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HYDRAULIC TURBINES

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HYDEAULIC TUEBINES

WITH A CHAPTER ON

CENTRIFUGAL PUMPS

R. L. DAUGHERTY, A. B., M. E.

PROFESSOR OF MECHANICAL AND HYDRAULIC ENGINEERING, CALIFORNIA

INSTITUTE OF TECHNOLOGY; FORMERLY PROFESSOR OF HYDRAULIC

ENGINEERING, RENSSELAER POLYTECHNIC INSTITUTE.

THIKD EDITION
REVISED, ENLARGED AND RESET

IMPRESSION

McGRAW-HILL BOOK COMPANY, INC.

NEW YORK: 239 WEST 39TH STREET

LONDON: 6 & 8 BOUVERIE ST., E. C. 4

1920

THE MAPLE PRE S B YORK PA

PREFACE TO THE THIRD EDITION

Since this book was first written, practice has changed to such
an extent that many statements, which were true at that time,
are not true today. These portions have been entirely rewritten
so as to present the very latest features in construction and
practice. Also practically every other chapter has been altered
and new matter and illustrations inserted, where it was thought
that greater clearness could be so attained.

The presentation of the theory has been quite carefully
considered and has been largely rewritten in order to be more
effective. An attempt has been made to so arrange this that
the fundamental principles could be grasped without going into
a lot of technical details. If desired, certain portions of the
latter, of theoretic interest only, can be omitted without brea ing
the continuity of thought.

Chapters have been added on turbine governors and on the
methods of turbine design. The latter has been inserted in
order to meet a demand for something on that subject. The
methods that are given are those employed by the best designers
at the present time. The procedure avoids the old " cut and
try" practice on .the one hand, as well as a highly theoretical
treatment, that is of no practical value, on the other. It is
rather a happy compromise between the two. The author is
still of the opinion that the greater number of engineers are
concerned with the construction and operating characteristics
of turbines rather than with the details of their design. But
there are some phases of turbine performance and construction
that can be understood more completely, if approached from the
view point of hydraulic design.

Questions and numerical problems have been added at the
end of every chapter, in order to increase the usefulness of the
book for instruction purposes. The questions are intended to
call attention to the most important features presented in the
chapter and also to bring out more clearly the thought there
expressed. The problems are arranged so as to afford applica-
tions of the principles stated and are hence quite limited in
character. If time available for the course permits, it is thought

415328

vi PREFACE

that problems of a more general character are desirable. But
it is believed that it is better for the instructor to prepare these
to suit his individual course, and to vary them from year to
year, rather than to incorporate such in the book.

The notation has been changed slightly in the present edition
in order to conform more closely to the standard notation
recommended by the Society for the Promotion of Engineering
Education.

The author is indebted to many teachers and students, who
have used the former editions, and also to engineers with whom
he has discussed these matters for numerous suggestions which
have been helpful to him in the preparation of the present volume.

R. L, D.
PASADENA, CALIF.,
February, 1920.

PREFACE TO THE SECOND EDITION

In addition to correcting typographical errors and rewriting
two articles, the issuing of a second edition has afforded an
opportunity to add new material which it is believed will increase
the sphere of usefulness of the book. The discussion of several
matters in the text has been amplified and there have been added
numerous questions and problems. This together with the 15
tables of test data in Appendix C will afford much suitable
material for instruction purposes.

The author wishes to acknowledge his indebtedness to Prof.
E. H. Wood of Cornell University for his careful criticism of the
first edition and to Prof. W. F. Durand of Leland Stanford
University for much valuable assistance.

ft. L. D.

ITHACA, N. Y.,

August, 1914.

PREFACE TO THE FIRST EDITION

The design of hydraulic turbines is a highly specialized
industry, requiring considerable empirical knowledge, which
can be aquired only through experience; but it is a subject in
which comparatively few men are interested, as a relatively
small number are called upon to design turbines. But with the
increasing use of water power many men will find it necessary
to become familiar with the construction of turbines, understand
their characteristics, and be able to make an intelligent selection
of a type and size of turbine for any given set of conditions.
To this latter class this book is largely directed. However, a
clear understanding of the theory, as here presented, ought to
be of interest to many designers, since it is desirable that Ameri-
can designs be based more upon a mathematical analysis, as in
Europe, and less upon the old cut and try methods.

The br.oad problem of the development of water power is
treated in a very general way so that the reader may understand
the conditions that bear upon the choice of a turbine. Thus the
very important items of stream gauging and rating, rainfall and
runoff, storage, etc., are treated very briefly, the detailed study
of these topics being left for other works.

The purpose of the text is to give the following: A general
idea of water-power development and conditions affecting the
turbine operation, a knowledge of the principal features of
construction of modern turbines, an outline of the theory and
the characteristics of the principal types, commercial constants,
means of selection of type and size of turbine, cost of turbines
and water power and comparison with cost of steam power.
A chapter on centrifugal pumps is also added. It is hoped that
the book may prove of value both to the student as a text and
to the practicing engineer as a reference. R. L. D.

CORNELL UNIVERSITY, ITHACA, N. Y.,
August, 1913.

viu

CONTENTS

PAGE

PREFACE V ..... . .".••.•• v

CHAPTER I

INTRODUCTION 1

Historical — The turbine — Advantage of turbine over water wheel —
Advantages of water wheel over turbine — Essentials of a water-
power plant — Questions.

CHAPTER II

TYPES OF TURBINES AND SETTINGS 7

Classification of turbines— Action of water — Direction of flow —
Position of shaft — Arrangement of runners — The draft tube —
Flumes and penstocks — Questions.

CHAPTER III

WATER POWER 15

Investigation — Rating curve — The hydrograph — Rainfall and
run-off — Absence of satisfactory hydrograph — Variation of head —
Power of stream — Pondage and load curve — Storage — Storage and
turbine selection — Power transmitted through pipe line — Pipe line
and speed regulation — Questions and problems.

CHAPTER IV

THE TANGENTIAL WATER WHEEL 30

Development — Buckets — General proportions — Speed regulation
— Conditions of use — Efficiency — Questions.

CHAPTER V

THE REACTION TURBINE 41

Development — Advantages of inward flow turbine — General pro-
portions of types of runners — Comparison of types of runners —
Runners — Speed regulation — Bearings — Cases — Draft tube con-
struction— Velocities — Conditions of use — Efficiency — Questions.

x CONTENTS

CHAPTER VI

TURBINE GOVERNORS 74

General principles — Types of governors — The compensated gover-
nor— Questions.

CHAPTER VII

GENERAL THEORY ...*..' 80

Equation of continuity — Relation between absolute and relative
velocities — The general equation of energy — Effective head on
wheel — Power and efficiency — Force exerted — Force upon moving
object — Torque exerted — Power and head delivered to runner —
Equation of energy for relative motion — Impulse turbine— Reac-
tion turbine — Effect of different speeds — Forced vortex — Free
vortex — Theory of draft tube — Questions and problems.

CHAPTER VIII

THEORY OF THE TANGENTIAL WATER WHEEL 105

Introductory — The angle a\— ^The ratio of the radii — Force exerted
— Power — The value of W — The value of k — Constant input —
Variable speed — Best speed — Constant speed — Variable input —
Observations on theory — Illustrative problem — Questions and
problems.

CHAPTER IX

THEORY OF THE REACTION TURBINE 121

Introductory — Simple theory — Conditions for maximum efficiency
— Determination of speed for maximum efficiency — Losses — Rela-
tion between speed and discharge — Torque, power, and efficiency —
Variable speed and constant gate opening — Constant speed and
variable input — Runner discharge conditions — Limitations of
theory — Effect of y — Questions and problems.

CHAPTER X

TURBINE TESTING ; 140

Importance — Purpose of test — Measurement of head — Measure-
ment of water — Measurement of output — Working up results —
Determination of mechanical losses — Questions and problems.

CHAPTER XI

GENERAL LAWS AND CONSTANTS 150

Head — Diameter of runner — Commercial constants — Diameter and
discharge — Diameter and power — Specific speed — Determintion of
constants — Illustrative case — Uses of constants — Numerical illustra-
tions— Questions and problems.

CONTENTS xi

CHAPTER XII

TURBINE CHARACTERISTICS 164

Efficiency as a function of speed and gate opening — Specific speed
an index of type — Illustrations of specific speed — Selection of a
stock turbine — Illustrative case — Variable load and head — Char-
acteristic curve — Use of characteristic curve — Questions and
problems.

r" CHAPTER XIII

SELECTION OF TYPE OF TURBINE . . . . . . . . . 177

Possible choice — Maximum efficiency — Efficiency on part-load —
Overgate with high-speed turbine — Type of runner as a function
of head — Choice of type for low head — Choice of type for medium
head — Choice of type for high head — Choice of type for very high
head — Questions and problems.

CHAPTER XIV

COST OF TURBINES AND WATER POWER 192

General considerations — Cost of turbines — Capital cost of water
power — Annual cost of water power — Cost of power per horse-
power-hour— Sale of power — Comparison with steam power —
Value of water power — Questions and problems.

CHAPTER XV

DESIGN OF THE TANGENTIAL WATER WHEEL 204

General dimensions — Nozzle design — Pitch of buckets — Design
of buckets — Dimensions of case — Questions and problems.

CHAPTER XVI

DESIGN OF THE REACTION TURBINE 211

Introductory — General dimensions — Profile of runner — Outflow
conditions and clear opening — Layout of vane on developed cones-
Intermediate profiles — Pattern maker's sections — The case and
speed ring — The guide vanes — Questions and problems.

CHAPTER XVII

CENTRIFUGAL PUMPS 230

Definition — Classification — Centrifugal action — Notation — Defini-
tion of head and efficiency — Head imparted to water — Losses —
Head of impending delivery — Relation between head, speed and
discharge — Defects of theory — Efficiency of a given pump —
Efficiency of series of pumps — Specific speed of centrifugal pumps
— Conditions of service — Construction — Questions and problems.

xii CONTENTS

APPENDIX A 249

The retardation curve

APPENDIX B . . . . . . 251

Stream lines in curved channels.

APPENDIX C . . .- . . . . 256

Test data.

INDEX . 279

NOTATION

A = total area of streams in square feet measured normal to absolute

velocity.
a = total area of streams in square feet measured normal to relative

velocity.

B = height of turbine runner in inches.
c = coefficient of discharge in general.

cc = coefficient of contraction.
cv = coefficient of velocity.
cr = coefficient of radial velocity.
cu = coefficient of tangential velocity.
D = diameter of turbine runner in inches.
e = efficiency.
eh = hydraulic efficiency.
em = mechanical efficiency.
cv = volumetric efficiency.
F = force in pounds.
/ = friction factor.

g = acceleration of gravity in feet per second per second.
H = total effective head = z + V*/2g + p/w.
H' = any loss in head in feet.
h = head in feet.

h' = head lost in friction in turbine or pump.
h" = head converted into mechanical work or vice versa.
K = any factor.
K\ = capacity factor.
Kz = power factor.
k = any coefficient of loss.
N = revolutions per minute.
Ne — speed for maximum efficiency.
Ns = specific speed = Ne\/}$.h.p./h
m = abstract number.
n = abstract number.
0 = axis of rotation. \ ;.•

p = power.

p = intensity of pressure in pounds per square foot.
Q = total quantity in cubic feet.
q = rate of discharge in cubic feet per second.
R = resultant force.
r = radius in feet.
T = torque in foot-pounds.
i = time in seconds.

xiii

xiv NOTATION

u = linear velocity of a point on wheel in feet per second.
V = absolute velocity (or relative to earth) of water in feet per second.
Vr — radial component of velocity = V sin or.
Frt = tangential component of absolute velocity = V cos a.

v = velocity of water relative to wheel in feet per second.
W = pounds of water per second = wq.
w = density of water in pounds per cubic foot.
x = rt/ri.
y = Ai/a2.

a = angle between V and u (measured between positive directions).
/3 = angle between v and u (measured between positive directions).
<£ = ratio Ui/\/2gh.

<j>e = value of <f> for maximum efficiency,
to = angular velocity = u/r.

The subscript (1) refers to the point of entrance and the subscript (2)
refers to the point of outflow in every case.

HYDRAULIC TURBINES

CHAPTER I
INTRODUCTION

1. Historical. — Water power was utilized many centuries
ago in China, Egypt, and Assyria. The earliest type of water
wheel was a crude form of the current wheel, the vanes of which
dipped down into the stream and were acted upon by the impact
of the current (Fig. 1). A large wheel of this type was used
to pump the water supply of London about 1581. Such a wheel
could utilize but a small per cent, of the available energy of
the stream. The current wheel, while very inefficient and
limited in its scope, is well suited for certain purposes and is
not yet obsolete. It is still in use in parts of the United States,

FIG. 1. — Current wheel.

FIG. 2. — Breast wheel.

in China, and elsewhere for pumping small quantities of water
for irrigation.

The undershot water wheel was produced from the current
wheel by confining the channel so that the water could not escape
under or around the ends of the vanes. This form of wheel was
capable of an efficiency of 30 per cent, and was in wide use up to
about 1800.

The breast wheel (Fig. 2) utilized the weight of the water
rather than its velocity with an efficiency as high as 65 per cent.
It was used up to about 1850.

2 HYDRAULIC TURBINES

The overshot water wheel (Fig. 3) also utilized the weight
of the water. When properly constructed it is capable of
an efficiency of between 70 and 90 per cent, which is as good
as the modern turbine. The overshot water wheel was exten-
sively used up to 1850 when it began to be replaced by the
turbine, but it is still used as it is well fitted for some conditions.

-~ \

FIG. 3. — I. X. L. steel overshot water wheel. (Made by Fitz Water Wheel Co.)

2. The Turbine. — The turbine will be more completely
described in a later chapter but in brief it operates as follows:
A set of stationary guide vanes direct the water flowing into
the rotating wheel and, as the water flows through the runner,
its velocity is changed both in direction and in magnitude.
Since a force must be exerted upon the water to change its
velocity in any way, it follows that an equal and opposite
force must be exerted by the water upon the vanes of the wheel.
A turbine may be defined as a water wheel in which a motion
of the water relative to its buckets is essential to its action.

INTRODUCTION 3

The term "water wheel" has several shades of meaning in
American usage. First it may be employed in its most general
sense to indicate any rotary prime mover operated by water.
It may thus be applied to the turbine, since the latter is a special
type of water wheel, according to the definition in the preceding
paragraph. Second it may be used to designate the types of
machines described in Art. 1 in order to distinguish them from

FIG. 4. — Francis turbine in flume.

the modern turbine. Third it may be understood to indicate
impulse turbines of the Pelton type as contrasted with turbines
of the reaction type. In this book the term is used in the first
or second sense only, the context making it clear which is meant
in any case.

The original inward flow turbine of James B. Francis (1849) is
shown in Figs. 4 and 5. In Fig. 6 are shown two views of an

4 HYDRAULIC TURBINES

inward flow runner of this general type which was constructed
about 1900. This style is now obsolete.

3. Advantage of Turbine over Water Wheel. — The water
wheel has been supplanted by the turbine because :

1. The latter occupies smaller space.

2. A higher speed may be obtained.

3. A wider range of speeds is possible.

FIG. 5. — Francis turbine.

4. It can be used under a wide range of head, whereas the
head for an overshot wheel should be only a little more than
the diameter of the wheel.

5. A greater capacity may be obtained without excessive
size. .

6. It can work submerged.

7. There is less trouble with ice.

8. It is usually cheaper.

4. Advantages of Water Wheel over Turbine. — For small
plants the turbine is often poorly designed, cheaply made,

INTRODUCTION 5

unwisely selected, and improperly set. It may thus be very
inefficient and unsatisfactory. In such cases the overshot
water wheel may be better. The latter has a very high efficiency
when the water supply is much less than its normal value.
It is adapted for heads which range from 10 to 40 ft. and for
quantities of water from 2 to 30 cu. ft. per second.1

An overshot wheel on the Isle of Man is 72 ft. in diameter
and develops 150 h.p. Another at Troy, N. Y., was 62 ft. in

FIG. 6. — Pure radial inward flow runner of the original Francis type.

diameter, 22 ft. wide, weighed 230 tons, and developed 550 h.p.
The latter is now in a state of ruin.

5. Essentials of a Water-power Plant. — A water-power plant
requires some or all of the following:

1. A Storage Reservoir. — This may hold enough water to run the
plant for several months or more. In many cases it may be
totally lacking.

lSee "Test of Steel Overshot Water Wheel," by C. R. Weidner, Eng.
News, Jan. 2, 1913, Vol. LXIX, No. 1. A later test of this I.X.L.^wheel
after ball bearings were substituted gave an efficiency of 92 per cent.

6 HYDRAULIC TURBINES

2. A Dam. — This may create most of the head available or it
may merely create a small portion of it and be erected primarily
to provide a storage reservoir or mill pond or to furnish a suit-
able intake for the water conduit. In some cases the dam may
be no more than a diversion wall to deflect a portion of the
current into the intake.

3. Intake Equipment. — This usually consists of racks or screens
to keep trash from being carried down to the wheels and of head
gates so that the water may be shut off, if need be.

4. The Conduit. — The water may be conducted by means of an
open channel called a canal or flume, or through a tunnel, or by
means of a closed pipe under pressure, which is called a penstock
if it leads direct to the turbines.

5. The Forebay. — A small equalizing reservoir is often placed
at the end of the conduit from the main intake and the water is
then led from this to the turbines through the penstock. This is
called the forebay and is also referred to as the headwater. In
the case of a plant without any storage reservoir the body of
water at the intake is often termed the forebay.

6. The Turbine. — The turbine with its case or pit and draft
tube, if any, comprise the setting.

7. The Tail Race. — The body of water into which the turbine
discharges is called the tail water. The channel conducting the
water away is the tail race.

6. QUESTIONS

1. What is a turbine? What is a water wheel?

2. Under what circumstances would a current wheel be used? Could
a turbine be used under the same conditions? What is the advantage of
the undershot wheel over the current wheel?

3. Under what circumstances would an overshot water wheel be used?
Could a turbine be used under the same conditions? Could an overshot
wheel replace any turbine?

4. What elements would be found in every water-power plant? What
elements may be in some and lacking in others? What is the difference
between a storage reservoir and a forebay?

I CHAPTER II
TYPES OF TURBINES AND SETTINGS

7. Classification of Turbines. — Turbines are classified ac-
cording to:

1. Action of Water

(a) Impulse (or pressureless) .
(6) Reaction (or pressure).

2. Direction of Flow

(a) Radial outward
(6) Radial inward

(d) Mixed (radial inward and axial).

3. Position of Shaft

(a) Vertical.
(6) Horizontal.

8. Action of Water. — In the impulse turbine the wheel pas-
sages are never completely filled with water. Throughout the
flow the water is under atmospheric pressure. The energy of the
water leaving the stationary guides and entering the runner is all
kinetic. During flow through the wheel the absolute velocity
of the water is reduced as the water gives up its kinetic energy
to the wheel. In Europe a type of impulse turbine commonly
used is called the Girard turbine. In the United States prac-
tically the only impulse turbine is the tangential water wheel
or impulse wheel, more commonly known as the Pelton wheel.
(See Fig. 7.)

In the reaction turbine the wheel passages are completely filled
with water under a pressure which varies throughout the flow.
The energy of the water leaving the stationary guide vanes and
entering the runner is partly pressure energy and partly kinetic
energy.1 During flow through the wheel both the pressure
and the absolute velocity of the water are reduced as the water
gives up its energy to the wheel.

1 Strictly speaking, the water possesses only kinetic energy but transmits
pressure energy.

8 HYDRAULIC TURBINES

Impulse and reaction turbines were so called because in primi-
tive types the force on the former was due to the "impulse"
of water striking it while the force on the latter was the " reac-
tion" of the streams leaving it. But these terms are not very
appropriate^for the forces in question, since in either case the
dynamic force is due to a change produced in the velocity of
the water and the distinction is largely artificial. And in
modern turbines the so-called " impulse" at entrance and
" reaction" at outflow may be effective in either type.

A far better classification is as pressureless and pressure
turbines. Another classification is as partial and complete
admission turbines, as in the former type the water is admitted

FIG. 7. — Tangential water wheel with deflecting nozzle.

at only a portion of the circumference while in the latter type it
js necessarily admitted around the entire circumference.

9. Direction of Flow. — Radial flow means that the path of a
particle of water as it flows through the runner lies in a plane
which is perpendicular to the axis of rotation. If the water enters
at the inner circumference of the runner and discharges at the
outer circumference we have an outwarcf flow type known as
the Fourneyron turbine. (See Fig. 78.)

If the water enters at the outer circumference of the runner
and discharges at the inner circumference we have an inward
flow type as in the original Francis turbine shown in Figs. 4
and 5.

TYPES OF TURBINES AND SETTINGS 9

If a particle of water remains at a constant distance from the
axis of rotation as it flows through the runner we have what is
known as axial or parallel flow. The type of turbine falling in
this class is commonly called the Jonval turbine and is used to
some extent in Europe.

If the water enters a wheel radially inward and then during
its flow through the runner turns and discharges axially we
have a mixed flow turbine. This is known as the American type
of turbine and is also called a Francis turbine, though it is not
identical with the one built by Francis.

Modern reaction turbines are practically all inward flow
turbines of the mixed flow type and to this type our discussion
will be confined.

10. Position of Shaft. — The distinction as to position of shaft is
obvious. The vertical shaft turbines are, however, further classi-
fied as right-hand or left-hand turbines according to the direction
of rotation. If, in looking down upon the wheel from above, the
rotation appears clockwise it is called a right-hand turbine. The
reverse of this is a left-hand turbine.

So far as efficiency of the runner alone is concerned there is
little difference between vertical and horizontal turbines. Other
things being equal, the hydraulic losses should be identical in
either case, but there might be some difference in the friction of
the bearings. As the latter is only a relatively small item, a
reasonable variation in its value would have but slight effect
on the efficiency.

But when we consider the runner and draft tube together, we
may find a difference, since the draft tubes are not necessarily
equally efficient in the two cases. The single-runner, vertical-
shaft turbine, as shown in Fig. 9, is readily seen to lend itself to a
more efficient draft tube construction than the horizontal-shaft
unit, as shown in Fig. 11, with the necessary sharp quarter turn
near the runner where the velocity of the water is still high.
If the velocity of discharge from the runner is low, the difference
in the two cases may be insignificant, but, where the velocity of
the water is relatively high, the draft tube for the vertical-shaft
wheel may be decidedly better.

In general a horizontal shaft is more desirable from the stand-
point of the station operator on account of greater accessibility
and less bearing trouble, but the latter is of less significance
in recent years due to the greater perfection that has been ob-

HYDRAULIC TURBINES

tained in the construction of suspension bearings for such service.
A vertical-shaft turbine occupies much less floor space, but often
requires more excavation and a higher building.

The vertical shaft turbine is used where it is necessary to set
the turbine down by the water while the generator or other
machinery that it drives must be above. Since such conditions
are usually met with in low-head plants, it will be found that
ordinarily the vertical setting is used only for low heads. (See
Fig. 8.)

FIG. 8. — Pair of vertical shaft turbines in open flume.

The horizontal shaft turbine is used where the turbine can be
set above the tail water level and if the generator or other machin-
ery that it drives can be set at the same elevation. This is almost
always the case with a high-head plant and is also quite frequently
the case with a low-head plant. (See Fig. 10.)

These statements are purely general and there are many
exceptions.

TYPES OF TURBINES AND SETTINGS

11. Arrangement of Runners.— A turbine may be mounted
up as an independent unit with its own bearings, usually two in
number, and connected to whatever it drives by means of a

OOOOOOOOOIII
OOOOOODOIIIII

HltOOOOO'OOO
IHOOOOOOOOO

Fio. 9. — Vertical shaft turbine with spiral case.

coupling, belt, or other device. But some horizontal-shaft
hydro-electric machines are set up as three-bearing units, so that
neither the turbine nor the generator are independent of each

HYDRAULIC TURBINES

other. Recent practice is to reduce this to two bearings, as it
is more compact and the problem of alignment is simplified.
The generator is mounted between the two bearings and the tur-
bine runner, which is relatively light, is overhung on the end of
the generator shaft. Sometimes there are two runners for one
generator and in this case one may be overhung on either end.
The former is called the single-overhung and the latter the
double-overhung construction.

FIG. 10. — Horizontal shaft turbine in case.

The double-overhung type is found only with horizontal-shaft
units and naturally requires two separate cases and two draft
tubes. On the other hand with either a horizontal or vertical
shaft we may have two runners discharge into a common draft
chest and tube as in Fig. 8. If open flume construction is not
employed this likewise requires two separate cases. We may
also have only one case and two separate draft tubes for a single
runner with a double discharge, as in Fig. 51, page 58.

TYPES OF TURBINES AND SETTINGS 13

Multiple runner units are used to some extent but present
practice favors single runners of larger size for vertical shaft
installations, as in Fig. 9. For horizontal shafts also four or
more runners have been employed but the preference is for one
runner of either the single or double discharge type or two runners
with separate draft tubes.

12. The Draft Tube. — Occasionally reaction turbines have
beer set so as to discharge above the tail water; in such cases the

FIG. 11. — Horizontal shaft turbine showing draft elbow.

fall from the point of discharge to the water was lost. To avoid
this loss turbines have been submerged below the tail water level
as in Fig. 4, page 3. By the use of a draft tube (or suction
tube), as in Fig. 8 and Fig. 10, it is possible to set the turbine
above the tail water without suffering any loss of head. This is
due to the fact that the pressure at the upper end of the draft
tube is less than the atmospheric pressure. This suction com-
pensates for the loss of pressure at the point of entrance to the
turbine guides.

As will be shown later, when the theory is presented, the use of
a draft tube that diverges or flares may result in a small increase
in efficiency. The chief advantage of the draft tube, however, is

14 HYDRAULIC TURBINES

that it allows the turbine to be set above the tail water where it is
more accessible and yet does not cause any sacrifice in head. It
is this that permits a horizontal shaft turbine to be installed with-
out any loss of head.

Since the wheel passages of an impulse turbine must be open
to the air it is readily seen that the use of a draft tube in the
usual sense of the word is not possible. However, as will be
seen later, the impulse turbine is better suited for comparatively
high heads so that the loss from the wheel to the tail water is
a relatively unimportant item.

13. Flumes and Penstocks. — If the turbine be used under a
head of about 30 ft. or less a flume may conduct the water to an
open pit as in Fig. 4 and Fig. 8. If the head is much greater
than this it becomes uneconomical and a penstock is used as in
Fig. 10. The turbine must then be enclosed in a water-tight case.
Various forms of cases will be described in Chapter V.

For penstocks where the pressure head is less than about 230
ft. (100 Ib. per square inch) wood-stave pipe is frequently used.
It is cheaper than metal pipe for similar service.

Cast-iron pipe is used for heads up to about 400 ft. It is not
good in large diameters nor for high pressures on account of
porosity, defects in casting, and low tensile strength. Its
advantages are durability and the possibility of readily obtaining
odd shapes if such are desired.

For high heads, steel pipe, either riveted or welded, is used.
It is cheaper than cast iron in large sizes but it corrodes more
rapidly.

14. QUESTIONS

1. In what ways may turbines be classified? How many of these are
found in current practice? Explain the features of each.

2. What are the differences between impulse and reaction turbines?
What types of each are now used? Explain the various directions of flow
that may be used.

3. What are the relative merits of horizontal and vertical shaft turbines?
When would each ordinarily be used?

4. What arrangements of runners may we have for vertical shaft units?
For horizontal shaft units? What is meant by single- and double-over-
hung construction?

6. What two functions does the draft tube fulfill? How does it prevent
loss of head?

CHAPTER III
WATER POWER

15. Investigation. — Before a water-power plant is erected a
careful study should be made of the stream to determine the
horse-power that may be safely developed. It is important to
know not only the average flow but also both extremes. The
extreme low-water stage and its duration will determine the
amount of storage or auxiliary power that may be necessary.
The extreme high-water stage will fix the spillway capacities of
dams, determine necessary elevations of machines, and other
facts essential to the safety and continuous operation of the plant.

16. Rating Curve. — The first step in such an investigation is
the establishment of a rating curve. (See Fig. 12.) To determine

Cross Section of Stream. Discharge

FIG. 12.— Rating curve.

the discharge of the stream a weir, current meter, floats, or other
means may be employed according to circumstances.1

By measuring the flow of the stream for different stages a
rating curve is readily drawn. This will not be a smooth curve
if there are abrupt changes in the area of the section. A given
gage height may really represent a range of flows depending upon
whether the river is rising or falling, the flow being greater if the
stream is rising and less if it is falling. This is because the
hydraulic gradient is different in the two cases.2 If possible,

1 Hoyt and Grover, "River Discharge."

Water Supply Papers No. 94 and No. 95 of the U. S. G. S.

2 Mead, "Water-power Engineering," p. 201.

HYDRAULIC TURBINES

the points for the rating curve should be taken when the river is
neither rising nor falling. The discharges from the rating curve
for gage readings taken under all conditions will be more or less
in error, but in the end such errors will usually balance each other
and be unimportant.

If the bed of the stream changes, as it frequently does in
sandy or alluvial soil, the rating curve will also change and must
be determined anew from time to time. Sometimes a special
permanent control station may be constructed to avoid this.

17. The Hydrograph. — When gage readings are taken regularly
and frequently for any length of time and the corresponding

Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov.

FIG. 13.— Hydrograph.

discharges secured from the rating curve a history of the flow may
be plotted as' in Fig. 13. Such a curve is called a hydrograph.
This curve is extremely useful in the study of a water-power
proposition. To be satisfactory it should cover a period of
several years since the flow will vary from year to year. Since
it is very important to know the extremes also, it should cover
both a very dry year and a very wet one as well as the more
normal periods.

18. Rainfall and Run-off. — Rainfall records are usually avail-
able for many years back and are a valuable aid in extending
the scope of the hydrograph taken, provided a relation between
rainfall and run-off can be estimated. If the ground be frozen,
or the slopes steep and stony, or the ground saturated and the

WATER POWER

rain violent nearly all the water that falls upon the drainage
basin may appear in the stream as run-off. On the other hand,
if the soil be dry and the land such that opportunity is given it,
all the rain may be absorbed and none of it appear in the stream.
Usually the conditions are such that the relation is between these
two extremes. In a general way it may be said to lie between the
two curves shown in Fig. 14. l

The relation between rainfall and run-off is very complicated
and only partially understood at present. For more information
consult Water Supply Papers of the U. S. G. S. and other sources.

10 20 30 40 50

Mean Annual Rainfall, Depth in Inches
FIG. 14. — Relation of rainfall to run-off.

19. Absence of Satisfactory Hydrograph. — If no hydrograph
of the stream is available and there is no time to secure one, a
study of the stream may be made by comparison with the hydro-
graphs of adjacent streams. It is well, however, to take a
hydrograph for a year, if possible, in order to be able to check
the comparison.

If no hydrographs of adjacent streams are available, it is neces-
sary to use the rainfall records and make a thorough study of
the physical conditions of the water shed. If the relation between
rainfall and run-off can be estimated, then fairly satisfactory
conclusions may be drawn, provided a hydrograph for one year
can be used to work from. Where there is not time to take a
year's record it is well to be very conservative and provide for
future extension of power if it is later found to be warranted.

20. Variation of Head. — Since the discharge of any stream is
usually a widely varying quantity, it follows that the water

1 F. H. Newell, Proc, Eng. Club of Phila., Vol. XII, 1895,
2

HYDRAULIC TURBINES

level at any point must vary. If the turbine be of the reaction
type set in the usual way, the total head acting upon the wheel
will be the fall from the surface of the head water to the surface
of the tail water with the pipe line loss deducted. If, in times
of high water, the head water level rose the same amount as the
tail water level the net head under which the turbine operated
would remain constant. But, under the usual conditions, the
tail water level rises more than the head water level and the net
head under which the turbine operates becomes less. This is
illustrated in Fig. 15 where three rates of flow are^shown.

At high water the horse-power of the stream may be large even
though the fall be reduced, owing to the increased quantity of

FIG. 15. — Decrease of available head at high water.

water. But the horse-power of the turbine may be seriously
diminished. A turbine is only a special form of orifice and there-
fore the discharge through it is proportional to the square root of
the head. If then the discharge through it be reduced due to the
lower head, the horse-power input to the turbine is decreased. If
the best efficiency is to be obtained, the speed also should vary as
the square root of the head. But usually the turbine is compelled
to run at constant speed and this causes a further reduction of the
power of the turbine since the efficiency is lowered. (The speed
should be the best for low water because economy of water is then
important.) It is thus seen that the decrease of the head at high
water causes a loss of power and a drop in efficiency. This
change of head will be an insignificant item for a high-head plant
but may be very serious for a low-head plant.

WATER POWER

21. Power of Stream. — If the conditions are such that there
is no appreciable change in head, the hydrograph with a suitable
scale may represent the power of the stream also. But if the head
varies to any extent with the flow then the power curve must be
computed from the hydrograph by using the heads that would be

Dec. Jan. leb. Mar. Apr. May June July Aug. Sept. Oct. Nov.
FIG. 16. — Power curve of a stream.

obtained at various stages of flow. Or the hydrograph itself may
still be used as a power curve if the power scale that is used is
made to vary as the head varies instead of being uniform.

If Fig. 16 represents the power curve of a stream then A-B
represents the greatest power that the stream can be counted
upon to furnish at all times.

Average Load

68 10

Time

FlG. 17.

6 8 10 12

22. Pondage and Load Curve. — By pondage is meant the
storing of a limited amount of water. If the plant be operated
24 hours on a steady load then pondage is of little value except for
equalizing the flow of water when the stream is low. But if the

20 HYDRAULIC TURBINES

plant be operated for only a portion of the 24 hours or if the load
be variable as shown by the load curve in Fig. 17, then the water
that is not used when the load is light may be stored and used
when the load is heavy. If the pondage be ample, the average
load carried by the plant may then be equal to A-B in Fig. 16,
while the peak load may be much greater.

23. Storage. — By storage is meant the storing of a consider-
able quantity of water, so that it varies from pondage in degree
only. Pondage indicates merely sufficient capacity to supply
water for a few hours or perhaps a few days, but storage implies
a capacity which can supply water needed during a dry spell of
several weeks or months or more. The effect of storage is to
enable the minimum power of the stream to be raised from A-B
to C-D (Fig. 16). The greater the storage capacity the higher
C-D is placed until it equals the average power of the stream.
The water for the turbines may be drawn direct from the storage
reservoir (in which case the head varies) or the reservoir may be
used as a stream feeder only.

A plant operating under a low head requires a relatively large
amount of water for a given amount of power. A storage basin
for such a plant would require a very large capacity if it were to
furnish power for any length of time. But a low head is usually
found in a fairly flat country where it is possible to construct a
storage reservoir of limited capacity only, and often none at all,
on account of flooding the surrounding country. But for a high
head the conditions are different as only a relatively small amount
of water is required so that the capacity of the storage reservoir
need not be excessive. The higher the head, the more valuable a
cubic foot of water becomes. The topography of a country where
a high head can be developed is usually such that storage reser-
voirs of large capacity can be constructed at reasonable cost. A
low-head plant usually possesses pondage only — a high-head
plant usually possesses storage.

24. Storage and Turbine Selection. — If a plant possesses
neither storage nor pondage, or the stream flow may not be inter-
rupted because of other water rights, the economy of water when
the turbine is running under part load is of no importance. The
efficiency at full load is all that is of interest. But if the plant
does have pondage or storage in any degree the economy of water
under all loads is of importance. The more extensive the
pondage the more valuable a high efficiency on all loads becomes.

WATER POWER

Thus the question of storage has an important bearing in turbine
selection.

25. Power Transmitted through Pipe Line. — Suppose that
a nozzle, whose area can be varied, is placed at the end of a pipe
line B-C (Fig. 18). With the nozzle closed we have a pressure
head at C of CX which is equal to the static head. The hydraulic

FIG. 18. — Varying rates of flow in pipe line.

gradient is then a horizontal line. If the nozzle be partially
opened, so that flow takes place, the losses in the pipe line as well
as the velocity head in the pipe cause the pressure to drop to
CY. A further opening of the nozzle would cause the pressure
to drop to a lower value. If the nozzle were removed the pres-

FIG. 19

««-— ^— ^—— ^^^^— «

Rate of Discharge
. — Head and power at end of pipe line.

sure at C is then atmospheric only, which we ordinarily call zero
pressure. The hydraulic gradient is then A-C.

Head is the amount of energy per pound of water. The head at
C is the elevation head, taken as zero, plus the pressure head, plus
the velocity head. When the discharge is zero the head is a
maximum, being equal to CX. When the nozzle is removed the
discharge is a maximum but the head at C is a minimum, being

22 HYDRAULIC TURBINES

only the velocity head. For any intermediate value of discharge
the head will be intermediate between these two extremes.

The power transmitted through the pipe line and delivered at C
is a function of both the quantity of water and the head. It is
zero when the discharge is zero and very small when the discharge
is a maximum. The power becomes a maximum for a discharge
between these two extremes as is shown in Fig. 19. Let the
rate of discharge through the pipe be denoted by q, the net
head at C by h, the loss of head by H', and the height CX by z.
If the loss of head in the pipe be assumed proportional to the
square of the velocity of flow we may write Hf = Kq2, where K
is a constant whose value depends upon the length, size, and
nature of the pipe. Then

Power = qh = q(z - H') = qz - Kq*
Differentiating d(PowQv)/dq = z - ZKq2 = 0
Or z = 3Kqz = 3H'.

Thus the power delivered by a given pipe line is a maximum when
the flow of water is such that one-third the head available is
used up in pipe friction, leaving the net head only two-thirds
of that available.

The efficiency of the pipe line is expressed by h/z. Thus in the
case where the pipe line is delivering its maximum power, its
efficiency is only 66% per cent. But if economy in the use of
water is an object the discharge through the pipe would be kept
at a lower value than this so as to prevent so much of the energy
of the water being wasted. For a given quantity of water, this
means that a larger pipe would be used, so that its efficiency
would be higher. In a similar manner, if a given amount of power
is required, the smallest pipe that can be used will be of such a
size that its efficiency is 66% per cent. As the pipe is made
larger than this, its efficiency rises and the amount of water
required decreases.1

The most economical size of pipe may be found as shown in
Fig. 20. One curve represents the annual value of the power lost

1 It should be noted that in this paragraph there are three separate cases
mentioned. First the size of the pipe is fixed and different rates of discharge
are assumed to flow through it. Second the quantity of water available is
fixed and the size of the pipe is the variable. Third the power delivered is
fixed and the size of the pipe is varied,

WATER POWER

in pipe friction, the other the annual fixed charge on the pipe.
This includes interest on the money expended, depreciation,
repairs, etc. The total cost of the pipe per year is the curve
whose ordinates are the sums of the other two. The size of pipe
for which this sum is a minimum is the most economical.

If the rate of discharge is not constant, careful study must
be made of the load curve in order to determine what value of
the rate of discharge will give the average power lost. For the
typical load curve this value may often be found to be about 80
per cent, of the maximum flow.

Size of Pipe
FIG. 20. — Determination of economic size of pipe.

It must be noted that this solution may not always be the most
practical because of other considerations. For instance the
velocity of the water may be too high and thus give rise to
trouble due to water hammer. Again if the loss of head is too
large a percentage of the head available, the variation of the net
head between full discharge and no discharge may be con-
siderable. This might cause trouble in governing the turbine.

26. Pipe Line and Speed Regulation. — A fundamental propo-
sition in mechanics is that

input = output -f losses + gain in energy.

If the speed of a turbine is to remain constant it follows that the
input must aways be equal to the power output plus the losses.
As the power output varies, therefore, the quantity of water sup-

24 HYDRAULIC TURBINES

plied to the turbine must vary. It is thus apparent that a turbine
does not run under an absolutely constant head at all loads.
By referring to Fig. 19 it is seen that when the turbine is using
only a small quantity of water the head will be higher than when
it is carrying full load.

If the load on a turbine is rapidly reduced the quantity of water
supplied to it must be very quickly decreased in order to keep the
speed variation small. This means that the momentum of the
entire mass of water in the penstock and draft tube must be sud-
denly diminished. If the penstock be long a big rise in pressure
may be produced so that momentarily the pressure may be greater
than the static pressure. This increase in pressure may be suffi-
cient to even cause an increase in the power input for a very brief
interval of time. On the other hand, if the load on the turbine be
suddenly increased, the water in the penstock and draft tube must
be accelerated and this causes a temporary drop in pressure below
the normal value, and for the time being the power input to the
turbine may be diminished below its former value. The longer
the pipe line and the higher the maximum velocity of flow, the
worse these effects become. It is thus seen that the speed regu-
lation depends upon the penstock and draft tube as well as upon
the governor and the turbine.1

If the velocity of the water is checked too suddenly a dangerous
water hammer may be produced. In order to avoid an excessive
rise in pressure, relief valves are often provided. Automatic re-
lief valves are analogous to safety valves on boilers; they do not
open until a certain pressure has been attained. Mechanically
operated relief valves are opened by the governor at the same time
the turbine gates are closed and afford the water a by-pass so that
there is no sudden reduction of flow. To prevent waste of water
these by-passes may be slowly closed by some auxiliary device.
Another means of equalizing these pressure variations is to place
near the turbine a stand pipe or a surge chamber, with compressed
air in its upper portion, or open to the atmosphere if it can be
made high enough. These have the advantage over the re-
lief valves that they are not only able to prevent the pressure in-

1 A case may be cited where the length of a conduit was 7.76 miles, the
average cross-section 100 sq. ft., and the maximum velocity 10 ft. per second.
The amount of water in the conduit was, therefore, 128,125 tons and with
the velocity of 10 ft. per second there would be in round numbers 200,000
ft .-tons of kinetic energy.

WATER POWER 25

crease from being excessive but they are able to supply water in
case of an increasing demand and thus prevent too big a pressure
drop.1

27. QUESTIONS AND PROBLEMS

1. Before a water power plant is built what information should be ob-
tained regarding the stream? How may this be determined?

2. What is the rating curve of a stream? How is it obtained? What use
is made of it? Is it always the same?

3. What is the hydrograph? How is it obtained? What is its use?

4. What use may be made of rainfall records, if a hydrograph of the stream
has been obtained by direct measurement? What use may be made of
rainfall records, if no hydrograph is in existence?

5. Is the head on a water power plant constant? What causes this?
Do the head water levels and the tail water levels change at the same rate?
Why? What effect does this have on the power and efficiency of the tur-
bine? What types of plants are most seriously affected?

6. How is the power of a stream to be determined? What effect does
pondage have upon this? What is the difference between pondage and
storage and how do they differ in their effects upon the extent of the power
development?

7. As the flow of water through a given pipe increases, how do the head
and power delivered change? How does the efficiency vary? For what
condition is the power a maximum? Is this desirable?

8. If a given rate of discharge is to be used for power, how may the
proper size of pipe be determined? Are there several factors that need to
be considered?

9. If a given amount of power is required and the water supply is ample,
how can the smallest size of pipe that would serve be found? What would
limit the largest size that might be used?

10. How does the head on a turbine change with the load the wheel
carries? What effect does the pipe line have upon speed regulation?

11. What devices are employed to care for the condition when the gover-
nor suddenly diminishes the water supply? What may be used to care for
a sudden demand?

12. The following table gives the results of a current meter traverse of a
stream: Velocity of water in ft. per second equals 2.2 times revolutions per
second of the metre plus 0.03.

From this data compute the area, rate of discharge, and mean velocity of
the stream. (The mean velocity in a vertical ordinate will be found at
about 0.6 the depth. The mean velocity is obtained with a slightly greater
degree of accuracy by taking the mean of readings at 0.2 and 0.8 the depth.

1 See "Control of Surges in Water Conduits," by W. F. Durand, Journal
A. S. M. E., June, 1911; "The Differential Surge Tank," by R. D. John-
son, Trans. A. S. C. E., Vol. 78, p. 760, 1915; and "Pressure in Penstocks
caused by the Gradual Closure of Turbine Gates," by N. R. Gibson, Proc.
A. S. C. E., Vol. 45, Apr., 1919.

HYDRAULIC TURBINES

Distance
from
initial
point

Depth
of
stream

Depth
of obser-
vation

Time
in
seconds

Revolu-
tions

Distance
from
initial
point

Depth
of
stream

Depth
of obser-
vation

Time
in
seconds

Revolu-
tions

0.0

1.4

0.28

0.7

0.42

1.12

1.0

0.60

1.2

0.24

1 .0

0.60

0.96

0.9

0.54

1.3

0.26

1.5

0.30

1.04

44'

1.20

1.4

0.28

1.7

0.34

1.12

1.36

1.2

0.24

1.9

0.38

0.96

1.52

1.3

0.26

1.8

0.36

1.04

1.44

100

1.1

0.22

1.7

0.34

0.88

1.36

105

1.0

0.20

1.6

0.32

0.80

1.28

110

1.2

0.24

1.5

0.30

0.96

1.20

115

1.2

0.24

1.6

0.32

0.96

1.28

120

0.8

0.48

1.4

0.28

125

0.9

0.54

1 .12

130

1 .1

0.66

135

1 .1

140

0.0

The area between two ordinates may be taken as the product of the distance
between them by half the sum of the two depths. The mean velocity in
such an area may be taken as half the sum of the mean velocities of the
ordinates. The product of area and mean velocity gives the discharge
through the area. The sum of all such partial areas and discharges gives
the total area and total discharge of the stream. The total discharge divided
by the total area gives the mean velocity of the stream.)

Ans. 171.8 sq. ft., 146.8 cu. ft. per second.

13. The traverse of the stream given in problem (12) was made May 14,
1913 when the gage height was 1.21 ft. Other ratings had been made as
noted.

Date

Width,
ft.

Area,
sq. ft.

Mean
velocity,
ft. per sec.

Gage
height,

ft.

Discharge,
sec.-ft.

November 3, 1906
May 10, 1908

138
138

485
345

3.10
2 32

1345

758

September 4, 1908

0.57

July 24, 1909

138

157

1.12

150

November 19, 1909

0.74

May 12, 1910

138

226

1.63

364

October 11, 1911 .

138

165

1 32

202

July 30, 1912. . .

0 70

May 14, 1913

138

172

1 21

WATER POWER

It will be noted that the data is not always consistent, due to changes in
the bed of the stream. From the data given draw to scale the probable out-
line of the cross-section of the stream. Plot values of area, mean velocity,
and discharge against gage height. (The area and velocity curves can be
extended with greater assurance than the discharge curve. By computing
values of discharge from these two, the discharge curve may be produced
beyond readings taken.)

14. The daily gage heights. of the stream of the preceding problem for
1912 are given below. Plot the hydrograph. Note values of maximum,
minimum, and average flow, and the duration of the minimum flow.

Day

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

1.10

.48

6.50

2.30

2.70

1.32

0.89

0.80

1.04

1.22

1.48

1.31

1.20

.46

5.70

2.90

2.30

1.25

0.85

0.74

1.25

1.24

.60

1.34

1.10

.48

4.60

3.80

2.20

2.30

0.82

0.76

1.34

1.16

.48

2.30

1.30

.38

2.00

2.90

2.16

0.90

0.79

1.14

1.12

.42

1.90

1.05

.35

1.80

2.60

1.90

.55

0.86

0.74

1.06

.14

.31

1.70

1.10

.32

1.55

2.40

1.85

.50

0.84

0.71

1.01

.09

.28

2.10

2.20

.30

1.15

2.35

1.95

.80

0.80

0.71

0.92

.01

.31

2.10

2.55

.52

1.04

2.90

2.15

.55

0.76

0.71

0.02

.95

.00

1.60

2.30

.58

1.90

2.45

2.40

.38

0.74

0.95

.14

.70

1.70

2.50

1.58

1.95

2.25

2.20

.29

0.72

0.79

0.91

.05

.55

1.65

1.85

1.44

1.70

2.15

1.95

.24

0.74

2.35

0.92

.09

.50

1.55

2.00

1.48

1.70

2.10

1.80

.21

0.74

1.50

0.90

.10

.41

1.49

1.85

1.52

3.60

2.05

1.80

.21

0.72

1.05

0.90

.11

.42

1.35

1.85

1.60

3.10

2.00

1.70

.15

0.88

1.20

0.85

.08

.50

1.48

2.00

1.50

3.20

2.05

2.00

.12

0.88

1.00

0.85

.01

.46

1.35

1.80

1.52

4.50

2.30

.15

0.83

0.95

0.88

0.95

.39

1.40

1.75

1.55

3.20

2.35

2.00

.09

0.80

1.42

0.86

0.89

.38

1.38

1.85

1.58

3.00

1.80

.11

1.00

1.15

0.94

0.91

.94

1.34

2.20

1.56

2.80

3.06

1.70

.05

0.98

1.42

1.15

1.01

.28

2.25

2.80

1.55

2.60

2.70

1.65

.02

0.84

1.30

1.14

0.96

.38

2.10

2.86

1.58

2.25

2.35

1.60

.06

0.85

1.12

1.01

0.96

.28

1.90

6.30

2.42

2.30

2.10

1.60

1.02

1.20

1.04

0.92

1.26

1.75

3.35

2.29

1.95

2.25

1.48

0.94

0.95

1.06

1.01

1.35

1.22

1.70

2.55

2.30

2.05

2.10

1.43

0.95

0.76

1.14

1.18

2.80

1.36

1.60

2.16

2.22

2.10

.45

0.95

0.84

1.04

2.05

2.60

1.70

1.55

1.95

2.20

1.95

1.90

.70

0.94

0.84

0.99

1.70

2.25

1.60

1.95

2.05

2.00

.55

0.98

0.81

0.96

1.45

.95

1.50

1.70

1.80

2.60

2.20

1.90

.40

0.88

0.80

1.04

1.32

.80

1.42

1.80

1 68

2 60

4.20

1.80

0.90

0.79

1.00

1.28

.70

1.40

1.65

1.60

3.60

3.20

.65

0.89

0.75

0.91

1.26

.60

1.35

1.70

1.62

3.00

.38

0.72

0.99

.50

2.50

15. The following table gives the rainfall record in a certain vicinity for

several years, and also the estimated run-off. The relation of rainfall to

run-off is not only different for different drainage basins, but for a given

drainage basin it varies according to the time of year and the extent of the

rainfall. There is thus no constant relation between the two in the table.1

With these records construct a hydrograph for the estimated average

1 See Kuichling's Rainfall- Run-off Diagrams in the report on the New

York State Barge Canal of 1900.

HYDRAULIC TURBINES

monthly rate of discharge of a stream with a drainage basin of 20 square
miles.

Rainfall,

inches

Run-off,
inches

Rainfall,
inches

Run-off,
inches

Rainfall,
inches

Run-off,
inches

Rainfall,
inches

5e"«
||

C o
3 fl
PS""

Rainfall,

inches

Run-off,
inches

3" 2

!§•§

c3 fi
P»--

*?I

C o

.2 a
tf"1

Jan

3.05

1.9

3.21

2.0

4.14

2.2

1.15

2.85

1.8

4.91

3.5

Feb

1.95

1.8

4.61

3.5

5.17

4.0

1.84

1.7

2.11

1.8

4.04

3.0

March. . . .

1.91

2.6

4.04

4.0

3.74

3.6

1.48

2.3

2.98

3.2

5.16

4.7

April

2.19

2.2

3.78

3.1

4.91

3.6

5.96

4.0

2.82

2.6

5.71

3.9

May

2.72

1.2

4 .98

2.0

2.94

1 .3

2.58

1.2

1.33

0.9

3.15

1 .3

June

2.73

0.8

1.53

0.6

3.50

0.9

3.47

0.9

7.98

2.3

1.32

0.6

July

2.74

0.4

3.44

0.4

1.86

0.3

2.00

0.3

3.03

0.4

3.14

0.4

Aug

2.55

0.4

2.66

0.4

3.66

0.5

2.80

0.4

5.70

0.9

6.30

1.0

Sept

6.88

1.7

4.04

0.8

2.73

0.5

3.38

0.6

3.57

0.7

4.49

0.9

Oct

4.69

1.7

1.40

0.5

1.28

0.4

1.20

0.4

5.33

1.2

3.56

0.8

Nov

4.70

1.6

2.51

0.8

1.75

0.6

3.15

1.0

3.06

1.0

2.32

0.8

Dec

4.88

2.5

0.00

0.5

2.93

1.4

1.93

1.1

3.20

1.6

4.02

2.0

16. The present capacity of the Lake Spaulding reservoir of the Pacific
Gas and Electric Co. is 2,000,000,000 cu. ft. (it will eventually be twice
this), the present flow is 300 cu. ft. per second, and the net head on the
power house is approximately 1300 ft. If the plant runs at full load con-
tinuously and there is no stream flow into the lake, how long would this
water last? If this same storage capacity were available for a plant of the
same power under a head of 40ft., what rate of discharge would be required
and how long would the water last? (It is worth noting that the surface
area of Lake Spaulding is 1.3 square miles and the total drop in the water
surface would be 56 feet if the sides were vertical. Actually the drop is
greater. No such drop in level would be found in connection with a plant
under a 40-ft. head. If we assume the drop in level to be 10 ft., for example,
the surface area of the storage reservoir would have to be 233 square miles.
Also the lowering of the head on the plant in the latter case would make
it necessary to use more water and hence shorten the time as computed.)

Ans. 772 days, 9750 cu. ft. per second, 2.4 days.

17. The difference in elevation between the surface of the water in a
storage reservoir and the intake to a turbine was 132.4 ft. During a test
the pressure at the latter point was 126.6 ft. and the discharge 44.5 cu. ft.
per second, giving a velocity head in a 30 in. intake of 1.3 ft. What was
the efficiency of the pipe line? What was the value of the power delivered?

Ans. 96.6 per cent., 647 h.p.

18. Assuming the loss of head to be proportional to the square of the rate
of discharge, what is the maximum power the pipe in problem (17) could
deliver? How many cubic feet of water per second are consumed per
horsepower in problems (17) and (18)? Ans. 1400 h.p., 0.069, 0.088.

WATER POWER 29

19. The pipe line in problem (17) was 5 ft. in diameter. From the test
data the loss of head may be computed as

Hf = 5.63 V2/d = 18,100/d5

where d is in feet and the rate of discharge is supposed to be 44.5 cu. ft. per
second in every case. Assume this expression to be true for similar pipes
of different sizes, the cost of 3, 4, 5, and 6 ft. riveted steel pipes to be $4.25,
$7.50, $12.50, and $18.00 per foot respectively, and the length of pipe to be
2000 ft. If the value of a horsepower per year is $20, the interest and de-
preciation rate 7 per cent., and the rate of discharge 44.5 cu. ft. per second,
what is the most economical size of pipe? Ans. 5 ft.

CHAPTER IV
THE TANGENTIAL WATER WHEEL

28. Development. — The tangential water wheel is the type of
impulse turbine used in this country. Its theory and charac-
teristics are precisely the same as those for the Girard impulse
turbine, used abroad, and the two differ only in appearance
and mechanical construction. It is used rather than the Girard
turbine, because of the advantages offered by its superior type

FIG. 21. — Doble ellipsoidal bucket.

of construction. The tangential wheel is also called an impulse
wheel or a Pelton wheel in honor of the man who contributed to
its early development. The use of the term "Pelton water
wheel" does not necessarily imply, therefore, that it is the
product of the particular company of that name.

The development of this wheel was begun in the early days in
California but the present wheel is a product of the last 20 years.
For the purpose of hydraulic mining in 1849 numerous water
powers of fairly high head were used, some of the jets being as

THE TANGENTIAL WATER WHEEL

much as"2000 h.p. When the gold was exhausted many of these
jets were then used for power purposes. The first wheels were
very crude affairs, often of wood, with flat plates upon which
the water impinged. The ideal maximum efficiency of a wheel
with flat vanes is only 50 per cent. The next improvement
was the use of hemispherical cups with the jet striking them
right in the center. A man by the name of Pelton was running
one of these wheels one day when it came loose on its shaft and
slipped over so that the water struck it on one edge and was dis-

FIG. 22. — Allis-Chalmers bucket. (Courtesy of Allis- Chalmers Mfg. Co.)

charged from the other edge. The wheel was observed to pick
up in power and speed and this led to the development of the
split bucket.

29. Buckets. — The original type of Pelton bucket may be seen
in Fig. 76, page 88, the Doble ellipsoidal bucket is shown in
Fig. 21, the Allis-Chalmers type in Fig. 22, while the recent
Pelton bucket may be seen in Fig. 23. In every case the jet
strikes the dividing ridge and is split into two halves. The
better buckets are made of bronze or steel, the cheaper ones for

32 HYDRAULIC TURBINES

low heads of cast iron. They are all polished inside and the
" splitter" ground to a knife edge so as to reduce friction and
eddy losses within the bucket. They may weigh as much as
430 Ib. apiece and be from 24 to 30 in. in width.

The buckets are bolted onto a rim. The interlocking chain
type is shown in Figs. 23 and 24. With this design each bolt
serves two buckets in such a fashion that the latter are connected

FIG. 23. — Pelton bucket. (Courtesy ofPelton Water Wheel Co.)

as a chain. The advantage gained is one of compactness, it
being possible to place the buckets somewhat closer together.

30. General Proportions. — It has been found that for the best
efficiency the area of the jet should not exceed 0.1 the projected
area of the bucket, or the diameter of the jet should not exceed
0.3 the width of the bucket. l If this ratio is exceeded the buckets
are crowded and the hydraulic friction loss becomes excessive.
It is evident also that there must be some relation between size
of jet and the size of the wheel. For a given size jet there is no

i W. R. Eckart, Jr., Proc. of Inst. of Mech. Eng. (London), Jan. 7, 1910.

THE TANGENTIAL WATER WHEEL 33

upper limit as to size of wheel so far as the hydraulics is concerned.
In special cases, where a low r.p.m. was desired, diameters as
large as 35 ft. have been used when the diameter of the jet was
only a few inches. But there is a lower limit for the ratio of
wheel diameter to jet diameter. Obviously, for instance, the
wheel could not be as small as the jet. The considerations which
influence this matter will be further considered in Chapter VII,
but for the present it will be sufficient to state that a' ratio as
low as 9 may be used without an excessive loss of efficiency.1
(The nominal diameter is that of a circle tangent to the center

FIG. 24. — Pelton tangential water wheel runner showing interlocking chain-
type construction. (Made by Pelton Water Wheel Co.)

line of the jet.) The more common value, and one which
involves no sacrifice of efficiency, is 12. From that we get a
very convenient rule that the diameter of the wheel in feet equals
the diameter of the jet in inches. The size of jet necessary to
develop a given amount of power under any head may be com-
puted and then the diameter of wheel necessary is known at once.
The r.p.m. of the wheel can be computed by taking the periph-
eral speed as 0.47 of the jet velocity or 0.45 -\/2gh.

XS. J. Zowski, "Water Turbines," published by Engineering Society,
Univ. of Mich., 1910.
3

HYDRAULIC TURBINES

The use of one jet only upon a single wheel is to be preferred
if it is possible. However, two jets are often used upon one
wheel though at some sacrifice of efficiency. For a given size
wheel the horsepower of one jet is limited by the maximum size
of the jet that may be employed. If a greater horsepower is
desired it is necessary to use two or more jets upon the one
wheel or to use a larger wheel with a single jet. The larger whool

FIG. 25. — Tangential water wheel unit with deflecting nozzle.

means a lower r.p.m. and a higher cost both of the wheel and the
generator if a direct connected unit is used. In case this addi-
tional expense is not justified by the increased efficiency of the
single jet wheel the duplex nozzle would be used.

The tangential water wheel is almost always set with a hori-
zontal shaft and, if direct connected to a generator, is overhung
so that the unit has only two bearings (Fig. 25). It is quite
common for two wheels to drive a single generator mounted
between them in which case we have the double-overhung type.

THE TANGENTIAL WATER WHEEL

31. Speed Regulation. — Various means have been adopted to
regulate the power input to the tangential water wheel but the
following are the only ones that are of any importance. The

FIG. 26. — 5286 h.p. Jet, from
velocity

in- needle nozzle. Head = 822 ft. Jet
227.4 ft. per second.

use of any throttle valve in the pipe line is wasteful as it destroys
a portion of the available head and thus requires more water to
be used for a given amount of power than would otherwise be the
case. The ideal mode of governing, so far as economy of water

FIG. 27. — Deflecting needle nozzle. (After drawing by Prof. W. R. Eckart, Jr.)

is concerned, would not affect the head but would merely vary
the water used in direct proportion to the power demanded.
The needle nozzle (Fig. 27) accomplishes this result very nearly.

36 HYDRAULIC TURBINES

As the needle is moved back and forth it varies the area of the
opening and thus varies the amount of water discharged. The
coefficient of velocity is a maximum when the nozzle is wide
open but it does not decrease very seriously for the smaller-nozzle
openings. (See Fig. 89.) Thus the velocity of the jet is very
nearly the same for all values of discharge. The efficiency of a
well-constructed needle nozzle is very high, being from 95 to 98
per cent.1 The needle nozzle is nearly ideal for economy of water
but may not always permit close speed regulation. If the pipe
line is not too long, the velocity of flow low, and the changes of
load small and gradual, the needle nozzle may be very satis-
factory. In case it is used the penstock is usually provided with
&_standpipe or a surge tank.

If the pipe line is long, the velocity of flow high, and the changes
of load severe, dangerous water hammer might be set up if the
discharge were changed too quickly. It fought therefore be
difficult to secure close speed regulation with the needier nozzle
as the governors would have to act slowly. The deflecting nozzle,
shown in Fig. 7, page 8, is much used for such cases. The
nozzle is made with a ball-and-socket joint so that the entire jet
can be deflected below the wheel if necessary. The governor
sets the nozzle in such a position that just enough water strikes
the buckets to supply the power demanded. The rest of the
water passes below the buckets and is wasted. Since there is no
change in the flow in the pipe line the governor may accomplish
any degree of speed regulation desired as there is little limit to
the rapidity with which the jet may be deflected. Such a nozzle
is usually provided with a needle also which is regulated by hand.
Fig. 27 is really of this type. In another type the body of the
nozzle is stationary and only the tip is moved. The needle
stem must be equipped with a guide in this moving part and
also be fitted with a universal joint so that the needle point may
always remain in the center of the jet. The station attendant
sets the needle from time to time according to the load that he
expects to carry. However, the device is wasteful of water
unless carefully watched. If other water rights prevent the flow
of a stream from being interfered with it may be satisfactory.

In some modern plants the operator can control the position
of the needle from the switchboard and by careful attention very

!W. R. Eckart, Jr., Inst. of Mech. Eng. (London), Jan. 7, 1910.
Bulletin No. 6, Abner Doble Co.

THE TANGENTIAL WATER WHEEL

little water is wasted. Since the experience is that loads in-
crease slowly, the operator need have little trouble in keeping
the unit up to speed.

The combined needle and deflecting nozzle may possess the
advantages of both of the above types, by having the needle
automatically operated. If the load on the wheel is reduced

M \ EIG. 28. — Deflecting needle nozzle for 8000 h.p. wheel.

the goveriftfr at once deflects the jet thus preventing any increase
of speed. .Then a secondary relay device slowly closes the needle
nozzle and,jas it does so, the nozzle is gradually brought back to
its original "position where all the water is used upon the wheel.
Thus close speed regulation is accomplished with very little
waste of water.

FIG. 29.— Needles and nozzle tips. (Courtesy ofPelton Water Wheel Co.)

The needle nozzle with auxiliary relief shown in Fig. 30 and Fig.
31 accomplishes the same results as the above. When the needle
of the main nozzle is closed the auxiliary nozzle underneath it is
opened at the same time. This discharges an equivalent amount
of water which does notjstrike the wheel. This auxiliary nozzle
is then slowly closed by means of a dash-pot mechanism. While

i HYDRAULIC TURBINES

both of these types relieve the pressure in case of a decreasing
load they are unable to afford any assistance in the case of a
rapid demand for water. The deflecting nozzle alone is the only
type that is perfect there.

32. Conditions of Use. — The tangential water wheel is best
adapted for high heads and relatively small quantities of water.
By that is meant that the choice of the type of turbine is a func-
tion of the capacity as well as the head. For a given head the
larger the horsepower, the less reason there is for using this
type of wheel.

FIG. 30. — Auxiliary relief needle nozzle.
(Made by Pelton Water Wheel Co.)

In Switzerland a head as high as 5412 ft. has been used for
5 wheels of 3000 h.p. each. The jets are 1.5 in. in diameter and
the wheels, which run at 500 r.p.m., are 11.5 ft. in diameter.
There are several installations in this country under heads of
about 2100 ft. There are numerous cases of heads between
1000 and 2000 ft. but probably the majority of the installations
are for heads of about 1000 ft.

The largest power developed by a single jet upon a single
wheel is 15,000 h.p. The jet is 8 in. in diameter and the wheel
runs at 375 r.p.m. under a head of about 1700 ft.

THE TANGENTIAL WATER WHEEL 39

The largest jet employed upon any Pelton wheel is about
in. in diameter. The net head is 506 ft. in this case.
There are a number of large jets of 9 in. or over used for heads
trom 900 to 1500 ft.

33. Efficiency. — The efficiency of the tangential water wheel
is about the same as that of the average reaction turbine. From
75 to 85 per cent, may reasonably be expected though lower
yafiSfare 6FtelTolDtamed7~due to poor design.

FIG. 31.— Auxiliary relief needle nozzle for use with 10,000 kw. tangential water
wheel. (Made by Pelton Water Wheel Co.)

34. QUESTIONS AND PROBLEMS

1. Of what materials are impulse wheel buckets constructed? How are
they secured to the rim? What is the advantage of "chain type" con-
struction? What should be the relation between the size of the jet and the
size of the bucket?

2. When would two or more jets be used upon a Pelton wheel? What
is the relation between the diameter of the wheel and the diameter of the
jet? How may the speed of rotation of a wheel of given diameter be com-
puted, if the head is known? What fixes the diameter of the jet that is
to be employed, assuming that it is not limited by any wheel size?

3. What is meant by single-overhung and double-overhung construction?
What is the advantage of the latter? How is the shaft usually placed?

4. What is the needle nozzle, the deflecting nozzle, and the deflecting

40 HYDRAULIC TURBINES

needle nozzle? What is the needle nozzle with auxiliary relief and how
does it operate?

5. What are the relative merits of the different methods of governing
the tangential water wheel?

6. What are the conditions of use of impulse wheels in regard to head,
power, size of jet, etc.? What efficiency should be expected?

7. It is desired to develop 3880 h.p. with a Pelton wheel under a head
of 900 ft. Assuming the efficiency of the wheel to be 82 per cent, and the
velocity coefficient of the nozzle to be 0.98, what will be the diameter of
the jet? What will then be a reasonable diameter for the wheel and its
probable speed of rotation? Ans. 6 in., 6 ft., 345 rev. per min.

8. How small could the wheel be made in the preceding problem ? What
would then be its speed of rotation ? If a higher speed than this is desired
for the same horsepower, what construction could be employed?

9. A Pelton wheel runs at a constant speed under a head of 625 ft. The
cross-section area of the jet is 0.200 sq. ft. and the nozzle friction loss is to
be neglected. Suppose a throttle valve in the pipe reduces the head at
the base of the nozzle from 625 ft. to 400 ft. Under these conditions the
efficiency of the wheel (the speed of the wheel no longer being proper for the
head) is known to be 50 per cent. Find the rate of discharge, power of jet,
and power output of wheel.

Ans. 32.08 cu. ft. per second, 1458 h.p., 729 h.p.

10. A Pelton wheel runs at a constant speed under a head of 625 ft.
The cross-section area of the jet is 0.200 sq. ft. and the nozzle friction loss
is to be neglected. Suppose the needle of the nozzle is so adjusted as to
reduce the area of the jet from 0.200 to 0.0732 sq. ft. Under these condi-
tions the efficiency of the wheel is known to be 70 per cent. Find the rate
of discharge, power of jet, and power output of wheel.

Ans. 14.67 cu. ft. per second, 1041 h.p., 729 h.p.

11. Compare the water consumed per horsepower output for the wheel
jn the preceding two problems. Compute the overall efficiency in each
case using the head of 625 ft. Ans. 32 per cent., 70 per cent.

CHAPTER V
THE REACTION TURBINE

35. Development. — The primitive type of reaction turbine
known as Barker's Mill is shown in Fig. 32. The reaction of the
jets of water from the orifices causes the device to rotate. In
order to improve the conditions of flow the arms were then curved
and it became known in this form as the Scotch turbine. Then
three or more arms were used in order to increase the power, and
with still further demands for power more arms were added and
the orifices made somewhat larger until the final result was a
complete wheel. In 1826 a French
engineer, Fourneyron, placed station-
ary guide vanes within the center to
direct the water as it flowed into the
wheel and we then had the outward
flow turbine. In 1843 the first Four-
neyron turbines were built in America. x ^ T-,

The axial flow turbine commonly _j_ (^ j) I

called the Jonval was also a Euro-
pean design introduced . into this ,.,

FIG. 32. — Barker s mill.

country in 1850.

An inward flow turbine was proposed by Poncelot in 1826 but
the first one was actually built by Howd, of New York, in 1838.
The latter obtained a patent and installed several wheels of crude
workmanship in the New England mills. In 1849 James B.
Francis designed a turbine under this patent but his wheel was
of superior construction. Furthermore he conducted accurate
tests, published the results, analyzed them, and formulated rules
for turbine runner design. He thus brought this type of wheel
to the attention of the engineering world and hence his name
became attached to it.

The original Francis turbine is shown in Fig. 5, page 4, and
in Fig. 6, page 5, may be seen photographs of a radial inward
flow runner of this type though of more recent date. As may be
seen in Fig. 4, page 3, the water has to turn and flow away

HYDRAULIC TURBINES

axially after its discharge and hence the original design was
gradually modified so that the water began to turn before its
discharge from the runner. The Swain turbine (1855) shows
this evolution and the McCormick runner (1876) carries it still
further. The latter is the prototype of the modern high speed
mixed flow runner. The nearest approach to the original Francis
runner in present practice is to be seen in Fig. 34, Type I, and in
Fig. 36. The pure radial flow turbine is no longer built, but
since all the modern inward mixed flow turbines may be said to
have grown out of it, they are today quite generally known as
Francis turbines.

Howd 1838

Francis 1849

Swain 1855

FIG. 33.-

McCormick 1876
-Evolution of the modern turbine.

The high-speed mixed flow runner, illustrated by the original
McCormick type in Fig. 33, arose as the result of a demand for
higher speed and power under the low falls first used in this
country. Higher speed of rotation could be obtained by using
runners of smaller diameter, but higher power required runners
of larger diameter, so long as the same designs were adhered to.
So in order to increase the capacity of a wheel of the same or
smaller diameter, the design was altered by making the depth
of the runner greater (i.e., the dimension B, Fig. 34, was in-
creased). The area of the waterway through the runner was

THE REACTION TURBINE

HYDRAULIC TURBINES

also increased slightly by using fewer vanes and it was then de-
sirable to extend these further in toward the center. As that
left a very small space in the center for the water to discharge
through, it was necessary for the runner to discharge the greater
part of the water axially. Type IV of Fig. 34 shows the high-
speed high-capacity runner of today.

FIG. 35. — Leffel turbine for open flume. (Made by James Leffel and Co.)

As civilization moved from the valleys, where the low falls were
found, up into the more mountainous regions, and as means of
transmitting power were introduced, it became desirable to
develop higher heads, and in 1890 a demand arose for high-
head wheels which American builders were not able to supply.
For a time European designs were used and then it was seen that

THE REACTION TURBINE 45

a type similar to the original Francis turbine was well suited to
those conditions. This is shown by Type I of Fig. 34.

FIG. 36. — 42" Francis runner. 8000 h.p., 600 ft. head.
(Made by Platt Iron Works Co.)

FIG. 37. — Turbine runners of the Allis-Chalmers Mfg. Co.

At present the range of common American practice is covered
by the four types shown in Fig. 34, though in a few cases extreme

46 HYDRAULIC TURBINES

designs have passed beyond these limits. American turbines
in the past were developed by "cut and try" methods, European
turbines largely by mathematical analysis. At the present
time the best turbines in this country are designed from rational
theory supplemented by experimental investigation.

36. Advantages of Inward Flow Turbine. — The Fourneyron
turbine has a high efficiency on full load and is useful in some cases
where a low speed is desired, but it has been supplanted by the
Francis turbine for the following reasons:

1. The inward flow turbine is much more compact, the runner
can be cast in one piece, and the whole construction is better
mechanically.

(From a photograph by the author.)
FIG. 38. — Construction of a built-up runner.

2. Since the turbine is more compact and smaller, the con-
struction will be much cheaper. The smaller runner will permit
of a higher r.p.m. and that means a cheaper generator can be
used.

3. The gates for governing are more accessible and it is easier
to construct them so as to minimize the losses. Thus the effi-
ciency of the turbine on part load is better than is the case with
the outward flow type.

4. It is easier to secure the converging passages that are neces-
sary through the runner.

5. A draft tube can be more conveniently and effectively used.

THE REACTION TURBINE

37. General Proportions of Types of Runners. — It has already
been seen how the need for varying the capacity of runners
without changing their diameters has led to altering the height
n and the general profile as illustrated in Fig. 34. The increased
volume of water through the higher capacity runners also re-
quires a larger diameter of draft tube, as well as a higher velocity

mm.

(From a photograph by the author.)
FIG. 39. — Double-discharge runner.

of flow at this section, and in extreme types the flow through the
runner is not merely inward and downward but for those particles
of water nearest the band or ring it is inward, downward, and
outward.

But the quantity of water which will flow through the runner
depends not only upon the area at inlet but also upon the velocity
of the water. If we confine our attention to the circumferential
area of the runner at entrance we are concerned with the velocity
normal to it and this is the radial component of velocity. Hence
we increase the capacity of the runner by making the radial

48 HYDRAULIC TURBINES

component of the velocity of the water larger. This causes the
angle a i to be increased as may be seen in Fig. 34. The angle a\
is determined by the guide vanes.

It is convenient to express the peripheral velocity of the runner
HI as equal to <t>\/2gh. The value of 0 which gives the most
efficient speed for a given turbine is denoted by <f>e and values of
<f>e for different turbines range from about 0.55 to about 0.90
according, to the design.1 If the value of <£ is higher than this
it is probable that the speed is higher than the best speed or
that the nominal diameter for which u\ is computed is larger than
the real diameter. Values of 4>e may be varied in the design by
altering certain angles and areas of the runner.

Since it is desirable, in general, to increase or decrease the
rotative speed and the capacity simultaneously, the custom is
to so proportion the runners that low values of (f>e are found with
turbines of Type I, Fig. 34, while high values are found with
those of Type IV. Thus a low-capacity runner also has a low
peripheral speed for a given head, while a high-capacity runner
would have a higher peripheral speed. Thus for a given diameter
of runner under a given head both power and speed of rotation
increase from Type I to Type IV. If, on the other hand, the
power is fixed, the diameter of runner of Type IV would be
much smaller than that of Type I. Hence the rotative speed of
the former would be higher due to the smaller diameter as well
as the increased linear velocity. For this reason this type is
called a high-speed runner, while Type I is a low-speed runner.
Both capacity and speed are involved in a single factor variously
known as the specific speed, characteristic speed, unit speed, and
type characteristic. It is N8 = Ne\/B.hp./h5/4:, the derivation of
which will be given later. (Ne is the speed for highest efficiency.)
As the capacity and speed increase, this factor increases. Hence
a " high-speed" turbine is really a high specific speed turbine and
a "low-speed" turbine is a low specific speed turbine. The
value of N, is an index of the type of turbine. Values of N8 for
reaction turbines range from 10 to 100, though the latter limit
is occasionally exceeded.

The vector diagrams of the velocities at entrance are drawn to
the same scale in Fig. 34 as if all four types were under the same
head. It may be seen that as we proceed from Type I to Type
IV, Ui, ai, and /3'i increase, while V\ decreases. Since the angle

1 "Water Turbines," by S. J. Zowski is the source of much of Fig. 34.

THE REACTION TURBINE

0i is the angle which the relative velocity of the water makes at
entrance, the vane angle fi'i should be made equal to it.

38. Comparison of Types of Runners. — As a means of illustrat-
ing the differences between the various types of runners the
following tables are presented:

Ring

FIG. 40. — Methods of specifying runner diameter.
TABLE 1. — COMPARISON OF 12-iN. WHEELS UNDER 30-FT. HEAD

Type

Discharge,
cu. ft. per

minute

H.p.

R.p.m.

Tangential water wheel

7.9

0 37

380

Reaction turbines:
Tvne I .

99.0

4.3

460

Type II

329 0

14 9

554

Type III

741 0

33 4

600

Type IV

1209.0

55 5

730

TABLE 2. — COMPARISON OF WHEELS TO DEVELOP 15 H.p. UNDER
30-FT. HEAD

Type

Diameter, in.

R.p.m.

Tangential water wheel

1 55

Reaction turbines:
Tvne I

274

Type II

554

Type III . .

900

Type IV..

1460

It will be seen that the tangential water wheel is a low-speed,
low-capacity type, while the reaction turbine of Type IV is a high-
speed, high-capacity runner. This may be contrary to the popu-
lar impression, but these terms as used here have only relative

HYDRAULIC TURBINES

meanings. Under high heads where the r.p.m. would naturally
be high the relatively lower speed of the tangential water wheel
is of advantage, while under the low heads the relatively higher
speed of the reaction turbine is of advantage. This difference of
speed exists even when the runners are of the same diameter as
seen by the first table. But when the diameters are made such as
to give the same power as in the second table the difference be-

FIG. 41. — 13,500 h.p. runner. Head = 53 ft., speed

I. P. Morris Co.)

94 r.p.m. (Made by

comes much greater. It must be understood that these tables do
not prove one type of wheel to be any better than another but
merely show what may be obtained. If the tangential water
wheel or Type I of the reaction turbines appear in an unfavorable
light it is only because the head and horsepower are not suitable
for them.

THE REACTION TURBINE

39. Runners. — Runners may be cast solid or built up, but the
majority are cast solid as the construction is more substantial.
Occasionally a very large runner may be cast in sections. Built
up wheels have the vanes shaped from steel plates andfthe
crown, hubs, and rings are cast to them, as shown in Fig.* 38.
The best runners are made of bronze. Cast steel is used for very

FIG. 42. — 10,000 h.p. runner at Keokuk, la. Head = 32 ft., speed = 57.7
r.p.m. (Made by Wellman-Seaver-M organ Co.)

high heads in some cases, while cheaper runners are made of
cast iron. Very naturally the large runners are made of the
latter metal.

Runners may be divided into two broad classes of single and
double discharge runners. Figs. 36 and 42 are of the first type
and Fig. 39 of the second. The latter is essentially two single

HYDRAULIC TURBINES

discharge runners placed back to back and requires two draft
tubes as the water is discharged from both sides. It is used
only for horizontal shaft units, while the single discharge runner
may be used for either horizontal or vertical shaft turbines.

Turbines are often rated according to the diameter of the
runner in inches. This diameter is easily fixed in many cases,
but for one of the type shown in Fig. 40 either one of four dimen-
sions may be used. Different makers follow different practices
in this regard but the usual method is to measure the diameter
at a point about halfway down the entrance height.

Position of Gate
when Closed

No. 1 Runner
No. 2 Chute Case
No. 3 Gate

FIG. 43. — Register gate.

40. Speed Regulation. — The amount of water supplied to the
reaction turbine is regulated by means of gates of which there are
three types.

The cylinder gate is shown in Fig. 5, page 4. It is the simplest
and cheapest form of gate and also the poorest, although, when
closed, it will not leak as badly as the others. When the gate
is partially closed there is a big shock loss in the water entering
the turbine runner due to the sudden contraction and the sudden
expansion of the stream that must take place. With this type of

THE REACTION TURBINE

gate the efficiency on part load is relatively low and the maximum
efficiency is obtained when the gate is completely raised.

A better type of gate is the register gate shown in Fig. 43. With
this type the guide vanes are made in two parts, the inner portion
next to the runner is stationary, the outer portion is on a ring
which may be rotated far enough to shut the water off entirely,
if necessary, as shown by the dotted lines. While this is more

(Courtesy of Allis-Chalmers Co.)
FIG. 44. — View of guide vanes and shifting ring.

efficient than the preceding type there is still a certain amount of
eddy loss that cannot be avoided. It is seldom used.

The wicket gate, also called the swing gate or the pivoted
guide vane, is shown in Fig. 45. This is the best type and also
the most expensive. As the vanes are rotated about their pivots
the area of the passages through them is altered. The vanes
may be closed up so as to shut off the water if necessary. Of
course the angle, «i, is altered and a certain amount of eddy loss
may also result but it is less than occasioned by either of the

HYDRAULIC TURBINES

FIG. 45. — AA'icket gate with all operating parts outside.

FIG. 46. — Wicket gates and runner in turbine made by Platt Ironworks

THE REACTION TURBINE

other forms. The maximum efficiency is obtained before the
gates are opened to the greatest extent.

The connecting rod from the relay governor operates a shifting
ring. This in turn, by means of links, rotates the vanes. These

FIG. 47.— 10,000 h.p. turbine at Keokuk, la.

Co.)

(Made by Wellman-Seaver-M organ

links are shown in Figs. 45, 47, and 48. Often the shifting ring
and links are inside the case, but the better, though more ex-
pensive, type has the working parts outside the case.

In order to prevent shock in the penstock when the governor
quickly closes the gates, many turbines .are provided with

56 HYDRAULIC TURBINES

mechanically operated relief valves, as in the left hand side of
Fig. 49. This valve is opened at the same time the gates are
closed, thus by-passing the water. The relief valve may be so
arranged with a dash-pot mechanism that it will slowly close.

41. Bearings. — For small vertical shaft turbines a step bearing
made of lignum vitse is used under water, as at the bottom of
the runner in Fig. 35. This wood gives good results for such
service and wears reasonably well. For larger turbines a thrust
bearing is usually provided to which oil is supplied under pressure.
Roller bearings are also used with the rollers running in an oil
bath, as in Fig. 50. Sometimes rollers are provided in the former
type but act only when the pressure fails, and again roller bear-

FIG. 48. — Shifting ring and links on a wicket gate spiral case turbine. (Made
by Platt Iron Works Co.)

ings may sometimes be supplied with oil under pressure between
two bearing surfaces in case the rollers fail. The Kingsbury
bearing is fitted with a number of metal shoes so mounted that
their bearing surfaces are not quite level. Thus as they advance
through the oil bath a wedge-shaped film of oil is forced in be-
tween these shoes and the other surface. Such a bearing is
preferably located at the top of the shaft in which case it is
called a suspension bearing, though it may be placed between the
generator and the runner.

A horizontal turbine set in an open flume often has lignum
vitae bearings as the water is a sufficient lubricant. However

THE REACTION TURBINE

(Courtesy of Pelt on Water Wheel Co.)
Fie;. 49. — Spiral case turbine with relief valve.

(Courtesy of Electric Machinery Co.)
FIG. 50. — Heavy duty suspension bearing. |

HYDRAULIC TURBINES

the water must be clear; gritty water would destroy the bearings.
If the turbine is in a case so that the bearings are accessible the
usual types of bearings are used. It must not be forgotten that
even though the shaft be horizontal a very considerable end thrust
must be allowed for due to the reaction of the streams discharged
from the runner. That is one reason for using runners in pairs.
Also a single runner is often used which has a double discharge.
(See Fig. 51.) Single discharge runners are often provided with
some form of automatic hydraulic balancing piston to equalize
the thrust.

FIG. 51. — Double discharge runner in spiral case.

As the leakage of water through the gates, when closed, may
be sufficient to keep the turbine running slowly under no-load,
large units are often provided with brakes so they can be stopped.

42. Cases. — For low heads turbines may be used in open
flumes without cases. Fig. 4, page 3, Fig. 8, page 10, and
Fig. 35, page 41, are of this character. Fig. 52 shows such a
type consisting of four wheels on a horizontal shaft.

Cases may also be used for very low heads and are always
used for high heads. The cheapest cases are the cylinder cases
(Fig. 10, page 12), and the globe cases (Fig. 53). These cases
are undesirable because they permit of considerable eddy loss
as the water flows into them and around in them to the guides.

THE REACTION \TURBINE

HYDRAULIC TURBINES

The cone case shown in Fig. 54 is a very desirable type. It can
be seen that the water suffers no abrupt changes in velocity as it
flows from the penstock to the guides, but instead is uniformly
accelerated.

The spiral case, illustrated by Fig. 55, is considered the best
tvpe. The area of the waterway decreases as the case encircles
the guides, because only a limited portion of the water flows clear

FIG. 53. — Turbine in globe case. (Made by James Leffel and Co.)

around to enter the further part of the circumference. Thus the
average velocity throughout the case is kept the same. The
ca.se is also designed to accelerate the water somewhat as it
leaves the penstock and flows to the guides.

Globe and spiral cases for low heads are made of cast iron. For
higher heads they are made of cast steel as in Fig. 56. Cylinder
cases (Fig. 10, page 12), are usually^made of riveted sheet steel.
Some very nice spiral cases are now made by several firms of

THE REACTION TURBINE

riveted steel, as may be seen in Fig. 57. Recent practice with
large vertical shaft units is to form the case of reinforced concrete
as in Fig. 58. The weight of the turbine and generator is carried
by the "pit liner/' which is set into the concrete, and this in turn

FIG. 54. — Cone case turbine.

rests upon the " speed ring." The latter consists of an upper and
a lower flange, as shown, which are joined together by vanes so
shaped as to conform to the free stream lines of the water flowing
from the case into the guide vanes. Thus they offer less re-

HYDRAULIC TURBINES

sistance to the flow of the water than the round columns that
were once employed. The speed ring vanes are also shown in

[ FIG. 55. — Spiral case turbine

FIG. 56. — Cast steel spiral casings at Niagara Falls. 14,000 h.p. at ISO ft. head.
(Made by Wellman-Seaver-M organ Co.)

Figs. 59 and 60. In some instances the case is of sheet metal
surrounded by concrete as in Fig. 60.

THE REACTION TURBINE

43. Draft Tube Construction. — It must be borne in mind that
the draft tube fulfills two distinct functions. First, it connects

(Courtesy of S. Morgan Smith Co.)
FIG. 57.— Spiral case of riveted steel plates.

FIG. 58. — Typical large vertical shaft unit.

the turbine runner with the tail water and, since it is air tight,
it prevents any loss of head due to the elevation of the wheel.

HYDRAULIC TURBINES

Second, it may be made to reduce the outflow loss and thus to
improve the efficiency of the plant.

For the first purpose alone the tube might be made of a uni-
form cross-section, but in practice it is always made diverging
so as to accomplish the second object as well. In fact, even if
the runner should be set below the tail-water level, a draft tube
would be of value for the second purpose. This was proven
many years ago when Francis tested an outward-flow turbine
with a "diffuser" surrounding the runner and found that the

(Courtesy of Allis-Chalmers Mfg. Co.)
FIG. 59. — Speed ring.

latter improved the efficiency by 3 per cent. As has been pointed
out_in Art. 37, the higher the capacity of a runner of given
diameter the greater the velocity of the water must be at the
point of outflow from it into the draft tube. This velocity
represents kinetic energy which would otherwise be carried away
and many modern wheels of an extreme high-capacity type
would not have favorable efficiencies at all if it were not for the
use of a suitable draft tube.

THE REACTION TURBINE

The reason for this gain in efficiency may be seen in either of
the following ways. First, the total power available is that due
to the fall from head-water level to tail-water level. The power
of the turbine is less than this by an amount equal to that lost
in the intake, penstock, and draft tube. Anything which re-
duces the loss outside the turbine adds just that much more to

(Courtesy of Allis-Chalmers Mfg. Co.)

FIG. 60.— Turbine for Niagara Falls Power Co., 37,500 h.p., 214 ft. head, 150

r.p.m.

the power which the water can give up within the turbine. The
velocity head with which the water leaves the lower end of the
draft tube represents kinetic energy which is lost, and the less
this discharge loss becomes, the better the efficiency of the tur-
bine. From another standpoint the pressure at the upper end
of the draft tube depends not only upon the elevation of this

HYDRAULIC TURBINES

point above the tail-water level but also upon the velocity of the
water at that section and the losses within the tube, including
the discharge loss at its mouth. The less this loss, the lower
the pressure at the upper end. And the less the pressure at the
point of outflow from the turbine runner, the better will be its
performance.

Draft tubes are usually made of riveted steel plates as in Fig.
10, page 12, or are moulded in concrete as in Fig. 8, page 10.

FIG. 61. — Draft tube with quarter turn.

The tube should preferably be straight but where the setting
does not permit of enough room for this without excessive cost
of excavation the tube is often turned so as to discharge hori-
zontally as in Fig. 61. If the tube is large in diameter it may be
necessary to make the horizontal portion of some other section
than circular as in Fig. 62, in order that the vertical dimension
may not be too great. A good form of section to use is oval.
The draft tube is commonly made as a frustum of a cone with a

•
THE REACTION TURBINF 67

vertex angle of 8°. If the section becomes some other shape,
the tube is so made that the area increases at a similar rate to
what it would if it were circular. The conical form has been
largely employed chiefly because of ease of manufacture, but
when draft tubes are moulded in concrete other forms may be
used. A form that is theoretically good is " trumpet shaped,"
somewhat as in Fig. 60, so that the velocity of the water may be
made to decrease uniformly along the length of the tube. In
any event the draft tube should be so made as to secure a gradual
reduction of velocity from the runner to the mouth.

(Courtesy of Wellman-Seaver-M organ Co.')
FIG. 62.— Mouth of draft tube at Cedar Rapids.

The most recent innovation in draft tube construction is shown
in Figs. 60 and 63. At the lower end of a comparatively short
draft tube is a conoidal portion through which the water passes
just before impinging on a circular plate which is concentric
with the tube. The water turns and flows out along this plate
around its entire circumference through an annular opening
into a collecting chamber and from thence through a horizontal
diverging tube to the tail race. As the water flows through
the conoidal portion of the tube and impinges on the plate, its
velocity is_greatly reduced. This portion is called by the in-

HYDRAULIC TURBINES

ventor, W. M. White, a "hydraucone." As the water turns
and flows through the annular opening, its velocity is increased
and is then decreased again as it enters the collecting chamber.
The velocity is still further decreased as the water flows to the tail
race through the horizontal tube. A design by Lewis F. Moody
differs from the above in that the collecting chamber is a spiral,
somewhat like the spiral case, so proportioned that the water is
continuously decelerated throughout the flow.

Bearing in mind that one function of the draft tube is to
efficiently convert velocity head into pressure head, we see the

FIG. 63. — Draft tube with hydraucone.

limitations of the ordinary construction. In order to secure the
diffusion desired, the length of the tube may be such that the
expense of excavation is prohibitive and hence the tube is turned
from vertical to horizontal with a bend of short radius. But such
a bend inevitably induces eddy losses which interfere with the
efficient performance of the tube. Furthermore velocity head
cannot usually be converted into pressure head without a great
deal of loss unless the flow of the water be smooth. Since the
discharge from a turbine runner is usually quite turbulent, this
alone would limit the value of a draft tube, even if it were
straight. If the device just described is properly proportioned,
then, as the result of hydrodynamic laws for which we have not

THE REACTION TURBINE

room here,1 the flow may be turned from vertical to horizontal
in a very small space with less loss of energy than otherwise.
Another peculiarity of the hydraucone is that it can convert
the velocity head of turbulent water into pressure head efficiently.
Then when this water is accelerated, as it leaves through the
annular opening, it flows away with smooth stream lines and is
in proper shape for the ultimate conversion in the diverging
horizontal tube. One of the things which limits the high-speed
high-capacity type of runner is the inability of the draft tube to
recover the kinetic energy of the water leaving the runner,
especially in view of the fact that with this type the water leaves
with some considerable "whirl." This new development may

r* Case **-*G-uldesH<5 H

j Runnei*'

FIG. 64. — Velocity and energy transformations in turbine.

make it possible to extend the present limits of turbine runner
design.

44. Velocities. — The velocities at different points are indicated
by Fig. 64. 2 The velocity of flow in the penstock is determined
by the consideration of the cost and other conditions in eace
case. The mean velocity of flow allowable in the turbine cash
is as follows:

If the case is cylindrical the velocity should be as low as 0.08
to O.l2\/2gh where h is the effective head. If a spiral case is
used the velocity may be from 0.15 to 0.24\/2gh. For heads of
several hundred feet the value of 0.15 is used to reduce wear on
the case, 0.20 is used for moderate heads, and 0.24 is used for
low heads.

The velocity at entrance to the turbine runner, Vi = 0.6 to

\The Journal of the Association of Engineering Societies, vol. 27, p. 39.

2_Mead's "Water Power Engineering."

70 HYDRAULIC TURBINES

Q.8\/2gh. The velocity at the point of discharge, V% = from
0. 10 to QAQ\/2gh. These values depend entirely upon the design
of the turbine and are not arbitrarily assigned.

The velocity at entrance to the upper end of the draft tube
should equal the velocity with which the water leaves the turbine,
otherwise a sudden change in velocity will take place. Velocity
of discharge from the lower end of the draft tube may be about
0.10 to Q.15\/2gh. The value of the latter is determined by the
value of the velocity at the upper end and by the length and
the amount of flare to be given the tube.

45. Conditions of Use. — The reaction turbine is best adapted
for a low head or a relatively large quantity of water. As was
stated in Art 32, the choice of a turbine is a function of capacity
as well as head. For a given head the larger the horse-power
the more reason there will be for using a reaction turbine.

The use of a reaction turbine under high heads is accompanied
by certain difficulties. It is necessary to build a case which is
strong enough to stand the pressure; also the case, guides, and
runner may be worn out in a short time by the water moving at
high velocities. This depends very much upon the quality of the
water. Thus a case is on record where a wheel has been operating
for six years under a head of 260 ft. with clear water and the tur-
bine is still in excellent condition. Another turbine made by the
same company and according to the same design was operated
under a head of 160 ft. with dirty water. In four years it was
completely worn out and was replaced with an impulse wheel.
The tangential water wheel has the advantage that the relative
velocity of flow over its buckets is less for the same head and thus
the wear is less. Also repairs can be more readily made.

The runners of reaction turbines and the buckets of impulse
wheels will not last long if their design is imperfect. This is due
to the fact that wherever there is an eddy or wherever there is a
point of extremely low pressure, the air that is in solution in the
water will always tend to be liberated at that point. And as
water tends to absorb more oxygen in proportion to nitrogen
than is in the air, the result is that the liberated mixture is rich
in oxygen and hence readily attacks and pits the metal. Fig.
6, page 5, shows a turbine runner that has had holes eaten in
it because of this reason. Thus a defective design not only
produces a runner of lower efficiency because of the eddy losses
within the water, but such eddies shorten the life of the wheel.

THE REACTION TURBINE 71

Also great care should be used in designing so that the velocity
along any stream line does not cause the pressure to approach the
vapor pressure of the water too closely, otherwise the same action
will take place. With the reaction turbine it is possible to de-
sign a runner free from eddies for one gate opening only. The
operation of reaction turbines at part gate for long periods of
time must inevitably shorten the life of the runner.

FIG. 65. —22,. 500 h.p. turbine for Pacific Coast Power Co.
(Made by Allis-Chatmers Mfg. Co.)

Reaction turbines are used under very low heads in som?
instances. The lowest head on record is 16 in. but several feet
is the usual minimum. The highest head yet employed for a
reaction turbine is 800 ft. The latter is used for two 22,500
h.p. units built by the Pelton Water Wheel Co.

The most powerful turbine in the world will develop 52,500

72 HYDRAULIC TURBINES

h.p. Two such units, built by the Allis-Chalmers Mfg. Co.,
will be installed at the Chippewa Development of the Hydro-
Electric Power Commission of Ontario. The head is 320 ft.

In Fig. 60 is shown ji turbine of 37,500 h.p., which is similar
in type to the above. This wheel runs under a head of 214 ft.
and is for the Niagara Falls Power Co.

There are at present quite a number of turbines in operation
whose power ranges from 20,000 to 30,000 h.p.

The power of a turbine depends not only upon its size but also
upon the head under which it operates. The turbines above are
the most powerful, but they are not the largest in point of size.
The largest turbines so far are the 10,800 h.p. turbines of the
Cedars Rapids (Canada) Mfg. and Power Co., which run at
56.6 r.p.m. under a head of 30 ft. The rated diameter is 143 in.,
but the maximum diameter' (see Fig. 40) is 17 ft. 8 in. The
runner weighs 160,000 lb., the revolving part of the generator
and the shaft, 390,000 lb., while the suspension bearing weighs
300,000 lb. The total weight of the entire unit is 1,615,000 lb.

The largest runners in this country are those of the Mississippi
River Power Co. at Keokuk, la. They develop 10,000 h.p.
at 57.7 r.p.m. under a head of 32 ft. They are slightly smaller
than those at Cedars Rapids. (See Figs. 42 and 47.) The I. P.
Morris Co. built eight of these wheels and the Wellman-Seaver-
Morgan Co. seven. These two concerns likewise built the
Cedars Rapids turbines.

46. Efficiency. — The efficiency obtained from the average
reaction turbine may be from 80 to 85 per cent. Under favorable
conditions with large capacities higher efficiencies up to about 90
per cent, or more may be realized. For small powers or un-
favorable conditions 75 per cent, is all that should be expected.

47. QUESTIONS

1. What was the origin of the Fourneyron turbine ? What is the Jonval
turbine? What was the origin of the Francis turbine?

2. What is the Swain turbine? What is the McCormick turbine? Why
were they developed? What is the modern Francis turbine? Why is this
name attached to all inward flow turbines at present?

3. Sketch the profiles of different types of modern turbine runners and
explain why they are so built.

4. Why has the inward flow turbine superseded the outward flow turbine ?

5. How does the angle a\ vary with different types of runner and why?
How does the factor <f>« vary and why?

THE REACTION TURBINE 73

6. For a given head and diameter of runner explain how the power varies
with different types. For a given head and power explain how the rotative
speed varies.

7. Draw typical vector diagrams for the velocities at entrance to the dif-
ferent types of turbine runners. Show how the vane angle varies.

8. How are turbine runners constructed? What materials are used?
What classes of runners are there ?

9. What are the different kinds of gates used for governing reaction tur-
bines, and what are their relative merits ?

10. What means are provided to save the penstock 'from water hammer,
when the gates of a reaction turbine are quickly closed ? How are the gates
of a turbine operated?

11. What kinds of bearings are used for horizontal shaft turbines? For
vertical shaft turbines? What means may be provided to take care of end

hrust in either type?

12. What types of cases are used for turbines? What are the cheapest
forms and what are the best? What are speed rings?

13. What is the purpose of a draft tube and how are they constructed ?

14. What different factors may cause a turbine runner to wear out?
Under what range of heads are reaction turbines now used?

16. What horsepower is developed by the most powerful turbine?
What is the largest in point of size? Why is not the largest one also the
most powerful? What efficiencies should be expected from reaction
turbines?

CHAPTER VI
TURBINE GOVERNORS

48. General Principles. — All governors depend primarily upon
the action of rotating weights. Thus the governor head in Fig. 66
is rotated by some form of drive so that its speed is directly
proportional to that of the machine which it regulates. The
higher the speed of rotation, the farther the balls stand from
the axis, and the higher will the collar be raised on the vertical
spindle. The collar in turn transmits motion to some element
of the mechanism which effects the speed regulation.

Let W be the weight of each ball, 2KW that of the center
weight, h the height of the "cone" in inches, x the ratio of the

FIG. 66. — Governor head.

velocity of the collar to the vertical velocity of the balls, and N
the revolutions per minute of the governor head. Also let the
force which opposes the motion of the collar, due to the friction
of the moving parts of the governor mechanism actuated by it,
be denoted by 2fW. Then the following equation may be found
to hold :

N2h = 35,200 [1 + x(K ± /)].

Considering the right hand member of the above as constant for
the moment, it may be seen that for every value of h there must
be a definite value of N. For different loads on the machine it

TURBINE GOVERNORS

is necessary that the gates and gate mechanism occupy different
positions and, if this requires different positions of the collar of
the governor head, it may be seen that the speed must decrease
as the load increases.1

If the change of speed from no load to full load be denoted by
AJV and N be interpreted as the average speed, the coefficient of
speed regulation is AN/N. This coefficient may be reduced to a
very small value by careful design of the governor. The essen-
tials of a good governor are:

1. Close regulation or a small value of &N/N.

2. Quickness of regulation.

3. Stability or lack of hunting.

4. Power to move parts or to resist disturbing forces.
To some extent certain of these requirements conflict with others

N
FIG. 67.

so that the final design is something of a compromise. Close
regulation may be obtained by so proportioning the arms that
# is a variable in such a way as to .permit h to change but little
for the different collar positions. Stability and power may be
secured by making the center weight, 2KW, sufficiently heavy.
This weight is often replaced by a spring, which exerts an
equivalent force. The importai ce of a large value of K is seen
when we consider its relation to the friction. The latter changes
sign according to the direction of motion, and may also change

1 A constant speed or asynchronous governor could be constructed by so
arranging it that h remained constant as the balls changed their position,
but such a governor would lack stability as it might be in equilibrium with
the collar in any position for a given speed. Then for a slight change in
speed the governor would move over to its extreme position which would
be limited by a stop. Such governors have been built, however, and are
practicable if a strong dash pot is used to prevent their "hunting."

76 HYDRAULI'C TURBINES

in value from time to time, so that the larger the value of K
the smaller will be the effect of friction, arid the closer together
will the two curves of Fig. 67 be. We can also see the great
necessity for keeping / as small as possible, and this requirement
leads us to the use of the relay governor.

The operation of the nozzles of impulse wheels or of the gates
of reaction turbines requires a considerable force to be exerted.
The governor head could not do this directly without being of
prohibitive size and hence it does nothing more than set some
relay device into action, the latter furnishing the power to
operate the regulating mechanism.

49. Types of Governors. — There are two fundamental types of
water wheel governors :

(a) Mechanical governors.

(6) Hydraulic governors.

With the first type the governor head causes some form of
clutch to be engaged so that the gates are operated by the power
of the turbine itself. This is the least expensive but has the
disadvantage that the operation of the gates adds just that much
more to a demanded load. The second type of governor costs
more but is always used in the best plants. The governor head
in this case merely operates a pilot valve which admits a liquid
under pressure to one side or the other of a piston in a cylinder.
This piston and cylinder is known as the servo-motor and
operates the gates.

The liquid used to operate the servo-motor is stored under air
pressure in a tank into which it is pumped. The power for
operating the gates also comes from the turbine in this instance
but it is spread over the entire period of operation instead of
being concentrated just when the load is changing. Oil is
commonly used as the working fluid and is very satisfactory
except for its cost. Some effort, which is meeting with success,
is being made to produce emulsions consisting principally of
water but which will be similar to oil in its action. If water
alone is used, it should be carefully filtered and circulated over
and over again. Occasionally water has been used under pen-
stock pressure and, of course direct from the penstock, but the
grit and sediment in it is very bad for the operating parts of
the governor.

50. The Compensated Governor. — It has already been pointed
out in Art. 26 that the inertia of the water in the penstock and

TURBINE GOVERNORS

draft tube makes close speed regulation difficult, since often the
immediate result of a change of gate position is directly opposite
to that desired. With the simple type of governor the latter
would continue to operate in the same direction under these
circumstances and would thus move far beyond the proper
point. Finally when the hydraulic conditions would readjust
themselves the governor would then be compelled to move back,
but would this time pass to the other side of the proper place.
Thus the governor would continually "hunt" and maintain a
constant oscillation of flow in the pipe line.

To Kegulate Speed

variation from No Load

to Full Load

FIG. 68. — Compensated floating lever governor.

To prevent such action a waterwheel governor is usually
"compensated" so that it will slowly approach its proper place
and practically remain there. Such a governor is also said to be
"dead beat." In Fig. 68 may be seen the essential features of
the floating-lever compensated governor. If, for example, the
wheel speed increases, the balls of the governor raise the collar,
C, to which is attached the floating lever. The latter for the
moment pivots about A and through the link at B raises the relay
valve, D. This action admits oil (or other fluid) to the left-
hand side of the servo-motor piston and exhausts it from the
right-hand side, thus compelling the piston to move to the right
and decrease the turbine gate opening. But at the same time

HYDRAULIC TURBINES

the bell-crank EFG, which is attached to the gate connecting
rod at E, is rotated about F so the arm G is lowered. This pulls
down the pivot A, which causes B to be lowered, thus closing the
relay valve ports and stopping the motion. Thus the governor
is prevented from over-travelling. Of course, if the gates have
not been moved far enough, this action can be repeated.

The dash pot, H , will not cause the pivot A to be moved unless
the governor acts quickly. If the governor changes slowly,
there is little need for the compensating action and hence the
dash pot does not then transmit the motion. But there is a

FIG. 69.

(Courtesy of Allis-Chalmers Mfg. Co.)
Hydraulic turbine governor.

second rod from G which is connected with the other vei tical rod
by springs at M. This will serve to stop the motion in such a
case though it does not move A as much, since it has a shorter
radius arm.

For a given speed of the governor head, and hence for a given
position of the collar C, moving A will tend to shift the relay
valve and hence change the position of the turbine gates. But
if the turbine gates are changed, without any corresponding
change, in load, the turbine speed will vary. The length of the
rod from G to M is adjustable by turning a wheel L into which
the two ends of the rods are fitted with right and left handed
threads. Hence the speed of the turbine can be varied within

TURBINE GOVERNORS 79

certain limits by L, which is convenient for synchronizing, for
instance. Generally L is turned by a very small electric motor
which can be operated from the switchboard.

At full-load on the turbine the servo-motor piston is at the
opposite end of the stroke from no-load and hence the pivot A
has a corresponding vertical travel. The amount of this travel
can be altered by changing the radius of the connection at G.
Considering B as fixed (as it must be if the relay valve is closed in
both cases) it is evident that changing the amount of travel of
A will change the amount of travel of the collar C. Remember-
ing that different positions of collar C correspond to definite
values of N, it is clear that changing the amount of travel of the
collar C from no-load to full-load will vary the speed regulation.

Other adjustments that can be made to secure the proper
degree of sensitiveness for the hydraulic conditions are to vary
the springs at M and to change the speed of the dash pot.

One of the recent changes in governor construction for vertical
type turbines is to mount the rotating weight on the turbine shaft
itself. This eliminates any lost motion between the turbine and
governor head.

51. QUESTIONS

1. With the usual type of governor, why must the speed vary to a slight
extent from no-load to full-load? Which way does the speed change as
the load increases? Why?

2. What qualification are essential in a good governor and how may
they be obtained? What is the effect of friction on the operation of the
governor?

3. Why is the speed range for a decreasing load different from that for
an increasing load ? What is the purpose of the center weight or the spring
loading in governors?

4. What is a relay governor? Why is it necessary for water wheels?
How is it operated?

5. What are the relative merits of different types of relay governors?
What are the relative merits of the fluids used in hydraulic governors?

6. What is the compensated governor? Why is it necessary? Describe
the action of one?

7. Describe the adjustments that can be made on a floating lever governor

CHAPTER VII
GENERAL THEORY

52. Equation of Continuity. — In a stream with steady flow
(conditions at any point remaining constant with respect to
time) the equation of continuity may be applied. This is that
the rate of discharge is the same for all cross-sections so that
q = AV = av = constant, and in particular

q = AiVi = aiVi = a&z (1)

53. Relation between Absolute and Relative Velocities. — The

absolute velocity of a body is its velocity relative to the earth.
The relative velocity of a body is its velocity relative to some
other body which may itself be in motion relative to the earth.
The absolute velocity of the first body is the vector sum of its
velocity relative to the second body and the velocity of the second
body. The relation between the three velocities u, v, V is shown

FIG. 70. — Relation between relative and absolute velocities.

by the vector triangles in Fig. 70. The tangential component of
Fis

Vu — V cos A = u + v cos a (2)

54. The General Equation of Energy. — Energy may be trans-
mitted across a section of a flowing stream in any or all of the
three forms known as potential energy, kinetic energy, or pressure
energy.1 Since head is the amount of energy per unit weight
of water, the total head at any section

<»>

M. Hoskins, "Hydraulics," Chapter IV.

GENERAL THEORY

There can be no flow without some loss of energy so that the total
head must decrease in the direction of flow by the amount of head
lost or

F! - #2 =. Head lost (4^

Suffixes (1) and (2) may here denote any two points.

In flowing through the runner of a turbine the water gives up
energy to the vanes in the form of mechanical work and a portion
of the energy is lost in hydraulic friction and is dissipated in the
form of heat. Thus the head'lost by the water equals In!' + hf.
And if suffixes (1) ; and (2) "are restricted to the points of
entrance to and discharge from the runner, equation (4) may
be written

(5)

55. Effective Read on Wheel. — Obviously the turbine should
not be charged up with head which is lost in the pipe line, so the

FIG. 71. — Effective head for tangential water wheel.

value of h should be the total fall available minus the penstock
losses. Thus if Z is total fall available from head water to tail
water, Hf the head lost in the penstock or other places outside
the water wheel, and h the net head actually supplied the turbine,
we have

h = Z - H' (6)

The head supplied to the impulse wheel in Fig. 71 is the head
measured at the base of the nozzle. Thus for the tangential
water wheel

(7)

The reaction turbine, shown in Fig. 72, is able to use the
total fall to the tail-water level by virtue of its employment of
the draft tube. Hence the total head supplied to the wheel at C

HYDRAULIC TURBINES

is measured above the tail-water level as a datum plane,
for the reaction turbine

Thus

(8)

The turbine with its draft tube, which in a sense is as much
an appurtenance of the runner as the guide vanes, is here charged
with the total amount of the energy supplied to it. The kinetic
energy of the water at discharge from the mouth of the draft tube
E is a loss for which the runner and draft tube may be said to be
responsible in part, though some loss there is inevitable, but the
trouble is that the setting of the turbine, over which the turbine
builder has little control, limits the design of the draft tube and

FIG. 72. — Effective head for reaction turbine.

hence the manufacturer may not be able to reduce this discharge
loss to a desired value. Two similar runners installed under
different settings might yield different efficiencies because of this.
Consequently turbine builders desire some method which will
make the measured efficiency of a runner independent of the
conditions of the setting over which their designers have no
control. This second method is to charge up the turbine with all
losses within the draft tube but to credit it with the velocity
head at the point of discharge. Thus

h = Hc — HE — zc

PC
W

v c

(9)

It is believed that equation (8) is rational and scientifically
correct, but that equation (9) may be commercially more de-
sirable.1 In general the actual numerical difference between
the values of h computed by these two methods will be small.

1 For discussion on this point see "Investigation of the Performance of a

GENERAL THEORY 83

56. Power and Efficiency. — Since head is the amount of energy
per unit weight of water it follows that by multiplying by the
total weight of water per unit time we have energy per unit time
and this is power. Thus

Power = WH = pounds per second X feet (10)

In this expression H may be interpreted as in (3) or it may be
replaced by h" or any other head according to what is wanted.

But also power equals force applied times the velocity of the
point of application. Thus

Power = Fu = pounds X feet per second (11)

where F represents the total force applied.
Torque, T, equals F X r and angular velocity o> =
Since then Fu = Tu it is evident that ,

Power = Tco = foot pounds per second I (12)

Any of these three expressions for power may be used according
to circumstances. While (11) is the most obvious to many, it
will be found that in hydraulics (10) is usually more convenient.

(The following simplifications for horsepower of a turbine are
convenient. Using the h of Art. 55,

B.h.p. = 62.5 qhe/550 = qhe/8.8.

For estimations, the value of the efficiency may be assumed as
0.80 in which case our expression becomes h.p. = qh/ll.

The word " efficiency" is always understood to mean total
efficiency. It is the ratio of the developed or brake power to
the power delivered in the water to the turbine based on the
head h of Art. 55.

Reaction Turbine," by R. L. Daugherty, Trans. Am. Soc. of C. E., vol. 78,
p. 1270 (1915).

It may be noted that it might be desirable under some circumstances to
eliminate the draft tube losses altogether and compute the efficiency of the
runner alone. This would necessitate the measurement of the head at D
in order to take the difference between it and the head at C. The practical
difficulty here is that, due to the turbulent and often rotary motion of the
water at this point, it is impossible to measure the pressure with any degree
of accuracy. Likewise the velocity head cannot be computed, since the actual
velocity under the conditions of flow will not be equal to the rate of discharge
divided by the cross-section area. This same consideration holds in regard
to the computation of the velocity head at E.

HYDRAULIC TURBINES

Mechanical efficiency is the ratio of the power delivered by
the machine to that delivered to its shaft by the runner. The
difference between these two powers is that due to mechanical
losses.

Hydraulic efficiency is the ratio of the power actually de-
livered to the shaft to that supplied in the water to the runner.
The difference between these two is due to hydraulic losses.

Volumetric efficiency is the ratio of the water actually passing
through the runner to that supplied. The difference between
these two quantities is the leakage through the clearance spaces.

The total efficiency is the product of these three. Thus

e = em X eh X e0.

57. Force Exerted. — Whenever the velocity of a stream of
water is changed either in direction or in magnitude, a force is
required. By the law of action and reaction an equal and
opposite force is exerted by the water upon the body producing
this change. This is called a dynamic force.

Let R be the resultant force exerted by any body upon the
water and Rx and Ry be its components parallel to x and y axes.
Also let us here consider a as the angle made by V with the x
axis. The force exerted by the water upon the body will be
denoted by F. Its value may be found in either of the two ways
shown below. The first depends upon the principle that the
resultant of all the external forces acting on a body is equal to

dV
the mass times the acceleration or R = m ^- The second is

based upon the principle that the resultant of all the external

GENERAL THEORY 85

forces acting on a body or system of particles is equal to the time
rate of change of momentum of the system or R = — -j- —

(a) Force Equals Mass Times Acceleration. — Let dR be the
force exerted upon the elementary mass shown in Fig. 73. Let
the time rate of flow be dm/dt, where m denotes mass. Then
in an interval of time dt there will flow past any section the mass
(dm/dt) dt, which will be the amount considered. Thus

dV dm,\dV dm /dV

Our discussion will be restricted to the case where the flow is
steady in which case dm/dt is constant and equal to W/g. There-
fore, since (dV/dt)dt = dV,

dR = W dv

The summation of all such forces along the vane shown will give
the total force. But, since integration is an algebraic and not a
vector summation and in general these various elementary forces
will not be parallel, it is necessary to take components along any
axes. Thus

Now at point (1) the value of Vx is Vi cos «i and at (2) it is V%
cos <*2. Inserting these limits and noting from Fig. 72 that
Vo cos «2 — V\ cos on = &VX, we have

W W

Rx = - (Vz cos «2 — Vi cos «i) = — AFs
9 0

(6) Force Equals Time Rate of Change of Momentum. — Consider
the filament of a stream in Fig. 74 which is between two cross-
sections M and N at the beginning of a time interval dt, and
between the cross-sections M' and N' at the end of the interval.
Denote by dsi and ds2 the distances moved by particles at M
and N respectively. Let A\ be the cross-section area at M,
Vi the velocity of the particles, and «i the angle between the
direction of V\ and any convenient x axis. Let the same letters
with subscript (2) apply at N.

At the beginning of the interval the momentum of the portion
of the filament under consideration is the sum of the momentum

86 HYDRAULIC TURBINES

of the part between M and M' and that of the part between
M' and N. At the end of the interval its momentum is the sum
of the momentum of the part between M' and N and that of the
part between N and N'. In the case of steady flow and with the
vane at rest or moving with a uniform velocity in a straight
line, the momentum of the part between M' and N is constant.
Hence the change of momentum is the difference between the
momentum of the part between N and N' and that of the part
between M and M'. Noting that wAidsi = wA2ds2) since the
flow is steady, the change in the x component of the momentum
during dt is then

/TT

( F 2 COS «2 — V'l COS Ofi).

FIG. 74.

If the rate of flow be denoted by W (Ib. per sec.), then

wAldsl = Wdt

and the time rate of change of the x component of the momentum
is

(Vz COS <* 2 — Vi COS «i).

Thus the x component of the resultant force is

W W

Rx = (Vz cos 0:2 — Vi cos «i) = — AFX.

9 9

This method has the advantage that it may be extended to
the case where the flow is unsteady, if desired. In this event
the two masses at the ends would be unequal and the momentum
of the portion from M' to N would be variable. In the case
of a series of vanes on a rotating wheel running at a uniform
angular velocity the momentum of the water on any one vane

GENERAL THEORY 87

will be changing. But for the wheel as a whole, the momentum
of the water on all the vanes will be constant so long as the flow
is steady.

The method in (a) pictures the total force as the vector sum of
all the elementary forces along the path of the stream. The
method in (b) shows that the total force is independent of the
path and depends solely upon the initial and terminal conditions.

Since the force exerted by the water upon the object is equal
and opposite to R, we have

W W

Fx = ™- (Fi cos ai - F2 cos «2) = - AF. (13)

a ^ __' , .,_— L-- A/. . J-T.- -x '

In a similar manner the y component of F will be

W W

Fy = — (Fi sin ai - F2 sin «2) = - • — AF, (14)

\J . yf

Since .F = VK^'+'/V and AF == x/AFg2 + AF,2,
"the value of the resultant force Is

F = ~ AF (15)

The direction of R will be the same as that of AF and the direction
of F will be opposite to it. It is because F and AT^ are in opposite
directions that the minus sign appears
in the last members of equations (13)
and (14). Note that AF is the vector
difference between V\ and F2.

58. Force upon Moving Object. —
The force exerted by a stream upon
any object may be computed by the
equations of tfye preceding article,
whether the object is stationary or mov-
ing. The principal difference is that in

the latter case the determination of AF may be more difficult.
Thus in Fig. 75 assume the initial velocity of the stream Fi,
the velocity of the object "ult the angle between them ai, and
the shape of the object to be given. The relative velocity v\
can be determined by the vector triangle. Its direction is also
determined by this triangle and is not necessarily the same as
that of the vane or object struck by the water. But the direc-
tion of the relative velocity of the water leaving is determined
by the shape of the object, since v2 is tangent to the surface

88 HYDRAULIC TURBINES

at this point. The magnitude of v2 may be some function of
»i. Then knowing u2, which is not necessarily equal to u\ how-
ever, the magnitude and direction of Vz can be computed from
the vector triangle. The AF desired is the vector difference be-
tween Vi and this F2.

In case the stream is confined so that its cross-section is
known, the magnitude of vz may be computed directly from the
equation of continuity.

The remaining difficulty is the one of determining the amount
of water acting upon the body per unit time. The rate of dis-
charge in the stream flowing upon the object is AiVi so that
W = wAiVi. But this may not be the amount of water striking
the object per second. For instance if the object is moving in

FIG. 76. — View showing action of jet on several buckets.

the same direction as the water and with the same velocity, it
is clear that none of the water will strike it. The amount of
water which will flow over any object is proportional to the
velocity of the water relative to the object itself. If we denote
by W the pounds of water striking the moving object per second,
and by a\ the cross-section area normal to Vi, then W = wa\v\.

If we consider a wheel with a number of vanes acted upon by
the water, the above is true for one vane only. The reason .that
less water strikes one vane per second than issues from the nozzle
in the same time is that the vane is moving away from the nozzle
and thus there is an increasing volume of water between the two.
But for the wheel as a whole the entire amount of water may be
used, since one vane replaces another so that the volume of water

GENERAL THEORY

from the nozzle to the wheel remains constant. If one vane
uses less water than is discharged from the nozzle in any given
time interval and yet the wheel as a whole uses the entire amount
of water, it means that the water must be acting upon more than
one bucket at the same instant. This is shown in Fig. 76.

59. Torque Exerted. — When a stream flows through a turbine
runner in such a way that its distance from the axis of rotation
remains unchanged, the dynamic force can be computed from the
principles of Art. 57. But when the radius to the stream varies,
it is not feasible to compute a single resultant force. And, if it
were, it would then be necessary to determine the location of its
line of action in order to compute the torque exerted by it.
Hence we find the total torque directly by other means.

Relative Path
of Water

Absolute Path
of Water

A fundamental proposition of mechanics is that the time rate of
change of the angular momentum" (moment of momentum) of
any system of particles with respect to any axis is equal to the
torque of the resultant external force on the system with respect
to the same axis.1 I i

In Fig. 77 let MN represent a vane of a wheel which may
rotate about an axis 0 perpendicular to the plane of the paper.
Water enters the wheel at M and, since the wheel is in motion,
by the time the water arrives at N on the vane that point of the
vane will have reached position N'. Thus the absolute path
of the water is really MN'.

1 See the author's "Hydraulics," Art. 112. This proposition is analogous
to force = time rate of change of momentum, but here we deal with moments
on both sides

90 HYDRAULIC TURBINES

Let us consider an elementary volume of water forming a hollow
cylinder, or a portion thereof, concentric with 0. Let the time
rate of mass flow be dm/dt. Then in an interval of time dt,
there will flow across any cylindrical section the mass (dm/dt)dt.
Let this be the mass of the elementary volume of water we are
to consider. Let the radius to this elementary cylinder be r.
Only the tangential component of the velocity will appear in a
moment equation, hence the angular momentum of this cylinder
of water will be mass X radius X tangential velocity or (dm/dt)
dt X r X V cos a, and the time rate of change of momentum,
which is equal to torque, will be

d(rV cos a)

In the case of steady flow (dm/dt) is constant and equal to W/g
and

W C2

Tf = - d(rVcosa).

Integrating between limits we have the value of the torque
exerted by the wheel upon the water, or by changing signs, the
value of the torque T exerted by the water upon the wheel.
Thus

T = — (rlVl cos ai - r2V2 cos «2) (16)

Representing the tangential* component of the velocity of the
water, often called the " velocity of whirl/' by FM, since it is in
the direction of u, we have

T = ^ (fi-7., - r2rM2) (17)

It is immaterial in the application of this formula whether the
water flows radially inward, as in Fig. 76, radially outward, or
remains at a constant distance from the axis. In any event r\
is the radius at entrance and r2 is that at exit.

A shorter method of proving the above is analogous to method
(b) of Art. 57. During an interval of time dt the wheel has
received angular momentum at M of dmriVi cos ai and given
up angular momentum at N' of dmr2V2 cos a2, assuming the
flow to be st< ady. And, since dm = (W/g)dt for steady flow,
the time ral,c of change of angular momentum is (W/g)(riVi cos
ai — r2Vz cos 0-2).

GENERAL THEORY 91

It is also possible to consider Fx of equation (13) to be made
up of two forces concentrated at the points of entrance and
exit. The former is (W/g)Vi cos «i at radius ri and the latter is
(W/g)Vz cos «2 at radius r2. Taking the moments of these two
tangential forces, we get equation (16) at once.

60. Power and Head Delivered to Runner. — If the flow is
steady and the speed of the wheel uniform, an expression for the
power developed by the water is readily obtained. From Art. 56

Power = Wh" = 7\o.
Using the value of T given by (16) and noting that no == u,

Power = Wh" = — (ftiVi cos «i — w2F2 cos t*i) (18)
y

This is the power actually developed on the runner by the
water. It is analogous to the indicated power of a steam engine.
The power output of the turbine is less than this by an amount
equal to the friction of the bearings and other mechanical losses,
such as windage or the disk friction of a runner in water in the
clearance spaces.

Eliminating the W from the equation above we have the head
actually utilized by the runner. Thus

h" = ehh = l (UlVul - u,Vu,) (19)

As just seen, the hydraulic efficiency is equal to h" '/h. The net
head h supplied to the turbine is used up in the following ways :

Of these items h" is the head converted into mechanical work, the
second term represents the energy dissipated in the form of heat
due to internal friction and eddy losses within the runner, the
_third_term is the kinetic energy lost at discharge, and the fourth
term represents the loss in the nozzle of a Pelton wheel or the
case and guided of a reaction turbine. The factor m in the
above may be unity in the case of an impulse wheel or a reaction
turbine without a diverging or proper draft tube. For a reac-
tion tur&ine with an efficient draft tube it will be less than unity.
61. Equation of Energy for Relative Motion. — Using the
value of h" given by (19) in (5) we have

(* + •+ 5) -

HYDRAULIC TURBINES

All of the absolute velocities will be replaced in terms of relative
velocities as follows:

COS 0i
COS 02

S] = Wj +
«2 = W2 +

COS 0
COS 0

The substitution of these values gives us

,2 _

This equation serves to establish a relation between points
(1) and (2). If the wheel is at rest u\ and w2 become zero, Vi and
vz become absolute velocities and equation (21) becomes the equa-
tion of energy in its usual form as in (4).

62. Impulse Turbine. — The following numerical solution is
given to illustrate the application of the foregoing principles.
This impulse turbine is of the outward flow type known as the

Absolute Path
of Water

Guide

Shaft • Runner

FIG. 78. — Outward flow turbine.

Exit

Girard turbine. Obviously the direction of flow makes no dif-
ference in the. theory.

By construction, on = 18°, 02 = 165°, n = 2.0 ft., rz = 2.5 ft.
The hydraulic friction loss in flow through the runner will be
taken as proportional to the square of the relative velocity so that

hr = k ~~-j where k is an empirical constant. Assume k = 0.4.

Suppose h = 350 ft., N = 260 r.p.m., q = 100 cu. ft. per second.
Find relative velocity at entrance to runner, relative velocity
and magnitude and direction of absolute velocity at exit from
runner, head utilized by wheel, hydraulic efficiency, losses, and
the horsepower. (See Fig. 77.)

GENERAL THEORY 93

Vi = V2gh = 8.025 V350 = 150 ft. per second.
HI = 2irri N/6Q = 54.4 ft. per second.
U2 = (r2/ri)ui = 68.0ft. per second.

By trigonometry Vi = 99.55ft. per second.

Suppose the flow is in a horizontal plane so that z\ = z2. Since
it is an impulse turbine the pressure throughout the runner will
be atmospheric. Thus pi = p2.
Equation (21) then becomes

1.4 v2* = 9910 + 4624 - 2960 = 11,574

v2 = 90.9 ft. per second.

By trigonometry V2 = 30.6 ft. per second, «2 = 130°

Vui = VL cos on - 150 X 0.951 = 143
Vu2 = az + P2 cos .& - 68.0 - 90.9 X 0.966 = -19.7
(Also Vu2 = V2 cos a2 30.6 - X (- 0.639) = -19.7)

= ^ (54.4 X 143 + 68 X 19.7; = 283 ft/

Oju . 2i

Hydraulic efficiency = h"/h = 283/350 = 0.81.
Hydraulic friction loss = k-^- = 0.4 ^ . = 51.3 ft.

y 2
Discharge loss = -^- = ^-^ = 14.6 ft.

Wh" 100 X 62.5 X 283
:^50: -550"

63. Reaction Turbine.1 — Another numerical case will be given
to illustrate the application of the foregoing principles to the
reaction turbine. The turbine used here is the Fourneyron or
outward flow type, though the theory applies to* any type.

By construction, ai = 18°, fc = 165°, ri = 2.0 ft., ra = 2.5 ft.,

Ai = 1.36sq.ft.,a2 = 1.425 sq.ft. Assumed = 0.2/ft' = fc^-)-

1 See Art. 8. If the area a2 is made small enough the wheel passages will
be completely filled with water under pressure. We then have a reaction
turbine. Note that

+ ' so that Vi is not equal to

94 HYDRAULIC TURBINES

Suppose h = 350 ft., N = 525 r.p.m., and q = 164.5 cu. ft. per
second. Find head utilized by turbine, hydraulic efficiency,
losses, pressure at guide outlets (entrance to turbine runner),
and the horsepower.

Since the wheel passages are completely filled the areas of the
streams, A i and az are known, thus

Vl = q/Ai = 164.5/1.36 = 121 ft. per second.
v% = (A i/ a*) Vi = 115.5 ft. per second.

For the above r.p.m. HI = 110 ft. per second, uz = 137.5 ft. per
second.

Vui = Vi cos ai = 115.

VU2 = u2 + v2 cos j8a = 137.5 - 115.5 X 0.966 = 26.0.

h" = - (UlVul - u,Vu2) = ^-= (110 X 115 - 137.5 X 26) =
g OA . *

282 ft.

Hydraulic efficiency = 282/350 = 0.805.

TT 1 T f • ^ 1 7 ^22 rk n 13350 . _ .,

Hydraulic friction loss = K-^- = 0.2— - = 41.5 ffc.

2g 64.4

By trigonometry V2 = 41 ft. per second.

Discharge loss = = = 26 ft.

Zg O4.4

Since v% is determined by the area a2 we do not have the use for
equation (21) that we did in the case of the impulse turbine.
By it, however, we can compute the difference in pressure between
entrance to and discharge from the runner. Thus from (21),
taking zi = z2,

w w 2g

(If the turbine discharges into the air then — - = 0 and -1 =122

w w

ft.) This pressure difference may also be computed from equa-
tion (5).

W h" 62.5 X 164.5 X 282

550

=5270h.p.

64. Effect of Different Speeds. — If a wheel is run at different
speeds under the same head, the quantities vi, v2) V2, a2, h",
efficiency, and power all vary. In Fig. 79 may be seen the
velocity diagrams for entrance and discharge from a wheel at five

GENERAL THEORY 95

different speeds from standstill to run-away and photographs1
of the wheel at these speeds. The relations for a reaction
turbine would be very similar to these for the impulse wheel.
The most important change is that in the quantities a2 and
F2. When the wheel is at rest, a2 = 02 and F2 =' t>2. As the
speed increases o:2 decreases, passes through 90°, and approaches
0°. The value of F2 decreases to a minimum and then increases
again.

From equation (20) it may be seen that, other losses being
equal, the maximum efficiency would be obtained when the dis-
charge loss is a minimum. It can be seen that F2 is very small
when either v% = uz or «2 = 90°. A means of computing the
speed necessary for this will be given later. Neither of these
gives the actual mathematical minimum but they are very
close to it.

The torque exerted on the wheel by the water may be seen to
decrease as the wheel speed increases. In equation (17) W and
FMi are practically constant, though they vary slightly in the
case of the reaction turbine. But Fw2 continuously increases
algebraically. It has its maximum negative value when the wheel
is at standstill, it is zero when the speed is such that «2 = 90°,
and it attains its maximum positive value when the turbine is
at run-away speed. This is the maximum speed which the wheel
can reach under a given head and is attained when all external
load is removed. Under these circumstances the torque exerted
by the water is just sufficient to overcome the mechanical losses
of the turbine. The run-away speed of the wheel is thus strictly
limited by hydraulic conditions.

In the ideal case the maximum possible speed of the Pelton
wheel would be when the velocity of the buckets equalled the
velocity of the jet. But under these conditions AF of equation
(15) would equal zero. Consequently the wheel must run at a
speed somewhat less than this as some power is required to over-
come the mechanical losses at this speed. For the impulse wheel
the maximum value of <£ usually attained is about 0.80 at run-

1 These photos also show the needle in the center of the jet. The piece
at the side of the buckets toward the lower right hand side of the case is
the "stripper," its function being to deflect water that might otherwise be
carried around with the wheel up into the upper part of the case. The
buckets pass through this with a relatively small clearance. A close inspec-
tion of the views of the wheels and the water leaving it will give one a fair
idea of the variation in velocities.

HYDRAULIC TURBINES

* o£

g, a

CO O>

1 I

I ~

GENERAL THEORY

away. For the reaction turbine it is about 1.3. The relations
in the latter case are somewhat more complex but are similar
to those for the Pelton type.

65. Forced Vortex. — A forced vortex is produced when a
liquid is compelled to rotate by means of external forces applied
to it. Thus, if the vessel X Y of Fig. 80, is rotated about the axis
0-0, the water filling the vessel will tend to rotate at the same
speed with it and we will have a forced vortex.

The pressure within this body of water will then vary as shown
by the curve CD. The law of variation may be found as follows

I'Af

FKJ. 80. — Forced and free vortices.

Consider an elementary volume, whose length along the radius
is dr and whose area normal to this is dA, and which rotates at
an angular velocity co at radius r. The difference in the pressures
on the two faces, which is the resultant force acting, is equal to
dp X dA, and the acceleration of the mass is co2r, directed toward
the axis of rotation. Thus

dpdA = (wdA dr/g) o>2r
dp = (wuz/g) rdr
p = (w <o2/#) r2/2 -}- constant.

To find the value of the constant of integration, let p0 be the pres-
sure when r equals zero. Thus the constant is equal to p0, and

98 HYDRAULIC TURBINES

p = rV2 p0
w 2g w

= "2 + V" (22)

2g w

From this it may be seen that the curve is a parabola. If the
vessel is open, but with sides high enough so that the water cannot
overflow, the surface of the water will become a paraboloid, since
the pressure variation along the radius is the same whether the
water be confined or not.

This equation is really a special case of equation (21) with v\
and vz equal to zero, since there is no flow ot water. If water
flows then equation (21) must be employed. Flow may occur
in either direction. It may be noted that the energy of the water-
is not constant along the radius, as both the pressure and the
velocity of the water increase. This is possible because, due to
the action of external forces, energy is being delivered to (or
taken from) the water.

An important application of equations (21) and (22) is in con-
nection with the centrifugal pump. The vessel XY of Fig. 78
may be said to be a crude illustration of the impeller of such a
pump. But the equations are also of value in determining the
conditions of flow through turbine runners, of either the impulse
or reaction type.

66. Free Vortex. — A free vortex is produced when a liquid
rotates by virtue of its own angular momentum, previously de-
rived from some source, and is free from the action of external
forces. Thus in Fig. 80, if the rotating vessel XY is surrounded
by a stationary vessel MN into which the water can pass from
XY, the water will still tend to rotate and, neglecting friction,
we will have a free vortex.

The pressure within the free vortex will vary as shown by the
curve DE. The law of variation may be found as follows: Since
no external forces are applied, the resultant torque exerted is
zero, and hence the angular momentum is constant (Art. 59).
Since angular momentum is proportional to rV cos a or rVu,
it follows that

rV cos a = rVu = constant (23)

the value of the constant being the value obtained from the
numerical value of these factors at the point of entrance.

Considering the radial component of the velocity Vr, we have

GENERAL THEORY 99

q = 2wrb X Vr = constant, from the equation of continuity.
Hence

rbV sin a = rbVr = constant (24)

the value of this constant being proportional to q.

The total velocity V is the resultant of these two components

so that .

72 = Fu2 + V2 (25)

Since no energy is delivered to (or taken from) the water, the
value of the effective head H must remain constant. Thus

V2 p
H = z -f o + = constant,

the value of this constant being determined by the total head of
the water initially. Taking the datum plane such that z = 0,
we have

<n V2 V 2 V 2

P - 77 _ * - H — --— — •• (26)

w 2g - 2g 2g

The flow of the water may be in either direction. Actual fric-
tion losses will modify the resulting values of the pressure and
also of Vu, but cannot alter Vr: If b is constant, Vu and Vr vary
in the same proportion, neglecting friction, so that a is constant
and the path of the water is the equi-angular or logarithmic
spiral.

The free vortex is found in the casing surrounding the im-
peller of some types of centrifugal pumps. It is also found in
the water in a spiral case approaching a turbine runner, and
the above equations have many applications.

For example equations (23), (24), and (25) show that the
velocity of the water varies inversely as the radius of curvature
of its path. Hence if the vanes of turbine runners are so
shaped that the stream lines have sharp curvatures, the velocity
of the water will be excessive and; from equation (26), it may
be |seen that the pressure will be -reduced. This may result
in the pressure becoming so low that erosion will result,
as mentioned in Art. 45. For the same reason it is undesirable
to let the water discharged from a turbine runner flow direct
toward the axis, as in the pure radial inward flow turbine.
For if the water leaving the runner had any " whirl" this would
increase as the axis was approached and, according to the
equation, would become infinity, while the pressure would be

100

HYDRAULIC TURBINES

minus infinity, when the radius equalled zero. While these
limits could not be reached, the vaporization of the water that
would actually take place would induce corrosion and also
cause additional eddy losses. Hence the water is turned at
discharge as has been shown in Fig. 34, page 43, and the
central space is often taken up with a cone.

67. Theory of the Draft Tube. — The flow of water through a
draft tube is no different in principle from the flow through any

'1111 ***--

-^-^- *- -4 -\-

FIG. 81.

other conduit and hence Bernoulli's theorem, otherwise known as
the general equation of energy of Art/54, may be applied to it.
Equation (4) however has been stated only for the case of
steady flow and for the present purpose we are concerned with
any condition of flow that may exist. Hence we shall add another
term called the acceleration head, which is the head necessary to
accelerate the velocity of the water when the rate of discharge
is*changed by the action of the governor. Let this head be
denoted by hacc, while the loss of head in friction, H', is divided
into its two separate factors, hf, the friction loss in the tube, and

GENERAL THEORY 101

Vs2/2g, the velocity head lost at discharge from the mouth of the
draft tube. Thus referring to Fig. 81, we have

0 +

where p2 denotes absolute pressure and pa denotes atmospheric
pressure. Then

HZ — H± = H' -\- hacc

The solution of this equation will give the allowable height of the
turbine runner above the tail- water level. Or the equation can
equally well be used to determine the pressure for any given
elevation. In the above,

- is governed by the altitude and local variations but is
approximately equal to 34 ft. of water.

— cannot be less than the vapor pressure of the water at that

temperature as determined from the steam tables and should be
from at least 2 to 4 ft. of water more.

Vz is a function of the design and type of the runner. The
higher the capacity and speed of the type the higher will be the
value of Vz- It is also a function of the head under which the
turbine runs, because all velocities vary as the square root of the
head. Also if a2 is not 90°, the value of Vz will be greater than
q divided by draft tube area.

hf depends upon the construction of the draft tube. Ordinarily
it may be assumed as about 15 to 25 per cent, of Vz2/2g.

F3 is controlled by the setting of the plant for that fixes the
allowable length of the draft tube. It is also a function of the
construction of the draft tube and the head under which the
turbine operates.

hacc is determined by the action of the governor and it may
be either plus or minus in value. The negative value is the
one to use in the above equation.

It can be seen from the foregoing that two types of turbines
with different discharge velocities would have different limiting

102 HYDRAULIC TURBINES

values of 22 under the same head. And the same runner would
also require a lower setting under a higher head because of the
change in this same item.

If the turbine is set higher than the limiting value, as deter-
mined by this equation, the efficiency of conversion in the
draft tube will be lost due to the vaporization and subsequent
recondensation of the water. Also corrosion of the runner will
take place because of the liberation of oxygen. Again if the
height is very close to the allowable limit for steady flow, the
sudden closure of the turbine gates by the governor may cause the
pressure at discharge from the runner to drop to such a low value
that the water vaporizes. But an instant later the water will
surge back up the draft tube, striking the runner a decided blow.

68. QUESTIONS AND PROBLEMS

1. How is the effective head to be measured on the Pelton wheel and on
the reaction turbine? Why are two values possible in the latter case?
What are the definitions of the various efficiencies that may be dealt with?

2. What is the procedure for computing the force exerted by a stream
upon a moving object? What are the reasons for the difference between
W and W"!

3. What becomes of the total energy supplied in the water to the wheel?
As the speed of a wheel varies, under a constant head, the torque exerted
on it, and consequently its power, varies. Since the power supplied in the
water is constant, what becomes of the difference between the two?

4. What are the fundamental differences between the solution of the prob-
lem of the impulse wheel and of the reaction turbine?

5. As the speed of a wheel changes how do V% and 0:2 vary? Of what
significance is this? What limits the maximum speed of a Pelton wheel
under a given head?

6. What is a forced vortex? How does the pressure vary in it? What
examples of it are found?

7. What is a free vortex? How does the velocity vary in it? How does
the pressure vary? What common examples of this are found?

8. What conclusions can one draw from the equations for the free vortex
that have an important practical application?

9. Derive the equation for the maximum allowable height of a turbine
runner and discuss the items that affect this value?

10. What is the effect of the action of the governor upon the conditions
within the draft tube? What will be the effect if a turbine runner is set
too high?

11. In the test of a reaction turbine the following readings were taken
(see Fig. 71) : Pressure at entrance, pc/w = 126.6 ft., zc = 12.6 ft., diameter
at C = 30 in., diameter at D = 60 in., and rate of discharge = 44.5 cu. ft.
per second. Compute the head on the turbine by each of the two methods
given. Ans. 140.5 ft.

GENERAL THEORY 103

12. A jet of water 2 in. in diameter and with a velocity of 100 ft. per sec-
ond issues from a nozzle on the end of a 6-in. pipe and strikes a flat plate
normally. Find: (a) Power of jet, (b) thrust exerted on pipe, (c) force ex-
erted on plate. Ans. (a) 38.4 h.p., (b) 376 lb., (c) 423 Ib.

13. Suppose the jet in problem (12) strikes a vane which deflects it 60°
without loss of velocity. Find (a) component of force in direction of jet,
(6) component normal to jet, (c) magnitude and direction of total force.
Solve also assuming the terminal velocity to be reduced to 80 ft. per second,
all other factors remaining the same.

Ans. (a) 211 lb., (b) 365 lb., (c) 422 lb. at 60° with Vi.
(a) 254 lb., (6) 293 lb., (c) 388 lb. at 49° 08':

14. Solve problem (13) assuming the angle of deflection to be 180°.
What difference does the angle make in the magnitude of each force? What
difference is there in the effect of friction in each instance ?

Ans. (a) 844 lb., (b) 0, (c) 844 lb. at 0° with Vi.
(6) 760 lb.

15. Suppose the vane in problem (14) moves in the same direction as the
jet with a velocity u, and that friction loss is such that v2 = O.Svi. When
u has values of 0, 30, 44.4, 70 and 100 ft. per second, find: (a) Values of the
lb. of water per second striking the vane, (6) values of absolute velocity at
discharge, (c) values of the force exerted.

Ans. (a) 136.3, 95.3, 75.7, 40.8, and 0 lb. per second.

(b) -80, -26, 0, +46, and +100 ft. per second.

16. Solve problem (15) if the jet is upon a wheel equipped with similar
vanes. Find the power delivered to the shaft at each speed. What be-
comes of the difference between this and the power of the jet?

Ans. 760, 532, 422, 228, and 0 lb.
0, 29.0, 34.0, 29.0, and 0 h.p.

17. For a turbine runner, Vi = 70 ft. per second, V-i = 20 ft. per second,
ri = 2 ft., r2 = 3 ft., an = 20°, «2 = 80°, and W = 300 lb. per second, (a)
Find torque exerted upon the wheel, (b) If u\ = 50 ft. per second, find
the power. Ans. (a) 1128 ft. lb., (b) 51.3 h.p.

18. For a turbine runner, Vi = 70 ft. per second, Vz — 20 ft. per second,
Pi/w = 25 f;-., pz/w = —25 ft. Assume friction loss (kv-22/2g} in flow
through runner as 5.78 ft. and that there,isno difference in elevation, (a)
Find head utilized by runner, (b) If W = 300 lb. per second, find the power.

Ans. (a) 94.2 ft., (b) 51.3 h.p.

19. For the impulse turbine in Art. 62 it will be found that v2 = u« when
u\ = 68.4 ft. per second. Find the r.p.m., efficiency, losses, and horse-
power. Compare with values given in Art. 62.

Ans. 326 r.p.m., e = 0.845, 3365 h.p.

20. For the reaction turbine in Art. 63 it will be found that «2 = 90° if
ui = 86.3 ft. per second. At that speed the rate of discharge will be found
(by method given later) to be 159 cu. ft. per second. Find the r.p.m.,
efficiency, losses, and horsepower. Compare with values given in Art. 63.

Ans. 412 r.p.m., e = 0.852, 5380 h.p.

21. Compare the best r.p.m. of the impulse turbine with the best r.p.m.
of the reaction turbine in Problems (19) and (20). Compare the values of v*

104 HYDRAULIC TURBINES

in Problems (19) and (20). Why are these different? What effect has this
upon the best speed?

22. Water enters the spiral case of a turbine with a velocity of 10 ft. per
second, (a) Considering this velocity as tangential at a radius of 9 ft.,
which is the distance from the runner axis to the center of the case near the
point of intake, what is the tangential component of the velocity at entrance
to the speed ring vanes, the outer radius of which is 7 ft.? (6) If the height
of the vanes at this point is 5 ft., find the radial component of the velocity
if the turbine discharges 900 cu. ft. per second, (c) What should be the
angle of the speed ring vanes at this point? (d) What should be the angle
of entrance to the turbine guide vanes, if the radius is 6 ft., and the height
is 3 ft.? Ans. (a) 12.85 ft. per second, (&) 4.09 ft. per second.

23. A turbine running under a head of 200 ft. is of such a design that
Vp/Zg = 7 per cent, of h and TV/20 = 1 per cent, of h. If the minimum
pressure allowable is 3 Ib. per sq. in., what is the maximum height the
runner may be set above the tail-water level, assuming the draft tube loss
to be 25 per cent, and the maximum negative acceleration head to be 50
per cent, of F22/20? What will be the result if this same turbine is used
under a head of 50 ft.? Ans. 11.6 ft., 23.2 ft.

24. A turbine running under a head of 50 ft. is of such a design that Vzz/2g
= 20 per cent, and V$zf2g = 2 per cent, of h. If the minimum pressure

allowable is 3 Ib. per sq. in., what is the allowable height the runner may be
set above tail-water level, assuming the draft tube loss to be 25 per cent,
of V^/Zg and the maximum negative acceleration head to be 50 per cent,
of F22/2<7? Compare with second part of preceding problem.

Ans. 15.6 ft.

CHAPTER VIII
THEORY OF THE TANGENTIAL WATER WHEEL

69. Introductory. — The tangential water wheel has been
classed as an impulse turbine with approximately axial flow.
The term tangential is applied because the center line of the jet
is tangent to the path of the centers of the buckets. In this
article the assumption will therefore be made that a\ = 0° and
that r1 = r2. It will be shown later that*these assumptions are
not entirely correct.

FIG. 82.

If the angle on be assumed equal to zero then HI and Vi are
in the same straight line and Vi = Vi — Ui. The conditions at
exit from the buckets are shown in Fig. 82. In applying equation
15 we desire to find only the component of the force tangential to
the wheel since that is all that is effective in producing rotation.
Therefore we shall find only the component of AF along the direc-
tion of Ui. Thus, if F here denotes tangential force,

F = — (Vi - V2 cos «2)

== -~- (Fi — u2 — v2 cos 182)
9

105

106 HYDRAULIC TURBINES

By equation (21) since zi = z2, pi = p2, Ui = u2, (1 + A;)y22

v\ Vi — HI

Substituting this value of #2 we obtain

- "(' -vrfth - '«>

A more exact value for the force exerted may be found in Art. 72.
The above is only an approximation.

Multiplying the force given above by the velocity of its point
of application, we have the power developed. Thus

P = Fui = W/l_™s^\(Vi _Mi)Mi (2g)

g \ VI + Ay

The power input to the wheel, including the nozzle, is Wh, where
h is determined as in Art. 55. The power in the jet is WVi2/2g
and is less than the former by the amount lost in friction in the
nozzle. •

. Equation (28) is the equation of a straight line. It shows that
F is a maximum when u\ is zero and that it decreases with the
speed until it becomes zero when m = Vi. Equation (29) is the
equation of a parabola. It shows that the power is zero when
Ui = 0 and again when u\ = V\. The vertex of the curve, which
gives the maximum power and hence the maximum efficiency, is
found when u\ = 0.5V i. Since in reality both of these curves
are altered somewhat, when all the factors are considered, some
of these statements require modification.

The actual speed for the highest efficiency has been found by
test to be such that <j>e = 0.45 approximately, while the value of
the efficiency is about 80 per cent. Both of these values vary
somewhat with the design of the wheel and the conditions of use.
But one can approximately compute the bucket speed and the
power of any impulse wheel, provided the head and sue of
jet are known. The bucket speed u\ = </>\/?<7S while the
velocity of the jet Vi = cv\/2g'i. For a good nozzle with full
opening, if equipped with a needle, the coefficient of velocity
should be about 0.98. Thus the rate of discharge is determined.
If the diameter of the wheel is known, or assumed as a function
of the size of the jet, the rotative speed can be computed.

The reasons for the modifications of the simple theory given

THEORY OF THE TANGENTIAL WATER WHEEL 107

above and an analysis oi the characteristics of an actual wheel are
given in the following parts of this chapter.

70. The Angle a\. — The angle ai is usually not zero as can be
seen from Fig. 83. One bucket will be denoted by B and the
bucket j list ahead of it by C. Different positions of these buckets
will be denoted by suffixes. The bucket enters the jet when it is
at Bi and begins to cut off the water from the preceding bucket Ci.
When the bucket reaches the position B2 the last drop of water
will have been cut off from C2, but there will be left a portion of
the jet, MPX Y, still acting upon it. The last drop of water X
will have caught up with this bucket when it reaches position C3.
Thus while the jet has been striking it the bucket has turned

FIG. 83.

through the angle BiOCs. The average value of ai will be taken
as the angle obtained when the bucket occupies the mea'n between
these two extreme positions. It is evident that position C3 will
depend upon the speed of the wljeel, and that the faster the wheel
goes the farther over will C3 be. Thus the angle a i decreases as
the speed of the wheel increases. The variation in the value of
ai as worked out for one particular case is shown in Fig. 85.

71. The Ratio of the Radii. — It is usually assumed that n = r%.
However inspection of the path of the water in Fig. 84 (a) will
show that when the bucket first enters the jet r2 may be less than
ri. When the bucket has gotten further along r2 may be greater

108

HYDRAULIC TURBINES

than ri. The value of x( = r2/Vi) depends upon the design of the
buckets, and its determination is a drafting-board problem which
is not within the scope of this book. It is evident that a value of
% must be a mean in the same way that a value of a i is a mean.
And just as on varies with the speed, so also does x vary with the
speed. A little thought will show that when the wheel is running
slowly compared with the jet velocity the value of x will be less
than when the wheel is running at a higher speed. This- may be
verified by actual observation. When the wheel is running at its
proper speed it is probably true that x is very nearly equal to
unity. In any case the variation of the value of x from unity
cannot be very great.

(a) (6)

FIG. 84. — Radii for different bucket positions.

72. Force Exerted. — The force acting on the wheel may be
determined by the principles of Art. 58, but, if the radii are qot
equal it will not be convenient to use equation (15) on account of
the difficulty of locating the line of action of the force. How-
ever we can use equation (17) and by it determine a force at
the radius r\ which shall be the equivalent of the real force.
Dividing (17) by 7*1 and letting F denote tangential force we
obtain

F = -(Vul-xVu,)

Vui =

COS Oil

cos

By equation (21)

By trigonometry

i2 - 2ViUi

THEORY OF THE TANGENTIAL WATER WHEEL 109

Substituting this value of Vi, and with u2 = xui,

(1 + &>22 = Fi2 -.2ViUi cos ai + Ai2.
Substituting ^2 from this in the expression for F«2 we obtain

TTT" r

= -

cos ai - zU!

(30)

Equation (30) is a true expression for the force exerted. No
great error is involved, however, by taking x = 1.0. If that is
done the expression under the radical becomes the value of v\
and may be found graphically. For the sake of simplicity and
ease in computation a\ may be taken equal to zero and the equa-
tion then reduces to (28), but an exact value of F will not be ob-
tained. There is little excuse for taking k = 0, as most writers
do, for equation (28) is not simplified to any extent and the re-
sults are entirely incorrect.

73. Power. — With F as obtained from (30) the power is given
by Fui. We may also compute h" and obtain the power by
multiplying by W.

Since h" = - Uj ( Vui —xVu2) it is evident that the expression for

«7

h" is the same as (30) if Wi be substituted for W. Thus the expres-
sion for power has the same value no matter from which basis it is
derived.

74. The Value of W.— W is the total weight of water striking
the wheel per second. It is obvious that the weight of water
discharged from the nozzle is

W = wAiVi.

Under normal circumstances all of this water acts upon the wheel.
However for high values of the ratio Ui/Vi a certain portion of
the water may go clear through without having had time to
catch up with the bucket before the latter leaves the field of action.
It is apparent, for instance, that if the buckets move as fast as the
jet none of the water will strike them at all. For all speeds less
than that extreme case a portion of the water only may fail to act.
Thus referring to Fig. 83, it can be seen that if the wheel speed is
high enough compared to the jet velocity the water at X may not
have time to catch up with bucket C. The variation of W with
speed is shown in a particular case by Fig. 85.

110

HYDRAULIC TURBINES

It may also be seen that the larger the jet compared to the
diameter of the wheel the 'lower the value of Ui/Vi at which
this loss will begin to occur and it is not desired to have it occur
until the normal wheel speed is exceeded. Thus there is a limit
to the size of jet that may be used for a given wheel, as stated
in Art. 30. For a given diameter of wheel, as the size of the
nozzle is increased, larger buckets must be used and they must
also be spaced closer together.

75. The Value of k. — The value of k is purely empirical and
must be determined by experiment. If the dimensions of the
wheel are known and the mechanical friction and windage losses
are determined or estimated, then from the test of the wheel the
horse-power developed by the water may be obtained. The
value of k is then the only unknown quantity and may be solved

100

400

500

fM) 300

R.P.M.
FIG. 85. — Values of aL and W for a certain wheel.

for. The value of k is probably not constant for all values of
Ui/Vi. Some theoretical considerations, which need not be given
here, have indicated that it could scarcely be constant and an
experimental investigation has shown the author that k decreased
as Ui/Vi increased. For a given wheel speed however it is
nearly constant for various needle settings unless the jet diameter
exceeds the limit set in Art. 30. The crowding of the bucket
then increases the eddy losses and would require a higher value of k.

The value of k may be as high as 2.0 but the usual range of
values is from 0.5 to 1.5.

76. Constant Input — Variable Speed. — The variation of torque
and power with speed for different needle settings is shown by
Fig. 86 and Fig. 87. With the wheel at rest the torque may

THEORY OF THE TANGENTIAL WATER WHEEL 111

vary within certain limits as is shown by the curve for full nozzle
opening. This is due to differences in 0:1 and in x for various
positions of the buckets. When running at a slow speed the
brake reading was observed to fluctuate between the limits shown.
At higher speeds this could not be detected. This action is here
shown for only one nozzle opening but it exists for all. With
a given nozzle opening the horsepower out put is fixed and constant.
The horsepower output varies with the speed. It will be noticed
that the maximum efficiency is attained .at slightly higher speeds

'0 100 200 300 400

R.r.M.

FIG. 86. — Relation between torque and speed.

500

for the larger nozzle openings than Tor the smaller. This is due,
in part, to the fact that the mechanical losses, which are practically
constant at any given speed, become of less relative importance
as the power output increases.

Fig. 88 shows the variation of the different losses for a constant
power input but a variable speed.1

77. Best Speed. — It is usually assumed that the best speed is
the one for which the discharge loss is the least. As shown in Art.
64, the latter will be approximately attained either when u2 = v2
or when «2 = 90°. In the case of the impulse turbine the former

1 The curves shown in this chapter are from the test of a 24-in. tangential
water wheel by F. G. Switzer and the author.

112

HYDRAULIC TURBINES

assumption gives an easier solution. It will be found that HZ = v%
if u\ is found from

fczW + 2ViUi cos A! - Fi2 = 0.*

An inspection of the curves in Fig. 88 will show that the highest
efficiency is not obtained when the discharge loss is the least. So

0 100 200 „ „ ... 300 400 500

K.l .M.

FIG. 87. — Relation between power and speed for different needle settings.

that, although the difference is not great, the above equation
does not give the best speed. The hydraulic friction losses and
*L. M. Hoskins,;' Hydraulics," Art. 198, Art. 208.

THEORY OF THE TANGENTIAL WATER WHEEL 113

the bearing friction and windage cause the total losses to become
a minimum at a slightly higher speed. It does not seem possible
to compute this in any simple way but it will be found that the
best speed is usually such that ui/V\ = 0.45 to 0.49.

The speed of any turbine is generally expressed as HI = <t>-\/2gh.
The coefficient of velocity of the nozzle will reduce the*above

H.P. at Nozzle

0 100 200 300 400 500 600
R.P.M.

FIG. 88. — Segregation of losses for constant input and variable speed.

values slightly, so that the best speed is usually such that
<t>e = 0.43 to 0.47

78. Constant Speed — Variable Input. — The case considered in
Art. 76 is valuable in showing us the characteristics of the wheel
but the practical commercial case is the one where the speed is
constant and the input varies with the load. From Fig. 87 it is

114

HYDRAULIC TURBINES

500

400

100 g

0 0

.26 .50 .75 1.00 1.25 1.50 1.75 2.00

Position of .Needle. Inches

FIG. 89. — Nozzle coefficients and'other data.

10 12 13.2

FIG. 90 — Relation of input to output and segregation of losses for variable
input and constant speed.

THEORY OF THE TANGENTIAL WATER WHEEL 115

seen that the best speed is 275 r.p.m. That value was taken
because the highest efficiency was obtained with the nozzle open
six turns. For that value of N the curves in Fig. 90 were plotted.
It will be noted that the relation between input and output is

100

100% = H.P. Delivered to Nozzle

12 13,2

B.H.P.

FIG. 91. — Efficiencies and per cent, losses at constant speed.

very nearly a straight line. Above six turns it bends up slightly
because the wheel is then slightly overloaded.

The friction and windage was determined by a retardation
run1 and was assumed to be constant at all loads. The hydraulic
losses were segregated by the theory already given (Art. 62).'
These results plotted in per cent, are shown in Fig. 91 and Fig. 92.

1 See Art. 101.

116 HYDRAULIC TURBINES

79. Observations on Theory. — The theory as presented in this
chapter is of value principally for the purpose of explaining the
actual characteristics of Pel ton wheels. Thus the determina-
tion of values of the angle a\ is a rather tedious process,1 and it is
open to question whether the average value as defined in Art. 70
is really the proper one to use. But the important fact is that the
angle is not zero and that it does vary. In similar manner the
determination of the amount of water acting upon the wheel at
speeds above normal, and the determination of the speed at which
this waste of water begins, is difficult. But the consideration of
the problem makes it clear why the curves for the force exerted are
not straight lines, as may be seen in Fig. 86, and why the right-
hand portion is steeper. In turn this explains why the actual
power curves of Fig. 87 are distorted parabolas with the right-
hand side much steeper than the left-hand side. Ideally the
maximum speed of the wheel should be such that u} = Vi, but
actually the run-away speed is such that <j> = 0.80 approximately.
This is due to the fact that the proportion of the water acting on
the wheel at higher speeds would become so small that the force
exerted would be less than that required to overcome the bearing
friction and windage loss.

The losses computad by theory and in part determined by
experiment are shown in Fig. 88. If it were not for the waste of
water mentioned above, the discharge loss from the buckets
would be as shown by the dotted line. Actually the loss of energy
in this water is shown by the solid curve to the left of this, while
the discharge loss from the buckets is only the intercept between
the latter curve and the one to its left.

The theory, as illustrated in Fig. 90, shows that for a wheel at
the proper speed the principle loss of energy is in the buckets.
This emphasizes the importance of close attention to the proper
design of the latter. The theory also shows that the individual
losses tend to follow straight line laws. This means that the
relation between input and output is also a straight line. When
the size ot the jet becomes too large for the particular wheel, the
bucket losses increase more rapidly and hence the curve bends
upward at this point, as shown. The relation between input and
output is not exactly a straight line for loads less than that for
maximum efficiency, but it is nearly so. This is of interest be-

1 See "Theory of the Tangential Water Wheel," by R. L, Daugherty, in
Cornell Civil Engineer, Vol, 22, p. 164 (1914),

THEORY OF THE TANGENTIAL WATER WHEEL 117

cause, if only a very few points are determined by test, the
complete curve can be drawn with a reasonable degree of accuracy.
The theory also shows that the hydraulic efficiency of the wheel
alone is nearly constant from no-load to full-load at constant
speed. And considering the efficiency of the wheel and nozzle

100 % =JB" P. in Jet

0 2 4 6^8 10 12 13.2

15. ii.i .

FIG. 92. — Efficiencies and per cent, losses at constant speed based upon power

in jet.

together the hydraulic efficiency does not begin to drop off rap-
idly until very small nozzle openings are reached. The reason
for this is that the vector velocity diagrams upon which the
theory is based are independent of the size of the jet. The
variations shown .in Fig. 92 are due to changes in cv and k. This

118 HYDRAULIC TURBINES

is of practical importance as showing why impulse wheels have
relatively flat efficiency curves.

80. Illustrative Problem. — Referring to Fig. 93 let the total
fall to the mouth of the nozzle be 1000 ft. Suppose BC =
5000 ft. of 30-in. riveted steel pipe and at C a nozzle be placed
whose coefficient of velocity = 0.97. Suppose the diameter of
the jet from the nozzle = 6 in. Let this jet act upon a tangential
water wheel of the following dimensions: Diameter = 6 ft.,
ai = 12°, fo = 170°. Assume k = 0.6, </> = 0.465, and assume
bearing friction and windage = 3 per cent, of power input to
shaft.

The problem of the pipe line is a matter of elementary
hydraulics and a detailed explanation will not be given of the steps
here employed. The coefficient of loss at B will be taken as

FIG. 93.

1.0, the coefficient of loss in the pipe will be assumed 0.03. The
loss in the nozzle will be given by( — ^ — Ij ^-, where cv = the

coefficient of velocity and Vi the velocity of the jet. If Vc = the
velocity in the pipe then the losses will be

y 2
Taking HA = 1000 ft. and HI = -—• then by equation (4) we

may solve for ^ = 1.38 ft. or ~ = 861 ft.
*g *g

Thus Vc = 9.42 ft. per second and Vi = 235.5 ft. per second.
Rate of discharge, q = 4.62 cu. ft. per second.

ff\

The pressure head at nozzle, — - = 914.5 ft.

The wheel speed ut ^0.465 X 8.025 V915.88 = 113 ft. per

second.

Therefore N = 360r.p.m.

THEORY OF THE TANGENTIAL WATER WHEEL 119

By methods illustrated in Art. 62, v\ = 126.7 ft. per second,
v% = 100 ft. per second, and Vu2 = 14.5, assuming x = 1.0.

Thus, h" = — (vul - Fw2 ) = 757 ft.

The means of obtaining the following answers will doubtless
be obvious.

Total head available, HA = 1000 ft.

Head at nozzle, Hc = 915.88 ft.

Head in jet, #1 = 861 ft.

Head utilized by wheel, h" = 757 ft.

Total power available at A = 5250 h.p.
Power at nozzle (C) = 4800 h.p.

Power in jet = 4520 h.p.

Power input to shaft = 3970 h.p.

Power output of wheel = 3851 h.p

Hydraulic efficiency of wheel = 0.878

Mechanical efficiency of wheel = 0.970

Gross efficiency of wheel = 0.852

Efficiency of nozzle = 0.941
Gross efficiency of wheel and nozzle = 0.801

Efficiency of pipe line BC = 0.915

Overall efficiency of plant = 0.733

81. QUESTIONS AND PROBLEMS

1. With the simple theory of the tangential wheel what are the relations
for torque and power as functions of speed ? How may the speed and power
of an impulse wheel be computed in practice ?

2. What are the true conditions of flow in the Pelton water wheel and
what assumptions are often made in order to simplify the theory ?

3. When may a portion of the water discharged from a nozzle fail to act
upon the wheel? Why? What changes in design will improve this
condition?

4. Why is the relation between input and output at a constant speed and
head not a straight line throughout its range? How does the hydraulic
efficiency vary from no-load to full-load at constant speed? Why?

5. Suppose the dimensions of a tangential water wheel are: /32 = 165°,
<t> — 0.45, k =0.5, and the velocity coefficient of the nozzle = 0.98. If
the diameter of the jet = 8 in. and the head on the nozzle 900 ft., compute
the value of the force exerted on the wheel, assuming ai •= 0° and x = 1.0.

6. Compute the force on the wheel in problem (5) assuming a\ = 20°.

120 HYDRAULIC TURBINES

7. Compute the hydraulic efficiency of the wheel in problem (5). Is this
dependent upon the value of the head?

8. Derive an equation for the hydraulic efficiency of a Pelton wheel,
giving the result in terms of wheel dimensions and factors such as <f> and cr.

9. Suppose it is desired to develop 2000 h.p. at a head of 600 ft. Assum-
ing an efficiency of 600 ft., what will be the size of jet required, and what
will be the approximate diameter and r.p.m. of the wheel?

CHAPTER IX
THEORY OF THE REACTION TURBINE

82. Introductory. — The main purpose of this chapter is to
explain the characteristics of reaction turbines. In turbine theory
there are many variables and one must assume some of these and
compute the rest, and, according to what is assumed as known,
the theory presented by various indididuals will differ. Also
there are matters of difference of detail. For instance one may
assume the hydraulic friction losses through the entire turbine,
including guides and runner, to be some function of the rate of
discharge, while another will attempt to analyze these losses and
compute them individually.

The turbine designer, desiring to obtain some definite perform-
ance, naturally assumes certain results and computes the dimen-
sions necessary. For our present purpose, we shall do exactly
the opposite of this and assume all the dimensions as known and
endeavor to determine the characteristics of the given turbine.

83. Simple Theory. — A very simple theory is possible by as-
suming certain factors to be known as the result of experience.
Thus, as in the case of the impulse wheel, the peripheral velocity
of the runner may be represented as Ui = <j>\/2gh. _And_ the
speed at which the efficiency is a maximum is given by values of
<t>e ranging from 0.55 to 0.90 according to the type of the runner
as in Fig. ,34, page 43. This differs from the Pelton wheel not
only in the numerical values of <f>e but also in the much greater
range that is possible.

The efficiency of the turbine may be assumed as from 80 to 90
per cent, according to the size and type of the runner, and hence
the power may be computed if the rate of discharge is known.
We here introduce another factor c such that V\ = c\/2gh.
(This is really a velocity coefficient but there is no need to draw
any distinction between it and the coefficient of discharge, since
here the coefficient of contraction is unity.) It may be noted
that in the reaction turbine the water is under pressure through-
out its flow and hence the total energy of the water entering the
runner is not all kinetic. Thus the coefficient c can never be unity.'
And again, since the water flows through a closed conduit all

121

122 HYDRAULIC TURBINES

the way from the case to the tail race it is evident that any loss
of head in any part must cause a change in the rate of discharge.
And since the losses within the turbine runner vary with the speed,
it is evident that the rate of discharge, and hence of c, must also
vary. This again is different from the impulse wheel, where the
action of the wheel has no effect upon the velocity of the water
from the nozzle. Thus the factor c is not only less than unity
but it depends upon the design and type of the runner, and fur-
thermore it varies with the speed of the latter. Because of this
variation with the speed, we shall here give only the values
obtained at speeds which result in the highest efficiency being
obtained from the wheel. In practice ce varies from 0.6 to 0.8
according to the type of runner. Then

q = A! X ce\/2gh~.

It may be of interest to note that as one proceeds from runners
of Type I to Type IV of Fig. 34, page 43, one gets farther away
from the impulse wheel in all respects. Not only are the resulting
operating characteristics and conditions of service more unlike
but the numerical factors are of increasing difference. Thus
values of (j>e for the reaction turbine are larger than for the im-
pulse turbine and they increase in the direction mentioned. The
pressure pi is zero for the impulse turbine, but not for the reac-
tion turbine. For the same head and setting, the value of p\
will increase from Type I to Type IV. But if pi increases, V^
must decrease. Hence high values of ce accompany low values
of <$>e and vice versa.

For the present we are assuming that values of <f>e and ee are
to be chosen according to the type of runner concerned.

84. Conditions for Maximum Efficiency.— To obtain the best
efficiency the water must enter the runner without shock and
leave with as little velocity as possible. In order to enter without
shock the vane angle must agree with the angle j8i determined by
the velocity diagram and, in the case of the reaction turbine, the
velocity vi as determined by the velocity diagram should be equal
in magnitude to that determined by the rate of discharge and the
runner area 01. In order to leave with as little velocity as pos-
sible the angle <x2 may be made equal to 90°, as has been shown in
Art. 64. In the early type of turbine as built by Francis such an
angle would make the water flow along a radius and hence such
a discharge was called "radial" discharge. With the develop-

THEORY OF THE REACTION TURBINE 123

ment of the mixed flow type, this term is no longer appropriate
but it is quite commonly used nevertheless. Such a condition is
also spoken of as " perpendicular " discharge, from the fact that
the absolute velocity of the water is normal to the linear velocity
of the vane, and the term " axial" flow is also usedjrom the fact
that with the high-speed type of runner the flow is approximately
parallel to the axis.

A further reason for the use of az = 90° as desirable for a
high efficiency of the read ion turbine is that otherwise the water
would enter the draft tube with a whirling motion which would
increase the losses within the latter.

85. Determination of Speed for Maximum Efficiency. — A
runner of rational design would be so proportioned that there
would be no shock at entrance for the same speed at which the
discharge velocity would be normal to the vane velocity.
That is a2 = 90° and p\ = pi at the same speed, where £'1 is
the angle of the runner vane and /3i the angle of Vi as determined
by the vector diagram. The following equations therefore
apply only to such a runner.

From the velocity diagrams we have, if p'i = $\.

Vi sin ai = Vi sin jS'j
Vi cos ai = Ui + Vi cos |8'i
Eliminating v\ between these two equations we have

_ sin (0'j - ai) , .

Ui~ ~^wr l

as the relation between u\ and V\ when there is no abrupt change
of velocity at entrance to the runner.

Since a2 = 90° and hence Vu^ = 0, we have from equation (19)

€kh = = (33)

9 ff

as the relation between HI and V\ for which the discharge loss is
a minimum.

Solving equations (32) and (33) simultaneously we have

jeh2ghrin (ff'i
\ 2 sin £'i cos

= /_ e*2gh sin jg;i
' \2sin (j8'i - «0

124 HYDRAULIC TURBINES

From this it follows that

leh sin (0'i - ai)
*e =: \2sin/J'1coB«1

/Z 6/t sin ff'i ,«KN

: \2 sin (/5'i - «0 cos <*!

It must be borne in mind that the preceding equations apply
only to a runner designed as stated. For any runner, whether of
rational design or not, the value of <f> necessary to make a2 =
90° can be determined by involving more dimensions than the
above, and such an expression will now be derived. It will be
assumed also that this is the most efficient speed for any
runner, though this may not be strictly true if the entrance loss
is not zero at this speed.

If 0:2 = 90°, VU2 = Uz -f v 2 cos 02 = 0. Since u2 = xui and
vz = (A i/ 0,2) Vi = yVi from the equation of continuity, we may
write

x ui + y Vi cos fa = 0 (36)

as the relation between ui and V\ for which a2 = 90°. Solving
this simultaneously with equation (33) we obtain

eh'2gh-y
' \ - 2x c

cos 02

y I €h'2gh'X

\ — 2y cos 02 cos
From this it follows that

cos

\-22/cos02cosa1 (38)

From equation (33) we may write Ui Vi = eh • 2gh/2cos a\ and
from this it follows that

<t>ece = eh/2 cos on (39)

Values of eh and ai change somewhat with different types of tur-
bines but this shows that the factors (f>e and ce vary approximately
inversely, as stated in Art. 83.

The equations of this article are all based upon assumptions
which prescribe special relationships between c and 0, and are

THEORY OF THE REACTION TURBINE 125

true only for a special value of </>. A method of determining c
for any value of <£ will be found in Art. 87.

86. Losses. — The net head supplied the turbine is used up in
two ways; in hydraulic losses and in mechanical work delivered
to the runner. The head utilized in mechanical work is h" =

~(uiVUl — u2VU2). In accordance with the usual method in hy-
draulics we may represent hydraulic friction loss in the runner by
k v22/2g, k being an experimental constant. If the turbine dis-
charges into the air or directly into the tail race the discharge
loss is V^/2g. In addition there may be a shock loss at en-
trance to the runner. The term shock is commonly applied here
but the phenomena are rather those of violent turbulence. This
turbulent vortex motion causes a large internal friction or eddy
loss.

Referring to Fig. 94, the value of vi and its direction are de-
termined by the vectors ui, and V\. Since the wheel passages

c c

FIG. 94.

are filled in the reaction turbine, the relative velocity just after
the water enters the runner is determined by the area «i and its
direction by the angle of the wheel vanes at that point. If all
loss is to be avoided, these values should agree with those deter-
mined by the vector diagram; but that is possible for only one
value of Ui for a given head. For any other condition the velocity
Vi at angle ft is forced to become v'\ at angle 0Y This causes a
loss of head which will be assumed to be equal to (CC')2/2g.
Since the area of the stationary guide outlets normal to the
radius should equal the area of the wheel passages at entrance
normal to the radius, the normal component (i.e., perpendicular
to ui) of vi, should equal that of V\. Therefore CCf is parallel
to ui and its value is easily found to be

nn, _ sin (|8'] — «i) T7
uo — u\ — - — -. — —, K i

126 HYDRAULIC TURBINES

If k' = sin (0'i — «i)/ sin 0'i, then

shock loss = - — » —

87. Relation between Speed and Discharge. — Equating the
net head to the sum of all these items we have

T,"** j. I?2
k+

2(u,F.1 -

All velocities can be expressed in terms of u\ and V\ as follows :
HZ = xu\, v2 = yVi, Vui = Vi cos on,
Vuz = u2 + v2 cos 02 = a?wi + 2/F]' cos 02;

cos 32 =

cos

i.o

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.8 1.4

Values of p

Peripheral Speed, u \ = ^> ~\l % gh

FIG. 95. — Comparison of the relation between c and 0 as determined by theory

and by test.

Making these substitutions and reducing we obtain,
[(1 + k)y2 + fc'*]Fi2 + 2(cos «,. - fc')FiWi + (1 - a;2)^]2= 2^.

From this equation Fi may be computed for any value of wheel
speed, MI. It is customary to express the wheel speed as ui =
<j>-\/2gh, and we may also say Vi = c\/2gh. The use of these
factors is more convenient in general. Introducing them our
equation becomes

f(l + fc)2/2+fc'2]c2+2(cos ai-fcOc^+Cl-s2)^!. (40)
From this equation c may be computed for any value of <£. For

THEORY OF THE REACTION TURBINE 127

the outward flow turbine (37) becomes a hyperbola concave up-
ward, for the inward flow turbine it becomes an ellipse concave
downward.

A comparison between the values of c as determined by this
equation and as determined by experiment is shown in Fig. 95.
One turbine was an outward flow turbine and the other was a
radial inward flow turbine. Considering the imperfections and
limitations of the theory, the agreement is remarkably close.

If the turbine discharges into an efficient draft tube the discharge
loss may be represented by mV^/Zg, where m is a factor less than
unity. If there were no internal friction and eddy looses within
the tube, the value of m would depend only upon the areas of ends
of the tube and would be equal to (Az/Az)2. Actually m is
greater than this due to hydraulic friction losses. And as the
speed of the turbine departs from the normal value, it is probable
that m increases still more and approaches unity. Introducing
the discharge loss as mV^/2g in the equation at the beginning of
this article, we obtain as a substitute for equation (40),

[(m + k)y2 + fc'2]c2 + 2[cos ai - k' - (1 - m)xy cos fc]c0

- [(2 - m)x2 - 1]02 = 1 (41)

It will be found that this equation will give slightly higher values
of c than equation (40), which is to be expected. Thus the use of
a diverging draft tube increases the power of the turbine not only
by increasing its efficiency but also by increasing the quantity of
water it can discharge.

If desired, equation (36), when put in terms of 0 and c, can be
solved simultaneously with equations (40) or (41), thus giving a
third method of computing the value of <j>e. Also it is possible to
derive a general equation for the efficiency of a reaction turbine
and by calculus find the value of <f> for which the efficiency is a
maximum. However the resulting equation is somewhat lengthy
and, because it is of no practical value, will not be given here.
Values of 0 determined by it will usually not differ much from
those determined by the simpler approximate method of assuming
that <*2 = 90°.

88. Torque, Power and Efficiency. — General equations for
torque, power, and efficiency were derived in Chapter VII and the
application of these illustrated by a numerical case in Art. 63.
In that article the speed and rate of discharge of the turbine were
assumed. In the present chapter methods are shown for com-

128 HYDRAULIC TURBINES

puting by theory the speed for maximum efficiency and the rate
of discharge for any speed. From this point on the procedure
is the same as in Chapter VII.

It is of course possible to make algebraic solutions for these
quantities and the resulting equations then express results in
terms of known factors and dimensions. Thus, to illustrate, the
hydraulic efficiency is in general

th = h" '/h = (uiVi cos «i — U2V2 cos a^/gh.

For the reaction turbine in particular u2 = xui and F2 cos a2 =
u2 + v2 cos /32 = xui + yVi cos /?2. Substituting in the above we
obtain e^ = [(cos a\ — xy cos j32) V\u\ — x2Ui2]/gh. From this it
follows that

eh = 2(cos ai - xy cos /32) c<j> - 2#202 (42)

A numerical result in a given case can be computed either by
substituting the known quantities in the above equation or by
computing the separate items of the general equation. The latter
usually involves no more labor.

Equation (42) is of interest because it involves no arbitrary
factors of loss. Thus if the relation between speed and discharge
is known, as by experiment, the hydraulic efficiency can be
computed, provided the proper wheel dimensions are known.
Actually it is so difficult, as will be explained later, to determine
the proper values of the runner dimensions, that the numerical
accuracy of the result is doubtful. The hydraulic efficiency can
probably be estimated more accurately than the separate factors
in these equations. The euqation is of very practical value
however in showing that the hydraulic efficiency is independent
of the head under which the turbine is run.

Equation (42) is perfectly general for any reaction turbine
and is not restricted to the maximum efficiency. The value of the
maximum efficiency will be obtained by using the values of <t>e and
ce in the equation. Of course, since Vu is assumed to be zero for
this case the value of the maximum efficiency can be computed
much more directly than by the use of the above.

89. Variable Speed — Constant Gate Opening. — Since c varies
with the speed the input for a fixed gate opening will not be con-
stant for all speeds as it is in the case of the impulse turbine. The
variation of the losses at full gate with the speed ranging from zero
up to its maximum value is shown by Fig. 96. The horse-power

THEORY OF THE REACTION TURBINE 129

in each case obtained by multiplying wq/55Q by the head lost as
given by Art. 86.

The curves for the impulse turbine in Fig. 88 may also represent
percentages by the use of a proper scale since the input is con-
stant. But for the reaction turbine the percentage curves will
be slightly different from those in Fig. 96. It will thus be true
that the speed at which the efficiency is a maximum will be slightly
different from the speed for which the power is the greatest.

The Francis turbine for which the curves in Fig. 95 and Fig. 96
were constructed had the following dimensions :

Francis Turbine at Full Gate

: 1 Ft Head x h 3/2 Peripheral Speed

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 - 1.2 1.3
Values of <?

FIG. 96. — Losses at full gate and variable speed.

= 13°, 0'! = 115°
.ft.,

2 = 165°, Ai = 5.87 sq. ft., a2 = 6.83 sq.
= 4.67 ft., r2 = 3.99 ft.

From this data x = 0.855, y = 0.860, k' = 1.08, and k = 0.5 (as-
sumed). Attention is called to the fact that the horse-power
output was determined by an actual brake test while the horse-
power input to shaft was computed from the theory given in the
preceding article. The two differ by the amount of power con-
sumed in bearing friction and other mechanical losses.

90. Constant Speed— Variable Input. — The relation between
input and output and the segregation of losses for a cylinder gate
turbine at constant speed is shown in Fig. 97. In the four tests
the gate was raised %, %, %, and % of its opening. With
the turbine running on full gate but at an incorrect speed there is

130

HYDRAULIC TURBINES

a shock loss at entrance as shown in Art. 86. This loss is due to
an abrupt change in the direction of the relative velocity of the
water. When the turbine is running at the normal speed but
with the gate partially closed there is a shock loss of a slightly
different nature. A partial closure of the gates increases the
value of Vi and the angle on may be affected somewhat. How-
ever q will be reduced while ai remains the same and thus the

3.53

1.0 2.0

B.H.P. under 1 Ft. Head

FIG. 97. — Losses for cylinder gate Francis turbine at constant speed.

velocity Vi must be suddenly reduced to v\. The loss of head due
to this may be roughly represented by (vi — v'i)2/2g. While this
expression may not give the exact value of the loss, yet it must
be true that it will be of the nature shown by the curves.

This loss is known to be less in the case of the swing gate
turbine than in the cylinder gate turbine. While both on and V\
are altered in the case of the wicket gate, the transition in the
runner is less abrupt and consequently the eddy losses are less.

There will always be a slight leakage through the clearance

THEORY OF THE REACTION TURBINE

131

spaces and such a loss is indicated in Fig. 97 though it was not
possible to compute it with any exactness. It was merely added
to show that it exists but it was not accounted for in Fig. 96.
These results on a percentage basis are shown in Fig. 98.

The cylinder gate turbine is rather inefficient on light loads due
to the big shock loss. The wicket gates do not occasion such
large shock losses and hence reduce the input curve to a line more
nearly parallel to the output line in Fig. 97, and thus improve the
part load efficiency of the turbine. Also the cylinder gate turbine
gives its best efficiency when the gate is completely raised and the

100

1.0 2.0

B.H.P. under 1 Ft. Head
FIG. 98. — Cylinder gate turbine at constant speed.

power output has its greatest value. But the wicket gate tur-
bine usually develops its best efficiency before the gates are fully
open. There is thus left some overload capacity.

91. Runner Discharge Conditions. — The following theory,
though open to certain objections, serves to explain the observed
phenomena at the discharge from a turbine runner. A low
specific speed type of runner, such as Type I in Fig. 34, page 43,
will have stream lines through it which differ but little from one
another, while a high specific speed type such as Fig. 99 will
have stream lines which differ considerably. Thus in Fig. 99
stream line (a) next to the crown will have smaller radii at both
entrance and outflow than stream line (c) next to the band.

132

HYDRAULIC TURBINES

Also the radii of points along the outflow edge vary considerably
more in the case of the high specific speed runner. The radius of
curvature of stream line (a) is much greater than that of line (c)
and in accordance with equation (23) the velocity along line (a)
will be less than along line (c). It must be noted that the lines
shown in the drawing are really " circular" projections of the
actual stream lines, by which is meant that the various points are
revolved about the axis of the runner until they lie in the plane of
the paper; and also a free vortex, while existing in the space be-

tween guides and runner, and
also in the draft tube, does not
exist within the runner. But,
considering the velocity com-
ponent in the plane of the paper
only, and considering the rota-
tion about the centers of curva-
ture of the lines drawn (and
not about the axis of the run-
ner), the equations of Art. 66
may be applied.

Since eh = h"/h = (uiVui —
we may write

~ gh" (43)

FIG. 99. riot only for the turbine as a

whole, but for each individual

stream line. It has been stated that usually a runner is so designed
that <*2 = 90°. With some runners observation shows that there
is a slight whirl of the water across the entire draft tube at the
point of maximum efficiency, but this might be expected, since the
assumption that Vuz should equal zero is a mere approximation.
With the low specific speed turbine it is possible to have a2 = 90°
for all stream lines at some speed which may or may not be exactly
the most efficient, but it is difficult to do this with the high
specific speed runner and still satisfy the equation above. Thus
suppose the discharge is normal to u2 for stream line (6) in Fig.
99. Then for this stream line UiVu\ = gh". Considering line
(c) both u\ and Vui are larger for the reasons stated in the first
paragraph. If Vuz is to be zero here also, h" must be larger.
But the conditions here are not favorable to as high an efficiency
as along line (b), because of the proximity to the boundary (which

THEORY OF THE REACTION TURBINE

133

in this instance is the band) and because of the sharper curvature.
Hence by no proportioning can the water be compelled to flow
as desired. Since UiVui is larger and In" is smaller than for line
(6) it follows that the right hand member of the equation must be
positive and hence there must be some whirl at the point of dis-
charge in the direction of rotation of the runner. In similar
fashion there may be a negative whirl at the point of discharge
from line (a), but since u\, Vui, and h" all decrease for this line,
as compared with (b) , it is possible that there may be little or no
whirl here. All this reasoning has been verified by experimental
observations.1 This whirl of the water near the band decreases
the efficiency of the draft tubes as constructed in the past and
points the reason for the development of a new type of tube if
turbines of higher specific speed are to be used. And unless more
effective draft tubes are used, this shows that this factor tends to
reduce the efficiency of the runner as the specific speed increases.

FIG. 100.

For the higher the specific- speed the greater the variation in r2
for stream lines (a) and (c), and this has been shown to be undesir-
able.

At part gate on any turbine the efficiency and hence h" are
known to be less than on full load, the latter being taken as the
load for which the efficiency is a maximum. And if wicket gates
are used the angle <*i is less than at full load so that Vi cos a\
would appear to be higher. Hence if the right hand side of equa-
tion (43) is equal to zero at full load, it would have a relatively
large positive value for a partial opening of the turbine gates.
Thus Vuz would be positive, which agrees with the vector dia-
gram, since with a smaller rate of discharge the velocity v2 would
be less while the wheel speed u2 is considered to be the same.

Since u2 Vu2 must have a large positive value for a small gate
opening and u2 varies with the radius, it follows that Vu2 is rela-
tively small near the band and relatively large near the crown

1 See Trans. A. S. C. E., Vol. LXVI, p. 378 (1910).

134 HYDRAULIC TURBINES

Since Vu2 = ^2 + v 2 cos /32 and w2 is fixed, the value of v2 must
decrease as the crown is approached (cos /32 is negative) and may
even become negative. This means that water is actually
pumped back into the runner near the crown and out near the
band. This shows the undesirability of a large variation in the
radii of the discharge edge of a runner, and this latter is character-
istic of the profile of the high specific speed runner. Hence this
theory presents a reason why the part gate efficiency of a reac-
tion turbine must be less as the specific speed increases.

The smaller the rate of discharge for a given head the less the
value of vz at any point on the outflow edge and hence the less
the wheel speed necessary to make «2 = 90° at this point. Thus
with any turbine the speed for which the efficiency is a maxi-
mum decreases as the gate opening decreases.

92. Limitations of Theory. — The defects of this theory or
any theory are as follows: In order to apply mathematics in
any simple way it is necessary to idealize the conditions of flow
by assuming that all the particles of water at any section move
in the same direction and with the same velocity. Such is
very far from being the case so what we use in our equations
is the average direction and the average velocity of all the
particles of water. That in itself could easily cause a discrep-
ancy between our theory and the fact, because the theory is
incomplete.

But even to determine accurately these average values that
are used in the equations is a matter involving some difficulty.
Thus, though the direction of the streams leaving the runner is
influenced by the vane angle at that point, it cannot be said that
the angle £2 is exactly equal to the vane angle at exit. In fact
the author has roughly proved by study of a test where some
special readings were observed that the two may differ by from
5 to 10 degrees, and that /32 varied regularly for different values
of <£. The same thing may be said about the area a2. Some
recent experiments in Germany1 have shown that there may be a
certain amount of contraction of the streams and that this con-
traction varies for different speeds. Thus the true value of
a2 may be slightly less than the area of the wheel passages.
These observations concerning 02 and a2 apply equally well to
other angles and areas.

i Zeitschrift des Vereins deutscher Ingenieure, May 13, 1911.

THEORY OF THE REACTION TURBINE

135

In computing the results plotted in Fig. 95 the coefficients of c
and <£ in equation (40) were treated as constant. It has just
been shown that the real values of the angles and areas may vary
slightly with the speed. Also it is stated in Art. 75 that the value
of k is not constant at all speeds for the impulse turbine. While
the conditions with the reaction turbine are very different, yet
it is doubtless true that k is not strictly constant here. If it were
known just how k and the dimensions used varied with the speed,
the theoretical curve could be made to more nearly coincide with
the actual curve. In addition the expression for shock loss is
only an approximation. But even as it is the discrepancy is not
serious.

By the use of the proper average dimensions the equations
given may be successfully applied to a radial flow turbine. For
the mixed flow turbine they will apply approximately. The

Runner

FIG. 101.

reasons for this are that with the mixed flow turbine the radius r2
varies through such a wide range of value that it is difficult to
fix a proper mean value; likewise the vane angle at exit and also
the area varies so radically that a mean value can scarcely be
obtained with any accuracy. Even if these mean values could
be obtained the theory would still be imperfect, for the reason
stated in the first paragraph of this article.

The value of the angle a\ may be taken as that of the angle
shown in Fig. 101, though it may be seen that this is a mean for
the various stream lines. The velocity of the water through the
guide vanes may be denoted by F0 but, since the space between
guides and runner is a free vortex, the velocity Vi is increased in

136 HYDRAULIC TURBINES

the ratio r0/ri. In all the discussion so far it has been inferred
that the point (1) coincided with the outer radius of the runner.
It is the velocity of this point that is given by the factor <f> and most
of our empirical computations will be based upon this, for ease
in computation. But if one wishes to compute certain results
by applying the laws of hydrodynamics, such as equations (40),
(41), and (42) for example, it is desirable to select the point (1)
such that the most suitable mean values for use in the equations
will be obtained.1 Such a point is often said to be the center
of the circle shown at entrance to the runner of Fig. 101. The
diameter of this circle is the shortest line that can be drawn from
the tip of one vane to the next vane.

A similar procedure should be followed for the point of dis-
charge if the turbine were a pure rad'al flow turbine with all
points on the outflow edge at the same radius. For the mixed
flow type of turbine it can be proven that the discharge may be
considered as concentrated at the center of gravity of the out-
flow area.

Because of the difficulties of applying the theory in a definite
case numerical results are of doubtfuraccuracy. But the theory
has other uses. Thus the theory shows why certain factors and
dimensions must vary with the specific speed of the runner.
It shows that the rate of discharge cannot be constant for a given
runner at different speeds under a constant head and gate open-
ing. It shows why certain conditions are desirable for efficiency
and how the proper speed may be approximately computed. It
explains the losses within a turbine and shows why certain
characteristics vary as they do. It serves to give the reasons
why there are fundamental differences in the operating charac-
teristics of turbines of different types. In other words it will
in general furnish the reason for any result found in practice.
And beyond explaining these characteristics, it indicates the
effect of any change in any direction.

93. Effect of y. — The ratio of Ai/a2 is expressed by y. If y
is small enough the turbine will be an impulse turbine, the value
of <f> giving the best speed will be about 0.45, pi/w = 0, and c
will equal 1.00 if the slight loss in the nozzle is neglected. (Ac-

i This procedure was not followed however in dealing with the Francis
turbine for which the curves in this chapter were drawn. But this turbine
is much more amenable to mathematical analysis than the runners of the
present day.

THEORY OF THE REACTION TURBINE

137

tually c will be the coefficient of velocity of the nozzle and will be
about 0.97). As the value of y increases, the turbine becomes a
reaction turbine, the value of <£ increases, p\/w increases, and c
decreases. The general tendency of these factors is shown in
Fig. 102.

It is thus seen that the design of a reaction turbine can be
varied so as to secure quite a range of results.

Values of |f (- A1
FlQ. 102.

1.2

94. QUESTIONS AND PROBLEMS

1. Given the diameter D and the height B of a turbine runner, how can
one approximately compute the speed and power for any head?

2. Why does the rate of discharge from a turbine runner vary with the
speed under a fixed head? Why is the velocity of the water entering the
runner less than V2g7i?

3. What are the conditions necessary for high efficiency of a reaction
turbine? What effect does the draft tube have upon this also?

4. In what two ways may <f>6 be computed? What are the fundamental
differences involved in these methods? Should the numerical results differ?

5. What are the various losses of the turbine and how may they be ex-
pressed? What is the effect of the draft tube in this?

6. How may a general equation between speed, discharge, and head be
derived?

7. How may a general equation for the hydraulic efficiency of a reaction
turbine be derived? What does it indicate?

8. From the curves, what are the differences between the variations in
the losses for impulse and reaction turbines?

138 HYDRAULIC TURBINES

9. Explain why the discharge conditions for a high specific speed runner
are less favorable than those for a low specific speed runner both being
assumed to be running at their points of maximum efficiency.

10. Explain why the discharge conditions at part gate are less favorable
for the high specific speed runner than for the low specific speed runner.

11. What are the limitations of turbine theory and why? What is the
value of the theory?

12. What effect does the change in the ratio of the area through the guide
vanes to that at outflow from the runner have upon the values of 0, c,
and pi/w!

13. A turbine runner 36 in. in diameter and 12 in. high at entrance will
run at what probable r.p.m. and develop what power under a head of 60 ft.?
(Assume value of «i.) Ans. N = 317, B.h.p = 890.

14. In problem (13) suppose the intake to the runner is at a height of
15 ft. above the tail-water level. What is the probable value of the pressure
head at this point? Ans. 29 ft.

15. The dimensions of the original Francis runner were on = 13°, ft\ =
115°, ft, = 165°, A! = 5.87 sq. ft., a2 = 6.83 sq. ft., n = 4.67 ft., and r2 =
3.99 ft. Compute the values of <£e and ce by the first method given, assum-
ing eh = 0.83. Do these answers give shockless entrance? Do they give
a2 = 90°? What dimensions could be changed to make both of these con-
ditions, be fulfilled at the speed computed? Ans. <j>e = 0.678, ce = 0.628.

16. ^Compute the values of <f>e and ce for the Francis turbine in the pre-
ceding problem by the second method given? Do these answers give shock-
less entrance? Do they give «2 = 90°? What dimensions could be
changed so as to fulfill both these conditions at this speed?

Ans. <j>e = 0.643, ce = 0.663.

17. Francis noted that his runner was not quite properly designed and
that there was some shock loss at entrance when running at the most effi-
cient speed. By test the actual value of 4>e was found to be 0.67. Compute
the corresponding value of ce and compare with the curve in Fig. 95. As-
sume k = 0.5. Ans. ce = 0.655.

18. Compute the hydraulic efficiency of the Francis turbine of problem
(15) using the values of <f> and c given in problem (17) and compare with
value given by curve in Fig. 98. Ans. 0.825.

19. If this turbine discharges into a draft tube of such dimensions that
m may be assumed equal to 0.3, compute the value of c for a value of <f>e
equal to 0.675. Compute the hydraulic efficiency. The value of <J>e has been
increased slightly here because of the presumption that the draft tube will
increase the efficiency of the turbine. Compare with problems (17) and (18).

Ans. ce = 0.66, eh = 0.833.

20. What is the percentage value of the discharge loss from the Francis
turbine of problem (15), assuming «2 = 90° and ce = 0.66? For this par-
ticular turbine, what is the possible gain in efficiency due to using a draft
tube which would reduce the velocity to zero without loss of energy? (Note.
F2 sin «2 = Vz for 90° and F2 sin «2 = v2 sin /32 = ycV2gh sin /32.)

Ans. 2.15 per cent.

21. If the turbine in'problem (17) is used under a head of 30 ft., find the

THEORY OF THE REACTION TURBINE 139

r.p.m., the quantity of water discharged, and the power delivered to the

shaft. Find similar results for problem (19).

Ans. N = 60.2, q = 169, h.p. = 475, N = 60.8, q = 170.2, h.p. = 484.

22. If the turbine in the preceding problem were to be run at the same
speed of 60.2 r.p.m., while the head decreased to 18 ft., find the rate of dis-
charge, hydraulic efficiency and power.

Ans. <f> = 0.864, c = 0.633 by (40), q = 126 5, eh = 0.755, 195 h.p.

CHAPTER X

TURBINE TESTING

95. Importance. — Testing is necessary to accompany theory
in order that the latter may be perfected until it becomes reliable
enough to be useful. Unless the theory agrees with the facts it
is not true theory but only an incorrect hypothesis. Only by
means of theory and testing working hand in hand can improve-
ments in design be readily brought about. Thus the ease of
testing is a measure of the rate of development of any machine.

Again, if we are to thoroughly understand turbines, it will be
necessary to make a thorough study of test data in order to appre-
ciate the differences between different types. Unfortunately
there is a scarcity of good and thorough test results.

The only public testing flume in the United States is the one at
Holyoke, Mass. Nearly 3000 runners ha\ _, oeen tested there and
it has been an important factor in the development of modern
turbines. The maximum head obtainable there is about 17 ft.,
also it is scarcely possible to test runners above 42 in. in diameter
because of the limitations imposed by the depth of the flume.

An acceptance test should always be made when a turbine is
purchased if it is possible to do so. Otherwise the purchaser will
have no assurance that the specifications have been fulfilled.
Thus a case may be cited where the power and efficiency of a
tangential water wheel were both below that guaranteed as can
be seen by the following:

Efficiency

Normal h.p.

Maximum h.p.

Guarantee

0 800

3500

5225

Test. ..'.

0 720

2300

3500

In this table the normal horse-power means the power at which
the maximum efficiency is obtained, any excess power over that
being regarded as an overload. The actual efficiency is 8 per
cent, less than that guaranteed and the wheel is* really a 2300-
h.p. wheel instead of a 3500-h.p. wheel. It is 'true that the

140

TURBINE TESTING

141

wheel could deliver 3500 h.p. but at an efficiency of only 67
per cent. Since that is the maximum power the 5225-h.p. over-
load could not be attained.

Another case may also be given where the facts are of a differ-
ent nature.1 A comparison of the guaranteed and test results
for a reaction turbine is shown in -Fig. 103. The efficiency se-
cured was higher than that guaranteed, but it was also attained at
a much higher horse-power. If the turbine were then run on the
load specified it would be operating on part gate all the time and
at a correspondingly low efficiency. This is a common failing
in "cut and try" practice. A turbine of excess capacity is

provided; it never lies down under any load put upon it and the
owner is satisfied. Quite frequently also a turbine which must
run at a certain speed is really adapted for a far different speed.
Thus under the given conditions its efficiency may be very low,
when the runner might really be excellent if operated at its proper
speed. A test would show up these defects, otherwise they may
remain unknown.

Another reason for making tests would be to determine the
condition of the turbine after length of service. The effect of
seven years' continuous operation upon a certain tangential water
wheel is seen in Fig. 104. This drop in efficiency is due to rough-
ening of the buckets, to wear of the nozzle, and to the fact that
end play of the shaft together with the worn nozzle caused the
jet to strike upon one side of the buckets rather than fairly in the
center. It might be noted however that a 7-ft. wheel of the
same make in the same plant showed no change in efficiency after

1 Trans. A. S. C. E., Vol. LXVI, p. 357.

142

HYDRAULIC TURBINES

the same length of service. With reaction turbines the guides
and vanes become worn and the clearance spaces also wear so as to
permit the leakage loss to increase.

As to whether efficiency is important or not depends upon cir-
cumstances. If there is an abundance of water in excess of the
demand the only requirement is that the turbine deliver the power
demanded. But where the water must be purchased, as it is in
some cases, or where vast storage reservoirs are constructed at
considerable expense it is desirable that water be used with the
utmost economy.

1&05

0 10 20 30 40 50 GO

B.H.P.

FIG. 104. — Effects of service upon a 42" tangential water wheel.

96. Purpose of Test. — The nature of the test will depend upon
the purpose for which it is made. In a general way there are four
purposes as follows :

1 . To Find Results for Particular Specified Conditions. This will
usually be an acceptance test to see if certain guarantees have
been fulfilled. • The guarantee will usually specify certain values
of efficiency obtained at certain loads at a fixed speed under a
given head. Occasionally several values of the head will be
specified.

2. To Find Best Conditions of Operation. Such a test will cover
a limited range of speed, load, and head; all of them, however,
being in the neighborhood of the maximum efficiency point.
A test of this nature will show what a given turbine is best
fitted for.

TURBINE TESTING 143

3. To Determine General Principles of Operation. — This test is
similar to the above except that it is more thorough. It should
cover all speeds from zero to the maximum possible under no load.
Various gate openings may be used and the head may also be
varied. Such a test will enable one to understand the turbine
better and could also be used to verify the theory.

4. To Investigate Losses. — This test would be similar to the
preceding except that a number of secondary readings of veloc-
ities, pressures, etc., at various points might be taken. Such a
test will be of interest chiefly to the designer.

97. Measurement of Head. — The head should be measured as
close to the wheel as possible in order to eliminate pipe-line losses.
The head to be used should be as specified in either equation (7)
or (8) or (9) of Art. 55, according to circumstances. The pressure
may be read by means of a pressure gage if it is high enough.
For lower heads, a mercury column or a water column will give
more accurate results. Care should be taken in making connec-
tions for the pressure reading so that the true pressure may be
obtained. The reading of any piezometer tube will be correct
only when the tube leaves at right angles to the direction of flow
and when its orifice is flush with the walls of the pipe. No tube
projecting within the pipe will give a true pressure reading, even
though it be normal to the direction of flow.1

98. Measurement of Water. — The chief difficulty in turbine
testing is the measurement of the water used. In some commer-
cial plants the circumstances are such that it is scarcely possible
to measure the water at all and in others the expense is prohibi-
tive. The necessity of cheap and accurate means of determining
the amount of water discharged is imperative.

The standard method of measurement is^by means of a weir.
For large discharges, however, the expense of constructing a
suitable weir channel may be excessive, and, in case the turbine
discharges directly into a river, it may be almost impossible to
construct it. In the case of a turbine operating under a low head
the increase in the tail-water level caused by the weir may cause
a serious decrease in head below that normally obtained. This
would make the test of little value. However, where it is feasible,
the use of a weir is a very satisfactory method and should be pro-
vided for when the plant is constructed. It should be remem-
bered, though, that all weir formulas and coefficients are purely

1 Hughes and Safford, " Hydraulics," p. 104.

144 HYDRAULIC TURBINES

empirical in their nature and that the discharge as determined by
them may be as much as 5 per cent, in error, unless standard
proportions are carefully adhered to.1

In order to avoid the increase in the tail-water level the use of
submerged orifices may be desirable in low-head plants. A sub-
merged orifice will produce a certain elevation of the tail-water
level, but it will not be as great as the weir. At present enough
experimental data had not been gathered to make this method
applicable in general, but perhaps in the future it may be used
with fair success.

Either in the tail race or in the head race a Pitot tube, current
meter j or floats may be used. These methods involve no dis-
turbance of the head under which the turbine ordinarily operates,
but they do require a suitable channel in which the observations
can be taken. These instruments should be in the hands of a
skilled observer who understands the sources of error attendant
upon their use.2

The Pitot tube consists of a tube with an orifice facing the
current. The impact of the stream against this orifice produces
a certain pressure which is proportional to the square of the
velocity. If h is this reading in feet of water and K and experi-
mental constant, then

V =

Since it would be very difficult to determine accurately the height
of the column of water in a tube above the level of the stream
it is customary to use two tubes and read the difference between
the two. For convenience in reading, the instrument is made so
that valves may be closed and the device lifted out of the water-
without changing the levels of the columns, or sometimes both
columns may be drawn up to a convenient place. The orifice
for this second tube is usually in a plane parallel to the direction
of flow and will thus give a lower reading than the other. It does
not, however, give the value of the pressure at that point, as
stated in Art. 97. For low velocities it is desirable to magnify
this difference in the two readings and for that purpose the orifice
of the second tube may be directed down stream. Its reading
will then be less than for the one parallel to the direction of

1 See "Weir Experiments, Coefficients, and Formulas," by R. E. Horton,
U. S. G. S. Water Supply and Irrigation Paper No. 150, Revised, No. 200.

2 See Hoyt and Grover, "River Discharge."

TURBINE TESTING 145

flow. Such an instrument is called a pitometer, and the value
of K for it is always less than 1 .0.

The current meter is an instrument having a little wheel which
is rotated by the action of the current, the speed of rotation
being proportional to the velocity of flow.

The Pitot tube may also be used in a pressure pipe. Since the
reading of the impact tube alone will be the sum of the pressure
head plus the velocity head it will be necessary to use two tubes
in the same manner as in the case of the open channel. The
value of h will be the difference between these two readings,
and the value of K must be determined experimentally. If,
however, only one orifice is used and the pressure is determined
by a piezometer tube with its orifice lying flush with the walls of
the pipe the difference between these two readings may be con-
sidered equal to the velocity head, that means the value of K = 1.0.

For the tangential water wheel the Pitot tube may also be
used to determine the jet velocity. In such a case only the
impact tube is required. While it is well to determine the value
of K experimentally, yet if the tube is properly constructed it
may be taken as 1 .0. A check on this may be obtained as follows :
It is probably true that the maximum velocity obtained at any
point in the jet is the ideal velocity. The latter can be computed
from the head back of the nozzle and the value of K should be
such as to make the two agree. Either in the pipe or the jet it is
desirable to take a velocity traverse across each of two diameters
at right angles to each other. In computing the average velocity
it is necessary to weight each of these readings in proportion
to the area affected by them.1

Chemical methods are often of value. If the pipe line is
sufficiently long a highly colored stain may be added to the water
at intake and the time noted that it takes the color to appear in
the tail race. From this and the pipe dimensions the rate of
discharge can be computed. A second chemical method is
to inject a salt solution into the water of known concentration and

3 See "Application of Pitot Tube to Testing of Impulse Water Wheels,"
by Prof. W. R. Eckart, Jr., Institution of Mechanical Engineers, Jan. 7, 1910.
Also printed in Engineering (London), Jan. 14, 21, 1910.

Engineering News, Vol. LIV, Dec. 21, 1905, p. 660. See also Zeitschrift des
ver. deut. Ing., Mar. 22, 29, and Apr. 5, 1913. For useful information re-
garding all the methods of measurement given here see Hughes and Safford,
Hydraulics.
10

146 HYDRAULIC TURBINES

at a known constant rate. Samples of the water, after thorough
mixing, are taken and analyzed. Knowing the amount of
dilution it is then possible to compute the rate of discharge.1

99. Measurement of Output. — The determination of the power
output of a turbine is also a matter of some difficulty. Perhaps
the most satisfactory method is to use some form of a Prony
brake or absorption dynamometer.2

The use of a simple brake is restricted to comparatively small
powers. For large powers it becomes rather expensive and diffi-
cult. A good absorption dynamometer may be used satisfac-
torily for fairly large powers but the drawback is one of initial
expense. In many cases also where turbines are direct connected
to electric generators it may be impossible to attach a brake of
any kind.

In such cases it is necessary to supply an electrical load for
the generator and determine the generator efficiency. However,
this method of testing involves a. number of instrument readings
which may be more or less in error and a rather complicated
process of computation. Nevertheless it can be done with very
satisfactory results. One drawback about it is that the speed
cannot be varied through the same range of values as in the brake
test. The output of the generator may be absorbed by a water
rheostat which will furnish an absolutely constant load. If it
is a direct-current machine this rheostat may simply consist of
a number of feet of iron wire wound on a frame and immersed in
water to keep it cool. This water should be running water or a
large pond so that its temperature may not change. The current
is shorted through this coil; the load is varied by changing the
length of wire in use. For a three-phase alternator the rheostat
may consist of three iron pipes at the vertices of an equilateral
triangle with a terminal connected to each. The load is varied
by changing the depth of immersion of the pipes in water.3

A good method recently employed in a hydro-electric plant
where there are two or more similar units is to let one alternator
drive the other as a synchronous motor. The second rotates
the impulse wheel or reaction turbine in the reverse direction.

1 B. F. Groat, " Chemi-hydrometry and precise Turbine Testing," Trans.
A. S. C. E., Vol. LXXX, p. 951 (1915).

2 C. M. Allen, " Testing of Water Wheels after Installation," Journal
A.S. M.E., April, 1910.

•Power, Vol. XXXVII, June 17, 1913, p. 857.

TURBINE TESTING 147

By running varying quantities of water through the latter it
is possible to supply any constant load desired.

100. Working Up Results. — In figuring up the results of test
data it should be borne in mind that any single reading may be
in error but that all of them should follow a definite law. Thus
a smooth curve should be drawn in all cases. Also if any readings
should follow a law which is any approach to a straight line it is
better to work from values given by that line rather than from
the experimental values themselves. Thus if a turbine be tested
at constant gate opening and at all speeds, the curve showing the
relation between speed and efficiency may be drawn at once from
the experimenal data. However, a more accurate curve can be
constructed by noting that the relation between speed and brake
reading is a fairly straight line. See Fig. 86. TlnV is not a
straight line but the curvature is not very marked so that it may
be drawn readily and accurately. Values given by this curve
may then be used for constructing the efficiency curve. Again,
when a turbine is tested at constant speed, it should be noted
that the relation between input and output is not a straight line
absolutely, but it is approximately so. If any point falls de-
cidedly off from a straight line it is probably in error. From
the line giving the relation between input and output the effi-
ciency curve may be constructed.

In computing the true power in a jet it might also be noted that
it is not that given by using the square of the average velocity but
something 1 or 2 per cent, higher than that. The reason is that
the velocity throughout the jet varies and the summation of the
kinetic energy of all the particles is not that obtained by using the
average velocity.1

101. Determination of Mechanical Losses. — With the tangen-
tial water wheel the mechanical losses will consist of bearing
friction and windage. With the reaction turbine they will
consist of the bearing friction and the disk friction due to the
drag of the runner through the water in the clearance spaces.
There are several ways of determining this but the retardation
method is probably as satisfactory as any. The turbine is
brought up to as high a speed as is possible or desirable and the
power shut off. As the machine slows down readings of instan-

1 L. M. Hoskins, "Hydraulics," p. 119.

L. F. Harza, Engineering News, Vol. LVII, Mar. 7, 1907, p. 272.
See also Prof. Eckart's paper previously mentioned.

148

HYDRAULIC TURBINES

taneous speed are taken every few seconds and a curve plotted
between instantaneous speed and time as shown in Fig. 105.
Instantaneous speed may be determined by a tachometer, by a
voltmeter, or by an ordinary continuous revolution counter.
With the latter the total revolutions are read every few seconds
without stopping it; the difference between two consecutive
readings will then enable us to find the average speed correspond-
ing to the middle of this time interval.

The power lost at any speed is equal to a constant times the
subnormal to the curve at that speed. If L equals the power lost
then

L = K X BD.

For the proof of this proposition see Appendix A, II.

B D B D

Time
FIG. 105. — Retardation curves.

To determine the value of the constant K a second run is
necessary with a definite added load. This load, which may be
small, may be obtained by closing the armature circuit on a
resistance if a generator is used in the test or by applying a
known torque if a Prony brake is used. With the first method
a watt-meter should be used and the load kept constant for a
limited range of speed, with the second method the torque should
be kept constant for a limited range of speed. If this known
added load be denoted by M we then have

L + M = K X B'D'.

Since L is the only unknown quantity except K it may be elimi-
nated from these two equations and we have

K =

B'D' - BD

TURBINE TESTING 149

In some work that the author has done this method has proven
to be very reliable and has checked with values of friction and
windage losses as determined by other methods.

102. QUESTIONS AND PROBLEMS

1. What is the value of testing a turbine upon installation? What is the
value of testing one that has been in operation for some time? What is
the value of a Holyoke test to the purchaser of another runner but of similar
type? Is it cheaper to increase the power output of a plant by additional
construction or by improving the efficiency?

2. What are the various purposes for which turbine tests may be con-
ducted? What conditions would be varied for each of these and what kept
constant?

3. What methods of measuring the rate of discharge are usually employed
in turbine tests?

4. In what ways may the power output of a waterwheel be absorbed and
measured?

5. A case is reported where tests conducted at an expense of $5000 resulted
in changes which improved the efficiency of the turbines 1 per cent. If the
capacity of the plant is 100,000 h.p. and 1 h.p. is worth $100, what would
be the value of the gain in efficiency, assuming the changes cost $20,000?

6. In the Cedars Rapids turbines the area of the water passages at en-
trance to the casing = 1080 sq. ft. per unit, elevation of section above tail
water = 10 ft., and pressure head at this point = 20 ft. The area of the
mouth of the draft tube = 1050 sq. ft. The test showed the power output
to be 10,800 h.p. with a rate of discharge of 3450 cu. ft. per second. Cal-
culate two values of the efficiency, using two values of the head.

7. In the test of a reaction turbine the water flowing over the weir in the
tail race was found to be 39.8 cu. ft. per second. The leakage into the tail
race was found to be 1 cu. ft. per second. The elevation of the center line
of the shaft above the surface of the tail water was 12.67 ft. The diameter
of the turbine intake was 30 in. and the pressure at this section was meas-
ured by a mercury U tube. The readings in the two sides of the mercury
U tube were 10.556 ft. and 0.900 ft., the zero of the scale being at a level
3.82 ft. below that of the center line of the turbine shaft. The generator
output was 391.8 kw., friction and windage 13.8 kw., iron loss 2.0 kw., and
armature loss 4.4 kw. The specific gravity of the mercury used was 13.57.
Find: Input to turbine, output of turbine (generator being excited from
another unit), efficiency of turbine, efficiency of generator, efficiency of set.

Ans. h = 141.80 ft., 625 w.h.p., 550 b.h.p., 0.880, 0.951, 0.837.

CHAPTER XI
GENERAL LAWS AND CONSTANTS

103. Head. — The theory that has been presented has made it
clear that the speed and power of any turbine depends upon the
head under which it is operated. The peripheral speed of any
runner may be expressed as HI = $\/2gh. It has also been
shown that for the best efficiency <j> must have a certain value de-
pending upon the design of the turbine. It is thus apparent that
the best speed of a given turbine varies as the square root of the
head.

The discharge through any orifice varies as the square root of
the head, and a turbine is only a special form of discharge orifice.
Since Vi = c\/2gh, and since a definite value of c goes with the
best value of </> as given above, it follows that the rate of discharge
of a given turbine varies as the square root of the head.

Since the energy of each unit volume of water varies as the
head, and since the amount of water discharged per unit time
varies as the square root of the head it must then be true that the
power input varies as the three halves power of the head.

In reality the rate of discharge through any orifice is not strictly
proportional to the square root of the head, that is, the coeffi-
cient of discharge is not strictly a constant but varies slightly
with the head. However, the variation in the coefficient is small
and inappreciable except for very large differences in the head.
Therefore the above statements are accurate enough for most
practical purposes.

The theory has also shown that the losses of head in any tur-
bine vary as the squares of the various velocities concerned.
This rests upon the assumption that the coefficient of loss k is
constant for all values of h as long as <f> remains constant. That
is probably not true, but may be assumed as true for all practical
purposes. Since these velocities vary as the square root of the
head their squares will vary as the first power of the head. The
amount of water varies as the square root of the head and, since
the power Ipst is the product of these two items, it follows that

150

GENERAL LAWS AND CONSTANTS

151

the various hydraulic losses vary as the three-halves power of
the head. As the hydraulic losses vary in just the same propor-
tion as the power input, the hydraulic efficiency will be independ-
ent of the value of the head. If the mechanical losses followed
the same law then the gross efficiency would also remain
unchanged. The mechanical losses really follow different laws
at different speeds, as can be seen in Fig. 106. The factors which
influence this are rather complicated and it does not seem possi-
ble to lay down any rule to express mechanical losses as a func-
tion of the speed. It is probably true, however, that these losses
increase faster than the first power of the speed but not much
faster than the square of the speed. Since the speed varies as

0.4

o.s

0.2

0.1

100

200

R.P.:

300

400

FIG. 106. — Friction and windage of a 24" tangential water wheel.

the square root of the head it is seen that if the friction losses
vary at the same rate as the hydraulic losses they must increase
as the cube of the speed. As they do not do so, it is apparent
that the gross efficiency will be higher the higher the head under
which the turbine operates. The change in the gross efficiency
with a change in head is most apparent when the latter is very
low. As the head increases the mechanical losses become of
smaller percentage value and the gross efficiency tends to approach
the hydraulic efficiency, which is constant, as a limit. Thus
there is little variation in efficiency unless the head is very low.
The mechanical losses are really comparatively small being
only from 2 to 5 per cent, usually and thus the change in gross
efficiency cannot be very great. For moderate changes in head

152 HYDRAULIC TURBINES

we may then state that the efficiency of a turbine remains con-
stant as long as the speed varies so as to keep <j> constant.
Therefore the power output of a given turbine varies approxi-
mately as the three-halves power of the head. ^

This proposition is rather important because it is often nec-
essary to test a turbine under a certain head which is different
from the head under which it is to be run. The question may
then arise as to how far the test results can be applied to the new
head. As long as the two heads are not radically different we
may state that they will apply directly. If there is a large differ-
ence in head we may expect that the efficiency under the higher
head may be one or two or more per cent, higher. This is borne
out by some tests made by F. G. Switzer and the author where the
head was varied from 30 ft. to 175 ft. and later from 9 ft. to 305 ft.

It is customary to state the performance of a turbine under
one foot head . Then by means of the above relations we may easily
tell what it will do under any head. If the suffix (1) denotes a
value for one foot head we may then write,

N = NiVh (44)

q = qiVh (45)

h.p. = h.p.ih3/2* (46)

It must be noted that these simple laws of proportion may be
applied only when the speed varies with the head in such a way
as to keep <£ constant. The value of <f> need not necessarily be
that for the highest, efficiency. But if the speed does not change
or if it varies in some other way so that <£ is different, the results
under the new head cannot be computed save by complex equa-
tions, such as those of Arts. 87 and 88, or by the use of test curves
such as those of Figs. 95 and 96.

104. Diameter of Runner. — When a certain type of runner has
been perfected a whole line of stock runners of that type may then
be built with diameters ranging from 10 to 70 in. or more. All
of these runners will be homologous in design, that is they will
have the same angles and the same values of the ratios x, y,
and B/D. Each runner will simply be an enlargement or
reduction of another. They will then have the same character-
istics, that is, the same values of <j>e and ce) and will therefore
follow certain laws of proportion.

*The easiest way to find h** is to note that h% = h\/h. This may be
found in one setting of the slide rule.

GENERAL LAWS AND CONSTANTS 153

Since for a given head HI will have the same value for all of
them, it follows that for a series of runners of homologous design
the best r.p.m. will be inversely proportional to the diameter.

Since the discharge through any runner is equal to Aic\/2gh,
and since c will have essentially the same value for all the runners
of such a series, the discharge will be proportional to the area A i.
But if the runners are strictly homologous the area AI will be
proportional to the square of the diameter. It will therefore be
true that the discharge of any turbine of the series will be pro-
portional to the square of the diameter.

Since the power is directly related to the discharge it also fol-
lows that the power of the turbine is proportional to the square
of the diameter.

These relations are of practical value because if the speed,
discharge, and power of any runner is^nown by accurate test,
predictions may then be made regarding the performance of any
other runner of the series. These laws may not hold absolutely
in all cases because the series may not be strictly homologous,
that is the larger runners may differ slightly from the smaller
ones. Also it will no doubt be true that the efficiency of the larger
runners will be somewhat higher than that of the smaller ones.
It may also be found that careful tests of two runners made from
the same patterns will not give exactly the same results due to
difference in finish or other imperceptible matters. Despite
these factors, however, the relations stated are true enough to be
used for most purposes.

105. Commercial Constants. — For a given turbine the maxi-
mum efficiency will be obtained only for a certain value of <t>.
All tables in catalogs of manufacturers as well as all values given
in this chapter are based upon the assumption that the speed will
be such as to secure this value of 4>. Substituting values of N
and D for HI in the expression HI = <i>\/2gh, we obtain

N = 184Q.frVfe (47)

where D is the diameter of the runner in inches. From this may
also be written

</> = 0.000543 —7= (48)

Since <t>e is constant for any series of runners of homologous

design, it follows from (48) that the expression — -r=- must remain

154 HYDRAULIC TURBINES

a constant. If, then, the best r.p.m. of any diameter of runner
under any head is determined, the proper r.p.m. of any other
runner of the series under any head may be readily computed.
For the tangential water wheel:

4>. = 0.43 to 0.47

-77 = 790 to 870.
Vh

For the reaction turbine :

4>e = 0.55 to 0.90.

= = 1050 to 1600.

If values outside these limits are met with it is because the speed
is not the best or because the nominal value of D is not the true
value.

106. Diameter and Discharge. Since, for any fixed gate open-
ing and a constant value of <£, the rate of discharge of any runner
is proportional to the square of its diameter and to the square
root of the head, we may write

q = KiD*Vh (49)

The value of K\ depends upon the velocity Vi and the area A\.
The former depends upon the value of c (Art. 83), and the latter
depends upon the diameter D, the height of the runner B (Fig.
34), the value of the angle «i, and also the number of buckets
and guides.

Since there are so many factors involved, it will be seen that a
given value of KI can be obtained in several ways. For some
purposes it might be convenient to express these items by sepa-
rate constants but for the present purpose it will be sufficient
to cover all of them by the one constant.

The lowest value of KI will be obtained for the tangential water
wheel with a single jet. For this type of wheel there is evidently
no minimum value of KI below which we could not go. The
maximum value of KI is, however, fixed by the maximum size of
jet that may be used. (See Art. 30 and Art. 74.) Using this
maximum size of jet we obtain a value of KI = 0.0005. How-
ever the more usual value is about KI = 0.0003. There is seldom
any reason for using a large diameter of wheel with a small jet
and so much lower values are rare.

GENERAL LAWS AND CONSTANTS 155

With the reaction turbine the lowest values of K\ would be
obtained with type / in Fig. 34 and the highest with type IV.
The value of the area AI is proportional to the sine of a\ and nor-
mally small values of on go with small values of the ratio B/D.
Taking the usual values that go with either extreme we- get a
minimum value of KI = 0.0010 and a maximum value KI =
0.050. These are not absolute limits but they cannot be ex-
ceeded very much and to do so at all would mean to extend our
proportions of design beyond present practice. For the usual
run of stock turbines values of KI vary from 0.005 to 0.025. To
summarize :

For the tangential water wheel K± = 0.0002 to 0.0005
For the reaction turbine KI = 0.001 to 0.050

107. Diameter and Power. — Since the power of any runner is
proportional to the square of Ihe diameter and to the three-halves
power of the head, we may write

h.p. = K2D2h* (50)

As the power is directly dependent upon the discharge it is
evident that the discussion in the preceding article will apply
equally well here. K2 may be computed directly from KI if the
efficiency is known, or it may be determined independently by
test.

For the tangential water wheel K2 = 0.000018 to 0.000045
For the reaction turbine K2 = 0.00008 to 0.00450

108. Specific Speed. — In Art. 105 we have the relation between
diameter and r.p.m.; in Art. 107 we have the relation between
diameter and power. It is now desirable to establish the rela-
tion between r.p.m. and power as follows:
From (47)

N
From (50)

Substituting the above value of D in the second expression we
have

/— 18400 V/i

VK* —

156 HYDRAULIC TURBINES

Letting N8 stand for the constant factors and rearranging we
have

This expression is a very useful factor and is called the specific
speed. It is also called unit speed or type characteristic or char-
acteristic speed by various writers. Its physical meaning can be
seen as follows: If the head be reduced to 1 ft. then Ns = N \/Ti~p.
By then varying the diameter of the runner the value of N
will change in an inverse ratio, but the square root of the horse-
power varies directly as D. Thus the product of the two or Ns
remains constant for all values of D as long as the series is homo-
logous. If a value of D be chosen which will make the h.p. =
1.0 when h = 1 ft., we then have N8 = N.

That is, the specific speed is the speed at which a turbine
would run under one foot head if its diameter were such that it
would develop 1 h.p. under that head. The specific speed is
also an excellent index of the class to which a turbine belongs and
hence the term type characteristic is very appropriate. There
is no standard symbol used by all to denote this constant though
Ns is quite common. Other notations are Nu, KT, and numerous
others. In Europe the specific speed will be expressed in metric
units; to convert from one to the other multiply N, in English
units by 4.45.

It should be noted that the power to be used in this formula is
the power output of the machine. Thus the efficiency is involved
in the value of Ns, though it does not appear directly. In the
case of a Pelton wheel with two or more nozzles, the power to be
used is that corresponding to only one jet. In the case of multi-
runner units, the specific speed should be computed for the
power of one runner.

For any turbine the value of N8 is a constant, so long as the
speed of the turbine is varied as the square root of the head.
For if N varies as \/h and the power varies as h?*, it is seen that
N\/h.p. varies as hy*. Also for a series of homologous runners the
square root of the power increases with D directly while the speed
N varies inversely. Thus the factor is a constant for all turbines
of the same type.

The value of the specific speed is ordinarily computed by equa-

GENERAL LAWS AND CONSTANTS 157

tion (51), since this involves the quantities with which the
engineer is most concerned. But the great practical value of this
factor in turbine work is such as to make it worth while to derive
this expression in other ways and in terms of other quantities.

Thus

M, = irDN/720 = <t>\/2gh

From which D = 72Q<f>\/2gh/TrN (52)

Also, if B = mD,

q = (0.957rBD/144)yrl = 0.95irroZ>V\/ty£/144 (53)

where 0.95 is a factor to compensate for the area taken up by the
runner vanes.
Since B.h.p. = wqhe/550

B.h.p. = 0.95 wir^/Zgm D2cr.h- e/144 X 550 (54)

Eliminating D between the simultaneous equations (52) and (54)
and reducing, we have (giving </> the special value <f>e)

N, = -.. _ 252*.v%XwX.« (55)

This equation shows how the value of the specific speed may be
varied in the design by means of the factors <£e, cr, and m.3
An instructive form, however, is that of Lewis F. Moody, in
which the diameter of the draft tube is represented as nD, and
the discharge velocity head V22/2g = Lh, where L is the frac-
tional part of the head h that is lost at discharge from the runner.
(Of course an efficient draft tube is relied upon to recover a part
of this). With these we may write

' (56)

Substituting this expression for q in that for horsepower, we
obtain

B.h.p. = ww\/2gn2D2\/Lhe/4: X 144 X 550 (57)

Eliminating D between the simultaneous equations (52) and (57)
and reducing, we have

N. = - - 129.5n*.VvEv7 (58)

/l/4

xln a similar manner the specific speed for a Pelton wheel may be shown
to be, N» = 1290e\/clle^ where d = jet diameter in inches. Since, for the
impulse wheel <f>e and cw are practically constant this may be reduced to
N. = 53.7 i

158 HYDRAULIC TURBINES

While the equation in this form shows how the specific speed may
be varied in design by changing the factors n, <j>e, and L, its chief
use is in showing the limit which the specific speed approaches.
Thus to increase the value of Na, the ratio n may be increased.
But it will soon reach a definite limit. The factor <j>e may also be
increased, but it also will reach a definite limit, which is something
under 1.0. The efficiency cannot readily be increased any more
than for lower specific speed runners and as a matter of fact, is
already decreasing. Thus after these factors have reached their
maximum limits, so that they may be assumed to be constant, the
only means of increasing Na any further would appear to be by
increasing L. Thus

Ns cc vVl or L oc N44 (59)

But after this limit is passed so that equation (59) applies,
the outflow loss increases much faster than the specific speed.
Even with the best of draft tubes a certain percentage of L
must be lost eventually and hence e is rapidly reduced. The
outflow conditions thus impose a maximum limit upon N,.

For the lower values of Ns the outflow loss becomes of small
consequence, but other factors then enter. The chief of these are
the leakage losses and the disk friction. For with small values
of the specific speed the runner becomes relatively large in
diameter and correspondingly narrow. The area of the spaces
through which water can leak becomes of greater percentage as
compared with the area through the runner. And the percentage
of the power consumed in rotating the large diameter runner
through the water in the clearance spaces becomes of increasing
importance. If we assume that the power lost in disk friction
varies as DbN3, it may be readily shown by combining this with
equations (51) and (52) that the power so lost varies as <l>b./N82.
After <t>e has been reduced to its minimum, which approaches
0.50 as a limit, any further decrease in N, increases the disk fric-
tion loss much more rapidly. Also as 4>e is decreased ce must in-
crease (approaching unity as a limit), as shown by equation
(39), and consequently pi decreases (approaching zero as a limit).
But this is undesirable, due to the danger of oxidation of parts of
the runner.

In view of these facts, it may be shown that the minimum allow-
able value for the specific speed of a reaction turbine is about 10,

GENERAL LAWS AND CONSTANTS* 159

though there are a few extreme cases of design that have 'carried
it as low as about 9. With present draft tube construction, the
maximum limit for the specific speed of a reaction turbine is
about 100, though values as high as 130 are attainable at some
sacrifice of efficiency. The usual range in practice varies from
about 20 to 80.

A very recent type of turbine runner proposed by Nagler is
of an axial flow type and is similar to a screw propeller. The
present specific speed of this type is 165 and it is possible that
this may be extended in the future.1

The impulse turbine runs in air and thus the disk friction loss
for it becomes windage loss, which is of less consequence. There
can be no leakage loss with this type and also the reduction
of the pressure to atmospheric gives rise to no trouble. Hence
this type of turbine is suitable for specific speeds below those for
the reaction turbine. For the tangential water wheel there is
no definite lower limit to its specific speed, save that the windage
loss affects it in a similar manner to the disk friction in the case
of the low-speed reaction turbine. But as the specific speed of a
Pelton wheel is increased the size of the jet must become larger
in proportion to that of the wheel and for the reasons already
given there is a limit to this. The further increase in ratio of
jet diameter to wheel diameter causes the efficiency to rapidly
decrease, due to loss of water past the buckets. There have been
cases of tangential wheels with specific speeds of less than 1 and
maximum values of 6, though the latter involves some sacrifice
of efficiency. The usual range in practice is from 3 to 4.5.

It will be seen that there is a gap in the values of N8 between
the tangential water wheel and the reaction turbine. Similar
gaps are also found for the values of <£e, KI, and K2. In'Europe a
few two-stage radial inward-flow reaction turbines have been
built and these could have lower values of the specific speed than
10. And by the use of two or more nozzles on one impulse wheel
runner, the value of N8 for the tangential wheel can be increased
above the 5 or 6 set 'as the limit for the single nozzle. Thus the
entire field can be covered.

To recapitulate:

For the tangential water wheel N8 = 3.5 to 4.5 (6 max.)
For the reaction turbine N8 = 10 to 100.

Jour, of the Amer. Soc. of Mech. Eng., Dec., 1919.

160

HYDRAULIC TURBINES

109. Determination of Constants. — The constants given in
this chapter may be computed from theory, but for practical
use should be secured from test data. The catalogs of turbine
manufacturers usually contain tables giving the discharge,
power, and speed of different diameters of runners under various
heads. As these tables are supposed to be based upon tests
they may be used for the determination of these factors. If
all the runners of the series were strictly homologous it would
be necessary to compute these constants for one case only.
Actually variations will exist with different diameters of runners
and thus there will be some variation in the values secured.
Since each manufacturer usually makes several lines of runners
so as to cover the field to better advantage, there will be as many
distinct values of these constants as he makes types of runners.
If the catalog tables are purely fictitious then the computations
based upon them will not be very reliable.

110. Illustrative Case. — In order to illustrate the preceding
article the following tables are given. For the sake of compari-
son only two firms out of many are chosen for this case. The
values given are based upon catalog tables. Since K% depends
upon Ki it has been omitted to save space.

TABLE 3. — JAMES LEPFEL AND Co.

Type

<i>

Standard

0 722-0.727

0.0061-0.0064

30.8-32.6

Special

0.750-0.779

0.0094-0.0097

41.6-43.2

Samson

0.838-0.844

0.0170-0.0171

61.5-61.9

Improved Samson ....

0.856-0 886

0 0220-0 0220

71.0-73.5

TABLE 4. — DAYTON GLOBE IRON WORKS Co.

Type

<t>

/Ci

High head type

0.578-0.585

0.0051-0.0064

22.8-26.0

American

0.662-0.704

0.0054-0.0080

25.0-32.3

Special New American
Improved New American. . .

0.697-0.727
0.886-0.944

0.0175-0.0205
0.0233-0.0263

50.0-57.4
78.2-80.5

This table shows the variation in constants that might be ex-
pected, and shows also how each firm attempts to cover the
ground. It will be noticed, however, that the two do not agree
in all respects. Thus suppose a turbine was desired whose speci-

GENERAL LAWS AND CONSTANTS 161

fie speed was 42. The "Special" turbine of the Leffel Co. would
fulfill the conditions, but the Dayton Globe Iron Works Co.
have no line of turbines that would exactly answer the require-
ment. The latter firm might furnish a turbine that would have
the required specific speed but it would have to be a special
design — it would not be a stock turbine, and would therefore
be more expensive.

111. Uses of Constants. — After these factors are determined
it will then be easy to find what results may be secured for any
size turbine of the same design under any head. Another use
for them is that when the limits are fixed they will enable one
to tell what is possible and what is not. In the next chapter
it will be shown how they are of direct use also in the selection
of a turbine.

112. NUMERICAL ILLUSTRATIONS

1. The test of a 16-in. runner under a 25-ft. head gave the following as
the best results: N = 400, q= 17.5 cu. ft. per second, h.p = 39.8. Find
the constants.

From (48) </> = 0.000543 16'X400 = 0.696

• ' • 1 1 • o

From (49) Kl = ^J^g = 0.01368

39 8
From (50) K* = lffl x 12g = 0.00124

From (51) N. = ***£*** = 45'2

2. Suppose that a 40-in. runner of the same design as in problem (1) is
used under a 150-ft. head. Compute the speed, discharge, and horse-power.

1840 X 0.696 X 12.25
From (47) N = - 4Q - = 392 r.p.m.

From (49) q = 0.01368 X 1600 X 12.25 = 268 cu. ft. per second
From (34) 0.00124 X 1600 X 1838 = 3650 h.p.

3. Suppose that turbines of the type in problem (1) were satisfactory for
a certain plant but that the number of the units (and consequently the
power of each) and the speed has not been decided upon. If the head is
150ft., then by (51)

N xVh'.p'. = 45.2 X 525 = 23,730.

By the use of different diameters of runners of this one type the following
results can be secured :

14, 100 h.p. at 200 r.p.m.

6,250 h.p. at 300 r.p.m.

3,520 h.p. at 400 r.p.m.

2,250 h.p. at 500 r.p.m.

162 HYDRAULIC TURBINES

1,560 h.p. at 600 r.p.m.

,. 1,150 h.p. at 700 r.p.m.

878 h.p. at 800 r.p.m.
695 h.p. at 900 r.p.m.

If the capacity of the plant were 25,000 h.p. it might then have 4 units at
300 r.p.m., 16 units at 600 r.p.m., or 36 units at 900 r.p.m. If none of the
possible combinations were suitable it would be necessary to use another
type of turbine — that is one with a different value of Ns.

By equation (50) the diameters are found to be 52.3 in., 26.2 in., and
17.5 in. for 300, 600, and 900 r.p.m. respectively.

4. Compute values of <f>, Ki, Kz, and N8 for each of the turbines whose
tests are given in Appendix C: (a) for the point of highest efficiency, (6)
for the point of maximum power.

113. QUESTIONS AND PROBLEMS

1. How do the speed, rate of discharge, power, and efficiency of a turbine
vary with the head, the value of 0 remaining constant? Why?

2 . Suppose the speed of a turbine remains constant while the head changes,
how will the rate of discharge, power and efficiency vary? What is neces-
sary in order to answer this question?

3. How do the speed, power, and efficiency vary with the diameter of a
series of homologous runners? Why? How do these quantities change
when both the head and diameter are different, the runners being of the
same type, however?

4. What is the physical meaning of the term "specific speed?" Why are
the terms "type characteristic" and "characteristic speed" also appropriate?
How may the value of this factor be changed in the design of the runner?

5. What limits, the maximum and minimum values of the specific speed
for reaction turbines? For impulse wheels? Why do the latter have
lower specific speeds than the former?

6. If a turbine gives an efficiency of 82 per cent, when tested under a
head of 10 ft. what would you estimate its efficiency to be if installed under
a head of 100 ft.? Under a head of 225 ft.? If the test of a 27-in. runner
under a head of 150 ft. gives, as the best results, N = 600, q = 40, h.p. =
550, what will be the speed, rate of discharge, and power of a 54-in. runner
of the same type under a head of 50 ft. ?

7. If a turbine is desired to run at 300 r.p.m. under a head of 60 ft., what
are the minimum and maximum diameters of runners that might be used?
If 30 cu. ft. of water per second is to be used under a head of 60 ft., what
range of diameters might be employed?

8. Suppose that a type of turbine, whose specific speed is 80, is suitable
for use in a certain plant where the head is 16 ft. What combinations of
h.p. and r.p.m. are possible?

9. If a tangential water wheel was desired to deliver 1000 h.p. under
150 ft. head, what r.p.m. could be used? How high a speed could be ob-
tained with a reaction turbine?

GENERAL LAWS AND CONSTANTS 163

10. Wouldtit be possible to obtain a 5000 h.p. turbine to run at 600 r.p.m.
under a 50-ft. head? What could be done to secure that power? To secure
600 r.p.m.?

11. A reaction turbine is designed so that <f>e = 0.72, B/D = 0.29, cr =
0.20, and the efficiency may be assumed = 0.88. What is the value of the
specific speed? Compute the probable values of the specific speeds for the
four types of turbines shown in Fig. 34, making whatever assumptions are
necessary.

12. The turbine, whose dimensions are given in problem (11) had a value
of n = 1.04. Find the per cent, of the total head that is equal to the velocity
head at discharge from the runner?

CHAPTER XII
TTRBINE CHARACTERISTICS

114. Efficiency as a Function of Speed and Gate Opening.
In Fig. 87, page 112, it has been shown how the power, and hence
the efficiency, of an impulse turbine varies with the speed for

Values of 0
FIG. 107. — Characteristics of high-speed runner.

Ng = 93.

any gate opening; and in Fig. 91, page 115, how the efficiency
varies with the power at different gate openings at a uniform
speed, the head being constant in both cases. Similar curves
for a reaction turbine are shown in Figs. 96 and 98.

It should be noted that the value of <£ for the highest efficiency

164

TURBINE CHARACTERISTICS

165

at one gate opening is not the same as that for any other gate
opening. This is best shown by Fig. 107, and reasons for it
are given in Arts. 76 and 91. Hence the speed that is most effi-
cient for one gate opening is not exactly the best for any other
gate opening. Also the values of efficiency vary for the different
gate openings and the maximum efficiency will be found at some-
thing less than "full" gate.1 Therefore, in general, the maxi-
mum efficiency and the maximum power are found at different
gate openings and different speeds.

0.8 1.0 1.2 1.4 1.6 L8 2.0 2.2 2.4 2.6 •
Brake Horse-Power under 1 Ft. Head
FIG. 108. — Efficiency-power curves for different speeds under same head.

Thus in Fig. 107 the maximum value of the efficiency is found
at 0.820 gate and at such a speed that <j>e = 0.780, but the maxi-
mum power is found at 1.103* gate and at such a speed that
0 = 1.03. The efficiency in the. former case is 0.88 and in the
latter 0.77. In Fig. 108 are shown efficiency curves as a function
of power for values of 0 = 0.64, 0.78, and 1.03.

1 This statement does not hold in the case of the cylinder gate turbine,
where maximum power and maximum efficiency coincide at full gate, but
this type is of little importance at present.

* The numbers indicating the extent of the gate opening are purely
arbitrary and 1.0 does not necessarily indicate the maximum gate opening.
This will be explained subsequently.

166

HYDRAULIC TURBINES

In selecting the proper speed for a given turbine a number of
operating factors must be taken into consideration. If the head
is constant and the load is constant, it may be possible to operate
the turbine near the point of maximum efficiency most of the
time. In this case it might be desirable to select the speed giv-
ing the maximum efficiency, or 0 = 0.78 in the case above. But
if the load is variable and especially if it is apt to be light for long
periods of time a lower value of the speed might give a higher
average efficiency, though the peak is not so high. On the other
hand it may be deemed worth while to sacrifice efficiency for the
sake of capacity and increased speed, which could be attained by
using the higher values of the speed. It should be borne in mind
that some of these results might be better attained with another
type of turbine, but the latter is a subject for consideration in

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Brake Horse Power

FIG. 109. — Power and efficiency of a turbine at constant speed under different

heads.

the next chapter. We are here studying the possibilities of
a single turbine or at- least a single type of turbine.
, It must be remembered, as explained in Art. 20, that the head
is apt to vary for many water power plants, especially those under
low head. If the head decreases in time of flood, the power
output of the turbine may be seriously reduced. Under these
circumstances the important consideration is the maximum
power output. Since there is a superabundance of water for the
time being, efficiency is a secondary consideration. While
efficiency under the normal head is of importance, it might be
sacrificed to some extent in favor of a speed which would be such
as to give the maximum power under flood conditions. It is
here assumed that, whatever speed be selected, the turbines

TURBINE CHARACTERISTICS 167

must run at that constant speed regardless of fluctuations in the
water. If the speed is constant, it is seen that (f>\^h = constant.
Thus if the head decreases, the value of <£ must increase. In
Fig. 109 may be seen the performance of a turbine at a confetant
speed under different heads. By making a different choice of
the speed for the normal "head, the results under all other con-
ditions will be altered, and careful study must be made of all
the variables to decide what is best.

115. Specific Speed an Index of Type. — Both the elements of
speed and capacity are involved in the specific speed. It was
stated in Art. 38 that both speed and capacity were merely
relative terms; that is, a high-speed turbine is not necessarily
one which runs at a high r.p.m., but one whose speed is high
compared with other turbines of the same power under the same
head. In like manner a high-capacity turbine is not necessarily
one of great power but merely one whose power is high compared
with others at the same speed under the same head. Since

Ns = - ,y^p' it is evident that a low-speed, low-capacity turbine

will be indicated by a low value of N8 and a high-speed high-ca-
pacity turbine by a high value of Na. As stated in Art. 108,
values of N9 for the tangential water wheel may run up as high
as 5 or 6, for the reaction turbine they range from 10 to 100.
Values in the neighborhood of 20 indicate a runner such as Type
I in Fig. 34, while values in the neighborhood of 80 indicate
Type IV. Thus when the speed and horsepower of any turbine
under a given head are specified the type of turbine necessary
is fixed.

Other things being equal, it is seen that a high head means a
comparatively low value of Na while a low head means a high
value. Aside from any structural features it is apparent that a
high head calls for a tangential water wheel or a low-speed re-
action turbine, while a low head demands a high-speed reaction
turbine. However, the head alone does not determine the value
of N». So far as the r.p.m. is concerned there may be consider-
able variation, yet neither a very low nor a very high r.p.m.
is desirable and for the present purpose we may suppose that it
is restricted within narrow limits. The value of N» will thus
be affected by the power of the turbine as well as the head. If
the head is high the value of Nt may still be high enough to
require a reaction turbine. Or if the head is very low and the

168 HYDRAULIC TURBINES

power is likewise low a low value of specific speed may result. It
is thus clear that the choice of the type of turbine is a function
of the power and speed as well as the head.

Since a given turbine under a fixed head may be run at different
speeds and gate openings, there are any number of values of N
and h.p. that may be substituted in equation (51), with a result-
ing variety of values of Ns for the parlicular turbine. It is thus
necessary to define the speed and power for which this factor is
to be computed, if it is to have a definite value for a given runner.
The current practice is to rate turbines at the maximum guar-
anteed capacity, the actual maximum capacity being usually
slightly greater than this, since the builder allows a small margin
to insure his meeting the guarantee. The nominal specific
speed is that corresponding to this rated capacity at a stated
speed. But under a given head the turbine speed might be
selected from a limited range of values, as explained in Art. 114.
It may be seen that, though the true maximum power of the
turbine is a definite value, the actual maximum power it can de-
liver at full gate, under the operating conditions, depends upon
the speed at which it is run. Hence the value of N8, as thus
computed, varies with the speed, and is not a perfectly definite
value. Despite this, the value of specific speed is usually so
computed because the rated capacity is often known when the
power and speed for maximum efficiency are not.

For accurate comparisons of one turbine with another and for
exact work, it is best to select the values of power and speed for
which the true maximum efficiency is obtained. The value of
Ns, so computed, may be called the true specific speed. Since
this is based upon a single definite point, there can be but one
value for the turbine.

116. Illustrations of Specific Speed.— For a turbine of 2000
h.p. at 1000 r.p.m. under 1600 ft. head the value of Ns is 4.42.
Thus a very low-speed turbine, the tangential water wheel, is
required. The actual r.p.m., however, is high.

For a 5000 h.p. turbine at 100 r.p.m. under 36 ft. head Ns
equals 80.3. Thus a high-speed reaction turbine is indicated,
though the actual r.p.m. may be relatively low.

Suppose that a 12-h.p. turbine is to be run at 100 r.p.m. under
a 36-ft. head, the value of the specific speed is 3.95, which means
a tangential water wheel. For the larger power under the same
conditions in the preceding example a reaction turbine was re-

TURBINE CHARACTERISTICS 169

quired. If the speed were 600 r.p.m., however, a low-speed re-
action turbine would be necessary for N8 would equal 23.6.

Suppose that a 20-h.p. turbine is to run at 300 r.p.m. under
a 60-ft. head. The value of N8 is 8.04 and that would require
a tangential water wheel with two nozzles.

If 10,000 h.p. is required at 300 r.p.m. under a 60-ft. head,
the value of N8 would be 179.5. As this is an impossible value
it would be necessary to reduce the speed or to divide the power
up among at least 4 units of 2500 h.p. each.

117. Selection of a Stock Turbine. — The choice of the type
of turbine will be taken up in the next chapter. For the present
suppose that required values of speed and power under the given
head are determined. The value of the specific speed can then
be computed and will indicate the type necessary. If the tur-
bine is to be built as a special turbine nothing more is to be done
except to turn the specifications over to the builders.

If, however, the turbine is to be selected from the stock run-
ners listed in the catalogs of tjie various makers, it will be
necessary to find out what firms are prepared to furnish that
particular type of runner. It would be a tedious matter to
search through a number of tables in numerous catalogs to find
the particular combination desired, but 1 he labor is avoided by
the use of the constants given in the preceding chapter. It will
be necessary merely to compute values of specific speeds of tur-
bines made by different manufacturers. This can be quite readily
done and such a table will always be available for future use.

A make of turbine should then be selected having a value of
N8 very near to the value desired. The value of N8 ought to
be as large as that required, otherwise the turbine may prove
deficient in power, and for the best efficiency under the usual
loads it should not greatly exceed the desired value. Having
selected several suitable runners in this way, bids may be called
for. These bids should be accompanied by official signed re-
ports of Holyoke tests of this size of wheel or the nearest sizes above
and below, if none of that particular size are available. This
is to enable us to check up the constants obtained from catalog
data and to verify the efficiencies claimed. Holyoke test data
is very essential if the conditions of the installation are such that
an accurate test is not feasible. In making a final choice other
factors would be considered such as efficiency on part load, and
efficiency and power under varying head.

170

HYDRAULIC TURBINES

118. Illustrative Case. — Suppose a turbine is required to
develop 480 h.p. at 120 r.p.m. under 20-ft. head. The value
of N8 is then 62.2. There are four makes of turbines which
approach this as follows :

TABLE 5

Maker

Type

Camden Water Wheel Wks. . .
James Leffel and Co

United States Turbine
Samson

64.7
61.8

0.00190
0.00158

Platt Iron Works

Victor Standard

63.0

0.00205

Trump Mfg. Co

Standard Trump

61.5

0.00210

It is thus apparent that any one of these manufacturers could
supply a turbine from their present designs which would nearly
fill the requirements. A number of other firms in this country
could not fit the case except with a special design or a modifica-
tion of an existing design. Thus inspection of the table for the
Dayton Globe Iron Works Co. in Art. 110 will show that the
nearest approach they have to it is their Special New American
with an average value of Ns of 53.7. They could supply a tur-
bine to run at 120 r.p.m. under the head specified, but it would
develop only 358 h.p. Or if they supplied a turbine capable
of delivering 480 h.p. it should run at 103.5 r.p.m.

Turning to the four cases presented in the table, it is apparent
that the Camden wheel is a little over the required capacity,
but it may not be enough to be objectionable. The Platt ron
Works wheel is very little over the required capacity and the
Leffel and Trump are a trifle under it. If there is a little margin
allowable in the power, any of these might be used. The value
of N8 according to which the wheel is rated should be the value
for the speed and power at which it develops its best efficiency.
In any plant the variation in the head produces a deviation from
the best value of <j>, if the wheel be run at constant speed, and
thus causes a drop in efficiency. The power of the wheel may
increase or decrease according to the way the head changes.
Thus in actual operation the conditions depart so much from
those specified in the determination of Ns that small discrep-
ancies in its value such as exist in the table are of little im-
portance.

If desired, the diameters of the runners may be determined
by means of K* For the four cases in the order given they will

TURBINE CHARACTERISTICS

171

1.00

be 53.2 in., 58.2 in., 51.2 in., 50.3 in. Actually standard sizes
will not agree precisely with these figures and thus a further
modification may be brought about. However, mathematical
exactness must not be expected in work of this nature. What
we attempt to do is merely to select a turbine the peak of whose
efficiency falls as near as possible to the conditions of head,
speed, and power chosen. Al-
though our conditions may be
such that we may rarely realize
the very highest efficiency of |
which the turbine is capable, *j
yet we should be very close ~
to it. §

119. Variable Load and J
Head. — In any plant the load '
is usually not constant but
varies over a considerable
range. In comparing turbines
for certain situations the aver- ^
age operating efficiency may J
be more important than the |
efficiency on full load only.
If the turbine is to run on full
load most of the time or if the
installation is such that the
pondage is limited or lacking
altogether, then the efficiency |
on part load is of little im- £
portance. But if the load is ~

= 1.00

R.P.M. 1 Ft. Head
FlG. 110.

variable and if water can be

stored up during the time the

wheel may be running under a

light load then the efficiency at

all times becomes of interest.

If the plant has a number of

units it is possible to shut down some of them at times so as to

keep the rest on full load.

In most low head plants the variation in head is a serious item
also and the turbines submitted should be compared as to their
efficiency and power under the range of head anticipated.

All turbines having the same specific speed are not necessarily

172

HYDRAULIC TURBINES

loo jj artQ Japun'fflM'

C> IA O tO O "3 O iOOOOt<3OO»u3

dO S £ «5 S to trt ^f ^l 1 t 1 M. *1 rt. ^

TURBINE CHARACTERISTICS 173

equally well suited for the same conditions. A detailed study of
the characteristics of each one is essential before the final choice
can be made. In many cases the best efficiency will be the
deciding factor. In others the average operating efficiency will
be more important, and sometimes capacity under varying head
will be the chief item.

These factors can be studied by means of curves such as are
shown in Fig. 110. Efficiency, discharge, and power for various
gate openings reduced to 1-ft. head are plotted against (f> or the
r.p.m. under 1-ft. head. The normal speed and power should be
that corresponding to the maximum efficiency. If the wheel is
run at constant speed a variation in head causes a change in <£.

120. Characteristic Curve. — For a thorough study of a turbine
the characteristic curve is a most valuable graphic aid. The
coordinates of such a curve are discharge under 1-ft. head
and 0 or r.p.m. under 1-ft. head. Values of the horsepower
input under 1-ft. head should also be laid off to correspond
to the values of the discharge. Lines should then be drawn on
the diagram to indicate the relation between speed and discharge
for various gate openings. Alongside of each experimental
point giving this relation, the value of the efficiency should be
written. When a number of such points are located, lines of
equal efficiency may be drawn by interpolation.

Another very good rnethod is to draw curves of efficiency as a
function of 0 for each gate opening. For any iso-efficiency curve
desired on this diagram it is possible to read off corresponding
values of <£.

If desired, lines of equal power may also be constructed. To
do so, assume the horsepower of the desired curve, then compute
the horsepower input for any efficiency by the relation, horse-
power input = horsepower output -f- e. This value of e on one
of the iso-efficiency curves together with the value of horse-
power input locates one or two points of an iso-power curve.

The characteristic curve for a 24-in. tangential water wheel is
shown in Fig. 111. This curve covers all the possible conditions
under which the wheel might run. The only way to extend the
field would be to put on a larger nozzle. Since the discharge of a
tangential water wheel is independent of the speed the lines for
the various gate openings will be straight. For the reaction
turbine they will be curved as seen in Fig. 112. The latter is a
portion of a chracteristic curve for a high-speed turbine.

174

HYDRAULIC TURBINES

Any marked irregularities in the characteristic curve are
indications of errors in the test. It is possible for there to be
only one peak in the efficiency curves and an indication of two,
as sometimes occurs, is due to incorrect data.

This method of plotting test data and also that shown in Fig.
110 were first given by Prof. D. W. Mead in his " Water Power
Engineering." Other diagrams have also been proposed by
various men, the object of all being to represent the fundamental
variables in the best form for the ready comparison of one turbine
with another.

1.4

H.E» Input under One Foot Head
1.6 1.8 2.0 . 2.2 2.4

2.6

1.00 —

0.95 =

0.70 -

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Discharge? in Cu.Ft. per Sec. under One Foot Head,.
FIG. 112. — Characteristic curve for a high speed reaction turbine.

121. Use of Characteristic Curve. — From the characteristic
curve it is apparent, at a glance, at what speed the turbine should
.run for the best efficiency at any gate opening. The best effi-
ciency in Fig. Ill is obtained when $ = 0.457 or Ni = 34, and
with the needle open 6 turns. With full nozzle opening the best
value of Ni is 35, with the needle open 3 turns the best speed is
such that Ni = 32. (With the reaction turbine these differences
would be greater.)

From the characteristic curves any other curves may be con-
structed. For constant speed follow along a horizontal line, for
a fixed gate opening follow ^long the curve for that relation.

If it is desired to investigate the effect of change of head when

TURBINE CHARACTERISTICS 175

the speed is kept constant, compute the new values of <£ or Ni.
Thus the curve in Fig. Ill was determined by a test under a head
of 65.5Jft. and the best speed was 275 r.p.m. That corresponded
to <f> = 0.457 or Ni = 34. (The value of D used for computing <£
was slightly different from the nominal diameter.) If the speed
is maintained at 275 r.p.m. when the head is 74 ft., then </> = 0.429
and Ni = 32. If h = 55 ft., 0 = 0.497 and Ni = 37. In the
last case the best efficiency would be 77 per cent., a drop of 1 per
cent.

The iso-efficiency curves represent contour lines on a relief
model and thus the point of maximum efficiency is represented
by a peak. It is apparent that for varying loads or heads a tur-
bine giving a diagram, that indicates a model with gentle slopes
from this point, would probably be better than a turbine for
which the peak might be higher and the slopes steeper. The rela-
tiVe increase in discharge capacity at full gate as 0 increases is also
apparent and indicates which turbine is better for operation at
reduced head but normal speed.

122. QUESTIONS AND PROBLEMS

1. For any turbine, how does the speed for highest efficiency vary with
the gate opening used? How does the efficiency vary with the gate opening
for any speed? At what speed and gate will maximum efficiency be found,
as compared with maximum power?

2. Should a turbine necessarily be run at the speed for maximum effi-
ciency? Why?

3. What happens to the power and efficiency of a turbine when the head
changes, but the speed is kept constant? In time of flood, what is the im-
portant consideration?

4. What is the difference between the true and the nominal specific
speed? What would be the general profile of a runner whose specific speed
was 10? What of one whose specific speed was 100?

6. When is efficiency on full load important and when is efficiency on
part load of more value? When is maximum power of principal interest?
When is maximum speed the chief object?

6. If a plant contains a number of units, what should be done if all of
them are carrying half load? Why? Would there be any object in shut-
ting down some of them if the supply of water was abundant?

7. If there is a great shortage of water so that the supply is inadequate
for all the wheels at full head, so that the water level falls considerably below
normal before equilibrium is attained, is it better to operate the plant with
all the wheels or shut down enough of them to keep the water level near the
crest of the spillway? Would it be better to shut down some of the wheels
to accomplish this or to operate all of them at part gate opening?

176 HYDRAULIC TURBINES

8. In Fig. 112, what is the maximum efficiency and what is the value of
the efficiency for maximum power? What is the ratio of the speeds for
each of these, and what is the ratio of the power outputs at full gate for
each of these speeds?

9. If the turbine, whose performance is shown by Fig. 112, is run at the
speed for best efficiency under a head of 50 ft., what will be its maximum
output and what the power at the point of maximum efficiency? If the
speed is kept at this same value while the head falls to 36.5 ft., what will
be the value of the maximum power delivered?

Ans. 755 h.p., 720 h.p., 498 h.p.

10. In problem (9) the second value of the head is 73 per cent, of its
initial value and the maximum power is 66 per cent, of its value in the first
case. For the impulse turbine in Fig. Ill, what would be the ratio of the
maximum power outputs, if the head dropped the same proportional amount
while the speed remained the same as for the maximum efficiency under the
initial head? Ans. 58.7 per cent.

11. Suppose that an impulse wheel, similar to that for which the curves
of Fig. Ill, were drawn, is made of such a size as to develop 5000 h.p.
under a head of 1200 ft. Find the diameter of the wheel, its r.p.m., and
plot a curve between efficiency and power for a constant speed.

CHAPTER 'XIII
SELECTION OF TYPE OF TURBINE

123. Possible Choice. — It has been shown that, if the speed
and power under a given head are fixed, the type of turbine
necessary is determined. If there is some leeway in these mat-
ters it may be possible to vary the specific speed through a
considerable range of values. Suppose turbines of a given power
may be run at 120 r.p.m., at 600 r.p.m., or at 900 r.p.m. Each
one of these would give us a different specific speed and thus a
different type of runner. Or, if the speed be fixed, the power,
such as 20,000 h.p. may be developed in a single unit, in two
units of 10,000 h.p. each, or in eight units of 2500 h.p. each.
Again we have different types of runners demanded. Both the
speed and power may be varied in some cases and the choice is
wider still. As an example, it may be required to develop
500 h.p. under 140-ft. head. Suppose this power is to be divided
up between two runners and the speed to be 120 r.p.m. The
value of Ns is then 4.12, showing that a double overhung tan-
gential water wheel is required. Or if the power be developed
in a single runner at 600 r.p.m., the value of N8 would be 29.2,
which would call for a reaction turbine.

It is customary to choose a speed between certain limits, as
neither a very low nor a very high r.p.m. is desirable. Also the
number of units into which a given power is divided is limited.
Nevertheless considerable latitude is left. It remains to be seen
what considerations would lead us to choose such values of speed
and power as would permit the use of a certain type of runner.

124. Maximum Efficiency. — The best efficiency developed
by a turbine will depend, to some extent, upon the class to which
it belongs. The impulse and reaction turbines are so different
in their construction and operation that the difference in effi-
ciency between them can be determined solely by experiment.
However, abstract reasoning alone will lead to certain conclusions
as to the relative merits of different types within each of these
two main divisions.

12 177

178 HYDRAULIC TURBINES

For the tangential water wheel it has been shown that, if the
highest efficiency is to be obtained, certain proportions must not
be exceeded. If we desire a specific speed higher than 4, it is
necessary to pass beyond these limits and thus a wheel whose
specific speed is as high as 5 or 6 will not have as high an efficiency
as the normal type. On the other hand too low a specific speed
is not conducive to efficiency, since the diameter of the wheel
becomes relatively large in proportion to the power developed,
so that the bearing friction and windage losses tend to become
too large in percentage value. The value of Ns for the highest
efficiency is about 4.

A low specific speed reaction turbine, such as Type I in Fig. 34
for example, will have a small value of the angle a\. A considera-
tion of the theory, especially equation (33), shows that this is
conducive to high efficiency. However this is more than offset
by other factors, such as the large percentage value of the disk
friction, as explained in Art. 108. In addition, the leakage area
through the clearance spaces becomes a greater proportion of
the area through the turbine passages, and also the hydraulic
friction through the small bucket passages is larger. The result
of all these factors is that the efficiency tends to be reduced as
very small values of the specific speed are approached.

A medium specific speed turbine runner would have a some-
what larger value of the guide vane angle but this slight dis-
advantage would be more than offset by the reduction in the
relative values of the disk friction, leakage loss, and hydraulic
friction loss within the runner. Thus this type would have a
higher efficiency than the former.

But when the high specific speed type is reached the inherently
large value of the discharge loss is such as to materially reduce
the efficiency. This reduction is aided also by the large value
given to the guide vane angle and opposed by the decreased disk
friction, leakage through the clearance spaces, and internal
hydraulic friction. However the effect of these latter factors
is not sufficient to offset the increased discharge loss. In other
words, efficiency has been sacrificed in favor of increased speed
and capacity, just as in the case of a high-speed impulse turbine.

This reasoning is borne out by the facts, as can be seen by Fig.
113, where efficiency is plotted as a function of specific speed.1
A number of test points were located and the curve shown was

i L. F. Moody, Trans. A. S. C. E., Vol. LXVI, p. 347.

SELECTION OF TYPE OF TURBINE

179

drawn through the highest on the sheet. It shows what has
actually been accomplished and it also shows how the maximum
efficiency varies with the type of turbine. It is apparent that if
one desires the highest efficiency possible a specific speed should
be chosen between 25 and 50.

It must not be thought that this curve represents the results
that one should expect in every case. It merely shows the rela-
tive merits of the different types. The actual efficiency obtained
depends not only upon the specific speed but also upon the ca-
pacity of the turbine and the head and other factors. The larger
the capacity of a turbine the higher the efficiency will be. In a
given case the efficiency obtained for a specific speed of 30, say,
might be only 83 instead of the 93 shown by the curve. But if
the specific speed had been 95 instead of 30 the efficiency realized
might have been only 73.

30 40 50 60 70

Values of the Specific Speed - M8

FIG. 113.

100

Higher efficiencies have been attained with reaction turbines
than with Pelt on wheels. The maximum recorded efficiency
for the former is 93.7 per cent, and quite a few large units
have shown efficiencies over 90 per cent, where conditions were
favorable. The highest reported value for an impulse wheel
is V89 per cent, but the usual maximum is about 82 per cent.
However the efficiency of a reaction turbine is a function of
its capacity, that is for small sizes the efficiency is relatively
low. As the larger sizes are reached this difference dis-
appears. The reason for this is that the clearance spaces
and hence leakage losses are a greater percentage with the
small sizes. The efficiency of the Pelton wheel is not depend-
ent on its size. Hence for smaller powers the tangential wheel
may have a higherj[maximum efficiency than the reaction
turbine.

125. Efficiency on Part-load. — Full-load will be defined as the
load under which a turbine develops its maximum efficiency.

180 HYDRAULIC TURBINES

Anything above that will be called an overload and anything
less than that will be known as part-load.

It has already been explained (Art. 84) that to obtain the high-
est efficiency the water must enter without turbulent vortex
motion (known as shock) and must leave with as little velocity
as possible. In order to obtain the former the vane angle jS'i
must agree with the angle of the relative velocity of the water as
determined by the vector diagram, and the quantity of water
should be such that its relative velocity t/i, as determined by the
equation of continuity, should agree with the velocity v\y as
determined by the vector diagram of velocities. In order to re-
duce the discharge loss to a minimum it has also been shown that
a2 should have a value of approximately 90°.

There is practically no additional loss at entrance to the buck-
ets of a Pelton wheel due to the reduction in the size of the jet
at part-load. If the jet and wheel velocities remained just the
same, the velocity diagrams would be identical at all loads.
Actually the jet velocity may vary slightly but the shape of the
buckets is such that there is no well denned vane angle at en-
trance. And since, in the impulse turbine, the relative velocity
through the runner is not determined by the equation of con-
tinuity, there can be no abrupt change in either the direction or
magnitude of the relative velocity of the water at entrance. But
this is not the case with the reaction turbine. The smaller gate
opening changes the angle a'\. This alters the entrance velocity
diagram. Hence the angle /3\ will no longer agree with the vane
angle 0'i. Since the quantity of water discharged per unit time
is less than before, it follows that the velocity t/i, as determined
by the area of the runner passages, is less than the value at full-
load. Thus when a reaction turbine runs at part-gate there are
eddy losses produced at entrance to the runner due to the abrupt
change in the direction and magnitude of the velocity of the
water through the wheel passages. No such losses occur with the
impulse turbine.

At the point of discharge the velocity diagram for the tangential
wheel is practically the same at all loads, provided the jet velocity
and bucket velocity are the same. f There may be slight increases
in the losses in flow over the bucket surfaces which would affect
this statement somewhat for very large or very small nozzle
openings, but for a reasonable range the statement is true. Thus
the discharge loss would be the same at all loads. But for the

SELECTION OF TYPE OF TURBINE 181

reaction turbine, since the water completely fills the bucket pas-
sages, a reduced rate of discharge requires a proportionate reduc-
tion in the relative velocity vz. Thus F2 and hence the discharge
loss are inevitably increased. The higher the specific speed of
the turbine the greater the discharge loss .at the normal gate
opening, and hence the greater the effect produced upon the
efficiency when this loss is increased at part-gate.

It is thus apparent that at part-load there are inherent losses
.within the reaction turbine that are not found with the Pelton
wheel.

In fact the hydraulic efficiency of the latter would appear
to be the same at all nozzle openings. In reality the re-
duction in the velocity coefficient of the nozzle, as the needle
closes the discharge area, together with some change in the bucket
friction, changes the efficiency slightly." It is the gross efficiency
with which we are really concerned, and of course the mechan-
ical losses due to friction and windage, which are constant at
constant speed, cause the efficiency to decrease as the gate
opening decreases. But the efficiency-load curve of the tangen-
tial water wheel is inherently a flat curve.

The losses within the reaction turbine runner are such that the
hydraulic efficiency must decrease as the gate is changed in
either direction from the position at full-load. Hence the effi-
ciency at part-load or overload tends to be less than that for
the impulse wheel, as shown in Fig. 114 (assuming both to be the
same at full-load), and the higher the specific speed the steeper
will the efficiency curve be.

for the tangential water wheel in Fig. Ill it can be seen that
the best speed is slightly different for different gate openings
and that it increases as the latter increases. This is also true
with the reaction turbine, but in a mofeTmarked degree as can
be seen in Fig. 112. If the speed is selected so as to give the best
efficiency at a certain gate opening it will not be correct for any
other gate opening and thus efficiency will be sacrificed at ail
gates except one.

This variation of the best speed with different gate openings
is found in all turbines, but not in the same degree. With the low-
speed reaction turbine it is small, approaching the tangential
water wheel in that regard. With the high-speed reaction tur-
bine it is very marked. There seems to be little difference be-

182

HYDRAULIC TURBINES

tween turbines, in this regard, for specific speeds less than 50;
but for specific speeds above that, it increases rapidly.1

If, then, a constant speed be selected which is the best for full-
load, there will be a sacrifice of efficiency on part-load, and this

Load
FIG. 114. — Relative efficiencies on part-load of impulse and reaction turbines.

100 1

10 20

50 60 70 80
Percent of Full Load

FIG. 115. — Typical efficiency curves.

100 110 120 130

sacrifice will be greater the higher the specific speed of the tur-
bine. These considerations, together with the facts given in the
preceding article, imply efficiency curves for the various types
i C. W. Larner, Trans. A. S. C. E., Vol. LXVI, p. 341 (1910).

SELECTION OF TYPE OF TURBINE

183

such as are shown in Fig. 115. To prevent confusion the effi-
ciency curve for a low-speed turbine is not shown, but its effi-
ciency on full-load would be about the same as that for the
tangential water wheel, while on part-load it would be a little
less.

It may also be noticed that there is less overload capacity
with the high-speed turbine than with the other types. This
is because the point 6f maximum efficiency is nearer full-gate
than with the other types. If the customary 25 per cent, over-
load must be allowed, then the normal load must be less than
the power for maximum efficiency with a further decrease in
operating efficiency.

0.2

0.4

1.0

1.2

1.4

0.6 0.8

Gate Opening

FIG. 116. — Relation between power and gate opening for same speed under differ-
ent heads.

Thus from the tangential wheel on the one hand to the high-
speed reaction turbine on the other the relative efficiency on part-
load decreases as the specific speed increases.

126. Overgate with High-speed Turbines.— With the wicket
or swing gates, as used today, there is no definite limit to their
opening save that imposed by an arbitrary mechanical stop. As
the gate opening increases the rate of discharge and hence the
power of the turbine increases, as shown by curve for <£ = 0.78 in
Fig. 116. But with too great an angle of the vanes the efficiency
decreases so much that the power output no longer continues to
increase and may even decrease. Ordinarily there is no advan-
tage gained by opening the gates any wider than that necessary
to secure maximum power, and hence the mechanism is usually
so constructed that it cannot move the gates any farther than

184

HYDRAULIC TURBINES

this position. This may be termed " full-gate op'ening."1 If the
construction is such that the gates can be opened wider than this
amount, the range from full-gate up to the maximum opening may
be termed " over-gate."

1.3

0,5

Values of 0
FIG. 117. — Characteristics of low-speed runner. N8 = 27.

If the normal speed be taken as the speed at which the wheel
develops its maximum efficiency, it may be seen in Fig. 112 that

1 This is a purely arbitrary definition, but there is at present no agreement
as to what the term "full-gate" really signifies, and so it will be here used
as defined above. It may be noted that the gate movement might also be
limited to something less than the position shown, and in such an event it
would be logical to denote the maximum opening as the " full-gate." The
effect of this construction would be to decrease the overload capacity or to
move the power for maximum efficiency nearer to the maximum power.
For the same maximum power this would require a slightly larger runner.

SELECTION OF TYPE OF TURBINE , 185

the power at full-gate increases as the speed increases above nor-
mal. This is a peculiarity of the high-speed turbine. With the
medium and low-speed turbine there is no such increase. In
fact, as shown in Fig. 117, there may be a reduction in power
with the low specific speed turbine at full-gate, if the speed is
increased above normal. The explanation of this difference
in the two types is that with the inward flow turbine the centrifugal
action opposes the flow of water, and hence the rate of discharge
tends to decrease as the speed increases, while with the outward
flow turbine the centrifugal action tends to increase the rate of
discharge with the speed. This may be seen in Fig. 95, page 126.
The low specific speed runner approaches the pure radial inward
flow type, while the high specific speed runner of the present with
inward, downward, and outward flow (the radius to the outer limit
of the discharge edge being often as much as one-third greater
than the radius to the entrance edge) approaches the outward
flow turbine in this characteristic. Thus, despite the decrease
in efficiency, as the speed departs from the normal, the in-
creased rate of discharge tends to increase the power output for
a certain range of speed above normal. This feature of the
high-speed turbine is of great value, as it especially fits it for
the class of service, to which it is otherwise adapted.

As has been explained, the maximum opening of the turbine
gate would usually be that at which no further increase in power
at normal speed could be obtained and this is termed " full-gate."
But with the high specific speed turbine it is found that, when
running at a speed above normal, the power continues to increase
for an opening of the gate beyond its usual maximum value, as
shown by curves for </> = 1.03 and 1.10 in Fig. 116. A turbine
so constructed that the gate can be opened wider than the maxi-
mum value necessary under normal conditions is said to be
"overgated. " This additional gate opening would be of no
value with a low-speed turbine under any circumstances, and it
would be of no value with a high-speed turbine under normal
conditions. But, not only does the power of the latter increase
at full-gate for speeds higher than the normal, but by opening
the gate wider than the usual value the power may be still
further increased as may be seen in Figs. 107 and 118. The
nominal full-gate opening is denoted by unity.

If the head on a water power plant decreases, as in time of
flood, the capacity of each turbine is reduced, with a resulting

186

HYDRAULIC TURBINES

shortage of power. If the wheels must run at a constant speed,
as is usually the case, the speed is no longer correct for the head,
and this causes a decrease in efficiency with a further reduction
of power. »j

Any feature which will improve the capacity of the turbine
under these circumstances is of value. Now since speed is
proportional to 0VX a constant speed under a reduced head
means an increase in <f> above its normal value. As has been seen,

47 49

51 53 55 57 59
R.P.M.-l 1't. Head
FIG. 118. — High-speed turbine

61 63 65 67

an increase in <£ above its normal value causes no increase in the
power of a low- or medium-speed runner, but with the high-speed
runner not only does the power at full-gate increase but, by
overgating, the power may be still further increased. The effect
of an overgate is to materially increase the capacity of the tur-
bine at a time when there is a shortage of power. This feature is
not possessed by lower speed runners.

The high specific speed turbine has a lower maximum efficiency,

SELECTION OF TYPE OF TURBINE

187

a lower part load efficiency, and less overload capacity than the
medium speed turbine when both are operated under a constant
head, but its higher speed is a decided advantage where the nor-
mal head is low. But it is with low head plants that flood con-
ditions are most serious in their effects upon the capacity of the
plant and, as has been seen, the characteristics of the high-speed
runner are such that it is able to deliver more power under these
circumstances.

The results obtained with "overgating" high-speed turbines
may be seen in Figs. 107 and 118. In the latter case full-gate
is denoted by 1.00 and the maximum gate opening by 1.077.
The normal speed is 52 r.p.m. under 1 ft. head. At that speed
any further gate opening would be of no advantage, and in fact
would merely cause a drop in efficiency. But at a higher speed,
such as 65 r.p.m., the overgate feature raises the power under 1ft.
head from 2.0 h.p. at full-gate to 2.35. Bearing in mind that a
medium speed runner for the same situation would deliver less
than 2 h.p. under these circumstances, it is seen how much supe-
rior the high specific speed turbine is for the particular condi-
tions of service.

The differences between the low- and high-speed runners are
brought out in the following table. The normal head is 15 ft.
and the wheels develop 100 h.p. In time of high water the head
will decrease to 10 ft., while the wheels are kept at their normal
speed.

TABLE 6. — HEAD = 15 Ft.

Type

Ar,

R.p.m.

Gate

H.p.

Efficiency

Low speed

35.2

104

0.70

1.00

100

High speed

77 7

232

0 81

1 00

100

TABLE 7.— HEAD = 10 Ft.

Type

JV.»

R.p.m.

<t>

Gate

H.p.

Efficiency

Low speed

35 2

104

0 86

1.00

47 3

High speed

77.7

232

0.99

1.00

49.3

High speed

232

0.99

1.077

58.0

1 These values of specific speed apply only when the turbine is developing
its best efficiency. Under the reduced head with an incorrect value of <f>,
the real specific speed is different. But as an index of the type the values
given are always appropriate.

188

HYDRAULIC TURBINES

By means of the overgate feature the high-speed turbine is seen
to be capable of developing 22J^ per cent, more power under the
lower head than the low-speed turbine. The efficiency is also
seen to be greater.

127. Type of Runner as a Function of Head. — It has been
stated that the choice of the type of turbine is a function of the
power and the speed desired, as well as the head. While this is
true, the value of the head does exert a predominating influence
and hence there is some justification for the presentation of a
relationship between the two, such as is given in Table 8. How-
ever, it should be noted that the figures given for specific speed
are merely limits. Thus a head of 100 ft. does not require a
turbine whose specific speed is 50 for example. The latter
is merely the maximum value found in current practice for such a
head, and a lower value of Ns might be used. Within this max-
imum limit the specific speed chosen would depend upon the
power and speed and a consideration of the characteristics desired.

TABLE 8. — RELATION BETWEEN HEAD AND SPECIFIC SPEED

Head ft.

Maximum
value of Na

Type of setting

100

Vertical shaft single runner units with

good draft tubes.

Single runner, either horizontal

or ver-

100

tical, or two runners discharging

into a

160

common draft chest.

350
600
800

30
20
10

Single or double discharge runner on
horizontal shaft.

1000

Impulse wheels.

2000

5000

128. Choice of Type for Low Head. — No definite rules can be
laid down for universal use because each case is a separate prob-
lem. Neither is it possible to draw any line between a high and
a low head. AllTthat can be done is to assume cases that are
typical and establish broad general conclusions. In any particu-

SELECTION OF TYPE OF TURBINE 189

lar case the engineer can then decide what considerations have
weight and what have not.

The average low-head plant has very little, if any, storage
capacity. In times of light load the water not used is generally
being discharged over the spillway of the dam. Economy of
water on part -load is thus of very little importance. The effi-
ciency on full load is of value as it determines the amount of power
that may be developed from the flow available.

Under a low head the r.p.m. is normally low and it is desirable
to have a runner with a small diameter and a high value of <£, in
order to secure a reasonable speed. A high speed means a
cheaper generator and, to some extent, a cheaper turbine. These
were the factors that brought about the development of the high-
speed turbine.

A low-head plant is also usually subjected to a relatively large
variation in the head under which it operates. When the head
falls below its normal value the overgate feature of the high-
speed turbine, enabling it to hold up the power, to some extent,
at a good efficiency, is a very valuable characteristic.

The only disadvantage- of the high-speed turbine for the typical
low-head plant is that its maximum efficiency under normal head
is not as good as that of the lower speed turbines. However, the
other advantages outweigh this so that it is undoubtedly the best
for the purpose.

129. Choice of Type for Medium Head. — With a somewhat
higher head a limited amount of storage capacity usually becomes
available and thus the efficiency on part-load becomes of interest
as well as the efficiency on full-load. The r.p.m. also approaches
a more desirable value so that the necessity for a high-speed run-
ner disappears. The variation in head will generally be less
serious also, so that the overgate feature of the high-speed turbine
becomes of less value. The high efficiency of the medium-speed
turbine fits it for this case. The high-speed turbine should not
be used unless the interest on the money saved is more 'than the
value of the power lost through the lower efficiency.

130. Choice of Type for High Head.— For high heads the
possibility of extensive storage increases and the average oper-
ating efficiency then becomes of more interest than the maximum
efficiency, especially if the turbine is to run under a variable load.
Since the normal speed under such a head is high, a runner with a
large diameter and a low value of 0 may be desirable, as it keeps

190 HYDRAULIC TURBINES

the r.p.m. down to a reasonable limit. The choice lies between a
medium-speed turbine, a low-speed turbine, or a tangential
water wheel.

If the wheel is to run on full-load most of the time, the high
full-load efficiency of the medium-speed turbine fits it for the
place. If the load is apt to vary over a wide range and be very
light a considerable portion of the time, the comparatively flat
efficiency curve of the tangential water wheel renders it suitable.
There is little difference between the characteristics of the low
and medium-speed wheels. The choice between them is largely
a matter of the r.p.m. desired, although there is some slight
difference in efficiency.

131. Choice of Type for Very High Head. — Within certain
limits there is a choice between the low-speed reaction turbine
and the tangential water wheel. The former might be chosen
in some cases because of its higher speed with a consequently
cheaper generator and the smaller floor space occupied by the
unit. The latter has the advantage of greater freedom from
breakdowns and the greater ease with which repairs may be
made. This consideration is of more value with the average
high-head plant than with the average low-head plant, since the
former is usually found in a mountainous region where it is
comparatively inaccessible, and is away from shops where ma-
chine work can be readily done.

For extremely high heads there is no choice. The structural
features necessary are such that the tangential water wheel is
the only type possible. Also the relatively low speed of the
tangential water wheel is of advantage where the speed is in-
herently high.

132. QUESTIONS AND PROBLEMS

1. For a given head and stream flow available at a certain power plant,
what quantities may be changed so as to permit the use of various types of
turbines? Which type of turbine will give the smallest number of units
in the plant? Which type will run at the lowest r.p.m.?

2. How do impulse wheels and reaction turbines compare as to the maxi-
mum efficiency attained by each? How does the efficiency of an impulse
wheel vary with its size? Why? How does that of a reaction turbine
vary with its size? Why?

3. For the same power under the same head compare impulse wheels and
reaction turbines with respect to efficiency, rotative speed, space occupied,
freedom from breakdown, ease of repairs, and durability with silt laden
water.

SELECTION OF TYPE OF TURBINE 191

4. How does the maximum efficiency of a reaction turbine vary with the
type of turbine? For what type is it the highest? Why? For what type
is it the lowest? Why?

6. What are the disadvantages of a very low specific speed reaction tur-
bine? What are its advantages?

6. How does the efficiency of the Pelton wheel vary with its specific speed?
Why?

7. What is meant by full-load? What affects the efficiency of a tangential
water wheel on part-load?

8. What affects the efficiency of a reaction turbine on part-load? Is the
part-load efficiency a function of specific speed?

9. What is meant by full-gate? By overgate? What types of turbines
are overgated?

10. What is the difference in the characteristics of low and high specific
speed reaction turbines when run at the same speed under a head less than
normal? Why?

11. What are the advantages and disadvantages of very high specific
speed turbine runners?

12. What types of turbines could be used under a head of 20 ft.? Under
200ft.? Under 1000ft.?

13. What are the advantages of a high-speed runner under very low
heads? What are the advantages of a medium speed runner under the
same conditions?

14. What are the especial merits of tangential water wheels for very high
heads? What are the disadvantages of a low-speed reaction turbine for
the same conditions?

15. The turbine runner for which the curves in Fig. 107 were plotted was
23 in. in diameter and had a specific speed of 93. The specific speed of the
runner for which the curves of Fig. 117 were drawn was 27 and the diameter
was 57 in. Suppose a turbine was required to deliver 1200 h.p. at full-gate
under a head of 25 ft., find the size and r.p.m. for a runner of each of these
types. Ans. 47.8 in., 150 in., 144 r.p.m., 43.5 r.p.m.

16. If the speeds remain as in problem (15) while the head decreases from
25 ft. to 16 ft., find the power of each turbine. Ans. 648 h.p., 465 h.p.

17. The average flow of a stream is 3000 cu. ft. per second and the
pondage is very limited. The normal head is 30 ft. but is at times as low
as 18 ft. What type of turbine should be employed, how many units should
there be, and at what speed will they run?

Ans. 4 units at 124 r.p.m. probably best.

18. The average flow of a stream is 3000 cu. ft. per second. The normal
head is 30 ft. which is decreased somewhat in times of flood. The stream
flow is fluctuating with long low water periods, but there is considerable
storage. The load on the plant also varies considerably. What type of
turbine should be used, how many units should there be, and at what speed
should they run?

19. A turbine is required to carry a constant load of 800 h.p. under a
head of 120 ft. There is considerable storage capacity and the stream has
periods of ^low run-off. The wheel is to drive a 60-cycle alternator. What
type of turbine should be used and what will be its speed?

CHAPTER XIV
COST OF TURBINES AND WATER POWER

133. General Considerations. — Since there are so many factors
involved, it is rather difficult to establish definite laws by which
the cost of a turbine may be accurately predicted. No attempt
to do so will be made here, but a discussion of the factors involved
and their affects will be given and the general range of prices
stated. A few actual cases are cited as illustrations.

A stock turbine will cost much less than one that is built to
order to fulfil certain specifications. This fact is illustrated by
the comparison of two wheels of about the same size and speed.
The specifications of the stock turbine were as follows: 550 h.p.
at 600 r.p.m. under a head of 134 ft., 26-in. double discharge
bronze runner, cast steel wicket gates, cast-iron split globe casing
5 ft. in diameter, and riveted steel draft tube. Weight about
11,500 Ib. Price $1750. The special turbine was as follows:
500 h.p. at 514 r.p.m. under a head of 138 ft., bronze runner,
spiral case, riveted steel draft tube, connections to header, relief
valve, and vertical type 5000 ft.-lb. Lombard governor. Price
$4000. The latter includes a governor, relief valve, and some
connections which the former did not, but the difference in cost
is more than the price of these.

The cost of the turbine is also affected by the quality and
quantity of material entering into it, the grade of workmanship,
and the general excellence of the design. With the $4000 tur-
bine cited in the preceding paragraph another may be compared
which is of superior design. The specifications for the latter
were as follows: 550 h.p. at 600 r.p.m. under 142-ft. head, single
discharge bronze runner, spiral case with 30-in. intake, cast steel
wicket gates, bronze bushed guide vane bearings, riveted steel
draft tube, lignum vitae thrust bearing, oil pressure governor
sensitive to 0.5 per cent. The guaranteed efficiencies were

83 per cent, at 410 h.p.

84 per cent, at 500 h.p.
83 per cent, at 550 h.p.

192

COST OF TURBINES AND WATER POWER 193

(Nothing was said about efficiency in the preceding case.)
Weight of turbine 30,000 lb., of governor 3000 Ib. Price $6000.
The turbine just quoted was similar to one previously built
and the patterns required only slight modification. Where an
entirely new design is called for the cost will be greater still, as
is evidenced by the bid of another firm, as follows: 550 h.p. at
600 r.p.m. under 142-ft. head, single discharge cast iron runner,
spiral case, cast steel guide vanes, cast steel flywheel, oil pres-
sure governor, connections to header, 30-in. hand-operated gate
valve, riveted steel draft tube, and relief valve. The guaranteed
efficiencies were

Per cent, of max. h.p.

81.5 per cent, at

100

84.5 per cent, at .

84.5 per cent at

82.5 per cent, at

79.5 per cent, at . .

Weight of turbine complete 38,000 lb. Price $8740. This last
turbine includes a few items that the former does not, but the
difference in cost cannot be accounted for by them. It will be
noted that a flywheel was deemed necessary here, while it was
not used on any of the others. Compare the weights and costs
of these last two turbines with the weight and cost of the stock
turbine first mentioned.

134. Cost of Turbines. — The cost of a turbine depends upon
its size and not upon its power. Since the power varies with
the head, it is apparent that the cost per h.p. is less as the head
increases. Thus a certain 16-in. turbine (weight = 7000 lb.)
without governor or any connections may be had for $1000.
Under various heads the cost per horsepower would be as
follows :

Head

H.p.

[ Cost per h.p.

30ft

$19 20

60 ft

148

6 75

100 ft

318

3 14

194 HYDRAULIC TURBINES

One would not be warranted in saying, however, that under
10-ft. head a turbine would cost $100 per horsepower because
the above would develop only 10 h.p. under that head. Neither
would one be justified in saying that, since this turbine would
develop 1650 h.p. under 300-ft. head, that the cost per horse-
power might be only $0.605. Under a 10-ft. head a much
lighter and cheaper construction would be entirely reasonable,
while under a 300-ft. head the turbine would have to be built
stronger and better than this one was.

For a given head, the greater the power of the turbine the less
the cost per horsepower will be. Also for a given head and
power, the higher the speed, the smaller the wheel, and conse-
quently the less the cost. Compare the 600-r.p.m. reaction
turbines in Art. 133 with the following, which is a double over-
hung tangential water wheel at 120 r.p.m. The horsepower
is 500 under 134-ft. head. Oil pressure governor is included, but
no connections to penstock are furnished. Weight 80,000 Ib.
Price $8900.

These last differences are very much magnified if we combine
the cost of the generator with that of the turbine. The follow-
ing are some generator quotations. The first is that of a gener-
ator at a special speed. The second is that of a generator of
somewhat better construction than the first but of a standard
speed. The others are all standard speeds.

150 kv.-a., 2400 volts, 3-phase, 60-cycle, 124 r.p.m. $4850.

150 kv.-a., 2400 volts, 3-phase, 60-cycle, 120 r.p.m. $3300.

(Weight 17,210 Ib.)

300 kv.-a., 2400 volts, 3-phase, 60-cycle, 120 r.p.m. $4700.

(Weight 25,520 Ib.)

350 kv.-a., 2400 volts, 3-phase, 60-cycle, 514 r.p.m. $2330.

350 kv.-a., 2400 volts, 3-phase, 60-cycle, 600 r.p.m. $2100.

Taking the highest priced 600-r.p.m.. turbine and combining it
with the 350-kv.-a. generator we get a total of $10,850. Adding
the cost of the 120-r.p.m. turbine to that of the 300-kv.-a. genera-
tor we get a total of $13,600 for a smaller amount of power.

Prof. F. J. Seery has derived the following empirical formula
based upon the list prices of 35 wheels made by 20 manufacturers.

Log X = A + D/B, in which X is the cost in dollars for a
single stock runner with gates and crown plates suitable for

COST OF TURBINES AND WATER POWER 195

setting in a flume. The value of A ranges from 1.09 to 2.17,
but the usual value is about 1.9. The value of B varies from
40 to 83 with a usual value of about 50. These prices are sub-
ject to discounts also. The cost of a draft chest for a twin run-
ner will be given by

X = 0.045 D2-25, in which X is in dollars and D is the
diameter of the runners in inches.

The cost of the casing increases these values very greatly,
as some spiral cases may cost much more than the runner. A
single case may be cited of a pair of 20-in. stock runners in a
cylinder case with about 30 ft. of 5 ft. steel penstock. Each
runner discharges into a separate draft tube about 3 ft. long.
The power is 150 h.p. under 30-ft. head. The cost was $2000.

A few quotations are here given. A reaction turbine to de-
velop 4000 h.p. at 600 r.p.m. unc(er 375 ft. head and weighing
90,000 Ib. would cost $14,000. Another reaction turbine of
10,000 h.p. under 565-ft. head cost $37,000. In the latter case
the governor, pressure regulator, and the generator were included.
The building, crane, transformer room, etc., cost $20,000 for
this installation. A tangential water wheel of 2500 h.p. under
1200-ft. head cost $12,000, while another of 4500 h.p. under
1700-ft. head cost $8,000.

As has been stated, the cost of a turbine varies between fairly
wide limits due to difference in design, workmanship, and com-
mercial conditions. The cost per h.p. is also less the higher the
head or the greater the power. In a general way it can be said to
vary between $2 and $30 per horse-power and according to the
following table:

Head

Cost per h.p.

Cost of building per h.p.

60 ft

$30-$7

$30-$4

100-600 ft

$12-$2

$ 7-$2

500-2000 ft

$ 8-$2

$ 7-$2

The cost of the turbine is usually only about 6 per cent, of
the total cost of the power plant. It scarcely pays, therefore,
to buy a cheap turbine when the money saved is such a small
portion of the entire investment.

135. Capital Cost of Water Power. — The capital cost of water
power includes the investment in land, water rights, storage

196

HYDRAULIC TURBINES

reservoirs, dams, head races or canals, pipe lines, tail race,
power house, equipment, transmission lines, interest on money
tied up before plant can be put into operation, and often the cost
of an auxiliary power or heating plant.

The capital cost per horsepower is less as the capacity of
the plant is greater. This is shown by the following table from
the report of the Hydro-Electric Power Commission of the
Province of Ontario. The proposed plant was to be located at
Niagara Falls.

TABLE 9

Items

50,000 h.p.

100,000 h.p.

Tunixel tail race

$1,250,000

$1 250 000

Headworks and canal
Wheel pit .

450,000
500,000

450,000
700,000

Power house

300,000

600 000

Hydraulic equipment

1,080,000

1,980,000

Electric equipment

760,000

1,400,000

Transformer- station and equipment
Office building and machine shop
Miscellaneous

350,000
100,000
75,000

700,000
100,000
75,000

Engineering etc 10 per cent

$4,865,000
485 000

$7,255,000
725 000

Interest 2 years at 4 per cent

$5,350,000
436,560

$7,980,000
651,168

Total capital cost

$5,786,560

$8,631,168

Capital cost per horsepower

$114

$86

The cost per unit capacity is usually less as the head increases.
This is illustrated by the following table taken from Mead's
" Water Power Engineering."

Capital cost per h.p.

Capacity
horsepower

Head

Without dam

With
dam

With dam
and electrical
equipment

With dam, electric
equipment, and
transmission line

8000

$63.50

115

150

8000

21.00

COST OF TURBINES AND WATER POWER 197

The capital cost may range from $40 to $200 per horsepower,
but the average value is about $100. l

136. Annual Cost of Water Power. — The annual cost of water
power will be the sum of the fixed charges and the operating
expenses. The former will cover interest on the capital cost,
taxes, insurance, depreciation, and any other items that are con-
stant. The latter includes repairs, supplies, labor, and any
other items that vary according to the load the plant carries.
The annual cost per horsepower is the total annual cost divided
by the horsepower capacity of the plant.

The total annual cost will vary with the number of hours the
plant is in service and also with the load carried. The cost will
be a maximum when the plant carries full load 24 hours per day
and 365 days per year. It will be a minimum when the plant is
shut down the entire year, being then only the fixed charges. (See

Hours per Year
FlG. 119.

8760

Fig. 119). It is evident that the annual cost per horsepower
depends upon the conditions of operation.

However, under the usual conditions of operation, the annual
cost may be said to vary from $10 to $30 per horsepower.

137. Cost of Power per Horsepower-hour. — In order to
have a. true value of the cost of power it is necessary to consider
both the load carried and the duration of the load. While the
annual cost per horsepower will be a maximum when the plant
carries full load continuously throughout the year, the cost per
horsepower-hour will be a minimum. Thus suppose the annual
cost per horsepower of a plant in continuous operation on full-
load is $20. The cost per horsepower-hour is then 0.228 cents.
Suppose that the plant is operated only 12 hours per day and that

1 For specific cases see Mead's "Water Power Engineering," p. 650.

198

HYDRAULIC TURBINES

the annual cost per horsepower then becomes $17, the cost of
power will be 0.388 cents per horsepower-hour. So far the load
has been treated as constant; we shall next assume that it varies
continuously and that it has a load factor of 25 per cent. By that
is meant that the averagejoad is 25 per cent, of the maximum.
If the plant be operated 12 hours per day as before, the annual cost
per maximum horsepower may still be $17, but the annual cost
per average horsepower will be $68. This latter divided by
4380 hours gives 1.55 cents per horsepower-hour. It is clear,
then, that the cost of power per horsepower-hour depends
very greatly upon the load curve. It may range anywhere from

Load Factor
FlG. 120.

0.40 cents to 1.3 cents per horsepower-hour and more if the load
factor is low. (See Fig. 120.)

138. Sale of Power. — If power is to be sold, one of the first
requirements generally is that the output of the plant should be
continuous and uninterrupted. Such a plant should possess at
least one reserve unit so that at any time a turbine can be shut
down for examination or repair. This adds somewhat to the
cost of the plant. The larger the units the more the added cost
of this extra unit will be. On the other hand small units are
undesirable since a large number of them make^the plant too
complicated. Also the efficiency of the smaller wheels will be
less than that of the larger sizes. Unless the water-supply is
fairly regular, storage reservoirs will be necessary and often auxil-

COST OF TURBINES AND WATER POWER 199

iary steam plants are essential in order that the service may not
be suspended either in time of high or low water.

A market for the power created is essential. If the demand for
the power does not exist at the time the plant is projected/ there
should be very definite assurance that the future growth of in-
dustry will be sufficient to absorb the'output of the plant.

If the plant is to be a financial success, the price at which
power is sold should exceed the cost of generation by a reasonable
margin of profit. The price for which the power may be sold
is usually fixed by the cost of its production in other ways. This
point should be carefully investigated and, if the cost from other
sources is less than the cost of the water power plus the profit,
the proposition should be abandoned.

139. Comparison with Steam Power. — It is necessary to be
able to estimate the cost of other sources of power in order to tell

Load Factor
FIG. 121. — Comparison of costs of steam and water power.

whether a water-power plant will pay or not. Also it is often
essential to figure on the cost of auxiliary power. As steam is
the most common source of power and is typical of all others, our
discussion will be confined to it.

In general the capital cost of a steam plant is less than that of a
water-power plant. It varies from $40 to $100 per horsepower,
with an average value of about $60 per horsepower. Deprecia-
tion, repairs, and insurance are at a somewhat higher rate but,
nevertheless, the fixed charges are less than for water power.

The amount of labor necessary is greater and this, together

200

HYDRAULIC TURBINES

with the cost of fuel and supplies, causes the operating expenses
to be higher than with the water power. The , total cost of
power for the two cases is compared in Fig. 121. As to whether
the cost of steam power in a given case is greater or less than that
of water power at 100 per cent, load factor it is impossible to
state without a careful investigation. But it is clear that, as a
rule, the cost of steam power is less when the plant is operated
but a portion of the year or when the load factor is low. Thus a
water-power plant is of the most value when operated at high
load factor throughout the year.

The annual cost of steam power per horsepower is very high
for small plants but for capacities above 500 h.p. it does not vary
so widely. Its value depends upon the capacity of the plant, the
load factor, and the length of time the plant is operated. It may
be anywhere from $20 to $70, though these are by*jnojneans
absolute limits. /

Since the operating expenses are of secondary importance in a
water-power plant, the annual cost per horsepower will not be
radically different for different conditions of operation. But
with a steam plant the annual cost per horsepower varies widely
for different conditions of operation on account of the greater
effect of the variable expenses. It is much better to reduce all
costs to cents per horsepower hour. The accompanying table
gives the usual values of the separate items that make up the
cost of steam power, reduced to cents per horsepower hour.

Items

Min., cents

Max., cents

Fuel

0.20

0.75

Supplies . . .

0 03

0.06

Labor ....

0.07

0.14

Administration

0 02

0.15

Repairs

0.05

0.10

Fixed charges . .

0 30

0.45

Total cost per horsepower hour '.

0.67

1.65

The following comparison is made by C. T. Main in Trans.
A. S. M. E., Vol. XIII, p. 140. The location was at Lawrence,
Mass. Fixed charges were estimated on the following basis :

COST OF TURBINES AND WATER POWER 201

Steam,
per cent.

Water,
per cent.

Interest . . .

5 0

Depreciation

3.5

Repairs

2.0

Insurance . . ...

2 0

Total

12.5

For a steam plant at that location the capital cost was taken
as $65 per horsepower. The annual cost per horsepower was as
follows :

Fixed charges 12.5 per cent $8. 13

Fuel 8.71

Labor 4.16

Supplies 0.80

Total annual cost per horsepower $21 .80

For a water plant the cost of the power house and equipment
was taken as $65 while the cost of dams and canals at that place
averaged $65 also, making a total capital cost of $130 per horse-
power. The annual cost per horsepower was as follows :

Fixed charges 9 per cent . . . , $11 . 70

Labor and supplies 2 . 00

Total annual cost per horsepower $ 13 . 70

However, for the case in question, a steam-heating plant was
necessary and its cost was divided by the horsepower of the
plant giving the capital cost of the auxiliary steam plant as $7.50
per horsepower of the power plant. The cost of its operation
based upon the power plant would ;be,

Fixed charges at 12.5 per cent $0 .94

Coal 3.26

Labor 1.23

Total cost of heating per horsepower of plant .... $5. 43

Adding this to the cost of the power we obtain the total cost
of the water power to be $19.13 per horsepower per year. Evi-

202 HYDRAULIC TURBINES

dently the difference in favor of the water power will be $2.67 per
horsepower.

The cost of any kind of power will evidently vary in different
portions of the country and it is impossible to lay down absolute
facts of universal application. In places near the coal fields the
cost of steam power will be a minimum and it may be impossible
for water power to complete with it. However where the cost
of fuel is high water power may be a paying proposition even
though its cost may be relatively high.

140. Value of Water Power. — The value of a water power is
somewhat difficult to establish as it depends upon the point of
view. However, the following statements seem reasonable:

An undeveloped water power is worth nothing if the power,
when developed, is not more economical than steam or other
power. If the power, when developed, can be produced cheaper
than other power, then the value of the water rights would be a
sum the interest on which would equal the total annual saving
due to the use of the latter. Thus, referring to the case of Mr.
Main cited in the preceding article, suppose the water supply is
capable of developing 10,000 h.p. The annual saving then due
to its use would be $26,700 as compared with steam. Its value
is then evidently a sum the interest on which would be $26,700
per year.

A power that is already developed must be considered on a dif-
ferent J)asis. If the power cannot be produced cheaper than that
from any other available source, the value of the plant is merely
its first cost less depreciation, or from another point of view the
sum which would erect another plant, such as a steam power
plant, of equal capacity. If the water power can be produced
cheaper than any other, the value of the plant will be its first
cost less depreciation added to the value of the water right as
given in the preceding paragraph.

141. QUESTIONS AND PROBLEMS

1. What are the general factors that affect the cost of a turbine of a
given speed and power. ?

2. What factors affect the cost of a turbine per h.p.?

3. What is meant by capital cost of water power? What items does it
include? How is this cost per h.p. affected by the head and by the size of
the plant?

4. What is meant by the annual cost of water power? How is it com-

COST OF TURBINES AND WATER POWER 203

puted? When will it be a maximum and when a minimum for a given
plant?

5. How is cost per h.p. hour computed? Upon what factors does it
depend? How does it vary as a function of load factor? When is it a
maximum for a given plant and when is it a minimum? What can its
maximum value be?

6. How do water and steam power compare in general as to capital cost
per h.p. and hence- as to fixed charges? How do they compare as to oper-
ating expenses. How do the total annual costs and the cost per h.p. hour
vary for each as functions of load factor?

7. How is it to be determined beforehand whether a water power plant
will pay or not?

8. How is the value of a water right to be determined?

9. How is the value of an existing water power plant to be computed?
Can there be any doubt about the correctness of the method?

10. Suppose you were called upon to make a report upon a water-power
development, the only information given being the head available and the
location for the plant, together with an assurance of a market for all power
produced. How would you determine: (a) Amount of power that can be
developed; (6) How much storage capacity should be provided; (c) Whether
the plant should be built at all; (d) Value of the water right; (e) Size of
penstock to be used; (/) Type of turbine to be used; (g) Number, size and
speeds of units to be used?

11. If steam power costs $20 per h.p. per year and water power can be
produced for $19 per h.p. per year, what would be the value of an undevel-
oped water right of 5000 h.p?

12. A water power plant cost $100 per h.p. and is estimated to have de-
preciated 15 per cent. If it costs $20 per h.p. per year to produce power
from it in a place where steam power would cost $23 per h.p. per year, what
is the value of the development?

CHAPTER XV
DESIGN OF THE TANGENTIAL WATER WHEEL

142. General Dimensions.1 — Assume that the head, speed, and
power for a proposed water wheel are known, these values being
so selected as to give the specific speed necessary for the type of
impulse wheel desired. It is to be understood that the head is
that at the base of the nozzle, and the power is the output corre-
sponding to one jet. The velocity of the jet is given by the
equation, V\ = cv\/2gh, where the value of the velocity coeffi-
cient may be taken as 0.98. (See Fig. 89, page 114.) Since
B.h.p. = qhe/8.8, we may write

8.8 X B.h.p. wd2 —

where d is the diameter of the jet in inches. From this the value
of d may be found to be

The diameter of the wheel may be found from equation (47),
which gives

1840 ^Vh

• v ; N

where D is the diameter in inches of the " impulse circle," which
is the circle tangent to the center line of the jet. The overall
diameter of the runner depends upon the dimensions of the buck-
ets. The value of <f>e is from 0.43 to 0.47.

143. Nozzle Design. — The nozzle tip and needle should be so
proportioned as to give a constantly decreasing stream area from
a point within the nozzle to a point in the jet beyond the tip of
the needle, so that the water may be continuously accelerated.
This must be so for every position of the needle. The curve of

1 In this book only the hydraulic features of design will be considered.
No space will be devoted to the determination of dimensions which can be
computed by the usual methods of machine design.

204

DESIGN OF THE TANGENTIAL WATER WHEEL 205

the needle must therefore change from convex to concave and
the point of inflection must be at a diameter greater than that
of the nozzle'tip, otherwise the water will tend to leave the needle
at the smaller openings with a resulting tendency to corrosion.
It is also desirable, for the sake of the governor action, that the
rate of discharge vary approximately in direct proportion to the
linear movement of the needle. (See Fig. 89.)

The diameter of the orifice of the nozzle tip must be greater
than the diameter of the jet, due to the contraction of the latter
and also to the space taken up by the needle tip, which is never
entirely withdrawn. At wide open setting the needle tip may

FIG. 122.

occupy 10 per cent, or more of the area of the orifice. The re-
maining area, through which the water passes, may be computed
from the area of the jet by the use of a coefficient of contraction,
typical values for which are given in Fig. 89, .page 114. In
reality the effective area of the nozzle is that perpendicular to
the stream lines and is the surface of the frustum of a cone, whose
elements are perpendiculars dropped from the edge of the orifice
to the needle. This area is slightly greater than that in the plane
of the orifice. The nozzle tip diameter shoud be computed for
a size of jet sufficiently large to carry the maximum load on the
wheel.

144. Pitch of Buckets. — In Fig. 122 the bucket A has just com-
pletely intercepted the jet. If the particle of water at C is to
hit bucket B it must do so before the latter reaches point E.

206

HYDRAULIC TURBINES

The time for a particle of water to go from C to E is t = l/V\.
If in the same time bucket B reaches E, we have, t = y/u.
But co = UQ/TQ, and equating the two values of t, we obtain
7 = (Wro) (l/Vi) = (UQ/VI) (//r0). Since l/r0 = 2 sin5 and
0 = 26 — r, we have

= 26 - 2 Y sin 8

(62)

But this value of the pitch angle would be such as to permit the
particle of water to merely touch the bucket before the latter
swung up out of its line of action. In order to permit the water
to flow over the bucket a closer spacing than this is required. The
time necessary for flow over the bucket may be represented by
t' = l'/vf, where V is the length of path and vf the velocity
relative to the bucket, a mean value being chosen between Vi

FIG. 123.

and t;2. It appears rather difficult to express this readily in a
simple formula and the practical procedure appears to be to
assume an approximate spacing for the buckets and then compute
the probable time required for a particle of water to complete
its flow. As a preliminary trial value we may assume the above
value to be reduced by 20 per cent., in which case the number of
buckets n may be found by

n = 27T/0.8 X 0 (63)

As noted above, this value should be checked by a numerical
computation.

So far the bucket has been considered as if all points on its lower
edge were at the same distance from the axis, whereas the buckets
of the present day have some form of notch in this edge, as may
be seen in Fig. 123. The water which strikes the bucket during

DESIGN OF THE TANGENTIAL WATER WHEEL 207

the early part of its course flows out along the arc ST, while that
which strikes during the latter part, when the bucket is quite
inclined to the axis of the jet, flows out across the portion X Y.
The effect of this prolongation of the bucket beyond the part MN
is to permit the water which strikes it just before it reaches point
E in Fig. 122 to fully act. In other words, referring to Fig. 122,
the entrance edge of the bucket describes the arc CE, but the
extreme discharge edge describes a larger arc. This permits the
use of fewer buckets on a wheel without involving any loss. It is
not desirable to extend the part MN to the same radius because
that would make conditions less favorable when the bucket first
enters the jet.

For a high specific speed wheel the runner diameter becomes
relatively smaller for the same jet diameter and this shortens
the length of the path CE (Fig. 122). In order that all the water
may be fully utilized it is necessary to reduce the time required
for the particle of water at C to catch up with bucket B. This
can be done by reducing the pitch.

But there is evidently a limit to this for mechanical reasons,
as a certain amount of metal is necessary in order that each bucket
may be securely fastened to the rim. Furthermore, the closer the
buckets are placed together the quicker must the water dis-
charged be gotten out of the way of the following bucket. This
means that it must leave with a higher residual velocity, which
means in turn that the kinetic energy lost af; discharge is greater.
This is one reason why the efficiency of an impulse wheel is
less, if the specific speed is too high. After the number of buckets
on a given wheel has been made a maximum, the only other means
of increasing the specific speed is to lengthen the buckets still
more, but this evidently soon reaches its limit.

The curve representing the end of the portion of the jet in-
tercepted by the bucket A may be drawn by plotting the path
of the tip of the bucket relative to the jet. By computing the
time necessay for bucket B to get to its extreme right hand posi-
tion and then by moving CF the distance the water would travel
in the same time interval, it is apparent whether any water
is not utilized or not, and also the amount wasted can be approxi-
mately determined.1

'See, "Theory of the Tangential Waterwheel," by R. L. Daugherty in
Cornell Civil Engineer, Vol. 22, p. 164 (1914).

208 HYDRAULIC TURBINES

145. Design of Buckets. — The general dimensions of the
buckets must bear some relation to the size of the- jet and experi-
ence shows that the width of the bucket should be at least three
times that of jet and the length about the same or a little more.1
The exact dimensions should be determined for individual cases
and naturally vary somewhat with the specific speed of the wheel.

In order to get the proper bucket shape, curves may be plotted
showing the path of the jet relative to the wheel, as in Fig. 124.
In the figure only one such curve is shown, that for the top of the
jet, and also we consider here only one section, that in the plane
of the paper; but other stream lines and other parallel planes
should also be used. It should be noted that this is the relative
path for the free jet only. As soon as the water flows over the
buckets its absolute velocity is altered, and consequently its

\
Path of Point on Wheel

Eelative to Jet

FIG. 124.

relative velocity and path are different. But as the function of
these curves is merely to aid in determining the entrance condi-
tions they are sufficient. As the bucket first enters the jet, the
water flows in over the lip in the center of the notch, as MN in
Fig. 123. It is only after the bucket has travelled somewhat
farther that the water strikes it fully on the "-splitter." As seen
in Fig. 124, the face of the bucket along the lip should be such
that the surface is approximately tangent to the relative path
of the water, in order, not to have any loss of energy at this
point. After the bucket has moved along to another position
where the jet strikes it in another place, the shape of that portion
will be determined in the same way, but of course another por-
tion of the curve will be used. Also the "splitter" should be
approximately perpendicular to the relative path. (See Fig. 23.)
It should be borne in mind that where the relative and absolute

1 See paper by Eckart to which reference is made in note on page 145,
and for some proportions see Marks' Mech. Eng. Handbook, page 1089.

DESIGN OF THE TANGENTIAL WATER WHEEL 209

paths coincide, the bucket, as at A (Fig. 124), is shown in its
true position in space. But a bucket as at B, if shown only in its
position relative to the jet, has its true position in space to the
right of this. This may be seen further in that the point where
the relative path cuts the circle to the right corresponds to the
actual position of the particle of water as it leaves the right hand
side- of the wheel in its absolute path. The use of these curves
will enable one to determine the best shape for the bucket along
the lip and along the splitter, as well as the best outline for the
notch.

But of equal importance with the design of the face of the
bucket is that of the shape of the back. As the bucket A enters
the jet in Fig. 124, its back should not intersect the curve of the
relative path of the water. If it does intersect it, it indicates
that the back of the bucket will strike the water in the jet and
it is obvious that this would result in a loss of efficiency. The
back of the bucket could strike the water, despite the higher
velocity of the latter, because they are not moving in the same
direction. Hence the back of the bucket should be no more
than tangent to the curve shown. It is obvious that this matter
should be investigated for other stream lines and other planes,
besides the one shown.

Ideally the water should be reversed by the bucket and dis-
charged backwards, relatively, at an angle of 180°. But this is
impractical because the water would then be unable to get out
of the way of the next bucket. Hence such an angle should be
used as will enable the water to be discharged with an absolute
velocity whose lateral component is sufficient. As has been
pointed out, the closer the buckets are placed, the greater must
be the value of this velocity and hence the more this angle must
be made to differ from 180°. The bucket angle used in practice
is about 170°.

If the shape of the bucket can be determined for the entrance
and discharge edges by the application of the preceding principles,
the bucket may be completed by joining these two portions with
any smooth surface of double curvature. There should be no
sharp curvature used nor anything which would tend to cause
any abrupt change in the path or velocity of the water.

146. Dimensions of Case. — The case should be made of suffi-
cient size to allow a reasonable clearance between it and the
buckets around the top portion. Usually the lower portion of

210 HYDRAULIC TURBINES

the wheel is below the floor level and so the case does not extend
to that section. But in any event there should be ample room
on either side of the buckets here so that the water discharged
from the buckets may not rebound back to the wheel. It is
apparent that the higher the head, the greater the discharge
velocity, and hence the more room there should be at this place,
otherwise water will be thrown back upon the wheel and thus
increase the so-called windage loss. This action is most marked
in many of the small laboratory wheels that have been made with
very narrow cases.

147. QUESTIONS AND PROBLEMS

1. How may the diameter of a Pelton wheel be found for a given head,
speed, and power? How may the diameter of jet be found?

2. What are the principles in the design of a needle nozzle? How may
the size of the nozzle tip be determined, if the jet diameter is given?

3. How may the necessary pitch for the buckets of an impulse wheel
be computed?

4. Why is the tangential water wheel bucket made as it is with a notch
in the edge? Would it be possible to have an efficient bucket without this?

6. How may the specific speed of a Pelton wheel be increased? What
limits the maximum value of the specific speed?

6. How may the shape of the bucket at the entrance edge be determined?
How is the shape of the entire bucket fixed?

7. Suppose an impulse wheel is required to deliver 5,000 h.p. at 300
r.p.m. under a head of 1200 ft. Find the diameter of jet and the diameter
of wheel necessary.

8. What would be the approximate diameter of the orifice of the nozzle
tip in problem (7) ?

9. What would be the probable pitch of the buckets in problem (7) and
how many of them would be used on the wheel?

10. The bucket for the wheel in problem (7) may be laid out on the
drafting board.

CHAPTER XVI
DESIGN OF THE REACTION TURBINE

148. Introductory. — Assume that the head, speed, and power
for a proposed turbine are known, the speed and power of the
runner having been so chosen as to give the specific speed neces-
sary for the type of turbine desired. The type of runner will
have been selected in accordance with the principles and considera-
tions of the preceding chapters, so that, as the problem comes to
the designer, it is merely a matter of designing a turbine to fit
the specified conditions.

It has been seen that practically all dimensions, factors, and
even characteristics can be expressed as functions of the specific
speed, hence the latter is the logical key to design. After the
specific speed of the desired unit is known, the proper factors may
be selected in the light of previous experience, and the necessary
dimensions computed.

The data given in this chapter is to be understood as merely
typical of present practice. It is perfectly possible to alter any
of the quantities given, within certain limits, providing other
related factors are changed also. Consequently runners of the
same specific speed may be built without their being identical in
all other respects. Also, of the numerous variables, certain ones
are assumed and the rest computed to correspond. It is appar-
ent that the practice of designers may vary according to what is
assumed and what is computed, and hence the procedure given
here is not the only one that may be followed.

149. General Dimensions. — As explained in Art. 37, the value
of <t>e increases in rational design as the specific speed increases.
Customary values of this factor for different values of N8 are
given by a curve in Fig. 126. As stated in the preceding article,
this curve is not intended to be followed precisely, but the varia-
tion from it should not be too great.

Having selected a suitable value of fa, for the type of runner
desired, the diameter may be computed from equation (52), which
reduces to

211

212

HYDRAULIC TURBINES

As has also been explained, the ratio B/D is a function of specific
speed, and, choosing a value of this from Fig. 126, the height of
the runner at entrance or the height of the guide vanes may
at once be determined.

From equations (53) and (55) we may find the value of cr,
using whichever form happens to be more convenient.

6.01g

Cr =

BD\/h
0.0000157N,2

(65)

(66)

As an illustration of the possible variation in procedure, it may be
noted that we compute cr after assuming the value of B. It

Low Ns

C -

FIG. 125.

would be equally proper to assume a value of cr and compute the
corresponding value of B. It should also be noted that the above
factors involve the assumption that 5 per cent, of the total area
is taken up by the runner vanes. After the design has progressed
to the point where the number and thickness of the vanes can be
determined, the above may be corrected if that refinement is
deemed necessary.

From equation (39), letting cu = ce cos on, we have

cu = eh/2<f>e. (67)

In the case of high specific speed runners this value needs to be
increased about 5 per cent., but it is substantially correct for
lower speed runners. See Art. 92.

DESIGN OF THE REACTION TURBINE

213

100

.150

Ns - Metric

200 250 300 350

40 50 60 70

Ns - English Units

FIG. 126.

100

0.10

130
120

100

<a* W

•o

« 5°
40

30
20-
10

40 50 60

Ns - English Units

FIG. 127.

Extreme-High-»-
400

100

214 HYDRAULIC TURBINES

The direction of the absolute velocity of the water entering the
runner may now be found since

tan ai = cr/cu (68)

and the direction of the relative velocity, which should also be
the direction of the runner vane, is given by

tan 0i = cr/(cu - <f>e). (69)

The number of guide and runner vanes to be used is decided
somewhat arbitrarily, but one fundamental principle to be
observed is that they should not be equal to each other nor any
simple multiple, otherwise pulsations will be set up. For sim-
plicity of design and shop reasons it is convenient to make the
guide vanes a multiple of 4. Zowski's rule is that the number
of guide vanes, n', may be found by

n' = K'^/D (70)

where K' = 2.5 for on = 10° to 20°, 3.0 for on = 20° to 30°,
3.5 for on = 30° to 40°. Although K' increases with the specific
speed, the diameter of the runner decreases for the same power so
that actually the number of vanes is often less.

In order to avoid any pulsations the runner vanes are often
made an odd number, though other designers prefer to use an
eveji number which is 2 less than the number of guide vanes.
Zowski's rule is that the number of runner vanes, n, may be
fouid by

>^ n = K\/D (71)

- j *"""""! "" i

where K = 3.7 for a low specific speed, 3.0 for a medium specific
speed, and 2.2 for a high specific speed.

160. Profile of Runner. — The profile of a runner is shown in
Fig., 125 and the notation applied to it is clearly indicated. By
Dz is meant the diameter of the circle passing through the center
of gravity of the outflow area. In Fig. 34 were shown a few
typical profiles, and a more complete set is shown in Fig. 128.

Tne_ exact shape of profile desired is determined largely by
experience, a shape being used that had been found satisfactory
for the: specific speed in question. But it is also a matter of the
whim or taste of the designer, as theory has little bearing on it
directly. But the theory (Art. 66) does indicate that a very
sharp radius of curvature is undesirable and an excessive curva-
ture near the band, as is often found with certain high specific

DESIGN OF THE REACTION TURBINE

215

speed runners, simply results in the water failing to follow the
path desired, which may result in eddy losses, thus not only re-
ducing the efficiency but also facilitating corrosion at such places.
It is also considered desirable to have the length of the path
along the crown approximately equal to that along the band,
though it is frequently a little more.

With the values given by the curves in Fig. 126 and the aid
of the samples shown in Fig. 128 and elsewhere, it is possible to
lay out a profile that should be satisfactory, But before draw-
ing in the outflow edge, it is necessary to consider the stream
lines. Let us assume that all particles of water flow with the

Ns=10

Ns=30

Ns = 70 Ns=85

FIG. 128. — Typical profiles.

Ns = 100

same velocities and that the total rate of discharge is divided into
equal portions. If then the water passages are divided into
portions of equal cross-section area, it follows that the boundary
lines between them must be stream lines. Hence the height B
at entrance may be divided into equal portions, and for our pur-
pose here we shall assume four. Also the draft tube may be
divided into concentric rings of equal area, as shown in Fig. 125,
the section CC being removed far enough from the runner for
the stream lines to have become parallel. Then the curves in
between may be sketched in by eye. It may be noted that, if at
an intermediate section a line be drawn normal to these stream
lines, the product r A b must be constant along this normal.

216 HYDRAULIC TURBINES

In reality the velocity is not the same for all stream lines,
those nearer the band having a higher velocity than those near
the crown, because of the smaller radius of curvature. The
difference is very small in the case of the low speed runner, but
for the high speed runner the velocity near the crown may be
about 30 per cent, below the mean and that near the band about
50 per cent, above the mean velocity. This lowers the flow lines
at entrance below the positions as determined above. Also the
water in the draft tube, as in any other pipe, tends to flow with
a higher velocity in the center than around the circumference.
In accordance with these considerations the tentative flow lines,
as first sketched in, may be modified, according to judgement.
If further refinement is desired, this second set of flow lines may
be checked and corrected by the method given in Appendix B.

The outflow edge may now be drawn by making it perpendicu-
lar to these various stream lines. But for the portion near the
crown this procedure may tend to bring the discharge edge too
close to the axis of rotation. The theory (Art. 91) indicates
that a large variation in the radii to points along the outflow
edge is undesirable, and that the discharge edge should really be
parallel to the axis of rotation. The latter is not practicable,
but for this portion of the outflow edge a compromise is effected
by making it about a mean between a line parallel to the axis
and one which would be normal to all the stream lines.

It should be noted that in these profiles we are dealing with
circular projection, by which is meant that points are rotated
about the axis of the runner until they lie in the plane of the paper.
Thus the actual stream lines are not as shown in such a view,
the lines drawn being merely circular projections of the actual
paths.

151. Outflow Conditions and Clear Opening. — In case a stream
line is not perpendicular to the outflow edge it indicates that the
relative velocity of the water leaving the runner is not really
normal to the outflow area, as the latter would ordinarily be
measured. It is thus more convenient to deal with components
of the velocity in a plane normal to the discharge edge at the point
in question. Referring to Fig. 129, let us assume that the out-
flow edge is actually in the plane of the paper. If <*2 = 90° be
assumed to be the conditions for which the runner is to be de-
signed, the absolute velocities of the water at all points along
the outflow edge also lie in the plane of the paper but have the

DESIGN OF THE REACTION TURBINE 217

various directions indicated by the stream lines. The magni-
tude of the absolute velocity in the draft tube may be computed
from the rate of discharge and the area of the tube, provided there
is no whirl. The absolute velocity of the water discharging from
the runner vanes is somewhat more than this because of the space
taken up by the vanes, hence the Vz at discharge from the runner
should be larger than that in the tube by a factor which may be
assumed to be about 15 per cent. The absolute velocity of the
water is ordinarily assumed to be uniform all along the outflow
edge, but actually there may be some variation in it in certain
types of runner because of the varying radii of curvature of the
different stream lines.

Values of V% may be laid off along their respective stream lines
as indicated in Fig. 129, and, where they are not perpendicular
to the stream line, components indicated by TV should be found.
The latter will be used in the diagram below.

The linear velocity of a point on the runner may be laid off at
any radius perpendicular to a line representing this radius and
a third line then drawn so as to form a triangle. It is often
convenient to lay off Ui at radius r1} as shown in Fig. 129. At
any other radius the peripheral velocity is given by the inter-
cept. Since a2 = 90° and F2 (or F2') is known for each stream
line, a velocity diagram may be drawn with u2 as a base, and as
many of these constructed as there are stream lines. It should
be noted that the crown and band of the runner form boundaries
and hence furnish stream lines also. The series of diagrams so
constructed, as in Fig. 129, give the values of the relative
velocities and the direction of the runner vane at outflow for
various points along its edge, if the absolute velocity of dis-
charge is to be at 90°.

The clear opening is the shortest distance from a point on the
discharge edge of one vane to the back of the next vane. This
is shown in Fig. 129 and it may be seen that the clear opening is
practically equal to pitch X sin £'2 — vane thickness. The pitch
at any radius is known, since the number of runner vanes are
known. It may conveniently be found by laying off a value for
the pitch at some radius, similar to the procedure for the veloci-
ties above, and then the pitch at any other radius may be found
by using the proper intercept. This diagram should be drawn
to the same scale as the runner. The angle /3'2 is that found in
the velocity diagrams constructed by using values of V2. In

218

HYDRAULIC TURBINES

case this is not necessary, the angle of course becomes merely /32.
Referring to Fig. 129 again, it may be seen that the clear opening
may be found graphically at any point by laying off a line from
the end of the pitch distance perpendicular to the vector t/2

Fia. 129. — Determination of clear opening at outflow.

(or v2). The vane thickness may be subtracted from this line
and the remainder is the clear opening.

The clear opening, determined for each one of the stream lines
by the method just described, may be laid off from the outflow
edge of the runner in the form of an arc. The curve enveloping

DESIGN OF THE REACTION TURBINE 219

these arcs represents the clear opening. The clear opening is not
actually in the plane of the paper but each element of it is as-
sumed to be rotated about the line representing the outflow edge
until it does lie in the plane of the paper. This may introduce
some error where the curvature of the outflow edge is very
marked. Since the angle through which it is rotated is usually
not very great, the error is small and hence this area in the
plane of the paper may be taken as the true outflow area between
two runner vanes.

The rate of discharge through any section of the outflow area
may now be determined by multiplying each sectional area by
the average value of the component of the relative velocity
through it. In reality the true outflow area is normal to the
true relative velocity, but in case the latter is not in the plane
perpendicular to outflow edge, we obtain the same product by
the method used.

The rate of discharge for the turbine may now be computed as

q = c nSt/2Aa2 (72)

where n denotes the number of vanes, v'z the average value of the
relative velocity for the section considered and in the plane
defined, and Aa2 the element of area between two stream lines.
The coefficient of discharge may be taken as 0.95 for low specific
speed runners, 0.90 for medium specific speed runners, and 0.85
for high specific speed runners. This is really a coefficient of
contraction, since the actual stream areas are less than the areas
at the end of the converging passages.

If the rate of discharge is not the exact quantity required, the
outflow area may be altered somewhat by shifting the position
of the crown or by changing the outflow edge until the desired
result is obtained. It may be noted that the friction in flow
through the runner passages will be less along the middle stream
lines than for those near the crown and band. The relative
velocity will thus be a little higher in the middle and to preserve
a " radial" discharge all along the edge it will be necessary to
decrease the angle 02, which can be done by increasing the clear
opening a trifle along the middle portion. By the same line of
reasoning the opening may be reduced slightly at each end.

152. Layout of Vane on Developed Cones. — By the methods
previously given, the conditions at entrance to and discharge
from a turbine runner may be determined. Theory does not

220

HYDRAULIC TURBINES

prescribe the form of vane between these two portions, except
that it must be a smooth surface of gradual curvature to avoid
eddy losses. In order to determine the form of this surface, it
is convenient to lay out the vanes on developed cones.

In Fig. 130 are shown several lines which are elements of cones
whose axes coincide with the axis of rotation of the runner.

These elements should be so taken as to cut both the discharge
and the entrance edges and it is desirable also to have them
approximately perpendicular to the outflow edge. If this re-
quirement in some instances causes the vertex of a cone to be
removed to too great a distance, a cylinder may be used instead,
the cylinder being a special case of the cone.

It is desirable to begin with the cone nearest the crown, as

DESIGN OF THE REACTION TURBINE

221

cone A in Fig. 130. This is developed in Fig. 131, the distances
OM and ON being equal respectively in both figures. Along the
arc through M is laid off a pitch distance MP, corresponding to
the actual pitch of the runner at M. (This is the pitch at the
radius of M from the axis of rotation and not at the radius OM) .

FIG. 131. — Layout of vanes on developed cones.

From P are described two arcs whose radii are equal to the clear
opening at N and the clear opening plus the vane thickness.
From M the two sides of a vane are then drawn tangent to these
arcs, as shown. The actual vane is sharpened on the end, so as
to minimize eddy losses. The front side of the vane from M
tangent to the outer arc should be practically a straight line. In

222

HYDRAULIC TURBINES

theory the portion along the back of the vane as far as point S
should be a spiral of the form r = constant + KB' (K being a
constant and 0' the angle subtended at 0), or an involute so as
to keep the cross-section area of the stream constant from PS
to the arc MP. Actually this is not usually convenient, but the
actual shape is between such a curve and a straight line. Thus
the shape of discharge ends of the vanes are completely fixed.
It may be noted that the clear opening PS is a constant distance
for any cone taken through point M in Fig. 130.

In the case of the entrance ends of the vanes we might proceed
in the same way, but it is usually more convenient to lay off the
vane angle instead. It may be noted however that the value of

FIG. 132.

the angle on the developed cone is different for different cones
through the same point. In the most general case the entrance
edge of a runner may be inclined to the axis of rotation at an
angle 7 as shown in Fig. 132, and also it may not be in the same
plane as the axis but in another plane at an angle e. Let the
elements of the cone make an angle 6 with the axis, while the
projected stream line at entrance makes an angle 6 with the
axis. It may be noted that the actual velocity diagram should
always be in the plane of the stream line, and is not necessarily
in a plane perpendicular to the axis. If the angle of the relative
velocity ft becomes 0"i on the developed coWe, it may be proven
by geometry and trigonometry that

sin (8 + T)

sin (B + 7)

tan /3"i = tan ft

sin (0 -

(73)

DESIGN OF THE REACTION TURBINE 223

If the angle e is zero, as it is in many turbine runners, the expres-
sion in the brackets becomes equal to unity and the above
formula is greatly simplified.

The angle (3"i may be laid off as'at Rf in^Fig. 131*and the vane
tip then moved along until it reaches such a position that a good
smooth curve may join this end of the vane with the portion MS.
The complete vane is now MR. The next vane PQ should be
drawn in order to find out if the cross-section area continuously
decreases from entrance to outflow.

A similar procedure may be gone through with for the other
cones or cylinders, except for one restriction. The relative
position of M and-R is purely arbitrary in the first cone used,
but for all the remainder this much is fixed. Psually runners
are so constructed that all points along the outflow edge are in
one plane and furthermore this plane contains the axis of rota-
tion. Our discussion will therefore be confined to this case,
though the method could readily be extended to the more general
treatment, if desired. If all points along the entrance edge are
in the same plane and this plane contains the axis (the angle e
being zero), the arc NR subtends an equal number of pitches or
fractions thereof in every cone or cylinder. This is most con-
veniently laid off on the drawing board by establishing an arc
at a fixed radius from -the axis of rotation and putting this in
every cone. Its length is the same in every -case. It is conven-
ient to take this arc through the point where the diameter D is
measured.

If the entrance edge is inclined, as indicated by the angle e
in Fig. 132, the runner vane near the crown subtends a greater
angle than that portion nearer the band, and a different length
of arc is used in different cones.

If it is difficult or impossible to secure proper vane curves
in some of the cones, it may be necessary to go back to cone A and
to change the position of R so that the vane MR subtends a
different angle. One advantage of inclining the entrance edge,
so that it makes the angle as shown, is that it permits of securing
better vane curves in all the cones, in some instances. This is
particularly true in the case of high specific speed runners.

153. Intermediate Profiles. — The various cones and cylinders
of Fig. 131 are next divided up into fractional pitches, preferably
quarter pitches. These lines represent a series of planes passing
through the axis and cutting the vane. The model, 'the photo-

224

HYDRAULIC TURBINES

graph of which is shown in Fig. 133, illustrates this very nicely
and shows the intermediate profiles cut out. For cone A the
distances from 0 to the intersections of planes I, II, III, IV, etc.
with the vane are transferred from Fig. 131 to Fig. 130, being laid
off from 0 along the line OMN. A similar procedure is followed
for all the cones and cylinders.

Since planes I, II, III, IV, etc. are the same in every cone,
it follows that profiles can be drawn through all the points with
the same number in Fig. 130. If the vane surface is proper,

FIG. 133. — Model constructed by Lewis F. Moody illustrating development of

vane surface.

these profiles will all be smooth curves, and will all be similar
but changing gradually from the'jentrance^edge to the discharge
edge. If the curves are not smooth and of the proper shape, it
will be necessary to change the vanes laid out on the developed
cones until both the profiles and the curves^on all thejcones are
satisfactory. Thus these profiles serve as^a check on the work,
and also are desirable in order to determine the pattern maker's
sections.

DESIGN OF THE REACTION TURBINE

225

154. Pattern Maker's Sections. — If a plane be passed through
Fig. 130 normal to the axis of the runner, the vane will cut a
curve in it which may be found as follows. In Fig. 134 a portion
of this plane is drawn and it is subdivided into the same fractional
pitches as the various developed cones. The distance XY may
be transferred from Fig. 130 to Fig. 134 and laid off in plane 7.

FIG. 134. — Patternmaker's sections.

The distance XZ may be transferred in similar manner and laid
off along plane //. Proceeding in this way the entire curve may
be established.

A series of parallel planes at fixed distances apart are used
and curves constructed for all of them in the same manner, but
on the same drawing, as in Fig. 134. Where the curvature of
the vane is sharper the planes are spaced closer together, as shown

226

HYDRAULIC TURBINES

in Fig. 133. These planes really represent the surfaces of boards
of different thicknesses and on each board the proper curve from
Fig. 134 may be laid out. These boards, when placed together,
as in Fig. 135, and the surface smoothed down, give the proper
shape of the vane surface. In this way the core box for the vane
may be formed.

Since the vane has both a front and a back surface which differ
slightly from each other, this entire proceeding is carried through

FIG. 135.-

(Courtesy of Wellman-Seaver-M organ Co.)
-Construction of pattern for rear face of core box for runner vane.

for both surfaces. It is desirable to draw the profiles and pattern
maker's sections for the front of the vane in full lines and for the
back of the vane as dotted lines. Also if the lines for the back
of the vane in Fig. 134 are placed one pitch distance away from
those for the front, one can more readily grasp the appearance of
the passageway between the two vanes.

DESIGN OF THE REACTION TURBINE

227

155. The Case and Speed Ring. — The velocity of the water in
a spiral case may range from 0.15 \/Zgh to 0.20 -\/2gh, the higher
factor being used for lower heads. For globe and cylinder cases
even lower velocities should be employed, because of the hydraulic
losses in such cases. A spiral case should be so proportioned that
equal quantities of water flow to equal portions of the runner, as
shown in Fig. 136. If the cross-section of the case is circular,
it may be seen that the radius of a point on the outer boundary
is given by r = \/cd + K where c and K are constants and 6
the subtended angle, as this curve will give an area which is
directly proportional to the angle. For any other change of
cross section, it is easy to determine the form necessary by apply-
ing the principle that the area must vary as the subtended angle
at the runner axis.

FIG. 136.

The'path of a free stream line in the case may be plotted by
the principles of Art. 66. If Vc be the velocity with which the
water enters the case and at radius rc, the tangential component
of the velocity at any other radius is given by

Vu = rc Vc/r

while the radial component is given by

Vr = q /2irrb

where r is the radius in feet to any point and b the height of the
water passage in feet. These two components give the direction
of the water at any radius, and by sketching in a series of tangents
it is easy to plot the path by a little trial.

If speed ring vanes are used, they should be so shaped as to

228

HYDRAULIC TURBINES

conform to these free stream lines. The number of speed ring
vanes should be half that of the number of guide vanes.

156. The Guide Vanes. — The guide vanes should be so shaped
that they are tangent to the free stream lines of the water entering.
If the turbine is set in an open flume or case where free stream lines
cannot readily be plotted, the guide vanes are made so as to
approach a radial direction at this point.

The direction of the water is changed during flow through the
guide passages from that of the free stream line to the direction
desired. After a particle of water passes point a in Fig. 137, its
direction should remain unchanged until it strikes the runner, as
it is now following a free stream line once more. Since the space
between guide vanes and runner vanes is one of a uniform height,
it may be shown that free stream lines are then equi-angular or

FIG. 137.

logarithmic spirals. Thus the portion of the vane from a to &
should be such a curve. The other side of the vane may be
either a straight line or another logarithmic spiral. The equation
of the equi-angular spiral is loge r = 0 tan a, where a is the angle
desired and 6 is the subtended angle.

There should be considerable clearance between the ends of
the guide vanes and the runner vanes so that the streams of water
from the guides may unite into a solid ring before entering the
runner. In particular the point of intersection o of Fig. 137
should be located outside the runner, so that no eddies maybe
produced unnecessarily in the latter.

The gates are sometimes pivoted near the point so that, if the
governor mechanism fails, the gates will drift shut. But this
places the vane shaft at a point where the section of the vane

DESIGN OF THE REACTION TURBINE 229

should be small, and also makes it necessary to exert a consider-
able torque to hold them wide open. The better practice is to
so locate the pivot that x/y = 3/2. The gates are then hydrauli-
cally balanced when about half way open and the torque resuired
for either extreme position is less.

157. QUESTIONS AND PROBLEMS

1. Given the head, speed and power for a reaction turbine, how may the
size of the runner, the height of the guide vanes, and the diameter of the
draft tube be determined?

2. For the case in problem (1), how would the guide vane angle and the
runner vane angle be determined? What principles are involved in deciding
upon the number of guide vanes and runner vanes?

3. How is the profile of a runner to be fully determined? How should the
stream lines be drawn in?

4. How may the clear opening of a turbine runner be determined?

5. How may the capacity of a runner be checked? What changes can
be made in order that its capacity may be exactly that desired?

6. How are runner vanes laid out on developed cones?

7. Having the vanes laid out on developed cones, how may the inter-
mediate profiles be constructed? What use is made of these?

8. How are pattern maker's sections drawn?

9. What is the object of plotting the free stream lines in a spiral case?

10. How should guide vanes be shaped? What other factors should be
considered in their -design?

11. A turbine runner is to deliver 4000 h.p. at 600 r.p.m. under a head of
305 ft. Determine D, B, Dt, D*, «i, /3'i, V2, number of guide vanes, and
number of runner vanes.

Ans. D = 37 in., B = 6.67 in., Dt = 34.4 in., Dd = 33.3 in., ttl = 16°,
p\ = 102°, F2 = 25.6 ft. per second, 24 guide vanes, and 22 runner vanes.

12. A turbine runner is to be designed for 2000 h.p. at 300 r.p.m. under a
head of 88 ft. Find same as in problem (11).

Ans. D = 43.2 in., B = 16.4 in., «i = 20°, ffl = 125°, 20 guide vanes,
and 18 runner vanes.

13. A runner is to be designed to deliver 3000 h.p. at 200 r.p.m. under a
head of 64 ft. Find the results called for in problem (11).

14. Find the allowable height above the tail water level for each of the
runners in the preceding three problems.

15. Draw profile, sketch tentative flow lines, construct velocity diagrams,
lay out vanes on cones, and draw patternmaker's sections for one of the
turbines given above.

CHAPTER XVII
CENTRIFUGAL PUMPS

158. Definition. — Centrifugal pumps are so called because of
the fact that centrifugal force or the variation of pressure due to
rotation is an important factor in their operation. However,
as will be shown later, there are other items which enter.

The centrifugal pump is closely allied to the reaction turbine
and may be said to be a reversed turbine in many respects.
Therefore it will be found that most of the general principles
given in Chapter VII will apply here also with suitable modifica-
tions. Energy is now given up by the vanes of the impeller

FIG. 138. — Turbine pump.

FIG. 139. — Volute pump.

to the water and we have to deal with a lift instead of a fall.
The direction of flow through the impeller is radially outward.
During this flow both the pressure and the velocity of the water
are increased and when the water leaves the impeller a large
part of its energy is kinetic. In any efficient pump it is neces-
sary to conserve this kinetic energy and transform it into pressure.
159. Classification. — Centrifugal pumps are broadly divided
into two classes:

1. Turbine Pumps.

2. Volute Pumps.

230

CENTRIFUGAL PUMPS

231

While there are other types besides these, the two given are
the most important, and only these will be considered in this-
chapter.

The turbine pump is one in which the impeller is surrounded by
a diffusion ring containing diffusion vanes. These provide
gradually enlarging passages whose function is to reduce the
velocity of the water leaving the impeller and efficiently trans-
form velocity head into pressure head. The casing surrounding
the diffuser may be either circular as shown in Fig. 138 or it may
be of a spiral form. This latter arrangement would be similar
to that of the spiral case turbine shown in Fig. 55.

The volute pump is one which has no diffusion vanes, but,
instead, the casing is of a spiral type so made as to gradually
reduce the velocity of the water as
it flows from the impeller to the dis-
charge pipe. (See Fig. 139.) Thus
the energy transformation is ac-
complished in a different way. The
spiral curve for such a case is usu-
ally called the volute, and from this
the pump receives its name.

The discussion of the volute pump
will apply equally well to all other
types without diffusion vanes. The
only difference will be that these
other types are less efficient and also it will probably be im-
possible to express the shock loss at exit in any satisfactory way.
Some of these other types have circular cases with the impeller
placed either concentric or eccentric within them. Their only
merit is cheapness.

160. Centrifugal Action. — If a vessel containing water or any
liquid is rotated at a unifrom rate about its axis, the water will
tend to rotate at the same speed and the surface will assume a
curve as shown in Fig. 140. This curve can be shown to be a
parabola such that h = Uz2/2g, where u^ = linear velocity of
vessel at radius r2. If the water be confined so that its surface
cannot change, the pressure will follow the same law, as shown
in Art. 65.

If, as in Fig. 141, we have the water in a closed chamber set in
motion by a paddle wheel, the pressure in an outer chamber
communicating with it will be greater than that in the center

FIG. 140.

232

HYDRAULIC TURBINES

by the amount u22/2g. If a piezometer tube be inserted in this
chamber, water will rise in it to a height h = uz*/2g. If the
height of the tube be somewhat less than this, water will flow
out, and we would have a crude centrifugal pump.

161. Notation. — The notation used will be essentially the same
as that for the turbine. To this, however, we shall add a'2
as the angle the diffusion vanes make with uz, and subscript (3) to
denote a point in the casing.

FIG. 141.

The actual lift of the pump will be denoted by h, while the head
that is imparted to the water by the impeller will be denoted by
h". If h' represents all the hydraulic losses within the pump and
eh represents the hydraulic efficiency, we may write

ehh" = h" - h'

(74)

It will also be found to be more convenient to express all veloc-
ities in terms of u2 and v%.

Whereas turbines are rated according to the diameter of the
runner, centrifugal pumps are rated according to the diameter
of the discharge pipe in inches. The usual velocity of flow at the
discharge is 10 ft. per second. From this the size of pump nee-

CENTRIFUGAL PUMPS 233

essary for a given capacity may be approximately estimated.
In some cases, however, the velocity may be twice this value.

162. Definition of Head and Efficiency. — In all cases the head
h under which the pump operates is the actual vertical height
the water is lifted plus all losses in the suction and discharge
pipes. It should be noted that the velocity head at the mouth
of the discharge pipe is a discharge loss which should be added.

The head may also be obtained in a test by taking the dif-
ference between the total heads (Equation 3) on the suction
and discharge sides of the pump. If the suffix (S) signifies a
point in the suction pipe and suffix (D) a point in the discharge
pipe we have

P° Pa VD2 TV

In this case PD/W represents the pressure gage reading reduced
to feet of water while PS/W represents the suction gage reading
reduced to feet of water. In general the latter pressure will be
less than that of the atmosphere. In such a case ps/w will be
negative in value.

The word efficiency without any qualification will always
denote gross efficiency, that is the ratio of the power delivered
in the water to the power necessary to run the pump. The hy-
draulic efficiency is the ratio of the power delivered in the water
to the power necessary to run the pump after bearing friction,
disk friction, and other mechanical losses are deducted. The
hydraulic efficiency is therefore equal to Wh/Wh" or h/h".
This latter expression is termed manometric efficiency by some
and is treated as something essentially different from hydraulic
efficiency. If the true value of h" could be computed, the value
of the hydraulic efficiency so obtained would be the same as
that obtained experimentally by deducting mechanical losses
from the power necessary to drive the pump. ActuallyA-the
ratio of h/h" will usually be less than this value but that Js due
to the fact that our theory is imperfect. (Art. 167.)

163. Head Imparted to Water. — By reversing equation (19)
in Art. 60, since we are now dealing with a'pump and not a tur-
bine, we may write

h" = 7 = \ (u*Vu* ~ MlVwi)'
"k y

This would be very appropriate, if the pump were fitted with

234 HYDRAULIC TURBINES

stationary guide vanes in the center of the impeller to direct the
water entering. Occasionally centrifugal pumps are so built,
but for the usual type of pump, we may say that whatever angu-
lar momentum the water has, as it enters the impeller, it has
received from the latter, through the medium of intervening
particles of water. This is proven by the fact that the water in
the eye of the impeller and even in the suction pipe may be set
into rotation. The effect is as if the vanes of the impeller,
extended to this space. For this reason we shall drop the last
term in the above equation and write

h" = - UtVu, = -2 (u2 + v2 cos /32) (76)

y y

By another line of reasoning, or by a slight transformation .of
equation (76), we may obtain

''-t'-i+g'

Sometimes one of these forms is more convenient than the other.

Inspection of equation (76) shows that if the pump is to do
positive work, VuZ must be positive. Thus the absolute velocity
of the water must be directed so as to have a component in the
direction of rotation. If the pump speed, w2, be assumed constant,
equation (76) will plot as a straight line for values of vz (or q) .
If j82 is less than 90°, the value of h" will increase as the rate of
discharge increases above zero. If /32 is equal to 90°, h" will be
independent of the rate of discharge and will plot as a horizontal
line for all values of v2. If £2 is greater than 90°, the value of
h" will decrease as the rate of discharge increases. (See Fig. 144.)

Since it is difficult to transform velocity head into pressure
head without considerable loss, it is desirable to keep the abso-
lute velocity of the water leaving the impeller as small as pos-
sible. For that reason the best pumps have vane angles as near
180° as possible in order that the relative velocity may be nearly
opposite to the peripheral velocity of the impeller.

164. Losses. — In accordance with the usual methods in
hydraulics, the friction loss in flow through the impeller may be
represented by kv22/2g, where k is an experimental constant.
A study of Fig. 142 would indicate that there is no abrupt change
of velocity at entrance to the impeller under any rate of flow;
there is then no marked shock loss at entrance that would re-

CENTRIFUGAL PUMPS

235

quire the use of a separate expression as whatever loss there is
may be covered by the value of k. Where the water leaves the
impeller, however, there is an important shock loss that follows a
different law from the friction loss.

For the turbine pump this shock loss is similar to that in the
case of the reaction turbine in Art. 86. Referring to Fig. 143, it

FIG. 142. — Velocity diagram for three rates of discharge.

may be seen that the velocity F2 and the angle az will be deter-
mined by the vectors HZ and v2. Since the vane angle a' 2 is
fixed there can be only one value of the discharge that does not
involve a shock loss. For any other value of the discharge the
velocity F2 will be forced to become V'% with a resultant loss

236

HYDRAULIC TURBINES

which may be represented by (CC')2/2g. Since the area of the
diffusion ring normal to the radius should equal the area of the
impeller normal to the radius, the normal component (i.e., per-
pendicular to u2) of V2 should equal that of V*. Therefore CC'
is parallel to u2 and its value may be found to be

= U2

If A;'

-a'8)

sn ft 2 - <

•—• - /
sm a'2

sin a 2
is approximately equal to

then for the turbine pump the shock loss

(u2 -

For the turbine pump the total hydraulic loss may be repre-
sented by

7 / 7 ^2 i V^2 *C ^2/ /^TO\

ft =fc2^+ ^7" (78)

Since the volute pump has no diffusion vanes, there will be no
abrupt change in the direction of the water at exit from the im-
peller, but there may be an abrupt change in the magnitude of
the velocity. The water leaves the impeller with a velocity V2
and enters the body of water in the case which is moving with a
velocity V$. In accordance with the usual law in hydraulics
this shock loss may be represented by

For the usual type of pump F2 will decrease as the discharge in-
creases, and in any case F3 must increase as the quantity of
water becomes greater. If the discharge becomes such that the
two are equal then there will be no shock loss. The value of V2
may be expressed in terms of u2 and v2, and if the ratio of (a2/A3)
be denoted by n, we have V$ = nv2. Making these substitutions

*L. M. Hoskins, "Hydraulics," p. 237.

CENTRIFUGAL PUMPS 237

the total hydraulic loss for the volute pump may be represented
by

, , _ , , 2 + 2u2v2 cos

Though the values of k may be different for the two types and
though the expressions for shock loss are unlike in appearance,
yet it can be seen that the losses in each case follow the same
general kind of a law. In the turbine pump we have a gradual
reduction of velocity but, except for one value of discharge, a
sudden change in direction as the water leaves the impeller.
With the volute pump we have no abrupt change of direction but;
a sudden change of velocity. The transformation of kinetic
energy into pressure energy is incomplete in either case, but it
is generally believed that the loss is somewhat greater in the
volute pump than in the turbine pump.

For an infinitesimal discharge the value of the velocity in the
case, Vs, would be practically zero. Therefore a particle of
water leaving the impeller with a velocity V2 and entering a body
of water at rest would lose all its kinetic energy. For such a
case, however, the value of v2 would be also practically zero so
that V2 would equal u2. Therefore for a very slight discharge the
shock loss would be hf = u22/2g. Such a value of h' may be ob-
tained from either (78) or (79) by putting v2 = 0.

165. Head of Impending Delivery. — The head developed by
the pump when no flow occurs is called the shut-off head or the
head of impending delivery. We are then concerned only with
the centrifugal head or the height of a column of water sustained
by centrifugal force. In Art. 160 this was shown to be equal to
u22/2g. The same result may be obtained from the principles
of Art. 163 and Art. 164. If v2 becomes zero, then by equation
(76), h" = u^/g = 2 u22/2g. But, as was shown in Art. 164, the
loss of head, h' = u22/2g. Therefore h = h" - In! = u22/2g.

Although ideally the head of impending delivery equals u22/2g,
we find that various pumps give values either above or below
that. This may be accounted for in a number of ways. In any
pump we never have a real case of zero discharge; for a small
amount of water, about 5 per cent, of the total rated capacity
perhaps, will be short circuited through the clearance spaces. A
pump is said to have a rising characteristic if, when run at con-
stant speed, the head increases as the discharge increases above
zero until a certain value is reached and then begins to decrease.

238 HYDRAULIC TURBINES

If the head continually decreases as the discharge increases above
zero, the pump is said to have a falling characteristic. Thus the
leakage through the clearance spaces will tend to make the shut-
off head greater or less than u22/2g according to whether the pump
has a rising or a falling characteristic. The more the vanes are
directed backward, the more tendency there is for internal eddies
to be set up and these tend to decrease the head. Also if the
water in the eye of the impeller is not set in rotation at the same
speed as the impeller the head may be further reduced. There
is also a tendency for the water surrounding the impeller to be
set in rotation but this, on the other hand, helps to increase the
head since the real value of rz is greater than the nominal value.

It will usually be found that actually the head of impending

u 2
delivery may be from 0.9 to 1.1 -~-"

166. Relation between Head, Speed and Discharge. — When
flow occurs the above relation no longer holds, for other factors
besides centrifugal force enter in. Due to conversion of velocity
head into pressure head when water flows, a lift may be obtained
which is greater than u22/2g. (See Fig. 144.)

This may be shown best by equation (77), when the losses are
introduced. The hydraulic friction loss in flow through the im-
peller may be represented by kv22/2g. Then at discharge from
the impeller a portion of the kinetic energy is lost within the
diffuser or within the volute case, and the remainder may be
represented by mV22/2g, where m is a factor less than unity.
Deducting the losses from the expression for h" in equation (77)
we have

5;|W;'^>||*W (80)

If the factor involving V2 is greater than that with v2 the head will
be greater than the shut-off head, while the reverse is true if it
is less.

In order to produce a pump with a rising characteristic, it is
not only necessary to conserve the kinetic energy of the water
discharged from the impeller, or in other words to keep the factor
m high, but it is also necessary to have V2 large and v2 small.
But a pump with a falling characteristic is not necessarily any less
efficient than the former type. The factor m may be high but yet
V2 may be made low and v2 high. In order to accomplish these
results, it may be seen that the vane angle /32 has some influence,

CENTRIFUGAL PUMPS

239

but.it is not the sole determining factor that many have supposed.
It must be noted that we are concerned with h, which is a very
different quantity from h", and hence the remarks in Art. 163
cannot be applied here. In fact it has been shown that pumps
with vane angles greater than 90,° and in fact as large as 154,°

Discharge
FIG. 144. — Ideal curves for a turbine punp.

may manifest decidedly rising characteristics, while certain im-
pellers with radial vanes have given steep falling characteristics
and not flat characteristics.1

While the above equation is very satisfactory in explaining
how the value of h may either increase or decrease, and in comput-

J See the author'^" Centrifugal Pumps."

240 HYDRAULIC TURBINES

ing its value for some specified condition, such as that for maxi-
mum efficiency, it is not the best form of equation for showing the
complete characteristic. This is largely due to the fact that the
factor m is a variable, ranging all the way from zero up to a
maximum of about 0.75 at the condition for highest efficiency,
and also the rate of discharge affects two different variables v2
and Vz, which are in reality related to each other. For some
purposes, therefore, it is better to derive the following forms
of equations.

The actual lift of the pump h may be obtained by subtracting
the losses hr from the head h" imparted by the impeller. The

, ,„ .„ i u2 (uz + v% cos 02) , c , ,

value of h will be taken as — - and the values of h

are given in equations (78) and (79).

Making these substitutions for the turbine pump we obtain
after reduction

U22 + 2 (kf + cos/32) u2v2 - (k + &'2) vz2 = 2gh. (81)
For the volute pump we obtain after rearranging

2nv2 Vw22 2uzv2 cos

These equations involve the relation between the three vari-
ables u2, vz, and h. Any one of these may be taken as constant
and the curve for the other two plotted. If the pump is to run
at various speeds under a constant head, the latter will then be
fixed and we may determine the relation between speed and
discharge. The more common case is for the pump to run at a
constant speed. For that case values of h may be computed for
different values of v2. The curves for a turbine pump run at
constant speed are shown in Fig. 144.

Although it will not be done here, it will be found convenient
to introduce ratios or factors as was done in the case of the tur-
bine. We may write uz = <t>\/2gh and v2 = c\/2gh and using
these in equations (81) and (82) we obtain relations between c
and (f> similar to equation (40). As in the case of the turbine
it will be found that the best efficiency will be obtained for a
certain value of <f> and c. It will thus be clear that the speed of
the pump should vary as the square root of the lift, and that the
best, value of the discharge will be proportional to the square

root of the lift. Since h = Tii it is apparent that the lift

CENTRIFUGAL PUMPS

241

varies as the square of the speed. If this value of h be substi-
tuted in v2 = c\/2gh we obtain v 2 = I— J uz, and this shows that the

best value of the discharge varies directly as- the speed.

Curves between c and </> will be of the same appearance as those
drawn for a constant value of h. To construct curves of the same
shape as those drawn for a constant speed it will be necessary to

plot values of (—^ and of I— ) •

Discharge
FIG. 145. — Actual curves for turbine pump.

The value of 4> for the maximum efficiency depends upon the
design of the pump. By choosing different values of @2 and either
a '2 or n, and different numbers of impeller vanes and other fac-
tors, a pump may be given a rising or a flat or a steep falling
characteristic. The values of <f> for the highest efficiency range
from about 1.30 down to about 0.90. This means that the
normal head is usually

h = 0.6 to l.l^2-

The value of cr, the coefficient of the radial velocity at the point
of outflow from the impeller, is usually from 0.05 to 0.15. All
formulas and values in this chapter are based upon the
head developed per stage.

242

HYDRAULIC TURBINES

167. Defects of Theory. — The discussion of the defects of
the theory of the reaction turbine in Art. 92 applies equally
well to the centrifugal pump. Probably one of the greatest
sources of error lies in the assumptions made regarding losses.
In particular the expressions for shock loss for either the tur-
bine or the volute pump must be regarded as only rough
approximations.

Discharge
FIG. 146. — Curves for pump run at different speeds.

While actual tests have shown curves similar to the ideal
curves given in Fig. 144, it is more common to find the relation
between head and discharge to be like that in Fig. 145. In many
cases also the pump has a falling characteristic so that the head
for any delivery is less than the shut-off head. But at the same
time the gross efficiency will be high and the hydraulic efficiency
must be still higher. Since h" = h/e it will be seen that the

CENTRIFUGAL PUMPS 243

actual h" must be of the form shown in Fig. 145. This accounts
for the discrepancy between the so-called manometric efficiency
and the true hydraulic efficiency.

The reasons for this are the same as those given for the reaction
turbine. In addition there is strong reason for believing that
there is a dead water space on the rear of each vane, thus the
actual area a2 will be less than the nominal area used in the
computations. This is probably a larger item than the contrac-
tion of the streams mentioned in connection with the turbine.
The ordinary pump has no guide vanes at entrance to the impeller
and the conditions of flow at that point are uncertain.

The more vanes the impeller has the more perfectly the water
is guided and the more nearly the actual curves approach the
ideal. It is necessary to have enough vanes to guide the water
fairly well but too many of them cause an excessive amount of
hydraulic friction. Within reasonable limits — say 6 to 24 — the
efficiency is but little affected. If the use of few vanes lowers
the value of h, the value of h" is lowered at about the same rate
so that the ratio of the two is but little altered.

168. Efficiency of a Given Pump. — If a given pump is run
at different speeds the lift should vary as the square of the speed,
the discharge as the speed, and the water h.p. as the cube of
the speed. If the efficiency of the pump remained constant the
horsepower necessary to run the pump would also vary as the
cube of the speed. It is probable that the hydraulic efficiency
is reasonably independent of the speed. The mechanical losses,
however, do not vary as the cube of the speed. For low speeds
the mechanical losses do not increase so fast and thus the gross
efficiency of the pump will improve as it is used under higher
heads at higher speeds. After a certain limit is reached, how-
ever, the mechanical losses follow another law and for very high
speeds they will increase faster than the hydraulic losses and
the efficiency will begin to decline. Thus for a given. pump
run at increasing speeds the maximum efficiency will increase
and then decrease again. It is thus clear that the head which
may be efficiently developed with a single stage is limited.
For higher heads it is necessary to resort to multi-stages.

These conclusions regarding efficiency are borne out by the
curves shown inJFig. 146 and Fig. 147. In the latter the highest
speed attained was not sufficient for the efficiency to begin to
decrease* again, though it had evidently reached its limit.

244

HYDRAULIC TURBINES

In a set of curves such as are given in Fig. 146, the operation
of the pump under a constant head can be determined by fol-
lowing a horizontal line. For a constant discharge follow a ver-
tical line, and to determine the conditions for maximum efficiency
follow the curved dotted line. The values of the maximum
efficiency will be given by the curve tangent to the peaks of all
the efficiency curves for the various speeds.

900 1000 1100 . 1200 1300 1400 1500 1600 1700

R.P.M.

FIG. 147. — Characteristic curve of a 4-stage turbine pump.

1800

169. Efficiency of Series of Pumps. — For a given pump the
speed and head are seen to have some influence upon its efficiency.
However, the capacity for which it is designed is the greatest
factor. Suppose we have a series of impellers of the same
diameter and same angles running at the same speed, the lift
will be approximately the same for all of them. Suppose, how-
ever, that the impellers are of different widths. The discharge
will then be proportional to the width and the water horsepower
is proportional to the discharge. But the bearing friction and the
disk friction are practically the same for all of them. In addition
the hydraulic friction in the narrow impellers will be greater
than that in the larger ones. It is therefore evident that the
efficiency of the high-capacity impellers will be much greater
than that of the low-capacity impellers. This is true to such

CENTRIFUGAL PUMPS

245

an extent that the efficiency of a centrifugal pump may be said
to be a function of the capacity. (See Fig. 148.)

Very large pumps have given efficiencies around 90 per cent.
Single suction pumps have slightly lower efficiencies than double
suction pumps.

170. Specific Speed of Centrifugal Pumps. — The specific speed
factor for centrifugal pumps is as useful as that for hydraulic
turbines. By it, we can at once determine the conditions that
are possible for a pump of existing design, and can also select the
most suitable combination of factors for a proposed pump for
any case. It also serves to classify pump impellers as to type
in the same way that it indicates the type of turbine runner.

100
90

I 8°
£ 70

I 60
I
W 50

40
30

2,000

10,000

4,000 6,000 8,000

Capacity in Gal. per Min.
FIG. 148. — Efficiency as a function of capacity,

12,000

14,000

Thus a low value of the specific speed indicates a narrow impeller
of large diameter, while the reverse is true for a high value.

For the centrifugal pump, however, it is more convenient to
use a different form for this factor than for the turbine. We
are not primarily concerned with the power required to drive
a pump, but have our attention centered first upon its capacity.
But since the capacity and power are directly related, it is seen
that we are merely expressing the specific speed in different
units. Since, as in the case of the turbine, q = KiD^^/h, and
N = 18400 \/h/D, we may eliminate D between the two equa-
tions and obtain

• -N't = 18400 V^i =

246

HYDRAULIC TURBINES

But centrifugal pumps are usually rated in gallons per minute
rather than in cubic feet per second and it may be more con-
venient to express the above as

N. =

N\/G.P.M.

(84)

Since 448 G.P.M. = 1 cu. ft. per second, it may be seen that
N. = 2l.2N'8.

For a single impeller, values of Na ordinarily range from 500
to 8000. This latter figure has been greatly exceeded in a few
cases of special types. Just as in the case of the turbine, the
efficiency may be expressed as a function of the specific speed,
as is shown in Fig. 149.

100

-60

1,000

2,000 3,000 4,000

Specific Speed, Ns -N)^

5,000

6,000

7,000

FIG. 149. — Efficiency as a function of specific speed.

171. Conditions of Service. — Centrifugal pumps are used for
lifting water to all heights from a few feet to several thousand.
Several pumps have been built to work against a head of 2000
ft., though these are all multi-stage pumps. The usual head per
stage is not more than 100 to 200 ft., though this figure has been
exceeded in. numerous instances.

The capacities of centrifugal pumps ranges all the way from
very small values up to 300 cu. ft. per second or 134,500 gal. per
minute. Rotative speeds range ordinarily from 30 to 3000
r.p.m. according to circumstances. All the above figures are for
ordinary practice, and are not meant to be the limiting values
that can be used.

CENTRIFUGAL PUMPS 247

172. Construction. — The design and construction of centrifugal
pumps is very similar in principle to that of reaction turbines.
Impellers are made either single suction or double suction, ac-
cording to whether water is admitted at only one or both sides
of the impeller. The latter construction permits of a smaller
diameter of impeller for the same capacity.

Water leakage from the discharge to the suction side is minim-
ized by the use of clearance rings, as in the case of turbines, and
sometimes labyrinth rings are used so as to provide a more tor-
tuous passage for the leakage water. The leakage of air along
the shaft on the suction side should be prevented by a water seal
in addition to the usual packing.

The end thrust is taken care of by a thrust bearing, by sym-
metrical construction, as in the case of the double suction pump or
a multi-stage pump with impellers set back to back, or by use of
an automatic hydraulic balancing piston. The majority of multi-
stage pumps are built with the impellers all arranged the same
way in the case as this permits the most direct flow from one
impeller to the next and also simplifies the mechanical
construction.

173. QUESTIONS AND PROBLEMS

1. Why is the centrifugal pump so called? How does the pressure and
velocity of the water vary as it flows through such a pump?

2. What classes of centrifugal pumps are there, and how do they differ?

3. What is the difference between the head imparted to the water and
the head developed by the pump? How is the latter measured in a test?

4. What are the important hydraulic losses in the centrifugal pump?
What is meant by the head of impending delivery? What is its approximate
value?

5. How may the head vary with the rate of discharge, the speed being
constant? Why is this?

6. If a given centrifugal pump is run at a different speed how will the
head, rate of discharge, power, and efficiency vary, assuming that the con-
ditions are such that <j> is constant?

7. Why is the efficiency of a centrigufal pump a function of its capacity?

8. What is meant by the specific speed of a centrifugal pump? What
is the use of such a factor?

9. The diameter of the impeller of a single-stage centrifugal pump is 6 in.
If it runs at 2000 r.p.m., what will be the approximate value of the shut-off
head and the head for the rate of discharge corresponding to maximum
efficiency? Ans. 42.5 ft. and 30 ft.

10. What may be the maximum and minimum limits of the capacity of
a series of pumps of the same diameter as the single pump in problem (9)
and running at the same speed ?

248 HYDRAULIC TURBINES

11. What will be the answers to problems (9) and (10) if the pump is a
four-stage pump, all other data remaining the same?

12. If a single-stage centrifugal pump is to develop a shut-off head of 200
ft., what must be its r.p.m., if the impeller diameter is 18 in.?

Ans. 1446 r.p.m.

13. If the above pump were a two-stage unit, what would be the necessary
speed? Ans. 1045 r.p.m.

14. Compute the specific speed for the pumps in problems (12) and (13).

15. Compute the factors by which u<?/2g must be multiplied to give the
shut-off head and the head for highest efficiency for the pumps whose tests
are given in Appendix C, Tables 14 and 15.

APPENDIX A
THE RETARDATION CURVE

Let the relation between instantaneous speed and time be
represented by the curve shown in the figure. Let
N = r.p.m.
t = seconds.

s = length of subnormals in inches.
x = distance in inches.
y = distance in inches.
m = seconds per inch.
n = r.p.m. per inch.

/ = moment of inertia of the rotating mass in ft.-lb. sec.2
units.

FIG. 150.

Thus y = N/n and x = t/m

co = radians per second = 2irN/60. da>/dt = (2<,r/6Q)dN/dt.

From mechanics, Torque = Idu/dt. Power = Iwdw/dt.

Power = (2ir/wyiNdN/dt

Tan 0 = dy/dx = (dN/n) + (dt/m).

But also tan <£ = s/y. Equating these two, dN/dt = ns/my.
Thus

Power = (27T/60)2 (n2/m)Is
= Ks
249

250 HYDRAULIC TURBINES

This gives the value of power in ft.-lb. per second. To obtain
horse-power or kilowatts it is necessary to introduce the proper
constants in computing K. If the moment of inertia could be
computed or determined experimentally the value of K could be
obtained from the above. Usually it is necessary to obtain K
by direct experiment.

APPENDIX B

STREAM LINES IN CURVED CHANNELS

The following theory is based upon certain assumptions which
are only approximately realized in practice, but yet there are
many cases which approach these conditions so closely that the
methods here given may be successfully applied.1 Assume that
across any section, such as AB in Fig. 151, the total head is
constant. This will be true if all particles of water, coming from
some source, have lost equal amounts of energy en route and thus
all reach the section AB with an equal store of energy. Actually
some particles of water may have lost more than others. But

FIG. 151.

if- it be assumed that the total head across the section is constant,
it follows that, if the pressure is higher at any point, the velocity
will be lower than at some other point and vice versa. Owing to
centrifugal action, the pressure at B will be greater than that at
A and hence the velocity will decrease along the line from A to
5, the line AB being normal to the stream lines.

In Fig. 151, let us consider an elementary volume of water

1 From notes by Lewis F. Moody.

251

252 HYDRAULIC TURBINES

whose length in the direction AB is dn, the area of the face per-
pendicular to AB being AA. The mass is then wAAdn/g. Let
the stream line in question be at a distance n from the wall at
A and have a radius of curvature of p. Considering forces along
the normal line AB, we have dp X A A due to the difference in
pressure on the two forces and wAAdn cos a as the component
of gravity. The normal acceleration is F2/P- Hence we may
apply the proposition that force equals mass times acceleration
and obtain

dp&A + wAAdn cos a = (w&Adn/g)(V2/p)

Letting dz represent the change in elevation corresponding to
dn, we have dz = dn cos a. Thus from the above we may write

gdp ,gdz=V^ _

wdn wdn p

-^- = constant

differentiate with respect to n and obtain

dH = dp
dn wdn

And from this we may write

(86)

p V2

Since H = - + z+-^- = constant along line A B} we may

dH = dp dz 2VdV =
dn wdn dn 2gdn

wdn dn dn
Combining equations (85) and (86) we obtain

P dn

This may be written as

dV _dn
V .' ' p

Integrating

VA =

V = ~^rdn = ^ (87)

+ I e

e J° p

; ( » '

In the above, VA is the velocity next to the wall at A, where
n = 0.

I STREAM LINES IN CURVED CHANNELS 253

In general, sketch in tentative flow lines. Measure the curva-
ture and plot values of 1/p as a function of n. The area between
the curve and the n axis is the value of the integral. Denote this
area by Yi. Let F2 = l/eY\ Then V = VAY2. Hence if the
velocity at A were known, the velocity along any other stream
line crossing AB could be determined.

The solution of the problem from this point depends upon the
variation in the cross-section perpendicular to the plane of the
paper. The remainder of the discussion will be confined to the
case where the boundary walls are planes passing through an
axis of rotation, as shown. The thickness of the elementary

Values of n
FlG. 152.

volume perpendicular to the plane of the paper will be rA0.
The rate of discharge through the channel between the wall at
A and the stream line at distance n will be

\ 0 . dn . V.

For the entire circumference around the axis from which r is
measured, we may substitute 2x for A0, and, inserting the value
of V given by equation (87), we have

Y3dn

(88)

In order to evaluate this, plot a curve for values of r/eYl = r
YZ = YZ as a function of n. Denote the area under this curve

254

HYDRAULIC TURBINES

by F4. The ordinates of the curve to F4 in Fig. 152 indicate
values of qf from the wall at A up to any stream line. When
n = AB, the value of q' = q, which is the known rate of discharge
through the entire channel. This final ordinate may then be
divided up into any number of equal parts desired and corre-
sponding values of n determined from the F4 curve. This fixes
the division points for the stream lines along the section AB.
A similar procedure may be gone through with for any other
sections. If a considerable change is effected in the tentative
stream lines first assumed, this may be repeated for the corrected

FIG. 153.

set and so on. However extreme accuracy is not warranted, so
that a reasonable approximation is quite sufficient.

As a final check on the work it may be noted that if a series
of normal lines be drawn, as in Fig. 153,

An X r

= constant

(89)

The proof of this is that if two flow lines are spaced dn apart, we
may write Asi/As2 = P/(P + dn). From the preceding treat-
ment, we have dV/V — — dn/p, from which may be obtained
(V + dV)/V = Vi/Vt= (P - dn)/p. Multiplying both num-
erator and denominator of the last term by (p -\- dn) and drop-
ping differentials of the second order, we have

or FAs = constant

(90)

This shows that the velocities along any normal line are inversely

STREAM LINES IN CURVED CHANNELS 255

proportional to the distances between the successive normal
lines.

If the distance between two bounding surfaces, measured
perpendicular to the plane of the paper is rA0, we may write

a' — AnrA0F = AnrA0 X - — , since VAs = constant.

If all stream lines are so spaced as to subdivide the total flow
into equal parts, we have for the entire channel Anr/As =
constant.

In the profile views of the turbine runner are shown only the
circular projections of the true stream lines. The application of
the preceding theory to this case is open to some uncertainty,
but the theory should apply rather closely to the stream lines from
the draft tube up to the discharge edge of the runner, since these
lines should be in the plane of the paper. The principal object
of the procedure is to determine the division points along the
outflow edge and the direction of the stream lines at these points,
and any uncertainty as to the stream lines within the runner will
have little effect upon this. Hence the method is acceptable.

APPENDIX C
TEST DATA

The following data will supply material from which a number
of curves may be constructed. Most of it will be found suitable
for plotting characteristic curves, if desired.

Tables 1 to 5 inclusive are Holyoke tests ofcfive reaction tur-
bines of different types, taken from " Characteristics of Modern
Hydraulic Turbines" by C. W. Lamer in Trans. A. S. C. E., Vol.
LXVI, p. 306. Table 6 contains the results of the test of an
I. P. Morris turbine in the hydro-electric plant of Cornell Uni-
versity.

Tables 7 to 11 inclusive are tests of the same Pelton-Doble
tangential water wheel under widely different heads. These
were made under the direction of the author by F. W. Hoyt and
H. H. Elmendorf, seniors in Sibley College. In general they con-
firm the conclusions in Art. 103. Within reasonable limits the
characteristic curve is about the same regardless of the head
under which the test was made. The results show that the
efficiency increases rather rapidly as the head is increased from
very low values, but, as the effect of mechanical losses becomes
relatively less for the higher heads, the efficiency increases but
slightly thereafter. It might be expected that the efficiency
would approach a certain value as a limit as the head was indefi-
nitely increased, provided the bearings were adapted to the higher
speeds. Such might be the case if it were not for another factor.
The absolute velocity of discharge, F2, varies as the square root
of the head. For low heads the water discharged from the buck-
ets strikes the case and falls into the tail race without interfering
with the wheel. For high heads it was observed that the water
was deflected back from the case with sufficient velocity to strike
the wheel and thus to greatly increase the values of friction and
windage over the values given in Table 12, where no water was
present. The head at which this interference began to take place
was lower as the nozzle opening was increased and varied from
160 ft. with the nozzle open three turns to 50 ft. with the nozzle

256

TEST DATA 257

wide open. This would account for the decrease in efficiency
found under the high heads and would also account to some
extent for the maximum efficiency being found at different nozzle
openings under the various heads. These last facts would have
no application for the reaction turbine.

Some test data taken for the Pelton Water Wheel Co. by the
J. G. White Co. will be found in Table 13. The results of tests
on two centrifugal pumps of widely different types are given in
Tables 14 and 15.

In the construction of characteristic curves, the following
method has been found to be very convenient. Construct curves
between efficiency and speed under 1-ft. head for the various
gate openings. For any given efficiency the speeds for the dif-
ferent gates can be obtained from these and the points thus
dermined location on the characteristic curve. The iso-effi-
ciency curves may be drawn through these points, thus eliminat-
ing the necessity of interpolation. Smooth efficiency curves,
however, should be drawn, since very slight errors in data appear
magnified on the characteristic curve.

258

HYDRAULIC TURBINES

TABLE 1. — TESTS OP A 32-iNCH R. H. WELLMAN-SEAVER-MORGAN
COMPANY TURBINE WHEEL, No. 1795

Date, February 18 and 19, 1909. Case No. 1794
Wheel supported by ball-bearing step. Swing-gate. Conical draft-tube

Number of experi-
ment

Porportional
part of

Head acting on wheel,
in feet

Duration of experi-
ment, in minutes

a
§ %

o 2
•43 o

Quantity of water
discharged by wheel,
in cubic feet per
second

Horse-power devel-
oped by wheel

Percentage of effi-
ciency of wheel

Percentage of
full opening of
speed-gate

Percentage of
full discharge
of wheel

•69

1.000

1.024

17.23

112.50

34.45

46.84

69.59

1.000

1.009

17.50

152.67

34.22

55.80

82.16

1.000

1.008

17.57

163.00

34.05

57.31

83.98

1.000

1.005

17.46

164.00

34.05

56.90

84.40

' .000

0.992

17.47

172.67

33.60

55.92

84.00

.000

0.979

17.48

181.33

33.17

54.53

82.93

.000

0.965

17.46

189.25

32.67

52.53

81.21

.000

0.946

17.47

213.00

32.06

49.27

77.57

.000

0.688

17.59

251.50

23.39

34.91

74.81

1.000

0.631

17.86

298.00

21.62

0.00

0.889

0.933

17.50

113.67

31.63

45.23

72.04

0.889

0.934

17.52

135.67

31.70

50.84

80.72

0.889

0.931

17.54

146.33

31.60

52.80

84.01

0.889

0.925

17.53

153.67

31.40

53.32

85.41

0.889

0.922

17.44

156.50

31.21

52.85

85.62

0.889

0.916

17.45

160.33

31.00

52.66

85.84

0.889

0.909

17.46

164 . 00

30.79

52.35

85.87

0.889

0.903

17.47

168.00

30.59

52.07

85.92

0.889

0.898

17.48

172.75

30.42

51.95

86.14

0.889

0.893

17.48

176.75

30.25

51.52

85.91

0.889

0.886

17.49

184.00

30.05

51.08

85.69

0.889

0.864

17.53

206 . 50

29.31

47.77

81.98

0.889

0.795

17.59

244 . 75

27.02

33.97

63.02

0.889

0.572

17.91

298.75

19.61

0.00

0.741

0.809

17.57

106.00

27.48

39.23

71.65

0.741

0.801

17.57

139.50

27.20

45.18

83.35

0.741

0.791

17.55

152.00

26.85

45.71

85.53

0.741

0.784

17.57

159.75

26.63

45.82

86.35

0.741

0.780

17.60

166.25

26.52

46.15

87.18

0.741

0.777

17.56

171.50

26.40

•46.02

87.53

0.741

0.770

17.64

180.75

26.22

45.99

87.68

0.741

0.758

17.67

191.50

25.82

44.30

85.61

0.741

0.727

17.73

210.67

24.81

38.99

78.15

0.741

0.693

17.79

231.33

23.68

32.11

67.20

0.741

0.503

18.03

298.00

17.31

0.00

0.593

0.669

17.84

101.00

22.89

32.71

70.63

0.593

0.660

17.84

122.67

22.61

36.32

79.40

0.593

0.644

17.86

145.00

22.07

37. 17

84.04

0.593

0.642

17.85

151.33

21.98

37.81

84.97

0.593

0.637

17.88

158.00

21.83

38.081

85.87

0.593

0.632

17.91

163.75

21.68

37.8

86.02

TEST DATA

259

TABLE 1. — (Continued]

•n

Proportional

"if

•c

S3 "aS <3

*fl>

part of

5 OT

11 J

*o o

O g)

•3

„_, ^ «

*!i

"3 »

II!

IJI

o'2

Jl
a

.2 'g

IJU

P. -0
1

a u

i |

|il

I 3*

§ a

£ a

11*1

0< -

AH *"

W ""*

A5 ft

Of'*- «

n °

AH °

0.593

0.622

17.91

173.00

21.33

36.82

84.98

0.593

0.606

17.99

188.00

20.82

34.79

81.90

0.593

0.537

18.16

239.00

18.54

22.11

57.92

0.593

0.406

18.05

281 . 00

13.98

0.00

0.444

0.505

17.91

94.00

17.32

23.92

67.99

0.444

0.489

17.92

130.00

16.78

27.06

79.36

0.444

0.486

17.94

136 . 00

16.70

27.68

81.48

0.444

0.486

17.92

139 . 25

16.66

27.70

81.82

0.444

0.481

17.94

145.33

16.50

27.57

82.12

0.444

0.472

17.94

153.00

16.21

26.90

81.56

0.444

0.465

17.95

161.40

15.98

26.13

80.34

0.444

0.450

17.99

178.00

15.46

24.71

78.32

0.444

0.402

18.04

231.50

13.85

16.07

56.70

0.444

0.325

18.17

281.25

11.22

0.00

0.296

0.325

18.19

89.67

11.22

15.35

66.32

0.296

0.320

18.21

107.00

11.06

16.58

72.60

0.296

0.317

18.23

114.67

10.97

16.98

74.85

0.296

0.316

18.22

118.33

10.94

16.97

75.07

0.296

0.313

18.24

122 . 67

10.82

17.03

76.07

0.296

0.308

18.25

130.75

10.68

16.94

76.62

0.296

0.304

18.26

137.75

10.54

16.57

75.91

0.296

0.300

18.30

146.50

10.41

16.27

75.29

0.296

0.295

18.31

154.75

10.23

15.75

74.15

0.296

0.291

18.32

164.00

10.10

15.17

72.31

0.296

0.283

18.33

181.20

9.82

14.25

69.81

0.296

0.276

18.36

191.25

9.58

13.27

66.53

0.296

0.266

18.33

208 . 00

9.23

11.55

60.18

0.296

0.225

18.37

261.00

7.83

0.00

NOTE. — For experiments Nos. 1, 15, 25, 35, 46 and 60, the jacket was loose.

During the above experiments, the weight of the dynamometer, and of that portion of
the shaft which was above the lowest coupling was 1,300 Ib.

With the flume empty, a strain of 1.0 Ib., applied at a distance of 2.4 ft. from the center
of the shaft, sufficed to start the wheel.

260

HYDRAULIC TURBINES

TABLE 2.— TESTS OF A 28-iNCH R. H. WELLMAN-SEAVER-MORGAN
COMPANY TURBINE WHEEL, No. 1796

Date, February 25, 1909
Wheel supported by ball-bearing step. Swing-gate. Conical draft-tube

Proportional

Q) *^ Q)

^1,

part of

ft J

"a 43 a

"o *o

•8 8,

a
o

o "3
|

* *1

73 1

"o 8

e 1

*o 2

lis

S3 -g

« ^

I ll

|i|

•£ -^

I**

'•3 43 o fl

S b

ft42

i -a

a g

o — • v

V £H ^

"3 «2

2 g

> 85

g g S

£ «

8 g

3 a

K ^3 <M

f-t *2 ^_,

£ a

3 a

S 2

^•^J3O

0 £

fc ^

PH "" "

AH *~

H<a

Q e

« ft

w °

1.077

0.971

17.11

153.00

97.00

125.20

66.52

1.077

.017

16.97

199 . 67

101.16

147.66

75.81

1.077

.036

16.94

224 . 33

102.98

156.38

79.04

1.077

.053

16.89

239 . 33

104 . 50

159 . 58

79.72

1.077

.061

16.87

247.33

105.22

161 . 17

80.06

1.077

.068

16.81

253.67

105.70

161.45

80.12

1.077

.072

16.80

259 . 00

106.08

161.71

80.01

1.077

.079

16.82

267.50

106.82

162.15

79.58

1.077

.026

17.05

294.67

102.27

125.03

63.23

1.000

0.913

17.28

147.00

91.65

120.29

66.98

1.000

0.957

17.16

190.50

95.70

144 . 34

77.50

1.000

0.972

17.14

211.67

97.10

152.69

80.89

1.000

0.981

17.13

225.00

98.03

156.85

82.36

1.000

0.990

17.11

232.67

98.90

157.96

82.31

1 . 000

0.996

17.09

240.25

99.43

160.19

83.13

1.000

1.003

17.07

247 . 33

100.07

161.17

83.19

1.000

1 .004

17.07

252 . 25

100.14

160,55

82.82

• 78

1.000

1.001

17.13

259 . 00

100.00

157.00

80.82

1.000

0.983

17.22

268.33

98.43

146.39

76.15

1.000

0.911

17.47

293 . 50

91.96

106.75

58.59

106

0.923

0.868

17.32

143.25

87.24

115.49

67.39

105

0.923

0.899

17.24

176.00

90.12

135.49

76.90

104

0.923

0.920

17.15

201.00

91.96

146.21

81.74

101

0.923

0.931

16.93

213.20

92.52

148.62

83.66

100

0.923

0.936

16.93

220.67

92.96

150.48

84.31"

0.923

0.942

16.91

227.25

93.51

151.52

84.50

102

0.923

0.945

16.93

232.00

93.82

151.88

84.31

103

0.923

0.945

17.04

235.50

94.14

152.74

83.96

0.923

0.945

16.93

237.75

93.82

151.32

84.00

0.923

0.924

17.02

254 . 50

92.04

138.84

78.15

0.923

0.823

17.25

288.25

82.47

87.36

54.15

0.923

0.870

17.17

146.33

87.02

116.20

68.57

0.923

0.895

17.10

170.00

89.37

130.87

75.51

0.923

0.921

17.04

202.75

91.80

146.25

82.44

0.923

0.928

16.99

208 . 67

92.35

147.99

83.17

0.923

0.932

17.03

216.25

92.81

150.75

84.10

0.923

0.937

16.97

220.00

93.20

151.36

84.38

0.923

0.939

17.01

223.75

93.44

152.58

84.65

0.923

0.940

17.02

226.25

93.66

150.86

83.45

0.923

0.944

16.97

238.33

93.90

151.69

83. 9t

0.923

0.921

17.13

256.25

92.05

139.80

78.18

0.923

0.823

17.27

288.00

82.54

87.29

53.99

TEST DATA

261

TABLE 2 . — (Continued)

Number of experi-
ment

Proportional
part of

Head acting on wheel,
in feet

Duration of experi-
ment, in minutes

Revolutions of wheel
per minute

Quantity of water
discharged by wheel,
in cubic feet per
second

Horse-power devel-
oped by wheel

Percentage of effi-
ciency of wheel

Percentage of
full opening of
speed-gate

Percentage of
full discharge
of wheel

0.923

0.730

17.50

334 . 75

73 . 75

0.00

0.846

0.824

17.46

158.67

83.15

120.23

73.02

0.846

0.836

17.46

175.67

84.35

129.91

77.78

0.846

0.861

17.34

202.20

86.50

143.40

84.30

0 . 846

0.865

17.33

209.00

86.95

145.69

85.25

0 . 846

0.868

17.33

215.00

87.24

147.27

85.89

0.846

0.870

17.34

219.25

87.47

148.19

86.15

0.846

0.869

17.32

221.25

87.32

147.53

86.01

0 . 846

0.866

17.33

227.75

87.02

144.96

84.76

0.846

0.858

17.36

231.75

86.25

140.48

82.73

0.846

0.845

17.39

243 . 75

85.11

132.98

79.22

0.846

0.828

17.44

256.50

83.44

124.39

75.37

0.846

0.754

17.59

282.00

76.31

85.47

56.15

0.769

0.750

17.38

141.00

75.52

102.56

68.90

'S6

0.769

0.766

17.35

166.00

77.02

115.72

76.36

0.769

0.779

17.32

183.00

78.24

124.24

80.84

0.769

0.789

17.31

194.00

79.25

129.36

83.15

0.769

0.793

17.25

200.75

79.48

131.42

84.52

0.769

0.792

17.26

206.00

79.40

131.11

84.36

0.769

0.773

17.38

226.25

77.82

123.43

80.47

0.769

0.735

17.49

251.33

74.17

106.64

72.49

-22

0.769

0.690

17.56

269 . 67

69.82

81.73

58,78

0.769

0.623

17.68

323.75

63.27

0.00

0.615

0.617

17.91

139.50

63.07

88.79

69.31

0.615

0.627

17.79

158.33

63.81

95.97

74.55

0.615

0.634

17.80

171.33

64.55

101.26

77.71

0.615

0.638

17.73

179 . 50

64.82

103.37

79.31

0.615

0.636

17.77

183.00

64.74

103.16

79.07

0.615

0.634

17.80

188.00

64.61

102.56

78.64

14 0.615

0.627

17.72

194.50

63.74

100.21

78.24

13 0.615

0.596

17.77

218.67

60.60

92.78

75.97

0.615

0.563

17.79

243.00

57.29

73.65

63.72

0.615

0.519

17.92

312.00

53.00

0.00

0.462

0.452

17.34

117.33

45.40

56.90

63.73

0.462

0.453

17.02

136.00

45.65

61.83

70.17

0 . 462

0.461

17.08

146.75

46.04

64.94

72.82

0.462

17.05

152.00

46.04

66.34

74.52

0.462

16.92

155.25

45.90

65.88

74.79

0.462

0.459

17.16

162.00

45.94

66.78

74.69

0.462

0.457

17.15

166.67

45.69

65.67

73.90

0.462

0.451

16.98

172.25

44.83

62 . 65

72.57

0.462

0.432

17.05

217.50

43.04

52.74

63.37

0.462

0.404

17.18

282.40

40.40

0.00

NOTE. — During the above experiments, the weight of the dynamometer, and of that
portion of the shaft which was above the lowest coupling, was 2,600 Ib.

With the flume empty, a strain of 0. 5 Ib., applied at a distance of 3. 2 ft. from the center
at the shaft, sufficed to start the wheel.

262

HYDRAULIC TURBINES

TABLE 3. — TESTS OF A 30-iNCH R. H. WELLMAN-SE AVER-MORGAN
COMPANY TURBINE WHEEL, No. 1797

Date, February 26 and 27, 1909
Wheel supported by ball-bearing step. Swing-gate. Conical draft-tube

Proportional

.£

fr-S|

•

part of

a S

* J3 ft

«*" •*-! «*« 0>

3. I

a
o

•3

..'I!

«*. «

I'g 8 I 2 •«

§ |

I >>

ill SU"8

^ fH

fill

fl W O

3-2 0 S

1 1

3 S

fir.-* 1 ^ °

W .2

P *

rt ^

§*.a S

w °

£'5

.000

0.942

17.06

149.00

94.89

126.41

68.85

.000

0.950

16.99

160.33

95.52

131.16

71.26

.000

0.963

17.00

190.50

96.85

144.30

77.28

.000

0.977

17.02

210.50

98.34

150.52

79.30

.000

0.984

16.95

220.00

98.81

153.31

80.72

.000

0.992

17.01

227 . 25

99.75

155.61

80.87

.000

0.994

16.94

229 . 50

99.75

155.76

81.28

.000

0.997

16.98

231.25

100.14

155.55

80.66

.000

1.002

16.90

239.67

100.45

156.85

81.47

1.000

1.003

16.94

254.75

100.70

154.37

79.80

1.000

0.871

17.13

287.00

87.93

86.96

50.91

. 1.000

0.728

17.44

317.67

74.17

0.00

0.923

0.875

17.08

131.33

88.14

111.42

65.26

0.923

0.896

16.95

156.33

89.97

127.89

73.95

0.923

0.913

16.93

188.25

91.66

142 . 59

81.02

0.923

0.920

16.89

201.25

92.20

146 . 34

82.86

0.923

0.927

16.85

211.50

92.81

148.67

83.83

0.923

0.931

16.86

217.00

93.20

149.91

84.12

0.923

0.934

16.83

220.50

93.42

150.99

84.68

0.923

0.936

16.83

223.75

93.60

151.86

85.00

0.923

0.937

16.83

226.33

93.74

150.87

84.32

0.923

0.929

16.88

241.75

93.13

146.49

82.17

0.923

0.788

17.21

276.25

79.76

83.70

53.77

0.923

0.670

17.44

312.50

68.26

0.00

0.846

0.868

17.07

192.75

87.47

142.50

84.15

0.846

0.875

17.10

205.00

88.22

147.83

86.41

0.846

0.876

17.06

209 . 50

88.22

148.53

87.02

0.846

0.876

17.11

212.00

88.37

149 . 02

86.91

0.846

0.877

17.05

213.00

88.30

148.43

86.94

0.846

0.876

17.13

213.80

88.37

147.70

86.03

0.846

0.875

17.13

215.80

88.30

146.46

85.38

0.808

0.810

17.33

143.50

82.24

114.78

71.01

0.808

0.834

17.25

177.00

84.44

135.14

81.81

0.808

0.843

17.24

194 . 80

85.35

142.83

85.59

0.808

0.847

17.21

201.00

85.73

144.94

86.62

0.808

0.848

17.19

206.00

85.73

146.05

87.39

0.808

0.847

17.19

208.25

85.65

145.12

86.91

0.808

0.845

17.19

210.00

85.41

143.80

86.36

0.808

0.840

17.22

216.67

85.05

141.80

85.37

0.808

0.825

17.25

224.75

83.53

136.19

83.34

0.808

0.807

17.29

236.50

81.80

128.98

80.41

TEST DATA

263

TABLE 3.— (Continued}

«ll

Proportional

•A

M _r fc(

jrj

part of

8
jfl

5 j s

•

•
O

•3 "3

o a

p
o

8 3

* *1

•si

.8 -

f.s <§
tj

rfi

*3 £

§ 3
'•3 .S

JS g

Ili,

A -^

6 g

§ £2 $

s ^ ^

"S ""

g 1 '

5 *«

g g I

2 S

g d

3 a

fe *2 ^*

fe *"3 **•*

g d

3 a

^ ^

o ft

t- fQ>

55 B

&< •**

OH "*"

W "*

Q *

PH ft

^TJ .„ 0)

a °

& °

0.808

0.695

17.51

264 . 25

70.97

80.06

56.81

0.769

0.755

17.29

118.50

76.53

93.35

62.21

0.769

0.776

17.21

146.50

78.47

110.97

72.46

0.769

0.800

17.03

174.75

80.49

127.07

81.74

0.769

0.806

17.03

190.75

81.13

135.24

86.31

0.769

0.805

17.10

197.25

81.21

137.46

87.28

0.769

0.805

17.15

200.50

81.28

137.29

86.84

0.769

0.803

16.83

199.50

80.34

132.98

86.72

0.769

0.788

16.93

211.75

79.04

128.32

84.55

0.769

0.745

16.93

232.50

74.72

112.71

78.56

0.769

0.660

17.36

257.50

67.02

78.02

59.13

0.769

0.579

17.60

306.00

. 59.27

0.00

0.615

0.610

17.69

100.67

62.60

69.54

55.37

0.615

0.621

17.64

127.67

63.60

85.10

66.89

0.615

0.643

17.60

167.33

65.76

104.44

79.57

0.615

0.644

17.57

175.50

65.83

107.41

81.89

0.615

0.643

17.61

178.00

65.76

107.86

82.13

0.615

0.641

17.59

179.67

65.55

106.70

81.60

0.615

0.482

17.96

296.67

49.84

0.00

0.615

0.634

17.88

155.00

65.37

101.44

76.53

0.615

0.634

17.80

151.40

65.22

98.17

74.56

0.615

0.639

17.69

160.33

65.50

102.98

78.37

0.615

0.644

18.08

174.20

66.80

111.89

81.69

0.615

0.643

18.11

180.00

66.73

112.35

81.97

0.615

0.638

18.07

183.50

66.19

111.20

81.98

0.615

0.636

18.03

185.00

65.90

109 . 86

81.53

0.615

0.630

17.96

190.00

65.15

109 . 38

82.43

0.615

0.609

17.81

201.00

62.66

103.53

81.80

0.615

0.574

17.79

216.75

59.07

91.94

77.15

0.615

0.553

17.54

245.20

56.46

74.29

66.15

0.615

0.483

17.71

294.60

49.58

0.00

0.462

0.475

17.82

104.33

48.95

58.16

58.80

0.462

0.483

17.75

139.75

49.58

71.98

72.12

0.462

0.482

17.76

147.00

49.51

73.93

74.14

0.462

0.481

17.80

156.75

49.45

75.99

76.12

0.462

0.477

17.93

162.00

49.22

76.57

76.51

0.462

0.472

17.97

167.00

48.78

75.90

76.35

0.462

0.464

17.88

173.60

47.80

73.64

75.97

0.462

0.436

18.05

216.50

45.13

65.60

71.01

0.462

0.415

18.08

246 . 25

43.02

44.77

50.75

0.462

0.381

18.27

280.50

39.75

0.00

NOTE.— The jacket was loose for Experiments Nos. 1, 11, 23, 30, 41, and 71.

During the above experiments, the weight of the dynamometer, and of that portion of
the shaft which was above the lowest coupling was 2,600 Ib.

With the flume empty, a strain of 0.5 Ib., applied at a distance of 3.2 ft. from the center
of the shaft, sufficed to start the wheel.

264

HYDRAULIC TURBINES

TABLE 4. — TESTS OF A 31-iNCH R. H. WELLMAN-SEAVER-MORGAN
COMPANY TURBINE WHEEL, No. 1799

Date, March 2 and 3, 1909
Wheel supported by ball-bearing steps. Swing-gate. Conical draft-tube

•g

Proportional

s ^ s

part of

. 1$

1"8

..•SB

£
a

!>
*0

£*

^ %£

73 — .

«*- *^
°.S

Number of
ment

Percentage
full opening
speed-gate

Percentage
full discha
of wheel

Head acting
in feet

Duration ol
ment, in ir

Revolutions
per minute

Quantity o
discharged 1
in cubic :
second

Horse-power
oped by. wh

Percentage
ciency of w

.000

1.049

17.15

134.67

79.75

115.81

74.66

.000

1.038

17.15

147.00

78.95

119.29

77.68

.000

1.025

17.19

165.67

78.02

123.40

81.13

.000

1.024

17.18

174.00

77.91

124 . 34

81.91

.000

1.018

17.19

178.33

77 . 52

124.19

82.18

1.000

1.013

17.19

183.33

77.16

124.35

82.66

1.000

0.012

17.19

186.25

77.09

124.07

82.56

1.000

1.009

17.18

189 . 50

76.82

123.94

82.81

.000

1.007

17.19

' 3

193.00

76.67

123.89

82.89

.000

1.006

17.12

195.50

76.44

123.13

82.96

.000

1.002

17.09

200.25

76.10

122.48

83.04

.000

0.999

17.10

206 . 25

75.87

122.41

83.19

.000

0.997

17.07

210.33

75.66

121.01

82.62

.000

0.993

17.09

219.00

75.38

119.36

81.70

.000

0.906

17.24

258.75

' 69.11

78.35

57.98

.000

0.744

17.45

302.25

57.07

0.00

0.883

0.909

17.18

129.33

69.23

101 . 03

74.90

0.883

0.904

17.20

154.25

68.82

111.16

82.81

0.883

0.901

17.19

165.00

68.60

113.91

85.18

0.883

0.896

17.20

172.33

68.25

114.80

86.23

0.883

0.892

17.21

181.50

67.97

116.51

87.82

0.883

0.888

17.25

188.00

67.77

117.27

88.45

0.883

0.886

17.30

194.00

67.71

117.49

88.44

0.883

0.885

17.23

197.25

67.43

117.06

88.85

0.883

17.31

201 . 75

67.49

116.07

87.61

0.883

0.857

17.28

213.25

65.42

109 . 77

85.62

0.883

0.818

17.31

227.00

62.53

96.23

78.39

0.883

0.761

17.42

244.25

58.32

73.96

64.19

0.883

0.620

17.65

291.75

47.83

0.00

0.750

0.835

17.28

124.75

63.73

91.41

73.19

0.750

0.839

17.23

148.75

63.93

103 . 59

82.93

0.750

0.834

17.23

168.75

63.60

109.35

87.99

' 0.750

0.830

17.24

179 . 25

63.26

110.72

89.52

0.750

0.829

17.23

182.00

63.20

110.77

89.69

0.750

0.828

17.25

186.25

63.12

111.66

90.43

0.750

0.824

17.23

188.67

62.85

110.83

90.24

0.750

0.821

17.29

191.75

62.72

110.32

89.70

0.750

0.809

17.30

197.50

61.81

107.64

88.76

0.750

0.796

17.34

203 . 25

60.86

104.62

87.42

0.750

0.768

17.38

212.50

58.80

96.52

83.28

0.750

0.693

17.55

237.00

53.31

71.76

67.64

0 . 750

0.573

17.75

287.00

44.32

0.00

O.'OO

TEST DATA

265

TABLE 4. — (Continued)

Proportional

g-s s

Efi

part of

a S

1 ^ a

"8 "8

•8 S,

Q> 2
g

^ 1

-3 |

o> .9 5

• -Si

a
$

"8 '§

a %
.1 S

S|i

1 ^

fl o *^

S^.3

•_2 _p-

"S '3

'£ j~ o £3

ft X!

a -^1

1 S

8 ~ *

2 §

"o .
> a3

§ g

|I 8

|-°

* £3

B "*

Q S

|lil

£= I'*

0.667

0.726

17.27

106.25

55.43

70.78

65.20

0.667

0.737

17.26

148.67

56.25

90.93

82.59

0.667

0.739

17.25

161.00

56.39

95.55

86.62

0.667

0.738

17.24

168.50

56.31

96.94

88.05

0.667

0.733

17.27

173.67

55.92

96.76

88.35

0.667

0 . 722

17.30

179 . 25

55.17

95.52

88.25

0.667

0.699

17.34

189.50

53.48

91.81

87 ,30

0.667

0.671

17.39

201 . 75

51.37

85.52

84 .42

0.667

0.508

17.63

280.75

39.17

0.00

0.500

0.554

17.82

117.00

42.95

60.23

69.39

0.500

0.546

17.91

135.00

42.46

65.40

75.84

0.500

0.548

17.67

151.00

42.28

68.58

80.95

0.500

0.548

18.05

157.50

42.72

71.54

81.80

0.500

0.547

17.71

157.00

42.28

69.41

81.73

0.500

0.539

18.12

167.60

42.11

71.05

82,10

0.500

0.512

18.13

187.00

40.02

67.95

82.58

0.500

0.488

.18.18

213.00

38.21

58.05

73.68

0.500

0.460

18.07

232 . 00

35.92

42.15

57.26

0.500

0.402

18.20

275.00

31.50

0.00

0.333

0.362

18.18

96.00

28.35

34.88

59.68

0.333

0.361

18.01

177.33

28.10

39.08

68.09

0.333

0.348

18.19

133.00

27.25

40.27

71.64

0.333

0.347

18.21

139.67

27.20

40.60

72.28

0.333

0.340

18.22

143.67

26.62

39.15

71.18

0.333

18.06

148.00

26.00

37.64

70.69

0.3S3

0.316

18.26

201.67

24.82

36.64

71.28

NOTE. — For Experiments Nos. 1, 11, 21, and 57, the jacket was loose.

During the above experiments, the weight of the dynamometer and of that portion of
the shaft which was above the lowest coupling was 2,600 Ib.

With the flume empty, a strain of 1.0 Ib., applied at a distance of 3.2 ft. from the center of
the shaft, sufficed to start the wheel.

266

HYDRAULIC TURBINES

TABLE 5. — TESTS OF A 31-iNCH R. H. WELLMAN-SE AVER-MORGAN
COMPANY TURBINE WHEEL, No. 1800

Date, March 4 and 5, 1909
Wheel supported on ball-bearing step. Swing-gate. Conical draft-tube

Proportional
part of

Head acting on wheel,
in feet

Duraton of experi-
ment, in minutes

Revolutions of wheel
per minute

fill

S g 3
3* .2 8

Horse-power devel-
oped by wheel

Percentage of effi-
ciency of wheel

Percentage of
full opening of
speed-gate

Percentage of
full discharge
of wheel

1.000

1.052

17.38

112.33

64.87

82.99

64.91

1.000

1.040

17.41

133.33

64.20

91.24

71.98

1.000

1.025

17.43

154.25

63.33

96.21

76.85

. 000

1.014

17.42

168.67

62.60

98.64

79.76

.000

.009

17.40

183.33

62.25

99.92

81.34

.000

.006

17.42

193.25

62.13

100.65

82.00

.000

.004

17.43

201.00

62.01

101 . 03

82.42

.000

.002

17.44

207.00

61.88

101 . 54

82.96

.000

17.35

211.00

61.65

100 . 95

83.22

.000

0.999

17.39

216.75

61.65

101.07

83.13

.000

0.998

17.36

221.50

61.53

100.60

83.05

.000 .

0.995

17.38

247.50

61.40

97.42

80.50

.000

0.951

17.45

266.75

58.80

80.77

69.41

.000

0.730

17.71

321.25

45.44

0.00

0.883

0.889

17.48

132.25

55.02

79.29

72.69

0.883

0.888

17.48

147.00

54.91

83.68

76.88

0.883

0.885

17.47

165.00

54.71

88.93

82.04

0.883

0.879

17.39

179.25

54.24

91.18

85.24

0.883

0.876

17.42

189.20

54.11

91.66

85.75

0.883

0.873

17.54

197.50

54.11

93.29

86.67

0.883

0.873

17.40

204.25

53.85

92.77

87.30

0.883

0.872

17.40

209.00

53.79

92.40

87.05

0.883

0.871

17.42

215.50

53.79

92.66

87.19

0.883

0.858

17.40

225.00

52.92

88.57

84.81

0.883

0.810

17.45

248.00

50.06

75.09

75.80

0.883

0.609

17.68

314.00

37.87

0.00

0.733

0.801

17.76

158.50

49.93

81.59

81.13

0.733

0.798

17.73

183.33

49.70

86.60

86.66

0.733

0.797

17.71

190.33

49.63

87.60

87.88

0.733

0.796

17.72

197.00

49.57

88.28

88.62

0.733

0.795

17.73

199.00

49.50

87.97

88.39

0.733

0.793

17.72

201.67

49.37

87.93

88.63

0.733

0.788

17.70

204.50

49.02

86.69

88.10

0.733

0.775

17.71

213.50

48.27

84.04

86.69

0.733 •

0.722

17.79

234 . 00

45.04

70.85

77.97

0.733

0.544

18.03

308.67

34.15

0.00

0.667

0.751

17.58

171.00

46.58

78.70

84.75

0.667

0.750

17.57

184.00

46.52

81.34

87.75

0.667

0.749

17.56

187.00

46.45

81.54

88.14

0.667

0.749

17.59

185.50

46.45

80.88

87.29

0.667

0.747

17.59

188.75

46.38

81.73

88.33

TEST DATA

267

TABLE 5. — (Continued)

Proportional

J""

« "o3 a>

part of

1 8

i-2 a

° i o>

P
a

X g

"S'l

•8
§ 5

•si!

M '3 t«

M g —

'Js

a 5

•I a

>> ^ -2

0 >>

2 "3

!«>

f II

I 1

0 "*

.tn 03 "g "0
"fl *« " O

1 1

1 *

Js &

|1°

g a

1 &

|«lj| 1

1°

£•3

0.667

0.742

17.58

192.50

46.02

80.44

87.67

0.667

0.731

17.59

198.25

45.37

78.04

86.22

0.667

0.753

17.70

127.67

46.87

68.04

72.32

0.667

0.752

17.68

144.67

46.76

72.72

77.56

0.667

0.752

17.66

164.25

46.76

77.59

82.85

0.667

0.753

17.68

178.67

46.82

81.15

86.44

0.667

0.750

17.62

189.00

46.57

82.41

88.56

0.667

0.741

17.62

193.67

46.02

80.93

88.00

0.667

0.731

17.64

201.00

45.44

79.12

87.04

0.667

0.725

17.63

204.00

45.05

77.83

86.41

0.667

0.709

17.64

213.33

44.05

74.93

85.03

0.667

0.678

17.70

224 . 50

42.23

67.98

80.19

0.667

0.506

17.85

303.33

31.61

0.00

0.500

0.566

17.72

124.00

35.27

50.31

70.98

0.500

0.560

17.74

136.33

34.87

52.84

75.32

0.500

0.554

17.71

146.50

34.50

53.23

76.82

0.500

0.543

17.72

155.75

33.82

52.82

77.72

0.500

0.535

17.74

166.00

33.31

52.27

78.00

0.500

0.521

17.77

176.67

32.47

51.36

78.48

0.500

0.507

17.79

189.00

31.61

50.36

78.97

0.500

0.494

17.83

200.25

30.87

48.51

77.71

0.500

0.481

17.86

214.00

30.08

45.36

74.45

0.500

0.464

17.89

228.75

29.02

41.56

70.58

0.500

0.380

18.00

299 . 50

23.87

0.00

0.333

0.360

18.09

112.00

22.65

30.52

65.68

0.333

0.351

18.12

127.00

22.10

31.53

69.43

0.333

0.350

18.13

136.00

22.02

32.12

70.95

0.333

0.346

18.22

141.50

21.84

31.71

70.26

0.333

0.340

18.24

148.00

21.47

31.37

70.63

0.333

0.332

18.31

154.75

21.03

30.93

70.82

0.333

0.323

18.25

165.25

20.42

30.02

71.04

0.333

0.307

18.17

212.50

19.33

28.31

71.08

0.333

0.300

18.15

231.33

18.93

25.22

64.72

0.333

0.248

18.35

287.25

15.74

0.00

NOTE. — The jacket was loose for Experiments Nos. 1, 11, 22, 40, 52, and 66.

During the above experiments, the weight of the dynamometer and of that portion of the
shaft which was above the lowest coupling was 2,600 Ib.

With the flume empty, a strain or 0.5 Ib., applied at a distance of 3.2 ft. from the center
of the shaft, sufficed to start the wheel.

268

HYDRAULIC TURBINES

TABLE 6. — TEST OF A 27-iNCH I. P. MORRIS Co. REACTION TURBINE
AT CORNELL UNIVERSITY l

By R. L. Daugherty, Feb., 1914

Proportional
part of
gate opening

Road,
ft.

Discharge
cu. ft. per
sec.

R.p.m.

Torque
ft. Ib.

B.h.p,

Efficiency,
per cent.

0.067

146.1

7.4

600

143

16.4

13.3

0.067

146.2

6.9

647

00.0

0.248

144.9

18.7

2760

00.0

0.248

145.4

16.4

600

1365

156

57.6

0.248

145.7

12.5

845

00.0

0.476

143.9

27.5

5390

00.0

0.476

144.3

25.4

600

2760

318

76.3

0.476

145.2

18.8

975

00.0

0.600

142.8

34.8

6550

00.0

0.600

143 . 1

31.8

600

3820

437

84.5

0.600

144.7

23.0

1022

00.0

0.772

141.8

40.2

7520

00.0

0.772

141.8

38.8

600

4820

550

88.0

0.772

144.0

26.8

1038

00.0

1.000

140.6

46.3

8130

00.0

1.000

140.5

44.5

600

5400

617

87.0

1.000

143.4

32.4

1060

00.0

xFor an account of this test see " Investigation of the Performance of a
Reaction Turbine," Trans. A. S. C. K, Vol. LXXVIII, p. 1270 (1915).

TEST DATA

269

TABLE 7. — TEST OF A 12-lNCH PELTON-DOBLE TANGENTIAL WATER

WHEEL UNDER A CONSTANT PRESSURE HEAD OF 8.93 FT. LENGTH OF

BRAKE-ARM = 14-iN.

Turns of
needle

Head,
ft.

Discharge,
cu. ft. per
sec.

R.p.m.

Brake
load,
lb.

B.h.p.

Efficiency,
per cent.

8.932

0.0175

0.48

0.00000

00.0

100

0.32

0.00712

40.0

150

0.24

0.00802

45.0

200

0.16

0.00712

40.0

250

0.06

0.00334

18.8

280

0.00

0.00000

00.0

8.936

0 . 0300

0.95

0.0000

00.0

100

0.71

0.0158

51.8

150

0.58

0.0194

63.7

200

0.43

0.0191

62.8

250

0.27

0.0150

49.2

320

0.00

0.0000

00.0

8.943

0.0448

1.50

0.0000

00.0

100

1.11

0.0247

54.2

150

0.90

0.0301

66.0

200

0.67

0.0298

65.3

250

0.42

0 . 0234

51.3

330

0.00

0.0000

00.0

8.951

0.0575

1.80

0.0000

00.0

100

1.38

0.0347

59.3

150

1.16

0 . 0388

66.3

200

0.90

0.0400

68.3

250

0.60

0.0334

57.0

340

0.00

0.0000

00.0

8.959

0 . 0670

2.00

0.0000

00.0

100

1.56

0.0347

50.8

150

1.32

0.0441

64.6

200

1.06

0.0472

69.2

250

0.82

0.0457

67.0

360

0.00

0.0000

00.0

8.966

0.0750

2.14

0.0000

00.0

100

1.69

0.0376

49.2

150

1.46

0.0488

63.8

200

1.19

0.0526

68.8

• | ; .«

250

0.90

0.0502

65.7

360

0.00

0.0000

00.0

7.85

8.978

0.0860

2.60

0.0000

00.0

100

1.89

0.0422

48.0

150

1.66

0.0555

63.2

200

1.34

0.0596

67.8

250

1.03

0.0574

65.0

360

0.00

0.0000

00.0

270

HYDRAULIC TURBINES

TABLE 8. — TEST OF A 12-iNCH PELTON-DOBLE TANGENTIAL WATER
WHEEL UNDER A CONSTANT PRESSURE HEAD OF 62.5 FT.

Turns
<- of
needle

Head,
ft.

Discharge,
cu. ft. per
sec.

R.p.m.

Brake
load,
Ib.

B.h.p.

Efficiency,
per cent.

62.52

0.041

100

3.10

0.069

22.8

200

2.85

0.127

42.0

300

2.50

0.167

55.2

400

2.10

0.187 .

61.8

450

1.90

0.191

63.0

500

1.70

0.189

62.5

600

1.15

0.153

50.7

700

0.55

0.086

28.4

775

0.00

0.000

00.0

62.55

0.081

100

6.70

0.149

25.9

200

6.10

0.276

48.0

300

5.40

0.361

62.7

400

4.70

0.419

72.8

500

3.85

0.429

74.5

600

3.00

0.400

69.5

700

1.80

0.284

49.2

800

0.70

0.125

21.6

880

0.00

0.000

00.0

62.58

0.117

100

9.60

0.214

25.6

200

8.70

0.388

46.5

300

7.80

0.522

62.5

400

6.80

0.606

72.6

500

5.85

0.652

78.0

600

4.75

0.635

76.0

700

3.30

0.515

61.7

800

1.65

0.294

35.2

930

0.00

0.000

00.0

62.63

0.150

100

12.30

0.274

25.6

200

11.25

0.510

47.6

300

10.05

0.672

62.8

400

8.85

0.790

73.8

500

7.80

0.870

81.3

600

6.45

0.862

80.6

700

4.75

0.740

69.1

800

2.60

0.463

42.3

900

0.85

0 . 160

15.0

950

0.00

0.000

00.0

TEST DATA

271

TABLE 8. — (Continued]

Turns
of
needle

Head,
ft.

Discharge,
cU. ft. per
sec.

R.p.m.

Brake
load,
Ib.

B.h.p.

Efficiency,
per cent.

62.68

0.174

100

14.10

0.314

25.3

200

12.90

0.575

46.4

300

11.50

0.768

62.0

400

10.20

0.908

73.3

500

9.05

1.010

81.5

600

7.30

0.975

78.7

700

5.45

0.850

68.6

800

3.30

0.515

41.6

900

1.20

0.241

19.4

970

0.00

0.000

00.0

62.74

0.200

100

15.25

0.340

23.8

200

14.00

0.624

43.7

300

12.65

0.846

59.5

400

11.25

1.011

71.0

500

9.95

1.110

78.0

600

8.20

1.098

77.0

700

6.00

0.935

65.6

800

3.60

0.561

39.4

900

1.20

0.302

21.2

985

0.00

0.000

00.0

7.85

62.81

0.230

100

17.40

0.388

23.7

200

16.10

0.717

43.8

300

14.70

0.982

60.0

400

12.80

1.140

69.5

500

11.10

1.235

75.5

600

9.30

1.242

76.0

700

6.90

1.080

66.0

800

4.20

0.655

40.0

900

1.85

0.370

22.6

985

0.00

0.000

00.0

272

HYDRAULIC TURBINES

TABLE 9. — TEST OF A 12-iNCH PELTON-DOBLE TANGENTIAL WATER
WHEEL UNDER A CONSTANT PRESSURE HEAD OF 130.5 FT.

Turns
of
needle

Head,
ft.

Discharge,
cu. ft. per
sec.

Brake
R.p.m. load,
Ib.

B.h.p.

Efficiency,
per cent.

130.68

0.062

100

6.8

0.152

16.4

300

6.2

0.414

44.8

500

5.2

0.579

62.8

700

4.0

0.624

67.6

900

2.2

0.440

47.7

1100

0.6

0.147

15.9

1190

0.0

0.000

00.0

130.74

0.118

100

14.2

0.316

18.0

300

12.8

0.855

48.6

500

10.8

1.201

68.3

700

8.8

1.372

78.0

900

6.2

1.242

70.6

1100

3.2

0.784

44.5

1300

0.5

0.144

08.2

1340

0.0

0.000

00.0

130.83

0.173

100

21.2

0.472

18.4

300

18.6

1.242

48.3

500

15.8

1.760

68.5

700

12.9

2.010

78.2

800

11.7

2.080

81.0

900

10.0

2.002

78.0

1100

5.4

1.320

51.4

1300

1.6

0.398

15.5

1395

0.0

0.000

00.0

130.95

0.222

100

26.6

0.592

17.9

300

24.0

1.602

48.6

500

20.6

2.295

69.5

700

16.8

2.620

79.5

800

15.0

2.670

81.0

900

12.8

2.660

80.6

1100

7.6

1.860

56.5

1300

2.6

0.752

' 22.8

1420

0.0

0.000

00.0

131.06

0.262

100

31.4

0.700

17.9

300

28.2

1.884

48.2

500

24.2

2.695

69.0

700

19.5

3.040

78.0

TEST DATA

273

TABLE 9.— (Continued)

Turns
of
needle

Head,
ft.

Discharge,
cu. ft. per
sec.

R.p.m.

Brake
load,
Ib.

Bh.p.

Efficiency,
per cent.

800

17.6

3.135

80.3

900

15.2

3.040

78.0 '

1100

8.8

2.155

55.2

x, -

1300

3.4

0.984

25.2

1450

0.0

0.000

00.0

131.19

0.300

100

34.2

0.762

17.3

300

31.0

2.070

47.0

500

26.6

2.960

67.4

700

21.4

3.335

75.8

800

19.1

3.400

77.3

900

16.5

3.300

75.0

1100

10.2

2.497

56.7

1300

4.2

1.215

27.6

1460

0.0

0.000

00.0

7.85

131 ,f4Q

0.356

100

39.2

0.874

16.4

300

35.0

2.340

44.0

500

29.8

3.315

62.2

700

24.6

3.830

72.0

800

21.8

3.880

73.0

900

18.7

3.740

70.3

1100

11.6

2.838

53.3

1300

4.8

1.390

26.2

1460

0.0

0.000

00.0

274

HYDRAULIC TURBINES

TABLE 10. — TEST OF A 12-iNCH PELTON-DOBLE TANGENTIAL WATER
WHEEL UNDER A CONSTANT PRESSURE HEAD OF 230 FT.

Turns
of
nozzle

Head,

ft.

Discharge,
cu. ft. per
sec.

R.p.m.

Brake
load,
Ib.

B.h.p.

Efficiency,
per cent.

230.2

0.081

100

11.4

0.254

12.0

300

10.3

0.687

32.4

500

9.2

1.025

48.4-

700

7.8

1.217

57.5

800

7.0

1.248

59.0

900

6.2

1.242

58.7

1100

4.3

1.054

49.7

1300

2.0

0.578

27.3

1440

0.0

0.000

00.0

230.3

0.163

100

24.6

0.548

12.8

300

22.3

1.490

34.9

500

19.7

2.190

51.2-

700

17.1

2.665

62.4

900

14.4

2.880

67.5

1100

11.4

'2.795

65.4

1300

8.0

2.320

54.3

1500

4.5

1.505

35.2

1700

1.0

0.378

12.5

0.0

0.000

00.0

230.5

0.231

100

31.3

0.698

11.5

300

32.7

2.185

36.2

500

28.3

3.150

52.1-

700

25.6

3.990

66.0

900

21.6

4.330

71.6

1100

17 A

4.268

70.6

1300

12.4

3.588

59.4

1500

7.0

2.340

38.7

1700

2.0

0.756

12.5

1760

0.0

0.000

00.0

230.6

0.291

100

45.2

1.005

13.2

300

41.6

2.780

36.4

500

37.4

4.160

54.5-

700

33.0

5.145

67.4

900

28.2

5.650

74.0

1100

23.1

5.665

74.3

1300

17.6

5.096

66.7

1500

9.5

3.180

41.7

1700

3.9

1.477

19.3

1790

0.0

0.000

00.0

TEST DATA

275

TABLE 10. — (Continued)

Turns
of
nozzle

Head,
ft.

Discharge,
cu. ft. per
sec.

R.p.m.

Brake
load,
Ib.

B.h.p.

Efficiency,
per cent.

230.8

0.343

100

54.2

1.211

13.5

300

49.6

3.315

36.8

500

44.4

4.940

54.9-

700

38.8

6.050

67.2

900

32.6

6.530

72.6

1100

26.6

6.512

72.4

1300

20.2

5.850

65.0

1500

12.8

4.280

47.6

1700

5.4

2.040

22.7

1880

0.0

0.000

00.0

231.1

0.379

100

61.0

1.360

13.6

300

55.5

3.710

37.2

500

49.5

5.510

55.4-

700

43.0

6.700

67.3

900

36.4,,

7.300

73.4

1000

33.2

7.390

74.2

1100

29.7

7.270

73.0

1300

32.4

6.480

65.0

1500

14.4

4.800

48.2

1700

6.4

2.420

24.3

1890

0.0

0.000

00.0

7.85

231.2

0.434

100

67.2

1.499

13.3

300

61.8

4.130

36.8

500

55.4

6.160

54.9-

700

47.8

7.450

66.3

900

40.0

8.015

71.4

1000

36.5

8.125

72.3

1100

32.9

8.063

71.8

1300

24.8

7 . 180

64.0

1500

16.3

5.450

48.5

1700

7.0

2.550

22.7

1890

0.0

0.000

00.0

276

HYDRAULIC TURBINES

TABLE 11. — TEST OF A 12-INCH PELTON-DOBLE TANGENTIAL WATEH
WHEEL UNDER A CONSTANT PRESSURE HEAD OF 305 FT.

Turns
of
needle

Head,

ft.

Discharge,
cu. ft. per
sec.

R.p.m.

Brake
load,
Ib.

B.h.p.

Efficiency,
per cent.

305.1

0.1025

18.0

0.00

00.0

400

15.8

1.41

39.6

800

12.6

2.24

63.2

1000

10.8

2.41

67.7

1200

8.4

2.24

63.2

14(30

6.0

1.87

52.7

1600

3.2

1.14

32.2

1800

0.4

0.16

4.5

1920

0.0

0.00

0.0

305.2

0.185

35.6

0.00

00.0

400

30.8

2.74

42.7

800

24.6

4.38

68.4

1000

21.0

4.67

72.7

1200

17.5

4.67

72.7

1400

13.4

4.18

65.1

1600

9.1

3.24

50.5

1800

4.4

1.76

27.4

2020

0.0

0.00

00.0

305.5

0.278

52.8

0.00

00.0

400

45.2

4.02

41.5

800

36.6

6.52

67.4

1000

32.0

7.12

73.7

1200

26.8

7.15

74.0

1400

20.7

6.45 66.8

1600

14.4

5.13

53.0

1800

7.8

3.12

32.2

2080

0.0

0.00

00.0

305.7

0.341

62.8 0.00

00.0

400

53.6 4.77

40.3

800

43.0

7.65

64.7

1000

38.4

* 8.55

72.2

1200

33.0

8.82

74.5

1400

26.0

8.10

68.4

1600

18.6

6.62

55.9

1800

10.0

4.00

33.9

2110

00.0

0.00

00.0

TEST DATA

277

TABLE 11.— (Continued]

Turns
of
needle

Head,
ft.

Discharge,
cu. ft. per
sec.

R.p.m.

Brake

load,
Ib.

B.h.p.

Efficiency,
per cent.

306.0

0.390

72.0

0.00

00.0

400

62.6

5.57

40.9

800

52.0

9.27

68.1

1000

45.8

10.20

75.0

1200

38.4

10.25

75.3

1400

30.2

9.40

69.0

1600

21.0

7.48

55.0

1800

11.0

4.40

32.4

2150

00.0

0.00

00.0

TABLE 12. — FRICTION AND WINDAGE OF 12-iNCH PELTON-DOBLE TAN

GENTIAL

WATER WHEEL

R.p.m.

H.p.

R.p.m.

H.p.

100

0.0025

800

0.1545

200

0.0089

900

0.2190

300

0.0146

1000

0.2660

400

0.0305

1100

0.3270

500

0.0515

1200

0.3910

600

0.0746

1300

0.4980

700

0.1135

1400

0.5970

1500

0.7020

TABLE 13. — TEST OF A PELTON-DOBLE TANGENTIAL WATER WHEEL NEAR

FRESNO, CAL.

Static Head = 1403.45 ft.

Head,
ft.

Discharge, cu.
ft. per sec.

H.p.
input

B.h.p.

Efficiency,
per cent.

1400.77 17.50

2790

2075

74.4

1398.25 22.10

3510

2767

78.7

1396.56 27.00

4280

3450

80.7

1394.15

31.90

5050

4120

81.7

1391.80

36.70

5800

4765

82.2

1389.65

42.00

6630

5480

82.7

1383.90

54.00

8475

6825

80.6

278

HYDRAULIC TURBINES

TABLE 14. — TEST OF A 2|-iNCH TWO-STAGE WORTHINGTON TURBINE PUMP
AT CORNELL UNIVERSITY

By R. L. Daugherty

Diameter of Impellers = 12 inches. Test made at a constant speed of

1700 r.p.m.

Discharge, cu. ft.
per sec.

Head,
ft.

B.h.p.

Efficiency,
per cent.

0.000

248.5

9.13

00.0

0.049

248.6

10.12

13.7

0.112

254.6

11.78

27.6

0 155

257.5

12.70

35.8

0.236

264.1

15.08

47.1

0.348

248.8

18.15

54.3

0.429

225.0

20.06

55.0

0.494

192.2

21.60

50.0

0.531

157.3

22.00

43.2

0.573

73.1

21.25

22.4

0.578

47.1

20.60

15.0

0.580

26.3

20.15

8.6

TABLE 15. — TEST OF A 6-lNCH SINGLE-STAGE DEL.AVAL CENTRIFUGAL
PUMP AT CORNELL UNIVERSITY

. By R. L. Daugherty
Volute Type. Diameter of Impeller = 9.11 inches. Speed 1700 r.p.m.

Discharge, cu. ft.
per sec.

Head,

ft.

B.h.p.

Efficiency,
per cent.

0.000

68.5

4.3

00.0

0.068

68.4

4.5

11.7

0.188

69.6

5.2

28.6

0.320'

69.3

6.0

42.0

0.606

69.2

7.8

61.0

0.873

65.8

9.3

70.2

1.063

62.7

10.3

73.5

1.315

55.7

11.3

73.7

1.632

47.3

12.0

73.2

1.968

35.7

11.8

67.7

2.090

28.1

11.5

58.0

2.240

22.3 .

11.2

50.7

INDEX

Absolute velocity, 80
Action of water, 7
American turbine, 9
Annual cost of power, 197
Arrangement of runners, 11
Axial flow, 9

Barker's mill, 41
Bearings, 56
Brake, 146
Breast wheel, 1
Buckets, 31

design of, 208

pitch of, 205

Capital cost, 195
Case, 58, 209
Centrifugal force, 97, 231

pump, 230

Chain type construction, 32
Characteristic curve, 173
Classification of turbines, 7
Clear opening, 216
Coefficient of nozzle, 114
Conduit, 6
Constants, 153

determination of, 160

uses of, 161
Cost, steam power, 200

turbines, 192

water power, 197
Current meter, 145

wheel, 1
Cylinder gate, 52

Defects of theory/ 134," 24 2
Deflecting nozzle, 36
Diameter of runner, 52

and discharge, 154

and power, 155

and speed, 152
Diffusion vanes, 231
Direction of flow, 8
Discharge curve, 15

measurements of, 143
Doble bucket, 30
Draft tube, 13, 63, 100
Dynamometer, 146

Eddy loss, 70, 125
Efficiency, 83

as function of speed and gate,
164

of impulse wheel, 39, 115

maximum, 177

on part load, 179

of reaction turbine, 72

relative, 182
Energy, 80

Falling stream, effect on flow, 15

Fitz water wheel, 2

Flow and head, 18

Flow, direction of, 8

Flume, 14

Force exerted, 84, 108

Forebay, 6

Fourneyron turbine, 8, 41, 93

Francis turbine, 3, 41

279

280

INDEX

Friction, 147

variation with speed, 151
Full gate, 183
Full load, 179

Gage heights, fluctuations in, 16

relation of, to flow, 15
Gates, cylinder, 52

design of, 228

wicket, 53

Girard turbine, 30, 92
Governing — see speed regulation.
Governors, 74
Gradient, 15
Guides — see gates.

Hand of turbine, 9
Head, -80

delivered to runner, 91

and discharge, 152

and efficiency, 152

measurement of, 143

net for pump, 233

net for turbine, 81

and power, 152

and speed, 152

variation of, 17

-water, 6

Holyoke testing flume, 140
Howd turbine, 41
Hydraucone, 68
Hydraulic gradient, 21
Hydrograph, 16

Impending delivery, 237
Impulse circle, 204
Impulse turbine, 7

advantages of, 180, 190

problem of, 92
Inward flow, 3, 8

advantages of, 46

Jet, use of two or more, 34

size of, to size of wheel, 33

Link, 55
Load "curve, 19

factor, 19

factor and cost, 198
Losses, centrifugal pump, 234

impulse turbine, 93, 113

reaction turbine, 94, 125

Manoinetric efficiency, 233
McCormick runner, 42
Mixed flow, 9

Needle nozzle, 35
Nozzle coefficients, 114
design, 204

Operating expenses, 197
Outward flow turbine, 8
Overgate, 183
Overhung wheel, 12
Overload, 180
Overshot wheel, 2

Part load, 180

Pelton wheel, 30

Penstock, 6

Pitot tube, 144

Pivoted guides, 53

Pondage, 19

Power, 83

delivered to runner, 91
effect of head on, 18
effect of pondage on, 19
effect of storage on, 20
measurement of, 146
through pipe line, 21
of stream, 19

INDEX

281

Race, 6
Radial flow, 8
Rainfall and run-off, 16

use of records of, 17
Rating curve, 15
Reaction turbine, 7

advantages of, 179, 185, 189

development of, 41

problem of, 93
Register gate, 53
Regulation, 52
Relative velocity, 80
Relief valve, 24, 37, 56
Retardation runs, 148, 249
Rheostats, 146
Runners, types of, 43

construction of, 51

design of, 211

diameter of, 52

Sale of power, 198
Scotch turbine, 41
Selection of turbine, 169, 177
Shaft, choice of, 10

position of, 9
Shifting rim, 55
Shock loss, 125, 236
Shut-off head, 237
Specific speed, 155, 167, 245
Speed, effect of, 94

ring, 61, 227
Speed regulation, 35, 52

effect of pipe on, 23
Splitter, 32
Stand pipe, 24
Steam power cost, 200
Step bearing, 56
Storage, 20

Stream flow, 16
gaging, 15

Suction tube — see draft tube.
Surge chamber, 24
Suspension bearing, 57
Swain turbine, 42

Tail race, 6

Tangential water wheel, 105

Testing, 140

Thrust of runnet, 58

Torque, 89

Turbines, advantages of, 4

classification of, 7

definition of, 2

types of, 7
Turbine pump, 230

Undershot water wheel, 1
V

Value of water power, 202
Vanes, number of, 214

layout of, 219
Velocity in case, 227

of pump discharge, 232
Volute pump, 230
Vortex, 97

Water hammer, 24
Water, measurement of, 15
Water power, 15
Water power plant, 5
Water wheels, 1, 3, 4
Wear of turbines, 70, 102
Wicket gate, 53

FOURTEEN DAY USE

RETURN TO DESK FROM WHICH BORROWED

This book is due on the last date stamped below, or

on the date to which renewed.
Renewed books are subject to immediate recall.

LIBRARY USE

EC'D LD

UUI 1*1960

LD 21-100m-2,'55
(B139s22)476

General Library

University of Californu

Berkeley

YC 19862

UNIVERSITY OF CALIFORNIA LIBRARY

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