-o
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=1^
^co
-.^;?r
■!'-'r,-A-.,
presenteC) to
Zbc Xibrar?
Of tbe
XDintvereit? of Toronto
bs
J^&.O. Kerry, Esq.
The D. Van Nostrand Company
intend this book to be sold to the Public
at the advertised price, and supply it to
the Trade on terms which will not allow
of discount.
THE FLOW OF WATEE
A NEW THEOEY
OF
THE MOTION OF WATER UNDER PRESSURE
AND IN OPEN CONDUITS AND ITS
PRACTICAL APPLICATION
BY
LOUIS SCHMEER
CIVIL AND IRRIGATION ENGINEER
Hi/,///
v/
\o
.6-
NEW YORK:
D. VAN NOSTRAND COMPANY
23 MURRAY AND 27 W-AKEEN STS.
1909
Copyright, 1909,
BY
D. VAN NOSTRAND COMPANY
NEW YORK
F. H. GILSON COMPANY
BOSTON, U.S.A.
PREFACE.
The present work is the outcome of a series of investigations
begun several years ago with the object of finding a simple
expression for the phenomenon of flow in irrigation channels.
The author hopes that his work will prove of interest and
value to the student and useful to the practical engineer.
He also hopes that it will stimulate further research and thus
tend to widen the field of hydraulic knowledge.
Louis Schmeer.
Los Gatos, California, October, 1909
m
NOTATION.
F, V = Velocity in feet per second.
72, r =
S^s =
c
/
z
M,K
H,h
The mean hydraulic radius of a conduit.
Area of cross section.
Wet Perimeter.
Diameter of circular or semicircular conduit.
r Diameter of a circular conduit.
■< Depth of Semi square or semi circle.
[ Depth of Water in a Channel.
Slope of Water Surface.
Head in feet.
Length of conduit in feet.
Fall of surface in feet
Distance in feet.
The variable coefficient in the formula v = c \/r . s.
The coefficient of friction, loss of head per unit area of surface at
unit velocity.
A coefficient indicating the resistance of an impediment to flow.
Coefficients indicating the degree of roughness of the wet peri-
meter.
A coefficient indicating the variation of the coefficient c with the
velocity of flow.
A vertical distance, a head of water.
Length of a conduit, a horizontal distance.
Width of surface of water.
iv
ERRATA
Page
Column
Une
For
Read
IV
1
V,v
V.v
3
1
L
R, r. etc.
1
13
1
R
R
2
E
AG
E
QG
15
v>"
Z/2
27
15
minute
second
31
22
Rv
RS
45
9
experimental
exponential
49
9
assistance
resistance
17
r
V
53
3
canals to
canals and
63
14
second )
\/T0 =3.163. \/T0 =
\/T0 = 3.163. VTO =
65
from >
bottom )
0.466
0.448
1
70
heading
9
1
Vi\
72
last four
heading
Loss of Head in Feet
per Unit Length
P , or Loss of Head
in Feet per 2 g feet
79
2
14
ppes
pipes
110
8
heading
r
V
liM
7
heading
r
V
124
Formula
(b)
(If
(If
18
76.69
73.27
21
53.33
55.33
136
15
same
some
145 )
Table C 1
7
next to
last
0.7425
0.9425
206
2
12
0.42
0.34
207
2
7
0.26
0.34
225
sixth from
Diameters of Veloci-
Diameters and Ve-
bottom
ties
locities
228
7
Thread and Mean
Velocity
Thread of Mean
Velocity
Digitized by the Internet Archive
in 2007 with funding from
IVIicrosoft Corporation
http://www.archive.org/details/cabotbibliographOOwinsuoft
TABLE OF CONTENTS.
PAGE
Introduction 1
Primary Laws of Pressure and Fall 4
Primary Laws of Fluid Friction 8
Distribution of Head 12
Distribution of Energy 14
The Coefficient c in the Formula v = c \^rs 15
Primary Determination of the Coefficient c 17
Variation of the Coefficient c
(a) with the roughness of the wet perimeter 21
(6) with the velocity of flow 24
Mathematical Expressions for the Variation of the Coefficient c
with the Velocity:
(a) for conduits under pressure 32
(b) for open conduits 43
(c) for channels in earth 47
The Resistance Due to Curves 55
The Resistances Due to Entrances, Elbows, etc 57
Riveted Conduits 59
Practical Application of the Formula 63
Values of a, the Coefficient of Variation of c 70
Values of the Coefficients c and /for Conduits under Pressure . . 71
Loss of Head in Welded Conduits 72
Diameters, Internal Areas, Radii and Their Roots 73
Roots of Mean Hydraulic Radii 74
Values of m and K, the Coefficients Indicating the Degree of
Roughness 76
Alphabetical List of Authorities 77
Experimental Data 82
Forms of Sections of Conduits. 113
Sewers 118
Exponential Equations 121
(a) for conduits under pressure 124
(6) for sewers 126
(c) for open conduits 1 29
Explanation of the Use of the Tables of Velocities and Quantities 1 36
Sines of Slopes and Their Roots 143
V
VI TABLE OF CONTENTS
PAGE
Powers of Diameters of Circular Conduits 145
Powers of Mean Hydraulic Radii or of Depths of Water in the
Form of Section Most Favorable to Flow 147, 151
Quantities of Discharge in Cubic Feet per Second of a Conduit
One Foot in Diameter 155
Velocity of Flow in a Semi Square 1 Foot Deep 159
Discharge of a Semi Square 1 Foot Deep 163
Weir Discharges:
(a) Francis' formula 167
(6) Bazin's formula 168
Weir Formula 173
Methods of Measurement:
(a) loss of head 183
(b) discharges 184
Surface, Mean and Bottom Velocities 193
Variation of the Coefficient c with the Slope 196
The Formula in Metric Measure 205
English and Metric Equivalents 209
Greatest Efficiency of a Conduit of a Given Diameter as a Trans-
mitter OF Energy 211
Most Economical Diameter of a Conduit under Pressure 212
THE FLOW OF WATEE.
INTRODUCTION.
There is no branch of the science of physics on which more
has been written than on hydraulics. The master minds of the
last four centuries have wrestled with the problem and thread
by thread they have torn away the veil of mystery that enveloped
the phenomenon of flow.
The universal mind of Leonardo da Vinci (1452-1519),
painter, sculptor, scientist and engineer, was the first to pierce
the darkness and although he did not give his thoughts on the
flow of water mathematical expression, we are to-day, with all
the knowledge and experience gained since his time, astounded
at his clear and comprehensive reasoning.
The great Galileo (1564-1642) admitted that he had less
trouble in finding the law of motion of the planets millions of
miles away than in discerning any law in the motion of water
in the stream flowing at his feet.
Torricelli (1608-1644), inventor of the barometer, investi-
gated the laws of falling bodies and found that the velocities
of bodies falling free vary with the square roots of the heights
fallen through, or with ^/IT.
Huygens (1629-1695) first found the numerical value of gf,
the acceleration due to gravity; and following him Bernoulli
was (in 1738) able to write the fundamental formula for the
velocities of bodies falling free.
On this general theoretical foundation our present system
of hydraulics has gradually been built. Brahms (Dyke and
1
2 THE FLOW OF WATER
other Hydraulic Constructions 1753) made the first step towards
a practical application of the then existing theories of motion
to the motion of water flowing in a channel. He found that
the motion of water flowing in a channel is not like the motion
of water falling free, or that of a body rolling down an inclined
plane continually accelerated in speed, but moves with a
uniform velocity, and that the resistance due to the friction of a
fluid against the walls of the conduit depends on the relation of
the wet perimeter to the area of the cross-section or on the mean
hydraulic depth.
Chezy (in 1776), gave the ideas of Brahms an elegant mathe-
matical expression by writing for the velocity of flow
V = c Vr . s
in which c is a coefficient, which Chezy assumed to be constant,
and r the mean hydraulic depth. This simple formula found
general application in practice and is still in use.
Subsequent writers occupied themselves chiefly with the
definition of variations of the coefficient c in the formula pro-
posed by Chezy.
Owing to the researches of Coulomb (1736-1806) on the
resistance of fluids to slow motions, the variation of the
coefficient c with the velocity of flow was the first to be
recognized and Weisbach and others found expressions for this
variation.
If Darcy was not the first to perceive the influence of the
degree of roughness of the walls of a conduit on the velocity of
flow, he at any rate was the first who thoroughly investigated
the subject. (Mouvement de Feau dans les tuyaux, Paris, 1851.)
Beginning his investigations on flow in conduits under pressure
he extended them to flow in open conduits and under the
auspices of the government of France constructed a special
test channel 596.5 meters (1956.5 feet) long and 2 meters wide.
This channel was successively lined with materials possessing
characteristic degrees, of roughness, the cross-section was given
various forms and the bottom various slopes. To regulate the
discharge two reservoirs were constructed at the head of the
INTRODUCTION 3
channel and the water admitted through carefully tested sharp-
edged orifices 20 centimeters square. The experiments were
extended also to flow in channels lined with masonry and to
flow in channels in earth.
Darcy's work was after his death completed by Bazin, his
successor in the office of Chief Engineer of Bridges and Roads
in France. Darcy-Bazin's experiments were made with the
utmost care and precision and the tabulated data (Darcy-
Bazin, Recherches HydrauUques, Paris, 1856) bear the stamp
of scientific exactness and truth; they are mines of reliable
information on all matters relating to flow.
Darcy's experiments on flow in pipes have since his time been
supplemented by many others. Hamilton Smith in California
carefully gauged the discharge of sheet-iron riveted pipes
under great pressures, and his data rank in reliability with
those of Darcy. Clemens Herschel gauged the discharge of
large steel-riveted pipes; Iben that of pipes coated with
tar; Adams and Noble the discharge of circular pipes of planed
boards.
Kutter, a Swiss engineer, extended the researches of Darcy-
Bazin on flow in open conduits to channels of greater slopes
and greater dimensions and published (in 1869) the results of
his investigations under the title '' Versuch zur Aufstellung
einer allgemeinen Formel," etc.
Kutter and Ganguillet elaborated a general formula intended
to define the variation of the coefficient c in the formula of
Chezy with the mean hydraulic radius, the degree of roughness
of the walls of the channel and also with the slope.
Despite its cumbrousness this formula found universal appli-
cation. It has, however, many defects and is no longer regarded
as embodying any true law of flow.
Bazin, in his memoir, '' Etudes sur les mouvements des eaux
dans les canaux decou verts " (Annales des Fonts et Chauss^es,
Paris, 1898), reviews the accumulated experimental data and
proposes a formula of great simplicity. It does not, however,
express the variation of c with the velocity or with the
slope.
4 THE FLOW OF WATER
PRIMARY LAWS OF PRESSURE AND FALL.
A. .
The physical laws relating to fluids at rest, which are of interest
in their relation to fluid motion, are briefly as follows :
1. The pressure of water on a surface is proportional to the
depth below the free surface.
Let H be the vertical distance of a horizontal plane below the
free surface,
G the weight of one cubic foot of water = 62.37 pounds.
P the pressure in pounds per square foot,
then p = GH = 62.37 H
and the pressure per square inch
P = ^T^ H = 0.433 H pounds.
144 ^
2. The pressure of water is the same at all points in a hori-
zontal plane irrespective of the horizontal distance of any point
in the plane from the free surface. No matter what the shape
of the vessel or the length of the conduit may be the pressure at
any point is always proportional to the vertical distance below
the free surface.
At the bottom of a stand pipe 80 feet below the free surface
of the water the pressure on the area of a circle 4 inches in
diameter will be
0.433 80.0 42 0.7854 = 435.2 pounds.
Let a 4-inch pipe 5 miles long be connected with the standpipe
at any point below the free surface, and the end of the pipe be
placed in the same horizontal plane as the bottom of the stand-
pipe, then, no matter how many curves or elbows there may be
in the length of the conduit, the pressure will be as before, equal
to 435.2 pound.
3. If a pressure be applied to the free surface of the water,
this pressure is transmitted equally and undiminished in all
directions, and to any distance, horizontal or vertical.
Into the upper end of a pipe 1 foot in diameter and filled
PRIMARY LAWS OF PRESSURE AND FALL 5
with water let a piston be inserted and a pressure of 100 pounds
applied. Then a pressure equal to = 129.5 pounds per
square foot will be exerted on any square foot of the inner
surface of the pipe, no matter how great the distance. Let the
depth of the water below the surface be 20 feet. Then the
total pressure per square foot will be
129.5 + (20 62.37) = 1403.9 pounds.
If Pj is the external pressure in pounds per square foot, the
total pressure will be, for any distance H,
P = P, +GH.
The external pressure due to the atmosphere is equal to 14.7
pounds per square inch. It is consequently equal to that of a
14 7
column of water _ = 33.9 feet in height.
U.4oo
B
Torricelli's fundamental theorem for the velocity of bodies
falling free is expressed by the equation :
1. V = gt .
2. v^ = 2gh
3. h = igP
Or:
1. The speed of fall is proportional to the time of fall.
2. The square of the speed is proportional to the distance
fallen through.
3. The distance fallen through is proportional to the square
of the time of fall.
The velocity of fall in feet per second is consequently:
At the end of the first second of fall equal to g = 32.2 ft.
At the end of the second second of fall equal to 2g = 64.4 ft.
At the end of the tenth second of fall equal to 10 gf = 322.0 ft.
6 THE FLOW OF WATER
The velocity of fall in feet per second is equal.
At the end of the first foot of space fallen through to
VW = 8.025.
At the end of the second foot of space fallen through to
V?^ = 11.34.
At the end of the tenth foot of space fallen through to
V20^= 25.35.
The distance fallen through is equal:
At the end of the first second of the time of fall to i g = 16.1 ft.
At the end of the second second of the time of fall to ^ g2^
= 64.4 ft.
At the end of the tenth second of the time of fall to ^ g 10^
= 1610.0 ft.
C.
The laws of fall thus stated apply to any body, solid or liquid
falling free in vacuo.
For bodies falling in the atmosphere, the resistance of the air
has to be considered. This resistance is proportionally the
greater, the less the density of the body. Disregarding the
resistance of the air, a jet of water issuing from a well-formed
orifice has a velocity proportional to the square root of the
height of the column of water above the centre of gravity of the
orifice.
Let h be the head of water above the centre of gravity of the
orifice.
h a coefficient of velocity differing with the nature of the
orifice, and the velocity of the jet will be
V = h \/2gh.
If the discharge is into free space the speed of the motion will
continue to increase with the distance fallen through, and if
hj^ be the vertical distance fallen through in the atmosphere,
the water will have, at the end of its journey, acquired a velocity
equal to
V = h V^gih + hj) nearly.
PRIMARY LAWS OF PRESSURE AND FALL 7
D.
The motion of a rigid body descending in an inclined plane
infinitely smooth is continually accelerated; the law of fall still
holds, only with this difference, that in the equation
g is replaced by g sin d, d being the angle which the inclined
plane makes with the horizon. The kinetic energy or living
force aquired by a body descending in a plane infinitely smooth
is equal to
Wh OY hmv^
W
in which m = — = the mass of the body. The weight of the
body W , divides into two components; one, equal to W sin d
acts parallel to the plane and produces motion ; the other, equal
to W cos d, acts at right angles to the plane.
When the frictional resistance between the plane and the
descending body is considered, the force that produces the
motion or W sin d reduces to W sin d, — zW cos d, z being a
coefficient of friction.
The acceleration of motion continues as long as W sin d is
greater than zW cos d. If they are equal, or if -%, or tangent
d is equal to z^ the coefficient of friction, the motion will cease.
Following the laws of motion of a rigid body, the motion of a
perfect fluid flowing down an inclined plane infinitely smooth
would be continually accelerated. Owing, however, to internal
friction, to its adhesive qualities, and the friction of the fluid
against the surface of the channel in which it flows, water soon
spends its accelerating force and the motion arrives at a state
of steadiness more or less approaching uniformity.
The motion of water is said to be steady, when at a given point
of the cross-section the fluid arrives with the same velocity and
in the same direction.
The motion is said to be uniform, if in following a given
course the mass of water has a constant velocity.
8 THE FLOW OF WATER
The motion is said to be varying, if in following a given course
the velocity varies from point to point.
In our subsequent discussions of flow we always assume the
motion to be uniform, or conditions to be such that there is no
acceleration of velocity with increase of the distance fallen
through, that the accelerating forces are equalized by frictional
resistances and that the velocity of flow at any point in a given
course remains constant as long as the slope remains constant.
PRIMARY LAWS OF FLUID FRICTION.
A plane surface moving in a still body of water is retarded in
its motion by a resistance due to the friction of the fluid against
the surface.
The subject of fluid friction was investigated by Coulomb
by rotating disks of greater or lesser diameters and having
surfaces of a greater or lesser degree of roughness with more or
less speed in a still body of water, at greater or lesser depths, and
ascertaining the work done under the various conditions.
The researches of Coulomb were extended by Froude in his
investigations on the resistance of the surfaces of ships (1870-
1874). For the rotating disks of Coulomb, Froude substituted
sharp-edged planks or metal plates of greater or lesser length
and coated with various substances. These he impelled to move
in a still body of water and ascertained the resistance by a suit-
able device.
The laws deduced from experiments made by these investi-
gators may be summed up as follows :
1. The pressure existing in any horizontal plane below the
free surface or in any part of a conduit under pressure has no
influence on the friction of the fluid against a solid surface.
Though the pressure in pounds per unit area may be much
greater in one part of a conduit than in another, the frictional
resistance of the area is not thereby increased. This is demon-
strated as follows:
A plank of suitable shape is immersed in a still body of water
just below the surface, impelled to move at a certain constant
PRIMARY LAWS OF FLUID FRICTION 9
speed, and the resistance to motion ascertained. If the plank
is subsequently placed at a greater depth and impelled to move
at the same constant speed, it is found that the resistance to
motion has not been increased. If a pipe of constant dimensions
is resting on an inclined plane, it can also be shown that the
loss of head due to the frictional resistance is for equal lengths
of the conduit the same in the lower part of the conduit
where the pressure is greatest,, as in the upper part, where it is
least.
2. The resistance to motion, due to the friction of a fluid
against a solid surface, is proportional to the area of the surface.
This is demonstrated as follows: A plank of a certain length
and width is impelled to move at a certain constant speed in a
still body of water and the work done in foot pounds noted.
If the width of the plank is subsequently doubled, thus doubling
the area of its surface, and it is impelled to move at the same
constant speed, it is found that the work done in foot pounds is
also doubled.
If water flows in a pipe running full it is found that the amount
of head consumed in overcoming the resistance of the walls is
proportional to the length of the pipe.
Let ^0 be the area of a surface in square feet: W the weight
in pounds required to move a plank in a still body of water
at a velocity of one foot per second; / the frictional resistance
in pounds per square foot of surface
W
then f =T'
and the total resistance to motion in pounds at any velocity
W = jAoV-,
X being the variable exponent of the power of ?;, to which the
resistance is proportional.
As the frictional resistance in pounds per square foot for a
velocity of one foot per second corresponds to an equal pressure
per square foot, the head corresponding to the resistance is
equal to h = -^^
10 THE FLOW OF WATER
The head equal to the resistance or -^, multiplied by 2gf,the accel-
eration due to gravity or
G
is termed the coefficient of friction and denoted by z. As /
= - — the total resistance of a surface in pounds is equal to
2 9
W :=zGA,^.
The velocity of flow remaining after the frictional resistance
is equalized acts through a distance equal to v. The total work
done in foot pounds in overcoming the frictional resistance of a
surface is consequently:
, . ,■! zGAoV''+'-
f Ao v'-^'' = ^ .
"^9
3. The resistance to motion due to the friction of a fluid
against a solid surface is for equal areas of the surface greater
for a short than for a long surface. This is demonstrated by
impelling two planks of equal areas but different lengths to move
at equal constant speeds in a still body of water. It will be
found that more power is consumed in moving the shorter
plank. There is a resistance due to the cutting edge of the
plank, this resistance is proportionally more apparent the
shorter the plank, because the total surface is proportionally less.
At the entrance of any kind of a conduit head is consumed by
a resistance due to shock. For short conduits this head is an
appreciable part of the total head consumed. With increasing
length of the conduit the head thus consumed becomes pro-
portionally less and less in comparison with the total loss of head
and becomes insignificant for very long conduits.
4. The resistance to motion due to the friction of a fluid
against a solid surface is increased by elbows, curves, etc.
Joessel, experimenting on the resistance of ships, found the
resistance of oblique planes to be equal to
^ ^ 0.39 + 0.61 sin. a 2g
PRIMARY LAWS OF FLUID FRICTION
11
in which / is a coefficient indicating the degree of roughness of
the surface, varying between 1.1 and 1.7, d the density of the
fluid, A the area of the surface, a the angle the plane makes with
the line of motion.
The resistance to motion in conduits is proportional to the
angle of deflection, the radius of a curve and its length.
5. The resistance to motion due to the friction of a fluid
against a soHd surface varies with the degree of roughness of
the surface. It increases rapidly as the roughness of the surface
increases. By impelling surfaces coated with different materials
to move in a still body of water Coulomb found the following
values of z, the coefficient of friction and /, the resistance in
pounds per square ft.
Description of Surface.
z
/
For a varnished surface
00258
00350
00362
00489
00418
00503
00250
For a planed and painted plank
00339
For the surface of iron ships
For a new painted iron plate ,
00351
00443
For a surface coated with fine sand
00405
For a surface coated with coarse sand
00488
6. The power of the velocity to which the frictional resistance
is proportional is not constant. It varies with the degree of
roughness of the surface; with the length of the surface in the
direction of motion: it is also influenced by angles, curves, etc.,
in the surface.
By impelling surfaces coated with various materials and of
various lengths in the direction of motion to move in a still
body of water Froude found the following values of x, the
exponent of the power of v to which the resistance is proportional :
Description of Surface.
Length of Surface in Feet.
2
8
20
50
Varnished surface
2.0
2.16
2.0
1.85
1.94
1.99
2.0
1.85
1.93
1.90
2.0
1 83
Surface coated with paraffin
Surface coated with tinfoil
Surface coated with sand
1.83
2 0
12 THE FLOW OF WATER
DISTRIBUTION OF HEAD.
Water issuing from a well-formed orifice flows with a velocity
directly proportional to the square root of the vertical distance
between the centre of gravity of the orifice and the free surface,
and the velocity will continue to increase if the discharge is into
free space.
A stream of water entering a conduit encounters various
frictional resistance tending to equalize the accelerating forces
and uniform motion ensues. The total head consumed in
producing this uniform motion may be resolved into several
components :
1. Head consumed in producing the velocity. This is always
equal to
' = 2T
and usually but a small fraction of the total head.
2. Head consumed in overcoming the frictional resistance
due to the entrance of the conduit. Let z^ be a coefficient
indicating the resistance due to the entrance and the head con-
sumed will be
ho = z,-
3. Head consumed in overcoming the frictional resistance of
the wet perimeter, or of the walls of the conduit.
We have previously seen that the energy expended in over-
coming the resistance of a surface is
E = z^ GAq — foot pounds.
^9
Replacing Aq, the area of the surface by its equivalent P, the
wet perimeter multiplied by L, the length of the conduit, this is
and since Q, the discharge, is equal to ^i, the area of the cross-
section multiplied by v, the velocity,
P v^
DISTRIBUTION OF HEAD
13
and as
P
A"
L
R
we have
E
AG
L ^2
''R 2g'
As E, the total force in
foot pounds, is
the product of height
of
fall, quantity and
weight we have
and consequently
29
4. Head consumed in overcoming the frictional resistances
due to curves, elbows, changes of section, etc.
If Zn is a coefficient indicating the resistances due to these
impediments to flow, the head consumed will be equal to
h - "^
Summing up all the components we have
H = h +h^ ■{■ h, + K
or
or
2g 2g
+ z
R2g
+ ^.
2g
i7 = (1 + ^0 + ^1^ + Zn) Y~
From this we have for the velocity
2gH
-\h
I -V Z^ + Z^--V ZUr,
This is on the assumption that the resistance of a surface is
proportional to the square of the speed. We have already
observed, however, that this is not always the case; it is in fact
an exception. But we are not yet in a position to give the true
indexes of the powers of v to which the resistance is pro-
portional.
14 THE FLOW OF WATER
DISTRIBUTION OF ENERGY.
A quantity of water, GQ, impounded at a vertical distance, Jf,
above a horizontal plane, possesses with reference to that plane,
a stored up or potential energy equal to
QGH.
If by means of a conduit of greater or lesser length the water
is transported to the horizontal plane at the vertical distance
H, below the free surface the stored-up energy is transformed
into work. The total stored-up energy resolves into several
components.
Let the difference of level between the free surface and the
horizontal plane be 80 feet, the length of the asphalt-coated
cast-iron conduit transporting the water 10,000 feet, and its
diameter one foot.
Assuming for zi the average value 0.00489 we have for the
velocity of flow from the data given
64.4 . 80
V =
+ 0.505 + 0.00489 ^^^^^
0.25
or r = 5.11 feet per second.
The discharge in cubic feet per second will be
Q = 5.11 d=^ 0.7854 = 4.013 cubic feet.
The total energy expended in transporting this quantity is
equal to
E = 4.013 . 62.4 . 80 = 20,033 foot pounds.
This total energy of 20,033 foot pounds is consumed as follows:
1. A quantity of work is done in producing the velocity of
flow. This is equal to
QG — = 4.013 . 62.4 4^ = 101.6 foot pounds.
2g 64.4
2. Another quantity of work is done in overcoming the
resistance at the entrance. This is equal to
QGzo^ = 4.013 . 62.4 . 0.505 =^ = 51.3
2^ 2g
This is on the assumption that Zq = 0.505.
DISTRIBUTION OF ENERGY . 15
3. The principal part of the work is done in overcoming the
frictional resistance of the interior surface of the conduit. This
is equal to
QGz ~'^ = 4.013 . 62.4 . 0.00489 3M^ ?^ = 19 880
^ R 2g 0.25 64.4
foot pounds.
The sum of the several quantities of work done in trans-
porting 4.013 cubic feet of water a vertical distance of 80 and a
horizontal distance of 10,000 feet is equal to
101.6 + 51.3 + 19,880 = 20,033 foot pounds,
or
Dividing both sides of the equation by QG we have as before
The Coeflacient C in the Formula v = C Vr^,
Neglecting the loss of head due to the velocity, the loss of
head due to the frictional resistance of the entrance, and the loss
of head due to the resistance of other obstructions to flow,
w^hich severally or combined, form but a small part of the total
head lost if the conduit is of a length of 4,000 times the mean
hydraulic depth or the velocity not great, we have
L v^
H = z. - :r- as the loss of head due to the frictional resistance
R2g
of the walls of the conduit. From this we have
V' HR
and V = V — V;
2g
r . s.
The term y — is equal to the coefficient c first introduced into
16 THE FLOW OF WATER
hydraulic calculations by Chezy, a French engineer (in 1776).
On account of its brevity, this term is almost exclusively used
to indicate the frictional resistance of long conduits of all
descriptions.
As . =^^
As z, ^
in which / = the frictional resistance in pounds per square foot of
surface,
G = the weight of one cubic foot of water = 62.4 we
may write
C =
■2^
2g/
G
and as -^ = head lost per unit area of surface at unit velocity,
G
I
G
we have finally
c_J .1
^ head lost per unit area at unit velocity.
Chezy and many of his followers up to the middle of the last
century considered the coefficient c to be a constant. The
researches of Coulomb, the investigations of Prony, Eytelwein,
Weisbach and others, however, revealed the fact, that it varies
with the velocity of flow. Later researches by Darcy and
Darcy-Bazin brought to light the astounding influence of the
degree of roughness of the walls of a channel and of the value of
the mean hydraulic radius on the value of c. The manifold
variations of c render the problem of its exact valuation one of
great difficulty. A mathematical expression embodying all
variations will necessarily be very complex; to be of practical
value, however, it should be as simple as possible. It is some-
what difficult to harmonize great exactness and great simplicity
without making sacrifices at one end or the other. On this
account two expressions are often found embodying the same
idea and rendering it with great exactitude or great sim-
pficity.
DISTRIBUTION OF ENERGY 17
We will now proceed to investigate the laws on which the
variation of c depends and to find suitable mathematical expres-
sions embodying these laws.
I. Primary Determination of the Coeflacient c.
Going back to first principles we may ask the question : To
what power of R, the mean hydraulic radius, is the velocity of
flow proportional? Using the exponential equation
V. \rJ
I.- u • log ^1 - log ^0
which gives x = , ^ ^ , ^ "
log R, - log R,
we find that the value of x is, in the case of channels in earth,
such as rivers and canals and with R varying between 1 and 50
feet in the majority of cases equal to
1 2 ^j. 3
1,333 2.666 ^^ 4
For this class of conduits we may consequently write:
V = yR^ VT,
in which y is variable, difTering with the degree of roughness and
with the slope of the conduit. Asv = c VrTsTand R^ = -v^r" Vr"
we have __
C = y Vr,
hence c increases directly with Vr.
c
Column 5, Table I, gives values of ?/ = -47= for conduits of
Vr
several degrees of roughness. It will be observed that the
formula gives fairly constant values of y only for large conduits,
such as rivers and canals,
For small conduits however y increases with increase of R if
the wet perimeter be smooth, but decreases with increase of R
if the contrary is the case. Applying the exponential equation
to other classes of conduits, the following values of x, the power
of R, to which the velocity is proportional were found.
18 THE FLOW OF WATER
For a semi-circular channel of fine cement x = 0.67
For a semi-circular channel of concrete x = 0.68
For a rectangular channel of rough boards x = 0.69
For a rectangular channel of rough masonry x = 0.75
For a channel carrying coarse detritus a; = 1.00
The conclusions to be drawn from these data may be summed
up as follows :
1. For rivers and canals the power of R, to which the velocity
is proportional, is approximately equal to |.
2 For small channels the power varies with the degree of
roughness of the perimeter and the form of the cross-section of
the conduit.
3. For small channels the power of R increases with increase
of roughness.
4. For the smoothest class of conduits the velocity is pro-
portional to R^'^"^ for the very roughest to R^'^. Hence the
rougher the wet perimeter, the more conditions are approached
resembling those pertaining to flow in permeable strata, in which
instance the velocity is proportional to the square of the diameter
of the channel.
5. No formula, based on any one single power of R can give
satisfactory results when applied to all classes of conduits.
VARIATION OF THE COEFFICIENT C
Table I.
19
Description of Conduit,
R
1000 s
V
Vr
Sudbury Conduit. Smooth hard brick well
0.5
0.189
1.134
138
pointed.
0.6
0.189
1.371
135
0.8
0.189
1.515
131
1.0
0.189
1.754
127
1.2
0.189
1.948
124
1.4
0.189
2.148
121
1.6
0.189
2.382
119
1.8
0.189
2.514
119
2.0
0.189
2.683
116
2.2
0.189
2.843
114
2.33
0.189
2.929
113
Semicircular channel lined with pebbles |
0.454
1.5
2.17
95.1
to I inch diameter.
0.546
1.5
2.50
95.3
0.619
1.5
2.69
92.5
0.681
1.5
2.93
92.3
0.731
1.5
3.05
91.3
0.784
1.5
3.22
90.4
0.826
1.5
3.33
88.4
0.900
1.5
3.54
87.6
0.968
1.5
3.73
85.8
1.012
1.5
3.95
87.9
Solani Embankment.
6.32
0.140
2.63
55.9
Jaoli Site.
6.53
0.144
2.70
55.0
Sides of brick set in mud, bottoms very-
6.79
0.145
2.80
54.9
rough.
7.05
0.146
2.81
53.7
7.46
0.160
2.94
51.5
Linth Canal, channel in earth, fairly
5.14
0.29
3.414
58.6
regular.
5.93
0.30
3.830
58.2
6.48
0.31
4.152
58.2
7.12
0.32
4.418
56.5
7.52
0.33
4.753
57.4
8.09
0.34
4.920
55.8
8.28
0.34
5.058
56.1
8.62
0.35
5.225
57.7
8.87
0.36
5.392
55.5
9.18
0.37
5.530
54.5
River Seine at Paris.
9.48
0.14
3.37
53.1
10.92
0.14
3.74
52.8
12.19
0.14
3.81
49.6
14.50
0.14
4.23
49.4
15.02
0.14
5.11
49.8
15.93
0.14
4.68
49.5
16.85
0.131
4.80
48.6
18.39
0.103
4.69
51.8
Mill race at Pricbam, Hungary.
0.316
2.2
0.389
20.0
0.336
2.2
0.588
28.4
Irregular channel lined with rubble
0.442
2.2
0.953
35.7
masonry.
0.548
2.2
1.135
37.7
0.560
2.2
1.190
39.1
0.566
2.2
1.270
41.3
20
THE FLOW OF WATER
Table II.
Description of Conduit.
^
c
a
I
New straight asphalt-coated wrought-iron
0.328
1 171
76.7
99 9
0.80
1 04
riveted pipe with screw joints.
3!ll7
108^4
l!l3
m = 0.94
R = 0.0677
6.148
10 .535
12 .786
117.1
124.0
124.3
1.219
1.289
1.291
II
2.78
139.1
1.122
Test pipe of clear cement.
3.65
139.2
1.139
4.20
139.5
1.140
m = 0.95
4.72
140.4
1.141
R = 0.658
4.79
141 .2
1.155
4.92
141.4
1.157
5.81
141 .4
1.157
6.58
142.5
1 .166
III
1.0
101 .2
1.0
New cyhnder joint asphalt -coated steel-riveted
2.0
108.3
1.09
pipe with ma,ny curves.
3.0
112.8
1 .113
3.5
113.4
1.119
m = 0.53
4.0
113.2
1.118
R = 1.0
5.0
112.0
1.105
6.0
111.6
1.091
IV
1.007
73.6
1.0
Old cast-iron pipe.
2.32
75.5
1.023
m = 0.45
5.075
75.1
1.02
R = 0.1995
6.801
75.2
1.02
12.576
75.3
1.02
V
Heavily in crusted cast-iron pipe. Twenty-
1.60
64.0
0.948
five years in use.
2.70
60.0
0.900
m = 0.30
R = 0.416
3.60
59.0
0.874
4.50
58.0
0.858
VI
Channel of dry rubble masonry of large stones,
8.442
57.4
0.890
bed somewhat damaged.
8.905
54.3
0.842
Six years old.
9.181
50.7
0.784
9.427
49.6
0.769
m = 0.30
10.145
47.2
0.731
R = 0.19
VII
0.50
126.9
1 .089
Short conduit.
1.0
116.6
1 .00
Wrought-iron riveted pipe, somewhat rusty.
1.5
111.9
0.959
2.0
109.4
0.938
2.5
109.0
0.934
Length 152.9 feet.
3.0
108.2
0.928
Diameter 8.58 feet.
3.5
107.0
0.917
m = 0.54
4.0
106.2
0.910
R = 2 .145
4.5
105.6
0.905
VARIATION OF THE COEFFICIENT C 21
Variation of the Coeflacient C with the Roughness of the Wet Perimeter
of a Conduit.
Although the primary formula v = y R^ VJ does not give
satisfactory results when appUed to all classes of conduits it
may be made the basis of formulae of general appUcation.
Regarding 2/ Vr as an approximate value of c , expressions
may be found defining the variation of c with the roughness of
the wet perimeter as depending on Vr. The primary value of
c from which its variations with the slope or the velocity of flow
must be derived is that value which corresponds to a velocity
of one foot per second.
In order to retain if possible a straight line formula we may
choose the expression
c = (y Vr) 1 + m,
m indicating the condition of the wet perimeter of the conduit.
For a primary determination of y and m Darcy's values of c
for clean iron pipes were selected. These data give c = 112.0
for i^ = 1.0 and c = 80.4 for R = 0.0208 (or a one-inch pipe).
These values of c are merely average values found by Darcy
from a great number of experiments on clean pipes, which, how-
ever, did not include pipes of great diameters. Taking 50 as a
trial value for y we find
112.0
— -i7= = 2.24, hence 1 + m = 1 + 1.24
50 vr
^^ji = 4.21, hence 1 + m = 1 + 3.21.
50 V
Dividing 1.24 by 3.21 that the quotient is 0.386. This is almost
equal to 0.38, the fourth root of 0.0208, the value of R for the
one-inch pipe. We have consequently in both instances
1 24
c = (50 v^r)l \-^
or in general c = (50 vr) 1 + tj^
^
Testing this formula by experimental data pertaining to flow
in conduits differing widely in their degree of roughness it did
22
THE FLOW OF WATER
not prove entirely satisfactory. As already stated, Darcy's
experiments were made on conduits of comparatively small
diameters and his coefficients for the larger conduits do not quite
agree with those found by recent experiments. For the final
determination of the value of y we choose the graphical method.
If
then
m
/ 4/- '"'
c = (2/ Vr) 1 + -47--
4/- -^ ^ 4,
2/ Vr Vr
This is the equation of a straight line (Fig. 1) having for
1 . C .
abscissae values of -rj=- , for ordinates values of —^- and having
Vr 2/Vr
i
a
2.5
-:::::
^
2.0
^^^
^^
:^
—
^
^
/
1.5
^
^::::i
I^
—
1.0
0.5
0
2
0
4
0
6
0
8 1
0
1
2
1
4
1
6
Values of
^;
Fig. 1.
1.0 as the common distance from the axis of abscissae where all
the lines intersect the axis of ordinates ; the tangent
C
y Vr
1.0
Vr
of the angle a b c will give the value of m. Identical values
will be obtained by putting
V
2/i2Ws
= 1 +
m
Vr
selecting data in which 2; = 1.0 foot per second.
VARIATION OF THE COEFFICIENT C 23
Experimental data giving values of c corresponding to a veloc-
ity of one foot per second are not numerous while those giving
values of c corresponding to a velocity of one metre per second
are quite abundant, this coming nearer to being an average
velocity. On this account data given in metric measure were
chosen, taken chiefly from the writings of Darcy-Bazin.
After numerous trials, and using all the reliable material
available, a constant value of y and corresponding values of
m were found, producing a straight line in every instance. As
our subsequent w^ork depends much on the reliability of this
constant, great pains were taken to find its exact value. In
metric measure its value is equal to 50.0 for which in English
measure we substitute 66.0. We have consequently for the
value of c corresponding to a velocity of one foot per second
c = 66 ^r 1 + 4^^
Vr
or, reducing
As
to
a straight line
c = 66 {^r + m). "
66 (^r + m) - c - y^^
and
(66(^r + m))^ = c2=^
wp hflvp
2^
(66 (^r + m)y
or
0.01478
Z = :
(Vr + my
As primary expressions for the velocity, in most instances
true only when the velocity is equal to one foot per second we
have now the formulae
V = 66 {^r + m) VFTs. (1)
2^
0.01478 L
(Vr + my R
(2)
24 THE FLOW OF WATER
In the formula c = 66 {\lr -{- m), when applied to calculations
of flow in channels in earth of a great degree of roughness of
the bed, the coefficient m, which indicates the degree of roughness
will have a negative value and c will in consequence vanish for
very small values of 'sir. To avoid this defect the formula may
be written, when applied to channels in earth, so that it reads
66 (A/r + 1)
66 (-Vr + Vr)
K
^ Vr + X
in which K is a coefficient increasing in value with increasing
roughness of the wet perimeter. The relation between m and
K is given by
1 + m
Variation of the Coefficient C with the Velocity of Flow.
A.
The characteristics which distinguish water from a perfect
fluid are its adhesive qualities, its viscosity. All fluids, includ-
ing gases, have these qualities in a greater or lesser degree. It
is even asserted that solids Uke ice become viscous under great
pressures. The adhesive qualities of tar or crude oil are apparent
to the eye, those of other fluids can only be inferred from their
effects.
To its viscosity is due the fact, that water flowing in a
channel perfectly smooth, is not, in accord with the law of
falling bodies, continually increasing in speed. The retarding
forces due to viscosity equalize the accelerating forces due to
gravity and distance fallen through, the speed of the water
shows no increase from point to point, in other words, the
motion is uniform.
The layer of water immediately in contact with the walls of
the channel in which it flows does not change except by diffusion;
VAEIATION OF THE COEFFICIENT G 25
it is held fast by surface adhesion. If the wall is perfectly
smooth there is consequently no friction between it and the
fluid directly in contact; the resistances to flow are entirely due
to shearing stresses between the infinitely fine film coating
the wall and the moving body of water.
Frictional resistances are always proportional to the areas of
the surfaces in contact; surface areas near the periphery of a
conduit are always greater than near the centre and the retarda-
tion will in consequence be greater and the velocity less.
This decrease of speed from the centre towards the periphery
is in a measure counteracted by difference of pressure. Greater
velocities are always accompanied by a corresponding fall of
pressure and the pressure in the centre is in consequence less
than near the wall. This difference of pressure continually
tends to draw the water towards the centre and thus to equalize
the speeds. When this equalizing tendency is for a moment
interrupted, we suddenly perceive a wave or flash-like motion,
clearly indicating the speed the water would acquire were it not
for the resistances near the periphery. In conduits having
smooth walls the equalization of velocities is performed so
rapidly that a difference of speed between the centre and the
periphery is scarcely perceptible. A wave-like rotation is set
up and the water glides through the conduit very much like a
bullet through a rifled channel.
Let R be the force required to keep up the flow of a liquid in
two parallel planes past each other, let the surface area of each
plane be A, let the respective distances of the two planes from a
common plane of reference be D^ and Dq, let the velocities be v^
and Vq and e a coefficient indicating the degree of viscosity of the
liquid and we have:
P eA (v^ - Vq)
or: the resistance is proportional to the degree of viscosity into
the area and the relative velocity v^ — Vq, the whole divided by
the difference in the distance of the two layers from a common
plane of reference.
26 THE FLOW OF WATER
For a circular conduit the total force required to set up motion
in a stream line is given by
„ Ael (v, - V,) , 2flv
li = ; ; — "T
^1 — Tq^ r^
in which r^ is the semi-diameter of the conduit, r© a distance from
the axis of the conduit, I its length, and / the coefficient indicating
the degree of roughness of the surface.
This indicates, that the resistance due to viscosity is least in
the centre or the axis of the conduit where r^^ = 0 and greatest
at the periphery where r^^ — r^^ = 0.
The last expression gives for the velocity of flow in a circular
conduit
^
/m + »
4el
(r^^ - To' + 2jr,y r,^-ro^ + 2^r,
from which, the coefficients e and / being known, the value of v
and the discharge may be computed. The coefficient e depends
for its value on the temperature of the liquid ; its value diminishes
rapidly with increase of temperature and is five times less for
water at the boiling point than for water at the freezing point.
According to Mayer its values are (in c. g. s. units) :
at 0.6° Celsius e = 0.0173
at 10° Celsius e = 0.0131
at 20° Celsius e = 0.010
at 45° Celsius e = 0.005833
at 90° Celsius e = 0.00339.
The influence of the temperature on the flow of water through
capillary tubes has been minutely studied by Poisseule.
Slichter has demonstrated the immense influence of the
temperature on the movement of water through permeable
strata and Saph and Shoder have shown its influence on the dis-
charge of pipes. A tube having a diameter of 0.5 millimetre
(0.02 inch) or less is considered to be a capillary tube.
VARIATION OF THE COEFFICIENT C
27
Poisseule's experiments demonstrated, that the velocity of
flow in such tubes is equal to
^2
V = gh
SeL
and the discharge to
Q = It gh
SeL
This shows, that in capillary tubes the velocity is proportional
to the head and not to the square root of the head, to the square
of the radius and not to its square root.
In investigations on the movement of water through porous
strata it has been found, that the velocity of flow is proportional
to the square of the diameter of the soil grains through which
the water percolates; from which it follows that it is also pro-
portional to the square of the voids between the soil grains.
The general equation for the movement of water in a per-
meable stratum may be written {v and Q per minute)
V =
ih + z)''
2L
0.0189^2 (0.7 + 0.03 fc)
Q = mvhh.
In these equations h is the elevation of the water table at the
point of efflux, h + z its elevation at the distance L, d the
Fig. 2.
diameter of the soil grains in millimetres, if (7, the temperature of
the water in degrees centigrade, m the percentage of the voids in
the material, b the breadth of the stratum.
28
THE FLOW OF WATER
The elevation of the water table at the distance x from the
point of efflux is equal to
V
-i
h' +
2Qx
hm 0.0198 d' h
The discharge of a well is given by
_ (h \^y -^' j:rn 0.0189 d' (0.7 + 0.03 f)
log L - log R
in which R is the semi-diameter of the well, and the logarithms
the Naperian
Fig. 3.
The elevation of the water table at a distance x from the well is
given by
Q
0.0189 ^^\RJ'
The surface is consequently a logarithmic curve. These
equations serve to illustrate that between the movement in
capillary tubes and in a porous stratum there is only this dif-
ference, that h is displaced by -- , the velocity is not proportional
to the head but to the square of the head.
On account of internal motions the phenomenon of flow in
VARIATION OF THE COEFFICIENT C 29
pipes and other channels is much more complex than in capillary
tubes or porous strata.
The equation for the velocity of flow in a cylindrical conduit
we have given above may he transformed so it will read
(r,'-r-„^ + 2^r,y r,^ - V + 2^r„
which shows that 9ne of the terms above the hne, denoting the
internal resistance, is directly proportional to the velocity, the
other to its square. This is also true of the two terms 2-T-rj
denoting the friction of the fluid against the walls of the conduit.
Moreover, f, the coefficient denoting the surface friction, depends
for its value on e, the coefficient denoting the internal friction;
its value is consequently modified by temperature. Even in
conduits having the smoothest walls there are always rotary and
wave-like motions tending to equalize pressures to speeds.
Wherever there are cross-currents there naturally is impact, one
stream impinging on the other. To this impact and the attend-
ing shearing stresses between a streamline and its surroundings
are due the increasing powers of the velocity to which the
resistances are proportional. Furthermore, if the walls of the
conduit are not perfectly smooth there are streamlines constantly
impinging on projections, however small they may be.
Conditions existing at the entrance, curves, elbows, changes
of section, etc., also affect the power of the speed to which the
total resistance is proportional. It was formerly assumed that
the resistances due to these impediments were proportional to
the square of the velocity.
From experiments made by Hubbel and Fenkell (Detroit) to
determine the resistances due to curves, the writer found,
neglecting curves the radius R of which is less than 2.5 diameters
of the conduit, that the resistance of a curve is equal to
4.9 c?
'(f
30 THE FLOW OF WATER
times the resistance of a tangent of equal length, and the excess
of frictional resistance in a curve equal to
i*"HW)
1.0
times the resistance in a tangent of equal length. The length
of tangent equal in frictional resistance to the resistance in a
curve of 90° is equal to
7? 13.
This evidently vanishes when — = (4.9) ' d! = 539.3 and is a
CL
7? 11
maximum when — = (4.9)^ d
Hubbel and FenkelFs experiments were made on 30'', 16'' and
12" conduits, and comparison showed that the influence of the
diameter on the resistance depends on the value of d^^. The
value of z, or /, the coefficient of friction is therefore for any
curve.
n' ^/ .-^ /d\^ 0.01478
360 \ \R/ \yr + m)V
in which n is equal to the number of degrees in the curve and
d"*^^ substituted for rfi3 for diameter less than a foot.
Hubbel and FenkelFs experiments were supplemented by those
of Saph and Schoder on 2-inch brass tubes and more recently
by those of Alexander on a IJ-inch wooden tube. Although we
cannot accept the formulae the latter deduced from his own ex-
periments and those of Hubbel and Fenkell, Saph and Schoder,
his experiments are valuable in indicating the powers of the
velocity to which resistances in curves are proportional. While
Alexander's experiments show that resistances in a curve are
proportional to the same powers of the speed as resistances in a
tangent, provided there is no shock, the experiments of Saph
and Schoder indicate that the power of the speed increases
rapidly with increasing values of — • Their data indicate that
VARIATION OF THE COEFFICIENT C 31
7? /?
for — = 10 the resistance is proportional to V^'^, for — =4 to
a a
V^'^\ How far this holds good for diameters greater than
2 inches we are not prepared to say. It is probable, however,
that with increasing diameter the force of the shock decreases
and the powers of v with it.
It is probable that the resistances due to right-angled entrances
right-angled elbows are also proportional to powers of v higher
than 2.0.
The effect of the temperature on the variation of the power
of V has so far not been determined with precision. Saph
and Schoder, experimenting with a 2-inch brass pipe, found for
a rise of 10° F. an increase in the discharge of 4%.
That the resistances to flow are not proportional to the square
of the speed was recognized long before Darcy and Bazin
demonstrated the great influence of the degree of roughness of the
walls of a channel on its discharge.
The laws of fluid friction were first investigated by Coulomb.
He states, that the total resistance to motion is a compound of
two factors, one being proportional to v, the other to v^. Dubois's
experiments on flow confirmed this view and from his data
Prony found for the resistance the expression (in metric measure)
Rv = 0.000044 V + 0.000309 ^;^
this corresponds to
2g r
Weisbach put Prony's formula into the form
// = 0.00741 (1+0:01^)^.=
\ V / r
which in our day is still used.
The relation of the power of the velocity, to which the resist-
ance is proportional, to the variation of the coeflScient c with the
velocity is such, that c remains constant for all velocities if the
resistance is proportional to v^; it increases with increase of
velocity if the resistance is proportional to v^^, and decreases if
the resistance is proportional to z;^"*"^.
32 THE FLOW OF WATER
B.
If the value of the coefficient c corresponding to any velocity
is divided by its value corresponding to a velocity of one foot per
second ; the quotient is a variable which we will call the coefficient
of variation of c and denote by (a). Hence
66 (-Vr + m)
While the term i
[66 (^/r + m)f
represents the frictional resistance per unit area of surface at
unit velocity, the term
c
a =
66 i^r + m)
indicates the power of the velocity to which the resistance is
proportional. We shall presently s^e, that under normal con-
ditions, that is if resistances proportional to different powers of
V do not enter, the coefficient a is merely a root of v.
An analysis of the values of a found in Column 4, Table II,
shows that its value does not entirely depend on the velocity,
but is affected by the degree of roughness of the walls of the
conduit, by its length and alignment, by conditions existing at
the entrance, by changes of section, etc. According to the
manner in which the coefficient a is affected we may classify
conduits as follows :
1. Long straight conduits without internal obstructions and
a great degree of smoothness of the wet perimeter.
2. Long conduits of a great degree of smoothness of the wet
perimeter but with some easy curves or other impediments, also
long straight conduits of a fair degree of smoothness of the wet
perimeter.
3. Long conduits of a great or fair degree of smoothness of
the wet perimeter but with sharp curves, angles or other impedi-
ments to flow.
4. Conduits whose walls are coated with rust, slimy or sticky
substances.
VARIATION OF THE COEFFICIENT C 33
5. Conduits of a great degree of roughness of the wet peri-
meter, badly tuberculated pipes, damaged masonry, channels
in earth with sharp bends, bars or other obstructions.
6. Short conduits.
For classes 1 and 2 the resistances are proportional to
powers of v less than 2.0 and the coefficients c and a continue
to increase in value with increasing velocity. For the third
class some resistances proportional to a power higher than
2.0 enter, a increases with increase of velocity and then
decreases. For class 4 the resistance is proportional to v^
or nearly so and a is constant. For classes 5 and 6 the resist-
ances are proportional to powers of v higher than 2.0 and a
continually decreases with increasing velocity.
C.
We have so far only found expressions for the value of c
corresponding to a velocity of one foot per second. These give
for the velocity
V =66 i^r + m) Vrs (1)
2gU
0.01478 L
(^r + my R
(2)
We will now proceed to find in what relation the value of v
as found from the formula
i; = 66 (Vr + m) Vr . s
stands to the true mean velocity in all cases where v is more
or less than unity or the value of a, the coefficient of variation,
is affected by the conditions we have enumerated. Using the
exponential equation
v^ [ (66 (^r + m) Vrg),T
(66 (y/r + m) Vrs) J
log (66 {^r + m) VrTs)^ - log (66 {Vr + m) Vrs)^
which gives
3. _ log ^1 - log ^0
34 THE FLOW OF WATER
we find from the experimental data given by Darcy ^nd Hamilton
Smith for straight or nearly straight clean cast-iron, wrought-
iron and sheet-iron riveted pipes of all diameters and for velocities
up to 20 feet per second
in other words, from the data given by Darcy and Hamilton
Smith we find, that the true mean velocity is equal to
2; = (66 ( Vr + m) Vy. . s)f (3>
which may be written
V = m (^r + m) Vr-s ^qq (^r + m) V^,
hence the coefficient a, indicating the variation of the coefficient
c with the velocity is equal to
a = 4/66 {-^r + m) Vr . s.
From Formula 3 we have also
V^ = (66 -^r + m) VV&
consequently
and v^ = ^'(mW^^m^'vrrr
Table III contains a number of experimental data relating to
flow in conduits under pressure. They are purposely selected
in order to show the variation of the coefficient c as affected by
various conditions of flow.
The values of the coefficient a found in columns 3, 6 and 9
show that for 1-inch pipes of tin and wrought iron, for sheet-iron
riveted pipes up to 2.43 feet in diameter, for new cast-iron pipes
up to 1.393 feet in diameter, for pipes of planed shares up to 4.5
feet in diameter the coefficient a is equal to 7^ or nearly so.
The fact that a = V^ holds good for a tin or wrought-iron pipe
1 inch in diameter, and also for a pipe of planed staves 54 inches
m diameter allows us to conclude, that it holds good also, be-
tween these limits for other conduits having walls of a similar
degree of roughness such as asphalt-coated, cast and wrought-
iron or cement-fined pipes.
VARIATION OF THE COEFFICIENT C 36
This, however, holds good only when the value of — , the ratio
(Jb
between the length of a pipe and its diameter, is at least 1,000.
For lesser values of — the value of (a) decreases with -^ • The
a a
experimental values given by Stearns and Fitzgerald relating to
flow in four-foot cast-iron pipes indicate this plainly. In the
case of the four-foot Sudbury conduit (Stearns) the ratio — is
equal to 439.
If the formula v = (66 (v'r + m) Vr . s)^ is put into the
form
2gH
1^
0.01478 L
the term
Nr+myR
0.01478 L
i^/r + my R
includes all the resistances, those due to the velocity itself,
those due to the entrance, and those due to the friction of the
fluid against the walls of the conduit.
The loss of head due to the velocity itself is proportional to
the square of the speed ; resistances due to the entrance are pro-
portional, according to the nature of the entrance all the way
from the square of the speed up to its cube. An average value is
probably 2.5.
The value of the coefficient of resistance due to the velocity
itself is equal to 1.0; the value of Zq, the coefficient representing
the resistance due to the entrance is, according to Weisbach, for
a well rounded entrance, equal to 0.505; hence the value of z^,
the coefficient representing the resistance due to the walls of
the conduit, is equal to
0.01478 L _^^p^^
-v'r + my R
36 . THE FLOW OF WATER
If the conduit is long, above 1,000 diameters in length, 1.505
is a quantity small in comparison with
0.01478 L , ^^^
Vr + my R
and does in consequence not appreciably affect the variation of
c. With decreasing length of the conduit, however, the ratio
between the two quantities changes at an increasing rate and
more and more affects the variation of c. In the case of the
four foot Sudbury conduit (Stearns), we have the following
data, taking m = 0.97 :
V = 3.738, H = 1.2421 ft., L 1747 ft.
0.01478 L .._
(l + 0.97r5^'-^^^
6.656 - 1.505 = 5.151 = z,.
(3.738)2 X 1.0 = 0.217 = h = loss of head due to velocity.
X 0.505 = 0.2119 = ho = loss of head due to entrance.
2^
(3.738)2-^
2^
0.217 + 0.2119 = 0.4289 = h + ho.
1.2421 - 0.4289 = 0.8182 = h, = loss of head due to friction in
the pipe itself.
Using the formula -^ = v'^ and inserting values we have
64.4 X 0.8142 ^ ^^^^^ ^ ^.^
5.151
Dividing log 10.17 = 1.0073209 by log 3.738 = 0.5736293
the quotient = 1.76 very near.
Consequently the frictional resistance in the pipe itself is
proportional to v^'''^, corresponding closely to V^, the value we
have found for long pipes.
The data relating to flow in a riveted flume 8.58 feet in diameter
and 152.9 feet long (Herschel, Holyoke Testing Flume) show the
great influence of the length of the conduit on the variation of c
most plainly.
VARIATION OF THE COEFFICIENT C
37
Table III.
Experimental Data Showing Extent of Variation of c with the
Velocity of Flow.
Tin pipe, straight.
Dubuat.
Wrought-iron
pipe.
Asphalt- coated riveted
Darcy,
pipe. Darcy.
d = 0.0888 ft.
d = 0.1296 ft.
d = 0.271 ft.
L not given.
L = 372 ft.
L = 365 ft.
m = 0.98
m = 0.83
m = 0.94
a =V^
a = V^
a =V^
V
c
a
V
c
a
V
c
0.141
67.6
0.751
0.205
76.9
0.932
0.328
76.7
0.772
82.8
0.92
0.858
82.3
0.995
1 .171
99.9
1.183
91.4
1.019
2.585
92.9
1.112
3.117
108.4
2.546
98.9
1.099
6.3
99.8
1.21
6.148
117.4
2.606
100.4
1.115
8.521
100.0
1.212
10 .535
124.0
5.223
111.4
1.237
12.786
124.3
0.80
1 .043
1.132
1.223
1.274
1 .298
Asphalt-coated riveted
Asphalt-coated riveted
Asphalt-coated riveted
pipe. Darcy.
pipe. H. Smith.
pipe. H. Smith.
d = 0.643 ft.
d = 0.911 ft.
d = 1.229 ft.
L = 365 ft.
L = 700 ft.
L = 700 ft.
m = 0.92
m = 0.68
m = 0.69
a =V^
a =V^
a =V^
V
c
a
V
c
d
V
c
a
0.591
104.1
1.013
4.712
107.1
1.19
4.283
111 .6
1 .181
1.529
106.2
1.035
6.094
110.6
1.229
6.841
117.8
1.246
1.53
115.6
1 .125
6.927
111 .5
1.24
7.314
119.1
1.261
5.509
125.4
1 .22
8.659
113.4
1.26
8.462
119.1
1.26
9.0
130.2
1.267
10 .021
115.5
1.283
10 .593
121 .6
1.285
19.72
141.0
1 .372
12.09
121.3
1.280
38
THE FLOW OF WATER
Table III. — Continued,
Asphalt-coated cast-
iron pipe. Darcy.
= 0.4495 ft.
= 366 ft.
= 0.90
Asphalt- coated cast-
iron pipe. Hubbel
& Fenkell.
d = 1.0 ft.
L not given.
m = 0.83
a =V^
Asphalt-coated cast-
iron pipe. Lampe.
d = 1.373 ft.
L = 26,000 ft.
m = 0.83
a =V^
0.489
2.503
5.625
11 .942
15 .397
94.1
108.4
112.5
113.5
112.2
0.96
1.107
1.15
1.16
1.15
1.0
2.0
3.0
4.0
5.0
101 .5
109.6
114.6
118.3
121 .3
1.0
1.06
1.13
1.166
1.196
V
c
a
1.577
110.5
1.072
2.489
114.1
1.107
2.709
114.6
1 .112
3.090
119.4
1.162
Redwood Stave Pipes.
Cedar Stave Pipe.
Cedar Stave Pipe.
A. L. Adams
Th. A. Noble
Th. A. Noble.
d = 1.166 ft.
d = 3.667 ft.
d = 4.5 ft.
L = 80,006 ft.
Lnot given.
Lnot given.
m = 0.93
m = 0.50
m = 58
a =V^
a =V'^
a =V'^
V
c
a
V
c
a
V
c
a
0.698
97
0.908
3.468
110.1
1.134
2.282
116.8
1 .095
0.698
101
0.926
3.522
108.6
1.12
2.276
115.5
1.086
0.751
104
0.953
3.685
110.9
1.144
2.650
119.9
1 .12
0.691
105
0.963
3.853
112.6
1.163
3 067
122.1
1.151
1.167
109
1.0
3 .964
112.9
1.164
3.045
121 .4
1 .138
1.531
112
1.027
3.972
113.1
1.164
3.408
123.7
1.164
1.181
113
1.036
4.415
113.7
1.164
3.724
125.2
1.176
4.635
114.9
1.183
3.929
126.2
1.179
4.831
115.5
1.190
4.688
129.0
1.205
VARIATION OF THE COEFFICIENT C
39
Table III. — Continued.
New steel-riveted pipe.
New steel-riveted pipe.
New steel-riveted pipe.
Herschel
Herschel
Marx-Wing.
d = 3.5 ft.
d = 4.0 ft.
d = 6.0 ft.
L = 81,339 ft.
L = 24,648 ft.
L = 4,367 ft.
m = 0.56
m = 0.47
m = 0.50
a = F-A
a = Fi-8
a = Vrs
V
c
a
V
c
a
V
c
a
1.0
101.0
1.0
1
97.1
1.0
1.07
103.5
1.0
2.0
104.3
1.032
2
101.3
1.043
1.67
108.0
1.024
3.0
106.4
1.053
3
102.2
1.052
2.14
113.0
1.091
4.0
107.8
1.067
4
104.2
1.073
2.50
108.0
1.024
5.0
108.4
1.073
5
105.1
1.083
3.0
112.0
1.082
6.0
108.5
1.074
6
105.2
1.084
3.84
113.0
1.091
Asphalt-coated cast-
iron pipe. Steams.
d = 4.0 ft.
L = 1,747 ft.
m = 0.97
a = Fi8
Cleaned cast-iron pipe.
Fitzgerald.
d = 4.0 ft.
L not given.
m = 0.98
Cedar stave pipe
Marx-Wing.
d = 6.0 ft.
L = 4,000 ft.
m = 0.66
a = F^
3.738
4.965
6.193
c
a
V
c
a
V
c
140.1
1.077
2.472
137.5
1.051
1.0
116.0
142.1
1.093
3.723
139.1
1.064
1.5
118.7
144.1
1.109
4.796
141.1
1.085
2.0
119.9
6.141
143.6
1.100
3.0
4.0
5.0
6.0
121.4
122.0
122.4
122.5
1.0
1.023
1.032
1.046
1.051
1.055
1.056
40 THE FLOW OF WATER
The experimental data relating to flow in riveted conduits
show great diversities both in the values of m and a.
The coefficient m is equal to 0.94 for a riveted pipe 0.270 feet
in diameter (Darcy) and equal to 0.51 for a butt- jointed riveted
pipe six feet in diameter (Marx -Wing). This great difference
in the values of m is mainly due to the size of the rivet heads.
In pipes of small diameters the rivet heads, especially when
coated with asphalt, do not offer an appreciable impediment to
flow. In large conduits, however, their size is such, that they
not only produce constriction of the section, but also vortex
motions, thus reducing the discharge in a twofold manner.
From data relating to flow in steel-riveted pipes exceeding three
feet in diameter, given by Herschel, and by him considered the
most reliable (see '' Herschel "115 Experiments), we find that
the coefficient of variation of c for these conduits is fairly, though
not precisely, equal to
a = ^tH = 7^
Consequently
= 4/66 (^r + m) Vr , s.
V = (66 (</r + m) VVTVy^ (5)
or V =
2gH
0.01478 L
r^
(6)
L(Vr + m)2 R.
This value of a = F'~^ we find to hold good also for flow in
rectangular pipes (Darcy).
The experimental data relating to flow in old iron pipes, those
not heavily incrusted or tuberculated show that the coefficient c
does not vary to any extent with variations in the value of v.
For this class of conduits we have consequently
a = 1.0.
The data relating to flow in badly incrusted or heavily tuber-
culated pipes indicate a decrease in the value of c with increasing
velocity. Using as before the equation
log i\ - log v^
X =
log (66 {Vr + m) Vrs)^ - (66 (^r -{- m) Vrs)o
X
VARIATION OF THE COEFFICIENT C 41
we find for incrusted pipes :
X = if, hence a = -^= , ,
and for very badly tuberculated pipes:
T^. hence a = —= .
V^ ^66 (Vr -h m) Vr .s
The experimental data relating to flow in a 12-foot brick
sewer at Milwaukee, a 7.5-foot brick sewer at Dorchester Bay, in
a siphon aqueduct of 119 feet cross-section at the river Elvo all
show a slight decrease in the value of c with increasing velocity.
This decrease is due, in the first two cases, partly to the fact
that these conduits are discharging under water against a
hydraulic counter pressure, partly it is due to the greater viscosity
of the sewerage and partly also to the relative shortness of these
conduits. In the case of the siphon aqueduct, its length is so
short comparatively, that it can only be considered as a short
pipe, conditions being much the same as in the case of the
Holyoke Testing Flume.
D.
The variation of the coefficient c as deduced from experi-
mental data relating to flow in conduits under pressure may be
summarized as follows :
1. For long, straight conduits fairly clean, such as pipes of
glass, tin, lead, galvanized iron, cast and wrought iron, .planed
staves, cement, riveted pipes up to 3 feet in diameter, the coeffi-
cient of variation of c is equal to
a= V"^
and the frictional resistance is proportional to V"^.
2. For pipes rectangular in section, for riveted pipes exceeding
3 feet in diameter, for those enumerated under (1) between 300
and 1,000 diameters in length, the coefficient of variation of c
is equal to
1 7
and the frictional resistance is proportional to"F*'.
42 THE FLOW OF WATER
3. For the classes of pipes enumerated under (1) and (2)
discharging against a hydraulic counterpressure, or between
100 and 300 diameters in length, for old pipes not inc rusted or
tuberculated the coefficient c does not appreciably vary with the
velocity, and consequently
a = 1.0
and the frictional resistance is proportional to V^.
4. For incrusted pipes and those enumerated under (1) and
(2) less than 100 diameters in length the coefficient of variation
of c is equal to
1
a = -^
and the frictional resistance is proportional to "FY.
3. For very heavily tuberculated pipes the coefficient of vari-
ation of c is equal to
1
a = — i
2 0
and the frictional resistance is proportional to F ^ .
In our collection of experimental data we find many instances
relating to flow in one and the same conduit which do not fit
any of the values of (a) enumerated above and which indicate :
1. First an increase in the value of c with increasing velocity
up to a certain critical velocity.
2. Then a decrease in the value of c with increasing velocity.
As instances of this kind we mention:
Two new steel riveted pipes at East Jersey, 3.5 and 4 feet in
diameter (Herschel).
A cement lined pipe with elbows (Fanning).
This peculiar variation of c indicates the presence of resistances
which are proportional to powers of the velocity greater than
2.0, that is resistances which produce shocks. In case of the
steel pipes the shocks are no doubt due to the rivet heads, in
the second to the elbows in the line of the conduit. This
peculiar variation of the coefficient c is also very plainly indicated
in the data relating to flow in channels of rough boards with
cleats nailed crosswise to bottom and sides of the channel
OPEN CONDUITS 43
(Darcy-Bazin, series 12-17). These cleats or laths were 1 centi-
metre thick and 2.5 centimetres wide. In one channel they
were spaced apart 1 centimetre, in the other 5.0. The data
relating to flow in the channel with the cleats spaced 1 centi-
metre indicate the highest values of both the coefficients m
and a, plainly showing the effect of the shock due to the wider
spacing of the cleats. In the first case the coefficients are m =
0.41, a = V^, in the second m = 0.03 a = -^, indicating that
the frictional resistance was proportional in the first case to
V^' \ in the second to 7 ' .
Open Conduits.
E.
An analysis of experimental data relating to flow in open
conduits of permanent cross-section, such as aqueducts, flumes,
etc., indicates, that the coefficient c is affected in its variation
with the velocity by the shape of the cross-section, or by the
depth of the water in the channel.
For semicircular or well rounded channels, for the semi-square
when flowing full, for all sections for which the mean hydraulic
radius is equal to half the depth, for the triangle with sides
inclined 45° the variation of the coeflficient c with the velocity
does not seem to be affected by slight variations in the value
of r. For rectangular channels, however, and others having very
steep side walls (excluding those mentioned above) the varia-
tion of c is affected by the depth of water in the conduit.
The coefficient a seems to have its normal value in all instances
when the depth of water is equal to one- half the mean width of the
channel, it increases in value as the depth of water decreases,
and decreases in value as the depth of water increases.
This peculiar influence of the steepness of the walls of a
conduit on the frictional resistance has been revealed by nu-
merous current metre observations in rectangular flumes and
aqueducts and other channels with steep side-walls. It has been
44 THE FLOW OF WATER
found that in such channels the position of the thread of max-
imum velocity is situated at a greater distance from the surface
than in channels . having side walls more inclined; thus clearly
indicating the retarding influence of the steepness of the walls.
Experimental data relating to flow in rectangular flumes fre-
quently indicate values of the coefficient (a) as high as vi for
small depths, its value is generally equal to vl when the mean
hydraulic radius is equal to one-fourth the width of the channel.
Its value is less than the normal when the depth exceeds one -half
the mean width of the channel. Applying the exponential equa-
tion
log^i - log Vo
/>• ^^ — — _
log (66 {^r + m) Vrs)^ - log (66 {^r -\- m) Vr * s)o
to data relating to flow in a semi -circular channel lined with
neat cement (Darcy-Bazin, series 24) we find
X ^ \l very near.
Applying the same equation to data relating to flow in
channels fined with rough boards, semicircular in section, we
find (Darcy-Bazin, series 26)
^ _ 13
thus indicating a slight decrease in the value of a with increasing
roughness of the conduit's wet perimeter. As a mean between
these two values and differing but slightly from either we may
take
^ _ 12
which corresponds to a = V^
or a =\/m (Vr + m) Vrs
and the frictional resistance is proportional to V'^' = V ' .
This value of the power of the velocity we observe is the identical
value Froude found for smooth plain surfaces in his investiga-
tions on the resistance of ships.
Besides the two series mentioned, given by Darcy-Bazin, we
find that
*^ 11
holds also good for the following :
OPEN CONDUITS 46
Darcy-Bazin, series 25, semicircular channel lined with smooth
concrete.
McDougall, Provo Canal Flume, semicircular channel of planed
staves.
Th. Horton, Conduit of North Metropolitan Sewage System of
Massachusetts. Brickwork washed with cement. Diameter
9 feet. Values of R up to 2.31 feet.
F.
Applying the experimental equation as indicated above to
data relating to flow in channels not semicircular in section
and Hned with cement or concrete, planed or rough boards,
brickwork and good ashlar masonry we find
^ _ 18
a = V^'
= i/66 ('vV + m)Vrs
and the frictional resistance is proportional to V^. This we
find to hold good for the following :
Darcy-Bazin, series 2, neat cement, section rectangular.
Darcy-Bazin, series 6, 7, 8, 9, 10, 11, 18, 19, 21, 22, and 23,
sawed boards, section rectangular, triangular or trapezoidal.
Darcy-Bazin series 32, 33, 39, channels lined with good ashlar
masonry, section trapezoidal.
Darcy-Bazin, series 3, rough brick work, section rectangular.
Darcy-Bazin, series 4, channel lined with pebbles up to | inch
in diameter, section rectangular.
Fteley and Stearns, Sudbury conduit, very good brickwork,
sides of channel nearly vertical, bottom flat arch.
Fairlie Bruce, Aqueduct of Glasgow, smooth concrete, sides of
channel nearly vertical, bottom flat arch.
Th. Horton, Conduit of North Metropolitan Sewage System of
Massachusetts, brickwork washed with cement, covered with
sewer slime, sides of conduit vertical, bottom flat arch.
46 THE FLOW OF WATER
Lippincott, San Bernardino Canal Trapezoidal channels in
earth, lined with concrete.
Kutter, Gontenbachschale, new and well built channel of dry
rubble masonry.
Passini and Gioppi, Aqueduct of the Cervo, Canal Cavour.
Floor of concrete, sides of brick, section rectangular. Values of
R up to 7.2 feet.
G.
Applying the exponential equation as indicated to data relating
to flow in channels having walls possessing a greater degree of
roughness than those enumerated above we find
X = 1.0
a = 1.0
and the frictional resistance is proportional to v^. This, amongst
others, holds good for the following :
Darcy-Bazin, series 1, 34, 35, channels lined with roughly
hammered stone masonry.
Darcy-Bazin, series 5, channel lined with pebbles IJ inch to
1^ inch in diameter.
Kutter, numerous channels lined with dry rubble masonry.
Perrone, Torlonia drain tunnel, channel in rockwork, partly
lined with rubble masonry.
We mention here also :
Cunningham, Aqueduct of the Solani, Ganges Canal. Floor of
brick, laid flat, sides of masonry, length 920 feet.
In this case the fact that c does not vary with the velocity of
flow is due to the shortness of the conduit. It ha^ no independent
slope and the movement of the water is influenced by the greater
resistance in the rough channel in earth downstream. This is
plainly indicated by the low value of the coefficient m.
Of open conduits, not channels in earth, there are few possess-
ing a degree of roughness still greater than those enumerated,
exceptional cases of old and damaged rubble masonry.
CHANNELS IN EARTH 47
H.
The variation of the coefficient c with the velocity of flow as
deduced from experimental data relating to flow in open conduits
not channels in earth may be briefly summarized as follows:
1. For semicircular channels lined with cement, concrete,
good brickwork, planed or rough boards, the value of the coeffi-
cient a is equal to F*^.
2. For rectangular, triangular or trapezoidal channels of the
same description, for channels lined with rough brickwork,
ashlar and very good rubble masonry, for channels lined with
pebbles up to | inch diameter the value of the coefficient a is
equal to 7"^.
3. For channels lined with roughly hammered stone or
common rubble masonry, for channels lined with pebbles up
to IJ inch in diameter, for channels in rockwork, for aqueducts
of any description discharging into channels in earth and having
no independent slopes, the value of the coefficient a is equal to 1.0.
4. For channels with obstructions producing shocks, such as
channels with cleats nailed crosswise to retard the flow, for
channels lined with old and damaged masonry the value of the
coefficient a is equal to — r 1 *
Channels in Earth.
I.
When we scrutinize the data relating to flow in rivers and
other channels in earth we perceive that these data contain many
irregularities and contradictions which make them appear doubt-
ful and untrustworthy. Even those given by the best authori-
ties are not entirely free from anomalies. These irregularities
and contradictions are occasionally the result of inaccurate
measurements; more often, however, they must be attributed
to the unstable character of the beds of these channels. This
instability of the bed of the channels makes the phenomenon
of flow a problem of great complexity. An exact valuation of
all the facts entering is as yet, with the incomplete data at
present available, out of the question. We here leave the path
48 THE FLOW OF WATER
of exactitude and enter a labyrinth, satisfied if we come out with
the gain of an increment of knowledge which may prove useful.
Natural and artificial channels in rock work or earth may be
divided according to the stability of their beds, into three classes :
1. Channels having beds in a regime of stability at velocities
exceeding the ordinary. Channels in rockwork, cemented gravel,
channels in earth protected by riprap or masonry side walls.
2. Channels in a regime of stability at ordinary velocities.
Channels in gravel, stiff clay, clayey loam, sandy soils with over
50 per cent clay.
3. Channels in a regime of instability at ordinary velocities.
Channels in sand, sand with fine gravel, sandy loam with less
than 50 per cent clay.
The beds of the second and third class are in a regime of
stability until the velocity becomes suflficiently great to erode
the bed.
The velocity at which erosion begins varies with the cohesion
of the material. In channels in sand, sandy gravel, sandy
soils with small percentages of clay, erosion begins at very low
velocities; these channels are consequently very unstable.
Omitting channels in firm rock or cemented gravel, the stability
of the bed depends mainly on the percentages of clay in the
material. According to W. A. Burr pure clay resists erosion up
to a velocity of 7.35 feet per second. The following table, based
chiefly on Burr's experiments, gives the mean velocities at which
erosion begins:
Nature of Material Forming the Bed.
Fine sand
Coarse sand, sand with pebbles up to pea size .
Sandy soil 15 per cent clay
Fine gravel up to ^ inch in diameter ....
Sandy loam, 45 per cent clay
Common loam, 65 per cent clay
Gravel or pebbles from ^ to 1 inch in diameter
Coarse gravel
Clayey loam, 85 per cent clay
Clay soil, 95 per cent clay, loose rock ....
Stratified rock, slaty rock
Hard rock
Mean
Velocity.
0.72
1 .10
1 .20
1 .50
1 .80
3.00
3.15
4.00
4.80
6.20
7.45
12.00
CHANNELS IN EARTH 49
In the process of erosion energy is consumed which varies
with the specific gravity and the cohesion of the material.
The erosive power of a current is proportional to the square
of its speed. Its transporting power, however, varies (according
to Le Conte) :
When the surface is constant with v^.
When the velocity is constant with the surface of the object or
with d^.
When both vary the assistance is equal to v"^ d^. But the
weight of the object is proportional to d^.
Hence, when the forces are in equilibrium or the weight equal
to the energy d^ = v^ d^.
Dividing by the surface or d^ we have d = v^.
Consequently when the forces are in equilibrium the resistance
is proportional to v^. In other words, the transporting power
of a current is proportional to the sixth power of the speed.
This indicates that powers of r ranging between 2 and 6 enter
the problem of flow when erosion begins.
With the beginning of erosion the destruction of the bed will
be the greater ; the less the cohesion of material the greater the
velocity. Changes and alterations in course and section generally
continue till a channel is formed which, owing to its greater
length, its deflections, curves and bars offers such resistances
that the power of the current is reduced and course and section
again become stable when force and resistance are in equi-
librium. A stream will pick up material in a narrow, deep
section of its course where the force of the current is great, and
deposit it in a wide and shallow section where the current is
feeble. At high water, the greater depth of the water in the
shallow section will result in greater velocities, the material
previously deposited will again be put in motion and carried to
a place where the current is feeble.
The work done during these processes of building and rebuild-
ing cannot be accurately measured, and on this account slope
formulae, when applied to flow in channels where erosion is going
on, are always more or less deficient. They cannot be depended
on in computing discharges ; this falls into the province of the
60 THE FLOW OF WATER
current metre and the rod float. They are useful, however, as a
guide to the engineer in the design of new conduits, alterations
in courses or sections, etc., etc.
The banks of channels having unstable beds are frequently
protected by riprap or masonry walls. Frequently the bottoms
of such channels are also protected by artificial bars made of
boulders or masonry.
Rittinger, Borneman, Epper, Cunningham, and others, have
given us data relating to flow in such channels. An analysis
of these data gives surprising results. Using the exponential
equation
^ log V, - log Vq
~ log r^ - log To
we find the following values of x, the power of the mean hydraulic
radius to which the velocity is proportional:
Rittinger, millrace of dry rubble side walls, bed very rough,
depth of water 0.40 to 0.90, x = 3.0.
Rittinger, mill race, bed sand and gravel, side walls of masonry,
depth 0.28 to 0.90 ft., x = 1.77.
Rittinger, Aqueduct in earth lined with dry rubble side walls,
depth 0.61 to 1.27 feet, x = 1.19.
Cunningham, Solani Embankment, sides of masonry built in
steps, bed of clay and boulders, with frequent artificial bars
to prevent erosion. Main site, width, 150 to 170 ft. ; depth of
water, 1.7 to 4.1 ft., x = 1.49; depth of water, 5.6 to 9.34 ft.,
a; = 0.9; Jaoli site, depth, 6.8 to 8.1 ft., x = 0.93.
Excluding extremes, the powers of R, to which the velocity
is proportional as expressed in these data, may be given by the
equation
a; = 1.8 - OAR
so that for R = 1.0 x = 1.7
R =2.0x = 1.6
R = 9.0 x = 0.9.
A high value of x indicates a low value of the coefficient c,
but a rapid increase in its value with increasing value of i2; a
CHANNELS IN EARTH 61
low value of x indicates a high value of c and a slow increase in
its value with increasing values of R.
The influence of the roughness of the bed is necessarily much
greater when the water in the channel is shallow than when it is
high; the diminishing values of x indicate a rapid decrease in the
relative influence of the character of the bed. But, on the
other hand, while the powers of R are abnormally high for
shallow water in rough channels, the powers of the sine of the
slope to which the velocity is proportional are abnormally low.
This may be illustrated by data deduced from experimental
values relating to flow in rough channels in earth. Amongst
others we find :
Wampfler, Simme Canal, coarse gravel and detritus,
pi. 104 00.23
La Nicca, Rhine in the Forest, coarse gravel and detritus,
depth 0.42 to 0.9 feet, i^'"' S '•*.
La Nicca, Plessur River, coarse gravel to detritus, depth 1.25
to 4.58 feet. R'''' S'-\
Darcy-Bazin, Grosbois Canal, Chazilly Canal, channels in earth,
with stones and vegetation, depth 1.5 to 3.0 feet.
Reich, River Salzach, gravel and detritus, depth 3.53 to 7.39 ft.
pO.8 00.333
Funk, Weser River, depth 4.5 to 11 ft. R'-'' S''\
Villevert, River Seine, depth 5.66 to 18.39 ft. R'''^ S''"'^
In general therefore, for shallow water in rough channels the
power of the sine of the slope to which the velocity is proportional
is equal to 0.4 and equal to 0.473 for depths exceeding 4 feet.
The variations in the powers of both r and S with the depth of
the water in the channel are chiefly due to the fact, that the
bottoms of such channels are in most cases much rougher than
the sides. In shallow water, the resistance due to the bottom
preponderates, with increasing depth the influence of the less
rough sides more and more reduces the mean resistance per
unit area of surface.
The powers of r vary not only with the degree of roughness
52 THE FLOW OF WATER
in general and with the depth of the water, but also with the
value of a, the coefficient of variation of c.
For the same degree of roughness, the powers of r have their
highest value for the highest value of a.
For
m = — 0.33 or K = 2.0 for instance,
and
a = 1.0 R^ = R'-''\
But for
a= V^^ R^ = R'-'"'
for
a= ^ R- = R^'-^^^
for
a = ^ R- ^ R'''\
This shows the great influence of bends, bars, or other impedi-
ments on the powers of R.
Our general equation expresses the variation of the powers of
r with the depth with a fair degree of accuracy. Greater
accuracy is obtained if the formula is put into the form
c = eef-Vr + ("2" 1 + ^^ )/ ^^^ giving m a negative value, as
for instance:
for K = 1.20 m = - 0.10
for K = 1.50 m = - 0.20
for K = 2.0m = - 0.33.
For values of R less than 1.0 foot the formula
66 (^r + Vr)
C = T7=
^r + K
gives slightly excessive results.
Amongst the mass of experimental data accumulated during
recent years those given by Fortier for irrigation channels are,
considered from the practical standpoint, the most valuable.
They relate to flow in channels possessing all possible degrees
of roughness and a minute description of the nature of the bed
is always given. Gaugings were, however, taken only for a single
depth and a single slope at each section and on this account
no deductions can be made in regard to the variation of the
coefficient c with the velocity.
CHANNELS IN EARTH 53
Besides these Dubuat, Darcy-Bazin, Legler, Cunningham,
Rittinger and others have given valuable data relating to flow in
canals to ditches; Funk, Villevert, Revy, Gordon and the U. S.
Engineers, interesting data relating to flow in rivers. After a
careful analysis of all the material available we come to the
following conclusions in regard to the variation of the coefficient
c with the velocity:
1. For channels of fairly regular cross-sections and courses
having tolerably smooth beds, such as channels in firm clay,
clayey loam, sandy soil with over 50 per cent clay, fine cemented
gravel, the coefficient c increases at ordinary velocities with the
velocity of flow. Under ordinary velocities in this sense we
understand velocities which do not cause erosion.
The increase in the value of c with increasing velocities is
equal to
a = V'^
for the smoothest down to
for the roughest channels of this class.
Examples:
S. Fortier, Bear River Canal Branch.
S. Fortier, Providence Canal.
S. Fortier, Solveron and Logan City Canals, Utah.
Darcy-Bazin, rectangular channel lined with pebbles up to
I inch diameter.
Epper, millrace, channel in earth, bottom covered with fine gravel.
Dubuat, Canal du Jard. Channel in earth.
Reich, River Salzach, reach very regular.
2. At velocities exceeding the ordinary, or when erosion
begins, the coefficient c decreases in value for the classes of
channels enumerated above. The decrease is usually such
that
1
a = — I--
T/rs
64 THE FLOW OF WATER
Examples:
Legler, Linth Canal. The coefficient c increases until v is equal
to 4.72 ft. per second, then decreases.
Gordon, Irrawaddi River. The coefficient c increases until v is
equal to 2.62 ft. per second, then decreases.
In the first case the bed is firm earth, in the second sand.
3. For channels of fairly regular cross-section and course in
rockwork, firm gravel up to 2 inches diameter, for channels in
firm earth or sand, or sand with gravel, with stones or vegeta-
tion, the coefficient c does not appreciably vary with the velocity
of flow. Consequently
a = 1.0.
Examples :
Perrone, Torlonia Drain tunnel, channel in rock work.
Darcy-Bazin, series 5, rectangular channel lined with pebbles up
to l^-inches diameter.
Darcy-Bazin, series 36, 37, 38, 41, 43, 47, 48, 50, Grosbois and
Chazilly Canals. Channels in earth of regular cross- section but
with stones or weeds.
La Nicca, Moesa River, coarse gravel.
La Nicca, Plessur River, coarse gravel.
Funk, Weser River.
Passini and Gioppi, Canal Cavour, below the Syphon of the Sesia.
4. For the class of channels enumerated under (3) the co-
efficient c decreases in value whenever the velocity becomes
sufficient to cause erosion. The decrease usually corresponds to
1
a = — r •
5. For channels with very rough beds, channels with boulders^
loose cobblestones, loose coarse gravel or detritus, for channels
with artificial bars to prevent scour, the coefficient c decreases
rapidly in value with increasing velocities. The decrease is equal
to
1
a = — i •
CHANNELS IN EARTH 55
Example :
Cunningham, Solani Embankment, bed in clay and boulders with
artificial bars to prevent erosion, sides of masonry.
Omitting the extremes, we may briefly sum up the variation of
the coefficient c with the velocity as follows :
1. For channels of very regular cross- sections and courses in
clay, clayey loam, sandy soils with large percentages of clay,
cemented gravel up to one inch in diameter, the coefficient of
variation of c is equal up to the eroding limit to
a = 7^"^.
2. For channels in rock work or cemented gravel exceeding one
inch in diameter, for ordinary channels in earth, channels with
some stones or vegetation, the coeflScient a is equal up to the
eroding limit to
a = 1.0.
3. For channels in sand at any velocity and for all others at
velocities exceeding the eroding fimit, the coefficient c decreases
in value with increasing velocities and the coefficient of variation
is fairly equal to
1
a = -— r •
K.
In a preceding chapter we have mentioned the experiments
made by Hubbel and Fenkell, Saph and Schoder to determine
the loss of head due to the resistance in curves. From data
given by them we computed, that, omitting values of -r less
than 2.5, the friction per unit length of curve, in terms of the
friction per unit length of tangent is equal to
4.9 d^
&■■
66
THE FLOW OF WATER
and the excess of friction per unit length of curve in terms of
tangent friction is equal to
(*"*g)")
1.0,
and the length of tangent equal in the amount of frictional
resistance to the frictional resistance in a curve of 90° equal to
0.5 71 R (4:,9 d^' (^''^ - 1.0.
7? 7?
This vanishes when -j -= 4.9 ^ d, it is a maximum when -r
d d
= 4.9 •'* d and the total excess of friction is greatest. The
loss of head due to any curve is consequently
).01478 v^
h=^2.RU.9d^(^-))-1.0j^-
360 \ W / i^i
r + mYr 2g
Table IV.
Friction in Curves.
R
Values of f4.9d^ -J J— 1.0. Diameters 1 to 72 Inches.
a
1''
0.375
2''
0.777
4''
6"
1.907
12"
18''
24''
30"
3.903
36"
48"
60"
72"
2.5
1.422
2.971
3.360
3.657
4.113
4.462
4.750
4.996
4
0.271
0.595
1.174
1.609
2.564
2.913
3.080
3.400
3.588
3.903
4.160
4.382
5
0.225
0.515
1.065
1.478
2.386
2.717
2.971
3.180
3.359
3.657
3.903
4.113
6
0.188
0.453
0.980
1.377
2.247
2.565
2.808
3.009
3.180
3.466
3.701
3.903
10
0.091
0.292
0.761
1.113
1.887
2.170
2.388
2.564
2.717
2.971
3.186
3.369
15
0.020
0.177
0.604
0.925
1.630
1.887
2.084
2.247
2.386
2.617
2.808
2.971
20
0.101
0.501
0.802
1.461
1.703
1.887
2.039
2.168
2.386
2.564
2.717
25
0.046
0.426
0.712
1.337
1.567
1.742
1 . 887
2.010
2.216
2.386
2.531
50
0 216
0 460
0 994
1.189
1 . 838
1.462
1,567
1 . 743
1.887
2.010
100
0.037
0.244
0.700
0.867
0.994
1.099
1.189
1.348
1.461
1.565
Values of z. Curve of 90 degrees, m = 0.95.
2.5
0.049
0.091
0.151
0.184
0.249
0.258
0.263
0.266
0.268
0.270
0.271
0.272
4
0.057
0.112
0.210
0.248
0.344
0.358
0.366
0.371
0.375
0.378
0.380
0.382
5
0.059
0.121
0.227
0.285
0.401
0.416
0.427
0.432
0.438
0.443
0.446
0.448
6
0.059
0.128
0.251
0.319
0.451
0.472
0.484
0.492
0.478
0.504
0.507
0.510
10
0.048
0.137
0.325
0.429
0.634
0.666
0.686
0.699
0.710
0.720
0.727
0.734
15
0.015
0.147
0.386
0.535
0.821
0.869
0.899
0.918
0.935
0.951
0.963
0.971
20
0.095
0.427
0.619
0.982
1.046
1.085
1.111
1.133
1.157
1.173
1.184
25
0.054
0.454
0.687
1.123
1.203
1.252
1.296
1.313
1.343
1.364
1.379
50
0.460
0.888
1.670
1.825
1 9?A
9, 001
2 047
2.113
2.157
2.190
100
0.158
0.931
2.352
2.836
2.858
2.995
3.107
3.268
3.340
3.410
CHANNELS IN EARTH
57
Table IV. A.
Weisbach's Coefficients for Resistances Due to Entrances, Elbows,
Curves, Changes of Section, etc., etc.
Values of z.
Description of Resistance.
0.054
Funnel-shaped or bell-mouthed entrance not pro-
truding into the reservoir.
0.505
Well rounded entrance not protruding into the
reservoir.
0.505
Funnel-shaped or bell-mouthed entrance protruding
into the reservoir.
1.957
Ordinary pipe protruding into the reservoir.
0.9457 sine^ ^ +
2.047 sine*^
Elbows d = angle of deflection.
0.131 -f 1
•-(oJ
Curves. Section circular, d = diameter, U
radius of curve.
0.124 -f 3
-<.4.y
Curves. Section rectangular, d = Width of side
parallel to R, the radius of the bend.
\am J
Constrictions, m
Section contracted
Section not contracted
a = 1.225 -f 1.45 m2 - 1.675 m.
ifr'J
Enlargements or Contractions, A^ = Section hot
contracted, A2 = Section contracted.
0.12
30
Bends of Rivers, n = Number of degrees in arc of
bend.
1.0874
(i-)
Obstructions in Rivers, m => Percentage not ob-
structed.
V" being equal to 7"^^, V'^~, V'^~, etc., according to the degree of
roughness of the conduit. The coefficient of frictional resistance
is given by
'^'^ c. r. f.r. . /dy\ , ^ 0.01478
— 2 7rR 4.9 d - ) - 1.0 -TT=
z =
(Vr + m)'
68 THE FLOW OF WATER
in these equations
n = number of degrees in curve.
TT = 3.1416.
d = diameter of conduit in feet.
R = radius of curve in feet.
X = -^^ for diameters greater than 1 foot.
X = 0.45 for diameters less than 1 foot.
y = J for a diameter of 1 inch.
y = is fo^ ^^y other diameter.
From the foregoing we draw the conclusion, that the value of
z depends :
1. On the value of -r and the value of d,
d
2. On the value of T^rrp: •
360
3. On the value of m.
For any arc, multiply the values of z, found in the table, by
the number of degrees and divide by 90.
For any degree of roughness multiply the values of z by
the following:
m = 0.95, multiply by 1.0.
m = 0.83, multiply by 1.166.
m = 0.68, multiply by 1.436.
m = 0.53, multiply by 1.802.
m = 0.45, multiply by 2.060.
m = 0.30, multiply by 2.717.
If in the formula for the loss of head due to a curve we
substitute
2 grs , .^ -1 4. 0.01478
tor its equivalent -7—=
V ( Vr + my
and L for the length of the curve the formula will read, after
reduction,
ff=(4.«.(D')
1.0 • L • s.
RIVETED CONDUITS 69
which simply expresses the theory outlined at the beginning of
this chapter that the excess loss of head due to a curve is
(4.9^ (!)')-,.„
times the loss due to an equal length of straight pipe; S being
the sine of the slope to which velocities in the tangent are due.
Riveted Conduits.
L.
Riveted conduits form a class apart in so far as the degree of
roughness varies with the diameter. Up to date the coefficients
for such conduits have been fairly well determined for diameters
up to 8.5 feet (Holyoke Testing Flumes); for larger sections
they are as yet problematical.
Fairly reliable values of the coefficients for riveted conduits
may be found by computing the losses of head due to the
resistance of rivet heads, or to enlargements and contractions
of the section as follows :
If in an 18-foot steel-riveted pipe we allow an internal pressure
of 140 pounds per square inch, in the steel a tension of 20,000
pounds per square inch; and if we assume the efficiency of the
riveted joints to be 70 per cent of the metal, we have for the
thickness of the metal in inches
_ 140 X diameter in inches,
0.7 X 40,00"0
which gives t = 1.08 inches.
It is usual to take for the diameter of the rivet in inches
d = 0.15 + 1.5 t,
and for the pitch of the rivets in a single row
Sj = 0.375 + 2 d,
and s^ = 0.75 + dd
for the pitch in a double row.
60 THE FLOW OF WATER
Hence in our case
d = 1.75,
s, = 3.875,
s, = 6.0.
The usual diameter of the rivet head is 1.8 c? and its depth 0.6 d.
This gives for the sectional area of the rivet at right angles to
the line of flow
3.15 X 1.05 = 3.3075 square inches nearly.
As the circumference of the conduit is 12 X 18x 3.14 = 678.25
inches and the spacing 3.875 inches, there will be 175 rivets in
the single circumferential row. The open space between the
rivets will only be 3.875 - 3.25 = 0.725 inches. The dis-
turbance in the motion in this narrow space will be such, that
it will be safe to consider the row of rivet heads as an unbroken
line of a depth 0.6 d = 1.05 inches. Weisbach gives for the loss
of head due to constrictions
^-{al, ^J 2g
J 2g
in which
Aj^ = section not constricted,
A^ = section constricted,
a = 1.225 + f'4-V - 1.695 ^
©■
2
A.
In our case A, = 18=^ X 0.7854 = 254.34,
A^ = (17.825)2 X 0.7854 = 249.5.
Inserting these values in Weisbach's formula we find
h = 00187489 ^.
Assuming the metal sheets to be 10 feet each way there will
be six sheets in the circumference, and as the pipe is double
riveted longitudinally there will be twelve longitudinal rows of
rivets, and allowing 13 d for the outside rim on each side there
RIVETED CONDUITS 61
will be twenty circumferential rows, the pitch being six inches.
The twelve rivets in each row will cause a constriction of 12 x
3.3075 = 39.69 square inches = 0.275 p. According to Weis-
bach's formula this constriction causes a loss of head equal to
/i = 00005936^,
and the twenty rows a loss equal to
h = 00011907^-
Adding the resistances due to all the circumferential rows in a
section of 9.5 feet we have
Z, = 00187489 + 00011907 = 00199396.
Assuming the conduit to be 20,000 feet long the total resistance
due to the rivet heads will be
^1= ^^ = 2105 X 00199396 = 4.196985.
y.o
To this must be added the resistance due to the enlargement
or contraction caused by the circumferential lap of the sheets.
As the thickness of the metal is 1.08 inches the diameter is
enlarged or contracted 2.16 inches at each lap. The loss of
head due to enlargements or contractions is, according to
Weisbach,
r y 216'' \^ 1^ V^ V^
hence in our case [ (^ig.si^j " ^J 2^ = 00041209 ^^-
The total resistance due to all the enlargements or contractions
is consequently
Z, == 2105 X 00041209 = 0.86755.
If the conduit had no rivet heads or enlargements and con-
tractions to increase the resistance, the value of the coefficient m
62 THE FLOW OF WATER
would be the same as for a cast-iron pipe, or equal to 0.83,
and the frictional resistance per unit area of surface would be
_ 0-01478 _
' ~ (1.456 + 0.83)^ - ^''^'^
and the total resistance of the wet perimeter
^3= 002829^ =12.473.
^ 4.5
Adding, we have for the sum of all the resistances
Zj + ^2 + ^3 = 17.5375.
This gives for the total frictional resistance per unit area of
surface
4.5
17.5325 X
20000
or / = 00394594;
hence the coefficient c is equal to V ' = 127.7, and m is
00394597
7
_ _ 1 4Kfi = n 4S
66
127.7
equal to -^ 1.456 = 0.48.
Practical Applications of the Formulse
M.
1. From the formula
V = (66 (Vr + m) V^)lf
we have
yTl= -J^ = 66 {</r + m) V7s
and s = — I 47= -r
(Vr+ m)
_ = r^/r
Ft^ 66 Vs
We have also , ,_ = (^/r + m)Vr = Ri + m Vr.
PRACTICAL APPLICATIONS OF THE FORMULA 63
Putting ^r = X and transposing we have
X' + mX^ + 0 - , ^ -=. = 0,
FtV 66 Vs
from which the value of a; = \^r is found by Horner's method.
We have also
m = — J y= - ^r,
yiV 66 \/Ts
If the coefficient of variation of c is equal to V^, V^^ — -j- etc.,
these values are substituted in the given equations.
Values of a =V^ 7^ V^^, Vt^, F^V yrV 7/3^
are found in Table V.
Example: Let it be required to find the slope for a rectangular
aqueduct of common brickwork or concrete 100 feet wide, 12.5
feet deep, the velocity to be 4 feet per second. The cross- sec-
tion is 1,250 p, the wet perimeter 125 /, hence R = 10.0. In
the table of roots of mean radii we find VlO = 3.163 VlO =
1.78. The value of m for common brickwork or concrete is
0.57. The value of a = V^^ for v = 4.0 is, according to Table V,
equal to 1.08. Inserting these values into our formula we have
for the slope
[r
08 X 66 X (1.78 + 0.57) X 3.163
V530 /
= 0.0000569.
Example: Let it be required to find the diameter of a semi-
circular channel lined with common ashlar or very good rubble
masonry, the slope being 1 in 1,000, and the permissible velocity
10 feet per second.
In this case m = 0.30
V7 = 0.0316
a = '^lo = 1.137.
64
THE FLOW OF WATER
Solving by Horner's method and inserting values we have
10 0.0
X' + 0.3 .
^ ' • 1.137 X 66 X 0.0316
X' + 0.3 X' + 0.0 - 4.217 = 0.0 |a; = 1.521
1.0
+ 1.3 + 1.300
1.3
+ 1.3 - 2.917
1.0
+ 2.3
2.3
+ 3.6
1.0
3.3
+ 3.6 - 2.917
0.5
+ 1.9 + 2.750
3.8
+ 5.5 - 0.167
0.5
+ 2.15
4.3
+ 7.65
0.5
4.8
+ 7.65 - 0.167
0.02
+ 0.096 + 0.1544
4.82
+ 7.746 - 0.0121
0.001 + 0.005 + 0.0077
4.821 + 7.751 - 0.0044.
This gives x = 1.521; hence the mean hydraulic radius r
(1.521)' = 5.352, and the diameter = 21.408 ft.
2. From the formula
we have
2gh
0.01478 L
r + m)2 R
(</
^
gh
yV
H =
0.01478_ L
m) 2 R
{^r
(^r +myR^ 0.01478 -^ —-
+
0.01478
H 2g
~'R2g
(-v^rH- m)Vr = R^ -\- mVr = y '
0.01478
L_
H
yV
PRACTICAL APPLICATIONS OF THE FORMULA 65
and putting x = ^/r we have
X' + mX' + 0.0 - V 0.01478 ^^ = 0.0,
n 2 g
which may be solved by Horner's method.
To faciUtate calculations it is well to remember
that yT^=7^X 7^.
and VV.(JLJ
Values of y ^ and F^ are found in Table V.
Resistances due to entrance and the velocity itself are included
in the term —^ and need not be further considered
i^r -i-my R
unless the length of the conduit is less than 1,000 diameters.
For pipes between 300 and 1,000 diameters in length (as also
for riveted pipes exceeding 3 feet in diameter), the coefficient of
variation is equal to a = V^^, and t\ and V are substituted in
the given equations for ^^ and y. If the pipe is between 100
and 300 diameters in length (or an old pipe not very clean) the
coefficient a is equal to 1.0, and i and 2.0 are substituted in
the given equations for t^b- and V. In case the conduit is less
than 100 diameters in length the coefficient a is equal to — - and
^^ and V are substituted for j\ and V^. Values of -—4: r^
are found in Table VI.
Example: Let it be required to find the velocity of flow in a
new steel riveted conduit 6 feet in diameter, 10,000 feet long,
the head to be 5 feet and the conduit to have 20 curves of 10°
each and a radius of 30 feet. In this casern = 0.53, R = 1.5,
Vl.5 = 1.107. For the curves we have the relation -r = -^r
d 6
= 5.0. In Table IV we find the coefficient z^ for - = 5.0 and a
curve of 90° to be equal to 0.466. As there are 20 curves of 10
, , „ 20 X 10 X 0.448 ^ not: -17 A fto
degrees we have Z^ = — = 0.995. For m = 0.53
66
THE FLOW OF WATER
this is to be multiplied by 1.802, which gives for the total resist-
ance due to curves Z^ = 1.782.
Inserting values into our formula we have
64.4 X 5.0
0.01478 X 10000
1.782
(1.107 + 0.53) 2 X 1.5
= (8.312) tV
Remembering that Ftt = y^ x Y'^'^ we first draw the square
root out of the quotient and multiply this by the seventeenth
root of the square root.
The quotient is 8.312, V8.312 = 2.884. In the table of roots
we find ^-^3 = 1.065, ^-^2.75 ^ 1.059. Interpolating we have
for ''n/2!884 = 1-062. Consequently?; = 2.884 X 1.062 = 3.0628
feet per second.
3. From the formula
V = (66 (Vr + m) Vrs)^
we have for the discharge of a circular conduit in cubic feet per
second
Q = (66 (^r + ^)\/^) t d' 0.7854.
From this we have for the head in feet
Q \t 1 rL
^d' 0.7854/ 66 (^r +m)l R
and for the diameter in feet
LV(i2 0.7854/
d =
L\0.7854/ [m{^r + m)]' H J
If a = FtV the index if is substituted for |, H for f , V for V
and ?^^ for ^S:. From this equation the value of d can only be
found by trial, assuming a value of -Vr in the term ^/r + m.
For a first trial a value of "^r = 1.0 will give good results.
From the formula
V =
2gH
0.01478 L
+ Z,+Zn
we have Q
PRACTICAL APPLICATIONS OF THE FORMULA 67
2gH
omm L^^^^^^
^ (P 0.7854
( Vr -\-m)^ R
0.01478 L ^ 5, ^ 5,
jj_( Q ^^ i^r + m)' R
\ d' 0.7854^ 2 g
From this equation also d can only be found by a second or
third trial, assuming a value of Vr and R.
For a = yi^ the index t\ is substituted for ^^, V for V and
^5 for 4^.
For a = 1.0 the index i is substituted for j%, 2 for V and
?\ = i for 4T.
Example: What will be the loss of head corresponding to a
discharge of 5 cubic feet per second, the conduit being a 2-foot
riveted pipe 20,000 feet long and having 30 curves of 15° each
and a radius of 20 feet.
In this case m = 0.68; R = 0.5; Vr = 0.84, consequently
0.01478 0.01478 ^_^^^^
i^/r + m)2 (0.84 + 0.68) ^^
In Table IV we find for the resistance of a 90° curve for the
relation — = — - = 10 Z^ = 0.686, consequently for 30 curves
of 15° each Z, = ^^ ^ ^^^ ^'^^^ -= 3.430. For m = 0.68 this
is to be multiplied by 1.436 which gives z^ = 4.925.
Inserting these values we have
r . -,,e [0.00638 X ^H^l +4.925
rr _[__£__> L O.O J
L4 X 0.7854J 64.4
or ^ = 9.24 feet.
68 THE FLOW OF WATER
5. The Kinetic energy or living force acquired by a body
falling free or descending in a plane infinitely smooth is equal to
E = imv' = Q.W.H.,
m = the mass of a body = -pz — -. — ,
*^ Gravity
Q = the discharge in cubic feet per second,
W = the weight of one cubic foot,
H = the total fall in feet.
Expressed in horsepowers the energy is equal to
Q.W.H.
H.P, =
550
or, in kilowatts, to
K.W. = «--
If a body of water is not falling free, the total head is reduced
by an amount which depends on the velocity, the length of the
conduit, its diameter and its degree of roughness.
The loss of head is equal to
v'f-
ri
V 0.01478
L
h =
29-
2g'
2g (^r +m)^
R
as the case may be. For conduits of equal length the loss is
evidently the least for the greatest diameter and for the lesser
speed of flow.
For a given diameter the efficiency of a conduit as a trans-
mitter of energy is greatest when the speed of flow is such, that
one-third of the total available head is consumed in overcoming
frictional resistances (see *' Adams and Gummel," Eng. News,
May 4, 1893) .
Example: A new four foot steel riveted conduit 2,000 feet
long, under a head of 300 feet is to deliver water to the gener-
ator at such a velocity that its efficiency will be a maximum.
What will be the discharge and the horsepower transmitted?
PRACTICAL APPLICATION OF THE FORMULA. 69
Allowing one-third of the total head to be spent in overcoming
frictional resistances we have v = = 0.05. For this value
of V the velocity will be
V = (66 (1 + 0.53) Vl X 0.05) T7 = 2.369 feet per second;
the discharge, Q = 2.369 X 16^ X -j =30.4 cubic feet per second;
., . rj T, 30.4 X 62.4 X 200 -^^ ^
the horsepower, H P = — = 708.0.
550
TABLE V.
Table V contains roots of velocities or values of (a), the
coefficient of variation of c.
To find the value of c corresponding to any velocity multiply
the value of 66 (\^r + m) by the value of (a) = V\ F^V^ yjh,
,as the case may be.
To find the velocity multiply the value of 66 (Vr + m)
by the value of
(66 ( Vr + m) Vr.s) ^ which in Table V is given as V^,
(66 {{/r + m) Vr.s) " which in Table V is given as V^^,
(66 ( Vr + m) Vr.s) ^^ which in Table V is given as V^^,
1 1
j-r= ^ . which in Table V is given as — t-
(66 {^r+ m) Vr.s) ^ FtV
as the case may be.
Also , to find the velocity, multiply the value of
r.i
1 y
?ff
' R
which in Table V is given as 7"'
70
THE FLOW OF WATER
Table V.
Roots of Velocities or Values of (a), the
COEFl-'ICIENT OP
Variation of c
V
yi
vl
V^
yA
7T^
yxV
v.h
1
1
0.25
0.841
0.857
0.882
0.891
0.917
0.925
0.925
1.081
1 .075
0.50
0.918
0.925
0.939
0.944
0 .958
0.959
0.961
1.040
1.037
0.75
0.964
0.964
0.974
0.976
0.982
0.981
0.982
1.018
1.015
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1 .0
1.0
1 .0
1.25
1.026
1.025
1 .0205
1.019
1.013
1 .013
1.013
0.987
0.989
1.50
1 .052
1 .046
1 .038
1.034
1.025
1.024
1.022
0.978
0.979
1.75
1 .072
1.064
1.052
1.048
1.035
1.033
1.031
0.97
0.971
2.0
1 .090
1.080
1.065
1 .0594
1.044
1 .042
1.039
0.962
0.964
2.25
1.107
1.095
1.077
1.070
1 .052
1.049
1.046
0.956
0.959
2.50
1.121
1 .107
1.087
1.080
1.059
1.056
1.052
0.95
0.953
2.75
1.135
1 .118
1.096
1 .086
1.065
1.061
1.058
0.945
0.948
3.0
1 .147
1 .130
1.105
1.096
1.071
1.067
1.063
0.940
0.943
3.25
1 .159
1.140
1.113
1 .103
1 .076
1.072
1.068
0.936
0.940
3.50
1 .170
1 .150
1.121
1 .110
1.081
1.077
1 .072
0.932
0.936
3.75
1.18
1.158
1.128
1.116
1 .086
1.081
1 .076
0.929
0 .933
4.0
1 .189
1 .166
1 .134
1.123
1 .090
1 .085
1.080
0.926
0.930
4.25
1 .198
1 .174
1.141
1 .128
1.094
1 .088
1.083
0.923
0.927
4.50
1 .207
1 .182
1.146
I .133
1 .098
1 .093
1.087
0.919
0.924
4.75
1.215
1.189
1.152
1.139
1.102
1.096
1.090
0.917
0.921
5.0
1 .223
1.196
1.158
1.144
1 .106
1.10
1.093
0.914
0.919
5.25
1 .231
1.203
1 .163
1 .149
1 .109
1.103
1.098
0.912
0.917
5.50
1.237
1.209
1 .168
1.153
1.112
1.106
1.099
0.910
0.914
6.75
1 .244
1 .215
1.173
1.159
1.115
1.108
1.102
0.907
0.912
6.0
1 .251
1.220
1.177
1.161
1.118
1.110
1.104
0.905
0.910
6.25
1 .258
1.226
1.181
1.165
1 .121
1.114
1.107
0.903
0.908
6.50
1.262
1.231
1 .186
1 .169
1.123
1.116
1.109
0.901
0.906
6.75
1.269
1.237
I .190
1 .173
1.126
1 .119
1 .112
0.899
0.904
7.0
1.275
1.241
1 .194
1.176
1.129
1.121
1 .114
0.897
0.902
7.25
1 .281
1.246
1.197
1.18
1.131
1.124
1.116
0.895
0.901
7.50
1 .286
1 .251
1.201
1.183
1.134
1.126
1.118
0.894
0.899
7.75
1.292
1 .256
1.205
1.186
1 .137
1.128
1 .12
0.893
0.898
8.0
1.297
1 .260
1.208
1 .189
1.139
1.130
1 .122
0.891
0.896
8.25
1 .302
1 .264
1.212
1.192
1.141
1.132
1.124
0.889
0.895
8.50
1 .307
1 .268
1.215
1 .195
1.143
1 .134
1.126
0.888
0.893
8.75
1 .311
1.272
1.218
1.198
1.145
1.136
1 .128
0.886
0.892
9.0
1.316
1.277
1.221
1.201
1.147
1.138
1.130
0.885
0.891
9.25
1.320
1 .280
1.224
1 .204
1.149
1 .140
1 .131
0.884
0.889
9.50
1 .325
1 .284
1.227
1 .207
1.151
1 .142
1 .133
0.882
0.888
9.75
1 .329
1.288
1.230
1.209
1.153
1.143
1 .135
0.881
0.887
10.0
1.333
1.292
1.233
1 .212
1.155
1.145
1 .137
0.880
0.886
10.5
1.341
1.298
1 .238
1.216
1 .158
1.149
1.139
0.877
0.884
11.0
1 .348
1.305
1.244
1.221
1.161
1.152
1.142
0.875
0.881
11.5
1.357
1 .310
1.249
1.226
1.165
1 .155
1.145
0.873
0.879
12.0
1.364
1.318
1.254
1.230
1 .169
1.158
1.148
0.871
0.877
14.0
1 .39
1.340
1.271
1.246
1 .179
1.168
1.158
0.863
0.870
16.0
1.414
1.365
1 .286
1.260
1 .189
1 .178
1 .169
0.855
0.864
18.0
1.435
1.38
1 .301
1.272
1.199
1 .185
1.175
0.851
0.859
20.0
1.450
1.395
1.311
1.288
1 .204
1 .193
1 .182
0.846
0.854
PRACTICAL APPLICATIONS OF THE FORMULA
71
Table VI .
Values of 66 (-s/r + m) and Corresponding Values of /, the
Coefficient of Friction. Conduits under Pressure.
m = 0.95
m = 0.83
m = 0.68
m = 0.53
w = 0.45
m = 0.30
c
/
00839
c
79.9
/
01007
c
70.0
/
01313
c
60.1
/
01780
c
54.8
/
c
44.9
/
1
87.3
02142
03190
2
92.4
00750
84.5
00901
74.6
01156
64.7
01534
59.4
01823
49.5
02625
3
95.7
00699
87.8
00834
77.9
01060
68.0
01391
62.7
01636
52.8
02307
4
97.1
00679
90.2
00790
80.3
00984
70.4
01298
65.1
01528
55.2
02111
6
102.0
00615
94.1
00726
84.2
00907
74.3
01166
69.0
01351
59.1
01841
8
104.9
00581
97.0
00684
87.1
00848
77.2
01080
71.9
01244
62.0
01673
10
106.3
00566
99.4
00651
89.5
00803
79.6
01015
74.3
01165
64.4
01551
12
109.4
00535
101.5
00624
91.6
00767
81.7
00964
76.4
01102
66.5
01454
14
111.2
00517
103.3
00603
93.4
00737
83.5
00922
78.2
01052
68.3
01379
16
112.9
00502
105.0
00583
95.1
00711
85.2
00886
79.9
01007
70.0
01313
18
114.4
00489
106.5
00567
96.6
00689
86.7
00856
81.4
00971
71.5
01258
20
115.9
00498
107.8
00554
97.9
00671
88.0
00829
82.7
0094
72.8
01214
22
117.0
00467
109.1
00540
99.2
00654
89.3
00807
84.0
00912
74.1
01171
24
118.2
00458
110.3
00528
100.4
00638
90.5
00785
85.2
00886
75.3
01134
26
119.4
00449
111.5
00517
101.6
00623
91.7
00765
86.4
00862
76.5
0110
28
120.4
00441
112.5
00507
102.4
00611
92.8
00747
87.5
00840
77.6
01068
30
121.4
00434
113.5
00499
103.6
00599
93.7
00733
88.4
00823
78.5
01044
32
122.2
00428
114.3
00492
104.4
00590
94.5
00720
89.2
00808
79.3
01023
34
123.1
00422
115.2
00485
105.3
00580
95.4
00707
90.1
00792
80.2
00100
36
124.0
00416
116.1
00477
106.2
00570
96.3
00693
91 .0
00777
81 .1
00978
38
124.9
00410
117.0
00470
107.1
00560
97.2
0068
91 .9
00762
82.0
00957
40
125.7
00405
117.8
00464
107.9
00552
98.0
0067
92.7
00748
82.8
00938
42
126.4
0040
118.5
00458
108.6
00545
98.7
0066
93.4
00737
83.5
00923
44
127.0
00396
119.1
00453
109.2
00539
99.3
00651
94 .2100728
84.1
00909
46
127.9
00391
120.0
00446
110.1
00531
100.2
00641
94.9
00714
85.0
00890
48
128.4
00386
120.8
00441
110.9
00523
101.0
00631
95.7
00702
85.8
00874
50
129.4
00382
121.5
00436
111.6
00516
101 .7
00622
96.4
00692
86.5
0086
52
130.0
00379
122.1
00431
112.2
00511
102.3
00614
97.0
00684
87.1
00848
54
130.7
00375
122.8
00426
112.9
00504
103.0
00606
97.7
00674
87.8
00834
56
131.2
00372
123.3
00423
113.4
00500
103.5
00600
98.2
00667
88.3
00826
58
131.9
00368
124.0
00418
114.1
00494
104.2
00592
98.9
00657
89.0
00812
60
132.5
00364
124.6
00414
114.7
00489
104.8
00586
99.5
00650
89.6
00801
62
133.1
00361
125.2
00410
115.3
00484
105.4
00579
100.1
00642
90.2
00791
64
133.6
00358
125.7
00407
115.8
00480
105.9
00573
100.6
00636
90.7
00782
66
134.2
00355
126.3
00403
116.4
00475
106.5
00567
101.2
00628
91.3
00772
68
134.7
00352
126.8
00400
116.9
00471
107.0
00562
101 .7
00622
91 .8
00763
70
135.2
00350
127.3
00397
117.4
00467
107.5
00557
102.2
00616
92.3
00755
72
135.8
00347
127.9
00393
118.0
00461
108.1
00551
102.8
00609
92.9
00745
78
137.2
00340
129.3
00385
119.4
00451
109.5
00536
104.2
00592
94.3
00723
84
138.5
00333
130.6
00374
120.7
00441
110.8
00524
105.5
00574
95.6
00704
90
139.9
00327
132.0
00369
122.1
00431
112.2
00511
106.9
00563
97.0
00684
96
141.2
00321
133.3
00362
123.4
00423
113.5
00499
108.2
00549
98.3
00666
102
142.2
00316
134.3
00356
124.4
00416
114.5
00490
109.2
00539
99.3
00652
108
143.2
00312
135.3
00351
125.4
00409
115.5
00482
110.2
00530
100.3
00639
114
144.5
00306
136.6
00344
126.7
00401
116.8
00472
111.5
00518
101.6
00624
120
145.7
00301
137.8
00339
127.9
00393
118.0
00462
112.7
00506
102.8
00609
126
146.7
00297
138.8
00334
128.9
00387
119.0
00454
113.7
00497
103.8
00597
132
147.7
00293
139.8
00329
129.9
00381
120.0
00446
114.7
00489
104.8
00590
138
148.6
00290
140.7
00325
130.8
00376
120.9
00440
115.6
00481
105.7
00576
144
149 .5
00286
141.6
00321
131.7
00371
121.8
00434
116.5
00474
106.6
00566
156
151 .4
00279
143.5
00313
133.6
00360
123.7
0042
118.4
00459
108.5
00557
168
153.0
00273
145.1
00305
135.2
00352
125.3
00410
120.0
00447
110.1
00546
180
154.6
00268
146.7
00299
136 .8
00343
126.9
00400 121 .6
00435
111.7
00538
72
THE FLOW OF WATER
Table VI. A.
WELDED PIPES.
Tubes of Brass, Galvanized Iron, Sheet Iron, Steel, Etc.
.-I i^
i
i
I
h
f
1
H
n
2
2h
3
3i
4
4i
5
6
7
8
9
10
11
12
13
14
15
Actual
Diam-
eter
in
feet.
.0225
.0303
.0411
.0516
.0686
.0873
.1150
.1341
.1722
.2056
.2556
.2956
.3356
.3756
.4204
.5056
.5857
.6651
.7449
.8348
.9166
.0
.1641
.1875
.2708
Area =
^20.7854
.00038
.00072
.00133
.00209
.00370
.00599
.01039
01412
02339
03320
05130
06863
08840
1168
1388
2008
2694
3474
4356
5473
6599
7854
9531
1075
2675
Values of 66 ( \/7^+ m)\/r
= c Vr
m =
0.95
.058
.111
.483
.646
.34
.02
.22
.65
24
34
24
61
44
40
48
26
61
73
74
97
87
70
99
70
29
m =
0.83
,467
423
.665
745
31
85
90
20
59
23
25
24
15
98
93
45
60
53
25
38
06
70
84
4
78
.722
.56
.677
.623
,006
.38
,22
39
,54
30
75
56
29
94
72
94
05
49
08
85
35
77
76
21
22
0.45
.584
.239
.139
.398
.018
.142
.646
.61
,40
,85
,90
42
,89
,39
80
54
00
30
52
91
08
18
66
72
66
^ Loss of Head in Feet
Ikijfc Length of-Gonduit
at Unit Velocity.
tn a. /«/t<t.7^
25
f='e
m =
0.95
.714
.274
.8949
.692
.5008
.3801
.2781
.2268
.1740
.1414
.1095
.0910
.0796
.0696
.0610
.0489
.0410
.0352
.0308
.0268
.0239
.0215
.0191
.0174
.0160
m =
0.83
.154
.561
.096
.842
.6063
.4587
.3331
.2790
.2081
.1936
.1302
.1095
.0941
.0823
.0703
.0575
.0481
.0412
.0351
.0303
.0279
.02505
.0222
.0202
.0186
m =
0.68
888
083
444
108
,794
597
431
359
267
215
165
139
119
103
090
0718
0598
0511
0444
0377
0343
0304
0272
0248
0227
m =
0.45
.025
.585
.439
.850
.308
.9496
.692
.572
.4155
.3354
.2548
.2122
.1806
.1565
.1355
.1069
.0883
.0750
.0648
.056
.0495
.0442
.039
.0353
.0323
PRACTICAL APPLICATIONS OF THE FORMULA
73
Table VII.
CIRCULAR CONDUITS.
Diameters, Internal Areas, Mean Hydraulic Radii and their Roots.
1
IS
^1
R
Vr
Vr
^
1 s
R
Vr
V^
Is
1— 1
1— (
i33
1 .0408
1 .060
1 .0801
1 .0202
1 .030
1 .0393
i
BRRA
TA
183
»
1 .0993
1 .118
1 .0485
1.057
Page 72,
Table VI
A, for
)66
1 .137
1.066
Loss of 1
iead in
Feet per Unit Length
of
133
'5
1 .415
1 .173
1 .0684
1.083
Condi
Jit at Ui
s
lit Velo
ubstitut
city
e
66
183
55
1 .1903
1 .2076
1 .226
1 .275
1.091
1.099
1.107
1.129
z
J
1 .323
1 .150
R'
or Lo«
>s of Head in Feet in a Length
of
'5
1 .369
1.170
1 .414
1.189
2 g Feet at Unit Velocity.
!5
1 .457
1.208
i
1 .5
1 .224
'5
1 .541
1 .581
1 .244
1 .257
'R
1 .620
1 .658
1 273
16
1.396
0 .3333
0 .5771
0.759
132
95.03
2.75
1.288
18
1.767
0 .3750
0 .6124
0.782
138
103 .87
2.875
1 .696
1 .302
20
2.234
0 .4166
0.645
0 .8031
144
113.10
3.0
1 .732
1.316
22
2.640
0 .4583
0 .6770
0 .8220
150
122 .72
3.125
1 .768
1.330
24
3.142
0.5
0 .7071
0.841
156
132 .76
3.25
1 .803
1 .343
26
3.687
0 .5416
0 .7360
0.859
162
143 .16
3 .375
1 .837
1 .355
28
4.275
0 .5833
0 .7637
0.874
168
153 .96
3.5
1 .871
1 .368
30
4.909
0.625
0 .7906
0.889
174
165.17
3.625
1 .904
1 .380
32
5.585
0 .6666
0 .8165
0.904
180
176 .70
3.75
1 .937
1.392
34
6.305
0 .7084
0 .8416
0.917
186
188 .70
3.875
1 .968
1.403
36
7.069
0.75
0.866
0.931
192
201 .03
4.0
2.0
1.414
38
7.876
0 .7916
0 .8898
0 .9435
198
213.8
4.125
2.031
1 .425
40
8.927
0 .8333
0 .9129
0 .9555
204
227.0
4.25
2.062
1.437
42
9.621
0.875
0 .9355
0.967
210
240.5
4.375
2.091
1 .446
44
10 .559
0 .9166
0 .9575
0 .9784
216
254.5
4.5
2.121
1 .456
46
11 .509
0 .9584
0 .9983
0 .9991
222
268.8
4.625
2.151
1 .466
48
12.566
1.0
1.0
1.0
228
283.5
4.75
2.180
1 .476
50
13 .635
1 .0416
1 .0206
1 .0103
240
314.2
5.0
2.236
1 .496
72
THE FLOW OF WATER
Table VI. A.
WELDED PIPES.
Tubes of Brass, Galvanized Iron, Sheet Iron, Steel, Etc.
Si
I"
1
n
n
2
21
3
3i
4
4i
5
6
7
8
9
10
11
12
13
14
15
Actual
Diam-
eter
in
feet.
.0225
.0303
.0411
.0516
.0686
.0873
.1150
.1341
.1722
.2056
.2556
.2956
.3356
.3756
.4204
.5056
.5857
.6651
.7449
.8348
.9166
.0
.1641
.1875
.2708
Area =
^20.7854
.00038
.00072
.00133
.00209
.00370
.00599
.01039
.01412
.02339
.03320
.05130
.06863
.08840
.1168
.1388
.2008
.2694
.3474
.4356
.5473
.6599
.7854
.9531
.1075
.2675
Valuef
/.4/-
X ^ /- \^ Loss of Head in Feet eeir
/^M4a-A
T ^T^ ort h of Conduit "^ ^
/=-^
m =
0.95
6.05
7.11
8.48
9.64
11.34
13 .0.^
15 .2:
16 .6.'
19 .2<
21 .3
24.2
26.6
28.44
30.40
32.48
36.26
39.61
42.73
45.74
48.97
51.87
54.70
57.99
60.70
63.29
26.15
27 .98
29.93
33.45
36 .60
39.53
42.25
45.38
48 .06
50.70
53.84
56.4
58.78
24.94
26.72
29 .94
33.05
35.49
38.08
40.85
43.35
45.77
49.76
50.21
53.22
lO .Ot7
20.39
21.80
24.54
27.00
29.30
31.52
33.91
36.08
38.18
40.66
42.72
44.66
0 .0696
0 .0610
0 .0489
0 .0410
0 .0352
0 .0308
0 .0268
0 .0239
0 .0215
0 .0191
0 .0174
0 .0160
.0823
.0703
.0575
.0481
.0412
.0351
.0303
.0279
,02505
.0222
0202
0186
.103
.090
.0718
.0598
.0511
.0444
.0377
.0343
.0304
.0272
.0248
.0227
.1566
.1355
.1069
.0883
.0750
.0648
.056
.0495
.0442
.039
.0353
.0323
PRACTICAL APPLICATIONS OF THE FORMULAE
73
Table VII.
CIRCULAR CONDUITS.
Diameters, Internal Areas, Mean Hydraulic Radii and their Roots.
— r O
1
•S
R
v7
Vr
Is
-*^ .2
R
Vr
Vr
52
1— (
i
0 .001364
0 .010417
0 .10206
0 .3195
14 .750
1 .0833
1 .0408
1 .0202
1
0 .005454
0 .02083
0 .1444
0.380
54
15 .904
1.125
1 .060
1.030
li
0 .012272
0 .03125
0 .1768
0 .4302
56
17 .106
1 .1666
1 .0801
1 .0393
2
0 .02182
0 .04166
0 .2039
0 .4516
58
18 .347
1 .2083
1 .0993
1 .0485
2h
0 .03163
0 .0502
0 .2240
0 .4733
60
19 .635
1.25
1 .118
1 .057
3
0 .04909
0 .0625
0.25
0.5
62
20 .964
1 .2966
1 .137
1.066
Si
0 .06681
0 .07292
0.270
0 .5196
64
22 .340
1 .3333
1 .415
1 .0684
4
0 .08726
0 .08333
0.291
0 .5370
66
23 .758
1 .375
1 .173
1.083
4^
0 .11045
0 .09375
0 .3062
0 .5533
68
25 .220
1 .4166
1 .1903
1.091
5
0 .1364
0 .10416
0 .3227
0 .5681
70
26 .725
1 .4583
1 .2076
1.099
6
0 .1963
0.125
0.3535
0.594
72
28.27
1 .5
1 .226
1.107
7
0 .2672
0 .1458
0 .3819
0 .6180
78
33.18
1 .625
1 .275
1.129
8
0 .3490
0 .1666
0 .4082
0.639
84
38.48
1 .75
1 .323
1.150
9
0 .4418
0 .1875
0.433
0.658
90
44.18
1 .875
1 .369
1.170
10
0 .5585
0 .2083
0 .4564
0.675
96
50.27
2.0
1 .414
1.189
11
0 .6599
0 .2297
0 .4787
0.692
102
56.75
2.125
1 .457
1 .208
12
0 .7854
0.25
0.5
0 .7071
108
63.62
2.25
1 .5
1.224
13
0 .9217
0 .2708
0 .5204
0.721
114
70.88
2.375
1 .541
1 .244
14
1 .0689
0 .2916
0.540
0.735
120
78.54
2.5
1 .581
1.257
15
1 .2272
0 .3125
0 .5339
0.748
126
86.59
2.625
1 .620
1 .273
16
1.396
0 .3333
0 .5771
0.759
132
95.03
2.75
1 .658
1 .288
18
1.767
0 .3750
0 .6124
0.782
138
103 .87
2.875
1 .696
1.302
20
2.234
0 .4166
0.645
0 .8031
144
113.10
3.0
1 .732
1 .316
22
2.640
0 .4583
0 .6770
0 .8220
150
122 .72
3.125
1 .768
1.330
24
3.142
0.5
0 .7071
0.841
156
132 .76
3.25
1 .803
1.343
26
3.687
0 .5416
0 .7360
0.859
162
143 .16
3 .375
1 .837
1.355
28
4.275
0 .5833
0 .7637
0.874
168
153 .96
3.5
1 .871
1.368
30
4.909
0.625
0 .7906
0.889
174
165.17
3.625
1 .904
1.380
32
5.585
0 .6666
0 .8165
0.904
180
176 .70
3.75
1 .937
1 .392
34
6.305
0 .7084
0 .8416
0.917
186
188 .70
3.875
1 .968
1 .403
36
7.069
0.75
0.866
0.931
192
201 .03
4.0
2.0
1 .414
38
7.876
0 .7916
0 .8898
0 .9435
198
213.8
4.125
2.031
1.425
40
8.927
0 .8333
0 .9129
0 .9555
204
227.0
4.25
2.062
1.437
42
9.621
0.875
0 .9355
0.967
210
240.5
4.375
2.091
1 .446
44
10 .559
0 .9166
0 .9575
0 .9784
216
254.5
4.5
2.121
1 .456
46
11 .509
0 .9584
0 .9983
0 .9991
222
268.8
4.625
2.151
1 .466
48
12.566
1.0
1.0
1.0
228
283.5
4.75
2.180
1 .476
50
13 .635
1 .0416
1 .0206
1 .0103
240
314.2
5.0
2.236
1 .496
74
THE FLOW OF WATER
Table VII. A.
Roots of Mean Hydraulic Radii.
R
v7
y-r
R
Vr
Tr
R
Vr
v^
0.05
0.224
0.473
2.75
1.658
1.287
5.9
2.429
1.558
0.1
0.316
0.562
2.80
1.673
1 .293
6.0
2.449
1.565
0.15
0.387
0.622
2.85
1.688
1.299
6.1
2.470
1.571
0.20
0.447
0.668
2.90
1.703
1 .305
6.2
2.490
1.578
0.25
0.5
0.707
2.95
1.718
1.311
6.3
2.510
1.584
0.30
0.548
0.740
3.0
1 .732
1.316
6.4
2.530
1.590
0.35
0.592
0.769
3.05
1.746
1 .322
6.5
2.550
1 .597
0.40
0 .632
0.803
3.10
1.761
1 .327
6.7
2.588
1.609
0.45
0.671
0.819
3.15
1.775
1.332
6.8
2.608
1 .615
0.5
0.707
0.841
3.20
1.789
1.338
6.9
2.627
1.621
0.55
0.742
0.861
3.25
1.803
1.343
7.0
2.644
1.624
0.60
0.775
0.881
3.30
1.817
1.347
7.1
2.665
1 .630
0.65
0.806
0.898
3.35
1.830
1.352
7.2
2.683
1.637
0.70
0.837
0.914
3.40
1.844
1.358
7.3
2.702
1.643
0.75
0.866
0.930
3.45
1.857
1.363
7.4
2.720
1.649
0.80
0.894
0.946
3.50
1.871
1.368
7.5
2.739
1.655
0.85
0.922
0.960
3.55
1.884
1.373
7.6
2.757
1.661
0.90
0.949
0.974
3.60
1.898
1.378
7.7
2.775
1.664
0.95
0.975
0.987
3.65
1.910
1.382
7.8
2.793
1.670
1.0
1.0
1.0
3.70
1.924
1.387
7.9
2.811
1.676
1 .05
1.025
1.012
3.75
1.936
1.392
8.0
2.828
1.682
1.10
1.049
1.024
3.80
1.949
1.396
8.1
2.846
1.688
1.15
1 .072
1.036
3.85
1.962
1.401
8.2
2.868
1.692
1.20
1 .095
1.047
3.90
1.975
1.405
8.3
2.881
1.697
1.25
1 .118
1.057
3.95
1.987
1.410
8.4
2.898
1.702
1 .30
1 .140
1.068
4.0
2.0
1.414
8.5
2.915
1.707
1.35
1.162
1.079
4.05
2.012
1.419
8.6
2 933
1.712
1.40
1.183
1.088
4.10
2.025
1.423
8.7
2.950
1.717
1.45
1 .204
1.097
4.15
2.037
1.427
8.8
2.966
1.722
1.50
1.225
1.107
4.20
2.049
1.432
8.9
2.983
1.727
1.55
1 .245
1.115
4.25
2.062
1.436
9.0
3.0
1.732
1.60
1.265
1 .125
4.30
2.074
1.440
9.1
3.017
1 .737
1.65
1 .285
1.133
4.35
2.086
1.444
9.2
3.043
1.741
1.70
1.304
1.142
4.40
2.098
1.448
9.3
3.056
1.746
1.75
1 .323
1.150
4.45
2.111
1.453
9.4
3.066
1.750
1.80
1 .342
1.158
4.50
2.121
1.457
9.5
3.082
1.755
1.85
1 .360
1 .166
4.55
2.133
1.461
9.6
3.098
1.760
1.90
1 .378
1 .174
4.60
2.145
1.466
9.7
3.114
1.764
1.95
1.396
1.182
4.65
2.156
1.469
9.8
3.130
1.769
2.0
1.414
1.189
4.70
2.168
1.473
9.9
3.146
1.773
2.05
1.432
1.196
4.75
2.179
1.476
10.0
3.162
1.777
2.10
1.449
1.204
4.80
2.191
1.480
10.5
3.240
1.800
2.15
1.466
1 .211
4.85
2.202
1.484
11.0
3.317
1.820
2.20
1.483
1.218
4.90
2.214
1.488
11.5
3.391
1.845
2.25
1 .5
1.225
4.95
2.225
1.492
12.0
3.464
1.860
2.30
1 .517
1.232
5.0
2.236
1.495
12.5
3.536
1 .880
2.35
1 .533
1.238
5.1
2.258
1.503
13.0
3.606
1.899
2.40
1 .549
1 .245
5.2
2.280
1.511
13.5
3.674
1.918
2.45
1 .565
1 .251
5.3
2.302
1.518
14.0
3.742
1.934
2.50
1 .581
1 .257
5.4
2.324
1.526
14.5
3.808
1.951
2.55
I .597
1 .263
5.5
2.346
1.532
15.0
3.873
1.968
2.60
1.612
1.270
5.6
2.366
1 .539
15.5
3.937
1 .984
2.65
1 .628
1.276
5.7
2.387
1 .548
16.0
4.0
2.0
2.70
1.643
1.282
5.8
2.408
1 .552
PRACTICAL APPLICATIONS OF THE FORMULA 75
TABLE VIII.
Table VIII contains the practically most useful coefficients
indicating the degree of roughness of a conduit.
In the design of a new conduit it is well to remember, that the
degree of roughness of a conduit is not a permanent quantity.
Conduits lined with cement, smooth concrete, good brickwork,
planed boards, metals, etc., gradually deteriorate and assume a
degree of roughness which closely resembles that of a sawed
board (m = 0.68) , in case of sewers that of common brick work
(m = 0.57) , in case of large riveted pipes that of very rough
brick work (m = 0.45) .
If the velocity is feeble, or the flow often interrupted, crypto-
gamic plants sooner or later appear on the walls of open conduits
and rust or calcareous matter coates the walls of pipes. In
such a condition the degree of roughness corresponds to that of
very rough brick work {m = 0.45) .
If left to themselves, channels in earth of all descriptions
likewise deteriorate and gradually assume a degree of roughness
corresponding to that of a natural channel {k = 1.93 in most
cases) .
For artificial channels in earth Table VIII gives values of
both m and k. Owing to the abnormally rapid decreases in the
value of c with the decrease of the depth of the water in rough
channels in earth a negative value of m gives better results
than k.
The k formula, however, gives good results in all cases where R
is greater than one foot.
The relation between K and m and the coefficient n of the for-
mula of Kutter is given by
_ 1 + i^ _ 0.02
^ ~ 100 1 + m *
76 THE FLOW OF WATER
Table VIII.
Values op m and k, the Coefficients Indicating the Degree of Rough-
ness. A. Conduits under Pressure.
Description of Conduit.
1.0
0.95
0.83
0.68
0.57
0.53
0.45
0.30
0.20
New, straight tin or plated pipes.
Pipes of planed boards or clean cement, new. Very smooth new
asphalt-coated cast and wrought-iron pipes. New asphalt-coated
riveted pipes not exceeding 6 inches in diameter.
Ordinary new asphalt-coated cast and wrought-iron pipes. Wrought-
iron pipes not coated, new. Glass and lead pipes. Pipes lined with
smooth concrete or cement plaster.
Pipes lined with cement or smooth concrete, pipes of planed or rough
boards, cast and wrought-iron pipes, coated or not coated, steel
and wrought-iron riveted pipes not exceeding 3 feet in diameter
(all some time in use but fairly clean).
Sewer pipe. Conduits lined with common brickwork or rough concrete.
New asphalt-coated steel-riveted pipe exceeding 3 feet in diameter.
Conduits lined with very rough brickwork or very rough concrete.
Steel-riveted pipe exceeding 3 feet in diameter, some years in use.
Old cast and wrought-iron pipes of all descriptions, not very clean.
Old steel-riveted pipe exceeding 3 feet in diameter.
Drain tile.
B. Open Conduits.
1.0
0.95
0.83
0.80
0.70
0.57
0.45
0.30
0.15
0.0
Description of Conduit.
Conduits lined with neat cement exceptionally smooth.
New conduits lined with neat cement or planed boards.
New brick conduits washed with cement, conduits smoothly dressed
with cement mortar.
New conduits lined with smooth concrete or very good brick work.
Conduits lined with sawed boards or fairly good brick work.
Aqueducts lined with neat cement, cement plaster, smooth concrete
very good brickwork, planed boards (all some time in use).
Channels lined with common brickwork, rough concrete or smoothly
dressed ashlar masonry. Sewers lined with neat cement, smooth
concrete, brickwork washed with cement or plastered Avith cement
mortar, fairly good and very good brickwork (all some time in use.)
Channels lined with very rough brickwork or concrete, fairly good
ashlar masonry.
Channels lined with common ashlar or very good rubble masonry.
Channels lined with roughly hammered stone masonry.
Channels lined with common rubble masonry. Channels in rockwork.
PRACTICAL APPLICATIONS OF THE FORMULiE
77
Table VIII. — Continued.
C, Channels in Earth.
m
k
Description of Conduit.
0.57
0.27
Channels of very regular cross-section in stiff clay or clayey
loam.
0.15
0.74
Channels of fairly regular cross-section in fine cemented
gravel.
0.0
1.0
Channels of fairly regular cross-section in coarse cemented
gravel.
Channels in rockwork.
-0.10
1.2
Fairly regular channels in sand or sand with gravel imbedded.
-0.20
1.5
Fairly regular channels in earth, tolerably free from stones
and plants.
-0.32
1.93
Ordinary channels in earth or gravel. Channels with stones
vegetation or other impediments to flow.
Natural channels, creeks, rivers.
Table IX.
Alphabetical List of Authorities Whose Experimental Data Are
Given in Table X, Methods of Gauging and Publication Containing
Original Record of Experiments.
Author.
Description of
Channel Gauged.
Method of Gauging.
Where Recorded.
Adams, A. L.
Wooden Stave
pipes.
Discharge meas-
ured by rise in
reservoir surface,
loss of head by
open standpipes.
Engineering News.
Sept. 1898.
Baumgarten . .
Aqueduct. Bot-
tom of cement,
sides of brick.
Piezometer.
Darcy-Bazin.
Recherches hy-
drauliques.
Benzenberg . .
Brick sewer.
Floats probably.
Trans. A. S. C. E.
Bossut ....
Tin and Lead
pipes.
Discharge meas-
ured in tanks.
Hamilton Smith,
Hydraulics, 1886.
Bruce
Aqueduct. Con-
crete.
Discharge meas-
ured by rise in
reservoir surface.
Proceedings of
Institute of C. E.
London, 1896.
Brush
Cast-iron pipes.
Quantities meas-
ured at pumps.
Quoted by Kutter.
''The Flow of
Water."
Clarke
Brick sewer.
Discharge meas-
ured by rise in
reservoir surface.
H. Smith. Hy-
draulics, 1886.
78
THE FLOW OF WATER
Table IX. — Continued.
Author.
Description of
Channel Gauged.
Method
of Gauging.
Where Recorded.
Cunningham . .
Aqueduct of
masonry. Ganges
Canal.
Velocities meas-
ured by one-inch
tin rod floats.
Roorkee Hydr. Ex-
periments, 1880.
Darcy
Pipes.
Discharge meas-
ured in tanks, loss
of head by piezo-
meter or mercury
column.
Experiments sur
le mouvement de
I'eau dans les
tuyaux.
Darcy-Bazin . .
Cement and con-
crete conduits.
Discharges meas-
ured by orifices
previously tested.
Recherches hydr.
Paris 1865.
Darcy-Bazin . .
Conduits of planed
and rough boards,
or lined with brick.
Discharges meas-
ured by orifices
previously tested,
20 centimeter
square.
Recherches hydr.
Paris, 1865.
Darcy-Bazin . .
Canal lined with
ashlar masonry.
Piezometer.
Recherches hydr.
Paris, 1865.
Darcy-Bazin . .
Tailrace lined with
ashlar masonry.
Discharge meas-
ured by orifice
50 centimeter
square.
Recherches hydr.
Paris, 1865.
Darcy-Bazin . .
Tunnel lined with
ashlar masonry.
Current meter
and reservoir.
Recherches hydr.
Paris, 1865.
Darcy-Bazin . .
Section of Gros-
bois canal lined
with masonry.
Current meter
and reservoir.
Recherches hydr.
Paris, 1865.
Darcy-Bazin
Chazilly Canal.
Piezometer, cur-
rent meter and
reservoir.
Recherches hydr.
Paris, 1865.
Darcy-Bazin
Grosbois Canal.
Piezometer, cur-
rent meter and
reservoir.
Recherches hydr.
Paris, 1865.
Dubuat
Canal du Jard.
Surface floats.
Principes hydr.
Paris, 1786.
Ehman ....
Galvanized and
cast-iron pipes.
Discharges meas-
ured by volumes.
Iben Druckho-
henverlust.
Fanning ....
Cement lined pipe.
Weir measure-
ment probably.
Water Supply
Engineering.
PRACTICAL APPLICATIONS OF THE FORMULA
79
Table IX. — Continued,
Author.
Description of
Channel Gauged.
Method
of Gauging.
Where Recorded.
Fteley & Stearns
Sudbury Conduit.
Brick coated with
cement and not
coated.
Weir measure-
ment.
H. Smith.
Hydraulics, 1886.
Fortier ....
Irrigation Chan-
nels.
Current meter.
U. S. Geol. Sur-
vey. Irr. Papers,
1901.
Hawks ....
Steel-riveted pipe.
Weir measure-
ment.
Tr. A. Soc. C. E.,
1899.
Herschel . . .
Steel-riveted pipes.
Discharges meas-
ured by Ventury
meter, loss of
head by Bourdon
gauges.
115 Experiments.
Horton ....
Brick sewers
washed with ce-
ment.
Weir measure-
ment probably.
Eng. News.
Hubbel & Fenkell
Cast-iron ppes.
Tr. A. Soc. C. E.,
1898.
Iben
Cast-iron pipes.
Pipes coated with
tar.
Discharges meas-
ured by volumes,
loss of head
by pressure
gauges.
Druckhohenver-
lust.
Kuichling . . .
Riveted pipes.
Cast-iron pipes.
Quantities meas-
ured by rise in
reservoir surface,
loss of head by
mercury gauges.
Marx-Wing &
Hoskins, Tr. A. S.
C. E., 1898-1899.
Kutter ....
Channels lined
with rubble ma-
sonry.
Surface floats.
Die neue Theorie.
La Nicca . . .
Alpine Streams.
Surface floats.
Kutter, Die neue
Theorie.
Lampe ....
Cast-iron pipes.
Discharges meas-
ured by reservoir
contents, loss of
head by pressure
gauges.
Iben.
Druckhohenver-
lust.
Legler . . .
Canals.
Rodfloats.
Hydrotechnische
Mittheilungen.
80
THE FLOW OF WATER
Table IX. — Continued.
Author.
Description of
Channel Gauged.
Method
of Gauging.
Where Recorded.
McDougall . .
Irrigation chan-
nels.
Current meter.
U. S. Geolog, Sur-
vey Irr, Papers,
Marx-Wing &
Hoskins . . .
Riveted pipe.
Stave pipe.
Discharges meas-
ured by Venturi
meters, loss of
head by mercury
gauges.
Trans, A. S. C. E„
1898.
Noble
Stave pipes.
Trans. A, S. C, E„
1902,
Passini & Gioppi
Aqueduct, Bot-
tom of concrete,
sides of brick.
Current meter.
Giomale del.
Genio Civile. Ro-
ma, 1893.
Passini & Gioppi
Syphon aqueduct
of brick.
"
"
Passini & Gioppi
Canal Cavour.
a
(C
Perrone ....
Aqueduct coated
with clean cement.
K
Zoppi: Sul Vol-
turno, Carte hy-
drographique d'
Perrone ....
Tunnel in rock-
work.
"
Rafter
Riveted pipes.
Discharge meas-
ured by rise in
reservoir surface,
loss of head by
piezo-meter.
Tr. A. S. C. Eng.,
XXVI.
Revy
La Plata and Pa-
rana Rivers.
Current meter.
Hydraulics of
great rivers, Lon-
don, 1881.
Rittinger . . .
Channels lined
with rubble
masonry.
Discharges meas-
ured in tanks.
Bornemann : Der
Civil Ingenieur.
Roff
Saalach River.
Piezometer.
Grebenau, Theorie
der Bewegung des
Wassers.
Rowland . . .
Wrought-iron
pipes.
Discharge meas-
ured by volumes.
Brush.
Smith, H. . . .
Riveted pipes.
Velocities meas-
ured by weirs
and Standard
orifices.
Tr. A, S, C, E.
"Hydraulics,"
N, Y,, 1886.
PRACTICAL APPLICATIONS OF THE FORMULA
81
Table IX. — Concluded.
Author.
Description of
Channel Gauged.
Method
of Gauging.
Where Recorded.
Smith, J. W. . .
Riveted pipes.
Discharge meas-
ured by weir,
loss of head by
piezometers.
Tr. A. S. C. E..
Vol. XXVI.
Steams ....
Brick aqueduct.
Current meter.
Report to the
New Croton Aque-
duct Com., 1895.
Schwartz . . .
Weser River.
Current meter.
Funk: Beit rag zur
allgem'einen
Wasserbaukunst
Lemgo, 1808.
Wampfler . . .
Canal.
Surface floats.
Kutter: Die neue
Theorie.
TABLE X.
Table X contains the most reliable experimental data from
which the general formula is deduced. For conduits under
pressure the numerical values of (a), the coefficient of variation
of c is generally given. This is done in order to show the details
of the variation of c. For open conduits the variation is generally
indicated by giving the velocity roots. These roots are found
from the formula
_ log» v^ _ - log- '^0 ^^^^
log (66 ^r + m) \/rs )^ — log. (66 ( ^r + ni)Vrs )q
The value of x being thus found the value of m is found by
putting
66 Vi
or
r . s
66 a
Vr =
- vr=
m
m
For artificial channels in earth the values of m have been
given in addition to the values of K. Owing to the abnormally
rapid decrease in the values of c with the decrease of the depth
of the water in very rough channels a negative value of m gives
better results than K. The K formula, however, gives good
results in all cases where R is greater than one foot.
82
THE FLOW OF WATER
Table X.
EXPERIMENTAL DATA.
I. Riveted Pipes
Description of Conduit.
m
L
365
d
R
1000 s
V
c
a
New straight asphalt-
0.94
0.271
0 .0677
0.27
0.328
76.7
0.80
coated wrought-iron
2.028
1.171
99.9
1.04
riveted pipe with
. . .
. . .
12.20
3.117
108.4
1.130
screw joints.
40.70
6.148
117.1
1.219
— Darcy.
106 .54
156 .05
10 .535
12 .786
124.0
124.3
1.287
1.291
Do.
0.92
365
0.643
0 .1607
0.20
1.29
5.80
12.0
29.7
121 .56
0.591
1.529
3.53
5.509
9.00
19.72
104.1
106.2
115.6
125.4
130.2
141.0
1.013
1.035
1.125
1.220
1.267
1.372
Do.
0.82
365
0.935
0.234
0.70
4.33
11 .90
28.07
1.296
3.868
6.673
10 .522
101.3
121.6
126.5
129.9
1.013
1.216
1.265
1.299
Sheet-iron riveted
0.68
700
0.911
0.228
8.50
4.712
107.1
1 .19
pipe with funnel
13.34
6.094
110.6
1 .229
mouthpiece 7.8 ft.
16.95
6.927
111.5
1.240
long.
— Hamilton Smith.
25.59
8.659
113.4
1.260
33.09
10 .021
115.5
1.283
Do.
0.68
700
1.056
0.264
6.68
4.595
109.4
1 .200
Coated with asphalt.
14.28
6.962
113.4
1.242
Funnel mouthpiece
22.19
8.646
113.0
1.237
12 ft. long.
...
...
...
...
33.18
10 .706
114.4
1.253
Do.
0.69
700
1.229
0.307
5.02
4.383
111.6
1.181
Funnel mouthpiece
10.97
6.841
119.8
1.246
14.8 ft. long.
...
12.27
16.46
24.70
32.31
7.314
8.462
10 .593
12.09
119.1
119.2
121.6
121.3
1.261
1.260
1.286
1.285
Do.
0.65
4440
1.416
0.354
66.72
20 .143
131 .1
1.395
Double riveted pipe
with some easy-
curves.
Do.
0.69
1200
2.154
0.538
16.41
12 .605
134.1
1.325
Do.
0.63
12800
2.43
0.607
11.72
10.78
127.8
1.30
Inverted syphon with
887 ft. depression.
DESCRIPTION OF CONDUIT, ETC.
88
Table X. — Continued,
Description of Conduit
Wrought-iron riveted
pipe with lap joints.
Paint coating worn
off, somewhat rusty
— Clemens- Herschel
0.54
Asphalt- coated steel-
riveted pipe. — A.
McL. Hawks.
Asphalt coated cylin-
der joint steel
pipe. — A.L.Adams.
0.55
0.65
152.9
16416 1 .33
R
8.58
1 .166
2.145
1000 s
0 .2915
0.333
0 .0079
0.032
0 .0837
0 .1557
0 .2453
0.354
0 .4991
0 .6619
0 .8470
0.50
126.9
116.6
111.9
109.4
109.0
108.2
107.0
106.2
105.6
0.4550 0.932
0 .584 1 .136
5.0
4.58
82.2
86.0
1.089
1.00
0.959
0.938
0.934
0.928
0.917
0.910
0.905
0.98
1.026
110.0
1.183
Asphalt- coated cyHn-
der joint steel-riveted
pipe with curves.
— E. Kuichling.
0.57
91641
3.166
0 .7915
1.01
0.99
3.23
3.27
114.0
116.6
1.14
1.166
Asphalt- coated taper
joint steel- riveted
pipe. New. — Clem-
ens-Herschel.
0.56
81139
3.5
0.875
0.112
1.0
2.0
3.0
4.0
5.0
6.0
101.0
104.3
106.4
107.8
108.4
108.5
1.0
1.032
1.053
1.067
1.073
1.074
Do.
0.54
5574
3.5
0.875
0.13
.0
.0
3.0
3.5
4.0
5.0
6.0
96.0
107.9
112.6
113.0
112.8
110.8
110.0
0.96
1.08
128
132
130
111
1.102
Do.
Cylinder joint, many
curves.
0.53
24000
4.0
1.0
0 .0976
1.0
2.0
3.0
3.5
4.0
5.0
6.0
101.2
108.3
112.8
113.4
113.2
112.0
111 .6
0
07
113
119
118
105
.091
Asphalt-coated cylin-
der joint steel-
riveted pipe. — J. W.
Smith.
0.80
0.74
39809
34176
2.916
2.75
0.584
0.55
1.31
1.31
3.52
3.96
126.8
123.2
1.15
1.166
84
THE FLOW OF WATER
Table X. — Continued.
Description of Conduit.
Asphalt-coated butt-
jointed steel-riveted
pipe with many
curves, — Marx- Wing.
m
0.50
L
4367
d
6.00
R
1000 s
V
c
1.5
0.07
1.08
108.0
0.16
1.57
114.0
0.24
2.14
113.0
0.559
2.59
110.0
0.495
3.02
112.0
. . .
0.776
3.84
113.0
1.0
1.055
1.046
1.018
1.037
1.046
II. Old Riveted Pipes.
Cylinder joint asphalt-
0.31
45400
2.0
0.50
3.83
3.32
76.0
1.0
coated steel- riveted
....
3.58
3.32
78.0
1.030
pipe. Fourteen
3.46
3.35
80.5
1.06
years in use.
—George W. Rafter.
Do.
0.29
45400
3.0
0.75
0.45
0.43
1.47
1.49
80.4
83.0
1.0
Do.
0.59
45400
3.17
0 .7915
1.59
3.88
109.3
1.08
One year in use. — E.
1.61
3.91
109.3
1.08
Kuichling
...
...
1.62
3.90
109.1
1.08
Taper joint, steel
0.26
24648
4.0
1.0
. 1.0
78.0
0.94
riveted. Four years
.
. . .
. 1.5
84.6
1.019
old . — Clemens-Her-
,
. 2.0
89.6
1.080
schel.
...
. 2.5
. 3.0
. 3.5
. 4.0
. 5.0
. 6.0
92.4
93.0
93.2
94.2
94.4
94.9
1 .113
1 .120
1 .121
1.135
1.137
1.143
Cylinder joint steel-
0.47
24600
4
1.0
. 1.0
97.2
1.0
riveted pipe, four
. 1.5
100.8
1.024
years in use. — C.
. 2.0
103.3
1.062
Herschel.
...
...
...
. 2.5
. 3.0
. 3.5
. 4.0
. 5.0
104.9
105.3
104.8
104.0
103.9
1.079
1.083
1.079
1.069
1.066
.
. 6.0
103.7
1.066
III. New Wrought-iron Pipes, not coated.
Straight
Darcy.
pipe.
0.83
372
0 .0873
0 .0218
0.33
10.15
43.48
105 .71
309 .52
0.19
1.207
2.612
4.203
7.166
70.7
81.1
84.8
87.5
87.2
0.877
1.006
1 .052
1.098
1.094
DESCRIPTION OF CONDUIT, ETC.
85
Table X. — Continued.
Description of
Conduit.
m
L
d
R
1000 s
V
c
76.9
a
Straight pipe.
0.83
372
0 .1296
0 .0324
0.22
0.205
0.824
— Darcy.
...
...
3.36
23.89
123.15
224 .08
0.858
2.585
6.300
8.521
82.3
92.9
99.8
100.0
0.989
1 .128
1.212
1 .215
Do.
0.83
31.0
0 .0833
0 .0208
6258 .6
36.1
100.0
1 .240
Rowland.
31.0
8935 .5
43.4
100.6
1 .248
31.0
10741 .9
48.1
101 .7
1.261
97.0
0 .0833
0 .0208
2000 .0
2855 .6
3432 .9
19.9
24.5
27.2
97.5
100.5
101 .7
1 .209
1.247
1.261
IV. Pipes Coated with Tar.
New cast-iron pipe.
0.66
415
0.335
0.084
1.98
1.0
79.0
1 .0
— Iben. 1876.
4.11
1 .70
92.0
1 .164
Quoted by Kutter.
6.56
7.83
2.10
2.30
90.0
90.0
1 .139
1 .139
...
11.07
2.80
91.0
1.152
Do.
0.56
1093
0.50
0.125
4.59
11.62
16.21
22.32
30.27
2.00
3.30
3.90
4.80
5.30
82.0
87.0
88.0
92.0
87.0
1.080
1 .144
1 .156
1 .210
1.144
Do.
0.65
...
1795
1.001
0.25
1.46
1.830
2.19
3.84
6.03
1.60
2.10
2.60
3.80
4.80
85.0
97.0
112.0
121.0
125.0
0.944
1.080
1 .244
1 .344
1.388
Do.
0.69
3514
1.667
. . .
0.417
0.12
0.48
0.76
1.21
0.70
1.60
1 .90
2.50
105.0
110.0
109 .0
109.0
1.077
1.125
1 .115
1.115
V. Asphalt-Coated, Wrought and Cast-Iron Pipe. New,
Asphalt-coated
0.89
60
0 .0875
0 .0218
26.93
2.22
91.6
1.095
wrought-iron pipe
52.19
3 .224
95.5
1.140
with funnel mouth-
103 .38
4.761
100.2
1 .189
piece, — Hamilton
130 .64
5.443
101 .9
1 .205
Smith.
Asphalt-coated cast-
0.90
366
0 .4495
0.1124
0.24
0.489
94.1
0.960
iron pipe. — Darcy.
...
4.25
22.25
98.52
167 .56
2.503
5.623
11 .942
15 .397
108.4
112.5
113.5
112.2
1 .107
1.150
1.160
1.150
86
THE FLOW OF WATER
Table X. — Continued.
Description
of Conduit.
m
L
d
R
1000 s
V
c
a
Asphalt-coated pipe
five years old, in
0.83
26000
1.373
0.343
0.594
1.577
110.5
1.072
1.376
2.479
114.1
1 .107
good condition.
. . .
1.63
2.709
114.6
1.112
— Lampe.
1.95
3.090
119.4
1.162
Cast-iron pipe. —
0.81
365
0 .6168
0 .1542
3.68
2.487
104.4
1 .107
Darcy.
22.50
109 .80
145 .91
6.342
14 .183
16.168
107.7
109.0
107.8
1.142
1.155
1.142
Asphalt-coated pipe,
four years in use.
0.80
810
0.662
0.166
0.367
0.73
92.7
0.984
0.850
1.12
94.7
1 .01
— Ehmann.
1.332
1.883
1.45
1.69
97.9
96.0
1 .039
1.017
Asphalt-coated cast-
0.83
1.0
0.25
1.0
101.5
1.0
iron pipe. — Hub-
bel and Fenkel.
2.0
109.6
1.08
3.0
114.6
1.13
4.0
118.3
1.166
...
5.0
121.5
1.196
Cast-iron pipe. —
0.77
365
1 .6404
0.41
0.45
1.472
108.4
1 .047
Darcy.
1.20
2.10
2.60
2.602
3.416
3 .674
117.3
116.4
112.5
1 .135
1.126
1.090
Cast-Iron Force
0.80
75000
1.667
0.417
0.733
2.0
114.4
1 .08
Main. Large num-
.. .
0.880
2.24
117.0
1 .105
ber of summits.
. . .
. .
1.026
2.36
114 .1
1.080
angles and curves.
1 .187
2.52
113.3
1.071
amongst which
1.333
2.68
113.7
1.071
there are four
1.493
2.76
110.6
1 .045
right angles and
ten quadrants of
1.64
2.92
111.7
1 .055
1.800
3.0
109.5
1 .035
30 ft. radius.—
Brush.
Asphalt-coated cast-
0.97
1747
4.0
1.0
0.318
2.616
146.7
r'^
iron pipe. Some
. . .
. . .
0.711
3.738
140.1
1.077
easy vertical cur-
1.221
4.965
142.1
1.093
ves. — Stearns.
1.849
6.195
144.1
1 .109
VI. Old Cast and Wrought-Iron Pipes.
Old cast-iron pipe
— Darcy.
0.52
366
0 .2628
0 .0657
0.84
0.458
62.0
7.25
1.463
67.3
16.10
2.224
68.7
45.35
3.777
68.9
0.94
1.04
1 .062
1.065
DESCRIPTION OF CONDUIT, ETC.
87
Table X. — Continued.
Description of
Conduit.
m
L
d
R
1000 s
V
0.633
2.014
2.835
5.007
c
a
Old cast-iron pipe,
cleaned. — Darcy.
0
85
...
...
...
0.84
7.23
15.57
44.73
85.2
92.4
88.6
93.4
1 .00
1 .08
1 .026
1.092
Old cast-iron pipe.
— Darcy.
0
45
365
0.798
0 .1995
0.94
4.73
22.90
41 .05
139 .81
1 .007
2.32
5.095
6.801
12 .576
73.6
75.5
75.1
75.2
75.3
1.0
1.023
1 .02
1.02
1.02
Old cast-iron pipe,
twelve years in use.
Slightly tubercu-
lated. — Iben.
0
45
541
1.0
0.25
2.24
2.84
1.79
2.03
75.7
76.2
1.0
1.01
Do., two years in use,
slightly incrusted.
0
56
2149
1.0
0.25
0.26
0.41
0.81
1.28
2.99
0.60
0.80
1.20
1.60
2.40
74.5
81.0
85.0
92.0
86.0
0.878
0.962
1.01
1.092
1.021
Do . , fourteen years in
use, slightly in-
crusted.
0
39
•
7179
1.00
0.25
0.42
1.65
4.44
9.43
0.70
1 .60
2.70
3.90
71.0
78.0
80.0
80.0
0.979
1 .075
1 .133
1 .133
Do., fifteen years in
use. Heavily in-
crustated.
0
30
1808
1.00
0.25
0.65
3.76
6.12
7.13
0.90
1.80
2.30
2.60
67
58
59
58
0.991
0.860
0.875
0.860
Do., twenty-two years
in use. Very heav-
ily incrustated.
0
05
1736
1 .00
0.25
1 .08
4.29
10.91
23.86
0.80
1 .50
2.40
3.50
50
44
45
46
1.041
0.179
0.937
0.958
New asphalt-coated
cast-iron pipe.
Rochester, N.Y.—
E.Kuichling,1895.
0
83
3
. . .
0.75
1.38
1 .50
1 .50
4.204
4.234
4.234
129.4
125 .25
125 .25
FtV
Do.
1897.
0
45
2.27
4.82
4.85
4.128
4.045
4.022
91.22
66.84
66.24
1.0
Do.
1898.
0
13
4.34
3.76
4.034
4.026
70.23
75.32
1 .0
Do.
1899.
0
28
...
...
...
3.44
3.25
4.084
4.079
79.79
81 .93
1.0
88
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
0.95
L
d
R
1000 s
V
c
a
Old cast-iron pipe
16yearsold,tuber-
cles removed. —
Fitzgerald.
...
4
1.0
0 .4167
1.241
1 .8283
3.723
4.973
6.141
139.1
141.1
143.6
vh
Cast iron intake
pipe at Erie, Pa.,
8 years in use.
0.48
8215
5
1.25
0.178
1.088
99.8
102.1
1.0*
Old cast-iron pipe in
good condition,
some easy bends.
— Jas. M. Gale.
0.70
19600
4
1
0.947
3.458
112.4
1.0
VII. Galvanized Pipes, Glass, Tin and Lead Pipes.
New wrought-iron 0 .£
12
301.8
0 .0842
0.021
7.61
1.11
87.1
1.016
galvanized pipe, . .
29.35
2.13
85.9
1.00
st ra ight. — Eh- . .
113.04
3.71
79.2
1.035
mann.
225.0
239 .13
5.80
5.90
84.5
83.2
0.984
0.970
Glass pipe with fun- 0 .8
M
63.9
0 .0764
0 .0191
25.01
1.955
89.5
1.078
nel-mouthpiece.— ..
50.77
2.945
94.6
1.140
Hamilton Smith.
75.30
102.6
129.18
3.685
4.383
5.009
92.2
99.3
100.8
1 .171
1 .199
1.219
Do., no funnel. 0 .8
^4
63.9
0 .0764
0 .0191
17.97
132 .51
1.398
4.373
83.6
96.3
1.030
1.185
Glass pipe, straight. 0 .^
54
147.0
0.163
0 .0407
0.96
0.502
80.3
0.930
— Darcy.
;::
7.71
57.62
111 .91
1.591
4.849
6.916
89.8
100.1
102.4
1.044
1.164
1.191
New lead pipe, 0 .fi
14
172
0 .0886
0 .0221
0.44
0.213
68.3
0.843
straight. — Darcy. . .
...
::;
8.14
54.36
146 .32
1 .089
3.35
5.509
81.1
96.5
96.8
1.00
1.191
1.195
New lead pipe, 0 .S
54
172
0 .1345
0 .0336
0.82
0.394
75.0
0.90
straight.— Darcy. ..
. . .
7.48
56.00
158 .82
1 .404
4.318
7.562
86.8
99.5
103.5
1.038
1.178
1.238
Tin pipe, straight. 0 .£
18
0 .0888
0 .0222
0.196
0.141
67.6
0.746
— Dubuat.
_
•
0.641
3.91
5.39
7.54
9.91
13.7
29.82
30.31
99.01
0.322
0.772
0.927
1.183
1.342
1.476
2.546
2.606
5 .223
85.3
82.8
84.8
91.4
90.5
92.9
98.9
100.4
111 .4
0.942
0.914
6.937
1.009
1.00
1 .015
1.092
1.109
1 .120
DESCRIPTION OF CONDUIT, ETC.
Table X. — Continued.
89
Description of
Conduit.
m
L
d
R
1000 s
V
t
a
Tin pipe. Straight.
— Bossut.
0.95
192
192
64
32
32
0 .1184
11
1 .0296
a
((
5.40
10.76
15.08
26.94
52.98
1.116
1.678
2.075
2.946
4.31
88.2
94.0
98.2
104.3
108.8
1.00
1.068
1.116
1.185
1.234
Do.
0.94
63
126
189
0.1184
IC
(C
0 .0296
a
113.4
113.5
113.4
6.143
6.15
6.157
106.0
106.1
106.2
1 .220
1 .220
1.220
VIII. Pipes and Open Conduits op Planed or Rough Boards.
Redwood stave pipe.
0.93
4-8000
1 .166
0.2C
)2 0 .17
0.698
99
0.908
LosAngles, Cal.—
. 0 .161
0.698
101
0.926
A.L.Adams,1898.
. 0.178
. 0 .145
0 .391
. 0 .638
. 1 .355
0.751
0.691
1 .167
1 .531
1.181
104
105
109
112
113
0.953
0.963
1.01
1 .027
1 .043
Do.
0.96
4188
1 .5
0.37
5 2.07
3.605
132.9
yi
At Astoria, Ore» —
A. L. Adams.
Wooden stave pipe,
0.50
3.67
0.91
7 1 .067
3.468
110.1
V-
at Cedar River,
•
. 1 .134
3.522
108.6
Wash. Long but
. 1 .191
3.685
110.9
easy curves. Sev-
. 1 .262
3.853
112.6
eral years in use.
. 1.33
3.964
112.9
Some slight de-
posits.— Theron A.
Noble.
. 1 .331
3.972
113.1
. 1 .401
4.072
112.9
. 1 .627
4.415
113.7
. 1 .757
4.595
113.8
. 1 .757
4.635
114.8
. 1 .888
4.831
115.6
Do.
0.58
4.5
1.12
5 0 .342
2.282
116.8
F^
. 0 .342
2.276
115.8
.
. 0 .436
2.65
119.4
. 0 .558
3.07
122.1
. 0 .557
. 0 .672
. 0 .783
. 0 .856
. 0 .983
. 1 .076
. 1 .162
3.05
3.41
3.724
3.924
4.215
4.42
4.69
121 .4
123.7
125 .2
126.2
126.5
126.7
129.2
90
THE FLOW OF WATER
Table X.
Continued.
Description of
Conduit.
m L
d
R
1000 s
V
c
a
Wooden stave pipe
0 .51 4 .000
6.0
1.5
1.40
110.5
vh
at Ogden, Utah.
. . .
1.68
112.5
Many easy curves,
2.14
115.0
two years in use
. . .
. . .
. . .
2.43
119.5
— Marx-Wing and
2.96
122.5
Hoskins.
3.59
3.63
126.5
124.5
Rectangular pipes
0.68 145.7
0.319
0.523
1.23
94.3
■|7tV
of unplaned board.
1.067
1.778
96.4
— Darcy.
1 .933
2.733
3.867
6.267
7.267
8.80
2.267
2.939
3.529
4.349
4.625
5.307
96.8
99.5
100.5
97.3
96.1
100.2
Rectanglar pipe of
0 .73 230
0.505
0.475
1.67
107.6
|7tV
unplaned boards.
—Darcy.
1 .076
2.52
108.1
... ...
1.90
3.37
108.9
...
2.91
4.23
110.2
4.27
5.07
109.1
5.06
5.52
109.3
5.76
5.91
109.7
6.61
6.37
110.3
Provo Canal Flume,
0 .85 ...
1.45
1.0
5.67
147.7
7TJ
7T2
Utah. Semicircu-
0.81 ...
1.46
1.0
5.37
141 .8
lar conduit of
planed staves.
several years in
use.— W. B. Mc-
Dougall.
...
Wooden trough,
W
0 .71 ...
0.24
0.159
34.3
8.26
111.9
7Ti
trapezoidal. Bot-
0.26
0.173
((
8.21
106.6
tom width 10.4
0.38
0.237
((
10.11
112.1
ft. Rittinger.
0.41
0.246
(I
10 .64
115.8
Rectangular test
0.94 0.328
0.04
0.029
4.7
0.90
76.5
vh
channel of planed
0.08
0.052
i(
1.30
83.0
boards. — Darcy-
0.11
0.066
({
1.58
89.4
Bazin. Series 28.
0.14
0.075
(C
1.74
92.7
0.17
0.084
il
1 .94
97.6
0.20
0.091
tc
2.11
102.1
0.22
0.093
it
2.16
103.2
DESCRIPTION OF CONDUIT, ETC.
91
Table X. — Continued.
Description of
Conduit.
m
M
r d
R
1000 s
V
c
a
Semicircular test
0.70
3.
16 0 .63
0.39
1.5
2.61
107.8
Vr\
channel of un-
3.(
32 0 .88
0.537
((
3.23
113.8
planed boards.
3.<
B9 1 .07
0.632
it
3.71
120.6
— Darcy-Bazin.
.
4.(
38 1 .24
0.717
{(
4.04
123.0
Series 26.
.
4.
24 1 .40
0,796
(C
4.25
123.2
4..
53 1 .55
0.856
(I
4.51
125.8
4.'
i3 1 .68
0.926
11
4,64
124.7
4.'
i8 1 .79
0.964
11
4,87
128.2
4.
53 1 .92
1.005
(I
5,0
128.2
4..
56 2 .02
1.054
CI
5.18
130.3
4.
59 2 .14
1.096
((
5.29
130.4
4.
59 2 .24
1.129
(I
5.45
132.3
4.,
59 2 .29
1.148
((
5.54
133.5
Rectangular test
0.66
6.,
53 0 .26
0,24
2.08
2.08
93.2
F'
V
channel of un-
0.41
0.363
ii
2.69
97.8
planed boards. —
Darcy-Bazin.
. 0.53
0.453
IC
3.16
102.8
.
. 0.63
0.528
(t
3.53
106.5
.
Series 6.
•
. 0.73
. 0.81
0.90
. 0.99
. 1.06
. 1.14
, 1.20
. 1.28
0.601
0.648
0.704
0.759
0.801
0.846
0.880
0.992
(I
t(
(C
(C
it
It
It
l(
3.78
4.13
4.34
4.51
4,72
4.88
5.09
5.21
106.9
112.5
113.5
113.5
115.8
116.3
119.0
118.9
•
Do.
0.70
6.,
53 0 .20
0.188
4.9
2.71
89.3
yA
Series 7.
. 0.30
. 0.38
. 0.46
. 0.53
. 0.60
. 0.66
. 0.72
. 0.78
. 0.83
, 0.89
0.272
0.342
0.402
0.453
0.504
0.547
0.587
0.628
0.662
0.698
{(
t(
It
tc
it
(t
It
tt
tt
3.70
4.35
4.85
5.29
5.61
5.93
6.23
6.45
6.71
6.90
101.2
106.2
109.4
112.2
113.0
114.5
116.1
116.4
117.8
117.9
0.94
0.727
tt
7,15
119.8
Rectangular test
0.71
6.,
53 0.15
0.147
8.24
3,52
100.4
yxV
channel of un-
. 0 .25
0.231
tt
4.42
101.4
planed boards.
. 0.32
0.289
tt
5.23
107.1
Darcy-Bazin.
. 0.38
0.341
tt
5.83
109,8
Series 8.
. 0.45
0.50
. 0.54
0.60
0.65
, 0.69
0.74
. 0.78
0.393
0.431
0.466
0.506
0.541
0.572
0.604
0.630
tt
tt
tt
tt
tt
tt
tt
tt
6.24
6.74
7.07
7.44
7.73
8.03
8.26
8.57
109.7
113.1
115.8
115.2
115.8
116.9
117.1
119.0
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
W
d
R
1000 s
V
c
a
Triangular test
0.69
1.85
0.92
0.327
4.9
4.13
103.1
FtV
channel of un-
2.39
1.19
0.422
<<
5.02
110.4
planed boards. —
Darcy-Bazin.
.
2.79
1.40
0.494
ti
5.56
113.0
3.10
1.55
0.549
ct
6.03
116.2
Series 23.
3.38
3.64
3.86
4.07
4.26
4.43
4.61
4.75
1.69
1.82
1.93
2.03
2.13
2.22
2.30
2.37
0 .597
0.643
0.683
0.719
0.752
0.783
0.814
0.839
a
(I
t(
((
6.36
6.59
6.83
7.03
7.23
7.40
7.54
7.75
117.6
117.3
118.0
118.4
119.0
119.5
119.4
120.9
Flume of Kem River
0.72
8.0
2.5
1.538
1.5
6.681
139.1
Vt\
Power Plant No.
3.0
1.714
«
6.968
137.4
1. Plain boards,
. .
. . .
3.5
1.866
<<
7.394
139.8
seams covered
4.0
2.00
((
7.920
144.8
with ^ inch bat-
tens. Sect. rect.
Length 1029. 6 feet.
Tunnel lined with
cement plaster,
1 cement to 2
sand of same sec-
tion and slope be-
low flume.
— F.C. Finkle.
IX. Pipes and Open Conduits of Cement or Concrete.
Cement lined pipe of
0.8:
L
5 8171
1.667
0.416
0.23
0.949
97.4
0.92
of wrought iron.
0.44
1.488
109.8
1.046
Three-stop valves
0.73
1 .925
110.7
1 .054
and two large
1.04
2*. 329
112.0
1.066
branches on line.
1 .34
2.598
110.1
1.046
— Fanning.
1.58
1.99
2.28
2.72
3.0
3.20
2.867
3.271
3.439
3.741
3.920
4.040
111 .7
113.5
111.7
111 .1
110.8
110.6
1.063
1.081
1.063
1 .056
1.052
1.051
Test pipe of clear
0.9^
) ...
2.624
0.656
0.625
2.78
137.1
yT2
cement. Diame-
1.05
3.65
139.2
1 .139
ter 0.8 meter. —
1.375
4.20
139.5
1.140
Dijon.
1 .725
4.72
140.4
1.141
Quoted by Bazin.
••
1.75
1.88 •
2.57
3.27
4.79
4.92
5.81
6.58
141.2
141.4
141 .4
142.5
1.155
1 .157
1.157
1.166
DESCRIPTION OF CONDUIT, ETC.
93
Table X. — Continued.
Description of
Conduit.
m
W
d
R
1000 s
V
c
a
Semicircular test-
1.0
2.874
0.59
0.366
1 .5
3.02
128.9
yT2
channel of clear
3.294
0.83
0.503
((
3.82
135.6
cement. — Darcy-
3.563
1 .03
0.605
"
4.16
138.0
Bazin. Series 24.
3.707
1.18
0.682
((
4.60
143.7
3.832
1.34
0.750
a
4.87
145.1
3.924
1 .47
0.809
"
5.12
147.1
3.97
1 .61
0.867
i(
5.29
146.7
4.05
1 .72
0.915
"
5.51
148.8
4.075
1 .83
0.949
i(
5.75
152.5
4.095
1 .94
0.992
(C
5.91
153.3
4.101
2.05
1 .029
tt
6.06
154.2
4.16
2.08
1 .034
((
6.11
155.1
Rectangular test
0.95
5.94
0.18
0.168
4.9
3.34
116.5
yA
channel of clear
0.28
0.251
4.39
125.1
cement. — Darcy-
0.36
0 .322
5.04
126.9
Bazin. Series 2.
•
••
0.43
0.56
0.56
0.63
0.69
0.76
0.80
0.86
0.91
0.375
0.43
0.475
0.518
0.558
0.595
0.632
0.665
0.696
5.68
6.06
6.51
6.83
7.12
7.41
7.63
7.86
8.07
132.4
132.4
135.1
135.5
136.2
137.2
137.2
137.8
138.2
Sudbury conduit.
0.94
8.6
3.071
1 .863
0 .1600
2.529
146.2
yxV
Plaster of pure
3.574
2.048
0 .1596
2.672
147.9
cement over brick
3.768
2.111
0 .1580
2.805
153.9
work. Sides
nearly vertical,
bottom flat seg-
mental arch. —
Fteley & Steams.
Aqueduct of the Se-
0.98
5.38
1.41
0.50
4.06
152 .5
yrV
rino, Naples. Pure
cement, polished.
Sides vertical, bot-
tom ellyptical
arch. — Perrone,
1896.
1
94
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
W
d
R
1000 s
V
c
a
Semicircular test
0.8
5 2.913
0.61
0.379
1.5
2.89
120.5
yj-2
channel of cement.
3.36
0.88
0.529
3.43
122.0
mortar. Two-
3.616
1.09
0.635
3.89
125.2
thirds cement,
3.760
1.24
0.706
4.30
132.1
one-third fine sand.
3.891
1.41
0.787
4.51
131 .3
— Darcy-Bazin.
3.963
1 .54
0.839
4.80
135 .3
Series 25.
4.029
4.068
4.088
4.095
4.095
4.095
1 .69
1.80
1 .92
1 .98
2.04
2.09
0.900
0.941
0.983
1 .006
1.022
1.038
4 .94
5.20
5.38
5.48
5.55
5.56
134.5
138.3
140.1
141.0
141.7
143.5
Conduit of North
0.8
3 ...
1.02
0.619
0 .333
1.58
110
yi.
Metropolitan Sew-
1.52
0.928
2.21
126
. . .
age System, East
2.04
1.208
2.70
134
Boston section.
2.45
1.408
3.03
139
Brickwork washed
3.16
1.830
3.48
141
with cement. Sec-
3.75
1.999
3.73
145
tion circular, di-
4.62
2.31
4.18
150
ameter 9 ft.— Th.
Horton. Experi-
ments of 1896. 10
months in use.
Do.
1
Experiments of 1897.
0.5
9 ...
2.15
1.28
0.333
2.55
123
71^
2.74
1.56
"
2.90
127
3.19
1.76
"
3.06
126
...
...
3.20
1.97
((
3.18
131
...
Do.
0.5
6 ...
1.99
1.12
0.333
2.38
119
yT2
Experiments of 1900.
2.83
1 .61
((
2.82
121
. • .
3.64
1.95
11
3.16
124
...
4.18
2.13
cc
3.30
124
...
Do.
0.6
5 6.0
1 .02
0.688
0.50
1.99
107
v^
Charlestown section.
1.44
0.958
(I
2.46
112
Sides vertical, bot-
1.91
1 .187
u
2.825
115
tom flat arch.
2.40
1.387
<(
3.33
118
Experiments of 1896.
2.89
1.539
(C
3.44
124
. '. '.
10 months in use.
Do.
0.4
0 ...
2.91
1.54
0.5
2.97
107
vh
Experiments of 1897.
3.29
1.64
((
3.16
111
•
DESCRIPTION OF CONDUIT, ETC.
95
Table X. — Continued.
Description of
Conduit.
m
0.3
W
d
R
1000 s
V
c
a
Conduit of North
Metropolitan Sew-
age System,
Charlestown sec-
tion (continued).
Experiments of 1900.
3 ...
2.29
2.78
3.20
1 .34
1.51
1.64
0.50
2.66
2.86
3.04
102
104
106
FtV
Aqueduct of Glas-
gow. Smooth
concrete. Nearly
rectangular, bot-
tom flat arch. —
Fairlie Bruce.
1896.
0.8
0 9.
6
■
•
1 .22
1 .47
1.47
1.49
1.50
1.50
1.55
1.60
1.61
1.61
1 .62
1.63
1.74
1.81
0.
182
1 .67
2.07
2.10
2.21
2.13
2.15
2.17
2.20
2.23
2.22
2.24
2.25
2.26
2.41
125.0
126.3
128.5
134.5
129.3
130.3
129.4
129.3
130.5
129.7
130.6
130.8
126.9
135.4
71
\
Millrace at Idria,
Hungary. Cement
mortar on rubble
masonry. — Rit-
tinger.
0.6
5 ...
2.04
0.977
0.5
2.523
114.1
7tV
Aqueduct of Roque
favour, canal of
Marseilles. Bottom
of clear cement,
sides of good brick
work. Rectangu-
lar.— Baumgarten .
0.7
2 6.88
1 .504
3.72
10.26
137
vA
Aqueduct of the
Cervo, Canal Ca-
vour. Bottom of
good concrete,
sides of brick.
Rectangular. —
Passini & Gioppi,
1892.
0.5
S66.0
...
5.12
5.76
7.20
0.11
it
11
3.52
3.76
4.38
148.5
149.3
155.8
vi^
96
THE FLOW OF WATER
Table X. — Continued,
Description of
Conduit.
m
W
d
R
1000 s
V
c
a
Gage Canal. San
Bernardino, Cal.
Channels in earth,
roughly coated
with cement plas-
ter, 1 part cement,
4 parts sand. —
U.S.Geol. Survey.
0.49
0.48
0.46
0.48
0.44
0.47
9.25
10.25
14.0
12.25
16.0
17.0
3.5
3.5
3.5
5.5
3.5
3.5
1.82
1.94
2.13
2.13
2.38
2.38
0.4
0.4
0.478
0.382
0.520
0.413
3.14
3.28
3.78
3.38
4.24
3.78
117
117
119
119
120
121
y.\
Canal of Verona.
Channel lined
with Baton ma-
sonry. Trape-
zoidal. Bottom
width about 20 ft.
0.00
5.12
0.31
4.2
107.8
W^
San Gabriel Tunnel
No. 15.
Coated with cement
mortar, 1 cement
to 3 sand. Sec-
tion rect. L. 446
ft. — Lippincott.
0.89
4.5
4.0
1.31
0.96
5.01
141.3
yA
San Gabriel Tunnel
No. 23.
L. 318 ft.
0.89
1.37
0.86
4.74
141.6
yA
Old Aqueduct of Los
Angeles.
Coated with cement
plaster on con-
crete. 4 years in
use. Covered.
0.95
0.90
...
0.817
0.830
0.51
0.51
2.71
2.81
132.6
136.7
vh
Colton Canal.
Channel Hnedwith
concrete. Bottom
clean, sides lightly
coated with moss.
0.26
0.98
20.70
2.27
86.7
yi.
Santa Ana Canal.
Channel lined with
concrete. Bottom
covered with sand
and gravel, sides
coated with plants
in places.
0.33
0.817
1.06
2.62
89.2
yA
DESCRIPTION OF CONDUIT, ETC.
97
Table X. — Continued.
Description of
Conduit.
Riverside canal.
Open conduit
coated with cement
mortar on concrete.
Bottom covered
with fine sand to
a depth of 1.5 to
2.5 ft. Trape-
zoidal.
m
W
d
R
1000 s
V
c
-0.27
...
...
1.49
0.92
1.96
52.9
-0.08
0.703
0.63
1.22
38.0
1
X.
Brick
Conduits.
Sudbury Conduit. 0 .80|
9.0
4.672
2.359
0 .0334
1.207
136.0
Vjh
Hard glazed brick,
4.972
2.417
0 .0488
1 .497
137.9
smoothly jointed,
3.319
1 .963
0 .0625
1 .512
136.5
fairly clean. Sides
2.561
1.648
0 .0948
1.616
129.3
nearly vertical,
2.998
1.838
0.1155
1.983
136.1
bottom flat arch.
3.369
1.981
0 .1356
2.255
137.6
Fteley & Steams.
2.192
1 .468
0 .1466
1.931
131.6
4.602
2.343
0 .1793
2.889
141.0
3.878
2.151
0 .2102
2.955
139.0
3.266
1 .943
0 .2389
2.957
137.3
1.799
1 .251
0 .2553
2.448
137.0
2.245
1 .495
0 .2580
2.687
138.5
2.707
1.714
0 .2602
2.886
136.6
2.881
1.789
0 .4604
4.163
142.9
3.437
2.005
0 .4913
4.913
140.8
••
New Croton Aque-
0.68
13.6
0.75
0 .1326
1.1
110.4
yh
duct, New York.
1.0
a
1.37
118.9
Good brickwork.
1.25
((
1.59
123.0
Sides nearly verti-
1.50
ti
1 .80
127.4
cal, bottom flat
1.75
It
1 .94
128.3
arch. — Fteley,
2.0
ii
2.1
129.2
1895.
2.25
2.50
2.75
3.0
3.50
3.81
It
tt
tt
tt
tt
2.27
2.40
2.52
2.65
2.89
3.02
131.2
132.1
132.1
133.0
134.0
134.0
Rectangular test
0.57
6.27
0.20
0.192
4.9
2.75
89.7
yh
channel of com-
0.31
0.284
it
3.66
98.3
mon brickwork,
0.41
0.365
tt
4.18
• 98.8
rather rough. —
0.49
0.424
tt
4.72
103.7
Darcy-Bazin.
0.57
0.481
tt
5.10
105.1
Series 3.
0.65
0.71
0.77
0.85
0.90
0.97
0.540
0.580
0.620
0.668
0.697
0.739
tt
tt
tt
tt
tt
tt
5.33
5.68
6.01
6.15
6.47
6.60
103.7
106.3
109.0
107.4
110.8
109.7
...
1
.0
4
0.779
tt
6.72
108.7
'. '. .
98
THE FLOW OF WATER
Table X.
— Continued.
Description of
Conduit.
m
0.57
L
d
R
1000 s
V
c
a
Brick sewer at Mil-
waukee, Wis.
Smooth brick, well
pointed. —G. H.
Benzenberg.
2,
534
12.0
3.0
0.523
0.547
0.814
0.821
0.793
1.046
1.046
1.040
1 .010
5.083
5.043
5.154
6.195
6.207
6.886
6.872
6.961
6.821
128.3
125.1
124.5
124.9
127.9
122.9
122.7
124.3
124.2
1.0
Brick sewer, Dor-
chester Bay Tun-
nel. Inverted sy-
phon. Hard brick,
well pointed.
Sewer slime. —
Clarke, 1895.
0.47
7166
7.5
1.875
0.513
0.554
0.581
3.769
3.798
3.929
121.0
118.0
119.0
1.0
Syphon Aqueduct of
the Elvo, Canal
Cavour. This con-
duit consists of 5
oval tubes, each
having a cross sec-
tion of 119.25 sq.
ft. — Passini &
Gioppi, 1892.
0.4C
1 5
81.5
••
2.78
0.067
0.107
0.152
0.208
0.276
0.361
0.462
0.586
.733
1.61
1.95
2.31
2.66
3.1
3.53
4.01
4.52
5.094
117.3
112.9
112.2
111 .6
111.8
111.6
111 .8
112.0
113.1
i'.d
XI. Channels Lined with Ashlar or Rubble Masonry.
Chazilly Canal. Ash-
0.57
W
4.04
0.50
0.41
8.1
5.73
100.0
yA
lar masonry.
4.10
0.78
0.57
u
7.52
111.0
smoothly dressed,
. . .
4.14
1.0
0.68
a
8.19
110.0
section trapezoi-
4.18
1.20
0.77
"
8.75
111.0
dal, very regular.
— Darcy-B a z i n.
Series 39.
Aqueduct of Crau,
0.59
8.5
3.0
1.774
0.84
5.55
125.0
^tV
Canal of Craponne.
Ashlar masonry,
smoothly dressed.
Section rectangu-
lar. — Darcy-Ba-
zin.
DESCRIPTION OF CONDUIT, ETC.
99
Table X. — Continited,
Description of
Conduit.
m
0.44
W
d
R
1000 s
V
c
112.8
a
Solani Aqueduct.
85.0
2.66
2.52
0.151
2.20
1.0
Ganges Canal, In-
.. .
. . .
2.88
2.72
0.145
2.54
117.9
1.04
dia. Rectangular
3.13
2.94
0.20
2.51
103.5
0.90
conduit consisting
3.12
2.94
0.208
2.79
112.8
0.984
of two sections,
3.18
2.99
0.253
3.20
116.4
1.016
separated by a
. . .
. . .
3.96
3.65
0.473
4.83
116.2
1 .0
central wall, len-
4.60
4.20
0.025
1.24
121.0
0.99
gth 920 ft. Floor
of brick, laid flat,
sides of masonry.
Some deposits
here and there.
Right section. —
Allan Cunning-
ham, 1880.
New and well built
0.32
6.0
0.50
0.32
42.35
9.45
80.5
v.h
channel of dry
0.32
46.42
10.49
85.4
rubble masonry of
. . .
. . .
0".57
0.37
42.35
9.89
82.5
. . .
large stones. Semi-
6.37
46.42
10.97
83.6
circular. — Kutter,
1867.
Old channel of dry
0.27
8.0
0.55
0.36
82.8
11 .81
68.6
1.0
rubble masonry of
8.0
0.55
0.38
99.3
13.32
68.4
((
large stones, bed
7.4
0.60
0.39
106.8
13.75
67.3
(<
somewhat dam-
10.6
0.90
0.58
82.8
15.54
70.6
<<
aged. Semicircu-
10.6
0.90
0.61
99.3
18.28
72.8
(I
lar. — Kutter,
9.0
1.0
0.65
106.8
19.17
72.8
0.97
1867.
Spillway of Grosbois
0.15
5.98
0.36
0.324
101 .0
12.29
67.9
v.h
Reservoir. Ash-
6.01
0.55
0.467
<<
16.18
74.5
lar with cement
6.05
0.71
0.580
«
18.68
77.2
joints, partly dam-
6.07
0.84
0.662
(C
21.09
81.6
. . .
aged, covered with
a sticky slime.
Rectangular. —
Dare y-B a z i n.
Series 32.
Do.
0.16
6.0
0.49
0.424
37.0
9.04
72.2
v^
Tailrace of Gros-
6.1
0.77
0.620
«
11 .46
75.7
bois Reservoir,
6.1
0.97
0.745
(C
13.55
81 .6
— Darcy-Bazin.
6.1
1.16
0.846
u
15.08
84.9
. . .
Series 33.
Tailrace of dry rub-
-0.03
0.42
0.289
2.5
1 .257
46.8
1 .012
ble masonry, pav-
.. .
0.56
0.359
«
1 .491
49.8
1 .015
ed, semicircular. —
0.69
0.419
<(
1.643
50.8
1.010
Rittinger, 1855.
Quoted by Kutter.
100
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
W
d
R
1000 s
V
c
a
Aqueduct of dry
rubble masonry,
paved. Rectan-
gular. Rittinger.
-0.03
0.24
0.65
0.81
0.213
0.439
0.486
4.5
1 .324
2.396
2.432
42.8
53.9
52.0
1.012
1.056
1 .0
Do.
Tailrace of dry rub-
ble masonry paved.
Trapezoidal. Ritt.
-0.02
...
0.35
0.47
0.56
0.278
0.351
0.403
3.6
((
<<
1.502
1.928
2.104
47.5
54.2
55.2
1 .022
1 .08
1.041
Headrace of dry
rubble masonry
paved. Trapezoid-
al. — Rittinger.
0.04
. . .
0.26
0.45
0.213
0.344
3.8
1 .369
1.828
48.1
50.6
1.012
0.988
Grosbois Canal.
Channel of rough-
ly hammered
stone masonry. —
Da rcy- Bazin .
Series 1.
0.15
3.9
3.6
3.5
3.5
1.6
1.5
1 .2
0.9
0.88
0.84
0.71
0.62
12.1
14.0
29.0
60.0
9.58
8.36
11.23
13.93
73.5
77.3
78.4
72.5
1 .0
1 .06
1.12
1.055
Do.
Masonry in bad con-
dition, mud and
stones in bed. —
Darcy- Bazin ,
Series 46.
-0.02
6.8
6.9
6.9
7.0
1.5
2.0
2.4
2.4
0.88
1.23
1 .40
1.42
0.648
0.671
0.683
0.683
1.47
2.02
2.34
2.78
62
70
76
87
1 .02
1.06
1.02
1.024
XII. Channels Lined
WITH Pebbles Held
IN
Place with Cement.
Semicircular test-
0.38
^
5 .1 0 .7
0.454
1.5
2.17
83.1
72V
channel lined with
;
5 .4 0 .9
0.546
(C
2.50
89.4
pebbles f to
|-inch diameter,
t
5 .5 1 .1
0.619
il
2.69
88.2
{
5 .7 1 .2
0.681
(
2.93
89.5
. . .
held in place with
[
5 .8 1 .3
0.731
I
3.05
92.1
. . .
cement. — Darcy-
{
3 .8 1 .4
0.784
i
3.22
93.9
Bazin. Series 27.
i
5 .9 1 .5
1 .0 1 .7
1 .0 1 .9
0.826
0.900
0.968
I
t
I
3.33
3.54
3.73
94.6
96.3
97.9
'
1 .0 2 .0
1.012
((
3.95
102.1
...
Do.
0.19
(
3 .0 0 .27
0.25
4.9
2.16
61.7
v.\
Section rectangular.
. . 0 .41
0.357
2.95
70.5
— Darcy-Bazin.
. . 0 .53
0.450
3.40
72.5
Series 4.
. . 0 .63
. . 0 .73
. . 0 .82
. . 0 .91
. . 0 .99
. . 1 .06
.. 1.15
. . 1 .23
. . 1 .30
0.520
0.588
0.644
0.740
0.746
0 785
0.832
0.871
0.910
3.84
4.14
4.43
4.64
4.88
5.12
5.26
5.43
5.57
76.1
77.2
78.8
79.3
80 7
82.6
82.4
83.1
83.4
DESCRIPTION OF CONDUIT, ETC.
101
Table X. — Continued,
Description of
Conduit.
m
W
d
R
1000 s
V
c
a
Rectangular test-
0.00
6.11
0.32
0.291
4.9
1.79
45.7
Tff
channel lined with
0.48
0.417
2.43
53.8
pebbles \\ to 1^-
0.61
0.510
2.90
58.0
inch diameter,
0.73
0.587
3.27
61 .1
held in place with
0.84
0.636
3.56
62.8
cement. — Darcy-
0.93
0.712
3.85
65.2
Bazin. Series 5.
1.03
0.772
4.03
65.5
1.13
0.823
4.23
66.6
1 .21
0.867
4.43
68.0
1 .29
0.909
4.60
69.0
1.37
0.946
4.78
70.3
1.46
0.987
4.90
70.4
XIII. Rectangular Test Channels with Cleats Nailed Crosswise.
Rectangular test-
0.41
6.43
0.33
0.302
1.5
1.65
77.4
73V
channel of boards,
0.51
0.442
2.17
84.5
with cleats 1 inch
0.89
0.634
2.86
91.0
by f inch nailed
1 .02
0.775
3.33
94.0
crosswise on bot-
1.23
0.889
3.68
97.0
tom and sides, |
.
1.42
0.986
3.98
99.0
inch apart. —
. . .
1.62
1.076
4.19
99.0
. ,
Darcy- Bazin.
Series 12.
.
Do.
6.43
0.22
0.205
5.9
2.50
71.8
yi^
Series 13.
0.33
0.302
((
3.34
79.0
0.51
0.442
(I
4.40
86.0
0.67
0.552
u
5.08
89.0
0.80
0.643
u
5.63
91 .4
0.92
0.716
11
6.14
94.5
1 .05
0.790
((
6.48
94.8
Do.
6.40
0.19
0.182
8.9
2.85
70.8
yh
Series 14.
0.30
0.273
(I
3.75
76.4
0.46
0.403
u
4.92
82.4
0.59
0.499
tl
5.77
86.8
0.71
0.582
it
6.38
88.9
0.83
0.658
tc
6.86
89.9
0.94
0.726
(C
7.26
90.5
Rectangular test-
0.03
6.43
0.43
0 .378
1.5
1.28
53.7
1
y\
channel of boards
0.66
0.550
1.68
58.6
with cleats nailed
1.02
0.777
2.21
64.8
crosswise to bot-
1.33
0.942
2.55
67.8
tom and sides,
1.61
1.073
2.81
70.1
cleats 1 by 1 inch,
. . .
1.91
1.197
2.97
70.0
2 inches apart. —
2.18
1.299
3.11
70.5
Darcy- B-azin.
Series 15.
1
102
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
W
d
R
1000 s
V
c
a
Do.
6.43
0.29
0.264
5.9
1.91
48.3
1
yi
Series 16.
...
0.44
0.67
0.87
1 .05
1 .21
0.384
0.553
0.686
0.791
0.882
2.56
3.37
3.88
4.31
4.65
53.7
59.0
61 .0
63.1
64.5
1.38
0.965
4.91
65.1
Do.
. . .
6.40
0.25
0.252
8.86
2.21
48.7
1
Series 17.
0.39
0.35
2.85
51.2
0.60
0.501
3.75
55.8
0.78
0.628
4.37
58.6
0.94
0.725
4.85
60.5
1 .09
0.812
5.22
61 .5
1.22
0.885
5.57
62.9
XIV. Channels in Rockwork.
Description of
Conduit.
m
K
W
d
R
1000 s
V
c
a
Torlonia Drain
Tunnel, Lake
Fucino, Italy.
Section oval,
13.12 ft. wide
and 18.9 ft.
high. Total
length, 20,666
ft., of which f is
in limestone
rock, the rest is
lined with free-
stone masonry.
— Perrone,1894.
-O.O
4
1.0'
J 13
.12
1.93
2.07
2.55
2.67
3.24
3.43
3.52
3.75
1.04
3.25
3.62
4.24
4.32
5.05
4.95
4.92
5.35
75.3
76.2
82.2
81.7
86.7
82.9
80.9
86.2
1 .0
1 .0
1.027
1.013
1.027
0.97
0.94
0.98
Beacon Street
Tunnel, Sud-
bury Aqueduct.
Length 4592 ft.
Bottom lined
with rough con-
crete, sides for
the greater part
unlined. Sec-
tion oval. —
Fteley & Steams
1878.
-o.c
12
1.0(
5 10
.0
2.21
0.281
1 .97
79.2
1.0
DESCRIPTION OF CONDUIT, ETC.
103
Table X. — Continued.
Description of
Conduit.
Turlock Rock
Canal. Exca-
vated along the
banks of Tur-
lock River, Cal-
ifornia.
m
K
1 .50
W
d
R
1000 s
V
c
-0.20
50.0
10.0
5.9
1.5
7.5
86.0
1.0
XV. Artificial Channels in Earth.
Experiments by
S. Fortier, U. S.
Geol. Survey,
1901.
Bear River Canal
branch. Well
rounded chan-
nel in clayey
loam, coated
with a fine sedi-
ment.
0.58
0.27
2.49
0.31
3.62
130.2
v.h
Do.
Providence Canal.
0.58
0.27
0.86
0.12
1 .04
102.6
Vt\
Do.
Logan & Hyde
Park Canal.
0.54
0.27
...
...
0.51
1.88
2.33
95.3
^tV
Channel in fine
hard gravel, ^
inch in diameter
0.20
0.66
...
1.0
0.32
1.44
80.8
yxV
Do.
0.12
0.77
1.20
0.83
2.58
81.5
yxV
Channel in clay
with fragments
of rock i inch
in diameter im-
bedded.
0.07
0.86
1.07
0.62
1.94
74.6
v.^
Channel in sand
some plants at
edges.
0.05
0.90
1.40
0.15
1.08
75.4
1.0
Channel in sand
with small peb-
bles imbedded.
0.05
0.90
. . .
0.52
0.56
1.01
59.2
1.0
404
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
K
W
d
R
1000 s
V
0.54
c
39.3
a
Experiments of S.
Fortier (contin-
ued).
Channel in sand
with small peb-
bles imbedded.
-0.01
1.04
...
...
0.14
1.35
1.0
Channel in ce-
mented gravel,
1,2 and 3 inches
in diameter.
0.0
1.0
. . .
. . .
1.52
0.77
2.49
73.0
1 .0
Channel in sand
with gravel itn-
bedded, no
vegetation.
-0.07
1.17
. . .
. . .
0.40
0.40
0.61
48.0
1.0
Do.
-O.09
1.22
...
1.48
6 A3
0.71
64.9
1.0
Do.
-0.08
1.18
...
0.65
0.75
1.19
54.0
1.0
Do.
-0 .10
1.24
0.71
1.75
1.09
53.8
1 .0
Channel in earth
with gravel im-
bedded, size i
to 2 inches.
-0.10
1.24
* • ■
0.55
1.16
2.93
50.4
1.0
Channel in earth
with fragments
of rock imbed-
ded, size i to 2
inches.
-0 .15
1.35
0.65
1.6
1.61
50.8
1.0
Channel in gravel
covered with
sediment, grav-
el up to 2^
inches in diam-
eter.
-0.16
1.37
1.62
0.60
1.94
63.2
1.0
Channel in earth,
bottom covered
with fragments
of rock.
-0.18
1.48
.
0.35
1.3
0.84
39.8
1.0
Channel in cob-
bles covered
with silt, edges
irregular.
-0.38
2.52
0.52
0.35
0.43
32.0
1.0
DESCRIPTION OF CONDUIT, ETC.
105
Table X. — Continued.
Description of
Conduit.
m
K
2.44
W
d
R
1000 s
V
1.35
c
a
Channel in loose
gravel up to 1 J
inches in diam-
eter.
-0.48
...
...
0.27
9.91
25.9
1.0
Channel in earth,
bottom and
sides coated
with fragments
of rock up to 3
inches in diam-
eter.
-0.42
2.90
0.20
12.2
1 .02
20.9
1.0
Rough channel in
coarse gravel
and cobbles.
-0.40
2.96
0.23
17.1
1.33
21.1
1.0
Do.
-0.49
3.62
0.23
17.0
1 .10
17.9
1.0
Experiments by
Darcy-Bazin.
Grosbois Canal.
Trapezoidal
channel in
earth. No vege-
tation. Series 49.
-0.12
1.28
10.7
11.9
14.1
15.7
1.4
1 .9
2.5
2.9
0.96
1 .32
1.57
1.78
0.25
0.275
0.246
0.275
0.89
1.34
1 .36
1 .49
57
70
69
66
1.0
0*98
Do.
Some vegetation.
Series 50.
-0.33
1.97
10.5
11 .4
13.8
15.5
1.5
2.1
2.7
3.1
1.05
1 .42
1.65
1.85
0.31
0.29
0.33
0.33
0.82
1 .26
1.30
1 .41
45
52
56
57
1 .0
Do.
Stony earth, lit-
tle vegetation.
Series 37.
-0.31
1 .89
9.1
11.4
12.6
13.3
1.5
2.0
2.4
2.7
0.96
1.26
1.41
1.56
0.792
0.806
0.858
0.842
1.23
1.67
1.81
2.0
45
53
52
55
1.0
Do.
Series 41.
-0.33
1.97
10.1
12.0
13.2
14.3
1 .6
2.3
2.9
3.0
1.04
1 .38
1 .57
1.71
0.445
0.450
0.455
0.441
0.96
1 .27
1 .40
1 .51
45
51
52
55
1.0
Do.
Covered with
vegetation at
many points.
Series 43.
-0.37
2.18
10.1
12.3
13.5
14.7
1 .7
2.4
2.8
3.1
1.06
1 .41
1 .60
1.76
0.42
0.47
0.43
0.45
0.89
1.18
1 .31
1 .39
42
46
49
49
1.0
106
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
-0.38
K
2.22
W
d
R
1000 s
V
c
42
53
51
58
a
Do.
Bottom and sides
coated with
mud, little
vegetation. Se-
ries 47.
9.9
11.1
12.5
14.0
1.7
2.2
2.7
2.9
1.09
1.38
1.63
1.71
0.464
0.450
0.479
0.493
0.82
1.32
1.43
1.68
1.0
Do.
Section well
rounded, little
vegetation. Se-
ries 48.
-0.38
2.22
9.5
10.7
11 .9
13.5
1.5
2.1
2.5
2.9
0.99
1.30
1.56
1.71
0.555
0.555
0.525
0.515
0.96
1.48
1.57
1 .75
41
55
55
59
1.0
Do.
Trapezoidal chan-
nel in eariih ;
vegetation at
many points.
Series 36.
-0.44
2.41
10.5
12.9
14.2
15.2
1.9
2.5
2.9
3.2
1.14
1.42
1.61
1.74
0.678
0.633
0.644
0.622
0.91
1.28
1.45
1.65
33
43
45
50
1.0*
Chazilly Canal.
Trapezoidal
channel in stony
earth; little
vegetation. Se-
ries 38.
-0.37
2.18
8.9
11.0
12.1
13.0
1.5
2.0
2.4
2.6
0.96
1.18
1.41
1.54
0.957
0.929
0.993
0.986
1.24
1.70
1.80
1.96
41
51
48
50
1.0
Canal of the Jard.
Channel in
earth ; no stones
or plants. —
Dubuat, 1779.
0.03
0.95
1.68
1.94
2.05
2.58
0 .0362
0 .0362
0 .0458
0 .0651
0.449
0.479
0.607
1.069
57.6
57.0
62.6
82.5
1.6'
Millrace of Kag-
iswyl. Very
regular channel
in earth, bottom
covered with
fine gravel. —
Epper, 1885.
-0.07
1.04
...
...
1.04
1.38
1.41
1.754
1.255
1.20
2.819
3.139
3.221
65.8
75.3
78.3
v.\
Canal at Real-
tore. Trape-
zoidal channel
in earth ; bot-
tom muddy. —
Darcy-Bazin.
-0.18
1.55
19.7
4.5
2.87
0.43
2.54
72.2
1.0
DESCRIPTION OF CONDUIT, ETC.
107
Table X. — Continued.
Description of
Conduit.
m
K
W
d
R
1000 s
V
3.98
c
92.8
a
Ganges Canal,
-0.22
1.54
184.2
9.7
8.35
0.22
1.0
Solani Embank-
ment. Trape-
zoidal channel
in earth ; bed
quite uniform.
— A. Cunning-
ham. Series 197.
Do.
Bed somewhat
-0.22
1.52
64.0
4.6
4.07
0.306
2.71
76.8
1.0
rough. Series
64.3
4.8
4.18
0.304
2.74
78.3
222-225.
64.8
5.1
4.37
0.297
2.79
77.4
...
...
65.2
5.3
4.50
0.291
2.82
78.8
...
Do.
-0.27
1.67
174.9
10.0
8.64
0.231
3.98
89.1
1.0
Bed uneven. Se-
ries 192.
Canal Cavour.
-0.24
1.60
5.16
0.29
3.10
80.0
1.0
Above the sy-
5.83
te
3.38
80.2
phon of the
Sesia. Bottom
7.32
a
3.70
80.7
width, 65.8 ft.;
side slopes, 1 : 1
— Passini &
Gioppi, 1892.
Do.
-0 .28
1.80
4.44
0.33
3.05
72.2
1.0
Below the sy-
5.25
<(
3.08
78.1
phon of the
5.62
Cl
3.40
79.0
Sesia.
Escher Canal.
-0.33
1.94
3.76
3.0
6.986
65.7
1.0
Coarse gravel
4.42
(t
8.364
70.6
and detritus. —
Legler.
Linth Canal at
-0.20
1.50
123
5.14
0.29
3.41
88.4
1.074
Grynau. Trape-
.
5.93
0.30
3.83
90.8
1.052
zoidal channel
.
6.48
0.31
4.15
92.6
in earth ; bot-
,
7.12
0.32
4.42
92.6
tom slightly
.
7.52
0.33
4.72
95.4
. . .
rounded. —
8.09
0.34
4.92
93.8
1.0
Legler.
••
••
•
8.28
8.62
8.87
9.18
0.34
0.35
0.36
0.37
5.06
5.22
5.39
5.53
95.3
95.1
95.5
94.9
o!98
108
THE FLOW OF WATER
Table X. — Continued.
Description of
Conduit.
m
K
W
d
R
1000 s
V
4.26
4.17
c
75.5
80.7
a
Do.
At Blase hen.
Trapezoidal
channel in
gravel ; bottom
slightly round-
ed.
-0.22
1.58
113.2
4.0
6.5
0.80
0.41
...
Simme Canal.
Very coarse
gravel and
stones. — Wam-
pfler, 1867.
-0.44
':''
...
1.82
1.87
1.36
1.32
6.5
7.0
11.6
17.0
4.92
5.37
5.49
5.99
45.1
46.9
43.6
39.8
1.0
Test channel in
river sand, —
Seddon, 1886.
0.48
0.57
0.065
0 .0465
3.5
3.7
4.1
5.2
6.2
0.86
0.87
0.84
0.83
0.86
67.4
67.8
61.6
54.0
52.3
1.0
Do.
0.48
0.83
0.040
0 .0316
7.9
8.0
8.6
9.7
11.3
0.93
0.89
0.88
0.91
0.91
58.9
55.7
53.2
54.0
51.1
1.0
Triangular chan-
nel in sand ;
very regular. —
Fresno.
0.27
0.57
• • ■
• • *
0.416
0.312
0.80
71.0
1.0.
Well rounded
irrigation chan-
nel in sandy
soil. — Fresno.
-0.03
1.04
. . .
1.0
0.06
0.50
64.5
1.0
XVI. Artificial Channels in Earth with Side Walls of
Masonry.
Millrace at Pris-
0.12
0.79
0.54
0.373
1.0
1.127
58.4
1.0
bram. Very
0.66
0.425
((
1.254
60.8
regular channel
in clay with
side walls of
masonry. Sec-
tion trapezoid-
al. — Rittinger,
1855.
Do.
-0.62
5.09
0.41
0.316
2.2
0.289
14.8
1 .0
Trapezoidal chan-
0.44
0.336
0.588
21.6
. . .
nel in earth
0.70
0.472
0.953
29.6
. . .
with dry rubble
. . .
. . .
0.80
0.548
1 .135
32.7
side walls. Very
0.86
0.560
1 .190
33.9
...
irregular.
-6.43
2.10
0.90
0.566
1.269
36.0
1 .0
DESCRIPTION OF CONDUIT, ETC.
Table X. — Continued.
109
Description of
Conduits.
m
K
2.79
0.74
W
d
R
1000 s
V
c
22.5
31.7
43.4
43.4
65.6
a
Millrace at Bezy-
banya. Trape-
zoidal channe:
in sand and
gravel with side
walls of mason-
ry. — Rittinger.
-0.42
6.02
.
0.28
0.35
0.56
0.73
0.90
0.242
0.282
0.407
0.483
0.561
5.0
0.782
1 .191
1 .956
2.134
3 .475
1 .0
1.6'
Millrace at Dio-
sgyor. Rectan-
gular channel in
clay with side
walls of dry
rubble masonry,
— Rittinger,
0.07
-0 .28
1.0
1.38
0.63
1.14
1.66
0.487
0.736
0.924
4.0
2.463
2.750
3.323
54.5
50.7
54.8
1.0
1.6"
Embankment of
Solani Ganges
Canal, India.
Main Site,
Built up channel
in clayey soil
with many arti-
ficial bars of
masonry and
boulders. Side
walls of ma-
sonry built in
steps, bed very
rough, masonry
damaged in
places. The first
four gagings in-
dicate the pre-
pondering influ-
ence of the
great roughness
of the bed. —
Allan Cunning-
ham,
-C
.20
1.37
150.0
u
tc
(C
152,3
157.0
159.3
161.3
164.0
166.3
168.7
170.1
1.7
2.3
3.9
4.1
5,6
6.8
7.6
8.2
9.1
9.9
10.7
11.0
1.69
2.26
3.86
4.07
5.39
6.18
6.78
7.26
7.84
8.42
8.96
9,34
0.090
0.148
0.088
0.215
0.155
0.171
0.221
0.214
0.215
0.227
11
0.44
0.87
1.35
1.79
2.40
3.05
3.39
3.22
3.43
3.58
3.71
4.02
35.7
45.9
73.2
66.5
83.0
93.8
87.5
81.7
83.6
83.6
82.3
87.3
1
-
Do.
Jaoli Site.
Side walls lined
with brick set
in clay; side
slopes 1 : 2.
-^.17
1 .32
190.9
191.2
191 .5
191.8
192.3
6.8
7.0
7.3
7.6
8.1 '
6.32
6.53
6.79
7.05
7.46
0.140
0.144
0.145
0.146
0.166
2.63
2.76
2.80
2.81
2.94
88.4
88.1
89.2
87.6
85 .ll
1
17 f 8
110
THE FLOW OF WATER
Table X. — Continued,
Description of
Conduit.
m
K
W
d
R
1000 s
r
3.07
3.01
3.12
3.17
c
75.4
74.7
74.7
76.4
a
Do.
Belra Site.
-0.45
1.97
187.3
187.5
188.0
188.4
8.6
8.7
9.5
9.6
7.96
8.21
8.72
9.02
0.208
0.198
0.200
0.191
1
XVII.
Natural Channels
IN Earth.
Description of
K
W
d
R
1000 s
r
c
a
Channel.
1.03
La Plata River,
16.22
0.007
1.391
128.3
1
Vr\
Catalina channel.
Width of channel
^ many miles. Bed
fine sand. Slopes
measured with
great accuracy for
a distance of 85
miles.— J.J. Revy.
Parana de las Pal-
1.12
1222
50.3
0.007
3.07
160.0
1
1
mas.
FTI
Do.
49.7
49.5
0 .0068
2.95
2.87
160.3
156.6
Parana. Rosario
1.18
2460
44.6
0 .0058
2.63
152.9
1
Section.
VT^
Do.
Seine River at Paris.
1.45
9.48
0.14
3.37
92.5
1
Section between
7tV
the bridges of Jena
10.92
((
3.74
95.6
and "The Inva-
12.19
(t
3.80
92.4
lides."— ViUe-
14.50
(t
4.23
94.0
vert.
15.02
15.93
16.85
18.39
0.173
0.131
0.103
4.51
4.68
4.80
4.69
98.3
89.5
102.1
107.6
'. '. '.
River Po at Fossa
1.96
9.5
0.119
2.^0
83.5
1
d'Albero. Com-
ftV
mission of Italian
10.1
0.12
2.92
88.1
Engineers, 1878.
••
11.0
9.8
11.8
0.12
0.103
0.104
3.15
3.25
3.28
112.0
96.0
100.0
Do. At Porto Mo-
12.4
0.093
3.37
95.0
rone.
8.5
0.165
3.04
81.5
. . .
DESCRIPTION 0¥ CONDUIT, ETC.
Ill
Table X. — Continued.
Description of
Channel.
K
W
d
R
1000 s
r
c
a
Weser River near
1.92
9.4
0 .2499
4.07
83.8
1.0
Minden.
9.82
"
4.37
87.7
— Schwartz, 1808.
10.57
11 .18
12.07
12.64
12.66
14.13
4.75
4.94
5.24
5.26
5.35
5.67
92.8
93.2
95.0
94.2
97.0
96.0
0
98
Do. near Minden.
1.98
4.50
5.33
6.14
6.66
7.59
8.14
8.61
10.23
10.45
10.71
0 .5032
<<
u
(I
It
((
(t
3.38
4.0
4.88
5.24
5.75
5.95
6.15
6.66
6.56
6.56
71.0
78.6
88.0
90.5
93.2
93.2
93.2
93.2
90.6
94.0
1
0
11.28
i(
6.94
92.0
1
01
Do. AtVlotow.
1.51
6.25
0 .5503
4.95
81 .0
1
7.41
<(
5.32
84.0
8.92
(I
6.30
90.0
9.3
11
6.52
91 .2
9.72
a
6.65
91.2
11 .70
(I
7.53
93.8
13.0
a
7.92
93.9
...
••
13.35
(<
7.90
92.1
Plessur River. Some
1.55
1 .25
9.65
6.6
54.7
1
.0
stones and gravel.
. . .
2.33
u
9.99
66.4
.
— La Nicca, 1839.
...
3.48
3.58
3.59
u
t(
10.19
13.58
13.94
66.4
72.4
74.8
;
Saalach River. Some
1.61
1.54
0.875
2.073
56.5
1
.0
stones. — Roff,
1.31
1.100
2.246
58.8
1854.
...
1.91
1.98
2.16
1 .242
1.240
3.660
3.077
3.385
5.474
63.0
68.2
64.3
River Rhine in Dom-
1.71
0.25
5.74
1.25
32.8
1
leschger Valley.
yll
Gravel and detri-
1 .32
7.73
4.75
47.0
tus. — La Nicca.
2.95
7.95
7.42
48.3
112
THE FLOW OF WATER
Table X. — Concluded.
Description of
K
W (
i R
1000 s
r
c
a
Channel.
Salzach River. From
1.94
.. 3.53
0.94
3.48
60.3
1
Be rghe im to
Wildshut. Gravel
. . 4 .20.
0.94
4.03
63.9
. . 7 .39
1.12
5 .786
63.4
and detritus. —
. . 3 .51
1 .55
4.10
55.4
Reich.
. . 4 .64
. . 3 .87
1.55
1 .79
4.67
4.45
67.5
53.4
...
. . 4 .26
1.79
5.15
58.8
...
Zihl River near Gott-
1.98
. . 3 .52
0.4
2.296
61.0
1
1
statt. Bed very
Vt\
irregular, covered
. . 5 .02
li
3.706
77.1
with mud and de-
.. 5 .53
((
4.625
69.1
tritus. — Trechsel,
1825.
Mississippi River at
4.14
3122
. . 64 .9
0.08
6.415
91 .9
1
Columbus, Ky., at
high water, 1895.
Rep. of Miss.
River Com. of
1896.
Do. At Helena, Ark.
2.91
5100
. . 40 .5
0.07
5.207
97.8
Do. At Arkansas City
4.31
3453
. . 65 .2
0.064
5.807
89.7
({
Do. At Wilson Point,
2.68
3944
. . 56 .4
0.054
6.145
111 .4
ti
La.
Do. At Natchez,
1.13
2173
.. 69.5
0 .0459
9.512
161.2
It
Miss.
Do. At Red River
1.59
4044
.. 57.2
0 .0284
5.636
140.9
it
Landing, La.
Do. At Carrollton,
1.22
2338
.. 71.0
0 .0219
6.494
162.3
It
La., at high water.
. . 72 .7
0 .0254
6.254
161.8
Bed sand.
Do. At low water.
1.51
2338
. . 65 .5
0 .0021
1.842
158.0
11
Irrawaddi River at
2.66
3395 :
35 16 .28
0 .0086
1 .007
85.1
1.0
Saiktha, Burmah.
«
39 18 .49
0 .0172
1.783
99.9
Bed sand with
i
13 19 .99
0 .0218
2.360
103.9
stones, right bank
i
17 21 .13
0 .0344
2.857
105.9
1 !l7
rocky in places. —
'.'.
53 26 .42
0 .0474
3.548
100.3
Gordon, 1873.
...
(
57 29 .80
33 35 .44
39 41 .01
0 .0559
0 .0688
0 .0817
3.993
4.052
5.382
97.8
94.2
92.9
...
. . .
'
r3 44 .47
0 .0904
6.147
97.0
6.84
FORMS OF SECTIONS OF CONDUITS 113
Forms of Sections of Conduits.
Relation of mean Hydraulic Radius to Wet Perimeter.
In the design of the form of cross-section of an artificial con-
duit two factors enter into consideration:
1. The material composing the walls of the channel.
2. The special purpose for which it is intended.
Conduits under pressure, whether constructed of metal,
wood, earthenware, concrete or masonry are nearly always
circular in section, because this form can best be given the
strength to resist internal and external pressures. The thickness
of the material forming the walls of a circular conduit is found
from the formula:
^= -^ + c
in which P is the pressure in pounds per square inch;
D the internal diameter in inches;
T the safe tensile strength of the material;
c a constant added to guard against defects in the
casting or the welding.
For such conduits as are subject to water ram a pressure of
100 pounds per square inch is allowed in addition to the pressure
due to the head which is equal to P = 0.434 h. The stresses
allowed in the material and the constants added are :
For cast iron
T = 4,000, c = 0.33
For wrought iron
T = 17,000, c = 0.06
For steel
T = 20,000
For lead
T = 450, c = 0.3
For concrete, 2 per cent steel
T = 480, c = 1.0
Since the advent of reinforced concrete, conduits constructed
of this material are coming more and more into favor. Steel-
concrete water pipes resisting pressures of heads exceeding 100
feet are now in use. The two aqueduct-syphons of Sosa have
internal diameters of 12.46 feet, and resist the pressure of a
head of 92 feet.
114
THE FLOW OF WATER
Fig. 4.
Forms of Sections of Masonry Conduits. The Numbers are Proportional.
FORMS OF SECTIONS OF CONDUITS
115
Steel concrete sewer pipe is now made in diameters from 15 to
120 inches.
Open conduits lined with concrete are most frequently made
semicircular in section. Wooden flumes which are acting simply
as aqueducts are generally made semi-square in section, if they
are intended to carry lumber or wood the triangular section is
used. For aqueducts lined with masonry a section is generally
preferred whose sides are vertical or nearly so, whose bottom is
a flat segmental arch, and whose top (if covered) is a semi-
circle. Very large aqueducts, those crossing valleys, streams
or other depressions are given a rectangular section.
In designing channels in earth the velocity enters into the
problem. In those of some dimensions the bottoms are well
rounded and the sides given slopes ranging from i- to 1 for
cemented gravel, to 3 to 1 for loose sand.
In a preceding chapter we have observed, that the form of the
cross-section of a conduit has an appreciable influence on the
power of the velocity to which the frictional resistance is pro-
portional and that the circular or semicircular form is the one
most favorable to flow. For rectangular conduits lined with
boards, for instance, we have found the value of the coefficient
a to be equal to FtV and equal to FtV for semicircular conduits
lined with the same material. The circular form has the
additional advantage of having a wet perimeter less in propor-
tion to the area of the section than any other form.
A F „ E D Let ^D (Fig. 5) be
the top width of a trap-
ezoidal channel, BC its
bottom width, and FB
the depth.
The area of the cross-
section will then be :
Fig. 5.
A =
C
AD + BC
FB,
116 THE FLOW OF WATER
and the wet circumference
P = AB + BC + C D,
Let BC = 6,
BF = d,
AD = t,
BF '•
Then the area A = db + W = d (b + Id);
the wet perimeter P = h + 2 d \/l + P;
d
A
the bottom width h = -=- — Id;
p
and the relation -^, the reciprocal of R,
P Id
Let the angle which the side of the conduit GD makes with
the horizontal be denoted by a, and we have for the conditions
P . . A
most favorable to flow or -j- a miminum, and -^r- a maximum,
A P
the depth d = t/A!HL^.
^ 2 — cos a'
A
the top width t=b + 2ld= — + d cotangent o;
tt
A
the bottom width b = — — d cotangent a;
Ob
XI, ^ J.' Wet Perimeter
the relation
Area of Section
P_ _h_ 2d
A A sin a
1 /2 — cos a)\ ,
d \ ^ sin a /
2 J, d
Consequently, for a given value of R and given side slopes the
area of section is least if R is equal to one-half the actual depth of
water.
For the semicircular section R is equal to one-half the radius,
hence this forms fulfils the conditions best and other forms of
SEWERS
117
section fulfil it the better the nearer they approach the semi-
circle. Table A contains values of R and areas of sections in
terms of the radius or semi-diameter for semicircular conduits.
TABLE A.
Depth of
Water in
Terms of
Value of R
Wetted Sec-
Depth of
Water in
Terms of
Value of R
Wetted Sec-
in Terms of
Radius.
tion in Terms
of Radius.
in Terms of
Radius.
tion in Terms
of Radius.
Radius.
Radius.
0.05
0.0321
0.0211
0.55
0.320
0.709
0.10
0.0524
0.0598
0.60
0.343
0.795
0.15
0.0963
0.1067
0.65
0.365
0.885
0.20
0.1278
0.1651
0.70
0.387
0.979
0.25
0.1574
0.228
0.75
0.408
1.075
0.30
0.1852
0.298
0.80
0.429
1.175
0.35
0.2142
0.370
0.85
0.439
1.276
0.40
0.2424
0.450
0.90
0.446
1.371
0.45
0.2690
0.530
0.95
0.484
1.470
0.50
0.2930
0.614
1.00
0.500
1.571
Table B contains proportions of channels of maximum values
of R, the mean hydraulic radius for a given area and given
side slopes.
Half the top width is the length of each side slope and the wet
perimeter is the sum of the top and bottom widths. The mean
hydraulic radius is equal to one-half the depth of water.
TABLE B.
Description of Form of
Section.
Inclination of
Sides to
Horizon.
Ratio of
Side
Slopes.
Area of
Section in
Terms of
Depth.
Bottom
Width in
Terms of
Depth of
Water.
Top
Width in
Terms of
Depth of
Water.
Semi Circle ....
Semi Hexagon . . .
Semi Square ....
Trapezoid
Do
Do
Do
Do
Do
Do
Do
Do
Do
Do
Do
60° '
90°
78° 58'
63° 26'
53° 8'
45.0
38° 40'
33° 42'
29° 44'
26° 34'
23° 58'
21° 48'
19° 58'
18° 26'
'3'
0
1
1
\
U
n
2
^
3
5
1.571^2
1 . 732 d^
1.812 d2
1.736^2
1.750^2
1.828^2
1.952^2
2.106^2
2.282 d2
2.472^2
2.674^2
2.885 d2
3.104^2
3.325^2
l.lbbd
2d
1.562d
1.236<^
d
0.828rf
0.702d
0.606d
0.532^
0.472rf
0.424d
0.385d
0.354d
0.325d
2.Z\'d
2d
2.062rf
2.236rf
2.50^
2.828d
3.022d
3.606d
4.032d
4.472d
4.924d
5.385d
5.854<^
6.325d
118
THE FLOW OF WATER
Sewers.
The forms of the cross-section most frequently adopted for
sewers are the circular and the oval or egg-shaped. Only sewers
of very great dimensions are given a rectangular section, the
roof being a flat segmental arch.
For sewers less than two feet in diameter glazed earthenware
pipe is mostly used, less frequently concrete pipe. Sewers con-
structed of brick, masonry or concrete are, however, found with
diameters down to two feet.
When the discharge of a sewer is estimated to be fairly con-
stant the circular section is preferred, when it varies con-
siderably, however, some form of an oval sewer is used.
Two forms of egg-shaped sewers are in general use. The one
most frequently adopted has the proportional parts as given in
the annexed figure. In the other form the lower circle has a
diameter of one-fourth the diameter of the upper circle only.
This form is used for sewers of small dimensions and greatly
varying in discharges.
The vertical diameter in both forms is always equal to 1§
diameters of the upper circle.
SEWERS 119
In the figure Tangent |^ = | = 0.75.
Hence the angle BCG = 36° 53',
and the angle BGC - 180° - (90° + 36° 53') = 53° 7',
hence, the angle FGD = 2 X 53° 7' = 106° 14'.
Using trigonometry, these data enable us to compute the
different parts of the area and the circumference with precision.
For the proportional parts given in the figure we find by this
method :
The area = 18.35 which is equal to 1.147 d?,
the circumference = 15.8488 which is equal to 3.9622 d,
the mean hydraulic radius = 1.1584, which is equal to 0.2896 d,
d being the horizontal diameter.
By the same method we may compute the value of the mean
hydraulic radius or the area for any depth of water in the sewer.
The mean hydraulic radius has its greatest value when the con-
duit is about 0.85 full. It is equal, being
0.85 full to 0.345 the horizontal diameter;
I full to 0.33 the horizontal diameter;
^ full to 0.28 the horizontal diameter;
i full to 0.20 the horizontal diameter.
The speed of flow necessary to prevent a deposit of sewage is
given, for all forms of the cross-section by the equation
V = 2 + ,
r
or, in exceptional cases, when the sewer is very well constructed
by
^ , 0.0625
V = 2 + ,
r
r being the mean hydraulic radius.
Hence for the very greatest section the least permissible
velocity is two feet per second. In order that the velocity
should not fall below the permissible limit the value of
66 ( ^/7 + m) V^
120 THE FLOW OF WATER
on which for equal slopes the velocity depends should not, for
any quantity of discharge vary greatly.
If we assume the horizontal diameter to be equal to 4 feet,
the value of r is, for the sewer running full, equal to 1.1584 feet,
for the sewer running j full to 0.80 feet.
Taking m = 0.57 (corresponding to common brickwork) the
value of 66 ( Vr + m) Vr is in the first case equal to 113.6, in the
second to 89.48.
For r = 0.8 the velocity necessary to prevent a deposit is
equal to
2 H — ^—^ = 2.156 ft. per second.
O.o
For this velocity the slope will be
For the same slope and the sewer running full the velocity will
be _ __
V = (66 ( Vr + m) Vrs)^^ = (113.6 X 0.02304)if = 2.575 feet
per second.
The cross-section is equal to 1.147 d"^ = 18.864 f , hence the
discharge, for the sewer running full
Q = 18.864 X 2.575 = 48.57 f
and Q = i^l^ X 2.516 = 9.09 f
for the sewer running I full. Thus, while the actual discharge in
cubic feet per second for the sewer running full is 5.34 times the
discharge of the sewer running k full, the difference in the speed
of flow is only 2.575 - 2.156 = 0.419 feet per second.
EXPONENTIAL EQUATIONS 121
EXPONENTIAL EQUATIONS.
General Relations between Diameters and Velocities or Quan-
tities. General Relations between Slopes and Velocities or
Quantities.
Long Circular Conduits Running Full.
A.
If in our general equation for the velocity of flow we substitute
d, the diameter of a conduit in feet, for r, its mean hydraulic
radius, the formula thus transformed will read:
V = 23.34 ( V5 + 1.414 m) VE
and for a = v\
V = (23.34 ("VS + 1.414 m) Vds)^
which may be written
V = 34.607 ( ^/5 + 1.414 m)^ d^ s^.
For the term (Vd + 1.414 m) and its variation with the
velocity we have no adequate substitute, containing as it does
two variables in an everchanging relation. This fact makes
the problem of finding an exponential equation, giving values as
exact as those found from the general formula an impossibility.
The powers of the diameter (or the mean hydraulic radius) to
which velocities and quantities are proportional are not con-
stant, even for the same degree of roughness, but vary with the,^
diameter (or the mean hydraulic radius) itself. On this account
exponential equations with constant values of the powers of
d or r are only approximations, fairly true between certain
limits, but the more incorrect the farther outside of these limits.
Such equations should only be considered as brief empirical
expressions, valuable only on account of their brevity; they
should never be substituted for the general formula when a
great degree of accuracy is desired.
Computing the velocities and discharges of two long straight
circular conduits of different diameters, but of the same degree of
roughness and having the same slope from the general equation,
we may by means of the data thus obtained find an expression for
the relation between the diameter and the velocity or the discharge
which holds good between the limits of the two values of d.
122 THE FLOW OF WATER
To find the exponents of the powers of d to which velocities
and quantities are proportional we may put
^ ^ log 7, - log V,
log di - log do
log Q, - log Q,
log d, - log d^
By means of these equations and for values of d between 1
and 50 inches for a = vh, and between 1 foot and 20 feet for
other values of a we find the following values of x and y.
a = vl m = 1.0 X = 0.67 y = 2.67
a = vl m = 0.95 x = 0.67 y = 2.67
a = vh m = 0.83 x = 0.68 y = 2.68
a = vh m = 0.68 x = 0.695 y = 2.695
a = vh m = 0.57 x = 0.70 y = 2.70
a = vj\ m = 0.53 a: = 0.70 y = 2.70
a = 1.0 m = 0.57 a: = 0.66 y = 2.66
a = 1.0 m = 0.45 x = 0.67 i/ = 2.67
a = 1.0 m = 0.30 x = 0.68 ?/ = 2.68.
Consequently, for m = 0.68 (pipes of planed staves, cast and
wrought iron, etc., all some time in use), velocities are pro-
portional to d^'^^^, quantities to d""'^^^ and we have between V
and d the relation
r 7 0.695
b^ = ^ .... (1)
J 0-695
^1 =^0 3^7695- («)
1.45S
and between Q and d
n /? 2.695
(2)
Q, _ d,'-"'
EXPONENTIAL EQUATIONS
123
Equations (a) 1 and 2 enable us to find velocities and
quantities for a diameter d^, provided velocities and quantities
for a diameter d^ are known.
From equations (6) 1 and 2 we may find the diameter c?j for
a velocity V^ or a quantity Q^, provided the diameter d^ for the
Velocity v^ or the discharge Q^ is known.
In the same manner we find for the relation between the slope
and velocities and quantities:
Sxft
s,= s.
v.\V
Q<, So* ■
«, = <f .
s, - s. (D""
Combining equation (1) and (3) we have:
v' \dj [sj
(3)
(a)
(h)
(4)
(«)
(&)
(5)
^' = %y fe) («)
<^1 = ^0
s. ==s.
S,\^
^
id)
,¥
(&)
(c)
124
THE FLOW OF WATER
fS\^
Combining (2) and (4) we have:
d.
Sj = s„
wv,.
I Q. y^'
Qc
©
(6)
(a)
(h)
(c)
By means of these equations we may find :
The value of V^ or Q^ for any value of d^ or S^.
The value of d^ for any value of V^, Q^, or S^.
The value of aS^ for any value of V^, Q^ or d^
provided values of V^, Q^, d^, S^ are known. In these equations
S^^ is substituted for S^ when a = v'^'^ and *S^ when a = 1.0.
Computing velocities and quantities of discharge for a con-
duit one foot in diameter and having different degrees of rough-
ness from the general formula we find for three values of (a)
the following equations:
1. For a = v^,
V = 93.25 - 59.84 (1 - m) d^ S^,
Q = 76. 69-47.0 {1 - m) dy S^.
2. For a = v^^'^,
V = 70.44 - 41 .85 (1 - m) d^ /S^^,
Q = 53.33 - 32.87 (1 - m) dy S^'\
3. For a = 1.0,
V = 56.33 -33.0 (1 - m) d- SK
Q =44.24 - 25.92 (1 - m) dy SK
In particular we have the following:
m = 1.0. New long straight conduits lined with clean cement
very smooth. Tin pipes. Plated pipes.
V =93.25 d'-'' S^',
Q = 7QM d'-'' S\
EXPONENTIAL EQUATIONS 125
m = 0.95. Very smooth new asphalt-coated cast or wrought-
iron pipes, also new asphalt-coated wrought-iron
and steel riveted pipes not exceeding 6'' in diame-
ter. New conduits of planed staves.
Q = 70.82 d'-^'>S*.
m = 0.83. Ordinary new asphalt- coated cast and wrought-iron
pipes. Wrought-iron pipes not coated. Glass and
lead pipes. Pipes lined with smooth, concrete,
Q = 65.01 d'-'' S^.
m = 0.68. Pipes lined with cement or smooth concrete, pipes
of planed or rough staves, cast and wrought-iron
pipes, coated or not coated, wrought-iron and
steel-riveted pipes not exceeding 36'' in diameter
V = 74,1 d'"''S^,
Q = 5S.2d'-'''S^.
(All some time in use but fairly clean.)
m = 0.57. Sewer pipe. Conduits lined with common brick-
work.
V = 52.5 d'-'' >St%
Q = 41.27 d'-' >ST7.
m = 0.53. New asphalt-coated steel-riveted pipe exceeding 36''
in diameter.
V = 50.77 d'-' S^^,
Q = 39.875 d'*' ^tV
M= 0.45. Old cast and wrought-iron pipes of all descriptions,
not very clean. Pipes of riveted steel exceeding
36" in diameter, in use for several years.
V =38.16 d''''S^,
Q =29.96 d''''S^.
M = 0.30. Old pipes of riveted steel exceeding 36" in diameter.
V = 33.23 d'-''SK
Q = 26.10 d'-'^SK
126 THE FLOW OF WATER
It will be observed that the powers of d to which Velocities
and Quantities are proportional vary with (a) the coefficient
of variation of c. The difference between the values of the
powers of d is equal to 0.04 between the successive values of (a).
For m = 0.68 we have for instance:
For a = v^ d'-"'\ d'-'^,
a = 1.0 d''''\ d''''\
a = -^ d'-
12.575
Sewers.
B.
For sewers of circular section the general equations for velocity
and quantity are:
for a = 7tV^
v = 71.41 -44.0 {l-m)d^S^'\
Q = 56.08 - 34.55 (1 - m) # S^'\
and for the egg-shaped section
v = 78.68 -47.7 {I - m) d^ S""^,
Q = 92.76 - 56.24 {1 - m) dy S^'\
d being the horizontal diameter.
The practically useful coefficients of roughness for sewers are
as follows:
m = 0.83, a: = 0.68, 2/ = 2.68, smooth concrete, very good
brickwork, brickwork washed with cement,
m = 0 . 70, a; = 0 . 69, 2/ = 2 . 69, good concrete, fairly good brick-
work, very well laid sewer pipe,
m = 0 . 57 common concrete or brickwork, common sewer pipe.
Sewers of all descriptions become in the course of a few years,
frequently in the course of a few months, coated with sewer
slime. Sewage, moreover, does not, on account of its greater
viscosity or stickiness, flow with the same velocity as pure water.
The most reliable data pertaining to flow in sewers of all descrip-
tions some time in use indicate, that a value of m greater than
0 . 57 cannot be safely taken.
SEWERS 127
For m = 0 . 57 we have in particular
a = v^^,
V = 52.55 d'-' S^"^,
Q = 4:1.27 d'-' SA
for the circular and
V = 57.4:3 d'-' S^"'',
Q = 67.70 d'-' S^"-^,
for the egg-shaped section.
For a = 1 . 0 we have for the same value of m,
i; = 42.19 d'-'' SK
Q = 33.14 d'-''SK
for the circular and
2; =46.30 d'-'' SK
Q = 53.11 c^^-'^ S^ for the egg-shaped section.
In all these equations the horizontal diameter is assumed to
be § of the vertical diameter or the vertical diameter 1^ the
horizontal diameter. For long sewers with easy curves the
equations corresponding to a = i^^^ should be taken, for sewers
with many sharp curves, sharp angles, etc., for sewers discharg-
ing under water against a hydraulic counterpressure the equa-
tions given under a = 1 . 0 give the best results.
Comparing the constants given for circular and egg-shaped
sewers we find for equal horizontal diameters and a = 1.0 the
relation
Velocity egg-shaped section _ 46.30 _ ^ ^q_
Velocity circular section 42.19
Discharge egg-shaped section _ 53.11 _ ^ ^^^
Discharge circular section 33.14
The velocity for the egg-shaped section is consequently 1 . 097
times and the discharge 1 . 602 times that of the circular section.
It would, however, be very erroneous to conclude that for an
equal velocity an egg-shaped sewer should have a horizontal
diameter of =0.911 and for an equal discharge a horizon-
tal diameter of = 0 . 622 times the diameter of the circular
sewer.
128 THE FLOW OF WATER
For equal velocities the relation between the horizontal
diameter of an egg-shaped section and the diameter of a circular
section is given by:
Horizontal diameter egg-shaped section
^42 . 19 rf"'^^ circular sectionV*^^^
('
46.30 /
and for equal discharges by:
Horizontal diameter egg-shaped section
^33.14 d'-'' circular sectionN*^'"'
-(
53.11 /
Assuming a circular sewer to have a diameter of 3 feet, an
egg-shaped sewer will have, for the same velocity a horizontal
diameter of
/42.19 X 2.065V-'^^
A1.515 ^
2.667 feet.
\ 46.3
or 0.889 the diameter of the circular sewer.
For an equal discharge the diameter of the egg-shaped sewer
will be
/33.14 X 18.59^
A0.376
) =2-
512 feet,
V 53.11
or 0.834 times the diameter of the circular section.
For a diameter of 12 feet we compute the relations, for equal
velocities
d = 10.42 egg-shaped section
12
for equal quantities
= 0.8701,
d = 10 . 10 egg-shaped section _ ^ ^^ .
I2 ^•^'^^•
Consequently we may say, that an egg-shaped sewer should
have for an equal velocity a diameter of 0 . 88 = | times the
diameter of the circular section and for an equal discharge a
OPEN CONDUITS
129
diameter of 0.834 = | times the diameter of the circular
section.
C. Open Conduits.
I. Form of the Cross-Section Most Favorable to Flow.
The form of cross-section most favorable to flow is one whose
top width is equal to the two side slopes, whose top and
bottom width together are equal to the wet perimeter, and
whose mean hydraulic radius is equal to one half the depth.
The semisquare is the simplest form fulfilling these condi-
tions, and the areas of the trapezoids may be expressed in
terms of the areas of this standard. Areas of trapezoids and
the semi-circle thus expressed are found in the following
table :
Form of Section.
Ratio of
Side slopes
Propor-
tional
Areas.
Bottom
width in
terms of
depth.
Semisquare
0:1
1.0
2d
Trapezoid
i-.l
0.906
1 .562d
h'-l
0.868
1 .236d
ti
f:l
0.875
\M
1:1
0.914
0 .828d
tt
U:l
0.976
0 .702d
tt
1^:1
1.053
0 .606d
tt
lf:l
1.141
0.532c«
tt
2:1
1.236
0 A12d
tt
2i:l
1.337
0 .424c?
tt
2h:l
1 .4425
0 .385d
tt
2f:l
1.525
0 .354d
•
3:1
1 .6625
0 .325d
Semicircle
0 .7854
_ __ __, - ^_
\ 1
1
/
\ 1
\ 1
j-— ^
1! X
/
N. 1
d\
N. 1
\i b
V
Fig. 7.
130 THE FLOW OF WATER
The area of a trapezoid is equal to
A = d(b + d tang a) ;
the wet perimeter to
P = h + 2dVl+ tang^ a;
the mean hydraulic radius to
P _ d(b -\- d tang a)
h + 2d Vl + tang' a
the bottom width to
, Area , .
0 = —7- — a tang a;
the top width to
T = —7- + d tang a.
OPEN CONDUITS
131
General Relations between the Velocity, the Discharge and
the Depth of Water in the Form of Section
most Favorable to Flow.
Computing the velocities of flow for two conduits having
mean hydraulic radii equal to 0.1 foot and 10.0 feet, respectively,
from the general formula
2; = [66 ( Vr + m) VrT~sY,
and using the values of v thus found in the equation
3, _ log ^1 - log ^0 .
log R^ - log R^
We find for the powers of R, to which the velocity is propor-
tional, the values tabulated below:
Power of R or d.
Values of m
otK.
a=VT\
a = FtV
a = 1.0
1
a— , a -
Vt-e
1
m = 1 .0 - 0 .95
0.69
0.67
m = 0 .83 - 0 .80
0.70
0.68
m = 0 .70 - 0 .68
0.71
0.69
m = 0 .57
0.70
m = 0 .45
0.715
m = 0 .30
0.735
w = 0.15
0.75
m = 0 .0 X = 1 .0
0
75
0
71
K = 1.25
0
765
0
725
K = 1.50
0
775
0
735 0
.70
K = 2.0
0
795
0
755 0
.71
Computing the velocities of flow for semisquares one foot in
depth from the general equation we find the following exponen-
tial equations:
a = 7^2^
1? = 129 - 73 (1 - m)d^S^^;
a = 7^'^
V = 111.6 - 63.3 (1 - m) d- S^'"^;
a = 1.0,
V = 85.86 - 46.61 (1 - m) d^ S^]
1
For
for
for
for
a =
yjV
V = 67.62 - 35.4 (1 - m)d' SA
132 THE FLOW OF WATER
2
In any of these equations the term 2 — may be sub-
stituted for its equivalent 1 — m. In particular we have for
the semicircle:
m = 1.0 Semicircular channels lined with clean cement,
V = 129 d'-'' aS".
m = 0.83 Semicircular channels of brickwork washed with
cement,
V = 116 d'-' aStt.
m = 0.70 Semicircular channels lined with rough boards,
V = 104.5 d^-^^ /STr.
For the semisquar^ we have:
m = 0.95 Channels lined with clean cement, planed boards,
V = 108.43 cZ«-^^>St\
Q = 216.86 ^^'^^^TT.
m = 0.80 Channels lined with smooth concrete, very good
brickwork,
v = 98.87 d'-^'S^'-^,
Q = 197 .74 d'-'' S^'\
m = 0 . 70 Channels lined with sawed boards or good brickwork,
V = 92ASd'-''S^'\
Q = 184.96 (i^-«^>ST\
m = 0.57 Channels lined with common brickwork, rough con-
crete or very good ashlar,
V = 84.23 d«-^«>S^
Q = 168.46 d'-^'S^'-^.
m = 0.45 Channels lined with rough brickwork, common
ashlar or very rough concrete,
V = 76.67 d'''''S^^,
Q = 153.34 d'-'"' S^"^.
m = 0.30 Channels lined with good rubble masonry,
V = 67.27 d'-'^'S^-^
e = 134.54 d^''''S^"\
OPEN CONDUITS 133
m = 0.15 Channels lined with rouglily hammered masonry,
channels in cemented gravel up to one inch in
diameter,
V = 57.8 d'-'' S-^^,
Q = 114.6 d'-'' S^'^,
a = 1.0.
m = 0.0 Channels lined with common rubble masonry, tunnels
K = 1.0 in rockwork, channels in cemented gravel exceeding
one inch in diameter,
i; = 39.24 d'-'' Si,
0 = 78.48 d'-'' SK
m = — 0.1 Fairly regular channels in loose sand, or sand with
K = 1.2 gravel imbedded
t; = 35.39 d'-'"' S\
Q = 70.78 d''-"' SK
m = — 0.2 Fairly regular channels in earth, free from debris or
K = 1.5 vegetation,
t; = 30.86 d'-'"' S^,
Q = 61.72 d'-'"' SK
m = — 0 . 32 Channels in earth with debris or vegetation,
K= 1.93
i; = 26.1 d'-'"' S\
Q = 52.2 d'-'"' SK
If the cross-section of the conduit is a trapezoid, the dis-
charges are multiplied by the proportional areas found in the
table given at the beginning of this chapter.
For a trapezoid having side slopes of 1: 1, for instance, the
discharges found from any of the equations given above are mul-
tiplied by 0 . 914, for side slopes of 2 : 1 by 1 . 236 etc. The depth
being the same, the velocity is not affected by the side slope.
Values of the powers of d relating to Velocities are found in
Table E ; those relating to Quantities of discharge in Table F.
Values of the sines of the slope and their roots are found in
Table C.
134 THE FLOW OF WATER
For channels in earth, in case the velocity exceeds the limit
where erosion begins, the following equations may be used:
m =
- 0.1
V
= 28.68
^0.725 ^fV,
K =
1.20
Q
= 57.36
^2.725 ^^
m =
-0.20
V
= 25.14
^0.735 ^A^
K =
1.50
Q
= 50.28
^2.735 gj%^
m =
-0.32
V
= 20.90
^0.755 gj%^
K =
1.93
Q
= 41.80
^2.755 ^t\
It is, however, more convenient to use the equations previ-
ously given, for which the powers of d and S are found in the
tables, and multiply the Velocities or Quantities found from the
formulas by the coefficient of variation of C, which in these
cases is equal to a = — r- • Values of a = — - are found in
^ y^^ yh
column 10, Table V.
II. General Equations.
In the design of cross-sections of channels it is not always
possible to use the form of section most favorable to flow. Other
forms are frequently required for special purposes, are constructed
at less cost, or offer other advantages.
For wooden flumes, for instance, the triangular section is fre-
quently adopted.
If the sideslopes of a triangle are 1 : 1, or the sides inclined 45°
(which is the usual sideslope for a flume), the area of its cross-
section, its mean hydraulic radius and consequently its velocity
and discharge are equal to those of a semisquare when
(1)
(2)
the depth = Varea of semisquare.
the top width = \/4 X area of semisquare.
In the design of channels in earth it is frequently necessary,
in order to keep the velocity below the eroding limit, to make
the sections wide and shallow, so that the frictional resistance
may be increased and the flow retarded. In many cases a
shallow section is also more easily constructed and at less cost.
OPEN CONDUITS 135
The general exponential equations, derived as previously
indicated, are as follows:
a = 7^
?; = 243 - 131 .6 (1 - m) r^ S^^
a = 7rV
V = 205.8 - 112 (1 - m) r^ S^,
a = 7t\
V = 176.3 - 93.5 (1 - m) r^^^,
a = 1.0,
V = 132 - 66 (1 - m) r^ S^,
1
a = — r-»
V = 100.6 - 48.3 (1 - m) r^ ,St\
2
In any of these equations 2 — — may be substituted for
(1 - m).
The practically most useful special equations are as follows :
a = V^'^ m = 1.0 V = 205.8 r'''' S^^
a = V^^ m = 0.83 v = 186.75 r°-^ .S"
a^Y^'' m = 0.70 i; = 172.3 r^'^' aS^t
a = V^^ m = 0.95 2; = 171.6 r«-" >St7
a = V^^ m = 0.80 v = 158.2 r«-«« ^S^^
a = 7^^ m = 0.70 2; = 148.1 r'-'^ 5r7
a = 7tV m = 0.57 7; = 136.1 r'-^ S^'^
a = V^ m = 0.45 t; = 121.9 r"-^^^ aS^^
a = 7^^ m = 0.30 ?; = 110.8 r^'^^^ .S^^
a = V^'^ m = 0.15 ?; = 96.8 r'''" S^'^
a = 1.0 m = 0.0 v= 66.0 r'*'' /S^
a = 1.0 K = 1.2 ?; = 60.0 r'-'"' S^
a = 1.0 K = 1.5 v= 52.8 r«-"^ aS^
a = 1.0 K = 1.93 v= 45.0 r'-''' S^
It will be observed that the constants of these equations are
equal to those given for the semisquare and the semicircle,
multiplied by 2^ = 2«-«^ 2«-««, 2°-«^ etc., as the case may be.
136 THE FLOW OF WATER
Explanation of the Use of the Tables of Velocities and
Quantities Q, H, and I.
Table G contains the quantities of discharge in cubic feet per
second of a conduit one foot in diameter, for seven different
degrees of roughness and 174 slopes.
For the discharge in gallons per second multiply the quanti-
ties found in the table by 7.48052. For the velocity of flow
1 14
multiply the quantities found in the table by = — or,
if the conduit is egg-shaped by _ = | nearly.
I.
To find the quantity of discharge for any diameter, and a
given slope, multiply the value of Q found on line with the given
slope under the value of m, which indicates the particular degree
of roughness of the conduit, by the value of d^ found in Table D
under the same value of m.
Example: What is the discharge of a cast-iron conduit same
time in use; the diameter being 36 inches, and the slope 1 : 10,000?
In Table G, in column under m = 0 . 68, in line with aS = 0 . 0001
we find Q = 0.3349 f per second.
In Table D, under d^ = d""'"'', we find for d = 36'', d'-'"' =
19.31. Hence Q = 0.3349 X 19.31 = 6.54 f per second.
II.
To find the loss of head or the slope corresponding to a given
discharge and a given diameter, divide the given discharge by
the value of d^ found in Table D as indicated above. The quo-
tient then found will indicate, in Table G, under the proper value
of m, the slope required to produce the given discharge.
Example: What is the loss of head corresponding to a dis-
charge of 55 f per second, the conduit being a new, asphalt-
coated, steel-riveted pipe 6 feet in diameter?
In Table D, under d^ = d^'\ we find for a diameter of 72"
d'-'^« = 126.18. Dividing 55 by 126.18 the quotient 0.436 is
OPEN CONDUITS 13T
the quantity of discharge of a conduit one foot in diameter for
the required slope.
In Table G, under m = 0.53, the value of Q coming nearest
to 0.436 is 0.439. This stands in line with S = 0.0002, conse-
quently S = 0.0002 is the slope required to produce the given
discharge.
III.
To find the diameter corresponding to a given discharge and
a given slope, divide the given discharge by the discharge of a
conduit one foot in diameter for the given slope as found in
Table G. The quotient is the value of d^ for which the diameter
is found in Table D.
Example: What will be the horizontal diameter of an egg-
shaped sewer, the discharge being 200 /^ per second, and the
slope 1 : 2500?
In Table G, in column headed ''Egg-shaped section," and in
line with S = 0.0004, we find Q = 1.0759. Dividing 200 by
1.0759 the quotient is 185.8.
In Table D, in column headed d^'''^ the nearest value above
185.8 is 191.3, which stands in line with d = 7 .0 feet; hence
7 feet is the horizontal diameter required to produce the given
discharge.
IV.
To find the loss of head or the slope required to produce a
given velocity for a given diameter, divide the given velocity
by the value of d^ found in Table D, as indicated above. The
quotient thus found, multiplied by 0.7854 = \^ will indicate,
in Table G, under the proper value of m, the slope required.
Example: What is the proper slope for an 8-inch sewer pipe?
For well constructed sewers the permissible velocity is
d
which gives for an 8-inch sewer v = 2.375 feet per second.
In Table D, in column headed d^ = d^'\ in line with d = S
inches, we find ^^ = 0.7529. Dividing 2.375 by 0.7529 the
quotient is 3.053, which multiplied by \\ gives Q = 2.477.
138 THE FLOW OF WATER
In Table G, in column headed m = 57, we find the nearest
value of Q above 2.477 to be 2.495, which is in line with
S = 0. 005, which is the slope required.
V.
In case the conduit is egg-shaped, proceed as before, but
instead of multiplying by 0.7854 = {^ multiply by 1.147 = f.
Example: What is the least permissible slope for an egg-
shaped sewer having a horizontal diameter of 10 feet?
0 25
In this case: v = 2 -\ — '-— = 2.025 feet per second.
a
In Table D, in column headed d^ = d^'"^ and in line with
d = 120 inches, we find d''' = 5.011. Dividing 2.025 by
5.011 the quotient is 0.404, which multiplied by 1.147 gives
Q = 0.473.
In Table G, in column headed "Egg-shaped section," we find
the nearest value of Q above 0.473 to be 0.4738, which is in
line with S = 0.000085, hence S = 0.000085 is the least per-
missible slope for a 10-foot egg-shaped sewer.
VL
In Table H we find velocities of flow in a semisquare one
foot in depth for the practically most useful values of m or K
and 174 slopes.
Table H applies to any trapezoid having the form of section
most favorable to flow.
To find the velocity of flow corresponding to any depth what-
soever, either in the semisquare, or the trapezoid having the
form of section most favorable to flow, multiply the values
found in the table by the values of d^ found in Table E.
Example: What is the velocity of flow in a channel in earth
having the form of section most favorable to flow, the bed of
the channel being covered with stones, the depth 10 feet and the
slope 1 : 10,000?
In Table H under Z = 1.93, and in line with S = 0.0001,
we find V = 0.261.
In Table E, in column headed K = 2.0, and in line with
d = 10, we find d'^''^' = 6.238. Multiplying the two quantities,
we find V = 1.628 feet per second.
OPEN CONDUITS 139
VII.
Remembering that in the form of section most favorable to
flow 2R = d; or the mean hydraulic radius equal to one half
the depth, it is plain that Table H can also be used to find the
velocity of flow in channels not having the form of section most
favorable to flow. It is only necessary always to consider
R = ^ d, and multiply or divide by the value of d, which corre-
sponds to the given value of R.
Example: What is the velocity of flow in a channel lined
with common brickwork, the slope being 1 : 10,000 and the
mean hydraulic radius 6 feet?
In Table H, in column headed m = 0.57, and in line with
S = 0.0001, we find v = 0.6424.
R = 6 is equal to d = 12.
In Table E, in column headed m = 0.57, and in line with
d = 12, we find Z)°-^ = 5.695. Multiplying the two quantities
we have v = 3.658 feet per second.
Example: What is the slope required for a velocity of 8 feet
per second, the conduit being a triangular flume of sawed boards
and the mean hydraulic radius equal to 2 feet?
R = 2 is equal to d = 4.0.
In Table E, in column headed m = 0.70 and in line with
d = 4:.0, we find d«-«« = 2.603. Dividing 8 by 2.603, the
quotient is 3.073, which is the value of v corresponding to a
depth of one foot.
In Table H, in column headed m = 0.70, we find the value
nearest to 3.073 to be 3.061, which is in line with S = 0.0016,
which is the required slope.
Example: What is the value of the mean hydraulic radius
required to produce a velocity of 2.8 feet per second, the slope
being 1 : 10,000 and the conduit a channel in earth in good con-
dition, free from stones and plants?
In Table H, in column headed K = 1.5 and in line with
S = 0.0001 we find v = 0.3086. Dividing 2.8 by 0.3086 the
quotient is 9.073.
140 THE FLOW OF WATER
In Table E, in column headed K = 1.5, we find the nearest
value above 9 . 073, to be 9 . 191, which stand in line with d = 17.5.
. 17 5
Hence R, the mean hydraulic radius required, is —^ =8 . 75 feet.
VIII.
As Table H holds good for all conduits having the form of
section most favorable to flow, it is evident that it may be
used to find velocities of flow in circular conduits running full.
To find velocities corresponding to any diameter, it is necessary
to keep in mind the fact, that the depth in a semicircle is one
half the diameter, and multiply or divide by the value of d
which corresponds to the semidiameter.
Example: What is the least permissible slope for a 6-inch
sewer pipe?
Here the semidiameter or the depth is 3 inches. The per-
0 25
missible velocity is 2; = 2 -f -^ = 2.5 feet per second.
d
In Table D, in column headed d^ = d^''', we find in line with
d = S inches, d'-^ = 0.3703. Dividing 2.5 by 0.3703, the
quotient is 6.751.
In Table H, under m = 0.57, the value of v coming nearest
to 6.751 is 6.757, which is in line with s = 0.0085. Hence
s = 0 . 0085 is the least permissible slope for a 6-inch sewer pipe.
IX.
For the classes of circular conduits whose degrees of roughness
are indicated by the coefficient m = 0.95, m = 0.83, m = 0.68.
Table H gives velocities also in case the conduit is between 300
and 1000 diameters in length, or has sharp elbows, such as city
mains.
Example: What will be the velocity of flow in a city main
3 feet in diameter, the slope being 1 : 200?
Here the semidiameter is 1.5 feet.
The difference in the powers of d to which the velocity is pro-
portional between a = V^ and a == V^^ is equal to 0.04. For
m = 0 . 68 or 0 . 70 and a = V^, we have for the power of d,
X = 0.705, consequently for a = 7i'« x = 0.665.
OPEN CONDUITS 141
In Table D, in line with d = 1.5, we find d'-'' = 1.307,
^0.67 _ 1^12, hence d''''' = 1.3095.
In Table H, in column headed m = 0.70 (which is sufficiently
equal to 0 . 68 to apply in such cases) and in line with S = 0. 005,
we find v = 5.595. Multiplying we have 1.3095 X 5.595
= 7.3266 feet per second. The discharge will be Q = 7.3266
X 9 X 0.7854 = 51.79 cubic feet per second.
X.
Table I gives the Quantities of discharge in cubic feet per
second of a semisquare one foot deep for the practically most
useful values of m or K and 174 slopes.
For the trapezoids or the semicircle the quantities given in
the table are to be multiplied by their proportional areas.
Example: What is the discharge of a channel lined with dry
rubble masonry, or a channel in rock work, or a channel in coarse
cemented gravel, the side slopes being one half to 1, the depth
12 feet, and the sine of the slope 0.0005?
In Table I, in column headed m = 0.0, and in line with
S = 0.0005 S, we find Q = 1.755.
In Table F, in column headed m = 0.0, and in line with
d = 12, we find d'-'' = 928.4.
For a sideslope of ^ : 1 the proportional area is 0 . 868. Mul-
tiplying the three quantities 1.755 X 928.4 X 0.868, we find
Q = 1414.2 f per second.
Example: What will be the dimensions of a channel in sand,
the discharge being 200 cubic feet per second, the slope 1 : 10,000
and the sideslopes 3:1? For a sideslope of 3 : 1 the propor-
tional area is 1.6625. Dividing 200 by 1.6625 we have for
the discharge of a semisquare of equal depth Q = 120.3 feet.
In Table I, in column headed K = 1.2 and in line with
S = 0.0001, we find Q = 0.708. Dividing 120.3 by 0.708
the quotient is 169.9.
In Table F, in column headed K = 1 . 25, we find the value
nearest to 169.9 to be 169.5, which stands in line with 6.4 feet,
which is the depth required.
The bottom width of the conduit will be 6.4 X 0.325 = 2.08
feet, the top width 6. 4x3x2 + 2. 08 =40. 48 feet.
142 THE FLOW OF WATER
The cross-section will be 6.4 X — ' — - — '■ — = 136.2 square
feet, and the velocity =1.46 feet per second.
lou . 2i
XI.
In the design of channels in earth, especially those in light
soils, it is necessary to keep the velocities of flow within the
eroding limits. For channels in light sandy soils a velocity
exceeding 1.5 feet per second should not be allowed, for chan-
nels in earth with some clay 2 . 5 feet per second should be the
limit.
To keep the velocities down two methods may be used :
(1) The slope may be reduced by means of weirs, dams and
drops.
(2) The mean hydraulic radius may be reduced by making
the channel wide and shallow.
Example: A channel in sandy soil is to carry 500 cubic feet
per second, the velocity is not to exceed 1.5 feet per second,
the sideslopes are to be 3 : 1 and the depth of the water 8 feet.
What will be the slope of the channel?
The area of the cross-section will be — -, = 333.3 feet.
1.0
The mean width — -^ = 41.66 feet.
8
The bottom width 41.66 - 8 X 3 = 17.66 feet.
The wet perimeter 17.66+2 VS' + (8 X 3)' = 68.26 feet.
333 3
The mean hydrauhc radius ^^ '^ = 4.883 feet.
*^ 68 . 26
R =4.883 corresponds to d = 9.766.
In Table E, in column headed K = 1 . 25, we find the value of
d«-^^-^for9.8tobe5.732and5.687for9.7,meanfor9.75 = 5.71.
Dividing 1.5 by 5.71 the quotient is 0.2627.
In Table H, in column headed K = 1.20, we find the value
coming nearest to 0.2627 to be 0.2625, which stands in line
with S = 0. 000055. This is equal to a fall of 0 . 29 feet per mile.
Example: A channel in sandy soil is to carry 500 cubic feet
per second at a velocity of 1.5 feet per second. The sideslopes
are to be 3 : 1 and the slope of the channel 0 . 5 feet per mile, or
OPEN CONDUITS
143
0 . 000095. What will be the depth of the channel and its bottom
width?
In Table H, in column headed K = 1.2 and in Hne with
S = 0.000095, we find v = 0.345. Dividing 1.5 by 0.345 the
quotient is 4.347.
In Table E, under K = 1.25, we find the value of d^*^^^ next
above 4.347 to be 4.383, which stands in line with d = 6.9.
Hence R = SA5.
If the channel is given a depth of 4 feet, its mean width will be
^-^ = 83.33 feet;
its bottom width 83.33 - 4 X 3 = 71.33 feet
its wet perimeter 71.33 + 2 V4' + (4 X 3)' = 96.53 feet;
333 3
its mean hydraulic radius . = 3 . 45 as above.
TABLE C.
Sines of Slopes and Roots of Sines of Scopes.
s
S^
sh
StV
Si
S
&^.
Sj\
SiS
Sh
000025
002580
00309
00366
005
00050
01390
01583
01779
02236
000030
00284
00341
00402
00548
00055
01467
01668
01882
02346
000035
00311
00371
00437
00591
00060
01541
01748
01970
02450
000040
00336
00399
00469
00631
00065
01612
01826
02054
02550
000045
00359
00425
0050
00671
00070
01680
01902
02136
02645
000050
00381
00451
00528
00707
00075
01747
01975
02216
02739
000055
00402
00475
00556
00741
00080
01806
02045
02287
02830
000060
00422
00498
00582
00775
00085
01874
02114
02369
02916
000065
00441
00520
00607
00806
00090
01935
02181
02441
030
000070
00461
00541
00631
00836
00095
01995
02247
02511
03082
000075
00478
00562
00655
00866
001
02047
0231
02581
03163
000080
00496
00583
00677
00895
0011
02166
02424
02714
03316
000085
00513
00602
0070
00922
0012
02276
02552
02842
03464
000090
00530
00621
00721
00948
0013
02380
02665
02965
03605
000095
00544
00640
00742
00975
0014
02481
02775
03084
03742
0001
00562
00658
00762
010
0015
02579
02878
03199
03873
000125
00637
00743
008572
01118
0016
02647
02985
03310
040
00015
00706
00821
00945
01225
0017
02768
03077
03428
04121
000175
00770
00893
01003
01323
0018
02858
03183
03523
04243
00020
00831
00960
01123
01414
0019
02946
03278
03624
04359
000225
00887
01024
01172
015
0020
03033
03374
03725
04472
00025
00941
01085
01248
01581
0021
03117
03462
03822
04583
000275
00993
01140
01303
01658
0022
0320
03552
03918
04691
00030
01043
0120
01365
01732
0023
03280
0363P
04011
04776
00035
01138
01301
01480
01871
0024
0336
03724
04103
04898
00040
01226
01442
01589
020
0025
03438
03808
04191
050
00045
01311
01474
01693
02122
0026
03515
03820
04280
05099
144
THE FLOW OF WATER
TABLE C — Continued.
s
Sh
St't
Sh
sh
S
S^
Sj\
Sj\
Sh
0027
03590
03941
04367
05196
0080
06614
07182
07760
08944
0028
03665
04051
04452
05292
00825
06731
07303
07888
09083
0029
03737
04165
04535
05384
0085
06843
07423
08013
09193
0030
03810
04206
04617
05478
00875
06956
07542
08139
09357
0031
0388
04282
04698
05567
0093
07067
07659
0826
09413
0032
03951
04357
04779
05656
00925
07177
07774
0838
09618
0033
04019
04430
04856
05744
0095
07286
07888
08499
09747
0034
04087
04503
04934
05831
00975
07393
08001
08611
09805
0035
04155
04595
05010
05916
01
07499
08111
08733
10
0036
04227
04646
05092
060
01025
07604
08222
08848
10117
0037
04287
04716
05159
06083
0105
07708
08330
08962
10247
0038
04352
04785
05233
06164
01075
07818
08440
09032
10369
0039
04415
04853
05305
06245
0110
07912
08544
09185
10488
0040
04479
04921
05377
06310
01125
08012
08650
09295
10600
0041
04541
04987
05446
06402
01150
08113
08753
09361
10724
0042
04603
05053
05519
06481
01175
08211
08857
09512
10840
0043
04665
05109
05586
06557
0120
08309
08959
09620
10954
0044
04726
05187
05655
06633
01225
08406
09061
09722
11068
0045
04786
05248
05722
06709
01250
08502
09161
09828
1118
0046
04845
05314
05789
06782
01275
08597
09261
09939
11292
0047
04904
05374
05855
06855
0130
08691
09360
10034
11368
0048
04962
05435
05922
06928
01325
08785
09457
10136
11511
0049
05020
05497
05986
070
01350
08878
09554
10236
11619
0050
05078
05567
06051
07071
01375
08970
09650
10337
11726
0051
05135
05618
06115
07141
0140
09060
09745
10436
11832
0052
05191
05678
06178
07212
01425
09152
09840
10535
11937
0053
05247
05738
06241
07280
01450
09242
09934
10632
12042
0054
05303
05795
06301
07347
01475
09342
10026
10729
12145
0055
05357
05854
06364
07415
0150
09420
1012
10825
12248
0056
05412
05912
06425
07483
01525
09508
10211
10920
12349
0057
05466
05970
06486
07548
01550
09595
10302
11014
12450
0058
05520
06026
06545
07616
01575
09682
10392
11108
1255
0059
05574
06083
06605
07681
016
09767
10480
11199
1265
0060
05627
06139
06664
07746
01625
09854
10571
11293
1275
0061
05680
06193
06723
07810
01650
09939
10659
11385
1285
0062
05731
0625
06781
07874
01675
10024
10748
11476
1294
0063
05783
06304
06839
07937
0170
10107
10834
11566
1304
0064
05834
06359
06895
080
01725
10190
1092
11656
1313
0065
05886
06413
06953
08062
01750
10273
11007
11745
1323
0066
05936
06466
07009
08124
01775
10332
11092
11834
1331
0069
05983
0652
07064
08185
018
10437
11177
11921
1342
0068
06036
06573
07121
08246
01825
10519
11261
12009
1351
0069
06086
06625
07176
08307
01850
10600
11346
12096
1360
0070
06136
06677
07231
08367
01875
1068
11429
12182
1369
0071
06186
06739
07285
08427
0190
1076
11512
12268
1378
0072
06234
06781
07339
08485
01925
1088
11594
12353
1388
0073
06283
06832
07393
08544
01950
1092
11676
12437
1396
0074
06331
06883
07447
08602
01975
10946
11757
1252
1406
0075
06379
06933
07500
08660
020
11074
11812
1261
1414
0076
06426
06983
07552
08718
0205
11228
120
1277
1432
0077
06478
07033
07605
08775
021
11383
12157
1294
1450
0078
06521
07083
07657
08832
0215
11534
12315
1310
1466
0079 06568 I
07133
07709
08888
0220
11658
1247
1336 1483
OPEN CONDUITS
145
TABLE C
— Concluded.
s
Si\
>sA
St't
Si
S
St\
St\
Sj\
2538
Si
0225
11842
1262
1343
150
075
2330
2434
2739
0230
11980
1228
1357
1517
080
2415
2521
2626
2828
0235
12118
1293
1373
1533
085
2497
2606
2712
2916
0240
12270
1308
1388
1550
090
2581
2689
2795
30
0245
12414
1323
1404
1565
095
2644
2750
2859
3082
0250
12550
1337
1419
1581
10
2739
2848
2956
3163
030
1391
1478
1562
1732
20
4044
4157
4265
4583
035
1517
1606
1695
1871
30
5080
5185
5287
5478
040
1635
1728
1820
20
40
5973
6066
6157
6310
045
1748
1842
1931
2122
50
6771
6852
6928
7071
050
1854
1950
2048
2236
60
7503
7567
7631
7746
055
1956
2073
2153
2346
70
8182
8232
8279
8367
060
2008
2155
2255
2449
80
8820
8854
8899
8944
065
2155
2251
2353
2550
90
0 .7425
0 .9442
0 .9457
0 .9487
070
2240
2344
2447
2646
1 .00
1 .0
1 .0
1.0
1 .0
TABLE D.
Powers of the Diameters of Circular, Semicircular and Egg-
shaped Conduits in Feet.
^0.66
^2.66
^.67
^2.67
dO.68
^2.68
^.695
^2.695
d«-7
d2-7
1
0.0194
0.00135
0.0189
0.00134
0.01844
0.00128
0.0178
0.00123
0.0176
0.00192
2
0.3064
0.0085
0.3010
0.00835
0.2956
0.0082
0.2878
0.0080
0.2852
0.60772
3
0.4005
0.0150
0.3950
0.0246
0.3807
0.0243
0.3729
0.02385
0.3703
0.0235
4
0.4842
0.0538
0.4790
0.0532
0.4738
0.0526
0.4660
0.0518
0.4635
0.1511
5
0.5611
0.0974
0.5562
0.0965
0.5514
0.0957
0.5460
0.0941
0.5418
0.0949
6
0.6329
0.1582
0.6285
0.1571
0.6241
0.1561
0.6177
0.1544
0.6155
0.1534
7
0.7006
0.2584
0.6968
0.2371
0.6932
0.2358
0.6875
0.2339
0.6857
0.2335
8
0.7652
0.840
0.7620
0.3386
0.7590
0.3373
0.7543
0.3353
0.7529
0.3343
9
0.8271
0.4656
0.8247
0.4644
0.8223
0.4630
0.8187
0.4610
0.8175
0.4602
10
0.8866
0.6187
0.8810
0.6145
0.8834
0.6134
0.8809
0.6917
0.8801
0.6112
11
0.9442
0.7931
0.9433
0.7926
0.9424
0.7919
0.9413
0.7908
0.941
0.7904
12
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
13
1.0542
1.2370
1.055
1.238
1.056
1.239
1.057
1.2404
1.0576
1.241
14
1.107
1.507
1.1088
1.509
1.1105
1.511
1.1131
1.515
1.1139
1.516
15
1.159
1.810
1.1613
1.814
1.1639
1.819
1.1678
1.825
1.1691
1.827
16
1.209
2.150
1.2126
2.156
1.2161
2.162
1.2213
2.171
1.2239
2.175
18
1 . 307
2.941
1.312
2.952
1.317
2.964
1.325
2.983
1.328
2.988
20
1.401
3.892
1.408
3.912
1.415
3.931
1.426
3.961
1.430
3.971
22
1.492
5.012
1.501
5.045
1.511
5.075
1.524
5.122
1.528
5.138
24
1.580
6.321
1.591
6.364
1.602
6.409
1.619
6.475
1.625
6.498
26
1.666
7.820
1.679
7.881
1.692
7.942
1.711
8.035
1.718
8.066
28
1 . 749
9.524
1.764
9.605
1.780
9.687
1.802
9.810
1.810
9.852
30
1.831
11.443
1.848
11.547
1.866
11.552
1.890
11.815
1.899
11.870
32
1.910
13.585
1.929
13.720
1.948
13.854
1.977
14.06
1.987
14.128
34
1.988
15.96
2.009
16.13
2.030
16.30
2.062
16.56
2.073
16.64
36
2.065
18.59
2.088
18.79
2.111
19.0
2.146
19.31
2.158
19.42
38
2.140
21.46
2.165
21.71
2.190
21.97
2.214
22.36
2.240
22.46
146
THE FLOW OF WATER
TABLE D. — Continued.
E ^
Q .S
^0.66
^2.66
^0.67
^2.67
^O.C8
^2.68
^ 0.t95
d 2.C95
rfO-7
d^--'
40
2.214
24.60
2.241
24.90
2.268
25.20
2.316
25.67
2.324
25.81
42
2.286
28.0
2.315
28.36
2.344
28.71
2.388
29.26
2.404
29.44
44
2.358
31.63
2.388
32.11
2.419
32.53
2.467
33.17
2.483
33.39
46
2.427
35.67
2.460
36.15
2.493
36.63
2.544
37.38
2.561
37.63
48
2.496
39.25
2.531
40.51
2.567
41.07
2.621
41.93
2.038
42.22
50
2.565
44.53
2.612
45.17
2.639
45.82
2.696
46.87
2.795
47.15
52
2.632
49.43
2.671
50.16
2.711
50.89
2.771
52.03
2.791
52.41
54
2.698
59.65
2.739
55.47
2.781
56.32
2.844
57.60
2.866
58.03
56
2.764
60.19
2.807
61.13
2.851
62.08
2.917
63.53
2.940
64.02
58
2.829
66.08
2.874
67.13
2.919
68.20
2.989
69.83
3.013
70.38
60
2.893
72.32
2.940
73.50
2.988
74.69
3.060
76.51
3.085
77.13
62
2.956
78.91
3.005
80.22
3.055
81.55
3.131
83.58
3.157
84.26
64
3.019
82.0
3.070
83.38
3.121
84.80
3.201
86.95
3.228
87.68
66
3.081
91.07
3.124
94.79
3.187
96.42
3.270
98.92
3.298
99.76
68
3.143
100.92
3.198
102.78
3.253
104.48
3.339
107.23
3.368
108.16
70
3.203
108.98
3.260
110.91
3.318
112.89
3.408
115.92
3.438
116.94
72
3.262
117.46
3.322
119.58
3.382
121.75
3.444
125.05
3.505
126.18
78
3.440
145.3
3.505
148.0
3.571
150.9
3.673
155.1
3.707
186.6
84
3.612
177.0
3.683
180.5
3.755
184.0
3.867
189.5
3.904
191.3
90
3.780
212.7
3.857
217.0
3.936
221.4
4.057
228.2
4.098
230.5
96
3.945
252.5
4.028
258.0
4.112
263.2
4.246
271.5
4.287
274.4
102
4.106
296.7
4.195
303.1
4.286
309.6
4.425
319.7
4.473
323.2
108
4.264
345.3
4.359
353.1
4.455
360.9
4.604
393.0
4.655
377.1
114
4.318
399.7
4.519
407.9
4.622
417.1
4.781
431.5
4.835
436.4
120
4.571
457.1
4.678
467.8
4.786
478.6
4.955
495.5
5.011
501.1
126
4.720
520.4
4.832
532.8
4.947
545.5
5.125
565.0
5.186
571.8
132
4.868
589.0
4.986
604.7
5.107
619.3
5.294
642.0
5.356
648.2
138
5.012
662.9
5.136
679.0
5.263
696.1
5.460
722.1
5.527
730.9
144
5.155
742.4
5.265
761.0
5.418
780.0
5.624
809.8
5.694
819.6
150
5.296
827.5
5.432
848.7
5.576
870.4
5.786
904.0
5.859
915.5
156
5.423
916.4
5.564
940.0
5.708
964.6
5.931
1002.5
6.008
1005.4
162
5.572
1015.5
5.719
1042.3
5.870
1069.8
6.103
1112.5
6.184
1125.1
168
5.708
1118.7
5.861
1148.6
6.017
1179.3
6.260
1226.9
6.343
1243.2
174
5.842
1228.1
6.0
1261.0
6.162
1259.6
6.264
1348.6
6.501
1366.6
180
5.974
1344.0
6.137
1381.0
6.306
1419.6
6.567
1478.0
6.656
1497.8
186
6.104
1467
6.274
1507
6.448
1549
6.716
1614
6.808
1636
192
6.233
1596
6.409
1641
6.589
1687
6.869
1758
6.964
1783
198
6.361
1732
6.542
1781
6.728
1831
7.017
1910
7.116
1937
204
6.488
1875
6.675
1939
6.867
1985
7.164
2071
7.266
2100
210
6.613
2025
6.815
2084
7.003
2145
7.310
2238
7.415
2291
216
6.737
2183
6.935
2247
7.138
2313
7.454
2415
7.563
2450
222
6.86
2348
7.064
2417
7.272
2489
7.597
2600
7.710
2638
228
6.982
2579
7.191
2596
7.466
2673
7.740
2794
7.855
2836
234
7.103
2700
7.317
2782
7.538
2866
7.881
2997
7.999
3041
240
7.222
2889
7.442
2977
7.668
3067
8.021
3208
8.142
3256
OPEN CONDUITS
147
TABLE E.
Powers of Depths of Water in the Form of Section most Favor-
able TO Flow. Powers of Mean Hydraulic Radii in General.
m =
m =
m =
m, =
m =
m =
K =
K =
K =
dOT
0.95
0.83
0.70
0.57
0.30
0.0
1.25
1 .5
2.00
T
in
2)0.6.7
2)0.68
2)0.69
2)0.70
2)0.735
2)0.75
2)0.765
2)0.775
2)0.796
Feet.
^0.67
2^0.68
2^0.69
2^0.70
2^0.735
2^0.75
2^0.765
2^0.775
2^0.795
0.05
0 .1344
0 .1304
0 .1265
0 .1228
0.1123
0 .1057
0.1010
0 .0981
0 .0924
0.10
0 .2138
0 .2090
0 .2044
0 .1996
0 .1841
0 .1779
0.1718
0 .1699
0 .1603
0.15
0 .2805
0 .2752
0 .2713
0 .2650
0 .2480
0 .2410
0 .2343
0 .2298
0 .2163
0.20
0 .3401
0 .3348
0 .3294
0 .3241
0 .3064
0 .2990
0 .2919
0 .2872
0 .2782
0.25
0 .3951
0 .3896
0 .3842
0 .3789
0 .3609
0 .3535
0 .3463
0 .3415
0 .3322
0.30
0 .4463
0 .4410
0 .4358
0 .4305
0 .4127
0 .4054
0 .3981
0 .3933
0 .3840
0.35
0 .4949
0 .4897
0 .4847
0 .4796
0 .4622
0 .4550
0 .4480
0 .4432
0 .4340
0.40
0 .5412
0 .5363
0 .5314
0 .5266
0.510
0 .5030
0 .4961
0 .4915
0 .4827
0.45
0 .5857
0 .5810
0 .5764
0 .5718
0 .5561
0 .5495
0 .5429
0 .5386
0 .5300
0.50
0 .6285
0 .6241
0 .6198
0 .6156
0 .6008
0 .5946
0 .5885
0 .5844
0 .5763
0.55
0.670
0.666
0.662
0 .6580
0 .6444
0 .6389
0 .6330
0 .6292
0 .6218
0.60
0 .7102
0 .7066
0 .7030
0 .6984
0 .6870
0 .6818
0 .6765
0 .6729
0 .6662
0.65
0 .7493
0 .7461
0 .7429
0 .7397
0 .7286
0 .7239
0 .7192
0 .7162
0 .7100
0.70
0 .7895
0 .7846
0 .7818
0 .7790
0 .7694
0 .7653
0 .7612
0 .7585
0 .7551
0.75
0 .8247
0 .8223
0 .8199
0 .8176
0 .8094
0 .8059
0 .8025
0 .8001
0 .7956
0.80
0 .8611
0 .8592
0 .8573
0 .8553
0 .8487
0 .8489
0 .8431
0.8411
0 .8374
0.85
0 .8968
0 .8955
0 .8939
0 .8924
0 .8874
0 .8853
0.8831
0 .8815
0 .8788
0.90
0 .9318
0 .9309
0.930
0 .9289
0 .9254
0 .9246
0 .9226
0 .9216
0 .9196
0.95
0 .9662
0 .9657
0 .9652
0 .9647
0 .9630
0 .9623
0 .9615
0 .9610
0 .9600
1.0
1.0
1.0
1.0
1.0
1.0
1 .0
1.0
1 .0
1.0
1.05
1 .0332
1 .0337
1 .0342
1 .0347
1 .0365
1 .0373
1 .0380
1 .0385
1 .0396
1.10
1 .0660
1.067
1 .068
1 .0690
1 .0726
1 .0741
1 .0756
1 .0768
1 .0787
1 .15
1 .0982
1 .0997
1 .1012
1 .1028
1 .1082
1 .1105
1 .1128
1 .1144
1.116
1.20
1.130
1.132
1.134
1 .1361
1 .1434
1 .1465
1 .1497
1 .1518
1.156
1.25
1 .1612
1 .1638
1 .1664
1 .1691
1 .1782
1 .1782
1 .1861
1 .1888
1 .1941
1.30
1.192
1.195
1.199
1 .2016
1 .2126
1 .2174
1 .2222
1 .225
1 .2319
1.35
1.223
1.227
1.230
1 .2338
1 .2468
1 .252
1.258
1 .262
1 .269
1.40
1.244
1.248
1.253
1 .265
1.281
1 .287
1.294
1.298
1.307
1.45
1 .283
1.287
1.292
1 .297
1.314
1.321
1.329
1.304
1.343
1.50
1 .312
1.318
1.323
1 .328
1.347
1.355
1.364
1.369
1.380
1 .55
1 .341
1.347
1 .353
1.359
1.380
1.389
1.398
1.405
1.417
1 .60
1.370
1.377
1.383
1 .389
1.413
1 .423
1 .433
1 .440
1.453
1.65
1 .399
1.406
1.413
1.420
1.445
1 .456
1 .467
1 .474
1.489
1.70
1.427
1.435
1.442
1.450
1.477
1 .489
1 .501
1 .509
1.525
1.75
1 .455
1.463
1.471
1.480
1.509
I .521
1 .534
1 .543
1 .561
1.80
1.483
1.491
1.500
1 .509
1.540
1 .554
1 .568
1 .577
1 .595
1.85
1.510
1.520
1.529
1.538
1.572
1.586
1 .601
1 .611
1 .631
1.90
1.537
1.547
1 .557
1.569
1.603
1.618
1 .634
1.644
1.665
1.95
1.564
1.575
1.586
1.596
1 .634
1.650
1 .667
1.717
1 .699
2.0
1 .589
1.600
1.611
1 .624
1.664
1.681
1 .700
1 .711
1 734
2.05
1.618
1 .629
1.641
1.653
1.695
1.713
1 .732
1 .744
1.769
2.10
1 .644
1 .657
1.669
1 .681
1.725
1 .745
1 .764
1 .777
1 .804
2.15
1.670
1.683
1 .696
1 .709
1 .755
1.776
1 .796
1.810
1 .837
2.20
1.696
1.710
1.723
1.736
1.785
1.809
1.828
1.842
1 .861
148
THE FLOW OF WATER
TABLE E. — Continued.
m =
m =
m =
m =
m =
m =
K =
K =
K =
dor
0.95
0.83
0.70
0.57
0.30
0.0
1.25
1.50
2.00
in
£)0.67
2)0.68
2)0.69
2)0.70
2)0.735
2)0.75
2)0.765
2)0.775
2)0.796
feet.
2^0.67
^0.68
2^0.69
2^0.70
220.735
220.75
220.765
220.775
220.795
2.25
1.718
1.732
1.746
1.764
1 .815
1.836
1.860
1.875
1.905
2.30
1 .747
1.762
1.777
1.792
1 .844
1.867
1.891
1.907
1 .939
2.35
1 .773
1.788
1.803
1.818
1.874
1.898
1 .922
1.939
1 .973
2.40
1.798
1.814
1.830
1.846
1.903
1.928
1.954
1 .971
2.006
2.45
1 .826
1.839
1 .856
1.872
1.932
1 .958
1.985
2.003
2.038
2.50
1.848
1.864
1.882
1.899
1.961
1 .960
2.016
2.034
2.069
2.55
1 .872
1.890
1 .908
1.926
1.990
2.018
2.046
2.065
2.104
2.60
1 .897
1.915
1.934
1 .952
2.019
2.047
2.077
2.097
2.137
2.65
1 .921
1.940
1.959
1.978
2.048
2.077
2.108
2.138
2.169
2.70
1 .946
1.965
1.984
2.004
2.075
2.106
2.136
2.160
2.203
2.75
1 .970
1.990
2.010
2.030
2.103
2.135
2.168
2.190
2.212
2.80
1 .994
2.014
2.035
2.056
2.131
2.165
2.198
2.221
2.267
2.85
2.017
2.038
2.060
2.081
2.159
2.194
2.228
2.252
2.306
2.90
2.041
2.063
2.085
2.107
2.187
2.223
2.258
2.282
2.331
2.95
2.064
2.089
2.110
2.132
2.215
2.251
2.288
2.313
2.363
3.0
2.088
2.111
2.134
2.157
2.242
2.279
2.317
2.343
2.395
3.05
2.111
2.135
2.159
2.183
2.270
2.308
2.347
2.373
2.426
3.10
2.134
2.159
2.183
2.208
2.297
2.337
2.376
2.403
2.458
3.15
2.159
2.182
2.207
2.233
2.324
2.364
2.406
2.433
2.488
3.20
2.180
2.205
2.231
2.258
2.351
2 .393
2.435
2.463
2.521
3.25
2.201
2.229
2.255
2.282
2.378
2.421
2.464
2.493
2 551
3.30
2.225
2.252
2.279
2.307
2.421
2.448
2.493
2.522
2.582
3.35
2.248
2.275
2.303
2.331
2.432
2.476
2.521
2.552
2.614
3.40
2.270
2.298
2.327
2.355
2.458
2.504
2.550
2.580
2.645
3.45
2.293
2.323
2.354
2.379
2.485
2.531
2.579
2.611
2.677
3.50
2.315
2.344
2.374
2.404
2.511
2.559
2.608
2.640
2.707
3.55
2.337
2 .367
2.397
2.428
2.538
2.586
2.636
2.666
2.738
3.60
2.359
2.389
2.420
2.451
2.564
2.613
2.664
2.699
2.769
3.65
2.381
2.412
2.443
2.476
2.590
2.641
2.693
2.725
2.799
3.70
2.403
2.434
2.467
2.499
2.616
2.668
2.720
2.755
2.830
3.75
2.424
2.457
2.489
2.522
2.642
2.695
2.749
2.785
2.858
3.80
2.446
2.479
2.512
2.546
2.668
2.722
2.777
2.814
2.890
3.85
2.468
2.501
2.536
2.590
2.693
2.748
2.804
2.843
2.920
3.90
2.489
2 .523
2.558
2.593
2.719
2.775
2.833
2.871
2.951
3.95
2.510
2.545
2.580
2.616
2.745
2.802
2.860
2.900
2.980
4.0
2.531
2.567
2.603
2.639
2.770
2.828
2.888
2.928
3.01
4.05
2.552
2.588
2.626
2.662
2.796
2.855
2.915
2.956
3.04
4.10
2.573
2.610
2.647
2.685
2.821
2.881
2.943
2.985
3.07
4.15
2.595
2.632
2.670
2.708
2.846
2.908
2.970
3.013
3.10
4.20
2.616
2.653
2.692
2.731
2.871
2.934
2.998
3.041
3.13
4.25
2.637
2.675
2.714
2.753
2.897
2.960
3.025
3.070
3.16
4.30
2.657
2.696
2.736
2.776
2.921
2.986
3.052
3.097
3.19
4.35
2.678
2.717
2.758
2.798
2.946
3.012
3.080
3.125
3.219
4.40
2.698
2.739
2.780
2.821
2.971'
3.038
3.107
3.153
3.247
4.45
2.719
2.760
2.801
2.844
2.997
3.064
3.134
3.180
3.277
4.50
2.740
2.781
2.823
2.866
3.021
3.090
3.160
3.208
3.306
4.55
2.760
2.802
2.845
2.888
3.045
3.115
3.187
3.235
3 .335
4.60
2.780
2.823
2.866
2.910
3.070
3.140
3.214
3.263
3.364
OPEN CONDUITS
149
TABLE E. — Continued.
m =
m =
m =
m =
m =
m =
K =
K =
K =
dor
0.95
0.83
0.70
0.57
0.30
0.0
1.25
1 .50
20
T
in
£)0.67
£)0.68
£)0.69
£)0.70
2)0.735
2)0.75
2)0.765
2)X).775
2)0.795
feet.
/20-67
2^0.68
2^0.69
^0.70
7^0.735
2^0.75
2^0.765
2^0.775
2^0.795
4.65
2.800
2.844
2.888
2.932
3.094
3.166
3.240
3. .290
3.393
4.70
2.820
2.864
2.909
2.954
3.119
3.192
3.267
3.318
3.422
4.75
2.840
2.885
2.930
2.976
3.143
3.218
3.294
3.345
3.451
4.80
2.860
2.906
2.952
2.998
3.163
3.243
3.320
3.373
3.480
4.85
2.880
2 .926
2.973
3.020
3.192
3.268
3.346
3.404
3.509
4.90
2.900
2.947
2.994
3.042
3.216
3.295
3.373
3.427
3.538
4.95
2.920
2.967
3.015
3.064
3.240
3.318
3.397
3.454
3.567
5.00
2.940
2.987
3.036
3.085
3.264
3.344
3.425
3.481
3.595
5.1
2.980
3.028
3.078
3.128
3.312
3.393
3.478
3.534
3.652
5.2
3.018
3.068
3.119
3.171
3.360
3.444
3.530
3.589
3.709
5.3
3.057
3.108
3.160
3.214
3.407
3.493
3.582
3.642
3.765
5.4
3.095
3.148
3.201
3.256
3.454
3.542
3.633
3.695
3.822
5.5
3.134
3.188
3.243
3.298
3.501
3.591
3.684
3.748
3.878
5.6
3.171
3.227
3.283
3.340
3.548
3.640
3.735
3.818
3.933
5.7
3.210
3.266
3.323
3.382
3.594
3.689
3.787
3.853
3.990
5.8
3.249
3.307
3.365
3.423
3.640
3.737
3.837
3.908
4.045
5.9
3.284
3.343
3.403
3.464
3.686
3.786
3.888
3.958
4.101
6.0
3.322
3.382
3.443
3.505
3.732
3.834
3.938
4.009
4.156
6.1
3.358
3.420
3.482
3.546
3.778
3.882
3.988
4.061
4.211
6.2
3.396
3.458
3.521
3.586
3.823
3.929
4.038
4.113
4.265
6.3
3.432
3.496
3.561
3.627
3.868
3.977
4.087
4.164
4.320
6.4
3.469
3.534
3.600
3.667
3.913
4.604
4.138
4.215
4.375
6.5
3.505
3.571
3.638
3.707
3.958
4.071
4.187
4.265
4.429
6.6
3.541
3.608
3.677
3.747
4.003
4.118
4.236
4.317
4.483
6.7
3.576
3.645
3.715
3.787
4.047
4.164
4.285
4.367
4.537
6.8
3.612
3.682
3.754
3.826
4.091
4.211
4.334
4.417
4.590
6.9
3.648
3.719
3.792
3.865
4.136
4.257
4.383
4.468
4.644
7.0
3.683
3.755
3.829
3.905
4.180
4.304
4.431
4.518
4.698
7.1
3.718
3.792
3.867
3.944
4.223
4.350
4.480
4.567
4.751
7.2
3.753
3.828
3.904
3.982
4.267
4.396
4.528
4.618
4.804
7.3
3.788
3.864
3.942
4.021
4.311
4.441
4.576
4.667
4.857
7.4
3.822
3.900
3.979
4.069
4.354
4.487
4.623
4.717
4.910
7.5
3.857
3.936
4.016
4.097
4.397
4.532
4.671
4.766
4.962
7.6
3.892
3.972
4.053
4.136
4.439
4.577
4.719
4.816
5.014
7.7
3.926
4.007
4.090
4.174
4.483
4.622
4.766
4.865
5.066
7.8
3.960
4.042
4.126
4.212
4.526
4.667
4.813
4.913
5.119
7.9
3.994
4.077
4.163
4.250
4.568
4.710
4.860
4.962
5.171
8.0
4.028
4.113
4.199
4.278
4.611
4.754
4.908
5.010
5.223
8.1
4.061
4.147
4.235
4.325
4.653
4.801
4.954
5.059
5.275
8.2
4.095
4.182
4.271
4.362
4.696
4.847
5.002
5.108
5.328
8.3
4.128
4.217
4.306
4.399
4.737
4.890
5.047
5.155
5.379
8.4
4.162
4.251
4.343
4.436
4.779
4.934
5.094
5.204
5.430
8.5
4.195
4.286
4.378
4.473
4.821
4.978
5.141
5.251
5.481
8.6
4.228
4.320
4.414
4.516
4.862
5.022
5.184
5.300
5.532
8.7
4.261
4.354
4.449
4.547
4.904
5.066
5.233
5.348
5.584
8.8
4.294
4.388
4.484
4.583
4.945
5.109
5.279
5.395
5.635
8.9
4.326
4.422
4.519
4.619
4.987
5.153
5.324
5.442
5.686
9.0
4.359
4.455
4.555
4.656
5.028
5.196
5.370
5.490
5.736
160
THE FLOW OF WATER
TABLE 'E. — Concluded.
m =
m =
m =
m =
m =
m =
K =
K =
K =
d or
0.95
0.83
0.70
0.57
0.30
0.0
1.25
1.50
2.00
r m
Feet.
£)0.67
2)0.68
2)0.69
2)0.70
2)0.735
2)0.75
2)0.765
2)0.775
2)0.795
^0.07
igo-o*
220.M
220.70
2^0.735
2^0.75
2^0.765
220.775
220.796
9.1
4.391
4.489
4.589
4.692
5.069
5.239
5.416
5.537
5.786
9.2
4.423
4.522
4.624
4.728
5.110
5.282
5.461
5.584
5.837
9.3
4.455
4.556
4.659
4.764
5.150
5.325
5.506
5.631
5.888
9.4
4.487
4.589
4.693
4.799
5.191
5.368
5.552
5.678
5.938
9.5
4.519
4.622
4.728
4.835
5.232
5.411
5.597
5.725
5.989
9.6
4.551
4.655
4.762
4.871
5.272
5.454
5.642
5.772
6.039
9.7
4.583
4.688
4.796
4.906
5.312
5.496
5.687
5.818
6.089
9.8
4.615
4.721
4.831
4.942
5.352
5.539
5.732
5.864
6.138
9.9
4.646
4.754
4.865
4.977
5.393
5.582
5.779
5.910
6.188
10.0
4.678
4.787
4.898
5.012
5.433
5.624
5.821
5.957
6.238
10.5
4.833
4.948
5.065
5.186
5.631
5.833
6.042
6.186
6.484
11.0
4.986
5.107
5.231
5.358
5.827
6.040
6.261
6.413
6.728
11.5
5.137
5.263
5.394
5.527
6.020
6.245
6.478
6.638
6.969
12.0
5.285
5.418
5.554
5.695
6.211
6.447
6.692
6.861
7.210
12.5
5.432
5.570
5.713
5.859
6.401
6.648
6.905
7.082
7.437
13.0
5.576
5.721
5.870
6.022
6.588
6.846
7.115
7.300
7.684
13.5
5.719
5.870
6.025
6.183
6.773
7.043
7.323
7.516
7.907
14.0
5.860
6.017
6.178
6.343
6.957
7.238
7.530
7.731
8.150
14.5
6.000
6.162
6.329
6.501
7.139
7.430
7.735
7.945
8.380
15.0
6.137
6.306
6.479
6.657
7.319
7.622
7.938
8.156
8.610
15.5
6.274
6.448
6.627
6.812
7.497
7.812
8.140
8.369
8.837
16.0
6.409
6.589
6.777
6.964
7.674
8.000
8.340
8.574
9.063
16.5
6.542
6.728
7.016
7.116
7.850
8.187
8.538
8.782
9.287
17.0
6.674
6.866
7.064
7.266
8.024
8.372
8.736
8.986
9.511
17.5
6.805
7.003
7.206
7.415
8.197
8.556
8.932
9.191
9.732
18.0
6.935
7.138
7.347
7.563
8.368
8.739
9.125
9.395
9.953
18.5
7.063
7.272
7.488
7.709
8.538
8.920
9.320
9.596
10.172
19.0
7.191
7.405
7.626
7.855
8.707
9.101
9.512
9.796
10.390
19.5
7.317
7.538
7.765
7.999
8.875
9.280
9.702
9.996
10.610
20.0
7.442
7.668
7.902
8.142
9.042
9.410
9.892
10.192
10.830
21.0
7.689
7.927
8.172
8.425
9.372
9.810
10.268
10.585
11.25
22.0
7.806
8.182
8.439
8.704
9.698
10.168
10.640
10.974
11.674
23.0
8.173
8.433
8.702
8.979
10.020
10.502
11.008
11.359
12.094
24.0
8.409
8.680
8.961
9.250
10.338
10.843
11.373
11.739
12.510
25.0
8.642
8.925
9.217
9.518
10.653
11.180
11.734
12.117
12.920
OPEN CONDUITS
151
TABLE F.
Powers of the Depths of Water in the Form of Section Most
Favorable to Flow.
d in
Feet.
m =
m =
m =
TO =
m =
m =
K=
K=
K=
0.95
0.83
0.70
0.57
0.30
0.0
1.25
1.5
2.0
^2.67
^2.68
^2.69
^2.70
^2.735
^2.75
^2.765
^2.775
^2.795
0.05
.000336
.000326
.000316
.000307
.000281
.000264
.000252
.000245
.000231
0.10
.00214
.00209
.00204
.00200
.001841
.00178
.001919
.00168
.001604
0.15
.00631
.00619
.00610
.00596
.00558
.00542
.00527
.00517
.00428
0.20
.01361
.01339
.01311
.01292
.01226
.01196
.01168
.01150
.01113
0.25
.02469
.02435
.02401
.02368
.02256
.02210
.02164
.02134
.02076
0.30
.04017
.03969
.03921
.03875
.03715
.03648
.03583
.03540
.03456
0.35
.06063
.06000
.05936
.05875
.05663
.05574
.05488
.05430
.05317
0.40
.08660
.08581
.08502
.08425
.08159
.08048
.07939
.07866
.07723
0.45
.1186
.1176
.1167
.1158
.1126
.1113
.1099
.1091
.1093
0.50
.1571
.1560
.1550
.1534
.1502
.1487
.1471
.1461
.1441
0.55
.2026
.2015
.2003
.1990
.1949
.1932
.1914
.1903
.1881
0.60
.2557
.2544
.2531
.2518
.2473
.2454
.2435
.2421
.2398
0.65
.3163
.3152
.3139
.3125
.3078
.3058
.3039
.3026
.3000
0.70
.3859
.3845
.3831
.3817
.3770
.3750
.3730
.3717
.3690
0.75
.4639
.4626
.4613
.4600
.4553
.4533
.4514
.4501
.4475
0.80
.5511
.5428
.5487
.5475
.5432
.5413
.5396
.5383
.5360
0.85
.6480
.6469
.6459
.6449
.6412
.6396
.6380
.6370
.6345
0.90
.7552
.7542
.7532
.7522
.7496
.7484
.7472
.7465
.7450
0.95
.8720
.8716
.8711
.8707
.8691
.8684
.8678
.8672
.8664
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.05
1.1391
1.1397
1.1402
1.1408
1.1428
1.1436
1 . 1444
1.1450
1.1461
1.10
1.290
1.291
1.292
1.294
1.298
1.300
1.302
1.303
1.305
1.15
1.452
1.454
1.456
1.458
1.465
1.469
1.472
1.474
1.478
1.20
1.627
1.630
1.633
1.636
1.647
1.651
1.656
1.659
1.665
1.25
1.814
1.819
1.823
1.827
1.841
1.847
1.853
1.858
1.866
1.30
2.015
2.020
2.025
2.030
2.050
2.059
2.066
2.071
2.082
1.35
2.228
2.235
2.242
2.249
2.272
2.283
2.293
2.30
2.314
1.40
2.456
2.464
2.472
2.481
2.510
2.523
2.535
2.544
2.561
1.45
2.697
2.707
2.717
2.727
2.763
2.778
2.794
2.804
2.825
1.50
2.952
2.964
2.976
2.988
3.031
3.050
3.068
3.081
3.106
1.55
3.222
3.236
3.251
3.265
3.316
3.338
3.359
3.377
3.404
1.60
3.508
3.524
3.540
3.557
3.616
3.641
6.668
3 . 685
3.720
1.65
3.808
3.827
3.846
3.866
3.934
3.954
3.994
4.014
4.054
1.70
4.124
4.146
4.168
4.190
4.268
4.303
4.337
4.360
4.407
1.75
4.456
4.481
4.506
4.531
4.621
4.660
4.699
4.726
4.778
1.80
4.804
4.832
4.860
4.889
4.991
5.035
5.080
5.110
5.170
1.85
5.168
5.200
5.232
5.265
5.379
5.429
5.479
5.513
5.582
1.90
5.550
5.585
5.621
5.658
5 . 786
5.842
5.899
5.937
6.013
1.95
5.962
6.002
6.042
6.068
6.212
6.274
6.338
6.380
6.466
2.0
6.356
6.400
6.444
6.498
6.643
6.727
6.797
6.845
6.940
2.05
6.798
6.847
6.896
6.947
7.123
7.198
7.278
7.330
7.436
2.10
7.250
7.304
7.358
7.413
7.608
7.693
7.780
7.837
7.955
2.15
7.720
7.779
7.839
7.899
8.114
8.208
8.302
8.366
8.495
2.20
8.209
8.273
8.339
8.405
8.640
8.743
8.847
8.917
9.059
2.25
8.696
8.767
8.838
8.931
9.188
9.301
9.415
9.464
9.646
2.30
9.243
9.321
9.398
9.477
9.757
9.880
10.004
10.111
10.256
2.35
9.790
9.873
9.958
10.044
10.348
10.482
10.616
10.708
10.892
152
THE FLOW OF WATER
TABLE F. — Continued.
d in
Feet.
m =
m =
m =
m —
m =
m =
K =
K =
K =
0.95
0.83
0.70
0.57
.0.30
0.0
1.25
1.5
2.0
^2.67
^2.68
^2.69
^2.70
^2.735
^2.75
(J2.765
^2.775
^2.795
2.40
10.355
10.446
10.538
10.631
10.952
11.106
11.253
11.352
11.553
2.45
10.141
11.040
11.139
11.265
11.597
11.754
11.913
12.021
12.276
2.50
11.548
11.654
11.761
11.870
12.256
12.426
12.600
12.710
12.950
2.55
12.175
12.289
12.405
12.52
12.92
13.12
13.21
13.43
13.69
2.60
12.82
12.95
13.07
13.20
13.34
13.84
14.04
14.18
14.45
2.65
13.49
13.63
13.76
13.89
14.37
14.59
14.80
14.94
15.27
2.70
14.18
14.32
14.47
14.63
15.13
15.35
15.59
15.74
16.06
2.75
14.90
15.05
15.20
15.35
15.91
16.15
16.40
16.56
16.90
2.80
15.63
15.79
15.95
16.12
16.71
16.97
17.23
17.41
17.77
2.85
16.88
16.56
16.73
16.91
17.54
17.82
18.10
18.29
18.68
2.90
17.16
17.35
17.53
17.72
18.39
18.69
18.99
19.19
19.61
2.95
17.97
18.16
18.36
18.56
19.27
19.59
19.91
20.12
20.56
3.0
18.79
19.00
19.21
19.42
20.18
20.51
20.86
21.09
21.56
3.05
19.64
19.86
20.08
20.31
21.11
21.47
21.83
22.08
22.57
3.10
20.51
20.75
20.98
21.22
22.09
22.45
22.84
23.10
23.62
3.15
21.40
21.65
21.90
22.15
23.06
23.46
23.87
24.14
24.70
3.20
22.32
22.58
22.85
23.12
24.08
24.50
24.93
25.22
25.81
3.25
23.27
23.54
23.82
24.10
25.11
25.56
26.02
26.33
26.96
3.30
24.24
24.53
24.82
25.11
26.19
26.67
27.14
27.47
28.14
3.35
25.23
25.53
25.84
26.16
27.29
27.79
28.30
28.64
29.34
3.40
26.26
26.59
26.98
27.23
28.42
28.95
29.48
29.84
30.58
3.45
27.29
27.65
28.02
28.32
29.58
30.13
30.69
31.08
31.86
3.50
28.36
28.72
29.00
29.44
30.76
31.34
31.94
32.34
33.16
3.55
29.45
29.83
30.21
30.59
31.98
32.60
33.22
33.60
34.50
3.60
30.57
30.97
31.37
31.77
33.23
33.87
34.53
34.97
35.88
3.65
31.72
32.13
32.55
32.97
34.50
35.18
35.87
36.34
37.29
3.70
32.89
33.33
33.76
34.21
35.81
36.52
37.25
37.74
38.74
3.75
34.09
34.55
35.01
35.47
37.15
37.89
38.65
39.17
40.22
3.80
35.32
35.80
36.26
36.80
38.53
39.30
40.10
40.63
41.73
3.85
36.57
37.07
37.57
38.08
39.93
40.74
40.62
42.14
43.29
3.90
39.86
38.37
38.90
39.43
41.36
42.21
43.08
43.67
44.88
3.95
39.17
39.71
40.26
40.82
42.82
43.42
44.62
45.24
46.50
4.0
40.51
41.07
41.64
42.22
44.34
45.25
46.20
46.85
48.17
4.05
41.87
42.46
43.06
43.67
45.85
46.82
47.82
48.49
49.87
4.10
43.27
43.88
44.50
45.14
47.42
48.43
49.47
50.17
51.61
4.15
44.89
45.54
46.19
46.64
49.02
50.08
51.15
52.13
53.39
4.20
46.14
46.81
47.48
48.17
50.65
51.76
52.88
53.65
55.20
4.25
47.62
48.38
49.02
49.74
52.32
53.46
54.64
55.46
57.06
4.30
49.00
49.85
50.59
51.33
54.02
55.21
56.44
57.26
58.96
4.35
50.67
51.42
52.19
53.06
55.95
57.00
58.27
59.12
60.89
4.40
52.24
5^.02
53.81
54.62
57.52
58.81
60.14
61.03
62.87
4.45
53.84
54.66
55.47
56.31
59.33
60.68
62.05
62.98
64.89
4.50
55.47
56.31
57.17
58.03
61.17
62.57
63.99
64.96
66.25
4.55
57.13
58.01
58.89
59.79
63.05
64.50
65. €8
66.98
69.05
4.60
58.83
59.73
60.65
61.58
64.96
66.46
68.00
69.05
71.19
4.65
60.44
61.37
62.32
63.41
66.91
68.47
70.07
71.02
73.37
4.70
62.30
63.28
64.26
65.26
68.89
70.51
72.17
73.29
75.60
4.75
64.09
65.09
66.12
67.15
70.92
72.76
74.31
75.48
77.87
4.80
65.91
66.95
68.00
69.08
72.98
74.72
76.49
77.70
80.18
OPEN CONDUITS
TABLE F. — Continued.
153
din
Feet.
m=
w=
m =
m=
m=
m =
K=
K=
K^
0.95
0.83
0.70
0.57
0.30
0.0
1.25
1.5
2.0
^2.67
^2.68
^2.69
^2.70
^2.735
^2.75
J2.765
^2.775
^2.795
4.85
67.75
68.83
69.93
71.04
75.08
76.88
78.72
80.06
82.54
4.90
69.65
70.75
71.88
73.05
77.21
79.08
80.98
82.28
84.94
4.95
71.54
72.70
73.87
75.07
79.39
81.31
83.29
84.63
87.38
5.0
73.50
74.69
75.90
77.00
81.60
83.59
85.63
87.03
89.87
5.0
77.49
78.77
80.06
81.37
86.14
88.27
90.46
91.92
94.99
5.2
81.61
82.96
84.34
85.25
90.85
93.12
95.44
96.85
100.3
5.3
85.87
87.31
88.78
90.27
95.69
98.12
100.61
102.30
105.77
5.4
90.26
91.80
93.36
94.94
100.71
103.29
105.94
107.89
111.44
5.5
94.79
96.43
98.09
99.76
105.90
108.67
111.46
113.37
117.31
5.6
99.46
101.20
102.95
104.74
111.25
114.16
116.65
119.74
123.34
5.7
104.08
105.80
107.79
109.87
116.77
119.85
123.02
125.2
129.6
5.8
109.50
111.44
113.42
115.15
122.45
125.7
129.1
121.7
126.1
5.9
114.3
116.4
118.5
120.6
128.3
131.8
135.5
137.8
142.7
6.0
119.6
121.7
123.9
126.2
134.4
138.0
141.8
144.3
149.6
6.1
125.0
127.2
129.6
132.0
140.6
144.4
148.4
150.9
156.7
6.2
130.5
132.9
135.3
137.9
147.0
151.0
155.2
158.1
164.0
6.3
136.2
128.8
141.7
144.0
153.5
157.8
162.3
165.3
171.5
6.4
142.0
144.7
147.4
150.2
160.3
164.8
169.5
172.6
179.2
6.5
148.1
150.9
153.7
156.7
167.2
172.0
176.9
180.2
187.1
6.6
154.2
157.2
160.1
163.2
174.4
179.4
184.1
188.0
195.2
6.7
160.5
163.5
166.8
170.0
181.7
187.0
192.4
195.9
203.6
6.8
167.0
170.3
173.6
176.9
189.2
194.7
200.4
204.3
212.2
6.9
173.7
177.1
180.4
184.1
196.9
202.7
213.5
212.7
221.1
7.0
180.5
184.0
187.6
191.3
204.8
210.9
217.1
226.5
230.2
7.1
187.4
191.1
194.9
198.8
212.9
219.2
225.8
230.2
239.5
7.2
193.7
197.5
202.4
206.5
221.2
227.9
234.7
239.4
249.0
7.3
201.9
205.9
210.0
214.2
229.7
236.7
243.8
248.7
258.8
7.4
209.3
213.6
217.9
222.3
238.4
245.7
253.2
258.3
268.9
7.5
217.0
221.4
225.9
230.5
247.3
254.9
262.8
268.1
279.1
7.6
224.8
229.4
234.1
238.9
256.5
264.4
272.5
278.1
289.8
7.7
232.8
237.6
242.4
247.5
265.8
274.1
282.6
288.5
300.4
7.8
240.9
245.9
251.0
256.3
275.3
284.0
292.7
298.9
311.5
7.9
249.3
254.5
259.8
265.2
285.1
294.1
303.4
309.7
322.7
8.0
257.8
263.2
268.7
274.4
295.1
304.2
314.1
321.1
334.3
8.1
266.5
272.1
277.9
283.7
305.3
315.0
325.1
331.9
346.1
8.2
275.3
281.2
287.2
290.8
315.8
325.9
330.6
343.4
358.3
8.3
284.4
290.4
296.7
303.1
326.3
336.9
347.7
355.2
370.5
8.4
293.6
300.0
306.4
313.0
337.2
348.2
359.5
367.2
383.1
8.5
303.0
309.6
316.3
323.2
348.3
359.6
371.4
379.4
396.0
8.6
312.7
319.5
326.4
333.5
359.6
371.4
383.6
391.1
409.2
8.7
322.5
329.6
336.8
344.1
371.2
383.4
396.0
404.7
422.6
8.8
332.5
339.8
347.3
354.9
383.0
395.7
408.8
417.8
436.4
8.9
341.9
349.4
358.0
365.9
395.0
408.2
421.7
431.1
450.3
9.0
353.
360.9
368.9
377.1
407.2
420.9
435.0
444.7
464.6
9.1
363.6
371.7
380.1
388.5
419.7
433.9
448.4
458.5
479.2
9.2
374.1
382.8
382.4
400.2
432.5
447.1
462.2
492.6
494.1
9.3
385.3
394.0
402.9
412.0
445.5
460.6
476.3
487.0
509.3
9.4
396.5
405.5
414.7
424.1
458.7
474.4
490.6
501.7
524.7
9.5
407.9
417.2
426.6
436.4
472.1
488.4
505.2
516.6
540.5
9.6
419.5
429.3
438.9
448.9
485.9
502.6
520.0
531.9
556.5
154
THE FLOW OF WATER
TABLE F. — Concluded.
din
Feet.
m =
m =
m =
m=
m=
m =
K =
K=
K =
0.95
0.83
0.70
0.57
0.30
0.0
1.25
1.5
2.0
^2.67
^2.68
^2.60
^2.70
^2.735
d^-^S
^.766
^2.775
^2.795
9.7
431.2
441.1
451.3
461.6
499.9
517.2
535.1
547.4
572.8
9.8
443.4
453.6
464.0
474.6
514.1
531.9
550.5
563.3
589.5
9.9
455.4
466.0
476.8
487.8
528.6
547.1
566.5
579.3
606.6
10.0
467.8
478.7
489.8
501.2
543.3
562.4
582.2
595.7
623.8
10.5
532.8
545.5
558.4
571.8
620.8
658.1
666.6
682.0
714.9
11.0
603.3
617.9
632.9
648.4
705.1
730.9
757.6
776.0
814.1
11.5
679.3
696.1
713.3
730.9
778.1
827.8
856.9
878.0
921.8
12.0
761.1
780.2
799.7
820.1
894.5
928.4
963.7
987.9
1038.3
12.5
848.7
870.4
892.6
915.5
1000.1
1038.7
1028.8
1106.4
1163.8
13.0
942.4
966.9
992.0
1017.8
1110.4
1157.0
1202.3
1233.7
1299.0
13.5
1042.3
1069.8
1098.0
1135.0
1234.4
1284.0
1335.0
1370.0
1443.0
14.0
1148.2
1179.3
1211.0
1243.0
1363.0
1419.0
1476.0
1516.0
1597.0
14.5
1261.0
1296.0
1330.0
1399.0
1501.0
1562.0
1626.0
1670.0
1762.0
15.0
1381.0
1419.0
1458.0
1498.0
1647.0
1715.0
1786.0
1835.0
1937.0
15.5
1509.0
1549.0
1592.0
1637.0
1801.0
1877.0
1956.0
2010.0
2125.0
16.0
1641.0
1687.0
1734.0
1783.0
1965.0
2048.0
2135.0
2195.0
2320.0
16.5
1781.0
1832.0
1884.0
1937.0
2137.0
2229.0
2325.0
2391.0
2528.0
17.0
1929.0
1971.0
2041.0
2100.0
2319.0
2420.0
2525.0
2597.0
2749.0
17.5
2084.0
2145.0
2207.0
2271.0
2569.0
2620.0
2135.0
2815.0
2980.0
18.0
2247.0
2313.0
2381.0
2450.0
2711.0
2832.0
2959.0
3044.0
3225.0
18.5
2417.0
2489 . 0
2563.0
2639.0
2922.0
3053.0
3264.0
3284.0
3481.0
19.0
2596.0
2673.0
2753.0
2836.0
3143.0
3285.0
3434.0
3536.0
3751.0
19.5
2782.0
2866.0
2952.0
3042.0
3375.0
3528.0
3689.0
3801.0
4033.0
20.0
2977.0
3067.0
3160.0
3257.0
3617.0
3871.0
3957.0
4079.0
4329.0
21.0
3391.0
3496.0
3604.0
3715.0
4133.0
4329.0
4528.0
4668.0
4962.0
22.0
3839.0
3960.0
4085.0
4213.0
4694.0
4917.0
5150.0
5312.0
5650.0
23.0
4323.0
4461.0
4603.0
4750.0
5300.0
5556.0
5823.0
6009.0
6395.0
24.0
4844.0
5000.0
5161.0
5328.0
5955.0
6245.0
6549.0
6762.0
7206.0
25.0
5401.0
5578.0
5761.0
5949.0
6658.0
6988.0
7334.0
7573.0
8077.0
OPEN CONDUITS
155
TABLE G.
Quantities of Discharge in Cubic Feet per Second of a
Conduit One Foot in Diameter.
Sec-
Section Circular.
tion
Egg-
shaped
Section Circular.
Sine of
a = vi
a = fT5
a = 1.0
the Slope.
m =
m =
m=
m=
m=
m =
m =
m=
0.95
0.83
0.68
0.57
0.57
0.53
0.45
0.30
.000025
0.1826
0.1675!o.l501
0.2479
0.1501
0.1460
0.1460
0.1305
.000030
0.2023
0.1856
0.1663
0.2730
0.1663
0.1608
0.1608
0.1430
.000035
0.2207
0.2024
0.1813
0.2962
0.1804
0.1744
0.1744
0.1544
.000040
0.2378
0.2182
0.1955
0.3179
0.1941
0.1872
0.1872
0.1651
.000045
0.2541
0.2331
0.2088
0.3383
0.2061
0.1993
0.1993
0.1751
.000050
0.2697
0.2473
0.2216
0.3577
0.2179
0.2107
0.2107
0.1846
.000055
0.2845
0.2609
0.2338
0.3763
0.2292
0.2216
0.2216
0.1936
.000060
0.2988
0.2740
0.2455
0.3940
0.2400
0.2321
0.2321
0.2021
.000065
0.3126
0.2867
0.2582
0.4110
0.2504
0.2421
0.2416
0.2104
.000070
0.3259
0.2989
0.2678
0.4275
0.2604
0.2518
0.2507
0.2183
.000075
0.3387
0.3107
0.2784
0.4434
0.2701
0.2612
0.2656
0.2260
.000080
0.3513
0.3222
0.2887
0.4588
0.2725
0.2702
0.2681
0.2334
.000085
0.3635
0.3333
0.2987
0.4738
0.2886
0.2790
0.2763
0.2406
.000090
0.3753
0.3442
0.3084
0.4884
0.2975
0.2876
0.2843
0.2496
.000095
0.3869
0.3554
0.3180
0.5024
0.3060
0.2959
0.2855
0.2544
.0001
0.3983
0.3652
0.3349
0.5164
0.3145
0.3071
0.2997
0.2610
.000125
0.4516
0.4141
0.3711
0.5803
0.3535
0.3418
0.3351
0.2851
.00015
0.5003
0 . 4588
0.4111
0.6400
0.3898
0.3769
0.3671
0.3271
.000175
0.5456
0.8004
0.4484
0.6944
0.4133
0.4090
0.3965
0.3452
.0002
0.5881
0.5408
0.4833
0.7453
0.4539
0.4390
0.4239
0.3691
.000225
0.6284
0.5749
0.5165
0.7932
0.4831
0.4672
0.4444
0.4006
.00025
0.6668
0.6258
0.5480
0.8387
0.5108
0.4940
0.4739
0.4127
.000275
0.7035
0.6453
0.5782
0.8821
0.5373
0.5196
0.4970
0.4328
.0003
0.7388
0.6776
0.6071
0.9225
0.5629
0.5433
0.5191
0.4520
.00035
0.8057
0.7390
0.6630
1.0023
0.6104
0.5903
0.5607
0.4663
.00040
0.8686
0.7785
0.6817
1.0759
0.6553
0.6337
0.5994
0.5220
.00045
0.9281
0.8512
0.7627
1 . 1449
0.6973
0.6743
0.6358
0.5536
.00050
0.9848
0.9030
0.8092
1.2105
0.7373
0.7130
0.6701
0.5836
.00055
1.0390
0.9529
0.8538
1.273
0.7755
0.7329
0.7048
0.6138
.00060
1.0911
1.0007
0.8967
1.333
0.8120
0.7853
0.7341
0.6393
.00065
1.1414
1.0468
0.9369
1.391
0.8475
0.8195
0.7641
0.6654
.00070
1.1899
1.0913
0.9779
1.447
0.8610
0.8520
0.7930
0.6905
.00075
1.2370
1.1345
1.0166
1.501
0.9139
0.8837
0.8208
0.7149
.00080
1.283
1.1765
1.0541
1.553
0.9456
0.9144
0.8477
0.7382
.00085
1.327
1.2173
1.0907
1.603
0.9765
0.9447
0.8738
0.7609
.00090
1.371
1.257
1.1248
1.653
1.0065
0.9733
0.8991
0.7830
.00095
1.413
1.296
1.1611
1.70
1.0357
1.0015
0.9238
0.8044
.001
1.454
1.334
1.1951
1.747
1.0642
1.0291
0.9477
0.8253
.0011
1.535
1.407
1.261
1.838
1.1193
1.0823
0.9940
0.8656
.0012
1.611
1.478
1.324
1.924
1.1721
1.1334
1.0382
0.9041
.0013
1.686
1.546
1.385
2.007
1.2228
1 . 1824
1.0806
0.9410
166
THE FLOW OF WATER
TABLE G. — Continued.
1
Sec-
Section Circular.
tion
Egg-
Section Circular.
shaped
Sine of
a = v'^
a = yTF
a=1.0
the Slope.
m=
m=
m —
m =
m=
7n =
m =
m—
0.95
0.83
0.68
0.57
0.57
0.53
0.45
0.30
.0014
1.757
1.612
1.444
2.088
1.272
1.2298
1.1214
0.9775
.0015
1.826
1.676
1.501
2.165
1.319
1.276
1.1608
1.0108
.0016
1.894
1.742
1.557
2.241
1.365
1.320
1.1988
1.0442
.0017
1.960
1.798
1.611
2.314
1.409
1.363
1.2357
1.0761
.0018 •
2.034
1.856
1.664
2.385
1.453
1.405
1.272
1.1083
.0019
2.089
1.914
1.715
2.454
1.495
1.446
1.306
1.1386
.0020
2.148
1.970
1.765
2.523
1.537
1.486
1.340
1.1672
.0021
2.208
2.025
1.814
2.588
1.576
1.524
1.373
1.1960
.0022
2.270
2.082
1.865
2.653
1.616
1.562
1.406
1.2242
.0023
2.323
2.131
1.909
2.715
1.654
1.599
1.437
1.252
.0024
2.385
2.183
1.955
2.777
1.692
1.636
1.468
1.279
.0025
2.435
2.234
2.001
2.838
1.729
1.672
1.498
1.305
.0026
2.489
2.283
2.046
2.898
1.765
1.707
1.528
1.331
.0027
2.543
2.332
2.085
2.956
1.801
1.741
1.557
1.356
.0028
2.595
2.381
2.133
3.014
1.835
1.775
1.586
1.381
.0029
2.647
2.428
2.175
3.070
1.870
1.808
1.614
1.405
.0030
2.698
2.474
2.217
3.126
1.904
1.841
1.642
1.430
.0031
2.748
2.520
2.259
3.180
1.937
1.873
1.669
1.453
.0032
2.798
2.566
2.299
3.234
1.970
1.905
1.695
1.476
.0033
2.847
2.610
2.339
3.288
2.002
1.936
1.722
1.500
.0034
2.895
2.655
2.379
3.340
2.034
1.967
1.748
1.522
.0035
2.942
2.699
2.418
3.391
2.066
1.996
1.773
1.544
.0036
2.989
2.742
2.457
3.442
2.097
2.028
1.798
1.566
.0037
3.036
2.784
2.495
3.493
2.127
2.057
1.823
1.588
.0038
3.082
2.826
2.532
3.542
2.158
2.086
1.848
1.608
.0039
3.126
2.867
2.569
3.591
2.187
2.115
1.872
1.628
.0040
3.172
2.909
2.606
3.640
2.217
2.144
1.896
1.651
.0041
3.216
2.950
2.643
3.688
2.246
2.172
1.919
1.671
.0042
3.260
2.990
2.679
3.735
2.275
2.200
1.943
1.691
.0043
3.304
3.030
2.715
3.782
2.304
2.227
1.965
1.711
.0044
3.346
3.069
2.750
3.829
2.332
2.255
1.988
1.731
.0045
3.389
3.109
2.786
3.875
2.360
2.282
2.011
1.751
.0046
3.431
3.147
2.820
3.920
2.387
2.308
2.033
1.770
.0047
3.473
3.185
2.854
3.964
2.415
2.335
2.055
1.789
.0048
3.514
3.223
2.888
4.009
2.442
2.361
2.076
1.808
.0049
3.555
3.261
2.922
4.053
2.468
2.387
2.098
1.827
.0050
3.596
3.298
2.955
4.096
2.495
2.413
2.119
1.845
.0051
3.636
3.335
2.988
4.140
2.521
2.438
2.140
1.864
.0052
3.676
3.372
3.021
4.182
2.547
2.463
2.161
1.882
.0053
3.716
3.408
3.053
4.225
2.573
2.488
2.182
1.900
.0054
3.755
3.444
3.086
4.265
2.598
2.517
2.202
1.917
.0055
3.794
3.480
3.118
4.308
2.623
2.538
2.223
1.936
.0056
3.833
3.515
3.150
4.350
2.649
2.562
2.243
1.953
.0057
3.871
3.550 I3.I8I
4.391 '2.674 12.586 12.262 '1.970
OPEN CONDUITS
157
TABLE G. — Continued.
Sec-
Section Circular.
tion
Egg-
Section Circular.
shaped
Sine of
the Slope.
a=F^
a=yT8
a=
1.0
m =
m =
m =
m =
m =
m =
m=
m=
0.95
0.83
0.68
0.57
0.57
0.53
0.45
0.30
.0058
3.909
3.585
3.212
4.431
2.699
2.610
2.282
1.987
.0059
3.947
3.620
3.244
4.471
2.724
2.634
2.302
2.005
.0060
3.984
3.655
3.274
4.511
2.748
2.658
2.322
2.022
.0061
4.022
3.689
3.305
4.551
2.772
2.681
2.341
2.038
.0062
4.060
3.723
3.335
4.590
2.796
2.704
2.360
2.055
.0063
4.095
3.756
3.365
4.630
2.820
2.727
2.379
2.072
.0064
4.132
3.789
3.395
4.668
2.843
2.750
2.398
2.088
.0065
4.168
3.823
3.425
4.707
2.867
2.772
2.417
2.104
.0066
4.204
3.856
3.455
4.745
2.890
2.795
2.435
2.120
.0067
4.240
3.887
3.484
4.783
2.913
2.817
2.453
2.136
.0068
4.275
3.921
3.513
4.820
2.936
2.839
2.472
2.152
.0069
4.311
3.953
3.543
4.858
2.959
2.861
2.490
2.168
.0070
4.345
3.985
3.571
4.895
2.982
2.883
2.508
2.184
.0071
4.380
4.017
3.600
4.932
3.004
2.905
2.525
2.199
.0072
4.415
4.049
3.629
4.969
3.026
2.926
2.543
2.215
.0073
4.449
4.081
3.656
5.005
3.049
2.948
2.561
2.230
.0074
4.483
4.112
3.684
5.041
3.071
2.969
2.580
2.245
.0075
4.517
4.143
3.712
5.077
3.093
2.990
2.596
2.261
.0076
4.551
4.174
3.740
5.113
3.114
3.011
2.613
2.275
.0077
4.588
4.208
3.770
5.148
3.136
3.032
2.630
2.290
.0078
4.619
4.235
3.795
5.184
3.157
3.057
2.647
2.305
.0079
4.651
4.266
3.822
5.219
3.179
3.074
2.664
2.320
.0080
4.684
4.296
3.849
5.254
3.200
3.094
2.681
2.334
.00825
4.767
4.372
3.917
5.340
3.254
3.145
2.722
2.370
.0085
4.847
4.445
3.983
5.425
3.304
3.195
2.763
2.406
.00875
4.927
4.518
4.048
5.509
3.355
3.244
2.804
2.441
.0090
5.005
4.591
4.113
5.592
3.406
3.293
2.843
2.476
.00925
5.083
4.662
4.177
5.673
3.456
3.342
2.882
2.510
.0095
5.160
4.732
4.240
5.254
3.505
3.389
2.921
2.544
.00975
5.236
4.802
4.304
5.834
3.553
3.436
2.960
2.577
.010
5.311
4.871
4.366
5.913
3.601
3.482
2.997
2.610
.01025
5.885
4.939
4.425
5.990
3.649
3.528
3.032
2.640
.0105
5.458
5.006
4.486
6.067
3.696
3.573
3.071
2.674
.01075
5.531
5.073
4.545
6.143
3.742
3.618
3.107
2.706
.011
5.603
5.139
4.605
6.218
3.788
3.663
3.143
2.737
.01125
5.675
5.204
4.663
6.293
3.833
3.706
3.179
2.968
.01150
5.745
5.269
4.721
6.367
3.878
3.750
3.214
2.799
.01175
5.815
5.333
4.779
6.439
3.746
3.793
3.249
2.829
.012
5.886
5.397
4.836
6.511
3.966
3.835
3.283
2.859
.01225
5.953
5.460
4.892
6.583
4.010
3 . 877
3.317
2.888
.0125
6.021
5.523
4.949
6.653
4.053
3.919
3.351
2.918
.01275
6.089
5.584
5.004
6.726
4.096
3.960
3.384
2.947
.0130
6.155
5.645
5.059
6.793
4.138
4.001
3.330
2.967
.01325
6.222
5.706
5.113
6.862
4.180
4.042
3.450
3.004
158
THE FLOW OF WATER
TABLE G. — Concluded.
Sec-
Section Circular.
tion
Egg-
shaped
Section Circular.
Sine of
a=v'^
a=FT5
a=1.0
the Slope.
m =
m =
m =
m= m =
m=
m =
m=
0.95
0.83
0.68
0.57 0.57
0.53
0.45
3.30
.0135
6.287
5.767
5.167
6.931
4.221
4.082
3.482
3.032
.01375
6.353
5.826
5.220
6.998
4.263
4.122
3.514
3.060
.014
6.417
5.886
5.274
7.065
4.303
4.160
3.546
3.088
.01425
6.482
5.945
5.326
7.132
4.344
4.201
3.578
3.116
.0145
6.545
6.003
5.379
7.198
4.384
4.240
3.609
3.143
.01475
6.609
6.062
5.431
7.263
4.424
4.278
3.640
3.170
.015
6.671
6.119
5.482
7.328
4.464
4.316
3.671
3.196
.01525
6.734
6.176
5.534
7.392
4.503
4.354
3.700
3.222
.0155
6.795
6.238
5.584
7.456
4.542
4.392
3.731
3.249
.01575
6.857
6.289
5.635
7.520
4.580
4.429
3.761
3.275
.016
6.917
6.344
5.684
7.582
4.618
4.466
3.791
3.302
.01625
6.978
6.400
5.735
7.645
4.657
4.503
3.821
3.327
.0165
7.038
6.456
5.784
7.707
4.695
4.540
3.850
3.352
.01675
7.099
6.510
5.833
7.769
4.732
4.576
3.880
3.379
.0170
7.158
6.565
5.882
7.830
4.769
4.612
3.909
3.404
.01725
7.217
6.619
5.931
7.891
4.806
4.648
3.936
3.428
.0175
7.276
6.673
5.979
7.951
4.843
4.683
3.965
3.453
.01775
7.334
6.726
6.027
8.011
4.879
4.718
3.993
3.477
.018
7.392
6.779
6.074
8.071
4.916
4.754
4.021
3.501
.01825
7.450
6.832
6.122
8.130
4.952
4.788
4.049
3.526
.0185
7.507
6.885
6.169
8.189
4.987
4.823
4.076
3.549
.01875
7.564
6.937
6.215
8.247
5.023
4.858
4.104
3.574
.019
7.620
6.989
6.262
8.305
5.059
4.892
4.131
3.598
.01925
7.676
7.040
6.309
8.363
5.094
4.926
4.158
3.621
.0195
7.732
7.092
6.354
8.420
5.128
4.959
4.185
3.644
.01975
7.788
7.142
6.40
8.477
5.163
4.993
4.212
3.668
.02
7.843
7.193
6.445
8.534
5.198
5.026
4.239
3.691
.0205
7.952
7.293
6.535
8.645
5.266
5.092
4.291
3.737
.021
8.061
7.394
6.625
8.757
5.334
5.158
4.343
3.782
.0215
8.169
7.492
6.713
8.867
5.401
5.222
4.395
3.827
.022
8.275
7.590
6.801
8.977
5.469
5.286
4.445
3.871
.0225
8.389
7.692
6.892
9.089
5.536
5.353
4.496
3.915
.023
8.485
7.782
6.972
9.189
5.597
5.412
4.545
3.958
.0235
8.582
7.871
7.052
9.294
5.661
5.474
4.594
4.001
.024
8.690
7.970
7.141
9.398
5.724
5.536
4.643
4.043
.0245
8.792
8.063
7.225
9.501
5.787
5.596
4.691
4.085
.025
8.893
8.155
7.307
9.604
5.850
5.657
4.739
4.127
.030
9.852
9.036
8.096
10.577
6.442
6.230
5.191
4.485
.040
11.583
10.623
9.518
12.317
7.502
7.255
5.994
5.221
.050
13.13
12.052
10.791
13.86
8.443
8.164
6.701
5.836
.060
14.55
13.34
11.957
15.27
9.298
8.992
7.341
6.393
.070
15.90
14.55
13.40
16.57
10.089
9.756
7,929
6.904
.080
17.11
15.69
14.05e
) 19.98
10.828
10.471
8.477
7.382
.090
18.28
16.76
15.02(
) 18.92
11.525
11.144
8.992
7.831
0.10
19.39
17.79
15.93f
)|20.0
12.18e
111.784
9.478
8.253
OPEN CONDUITS
159
TABLE H.
Velocities of Flow in a Semisquare One Foot in Depth.
a=V^^
a=1.0
Sine of
the
Slope.
m^
m =
m =
m =
m =
m =
K =
K =
K =
0.95
0.80
0.70
0.57
0.30
0.0
1.2
1.5
1.93
.000025
0.397
0.362
0.3386
0.3084
0.2479
0.1962
0.1770
0.1543
0.1305
.000030
0.4372
0.3986
0.3730
0.3396
0.2730
0.2150
0.1939
0.1691
0.1430
.000035
0.4744
0.4326
0 . 4046
0.3685
0.2962
0.2322
0.2095
0.1826
0.1544
.000040
0.5092
0.4642
0.4342
0.3955
0.3179
0.2482
0.2239
0.1952
0.1651
.000045
0.5419
0.4941
0.4621
0.4192
0.3383
0.2642
0.2375
0.2070
0.1751
.000050
0.5863
0.5346
0.5001
0.4451
0.3577
0.2775
0.2503
0.2182
0.1846
.000055
0.6027
0.5495
0.5139
0.4681
0.3763
0.2911
0.2625
0.2289
0.1936
.000060
0.6311
0.5754
0.5382
0.4902
0.3940
0.3040
0.2742
0.2391
0.2021
.000065
0.6584
0.6003
0.5615
0.5114
0.4110
0.3164
0.2854
0.2488
0.2104
.000070
0.6847
0.6243
0.5839
0.5318
0.4275
0.3283
0.2962
0.2582
0.2183
.000075
0.7102
0.6475
0.6057
0.5516
0.4434
0.3399
0.3066
0.2673
0 . 2260
.000080
0.7349
0.6701
0.6267
0.5708
0.4588
0.3510
0.3166
0.2761
0.2334
.000085
0.7588
0.6919
0.6472
0.5894
0.4738
0.3618
0.3264
0.2845
0.2406
.000090
0.7822
0.7131
0.6671
0.6075
0.4884
0.3723
0.3358
0.2928
0.2476
.000095
0.8048
0.7338
0.6863
0.6251
0.5024
0.3825
0.3450
0.3008
0.2544
.0001
0.8720
0.7541
0.7053
0.6424
0.5164
0.3924
0.3540
0.3086
0.2610
.000125
0.9295
0.8475
0.7927
0.7220
0.5803
0.4388
0.3958
0.3450
0.2851
.00015
1.025
0.9346
0.8742
0.7961
0.646
0.4806
0.4437
0.3868
0.3271
.000175
1.1122
1.0141
0.9485
0.8638
0.6944
0.5191
0.4683
0.4083
0 . 3452
.0002
1.1937
1.0884
1.0180
0.9272
0.7453
0.555
0.5006
0.4365
0.3691
.000225
1.270
1.1584
1.0835
0.9869
0.7932
0.5885
0.5433
0.4737
0.4006
.00025
1.3404
1.2248
1.1457
1.0423
0.8387
0.6205
0.5597
0.4880
0.4127
.000275
1.413
1.289
1.2044
1.0975
0.8821
0.6508
0.5870
0.5118
0.4328
.0003
1.477
1.347
1.260
1.1476
0.9225
0.6797
0.6131
0.5345
0.4520
.00035
1.605
1.464
1.369
1.2444
1.0023
0.7342
0.6623
0.5774
0 . 4663
.00040
1.723
1.571
1.470
1.338
1.0759
0.7852
0.7080
0.6172
0 . 5220
.00045
1.834
1.672
1.564
1.424
1.1449
0.8325
0.7509
0.6547
0.5536
.0005
1.939
1.768
1.654
1.506
1.2125
0.8775
0.7915
0.6901
0.5836
.00055
2.039
1.859
1.739
1.584
1.273
0.9229
0.8325
0.7258
0.6138
.0006
2.135
1.947
1.821
1.659
1.333
0.9613
0.8671
0.7559
0 . 6393
.00065
2.229
2.032
1.901
1.731
1.391
1.0005
0.9025
0.7868
0 . 6654
.0007
2.317
2.113
1.976
1.800
1.447
1.0383
0.9366
0.8165
0 . 6905
.00075
2.403
2.191
2.050
1.867
1.501
1.0750
0.9697
0.8454
0.7149
.0008
2.487
2.269
2.121
1.932
1.553
1.110
1.0012
0.8729
0.7382
.00085
2.568
2.341
2.192
1.995
1.603
1.1441
1.032
0.8998
0 . 7609
.0009
2.647
2.413
2.257
2.056
1.653
1.1774
1.062
0.9259
0.7830
.00095
2.724
2.483
2.323
2.116
1.70
1.2096
1.091
0.9512
0 . 8044
.001
2.799
2.551
2.388
2.174
1.747
1.241
1.1194
0.9759
0.8253
.0011
2.943
2.684
2.510
2.286
1.838
1.302
1.174
1.0236
0.8656
.0012
3.082
2.810
2.629
2.394
1.924
1.360
1.2262
1.0691
0 . 9041
.0013
3.215
2.932
2.742
2.498
2.007
1.404
1.276
1.1127
0.9410
.0014
3.344
3.049
2.852
2.598
2.088
1.468
1.325
1.1548
0.9775
.0015
3.469
3.163
2.958
2.694
2.165
1.520
1.371
1.1953
1.0108
.0016
3.589
3.273
3.061
2.788
2.241
1.570
1.416
1.2347
1 . 0442
.0017
3.706
3.379
3.161
2.879
2.314
1.621
1.460
1.267
1.0761
.0018
3.820
3.483
3.258
2.967
2.385
1.665
1.502
1.309
1.1083
.0019
3.931
3.584
3.353
3.053
2.454
1.711
1.543
1.346
1.1388
160
THE FLOW OF WATER
TABLE H. — Continued.
a^V^-^
a=1.0
Sine of
the
Slope.
m=
m=
m=
m =
m =
m=
K=
K =
K =
0.95
0.80
0.70
0.57
0.30
0.0
1.2
1.5
1.93
.0020
4.041
3.685
3.446
3.139
2.523
1.755
1.583
1.380
1.1672
.0021
4.145
3.779
3.535
3.220
2.588
1.798
1.622
1.414
1.1960
.0022
4.248
3.874
3.623
3.300
2.653
1.841
1.660
1.447
1.2242
.0023
4.349
3.966
3.709
3.378
2.715
1.882
1.698
1.480
1.252
.0024
4.449
4.056
3.794
3.456
2.777
1.923
1.734
1.521
1.279
.0025
4.546
4.145
3.877
3.532
2.838
1.962
1.770
1.543
1.305
.0026
4.641
4.232
3.958
3.605
2.898
2.001
1.805
1.574
1.331
.0027
4.735
4.317
4.038
3.678
2.956
2.039
1.839
1.604
1.356
.0028
4.827
4.401
4.117
3.749
3.014
2.076
1.873
1.633
1.381
.0029
4.918
4.484
4.194
3.820
3.070
2.114
1.906
1.662
1.405
.0030
5.008
4.566
4.271
3.890
3.126
2.149
1.939
1.690
1.430
.0031
5.094
4.645
4.345
3.957
3.180
2.185
1.971
1.718
1.453
.0032
5.180
4.724
4.418
4.024
3.234
2.220
2.003
1.746
1.476
.0033
5.265
4.801
4.491
4.090
3.288
2.254
2.034
1.773
1.500
.0034
5.350
4.898
4.562
4.155
3.340
2.288
2.064
1.800
1.522
.0035
5.432
4.953
4.633
4.219
3.391
2.322
2.094
1.826
1.544
.0036
5.514
5.027
4.703
4.283
3.442
2.355
2.124
1.852
1.566
.0037
5.594
5.101
4.771
4.346
3.493
2.389
2.153
1.877
1.588
.0038
5.674
5.173
4.839
4.407
3.542
2.419
2.182
1.902
1.608
.0039
5.753
5.245
4.906
4.468
3.591
2.451
2.211
1.928
1.628
.0040
5.830
5.316
4.972
4.528
3.640
2.482
2.239
1.952
1.651
.0041
5.907
5.386
5.037
4.588
3.688
2.513
2.267
1.976
1.671
.0042
5.982
5.455
5.102
4.647
3.735
2.543
2.294
2.000
1.691
.0043
6.057
5.523
5.166
4.705
3.782
2.573
2.321
2.024
1.711
.0044
6.132
5.591
5.229
4.763
3.829
2.603
2.348
2.047
1.731
.0045
6.206
5.659
5.293
4.820
3.875
2.633
2.375
2.070
1.751
.0046
6.278
5.724
5.354
4.876
3.920
2.662
2.401
2.093
1.770
.0047
6.350
5.790
5.415
4.932
3.964
2.691
2.427
2.116
1.789
.0048
6.421
5.854
5.476
4.987
4.009
2.719
2.452
2.138
1.808
.0049
6.491
5.919
5.536
5.042
4.053
2.747
2.448
2.160
1.827
.0050
6.561
5.982
5.595
5.096
4.096
2.775
2.503
2.182
1.845
.0051
6.630
6.045
5.654
5.150
4.140
2.802
2.528
2.204
1.864
.0052
6.698
6.108
5.713
5.203
4.182
2.830
2.553
2.226
1.882
.0053
6.767
6.170
5.771
5.256
4.225
2.857
2.577
2.247
1.900
.0054
6.832
6.230
5.827
5.307
4.265
2.884
2.601
2.268
1.917
.0055
6.901
6.292
5.885
5.360
4.308
2.910
2.625
2.289
1.936
.0056
6.967
6.352
5.941
5.412
4.350
2.937
2.649
2.310
1.953
.0057
7.032
6.412
5.997
5.462
4.391
2.962
2.672
2.330
1.970
.0058
7.097
6.471
6.053
5.513
4.431
2.989
2.695
2.350
1.987
.0059
7.162
6.530
6.108
5.563
4.471
3.014
2.719
2.370
2.005
.0060
7.226
6.588
6.162
5.613
4.511
3.040
2.742
2.390
2.022
.0061
7.289
6.646
6.217
5.662
4.551
3.065
2.765
2.410
2.038
.0062
7.352
6.704
6.270
5.711
4.590
3.090
2.787
2.430
2.055
.0063
7.415
6.761
6.324
5.759
4.630
3.115
2.810
2.450
2.072
.0064
7.477
6.818
6.377
5.808
4.668
3.140
2.832
2.469
2.088
.0065
7.539
6.873
6.429
5.856
4.707
3.164
2.854
2.488
2.104
.0066
7.600
6.930
6.481
5.903
4.745
3.188
2.876
2.507
2.120
.0067
7.660
6.985
6.533
5.951
4.783
3.212
2.897
2.526
2.136
OPEN CONDUITS
161
TABLE H. — Continued.
-•
a= yTs
a=
= 1.0
Sine of
the
Slope.
m=
m=
m =
m=
TO =
m =
K =
K==
A'=
0.95
0.80
0.70
0.57
0.30
0.0
1.2
1.5
1.93
.0068
7.721
7.040
6.585
5.997
4.820
3.236
2.919
2.545
2.152
.0069
7.781
7.095
6.636
6.044
4.858
3.260
2.940
2.564
2.168
.0070
7.840
7.149
6.686
6.090
4.895
3.283
2.962
2.582
2.184
.0071
7.899
7.202
6.737
6.136
4.932
3.307
2.983
2.601
2.199
.0072
7.958
7.256
6.787
6.181
4.969
3.330
3.004
2.619
2,215
.0073
8.017
7.309
6.837
6.227
5.005
3.353
3.024
2.637
2.230
.0074
8.075
7.362
6.886
6.272
5.041
3.376
3.045
2.655
2.245
.0075
8.132
7.414
6.935
6.321
5.077
3.399
3.067
2.673
2.261
.0076
8.189
7.467
6.984
6.361
5.113
3.421
3.086
2.690
2.275
.0077
8.246
7.519
7.033
6.405
5.148
3.444
3.106
2.708
2.29
.0078
8.303
7.571
7.081
6.449
5.184
3.466
3.126
2.726
2.305
.0079
8.359
7.622
7.129
6.493
5.219
3.488
3.146
2.743
2.320
.0080
8.415
7.672
7.176
6.536
5.254
3.510
3.166
2.760
2.334
.00825
8.553
7.799
7.294
6.644
5.340
3.565
3.215
2.803
2.370
.0085
8.689
7.923
7.410
6.757
5.425
3.618
3.264
2.845
2.406
.00875
8.824
8.045
7.525
6.853
5.509
3.671
3.311
2.887
2.411
.0090
8.957
8.167
7.639
6.957
5.592
3.723
3.358
2.928
2.476
.00925
9.087
8.286
7.749
7.058
5.673
3.774
3.405
2.968
2.510
.0095
9.216
8.403
7.860
7.159
5.754
3.825
3.450
3.008
2.544
.00975
9.344
8.520
7.969
7.258
5.834
3.895
3.495
3.048
2.577
.01
9.476
8.635
8.076
7.356
5.913
3.924
3.540
3.086
2.610
.01025
9.594
8.748
8.182
7.453
5.990
3.970
3.581
3.122
2.640
.0105
9.718
8.860
8.288
7.548
6.067
4.021
3.627
3.163
2.674
.01075
9.839
8.971
8.391
7.643
6.143
4.069
3.670
3.200
2.706
.011
9.960
9.082
8.494
7.736
6.218
4.116
3.713
3.237
2.737
.01125
10.079
9.188
8.596
7.829
6.293
4.163
3.755
3.274
2.768
.0115
10.197
9.298
8.696
7.921
6.367
4.208
3.796
3.310
2.799
.01175
10.314
9.404
8.796
8.011
6.439
4.254
3.837
3.345
2.829
.012
10.430
9.510
8.895
8.101
6.511
4.289
3.869
3.381
2.859
.01225
10.544
9.614
8.992
8.190
6.583
4.343
3.918
3.415
2.888
.0125
10.658
9.717
9.089
8.278
6.653
4.388
3.958
3.450
2.918
.01275
10.772
9.822
9.187
8.367
6.726
4.431
3.997
3.485
2.947
.013
10.881
9.921
9.280
8.452
6.793
4.461
4.024
3.509
2.967
.01325
10.991
10.022
9.374
8.537
6.862
4.517
4.074
3.552
3.004
.0135
11.10
10.121
9.467
8.622
6.931
4.560
4.113
3.586
3.032
.01375
11.209
10.220
9.559
8.706
6.998
4.602
4.151
3.619
3.060
.014
11.316
10.318
9.652
8.790
7.065
4.643
4.188
3.652
3.088
.01425
11.423
10.415
9.742
8.873
7.132
4.685
4.226
3.684
3.116
.0145
11.528
10.511
9.832
8.954
7.198
4.726
4.263
3.716
3.143
.01475
11.633
10.607
9.921
9.036
7.263
4.766
4.299
3.748
3.170
.015
11.737
10.702
10.01
9.117
7.328
4.806
4.336
3.780
3.196
.01525
11.840
10.796
10.098
9.197
7.392
4.845
4.370
3.810
3.222
.0155
11.943
10.890
10.185
9.276
7.456
4.886
4.407
3.842
3.249
.01575
12.025
10.981
10.272
9.373
7.520
4.925
4.442
3.873
3.275
.016
12.230
11.073
10.357
9.433
7.582
4.965
4.479
3.905
3.302
.01625
12.274
11.165
10.443
9.512
7.645
5.003
4.513
3.934
3.327
.0165
12.345
11.256
10.522
9.589
7.707
5.041
4.547
3.964
3.352
.01675
12.443
11.346
10.612
9.665
7.769
5.080
4.582
3.995
3.376
.017
12.54
11.435
10.696
9.742
7.830
5.118
4.617
4.025
3.404
162
THE FLOW OF WATER
TABLE H. — Concluded.
a=V^'^
a=1.0
Sine of
the
Slope
m=
m =
m =
m =
m=
m =
K =
K =
K=
0.95
0.80
0.70
0.57
0.30
0.0
1.2
1.5
1.93
.01725
12.64
11.524
10.779
9.817
7.891
5.154
4.649
4.053
3.428
.0175
12.74
11.612
10.861
9.892
7.951
5.191
4.683
4.083
3.453
.01775
12.83
11.700
10.943
9.967
8.011
5.228
4.716
4.112
3.477
.018
12.92
11.786
11.024
10.041
8.091
5.265
4.749
4.141
3.501
.01825
13.02
11.873
11.105
10.115
8.130
5.302
4.782
4.169
3.526
.0185
13.11
11.959
11.185
10.188
8.189
5.338
4.815
4.198
3.549
.01875
13.21
12.044
11.265
10.260
8.247
5.374
4.847
4.226
3.574
.019
13.33
12.157
11.344
10.332
8.305
5.409
4.879
4.254
3.598
.01925
13.40
12.213
11.423
10.404
8.363
5.445
4.911
4.282
3.621
.0195
13.48
12.297
11.502
10.475
8.420
5.480
4.943
4.310
3.644
.01975
13.58
12.380
11.579
10.546
8.477
5.515
4.975
4.337
3.668
.020
13.67
12.46
11.657
10.616
8.534
5.550
5.006
4.365
3.691
.0205
13.85
12.62
11.809
10.756
8.645
5.619
5.668
4.419
3.737
.021
14.03
12.79
11.962
10.895
8.757
5.687
5.129
4.472
3.782
.0215
14.20
12.95
12.112
11.031
8.867
5.754
5.191
4.525
3.827
.022
14.38
13.11
12.263
11.169
8.977
5.821
5.251
4.578
3.871
.0225
14.56
13.27
12.415
11.308
9.089
5.887
5.310
4.630
3.915
.023
14.72
13.42
12.55
11.432
9.189
5.952
5.369
4.680
3.958
.0235
14.89
13.57
12.70
11.563
9.294
6.016
5.427
4.731
4.001
.024
15.05
13.72
12.84
11.693
9.398
6.080
5.484
4.781
4.043
.0245
15.22
13.88
12.98
11.821
9.501
6.143
5.541
4.831
4.085
.025
15.38
14.03
13.12
11.948
9.604
6.205
5.597
4.880
4.127
.030
16.94
15.45
14.45
13.160
10.577
6.797
6.131
5.345
4.485
.040
19.73
17.99
16.83
15.32
12.317
7.851
7.082
6.174
5.221
.050
22.20
20.24
18.93
17.25
13.860
8.775
7.915
6.901
5.836
.060
24.45
22.30
20.85
18.99
15.27
9.613
8.671
7.559
6.393
.070
26.53
24.19
22.63
20.61
16.57
10.383
9.366
8.165
6.904
.080
28.48
25.96
24.28
22.12
17.78
11.10
10.012
8.729
7.382
.090
30.31
27.63
26.45
22.48
18.92
11.762
10.621
9.259
7.831
.100
32.04
29.22
27.33
24.89
20.00
12.41
11.194
9.760
8.253
OPEN CONDUITS
163
TABLE I.
Quantities of Discharge in Cubic Feet per Second of a Semi-
square One Foot in Depth.
a=V^'^
a=1.0
Sine of
the
Slope
m —
m =
m=
m =
m=
m =
K =
K =
K =
0.95
0.80
0.70
0.57
0.30
0.0
1.2
1.5
1.93
.000025
0.794
0.724
0.6772
0.6168
0.4958
0.3924
0.3540
0.3086
0.2610
.000030
0.8744
0.7972
0.7560
0.6792
0.5460
0.4300
0.3878
0.3382
0.2860
.000035
0.9488
0.8652
0.8092
0.7370
0.5924
0.4644
0.4190
0.3652
0.3088
.000040
1.0184
0.9284
0.8684
0.7910
0.6358
0.4964
0.4478
0.3904
0.3302
.000045
1.0838
0.9982
0.9242
0.8384
0.6766
0.5284
0.4750
0.4140
0.3502
.000050
1.1726
1.0692
1.0002
0.8902
0.7154
0.5550
0.5006
0.4364
0.3692
.000055
1.2054
1.0990
1.0278
0.9362
0.7526
0.5822
0.5250
0.4578
0.3892
.000060
1.2622
1.1508
1.0^64
0.9804
0.7880
0.6080
0.5484
0.4782
0.4042
.000065
1.2168
1.2006
1.1230
1.0228
0.8220
0.6328
0.5708
0.4976
0.4208
.000070
1.3694
1.2486
1.1678
1.0636
0.8550
0.6566
0.5924
0.5164
0.4366
.000075
1.4204
1.3950
1.2114
1.1032
0.8868
0.6798
0.6132
0.5346
0.4520
.000080
1.4698
1 . 3402
1.2534
1.1416
0.9176
0.7020
0.6332
0.5522
0.4668
.000085
1.5176
1.3938
1 . 2944
1.1788
0.9476
0.7236
0.6528
0.5690
0.4812
.000090
1.5644
1 . 4262
1.3342
1.2150
0.9768
0.7446
0.6716
0.5856
0.4952
.000095
1.6096
1.4676
1.3726
1.2502
1.0048
0.7650
0.6900
0.6016
0.5088
.0001
1.6440
1.5082
1.4106
1.2848
1.0328
0.7848
0.7080
0.6172
0.5220
.000125
1.8590
1.6950
1.5854
1.4440
1.1606
0.8776
0.7916
0.6900
0.5702
.00015
2.050
1.8692
1.7484
1.5922
1.280
0.9612
0.8874
0.7736
0 . 6542
.000175
2.2244
2.0282
1.8970
1.7276
1.3888
1.0382
0.9366
0.8166
0.6904
.0002
2.3894
2.1768
2.0360
1.8544
1 . 4906
1.110
1.0012
0.8730
0.7382
.000225
2.540
2.3168
2.1670
1.9738
1.5864
1.1770
1.0866
0.9474
0.8002
.00025
2.6808
2.4496
2.2914
2.0866
1.6774
1.2410
1.1194
0.9760
0.8254
.000275
2.826
2.578
2.4088
2.1950
1.7642
1.3016
1.1740
1.0236
0.8656
.0003
2.954
2.694
2.520
2.2952
1.8450
1.3594
1.2262
1.0690
0.9040
.00035
3.210
2.928
2.738
2.4888
2.0046
1.4684
1.3246
1.1548
0.9326
.00040
3.446
3.142
2.940
2.676
2.1518
1.5704
1.416
1.2344
1.044
.00045
3.668
3.344
3.128
2.848
2.2898
1.6650
1.5018
1.3094
1.1072
.0005
3.878
3.536
3 . 308
3.012
2.4250
1.7550
1.5830
1.3802
1.1672
.00055
4.078
3.718
3.478
3.168
2.546
1.8458
1.6650
1.4516
1.2276
.0006
4.270
3.894
3.642
3.318
2.666
1.9226
1.7342
1.5118
1.2786
.00065
4.458
4.064
3.802
3.462
2.782
2.001
1.8050
1.5936
1.3308
.0007
4.634
4.226
3.952
3.600
2.894
2.0766
1.8732
1.6330
1.3810
.00075
4.806
4.382
4.100
3.734
3.002
2.150
1.9394
1.6908
1.4298
.0008
4.974
4.534
4.242
3.864
3.106
2.220
2.0024
1.7458
1.4764
.00085
5.136
4.682
4.394
3.990
3.206
2.2882
2.064
1.7996
1.5218
.0009
5.294
4.826
4.514
4.112
3.306
2.3548
2.124
1.8518
1.5660
.00095
5.448
4.966
4.646
4.232
3.40
2.4192
2.182
1.9024
1.6088
.001
5.598
5.102
4.776
4.348
3.494
2.482
2.2398
1.9518
1.6506
.0011
5.886
5.368
5.020
4.572
3.676
2.604
2.348
2.0472
1.7312
.0012
6.164
5.620
5.258
4.798
3.848
2.720
2.4524
2.1382
1.8082
.0013
6.430
5.864
5.484
4.996
4.014
2.808
2.552
2.2254
1.882
.0014
6.688
6.098
5.704
5.196
4.176
2.936
2.650
2.3096
1.9550
.0015
6.938
6.326
5.916
5.388
4.330
3.04
2.742
2.3906
2.0216
.0016
7.178
6.546
6.122
5.576
4.482
3.14
2.832
2.4694
2.0884
.0017
7.412
6.758
6.322
5.758
4.628
3.242
2.92
2.534
2.1562
164
THE FLOW OF WATER
TABLE I. — Continued.
a=V^^
a=1.0
Sine of
the
Slope
m=
m =
m=
m=
w =
m =
K =
K =
K =
0.95
0.80
0.70
0.57
5.934
0.30
0.0
1.2
1.5
1.93
.0018
7.640
6.966
6.516
4.770
3.330
3.004
2.618
2.2166
.0019
7.862
7.168
6.706
6.106
4.908
3.422
3.086
2.692
2.2772
.0020
8.082
7.370
6.892
6.278
5.046
3.510
3.166
2.760
2.3344
.0021
8.290
7.558
7.070
6.440
5.176
3.596
3.244
2.828
2.3920
.0022
8.496
7.948
7.246
6.600
5.306
3.682
3.320
2.894
2.4484
.0023
8.698
7.932
7.418
6.756
5.430
3.764
3.396
2.960
2.504
.0024
8.898
8.112
7.588
6.912
5.554
3.846
3.468
3.042
2.558
.0025
9.092
8.290
7.754
7.064
5.676
3.924
3.54
3.086
2.610
.0026
9.282
8.464
7.916
7.210
5.796
4.002
3.610
3.148
2.662
.0027
9.470
8.634
8.076
7.356
5.912
4.078
3.678
3.208
2.712
.0028
9.654
8.802
8.234
7.498
6.028
4.152
3.746
3.266
2.762
.0029
9.836
8.968
8.388
7.64
6.14
4.228
3.812
3.324
2.810
.0030
10.016
9.132
8.542
7.78
6.252
4.298
3.878
3.38
2.86
.0031
10.188
9.390
8.690
7.914
6.36
4.370
3.942
3.436
2.906
.0032
10.360
9.448
8.836
8.048
6.464
4.440
4.006
3.492
2.952
.0033
10.530
9.602
8.982
8.18
6.576
4.508
4.068
3.546
3.0
.0034
10.70
9.956
9.124
8.310
6.68
4.576
4.128
3.60
3.044
.0035
10.864
9.906
9.266
8.438
6.782
4.644
4.188
3.652
3.088
.0036
11.028
10.054
9.406
8.566
6.884
4.710
4.248
3.704
3.132
.0037
11.188
10.202
9.542
8.692
6.986
4.774
4.306
3.754
3.176
.0038
11.348
10.346
9.678
8.814
7.084
4.838
4.364
3.804
3.216
.0039
11.506
10.490
9.812
8.936
7.182
4.902
4.422
3.856
3.256
.0040
11.66
10.632
9.944
9.056
7.28
4.964
4.478
3.904
3.302
.0041
11.814
10.772
10.074
9.176
7.376
5.026
4.434
3.952
3.342
.0042
11.964
10.910
10.204
9.294
7.570
5.086
4.588
4.0
3.382
.0043
12.114
11.046
10.332
9.410
7.564
5.146
4.642
4.048
3.422
.0044
12.264
11.182
10.458
9.526
7.658
5.206
4.696
4.094
3.462
.0045
12.412
11.318
10.586
9.64
7.750
5.266
4.750
4.140
3.502
.0046
12.556
11.448
10.708
9.752
7.84
5.324
4.802
4.186
3.54
.0047
12.70
11.580
10.830
9.864
7.928
5.382
4.854
4.232
3.578
.0048
12.842
11.708
10.952
9.974
8.018
5.438
4.904
4.276
3.616
.0049
12.982
11.838
11.072
10.084
8.106
5.494
4.956
4.320
3.654
.005
13.122
11.964
11.190
10.192
8.192
5.550
5.006
4.364
3.690
.0051
13.26
12.090
11.308
10.300
8.280
5.604
5.056
4.408
3.628
.0052
13.396
12.216
11.426
10.406
8.364
5.660
5.106
4.452
3.764
.0053
13.534
12.34
11.542
10.512
8.450
5.714
5.154
4.494
3.80
.0054
13.664
12.46
11.654
10.614
8.530
5.768
5.202
4.536
3.834
.0055
13.802
12.584
11.770
10.720
8.616
5.82
5.250
4.578
3.872
.0056
13.934
12.704
11.882
10.824
8.700
5.874
5.298
4.620
3.906
.0057
14.064
12.824
11.994
10.924
8.782
5.924
5.344
4.660
3.94
.0058
14.194
12.942
12.106
11.026
8.862
5.978
5.390
4.70
3.974
.0059
14.324
13.060
12.216
11.126
8.942
6.028
5.438
4.74
4.010
.006
14.452
13.176
12.324
11.226
9.022
6.080
5.484
4.78
4.044
.0061
14.578
13.392
12.434
11.324
9.102
6.130
5.530
4.82
4.076
.0062
14.704
13.408
12.54
11.422
9.180
6.180
5.574
4.86
4.110
.0063
14.830
13.522
12.648
11.518
9.26
6.230
5.620
4.90
4.144
.0064
14.954
13.636
12.754
11.616
9.336
6.28
5.664
4.938
4.176
.0065
15.078
13.746
12.858
11.712
9.414
6.328
5.708
4.976
4.208
.0066
15.2
13.860
12.962
11.806
9.490
6.376
5.752
5.014
4.240
OPEN CONDUITS
165
TABLE I. — Continued.
a=FTV
a=1.0
Sine of
the
Slope
m =
m =
m=
m =
rre =
m =
K =
K =
K=
0.95
0.80
0.70
0.57
0.30
0.0
1.2
1.5
1.93
.0067
15.32
13.970
13.066
11.902
9.566
6.424
5.794
5.052
4.272
.0068
15.442
14.080
13.170
11.994
9.64
6.472
5.838
5.090
4.304
.0069
15.562
14.190
13.272
12.088
9.716
6.520
5.88
5.128
4.336
.0070
15.680
14.298
13.372
12.18
9.790
6.566
5.924
5.164
4.368
.0071
15.798
14.404
13.474
12.272
9.864
6.614
5.966
5.202
4.398
.0072
15.916
14.512
13.574
12.362
9.938
6.660
6.008
5.238
4.430
.0073
16.034
14.618
13.674
12.454
10.010
6.706
6.048
5.274
4.460
.0074
16.150
14.724
13.772
12.542
10.082
6.752
6.090
5.310
4.490
.0075
16.264
14.828
13.870
12.654
10.154
6.798
6.134
5.346
4.522
.0076
16.378
14.934
13.968
12.722
10.226
6.842
6.172
5.380
4.550
.0077
16.492
15.038
14.066
12.810
10.296
6.888
6.212
5.416
4.580
.0078
16.606
15.142
14.162
12.898
10.368
6.932
6.252
5.452
4.610
.0079
16.718
15.244
14.258
12.986
10.438
6.976
6.292
5.486
4.640
.008
16.830
15.344
14.352
13.072
10.508
7.020
6.332
5.520
4.668
.00825
17.106
15.598
14.588
13.288
10.680
7.130
6.430
5.606
4.740
.0085
17.378
15.846
14.820
13.514
10.850
7.236
6.528
5.690
4.812
.00875
17.648
16.090
15.050
13.706
11.018
7.342
6.622
5.774
4.822
.009
17.914
16.334
15.278
13.914
11.184
7.446
6.716
5.856
4.952
.00925
18.174
16.572
15.498
14.116
11.346
7.548
6.810
5.936
5.020
.0095
18.432
16.806
15.720
14.318
11.508
7.650
6.900
6.016
5.088
.00975
18.688
17.04
15.938
14.516
11.668
7.750
6.990
6.096
5.154
.01
18.94
17.270
16.152
14.712
11.826
7.848
7.080
6.172
5.220
.01025
19.198
17.496
16.364
14.906
11.980
7.940
7.162
6.244
5.280
.0105
19.436
17.72
16.576
15.096
12.134
8.042
7.254
6.326
5.348
.01075
19.678
17.942
16.782
15.286
12.286
8.138
7.340
6.400
5.412
.011
19.920
18.164
16.988
15.472
12.436
8.232
7.426
6.474
5.474
.01125
20.158
18.376
17.192
15.658
12.586
8.326
7.510
6.548
5.536
.0115
20.394
18.596
17.392
15.842
12.734
8.416
7.592
6.620
5.598
.01175
20.628
18.408
17.592
16.022
12.878
8.508
7.674
6.690
5.658
.012
20.860
19.020
17.770
16.202
13.022
8.578
7.738
6.762
5.718
.01225
21.088
19.228
17.984
16.380
13.166
8.686
7.836
6.830
5.776
.0125
21.316
19.434
18.178
16.556
13.306
8.776
7.916
6.900
5.836
.01275
21.544
19.644
18.374
16.734
13.452
8.862
7.994
6.970
5.894
.013
21.762
19.842
18.560
16.904
13.586
8.922
8.048
7.018
5.934
.01325
21.982
20.044
18.748
17.074
13.724
9.034
8.148
7.104
6.008
.0135
22.20
20.242
18.934
17.244
13.862
9.120
8.226
7.172
6.064
.01375
22.418
20.440
19.118
17.412
13.996
9.204
8.302
7.238
6.120
.014
22.632
20.636
19.304
17.580
14.130
9.286
8.376
7.304
6.176
.01425
22.846
20.830
19.484
17.746
14.264
9.370
8.452
7.368
6.232
.0145
23.056
21.022
19.664
17.908
14.396
9.452
8.526
7.432
6.286
.01475
23.266
21.214
19.842
18.072
14.526
9.532
8.598
7.496
6.340
.015
23.474
21.404
20.02
18.234
14.656
9.612
8.672
7.560
6.392
.01525
23.680
21.592
20.196
18.394
14.784
9.690
8.740
7.620
6.444
.0155
23.886
21.780
20.370
18.552
14.912
9.772
8.814
7.684
6.498
.01575
24.050
21.962
20.544
18.746
15.040
9.850
8.884
7.746
6.550
.016
24.460
22.146
20.714
18.866
15.164
9.930
8.958
7.810
6.604
.01625
24.548
22.330
20.886
19.024
15.290
10.006
9.026
7.868
6.654
.0165
24.690
22.512
21.044
19.178
15.414
10.082
9.094
7.928
6.704
.01675
24.886
22.692
21.224
19.330
15.538
10.160
9.164
7.990
6.758
166
THE FLOW OF WATER
TABLE I. — Concluded.
a=
v-h
a=1.0
Sine of
the
Slope
m =
m=
m=
m =
m —
OT =
K^
K =
K =
0.95
0.80
0.70
0.57
0.30
0.0
1.2
1.5
1.93
.017
25.08
22.870
21.392
19.484
15.660
10.236
9.234
8.050
6.808
.01725
25.28
23.048
21.558
19.634
15.782
10.308
9.298
8.106
6.856
.0175
25.48
23.224
21.722
19.784
15.902
10.382
9.366
8.166
6.906
.01775
25.66
23.400
21.886
19.934
16.022
10.456
9.432
8.224
6.954
.018
25.84
23.572
22.048
20.082
16.142
10.530
9.498
8.282
7.002
.01825
26.04
23.746
22.210
20.230
16.260
10.604
9.564
8.338
7.052
.0185
26.22
23.918
22.370
20.376
16.378
10.676
9.630
8.396
7.098
.01875
26.42
24.088
22.430
20.520
16.494
10.748
9.694
8.452
7.148
.019
26.66
24.314
22.688
20.664
16.610
10.818
9.758
8.508
7.196
.01925
26.80
24.426
22.846
20.808
16.726
10.890
9.822
8.564
7.242
.0195
26.96
24.594
23.004
20.950
16.840
10.960
9.886
8.620
7.288
.01975
27.16
24.760
23.158
21.092
16.954
11.030
9.950
8.674
7.336
.020
27.34
24.92
23.314
21.232
17.068
11.10
10.012
8.730
7.382
.0205
27.70
25.24
23.618
21.512
17.390
11.238
10.136
8.838
7.474
.021
28.06
25.58
23.924
21.790
17.514
11.374
10.258
8.944
7.564
.0215
28.40
25.90
24.224
22.062
17.734
11.508
10.382
9.050
7.654
.022
28.76
26.22
24.526
22.338
17.954
11.642
10.502
9.156
7.742
.0225
29.12
26.54
24.830
22.616
18.178
11.774
10.626
9.260
7.830
.023
29.44
26.84
25.10
22.864
18.378
11.904
10.738
9.360
7.916
.0235
29.78
27.14
25.40
23.126
18.588
12.032
10.854
9.462
8.002
.024
30.10
27.44
25.68
23.386
18.796
12.160
10.968
9.562
8.086
.0245
30.44
27.76
25.96
23.642
19.002
12.246
11.082
9.662
8.170
.025
30.76
28.06
26.24
23.896
19.208
12.410
11.194
9.760
8.254
.030
33.88
30.90
28.90
26.32
21.154
13.594
12.262
10.690
8.970
.040
39.46
35.98
33.66
30.64
24.634
15.702
14.164
12.348
10.442
.050
44.40
40.48
37.96
34.50
27.62
17.550
15.830
13.802
11.672
.060
48.90
44.60
41.70
37.98
30.54
19.226
17.342
15.118
12.786
.070
53.06
48.38
45.26
41.22
33.14
20.766
18.732
16.330
13.808
.080
56.96
51.92
48.56
44.24
35.56
22.20
20.024
17.458
14.764
.090
60.62
55.26
52.90
44.96
37.84
23.524
21.242
18.518
15.662
0.100
64.08
58.44
54.66
49.78
40.0
24.82
22.388
19.526
16.706
WEIR DISCHARGES 167
Weir Discharges.
Francis' Formula.
The discharge of a sharp-edged measuring weir is usually
computed from Francis' formula, which reads:
Q =3.33 {h-nOAH) {H + hf^ - h^,
in which Q = discharge in cubic feet per second;
h = breadth of weir in feet;
n = number of end contractions;
H = the vertical distance between the crest of the
weir and the surface of the still water in the
reservoir or the channel;
h = the head due to the velocity of approach.
The head due to the velocity of approach is found from the
equation
i!
in which Q = discharge found from the formula given above,
neglecting the velocity of approach;
A = cross-section of the channel or reservoir parallel
to the weir, at the point where the surface
of the water begins to slope towards the weir.
If the discharge of the weir is small in comparison with the
width and depth of the channel or the contents of the reser-
voir the velocity of approach and the head due to it may be
neglected.
Table K contains values of 3.33 H^.
The table is used as follows:
Let the depth of the water from the crest of the weir to the
still surface be 3 feet.
168
THE FLOW OF WATER
Let the head due to the velocity of approach be 0.1 foot.
Then 3.33 {H + hf - 3.33 h^ = 3.33 (3.1)^ - 3.3 (0.1)^
= 18.176 -0.1053 = 18.0707.
Let the breadth of the weir be 10 feet and we have :
Q = (10 - 2 X 0.3) X 18.0707 = 169.86458 cubic feet per
second.
TABLE K.
H
3.33 H^
H
3.331/^
H
3.33^2
H
3.33//^
H
3.33^^
0.01
0.00333
0.30
0.5472
0.78
2.294
1.65
7.025
2.85
16.022
0.02
0.009406
0.32
0.6028
0.80
2.383
1.70
7.381
2.90
16.445
0.03
0.01722
0.34
0.6602
0.82
2.478
1.75
7.709
2.95
16.872
0.04
0.02664
0.36
0.7193
0.84
2.564
1.80
8.042
3.0
17.307
0.05
0.03898
0.38
0.7792
0.86
2.656
1.85
8.379
3.05
17.736
0.06
0.05125
0.40
0.8425
0.88
2.749
1.90
8.721
3.10
18.177
0.07
0.06167
0.42
0.9064
0.90
2.843
1.95
9.068
3.15
18.61
0.08
0.07535
0.44
0.9719
0.92
2.938
2.0
9.418
3.20
19.06
0.09
0.08991
0.46
1.0389
0.94
3.035
2.05
9.774
3.25
19.50
0.10
0.1053
0.48
1.1074
0.96
3.132
2.10
10.144
3.30
19.96
0.11
0.1215
0.50
1.1773
0.98
3.231
2.15
10.498
3.35
20.42
0.12
0.1383
0.52
1.2486
1.0
3.330
2.20
10.866
3.40
20.88
0.13
0.1561
0.54
1.3215
1.05
3.583
2.25
11.239
3.45
21.34
0.14
0.1744
0.56
1.3958
1.10
3.842
2.30
11.616
3.50
21.81
0.15
0.1934
0.58
1.4708
1.15
4.107
2.35
11.996
3.55
,22.27
0.16
0.2133
0.60
1.5476
1.20
4.377
2.40
12.381
3.60
22.75
0.17
0.2333
0.62
1.6260
1.25
4.654
2.45
12.770
3.65
23.22
0.18
0.2543
0.64
1.7050
1.30
4.936
2.50
13.163
3.70
23.70
0.19
0.2758
0.66
1.7855
1.35
5.223
2.55
13.560
3.75
24.18
0.20
0.2978
0.68
1.8672
1.40
5.516
2.60
13.960
3.80
24.67
0.22
0.3436
0.70
1 .y502
1.45
5.815
2.65
14.365
3.85
25.15
0.24
0.3915
0.72
2.0344
1.50
6.117
2.70
14.773
3.90
25.65
0.26
0.4413
0.74
2.1197
1.55
6.426
2.75
15.186
3.95
26.14
0.28
0.4938
0.76
2.206
1.60
6.739
2.80
15.602
4.0
26.64"
THE FORMULA OF BAZIN.
The weir formula of Francis is based on experiments made
with heads ranging between 5 and 19 inches and with weir crests
up to 10 feet in length.
The accuracy of the formula when appHed to flow over weirs
having end contractions has been demonstrated; it also gives
fairly good results when applied to flow over weirs whose sides
are flush with the walls of the channel of approach. In that
case n = 0. The difficulty in the application of the formula of
WEIR DISCHARGES 169
Francis consists in the fact, that it is frequently impossible to
evaluate properly the head due to the velocity of approach.
If the formula of Bazin is used the head due to the velocity of
approach does not enter directly into calculations; it is replaced
by a coefficient which depends for its value on the relation
between the head and the vertical distance between the crest and
the floor of the channel of approach.
Bazin conducted his experiments with weirs 0.5, 1.0 and 2.0
meters wide, the heads ranging between 0.05 meters (2 inches)
and 0.6 meters (24 inches). The crests of the weirs were raised
to various heights above the floors of the channels of approach
and the sides were flush with the walls of the channels. The
formula of Bazin reads :
Q = imh+ 0.55 (— ^T Lh V2^
• u- u A ^n7tt > 0.01418
m which m = 0.6075 + ; ,
h
h = the head above the crest to the surface of th3 still
water;
p = the depth of the water below the crest to the floor
of the channel of approach.
The formula as given holds good for any system of measure.
For Enghsh measure it may be written :
Q = (3.2485 h^ + 0.07914 Vh) L \l +0.55 7— ^J •
L {p + hfj
Values of 1 + —^ — 7- are found in Table L.a. It will
L (p + hfj
be observed, that the value of this factor diminishes rapidly as
h
V
the relation - diminishes in value
It is equal to 1.2444 for - = ^ . 1.0220 for - = y
pi p 4
• 1.1375 for - = i . 1.0068 for - = i
pi p 8
1.0611 for - = i .
p 2
170
THE FLOW OF WATER
TABLE L.a.
h:p
0.55 h?
h:v
0.55^3
1 : 25
1.0008
1 : 3
1.0344
20
1.0012
2.75
1.0391
15
1.0021
2.5
1.0449
10
1.0045
2.25
1.0529
9.5
1.0050
2
1.0611
9
1.0055
1.75
1.0727
8.5
1.0061
1.5
1.0880
8
1.0068
1.25
1.1086
7.5
1.0076
1.1375
7
1.0086
1.25
1.1698
6.5
1.0097
1.5
1.1979
6
1.0112
1.25
J
1.2228
5.5
1.0130
2
1.2444
5
1.0153
2.25
1.2638
4.75
1.0166
2.5
1.2806
4.5
1.0181
2.75
1.2960
4.25
1.0199
3
1.3095
4
1.0220
3.25
1.3220
375
1.0244
3.5
1.3331
*
3.5
1.0276
3.75
1.3431
i
3.25
1.0304
4
1 . 3520
WEIR DISCHARGES
171
TABLE L.b.
Values of q= 3.2485 h^ + 0.07914 Vh.
Q
Q
Q
Q
h
In Cu. Ft.
h
In Cu. Ft.
h
In Cu. Ft.
h
In Cu. Ft.
per Sec.
per Sec.
per Sec.
per Sec.
0.01
0.0111
0.82
2.4900
2.60
13.7481
4.65
32.7513
0.02
0.0204
0.84
2.5738
2.65
14.1393
4.70
33.2832
0.04
0.0418
0.86
2.6646
2.70
14.5405
4.75
33.8031
0.06
0.0695
0.88
2.7565
2.75
14.9417
4.80
34.3340
0.08
0.0960
0.90
2.8493
2.80
15.3529
4.85
34.8749
0.10
0.1279
0.92
2.9432
2.85
15.7641
4.90
35.4158
0.12
0.1625
0.94
3.0380
2.90
16.1753
4.95
35.9561
0.14
0.1998
0.96
3.1338
2.95
16.5864
5.0
36.4976
0.16
0.2304
0.98
3.2308
3.00
17.0176
5.10
37.5894
0.18
0.2819
1.0
3.3276
3.05
17.4387
5.2
38.7011
0.20
0.3261
1.05
3.5763
3.10
17.8698
5.3
39.8229
0.22
0.3724
1.10
3.8313
3.15
18.3010
5.4
40.9446
0.24
0.4208
1.15
4.0911
3.20
18.7421
5.5
42.0863
0.26
0.4712
1.20
4.3570
3.25
19.1732
5.6
43.2380
0.28
0.5238
1.25
4.6288
3.30
19.6143
5.7
44.3996
0.30
0.5773
1.30
4.9065
3.35
20.0654
5.8
45.5713
0.32
0.6330
1.35
5.1873
3.40
20.5165
5.9
46.7530
0.34
0.6904
1.40
5.4750
3.45
20.9675
6.0
47.9440
0.36
0.7493
1.45
5.7676
3.50
21.4186
6.2
50.3478
0.38
0.8090
1.50
6.0653
3.55
21.8797
6.4
52.8010
0.40
0.8720
1.55
6.3682
3.60
22.3407
6.6
55.2940
0.42
0.9356
1.60
6.6755
3.65
22.8018
6.8
57.8071
0.44
1.0080
1.65
6.9560
3.70
23.2728
7.0
60.3701
0.46
1.0673
1.70
7.3046
3.75
23.7438
7.2
62.9731
0.48
1.1353
1.75
7.6261
3.80
24.2248
7.4
65.6061
0.50
1.2047
1.80
7.9515
3.85
24.6959
7.6
68.2791
0.52
1.2753
1.85
8.2820
3.90
25.1769
7.8
70.9918
0.54
1.3473
1.90
8.6175
3.95
25.6598
8.0
73.7341
0.56
1.4204
1.95
8.9569
4.00
26.1463
8.2
76.5075
0.58
1.4955
2.0
9.3023
4.05
26.6398
8.4
79.3202
0.60
1.5715
2.05
9.6483
4.10
27.1308
8.6
82.1629
0.62
1.6485
2.10
10.0011
4.15
27.6218
8.8
85.0157
0.64
1.7265
2.15
10.3575
4.20
28.1228
9.0
87.9483
0.66
1.8065
2.20
10.7188
4.25
28.6237
9.2
90.8909
0.68
1.8865
2.25
11.0830
4.30
29.1347
9.4
94.0335
0.70
1.9694
2.30
11.4524
4.35
29.6357
9.6
96.8661
0.72
2.0524
2.35
11.8247
4.40
30.1466
9.8
99.9187
0.74
2.1363
2.40
12.2010
4.45
30.6675
10.0
102.9812
0.76
2.2212
2.45
12.5823
4.50
31.1785
0.78
2.3081
2.50
12.9656
4.55
31.6994
0.80
2.3950
2.55
13.3569
4.60
32.2204
172
THE FLOW OF WATER
If accurate results are desired from the application of this
formula the depth 'p as well as the length L should never be less
than 2 h, and the width of the channel of approach should
increase up stream from the crest.
Heads are most conveniently and accurately ascertained by
means of a plumb-bob, the string of which is hung over a nail
driven horizontally and pulled horizontally along a board to
which a graduated scale is attached. A datum reading is taken
and laid off on the scale when the surface of the water is just
flush with the crest and the point of the plumb-bob grazes the
surface when it is made to swing to and fro.
Of weirs not originally constructed to be measuring devices
those most frequently found are the sharp-crested triangular
weir, the triangular weir with a quarter round cjest and the
rectangular, broad-crested weir. The factors of proportional
discharge for these shapes, for which we are indebted to Bazin,
the Cornell Engineers, G. W. Rafter and others, are as follows,
the down stream face being in all cases vertical, air admitted
under the descending sheet of water and the relation between h
and p being the same as for the sharp edged measuring weir:
Description of Shape of Gross-Section.
Head on Sill in Feet.
0.5
1.0
1.5
2.0
3.0
4.0
Crest triangular, up stream face
inclined 1:1.
Crest quarter round of circle, di-
ameter 1 meter, up stream face
inclined 1:1.
Crest rectangular, both faces ver-
tical. Thickness of wall:
0 . 5 feet
1.06
0.971
0.902
0.830
0.819
0.797
0.785
0.783
0.783
1.079
0.983
0.972
0.904
0.879
0.812
0.800
0.798
0.792
1.092
1.012
1.0
0.957
0.910
0.821
0.807
0.803
0.797
1.094
1.040
1.0
0.989
0.925
0.821
0.805
0.800
0.797
1.082
1.072
1.0
1.0
0.928
0.813
0.796
0.797
0.784
1.072
1.097
1 0
1.0 feet
1 0
1 . 5 feet
0 947
3.0 feet
0 808
6.0 feet
0 790
9 . 0 feet
0 783
16.0 feet
0.777
WEIR FORMULiE 173
Weir Formulae.
Weirs are constructed for the following purposes:
(1) To measure the discharge of a conduit.
(2) To regulate the discharge of a conduit.
(3) To serve as impounding and regulating dams for the
storage of water.
(4) To raise the surface of the water at a certain point to a
certain level.
According to the manner of outflow weirs are classified as
follows :
(1) Complete overflow weirs, when the crest of the weir is
above the surface of the run-off water.
(2) Incomplete weirs, when the crest of the weir is below the
surface of the runoff water.
(3) Discontinuous weirs (wing dams, bridge piers, etc.)
when the weir does not extend the whole width of the channel.
(4) Sluice weirs (water-gate, head-gate, regulating weir,
needle weir, etc.), when the water flows out through an orifice.
Theoretically the discharge through a rectangular sharp-
edged orifice is found as follows:
Let h be the breadth of a rectangular jet,
h^ the depth of its upper,
h^ the depth of its lower surface below the surface of the
stiH water (Fig. 8).
An infinitesimal thin layer of the jet between its surface and
an infinitesimal depth h has a section equal to hdh.
The velocity of flow in this infinitesimal layer bdh is equal to-
V = V2gh.
The discharge will consequently be
h \/2 gh dh,
which integrated, gives for the discharge of the whole jet
= r
q^lh\/2g(hi -hi).
174
THE FLOW OF WATER
Let B be the breadth of the orifice,
H^ the depth of its upper,
H^ the depth of its lower edge below the free surface,
we then have for the coefficient of discharge
^_ h (h,^ - h,^)
B {H} - H^i)
and for the discharge in terms of the orifice
(1)
iiiiiiiiiiiii;
iiiiiii
""lllillll!
III!
ililllmili
Fig. 8.
The value of C, the coefficient of discharge, differs with the
nature of the orifice, and must be found by experiment. (For
sharp-edged orifices and weirs C = 0.622, for broad-crested
weirs C = 0.577).
For the discharge through a rectangular sharp-edged notch we
have, since there is in this case no head H^
Q = iCB^2~g H}. (2)
The discharge through rectangular notches and over sharp-
crested weirs, has been minutely investigated by Bazin, Francis,
and others. Bazin found for the discharge the expression,
2
3L W
0.0148
« = l[^^ + °-^K^J>^'^'
(3)
in which
c =0.6075 +
p = height of crest above bottom of channel.
WEIR FORMULA 175
Francis found that the loss of discharge due to end contraction
is equal to -^^ the height of the submerged opening for each
contraction. The discharge is consequently
Q = ic (h -0.1mH)\/2g H'^,
in which m is the number of end contractions.
If Francis' formula is used and the discharge is relatively
large compared with the dimensions of the conduit, the head
due to the velocity of approach must also be considered. This
head is equal to
v'
the velocity of approach is equal to
60 and h^ being the breadth and depth of the channel at the
point where the surface of the water begins to drop towards the
crest of the opening. Making this correction for the head due
to the velocity of approach, Francis' formula becomes
Q = 2^(6 -0.1H)V2~g[(H + a)^ -J],
in which c = 0.622 for sharp-edged orifices.
Putting f 0.622 V2^ = 3.33, Francis' formula reads
Q = 3.33 (h - mO.l H) [{H + a)^ - cfi]. (4)
Bazin's and Francis' formulae give equally good results; the
latter is the one most frequently used in this country.
To measure the discharge of a small stream (pipe line, flume,
etc.), a temporary weir of planks is usually constructed. In
order to arrive at accurate results care must be taken that the
sill or the crest of the weir is perfectly level, that it is at right
angles to the line of flow, and that it is above the surface of the
run-off water. The head on the sill should not be less than one-
half, nor more than 2 feet, and the depth of water in the channel
should be at least three times the head on the crest. In order
to measure the head on the sill a stake is driven in the bed of
the stream a short distance above the weir. The top of the
176
THE FLOW OF WATER
stake must be on a perfect level with the crest of the weir. A
thin-edged graduated scale fastened vertically to the top of the
stake is very convenient. On this gauge the height H, the
head on the sill is read off to the surface of the still water.
Fig. 9. Measuring Weir.
Weirs intended to regulate the discharge of a conduit, or to
raise the surface of the water at a certain point to a certain
level are constructed of various materials and in various forms,
A complete overflow weir, when constructed of masonry in the
bed of a stream, is usually of the form shown in Fig. 10.
_^ M
Fig. 10. Complete Overflow Weir.
The height of the weir necessary to raise the surface of the
water to a given height h, is found as follows :
WEIR FORMULA
177
Let-H" be the head on the sill, measured to still water,
b the breadth of the channel,
a the height due to the velocity of approach,
h the difference of level between the surface of the water
down stream and up stream, or the swell {MO, Fig. 10)
and the discharge is
Q = lcbV2~g[{H + aY
_ nh
from which we find for the head on the sill
\cbV2g I
,3.336
Denoting the height of the weir above the bottom of the
channel by x and the depth of the run-off water down stream
by /, we have
X + H = h + f,
hence x = (f + h) - H,
or x = f + h- ( ^ ^Y
\ch \/2 g I
when the velocity of approach is small.
Incomplete Weir.
For an incomplete weir the discharge and the height of crest
necessary to raise the water to a given level are found as follows:
The head on the sill (MiV, Fig. 11) is greater than the swell-
178
THE FLOW OF WATER
head (MO), therefore only the water above 0 flows off freely,
while the water below '0 flows off under the head (MO) = h.
The discharge through (MO) is
q^ = ichV2^[(h + a)t -a^
and that through (ON) = H - his
q^ = ch (H - h)V2g (h + a);
hence the whole discharge:
Q = c6 \/2^ f [(/i + a)^ -a^]+ H-h V (h + a).
From the discharge Q and the height h (MO) to which the
water is raised, we find for the height of water above the crest,
or the head on the sill,
H = h +
Q
2 (h + a)^ -J.
chV2g (h + a) 3 V (h + a)
hence we have for the necessary height of the crest of the weir
above the bottom of the channel (NP),
(NP) =x=(f + h)-H = [(OP) + (MO)] - (MN).
Neglecting the velocity of approach we have for the height of
the weir the simple expression
(iVP)=x = / + H-— ^.
cb V 2 gh
Fig. 12. Discontinuous Weir.
WEIR FORMULA
179
Wing dams are built whenever an obstruction extending the
whole width of the stream is either on account of navigation
not permissible, or on account of the form of cross-section of
the channel not feasible or necessary.
Fig. 13. Wing Dam.
While for overflow weirs the usual problem consists in finding
the height of sill or crest necessary to raise the water to a given
level, the problem for wing dams consists in finding the breadth
of channel required to be closed in order to raise the surface of
the water to the given level.
Let QR = h = breadth of efflux (Fig. 13),
* MN = h = the height of swell,
NO = f = the undercurrent,
and we have for the quantity of water flowing off freely above /
the undercurrent
and for the undercurrent /
q^= chf\/2gh;
the whole discharge is consequently
Q = c6 \/2^ (f /i + /),
from which we find, for the breadth of efflux,
180
THE FLOW OF WATER
If the velocity of flow in the stream is great, or the swell h
comparatively small, it will be necessary to consider the velocity
of approach. Denoting as before by a the head due to the
velocity of approach, we have for the water flowing off freely
q^ = fc&\/2^[(/i + a)t-a^]
and for the undercurrent
q, = chf V2g (h + a),
for the whole discharge
and finally for the breadth of efflux
Q
a^] + f Vh + a,
h =
cV2g m(h + a)^ - a^ + f Vh + a
This formula may be applied to discontinuous weirs of any
description, such as bridge piers, etc., etc. Denoting by b the
Fig. 14. Sluice Weir.
sum of the openings between bridge piers the swell may for
instance be found by putting
swell
<chV2gf
The coefficient of efflux for discontinuous weirs is very high,
usuafly only the end contraction needs to be considered.
For wing dams c = 0.98 will give good results. For well-
rounded bridge piers c may be taken equal to 0.90, for those
WEIR FORMULA
181
forming acute angles c = 0.95, and for those of elliptical cross-
section c = 0.97.
Sluice weirs are constructed to regulate the discharge of a
conduit or reservoir as well as to raise the surface of the water
to a given level.
In computations of the discharge of sluice weirs, the head H
is measured from the free surface to the center of the opening.
If the water flows off freely we have for the discharge
Q = cfh V2gH,
and
H 1 ^^Y
2g\cfb/
in which
/ is the height of the opening.
b = the breadth
and
c = 0.60.
-— ~—
M
N
^___^ ^^
^ ,,:,.,^
_
E^ __^ -:=::
Fig. 15.
For a given discharge and a given head H the height of the
opening is given by
/ =
Q
cb\/2gH
In case the surface of the run-off water down stream rises
above the sluice opening, the effective head reduces to the dis-
tance M N (Fig. 15) and we have for the height of opening
Q
1
cbV2g (MN)
182
THE FLOW OF WATER
If as in Fig. 16 the surface of the run-off water downstream
lies somewhere within the opening, a part of the water runs off
under water, while the rest flows off freely.
Let MO = H
NO = U
OP = U,
and the discharge through NO is
q,^c}fiV2gH -0.5f„
and the discharge through OP
q, = cfjy V2iH,
therefore the whole discharges through NP
Q = ch V2g (/, VH -0.5/, + /, Vh).
M
^r =:^== ~
^ . , _^
N
Fig. 16.
For a given discharge Q, a given effective head H(MO) and
a given height f^ of the sill below the surface of the run-off water,
the height /,, or the distance of the lower edge of the sluice board
above the surface of the runoff water may be found by putting
Q
h
cbV2g
-jyti
VH -0.5f,
LOSS OF HEAD 183
METHODS OF MEASUREMENT.
Loss of Head.
A.
When a conduit discharges into an open tank or reservoir,
or into the open air, the loss of head is ascertained by levelUng
between the surface of the source of supply, and the surface of
the discharge tank, reservoir or outflowing stream. When
this is not the case, or when the loss in part of the conduit only
is to be found, other methods must be employed. Where the
pressures are not great, open stand pipes or piezometers are most
convenient, otherwise the pressures are measured by means of
manometers. A mercury manometer of the form generally
used has the following essentials: A cast-iron mercury reservoir
into one side of which a glass plate is fitted through which the
height of the mercury within may be observed. A metal tube
with a gate valve connects the top of the reservoir with the main
pipe at the point at which the pressure is to be measured. At
its highest point, this tube has an air valve. Into the mercury
reservoir, which is about half filled with mercury, a vertical tube
is placed, nearly reaching to the bottom. This tube, usually
one quarter of an inch in diameter, is of brass or wrought iron in
its lower part and of glass in its upper part. To the glass tube
a graduated scale is attached. As mercury is very sensitive to
changes of temperature the tube is surrounded by a water-
jacket, in its upper parts also of glass. When the gauge is to
be used the air valve in the connecting tube is opened and also,
by degrees, the gate valve. When the air is wholly removed
the air valve is closed and the gate valve fully opened. The
pressure of the water in the reservoir depresses the surface of
the mercury and causes it to rise in the tube. The height of the
mercury column above the surface of the mercury in the reservoir
is read on the graduated scale, both at times of discharge and
times of no discharge.
If a is the difference of the heights of the mercury columns at
two sections at times of no discharge, and A the difference at
184 THE FLOW OF WATER
times of discharge, the loss of head between the two sections
whose pressures are measured is equal to
H = 13.6 A - a,
13.6 being the specific gravity of mercury.
When the conduit is of great length and the difference between
the pressures at two sections considerable, a form of the manom-
eter known as the Bourdon gauge, is used with good results. The
essential parts of this instrument, universally used as a steam-
gauge, consist of a hollow curved metal spring, one end of which
is free to curve, while the other is fastened to the case of the
instrument. A pipe connects the interior of the tube, which
is oval in cross-section, with the main pipe at the point where
the pressure is to be measured. The pressure of the liquid
expands the spring, the free end moves and by a lever the move-
ment is transmitted to a toothed bar lever, which again transmits
the motion to a toothed wheel. The movement of the spring,
thus converted into rotary motion, is, by a pointer, indicated
upon a graduated circular scale. The pressure is indicated in
pounds per square inch. This is converted into feet of pressure
by dividing it by 0.434.
If A is the difference between the indicated pressures at two
sections at times of discharge and a the difference at times of
no discharge, the loss of head between the two sections is equal
to
A - a
H =
0.434
Discharge of Conduits under Pressure.
B.
Discharges are measured by means of vessels, tanks, by the
rise in the surface of a reservoir, or the overflowing stream is
measured by a weir, an orifice or the current meter. When
these methods are not feasible, some form of water meter is
used. The best known devise of this kind is the Venturi meter,
invented by Herschel and named for a celebrated Italian
hydraulician.
DISCHARGE OF CONDUITS UNDER PRESSURE
185
The theory of the Venturi meter is based on the principles
enunciated by BernouilU:
'The fall of the free surface level between two sections of a
conduit is equal to the difference of the heights due to the
velocities at the sections."
If p^ is the pressure at one section of a conduit and v^ the
velocity and p^ and v^ the pressure and velocity at another sec-
tion and y and 2/1 elevations above datum, then
Pi
G
§.»
2/1
2g
In Fig. 17 the line p^, p^, p^, shows the theoretical variation
of the free surface level due to the contraction and subsequent
enlargement of a conduit. The line p^, p^, p^, shows the actual
variation, the difference being due to the pressure expended
in overcoming the frictional resistance of the walls of the con-
duit. It will be observed that this difference increases with
the distance.
Fig. 17.
Differences of pressure in sections of conduits not far apart
are most conveniently measured by mercury difference gauges.
In Fig. 18 is shown a gauge of this kind connected to sections,
the pressures at which are to be compared.
186
THE FLOW OF WATER
The bottom of the gauge is filled with mercury. When the
gate valves are opened, the pressure of the water causes the
mercury to rise or fall to heights which indicate the pressures
at the points of the main to which the gauge is attached. A
graduated scale allows a comparison of the pressures.
PRESSURE FROM
LOWER END OF.
VENTUni METER
PRESSURE FRO^
THROAT OF
VENTURI METER
:jm~m~w^
u^mP
-
=it4
N
-
T M-j
hi
-
■j i— i-J
I-
4 tl.OfM
ii.ntu]
^
-f |— 9-i
F^-]
[—
-3 r 8 ~!
1—8 -4
[_
H r -i
h
~Ii C Tl
C ^ Tj
r"
-I h- 0 — :
-6-j
—
-| ^ _ -1
-1 -5-1
I— 5-1
? —
-^ h*H
h^H
P-'
-1 1— 3-H
t-3 J
■Z_
-ill — 2—1
h 2H
\ —
"Jit"* — 1
pojj
t
:□
r
Fig. 18.
The difference between the pressures at the full section and
the section most contracted indicates the difference between
the velocities at the two points; the difference between the
pressures at the full sections above and below the contraction
corresponds to the loss of head between the two points.
Denoting the area of the section not contracted by A^ the
area of the section most contracted by a, and the difference
between the pressures converted into feet of head of water by
K^ the theoretical quantity passing through the section most
contracted per second is given by the equation
Q =
V^2- a^
\/2gH.
For the actual discharge this is multiplied by a coefficient,
which, however, differs little from unity. In the Venturi meter,
DISCHAKGE OF CONDUITS UNDER PRESSURE 187
as usually constructed, the area of the throat is contracted to
one ninth the area of the full section of the main. Its length is
from eight to sixteen times the diameter of the full section. It
has a registering device which mechanically converts differences
of pressure into corresponding velocities and these, for a given
diameter and a certain interval of time (10 minutes), into
gallons of discharge.
The meter is made in sizes from 2 up to 100 inches in diameter.
The loss of head is insignificant and the condition of the water
does not affect its working.
The discharge from vertical tubes was recently determined
by Lawrence and Braunworth and formulae deduced, which not
only will prove to be of great value in computing the discharge
of artesian wells, but furnish another method to determine the
discharge of any conduit under pressure with a fair degree
of accuracy. To do this, it will simply be necessary to give
the end of the conduit a vertical direction and observe the
elevation of the crest of the outflow above the rim of the
conduit.
The investigators mentioned experimented with tubes rang-
ing between 2 and 12 inches in diameter and 15 feet long and
three conditions of out- flow were observed, depending on the
pressure head. Under a feeble head the water flows simply
over the rim of the conduit as it does over a sharp edged weir
and the discharge is equal to
When the issuing water forms a jet the discharge is equal to
in which Q = cub. ft. per sec.
d = actual internal diameter in feet.
h = elevation of crest of water above the rim of the
conduit, in feet, determined by sighting rod. For the condition
intermediate between the weirflow and the jetflow no formula
was deduced.
188 THE FLOW OF WATER
Discharge of Open Conduits.
c.
When the discharge of an open conduit cannot be measured
by a weir or an orifice, it is necessary to find the mean velocity
of flow.
The mean velocity in a vertical section is ascertained directly
by means of rod-floats or by making measurements at the point
where the thread of mean velocity is found, either with a current
meter or with a double float. Indirectly the mean velocity is
found by means of surface floats or by current meter observa-
tions at different points in the vertical section.
If the channel is narrow, measurements in one vertical section
are generally sufficient, especially if a rod-float is used. With
increasing width of the channel observations in two or more
vertical sections are necessary.
When the mean velocity of flow in a river is to be ascertained,
the channel is divided, at right angles to the line of flow, into
sections 5, 10, 20 or more feet wide, the distance depending on
the degree of accuracy desired.
The mean velocity at each section is found by means of rod-
floats, by observations at the surface, at mid-depth, at the
position of the thread of mean velocity or at points of propor-
tional depth. The mean velocity for the whole channel is found
by taking the mean of the mean velocities of all the sections.
For the discharge of the whole channel the mean velocity of
each section is multiplied by its area and the discharges of all
the sections summed up. If floats are used, the stretch over
which the float is to pass should be carefully measured and
staked off. If possible ropes or wires should be stretched across
the stream, at right angles to the line of flow. The float should
be started some distance above the rope and the time of its
passage carefully observed.
The distance measured out may be 250 to 500 feet for swift
streams ; 50 feet will suffice if the current is feeble. The longer
the stretch the more reliable the time observation. On the
other hand, if the stretch is long it is often exceedingly difficult
DISCHARGE OF OPEN CONDUITS
189
to keep the float in a position parallel to the axis of the stream.
This is especially so near the banks. On this account it may
be necessary to measure stretches as short as 20 feet.
A surface float may be a ball of wood or some other light
material, or else a watertight metal cylinder, so loaded as to
float flush with the surface of the stream. A small flag will
render the float more visible.
Double floats are used to find the velocities at different depths
below the surface. They consist of light surface floats con-
nected by a fine strong cord, to a large sub-surface float. A
ball of wood or a flat watertight metal box makes a good surface
float, a watertight metal cylinder, heavy enough to keep the
cord in tension, but not to drag it below the surface is an excellent
sub-surface float. The speed of the surface float is identical
with that of the larger float and observations of its passage will
give the speed of the latter.
Usually the subsurface float is placed at the point where the
thread of mean velocity is found. The use of double floats
generally leads to trouble of one kind or another; they are rarely
used, except to measure velocities in very deep channels.
A cylindrical wooden pole two inches in diameter and loaded
at the bottom, so that it will float vertically, makes an excellent
rod-float. It may be made in sections and screwed together.
A brass cylinder screwed to the bottom makes an excellent
weight. Into it shot may be placed to suit the weight to all
requirements. Watertight tin tubes also make good rod-
^Cl^l^^^^RF^
Fig. 19. Channel of River Divided into Vertical Sections.
floats. Rod floats should be loaded so that they nearly reach
to the bottom of the channel, but never touch it. On the other
hand they must not be too short, or else they will travel with
a speed exceeding the mean velocity.
190
THE FLOW OF WATER
The rod-float is the ideal instrument to measure the velocity
in a flume or aqueduct. The fact that it integrates the velocity
of the whole section and thus indicates the mean velocity directly
is an advantage not possessed by any other measuring device.
If properly used it gives results whose accuracy cannot be ques-
tioned. However, if the bottom of the channel is very rough,
covered with plants or else very deep, its use is not indicated.
Fig. 20.
Velocity measurements are made in the centre of each sec-
tion. Depths are taken by soundings.
Line (1) indicates the position of the thread of maximum
velocity in each section, line (2) the position of the thread of
mean velocity in each section, and line (3) the position of the
thread of mean velocity for the whole section.
The current meter, like so many other hydraulic measuring
devices, originated centuries ago in the Valley of the Po, Italy,
the cradle of hydraulics. The earliest form consisted of a small
paddle wheel mounted in a floating frame. It could only be
used at the surface.
When Woltman, in 1790, added a recording device the instru-
ment could be used at any depth. The recording mechanism
consisted of an endless screw fitted to the horizontal axis, and a
series of toothed wheels which transmitted the motion of the
axis to a register. The recording mechanism was thrown in and
out of gear by a string, attached to a lever. The instrument
was fitted and clamped to a one-inch pole on which it could be
DISCHARGE OF OPEN CONDUITS 191
slid up and down. To read the number of revolutions recorded
the current meter had to be taken out of the water. The instru-
ment was generally known as ^'Woltman's Tachometer."
Many modifications of this instrument appeared, mostly of
the windmill pattern, with propellers and vanes. Some have
the axis of the propeller horizontal, others vertical, and the
shape of the propellers is variable. The general form of the
instrument is, however, always the same. The present day
current meter has an electrical signalling or registering device.
The best known patterns are those of Harlacher in Europe, and
those of Price and Ritchie-Haskell in the United States.
The Harlacher meter is of the windmill pattern; its propeller
has four blades. A vane about 12 inches long and 5 inches wide
is fitted to a prolongation of the axis of the wheel. This direct-
ing device keeps the face of the wheel at right angles to the line
of flow. To the axis of the propeller is fitted an endless screw,
operating a toothed wheel. A pin in the side of the wheel
strikes an electric wire at each revolution, thus completing an
electrical circuit. The battery with the registering or sounding
device is kept at the surface. The meter slides up or down on
a vertical rod. To move the meter up and down with a uniform
speed an apparatus consisting of ropes, pulleys and weights is
often used.
The propeller of the Price current meter has four cup-shaped
wings; its axis is vertical and its revolutions are indicated by an
electrical buzzer. The instrument is generally used without a
rod; it is kept vertical by a weight attached to the frame and
moved up and down by a cord. Its vane consists of two blades,
one horizontal, the other vertical, intersecting in the middle
at right angles. It is made in two sizes. The small meter
measures velocities as low as 0.2 feet per second with a fair degree
of accuracy; the large meter gives good results down to velocities
of 0.5 foot per second.
The latest design in the line of meters is the Ritchie-Haskell
so-called ''direction current meter.'' Like the Harlacher and
the Price this instrument has a device recording the number of
revolutions of the propeller electrically. It has also a device
192 THE FLOW OF WATER
indicating the direction of the current. The body of the instru-
ment is a compass with a magnetic needle. An electrical circuit
measures the angle between the direction of the needle and the
direction in which the vane points and indicates the angle on a
graduated dial.
Current meters must be rated; that is, the relation between
the velocities and the number of revolutions of the propeller
must be ascertained. This is done by pulling the meter at
various constant speeds through a still body of water, and deter-
mining the relation between speeds and revolutions.
Current meters as furnished by the makers are always rated,
but they must subsequently be rerated at frequent intervals,
if good results are desired. As with floats, measurements with
the current meter are made in various ways. The best method
is no doubt the one adopted by Harlacher of sliding the instru-
ment by means of a mechanism at a uniform speed up and down
on a pole. By this process the velocity of the whole section is
integrated and a very good mean value found. If no pole is
used the instrument is most conveniently moved up or down by
means of a cord thrown over a small pulley.
A good current meter, properly rated and carefully handled,
surpasses any other instrument in the facility and extent of its
application; it gives results nearly as trustworthy as the rod-
float, and for average velocities nearly as accurate as a weir.
The Darcy gauge, an instrument formerly in great favor, is at
present, owing to the great perfection of the current meter,
but rarely used. The instrument consists of a combination of
two Pitot tubes, fastened to a supporting frame.
A Pitot tube is a vertical glass tube with a right-angled bend.
If such a tube is placed into ^ stream, with its mouth facing up-
stream and at right angles to the line of motion, the water will
ascend in the tube to a height which is equal to
h = —-, nearly.
2g
If the mouth of the tube faces the bank of the stream, and is
in line with the line of motion, there will be no difference of
level between the surface of the water in the tube and the
surface of the stream.
SURFACE MEAN AND BOTTOM VELOCITIES 193
If the mouth of the tube faces downstream and is at right
angles to the Hne of motion, the surface of the water in the tube
will be below the surface of the stream, the difference being
equal to
In this case the velocity is somewhat modified by the retard-
ing influence of the tube. Darcy combined two tubes having
their mouths at right angles, and provided their lower parts
with stopcocks, which can be operated, when the instrument is
in the water, by means of a string. If the cocks are open and
the mouth of one of the tubes faces upstream at right angles to
the line of motion the water will ascend in it while it will not
ascend in the other tube. If the corks are then closed, the
instrument may be lifted out of the water and the difference of
level in the two tubes read off on a graduated scale.
Surface Mean and Bottom Velocities.
Position of Thread of Mean Velocity.
From 82 observations of flow in small channels Bazin deduces
the following:
Mean Velocity = Maximum Velocity — 25.4 Vr.s
Bottom Velocity = Maximum Velocity — 36.3 Vr.s
Bottom Velocity = Mean Velocity - 10 . 87 VrT
From this we have
V mean + 25.4 Vrl . _
and as
we have also
and likewise
V max.
— X .
V
mean
1
V
max. 1 +
25.4 Vris
1
c,
V mean
1
V max. 25.4
c
V bottom 1
V max. ^ 36.3
c
194 THE FLOW OF WATER
Comparison of values of^— = — — - with values of
V max. ^ 25.4
c
— found by observations of flow in a great variety of
V max.
channels shows that Bazin's formula is not of general application.
It fails because the influence of the value of the total depth of
the channel is not considered.
The following values of — — are given by the most reliable
V surface
authorities :
V mean
V surface
Revy, Parana de las Palmas, La Plata 0.835
Harlacher, Bohemian Rivers, 28 observations 0.838
Swiss Engineers, Swiss Rivers, 200 observations 0.835
Lippincott, Sacramento River, Cal., Depth, 3-5 feet ... 0.88
Lippincott, Tuolumne River, Cal., Depth, 1.12-1.84 feet 0.88
Lippincott, San Gabriel and Santa Anna, Rough channels,
10-20 feet wide. Depth, 0.25-1.0 feet 0.92
Pressey, Catskill Creek, partial section 0.82
Pressey, Fishkill Creek, partial section 0.93
Pressey, Mean of 28 observations of flow in rivers with
rough bottoms, Average depth, 5.05 feet 0.80
Prony, Small wooden channels 0.8164
Prony and Destrem, Neva River, Russia 0.78
Boileau, Canals 0.82
Baumgartner, Garrone River, France 0.80
Cunningham, Solani Aqueduct 0.823
Humphreys & Abbot, Mississippi 0.79-0.82
From these and other data given by Murphy (Cornell testing
flume) and others, the writer found that the relation between
the surface velocity and the mean velocity may be expressed by
the equation
Mean velocity = ^^ surface velocity (1)
1 +n^'
in which n is a coefficient ranging in value between 0 . 25 for the
roughest and 0.35 for the smoothest classes of conduits.
Its value is
n = 0.32 for K = 1.25
n = 0.30 for K = 1.75
n = 0.27 for K = 2.25
For the velocity at any point x, depth d, in the vertical section
SURFACE MEAN AND BOTTOM VELOCITIES
195
we found from data relating to flow in channels with rough
bottoms, such as rivers with detritus or coarse gravel,
1
Vx =
1 +
(f)'
(2)
in which D is the total depth. This is on the assumption that
the bottom velocity is equal to one half the surface velocity, a
relation which holds good only for channels with rough bottoms.
Bazin found from observations of flow in small artificial channels
that the difference between the surface and the bottom velocity
ranges between 0 . 25 and 0 . 5 of the surface velocity, the differ-
ence increasing in value with the roughness of the walls. In
canals and rivers with comparatively smooth bottoms the
difference ranges between 0.3 and 0.4, the average difference
being 0.35 of the surface velocity.
Combining the two equations (1) and (2), we have for the
position of the thread of mean velocity in the vertical section
of rivers and canals with somewhat rough bottoms and whose
width is several times the depth
. = Z)(n|y (3)
as the depth below the surface at which the thread of mean
velocity is found. The formula does not apply to flumes and
other narrow, deep channels.
From equations (1) and (3) we find the following values of
the relation
V mean
and the relative position of the thread
V surface
of mean velocity in a vertical section, assuming K = 1.0 and
V = 3 feet.
R
V mean
Relative
R
V mean
Relative
V surface
Depth.
V surface
Depth.
1.0
0.898
0.538
10.0
0.854
0.604
2.5
0.881
0.563
15.0
0.842
0.616
5.0
0.369
0.583
25.0
0.832
0.631
7.5
0.859
0.596
30.0
0.813
0.656
APPENDIX I.
Variation of the Coefficient c with the Slope.
In the preceding chapters we have defined the variation of
the coefficient c with the mean hydrauHc radius, with the degree
of roughness of the wet perimeter and with the velocity of flow.
We will now proceed to investigate if it is possible to find a
true expression for the variation of the coefficient c with the
slope by the graphical method. From Formula III we have
66 ( ^/r + m) 7* = c,
7^= "
66 ( Vr + ?7i) '
V66 ( Vr + m)/ '
or substituting for v its equivalent
(66(^7^+ m)VrTsj^= (■
hence 66 ( Vr + m) VrTs
\m(^+ m)l
and (66 ( Vr + m))' Vr . s = c«;
consequently
(66(Vr +m))^(r.s)i'^= c;
or 66 ( Vr +m) (66 ( Vr + m))^ (r .s)^ = c.
This goes to show that c increases with {rs)^^) consequently the
variation of the coefficient c with the slope depends on the
value of R.
The variation of the coefficient c follows the law of the para-
bola. If values of the coefficients a =V^ and 7^^ are plotted as
ordinates to values of v as abscissae, the points so found He in
curves which are parabolas of the ninth or eighteenth order. A
curve somewhat resembhng a parabola is the equilateral hyper-
196
APPENDIX I
197
bola, and it is possible to draw a curve of this kind which nearly
coincides with the parabola.
The equation of the equilateral hyperbola concave towards
the axis of abscissae may be put into the simple form
c =
1 +
R
The curve in Fig. 1 represents the hyperbola of this equation.
In the figure ZO is the vertical asymptote,
Zd the horizontal asymptote,
YK the axis of ordinates,
KX the axis of abscissae,
Zg the axis of the hyperbola,
X the distance between the vertical asymptote
ZO and the axis of ordinates YK,
c the ordinate of any point in the curve.
2 • Y n
d
\\
K
\
\^
^^^-''"'
t-----^ ;r-
f
\
((J ^s^^
0
\
1
1
1
1
1
1
1
1
1
X
/
/
/
/
/
/
/
/
/
/
/
(
) \
<
Fig. 1.
)
I
The area of the rectangle ZOKY is the constant which deter-
mines the hyperbola. It is equal to the square zfgh or the area
of any rectangle comprised between the asymptotes and per-
pendiculars drawn to them from any point in the curve.
198 THE FLOW OF WATER
Consequently, if lines are drawn from the center z to points U
on the axis of abscissse, these lines will intersect the axis of
ordinates in points which give the values of c corresponding to
the values of 22. In this way the hyperbola may be easily
constructed.
Bazin in his paper, '^ Etude d'une nouvelle formule," etc., put
the equation for the coefficient c into the form
c = -L
1 + ^
Vr
in which y is constant and equal to 157.5 in English measure,
and gf, a variable, indicating the degree of roughness.
Dividing by y we have
c = ' substituting x for g,
y yVr
Transposing we have
1 _1 x_l_
c y y Vr
This is the equation of a straight line having values of — ■_ as
Vr
abscissae, values of - as ordinates. If this equation would hold
good, points of values of - pertaining to one slope would lie in
straight lines intersecting the axis of ordinates in a point-. If,
however, values of - and -— are plotted as indicated it appears
c Vr
that only those points - pertaining to data of flow in old pipes
or fairly regular channels in earth lie in straight lines, while
those pertaining to data of flow in very smooth conduits He in
APPENDIX I
199
curved lines convex towards the axis of abscissse, and those
pertaining to data of flow in very irregular channels He in curved
lines concave towards the axis of abscissae.
If straight lines are drawn averaging between the points as
much as possible, these hues will intersect the axis of ordinates
Fig. 2.
in points giving values of - for the greatest value of v and the
greatest value of R included in the series plotted. These lines
will also intersect the axis of abscissae in points which give the
value of - pertaining to each value of - . In Fig. 2 we thus plotted
the experimental data of Darcy-Bazin, series 7, 8, 9 and one
series given by Rittinger (s = 0.0343), all pertaining to flow in
testing channels of rough boards.
It will be observed that the lines pertaining to the steeper
slopes intersect each other in a point whose abscissa for — _ is
vr
1 . 0. This is due to the fact that for the greater slopes s = 0.0049,
0.00824, and 0.0343, the velocity is so high that c varies but
very little, while it varies much for the feebler slope s = 0.0015.
200 THE FLOW OF WATER
The highest value of -corresponding to —-z = 1.0 is 0.0084, the
c Vr
lowest 0.0080, average 0.0082. Denoting the abscissa of the
point of intersection by a and the average ordinate by K we
have
y y a
and X = Kay - a,
and considering -- = Ka as a tangent and denoting it by Z,
we have x = ly — a.
Consequently in our case
x = 0.0082 2/ - 1.0,
which gives values of x very nearly equal to those found graph-
ically. This formula will, however, only hold good for the
values of R included in the series, the highest of which is 1.0.
In Fig. 3 the values of y found graphically from Fig. 2 are
plotted as ordinates to values of - as abscissse. The points y
are seen to lie in a curved line, intersecting the axis of ordinates
at a point B = 131.0 nearly. If the line CD is produced, it
will intersect the axis of ordinates in y = 157.5, which is the
constant in the formula of Bazin mentioned above. The tan-
gent of the angle CEF (in this instance 0.29) corresponds to
Bazin's coefficient, g, indicating the degree of roughness. The
value of m obtained from the given data is 0.70; hence the value
of ^r + m, for the highest value of i? is 1 . 70. Dividing 131 . 0
by 1 . 70 we have
2/' = 77 ( Vr + m), nearly,
as the value of y corresponding to the highest velocity included
in the series plotted.
If from the point B = 131. 0 = 77 (Vr + m) a line is drawn
parallel to the axis of abscissae, any increase in the value of
APPENDIX I
201
77 {y/r + m) due to any slope less than 0.0343 will appear as
an ordinate above this line BG.
It will be observed that values of y\ y''\ '}/'', etc., increase
with the decrease of the slope or increasing values of - • We
may therefore put 2/'', y''\ etc., = 77 ( 's/r + m) + - in which
o
0 is a coefficient still to be determined.
E - — '^:^^
C
F
..^^
^ ^
G
y'
yll
ym
Values of "F
^IV
Fig. 3.
The line BDC is evidently a parabola. If a line is drawn
from B io C the tangent of the angle CBG will be equal to z,
for s = 0.0015. For this slope we have from the figure
y = 198.0
2/' = 131.0;
z
therefore
or
Hence
- = 198 - 131 = 67 and z = 0.0015 X 67,
s
2 = 0.1005.
0.1
= 77 ( Vr + m) +
From experimental data pertaining to flow in small channels
in earth, R ranging between 1 and 1 . 75 (Darcy-Bazin, Grosbois
202 THE FLOW OF WATER
canal) which are, however, somewhat doubtful, we found
z = 0. 0936, while from data pertaining to flow in the La Plata
and its tributaries we found z = 0.00293. From this it is
evident that 0 is a variable and that its value depends on the
value of R. Having found an expression for y, the value of x
may be found from experimental data without resorting to
graphical methods.
No. 12, series 6, Darcy-Bazin gives
R = 0.922
s = 0.00208
c = 118.9.
Hence , = 77(^0:922 + 0.68)+^
or y = 127.87 +47.6 = 175.47.
Dividing 175.47 by c = 118.9 we find x = 1.475,
= 1 +0.475. But 0.475 is equal to 0.01, ^t^t;^ or 0.01,47.6.
Denoting the term 0.01, which is variable, by I, we have from
the given data for the variation of the coefficient c with the
slope the expression
77 ( 's/r + m) + —
s
c =
1 + ^4-
which, within certain limits corresponds to
c = 66 ( -Vr + m) 7^^.
From data relating to flow in a semicircular channel of rough
boards (Darcy-Bazin, series 26) we find
2/ + — = 210, hence y = 210 - 67,
o
which is equal to
84 ( Vr + m) + —
c =
1 + ^4-
APPENDIX I 203
and which corresponds within certain limits to
c = 66 ( Vr + m) V^'\
Dividing 84 by 66 the quotient 1.272 is the value of the
coefficient of variation of c for f = 18.0. Hence v = 18.0 is
the limit up to which the formula holds good.
The formula apparently gives good values of c up to the
limit indicated. By trial we find, however, that it does not
hold for values of R greater than 1.0; unless rs is substituted in
the equation for s. Consequently the variation of c with the
slope is dependent on the value of R, a fact we demonstrated at
the beginning of this chapter.
The facts related plainly show that a formula derived in the
manner indicated can only be of limited application. It holds
good only within the range of values of R, s and m included
in the series of data from which it is derived. In other words :
We cannot get out of a formula what we do not put into it.
A general formula, like that of Ganguillet and Kutter, derived
by the methods we have indicated, cannot embody true laws of
flow, it naturally must be deficient in one respect or the other.
The more so, if the data on which the formula is based are
erroneous. The experimental data derived from observations
of flow in the lower Mississippi by Humphreys and Abbot and
embodied by Ganguillet and Kutter in their formula have been
found to be incorrect, greatly at variance with those time and
again found by the United States engineers. The contention of
Ganguillet and Kutter, that, if values of - are plotted as ordinates
c
to values of -— r as abscissae and lines drawn through all the points
vr
- these lines will intersect each other in a point — ^ = 1 meter,
c vr
and that therefore c will increase with increasing values of s if
R is less than 1 meter, and decrease if R is greater than 1 meter,
is also plainly a fallacy.
If values of - and — derived from the numerous series given
c vr
204 THE FLOW OF WATER
by Darcy-Bazin are plotted as indicated, it will be observed
that for many of the series the lines intersect at -p = 1 foot.
vr
It would be absurd, however, to draw the conclusion there-
from that c will increase or decrease with increase of the slope
if R is less or greater than 1 foot. The intersection of the lines
at — = 1 foot is due to the fact, that for the greater slopes
values of c are nearly constant for values of R equal for 1 foot or
more, because the value of F^^ increases slowly at high velocities.
APPENDIX II.
The Formula in Metric Measure.
The general equation for the velocity of flow reads, for
Metric measure,
7 = 50 ( -N/r + m) vTTi
0.007844 L
{-Jr + mf R_
The coefficients of variation of c are equal, as for English
measure, to
a=V^ V^"'^ 1.0 \,\',
a=V^ holds good also for semicircular open conduits.
Values of the coefficient m, indicating the degree of rough-
ness, are found in the following table, mE signifying the English
and mM the Metric values.
Exponential Equations.
The constants of the exponential equations which we have
found for English measure are converted into Metric equivalents
by putting
log constant Metric measure = log constant Enghsh measure
+ '' 3.281X
- " 3.281.
X being the variable power of R or D.
The equations for conduits under pressure are as follows,
diameters being in meters, velocities in meters per second and
quantities in cubic meters (1000 liters) per second :
205
206
THE FLOW OF WATER
Values of mE which apply in the English and Values of mM
WHICH Apply in the Metric System.
mE
0.68
0.57
1.0
0.85
0.95
0.80
0.85
0.70
0.83
0.75
0.80
0.65
0.70
0.62
0.60
0.48
0.53
0.50
0.45
0.45
0.47
0.42
0.30
0.25
0.20
0.20
0
0
0.10
-0.1
0.20
-0.2
0.27
-0.27
0.32
-0.32
Description of Conduits.
Semicircular and circular conduits lined with pure
cement. Long straight brass, tin, nickel and glass
pipes.
Rectangular conduits lined with pure cement. New
pipes of planed boards and very smooth asphalt-
coated cast iron.
Semicircular conduits lined with cement plaster, 1 part
cement, 2 parts sand.
Ordinary new straight asphalt-coated cast, wrought
iron welded and wrought iron riveted pipes with
screw joints, common lead, tin, glass, brass and
galvanized pipes.
Rectangular conduits lined with cement plaster,
smooth concrete or very good brickwork.
Semicircular channels lined with rough boards. Chan-
nels lined with fairly good brickwork or fairly
smooth concrete.
Rectangular channels lined with rough boards.
Sewer pipe very well laid.
Pipes of planed boards, asphalt-coated cast and
wrought iron, riveted wrought iron pipes of small
diameters or with screw joints, pipes coated with
tar or lined with cement or smooth concrete, all
some time in use.
Common brickwork or concrete. Very good ashlar
masonry. Ordinary sewer pipe.
Asphalt-coated riveted pipe above 3 feet in diameter.
Channels in earth roughly lined with cement mortar.
Old pipes of all descriptions, fairly clean. Channels
lined with rough brickwork or rough concrete.
Old riveted pipes over 3 feet in diameter. Ordinary
ashlar and very good rubble masonry.
Channels of regular cross-section in fine cemented
gravel. Tile drains.
Channels of regular cross-section in coarse cemented
gravel or rockwork.
Channels of fairly regular cross-section in firm sand
or sand with pebbles, no vegetation.
Channels in earth somewhat above the average in
regularity and condition, no stones or vegetation.
Ordinary channels in earth, with stones or vegetation
here and there.
Channels of irregular cross-sections or channels of
fairly regular cross-sections but with stones or
plants.
The values of K corresponding to m = — 0.1, — 0.2,
-0.27,-0.32 are 1.2, 1.5, 1.75, 1.93.
APPENDIX II
207
mE
mM
V in Meters per Second.
Q in Cubic Meters per Second.
0.95
0.83
0.68
0.83
0.75
0.60
60.92
56.54
51.28
2)0.67 ^A
2)0.68 a
2)0.69 u
47.85
44.41
40.28
2)2-67 ^T5
2)2.68
2)2.69 a
0.57
0.57
0.48
0.48
36.76
40.21
2)0.7 gT7
D"-^ Egg-shaped
28.87
47.40
2)2.7 ^1^
Z>2-^ Egg-shaped
2)2.7 5^
0.53
0.45
35.55
2)0.7 ^T7
27.92
0.45
0.30
0.26
0.22
25.48
22.45
2)0.66 gh
2)0.67 ti
20.0
17.64
2)2.66 ^1
2)2.67 a
Values of D^'^'', etc., are found in Table E, values of D^'^'', etc.,
in Table F.
These tables give the values of the powers of diameters for
diameters of 0.05, 0.10, 0.15, 0.20, 0.25 meters, etc. These
correspond closely to 2, 4, 6, 8, 10 inches, one foot being 0.3048
meters, one meter 39.4 inches. In order that the powers of the
diameters found in Tables E and F may apply to a greater range
of diameters we shall find equations in which the unit is 1 deci-
meter = 0.1 meter, so that diameters must be taken in deci-
meters and fractions thereof. The results will be velocities in
decimeters per second and quantities in cubic decimeters or
liters per second.
The diameters found in the tables as 0.05, 0.10, 0.15, 0.20,
0.25, 0.50, 0.75 when taken as fractions of a decimeter corre-
spond to 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 3.0 inches respectively.
The discharge of a new wrought iron pipe (m = 0.83) of one
inch diameter (0.025 meter or 0.25 decimeter) for a slope of
1 : 100 is for instance :
Q = 92.78 (0.25)2-«« (0.01)^
= 92.78 . 0.02435 . 0.075 = 0.1694 liters per second or 10.164
liters per minute.
mE
mM
Velocity in Decimeters per
Second.
Discliarge in Liters per Second.
0.95
0.83
0.68
0.83
0.75
0.60
130.2 D»" ^T^
118.1 Z)«-«8 "
104.7 Z)°«9 "
102.3 D2" S'^'
92.78 i)2-«8 "
82.23 i)2.69 it
0.57
0.57
0.53
0.48
0.48
0.45
73.34 Do-^ 5T7
80.23 Z)o-7 Egg-shaped
70.93 Z)°-^ 5T7
57.60 D2-' iSi'
94.58 D^'' Egg-shaped
55.71 D^' '^i'^
0.45
0.30
0.36
0.22
55.74 Do«« S^
48.0 Do-«7 "
43.78 Z)2-e« S^
37.7 D2" "
208
THE FLOW OF WATER
Exponential Equations Relating to Flow in Open Conduits.
Of the following sets of equations the first three relate to flow
in the semicircle, the rest to flow in the semisquare, the depth
being in meters, the velocities in meters per second, the dis-
charges in cubic meters per second :
mE
mM
V in Meters per
Second.
Q in Cubic Meters per Second.
1.0
0.85
89.25 !)«•««
5^
70.12 i)2.68
^A
0.85
0.70
83.0 D"*"
U
65.1 D2.68
(<
0.70
0.62
74.0 i)°-^°
((
58.2 2)2-70
"
0.95
0.80
73.3 D""
S-r'r
146.6 D2-67
sA
0.80
0.65
67.6 D°««
it
135.2 Z)2-«8
<<
0.70
0.62
64.0 Z)°«»
"
128.0 i)2.69
<<
0.57
0.48
59.0 D'^-^
(<
118.0 D'-'
((
0.50
0.49
57.0 Z)o-^^5
It
114.0 1)2.715
<'
0.45
0.42
54.7 7)°-^»5
tt
109.4 D'-'"^
(<
0.30
0.25
49.1 D°"^
tt
98.2 D'-'"'
(<
0.20
0.20
44.6 D'-'^^
tt
89.2 D2-735
<<
0
0
29.2 D°'^5
si
58.4 2)2-75
si
K
K
1.2
1.2
26.75 D°-'«5
tt
53.5 2)2-'85
<<
1.5
1.5
23.6 Z)°"5
tt
47.2 2)2-"5
tt
1.75
1.25
21.6 i)o-^«5
"
43.2 2)2-785
tt
1.93
1.93
20.5 Z)°-^«5
tt
41.0 D2-^«5
tt
Of the following equations the first three apply to any depth
of water in the semicircular section, the rest to any depth of
water in any other form of section, R being in meters, velocities
in meters per second.
mE
mM
Velocities in Meters per
Second.
mE
mM
Velocit
ies in Meters per
Second.
1.0
0.85
142.4
2^0.68 ^TT
0.30
0.25
81.0
2^0.735 ^tV
0.85
0.70
134.0
2^0.69 tt
0.20
0.20
74.0
^0.735 tt
0.70
0.62
122.0
^0.70 tt
^0.67 5T7
0
0
49
720.75 S^
0.95
0.80
116
K
K
0.80
0.65
108
^0.68 tt
1.2
1.2
45.4
7^0.765 cc
0.70
0.62
102.5
7^0.69 tt
1.5
1.5
40.4
j^.m u
0.57
0.48
95.3
J^.IO tt
1.75
1.75
37.2
^0.785 ti
0.50
0.47
92.3
^0.715 tt
1.93
1.93
35.3
2^0.795 u
0.45
0.42
89.0
2^0.715 tt
APPENDIX II 209
English and Metric Equivalents.
The following relations between the units of the English and
the Metric Systems of Measurements are of interest in their
relation to the flow of water.
1 meter = 10 decimeters = 100 centimeters = 1000 millimeters.
1 sq. meter = 100 sq. decimeters = 10,000 sq. centimeters.
1 cu. meter = 10 hectoliters = 1000 liters.
1 liter of water at 4 degrees centigrade weighs 1 kilogram.
1 kilogram = 1000 grams,
1 meter = 3.280899 feet = 39.37079 inches.
1 foot = 0.304794 meter = 30.4794 centimeters.
1 inch = 25.3995 millimeters = 2.53995 centimeters.
= 0.253995 decimeter = 0.0253995 meter.
1 sq. meter = 10.7643 sq; feet = 1550 sq. inches.
1 sq. foot = 0.0928997 sq. meter = 928.997 sq. centimeters.
1 sq. inch = 6.451368 sq. centimeters.
1 cu. meter = 35.316585 cu. feet = 264.1863 gallons.
1 liter = 0.035316585 cu. feet; = 0.2641863 gallons.
1 cu. foot = 0.0283153 cu. meters, = 28.3153 liters.
1 cu. inch = 0.0163861 liters, = 16.38618 cu. centimeters.
1 gallon = 3.7852 liters.
1 liter weighs 2.204672 English pounds.
1 cu. foot weighs 62.425 English pounds.
1 gallon weighs 8.3448 English pounds.
1 gallon = 231 cubic inches.
The pressure of water in kilograms is equal
per square meter to 1000 h {h in meters)
" " decimeter '' 10 h
centimeter " 0.1 h
" " milKmeter " 0.001 h.
A pressure of one pound per square inch is equal to
a pressure of 0.07031 kilo per square centimeter
" " " 0.0007031 " " " millimeter.
The tensile, shearing, or compressive strength of any material
in pounds per square inch multipHed by 0.0007031 gives the
value in kilos per square millimeter and multiplied by 0.07031,
the value in kilos per square centimeter.
A pressure of 1 atmosphere = 14.7 pounds per square inch
corresponds to a pressure of 1.03296 kilos per square centimeter
or a head of 10.3296 meters. 2g = 19.61.
210
THE FLOW OF WATER
Thickness of walls of conduits :
t^^ + c,
m
t, D, C and m in millimeters.
P in kilos per square millimeter = 0.001 /i.
Material.
m
C
Cast iron
2.8
12.0
14.0
0.3
7.6
Wrought iron
1.5
Steel
Lead
7.6
APPENDIX III.
Greatest Efficiency of a Conduit of a Given Diameter as a Transmitter
of Energy.
Most Economical Diameter of a Conduit Transmitting Energy
under Pressure,
I.
In a preceding chapter the ratio between the total head and
the head lost in overcoming frictional resistances, which for a
conduit of a given diameter under a given head corresponds to a
maximum of efficiency, has been mentioned.
The potential energy of Qf^ of water delivered per second at
a vertical distance H above the generator is equal to
Q 62.4 H foot-pounds,
or 0.1134 QH horsepowers.
The discharge of a steel-riveted conduit in P per second is equal
to
Q = 40 d'-'S^^
which gives for the loss of head,
1062 d'-'
Consequently the net energy transmitted to the generator is
equal to
H.P.= 0.1134 0/^ -2:^1^
= 0.1134 Qff-^^^-^Q-Vf^.
^ 1062 d'-'
This is to be a maximum.
Regarding Q as the variable and equating the first differential
coefficient to zero we have
0.1134 H - ^6 Q'^^^ 0.1134 ^
9 1062 d'
211
212 THE FLOW OF WATER
hence 0.1134 i? = ?5-S%Mlf4 .
9 1062 d^-^
The root of this equation corresponds to a maximum. We
have consequently, for the state of maximum efficiency
9H _ _QJL_.
26 10Q2d'-''
or, — - is the head sacrificed in overcoming frictional resistances
when the conduit is in a state of maximum efficiency as a trans-
mitter of energy.
We have also for the discharge which corresponds to ,
1062^^-1 -^^1
26 L/
= 40 ^2.7 0.57 S^.
"'{
Hence the efficiency of the conduit is greatest when the velocity
and the discharge are 0.57 times the velocity and discharge
corresponding to the total head H,
II.
Of much greater importance is the quest after the most eco-
nomical diameter of a conduit for a given discharge and under a
given head, a subject recently investigated by A. L. Adams.
The function of a pressure pipe is the transmission of energy
with a minimum of loss; the usefulness of a power plant as a
whole depends on several factors, chief amongst which is the
amount of revenue derived from its operation.
In comparison with the power transmitted the cost of a con-
duit transmitting all or nearly all the energy would be exces-
sive. The conduit having a diameter just sufficient to carry the
given quantity of water under the given head delivers but a
small percentage of the gross energy and its cost per horsepower
transmitted is equally excessive as the cost of the conduit deliver-
ing all the energy. The diameter of a conduit just sufficient to
carry a given quantity under a given head is equal to
APPENDIX III 213
The diameter necessary to carry the same quantity with a loss
of T^Vo of the gross energy is equal to
40(0.001>S)-
1£
= (1000)^^ = 3.875 times the diameter,
just sufficient to carry the given quantity.
A quantity of 100 /^ of water delivered at an elevation of 1000
feet above the generator possesses a potential energy of
100 X 1000 X 0.1134 = 11,340 H.P.
The diameter of a vertical steel-riveted conduit just sufficient
to carry the given quantity,
.=g)-^L ,404 feet.
The velocity corresponding to this diameter is equal to
V = 50.8 X (1.404)«-7 = 64.38 feet per second.
The energy transmitted is
(^^•^^y X 0.1134 X 100 = 729.9 H.P.
This is 6.43 per cent of the gross energy. The percentage
transmitted by the conduit just sufficient is not constant but
decreases with decreasing quantities and slopes.
For Q = 10, H = 100, L = 1000, for instance, the gross energy
is 113.4 H.P. and the energy transmitted 3.635 H.P., which
is 3.21 per cent.
The diameter corresponding to a loss of y oV(j oi the gross energy
is, for a vertical steel-riveted pipe carrying 100 /^,
3.875 X 1.404 = 5.437 feet.
A diameter of 5.407 feet transmits
.. o,^ (100)'^' X 1000 X 0.1134
1062 X (5.437)^-1
= 11,328.7 H.P.
The efficiency of the two conduits of 1.404 and 5.437 feet
diameter is consequently as 729.9 to 11,328.7 or 1 to 15.52. .
214 THE FLOW OF WATER
At n dollars per H.P. the value of the energy transmitted is
equal to
The thickness of the shell of riveted pipes is made equal to
0.434 hd r , . • 1
t = tor a m mches.
20,000
Hence the cubic contents of a shell one foot long
/.g _ td 12 7t
•^ ~ 1728 '
and its weight (specific gravity 7.854) per foot
0.434 hd:" 12 n 490
20,000 1728 '
which reduces to
w = 0.0334 hd'' for d in feet.
The weights of finished pipes indicate that the additional
weight due to rivets, laps and straps is sensibly equal to
w = 0.00607 hd\
so that the total weight of a finished pipe amounts to
w = 0.03947 hd\
At m dollars a pound for steel the cost of a finished pipe will
be
Di = 0.03947 hd?m.
If we now compare the cost of the two conduits of 1.404 and
5.437 feet with the value of the power lost and the respective
cost of the pipes per horsepower delivered we find, taking n = 100
and m = 0.06,
d
Cost
Value of energy lost.
Cost of pipe per h.p.
1.404 ft.
2318
1,061,000
3.171
5.437 ft.
34,803
1134
3.072
Between the two extremes, the conduit delivering but a small
percentage of the gross energy, and the conduit transmitting
nearly the whole, both delivering the energy at an equally high
expense per horsepower transmitted, there is evidently a con-
APPENDIX III
215
dition more favorable to economy and it is evident that the
greatest economy exists when
T Cost of Conduit • .
I. -— ;; : = a mmimum.
Value of energy transmitted
II. Value of Energy lost + Cost of conduit = a minimum.
The value of the energy lost + cost of conduit is
0.1134 ^'^'Lti
1062 c^«-i
+ 0.03947 hd'Lm.
Equating the first differential coefficient with regard to d to
zero we have
5.1 X 0.1134 Q'^Ln
1062 d'-'
2 X 0.03947 hdLm = 0,
which gives
= (:
5.1 X 0.1134 Q^n ^
2 X 1062 X 0.03947 mhj
M
= 0.4962
(Fl)
Values of Q^-"^^^, H'"^ and n^° are found in the table below.
For steel at c cents per pound values of ]^ and 0.4962
/lOOV^ r „
I are as follows
m'
0.4962
/100\71
0.4962 (^) •
c.
m^? *
5.0
0.7566
1.447
5.5
0.7463
1.428
6.0
0.7381
1.410
6.5
0.7291
1.395
7.0
0.7213
1.381
7.5
0.7146
1.367
8.0
0.7079
1.354
If we again take the previous example of the vertical conduit
1000 feet long and carrying 100 P per second, we find for n =
100, m = 0.06 and H a mean value of 500, from the tables,
d = 1.410^^^= 3.831 feet
2.40
= 46 inches; very near.
216
THE FLOW OF WATER
The loss of energy corresponding to this diameter is
= 67.8
and the net horsepower = 11,340 - 69.8 = 11,272.2. The three
conduits we have taken as examples compare as follows:
d in
Feet.
Cost of
Conduit.
Value of
Energy Lost.
Cost of Con-
duit + Value
of Energy
Lost.
Cost of
Conduit per
net
Horsepower.
Efficiency.
1.404
3.831
5.437
2,318
17,781
34,803
1,061,000
6,380
1,134
1,063,318
24,161
35,937
3.171
1.574
3.072
1.0
15.45
15.52
Formula I as given above gives best results when applied to
conduits of riveted steel under high pressures.
The experiments of Darcy, Hamilton, Smith, C. Hershel indicate
that for riveted pipes up to 4 feet in diameter the mean value of
the coefficient c corresponding to a velocity of one foot per second
is equal to 101.1, or nearly so. Taking the value of the coefficient
of variation of c equal to a = F^ the exponential equation
corresponding to the given values of c and a reads,
F = 63.66 d^-^/S^,
Q = SOd^f^A
rr_ Q'^L
^ n •
1622 d ^
This gives for the most economical diameter
d = 0.4465
H^
L (VlY
\ [mj
(FIT)
For steel at c cents a pound and n = 100 dollars the values of
-]'' are as follows:
c= 0.4465
5.0= 1.371
5.5= 1.351
6.0= 1.334
C-)^
6.5= 1.319
7.0= 1.304
7.5 = 1.291
8.0= 1.279
APPENDIX III
217
Formula II is best suited to small quantities of discharge and
low heads. For our previous example, Q = 100, ^ = 0.5 X 1000,
the formula gives d = 3.797, hence 0.034 feet or 0.408 inch less
than Formula I. The difference increases with the decrease of the
quantity; for small diameters the difference amounts to as much
as one inch. It will be observed that according to Formula I,
the value of the energy lost is equal to %^ = 0.392, the cost of the
pipe. Formula II gives if = 0.418.
VALUES OF Q AND Q''-"'^
Q
Q0.407
Q
qO.407
Q
qO.407
Q
n0.407
Q
g0.407
0.5
0.737
15.5
3.051
36
4.300
82
6.011
210
8.813
1.0
1.000
16.0
3.091
37
4.348
84
6.071
220
8.982
1.5
1.179
16.5
3.130
38
4.395
86
6.128
230
9.146
2.0
1.326
17.0
3.168
39
4.442
88
6.185
240
9.305
2.5
1.452
17.5
3.206
40
4.488
90
6.243
250
9.462
3.0
1.564
18.0
3.242
41
4.533
92
6.299
260
9.614
3.5
1.665
18.5
3.279
42
4.578
94
6.354
270
9.763
4.0
1.758
19.0
3.315
43
4.622
96
6.408
280
9.907
4.5
1.844
19.5
3.350
44
4.665
98
6.463
290
10.063
5.0
1.912
20.0
3.385
45
4.708
100
6.516
300
10.190
5.5
2.001
20.5
3.419
46
4.751
105
6.647
310
10.327
6.0
2.072
21.0
3.453
47
4.792
110
6.772
320
10.413
6.5
2.142
21.5
3.486
48
4.834
115
6.898
330
10.594
7.0
2.202
22.0
3.519
49
4.874
120
7.018
340
10.723
7.5
2.211
22.5
3.551
50
4.914
125
7.136
350
10.850
8.0
2.331
23.0
3.575
52
4.994
130
7.251
360
10.978
8.5
2.389
23.5
3.614
54
5.098
135
7.363
370
11.098
9.0
2.446
24.0
3.645
56
5.146
140
7.472
380
11.219
9.5
2.500
24.5
3.676
58
5.221
145
7.580
390
11.339
10.0
2.553
25.0
3.707
60
5.293
150
7.684
400
11.455
10.5
2.604
26
3.766
62
5.364
155
7.789
410
11.572
11.0
2.654
27
3.824
64
5.434
160
7.890
420
11.686
11.5
2.702
28
3.881
66
5.502
165
7.990
430
11.799
12.0
2.759
29
3.937
68
5.570
170
8.087
440
11.909
12.5
2.795
30
3.992
70
5.636
175
8.184
450
12.018
13.0
2.840
31
4.046
72
5.701
180
8.277
460
12.127
13.5
2.885
32
4.098
74
5.765
185
8.370
470
12.223
14.0
2.929
33
4.150
76
5. -828
190
8.463
480
12.388
14.5
2.969
34
4.201
78
5.887
195
8.551
490
12.443
15.0
3.011
35
4.251
80
5.951
200
8.640
500
12.545
218
THE FLOW OF WATER
VALUES OF H AND i?H, N AND iV^l
H
hH
H
h\1
H
H^l
H
nVt
H
H^
5
1.254
95
1.899
270
2.200
500
2.400
860
2.589
10
1.383
100
1.913
280
2.211
520
2.413
880
2.598
15
1.464
110
1.939
290
2.222
540
2.426
900
2.606
20
1.525
120
1.966
300
2.233
560
2.438
920
2.615
25
1.574
130
1.985
310
2.243
580
2.456
940
2.623
30
1.614
140
2.006
320
2.253
600
2.462
960
2.631
35
1.650
150
2.025
330
2.263
620
2.473
980
2.638
40
1.681
160
2.044
340
2.273
640
2.485
1000
2.646
45
1.709
170
2.061
350
2.282
660
2.495
1050
2.664
50
1.735
180
2.098
360
2.291
680
2.506
1100
2.681
55
1.758
190
2.094
370
2.300
700
2.516
1150
2.698
60
1.780
200
2.109
380
2.309
720
2.526
1200
2.717
65
1.800
210
2.123
390
2.319
740
2.536
1250
2.730
70
1.819
220
2.138
400
2.325
760
2.545
1300
2.745
75
1.837
230
2.151
420
2.341
780
2.555
1350
2.760
80
1.854
240
2.164
440
2.357
800
2.564
1400
2.774
85
1.870
250
2.179
460
2.371
820
2.573
1450
2.788
90
1.885
260
2.189
480
2.396
840
2.581
1500
2.801
III.
CONDUITS OF PLANED STAVES.
Circular conduits of planed staves are occasionally used for
heads up to 200 feet. The thickness of the shell of such con-
duits is usually made equal to 2 or 2.5 inches. For these dimen-
sions a conduit one foot in internal diameter will contain 9 or 12
feet board measure. Owing, to the constant addition of 4 or 5
inches to the internal diameter the number of feet board measure
does not increase with d but with d^, very near. At I dollars a
foot board measure the cost of the wooden shell put in place will
be equal to
9 or 12 d^ LI respectively.
Likewise the length of the tension rods, usually f-inch steel,
increases with d^', their weight is therefore proportional to d^ .
It is safe to allow a stress of 15,000 pounds per square inch in these
rods and as the outside diameter of the one foot pipe is equal to
1.333 or 1.416 the inside diameter; the cost of the metal will be
equal, at m dollars a pound, to
0.0338 or 0.0359 d^^ hLm.
APPENDIX III 219
For pipes of planed staves a mean value of the coefficient c
corresponding to a velocity of 1 foot per second is equal to 108, or
nearly so. Taking the coefficient of variation of c equal to V^
this gives the exponential equation
Q = 53.63 d%^,
H =
1848/^'
Value of power lost at n dollars per horsepower,
q'^'l 0.1134 n
1848 d"^'
Equating the first differential coefficient of value of power lost
plus cost of pipe to zero we have (for t = 2 inches)
- 43 Q^^L 0.1134 n + iZo.0338 dhLm+ | ^d'^Ll = 0;
■^ 9 9
9 1848 d
hence
43 Q^^^ 0.1134 n ^ ^ ^^^^ ^^^^ ^V;.
9 1848
It is possible to solve this equation by Horner's, or some other
method of approximation. Fairly good results are obtained by
taking a mean of the exponents of d.
The formula will then read, after reduction,
. _ /a0002932Q^Y3
for ^ = 2 inches. For t = 2.5 inches 0.0678 is substituted for
0.0638 and 10.8 I for 8 l. This formula gives results sometimes
above, sometimes below the true value. Where great accuracy
is desired, the value of d obtained from the formula may be
tested by putting its ninth root into the expression 0.0638, or
'0.0678 X^ hm + 8, or 10.8 Z^S and increasing or diminishing
the value of X till this expression is equal in value to
43 q'^' 0.1134 n
9 1848
It is to be observed that for this class of conduits the ratio
between the value of the power lost and the cost of the pipe which
220 THE FLOW OF WATER
corresponds to a maximum of economy is not the same as for
metal conduits. If the tension rods did not enter the problem the
ratio would be as 8 to 43; as it is the ratio is variable but usually
in the neighborhood of 11 or 12 to 43 or 0.25 to 0.28 to 1.0.
IV.
CONDUITS LINED WITH PLAIN OR ARMORED CONCRETE.
Concrete, plain or armored, is coming more and more into
favor as a material forming the shells of conduits of all descrip-
tions. Over metal and wood this substance possesses the great
advantage not to be subject to corrosion and decay; it is prac-
tically indestructible. Experiments have brought to light the
fact that plain concrete conduits under internal pressure fail
when the stress in the material reaches 168 pounds per square
inch or nearly so. Under external pressures, however, they fail
only when the stress reaches 1500 pounds per square inch or
nearly so. Plain concrete is therefore not economical where inter-
nal pressures enter the problem. But the material may be used
to great advantage when great quantities of water are to be
delivered under low heads.
The thickness of the shell of plain concrete conduits as com-
monly used for sewers and other conduits not subject to internal
pressures is usually made equal to
2 inches for d = 1 foot.
4 inches for c? = 3 feet.
8 inches ior d = 9 feet.
12 inches for d = 18 feet.
These conduits will fail when the 1 foot pipe is under a head
of 127 feet or the 18 foot pipe under a head of 43 feet.
The thickness of the shell in inches of such conduits is propor-
tional to 2 (i<^•«^ the cubic contents of the shell to 0.611 c^^-^y^
At c dollars per P of concrete put into place the cost of such a
conduit will consequently be
0.611 d^-^^Lc.
Taking the coefficient of variation of c equal to F^* the exponen-
tial equations which apply to flow in conduits lined with concrete
are as follows:
m = 0.95, conduits smoothly dressed with neat cement,
Q = 54.3 ^2.66^1^^
APPENDIX III 221
m = 0.83, conduits lined with cement plaster, 1 part cement,
2 parts sand; plain concrete washed with neat cement,
Q = 50.2 c^2.67/^il,
m = 0.57, conduits lined with plain concrete,
Q = 41.2 d^'^'^S^.
For m = 0.95 the value of the power lost plus the cost of the
conduit will be equal to
'Q"L 0.1134 r.g^^^,.e3j^
1893 (i'-"24
of which the first differential coefficient equated to zero
_ 5.024 O^^^L 0.1134 r^ ^^^3 ^ ^^^ d-^^Lc=0,
1893 d«-«24
The most economical diameter will be equal
for m = 0.95 to d = 0.2967 Q^' (-X\
for m = 0.83 io d = 0.3044 Q^^(~\ "",
3
for m = 0.57 to 6^ = 0.3247 Q^^ /-V' .
It is to be observed that this class of conduits is in the state
of greatest economy when the value of the power lost is equal
(for m = 0.83) to ^^ = 0.323 of the cost of the conduit.
^ 5.043
Concrete beams armored with 1.75 to 2 per cent steel fail when
the modulus of rupture equals 2400 pounds per square inch or
nearly so. Taking 10 as a factor of safety the working stress
for internal pressures will be 240 pounds per square inch and the
thickness of the shell will be equal to
, OAS^hd ,
480 '
z being equal to 1 inch ior h = 1 to h = 100, and vanishing for
h = 1000. Accordingly the thickness of the shell of a conduit
1 foot in internal diameter for h = 1000 will be 10.4 inches and the
cubic contents of the shell for any diameter and any head will be
(0.611 + 0.0035 h)d'''T-
222 THE FLOW OF WATER
Cracks in armored concrete begin to appear when the stress
in the steel equals 12 to 15,000 pounds per square inch. A safe
working stress will therefore be 10,000 pounds per square inch.
Allowing one-sixth for the increase of the diameter where the
armoring is placed in the 1 foot pipe and also one-sixth for the
laps of the armoring, the weight of the metal in the 1 foot pipe
will be ior h = 1 equal to 0.0445 pounds.
But the thickness of the shell increases with h and conse-
quently the length of the circumference where the armoring is
placed. The necessary increase in the amount of the armoring
is proportional to V^ very near, so that for any head and any
diameter the weight of the armoring will be
0.0445 h^^ d'-^' L.
Using these values we find for the most economical diameter,
m = 0.95,
_/ 0.0003031 Q^n V
\0.0445 h^'^m + (0.611 + 0.0035 h) J
0445 h^'^m + (0.611 + 0.0035 h) d
m = 0.83,
d
m = 0.57,
d
'k
=(.:
0.0003523 Q^n
0445 h m + (0.611 + 0.0035 h) d
0.000514 Q^^n ^'^
0445 h^^m + (0.611 + 0.0035 K) J
In these equations n = value of 1 horsepower,
m = value of 1 pound of steel,
c = value of 1 cubic foot of concrete.
APPENDIX III 223
V.
THE MOST ECONOMICAL DIAMETER FOR METRIC MEASURE.
The exponential equation for steel riveted pipes reads,
e = 28 ci2.7^^.
H =
Q'^L
Value of power lost at n dollars per kilowatt
9.81 Q^^'Ln
541.4(^^-1 •
Allowing a tension of 14 kgm. per square millimeter in the
steel the weights of the shell in kilograms will be
J^^3.1416ci7854 = 1.76246 <^^;.,
and allowing for rivets, laps and straps, the cost of the conduit
at m dollars per kilogram will be equal to
2.083 d?hm,
which gives for the most economical diameter
X 9.81 q'^'n \^^
< 2 X 2.083 hm)
(LQ.
A mean value of the coefficient c corresponding to a velocity
of 1 meter per second found from data relating to flow in 17
riveted conduits including the largest as well as the smallest is
equal to 61.93. Taking a = F^ this corresponds to the exponen-
tial equation,
Q = 29.76 #>Si^,
from which we find for the most economical diameter
''^(siift
d = 0.5551
""(s:)"
In these equations d and h are in meters,
Q in m' per second,
n the value of 1 kilowatt,
m the value of 1 kilogram of steel.
INDEX.
PAGE
Alexander's experiments 30
Authorities of experimental data 77
Bazin formula 168
Bourdon gauge 184
Brass tubes 72
Channels in earth 47
Channels, proportions of 117
Circular conduits 73
Coefficient C 15
primary determination 17
variation of 19, 196
Coefficient of friction 71
Conduits, circular 73
classification according to coefiicient a 32
forms of sections 113
greatest efficiency 211
long, circular 121
masonry 115
planed staves 218
open 43, 129
discharge of 188
powers of diameters 145
quantities of discharge 155
riveted 59
Coulomb's investigations 31
Cross-section most favorable to flow 129
Current meters 190
Curve, friction of 56
Darcy and Hamilton Smith's experiments 34, 216
Darcy gauge 192
Depths of water, powers. 147
Determination of coefficient C 17
Diameters of velocities, general relations 121
Diameter of conduits, most economical 211
for metric measure 223
powers of 145
Direction current meter 191
Discharge of conduits 155, 184, 188
225
226 INDEX
PAGE
Discharge, quantities, of semisquare 163
weir 167
Distribution, energy 14
head 12
Double float 189
Energy, distribution of 14
EngHsh and metric equivalents 209
Erosion, resistance to 48
Exponential equations 121
metric measure '. 205
Fall, primary laws 4
Floats 189
Flow, velocities in semisquare 159
velocity of 24
Fluid friction, primary laws 8
Formulae, metric measure 205
practical applications 62
Francis' formula 167
Friction in curves 56
Galvanized iron tubes 72
Ganguillet and Kutter's formula 203
Harlacher meter 191
Head, distribution of 12
loss of 183
Hubbel and Fenkell's experiments 29
Kinetic energy 68
Lawrence and Braunworth's experiments 187
Loss of head 183
Masonry conduits, forms of sections 115
Mean hydraulic radius, relation to wet perimeter 113
Mean hydraulic radii, roots of 74
Measurement, loss of head 183
Meters 191
Metric equivalents 209
Metric measure, formulae and equations 205
most economical diameter 223
Notation ". . . iv
Open conduits 43, 129
Pipes, welded 72
Pitot tube 192
Powers of depth of water 147, 151
Pressure, primary laws 4
INDEX 227
PAGE
Price current meter 191
Primary laws, fluid friction 8
pressure and fall 4
Prony's formula 31
Quantities, use of tables of 136
Resistance due to entrances and elbows 57
Ritchie-Haskell current meter 191
Riveted conduits 59
Rod floats 189
Roughness, degree of 76
Saph and Schoder's experiments 30
Sections, areas of 117
Semisquare, quantities of discharge 163
velocities of discharge 159
Sewers 118
Sheet iron tubes 72
Slopes, table of sines of 143
Stearns and Fitzgerald's experiments 35
Subsurface float 189
Surface float 189
Surface mean and bottom velocities 193
TABLES:
I. II. Variation of coefficient C 19
III. Experimental data showing extent of variation of C with the
velocity of flow 37
IV.A. Weisbach's coefficients for resistance due to entrances, elbows,
etc 57
IV. Friction in curves 56
V. Roots of velocities 70
VI. Values of 66 {^r + m) 71
VI.A. Welded pipes 72
VII. Circular conduits 73
VILA. Roots of mean hydraulic radii 74
A. Values of R and areas of sections in terms of radius 117
B. Proportions of channels of maximum values of i? 117
C. Sines of slopes and roots of sines of slopes 143
D. Powers of diameters of conduits 145
E. Powers of mean hydraulic radii 147
F. Form of section most favorable to flow 151
G. Quantities of discharge of conduits 155
H. Velocities of flow in semisquare 159
I. Quantities of discharge of semisquare 163
K. Values of 3.33 H^ 168
L.a, Value of constant in Bazin's formula 170
L.b, Value of Q in Bazin's formula 171
228 INDEX
Tables {continued). page
VIII. Values of coefficients indicating degree of roughness 76
IX. List of authorities whose experimental data are given 77
X. Experimental data 82
Tables of velocities, use of 136
Tachometer, Woltman's 191
Thread and mean velocity 193
Variation of coefficient C, extent of 37
with slope 196
Velocities, roots of 70
surface, mean and bottom 193
tables of 136
Velocity, discharge and depth of water, relations between 131
Velocity measurements 190
Velocity of flow, variation of coefficient C 24
Venturi meter, theory of 185
Weir discharges 167
formulae 173
Weisbach's coefficients of resistance 57
Welded pipes 72
Wet perimeter, relation to mean hydraulic radius 113
roughness of 21
Woltman's tachometer 191
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10 STANDARD TEXT BOOKS.
PRESCOTT, A. B., Prof. Organic Analysis. A Manual of the
Descriptive and Analytical Chemistry of Certain Carbon Compounds in
Common Use; a Guide in the Qualitative and Quantitative Analysis of
Organic Materials in Commercial and Pharmaceutical Assays, in the Esti-
mation of Impurities under Authorized Standards, and in Forensic Exami-
nations for Poisons, with Directions for Elementary Organic Analysis.
Sixth edition. 8vo, cloth $5.00
and SULLIVAN, E. C. First Book in Qualitative Chemistry.
Eleventh edition. 12mo, cloth net, $1.50
and OTIS COE JOHNSON. Qualitative Chemical Analysis. A
Guide in the Practical Study of Chemistry and in the Work of Analysis.
Sixth revised and enlarged edition, entirely rewritten, with an
Appendix by H. H. Willard containing a few improved methods of Analysis.
8vo, cloth net, $3. 50
RANKINE, W. J. MACQUORN, C.E., LL.D., F.R.S. Machinery and
Mill-work. Comprising the Geometry, Motions, Work, Strength, Con-
struction, and Objects of Machines, etc. Illustrated with nearly 300
woodcuts. Seventh edition. Thoroughly revised by W. J. Millar. 8vo,
cloth $5.00
The Steam-Engine and Other Prime Movers. With diagram of
the Mechanical Properties of Steam. With folding plates, immerous
tables and illustrations. Fifteenth edition. Thoroughly revised by W. J.
MUlar. 8vo, cloth $5.00
Useful Rules and Tables for Engineers and Others. With
appendix, tables, tests, and formulae for the use of Electrical Engineers.
Comprising Submarine Electrical Engineering, Electric Lighting, and
Transmission of Power. By Andrew Jamieson, C.E., F.R.S. E. Seventh
edition, thoroughly revised by W. J. Millar. 8vo, cloth $4.00
A Mechanical Text-Book. By Prof. Macquom Rankine and E. E.
Bamber, C.E. With numerous illustrations. Fifth edition. 8vo,
cloth $3.50
RANKINE, W. J. MACQUORN, C.E., LL.D., F.R,S. Applied Me-
chanics. Comprising the Principles of Statics and Cinematics, and Theory
of Structures, Mechanics, and Machines. With numerous diagrams.
Eighteenth edition. Thoroughly revised by W.J.Millar. 8 vo, cloth $5.00
Civil Engineering. Comprising Engineering, Surveys, Earthwork,
Foundations, Masonry, Carpentry, Metal-Work, Roads, Railways, Canals,
Rivers, Water-Works, Harbors, etc. With numerous tables and illus-
trations. Twenty-third edition. Thoroughly revised by W. J. Millar.
8vo, cloth $6.50
RATEAU, A. Experimental Researches on the Flow of Steam
Through Nozzles and Orifices, to which is added a note on The Flow of
Hot Water. Authorized translation by H. Boyd Brydon. 12mo, cloth.
Illustrated net, $1.50
RAUTENSTRAUCH, W., Prof., and WILLIAMS, J. T. Machine
Drafting and Empirical Design. A Textbook for Students in Engineering
Schools and Others Who are Beginning the Study of Drawing as Applied
to Machine Design. Part I. Machine Drafting. Illustrated, 70 pp.,
8vo, cloth net, $1.25
Complete in Two Parts. Part II in preparation.
STANDARD TEXT BOOKS, 11
RAYMOND, E. B. Alternating Current Engineering Practically
Treated. Tiiird edition, revised and enlarged, with an additional
chapter on "The Rotary Converter." 12mo, cloth. Illustrated. 232 pages.
net, $2.50^
REINHARDT, CHAS. W. Lettering for Draughtsmen, Engineers and
Students. A Practical System of Free-hand Lettering for Working Draw-
ings. New and revised edition. Thirty-first thousand. Oblong boards.
$1.00-
RICE, J. M., Prof., and JOHNSON, W. W., Prof. On a New Method
of Obtaining the Differential of Functions, with especial reference to the
Newtonian Conception of Rates of Velocities. 12mo, paper $0.50
RIPPER, WILLIAM. A Course of Instruction in Machine Drawing
and Design for Technical Schools and Engineer Students. With 52 plates
and numerous explanatory engravings. Second edition. 4to, cloth. $6.00
ROBINSON, J. B. Architectural Composition. An attempt to order
and phrase ideas which hitherto had been only felt by the instinctive taste
of designers. 233 pp., 173 illustrations. 8vo, cloth net, $2.50
ROGERS, ALLEN. A Laboratory Guide of Industrial Chemistry.
Illustrated. 170 pp. 8vo, cloth net, $1.50
SCHMALL, C. N. First Course in Analytic Geometry, Plane and
Solid, with Numerous Examples. Containing figures and diagrams. 12mo^
half leather, illustrated net, $1.75
and SHACK, S. M. Elements of Plane Geometry. An Elemen-
tary Treatise. With many examples and diagrams. r2mo, half leather,
illustrated net, $1.25
SEATON, A. E., and ROUNTHWAITE, H. M. A Pocket-book of
Marine Enghieering Rules and Tables. For the Use of Marine Engineers
and Naval Architects, Designers, Draughtsmen, Superintendents and all
engaged in the design and construction of Marine Machinery, Naval and
Mercantile. Seventh edition, revised and enlarged. Pocket size.
Leather, with diagrams $3.00
SEIDELL, A. (Bureau of Chemistry, Wash., D. C). Solubilities of
Inorganic and Organic Substances. A handbook of the most reliable
Quantitative Solubility Determinations. 8vo, cloth, 367 pp net, $3.00
SEVER, Prof. G. F. Electrical Engineering Experiments and Tests
on Direct-Current Machinery. With diagrams and figures. Second
edition, thoroughly revised and enlarged. 8vo, pamphlet. Illus-
trated net, $1.00
^^and TOWNSEND, F. Laboratory and Factory Tests in Elec-
trical Engineering. Second Edition, tliorougiily revised and enlarged*
8vo, cloth. Illustrated. 236 pages , net, $2.50
SHELDON, S., Prof., and MASON, HOBART, B.S. Dynamo Elec-
tric Machinery; its Construction, Design, and Operation. Direct-Current
Machines. Seventh edition, revised. 12mo, cloth. Illustrated.
net, $2.50
12 STANDARD TEXT BOOKS.
SHELDON, S., MASON, H., and HAUSMANN, E. Alternating Current
Machines. Being the second volume of the authors' "Dynamo Electric
Machinery; its Construction, Design, and Operation." With many dia-
grams and figures. (Binding uniform with volume I.) Seventh edition,
completely rewritten. 12mo, cloth. Illustrated. net, $2.50
SHIELDS, J. E. Notes on Engineering Construction. Embracing
Discussions of the Principles involved, and Descriptions of the Material
employed in Tunneling, Bridging, Canal and Road Building, etc. 12mo,
cloth $1.50
SHUNK, W. F. The Field Engineer. A Handy Book of Practice
the Survey, Location and Track-work of Railroads, containing a large
collection of Rules and Tables, original and selected, applicable to both the
standard and Narrow Gauge, and prepared with special reference to the
wants of the young engineer. Nineteenth edition, revised and enlarged.
With addenda. 12mo, morocco, tucks $2.50
SMITH, F. E. Handbook of General Instruction for Mechanics.
Rules and formulae for practical men. 12mo, cloth, illustrated. 324 pp.
net, $1.50
SOTHERN, J. W. The Marine Steam Turbine. A practical descrip-
tion of the Parsons Marine Turbine as now constructed, fitted and run,
intended for the use of students, marine engineers, superintendent engineers
draughtsmen, works managers, foremen, engineers and others. Third
edition, rewritten up to date and greatly enlarged. ISO illustrations
and folding plates, 352 pp. 8vo, cloth net, $5.00
STAHL, A. W.,and WOODS, A. T. Elementary Mechanism. A Text-
Book for Students of Mechanical Engineering. Sixteenth edition, en-
larged. 12mo, cloth $2.00
STALEY, CADY, and PIERSON, GEO. S. The Separate System of
Sewerage; its Theory and Construction. With maps, plates, and illus-
trations. Third edition, revised and enlarged, with a chapter on
" Sewage Disposal." 8vo, cloth $3.00
STODOLA, Dr. A. The Steam-Turbine. With an appendix on Gas
Turbines and the future of Heat Engines. Authorized Translation from
the Second Enlarged and Revised German edition by Dr. Louis C. Loewen-
stein. 8vo, cloth. Illustrated. 434 pages net, $4.50
SUDBOROUGH, J. J., and JAMES, T. C. Practical Organic Chem-
istry. 92 illustrations. 394 pp., 12mo, cloth net, $2.00
SWOOPE, C. \TALTON. Practical Le.esons in Electricity. Princi-
ples, Experiments, and Arithmetical Problems. An Elementary Text-
Book. "V\ ith numerous tables, formulse, and two large instruction plates.
Tenth edition. 12mo, cloth. Illustrated net, $2.00
TITHERLEY, A. W., Prof. Laboratory Course of Organic Chemistry,
including Qualitative Organic Analysis. With figures. 8vo, cloth. Illus-
trated net, $2.00
THURSO, JOHN W. Modern Turbine Practice and Water-Power
Plants. Second edition, revised. 8vo, 244 pages. Illustrated.
net, $4.00
STANDARD TEXT BOOKS. 13
TOWNSEND, F. Short Course in Alternating Current Testing. 8vo,
pamphlet. 32 pages net, $0.75
URQUHART, J. W. Dynamo Construction. A practical handbook
for the use of Engineer-Constructors and Electricians in charge, embracing
Framework Building, Field Magnet and Armature Winding and Group-
ing, Compounding, etc., with examples of leading English, American,
and Continental Dynamos and Motors. With numerous illustrations.
12mo, cloth $3.00
VAN NOSTRAND'S Chemical Annual, based on Biederman's " Chiem-
ker Kalender." Edited by Prof. J. C. Olsen, with the co-operation of
Eminent Chemists. Revised and enlarged. Second issue 1909. 12mo,
cloth net, $2.50
VEGA, Von (Baron). Logarithmic Tables of Numbers and Trig-
onometrical Functions. Translated from the 40th, or Dr. Bremiker's
thoroughly revised and enlarged edition, by W. L. F. Fischer, M.A., F.R.S.
Eighty-first edition. Svo, half morocco $2.50
WEISBACH, JULIUS. A Manual of Theoretical Mechanics. Ninth
American edition. Translated from the fourth augmented and im-
proved German edition, with an Introduction to the Calculus by Eckley B.
Coxe, A.M., Mining Engineer. 1100 pages, and 902 woodcut illustrations.
Svo, cloth $6.00
Sheep $7.50
and HERRMANN, G. Mechanics of Air Machinery. Author-
ized translation with an appendix on American practice by Prof. A.
Trowbridge. Svo, cloth, 206 pages. Illustrated net, $3.75
WESTON, EDMUND B. Tables Showing Loss of Head Due to
Friction of Water in Pipes. Fourth edition. 12mo, full leather. . .$1.50
WILLSON, F. N. Theoretical and Practical Graphics. An Educational
Course on the Theory and Practical Applications of Descriptive Geometry
and Mechanical Drawing. Prepared for students in General Science,
Engraving, or Architecture. Third edition, revised. Ito, cloth,
illustrated net, $4.00
Descriptive Geometry, Pure and Applied, with a chapter on
Higher Plane Curves, and the Helix. 4to, cloth, illustrated net, $3.00
WILSON, GEO. Inorganic Chemistry, with New Notation. Revised
and enlarged by H. G. Madan. New edition. 12mo, cloth $2.00
WINCHELL, N. H., and A. N. Elements of Optical Mineralogy.
An introduction to microscopic petrography, with descriptions of all
minerals whose optical elements are known and tables arranged for their
determination microscopically. 354 illustrations. 525 pages. Svo,
cloth net, $3.50
WRIGHT, T. W., Prof. Elements of Mechanics, including Kinematics,
Kinetics, and Statics. Seventh edition, revised. Svo, cloth $2.50
and HAYFORD, J. F. Adjustment of Observations by the
Method of Least Squares, with applications to Geodetic Work. Second
edition, rewritten. 8vo, cloth, illustrated net, $3.00
ZEUNER, A., Dr. Technical Thermodynamics. Translated from the
Fifth, completely revised German edition of Dr. Zeuner's original treatise
on Thermodynamics, by Prof. J. F. Klein, Lehigh University. 8\(). cloth,
two volumes, illustrated, 900 pages m L. $8.00
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