boat is at the 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 v F ' of the stream lines between the boat and the nearer 
bank, Fig. 41, will be higher than the velocity v v on the other 
side. But for each side of the boat 

PE = PF + v^ = pr + v^ 
w w 2# w 2g ' 

And since vy is greater than t? P , the pressure head p F is 
greater than >r, 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 B. 

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 difficulty 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 Bernoulli'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 

*** 

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 



w 2g w 2g 

EXAMPLES. 

(1) The diameter of the throat of a Venturi meter is | inch, and of 
the pipe to which it is connected If inches. The discharge through the 
meter in 20 minutes was found to be 314 gallons. 

The difference in pressure head at the two gauges was 49 feet. 
Determine the coefficient of discharge. 

(2) A Venturi meter has a diameter of 4 ft. in the large part and 
1-25 ft. in the throat. With water flowing through it, the pressure head is 
100 ft. in the large part and 97 ft. at the throat. Find the velocity in the 
small part and the discharge through the meter. Coefficient of meter 
taken as unity. 



FLUIDS IN MOTION 49 

(3) A pipe AB, 100 ft. long, has an inclination of 1 in 5. The head due 
to the pressure at A is 45 ft., the velocity is 3 ft. per second, and the section 
of the pipe is 3 sq. ft. Find the head due to the pressure at B, where the 
section is 1 sq. ft. Take A as the lower end of the pipe. 

(4) The suction pipe of a pump is laid at an inclination of 1 in 5, and 
water is pumped through it at 6 ft. per second. Suppose the air in the 
water is disengaged if the pressure falls to more than 10 Ibs. below 
atmospheric pressure. Then deduce the greatest practicable length of 
suction pipe. Friction neglected. 

(5) Water is delivered to an inward-flow turbine under a head of 100 feet 
(see Chapter IX). The pressure just outside the wheel is 25 Ibs. per 
sq. inch by gauge. 

Find the velocity with which the water approaches the wheel. Friction 
neglected. 

(6) A short conical pipe varying in diameter from 4' 6" at the large end 
to 2 feet at the small end forms part of a horizontal water main. The 
pressure head at the large end is found to be 100 feet, and at the small end 
96-5 feet. 

Find the discharge through the pipe. Coefficient of discharge unity. 

(7) Three cubic feet of water per second flow along a pipe which as it 
falls varies in diameter from 6 inches to 12 inches. In 50 feet the pipe 
falls 12 feet. Due to various causes there is a loss of head of 4 feet. 

Find (a) the loss of energy in foot pounds per minute, and in horse- 
power, and the difference in pressure head at the two points 50 feet apart. 
(Use equation 1, section 35.) 

(8) A horizontal pipe in which the sections vary gradually has sections 
of 10 square feet, 1 square foot, and 10 square feet at sections A, B, and C. 
The pressure head at A is 100 feet, and the velocity 3 feet per second. 
Find the pressure head and velocity at B. 

Given that in another case the difference of the pressure heads at A 
and B is 2 feet. Find the velocity at A. 

(9) A Venturi meter in a water main consists of a pipe converging to 
the throat and enlarging again gradually. The section of main is 9 sq. ft. 
and the area of throat 1 sq. ft. The difference of pressure in the main and 
at the throat is 12 feet of water. Find the discharge of the main per hour. 

(10) If the inlet area of a Venturi meter is n times the throat area, and 
v and p are the velocity and pressure at the throat, and the inlet pressure 
is mp, show that 



and show that if p and mp are observed, v can be found. 

(11) Two sections of a pipe have an area of 2 sq. ft. and 1 sq. ft. 
respectively. The centre of the first section is 10 feet higher than that of 
the second. The pressure head at each of the sections is 20 feet. Find 
the energy lost per pound of flow between the two sections, when 10 c. ft. 
of water per sec. flow from the higher to the lower section. 

L. 11. 4 



CHAPTER TV. 

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 determining 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 is 
constant, the effect of small changes of temperature and pressure 
in altering the density being thus neglected. 

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




Fig. 42. 



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

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

Let the velocities at E and F be V E and V F , the pressure heads 
h E and 7i F , and the position heads above some datum, Z E and z p , 
respectively. 



FLOW THROUGH ORIFICES 



51 



Then, applying Bernoulli's theorem to the stream line EF, 
If v f is zero, then 



VK ^ Vv 

+ ^T- + ZE = ft? + or + 



7^ = Tip - & E + z 
But from the figure it is seen that 

is equal to h, and therefore 



or 



V E = 



Since h% is the pressure head at E, the water would rise in 
a tube having its end open at E, a height h E , and h may thus 
be called following Thomson the fall of "free level for the 
point E." 

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 small vertical orifices, and 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. 




52 



HYDRAULICS 



Other experiments described on pages 5456, also show that, 
with carefully constructed orifices, the mean velocity through the 
orifice differs from v/2#H by a very small quantity. 

37. Coefficient of contraction for sharp-edged orifice. 

If an orifice is cut in the flat side, or in the bottom of a vessel, 
and has a sharp edge, as shown in Figs. 44 and 45, the stream lines 
set up in the water approach the orifice in all directions, as shown 
in the figure, and the directions of flow of the particles of water, 
except very near the centre, are not normal to the plane of the 
orifice, but they converge, producing a contraction of the jet. 




Fig. 44. 



Fig. 45. 



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 <o 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 



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 has 
no minimum value. Whether a minimum occurs for square orifices 
is doubtful. 

The diminution in section for a greater distance than that 
given by Weisbach is to be expected, for, as the jet moves away 
from the orifice the centre of the jet falls, and the theoretical 
velocity becomes \/2g (H + y),y being the vertical distance between 
the centre of the orifice and the centre of the jet. 



FLOW THROUGH ORIFICES 53 

At a small distance away from the orifice, however, the stream 
lines are practically parallel, and very little error is introduced in 
the coefficient of contraction by measuring the stream near the 
orifice. 

Poncelet and Lesbros in 1828 found, for an orifice '20 m. square, 
a minimum section of the jet at a distance of *3 m. from the orifice 
and at this section c was '563. M. Bazin, in discussing these 
results, remarks that at distances greater than 0'3 m. the section 
becomes very difficult to measure, and although the vein appears 
to expand, the sides become hollow, and it is uncertain whether 
the area is really diminished. 

Complete contraction. The maximum contraction of the jet 
takes place when the orifice is sharp edged and is well removed 
from the sides and bottom of the vessel. In this case the contrac- 
tion is said to be complete. Experiments show, that for complete 
contraction the distance from the orifice to the sides or bottom of 
the vessel should not be less than one and a half to twice the least 
diameter of the orifice. 

Incomplete or sn/ppressed contraction. An example of incom- 
plete contraction is shown in Fig. 46, the lower edge of the 
rectangular orifice being made level with the bottom of the vessel. 
The same effect is produced by placing a horizontal plate in 
the vessel level with the bottom of the orifice. The stream 
lines at the lower part of the orifice are normal to its plane 
and the contraction at the lower edge is consequently suppressed. 





Fig. 46. Fig. 47. 

Similarly, if the width of a rectangular orifice is made equal 
to that of the vessel, or the orifice abed is provided with side walls 
as in Fig. 47, the side or lateral contraction is suppressed. In any 
case of suppressed contraction the discharge is increased, but, as 
will be seen later, the discharge coefficient may vary more than 
when the contraction is complete. To suppress the contraction 
completely, the orifice must be made of such a form that the 
stream lines become parallel at the orifice and normal to its plane. 



54 



HYDRAULICS 




Fig. 49. 



Experimental determination of c. The section of the stream 
from a circular orifice can be obtained with considerable accu- 
racy by the apparatus shown in Fig. 49, which consists of a 
ring having four radial set 
screws of fine pitch. The 
screws are adjusted until the 
points thereof touch the jet. 
M. Bazin has recently used an 
octagonal frame with 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. Fig. 48. 

The screws were adjusted 
until they just touched the jet. The frame was then placed upon 
a sheet of paper and the positions of the ends of the screws 
marked upon the paper. The forms of the sections could then 
be obtained, and the areas 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. Coefficient of velocity for sharp-edged orifice. 

The theoretical velocity through the contracted section is, as 
shown in section 36, equal to \/2#H, but the actual velocity 
t?i is slightly less than this due to friction at the orifice. The 

ratio - L = Jc is called the coefficient of velocity. 

Experimental determination of k. There 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 o> 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 



Second method. An orifice, Fig. 50, is formed in the side of a 
vessel and water allowed to flow from it. The water after leaving 
the orifice flows in a parabolic curve. Above the orifice is fixed 
a horizontal scale 011 which is a slider carrying a vertical scale, 
to the bottom of which is clamped a bent piece of wire, with a sharp 




FLOW THROUGH ORIFICES 



55 



point. The vertical scale can be adjusted so tliat 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 = vj .................................... (1). 

The vertical distance is 

v = \gf ................................. (2). 

X* 

Therefore y = \g 3 

Vi 

and Vl = x V jfy' 

The theoretical velocity of flow is 



Therefore 



& 




It 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. 

If the orifice has its plane inclined at an angle to the 
vertical, the horizontal component of the velocity is Vi cos and 
the vertical component Vi sin 0. 

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

X = ViCOS0t ........................... (1), 

and the vertical movement is, 

y = v 1 *m0t + lgp .................... .(2). 

After a time ti seconds Xi = v i co&0t l ........................... (3), 

y l = vi sin 6^ + \gt? .................... (4). 



56 HYDRAULICS 

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




and, m - 

Prom (5), 

# y-xtanO 

Substituting for v* in (6), 

tan^^'-^y. ..(8). 

XX l (X - 0?i) 

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



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

In Table IY are given values of k 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 was 
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 k 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 



The coefficients given in column 3 were determined by dividing 
the actual mean velocity through different sections of the jet by 
\/2#H, 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 THROUGH ORIFICES 



57 



TABLE IV. 

Sharp-edged 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 \/2^H and to \/2g . (H + 7/), 
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 



Distance of the section 

from the plane of the 

orifice in metres 

0-08 

0-13 

0-17 

0-235 

0-335 

0-515 



Area of Jet 

Area of Orifice 

c 

6079 
5971 
5951 
5904 
5830 
5690 



Mean Velocity Mean Velocity 



0-983 
1-001 
1-004 
1-012 
1-025 
1-050 



998 

999 

1-003 

1-007 

1-010 



Horizontal circular orifice 0*20 m. ('656 feet) diameter, 
= '975m. (3198 feet). 

m = 0-6035. 



0-075 
0-093 
0-110 
0-128 
0-145 
0-163 



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



1-005 
1-016 
1-036 
1-053 
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 
0-175 
0-210 
0-248 
0-302 
0-350 



0-6052 
0-6029 
0-5970 
0-5930 
0-5798 
0-5783 



1-002 
1-006 
1-016 
1-023 
1-046 
1-049 



997 
1-000 
1-007 
1-010 
1-027 
1-024 



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

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

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

* See section 42 and Appendix 1. 



58 HYDRAULICS 

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, was 
-sV greater than the theoretical value. 

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

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 horizontal filaments 
of the jet are different. 

Theoretically the velocity in the lower part of the jet is greater 
than \/2<jr (H + y), and in the upper part less than \/2g (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 \/2#, and the discharge 
assuming this velocity for the whole section is 

6 x -04 x *Jfy = '024 J2g cubic metres. 

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

3Q = 01549 x<ta 
and the total discharge is 

rl -0775 

Q = 01549 



The theoretical discharge, taking account of the varying heads 
is, therefore, 1*004 times the discharge calculated on the assumption 
that the head is constant. 

As the head is increased this difference diminishes, and when 
the head is greater than 5 times the depth of the orifice, is very 
small indeed. 

The assumed data agrees very approximately with that given 
in Table IY for a square orifice, where the value of A; is given as 
1-006. 



FLOW THROUGH ORIFICES 59 

This partly then, explains the anomalous values of fe, but it 
cannot be looked upon as a complete explanation. 

The conditions in the actual jet are not exactly those assumed, 
and the variation of velocity normal to the plane of the section is 
probably much more complicated than here assumed. 

As Bazin further points out, it is probable that, in jets like 
those from the square orifice, which, as will be seen later when the 
form of the jet is considered, are subject to considerable deformation, 
the divergence of some of the filaments gives rise to pressures less 
than that of the atmosphere. 

Bazin has attempted to demonstrate this experimentally, and 
his instrument, Fig. 150, registered pressures less than that of the 
atmosphere; but he doubts the reliability of the results, and 
points out the extreme difficulty of satisfactorily determining the 
pressure in the jet. 

That the inequality of the velocity of the filaments is the 
primary cause, receives support from the fact that for the 
horizontal orifice, discharging downwards, the coefficient Jc is 
always slightly less than unity. In this case, in any horizontal 
section below the orifice, the head is the same for all the stream 
lines, and the velocity of the filaments is practically constant. 
The coefficient of velocity is never less than '96, so that the loss 
due to the internal friction of the liquid is very small. 

40. Distribution of velocity in the plane of the orifice. 
Bazin has examined the distribution of the velocity in the 

various sections of the jet by means of a fine Pitot tube (see 
page 245). In the plane of the orifice a minimum velocity 
occurs, which for vertical orifices is just above the centre, but at a 
little distance from the orifice the minimum velocity is at the top 
of the jet. 

For orifices having complete contraction Bazin found the 
minimum velocity to be '62 to '64 N/20H, and for the rectangular 
orifice, with lateral contraction suppressed, 0'69 N/20H. 

As the distance from the plane of the orifice increases, the 
velocities in the transverse section of the jets from horizontal 
orifices, rapidly become uniform throughout the transverse section. 

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

41. Pressure in the plane of the orifice. 

M. Lager j elm stated in 1826 that if a vertical tube open at 
both ends was placed with its lower end near the centre, and not 
perceptibly below the plane of the inner edge of a horizontal 



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 experiment and found, 
that the water in the tube did not rise to the level of the water in 
the reservoir. 

If Lager j elm'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. as/2011. 
Or, calling m the coefficient of discharge, 



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

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 emoted 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 IY 
Bazin's values for m are given. 

The values as given in Tables Y 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.H, deduces the values of m given in Table YL 

* Annales des Fonts et Chaussees, October, 1888. 

f Flow through Vertical Orifices. 

j Mechanics of Engineering. 

% Experiments upon the Contraction of the Liquid Vein. Bazin translated by 
Trautwine. Also see Appendix and the Bulletins of the University of Wisconsin. 

|| The Flow of Water through Orifices and over Weirs and through open Conduits 
and Pipes, Hamilton Smith, junr., 1886. 



FLOW THROUGH ORIFICES 

TABLE V. 



61 



Experimenter 


Particulars of orifice 


Coefficient of 
discharge 771 


Bazin 


Vertical square orifice side of square 0'6562 ft. 


0-606 


Poncelet and 




f\.ar\K 


Lesbros 


> 


(J DUO 


Bazin 


Vertical Kectangular orifice '656 ft. high x 2'624 
ft. wide with side contraction suppressed 


0-627 


H 


Vertical circular orifice 0*6562 ft. diameter 


0-598 


j 


Horizontal 


0-6035 





0-3281 


0-6063 



TABLE VI. 

Circular orifices. 



Diameter of 
orifice in ft. 

m 


0-0197 
0-627 


0-0295 
0-617 


0-039 
0-611 


0-0492 
0-606 


0-0984 
0-603 


0-164 
0-600 


0-328 
0-599 


0-6562 
0-598 


0-9843 
0-597 



Square orifices. 



Side of square 
in feet 

m 


0-0197 
0-631 


0-0492 
0-612 


0-0984 
0-607 


0-197 
0-605 


0-5906 
0-604 


0-9843 
0-603 



TABLE VII. 

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





Width of orifice -6562 feet 


Width of orifice 
j-968 feet 


Head of water 






above the top 




of the orifice 


Depth of orifice in feet 


in feet 






0328 


0656 


0984 


1640 


3287 


6562 


0656 


6562 


0328 


701 


660 


630 


607 










0656 


694 


659 


634 


615 


596 


572 


643 




1312 


683 


658 


640 


623 


603 


582 


642 


595 


2624 


670 


656 


638 


629 


610 


589 


640 


601 


3937 


663 


653 


636 


630 


612 


593 


638 


603 


6562 


655 


648 


633 


630 


615 


598 


635 


605 


1-640 


642 


638 


630 


627 


617 


604 


630 


607 


3-281 


632 


633 


628 


626 


615 


605 


626 


605 


4-921 


615 


619 


620 


620 


611 


602 


623 


602 


6-562 


611 


612 


612 


613 


607 


601 


620 


602 


9-843 


609 


610 


608 


606 


603 


601 


615 


601 



62 HYDRAULICS 

The heads for which Bazin determined the coefficients in 
Tables IY and V varied only from 2'6 to 3'3 feet, but, as will be 
seen from Table VII, deduced from results given by Poncelet and 
Lesbros* in their classical work, when the variation of head is not 
small, the coefficients for rectangular and square orifices vary 
considerably with the head. 

43. Effect of suppressed contraction on the coefficient 
of discharge. 

Sharp-edged orifice. When some part of the contraction of a 
transverse section of a jet issuing from an orifice is suppressed, 
the cross sectional area of the jet can only be obtained with 
difficulty. 

The coefficient of discharge can, however, be easily obtained, 
as before, by determining the discharge in a given time. The 
most complete and accurate experiments 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 orifices when the lateral contraction is suppressed. The 
reason being, that whatever the head, the variation in the section 
of the jet is 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 Bidone, if x 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, 

mi = m (1 + *15#), 
for rectangular orifices, and 

m l = m(l + *13aj) 
for circular orifices. 

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

His formulae are, however, 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 as '606, 
for mi to be unity it would require to have the value, 
mi = m (1 + *65oj). 

For accurate measurements, either orifices with perfect con- 
traction or, if possible, rectangular or square orifices with the 
lateral contraction completely suppressed, should be used. It will 

* Experiences hydrauliques sur Us lois de Cecoulement de Veau a travers lea 
orifices, etc., 1832. ' Poncelet and Lesbros. 



FLOW THROUGH ORIFICES 



63 



generally be necessary to calibrate the orifice for various heads, 
but as shown above the coefficient for the latter kind is more 
likely to be constant. 

TABLE VIII. 

Table showing the effect of suppressing the contraction on the 
coefficient of discharge. Lesbros*. 

Square vertical orifice 0*656 feet square. 



Head of water 
above the upper 
edge of the orifice 


Sharp-edged 


Side con- 
traction 
suppressed 


Contraction 
suppressed at 
the lower edge 


Contraction 
suppressed at 
the lower and 
side edges 


0-06562 


0-572 




0-599 




0-1640 


0-585 


0-631 


0-608 




0-3281 


0-592 


0-631 


0-615 




0-6562 


0-598 


0-632 


0-621 


0-708 


1-640 


0-603 


0-631 


0-623 


0-680 


3-281 


0-605 


0-628 


0-624 


0-676 


4-921 


0-602 


0-627 


0-624 


0-672 


6-562 


0-601 


0-626 


0-619 


0-668 


9-843 


0-601 


0-624 


0-614 


0-665 



Fig. 51. Section of jet from 
circular orifice. 



44. The form of the jet from sharp-edged orifices. 

From a circular orifice the jet emerges like a cylindrical rod 
and retains a form nearly cylindrical for some distance from the 
orifice. 

Fig. 51 shows three sections of a jet from a vertical circular 
orifice at varying distances from the 
orifice, as given by M. Bazin. 

The flow from square orifices is 
accompanied by an interesting and 
curious phenomenon called the in- 
version of the jet. 

At a very small distance from 
the orifice the section becomes as 
shown in Fig. 52. The sides of the 
jet are concave and the corners are 
cut off by concave sections. The 
section then becomes octagonal as in 
Fig. 53 and afterwards takes the form of a square with concave 
sides and rounded corners, the diagonals of the square being 
perpendicular to the sides of the orifice, Fig. 54. 




Figs. 52 54. Section of jet from 
square orifice. 



Experiments liydrauliques sur les lois de Vecoulement de Veau. 



64 



HYDRAULICS 



45. Large orifices. 

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 coefficient of velocity &, 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. 55. 

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, k\/2gh, and the discharge is 



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



Q = 



hr, 



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



FLOW THROUGH ORIFICES 



65 



By replacing h , hi and b by H , Hi and B an empirical 
formula of the same form is obtained which, by introducing a 
coefficient c, can be made to agree with experiments. Then 



or replacing |c by n t 



(1). 



The coefficient n varies with the head H , and for any orifice 
the simpler formula _ 

Q=m.a.v/2^H .............................. (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 various values of H , m will also be known. 

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

Rectangular sluices. If the lower edge of a sluice opening is 
some distance above the bottom of the channel the discharge 
through it will be practically 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. 

When 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. 




Pfe, 63. 



L. H. 



66 



HYDRAULICS 



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

' 



or 



or taking a coefficient of velocity k 




which, since H is constant, is the same for all filaments of the 
orifice. 

If the coefficients of discharge and contraction are c and m 
respectively the whole discharge through the orifice is then 

Q = cka v 2<?H = wi . a . v 2yH. 
*The coefficient m may be taken as 0'6. 
47. Partially drowned orifice. 

H 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 



B 



Kg. 57. 



48. Velocity of approach. 

It is of interest to consider the effect 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 VF. 

Bernoulli's equation for the stream line drawn is then 



and 

which is again constant for all filaments of the orifice. 
Then Q = m.c 

* Bulletins of University of Wisconsin, Nos. 216 and 270. 



SUDDEN ENLARGEMENT OF A STREAM 67 

49. Effect of velocity of approach on the discharge 
through a large rectangular orifice. 

If the water approaching the large orifice, Fig. 55, has 
a velocity of approach v\ 9 Bernoulli's equation for the stream line 
passing through the strip at depth h y will be 



w 2g w 

p a being the atmospheric pressure, or putting in a coefficient of 
velocity, 



The discharge through the orifice is now, 

t>,2 



50. Coefficient of resistance. 

In connection with the flow through orifices, and hydraulic 
plant generally, the term " coefficient of resistance " is frequently 
used. 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 



51. Sudden enlargement of a current of water. 

It seems reasonable to proceed from the consideration of flow 
through 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.) 

At the enlargement of the pipe, the 
stream suddenly enlarges, and, as shown 
in the figure, in the corners of the large 
pipe it may be assumed that eddy motions 
are set up which cause a loss of energy. 

52 




68 HYDRAULICS 

Consider two sections aa and dd at such a distance from 66 
that the flow is steady. 

Then, the total head at dd equals the total head at aa minus 
the loss of head between aa and dd, or if h is the loss of head due 
to shock, then 

fe + .a + f + fc 

w zg w 2g 

Let A and A^ be the area at aa and dd respectively. 
Since the flow past aa equals that past dd, 



t Then, assuming that each filament of fluid at aa has the 
velocity v a , and v d at dd, the momentum of the quantity of water 






which passes aa in unit time is equal to A^ a 2 , and the momentum 
of the water that passes dd is 



the momentum of a mass of M pounds moving with a velocity 
v feet per second being ~M.v pounds feet. 
The change of momentum is therefore, 



The forces acting on the water between aa and dd to produce 
this change of momentum, are 

paAa acting on aa, pd^d acting on dd, 

and, if p is the mean pressure per unit area on the annular ring 
66, an additional force p(Ad A). 

There is considerable doubt as to what is the magnitude of the 
pressure p, but it is generally assumed that it is equal to p, 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.) 

If this assumption is correct, then it is to be expected that the 
pressure throughout this still water will be practically equal at all 
points and in all directions, and must be equal to the pressure in 
the stream at the section 66, or the pressure p is equal to p a . 

Therefore 

- A a ) - PA a - W 



y 
A v 
from which (p* - p a ) A<i = w (v a - v<i 



SUDDEN ENLARGEMENT OF A STREAM 69 

and since A. a v a 

,1 f 'Pa Pd Isa^a ^a, 

therefore - + . 

w w g g 

Adding -?j- to both sides of the equation and separating 
into two parts, 



4. = 4., 

w 2g w 2g 



or h the loss of head due to shock is equal to 



29 

According to St Yenant this quantity should be increased by 

*1 2 

an amount equal to ~ ~ , but this correction is so small that as 
a rule it can be neglected. 

52. Sudden contraction of a current of water. 

Suppose a pipe partially closed by means of a diaphragm as in 
Fig. 59. 

As 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 sudden 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 




* The pressure at the section cc will be less than in the pipe to the left of the 
diaphragm. From Bernoulli's equation an expression similar to eq. 1 p. 46 can be 
obtained for the discharge through the pipe, and such a diaphragm can be used as 
a meter. Proc. Inst. C.E. Vol. cxcvii. 



70 HYDRAULICS 

If the pipe simply diminishes in diameter as in Fig. 58, the 
section of the stream enlarges from the contracted area ca to fill 
the pipe of area a, therefore the loss of head in this case is 



Or making St Yenant correction 



* Value of the coefficient c. The mean value of c for a sharp-edged 
circular orifice is, as seen in Table IV, about 0'6, and this may be 
taken as the coefficient of contraction in this formula. 

Substituting this value in equation (1) the loss of head is 

found to be -~ , and in equation (2), -^ , v being the velocity in 

the small pipe. It may be taken therefore as -~ . Further 
experiments are required before a correct value can be assigned. 

53. Loss of head due to sharp-edged entrance into a pipe 
or mouthpiece. 

When water enters a pipe or mouthpiece from a vessel through 
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 -~ 

and this agrees with the experimental value of ~ - given by 

Weisbach. 

This value is probably too high for small pipes and too low for 
large pipes t. 

54. Mouthpieces. Drowned Mouthpieces, 

If an orifice is provided with a short pipe or mouthpiece, through 
which the liquid can flow, the discharge may be very different 
from that of a sharp-edged orifice, the difference depending upon 
the length and form of the mouthpiece. If the orifice is cylindrical 
as shown in Fig. 60, being sharp at the inner edge, and so short 
that the stream after converging at the inner edge clears the 
outer edge, it behaves as a sharp-edged orifice. 

J Short external cylindrical mouthpieces. If the mouthpiece is 
cylindrical as ABFE, Fig. 61, having a sharp edge at AB and 
a length of from one and a half to twice its diameter, the jet 

* Proc. Inst. C.E. Vol. cxcvn. 

f See M. Bazin, Experiences nouvelles sur la distribution des vitesses dans 
les tuyaux. { See Bulletins Nos. 216 and 270 University of Wisconsin. 

Shorter mouthpieces are unreliable. 



FLOW THROUGH MOUTHPIECES 



71 



contracts to CD, and then expands to fill the pipe, so that at EF 
it discharges full bore, and the coefficient of contraction is then 
unity. Experiment shows, that the coefficient of discharge is 





Fig. 60. 



Fig. 61. 



from 0'80 to 0'85, the coefficient diminishing with the diameter 
of the tube. The coefficient of contraction being unity, the 
coefficients of velocity and discharge are equal. Good mean 
values, according to Weisbach, are 0*815 for cylindrical tubes, 
and 0'819 for tubes of prismatic form. 

These coefficients agree with those determined on the assump- 
tion that the only head lost in the mouthpiece is that due to 
sudden enlargement, and is 



2g ' 
v being the velocity of discharge at EF. 

Applying Bernoulli's theorem to the sections CD and EF, and 

taking into account the loss of head of -- , and p a as the atmo- 
spheric pressure, 

PCD ^CD 2 = Pa.tf_. '5^ 2 p. Pa 

"" -<7 w 2g + 2g ^ + - ' 



w 



W 



or 



= H. 



Therefore 



and v- 

The area of the jet at EF is a, and therefore, the discharge 
per second is 



Or m, the coefficient of discharge, is 0'812. 
The pressure head at the section CD. Taking the area at CD 
as 0'606 the area at EF, 

tto~ l*65v. 




72 HYDRAULICS 

Therefore P=& + ^-2^ = _ _ 

w w 2g 2g w 2g 

or the pressure at C is less than the atmospheric pressure. 

If a pipe be attached to the mouthpiece, as in Fig. 61, and the 
lower end dipped in water, the water should rise to a height of about 

o feet above the water in the vessel. 

55. Borda's mouthpiece. 

A short cylindrical mouthpiece projecting into the vessel, as in 
Fig. 62, is called a Borda's mouthpiece, and is of interest, as the 
coefficient of discharge upon certain assumptions can be readily 
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 BF the 
hydrostatic pressure will, therefore, simply 
depend upon the depth below the free 

surface AB. Imagine the mouthpiece produced to meet the 
face BC in the area IK. Then the hydrostatic pressure on AD, 
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 which 
gives momentum to the water escaping through the orifice, over- 
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 t through the orifice, is vfylL 

Let a be the area of the orifice and w the area of the transverse 
section of the jet. The discharge per second will be w . w V20H Ibs. 

The hydrostatic pressure on IK is 

pa + waS. Ibs. 

The hydrostatic pressure on EF is pa Ibs. 

The momentum given to the issuing water per second, is 

M = -. 

Therefore pa + <o 2#H = pa + walL, 

and w = a. 



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 coefficient being slightly greater 
than J. 

56. Conical mouthpieces and nozzles. 

These are either convergent as in Fig. 63, or divergent as in 
Fig. 64. 




Fig. 63. 



Fig. 64. 



Calling the diameter of the mouthpiece the diameter at the 
outlet, a divergent tube gives a less, and a convergent 
tube a greater discharge than a cylindrical tube of the 
same diameter. 

Experiments show that the maximum discharge for a 
convergent tube is obtained when the angle of the cone 
is from 12 to 13J degrees, and it is then 0'94 . a . J2gh. 
If, instead of making the convergent mouthpiece conical, 
its sides are curved as in Fig. 65, so that it follows as 
near as possible the natural form of the stream lines, the 
coefficient of discharge may, with high heads, approxi- 
mate very nearly to unity. 

Weisbach*, using the method described on page 55 
to determine the velocity of flow, obtained, for this 
mouthpiece, the following values of k. Since the mouth- 
piece discharges full the coefficients of velocity k and 
discharge m are practically equal. 



Fig. 65. 



Head in feet 
k and m 



0-66 
959 



1-64 
967 



11-48 
975 



55-8 
994 



338 
994 



According to Freeman t, the fire-hose nozzle shown in Fig. 66 
has a coefficient of velocity of '977. 

* Mechanics of Engineering. 

f Transactions 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 EF is then nearly equal to \/2#H 
and the discharge is ra . a . >/2#H, a being 
the area of EF, and 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 




Fig. 67. 



increases, the velocity 
decreases due to the friction of the sides of the tube, and further, 
as the discharge increases, the velocity through the contracted 
section CD increases, and the pressure head at CD consequently 
falls. 

Calling p a the atmospheric pressure, pi the pressure at CD, 
and Vi the velocity at CD, then 



w 2g 



w 



and 



w w g 

If ~- is greater than H + , pi becomes negative. 

As pointed out, however, in connection with Froude's apparatus, 
page 43, if continuity is to be maintained, the pressure cannot be 
negative, and in reality, if water is the fluid, it cannot be less 
than -- the atmospheric pressure, due to the separation of the air 
from the water. The velocity v\ cannot, therefore, be increased 
indefinitely. 



FLOW THROUGH MOUTHPIECES 75 

Assuming the pressure can just become zero, and taking the 
atmospheric pressure as equivalent to a head of 34 ft. of water, the 
maximum possible velocity, is 



and the maximum ratio of the area of EF to CD is 



34ft. 

TT~* 

Practically, the maximum value of Vi may be taken as 



and the maximum ratio of EF to CD as 



The maximum discharge is 



The ratio given of EF to CD may be taken as the maximum 
ratio between the area of a pipe and the throat of a Venturi meter 
to be used in the pipe. 

5 7. Plow through orifices and mouthpieces under constant 
pressure. 

The head of water causing flow through an orifice may be 
produced by a pump or other mechanical means, and the discharge 
may take place into a vessel, such as the condenser of a steam 
engine, in which the pressure is less than that of the atmosphere. 

For example, suppose water to be discharged from a cylinder 
A, into a vessel B, Fig. 68, through 
an orifice or mouthpiece by means 
of a piston loaded with P Ibs., and 
let the pressure per sq. foot in B 
be po Ibs. 

Let the area of the piston be 
A square feet. Let h be the height 
of the water in the cylinder above 
the centre of the orifice and 7i of 
the water in the vessel B. The 
theoretical effective head forcing water through the orifice may 
be written 




76 HYDRAULICS 

If P is large h and h will generally be negligible. 

At the orifice the pressure head is 7& + , and therefore for 

any stream line through the orifice, if there is no friction, 

+ ft, + a P .* 

2g w A.W 



w 

The actual velocity will be less than v, due to friction, and if Jc 
is a coefficient of velocity, the velocity is then 

v = Jc.*/2gH., 
and the discharge is Q = m . a\/2gIL. 

In practical examples the cylinder and the vessel will generally 
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 will be understood after the 
chapter on flow through pipes has been read. 

Example. Water is discharged from a pump into a condenser in which the 
pressure is 3 Ibs. per sq. inch through a short pipe 3 inches diameter. 
The pressure in the pump is 20 Ibs. per sq. inch. 

Find the discharge into the condenser, taking the coefficient of discharge 0'75. 
The effective head is 

H _ 20x144 3x144 

62-4 02-4 

= 39 2 feet. 



Therefore, Q= -75 x -7854 x ^ x ^64-4 x 39-2 cubic feet per seo. 
= 1*84 cubic ft. per see. 

58. Time of emptying a tank or reservoir. 

Suppose a reservoir to have a sharp-edged horizontal orifice 
as in Fig. 44. It is required to find the time taken to empty 
the reservoir. 

Let the area of the horizontal section of the reservoir at any 
height h above the orifice be A sq. feet, and the area of the 

orifice a sq. feet, and let the ratio be sufficiently large that the 

a 

velocity of the water in the reservoir may be neglected. 

When the surface of the water is at any height h above the 
orifice, the volume which flows through the orifice in any time ot 
will be ma \/2gh . dt. 



FLOW THROUGH MOUTHPIECES 77 

The amount dh by which the surface of water in the reservoir 
falls in the time dt is 

g, _ ma \J2ghdt 



ma \/2gh ' 
The time for the water to fall from a height H to H! is 



H A ^_ = 1 _ ( H Adh 
H, ma \l2gh a \/2g J H, 



_ 

\/2g 

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 



_ 1 f H Adh 
ma \/2g J H, h% 



ma 
and the time to empty the vessel is 



= 

ma \/2gr ' 

or is equal to twice the time required for the same volume of 
water 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 
in the lock at any instant above the down-stream water. 

Let A be the sectional area of the lock, at the level of the 
water in the lock, a the area of the sluice, and m its coefficient of 
discharge. 

The discharge through the sluice in time dt ia 

9Q = m . a \l2gx . dt. 

If da? is the distance the surface falls in the lock in time fit, then 
Ada? = ma \/2gxdt t 

or ot = 

ma 

To reduce the level by an amount H, 



o ma 



78 HYDRAULICS 

If m and A are constant, 

2A N/H 



ma \/2gr " 

Example. A reservoir, 200 yards long and 150 yards wide at the bottom, and 
having side slopes of 1 to 1, has a depth of water in it of 25 feet. A short pipe 
3 feet diameter is used to draw off water from the reservoir. 

Find the time taken for the water in the reservoir to fall 10 feet. The 
coefficient of discharge for the pipe is 0-7. 

When the water has a depth h the area of the water surface is 

A = (600 + 2/i) (450 + 2/i). 
The area of the pipe is a=7'068 sq. feet. 

Therefore . - ' - /* (+) (+*) 
0-70 V20- 7-068J 15 fci 

= -^ P 5 2 x 270000*4 + 1 x 2100** + 
39'b Ll5 

- * (610200 + 93800 + 3606) 

' 



= 17,850 sees. 
= 4-95 hours. 

Example. A horizontal boiler 6 feet diameter and 30 feet long is half full of 
water. 

Find the time of emptying the boiler through a short vertical pipe 3 inches 
diameter attached to the bottom of the boiler. 

The pipe may be taken as a mouthpiece discharging full, the coefficient of 
velocity for which is 0'8. 

Let r be the radius of the boiler. 

When the water has any depth h above the bottom of the boiler the area A is 

=30x2 s /r 2 -(r-*) 2 
= 30x2 N /2r*-* 2 . 

The area of the pipe is 0-049 sq. feet. 
2x30 



8x0-049^ 
\2r-h)*dh 



_, t 

Therefore t= 



= 127-4x9-5 
= 1210 sees. 

EXAMPLES. 

(1) Find the velocity due to a head of 100 ft. 

(2) Find the head due to a velocity of 500 ft. per see. 

(3) Water issues vertically from an orifice under a head of 40 ft. To 
what height will the jet rise, if the coefficient of velocity is 0'97 ? 

(4) What must be the size of a conoidal orifice to discharge 10 c. ft. 
per second under a head of 100 ft.? w='925. 



FLOW THROUGH ORIFICES AND MOUTHPIECES 79 

(5) A jet 3 in. diameter at the orifice rises vertically 50 ft. Find its 
diameter 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 
contraction. 

(7) The pressure in the pump cylinder of a fire-engine is 14,400 Ibs. 
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 Ibs. per sq. inch; 
the jet rises to a height of 150 ft. Find the coefficient of velocity. 

(9) A horizontal jet issues under a head of 9 ft. At 6 ft. from the 
orifice it has fallen vertically 15 ins. Find the coefficient of velocity. 

(10) Required the coefficient of resistance corresponding to a coefficient 
of velocity =0-97. 

(11) A fluid of one quarter the density of water is discharged from a 
vessel in which the pressure is 50 Ibs. per sq. in. (absolute) into the 
atmosphere where the pressure is 15 Ibs. per sq. in. Find the velocity of 
discharge. 

(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. 

(13) 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 
level fell 7 ft. in one minute. Find the coefficient of discharge. 

(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 6| ins. Find 
the supply of water per hour in gallons. Coefficient of discharge 0'62. 

(15) A vessel fitted with a piston of 12 sq. ft. area discharges water 
under a head of 10 ft. What weight placed on the piston would double the 
rate of discharge? 

(16) An orifice 2 inches square discharges under a head of 100 feet 
T338 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 
diameter, under a constant pressure of 34 Ibs. 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 
area is driven by a force of 9542 Ibs. and the jet is observed to rise to a 
height 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 
above its centre on the up-stream side, and the backwater on the other 
side is at the level of the centre of the orifice. Find the discharge if 



80 HYDRAULICS 

(20) Ten c. ft. of water per second flow through a pipe of 1 sq. ft. area, 
which suddenly enlarges to 4 sq. ft. area. Taking the pressure at 100 Ibs. 
per sq. ft. in the smaller part of the pipe, find (1) the head lost in shock, 
(2) the pressure in the larger part, (3) the work expended in forcing the 
water through the enlargement. 

(21) A pipe of 3" diameter is suddenly enlarged to 5" diameter. A U 
tube containing mercury is connected to two points, one on each side of the 
enlargement, at points where the flow is steady. Find the difference in 
level in the two limbs of the U when water flows at the rate of 2 c. ft. per 
second from the small to the large section and vice versd. The specific 
gravity of mercury is 13'6. Lond. Un. 

(22) A pipe is suddenly enlarged from 2 inches in diameter to 3 
inches in diameter. Water flows through these two pipes from the smaller 
to the larger, and the discharge from the end of the bigger pipe is two 
gallons per second. Find : 

(a) The loss of head, and gain of pressure head, at the enlarge- 
ment. 

(&) The ratio of head lost to velocity head in small pipe. 

(23) The head and tail water of a vertical-sided lock differ in level 
12 ft. The area of the lock basin is 700 sq. ft. Find the time of emptying 
the lock, through a sluice of 5 sq. ft. area, with a coefficient 0*5. The 
sluice discharges below tail water level. 

(24) A tank 1200 sq. ft. in area discharges through an orifice 1 sq. ft. 
in area. Calculate the time required to lower the level in the tank from 
50 ft. to 25 ft. above the orifice. Coefficient of discharge 0'6. 

(25) A vertical-sided lock is 65 ft. long and 18 ft. wide. Lift 15 ft. 
Find the area of a sluice below tail water to empty the lock in 5 minutes. 
Coefficient 0'6. 

(26) A reservoir has a bottom width of 100 feet and a length of 125 
feet. 

The sides of the reservoir are vertical. 

The reservoir is connected to a second reservoir of the same dimensions 
by means of a pipe 2 feet diameter. The surface of the water in the first 
reservoir is 17 feet above that in the other. The pipe is below the surface 
of the water in both reservoirs. Find the time taken for the water in the 
two reservoirs to become level. Coefficient of discharge 0'8. 

59. Notches and Weirs. 

When the sides of an orifice are 
produced, 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 figures and are largely used 
for gauging the flow of water. 




FLOW OVER WETRS 



81 



For example, if the flow of a small stream is required, a dam is 
constructed across the stream and the water allowed to pass 
through a notch cut in a board or metal plate. 




Fig. 70. Rectangular Notch. 

They can conveniently be used for measuring the compensation 
water to be supplied from collecting reservoirs, and also to gauge 
the supply of water to water wheels and turbines. 

The term weir is a name given to a structure used to dam up 
a stream and over which the water flows. 

The conditions of flow are practically the same as through 
a rectangular notch, and hence such notches are generally called 
weirs, and in what follows the latter term only is used. The top 
of the weir corresponds to the horizontal edge of the notch and is 
called the sill of the weir. 

The sheet of water flowing over a weir or through a notch is 
generally called the vein, sheet, or nappe. 

The shape of the nappe depends upon the form of the sill and 
sides of the weir, the height of the sill above the bottom of the 
up-stream channel, the width of the up-stream channel, and the 
construction of the channel into which the nappe falls. 

The effect of the form of the sill and of the down-stream 
channel will be considered later, but, for the present, attention 
will be confined to weirs with sharp edges, and to those in which 
the air has free access under the nappe so that it detaches itself 
entirely from the weir as shown in Fig. 70. 

60. Rectangular sharp-edged weir. 

If the crest and sides of the weir are made sharp-edged, as 
shown in Fig. 70, and the weir is narrower than the approaching 
channel, and the sill some distance above the bed of the stream, 
there is at the sill and at the sides, contraction similar to that at 
a sharp-edged orifice. 

The surface of the water as it approaches the weir falls, taking 
a curved form, so that the thickness h S) Fig. 70, of the vein over 
the weir, is less than H, the height, above the sill, of the water at 



L. H. 



6 



82 HYDRAULICS 

some distance from the weir. The height H, which is called the 
head over the weir, should be carefully measured at such a distance 
from it, that the water surface has not commenced to curve. 
Fteley and Stearns state, that this distance should be equal to 
2| times the height of the weir above the bed of the stream. 

For the present, let it be assumed that at the point where H is 
measured the water is at rest. In actual cases the water will 
always have some velocity, and the effect of this velocity will have 
to be considered later. H may be called the still water head over 
the weir, and in all the formulae following it has this meaning. 

Side contraction. According to Fteley and Stearns the amount 
by which the stream is contracted when the weir is sharp-edged 
is from 0'06 to 0'12H at each side, and Francis obtained a mean of 
O'lH. A wide weir may be divided into several bays by parti- 
tions, and there may then be more than two contractions, at each 
of which the effective width of the weir will be diminished, if 
Francis' value be taken, by O'lH. 

If L is the total width of a rectangular weir and N the number 
of contractions, the effective width Z, Fig. 70, is then, 

(L-O'INH). 

When L is very long the lateral contraction may be neglected. 

Suppression of the contraction. The side contraction can be 
completely suppressed by making the approaching channel with 
vertical sides and of the same width as the weir, as was done for 
the orifice shown in Fig. 47. The width of the stream is then 
equal to the width of the sill. 

61. Derivation of the weir formula from that of a large 
orifice. 

If in the formula for large orifices, p. 64, h is made equal to 
zero and for the effective width of the stream the length I is 
substituted for 6, and k is unity, the formula becomes 

If instead of hi the head H, Fig. 70, is substituted, and 
a coefficient C introduced, 



The actual width I is retained instead of L, to make allowance 
for the end contraction which as explained above is equal to O'lH 
for each contraction. If the width of the approaching channel is 
made equal to the width of the weir I is equal to L. 

With N contractions I =^L - 01NH), 
and Q = f C v/2^ . (L - O'INH) Hi 

If C is given a mean value of 0'625, and L and H are in feet, 
the discharge in cubic feet per second is 

Q = 3'o3(L- O'INH) H 1 (2). 



FLOW OVER WEIRS 83 

This is the well-known formula deduced by Francis* from 
a careful series of experiments on sharp-edged weirs. 

The formula, as an empirical one, is approximately correct and 
gives reliable values for the discharge. 

The method of obtaining it from that for large orifices is, 
however, open to very serious objection, as the velocity at F on 
the section EF, Fig. 70, is clearly not equal to zero, neither is the 
direction of flow at the surface perpendicular to the section EF, 
and the pressure on EF, as will be understood later (section 83) 
is not likely to be constant. 

That the directions and the velocities of the stream lines are 
different from those through a section taken near a sharp-edged 
orifice is seen by comparing the thickness of the jet in the two 
cases with the coefficient of discharge. 

For the sharp-edged orifice with side contractions suppressed, 
the ratio of the thickness of the jet to the depth of the orifice is not 
very different from the coefficient of discharge, being about 0*625, 
but the thickness EF of the nappe of the weir is very nearly 0'78H, 
whereas the coefficient of discharge is practically 0'625, and the 
thickness is therefore 1*24 times the coefficient of discharge. 

It appears therefore, that although the assumptions made in 
calculating the flow through an orifice may be justifiable, providing 
the head above the top of the orifice is not very small, yet when 
it approaches zero, the assumptions are not approximately true. 

The angles which the stream lines make with the plane of EF 
cannot be very different from 90 degrees, so that it would appear, 
that the error principally arises from the assumption that the 
pressure throughout the section is uniform. 

Bazin for special cases has carefully measured the fall of the 
point F and the thickness EF, and if the assumptions of constant 
pressure and stream lines perpendicular to EF are made, the 
discharge through EF can be calculated. 

For example, the height of the point E above the sill of the 
weir for one of Bazin's experiments was 0'112H and the thickness 
EF was 0'78H. The fall of the point F is, therefore, O'lOSH. 
Assuming constant pressure in the section, the discharge per foot 
width of the weir is, then, 



L 



0-108H 



= 53272^. H*. 

Lowell, Hydraulic Experiments, New York, 1858. 



62 



HYDRAULICS 



The actual discharge per foot width, by experiment, was 

q = 0-433 x/2<7.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 *." 

Let A and B, Figs. 71 and 72, be 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 GI. 

Then, since it has been shown (page 51) that the velocity in 
any stream line is proportional to the square root of the fall of 
free level, 

.*. v : Vi :: *Jn : 1. 

Again the area at c is n a times the area at Ci and, therefore, 
the discharge through c 
the discharge through d = n 
which proves the principle. 

* British Association Keports 1858, 1876 and 1885. 



n 



FLOW OVER WEIRS 85 

63. Discharge through a triangular notch by the 
principle of similarity. 

Let ADC, Figs. 73 and 74, be a triangular notch. 




Let the depth of the flow through the notch at one time be H 
and at another n . H. 

Suppose the area of the stream in the two cases to be divided 
into the same number of horizontal elements, such as ab and aj)i. 

Then clearly the thickness of ab will be n times the thickness 
of aj)i . 

Let a$i be at a distance x from the apex B, and ab at a 
distance nx ; then the width of ab is clearly n times the width of 
aA, and the area of ab will therefore be n* times the area of aj>i. 

Again, the head above ab is n times the Jiead above afo and 
therefore the velocity through ab will be >Jn times the velocity 
through aA and the discharge through ab will be n* times 
that through Oi&i. 

More generally Thomson expresses this as follows : 

" If two triangular notches, similar in form, have water flowing 
through them at different depths, but with similar passages of 
approach, the cross sections of the jets at the notches may be 
similarly divided into the same number of elements of area, and 
the areas of corresponding elements will be proportional to the 
squares of the lineal dimensions of the cross sections, or pro- 
portional to the squares of the heads." 

As the depth h of each element can be expressed as a fraction 
of the head H, the velocities through these elements are propor- 
tional to the square root of the head, and, therefore, the discharge 
is proportional to H^. 

Therefore Q oo H*, 

or Q = C.H', 

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

From experiments with a sharp-edged notch having an angle 
at the vertex of 90 degrees, he found C to be practically constant 
for all heads and equal to 2 '535. Then, H being measured in feet, 
the discharge in cubic feet per second is 

Q = 2-535.H* (3). 



86 HYDRAULICS 

64. Flow through a triangular notch. 

The flow through a triangular notch is frequently given as 



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 head 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^ is " ' . 

If the velocity through this strip is assumed to be v = k^/2gh, the width of the 

stream through o 1 & 1 , - - - , and the thickness dh t the discharge through it is 
H 



The section of the jet just outside the orifice is really less than the area EFD. 
The width of the stream through any strip Oj&j is less than a^, the surface is lower 
than EF, and the apex of the jet is some distance above B. 

The diminution of the width of Oj&j has been allowed for by the coefficient c, and 
the diminution of depth might approximately be allowed for by integrating between 
fc=0 and /j = H, and introducing a third coefficient Cj. 

Then - 



Replacing cc^k by n 

Qrr^.nVV.BH* ....................................... (4). 

Calling the angle ADC, 0, 



and Q = T 8 7 

When B is 90 degrees, B is equal to 2H, and 



Taking a mean value for n of 0-5926 

Q = 2-535 . IT* for a right-angled notch, 
and Q = 1-464^ for a 60 degrees notch, 

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 the 
stream lines normal to the section, and k 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 rectangular weir by the 
principle of similarity. 

The discharge through a rectangular weir can also be obtained 
by the principle of similarity. 



FLOW OVER 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 






A, 


^ 





H 

f 


'~ 




Fig. 7 


5. 


c 



Fig. 75. 

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 
parts the flow through each of these parts in any one of the weirs 
will be the same. 

Suppose the first weir to be divided into N equal parts. If 



then, the second weir is divided into 



N.H 



equal parts, the parts 



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 

as 7 , and the total discharge through the first weir is to the 



discharge through the second as 
N.H* 



Kj n*' 



Instead of two separate weirs the two cases may refer to the 
same weir, and the discharge for any head H is, therefore, pro- 
portional 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 JtH, Jc being about 0*1. 
* gee Example 3, page 260. 



88 HYDRAULICS 

Let the total width of each, weir be now divided into three 
parts, the width of each end part being equal to n . k . H. The 
width of the end parts of the transverse section of the stream will 
each be (n - 1) k . H, and the width of central part L - 2?iA;H. 

The flow through the central part of the weir will be equal to 



Now, whatever the head on the weir, the end pieces of the 
stream, since the width is (n 1) &H and A; is a constant, will be 
similar figures, and, therefore, the flow through them can be 
expressed as 



The total flow is, therefore, 

Q = C (L - 2wfcH) H* + 20j (n - 
If now Ci is assumed equal to C 

Q = 0(L-2fcH)H*. 

If instead of two there are N contractions, due to the weir 
being divided into several bays by posts or partitions, the formula 
becomes 

Q = 0(L-N01.H)H*. 

This is Francis' formula, and by Thomson's theory it is thus 
shown to be rational. 

67. Bazin's* formula for the discharge of a weir. 

The discharge through a weir with no side contraction may be 
written 




or 

the coefficient ra being equal to 

Taking Francis' value for C as 3'33, ra is then 0*415. 
From experiments on sharp-crested weirs with no side con- 
traction Bazin deduced for rat the value 

n ., n - -00984 
ra = 405 + ^ . 
1 

In Table IX, and Fig. 77, are shown Bazin's values for ra for 
different heads, and also those obtained by Rafter at Cornell upon 
a weir similar to that used by Bazin, the maximum head in the 
Cornell experiments being much greater than that in Bazin's 
experiments. In Fig. 77 are also shown several values of ra, as 
calculated by the author, from Francis' experimental data. 

* Annales des Fonts et Chaussees, 18881898. 

t " Experiments on flow over Weirs," Am.S,C.E, Vol. xxvu, 



Bazin. 


0-164 


0-328 


0-656 


0-984 


1-312 


1-64 


1-968 


0-448 


0-432 


0-421 


0-417 


0-414 


0-412 


0-409 




41 




0-00984 









FLOW OVER WEIRS 89 

TABLE IX. 

Values of the coefficient m in the formula Q = wL \/2gr H^ 
Weir, sharp-crested, 6'56 feet wide with free overfall and lateral 
contraction suppressed, H being the still water head over the weir, 
or the measured head h* corrected for velocity of approach. 



Head in feet 



"* .v w iw i - T j- 

1 

Rafter. 
Head in feet m G 

0-1 0-4286 3-437 

0-5 0-4230 3-392 

1-0 0-4174 3-348 

1-5 0-4136 3-317 

2-0 0-4106 3-293 

2-5 0-4094 3-283 

3-0 0-4094 3-283 

3-5 0-4099 3-288 

4-0 0-4112 3-298 

4-5 0-4125 3-308 

5-0 0-4133 3-315 

5-5 0-4135 3-316 

6-0 0-4136 3-317 

68. Bazin's and the Cornell experiments on weirs. 

Bazin's experiments were made on a weirt 6'56 feet long 
having the approaching channel the same width as the weir, so 
that the lateral contractions were suppressed, and the discharge 
was measured by noting the time taken to fill a concrete trench of 
known capacity. 

The head over the weir was measured by means of the hook 
gauge, page 249. Side chambers were constructed and connected 
to the channel by means of circular pipes O'l m. diameter. 

The water in the chambers was very steady, and its level 
could therefore be accurately gauged. The gauges were placed 
5 metres from the weir. The maximum head over the weir in 
Bazin's experiments was however only 2 feet. 

The experiments for higher heads at Cornell University were 
made on a weir of practically the same width as Bazin's, 6'53 feet, 
the other conditions being made as nearly the same as possible ; 
the maximum head on the weir was 6 feet. 

* See page 90. 

f Annales des Pouts et Chaussees, p. 445, Vol. 11. 1801. 



HYDRAULICS 



The results of these experiments, Fig. 77, show that the 
coefficient m diminishes and then increases, having a minimum 
value when H is between 2*5 feet and 3 feet. 



"* 


























1 


"5 




























_q 
























































cxp 




























s: no. 




























> -w* 




























E 




























II 


\ 


























^v 


\ 


























^^ 




























o 


\ 


























i3 


V 


























3 '/IQ 


\ 


























5 *^ 


v \ 




























\ i 


























c* 


\ \ 



























c 


\ % 




























\ \ 


























^ 


\ i 


























t. 


\ 


























Qj 


^ 






























EL 
























^ ^/7,O 




\ 
























*3 T^ 




\V 




















































e 




\s 


J 7 


rFSi 


,^-^j 


^^fei 


v ^/ 


















A 


ss^ 














< 


j 


^_ 




N 






v: 


Si 










J-- ^ < 











J 4? 








^~*> 




















Is 





























































i 


r 


.* 




1 J 




4 


i 




i 


< 


j 



Head in, Feet. 

Mean, coefHcuenb curves for Sharp -edged, Weirs 
+ jBo^t/uy Kccpervmjents 
o Corneli 

A Fronds' (Deduced by the author) 

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 head over the weir a 1 inch galvanised pipe 
with holes Jinch diameter and opening downwards, 6 inches 
apart, was laid across the channel. To this pipe were connected 
f inch pipes passing through the weir to a convenient point below 
the weir where they could be connected to the gauges by rubber 
tubing. The gauges were glass tubes f inch diameter mounted 
on a frame, the height of the water being read on a scale 
graduated to 2mm. 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, 



FLOW OVER WETT5S 91 

In actual cases the water where the head is measured will have 
some velocity, and due to this, the discharge over the weir will be 
increased. 

If Q is the actual discharge over a weir, and A is the area of 
the up-stream channel approaching the weir, the mean velocity in 

the channel is v = -f . 
.A. 

There have been a number of methods suggested to take into 
account this velocity of approach, the best perhaps being that 
adopted by Hamilton Smith, and Bazin. 

This consists in considering the equivalent still water head H, 
over the weir, as equal to 



a being a coefficient determined by experiment, and h the 
measured head. 

The discharge is then 



(5), 



or 



Expanding (5), and remembering that =r-, is generally a small 



quantity, 



The velocity v depends upon the discharge Q to be determined 
and is equal to -^ . 



Therefore Q = mL hJSgh 1 + -, .................. (6). 



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 



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. 



92 HYDRAULICS 

Substituting this value of Q in the above formula, 



(7). 



Let few 3 be called Jc. 
Then Q = m 



/ Z-T 2 7i 2 \ 

( 1 + ^f ) 
\ A. / 



Or, when the width of channel of approach is equal to the 
width of the weir, and the height of the sill, Fig. 78, is p feet above 
the bed of the channel, and h the measured head, 



and 




2 (8). 



Fig. 78. 

The mean value given to the coefficient k by Bazin is 0'55, 
so that 



This may be written 



Q 



in which 



, 
= m f 1 + 



Substituting for m the value given on page 88, 



mi may be called the absolute coefficient of discharge. 

The coefficient given in the Tables. 

It should be clearly understood that in determining the values 
of m as given in the Tables and in Fig. 77 the measured head h 
was corrected for velocity of approach, and in using this 



FLOW OVER WEIRS 93 

coefficient to determine Q, h must first be corrected, or Q 
calculated from formula 9. 

Rafter in determining the values of m from the Cornell ex- 
periments, increased the observed head h by x- only, instead of 

by 1-66 g. 

Fteley and Stearns*, from their researches on the flow over 
weirs, found the correction necessary for velocity of approach to 
be from 

1-45 to T5 |^. 

Hamilton Smith t adopts for weirs with end contractions 
suppressed the values 

T33 to T40 1^, 

and for a weir with two end contractions, 
1*1 to 1*25 1~. 

TABLE X. 
Coefficients n and m as calculated by Bazin from the formulae 

Q= 

and Q = 

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 
the coefficients n and m from the result of an experiment, and the 
difference between them for a special case. 

Example. In one of Bazin's experiments the width of the weir and the 
approaching channel were both 6*56 feet. The depth of the channel approaching 
the weir measured at a point 2 metres up stream from the weir was 7'544 feet and 
the head measured over the weir, which may be denoted by ft, was 0-984 feet. The 
measured discharge was 21-8 cubic ft. per second. 

* Transactions Am.S.G.E., Vol. xn. 
t Hydraulics. 



94 



HYDRAULICS 



The velocity at the section where h was measured, and which may be called the 
velocity of approach was, therefore, 

Q 21-8 

7-544x6-56' "7-544x6-56 
=0-44 feet per second. 
If now the formula for discharge be written 



and n is calculated from this formula by substituting the known values of 
Q, L and h 

n= 0-421. 
Correcting h for velocity of approach, 



= 9888. 
Then 



from which m = - ' 8 =fKiig. 

6 -56 V20.-9888 

It will seem from Table X that when the height p of the sill of the weir above 
the stream bed is small compared with the head, the difference may be much 
larger than for this example. 

When the head is 1-64 feet and larger than p, the coefficient n is eighteen 
per cent, greater than m. In such cases failure to correct the coefficient will lead 
to considerable inaccuracy. 

70. Influence of the height of the weir sill above the bed 
of the stream on the contraction. 

The nearer the sill is to the bottom of the stream, the less the 
contraction at the sill, and if the depth is small compared with H, 
the diminution on the contraction may considerably affect the 
flow. 

When the sill was 1'15 feet above the bottom of a channel, 

of the same width as the weir, Bazin found the ratio ^ (Fig. 85) 
to be 0'097, and when it was 3'70 feet, to be 0*112. For greater 

p 

heights than these the mean value of ^ was 0'13. 

71. Discharge of a weir when the air is not freely 
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, 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 may 
assume one of three distinct forms, and that the discharge for 
* Annales des Fonts et Chaussees, 1891 and 1898. 



FLOW OVER WEIRS 



95 



one of them may be 28 per cent, greater than when the air is 
freely admitted, or the nappe is "free." Which of these three 
forms the nappe assumes and the amount "by which the discharge 
is greater than for the "free nappe," depends largely upon the 
head over the weir, and also upon the height of the weir above 
the water in the down-stream channel. 

The phenomenon is, however, very complex, the form of the 
nappe for any head depending to a very large extent upon 
whether the head has been decreasing, or increasing, and for a 
given head may possibly have any one of the three forms, so that 
the discharge is very uncertain. M. Bazin distinguishes the forms 
of nappe as follows : 

(1) Free nappe. Air under nappe at atmospheric pressure, 
Figs. 70 and 78. 

(2) Depressed nappe enclosing a limited volume of air at a 
pressure less than that of the atmosphere, Fig. 79. 

(3) Adhering nappe. No air enclosed and the nappe adher- 
ing to the down-stream face of the weir, Fig. 80. The nappe in this 
case may take any one of several forms. 



Top of ChanrvdK 




Fig. 80. 

(4) Drowned or wetted nappe, Fig. 81. No air enclosed but 
the nappe encloses a mass of turbulent water which does not move 
with the nappe, and which is said to wet the nappe. 



Fig. 79. 
Drowned or wetted nappe, Fig. 81. 




Fig. 81. 



96 



HYDRAULICS 



72. Depressed nappe. 

The air below the nappe being at less than the atmospheric 
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-stream 
channel under the nappe. 

The discharge is slightly greater than for a free nappe. On a 
weir 2*46 feet above the bottom of the up-stream channel, the 
nappe was depressed for heads below 0*77 feet, and at this head 
the coefficient of discharge was 1'08 mi, mi being the absolute 
coefficient for the free nappe. 

73. Adhering nappes. 

As the head for this weir approached 0*77 feet the air was 
rapidly expelled, and the nappe became vertical as in Fig. 80, its 
surface having a corrugated appearance. The coefficient of dis- 
charge changed from 1*08 mi to l'28mi. This large change in 
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. Drowned or wetted nappes. 

As the head was further increased, and approached 0*97 feet, 
the nappe came away from the weir face, assuming the drowned 
form, and the coefficient suddenly fell to 119 mi. As the head 
was further increased the coefficient diminished, becoming 112 
when the head was above 1*3 feet. 

The drowned nappes are more stable than the other two, but 
whereas for the depressed and adhering nappes the discharge is 
not affected by the depth of water in the down-stream channel, 
the height of the water may influence the flow of the drowned 
nappe. If when the drowned nappe falls into the down stream 
the rise of the water takes place at a distance from the foot of the 
nappe, Fig. 81, the height of the down-stream water does not affect 
the flow. On the other hand if the rise encloses the foot of the 
nappe, Fig. 82, the discharge is affected. Let h- 2 be the difference 







Fig. 82. 



FLOW OVER WEIRS 97 

of level of the sill of the weir and the water below the weir. The, 
coefficient of discharge in the first case is independent of h^ but is 
dependent upon p the height of the sill above the bed of the up- 
stream channel, and is 



(11). 

Bazin found that the drowned nappe could not be formed if h 
is less than 0*4 p and, therefore, ? cannot be greater than 2*5. 
Substituting for m x its value 



from (10) page 92 

m = 0-470 + 0-0075^ .................. (12). 

In the second case the coefficient depends upon h*, and is, 

l -06 -t-0'16 --0'05 ............ (13), 



for which, with a sufficient degree of approximation, may be 
substituted the simpler formula, 



(14). 



The limiting value of m is 1*2 mi, for if h^ becomes greater 
than h the nappe is no longer drowned. 

Further, the rise can only enclose the foot of the nappe when 
h$ is less than (f p - h). As h 2 passes this value the rise is pushed 
down stream away from the foot of the nappe and the coefficient 
changes to that of the preceding case. 

75. Instability of the form of the nappe. 

The head at which the form of nappe changes depends upon 
whether the head is increasing or diminishing, and the depressed 
and adhering nappes are very unstable, an accidental admission 
of air or other interference causing rapid change in their form. 
Further, the adhering nappe is only formed under special circum- 
stances, and as the air is expelled the depressed nappe generally 
passes directly to the drowned form. 

If, therefore, the air is not freely admitted below the nappe 
the form for any given head is very uncertain and the discharge 
cannot be obtained with any great degree of assurance. 

With the weir 2'46 feet above the bed of the channel and 6'56 
feet long Bazin obtained for the same head of 0*656 feet, the four 
kinds of nappe, the coefficients of discharge being as follows : 
L. H. 7 



98 HYDRAULICS 

Free nappe, 0'433 

Depressed nappe, 0'460 
Drowned nappe, level of water down stream 

0*41 feet below the crest of the weir, 0*497 

Nappe adhering to down-stream face, 0'554 
The discharge for this weir while the head was kept constant, 
thus varied 26 per cent. 

76. Drowned weirs with sharp crests*. 

When the surface of the water down stream is higher than the 
sill of the weir, as in Fig. 83, the weir is said to be drowned. 



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, which 
in its simplest form is, 



A 2 being the height of the down-stream water above the sill of 
the weir, h the head actually measured above the weir, p the 
height of the sill above the up-stream channel, and Wi the 
coefficient ((10), p. 92) for a sharp-edged weir. This expression 
gives the same value within 1 or 2 per cent, as the formulae (13) 
and (14). 

Example. The head over a weir is 1 foot, and the height of the sill above the 
up-stream channel is 5 feet. The length is 8 feet and the surface of the water 
in the down-stream channel is 6 inches above the sill. Find the discharge. 

From formula (10), page 92, the coefficient 7% for a sharp-edged weir with free 
nappe is 



* Attempts have been made to express the discharge over a drowned weir as 
equivalent to that through a drowned orifice of an area equal to Lft 2 , under a head 
h-h%, together with a discharge over a weir of length L when the head is h - }i%. 

The discharge is then 

n V^IA (h-h^+m JZgl* (h-h$ 9 
n and m being coefficients. Du Buat gave the formula 



and Monsieur Mary Q = 8/ig \/2g (h - /' 2 + head due to velocity of stream). 



FLOW OVER WEIRS 99 

Therefore m = -4215 [1 -05 (1 + -021) 0-761] 

= 3440. 
Then Q = -344 x 8 </2g . it 

= 22-08 cubic ft. per second. 

77. Vertical weirs of small thickness. 

Instead of making the sill of a weir sharp-edged, it may 
have a flat sill of thickness c. This will frequently be the case in 
practice, the weir being constructed of timbers of uniform width 
placed one upon the other. The conditions of flow for these weirs 
may be very different from those of a sharp-edged weir. 

The nappes of such weirs present two distinct forms, according 
as the water is in contact with the crest of the weir, or becomes 
detached at the up-stream edge and leaps over the crest without 
touching the down-stream edge. In the second case the discharge 
is the same as if the weir were sharp-edged. When the head h 
over the weir is more than 2c this condition is realised, and may 
obtain when h passes f c. Between these two values the nappe is 
in a condition of unstable equilibrium ; when h is less than f c the 
nappe adheres to the sill, and the coefficient of discharge is 



0185 



^), 



any external perturbation such as the entrance of air or the 
passage of a floating body causing the detachment. 

If the nappe adheres between f c and 2c the coefficient m varies 
from *98wi to l'07mi, but if it is free the coefficient m^^m^. 
When H = Jc, m is '79rai. If therefore the coefficients for a 
sharp-edged weir are used it is clear the error may be con- 
siderable. 

The formula for m gives approximately correct results when 
the width of the sill is great, from 3 to 7 feet for example. 

If the up-stream edge of the weir is rounded the discharge is 
increased. The discharge* for a weir having a crest 6*56 feet 
wide, when the up-stream edge was rounded to a radius of 4 inches, 
was increased by 14 per cent., and that of a weir 2*624 feet wide 
by 12 per cent. 

The rounding of the corners, due to wear, of timber weirs of 
ordinary dimensions, to a radius of 1 inch or less, will, therefore, 
affect the flow considerably. 

78. Depressed and wetted nappes for flat-crested weirs. 

The nappes of weirs having flat sills may be depressed, and 
may become drowned as for sharp-edged weirs. 

* Amiales de* Fonts et Chausstes, Vol. u. 1896. 

72 



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 ra may 
be taken the same as when the nappe is free, or 

/' ^ 0'185/A 
w = Wj (0 70 + - J . 

When the nappe is free from the sill and becomes drowned, 
the same formula 



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 falls 
towards the weir, but the stream lines, as they pass over the weir, 
are practically parallel to the top of the weir. 

Let H be the height of the still water surface, and h the depth 
of the water over the weir, Fig. 84. 



' _^-=^^ ^~**77^%77S7777?7\ c^j!' 




Fig. 84. 

Then, assuming that the pressure throughout the section of the 
nappe is atmospheric, the velocity of any stream line is 

v = \/20 (H - h), 
and if L is the length of the weir, the discharge is 

Q = J&JLh x/CHT^TO (16). 



FLOW OVER WEIRS" J01 

For the flow to be permanent (see 'page 106) XJ> mist be a 

maximum for a given value of h, or -~ must equal zero. 

QLrit 

Therefore 




From which 2 (H - ft) - h = 0, 

and h = f H. 

Substituting for h in (16) 



= 0-385L 2^H . H = 3-08L x/H . 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 
mwasO'373, or C = 2'991. 

Lesbros' experiments on weirs sufficiently wide to approximate 
to the conditions assumed, gave '35 for the value of the co- 
efficient m. 

In Table XI the coefficient C for such weirs varies from 2'66 
to 310. 

81. Plow over dams. 

Weirs of various forms. M. Bazin has experimentally investi- 
gated the flow over weirs having (a) sharp crests and (&) flat 
crests, the up- and down-stream faces, instead of both being vertical, 
being 

(1) vertical on the down-stream face and inclined on the 
up-stream face, 

(2) vertical on the up-stream face and inclined on the down- 
stream face, 

(3) inclined on both the up- and down-stream faces, 
and (c) weirs of special sections. 

The coefficients vary very considerably from those for sharp- 
crested vertical weirs, and also for the various kinds of weirs. 
Coefficients are given in Table XI for a few cases, to show the 
necessity of the care to be exercised in choosing the coefficient for 
any 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 
obtained, 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- 

* Annalex des Fonts et Chaussges, 1898. 

t Transactions oj tlie Am.S.C.E., Vol. xuv., 1900. 



102 



HYDRAULICS 





f 


L 


*i -ci^ -t it<-\ 


i ..' 


TABLE XL 

Values of the coefficient C in the formula Q = CL . h*, for weirs 
of the sections shown, for various values of the observed head h. 

Bazin. 


Section of 
weir 


Head in feet 


0-3 


0-5 


1-0 


1-3 


2-0 


3-0 


4-0 


5-0 


6-0 




I>31$ 


__ 


2-66 


2-66 


2-90 


3-10 














i 

I 
i 

V 








I 




3-61 


3-80 


4-01 


3-91 












-r^\ 


4-02 


4-15 


4-18 


4-15 













' 


1 


i 


3-46 


3-57 


3-86 


3-80 










*T^ 

1 ^V 


3-46 


3-49 


3-59 


3-63 














jg^v; 


3-08 


3-08 


3-19 


3-22 















FLOW OVER WEIRS 



103 



TABLE XI (continued). 
Bazin. 



Section of 
weir 



66' 



wr a 



Head in feet 



0-3 



3-10 



2-75 



0-5 



3-27 



3-05 



1-0 1-3 



3-73 



3-52 



3-90 



3-73 



2-0 3-0 4-0 5-0 6-0 



Rafter. 



Section of 
weir 





Head in feet 



0-3 0-5 10 13 2-0 3-0 4-0 5-0 6-0 



3-35 



314 



3-68 



3-42 



3-83 



3-52 



3 ! 77 



3-61 



3-68 



3-66 



3-70 



3-66 



3-71 



3-64 



3-71 



3-63 




2-95 



3-16 



3-27 



3-45 



3-56 



3-61 



3-65 



3-67 



HYDRAULICS 

ments made at Cornell University on the discharge of weirs, similar 
to those used by Bazin and for heads higher than he used, and 
also weirs of sections approximating more closely to those of 
existing masonry dams, used as weirs. From Bazin's and Rafter's 
experiments, curves of discharge for varying heads for some of 
these actual weirs have been drawn 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 the channel 
and is not sharp-edged, and the instability 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 required, the weir should 
comply with the following four conditions, as laid down by 
Bazin. 

(1) The sill of the weir must be made as high as possible 
above the bed of the 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 nappe 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 
sill, and also due to the falling curved surface. 

If the top of the sill is well removed from the bottom of the 
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 experiments to be 
some fraction of the head H on the weir. 

Let CD, Fig. 85, be the section of the vein at which the 
maximum rise 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 curvature 0. 

* Comptes Bendus, 1867 and 1889. 



FLOW OVER WEIRS 



105 



Let H be the height of the surface of the water up stream 
above the sill. Let R be the radius of the stream line at any 
point B in CD at a height x above S, and RI and R 2 the radii of 
curvature at D and C respectively. Let Y, YI and Y 2 be the 
velocities 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 SR and the horizontal area is a. If w 
is the weight of unit volume, the weight of the element is w . aSR. 

Since the element is moving in a circle of radius R the centri- 



fugal force acting on the element is wa 



Y 2 3B 



Ibs. 



The force acting on the element due to gravity is wa 8R Ibs. 
Let p be the pressure per unit area on the lower face of the 
element and p + &p on the upper face. 

Then, equating the upward and downward forces, 



From which 



f * N JVT3 

(p + op) a + waoii = pa+ 



-1 + 




(1). 



wdR gR, " 

Assuming now that Bernoulli's theorem is applicable to the 
stream line at EF, 



w 



Differentiating, and remembering H is constant, 

J" VdY =Q 



dx 



w 



9 



1 dp = 1 VdV 
w dx g . dx * 



106 HYDRAULICS 

And since 



therefore 
or 



- , 
dx dli ' 

V 2 ' YdY 



Integrating, YR = constant. 

Therefore YR = ViRi = V 2 R.. 

At the upper and lower surfaces of the vein the pressure is 
atmospheric, and therefore, 



Since YR = YiRi, and R from the figure is (Ri + x - e), therefore, 

.................. (2). 



The total flow over the weir is 



- e 
.................. (3). 



Now if the flow over the weir is permanent, the thickness h Q of 
the nappe must adjust itself, so that for the given head H the 
discharge is the maximum possible. 

The maximum flow however can only take place if each 
filament at the section GrF has the maximum velocity possible to 
the conditions, otherwise the filaments will be accelerated; and 
for a given discharge the thickness h is therefore a minimum, or 
for a given value of h the discharge is a maximum. That is, when 

Q is a maximum, -jS^ = 0. 

CLn,Q 

If therefore RI can be written as a function of h Q} the value of 
ho, which makes Q a maximum, can be determined by differ- 

entiating (3) and equating ^ to zero. 



Then, since 



-, 
and 



FLOW OVER WEIRS 107 

Therefore, h = (H - e) (1 - n*), 

and Bi = w(l + n) (H-e). 

Substituting this value of RI in the expression for Q, 



n 



which, since Q is a maximum when ~rjr = 0, & n( l ^ is a function 

of n, is a maximum when -jr = Q. 
Differentiating and equating to zero, 



the solution of which gives 

71 = 0-4685, 



and therefore, Q - 0'5216 N/2^ (H - 

= 0-5216 ^(l-g 

= 0'5216 l - * x/2 . H/ 



= ra 
the coefficient m being equal to 

0-5216 (l - g) 1 . 

M. Bazin has found by actual measurement, that the mean 
value for 4-, when the height of the weir is at considerable 
distance from the bottom of the channel, is 0'13. 

Then, l- f - 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 



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 velocity 
of approach is unknown. 

A simple method of determining the discharge over a weir 
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 = mLJi *j2gh. 
The approximate velocity of approach is, then, 

= a , 

and H is approximately 



A nearer approximation to Q can then be obtained by sub- 
stituting H for h } 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 head 
as measured on the weir is 2 feet and the depth of the channel of approach below 
the sill of the weir is 10 feet. Find the discharge. 



m= 0-405 + = . 4099. 

Therefore C = 328. 

Approximately, Q=3-28 2^.16 

= 148 cubic feet per second. 

The velocity v = -^ fg= '77 ft. per sec. , 

and i^?= -0147 feet. 

A second approximation to Q is, therefore, 

Q = 3-28 (2-0147)1 16 

= 150 cubic feet per second. 

A third value for Q can be obtained, but the approximation is sufficiently near 
for all practical purposes. 

In this case the error in neglecting the velocity of approach altogether, is 
probably less than the error involved in taking m as 0-4099. 



FLOW OVER WEIRS 109 

85. Time required to lower the water in a reservoir a 
given distance by means of a weir. 

A reservoir has a weir of length L feet made in one of its sides, 
and having its sill H feet below the original level of the water in 
the reservoir. 

It is required to find the time necessary for the water to fall to 
a 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 surface of the water is at any height h above the sill 
the flow in a time dt is 



Let A be the area of the water surface at this level and dh the 
distance the surface falls in time dt. 

Then, 
and 



The time required for the surface to fall (H-H ) feet is, 
therefore, 

t&j*'** 

L./H 



The coefficient may be supposed constant and equal to 3*34. 
If then A is constant 

A f H dh 



_ 
"CLWSo 

To lower the level to the sill of the weir, H 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 
weir, which causes the time of emptying to be less than that 
given by the above formula. But although the actual time is 
not infinite, it is nevertheless very great. 

O A 

When Ho is JH, * 

When Ho is T VH, t 



So that it takes three times as long for the water to fall from 
to T VH as from H to iH. 



110 HYDKAULTCS 

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 reduce the level of 
the water 1' 11". 



Therefore ' (3 ' 46 " ' 7 8) ' 



= 89,000 sees. 
= 24-7 hours. 

So that, neglecting velocity of approach, there will be only one inch of water on 
the weir after 24 hours. 

Example 2. To find hi the last example the discharge from the reservoir in 
15 hours. 



Therefore 54,000=^ (-^ - -^) . 

From which N / H o=' 421 , 

H = 0-176 feet. 
The discharge is, therefore, 

(2-0-176) 540,000 cubic feet 
= 984,960 cubic feet. 



EXAMPLES. 

(1) A weir is 100 feet long and the head is 9 inches. Find the discharge 
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 water head is 2| inches. Find the effective 
length of the weir, m = '43. 

(3) A weir is 15 feet long and the head over the crest is 15 inches. 
Find the discharge. If the velocity of approach to this weir were 5 feet 
per second, what would be the discharge ? 

(4) Deduce an expression for the discharge through a right-angled 
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 day 
(1 gallon =10 Ibs.), with a normal head of 15 ins. Find the length of the 
weir. Choose a coefficient, stating for what kind of weir it is applicable, 
or take the coefficient C as 3'33. 

(6) What is the advantage in gauging, of using a weir without end 
contractions ? 

(7) Deduce Francis' formula by means of the Thomson principle of 
similarity. 

Apply the formula to calculate the discharge over a weir 10 feet wide 
under a head of 1-2 feet, assuming one end contraction, and neglecting the 
effect of the velocity of approach. 



FLOW OVER WEIRS 111 



(8) A rainfall of ^ * nc h P er hour is discharged from a catchment area 
of 5 square miles. Find the still water head when this volume flows over 
a weir with free overfall 30 feet in length, constructed in six bays, each 
5 feet wide, taking O415 as Bazin's coefficient. 

(9) A district of 6500 acres (1 acre =43,560 sq. ft.) drains into a large 
storage reservoir. The maximum rate at which rain falls in the district is 
2 ins. in 24 hours. When rain falls after the reservoir is full, the water 
requires to be discharged over a weir or bye-wash which has its crest at 
the ordinary top-water level of the reservoir. Find the length of such a 
weir for the above reservoir, under the condition that the water in the 
reservoir shall never rise more than 18 ins. above its top-water level. 

The top of the weir may be supposed flat and about 18 inches wide 
(see Table XI). 

(10) Compare rectangular and V notches in regard to accuracy and 
convenience when there is considerable variation in the flow. 

In a rectangular notch 50" wide the still water surface level is 15" above 
the sill. 

If the same quantity of water flowed over a right-angled V notch, what 
would be the height of the still water surface above the apex ? 

If the channels are narrow how would you correct for velocity of 
approach in each case? Lon. Un. 1906. 

(11) The heaviest daily record of rainfall for a catchment area was 
found to be 42*0 million gallons. Assuming two-thirds of the rain to reach 
the storage reservoir and to pass over the waste weir, find the length of 
the sill of the waste weir, so that the water shall never rise more than two 
feet above the sill. 

(12) A weir is 300 yards long. What is the discharge when the head 
is 4 feet ? Take Bazin's coefiicient 

-00984 



(13) Suppose the water approaches the weir in the last question in a 
channel 8' 6" deep and 500 yards wide. Find by approximation the dis- 
charge, taking into account the velocity of approach. 

(14) The area of the water surface of a reservoir is 20,000 square 
yards. Find the time required for the surface to fall one foot, when the 
water discharges over a sharp-edged weir 5 feet long and the original head 
over the weir is 2 feet. 

(15) Find, from the following data, the horse-power available in a given 
waterfall : 

Available height of faU 120 feet. 

A rectangular notch above the fall, 10 feet long, is used to measure 
the quantity of water, and the mean head over the notch is found to be 
15 inches, when the velocity of approach at the point where the head 
is measured is 100 feet per minute. Lon. Un. 1905. 



CHAPTER V. 

FLOW THROUGH PIPES. 

86. Resistances 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. 

Loss 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 + =r- + z foot pounds. 
w w 2g 

If now water flows along a pipe and, due to any cause, Ti 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 through 
a pipe of uniform diameter and of considerable length, the end B 
being open to the atmosphere. 



~T 




Fig. 86. Loss of head by friction in a pipe. 



FLOW THROUGH PIPES 113 

Let - be the head due to the atmospheric pressure. 

Then if there were no resistances and assuming stream line 
flow, Bernoulli's equation for the point B is 



*>B ' f) uy i 

w 2g w 

2 

from which ^- = Z P - Z B = H, 

or v B = \/2<;H. 

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 V B , and the kinetic 

energy would be ~- . 

A head h = -pr rr- = H ^~ 

2g 2g 2g 

has 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 v = * 



Q3 

= -- ~ * 



and therefore h = 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 

i; 2 
surface of the water in the tank by an amount equal to ~-. 

It is found by experiment, however, that the water does not 
rise to the same height in the three tubes, but is lower in 2 than 
in 1 and in 3 than in 2 as shown in the figure. As the fluid moves 
along the pipe there is, therefore, a loss of head. 

The difference of level h 2 of the water in the tubes 1 and 2 is 
called the head lost by friction in the length of pipe 1 2. 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 were 
a solid body sliding along the pipe, but is really the sum of the 
losses of energy, by friction along the surface, and due to relative 
motions in the mass of water. 

It will be shown later that, as the water flows along the pipe, 
there is relative motion between consecutive filaments in the pipe, 
and that, when the velocity is above a certain amount, the water 
has a sinuous motion along the pipe. Some portion of this head h z 
is therefore lost by the relative motion of the filaments of water, 
and by the eddy motions which take place in the mass of the 
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 E just inside the pipe, Bernoulli's equation is 

+ - + head lost at entrance to the pipe = h + , 
w Zg w 

being the absolute pressure head at B. 

The head lost at entrance has been shown on page 70 to be 
about -Q^- 3 and therefore, 

E p a , 




A ^ . 
w w 2g 

That is, the point C on the hydraulic gradient vertically above 

1 * 5?j 2 
E, is -~ below the surface FD. 



FLOW THROUGH PIPES 



115 



If the pipe is bell-mouthed, there will be no head lost at entrance, 
and 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 
tube, is called the hydraulic gradient, and the angle i which it 
makes with the horizontal is called the slope of the hydraulic 
gradient, or the virtual slope. The angle i is generally small, and 

sin i may be taken therefore equal to i, so that y = t. 

In what follows the virtual slope y is denoted by . 

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, 
when the head is lost uniformly along the pipe. 

If the pressure head is measured above the atmospheric 
pressure, the hydraulic gradient in Fig. 87 is AD, but if above 
zero, AiDi is the hydraulic gradient, the vertical distance between 

v 144 
AD and AiDi being equal to , p a being the atmospheric 

pressure per sq. inch. 




Fig. 87. Pipe rising above the Hydraulic Gradient. 

If the pipe rises above the hydraulic gradient AD, as in Fig. 87, 
the pressure in the pipe at C will be less than that of the atmosphere 
by a head equal to CB. If the pipe is perfectly air-tight it will 
act as a siphon and the discharge for a given length of pipe will 
not be altered. But if a tube open to the atmosphere be fitted at 

82 



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 resist- 
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, yet 
air will tend to accumulate and spoil the siphon action. 

As long as the point C is below the level of the water in the 
reservoir, water will flow along the pipe, but any accumulation of 
air at C tends to diminish the flow. In an ordinary pipe line it is 
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 friction. 
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. Beynolds' apparatus for determining loss of bead 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 the 
main and the pipe. 

At two points 5 feet apart near the end B, and thus at a distance 
sufficiently removed from the point at which the water entered 
the pipe, that any initial eddy motions might be destroyed and a 
steady regime established, two holes of about 1 mm. diameter were 
pierced into the pipe for the purpose of gauging the pressure, at 
these points of the pipe. 

Short tubes were soldered to the pipe, so that the holes 
communicated with these tubes, and these were connected by 

* Phil. Trans. 1883, or Vol. n. Scientific Papers, Keynolds. 



FLOW THROUGH PIPES 117 

indiarubber pipes to the limbs of a siphon gauge Gr, made of glass, 
and which contained mercury or bisulphide of carbon. Scales 
were fixed behind the tubes so that the height of the columns 
in each limb of the gauge could be read. 

For very small differences of level a cathetometer was used*. 
When water was made to flow through the pipe, the difference in 
the heights of the columns in the two limbs of the siphon measured 
the difference of pressure at the two points A and B of the pipe, 
and thus measured the loss of head due to friction. 

If s is the specific gravity of the liquid, and H the difference 
in height of the columns, the loss of head due to friction in feet of 
water is h = H (s - 1). 

The quantity of water flowing in a time t was obtained by 
actual measurement in a graduated flask. 

Calling v the mean velocity in the pipe in feet per second, Q 
the discharge in cubic feet per second, and d the diameter of the 
pipe in feet, 



_ 

The loss of head at different velocities was carefully measured, 
and the law connecting head lost in a given length of pipe, with 
the velocity, determined. 

The results obtained by Keynolds, and others, using this 
method of experimenting, will be referred to later. 

91. Equation of flow in a pipe of uniform diameter 
and determination of the head lost due to friction. 

Let dl be the length of a small element of pipe of uniform 
diameter, Fig. 89. 

A 




C 

Fig. 89. 

Let the area of the transverse section be o>, P the length of 
the line of contact of the water and the surface on this section, or 
the wetted perimeter, a the inclination of the pipe, p the pressure 
per unit area on AB, and p dp the pressure on CD. 
* p. 258, Vol. i. Scientific Paper*, Eeynolda. 



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 = . v . w . 9a. 

The work done on ABCD by the pressure acting upon the area 
AB 

= p . w . v f t. Ibs. per sec. 

The work done by the pressure acting upon CD against the 
flow 

= (p dp) . o> . v ft. Ibs. per sec. 

The frictional force opposing the motion is proportional to the 
area of the wetted surface and is equal to F . P . dl, where F is some 
coefficient which must be determined by experiment and is the 
frictional force per unit area. The work done by friction per sec. 
is, therefore, F , P . 9Z . v. 

The velocity being constant, the velocity head is the same at 
both sections, and therefore, applying the principle of the con- 
servation of energy, 

p.w.t; + a>.i;.i0.92! = (p dp) w . V + F . P . dl . V. 

Therefore w . w . dz = - dp . w + F . P . 9Z, 

, dp Y.P.dl 

or dz = - + . 

W W . <*> 

Integrating this equation between the limits of z and z ly p and 
PI being the corresponding pressures, and I the length of the pipe, 

= Pl p. F.P I 

1 W W W o> * 

FP I 



Therefore, + z 

w w w w 

FPZ 
The quantity is equal to h/ of equation (1), page 48, and is 

the loss of head due to friction. The head lost by friction is 
therefore proportional to the area of the wetted surface of the pipe 
Pl } 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 hydraulic 

mean depth. 

If then this quantity is denoted by m, the head h lost by 
friction, is 

~ w .m* 



FLOW THROUGH PIPES 119 

The quantity F, which has been called above the friction per 
unit area, is found by experiments to vary with the density, 
viscosity, and velocity of the fluid, and with the diameter and 
roughness of the internal surface of the pipe. 

In Hydraulics, the fluid considered is water, and any variations 
in density or viscosity, due to changes of temperature, are generally 
negligible. F, therefore, may be taken as proportional to the 
density, or to the weight w per cubic foot, to the roughness of the 
pipe, and as some function, f(v) of the mean velocity, and f(d) of 
the diameter of the pipe. 

Then, h 



m 

in which expression ^ may be called the coefficient of friction. 

It will be seen later, that the mean velocity v is different from 
the relative velocity u of the water and the surface of the pipe, 
and it probably would be better to express F as a function of u, 
but as u itself probably varies with the roughness of the pipe and 
with other circumstances, and cannot directly be determined, it 
simplifies matters to express F, and thus h, as a function of v. 

93. Empirical formulae for loss of head due to friction. 

The difficulty of correctly determining the exact value of 
f(v) /(d), has led to the use of empirical formulae, which have 
proved of great practical service, to express the head h in terms of 
the velocity and the dimensions of the pipe. 

The simplest* formula assumes that the friction simply varies as 
the square of the velocity, and is independent of the diameter of 
the pipe, or /(?;) f(d) = av*. 



^ .............................. (1), 

or writing p- 2 for a, 



from which is deduced the well-known t Chezy formula, 



or v = C "Jmi. 

Another form in which formula (1) is often found is 

\v*l 



* See Appendix 9, t See also pages 231-233. 



120 



HYDRAULICS 






or since m = 7 for a circular pipe full of water, 



2g.d 



(3), 



in which for a of (1) is substituted f- . 

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 = */ -% . 

Values of these constants are shown in Tables XII to XIV for 
different kinds and diameters of pipes and different velocities. 



TABLE XII. 

Values of C in the formula v = C *Jmi for new and old cast-iron 



pipes. 





New cast-iron pipes 


Old cast-iron pipes 


Velocities in ft. per second 


1 


3 


6 


10 


1 


3 


6 


10 


Diameter of pipe 


















3" 


95 


98 


100 


102 


63 


68 


71 


73 


6" 


96 


101 


104 


106 


69 


74 


77 


79 


9" 


98 


105 


109 


112 


73 


78 


80 


84 


12" 


100 


108 


112 


117 


77 


82 


85 


88 


15" 


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 


136 


98 


103 


106 


109 


36" 


124 


131 


136 


140 


103 


108 


111 


114 


42" 


130 


136 


140 


144 


105 


111 


114 


117 


48" 


135 


141 


145 


148 


106 


112 


115 


118 


60" 


142 


147 


150 


152 











For method of determining the values of C given in the tables, 
see page 132. 

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 varies with the velocity, and the diameter of 
the pipe, suggests that the coefficient is itself some function of 
the velocity of flow, and of the diameter of the pipe, and that 
does not, therefore, equal av*. 



TABLE XIII. 

Values of / in the formula 

, 4/yj 





New cast-iron pipes 


Old cast-iron pipes 


Velocities in 
ft. per second 


1 


3 


6 


10 


1 


3 


6 


10 


Diam. of pipe 


















3" 


0071 


0067 


0064 


0062 


0152 


0139 


0128 


0122 


6" 


007 


0063 


006 


0057 


0135 


0117 


0108 


0103 


9" 


0067 


0058 


0055 


0051 


0122 


0105 


010 


0092 


12" 


0064 


0056 


0051 


0048 


0108 


0096 


0089 


0084 


15" 


0062 


0053 


0048 


0043 


0099 


0087 


0081 


0078 


18" 


0058 


0051 


0045 


0041 


0087 


0078 


0073 


0069 


24" 


0053 


0045 


0040 


0037 


0076 


0067 


0063 


0060 


80" 


0046 


0040 


0037 


0035 


0067 


0061 


0057 


0055 


36" 


0042 


0037 


0035 


0033 


0061 


0056 


0052 


0050 


42" 


0038 


0035 


0033 


0031 


0058 


0052 


005 


0048 


48" 


0036 


0032 


0031 


0029 


0057 


0051 


0049 


0046 


60" 


0032 


0030 


0029 


0028 











TABLE XIV. 

Values of C in the formula v = C Jmi for steel riveted pipes. 



Velocities in ft. per second 


1 


3 


5 


10 


Diameter of pipe 










3" 


81 


86 


89 


92 


11" 


92 


102 


107 


115 


llf" 
15" 


93 
109 


99 
112 


102 
114 


105 
117 


38" 


113 


113 


113 


113 


42" 


102 


106 


108 


111 


48" 


105 


105 


105 


105 


72"* 


110 


110 


111 


111 


72" 


93 


101 


105 


110 


103" 


114 


109 


106 


104 



* See pages 124 and 137. 



122 HYDRAULICS 

94. Formula of Darcy. 

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 experiments 
upon pipes of various diameters and of different materials, 
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 

7 O'l 2 

h = . ^ 
m 

was of the form a - a + - , 

r being the radius of the pipe. 

For new cast-iron, and wrought-iron pipes of the same 
roughness, Darcy's values of and ft when transferred to English 
units are, 

a = 0-000077, 
= 0'000003235. 

For old cast-iron pipes Darcy proposed to double these values. 
Substituting the diameter d for the radius r, and doubling /?, for 
new pipes, 

>- 0-000077 



or 



= 0-00000647 

m 



Substituting for m its value ^ and multiplying and dividing 



For old cast-iron pipes, 

& = 0-00001294 



0-01 4 - - 

l ( l+ I2d) 2g 'd 
* Eeclierclies Experiment ales. 



FLOW THROUGH PIPES 123 



Or, *-^ 8 V l^ffl 




As the student cannot possibly retain, without unnecessary 
labour, values of / and C for different diameters it is convenient 
to remember the simple forms, 



for new pipes, and 



for old pipes. 

According to Darcy, therefore, the coefficient in the 
formula varies only with the diameter and roughness of the pipe. 

The values of C as calculated from his experimental results, for 
some of the pipes, were practically constant for all velocities, and 
notably for those pipes which had a comparatively rough internal 
surface, but for smooth pipes, the value of varied from 10 to 
20 per cent, for the same pipe as the velocity changed. The 
experiments of other workers show the same results. 

The assumption that p>f(v)f(d)=av* in which a is made to 
vary only with the diameter and roughness, or in other words, the 
assumption that h is proportional to v 2 is therefore not in general 
justified by experiments. 

95. As stated above, the formulae given must be taken as 
purely empirical, and though by the introduction of suitable 
constants they can be made to agree with any particular experi- 
ment, or even set of experiments, yet none of them probably 
expresses truly the laws of fluid friction. 

The formula of Chezy by its simplicity has found favour, and 
it is likely, that for some time to come, it will continue to be used, 
either in the form v = C Vrai, or in its modified form 

.I 



In making calculations, values of C or f y which most nearly suit 
any given case, can be taken from the tables. 

96. Variation of C in the formula v = C >/mi with service. 

It should be clearly borne in mind, however, that the dis- 
charging capacity of a pipe may be considerably diminished after 
a few years' service. 

Darcy's results show that the loss of head in an old pipe may 
be double that in a new one, or since the velocity v is taken as 



124 HYDRAULICS 

proportional to the square root of h, the discharge of the old pipe 
for the same head will be j=. times that of the new pipe, or about 

30 per cent. less. 

An experiment by Sherman *on a 36-inch cast-iron main showed 
that after one year's service the discharge was diminished by 
23 per cent., but a second year's service did not make any further 
alteration. 

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 % on a cast-iron main, coated with tar, 
which had been in use for 16 years, showed that cleaning increased 
the discharge by nearly 40 per cent. Fitzgerald also found that 
the discharge of the Sudbury aqueduct diminished 10 per cent, in 
one year due to accumulation of slime. 

The experiments of Marx, Wing, and Hoskins on a 72-inch steel 
main, when new, and after two years' service, showed that there 
had been a change in the condition of the internal surface of the 
pipe, and that the discharge had diminished by 10 per cent, at low 
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 head at least 
from 10 to 30 per cent, more than the calculated value. 

97. Ganguillet and Kutter's formula. Bazin formula. 

Ganguillet and Kutter endeavoured to determine a form for 
the coefficient C in the Chezy formula v = C Jmi, 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, 



(10). 



Vra 

This formula is very cumbersome to use, and the value of the 
coefficient of roughness n for different cases is uncertain; tables 
and diagrams have however been prepared which considerably 
facilitate its use. A simpler form has been suggested for channels 
by Bazin (see page 185) which, by changing the constants, can be 
used for pipes ||. 

* Trans. Am.S.C.E. Vol. XLIV. p. 85. f Trans. Am.S.C.E. Vol. XLIV. p. 56. 
J Trans. Am.S.C.E. Vol. xuv. p. 87. See Table No. XIV. || Proc. Inst. C.E. 1919. 



FLOW THROUGH PIPES 125 

Values of n in Ganguillet and Kutter's formula. 
Wood pipes = "01, may be as high as '015. 

Cast-iron and steel pipes = '011, '02. 
Grlazed earthenware = '013. 

98. Reynolds' experiments and the logarithmic formula. 

The formulae for loss of head due to friction previously given 
have all been founded upon a probable law of variation of h 
with v, but no rational basis for the assumptions has been adduced. 

It has been stated in section 93, that on the assumption that h 
varies with -u 2 , the coefficient C in the formula 



is itself a function of the velocity. 

The experiments and deductions of Reynolds, and of later 
workers, throw considerable light upon this subject, and show that 
h is proportional to v n , where n is an index which for very small 
velocities* as previously shown by Poiseuille by experiments on 
capillary tubes is equal to unity, and for higher velocities may 
have a variable value, which in many cases approximates to 2. 

As Darcy's experiments marked a decided advance, in showing 
experimentally that the roughness of the wetted surface has an 
effect upon the loss due to friction, so Reynolds' work marked 
a further step in showing that the index n depends upon the state 
of the internal surface, being generally greater the rougher the 
surface. 

The student will be better able to follow Reynolds, by a brief 
consideration of one of his experiments. 

In Table XY are shown the results of an experiment made 
by Reynolds with apparatus as illustrated in Fig. 88. 

In columns 1 and 5 are shown the experimental values of 

i = j , and v respectively. 

The curves, Fig. 90, were obtained by plotting v as abscissae 
and i as ordinates. 

For velocities up to 1*347 feet per second, the points lie very close 
to a straight line and i is simply proportional to the velocity, or 

i = hv (11), 

&i being a coefficient for this particular pipe. 

Above 2 feet per second, the points lie very near to a continuous 
curve, the equation to which is 

i = Jcv n (12). 

Phil. Trans. 1883. 



126 HYDRAULICS 

Taking logarithms, 

log i = log k + n log v. 



Curve N?2 is the part Aft of 



Curve N?l drawn to laraer 
ScaleL * 




Velocity. 

Fig. 90. 



The curve, Fig. 90 a, was determined by plotting log i as 
ordinate and logv as abscissae. Eeynolds calls the lines of this 
figure the logarithmic homologues. 

Calling log i, #, and log v, x, the equation has the form 



which is an equation to a straight line, the inclination of which to 
the axis of x is 

= tan" 1 ^, 

or n = tan 0. 

Further, when x = 0, y = Jc, so that the value of Jc can readily be 
found as the ordinate of the line when x or log v = 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 equal to unity is 0*038, so that for the 
first part of the curve ~k = '038, and i = 'OoSv. 



FLOW THROUGH PIPES 



127 



Above the velocity of 2 feet per second the points lie about 
a second straight line, the inclination of which to the axis of v is 

= tan' 1 T70. 

Therefore log i = 1 '70 log v + Je. 

The ordinate when v equals 1 is 0*042, so that 

fc = 0-042, 
and t 



-3-0 
20 



-1-0 

-8 

-7 

6 

-5 



--Z 





vetoctty 



3 4 S 6 755/0 



Fig. 90 a. Logarithmic plottings of i and v to determine the index n in 
the formula for pipes, i = kv n . 

In the table are given values of i as determined experimentally 
and as calculated from the equation i = Jc . v n . 

The quantities in the two columns agree within 3 per confc. 



123 



HYDRAULICS 



TABLE XV. 

Experiment on Resistance in Pipes. 
Lead Pipe. Diameter 0'242". Water from Manchester Main. 



Slope 


. h 
~T 


k 


n 


Velocity ft. per 
second 


Experimental value 


Calculated from 
i=kv n 








0086 


0092 


038 


1 


209 


0172 


0172 


038 


1 


451 


0258 


0261 


038 


1 


690 


0345 


0347 


038 


1 


914 


0430 


0421 


038 


1 


1-109 


0516 


0512 


038 


1 


1-349 


0602 


. . 


, 


... 


1-482 


0682 


, t 







1-573 


0861 


. . 




, 


1-671 


1033 




, 




1-775 


1206 








1-857 


-1378 


1352 


042 


1-70 


1-987 


1714 


1610 


042 


1-70 


2-203 


3014 


2944 


042 


1-70 


3-141 


4306 


4207 


042 


1-70 


3-93 


8185 


8017 


042 


1-70 


5-66 


1-021 


1-033 


042 


1-70 


6-57 


1-433 


1-476 


042 


1-70 


8-11 


2-455 


2-404 


042 


1-70 


10-79 


3-274 


3-206 


042 


1-70 


12-79 


3-873 


3-899 


042 


1-70 


14-29 



NOTE. To make the columns shorter, only part of Keynolds' results are given. 

99. Critical velocity. 

It appears, from Reynolds' experiment, that up to a certain 
velocity, which is called the Critical Velocity, the loss of head is 
proportional to v, but above this velocity there is a definite change 
in the law connecting i and v. 

By experiments upon pipes of different diameters and the 
water at variable temperatures, Reynolds found that the critical 
velocity, which was taken as the point of intersection of the two 
straight lines, was 

0388P 



the value of P being 



(13), 



1+ 0*0336 T + -000221T 2 
T being the temperature in degrees centigrade and D the diameter 
of the pipe. 



FLOW THROUGH PIPES 129 

100. Critical velocity by the method of colour bands. 

The existence of the critical velocity has been beautifully 
shown by Reynolds, by the method of colour bands, and his 
experiments also explain why there is a sudden change in the law 
connecting i and v. 

"Water was drawn through tubes (Figs. 91 and 92), out of 
a large glass tank in which the tubes were immersed, and in 
which the water had been allowed to come to rest, arrangements 
being made as shown in the figure so that a streak or streaks of 
highly coloured water entered the tubes with the clear water." 



Fig. 91. 



Fig. 92. 

The results were as follows : 

" (1) When the velocities were sufficiently low, the streak 
of colour extended in a beautiful straight line through the tube " 
(Fig. 91). 

"(2) As the velocity was increased by small stages, at 
some point in the tube, always at a considerable distance from the 
trumpet-shaped intake, the colour band would all at once mix up 
with the surrounding water, and fill the rest of the tube with 
a mass of coloured water" (Fig. 92). 

This sudden change takes place at the critical velocity. 

That such a change takes place is also shown by the apparatus 
illustrated in Fig. 88; when the critical velocity is reached there is 
a violent disturbance of the mercury in the U tube. 

There is, therefore, a definite and sudden change in the con- 
dition of flow. For velocities below the critical velocity, the flow 
is parallel to the tubes, or is " Stream Line " flow, but after the 
critical velocity has been passed, the motion parallel to the tube is 
accompanied by eddy motions, which cause a definite change to 
take place in the law of resistance. 

Barnes and Coker* have determined the critical velocity by 
noting the sudden change of temperature of the water when its 
motion changes. They have also found that the critical velocity, 
as determined by noting the velocity at which stream-line flow 

* Proceedings of the Royal Society , Vol. LXXIV. 1904; Phil. Transactions, 
Eoyal Society, Vol. xx. pp. 4561. 

L. H. 9 



130 HYDRAULICS 

breaks up into eddies, is a nrncli more variable quantity than 
that determined from the points of intersection of the two lines 
as in Fig. 90. In the former case the critical velocity depends 
upon the condition of the water in the tank, and when it is 
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 Coker have called the critical velocity obtained by 
the method of colour bands the upper limit, and that obtained by 
the intersection of the logarithmic homologues the lower critical 
velocity. The first gives the velocity at which water flowing from 
rest in stream-line motion breaks up into eddy motion, while the 
second gives the velocity at which water that is initially disturbed 
persists in flowing with eddy motions throughout a long pipe, or 
in other words the velocity is too high to allow stream lines 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 Gr. H. 
Benzenberg* on the discharge through a sewer 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 
9 feet. As will be seen by reference to section 130, the velocities 
of translation of the particles on any cross section at any instant 
are very different, and if the motion were stream line the colour 
must have been spread out over a much greater length. 

101. Law of frictional resistance for velocities above the 
critical velocity. 

As seen from Reynolds' formula, the critical velocity except 
for very small pipes is so very low that it is only necessary in 
practical hydraulics to consider the law of frictional resistance for 
velocities above the critical velocity. 

For any particular pipe, 

i = Jcv n , 

and it remains to determine k and n. 

From the plottings of the results of his own and Darcy's 

* Transactions Am.S.C.E. 1893; and also Proceedings Am.S.C.E., Vol. xxvu. 
p. 1173. 



FLOW THROUGH PIPES 



131 



experiments, Reynolds found that the law of resistance " for all 
pipes and all velocities " could be expressed as 

AD 3 . /BD V 

~P rl = (~P~ v ) 

"DWT\W nM T)2 

Transposing, i ' ' 



AP".D 3 



(15), 



and 



K ~7 



A D 3 - n * 

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 unit length, 

A = 67,700,000, 
B - 396, 



or 



_ 
" 



L + -0036T + -000221T 3 ' 

B re . V n . P 2 ~ n y . v* 

67,700,000 D 3 - = D 33 " 



.(16), 



in which 



y 67,700,000' 
Values of y when the temperature is 10 C. 



n 


7 


1-75 


0-000265 


1-85 


0-000388 


1-95 


0-000587 


2-00 


0-000704 



The values for A and B, as given by Reynolds, are, however, 
only applicable to clean pipes, and later experiments show that 
although 



- DP > 

it is doubtful whether 

p = 3 n y 

as given by Eeynolds, is correct. 

Value of n. For smooth pipes n appears to be nearly 1*75. 
Reynolds found the mean value of n for lead pipes was T723. 

Saph and Schoder*, in an elaborate series of experiments 
carried out at Cornell University, have determined for smooth 

* Transactions of the American Society of Civil Engineer*, May, 1903. See 
exercise 31, page 172. 

92 



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 surface, 
such as cannot be seen by the unaided eye, make a considerable 
difference in the value of n, as is seen by reference to the values 
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 XII. 

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 



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 n 

and in the Chezy formula 

v = C *Jmi, 



or 



mC*' 

v* 
therefore p- 2 = kv n 

and 21ogC = 21ogv- (log ra + log fc + w log v) (17). 

The index n and the coefficient k were determined for a 
number of cast-iron pipes. 



FLOW THROUGH PIPES 133 

Yalues of C for velocities from 1 to 10 were calculated. Curves 
were then plotted, for different velocities, having C as ordinates 
and diameters as abscissae, and the values given in the table were 
deduced from the curves. 

The values of C so interpolated differ very considerably, in 
some cases, from the experimental values. The difficulties 
attending the accurate determination of i and v are very great, 
and the values of C, for any given pipe, as calculated by substi- 
tuting in the Chezy formula the losses of head in friction and the 
velocities as determined in the experiments, were frequently 
inconsistent with each other. 

As, for example, in the pipe of 3'22 ins. diameter given in 
Table XVI which was one of Darcy^s pipes, the variation of C as 
calculated from Ji and v given by Darcy is from 78'8 to 100. 

On plotting log/I and log-u and correcting the readings so 
that they all lie on one line and recalculating C the variation was 
found to be only from 95'9 to 101. 

Similar corrections have been made in other cases. 

The author thinks this procedure is justified by the fact that 
many of the best experiments do not show any such inconsistencies. 

An attempt to draw up an interpolated table for riveted pipes 
was not satisfactory. The author has therefore in Table XIV 
given the values of C as calculated by formula (17), for various 
velocities, and the diameters of the pipes actually experimented 
upon. If curves are plotted from the values of C given in 
Table XIV, it will be seen that, except for low velocities, the 
curves are not continuous, and, until further experimental evidence 
is forthcoming for riveted pipes, the engineer must be content 
with choosing values of C which most nearly coincide, as far as 
he can judge, with the case he is considering. 

103. Variation of k, in the formula i = kv n , with the 
diameter. 

It has been shown in section 98 how the value of &, for a 
given pipe, can be obtained by the logarithmic plotting of i and v. 

In Table XVI, are given values of &, as determined by the 
author, by plotting the results of different experiments. Saph 
and Schoder found that for smooth hard-drawn brass pipes 
of various sizes n varied between 1*73 and 1'77, the mean value 
being 1*75. 

By plotting logd as abscissae and log A; as ordinates, as in 
Fig. 93, for these brass pipes the points lie nearly in a straight line 
which has an inclination with the axis of d, such that 

tan = - 1*25 



184 



HYDRAULICS 



and the equation to the line is, therefore, 

log k = log y-p log d, 
where p = 1*25, 

and log y = log k 

when d \. 

From the figure 

y = G'000296 per foot length of pipe. 






-01 



Equation to line 
log.lo-Log y -WSLog d 

twvO 725 



02 -03 -04- OG 08 ho -2O -3 \5 -6 -8 WO* t ' < t, 

Logd, X 




0031 



Fig. 93. Logarithmic plottings of fc and d to determine the index p in the formula 



On the same figure are plotted logd and logfc, as deduced 
from experiments on lead and glass pipes by various workers. It 
will be seen that all the points lie very close to the same line. 

For smooth pipes, therefore, and for velocities above the 
critical velocity, the loss of head due to friction is given by 



the mean value for y being 0'000296, for n, T75, and for p T25. 

From which, v = 104i' 672 ^ 715 , 

or log v = 2-017 + 0-572 log t + 0715 log d. 



FLOW THROUGH PIPES 135 

The value of p in this formula agrees with that given by 
Reynolds in his formula 

. yv n 



Professor Unwin* in 1886, by an examination of experiments 
on cast-iron pipes, deduced the formula, for smooth cast-iron 
pipes, 



_ 



-,, , . . '0007?; 2 

and for rough pipes, i - , ri . 

M. Flamantt in 1892 examined carefully the experiments 
available on flow in pipes and proposed the formula, 



yy 175 



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 k as abscissae, it will be found, that the points all lie between 
two straight lines the equations to which are 

log k = log '00069 - 1'25 log d, 
and log k = log '00028 - 1'25 log d. 

Further, the points for any class of pipes not only lie between 
these two lines, but also lie about some line nearly parallel to 
these lines. So that p is not very different from 1'25. 
From the table, n is seen to vary from 1*70 to 2'08. 
A general formula is thus obtained, 

, -00028 to '00069^' torflg I 

d 1 * 

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 000418i? r84tor97 Z 

h= -~a~ 

If the pipes are lined with bitumen the smaller values of y and 
n may be taken. 

* Industries, 1886. 

f Ann-ales des Fonts et Chausstes, 1892, Vol. n. 



136 



HYDRAULICS 



For new, steel, riveted pipes, 

, -0004to'00054t; r93to2 - 08 Z 

h r^ ...... . ... 

Fig. 94 shows the result of plotting log k and log d 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*25, 
and the ordinate of which, when d is 1 foot, is '000364. 

Therefore h = -~^ or t> = 59i 518 cZ' 647 

very approximately expresses the law of resistance for particular 
pipes of wood, new cast iron, cleaned cast iron, and galvanised 
iron. 




Locjk. 



Logarithmic plottings of log h 

and log d from, Table 76, 

to detefyiine the indea> p ttttfie 

-, vihvn. IL Is Obouutl 93 



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. 



Experimenter 


Kind of pipe 


Diameter 
(in ins.) 


Velocity in 
ft. per sec. 
from to 


Value of n 
in formula 
i = kv n 


Value of k 
in formula 
i = kv n 


Noble 


Wood 


44 


3-46 4-415 


1-73 


0001254 








54 


2-28 4-68 


1-75 


000083 


Marx, Wing ) 





72-5 


1 4 


1-72 


000061 


and Hoskins } 


> 


72-5 


1 5-5 


1-93 


000048 


Galtner Kitcham 


Riveted 


3 




1-88 


00245 


H. Smith 


Wrought 


11 




1-81 


000515 


99 


iron or steel 


11| 




1-90 


000470 


99 


i) 


15 




1-94 


000270 


Kinchling 


tt 


38 


505 1-254 


2-0 


000099 


Herschel 


11 


42 


2-10 4-99 


1-93 


00011 


M 


5> 


48 


2 5 (?) 


2-0 


000090 


Marx, Wing ) 


J> 


72 


1 4 


1-99 


000055 


and Hoskins j 


H 


72 


1 5-5 


1-85 


000077 


Herschel 





103 


1 4-5 


2-08 


000036 


Darcy 


Cast iron 


3-22 


28910-71 


1-97 


00156 


H 


new 


5-39 


48 15-3 


1-97 


OOQ79 





n 


7-44 


67316-17 


1-956 


00062 


M 


H 


12 




1-779 


000323 


Williams 





16-25 




1-858 


000214 


Lampe 


> 


16-5 


2-48 3-09 


1-80 


000267 


99 





19-68 


1-38 3-7 


1-84 


00022 


Sherman 





36 


4 7 


2* 


000062 


Stearns 


H 


48 


1-243 3-23 


1-92 


0000567 


Hubbell&Fenkell 


> 


30 




2 


00003 


Darcy 


Cast iron 


1-4136 


167 2-077 


1-99 


0098 


i 


old and 


3-1296 


403 3-747 


1-94 


0035 


H 


tuberculated 


9-575 


1-00712-58 


1-98 


0009 


Sherman 


n 


20 


2-71 5-11 






11 





36 


1-1 4-5 


2 


000105 


Fitzgerald 


n 


48 


1-176 3-533 


2-04 


000083 


>> 


tt 


48 


1-135 3-412 


2-00 


OOC085 


Darcy 


Cast-iron 


1-4328 


371_ 3-69 


1-85 


0041 


11 


old pipes 


3-1536 


633 5-0 


1-97 


00185 


H 


cleaned 


11-68 


8 10-368 


2-0 


000375 


Fitzgerald 


11 


48 


3-67 5-6 


2-02 


000082 


H 


11 


48 


395 7-245 


1-94 


000059 


Darcy 


Sheet-iron 


1-055 


098 8-225 


1-73 


0074 


11 


11 


3-24 


32812-78 


1-81 


00154 


n 


11 


7-72 


59119-72 


1-78 


00059 


11 


11 


11-2 


1-29610-52 


1-81 


00039 





Gas 


48 


113 3-92 


1-83 


0278 





11 


1-55 


205 8-521 


1-86 


00418 


?> 


11 






1-91 


0072 


Saph and Schoder 


Galvanised 


364 




1-96 


0352 


M 


n 


494 




1-91 


0181 


I) 


11 


623 




1-86 


0132 


M 


11 


824 




1-80 


0095 


M 





1-048 




1-93 


0082 




Hard-drawn 


15 pipes 




* m rrK 


00025 to 





brass 


up to 1-84 




1 75 


00035 


Reynolds 


Lead 






1-732 




Darcy 


> 


55 




1-761 


0126 


" 





1-61 




1-783 


00425 



138 



HYDRAULICS 



TABLE XVII. 

Showing reasonable values of y, and n, for pipes of various 
kinds, in the formula, 

,_n 





Reasonable 




values for 




7 


n 


7 


n 


Clean cast-iron pipes 


00029 to -000418 


1-80 to 1-97 


00036 


1-93 


Old cast-iron pipes 


00047 to -00069 


1-94 to 2-04 


00060 


2 


Riveted pipes 


00040 to -00054 


1-93 to 2-08 


00050 


2 


Galvanised pipes 


00035 to -00045 


1-80 to 1-96 


00040 


1-88 


Sheet-iron pipes cover- 
ed with bitumen 


00030 to -00038 


1-76 to 1-81 


00034 


1-78 


Clean wood pipes 
Brass and lead pipes 


00056 to -00063 


1-72 to 1-75 


00060 
00030 


1-75 
1-75 



When further experiments have been performed on pipes, of 
which the state of the internal surfaces is accurately known, and 
special care taken to ensure that all the loss of head in a given 
length of pipe is due to friction only, more definiteness may be 
given to the values of y, n, and p. 

Until such evidence is forthcoming the simple Chezy formula 
may be used with almost as much confidence as the more 
complicated logarithmic formula, the values of C or / being taken 
from Tables XII XIV. Or the formula h = kv n 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. 

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 
plotted, from the records of experiments, that, although the results 
seemed consistent amongst themselves, yet compared with other 
experiments, they seemed of little value. 



FLOW THROUGH PIPES 



139 



The value of n for one of Couplet's* experiments on a lead and 
earthenware pipe being as low as 1*56, while the results of an 
experiment by Simpson t on a cast-iron pipe gave n as 2'5. In the 
latter case there were a number of bends in the pipe. 

In making experiments for loss of head due to friction, it is 
desirable that the pipe should be of uniform diameter and as 
straight as possible between the points at which the pressure head 
is measured. Further, special care should be taken to ensure the 
removal of all air, and it is most essential that a perfectly steady 
flow is established at the point where the pressure is taken. 

105. Piezometer fittings. 

It is of supreme importance that the 
piezometer connections shall be made 
so that the difference in the pressures 
registered at any two points shall be 
that lost by friction, and friction only, 
between the points. 

This necessitates that there shall 
be no obstructions to interfere with the 
free flow of the water, and it is, there- 
fore, very essential that all burrs shall 
be removed from the inside of the pipe. 

In experiments on small pipes in 
the laboratory the best results are no 
doubt obtained by cutting the pipe 
completely through at the connection 
as shown in Fig. 95, which illustrates 
the form of connection used by Dr 
Coker in the experiments cited on 
page 129. The two ends of the pipe are not more than 
of an inch apart. 

Fig. 96 shows the method adopted by Marx, Wing and Hoskins 
in their experiments on a 72-inch wooden pipe to ensure a correct 
reading of the pressure. 

The gauge X was connected to the top of the pipe only while 
Y was connected at four points as shown. 

Small differences were observed in the readings of the two 
gauges, which they thought were due to some accidental circum- 
stance affecting the gauge X only, as no change was observed 
in the reading of Y when the points of communication to Y were 
changed by means of the cocks. 

* Hydraulics, Hamilton Smith, Junr. 

t Proceedings of the Institute of Civil Engineers, 1855. 




Fig. 95. 



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 effect 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 2 ~" is a small quantity compared with 
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. n. 

t See also Barnes and Coker, Proceedings of the Royal Society, Vol. LXX. 1904 ; 
Coker and Clements, Transactions of the Royal Society, Vol. cci. Proceedings 
Am.S.C.E. Vol. xxix. 



FLOW THROUGH PIPES 141 

10 degrees F. With galvanised pipes the correction appears to 
be from 1 per cent, to 5 per cent, per 10 degrees F. 

Since the head lost increases, as the temperature falls, the 
discharge for any given head diminishes with the temperature, 
but for practical purposes the correction is generally negligible. 

107. *Loss 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 
in the straight part of the pipe, that it can be neglected, and 
consequently the experimental determination of this quantity has 
not received much attention. 

Weisbacht, from experiments on a pipe 1J inches diameter, 
with bends of various radii, expressed the loss of head as 

. *923r\ v* 

+ -- 



r being the radius of the pipe, R the radius of the bend on the 
centre line of the pipe and v the velocity of the water in feet per 
second. If the formula be written in the form 

7^ ^_ 

the table shows the values of a for different values of ^ , 

A 

r 

B 

1 -157 

2 -250 

5 -526 

St Tenant J has given as the loss of head & B at a bend, 
TIB = '00152 j~ y/ 1 v 2 =0'l^ g y^ nearly, 

Z being the length of the bend measured on the centre line of the 
bend and d the diameter of the pipe. 
When the bend is a right angle 

L /I = * /I 

RV R 2 V R* 
When | = 1, '5, '2, 



V R~ 



111, '702 



See page 525. t Mechanics of Engineering. 

$ Comptes Rendus, 1862. 



142 



HYD-RAULICS 



Recent experiments by Williams, Hubbell and Fenkell* on cast- 
iron pipes asphalted, by Saph and Schoder on brass pipes, and 
others by Alexander t on wooden pipes, show that the loss of head 
in bends, as in a straight pipe, can be expressed as 

n being a variable for different kinds of pipes, while 

..-;:'' . . *sr 

y being a constant coefficient for any pipe. 

For the cast-iron pipes of Hubbell and Fenkell, y, n, m, and p 
have approximately the following values. 



Diameter of pipe 


7 


m 


n 


P 


12" 


0040 


0-83 


1-78 


1-09 


16" 


i) 





1-86 





30" 


M 


N 


2-0 






When v is 3 feet per second and jr is i, the bend being a right 

angle, the loss of head as calculated by this formula for the 

-lo-i '2068u a , , ,, OA . , . *238v 2 
12-inch pipe is ~ - , and for the 30-inch pipe -^ . 

For the brass pipes of Saph and Schoder, 2 inches diameter, 
Alexander found, 



and for varnished wood pipes when =5- is less than 0'2, 

h* = "008268 (5) "to 1 ", 
and when ^ is between 0'2 and 0*5, 

A 



He further found for varnished wood pipes that, a bend of 
radius equal to 5 times the radius of the pipe gives the minimum 
loss of head, and that its resistance is equal to a straight pipe 3'38 
times the length of the bend. 

Messrs Williams, Hubbell and Fenkell also state at the end of 
their elaborate paper, that a bend having a radius equal to 2J 

* Proc. Amer. Soc. Civil Engineers, Vol. xxvn. f Proc. Inst. Civil Engineers, 
Vol. CLIX. See also Bulletin No. 576 University of Wisconsin. 



FLOW THROUGH PIPES 143 

diameters, offers less resistance to the flow of water than those of 
longer radius. It should not be overlooked, however, that although 
the loss of head in a bend of radius equal to * 2 diameters of the 
pipe is less than for any other, it does not follow that the loss of 
head per unit length of the pipe measured along its centre line 
has its minimum value for bends of this radius. 



108. Variations of the velocity at the cross section of a 
cylindrical pipe. 

Experiments show that when water flows through conduits of 
any form, the velocities are not the same at all points of any 
transverse section, but decrease from the centre towards the 
circumference. 

The first experiments to determine the law of the variation of 
the velocity in cylindrical pipes were those of Darcy, the pipes 
varying in diameter from 7'8 inches to 19 inches. A complete 
account of the experiments is to be found in his Recherches 
Experimentales dans les tuyaux. 

The velocity was measured by means of a Pitot tube at five 
points on a vertical diameter, and xx ^^^^^^^ 

the results plotted as shown in 
Fig. 97. 

Calling V the velocity at the 
centre of a pipe of radius R, u the 
velocity at the circumference, v m 
the mean velocity, v the velocity 
at any distance r from the centre, 
and i the loss of head per unit 
length of the pipe, Darcy deduced the formulae 



1-33 




and v m = 



When the unit is the metre the value of Jc is 11 '3, and 20*4 when 
the unit is the English foot. 

Later experiments commenced by Darcy and continued by 
Bazin, on the distribution of velocity in a semicircular channel, 
the surface of the water being maintained at the horizontal 
diameter, and in which it was assumed the conditions were similar 
to those in a cylindrical pipe, showed that the velocity near the 
surface of the pipe diminished much more rapidly than indicated 
by the formula of Darcy. 

* See Appendix 3. 



144 HYDRAULICS 

Bazin substituted therefore a new formula, 

........................ (1), 



or snce 



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 feet 
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 



(3), 
...... (4), 

Y - v = VRi SS^l-^/l- '95 () 2 } ............... (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 Williams, Hubbell and Fenkell*. 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 experiments 
on the flow through brass tubes 2 inches diameter, the total 

* " Memoire de 1' Academic des Sciences de Paris, Kecueil des Savants Etrangeres," 
Vol. xxxn. 1897. Proc. Am.S.C.E. Vol. xxvn. p. 1042. 

+ See page 241. 

J "Experiments at Detroit, Mich., on the effect of curvature on the flow of 
water in pipes," Proc. Am.S.C.E. Vol. xxvn. p. 313. 

See page 246. 



FLOW THROUGH PIPES 145 

discharge being determined by weighing, and the mean velocity 
thus determined. From the results of their experiments they 
came to the conclusion that the curve of velocities should be an 
ellipse to which the sides of the pipe are tangents, and that the 
velocity at the centre of the pipe Y is l'I9v m , v m being the mean 
velocity. 

These results are consistent with those of Bazin. His experi- 

y 
mental value for for the cement pipe was T1675, and if the 

^m 

constant *95, in formula (5), be made equal to 1, the velocity curve 
becomes an ellipse to which the walls of the pipe are tangents. 

The ratio can be determined from any of Bazin's formulae. 



Substituting ^p for >/E5 in (1), (3), (4) or (5), the value of 
v at radius r can be expressed by any one of them as 

'r 



C 

Then, since the flow past any section in unit time is v,?rR a , and 
that the flow is also equal to 

Zvrdr.v, 



f E f v2o /rM 
therefore v m 7rR 2 = 27r I JV P^VP )| r ^ r * 

/ v \ ^ftr 3 

Substituting for f (^ ) > ifa value -5-3- from equation (1), and 

\.Q// K> 

integrating, 

^i" 1 + "C~ W 

and by substitution of ft ^J from equation (4), 

= 1 + (8) 

V m C 



so that the ratio is not very different when deduced from the 
v m 

simple formula (2) or the more complicated formula (4). 
When C has the values 

= 80, 100, 120, 

from (8) = 1-287, 1'23, T19. 

V m 

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 
L. H. 10 



146 HYDRAULICS 

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 surface of a pipe. Assuming that the 
velocity curve is an ellipse to which 
the sides of the pipe are tangents, as MB 
in Fig. 98, and that Y=l'19v m , 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. 



v 



Let the equation to the ellipse be Fi S- 98 



in which x = 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 f*rB, a 6, and the volume of 
discharge per second will be 
/R 

irR 2 t? m = I Zirrdr . V = 7rR a . U + 



/ 

I 



and u = "621 -y m . 

Using Bazin's elliptical formula, the values of for 

= 80, 100, 120, 
are - = '552, '642, '702. 

Dm 

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 necessary to give the mean velocity v m to 
the water in the pipe. 

It is generally assumed that the head necessary to give a mean 

2 

velocity v m to the water flowing in a pipe is |p-, which would be 

correct if all the particles of water had a common velocity v m . 

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 Ib. of 

v * 
water flowing in the pipe, and this is slightly greater than -- . 



FLOW THROUGH PIPES 147 

As before, let v be the velocity at radius r. 
The kinetic energy of the quantity of water which flows past 
any section per second 

R -y2 

w.%Trrdr.v . ~-, 
o 20' 

w being the weight of 1 c. ft. of water. 
The kinetic energy per lb., therefore, 



i 



w . 2-n-rdrv 
o 



The simplest value for / ( ^ ) is that of Bazin's formula (1) 
above, from which 



21'5 



and 



Substituting these values and integrating, the kinetic energy 
per Ib. is , and when 

C is 80, 100, 
a is 112, T076. 

On the assumption that the velocity curve is an ellipse to which 
the walls 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 v m to the water in 

the pipe may therefore "be taken to be ^~ , the value of a being 

about 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 1*1346. 

110. 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 
probjems, it will be convenient to tabulate them. 
* Flamant's Hydraulique. 

102 



148 HYDRAULICS 

NOTATION. 
h = loss of head due to friction in a length I of a straight pipe. 

i = the virtual slope = j . 
I/ 

v = the mean velocity of flow in the pipe. 
d = the diameter. 
m = the hydraulic mean depth 

A Tp*p A fj 

= Wetted Perimeter = P = 4 when the pipe is c y lindrioal and ful1 

Formula 1. h ^ = 4M 

Cm C d 

This may be written y = , 



or v 

The values of C for cast-iron and steel pipes are shown in 
Tables XII and XIV. 

Formula 2. ^ = 2^5' 

^- in this formula being equal to ^ of formula (1). 

Values of /are shown in Table XIII. 

Either of these formulae can conveniently be used for 
calculating h, v, or d when /, and Z, and any two of three 
quantities h, v y and d t 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 



12<2/20. d 
r=19 Vl2JTI^ 

= 394 N/l2iVl^- 
For rough and dirty pipes 

1 \ k?l 



IZdJZg.d* 
or vssm ^-*Ja 

= 278 v /j2| ri x. 



FLOW THROUGH PIPES 149 

If d is the unknown, Darcy's formulae can only be used to solve 
for d by approximation. The coefficient 1 1 + T^JJ 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. 

, yi 

d" ' 



h . y.v" 

=t= 



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 Z, v, and d are known. 
If h t l f and d are known, by writing the formula as 

n log v - log h log I - log y + p log eZ, 
v can be found. 

If h, I, 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 

the water flowing along the pipe is about ~ , but it is generally 

v 9 
convenient and sufficiently accurate to take this head as 5-, 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 

\/jj^ 

pipe is about -g and is generally negligible. 

Formula 7. The loss of head due to a sudden enlargement in 
a pipe where the velocity changes from v l to t? a is ^ Vl ~ V2 ' . 

Formula 8. The loss of head at bends and elbows is a very 
variable quantity. It can be expressed as equal to in which 
a varies from a very small quantity to unity. 

Problem 1. The difference in level of the water in two reservoirs is h feet, 
Fig. 99, and they 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 cubic 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 aud the fractional resistances. 




Fig. 99. Pipe connecting two reservoirs. 

Let v be the mean velocity of the water. The head necessary to give the water 

l*12y 2 
this mean velocity may be taken as ~ , and to overcome the resistance at the 



~ 



entrances 



Then 



Or using in the expression for friction, the coefficient 0, 
A=-0174v 3 + -0078t; 2 +^ 



= -025t; 2 + 



C 2 d' 



If - is greater than 300 the head lost due to friction is generally great compared 
a 

w th the othjr quantities, and these may be neglected. 
4 fto 2 4lv' 2 

Then h== > 



and 



_ 

C /Jh 
2~VT* 



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 



0=394 



= 278 




for clean pipes, 
if the pipe is dirty, 



and 

can be taken. 

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=^d 2 .v. 

The velocity can be deduced directly from the logarithmic formula h=^^, 
provided y and n are known for the pipe. 



FLOW THROUGH PIPES 151 

The hydraulic gradient is EF. 

At any point C distant x from A the pressure head is equal to the distance 
between the centre of the pipe and the hydraulic gradient. The pressure head 
just inside the end A of the pipe is h -- - , and at the end B the pressure head 
must be equal to /IB- The head lost due to friction is h, which, neglecting the 
small quantity - , is equal to the difference of level of the water in the two 
tanks. 

Example 1. A pipe 3 inches diameter 200 ft. long connects two tanks, the 
difference of level of the water in which is 10 feet, and the pressure is atmospheric. 
Find the discharge assuming the pipe dirty. 



Using Darcy's coefficient 
V=278 



3l 
= 3-88 ft. per sec. 

For a pipe 3 inches diameter, and this velocity, C from the table is about 69, so 
that the approximation is sufficiently near. 

OOOGivi-w. I 
Taking h= - jf^ , 

v=3-88 ft. per sec., 



- # ' 
gives v = 3'85 ft. per sec. 

Example 2. A pipe 18 inches diameter brings water from a reservoir 100 feet 
above datum. The total length of the pipe is 15,000 feet and the last 5000 feet 
are at the datum level. For tbis 5000 feet the water is drawn off by service pipes at 
the uniform rate of 20 cubic feet per minute, per 500 feet length. Find the pressure 
at the end of the pipe. 

The total quantity of flow per minute is 
00x 
oOU 

Area of the pipe is 1'767 sq. feet. 
The velocity in the first 10,000 feet is, therefore, 
200 



The head lost due to friction in this length, is 

4./. 10,000.1-888* 
2p.l-5 --- 
In the last 5000 feet of the pipe the velocity varies uniformly. At a distance 

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 2 .a; 2 ds 
20.T5.5000 2 ' 
and the total loss by friction is 

4/. 1-888 2 /"MM 2 _4/. (l-888) a 5000 
~2<7.1-5.5000 2 Jo ' 20.1-5 ' 3 ' 

The total head lost due to friction in the whole pipe is, therefore, 



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. 

Bequired the diameter of a pipe of length I feet which will discharge Q cubic feet 
per second between the two reservoirs of tbe last problem. 
Let v be the mean velocity and d the diameter of the pipe. 



and 

Therefore, 

Squaring and transposing, 
I 
If I is long compared with d, 

and 




(1), 



0-040G.Q 2 d 
~~ 



JL=o /** 

7T ,. V 4Z ' 






Since v and d are unknown C is unknown, and a value for C must be pro- 
visionally assumed. 

Assume C is 100 for a new pipe and 80 for an old pipe, and solve equation (3) 
ford. 

From (1) find v, and from the tables find the value of G corresponding to the 
values of d and v thus determined. 

If C differs much from the assumed value, recalculate d and v using this second 
value of C, and from the tables find a third value for C. This will generally be 
found to be sufficiently near to the second value to make it unnecessary to calculate 
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. 

yv n l 
Logarithmic formula. If the formula 'h, -^ be used, d can be found direct, 

from 

p log d=n log u + log7 + log I - log h. 

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 C = 110. 

From equation (3) 

5 _ 2-55. 14 /528Q 

or $ log d- log 16 -63, 

therefore d = 3-08 feet. 

For a thirty-eight inch pipe Kuichling found C to be 113. 

The assumption that C is 110 is nearly correct and the diameter may be taken 
as 37 inches. 

Using the logarithmic formula 



FLOW THROUGH PIPES 



153 



and substituting for v the value 2- 



000450^ 

h /_\ 1-95 

( - ) d 5 ' 15 

from which 

5 -15 log d = log -000 45-1 -95 log 0-7854 + 1-95 log 14 + log 2640, 
and d = 3-07 feet. 

Short pipe. If the pipe is short so that the velocity head and the head lost at 
entrance are not negligible compared with the loss due to friction, the equation 

. -0406Q 2 d 6-5ZQ 3 



when a value is given to C, can be solved graphically by plotting two curves 



and 



040GQ 2 6-5ZQ 2 

~~~ ~ 



The point of intersection of the two curves will give the 
diameter d. 

It is however easier to solve by approximation in the 
following manner. 

Neglect the term in d and solve as for a long pipe. 

Choose a new value for C corresponding to this ap- 
proximate diameter, and the velocity corresponding to it, 
and then plot three points on the curve y = d 5 , choosing 
values of d which are nearly equal to the calculated value 
of d, and two points of the straight line 

0406Q 2 d 
2/i=- 





<5 

Fig. 100. 



The curve y = d 5 between the three points can easily * 
be drawn, as hi Fig. 100, and where the straight line cuts 
the curve, gives the required diameter. 

Example 4. One hundred and twenty cubic feet of water are to be taken 
per minute from a tank through a cast-iron pipe 100 feet long, having a square- 
edged entrance. The total head is 10 feet. Find the diameter of the pipe. 

Neglecting the term in d and assuming C to be 100, 



and 



Therefore 



100.100.10 
d= -4819 feet. 
2 



v=- 



10-9 ft. per seo. 



From Table XII, the value of C is seen to be about 106 for these values of 
d and v. 

A second value for d 6 is 



from which d= '476'. 

The schedule shows the values of d 5 and y for values of d not very different 
from the calculated value, and taking C as 106. 

d -4 -5 -6 

d 6 -01024 -03125 -0776 

y l -0297 -0329 

The line and curve plotted in Fig. 100, from this schedule, intersect atp for which 

d= -4*98 feet. 



154 HYDRAULICS 

It is seen therefore that taking 106 as the value of C, neglecting the term in d, 
makes an error of -022' or -264". 

This problem shows that when the ratio -r is about 200, and the virtual slope is 
even as great as j^, for all practical purposes, the friction head only need be con- 
sidered. For smaller values of the ratio the quantity '0250 2 may become im- 
portant, but to what extent will depend upon the slope of the hydraulic gradient. 

The logarithmic formula may be used for short pipes but it is a little more 
cumbersome. 

Using the logarithmic formula to express the loss of head for short pipes with 
square-edged entrance, 

-*+ 




When suitable values are given to 7 and n, this can be solved by plotting the 
two curves 



and 

* 



U) 



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 Z a the pipe i 

of diameter d is divided into two branches j | 

laid side by side having diameters d-, and rf. t< Z^ --- >K- --- L --- H 
Fig- 101. 4> * j 

Assume all the head is lost in friction. A _ *< ^s/^ *%* ' 

Let Qj be the discharge in cubic feet. j y ( 

Then, since both the branches BC and BD * -- >v> - -* - , -~ 

are connected at B and to the same reservoir, j x> -- ^ - 1 1) 

the head lost in friction must be the same in I , 

BC as in BD, and if there were any number ; ~ 
of branches connected at B the head lost in -pj g 101 

them all would be the same. 

The case is analogous to that of a conductor joining two points between which 
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 v be the velocity in AB, v 1 in BC and v z in BD. 

Then vd^^v^ + v^ .................................... (1), 

and the difference of level between the reservoirs 



h=?2L + ^ l (2). 

And since the head lost in BC is the same as in BD, therefore, 



f (3). 

'2 



If provisionally Gj be taken as equal to C 2 , 



FLOW THROUGH PIPES 



155 



Therefore, 



and 



v.d? 



.(4). 



From (2), v can be found by substituting for Vj from (4), and thus Q can 
be determined. 

If AB, BC, and CD are of the same diameter and ^ is equal to ,, then 

and h 



Problem 4. Pipes connecting three reservoirs. As in Fig. 102, let three pipes 
AB, BC, and BD, connect three reservoirs A, C, D, the level of the water in each 
of which remains constant. 

Let t>j, v 2 , and t> 8 be the velocities in AB, BC, and BD respectively, Q 
and Q 3 the quantities flowing along these pipes in cubic feet per sec., l lt J a , and' 
the lengths of the pipes, and d^ , d a aud d s their diameters. 




Fig. 105!. 

Let t , 2 > and z 3 be the heights of the surfaces of the water in the reservoirs, 
and z the height of the junction B above some datum. 
Let h Q be the pressure head at B. 

Assume all losses, other than those due to friction in the pipes, to be negligible. 
The head lost due to friction for the pipe AB is 

(1), 



(2), 



and for the pipe BC, 

the upper or lower signs being taken, according as to whether the flow is from, or 
towards, the reservoir C. 

For the pipe BD the head lost is 



(3). 



Since the flow from A and C must equal the flow into D, or else the flow 
from A must equal the quantity entering C and D, therefore, 

Q 1 iQ 2 =Q 8 , 

or t^AtvV-tyV .................................... (4). 

There are four equations, from which four unknowns may be found, if it is 
further known which sign to take in equations (2) and (4). There are two cases to 
consider. 



156 HYDRAULICS 

Case (a). Given the levels of the surfaces of the water in the reservoirs and 
of the junction B, and the lengths and diameters of the pipes, to find the quantity 
flowing along each of the pipes. 

To solve this problem, it is first necessary to obtain by trial, whether water flows 
to, or from, the reservoir C. 

First assume there is no flow along the pipe BC, that is, the pressure head /? at 
B is equal to z z - z . 

Then from (1), substituting for v l its value 







from which an approximate value for Q x can be found. By solving (3) in the same 
way, an approximate value for Q 3 , is, 



(6). 



If Q 3 is found to be equal to Q a , the problem is solved ; but if Q 3 is greater than 
Qu the assumed value for h 9 is too large, and if less, h is too small, for a diminu- 
tion in the pressure head at B will clearly diminish Q 3 and increase Qj, and will 
also cause flow to take place from the reservoir C along CB. Increasing the 
pressure head at B will decrease Q 1} increase Q 3 , and cause flow from B to C. 

This preliminary trial will settle the question of sign in equations (2) and (4) 
and the four equations may be solved for the four unknowns, v lt v%, v 9 and h . It 
is better, however, to proceed by "trial and error." 

The first trial shows whether it is necessary to increase or diminish h 9 and new 
values are, therefore, given to h until the calculated values of v lt v 2 and v, satisfy 
equation (4). 

Case (b). Given Qj, Q 2 , Q 3 , and the levels of the surfaces of the water in 
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, therefore, 
only three equations are given from which to calculate the four unknowns d lt 
dg, d 3 and h . For a definite solution a fourth equation must consequently be 
found, from some other condition. The further condition that may be taken is 
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 and 
diameter, and if, therefore, Iid l + l 2 d 2 + l s d s is made a minimum, the cost of the 
pipes will be as small as possible. 

Differentiating, with respect to k Q , the condition for a minimum is, that 



Substituting in (1), (2) and (3) the values for v lt v a and v a , 



differentiating and substituting in (7) t . 



FLOW THROUGH PIPES 157 

Putting the values of Q a , Q 2 , and Q 3 in (1), (2), (3), and (8), there are four 
equations as before for four unknown quantities. 

It will be better however to solve by approximation. 

Give some arbitrary value to say d. 2 , and calculate /? from equation (2). 

Then calculate d\ and d a by putting h n in (1) and (3), and substitute in 
equation (8). 

If this equation is satisfied the problem is solved, but if not, assume a second 
value for d a and try again, and so on until such values of d l1 d! 2 , d s are obtained 
that (8) is satisfied. 

In this, as in simpler systems, the pressure at any point in the pipes ought not 
to fall below the atmospheric pressure. 

Flow through a pipe of constant diameter when the flow is diminishing at a 
uniform 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 
by friction. 

Let Q be the number of cubic feet per second that enters the pipe at a section A, 
and Q! the number of cubic feet that passes the section B, I feet from A, the 
quantity Q - Q x being taken from the pipe, by branches, at a uniform rate of 

Q~Q* cubic feet per foot. 

Then, if the pipe is assumed to be continued on, it is seen from Fig. 103, that 
if the rate of discharge per foot length of the 
pipe is kept constant, the ^ whole of Q will be 
discharged in a length of pipe, 



L= 



'(Q-Qi)' 

The discharge past any section, x feet from 
C, will be 

*~ L ~ l *' Fig. 103. 

The velocity at the section is 




Assuming that in an element of length dx the loss of head due to friction is 
and substituting for v x its value 

Q* 



the loss of head due to friction in the length I is 

x n dx 



t _[ L 7 /iQ\ 
"/M 7 VSBPj 



/i 
_ _y_ 

n + 
If Qi is zero, I is equal to L, and 



The result is simplified by taking for 9& the value 



and assuming C constant. 
Then 




158 HYDRAULICS 

Problem 5. *Pumping water tJirough long pipes. Kequired the diameter of a 
long pipe to deliver a given quantity of water, against a given effective head, in 
order that the charges on capital outlay and working expenses shall be a minimum. 

Let I be the length of the pipe, d its diameter, and h feet the head against which 
Q cubic feet of water per second is to be pumped. 

Let the cost per horse-power of the pumping plant and its accommodation 
be N, and the cost of a pipe of unit diameter n per foot length. 

Let the cost of generating power be m per cent, of the capital outlay in the 
pumping station, and the interest, depreciation, and cost of upkeep of the pumping 
plant, taken together, be r per cent, of the capital outlay, and that of the pipe line 
?*! per cent. ; r^ will be less than r. The horse-power required to lift the water 
against a head h and to overcome the frictional resistance of the pipe is 

60. Q. 62-4 ( 4t;*J 
Hr - 33,000 < h +^ 



Let e be the ratio of the average effective horse-power to the total horse-povrer, 
including the stand-by plant. The total horse-power of the plant is then 
T 0-1186Q 



The cost of the pumping plant is N times this quantity. 
The total cost per year, P, of the station, is 
m+r N.Q/ 



Assuming that the cost of the pipe line is proportional to the diameter and to 
the length, the capital outlay for the pipe is, nld, and the cost of upkeep and 

.. . . 
interest is 




is to be a minimum. 

Differentiating with respect to d and equating to zero, 



, 

That is, d is independent of the length I and the head against which the water 
is pumped. 



Taking C as 80, e as 0-6 and v "*" ; as 50, then 

WTj 



If ( m+r ) N is 100, 



3-68 x 50 
80 x 80 x -6 
= 0-603 VOT 



d= -675^/01 



Mr, 

^ 

Problem 6. Pipe with a nozzle at the end. Suppose a pipe of length I and 
diameter D has at one end a nozzle of diameter d, through which water is dis- 
charged from a reservoir, the level of the water in which is h feet above the centre 
of the nozzle. 

Required the diameter of the nozzle so that the kinetic energy of the jet is 
a maximum, 

* See also example 61, page 177. 



FLOW THROUGH PIPES 159 

Let V be the velocity of the water in the pipe. 

Then, since there is continuity of flow, v the velocity with which the water 

V.D 2 

leaves the nozzle is ^ . 
a* 

The head lost by friction in the pipe is 

4/ V 2 l 4/r 2 Z . d* 
2g.D~ 2#D 5 * 

2 

The kinetic energy of the jet per Ib. of flow as it leaves the nozzle is - . 



Therefore *~* .............................. -f-* 

from which by transposing and taking the square root, 

/ frD.fc \i 







The weight of water which flows per second =j d 2 . v . w where M> = the weight of 

a cubic foot of water. 

Therefore, the kinetic energy of the jet, is 



This is a maximum when -rr=6. 
M 

Therefore 



4 



~ 4 * * 2 5* 



from which D 5 + 4/Zd 4 = 12/Zd 4 , 

and D 



or 



If the nozzle is not circular but has an area a, then since in the circular nozzle 
of the same area 

jd2=a, 

v u i 16a2 

from which d*= p. 

Therefor, D'=i^, 



and 



J- 

V /t 



By substituting the value of D 8 from (5) in (1) it is at once seen that, for 
mazimum kinetic energy, the head lost in friction is 



Problem 7. Taking the same data as in problem 6, to find the area of the 
nozzle that the momentum of the issuing jet is a maximum. 

The momentum of the quantity of water Q which flows per second, as it leaves 

the nozzle, is W ' ^ V Ibs. feet. The momentum M is, therefore, 
9 



Substituting for * from equation (1), problem 6, 



160 HYDRAULICS 

Differentiating, and equating to zero, 

/ 
4 /D5 



If the nozzle has an area a, D 5 = - 



and 




a = -392 



Substituting for D 5 in equation (1) it is seen that when the momentum is a 
maximum half the head h is lost in friction. 

Problem 6 has an important application, in determining the ratio of the size 
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 plane 
perpendicular to the jet should be a maximum. 

Problem 8. Loss of head due to friction in a pipe, the diameter of which varies 
uniformly. Let the pipe be of length I and its diameter vary uniformly from d 
to d l . 

Suppose the sides of the pipe produced until they meet in P, Fig. 104. 



The diameter of the pipe at any distance x from the small end is 



The loss of head in a small element of length dx is - 8 , v being the velocity 
when the diameter is d. 







Fig. 104. 

If Q is the flow in cubic ft. per second 

Q 4 Q 



The total loss of head h in a length I is 
64Q 2 . dx 

64 . 



= /"* 



16Q 2 . S 5 



/ J_ __ 1_ \ 

\S 4 (8 + /) 4 / 



Substituting the value of S from equation (1) the loss of head due to friction 
can be determined. 

Problem 9. Pipe line consisting of a number of pipes of different diameters. In 
practice only short conical pipes are used, as for instance in the limbs of a Venturi 
meter. 

If it is desirable to diminish the diameter of a long pipe line, instead of using 
a pipe the diameter of which varies uniformly with the length, the line is made up 
of a number of parallel pipes of different diameters and lengths. 



FLOW THROUGH PIPES 



161 



Let 7 lf Z 2 , 1 3 ... be the lengths and d lt d z ,d 3 ... the diameters respectively, of 
the sections of the pipe. 

The total loss of head due to friction, if C be assumed constant, is 



*i**a+*-). 

The diameter d of the pipe, which, for the same total length, would give the 
same discharge for the same loss of head due to friction, can be found from ^he 
equation 

The length L of a pipe, of constant diameter D, which will give the same 
discharge for the same loss of head by friction, is 



Problem 10. Pipe acting as a siphon. It is sometimes necessary to take a 
pipe line over some obstruction, such as a hill, which necessitates the pipe rising, 
not only above the hydraulic gradient as in Fig. 87, but even above the original 
level of the water in the reservoir from which the supply is derived. 

Let it be supposed, as in Fig. 105, that water is to be delivered from the reservoir 
B to the reservoir C through the pipe BAG, which at the point A rises fy feet above 
the level of the surface of the water in the upper reservoir. 




Fig. 105. 

Let the difference in level of the surfaces of the water in the reservoirs 
be fc 2 feet. 

Let h a be the pressure head equivalent to the atmospheric pressure. 

To start the flow in the pipe, it will be necessary to fill it by a pump or other 
artificial means. 

Let it be assumed that the flow is allowed to take place and is regulated so that 
it is continuous, and the velocity v is as large as possible. 

Then neglecting the velocity head and resistances other than that due to friction, 

4/v 8 L /Zgdh* 

*- V or "=v in? 

L and d being the length and diameter of the pipe respectively. 

The hydraulic gradient is practically the straight line DE. 

Theoretically if AF is made greater than h at which is about 34 feet, the pressure 
at A becomes negative and the flow will cease. 

Practically AF cannot be made much greater than 25 feet. 

To find the maximum velocity possible in the rising limb AB, so that the pressure 
head at A shall just be zero. 

Let v m be this velocity. Let the datum level be the surface of the water in C. 



L. II, 



11 



162 HYDRAULICS 



Then 
But 

Therefore 



If the pressure head is not to be less than 10 feet of water, 



If v m is less than v, the discharge of the siphon will he determined by this 
limiting velocity, and it will be necessary to throttle the pipe at C by means of a 
valve, so as to keep the limb AC full and to keep the " siphon " from being broken. 

In designing such a siphon it is, therefore, necessary to determine whether the 
flow through the pipe as a whole under a head h 2 is greater, or less than, the flow 
in the rising limb under a head h a h^. 

If AB is short, or h^ so small that v m is greater than v, the head absorbed by 
friction in AB will be 



2nd ' 

If the end C of the pipe is open to the atmosphere instead of being connected to 
a reservoir, the total head available will be h s instead of 7^. 

111. Velocity of flow in pipes. 

The mean velocity of flow in pipes is generally about 3 feet 
per second, but in pipes supplying water to hydraulic machines it 
may be as high as 10 feet per second, and in short pipes much 
higher velocities are allowed. If the velocity is high, the loss of 
head due to friction in long pipes becomes excessive, and the risk 
of broken pipes and valves through attempts to rapidly check 
the flow, by the sudden closing of valves, or other causes, is 
considerably increased. On the other hand, if the velocity is too 
small, unless the water is -very free from suspended matter, 
sediment* tends to collect at the lower parts of the pipe, and 
further, at low velocities it is probable that fresh water sponges 
and polyzoa will make their abode on the surface of the pipe, and 
thus diminish its carrying capacity. 

112. Transmission of power along pipes by hydraulic 
pressure. 

Power can be transmitted hydraulically through a considerable 
distance, with very great efficiency, as at high pressures the per 
centage loss due to friction is small. 

Let water be delivered into a pipe of diameter d feet under a 
head of H feet, or pressure of p Ibs. per sq. foot, for which the 

equivalent head is H = - feet. 

* An interesting example of this is quoted on p. 82 Trans. Am.S.C.E. 

Vol. XLIV. 



FLOW THROUGH PIPES 163 

Let the velocity of flow be v feet per second, and the length of 
the pipe L feet. 

The head lost due to friction is 



2g.d ................... > 

and the energy per pound available at the end of the pipe is, 
therefore, 



Mr 

The efficiency is 



The fraction of the given energy lost is 

h 

m = H- 

For a given pipe the efficiency increases as the velocity 
diminishes. 

If / and L are supposed to remain constant, the efficiency is 

v* 
constant if -TFT is constant, and since v is generally fixed from 

other conditions it may be supposed constant, and the efficiency 
then increases as the product dH. increases. 

If W is the weight of water per second passing through the 
pipe, the work put into the pipe is W . H foot Ibs. per second, the 
available work per second at the end of the pipe is W (H - Ji) t and 
the horse-power transmitted is 

W.(H-fr) WH n , 
~" = 550" (1 " m) ' 



Since 

Tj 
*the horse-power = -- (H - 



From (1) mH 

therefore, v = 4*01 i\J ~j^ > 

and the horse-power = 0'357 J j^ dK* (1 - m). 
- * See example 60, page 177. 



164 HYDRAULICS 

If p is the pressure per sq. inch 



and the horse-power = 1'24 ^ -^ <2*p* (1 - ra). 

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 d 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 37*5^^ Ibs., approximately. 

Assuming the thickness of the pipe to be proportional to the 
pressure, i.e. to the head H, 

= fcp=&H, 

and the weight per foot may therefore be written 

w - fad . H. 

The initial cost of the pipe per foot will then be 

C=feJWH = K.d.H, 

and since the cost of laying is approximately proportional to d, 
the total cost per foot is 

p=K.d.n+K l d. 

And since the horse-power transmitted is 

HP = '357 ^/^ <#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 d*H* (1 - m) 



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 Ibs. per sq. inch, but for special purposes, it is 
sometimes as high as 3000 Ibs. per sq. inch. These high pressures 
are, however, frequently obtained by using an intensifier (Ch. XI) 
to raise the ordinary pressure of 700 Ibs. to the pressure required. 
* See example 61, page 177. 



FLOW THROUGH PIPES 165 

The demand for hydraulic power for the working of lifts, etc. 
has led to the laying down of a network of mains in several of the 
large cities of Great Britain. In London a mean velocity of 4 feet 
per second is allowed in the mains and the pressure is 750 Ibs. 
per sq. inch. In later installations, pressures of 1100 Ibs. per 
sq. 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 pressure is high. 

If p is the safe internal pressure per sq. inch, and s the safe 
stress per sq. inch of the metal, and TI and r a the internal and 
external radii of the pipe, 



p= 



For a pressure p = 1000 Ibs. per sq. inch, and a stress s of 
3000 Ibs. per sq. inch, r a is 5'65 inches when n is 4 inches, or the 
pipe requires to be 1'65 inches thick. 

If, therefore, the internal diameter is greater than 8 inches, the 
pipe becomes very thick indeed. 

The largest cast-iron pipe used for this pressure is between 
7" and 8" internal diameter. 

Using a maximum velocity of 5 feet per second, and a pipe 
7 J inches diameter, the maximum horse-power, neglecting friction, 
that can be transmitted at 1000 Ibs. per sq. inch by one pipe is 
4418x1000x5 

-55Q- 

= 400. 

The following example shows that, if the pipe is 13,300 feet 
long, 15 per cent, of the power is lost and the maximum power 
that can be transmitted with this length of pipe is, therefore, 
320 horse-power. 

Steel mains are much more suitable for high pressures, as the 
working stress may be as high as 7 tons per sq. inch. The greater 
plasticity of the metal enables them to resist shock more readily 
than cast-iron pipes and slightly higher velocities can be used. 

A pipe 15 inches diameter and \ inch thick in which the 
pressure is 1000 Ibs. per sq. inch, and the velocity 5 ft. per second, 
is able to transmit 1600 horse-power. 

Example. Power is transmitted along a cast-iron main 7$ inches diameter at 
a pressure of 1000 Ibs. per sq. inch. The velocity of the water is 5 feet per second. 

Find the maximum distance the power can be transmitted so that the efficiency 
is not less thanS5/ . 

* Swing's Strength of Materials. 



166 HYDRAULICS 

d = 0-625feet, 



therefore h= 0-15x2300 

= 345 feet. 
Then 34y= 4x 0-0104 x 25 .j. 

2g x 0-625 
345 x 64-4 x 0-625 



from which L = 



0-0104x100 
13,300 feet. 




114. Pressures on pipe bends. 

If a bent pipe contain a fluid at rest, the intensity of pressure 
being the same in all directions, 
the resultant force tending to move 
the pipe in any direction will be 
the pressure per unit area multiplied 
by the projected area of the pipe 
on a plane perpendicular to that 
direction. 

If one end of a right-angled 
elbow, as in Pig 106 be bolted to 
a pipe full of water at a pressure p 

pounds per sq. inch by gauge, and on the other end of the elbow 
is bolted a flat cover, the tension in the bolts at A will be the 
same as in the bolts at B. The pressure on the cover B is clearly 
'7854pd 2 , d being the diameter of the pipe in inches. If the elbow 
be projected on to a vertical plane the projection of ACB is daefc, 
the projection of DEF is dbcfe. The resultant pressure on the 
elbow in the direction of the arrow is, therefore, p . abed = *7854pd 2 . 

If the cover B is removed, and water flows through the pipe 
with a velocity v feet per second, the horizontal momentum of the 
water is destroyed and there is an additional force in the direction 
of the arrow equal to '7854wcV/144<7. 

When flow is taking place the vertical force tending to lift the 
elbow or to shear the bolts at A is A 



\ v\ 
If the elbow is less than a right \\ 

angle, as in Fig. 108, the total \VLl'' 

tension in the bolts at A is ^ "*" 



T = p (daehgc - aefgc) + '^* " (1 - cos 0), 
and since the area aehgcb is common to the two projected areas, 



FLOW THROUGH PIPES 



167 



Consider now a pipe bent as shown in Fig. 109, the limbs AA 
and FF being parallel, and the water being supposed at rest. 

The total force acting in the direction AA is 

P = p {dcghea - aefgcb + d'cg'tie'd - a'ef'g'c'b'}, 
which clearly is equal to 0. 




If now instead of the fluid being at rest it has a uniform 
velocity, the pressure must remain constant, and since there is no 
change of velocity there is no change of momentum, and the re- 
sultant pressure in the direction parallel to AA is still zero. 

There is however a couple acting upon the bend tending to 
rotate it in a clockwise direction. 

Let p and q be the centres of gravity of the two areas daehgc 
and aefgcb respectively, and m and n the centres of gravity of 
d'de'h'g'c and aefgcb'. 

Through these points there are parallel forces acting as shown 
by the arrows, and the couple is 

M = P' . mn P . pq. 

The couple P . pq is also equal to the pressure on the semicircle 
adc multiplied by the distance between the centres of gravity of 
adc and efg, and the couple P' . mn is equal to the pressure on a'd'c 
multiplied by the distance between the centres of gravity of a'd'c 
and efg. 

Then the resultant couple is the pressure on the semicircle efg 
multiplied by the distance between the centres of gravity of efg 
and efg. 

If the axes of FF and AA are on the same straight line the 
couple, as well as the force, becomes zero. 

It can also be shown, by similar reasoning, that, as long as the 
diameters at F and A are equal, the velocities at these sections 
being therefore equal, and the two ends A and F are in the same 
straight line, the force and the couple are both zero, whatever the 
form of the pipe. If, therefore, as stated by Mr Froude, "the 



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 flowing water, with 
its plane perpendicular to the direction of flow, 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 p. a & 

the plate will be contracted, and the 
section of the stream on a plane gd will 
be c(A-a), c being some coefficient of 
contraction. 

It has been shown on page 52 that 
for a sharp-edged orifice the coefficient 
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 coefficient of contraction is larger. 

If a coefficient 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 




Let the pressures at the sections a&, gd, ef be p, pi and p 2 pounds 
per square foot respectively. 

Bernoulli's equations for the three sections are then, 



to 20 w 2g 

and ^ + -S = ^ + +( \/ ) (2) - 

Adding (1) and (2) 

(P ffA (Yi-V) a t 
Vw w 2gr 

The whole pressure on the plate in the direction of motion is then 

2# 
V 2 / A ^ 



FLOW THROUGH PIPES 



169 



If a = J A, 



Y 2 

P = 4iiva 7p nearly. 

0-46. a. Y 2 



116. Pressure on a cylinder. 

If instead of a thin plate a cylinder be placed in the pipe, 
with its axis coincident with the axis of the pipe, Fig. Ill, there 
are two enlargements of the section of the water. 

As the stream passes the up-stream edge of the cylinder, it 
contracts to the section at cd, and then enlarges to the section 
ef. It again enlarges at the down-stream end of the cylinder 
from the section ef to the section gh. 

<JU 













<7 


~ 


zz^z^^ 




E~cuz--z. ~-r^r--. 




r_ 


>-=- 










===r * 






?~ 









h, 



Fig. 111. 



Let 0i, 2 , 03, 04 be the velocities at a&, cd, ef and 
spectively, v 4 and 0! being equal. 

Between cd and e/ there is a loss of head 

(02 - 3 ) 2 
2<7 ' 
and between e/ and gh there is a loss of 

fo-0i) a 
*] 
The Bernouilli's equations for the sections are 



re- 



w w 



Adding (2) and (3), 



P? + ^L = ^ + !!L 
w 2g w 2g 



.ax 

.(2), 
(3). 



LZB = 

10 



fa - 



170 HYDRAULICS 

If the coefficient of contraction at cd is c, the area at cd 

A-a 



c 



A 



m-i Vi.A. Vi A 

Then V 2 = 7-1 r and v s = 



/A \ BMJWA "3 A 

c . (A a) A-a 

Therefore 

v wvS |Y a \ 2 / A A \ 2 ) 

(pi - p 4 ) = -g I ( v ^i^ J + ( (^ _ a ) ~ (A - a)/ J ' 

and the pressure on the cylinder is 



=i-t .a. 



EXAMPLES. 

(1) A new cast-iron pipe is 2000 ft. long and 6 ins. diameter. It is to 
discharge 50 c. ft. of water per minute. Find the loss of head in friction 
and the virtual slope. 

(2) What is the head lost per mile in a pipe 2 ft. diameter, discharging 
6,000,000 gallons in 24 hours ? /= -007. 

(3) A pipe is to supply 40,000 gallons in 24 hours. Head of water 
above point of discharge =36 ft. Length of pipe = 2^ miles. Find its 
diameter. Take C from Table XII. 

(4) A pipe is 12 ins. in diameter and 3 miles in length. It connects 
two reservoirs with a difference of level of 20 ft. Find the discharge per 
minute in c. ft. Use Darcy's coefficient for corroded pipes. 

(5) A water main has a virtual slope of 1 in 900 and discharges 35 c. ft. 
per second. Find the diameter of the main. Coefficient / is 0'007. 

(6) A pipe 12 ins. diameter is suddenly enlarged to 18 ins., and then to 
24 ins. diameter. Each section of pipe is 100 feet long. Find the loss of 
head in friction in each length, and the loss due to shock at each enlarge- 
ment. The discharge is 10 c. ft. per second, and the coefficient of friction 
/= -0106. Draw, to scale, the hydraulic gradient of the pipe. 

(7) Find an expression for the relative discharge of a square, and a 
circular pipe of the same section and slope. 

(8) A pipe is 6 ins. diameter, and is laid for a quarter mile at a slope 
of 1 in 50; for another quarter mile at a slope of 1 in 100; and for a third 
quarter mile is level. The level of the water is 20 ft. above the inlet end, 
and 9 ft. above the outlet end. Find the discharge (neglecting all losses 
except skin friction) and draw the hydraulic gradient. Mark the pressure 
in the pipe at each quarter mile. 

(9) A pipe 2000 ft. long discharges Q c. ft. per second. Find by how 
much the discharge would be increased if to the last 1000 ft. a second pipe 
of the same size were laid alongside the first and the water allowed to flow 
equally well along either pipe. 



FLOW THROUGH PIPES 171 

(10) A reservoir, the level of which is 50 ft. above datum, discharges 
into a second reservoir 30 ft. above datum, through a 12 in. pipe, 5000 ft. 
in length ; find the discharge. Also, taking the levels of the pipe at the 
upper reservoir, and at each successive 1000 ft., to be 40, 25, 12, 12, 10, 15, 
fb. above datum, write down the pressure at each of these points, and 
sketch the position of the line of hydraulic gradient. 

(11) It is required to draw off the water of a reservoir through a 
pipe placed horizontally. Diameter of pipe 6 ins. Length 40 ft. Ef- 
fective head 20 ft. Find the discharge per second. 

(12) Given the data of Ex. 11 find the discharge, taking into account 
the loss of head if the pipe is not bell-mouthed at either end. 

(18) A pipe 4 ins. diameter and 100 ft. long discharges \ c. ft. per 
second. Find the head expended in giving velocity of entry, in overcoming 
mouthpiece resistance, and in friction. 

(14) Kequired the diameter of a pipe having a fall of 10 ft. per mile, 
and capable of delivering water at a velocity of 3 ft. per second when dirty. 

(15) Taking the coefficient / as O'Ol (l + ^Y find how much water 

would be discharged through a 12-inch pipe a mile long, connecting two 
reservoirs with a difference of level of 20 feet. 

(16) Water flows through a 12-inch pipe having a virtual slope of 3 feet 
per 1000 feet at a velocity of 3 feet per second. 

Find the friction per sq. ft. of surface of pipe in Ibs. 

Also the value of / in the ordinary formula for flow in pipes. 

(17) Find the relative discharge of a 6-inch main with a slope of 
1 in 400, and a 4-inch main with a slope of 1 in 50. 

(18) A 6-inch main 7 miles in length with a virtual slope of 1 in 100 
is replaced by 4 miles of 6-inch main, and 3 miles of 4-inch main. Would 
the discharge be altered, and, if so, by how much ? 

(19) Find the velocity of flow in a water main 10 miles long, con- 
necting two reservoirs with a difference of level of 200 feet. Diameter of 
main 15 inches. Coefficient /=0'009. 

(20) Find the discharge, if the pipe of the last question is replaced for 
the first 5 miles by a pipe 20 inches diameter and the remainder by a pipe 
12 inches diameter. 

(21) Calculate the loss of head per mile in a 10-inch pipe (area of cross 
section 0'54 sq. ft.) when the discharge is 2^ c. ft. per second. 

(22) A pipe consists of J a mile of 10 inch, and a mile of 5 -inch pipe, 
and conveys 2| c. ft. per second. State from the answer to the previous 
question the loss of head in each section and sketch a hydraulic gradient. 
The head at the outlet is 5 ft. 

(23) What is the head lost in friction in a pipe 3 feet diameter 
discharging 6,000,000 gallons in 12 hours? 

(24) A pipe 2000 feet long and 8 inches diameter is to discharge 85 c. ft. 
per minute. What must be the head of water ? 



172 HYDRAULICS 

(25) A pipe 6 inches diameter, 50 feet long, is connected to the bottom 
of a tank 50 feet long by 40 feet wide. The original head over the open 
end of the pipe is 15 feet. Find the time of emptying the tank, assuming 
the entrance to the pipe is sharp-edged. 

If ft = the head over the exit of the pipe at any moment, 

v^ -5v z 4fv*5W 
"20 + 20 + 20x0-5' 

from which, 

In time dt, the discharge is 

28-27 

v T44 

In time dt the surface falls an amount dh. 



Integrating, 

_2000 (1-5 + 400/) / _ 79000 (1-5 +400/) 

I -- - tu \/ J.O - 7- - - SGCS- 

0-196 \/20 A/20 

(26) The internal diameter of the tubes of a condenser is 0*654 inches. 
The tubes are 7 feet long and the number of tubes is 400. The number of 
gallons per minute flowing through the condenser is 400. Find the loss of 
head due to friction as the water flows through the tubes. /=0'006. 

(27) Assuming fluid friction to vary as the square of the velocity, find 
an expression for the work done in rotating a disc of diameter D at an 
angular velocity a in water. 

(28) What horse-power can be conveyed through a 6-in. main if the 
working pressure of the water supplied from the hydraulic power station is 
700 Ibs. per sq. in.? Assume that the velocity of the water is limited to 
3 ft. per second. 

(29) Eighty-two horse-power is to be transmitted by hydraulic pressure 
a distance of a mile. Find the diameter of pipe and pressure required for 
an efficiency of '9 when the velocity is 5 ft. per sec. 

The frictional loss is given by equation 

Mi.*. 

20 d 

(30) Find the inclination necessary to produce a velocity of 4| feet per 
second in a steel water main 31 inches diameter, when running full and 
discharging with free outlet, using the formula 

. -0005 v 1 " 94 
dn* 

(31) The following values of the slope i and the velocity v were 
determined from an experiment on flow in a pipe '1296 ft. diam. 

i -00022 -00182 -00650 -02389 -04348 -12315 -22408 
v -205 -606 1-252 2-585 3'593 6-310 8-521 



FLOW THROUGH PIPES 173 

Determine k and n in the formula 

i=kv n . 

Also determine values of C for this pipe for velocities of *5, 1, 8, 5 and 
7 feet per sec. 

(32) The total length of the Coolgardie steel aqueduct is 307 miles 
and the diameter 30 inches. The discharge per day may be 5,600,000 
gallons. The water is lifted a total height of 1499 feet. 

Find (a) the head lost due to friction, 

(6) the total work done per minute in raising the water. 

(33) A pipe 2 feet diameter and 500 feet long without bends furnishes 
water to a turbine. The turbine works under a head of 25 feet and uses 
10 c. ft. of water per second. What percentage of work of the fall is lost 
in friction in the pipe ? 

Coefficient /= "007 ( 1 + 



(34) Eight thousand gallons an hour have to be discharged through 
each of six nozzles, and the jet has to reach a height of 80 ft. 

If the water supply is 1 miles away, at what elevation above the 
nozzles would you place the required reservoir, and what would you 
make the diameter of the supply main ? 

Give the dimensions of the reservoir you would provide to keep a 
constant supply for six hours. Lond. Un. 1903. 

(35) The pipes laid to connect the Vyrnwy dam with Liverpool are 
42 inches diameter. How much water will such a pipe supply in gallons 
per diem if the slope of the pipe is 4^ feet per mile ? 

At one point on the line of pipes the gradient is 6| feet per mile, and the 
pipe diameter is reduced to 39 inches ; is this a reasonable reduction in the 
dimension of the cross section ? Lond. Un. 1905. 

(36) Water under a head of 60 feet is discharged through a pipe 
6 inches diameter and 150 feet long, and then through a nozzle the area of 
which is one-tenth the area of the pipe. Neglecting all losses except friction, 
find the velocity with which the water leaves the nozzle. 

(37) Two rectangular tanks each 50 feet long and 50 feet broad are 
connected by a horizontal pipe 4 inches diameter, 1000 feet long. The 
head over the centre of the pipe at one tank is 12 feet, and over the other 
4 feet when flow commences. 

Determine the time taken for the water in the two tanks to come to the 
same level. Assume the coefficient C to be constant and equal to 90. 

(38) Two reservoirs are connected by a pipe 1 mile long and 10" 
diameter; the difference in the water surface levels being 25 ft. 



Determine the flow through the pipe in gallons per hour and find by 
how much the discharge would be increased if for the last 2000 ft. a second 
pipe of 10" diameter is laid alongside the first. Lond. Un. 1905. 

(39) A pipe 18" diameter leads from a reservoir, 300 ft. above the 
datum, and is continued for a length of 5000 ft. at the datum, the length 
being 15,000 ft. For the last 5000 ft. of its length water is drawn off by 



174 HYDRAULICS 

service pipes at the rate of 10 c. ft. per min. per 500 ft. uniformly. Find 
the pressure at the end of the pipe. Lond. Un. 1906. 

(40) 350 horse-power is to be transmitted by hydraulic pressure a 
distance of 1^ miles. 

Find the number of 6 ins. diameter pipes and the pressure required for 
an efficiency of 92 per cent. /='01. Take v as 3 ft. per sec. 

(41) Find the loss of head due to friction in a water main L feet long, 
which receives Q cubic feet per second at the inlet end and delivers 

Q 

=- cubic feet to branch mains for each foot of its length. 

What is the form of the hydraulic gradient ? 

(42) A reservoir A supplies water to two other reservoirs B and C. 
The difference of level between the surfaces of A and B is 75 feet, and 
between A and C 97*5 feet. A common 8-inch cast-iron main supplies for 
the first 850 feet to a point D. A 6-inch main of length 1400 feet is then 
carried on in the same straight line to B, and a 5 -inch main of length 
630 feet goes to 0. The entrance to the 8-inch main is bell-mouthed, and 
losses at pipe exits to the reservoirs and at the junction may be neglected. 
Find the quantity discharged per minute into the reservoirs B and C. 
Take the coefficient of friction (/) as '01. Lond. Un. 1907. 

(43) Describe a method of finding the "loss of head" in a pipe due to 
the hydraulic resistances, and state how you would proceed to find the 
loss as a function of the velocity. 

(44) A pipe, I feet long and D feet in diameter, leads water from a 
tank to a nozzle whose diameter is d, and whose centre is h feet below 
the level of water in the tank. The jet impinges on a fixed plane 
surface. Assuming that the loss of head due to hydraulic resistance is 
given by 



show that the pressure on the surface is a maximum when 



(45) Find the flow through a sewer consisting of a cast-iron pipe 
12 inches diameter, and having a fall of 3 feet per mile, when discharging 
full bore. c=100. 

(46) A pipe 9 inches diameter and one mile long slopes for the first 
half mile at 1 in 200 and for the second half mile at 1 in 100. The pressure 
head at the higher end is found to be 40 feet of water and at the lower 
20 feet. 

Find the velocity and flow through the pipe. 

Draw the hydraulic gradient and find the pressure in feet at 500 feet 
and 1000 feet from the higher end. 

(47) A town of 250,000 inhabitants is to be supplied with water. Half 
the daily supply of 32 gallons per head is to be delivered in 8 hours. 

The service reservoir is two miles from the town, and a fall of 10 feet 
per mile can be allowed in the pipe. 

What must be the size of the pipe ? = 90. 



FLOW THROUGH PIPES 175 

(48) A water pipe is to be laid in a street 800 yards long with houses 
on both sides of the street of 24 feet frontage. The average number of 
inhabitants of each house is 6, and the average consumption of water for 
each person is 30 gallons in 8 hrs. On the assumption that the pipe is laid 
in four equal lengths of 200 yards and has a uniform slope of jfoj, and that 
the whole of the water flows through the first length, three-fourths through 
the second, one half through the third and one quarter through the fourth, 
and that the value of C is 90 for the whole pipe, find the diameters of the 
four parts of the pipe. 

(49) A pipe 3 miles long has a uniform slope of 20 feet per mile, and is 
18 inches diameter for the first mile, 30 inches for the second and 21 
inches for the third. The pressure heads at the higher and lower ends of 
the pipe are 100 feet and 40 feet respectively. Find the discharge through 
the pipe and determine the pressure heads at the commencement of the 
30 inches diameter pipe, and also of the 21 inches diameter pipe. 

(50) The difference of level of two reservoirs ten miles apart is 80 feet. 
A pipe is required to connect them and to convey 45,000 gallons of water 
per hour from the higher to the lower reservoir. 

Find the necessary diameter of the pipe, and sketch the hydraulic 
gradient, assuming /=0'01. 

The middle part of the pipe is 120 feet below the surface of the upper 
reservoir. Calculate the pressure head in the pipe at a point midway 
between the two reservoirs. 

(51) Some hydraulic machines are served with water under pressure 
by a pipe 1000 feet long, the pressure at the machines being 600 Ibs. per 
square inch. The horse-power developed by the machine is 300 and the 
friction horse -power in the pipes 120. Find the necessary diameter of the 

I v 2 

pipe, taking the loss of head in feet as 0*03 -5 x ^- and "43 Ib. per square 

a zg 

inch as equivalent to 1 foot head. Also determine the pressure at which 
the water is delivered by the pump. 

What is the maximum horse -power at which it would be possible to 
work the machines, the pump pressure remaining the same ? Lond. Un. 
1906. 

(52) Discuss Reynolds' work on the critical velocity and on a general 
law of resistance, describing the experimental apparatus, and showing the 
connection with the experiments of Poiseuille and D'Arcy. Lond. Un. 
1906. 

(53) In a condenser, the water enters through a pipe (section A) at the 
bottom of the lower water head, passes through the lower nest of tubes, 
then through the upper nest of tubes into the upper water head (section B). 
The sectional areas at sections A and B are 0'196 and 0'95 sq. ft. respec- 
tively ; the total sectional area of flow of the tubes forming the lower nest 
is 0-814 sq. ft., and of the upper nest 0'75 sq. ft., the number of tubes being 
respectively 353 and 326. The length of all the tubes is 6 feet 2 inches. 
When the volume of the circulating water was 1-21 c. ft. per sec., the 
observed difference of pressure head (by gauges) at A and B was 6'5 feet. 
Find the total actual head necessary to overcome frictional resistance, and 



176 HYDRAULICS 

the coefficient of hydraulic resistance referred to A. If the coefficient of 
friction (4/) for the tubes is taken to be '015, find the coefficient of hydraulic 
resistance for the tubes alone, and compare with the actual experiment. 
Lond. Un. 1906. (C r = head lost divided by vel head at A.) 

(54) An open stream, which is discharging 20 c. ft. of water per 
second is passed under a road by a siphon of smooth stoneware pipe, the 
section of the siphon being cylindrical, and 2 feet in diameter. When the 
stream enters this siphon, the siphon descends vertically 12 feet, it 
then has a horizontal length of 100 feet, and again rises 12 feet. If all the 
bends are sharp right-angled bends, what is the total loss of head in the 
tunnel due to the bends and to the friction ? C = 117. Lond. Un. 1907. 

(55) It has been shown on page 159 that when the kinetic energy of a 
jet issuing from a nozzle on a long pipe line is a maximum, 



Hence find the minimum diameter of a pipe that will supply a Pelton 
Wheel of 70 per cent, efficiency and 500 brake horse-power, the available 
head being 600 feet and the length of pipe 3 miles. 

(56) A fire engine supplies water at a pressure of 40 Ibs. per square 
inch by gauge, and at a velocity of 6 feet per second into a pipe 8 inches 
diameter. The pipe is led a distance of 100 feet to a nozzle 25 feet above 
the pump. If the coefficient/ (of friction) in the pipe be '01, and the actual 
lift of the jet is f of that due to the velocity of efflux, find the actual height 
to which the jet will rise, and the diameter of the nozzle to satisfy the 
conditions of the problem. 

(57) Obtain expressions (a) for the head lost by friction (expressed in 
feet of gas) in a main of given diameter, when the main is horizontal, and 
when the variations of pressure are not great enough to cause any important 
change of volume, and (b) for the discharge in cubic feet per second. 

Apply your results to the following example: 

The main is 16 inches diameter, the length of the main is 300 yards, 
the density of the gas is 0'56 (that of air=l), and the difference of pressure 
at the two ends of the pipe is inch of water ; find : 
(a) The head lost in feet of gas. 
(fc) 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 water 
62-4 Ibs. ; coefficient / (of friction) for the gas against the walls of the pipe 
0-005. Lond. Un. 1905. 

(See page 118 ; substitute for w the weight of cubic foot of gas.) 

(58) Three reservoirs A, B and are connected by a pipe leading 
from each to a junction box P situated 450' above datum. 

The lengths of the pipes are respectively 10,000', 5000' and 6000' and the 
levels of the still water surface in A, B and are 800', 600' and 200' above 
datum. 

Calculate the magnitude and indicate the direction of mean velocity in 
each pipe, taking v = WQ\ / mi t the pipes being all the same diameter, 
namely 15". Lond. Un. 1905. 



FLOW THROUGH PIPES 177 

(59) A pipe 3' 6" diameter bends through 45 degrees on a radius of 
25 feet. Determine the displacing force in the direction of the radial line 
bisecting the angle between the two limbs of the pipe, when the head of 
water in the pipe is 250 feet. 

Show also that, if a uniformly distributed pressure be applied in the 
plane of the centre lines of the pipe, normally to the pipe on its outer 
surface, and of intensity 

49ftd 2 
R+l-7* 
per unit length, the bend is in equilibrium. 

E= radius of bend in feet. d = diameter of pipe. 
h = head of water in the pipe. 

(60) Show that the energy transmitted by a long pipe is a maximum 
when one-third of the energy put into the pipe is lost in friction. 

The energy transmitted along the pipe per second is 

7T 7T 4 "fl) ^Z 

p being the pressure per sq. foot at the inlet end of pipe. 
Differentiating and equating to zero 



dv 
or, head lost by friction = J . 

(61) For a given supply of water delivered to a pipe at a given 
pressure, the cost of upkeep of the pipe line may be considered as made up 
of the capital charges on initial cost, plus repairs, plus the cost of energy 
lost in the pipe line. The repairs will be practically proportional to the 
original cost, i.e. to the capital charges. The original cost of the pipe line 
may be assumed proportional to the diameter and to the length. The 
annual capital charges P are, therefore, proportional to Id, or 

P=mld. 

If W is the weight of water pumped per annum, the energy lost per 
year is proportional to 



20. d" 

or, since v is proportional to W divided by the area of the pipe, the total 
annual cost PI may be written, 



For P! to be a minimum, - should be zero. 



Therefore -=ml-5m 1 - = 0, 



That is, the annual cost due to charges and repairs should be equal to 
5 times the cost due to loss of energy. 

If the cost of pipes is assumed proportional to d 2 , P x is a minimum 
when the annual cost is \ of the cost of the energy lost. 

L. H. 12 



CHAPTER VI. 

FLOW IN OPEN CHANNELS. 

117. Variety of the forms of channels. 

The study of the flow of water in open channels is much irore 
complicated than in the case of closed pipes, because of tne 
infinite variety of the forms of the channels and of the different 
degrees of roughness of the wetted surfaces, varying, as they do, 
from channels lined with smooth boards or cement, to the irregular 
beds of rivers and the rough, pebble or rock strewn, mountain 
stream. 

Attempts have been made to find formulae which are applicable 
to any one of these very variable conditions, but as in the case of 
pipes, the logarithmic formulae vary with the roughness of the 
pipe, so in this case the formulae for smooth regular shaped channels 
cannot with any degree of assurance be applied to the calculation 
of the flow in the irregular natural streams. 

118. Steady motion in uniform channels. 

The experimental study of the distribution of velocities of 
water flowing in open channels reveals the fact that, as in the 
case of pipes, the particles of water at different points in a cross 
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 point 
it is probable that the condition of flow is continually changing. 
In a channel of uniform section and slope, and in which the total 
flow remains constant for an appreciable time, since the same 
quantity of water passes each section, the mean velocity v in the 
direction of the stream is constant, and is the same for all the 
sections, and is simply equal to the discharge divided by the area 
of the cross section. This mean velocity is purely an artificial 
quantity, and does not represent, either in direction or magnitude, 
the velocity of the particles of water as they pass the section. 



FLOW IN OPEN CHANNELS 



179 



Experiments with current meters, to determine the distribution 
of velocity in channels, show, however, that at any point in the 
cross section, the component of velocity in a direction parallel to 
the direction of flow remains practically constant. The considera- 
tion of the motion is consequently simplified by assuming that 
the water moves in parallel fillets or stream lines, the velocities in 
which are different, but the velocity in each stream line remains 
constant. This is the assumption that is made in investigating 
so-called rational formulae for the velocity of flow in channels, 
but it should not be overlooked that the actual motion may be 
much more complicated. 

119. Formula for the flow when the motion is uniform 
in a channel of uniform section and slope. 

On this assumption, the conditions of flow at similarly situated 
points C and D in any two cross sections AA and BB, Figs. 112 
and 113, of a channel of uniform slope and section are exactly the 
same ; the velocities are equal, and since C and D are at the same 
distance below the free surface the pressures are also equal. For 
the filament CD, therefore, 

PC + W 5 = PD + V 
w 2g w 2g* 

and therefore, since the same is true for any other filament, 



t w 2g 
is constant for the two sections. 




Fig. 112. 



Let v be the mean velocity of the stream, i the fall per foot 
length of the surface of the water, or the slope, dl the length 
between AA and BB, to the cross sectional area EFGrH of the 
stream, P the wetted perimeter, i.e. the length EF + FGr + GrH, 
and w the weight of a cubic foot of water. 

Let p = m be called the hydraulic mean depth. 

Let dz be the fall of the surface between A A and BB. Since 
the slope is small dz = i.dl. 

12 2 



180 HYDRAULICS 

If Q cubic feet per second fall from AA to BB, the work done 
upon it by gravity will be : 



Then, since 3 + - 

\w 2g 

is constant for the two sections, the work done by gravity must 
be equal to the work done by the frictional and other resistances 
opposing the motion of the water. 

As remarked above, all the filaments have not the same velocity, 
so that there is relative motion between consecutive filaments, 
and since water is not a perfect fluid some portion of the work 
done by gravity is utilised in overcoming the friction due to this 
relative motion. Energy is also lost, due to the cross currents or 
eddy motions, which are neglected in assuming stream line flow, 
and some resistance is also offered to the flow by the air on the 
surface of the water. 

The principal cause of loss is, however, the frictional resistance 
of the sides of the channel, and it is assumed that the whole of 
work done by gravity is utilised in overcoming this resistance. 

Let F . v be the work done per unit area of the sides of the 
channel, v being the mean velocity of flow. F is often called the 
frictional resistance per unit area, but this assumes that the relative 
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.8Z. 

Then, wwidl = J?vPdl, 

CO . F 

therefore P l== w* 

F 

or vni = . 

w 

F is found by experiment to be a function of the velocity and 
also of the hydraulic mean depth, and may be written 



b being a numerical coefficient. 

Since for water w is constant may be replaced by Tc and 

therefore, mi = Jc.f (v) f (m) . 

The form of f(v) f(m) 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 

with precision were probably those of Chezy made on an earthen 
canal, at Coupalet in 1775, from which he concluded that 



and therefore mi = kv~ (1). 

Writing C for 4= 

v = C ,Jmij 

which is known as the Chezy formula, and has already been given 
in the chapter on pipes. 

121. Formulae of Prony and Eytelwein. 

Prony adopted the same formula for channels and for pipes, and 
assumed that F was a function of v and also of v a , and therefore, 

mi = av + bv*. 

By an examination of the experiments of Chezy and those of 
Du Buat* made in 1782 on wooden channels, 20 inches wide and 
less than 1 foot deep, and others on the Jard canal and the river 
Hayne, Prony gave to a and b the values 

a = '000044, 
b = '000094. 
This formula may be written 

mi=(-~ ) + b)v\ 
1 



or v = 



/ 

V v 



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 +, 
gave values to a and b of 

a = '000024, 

6 = '00011 14. 

Neglecting the term containing a, 
v = 95 \lrni. 

* Principes d'hydraulique. See also pages 231 233. 

f Experiments by Funk, 1803-6. 

$ Experiments by Brauings, 1790-92. 



182 HYDRAULICS 

As in the case of pipes, Prony and Eytelwein incorrectly 
assumed that the constants a and 6 were independent of the 
nature of the bed of the channel. 

122. Formula of Darcy and Bazin. 

After completing his classical experiments on flow in pipes 
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 with which the 
channels were lined and also with the form of the channel. 

Experimental channels of semicircular and rectangular section 
were constructed at Dijon, and lined with different materials. 
Experiments were also made upon the flow in small earthen 
channels (branches of the Burgoyne canal), earthen channels lined 
with stones, and similar channels the beds of which were covered 
with mud and aquatic herbs. The results of these experiments, 
published in 1858 in the monumental work, Recherches Hydrau- 
liqueSj very clearly demonstrated the inaccuracy of the assump- 
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 
the coefficient &, section 120, the form used by Darcy for pipes, 

*=(+), 

\ m/' 

a and being coefficients both of which depend upon the nature 
of the lining of the channel. 

Thus, mi = ( a. + j-u 3 

\ mj 

. 1 




The coefficient in the Chezy formula is thus made to vary 
with the hydraulic mean depth m, as well as with the roughness 
of the surface. 

It is convenient to write the coefficient k as 



Taking the unit as 1 foot, Bazin's values for a and /?, and 
values of k are shown in Table XVIII. 

It will be seen that the influence of the second term increases 
very considerably with the roughness of the surface. 

123. Ganguillet and Kutter, from an examination of Bazin's 



FLOW IN OPEN CHANNELS 



183 



experiments, together with some of their own, found that the 
coefficient C in the Chezy formula could be written in the form 



, 

6 + vra/ 

in which a is a constant for all channels, and 6 is a coefficient of 
roughness. 

TABLE XVIII. 

Showing the values of a, /?, and Jc in Bazin's formula for 
channels. 





a 


c 


k 


Planed boards and smooth 
cement 


0000457 


0000045 


0000157 (l I' 98N 1 


\ m J 


Rough boards, bricks and 
concrete 


0000580 


0000133 


000058 (l+\ 

\ Tfl J 


Ashlar masonry 


0000730 


00006 


(QO\ 
1 + ) 
mj 


Earth 


0000854 


00035 


0000854 (l+^Y 


Gravel (Ganguillet and 
Kutter) 


0001219 


00070 


0001219(l+5-I5) 

\ / 



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 

0'00281 






= 



n 



1+41-6 



00281 \ n ' 



in 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 Mississippi River, 1861 j Flow of water in 
fivers and canals, Trautwine and Bering, 1893, 



184 HYDRAULICS 

TABLE XIX. 

Showing values of n in the formula of Ganguillet and Kutter. 
Channel H 

Very smooth, cement and planed boards 009 to '01 

Smooth, boards, bricks, concrete ... ... ... ... '012 to '013 

Smooth, covered with slime or tuberculated -015 

Hough ashlar or rubble masonry '017 to -019 

Very firm gravel or pitched with stones -02 

Earth, in ordinary condition free from stones and weeds ... -025 

Earth, not free from stones and weeds -030 

Gravel in bad condition '035 to '040 

Torrential streams with rough stony beds -05 

TABLE XIX A. 

Showing values of n in the formula of Ganguillet and Kutter, 
determined from recent experiments. 

n 

Rectangular wooden flume, very smooth -0098 

Wood pipe 6 ft. diameter -0132 

Brick, washed with cement, basket shaped sewer, 6'x6'8". nearly 

. new -0130 

Brick, washed with cement, basket shaped sewer, 6'x6'8", one 

year old -0148 

Brick, washed with cement, basket shaped sewer, 6'x6'8", four 

years old -0152 

Brick, washed with cement, circular sewer, 9 ft. diameter, nearly 

new -0116 

Brick, washed with cement, circular sewer, 9 ft. diameter, four 

years old -0133 

Old Croton aqueduct, lined with brick -015 

New Croton aqueduct* '012 

Sudbury aqueduct ... ... ... ... ... ... ... -01 

Glasgow aqueduct, lined with cement -0124 

Steel pipe, wetted, clean, 1897 (mean) -0144 

Steel pipe, 1899 (mean) -0155 

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 formula, that when m equals unity 

then C = - and is independent of the slope ; and also when m is 

large, C increases as the slope decreases. 

It is also of importance to notice that later experiments upon 
the Mississippi by a special commission, and others on the flow of 
the Irrawaddi and various European rivers, are inconsistent with 

New York Aqueduct Commission, 



FLOW IN OPEN CHANNELS 185 

the early experiments of Humphreys and Abbott, to which 
Ganguillet and Kutter attached very considerable importance in 
framing their formula, and the later experiments show, as described 
later, that the experimental determination of the flow in, and the 
slope of, large natural streams is beset with such great difficulties, 
that any formula deduced for channels of uniform section and 
slope cannot with confidence be applied to natural streams, and 
vice versa. 

The application of this formula to the calculation of uniform 
channels gives, however, excellent results, and providing the value 
of n is known, it can be used with confidence. 

It is, however, very cumbersome, and does not appear to give 
results more accurate than a new and simpler formula suggested 
recently by Bazin and which is given in the next section. 

124. M. Bazin's later formula for the flow in channels. 

M. Bazin has recently (Annales des Pouts et Chaussees, 1897, 
Vol. IV. p. 20), made a careful examination of practically all the 
available experiments upon channels, and has proposed for the 
coefficient C in the Chezy formula a form originally proposed by 
Ganguillet and Kutter, which he writes 




or 



in which a is constant for all channels and {! is a coefficient of 
roughness of the channel. 

Taking 1 metre as the unit a = '0115, and writing y for , 

c=-2Z_ a)j 

or when the unit is one foot, 

(2), 



the value of y in (2) being 1'Slly, in formula (1). 

The values of y as found by Bazin for various kinds of channels 
are 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 few 
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 values 
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 three 
columns for m less than 1, are obtained on the assumption, that 
Bazin's formula is true for all values of m within the limits of the 
table. 

TABLE XX. 

Values of y in the formula, 

c 157 - 5 



unit metre unit foot 

Very smooth surfaces of cement and planed boards ... -06 -1085 

Smooth surfaces of boards, bricks, concrete '16 *29 

Ashlar or rubble masonry '46 '83 

Earthen channels, very regular or pitched with stones, 

tunnels and canals in rock *85 1/54 

Earthen channels in ordinary condition 1/30 2'35 

Earthern channels presenting an exceptional resistance, 
the wetted surface being covered with detritus, 

stones or weed, or very irregular rocky surface 1'7 3'17 

125. Glazed earthenware pipes. 

Vellut* from experiments on the flow in earthenware pipes has 
given to C the value 



in which 
or 



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. 




FLOW IN OPEN CHANNELS 



187 



TABLE XXI. 

Values of in the formula v = C *J- mi calculated from Bazin's 
formula, the unit of length being 1 foot, 

157-5 



C 



1 + 





Channels 






Smooth 






Earth canals 






Hydraulic 
mean 
depth 


Very smooth 
cement and 
planed 
boards 


boards, brick, 
concrete, 
glazed 
earthenware 


Smooth 
but dirty 
brick, 
concrete 


Ashlar 
masonry 


in very good 
condition, 
and canals 
pitched with 


Earth canals 
in ordinary 
condition 


Earth canals 
exceptionally 
rough 


m. 




pipes 






stones 








7 = '1085 


y = -29 


7 = -50 


7 = -83 


-y = l-54 


7 = 2-35 


7=3-17 


2 


127 


96 


74 


55 


35 


25 


19 


3 


131 


103 


82 


63 


41 


30 


23 


4 


135 


108 


88 


68 


46 


32 


26 


5 


137 


112 


92 


72 


50 


37 


29 


6 


139 


116 


96 


76 


53 


39 


31 


8 


141 


119 


101 


82 


58 


43 


35 


1-0 


142 


122 


105 


86 


62 


47 


38 


1-3 


144 


126 


109 


91 


67 


51 


42 


1-5 


145 


128 


112 


94 


70 


54 


44 


1-75 


146 


130 


114 


97 


73 


57 


46 


2-0 


147 


132 


116 


99 


76 


59 


49 


2-5 


148 


134 


119 


103 


80 


64 


53 


3'0 


149 


136 


122 


107 


84 


67 


56 


4-0 


150 


138 


126 


111 


89 


72 


61 


5-0 


151 


140 


129 


115 


94 


77 


65 


6-0 


151 


142 


131 


118 


98 


80 


69 


8-0 


152 


144 


134 


122 


102 


86 


74 


10-0 


153 


145 


136 


125 


106 


90 


79 


12-0 










109 


94 


82 


15-0 










113 


98 


87 


20-0 










117 


103 


92 


30-0 










123 


110 


100 


50-0 










129 


119 


108 



126. Bazin's method of determining a and J&. 
The method used by Bazin to determine the values of a and /? 
is of sufficient interest and importance to be considered in detail. 



He first calculated values of -j= and 

vra 



from experimental 



data, and plotted these values as shown in Fig. 114, -= as 



_ 

abscissae, and - - as ordinates. 
v 






188 



HYDRAULICS 



As will be seen on reference to the figure, points have been 
plotted for four classes of channels, and the points lie close to four 
straight lines passing through a common point P on the axis 
of y. 




The equation to each of these lines is 
y = a + fix, 



FLOW IN OPEN CHANNELS 189 



or - = a + T-- , 

v vm 

ct being the intercept on the axis of y, or the ordinate when r= is 

Jm 
zero, and /?, which is variable, is the inclination of any one of 

these lines to the axis of x : for when /= is zero, - - = a, and 

vm v 

transposing the equation, 

\frni 



which is clearly the tangent of the angle of inclination of the line 
to the axis of x. 

It should be noted, that since - = p , the ordinates give 

actual experimental values of ~ , or by inverting the scale, values 

of C. Two scales for ordinates are thus shown. 

In addition to the points shown on the diagram, Fig. 114, 
Bazin plotted the results of some hundreds of experiments for all 
kinds of channels, and found that the points lay about a series of 
lines, all passing through the point P, Fig. 114, for which a is '00635, 

and the values of - , i.e. y, are as shown in Table XX. 
Bazin therefore concluded, that for all channels 



v vm 

the value of ft depending upon the roughness of the channel. 

For very smooth channels in cement and planed boards, Bazin 
plotted a large number of points, not shown in Fig. 114, and the 
line for which y = '109 passes very nearly through the centre of 
the zone occupied by these points. 

The 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 may be as high as 0*4. 

It is further of interest to notice, that where the surfaces and 
sections 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 
difference being probably due to the pointing of the sides and 
arch of the New York aqueduct not being so carefully executed 
as for the Boston aqueduct. By simply washing the walls of the 
latter with cement, Fteley found that its discharge was increased 
20 per cent. 



190 HYDRAULICS 

y is also greater for rectangular-shaped channels, or those 
which approximate to the rectangular form, than for those of 
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 very 
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 experiments, which show clearly the effect of slime and 
tuberculations, in increasing the resistance of very smooth channels. 
The value of y for the basket-shaped sewer lined with brick, 
washed with cement, rising from '4 to '642 during 4 years' service. 

127. Variations in the coefficient C. 

For channels lined with rubble, or similar materials, some of 
the experimental points give values of C differing very considerably 
from those given by points on the line for which y is 0'83, Fig. 114, 
but the values of C deduced from experiments on particular 
channels show similar discrepancies among themselves. 

On reference to Bazin's original paper it will be seen that, for 
channels in earth, there is a still greater variation between the 
experimental values of C, and those given by the formula, but the 
experimental results in these cases, for any given channel, are 
even more inconsistent amongst themselves. 

An apparently more serious difficulty arises with respect to 
Bazin's formula in that C cannot be greater than 157*5. The 
maximum value of the hydraulic mean depth m recorded in 
any series of experiments is 74*3, obtained by Humphreys and 
Abbott from measurements of the Mississippi at Carroll ton in 1851. 
Taking y as 2'35 the maximum value for C would then be 124. 
Humphreys and Abbott deduced from their experiments values 
of C as large as 254. If, therefore, the experiments are reliable 
the formula of Bazin evidently gives inaccurate results for excep- 
tional values of m. 

The values of C obtained at Carrollton are, however, incon- 
sistent with those obtained by the same workers at Yicksburg, 
and they are not confirmed by later experiments carried out at 
Carrollton by the Mississippi commission. Further the velocities 
at Carrollton were obtained by double floats, and, according to 
Gordon*, 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, Proceedings Inst. Civil Eng., 1893. 



FLOW IN OPEN CHANNELS 191 

Humphreys and Abbott at Vicksburg and also to those obtained 
by the Mississippi Commission at Carrollton, and shows, that the 
maximum value for C is then, probably, only 122. 

That the values of C as deduced from the early experiments on 
the Mississippi are unreliable, is more than probable, since the 
smallest slope, as measured, was only '0000034, which is less than 
j inch per mile. It is almost impossible to believe that such small 
differences of level could be measured with certainty, as the 
smallest ripple would mean a very large percentage error, and 
it is further probable that the local variations in level would be 
greater than this measured difference for a mile length. Further, 
assuming the slope is correct, it seems probable that the velocity 
under such a fall would be less than some critical velocity similar 
to that obtained in pipes, and that the velocity instead of being 
proportional to the square root of the slope i, is proportional 
to i. That either the measured slope was unreliable, or that the 
velocity was less than the critical velocity, seems certain from the 
fact, that experiments at other parts of the Mississippi, upon the 
Irrawaddi by Gordon, and upon the large rivers of Europe, in no 
case give values of C greater than 124. 

The experimental evidence for these natural streams tends, 
however, clearly to show, that the formulae, which can with 
confidence be applied to the calculation of flow in channels of 
definite form, cannot with assurance be used to determine the 
discharge of rivers. The reason for this is not far to seek, as 
the conditions obtaining in a river bed are generally very far 
removed from those assumed, in obtaining the formula. The 
assumption that the motion is uniform over a length sufficiently 
great to be able to measure with precision the fall of the surface, 
must be far from the truth in the case of rivers, as the irregu- 
larities in the cross section must cause a corresponding variation 
in the mean velocities in those sections. 

In the derivation of the formula, frictional resistances only 
are taken into account, whereas a considerable amount of the 
work done on the falling water by gravity is probably dissipated 
by eddy motions, set up as the stream encounters obstructions in 
the bed of the river. These eddy motions must depend very 
much on local circumstances and will be much more serious in 
irregular channels and those strewn with weeds, stones or other 
obstructions, than in the regular channels. Another and probably 
more serious difficulty is the assumption that the slope is uniform 
throughout the whole length over which it is measured, whereas 
the slope between two cross sections may vary considerably 
between bank and bank. It is also doubtful whether locally 



192 HYDRAULICS 

there is always equilibrium between the resisting and accelerating 
forces. In those cases, therefore, in which the beds are rocky or 
covered with weeds, or in which the stream has a very irregular 
shape, the hypotheses of uniform motion, slope, and section, will 
not even be approximately realised. 

128. Logarithmic formula for the flow in channels. 

In the formulae discussed, it has been assumed that the f rictional 
resistance of the channel varies as the square of the velocity, and 
in order to make the formulae fit the experiments, the coefficient C 
has been made to vary with the velocity. 

As early as 1816, Du Buat* pointed out, that the slope i 
increased at a less rate than the square of the velocity, and 
half a century later St Tenant proposed the formula 

mi = '000401 lA 

To determine the discharge of brick-lined sewers, Mr Santo 
Crimp has suggested the formula 



and experiments show that for sewers that have been in use some 
time it gives good results. The formula may be written 

. 0-00006*; 2 ' 

- .--_. _____ T 

- 1 '^J. 

m !34 

An examination of the results of experiments, by logarithmic 
plotting, shows that in any uniform channel the slope 

. bo* 
*=^> 

k being a numerical coefficient which depends upon the roughness 
of the surface of the channel, and n and p also vary with the 
nature of the surface. 

Therefore, in the formula, 



From what follows it will be seen that n varies between 1*75 
and 2'1, while p varies between 1 and 1'5. 

Jcv n 
Since m is constant, the formula i = ^ may be written i = fo n , 

& 
b being equal to ^ . 

Therefore log i = log b + n log v. 

* Principes d'Hydr antique, Vol. r. p. 29, 1810. 



FLOW IN OPEN OTTAXNELS 



193 



In Fig. 115 are shown plotted the logarithms of i and v 
obtained from an experiment by Bazin on the flow in a semi- 
circular cement-lined pipe. The points lie about a straight line, 
the tangent of the inclination of which to the axis of v is 1'96 
and the intercept on the axis of i through v = 1, or log v = 0, is 
0000808. 




Fig. 115. Logarithmic plottings of i and v to determine the index n in 

the formula for channels, i = -. 
in" 

For this experimental channel, therefore, 

i = '00008085 v. 

In the same figure are shown the plottings of logi and logv for 
the siphon-aqueduct* of St Elvo lined with brick and for which 
m is 278 feet. In this case n is 2 and b is '000283. Therefore 

i = -000283v 9 . 

If, therefore, values of v and i are determined for a channel, 
while m is kept constant, n can be found. 



Annales des Fonts et Chaussees, Vol. iv. 1897. 



L, H. 



13 



194 HYDRAULICS 



To determine the ratio - . The formula, 



m j 
may be written in the form, 



k\* 



or log m = log (- J + - log v. 

By determining experimentally m and v, 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 



n 



inclination of which to the axis of v is equal to - . and the 

P 
intercept on the axis of m is equal to 



'" 



In Fig. 116 are shown the logarithmic plottings of m and v for 
a number of channels, of varying degrees of roughness. 

4? 

The ratio - varies considerably, and for very regular channels 

increases with the roughness of the channel, being about 1*40 for 
very smooth channels, lined with pure cement, planed wood or 
cement mixed with very fine sand, 1*54 for channels in unplaned 
wood, and 1*635 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 

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 channels similarly lined, and of different slopes, but here 



FLOW IN OPEN CHANNELS 



195 




Log. 



v. 



Fig. 116. Logarithmic plottings of m and v to determine the 

ratio - in the formula i= - . 
p mP 

TABLE XXII. 
Particulars of channels, plottings for which are shown in Fig. 116. 



1. 

2. 


Semicircular channel, very smooth, lined with wood 
,, ,, cement mixed with 


n 
P 
1-45 

1-36 


3. 

4. 
5. 
6, 


Rectangular channel, very smooth, lined with cement 
, wood, 1' 1" wide 
smooth , ,, ,, slope -00208 
, -0043 


1-44 
1-38 
1-54 
1-54 


7. 
8. 
9. 


> > > "01)49 
'00824 
New Croton aqueduct, smooth, lined with bricks (Report New York 
Water Supply) 


1-54 
1-54 

1-74 


10. 

11. 


Glasgow aqueduct, smooth, lined with concrete (Proc. I. C. E. 1896) 
Sudbury ,, ,, ,, brick well pointed (Tr. Am. 
S.C.E. 1883) 


1-635 
1-635 


12. 


Boston sewer, circular, smooth, lined with brick washed with cement 
(Tr. Am.S. C. E. 1901) 


1-635 


13. 

15! 
15a. 
156. 


Rectangular channel, smooth, lined with brick 
> wood ... ... ... 
,, ,, ,, ,, small pebbles 
Rectangular sluice channel lined with hammered ashlar 


1-635 
1-655 
1-49 
1-36 
1-36 


16. 




1-29 


17. 


Torlonia tunnel, rock, partly lined 


1-49 


18. 


Ordinary channel lined with stones covered with mud and weeds ... 


1-18 
94 


20. 


River Weser 


1-615 


21. 




1-65 


22. 




2*1 


23. 


Earth channel. Gros bois ... ... 


1-49 


24. 


Cavour canal 




25. 




1-37 



132 



196 



HYDRAULICS 



again, as will appear in the context, a difficulty is encountered, as 
even with similarly lined channels, the roughness is in no two 
cases exactly the same, and as shown by the plottings in Fig. 116, 
no two channels of any class give exactly the same values 



n 



for - , but for certain classes the ratio is fairly constant. 

Taking, for example, the wooden channels of the group (Nos. 4 



n 



to 8), the values of - are all nearly equal to 1'54. 

The plottings for these channels are again shown in Fig. 117. 
The intercepts on the axis of m vary from 0'043 to 0'14. 



I 

1-0 
09 
08 
07 

06 



05 




Lea v 



Fig. 117. Logarithmic plottings to determine the ratio - for smooth channels. 
Let the intercepts on the axis of m be denoted by y t then, 



FLOW IN OPEN CHANNELS 



197 



1 1 

& p p 

If k and p are constant for these channels, and log* and 
log y are plotted as abscissae and ordinates, the plottings should lie 
about a straight line, the tangent of the inclination of which to the 

axis of i is - , and when log y = 0, or y is unity, the abscissa i = &, 

i.e. the intercept on the axis of i is k. 

In Fig. 118 are shown the plottings of log i and log y for these 
channels, from which p=l'14 approximately, and k = '00023. 

Therefore, n is approximately 1*76, and taking - as 1'54 

00023u 176 



01 



-OU5 





V 


































\ 




































\ 


































\ 





































\ 




































\ 


\ 





































\ 


\ 






















1 














\ 


\ 


















' ' 
















\ 


\ 






















































\ s 














































) 






















\ 




































\ 




























tan/ (L 


'P 




_za 


\ 








y fa- 


OOO23. 


















i 




S 




V2 -0005, -OO1 '002 -005 '01 

Log. i/ 



Fig. 118. Logarithmic plottings to determine the value of p for smooth 
channels, in the formula i = . 

41 

Since the ratio - is not exactly 1*54 for all these channels, the 

values 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 values of v as observed and as calculated by the formula, 
the calculated and observed values of v agree very nearly. 



198 



HYDRAULICS 

TABLE XXIII. 



Values of v, for rectangular channels lined with wood, as 
determined experimentally, and as calculated from the formula 



; = '00023 



m 



ri4' 



Slope '00208 


Slope -0049 


Slope -00824 




v ob- 


v calcu- 




v ob- 


v calcu- 




v ob- 


v calcu- 


m in 


served 


lated 


m in 


served 


lated 


m in 


served 


lated 


metres 


metres 


metres 


metres 


metres 


metres 


metres 


metres 


metres 




per sec. 


per sec. 




per sec. 


per sec. 




per sec. 


per sec. 


0-1381 


0-962 


0-972 


0-1042 


1-325 


1-314 


0882 


1-594 


1-589 


1609 


1-076 


1-07 


1224 


1-479 


1-459 


1041 


1-776 


1-764 


1832 


1-152 


1-165 


1382 


1-612 


1-58 


1197 


1-902 


1-932 


1976 


1-259 


1-223 


1535 


1-711 


1-690 


1313 


2-053 


2-051 


2146 


1-324 


1-290 


1668 


1-818 


1-782 


1420 


2-186 


2-158 


2313 


1-374 


1-354 


1789 


1-898 


1-858 


1543 


2-268 


2-275 


2441 


1-440 


1-402 


1913 


1-967 


1-947 


1649 


2-357 


2-377 


2578 


1-487 


1-452 


2018 


2-045 


2-014 


1744 


2-447 


2-460 


2681 


1-552 


1-49 


2129 


2-102 


2-089 


1842 


2-518 


2-553 


2809 


1-587 


1-552 


2215 


2-179 


2-143 


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 in 
hammered ashlar, for which the logarithmic plottings of m and v 

are shown in Fig. 116, Nos. 15 a and 15 &, and - is 1'36. 

The slopes of these channels are '101 and '037. By plotting 
log* and log y, p is found to be 1'43 and k '000149. So that for 

these two channels 

. '000149^ r98 

m 1 ' 43 

The calculated and observed velocities are shown in Table XXXI 
and agree remarkably well. 

Very smooth channels. The ratio - for the four very smooth 

channels, shown in Fig. 116, varies between 1'36 and 1'45, the 
average value being about 1*4. On plotting logy and log* the 
points did not appear to lie about any particular line, so that p 
could not be determined, and indicates that k is different for the 
four channels. Trial values of n = 1*75 and p = 1'25 were taken, or 

k.v 



and values of k calculated for each channel. 



FLOW IN OPEN CHANNELS 199 

Velocities as determined experimentally and as calculated for 
three of the channels are shown in Table XXIII from which it will 
be seen that k varies from '00006516 for the channel lined with 
pure cement, to '0001072 for the rectangular shaped section lined 
with carefully planed boards. 

It will be seen, that although the range of velocities is con- 
siderable, there is a remarkable agreement between the calculated 
and observed values of v, so that for very smooth channels the 
values of n and p taken, can be used with considerable confidence. 

Channels moderately smooth. The plottings of log m and logv 
for channels lined with brick, concrete, and brick washed with 
cement are shown in Fig. 116, Nos. 9 to 13. 

It will be seen that the value of - is not so constant as for the 

P 
two classes previously considered, but the mean value is about 

M 

1'635, which is exactly the value of - for the Sudbury aqueduct. 

For the New Croton aqueduct - is as high as 1'74, and, as shown 
in Fig. 114, this aqueduct is a little rougher than the Sudbury. 

The variable values of -- show that for any two of these 

P 

channels either n> or p, or both, are different. On plotting logi 
and logi; as was done in Fig. 115, the points, as in the last case, 
could not be said to lie about any particular straight line, and the 
value of p is therefore uncertain. It was assumed to be 1'15, and 

4? 

therefore, taking - as 1'635, n is 1*88. 

Since no two channels have the same value for - , it is to be 

P 
expected that the coefficient Jc will not be constant. 

In the Tables XXIV to XXXIII the values of v as observed 
and as calculated from the formula 

._/b r88 
~ m ri > 
and also the value of Jc are given. 

It will be seen that Jc varies very considerably, but, for the 
three large aqueducts which were built with care, it is fairly 
constant. 

The effect of the sides of the channel becoming dirty with 
time, is very well seen in the case of the circular and basket- 
shaped sewers. In the one case the value of k, during four years' 
service, varied from '00006124 to '00007998 and in the other from 
'00008405 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 is 
the same as given by Bazin's formula, when 7 is '29, and when k is 
'0001096, as in the case of the dirty basket-shaped sewer, the value 
of y is '642, which agrees with that shown for this sewer on 
Fig. 114 

Channels in masonry. Hammered ashlar and rubble. Attention 
has already been called, page 198, to the results given in 
Table XXXI for the two channels lined with hammered ashlar. 

The values of n and p for these two channels were determined 
directly from the logarithmic plottings, but the data is insufficient 
to give definite values, in general, to n, p, and k. 

In addition to these two channels, the results for one of 
Bazin's channels lined with small pebbles, and for other channels 
lined with rubble masonry and large pebbles are given. The 

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 1'96, while p varies 
from 1'36 to 1'5. 

Earthen channels. A. very large number of experiments have 
been made on the flow in canals and rivers, but as it is generally 

impracticable to keep either i or m constant, the ratio - can only 

be determined in a very few cases, and in these, as will be seen 
from the plottings in Fig. 116, the results are not satisfactory, and 

appear to be unreliable, as - varies between '94 and 2*18. It seems 

probable that p is between 1 and 1*5 and n from 1*96 to 2' 15. 
Logarithmic formulae for various classes of channels. 
Very smooth channels, lined with cement, or planed boards, 

fl,V75 

i = ('000065 to '00011) ^ . 

Smooth channels, lined with brick well pointed, or concrete, 
t = '000065 to '00011 ~^. 

Channels lined with ashlar masonry, or small pebbles, 

-w 1 ' 96 
t = '00015^-4. 

Channels lined with rubble masonry, large pebbles, rock, and 
exceptionally smooth earth channels free from deposits, 

t-m 
t = '00023 



m 



FLOW IN OPEN CHANNELS 201 

Earth channels, 



k varies from '00033 to '00050 for channels in ordinary condition 
and from '00050 to '00085 for channels of exceptional resistance. 

129. Approximate formula for the flow in earth 
channels. 

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 

v = C v m% i y 

may, therefore, be taken for earth channels, in which C is about 
50 for channels in ordinary condition. 

In Table XXXIII are shown values of v as observed and 
calculated from this formula. 

The hydraulic mean depth varies from '958 to 14*1 and for all 
values between these external limits, the calculated velocities 
agree 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 -^ 
w 1 - 5 ' 

log & = 4'0300. 

v ft. per sec. v ft. per sec. 
m feet observed calculated 

2372 3'55 3'57 

2811 4-00 4-03 

3044 4-20 4-26 

3468 4-67 4'68 

3717 4-94 4-94 

3930 5-11 5-12 

4124 5-26 5-30 

4311 5-49 5-47 



202 HYDRAULICS 

TABLE XXIV (continued). 

Pure cement, semicircular. 

. _ fa; 1 " 75 
~m r23 ' 

-w 173 
00006516 ~ 5i 

log & = 5-8141. 

m v observed v calculated 

503 3-72 3-66 

682 4-59 4-55 

750 4-87 4-87 

915 5-57 5-62 

1-034 6-14 6-14 

Cement and very fine sand, semicircular. 



log & = 5-8802. 

v ft. per sec. v ft. per sec. 

ire feet observed calculated 

379 2-87 2-74 

529 3-44 8-49 

636 3-87 3-98 

706 4-30 4-30 

787 4-51 4-59 

839 4-80 4-84 

900 4-94 5-10 

941 5-20 5-26 

983 5-38 5-43 

1-006 5-48 5-53 

1-02 5-55 5-58 

1-04 5-66 5-66 



TABLE XXY. 



Boston circular sewer, 9 ft. diameter. 

Brick, washed with cement, i = CTHHT (Horton). 



i = '00006124^5, 

log v = '6118 log m + '5319 log i + 2'2401, 

v ft. per sec. v ft. per sec. 
TO feet observed calculated 

928 2-21 2-34 

1-208 2-70 2-76 

1-408 3-03 3-03 

1-830 3-48 3-56 

1-999 3-73 3-75 

2-309 4-18 4-10 



FLOW IN OPEN CHANNELS 203 

TABLE XXY (continued). 
The same sewer after 4 years' service. 

i = '00007998^, 
log v = '6118 logm + '5319 logi + 2*1795. 

m v observed v calculated 
1-120 2-38 2-29 

1-606 2-82 2-78 

1-952 3-16 3-22 

2-130 3-30 3-39 

TABLE XXVI. 
New Croton aqueduct. Lined with concrete. 

v 1 ' 88 
i = -000073^, 

logv = '6118 log m + '5319 log i+ 2'200. 





v ft. per sec. 


v ft. per sec. 


m feet 


observed 


calculated 


1-000 


1-37 


1-37 


1-250 


1-59 


1-57 


1-499 


1-79 


1-76 


1-748 


1-95 


1-93 


2-001 


2-11 


2-10 


2-250 


2-27 


2-26 


2-500 


2-41 


2-40 


2-749 


2-52 


2-55 


2-998 


2-65 


2-68 


3-251 


2-78 


2-82 


3-508 


2-89 


2-96 


3-750 


3-00 


3-08 


3-838 


3-02 


3-12 



TABLE XXVII. 
Sudbury aqueduct. Lined with well pointed brick. 

i = -00006427^, 
log v = '6118 log m + '5319 log i + 2'23. 

v ft. per sec. v ft. per sec. 
TO feet observed calculated 

4987 1-135 1-142 

6004 1-269 1-279 

8005 1-515 1-525 

1-000 1-755 1-752 

1-200 1-948 1-954 

1-400 2-149 2-147 

1-601 2-332 2-331 

1-801 2-513 2-511 

2-001 2-651 2-672 

2-201 2-844 2-832 

2-336 2-929 2-937 



204 HYDRAULICS 

TABLE XXVIII. 

Rectangular channel lined with brick (Bazin). 

* = '000107^. 
m 11 - 

v ft. per sec. v ft. per sec. 
m feet observed calculated 

1922 2-75 2-90 

2838 3-67 3'68 

3654 4-18 4-30 

4235 4-72 4'71 

4812 5-10 5-09 

540 5-34 5-46 

5823 5-68 577 

6197 6-01 5-94 

6682 6-15 6-22 

6968 6-47 6'39 

7388 6-60 6-62 

7788 6-72 6'83 

Glasgow aqueduct. Lined with concrete. 

i = '0000696 ^pa, 
log v = *6118 log m + '5319 log i + 2'2113. 





v ft. per sec. 


v ft. per sec. 


m feet 


observed 


calculated 


1-227 


1-87 


1-89 


1-473 


2-07 


2-11 


1-473 


2-106 


2-11 


1-489 


2-214 


2-13 


1-499 


2-13 


2-14 


1-499 


2-15 


2-14 


1-548 


2-18 


2-22 


1-597 


2-21 


2-23 


1-607 


2-23 


2-23 


1-610 


2-22 


2-24 


1-620 


2-24 


2-24 


1-627 


2-25 


2-27 


1-738 


2-26 


2-33 


1-811 


2-47 


2-40 



TABLE XXIX. 

Charlestown basket-shaped sewer 6' x 6' 8". 
Brick, washed with cement, i -s^inf (Horton). 



i= '00008405^, 

logv = '6118 log m + '5319 log i + 21678. 

v ft. per sec. v ft. per sec. 
m feet observed calculated 

688 1-99 2-05 

958 2-46 2-52 

1-187 2-82 2-87 

1-539 3-44 3-36 



FLOW IN OPEN CHANNELS 



205 



TABLE XXIX (continued). 
The same sewer after 4 years' service, 

v 1 ' 8 * 
; = -0001096 






log v = '6118 log m + *5319 log i + 21065. 



m feet 
1-342 
1-508 
1-645 



v ft. per sec. 
observed 

2-66 
2-86 
3-04 



v ft. per sec. 
calculated 

2-68 
2-88 
3-04 



TABLE XXX. 

Left aqueduct of the Solani canal, rectangular in section, lined 
with rubble masonry (Cunningham), 



i 

000225 
000206 
000222 
000207 
000189? 



,.1-96 

t = '00026 ^j. 
m 14 

v ft. per sec. v ft. per seo. 
m feet observed calculated 



Right aqueduct, 



6-43 

6-81 

7-21 

7-643 

7-94 



* = '0002213 



3'46 
3-49 
3-70 
3'87 
4-06 



^ 
m 1 



i 

000195 
000225 
000205 
000193 
000193 
000190 



3-42 
5'86 
6-76 
7-43 
7'77 
7-96 



v observed 
2-43 
3'61 
3'73 
3-87 
3-93 
4-06 



3'50 
3'47 
3'84 
3'83 
3'83 



v calculated 
2'26 
3'58 
3'76 
3'89 
4'04 
4'06 



Torlonia tunnel, partly in hammered ashlar, partly in solid 

rock, 

i= '00104, 

-.ITO 



i = -00022 



m 1 



1-932 
2-172 
2-552 
2-696 
3-251 
3-438 
3-531 
3-718 



v observed 


v calculated 


8-382 


3-45 


3-625 


3-73 


4-232 


4-16 


4-324 


4-32 


5-046 


4-90 


4-965 


5-08 


4-908 


5-18 


5-358 


6-37 



206 



HYDRAULICS 



TABLE XXXI. 



Channel lined with hammered ashlar, 

2,1-36, 

P 

i = -000149 ^L 



log fc = 41740. 



t'=-101 



' 


v ft. per sec. 


v ft. per sec. 


m feet 


observed 


calculated 


324 


12-30 


12-30 


467 


16-18 


16-18 


580 


18-68 


18-97 


562 


21-09 


20-8 



=037 





v ft. per sec. 


v ft. per sec. 


m feet 


observed 


calculated 


424 


9-04 


9-02 


620 


11-46 


11-86 


745 


13-55 


13-52 


852 


15-08 


14-93 



Channel lined with small pebbles, i = '0049 (n = 1'96, p 
will give equally good results). 



1-32 



P 

000152 



m 



log k = 41913. 



m feet 

250 
357 
450 
520 
588 
644 
700 
746 
785 
832 
871 
910 



v ft. per sec. 
observed 

2-16 
2-95 
3-40 
3-84 
4-14 
4-43 
4-64 
4-88 
5-12 
5-26 
5-43 
5-57 



v ft. per sec. 
calculated 

2-34 
2-97 
3-47 
3-82 
4-15 
4-43 
4-66 
4-88 
5-05 
5-25 
5-43 
5-58 



FLOW IN OPEN CHANNELS 



207 



TABLE XXXII. 

Channel lined with large pebbles (Bazin), 

i = -000229^ 
m 16 ' 



m feet 

291 
417 
510 
587 
656 
712 
772 
823 
867 
909 
946 
987 



log & = 4-3605. 



v ft. per sec. 
observed 

1-79 
2-43 
2-90 
3-27 
3-56 
3-85 
4-03 
4-23 
4-43 
4-60 
4-78 
4-90 



v ft. per sec. 
calculated 

1-84 
2-44 
2-90 
3-18 
3-45 
3-67 
3-91 
4-33 
4-53 
4-69 
4-84 
5-00 



TABLE XXXIII. 

Velocities as observed, and as calculated by the formula 
v=C^mN. = 50. 

Ganges Canal. 



t 

000155 
000229 
000174 
000227 
000291 



m feet 

5-40 
8-69 
7-82 
9-34 
4-50 



v ft. per sec. 
observed 


v ft. per sec. 
calculated 


2-4 


2-34 


3-71 


3'80 


2-96 


3-08 


4-02 


4-00 


2-82 


2-63 



i 

0005503 
0005503 
0002494 
0002494 



0001183 
0001782 
0001714 
0002180 



River Weser. 

m v observed 



8-93 
13-35 
14-1 
10-5 



6-29 
7-90 
5-69 
4-75 



Missouri, 
n v observed 



10-7 
12-3 
15-4 
17-7 



3-6 
4-38 
5-03 
6-19 



v calculated 

6-0 

8-18 
5-70 
4-78 



v calculated 

3-23 
4-37 
4-80 



208 



II YDRAULICS 



i 

00029 
00029 
00033 
00033 



Cavour Canal, 
m v observed 



7-32 
5-15 
5-63 
4.74 



3-70 
3-10 
3-40 
3-04 



v calculated 

3-80 
2-92 
3-14 
2-91 



Earth channel (branch of Burgoyne canal). 
Some stones and a few herbs upon the surface. 

0*48. 



I 

000957 
000929 
000993 
000986 
000792 
000808 
000858 
000842 



v ft. per sec. v ft. per sec. 
m feet observed calculated 



958 
1-181 
1-405 
1-538 

958 
1-210 
1-436 
1-558 



1-243 
1-702 
1-797 
1-958 
1-233 
1-666 
1-814 
1-998 



1-30 
1-66 
1-94 
2-06 
1-25 
1-56 
1-79 
2-08 



130. Distribution of the velocity in the cross section 
of open channels. 

The mean velocity of flow in channels and pipes of small cross 
sectional area can be determined by actually measuring the weight 
or the volume of the water discharged, as shown in Chapter VII, 
and dividing the volume discharged per second by the cross 
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 Pi tot 
tubes t. If the bed of the stream is carefully sounded, the cross 
section can be plotted and divided into small areas, at the centres 
of which the velocities have been observed. If then, the observed 
velocity be assumed equal to the mean velocity over the small 
area, the discharge is found by adding the products of the areas 
and velocities. 

Or Q = 2a.i>. 

M. Bazint, with a thoroughness that has characterised his 
experiments in other branches of hydraulics, has investigated the 
distribution of velocities in experimental channels and also in 
natural streams. 

In Figs. 119 and 120 respectively are shown the cross sections 
of an open and closed rectangular channel with curves of equal 



See page 238. 

Bazin, Eecherches Hydraulique. 



t See page 241. 



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 also shown. 




Curves of equal Velocity 
fbr Rectangular Channel/. 

Fig. 119. 



on 
Vertical Sections. 




i f /'' N \i 

' / -i i 

i / VeLodties orb \. | 

[/ Horizontal Sections. NJ 

! ! 



a, 5 e < e> 


f 







? f f 










// x" x'" x^ 


^ 














p 


^ // / X / 




X 








\ 


\ 




I ' ' /' / / 

S .'/ / x ' <7 






/" 




\ 








50 ill / / 

11 \\ < : 


! 


1 




/7? //? 


- 




i 


9 
K 
L 


\\\. \ \ 

V\ \ N \ 


a 


6 


V 


c <i. & 


j 








\ V X N N N ' 




N^. 








y 


/ 


. 


V \l\_ NV -W ^ ^ 




^ 








^-- 




,/ 



Fig. 120. 

Tt will be seen that the maximum velocity does not occur in 
the free surface of the water, but on the central vertical section 
at some distance from the surface, and that the surface velocity 
may be very different from the mean velocity. As the maximum 
velocity does not occur at the surface, it would appear that in 

L. H. H 



HYDRAULICS 

assuming the wetted perimeter to be only the wetted surface of 
the channel, some error is introduced. That the air has not the 
same influence as if the water were in contact with a surface 
similar to that of the sides of the channel, is very clearly 
shown by comparing the curves of equal velocity for the closed 
rectangular channel as shown in Fig. 119 with those of Fig. 120. 
The air resistance, no doubt, accounts in some measure for the 
surface velocity not being the maximum velocity, but that it does 
not wholly account for it is shown by the fact that, whether the 
wind is blowing up or down stream, the maximum velocity is still 
below the surface. M. Flamant* suggests as the principal reason 
why the maximum velocity does not occur at the surface, that 
the water is less constrained at the surface, and that irregular 
movements of all kinds are set up, and energy is therefore 
utilised in giving motions to the water not in the direction of 
translation. 

Depth on any vertical at which the velocity is equal to the mean 
velocity. Later is discussed, in detail, the distribution of velocity 
on the verticals of any cross section, and it will be seen, that if u 
is the mean velocity on any vertical section of the channel, the 
depth at which the velocity is equal to the mean velocity is about 
0'6 of the total depth. This depth varies with the roughness of 
the stream, and is 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, having beds of fine sand, and from *55 
to '66 in large rivers from 1 to 3j- feet deep and having strong 
bedst. 

As the banks of the stream are approached, the point at which 
the mean velocity occurs falls nearer still to the bed of the stream, 
but if it falls very low there is generally a second point near the 
surface at which the velocity is also equal to the mean velocity. 

When the river is covered with ice the maximum velocity of 
the current is at a depth of '35 to '45 of the total depth, and the 
mean velocity at two points at depths of '08 to '13 and '68 to '74 
of the total depth J. 

If, therefore, on various verticals of the cross section of a stream 
the velocity is determined, by means of a current meter, or Pitot 
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 the 
vertical. 

'* Hydrauliqne. 

t Le Genie Civil, April, 1906, " Analysis of a communication by Murphy to 
the Hydrological section of the Institute of Geology of the United States." 
J Cunningham, Experiments on the Ganges Canal. 



FLOW IN OPEN CHANNELS 



211 



The total discharge can then be found, approximately, by 
dividing the cross section into a number of rectangles, such as 
abcdy Fig. 120 a, and multiplying the area of the rectangle by the 
velocity measured on the median line at 0'6 of its depth. 



cu d 




Fig. 120 a. 

The flow of the Upper Nile has recently been determined in 
this way. 

Captain Cunningham has given several formulae, for the mean 
velocity u upon a vertical section, of which two are here quoted. 

(1), 
(2), 

V being the velocity at the surface, v 3. the velocity at f of the depth, 
v at one quarter of the depth, and so on. 

131. Form of the curve of velocities on a vertical 
section. 

M. Bazin* and Cunningham have both taken the curve of 
velocities upon a vertical section as a parabola, the maximum 
velocity being at some distance h m 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 
or 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 
h t and H the depth of the stream. 

Bazin found that, if the stream is wide compared to its depth, 
the relationship between v, Y, h, and i the slope, is expressed by 
the formula, 



=v-ft()vm ax 

k being a numerical coefficient, which has a nearly constant value 
of 36'2 when the unit of length is one foot. 

* Recherches Hydraulique, p. 228 ; Annales des Fonts et Chaussges, 2nd Vol.. 

1875. 



142 



212 HYDRAULICS 

To determine the depth on any vertical at which the velocity is 
equal to the mean velocity. Let u be the mean velocity on any 
vertical section, and h u the depth at which the velocity is equal to 
the mean velocity. 

The discharge through a vertical strip of width dl is 

rH 

v .dh. 
o 

/H / i 
Therefore uTL 



and A = V-<sH* (2). 

Substituting u and h u in (1) and equating to (2), 



and h u 

This depth, at which the velocity is equal to the mean velocity, 
is determined on the assumption that Jc is constant, which is only 
true for sections very near to the centre of streams which are 
wide compared with their depth. 

It will be seen from the curves of Fig. 120 that the depth at 
which the maximum velocity occurs becomes greater as the sides 
of the channel are approached, and the law of variation of velocity 
also becomes more complicated. M. Bazin also found that the 
depth at the centre of the stream, at which the maximum velocity 
occurs, depends upon the ratio of the width to the depth, the 
reason apparently being that, in a stream which is wide compared 
to its depth, the flow at the centre is but slightly affected by the 
resistance of the sides, but if the depth is large compared with the 
width, the effect of the sides is felt even at the centre of the 
stream. The farther the vertical section considered is removed 
from the centre, the effect of the resistance of the sides is 
increased, and the distribution of velocity is influenced to a 
greater degree. This effect of the sides, Bazin expressed by 
making the coefficient k to vary with the depth h m at which 
the maximum velocity occurs. 

The coefficient is then, 

36'2 



Further, the equation to the parabola can be written in terms 
of v m , the maximum velocity, instead of V. 



FLOW IN OPEN CHANNELS 213 

Thu3 , ^_36-27-p (3). 



The mean velocity u, upon the vertical section, is then, 
= i [*vdh 
36'2 



= m ~ 



Therefore 

36'2 



TT2 / 1 \ 

1 \f W 

When v = u, Ji = h u> 

-i it c J. fljn fT/u ^'ibu'l'm 

and therefore, o ~ TJ = xfa TTT~ 

o 3 M H 

The depth h m at which the velocity is a maximum is generally 
less than *2H, except very near the sides, and h u is, therefore, not 
very different from *6H, as stated above. 

Ratio of maximum velocity to the mean velocity. From 
equation (4), 

v m =u + 



/i_M 2 V3 H 

V H/ 

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 
from H, and since the mean velocity of flow through the section is 
C \/m? and is approximately equal to u, therefore, 

36-2 /I h m h m *\ 
h m \ 2 \3 H HV* 



u 



Assuming h m to vary from to "2 and C to be 100, varies 

u 

from 1'12 to 1'09. The ratio of maximum velocity to mean 
velocity is, therefore, probably not very different from 1*1. 

132. The slopes of channels and the velocities allowed 
in them. 

The discharge of a channel being the product of the area and 
the velocity, a given discharge can be obtained by making the 
area small and the velocity great, or vice versa. And since the 
velocity is equal to Cvwt, a given velocity can be obtained by 



214 HYDRAULICS 

varying either m or i. Since m will in general increase with the 
area, the area will be a minimum when i is as large as possible. 
But, as the cost of a channel, including land, excavation and 
construction, will, in many cases, be almost proportional to its 
cross sectional area, for the first cost to be small it is desirable 
that i should be large. It should be noted, however, that the 
discharge is generally increased in a greater proportion, by an 
increase in A, than for the same proportional increase in i. 

Assume, for instance, the channel to be semicircular. 

The area is proportional to d?, and the velocity v to \/d . i. 

Therefore Q oc d? *Jdi. 

IfjZ_is kept constant and i doubled, the discharge is increased 
to \/2Q, but if d is doubled, i being kept constant, the discharge 
will be increased to 5'6Q. The maximum slope that can be given 
will in many cases be determined by the diif erence in level of the 
two points connected by the channel. 

When water is to be conveyed long distances, it is often 
necessary to have several pumping stations en route, as sufficient 
fall cannot be obtained to admit of the aqueduct or pipe line being 
laid in one continuous length. 

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. The 
slope may be as high as 1 in 1000, but should not, only in excep- 
tional circumstances, be less than 1 in 10,000. 

In Table XXXIY are given the slopes and the maximum 
velocities in them, of a number of brick and masonry lined 
aqueducts and earthen channels, from which it will be seen that 
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 and 
masonry lined aqueducts, and from 1 in 300 to 1 in 20,000 for the 
earth channels. The slopes of large natural streams are in some 
cases even less than 1 in 100,000. If the velocity is too small 
suspended matter is deposited and slimy growths adhere to the sides. 

It is desirable that the smallest velocity in the channel shall be 
such, that the channel is "self-cleansing," and as far as possible 
the growth of low forms of plant life prevented. 

In sewers, or channels conveying unfiltered waters, it is 
especially desirable that the velocity shall not be too small, and 
should, if possible, not be less than 2 ft. per second. 

TABLE XXXIY. 

Showing the slopes of, and maximum velocities, as determined 
experimentally, in some existing channels. 



FLOW IN OPEN CHANNELS 



215 



Smooth aqueducts 




Slope 


Maximum velocity 


New Croton aqueduct -0001326 


3 ft. per second 


Sudbury aqueduct '000189 


2-94 


Glasgow aqueduct '000182 


2-25 


Paris Dhuis '000130 




Avre, 1st part -0004 




2nd part '00033 




Manchester Thirlmcre '000315 




Naples -00050 


4-08 


Boston Sewer '0005 


3'44 


000333 


4-18 ... 


Earth channels. 




Slope Maximum velocity Lining 


Ganges canal -000306 4-16 ft. 


per second earth 


Escher -003 4'08 





Linth -00037 5'53 


( gravel and 


Cavour '00033 3'42 


\ some stones 


Simmen '0070 3'74 


earth 


Chazilly cut '00085 1'70 


( earth, stony, 


MarseiUes canal '00043 1'70 


( few weeds 


Chicago drainage canal 




(of the bottom of the canal) '00005 3 


> j> 



TABLE XXXV. 

Showing for varying values of the hydraulic mean depth m, the 
minimum slopes, which brick channels and glazed earthenware 
pipes should have, that the velocity may not be less than 2 ft. 
per second. 

m feet slope 



1 


1 i 


Q 93 


2 


1 


275 


3 


1 


510 


4 


1 


775 


5 


1 


1058 


6 


1 


1380 


8 


1 , 


, 2040 


1-0 


1 


, 2760 



mfeet 




slope 


1-25 


1 i 


n 3700 


1-5 


1 


4700 


1-75 


1 


5710 


2-0 


1 


6675 


2-5 


1 


9000 


3'0 


1 


11200 


4-0 


1 


15850 



The slopes are calculated from the formula 

157-5 




The value of y is taken as 0'5 to allow for the channel becoming 
dirty. For the minimum slope for any other velocity v, multiply 

(2\ 2 
-j . For example, the minimum slope 

for a velocity of 3 feet per second when m is 1, is 1 in 1227. 



210 HYDRAULICS 

Velocity of flow in, and slope of earth channels. If the velocity 
is high, in earth channels, the sides and bed of the channel are 
eroded, while on the other hand if it is too small, the capacity of 
the channel will be rapidly diminished by the deposition of sand 
and other suspended matter, and the growth of aquatic plants. 
Du Buat gives '5 foot per second as the minimum velocity that 
mud shall not be deposited, while Belgrand allows a minimum 
of '8 foot per second. 

TABLE XXXVI. 

Showing the velocities above which, according to Du Buat, 
and as quoted by Rankine, erosion of channels of various materials 
takes place. 

Soft clay 0-25 ft. per second 

Fine sand 0'50 

Coarse sand and gravel as large as peas 0*70 

Gravel 1 inch diameter 2'25 

Pebbles 1| inches diameter 3'33 

Heavy shingle 4-00 

Soft rock, brick, earthenware 4'50 

Rock, various kinds 6*00 and upwards 

133. Sections of aqueducts and sewers. 

The forms of sections given to some aqueducts and sewers are 
shown in Figs. 121 to 131. In designing such aqueducts and 
sewers, consideration has to be given to problems other than the 
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 aqueduct 
is to be lined, or cut in solid rock, must be considered. In many 
cases it is desirable that the aqueduct or sewer should have such 
a form that a man can conveniently walk along it, although its 
sectional area is not required to be exceptionally large. In 
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 diminishes, and the velocity remains nearly constant 
for all, except very small, discharges. This is important, 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 



aqueduct taken under a stream or other obstruction, and the 
aqueduct 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 






Fig. 121. 



Fig. 122. 



Fig. 123. 




Fig. 127. 



Fig. 128. 



Fig. 129. 





Fig. 130. 



Fig. 131. 



218 HYDRAULICS 

long, is made up of two parts. The first is a masonry conduit of 
the section shown in Fig. 121, 23'9 miles long and having a slope 
of '0001326, the second consists almost entirely of a brick lined 
siphon 6'83 miles long, 12' 3" diameter, the maximum head in 
which is 126 feet, and the difference in level of the two ends is 
6*19 feet. In such cases, however, the siphon is frequently made 
of steel, or cast-iron pipes, as in the case of the new Edinburgh 
aqueduct (see Fig. 131) which, where it crosses the valleys, is 
made of cast-iron pipes 33 inches diameter. 

135. The best form of channel. 

The best form of channel, or channel of least resistance, is 
that which, for a given slope and area, will give the maximum 
discharge. 

Since the mean velocity in a channel of given slope is propor- 

i 
tional to p , and the discharge is A . v, the best form of channel for 

a given area, is that for which P is a minimum. 

The form of the channel which has the minimum wetted peri- 
meter for a given area is a semicircle, for which, if r is the radius, 

7* 

the hydraulic mean depth is ~. 

More convenient forms, for channels to be excavated in rock 
or earth, are those of the rectangular or trapezoidal section, 
Fig. 133. For a given discharge, the best forms for these 
channels, will be those for which both A and P are a minimum ; 
that is, when the differentials dA and dP are respectively equal to 
zero. 

Rectangular channel. Let L be the width and Ji the depth, 
Fig. 132, of a rectangular channel ; it is required to find the ratio 

y that the area A and the wetted perimeter P may both be a 

ri 

minimum, for a given discharge. 

A-Lfc, 

therefore 8A = /t . 8L + L3ft = (1), 

P-L+2&, 

therefore dP = dL + 2dh = (2). 

Substituting the value of 3L from (2) in (1), 

~L = 2h. 

2tf h 
Therefore m = ~4fa = 2' 

Since L = 2h t the sides and bottom of the channel touch a circle 
having h as radius and the centre of which is in the free surface 
of the water. 



FLOW IN OPEN CHANNELS 



219 



Earth channels of trapezoidal form. In Fig. 133 let 

Z be the bottom width, 

h the depth, 

A the cross sectional area FBCD, 

P the length of FBCD or the wetted perimeter, 

i the slope, 

and let the slopes of the sides be t horizontal to one vertical; CG 
is then equal to th and tan CDGr = t. 





-H 



Fig. 132. 



Fig. 133. 



Let Q be the discharge in cubic feet per second. 
Then A. 



(3), 
(4), 



and 



For the channel to be of the best form dP and dA. both equal 
zero. 

From (3) A = hl+th 2 , 

and therefore dA. = hdl + ldh + 2thdh = Q (6). 

From (4) P = I + 2hJt 2 + l 
and dP = dl + 2<J& + ldh = (7). 

Substituting the value of dl from (7) in (6) 

l = 2h>J^l-2th (8). 



Therefore, 



m 



l-2ht 



h 

2' 



Let be the centre of the water surface FD, then since from (8) 

I + th = Wf + 1, 
therefore, in Fig. 133 CD = EG - OD. 



220 HYDRAULICS 

Draw OF and OE perpendicular to CD and BC respectively. 

Then, because the angle OFD is a right angle, the angles CDG 
and FOD are equal ; and since OF = OD cos FOD, and DG = OE, 
and DG = CDcosCDG, therefore, OE = OF; and since OEC and 
OFC are right angles, a circle with as centre will touch the sides 
of the channel, as in the case of the rectangular channel. 

136. Depth of flow in a channel of given form that, 
(a) the velocity may be a maximum, (b) the discharge may 
be a maximum. 

Taking the general formula 

. k.v* 

l = ~^~ 

i P 

and transposing, v = j- 

For a given slope and roughness of the channel v is, therefore, 
proportional to the hydraulic mean depth and will be a maximum 
when m is a maximum. 

That is, when the differential of ^ is zero, or 



(1). 

For maximum discharge, A.V is a maximum, and therefore, 
P 

A /A\". 

A . ( p ] is a maximum. 

Differentiating and equating to zero, 

Q... ...(2). 



n n 

Affixing values to n and p this differential equation can be 
solved for special cases. It will generally be sufficiently accurate 
to assume n is 2 and p = 1, as in the Chezy formula, then 

n + p_S 
n ~2> 
and the equation becomes 

3PdA-AdP = ........................... (3). 

137. Depth of flow in a circular channel of given 
radius and slope, when the velocity is a maximum. 

Let r be the radius of the channel, and 2< the angle subtended 
by the surface of the water at the centre of the channel, Fig. 134. 



FLOW IN OPEN CHANNELS 221 

Then the wetted perimeter 



and dP = 2rd<f>. 

The area A = r 2 0-r 2 sin0 cos = 
and dA. = r*d<l>- r 2 cos 20 d0. 

Substituting these values of dP and dA. in equation (3), 
section 136, 

tan 20 = 20. 

The solution in this case is obtained 
directly as follows, 



m 



A_r /- sin 20\ 
P~2V 1 ~ 2+ }* 




This will be a maximum when sin 20 
is negative, and 

sin 20 

20 

is a maximum, or when Fig. 134. 

d /sin20\ 
d+\ 2<f> /~ U ' 
.'. 20 cos 20 -sin 20 = 0, 
and tan 20 = 2^. 

The solution to this equation, for which 20 is less than 360, is 

20 = 257 27'. 

Then A = 2'73V, 

P = 4'494r, 
m = '608r, 
and the depth of flow d = T626r. 

138. Depth of flow in a circular channel for maximum 
discharge. 

Substituting for dP and dA in equation (3), section 136, 

6^0(^0 - 6^0 cos 20d0 - 2r 3 0d0 + r 3 sin 20d0 = 0, 
from which 40 - 60 cos 20 + sin 20 = 0, 

and therefore = 154. 

Then A = 3'08r 2 , 

P = 5'30r, 



and the depth of flow d = l'899r. 

Similar solutions can be obtained for other forms of channels, 
and may be taken by the student as useful mathematical exercises 
but they are not of much practical utility. 



222 



HYDRAULICS 



139. Curves of velocity and discharge for a given 
channel. 

The depth of flow for maximum velocity, or discharge, can be 
determined very readily by drawing curves of velocity and dis- 
charge for different depths of flow in the channel. This method 
is useful and instructive, especially to those students who are not 
familiar with the differential calculus. 

As an example, velocities and discharges, for different depths 
of flow, have been calculated for a large aqueduct, the profile of 
which is shown in Fig. 135, and the slope i of which is (V0001326. 
The velocities and discharges are shown by the curves 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. 

The velocities were calculated by the formula 

v = C *Jmi, 

using values of C from column 3, Table XXI. 

It will be seen that the velocity does not vary very much for 
all depths of flow greater than 3 feet, and that neither the velocity 
nor the discharge is a maximum 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 much more rapidly 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 IN OPEN CHANNELS 223 

velocity. A circle is shown on the figure which gives the same 
maximum discharge. 

The student should draw similar curves for the egg-shaped 
sewer or other form of channel. 

140. Applications of tne formula. 

Problem 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 
logarithmic formula or by Bazin's formula. 

The only difficulty that presents itself, is to affix values to k, n, and p in the 
logarithmic formula or to y in Bazin's formula. 

(1) By the logarithmic formula. 

First assign some value to fc, n, and p by comparing the lining of the channel 
with those given in Tables XXIV to XXXIII. Let w be the cross sectional area of 
the water. 

k v n 

Then since i = , 

mP ' 

log v = - log i + log m - - log fe, 

and Q = b).v, 

or logQ = logw + -logi+^logm log k. 

(2) By the Chczy formula, using Bazin's coefficient. 

The coefficient for a given value of m must be first calculated from the formula 



or taken from Table XXI. 
Then 

and 

Example. Determine the flow in a circular culvert 9 ft. diameter, lined with 
smooth brick, the slope being 1 in 2000, and the channel half full. 

Area _ d_ 

-Wetted perimeter -4- 
(1) By the logarithmic formula 




Therefore, log = j log -0005 + log 2-25 - - log '00007, 
v = 4'55 ft. per sec., 
w = 7 Ll^ = 31-8 sq.ft., 

Q = 145 cubic feet per sec. 
(2) By the Chezy formula, using Bazin's coefficient, 




-43 ft. per sec. 
Q = 31-8 x 4-43 = 141 cubic ft. per sec. 



224 HYDRAULICS 

Problem 2. To find the diameter of a circular channel of given slope, for which 
the maximum discharge is Q cubic feet per second. 

The hydraulic mean depth m for maximum discharge is '573r (section 138) and 
A = 3-08r 2 . 

Then the velocity is v=-757C*JrT, 

and Q = 2-37 Cr*^, 

1 /O 2 " 

therefore r - \/^-., 



and the diameter D = 1-42 . . 

The coefficient C is unknown, hut by assuming a value for it, an approximation 
to D can be obtained ; a new value for C can then be taken and a nearer approxi- 
mation to D determined ; a third value for C will give a still nearer approximation 
to D. 

Example. A circular aqueduct lined with concrete has a diameter of 5' 9" and 
a slope of 1 foot per mile. 

To find the diameter of two cast-iron siphon pipes 5 miles long, to be parallel 
with each other and in series with the aqueduct, and which shall have the same 
discharge; the difference of level between the two ends of the siphon being 12-5 feet. 

The value of m for the brick lined aqueduct of circular section when the 
discharge is a maximum is 573r = l-64 feet. 

The area A = 3-08^= 25 sq. feet. 

Taking C as 130 from Table XXI for the brick culvert and 110 for the cast-iron 
pipe from Table XII, then 



TO , 
Therefore 




d=4-00 feet. 



Problem 3. Having given the bottom width I, the slope t, and the side slopes t 
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 v = C 

and Q=A.v. 

A convenient formula to remember is the approximate formula for ordinary 
earth channels 




For values of m greater than 2, v as calculated from this formula is very nearly 
equal to v obtained by using Bazin's formula. 

, -00037V 2 ' 1 

The formula * = - rl 

m 15 

may also be used. 



FLOW IN OPEN CHANNELS 225 

Example. An ordinary earth channel has a width 1= 10 feet, a depth d = 4i'eet, 
and a slope i = ^oVi7' Side slopes 1 to 1. To find Q. 
A =56 sq. ft., 
P = 21-312 ft., 
= 2-628 ft., 




v = l'91 ft. per sec., 
Q = 107 cubic ft. per sec. 



From the formula 



v=l-8S ft. per sec., 
Q = 105-3 cubic ft. per sec. 
From the logarithmic formula 



r = l'9ft. per sec., 

Q = 106 4 cubic feet per sec. 

Problem 4. Having given the flow in a canal, the slope, and the side slopes, to 
find the dimensions of the profile and the mean velocity of flow, 
(a) When the canal is of the best form. 
(6) When the depth is given. 

In the first case m = - , and from equations (8) and (4) respectively, section 136 

6 



Therefore 
Substituting - for m 



and A 2 = fc 4 (2 

But t? = j = 

A. 



Therefore C 3 i = 

2 

and fc 5 = - = - .... ........ (1). 



A value for C should be chosen, say 0=70, and h calculated, from which a mean 
value for m = - can be obtained. 

A nearer approximation to h can then be determined by choosing a new value of C, 
from Table XXI corresponding to this approximate value of m, and recalculating 
h from equation (1). 

Example. An earthen channel to be kept in very good condition, having a slope 
of 1 in 10,000, and side slopes 2 to 1, is required to discharge 100 cubic feet 
per second ; to find the dimensions of the chaunel ; take C = 70. 

L. H. 15 



226 HYDRAULICS 

20,000 
Then ft 5 - 



20,000 
~ -49 x 6-1 
= 6700, 

and ft =5-4 feet. 

Therefore m = 2-l. 

From Table XXI, = 82 for this value of m, therefore a nearer approximation 
to ft is now found from 

., 20,000 20,000 



S'J- 



10,000 
from which h = 5'22 ft. and m = 2'61. 

The approximation is now sufficiently near for all practical purposes and may 
be taken as 5 feet. 

Problem 5. Having given the depth d of a trapezoidal channel, the slope i, and 
the side slopes t, to find the bottom width I for a given discharge. 
First using the Chezy formula, 

v = C*Jmi 



and 

The mean velocity 

Therefore ^ + ^ 

In this equation the coefficient C is unknown, since it depends upon the value 
of m which is unknown, and even if a value for C be assumed the equation cannot 
very readily be solved. It is desirable, therefore, to solve by approximation. 

Assume any value for m, and find from column 4, Table XXI, the corresponding 
value for C, and use these values of m and C. 

Then, calculate v from the formula 

Since T =V > 

A. 

and 

Therefore dl + td' 2 = - (1). 

v 

From this equation a value of I can be obtained, which will probably not be the 
correct value. 

With this value of I calculate a new value for m, from the formula 




For this value of m obtain a new value of G from the table, recalculate u, and 
by substitution in formula (1) obtain a second value for I. 

On now again calculating m by substituting for d in formula (2), it will generally 
be found that m differs but little from in previously calculated ; if so, the approxi- 
mation has proceeded sufficiently far, and d as determined by using this value of m 
will agree with the correct value sufficiently nearly for all practical purposes. 

The problem can be solved in a similar way by the logarithmic formula 



The indices u &nd p may be taken as 2-1, and 1'5 respectively, and k as '00037. 



FLOW IN OPEN CHANNELS 227 

Example. The depth of an ordinary earth channel is 4 feet, the side slopes 
1 to 1, the slope 1 in GOOO and the discharge is to be 7000 cubic feet per minute. 
Find the bottom width of the channel. 
Assume a value for m, say 2 feet. 
From the logarithmic formula 

2 -1 log v = log i+ 1-5 log m- 4-5682 ........................... (3), 

v = 1-122 feet per sec. 

Thcn A 

But 



Substituting this value for I in equation (2) 

6 



Becalculating v from formula (3) 

v = 1-556. 
Then A = 75 feet, 

1= 14-75 feet, 
and m = 2-88 feet. 

The first value of Zis, therefore, too large, and this second value is too small. 
Third values were found to be v = l'455, 

A = 80- 2, 
1 = 16-05, 
m=2-935. 
This value of I is again too large. 

A fourth calculation gave v = 1-475, 

A=79-2, 

J=15'8, 

m=2-92. 

The approximation has been carried sufficiently far, and even further than is 
necessary, as for such channels the coefficient of roughness k cannot be trusted to 
an accuracy corresponding to the email difference between the third and fourth 
values of I. 

Problem 6. Having given the bottom width I, the slope t and the side slopes of 
a trapezoidal channel, to find the depth d for a given discharge. 

This problem is solved exactly as 5 above, by first assuming _a_value for m, and 
calculating an approximate value for v from the formula v = C^Jmi. 

Then, by substitution in equation (1) of the last problem and solving the 
quadratic, 



oy substituting this value for d in equation (2), a new value for m can be found, 
and hence, a second approximation to d, and so on. 

Using the logarithmic formula the procedure is exactly the same as for 
problem 5. 

Problem 1 *. Having a natural stream BC, Fig. 135 a, of given slope, it is required 
to determine the point C, at which a canal, of trapezoidal section, which is to 
deliver a definite quantity of water to a Riven point A at a given level, shall be 
made to join the stream so that the cost of the canal is a minimum. 

* The solution here given is practically the same as that given by M. Flamant 
in his excellent treatise Hydraulique. 

152 



223 HYDRAULICS 

Let I be the slope of the stream, i of the canal, h the height above some datum 
of the surface of the water at A, and ft, of the 
water in the stream at B, at some distance L 
from C. 

Let L be also the length and A the !T> \s~* 

sectional area of the canal, and let it be j j ~* 

assumed that the section of the canal is of the A *- ' j 

most economical form, or m = - . 

& -fig- loo a. 

The side slopes of the canal will be fixed 

according to the nature of the strata through which the canal is cut, and may be 
supposed to be known. 

Then the level of the water at C is 



h 7?i 
Therefore L = . . 

Let I be the bottom width of the canal, and t the slope of the sides. The cross 
section is then dl + td?, and 

_A_ dl + tcP 
~~ 



Substituting 2m for d, 

I = 4/ A/ 2 + 1 - 4tm, 

and therefore 



4m A/ t 2 H 
from which tn 2 = .- 



The coefficient C in the formula v = G*Jrni may be assumed constant. 
Then t? 2 =C 2 w, 

and v 4 =C 4 i 2 i 2 . 

For t? substituting ~ , and for w 2 the above value, 

_ C 4 Ai 2 
A 4 "^ 

and 

Therefore 

The cost of the canal will be approximately proportional to the product of the 
length L and the cross sectional area, or to the cubical content of the excavation. 
Let k be the price per cubic yard including buying of land, excavation etc. Let x 
be the total cost. 

Then * = &. L. A 




This will be a minimum when -^-=0. 
di 

Differentiating therefore, and equating to zero, 

I^IK-I, 

and i = ?I. 

The most economical slope is therefore $ of the slope of the natural stream. 

If instead of taking the channel of the best form the depth is fixed, the, 
slope 1 = ^.1. 



FLOW IN OPEN CHANNELS 229 

There have been two assumptions made in the calculation, neither of which is 
rigidly true, the first being that the coefficient C is constant, and the second that 
the price of the canal is proportional to its cross sectional area. 

It will not always be possible to adopt the slope thus found, as the mean 
velocity must be maintained within the limits given on page 216, and it is not 
advisable that the slope should be less than 1 in 10,000. 

EXAMPLES. 

(1) The area of flow in a sewer was found to be 0*28 sq. feet; tb.3 
wetted perimeter 1 '60 feet; the inclination 1 in 38*7. The mean velocity 
of flow was 6'12 feet per second. Find the value of G in the formula 



(2) The drainage area of a certain district was 19*32 acres, the whole 
area being impermeable to rain water. The maximum intensity of the 
rainfall was 0*360 ins. per hour and the maximum rate of discharge regis- 
tered in the sewer was 96% of the total rainfall. 

Find the size of a circular glazed earthenware culvert having a slope of 

1 in 50 suitable for carrying the storm water. 

(3) Draw a curve of mean velocities and a curve of discharge for an 
egg-shaped brick sewer, using Bazin's coefficient. Sewer, 6 feet high by 
4 feet greatest width ; slope 1 in 1200. 

(4) The sewer of the previous question is required to join into a main 
outfall sewer. To cheapen the junction with the main outfall it is thought 
advisable to make the last 100 feet of the sewer of a circular steel pipe 
3 feet diameter, the junction between the oval sewer and the pipe being 
carefully shaped so that there is no impediment to the flow. 

Find what fall the circular pipe should have so that its maximum 
discharge shall be equal to the maximum discharge of the sewer. Having 
found the slope, draw out a curve of velocity and discharge. 

(5) A canal in earth has a slope of 1 foot in 20,000, side slopes of 

2 horizontal to 1 vertical, a depth of 22 feet, and a bottom width of 
200 feet; find the volume of discharge. 

Bazin's coefficient -y=2*35. 

(6) Give the diameter of a circular brick sewer to run half -full for a 
population of 80,000, the diurnal volume of sewage being 75 gallons per 
head, the period of maximum flow 6 hours, and the available fall 1 in 1000. 

Inst. C. E. 1906. 

(7) A channel is to be cut with side slopes of 1 to 1 ; depth of water, 

3 feet; slope, 9 inches per mile: discharge, 6,000 cubic feet per minute. 
Find by approximation dimensions of channel. 

(8) An area of irrigated land requires 2 cubic yards of water per hour 
per acre. Find dimensions of a channel 3 feet deep and with a side slope 
of 1 to 1. Fall, 1 feet per mile. Area to be irrigated, 6000 acres. (Solve 
by approximation.) y=2*35. 

(9) A trapezoidal channel in earth of the most economical form has a 
depth of 10 feet and side slopes of 1 to 1. Find the discharge when the 
slope is 18 inches per mile. y=2*35. 



230 HYDRAULICS 

(10) A river has the following section : top -width, 800 feet ; depth of 
water, 20 feet ; side slopes 1 to 1 ; fall, 1 foot per mile. Find the discharge, 
using Bazih's coefficient for earth channels. 

(11) A channel is to be constructed for a discharge of 2000 cubic feet 
per second ; the fall is 1^ feet per mile ; side slopes, 1 to 1 ; bottom width, 
10 times the depth. Find dimensions of channel. Use the approximate 



formula, v= 

(12) Find the dimensions of a trapezoidal earth channel, of the most 
economical form, to convey 800 cubic feet per second, with a fall of 2 feet 
per mile, and side slopes, 1 to 1. (Approximate formula.) 

(13) An irrigation channel, with side slopes of l to 1, receives 600 
cubic feet per second. Design a suitable channel of 3 feet depth and 
determine its dimensions and slope. The mean velocity is not to exceed 
2| feet per second. y=2'35. 

(14) A canal, excavated in rock, has vertical sides, a bottom width of 
160 feet, a depth of 22 feet, and the slope is 1 foot in 20,000 feet. Find the 
discharge, y = 1*54. 

(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 line 
is 290 feet and its depth 22 feet. 

Find the slope this portion of the canal should have, taking y as 2'35. 

(16) An aqueduct 95| miles long is made up of a culvert 50 miles 
long and two steel pipes 3 feet diameter and 45 miles long laid side by side. 
The gradient of the culvert is 20 inches to the mile, and of the pipes 2 feet 
to the mile. Find the dimensions of a rectangular culvert lined with well 
pointed brick, so that the depth of flow shall be equal to the width of the 
culvert, when the pipes are giving their maximum discharge. 

Take for the culvert the formula 

._ -000061 yw 
m ' 
and for the pipes the formula 

, -00050. v 2 



(17) The Ganges canal at Taoli was found to have a slope of 0*000146 
and its hydraulic mean depth m was 7'0 feet ; the velocity as determined 
by vertical floats was 2'80 feet per second; find the value of C and the 
value of y in Bazin's equation. 

(18) The following data were obtained from an aqueduct lined with 
brick carefully pointed : 

m i v 

in metres in metres per sec. 

229 0-0001326 '336 

381 '484 

533 '596 

686 "691 

838 '769 

991 '848 

1-143 -913 

1-170 -922 



FLOW IN OPEN CHANNELS 231 

Plot -j-= as ordinates, as abscissae ; find values of a and /3 in Bazin's 

Vm v 

formula, and thus deduce a value of y for this aqueduct. 

(19) An aqueduct 107 miles long consists of 13 miles of siphon, and 
the remainder of a masonry culvert 6 feet 10 ^ inches diameter with a gradient 
of 1 in 8000. The siphons consist of two lines of cast-iron pipes 43 inches 
diameter having a slope of 1 in 500. Determine the discharge. 

(20) An aqueduct consists partly of the section shown in Fig. 131, 
page 217, and partly (i.e. when crossing valleys) of 33 inches diameter cast- 
iron pipe siphons. 

Determine the minimum slope of the siphons, so that the aqueduct 
may discharge 15,000,000 gallons per day, and the slope of the masonry 
aqueduct so that the water shall not be more than 4 feet 6 inches deep in 
the aqueduct. 

(21) Calculate the quantity delivered by the water main in question (30) , 
page 172, per day of 24 hours. 

This amount, representing the water supply of a city, is discharged into 
the sewers at the rate of one-half the total daily volume in 6 hours, and is 
then trebled by rainfall. Find the diameter of the circular brick outfall 
sewer which will carry off the combined flow when running half full, the 
available fall being 1 in 1500. Use Bazin's coefficient for brick channels. 

(22) Determine for a smooth cylindrical cast-iron pipe the angle 
subtended at the centre by the wetted perimeter, when the velocity of flow 
is a maximum. Determine the hydraulic mean depth of the pipe under 
these conditions. Lond. Un. 1905. 

(23) A 9-inch drain pipe is laid at a slope of 1 in 150, and the value of 
c is 107 (v=cvW). Find a general expression for the angle subtended at 
the centre by the water line, and the velocity of flow; and indicate how the 
general equations may be solved when the discharge is given. Lond. Un. 
1906. 

141. Short account of the historical development of the pipe and channel formulae. 
It seems remarkable that, although the practice of conducting water along pipes 
and channels for domestic and other purposes has been carried on for many 
centuries, no serious attempt to discover the laws regulating the flow seems 
to have been attempted until the eighteenth century. It seems difficult to realise 
how the gigantic schemes of water distribution of the ancient cities could have been 
executed without such knowledge, but certain it is, that whatever information they 
possessed, it was lost during the middle ages. 

It is of peculiar interest to note the trouble taken by the Roman engineers in 
the construction of their aqueducts. In order to keep the slope constant they 
tunnelled through hills and carried their aqueducts on magnificent arches. The 
Claudian aqueduct was 38 miles long and had a constant slope of five feet per mile. 
Apparently they were unaware of the simple fact that it is not necessary for a pipe 
or aqueduct connecting two reservoirs to be laid perfectly straight, or else they 
wished the water at all parts of the aqueducts to be at atmospheric pressure. 

Stephen Schwetzer in his interesting treatise on hydrostatics and hydraulics 
published in 1729 quotes experiments by Marriott showing that, a pipe 1400 yards 
long, 1| inches diameter, only gave of the discharge which a hole If inches diameter 
in the side of a tank would give under the same head, and also explains that the 
motion of the liquid in the pipes is diminished by friction, but he is entirely 
ignorant of the laws regulating the flow of fluids through pipes. Even as late as 



232 HYDRAULICS 

1786 Du Buat* wrote, "We are yet in absolute iterance of the laws to which the 
movement of water is subjected." 

The earliest recorded experiments of any valne on long pipes are those of 
Couplet, in which he measured the flow through the pipes which supplied the 
famous fountains of Versailles in 1732. In 1771 Abb Bossut made experiments on 
flow in pipes and channels, these being followed by the experiments of Du Buat, who 
erroneously argued that the loss of head due to friction in a pipe was independent 
of the internal surface of the pipe, and gave a complicated formula for the velocity 
of flow when the head and the length of the pipe were known. 

In 1775 M. Chezy from experiments upon the flow in an open canal, came to 
the conclusion that the fluid friction was proportional to the velocity squared, and 
that the slope of the channel multiplied by the cross sectional area of the stream, 
was equal to the product of the length of the wetted surface measured on the cross 
section, the velocity squared, and some constant, or 

iA=Pat> 2 (1), 

t being the slope of the bed of the channel, A the cross sectional area of the stream, 
P the wetted perimeter, and a a coefficient. 

From this is deduced the well-known Chezy formula 



Prony f, applying to the flow of water in pipes the results of the classical experi- 
ments of Coulomb on fluid friction, from which Coulomb had deduced the law that 
fluid friction was proportional to av + bv 2 , arrived at the formula 



\ v 
This is similar to the Chezy formula, ( - + /3 j being equal to ^. 

By an examination of the experiments of Couplet, Bossut, and Du Buat, Prony 
gave values to a and which when transformed into British units are, 

a = -00001733, 
/S= -00010614. 

For velocities, above 2 feet per second, Prony neglected the term containing the 
first power of the velocity and deduced the formula 



He continued the mistake of Du Buat and assumed that the friction was in- 
dependent of the condition of the internal surface of the pipe and gave the following 
explanation : " When the fluid flows in a pipe or upon a wetted surface a film of 
fluid adheres to the surface, and this film may be regarded as enclosing the mass 
of fluid in motion J." That such a film encloses the moving water receives support 
from the experiments of Professor Hele Shaw. The experiments were made upon 
such a smafl scale that it is difficult to say how far the results obtained are indica- 
tive of the conditions of flow in large pipes, and if the film exists it does not seem 
to act in the way argued by Prony. 

TT 

The value of t in Prony's formula was equal to , H including, not only the 

loss of head due to friction but, as measured by Couplet, Bossut and Du Buat, 
it also included the head necessary to give velocity to the water and to overcome 
resistances at the entrance to the pipe. 

Eytelwein and also Aubisson, both made allowances for these losses, by sub- 

tracting from H a quantity ^- , and then determined new values for a and b in the 
formula 



Le Discours prgliminaire de ses Principes d'hydraiiJique. 

t See also Girard's Movement des fluids dans Ics tubes capillaires, 1817. 

J Traite d'hydraulique. Engineer, Aug. 1897 and May 18U8. 



FLOW IN OPEN CHANNELS 233 

They gave to a and 6 the following values. 

Eytelwein a = -000023534, 

6= 000085434. 

Anbisson* o =-000018837, 

6= -000104392. 

By neglecting the term containing v to the first power, and transforming the 
terms, Aubisson's formula reduces to 



Young, in the Encyclopaedia Britannica, gave a complicated formula for v when 
FT and d were known, but gave the simplified formula, for velocities such as 
are generally met with in practice, 



St Tenant made a decided departure by making - proportional to V instead of 

to r 2 as in the Chezy formula. 

When expressed "in English feet as units, his formula becomes 

v= 206 (mi) A, 
Weisbach by an examination of the early experiments together with ten others by 

himself and one by M. Gueynard gave to the coefficient a in the formula h= 

the value 



that is, he made it to vary with the velocity. 

Then, mi 



the values of a and being a =0*0144, 

0=0-01716. 

From this formula tables were drawn up by Weisbach, and in England by 
Hawkesley, which were considerably used for calculations relating to flow of 
water in pipes. 

Darcy, as explained in Chapter V, made the coefficient a to vary with the 
diameter, and Hagen proposed to make it vary with both the velocity and the 
diameter. 

Lis formula then became m* = i 



The formulae of Ganguillet and Kutter and of Basin have been given in 
Chapters V and VI. 

Dr Lampe from experiments on the DanUig mains and other pipes proposed 

the formula 



thus modifying St Tenant's formula and anticipating the formulae of Beynoldi, 
Flamant and Unwin, in which, 

7** 

*~*> 9 
u and f being variable coefficients. 

* Iruite 



CHAPTER VII. 

*GAUGING THE FLOW OF WATER 

142. Measuring the flow of water by weighing. 

In the laboratory or workshop a flow of water can generally 
be measured by collecting the water in tanks, and either by 
direct weighing, or by measuring the volume from the known 
capacity of the tank, the discharge in a given time can be 
determined. This is the most accurate method of measuring 
water and should be adopted where possible in experimental 
work. 

In pump trials or in measuring the supply of water to boilers, 
determining the quantity by direct weighing has the distinct 
advantage that the results are not materially affected by 
changes of temperature. It is generally necessary to have two 
tanks, one of which is filling while the other is being weighed 
and emptied. For facility in weighing the tanks should stand 
on the tables of weighing machines. 

143. Meters. 

Linert meter. An ingenious direct weighing meter suitable for 
gauging practically any kind of liquid, is constructed as shown in 
Figs. 136 and 137. 

It consists of two tanks A 1 and A 2 , each of which can swing 
on knife edges BB. The liquid is allowed to fall into a shoot F, 
which swivels about the centre J, and from which it falls into 
either A 1 or A 2 according to the position of the shoot. The tanks 
have weights D at one end, which are so adjusted that when a 
certain weight of water has run into a tank, it swings over into 
the dotted position, Fig. 136, and flow commences through a 
siphon pipe 0. When the level of the liquid in the tank has 
fallen sufficiently, the weights D cause the tank to come back to 
its original position, but the siphon continues in action until the 
tank is empty. As the tank turns into the dotted position 

* See Appendix 10. 



GAUGING THE FLOW OF WATER 



235 



it suddenly tilts over the shoot F, and the liquid is discharged 
into the other tank. An indicator H registers the number of 
times the tanks are filled, and as at each tippling a definite weight 
of fluid is emptied from the tank, the indicator can be marked 
off in pounds or in any other unit. 





Fig. 13C. Fig. 137. 

Linert direct weighing meter. 

144. Measuring the flow by means of an orifice. 

The coefficient of discharge of sharp-edged orifices can be 
obtained, with considerable precision, from the tables of Chapter IV, 
or the coefficient for any given orifice can be determined for 
various heads by direct measurement of the flow in a given time, 
as described above. Then, knowing the coefficient of discharge at 
various heads a curve of rate of discharge for the orifice, as in 
Fig. 138, may be drawn, and the orifice can then be used to 
measure a continuous flow of water. 

The orifice should be made in the side or bottom of a tank. If 
in the side of the tank the lower edge should be at least one and 
a half to twice its depth above the bottom of the tank, and the 
sides of the orifice whether horizontal or vertical should be at 
least one and a half to twice the width from the sides of the tank. 
The tank should be provided with baffle plates, or some other 
arrangement, for destroying the velocity of the incoming water 
and ensuring quiet water in the neighbourhood of the orifice. The 
coefficient of discharge is otherwise indefinite. The head over the 
orifice should be observed at stated intervals. A head-time curve 
having head as ordinates and time as abscissae can then be plotted 
as in Fig. 139. 

From the head-discharge curve of Fig. 138 the rate of discharge 
can be found for any head h, and the curve of Fig. 139 plotted. 
The area of this curve between any two ordinates AB and CD, 



236 



HYDRAULICS 



which is the mean ordinate between AB and CD multiplied by the 
time t y gives the discharge from the orifice in time t. 

The head h can be measured by fixing a scale, having its zero 
coinciding with the centre of the orifice, behind a tube on the side 
of the tank. 



if^L 





Fig. 138. 




B Tbne 

Fig. 139. 





B 


A 




E 


D 






' X v 


A 



Fig. 140. 



145. Measuring the flow in open channels. 

Large open channels : floats. The oldest and simplest method 
of determining approximately the discharge in an open channel is 
by means of floats. 

A part of the channel as straight as possible is selected, and in 
which the flow may be considered as uniform. 

The readings should be taken on a calm day as a down-stream 
wind will accelerate the floats and an up-stream wind retard them. 

Two cords are stretched across the channel, as near to the 
surface as possible, and perpendicular to the direction of flow. The 
distance apart of the cords should be as great as possible consistent 
with uniform flow, and should not be less than 150 feet. From a 
boat, anchored at a point not less than 50 to 70 feet above stream, 
so that the float shall acquire before reaching the first line a 
uniform velocity, the float is allowed to fall into the stream and 



GAUGING THE FLOW OF WATER 237 

the time carefully noted by means of a chronometer at which it 
passes both the first and second line. If the velocity is slow, the 
observer may walk along the bank while the float is moving from 
one cord to the other, but if it is greater than 200 feet per minute 
two observers will generally be required, one at each line. 

A better method, and one which enables any deviation of the 
float from a path perpendicular to the lines to be determined, is, 
for two observers provided with box sextants, or theodolites, to be 
stationed at the points A and B, which are in the planes of the 
two lines. As the float passes the line AA at D, the observer 
at A signals, and the observer at B measures the angle ABD 
and, if both are provided with watches, each notes the time. 
When the float passes the line BB at E, the observer at B signals, 
and the observer at A measures the angle BAE, and both 
observers again note the time. The distance DE can then be 
accurately determined by calculation or by a scale drawing, and 
the mean velocity of the float obtained, by dividing by the time. 

To ensure the mean velocities of the floats being nearly equal 
to the mean velocity of the particles of water in contact with 
them, their horizontal dimensions should be as small as possible, 
so as to reduce friction, and the portion of the float above the 
surface of the water should be very small to diminish the effect of 
the wind. 

As pointed out in section 130, the distribution of velocity in 
any transverse section is not by any means uniform and it is 
necessary, therefore, to obtain the mean velocity on a number of 
vertical planes, by finding not only the surface velocity, but also 
the velocity at various depths on each vertical. 

146. Surface floats. 

Surface floats may consist of washers of cork, or wood, or 
other small floating bodies, weighted so as to just project above 
the water surface. The surface velocity is, however, so likely to 
be affected by wind, that it is better to obtain the velocity a 
short distance below the surface. 

147. Double floats. 

To measure the velocity at points below the surface double 
floats are employed. They consist of two bodies connected by 
means of a fine wire or cord, the upper one being made as small 
as possible so as to reduce its resistance. 

Gordon*, on the Irrawaddi, used two wooden floats connected 
by a fine fisning line, the lower float being a cylinder 1 foot long, 

* Proc. inst. C. E. t 1&93. 



238 HYDRAULICS 

and 6 inches diameter, hollow underneath and loaded with clay to 
sink it to any required depth ; the upper float, which swam on the 
surface, was of light wood 1 inch thick, and carried a small flag. 

The surface velocity was obtained by sinking the lower float 
to a depth of 3J feet, the velocity at this depth being not very 
different from the surface velocity and the motion of the float more 
independent of the effect of the wind. 




Fig. 141. Gurley's current meter. 

Subsurface velocities were measured by increasing the depths 
of the lower float by lengths of 3$ feet until the bottom was 
reached. 



GAUGING THE FLOW OF WATER 239 

Gordon has compared the results obtained by floats with those 
obtained by means of a current meter (see section 149). For 
small depths and low velocities the results obtained by double 
floats are fairly accurate, but at high velocities and great depths, 
the velocities obtained are too high. The error is from to 10 
per cent. 

Double floats are sometimes made with two similar floats, of 
the same dimensions, one of which is ballasted so as to float at any 
required depth and the other floats just below the surface. The 
velocity of the float is then the mean of the surface velocity 
and the velocity at the depth of the lower float. 

148. Rod floats. 

The mean velocity, on any vertical, may be obtained ap- 
proximately by means of a rod float, which consists of a long rod 
having at the lower end a small hollow cylinder, which may be 
filled with lead or other ballast so as to keep the rod nearly 
vertical. 

The rod is made sufficiently long, and the ballast adjusted, so 
that its lower end is near to the bed of the stream, and its upper 
end proj ects slightly above the water. Its velocity is approximately 
the mean velocity in the vertical plane in which it floats. 

149. The current meter. 

The discharge of large channels or rivers can be obtained most 
conveniently and accurately by determining the velocity of flow 
at a number of points in a transverse section by means of a current 
meter. 

The arrangement shown in Fig. 141 is a meter of the anemo- 
meter type. . A wheel is mounted on a vertical spindle and has 
five conical buckets. The spindle revolves in bearings, from 
which all water is excluded, and which are carefully made so 
that the friction shall remain constant. The upper end of the 
spindle extends above its bearing, into an air-tight chamber, and 
is shaped to form an eccentric. A light spring presses against 
the eccentric, and successively makes and breaks an electric 
circuit as the wheel revolves. The number of revolutions of the 
wheel is recorded by an electric register, which can be arranged 
at any convenient distance from the wheel. When the circuit is 
made, an electro-magnet in the register moves a lever, at the end 
of which is a pawl carrying forward a ratchet wheel one tooth 
for each revolution of the spindle. The frame of the meter, which 
is made of bronze, is pivoted to a hollow cylinder which can be 
clamped in any desired position to a vertical rod. At the right- 



240 HYDRAULICS 

hand side is a rudder having four light metal wings, which 
balances the wheel and its frame. When the meter is being used 
in deep waters it is suspended by means of a fine cable, and to 
the lower end of the rod is fixed a lead weight. The electric 
circuit wires are passed through the trunnion and so have no 
tendency to pull the meter out of the line of current. When 
placed in a current the meter is free to move about the horizontal 
axis, and also about a vertical axis, so that it adjusts itself to 
the direction of the current. 

The meters are rated by experiment and the makers recommend 
the following method. The meter should be attached to the bow 
of a boat, as shown in Fig. 142, and immersed in still water not 
less than two feet deep. A thin rope should be attached to the 
boat, and passed round a pulley in line with the course in which 
the boat is to move. Two parallel lines about 200 feet apart 
should be staked on shore and at right angles to the course of the 
boat. The boat should be without a rudder, but in the boat with 
the observer should be a boatman to keep the boat from running 




Fig. 142. 

into the shore. The boat should then be hauled between the two 
ranging lines at varying speeds, which during each passage should 
be as uniform as possible. With each meter a reduction table is 
supplied from which the velocity of the stream in feet per second 
can be at once determined from the number of revolutions recorded 
per second of the wheel. 

The Haskell meter has a wheel of the screw propeller type 
revolving upon a horizontal axis. Its mode of action is very 
similar to the one described. 

Comparative tests of the discharges along a rectangular canal 
as measured by these two meters and by a sharp-edged weir which 
had been carefully calibrated, in no case differed by more than 
5 per cent, and the agreement was generally much closer*. 

* Murphy on current Meter and Weir discharges, Proceedings Am.S.C.E., 
VoL xxvii., p. 779. 



GAUGING THE FLOW OF WATER 



241 



150. *Pitot tube. 

Another apparatus which can be used for determining the 
velocity at a point in a flowing stream, even when the stream is of 
small dimensions, as for example a small pipe, is called a Pitot 
tube. 

In its simplest form, as originally proposed by Pitot in 1732, 
it consists of a glass tube, with a 
small orifice at one end which may 
be turned to receive the impact of 
the stream as shown in Fig. 143. 
The water in the tube rises to a 
height h above the free surface of 
the water, the value of h depending 
upon the velocity v at the orifice of 
the tube. If a second tube is placed 




Fig. 143. Pitot tube. 



beside the first with an orifice parallel to the direction of flow, 
the water will rise in this tube nearly to the level of the free 
surface, the fall hi being due to a slight diminution in pressure 
at the mouth of the tube, caused probably by the stream lines 
having their directions changed at the mouth of the tube. A 
further 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 
pounds. 

The momentum is therefore - . a . v* pounds feet. 

y 

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 



and the pressure per unit area is 

wv 2 

9 
The equivalent head 

&=-- =- . 
wg 9 

According to this theory, the head of water in the tube, due to 
the impact, is therefore twice ~- , the head due to the velocity v, and 

' * See page 526. 



L. H. 



16 



242 



HYDRAULICS 



the water should rise in the tube to a height above the surface 
equal to h. Experiment shows however that the actual height 
the water rises in the tube is practically equal to the velocity 
head and, therefore, the velocity v of a mass of water w . a . v Ibs. 
is not destroyed by the pressure on the area of the tube. The 
head h is thus generally taken (see Appendix 4) as 

, cv* 



c being a coefficient for the tube, which experiment shows for 
a properly formed tube is constant and practically equal to 
unity. 

f* '?*^ C 75 

Similarly for given tubes hi =-~- and 7& 2 = -ST 

The coefficients are determined by placing the tubes in streams 
the velocities of which are known, or by attaching them to some 
body which moves through still water with a known velocity, and 
carefully measuring h for different velocities. 




B 





Fig. 144. 



Fig. 145. 



Darcy* was the first to use the Pitot tube as an instrument of 
precision. His improved apparatus as used in open channels con- 
sisted of two tubes placed side by side as in Fig. 144, the orifices 
in the tubes facing up-stream and down-stream respectively. The 



* Recherche* Hydrauliques, etc., 1857. 



GAUGING THE FLOW OF WATER 243 

two tubes were connected at the top, a cock C 1 being placed in the 
common tube to allow the tubes to be opened or closed to the 
atmosphere. At the lower end both tubes could be closed at the 
same time by means of cock C. When the apparatus is put into 
flowing water, the cocks C and C 1 being open, the free surface 
rises in the tube B a height hi and is depressed in D an amount 
hi. The cock C 1 is then closed, and the apparatus can be taken 
from the water and the difference in the level of the two columns, 

h = hi + hq, 
measured with considerable accuracy. 

If desired, air can be aspirated from the tubes and the columns 
made to rise to convenient levels for observation, without moving 
the apparatus. The difference of level will be the same, whatever 
the pressure in the upper part of the tubes. 

Fig. 145 shows one of the forms of Pitot tubes, as experimented 
upon by Professor Gardner Williams*, and used to determine 
the distribution of velocities of the water flowing in circular pipes. 

The arrangement shown in Fig. 146, is a modified form of the 
apparatus used by Freeman t to determine the distribution of 
velocities in a jet of water issuing from a fire hose under con- 
siderable pressure. As shown in the sketch, the small orifice 
receives the impact of the stream and two small holes Q are drilled 
in the tube T in a direction perpendicular to the flow. The lower 
part of the apparatus OY, as shown in the sectional plan, is made 
boat-shaped so as to prevent the formation of eddies in the 
neighbourhood of the orifices. The pressure at the orifice is 
transmitted through the tube OS, and the pressure at Q through 
the tube QR. To measure the difference of pressure, or head, 
in the two tubes, OS and QE were connected to a differential 
gauge, similar to that described in section 13 and very small 
differences of head could thus be obtained with great accuracy. 

The tube shown in Fig. 145 has a cigar-shaped bulb, the 
impact orifice being at one end and communicating with the 
tube OS. There are four small openings in the side of the bulb, 
so that any variations of pressure outside are equalised in the 
bulb. The pressures are transmitted through the tubes OS and 
TR to a differential gauge as in the case above. 

In Fig. 147 is shown a special stuffing-box used by Professor 
Williams, to allow the tube to be moved to the various positions in 

* For other forms of Pitot tubes as used by Professor Williams, E. S. Cole and 
others, see Proceedings of the Am.S.C.E., Vol. xxvn. 
f Transactions of the Am.S.C.E., Vol. xxi. 

162 



244 



HYDRAULICS 



the cross section of a pipe, at which it was desired to determine 
the velocity of translation of the water*. 

Mr E. S. Colet has used the Pitot tube as a continuous meter, 
the arrangement being shown in Fig. 148. The tubes were con- 
nected to a U tube containing a mixture of carbon tetrachloride 
and gasoline of specific gravity 1*25. The difference of level of 
the two columns was registered continuously by photography. 





Gauge 




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 issuing 

* See page 144. 

t Proc. A.M.S.C.E., Vol. xxvii. See also experiments by Murphy and Torranee 
in same volume. 



GAUGING THE FLOW OF WATER 



245 



from orifices, and in the interior of the nappes of weirs. Each 
tube consisted of a copper plate 1*89 inches wide, by 1181 inch 
thick, sharpened on the upper 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. 




Fig. 149. 



Fig. 150. 



151. Calibration of Pitot tubes. 

Whatever the form of the Pitot tube, the head h can be 
expressed as 



cv 



or 



k being called the coefficient of the tube. This coefficient k in 
special cases may have to be determined by experiment, but, as 
remarked above, for tubes carefully made and having an impinging 
surface which is a surface of revolution it is unity. 

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 
T034. 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. 

(6) The tube was placed in a stream, the velocity of which 
was determined by floats. The coefficient was 1*006. 

(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 



246 HYDRAULICS 

into areas, and about the centre of each a reading of the tube 
was taken. Calling a the area of one of these sections, and h the 
reading of the tube, the coefficient 



and was found to be "993. 

Darcy* and Bazin also found that by changing the position of 
the orifice in the pressure tube the coefficients changed con- 
siderably. 

Williams, Hubbell and Fenkell used two methods of calibration 
which gave very different results. 

The first method was to move the tubes through still water at 
known velocities. For this purpose a circumferential trough, 
rectangular in section, 9 inches wide and 8 inches deep was built of 
galvanised iron. The diameter of its centre line, which was made 
the path of the tube, was 11 feet 10 inches. The tube to be rated 
was supported upon an arm attached to a central shaft which was 
free to revolve in bearings on the floor and ceiling, and which also 
supported the gauge and a seat for the observer. The gauge was 
connected with the tube by rubber hose. The arm carrying the 
tube was revolved by a man walking behind it, at as uniform a 
rate as possible, the time of the revolution being taken by means 
of a watch reading to ^ of a second. The velocity was main- 
tained as nearly constant as possible for at least a period of 
5 minutes. The value of k as determined by this method was '926 
for the tube shown in Fig. 145. 

In the second method adopted by these workers, the tube was 
inserted into a brass pipe 2 inches in diameter, the discharge 
through which was obtained by weighing. Readings were taken 
at various positions on a diameter of the pipe, while the flow in the 
pipe was kept constant. The values of \/2gh t which may be called 
the tube velocities, could then be calculated, and the mean value 
V m of them obtained. It was found that, in the cases in which the 
form of the tube was such that the volume occupied by it in the pipe 
was not sufficient to modify the flow, the velocity was a maximum 
at, or near, the centre of the pipe. Calling this maximum velocity 

V c , the ratio W 2 for a given set of readings was found to be '81. 

Vc 
Previous experiments on a cast-iron pipe line at Detroit having 

shown that the ratio Jt 2 was practically constant for all velocities, 

Vm 

a similar condition was assumed to obtain in the case of the brass 
Recherches Hydrauli^ues, 



GAUGING THE FLOW OF WATER 247 

pipe. The tube was then fixed at the centre of the pipe, and 
readings taken for various rates of discharge, the mean velocity 
U, as determined by weight, varying from J to 6 feet per second. 

For the values of h thus determined, it was found that 



was practically constant. This ratio was '729 for the tube shown 
in Fig. 145. 

Then since for any reading h of the tube, the velocity v is 



the actual mean velocity 
r 




7 _ ratio of U to Y c _'729_ Q0 
""'-" 



But 
Therefore 



For the tube shown in Fig. 146, some of the values of Tc ae 
determined by the two methods differed very considerably. 

It will be seen that the value of k determined by moving the 
tube through still water, according to the above results, differs 
from that obtained from the running water in a pipe. Other 
experiments, however, on tubes the coefficients for which were 
obtained by moving through still water and by being placed in 
jets of water issuing from sharp-edged orifices, show that the 
coefficient is unity in both cases. Professor Gregory* using a 
tube (Fig. 373, Appendix 4), consisting of an impact tube i inch 
diameter surrounded by a tapering tube of larger diameter in which 
were drilled the static openings at a mean distance of 12'5 inches 
from the impact opening, found that the coefficient was unity 
when moved through still water, or when it was placed in flowing 
water in a pipe. With tubes having impact openings of the form 
shown in Fig. 144, or in Figs. 371 and 372, and the pressure 
openings well removed from the influence of eddy motions it may 
be taken that the coefficient is unity, and a properly designed Pitot 
tube can with care, therefore, be used with confidence to measure 
velocities of flow. 

152. Gauging by a weir. 

When a stream is so small that a barrier or dam can be easily 
constructed across it, or when a large quantity of water is required 
to be gauged in the laboratory, the flow can be determined by 
means of a notch or weir. 

* See Appendix 4, p. 528 ; Trans. 4m. S,M.E. 1904, 



248 



HYDRAULICS 




The channel as it approaches the weir should be as far as 
possible uniform in section, and it is desirable for accurate 
gauging*, that the sides of the channel be made vertical, and the 
width equal to the width of the weir. The sill should be sharp- 
edged, and perfectly horizontal, and as high as possible above the 
bed of the stream, and the down-stream channel 
should be wider 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 measured. This is best done by a Boyden 
hook gauge. 

153. The hook gauge. 

A simple form of hook gauge as made by G-urley 
is shown in Fig. 151. In a rectangular groove formed 
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 level 
staff. To the lower end of the scale is connected a 
hook H, 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, either up or down, by means of the milled 
nut. A vernier V is fixed to the frame by two small 
screws 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 to 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 zero of the scale, and the slider 
again raised until the hook approaches the surface of the water. 
By means of the screw, the hook is raised slowly, until, by piercing 



v.m- 



* See section 82. 



GAUGING THE FLOW OF WATER 



249 




Fig. 152. Bazin's Hook Gauge. 



250 



HYDRAULICS 



the surface of the water, it causes a distortion of the light reflected 
from the surface. On moving the hook downwards again very 
slightly, the exact surface will be indicated when the distortion 
disappears. 

A more elaborate hook gauge, as used by Bazin for his experi- 
mental work, is shown in Fig. 152. 

For rough gaugings a post can be driven into the bed of the 
channel, a few feet above the weir, until the top of the post is 
level with the sill of the weir. The height of the water surface 




Fig. 154. .Recording Apparatus Kent Venturi Meter, 



GAUGING THE FLOW OF WATER 



251 



above the top of the post can then be measured by any convenient 
scale. 

154. Gauging the flow in pipes; Venturi meter. 

Such methods as already described are inapplicable to the 
measurement of the flow in pipes, in which it is necessary that 
there shall be no discontinuity in the flow, and special meters have 
accordingly been devised. 

For large pipes, the Venturi meter, Fig. 153, is largely used in 
America, and is coming into favour in this country. 

The theory of the meter has already been discussed (p. 44), 
and it was shown that the discharge is proportional to the square 
root of the difference H of the head at the throat and the head in 
the pipe, or 



Jc* being a coefficient. 

For measuring the pressure heads at the two ends of the cone, 
Mr W. G. Kent uses the arrangement shown in Fig. 154. 




JFig. 155. Recording drum of the Kent Venturi Meter. 
* See page 46. 



252 



HYDRAULICS 



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 mercury. 





Fig. 156. Integrating drum of the Kent Venturi Meter. 

When no water is passing through the meter, the mercury in the 
two cylinders stands at the same level. When flow takes place 
the mercury in the left cylinder rises, and that in the right 
cylinder is depressed until the difference of level of the surfaces 



GAUGING THE FLOW OF WATER 



253 



TT 

of the mercury is equal to j^j, s being the specific gravity of the 

mercury and H the difference of pressure head in the two 
cylinders. The two tubes 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 \/H, 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 




Ci 



Fig. 157. Kent Venturi 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 the total 
flow. Concentric with the diagram drum shown in Fig. 155, and 
within it, is a second drum, shown in Fig. 156, which also rotates 
at a uniform rate. Fig. 157 shows this internal drum developed. 
The surface of the drum below the parabolic curve FEGr is recessed. 
If the right-hand carriage is touching the drum on the recessed 



254 



HYDRAULICS 



portion, the counter gearing is in action, but is put out of action 
when the carriage touches the cylinder on the raised portion 
above FGr. Suppose the mercury in the right cylinder to fall a 
height proportional to H, then the carriage will be in contact 
with the drum, as the drum rotates, along the line CD, but the 
recorder will only be in operation while the carriage is in 
contact along the length CE. Since FGr is a parabolic curve the 
fraction of the circumference CE = ra . \/H, ra being a constant, 
and therefore for any displacement H of the floats the counter for 
each revolution of the drum will be in action for a period propor- 
tional to \/H. When the float is at the top of the right cylinder, 
the carriage is at the top of the drum, and in contact with the 
raised portion for the whole of a revolution and no flow is 
registered. When the right float is in its lowest position the 
carriage is at the bottom of the drum, and flow is registered 
during the whole of a revolution. The recording apparatus can 
be placed at any convenient distance less than 1000 feet from 
the meter, the connecting tubes being made larger as the distance 
is increased. 

155. Deacon's waste- water meter. 

An ingenious and very simple meter designed by Mr Gr. F. 
Deacon principally for detecting the leakage of water from pipes 
is as shown in Fig. 158. 




Fig. 158. Deacoii waste-water meter. 

The body of the meter which is made of cast-iron, has fitted 
into it a hollow cone C made of brass. A disc D of the same diameter 
as the upper end of the cone is suspended in this cone by means of 
a fine wire, which passes over a pulley not shown; the other end 
of the wire carries a balance weight. 



GAUGING THE FI,OW OF WATER 255 

When no water passes through the meter the disc is drawn to 
the top of the cone, but when water is drawn through, the disc is 
pressed downwards to a position depending upon the quantity of 
water passing. A pencil is attached to the wire, and the motion 
of the disc can then be recorded upon a drum made to revolve by 
clockwork. The position of the pencil indicates the rate of flow 
passing through the meter at any instant. 

When used as a waste-water meter, it is placed in a by-pass 
leading from the main, as shown diagrammatically in Fig, 159. 

"tBr 




B 

r 


S.V. 


A 


s.v 


D 



Fig. 159. 

The valve A is closed and the valve C opened. The rate of 
consumption in the pipe AD at those hours of the night when the 
actual consumption is very small, can thus be determined, and an 
estimate made as to the probable amount wasted. 

If waste is taking place, a careful inspection of the district 
supplied by the main AD may then be made to detect where the 
waste is occurring. 

156. Kennedy's meter. 

This is a positive meter in which the volume of water passing 
through the meter is measured by the displacement of a piston 
working in the measuring cylinder. 

The long hollow piston P, Fig. 160, fits loosely in the cylinder 
Co, but is made water-tight by means of a cylindrical ring of 
rubber which rolls between the piston and the inside of the 
cylinder, the friction being thus reduced to a minimum. At each 
end of the cylinder is a rubber ring, which makes a water-tight 
joint when the piston is forced to either end of the cylinder, so 
that the rubber roller has only to make a joint while the piston is 
free to move. 

The water enters the meter at A, Fig. 161 5, and for the 
position shown of the regulating cock, it flows down the passage 
D and under the piston. The piston rises, and as it does so the 
rack E/ turns the pinion S, and thus the pinion p which is keyed 
to the same spindle as S. This spindle also carries loosely 
a weighted lever W, which is moved as the spindle revolves by 
either of two projecting fingers. As the piston continues to 
ascend, the weighted lever is moved by one of the fingers until its 



256 



HYDRA LTL1CS 



centre of gravity passes the vertical position, when it suddenly 
falls on to a buffer, and in its motion moves the lever L, which 
turns the cock, Fig. 161 6, into a position at right angles to that 




Rubber Seating 



Rubber Rclting 
Pcuckmg 



Robber Seating 



rig. ico. 



GAUGING THE FLOW OF WATER 



257 



shown. The water now passes from A through the passage C, 
and thus to the top of the cylinder, and as the piston descends' 




Fig. 161o. 




L. II. 



258 



HYDRAULICS 



the water that is below it passes to the outlet B. The motion of 
the pinion S is now reversed, and the weight W lifted until it 
again reaches the vertical position, when it falls, bringing the 
cock C into the position shown in the figure, and another up 





Fig. 161 c. 

stroke is commenced. The oscillations of the pinion p are trans- 
ferred to the counter mechanism through the pinions p t and p 2 , 
Fig. 161 a, in each of which is a ratchet and pawl. The counter 
is thus rotated in the same direction whichever way the piston 
moves. 

157. Gauging the flow of streams by chemical means. 

Mr Stromeyer* has very successfully gauged the quantity of 
water supplied to boilers, and also 
the flow of streams by mixing 
with the stream during a definite 
time and at a uniform rate, a 
known quantity of a concentrated 
solution of some chemical, the 
presence of which in water, even 
in very small quantities, can be 
easily detected by some sensitive 
reagent. Suppose for instance 
water is flowing along a small 
stream. Two stations at a known 
distance apart are taken, and the 
time determined which it takes 
the water to traverse the dis- 
tance between them. At a stated 
time, by means of a special ap- 
paratus Mr Stromeyer uses the 
arrangement shown in Fig. 162 
sulphuric acid, or a strong salt 
solution, say, of known strength, is run into the stream at a known 

* Transactions of Naval Architects, 1896; Proceedings Inst. C.E., Vol. CLX. and 
" Jaugeages par Titrations " by Collet, Mellet and Liitschg. Swiss Bureau of 
Hydrography. 



Fig. 162. 



S 



f 



Iv 



GAUGING THE FLOW OF WATEB, 259 

rate, at the upper station. While the acid is being put into the 
stream, a small distance up-stream from where the acid is introduced 
samples of water are taken at definite intervals. At the lower 
station sampling is commenced, at a time, after the insertion of the 
acid at the upper station is started, equal to that required by the 
water to traverse the distance between the stations, and samples 
are then taken, at the same intervals, as at the upper station. 
The quantity of acid in a known volume of the samples taken 
at the upper and lower station is then determined by analysis. 
In a volume Y of the samples, let the difference in the amount of 
sulphuric acid be equivalent to a volume v & of pure sulphuric 
acid. If in a time t, a volume Y of water, has flowed down the 
stream, and there has been mixed with this a volume v of pure 
sulphuric acid, then, if the acid has mixed uniformly with the 
water, the ratio of the quantity of water flowing down the stream 
to the quantity of acid put into the stream, is the same as the 
ratio of the volume of the sample tested to the difference of the 
volume of the acid in the samples at the two stations, or 



Mr Stromeyer considers that the flow in the largest rivers can 
be determined by this method within one per cent, of its true value. 

In large streams special precautions have to be taken in 
putting the chemical solution into the water, to ensure a uniform 
mixture, and also special precautions must be adopted in taking 
samples. 

For other important information upon this interesting method 
of measuring the flow of water the reader is referred to the papers 
cited above. 

An apparatus for accurately gauging the flow of the solution 
is shown in Fig. 162. The chemical solution is delivered into 
a cylindrical tank by means of a pipe I. On the surface of the 
solution floats a cork which carries a siphon pipe SS, and a balance 
weight to keep the cork horizontal. After the flow has been 
commenced, the head h above the orifice is clearly maintained 
constant, whatever the level of the surface of the solution in the 
tank. 



172 



260 



HYDKAULICS 



EXAMPLES. 



(1) Some observations are made by towing a current meter, with the 
following results : 



Speed in ft. per sec. 
1 
5 
Find an equation for the meter. 



Revs, of meter per min. 

80 

560 



(2) Describe two methods of gauging a large river, from observations 
in vertical and horizontal planes; and state the nature of the results 
obtained. 

If the cross section of a river is known, explain how the approximate 
discharge may be estimated by observation of the mid-surface velocity 
alone. 

(8) The following observations of head and the corresponding discharge 
were made in connection with a weir 6'53 feet wide. 



Head in feet ... ... O'l 

Discharge in cubic feet per 



sec. per foot width 



0-17 



0-5 
1-2 



1-0 

3-35 



1-5 



6-1 



2-0 
9-32 



2-5 

13-03 



3-0 

17-03 



3-5 

21-54 



4-0 

26-4 



Assuming the law connecting the head h with the discharge Q as 

Q=mL.7i n , 
find ra and n. (Plot logarithms of Q and h.) 

(4) The following values of Q and h were obtained for a sharp-edge 
weir 6'53 feet long, without lateral contraction. Find the coefficient of 
discharge at various heads. 



Head h ... 
Q per foot- 
length ... 



17 



1-56 



2-37 



1-0 
3-35 



2-0 

9-32 



2-5 I 3-0 



13-03 17-03 



3-5 
21-54 



4-0 

26-4 



4-5 
31-62 



5-0 

37-09 



5-5 

42-81 



(5) The following values of the head over a weir 10 feet long were 
obtained at 5 minutes intervals. 

Head in feet '35 -36 '37 '37 '38 -39 '40 -41 -42 -40 '39 -41 
Taking the coefficient of discharge C as 3'36, find the discharge in 
one hour. 

(6) A Pitot tube was calibrated by moving it through still water in a 
tank, the tube being fixed to an arm which 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 



1-738 
663 



2-275 
1-02 



2-713 
1-69 



3-235 
2-07 



3-873 



2-88 5-40 



4-983 



5-584 



6-142 



6-97 18-51 



Determine the coefficient of the tube. 

For examples on Venturi meters see Chapter II, 



CHAPTER VIII. 

IMPACT OF WATER ON VANES. 

158. Definition of a vector. A right line AB, considered as 
having not only length, but 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 
south, but the sense of the vector is definite, and is from A to B, 
that is from south 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 
c , 





Fig. 163. 

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, Quaternions. 



262 



HYDRAULICS 



length to the velocity of the body is the velocity vector ; the sense 
is from A to B. 

159. * Sum of two vectors. 

If a and /?, Fig. 163, are two vectors the sum of these vectors 
is found, by drawing the vectors, so that the beginning of ft is at 
the end of a, and joining the beginning of a to the end of ft. 
Thus y is the vector sum of a and ft. 

160. Resultant of two velocities. 

When a body has impressed upon it at any instant two 
velocities, the resultant velocity of the body in magnitude and 
direction is the vector sum of the two impressed velocities. This 
may be stated in a way that is more definitely applicable to the 
problems to be hereafter dealt with, as follows. If a body is 
moving with a given velocity in a given direction, and a second 
velocity is impressed upon the body, the resultant velocity is the 
vector sum of the initial and impressed velocities. 

Example. Suppose a particle of water to be moving along a vane DA, Fig. 164, 
with a velocity V r , relative to the vane. 

If the vane is at rest, the particle will leave it at A with this velocity. 

If the vane is made to move in the direction EF with a velocity v, and the 
particle has still a velocity V r relative to the vane, and remains in contact with the 
vane until the point A is reached, the velocity of the water as it leaves the vane at 
A, will be the vector sum 7 of a and p, i.e. of V r and v, or is equal to u. 

161. Difference of two vectors. 

The difference of two vectors a and ft is found by drawing both 
vectors from a common origin A, and joining the end of ft to the 
end of a. Thus, CB, Fig. 165, is the difference of the two vectors 
a and ft or y = a /3, and BC is equal to ft - a, or ft - a = - y. 





162. Absolute velocity. 

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. 



Ilenrici and Turner, Vectors and Rotors. 



IMPACT OP WATER ON VANES 263 

To avoid repetition of the word absolute, the adjective is 
frequently dropped and " velocity " only is used. 

163. When a body is moving with a velocity U, Fig. 166, in 
any direction, and has its velocity changed to U' in any other 
direction, by an impressed force, the change in velocity, or the 
velocity that is impressed on the body, is the vector difference of 
the final and the initial velocities. If AB is U, and AC, U', the 
impressed velocity is BC. 

By Newton's second law of motion, the resultant impressed 
force is in the direction of the change of velocity, and if W is the 
weight of the body in pounds and t is the time taken to change 
the velocity, the magnitude of the impressed force is 

W 

P = (change of velocity) Ibs. 
Qt 

This may be stated more generally as follows. 
The rate of change of momentum, in any direction, is equal to 
the impressed force in that direction, or 

r> W <fc>u, 
P = .37 Ibs. 

g dt 

In hydraulic machine problems, it is generally only necessary 
to consider the change of momentum of the mass of water that 
acts upon the machine per second. W in the above equation then 
becomes the weight of water per second, and t being one second, 

W 

P = (change of velocity). 
y 

164. Impulse of water on vanes. 

It follows that when water strikes a vane which is either 
moving or at rest, and has its velocity changed, either in magni- 
tude or direction, pressure is exerted on the vane. . 

As an example, suppose in one second a mass of water, weighing 
W Ibs. and moving with a velocity U feet per second, strikes a 
fixed vane AD, and let it glide upon the vane at A, Fig. 167, and 
leave at D in a direction at right angles to its original direction 
of motion. The velocity of the water is altered in direction but 
not in magnitude, the original velocity being changed to a velocity 
at right angles to it by the impressed force the vane exerts upon 
the water. 

The change of velocity in the direction AC is, therefore, 

equal to U, and the change of momentum per second is .U 
foot Ibs. 



264. 



HYDRAULICS 



Since W Ibs. of water strike the vane per second, the pressure 
P, acting in the direction CA, required to hold the vane in position 
is, therefore, 




Fig. 167. 

Again, the vane has impressed upon the water a velocity U in 
the direction DF which it originally did not possess. 
The pressure PI in the direction DF is, therefore, 

W 



The resultant reaction of the vane in magnitude and direction 
is, therefore, E, the resultant of P and PI. 

This resultant force could have been 
found at once by finding the resultant 
change in velocity. Set out ac, 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 
ad. 




Fig. 168. 



The impressed velocity cd is V = \/U 2 + U 2 , and the total 
impressed force is 

-^To N/2W , 



IMPACT OF WATER ON VANES 265 

It at once follows, that if a jet of water strikes a fixed plane 
perpendicularly, with a velocity U, and glides along the plane, the 

w 

normal pressure on the plane is U. 

Example. A stream of water 1 sq. foot in section and having a velocity of 
10 feet per second glides on to a fixed vane in a direction making an angle of 
30 degrees with a given direction AB. 

The vane turns the jet through an angle of 90 degrees. 

Find the pressure on the vane in the direction parallel to AB and the resultant 
pressure on the vane. 

In Fig. 167, AC is the original direction of the jet and DF the final direction. 
The vane simply changes the direction of the water, the final velocity being equal 
to the initial velocity. 

The vector triangle is acd, Fig. 168, ac and ad being equal. 

The change of velocity in magnitude and direction is cd, the vector difference of 
ad and ac ; resolving cd parallel to, and perpendicular to AB, ce is the change of 
velocity parallel to AB. 

Scaling off ce and calling it v lt the force to be applied along BA to keep the 
vane at rest is, 



But cd=j2.10 

and ce = cdcosl5 

-J2. 10. 0-9659; 



therefore, PBA= "o o x 13 ' 65 

Oif*9) 

= 264 Ibs. 
The pressure normal to AB is 



- ^ . 10 sin 15 =72 Ibs. 

, .. x . 10.62-4 , 100 J2. 62-4 ( 
The resultant is B= Q0 cd= ^r = 274 Ibs. 

O4'4 O&'a 

165. Relative velocity. 

Before going on to the consideration of moving vanes it is 
important that the student should have clear ideas as to what is 
meant by relative velocity. 

A train is said to have a velocity of sixty miles an hour when, 
if it continued in a straight line at a constant velocity for one 
hour, it would travel sixty miles. What is meant is that the train 
is moving at sixty miles an hour relative to the earth. 

If two trains run on parallel lines in the same direction, one 
at sixty and the other at forty miles an hour, they have a 
relative velocity to each other of 20 miles an hour. If they move 
in opposite directions, they have a relative velocity of 100 miles 
an hour. If one of the trains T is travelling in the direction AB, 
Fig. 169, and the other Ti in the direction AC, and it be supposed 
that the lines on which they are travelling cross each other at A, 



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 

of AC and AB, that is, the velocity of 

vector difference of AC and AB. 




T, 

Fig. 169. 
relative to T is the 



166. Definition of relative velocity as a vector. 

If two bodies A and B are moving with given velocities v and 
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, the 
plane of which is perpendicular to the direction of the jet, and 
which is moving in the same direction as the jet with a velocity v, 




Fig. 170. 



Fig. 171. 



The relative velocity of the water and the vane is U - v, tho 
vector difference of U and v, Fig. 170. If the water as it strikes 
the vane is supposed to glide along it as in Fig. 171, it will do 



IMPACT OF WATER ON VANES 267 

so 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 
vane. Instead of the water gliding along the vane, the velocity 
U-v may be destroyed by eddy motions, but the water will still 
have a velocity v in the direction of the vane. The change in 
velocity in the direction of motion is, therefore, the relative 
velocity U-v, Fig. 170. 

For every pound of water striking the vane, the horizontal 

change in momentum is - - , and this equals the normal pressure 

P on the vane, per pound of water striking the vane. 
The work done per second per pound is 



9 
The original kinetic energy of the jet per pound of water 

U 2 
striking the vane is -~- , and the efficiency of the vane is, therefore, 






U 2 ' 

which is a maximum when v is |TJ, and e = J. An application of 
such vanes is illustrated in Fig. 185, page 292. 

Nozzle and single vane. Let the water striking a vane issue 
from 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 
the end of one second the front of the jet, if perfectly free to 
move, would have arrived at B and the vane at C. Of the water 
that has issued from the jet, therefore, only the quantity BC will 
have hit the vane. 

! U ---- V-~ - -H! i 



T C[ 

j<_ . JJ H 

Fig. 172. 
The discharge from the nozzle is 

W = 62'4.a.U, 

and the weight that hits the vane per second is 

W.QJ-'u) 

U 
The change of momentum per second is 



268 HYDRAULICS 

and the work done is, therefore, 



U.g 

Or the work done per Ib. of water issuing from the nozzle is 



U.g 

hypothetical 



case and has no practical 



This is purely 
importance. 

Nozzle and a number of vanes. If there are a number of 
vanes closely following each other, the whole of the water issuing 
from the nozzle hits the vanes, and the work done is 

W(U-v)v 



The efficiency is 



2v (U - v) 
IP 



and the maximum efficiency is 

It follows that an impulse water wheel, with radial blades, as 
in Fig. 185, cannot have an efficiency of more than 50 per cent. 

168. Impact of water on a vane when the directions of 
motion of the vane and jet are not parallel. 

Let U be the velocity of a jet of water and AB its direction, 
Fig. 173. 

A, 




Fig. 173. 

Let the edge A of the vane AC be moving with a velocity v ; 
the relative velocity V r of the water and the vane at A is DB. 
From the triangle DAB it is seen that, the vector sum of the 
velocity of the vane and the relative velocity of the jet and the 
vane is equal to the velocity of the jet; for clearly U is the vector 
sum of v and V P . 

If the direction of the tip of the vane at A is made parallel to 
DB the water will glide on to the vane in exactly the same way 



IMPACT OF WATER ON VANES 269 

as if it were at rest, and the water were moving in the direction 
DB. This is the condition that no energy shall be lost by shock. 

When the water leaves the vane, the relative velocity of the 
water and the vane must be parallel to the direction of the 
tangent to the vane at the point where it leaves, and it is equal to 
the vector difference of the absolute velocity of the water, and 
the vane. Or the absolute velocity with which the water leaves 
the vane is the vector sum of the velocity of the tip of the vane 
and the relative velocity of the water to the vane. 

Let CGr be the direction of the tangent to the vane at C. Let 
CE be Vij the velocity of C in magnitude and direction, and let CF 
be the absolute velocity Ui with which the water leaves the vane. 

Draw EF parallel to CGr to meet the direction OF in F, then 
the relative velocity of the water and the vane is EF, and the 
velocity with which the water leaves the vane is equal to OF. 

If Vi and the direction CGr are given, and the direction in which 
the water leaves the vane is given, the triangle CEF can be 
drawn, and OF determined. 

If on the other hand Vi is given, and the relative velocity v r is 
given in magnitude and direction, CF can be found by measuring 
off along EF the known relative velocity v r and joining CF. 

If Vi and Ui are given, the direction of the tangent to the vane 
is then, as at inlet, the vector difference of Ui and VL 

It will be seen that when the water either strikes or leaves the 
vane, the relative velocity of the water and the vane is the vector 
difference of the velocity of the water and the vane, and the actual 
velocity of the water as it leaves the vane is the vector sum of the 
velocity of the vane and the relative velocity of the water and 
the vane. 

Example. The direction of the tip of the vane at the outer circumference of a 
wheel fitted with vanes, makes an angle of 165 degrees with the direction of motion 
of the tip of the vane. 

The velocity of the tip at the outer circumference is 82 feet per second. 

The water leaves the wheel in such a direction and with such a velocity that the 
radial component is 13 feet per second. 

Find the absolute velocity of the water in direction and magnitude and the 
relative velocity of the water and the wheel. 

To draw the triangle of velocities, set out AB equal to 82 feet, and make the 
angle ABC equal to 15 degrees. BC is then parallel to the tip of the vane. 

Draw EC parallel to AB, and at a distance from it equal to 13 feet and 
intersecting BG in C. 

Then AC is the vector sum of AB and BC, and is the absolute velocity of the 
water in direction and magnitude. 

Expressed trigonometrically 

AC 2 = (82 - 13 cot 15) 2 + 13 2 

= 38-6* + 13 8 and AC = 36-7 ft. per sec. 

sin BAG =^ = -354. 
AC/ 

Therefore BAG = 20 45'. 



270 



HYDRAULICS 



169. Conditions which the vanes of hydraulic machines 
should satisfy. 

In all properly designed hydraulic machines, such as turbines, 
water wheels, and centrifugal pumps, in which water flowing in 
a definite direction impinges on moving vanes, the relative velocity 
of the water and the vanes should be parallel to the direction of 
the vanes at the point of contact. If not, the water breaks into 
eddies as it moves on to the vanes and energy is lost. 

Again, if in such machines the water is required to leave the 
vanes with a given velocity in magnitude and direction, it is only 
necessary to make the tip of the vane parallel to the vector 
difference of the given velocity with which the water is to leave 
the vane and the velocity of the tip of the vane. 

Example (1). A jet of water, Fig. 174, moves in a direction AB making an angle 
of 30 degrees with the direction of motion AC of a vane moving in the atmosphere. 
The jet has a velocity of 30 ft. per second and the vane of 15 ft. per second. To find 
(a) the direction of the vane at A so that the water may enter without shock; (6) the 
direction of the tangent to the vane where the water leaves it, so that the absolute 
velocity of the water when it leaves the vane is in a direction perpendicular to AC ; 
(c) the pressure on the vane and the work done per second per pound of water 
striking the vane. Friction is neglected. 




K 



'U, 



Change orV&oribf irv the 
direction, ofmotiori. 



o, 



Fig. 174. 



The relative velocity V r of the water and the vane at A is CB, and for no shock 
the vane at A must be parallel to CB. 

Since there is no friction, the relative velocity V r of the water and the vane 
cannot alter, and therefore, the triangle of velocities at exit is ACD or FAjCj . 

The point D is found, by taking C as centre and CB as radius and striking the 
arc BD to cut the known direction AD in D. 

The total change of velocity of the jet is the vector difference DB of the initial 
and final velocities, and the change of velocity in the direction of motion is BE. 
Calling this velocity V, the pressure exerted upon the vane in the direction of 
motion is 

Ibs. per Ib. of water striking the vane. 
9 

The work done per Ib. is, therefore, ft. Ibs. and the efficiency, since there is 
no loss by friction, or shock, is 

Hgr- 



IMPACT OF WATER ON VANES 271 

The change in the kinetic energy of the jet is equal to the ivork done by the jet. 
The kinetic energy per Ib. of the original jet is and the final kinetic energy is 

iy 

2<7 ' 

The work done is, therefore, -= ~- ft. Ibs. and the efficiency is 




It can at once be seen from the geometry of the figure that 
Vv _ U 2 Uj 2 
g ~2g"2g' 

For AB 2 =AC 2 +CB 2 + 2AC.CG, 

and since CD = CB and 

therefore, AB 2 - AD 2 = 2 AC (AC + CG) 

But 

A , , 
therefore, 

If the water instead of leaving the vane in a direction perpendicular to v, leaves 
it with a velocity Uj having a component V x parallel to v t the work done on the 
vane per pound of water is 



If Uj be drawn on the figure it will be seen that the change of velocity in the 

V- V 

direction of motion is now (V- VJ, the impressed force per pound is - - 1 , and 

/ V V \ 
the work done is, therefore, ( * ) ^ ft. Ibs. per pound. 

As before, the work done on the vane is the loss of kinetic energy of the jet, and 
therefore, 




9 20 

The work done on the vane per pound of water for any given value of Uj , is, 
therefore, independent of the direction of U 1 . 

Example (2). A series of vanes such as AB, Fig. 175, are fixed to a (turbine) 
wheel which revolves about a fixed centre C, with an angular velocity u. 

The radius of B is R and of A, r. Within the wheel are a number of guide 
passages, through which water is directed with a velocity U, at a definite inclination 
6 with the tangent to the wheel. The air is supposed to have free access to the 
wheel. 

To draw the triangles of velocity, at inlet and outlet, and to find the directions 
of the tips of the vanes, so that the water moves on to the vanes without shock and 
leaves the wheel with a given velocity U,. Friction neglected. 

In this case the velocity relative to the vanes is altered by the whirling of the 
water as it moves over the vanes. It will be shown later that the head impressed 

- *-S-+3?-S- 

The tangent AH to the vane at A makes an angle <f> with the tangent AD to the 
wheel, so that CD makes an angle with AD. The triangle of velocities ACD at 
inlet is, therefore, as shown in the figure and does not need explanation. 

To draw the triangle of velocities at exit, set out BGr equal to vj and perpen- 



272 



HYDRAULICS 



dicular to the radius BO, and with B and G as centres, describe circles with U x and 
v r as radii respectively, intersecting in B. Then GE is parallel to the tangent to 
the vane at B. 

(See Impulse turhines.) 

Work done on the wheel. Neglecting friction etc. the work done per pound of 
water passing through the wheel, since the pressure is constant, being equal to the 
atmospheric pressure, is the loss of kinetic energy of the water, and is 



The work done on the wheel can also be found from the consideration of the 
change of the angular momentum of the water passing through the wheel. Before 
going on however to determine the work per pound by this method, the notation 
that has been used is summarised and several important principles considered. 




Notation used in connection with vanes, turbines and centrifugal 
pumps. Let U be the velocity with which the water approaches 
the vane, Fig. 175, and v the velocity, perpendicular to the radius 
AC, of the edge A of the vane at which water enters the wheel. 

Let Y be the component of U in the direction of v, 

u the component of U perpendicular to v, 

Y r the relative velocity of the water and vane at A, 

Vi the velocity, perpendicular to BC, of the edge B of the vane 
at which water leaves the wheel, 

Ui the velocity with which the water leaves the wheel, 

Yi the component of Ui in the direction of v it 



IMPACT OF WATER ON VANES 273 

U] the component of Ui perpendicular to Vi, or along BC, 
v r the relative velocity of the water and the vane at B. 
Velocities of whirl. The component velocities V and Vi are 

called the velocities of whirl at inlet and outlet respectively. 

This term 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. 
9 

171. Change of angular momentum. 

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 UiSij the change of angular 

momentum in time t is 

W 



and the rate of change of angular momentum is 





Fig. 176. Fig. 177. 

172. Two important principles. 

(1) Work done by a couple, 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 To> foot pounds per second, 

and the horse-power is -=^ . 



L. H. 18 



HYDRAULICS 

Suppose a body rotates about a fixed centre C, Fig. 177, and 
a force P Ibs. 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 <o in one second, the 
distance moved through by the force P is o> . S, and the work 
done by P in foot pounds is 

P<oS=To>. 

And since one horse-power is equivalent to 33,000 foot pounds 
per minute or 550 foot pounds per second the horse-power is 

HP T 

= 



(2) The rate of change of angular momentum of a "body 
rotating about a fixed centre is equal to the couple acting upon 
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 forces are 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 'U is destroyed in a 
time oti then a force will be exerted at the point A equal to 

P_W U 
~ g 'tt' 

and the moment of this force about C is P . S. 

At the end of the time dt, 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 1 
'~ g dt 

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 causing the 
velocity of W to change, the couple acting on the wheel is 



(1). 



When US is greater than UiSi, the body has done work on the 
wheel, as in water wheels and turbines. When UiSi is greater 
than US, the wheel does work on the body as in centrifugal pumps. 

Let the wheel of Fig. 175 have an angular velocity w. 



IMPACT OF WATER ON VANES 275 

In a time 3t the angle moved through by the couple is wdt, 
and therefore the work done in time dt is 

W 
T.oO* = eodJS-TLSO .................. (2). 

Suppose now W is the weight of water in pounds per second 
which 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 



and the work done per second per pound of water entering the 
wheel is 



This result, as will be seen later (page 337), is entirely inde- 
pendent of the change of pressure as the water passes through the 
wheel, or of the direction in which the water passes. 

173. Work done on a series of vanes fixed to a wheel 
expressed in terms of the velocities of whirl of the water 
entering and leaving the wheel. 

Outward flow turbine. If water enters a wheel at the inner 
circumference, as in Fig. 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_TJ 

s~v 

and for a similar reason 

R Ux 

STV/ 

Again the angular velocity of the wheel 



therefore the work done per second is 



and the work done per pound of flow is 

Yt? 



y 9 

Inward flow turbine. If the water enters at the outer cir- 
cumference of a wheel with a velocity of whirl V, and leaves at 
the inner circumference with a velocity of whirl Vi, the velocities 

182 



276 



HYDRAULICS 



of the inlet and outlet tips of the vanes being v and 
the work done on the wheel is still 

Yt> 



respectively 



9 9 
The flow in this case is said to be inward. 

Parallel flow or axial flow turbine. If vanes, such as those 
shown in Fig. 174, are fixed to a wheel, the flow is parallel to the 
axis of the wheel, and is said to be axial. 

For any given radius of the wheel, Vi is equal to v, 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 v, and a stream with a greater velocity U in the 
same direction. 

The relative velocity is 

V r =(U-). 

Neglecting friction, the relative velocity Y r will remain con- 
stant, and the water will, therefore, leave the cup at the point B 
with a velocity, Y r , relative to the cup. 





Fig. 178. 



Fig. 179. 



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 Y r , and is therefore Ui. 
The work done on the cups is then 



IMPACT OF WATER ON VANES 277 



per Ib. of water, and the efficiency is 
U 2 Ui 2 



For Ui, the value 



H! = >J{v - (U - f>) cos BY + (U - v) 2 sin & 

can be substituted, and the efficiency thus determined in terms of 
v, U and 0. 

Pelton wheel cups. If is zero, as in Fig. 178, and U v is 
equal to v, or U 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. 

The work done determined from consideration of the change of 
momentum. The component of Ui, Fig. 178, in the direction of 
motion, is 

v(U v) cos 0, 

and the change of momentum per pound of water striking the 
vanes is, therefore, 



9 
The work done per Ib. is 



and the efficiency is 






U 2 
When is 0, cos is unity, and 



which is a maximum, and equal to unity, when v is -^ . 

175. Force tending to move a vessel from which water 
is issuing 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 orifice, the 



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.fclbs., 
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 



and the quantity discharged per second in cubic feet is 

The momentum given to the water per second is 
-., _ w . a . v* 
9 

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 

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 v t there 
is, therefore, a pressure on his hand equal to 

2wav' 2 _ wav* 

20 g 

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 



9 
The work done per second is 

. -x-r wav V (v V) P . ,, 
F . V = ^ '- foot Ibs., 

or = Y(t? " V) foot Ibs. 

9 
per Ib. of flow from the nozzle. 

V (v - V) 
The efficiency is e = ~^~~ 

2YQ-V) 

tf 
which is a maximum, when 

v = 2Y 

and =i- 



IMPACT OF WATEE ON VANES 279 

176. The propulsion of ships by water jets. 

A method of propelling ships by means of jets of water issuing 
from orifices at the back of the ship, has been used with some 
success, and is still employed to a very limited extent, for the 
propulsion of lifeboats. 

Water is taken by pumps carried by the ship from that 
surrounding the vessel, and is forced through the orifices. Let 
v be the velocity of the water issuing from the orifice relative 

to the ship, and Y the velocity of the ship. Then ~ is the 

head 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. 

Imagine the ship to be moving through the water and having 
a pipe 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 

Y 2 
velocity V relative to the ship, and having a kinetic energy ~- 

per pound. If friction and other losses are neglected, the work 
that the pumps will have to do upon each pound of water to eject 
it at the back with a velocity v is, clearly, 

v Y 2 



As in the previous example, the velocity of the water issuing 
from the nozzles relative to the water behind the ship is 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 



y 
and the work done is 



9 
The efficiency is the work done on the ship divided by the 

work done by the engines, which equals wav(~-~^\ and, 

,, - \47 ty' 

therefore, 

_2YQ-Y) 



2Y 



280 HYDRAULICS 

which can be made as near unity as is desired by making v and 
V approximate to equality. 

But for a given area a of the orifices, and velocity v, the nearer 
v approximates to V the less the propelling force F becomes, and 
the size of ship that can be driven at a given velocity V for the 
given area a of the orifices diminishes. 

If vis 2V, e = |. 



EXAMPLES. 

(1) Ten cubic feet of water per second are discharged from a stationary 
jet, the sectional area of which is 1 square foot. The water impinges nor- 
mally on a flat surface, moving in the direction of the jet with a velocity 
of 2 feet per second. Find the pressure on the plane in Ibs., and the work 
done on the plane in horse-power. 

(2) A jet of water delivering 100 gallons per second with a velocity of 
20 feet per second impinges perpendicularly on a wall. Find the pressure 
on the wall. 

(3) A jet delivers 160 cubic feet of water per minute at a velocity of 
20 feet per second and strikes a plane perpendicularly. Find the pressure 
on the plane (1) when it is at rest ; (2) when it is moving at 5 feet per 
second in the direction of the jet. In the latter case find the work done 
per second in driving the plane. 

(4) A fire-engine hose, 3 inches bore, discharges water at a velocity of 
100 feet per second. Supposing the jet directed normally to the side of a 
building, find the pressure. 

(5) Water issues horizontally from a fixed thin-edged orifice, 6 inches 
square, under a head of 25 feet. The jet impinges normally on a plane 
moving in the same direction at 10 feet per second. Find the pressure on 
the plane in Ibs., and the work done in horse-power. Take the coefficient 
of discharge as "64 and the coefficient of velocity as '97. 

(6) A jet and a plane surface move in directions inclined at 30, with 
velocities of 30 feet and 10 feet per second respectively. What is the 
relative velocity of the jet and surface ? 

(7) Let AB and BC be two lines inclined at 30. A jet of water moves 
in the direction AB, with a velocity of 20 feet per second, and a series of 
vanes move in the direction CB with a velocity of 10 feet per second. Find 
the form of the vane so that the water may come on to it tangentially, and 
leave it in the direction BD, 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. 



IMPACT OF WATER ON VANES 281 

(8) A curved plate is mounted on a slide so that the plate is free to 
move along the slide. It receives a jet of water at an angle of 30 with a 
normal to the direction of sliding, and the jet leaves the plate at an angle 

of 120 with the same normal. Find the force which must be applied to 
the plate in the direction of sliding to hold it at rest, and also the normal 
pressure on the slide. Quantity of water flowing is 500 Ibs. per minute 
with a velocity of 35 feet per second. 

(9) A fixed vane receives a jet of water at an angle of 120 with a 
direction AB. Find what angle the jet must be turned through in order 
that the pressure on the vane in the direction AB may be 40 Ibs., when the 
flow of water is 45 Ibs. per second at a velocity of 30 feet per second. 

(10) Water under a head of 60 feet is discharged through a pipe 6 inches 
diameter and 150 feet long, and then through a nozzle, the area of which 
is one-tenth the area of the pipe. 

Neglecting all losses but the friction of the pipe, determine the pressure 
on a fixed plate placed in front of the nozzle. 

(11) A jet of water 4 inches diameter impinges on a fixed cone, the 
axis coinciding with that of the jet, and the apex angle being 30 degrees, 
at a velocity of 10 feet per second. Find the pressure tending to move the 
cone in the direction of its axis. 

(12) A vessel containing water and having in one of its vertical sides 
a circular orifice 1 inch diameter, which at first is plugged up, is 
suspended in such a way that any displacing force can be accurately 
measured. On the removal of the plug, the horizontal force required to 
keep the vessel in place, applied opposite to the orifice, is 3'6 Ibs. By the 
use of a measuring tank the discharge is found to be 31 gallons per minute, 
the level of the water in the vessel being maintained at a constant height 
of 9 feet above the orifice. Determine the coefficients of velocity, con- 
traction and discharge. 

(13) A train carrying a Ramsbottom's scoop for taking water into the 
tender is running at 24 miles an hour. What is the greatest height at 
which the scoop will deliver the water ? 

(14) A locomotive going at 40 miles an hour scoops up water from a 
trough. The tank is 8 feet above the mouth of the scoop, and the delivery 
pipe has an area of 50 square inches. If half the available head is wasted 
at entrance, find the velocity at which the water is delivered into the tank, 
and the number of tons lifted in a trench 500 yards long. What, under 
these conditions, is the increased resistance ; and what is the minimum 
speed of train at which the tank can be filled ? Lond. Un. 1906. 

If air is freely admitted into the tube, as in Fig. 179 A, the water will 



282 HYDRAULICS 




move into the tube with a velocity v relative to the 
tube equal to that of the train. (Compare with 
Fig. 167.) The water will rise in the tube with a 
diminishing velocity. The velocity of the train being 
58'66 ft. per sec., and half the available head being 
lost, the velocity at inlet is 



The velocity at a height h feet is 

179i ' W4TP^78 

= 34-8 ft. per sec. 
If the tube is full of water the velocity at inlet is 34'8 ft. per sec. 

(15) A stream delivering 3000 gallons of water per minute with a 
velocity of 40 feet per second, by impinging on vanes is caused freely to 
deviate through an angle of 10, the velocity being diminished to 35 feet 
per second. Determine the velocity impressed on the water and the 
pressure on the vanes due to impact. 

(16) Water flows from a 2-inch pipe, without contraction, at 45 feet per 
second. 

Determine the maximum work done on a machine carrying moving 
plates in the following cases and the respective efficiencies : 

(a) When the water impinges on a single flat plate at right angles and 
leaves tangentially. 

(5) Similar to (a) but a large number of equidistant flat plates are 
interposed in the path of the jet. 

(c) When the water glides on and off a single semi-cylindrical cup. 

(d) When a large number of cups are used as in a Pelton wheel. 

(17) In hydraulic mining, a jet 6 inches in diameter, discharged under 
a head of 400 feet, is delivered horizontally against a vertical cliff face. 
Find the pressure on the face. What is the horse-power delivered by the 
jet? 

(18) If the action on a Pelton wheel is equivalent to that of 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 3 square feet in aggregate area, and dis- 
charges through the jets 100 cubic feet of water per second. The speed of 
the ship is 15 feet per second. Find the propelling force of the jets, the 
efficiency of the propeller, and, neglecting friction, the horse-power of the 
engines. 



CHAPTER IX. 

WATER WHEELS AND TURBINES. 

Water wheels can be divided into two classes as follows. 

(a) Wheels upon which the water does work partly by 
impulse but almost entirely by weight, the velocity of the water 
when it strikes the wheel being small. There are two types of 
this class of wheel, Overshot Wheels, Figs. 180 and 181, and 
Breast Wheels, Figs. 182 and 184. 

(6) Wheels on which the water acts by impulse as when 
the 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 
the water comes in contact with the wheel. In most impulse 
wheels the water is made to flow under the wheel and hence 
they are called Undershot Wheels. 

It will be seen that in principle, there is no line of demarcation 
between impulse water wheels and impulse turbines, the latter 
only differing from the former in constructional detail. 

177. Overshot water wheels. 

This type of wheel is not suitable for very low or very high 
heads as the diameter of the wheel cannot be made greater than 
the head, neither can it conveniently be made much less. 

Figs. 180 and 181 show two arrangements of the wheel, the 
only difference in the two cases being that in Fig. 181, the top of 
the wheel is some distance below the surface of the water in the 
up-stream channel or penstock, so that the velocity v with which 
the water reaches the wheel is larger than in Fig. 180. This has 
the advantage of allowing the periphery of the wheel to have a 
higher velocity, and the size and weight of the wheel is conse- 
quently diminished. 

The buckets, which are generally of the form shown in the 
figures, or are curved similar to those of Fig. 182, are con- 
nected to a rim M coupled to the central hub of the wheel by 



284 



HYDRAULICS 



suitable spokes or framework. This class of wheel has been 
considerably used for heads varying from 6 to 70 feet, but is now 
becoming obsolete, being replaced by the modern turbine, which 
for the same head and power can be made much more compact, 
and can be run at a much greater number of revolutions per unit 
time. 

E D K 




Fig. 180. Overshot Water Wheel. 




Fig. 181. Overshot Water Wheel. 

The direction of the tangent to the blade at inlet for no shock 
can be found by drawing the triangle of velocities as in Figs. 180 
and 181. The velocity of the periphery of the wheel is v and the 
velocity of the water U. The tip of the blade should be parallel 
to V r . The mean velocity U, of the water, as it enters the wheel 



WATER WHEELS 285 

in Fig. 181, will be v + k \/2(/H, v being the velocity of approach 
of the water in the channel, H the fall of the free surface and k 
a coefficient of velocity. The water is generally brought to the 
wheel along a wooden flume, and thus the velocity U and the 
supply to the wheel can be maintained fairly constant by a simple 
sluice placed in the flume. 

The best velocity v for the periphery is, as shown below, 
theoretically equal to |U cos 0, but in practice the velocity v is 
frequently much greater and * experiment shows that the best 
velocity v of the periphery is about 0'9 of the velocity U of the 
water. 

If U is to be about 1'lv the water must enter the wheel at 
a depth not less than 

U 2 = r2^ 

2^ 2g 
below the water in the penstock. 

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 

i l'2v* 

fls Ct ~ * 

20 

Since U is equal to v 2^H, for given values of U and of h, 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 
head 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 minimum flow, when the head increases, and there is a greater 
quantity of water available, a loss in efficiency will not be 
important. 

The horse-power 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 2J to 3D. 

Let Q be the volume of water in cubic feet of water supplied 
per second. 

Let <o 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 r a - TI , be the depth of the shroud, which 
on actual wheels is from 10" to 20". 

* Theory and test of an Overshot Water Wheel, by C. E. Weidner, Wisconsin, 1913. 



286 HYDRAULICS 

Whatever the form of the buckets the capacity of each bucket is 

bd . -^- , nearly. 
The number of buckets which pass the stream per second is 

If a fraction k of each bucket is filled with water 



or 



llie fraction Jc in actual wheels is from ^ to . 
If h is the fall of the water to the level of the tail race and & 
the efficiency of the wheel, the horse-power is 



550 ' 
and the width b for a given horse-power, HP, is 



6 = 



1100HP 



= 17'6 



HP 



of centrifugal forces. As the wheel revolves, the surface 
of the water in the buckets, due to centrifugal forces, takes up a 
curved form. 

Consider any particle of water of mass w Ibs. at a radius r 
equal to CB from the centre of the wheel and in the surface of 




F 

Fig. 181 a. 

the water. The forces acting upon it are w due to gravity and 



w 



the centrifugal force - w 2 r acting in the direction CB, 



being the 
angular velocity of the wheel. The resultant BGr (Fig. 181 a) of 



WATER WHEELS 287 

these forces must be normal to the surface. Let BG- be produced 
to meet the vertical through the centre in A. Then 

AC AC w 
CB r w 2 

(D T 

g 
AC = 5. 

That is the normal AB always cuts the vertical through C in 
a fixed point A, and the surface of the water in any bucket lies 
on a circle with A as centre. 

Losses of energy in overshot wheels. 

(a) The whole of the velocity head - is lost in eddies in the 

buckets. 

In addition, as the water falls in the bucket through the 
vertical distance EM, its velocity will be increased by gravity, 
and 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 
V r the water will enter with shock, and a further head will be 
lost. The total loss by eddies and shock may, therefore, be 
written 



U 2 

or hi + h - , 

k and hi 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 m is the mean 
height above the tail level at which the water leaves the buckets, 
a head equal to h m is lost. By fitting an apron GrH in front of the 
wheel the water can be prevented from leaving the wheel until it 
is 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 



head ~- is 

(d) If the level of the tail water rises above the bottom of 
the wheel there will be a further loss due to, (1) the head h equal to 
the height of the water above the bottom of the wheel, (2) the 
impact of the tail water stream on the buckets, and (3) the 
tendency for the buckets to lift the water on the ascending side of 
the wheel. 



288 HYDRAULICS 

In times of flood there may be a considerable rise of the 
down-stream, and h may then be a large fraction of h. If on 
the other hand the wheel is raised to such a height above the tail 
water that the bottom of the wheel may be always clear, the 
head h m will be considerable during dry weather now, and the 
greatest possible amount of energy will not be obtained from the 
water, just when it is desirable that no energy shall be wasted. 

If h is the difference in level between the up and down-stream 
surfaces, the maximum hydraulic efficiency possible is 

..'-(";?*) ..................... ,, 

and the actual hydraulic efficiency will be 



h - i m 

. e= ; - 5 g Sf 

k, fa and Jc being coefficients. 

The efficiency as calculated from equation (1), for any given 
value of h m , is a maximum when 

Y r 2 v* . 

-~ H - is a minimum. 

From the triangles EKF and KDF, Fig. 180, 
(U cos - v) a + (U sin 0)* = Y r 2 . 
Therefore, adding v 2 to both sides of the equation, 
Y r 2 + v* = IP cos 2 6 - 2Uv cos 6 + 2v z + U 2 sin 2 0, 

which is a minimum for a given value of U, when 2Uv cos 6 2i> 2 
is a maximum. Differentiating and equating to zero this, and 
therefore the efficiency, is seen to be a maximum, when 

v - -ff cos 0. 
ft 

The actual efficiencies obtained from overshot wheels vary 
from 60 to 89* 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 that 
the water enters the wheel without shock. The water is retained 

* Theory and test of Overshot Water Wheel. Bulletin No. 529 University of 
Wisconsin. 



WATER WHEELS 



289 



in the bucket, by the breast, until the bucket reaches the tail race, 
and a greater fraction of the head is therefore utilised than in 
the overshot wheel. In order that the air may enter and leave 
the buckets freely, they are partly open at the inner rim. Since 
the water in the tail race runs in the direction of the motion of 
the bottom of the wheel there is no serious objection to the tail 
race level being 6 inches above the bottom of the wheel. 

The losses of head will be the same as for the overshot wheel 
except that h m will be practically zero, and in addition, there will 
be loss by friction in the guide passages, by friction of the water 
as it moves over the breast, and further loss due to leakage 
between the breast and the wheel. 




Fig. 182. Breast Wheel. 

According to Rankine the velocity of the rim for overshot and 
breast wheels, should be from 4J to 8 feet per second, and the 
velocity U should be about 2v. 

The depth of the shroud which is equal to r 2 - n is from 1 to 
If feet. Let it be denoted by d. Let H be the total fall and let 
it be assumed that the efficiency of the wheel is 65 per cent. Then, 
L. H. 19 



290 



HYDRAULICS 



the quantity of water required per second in cubic feet for a 
given horse-power N is 

N.550 

" 62-4xHxO'G5 



H 

From | to | 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 

JABCD.fe. 
Therefore 

/~ i ~ \ 

vdb. 




2r 2 

Equating this value of Q to the above value, the width 6 is 

27ND 



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 breast wheel 20 feet diameter and 6 feet wide, working on a fall 
of 14 feet and having a depth of shroud of 1' 3", has its buckets full The mean 
velocity of the buckets is 5 feet per second. Find the horse-power of the wheel, 
assuming the efficiency 70 per cent. 



= 26-1. 

The dimensions of this wheel should be compared with those calculated for an 
inward flow turbine working under the same head and developing the same horse- 
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, never 
exceeding 2^ to 3 feet per second, so that the loss due to the water 
leaving the wheel with this velocity and due to leakage between 
the wheel arid breast is small. 



WATER WHEELS 



291 



An efficiency of over 80 per cent, has been obtained with 
these wheels. 

The water enters the wheel in a horizontal direction with 
a velocity U equal to that in the penstock, and the triangle of 
velocities is therefore ABC. 

If the bucket is made parallel to Y r the water enters without 
shock, while at the same time there is no loss of head due to 
friction of guide passages, or to contraction as the water enters or 
leaves them ; moreover the direction of the stream has not to be 
changed. 




Fig. 184. Sagebien Wheel. 

The inclined straight bucket has one disadvantage ; when the 
lower part of the wheel is drowned, the buckets as they ascend are 
more nearly perpendicular to -the surface of the tail water than 
when the blades are radial, but as the peripheral speed is very 
low the resistance due to this cause is not considerable. 

180. Impulse wheels. 

In Overshot and Breast wheels the work is done principally 
by the weight of the water. In the wheels now to be considered 
the whole of the head available is converted into velocity before 
the water strikes the wheel, and the work is done on the wheel 
by changing the momentum of the mass of moving water, or in 
other words, by changing the kinetic energy of the water. 

192 



292 



HYDRAULICS 



Undershot wheel with flat blades. The simplest case is when 
wheel with radial blades, similar to that shown in Fig. 185, is 
into a running stream. 
If b 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 is b . d . w U Ibs., and 
the energy available per second is 

U 3 

b . d . w 2~ foot Ibs. 

Let v be the mean velocity of the blades. 

The radius 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 U and the maximum possible 
efficiency of the wheel is 0'5. 




-f 

Fig. 185. Impulse 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 assumed 
for the large number of flat vanes, and the maximum possible 
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 water 
leaves the penstock at ab is U = k v 2a/&. Let the depth of the 



WATER WHEELS 293 

stream at rib be t. The velocity with which the water leaves the 
wheel at the section cd is v, the velocity of the blades. If the 
width of the stream at cd is the same as at ab and the depth 
is h 0y then, 

Ji Q x v = t x Uj 

i W 
h = . 

Since IT is greater than v, h is greater than t, as shown in 
the figure. 

The hydrostatic pressure on the section cd is ^hfbw and on 
the section ab it is %t* bw. 

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 Ibs., 
then, 

P + Ihfbw - & 2 bw = 2^ (U - i>). 
Substituting for Ji , , the work done per second is 



Or, since Q = b . t . U, 



The efficiency is then, 

-tQ _ _ 

2\v 



t?(U-tQ _ t_ /U _ v\ 



IP 

29 

which is a maximum when 

2u 2 U a - 4y 3 U + ^^U 2 + gtv* = 0. 

The best velocity, v, for the mean velocity of the blades, has 
been found in practice to be about 0'4U, 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 on 



294 



HYDRAULICS 



a frame which may be raised or lowered as the stream rises, or the 
axle carried upon pontoons so that the wheel rises automatically 
with the stream. 

181. Poncelet wheel. 

The efficiency of the straight blade impulse wheels is very 
small, due to the large amount of energy lost by shock, and to the 
velocity with which the water leaves the wheel in the direction of 
motion. 

The efficiency of the wheel is doubled, if the blades are of such 
a form, that the direction of the blade at entrance is parallel to 
the relative velocity of the water and the blade, as first suggested 
by Poncelet, and the water is made to leave the wheel with no 
component in the direction of motion of the periphery of the 
wheel. 

Fig. 186 shows a Poncelet wheel. 




tangle of 
Velocities 
atEcut, 



E 

Fig. 186. Undershot Wheel. 

Suppose the water to approach the edge A of a blade with a 
velocity U making an angle with the tangent to the wheel at A. 

Then if the direction of motion of the water is in the direction 
AC, the triangle of velocities for entrance is ABC. 

The relative velocity of the water and the wheel is V r , and ii 
the blade is made sufficiently deep that the water does not overflow 
the upper edge and there is no loss by shock and by friction, a 
particle of water will rise up the blade a vertical height 

h _yj 

1 20 ' 



WATER WHEELS 295 

It then begins to fall and arrives at the tip of the blade with the 
velocity V r relative to the blade in the inverse direction BE. 

The triangle of velocities for exit is, therefore, ABE, BE being 
equal to BC. 

The velocity with which the water leaves the wheel is then 



It has been assumed that no energy is lost by friction or by 
shock, and therefore the work done on the wheel is 



and the theoretical hydraulic efficiency* is 

IP W 



20 

-1 Ul " 
' 



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 6 are constant, CD is constant 
for all values of v, 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 



For maximum efficiency, therefore, 



* In what follows, the terms theoretical hydraulic efficiency and hydraulic 
efficiency will be frequently used. The maximum work per Ib. that can be utilised 
by any hydraulic machine supplied with water under a head H, and from which 

it? 
the water exhausts with a velocity u is H - . The ratio 



is the theoretical hydraulic efficiency. If there are other hydraulic losses in the 
machine equivalent to a head h/ per Ib. of flow, the hydraulic efficiency is 




The actual efficiency of the machine is the ratio of the external work done per Ib. 
of water by the machine to H. 



296 HYDRAULICS 

The efficiency can also be found by considering the change of 
momentum. 

The total change of velocity impressed on the water is CE, and 
the change in the direction of motion is 
therefore FD, Fig. 186. 

And since BE is equal to BC, FB is 
equal to BD, and therefore, 

FD = 2(Ucos0-t>). 

The work done per Ib. is, then, 

2(Ucosfl-i?) 

9 ' V ' 

and the efficiency is 

TJ, 2(Ut; cos - v* 

& TT 




U 2 ........................ *r 

Differentiating with respect to v and equating to zero, 

Ucos0-2i;=0, 
or v = |U cos 0. 

The velocity Uj with which the water leaves the wheel, is then 
perpendicular to v and is 

Ui = Usin0. 

Substituting for v its value JU cos in (2), the maximum efficiency 
is cos 2 0. 

The same result is obtained from equation (1), by substituting 
forU^Usinfl. 

The maximum efficiency is then 



A common value for is 15 degrees, and the theoretical 
hydraulic efficiency is then 0*933. 

This increases as diminishes, and would become unity if 
could be made zero. 

If, however, is zero, U and v are parallel and the tip of the 
blade will be perpendicular to the radius of the wheel. 

This is clearly the limiting case, which practically is not 
realisable, without modifying the construction of the wheel. The 
necessary modification is shown in the Pelton wheel described on 
page 377. 

The actual efficiency of Poncelet wheels is from 55 to 65 per 
cent. 



WATER WHEELS 297 

Form of the bed. Water enters the wheel at all points between 
Q and R, and for no shock the bed of the channel PQ should be 
made of such a form that the direction of the stream, where it 
enters the wheel at any point A between R and Q, should make 
a constant angle 6 with the radius of the wheel at A. 

With as centre, draw a circle touching the line AS which 
makes the given angle with the radius AO. Take several 
other points on the circumference of the wheel between R and 
Q, and draw tangents to the circle STY. If then a curve 
PQ is drawn normal to these several tangents, and the stream 
lines are parallel to PQ, the water entering any part of the 
wheel between R and Q, will make a constant angle with the 
radius, and if it enters without shock at A, it will do so at all 
points. The actual velocity of the water U, as it moves along the 
race PQ, will be less than \/2grH, due to friction, etc. The 
coefficient of velocity Jc v in most cases will probably be between 
0'90 and 0'95, so that taking a mean value for Jc v of 0'925, 

U = 0'925 V2<7H. 

The best value for the velocity v taking friction into account. 
In determining the best velocity for the periphery of the wheel no 
allowance has been made for the loss of energy due to friction in 
the wheel. 

If Y r is the relative velocity of the water and wheel at entrance, 
it is to be expected that the velocity relative to the wheel at exit 
will be less than Y r , due to friction and interference of the rising 
and falling particles of water. 

The case is somewhat analogous to that of a stone thrown 
vertically up in the atmosphere with a velocity v. If there were 
no resistance to its motion, it would rise to a certain height, 



and then descend, and when it again reached the earth it would 
have a velocity equal to its initial velocity v. Due to resistances, 
the height to which it rises will be less than hi, and the velocity 
with which it reaches the ground will be even less than that due 
to falling freely through this diminished height. 

Let the velocity relative to the wheel at exit be riV r , n being 
a fraction less than unity. 

The triangle of velocities at exit will then be ABB, Fig. 188. 
The change of velocity in the direction of motion is GrH, which 
equals 



(Ucos0-t>). 



298 HYDRAULICS 

If the velocity at exit relative to tlie wheel is only riV r , there 
must have been lost by friction etc., a head equal to 



The work done on the wheel per Ib. of water is, therefore, 

-p)} V P 
-2jV- n >' 

tr c 




H 



Fig. 188. 

Let (1 - w 2 ) be denoted by /, then since 

V r 2 = BH 2 + CH 2 = (U cos B - vY + U 2 sin 2 0, 
the efficiency 



I 



Differentiating with respect to v and equating to zero, 
2 (1 +ri) Ucos^ -4 (1 + ri) v + 2U/cos 0-2vf=0 t 
from which 

_ 



/+ 



If /is now supposed to be 0'5, i.e. the head lost by friction, etc. 
is ^^ , n is 0'71 and 

v = -56U cos 0. 
If /is taken as 0*75, 

v - 0'6U cos 0. 

Dimensions of Poncelet wheels. The diameter of the wheel 
should not be less than 10 feet when the bed is curved, and not 
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 R 
and Q, Fig. 186. The water will rise on the buckets to a height 



WATER WHEELS 299 

V r 2 

nearly equal to -^- , and since the water first enters at a point R, 

the blade depth d must, therefore, be greater than this, or the 
water will overflow at the upper edge. The clearance between 
the bed and the bottom of the wheel should not be less than f ". 
The peripheral distance between the consecutive blades is taken 
from 8 inches to 18 inches. 

Horse-power of Poncelet wheels. If H is the height of the 
surface of water in the penstock above the bottom of the wheel, 
the velocity U will be about 



and v may be taken as 

0'55 x 0-92 V2^H = 0'5 JZgK. 

Let D be the diameter of the wheel, and b the breadth, and let 
t be the depth of the orifice RP. Then the number of revolutions 
per minute is 

0-5-X/205 
n - f^ . 

7T.D 

The coefficient of contraction c for the orifice may be from 0'6, 
if it is sharp-edged, to 1 if it is carefully rounded, and may be 
taken as 0'8 if the orifice is formed by a flat-edged sluice. 

The quantity of water striking the wheel per second is, then, 



If the efficiency is taken as 60 per cent., the work done per 
second is 0'6 x 62'4QH ft. Ibs. 
The horse-power N is then 



550 

182. Turbines. 

Although the water wheel has been developed to a considerable 
degree of perfection, efficiencies of nearly 90 per cent, having been 
obtained, it is being almost entirely superseded by the turbine. 

The old water wheels were required to drive slow moving 
machinery, and the great disadvantage attaching to them of 
having a small angular velocity was not felt. Such slow moving 
wheels are however entirely unsuited to the driving of modern 
machinery, and especially for the driving of dynamos, and they 
are further quite unsuited for the high heads which are now 
utilised for the generation of power. 

Turbine wheels on the other hand can be made to run at either 
low or very high speeds, and to work under any head varying 



300 HYDRAULICS 

from 1 foot to 2000 feet, and the speed can be regulated with 
much greater precision. 

Due to the slow speeds, the old water wheels could not develope 
large power, the maximum being about 100 horse-power, whereas 
at Niagara Falls, turbines of 10,000 horse-power have recently 
been installed. 

Types of Turbines. 

Turbines are generally divided into two classes; impulse, or 
free deviation turbines, and reaction or pressure turbines. 

In both kinds of turbines an attempt is made to shape the 
vanes so that the water enters the wheel without shock ; that is 
the direction of the relative velocity of the water and the vane is 
parallel to the tip of the vane, and the direction of the leaving 
edge of the vane is made so that the water leaves in a specified 
direction. 

In the first class, the whole of the available head is converted 
into velocity before the water strikes the turbine wheel, and the 
pressure in the driving fluid as it moves over the vanes remains 
constant, and equal to the atmospheric pressure. The wheel and 
vanes, therefore, must be so formed that the air has free access 
between the vanes, and the space between two consecutive vanes 
must not be full of water. Work is done upon the vanes, or in 
other words, upon the turbine wheel to which they are fixed, in 
virtue of the change of momentum or kinetic energy of the 
moving water, as in examples on pages 270 2. 

Suppose water supplied to a turbine, as in Fig. 258, under an 
effective head H, which may be supposed equal to the total head 
minus losses of head in the supply pipe and at the nozzle. The 
water issues from the nozzle with a velocity U = j2gH i and the 
available energy per pound is 



Work is done on the wheel by the absorption of the whole, or 
part, of this kinetic energy. 

If Ui is the velocity with which the water leaves the wheel, 
the energy lost by the water per pound is 



and this is equal to the work done on the wheel together with 
energy lost by friction etc. in the wheel. 

In the second class, only part of the available head is con- 
verted into velocity before the water enters the wheel, and the 



TURBINES 801 . 

velocity and pressure both vary as the water passes through the 
wheel. It is therefore essential, that the wheel shall always be 
kept full of water. Work is done upon the wheel, as will be seen 
in the sequence, partly by changing the kinetic energy the water 
possesses when it enters the wheel, and partly by changing its 
pressure or potential energy. 

Suppose water is supplied to the turbine of Fig. 191, under 
the effective head H ; the velocity U with which the water enters 
the wheel, is only some fraction of v/2^H, and the pressure head 
at the inlet to the wheel will depend upon the magnitude of U 
and upon the position of the wheel relative to the head and tail 
water surfaces. The turbine wheel always being full of water, 
there is continuity of flow through the wheel, and if the head 
impressed upon the water by centrifugal action is determined, as 
on page 335, the equations of Bernoulli * can be used to determine 
in any given case the difference of pressure head at the inlet and 
outlet of the wheel. 

If the pressure head at inlet is and at outlet , and the 

w w ' 

velocity with which the water leaves the wheel is Ui, the work 
done on the wheel (see page 338) is 

i ~ + 2^ ~ w per pound of water > 

or work is done on the wheel, partly by changing the velocity 
head and partly by changing the pressure head. Such a turbine 
is called a reaction turbine, and the amount of reaction is measured 
by the ratio 

P-Pl 
w_ w 

~H~- 

Clearly, if p is made equal to pi, the limiting case is reached, 
and the turbine becomes an impulse, or free-deviation turbine. 

It should be clearly understood that in a reaction turbine no 
work is done on the wheel merely by hydrostatic pressure, in the 
sense in which work is done by the pressure on the piston of a 
steam engine or the ram of a hydraulic lift. 

183. Reaction turbines. 

The oldest form of turbine is the simple reaction, or Scotch 
turbine, which in its simplest form is illustrated in Fig. 189. 

A vertical tube T has two horizontal tubes connected to it, the 
outer ends of which are bent round at right angles to the direction 

* fciee page 334 



302 



HYDRAULICS 



of length of the tube, or two holes and Oi are drilled as in the 
figure. 

Water is supplied to the central tube 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 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. 




Fig. 189. Scotch Turbine. 




Fig. 190. Whitelaw Turbine. 

To understand the action of the turbine it is first necessary to 
consider the effect of the whirling of the water in the arm upon 



TURBINES 303 

the discharge from the wheel. Let v be the velocity of rotation 
of the orifices, and h the head of water above the orifices. 

Imagine the wheel to be held at rest and the orifices opened ; 
then the head causing velocity of flow relative to the arm is 
simply h, and neglecting friction the water will leave the nozzle 
with a velocity 

t? = \/2gh. 

Now suppose the wheel is filled with water and made to rotate 
at an angular velocity w, the orifices being closed. There will 
now be a pressure head at the orifice equal to h plus the head 
impressed 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 

s - w . a . o>V3r 
dr = - . 

9 

The pressure per unit area at the outer periphery is, therefore, 

1 f R 
p = - 

a Jo g 



and the head impressed on the water is 

p = o> 2 R 2 
w 2g' 

Let v be the velocity of the orifice, then v = o>R, and therefore 

p _ v* 
w~2g' 

If now the wheel be assumed frictionless and the orifices are 
opened, and the wheel rotates with the angular velocity <o, the 
head causing velocity of flow relative to the wheel is 



Let Y r be the velocity relative to the wheel with which the 
water leaves the orifice. 



The velocity relative to the ground, with which the water 
leaves the wheel, is V r v, the vector sum of V r and v. 



304 HYDRAULICS 

The water leaves the wheel, therefore, with a velocity relative 
to the ground of /* = V r v, and the kinetic energy lost is 

^ per pound of water. 

The theoretical hydraulic efficiency is then, 
7 ~' 

fl 



V r 2 -t; 2 
2v 

Vr + V* 

Since from (2), V r becomes more nearly equal to v as v 
increases, the energy lost per pound diminishes as v increases, 
and the efficiency E, therefore, increases with v. 

The efficiency of the reaction wheel when friction is considered. 
As before, 



Assuming the head lost by friction to be -- 1 - , the total head 

*9 
must be equal to 



The work done on the wheel, per pound, is now 

, 

; 

and the hydraulic efficiency is 



, r /x 

20 20* 



2g 




Substituting for h from (4) and for /*, V r v, 



(l + ^Vr 2 -^ 2 
Let V r = nv, 

then 6= (l+Aj)w f -l' 

Differentiating and equating to zero, 



TURBINES 

ft 



305 



From which 



Or the efficiency is a maximum when 



k' 



and 




Fig. 191. Outward Flow Turbine. 



L. H. 



20 



306 HYDRAULICS 

184. Outward flow turbines. 

The outward flow turbine was invented in 1828 by Four- 
neyron. A cylindrical wheel W, Figs. 191, 192, and 201, having 
a number of suitably shaped vanes, is fixed to a vertical axis. 
The water enters a cylindrical chamber at the centre of the 
turbine, and is directed to the wheel by suitable fixed guide 
blades Gr, and flows through the wheel in a radial direction 
outwards. Between the guide blades and the wheel is a cylindri- 
cal sluice R which is used to control the flow of water through 
the wheel. 

*. 




Fig. 191 a. 

This method of regulating the flow is very imperfect, as when 
the gate partially closes the passages, there must be a sudden 
enlargement as the water enters the wheel, and a loss of head 
ensues. The efficiency at "part gate" is consequently very 
much less than when the flow is unchecked. This difficulty is 
partly overcome by dividing the wheel into several distinct 
compartments by horizontal diaphragms, as shown in Fig. 192, 
so that when working at part load, only the efficiency of one 
compartment is affected. 

The wheels of outward flow turbines may have their axes, 
either horizontal or vertical, and may be put either above, or 
below, the tail water level. 

The "suction tube" If placed above the tail water, the 
exhaust must take place down a " suction pipe," as in Fig. 201, 
page 317, the end of which must be kept drowned, and the pipe 
air-tight, so that at the outlet of the wheel a pressure less than 
the atmospheric pressure may be maintained. If hi is the height 
of the centre of the discharge periphery of the wheel above the 
tail water level, and p a is the atmospheric pressure in pounds per 
square foot, the pressure head at the discharge circumference is 

fc-fc-84-fc. 



TURBINES 



307 



The wheel cannot be more than 84 feet above the level of the tail 
water, or the pressure at the outlet of the wheel will be negative, 
and practically, it cannot be greater than 25 feet. 

It is shown later that the effective head, under which the 
turbine works, whether it is drowned, or placed in a suction tube, 
is H, the total fall of the water to the level of the tail race. 




Fig. 192. Fourneyron Outward Flow Turbine. 

The use of the suction tube has the advantage of allowing the 
turbine wheel to be placed at some distance above the tail water 
level, so that the bearings can be readily got at, and repairs can 
be more easily executed. 

By making the suction tube to enlarge as it descends, the 
velocity of exit can be diminished very gradually, and its final 

202 



308 HYDRAULICS 

value kept small. If the exhaust takes place direct from the 
wheel, as in Fig. 192, into the air, the mean head available is the 
head of water above the centre of the wheel. 

Triangles of velocities at inlet and outlet. For the water to 
enter the wheel without shock, the relative velocity of the water 
and the wheel at inlet must be parallel to the inner tips of the 
vanes. The triangles of velocities at inlet and outlet are shown 
in Figs. 193 and 194 



V ---------- * 




Fig. 194. 

Let AC, Fig. 193, be the velocity U in direction and magnitude 
of the water as it flows out of the guide passages, and let AD be 
the velocity v of the receiving edge of the wheel. Then DC is V r 
the relative velocity of the water and vane, and the receiving 
edge of the vane must be parallel to DC. The radial component 
GC, of AC, determines the quantity of water entering the wheel 
per unit area of the inlet circumference. Let this radial velocity 
be denoted by u. Then if A is the peripheral area of the inlet 
face of the wheel, the number of cubic feet Q per second entering 
the wheel is 

Q = A. W , 

or, if d is the diameter and b the depth of the wheel at inlet, and 
t is the thickness of the vanes, and n the number of vanes, 

Q. = (nd n.i).b.u. 

Let D be the diameter, and AI the area of the discharge peri- 
phery of the wheel. 

The peripheral velocity v t at the outlet circumference is 

v.T> 



TURBINES 309 

Let 1*1 be the radial component of velocity of exit, then what- 
ever the direction with which the water leaves the wheel the 
radial component of velocity for a given discharge is constant. 

The triangle of velocity can now be drawn as follows : 

Set off BE equal to Vi, Fig. 194, and BK radial and equal 
to UL 

Let it now be supposed that the direction EF of the tip of the 
vane at discharge is known. Draw EF parallel to the tip of the 
vane at D, and through K draw KF parallel to BE to meet EF 
in F. 

Then BF is the velocity in direction and magnitude with which 
the water leaves the wheel, relative to the ground, or to the fixed 
casing of the turbine. Let this velocity be denoted by Ui. If, 
instead of the direction EF being given, the velocity TJi is given 
in direction and magnitude, the triangle of velocity at exit can be 
drawn by setting out BE and BF equal to Vi and Ui respectively, 
and joining EF. Then the tip of the blade must be made parallel 
toEF. 

For any given value of Ui the quantity of water flowing 
through the wheel is 

Q = AiUi cos ft = AiWi. 

Work done on the wheel neglecting friction, etc. The kinetic 
energy of the water as it leaves the turbine wheel is 

2^- per pound, 

and if the discharge is into the air or into the tail water this 
energy is of necessity lost. Neglecting friction and other losses, 
the available energy per pound of water is then 

H-^Lfootlbs., 
and the theoretical hydraulic efficiency is 




and is constant for any given value of Ui, and independent of the 
direction of Ui. This efficiency must not be confused with the 
actual efficiency, which is much less than E. 

The smaller Ui, the greater the theoretical hydraulic efficiency, 
and since for a given flow through the wheel, Ui will be least 
when it is radial and equal to t*i, the greatest amount of work 
will be obtained for the given flow, or the efficiency will be a 
maximum, when the water leaves the wheel radially. If the 



310 HYDRAULICS 

water leaves with a velocity Ui in any other direction, the 
efficiency will be the same, but the power of the wheel will be 
diminished. If the discharge takes place down a suction tube, 
and there is no loss between the wheel and the outlet from the 
tube, the velocity head, lost then depends upon the velocity Ui 
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 
U, Fig. 193, in the direction of v is the velocity of whirl at inlet, 
and the component of Ui, Fig. 194, in the direction of v i9 is the 
velocity of whirl at exit. 

Let V and Vi be the velocities of whirl at inlet and outlet 
respectively, then 



and Vi = Ui sin /? = u t tan ft. 

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 U, and leaves it with velocity 
Ui, the component Vi of which is in the same direction as Vi, the 
work done on the wheel is 

per pound, 



9 9 
and therefore, neglecting friction, 

_TT W 

" 



This is a general formula for all classes of turbines and should 
be carefully considered by the student. 
Expressed trigonometrically, 

vU cos ^Mjtanff _ TT _ UL re>\ 

^^ _Li _. .. ( u) . 

g 9 %g 

If F is to the left of BK, V! is negative. 

Again, since the radial flow at inlet must equal the radial flow 
at outlet, therefore 

AUsin0 = AiTJiCos0 ..................... (3). 

When Ui is radial, Vi is zero, and Ui equals v l tan a. 

-H-' ........................... (4), 



, . , TJ i 

from which - =H -- ^ - ..................... ( 5 )> 

andfrom(3) AU sin 6 = Aj^ tan <* . .................... (6). 



TURBINES 311 

If the tip of the vane is radial at inlet, i.e. V r is radial, 



, 

and 



V a v* 



(8). 



In actual turbines is from '02H to '07H. 



Example. An outward flow turbine wheel, Fig. 195, has an internal diameter of 
6-249 feet, and an external diameter of 6-25 feet, and it makes 250 revolutions per 
minute. The wheel has 32 vanes, which may be taken as f inch thick at inlet and 
1 inches thick at outlet. The head is 141*5 feet above the centre of the wheel and 
the exhaust takes place into the atmosphere. The effective width of the wheel face 
at inlet and outlet is 10 inches. The quantity of water supplied per second is 
215 cubic feet. 

Neglecting all frictional losses, determine the angles of the tips of the vanes at 
inlet and outlet so that the water shall leave radially. 

The peripheral velocity at inlet is 

v = TT x 5-249 x Ytf 1 = 69 ft- per see., 
and at outlet v, = TT x 6-25 x 3f = 82 ft. 




Fig. 195. 

The radial velocity of flow at inlet is 

215 



TT x 5-249 x H - if 
= 18-35 ft. per sec. 
The radial velocity of flow at exit is 

215 



Therefore, 



= 16-5 ft. per seo. 
^=4-23 ft. 



312 



HYDRAULICS 



Then 



and 



= 141-5 -4-23 
9 

= 137-27 ft. 
T7 137-27 x 32-2 
V= 69~ 



: 64 ft. per sec. 



To draw the triangle of velocities at inlet set out v and u at right angles. 

Then since V is 64, and is the tangential component of U, and n is the radial 
component of U, the direction and magnitude of U is determined. 

By joining B and C the relative velocity V r is obtained, and BC is parallel to the 
tip of the vane. 

The triangle of velocities at exit is DEF, and the tip of the vane must be parallel 
toEF. 




Fig. 196. 




Pig. 197. 



The angles 0, <f>, and a can be calculated; for 



tan 0=- 



- 3-670 



and 

and, therefore, 



= 105 14', 
a = 11 23'. 



It will be seen later how these angles are modified when friction is considered. 

Fig. 198 shows the form the guide blades and vaues of the wheel would 
probably take. 

The path of the water through the wheel. The average radial velocity through 
the wheel may be taken as 17-35 feet. 

The time taken for a particle of water to get through the wheel is, therefore, 



The angle turned through by the wheel in this time is 0-39 radians. 

Set off the arc AB, Fig. 198, equal to -39 radian, and divide it into four equal 
parts, and draw the radii ea,fb, gc and Ed. 

Divide AD also into four equal parts, and draw circles through A 1 , A 2 , and A v 

Suppose a particle of water to enter the wheel at A in contact with a vane and 
suppose it to remain in contact with the vane during its passage through the wheel. 
Then, assuming the radial velocity is constant, while the wheel turns through tbe 
arc A.e the water will move radially a distance AA A and a particle that came on to 



TURBINES 



313 



the vane at A will, therefore, be in contact with the vane on the arc through A t . 
The vane initially passing through A will be now in the position el, al being 
equal to hJ and the particle will therefore be at 1. When the particle arrives on 
the arc through Ag the vane will pass through /, and the particle will consequently 
be at 2, 62 being equal to win. The curve A4 drawn through Al 2 etc. gives the 
path of the water relative to the fixed casing. 




Fig. 198. 

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

The losses of head may be enumerated as follows : 

(a) Loss by friction at the sluice and in the .penstock or 
supply pipe. 

If v is the velocity, and h a the head lost by friction in 
the pipe, 

h a = ~ 
2gm 

(6) As the water enters and moves through the guide 
passages there will be a loss due to friction and by sudden changes 
in the velocity of flow. 

This head may be expressed as 



being a coefficient. 



* Bee page 119. 



314 



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 

he = Jcil 2^> 

that is, it is made to depend upon V r the relative velocity of the 
water, and the tip of the vane. 

(d) In the wheel there is a loss of head h d) due to friction, 
which depends upon the relative velocity of the water and the 
wheol. 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 YI is diminished the relative velocity of flow v r at 
exit increases, but the relative velocity V r at inlet passes through 
a minimum when V is equal to v, or the tip of the vane is radial. 
If V 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, 



& 2 being a third coefficient. 

If there is any sudden change of velocity as the water passes 
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 




Rotor 



Boyden MSfuser 



fixed 




Fig. 199, 



TURBINES 315 

velocity with which the water leaves the turbine the effective 
head is 

U 2 
H- 2T-&/-H/. 

In well designed inward and outward flow turbines 



varies from O'lOH to '22H and the hydraulic efficiency is, therefore, 
from 90 to 78 per cent. 

The efficiency of inward and outward flow turbines including 
mechanical losses is from 75 to 88 per cent. 

Calling the hydraulic efficiency e, the general formula (1), 
section 184, may now be written 



9 9 

= '78to'9H. 

Outward flow turbines were made by Boy den* about 1848 for 
which he claimed an efficiency of 88 per cent. The workmanship 
was of the highest quality and great care was taken to reduce 
all losses by friction and shock. The section of the crowns of the 
wheel of the Boyden turbine is shown in Fig. 199. Outside of 
the turbine wheel was fitted a "diffuser" through which, after 
leaving the wheel, the water moved radially with a continuously 
diminishing velocity, and finally entered the tail race with a 
velocity much less, than if it had done so direct from the wheel. 
The loss 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. 

Double outward flow turbines. The general arrangement of an 
outward flow turbine as installed at Chevres is shown in Fig. 200. 
There are four wheels fixed to a vertical shaft, two of which 
receive the water from below, and two from above. The fall 
varies from 27 feet in dry weather to 14 feet in time of flood. 

The upper wheels only work in time of flood, while at other 
times the full power is developed by the lower wheels alone, the 
cylindrical sluices which surround the upper wheels being set in 
such a position as to cover completely the exit to the wheel. 

The water after leaving the wheels, diminishes gradually in 
velocity, in the concrete passages leading to the tail race, and the 
loss of head due to the velocity with which the water enters the 

* JLoivell Hydraulic Experiments, J. B. Francis, 1855. 



316 



HYDRAULICS 



tail race is consequently small. These passages serve the same 
purpose as Boyden's diffuser, and as the enlarging suction tube, 
in that they allow the velocity of exit to diminish gradually. 




Fig. 200. Double Outward Flow Turbine. (Escher Wyss and Co.) 

Outward flow turbine with horizontal axis. Fig. 201 shows a 
section through the wheel, and the supply and exhaust pipes, of an 
outward flow turbine, having a horizontal axis and exhausting 
down 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 Gr 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 



317 



were outward flow turbines of the type shown in Figs. 202 and 
203. 

There are two wheels on the same vertical shaft, the water 
being brought to the chamber between the wheels by a vertical 
penstock 7' 6" diameter. The water passes upwards to one wheel 
and downwards to the other. 




Fig. 201. Outward Flow Turbine with Suction Tube. 

As shown in Fig. 202 the water pressure in the chamber is 
prevented from acting on the lower wheel by the partition MN, 
but is allowed to act on the lower side of the upper wheel, the 
upper partition HK having holes in it to allow the water free access 
underneath the wheel. The weight of the vertical shaft, and of 
the wheels, is thus balanced, by the water pressure itself. 

The lower wheel is fixed to a solid shaft, which passes through 
the centre of the upper wheel, and is connected to the hollow 
shaft of the upper wheel as shown diagrammatically in Fig. 202. 
Above this connection, the vertical shaft is formed of a hollow 



318 



HYDRAULICS 



tube 38 inches diameter, except where it passes through the 
bearings, where it is solid, and 11 inches diameter. 

A thrust block is also provided to carry the unbalanced 
weight. 

The regulating sluice is external to the wheel. To maintain a 
high efficiency at part gate, the wheel is divided into three separate 
compartments as in Fourneyron's wheel. 

Gonrvee&on, for 
HoUUvw and 
Solid Shaft 



Water adsrvilted, 
uruler th& upper 
wheel to support 

eight 
of the- shaft 






Fig. 202. Diagrammatic section of Outward Flow Turbine at Niagara Falls. 

A vertical section through the lower wheel is shown in Fig. 
203, and a part sectional plan of the wheel and guide blades in 
Fig. 195. 

(Further particulars of these turbines and a description of the 
governor will be found in Cassier's Magazinej Yol. 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. 



TURBINES 



319 



As improved by Francis*, in 1849, the wheel was of the form 
shown in Fig. 204 and was called by its inventor a "central vent 
wheel." 




I 

fc 

s 



The wheel is carried on a vertical shaft, resting on a footstep, 
and supported by a collar bearing placed above the staging S. 

* Lowell Hydraulic Experiments, F. B. Francis, 1855. 



320 



HYDRAULICS 



Above tlie wheel is a heavy casting C, supported by bolts 
from the staging S, which acts as a guide for the cylindrical 
sluice F, and carries the bearing B for the shaft. There are 
40 vanes in the wheel shown, and 40 fixed guide blades, the former 
being made of iron one quarter of an inch thick and the latter 
three-sixteenths of an inch. 




Fig. 204. Francis' Inward flow or Central vent Turbine. 

The triangles of velocities at inlet and outlet, Fig. 205, are 
drawn, exactly as for the outward flow turbine, the only difference 
being that the velocities v, U, V, Y r and u refer to the outer 



TURBINES 



321 



periphery, and v,, Ui, V : , -y r and %! to the inner periphery of the 
wheel. 

The work done on the wheel is 



iVi -, ,, ,, 

--- - ft. Ibs. per lb., 
y y 



and neglecting friction, 



g 9 2g' 

For maximum efficiency, for a given flow through the wheel, 
Ui should be radial exactly as for the outward flow turbine. 




Fig. 205. 

The student should work the following example. 

The outer diameter of an inward flow turbine wheel is 7*70 feet, and the inner 
diameter 6-3 feet, the wheel makes 55 revolutions per minute. The head is 
14-8 feet, the velocity at inlet is 25 feet per sec., and the radial velocity may be 
assumed constant and equal to 7*5 feet. Neglecting friction, draw the triangles of 
velocities at inlet and outlet, and find the directions of the tips of the vanes at 
inlet and outlet so that there may be no shock and the water may leave radially. 

Loss of head by friction. The losses of head by friction are 
similar to those for an outward flow turbine (see page 313) and 
the general formula becomes 



When the flow is radial at exit, 



The value of e varying as before between 078 and 0*90. 

Example (1). An inward flow turbine working under a head of 80 feet has 
radial blades at inlet, and discharges radially. The angle the tip of the vane 
makes with the tangent to the wheel at exit is 30 degrees and the radial velocity 
is constant. The ratio of the radii at inlet and outlet is 1-75. Find the velocity 
of the inlet circumference of the wheel Neglect friction. 



L. H. 



21 



322 HYDRAULICS 

Since the discharge is radial, the velocity at exit la 



Then 



= pjg tan 30. 

v z tan a 30 



and since the blades are radial at inlet V is equal to r, 

therefore v*=g . 80 v ^22! 

1-75 2 2 ' 



from which 



V 



32x80 



1-0543 ' 
;49'3 ft. per see. 




Trijcungle cfPelociti^y 



Fig. 206. 



Example (2) The outer diameter of the wheel of an inward flow turbine of 
200 horse-power is 2-46 feet, the inner diameter is 1-968 feet. The wheel makes 
300 revolutions per minute. The effective width of the wheel at inlet = 1-15 feet. The 
head is 39 '5 feet and 59 cubic feet of water per second are supplied. The radial 
velocity with which the water leaves the wheel may be taken as 10 feet per second. 

Determine the theoretical hydraulic efficiency E and the actual efficiency e l of 
the turbine, and design suitable vanes. 

200x550 



39-5x59x62-5 



Theoretical hydraulic efficiency 




The radial velocity of flow at inlet, 



= 6-7 feet per sec. 



TURBINES 323 

The peripheral velocity 

v = 2-46 . TT x 8 ^}- = 38-6 feet. 

The velocity of whirl V. Assuming a hydraulic efficiency of 85 %, from 
the formula 



v _ 39-5 x 32-2 x -85 

38-6 

= 28-0 feet per sec. 
The angle 9. Since w = 6'7 ft. per sec. and V = 28'0 ft. per sec. 

tan = ^=0-239, 

Jo 

= 13 27'. 
The angle <f>* Since V is less than v, <f> is greater than 90. 



and = 152. 

For the water to discharge radially with a velocity of 10 feet per seo. 



and a = 18 nearly. 

The theoretical vanes are shown in Fig. 206. 
Example (3). Find the values of <j> and a on the assumption that e is 0-80. 

Thomson's inward flow turbine. In 1851 Professor James 
Thomson invented an inward flow turbine, the wheel of which 
was surrounded by a large chamber set eccentrically to the wheel, 
as shown in Figs. 207 to 210. 

Between the wheel and the chamber is a parallel passage, in 
which are four guide blades Gr, pivoted on fixed centres C and 
which can be moved about the centres C by bell crank levers, 
external to the casing, and connected together by levers as shown 
in Fig. 207. The water is distributed to the wheel by these guide 
blades, and by turning the worm quadrant Q by means of the 
worm, the supply of water to the wheel, and thus the power of 
the turbine, can be varied. The advantage of this method of 
regulating the flow, is that there is no sudden enlargement from 
the guide passages to the wheel, and the efficiency at part load 
is not much less than at full load. 

Figs. 209 and 210 show an enlarged section and part sectional 
elevation of the turbine wheel, and one of the guide blades Gr. 
The details of the wheel and casing are made slightly different 
from those shown in Figs. 207 and 208 to illustrate alternative 
methods. 

The sides or crowns of the wheel are tapered, so that the 
peripheral area of the wheel at the discharge is equal to the 
peripheral area at inlet. The radial velocities of flow at inlet 
and outlet are, therefore, equal. 

212 



324 HYDRAULICS 

The inner radius r in Thomson's turbine, and generally in 
turbines of this class made by English makers, is equal to one-half 
the external radius E. 




Fig. 207. Guide blades and casing of Thomson Inward Flow Turbine. 

The exhaust for the turbine shown takes place down two 
suction tubes, but the turbine can easily be adapted to work below 
the tail water level. 

As will be seen from the drawing the vanes of the wheel are 
made alternately long and short, every other one only continuing 
from the outer to the inner periphery. 



TURBINES 



325 



The triangles of velocities for the inlet and outlet are shown in 
Pig. 211, the water leaving the wheel radially. 

The path of the water through the wheel, relative to the fixed 
casing, is also shown and was obtained by the method described 
on page 312. 

Inward flow turbines with adjustable guide blades, as made by 
the continental makers, have a much greater number of guide 
blades (see Fig. 233, page 352). 




Fig. 208. Section through wheel and casing of Thomson Inward Flow Turbine. 

188. Some actual inward flow turbines. 

A later form of the Francis inward flow turbine as designed by 
Pictet and Co., and having a horizontal shaft, is shown in Fig. 212. 

The wheel is double and is surrounded by a large chamber 
from which water flows through the guides G to the wheel W. 
After leaving the wheel, exhaust takes place down the two suction 
tubes S, thus allowing the turbine to be placed well above the 
tail water while utilising the full head. 

The regulating sluice F consists of a steel cylinder, which 
slides in a direction parallel to the axis between the wheel and 
guides. 



326 



HYDKAULICS 



irvGwi/cLe. 




Fig. 209. Fig- 210. 

Detail of wheel and guide blade of Thomson Inward Flow Turbine. 



- - v 




. 211. 



TURBINES 



327 



The wheel is divided into five separate compartments, so that 
at any time only one can be partially closed, and loss of head by 
contraction and sudden enlargement of the stream, only takes 
place in this one compartment. 




328 



HYDRAULICS 



The sluice F is moved by two screws T, which slide through 
stuffing boxes B, and which can be controlled by hand or by the 
governor B. 

Inward flow turbine for low falls and variable head. The 
turbine shown in Fig. 213 is an example of an inward flow turbine 
suitable to low falls and variable head. It has a vertical axis and 
works drowned. The wheel and the distributor surrounding the 
wheel are divided into five stages, the two upper stages being 
shallower than the three lower ones, and all of which stages can 




Fig. 213. Inward Flow Turbine for a low and variable fall. (Pictet and Co.) 



TURBINES 329 

be opened or closed as required by the steel cylindrical sluice CO 
surrounding the distributor. 

When one of the stages is only partially closed by the sluice, 
a loss of efficiency must take place, but the efficiency of this one 
stage only is diminished, the stages that are still open working 
with their full efficiency. With this construction a high efficiency 
of the turbine is maintained for partial flow. With normal flows, 
and a head of about 6*25 feet, the three lower stages only are 
necessary to give full power, and the efficiency is then a 
maximum. In times of flood there is a large volume of water 
available, but the tail water rises so that the head is only about 
4'9 feet, the two upper stages can then be brought into operation 
to accommodate a larger flow, and thus the same power may be 
obtained under a less head. The efficiency is less than when the 
three stages only are working, but as there is plenty of water 
available, the loss of efficiency is not serious. 

The cylinder C is carried by four vertical spindles S, having 
racks R fixed to their upper ends. Gearing with these racks, are 
pinions p, Fig. 213, all of which are worked simultaneously by the 
regulator, or by hand. A bevel wheel fixed to the vertical shaft 
gears with a second bevel wheel on a horizontal shaft, the velocity 
ratio being 3 to 1. 

189. The best peripheral velocity for inward and outward 
flow turbines. 

When the discharge is radial, the general formula, as shown on 
page 315, is 

= eH>0-78toO'90H ....... .............. (1). 

If the blades are radial at inlet, for no shock, v should be equal 
to Y, and 



or v = V = G'624 to 

This is sometimes called the best velocity for v, but it should be 
clearly understood that it is only so when the blades are radial at 
inlet. 

190. Experimental determination of the best peripheral 
velocity for inward and outward flow turbines. 

For an outward flow turbine, working under a head of 14 feet, 
with blades radial at inlet, Francis* found that when v was 

0'626 



Lowell, Hydraulic Experiments, 



3 30 HYDRAULICS 

the efficiency was a maximum and equal to 79'87 per cent. The 
efficiency however was over 78 per cent, for all values of v 
between 0'545 *j2gK and '671 \/2#H. If 3 per cent, be allowed 
for the mechanical losses the hydraulic efficiency may be taken 
as 82'4 per cent. 

~Vv 
From the formula = *S24H, and taking V equal to v, 

v = '64 V2^H, 

so that the result of the experiment agrees well with the formula. 
For an inward flow turbine having vanes as shown in Fig. 205, 
the total efficiency was over 79 per cent, for values of v between 
0-624 \/2#H and 0'708 \/2#H, the greatest efficiency being 79'7 
per cent, when v was 0'708 v2#H and again when v was 
637 



It will be seen from Fig. 205 that although the tip of the vane 
at the convex side is nearly radial, the general direction of the 
vane at inlet is inclined at an angle greater than 90 degrees to 
the direction of motion, and therefore for no shock Y should be 
less than v. 

When v was '708 N/20H, V, Fig. 205, was less than v. The 
value of Y was deduced from the following data, which is also 
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 oi 
fixed guides external to the wheel. 

The minimum width of each guide passage was 0'1467 foot and 
the depth T0066 feet, 

The quantity of water supplied to the wheel per second was 
112*525 cubic feet, and the total fall of the water was 13'4 feet. 
The radial velocity of flow u was, therefore, 3*86 feet per second. 

The velocity through the minimum section of the guide passage 
was 19 feet per second. 

When the efficiency was a maximum, v was 20'8 feet per sec. 
Then the radial velocity of flow at inlet to the wheel being 
3'86 feet, and U being taken as 19 feet per second, the triangle 
of velocities at inlet is ABC, Fig. 205, and Y is 18'4 feet per sec. 

If it is assumed that the water leaves the wheel radially, then 

eH= = H'85 feet. 
9 

1 1*85 
The efficiency e should be =88'5 per cent., which is 9 per 



cent, higher than the actual efficiency. 



TURBINES 331 

The actual efficiency however includes not only the fluid losses 
but also the mechanical losses, and these would probably be from 
2 to 8 per cent., and the actual work done by the turbine on the 
shaft is probably between 80 and 86*5 per cent, of the work done 
by the water. 

Vv 

191. Value of e to be used in the formula = eH. 

g 

In general, it may be said that, in using the formula = eH, 

the value of e to be used in any given case is doubtful, as even 
though the efficiency of the class of turbines may be known, it is 
difficult to say exactly how much of the energy is lost mechanically 
and how much hydraulically. 

A trial of a turbine without load, would be useless to deter- 
mine the mechanical efficiency, as the hydraulic losses in such a 
trial would be very much larger than when the turbine is working 
at full load. By revolving the turbine without load by means of 
an electric motor, or through the medium of a dynamometer, the 
work to overcome friction of bearings and other mechanical losses 
could be found. At all loads, from no load to full load, the 
frictional resistances of machines are fairly constant, and the 
mechanical losses for a given class of turbines, at the normal load 
for which the vane angles are calculated, could thus approximately 
be obtained. If, however, in making calculations the difference 
between the actual and the hydraulic efficiency be taken as, say, 
5 per cent., the error cannot be very great, as a variation of 5 per 
cent, in the value assumed for the hydraulic efficiency e t will only 
make a difference of a few degrees in the calculated value of 
the angle <. 

The best value for e, for inward flow turbines, is probably 0'80, 
and experience shows that this value may be used with confidence. 

Example. Taking the data as given in the example of section 184, and assuming 
an efficiency for the turbine of 75 per cent., the horse-power is 
215 x 62-4 x 141-5 x -75 x 60 

33,000 

=2600 horse-power. 

If the hydraulic efficiency is supposed to be 80 per cent., the velocity of 
whirl V should be 

Sg.H^O-8.32.1415 

v 69 

=52 feet per sec. 

Then tan - 18 ' 35 - - 1835 

and 0=132 47'. 

Now suppose the turbine to be still generating 2600 horse-power, and to have 
an efficiency of 80 per cent., and a hydraulic efficiency of 85 per cent. 



332 



HYDRAULICS 



Then the quantity of water required per second, is 
215 x 0-75 



0-8 



: 200 cubic feet per sec. 



and the radial velocity of flow at inlet will be 

1835x200 . 
u= =17'1 ft. per sec. 

. -85.32.141-5 



Then 



tan 



69 
17-1 

"55-4- 69 : 
= 128. 24'. 



:55 - 4 ft. per sec. 



-17-1 
13-6 



192. The ratio of the velocity of whirl V to the velocity 
of the inlet periphery v. 

Experience shows that, consistent with Vu satisfying the general 

formula, the ratio ^ 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 values 




Fig. 214. 



TURBINES 



833 



v 

of /s = ff > V being calculated from = 0'8H. The corresponding 

variation in the angle <, Fig. 214, is from 20 to 150 degrees. 

For a given head, v may therefore vary within wide limits, 
which allows a very large variation in the angular velocity of the 
wheel to suit particular circumstances. 

TABLE XXXVII. 

Showing the heads, and the velocity of the receiving circum- 
ference v for some existing inward and outward, and mixed flow 
turbines. 











Katio 




v 

Katio 






Hfeet 


v feet 
per sec. 


N/20H 


v 


H.P. 


V being calculated 
Vv 


/O TT 










^ offtL 




from = -8H 














9 


Inward flow : 














Niagara Falls* 
Rheinfelden 


146 
14-8 


70 
22 


96-8 
30-7 


0-72 
0-71 


5000 
840 


0-555 
0-565 


By Theodor ) 
BeU and Co. j 


28-4 


39 


42-6 


0-91 




0-44 




60-4 


32-2 


62-3 


0-52 




0-77 


Pictet and Co. 


183-7 


51-1 


76-8 


0-47 


300 


0-85 


M 


134-5 


46-6 


65-6 


0-505 


300 


0-79 


M 


6-25 


16-6 


20 


0-83 




0-48 




30 


25-75 


44 


0-58 


700 


0-69 






38-5 


50-3 


0-77 


200 


0-52 


Ganz and Co. 


112 


64-3 


84-6 


0-54 




0-74 


}j 


225 


64-7 


120 


0-54 


682 


0-58 


Rioter and Co. 


10-66 


15-2 


26 


0-585 


30 


0-69 


Outward flow : 














Niagara Falls 
Pictet and Co. 


141-5 
130-5 


69 
69 


95-2 
91-6 


0-725) 
0-750) 


5000 


0-55 
0-53 


Ganz and Co. 


95-1 


38-7 


78-0 


0-495 


290 


0-81 


M 


223 


55-6 


120-0 


0-46 


1200 


0-87 



* Escher Wyss and Co. 

For example, if a turbine is required to drive alternators 
direct, the number of revolutions will probably be fixed by the 
alternators, while, as shown later, the diameter of the wheel is 
practically fixed by the quantity of water, which it is required to 
pass through the wheel, consistent with the peripheral velocity of 
the wheel, not being greater than 100 feet per second, unless, as 
in the turbine described on page 373, special precautions are 
taken. This latter condition may necessitate the placing of two 
or more wheels on one shaft. 



334 HYDRAULICS 

Suppose then, the number of revolutions of the wheel to be 
given and d is fixed, then v has a definite value, and V must be 
made to satisfy the equation 

Vv 
=eH. 
9 

Fig. 214 is drawn to illustrate three cases for which Yv is 
constant. The angles of the vanes at outlet are the same for all 
three, but the guide angle and the vane angle </> at inlet vary 
considerably. 

193. The velocity with which water leaves a turbine. 
In a well-designed turbine the velocity with which the water 

leaves the turbine should be as small as possible, consistent with 
keeping the turbine wheel and the down-take within reasonable 
dimensions. 

In actual turbines the head lost due to this velocity head 
varies from 2 to 8 per cent. If a turbine is fitted with a 
suction pipe the water may be allowed to leave the wheel itself 
with a fairly high velocity and the discharge pipe can be made 
conical so as to allow the actual discharge velocity to be as small 
as desired. It should however be noted that if the water leaves 
the wheel with a high velocity it is more than probable that there 
will be some loss of head due to shock, as it is difficult to ensure 
that water so discharged shall have its velocity changed gradually. 

194. Bernouilli's equations applied to inward and out- 
ward flow turbines neglecting friction. 

Centrifugal head impressed on the water by the wheel. The 
theory of the reaction turbines is best considered from the point 
of view of Bernoulli's equations ; but before proceeding to discuss 
them in detail, it is necessary to consider the " centrifugal head " 
impressed on the water by the wheel. 

This head has already been considered in connection with the 
Scotch turbine, page 303. 

Let r, Fig. 216, be the internal radius of a wheel, and R the 
external radius. 

At the internal circumference let the wheel be covered with a 
cylinder c so that there can be no flow through the wheel, and let 
it be supposed that the wheel is made to revolve at the angular 
velocity w which it has as a turbine, the wheel being full of water 
and surrounded by water at rest, the pressure outside the wheel 
being sufficient to prevent the water being whirled out of the 
wheel. Let d be the depth of the wheel between the crowns. 
Consider any element of a ring of radius r and thickness dr, and 
subtending a small angle 6 at the centre 0, Fig. 216. 



TURBINES 



335 



The weight of the element is 

wr . dr .d, 
and the centrifugal force acting on the element is 

wr Q . dr . d . wV ,, 
Ibs. 

g 

Let p be the pressure per unit area on the inner face of the 
element and p + dp on the outer. 

wr .dr.d. wV 



Then 



g.r.e.d 




Fig. 215. 



Fig. 216. 



The increase in the pressure, due to centrifugal forces, between 
r and B, is, therefore, 



w 



t PC _ U) /T>2 2\ _ V V l 

For equilibrium, therefore, the pressure in the water surround- 
ing the wheel must be p c . 

If now the cylinder c be removed and water is allowed to flow 
through the wheel, either inwards or outwards, this centrifugal 
head will always be impressed upon the water, whether the wheel 
is driven by the water as a turbine, or by some external agency, 
and acts as a pump. 

Bernoulli's equations. The student on first reading these 
equations will do well to confine his attention to the inward flow 
turbine, 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 p a 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 



w 



(i). 



Between B and A the wheel impresses upon the water the 
centrifugal head 



v being greater than 
outward flow. 



_ 
2g 2g> 

for an inward flow turbine and less for the 




Fig. 217. 

Consider now the total head relative to the wheel at A and B. 
The velocity head at A is - and the pressure head is , and 

at B the velocity and pressure heads are - and respectively. 

If no head were impressed on the water as it flows through 
the wheel, the pressure head plus the velocity head at A and B 
would be equal to each other. But between A and B there is 
impressed on the water the centrifugal head, and therefore, 



_- 
w 2g 2g 2g w 2g 



TURBINES 337 

This equation can be used to deduce the fundamental equation, 

Y!?_!^ = ^. ...(3). 

9 
From the triangles ODE and ADE, Fig. 218, 

Y r 2 =(Y-<y) 2 + ^ 2 andY 2 + ^ 2 = IP, 
and from the triangle BFGr, Fig 219, 

Vr = (vi - Yi) 2 + U? and Yi 2 + u* = Ui 2 . 
Therefore by substitution in (2), 

Pi , fa-Yi)' + v* _t?i a , u? = P i (V-vY { u* 



2(7 



2g 



From which 



U 



w g 2g w 2g g ' 



and 



g g w w 2g 2g 

JT2 

Substituting for - + ~- from (1) 



.(5). 




Fig. 218. 



Wheel in suction tube. If the centre of the wheel is 7i 
above the surface of the tail water, and Uo is the velocity with 
which the water leaves the down-pipe, then 



w 



Substituting f or ^ + i- in (6), 



SI.ftSl.Hi+fis-***.-^ 

g g w w zg 

= H-P. 



L. U. 



23 



338 HYDRAULICS 

IfVisO, ^H-^U. 

9 20 

The wheel can therefore take full advantage 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 Jh is the depth of the centre of the 
wheel below the tail race level, 

fc-n,+*, 

w w ' 

and the work done on the wheel per pound of water is again 

vV ViV! _ W , 
--- = JbL s~~ = fi. 
99 20 

vV 
IfVjisO, g =h ' 

From equation (5), 

vV t?iYi = p Pi , H* Hi 3 
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 done 
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 p 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 

lL = ^L 2 + ^_^L 

20 2g 20 2g' 

and if at exit v r is made equal to Vi, or the triangle BFG, 
Fig. 219, is isosceles, 

V,' = ^ 
20 20' 

and the triangle of velocities at entrance is also isosceles. 
The pressure head at entrance is 

' 



and at exit is either + fei , or - h . 



TURBINES 339 

Therefore, since the pressures at entrance and exit are equal, 

U 2 

2^ = H -^ = H, 

or else H + 7io = H. 

The water then enters the wheel with a velocity equal to that 
due to the total head H, and the turbine becomes a free-deviation 
or impulse turbine. 

195. Bernoulli's equations for the inward and outward 
flow turbines including friction. 

If H/ is the loss of head in the penstock and guide passages, 
hf the loss of head in the wheel, h t the loss at exit from the wheel 
and in the suction pipe, and Ui the velocity of exhaust, 

+-*+*-* ........................ a), 



and = + fc e -fci ........................... (3), 

w w 

from which = H-(^ + h f + H/+k) .................. (4). 

If the losses can be expressed as a fraction of H, or equal to KH, 
then 

= (l-K)H=eH 

= 0'78H to 0-90II*. 

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 
by the turbine. 

Let E be the theoretical hydraulic efficiency. 

Let e be the hydraulic efficiency. 

Let e m be the mechanical efficiency. 

Let 61 be the actual efficiency including mechanical losses. 

Let Ui be the radial velocity with which the water leaves the 
wheel. 

Let D be the diameter of the wheel in feet at the inlet circum- 
ference and d the diameter at the outlet circumference. 

Let B be the width of the wheel in feet between the crowns 
at the inlet circumference, and b be the width between the crowns 
at the outlet circumference. 

Let N be the horse-power of the turbine. 
* See page 315. 

22-2 



340 HYDRAULICS 

The number of cubic feet per second required is 

N.33 ? 000 
* eiH. 62'4. 60 ' 

A reasonable value for e\ is 75 per cent. 

The velocity U with which the water leaves the turbine, since 

U 2 




_ 

is U =\/20(l-E)Hft. per sec ................ (2). 

If it be assumed that this is equal to u\ 9 which would of 
necessity be the case when the turbine works drowned, or 
exhausts into the air, then, if t is the peripheral thickness of the 
vanes at outlet and m the number of vanes, 



If Uo is not equal to Ui, then 

(ird-mt) 1^1 = 0, ........................ (3). 

The number of vanes m and the thickness t are somewhat 
arbitrary, but in well-designed turbines t is made as small as 
possible. 

As a first approximation mt may be taken as zero and (3) 
becomes 

wd6wi = 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 Fig. 208, 
and the diameter of the shaft is d , the axial velocity is 



UQ - - j - ft. per sec. 



The diameter d 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 %. Then 

j /^Q> fi. ffc\ 

a = A/^ it \oj. 

From (4) and (5) b and d can now be determined. 

A ratio for -T having been decided upon, D can be calculated, 

d 

and if the radial velocity at inlet is to be the same as at outlet, 
and i is the thickness of the vanes at inlet, 

v^ / j /\ 7 /^5\ 

(TT -m ; - Ui -V* m 



TURBINES 341 



For rolled brass or wrought steel blades, t may be very small, 
and for blades cast with the wheel, by shaping them as in Fig. 227, 
to is practically zero. Then 



7TUL) 

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 



P er sec 



Vv 

Then = eR. 

9 

and if e is given a value, say 80 per cent., 

V= -^ ft. per sec (8). 

Since u, V, and v are known, the triangle of velocities at inlet 
can be drawn and the direction of flow and of the tip of vanes 
at inlet determined. Or and <, Fig. 214, can be calculated from 



(9) 



and tan< = y ........................ (10). 

Then IT, the velocity of flow at inlet, is 



irn f . 
At exit t?i = -gQ- it. per sec., 

and taking u^ as radial and equal to u, the triangle of velocities 
can be drawn, or a calculated from 

tan a = - . 
v\ 

If Ho is the head of water at the centre of the wheel and H/ 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 6U feet, and to run at 250 revolutions per minute. 
To determine the leading dimensions of the turbine. 
Assuming c^ to be 75 per cent., 

300 x 33,000 
^~ -75x60x62-4x60 
= 58-7 cubic feet per sec. 



342 HYDRAULICS 

Assuming E is 95 per cent., or five per cent, of the head is lost by velocity 
of exit and Uj = u, 

| = -05.60 

and w= 13-8 feet per sec. 

Then from (5), page 340, 



= 1-65 feet, 
say 20 inches to make allowance for shaft and to keep even dimension. 

Then from (4) , b = ^ = -82 foot 

JL'OD 

= 9 inches say. 
Taking - as 1-8, D=3-0 feet, and 

v = TT . 3 . *f = 39-3 feet per sec., 
and B = 5 inches say. 

Assuming e to be 80 per cent., 

T7 -80 x 60 x 32 , 

- 3^3 - per sec< 

13-8 

, 

and 0=19 30', 

13-8 



and 0=91 15'. 

13-8x1-8 



and a =32 18'. 

The velocity U at inlet is 



= 41-3 ft. per sec. 
The absolute pressure head at the inlet to the wheel is 

2- = H +^ -- - -- hf, the head lost by friction in the down pipe 

= H + 34 -26-5 -ft/. 

The pressure head at the outlet of the wheel will depend upon the height of the 
wheel above or below the tail water. 

197. Parallel or axial flow turbines. 

Fig. 220 shows a double compartment axial flow turbine, the 
guide blades being placed above the wheel and the flow through 
the wheel being parallel to the axis. The circumferential section 
of the vanes at any radius when turned into the plane of the 
paper is as shown in Fig. 221. A plan of the wheel is also shown. 

The triangles of velocities at inlet and outlet for any radius 
are similar to those for inward and outward flow turbines, the 
velocities v and v i9 Figs. 222 and 223, being equal. 



TURBINES 

The general formula now becomes 



343 



U, 1 



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. Double Compartment Parallel Flow Turbine. 





Figs. 221, 222, 223. 
Then, taking friction and other losses into account, 



9 



344 



HYDRAULICS 



The velocity v will be proportional to the radius, so that if the 
water is to enter and leave the wheel without shock, the angles 0, 
<, and a must vary with the radius. 

The variation in the form of the vane with the radius is shown 
by an example. 

A Jonval wheel has an internal diameter of 5 feet and an 
external diameter of 8' 6". The depth of the wheel is 7 inches. 
The head is 15 feet and the wheel makes 55 revolutions per 
minute. The flow is 300 cubic feet per second. 

To find the horse-power of the wheel, and to design the wheel 
vanes. 

Let TI be any radius, and r and r 2 the radii of the wheel at the 
inner and outer circumference respectively. Then 

r = 2'5 feet and v = 2-n-r |f = 14*4 feet per sec., 
TI = 3'75 feet and Vi = 2irri f = 21'5 feet per sec., 
r- 2 = 4*25 feet and v 2 = 27rr 2 f = 24*5 feet per sec. 
The mean axial velocity is 

300 
u = = 8 ' 15 f 




Fig. 224. Triangles of velocities at inlet and outlet at three different 
radii of a Parallel Flow Turbine. 



Taking e as 0'80 at each radius, 

T7 _0'8. 32'2.15 385 
V = 14'4 



= 26'7 ft. per sec., 

JL"J? ~a? j.~x ~x 

Vi = ni^c = 17'9 ft. per sec., 

V 2 = oTTc = 15*7 ft. per sec. 

Inclination of the vanes at inlet. The triangles of velocities 
for the three radii r, ri, r 2 are shown in Fig. 224. For example, 
at radius r, ADC is the triangle of velocities at inlet and ABC the 



TURBINES 345 

triangle of velocities at outlet. The inclinations of the vanes at 
inlet are found from 

O.-l K 

tan <ft = . _ . > fr m which < = 3330', 



8'15 

= 113 50, 



8'15 
tan < 2 = 15 . 7 _ 24-5 > from wnicn ^ = 137 6'. 

The inclination of the guide blade at each of the three radii. 
8 ' 15 



from which = 17, 

tan^ji^ and ^ = 24 30', 

tan<9 2 = f^ and 2 = 27 30'. 
lo / 

The inclination of the vanes at exit. 



tan a, = = 20 48', 

Zi O 

tan -,= |^> = 18 22'. 

<u4 O 

If now the lower tips of the guide blades and the upper tips 
of 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 
difficulty 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 
upper 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 
blades. Let MN be the plan of the upper edge of a guide blade 
and let DGr be the plan of the lower edge, DGr being parallel to 
MN. Then as the water runs along the guide at D, it will leave 
the guide in a direction perpendicular to OD. At Gr it will leave 
in a direction HGr perpendicular to OGr. Now suppose the guide 
at the edge DGr to have an inclination ft to the plane of the paper. 
If then a section of the guide is taken by a vertical plane XX 
perpendicular to DG , the elevation of the tip of the vane on this 
plane will be AL, inclined at ft to the horizontal line AB, and AC 



346 



HYDRAULICS 



will be the intersection of the plane XX with the plane tangent 
to the tip of the vane. 

Now suppose DE and GH to be the projections on the plane 
of the paper of two lines lying on the tangent plane AC and 
perpendicular to OD and OG respectively. Draw EF and HK 
perpendicular to DE and GH respectively, and make each of 
them equal to BC. Then the angle EDF is the inclination of the 
stream line at D to the plane of the paper, and the angle HGK is 
the inclination of the stream line at G to the plane of the paper. 
These should be equal to and a . 




flarb of lower edge of guide, 
blade'& of upper edge* of* 



Fig. 225. Plan of guide blades and vanes of Parallel Flow Turbines. 

Let y be the perpendicular distance between MN and DGr. 
Let the angles GOD and GOH be denoted by < an a respectively. 

Since EF, BC and HK are equal, 

ED tan B = y tan /? (1), 

and GH tan a = # tan /3 (2). 



But 

and 

Therefore 
and 

Again, 



cos (a + 



= cos(a + )tan/3 ..................... (3), 

tan 2 = cos a tan p ........................... (4). 



sin a = 

XV 



(5). 



There are thus three equations from which a, <f> and P can be 
determined. 

Let x and y be the coordinates of the point D, being the 
intersection of the axes. 



TURBINES 



347 



Then 



and from (5) 



cos (a 



cos a 



- /ITZ 

-V ] R 2 ' 



Substituting for cos (a + <) and cos a and the known values of 
tan and tan 2 in the three equations (35), three equations are 
obtained with x, y, and ft as the unknowns. 

Solving simultaneously 

x = 1*14 feet, 

y = 2'23 feet, 

and tan p = 0'67, 

from which p = 34. 




Fig. 226. 




Fig. 228. 

The length of the guide blade is thus found, and the constant 
slope at the edge DG so that the stream lines at D and Gr shall 
have the correct inclination. 

If now the upper edge of the vane is just below DG, and the 
tips of the vane at D and G- are made as in Figs. 226 228, < and 



HYDRAULICS 

< 2 being 33 30' and 137 6' respectively, the water will move on to 
the vane without shock. 

The plane of the lower edge of the vane may now be taken as 
D'G-', Fig. 225, and the circular sections DD', PQ, and GGT at the 
three radii, r, r 1} and r 2 are then as in Figs. 226 228. 

198. Regulation of the flow to parallel flow turbines. 

To regulate the flow through a parallel flow turbine, Fontaine 
placed sluices in the guide passages, as in Fig. 229, connected to 
a ring which could be raised or lowered by three vertical rods 
having nuts at the upper ends fixed to toothed pinions. When 




Fig. 229. Fontaine's Sluices. 




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 pinions 
carrying the nuts. Fontaine fixed the turbine wheel to a hollow 
shaft which was carried on a footstep above the turbine. In some 
modern parallel flow turbines the guide blades are pivoted, as in 
Fig. 230, so that the flow can be regulated. The wheel may be 
made with the crowns opening outwards, in section, similar to 
the Grirard turbine shown in Fig. 254, so that the axial velocity 
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 

power. During times of flood, and when there is plenty of water, 
the head falls to 2 feet, and the sluices of the inner ring are 
opened. A larger supply of water at less head can thus be 
allowed to pass through the wheel, and although, due to the shock 
in the guide passages of the inner ring, the wheel is not so efficient, 
the abundance of water renders this unimportant. 

Example. A double compartment Jonval turbine has an outer diameter of 
12' 6" and an inner diameter of 6 feet. 

The radial width of the inner compartment is 1' 9" and of the outer compart- 
ment r 6". Allowing a velocity of flow of 3-25 ft. per second and supposing the 
minimum fall is V 8", and the number of revolutions per minute 14, find the horse- 
power of the wheel when all the guide passages are open, and find what portion of 
the inner compartment must be shut off so that the horse-power shall be the same 
under a head of 3 feet. Efficiency 70 per cent. 

Neglecting the thickness of the blades, 

the area of the outer compartment = - (12-5 a -9-5 2 ) = 52 < 6 sq. feet. 

inner = (9'5 2 -6 2 )=42-8 sq. feet. 

Total area = 95 -4 sq. feet. 

The weight of water passing through the wheel is 

W=95-4 x 62-4 x 3-25 Ibs. per sec. 

= 19,3001bs. per seo. 
and the horse-power is 

Hp = l<l800xl*6x0.7 

550 

Assuming the velocity of flow constant the area required when the head 
is 3 feet is 

40-8x33,000 

k ~GOx62-5x3x-7 
= 55-6sq. feet, 
or the outer wheel will nearly develop the horse-power required. 

199. Bernouilli's equations for axial flow turbines. 

The Bernouilli's equations for an axial flow turbine can be 
written down in exactly the same way as for the inward and 
outward flow turbines, page 335, except that for the axial flow 
turbine there is no centrifugal head impressed on the water 
between inlet and outlet. 

Then, + F = -' + ^ + A /. 

w 2g w 2g 

from which, since v is equal to v if 

p Y 2 -2Yi; + i; 2 <u? p, . ^-2V 1 t?+Vi a , V , R 

+ ~~ + ~~ 



p V 2 v u pi _,_ i 
therefore -+5 --- + ^- = + -Q- 
w 2g g 2g w 2g 



, p u 2 u, 2 P! 

and. --- = H n --- ^ 

g g w 2g 2g w 



350 



HYDRAULICS 



But in Fig. 220, 



w 



and 



w 



~\7"<j) V it TL 2 

Therefore, -^- -^ = H-^ -Hr-fc, 

i/ I/ ^!/ 

If Ui is axial and equal to w, as in Fig. 223, 



U 



-p. -p- , 

1 ^ i/ fit 



200. Mixed flow turbines. 

By a modification of the shape of the vanes of an inward flow 
turbine, the mixed flow turbine is obtained. In the inward and 
outward flow turbine the water only acts upon the wheel while it 
is moving in a radial direction, but in the mixed flow turbine the 
vanes are so formed that the water acts upon them also, while 
flowing axially. 




Fig. 231. Mixed Flow Turbine. 

Fig. 231 shows a diagrammatic section through the wheel of 
a mixed flow turbine, the axis of which is vertical. The water 



TURBINES 



351 



enters the wheel in a horizontal direction and leaves it vertically, 
but it leaves the discharging edge of the vanes in different 
directions. At the upper part B it leaves the vanes nearly 
radially, and at the lower part A, axially. The vanes are spoon- 
shaped, as shown in Fig. 232, and should be so formed, or in other 
words, the inclination of the discharging edge should so vary, 
that wherever the water leaves the vanes it should do so with no 
component in a direction perpendicular to the axis of the turbine, 
i.e. with no velocity of whirl. The regulation of the supply to 
the wheel in the turbine of Fig. 231 is effected by a cylindrical 
sluice or speed gate between the fixed guide blades and the wheel. 




Fig. 232. Wheel of Mixed Flow Turbine. 

Fig. 233 shows a section through the wheel and casing of a 
double mixed flow turbine having adjustable guide blades to 
regulate the flow. Fig. 234 shows a half longitudinal section of 
the turbine, and Fig. 235 an outside elevation of the guide blade 
regulating gear. The guide blades are surrounded by a large 



352 



HYDRAULICS 



vortex chamber, and the outer tips of the guide blades are of 
variable shapes, Fig. 233, 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 R 
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 of 
the other guide blades is not interfered with, as the spring con- 
nected to the locked blade by its elongation will allow the ring 
to rotate. 

As with the inward and outward flow turbine, the mixed 
flow turbine wheel may either work drowned, or exhaust into a 
"suction tube." 



TURBINES 



353 



For a given flow, and width of wheel, the axial velocity 
with which the water finally flows away from the wheel being the 
same for the two cases, the diameter of a mixed flow turbine can 
be made less than an inward flow turbine. As shown on page 340, 
the diameter of the inward flow turbine is in large measure fixed 




Fig. 234. Half-longitudinal section of Mixed Flow Turbine. 

by the diameter of the exhaust openings of the wheel. For the 
same axial velocity, and the same total flow, whether the turbine 
is an inward or mixed flow turbine, the diameter d of the exhaust 
openings must be about equal. The external diameter, therefore, 
of the latter will be much smaller than for the former, and the 
L.IL 23 



354 



HYDRAULICS 



general dimensions of the turbine will be also diminished. For 
a given head H, the velocity v of the inlet edge being the same in 
the two cases, the mixed flow turbine can be run at a higher 
angular velocity, which is sometimes an advantage in driving 
dynamos. 



m 




TURBINES 



355 



Form oftlie vanes. At the receiving edge, the direction of the 
blade is found in the same way as for an inward flow turbine. 

ABC, Fig. 236, is the triangle of velocities, and BO is parallel 
to the tip of the blade. This triangle has been drawn for the data 
of the turbine shown in Figs. 233 235 ; v is 46'5 feet per second, 
and from 



Y = 33'5 feet per second. 
The anglo < is 139 degrees. 



'-\ w 



/"" v.'A" -""- 




<fc Triangle of Velocities 
^ at receiving edge. 



Fig. 236. 



The best form for the vane at the discharge is somewhat 
difficult to determine, as the exact direction of flow at any point 
on the discharging edge of the vane is .not easily found. The 
condition to be satisfied is that the water must leave the wheel 
without any component in the direction of motion. 

The following construction gives approximately the form of 
the vane. 

Make a section through the wheel as in Fig. 237. The outline 
of the discharge edge FGrH is shown. This edge of the vane is 
supposed to be on a radial plane, and the plan of it is, therefore, 
a radius of the wheel, and upon this radius the section is taken. 

It is now necessary to draw the form of the stream lines, as 
they would be approximately, if the water entered the wheel 
radially and flowed out axially, the vanes being removed. 

Divide 04, Fig. 237, at the inlet, into any number of equal 
parts, say four, and subdivide by the points a, 6, d, e. 

Take any point A, not far from c, as centre, and describe 
a circle MM a touching the crowns of the wheel at M and MI. 
Join AM and AMi. 

Draw a flat curve Mi Mi touching the lines AM and AM a in M 
and MI respectively, and as near as can be estimated, perpendicular 

233 



356 



HYDRAULICS 



to the probable stream lines through a, 6, d, e, which can be 
sketched in approximately for a short distance from 04. 

Taking this curve MMi as approximately perpendicular to the 
stream lines, two points / and g near the centres of AM and 
are taken. 




Fig. 237. 

Let the radius of the points g and / be r and r L respectively. 
If any point Ci on MMi is now taken not far from A, the 
peripheral area of Mci is nearly 2wrMci, and the peripheral area 
of MiCi is nearly SSwriM^. 

On the assumption that the mean velocity through MiM is 
constant, the flow through Md will be equal to that through 
when, 



TUKBTNES 857 

If, therefore, MMi is divided at the point Ci so that 



the point d will approximately 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 Ci cannot be much in error. 

A nearer approximation to Ci can be found by taking new values 
for r and n, obtained by moving the points / and g so that they 
more nearly coincide with the centres of CiM and CiMi. If the 
two curves are not perpendicular, the curve MMi and the point Ci 
are not quite correct, and new values of r and n will have to be 
obtained by moving the points / and g. By approximation Ci can 
be thus found with considerable accuracy. 

By drawing other circles to touch the crown of the wheels, the 
curves M 2 M 3 , M 4 M 5 etc. normal to the stream lines, and the points 
Ca, c 3 , etc. on the centre stream line, can be obtained. 

The curve 22, therefore, divides the stream lines into equal 
parts. 

Proceeding in a similar manner, the curves 11 and 33 can be 
obtained, dividing the stream lines into four equal parts, and 
these again subdivided by the curves aa, 66, dd, and ee, which 
intersect the outlet edge of the vane at the points F, Gr, H and e 
respectively. 

To determine the direction of the tip of the vane at points on the 
discharging edge. At the points F, Gr, H, the directions of the 
stream lines are known, and the velocities U F) U Q , UH can be found, 
since the flows through 01, 12, etc. are equal, and therefore 



at = u G ~R 2 mn = i . 

O7T 

Draw a tangent FK to the stream line at F. This is the inter- 
section, with the plane of the paper, of a plane perpendicular to 
the paper and tangent to the stream line at F. 

The point F in the plane of FK is moving perpendicular to the 
plane of the paper with a velocity equal to w.R , w being the 
angular velocity of the wheel, and R the radius of the point F. 

If a circle be struck on this plane with K as centre, this circle 
may be taken as an imaginary discharge circumference of an 
inward flow turbine, the velocity v of which is u>R , and the tip of 
the blade is to have such an inclination, that the water shall 
discharge radially, i.e. along FK, with a velocity up. Turning this 
circle into the plane of the paper and drawing the triangle of 
velocities FST, the inclination a r of the tip of the blade at F in 
the piano FK is obtained. 



358 



HYDRAULICS 



At Gr the stream line is nearly vertical, but wRg can be set out 
in the plane of the paper, as before, perpendicular to U Q and the 
inclination a , on this plane, is found. 

At H, dtn 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. 233. 




Fig. 239. 

Sections of the vane "by planes 0Gb, 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 wheel 
at b is the angle <, Fig. 236. Let BC of the same figure be the 



TURBINES 359 

plan of a horizontal line lying in this plane, and BD the plan of 
the radius of the wheel at 6. The angle between these lines is y. 

Let ft be the inclination of the plane OG-fe to the horizontal. 

From D, Fig. 236, set out DE, inclined to BD at an angle /?, 
and intersecting AB produced in E; with D as centre* and DE 
as radius draw the arc EG intersecting DB produced in G. 
Join CG. 

The angle CG-D is the angle 71, which the line of intersection, 
of the plane 0Gb, Fig. 237, with the plane tangent to the inlet tip 
of the vane, makes with the radius 0&; and the angle CGF is 
the angle on the plane 0GB which the tangent to the vane 
makes with the direction of motion of the inlet edge of the 
vane. 

In Fig. 238 the inclination of the inlet tip of the blade is yi as 
shown. 

To determine the angle a at the outlet edge, resolve U Q , Fig. 
237, along and perpendicular to OG, U O Q being the component 
along OG. 

Draw the triangle of velocities DEF, Fig. 238. 

The tangent to the vane at D is parallel to FE. 

In the same way, the section on the plane Hd, Fig. 237, may be 
determined; the inclination at the inlet is y 2 , Fig. 239. 

Mixed flow turbine working in open stream. A. double turbine 
working in open stream and discharging through a suction tube 
is shown in Fig. 240. This is a convenient arrangement for 
moderately low falls. Turbines, of this class, of 1500 horse- 
power, having four wheels on the same shaft and working under 
a head of 25 feet, and making 150 revolutions per minute, have 
recently been installed by Messrs Escher Wyss at Wangen an der 
Aare in Switzerland. 

201. Cone turbine. 

Another type of inward flow turbine, which is partly axial and 
partly radial, is shown in Fig. 241, and is known as the cone 
turbine. It has been designed by Messrs Escher Wyss to meet 
the demand for a turbine that can be adapted to variable flows. 

The example shown has been erected at Gusset near Lyons and 
makes 120 revolutions per minute. 

The wheel is divided into three distinct compartments, the 
supply of water being regulated by three cylindrical sluices S, Si 
and S 2 . The sluices S and Si are each moved by three vertical 
spindles such as A and AI which carry racks at their upper ends. 
These two sluices move in opposite directions and thus balance 
each other. The sluice S 2 is normally out of action, the upper 



360 



HYDRAULICS 



compartment being closed. At low heads this upper compartment 
is allowed to come into operation. The sluice S 2 carries a rack 
which engages with a pinion P, connected to the vertical shaft T. 




Feet 6 



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 S a , it is revolved 
by means of the pinion P until the arms F come between collars 
D and E on the spindles carrying the sluice Si, and the sluice S 2 
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 R 
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. 



TURBINES 



361 




Fig. 241. Cone Turbine. 



362 



HYDRAULICS 



202. Effect of changing the direction of the guide blade, 
when altering the flow of inward flow and mixed flow 
turbines. 

As long as the velocity of a wheel remains constant, the 
backward head impressed on the water by the wheel is the same, 
and the pressure head, at the inlet to the wheel, will remain 
practically constant as the guides are moved. The velocity of 
flow U, through the guides, will, therefore, remain constant; 
but as the angle 0, which the guide makes with the tangent to the 
wheel, diminishes the radial component u y of U, diminishes. 




Fig. 242. 



Let ABC, Fig. 242, be the triangle of velocities for full opening, 
and suppose the inclination of the tip of the blade is made parallel 
to BC. On turning the guides into the dotted position, the incli- 
nation being <'i, the triangle of velocities is ABCi, and the relative 
velocity of the water and the periphery of the wheel is now Bd 
which is inclined to the vane, and there is, consequently, loss due 
to shock. 

It will be seen that in the dotted position the tips of the guide 
blades are some distance from the periphery of the wheel and it is 
probable that the stream lines on leaving the guide blades follow 
the dotted curves SS, and if so, the inclination of these stream 
lines to the tangent to the wheel will be actually greater than <'i, 
and BCi will then be more nearly parallel to BC. The loss may 
be approximated to as follows : 

As the water enters the wheel its radial component will remain 
unaltered, but its direction will be suddenly changed from Bd to 
BC, and its magnitude to BC 2 ; dC 2 is drawn parallel to AB. 
A velocity equal to dCa has therefore to be suddenly impressed on 
the water. 

On page 68 it has been shown that on certain assumptions the 



TURBINES 363 

head lost when the velocity of a stream is suddenly changed 
from Vi to v* is 



that is, it is equal to the head due to the relative velocity of 
Vi and v 2 . 

But CiC 2 is the relative velocity of BCi and BC 2 , and therefore 
the head lost at inlet may be taken as 



k being a coefficient which may be taken as approximately unity. 

203. Effect of diminishing the flow through turbines on 
the velocity of exit. 

If water leaves a wheel radially when the flow is a maximum, 
it will not do so for any other flow. 

The angle of the tip of the blade at exit is unalterable, and if 
u and u Q are the radial velocities of flow, at full and part load 
respectively, the triangles of velocity are DEF and DEFi, Fig. 243. 

For part flow, the velocity with which the water leaves the 
wheel is Ui. If this is greater than u, and the wlieel is drowned, 
or the exhaust takes place into the air, the theoretical hydraulic 
efficiency is less than for full load, but if the discharge is down a 
suction tube the velocity with which the water leaves the tube is 
less than for full flow and the theoretical hydraulic efficiency is 
greater for the part flow. The loss of head, by friction in the 
wheel due to the relative velocity of the water and the vane, 
which is less than at full load, should also be diminished, as also, 
the loss of head by friction in the supply and exhaust pipes. 
The mechanical losses remain practically constant at all loads. 





The fact that the efficiency of turbines diminishes at part loads 
must, therefore, in large measure be due to the losses by shock 
being increased more than the friction losses are diminished. 

By suitably designing the vanes, the greatest efficiency of 
inward flow and mixed flow turbines can be obtained at some 
fraction of full load. 



204. Regulation of the flow by cylindrical gates. 

When the speed of the turbine is adjusted by a gate between 
the guides and the wheel, and the flow is less than the normal, the 
velocity U with which the water leaves the guide is altered in 
magnitude but not in direction. 

Let ABC be the triangle of velocities, Fig. 244, when the flow is 
normal. 

Let the flow be diminished until the velocity with which the 
water leaves the guides is U , equal to AD. 

Then BD is the relative velocity of U and v, and u is the 
radial velocity of flow into the wheel. 

Draw DK parallel to AB. Then for the water to move along 
the vane a sudden velocity equal to KD must be impressed on 

& (KD) 2 
the water, and there is a head lost equal to ^ --. 

To keep the velocity U more nearly constant Mr Swain has 
introduced the gate shown in Fig. 245. The gate g is rigidly 
connected to the guide blades, and to adjust the flow the guide 
blades as well as the gate are moved. The effective width of the 
guides is thereby made approximately proportional to the quantity 
of flow, and the velocity TJ remains more nearly constant. If the 
gate is raised, the width b of the wheel opening will be greater 
than bi the width of the gate opening, and the radial velocity u 




Fig. 245. Swain Gate. 



Fig. 246. 



TURBINES 



365 



into the wheel will consequently be less than the radial velocity u 
from the guides. If U is assumed constant the relative velocity of 
the water and the vane will suddenly change from BC to BOi, 
Fig. 246. Or it may be supposed that in the space between the 
guide and the wheel the velocity U changes from AC to ACi. 



The loss of head will now be 



k (COO 2 
29 



205. The form of the wheel vanes between the inlet and 
outlet of turbines. 

The form of the vanes between inlet and outlet of turbines 
should be such, that there is no sudden change in the relative 
velocity of the water and the wheel. 

Consider the case of an inward flow turbine. Having given 
a form to the vane and fixed the width between the crowns of the 
wheel the velocity relative to the wheel at any radius r can be 
found as follows. 

Take any circumferential section ef at radius r, Fig. 247. Let 
b be the effective width between the crowns, and d the effective 
width ef between the vanes, and let q be the flow in cubic feet 
per second between the vanes Ae and B/. 




lug. 247. Relative velocity of tlie water and the vanes. 



Fig. 248. 



366 HYDRAULICS 

The radial velocity through e/is 



Find by trial a point near the centre of ef such that a circle 
drawn with as centre touches the vanes at M and MI. 

Suppose the vanes near e and / to be struck with arcs of circles. 
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 ef, but a second trial will probably 
make it do so. 

If then, b is the effective width between the crowns at C, 

6 . MMi . v r = q. 
MMi can be scaled off the drawing and v r 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 h. By trial, the vanes 
should be made so that the variation of velocity is as uniform 
as possible. 

If the vanes could be made involutes of a circle of radius E , 



TURBINES 367 

as in Fig. 249, and the crowns of the wheel parallel, the relative 
velocity of the wheel and the water would remain constant. 
This form of vane is however entirely unsuitable for inward 
flow turbines and could only be used in very special cases for 
outward flow turbines, as the angles < and which the involute 
makes with the circumferences at A and B are not independent, 
for from the figure it is seen that, 



, . . Jtio 

and sin $ - rf 

sin0 R, 

or -r r = - . 

sin 9 r 

The angle must clearly always be greater than <. 

206. The limiting head for a single stage reaction 
turbine. 

Eeaction turbines have not yet been made to work under heads 
higher than 430 feet, impulse turbines of the types to be presently 
described being used for heads greater than this value. 

From the triangle of velocities at inlet of a reaction turbine, 
e.g. Fig. 226, it is seen that the whirling velocity V cannot be 
greater than 

v + u cot <. 

Assuming the smallest value for <f> to be 30 degrees, and the 
maximum value for u to be 0'25 V2grH, the general formula 

S..B 

9 

becomes, for the limiting case, 



If v is assumed to have a limiting value of 100 feet per second, 
which is higher than generally allowed in practice, and e to 
be 0'8, then the maximum head H which can be utilised in a one 
stage reaction turbine, is given by the equation 

25-6H- 346 VS = 10,000, 
from which H = 530 feet. 

207. Series or multiple stage reaction turbines. 

Professor Osborne Eeynolds 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 



3Go 



HYDRAULICS 



and fixed blades be arranged alternately as shown in Fig. 250*. 
This arrangement, although not used in water turbines, is very 
largely used in reaction steam turbines. 



Fig. 250. 

f^""^ 

mrnmm^ 




Toothed 
quadrant: 

Figs. 251, 252. Axial Flow Impulse Turbine. 
* Taken from Prof. Reynolds' Scientific Papers, Vol. x. 



TURBINES 



369 



208. Impulse turbines. 

Girard turbine. To overcome the difficulty of diminution of 
efficiency with diminution of flow, 
Girard introduced, about 1850, the 
free deviation or partial admission 
turbine. 

Instead of the water being 
admitted to the wheel throughout 
the whole circumference as in the 
reaction turbines, in the Girard 
turbine it is only allowed to enter 
the wheel through guide passages 
in two diametrically opposite 
quadrants as shown in Figs. 252 
254. In the first two, the flow is 
axial, and in the last radial. 




Fig. 253. 



In Fig. 252 above the guide crown are two quadrant-shaped 
plates or gates 2 and 4, which are made to rotate about a vertical 
axis by means of a toothed wheel. When the gates are over the 
quadrants 2 and 4, all the guide passages are open, and by turning 
the gates in the direction of the arrow, any desired number of the 
passages can be closed. In Fig. 254 the variation of flow is 
effected by means of a cylindrical quadrant-shaped sluice, which, 
as in the previous case, can be made to close any desired number 
of the guide passages. Several other types of regulators for 
impulse turbines were introduced by Girard and others. 

Fig. 253 shows a regulator employed by Fontaine. Above the 
guide blades, and fixed at the opposite ends of a diameter DD, 
are two indiarubber bands, the other ends of the bands being 
connected to two conical rollers. The conical rollers can rotate 
on journals, formed on the end of the arms which are connected 
to the toothed wheel TW. A pinion P gears with TW, and by 
rotating the spindle carrying the pinion P, the rollers can be made 
to unwrap, or wrap up, the indiarubber band, thus opening or 
closing the guide passages. 

As the Girard turbine is not kept full of water, the whole of 
the available head is converted into velocity before the water 
enters the wheel, and the turbine is a pure impulse turbine. 

To prevent loss of head by broken water in the wheel, the air 
should be freely admitted to the buckets as shown in Figs. 252 
and 254. 

For small heads the wheel must be horizontal but for large 
heads it may be vertical. 

This class of turbine has the disadvantage that it cannot 
L. H. 24 



370 



HYDRAULICS 



run drowned, and hence must always be placed above the tail 
>vater. For low and variable heads the full head cannot therefore 
be utilised, for if the wheel is to be clear of the tail water, an 
amount of head equal to half the width of the wheel must of 
necessity be lost. 




Fig. 254. Girard Eadial flow Impulse Turbine. 

To overcome this difficulty Grirard placed the wheel in an air- 
tight tube, Fig. 254, the lower end of which is below the tail water 
level, and into which air is pumped by a small auxiliary air-pump, 
the pressure being maintained at the necessary value to keep the 
surface of the water in the tube below the wheel. 



TUKBINES 371 

Let H be the total head above the tail water level of the supply 
water, the pressure head due to the atmospheric pressure, H 

the distance of the centre of the wheel below the surface of the 
supply water, and h the distance of the surface of the water in 
the tube below the tail water level. Then the air-pressure in 
the tube must be 



, 

W 

and the head causing velocity of flow into the wheel is, therefore, 



W W 

So that wherever the wheel is placed in the tube below the tail 
water the full fall H is utilised. 

This system, however, has not found favour in practice, owing 
to the difficulty of preserving the pressure in the tube. 

209. The form of the vanes for impulse turbines, neg- 
lecting friction. 

The receiving tip of the vane should be parallel to the relative 
velocity Y r of the water and the edge of the vane, Fig. 255. 

For the axial flow turbine Vi equals v and the relative velocity v r 
at exit, Fig. 255, neglecting friction, is equal to the relative 
velocity V r at inlet. The triangle of velocities at exit is AG-B. 

For the radial flow turbine, Figs. 254 and 258, there is a 

centrifugal head impressed on the water equal to ^ - - and, 

2 ~\7 2 2 2 

neglecting friction, ^j- = -- + ^- - |- . The triangle of velocities 

at exit is then DEF, Fig. 256, and Ui equals DF. 

If the velocity with which the water leaves the wheel is Ui, 
the theoretical hydraulic efficiency is 




H 

and is independent of the direction of Ui . 

It should be observed, however, that in the radial flow turbine 
the area of the section of the stream by the circumference of the 
wheel, for a given flow, will depend upon the radial component of 
Ui, and in the axial flow turbine the area of the section of the 
stream by a plane perpendicular to the axis will depend upon the 
axial component of Ui . That is, in each case the area will depend 
upon the component of Ui perpendicular to Vi . 

242 



372 



HYDRAULICS 



Now the section of the stream must not fill the outlet area of 
the wheel, and the minimum area of this outlet so that it is just 
not filled will clearly be obtained for a given value of Ui when Ui 
is perpendicular to v*, or is radial in the outward flow and axial in 
the parallel flow turbine. 

For the parallel flow turbine since BC and BG-, Fig. 255, are 
equal, Ui is clearly perpendicular to v l when 



and the inclinations a. and </> of the tips of the vanes are equal. 




Figs. 255, 256. 




Fig. 257. 

If H and r are the outer and inner radii of the radial flow 
turbine respectively, 



* It is often stated that this is the condition for maximum efficiency but it only 
is so, as stated above, for maximum flow for the given machine. The efficiency 
only depends upon the magnitude of T^ and not upon its direction. 



TURBINES 373 

For Ui to be radial 

v r = Vi sec a 

.E 

= -- sec a. 
r 

"Y 

If for the parallel flow turbine v is made equal to -^ , Y r from 

y 
Fig. 255 is equal to -^sec^, and therefore, 

r 
sec a = ^ sec <f>. 

210. Triangles of velocity for an axial flow impulse tur- 
bine considering friction. 

The velocity with which the water leaves the guide passages 
may be taken as from 0'94 to 0*97 V20H, and the hydraulic losses 
in the wheel are from 5 to 10 per cent. 

If the angle between the jet and the direction of motion of the 
vane is taken as 30 degrees, and U is assumed as 0'95 \/2#H, and v 
as 0'45\/2#H, the triangle of velocities is ABC, Fig. 257. 

Taking 10 per cent, of the head as being lost in the wheel, the 
relative velocity v r at exit can be obtained from the expression 



__ 

H now the velocity of exit Ui be taken as 0*22N/2#H, and 
circles with A and B as centres, and Ui and v r as radii be 
described, intersecting in D, ABD the triangle of velocities at exit 
is obtained, and Ui is practically axial as shown in the figure. 
On these assumptions the best velocity for the rim of the wheel is 
therefore '45 \/20H instead of *5 x/2#H. 

The head lost due to the water leaving the wheel with velocity 
u is '048H, and the theoretical hydraulic efficiency is therefore 
95'2 per cent. 

The velocity head at entrance is 0*9025H and, therefore, "097H 
has been lost when the water enters the wheel. 
The efficiency, neglecting axle friction, will be 
H - 01H - 0-048H - 0-097H 

T- 
= 76 per cent, nearly. 

211. Impulse turbine for high heads. 

For high heads Girard introduced a form of impulse turbine, 
of which the turbine shown in Figs. 258 and 259, is the modern 
development. 

The water instead of being delivered through guides over an 
arc of a circle, is delivered through one or more adjustable nozzles. 




Bp 

s 



TT 




TURBINES 375 

In the example shown, the wheel has a mean diameter of 6'9 feet 
and makes 500 revolutions per minute; it develops 1600 horse- 
power under a head of 1935 feet. 

The supply pipe is of steel and is 1*312 feet diameter. 

The form of the orifices has been developed by experience, and 
is such that there is no sudden change in the form of the liquid 
vein, and consequently no loss due to shock. 

The supply of water to the wheel is regulated by the sluices 
shown in Fig. 258, which, as also the axles carrying the same, 
are external to the orifices, and can consequently be lubricated 
while the turbine is at work. The sluices are under the control 
of a sensitive governor and special form of regulator. 

As the speed of the turbine tends to increase the regulator 
moves over a bell crank lever and partially closes both the orifices. 
Any decrease in speed of the turbine causes the reverse action to 
take place. 

The very high peripheral speed of the wheel, 205 feet per 
second, produces a high stress in the wheel due to centrifugal 
forces. Assuming the weight of a bar of the metal of which the 
rim is made one square inch in section and one foot long as 
3'36 Ibs., the stress per sq. inch in the hoop surrounding the 
wheel is 

3-36. tf 

9 
= 4400 Ibs. per sq. inch. 

To avoid danger of fracture, steel laminated hoops are shrunk 
on to the periphery of the wheel. 

The crown carrying the blades is made independent of the disc 
of the wheel, so that it may be replaced when the blades become 
worn, without an entirely new wheel being provided. 

The velocity of the vanes at the inner periphery is 171 feet per 
second, and is, therefore, 0*484 \/2#H. 

If the velocity U with which the water leaves the orifice is 
taken as 0*97 \/2#H, and the angle the jet makes with the tangent 
to the wheel is 30 degrees, the triangle of velocities at entrance is 
ABC, Fig. 260, and the angle </> is 53*5 degrees. 

The velocity Vi of the outer edges of the vanes is 205 feet per 
second, and assuming there is a loss of head in the wheel, equal to 
6 per cent, of H, 

^p + 205* m* 

2g 2g 2g 2g 
and v r = 220 ft. per second, 



376 



HYDKAUL1CS 



If then the angle a is 30 degrees the triangle of velocities at 
exit is DEF, Fig. 261. 

The velocity with which the water leaves the wheel is then 
Ui = 111 feet per sec., and the head lost by this velocity is 191 feet 
or '099H. 




Fig. 260. 



Fig. 261. 



The head lost in the pipe and nozzle is, on the assumption 
made above, 



and the total percentage loss of head is, therefore, 

6 + 9-9 + 6-20-5, 
and the hydraulic efficiency is 78' 1 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.; allowing for mechanical losses 
the hydraulic losses were less than in the example. 

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 
in feet 


Diameter 
of wheel 
(two wheels) 


Kevolutions 
per minute 


V 


U 


H. P. 

j 


262 


39-4" 


375 


64-5 


129 


500 


*233' 


7" 


2100 


64 


125 


5 


*197 


20" 


650 


56-5 


112 


10 


722 


39" 


650 


111 


215 


167 


382 


60" 


300 


79 


156 


144 


*289 


54" 


310 


73 


136 


400 


508 


90" 


200 


79 


180 


300 



* Picard Pictet and Co., the remainder by Escher Wyss and Co. 

213. Oil pressure governor or regulator. 

The modern applications of turbines to the driving of electrical 
machinery, has made it necessary for particular attention to be 
paid to the regulation of the speed of the turbines. 

The methods of regulating the flow by cylindrical speed gates 
and moveable guide blades have been described in connection with 



378 



HYDRAULICS 



various turbines but the means adopted for moving the gates and 
guides have not been discussed. 

Until recent years some form of differential governor was 
almost entirely used, but these have been almost completely 
superseded by hydraulic and oil governors. 

Figs. 263 and 264 show an oil governor, as constructed by 
Messrs Escher Wyss of Zurich. 





Figs. 263, 264. Oil Pressure Kegulator for Turbines. 

A piston P having a larger diameter at one end than at the 
other, and fitted with leathers I and Zi, fits into a double cylinder 
Ci . Oil under pressure is continuously supplied through a pipe S 
into the annulus A between the pistons, while at the back of the 
large piston the pressure of the oil is determined by the regulator. 



TURBINES 



379 




Fig. 265. 



Suppose the regulator to be in a definite position, the space 
behind the large piston being full of oil, and the 
turbine running at its normal speed. The valve V 
(an enlarged diagrammatic section is shown in 
Fig. 265) will be in such a position that oil cannot 
enter or escape from the large cylinder, and the 
pressure in the annular ring between the pistons 
will keep the regulator mechanism locked. 

If the wheel increases in speed, due to a 
diminution of load, the balls of the spring loaded 
governor Gr move outwards and the sleeve M 
rises. For the moment, the point D on the lever 
MD is fixed, and the lever turns about D as a 
fulcrum, and thus raises the valve rod NY. This 
allows oil under pressure to enter the large 
cylinder and the piston in consequence moves to 
the right, and moves the turbine gates in the manner described later. 
As the piston moves to the right, the rod R, which rests on the 
wedge W connected to the piston, falls, and the point D of the 
lever MD consequently falls and brings the valve Y back to its 
original position. The piston P thus takes up a new position 
corresponding to the required gate opening. The speed of the 
turbine and of the governor is a little higher than before, the 
increase in speed depending upon the sensitiveness of the governor. 
On the other hand, if the speed of the wheel diminishes, the 
sleeve M and also the valve Y falls and the oil from behind the 
large piston escapes through the exhaust E, the piston moving 
to the left. The wedge W then lifts the fulcrum D, the valve Y 
is automatically brought to its central position, and the piston P 
takes up a new position, consistent with the gate opening being 
sufficient to supply the necessary water required by the wheel. 

A hand wheel and screw, Fig. 264, are also provided, so that 
the gates can be moved by hand when necessary. 

The piston P is connected by the connecting rod BE to a crank 
EF, which rotates the vertical shaft T. A double crank KK is 
connected 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 adjust- 
ment of the speed gates. 

214. Water pressure regulators for impulse turbines. 
Fig. 266 shows a water pressure regulator as applied to regulate 
the flow to a Pel ton wheel. 

The area of the supply nozzle is adjusted by a beak B which 



380 



HYDRAULICS 



M A 1 N 





Figs. 266, 267. Pelton Wheel and Water Pressure Regulator. 



TURBINES 



381 



rotates about the centre O. The pressure of the water in the 
supply pipe acting on this beak tends to lift it and thus to open 
the orifice. The piston P, working in a cylinder C, is also acted 
upon, on its under side, by the pressure of the water in the supply 
pipe and is connected to the beak by the connecting rod DE. 
The area of the piston is made sufficiently large so that when the 
top of the piston is relieved of pressure the pull on the connecting 
rod is sufficient to close the orifice. 

The pipe p conveys water under the same pressure, to the 
valve V, which maybe similar to that described in connection with 
the oil pressure governor, Fig. 265. 

A piston rod passes through the top of the cylinder, and carries 
a nut, which screws on to the square thread cut on the rod. A 
lever eg, Fig. 268, which is carried on the fixed fulcrum e, is made 
to move with the piston. A link /A connects ef with the lever 
MN, one end M of which moves with the governor sleeve and the 
other end N is connected to the valve rod NV. The valve V is 
shown in the neutral position. 



M 




Fig. 268. 

Suppose now the speed of the turbine to increase. The 
governor sleeve rises, and the lever MN turns about the fulcrum 
A which is momentarily at rest. The valve V falls and opens the 
top of the cylinder to the exhaust. The pressure on the piston 
P now causes it to rise, and closes the nozzle, thus diminishing 
the supply to the turbine. As the piston rises it lifts again the 
lever MN by means of the link A/ ; and closes the valve V. A 
now position of equilibrium is thus reached. If the speed of the 



382 



HYDRAULICS 



governor decreases the governor sleeve falls, the valve Y rises, 
and water pressure is admitted to the top of the piston, which is 
then in equilibrium, and the pressure on the beak B causes it to 
move upwards and thus open the nozzle. 

Hydraulic valve for water regulator. Instead of the simple 
piston valve controlled mechanically, Messrs Escher Wyss use, for 
high heads, a hydraulic double-piston valve Pp, Fig. 269. 

This piston valve has a small bore through its centre by means 
of which high pressure water which is admitted below the valve 
can pass to the top of the large piston P. Above the piston is a 
small plug valve Y which is opened and closed by the governor. 




Fig. 269. Hydraulic valve for automatic regulation. 

If the speed of the governor decreases, the valve Y is opened, 
thus allowing water to escape from above the piston valve, and the 
pressure on the lower piston p raises the valve. Pressure water is 
thus admitted above the regulator piston, and the pressure on the 
beak opens the nozzle. As the governor falls the valve Y closes, 
the exhaust is throttled, and the pressure above the piston P rises. 
When the exhaust through Y is throttled to such a degree that 
the pressure on P balances the pressure on the under face of the 
piston p, the valve is in equilibrium and the regulator piston is 
locked. 



TURBINES 383 

If the speed of the governor increases, the valve Y is closed, 
and the excess pressure on the upper face of the piston valve 
causes it to descend, thus connecting the regulator cylinder to 
exhaust. The pressure on the under face of the regulator piston 
then closes the nozzle. 

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. 

Within the cylinder, on a hexagonal frame, is stretched a 
piece of canvas. The 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. 




Figs. 270, 271. Water Filter for Impulse Turbine Regulator. 

To clean the filter while at work, the canvas frame is revolved 
by means of the handle shown, and the cock R is opened. Each 
side of the hexagonal frame is brought in turn opposite the 
chamber A, and water flows outwards through the canvas and 
through the cock E, carrying away any dirt that may have 
collected outside the canvas. 

Auxiliary valve to prevent hammer action. When the pipe line 
is long an auxiliary valve is frequently fitted on the pipe near to 
the nozzle, which is automatically opened by means of a cataract 
motion* as the nozzle closes, and when the movement of the nozzle 
beak is finished, the valve slowly closes again. 

If no such provision is made a rapid closing of the nozzle 
means that a large mass of water must have its momentum 
quickly changed and very large pressures may be set up, or in 
other words hammer action is produced, which may cause fracture 
of the pipe. 

When there is an abundant supply of water, the auxiliary 
valve is connected to the piston rod of the regulator and opened 
and closed as the piston rod moves, the valve being adjusted so 
that the opening increases by the same amount that the area of 
the orifice diminishes. 

* See Engineer, Vol. xc., p. 255. 



384 HYDRAULICS 

If the load on the wheel does not vary through a large range 
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 



Let the velocity change by an amount dv in time dt. Then the 

rate of change of momentum is TT-, an( l n & cross section of 

got 

the lower end of the column of water in the pipe a force P must 
be applied equal to this. 

mi P T\ ^ wljd* dv 

Therefore P = 7 -- 57 . 

4 g dt 

Referring to Fig. 266, let b be the depth of the orifice and da its 
width. 

Then, if r is the distance of D from the centre about which the 
beak turns, and n is the distance of the closing edge of the beak 
from this centre, and if at any moment the velocity of the piston 
is t? feet per second, the velocity of closing of the beak will be 



In any small element of time dt the amount by which the 
nozzle will close is 



Let it be assumed that U, the velocity of flow through the 
nozzle, remains constant. It will actually vary, due to the 
resistances varying with the velocity, but unless the pipe is very 
long the error is not great in neglecting the variation. If then v 
is the velocity in the pipe at the commencement of this element of 
time and v - dv at the end of it, and A the area of the pipe, 

v.A=fe.<Z,.TJ .............................. (1) 

and (v-dtOA^fc- dA.di.IJ . ...(2). 

\ T / 

Subtracting (2) from (1), 






TURBINES 385 

If W is the weight of water in the pipe, the force P in pounds 
that will have to be applied to change the velocity of this water 
by dv in time Ct is 

g ot " 
Therefore P = ~T- > 

*/ 

and the pressure per sq. inch produced in the pipe near the 
nozzle is 

W 



~ g r A 2 " 

Suppose the nozzle to be completely closed in a time t seconds, 
and during the closing the piston P moves with simple harmonic 
motion. 

Then the distance moved by the piston to close the nozzle is 

br 

and the time taken to move this distance is t seconds. 
The maximum velocity of the piston is then 



u 

and substituting in (3), the maximum value of r is, therefore, 

dv 
dt~ 
and the maximum pressure per square inch is 

vWb.d 1 .'U *.W.Q ?r Wv 
Pm 2^A 2 2g.t. A 2 2t' gA.' 

where Q is the flow in cubic feet per second before the orifice 
began to close, and v is the velocity in the pipe. 

Example. A 500 horse-power Pelton Wheel of 75 per cent, efficiency, and working 
under a head of 260 feet, is supplied with water by a pipe 1000 feet long and 
2' 3" diameter. The load is suddenly taken off, and the time taken by the 
regulator to close the nozzle completely is 5 seconds. 

On the assumption that the nozzle is completely closed (1) at a uniform rate, 
and (2) with simple harmonic motion, and that no relief valve is provided, 
determine the pressure produced at the nozzle. 

The quantity of water delivered to the wheel per second when working at full 
power is 

500x33.000 



The weight of water in the pipe is 

W=62-4x^. (2-25) 2 xlOOO 

= 250,000 Ibs. 
L. H. 25 



6 HYDRAULICS 

21*7 

Tne velocity is -7^ = 5-25 ft. per sec. 
D *y u 

In case (1) the total pressure acting on the lower end of the column of water in 
e pipe is 

250,000x5-25 



g x 5 

= 8200 Ibs. 
The 



386 HYDRAULICS 

Tne velc 

In case I 
the pipe is 

= 8200 lbs. 
pressure per sq. inch is 

8200 
p = - = 14-5 Ibs. per sq. inch. 

TT W . v 
In case (2) p m = ^ L -^=22-8 lbs. per sq. inch. 



EXAMPLES. 

(1) Find the theoretical horse-power of an overshot water-wheel 22 feet 
diameter, using 20,000,000 gallons of water per 24 hours under a total head 
of 25 feet. 

(2) An overshot water-wheel has a diameter of 24 feet, and makes 3 '5 
revolutions per minute. The velocity of the water as it enters the buckets 
is to be twice that of the wheel's periphery. 

If the angle which the water makes with the periphery is to be 15 
degrees, find the direction of the tip of the bucket, and the relative velocity 
of the water and the bucket. 

(3) The sluice of an overshot water-wheel 12 feet radius is vertically 
above the centre of the wheel. The surface of the water in the sluice 
channel is 2 feet 3 inches above the top of the wheel and the centre of the 
sluice opening is 8 inches above the top of the wheel. The velocity of the 
wheel periphery is to be one-half that of the water as it enters the buckets. 
Determine the number of rotations of the wheel, the point at which the 
water enters the buckets, and the direction of the edge of the bucket. 

(4) An overshot wheel 25 feet diameter having a width of 5 feet, and 
depth of crowns 12 inches, receives 450 cubic feet of water per minute, and 
makes 6 revolutions per minute. There are 64 buckets. 

The water enters the wheel at 15 degrees from the crown of the wheel 
with a velocity equal to twice that of the periphery, and at an angle of 20 
degrees with the tangent to the wheel. 

Assuming the buckets to be of the form shown in Fig. 180, the length 
of the radial portion being one-half the length of the outer face of the 
bucket, find how much water enters each bucket, and, allowing for centri- 
fugal forces, the point at which the water begins to leave the buckets. 

(5) An overshot wheel 32 feet diameter has shrouds 14 inches deep, and 
is required to give 29 horse-power when making 5 revolutions per minute. 

Assuming the 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 total 
fall is 32 feet and the efficiency 60 per cent. 



TURBINES 387 

Assuming the velocity of the water in the penstock to be If times that 
of the wheel's periphery, and the bottom of the penstock level with the top 
of the wheel, find the point at which the water enters the wheel. Find also 
where water begins to discharge from the buckets. 

(6) A radial blade impulse wheel of the same width as the channel in 
which it runs, is 15 feet diameter. The depth of the sluice opening is 
12 inches and the head above the centre of the sluice is 3 feet.. Assuming 
a coefficient of velocity of 0'8 and that the edge of the sluice is rounded so 
that there is no contraction, and the velocity of the rim of the wheel is 0*4 
the velocity of flow through the sluice, find the theoretical efficiency of 
the wheel. 

(7) An overshot wheel has a supply of 30 cubic feet per second on a fall 
of 24 feet. 

Determine the probable horse-power of the wheel, and a suitable 
width for the wheel. 

(8) The water impinges on a Poncelet float at 15 with the tangent to 
the wheel, and the velocity of the water is double that of the wheel. Find, 
by construction, the proper inclination of the tip of the float. 

(9) In a Poncelet wheel, the direction of the jet impinging on the floats 
makes an angle of 15 with the tangent to the circumference and the tip of 
the floats makes an angle of 30 with the same tangent. Supposing the 
velocity of the jet to be 20 feet per second, find, graphically or otherwise, 
(1) the proper velocity of the edge of the wheel, (2) the height to which the 
water will rise on the float above the point of admission, (3) the velocity 
and direction of motion of the water leaving the float. 

(10) Show that the efficiency of a simple reaction wheel increases 
with the speed when frictional resistances are neglected, but is greatest 
at a finite speed when they are taken into account. 

If the speed of the orifices be that due to the head (1) find the efficiency, 
neglecting friction ; (2) assuming it to be the speed of maximum efficiency, 
show that f of the head is lost by friction, and ^ by final velocity of water. 

(11) Explain why, in a vortex turbine, the inner ends of the vanes are 
inclined backwards instead of being radial. 

(12) An inward flow turbine wheel has radial blades at the outer 

periphery, and at the inner periphery the blade makes an angle of 30 with 

T> 
the tangent. The total head is 70 feet and r=. Find the velocity of the 

rim of the wheel if the water discharges radially. Friction neglected. 

(13) The inner and outer diameters of an inward flow turbine wheel 
are 1 foot and 2 feet respectively. The water enters the outer circumference 
at 12 with the tangent, and leaves the inner circumference radially. The 
radial velocity of flow is 6 feet at both circumferences. The wheel makes 
3 -6 revolutions per second. Determine the angles of the vanes at both 
circumferences, and the theoretical hydraulic efficiency of the turbine. 

(14) Water is supplied to an inward flow turbine at 44 feet per second, 
and at 10 degrees to the tangent to the wheel. The wheel makes 200 

252 



388 HYDRAULICS 

revolutions per minute. The inner radius is 1 foot and the outer radius 
2 feet. The radial velocity of flow through the wheel is constant. 

Find the inclination of the vanes at inlet and outlet of the wheel. 

Determine the ratio of the kinetic energy of the water entering the 
wheel per pound to the work done on the wheel per pound. 

(15) The supply of water for an inward flow reaction turbine is 500 
cubic feet per minute and the available head is 40 feet. The vanes are 
radial at the inlet, the outer radius is twice the inner, the constant 
velocity of flow is 4 feet per second, and the revolutions are 350 per 
minute. Find the velocity of the wheel, the guide and vane angles, 
the inner and outer diameters, and the width of the bucket at inlet and 
outlet. Lond. Un. 1906. 

(16) An inward flow turbine on 15 feet fall has an inlet radius of 1 foot 
and an outlet radius of 6 inches. Water enters at 15 with the tangent to 
the circumference and is discharged radially with a velocity of 3 feet per 
second. The actual velocity of water at inlet is 22 feet per second. The 
circumferential velocity of the inlet surface of the wheel is 19^ feet per 
second. 

Construct the inlet and outlet angles of the turbine vanes. 
Determine the theoretical hydraulic emciency of the turbine. 
If the hydraulic emciency of the turbine is assumed 80 per cent, find the 
vane angles. 

(17) A quantity of water Q cubic feet per second flows through a 
turbine, and the initial and final directions .and velocities are known. 
Apply the principle of equality of angular impulse and moment of 
momentum to find the couple exerted on the turbine. 

(18) The wheel of an inward flow turbine has a peripheral velocity of 
50 feet per second. The velocity of whirl of the incoming water is 40 feet 
per second, and the radial velocity of flow 5 feet per second. Determine 
the vane angle at inlet. 

Taking the flow as 20 cubic feet per second and the total losses as 
20 per cent, of the available energy, determine the horse-power of the 
turbine, and the head H. 

If 5 per cent, of the head is lost in friction in the supply pipe, and the 
centre of the turbine is 15 feet above the tail race level, find the pressure 
head at the inlet circumference of the wheel. 

(19) An inward flow turbine is required to give 200 horse -power under 
a head of 100 feet when running at 500 revolutions per minute. The 
velocity with which the water leaves the wheel axially may be taken as 
10 feet per second, and the wheel is to have a double outlet. The diameter 
of the outer circumference may be taken as If times the inner. Determine 
the dimensions of the turbine and the angles of the guide blades and 
vanes of the turbine wheel. The actual efficiency is to be taken as 75 per 
cent, and the hydraulic efficiency as 80 per cent. 

(20) An outward flow turbine wheel has an internal diameter of 5*249 
feet and an external diameter of 6'25 feet. The head above the turbine is 
141-5 feet. The width of the wheel at inlet is 10 inches, and the quantity 



TURBINES 389 

of water supplied per second is 215 cubic feet. Assuming the hydraulic 
losses are 20 per cent., determine the angles of tips of the vanes so that 
the water shall leave the wheel radially. Determine the horse-power of 
the turbine and verify the work done per pound from the triangles of 

velocities. 

(21) The total head available for an inward-flow turbine is 100 feet. 

The turbine wheel is placed 15 feet above the tail water level. 

When the flow is normal, there is a loss of head in the supply pipe of 
3 per cent, of the head ; in the guide passages a loss of 5 per cent. ; in the 
wheel 9 per cent. ; in the down pipe 1 per cent. ; and the velocity of flow 
from the wheel and in the supply pipe, and also from the down pipe is 
8 feet per second. 

The diameter of the inner circumference of the wheel is 9^ inches and 
of the outer 19 inches, and the water leaves the wheel vanes radially. 
The wheel has radial vanes at inlet. 

Determine the number of revolutions of the wheel, the pressure head in 
the eye of the wheel, the pressure head at the circumference to the wheel, 
the pressure head at the entrance to the guide chamber, and the velocity 
which the water has when it enters the wheel. From the data given 



9 

(22) A horizontal inward flow turbine has an internal diameter of 
5 feet 4 inches and an external diameter of 7 feet. The crowns of the 
wheel are parallel and are 8 inches apart. The difference in level of the 
head and tail water is 6 feet, and the upper crown of the wheel is just below 
the tail water level. Find the angle the guide blade makes with the tangent 
to the wheel, when the wheel makes 32 revolutions per minute, and the 
flow is 45 cubic feet per second. Neglecting friction, determine the vane 
angles, the horse-power of the wheel and the theoretical hydraulic efficiency. 

(23) A parallel flow turbine has a mean diameter of 11 feet. 

The number of revolutions per minute is 15, and the axial velocity of 
flow is 3*5 feet per second. The velocity of the water along the tips of the 
guides is 15 feet per second. 

Determine the inclination of the guide blades and the vane angles that 
the water shall enter without shock and leave the wheel axially. 

Determine the work done per pound of water passing through the wheel. 

(24) The diameter of the inner crown of a parallel flow pressure turbine 
is 5 feet and the diameter of the outer crown is 8 feet. The head over the 
wheel is 12 feet. The number of revolutions per minute is 52. The radial 
velocity of flow through the wheel is 4 feet per second. 

Assuming a hydraulic efficiency of 0'8, determine the guide blade angles 
and vane angles at inlet for the three radii 2 feet 6 inches, 3 feet 3 inches 
and 4 feet. 

Assuming the depth of the wheel is 8 inches, draw suitable sections of 
the vanes at the three radii. 

Find also the width of the guide blade in plan, if the upper and lower 
edges are parallel, and the lower edge makes a constant angle with the 



390 HYDRAULICS 

plane of the wheel, so that the stream lines at the inner and the outer 
crown may have the correct inclinations. 

(25) A parallel flow impulse turbine works under a head of 64 feet. 
The water is discharged from the wheel in an axial direction with a 
velocity due to a head of 4 feet. The circumferential speed of the wheel 
at its mean diameter is 40 feet per second. 

Neglecting all frictional losses, determine the mean vane and guide 
angles. Lond. Un. 1905. 

(26) An outward flow impulse turbine has an inner diameter of 5 feet, 
an external diameter of 6 feet 3 inches, and makes 450 revolutions per 
minute. 

The velocity of the water as it leaves the nozzles is double the velocity 
of the periphery of the wheel, and the direction of the water makes an 
angle of 80 degrees with the circumference of the wheel. 

Determine the vane angle at inlet, and the angle of the vane at outlet so 
that the water shall leave the wheel radially. 

Find the theoretical hydraulic efficiency. If 8 per cent, of the head 
available at the nozzle is lost in the wheel, find the vane angle at exit that 
the water shall leave radially. 

What is now the hydraulic efficiency of the turbine ? 

(27) In an axial flow Girard turbine, let V be the velocity due to the 
effective head. Suppose the water issues from the guide blades with the 
velocity 0'95V, and is discharged axially with a velocity '12 V. Let the 
velocity of the receiving and discharging edges be 0'55 V. 

Find the angle of the guide blades, receiving and discharging angles of 
wheel vanes and hydraulic efficiency of the turbine. 

(28) Water is supplied to an axial flow impulse turbine, having a mean 
diameter of 6 feet, and making 144 revolutions per minute, under a head of 
100 feet. The angle of the guide blade at entrance is 30, and the angle the 
vane makes with the direction of motion at exit is 30. Eight per cent, of 
the head is lost in the supply pipe and guide. Determine the relative 
velocity of water and wheel at entrance, and on the assumption that 10 per 
cent, of the total head is lost in friction and shock in the wheel, determine 
the velocity with which the water leaves the wheel. Find the hydraulic 
efficiency of the turbine. 

(29) The guide blades of an inward flow turbine are inclined at 30 
degrees, and the velocity U along the tip of the blade is 60 feet per second. 
The velocity of the wheel periphery is 55 feet per second. The guide blades 
are turned so that they are inclined at an angle of 15 degrees, the velocity 
U remaining constant. Find the loss of head due to shock at entrance. 

If the radius of the inner periphery is one-half that of the outer and the 
radial velocity through the wheel is constant for any flow, and the water 
left the wheel radially in the first case, find the direction in which it leaves 
in the second case. The inlet radius is twice the outlet radius. 

(30) The supply of water to a turbine is controlled by a speed gate 
between the guides and the wheel. If when the gate is fully open the 
velocity with which the water approaches the wheel is 70 feet per second 



TURBINES 391 

and it makes an angle of 15 degrees with the tangent to the wheel, find 
the loss of head by shock when the gate is half closed. The velocity of 
the inlet periphery of the wheel is 75 feet per second. 

(31) A Pelton wheel, which may be assumed to have semi-cylindrical 
buckets, is 2 feet diameter. The available pressure at the nozzle when it 
is closed is 200 Ibs. per square inch, and the supply when the nozzle is 
open is 100 cubic feet per minute. If the revolutions are 600 per minute, 
estimate the horse -power of the wheel and its efficiency. 

(32) Show that the efficiency of a Pelton wheel is a maximum 
neglecting frictional and other losses when the velocity of the cups equals 
half the velocity of the jet. 

25 cubic feet of water are supplied per second to a Pelton wheel through 
a 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 imparting 
energy to fluids. For example, when a mine has to be drained 
the water niay 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 h, it is called a force pump. 

216. * Centrifugal 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 shown 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 

* See Appendix. 



CENTRIFUGAL PUMPS 



393 



for this can be easily seen in a general way from the following 
consideration. The water approaches a turbine wheel with a 
high velocity and in a direction making a small angle with the 
direction of motion of the inlet circumference of the wheel, and 




Fig. 272. Diagram of Centrifugal 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. 
In the centrifugal pump these conditions are entirely reversed; 
the water enters the wheel with a small velocity, and leaves 



394 



HYDRAULICS 



it with a high velocity. If the case surrounding the wheel 
admits of this velocity being diminished gradually, the kinetic 
energy of the water is converted into useful work, but if not, it is 
destroyed by eddy motions in the casing, and the efficiency of the 
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 Pnmp with spiral casing. 

The casing of Fig. 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 the 
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 

* See page 542. 



CENTRIFUGAL PUMPS 



395 



by allowing the water as it leaves the wheel to enter guide 
passages, similar to those used in a turbine to direct the water 
to the wheel. The area of these passages gradually increases 
and a considerable portion of the velocity head is thus converted 
into pressure head and is available for lifting water. 

This class of centrifugal pump is known as the turbine pump. 




Fig. 274. Diagram of Centrifugal Pump with Whirlpool Chamber. 

217. Starting centrifugal or turbine pumps. 

A centrifugal pump cannot commence delivery unless the wheel, 
casing, and suction pipe are full of water. 

If the pump is below the water in the well there is no difficulty 
in starting as the casing will be maintained full of water. 

When the pump is above the water in the well, as in Fig. 272, 
a non-return valve Y must be fitted in the suction pipe, to prevent 
the pump when stopped from being drained. If the pump becomes 
empty, or when the pump is first set to work, special means have 
to be provided for filling the pump case. In large pumps the air 
may be expelled by means of steam, which becomes condensed and 
the water rises from the well, or they should be provided with 



396 



fiYDRAULICS 



an air-pump or ejector as an auxiliary to the pump. Small pumps 
can generally be easily filled by hand through a pipe such as 
shown at P, Fig. 276. 

With some classes of pumps, if the pump has to commence 
delivery against full head, a stop valve on the rising main, 
Fig. 296, is closed until the pump has attained the speed necessary 
to commence delivery*, after which the stop valve is slowly 
opened. 




Fig. 275. 



Turbine Pump. 



Fig. 276. 



It will be seen later that, under special circumstances, other 
provisions will have to be made to enable the pump to commence 
delivery. 

218. Form of the vanes of centrifugal pumps. 

The conditions to be satisfied by the vanes of a centrifugal 
pump are exactly the same as for a turbine. At inlet the direction 
of the vane should be parallel to the direction of the relative 
velocity of the water and the tip of the vane, and the velocity 
with which the water leaves the wheel, relative to the pump case, 
is the vector sum of the velocity of the tip of the vane and the 
velocity relative to the vane. 

* See page 409- 



CENTRIFUGAL PUMPS 397 

Suppose the wheel and casing of Fig. 272 is full of water, and 
the wheel is rotated in the direction of the arrow with such a 
velocity that water enters the wheel in a known direction with a 
velocity U, Fig. 277, not of necessity radial. 

Let v be the velocity of the receiving edge of the vane or inlet 
circumference of the wheel; Vi the velocity of the discharging 
circumference of the wheel ; Ui the absolute velocity of the water 
as it leaves the wheel ; Y and Vi the velocities of whirl at inlet 
and outlet respectively; Y r and v r the relative velocities of the 
water and the vane at inlet and outlet respectively ; u and u the 
radial velocities at inlet and outlet respectively. 

The triangle of velocities at inlet is ACD, Fig. 277, and if the 
vane at A, Fig. 272, is made parallel to CD the water will enter 
the wheel without shock. 





_ 

A * C B * E 

Triangle oC velocities Triangle of velocities 

cut inlet. at exit. 

Fig. 277. Fig. 278. 

The wheel being full of water, there is continuity of flow, and 
if A and AI are the circumferential areas of the inner and outer 
circumferences, the radial component of the velocity of exit at the 
outer circumference is 



If the direction of the tip of the vane at the outer circum- 
ference is known the triangle of velocities at exit, Fig. 278, can be 
drawn as follows. 

Set out BG radially and equally to HI, and BE equal to VL 

Draw GF parallel to BE at a distance from BE equal to Ui, 
and EF parallel to the tip of the vane to meet GF in F. 

Then BF is the vector sum of BE and EF and is the velocity 
with which the water leaves the wheel relative to the fixed casing. 

219. Work done on the water by the wheel. 

Let B and r be the radii of the discharging and receiving 
circumferences respectively. 

The change in angular momentum of the water as it passes 
through the wheel is ViB/$rVr/0 per pound of flow, the plus 
sign being used when V is in the opposite direction to Y J; as in 
Figs. 277 and 278. 



398 HYDRAULICS 

Neglecting frictional and other losses, the work done by the 
wheel on the water per pound (see page 275) is 



9 ' 9 ' 

If U is radial, as in Fig. 272, Y is zero, and the work done on 
the water by the wheel is 



- foot Ibs. per Ib. flow. 

J/ 

If then H , Fig. 272, is the total height through which the water 
is lifted from the sump or well, and u d is the velocity with which 
the water is delivered from the delivery pipe, the work done on 
each, pound of water is 

' 



and therefore, 

1 

9 ' 2# 

Let (180 -* <) be the angle which the direction of the vane at 
exit makes with the direction of motion, and (180 -,0) the angle 
which the vane makes with the direction of motion at inlet. Then 
ACD is and BEF is <f>. 

In the triangle HEF, HE = HF cot <, and therefore, 

Vi = Vi -MI cot <#>. 
The theoretical lift, therefore, is 



29 9 

If Q is the discharge and AI the peripheral area of the dis- 
charging circumference, 



v\ Vi -- cot <f> 
and H = - =1 - ........................ (1). 

y 

If, therefore, the water enters the wheel without shock and all 

p 
resistances are neglected, the lift is independent of the ratio , and 

depends only on the velocity and inclination of the vane at the 
discharging circumference. 

220. Ratio of V x to v r 

As in the case of the turbine, for any given head H, Vi and Vi 
can theoretically have any values consistent with the product 



CENTRIFUGAL PUMPS 



399 



being equal to #H, the ratio of V x to v l simply depending upon 
the magnitude of the angle <j>. 

The greater the angle <j> is made the less the velocity ^ 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 <j> is given three values, 
30 degrees, 90 degrees and 150 degrees, and the product V^i and 
also the radial velocity of flow % 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 HYDRAULICS 

U 2 
the wheel. This kinetic energy is equal to 7^-, 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 being accompanied by a 
considerable loss by eddy motions. If it be assumed that the same 

Ui 2 
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 centrifugal pump. 

Let h a be the actual height through which water is lifted; 
h s the head lost in the suction pipe ; Tid the head lost in the delivery 
pipe ; and u d the velocity of flow along the delivery pipe. 

Any other losses of head in the wheel and casing are incident 

to the pump, but h s , hd, and the head ^ should be considered as 

30 

external losses. 

The gross lift of a pump is then 



and this is always less than H. 

223. Efficiencies of a centrifugal pump. 
Manometric efficiency. The ratio g , or 

g .h 

e ~ " Q~~~ """' 

Ui 2 Vi -r- cot <}> 
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 may 
be taken as the ratio of the gross lift at that discharge to the 
theoretical head 

tf-^-Scot* 



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 efficiency. The hydraulic efficiency of a pump is 
the ratio of the gross work done by the pump to the work done 
on the pump wheel. 

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 Bh = the hydraulic efficiency. Then 

W.h 

e =~w 

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. 

Actual efficiency. From a commercial point of view, what is 
generally 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 s be the energy given to the pump shaft per sec. and 
e m the mechanical efficiency of the pump, then 

E-E s .e OT , 
and the actual efficiency 

W.h a 



Gross efficiency of the pump. The gross efficiency of the pump 
itself, including mechanical as well as fluid losses, is 

_W.h 
e g - Es 

224. Experimental determination of the efficiency of a 
centrifugal pump. 

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. ii. 26 



402 HYDRAULICS 

discharged, and the energy E s given to the pump shaft in unit 
time. 

A very convenient method of determining E, with a fair 
degree of accuracy is to drive the pump shaft direct by an electric 
motor, the efficiency curve* for which at varying loads is known. 
A better method is to use some form of transmission dynamo- 
meter t. 

225. Design of pump to give a discharge Q. 

If a pump is required to give a discharge Q under a gross 
lift h, and from previous experience the probable manometric 
efficiency e at this discharge is known, the problem of determining 
suitable dimensions for the wheel of the pump is not difficult. 
The difficulty really arises in giving a correct value to e and in 
making proper allowance for leakage. 

This difficulty will be better appreciated after the losses in 
various kinds of pumps have been considered. It will then be 
seen that e depends upon the angle <, the velocity of the wheel, 
the dimensions of the wheel, the form of the vanes of the wheel, 
the discharge through the wheel, and upon the form of the casing 
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 wheel of a pump for a given discharge under a 
given head. If a pump is required to give a discharge Q under an 
effective head h aj the gross head h can only be determined if h sj 

h d , and |^- , are known. 

Any suitable value can be given to the velocity Ud. If the 
pipes are long it should not be much greater than 5 feet per second 
for reasons explained in the chapter on pipes, and the velocity u 8 
in the suction pipe should be equal to or less than u*. The 
velocities u s and u& having been settled, the losses h 8 and ha can be 
approximated to and the gross head h found. In the suction pipe, 
as explained on page 395, a foot valve is generally fitted, at which, 
at high velocities, a loss of head of several feet may occur. 
The angle < is generally made from 10 to 90 degrees. Theoreti- 
cally, as already stated, it can be made much greater than 
90 degrees, but the efficiency of ordinary centrifugal pumps might 
be very considerably diminished as <f> is increased. 

The manometric efficiency e varies very considerably ; with 
radial blades and a circular casing, the efficiency is not generally 

* See Electrical Engineering, Thomaleu-Howe, p. 195. 
t See paper by Stanton, Proc. Inst. Mech. Engs. y 1903. 



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 <j> 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 
smaller, as by so doing the frictional losses might be considerably 
reduced. The radial velocity Ui may be taken from 2 to 10 feet 
per second. 

Having given suitable values to u t and to any two of the three 
quantities, e, v, and <, the third can be found from the equation 

, e (vi Viiii cot <) 
ri - . 
9 

The internal diameter d of the wheel will generally be settled from 
consideration of the velocity of flow u% 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 ^ 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. 

Then & = -- 

and E 

If the water moves toward the vanes at inlet radially, the 
inclination of the vane that there shall be no shock is such that 

a u 
tan = - . 
v 9 

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 v l is found from 

Ui 

tan a = y- . 

, The guide passages should be so proportioned that the velocity 
Ui is gradually diminished to the velocity in the delivery pipe. 

262 



404 HYDRAULICS 

Limiting velocity of the rim of the wheel. Quite apart from 
head lost by friction in the wheel due to the relative motion of 
the water and the wheel, there is also considerable loss of energy 
external to the wheel due to the relative motion of the water and 
the wheel. Between 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 
the square of the velocity of the wheel periphery and to the area 
of the wheel crowns. An attempt is frequently made to diminish 
this loss by fixing the vanes to a central diaphragm only, the 
wheel thus being without crowns, the outer casing being so 
formed that there is but a small clearance between it and the 
outer edges of the vanes. At high velocities these frictional resist- 
ances may be considerable. To keep them small the surface of 
the wheel crowns and vanes must be made smooth, and to this 
end many high speed wheels are carefully finished. 

Until a few years ago the peripheral velocity of pump wheels 
was generally less than 50 feet per second, and the best velocity 
was supposed to be about 30 feet per second. They are now, how- 
ever, run at much higher speeds, and the limiting velocities are 
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 Eateau has constructed small pumps 
with a peripheral velocity of 250 feet per second*. 

Example. To find the proportions of a pump with radial blades at outlet 
(i.e. = 90) to lift 10 cubic feet of water per second against a head of 50 feet. 

Assume there are two suction pipes and that the water enters the wheel from 
both sides, as in Fig. 273, also that the velocity in the suction and delivery pipes 
and the radial velocity through the wheel are 6 feet per second, and the manometric 
efficiency is 75 per cent. 

First to find Vj. 

Since the blades are radial, *75 = 50, 

y 

from which 1^=46 feet per sec. 

To find the diameter of the suction pipes. 
The discharge is 10 cubic feet per second, therefore 



from which <Z = l-03'=12f". 

If the radius R of the external circumference be taken as 2r and r is taken equal 
to the radius of the suction pipes, then B = 12f", and the number of revolutions 
per second will be 



The velocity of the inner edge of the vane is 
v = 23 feet per sec. 

* Engineer, 1902. 



CENTRIFUGAL PUMPS 405 

The inclination of the vane at inlet that the water may move on to the vane 
without shock is 



and the water when it leaves the wheel makes an angle a with v x such that 



If there are guide blades surrounding the wheel, a gives the inclination of these 
blades. 

The width of the wheel at discharge is 

M> = 7r.D.6 / = 7r.2-0 
= 3 inches about. 
The width of the wheel at inlet =6^ inches. 

226. The centrifugal head impressed on the water by 
the wheel. 

Head against which a pump will commence to discharge. As 
shown on page 335, the centrifugal head impressed on the water as 
it passes through the wheel is 

, _V v* 

hc ~2g-W 

but 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 
pressure at inlet and outlet when the pump is first set in motion ; 
or it is the statical head which the pump will maintain when 

running 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 



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, h c was 

Q , and with curved vanes, <f> being 30 degrees, h was ^ . 

For a pump with a spiral case surrounding the wheel, the 
centrifugal head h c 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. E. t 1903. 



406 HYDRAULICS 

Parsons found for a pump having a wheel 14 inches diameter 
with radial vanes at outlet, and running at 300 revolutions per 

minute, that the head maintained without discharge was 9, 
and with an Appold* wheel running at 320 revolutions per minute 
the statical head was ~ . For a pump, with spiral casing, 

having a rotor 1*54 feet diameter, the least velocity at which 
it commenced to discharge against a head of 14*67 feet was 

OK 2 

392 revolutions per minute, and thus h c was ^ l , and the least 
velocity against a head of 17*4 feet was 424 revolutions per 
minute or h c was again ~ - . For a pump with circular casing 

larger than the wheel, h c was ~-^- . For a pump having guide 

passages surrounding the wheel, and outside the guide passages 
a circular chamber as in Fig. 275, the centrifugal head may also 

2 

be larger than ~; the mean actual value for this pump was 



found to be T087. 

Stanton found, when the seven inches diameter wheels mentioned 
above discharged into guide passages surrounded by a circular 

chamber 20 inches diameter, that h c was - ; - when the vanes of 

^9 

the wheel were radial, and -^ - when < was 30 degrees. 

*9 

That the centrifugal head when the wheel has radial vanes is 
likely to be greater than when the vanes of the wheel are set back 
is to be seen by a consideration of the manner in which the water 
in the chamber outside the guide passages is probably set in 
motion, Fig. 280. Since there is no discharge, this rotation cannot 
be caused by the water passing through the pump, but must be 
due to internal motions set up in the wheel and casing. The 
water in the guide chamber cannot obviously rotate about the 
axis 0, but there is a tendency for it to do so, and consequently 
stream line motions, as shown in the figure, are probably set 
up. The layer of water nearest the outer circumference of the 
wheel will no doubt be dragged along by friction in the direction 
shown by the arrow, and water will flow from the outer casing to 
take its place ; the stream lines will give motion to the water in 
the outer casing. 

* See page 415. 



CENTRIFUGAL PUMPS 



407 



When the vanes in the wheel are radial and as long as a vane is 
moving between any two guide vanes, the straight vane prevents 
the friction between the water outside the wheel and that inside, 
from dragging the water backwards along the vane, but when the 
vane is set back and the angle < is greater than 90 degrees, there 
will be a tendency for the water in the wheel to move backwards 
while that in the guide chamber moves forward, and consequently 
the velocity of the stream lines in the casing will be less in the 
latter case than in the former. In either case, the general 
direction of flow of the stream lines, in the guide chamber, will 
be in the direction of rotation of the wheel, but due to friction 
and eddy motions, even with radial vanes, the velocity of the stream 




Fig. 280. 

lines will be less than the velocity i\ of the periphery of the wheel. 
Just outside the guide chambers the velocity of rotation will be 
less than VL In the outer chamber it is to be expected that the 
water will rotate as in a free vortex, or its velocity of whirl will 
be inversely proportional to the distance from the centre of the 
rotor, or will rotate in some manner approximating to this. 

The head which a pump, with a vortex chamber, will theoreti- 
cally maintain when the discharge is zero. In this case it is 
probable that as the discharge approaches zero, in addition to the 
water in the wheel rotating, the water in the vortex chamber will 
also rotate because of friction, 



408 HYDRAULICS 

The centrifugal head due to the water in the wheel is 



If K = 2r, this becomes j ^- . 
4 Zg 

The centrifugal head due to the water in the chamber is, 
Fig. 281, 

v<?dr 



r and VQ being the radius and tangential velocity respectively of 
any ring of water of thickness dr. 




Fig. 281. 

If it be assumed that v r is a constant, the centrifugal head 
due to the vortex chamber is 

tfrV P*dr = vfr? /_! J_\ 

g !r w r 3 2g W R.V' 
The total centrifugal head is then 



i, -^.^o-^LYi- _L\ 
~2^ 2^ + 2^ W R.V* 



If r; is 2r and R, is 



The conditions here assumed, however, give h c too high. In 
Stanton's experiments h was only ~ . Decouer from experi- 



CENTRIFUGAL PUMPS 409 

ments on a small pump with a vortex chamber, the diameter being 

I .q 2 

about twice the diameter of the wheel, found h c to be ~ 1 - . 

Let it be assumed that h c is -~-*- in any pump, and that the lift 

*9 
of the pump when working normally is 

, _ eViVi _ e fa 2 - vMi cot <fr) 





9 mv* g 
Then if h is greater than -^, the pump will not commence to 

discharge unless speeded up to some velocity v z such that 
my* e (vi - ViUi cot <ft) 
2g ' ~T 

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 v z at which delivery 
commences decreases with the angle <. If <f> is 90 or greater than 
90 degrees, and m is unity, the pump will only commence to 
discharge against the normal head when the velocity is v i9 if the 
manometric efficiency e is less than 0*5. If < is 30 degrees and m 
is unity, v z is equal to Vi when e is 0'6, but if </> is 150 degrees v 2 
is equal to Vi 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 gross 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 < 
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, 419 and 542. 



410 



HYDRAULICS 



of the suction pipe a vacuum gauge. Start the pump 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 pipe, then - 
is the total centrifugal head h . 



60- 



,4V 




WOO 1800 2000 2200 

Revolutions per Minute. ' 
Fig. 282. 

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 against 
this head at this speed it cannot deliver less than 12 cubic feet 
per minute. 

228, Variation of the discharge of a centrifugal pump 
with the head when the speed is kept constant*. 

Head-discharge curve at constant velocity. If the speed of a 
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. Rateau on a pump having 
a wheel 11*8 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. 

701 




20 

3 4- 

Radii Velocity dffiow from. Wheel. 
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 
the pump will be however very different at the two discharges, 
and, as shown by the curves of efficiency, the actual efficiencies 
for the two discharges are very different. At the given velocity 
therefore and at 80 feet head, the flow is ambiguous and is 
unstable, and may suddenly change from one value to the other, 
or it may actually cease, in which case the pump would not start 
again without the velocity ^ being increased to 707 feet per 
second. This value is calculated from the equation 



Proceedings last. Mech. Engs. t 1903. 



412 



HYDRAULICS 



the coefficient m for this pump being 1'02. For the flow to be 
stable when delivering against a head of 80 feet, the pump should 
be run with a rim velocity greater than 70'7 feet per second, in 
which case the discharge cannot be less than 4J cubic feet per 
minute, as shown by the velocity-discharge curve of Fig. 287. 
The method of determining this curve is discussed later. 


Pump Wheel fl-Sctiam/. 



90 

60 

* 10 
%60 

> 

'* W 

1" 

#20 
*/> 










^~ ; 




~~-^ 


^^ 






^ 




ad-Disc) 


large Gar 


r e 


i) 










v,*=66' 


oersec. 




























/</ 











Fig. 235. 



3 4> 

Discharge in c.fl. per mm/. 



10 



^Y 



Fig. 286. 



75 



-pischarge 
f jonstarub= 



Carve 



Fig. 287. 



Example. A centrifugal pump, when discharging normally, has a peripheral 
velocity of 50 feet per second. 

The angle at exit is 30 degrees and the manometric efficiency is 60 per cent. 
The radial velocity of flow at exit is 2 ^//i. 

Determine the lift h and the velocity of the wheel at which it will start delivery 
under full head. 



V = 50 -(2 cos 130 
* 60- 1-73 *. 



CENTRIFUGAL PUMPS 413 



Therefore 



from which h = 37 feet. 

Let ? 2 be the velocity of the rim of the wheel at which pumping commences. 
Then assuming the centrifugal head, when there is no discharge, is 



v 2 =48-6 ft. per sec. 

229. Bernoulli's equations applied to centrifugal pumps. 

Consider tlie motion of the water in any passage between two 
consecutive vanes of a wheel. Let p be the pressure at inlet, pi at 
outlet and p a 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 
Bernoulli's equations (see Figs. 277 and 278), 



w 2g w 2g 

But since, due to the rotation, a centrifugal head 



is impressed on the water between inlet and outlet, therefore, 



_ 
w 2g w 2g 2g 2g 

p, p , * V r 2 v? 
w-w=2g-2- g + 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 u, 



> 

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 
u d , and if frictional losses be neglected, the height to which the 
water can be lifted above the centre of the pump is, by Bernoulli's 
equation, 

h = P 1+ W_P_uJ (?)> 

w 2g w 2g 

If the centre of the wheel is h feet above the level of the water 
in the sump 01 well, and the water in the well is at rest, 

& = *. + *+ ...(8). 

w w 2g 



414 HYDRAULICS 

Substituting from (7) and (8) in (6) 



9 2 2gr 

= H,+ g = H ..................... (9). 

This result verifies the fundamental equation given on page 398. 
Further from equation (6) 

^1^IZL_2_^L = TT +- d - 
w 2g w 2g 2g* 

Example. The centre of a centrifugal pump is 15 feet above the level of the 
vater in the sump. The total lift is 60 feet and the velocity of discharge from the 
delivery pipe is 5 feet per second. The angle <j> at discharge is 135 degrees, and 
the radial velocity of flow through the wheel is 5 feet per second. Assuming there 
are no losses, find the pressure head at the inlet and outlet circumferences. 

At inlet *S4' -!?-. 

w 64 

= 18-6 feet. 
The radial velocity at outlet is 

! = 5 feet per second, 



and = ll = 60 25 

9 <J 64' 

and therefore, v 1 2 + 5v 1 =1940 ....................................... (1), 

from which Vj = 41 -6 feet per second, 

and V = 46-6 



The pressure head at outlet is then 



w w 

= 45 feet. 
To find the velocity v when <f> is made 30 degrees. 

cot 0=^/3, 

therefore (1) becomes vf - 5 */3 . v l = 1940, 

from which v 1 = 48 > 6 ft. per sec. 

and V 1 = 40 

Then ^L = 25-4 feet, and % = 53-6 feet. 

2g w 

230. Losses in centrifugal pumps. 

The losses of head in a centrifugal pump are due to the same 
causes as the losses in a turbine. 

Loss 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 diminish 
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, 

U 2 
the whole of the velocity head ^- was lost, no special precautions 

*9 

being taken to diminish it gradually and the efficiency was 
constantly very low, being less than 40 per cent. 

The effect of the angle < on the efficiency of the pump. To 
increase the efficiency Appold suggested that the blade should be 
set back, the angle < being thus less than 90 degrees, Fig. 272. 

Theoretically, the effect on the efficiency can be seen by 
considering the three cases considered in section 220 and illustrated 

TJ 2 
in Fig. 279. When < is 90 degrees -~- is *54H, and when < is 

U 2 

30 degrees -^- is *36H. If, 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. 
respectively. 

In general when there is no precaution taken to utilise the 
energy of motion at the outlet of the wheel, the theoretical lift is 



and the maximum possible manometric efficiency is 



Substituting for Vi, i - Ui cot <}>, and for Ui 2 , Vi 8 + u*, 

Ht =lrS 2cosec '*' 

, _- fa - U, COt <ft) 2 + U? 

' 



v Ui cosec < 
~ 2vi (vi Ui cot <#>) " 

When v l is 30 feet per second, Ui 5 feet per second and < 
30 degrees, e is 62'5 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 <f>. 

Parsons* found that when < was 90 degrees the efficiency of a 
pump in which the wheel was surrounded by a circular casing 
was nearly 10 per cent, less than when the angle < was made 
about 15 degrees. 

* Proceedings Inst. C. E. t Vol. XLVH. p. 272. 



416 HYDKAULICS 

Stanton found that a pump 7 inches diameter having radial 
vanes at discharge had an efficiency of 8 per cent, less than when 
the angle $ at delivery was 30 degrees. In the first case the 
maximum actual efficiency was only 39'6 per cent., and in the 
second case 50 per cent. 

It has been suggested by Dr Stanton that a second reason for 
the greater efficiency of the pump having vanes curved back at 
outlet is to be found in the fact that with these vanes the variation 
of the relative velocity of the water and the wheel is less than 
when the vanes are radial at outlet. It has been shown experi- 
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 the 
sections of least pressure. These return stream lines cause a loss 
of energy by eddy motions. Now in a pump, when the vanes are 
radial, there is a greater difference between the relative velocity 
of the water and the vane at inlet and outlet than when the angle 
</> is less than 90 degrees (see Fig. 279), and it is probable there- 
fore that there is more loss by eddy motions in the wheel in the 
former case. 

Loss of head at entry. To avoid loss of head at entry the vane 
must be parallel to the relative velocity of the water and the 
vane. 

Unless guide blades are provided the exact direction in which 
the water approaches the edge of the vane is not known. 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 pipe 
are parallel to the sides of the pipe, would be simply turned to 
approach the vanes radially. 

It has already been seen that when there is no flow the water 
in the eye of the wheel is made to rotate by friction, and it is 
probable that at all flows the water has some rotation in the eye 
of the wheel, but as the delivery increases the velocity of rotation 
probably diminishes. If the water has rotation in the same 
direction as the wheel, the angle of the vane at inlet will clearly 
have to be larger for no shock than if the flow is radial. That 
the water has rotation before it strikes the vanes seems to be 
indicated by the experiments of Mr Livens on a pump, the vanes 
of which were nearly radial at the inlet edge. (See section 236.) 
The efficiencies claimed for this pump are so high, that there 
could 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 

vane can only be parallel to the tip of the vane for one discharge, 
and at other discharges in order to make the water move along 
the vane a sudden velocity must be impressed upon it, which 
causes a loss of energy. 

Let u. 2) Fig. 288, be the velocity with which the water enters a 
wheel, and and v the inclination 
and velocity of the tip of the vane \*- us ->j 
at inlet respectively. 

The relative velocity of u 2 and v 
is V/, the vector difference of u* 
and v. 

The radial component of flow 
through the opening of the wheel 
must be equal to the radial com- 
ponent of u 2 , and therefore the 
relative velocity of the water along the tip of the vane is V r . 

If Uz is assumed to be radial, a sudden velocity 

u 8 - v - u* cot 
has thus to be given to the water. 

If Us has a component in the direction of rotation u a will be 
diminished. 

It has been shown (page 67), on certain assumptions, that if 
a body of water changes its velocity from v a to v* suddenly, the 

head lost is ^-^ , or is the head due to the change of velocity. 

*9 
In this case the change of velocity is u s , and the head lost may 

ku 2 
reasonably be taken as -^- . If k is assumed to be unity, the 

*9 
effective work done on the water by the wheel is diminished by 

Ug__ (V-Uy COt #) 2 

2<r 2 3 

If now this loss takes place in addition to the velocity head 
being lost outside the wheel, and friction losses are neglected, 
then 



20 



V? Q 2 z , 

- Si 2 cosec 9 



20 

- " 2 cot. a 



t. H. 27 



418 



HYDRAULICS 



Example. The radial velocity of flow through a pump ia 5 feet per second. 
The angle is 80 degrees and the angle Q is 15 degrees. The velocity of the 
outer circumference is 50 feet per sec. and the radius is twice that of the inner 
circumference. 

Find the theoretical lift on the assumption that the whole of the kinetic energy 
is lost at exit. 

v,* 5 2 (25 - 5 cot 15)* 

h = cosec 2 30 - s 

2g 2g 2g 

= 37-0 feet. 

The theoretical lift neglecting all losses is 64-2 feet, and the manometric 
efficiency is therefore 58 per cent. 

231. Variation of the head with discharge and with the 
speed of a centrifugal pump. 

It is of interest to study by means of equation (1), section 230, 
the variation of the discharge Q with the velocity of the pump 
when h is constant, and the variation of the head with the 
discharge when the velocity of the pump is constant, and to 
compare the results with the actual results obtained from 
experiment. 

The full curve of Fig. 289 shows the variations of the head 
with the discharge when the velocity of a wheel is kept constant. 
The data for which the curve has been plotted is indicated in 
the figure. 




13 

I. 

i" 

|M 




















1 


>< 


St 




^ 


\ 






/_ 






N 




\ 




/ 


v t -30Ft,.p< 
v = 15 ' o * " 

fi i 1 


zrSet 

" 


z. 


\ 




\ 


' 








^ 


. Normal radial veloct 

\1 \Z \3 |4 


f-" 


FT 



RacbLaJL velocity of Fkw= H 
A, 

Fig. 289. Head-discharge curve at constant velocity. 

When the discharge is zero 

h = pr PT = 10*5 feet. 
2g 2g 

The velocity of flow -~ at outlet has been assumed equal to 

-5r at inlet. 
A. 

Values of 1, 2, etc. were given to ~ and the corresponding 
values of h found from equation (1). 



CENTRIFUGAL PUMPS 419 

When the discharge is normal, that is, the water enters the 

wheel without shock, ~ is 4 feet and h is 14 feet. The theoretical 

JL 

head assuming no losses is then 28 feet and the manometric 
efficiency is thus 50 per cent. For less or greater values of -f 

_X 

the head diminishes and also the efficiency. 

The curve of Fig. 290 shows how the flow varies with the 
velocity for a constant value of ft, which is taken as 12 feet. 




Radial Velocity through, Wheei. 
Fig. 290. Velocity-discharge curve at constant head for Centrifugal Pnrap. 

It will be seen that when the velocity t?i is 31*9 feet per second 
the velocity of discharge may be either zero or 8'2 feet per second. 
This means that if the head is 12 feet, the pump, theoretically, 
will only start when the velocity is 31*9 feet per second and the 
velocity of discharge will suddenly become 8*2 feet per second. 
If now the velocity Vi is diminished the pump still continues to 
discharge, and will do so as long as Vj. is greater than 26*4 feet per 
second. The flow is however unstable, as at any velocity v it may 
suddenly change from CB to CD, or it may suddenly cease, and it 
will not start again until ^ is increased to 31*9 feet per second. 

232. The effect of the variation of the centrifugal head 
and the loss by friction on the discharge of a pump. 

If then the losses at inlet and outlet were as above and were 
the only losses, and the centrifugal head in an actual pump was 
equal to the theoretical centrifugal head, the pump could not be 
made to deliver water against the normal head at a small velocity 
of discharge. In the case of the pump considered in section 231, 
it could not safely be run with a rim velocity less than 31*9 ft. 
per sec., and at any greater velocity the radial velocity of flow 
could not be less than 8 feet per second, 

272 



420 HYDRAULICS 

In actual pumps, however, it has been seen that the centrifugal 
head at commencement is greater than 



There is also loss of head, which at high velocities and in small 
pumps is considerable, due to friction. These two causes consider- 
ably modify the head-discharge curve at constant velocity and the 
velocity-discharge curve at constant head, and the centrifugal 
head at the normal speed of the pump when the discharge is zero, 
is generally greater than any head under which the pump works, 
and many actual pumps can deliver variable quantities of water 
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 diminish and the centrifugal head will therefore 
diminish. 

Let it be assumed that when the velocity of flow is u (supposed 
constant) the centrifugal head is 



7, _^L_^ - 

flc ~'2g 20 20 

and n being constants which must be determined by experiment. 
When u is zero 



v\ 



and if m is known Jc can at once be found. 

Let it further be assumed that the loss by friction* and eddy 

cV 
motions, apart from the loss at inlet and outlet is -~- . 

* The loss of head by friction will no doubt depend not only upon u but also 
upon the velocity v l of the wheel, and should be written as 



Cu 2 o 
or, as 27 + -!r + 

If it be supposed it can be expressed by the latter, then the correction 
fcV 2nku 1 v 1 ^ 
~2g'~ 2g 2g ' 

if proper values are given to &, n^ and k^ , takes into account the variation of the 
centrifugal head and also the friction head v l . 



CENTRIFUGAL PUMPS 421 

The gross head h is then, 

2vu cot 9 A 
~^- ~*<*>** 

nu* cV 



2g -~2j ............... ' 

If now the head h and flow Q be determined experimentally, 
the difference between h as determined from equation (1), page 4J 7, 
and the experimental value of h, must be equal to 

V 2nkuv l 




2g 
hi being equal to (c 2 -w 2 ). 

The coefficient Jc being known from an experiment when u is 
zero, for many pumps two other* experiments giving corresponding 
values of h and u will determine the coefficients n and fa. 

The head-discharge curve at constant velocity, for a pump such 
as the one already considered, would approximate to the dotted 
curve of Fig. 289. This curve has been plotted from equation (2), 
by taking k as 0'5, n as 7*64 and fa as - 38. 

Substituting values for fa n, fa, cosec < and cot <, equation (2) 
becomes 



C and Ci being new coefficients ; or it may be written 



Q being the flow in any desired units, the coefficients C 2 and C 8 
varying with the units. If * equation (4) is of the correct form, 
three experiments will determine the constants m, C 2 and C 3 
directly, and having given values to any two of the three 
variables h, v, and Q the third can be found. 

233. The effect of the diminution of the centrifugal head 
and the increase of the friction head as the flow increases, on 
the velocity-discharge curve at constant head. 

Using the corrected equation (2), section 232, and the given 
values of k, rh and fa the dotted curve of Fig. 290 has been plotted. 

From the dotted curve of Fig. 289 it is seen that u cannot 
be greater than 5 feet when the head is 12 feet, and therefore the 
new curve of Fig. 290 is only drawn to the point where u is 5. 

The pump starts delivering when v is 27*7 feet per second and 
the discharge increases gradually as the velocity increases. 

* See page 544. 



422 HYDRAULICS 

The pump will deliver, therefore, water under a head of 
12 feet at any velocity of flow from zero to 5 feet per second. 

In such a pump the manometric efficiency must have its 
maximum value when the discharge is zero and it cannot be 
greater than 



COt ' 



9 

This is the case with many existing pumps and it explains why, 
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 
head ^- with which the water leaves the wheel into pressure 

head. 

The methods for converting the velocity head with which the 
water leaves the wheel into pressure head have been indicated on 
page 394. They are now discussed in greater detail. 

Thomson's vortex or whirlpool chamber. Professor James 
Thomson first suggested that the wheel should be surrounded by 
a chamber in which the velocity of the water should gradually 
change from Ui to u d the velocity of flow in the pipe. Such a 
chamber is shown in Fig. 274. In this chamber the water forms 
a free vortex, so called because no impulse is given to the water 
while moving in the chamber. 

Any fluid particle ab, Fig. 281, may be considered as moving 
in a circle of radius r with a velocity v 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 

perpendicular to the radius is - and the moment of mo- 

~\/\ / H 7* 

mentum or angular momentum about the centre C is - . 

y 

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 
upon the body. 

But since no turning effort, or couple, acts upon the element 
after leaving the wheel its moment of momentum must be 
constant. 




CENTRIFUGAL PUMPS 423 

Therefore, 

is constant or V r = 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 u for any ring will be inversely proportional to r , and there- 
fore, the ratio is constant, or the direction of motion of any 

element with its radius r is constant, and the stream lines are 
equiangular spirals. 

If no energy is lost, by friction and eddies, Bernoulli's theorem 
will 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 u is constant 

?r" + = constant. 
2g w 

Let the outer radius of the whirlpool chamber be R, and 
the inner radius r w . Let v fw and v Rw be the whirling velocities 
at the inner and outer radii respectively. 

Then since v ^o is a constant, 

and - ? + - = constant, 

w 2g 



w w 2g 2g 

= ^ + w( l ~^' 
When U w = 2r w , 



w w 4* 2g 

If the velocity head which the water possesses when it leaves 
the vortex chamber is supposed to be lost, and hi is the head of 
water above the pump and p a the atmospheric pressure, then 
neglecting friction 



u d * 



or 



- = i ?i -- " 

w 2g w 

, PR W UA Pa 



424 HYDRAULICS 

If then h is the height of the pump above the well, the total 
lift h% is hi + ho. 
Therefore, 

/, -7, + P + v '- 
k-^ + + 

But ^zfc.p * 

to 10 z# 
also Pr; = pi, ?*> = R, and v ru , = V lt 

Therefore 

, _pi_ p_^ V^/, R 2 \ _^ 
^ w 2g 2g V " R.V 2<? ' 

But from equation (6) page 413, 



tt; w 2g g 2g 
Therefore 



Ui V^A R 2 \ 

^ ^A B2/' 



This might have been written down at once from equation (1), 
section 230. For clearly if there is a gain of pressure head 

V 2 / R 2 \ 
in the vortex chamber of -^- fl ~-p~2J> ^ ne velocity head to 

be lost will be less by this amount than when there is no vortex 
chamber. 

Substituting for Vi and Ui the theoretical lift h is now 

, _V*-V 1 Ui COt<j> U? fa - Ui COt <^>) a R 2 ^ n ^ 

g -fy~ ~W -'&, 

When the discharge or rim velocity is not normal, there is a 
further loss of head at entrance equal to 



, 

and 

.-. cot* 



(2). 



When there is no discharge v rtt , is equal to Vi and 

J, = ^1_^ 



CENTRIFUGAL PUMPS 425 

R = 2 RW and v = 2^1 






Correcting equation (1) in order to allow for the variation of 
the centrifugal head with the discharge, and the friction losses, 

, _ Vi - ViUi COt < Ui (Vi Ui COt <ft) 2 R 2 

~~ "" 



(v u cot 0)* k?v* 2nkuvi 



which reduces to h - 



The experimental data on the value of the vortex chamber 
per se, 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 
diameter, the efficiency of the chamber in converting velocity head 
to pressure head was about 40 per cent. It is however questionable 
whether the design of the pump was such as to give the best results 
possible. 

So far as the author is aware, centrifugal pumps with vortex 
chambers are not now being manufactured in England, but it 
seems very probable that by the addition of a well-designed 
chamber small centrifugal pumps might have their efficiencies 
considerably increased. 

235. Turbine pumps. 

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 
of Uj. 

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 mechanical difficulties, would militate against their adoption. 

Single wheel pumps of this type can be used up to a head of 
100 feet with excellent results, efficiencies as high as 85 per cent. 
* Proceedings Inst. C.E., 1903. See also page 542. 



426 HYDRAULICS 

having been claimed. They are now being used to deliver water 
against heads of over 350 feet, and M. Rateau has used a single 
wheel 3'16 inches diameter running at 18,000 revolutions per 
minute to deliver against a head of 936 feet. 

Loss of head at the entrance to the guide passages. If the 
guide blades are fixed, the direction of the tips can only be correct 
for one discharge of the pump. For any other discharge than the 
normal, the direction of the water as it leaves the wheel is not 
parallel to the fixed guide and there is a loss of head due to 
shock. 

Let a be the inclination of the guide blade and < the vane 
angle at exit. 

Let Ui be the radial velocity of 
flow. Then BE, Fig. 291, is the 
velocity with which the water leaves 
the wheel. 

The radial velocity with which 
the water enters the guide passages must be Ui and the velocity 
along the guide is, therefore, BF. 

There is a sudden change of velocity from BE to BF, and on 
the assumption that the loss of head is equal to the head due to the 
relative velocity FE, the head lost is 

fa - -MI cot <ft - Uj cot cp a 

%r 

At inlet the loss of head is 

(v-u cot (9) 2 

20 
and the theoretical lift is 

cot cfr (v-u cot 0) a fa Ui cot <j> - u\ cot a) a 
~ 




= ~W 2g 

_ v* v 9 2v 1 u l cot a 2vu cot 
= 2g~2~g* ~~2g~ 20 

Ui (cot <f> + cot a) 2 u? cot 2 m 

~^~ % 

To correct for the diminution of the centrifugal head and to 

allow for friction, 

tfv* _Zkvin^Ui _ -, u? 
29 " 2^ Cl 2g> 

must be added, and the lift is then 
, _ Vi v 2 2viUi cot a 2vu cot U* (cot <ft + cot a) 9 
h= 2g-*} + 20 2g 2g 

u* cot 2 feV ^ ZJcnViUj TKU? 
2g h "20 20 " 20 ' 



CENTRIFUGAL PUMPS 



427 



which, since u can always be written as a multiple of Ui, reduces 
to the form 

2gh = mv*+ CuiVi + du* (2). 

Equations for the turbine pump shown in Fig. 275. Character- 
istic curves. Taking the data 

= 5 degrees, cot = 11 '43 
= 1732 



equation (2) above becomes 
20fc = ' 



cot a =19*6 



- 587 



eo- 



(3) 




3)i<$cftarge, in/ Cubic Feet per J&nute. ,^ 

_l L_ i i_ __J i 1.1 



f 23 

Velocity cub Exit/ fronv the Wlieet/. Feet fer Second/. 

Fig. 292. Head-discharge curves at constant speed for Turbine Pump. 

From equation (3) taking Vi 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 series of head-discharge curves at 



428 



HYDRAULICS 



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 = l-087t>! a + 2'26tM>! - 62' W, 

obtained by substituting for m, C and d in equation (2) the values 
1*087, 2'26 and - 62*1 respectively. The value of m was obtained 
by determining the head h, 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 Ui 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. 




ZOOO 2100 

Speed* Revolutions per Alutute,. 
Fig. 293. Velocity-Discharge curves at Constant Head. 

No attempt has been made to draw the actual mean curves in 
the figures, as in most cases the difference 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 curves 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 purnp 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 21| inches diameter an efficiency of 82 per 
cent, was claimed. 

* Proceedings Inst. Mech. Engs., 1903. 



430 HYDRAULICS 

The * author finds the equation to the head- discharge curve for 
the 19 inches diameter pump from Mr Livens' data to be 

118v 1 2 + 3^1-142^ = 2gh .................. (1), 

and for the 21 inches diameter pump 

I'18v l *-4,'5u l v 1 = 2gh ..................... (2). 

The velocity of rotation of the water round the wheel will be 
less than the velocity with which the water leaves the wheel and 
there will be a loss of head due to the sudden change in velocity. 

k U 2 
Let this loss of head be written -75-^ . The head, when Ui is the 

*9 

radial velocity of flow at exit and assuming the water enters the 
wheel radially, is then 

, tti 2 -j;i^icot< /CsTJi 8 (v-ucotOy 

g ' 2g 2g 

Taking friction and the diminution of centrifugal head into 
account, 

, _ v* ViUiCoi<j> _ fe 3 Ui a _ (v ucotOy Jcv* __ nJm } v l _ Jc^t? 

g ' 2g ' 2g 2*7 ~ ~2<T "~2g' 

which again may be written 

7, = mv * + C ^i^i , Gi^i a 
" 20 2g 2g ' 

The values of m, C and Ci are given for two pumps in equations 
(1) and (2). 

237. General equation for a centrifugal pump. 
The equations for the gross head h at discharge Q as determined 
for the several classes of pumps have been shown to be of the form 



_ 
= 



2g 2g > 
or, if u is the velocity of flow from the wheel, 

Cuv 



in which m varies between 1 and 1'5. The coefficients C 2 and C 3 
for any pump will depend upon the unit of discharge. 

As a further example and illustrating the case in which at 
certain speeds the flow may be unstable, the curves of Figs. 
285287 may be now considered. When v l is 66 feet per second 
the equation to the head discharge curve is 

. 15-5Qt?i _ 236Q a 



Q being in cubic feet per minute. 

* See Appendix 11. 



CENTRIFUGAL PUMPS 431 

The velocity-discharge curve for a constant head of 80 feet as 
calculated from this equation is shown in Fig. 287. 

To start the pump against a head of 80 feet the peripheral 
velocity has to be 70' 7 feet per second, at which velocity the 
discharge Q suddenly rises to 4'3 cubic feet per minute. 

The curves of actual and manometric efficiency are shown in 
Fig. 286, the maximum for the two cases occurring at different 
discharges. 

238. The limiting height to which a single wheel centri- 
fugal pump can be used to raise water. 

The maximum height to which a centrifugal pump can raise 
water, depends theoretically upon the maximum velocity at which 
the rim of the wheel can be run. 

It has already been stated that rim velocities up to 250 feet 
per second have been used. Assuming radial vanes and a mano- 
metric efficiency of 50 per cent., a pump running at this velocity 
would lift against a head of 980 feet. 

At these very high velocities, however, the wheel must be of 
some material such as bronze or cast steel, having considerable 
resistance to tensile stresses, and special precautions must be 
taken to balance the wheel. The hydraulic losses are also 
considerable, and manometric efficiencies greater than 50 per 
cent, are hardly to be expected. 

According to M. Eateau *, the limiting head against which it is 
advisable to raise water by means of a single wheel is about 
100 feet, and the maximum desirable velocity of the rim of the 
wheel is about 100 feet per second. 

Single wheel pumps to lift up to 350 feet are however being 
used. At this velocity the stress in a hoop due to centrifugal forces 
is about 7250 Ibs. per sq. incht. 

239. The suction of a centrifugal pump. 

The greatest height through which a centrifugal or other class 
of pump will draw water is about 27 feet. Special precaution has 
to be taken to ensure that all joints on the suction pipe are perfectly 
air-tight, and especially is this so when the suction head is greater 
than 15 feet; only under special circumstances is it therefore de- 
sirable for the suction head to be greater than this amount, and it 
is always advisable to keep the suction head as small as possible. 

* "Pompes Centrifuges," etc., Bulletin de la Societe de I'Industrie minfrale, 
1902 ; Engineer, p. 236, March, 1902. 

t See Swing's Strength of Materials ; Wood's Strength of Structural Members ; 
The Steam Turbine Stodola. 



432 



HYDRAULICS 




CENTRIFUGAL PUMPS 



433 



240. Series or multi-stage turbine pumps. 

It has been stated that the limiting economical head for a single 
wheel pump is about 100 feet, and for high heads series pumps 
are now generally used. 




Fig. 296. General Arrangement of Worthington Multi-stage Turbine Pump. 

By putting several wheels or rotors in series on one shaft, each 
rotor giving a head varying from 100 to 200 feet, water can be 
lifted to practically any height, and such pumps have been 
L. n, 28 



434 



HYDRAULICS 



constructed to work against a head of 2000 feet. The number 
of rotors, on one shaft, may be from one to twelve according 
to the total head. For a given head, the greater the number of 
rotors used, the less the peripheral velocity, and within certain 
limits the greater the efficiency. 

Figs. 295 and 296 show a longitudinal section and general 
arrangement, respectively, of a series, or multi-stage pump, as 
constructed by the Worthington Pump Company. On the motor 
shaft are fixed three phosphor-bronze rotors, alternating with fixed 
guides, which are rigidly connected to the outer casing, and to 
the bearings. The water is drawn in through the pipe at the left 
of the pump and enters the first wheel axially. The water leaves 
the first wheel at the outer circumference and passes along an 
expanding passage in which the velocity is gradually diminished 
and enters the second wheel axially. The vanes in the passage 
are of hard phosphor-bronze made very smooth to reduce friction 
losses to a minimum. The water passes through the remaining 
rotors and guides in a similar manner and is finally discharged 
into the casing and thence into the delivery pipe. 




'///////////////////////////w 
Fig. 297. Sulzer Multi-stage Turbine Pump. 

The difference in pressure head at the entrances to any two 
consecutive wheels is the head impressed on the water by one 
wheel. If the head is h feet, and there are n wheels the total 
lift is nearly nh feet. The vanes of each wheel and the directions 
of the guide vanes are determined as explained for the single 
wheel so that losses by shock are reduced to a minimum, and 
the wheels and guide passages are made smooth so as to reduce 
friction. 

Through the back of each wheel, just above the boss, are 
a number of holes which allow water to get behind part of the 
wheel, under the pressure at which it enters the wheel, to balance 
the end thrust which would otherwise be set up. 



CENTRIFUGAL PUMPS 435 

The pumps can be arranged to work either vertically or 
horizontally, and to be driven by belt, or directly by any form 
of motor. 

Fig. 297 shows a multi-stage pump as made by Messrs Sulzer. 
The rotors are arranged so that the water enters alternately 
from the left and right and the end thrust is thus balanced. 
Efficiencies as high as 84 per cent, have been claimed for multi- 
stage pumps lifting against heads of 1200 feet and upwards. 

The Worthington Pump Company state that the efficiency 
diminishes as the ratio of the head to the quantity increases, the 
best results being obtained when the number of gallons raised 
per minute is about equal to the total head. 

Example. A pump is to be driven by a motor at 1450 revolutions per minute, and 
is required to lift 45 cubic feet of water per minute against a head of 320 feet. 
Required the diameter of the suction, and delivery pipes, and the diameter and 
number of the rotors, assuming a velocity of 5 '5 feet per second in the suction and 
delivery pipes, and a manometric efficiency at the given delivery of 50 per cent. 

Assume provisionally that the diameter of the boss of the wheel is 3 inches. 

Let d be the external diameter of the annular opening, Fig. 295. 

Then, f(^-3 2 ) ^ 

144 = 60 x 5-5 * 
from which eZ=6 inches nearly. 

Taking the external diameter D of the wheel as 2d, D is 1 foot. 

1450 
Then, t?i = -^- x v - 76 feet per sec. 

Assuming radial blades at outlet the head lifted by each wheel is 



=90 feet. 
Four wheels would therefore be required. 

241. 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 
reciprocating pumps, dirty water and even water containing com- 
paratively large floating bodies can be pumped without greatly 
endangering the pump. 

Another advantage is that as delivery from the wheel is 
constant, there is no fluctuation of speed of the water in the 
suction or delivery pipes, and consequently there is no necessity 
for air vessels such as are required on the suction and delivery 
pipes of reciprocating pumps. There is also considerably less 
danger of large stress being engendered in the pipe lines by 
"water hammer*." 

Another advantage is the impossibility of the pressure in the 
See page 384. 

282 



436 HYDRAULICS 

pump casing rising above that of the maximum head which the 
rotor is capable of impressing upon the water. If the delivery 
is closed the wheel will rotate without any danger of the pressure 
in the casing becoming greater than the centrifugal head (page 
335). This may be of use in those cases where a pump is de- 
livering into a reservoir or pumping from a reservoir. In the first 
case a float valve may be fitted, which, when the water rises to 
a particular height in the reservoir, closes the delivery. The 
pump wheel will continue to rotate but without delivering water, 
and if the wheel is running at such a velocity that the centri- 
fugal head is greater than the head in the pipe line it will start 
delivery when the valve is opened. In the second case a similar 
valve may be used to stop the flow when the water falls below a 
certain level. This arrangement although convenient is uneco- 
nomical, as although the pump is doing no effective work, the 
power required to drive the pump may be more than 50 per cent, 
of that required when the pump is giving maximum discharge. 

It follows that a centrifugal pump may be made to deliver 
water into a closed pipe system from which water may be taken 
regularly, or at intervals, while the pump continues to rotate at a 
constant velocity. 

Pump delivering into a long pipe line. When a centrifugal 
pump or air fan is delivering into a long pipe line the resistances 
will vary approximately as the square of the quantity of water 
delivered by the pump. 

Let > 2 be the absolute pressure per square inch which has 
to be maintained at the end of the pipe line, and let the 
resistances vary as the square of the velocity v along the pipe. 
Then if the resistances are equivalent to a head hs=kv*, the 

pressure head at the pump end of the delivery pipe must be 

ES-fc+W 

w w 

-p* + fcQ! 

"w A 2 ' 

A being the sectional area of the pipe. 

Let - be the pressure head at the top of the suction pipe, then 

w 
the gross lift of the pump is 

h== Pl-P = P? + l_P f 
www A. 2 w 

If, therefore, a curve, Fig. 298, be plotted having 
fej^p) W 
w A 3 



CENTRIFUGAL PUMPS 



437 



as ordinates, and Q as abscissae, it will be a parabola. If on 
the same figure a curve having h as ordinates and Q as abscissae 
be drawn for any given speed, the intersection of these two 
curves at the point P will give the maximum discharge the pump 
will deliver along the pipe at the given speed. 




Discharge in/ C. Ft/, per Second/. 
Fig. 298. 

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, Bernoulli's equation is 



w 



From which, as in the centrifugal pump, 



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 are properly 
chosen, V r and v r 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 



HYDRAULICS 



Such a pump is similar to a reversed impulse turbine, the 
guide passages of which are kept full. The velocity with which 
the water leaves the wheel would however be great and the lift 
above the pump would depend upon the percentage of the velocity 
head that could be converted into pressure head. 




Fig. 299. 

Since there is no centrifugal head impressed upon the water, 
the parallel-flow pump cannot commence discharging unless the 
water in the pump is first set in motion by some external means, 
but as soon as the flow is commenced through the wheel, the full 
discharge under full head can be obtained. 




Fig. 300. 



Fig. 301. 



To commence the discharge, the pump would generally have to 
be placed below the level of the water to be lifted, an auxiliary 
discharge pipe being fitted with a discharging valve, and a non- 
return valve in the discharge pipe, arranged as in Fig. 300. 



CENTRIFUGAL PUMPS 439 

The pump could be started when placed at a height h above 
the water in the sump, by using an ejector or air pump to exhaust 
the air from the discharge chamber, and thus start the flow 
through the wheel. 

243. Inward flow turbine pump. 

Like the parallel flow pump, an inward flow pump if constructed 
could not start pumping unless the water in the wheel were first 
set in motion. If the wheel is started with the water at rest 
the centrifugal head will tend to cause the flow to take place 
outwards, but if flow can be commenced and the vanes are 
properly designed, the wheel can be made to deliver water at its 
inner periphery. As in the centrifugal and parallel flow pumps, 
if the water enters the wheel radially, the total lift is 



g w w g g 
From the equation 

p_ Vr 8 _ PI v* v* v\ 

w 2g w 2g 2g 2g' 

it will be seen that unless V r 2 is greater than 

?L. 4. ^- _ ^L 
2g + 2g 2g> 

U 2 
Pi is less than p, and ^- will then be greater than the total 

lift H. 

Yery special precautions must therefore be made to diminish 
the velocity U gradually, or otherwise the efficiency of the pump 
will be very low. 

The centrifugal head can be made small by making the 
difference of the inner and outer radii small. 

f-f ?L. + *. - ^ 

2g + 2g 2g 

Y a 

is made equal to 7p- , the pressure at inlet and outlet will be the 

^9 
same, and if the wheel passages are carefully designed, the 

pressure throughout the wheel may be kept constant, and the 
pump becomes practically an impulse pump. 

There seems no advantage to be obtained by using either 
a parallel flow pump or inward flow pump in place of the centri- 
fugal pump, and as already suggested there are distinct dis- 
advantages. 

244. Reciprocating pumps. 

A simple form of reciprocating force pump is shown dia- 
grammatically in Fig. 301. It consists of a plunger P working in 



440 



HYDllAULICS 




Fig. 301 a. Vertical Single-acting Keciprocating Pump. 



RECIPROCATING PUMPS 441 

a cylinder C and has two valves Y s and Y D , known as the suction 
and delivery valves respectively. A section of an actual pump 
is shown in Fig. 301 a. 

Assume for simplicity the pump to be horizontal, with the 
centre of the barrel at a distance h from the level of the water 
in the well; h may be negative or positive according as the 
pump is above or below the surface of the water in the well. 

Let B be the height of the barometer in inches of mercury. 
The equivalent head H, in feet of water, is 

H13'596 . B -I.IQQ-R 
~12~ 183B ' 

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 Y D is closed by the head 
of water above it, and the water in the suction pipe is sustained by 
the atmospheric pressure. 

Let ho be the pressure head in the cylinder, then 

ho = H h, 
or the pressure in pounds per square inch in the cylinder is 

p = '43(H-/i), 

p cannot become less than the vapour tension of the water. At 
ordinary temperatures this is nearly zero, and h cannot be greater 
than 34 feet. 

If now the plunger is moved outwards, very slowly, and there 
is no air leakage the valve Y s opens, and the atmospheric pressure 
causes water to rise up the suction pipe and into the cylinder, 
h remaining practically constant. 

On the motion of the plunger being reversed, the valve YS 
closes, and the water is forced through Y D into the delivery 
pipe. 

In actual pumps if h Q is less than from 4 to 9 feet the 
dissolved gases that are in the water are liberated, and it is there- 
fore practically impossible to raise water more than from 25 to 
30 feet. 

Let A be the area of the plunger in square inches and L the 
stroke in feet. The pressure on the end of the plunger outside the 
cylinder is equal to the atmospheric pressure, and neglecting 
the friction between the plunger and the cylinder, the force neces- 
sary to move the plunger is 

P = '43 {H - (H - h)} A = -43fc . A Ibs., 
and the work done by the plunger per stroke is 
E = '43h. A. L ft. Ibs. 



442 HYDRAULICS 

If Y is the volume displacement per stroke of the plunger 
in cubic feet 

E = 62-4/i. Y ft. Ibs. 

The weight of water lifted per stroke is *43AL Ibs., and the 
work done per pound is, therefore, h foot pounds. 

Let Z be the head in the delivery pipe above the centre of the 
pump, and Ud the velocity with which the water leaves the delivery 
pipe. 

Neglecting friction, the work done by the plunger during the 

2 

delivery stroke is Z + ^ foot pounds per pound, and the total work 

in the two strokes is therefore h + Z + ~ foot pounds per pound. 

The actual work done on the plunger will be greater than this 
due to mechanical friction in the pump, and the frictional and 
other hydraulic losses in the suction and delivery pipes, and at the 
valves; and the volume of water lifted per suction stroke will 
generally be slightly less than the volume moved through by the 
plunger. 

Let W be the weight of water lifted per minute, and lit the 
total height through which the water is lifted. 

The effective work done by the pump is W . h t foot pounds per 
minute, and the effective horse-power is 

HP 



33,000' 

245. Coefficient of discharge of the pump. Slip. 

The theoretical discharge of a plunger pump is the volume 
displaced by the plunger per stroke multiplied by the number of 
delivery strokes per minute. 

The actual discharge may be greater or less than this amount. 
The ratio of the discharge per stroke to the volume displaced by 
the plunger per stroke is the Coefficient of discharge, and the 
difference between these quantities is called the Slip. 

If the actual discharge is less than the theoretical the slip is 
said to be positive, and if greater, negative. 

Positive slip is due to leakage past the valves and plunger, 
and in a steady working pump, with valves in proper condition, 
should be less than five per cent. 

The causes of negative slip and the conditions under which it 
takes place will be discussed later*. 

* See page 461. 



RECIPROCATING PUMPS 



443 



246. Diagram of work done by the pump. 

Theoretical Diagram. Let a diagram be drawn, Fig. 302, the 
ordinates representing the pressure in the cylinder and the abscissae 
the corresponding volume displacements of the plunger. The 
volumes will clearly be proportional to the displacement of the 
plunger from the end of its stroke. During the suction stroke, 
on the assumption made above that the plunger moves very 
slowly and that therefore all frictional resistances, and also the 
inertia forces, may be neglected, the absolute pressure behind the 
plunger is constant and equal to H - h feet of water, or 62'4 (H h) 
pounds per square foot, and on the delivery stroke the pressure is 

(2\ 
Z + H + 2^- J pounds per square foot. 

The effective work done per suction stroke is ABCD which equals 
62*4 . h . V, and during the delivery stroke is EADF which equals 

iCtA 1 T7 'M'd \ 

62 4 Z + ^ ) , 
\ 2g / ' 

and EBCF is the work done per cycle, that is, during one suction 
and one delivery stroke. 

E F 



so 


f 




40 


3j 






* ' 




30 


Z 




20- 
AtTTL 


A i B 


Freesu 


$ 


t | 


c 


J"f r *f 


a 


i Tie 





SccuLe, of 



Fig. 302. Theoretical diagram of pressure in a Eeciprocating Pump. 



Strokes per 



Fig. 303. 

Actual diagram. Fig. 303 shows an actual diagram taken by 
means of an indicator from a single acting pump, when running 
at 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 suction stroke, 
pq in feet of water is equal to h plus the losses in the suction 
pipe, including loss at the valve, plus the head required to 
accelerate the water in the suction pipe, and qr is the head 
required to lift the water and overcome all losses, and to accelerate 
the water in the delivery pipe. The velocity of the plunger being 
small, these correcting quantities are practically inappreciable. 

The area of this diagram represents the actual work done on 
the water per cycle, and is equal to W (Z + h), together with the 
head due to velocity of discharge and all losses of energy in the 
suction and delivery pipes. 

It will be seen later that although at any instant the pressure 
in the cylinder is effected by the inertia forces, the total work 
done in accelerating the water is zero. 

247. The accelerations of the pump plunger and of the 
water in the suction pipe. 

The theoretical diagram, Fig. 302, has been drawn on the 
assumption that the velocity of the plunger is very small and 
without reference to the variation of 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 delivery pipes. To realise how the velocity and acceleration 
of the plunger vary, suppose it to be driven by a crank and 
connecting rod, as in Fig. 304, and suppose the crank rotates with 
a uniform angular velocity of w radians per second. 




Fig. 304. 

If r is the radius of the crank in feet, the velocity of the crank 
pin is V = wr feet per second. For any crank position OC, it is 
proved in books on mechanism, that the velocity of the point B is 



By making BD equal to OK a diagram of velocities 



Y.OK 

00 
EDF is found. 

When OB is very long compared with CO, OK is equal to 
00 sin 0, and the velocity v of the plunger is then Ysin#, 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. 304, and there is to be 
continuity 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 be 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 : . 




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 = o>V (cos 



j-cos 20) . 



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 0, 

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 



v and the acceleration 
a 



F.A 



* See Balancing of Engines, W. E. Dalby. 



44C HYDRAULICS 

248. The effect of acceleration of the plunger on the 
pressure in the cylinder during the suction stroke. 

When the velocity of the plunger is increasing, F is positive, 
and to accelerate the water in the suction pipe a force P is 
required. The atmospheric pressure has, therefore, not only to 
lift the water and overcome the resistance in the suction pipe, 
but it has also to provide the necessary force to accelerate the 
water, and the pressure in the cylinder is consequently diminished. 

On the other hand, as the velocity of the plunger decreases, 
F is negative, and the piston has to exert a reaction upon the 
water to diminish its velocity, or the pressure on the plunger is 
increased. 

Let L be the length of the suction pipe in feet, a its cross- 
sectional area in square feet, f a the acceleration of the water in 
the pipe at any instant in feet per second per second, and w the 
weight of a cubic foot of water. 

Then the mass of water in the pipe to be accelerated is w . a . L 
pounds, and since by Newton's second law of motion 
accelerating force = mass x acceleration, 
the accelerating force required is 



The pressure per unit area is 

f-s^./.n. 

and the equivalent head of water is 

, _L 

9 ' "' 

, F.A 

or since f a = , 



g.a 
This may be large if any one of the three quantities, L, , or 

B 
F is large. 

Neglecting friction and other losses the pressure in the 
cylinder is now 

H Ji h a , 

and the head resisting the motion of the piston is h + h a . 

249. Pressure in the cylinder during the suction stroke 
when the plunger moves with simple harmonic motion. 

If the plunger be supposed driven by a crank and very long 



RECIPROCATING PUMPS 



447 



connecting rod, the crank rotating uniformly with angular velocity 
u> radians per second, for any crank displacement 0, 

F = w 2 r cos 0, 



and 



, L.A.<o 2 r 

tla = . COS 



^6 

The pressure in the cylinder is 

TT 7 L AwV cos 

ga 

When is zero, cos is unity, and when is 90 degrees, cos 
is zero. For values of between 90 and 180 degrees, cos# is 
negative. 

The variation of the pressure in the cylinder is seen in 
Fig. 306, which has been drawn for the following data. 



G 

A 
Ef 



ALPr. 




B 



Fig. 306. 



Diameter of suction pipe 3J inches, length 12 feet 6 inches. 
Diameter of plunger 4 inches, length of stroke 7| inches. 

Number of strokes per minute 136. Height of the centre of 
the pump above the water in the sump, 8 feet. The plunger is 
assumed to have simple harmonic motion. 

The plunger, since its motion is simple harmonic, may be 
supposed to be driven by a crank 3f inches long, making 68 revo- 
lutions per minute, and a very long connecting rod. 

The angular velocity of the crank is 

27T.68 h -. 1 -,. , 

w = =71 radians per second. 

The acceleration at the ends of the stroke is 

E2 M I 7*12 v /VQ1O 
= o> . r = / 1 x (j 6\.i 

- 15*7 feet per sec. per sec., 



and 



12-5. 15-7. 1-63 
32 



10 feet. 



448 HYDRAULICS 

The pressure in the cylinder neglecting the water in the 
cylinder at the beginning of the stroke is, therefore, 

34 -(10 + 8) =16 feet, 

and at the end it is 34-8+ 10-36 feet. That is, it is greater 
than the atmospheric pressure. 

When is 90 degrees, cos is zero, and h a is therefore zero, 
and when is greater than 90 degrees, cos is negative. 

The area AEDF is clearly equal to GADH, and the work done 
per suction stroke is, therefore, not altered by the accelerating 
forces; but the rate at which the plunger is working at various 
points in the stroke is affected by them, and the force required to 
move the plunger may be very much increased. 

In the above example, for instance, the force necessary to 
move the piston at the commencement of the stroke has been 
more than doubled by the accelerating force, and instead of 
remaining constant and equal to '43. 8. A during the stroke, it 
varies from 

P = -43 (8 + 10) A 
to P = '43 (8 -10) A. 

Air vessels. In quick running pumps, or when the length 
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 violent hammer actions may be set up. To reduce 
the effects of the accelerating forces, air vessels are put on the 
suction and 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 pipe, but, if no 
Y/air vessel is provided, it has also to accelerate the water in the 
cylinder and the delivery pipe. Let D be the diameter, a x 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 

ha == - * 

Wi 

and neglecting head lost by friction etc., and the water in the 
cylinder, the head resisting motion is 



U d * 



If F is negative, h a is also negative. 



RECIPROCATING PUMPS 



449 



When the plunger moves with simple harmonic motion the 
diagram is as shown in Fig. 307, which is drawn for the same 
data as for Fig. 306, taking Z as 20 feet, LI as 30 feet, and the 
diameter D as 3J inches. 




Fig. 307. 

The total work done on the water in the cylinder is NJKM, 
which is clearly equal to HJKL. If the atmospheric pressure is 
acting on the outer end of the plunger, as in Fig. 301, the nett 
work done on the plunger will be SNRMT, which equals HSTL. 

251. Variation of pressure in the cylinder due to friction 
when there is no air vessel. 

Head lost by friction in the suction and delivery pipes. If v is 
the velocity of the plunger at any instant during the suction 
stroke, d the diameter, and a the area of the suction pipe, the 
velocity of the water in the pipe, when there is no air vessel, is 

, and the head lost by friction at that velocity is 
a 



2gda* ' 

Similarly, if Oi, D, and L x are the area, diameter and length 
respectively of the delivery pipe, the head lost by friction, when 
the plunger is making the delivery stroke and has a velocity v, is 



When the plunger moves with simple harmonic motion, 
v = wr sin 0, 



and 



L. H. 



29 



450 



HYDRAULICS 



If the pump makes n strokes per second, or the number of 

revolutions of the crank is ~ per second, and I* is the length of 

& 

the stroke, 

iD = 7rn, 

and I, = 2r. 

Substituting for <*> and r, 



Plotting values of h f at various points along the stroke, the 
parabolic curve EMF, Fig. 308, is obtained. 

When is 90 degrees, sin# is unity, and h f is a maximum. 
The mean ordinate of the parabola, which is the mean frictional 
head, is then 

2 /AVVLZ,' 
' 



E 



3 2gda? 
M~~"~ 



Fig. 308. 

and since the mean frictional head is equal to the energy lost per 
pound of water, the work done per stroke by friction is 



all dimensions being in feet. 




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 
there is no air vessel on the suction pipe, is 



d* 

The pressure in the cylinder for any position of the plunger 
during the suction stroke is now, Fig. 309, 

ho = H h h a h/. 

At the ends of the stroke h/ is zero, and for simple harmonic 
motion h a is zero at the middle of the stroke. 

The work done per suction stroke is equal to the area 
AEMFD, which equals 

ARSD + EMF = 62-4W + 
Similarly, during the delivery stroke the work done is 



The friction diagram is HKGr, Fig. 309, and the resultant 
diagram of total work done during the two strokes is EMFGrKH. 

252. Air vessel on the suction pipe. 

As remarked above, in quick running pumps, or when the 
lengths of the pipes are long, the effects of the accelerating forces 
become serious, and air vessels are put on the suction and delivery 
pipes, as shown in Figs. 310 and 311. By this means the velocity ^ 
in the part of the suction pipe between the well and the air 
vessel is practically kept constant, the water, which has its 
velocity continually changing as the velocity of the piston \ 
changes, 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, 
and the friction head also remains nearly constant. 

Let Z t be the length of the pipe between the air vessel and 
the cylinder, I the length from the well to the air vessel, a the 
cross-sectional area of each of the pipes and d the diameter of the 
pipe. 

Let h v be the pressure head in the air vessel and let the air 
vessel be of such a size that the variation of the pressure may for 
simplicity be assumed negligible. 

Suppose now that water flows from the well up the pipe AB 
continuously and at a uniform velocity. The pump being single 
acting, while the crank makes one revolution, the quantity of 
water which flows along AB must be equal to the volume the 
plunger displaces per stroke. 

292 



452 



HYDRAULICS 



The time for the crank to make one revolution is 

27T 

t = sees., 

therefore, the mean velocity of flow is 

_ A 2rw_ Awr 

(For a double acting pump v m = . ) 

\ a TT / 

During the delivery stroke, all the water is entering the air 
vessel, the water in the pipe BC being at rest. 




Fig. 310. 

Then by Bernoulli's theorem, including friction and the velocity 
head, other losses being neglected, the atmospheric head 

H = x lh , A 2 coV t 4/AWZ a) 

The third and fourth quantities of the right-hand part of the 
equation will generally be very small and h v is practically equal 
to H-OJ. 

When the suction stroke is taking place, the water in the pipe 
BC has to be accelerated. 

Let H B be the pressure head at the point B, when the velocity 
of the plunger is v feet per second, and the acceleration F feet per 
second per second. 



RECIPROCATING PUMPS 453 

Let hf be the loss of head by friction in AB, and h/ the loss in 
BC. The velocity of flow along BC is , and the velocity of 
flow from the air vessel is, therefore, 

v . A Awr 

__ ^ 

a no, 

Then considering the pipe AB, 

H , AW 

HB ~ ~ h ~2^~ hf > 

and from consideration of the pressures above B, 

/vA. A "~^ 2 



7TO, J 



Neglecting losses at the valve, the pressure in the cylinder is 
then approximately 



, AW 
fl TV 5 s 



Neglecting the small quantity 



For a plunger moving with simple harmonic motion 



By putting the air vessel near to the cylinder, thus making 
Z t small, the acceleration head becomes very small and 

h Q = TL h-hf nearly, 
and for simple harmonic motion 

j, -H 7, 4 > VA ' * 
^- n - h --^NT^- 

The mean velocity in the suction pipe can very readily be 
determined 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 

constant v m = and the friction head is 



454 



HYDRAULICS 



Wlien the pump is single acting and there are n strokes per 
second. 



and therefore, 



and 



A . l s . n 



. 



If the pump is double acting, 

h = /AWZ 

ga?d 

For the same length of suction pipe the mean friction head, 
when there is no air vessel and the pump is single acting, is -7r 2 
times the friction head when there is an air vessel. 

253. Air vessel on the delivery pipe. 

An air vessel on the delivery pipe serves the same purpose 
as on the suction pipe, in diminishing the mass of water which 
changes its velocity as the piston velocity changes. 




Fig. 311. 

*As the delivery pipe is generally much longer than the suction 
pipe, the changes in pressure due to acceleration may be much 
greater, and it accordingly becomes increasingly desirable to 
provide an air vessel. 

Assume the air vessel so large that the pressure head. in it 
remains practically constant. 



RECIPROCATING PUMPS 455 

Let Z 2 , Fig. 311, be the length of the pipe between the pump and 
the air vessel, Id be the length of the whole pipe, and i and D the 
area and diameter respectively of the pipe. 

Let hz be the height of the surface of the water in the air vessel 
above the centre of the pipe at B, and let H be the pressure head 
in the air vessel. On the assumption that Ht, 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 



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 w + 7i 2 nearly, therefore, 

-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 H r 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. 

rm, P TT TT L 4/^A 2 F.A.I, 

Therefore, H r 



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 v + 7^ from (1), 



If the pump is single acting and the plunger moves with simple 
harmonic motion and makes n strokes per second, 



and 

a, 



456 HYDRAULICS 

Therefore, 



0,1 g 

Neglecting the friction head in Z 2 and assuming 1 2 small com- 
pared with Idj 

4/rWA 2 Z d AZ 2 , /j 

H r = Z + H + -r. + - w r cos ft 



254. Separation during the suction stroke. 

In reciprocating pumps it is of considerable importance that 
during the stroke no discontinuity of flow shall take place, or 
in other words, no part of the water in the pipe shall separate 
from the remainder, or from the water in the cylinder of the pump. 
Such separation causes excessive shocks in the working parts of 
the pump and tends to broken joints and pipes, due to the hammer 
action caused by the sudden change of momentum of a large mass 
of moving water overtaking the part from which it has become 
separated. 

Consider a section AB of the pipe, Fig. 301, near to the inlet 
valve. For simplicity, neglect the acceleration of the water in the 
cylinder or suppose it to move with the plunger, and let the 
acceleration of the plunger be F feet per second per second. 

If now the water in the pipe is not to be separated from that in 
the cylinder, the acceleration / of the water in the pipe must not 

FA 

be less than - feet per second per second, or separation will not 

FA 
take place as long as - / a . 

FA 
If f a at any instant becomes equal to - , and f a is not to be- 

FA 

come less than , the diminution 8/of / a , when F is diminished 

^ 

by a small amount 9F, must not be less than dF, or in general 

a 

A 

the differential of f a must not be less than times the differential 

of F. 

The general condition for no separation is, therefore, 

fdF<3/ .............................. (1). 

Perhaps a simpler way to look at the question is as follows. 

Let it be supposed that for given data the curve of pressures 
in the cylinder during the suction stroke has been drawn as in 
Fig. 309. In this figure the pressure in the cylinder always remains 
positive, but suppose some part of the curve of pressures EF to 



RECIPROCATING PUMPS 



457 



come below the zero line BC as in Fig. 312*, The pressure in the 
cylinder then becomes negative; but it is impossible for a fluid 
to be in tension and therefore discontinuity in the flow must 
occur t. 

In actual pumps the discontinuity will occur, if the curve EFGr 
falls below the pressure at which the dissolved gases are liberated, 
or the pressure head becomes less than from 4 to 10 feet. 




Fig. 312. 

At the dead centre the pressure in the cylinder just becomes 
zero when h + h a = H, and will become negative when h + h a > H. 
Theoretically, therefore, for no separation at the dead centre, 



- or 



ga 



If separation takes place when the pressure head is less than 
some head /&,,, for no separation, 

li a 2i H h m h, 

and 



a 



I 



Neglecting the water in the cylinder, at any other point in the 
stroke, the pressure is negative when 



FAL + /i + ^A 2 >H 
a g f 2g a? 
And the condition for no separation, therefore, is 



rm,' . T JD^V.U 7 V 

That is, when h + + h f + ~- 



FA 

a 



(2). 



See also Fig. 315, page 459. 

t Surface tension of fluids at rest is not alluded to. 



458 HYDRAULICS 

255. Separation during the suction stroke when the 
plunger moves with simple harmonic motion. 

When the plunger is driven by a crank and very long con- 
necting rod, the acceleration for any crank angle is 

F = co 2 r cos 0, 
or if the pump makes n single strokes per second, 



and F = ^n* . r cos = -^- . 1 8 cos 0, 

l s being the length of the stroke. 

F is a maximum when is zero, and separation will not take 
place at the end of the stroke if 



a L 

and will just not take place when 
A T S A 27 



The minimum area of the suction pipe for no separation is, 
therefore, 



= 



and the maximum number of single strokes per second is 



A.Z..L 

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 formula. 

Example. A single acting pump has a stroke of 1\ inches and the plunger is 
4 inches diameter. The diameter of the suction pipe is 3-J- inches, the length 
.12-5 feet, and the height of the centre of the pump above the water in the well is 
10 feet. 

To find the number of strokes per second at which separation will take place, 
assuming it to do so when the pressure head is zero. 

H- ft = 24 feet, 



and, therefore, * / ^4x24x12 

TT V 1-63 x 7-5x12-5 

~~ 7T ~~ 

= 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 



rounded 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 yielding of the air compressed in the water 
at the end of the delivery stroke. 



Delivery 




Line 



Fig. 315. 



The diagrams Figs. 303 and 313 315, taken from a single-acting 
pump, having a stroke of 7J inches, and a ram 4 inches diameter, 
illustrate the effect of the rounding of the corner in producing 
separation at a less speed than that given by equation (4). 

Even at 59 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 



460 HYDRAULICS 

per minute at the point e in the stroke the available pressure is 
clearly less than ef the head required to lift the water and to 
produce acceleration, and the water lags behind the plunger. 
This condition obtains until the point a is passed, after which 
the water is accelerated at a quicker rate than the piston, and 
finally overtakes it at the point 6, when it strikes the plunger and 
the indicator spring receives an impulse which makes the wave 
form on the diagram. At 230 strokes per minute, the speed being 
greater than that given by the formula when h m is assumed to 
be 10 feet, the separation is very pronounced, and the water does 
not overtake the piston until *7 of the stroke has taken place. It 
is interesting to endeavour to show by calculation that the water 
should overtake the plunger at b. 

While the piston moves from a to b the crank turns through 
70 degrees, in T 6 T 5- . ^$7 seconds = '101 seconds. Between these two 
points the pressure in the cylinder is 2 Ibs. per sq. inch, and 
therefore the head available to lift the water, to overcome all 
resistances and to accelerate the water in the pipe is 29'3 feet. 

The height of the centre of the pump is 6' 3" above the water 
in the sump. The total length of the suction pipe is about 
12'5 feet, and its diameter is 3 inches. 

Assuming the loss of head at the valve and due to friction etc., 
to have a mean value of 2'5 feet, the mean effective head accele- 
rating the water in the pipe is 20*5 feet. The mean acceleration 
is, therefore, 

- 20'5 x 32 Crt K 
f a - o* = 52'5 feet per sec. per sec. 



When the piston is at g the water will be at some distance 
behind the piston. Let this distance be z inches and let the 
velocity of the water be u feet per sec. Then in the time it 
takes the crank to turn through 70 degrees the water will move 
through a distance 

S = Ut + %f a t* 

= 0101tt + J52'5x -0102 feet 
= l'2u + 3'2 inches. 

The horizontal distance ab is 4*2 inches, so that z + 4'2 inches 
should be equal to l'2u + 3*2 inches. 

The distance of the point g from the end of the stroke is 
"84 inch and the time taken by the piston to move from rest to g, 
is 0'058 second. The mean pressure accelerating the water during 
this time is the mean ordinate of akm when plotted on a time 
base ; this is about 5 Ibs. per sq. inch, and the equivalent head is 
12'8 feet. 



RECIPROCATING PUMPS 4G1 

The frictional resistances, which vary with the velocity, will be 
small. Assuming the mean frictional head to be '25 foot, the head 
causing 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, 

- 12-55 x 32 Q0 , 

f a = TOTE = 32 feet per sec. per sec. 

The velocity in the pipe at the end of 0*058 second, should 
therefore be 

v = 32 x -058 = 1*86 feet per sec. 

and the velocity in the cylinder 

1*86 i irk P 

u= T^Q = * ** * ee * P er sec> 



Since the water in the pipe starts from rest the distance it 
should move in 0*058 second is 

12.j32.(*058) 2 =*65in., 
and the distance it should advance in the cylinder is 

0*65 . 

.pgo ins. = *4 in. ; 

so that z is 0*4 in. 

Then z + 4*2 ins. = 4*6, 

and l'2u + 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 coefficient 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 9 



is o>V cos 0, and of the water in the pipe is - cos 0, and therefore 
for no separation at any crank angle 



462 HYDRAULICS 

Putting in the value of h f) and differentiating both sides of the 
equation, and using the result of equation (1), page 456, 



from which aL A (l + -~ \ r cos 0. 

Separation will just not take place if 






Since cos cannot be greater than unity, there is no real 
solution to this equation, unless Ar ( 1 + ~^~j is equal to or 

greater than al. 

4/7 
If, therefore, -4- is supposed equal to zero, and aL the volume 

of the suction pipe is greater than half the volume of the cylinder, 
separation cannot take place if it does not take place at the dead 
centre. 

In actual pumps, aL is not likely to be less than Ar, and 
consequently it is only necessary to consider the condition for no 
separation at the dead centre. 

258. Separation with a large air vessel on the suction pipe. 

To find whether separation will take place with a large air 
vessel on the suction pipe, it is only necessary to substitute in 
equations (2), section 254, and (3), (4), section 255, h v of Fig. 310 
for H, li for L, and hi for h. In Fig. 310, hi is negative. 

For no separation when the plunger is at the end of the stroke 
the minimum area of the pipe between the air vessel and the 
cylinder is 






g v-m-i 
Substituting for h v its value from equation (1), section 253, and 

o V . A . li 



If the velocity and friction heads, in the denominator, be 
neglected as being small compared with (H - h) } then, 



a = 



RECIPROCATING PUMPS 



463 



The maximum number of strokes is 



KH-fe-Ma 
AM, 

A pump can therefore be run at a much greater speed, without 
fear of separation, with an air vessel on the suction pipe, than 
without one. 

259. Separation in the delivery pipe. 

Consider a pipe as shown in Fig. 316, the centre of CD being at 
a height Z above the centre of AB. 

Let the pressure head at D be H , which, when the pipe 
discharges into the atmosphere, becomes H. 

Let Z, Zi and Z 2 be the lengths of AB, BC and CD respectively, 
hf, h/ t and /i/ a the losses of head by friction in these pipes when the 
plunger has a velocity v, and h m the pressure at which separation 
actually takes place. 





/t 


^ 


1 


L DJ 




/ * 


X- 




1 f 










{ 










\ 








^ t> 


\ 

a 








J * 


' 1 

I 










1 










I 










i 


/ 






_- _ 


) 


L Bi 


J 




L' 1 


' 



s' 



Fig. 316. 

Suppose now the velocity of the plunger is diminishing, and its 
retardation is F feet per second per second. If there is to be 

Tjl A 

continuity, the water in the pipe must be also retarded by ^~ 

feet per second per second, and the pressure must always be 
positive and greater than h m . 

Let H c be the pressure at C ; then the head due to acceleration 

in the pipe DC is 

FAZ 2 

ga 
and if the pipe CD is full of water 



which becomes negative when 

FAZ 2 
ga 



ga 



464 HYDRAULICS 

The condition for no separation at C is, therefore, 

w . i > FAZ 2 

Ho - h m + h f - , 

or separation takes place when 

FAZ, -p- , , 
-^" Ho -* + *,. 

At the point B separation will take place if 

^(A >Ho -^ + ^ + A, I+ Z, 
and at the point A if 

tAW > HO + z - h m + n f + n fl + & 



At the dead centre v is zero, and the friction head vanishes. 
For no separation at the point C it is then necessary that 

> FAZ 2 

B -~ hm = ~^ t 
for no separation at B 



and for no separation at A 

_ fc .FAft 



ga 

For given values of H , F and Z, the greater Z 2 , the more likely 
is separation to take place at C, and it is therefore better, for 
a given total length of the discharge pipe, to let the pipe rise near 
the delivery end, as shown by dotted lines, rather than as shown 
by the full lines. 

If separation does not take place at A it clearly will not take 
place at B. 

Example. The retardation of the plunger of a pump at the end of its stroke 
is 8 feet per second per second. The ratio of the area of the delivery pipe to the 
plunger is. 2, and the total length of the delivery pipe is 152 feet. The pipe is 
horizontal for a length of 45 feet, then vertical for 40 feet, then rises 5 feet on 
a slope of 1 vertical to 3 horizontal and is then horizontal, and discharges into 
the atmosphere. Will separation take place on the assumption that the pressure 
head cannot be less than 1 feet ? 

Ans. At the bottom of the sloping pipe the pressure is 
39 feet -|^=5-5 feet. 

(I O-i 

The pressure head is therefore less than 7 feet and separation will take place. 
The student should also find whether there is separation at any other point. 



RECIPROCATING PUMPS 4G5 

260. Diagram of pressure in the cylinder and work done 
during the suction stroke, considering the variable quantity of 
water in the cylinder. 

It is instructive to consider the suction stroke a little more in 
detail. 

Let v and F be the velocity and acceleration respectively of 
the piston at any point in the stroke. 

As the piston moves forward, water will enter the pipe from the 

well and its velocity will therefore be increased from zero to 

j^ 
v.'j the head required to give this velocity is 



On the other hand water that enters the cylinder from the pipe 
is diminished in velocity from - to v, and neglecting any loss due 

to shock or due to contraction at the valve there is a gain of 
pressure head in the cylinder equal to 



The friction head in the pipe is 

4 



The head required to accelerate the water in the pipe is 

^ ........................... >. 

The mass of water to be accelerated in the cylinder 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 - - Ibs. and the force 
required to accelerate it is 

P=^.P, 

and the equivalent head is 

P = p.F 
wA. g 

The total acceleration head is therefore 



F/ LAN 

(x + ), 

g \ a J 



L. n. 30 



466 HYDRAULICS 

Now let Ho be the pressure head in the cylinder, then 
H = H-7i- + - - 4 / LAV _ JV | LA 



(5). 
g a 

When the plunger moves with simple harmonic motion, and is 
driven by a crank of radius r rotating uniformly with angular 
velocity a>, the displacement of the plunger from the end of the 
stroke is r (1 - cos 0), the velocity wr sin & and its acceleration is 
wV cos 0. 

Therefore 

TT _ TT 7, "V sin 2 4/LAV 

J~l o .tL ~~ fl ~ ~ ~~ p: 7 ^~ 

2gr 2srda 2 

L A /, wV 2 cos ^ wV 2 cos 2 



. 
...(6). 

^ g g 

Work done during the suction stroke. Assuming atmospheric 
pressure on the face of the plunger, the pressure per square foot 
resisting its motion is 

(H-H ) w. 

For any small plunger displacement dx, the work done is, 
therefore, 

A(H-Ho).a0, 
and the total work done during the stroke is 

B = ( A (H - Ho) w . dx. 
Jo 

The displacement from the end of the stroke is 

x = r (1 - cos 0), 
and therefore dx = r sin QdQ, 

and E = (*w . A (H - H ) r sin OdO. 

Jo 

Substituting for H its value from equation (6) 



^ / 4/LAW sm 2 o>Vsm 2 

E = w . Ar & + -* , a + 5 

7o 2acZa 2 2a 



2^cZa 2 2^ 

') si 



VooW LA^ 



^ 9 9 a 

The 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 



RECIPROCATING PUMPS 



467 



Or the work done is that required to lift the water through 
a height h together with the work done in overcoming the 
resistance in the pipe. 

Diagrams of pressure in the cylinder and of work done per 
stroke. The resultant pressure in the cylinder, and the head 
resisting the motion of the piston can be represented diagram- 
matically, by plotting curves the ordinates of which are equal to 
Ho and H-Ho as calculated from equations (5) or (6). For 
clearness the diagrams corresponding to each of the parts of 
equation (6) are drawn in Figs. 318 321 and in Fig. 317 is shown 
the combined diagram, any ordinate of which equals 




Fig. 317. 






Figs. 318, 319, 320. Figs. 321, 322. 

In Fig. 318 the ordinate cd is equal to 

4/T.A 2 2 . 2 A 
- ' , 2 o>V sin 2 0, 
2gda? 

and the curve HJK is a parabola, the area of which is 

2 4/LA 2 2 27 



302 



468 HYDRAULICS 

In Fig. 319, the ordinate e/is 

V . 2/} 
"2^ S *' 
and the ordinate gh of Fig. 320 is 

+ cos 2 0. 

g 

The areas of the curves are respectively 
2 <uV t , 1 o>V 

and are therefore equal; and since the ordinates are always of. 
opposite sign the sum of the two areas is zero. 
In Fig. 322, Jem is equal to 

o>V cos 

9 ' 
and Jcl to 

<uV a ( L.A\ 

cos (x + ) . 

g a / 

Since cos is negative between 90 and 180 the area WXY is 
equal to YZU. 

Fig. 321 has for its ordinate at any point of the stroke, the 
head H-H resisting the motion of the piston. 

This equals h + Jcl + cd + efgh, 

and the curve NFS is clearly the curve GFE, inverted. 

The area VNST measured on the proper scale, is the work done 
per stroke, and is equal to VMET + HJK. 

The scale of the diagram can be determined as follows. 

Since h feet of water = 62'4/& Ibs. per square foot, the pressure 
in pounds resisting the motion of the piston at any point in the 
stroke is 

62-4. A. Tilbs. 

If therefore, VNST be measured in square feet the work done 
per stroke in ft.-lbs. 

= 62'4 A. VNST. 

261. Head lost at the suction valve. 

In determining the pressure head H in the cylinder, no account 
has been taken of the head lost due to the sudden enlargement 
from the pipe into the cylinder, or of the more serious loss of head 
due to the water passing through the valve. It is probable that the 

v 2 A 2 
whole of the velocity head, ~ $ , 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 will 



RECIPROCATING PUMPS 469 

also have to give a velocity head g- to any water entering the 

cylinder from the pipe. 

The pressure head H in the cylinder then becomes 

rr H r v* A 2 v* 4/1VA 2 P/ ZA 
= -~"~-- - 



262. Variation of the pressure in hydraulic motors due 
to 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 

"HI A 

of the water in the pipe is ! and the head required to accelerate 

the water in the pipe is 

, FAL 

fl a = - . 

ga 

If p is the pressure per square foot at the inlet end of the 
supply pipe, and h f 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 



When the velocity of the piston is diminishing, F is negative, 
and the inertia of the water in the pipe increases the pressure on 
the 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 
with simple harmonic motion. The centre of the pump is 18 feet above the water 
in 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, 
assuming 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 
by a crank of 7 inches radius and a very long connecting rod, the angular 
velocity of the crank being 27r40 radians per minute. 

The acceleration at the end of the stroke is then 



Therefore, || ^ x 40* x 5 ^=34' - 20', 

from which - = 1'64. 



470 HYDRAULICS 

Therefore ? = 1'28 

a 

and d=9-4". 

Ar is clearly less than al, therefore separation cannot take place at any other 
point iii the stroke. 

Example (2). The pump of example (1) delivers water into a rising main 
1225 feet long and 5 inches diameter, which is fitted with an air vessel. 

The water is lifted through a total height of 220 feet. 

Neglecting all losses except friction in the delivery pipe, determine the horse- 
power required to work the pump. /=-Ol05. 

Since there is an air vessel in the delivery pipe the velocity of flow u will be 
practically uniform. 

Let A and a be the cross-sectional areas of the pump cylinder and pipe respect- 
ively. 

, A.2r.80 D22r.80 

Then > =-6ito * -60- 

12 2 10 80 , 
= 25'T'60 = 9 ' 6 
The head h lost due to friction is 

042 x 9-6 2 x 1225 

.* 

= 176-4 feet. 
The total lift is therefore 

220 + 176-4=396-4 feet. 
The weight of water lifted per minute is 

. i . 80 x 62-5 lbs.=4900 Ibs. 



Therefore, H , 

Example (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 the 
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. 
Therefore, H v + 2' = 207 + 176-4, 

H u =:381-4feet, 

381-4 x 62-5 
P= 144 
= 165 Ibs. per sq. inch. 

Example (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 is 
supplied with water from an accumulator, see Fig. 339, at a constant pressure of 
300 Ibs. 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, 

25. | d 2 = 8-725 cubic feet. 

At the commencement of the stroke the acceleration is v 2 ~ 2 r, and the velocity 
in the supply pipe is zero. 



RECIPROCATING PUMPS 471 

The head required to accelerate the water in the pipe is, therefore, 

_7r 2 .50 2 .1.8 2 .500 
~ 60 2 .2.3 2 .32 

= 380 feet, 
which is equivalent to 165 Ibs. per sq. inch. 

The effective pressure on the piston is therefore 135 Ibs. per sq. inch. 
At the end of the stroke the effective pressure on the piston is 465 Ibs. 
per sq. inch. 

At the middle of the stroke the acceleration is zero and the velocity of the 
piston is 

$ irr=l-31 feet per second. 
The friction head is then 

04. l-BP.S^SOO' 

20. 3*. 
= 108 feet. 
The pressure on the plunger at the middle of the stroke is 

300 Ibs. - . *J =253 Ibs. per sq. inch. 

The mean friction head during the stroke is f . 108 = 72 feet, and the mean loss 
of pressure is 31 '3 Ibs per sq. inch. 

The work lost by friction in the supply pipe per stroke is 31 '3 . j . 8 2 . l t 
= 1570 ft. Ibs. 

The work lost per minute = 39250 ft. Ibs. 

The net work done pei minute neglecting other losses is 

(300 Ibs. -31-3).^. Z,.8 2 .25 

. =337, 700 ft. Ibs., 
and therefore the work lost by friction is about 10*4 per cent, of the energy supplied. 

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 
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 Ibs. 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 



472 



HYDRAULICS 




RECIPROCATING PUMPS 



473 



drawn into the cylinder is -: D 2 . L cubic feet, D being the dia- 
meter of the piston and L the length of the stroke. The quantity 
of water forced into the delivery pipe through the valve VD is 



j (Do 2 -<2 2 )L cubic feet, 

d being the diameter of the small part of the 
plunger. 

On the in-stroke, the suction valve is 
closed and water is forced through the 
delivery valve; part of this water enters 
the delivery pipe and part flows behind the 
piston through the port P. 

The amount that flows into the delivery pipe is 




Fig. 324. 



If, therefore, (D 2 - d 2 ) is made equal to d 2 , or D is */2d, the 
delivery, during each stroke, is ^ Do 2 L cubic feet, and if there are 

n strokes per minute, the delivery is 42'45D 2 Lw gallons per 
minute. 




Fig. 325. Tangye Duplex Pump. 

264. Duplex feed pump. 

Fig. 325 shows a section through one pump and steam cylinder 
of a Tangye double-acting pump. 



474 



HYDRAULICS 



There are two steam cylinders side by side, one of which only 
is shown, and two pump cylinders in line with the steam cylinders. 

In the pump the two lower valves are suction valves and the 
two upper delivery valves. As the pump piston P moves to the 
right, the left-hand lower valve opens and water is drawn into the 
pump from the suction chamber C. During this stroke the right 
upper valve is open, and water is delivered into the delivery d. 
When the piston moves to the left, the water is drawn in through 
the lower right valve and delivered through the upper left valve. 

The steam engine has double ports at each end. As the piston 
approaches the end of its stroke the steam valve, Fig. 326, is at rest 
and covers the steam port 1 while the inner steam port 2 is open 
to exhaust. When the piston passes the steam port 2, the steam 
enclosed in the cylinder acts as a cushion and brings the piston 
and plunger gradually to rest. 





Fig. 326. 



Fig. 327. 



Let the one engine and pump shown in section be called A and 
the other engine and pump, not shown, be called B. 

As the piston of A moves from right to left, the lever L, Figs. 
325 and 327, rotates a spindle to the other end of which is fixed a 
crank M, which moves the valve of the cylinder B from left to 
right and opens the left port of the cylinder B. Just before the 
piston of A reaches the left end of its stroke, the piston of B, 
therefore, commences its stroke from left to right, and by a lever 
LI and crank Mi moves the valve of cylinder A also from left to 
right, and the piston of A can then commence its return stroke. 
It should be noted that while the piston of A is moving, that of 
B is practically at rest, and vice versa. 

265. The hydraulic ram. 

The hydraulic ram is a machine which utilises the momentum 
of a stream of water falling a small height to raise a part of the 
water to a greater height. 

In the arrangement shown in Fig. 328 water is supplied from a 
tank, or stream, through a pipe A into a chamber B, which has two 



PUMPS 



475 



valves V and Vi. When no flow is taking place the valve V falls 
off its seating and the valve YI rests on its seating. If water is 
allowed to flow along the pipe A it will escape through the open 
valve V. The contraction of the jet through the valve opening, 
exactly as in the case of the plate obstructing the flow in a pipe, 
page 168, causes the pressure to be greater on the under face of 
the valve, and when the pressure is sufficiently large the valve 
will commence to c]pse. As it closes the pressure will increase 
and the rate of closing will be continually accelerated. The rapid 
closing of the valve arrests the motion of the water in the pipe, 
and there is a sudden rise in pressure in. B, which causes the 
valve YL to open, and a portion of the water passes into the air 
vessel C. The water in the supply pipe and in the vessel B, after 
being "brought to rest, recoils, like a ball thrown against a wall, 
and the pressure in the vessel- is again diminished, allowing the 
water to once more escape through the valve Y. The cycle of 
operations is thear-repeated, more water being forced into the air 
chamber C, in which the air is compressed, and water is forced up 
the delivery pipe to any desired height. 




Fig. 328. 



Let h be the height the water falls to the ram, H the height to 
which the water is lifted. 

If W Ibs. of water descend the pipe per second, the work 
available per second is Wh foot Ibs., and if e is the efficiency of the 
ram, the weight of water lifted through a height H will be 

e.W.h 



w 



H 



The efficiency e diminishes as H increases and may be taken as 
60 per cent, at high heads. (See Appendix 7.) 

Fig. 329 shows a section through the De Cours hydraulic 
ram, the valves of which are controlled by springs. The springs 



476 



HYDRAULICS 



can be regulated so that the number of beats per minute is com- 
pletely under control, and can be readily adjusted to suit varying 
heads. 

With this type of ram Messrs Bailey claim to have obtained at 
low heads, an efficiency of more than 90 per cent., and with H 
equal to 8h an efficiency of 80 per cent. 




Fig. 329. De Cours Hydraulic Earn. 

As the water escapes through the valve Vi into the air vessel C, 
a little air should be taken with it to maintain the air pressure in 
C constant. 

This is effected in the De Cours ram by allowing the end of the 
exhaust pipe F to be under water. At each closing of the valve 



PUMPS 



477 



V, the siphon action of the water escaping from the discharge 
causes air to be drawn in past the spindle of the valve. A cushion 
of air is thus formed in the box B every stroke, and some of this 
air is carried into C when the valve Vi opens. 

The extreme simplicity of the hydraulic ram, together with 
the ease with which it can be adjusted to work with varying 
quantities of water, render it particularly suitable for pumping 
in out-of-the-way places, and for supplying water, for fountains 
and domestic purposes, to country houses situated near a stream. 

266. Lifting water by compressed air. 

A very simple method of raising water from deep wells is by 
means of compressed air. A delivery pipe is sunk into a well, 
the open end of the pipe being placed at a considerable distance 
below the surface of the water in the well. 



AirTuUbe 




( s 'Wai4 



-*&m 




Fig. 330. 



Fig. 331. 



In the arrangement shown in Fig. 330, there is surrounding the 
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. 331. 

Whether the air acts as a piston and pushes the water in front 
of it, or forms a mixture with the water, according to Kelly*, 
depends very largely upon the rate at which air is supplied to the 
pump. 

In the pump experimented upon by Kelly, at certain rates of 

* Proc. Inst. C. E. Vol. CLXIII. 



478 HYDRAULICS 

working the discharge was continuous, 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 commenced 
slowly; the velocity then gradually increased until the pipe 
discharged full bore; this was followed by a rush of air, after 
which 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 was further diminished, the water rose 
up the delivery tube, but not sufficiently high to overflow, and the 
air escaped without doing useful work. 

The efficiency of these pumps is very low and only in exceptional 
cases does it reach 50 per cent. The volume v of air, in cubic feet, 
at atmospheric pressure, required to lift one cubic foot of water 
through a height h depends upon the efficiency. With an ef- 
ficiency of 30 per cent, it is approximately v = o7T, and with an 

zu 

efficiency of 40 per cent, v = ~v approximately. 

zo 

It is necessary that the lower end of the delivery be at a greater 
distance below the surface of the water in the well, than the height 
of the lift above the free surface, and the well has consequently to 
be made very deep. 

On the other hand the well is much smaller in diameter than 
would be required for reciprocating or centrifugal pumps, and the 
initial cost of constructing the well per foot length is considerably 



EXAMPLES. 

(1) Find the horse-power required to raise 100 cubic feet of water per 
minute to a height of 125 feet, by a pump whose efficiency is ^. 

(2) A centrifugal pump has an inner radius of 4 inches and an outer 
radius of 12 inches. The angle the blade makes with the direction of 
motion at exit is 153 degrees. The wheel makes 545 revolutions per minute. 

The discharge of the pump is 3 cubic feet per second. The sides of the 
wheel are parallel and 2 inches apart. 

Determine the inclination of the tip of the blades at inlet so that there 
shall be no shock, the velocity with which the water leaves the wheel, and 
the theoretical lift. If the head due to the velocity with which the water 
leaves the wheel is lost, find the theoretical lift. 

(3) A centrifugal pump wheel has a diameter of 7 inches and makes 
1358 revolutions per minute. 

The blades are formed so that the water enters and leaves the wheel 
without shock and the blades are radial at exit. The water is lifted by the 
pump 29'4 feet. Find the manometric efficiency of the pump. 



PUMPS 479 

(4) A centrifugal pump wheel 11 inches diameter which runs at 1203 
revolutions per minute is surrounded by a vortex chamber 22 inches 
diameter, and has radial blades at exit. The pressure head at the circum- 
ference of the wheel is 23 feet. The water is lifted to a height of 43'5 
feet above the centre of the pump. Find the efficiency of the whirlpool 
chamber. 

(5) The radial velocity of flow through a pump is 5 feet per second, and 
the velocity of the outer periphery is 60 feet per second. 

The angle the tangent to the blade at outlet makes with the direction 
of motion is 120 degrees. Determine the pressure head and velocity head 
where the water leaves the wheel, assuming the pressure head in the eye 
of the wheel is atmospheric, and thus determine the theoretical lift. 

(6) A centrifugal pump with vanes curved back has an outer radius of 
10 inches and an inlet radius of 4 inches, the tangents to the vanes at outlet 
being inclined at 40 to the tangent at the outer periphery. The section of 
the wheel is such that the radial velocity of flow is constant, 5 feet per 
second ; and it runs at 700 revolutions per minute. 

Determine : 

(1) the angle of the vane at inlet so that there shall be no shock, 

(2) the theoretical lift of the pump, 

(3) the velocity head of the water as it leaves the wheel. Lond. 
Un. 1906. 

(7) A centrifugal pump 4 feet diameter running at 200 revolutions per 
minute, pumps 5000 tons of water from a dock in 45 minutes, the mean 
lift being 20 feet. The area through the wheel periphery is 1200 square 
inches and the angle of the vanes at outlet is 26. Determine the hydraulic 
efficiency and estimate the average horse-power. Find also the lowest 
speed to start pumping against the head of 20 feet, the inner radius being 
half the outer. Lond. Un. 1906. 

(8) A centrifugal pump, delivery 1500 gallons per minute with a lift of 
25 feet, has an outer diameter of 16 inches, and the vane angle is 30. All 
the kinetic energy at discharge is lost, and is equivalent to 50 per cent, of 
the actual lift. Find the revolutions per minute and the breadth at the 
inlet, the velocity of whirl being half the velocity of the wheel. Lond. 
Un. 1906. 

(9) A centrifugal pump has a rotor 19^ inches diameter ; the width of 
the outer periphery is 3 T 7 g- inches. Using formula (1), section 236, deter- 
mine the discharge of the pump when the head is 30 feet and Vi is 50. 

(10) The angle $ at the outlet of the pump of question (9) is 13. 
Find the velocity with which the water leaves the wheel, and the 

minimum proportion of the velocity head that must be converted into work, 
if the other losses are 15 per cent, and the total efficiency 70 per cent. 

(11) The inner diameter of a centrifugal pump is 12^ inches, the outer 
diameter 21 f inches. The width of the wheel at outlet is 3| inches. Using 
equation (2), section 236, find the discharge of the pump when the head is 
21'5 feet, and the number of revolutions per minute is 440. 



480 HYDRAULICS 

(12) The efficiency of a centrifugal pump when running at 550 revolu- 
tions per minute is 70 per cent. The mean angle the tip of the vane makes 
with the direction of motion of the inlet edge of the vane is 99 degrees. 
The angle the tip of the vane makes with the direction of motion of the 
edge of the vane at exit is 167 degrees. The radial velocity of flow is 3'6 
feet per second. The internal diameter of the wheel is 11^ inches and the 
external diameter 19^ inches. 

Find the kinetic energy of the water when it leaves the wheel. 

Assuming that 5 per cent, of the energy is lost by friction, and that one- 
half of the kinetic energy at exit is lost, find the head lost at inlet when the 
lift is 30 feet. Hence find the probable velocity impressed on the water as 
it enters the wheel. 

(13) Describe a forced vortex, and sketch the form of the free surface 
when the angular velocity is constant. 

In a centrifugal pump revolving horizontally under water, the diameter 
of the inside of the paddles is 1 foot, and of the outside 2 feet, and the 
pump revolves at 400 revolutions per minute. Find approximately how 
high the water would be lifted above the tail water level. 

(14) Explain the action of a centrifugal pump, and deduce an expression 
for its efficiency. If such a pump were required to deliver 1000 gallons an 
hour to a height of 20 feet, how would you design it ? Lond. Un. 1903. 

(15) Find the speed of rotation of a wheel of a centrifugal pump which 
is required to lift 200 tons of water 5 feet high in one minute ; having given 
the efficiency is 0'6. The velocity of flow through the wheel is 4'5 feet per 
second, and the vanes are curved backward so that the angle between their 
directions and a tangent to the circumference is 20 degrees. Lond. Un. 
1905. 

(16) A centrifugal pump is required to lift 2000 gallons of water per 
minute through 20 feet. The velocity of flow through the wheel is 7 feet 
per second and the efficiency 0'6. The angle the tip of the vane at outlet 
makes with the direction of motion is 150 degrees. The outer radius of the 
wheel is twice the inner. Determine the dimensions of the wheel. 

(17) A double-acting plunger pump has a piston 6 inches diameter 
and the length of the strokes is 12 inches. The gross head is 500 feet, 
and the pump makes 80 strokes per minute. Assuming no slip, find the 
discharge and horse-power of the pump. Find also the necessary diameter 
for the steam cylinder of an engine driving the pump direct, assuming the 
steam pressure is 100 Ibs. per square inch, and the mechanical efficiency 
of the combination is 85 per cent. 

(18) A plunger pump is placed above a tank containing water at a 
temperature of 200 F. The weight of the suction valve is 2 Ibs. and its 
diameter 1 inches. Find the maximum height above the tank at which 
the pump may be placed so that it will draw water, the barometer standing 
at 30 inches and the pump being assumed perfect and without clearance. 
(The vapour tension of water at 200 F. is about 11*6 Ibs. per sq. inch.) 

(19) A pump cylinder is 8 inches diameter and the stroke of the plunger 
is one foot. Calculate the maximum velocity, and the acceleration of the 



PUMPS 481 

water in the suction and delivery pipes, assuming their respective diameters 
to be 7 inches and 5 inches, the motion of the piston to be simple harmonic, 
and the piston to make 36 strokes per minute. 

(20) Taking the data of question (19) calculate the work done on the 
suction stroke of the pump, 

(1) neglecting the friction in the suction pipe, 

(2) including the friction in the suction pipe and assuming that the 

suction pipe is 25 feet long and that /= 0*01. 

The height of the centre of the pump above the water in the sump is 
18 feet. 

(21) If the pump in question (20) delivers into a rising main against 
a head of 120 feet, and if the length of the main itself is 250 feet, 
find the total work done per revolution. Assuming the pump to be double 
acting, find the i. H. p. required to drive the pump, the efficiency being '72 
and no slip in the pump. Find the delivery of the pump, assuming a slip 
of 5 per cent. 

(22) The piston of a pump moves with simple harmonic motion, and it 
is driven at 40 strokes per minute. The stroke is one foot. The suction 
pipe is 25 feet long, and the suction valve is 19 feet above the surface of the 
water in the sump. Find the ratio between the diameter of the suction 
pipe and the pump cylinder, so that no separation may take place at the 
dead points. Water barometer 34 feet. 

(23) Two double-acting pumps deliver water into a main without an 
air vessel. Each is driven by an engine with a fly-wheel heavy enough to 
keep the speed of rotation uniform, and the connecting rods are very long. 

Let Q be the mean delivery of the pumps per second, Q x the quantity of 
water in the main. Find the pressure due to acceleration (a) at the begin- 
ning of a stroke when one pump is delivering water, (5) at the beginning 
of the stroke of one of two double-acting pumps driven by cranks at right 
angles when both are delivering. When is the acceleration zero ? 

(24) A double-acting horizontal pump has a piston 6 inches diameter 
(the diameter of the piston rod is neglected) and the stroke is one foot. 
The water is pumped to a height of 250 feet along a delivery pipe 450 feet 
long and 4 inches diameter. An air vessel is put on the delivery pipe 
10 feet from the delivery valve. 

Find the pressure on the pump piston at the two ends of the stroke 
when the pump is making 40 strokes per minute, assuming the piston 
moves with simple harmonic motion and compare these pressures with the 
pressures when there is no air vessel. /='0075. 

(25) A single acting hydraulic motor makes 160 strokes per minute and 
moves with simple harmonic motion. 

The motor is supplied with water from an accumulator in which the 
pressure is maintained at 200 Ibs. per square inch. 

The cylinder is 8 inches diameter and 12 inches stroke. The delivery 
pipe is 200 feet long, and the coefficient, which includes loss at bends, etc. 
may be taken as /= 0'2. 

L. H. 31 



482 HYDRAULICS 

Neglecting the mass of the reciprocating parts and of the variable 
quantity of water in the cylinder, draw a curve of effective pressure on the 
piston. 

(26) The suction pipe of a plunger pump is 35 feet long and 4 inches 
diameter, the diameter of the plunger is 6 inches and the stroke 1 foot. 

The delivery pipe is 2| inches diameter, 90 feet long, and the head at 
the delivery valve is 40 feet. There is no air vessel on the pump. The 
centre of the pump is 12 feet 6 inches above the level of the water in the 
sump. 

Assuming the plunger moves with simple harmonic motion and makes 
50 strokes per minute, draw the theoretical diagram for the pump. 

Neglect the effect of the variable quantity of water in the cylinder and 
the loss of head at the valves. 

(27) Will separation take place anywhere in the delivery pipe of the 
pump, the data of which is given in question (26), if the pipe first runs 
horizontally for 50 feet and then vertically for 40, or rises 40 feet im- 
mediately from the pump and then runs horizontally for 50 feet, and 
separation takes place when the pressure head falls below 5 feet ? 

(28) A pump has three single-acting plungers 29|- inches diameter 
driven by cranks at 120 degrees with each other. The stroke is 5 feet and 
the number of strokes per minute 40. The suction is 16 feet and the length 
of the suction pipe is 22 feet. The delivery pipe is 3 feet diameter and 
350 feet long. The head at the delivery valve is 214 feet. 

Find (a) the minimum diameter of the suction pipe so that there is no 
separation, assuming no air vessel and that separation takes place when 
the pressure becomes zero. 

(6) The horse-power of the pump when there is an air vessel on the 
delivery very near to the pump. /= -007. 

[The student should draw out three cosine curves differing in phase by 
120 degrees. Then remembering that the pump is single acting, the 
resultant curve of accelerations will be found to have maximum positive 

and also negative values of o~~~ every 60 degrees. The maximum 

i j-i -, AI T , o)V . AL 
acceleration head is then h a = - 



2ga 

47rVLA ~| 
For no separation, therefore, a = - . 

I8g (34-10) J 



(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 feet 
long and it is fitted with an air vessel 8 feet from the pump cylinder. The 
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 piston 
at the beginning and middle of its stroke and the horse-power of the pump 
when it makes 80 strokes per minute. Neglect the effect of the variable 
quantity of water in the cylinder. Lond. Un. 1906. 



PUMPS 483 

(30) The plunger of a pump moves with simple harmonic motion. 
Find the condition that separation shall not take place on the suction 
stroke and show why the speed of the pump may be increased if an air 
vessel is put in the suction pipe. Sketch an indicator diagram showing 
separation. Explain " negative slip." Lond. Un. 1906. 

(31) In a single-acting force pump, the diameter of the plunger is 
4 inches, stroke 6 inches, length of suction pipe 63 feet, diameter of suction 
pipe 2 1 inches, suction head 0'07 ft. When going at 10 revolutions per 
minute, it is found that the average loss of head per stroke between the 
suction tank and plunger cylinder is 0*23 ft. Assuming that the frictional 
losses vary as the square of the speed, find the absolute head on the suction 
side of the plunger at the two ends and at the middle of the stroke, the 
revolutions being 50 per minute, and the barometric head 34 feet. Draw a 
diagram of pressures on the plunger simple harmonic motion being 
assumed. Lond. Un. 1906. 

(82) A single-acting pump without an air vessel has a stroke of 
7 inches. The diameter of the plunger is 4 inches and of the suction 
pipe 3 inches. The length of the suction pipe is 12 feet, and the centre 
of the pump is 9 feet above the level in the sump. 

Determine the number of single strokes per second at which theoreti- 
cally separation will take place, and explain why separation will actually 
take place when the number of strokes is less than the calculated value. 

(33) Explain carefully the use of an air vessel in the delivery pipe of a 
pump. The pump of question (32) makes 100 single strokes per minute, 
and delivers water to a height of 100 feet above the water in the well 
through a delivery pipe 1000 feet long and 2 inches diameter. Large air 
vessels being put on the suction and delivery pipes near to the pump. 

On the assumption that all losses of head other than by friction in 
the delivery pipe are neglected, determine the horse-power of the pump. 
There is no slip. 

(34) A pump plunger has an acceleration of 8 feet per second per 
second when at the end of the stroke, and the sectional area of the plunger 
is twice the sectional area of the delivery pipe. The delivery pipe is 152 
feet long. It runs from the pump horizontally for a length of 45 feet, then 
vertically for 40 feet, then rises 5 feet, on a slope of 1 vertical to 3 hori- 
zontal, and finally runs in a horizontal direction. 

Find whether separation will take place, and if so at which section 
of the pipe, if it be assumed that separation takes place when the pressure 
head in the pipe becomes 7 feet. 

(35) A pump of the duplex kind, Fig. 325, in which the steam piston is 
connected directly to the pump piston, works against a head of h feet of 
water, the head being supplied by a column of water in the delivery pipe. 
The piston area is A , the plunger area A, the delivery pipe area a, the 
length of the delivery pipe I and the constant steam pressure on the piston 
PQ Ibs. per square foot. The hydraulic resistance may be represented by 

Fv 2 

g , v being the velocity of the plunger and F a coefficient. 

312 



484 HYDRAULICS 

Show that when the plunger has moved a distance x from the beginning 
of the stroke 

O^. /nn A \ TfflV 

Lond. Un. 1906. 

Let the piston be supposed in any position and let it have a velocity v. 
Then the velocity of the plunger is v and the velocity of the water in the 

pipe is ' . The kinetic energy of the water in the pipe at this velocity is 



If now the plunger moves through a distance dx, the work done by the 
steam is p A. Q dx ft. Ibs.; the work done in lifting water is w . h . Ada;; the 



work done by friction is -^ w.A.dx; and, therefore, 




Let =E, =Z and Fw?A=/. 



^ITI 

Then /E + Z - ,- 

dx 

/ cZE 

z E+ dS = 

The solution of this equation is 



~ 

(36) A pump valve of brass has a specific gravity of 8 with a lift of 
J$ foot, the stroke of the piston being 4 feet, the head of water 40 feet and 
the ratio of the full valve area to the piston area one-fifth. 

If the valve is neither assisted nor meets with any resistance to closing, 
find the time it will take to close and the "slip" due to this gradual closing. 

Time to close is given by formula, S = $ft 2 . /= x 32-2. Lond. Un. 1906. 



CHAPTER XL 



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. 333. 



Fig. 834. 



486 



HYDRAULICS 



wfought-iron pipes, no packing being required. In Fig. 333 the 
packing material is a gutta-percha ring. Fig. 336 shows an 
ordinary socket joint for a cast-iron hydraulic main. To make 
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 socket 
with a caulking tool. Fig. 335 shows various forms of packing 
leathers, the applications of which will be seen in the examples 
given of hydraulic machines. 



Neck leather 




Cup leather 
Fig. 335. 




Fig. 336. 



Hemp twine, carefully plaited, and dipped in hot tallow, 
makes a good packing, when used in suitably designed glands 
(see Fig. 339) and is also very suitable for pump buckets, Fig. 323. 
Plaited Asbestos or cotton may be substituted for hemp, and 
metallic packings are also used as shown in Figs. 337 and 338. 




Fig. 337. 



Fig. 338. 



268. The accumulator. 

The accumulator is a device used in connection with hydraulic 
machinery for storing energy. 

In the form generally adopted in practice it consists of a long 
cylinder C, Fig. 339, in which slides a ram R and into which water 
is delivered from pumps. At the top of the ram is fixed a rigid 
cross head which carries, by means of the bolts, a large cylinder 
which can be filled with slag or other heavy material, or it may 
be loaded with cast-iron weights as in Fig. 340. The water is 



HYDRAULIC MACHINES 



487 




Fig. 339. Hydraulic Accumulator. 



488 HYDRAULICS 

admitted to the cylinder at any desired pressures through a pipe 
connected to the cylinder by the flange shown dotted, and the 
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 Ibs. 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 



and the energy that can be stored in the accumulator is 



The principal object of the accumulator is to allow hydraulic 
machines, or lifts, which are being supplied with hydraulic power 
from the pumps, to work for a short time at a much greater rate 
than the pumps can supply energy. If the pumps are connected 
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 by 
friction, etc., and the pump must be of such a capacity as to supply 
energy at the greatest rate required by the machines, and the 
frictional resistances. If the pump supplies water to an accumu- 
lator, it can be kept working at a steady rate, and during the time 
when the demand is less than the pump supply, energy can be 
stored in the accumulator. 

In addition to acting as a storer of energy, the accumulator 
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 varying, 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 pump. 
With an accumulator on the pipe line, unless the ram is 
descending and is suddenly brought to rest, the pressure cannot 
rise very much higher than the pressure p which will lift the ram. 

To start and stop the pump automatically, the ram as it 
approaches the top of its stroke moves a lever connected to 
a chain which is led to a throttle valve on the steam pipe of the 
pumping engine, and thus shuts off steam. On the ram again 
falling below a certain level, it again moves the lever and opens 
the throttle valve. The engine is set in motion, pumping re- 
commences, and the accumulator rises. 



HYDRAULIC MACHINES 489 

Example. A hydraulic crane working at a pressure of 700 Ibs. per sq. inch has 
to lift 30 cwts. at a rate of 200 feet per minute through a height of 50 feet, once 
every 1 minutes. The efficiency of the crane is 70 per cent, and an accumulator 
is provided. 

Find the volume of the cylinder of the crane, the minimum horse-power for the 
pump, and the minimum capacity of the accumulator. 

Let A be the sectional area of the ram of the crane cylinder in sq. feet and L 
the length of the stroke in feet. 

Then, p .144. A. Lx 0-70 = 30 x 112 x 50', 

AT-V- 30x112x50 

"0-70x144x700 
= 2-38 cubic feet. 
The rate of doing work in the lift cylinder ia 



and the work done in lifting 50 feet is 240,000 ft. Ibs. Since this has to be done 
once every one and half minutes, the work the pump must supply in one and half 
minutes is at least 240,000 ft. Ibs. , and the minimum horse-power is 

240.000 

-33,000x1-5 = 

The work done by the pump while the crane is lifting is 
240,000 *0-* 



The energy stored in the accumulator must be, therefore, at least 200,000 ft. Ibs. 
Therefore, if V a is its minimum capacity in cubic feet, 

V a x 700 x 144 = 200,000, 
or V =2 cubic feet nearly. 

269. Differential accumulator*. 

Tweddell's differential accumulator, 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 
with heavy cast-iron weights slides upon the ram, water-tight 
joints being made by means of the cup leathers shown. Water 
is pumped into the cylinder through a pipe, and a passage drilled 
axially along the lower part of the ram. 

Let p be the pressure in Ibs. per sq. inch, d and di the dia- 
meters of the upper and lower parts of the ram respectively. 
The weight lifted (neglecting friction) is then 



and if h is the lift in feet, the energy stored is 

. foot Ibs. 



The difference of the diameters d^ 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 small. 
This is of advantage when being used in connection with 

* Proceedings Inst. Mech. Engs., 1874. 



490 



HYDRAULICS 




Fig. 340. 



311. Hydraulic Intensitier. 



HYDRAULIC MACHINES 491 

hydraulic riveters, as when a demand is made upon the ac- 
cumulator, the ram falls quickly, but is suddenly arrested when 
the ram of the riveter comes to rest, and there is a consequent 
increase in the pressure in the cylinder of the riveter which 
clinches the rivet. Mr Tweddell estimates that when the ac- 
cumulator is allowed to fall suddenly through a distance of from 
18 to 24 inches, the pressure is increased by 50 per cent. 

270. Air accumulator. 

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 air 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 is pumped into 
the vessel, the volume of air is (V v). 

Assuming the temperature remains constant, the pressure pi in 
the vessel will now be 

p.V 

tt-v^V 

If V is the volume of air, and a volume of water v is taken out 

of the vessel, 



271. Intensifiers. 

It is frequently desirable that special machines shall work at 
a higher pressure than is available from the hydraulic mains. To 
increase the pressure to the desired amount the intensifier is used. 

One form is shown in Fig. 341. A large hollow ram works in 
a fixed cylinder C, the ram being made water-tight by means of a 
stuffing-box. Connected to the cylinder by strong bolts is a cross 
head which has a smaller hollow ram projecting from it, and 
entering the larger ram, in the upper part of which is made a 
stuffing-box. Water from the mains is admitted into the large 
cylinder and also into the hollow ram through the pipe and 
the lower valve respectively shown in Fig. 342. 

If p Ibs. per sq. 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 2 pounds, and the pressure inside the hollow ram rises to 

pL pounds per sq. inch, D and d being the external diameters 
oi: the large ram and the small ram respectively. 



492 



HYDRAULICS 



The form of intensifier here shown is used in connection with 
a large flanging press. The cylinder of the press and the upper 
part of the intensifier are filled with water at 700 Ibs. per sq. inch 
and the die brought to the work. Water at the same pressure is 
admitted below the large ram of the intensifier and the pressure 
in the upper part of the intensifier, and thus in the press cylinder, 
rises to 2000 Ibs. per sq. inch, at which pressure the flanging 
is finished. 




To Small 

to/Under 

oflntensifier 



Tb Larqe fyUndef of Intensifier 



Won, Return VaJbves for 
Intensifier. 

Fig. 342. 



ouilOOVbs. 
persq. 



272. Steam intensifies. 

The large cylinder of an intensifier may be supplied with 
steam, instead of water, as in Fig. 343, which shows a steam in- 
tensifier used in conjunction with a hydraulic forging press. These 
intensifies have also been used on board ship* in connection with 
hydraulic steering gears. 

273. Hydraulic forging press, with steam intensifier and 
air accumulator. 

The application of hydraulic power to forging presses is illus- 
trated in Fig. 343. This press is worked in conjunction with a 
steam intensifier and air accumulator to allow of rapid working. 
The whole is controlled by a single lever K, and the press is 
capable of making 80 working strokes per minute. 

When the lever K is in the mid position everything is at rest ; 

on moving the lever partly to the right, steam is admitted into the 

cylinders D of the press through a valve. On moving the lever to 

its extreme position, a finger moves the valve M and admits water 

* Proceedings List. Mech. Engs., 1874. 



HYDRAULIC MACHINES 



493 



under a relay piston shown at the top of the figure, which opens 
a valve E at the top of the air vessel. In small presses the valve 
E is opened by levers. The ram B now ascends at the rate of 




about 1 foot per second, the water in the cylinder c being forced 
into the accumulator. On moving the lever K to the left, as soon 
as it has passed the central position the valve L is opened to 



494 



HYDRAULICS 



exhaust, and water from the air vessel, assisted by gravity, forces 
down the ram B, the velocity acquired being about 2 feet per 
second, until the press head A touches the work. The movement 
of the lever K being continued, a valve situated above the valve 
J is opened, and steam is admitted to the intensifier cylinder H ; 
the valve E closes automatically, and a large pressure is exerted 
on the work under the press head. 

If only a very short stroke is required, the bye-pass valve L is 
temporarily disconnected, so that steam is supplied continuously 
to the lifting cylinders I). The lever K is then simply used to 
admit and exhaust steam from the intensifier H, and no water 
enters or leaves the accumulator. An automatic controlling gear 
is also fitted, which opens the valve J sufficiently early to prevent 
the intensifier from overrunning its proper stroke. 




W/7///M. 

Fig. 346. 



347. 



Fig. 344. Fig. 345. 

274. Hydraulic cranes. 

Fig. 344 shows a section through, and Fig. 345 an elevation 
of, a hydraulic crane cylinder. 



HYDRAULIC MACHINES 495 

One end of a wire rope, or chain, is fixed to a lug L on the 
cylinder, and the rope is then passed alternately round the upper 
and lower pulleys, and finally over the pulley 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 six ropes. The weight lifted is therefore one- 
sixth of the pressure on the ram, but the weight is lifted a distance 
equal to six times the movement of the ram. 

Let po and p be the pressures per sq. inch in the crane valve 
chest and in the cylinder respectively, d the diameter and A the 
area of the ram in inch units, a the area of the valve port, and 
v and Vi the velocities in ft. per sec. of the ram and the water 
through the port respectively. Then 

w vi-v '433^ A 3 



The energy supplied to the crane per cubic foot displacement 
of the ram is 144p ft. Ibs., and the work done on the ram is 
144p ft. Ibs. For a given lift, the number of cubic feet of water 
used is the same whatever the load lifted, and at light loads the 
hydraulic efficiency p/p is consequently small. If there are n/2 
pulleys on the end of the ram, arranged as in Fig. 347, and e is 
the mechanical efficiency of the ram packing and BI of the pulley 
system, the actual weight lifted is 



When the ram is in good condition the efficiency of cup 
leather packings is from '6 to '78, of plaited hemp or asbestos 
from *7 to '85, of cotton from *8 to '96, and the efficiency of each 
pulley is from '95 to '98. When the lift is direct acting n in (2) 
is replaced by unity. To determine the diameter of the ram to 
lift a given load, at a given velocity, with a given service pressure 
>o, the ratio of the ram area to port area must be known so that p 
can be found from (1). If Wi is the load on the ram when the 
crane is running light, the corresponding pressure p l in the 
cylinder can be found from (2), and by substituting in (1), the 
corresponding velocity v z of lifting can be obtained. If the valve 
is to be fully open at all loads, the ratio of the ram area to the 
port area should be fixed so that the velocity v a does not become 
excessive. The ratio of v 2 to v is generally made from 1*5 to 3. 

275. Double power cranes. 

To enable a crane designed for heavy work to lift light loads 
with reasonable efficiency, two lifting rams of different diameters 
are employed, the smaller of which can be used at light loads. 



496 



HYDRAULICS 



A convenient arrangement is as shown in Figs. 348 and 349, 
the smaller ram B/ working inside the large ram R. 

When light loads are to be lifted, the large ram is prevented 
from moving by strong catches C, and the volume of water used 
is only equal to the diameter of the small ram into the length of 
the stroke. For large loads, the catches are released and the 
two rams move together. 




HYDRAULIC MACHINES 



497 



Another arrangement is shown in Fig. 350, water being ad- 
mitted to both faces of the piston when light loads are to be 
lifted, and to the face A only when heavy loads are to be raised. 

For a given stroke s of the ram, the ratio of the energy supplied 
in the first case to that in the second is (D 2 - d 2 )/D\ 




Fig. 350. Armstrong Double-power Hydraulic Crane Cylinder. 

276. Hydraulic crane valves. 

Figs. 351 and 352 show two forms of lifting and lowering 
valves used by Armstrong, Whitworth and Co. for hydraulic 
cranes. 

In the arrangement shown in Fig. 351 there are two inde- 
pendent valves, the one on the left being the pressure, and that 
on the right the exhaust valve. 




Fig. 351. Armstrong-Whitworth 
Hydraulic Crane Valve. 

L. H 



Fig. 352. Armstrong-Whitworth 
Hydraulic Crane Slide Valve. 

32 



In the arrangement shown in Fig. 352 a single D slide valve is 
used. Water enters the valve chest through the pressure passage 
P. The valve is shown in the neutral position. If the valve 
is lowered, the water enters the cylinder, but if it is raised, 
water escapes from the cylinder through the port of the slide 
valve. 

277. Small hydraulic press. Fig. 353 is a section through 
the cylinder of a small hydraulic press, used for testing springs. 

The cast-iron cylinder is fitted with a brass liner, and axially 
with the cylinder a rod P r , having a piston P at the free end, 
is screwed into the liner. The steel ram is hollow, the inner 
cylinder being lined with a brass liner. 

Water is admitted and exhausted from the large cylinder 
through a Luthe valve, fixed to the top of the cylinder and 
operated by the lever A. The small cylinder inside the ram is 
connected directly to the pressure pipe by a hole drilled along the 
rod P r , so that the full pressure of the water is continuously 
exerted upon the small piston P and the annular ring RR. 

Leakage to the main cylinder is prevented by means of a 
gutta-percha ring Gr and a ring leather c, and leakage past the 
steel ram and piston P by cup leathers L and LI. 

When the valve spindle is moved to the right, the port p is 
connected with the exhaust, and the ram is forced up by the 
pressure of the water on the annular ring RR. On moving the 
valve spindle over to the left, pressure water is admitted into the 
cylinder and the ram is forced down. Immediately the pressure 
is released, the ram comes back again. 

Let D be the diameter of the ram, d the diameter of the 
rod P r , di the diameter of the piston P, and p the water pressure 
in pounds per sq. inch in the cylinder. 

The resultant force acting on the ram is 



and the force lifting the ram when pressure is released from the 
main cylinder is, 



The cylindrical valve spindle S has a chamber C cast in it, 
and two rings of six holes in each ring are drilled through 
the external shell of the chamber. These rings of holes are at 
such a distance apart that, when the spindle is moved to the 
right, one ring is opposite to the exhaust and the other opposite 
to the port p, and when the spindle is moved to the left, the holes 



HYDRAULIC MACHINES 



499 



are respectively opposite to the port p and the pressure water 
inlet. 

Leakage past the spindle is prevented by the four ring leathers 
shown. 




Fig. 353. Hydraulic Press with Luthe Valve. 

278. Hydraulic riveter. 

A section through the cylinder and ram of a hydraulic riveter 
is shown in Fig. 354. 

323 



500 



HYDRAULICS 




tfig. 354. Hydraulic Kiveter. 




Sprung 
for closing 



Fig. 355. Valves for Hydraulic Biveter. 



HYDRAULIC MACHINES 



501 



The mode of working is exactly the same as that of the small 
press described in section 277. 

An enlarged section of the valves is shown in Fig. 355. On 
pulling the lever L to the right, the inlet valve Y is opened, and 
pressure water is admitted to the large cylinder, forcing out 
the ram. When the lever is in mid position, both valves are 
closed by the springs S, and on moving the lever to the left, the 
exhaust valve Yi is opened, allowing the water to escape from the 
cylinder. The pressure acting on the annular ring inside the 
large ram then brings back the ram. The methods of preventing 
leakage are clearly shown in the figures. 

279. Hydraulic engines. 

Hydraulic power is admirably adapted for machines having a 
reciprocating motion only, especially in those cases where the load 
is practically constant. 




Fig. 356. Hydraulic Capstan. 



502 



HYDRAULICS 



It has moreover been successfully applied to the driving of 
machines such as capstans and winches in which a reciprocating 
motion is converted into a rotary motion. 

The hydraulic-engine shown in Figs. 356 and 357, has three 
cylinders in one casting, the axes of which meet on the axis of the 
crank shaft S. The motion of the piston P is transmitted to the 
crank pin by short connecting rods R. Water is admitted and 
exhausted through a valve Y, and ports p. 




Fig. 357. 



The face of the valve is as shown in Fig. 358, E being the 
exhaust port connected through the centre of the valve to the 
exhaust pipe, and KM the pressure port, connected to the supply 
chamber H by a small port through the side of the valve. The 
valve seating is generally made of lignum-vitae, and has three 
circular ports as shown dotted in Fig. 358. The valve receives its 
motion from a small auxiliary crank T, revolved by a projection 
from the crank pin Gr. When the piston 1 is at the end of its 
stroke, Fig. 359, the port pi should be just opening to the pressure 
port, and just closing to the exhaust port E. The port p 3 should 
be fully open to pressure and port p 2 fully open to exhaust. 
When the crank has turned through 60 degrees, piston 3 will 



HYDRAULIC MACHINES 



503 



be at the inner end of its stroke, and the edge M of the pressure 
port should be just closing to the port p 3 . At the same instant the 
edge N of the exhaust port should be coincident with the lower 
edge of the port p s . The angles QOM, and LON, therefore, 
should each be 60 degrees. A little lead may be given to the 
valve ports, 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 type 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 inefficient. 
There have been many attempts to overcome this difficulty, 
notably as in the Hastie engine t, and Eigg engine. 

* See text book on Steam Engine. 

f Proceedings Inst. Mech. Engs. , 1874. 



504 



HYDRAULICS 



280. Rigg hydraulic engine. 

To adapt the quantity of water used to the work done, Rigg * 
has modified the three cylinder engine by fixing the crank pin, and 
allowing the cylinders to revolve about it as centre. 

The three pistons PI, P 2 and P 3 are connected to a disc, 
Fig. 360, by three pins. This disc revolves about a fixed centre A. 
The three cylinders rotate about a centre Gr, which is capable of 
being moved nearer or further away from the point A as desired. 
The stroke of the pistons 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. Eigg Hydraulic Engine. 

The alteration of the length of the stroke is effected by means 
of the subsidiary hydraulic engine, shown in Fig. 361. There are 
two cylinders C and Ci, in which slide a hollow double ended 
ram PPi which carries the pin G-, Fig. 360. Cast in one piece with 
the ram is a valve box B. E. is a fixed ram, and through it water 
enters the cylinder Ci, in which the pressure is continuously 
maintained. The difference between the effective areas of P and 
Pa when water is in the two cylinders, is clearly equal to the area 
of the ram head EI. 

* See also Engineer, Vol. LXXXV, 1898. 



HYDRAULIC MACHINES 



ro.5 



From the cylinder Ci the water is led along the passages 
shown to the valve V. On opening this valve high-pressure 
water is admitted to the cylinder C. A second valve similar to 
V, but not shown, is used to regulate the exhaust from the 
cylinder C. When this valve is opened, the ram PPi moves to 
the left and carries with it the pin Gr, Fig. 360. On the exhaust 
being closed and the valve V opened, the full pressure acts upon 
both ends of the ram, and since the effective area of P is greater 
than PI it is moved to the right carrying the pin Gr. If both 
valves are closed, water cannot escape from the cylinder C and 
the ram is locked in position by the pressure on the two ends. 




Water 



Fig. 361. 



EXAMPLES. 

(1) The ram of a hydraulic crane is 7 inches diameter. Water is 
supplied to tlie crane at 700 Ibs. per square inch. By suitable gearing the 
load is lifted 6 times as quickly as the ram. Assuming the total efficiency 
of the crane is 70 per cent., find the weight lifted. 

(2) An accumulator has a stroke of 23 feet ; the diameter of the ram is 
23 inches; the working pressure is 700 Ibs. per square inch. Find the 
capacity of the accumulator in horse-power hours. 

(3) The total weight on the cage of an ammunition hoist is 3250 Ibs. 
The velocity ratio between the cage and the ram is six, and the extra load 
on the cage due to friction may be taken as 30 per cent, of the load on the 
cage. The steady speed of the ram is 6 inches per second and the available 
pressure at the working valve is 700 Ibs. per square inch. 

Estimate the loss of head at the entrance to the ram cylinder, and 
assuming this was to be due to a sudden enlargement in passing through 
the port to the cylinder, estimate, on the usual assumption, the area of the 
port, the ram cylinder being 9g inches diameter. Lond. Un. 1906. 

3250 x 1-3 x 6 
The elective pressure p=* . 



506 HYDRAULICS 

^(700-.p).144 = (i;--5) a 

w 2<7 

v= velocity through the valve. 



Loss of head 



Area of port = . 

(4) Describe, with sketches, some form of hydraulic accumulator suit- 
able for use in connection with riveting. Explain by the aid of diagrams, 
if possible, the general nature of the curve of pressure on the riveter ram 
during the stroke ; and point out the reasons of the variations. Lond. Un. 
1905. (See sections 262 and 269.) 

(5) Describe with sketches a hydraulic intensifier. 

An intensifier is required to increase the pressure of 700 Ibs. per square 
inch on the mains to 3000 Ibs. per square inch. The stroke of the intensi- 
fier is to be 4 feet and its capacity three gallons. Determine the diameters 
of the rams. Inst. C. E. 1905. 

(6) Sketch in good proportion a section through a differential hydraulic 
accumulator. What load would be necessary to produce a pressure of 1 ton 
per square inch, if the diameters of the two rams are 4 inches and 4^ inches 
respectively ? Neglect the friction of the packing. Give an instance of the 
use of such a machine and state why accumulators are used. 

(7) A Tweddell's differential accumulator is supplying water to riveting 
machines. The diameters of the two rams are 4 inches and 4 inches 
respectively, and the pressure in the accumulator is 1 ton per square inch. 
Suppose when the valve is closed the accumulator is falling at a velocity 
of 5 feet per second, and the time taken to bring it to rest is 2 seconds, find 
the increase in pressure in the pipe. 

(8) A lift weighing 12 tons is worked by water pressure, the pressure 
in the main at the accumulator being 1200 Ibs. per square inch ; the length 
of the supply pipe which is 3 inches in diameter is 900 yards. What is 
the approximate speed of ascent of this lift, on the assumption that the 
friction of the stuffing-box, guides, etc. is equal to 6 per cent, of the gross 
load lifted and the ram is 8 inches diameter ? 

(9) Explain what is meant by the " coefficient of hydraulic resistance " 
as applied to a whole system, and what assumption is usually made regard- 
ing it ? A direct acting lift having a ram 10 inches diameter is supplied 
from an accumulator working under a pressure of 750 Ibs. per square inch. 
When carrying no load the ram moves through a distance of 60 feet, at a 
uniform speed, in one minute, the valves being fully open. Estimate the 
coefficient of hydraulic resistance referred to the velocity of the ram, and 
also how long it would take to move the same distance when the ram 
carries a load of 20 tons. Lond. Un. 1905. 

/r* 750 x 144 \ 

( -^ head lost = ^^ . Assumption is made that resistance varies as v 2 . ) 



CHAPTER XTI. 

RESISTANCE TO THE MOTION OF BODIES IN WATER 

281. Froude's* experiments to determine frictional re- 
sistances of thin boards when propelled in water. 

It has been shown that the frictional resistance to the flow of 
water along pipes is proportional to the velocity raised to some 
power n, which approximates to two, and Mr Froude's classical 
experiments, in connection with the resistance of ships, show that 
the resistance to motion of plane vertical boards when propelled 
in water, follows a similar law. 




Fig. 362. 

The experiments were carried out near Torquay in a parallel 
sided tank 278 feet long, 36 feet broad and 10 feet deep. A light 
railway on "which ran a stout framed truck, suspended from the 
axles of two pairs of wheels," traversed the whole length of the 
tank, about 20 inches above the water level. The truck was pro- 
pelled by an endless wire rope wound on to a barrel, which could 
be made to revolve at varying speeds, so that the truck could 
traverse the length of the tank at any desired velocity between 
100 and 1000 feet per minute. 

* Brit. Ass. Reports, 1872-4. 



508 HYDRAULICS 

Planes of wood, about f'V inch thick, the surfaces of which were 
covered with various 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 upper edge being 
about 1 J inches below the surface. The lengths were varied from 
2 to 50 feet. 

The apparatus as used by Froude is illustrated and described 
in the British Association Reports for 1872. 

A later adaptation of the apparatus as used at Haslar for 
determining the resistance of ships' models is shown in Fig. 362. 
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 fixed 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 recording 
cylinder. This cylinder is made to revolve by means of a worm 
and wheel, the worm being driven by an endless belt from the axle 
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 moved. 
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 S, weights W are hung from a knife 
edge, which is exactly at the same distance from H as the points 
of attachment of L and the spring S. 

From the results of these experiments, Mr Froude made the 
following deductions. 

(1) The frictional resistance varies very nearly with the 
square of the velocity. 

(2) The mean resistance per square foot of surface for lengths 
up to 50 feet diminishes as the length is increased, but is prac- 
tically constant for lengths greater than 50 feet. 

(3) The frictional resistance varies very considerably with 
the roughness of the surface. 

Expressed algebraically the frictional resistance to the motion 
of a plane surface of area A when moving with a velocity v feet 
per second is 



/ being equal to 



RESISTANCE TO THE MOTION OF BODIES IN WATER 



509 



TABLE XXXIX. 

Showing the result of Mr Froude's experiments on the frictional 
resistance to the motion of thin vertical boards towed through 
water in a direction parallel to its plane. 

Width of boards 19 inches, thickness -f$ inch. 

n = power or index of speed to which resistance is approxi- 
mately proportional. 

/ = the mean resistance in pounds per square foot of a surface, 
the length of which is that specified in the heading, when the 
velocity is 10 feet per second. 

/i = the resistance per square foot, at a distance from the 
leading edge of the board, equal to that specified in the heading, 
at a velocity of 10 feet per second. 

As an example, the resistance of the tinfoil surface per square 
foot at 8 feet from the leading edge of the board, at 10 feet per 
second, is estimated at 0'263 pound per square foot; the mean 
resistance is 0'278 pound per square foot. 





Length of planes 




2 feet 


8 feet 


20 feet 


50 feet 


Surface 
covered with 


n 


/ 


/i 


71 


/o 


/i 


n 


/o 


/i 


n 


/o 


/i 


Varnish 


2-0 


0-41 


0-390 


1-85 


0-325 


0-264 


1-85 


0-278 


0-240 


1-83 


0-250 


0-226 


Tinfoil 


2-16 


0-30 


0-295 


1-99 


0-278 


0-263 


1-90 


0-262 


0-244 


1-83 


0-246 


0-232 


Calico 


1-93 


0-87 


0-725 


1-92 


0-626 


0-504 


1-89 


0-531 


0-447 


1-87 


0-474 


0-423 


Fine sand 


2-0 


0-81 


0-690 


2-0 


0-583 


0-450 


2-0 


0-480 


0-384 


2-06 


0-405 


0-337 


Medium sand 


2-0 


0-90 


0-730 


2-0 


0-625 


0-488 


2-0 


0-534 


0-465 


2-00 


0-488 


0-456 


Coarse sand 


2-0 


1-10 


0-880 


2-0 


0-714 


0-520 


2-0 


0-588 


0-490 









The diminution of the resistance per unit area, with the length, 
is principally due to the relative velocity of the water and the 
board not being constant throughout the whole length. 

As the board moves through the water the frictional resistance 
of the first foot length, say, of the board, imparts momentum to 
the water in contact with it, and the water is given a velocity in 
the direction of motion of the board. The second foot length will 
therefore be rubbing against water having a velocity in its own 
direction, and the frictional resistance will be less than for the 
first 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 



HYDRAULICS 



282. Stream line theory of the resistance offered to the 
motion of bodies in water. 

Resistance of ships. In considering the motion of water along 
pipes and channels of uniform section, the water has been assumed 
to move in " stream lines," which have a relative motion to the 
sides of the pipe or channel and to each other, and the resistance 
to the motion of the water has been considered as due to the 
friction between the consecutive stream lines, and between the 
water and the surface of the channel, these frictional resistances 
above certain speeds being such as to cause rotational motions in 
the mass of the water. 







Fig. 363. 




Fig. 364. 

It has also been shown that at any sudden enlargement of a 
stream, energy is lost due to eddy motions, and if bodies, such 
as are shown in Figs. 110 and 111, be placed in the pipe, there is 
a pressure acting on the body in the direction of motion of the 
water. The origin of the resistance of ships is best realised by 
the "stream line" theory, in which it is assumed that relative to 
the ship the water is moving in stream lines as shown in Figs.. 
363, 364, consecutive stream lines also having relative motion. 



RESISTANCE TO THE MOTION OF BODIES IN WATER 511 

According to this theory the resistance is divided into three 
parts. 

(1) Fractional resistance due to the relative motions of con- 
secutive stream lines, and of the stream lines and the surface 
of the ship. 

(2) Eddy motion resistances due to the dissipation of the 
energy of the stream lines, all of which are not gradually brought 
to rest. 

(3) Wave making resistances due to wave motions set up at 
the surface of the water by the ship, the energy of the waves 
being dissipated in the surrounding water. 

According to the late Mr Froude, the greater proportion of 
the resistance is due to friction, and especially is this so in long 
ships, with fine lines, that is the cross section varies very gradually 
from the bow towards midships, and again from the midships 
towards the stern. At speeds less than 8 knots, Mr Froude has 
shown that the frictional resistance of ships, the full speed of 
which is about 13 knots, is nearly 90 per cent, of the whole 
resistance, and at full speed it is not much less than 60 per cent. 
He has further shown that it is practically the same as that 
resisting the motion of a thin rectangle, the length and area of 
the two sides of which are equal to the length and immersed 
area respectively of the ship, and the surface of which has the 
same 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, Y the velocity of the ship in feet per second, the 
resistance due to friction is 

r/./.A.V, 

the value of the index n approximating to 2. 

The eddy resistance depends upon the bluntness of the stern of 
the boat, and can be reduced to a minimum by diminishing the 
section of the ship gradually, as the stern is approached, and by 
avoiding a thick stern and stern post. 

As an extreme case consider a ship of the section shown in 
Fig. 364, and suppose the stream lines to be as shown in the 
figure. At the stern of the boat a sudden enlargement of the 
stream lines takes place, and the kinetic energy, which has been 
given to the stream lines by the ship, is dissipated. The case is 
analogous to that of the cylinder, Fig. Ill, p. 169. Due to the 
loss of energy, or head, there is' a resultant pressure acting upon 
the ship in the direction of flow of the stream lines, and con- 
sequently opposing its motion. 



5 1 2 HYDRAULICS 

If the ship has fine lines towards the stern, as in Fig. 363, 
the velocities of the stream lines are diminished gradually and the 
loss of energy by eddy motions becomes very small. In actual 
ships it is probably not more than 8 per cent, of the whole 
resistance. 

The wave making resistance depends upon the length and the 
form of the ship, and especially upon the length of the "entrance" 
and "run." By the "entrance" is meant the front part of the 
ship, which gradually increases in section* until the middle body, 
which is of uniform section, is reached, and by the "run," the 
hinder part of the ship, which diminishes in section from the 
middle body to the 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 Froude found that for the 
S.S. "Merkara" the wave making resistance at 13 knots, the 
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 knots. 

An approximate formula for the critical speed V in knots is 



L being the length of entrance, and Li the length of the run in 
feet. 

The mode of the formation by the ship of waves can be partly 
realised as follows. 

Suppose the ship to be moving in smooth water, and the stream 
lines to be passing the ship as in Fig. 363. As the bow of the 
boat strikes the dead water in front there is an increase in 
pressure, and in the horizontal plane SS the pressure will be 
greater at the bow than at some distance in front of it, and 
consequently the water at the bow is elevated above the normal 
surface. 

Now let AA, BB, and CO be three sections of the ship and the 
stream lines. 

Near the midship section CO the stream lines will be more 
closely packed together, and the velocity of flow will be greater, 
therefore, than at A A or BB. Assuming there is no loss of energy 
in a stream line between AA and BB and applying Bernoulli's 
theorem to any stream line, 

PA + V = PC + ^l = ? + ^ 
w 2g w 2g w 2g' 

* See Sir W. White's Naval Architecture, Transactions of Naval Architects, 
1877 and 1881. 



RESISTANCE TO THE MOTION OF BODIES IN WATER 513 

and since V A and V B are less than v c , 

^ and ? are greater than ^. 
w w w 

The surface of the water at A A and BB is therefore higher 
than at CO and it takes the form shown in Fig. 363. 

Two sets of waves are thus formed, one by the advance of the 
bow and the other by the stream lines at the stern, and these 
wave motions are transmitted to the surrounding water, where 
their energy is dissipated. This energy, as well as that lost in 
eddy motions, must of necessity have been given to the water by 
the ship, and a corresponding amount of work has to be done by 
the ship's propeller. The propelling force required to do work 
equal to the loss of energy by eddy motions is the eddy resist- 
ance, and the force required to do work equal to the energy of 
the waves set up by the ship is the wave resistance. 

To reduce the wave resistance to a minimum the ship should 
be made very long, and should have no parallel body, or the 
entire length of the ship should be devoted to the entrance and 
run. On the other hand for the frictional resistance to be small, 
the area of immersion must be small, so that in any attempt 
to design a ship the resistance of which shall be as small as 
possible, two conflicting conditions have to be met, and, neglecting 
the eddy resistances, the problem resolves itself into making the 
sum of the frictional and wave resistances a minimum. 

Total resistance. If R is the total resistance in pounds, r/ the 
frictional resistance, r e the eddy resistance, and r w the wave 
resistance, 

~R = r/ + r e + r w . 

The frictional resistance r/ can easily be determined when the 
nature of the surface is known. For painted steel ships / is 
practically the same as for the varnished boards, and at 10 feet 
per second the frictional resistance is therefore about J Ib. per 
square foot, and at 20 feet per second 1 Ib. per square foot. The 
only satisfactory way to determine r e and r w for any ship is to 
make experiments upon a model, from which, by the principle of 
similarity, the corresponding resistances of the ship are deduced. 
The horse-power required to drive the ship at a velocity of Y feet 
per second is 

RV 



To determine the total resistance of the model the apparatus 
shown in Fig. 362 is used in the same way as in determining the 
frictional resistance of thin boards. 

L. H. 33 



514 HYDRAULICS 

283. Determination of the resistance of a ship from, the 
resistance of a model of the ship. 

To obtain the resistance of the ship from the experimental 
resistance of the model the principle of similarity, as stated by 
Mr Froude, is used. Let the linear dimensions of the ship be I) 
times those of the model. 

Corresponding speeds. According to Mr Fronde's theory, for 
any speed Y m of the model, the speed of the ship at which its 
resistance must be compared with that of the model, or the 
corresponding speed Y a of the ship, is 



Corresponding resistances. If R m is the resistance of the model 
at the velocity V m , and it be assumed that the coefficients of 
friction for the ship and the model are the same, the resistance R/ 8 
of the ship at the corresponding speed V is 



As an example, suppose a model one-sixteenth of the size 
of the ship ; the corresponding speed of the ship will be four times 
the speed of the model, and the resistance of the ship at corre- 
sponding speeds will be 16 3 or 4096 times the resistance of the 
model. 

Correction for the difference of the coefficients of friction for the 
model and shvp. The material of which the immersed surface of 
the model is made is not generally the same as that of the ship, 
and consequently R a must be corrected to make allowance for the 
difference of roughness of the surfaces. In addition the ship is 
very much longer than the model, and the coefficient of friction, 
even if the surfaces were of the same degree of roughness, would 
therefore be less than for the model. 

Let A,n be the immersed surface of the model and A* of 
the ship. 

Let f m be the coefficient of friction for the model and /, for the 
ship, the values being made to depend not only upon the roughness 
but also upon the length.' If the resistance is assumed to vary as 
V 2 , the frictional resistance of the model at the velocity V m is 



and for the ship at the corresponding speed V, the frictional 
resistance is 



But 
and 



RESISTANCE TO THE MOTION OF BODIES IN WATER 515 

and, therefore, r s =/ 8 A w V m 2 D 3 



Then the resistance of the ship is 



^ 

Determination of the curve 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 
velocities 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 




4CO 



D 



Fig. 365. 



Mr Froude's method of correcting the curve for the difference of 
the coefficients of friction for the ship and the model. From the 
formula 



332 



516 HYDRAULICS 

the frictional resistance of the model for several values of V,,, 
is calculated, and the curve FF plotted on the same scale as used 
for the curve RR. The wave and eddy making resistance at any 
velocity is the ordinate between FF and RR. At velocities of 
200 feet per minute for the model and 800 feet per minute for 
the ship, for example, the wave and eddy making resistance is 6c, 
measured on the scale BG- for the model and on the scale EH for 
the ship. 

The frictional resistance of the ship is now calculated from the 
formula r s = /,AsV8 n , and ordinates are set down from the curve 
FF, equal to r 8) to the scale for ship resistance. A third curve is 
thus obtained, and at any velocity the ordinate between this curve 
and RR is the resistance of the ship at that velocity. For example, 
when the ship has a velocity of 800 feet per minute the resistance 
is ac, measured on the scale EH. 



EXAMPLES. 

(1) Taking skin friction to be 0'4 Ib. per square foot at 10 feet per 
second, find the skin resistance of a ship of 12,000 square feet immersed 
surface at 15 knots (1 knot = T69 feet per second). Also find the horse-power 
to drive the ship against this resistance. 

(2) If the skin friction of a ship is 0*5 of a pound per square foot of 
immersed surface at a speed of 6 knots, what horse-power will probably 
be required to obtain a speed of 14 knots, if the immersed surface is 18,000 
square feet ? You may assume the maximum speed for which the ship is 
designed is 17 knots. 

(3) The resistance of a vessel is deduced from that of a model ^th the 
linear size. The wetted surface of the model is 29'4 square feet, the skin 
friction per square foot, in fresh water, at 10 feet per second is 0*3 Ib., and 
the index of velocity is T94. The skin friction of the vessel in salt water 
is 60 Ibs. per 100 square feet at 10 knots, and the index of velocity is T83. 
The total resistance of the model in fresh water at 200 feet per minute is 
T46 Ibs. Estimate the total resistance of the vessel in salt water at the 
speed corresponding to 200 feet per minute in the model. Lond. Un. 1906. 

(4) How from model experiments may the resistance of a ship be 
inferred? Point out what corrections have to be made. At a speed of 
300 feet per minute in fresh water, a model 10 feet in length with a wet 
skin of 24 square feet has a total resistance of 2*39 Ibs., 2 Ibs. being due to 
skin resistance, and '39 Ib. to wave-making. What will be the total resist- 
ance at the corresponding speed in salt water of a ship 25 times the linear 
dimensions of the model, having given that the surface friction per square 
foot of the ship at that speed is 1-3 Ibs. ? Lond. Un. 1906. 



CHAPTER XIII. 

STREAM LINE MOTION. 

284. Hele Shaw's experiments on the flow of thin 
sheets of water. 

Professor Hele Shaw* has very beautifully shown, on a small 
scale, the form of the stream lines in moving masses of water 
under varying circumstances, and has exhibited the change from 
stream line to sinuous, or rotational flow, by experiments on the 
flow of water at varying velocities between two parallel glass 
-plates. In some of the experiments obstacles of various forms 
were placed between the plates, past which the water had to flow, 
and in others, channels of various sections were formed through 
which the water was made to flow. The condition of the water 
as it floAved between the plates was made visible by 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 flowed, the water persisted 
in stream line flow. When the channel between the plates was 
made 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 
of loss of head. When the velocity was sufficiently high for the 
flow to become sinuous, the eddy motions were very marked. 
When the motion was sinuous and the water was made to flow 
past obstacles similar to those indicated in Figs. 110 and 111, the 
water immediately in contact with the down-stream face was 
shown to be at rest. Similarly the water in contact with the 
annular ring surrounding a sudden enlargement appeared to be 
at rest and the assumption made in section 51 was thus justified. 

* Proceedings of Naval Architects, 1897 and 1898. Engineer, Aug. 1897 and 
May 1898. 



518 HYDRAULICS 

When the flow was along channels and sinuous, the sinuously 
moving water appeared to be separated from the sides of the 
channel by a thin film of water, which Professor Hele Shaw 
suggested was moving in stream lines, the velocity of which in 
the film diminish as the surface of the channel is approached. 
The experiments also indicated that a similar film surrounded 
obstacles of ship-like and other forms placed in flowing water, 
and it was inferred by Professor Hele Shaw that, surrounding 
a ship as it moves through still water, there is a thin film moving 
in stream lines relatively to the ship, the shearing forces between 
which and the surrounding water set up eddy motions which 
account for the skin friction of the ship. 

285. Curved stream line motion. 

Let a mass of fluid be moving in curved stream lines, and let 
AB, Fig. 366, be any one of the stream lines. 

At any point c let the radius of curvature of the stream line 
be r and let be the centre of curvature. 

Consider the equilibrium of an element abde surrounding the 
point c. 

Let W be the weight of this element. 

p be the pressure per unit area on the face bd. 

p + dp be the pressure per unit area on the face ae. 

6 be the inclination of the tangent to the stream line at c 

to the horizontal. 

a be the area of each of the faces bd and ae. 
v be the velocity of the stream line at c. 
dr be the thickness ab of the stream line. 

If then the stream line is in a vertical plane the forces acting 
on the element are 

(1) W due to gravity, 

WV* 

(2) the centrifugal force --- acting along the radius away 

from the centre, and 

(3) the pressure adp acting along the radius towards the 
centre of curvature 0. 

Resolving along the radius through c, 



~, , TT A r, 

(top -- + W cos & = 0. 
9r 

or since W = wadr, 

dp wv* a f ^ 

-~ = -- w cos ........................ (1). 

dr gr 

If the stream line is horizontal, as in the case of water flowing 



STREAM LINE MOTION 



519 



round the bend of a river, Oc is horizontal and the component of 
W along Oc is zero. 

- .............................. (2). 



Integrating between the limits R and R! the difference of 
pressure on any horizontal plane at the radii R and RI is 



*--f)M* (s) > 



9 

which can be integrated when v can be written as a function of r. 
Now for any horizontal stream line, applying Bernoulli's 
equation, 

+ jj- is constant, 



or 



Differentiating 

A 



w 2g 

w + 2g~ ' 
!_ dp vdv _ dK 
wdr gdr~~dr V*' 1 





Fig. 366. 



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. 



Then 



_1 dp _ -vdv 
w dr~ gdr 



(5). 



Substituting the value of -f- from (5) in (2) 

dr 

wv dv _ w V? 
g dr ~ g ' r ' 

from which rdv + vdr = 0, 

and therefore by integration 

vr = constant = C 



520 HYDRAULICS 

Equation (3) now becomes 

Pi p _ CP [* l dr 
w ~ g JR r 3 



_ 

20 VR 2 

Forced vortex. If, as in the turbine wheel and centrifugal 
pump, the angular velocities of all the stream lines are the same, 
then in equation (3) 



-, i- <> , 

and - = - I rdr 



Scouring 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 opposite 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 
James Thomson 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. 



APPENDIX. 



1. Coefficients of discharge : 
(a) for circular sharp-edged orifices. 

Experiments by Messrs Judd and King at the Ohio University 
on the flow through sharp-edged orifices from f inch to 2J inches 
diameter showed that the coefficient was constant for all heads 
between 5 and 92 feet, the values of the coefficients being as 
follows. (Engineering News, 27th September, 1906.) 



Diameter of 




orifice in 


Coefficients 


inches 




H 


0-5956 


2 


0-6083 


H 


0-6085 


i 


0-6097 


I 


0-6111 



The results in the following table have been determined by 
Bilton (Victorian Institute of Engineers, Library Inst. C. E. 
Tract, 8vo. Vol. 629). Bilton claims that above a certain "critical" 
head the coefficient remains constant, but below this head it 
increases. 

Coefficients of discharge for standard circular orifices. 





Diameter of orifices in inches 


inches 


















2 and 
over 


2 


11 


1 


1 


i 


i 


45 and) 


0-598 


0-599 


0-603 


0-608 


0-613 


0-621 


0-628 


over ( 
















22 












0-621 


0-638 


18 










0-613 


0-623 


0-643 


17 


0-598 


0-599 


0-603 


0-608 


0-614 


0-625 


0-645 


12 


0-600 


0-601 


0-606 


0-612 


0-618 


0-630 


0-653 


9 


0-604 


0-606 


0-612 


0-619 


0-623 


0-637 


0-660 


6 


0-610 


0-612 


0-618 


0-626 


0-632 


0-643 


0-669 


3 








0-640 


0-646 


0-657 


0-680 


2 












0-663 


0-683 



522 



HYDRAULICS 



(b) for triangular notches. 

Recent experiments by Barr (Engineering, April 1910) on the 
flow through triangular notches having an angle of 90 degrees 
showed that the coefficient C (page 85) varies, but the mean value 
is very near to that given by Thomson. 

The coefficients as determined by Barr are given in the 
following table : 



Head 


2" 


2*"- 


3" 


*r 


4" 


7" 


10" 


Coefficient C 


2-586 


2-564 


2-551 


2-541 


2-533 


2-505 


2-49 



2. The critical velocity in pipes. Effect of temperature. 

A simple apparatus, Fig. 368, which can be made in any 
laboratory and a description of which it is hoped may be of value 
to teachers, has been used by the author for experiments on the 
flow of water in pipes. 

Three carefully selected pieces of brass tubing 0'5 cms. 
diameter, each about 6 feet long, were taken, and the diameters 
measured by filling with water at 60 F. The three tubes were 
connected at A A. by being sweated into brass blocks, holes 
through which were drilled of the same diameter as the outsides 
of the tubes. Between the two ends of the tubes, while being 
soldered in the blocks, was inserted a piece of thin hard steel 
about 2^th of an inch in thickness. The tubes were thus fixed 
in line, while at the same time a connection is made to the gauge Gr 
from each end of the tube AA. 

To the ends of each of the end tubes were fixed other blocks B 
into which were inserted tubes T. Inside each of these tubes was 
placed a thermometer. Flow could take place through the 
tubes T into vessels Y and Vi. During any experiment a con- 
stant head was maintained by allowing the water to flow into the 
tank S at such a rate by the pipe P that there was also a slight 
overflow down the pipe P'. 

Between the tank and the pipe was a coil which was surrounded 
by a tank in which was a mass of water kept heated by bunsen 
burners, or by the admission of steam. 

Flow from the tank could be adjusted by the cock C or by the 
pinch taps (1) to (4). 

The pinch tap (4) was found very useful in that by opening 
and closing, the quantity of water flowing through the coil could 
be kept constant while the flow through the pipe was changed. 



524 



HYDRAULICS 



The loss of head was measured at the air gauge G in cms. of 
water. 

The results obtained at various temperatures are shown plotted 
in Fig. 369. 




LogV 



10 



20 



40 50 60 70 80 90 100 
Velocity m cms. per second. 

Fig. 369. 



200 



At any temperature, for velocities below the critical velocity, 
the columns of water in the gauge were very steady, oscillations 
scarcely being perceivable with the cathetometer telescope. At 
the critical velocity the columns in the gauge become very unsteady 
and oscillate through a distance of two or three centimetres. 
When the upper critical velocity is passed the columns again 
become steady. 



APPENDIX 



525 



3. Losses of head in pipe bends. 

The experimental data, as remarked in the text, on losses of 
head in pipe bends are not very complete. The following table 
gives results obtained by Schoder* from experiments on a series 
of 6 inches diameter bends of different radii. The experiments 
were carried out by connecting the bends in turn to two lengths 
of straight pipe 6 inches diameter, the head lost at various 
velocities in one of the lengths having been previously carefully 
determined. The bend being in position the loss of head in the 
bend and in the straight piece was then found and the loss caused 
by the bend obtained by difference. 

In the table the length of straight pipe is given in which the 
loss of head would be the same as in the bend. 

Losses of head caused by 90 degree bends expressed in terms of 
the length of straight pipe of the same diameter in which a loss 
of head would occur equal to the loss caused by the bend. 

Diameter of all bends 6" (very nearly). 













Equivalent lengths of pipe 


No. of 
curve 


Material 


Radius 
in feet 


Eadius 
in pipe 
diameters 


Length 
of centre 
line in 
feet 


on centre lines 


Velocity in feet per second 












3 


5 


10 


16 




Wrought iron 


10 
7-50 


20 
15 


16-77 
12-84 


8-4 
3-2 


6-7 
1-6 


4-4 
0-2 


3-2 






5-00 


10 


9-01 


5'0 


3-5 


2-1 


1-4 




j) 


4-00 


8 


7-34 


6-8 


5-2 


3-9 


3-0 




|| 


3-00 


6 


5-89 


6-8 


5-1 


3-9 


3-2 




99 


2-50 


5 


5-08 


3-0 


2-5 


2-1 


2-2 






2-00 


4 


3-64 


5-6 


4-3 


3-5 


2-7 






1-50 


3 


2-86 


4-8 


4-1 


3-5 


2-7 






1-08 


2-16 


2-54 


5-2 


4-4 


3-9 


3-0 






0-95 


1-9 


1-75 


6-0 


5-1 


4-6 


3-8 






0-88 


1-76 


3-62 


5-8 


5-8 


5-6 


5-7 






0-67 


1-34 


1-05 


9-8 


8-6 


7-7 


7-0 



Fig. 370 shows the loss of head due to 90 degree bends in pipes 
3 inches and 4 inches diameter as obtained by Dr Brightmoret. 
The forms of the curves are very similar to the curves obtained 
by Schoder for the 6 inch bends quoted above. Brightmore found 
that the loss of head caused by square elbows in 3 inches and 

* Proc. Am. S.C.E. Vol. xxxiv. p. 416. 
t Proc. Imt. C.E. Vol. CLXIX. p. 323. 



526 



HYDRAULICS 



4 inches diameter pipes was the same and was equal to - 
v being the velocity of the water in the pipe in feet per second. 



Inchss. 




12 14 

Fig. 370. Loss of head due to bends in pipes 3" and 4" in diameter. 



4 6 8 10 

Radios of Bend in Diameters. 



Davies* gives the loss of head iri a 2 T V diameter elbow as 
0'0113v 2 and in a 2|" diameter elbow with short turn as 0'0202t; 3 . 

4. The Pitot tube. 

There has been considerable controversy as to the correct 
theory of the Pitot tube, some authorities contending that the 
impact head h produced by the velocity of the moving stream 
impinging on the tube with the plane of its opening facing up 
stream should be expressed as 

, 7b 2 

*">; 

and others contending that it should be expressed as 



In the text it is shown that if the momentum of the water per 
* Proc. Am. S.C.E., Sep. 1908, Vol. xxxiv. p. 1037. 



APPENDIX 527 

second which, would flow through an area equal to the area of the 
impact orifice is destroyed the pressure on the area is equal to 



wa 



9 

and the height of the column of water maintained by this pressure 
would be 



v 
Experiment shows that the actual height is equal to ~- > an( l ^ 

has therefore been contended that the destroyed momentum of 
the mass should not be considered as producing the head, but 
rather the " velocity head." Those who maintain this position do 
not recognise the simple fact that when it is stated that the 
kinetic energy of the stream is destroyed, it is exactly the 
same thing as saying that the momentum of the stream is 

v 2 
destroyed, and that the reason why the head is not equal to is 

that the momentum of a mass of water equal to the mass which 
passes through an area equal to the area of the impact surface 
is not destroyed. 

Experiments by White*, the author and others show that 
when a jet of water issuing from an orifice is made to impinge on 
a plate having its plane perpendicular to the axis of the jet, the 
pressure on the plate is distributed over an area much greater 
than the area of the original jet, and the maximum intensity of 
pressure occurs at a point on the plate coinciding with the axis of 
the jet; and is equal to one-half the intensity of pressure that 
would obtain if the whole pressure was distributed over an area 
equal to the area of the jet. In this case the whole momentum is 
destroyed on an area much greater than the area of the jet. The 
total pressure on the plate however divided by the area of the jet 
is equal to 

v* 

g ' 

When a Pitot tube is placed with its opening perpendicular to 
a stream, the water approaching the tube is deflected into stream 
lines which pass the tube with only part of their velocity per- 
pendicular to the tube destroyed. To obtain a complete theory 
of the Pitot tube it would be necessary that the conditions of flow 
in the neighbourhood of the tube should be completely under- 

* Journ. of the Assoc. of Eng. Soc. August 1901. 



528 



HYDRAULICS 



stood. The fact therefore that the head in the impact tube of 

v* 
a Pitot is equal to 5- cannot be said at present to be a theoretical 

deduction but simply an experimental result, and the formula 

2 

Ji = k i n the present state of knowledge must be looked upon as 

*9 
an empirical formula rather than a theoretical one. 

Fig. 371 shows a number of Pitot tubes impact surfaces, for 
which Mr W. M. White has determined the coefficients by 
measuring the height of a column of water produced by a jet 
issuing from a horizontal orifice, and also by moving them through 
still water. In all cases the coefficient k was unity. Fig. 372 



1-5mm 



2-5/nmi 





Copper Tubes. 
Fig. 371. Fig. 372. 

shows impact surfaces for which the author has determined the 
coefficients by inserting them in a jet of water issuing from a 
vertical orifice, the coefficient of velocity for which at all heads 




Fig. 373. Gregory Pitot tube having a coefficient of unity. 

was carefully determined by the method described on page 55. 
Fry and Tyndall by experiments on Pitot tubes revolving in air 
found a value for k equal to unity, and Burnham*, using a tube 
consisting of two brass tubes one in the other, the inner one 
^ inch outside diameter and -^ inch thick, forming the impact 
tube, and the outer pressure tube made of f inch diameter tube 



* Eng. News, Dec. 1905. 



APPENDIX 529 

^V inch thick, provided with a slit 1 J inches long by T V inch wide 
for transmitting the static pressure, also found k to be constant 
and equal to unity. If the walls of the impact tube are made 
very thin the constant may differ perceptibly from unity. Fry 
and Tyndall found that a tube '177 mm. diameter with walls 
"027 mm. thick gave a value of k several per cent, above unity, 
but when a small mica plate 2 mm. diameter was fitted on the 
end of the tube ~k was unity. The position of the pressure holes 
in the static pressure tube also affects the constant, and if the 
constant unity is to be relied upon they should be removed some 
distance from the impact face. The author has found in experi- 
menting on the velocity of flow in jets issuing from orifices, that, 
by using two small aluminium tubes side by side and their ends 
flush with each other, one of which had the end plugged and the 
other open, the plugged one having small holes pierced through 
the tube perpendicular to the axis of the tube very near to the 
end, the coefficient Jc was with some of the tube combinations as 
much as 10 per cent, greater than unity, but when the impact tube 
was used alone the coefficient was exactly equal to unity, indicating 
that the variation of ~k was due to uncertain effects on the static 
pressure openings. 

5. The Herschel fall increaser. 

This is an arrangement suggested by Herschel for increasing 
the head under which a turbine works when the fall is small, and 
thus making it possible to run the wheel at a higher velocity, or 
for keeping the head under which a turbine works constant when 
the difference of level between the head and tail water of a low 
fall varies. In times of heavy flow the difference of level between 
the head and tail water of a stream supplying a turbine may be 
considerably less than in times of normal flow, as shown in the 
examples quoted on pages 328 and 349, and if the power given by 
the turbine is then to be as great as when the flow is normal, 
additional compartments have to be provided so that a larger 
volume is used by the turbine to compensate for the loss of head. 
Instead of additional compartments, as in the examples cited, 
stand by plant of other types is sometimes provided. In all such 
arrangements expensive plant is useless in times of normal flow, 
and the capital expenditure is, therefore, high. 

The increased head is obtained by an application of the 
Venturi principle, the excess water not required by the turbines 
being utilised to create in a vessel a partial vacuum, into which 
the exhaust can take place instead of directly into the tail-race. 

L. H. 34 



530 



HYDRAULICS 



In Fig. 374, which is quite diagrammatic, suppose the turbine 
is working in a casing as shown and is discharging down a tube 
into the vessel V ; and let the water escape from V along the pipe 
EDF, entering the pipe by the small holes shown in the figure. 




Fig. 374. Diagram of Fall Increases 

When there is a plentiful supply of water, some of it is allowed to 
flow along the pipe EDF, entering at E where it is controlled by 
a valve. The pipe is diminished in area at D, like a Venturi 
meter, and is expanded as it enters the tail-race. When flow is 
taking place the pressure at D will be less than the pressure at 
F, and the head under which the turbine is working is thereby 
increased. Mr Herschel states that by suitably proportioning the 
area of the throat D of the pipe, and the area of the admission 
holes in D, the head can easily be increased by 50 per cent. Let 
Ji be the difference of level of the up and down streams. Then 
without the fall increaser the discharge of the turbine is pro- 
portional to \/h and the horse-power to h*Jh. 

Let hi be the amount by which the head at D is less than at F, 
or is the increase of head by the increaser. 

The work done without the increaser is to the work done with 
the increaser 



APPENDIX 531 



If Qi is the discharge through the turbine when the increaser 
is used, the work gained by the increaser 



The efficiency of the increaser is this quantity divided by 
h x weight of water entering at E. 

Mr Herschel found by experiment that the maximum value 
of this efficiency was about 30 per cent. 

The arrangement was suggested by Mr Herschel, and accepted, 
in connection with a new power house to be erected for the further 
utilisation of the water of Lake Leman at Geneva; one of the 
conditions which had to be fulfilled in the designs being that at 
all heads the horse-power of the turbines should be the same. 
When the difference between the head and tail water is normal 
the increaser need not be used, but in times of heavy flow when 
the head water surface has to be kept low to give sufficient slope 
to get the water away from up stream and the tail water surface 
is high, then the increaser can be used to make the head under 
which the turbine works equal to the normal head. 

6. The Humphrey internal combustion pump. 

An ingenious, and what promises to be a very efficient pump 
has recently been developed by Mr H. A. Humphrey, which is 
both simple in principle and in construction. The force necessary 
for the raising of the water being obtained by the explosion of a 
combustible mixture in a vessel above the surface of the water in 
the vessel. All rotating and reciprocating parts found in ordinary 
pumps are dispensed with. The idea of exploding such a mixture 
in contact with the water did not originate with Mr Humphrey, 
but the credit must remain with him of having evolved on a 
large scale a successful pump and of having overcome the serious 
difficulties to be faced in an ingenious and satisfactory manner. 

The pump in its simplest form is shown in Fig. 375. C is a 
combustion chamber, into which is admitted the combustible 
charge through the valve F. B is the exhaust valve. These two 
valves are connected by an interlocking* gear, so arranged that 
when the admission opens and closes it locks itself shut and 
unlocks the exhaust valve ready for the next exhaust stroke. 
When the exhaust valve closes it locks itself, and releases the 

* Proc. Inst. Mech. Engs. 1910. 

342 



532 



HYDRAULICS 



admission valve, which is then ready to admit a fresh charge, 
when the suction stroke occurs. A sparking plug, not shown in the 
figure, is used to explode the combustible mixture. 



f 



Fig. 375. 

The delivery pipe D is connected directly to the combustion 
chamber C and to the supply tank ET. W is the water valve box 
having a number of small valves Y, instead of one big one, opening 
inwards, each held on its seat by a light spring, and through 
which water enters the delivery pipe from the supply tank. 
Suppose a compressed charge to be enclosed in the chamber C and 
fired by a spark. The increase of pressure sets the water in C 
and in the pipe D in motion, a quantity of water entering the tank 
ET. The velocity of the water in D increases as long as the 
pressure of the gases in C is greater than the head against which 
the pump is delivering together with the head lost by friction, etc. 

Eeferring to the diagram, Fig. 375, let h be the head of water, 
supposed for simplicity constant, against which the pump is 
delivering; let H be the atmospheric pressure in feet of water, 
and p the pressure per sq. foot at any instant in the combustion 
chamber. Let v be the velocity of the column of water at any 
instant, and let the friction head plus the head lost by eddies as 

the water enters the supply tank at this velocity be -5 . As long 

40 

as is greater than H + h + -~ the mass of water in D will be 

accelerated positively and the maximum velocity v m of the water 
will be reached when 



w 
The water will have acquired a kinetic energy per Ib. equal to 

2 

7- , and will continue its motion towards the tank. As it does so 



APPENDIX 533 

the pressure in C falls below the atmospheric pressure and the 
exhaust valve E opens. The pressure in C plus the height of the 
surface of the water in C above the centre of W will give the 
pressure in W, and when this is less than the atmospheric pressure 
plus the head of water in ST the valves V will open and allow 
water to enter D. 

When the kinetic energy of the moving column has expended 
itself by forcing water into the tank ST, the water will begin to 
return and will rise in the chamber C until the surface hits the 
valve E and shuts the exhaust, the exhaust valve becoming 
locked as explained above while the inlet valve is released, 
and is ready to open when the pressure in C falls below the 
atmospheric pressure. A portion of the burnt gases is enclosed 
in the upper part of C, and the energy of the returning column is 
used to compress this gas to a pressure which is greater than 
h + IL. When the column is again brought to rest a second 
movement of the column of water towards D takes place, the 
pressure in C falling again below the atmospheric pressure and a 
fresh charge of gas and air is drawn in. Again the column begins 
to return and compresses the mixture to a pressure much greater 
than that due to the static head, when it is ignited and a fresh 
cycle begins. 

The action of the pump is unaltered if it discharges into an air 
vessel, as in Fig. 376, instead of into an elevated tank, this arrange- 
ment being useful when a continuous flow is required. 



Fig. 376. 

Figs. 377 and 377 a show other arrangements of the pump. In 
the two papers cited above other types and modifications of the 
cycle of operations for single and two barrel pumps are described, 
showing that the pump can be adapted to almost any conditions 
without difficulty. 

An important feature of the pump is in the use that is made of 
the " fly-wheel " effect of the moving column of water to give high 
compression, which is a necessity for the efficient working of an 
internal combustion engine*. 

* See works on gas and oil engines. 



534 



HYDRAULICS 



To start the pump from rest, a charge of air is pumped into 
the chamber C by a hand pump or small compressor, and the 
exhaust valve is opened by hand. This starts the oscillation of the 
column, which closes the exhaust valve, and compresses the air 
enclosed in the clearance. 




u 



Fig. 377. 





Fig. 377 a. 

This compressed air expands below the atmospheric pressure 
and a charge of gas and air is drawn into the cylinder, which is 
compressed and ignited and the cycles are commenced. 

For a given set of conditions the length of the discharge pipe 
is important in determining the periodicity of the cycles and thus 
the discharge of the pump. 

Lot the volume of gases when explosion takes place (Fig. 378) 
be po Ibs. per sq. foot absolute, and let the volume occupied by the 
gases be V cubic feet. Let A be the cross-sectional area of the 
explosion chamber, h the head against which the pump works in 
feet of water, H the atmospheric pressure in feet of water. Let 
the delivery pipe be of length L and of the same diameter as the 
explosion chamber. As the expansion of the gases takes place 
let the law of expansion be pV n = constant. 



APPENDIX 



535 



The volume V of the expanding gases when the surface of the 
water has moved a distance a? will be Vi = Vo + A and the pressure 




Fig. 378. 

If p is the pressure at any instant during expansion the work 
done by the expanding gases is 

Af%cfo= pteYo ""? lYl . 
7v n1 

This energy has had to give kinetic energy to the water in the 
pipe, to lift a quantity of water equal to AOJ into the tank, and to 
overcome friction. If the delivery pipe is not bell-mouthed the 
water as it enters the tank with a velocity v will have kinetic 

energy per Ib. of - ft. Ibs. 

The kinetic energy of the water in the pipe at any velocity v is 



pi a 

Let the friction head at any velocity be h/= -^ . 

Then 



-~ = I pA.dx w(h+ H) Adx ^ . dx 



. A.v*dx 



,<. , 
...(1). 



536 HYDRAULICS 

Or from the diagram let AB be the expansion curve of the 
exploded gases. Let h be the head against which the pump is 
lifting, and H the atmospheric pressure expressed in feet of water. 
If there is no friction in the pump, or other losses of head, the 
pressure in the chamber becomes equal to the absolute head 
against which water is being pumped when the volume is V 4 . 

Up to this point the velocity of the water is being increased, 



The actual velocity will be less than v 4 as calculated from this 
formula, due to the losses of head. 

Let it be assumed that the total loss of energy per Ib. at any 

TjV 2 

velocity v is -^- , this including frictional losses and losses by 

eddy motions as the water enters the supply tank. 
Then if EK be made equal to 

Yvf 
2<7 

and the parabolic arc FK be drawn, the frictional head at any 
other volume will be approximately 6c. The curve AB now cuts 
the curve FK at Gr, and Yi is a nearer approximation to the 
volume at which the maximum velocity occurs. 
Let v m be this maximum velocity. 

Then ^=AFcG. 

The friction head can now be corrected if thought desirable 
and v m re-calculated. At any volume V the velocity is given by 

A1? , 
^ ' 

Let the exhaust valve be supposed to open when the pressure 
falls to p B (say 14*5 Ibs. per sq. inch). 

Then the velocity when the exhaust opens is given by 



For further movements of the column of water the pressure 
remains constant, and if the energy of water entering through the 
valves Y is neglected the water will come to rest when 

ACQRSBA = FGTRC, 
or if the mean loss of head is taken as f of the maximum, when 



APPENDIX 537 

From this equation V 3 can be calculated or by trial the two 
areas can be made equal. 

By calculating the velocity at various points along the stroke 
a velocity curve, as shown in the figure, can be drawn. 

The time taken for the stroke OR can then be found by 

V V 

dividing the length ^-r - by the mean ordinate of the velocity 

diagram. 

On the return cushioning stroke the exhaust valve will close 
when the volume Y 3 is reached and the gases in the cylinder will 
then be compressed. The compression curve can be drawn and 
the velocities at the various points in the stroke calculated. The 
velocity at B for instance in the return stroke will be approxi- 
mately given by 

wKLv* 
2 B =BMTS-NMT, 

the area NMT being subtracted because friction will act in 
opposition to the head h which is creating the velocity. 

7. The Hydraulic Ram. 

In the text no theory is attempted of the working of this 
interesting apparatus, only a very imperfect and elementary 
description of the mode of working being attempted. Those 
interested are referred to an able and voluminous paper by Leroy 
Francis Harza (Bulletin of the University of Wisconsin) in which 
the Hydraulic Ram is dealt with very fully from both an experi- 
mental and theoretical point of view. 

8. Circular Weirs. 

If a vertical pipe, Fig. 379, with the horizontal end AB 
carefully faced is placed in a tank and water, having its surface 
a reasonable distance above AB, flows down the pipe as indicated 
in the figure, Grurley* has shown that the flow in cubic feet per 
second can be expressed in terms of the head H and the circum- 
ferential length of the weir by the formula 



in which n is 1*42, H and L are in feet, and K for different 
diameters has the values shown in the table : 

* Proc. Inst. C.E. Vol. CLXXXIV. 



538 HYDRAULICS 

Circular Weirs. Values of K in formula Q = KLH n . 



Diameter 




of Pipe, 


K 


inches 




6-91 


2-93 


10-08 


2-94 


13-70 


2-97 


19-40 


2-99 


25-90 


3-03 




Fig. 379. 

For reliable results H should not be greater than Jth of the 
diameter of the pipe, and as long as H is large enough for the 
water to leap clear of the inside of the pipe the thickness of 
the pipe is immaterial. The air must be freely admitted below 
the nappe. The flow is affected by the size of the chamber, but 
not to any very considerable extent, as long as the chamber 
is large. 



APPENDIX 539 

9. General formula for friction in smooth pipes. 

Careful investigations of the flow of air, oil and water through 
smooth pipes of diameters varying from 0*361 cms. to 12*62 cms. 
have been carried out at the National Physical Laboratory during 
recent years *. 

The loss of energy at varying temperatures and for velocities 
varying from 5 cms. to 5000 cms. per second have been determined 
in the case of water, and the distribution of velocity in pipes of 
moving air and water have also been carefully determined. These 
latter experiments have shown that if v is the mean velocity of the 
fluid in the pipe, d the diameter of the pipe and v the dynamical 
viscosity of the fluid, the velocity curves are similar for different 
fluids as long as vd/v is constant. If now R is the resistance of the 
pipe per unit area and p the density of the fluid flowing through 
the pipe, the Principle of Dynamical Similarity demands that when 
for various fluids and conditions of flow vd/v is constant thenf or these 
cases R//w 2 must also be constant. By plotting points therefore 
having R/pi; 2 as ordinates and vd/v as abscissae all cases of motion 
in smooth pipes should be represented by a smooth curve, and by 
plotting the logarithms of these quantities a straight line should 
be obtained. The plottings of the logs of these quantities obtained 
from the experiments at the National Physical Laboratory and 
those obtained by other experiments show however that the points 
do not lie about a straight line, but Professor Leest has shown 
that if points be plotted having 



log (-^2 - 0*0009^ as ordinates, 

and log vd/v as abscissae the points do lie on the straight line 
log (-^ - 0*0009) + 0*35 log ^ = log 0*0765, 

' 







which satisfies the Principle of Dynamical Similarity. 

0*017756 



. 
The value of v for water in dynes is 



+ o. 03368T + Q-QOQ221T 2 
which at 15 deg. Cent, is 0*0114 and the density is nearly unity. 

* Stanton, Proc. R.S. Vol. LXXXV. p. 366; Stanton and Pannell, Phil. Trans. 
A. Vol. ccxiv. p. 299. 

t Proc. R.S. A. Vol. xci. 



540 HYDRAULICS 

Then the resistance R in dynes per sq. cm. is 



If p and pi are the pressures in dynes per sq. cm. at two 
sections I cm. apart, 



0'0036 
and 



If p and pi are in pounds per sq. foot and d and I in feet, 

0'006981i 



If the difference of pressure is measured in feet of water 7&, 
then 

lim 0'000112t 



# 

For air at a temperature of 15 C. and at a pressure of 760 mm. 
of mercury, the difference of pressure p in pounds per sq. foot at 
sections a distance I feet apart is 

0-0000332<u 165 0-00000857A 



#* - 

If the pressure difference is measured in inches of water h, 
then 

, /0'00000637<?; r65 0'00000163tA 



10. The moving diaphragm method of measuring the 
flow of water in open channels. 

The flow of water along large regular-shaped channels can be 
measured expeditiously and with a considerable degree of accuracy 
by means of a diaphragm fixed to a travelling carriage as in 
Fig. 380. The apparatus is expensive, but in cases where it is 
difficult to keep the flow in the channel steady for any considerable 
length of time, as for example in the case of large turbines under 
test, and there is not sufficient head available to allow of using a 
weir, the rapidity with which readings can be taken is a great 
advantage. The method has been used with considerable success 
at hydro-electric power stations in Switzerland, Norway, and 
the Berlin Technische Hochschule. A carefully formed channel is 
required so that a diaphragm can be used with only small clearance 
between the sides and bottom of the channel ; the channel should 



542 HYDRAULICS 

be as long as convenient, but not less than 30 feet in length, as 
the carriage has to travel a distance of about 10 feet before it 
takes up the velocity of the water in the channel. The carriage 
shown in the figure weighs only 88 Ibs. and is made of thin steel 
tubing so as to get minimum weight with maximum rigidity. The 
diaphragm is of oiled canvas attached to a frame of light angles. 
The frame is suspended by the two small cables shown coiled 
round the horizontal shaft which can be rotated by the hand 
wheels N; the guides K slide along the tubes S; two rubber 
buffers P limit the descent and the hand brake E, prevents the 
frame falling rapidly. The clutch k holds it rigidly in the vertical 
position ; when k is released the diaphragm swings into the position 
shown in the figure. 

To make a gauging the car is brought to the upstream end of 
the channel with the diaphragm raised and locked in the vertical 
position. At a given signal the diaphragm is dropped slowly, 
being controlled by the brake, until it rests on the buffers which 
are adjusted so that there is only a small clearance between the 
diaphragm and the bottom of the channel. The car begins to 
move when the diaphragm is partly immersed but after it has 
moved a distance of about 10 feet the motion is uniform. The 
time taken for the car to travel a distance of, say, 20 feet is now 
accurately determined by electric* or other means. The mean 
velocity of the stream is taken as being equal to the mean velocity 
of the car. The Swiss Bureau of Hydrography has carried out 
careful experiments at Ackersand and has checked the results 
given by the diaphragm with those obtained from a weir and by 
chemical * means. The gaugings agree within one per cent. 

11. 1. The Centrifugal Pump. 

The effect of varying the form of the chamber surrounding 
the wheel of a centrifugal pump has been discussed in the text 
and it is there stated, page 402, that the form of the casing is 
more important than the form of the wheel in determining the 
efficiency of the pump. Kecent experiments, Bulletin Nos. 173 
and 318, University of Wisconsin, carried out to determine the 
effect of the form of the wheel show that, as is to be expected, the 
form of the vane of the wheel has some effect, but as in these 
experiments the form of the casing was not suitable for converting 
the velocity head of the water leaving the wheel into pressure 
head, the highest efficiency recorded was only 39 per cent., while 

* Sonderabdruck aus der Zeitschrift des Vereines deutscher Ingenieure, Jahrgang 
1908, and Bulletin of the University of Wisconsin, No. 672. See p. 258. 



APPENDIX 



543 



the highest efficiency for the worst form of wheel was less than 
31 per cent. Anything like a complete consideration of the effect 
of the whirlpool or free vortex chamber or of the spiral casing 
surrounding the wheel has not been attempted in the text, but 
experiment clearly shows that by their use the efficiency of the 
centrifugal pump is increased. 

In Figs. 381 and 382 are shown particulars of a pump with a free 
vortex chamber C and a spiral chamber B surrounding the wheel. 
The characteristic equation for this pump is given later. Tests 
carried out at the Des Arts et Metier, Paris, gave an overall 
efficiency of 63 per cent, when discharging 104 litres per second 
against a head of 50 metres. The vanes are radial at exit. The 
normal number of revolutions per minute is 1500. The peripheral 
velocity of the wheel is 31*4 metres per second and the theoretical 
lift is thus 

31 '4 2 
Y.QI = 100 metres, nearly, 



or the manometric efficiency is 50 per cent. 




'f 330 4 

Fig. 381. 



544. 



HYDRAULICS 



Radial 




I 

Fig. 382. Schabauer Centrifugal Pump Wheel with 8 blades, 

to prevent leakage. 



grooves 



2. Characteristic equations for Centrifugal Pumps. In- 
stability. 

The characteristic equations for centrifugal pumps have been 
discussed in the text, and for the cases there considered they have 
been shown to be of the form 
, _ mv 2 
~~ 



or since v is proportional to the number of revolutions per minute 
and u to the quantity of water delivered, the equations can be 
written in the form 



An examination of the results of a number of published experi- 
ments shows that for many pumps, by giving proper values to the 
constants, such equations express the relationship between the 
variables fairly accurately for all discharges, but for high efficiency 
pumps, with a casing carefully designed to convert at a given 
discharge a large proportion of the velocity head into pressure 
head, a condition of instability arises and the head-discharge 
curves are not continuous. This will be better understood on 
reference to Figs. 383-384, which have been plotted from the 
results of the experiments on a Schwade pump *, the construction 
of which is shown in Fig. 385. 

* Zeitschrift filr das Gesamte Turbinenwesen, 1908. 



L. H. 



CURVES FOR THREE FIXED POSITIONS 

N* 1 2 AND 3 OF THE VAL VE ON THE 

RISING MAIN. 




546 



HYDRAULICS 




APPENDIX 547 

A " forced vortex " chamber with, fixed guide vanes surrounds 
the wheel and surrounding this a spiral chamber. The diameter 
of the rotor is 420 mm. The water enters the wheel from both 
sides, so that the wheel is balanced as far as hydraulic pressures 
are concerned. The vanes of the wheel are set well back, the 
angle Q being about 150 degrees. The wheel has seven short and 
seven long vanes. The fixed vanes in the chamber surrounding 
the wheel are so formed that the direction of flow from each 
passage in this chamber is in the direction of the flow taking place 
in the spiral chamber toward the rising main. This is a very 
carefully designed pump and under the best conditions gave an 
efficiency of over 80 per cent. The performances of this pump 
at speeds varying from 531 to 656 revolutions per minute, the 
head varying from 7*657 to 13*86 metres and the discharge from 
to 275 litres per second, have been determined with considerable 
precision. In Tables XL, XLI and XLII are shown the results 
obtained at various speeds, and in Figs. 383-4 are shown head- 
discharge curves^ for speeds of 580 and 650 revolutions per 
minute. In carrying out experiments on pumps it is not easy to 
run the pumps exactly at a given speed, and advantage has been 
taken of simple reduction formulae to correct the experimental 
values of the head and the discharge obtained at a speed near to 
580 revolutions per minute or to 650 revolutions per minute 
respectively as follows. For small variations of speed the head 
as measured by the gauges is assumed to be proportional to the 
speed squared and the quantity to the speed. Thus if H , see 
page 414, is the measured head at a speed of N revolutions per 
minute and Q is the discharge, then the reduced discharge at a 
speed Ni nearly equal to N is 



and the reduced head HI is 

"\r a 

H-- * TT 
1 ~ JJ2 4 ' 

Before curves at constant speed are plotted it is desirable to 
make these reductions. Also if S is the nett work done on the 
shaft of the pump at N revolutions per minute the reduced nett 
work at Ni revolutions is taken as 



352 



548 



HYDRAULICS 



It will be seen on reference to the head-discharge curve at 650 
revolutions per minute that when the discharge reaches 120 litres 
per second the head very suddenly rises, or in other words an 
unstable condition obtains. A similar sudden rise takes place 
also at 580 revolutions per minute. The curves of Fig. 384 also 
illustrate the condition of instability. The explanation would 
appear to be that as the velocity of flow through the pump 
approaches that for which the efficiency is a maximum a sudden 
diminution in the losses by shock takes place, which is accompanied 
by a rather sudden change in the efficiency, as shown in Fig. 383. 




70 



80 90 100 110 

Quantity,- Litres per Second. 



Fig. 386. Quantity-speed curves for constant head of French pump. 

The parts of the head-discharge curves, from no discharge to 
the unstable portion, are fairly accurately represented by the 
equation 

10 5 H = 2'6N 2 + 31NQ - 16'5Q 2 



or 



H 



0-mvu - 0'904w a , 



APPENDIX 549 



and the second part of the curves by 

10 5 H - 1-46N 2 + 147NQ - 30Q 2 



or H = 

The agreement of the experimental values and the calculated 
values as obtained from these equations are seen in Tables 
XL-XLIL 

The quantity-speed curves for the pump shown in Figs. 381-2 
are shown in Fig. 386. The plotted points are experimental values 
while the curves have been plotted from the equation 

10 5 H = 2'216N 2 + 11-485NQ - 112'9Q 2 . 

The curves agree with the experimental values equally as well as 
the latter appear to agree amongst themselves. 

3. The power required to drive a pump. 

The theoretical work done in raising Q units of volume through 
a height H is 

E = w . Q . H. 
If e is the hydraulic efficiency of the pump, the work done on the 

wheel is 

w.Q.H 

e 

On reference to the triangles of velocities given on page 399 
it will be seen that when the angle of exit from the wheel is fixed 
the velocity HI is proportional to Vi and since the head is propor- 
tional to Vi the work done E is proportional to v-f or 

oo N 3 . 

The power required to drive a perfect pump would, therefore, be 
proportional to N :J , and as stated above for small changes in N the 
power required to drive an actual pump may be assumed propor- 
tional to N 3 . 

The loss of head in the pump has been shown, p. 420, to depend 
on both the velocity of the wheel and the flow through the pump, 
and it might be expected therefore that the power required to 
drive the pump can be expressed by 

S = DN 3 + Q (FNQ + GIQ 2 ), D, F and & being constants, 
or by 

S = AN 8 + N (F.NQ + GM2 2 ). 

The plotted points in Fig. 387 were obtained experimentally 
while the curves were plotted from the equation 

10 9 S = 0-852N 3 + 23'05N 2 Q + 67'7NQ 2 . 



HYDRAULICS 




POINTS OBTAINED FROM EXPERIMENTAL DATA. 
CURVES PLOTTED FROM EQUATION: 
!0 9 H.P.-0-85Z2N 3 +23-05N z Q+67-77NQ z 



60 



70 80 90 100 110 

Discharge, -Litres per Second 



130 



Fig. 387. Power Quantity Curves at various heads for Centrifugal Pump shown in 
Figs. 381, 382. 

Normal Head 50 m. 

Normal Discharge 100 L. per second. 

The equation gives reasonable values, for the heads indicated 
in the figure, up to a discharge of 130 litres per second, the values 
of N corresponding to any value of Q being taken from the 
curves, Fig. 386. In Fig. 383 the shaft-horse power at 580 and 
650 revolutions per minute respectively for various quantities of 
flow are shown. It will be seen that in each case the points lie 
very near to a straight line of which the equation is 

10 5 S = W (2'59 + 0'38Q). 

In Table XL are shown the horse-power as calculated by this 
formula and as measured by means of an Almsler transmission 
dynamometer. Closer results could, however, be probably obtained 
by taking two expressions, corresponding to the parts below and 
above the critical condition respectively, of the more rational form 
given above. 



APPENDIX 



551 



TABLE XL. 

H calculated from 10 5 H = 1'46N 2 h 147NQ -30Q 2 . 

S 



Eevs. 
per min. 

N 


Discharge 
Q litres 
per sec. 


Head 
metres 
Measured 
Hp 


Head 

metres 
Calculated 
H 


Shaft horse-power 


Measured 
So 


Calculated 
S 


652 


158 


13-799 


13-80 


36-75 


36-40 


635 


148-5 


13-03 


13-14 


32-40 


33-3 


616'3 


137-0 


12114 


12-27 


28-95 


28-6 


588-3 


88-3 


9-327 


10-36* 


18-69 


20-53 


558-0 


64-2 


8-44* 


8-59* 


15-37 


15-69 


655-7 


183-0 


13-904 


13-89 


42-3 


41-2 


633-7 


169-6 


13-02 


13-06 


35-6 


36-2 


621-3 


162-2 


12-55 


12-56 


33-35 


33-7 


597-7 


147-5 


11-62 


11-63 


28-75 


29-3 


572-0 


126-5 


10-43 


10-59 


23-45 


24-1 


555-9 


62-8 


8-35* 


8-46* 


15-58 


15-4 


531-3 


39-5 


7-65* 


7-16* 


10-94 


11-5 


677-7 


202-0 


14-56 


14-47 


47-75 


47-2 


652-7 


205-2 


13-15 


13-30 


44-40 


44-0 


627-0 


189-9 


12-30 


12-40 


38-2 


38-5 


602-7 


174-0 


11-55 


1163 


33-0 


33-50 


574-0 


156-0 


10-68 


10-65 


27-5 


28-10 


543-0 


124-0 


9-41 


9-56 


20-65 


21-50 


579-0 


159-0 


10-85 


10-82 


28-2 


28-90 


622-3 


187-0 


12-17 


12-25 


37-07 


37-60 



* These results are included although it is doubtful whether they would come 
on the part of the head- discharge curve given by the above equation. 

TABLE XLI. 
H calculated from H - 2'6N 2 + 31NQ - 16'5Q 2 

or H = ^ + 0104^ - 0-904^ 2 . 







1 


Eevs. 
per min. 

N 


Discharge 
Q litres 
per sec. 


H 

Measured 


H 

Calculated 


650 





11-01 


11 


650 


67-5 


11-469 


11-51 


650 


104-5 


11-41 


11-3 


580 





8-65 


8-75 


582-5 


22-1 


9-06 


9-17 


582 


72-9 


918 


9-26 


583 


91-3 


9-14 


9-14 


588-3 


88-3 


9-32 


9-32 


593 





9-07 


9-17 








J 



552 



HYDRAULICS 

TABLE XLIL 

H calculated from 10 5 H = 1'46N 2 + 14'7NQ - 30Q 2 . 
Revolutions per min. N = 580. 



Discharge 
Q litres 


H 

Measured 


H 

Calculated 


per sec. 






161-7 


10-87 


10-85 


217-9 


9-05 


9-25 


183-4 


10-40 


10-45 


203-6 


9-78 


9-85 


168-0 


10-96 


10-75 


143-4 


10-93 


10-94 


133-9 


10-92 


10-92 


215-9 


9-10 


9-30 


215-1 


9-32 


9-35 


221-5 


9-13 


9-05 


188-8 


10-16 


10-26 



Note : The results given in the table have been chosen haphazard from a very 
large number of experimental values. 



553 



ANSWEBS TO EXAMPLES. 



Chapter I, page 19. 

<1) 8900 Ibs. 9360 Ibs. (2) 784 Ibs. (8) 200'6 tons. 

(4) 176125 Ibs. (5) 17'1 feet. (6) 19800 Ibs. 

(7) P= 532459 Ibs. X= 13-12 ft. (8) '91 foot. (9) '089 in. 

(10) 15-95 Ibs. per sq. ft. (11) 6400 Ibs. 

(12) 89850 Ibs. 81320 Ibs. 

Chapter II, page 35. 

(1) 35,000 c. ft. (3) 2-98 ft. 

(4) Depth of C. of B. = 21-95 ft. BM= 14-48 ft. (5) 19-1 ft. 6'9 ft. 

(6) Less than 13-8 ft. from the bottom. (7) 1'57 ft. (8) 2'8ms. 

Chapter III, page 48. 

(1) -945. (2) 14-0 ft. per sec. 17'1 c. ft. per sec. (3) 25*01 ft. 

(4) 115 ft. (5) 53-3 ft. per sec. (6) 63 c. ft. per sec. 

(7) 44928ft. Ibs. 1'36 H.P. 8-84 ft. (8) 86'2 ft. 11'4 ft. per sec. 
(9) 1048 gallons. (11) 8-836. 

Chapter IV, page 78. 

(1) 80-25. (2) 3906. (3) 37'636. (4) 5 ins. diam. 

(5) 3-567 ins. (6) -763. (7) 86 ft. per sec. 115 ft. 

(8) -806. (9) -895. (10) -058. (11) 144-3 ft. per sec. 

(12) 2-94 ins. (13) '60. (14) 572 gallons. (15) 22464 Ibs. 
(16) -6206. (17) 14 c. ft. (18) '755. (19) 102 c. ft. 

(20) -875 ft. 136 Ibs. per sq. foot. 545 ft. Ibs. 

(21) 10-5 ins. 29-85 ins. (22) -683ft. (23) 4 52 minutes. 
(24) 17-25 minutes. (25) 6-29 sq. ft. (26) 1 -42 hours. 

Chapter IV, page 110. 

(1) 13,026 c. ft. (2) 4-15 ft. 

(3) 69-9 c. ft. per sec. 129-8 c. ft. per sec. (4) 2-535. 

(5) 4. (7) 43-3 c. ft. per sec. (8)' 1-675 ft. 

(9) 89-2 ft. (10) 2-22 ft. (11) 5'52 ft. (12) 23,500 c. ft. 

(13) 24,250 c. ft. (14) 105 minutes. (15) 640 H.P. 



554 ANSWERS TO EXAMPLES 



Chapter V, page 170. 
V 

(1) 27-8 ft. -0139. (2) 14-2 ft. (4) 65. (5) 3'78 ft. 

(6) 10-75. 1-4 ft. -33ft. -782ft. -0961ft. 

(8) -61 c. ft. 28-54 ft. 25-8 ft. 9 ft. (9) 26 per cent. 

(10) 1-97. 21 ft. 30 ft. 26 ft. 24 ft. 15 ft. (11) 3'64 c. ft. 

(12) 3-08 c. ft. (13) -574ft. -257ft. 7'72 ft. (14) 2'1 ft. 

(15) 1-86 c. ft. per sec. (16) F = -0468 Ibs. /='0053. 

(17) 1-023. (18) -704. (19) 2-9 ft. per sec. 

(20) 4-4 c. ft. per sec. (21) If pipe is clean 46 ft. 

(22) 23 ft. 736 ft. (23) Dirty cast-iron 6'1 feet per mile. 

(24) 8-18 feet. (26) 1 foot. 

^27) ' "A F= friction per unit area at unit velocity. 

(28) 108 H.P. (29) 1430 Ibs. 3 ins. (30) -002825. 

(31) fc=-004286. n=l-84. (32) (a) 940ft. (6) 2871 H.P. (33) '0458. 

(34) If cZ=9", v = 5 ft. per sec., and /= -0056, ft =92 and H = 182. 

(35) 1487xl0 4 . Yes. (36) 58-15 ft. (37) 54'5 hrs. 
(38) 46,250 gallons. Increase 17 per cent. (39) 295*7 feet. 
(40) 6 pipes. 480 Ibs. per sq. inch. 

(42) Velocities 6-18, 5 -08, 8-15 ft. per sec. Quantity to B = 60 c. ft. per min. 
Quantity to C= 66*6 c. ft. per min. (45) -468 c. ft. per sec. 

(46) Using formula for old cast-iron pipes from page 138, v=3'62 ft. per sec. 

(47) 2-91 ft. (48) d=3-8ins. ^ = 3-4 ins. d 2 =2-9 ins. cZ 3 =2-2 ins. 
(49) Taking C as 120, first approximation to Q is 14-4 c. ft. per sec. 

(51) d= 4-13 ins. v= 20*55 ft. per sec. p = 840 Ibs. per sq. inch. 

(53) 7-069 ft. 3-01 ft. C r =ll'9. C r for tubes = 5 -06. 

(54) Loss of head by friction = -73 ft. 

v 2 
A head equal to ^- will probably be lost at each bend. 

(56) 43-9 ft. -936 in. 

(57) ft =58'. Taking -005 to be / in formula h=-j^ , v= 16-6 ft. per sec. 

(58) V! = 8-8 ft. per sec. from A to P. v 2 = 4'95 ft. per sec. from B to P. 

V 3 = 13-75 ft. per sec. from P to C. 



Chapter VI, page 229. 

(1) 88-5. (2) 1-1 ft. diam. 

(3) Value of m when discharge is a maximum is 1'357. o>=17'62. C = 127, 

Q = 75 c. ft. per sec. 

(4) -0136. (5) 16,250 c. ft. per sec. (6) 3 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 7-35 ft. (12) Depth 10'7 ft. 

(13) Bottom width 75 ft. Slope -00052. (17) C = 87'5. 



ANSWERS TO EXAMPLES 555 

Chapter VIII, page 280. 

(1) 124-8 Ibs. -456H.P. (2) 623 Ibs. 

(3) 104 Ibs. 58-7 Ibs. 294 ft. Ibs. (4) 960 Ibs. 

(5) 261 Ibs. 4-7 H.P. (6) 21'8. (7) 57 Ibs. 

(8) 12-4, 3'4 Ibs. (9) Impressed velocity = 28'5 ft. per sec. Angle = 57. 
(10) 131 Ibs. 18-6 Ibs. (12) -93. '678. '63. (13) 19'2. 

(14) Vel. into tank =34-8 ft. per sec. Wt. lifted=10'3 tons or 8*65 tons. 

Increased resistance = 2330 Ibs. 

(15) 129 Ibs. 8-3 ft. per sec. 

(16) Work done, 575, 970, 1150, 1940 ft. Ibs. Efficiencies ^ , '50, f, 1. 

(17) 1420 H.P. (18) -9375. (19) 32 H.P. 

(20) 3666 Ibs. 161 H.P. 62 per cent. 
> 

Chapter IX, page 386. 

(1) 105 H.P. (2) = 29. V r = 4-7 ft. per sec. 

(3) 10 per minute. 11 from the top of wheel. 0=47. 

(4) 1-17 c. ft. (5) 4-1 feet. (8) 29 5'. 

(9) 10-25 ft. per sec. 1'7 ft. 6'3 ft. per sec. 19 to radius. 
(12) v = 24-7 ft. per sec. (13) 0=47 30'. a = 27 20'. 

(14) 79 15'. 19 26'. '53. 

(15) 35-6 ft. per sec. 6 24'. 23| ins. llfins. 12 39'. 16|ins. 32|ins. 

(16) 99 per cent. 0=73, a = 18. $ = 120, a = 18. 

(18) = 15323'. H = 77'64ft. H.p. = 14M6. Pressure head = 67 '3 ft. 

(19) d=l'22ft. D = 2-14ft. Angles 12 45', 125 22', 16 4'. 

(20) = 134 53', 6 = 16 25', a = 9 10'. H. p. = 2760. 

(21) 616. Heads by gauge, - 14, 35-6, 81. U = 51'5 ft. per sec. 

(22) = 153 53', a = 25. H.p. = 29'3. Eff. = -957. 

(23) Blade angle 13 30'. Vane angle 30 25'. 3'92 ft. Ibs. per Ib. 

(24) At 2' 6" radius, 6 = 10, = 23 45', a = 16 24'. At 3' 3" radius, 8 = 12 11' 

= 78 47', a = 12 45'. At 4' radius, 6 = 15 46', < = 152 11', a = 10 2] . 

(25) 79 30'. 21 40'. 41 30'. 

(26) 53 40'. 36. 24. 86'8 per cent. 87 per cent. 

(27) 12 45'. 62 15'. 31 45'. 

(28) v = 45'35. U=77. V r =44. v r =36. ^ = 23. e=73'75 per cent. 

(29) -36ft. 40 to radius. (30) About 22 ft. 
(31) H.P. =80-8. Eff.= 92-5 per cent. 

Chapter X, page 478. 

(1) 47-4 H.P. (2) 25. 53-1 ft. per sec. 94ft. 50ft 

(3) 55 per cent. (4) 52'5 per cent. 

(5) 1^ = 106 ft. g=51 ft. -55 ft. 

(6) 11 36'. 105ft. 47-4 ft. 

(7) 60 per cent. 251 H.P. 197 revs, per min. 

(8) 700 revs, per min. -81 in. Radial velocity 14-2 ft. per sec. 
(12) 15-6 ft. Ibs. per Ib. 3'05 ft. 14 ft. per sec. 



556 ANSWERS TO EXAMPLES 

(15) v=23-64ft. per sec. V=11'3. 

(16) d=9 ins. D = 19 ins. Revs, per min. 472 or higher. 

(17) 15 H. P. 9-6 ins. diam. (18) 4 -5 ft. 

(19) Vels. 1-23 and 2-41 ft. per sec. Max. accel. 2-32 and 4-55 ft. per sec. 

per sec. 

(20) 393 ft. Ibs. Mean friction head = -0268, therefore work due to friction 

is very small. 

(21) 4-61 H. P. 11-91 c. ft. per min. (22) -338. 

(23) p= 4 . Acceleration is zero when 0=(M + 2), m being any 



integer. 

(27) Separation in second case. 

(29) 67'6 and 66-1 Ibs. per sq. inch respectively. H. P. =3*14. 
(31) 7'93 ft. 25-3 ft. 59-93 ft. (32) 3'64 (33) '6. 

(34) Separation in the sloping pipe. 



Chapter XI, page 505. 

(1) 3150 Ibs. (2) 3-38 H.P. hours. (5) 4'7 ins. and 9'7 ins. 

(6) 3-338 tons. (7) 175 Ibs. per sq. inch. 

(8) 2-8 ft. per sec. (9) 2*04 minutes. 

Chapter XII, page 516. 

(1) 30,890 Ibs. 1425 H.P. (2) 3500 H.P. 

(3) 4575 Ibs. (4) 25,650 Ibs. 



557 



INDEX. 



[All numbers refer to pages.'] 



Absolute velocity 262 

Acceleration in pumps, effect of (see 

^ Reciprocating pump) 
Accumulators 

air 491 

differential 489 

hydraulic 486 
Air gauge, inverted 9 
Air vessels on pumps 451, 455 
Angular momentum 273 
Angular momentum, rate of change of 

equal to a couple 274 
Appold centrifugal pump 415 
Aqueducts 1, 189, 195 

sections of 216 
Archimedes, principle of 22 
Arm strong double power hydraulic crane 

497 
Atmospheric pressure 8 

Bacon 1 

Barnes and Coker 129 

Barometer 7 

Bazin's experiments on 

calibration of Pitot tube 245 
distribution of pressure in the plane 

of an prince 59 
distribution of velocity in the cross 

section of a channel 208 
distribution of velocity in the cross 

section of a pipe 144 
distribution of velocity in the plane of 

an orifice 59, 244 
flow in channels 182, 185 
flow over dams 102 
flow over weirs 89 
flow through orifices 56 
form of the jet from orifices 63 
Bazin's formulae for 
channels 182, 185 
orifices, sharp-edged 57, 61 
velocity at any depth in a vertical 

section of a channel 212 
Telocity at any point in the cross 

section of a pipe 144 
weir, flat crested 99 
weir, sharp-crested 97-99 
weir, sill of small thickness 99 



Bends, loss of head due to 140, 525 
Bernoulli's theorem 39 

applied to centrifugal pumps 413, 
423, 437, 439 

applied to turbines 334, 349 

examples on 48 

experimental illustrations of 41 

extension of 48 
Borda's mouthpiece 72 
Boussinesq's theory for discharge of a 

weir 104 

Boyden diffuser 314 
Brotherhood hydraulic engine 501 
Buoyancy of floating bodies 21 

centre of 23 

Canal boats, steering of 47 

Capstan, hydraulic 501 

Centre of buoyancy 23 

Centre of pressure 13 

Centrifugal force, effect of in discharge 

from water-wheel 286 
Centrifugal head 

in centrifugal pumps 405, 408, 409, 

419, 421 

in reaction turbines 303, 334 
Centrifugal pumps, see Pumps 
Channels 
circular, depth of flow for maximum 

discharge 221 
circular, depth of flow for maximum 

velocity 220 

coefficients for, in formulae of 
Bazin 186, 187 
Darcy and Bazin 183 
Ganguillet and Kutter 184 
coefficients for, in logarithmic for- 

mulae 200-208 
coefficients, variation of 190 
curves of velocity and discharge for 222 
dimensions of, for given flow deter- 
mined by approximation 225-227 
diameter of, for given maximum dis- 
charge 224 

distribution of velocity in cross sec- 
tion of 208 

earth, of trapezoidal form 219 
erosion of earth 216 



558 



INDEX 



Channels (cont.) 
examples on 223-231 
flow in 178 
flow in, of given section and slope 

223 
forms of 

best 218 

variety of 178 
formula for flow in 

applications of 223 

approximate for earth 201, 207 

Aubisson's 233 

Bazin's 185 

Bazin's method of determining the 
constants in 187 

Chezy 180 

Darcy and Bazin's 182 

Eyteiwein's 181, 232 

Ganguillet and Kutters 182, 184 

historical development of 231 

logarithmic 192, 198-200 

Prony 181, 232 
hydraulic mean depth of 179 
lined with 

ashlar 183, 184, 186, 187, 200, 206 

boards 183, 184, 187, 195, 201 

brick 183, 184, 187, 193, 195, 197, 
203 

cement 183, 184, 186, 187, 193, 202 

earth 183, 184, 186, 187, 201, 207 

gravel 183, 184 

pebbles 184, 186, 187, 206 

rubble masonry 184, 186, 187, 205 
logarithmic plottings for 193-198 
minimum slopes of, for given velocity 

215 

particulars of 195 
problems 223 (see Problems) 
sections of 216 
siphons forming part of 216 
slope of for minimum cost 227 
slopes of 213, 215 
steady motion in 178 
variation of the coefficient for 190 
Coefficients 

for orifices 57, 61, 63, 521 
for mouthpieces 71, 73, 76 (see 

Mouthpieces) 

for rectangular notches (see Weirs) 
for triangular notches 85, 522 
for Venturi meter 46 
for weirs, 88, 89, 93, 537 (see Weirs) 
of resistance 67 
Condenser 6 
Condition of stability of floating bodies 

24 

Contraction of jet from orifice 53 
Convergent mouthpiece 73 
Couple, work done by 274 
Cranes, hydraulic 494 
Crank effort diagram for three cy Under 

engine 503 
Critical velocity 129 



Current meters 239 

calibration of 240 

Gurley 238 

Haskell 240 

Curved stream line motion 518 
Cylindrical mouthpiece 73 

Dams, flow over 101 

Darcy 

experiments on flow in channels 182 
experiments on flow in pipes 122 
formula for flow in channels 182 
formula for flow in pipes 122 

Deacon's waste-water meter 254 

Density 3 

of gasoline 11 
of kerosine 11 
of mercury 8 
of pure water 4, 11 

Depth of centre of pressure 13 

Diagram of pressure on a plane area 
16 

Diagram of pressure on a vertical circle 
16 

Diagram of work done in a reciprocating 
pump 443, 459, 467 

Differential accumulator 489 

Differential gauge 8 

Discharge 

coefficient of, for orifices 60 (see 

Orifices) 

coefficient of, for Venturi meter 46 
of a channel 178 (see Channels) 
over weirs 82 (see Weirs) 
through notches 85 (see Notches) 
through orifices 50 (see Orifices) 
through pipes 112 (see Pipes) 

Distribution of velocity on cross section 
of a channel 208 

Distribution of velocity on cross section 
of a pipe 143 

Divergent mouthpieces 73 

Dock caisson 181, 192, 216 

Docks, floating 31 

Drowned nappes of weirs 96, 100 

Drowned orifices 65 

Drowned weirs 98 

Earth channels 

approximate formula for 201, 207 

coefficients for in Bazin's formula 
187 

coefficients for in Darcy and Bazin's 
formula 183 

coefficients for in Ganguillet and 
Kutter's formula 184 

erosion of 216 

Elbows, loss of head due to 140 
Engines, hydraulic 501 

Brotherhood 501 

Hastie 503 

Rigg 504 
Erosion of earth channels 216 



INDEX 



559 



Examples, solutions to which are given 

in the text 
Boiler, time of emptying through a 

mouthpiece 78 
Centrifugal pumps, determination of 

pressure head at inlet and outlet 

410 
Centrifugal pumps, dimensions for a 

given discharge 404 
Centrifugal pumps, series, numher 

of wheels for a given lift 435 
Centrifugal pumps, velocity at which 

delivery starts 412 
Channels, circular diameter, for a 

given maximum discharge 224 
Channels, diameter of siphon pipes 

to given same discharge as an 

aqueduct 224 
Channels, dimensions of a canal for 

a given flow and slope 225, 226, 227 
Channels, discharge of an earth 

channel 225 
Channels, flow in, for given section 

and slope 223 
Cranes 12, 489 
Floating docks, height of metacentre 

of 34 
Floating docks, water to be pumped 

from 33 

Head of water 7 
Hydraulic machinery, capacity of 

accumulator for working a by- 

draulic crane 489 
Hydraulic motor, variation of the 

pressure on the plunger 470 
Impact on vanes, form of vane for 

water to enter without shock and 

leave in a given direction 271 
Impact on vanes, pressure on a vane 

when a jet in contact with is turned 

through a given angle 267 
Impact on vanes, turbine wheel, 

form of vanes on 272 
Impact on vanes, turbine wheel, 

water leaving the vanes of 269 
Impact on vanes, work done on a 

vane 271 
Metacentre, height of, for a floating 

dock 34 

Metacentre, height of, for a ship 26 
Mouthpiece, discharge through, into 

a condenser 76 
Mouthpiece, time of emptying a 

boiler by means of 78 
Mouthpiece, time of emptying a 

reservoir by means of 78 
Pipes, diameter of, for a given dis- 
charge 152, 153 
Pipes, discharge along pipe connecting 

two reservoirs 151, 154 
Pipes in parallel 154 
Pipes, pressure at end of a service 

pipe 151 



Examples (cont.) 

Pontoon, dimensions for given dis- 
placement 29 

Pressure on a flap valve 13 

Pressure on a masonry dam 13 

Pressure on the end of a pontoon 13 

Eeciprocating pump fitted with an 
air vessel 470 

Reciprocating pump, horse-power of, 
with long delivery pipe 470 

Eeciprocating pump, pressure in an 
air vessel 470 

Reciprocating pump, separation in, 
diameter of suction pipe for no 469 

Reciprocating pump, separation in 
the delivery pipe 464 

Reciprocating pump, separation in, 
number of strokes at which sepa- 
ration takes place 458 

Reciprocating pump, variation of 
pressure in, due to inertia forces 470 

Reservoirs, time of emptying by weir 
108 

Reservoirs, time of emptying through 
orifice 78 

Ship, height of metacentre of 26 

Transmission of fluid pressure 12 

Turbine, design of vanes and de- 
termination of efficiency of, con- 
sidering friction 331 

Turbine, design of vanes and de- 
termination of efficiency of, fric- 
tion neglected 322 

Turbine, dimensions and form of 
vanes for given horse-power 341 

Turbine, double compartment parallel 
flow 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 head 321 

Venturi meter 46 

Water wheel, diameter of breast 
wheel for given horse-power 290 

Weir, correction of coefficient for 
velocity of approach 94 

Weir, discharge of 94 

Weir, discharge of by approximation 
108 

Weir, time of emptying reservoir by 
means of 110 

Fall Increaser 529 
Fall of free level 51 
Fire hose nozzle 73 
Flap valve, pressure on 18 

centre of pressure 18 
Floating bodies 

Archimedes, principle of 22 

buoyancy of 21 

centre of buoyancy of 23 



560 



INDEX 



Floating- bodies (cont.) 

conditions of equilibrium of 21 

containing water, stability of 29 

examples on 34, 516 

inetacentre of 24 

resistance to the motion of 507 

small displacements of 24 

stability of equilibrium, condition of 
24 

stability of floating dock 33 

stability of rectangular pontoon 26 

stability of vessel containing water 29 

stability of vessel wholly immersed 
30 

weight of fluid displaced 22 
Floating docks 31 

stability of 33 
Floats, double 237 

rod 239 

surface 237 
Flow of water 

definitions relating to 38 

energy per pound of flowing water 38 

in open channels 178 (see Channels) 

over dams 101 (see Dams) 

over weirs 81 (see Weirs) 

through notches 80 (see Notches) 

through orifices 50 (see Orifices) 

through pipes 112 (see Pipes) 
Fluids (liquids) 

at rest 3-19 

examples on 19 

compressible 3 

density of 3 

flow of, through orifices 50 

incompressible 3 

in motion 37 

pressure in, is the same in all direc- 
tions 4 

pressure on an area in 12 

pressure on a horizontal plane in, is 
constant 5 

specific gravity of 3 

steady motion of 37 

stream line motion in 37, 517 

transmission of pressure by 11 

used in U tubes 9 

viscosity of 2 

Forging press, hydraulic 492 
Fourneyron turbine 307 
Friction 

coefficients of, for ships' surfaces 509, 
515 

effect of, on discharge of centrifugal 
pump 421 

effect of, on velocity of exit from Im- 
pulse Turbine 373 

effect of, on velocity of exit from 
Poncelet Wheel 297 

Froude's experiments on fluid 507 

in centrifugal pumps 400 

in channels 180 

in pipes 113, 118 



Friction (cont.) 

in reciprocating pumps 449 
in turbines 313, 321, 339, 373 

Ganguillet and Kutter 

coefficients in formula of 125, 184 

experiments of 183 

formula for channels 184 

formula for pipes 124 
Gasoline, specific gravity of 11 
Gauges, pressure 

differential 8 

inverted air 9 

inverted oil 10 
Gauging the flow of water 234 

by an orifice 235 

by a weir 247 

by chemical means 258 

by floats 239 (see Floats) 

by meters 234, 251 (see Meters) 

by Pitot tubes 241 

by weighing 234 

examples on 260 

in open channels 236, 540 

in pipes 251 

Glazed earthenware pipes 186 
Gurley's current meter 238 

Hammer blow in a long pipe 384 
Haskell's current meter 240 
Hastie's engine 503 
Head 

position 39 

pressure 7, 39 

velocity 39 

High pressure pump 471 
Historical development of pipe and 

channel formulae 231 
Hook gauge 248 
Hydraulic accumulator 486 
Hydraulic capstan 501 
Hydraulic crane 494 

double power 495 

valves 497 

Hydraulic differential accumulator 490 
Hydraulic engines 501 

crank effort diagram for 503 
Hydraulic forging press 492 
Hydraulic gradient 115 
Hydraulic intensifier 491 
Hydraulic machines 485 

conditions which vanes of, must 
satisfy 270 

examples on 489, 505 

joints for 485 

maximum efficiency of 295 

packings for 485 
Hydraulic mean depth 119 
Hydraulic motors, variations of pressure 

in, due to inertia forces 469 
Hydraulic ram 474, 537 
Hydraulic riveter 499 
Hydraulics, definition of 1 



INDEX 



561 



Hydrostatics 4-19 

Impact of water on vanes 261 (see Vanes) 
Inertia forces in hydraulic motors 469 
Inertia forces in reciprocating pumps 

445 

Inertia, moment of 14 
Inverted air gauge 9 
Inverted oil gauge 9 
Intensifies, hydraulic 491 

non-return valves for 492 
Intensifiers, steam 493 
Inward flow turbines 275, 318 (see 

Turbines) 

Joints used in hydraulic work 485 

Kennedy meter 255 
Kent Venturi meter 253 
Kerosene, specific gravity of 11 

Leathers for hydraulic -packings 486 
Logarithmic formulae for flow 

in channels 192 

in pipes 125 
Logarithmic plottinga 

for channels 195 

for pipes 127, 133 
Luthe valve 499 

Masonry dam 17 
Mercury 

specific gravity of 8 

use of, in barometer 7 

use of, in U tubes 8 
Metacentre, height of 24 
Meters 

current 239 

Deacon's waste water 254 

Kennedy 255 

Leinert 234 

Venturi 44, 75, 251 
Moment of inertia 14 

of water plane of floating body 25 

table of 15 

Motion, second law of 263 
Mouthpieces 54 

Borda's 72 

coefficients of discharge for 
Borda's 73 
conical 73 
cylindrical 71, 76 
fire nozzle 73 

coefficients of velocity for 71, 73 

conical 73 

convergent 73 

cylindrical 73 

divergent 73 

examples on 78 

flow through, under constant pressure 
75 

loss of head at entrance to 70 

time of emptying boiler through 78 

L. H. 



Mouthpieces (cont.) 

time of emptying reservoir through 
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 263 
Notation used in connection with vanes, 
turbines and centrifugal pumps 272 
Notches 

coefficients for rectangular (see Weirs) 

coefficients for triangular 85 

rectangular 80 (see 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 377 
Orifices 

Eazin's coefficients for 57, 61 
Bazin's experiments on 56 
coefficients of contraction 52, 56 
coefficients of discharge 57, 60, 61, 

63, 521 

coefficients of velocity 54, 57 
contraction complete 53, 57 
contraction incomplete or suppressed 

53, 63 
distribution of pressure in plane of 

59 

distribution of velocity in plane of 69 
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 from 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, 368 
Parallel flow turbine pump 437 
Pelton wheel 276, 377, 380 
Piezometer fittings 139 
Piezometer tubes 7 

36 



562 



INDEX 



Pipes, flow of air in 539 

bends, loss of head due to 141, 525 

coefficients 

C in formula v 



and 



"2gd 



for cast iron, new and old 120, 

121, 122, 123, 124 
for steel riveted 121 
for Darcy's formula 122 
for logarithmic formulae 
brass pipes 133, 138 
cast iron, new and old 125, 137, 

138 

glass 135 
riveted 137, 138 
wood 135, 138 
wrought iron 122, 135, 138 
n in Ganguillet and Kutter's formula 
cast iron, new and old 125 
for glazed earthenware 125 
for steel riveted 184 
for wood pipes 125, 184 
variation of, with service 123 
connecting three reservoirs 155 
connecting two reservoirs 149 
connecting two reservoirs, diameter of 

for given discharge 152 
critical velocity in 128, 522 
Darcy's formula for 122 
determination of the coefficient C, 
as given in tables by logarithmic 
plotting 132 
diameter of, for given discharge 

152 

diameter for minimum cost 158, 177 
diameter varying 160 
divided into two branches 154 
elbows for 141 

empirical formula for head lost in 119 
empirical formula for velocity of flow 

in 119 

equation of flow in 117 
examples on flow in 149-162, 170 
experimental determination of loss of 

head by friction in 116 
experiments on distribution of velocity 

in 144 
experiments on flow in, criticism of 

138 
experiments on loss of head at bends 

142 
experiments on loss of head in 122, 

125, 129, 131, 132, 136,539 
experiments on loss of head in, 

criticism of 138 
flow through 112 
flow diminishing at uniform rate in 

157 

formula for 
Chezy 119 
Darcy 122 



Pipes (cont.) 

formula for (cont.) 

logarithmic 125, 131, 133, 137-138 

Reynolds 131 

nummary of 148,539 

velocity at any point in a cross 

section of 143 
friction in, loss of head by 113 

determination of 116 
Ganguillet and Kutter's formula for 

124 

gauging the flow in 251 
hammer blow in 384 
head lost at entrance of 70, 114 
head lost by friction in 113 
head lost by friction in, empirical 

formula for 119 
head lost by friction in, examples on 

150-162, 170 
head lost by friction in, logarithmic 

formula for 125, 133 
head required to give velocity to 

water in the pipe 146 
head required to give velocity to water 

in the pipe, approximate value 113 
hydraulic gradient for 115 
hydraulic mean depth of 118 
joints for 485 
law of frictional resistance for, above 

the critical velocity 130 
law of frictional resistance for, below 

the critical velocity 125 
limiting diameter of 165 
logarithmic formula for 125 
logarithmic formula for, coefficients 

in 138 

logarithmic formula, use of, for prac- 
tical calculations 136 
logarithmic plottings for 126 
nozzle at discharge end of, area of 

when energy of jet is a maximum 
159 

when momentum of jet is a maxi- 
mum 159 

piezometer fittings for 139 
pressure on bends of 166 
pressure on a cylinder in 169 
pressure on a plate in 168 
problems 147 (see Problems) 
pumping water through long pipe, 

diameter for minimum cost 158, 177 
resistance to motion of fluid in 112 
rising above hydraulic gradient 115 
short 153 
siphon 161 
temperature, effect of, on velocity of 

flow in 131, 140, 524 
transmission of power along, by hy- 
draulic pressure 158, 162, 177 
values of C in the formula v = C*Jmi 

for 120, 121 

variation of C in the formulav = 
for 123 



INDEX 



563 



Pipes (cont.) 

variation of the discharge of, with 

service 123 

velocity of flow allowable in 162 
velocity, head required to give velocity 

to water in 146 
velocity, variation of, in a cross section 

of a pipe 143 
virtual slope of 115 
Pitot tube 241, 526 
calibration of 245 
Poncelet water wheel 294 
Pontoon, pressure on end of 18 
Position head 29 
Press, forging 493 
Press, hydraulic 493, 498 
Pressure 

at any point in a fluid 4 
atmospheric, in feet of water 8 
gauges 8 
head 7 

measured in feet of water 7 
on a horizontal plane in a fluid 5 
on a plate in a pipe 168 
on pipe bends 166 
Principle of Archimedes 19 
Principle of similarity 84 
Problems, solutions of which are given 

in the text 
channels 
diameter of, for a given maximum 

discharge 224 
dimensions of, for a given flow 

225-227 

earth discharge along, of given di- 
mensions and slope 224 
flow in, of given section and slope 

223 

slope of, for minimum cost 227 
solutions of, by approximation 

225-227 
pipes 

acting as a siphon 161 
connecting three reservoirs 155 
connecting two reservoirs 149 
diameter of, for a given discharge 

152 

divided into two branches 154 
head lost in, when flow diminishes 

at uniform rate 157 
loss of head in, of varying diameter, 

160, 161 
pumping water along, diameter of, 

for minimum cost 158, 177 
with nozzle at the end 158, 159 
Propulsion of ships by water jets 279 
Pumping water through long pipes 158 
Pumps 

centrifugal 392 and 542 
advantages of 435 
Appold 415 

Bernoulli's equation applied to 
413 



Pumps (cont.) 
centrifugal (cont.) 

centrifugal head, effect of variation 

of on discharge 421 
centrifugal head, impressed on the 

water by the wheel 405 
design of, for given discharge 402 
discharge, effect of the variation 

of the centrifugal head and loss 

by friction on 419 
discharge, head-velocity curve at 

zero 409 
discharge, variation of with the 

head at constant speed 410 
discharge, variation of with speed 

at constant head 410 
efficiencies of 401 
efficiencies of, experimental de- 
termination of 401 
examples on 404, 412, 414, 418, 

435, 478 

form of vanes 396 
friction, effect of on discharge 419, 

421 
general equation for 421, 425, 428, 

430 

gross lift of 400 
head-discharge curve at constant 

velocity 410, 412, 427 
head lost in 414 
head, variation of with discharge 

and speed 418 
head-velocity curve at constant 

discharge 429 

head-velocity curve at zero dis- 
charge 409 
Horse-Power 549 

kinetic energy of water at exit 399 
limiting height to which single 

wheel pump will raise water 431 
limiting velocity of wheel 404 
losses of head in 414 
multi-stage 433 
series 433 

spiral casing for 394, 429 
starting of 395 
suction of 431 
Sulzer series 434 
Thomson's vortex 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 constant 

head 411, 412, 421, 428 
velocity, head-discharge curve for at 

constant 410 
velocity head, special arrangement 

for converting into pressure head 

422 
velocity, limiting, of rim of wheel 

404 



564 



INDEX 



Pumps (cont.) 
centrifugal (cont.) 

velocity of whirl, ratio of, to velocity 

of outlet edge of vane 398 
vortex chamber of 397, 407, 422 
with whirlpool or vortex chamber 

397, 407, 422 

work done on water by 397 
compressed air 477 
duplex 473 

examples on 458, 464, 469, 478 
force 392 
high pressure 472 
Humphrey Gas 531 
hydraulic ram 476 
packings for plungers of 472, 486 
reciprocating 439 

acceleration, effect of on pressure 

in cylinder of a 446, 448 
acceleration of the plunger of 444 
acceleration of the water in delivery 

pipe of 448 
acceleration of the water in suction 

pipe of 445 

air vessel on delivery pipe of 454 
air vessel on suction pipe of 451 
air vessel on suction pipe, effect of 

on separation 462 
coefficient of discharge of 442 
diagram of work done by 443, 450, 

459, 467 

discharge, coefficient of 443 
duplex 473 

examples on 458, 464, 469, 470, 480 
friction, variation of pressure in the 

cylinder due to 449 
head lost at suction, valve of 468 
head lost by friction in the suction 

and delivery pipes 449 
high pressure plunger 471 
pressure in cylinder of when the 

plunger moves with simple har- 
monic motion 446 
pressure in the cylinder, variation 

of due to friction 449 
separation in delivery pipe 463 
separation during suction stroke 

456 
separation during suction stroke 

when plunger moves with simple 

harmonic motion 458, 461 
slip of 442, 461 
suction stroke of 441 
suction stroke, separation in 456, 

461, 462 

Tangye duplex 473 
vertical single acting 440 
work done by 441 
work done by, diagram of 443, 459, 

467 

turbine 396, 425 
head-discharge curves at constant 

speed 427,545 



Pumps (cont.) 
turbine (cont.) 
head-velocity curves at constant 

discharge 429 
inward flow 439 
multi-stage 433 
parallel flow 437 
velocity-discharge curves at constant 

head 428, 548 
Worthington 432 
work done by 443 
work done by, diagram of (see Ee- 

ciprocating pumps) 
work done by, series 433 

Reaction turbines 301 
limiting head for 367 
series 367 
Eeaction wheels 301 

efficiency of 304 

Eeciprocating pumps 439 (see Pumps) 
Eectangular pontoon, stability of 26 
Eectangular sharp-edged weir 81 
Eectangular sluices 65 
Eectangular weir with end contrac- 
tions 88 
Eegulation of turbines 306, 317, 318, 

323, 348 
Eegulators 

oil pressure, for impulse turbine 377 
water pressure, for impulse turbine 

379 
Eelative velocity 265 

as a vector 266 
Eeservoirs, time of emptying through 

orifice 76 
Eeservoirs, time of emptying over weir 

109 

Eesistance of ship 510 
Eigg hydraulic engine 503 
Eivers, flow of 191, 207, 211 
Eivers, scouring banks of 520 
Eiveter, hydraulic 500 

Scotch turbine 301 

Second law of motion 263 

Separation (see Pumps) 

Sharp-edged orifices 
Bazin's experiments on 56 
distribution of velocity in the plane of 59 
pressure in the plane of 59 
table of coefficients for, when con- 
traction is complete 57, 61, 521 
table of coefficients for, when con- 
traction is suppressed 63 

Sharp-edged weir 81 (see Weirs) 

Ships 

propulsion of by water jets 279 
resistance of 510 
resistance of, from model 515 
stream line theory of the resistance 
of 510 

Similarity, principle of 84 



INDEX 



565 



Siphon, forming part of aqueduct 216 

pipe 161 

Slip of pumps 442, 461 
Sluices 65 

for regulating turbines (see Turbines) 
Specific gravity 3 

of gasoline 11 

of kerosene 11 

of mercury 8 

of oils, variation of, with temperature 
11 

of pure water 4 

variation of, with temperature 11 
Stability of 

floating body 24, 25 

floating dock 31 

floating vessel containing water 29 

rectangular pontoon 26 
Steady motion of fluids 37 
Steam intensifier 493 
Stream line motion 37. 129^ 517 

curved 518 

Hele Shaw's experiments on 284 
Stream line theory of resistance of 

ships 510 

Suction in centrifugal pump 431 
Suction in reciprocating pump 441 
Suction tube of turbine 306 
Sudden contraction of a current of 

water 69 
Sudden enlargement of a current of 

water 67 

Sulzer, multi-stage pump 434 
Suppressed contraction 53 

effect of, on discharge from orifice 
62 

effect of, on discharge of a weir 82 

Tables 

channels, sewers and aqueducts, par- 
ticulars of, and values of in 

to- p 

formula i= - 195 

channels 

slopes and maximum velocities of 

flow in 215 
values of a and ft in Bazin's formula 

183 

values of v and i as determined 
experimentally and as calculated 
from logarithmic formulae 198, 
201-208 

coefficients for dams 102 
coefficients for sharp-edged orifice, 

contraction complete 57, 61 
coefficients for sharp-edged orifice, 

contraction suppressed 63 
coefficients for sharp-edged weirs 89, 

93 

coefficients for Venturi meters 46 
earth channels, velocities above which 
erosion takes place 216 



Tables (cont.) 

minimum slopes for varying values 
of the hydraulic mean depth of 
brick channels that the velocity 
may not be less than 2 ft. per 
second 215 

moments of Inertia 15 

Pelton wheels, particulars of 377 

pipes 
lead, slope of and velocity of flow 

in 128 
reasonable values of y and n in 



the formula h = 



138 



values of C in the formula 

v = G\lmi 120, 121 
values of / in the formula 

121 



2gd 
values of n in Ganguillet and 

Kutter's formula 125, 184 
values of n and k in the formula 

i = kv n 137 
resistance to motion of boards in 

fluids 509 

turbines, peripheral velocities and 
heads of inward and outward flow 
333 

useful data 3 
Thomson, centrifugal pump, vortex 

chamber for 397, 407, 422 
principle of similarity 62 
turbine 323 
Time of emptying tank or reservoir by 

an orifice 76 
Time of emptying a tank or reservoir 

by a weir 109 
Torricelli's theorem 1 

proof of 51 
Total pressure 12 
Triangular notches 80, 522 

discharge through 85 
Turbines 

axial flow 276, 342 

axial flow, impulse 368 

axial flow, pressure or reaction 342 

axial flow, section of the vane with 

the variation of the radius 344 
Bernoulli's equations for 334 
best peripheral velocity for 329 
central vent 320 
centrifugal head impressed on water 

by wheel of 334 
cone 359 

design of vanes for 346 
efficiency of 315, 331 
examples on 311, 321, 323, 331, 341, 

349, 385, 387 
fall increaser for 529 
flow through, effect of diminishing, by 
means of moveable guide blades 362 
flow through, effect of diminishing 
by means of sluices 364 



566 



INDEX 



Turbines (cont.) 
flow through, effect of diminishing 

on velocity of exit 363 
Fontaine, regulating sluices 348 
form of vanes for 308, 347, 365 
Fourneyron 306 
general formula for 31 
general formula, including friction 

315 
guide blades for 320, 326, 348, 352, 

362 
guide blades, effect of changing the 

direction of 362 
guide blades, variation of the angle 

of, for parallel flow turbines 344 
horse power, to develop a given 

339 
impulse 300, 369-384 

axial flow 368 

examples 387 

for high heads 373 

form of vanes for 371 

Girard 369, 370, 373 

hydraulic efficiency of 371, 373 

in airtight chamber 370 

oil pressure regulator for 377 

radial flow 370 

triangles of velocities for 372 

triangles of velocities for considering 
friction 373, 376 

water pressure regulator for 379 

water pressure regulator, hydraulic 
valve for 382 

water pressure regulator, water filter 
for 383 

work done on wheel per Ib. of water 

272, 277, 323 
inclination of vanes at inlet of wheel 

308, 321, 344 
inclination of vanes at outlet of wheel 

308, 321, 345 
in open stream 360 
inward flow 275, 318 

Bernouilli's equations for 334, 339 

best peripheral velocity for, at 
inlet 329 

central vent 320 

examples on 321, 331, 341, 387 

experimental determination of the 
best velocity for 329 

for low and variable falls 328 

Francis 320 

horizontal axis 327 

losses in 321 

Thomson 324 

to develop a given horse- power 
339 

triangles of velocities for 322, 326, 
332 

work done on the wheel per Ib. of 

water 321 
limiting head for reaction turbine 

367 



Turbines (cont.) 

loss of head in 313, 321 
mixed flow 350 

form of vanes of 355 

guide blade regulating gear for 
352-354 

in open stream 360 

Swain gate for 374 

triangles of velocities for 355- 
356 

wheel of 351 
Niagara falls 318 
oil pressure regulator for 377 
outward flow, 275, 306 

Bernouilli's equations for 334, 
339 

best peripheral velocity for, at inlet 
329 

Boyden 314 
diffuser for 314 

double 316 

examples on 311, 387 

experimental determination of the 
best velocity for 329 

Fourneyron 307 

losses of head in 313 

Niagara falls 318 

suction tube of 308, 317 

triangles of velocities for 308 

work done on the wheel per Ib. of 

water 310, 315 
parallel flow 276, 342 

adjustable guide blades for 348 

Bernouilli's equations for 348 

design of vanes for 344 

double compartment 343 

examples on 349, 387 

regulation of the flow to 348 

triangles of velocities for 344 
reaction 301 

axial flow 276-342 

cone 359 

inward flow 275, 318 

mixed flow 350 

outward flow 306 

parallel flow 276-342 

Scotch 302 

series 368 
regulation of 306, 317, 318, 323, 348, 

350, 352, 360, 362, 364 
Scotch 301 
sluices for 305, 307, 316, 317, 319, 

327, 328, 348, 350, 361, 364 
suction tube of 306 
Swain gate for 364 
Thomson's inward flow 323 
to develop given horse-power 339 
triangles of velocities at inlet and 

outlet of impulse 372, 376 
triangles of velocities at inlet and 

outlet of inward flow 308 
triangles of velocities at inlet and 

outlet of mixed flow 356 



INDEX 



567 



Turbines (cont.) 

triangles of velocities at inlet and 

outlet of outward flow 344 
triangles of velocities at inlet and 

outlet of parallel flow 344 
types of 300 
vanes, form of 

between inlet and outlet 365 

for inward flow 321 

for mixed flow 351, 356 

for outward flow 311 

for parallel flow 344 
velocity of whirl 273, 310 

ratio of, to velocity of inlet edge 

of vane 332 

velocity with which water leaves 334 
wheels, path of water through 312 
wheels, peripheral velocity of 333 
Whitelaw 302 
work done on per Ib. of flow, 275, 

304, 315 

Turning moment, work done by 273 
Tweddell's differential accumulator 489 

U 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 

head 39 

of approach to orifices 66 

of approach to weirs 90 

relative 265 

Venturi meter 44, 75, 251 
Virtual slope 115 
Viscosity 2,539 

Water 

definitions relating to flow of 38 



Water (cont.) 
density of 3 
specific gravity of 3 
viscosity of 2 
Water wheels 
Breast 288 
effect of centrifugal forces on water 

286 

examples on 290, 386 
Impulse 291 
Overshot 283 
Poncelet 294 
Sagebien 290 

Undershot, with flat blades 292 
Weirs 

Bazin's experiments on 89 
Boussinesq's theory of 104 
circular 537 
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 ot 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 
gauging flow of water by 247 
nappe of 

adhering 95, 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 



568 INDEX 

Weirs (conf.) Weirs (conf.) 

time required to lower water in velocity of approach, effect of on 

reservoir by means of 109 discharge 90 

various forms of 101 wide flat-crested 100 

velocity of approach, correction of Whitelaw turbine 302 

coefficient for 92 Whole pressure 12 

velocity of approach, correction of Worthington multi-stage pump 433 

coefficient for, examples on 94 



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