UNIVERSITY OF CALIFORNIA
AT LOS ANGELES

SCLLIVflN'S

NEW HYDRAULICS

Consisting of

New Hydraulic Formulas

and

The Rational Law of Variation of
Coefficients

Plow and Resistance to Flow in all Classes of Rivers, Canals,
Flumes, Aqueducts, Sewers, Pipes, Fire Hose. Hy-
draulic Giants, Power Mains, Nozzles,
Reducers, etc. with Extensive Ta-
bles and Data of Cost of Pipes
and Trenching and Pipe
Line Construction.

MARVIN E. SULLIVAN, B. PH., L.L.B,
Hydraulic Engineer.

DENVER

MINING REPORTER
1900

BY
MARVIN E. SULLIVAN

SULLIVAN'S NEW HYDRAULICS.

Page 36 Under table of channels, add the following:
Divisor : quotient :: Dividend : quotient squared.

P : r :: a : r 8
or r : r 8 :: p : a, in any possible case,

Page 40 9th line from bottom. "If the value of the co-
efficient etc." This should be corrected so as to read:
As the loss of effective head or slope is inversely as v/d 3
or i/r 3 for any constant head,it is evident that a change in
the value of d or r cannot affect the value of the coeffi-
cient n, for as the loss of head per foot length, S", de-
creases directly as -\/d 3 or -y/r 3 increases, the effective
head or slope will increase as /d 3 or ^/r 3 and this will re-
sult in a like increase of v 2 . As n is the ratio of
SVd 8

, and as S" varies inversely with v/d s , it is evi-

dent that v s varies directly with v/d s or yr 3 . Hence

the ratio of 5- n, will be constant for all diameters
v 2

and all slopeu and velocities, and will not be affected by
anything except a change in roughness of perimeter. A
similar correction should be applied to similar errors oc-
curring from page 40 to page 51. See for a general cor-
rection of such errors, pages 237 to 241.

42 /_ L= / ^ should be C= '-L
V m vS-i/r 3 A m

' C =4^T_

Page 42 v=Cf/r 3 y<S should be v=C t/r 3 Xi/S
Page42 9th line from bottom, "Varies with {/r 3 " should be

7,,

SULLIVAN'S NEW HYDRAULICS.
"Varies inversely with $/r 8 . But see pages 237 241.

Page 45 18th line from top. "His n might be made & o"
should be "His C might be made to vary &c.

Page 46 Bottom line, v 3 should be v 8 .

Page 48 9th line from top. S/r 2 should be S/r 8 .

Page 52 Equation (18) should be m= s '+ Sv Xy/r 8

Page 85 River Elbe. C=49.80 should be 0=32.51, and C=
52 should be 0=38.40.

Page 86 Top line. 1885 should be 1855.

Page 109, 110 Remark. Add the following: As large cast
iron pipes are full of swellings and contractions, a 48 inch
pipe is not really 48 inches diameter. As the effective
value of any constant head or slope varies directly with
the actual value of ;/d 3 , if we credit the pipe as being 48
inches, when in fact it is not, the result is to make the
value of Sv/d 3 , apparently too large for the correspond-
ing value of v a . Hence m will be too large or C will be
too small for a pipe which is really 48 inches diameter
throughout. The irregularity of diameter in large cast
iron pipes reduces the value of C by from 3 to 5 per cenf
for diameters greater than 36 inches. It is not the fault
of the formula, nor is it a peculiarity in the law of flow.
It is simply the fault in casting pipes of large diameters.

Page 114 4th line from bottom. ACXt/d 3 -3- should

be A CXt/d 3 = -5L

Page 148-d=U/ gMgOBXq* 2632^^- shouM be

PREFACE

|_|YDRAULICIANS and engineers have long been aware
* that there is some element or law governing the flow
and resistance to flow of water which is not provided for in
any of the formulas presented up to this time. This is made
evident by the fact that the results as computed by formula
do not agree with actual results, and by the further fact that
no two formulas will give the same result for like conditions.
Nearly all writers give one theory of flow and resistance to
flow in pipes and closed channels, and an entirely different
theory and formula for flow in open channels. The usual
formulas for flow in pipes give results too high for all diam-
eters smaller than about fourteen inches, and too low for
all greater diameters.

The loss of head or pressure by resistance as computed
by the ordinary formulas is much too small in small pipes and
greatly in excess of the truth in large pipes. The reason of
these erroneous results is that the coefficients were deter-
mined for pipes of medium diameter and do not vary correctly
so as to meet the requirements of varying diameters. Hence
the greater the diameter varies from the medium, either be-
low or above, the greater will be the error in the computed
result. The usual formulas for flow in open channels are
equally faulty, but not in the same way.

The acknowledged fault in all the formulas so far pre-
sented is due to the failure of hydraulicians to discover the
rational law of variation of the coefficient. This law has
been sought in vain since the beginning of the sixteenth cen-
tury, at which time Galileo discovered the law of gravity and
undertook to formulate rules for flow in rivers. The failure
to discover the true law of variation of the coefficient has
been generally conceded by all, and the possibility of its dis-
covery has been denied by macy. Ganguillet and Kutter ob-

4 SULLIVAN'S NEW HYDRAULICS.

serve that "more than a century ago, Michelotti and Bossut
established the true principle that the formulae for the move-
ment of water must be ascertained from the results of observa-
tion, and not by abstract reasoning." In the Translator's
preface (Hering& Trautwine) to the work of Ganguillet and
Kutter the following observation occurs:

"As V=C i/RS will most likely remain the fundamental
expression for such formulae, the attention of hydraulicians
will be turned chiefly to the more accurate determination of
the variable coefficient C. A number of authors have en-
deavored to establish laws for its variation, and among them
Ganguillet and Kutter appear so far to have been the most
successful." Ganguillet and Kutter, however, do not claim to
have discovered the true law of variation, as many unthink-
ing persons have supposed, but on the contrary they announce
the belief that it cannot be discovered by abstract reasoning,
and in default of its discovery they propose a formula which
is entirely empirical. They observe (page 105): "The form-
ula (Kutter's) rests only upon actual guagings. * * * Be-
an empirical formula, it is confined to the limits occurring in
nature and makes no claim whatever to absolute perfection.
In spite of the large number of available guagings, it cannot
be denied that our knowledge of the elements and laws of the
motion of water still need extension and correction."

Webster's definition of the word empirical is, "used and
applied without science." The Ganguillet and Kutter formula
does not claim to be scientific or rational, and yet it is a con-
siderable improvement on some of the older formulas, but is
quite complicated and not simple and easy of application. In
the search for the true law of variation of the coefficient
the greatest mistake has been made in assuming that V=
C^/RS is the fundamental expression for the formula. The
I/ S is the factor which expresses the unimpeded and con-
stant effect of gravity, while R is an expression for a factor
which modifies the effect of gravity. The constant effect of
gravity should be expressed separately and should not be con-
fused in the same expression with other factors which are

SULLIVAN'S NEW HYDRAULICS. 6

variable and which impede or modify the effect of gravity.
All the variable factors should be included in the coefficient
formula for C, and then all elements which" affect or modify
the law of gravity will be included in the value of C, and the
fundamental form of the formula then becomes simply V=

(V~s.

This is evidently true, because the value of B or D ha&
no connection with the law of gravity, which is constant.
The value of R or D has to do directly with the law of resist-
ance, and as R or D varies the resistance to flow varies. C is
supposed to include this resistance, and hence the value of R
or D should be included in the formula for C. If there were
no resistance_to flow whatever, then the velocity would be di-
rectly as i/ "S~ regardless of the value of R or D.But as the re-
sistance to flow does vary with the value of R or D, it is evi-
dent that C must vary with R or D, and if we write V=Cy ~S~
simply, we thereby have all modifying factors included in
the value of C, while the constant law of gravity is expressed
by i/S. Thus we prevent confusing the opposing laws of
gravity and of resistance in one combined factor, and clear
the way for ascertaining the true law which governs the var-
iation of the value of C.

By the law of gravity we know that if there were no re-
sistance whatever the velocity would be equal in all diame-
ters, regardless of dimensions, where the values of ^/ S
were equal. But by experiment we tind that the smaller the
diameter becomes, the smaller the velocity becomes for equal
values of |/ S . It is therefore evident that the resistance
to flow must vary with some function of the diameter or of
the hydraulic mean radius. As the resistance to flow is the
variable factor, and is separate and distinct from and direct-
ly opposed to, the acceleration of gravity which is a con-
stant, we know that -\/ S has nothing to do with the value
or variation of C. We therefore narrow the field of investiga-
tion by writing V=C^ S, and then searching for the law of
variation in C, which we know is some function of the diam-
eter or hydraulic radius. After years of diligent experiment

6 SULLIVAN'S NEW HYDRAULICS.

and observation the writer discovered that for any constant
degree of roughness of wetted perimeter, either in pipes or
open channels, the value of C varied asi/R 3 or as f/D 3
In other words, if K is a constant which represents the given
degree_p roughness of perimeter, then C=KX t/R 3 ,and V
= GI/S. But if we confuse the element of resistance,D or R
with the element of acceleration |/~S7by writing V=Cv/RS
then we find C varies as f/R~ simply, and if K represents
the degree of roughness, and we write V=Ci/BS, then we
have C=KXf / T^andV=<VRS.

As all hydraulic formulas are necessarily based on the
laws of gravity and of friction, the correctness of such formu-
las must depend upon their accordance with these laws. If
our present understanding and acceptance of these laws is
correct, then any formula which violates either of these laws
must necessarily be incorrect. It is the object of thie vol-
ume to present the rational law of variation of the coefficient
in accordance with our present understanding of the laws of
gravity and friction. If those laws are yet unknown it must
remain for some future investigator to supply a theoretically
correct hydraulic formula. The discussion in the following
pages is based on the assumption that those laws are cor-
rectly known, or nearly so.

In the case of open channels and rivers of irregular cross
section, and where the banks alternately diverge and con-
verge, and the perimeter varies in roughness at different
depths of flow, the correct application of any formula will be
difficult. In sections 13 and 83 methods are pointed out for
ascertaining the value of C in such cases.

The law of resistance in nozzles and convergent pipes, as
herein stated has been very thoroughly tested and its cor-
rectness established by hundreds of experiments. This law
will be found cf great service to hydraulic miners and fire-
men and also in determining the coefficient for flow in con-
verging reaches of rivers and other channels.

In the course of experiments of the writer which has
extended over a period of six years, it was discovered that the

SULLIVAN'S NEW HYDRAULICS. 7

discharges over weirs and through orifices, as computed by
the usual formulas, and with the tabulated direct coefficient
were frequently erroneous, especially if the weir used did not
correspond exactly in breadth and depth with that from
which the coefficient was determined. Interpolation for in-
termediate conditions was certain to result in error. On in-
vestigation it was found that the difficulty lay in the fact that
the law of variation of the c;efficient of contraction has never
been discovered. In order to avoid this difficulty until the
law of contraction shall be understood, an appendix has
been added to this volume in which the difficulties are
pointed out and the suggestion made that the position of the
weir be reversed in order to prevent contraction from taking
place at all. New weir and orifice formulas are proposed and
the writer hopes that other experimentalists will perfect the
theories there suggested. It is proper that attention should
here be called to the fact that our coefficient m or C, as used
for determining the flow in pipes includes all resistances to
flow, including the resistance to entry into the pipe. No sep-
arate provision was made in the formula for the resistance
to entry because it is a matter of no practical consequence
under ordinary circumstances, or in any case except for high
velocities in very short pipes. (See remarks under Group No.
1, 14.) The writer hopes that the theory of coefficients and
the law of their variation as herein presented may contribute
something new and valuable to hydraulics as a science.

For each degree of roughness of perimeter there is a unit
value of the coefficient from which unit value the coefficient
varies as f/Rs", or as f/D^when V=Cy~S; or C varies as
t/~R~or f/ D~if we write V=CyRS. At section 20 the var-
ious methods of writing the formula are given. As the old
theories and formulas are generally admitted to be erroneous
they have been given no space in this volume except in a few
instances, where the defects in the best of them have been
pointed out in the course of demonstrating the new princi-
ples herein presented.

Perhaps the great variety of theories of variation of the

8 SULLIVAN'S NEW HYDRAULICS.

coefficient, and of formulas for flow will beet exhibit the pres-
ent confuBed and uncertain knowledge of hydraulics. A few
of the leading formulas for flow in pipes, and a few of the
leading formulas for flow in open channels are here given in
order to illustrate the general confusion with which every
student of hydraulics has met. It will be noted that all, or
nearly all, these formulas may be reduced to the form V^Cv/KS.
Hence the main difference in them lies in the different theor-
ies of the variation of C. We find in most cases the same
author gives a different law of variation in C for pipes from
that given for open channels, as though the law depended
upon'the form of the channel, and changed with the change
in form of channel. Others adopt a constant unit value of C
for all classes and degrees of roughness of both open channels
and pipes, and make C vary with ^/TTonly. Hence the same
value of i/RS will give the same result by such formulas for open
channels as for pipes, and for rough as for smooth perimeter.
If, in the following formulas, the cofficient formula for C be
separated from the formula for V, the various theories for
variation in C at once appear, and we at once see why it is
that scarcely any two formulas will give like results for
the same conditions. It is evident, therefore, that if one is
right all the others are wrong.

for pipes (Fanning)

.000077^+

00000162 for Pips' 3 (D'Arcy)

SULLIVAN'S NEW HYDRAULICS.

V=105.926 (RS)H for pipes (Saint Vennant)

V=(9579 RS + .00813) * 0.0902 for pipes

(D'Aubuisson)

V=47.913 ^Sd^for pipes (Black well)

V=100 ^RS for pipes (Leslie)

V =/(! 1703.95 RS + .01698) 01308 for pipes .......

..................................... (Eytelwein)

til

for pipes ........... (Hawkeslej )

/+54d

V=v/ (9978.76 RS + .02375) 0.15412 for pipes ......

................... ..................... (Prony)

V=J _ _ for pi pet- ...... (Neville)

A/.0234 R + 0001085 I

1.811 .00281

//R/
channels ............................... (Kutter)

In the Kutter formula n represents the degree of rough-
ness, and

In the formula for n
1=1.811; C=-5L=; B =

It is to be observed in regard to the variation of Ganguil-

let and Kutter's C that

First In all cases of pipes or open channels, where the hy-
draulic mean radius (R) is less than 3.281 feet, an in-
crease in slope (S) will increase the value of C.

Second In all cases of pipes or open channels where R is
greater than 3.281 feet, an increase in S will cause C
to decrease.

10 SULLIVAN'S NEW HYDRAULICS.

Third Where R=3.28l feet exactly, the value of C will be

1 fil 1

constant for all slopes and will equal

At page 106 of Ganguillet and Kutters work (Hering
and Trautwine'stranslation)an unsatisfactory attempt
is made to explain this reverse variation of the coeffi-
cient. In the translator's preface it is pertinently
stated "that the laws of flowing water must be the
same whether the channel is large or small, slightly
inclined or precipitous." In this remark the writer
fully concurs. The above variations of C would lead
to the conclusion that the law of gravity reversed it-
self at the point where R=3 281 feet. Such variation
is clearly erroneous and is unsupported by any sound
theory or facts.

In order that the various theories of flow in open channels
may be compared with the theories of flow in pipes, and their
differences noted, a collection of the most prominent formu-
las for flow in open channels will here be given:

l/T = Af R~Xt/~S~ (Gauckler.)

In Gauckler's formula A is supposed to be constant for
any given roughness, and the coefficient varies as tf~R.

V=92.20 t/RS (Brahms & Eytelwein) .

In Brahm's formula the coefficient varies only as -^ R

V=4.9 R e/~S~for small streams (Hagen.)

Here the coefficient varies directly with R.
V=3.34 y~R X e/S~ for large streams. . .(Hagen)

Hew the coefficient varies with y' ~R

(Bazin)

In Bazin's [formula A and B are constant for any given
degree of roughness of perimeter.

SULLIVAN'S NEW HYDRAULICS. 11

V=( _ 1000e _ \l ..... (D'Arcy and Bazin)
V-08534R + .35 )

y _ |2gRS ............................ (Fanning)

In Panning's formula m = y a It is a direct coefficient

which decreases asV* increases or as the roughness decreases
and also varies with R. The mean values of m for channels in
earth of ordinary roughness vary from 0.05 for R= 0.25 to m
=0.002 for R=25.00. For very rough channels the value of m
would be greater because V* would decrease as the rough -
cess increased. To show the theory of this formula it should

be written VJL X V S and m=x'2 g R The

g
value of -yg depends upon the degree of roughness only and

varies with R instead of / R^ as it should.

V=140v/RS 11 f RS . .... ................. (Neville)

V=v/ 1067.02 RS +0-0556 0.236 (Prony)

V=v/10567.80RS + 2671.64 (Girard)

V=v/8976 50 RS + 0.0120.109 (D'Aubuisson)

V=-/ 8975.43 RS+0.011589 0.1089 ...... (Eytelwein)

V=1(KVRS ................ (Pole, Leslie, Beardmore)

^225 R v

(Humphreys & Abbot)

In Humphreys & Abbot's formula R ^y et p_L. \yidth an( * m
1.69

~!/R +1. 5

It will be noted that a majority of these formulas make

12 SULLIVAN'S NEW HYDRAULICS.

no provision whatever for different degrees of roughness
or different classes of wet perimeter, and hence the computed
results will be the same for rough, stony channels of irreg-
ular cross section as for smooth channels in firm earth with
uniform cross sections. The experimental coefficients devel-
oped from actual guagings of different clasees of streams, and
tabulated in the following pages of this work show that the
value of C varies from 27 to 75 in channels of like dimensions
and slope, but of different degrees of roughness of wet peri-
meter. The different classes of perimeter must therefore be
classified and the unit value of C for each class must be es-
tablished experimentally by actual guagings. When the
proper unit value of the coefficient for each degree of rough-
ness is thus established, it must thereafter vary correctly
with all changes of conditions as to slope, hydraulic mean
depth, ete,,so that the one unit coefficient will accurately apply
to all channels in the same class of roughness, regardless of
dimensions and slope or velocity of flow.

Ganguillet & Kutter recognized the necessity of classify-
ing the degrees of roughness of wet perimeter andof.establish-
ing the unit value of th^ coefficient for each class. They
adopted the unit value of their coefficient of roughness, n,
for each degree of roughness, and this value of n was to ap-
ply to all perimeters of like roughness, regardless of the di-
mensions of the channel, but they failed to discover the law
which governs the true variation of n and hence the value of
their C will not vary correctly with changing conditions.

Mr. J. T. Panning also recognized the necessity of classi-
fying coefficients in accordance with the degree of roughness
of the perimeter, but he adopted a system of direct coeffi-
cients for each velocity and for each hydraulic mean depth,
instead of determining the unit value for each class of peri-
meter. The difficulty with such direct coefficients is that
they will apply only to cases exactly similar to the conditions
under which they were determined. The result is that we
must have a separate coefficient for each velocity in the same
channel, and for each change in hydraulic mean depth, and

SULLIVAN'S NEW HYDRAULICS. 13

for each degree of roughness. It is a fixed, inflexible quan-
tity whose value must be ascertained by experiment for ev-
ery change in any of the conditions. When the student of
hydraulics investigates and compares the conflicting theories
of flow and of the variation of the coefficient as set forth in
the old formulas, he is simply bewildered and discouraged,
for he can discover no satisfactory reason for adopting any
one of them in preference to another. The writer therefore
hopes to be pardoned for offering what he conceives to be the
rational solution of these difficulties.

MARVIN E. SULLIVAN.

CONTENTS.

Introductory.--Evolution of the Formula.
Discussion of Present Data of Flow.
Reverse Variation of Coefficients Explained.

CHAPTER I.

SECTION 1. The law of falling bodies.
" 2. The laws of fluid friction.

CHAPTER II.

SECTION 3. Properties of the circle and of open channels.
" 4. Coefficients of resistance.
" 5. Coefficients of velocity.
' 6. The law of variation of coefficients.
" 7. Deduction of general formulas for flow.
" 8. Variation in the coefficient illustrated.
' 9. Practical determination of coefficients of resist-

ance.

" 10, Conversion of coefficients.
' 11. Determination of coefficients of velocity.
" 12. Value of coefficient affected by density of peri-

meter.
" 13. To determine the value of the coefficient in cases

where the flow is in contact with different

classes of perimeter.
14. Tables of cofficients deduced from data of flow

in all classes of pipes, conduits, flumes, canals

and rivers, with remarks and discussions of

each group.
" 15. Roughness of wet perimeter defined.

CHAPTER III.
SECTION 16. General formulas in terms ot diameter in feet.

SULLIVAN'S NEW HYDRAULICS. 15

SECTION 17. General formulas in terms of pressure, quantity

and diameter.
" 18. General formulas in terms of hydraulic mean

radius.

" 19. Application and limitation of formulas.
" 20, General formulas using C instead of m.
" 21. General formulas in terms of hydraulic radius

using C.
" 22. Special formula for vertical pipee.

CHAPTER IV.

SECTION 23. Table for ascertaining discharge or velocity in
cast iron pipes. Table No. 2 for discharge of
asphaltum coated pipes. .

" 24. Table for velocity and discharge of brick lined
circular sewers and conduits flowing full.

" 25. Egg shaped brick sewers, elementary dimensions
of. Table for velocity and discharge of egg
shaped sewers.

" 26. Short formulas for use with foregoing tables.
27. Table No. 5. Value of d, and r with roots and
powers.

" 28. Trapezoidal canale. To find mean radius and
area. Tables for velocity and discharge of trap-
ezoidal canals.

" 89. Rectangular canals, flumes and conduits. Tables
for velocity and discharge.

" 30. Table showing fall per mile, distance in which
there is a fall of one foot, together with values
of S and v/ST

" 31. Table and rules for finding required slope of
cast iron pipe to generate any required velocity.

" 32. Head in feet lost by friction in cast iron pipe.
Table No. 17 for finding friction loss for any ve-
locity.

" 33. Formulas and table for finding loss by friction in
any class of pipe for a given discharge.

16 SULLIVAN'S NEW HYDRAULICS.

SECTION 34. Formulae and table for friction loss in asphaltum
coated pipe for any velocity.

" 35. Flow and resistance in tire hose, with formulas.

" 36. Pressure at hydrant or steamer required to force
the discharge of a given number of gallons per
minute.

" 37. Friction loss in fire nozzles. Formulas and dis-
cussion. Table No. 20, showing friction loss in
fire nozzles.

" 38. Friction loss in ring fire nozzles.

" 39. Friction loss in hydraulic power nozzles, re-
ducers, &c.

" 40. Multipliers for finding friction loss in cast iron
giants.

" 42. Total head or slope of cast iron pipe to cause
given discharge.

" . 43. To find diameter of pipe required.

" 44. Loss by friction in cast imn pipe for given dis-
charge.

' 45. Table of multipliers for finding friction loss in
cast iron pipe for any given discharge.

" 46. Friction loss for given discharge of asphaltum
coated pipe. Table No. 23 for friction loss per
100 feet length.

" 47. To find the discharge from the amount of fric-
tion lose.

" 48. To find the discharge from slope and diameter.

11 49. Power at pump required to force a given dis-
charge.

" 50. To find the discharge from the pressure.

" 51. Lbs. pressure lost by friction for a given dis-
charge.

" 52. Slope required to cause a given discharge. Ta-
ble No. 24 for finding any required slope.

CHAPTER IV.
SECTION 53. Wooden stave pipe.

SULLIVAN'S NEW HYDRAULICS. 17

SECTION 54. Wooden pipes compared with other classes of
pipe.

" 55. Earthware or vitrified pipe.

" 56. Table No. 56, elementary dimensions of pipes.

" 57. Length in feet of small pipes to hold one U. S
gallon.

" 58. Decimal equivalents to fractional parts of a
lineal inch. Fractional inches in equivalent
decimals of a foot. Tenths of a foot in equiva-
lent inches.

" 59. Conversion tables of weights and measures for
converting U. S. to metrical and metrical to U.
S. measures.

CHAPTER V.

SECTION 60. Work, power and horse-power defined. Form-
ulas for finding the horse-power of a stream.

" 61. To find cubic feet of water required to gener-
ate a given power.

" 62. To find the net head required to generate a
given power. Efficiency of water wheels de-
fined.
63. Head of water defined.

" 64. To find the diameter of a pipe which will carry
a given quantity of water with a given loss of
head by friction.

" 65. To find the required area and diameter of noz-
zle to discharge a given quantity.

" 66, Pipe lines of irregular diameter.

" 67. A power main with nozzle and water wheel to
run at a given speed and develop a given power.

" 68. Table No. 36. Eleventh roots and powers.
Converse Lock Joint pipe. Remark 3.

" 69. Friction loss at bends in pipes; formulae.

" 70. Weisbach and Rankine's formulas for friction at
bends.

" 71. Results of different formulas compared.

18 SULLIVAN'S NEW HYDRAULICS.

SECTION 72. Resistance at bends. Ronnie's experiments.

" 73. Thickness of pipe shell proportional to diame-
ter and pressure.

" 74. Value of S in water pipe formulas.

" 75. Riveting and riveted pipe.

76. Proportions of single and double riveted lap
joints.

" 77. Table of decimal equivalents to fractional
parts of an inch.

" 78. Weight and thickness of sheet iron and sheet
steel.

" 79. Rules for finding weight of riveted pipe.

" 80. Tests for strength of lap riveted joints.

" 81. Testing iron and steel plates for defects.

" 82. Pipe joints how made.

" 83. Formula for proportions of reducers for joining
large to small pipes.

CHAPTER VI.

SECTION 84. Flow in open channels permanent and uniform
flow.

" 85. Resistances and net mean head in open chan-
nels.

" 86. Ratio of surface, mean and bottom velocities.

" 87. The eroding velocity.

" 88. Eroding velocity in straight canals of uniform
section.

" 89. Slope required to generate given bottom veloc-
ity.

" 90. Stability of channel bed.

" 91. Adjustment of grade, bottom velocity and side
slopes.

" 92. Dimensions of canals to discharge given
quantities.

" 93. Allowance in cross-eection of canal for leakage
and evaporation.

" 94. Flume forming part of canal.

" 95. Float measurement of mean velocity in canals.

SULLIVAN'S NEW HYDRAULICS. 19

APPENDIX I.

SECTION 96. Weirs and weir coefficients. Discussion.

APPENDIX II.
SECTION 97. Water works rules and data.

" 98. Quantity of water required per person.

" 99. Consumption and cost per 1,000 gallons in var-
ious cities.

" 100. Formulas and tables for diameter of pipe re-
quired.

" 101. Formulas for conduits, pipes, etc.

' 102. Formulas for diameter to maintain a given
pressure while discharging.

" 103. Friction heads,velocities and discharge for given
diameters and slopes.

' 104. Formulas for thickness and weight of cast iron
pipe.

" 105. Weight and dimensions of cast iron pipe made
by the Colorado Fuel and Iron Company.

" 106. Weight per foot length of cast iron pipe for
150 and 200 pounds pressure.

" 107. "Standard" cast iron pipe.

108. Cost per 100 feet of pipe and laying in Denver,
Colorado.

" 109. Cost per foot of pipe and laying in Boston,
Massachusetts.

" 110. Cost of trenching, laying, calking, etc., in Om-
aha, Nebraska.

" 111. Weston's table for estimating cost of pipe lay-
ing.

112. Table showing cubic yards of excavation in
trench per foot.

113. Bell-holes in pipe trench.

" 114. Proper depth of trench for pipe laying.

" 115. Amount of trenching and laying per man per

day.
" 116. Amount of lead required per joint for cast iron

pipes.

"There is in this world but one work worthy of a man,
the production of a truth, to which we devote ourselves, and
in which we believe." Taine.

INTRODUCTORY.

The evolution of the formula and discussion of the present
available data of flow, with an explanation of the reverse
variation of coefficients,

The general formula for flow as herein finally presented
may justly be called the result of the combined labors of
Galileo and all subsequent writers and experimentalists, in-
cluding the present writer. The present writer has accepted
and adopted from all former writers on hydraulics such prin-
ciples and theories as have been thoroughly proven true and
general, and has rejected all theories of doubtful or uncertain
value and supplied the deficiencies thus arising by original
investigations and experiments. The foundation of the form-
ula was the discovery, early in the seventeenth century, of
the law of falling bodies by Galileo. In his investigations of
flow in rivers Galileo failed to recognize the nature of the
resistance of the solid wet perimeter and the difference be-
tween the resistance of a solid in contact with a liquid, and
that of two solid bodies in contact. His investigations there-
fore resulted in failure. Later it was discovered by Torri-
celli, a pupil of Galileo, that resistances aside, the square of
the velocity is directly proportional to the head or inclination,
or in other words, that the velocity would be as the square
root of the head or elope.

Brahms discovered that the acceleration which
would occur according to the law of gravity did not actually
occur, but that the velocity of flow became constant. His
investigations established the fact that the solid wet peri-

22 SULLIVAN'S NEW HYDRAULICS.

meter offered a resistance to the flow which opposed the ac-
celeration that would otherwise occur, and he assumed tht
this resistance was directly proportional to the hydraulic
mean radius, or to the area of cross-section of the column of
water divided by the wet girth. In the latter part of the
eighteenth century, Du Buat instituted a series of experi-
ments from the results of which he discovered that the
velocity of flow depended upon the slope of the water surface
or head, and that in channels of uniform area and grade,
when equilibrium was attained, the flow became uniform and
the resistance equalled the acceleration of gravity. Thus
each investigator has contributed some valuable discovery or
fact which has been able to stand, while many of their as-
sumptions have been proven wholly erroneous.

Du Buat also discovered that the resistance of a solid in
contact with a liquid, was in no manner increased or de-
creased by a change of pressure between the liquid surface
and the solid surface. In other words he discovered that
the pressure with which a liquid is pressed upon a solid does
not affect the friction between them. Du Buat and Prony
each discovered, as a consequence of the law of gravity, that
the head or slope had no influence whatever upon the value
or variation of the coefficient, but they erroneously assumed
also that the character of the wet perimeter had no influence
upon the coefficient. It was the opinion of Du Buat (and
adopted by Prony) that the nature of the walls and bottom of
a channel could not affect the coefficient because, as Du Buat
observed, "A layer of water adheres to the walls, and is there
fore to be considered as the wall proper which surrounds the
flowing mass." With this view, he supposed all perimeters
to be practically "water perimeters",|and consequently equally
smooth. It remained for D'Arcy and Bazin to demonstrate
by many practical experiments that this latter assumption of
Du Buat and Prony was entirely without foundation.

As a result of the experiments of D'Arcy and Bazin the
fact was established that the coefficient of flow would vary
directly as the degree of roughness or smoothness of wet

SULLIVAN'S NEW HYDRAULICS. 23

perimeter. Bazin stated also that the coefficient varied with
the value of the hydraulic mean radius, thus confirming the
observations of Brahms. D'Arcy and Bazin recognized that
the slope or head had no influence upon the value or variation
of the coefficient, and hence omitted that feature in their
formula, and assumed with Brahms that the total resistance
for any given degree of roughness would be directly propor-
tional to R. the hydraulic mean radiue. They were correct
in their assumptions thus far, but they failed to go one step
farther and provide for the acceleration as well as the re-
sistance, or in other words to ascertain the mean resistance of
all the particles of the entire cross-section by taking the
product of total retardation by total acceleration. They
adopted and embodied in their formula simply the feature
of total retardation without modification by the acceleration.
Hence they failed to ascertain the mean resistance of the
entire cross-section, and as a necessary result of adopting
total resistance instead, their formula gives results too low
in large pipes or channels and the larger the pipe or channel,
the greater the error will become.

Ganguillet and Kutter proposed a formula based partly
on Bazin's formula and partly upon the results of some ill
assorted guagings. While this latter formula has become
popular and is considered as standard by many engineers, it
is really based upon theories which are directly contradic-
tory of both the laws of friction and of gravity,and a short in-
vestigation will expose the fact that it can be applied with
accuracy only to open channels of very slight inclination, and
whose mean radii approach closely to unity. The discussion
of coefficients will point out the reasons why this is true.

The writer would also remark that the published tables
of data in relation to flow in pipes and open channels are, in
a large majority of cases, wholly unreliable, as many contract-
ing engineers have recently discovered to their great cost.

Coefficients should never be based upon data of uncer-
tain value, as the results must depend upon the correctness
of the coefficient used.

24 SULLIVAN'S NEW HYDRAULICS.

The data of guagings of rivers at different stages and for
various depths of flow are nearly all worthless for scientific
purposes for one or more of the following reasons:

1 . The data fail to show whether the stream was rising
or falling or stationary when the mean velocity and
corresponding alope of water surface were ascer-
tained.

Where a stream is either rising or falling with
considerable rapidity, there is little or no relation be-
tween the slope of the water surface and the actual
mean velocity then prevailing. The same depth of
flow or guage height at the same point does not
necessarily always produce the same slope of water
surface. The slopes are usually thus recorded as
corresponding to a certain guage height without
actual measurement. The same slope and guage
height on a rising river will cause a much greater ve-
locity and give a higher coefficient of flow than upon
a falling river. The difference in value of the veloc-
ity of flow and of the coefficient will depend upon the
magnitude of the freshet the distance it extends up
stream the suddenness and rapidity of the rise or
fall. In a rising or falling stream equilibrium is lost
and the actual effective slope which is generating the
velocity at such times is very different from the ap-
parent slope, and can be ascertained only from the
mean velocity actually existing at the time, and from
a previous knowledge of the degree of roughness of
the stream in that locality. The effective slope may
then be found by formula for S.

2. Guaging stations are always located at narrow, deep
sections of the stream, and the hydraulic radius thus
roughly measured is given as the mean hydraulic
radius of the stream. This is never the true nor
even scarcely approximate, mean hydraulic radius,
except at the particular point where measured.

SULLIVAN'S NEW HYDRAULICS. 25

3. In many cases the mean velocities tabled are deduced
from the surface velocities by some absurd formula
which is basad upon the theory that there is a con-
stant ratio between the maximum surface velocity
and the mean velocity, and that this ratio is the same
in all classes and dimensions and degrees of rough-
ness of channels.

4. The general slope is usually taken and is assigned as
the local slope. They are usually very different ex-
cept at extreme high water, when the general and
local slope are nearly equal.

5. In turbulent streams, or in very large streams it is
impossible, for many reasons to ascertain the slope of
water surface, especially the high water slope. The
slopes assigned in such cases are simply the record
of a guess, and have no value for scientific purposes.

6. The method of ascertaining the mean velocity as
finally tabled, is frequently by mid-depth floats.
These floats vary in the time of passing over the same
course by as much as 25 per cent, depending upon
the number of whirls, boils and cross-currents en-
countered. The mean is taken as the actual mean
velocity, or as bearing a given fixed ratio thereto, re-
gardless of formation of the channel bed. The in-
equalities of the bottom of the stream make it impos-
sible to adjust a float to mid depth. If the mid depth
velocity were absolutely known, it is not known
what relation it bears to the actual mean velocity.
The mean velocity may be either above or below mid-
depth. That will depend upon the magnitude and
roughness of the channel at the given place.

7. Many rivers are affected by gulf tides, as the Missis-
sippi river as far up as Donaldsonville, and the river
Seine at Poissy, Triel and Meulon, where the water
surface fluctuated as much as two feet during the
time of the guagings there. This is so marked on

36 SULLIVAN'S NEW HYDRAULICS,

the Mississippi river at Carrollton, La., as to actually
produce reverse slopes, as noted by the river engi-

8. Data are frequently published of the gaugings of a
channel at different stages where the value of the
hydraulic mean radius increases four or five hundred
per cent., and the discharge increases by a thousand
per cent, or more, and yet the same slope is assigned
for all stages! As an example of this kind, see the
ten gaugings of the Saone under the direction of M.
Leveille, 1858-9. Here the hydraulic radius varies
from R=3.88 to R=15.83, and yet the slope of water
surface is given as S=.00004 for each of the ten
gaugings. The gaugings of the Weser by Funk are
of no value. Bazin remarks; '-It is to be noted
that Funk has almost always adopted the same
slope for an entire group of experiments." Bazin also
states that in the experiments of Brunning on the
Rhine, "the slopes were not measured at all, but
subsequently computed so as to make the results
accord with the formula."

It is well known that the gaugings of the Mississippi
River under the direction of Humphreys and Abbot
during the year 1858 are of very doubtful value. The
areas had been measured at the gauging stations
seven years before, and were assumed to have remain-
ed constant ever afterward, when in fact the area at
a given point in that river is frequently altered by as
much as 14,000 square feet within twenty-four hours
by scour. In 1858 the velocities were taken at a
depth of only five feet below surface, and the mean
velocities were calculated by an empirical formula of
no value. Du Buat's mean velocities as tabled for
the Canal du Jard, were deduced from surface float
velocity by Du Buat's formula. Du Buat's formula
for ascertaining the mean from the surface velocity
was based on his experiments on a very small wooden

SULLIVAN'S NEW HYDRATLICS. 27

trough of smooth perimeter, and has long since been
discarded as being of no value.

If the present available tables of data were assorted
carefully and the worthless were rejected, there
would little remain. These remarks are made here
in order to call attention to the need of obtaining
new and accurate data, and to prevent too great
reliance upon the value of such data as are now
available. The data relating to flow in pipes and
conduits are equally bad and untrustworthy. The
data relating to asphaltum coated pipes except those
of Hamilton Smith Jr., and to wrought iron pipes
are especially of uncertain value, and no expensive
enterprise should be based upon them without ad-
ditional experiment. Some of the more recent data
relating to flow at different depths in large masonry
conduits of comparatively short length, show by the
value of the slope of water surface as compared with
the slope of the bottom of the conduit, that there
would have been no water in the upper end of the
conduit at all. It is probable that equilibrium had
not bem attained at the time of gauging. Any
other explanation renders the data absurd, and this
explanation renders them worthless. In a channel
of uniform grade, croes-section and roughness of
perimeter, the slope of the water surface will be the
same as the slope of the channel bed as soon as uni-
form flow is established. If this were not the case,
uniform flow could never occur, because the water
would be of greater depth at one point than at
another, and the velocities would be inversely as the
depths or wetted areas. An investigation of the data
of flow which is now available is discouraging to a
degree. It is a misfortune common to us all. In
large rivers where the roughness of perimeter and
the area of cross-section vary at almost every foot in
length, and where scour or fill is constantly in pro-

SULLIVAN'S NEW HYDRAULICS,

gross, it is impossible that equilibrium in its true
sense, should ever be established. The flow is alter-
nately checked by rough perimeter and increased
area due to scour, and accelerated by reaches of
straight, smooth perimeter where the area is con
tracted. The flow is similar to that in a compound
pipe made up of lengths alternately large and small,
and alternately smooth and rough. The velocity is
necessarily inversely as the areas, in case the supply
of water is constant.

For these reasons the local slope over a very short
length of channel, at normal stages of the stream, is
the slope that must be used in the application of a
slope formula. Otherwise the result by formula will
be of no value.

The value of the coefficient will usually decrease with
increase in depth of flow at any given point along a
natural channel, after the depth exceeds the usual
dpth of flow. This is due to the simple fact that
the bed is silted and worn smoother up to the depth
of ordinary flow than the sides above the usual flow.
Hence as depth of flow increases the proportion of
the rougher side wall perimeter increases also, and
thereby decreases the value of the coefficient as depth
of flow and ratio of rough perimeter increase. It
frequently occurs, however, that the reverse of this
is true, as for example in channels having very
rough, stony bottoms and comparatively regular
side walls, In this latter case the coefficient will
increase as depth of flow increases because with
each gain in depth there is a gain of the smooth
over the rough perimeter, and the mean of the
roughness of the perimeter considered as a whole
becomes less and less at each successive increase of
depth. In either class of channels the value of the
coefficient must vary directly with the mean of the
roughness of the entire wet perimeter taken as a

SULLIVAN'S NEW HYDRAULICS. 29

whole. It therefore follows that if the whole
perimeter be of equal and uniform roughness or
smoothness throughout, the coefficient will remain
absolutely constant for all depths of flow.

Large masonry conduits, not being adapted to
withstand pressure from within, are built on small
grades or inclinations and given free discharge. It is
nearly always found that the coefficient in such con-
duits is greater for very small depths of flow than for
greater depths. This is explained by the fact that
much cement or mortar is dropped upon the invert or
bottom during construction and is ground into the
pores and joints of the brick by the tramping of the
masons. The floor is worn smooth by reason of this,
and the slight inclination of the conduit and low ve-
locity permit of the deposit of a very fine, dense silt
upon the bottom which settles in and fills up all the
irregularities and depressions along the bottom thus
presenting a smooth, uniform, contiguous bottom peri-
meter to the flow. The side walls, although of the
same material, are much rougher than the invert or
bottom, because the rough projections of sand along
the sides of the brick are not rubbed off, and the
pores and small cavities are not filled and plugged by
mortar tramped in under pressure, and by the deposit
and settlement of fine silt, as occurs on the bottom.
This is, however, not true of open canals with paved
bottoms and masonry side walls where the slope is
sufficiently great to generate a velocity at the bottom
sufficient to prevent the deposit of silt or to scour out
the joints of the masonry floor. In this latter case
the bottom has no advantage of the sides so far as re-
lates to roughness, unless it is better constructed, or
is composed of smoother material, or is in better re-
pair than the side walls. In any given case the value
of the coefficient will be directly as the mean rough-
ness of the entire wetted portion of the perimeter, or

30 SULLIVAN'S NEW HYDRAULICS,

as the ratio of smooth to rough perimeter as the

depth of the flow varies.

With these general introductory remarks upon the evolu-
tion of the formula for flow, the uncertain value of the present
available data of flow, and the causes of contrary variation in
the value of the coefficient, the reader will be better prepared
to understand what follows.

CHAPTER I.

Of the Laws of Gravity and the Laws of Friction Between a
Liquid and a Solid.

/. The Law of Falling Bodies Ae the law of falling bod-
ies, or of gravity, and the law of friction between a liquid and
a solid include all the elements of the flow of water, it is of
prime importance that these laws should never be lost sight
of in any investigation or application of hydraulic formulas.
The moment that a theory or a formula deviates from the re-
quirements of any one of these natural laws, that moment it
must fail. These laws must be observed in their entirety.
No provision must be either excluded or violated. The pen-
alty is certain failure to the extent of the evasion or violation.

Let g=feet per second by which gravity will accelerate
the descent of a falling body. g=32.2 at sea level.

2g=64.4.

v =velocity in feet per second.

H=height in feet, total fall in feet, or total head in feet.

t=time in seconds.

A body falling freely from rest will descend 16.1 feet in
the first second of time,(t) and will have acquired a velocity, at
the end of the first second, of 32.2 feet per second, and will
be accelerated in each succeeding second 32.2 feet, so that for
each additional second of time consumed in falling, there will
be a gain in the rate of descent equal to 32.2 feet. At the end
of the first second the rate of velocity will be 32.2 feet per
second, at the end of the second second of time, the rate of
velocity will be 64.4 feet per second, and BO on for any num-
ber of seconds, adding 32.2 feet to the rate of velocity for each
second of time.

Velocity is the rate of motion. Acceleration is the gain

32 SULLIVAN'S NEW HYDRAULICS.

in this rate. The acceleration or gain in rate of motion in
the case of a body falling freely, is 32.2 feet per second, and
consequently the velocity, (v) at any time (t) is equal to gXt,
or to the acceleration per second (g) multiplied by the num-
ber of seconds of time (t). The distance in feet (H) fallen
through by a body in the first second is 16.1 feet, or one half
g, and the distance fallen in any given time(t)is as the square
of that time (t 8 ). Consequently the velocities are as the
square roots of the distances or vertical heights fallen
through, or as -j/ H. As gravity produces the velocity of 2
in falling through the height 1, the height in feet fallen mul-
tiplied by 2g will equal the square of the velocity in feet per
second or

v*=2gH (1)

From this fundamental law of gravity we have

v=-/2 gH=/ 64.4H =8.025 V / H .......... (2)

v=gt ...................................... (4)

g=I =JZl=32.2 at sea level ................ (5)

t 2H

......... (6) 1

The velocity head, or that portion of the head which is
producing the velocity of flow in any case is therefore

-=r ....... ..... ; ....................... (7 >

And the velocity generated by any velocity head is equal
v= /2g hv=8.025 y "hv .................... (8)

2. The Laws of Friction as Applied to a Liquid in con-
tact with a Solid Surface The results of many very careful
experiments establish the correctness of the following rules:
I. The friction on auy given unit of surface will be directly

SULLIVAN'S NEW HYDRAULICS. 33

as the roughness or smoothness of that surface.

II. The total resistance will be as the total number of unite

of friction surface.

III. The friction on any given unit of surface will be aug-
mented as the square of the velocity with which the
liquid is impelled along that surface.

IV. The friction between the molecules or particles of the
liquid themselves, is infinitely small, and may be en-
tirely neglected.

V. The friction between a liquid and a solid is not affected
by the pressure with which the liquid is pressed per-
pendicularly upon the solid. The friction is entirely
independent of the amount of radial pressure.

VI. The mean resistance of all the particles of the entire
cross section of the liquid vein considered as a whole,
will be as the total retardation or loss of velocity by re-
sistance, as modified by the total acceleration or free
and unretarded flow, or as the product of total retard-
ation by total acceleration. Total acceleration will be
as the square root of the net free head. Total retarda-
ation will be as the square root of the head consumed
or loet by resistance.

The mean resistance, or mean loss of head, of all the
particles of the entire cross section taken as a whole
will be as the product of total retardation by total ac-
celeration.

The mean velocity of all the particles in a cross-section
will be as the square root of the mean head of all the
particles.

CHAPTER II.

Of Coefficients and their Variation-

3. Properties of the Circle IK order to exhibit the
properties of the circle, and the relations of area to friction
surface in both open and closed channels, and the relations
common to both open channels of any form and to circular
closed channels or pipes, and to also show the relation of
theee common properties to the value and variation of the
coefficients, the following tables of circles and of open chan-
nels of various forms will he referred to. The notation here
given will be followed throughout:

H = total head in feet.

h"=friction head, or head required to balance the total
resistence.

h v =velocity head in feet in the total length /.

Z=1ength in feet of pipe or channel.

v=mean velocity in feet per second.

d=diameter in feet.

a=area in square feet.

p=wet perimeter in lineal feet, or friction surface.

r= ^hydraulic mean radius in feet.
n=coefficient of friction or of resistance.
m=ccefficient of flow or of velocity.

total head in feet H

S= total length in feet= r =8ine of lo P e = to * al head P*
foot.

h"
S'= =friction head per foot length of channel or pipe

Sy = Velocity head per foot length of pipe or channel.

SULLIVAN'S NEW HYDRAULICS,

In full pipes or circular closed channels r= = d X -25,
and d =4r. a=d s X.7854, or a=(4r)X-7854=r 8 Xl2.5664.
TABLE OF CIRCLES.

FEET

v/d

FEET

SQ FEET

FEET

BELATI'N
OF p to a

0.50
1.00
2.00
4.00
8.00
16.00
32.00

0.707
1.000
1.4142
2.000
2.828
4.0CO
5.657

0.125
0.25
0.50
1.00
2.00
4 00
8.00

0.1C635
0.78o4
3.1416
12.5664
50.2656
201.0624
804.2196

1.5708
3.1416
6.2832
12.5664
25.1328
50.2656
100.5312

p=8a
p=4a
p=2a
p= a
p=y,a
p=&a
P=Ha

It will be observed that the diameter d, is doubled here
each time, and that the result of doubling d is to also double
r and p. It follows therefore that in circular closed channels
and pipes d, r and p vary in the same ratio. As d, r and p
all vary exactly in the rame ratio, and as the area varies as d a
or r*, and as d or r must therefore vary as the square root of
the area, it follows that the friction surface p, must also vary
as the square root of the area. It will be observed that when
d, r or p is doubled, the area is increased four times. Hence if
the friction surface p is doubled the area is increased four
times. The right hand column of the table shows how rap-
idly the friction surface p gains on the area as the diameter
is reduced from four feet, and also the reverse gain of area
over friction surface as the diameter is increased above four
feet. In pipes or circular closed channels flowing full, the
the same value of d or r always represents the same length
of wet perimeter, because the parimeter or circumference of
a circular closed channel or pipe is always equal to dX3.1416,
or to rX12.5P64. As the area is always as d 2 or r 8 and as the
friction surface varies as d, r, p or ^/ a , it follows that the
same value of d, r or p in circular closed channels and pipes
flowing full, will always represent the same value of the area
and of the wet perimeter or circumference. Such circular
full channels may therefore be compared one with another,
by simple proportion, because in such channels,

SULLIVAN'S NEW HYDRAULICS.

a : a : : d 1 : d*, or a : a : : r* : r*

r:r::d:d, orr:r::p:p

p :p::d : d, or j/a : >/ a::p:p

In open channels, however, r is not necessarily an index
of either the extent of the area or of the perimeter, and there-
fore open channels of different forms cannot be thus compared
one with another, but in all cases r expresses the ratio of a to
p in the given case.

TABLE OF OPEN CHANNELS.

Channel Area, a

Peri-
meter, p

a
r=
P

Flume 10- x20'
Mississippi River...
Lamer Canal
River Seine
Chaz illy Canal

200.00
15911.00
50.40

i-o22 00
11.30

40.00
1612.00
31.00
518.00
10.80

5.00
9.87
1.81985

i- :^~2
1.0462

25. (C
97.427
3.31

m n

1.09453

While we cannot compare these open channels one with
another as in the case of pipes, yet if we take the data for any
one pipe or for any one open channel it will be found in any
case that

p : r : : a : r*

In other words the friction e irface p varies as r, and the area
varies as r* in any possible shape or form of open or closed
channel. As p : r : : a : r* in any form of channel it follows that
r bears the same relation to a and also to p in any given chan-
nel or pipe, as it does in any other channel or pipe, for r= a

in any given case. The properties which are common to all
shapes and classes of channels and pipes are that, in any
given case, the area varies as r" and the friction surface var-
ies as r or as y a, regardless of the shape of the pipe or chan-
nel. These properties which are common to all classes acd
forms of channels and pipes are the only two which affect the
coefficients. Hence the form or shape or size of the pipe or
channel does not affect the application of the coefficients
which vary with these properties which are common to all

SULLIVAN'S NEW HYDRAULICS. 37

possible forms of waterways. The tables were given not only
to illustrate the above facts, but for other reasons which will
be referred to when the application of certain formulas to open
channels is discussed.

Brahms discovered and announced these common re-
lations as early as the middle of the eighteenth century,
but like his successors, he mistook the total resistance for the
mean resistance or in other words he did not modify the
effects of total resistance by that of total acceleration. Since
that time, coefficients have been made to vary either as r or
asi/~F~~, and also as some other factor such as slope or velocity.

4. Coefficients of Friction or of Resistance In general
terms a coefficient may be defined as the constant amount or
per cent by which the head per foot length of pipe or channel
must be reduced on account of loss by frictional resistances.

In any constant diameter, or in any open channel of con-
stant hydraulic radius, the friction will be directly as the
total friction surface and directly as the roughness of that
surface, and will increase directly as the square of the
velocity. As the amount of resistance per foot length of pipe
or channel is always directly proportional to v s in any given
case, it follows that the amount of friction head per foot
length (S") required to balance it must also always be direct-
ly proportional to v s otherwise one could not balance the
other, and uniform flow could never occur. In any given
pipe or channel, if the head or slope increases, the square of
the velocity, and consequently the friction, will increase in
the same ratio, for v 8 is always directly proportional to the
header slope. Hence the ratio of friction head per foot
length S", to the square of the velocity, v 8 , is necessarily a
constant for all heads, slopes and velocities. The coefficient
of resistance n, in any given diameter or hydraulic radius is
simply the expression of this ratio of S" to v 8 . It follows
therefore, as this ratio is necessarily a constant, that a
change of slope or velocity can have no possible effect upon
the value of the coefficient. As long as the diameter or
mean hydraulic radius remains constant, the coefficient of

210991

38 SULLIVAN'S NEW HYDRAULICS,

Q It

resistance is n= 5- for all slopes and velocities, and will be

constantjfor all elopes and velocities because S" and v 2 must
always vary exactly at the same rate. It is evident that any
formula which causes the coefficient to vary in any manner,
or to any extent whatever, with a change of head, slope or
velocity, violates the law of gravity which shows that H or S
must be directly proportional to v 2 in all cases. It also
violates the law of friction which declares the friction to be
always proportional to v 8 . The results computed by such a
formula for any given diameter with different slopes, or for
any given open channel with different elopes must necessarily
be erroneous at least to the extent that the value of the
coefficient was made to vary with changing slopes. The
value of the coefficient for any given diameter or for any
given hydraulic radius will depend upon the degree of rough-
ness of the wet perimeter. A rough perimeter will offer
great resistance to the flow and will require a considerable
head or inclination to generate a small velocity. In such

S"
case the ratio (n) of rfwill be large, because S" will be large

and v* will be small. This ratio will, however, be constant
for any given degree of roughness in a pipe or channel where
r is constant.

5. Coefficient of Velocity Where the discharge is free
in a pipe or channel, and the value of r remains constant,
the totalhead will be consumed in balancing the resistance
and in generating the velocity of flow. The resistance must
be balanced before flow can ensue. The resistance being as
v a in all cases, the coefficients of velocity represents the ratio
of total head H, to v 8 , or rather Sto v a if the discharge is free
If the discharge is throttled so that a portion of the head is
converted into radial pressure, then this pressure head is
neither converted into velocity nor lost by resistance. In this

latter case the coefficient of velocity is the ratio oflttpl, in

SULLIVAN'S NEW HYDRAULICS. 39

which S v is the velocity head per foot length, and S" is the
friction head per foot length. Where the discharge is free,

S= total head per foot length, and the coefficient of velocity,

g
m -. As H or S is always directly proportional to v 8

(v 8 2gH) it follows that if S be increased in any given di-
ameter or hydraulic radius, v 8 will also increase at the same

o
rate, and hence m= -will necessarily be a constant in any

given hydraulic radius regardless of the value of the slope or
velocity. If it be admitted that the fundamental laws of
gravity, V 8 =2 g H, and V=-j/ 2g H are correct, and that
the frictional resistances on any given surface will increase
as v a , it must also be admitted that the ratio (m) of S tn v 8 is
always constant after equilibrium is attained, and that as a
necessary result, the head, slope or velocity can have no pos-
sible effect upon the value of the coefficient of velocity.

6. The Law of Variation of CoefficientsThe coefficient
of resistance n, and the coefficient of velocity m, have so far
been considered only as applied to a constant hydraulic rad-
ius or constant diameter, and it has been shown that in no
possible case can the slope or velocity affect the value of the
coefficients in any diameter or hydraulic radius. The effect of
variation of hydraulic radius or of diameter, upon the value
of the coefficient will next be investigated.

As the area in any given open channel varies with r 8 and
in any given circular closed channel or pipe as r s or d 8 , and
as the wet perimeter or friction surface in any shape of pipe
or open channel varies as r, d or p, it follows that the friction
surface p (which is as r or d) must vary in any given case as
I/!L Area is as d 8 or r 8 . Consequently / a is as d or r.
The total resistance will be directly as the total friction sur-
face which is as d, r or ^^7 The total area or free flow in any
given case varies with d 8 orr 8 . As area gains over wet peri-
meter a greater number of particles of water are set free from
the resistance of the perimeter, or acquire a head equal to th e

40 SULLIVAN'S NEW HYDRAULICS.

elope of the channel or water surface. This does not increase
the head of the particles already free of resistance, but simply
adds to their number. The number of these unresisted parti"
cles will increase directly as the area, or as d 8 or r s . Total
acceleration will increase as the square root of the net free
head or free flow, which is therefore as d or r. Total retard-
ation will be as the square root of the total head lost by re-
sistance. The head lost by resistance will be directly as the
amount of friction surface, which is as d or r, but the total
retardation or loss of acceleration due to this loss of head,
will be as the square root of the head lost, or as ^"d" r i/~rT
Then total acceleration ie directly proportional to the
square root of the area or net free head, or to d or r, or y&.
Total retardation or loss of velocity is directly proportional
to the square root of the loss of head, or to ^ d or ^/ r 4 y a.
It follows therefore that the mean loss of head, or mean re-
sistance, of all'the particles of the cross-section taken as a
whole, will be as the total acceleration modified by the total
retardation, or the mean loss of head wi'l be inversely as
d /~d~ or V T~= /"r 5 ^ or i/~&*.

The mean loss of head will be inversely as \/~r* or
because the acceleration increases as i/ a or d or r while re-
tardation is only as -j/ d or i/ r, or 4 j/ a. If the value of
the coefficient of resistance n be found for any degree of
roughness when d or r=l, then it will vary from this unit
point inversely as -/ r 8 or -/ d 3 , and the general formula for
the value of the coefficient becomes, n=___Xi/ d 3 , or n =

These expressions are equivalent to n= r ^ r ' or

n _h"d 1 /~d"

Z v*
It is apparent that the coefficient of resistance or friction

SULLIVAN'S NEW HYDRAULICS, 41

n, will vary for any given degree of roughness, only with the
variation in the value of -^ cl^or/ ~r". For the same de-
gree of roughness, no other factor will affect its value in any
manner.

Now as the mean loss of head is inversely as ^/ d 8 or
I/ r 8 the mean gain in head of the entire cross section taken
as a whole, will be directly as ^/ d 8 " or -/ T 8 ^ The mean veloc-
ity of the entire cross section, or the gain in the mean veloc-
ity, will be directly proportional to the square root of the net
mean head, or to the square root of the gain in the net mean

I y-^

head,or-yf "

The friction head h"is therefore inversely as i/r~*~OT v/lT 8 "

The velocity head is directly as v/T 1 or ,/~d.

The velocity of flow is directly as f/'d 8 " or f/T 8 ". It
is understood that in these cases the elope remains constant.

The coefficient of velocity, m, if determined for any given
degree of roughness when d or r = 1, will therefore vary
from this unit point directly as v /~r^~ or i/~d*.

(9)

n=J 7^=^ X ' /dl U0)

^^-^X^ <>

(12)

AB these coefficients for any given degree of roughness
vary only with ^"d 8 or v/T^.the value of d or r may be in

inches or feet, while the constant g- may remain in feet
in either case, or all the values may be expressed in metres.

42 SULLIVAN'S NEW HYDRAULICS,

The value of any given constant slope is made more effec-
tive directly as v/r~ increases, because the mean gain in
head increases directly as ,/ r 3 or the number of unre-
sisted particles increases as v/T^T The mean head and mean

velocity both gain without a change in slope. This does not

affect the constant ratio g because as S is made more effec-
tive by an increase in ^/ r 3 , v 2 increases in the same
ratio.

7. Formula for Mean Velocity Ry transposition in
equation (11) we have

V H x-V/= V m -c 18 )

But for each given degree of roughness of friction sur-
face m is a constant equal to the ratio-^j, and varies only as
l/r 8 . Hence we may take the square root of the reciprocal

/J_ / v8
and write>-\/ m = \8./^i~ whence

73 (14)

In this case C is a constant which applies to all pipes or open
channels of the given degrees of roughness represented by C.

V 8

In other words C represents -^-g-which is constant for any

given roughness and only varies with 4 ;/r 3 . If we replace C
by K in equation (14) and reduce the formula to the Chezy
form, that is, if we write

v=C v /rsT (15)

then, C=K 4 i/r , which, when multiplied by y^rs equals
C Vr T ~ 1 /S7~In the Chezy form it is seen that C=K Vr~I
or in other words, that for any given degree of roughness of
friction surface represented by K, the coefficient C will vary
only as 4 ^/r .

SULLIVAN'S NEW HYDRAULICS. 43

If we write simply. v=CVS, then C=K 4 1 /r.

The result is the same in either case.

The value of C in any formula in the Chezy form must
yary only as the roughness and as J/r or $/d.

If a series of pipes or open channels of equal roughness
be selected, it will be found that C: C: : /r~: e/r~regardless
of the slopes or dimensions of the channels. If C fails to vary
only as J/r in the Chezy form of formula for a series of chan-
nels of equal roughness, then it will be found that C y/rs will
not equal v. This will be illustrated by the following pairs
of open channels each pair being nearly equal in roughness,
but varying in the values of the slope and hydraulic radius.

8. Variation of C Illustrated In the following tables
we shall give the values of our C as found by the formula

C=-l => for each channel. We will also give the value

of the Chezy C for each channel which is required to make
C v/rs=v. It will be found in each case that barring the
slight difference in the degree of roughness in each pair of
channels the Chezy C which will cause C ^/rs to equal v,
will vary only as J/r , and their values in such class of chan-
nels may be found or compared by the simple proportion
C : C :: t/F~: J/rT The values of the Chezy C were deter-
mined by the formula, C= in all the following tables'

V rs

The values as given in the translation of Ganguillet and Kut-
ter are only approximate, being sometimes in error by as
much as 12 or 15. The values of Kutter's n, or constant co-
efficient of roughness, are also given for each channel. These
values of n are transcribed from Hering & Trautwines Trans-
lation, second edition. These values of n show that it is not
a constant, but is an auxiliary quantity which must be used
as r and s vary in order to balance the erroneous variation of
C with the slope.

SULLIVAN'S NEW HYDRAULICS.

Pair No. 1. Slopes equal. Radii vary slightly. R> 1
meter.

NAME OF
CHANNEL

FEET

f/R~

FEET

S
SLOPE

V
FEET
SEC.

SULLI-
VAN'S

KUT-

TER'S
n

CHEZY
C

Seine
(PariB)
Seine
(Paris)

9.50
10.90

1.7556
1.8170

.00014

3.37
3.741

52.85
52.71

.0240
.0238

92.40
95.77

The channel where R=10.9G is very slightly rougher, as
sbown by Sullivan's C, than the firet channel. *As the slopes
are equal, Kutter's n has to vary only with the slight differ-
ence in values of r,J order that c ^/rs" will equal v, and in
order that c: c: : /~r~: $/r~.

Pair No. 2. Slopes nearly equal. Small difference in R.
R > 1 meter.

NAME OF
CHANNEL

FEET

t/R

FEET

SLOPE

FEET
SEC.

SULLI

VAN'S
C

KUT-
TER'S
n

CHEZY

Seine
(Triel)
Seine
(Poissy)

12.40
15.90

1.876
2.000

.00060
.00062

2.359
2.911

46.12
46.43

.0295
.0285

86.43
92.70

The Chezy Ovaries only as /r in Pair 2. It is not af-
fected by the slight difference in elope. Where R=15.90 the
channel is very slightly smoother than where R=12.40. Yet
Kutter's n must be reduced because of the slight increase of
slope and hydraulic radius. As Kutter's C will increase with
decrease in elope where R is greater than 1 meter, or 3281
feet, n must be increased where R=12.40 because this slope is
least, and if n were taken as a constant for both channels, it
would make C too great for the first channel.

Pair No 3.
equal nearly.

R > 1 meter. Rand S vary. Roughness

NAME OF
CHANNEL

FEET

S
SLOPE

V
FEET
SEC.

SULLI-
VAN'S
C

KUT-
TER'S

CHEZY
C

La
Fourche

15.70

1988

.0000438

2.798

53.40

.0205

106.30

Mississip-
pi Rmr

72.00

2.913

.0000205

5.929

53.00

.0277

154.30

SULLIVAN'S NEW HYDRAULICS. 45

Note the difference in value of Kutter'e n for these two
channels of equal roughness.

In Pair No. 3 R is greater than one meter, and in this
case Kutter's C will increase as slope decreases. As the
slope of the Mississippi river is much less than that of
Bayou La Fourche, if n were used as a constant for both, the
value of Kutter's C would- be greatly too large for the Mis-
sissippi. Therefore, in order to balance the error of increase
in Kutter's C with decrease in slope, the value of n must be
increased in proportion as slope decreases. Otherwise his
GI/ rs will not equal v. It is seen from Pair No. 3 that when
the required value of the Chezy or Kutter C is obtained
which will make C v /~re~=\, then C:C: : V~T : V~, re-
gardless of the difference in slope. Kutter admits, in his
work on Hydraulics (pages 99 and 132) that n is not a con-
stant for the same degree of roughness if there is much vari-
ation in the dimensions of the channels to which it is applied.
His n might be made a constant like our C for each degree
of roughness, and regardless of the dimensions of the chan-
nels, if it were made to vary only as J/TT, for all slopes and
all dimensions of channels, whether R were greater or less
than one meter. It is absurd that C, and consequently the
velocity, should be proportionately less for a steep slope in a
large channel than for a small slope. Of course the value of
I/ S remains in any case, but decrease in C as S increases in
large channels amounts to reducing the actual value of S by
the amount that C is there made to decrease. It cannot be
justified upon any sound theory, and the above tables show
that it is not sustained by fact. It is equally erroneous that
C will increase with an increase in slope in small chan-
nels where R is less than one meter, and in which the
ratio of friction surface to the quantity of water passed
is much greater than in large channels. The laws of grav-
ity and of friction do not reverse themselves at the point
where R=l meter, nor at any other value of R. As Kut-
ter's n is not a constant for the same degree of roughness
where the slopes vary or where R varies, it is very mislead-

SULLIVAN'S NEW HYDRAULICS.

ing when viewed as an index of roughness, which is sup-
posed to be its special function.
Pair No. 4. R< 1 Meter. Roughness Equal. R and S Vary

Name of

VK~

Sulli

Kut-

Chezy

Channel

Feet

Slope

Ft Sec

van's C

ter's n

Rhine Forest

0.42

.8051

.0142

2.332

37.50

.0337

30.13

Simme

Canal

1.32

1.072

.0170

5.993

37.37

.0361

40.06

In Pair No. 4, R is lesa than one meter in either channel.
For this reason Kutter's C will increase with increase of
slope. Hence the steeper the slope becomes where R is less
than one meter, the greater we must increase the value of
his n in order to cut down this unnatural increase in C. We
find by simple proportion in Pair No. 4. as in all other cases
where the roughness is equal, that C:C: : *|/r:*- l /r, simply,
and regardless of difference in slope. Kutter's n must be
trimmed or increased in such manner as to cause C to vary
only as *y'r, otherwise his C^/rs will not equal v. It is there-
fore neither a constant nor an index of roughness, but is an
uncertain and misleading quantity. See Kutter'e discussion
of the variation of his n at pages 99, 110 and 132 of Hering
and Trautwine's edition of Kutter'e work. Also see Trans-
lators preface.
Pair No. 5. R and S vary. Roughness Equal.

Name of
Channel

Feet

t/r
Feet

S
Slope

Ft Sec

Sulli-
van's C

Kut-
ter's r

Chezy
C

Grosbois Ca-
nal
Seine (Paris)

1.71
14.50

1.143
1.951

.000441
.00014

1.51
4.232

48.08
48.12

.0284
.0255

55.50
93.92

In Pair No. 5, the value of R is less than 1 meter in one
case, and greater in the other, and there is a difference in
slope also. Notwithstanding both these facts, C must vary
only as 4 |/r as shown in the table, or C^/rs will not equal v.

Sullivan's C in all the above tables is C=A/-; =^; and ap-

SULLIVAN'S NEW HYDRAULICS. 47

plies in the formula, v=C$/r*~ ^ST" Its unit value is con-
stant for all slopes and all dimensions of pipes or open
channels of the same degree of roughness. It is
simply the square root of the reciprocal of m. It
has been shown that slope or velocity cannot affect the value

of m, as it is the expression of the ratio-^j I tB numer-
ical value depends only upon the degree of roughness of peri-
meter. The formula for m or n or C as heretofore given, will
give the unit value of the coefficient directly, that is, its value
for r or d=l. It therefore does not matter whether the for-
mula for ascertaining the coefficient is applied to the data of
a very small or very large channel, the result will be the value
of the coefficient for r=l, or d=l, as the case may be. From
this unit point the coefficient varies with the inverse value
of i/ r 8 or y' d 8 if it is n that is sought. The coefficient m
of velocity, varies from the unit value as found by formula

form, directly as i/~d*or l /~r^. The variation of C will be
as the {/T~if the formula is written v=C 1 /~r6, or if it
written v=C X V~ i/~rs. If we write v=C ,/S" then C
must vary as f/r 3 . This latter form is equivalent to the
form v=C X V~** /~S. in which C is the constant for any
given degree of roughness of perimeter. This last form has
been adopted in all the foregoing and following tables. For
the reason that m or C, as found by formula from the data
of guagings will be the unit value, and will differ in value
only as the degree of roughness differs, the mere develop-
ment of the unit values of the coefficient for a series of pipes
or open channels will at once classify such pipes or chan-
nels, and exhibit their relative degrees of roughness. Those
which give like values of the coefficient are of similar degrees
of roughness, because the unit value of the coefficient is not
affected by any element or factor except the degree of rough-
ness.

The coefficient C or m does not, and should not, vary, ex-

48 SULLIVAN'S NEW HYDRAULICS.

cept as the roughness of perimeter varies. For this reason
our in or G is an absolute index of the roughness for it cannot
vary with any other factor. We have shown that the effect-
ive value of the slope S is increased as -^ r 3 increases, because
the net meau head, or net gain in area over friction surface
is as >/ r 3 . But whatever increases or makes the mean head,
or S, more effective, must alao increase the value of v* in the
same ratio.

The effective slope S, is as S i/T 5 " , and the mean veloc-

ity is as,. /S i/ r 3 . Now in the formula for m or C. m= - ^
and C= | Y . In either formula an increase in the value

= | Y

^S/r 3

r 3 will cause the value of v 8 to increase in the same
ratio. It is then apparent that where the values of Sv/T 3 " are
equal, the velocities must be equal unless the resistances
caused by roughness of perimeter are greater in the one case
than in the other. It is also apparent from an inspection of
the formula for m or G that as v s will increase in the same
ratio as Sv/~r*" increases, m or C will be constant for all val-
ues of r or d if the roughness of perimeters is the same.

In the velocity formula, v=CV~r T Xv/~ST we see that the
mean velocity increases not only as ,/ S but also as the
square root ot ^/ r 3 , which is $/ r 3 , not because m or C var-
ies, but because the value of S is made more effective as v/~r^
increases.

9. Practical Determination of Coefficients of Resistance.

The resistance to flow, or loss of head by friction, is exactly
equal to the amount of head, pressure, or force required to
balance it. In a pipe of uniform diameter and roughness the
friction will be the same in one foot length of pipe as in any
other foot length, hence the total friction will be directly as
the length and roughness of the pipe. Friction in any given
diameter and roughness of pipe will increase with the square

SULLIVAN'S NEW HYDRAULICS. 49

of the velocity. Hence the head lost by friction, or the head
which is consumed in balancing friction, must also increase
as the square of the velocity. The friction or loss of head for
any given velocity in different diameters will be inversely as
y' d 8 or v/r 8 , because total acceleration is proportional to
the square root of the area, or to d or r, while total retarda-
tion is proportional only to j/cf or i/F". Hence the mean loss
of head of all the particles of water will be inversely propor-
tional to the resultant of total acceleration and total retarda-
tion, or to dy/ d = i/d 8 , or r i/ r = i/r 8 - (See columns
headed d.y/d, and "Relation of P to A," in table of circles,
ante, 3).

The mean of many experiments shows that a cast iron
pipe of ordinary density or specific gravity, one foot in diam-
eter and clean, will require a total head of one foot in a length
of 2,500 feet, in order to cause it to generate a velocity of one
foot per second. The discharge being free, it is evident that
the total head of one foot has been lost by resistance except
that part of the one foot head which remained to generate
the mean velocity of one foot per second. As the velocity
head is not lost by resistance, and as we wish to determine
the numerical value of the coefficient of resistance n, the ve-
locity head must be deducted from the total head of one foot
in order to find the total head lost by friction. By the law of
gravity we find that the head which generates any given ve-

In the case we are now considering v ff =l, and conse-

100
quentlv the velocity head hv=-gj^=.01552795 feet. Deduct-

ing this velocity head, which was not lost, from the total
head of one foot, and we find that the total head lost by fric-
tion in the 2,500 feet of 12-inch pipe while v*=l was equal to
1.00 .01552795=.98447205 feet. Therefore the head lost per

.98447205
foot length of pipe while v j =l, and d=l, was 2500

753

50 SULLIVAN'S NEW HYDRAULICS.

=.00039379 feet=n. As the friction will be as the number
of feet length of the constant diameter, and will increase as
V T , then, as long as d remains constant, the total head in feet
lost by friction, h"=n X I Xv 8 . But if the value of d
changes, or the formula is to be applied to a pipe of like
roughness, but of a different diameter, we have seen that
the friction will be inversely as ^/ d 3 . Henee the general
formula which will apply equally to all diameters of this
given degree of roughness will be

n * y8

v/d 3 )/ d 8

We might have found the value of n directly by applying
formula (10) ( 6).

Xv/^ F =mX-9845 (10)

v 2

Q //

As the ratio otj is always constant for any given de-
gree of roughness, regardless of slope or velocity, and as it
varies from the unit point, or d=l, and v*=l, only
as v/cl 1 varies, we may find the unit value of the co-
efficient from any diameter and velocity whatever.

S"
It is simply necessary to find the ratio-^ in any

case, and when the value of-|j is multiplied by l /~d f , the re-
sult will be the unit value of n. When this unit value of n
is inserted in formula (16) it is made to vary inversely as ^/ d*
as exhibited in formula (16). To make it appear more clearly
we write

It consequently does not matter what head, diameter or
velocity we may select for the purpose of finding the unit

SULLIVAN'S NEW HYDRAULICS. 51

value of n. The formula for n will always give the unit
value, regardless of the size of the pipe to which the for-
mula is applied. As the unit value of n is not affected by
any factor except the degree of roughness, it is a faithful in-
dex of roughness, and when the value of n for a series of dif-
ferent classes of perimeter has been found, it exhibits the
direct difference in roughness per unit of perimeter, between
the different classes.

10. Conversion of the Coefficient. The coefficient
may be determined in terms of diameter in feet, or diameter
in inches, or in terms of r instead of d, or in terms of cubic
feet or gallons. If the value of n has been found for any
given degree of roughness, it may be converted to any de-
sired terms. Thus, if the value of n has been found in terms
of d in feet, as above, it may, be converted to terms of r in
feet by simply multiplying it by 0.125 or dividing by eight.
If n was originally found in terms of r, and it is desired to
convert it to terms of d in feet, multiply by eight. If n is in
terms of d in feet, it may be converted to terms of d in inches
by multiplying by v /(r2) ff =41.5692. ^s n, for any given de-
gree of roughness, varies only with ^/"d^ the value of d may
be in meters, inches or feet, as may be most convenient, h",
I and v* may remain in feet or meters.

m=-;and n = mX-9845, for any given degree of rough-

//. Determination of Coefficients of Velocity. We have
just seen that a coefficient of resistance (n) represents only
the head per foot length of pipe which is lost or consumed in
balancing the resistance to flow. A coefficient of velocity,
however, must represent not only the head per foot length
required to balance the resistance, but also the head per
foot length required to generate the velocity of flow, or it
must represent S"-|-Sv in any case. If the diameter or
hydraulic radius is constant, and the discharge is free and
full bore, the total head per foot length S, will be converted

52 SULLIVAN'S NEW HYDRAULICS.

into velocity of flow except that part of S which is consumed
in balancing friction. In this case, S"+Sv=S, and S must
be used in the formula for determining the value of m the
coefficient of velocity. Where the discharge is partially
throttled, as by a reducer at discharge, or by a valve partly
closed, only a part of the total head per foot length will be
consumed by resistances and in generating velocity, and the
remainder of the head will remain as radial pressure within
the pipe. As the head due to this pressure is neither lost
by resistance nor engaged in generating velocity of flow, it
has no connection with the value of the coefficient of velocity
m. If the discharge is free, then

H v/r S S,/r n

m= ^s ^iXv/r= -^I-=;9845 < 17 >
If the discharge is throttled, then

For the ordinary cast iron pipe described in section 9,
the coefficient of velocity would be

-WM.m terms of d in feet.

The coefficient m may be converted to terms of d in
inches, or r in feet or to any other terms in the same manner,
and by the use of the same multipliers, as n may be con-
verted. (See 10)

The velocity coefficient m applies to open or closed chan-
nels alike and its unit value depends only on the degree of
roughness of perimeter. The value of m as found by the form-
ula is always the unit value, and is equally as accurate an
index of roughness as is the coefficient n. The remarks in
regard to n in this respect ( 9)apply to m with equal force.

The coefficient m is to be used in the formula,

SULLIVAN'S NEW HYDRAULICS. 53

If m was determined in terms of r, it must not be used
in formula (20) which is in terms of d, until it has been con-
verted to like terms with those in the formula. If m is in
terms of d in inches, then d in the formula must also be in
inches. In other words m must be in the bame terms as the
formula in which it is used is expressed.

The value of m in terms of d in feet for average cast iron
pipe is m=.0004. If it is desired to use C instead of m then

= 50.00 and

The value of C may be found directly and without refer-
ence to m by the formula

0=4

or C =

'8,/r*

This will give the unit value of C directly, and C is a
constant like m or n, which depends on the roughness of
perimeter.

If we have m=.0004 for ordinary cast iron pipe, in terms
of diameter in feet, we may convert it to terms of r in feet by

0004
simply dividing by 8. We then have -^ = .00005 = m in

terms of r in feet. We may convert m to C in terms of r in
feet by taking the square root of its reciprocal in terms of r,
and we have

v/20000 = 141.42 = C in terms of r.

Then, v = C {/r^ ,/S.

The unit values of n, m and C may be found in all classes
of pipes and channels, and may be converted at pleasure as
shown. The law governing the flow of water and the value
and variation of the coefficients, is exactly the same in open

54 SULLIVAN'S NEW HYDRAULICS,

channels as in pipes. The same formulas apply to all equally
well BO far as the coefficients and the formulas for flow are
concerned. Of course the unit value of the coefficient must
be found experimentally for each class or degree of Toughness
of friction surface. When the unit value of the coefficient is
determined for any given degree of roughness, it then applies
to all forms and dimensions of pipes and channels which fall
within that degree of roughness. These remarks apply to n,
m and C alike. The roughness or smoothness of perimeter
affects the flow in a large river in the same manner as in a
email canal. In a large, deep river the area of the cross-
section of the column of water is greater in proportion to the
wet perimeter than in a small stream, and hence the ratio of
free particles of water is greater than in small channels, but
the effect of roughness of perimeter is the same in both cases.
The unit value of m and C distinctly establish these facts.
It is the influence of the great values of r in large rivers that
has led some hydraulicians to conclude that the character of
the perimeter does not materially affect the flow in such
streams.

12. Coefficients Affected by Specific Gravity, or Den-
sity of Material. In a series of experiments with new, clean
cast iron pipes the writer was perplexed by the fact that one
12 inch new, clean pipe would not generate the same mean
velocity as another new, clean 12 inch pipe, when the con-
ditions were exactly the same in each case. The difference
was so great in the case of one pair of 12 inch new pipes,
that the experiment was repeated a number of times, but
always with the same result. As no other explanation could
be given the writer concluded to ascertain if it was caused by
the difference in density or specific gravity of the two pipes,
which were from different foundries. The shells were of
equal thickness, but on weighing a few lengths of the pipe
from each lot, it was found that the pipe which generated
the least velocity was much lighter than the other. The
investigation thus begun led to experiments with pipes of
different metals and different specific gravities. The results

SULLIVAN'S NEW HYDRAULICS, 55

then obtained seem to confirm the correctness of the view
that the density of the friction surface has a marked influ-
ence upon the flow and upon the value of the coefficient.
There may be some difference also between the values of the
coefficient for a surface of granular metal and a surface cf
fibrous metal, although the specific gravities of the two
metals may be equal. It appears that the flow over earthen
perimeters of equal regularity of cross-section will be affected
by the nature and specific gravity of the particular kind of
earth. The flow in a cement lined pipe or channel which is
clean and free of fine silt, will be affected by the fineness of the
cement and also of the sand used, as well as by the propor-
tion of sand to cement in the mortar lining. Even in pure
cement linings, it is noticed that the flow will be affected by
the quality and fineness of the cement used. Classification of
perimeters is therefore difficult.

It is stated by Professor Merriman that "it is proved by
actual gaugings that a pipe 10,000 feet long and one foot in
diameter discharges about 4.25 cubic feet per second under
a head of 100 feet. The mean velocity then is

v= -=5.41 feet per second." ("Treatise on hy-

draulics." page 165, fifth edition.) It will be noted that the
character of the pipe, whether cast iron, wrought iron, riveted
or welded, coated or uncoated, is not mentioned. It was
certainly a remarkably smooth pipe. If the value of the co-
efficient m is developed for this pipe we shall have

m= -Xv/ d 3 =.00034165, in terms of d in feet.

m= ^-Xi/ r=.00004270625, in terms of r in feet.

The average value of m for clean cast iron pipe is

m=.01662768, in term of d in inches.

m=.000i in terms of d in feet.

m=.00005 in terms of r in feet.
The writer made a numberof experiments with 6", 12'

56 SULLIVAN'S NEW HYDRAULICS.

and 24" cast iron pipes which were new and absolutely clean
and of the greatest density that the writer has ever dis-
covered before or since in cast iron pipes. The water was
pure mountain water from the melting snow on the granite
hills. The pipes were laid straight and perfectly jointed,
and the discharge was perfectly free, into a large measuring
tank. Under these perfect experimental conditions, the value
of m as developed by the three pipes was
m=. 000368 in terms of d in feet.
m=.000046 in terms of r in feet.

Such favorable conditions as these scarcely ever occur in
actual water works building, and do not continue if they
originally exist.

In later experiments with new clean cast iron pipes of in-
ferior quality and very low specific gravity, the values of the
coefficient of flow developed were

m=.01721 in terms of d in inches: C=7.622.

m=.000414 in terms of d in feet: C=49.14.

New, clean cast iron pipe of average weight per cubic
unit as long as it remains clean gives,

m=.01663 in terms of d in inches: C=7 .755.

m=.0004 in terms of d in feet: C=50.00.

m=.00005 in terms of r in feet: C=141.42.

It is therefore evident that where the pipes are made of
the same class of metal and are new and clean, the value of
the coefficient will bear a close relation to the specific grav-
ity, or density, of the pipe metal. The fact that clean leaden
or brass pipe will generate a much greater velocity of flow
under the same conditions than will a clean iron pipe of
equal diameter can be accounted for in no othei manner than
the difference in specific gravity of the different metals.

These facts demonstrate the important influence of even
very small degrees of roughness of perimeter upon the flow
and consequently upon the value of the coefficients. Low
specific gravity in metal indicates that it is porous and its
surface is affected by innumerable small cavities, rendering it

SULLIVAN'S NEW HYDRAULICS. 57

irregular. The specific gravity of cast iron varies from 6.90
to 7.50; of steel, from 7.70 to 7.90; of wrought iron from 7.60
to 7.90.

While the specific gravity of a metal, or of stone or brick,
01- earth where the cross section is equally uniform, undoubt-
edly affects the flow, yet other substances of much less spec-
ific gravity, when applied as a lining or coating, will greatly
increase the flow. Thus the specific gravity of asphaltum
varies from 1 to 1.80 according to its purity, and an asphaltum
coated pipe will generate a much higher velocity of flow than
a clean iron pipe. The coefficients developed by asphaltum
coated pipes, however, vary like cement lined pipes, with the
quality of the material, or the proportion of pure asphaltum
to the other ingredients u^ed in the manufacture of the coat-
ing comtound. It would appear therefore that while the
specific gravity of one metal may be compared with that of
another metal, or the specific gravity of one class of as-
phaltum coating compound may be compared with another,
as to its probable resistance to flow, we cannot compare ma-
terials of wholly different natures with each other, and judge
of the relative resistance by the respective densities. The
values of m for asphaltum coated double riveted wrought iron
pipe when new varies with quality of the coating as follows:
m=.000036 in terms of r in feet, to m=.000044.
m=.000288 in terms of d in feet, to m=. 000352.

The average value of m for such coating while in prime
condition may be taken as m=.00033, in terms of d in feet.
The average value of the coefficient of resistance in pipe thus
coated is about n=.000325 in terms of d in feet. The average
value of n for common cast iron pipe while clean is n=.0003938
in terms of d in feet.

Ordinary lead pipe gives m=.000135 in terms of d in feet,
or 0=86.07. In terms of r in feet, ordinary lead pipe gives
m=.000016875, or C=243.20. Lead pipe varies in specific grav-
ity, and the coefficient varies with the specific gravity. Very
dense, smooth lead pipe gives values of C in terms of r as high
as C=297.00 before the pipe becomes incrusted or scaled.

58 SULLIVAN'S NEW HYDRAULICS

13 Value of C Where the Flow is in Contact with Dif-
ferent Classes of Perimeter at the Same Time.

The sides of a channel may be rough and covered with
vegetation while the bottom is smooth and clean. In such
case the value of C will decrease as depth of flow increases,
because of the gain in ratio of rough to smooth perimeter as
depth increases. On the contrary the bottom may be rough,
stony and irregular, while the sides are smooth, clean and
regular. In the latter case the value of C will increase as
depth of flow increases, because of the gain in ratio of smooth
to rough perimeter as depth of flow increases. In all such
cases it is necessary to arrive at the mean or the average
roughness of the combined classes of perimeter. If the flow
is two feet deep in a canal six feet wide on the bottom and
the sides are smooth and vertical, while the bottom is rough
and stony, let us suppose that the sides correspond with C
=60, and the bottom with C=30. Then we have the two
smooth sides equal 4 feet and the rough bottom equal 6 feet
and the whole perimeter equal 10 feet.

Then, l_=Smooth perimeter where C=60.

6
-TQ = rough uerimeter where C=30.

4X60 240 6X30 180

And ~io~ = lo = 24> ~io-=To~ =18 - And 24

+18= 42.

The value of C for this combination of perimeters would
be 42.

14. Tables of Coefficients. In the following tables
of coefficients as developed from the published data of exper-
iments, the groups are arranged with reference to smoothness
or roughness of wet perimeter. The remarks in regard to the
available data for this purpose, which were made in the in-
troductory to this volume, should not be forgotten. Only a
part of the available data have been used, and that was sim-
ply a choice between evils in many cases. The writer is in-

SULLIVAN'S NEW HYDRAULICS.

debted to Mr. Charles D. Smith, C. E., of Visalia, California,
for the data of the guagings by him of sixteen canals in the
vicinity of Visalia, California. It is believed that these data,
all of which are given the common name of "Visalia Canal,"
are good and reliable. The writer is also indebted to Mr. J.
T. Fanning for a diagram of the results of experiments by
him on cast iron pipes of diameters ranging from 4 inches
to 96 inches, and exhibiting the average value of the coeffi-
cient in such pipes; and for guagings of the New Croton
aqueduct recently, made by Mr. Pteley, and for numerous
valuable suggestions. The writer is indebted to Mr. Otto
Von Geldern,C. E., of San Francisco, for the guagings of the
Sacramento river by C. E. Grunsky, C. E.

GROUP No. 1, STRAIGHT LEAD PIPE. (Rennie.)

L'GTH

DIAM.

COEFFICIENT

yd'

FEET

SLOPE

FEET

m ^Xi/d 3

'c r*

FEET

~V Sl/d3

"15:00

0.0417

.26666

5.00

.0000908

105.00

.008515

Straight lead pipe. (W. A. Provis.)

100.00

0.125

.02917

3.09

.0001350

86.07

.04119

80 00

125

.03646

H.396

.' 001397

81.60

0*419

60.00

0.125

.04861

3.903

.0001410

84.21

.04419

The coefficients for pipes are in terms of diameter in feet.
Straight Lead Pipe. (W. A. Provis.)

L'GTH

DIAM.

COEFFICIENT

/d 8

FEET

SLOPE

FEET

SEC.

m = V ,XV d3

' = ^S/d3

40.
20.

0.125
0.125

.07292
.14583

4.759
6.150

.0001422
.0001703

83. b6
76.55

.04419
.04419

REMARK. The coefficient m or C , includes all resistances
to flow, including the resistance to entry into the pipe. In
such very short pipes, where the velocity is considerable, the
effect of resistance to entry will materially affect the coeffi-
cient. For this reason a general pipe formula for ordinary
lengths of pipe will not apply with accuracy to short tubes
or very short pipes. A special formula for short pipes or
tubes should be applied in such cases. It is not known

SULLIVAN'S NEW HYDRAULICS.

whether all the above lead pipes of different lengths were of
the same quality and in the same condition or not. It is
probable that they were, and that the decrease in length
of pipe and increase in velocity greatly affected the resist-
ance to entry. The resistance to entry of a pipe cut off
square and flush with the inner walls of the reservoir is al
ways equal to .505 of the head generating the velocity of
flow through such pipe. Hence in order to obtain the true
coefficient of flow due only to the resistance of the inner cir
cumference of the pipe, the entry head should first be de-

v 8
ducted. The entry head=-^-X-505.

The data of experiments on very short pipes are not re-
liable, and should never be relied upon. They have no appli-
cation to long pipes.

Lead Pipe (Iben) Example of erroneous data.

L'GTH

FEET

TOTAL
HEAD
FEET

DIAM.
FEET

!/d

FEET

AL-
LEGED
VELOC
ITIES

COEFFICIENT

s
m =VirXi/d3

COEFFICIENT

=A/ ~

^S/ d 8

350.30
350.30

17.71
122.01

0.082
0.082

.02384
.02384

2.70
9.11

.000162^4
.0001000447

78.36
99.97

REMARK Here are the alleged results of two experiments
on the same pipe the only difference in conditions being a
change of head. As the length, diameter and roughness
were absolutely the same in both cases, the only possible ef-
fect of varying the head would be that the velocity would
vary directly as the square root of the head varied, and
nothing else.

Where all the other conditions are constant,the velocity will
vary directly as the square root of the head > and the resist-
ance, or loss of head by friction, will vary directly as the
square of the velocity. If this is not true, then the law of
gravity and the law of friction as accepted by the scientists
are necessarily erroneous, and all scientific calculations based
upon those laws must fail.

In the first experiment with this pipe of constant length,
diameter and roughness, the head was 17.71 feet, and velocity
was 2.70 feet per second. As all conditions remained constant
except an increase in head, then by the law of gravity and of

SULLIVAN'S NEW HYDRAULICS. 61

friction we would have

V/H : ^/H :: v : v; or 4.148 : 11.08 :: 2.70 : 721
In the last experiment, Iben makes v==9.11 instead of
7.21.

If the velocity was correct in the first experiment, or
v=2.70, then the head lost by friction for this velocity was
equal to the total head minus the head which remained to
generate the 2 70 feet velocity. The head required to generate

2.70 feet per second velocity was hv= V * = (2 70)2 =0.1132 ft.

64.4 64.4

The head lost by friction at this velocity was therefore 17.71
.1132=17.59 feet, and v s =7.29. Now, if the law of friction
is correct, to wit, that friction will increase in a constant di-
ameter and length as the square of the velocity, then the loss
of head in feet by friction in this pipe when the velocity in-
creased to 9 11 feet per second, would be
v 8 : v 8 ::head lost : head lost, or
7.29 : 83.00:: 17.59 : 211.00.

In other words in Iben's second experiment where the
total head was only 122.00 feet, he was able to lose 211.00 feet
by friction, and still have remaining 1.29 feet head to generate
the 9.11 feet per second velocity, which is alleged to have oc-
curred. It is conclusive that the laws of friction and of grav-
ity are absurd, or such data are in error.

All correct experimental data of flow for the same length,
diameter and roughness of pipe will necessarily develop the

same value of either of the coefficients, n, m or C, regardless

of all changes in head or velocity, because the ratio -^ is nec-
essarily constant in any given pipe. The foregoing illustra-
tration is given as a suggestion of a correct method of testing
the value of such published data of flow as are now available.
Most of such data are furnished by experiments of a century
or more ago, and have been translated from one language to
another and reduced from one system to another, and printed
and reprinted until the accumulated errors, added to the
original crude methods in vogue a century ago, render them

62 SULLIVAN'S NEW HYDRAULICS.

of very uncertain value. The writer is aware that Panning
and other very eminent hydraulicians have been of opinion
that m will decrease or C increase with the velocity in a
constant diameter, but this theory is not sustained by the re-
sults of Fanning's experiments on a constant diameter (See
Group No. 4) nor by the results of experiments by the writer
(Group No. 3). That theory cannot be accepted without first
rejecting the law of gravity and of resistance as now generally
accepted. If C increases with an increased velocity in a con-
stant diameter, it is obvious that resistance does not increase

S
as rapidly as v 8 , and hence the ratioy would not be constant

but would vary with the velocity. If that is true, then v* =
2gH the fundamental law of gravity is necessarily untrue,
and all our learned discussions of equilibrium and of uniform
flow are mere theoretical myths and rubbish. Either that
theory or the law of gravity and resistance must be rejected,
for both cannot stand. The experimental data now available
afford as much evidence to sustain an opposite theory as to
sustain the above theory, and hence these opposite results
destroy both theories, and prove only the erroneousness of the
data. The evidence to sustain one theory destroys that which
sustains the opposite theory, and the laws of gravity and of
resistance positively refute both theories, and establish the
theory that m or C is constant for all velocities in a constant
diameter, except as slightly affected by the resistance to
entry into the pipe. If the entry to the pipe is in the form of
the vena contraota, then the velocity cannot affect the value
of C or m at all.

SULLIVAN'S NEW HYDRAULICS.

GROUP No. 2.
Asphaltum Coated Pipe.

COATED PIPES.
[Hamilton Smith Jr.]

Lgth.
Feet

Diam.
Feet

v/d 3
Feet

Slope

Feet
Sec.

Coefficient.
S/d*

m y8

Coefficient

C P~
A/S^/d 8

1200.00

2.154

3J61

.01641

12.605

.0003265

55.34

700.00

1.056

1.085

.00668

4.595

.0003432

54.00

700.00

1.056

1.085

.01428

6.962

.0003200

55.90

700.00

1.0b6

1.085

.02219

8.646

.0003220

55.73

700.00

1.056

1.085

.03319

10.706

.0003142

56.40

4440.00

1.416

1.685

.06672

20.143

.0002771

60.07

700.00

0.911

0.8695

.0085

4.712

.0003330

54.80

700.00

0.911

0.8695

.01334

6.094

.0003123

56.58

700.00

0.911

8695

.01695

6.927

.0003072

57.05

700.00

0.911

0.8695

.02559

8.659

.0003000

57.73

700.00

.230

1.364

.01097

6.841

.00032000

55.90

700.00

.230

1.364

.01227

7.314

.00031264

56.56

700.00

.230

1.364

.01646

8.462

.000313.^6

56.48

700.00

.230

1.364

.02470

10.593

.00030025

57.71

700.00

.230

1.364

.03231

12.090

.00030150

57.58

REMARK. The slight variation of C or m in the same
diameter and length is due to errors in weir or orifice coeffi-
cients used in determining the velocities. The above pipes
were double riveted lap seam wrought iron pipes put together
like stove-pipe joints. Some of the velocities were determin-
ed by weir and others by orifice measurement. The differ-
ence in value of C for different diameters is due to difference
in quality of the coating. (See 12). In applying the above
coefficients it should be remembered that these pipes were
new and laid straight, and had free discharge and high veloci-
ties which would prevent any deposit in them. The propor-
tion of asphaltum in the coating is not stated. This is im-
portant and should be known.

Cast Iron Asphaltum Coated Pipe. [Lampe].

Legth
Feet.

Diam.
Feet

V/d 3
Feet

Slope.

Feet
Sec.

Coefficient
S/d 3

Coefficient

r~v*~
P. ./ v

- v *

"Vs^/d*

26,000
26,000
26000
26.000

1.373
1.373
1.373
1.373

1.609
1.609
1.609
1.609

.000594
.001376
.00163
.00195

1.577
2.479
2.709
3.090

.0003840

.(1003601)

.0003574

.0003300

51.03
52.69
52.91
55.04

REMARK. This pipe had been in use five years. Velocity
was judged of by reservoir contents and pressure guage.
The last coefficient is probably the true one. As the veloci-
ties tabled in the constant length and diameter do not cor-
respond with the slopes tabled, it is impossible to ascertain
whether either of the coefficients are correct or not. Only

SULLIVAN'S NEW HYDRAULICS.

one of them can be correct. The last one IB about the aver-
age value of the coefficient for such coated pipes.
Cast Iron Asphal turn Coated Pipe. [D'Arcyl.

L'gth.
Feet

Diam.
Feet

1/d*
Feet

Slope

Feet
Sec

Coefficient
Sv/d"

Coefficient

C= IVv*

^Sy^d 8

365.00
365.00
365.00
365.00
365.00

0.6168
0.6168
0.6168
0.6168
0.6168

0.4844
0.4844
0.4844
0.4844
0.4844

.00027
.00368
.02250
.10980
.14591

0.673

2.487
6.342
14.183
16.168

.00028SO
.0002882
.0002710
.0002644
.0002704

58.82
58.90
60.74
61.50
60.95

REMARK. Velocities determined by orifice. Variation
in C is due to error in orifice coefficients used. This pipe was
quite short, and must have had a remarkably smooth coating.
The coefficients developed by this pipe are too high for safe
use in ordinary practice. Lap welded wrought iron pipe in
long lengths with few joints, when coated with asphaltum
and oil, give C=60.00. It will be noted that D'Arcy's data
generally give the value of C too high. As would be expect-
ed from a series of experiments especially planned with ref-
erence to the most favorable conditions.

The weir and orifice coefficients should be standardized
in the same manner as m or C, so that a given form of weir
or orifice would have a unit coefficient which would vary
with i/r s for any dimensions of weir notch or orifice. The
results would then be uniform and correct.

Such weir formula might take the form, q=A ~5~-\/

" * m

The value of m would depend upon the form of the weir
only, and would, apply to all dimensions of weirs of that
given form. Before this kind of a weir formula could be
successfully adopted, however, it would be necessary to so
construct the weir as to suppress all contraction of the dis-
charge, for the contraction seems to follow no law. (See
Apendix.)

The Loch Katrine Cast Iron Pipe . Coated with Dr. Smith's
Coal Pitch. [Gale].

Lgth

3M
Miles

Diam.

Feet

/d
Feet

S
Slope

V
Feet
Sec.

Coefficient
ST/d 8

H):=: "

Coefficient

c \! v "

^Sv/ds

33 m.

4.00

8.00

.000947

3.458

.0006344

39.70

This pipe probably had large deposits of gravel in it. It

SULLIVAN'S NEW HYDRAULICS.

was evidently very rough from some cause. We give its
coefficient here simply because this particular pipe has been
the subject of so much discussion. See Flynn'e "Flow of
Water," page 34, for remark of Rankine and Humber on this
pipe.

GROUP No. 3.
Clean cast iron pipes not coated. (See 12.) (Sullivan.)

L'gth
Feet

2,800
2,800

Diam.
Feet.

Feet

Slope

Feet
Sec.

1.648
2.771

3.296
5.540

Coefficient

52.08
52.11
52.10
52.11
52.11

REMARK. These experiments were the foundation of the
writer's formula. They were made with the greatest possible
care. The writer being aware that a weir or orifice coeffici-
ent determined by the use of one degree of convergence of the
edges of the plate would not apply to another degree of con-
vergence or divergence, and having discovered discrepancies
of several per cent, in velocities thus determined,
did not rely on such measurements in the above ex-
periments, but erected a large measuring tank into which
the pipe discharged. The velocities were then determined

by the formula v= cubic feet 8ecopd . The pipes were remark-
area in sq. feet

ably dense and smooth, and had never before been wet.
They were laid straight and perfectly jointed. In doubling
the diameters and increasing the head four times, as will be
observed in the above table, it was the purpose to test the
law of gravity as well as to test the effect upon the flow of
doubling the diameter while the head remained constant.
A study of the results thus obtained resulted in the form-
ula for flow herein presented.

It may be remarked here that the coefficients developed
by the experiments under these exceedingly favorable cir-
cumstances with absolutely clean, very dense, straight pipes,
are not to be relied on for average weight cast iron pipes laid
in the ordinary manner. For average weight new cast iron
pipe, as long as it remains clean, m=.0004, and C=50.

The nature of the water which flows in a pipe which is
not coated may materially roughen the walls and reduce the

SULLIVAN'S NEW HYDRAULICS.

value of the coefficient in a very short time. Allowance should
always be made for this deterioration by adopting diameters
amply large.

GROUP No. 4.
Cement mortar lined wrought iron pipes, (Fanning.)

L'gth

Diam-

yd 8

Coefficient

Feet

eter
Feet

Feet

Slope

Sec.

o~i/ d"

C /

v a

8171 00

1.667

2.153

.00044

1.488

.0004300

48.22

8171.00

1.667

2.153

.00073

1.925

.0004241

48.56

8171.00

1.667

2.153

.00104

2.329

.0004130

49 20

8171.00

1.667

2.153

.00134

2.598

.0004274

48.38

8171.00

1.667

2.153

.00158

2.867

.0004139

49.15

8171.00

1.667

2.153

.00199

3.271

.0004004

49 97

8171.00

1.667

2.153

.00228

3.439

.0004151

49.08

8171.00

1.667

2.153

.00272

3.741

.0004183

48.92

8171.00

1.667

2.153

.00300

3.920

.0004203

48.78

8171.00

1.667

2.153

.00313

4.000

.0004212

48.72

8171.00

1 667

2.153

.00320

4.040

.0004221

48.67

REMARK. This was a force main, and velocities were
measured at the pump. Considering slight errors in calcula-
tions of slip, it is seen how nearly constant the coefficients
are. If there were no errors of slip, &c., there would result
but one constant value of m and C throughout, The above
guagings were remarkably accurate if the conditions under
which they were made be considered. They show great care
and excellent judgment on the part of the experimentalist.
Under more favorable conditions, still closer results would
have been had. From the values of the coefficient it is prob-
able that the lining of this pipe was one third sand and two-
thirds cement. Neat cement linings develop higher values of
C than the above, while the above coefficients agree closely
with those for linings of one-third sand and two-thirds ce-
ment.

The value of C does not increase with an increased veloc-
ity in a constant diameter, as has been claimed by some au-
thors. If so, the last value of C in the above table should be
the greatest.

SULLIVAN'S NEW HYDRAULICS.
GROUP No. 5.

Wooden conduits, planed poplar, closely jointed. (D'Arcy
& Bazin.)

L'gth
Feet

Feet

J/R3

Feet

S
Slope Feet Sec,

230.58
230.58
280.58
230.58

230.58

0.505
0.5C5
0.505
0.505

o.r,or,
0. :,(>:,
o.-,o:,
0.505

.000475
.001076

< ).:<>! i

.002911
.00(072

.00576
.006614

1.666
2.519
3.372
4.225
5.068
5.527
5.914
6.373

.00006143

.00005S53

.(KKHWITO
.00005948

.(MHKMilK.KI

.oooo:>.vi:>

129.10
130.60

129.75
129.10
130.70

REMARK. This conduit had a bottom width of 2.624 feet
and was 1.64 feet in depth. The velocities were determined
by weir measurement. The values of C developed illus-
trate the uncertain application of weir coefficients even in
the same small channel and for small differences in head, and
when applied by persons of great experience and sound judg-
ment. The value of the true coefficient in this conduit was
probably C=129.00 in each case. The value of the coefficient
for planed wood surfaces will doubtless vary with the density
of the wood. The coefficient will be greater in conduits in
which the boards are laid parallel to the flow than where the
flow is across the grain of the wood and the joints. Assum-
ing that m=.00006 is the true coefficient in terms of r in feert
for planed hard wood surfaces, we may reduce to terms of d
in feet (See 10) by multiplying by 8, and we have m=.00048
or. C=45.64 in terms of d in feet. This permits of a direct
comparison of the relative degrees of resistance to flow in
wooden pipes of planed staves closely jointed, and in iron
pipes, Thus

Lead pipes C=85.00 }

Asphaltum coated pipes, C=56.00

Clean cast iron pipes, C=50.00 ^All in terms of diam-

Clean planed hard wood, C=45.64 ] eter in feet.

Cement (one third sand) C=48.50 I

SULLIVAN'S NEW HYDRAULICS.

Wooden conduits, Planed boards. (D'Arcy & Bazin)

Surface
Width
Feet

Feet

v/R 3
Feet

S
Slope

Feet
Sec.

Coefficient
m _Sv/71T

Coefficient

n_ / v"

v *

^S,/^

3.16
3.62
3.89
4.08
4.53
4.59

0.390
.537
.632
.717
1.015
1.148

0.24355
.39350
.50240
.60710
1.0225
1.2300

.0015
.0015
.0015
.0015
.0015
.0015

2.61
3.23
3.71
4.04
5.00
5.54

.00005363
.00005660
.00005475
.C0005580
.00006131
.00006011

136.55
132.95
135.35
134.80
127.70
128.90

REMARK. The velocities in this table were determined by
surface floats and Pitot-D'Arcy tube measurements. The ve-
locities thus determined are undoubtedly too high. The weir
measurements given in the preceding table are more nearly
correct. A large majority of the guagings by D'Arcy and
Bazin were made by surface float and Pitot tube measure-
ments of velocity. They are not reliable when so made. This
table is introduced here to show that velocities thus deter-
mined are too high, and the fluctuating values of C show that
this method of guaging is not at all reliable. Data of flow de-
termined by such methods should be avoided. It is not in-
tended to convey the idea that all of D'Arcy and Bazin's
guagings are unreliable, but to show that such guagings as
are made by surface floats or by Pitot tube are worthless,
whether made by them or any one else. Some of D'Arcy's data
are good. Actual tank measurement of the discharge is the
only really accurate method of determining the velocity which
has so far been adopted. Weir measurement can be made
accurate by adopting unit coefficients for weirs similar to mor
C as suggested in a remark under Group No, 2, and the Ap-
pendix I.

Uiiplaned boards, well jointed and without battens.

The average value of m = . 000070 in terms of r in feet.
C=119.60 in terms of r in feet.

Ordinary Flume 6X5 feet Straight.

Clarke

Length.
Feet.

Feet

jiA 8

jFeet

S
Slope
.000435

\Feet
jSec

Coefficient

i Coefficient

! c p 1 "

V s

' V Sl /r 3

2500.

1.45

jl.746

.000088 ; 106.30

sewage. The grease and slime may affect the flow consider-
ably, ae well as the solid matter mixed with the sewage.

SULLIVAN'S NEW HYDRAULICS. 69

Bough Irrigation Flume. Highline Flume, Colorado. (Wilson)

Length
Feet

R [v' r8
Feet jFeet

S V

Feet
Slope iSec.

Coefficient
v*~~

ICoefficient

3000

4.50 J9.546

00099432:6.7657

.00020733

j 69.50

REMARK This is a rough bench flume with many abrupt
bends. For a cut and description of this flume see "Irriga-
tion Engineering" by Herbert M. Wilson, C. E., pages 173
and 174. The bends reduce the value of C considerably be-
low its value for a straight flume.

GROUP No. 6,

Stone and brick lined Channels. Chazilly Canal. D'Arcy
and Bazin.

Depth

Coefficient

V'iath

Feet

m- 8 ^' 3

r / v '

Feet

Slope

Sec.

V s

-Vs v /7T-

4.04
4.10
4.14

4.18

0.50
0.70
1.00

1.20

0.41
0.57
0.68
0.77

.0081
.0081
.0081
.0081

5.73

7.52
8.19
8.75

.000064765
.OOOOtJ]i!34
.OU0067713
.0 0071483

124.29
127.37
121.52
118.30

REMARK This canal is lined with smooth ashlar or cut
stone. The gaugings wera probably by surf ace floats or Pitot
tube which accounts for the fluctuating values of C developed.
If this is not the true cause, then the bottom and the sides
to a depth of .70 feet must be very much smoother than the
walls are above that depth. The last value of C is probably
nearest the correct value. See 13.

Roquefavour Aqueduct.
sides. {D'Arcy & Bazin.)

Neat cement bottom. Brick

Surface
Width
Feet

Depth
Feet

Feet

S
Slope

Feet
Sec.

Coefficient

Sy'r*

Coefficient
C- / v *

m Y
v a

A/Sv/ r*

7.40

2.50 1 1.504

.00372

10.26

.(WOJ652

123.85

REMARK. This aqueduct is nearly rectangular and at this
depth of flow the smooth cement bottom forms more than
half the wet perimeter. It should therefore develop a greater
value of C than the stone lined Chazilly canal of the preced-
ing table. In a smooth bottomed canal similar to this aque-
duct where the bottom is much smoother than the sides, the

SULLIVAN'S NEW HYDRAULICS.

value of C should be greatest for the least depths of flow, be-
cause as depth increases the proportion of the rougher side
perimeter becomes greater.

Aqueduct de Crau. Hammer dressed stone. (D'Arcy &
Bazin.

Surface
Width
Feet

Depth
Feet

R
?eet

Slope

Feet.
Sec.

Coefficient
tn S ' /TF

Coefficient

C-J V *

^Sv/ r"

8.50

3.00

1.774

.00084

5.55

.0000668

122.57

Sudbury Conduit. Hard brick, well jointed. (Fteley &
Stearns, 1880.)

L'gth
Feet

Greatest
Depth
Feet

Feet

S
Slope

Feet.
Sec.

Coefficient
m- S ^

Coefficient

C- / v *

4,200

1.518
2.037
2.519
3.561

1.078
1.385
1.628
2.049

.0001928

.noiniej

. 00019 M
.1001929

1.827
2.139
2.372
2.72J

.00006460
.00006851
.00007115
.00; 07648

124.33
120.82
118.60
114.35

REMARK. Velocities measured by weir. Only four of
these guagings are given because the slopes of water surface
in the others are so different from the slope of the conduit
and from each other as to show that equilibrium and uniform
flow had not ensued when the guagings were made. The co-
efficients are remarkably high for a plain brick perimeter.
The silt deposit on the bottom also affects the flow.

New Croton
(Fteley, 1895.)

Aqueduct. Hard brick, well jointed

Depth

Area

Coefficient

above

Feet

O / 3

/ V 2

center of
Invert.

oq.

Feet

Slope

Sec.

V-2

\Sv/r 3

1.10

9.24

0.7434

00013257

1.0969

.0000706267

118.97

1.50

14.12

1.0656

.00013257

1.4338

.0000709351

118.71

. 2.10

21.57

1.4886

.00013257

1.7731

.0000766000

114.26

3.00

33.04

2.0236

.00013257

2.1281

.0000842174

108.97

4.00

46.14

2.5137

00013257

2.4102

.0000906000

105.08

5.20

62.20

2.9947

00013257

2.6560

.0000974027

101.35

6.80

83.89

3.4998

.00013257

2.8894

.000103930

98.09

9.20

115.78

4.0062

00013257

3.0989

.000110689

95.05

11.00

136.93

4.1417

.00013257

3.1519

.000112500

94.28

12.50

150.55

4.0031

00013257

3.0977

.000110600

95.09

12.842

152.81

3.9161

'00013357

3.0625

.000109530

95.55

SULLIVAN'S NEW HYDRAULICS.

REMARK. Mr. Fteley states in his report that the veloc-
ities for depths below 1.90 feet are not as accurate as those
for greater depths, as the bottom or invert has slight silt de-
posits. It is evident that the bottom is very much smoother
than the sides, or the guaging apparatus was greatly at
fault. With the assistance of a very smooth silted bottom
the side walls and arch are apparently so rough as to run the
value of C below its value for common brick masonry. This
is a conduit of the horse shoe form and the velocities were
measured by meter. The result does not commend meter
guagings. From the slope of water surface it appears that
uniform flow was attained in each case before the guagings
were made. See 13.

First Class Brick Conduits Washed Inside With Cement.*
(Fteley.)

Name of
Conduit

Feet

Slope

Feet
Sec.

Coefficient

C =V^

Sudbury
Cochituate

2.4588
1.4170

.00020
.0000496

3.029
1.000

.000084000
.000083637

109.11
109.34

*See "Water Supply Engineering" by J. T. Fanning, p. 445,
Ninth Edition.

Washington, D. C., Aqueduct, Brick Conduit. Completed
1859. See Fanning, P 445.)

Feet

Slope

v
Feet Sec.

Coefficient
m Sl/7F

Coefficient

c r^~

-^S/T'

1.8735

.00015

1.893

.000107218

96.60

REMARK. This is about the correct value of C for ordi-
nary brick perimeters after several year's use. Where spe-
cially smooth or scraped brick are used or a cement wash is
applied the value of C will be greater. Pure cement linings
in channels of uniform cross section and good alignment de-
velop an average value of C=150.00 in terms of r in feet. The
value of C will vary somewhat with different qualities acd
fineness of pure cement linings, and uniformity of the walls.

SULLIVAN'S NEW HYDRAULICS.

Sudbury Conduit. Hard Brick With Surfaces Scraped.
(Fteley & Stearns 1880.)

Greatest

Coefficient

Depth

Feet

Slope

Ft. Sec.

Sv/r 3

0.719
1.055
1.076
1.187
1.224
1.328
1.415

0.493
0.762
0.778
0.858
0.885
0.957
1.016

.0001640
.0001742
.0000983
.0000246
.0001715
.0000746
.0000140

1.079
1.423
1.098
0.550
1.577
1.064
0.443

.000048763
.000057200
.000005950
.000064450
.000057600
.000061700
.000073000

143.21
132.28
133.65
124.85
131.79
127.75
117.10

REMARK. This conduit is of the horse shoe form and 600
feet in length. Velocities were determined by weir. The
conduit has a grade S=.00016. Compare the slopes in the
above table with that of the couduit. Also compare the
depths of flow with the corresponding velocities tabled. It
is quite remarkable to note the great changes in S for such
very small changes in R in a uniform channel with a grade
S=.00016. As the slope of water surface is so different from
that of the bottom of the conduit, it necessarily follows that
the depth of flow must have been different at each successive
point along the conduit, and the value of r was different at
each different point. The velocities were inversely as the
depths or wetted cross sections and hence were greatest
where the depths were least. Uniform flow had not occurred
and hence the effective value of S could not be known.

A comparison of the values of C for this conduit with
those for carefully dressed poplar conduits (Group No. 5) and
for average weight clean cast iron pipes would show this
brick surface to be smoother than either of the others. This
is, of course, not the fact. Because of the great number of
joints and resulting small irregularities of a brick wall, it is
scarcely possible that such wall should be more uniform and
smooth than a carefully constructed conduit of unplained
boards of hard wood, unless the wall were coated. In the
latter case the wetted perimeter would consist of the coating
and not of brick. Such data, although from eminent author-
ity, cannot be accepted. The last value is nearest correct.

SULLIVAN'S NEW HYDRAULICS. 73

Brick lined channel. (D'Arcy and Bazin)

Area

Coefficient

m- 8 ^ 13

r / V "

Sq. Feet

Feet

Slope

Feet Sec.

^vi2E

6.22

0.7554

.0049

6.69

118.67

REMARK. Velocity determined by surface float and Pitot
tube which almost invariably gives the mean velocity much
too high. This error results in giving too great a value to C
for ordinary plain brick perimeters.

Croton Aqueduct. Brick. Completed 1842.
Page 445).

(See Fanning,

2.3415 .00021

.0001677

77.50

REMARK. This conduit is of the horse shoe form. It
probably contains deposits of gritty material which reduce C
to so low a value.

Brooklyn Conduit. Brick. Completed 1859.
Page 445)

(See Fanning

2.5241 .00010

REMARK. As masonry conduits are permanent invest-
ments it is best to adopt a coefficient value low enough to
allow for deposits and future deterioration of perimeter.

Concrete Conduits. Old. Different stages of ruin. (See
Fanning, page 445).

Name of
Conduit

R
Feet

S
Slope

Feet Sec.

Coefficient
m= S J Al

V 2

Coefficient
r 1 v *

-Vsv/73-

94.87

84.52
104.77

117.00

Metz
Pont du
Gard
Pont Pyla
Mont-
pellier

0.915

1.250
0.6109

0.25

.00100

.00040
.00166

.00030

2.783

2.000
2.950

0.716

.00011175

.00014000
.00009110

.00007310

REMARK. In response to a recent inquiry of the writer
Mr. Fanning states that he visited these conduits a few years
ago and that some of them appeared to be in excellent re-
pair. They are constructed of hydraulic concrete, and are
rectangular in form.

SULLIVAN'S NEW HYDRAULICS.

Spillway of Grosbois Reservoir. Ashlar laid in Cement.
(D'Arcy and Bazin)

Surface

Depth

Coefficient

Width
Feet

Feet

Slope

Feet
Sec.

S,/r
m=

C =A!ST^

5.98
6.01
6.05
6.07

0.36
.55
.71

.84

0.324
.467
.580
.662

.101
.101
.101
.101

12.29
16.18
18.68
21.09

.00012331
.00012313
.00012786
.00012230

90.05
90.12

88.45
90.42

Covered with a slimy deposit.

Tail race Grosbois reservoir.
(D'Arcy & Bazin.)

Ashlar laid in cement.

Surface
Width
Feet

Dep'h
Feet

Feet

S
Slope

Feet
Sec.

Coefficient
ST/~T

Coefficient

r i~~**

m - v *

-V Sv /rT

6.00
6.10
6.10
6.10

0.49
.77
.97
1.16

0.424
.620
.745
.852

.037
.037
.037
.037

9.04
11.46
13.55
15.08

.00012499
.00013750
.00012958
.00012800

89.45
85.28
87.85
88.30

Covered with a light slimy deposit. Joints partly damaged.
Surface float.

Grosbois Conduit. Horseshoe form,
set in mortar. (D'Arcy & Bazin.)

Stone masonry

6.46
6.50

2.21
2.75
3.12

0.98
1.29
1.49
1.60

1.32
1.90
2.12
2.47

.000115118

86.50
93.27

REMARK. Bottom is rougher than sides. No deposit.
Joints not damaged. As D'Arcy & Bazin nearly always give
the slope of the bottom of the conduit, it is probable that
these guagings were made at different places along the con-
duit, as the slopes are different. The values of C may be at-
tributed to the rough bottom and smooth sides and also to er-
rors in guaging with Pitot tube and floats. For ascertaining
the correct value of C for any given depth in such channels
see 13.

Groisbois Canal.
(D'Arcy &Bazin.)

SULLIVAN'S NEW HYDRAULICS, 75

Roughly hammered stone masonry.

Sur-
face
Wdth
Feet

D'pth

Feet

Slope

Feet
Sec.

Coefficient

m v 2

A/S,/r

3.50
3.50
3.60
3.90

0.90
1.20
1.30
1.60

0.62
.71

.0600
.0290
.0141
.0121

13.93
11.23
8.36
7.58

.00015610
.00013757
.00015420
.00017400

80.04
85.26
80.53
75.82

REMARK. From the difference in slope it is probable
that these guagings were at different places where the rough-
ness was different. Otherwise the guagings are at fault. C
should be constant, unless the roughness of perimeter was
different at different depths of flow.

Qrosbois Canal. Stone Masonry. Broken Stones on
the Bottom. (D'Arcy & Bazin.)

Sur-
face
Wdth
Feet

D'pth
Feet

Feet

S
Slope

Feet
Sec.

Coefficient

=Vs^

6.80
6.90
6.90
7.00

1.50
2.00
2.40
2.70

0.88
1.23
1.40
1.50

.000648
.000671
.000683
.000683

1.47

2.02
2.34

2.78

.00024755
.00022434
!<KX)206G
.0001624

63.55
66.76
69.57
78.45

REMARK. The effect of the loose, broken stones and mud
deposits on the bottom, is to reduce the value of the coeffi-
cient C in the ratio that the mean of the different degrees of
roughness increases. If the depth of flow were reduced to .50
foot, the value of C would not exceed 45, because the rough
bottom perimeter would controJ. It is probable that if the
sides alone were considered apart from the rough bottom
the value of C would be 90.00. The value of C for different
depths of tlow in such channels will be different for each
depth, and may be determined by the rule given in 13.
Compare with New Croton Aqueduct.

SULLIVAN'S NEW HYDRAULICS.

Grosbois Canal. Stone Masonry in Bad Order. (D'Arcy and
Bazin)

Surface
Width
Feet

6.80
6.90
7.00
7.00

Depth
Feet

1.60
2.40
2.90
3.30

R
Feet Slope

Feet
Sec.

Coefficient

.00017610
.00014115

Coefficient

C=Jo-V

61.58
70.95
75.36

84.18

REMARK. Broken stones and mud on the bottom.
Sides fairly smooth. Rule given in 13 applies.

Solani Embankment. Sides of Stone Masonry. Stepped.
(Cunningham).

Surface
Width

Depth | R

1 S ! V

iFeet

Coefficient
Sv/r 3

Coefficient

Feet

Feet iFee

iSlope i Sec,

ra v s

~\Sv/r 3

150.00
150.00
160.00
164.00
170.10

1.50 ;i.69
2.30 :2.26
4.10 14.07
9.10 17.84
11.00 19.31

i. 000090 ! 0.44
i. 000148 i 0.87
i. 000215 i 1.79
J.UC0215 i 3.43
-.000227 i 4.02

.00102128
.00066400
.00055090
.000401162
.000400944

31.28
38.81
42.61
49.92
49.94

REMARK This channel has a bottom width of 150 feet
The side slopes are of stone masonry, built in steps. The
steps are broken and sunken in many places. The bottom is
of clay and boulders, very irregular, with bars of brick and
boulders built across at frequent intervals to prevent scour.
As depth of flow increases the ratio of smoother side peri-
meter increases, and the mean of the different degrees of
roughness becomes of a lesser degree of roughness. Hence
the value of C will increase with each increase in depth.

GROUP No. 7, RUBBLE AND RIP RAP.

Chazilly canal. Bottom of earth. One side wall of
mortar rubble and the other side wall of dry laid rubble.
(D'Arcy & Bazin.)

Surf.
Wdth

Feet

D'pth
Feet

Feet

S
Slope

Feet Sec.

Coefficient
m4?

Coefficient

r / v *

V 8

A/Sv/r*

8-50
9.50
9.80
10.20

1.30
2.00
2.40
2.70

1.00
1.36
1.54
1.67

.000525
.000450
.000462
.000487

1.01
1.38
1.58
1.74

.00051465
.00037470
.00035347
.00034730

44.08
51.66
53.20
53.66

SULLIVAN'S NEW HYDRAULICS,

REMARK If these guagings were all made at the same
point, the values of C show that the earth bottoms and rub-
ble footings were much rougher than the masonry side walls
higher up. See 13.

Turlock canal rock cut. Partly excavated in slate rock,
other parts of dry laid rubble and rip rap. ("IrrigationEogi-
neering" by H. M. Wilson, p. 82-83.)

Surf.
Wdth
Feet

Dpth
Feet

Feet

Slope

Feet Sec.

Coefficient

SV/TS"

*=-?r-

Coefficient

50.00

10.00

5.90

.0015

7.50

.000382

51.18

This rock cut is 6,200 feet in length and forms part of the
Turlock canal, California.

River Waal. Gravel bottom,
laid rubble. (Krayenhoff.)

Sides revetted with dry

17.25 11.10 .0001044 3.165 I

Description of this rubble,etc. from Beaumont's Geology,
Paris, 1845.

Head Race, Kapnikbanya, Hungary. (Rittinger)
Bottom is paved. Sides are of dry laid rubble not smooth as
bottom.

Depth
Feet

Feet

S
Slope

Feet
Sec.

Coefficient

C I

0.26
0.45

0.213
0.344

.0038
.0038

1.37
1.83

.000200
.000229

70.71
66.08

REMARK. The value of C for rubble will depend on the
size and shape of the stones. If the stones are small and
laid closely the coefficient will be much greater than if large,
irregular stones are laid with large spaces and projections.

Tail Race. Staukau. Hungary, Bottom paved. Sides of
dry laid rubble. (Rittinger)

Depth
Feet

Feet

S
Slope

V
Feet
Sec.

1.257
1.491
1.643

Coefficient
Sv/r 3

Coefficient

P / V "

m - v *

'-A/Sv/.o

0.42
0.56
0.69

0.289
iMr.ii
0.419

.0025
.0025
.0025

.0002457
.0002390
.0002500

63.80
64.73
63.25

SULLIVAN'S NEW HYDRAULICS.

Mill Race. Pricbram, Hungary. Very rough, irregular
bottom of earth.

Side walls of dry laid rubble. Bottom 2.07 feet width. (Rit-
tinger).

Bottom
Width
Feet

Depth
Feet

Feet

Slope

Feet
Sec.

Coefficient
Sv/r 8

Coefficient
r- / y8

-VSv/r"

2.07
2.07
2.07
2.07
2.07
2.07

0.41
0.44
0.70
0.80
0.86
0.90

0.316
0.336
0.472
0.548
0.560
0.566

.0022
.0022
.0022
.0022
.0022
.0022

A 9

0.588
1.953
1.135
1.190
I 269

.00258236
.001239473
.0007855758
.000692858
.000651000
.000581962

19.675
28.400
35.670
37.987
39.200
41.450

REMARK. This is a good illustration of the effect of a
combination of perimeters of different degrees of roughness,
which is referred to in 13, The rough bottom and large
rough rubble footings at the bottom of the side walls almost
prevented any bottom velocity of flow.

Grosbois Canal. Rough, trapezoidal canal. The bottom
of earth; one Bide slope rip-rapped, the other of earth with
some little vegetation.

The bottom is covered with stones and loose boulders.
(D'Arcy and Bazin)

Surface
Width
Feet

Depth
Feet

Feet

S
Slope

Feet
Sec.

Coefficient
S,/r 3

Coefficient

c / v2

m V 2

ViVr"

9.10
11.20
12.40
13.40

1.70
2.30
2.60
2.90

1.05
1.37
1.52
1.64

.000936
.000936
.000957
.000964

1.08
1.37
1.56
1.71

.00087200
.00080000
.00073100
.00069231

33.86
35.36
36.98
38.00

GROUP No. 8. IRRIGATION CANALS.

A series of Calif ornia Irrigation Canals carefully guaged.
By Charles D. Smith, C. E., of Visalia, California. (1895).
No. 1. A new canal in common loam just completed. Weir
measurement.

Feet

S
Slope

Feet
Sec.

Coefficient
S,/r 8

Coefficient

c- / v8

m - v *

C ' -VSv/r 8

0.99

.0006

1.33

.000334

54.72

See Group No. 11.

SULLIVAN'S NEW HYDRAULICS.

No. 2. An old canal in common loam. In only fair conditi on.
By meter.

R
Feet

S
Slope

v
Feet
Sec.

Coefficient
Sv/r"

*=-^r-

Coefficient

/ v

c =VsTvr

0.85

.001

1.52

.0003392

54.30

No. 3, An old canal in common loam recently cleaned but
not punned. By weir.

No. 4. A canal in sandy gravel, good repair. Firm gravel.
By meter.

1.40

4.02

56.87

No 5. A new canal. In river sand.

By meter.

1.16

.00175

49.83

Nos. 6 and 7. Canals in clay with loose gravel on the bottom,
otherwise clean. By meter.

.00177
.00194

3.46
3.51

66.04
64.73

Nos. 8, 9 and 10. Canal in firm earth with clay bottoms.
Good condition. By meter.

2.00

.0004

2.255

.00022249

67.08

3.35

.00001

0.531

.00021740

67.22

3.34

.0000375

1.032

.00021486

68.22

Nos. 11, 12 and 13. Canals in very heavy, smooth earth,
recently cleaned, trimmed and punned and put in excellent
order. Guaged by weir.

.000184553
.000195000

Nos. 14 and 15. Old canals grown up with weeds reaching
nearly to the surface. Weir and meter,

1.13
1.77

.00060

0.868
0.845

.001154

No. 16 Very crooked old slough in firm earth. By meter.

0.91 I .0030 | 3.086 | .00027354 | 60.47

SULLIVAN'S NEW HYDRAULICS.

GROUP. No. 9. SMOOTH CANALS IN EARTH.

Mill race, Kagiswyl, Switzerland. Side slopes of firm
earth, smooth. Bottom covered with fine gravel. Guaged
by meter. (Epper.)

Feet

Slope

Feet Sec.

Coefficient
m-SiA 8

V 2

Coefficient

c- / y8

A/S^r"

1.040
1.387
1.410

.001754
.001255
.001200

2.817
3.139
3.221

.00023447
.00020774
.000193711

65.30
69.40
71. 85

REMARK. Gravelly bottom reduces C as depth decreases. .
See 13.

Clean caual in firm earth and in beet order. Straight. (Watt.)

Surf.
Wdth
Feet

Dpth
Feet

Feet

S
Slope

Feet Sec.

Coefficient
m- S ^ r3

Coefficient
0= / v8

VST/r*

18.00

4.00

2.40

.0000631

1.134

.00018216

74.10

Mill race. Flachau, Hungary. Clean ditch in firm earth.
(Rittinger.)

0.55
0.86

0.467
0.703

.0020
.0020

1.953
2.199

.00016742
.00024350

77.46
64.58

REMARK. This is an example of remarkably bad guaging.
The writer has never known the smoothest and firmest peri-
meters of earth with best alignment to develop a higher value
of C than 75.00. In a clean canal with earth perimeter there
should be very little variation in C, especially for BO small a
change in depth. Rittinger's experimental data of flow are
usually much better than the average of such data, but the
above is evidently untrustworthy.

Linth canal. Grynau. Clean canal in common loam.
Side slopes a little irregular. Rod floats. (Legler.)

Surf.
Wdtb
Feet

Dpth
Feet

R
Feet

S
Slope

Feet Sec.

Coefficient

--*

Coefficient

^jg

123.00

5.14
7.12

8.87
9.18

.00029
.00032
.00036
.00037

3.414
4.418
5.40
5.53

.0002900
.0003115
.0003278
.00033648

58.74
56.70
55.30
54.52

REMARK. The slight irregularities of the side slopes
cause the value of C to decrease as the depth increases and

SULLIVAN'S NEW HYDRAULICS.

includes a greater proportion of side elope perimeter which is
rougher than the bottom. The results of guagings in such
channels as this are probably what led Kutter to suppose
that the value of C would decrease with an increase in slope
where RT>1 meter: The perimeter above the usual depth of
flow in a channel of any size whatever is exposed to freezing
and thawing, the burrowing of insects and the growth of veg-
etation. The change of slope or of hydraulic radius has no
effect upon the roughness. The value of C depends upon the
mean of the different degrees of roughness. See NOB. 8, 9, 10,
Group No. 8, and Solani Embankment, Group No. 6, where
the hydraulic radii are both less and greater than one meter
or 3.281 feet, and where the slopes increase with R. It will
be seen that it is the roughness of perimeter alone that af-
fects the unit value of C and that C varies with f/ r 8 only,
from its unit value as tabled for the same degree of rough-
ness. Kutter's C should vary only as {/r for any given de-
gree of roughness, and for different degrees, it should vary
as the mean of the roughness and as J/r, but should not be

affected by the slope at all, because-^- is necessarily con-
stant for all slopes. The recent gaugings of the Mississippi
entirely explode Kutter's theory.

GROUP No. 10. RIVERS.

Mississippi River, Carrolton, La. Bottom is fine sand
and the sides of alluvium, fairly stable. (Miss. River Com.
Report, 1882.)

Surface
Width
Feet

Depth
Feet

R
Feet

Slope

Feet
Sec.

Coefficient
m _Sy/r*
y

Coefficient

0=J v*

V Sv/r3

2647.00
2565.00
2582.00
2359.00
2423.00

93.00
90.00
92.00
86.00
89.00

63.10
63.40
57.20
57.60
57.70

.0000165
.0000127
! 0000139
.0000097

.0000112

5.90
5.08
4.46
2.95
3.73

.00024000
.00025000
.00032220
.00048745
.00035300

64.54
63.25
58.31
44.78
59.72

REMARK. The values of R were taken as nearly equal as
could be selected from the Report so that slope alone would
show its effect in connection with the various degrees of rough-
ness at different depths. The writer acknowledges that he
has little confidence in the correctness of these guagings, but
the various slopes and velocities tabled probably bear some re-
lation to the actual slopes and velocities. It does not appear
that fhe value of either Kutter's C or that of the writer de-
creases as slope increases. The values of Kutter's C for the

SULLIVAN'S NEW HYDRAULICS.

above guagings are given below, as transcribed from hie
work.

Surface
Width
Feet

Depth
Feet

Feet

S
Slope

Feet
Sec.

Kutter'e
C

Kutter's
n

2359.00
2423.00
2582.00
2565.00
2647.00

86.00
89.00
92.00
90.00
93.00

57.60
57.70
57.20
63.40
63.10

.0000097
.(1000112
.0000139
.0000127
.0000165

2.95
3.73
4.46
5.08
5.

124.8
146.7
158.2
179.0
182.9

.0452
.0354
.0290
.0261
.0218

REMARK. The value of the writer's C for a depth of 86
feet, in the first table above, corresponds with the average
value of C for rough, sandy perimeters in rivers, and is prob-
ably about the true value for this place. These are double
float, or mid-depth guagings. Kutter's n is not as constant or
as good an index of roughness as is chaimed for it.

Sacramento River, Freeport, California. Bottom of shift-
ing sand. Sides of earth. Straight reach. Guaged by me-
ter. (C. E. Grunsky.)

Date of
Guag-
ings

Area
Sq.
Feet

Feet

S
Slope

Feet
Sec.

Coefficient
n- 8 ^ 8

Coefficient

c =Vs^

March llrh
14th

17th
18th
19th
28th
May 26th

14,540
14,920
14.880
14.750
14.690
14,570
12,160

23.45
23.99
23.92
23.79
23.69
23.54
19.93

.0000744
.0000786
.0000675
.0000750
.0000713
.0000778
.0000580

3.994
4.157
3.974
3.897
3.741
3.383
2.879

.00053000
.00052867
.00050000
.00057290
.00058720
.00077637
.00062246

43.45
43.49
44.72
41.78
41.27
35.89
40.08

REMARK. The low water area at this place is 4,590 square
feet. From the dates and areas given it will be seen that the
guagings were made during high water, and that the river
was not stationary, or that continual scour or fill was going
on. Mr. Grunsky says in his report: "The river bottom is
sand. The river is there (at Freeport) surcharged with sand
brought in by its tributaries in quantities greater than the
water can assort, according to volume and yelocitiy of flow.
At the high stages of the river the changes in the contours of
the bottom are rapid and sometimes sudden. Boils are of fre-
quent occurrence. The river is full of whirls " (Report, p.
86.) At pages 96, 97 of his report Mr. Grunsky says: To pre-
pare a scale of discharge representing the volume of the
river's flow at various elevations of the water surface, for a

SULLIVAN'S NEW HYDRAULICS.

locality such as Freeport, was, in view of the shifting position
of the river bottom, an uncertain undertaking. * * *
Neither could any reasonably correct relation between water
surface elevation and velocity be established." Report Com.
Public Works to Governor of California, 1894.

River Rhine in Rhine Forest. Bed of coarse gravel.
(La Nicca.)

R
Feet

S
Slope

Feet
Sec.

Coefficient
ro-Sv/r"

Coefficient

n- / V

V 2

VSv/r

0.42
0.76
1.21

.0142
.0142
.0142

2.332
4.526
6.032

.000710
.000483
.000520

37.5">
45.50
44.47

Simme Canal. Canton Berme. Very coarse gravel.
(Wampfler.)

1.32
1.36

1.82
1.87

.0170
.0116
.0065
.0070

5.993
4.491
4.92
5.373

37.37

40.38
38.94
40.16

Mississippi River. Columbus, Ky. Rocky bluffs and
gravel. (Humphreys & Abbott.)

65.90 | 0000658 I 6.957 .OX)7516 36.48

Mississippi River, Vicksburg, Miss. Rocky bluffs and
Gravel. (Humphreys & Abbot.

.000678

River Izar. Coarse gravel bed. (G. ebenau.)

River Rhine. Boulders and gravel. Large stones on the
bottom.

21.65 .001 I 8.858 .001284 27.92

84 SULLIVAN'S NEW HYDRAULICS.

River Seine at Paris. Fairly regular reach,
while rising. (Poiree.)

Guaged

Area

Coefficient

Sq. Feet

Feet

Slope

Feet Sec.

Sv/r 8

p _ / V*

V 8

V SvA 8

1978

5.70

.000127

2.094

.0003940

50.38

2570

7.10

.000133

2.264

.0004909

45.13

3176

8.40

.000135

2.418

.0005620

42.18

3692

9.50

.000140

3.370

52.85

4421

10 90

.000140

3.741

! 0003600

52.71

5108

12.20

.000140

3.816

.0004090

49.44

6372

14.50

.000140

4.232

.0004316

48.12

6929

15.00

.000140

4.512

.0004000

50.00

8034

15.90

.000172

4.682

.0004974

44.84

8668

16.80

.000131

4.800

.0003915

50.54

9522

18.40

.000103

4.689

.0003700

51.98

REMARK. The guagings were made by floats, bazin says
they are good. It is seen from the areas recorded that the
river was rising. Considering tb.3 different degrees of rough-
ness of the sides as the water rose above its usual depth of
flow, and the great difficulty of ascertaining the true slope
on a rapidly rising river the results are quite satisfactory
The slope for r=15.90 is probably an error. See the discus-
sion of these and other data by Gen. H. L. Abbot in the
Journal of the Franklin Institute for May, June, July, 1873.
The guagings of the Seine at Triel, Menlon and Poissy have
been condemned because the water surface was affected by
tidal oscillations as great as two feet while the guagings and
slopes were taken. It was also rising at that time as shown
by the areas. The slope of the water surface under such con-
ditions could not be determined with any accuracy.

Ohio River, Point Pleasant, W. Va., Mid-depth floats. (Ellet).

Area
Sq. Feet.

R
Feet

Slope

v
Ft Sec

Coefficient
Sr/r 3
m=-V-

Coefficient

r / v *

^-AMVr"

7218.00

6.72

.0000933

2.515

.000257

62.38

Great Nevka River, Surface floats. 8-10 rule applied.
(Destrem).

15554.00

17.40 |

.0000149

2.049

.00025748 |

62.32

Mississippi River. Quincy,
Clarke).

111. Sandy alluvium. (T. C.

15911.00
51610.00

9.87
16.27

.00007434
.00007434

2.941

3.898

.00026526
.00032135

61.40
55.78

SULLIVAN'S NEW HYDRAULS.

REMARK. The first gauging at Quincy was at low water
when the flow was entirely in contact with its usual peri
meter which is somewhat smoother and less irregular than
the banks above the usual lo?7 water depth. The second
gauging was at high water after permanent high water con-
ditions had obtained. The slope of water surface was the
same for both stages of the river, showing that stationary
conditions had occurred.

Speyerbach Creek. Firm earth bed. (Grebenau)

Area
Sq. Feet

R
Feet

S
Slope

V
Feet Sec.

Coefficient

m =^l.

V s

' v a

r< /

-Vs^Ti-

30.20

1.54

.0004666

1.814

.00026931

60.93

River Neva. Surface floats. 8 10 rule applied-
43461.00 35.40 .000139? I 3.23 I

(Destrem).
59/70

River Elbe. Steep banks. Coarse gravel and small
boulders. (Harlacher.)

Surface
Width
Feet

Depth
Feet

R
Feet

S
Slope

V
Feet
Sec.

Coefficient

S^r"
m= ^

V 8

Coefficient

C =A/sr

343 00

452.00

6.20
11.80

3.51

7.77

.00038
.00041

2.49
4.95

0004030
.0003708

49.80
52.00

REMARK. In a channel like this with gravel and small
bouldeis on the bottom the value of Cfor a depth of 1.50 feet
would not exceed 40 if the channel were narrow. In a wide
bottomed rough channel with steep banks smoother than the
bottom, it will require a considerable depth of flow to include
sufficient side wall to balance the rougher bottom perimeter.
The above guagings were by meter.

River Salzach, Bavaria. At different places and stages.
Meter. (Reich.)

R
Feet

Slope

V
Feet
Sec.

Coefficient
m=^ r *

V 2

Coefficient

C= A/S7FT

3.45
3.52
4.96
5.00
5.20
7.00

.000280
.000348
.000290
.000607
.000410
.00036

2.686
3.618
3.510
5.543
5.094
4.118

.0002487
.0001760
.0002600
.0002190
.0001800
.000393

63.40
75.40
62.00
67.60
74.53
50.44

SULLIVAN'S NEW HYDRAULICS.

REMARK. These guagings were made in 1885 with the
meters then in use. The nature of the perimeter is not stated,
but it is safe to state that no natural channel will develop a
value of C as high as 75. The mill race at Pricbratu with its
masonry side walls and smooth clay bottom, and the smooth-
est, best aligned canals in firm earth and in perfect order, only
give C=T5. See next group.

GROUP No. 11 CANALS.

Mill race at Pricbram, Hungary. Side walls of masonry.
Smooth clay bottom 1.88 feet width. Trapezoidal. (Rittin-
ger.)

Depth
Feet

R
Feet

S
Slope

Feet Sec.

Coefficient
m =Ji^

V s

Coefficient

n_ / V

A/S;/r

0.54
0.66

0.373
0.425

.0010
.0010

1.127
1.254

.00017937
.00017621

74.67
75.33

See Nos. 11, 12 and 13 in group No. 8, also see
group No. 9.

Realtore canal. Common loam bed in only fair condi-
tion. (D'Arcy & Bazin.)

Surface
Wdth
Feet.

Dpth
Feet

R
Feet

S
Slope

Feet
Sec.

Coefficient 1 Coefficient

m _S;/r | c / v*

v V Sv/r*

19.70

4.50

287

.00043

2.54 ! .000324667 ! 55.51

Marseilles canal. Common loam bed in only fair condi
tion. (D'Arcy & Bazin.)

2.90 I .00043 2.536

Henares canal, Spain. Common loam bed in fair condi-
tion. (See Fanning.)

4.92 | 2.95 |. 000326 | 2.2% | .000313328

Lauter canal. Gravelly soil. Bed in fair condition.
(Strauss.)

29.50

1.82

.000664

2 106

.00036758

52.16

See Group No. 8, California canals.

Rivers, creeks and canals grouped according to roughness

SULLIVAN'S NEW HYDRAULICS.

of perimeter at the given depths. Areas are given in all cases
where known. Perimeters are described as fully as available
information will permit. The guagings are good, bad and
worthless. It is difficult to separate them without more pre-
cise knowledge of the exact conditions under which they were
made. The slopes of some rivers were measured while the
stream was rising and the velocities were taken when the
stream was falling. A fair average value of C may be arrived
at for each class of perimeters from what has already been
shown together with the following groups.

GROUP No. 12.

Shallow canals grown up in weeds reaching nearly to the
water surface.

Name of
Channel

Area I R
J^etJFeet

Slope

Feet
Sec.

Coefficient
Sv'r*

m = .-*-
V s

Coefficient

=V^

Visalia
Canal
Viealia
Canal

! 1.77
j 1.13

.00035
.00060

0.845
0.868

.001154
.00095658

30.00
32.33

GROUP No. 13.

Large canals with quantities of weeds and bushes on the
margins and shallow places.

Cavour
Canal
C. &0. C.
Feeder
C.&O. C.
Feeder

799.10
119.00
121.00

5.58
3.70
3.70

.000357
.0006985
.0006985

2.60
2.723
3.032

.0006964
.0006700
.0005410

37.85
38.64
43.00

GROUP No. 14.

Large streams with very rough banks and with large
stones and gravel on the bottom .

River
Rhine
River
Rnine
River Izar

13725.50

4650.10
1063.40

21.65

7.67
6.04

.00100

.00125
.00250

8.858

4.921
7.212

.001284

.001096
.000710

27.92

30.20
37.54

88 SULLIVAN'S NEW HYDRAULICS.

GROUP No. 15.

Shallow channels with very coarse gravel and email
boulders on the bottom.

Name of
Channel

Dpth

Feet

S
Slope

Feet
Sec.

Coefficient

V s

Coefficient

c ' v

\ fcn/rs

Schwarza
River

0.95

0.80

.0090

2.528

.001007

31.47

Schwarza

River

1.20

0.99

.0052

2.544

.0007914

35.55

Solani Em-

bankment

1.50

1.69

.000090

0.440

.00102128

31.28

Bimme

Canal

1.32

.0170

5.993

.00072

37.37

River
Rhine

0.42

.0142

2.332

.00071

37.50

GROUP No. 16.

Channels with one rough, stony bank and with gravel
bottoms. One bank of earth.

Name of
Channel

Dpth

Feet

Slope

Feet

Coefficient

V 2

Coefficient

M'ssissippi

River*

88.00

65.90

.0000680

6.957

.0007516

36.48

M ssissippi

Riverf

100.00

64.10

.0000638

6.949

.0006780

38.40

Grosbois

CanalJ

1.70

1.05

.0039360

1.080

.0008720

33.86

Grosbois

Canal

2.30

1.37

.0009360

1.370

.0007997

35.36

Grosbois

Canal

2.60

1.52

.000957

1.560

.0007310

36.98

Grosbois

Canal

2.90

1.64

.000964

1.710

.0006923

38.00

*At Columbus, Kentucky. Blutf on left bank composed
of strata of coarse sand, coarse brown clay, blue clay, fine
sand, coarse gravel, limestone, pudding stone, iron ore.

f At Vicksburg, Miss. Bluff forms left bank and is com-
posed of strata of blue clay, logs, carbonized wood, marine
shells, sand full of shells, sandstone. See "Levees of the Mis-
sissippi River," by Humphreys & Abbot, pages 28,29. (1867.)

JGravel and pebbles on the bottom. One side slope rip-
rapped with rough stone the other side slope of earth.

SULLIVAN'S NEW HYDRAULICS. 89

GROUP No. 17.

Channels in firm earth with low stumps and roots on the
bottom.

Name of
Channel

Area

Feet

Slope

Feet
Sec.

Coefficient

Coefficient
.

A/fcVr"

Bayou Pla-
quemine
Bayou Pla-
quemine

3560.00
4259.00

18.30
15.30

.0002064
.0001437

5.198
3.959

.000598
.00054875

40.88
42.69

REMARK. This bayou was guaged by Humphreys & Ab-
bot with mid-depth floats. It is simply an overflowed coule,
which was formerly covered by a thick forest of cypress
trees. These trees were cut down and the water brought
into the Plaquemine in 1770 by means of a small canal con-
necting with the Mississippi river. As the dirt washed from
around the stumps the Navigation company had them recut.
See "Levees of the Mississippi River," page 204, note. This
bayou varies in width from 200 to 300 feet, and in depth from
20 to 35 feet. There is luxuriant plant growth along the mar-
gins.

GROUP No. 18.

Grosbois canal. Earth bed in bad repair, with many
patches of vegetation.

Surface
Width
Feet

Dpth
Feet

Feet

Slope

Feet
Sec.

Coefficient
m - S^r

V s

Coefficient

-J&

10.10
12.30
13.50
14.70

1.70
2 40
2.80
3 10

1.06
1.41
1.60
1.76

.00042
.00047
.00047
.00045

0.89
1.18
1.31
1.39

.0005785
.000u6505
.000554326
.000o44

40 95
42.06
42 47

42.86

SULLIVAN'S NEW HYDRAULICS.

GROUP No, 19.
Channels with gravelly bottoms and rough, irregular banks

Name of
Channel

Area
Sq.
Feet

R
Feet

S
Slope

Feet
Sec.

Coefficient
Sv/r"

Coefficient

m V 8

Mississippi
River
Feeder
Chazilly
Feeder
Chazilly
River Rhine
Seine (Poissy)

150365.00
11.30

18.80
19135.00
10400.00

57.40
1.04

1.41

16 50
17.80

.0000481
.0004450
.0009930

.(HXHIWT
.0000750

6.310
0.962

1.789
3.575
3.330

.00052400
.00053100

.00051450
.00051235
.00050800

43.69
43.40

44.10
44.17
44.37

GROUP No. 20.

Rivers and Canals with beds of sand and with irregular side
Slopes of earth.

Name of
Channel

Area
Sq.
Feet

Feet

S
Slope

Feet
Sec

Coefficient

V 2

Seine

8034.00

15.90

000172

4.682

.0004974

44.83

^Pari.)

2570. 00

7.10

000133

2.264

.0004909

45.13

Chazilly

22.20

1.54

000986

1.959

.00049

45.16

Feeder

Chazilly

9.50

0.96

000792

1.234

.0004878

45.28

Feeder

Grosbois

23.00

1.63

000479

1.434

.00048456

45.40

Mississip-

pi River
Seine

179502.00

64.50

.0000436

6.825

.0004845

45.42

(Poissy)

9733.00

16.80

.0000670

3.101

.00048

45.65

Seine

(Triel)
Seine

6375.00

12.40

.0000600

2.359

.0 047

46.12

(Poissy)

8996.00

15.90

.0000620

2.911

.0004638

46.43

Feeder

Chazilly

22.90

1.57

.0004350

1.401

.000456

46.83

M?s^ 6 River

5010.10

10.85

.0000289

1.509

.0004536

46.95

Saalach

96.76

1.34

.001164011.970

.00045238

47.02

Seine

(Poisay)

7475.00

13.60

.00005002.372

.0004457

47.34

See Sacramento River, Group No, 10.

SULLIVAN'S NEW HYDRAULICS, 91

GROUP No. 21.

Natural Channels with sandy gravel bottoms and fairly
regular sides of earth. Canals in common loam in bad repcir

Name of

Area

Coefficient

q '

Feet

Sv/r 3

c / v *

Channel

Feet

Slope

Sec

v a

\ a

Feeder

Grosbois

19.40

1.41

.000858

1.815

.00043515

47.90

Feeder

Grosbois

18.10

1.38

.00045

1.296

.00043280

48.07

Feeder

Grosbois

27.20

1.71

.000441

1.510

.00043230

48.08

Seine

(Paris)

6372.00

14.50

.00014

4.232

.0 04316

48.12

Feeder

Grosbois

25.40

1.69

.00033

1.296

t . 00043155

48.14

Seine

(Poissy.

7952.00

14.20

.000054

2.595

.00042900

48.28

Izar

300.10

1.85

.0025

3.997

.00042430

48.54

Feeder

Grosbois

17.20

1.38

.00045

1.326

.000414

49.18

Feeder

Grosbois

20.90

1.56

.000525

1.575

.0004138

49.23

Feeder

Grosbois

32.00

1.85

.00 '33

1.411

.0004168

49.00

Feeder

Grosbois

22.90

1.56

.000842

1.998

.00041

49.39

Feeder

Chazilly

14.10

1.18

.000929

1.703

.00041068

49.34

Seine

(Paris)

5108. CO

12.20

.00014

3.816

.0004 9

49.44

Visalia

Canal

1.16

.00175

2.33

.0004026

49.83

Rhine

14149.80

9.72

.OOJ112

2.91

.0004008

49.95

Seine

(Paris

6929.00

15.00

.00014

4.512

.0004

50.00

Seine

(Paris)

1978.00

5.70

.000127

2.094

.000394

50.38

Salzacb

40 5.60

7.00

.00036

4.118

. 00393

50.44

Seine
(Paris)

8668.00

W.80

.000131

4.80

.0003915

50.54

Feeder

Grosbois

26.80

1.71

.000493

1.683

.00039

50.64

Feeder

Grosbois

14.90

1.21

.000808

1.667

.000387

50.83

Feeder

Grosbois

25.90

1.71

.000515

1.746

.0003775

51.46

Feeder

Grosbois

15.40

1.30

.000555

1.480

.00037557

51.60

Seine

(Paris)

9522.00

18.40

.(XWlCtt

4.689

.00037

51.98

Lauter

Canal

56.40

1.82

.000664

2.106

.00036756

52 16

Saalach

86.90

1.38

.0010357

2.155

.000362

52.56

Seine
(Paris)
Seine

3692.00

9.50

.00014

3.37

.0003578

52.85

(Paris)

4421.00

10.90 j. 00014

3.741

.00036

52.71

92 SULLIVAN'S NEW HYDRAULICS.

GROUP No. 22.

Channels revetted with rough angular rubble, dry laid;
channels in firm earth with rough, uneven bottoms and irreg-
ular side slopes.

Name of
Channel

Area
Square
Feet

Feet

Slope

Feet
Sec.

Coefficient

m-Syr

Coefficient

C= / V

V 8

VSv/r*

River

Waal*

14782.00

11.10

.0001044

3.165

.00038607

50.88

Turlock

Rock Cut

5.90

.0015

7.50

.000382

51.18

Mississippi

River

134942.00

52.10

.0000303

5.558

.0003693

52.03

Mississippi
River

193968.00

72.00

.0000205

5.929

.000356

53.00

Bayou La

Fourche

286'. 00

15.70

.0000438

2.789

.000352

53.30

Bear River

Canal

5.63

.00018939

2.67

.00035487

53.00

River

Rhine

6304.00

11.20

.0000999

3.277

.0003486

53.57

Seine

LMeulan]

6488.00

7.70

.000087

2.313

.00034734

53.66

*See Group No. 7.

GROUP No. 23.

New canals in loam, or light soil, just completed; old can-
als in similar soil with weeds along the margin; canals in
fairly go 3d condition but with pebbles on the bottom.

Name of
Channel

Area
Square
Feet

Feet

S
Slope

Feet
Sec.

Coefficient

Coefficient
n_ pH

m v

VsT/r"~

Visalia
Canal

Visalia

Idaho !? 81
&I Canal
Feeder'
Grosbois

11.80

0.85
0.99
7.232
1.050

.0010
.'006
.00037878
.00031

1.52
1.33
4.70
0.817

.0003392
.000334
.00033347
.000343

54.30
54.72
54.77
54.03

REMARK The flow in a new canal is never as great at
first as it will become after the bed is saturated with water
and the loose material is dissolved and deposited in the pores
of the earth and in the little depressions and irregularities
along the perimeter.

SULLIVAN'S NEW HYDRAULICS,

GROUP No. 24.

Clean channels with bottoms of fine gravel and sand
well settled, and with banks of sandy loam; channels in
sandy alluvium; canals in ordinary loam in fair condition but
not recently cleaned. (Average for canals in ordinary con
dition).

Name of
Channel

Area
Sq.
Feet

Feet

S
Slope

Feet
Sec.

Coefficient.
Sj/r 3

Coefficient

m v s

Marseilles
Canal

66.00

2.90

.00043

2.536

.00033

55.04

Eealtore

Canal

2.87

.00043

2.540

.000324667

55.51

Eiver

Tiber

2355.00

9.40

.001306

3.413

.00032335

55.62

Seine

(Meulan)

5982.00

7.10

.00009

2.31

.000318876

55.75

Eiver
Ha me

306.40

5.70

.0001559

2.558

.0003213

55.71

Miss. River

51610.00

16.27

.00007434

3.898

.00032135

55.78

Henares

Canal

2.95

.000326

2.296

.000313328

56.40

Mississip-

pi Eiver

78828.00

31.20

.0000223

3.523

.0003132

56.49

Visalia

Canal

1.40

.00302

4.02

.00030951

56.87

Feeder

Grosbois

18.00

1.42

.00029

1.26

.000309

56.89

* At Quincy, Illinois, gauged by Thomas C. Clarke, C. E.
GROUP No. 25.

Rivers and canals in good condition, having beds of fine
sand and small pebbles, with fairly regular banks of firm
loam.

Name of

Area

Coefficient

Sq.

Feet

S;/ r 8

Channel

Feet

Slope

Sec.

V 2

Hooken-

bach Creek
Hocken-

10.30

0.88

.0007966

1.463

.000307287

57.00

bach Creek

10.50

0.87

.0007783

1.440

.00030463

57.28

Misssissip-
pi Eiver

195349.00

72.40

.0000171

5.887

.000304

57.35

Upper*

Miss. Eiver

3441.00

4.42

.0002227

2.611

.00030355

57.40

Feeder

Grosbois

30.80

1.78

.000275

1.467

.0003034

57.48

Feeder

Grosbois

10.90

0.96

.00025

0.886

.0003

57.74

SULLIVAN'S NEW HYDRAULICS.

* At Fort Snelling, Minnesota. It is apparent from the
areas and hydraulic mean radii in Group 25, that the effect
of rough or smooth perimeter is the same in a very large
channel as in a very small one. These perimeters are almost
exactly alike, and develop like coefficients, regardless of size
or slope.

GROUP No. 26.

Rivers and canals in alluvial soil, or firm earth mixed with
tine eand, in good condition, and free of stones and weeds.

Name of

Area

Coefficient

Channel

Square
Feet

Feet

Slope

Feet
Sec.

m= *v;'

c =Vs^

Bayou La

Fourche

3738.00

15.70

.0000447

3.076

.0002948

58.24

Visalia

Canal

0.93

.(004

1.110

.000291

58.62

Eiver

Rhine

5341.00

7.60

.0001174

2.917

.00028905

58.83

Huben-

graben

3.80

0.59

.0013

1.424

.00029

58.74

River

Neva

43461.00

35.40

.0000139

3.230

.00028065

59.70

Marseilles

Canal

3.386

.000333

2.720

.00028065

59.70

GROUP No. 27.

Canals in heavy loam in excellent repair; natural chan
nels with very fine sand on firm and regular bottoms with
sandy loam banks in excellent condition, free of weeds and
stones.

Name o f
Channel

Area
Square
Feet

Feet

Slope

V j
Feet!
Sec.j

Coefficient
B _Sv/r

Coefficient

v 8

V s

\Si/r 3

Speyer-
bach Creek

30.20

1.54

.0004666

1.814!

.00026931

60.93

15911.00

9.87

.00007434

2.941;

.00026526

61.40

Feeder

Grosbois

24.20

1.57

.000246

1.362!

.00026080

61.88

Great

Nevka

15554.00

17 40

.0000149

2.049!

.00025748

62.32

Ohio

River

7218.00

672

.0000933

2.515!

.00025700

62.38

SULLIVAN'S NEW HYDRAULICS. 95

GROUP No. 28.
Canals in smooth clay with loose pebbles on the bottom.

Name of

Area
Square

VPont

CJIona

Feet

Coefficient
m- 8 ^"

Coefficient

c / v "

Feet

Sec.

AWr 8

Visalia*
Canal
Feeder
Grosbois
Viealia
Canal

17.10

1.32
1.32
1.34

.00194

.000275
.00177

3.510
1.336

3.460

.00023862
.00023366
000229-254

64.73
65.37
66.04

*A few weeds along the margin in patches.

GROUP No. 29.

Canals in very firm, heavy soil, with clay bottoms worn
smooth, but not recently trimmed and punned; natural
streams of good alignment with clay bottoms, and fine grain-
ed, firm and uniform alluvial banks, free of stones and vege-
tation .

Name of

Area

Coefficient

Sq-

Feet

m Sv/r 8

IV V *

Channel

Feet

Slope

Sec.

V B

~^Sv/r 8

Visalia

Canal
Visalia

2.00

.000400

2.255

.00022249

67.08

Canal
Yssel

3.35

.000010

0.531

.0002174

67.82

T,o,h EiVer

1930.00

15.90

.0001166

2.773

.00021728

67.84

Katrine

2.525

.0001578

1.7126

.0002158

68 05

Visalia
Canal
Bayou La

3.34

.0000375

1.032

.00021486

68.22

Fourche
Bayou La

3025.00

13.00

.0000373

2.843

.0002145

68.27

Fourche

2957.00

12.80

.0000366

2.807

.0002126

68.53

REMARK The bottom of this portion of Bayou La Four-
che is clay, and the banks are leveed. The banks are of
heavy, alluvial soil. Its bends are few and gentle. There
are no boils, whirls, nor eddies. It resembles an artificial
channel very much. For a general description see "Levees of
the Mississippi River", page 198.

96 SULLIVAN'S NEW HYDRAULICS.

GROUP No. 30.

Canals in very firm, smooth, dense earth, recently cleaned,
trimmed and punned, and put in the best condition.

Name of
Channel

Area
Sq.
Feet

Feet

Slope

Feet
Sec.

Coefficient

V s

^"""VSv/r 8

Visalia
Canal
Visalia
Canal
Visalia
Canal
English
Canal

50.00

1.13
1.09
0.92
2.40

.00060
.00060
.001165
.0000631

1.88
1.87
2.36
1.134

.00020396
.000195
.000184553
.00018216

70.02
71.61
73.73
74.10

IS Roughness of Perimeter Defined. The foregoing
tables of coefficients might be greatly enlarged by the ad-
dition thereto of the data of many other pipes and channels,
but such matter would be simply cumulative. It is believed
that the tables given cover all cases as accurately as the pub-
lished data will permit, and it was not deemed neccessary to
give but a few examples of each class in order to assist in
selecting the value of the coefficient in any ordinary case.
The inaccuracies which abound in the data of flow in all
classes of pipes and channels are due in great part to the
failure of weir and orifice coefficients. The writer is aware
that a general belief in the accuracy of weir measurement has
become very great, but the fact remains that such measure-
ments are very frequently erroneous to a very considerable
extent. Meter measurements of velocity are still less reliable
When better methods are discovered and adopted, we shall
have more reliable data than we now have. In the consid-
eration of the degree of roughness of any given channel, the
alignment, uniformity of cross-section, freedom from grit
gravel, stones and vegetation, are not, by any means, all that
are to be considered. The nature of the material in contact
with the flow, as to density and compactness, is as important
as any or all other features. The coefficients show that for
clean canals in earth, the value varies from about 56 to 75 as
the nature of the earth forming the perimeter varies. The

SULLIVAN'S NEW HYDRAULICS. 97

amount of sand, and whether coarse or fine, which enters in-
to the majority of different classes of earth, hae a great effect
upon the flow and upon the value of the coefficient. Every
table of data of open channels abundantly proves the incor-
rectness of the idea that the character of the perimeter has
no influence upon flow in very large channels. The Missis-
sippi river at Columbus and Vicksburg with depths of 88 and
100 feet respectively, develop the same value of the coeffi-
cient as very small ditches having the same kind of perime-
ter. (See Group No. 16). There is no reason why this should
not be the case, and it would be strange if it were not the
the case. The flow in large rivers is nearly always overesti-
mated, especially where meters or surface floats are used for
determining the velocity. Insufficient attention has been
given the character of the perimeter and its effect upon the
flow of the water in contact therewith and affected by the re-
actions therefrom. The velocity of the film of water in con-
tact with and affected by the sides and bottom has never
been considered of great importance in determining the mean
velocity of the whole cross section in large streams, and yet
if this outer layer of water thus affected were deducted from
the whole, at least one fourth the total area would be sub-
tracted.

CHAPTER III,

Of the Deduction of the General Formulas.

16 Formulas in Terms of Diameter in Feet. For large
pipes and circular channels flowing full, a set of formulas in
terms of diameter in feet will be most convenient. For
small pipes the coefficients may be in terms of diameter in
inches.

FORMULA FOB Loss OP HEAD BY FRICTION.

By formula (10) 6, the coefficient of friction n, is

(10)

Hy transposition in (10) we have the formula for loss of
head in feet by friction

_ n

h == ~ lv

In which, h"= total head in feet lost in the length in feet, I
d=diameter of pipe in feet.
n=coefficient in terms of diameter in feet.
Z=length of pipe in feet.
v=velocity in feet per second.

FORMULA FOR HEAD IN FEET LOST PER FOOT LENGTH OF

PIPE.
S"
As n =^jf-X i/d 8 , we have by transposition,

S "=VcT*Xv*....^ ............................ (22)

In which,

S"=head in feet lost by friction per foot length of pipe.
n=unit value of the coefficient of resistance which in-
creases as v s , and is inversely as i/d s .

FORMULA FOR MEAN VELOCITY OF FLOW.
By equation (12) the coefficient of velocity m, is

SULLIVAN'S NEW HYDRAULICS.
Hd^d S ff _ S^/d 8

And by transposition in (12) we have,

In which,

v mean velocity in feet per second.

H=total head in feet, where discharge is free.

fl=h"+bv where discharge is throttled ( 5).

1= length of pipe in feet.

d=diameter in feet.

m coefficient of velocity determined in terms of d in
feet.

Where the altitude is sufficient to affect the value of g,
the formula may be written,

, when m- / y8 ; H:=

In this case the value of m must be found according to
the value of 2gH at the given altitude.

FORMULA FOR TOTAL HEAD REQUIRED TO GENERATE A

GIVEN VELOCITY.
miv* mlv*

The slope (S) required to generate a given mean velocity
is

S = m -^n ^-^rrXd* (25)

To find the length in feet I, in which there must be the
given head in feet H, or fall in feet equal H, in order to
generate the given mean velocity v:

Hdy/d_H y yd* (26)

m v s v s m

100 SULLIVAN'S NEW HYDRAULICS.

FORMULAS IN TERMS OP CUBIC FEET PER SECOND AND DI-

AMETER IN FEET.

Letq=cubic feet per second discharged,=AreaXvelocityi
a= area of pipe in square feet.

Then a=d s X-7584, and v=/ X~ whence

q v/T~m=d 2 .785VH ^d 8 or q 1 /m=d ir .7854 v /S 1 /d

whence

(28)

I .61685 IS i/d*i_ / .6165 y /a /HII . . .(29)
9 \ m * m

Hence the diameter in feet required to cause the dis-
charge of a given number of cubic feet per second is

(28)
If total loss of head is predetermined then

And the slope required to cause a given diameter to dis-
charge a given quantity in cubic feet per second will be
m q* m q*

And,

qm? ._JE_xx__il
[= 61685^/c

SULLIVAN'S NEW HYDRAULICS. 101

_ , q* v i . . /am

.616853/d" - .61(
hVd" .61685

H .GieSSy'd 1 !
m = ; s :

_H_JP1685j/dij_
m q*

/Hyd".61685_ /S|/d"X. 61685 (36)

I/ " m { K " m

As v=-^- it appears from an inspection of (29) and (36)

that the relative discharges will be ao f/d 11 .

The slopes, or heads and lengths being equal, then

q : q : : ^/d 11 : fc/d 11 , provided the roughnesses are
equal.

(See Table No. 18, 33.)

17 Formulas in Terms of Pressure, Diameter and
Quantity. Head in feet and pressure in pounds per square
inch are convertible terms. Pressure increases directly
as head increases, and the velocity will be proportional to
either ,/H or /P. When H=2.3041 feet, P=l Ib. per square
inch. Hence the coefficients determined in terms of H will
not apply in a formula in terms of P. The coefficients may,
however, be converted from terms of H or S to terms of P as
pointed out in 10 and as follows:--
p

P=HX-434, and H=-jg|-=PX23041. Hence if we have

the value of n or m in terms of H and d in feet, and wish to con-
vert to terms of pressure in pounds per square inch P, and
diameter in feet d, we divide the value of n or m in terms of
H and d, by 2.3041, and the result is the value of n or m in
terms of Pand d.

102 SULLIVAN'S NEW HYDRAULICS.

Tt ,_ n Zv* nlv*

.(38)

V/d 8 '" dj/d
In which,

P'=total Ibo. per squan inch pressure lost by friction.
n=coefficient of resistance in terms of P' and d.
Let P=total pressure in Ibs. per square inch.
P'=total pressure lost by friction.
Pv=velocity pressure.
When the discharge is free, P =P Pv,
To find the pressure in pounds per square inch required
to balance the friction and generate a mean velocity v, in
feet per second:

q=av=$/d 11 \/ p X -7854 -H ^

q^nTT^Kd'VPX .7854,whence

To find the diameter in feet to discharge the quantity q,
in cubic feet per second, through the length I, when the
total pressure is P:

> 8 q* 11 / m 8 ll/Z 8 q*

d*= K -3S66- X I/pi"' " (43)

To find the total pressure required to balance the re-
sistance and force the discharge of q cubic feet per second
through a pipe of given length and diameter in feet (Lifting
weight of water not included).

I m q 8 m q 8

~".616853i/d TT = .616853 X ^d 11 ^ * )

To find the pressure lost by friction while discharging a
given quantity in cubic feet per second:

SULLIVAN'S NEW HYDRAULICS. 103

P '=.6mWdii (45)

The length of pipe in feet through which a given pres-
sure in pounds per square inch will force a given quantity
of discharge is,

P/d" X .61685

mq 2

To find the coefficient of resistance n, in terms of q, d
and P:

. 61685

To find the coefficient of velocity in terms of q, d and
^~ (48)

**1

18. Formulas In Terms of Hydraulic Radius (r) and
Slope (S).

m I v 8 m I v 8

Hy/r 3 Hry/r

mv* mv*

mv mv* ' ( 5 )

Hrv/r Sv/r 8 S

m =-T7"~ = -^~ = -v^Xv^ 3 (51)

h"rvT 8

I v 8 = ~ v s ~~ v

= ^1-Xl/r" I

The length in which there must be a fall of one foot in
order to generate any given mean velocity v, is

^=^-=-m7 2 - <)

/ TT / m ^ \ /H T/r 3 /Sr/r

v =i/ H ^ ("vr^ i/ x ^-= v-^~-

V** X \/-^r (54)

104 SULLIVAN'S NEW HYDRAULICS.

Area in square feet =12.5664Xr 2 . q=av.

q=12.5664rXe/' 8 Xl/ / - a 7 (55)

qT/m^f/riiXi/S X 12.5664, whence,
i X 157.91 44

q 2

n JLi^=. ...(57)

S " = "l57.9U4Vr rf " = 1573144" X 7^ (58)

S= 157.914Vri'i (59)

H =i57:9WFn- >

157.9144^11

* 8

....(61)
...(62)

-v' - ' T 7 ' 91i4 <)

'-V SBSOSg-xB* ' (64>

117 q*m 2

V 249\36:958"XS 2 " '"^
^157^/rii ^

19. Application and Limitations of the Foregoing
Formulas. As heretofore noted under the table of circles
and of open channels (3) r is not necessarily an index of ca-
pacity in open channels as it is in pipes and circular channels
flowing full. Hence in open channels the formulas (63) and
(55) for q will not necessarily give accurate results, unless the
value of r was originally determined in terms of q when the

SULLIVAN'S NEW HYDRAULICS. 105

channel was designed. In channels having side slopes of 2 to
1 the formula for q will usually apply quite accurately. For
the same reason the formula for r does not apply to open
channels in general, but only to those in which the value of r
was determined, or is to be determined in terms of q. All the
formulas apply with exactness to pipes and circular closed
channels flowing full. All the formulas in terms of r, except
those just noted, apply to all forms of open or closed chan-
cels. These exceptions in the case of open channels, do not,
however, affect the general application of the coefficients, be-
cause the coefficient depends upon the relations of a to p
which is always expressed in any given case by r which is

equal in all cases, and the friction surface p, always bears

the same relation to r that r* bears to the area in any given
case whether r is an index of capacity or of length of peri-
meter or not.

20 Formulas in Which C is Used Instead of m.

In the tables of coefficients heretofore developed the
values of both m and C were given in order that either form
of the formula might be applied in any case at pleasure. All
the formulas using the coefficient C instead of m may be de-
duced from the following:

v=C t/r !/r /S (67)

v=C f/r t/rS (68)

v=C e/i 8 i/S=C 1 /S7' :I " ( 69 )

v=C t/d !/d v/S (70)

v=C e/d !/dS". (71)

v=C t/d 8 1 /S=C 1 /S7d r , (72)

v=C S/d 8 !/?-!// : (73)

v=C /r 3 /H -r- i/l (74)

Area in square feet, A = d 8 X -354

Area in square feet, A = r 8 X 12.5664

q=Av. The same limitations mentioned in the preced-

106 SULLIVAN'S NEW HYDRAULICS

ing section will also be observed in the formulas in this form
relating to q and r in open channels.

FORMULAS IN TERMS OF DIAMETER IN FEET USING C.
By transposition in formula (72) we have,

v*-c t/d 8 "

"(YD)

J -CVd 8

r /"HZ v

(?())

~V S!/d 8 ~ ^3X^8
M

(70

2 Xi/d 8
C 2 H v/d 8

(78)

q = d s X -7854 XCX t/d 8 X i/S = t/d 11
X-7854 (80)

(79)
v/S X

t/a q di' q4

C^S-X.7854' S 2 C*.3805

, 11 / q 4

21. Formulas in Terms of Hydraulic Radius in Feet
Using C.

By transposition in formulas (69) and (74) we have

(82)

C =l/S7F

C H

q=Av=12.5664Xr 8 XCX*/r 8 X

t/r^xcysxia.sees

SULLIVAN'S NEW HYDRAULICS, 107

11 / q 4

I/ 24936.958XC 4

" 157.9144
I q 8

7.9144 " " <(90)

C* 1 /r^ 157.9144 "
|= 157.9144 y"0'H ........................ (92)

C== 125664 ,/SX t/r 11

A set of formulas in terms of pressure in pounds per
square inch and diameter in feet or inches may be deduced
in like manner from equation (73). It is not deemed neces-
sary to deduce the formula in all its possible forms and
terms, as that is a simple matter which may be performed
at the pleasure of the person using it, and would require un-
necessary space here.

22. Special Formula for Vertical Pipes. Because
of the relation of H to I in all formulas the ordinary
formulas for flow will not apply to a pipe in a vertical
or nearly vertical position. In such case H and I in-

crease at tne same rate, and hence -y=l, regardless

of the head or length. On account of this fact all the
formulas of the different writers on hydraulics will give
the same velocity for a head of a hundred feet as for a head
of 1,000 feet. It is therefore necessary to use a special form-
ula in such case. In a vertical pipe the water is supported
at no point whatever by any portion of the pipe walls. The
effect of gravity is not impeded except by the roughness of
the pipe walla. In such vertical pipe there is a gain of one
foot head for each foot of length. The resistance to entry
and the pipe wall friction will be the only loss of head. Hence

108 SULLIVAN'S NEW HYDRAULICS.

if the sum of their effects be deducted from the total head,
the velocity should equal that due to the remainder of the
head. On this theory the following formula is proposed:

TH- (-=-

The head, slope, velocity, or quantity may be found by
the principles given in 52, and table No. 24.

CHAPTER IV.

Of Tables for Rapid Calculation of Velocity and Discharge
in Open and Closed Channels, Friction Loss, &c.

23. Table for Velocity and Discharge. Clean, Average
Weight Cast Iron Pipes, Not Coated. In tables No. 1 and
No. 2 the diameters are given in inches, the areas in square
feet, and the discharge in cubic feet per second.
How to use Tables No. 1 and No. 2.

To find the mean velocity in feet per second: Multiply
the quantity in column No. 5 opposite the given diameter in
inches by ,/sT For v/S^ see table No. 15, 30. For S, see
Table No. 16, 31.

To find the discharge in cubic feet per second: Multiply
the quantity in column No. 6 opposite the given diameter by
,/S.

v=C X V& v/~ST q = AV= AC X t/d s " l /sT Take
d in inches.

For average weight clean cast iron pipe, C = 7.756 when
d = inches.

SULLIVAN'S NEW HYDRAULICS.

TABLE. No. 1.
Clean cast iron pipe, not coated.

109

Col. 1
Diam.
Inch's

Col. 2
Inches

Col. 3

t/d 8

Inches

Col. 4
Area
Sq. Feet

Col. 5
For
Velocity

Col. 6
For
Discharge
ACXK d8

Vt
&

0.35355
0.65227

0.5946
0.8059

.001366
.003068

4.6117
6.2505

.006300
.019176

1.00

.005454

7.7560

.042301

1.34

1.3975

1.1820

.008522

9.1675

.078125

1.8360

1.3550

.01227

10.5093

.128949

i'.

2.3152

1.5210

.01670

11.7968

.197006

2.8284

1.6810

.02232

13.0378

.291003

5.1961

2.2790

.04909

17.6759

.867709

2.8284

.08726

21.9370

1.91422

11.1803

3.3439

.13630

25.9352

3.53496

14.6969

3.8340

.19635

29.7365

5.83876

18.5202

4.3040

.26730

8.92295

22.6274

4.7570

.34910

36 '.8952

12.88014

27.

5.1960

.44180

40.3001

17.80458

31.6227

5.6231

.54540

43.6127

23.78636

36.4828

6.0400

.66000

46.8462

30.91849

41.5692

6.4470

.7854

50.0029

39.27000

46.8721

6.8460

.9218

53.0975

48.94527

52.3832

7.237

1.069

56.1301

60.00307

58.0747

7.622

1.227

59.1162

72.53557

64.

1.396

62.0480

86.61900

70.0927

8.372

1.576

64.9332

102.3347

76.3675

8.738

1.767

67.7719

119.7529

82.8190

9.100

1.969

70.5796

138.8300

89.4427

9.457

2.182

73.3484

160.0462

96.2340

9.810

2.405

76.0863

182.9876

103.189

10.158

2.640

78.7854

208.0000

110.304

10.504

2.885

81.4690

235.0380

117.575

10.844

3.1416

84.1060

264.2274

125.

11.180

3.409

86.7120

295.6012

132.574

11.514

3.687

89.3025

329.2583

140.296

11.844

3.976

91.8620

365.2433

148.162

12.172

4.276

94.4060

403.6800

156.169

12.496

4.587

96.9189

444.5670

164.316

12.820

4.909

99.4319

488.1112

172.600

13.139

5.241

101.9061

534.0898

181.0193

13.456

5.585

104.3647

582.8768

189.5705

13.768

5.940

106.7846

634.3005

216.0000

14.698

7.069

113.9977

805.8497

252.8822

15.907

8.726

123.3747

1076.5676

291.8629

17.086

10.558

132.5190

1399.1356

332.5537

18.237

12.567

141.4461

1777.5531

396.8173

19.920

15.905

154.4995

2457.3145

464.7580

21.560

19.635

167.2193

3283.3509

606.9402

24.710

29.607

191.6507

5674.2022

769.8727

27.746

38.484

215.1979

8281.6759

940.6040

30.670

50.265

237.8765

11956.8622

120

1314.5341

36.250

78.540

281.1550

22081.9137

REMARK. In large cast iron pipes, or in thick small pipes
there is great liability to blow holes and rough places. The

110

SULLIVAN'S NEW HYDRAULICS.

thicker the pipe shell is, the more liable it is to be rough. It
might be well to take C=7.65 in terms of diameter in inches
for cast iron pipes of 48 inches diameter or greater, and for
other and smaller diameters that are equally thick as 48 inch
pipe. Large pipes are never as perfect or as smooth as med-
ium diameters and thicknesses.

This fact has led some engineers to conclude that the law
of friction in pipes was slightly different in large pipes from
what it is in medium diameters. It is claimed that this
change occurs at about a diameter of 48 inches. It is due
simply to the rougher casting of large pipes which require
thickness. There is no change in the law of friction at any
diameter whatever. Very small cast iron pipes are also cast
thick to prevent breakage in handling and are usually as
rough as the very large diameters. Pipes less than six inches
diameter and over 36 inches diameter, are usually rougher to
some extent than the intermediate diameters.

q=A ~~

TABLE No. 2.

Asphaltum coated pipes. C=8.b7, d=inches.

Col. 1

Col. 2

Col. 3

Col. 4

Col. 6

Diam

s~fta

$/~d*

Area

cxt/in^

ACX/7F*-

Inch's

Inches

Sq.

For

Feet

Velocity.

Discharge

14.6969

3.834

.19635

33.2407

6.42681

18.5202

4.304

.2673

37.3156

9.97445

22.6274

4.757

.3491

41.2432

14.39800

27.

5.196

.4418

44.9493

19.85860

31.6227

5.623

.5454

48.7522

26.58945

36.4828

6.040

.6600

52.3668

34.56208

41. 5692

6.447

.7854

55.8955

43.90032

46.8721

6.846

.9213

59.3548

54.71325

52.3832

7.237

1.069

62.7448

67.07419

58.0747

7.622

1.227

66.0827

81.08347

64.

1.396

69.3600

96.82656

70.0927

8.372

1.576

72.5852

lit. 39427

76.3675

8.738

1.767

75.7584

133 . 86509

82.8190

9.100

1.969

78.8970

155.34819

89.4427

9.457

2.182

81.9922

178.90698

96.2340

9.810

2.405

85. '527

204.55174

103.189

10.158

2.640

88.0698

232.50443

110.304

10.504

2.885

91.0697

262.73603

117.575

10.844

3.1416

94.0L74

295.36541

125.

11.180

3.409

96.9306

330.43641

132.574

11.514

3.687

99.8264

368.05986

140.296

11.844

3.976

102.6874

407.88542

148.162

12.172

4.276

105.4312

450.82398

156.169

12.496

4.587

108.3403

496.957C4

164.316

12.820

4.909

111.1494

545.63240

SULLIVAN'S NEW HYDRAULICS.

Ill

TABLE No. 2 Continued.

Col. 1

Diam.
Inch's

Col. 2
v/d 8
Inches

Col. 3

e/d s

Inches

Col. 4
Area Sq.
Feet.

Col. 5
CXt/d 8
Velocity

Col. 6

ACXf/d 8
For Disch'g.

172.600

13.139

5.241

113.9151

597.02919

181.0193

13.456

5.585

116.6635

651.56576

189.5705

13.768

5.940

119.4685

709.64324

198.2523

14.081

6.305

122.0822

769.72871

35 '

207.U62S

14.390

6.681

124.7613

833.52624

216.

14.698

7.069

127.431B

900.80267

225.0622

15.002

7.467

130.0673

963.93283

234.2477

15.300

7.876

132.6510

1044.75927

252.8822

15.907

f-OC

8. /26

137.9137

1203.43495

291.8629

17.086

10.558

148.1356

1564.01587

332.5537

18.237

12.567

158.1148

1987.02869

3%. 8173

19.920

15.905

172.7064

2746.89529

464.7580

21.560

19.635

186.9252

3670.27630

6f!94<r>

24.710

29.607

214.2357

6342.87637

769.8727

27.746

38.484

240.5478

9257.24164

940.6040

80.670

50.265

265.9089

1S365. 41086

REMARK This table relates to asphaltum coated pipes
not to pipes coated with coal tar, nor to compound coatings
made of only one part asphaltum. What is meant by as-
phaltum coated pipes is that class of pipes which have been
properly coated with a compound composed of 18 to 20 per
cent of crude petroleum and the remainder of asphaltum.
The coating compound to be heated to 300 degrees, Fahr.,
and the pipe to remain submerged in the hot bath until the
pipe metal attains the same temperature as that of the bath.
Coal tar coatings do not form quite as smooth a surface as
the above described coating, and hence do not develop as
high values of C. If d is taken in feet, then m =.00032, and
C=55.90 as the average value of the coefficients for asphal-
tum and oil coated pipes. The value of C or m will vary
slightly with the quality or purity of the asphaltum used.
(See group No. 2.)

24. Table for Velocity and Discharge of Brick Lined
Circular Conduits or Sewers Flowing Full. In the follow-
ing Table No. 3 the diameters are in feet, the areas in square
feet, and the discharge in cubic feet per second. The coeffi-
cient is in terms of diameter in feet and is based upon the
discharge of Washington, D. C., aqueduct,. (See Group 6.)

112

SULLIVAN'S NEW HYDRAULICS.

;or v=

m=.0008577; C=

34.00 in terms of diameter in feet. q=A CXt/d* >/!

TABLE No. 3
Circular brick conduits and sewers. C=34.00.

Col. 1
Diana.
Feet

Col. 2
v/d 3
Feet

Col. 3

t/d s
Feet

Col. 4
Area
Sq Feet

Col. 5
For Vel.
CXt/d"

Col. 6
For Disch'g
ACXt/d s

1.50

1.837

1.355

1.767

46.070

81.4057

2.00

2.8-28

1.681

3.142

57.154

179.5778

2.50

3.953

1.988

4.909

67.592

331.8091

3.00

5.196

2.279

7.068

77.486

537.6711

4.00

2.828

12.566

96.166

1208.8016

5.00

11.180

3.344

19.635

113 6%

2232.4210

6.00

14.697

3>34

28.274

130.356

3685.6855

7.00

18.5^0

4.304

38.485

146.336

5631.7410

8.00

22.627

4.757

50 266

161.738

8129.9223

9.00

27.

5.1%

63.617

176.664

11238.8337

10.00

31.623

5.623

78.540

191.182

15015.4343

11.00

36.4b3

6.040

95.033

205.360

19515.9769

12.00

41.569

6.447

113.100

219.198

24791.2938

13.00

46.872

6.846

132.730

232 764

30894 . 7657

14.00

52.38<

7.237

153.940

246.058

37878.1685

NOTE. Compare the values of the coefficients of the new
Croton aqueduct for a depth of 9 feet with those of the Wash-
ington aqueduct both in group No. 6. The above value of
C in terms of diameter in feet is about correct for plain brick.

25. Egg Shaped Brick Sewers and Conduits. IK egg

shaped sewers the vertical diameter is one and one-half times
the horizontal or greatest transverse diameter. Radius of in-
vert, % vertical diameter. Radius of sides equal vertical diam-
eter. Let d=greatest transverse diameter in feet.

a=area in square feet.

p=wetted perimeter in lineal feet.

r=hydraulic mean depth=L
P
Then, in egg shaped sewers and conduits,

a=d*X-284 for \ full depth

a=d 8 X-755825 for f full depth.

a=d*Xl-148525 for full depth.
The wet perimeter in lineal feet will be,

p=dXl-3747 for \ full depth.

SULLIVAN'S NEW HYDRAULS. 113

p=dX23941 for | full depth.
p=dX3.965 for full depth.

The mean hydraulic depths, ' r~=r, will be,

r=dX-2066 for full depth.

r=dX-3157 for | full depth.

r=dX-2897 for full depth.

See "Hydraulic Tables" by P. J. Plynn; Van Nostrand'e Sci-
ence Series No. 67, and also see "Treatise on Hydraulics" by
Prof. Merriman, p. 235. (5th. Edition.)

TABLE FOR VELOCITIES AND DISCHARGES OP EGG SHAPED

BRICK CONDUITS AND SEWERS PLOWING TWO-THIRDS

FULL DEPTH.

As this class of conduits are not circular in form, the
coefficient is in terms of hydraulic mean depth (r) in feet,
and the value of the coefficient used in the following table
is that developed by the Washington, D. C., aqueduct, (See
Group No. 6). This table is to be used in the same manner
as Tables Nos. 1,2 and 3.

v=CX t/r 8 ,/S, and q=ACXt/r 8 i/S. C=96.00
TABLE No. 4.

Areas, hydraulic depths, velocities and discharges for %
Full Depth. 0=96.

Col. 0.

Col. ICol. 1

Col. 2

Col. 4

Column 5.

Column C

Trans.
Diam.

r
Feet

j/r'

Feet

Area
Sq.Ft

For velocity
CXKr 8

For Dischg
AC X V**

1.50

0.474

0.3263

0.5713

1.701

54.8443

93.2910

2.00

0.631

0.5'U2

0.70.O

3.025

67.9680

205.6032

2.50

0.789

0.7008

0.8372

4.724

80.3712

379.3535

3.00

0.947

0.9216

0.0600

6.802

92.1600

626.8723

3.50

1.105

1.1615

1.0780

9.259

103.4880

958.1954

4.00

1.263

1.419

1.19LO

12.093

114.3360

1382.6653

4.50

1.421

1.694

1.302

15.305

124.9920

1913.0025

5.00

1.579

J.984

1.408

18.895

135.1680

2555 . 3994

6.00

1.894

2.606

1.614

27.210

154.9440

4216.0262

7.00

2.210

3.285

1.812

37.035

173.9520

6442.3123

8.00

2.526

4.015

2.004

48.373

192.3840

9306.1912

9.00

2.841

4.789

2.188

61.222

210.0480

12859.5586

10.00

3.157

5.610

2.368

75.583

227.3280

17182.1322

11.00

3.473

6.472

2.544

91.455

244.2240

22335.5059

Small sewers should be circular in form. See 55.

114 SULLIVAN'S NEW HYDRAULICS.

26 Formulas for Use in Connection With the Fore-
going Tables.

In the tables for pipes and conduits are the tabular val-
ues of

CXKd 8 , CX^r 8 , A C XS/d* and A
Now

If the slope and mean velocity have been decided upon,

then the value of CX^ / d 8 = TQ- and opposite this value of

is the required diameter to generate the given veloc-
ity.

If a given diameter is required to discharge a given
number of cubic feet per second, then the grade or slope may
be found thus:

/ q

V b A CXf/d

The grade to generate a given velocity in feet per second
may be found thus:

If the quantity to be discharged and the grade are given,
then the required diameter will be found thus:

ACXv/d^yg. Look for the diameter which corresponds

to the value of A CXv/'d 8 in the table.

The general formulas already given are so simple that re-
sort to these formulas is not necessary.

SULLIVAN'S NEW HYDRAULICS. 115

27 General Table of Values of r or d, With Roots-

TABLE No. 5

r or d

V^r or^/d

y'r 8 or v/d

^'r or t/d

0.20

0.4472

0.089440

0.2990

.22

.4690

.103180

.3212

.24

.4899

.117576

.3429

.26

.5099

.132574

.3641

.5291

.5477

.148148
.164310

.3849
.4053

.32

.5656

.180992

.4244

.34

.5831

.19S254

.4452

.36

.6000

.216000

.4647

.38

.6164

.234232

.4840

.40

.6324

.252960

.5030

.42

.6481

.272202

.5217

.44

.6633

.291852

.5402

.46

.6782

.311972

.5585

.48

.6928

.332544

.5767

.50

.7071

354550

.5946

.52

.7211

.374972

.6124

.54

.7348

.396792

.6299

.56

.74<<3

.419048

.6473

.58

.7616

.441728

.6646

.60

.7746

.464760

.6817

.62

.7874

.488188

.6987

.64

.512

.7155

.66

.8124

.536184

.7322

.68

.8246

: 560728

.7488

.70

.8366

! 585620

.7653

.72

.8485

.610920

.7816

.74

.8602

.636548

.7978

.76

.8718

.662568

.8139

.78

8832

.688896

8300

.80

!8944

.715520

.8459

.82

.9055

.742510

.8617

.84

.9155

.769020

.8774

.86

.9273

.797478

.8930

.88

.9380

.825440

.9086

.90

.9487

.853830

.9240

.92

.9591

.882872

.9394

.94

.96

.9695
.9798

.911330

.940308

.9546
.9698

.98
1.00

.9899
1.

.970102
1.

.9849
1.

1.02

1.010

1.0302

1 015

1.04

1.020

1.0600

1.029

1.06

1.029

1.0907

1.045

1.08

1.039

1.1521

1 059

.10

1.049

1.1540

1.074

.12

1.058

1.1850

1.089

.14

1.068

1.2175

1.104

.16

1.077

1.2 94

1.118

.18

1.086

1.2815

1.132

.20

1.095

1.3140

1.146

.22

1.104

1.3469

1.160

.24

1.114

1.3800

1.175

.26

1.123

1.4150

1.189

116 SULLIVAN'S NEW HYDRAULICS.

TABLE No. 5. Continued.

r or d

y/r or i/d

v/r 8 or !/d

e/r" or f/d

1.28

131

1.4480

.203

1.30

.144

1.4872

.217

1.32

.150

1.5180

.239

1.35

.161

1.5673

.252

1.40

.183

1.6562

.287

1.45

.214

1.7603

.326

1.50

.225

1.8375

.355

1.55

.245

1.9297

.399

1.60

.265

2.0240

.422

1.65

.284

2.1186

.455

1.70

.304

2.2168

.488

1.75

.323

2.2852

.511

1.80

.341

2.4138

.554

1.85

.360

2.5160

.568

1.90

.378

2.6182

.618

1.95

.396

2.7122

.647

.414

2.8284

.663

2.05

.431

2.9335

.713

2.10

.459

3.0639

.750

2.15

.466

3.1519

.775

2.20

.483

3.2626

.806

2.25

.500

3.3750

.837

2.30

.526

3.5098

.873

2.35

.533

3.6025

.904

2.40

.549

3.7176

.928

2.45

.565

3.8342

1.958

2.50

.581

3.9525

1.988

2.55

.597

4.0723

2.018

2.60

.612

.1912

2.047

2.65

.638

.3407

2.083

2.70

.643

.4361

2 106

2.75

.658

.5595

2.135

2.80

.673

.6844

2.164

i:S

.688
.703

.8108
.9387

2.193
2.222

2.95

.717

5.0651

2.250

.732

5.1960

2279

3.05

.746

5.3253

2.308

3.10

.761

5.4591

2.336

3.15

.775

5.5912

2 364

3.20

: .789

5.7248

2.392

3.25

5.8597

2.421

3.30

!816

5.9928

2.448

3.35

.830

6.1305

2.476

3.40

.844

6.2696

2.504

3.45

.857

6.4066

2.531

3.50

.871

6.5485

2.559

3.55

.884

6.6882

2.586

3.60

.897

6.8292

2.613

3.65

.910

6.9715

2.640

3.70

.923

7.1151

2.667

3.75

.936

7.26

2.694

3.80

.949

7.4062

2.721

3.85

.962

7 5537

2.746

3.90

.975

7.7025

2.775

3.95

1.987

7.8486

2.801

SULLIVAN'S NEW HYDRAULICS.

TABLE NO. 5. Continued.

117

r or d

y'r or y'd.

,/f 8 or ,/d 8

t/r 8 or e/d 8

2.828

4.05

2.012

8.1486

2.854

4.10

2.024

8.2984

2.881

4.15

2.037

8.4535

2.908

4.20

2.049

8.6058

2.933

4.25

2.061

8.7592

2.960

4.30

2.073

8.9139

2.986

4.35

2.085

9.0698

3.012

4.40

2.097

9.2268

3.037

4.45

2.109

9.3850

3.064

4.50

2.121

9.5445

3.089

4.55

2.133

9.7051

3.115

4.60

2.144

9.8624

3.140

4.65

2.156

10.0254

3.165

4.70

2.168

10.1896

3.192

4.75

2.179

10.3502

3.218

4.80

2.191

10.5168

3.243

4.16

2.202

10.6797

3 268

4.90

2.213

10.8437

3.293

4.95

2.225

11.0137

3.319

2.236

11.1800

3.344

5.05

2.247

11.3473

3.369

5.10

2.258

11.5158

3.393

5.15

2.269

11.6853

3.419

5.20

2.280

11.8560

3.444

5.25

2.291

12.0277

3 468

5.30

2.302

12.2006

3 493

5.35

2.313

12.3745

3.518

5.40

2.823

12.5442

3.542

5.45

2.334

12.7203

3 567

5.50

2.845

12.8975

3.592

5.55

2 356

13 0758

3 616

5.60

2.366

13.2496

3^640

5.65

2.377

13.4300

3 665

5.70

2.388

13.6116

3.689

5.75

2.398

13.7885

3.713

5.80

13.9664

3 737

5.85

2 412

14.1453

3.761

5.90

2^429

14.3311

3.786

5.95

2.439

14.5120

3 810

2.449

14.6940

3.834

6.05

2.460

14.8830

3 858

6.10

2.470

15.0670

3 881

6.15
6.20

2.480
2.490

15.2520
15.4380

31905
3 929

6.25

2.500

15.6250

3 953

6.30
6.35

2.510
2.520

15.8130
16.0020

3.977
4.000

6.40

2.530

16.1920

4 024

6.45

2.540

16.3830

4 047

6.50

2.550

16.5750

4.071

6.55

2.560

16.7680

4 094

6.60

2.569

16.9554

4l 118

6.65

2.579

17 . 1503

4 140

6.70

2.588

4 164

6.75

2.598

1715365

4.188

118 SULLIVAN'S NEW HYDRAULICS.

TABLE No. 5 Continued.

r or d

l/r or \/d

j/r* or -j/d 8

V* or yd*

6.80

2.607

17.7276

4.211

6.85

2.617

17.9264

4.234

6.90

2.627

18.1263

4.257

6.95

2.636

18.3202

4.280

2.645

18.5150

4.304

7.05

2.655

18.7177

4.327

7.10

2.665

18.9215

7.15

2.6H

19.1191

4^373

7.20

2.683

19.3176

4.395

7.25

2.692

19.5170

4.418

7.30

2.702

19.7246

4441

7.35

2.711

19.9258

4.464

7.40

2.720

20.1280

4.487

7.45

2.729

20.3310

4.510

7.50

2.739

20.5425

4.532

7.55

2.748

20.7474

4.555

7.60

2.756

20.9456

4.578

7.65

2.766

21.1600

4.600

7.70

2.775

21.3675

4.622

7.75

2.784

21.5760

4.642

7.80

2.793

21.7854

4.668

7.85

2.802

22.0000

4.690

7.90

2.811

22.2069

4.712

7.95

2.819

22.411Q

4.735

2.828

22.6240

4.759

8.05

2.837

22.8378

4.779

8.10

2.846

23.0526

4.801

8.15

2.855

23.2682

4.823

8.20

2.864

23.4848

4.846

8.25

2.872

23.6940

4.868

8.30

2.881

23.9123

4.890

8.35

2.890

24.1315

4.912

8.40

2.898

24.3432

4.934

8.45

2.907

24.5641

4.950

8.50

2.915

24.7775

4.978

8.55

2.924

25.

8.60

2.932

25.2152

5.013

8.65

2.941

25.4396

5.044

8.70

2.949

25.6563

5 066

8.75

2.958

25.8825

.088

8.80

2.966

26.1008

.109

8.85

2.975

26.3300

.131

8.90

2.983

26.5487

.153

8.95

2.992

26.7784

.175

27.

196

9.05

3.008

27.2224

'217

9.10

3.016

27.4456

5 239

9.15

3.025

27.6787

5 261

9.20

3.033

27.9036

5^283

9.25

3.041

28.1293

5.304

9.30

3.049

28.3*57

9.35

3.058

28.5923

5i348

9.40

3.066

28.8204

5 369

9.45

3.074

29.0493

5.390

9.50

3.082

29.2790

5.411

9.55

3.091

29.5190

5.433

SULLIVAN'S NEW HYDRAULICS.
TABLE No. 5 Continued.

119

r or d

y'r or y'd

y'r 8 or -j/d 8

f/r or f/d 8

9.60
9.65
9.70
9.75
9.80
9.85
9.90
9.95
10.

3.098
3 106
3.114
3.123
3.131
3.138
3.146
3.154
3.162

29.7408
29.9729
30.2058
30.4492
30.6838
30.9093
81.1454

31 ! 6200

5.454
5.475
5.496
5.518

5^560
5.581
5.602
5.623

28. Tables for Velocity and Discharge of Trapezoidal
Canals. In Pig. 1, let A, E, F, D, equal the width of the
water surface in feet. Let B C equal bottom width of canal
in feet, and E B or P C, equal greatest depth of water in
feet.

TO FIND THE AREA IN SQUARE FEET.

Multiply E D by E B, or F A by B 1 C. Or secondly: Add
together the width of water surface and the bottom width in
feet, and divide the sum by 2. Then multiply the quotient
by the depth F C or E B in feet. In either case the result
will equal the area in square feet.

TO FIND THE LENGTH A E OR F D IN FEET.

If the side elopes A B and D C are 1 to 1, then AE=E B

120 SULLIVAN'S NEW HYDRAULICS.

and P D=F C. If the side slopes are \y z horizontal to 1 vert,
ical, then A E=E BXl-50. If the side slopes are 2 horizontal
to 1 vertical, then A E=E BX2.00.

TO FIND THE WETTED PERIMETER IN LINEAL FEET.

The length of B C, or of the bottom width in feet, is, of
course, always known. It is, therefore, only required to find
the length in feet of the side slopes A B and D C which when
added to B C, will equal the total wetted girth or perimeter-
If the side slopes are 1 to 1, then the length A B or D C is
equal to the diagonal of a square, or equal to the depth of
water E BX 1.41421.

The length of either side slope for any rate of slope what-
ever is the same as the hypotenuse of a right angled triangle,

and A B=V(AE) 8 +(E B) 8 or D C=v/(F D) 8 +(P C) 8 .

Adding together the lengths in A B, B C, and C D, we
have the wetted perimeter (p) in feet. The hydraulic mean

depth in feet is then r= area ip Bquare feet =4
wet perimeter in feet p

In the following tables of trapezoidal canals the value of
the area in square feet, and the hydraulic mean depth r, and
of $/r 3 for each additional half foot depth of water in each
canal is given, so that the velocity and discharge for each
depth of flow may be readily ascertained. The value of m or
C will depend upon the material forming the wetted perimeter,
and the condition of the canal a& to good or bad repair. The
value of m or G may be selected from the tables of values de-
veloped in the groups of rivers and canals heretofore given.
The following tables show the area for each depth of water.
The discharge for any given depth will equal the area for that
depth multiplied by the mean velocity. The slope required to
generate any desired mean velocity in feet per second for any
depth of flow will be

SULLIVAN'S NEW HYDRAULICS.

121

The distance or length in feet (I) of canal in which there
must be a fall of 1 foot in order to generate a given mean vel-
ocity in feet per second may be found by the formula,

HOW TO USE THE FOLLOWING TABLES.

To find the mean velocity for any given depth, multiply

t/r 8 for that depth by <V S, or multiply t/r 8 by J

To find the discharge in cubic feet per second for any
given depth of flow, multiply t/r 3 by areaXCVS, or multiply

t/r 8 by areaX-J-. For values of v/S, see 30, Table No. 15.

\ m

tol.

Trapezoidal canal.

TABLE No. 6.
Bottom width 2 feet.

Side slopes 1

Depth of
Water in ft.

Area
Square Feet

Feet

V/r
Feet

t/r
Feet

1.00
1.50
2.00
2.50
3.00

3.00
5.25
8.00
11.25
15.00

0.6213
0.8400
1.0450
1.240
1.430

0.4897
0.7699
1.068
1.380
1.710

0.6998
0.8774
1.034
1.177
1.308

REMARK. In climates where the earth freezes in winter,
side slopes of earth will not stand if they are steeper than 1%
to 1 even in very firm earth. In lighter soil in frosty climates
the side slopes should vary from 1% to 1 to 3 to 1, according to
the nature of the soil.

122 SULLIVAN'S NEW HYDRAULICS.

TABLE No. 7.
Trapezoidal canal. Bottom width 4 feet. Side slopes 1 to 1.

Depth of
Water in ft.

Area
Square Feet

Feet

V* 3
Feet

t/r
Feet

l.0

5.00

0.732

0.6263

0.7914

1.50

8.25

.000

1.000

.000

2.00

12.00

.2426

1.385

.176

2.50

16.25

.4700

1.782

.335

3.00

21.00

.762

2.339

.530

.50

26.25

.888

2.594

.611

.00

32.00

2.089

3.019

.737

.50

38.25

2.280

3.443

1.855

.00

45.00

2.480

3.905

1.976

.50

52.25

2.670

4.383

2.089

6.00

60. 00

2.861

4.836

2.199

6.50

68.25

3.050

5.327

2.308

7.00

77.00

3.235

5.819

2.412

8.00

96.00

3.605

6.844

2.616

TABLE No. 8.
Trapezoidal canal. Bottom width 4 feet. Side slopes 1*^ to 1.

Depth of
Water in ft

Area Square
Feet

Feet

3/r

Feet.

f/r"
Feet.

1.00

5.500

0.7231

0.6149

0.7841

1.50

9.375

.0000

1.0000

.000

2.00

14.000

.2488

1.3960

.181

2.50

19.375

.4880

1.815

.347

3.00

25.500

.7200

2.256

.502

3.50

32.375

.8370

2.490

.578

4.00

40.000

2.1710

3.099

.788

4.50

48.375

2.4000

3.718

.928

5.00

57.500

2.6100

4.216

2.053

6.00

78.000

3.0420

5.306

2.303

7.00

101.500

3.4730

6.472

2.544

8.00

128.000

3.9000

7.702

2.775

Trapezoidal canal.

TABLE No. 9.
Bottom width 6 feet.

Side slopes 1 to 1.

Depth of
Water in ft.

Area Square
Feet

Feet

Felt.

Feet.

1.00

7.00

0.7929

0.7060

0.8403

1.50

11.25

1.0980

1.1510

.0725

2.00

16.00

1.3726

1.6090

.2680

2.50

21.25

1.6280

2.077

.4420

3.00

27.00

1.8639

2.545

.595

3.50

33.25

2.0900

3.022

.738

4.00

40.00

2.3100

3.511

.874

4.50

47.25

2.5200

4.000

2.000

5.00

52.00

2.5800

4.144

2.036

5.50

63.25

2.9340

5.025

2.242

6.00

72.00

3.1345

5.548

2.355

7.00

91.00

3.5270

6.624

2.574

8.00

112.00

3.9100

7.731

2.780

SULLIVAN'S NEW HYDRAULICS.

123

Trapezoidal canal.

TABLE No. 10.
Bottom width 8 feet.

Side elopes 2 to 1.

Depth of
Water Feet

Area
Square Feet

Feet

V/r'
Feet

Feet

1.00

10.00

0.8018

0.7179

0.8473

1 50

16.50

1.1210

1.187

1.089

2.00

24.00

1.4164

1.685

1.298

2.50

32.50

1.7000

2.216

1 489

3 00

42.00

1.9600

2.744

1.656

3 50

52.50

2.2200

1.819

4.00

64.00

2.470

3 882

1.970

4.50

76.50

2.720

4^486

2.118

5.00

90.00

2.964

5.103

2.259

6.00

120.00

3.445

6.394

2.528

7.00

154.00

3.910

7.732

2.781

Trapezoidal canal.

TABLE No. 11.
Bottom width 8 feet.

Side slopes 1 to 1.

Depth of
Water Feet

Area
Square Feet

r
Feet

fr/r
Feet

1.00

9.00

0.831

0.7576

0.8704

1.50

14.25

1.164

1.256

1.121

2.00

20.00

1.464

1.771

1.331

2.50

26.25

1.741

2.297

1.516

3.00

33.00

2.000

2.828

1.682

3.50

40.25

2.248

3.370

4.00

48.00

2.485

3.917

1.979

4.50

56.25

2.710

4.461

2 112

5.00

65.00

2.935

5.029

2^242

5.50

74.25

3.153

5.598

2.366

6.00

84.00

3.364

6.170

2.484

7.00

105.00

3.777

7.341

2.709

Trapezoidal canal.

TABLE No. 12.
Bottom Width 10 feet.

Side elopes 1 to 1

Depth of
Water Feet

Area
Square Feet

Fe'et

Feet

1.00

11.00

O.a574

0.7939

0.891

2.00

24.00

1.5320

1.896

1.377

2.50

31.25

1.8300

2.475

1.574

3.00

39.00

2.1100

3.065

1.750

3.50

47.25

2.3743

3.658

1.912

4.00

56.00

2.6270

4.258

2.063

4.50

65.25

2.8700

4.862

2.205

5.00
5.50

75.00
85.25

3.1060
3.3350

5.474
6.090

2^468

6.00

96.00

3.5600

6.717

2.591

7.00

119.00

3.9930

7.979

2.825

8.00

144.00

4.4130

9.271

3.045

124

SULLIVAN'S NEW HYDRAULICS.

Trapezoidal canal.
2tol.

TABLE No. 13.
Bottom width 10 feet.

Side elopes

Depth of
Water Feet

Area
Square Feet

Fe^t

Feet

t/r"
Feet

1.00

12.00

0.8222

0.7455

0.8634

2.00

28.00

1.478

1.797

1.340

2.50

37.50

1.774

2.363

1.537

3.00

48.00

2.049

2.933

1.713

3.50

59.50

2.320

3.534

1.880

4.00

72.00

2.581

4.146

2.036

4.50

&5.50

2.840

4.786

2.187

5.00

100.00

-3.093

5.440

2.332

5.50

115.50

3.340

6.104

2.471

6-00

132.00

3.584

6.785

2.605

7.00

168.00

4.068

8.493

2.914

8.00

208.00

4.543

9.683

3.111

29. Table for Velocity and Discharge of Rectangular
Channels, Flumes, Masonry Conduits etc. The value of the
coefficient to be used with the following table will depend
upon the nature and condition of the lining of the flume or
channel. According to the experiments of D'Arcy and Bazin,
the average value of C for unplaned board flumes, well joint-
ed, and without strips or battens on the inside is C=U9.00, or
m=.00007. For nicely dressed lumber flumes, well jointed
and without battens on the inside, their experiments give C
=128.00 as an average. If we refer to the last two flumes in
Group No. 5, one at Boston gives C=106.30, and the High-
line in Colorado gives C=70.00. The data of flow in wooden
conduits are very unsatisfactory. The density of the wood,
the closeness of joints, the alignment of the flume, gritty de-
posits etc.. all affect the value of C in any case. Where the
flume is constructed of rough, very knotty, lumber and has
battens on the inside to cover the joints, it is proable that

SULLIVAN'S NEW HYDRAULICS.

125

the value of the coefficient will be about C=80.00, if the
alignment of the flume is fairly direct. For channels lined
with brick, ashlar, rubble etc., see the groups of such chan-
nels for value of C. See table No. 15 for value of /S.

TABKE No. 14
Flumes and other rectangular channels.

Width
Feet

Depth of
Water feet.

Area

Sq. Feet.

Feet.

t/r3
Feet.

1.5

0.50

0.75

0.300

0.1643

0.4054

1.5

1.00

1.50

.429

0.2810

0.5301

2.0

0.75

1.50

.429

0.2810

0.5301

2.0

1.50

3.00

.600

0.4647

0.6817

3.0

1.00

3.00

.600

0.4647

0.6817

3.0

1.50

4.50

.750

0.6495

0.8059

3.0

2.00

6.00

.860

0.7975

0.8930

4.0

1.50

6.00

.860

0.7975

0.8930

4.0

2.00

8.00

1.000

1.0000

5.0

1.50

7.50

.937

0.9070

0.9524

5.0

2.00

10.00

.111

.171

1.082

5.0

3.00

15.00

.363

591

.261

6.0

2.00

12.00

.200

.314

.147

6.0

2.50

15.00

.363

.591

.261

6.0

3 00

18.00

.500

.837

.355

6.0

4.00

24.00

.714

2.244

.498

8.0

3.00

24.00

.714

2.244

.498

8.0

4.00

32.00

2.000

2.828

.682

8.0

5.00

40.00

2.222

3.312

.820

8.0

6.00

48.00

2.400

3.718

1.928

10.0

4.00

40.00

2.222

3.312

1.820

10.0

5.00

50.00

2.500

3.953

1.988

10.0

6.00

60.00

2.727

4.503

2.122

10.0

7.00

70.00

2.916

4.979

2.231

12.0

4.00

48.00

2.400

3.718

1.928

12.0

5.00

60.00

2.727

4.503

2.122

12.0

6.00

72.00

3.000

5.196

2.279

12.0

7.00

84.00

3.230

5.805

2.410

14.0

5.00

70.00

2.916

4.979

2.231

14.0

6.00

84.00

3.230

5.805

2.410

30. Table of Values of Slopes S and ^S.

The distance in feet I, in which there is a fall of one foot is

126 SULLIVAN'S NEW HYDRAULICS.

TABLE No. 15.
Value of S and i/S.

Fall per
Mile in Feet

Fall
One In

Slope
S

0.50

10560.00

.0000947

.009731

0.75

7042.25

.0001420

.011915

1.00

5280.00

.0001894

.013762

1.76

3000.00

.0003333

.018255

2.00

2640.00

.0003788

.019463

2.64

2000.00

.0005000

.022361

3.00

1760.00

.0005682

.023836

3.18

1660.00

.0006024

.024544

3.30

1600.00

.0006250

.025000

3.38

1560.00

.0006410

.025318

3.52

1500.00

.0006667

.025820

3.62

1460.00

.0006849

.026171

3.70

1427.00

.0007007

.026472

3.75

1408.00

.0007102

.026650

3.80

1389.00

.0007199

.026832

8.85

1371.00

.0007294

.027007

3.90

1354.00

.0007385

.027176

4.00

1320.00

.0007576

.027524

.20

1257.00

.0007955

.028205

.40

1200.00

.0008333

.028868

.50

1173.00

.0008525

.029198

.60

1148.00

.0008710

.029514

.70

1123.00

.0008905

.029841

.75

1111.00

.0009000

.030001

.80

1100. 00

.0009090

.030151

4.90

1078.00

.0009276

.030457

5.00

1056.00

.0009469

.030773

5.10

1035.00

.0009662

.031083

5.20

1015.00

.0009852

.031388

5.28

1000.00

.0010000

.031623

6.00

880.00

.0011364

.033710

7.00

754.30

.0013258

.036411

8.00

660.00

.0015151

.038925

9.00

586.60

.0017044

041286

10 00

528.00

.0018940

.043519

11.00

480.00

.0020833

.045643

12.00

440.00

.0022730

.047673

13 00

406.10

.0024621

.049620

14.00

377.10

.0026515

.051493

15.00

352.00

.0028409

.053300

16 00

330.00

.0030303

.055048

17.00

310.60

.0032197

.056742

18.00

293.30

.0034090

058388

19.00

277 TO

.0035985

.059988

20 00

264.00

.0037878

.061546

21.00

251.40

.0039773

.063066

22.00

240.00

.0041666

.064549

23.00

229.60

.0043560

.066000

24.00

220.00

.0045454

.067419

25.00

211.20

.0047348

.068810

26.

203.10

.0049242

.070173

27.

195.20

.0051136

.071510

SULLIVAN'S NEW HYDRAULICS.
TABLE No. 15. Continued.

127

Fall Per
Mile in Feet

Fall,
One In

Slc g pe

51.

52.
52.80
55.80
60.00
70.00
80.00
90.00
100.00
120.00
140.00
160.00
180.00
200.00
240.00
280.00
320.00
360.00
400.00
450.00
500.00

700.00
800.00

188.60
182.10
176.00
170.30
165.00

155.30
150.90
146.60
142 70

135.40

132.00

128.80

125.70

122.80

120.00

117.30

114.80

112.30

110.00

107.70

105.60

103.50

101.50

100.00

96.00

88.00

75.43

6600

58.66

52.80

44.00

37.71

33.00

?933

26.40

22.00

18.86

16.50

14.66

13.20

11.73

10.56

880

7.54

.0053030

.U ).->-! '.KM
.0056818
.0058712

.0062500
.0064394

.0068182

.0070075

.0075757
.0077651
.0079545

.0085227
.0087121

.0098485

.01

.0104167

.0132576
.0151515
.0170455

.022727
.0265151
.0303030

.0378787
.0416667

.0681818
.0757575
.0852273

.1136364
.1325757
.1515151

.072822
.074111
.075378

.081417
.082572
.083711

.085944
.087039
.088120
.089188
.090244

.097312

.098281

.099241

.10

.102060

.106600

.115141

.137620
.150756

.174077
.184637
.194625

.261116

.275241

.307729
.337100
.364109

31 Table of Slopes tor Average Weight Clean Cast
Iron Pipes, Showing the Inclination Required in Each Di-
ameter to Generate a Mean Velocity of One Foot per Sec-

128 SULLIVAN'S NEW HYDRAULICS.

and, from which the Slope Required to Generate any
other Mean Velocity may be Found.

fHdt/d j f m tl3 j s genera i formula we assign H 1 foot
mv*

and v*=l foot, we have, 2=1^1 as the formula for finding

the length in which there must be a total head, fall or slope of
one foot to generate a velocity of one foot per second. For
this class of pipe m is a constant, and in terms of diameter in
feet m=.0004, or in terms of diameter in inches m=.0004X
V/(l?) 3 =.01662768. Hence the length in feet I, in which there
must be a head or fall of one foot in order to generate a mean
velocity of one foot per second will be

I = 1/dS if d is taken in feet, and s=
.0004

The length in feet in which there must be a head or fall
of one foot in order to generate any given or desired mean
velocity in feet per second is,

d-^/d i/d 8 m v* /m v*

In which v s is the square of the given or desired velocity
in feet per second. The coefficient m may be in feet or in
inches as above but the mean velocity will be in feet per
second in either case.

The required slope S, to generate a mean velocity of one
foot is,

m m

S= , j 3 , and to generate any velocity is S= , ~ 3 X v*.

Hence if the slope for any diameter, which causes v=l
be taken from the following table, the required slope to
cause any other velocity may be found at once by multiply-
ing this slope from the table by the square of the desired
velocity, v*.

SULLIVAN'S NEW HYDRAULS. 12

EXAMPLE.

From Table No. 16 it is seen that a slope of S=.0004 for
a pipe 12 inches diameter, will generate a mean velocity of
one foot per second. Required, the slope of a 12 inch pipe to
generate 5 feet per second velocity:

SOLUTION; From Table 16, take the slope for 1 foot
velocity, S .0004. Multiply this slope by the square of the
required velocity, and we have,

S=.0004X(o) 8 =.01,and l=-^-= ~= 100 feet. In other

words there must be a fall of one foot in a length of 100
feet.

TABLE No. 16.

Table giving the required slope to generate a mean
velocity of one foot per second in average weight clean cast
iron pipes.

.016628 1 T/d

For v=l, S=

Diameter
Inches

/d 8
Inches

Diameter
Inches.

T/d*
Inches

5.1961

.003200000

132.5740

.0001254242

.002078500

140.2960

.0001185208

11.1803

.001487250

148.1620

.oooir^x,

14.6969

.001131156

156.1690

.0001064743

18.5202

.0008975830

164.3160

. 000101 -i.XX)

22.6274

.000734861

172.6000

.OCO(i<v-: j ,:>:;:>

27.

.0006158&2

181.0193

31.6227

.000525825

189.5705

!OOOU8771. > I}

36.4828

.0004=57764

198.2523

.000083S0790

41.5692

.000400000

207.0628

.OOUO!S<-:'4]4

46.8721

.000354752

216.

.00007f.'.ixl(K)

52.3832

225.0822

.000073SXI70

58.0747

^ 000286320

234.2477

.00007098(80

64.

.000260000

252.8222

.00006.-75390

70.0927

.000237228

291.8629

76.3675

.0002177366

332.5537

JQBKOOOOM

82.8190

.0002007753

396.8173

. 00004 1'.':!41

89.4427

.0001859069

464.7580

.0000357777*5

96.2340

.0001727870

606.9402

.0000274. IO

103.1890

.0001611411

769.8727

.00002160000

110.3040

.0001516536

940.6040

:000017r,7,xm

117.5750

.0001414254

120

1314.5341

.00001264935

125.

.0001330240

130 SULLIVAN'S NEW HYDRAULICS.

32. Head in Feet Lost by Friction in Average Weight
Clean Cast Iron Pipes for Different Velocities of Flow.

By equation (10) the coefficient of resistance or friction is
b" di/d S"

(10)

From which the formula for head lost by friction h", is
n I v 8 n I v* n

h "=-d7o- = -7d^=v^x' v ' ................. < lb)

For a constant diameter and velocity the friction loss
will be directly as the length in feet ( I )of pipe, and will vary
as v* for different velocities. For constant degrees of
roughness of pipe n is a constant

As the friction loss is inversely as |/d 8 and directly as the
length and as v*, the loss in one foot length of any diameter

when v 2 =l, will be S" = / ^ 8 and for any other velocity it

u
will be S"= / j g X v>aQ d for any length in feet of pipe it

ii
will be /j 8 X*Xv*. Hence if we form a table which shows the

loss of head in feet for one foot length of pipe and for a velocity
of one foot per second, the loss for any other length in feet
will be found by multiplying the tabular quantity by the
given length in feet /, and the loss for any velocity will be
found by multiplying by the square of that velocity, v*. (See
9 and 10.)

TABLE No. 17.

Table showing the loss of head in feet by friction in one
foot length of clean cast iron pipe when v*=l.

SULLIVAN'S NEW HYDRAULICS. 131

.01637

-, when

When v z =l, the loss per foot length =- i =
d=incheg.

Diam.

Inchet

T/d 3
Inches.

Bead lost in
! eet per foot
length

Diam.
Inches

;/d 8
Inches.

Hd lost in ft
per ft length

5.1961

.003150440

125.

.00013U96000

.002046250

132.5740

.00 '12347820

11.1803

.0014641820

140.2960

.000116681710

14.6969

.0011138327

148.1620

.000110487170

18.5202

.0008839000

156.1690

.000104822340

22.6274

.0007234590

164.3160

.OOU0996251126

27.

.0006062963

172.6000

.(XX '094843569

31.6227

.0005176661

181.0193

.1 00(190435330

36.4828

.000448704CO

189.5705

.000' 186353094

41.5692

.0003937900

198.2523

.000082571551

46.8721

.00034924806

207.0628

.000079058190

52.3832

.000312:.(!177L'

216.

.000075787037

58.0747

.000281878341)

225.0622

.000072735444

64.

. 000255 7M -':>( i

234.2477

.000069883290

70.0927

. 0002335 47.V.M

252.8822

.0-10064 730000

76.3675

.000214358200

291.8629

.00005608800

82.8190

.00019766 OCO

332.5537

.0000492 '3100

89.4427

.000183022203

396.8173

.000041253239

96.2340

.00017016620(1

464.7580

.000035222632

103.1890

.00015Si;iU!'4(l

606.9402

.0000270000

110.3(MO

.00014810NW

769.8727

.000 212632556

117.5750

.000139230278

940.6D40

. (10001 74037000

REMARK. As the loss here tabulated is for one foot
length only and for a velocity of only one foot per second, none
of the decimals should be cut off especially in case the pipe is
of considerable length and the velocity is high, because the
losa increases directly as the number of feet in length and
also asv 8 .

How TO USE TABLE No. 17.

The table shows the loss of head in feet by friction for
oue foot length of pipe of each diameter, and for a velocity of
one foot per second. If the pipe is several hundred feet in
length, then move the decimal point two places to the right-
This will be equivalent to multiplying by 100, and will show
the loss of head in feet per 100 feet length for v"=l. Mul-
tiply this result by the square of the actual or proposed ve-
locity in feet per second and the result is the actual loss per
100 feet length for that velocity. If the pipe is several thousand

132 SULLIVAN'S NEW HYDRAULICS.

feet in length then take out from the table the loss for one
foot length and v 2 =1, and move the decimal point three
places to the right. Multiply by the square of the actual or
proposed velocity in feet per second. The result will be the
actual loss of head in feet per 1,000 feet length of pipe. The
loss of head in feet per mile (5280 feet) of pipe equals the loss
for 1,000 feet multiplied by 5.28.

EXAMPLE.

What is the loss of head in feet in an 8 inch cast iron
pipe 750 feet in length when the velocity is six feet per sec-
ond ?

SOLUTION.

In table 17, opposite a diameter of 8 inches and in the
third column the tabular loss for one foot length of 8 inch
pipe when v s =l is .000723459. Multiplying this by 100 feet
length by moving the decimal point two places to the right,
and the loss for 100 feet =.0723459 when v s =l. As the act-
ual velocity is to be six feet per second, and as the loss varies
as v 2 in any given diameter the last result must be multiplied
by (Q) s =36, and we have the actual loss per 100 feet length=
.0723459X36=2.60445 feet, and for 750 feet the loss will be
2 60445X? 5=19 5334 feet.

33. Formula and Table for Ascertaining the Loss of
Head in Feet In any Class of Pipe While Discharging a
Given Quantity In Cubic Feet Per Second.

Let h"= total head in feet lost by friction in the length
I

d=diameter of pipe in feet.

q= cubic feet per second discharged.

Then the coefficient of resistance is

hyd" X .616853 _ Syd"X.616853

Zq* q*

And the head in feet lost by friction is

T- < See e< l uation 32 ->

SULLIVAN'S NEW HYDRAULICS.

133

If I be taken =1 foot length of pipe, then n is constant
for any given class of pipe, and we may take the quotient of

n Q '

gifigK as a constant, which, when multiplied by yfp will

equal the loss of head in feet per foot of pipe for the given
discharge q. As q s ond V s are convertible terms we use the
same coefficient value in terms of either q or v.

The value of n in terms of diameter in feet for ordinary
cast iron pipes is n=.00039380.

The loss of head in one foot length is h"= fi1ft QgQ X

Then-

.00039380

!/d" ' '" .616853
h" = .OC063840 X:

.616853

=;.00063840. Whence

The following table gives

value of^/d 11 .

The slope required to cause a given diameter to discharge

q cubic feet, S= ^353 X^p

From tables Nos. 1 and 2, q=a cXt/d'Xv/S, and S=

TABLE No. 18.
" when d is taken in feet. (See 44. 45.)

Values of

Diameter
Inches

Diameter
Feet

i/d"

Feet

Diameter
Inches

Diameter
Feet

v/d"
Feet

0.2500

.0004883

2.00

45.25

0.3333

.002375

2.083

56.60

0.4167

.00811

2.166

70.17

0.5

.0221

2.25

86.50

0.5833

.05157

2.333

105.55

0.6667

.1075

2.416

128.00

0.75

.2055

2.50

154.40

0.8333

.3668

2.584

185.20

0.9167

.6198

2.666

219.90

1.0000

1.000

2.75

260.80

1.083

1.55

2.834

307.80

1.167

2.338

2.916

360.00

1.25

3.412

3.00

420.90

1.383

4.859

3.166

566.00

1.417

6.800

3.333

750.90

1.5

9.301

3.50

982.60

1.583

12.51

3.666

1268.00

1.667

16.62

4.00

2048.00

1.75

21.71

4.50

3914.00

1.833

28.01

5.00

6979.00

1.917

35.84

6.00

19050.00

134 SULLIVAN'S NEW HYDRAULICS.

For asphaltum coated pipes take n=. 000325 in terms of
diameter in feet. Then for such coated pipes,

h "=-6^5lT X 7Sl-X l =--00051864 X ^TX / =
.00051861

q=cubic feet discharged per second.
Z=length of pipe in feet.
d=diameter of pipe in feet. See 44.

34. Asphaltum Coated Pipes. Table for Ascertaining
the Loss of Head in Feet for any Velocity.

By formula (16)

n I v* n Z n

h = -avdr=7d-*= x < V * = 7H* XV

The average value of n for this class of pipe is n = . 00032
in terms of diameter in feet, or n=.013302 in terms of diame-
ter in inches. In order to find the loss of head in feet by
friction per 100 feet length of pipe for any velocity, make Z=
100, and insert the value of n, and we have

1 00 X. 01 3302
Head lost per 100 feet length

Xv^-V rF~Xv s , if d is in inches, or'- :: 7^Xv F , if d is in feet.
yd 3 yd 3

TABLE No. 19.

Table showing loss of head in feet per 100 feet length of
asphaltum coated pipe when v s =l. To find the loss for any

SULLIVAN'S NEW HYDRAULICS,

135

other velocity multiply the tabular loss by the square of that
velocity in feet per second.

Diam-
eter
In.

T/d

Inches

Head in Ft.
Lost per 100
Feet

Diam-
eter
In.

!/d 8
Inches

Head in Feet
Lost per 100
Feet

5.1961

.2560000

110.3040

.0120594

.1662800

117.5750

.01131363

11.1803

.1190000

125.

.01064160

14.6969

.0905080

132.5740

.01003364

18.5202

.0718242

140.2960

.009481382

22.6274

.0588000

148.1620

.009000000

27.

.0492667

156.1690

.008517700

31.6227

.0420600

164.3160

.008095377

36.4828

.0364610

172.6000

.0077068366

41.5692

.0320000

181.0193

.0073483880

46.8721

.0283800

189.5705

.0070700000

52.3832

.0254000

198.2523

.0067096300

58.0747

.0229050

207.0628

.0064241400

64.0000

.02078436

216.

.006160000

70.0927

.01897770

234.2477

.005680000

76.3675

.01741840

252.8822

.005460000

82.8190

.01606140

272.2500

.004885950

89.4427

.01487200

291.8629

.004557600

96.2340

.01382250

332.5537

.004000000

103.1890

.01289000

396.8173

.003352000

What is the loss of head in feet by friction in a 22 inch
coated pipe 2500 feet in length, when the velocity is six feet
per second?

SOLUTION.

From table 19 we see that the loss in one hundred feet
length of 22 inch pipe is .01289 feet head when v=l. If v=6,
then v*=36,and .01289X36=.46404 feet lost per 100 feet length
of pipe. As there are 2,500 feet of pipe the total loss in the
whole length will equal the loss for 100 feet length multiplied
by the number of 100 feet, or 25, and we have .46404X25=
11.601 feet head lost in 2500 feet length when the velocity is
6 feet per second.

If this asphaltum coated pipe were replaced by an aver-
age weight clean cast iron pipe 22 inches in diameter, what
would be the loss of head in the cast iron pipe for 6 feet ve-
locity, and what slope would be required to cause the latter
pipe to generate 6 feet per second velocity?
SOLUTION.

From table No. 17 the loss of head per one foot length of

136 SULLIVAN'S NEW HYDRAULICS.

22 inch cast iron pipe when v a =l is .00015864094. The loss
per 100 feet =.015864094, and when v=6 the loss per 100 feet
will be.015864094X(6) 2 =.571107384, and for 2500 feet,
.571107384X25=14.2777. The slope or fall in the 2500 feet must
therefore be 14.277711.601=2.6767 feet greater for the cast
iron pipe than for the aephaltum coated pipe.

The slope in either pipe which is required to generate the
given velocity is

m=.0004 for cast iron
m=.00033 for asphaltum coating

These values of m are in terms of diameter in feet. The
value of m may be converted to terms of diameter in inches
by multiplying by /(1 2) 8 =41.5692. (See 10, 12 and Group
No. 2, 14.)

The slopes to generate any given velocity may be found
from Tables No. 1 and No. 2 by the formula

s =( - - s =

Table No. 18 gives the different values of v/d 11 . TaDles
No. 1 and 2 give the values of f/d 3 and also of ACX W 'or
each diameter and class of pipe. When d=feet, the slope re-
quired to cause a cast iron pipe to discharge a given number
of cubic feet per second q, is

S= .000648456X-^ r - (See 42, 43).

From which the diameter in feet required to discharge a
given quantity when the slope is given, is

d= |/ / .0000004205X^/ / ^1, for clean cast iron, or

d=-i / 1 when d=inches.

See Tables Nos. 1 and 2 for value of AC, and see formulas

SULLIVAN'S NEW HYDRAULICS. 137

35. Plow and Friction In Fire Hose. Fire hose is made
of different material, such as woven hose, lined with rubber,
or hose made entirely of leather. The resistance to flow will
depend upon the nature of the material which forms the lin-
ing. The resistance to flow in rubber lined hose is much
smaller than in leather hose, or in iron pipes of equal diam-
eter. Fire hose of all classes are made 2^ inches in diame-
ter, and therefore the area and friction surface are constan-t.
Head in feet and pressure in Ibs per square inch increase or
vary at the same rate. The quantity discharged per second
by a hose of constant diameter increases directly as the ve-
locity. In a constant diameter the velocity or quantity in-
creases as the square root of the head in feet, or as the
square root of the pressure in Ibs per square inch. The
friction increases as v a or q 8 in a constant diameter. The
pressure or the head is as v a or q s . The coefficient may there-
fore be determined in terms of head in feet or in terms of
pressure in Ibs per square inch and in terms of v* or q a . The
friction loss will then vary as the head or pressure or as v fl or
q 1 in the constant diameter. As fire hose are all 2% inches
diameter,we may use the direct value of the coefficients m and
n instead of the unit values. It is more convenient to have
the discharge of fire hose in gallons per minute than in cubic
feet per second, hence the formulas will be given in terms of
pressure in Ibs per square inch and discharge in gallons per
minute.

Let P=total guage pressure at hydrant or steamer.

P'^=total pressure lost by friction in the length I, in feet
of hose.

q=gallons per minute discharged by the hose.

n=coefficient of friction.

As the diameter is constant, the direct value of n will be
P' d n

n= rq^~' and p ' =-d~Xq 5 X i.

If 200 feet of rubber lined woven hose 2^ inches diame-
ter be laid out straight on a level with one end attached to a
hydrant or steamer, and with a smooth nozzle one inch chain-

138 SULLIVAN'S NEW HYDRAULICS.

eter and 18 inches in length at the other end, and a
pressure guage at the hydrant end registers 50 pounds,
pressure per square inch, another guage attached at the butt
of the nozzle on the other end will register only 35 Ibs per
square inch, and the discharge will be 145 gallons per minute.
The pressure lost in the 200 feet of hose, (not including the
nozzle), was therefore P' =50 35=15 Ibs. Then,

And P'^' X q 2 X I =.000003567Xq 2 X I

q= gallons per minute.

Z=length in feet of hose.

The loes of pressure in Ibs per square inch in 2^ inch
rubber lined woven hose of any length and for any discharge
in gallons per minute will therefore be

P'=.000003567Xq 8 XZ-

In experiments with this class of hose the writer has ob-
served that the friction increases very slightly for low pres-
sures and decreases slightly for high pressures, be-
cause as the pressure within the hose becomes in-
tense, the rubber lining is compressed, enlarging the diame-
ter slightly and also causing the hose to straighten. An ex-
periment on 300 feet length of rubber lined hose with a
guage pressure of 156 Ibs per square inch at the hydrant end,
showed a pressure of 95 pounds at the butt of the nozzle, or a
loss by friction of 61 Ibs in 300 feet of hose while the dis-
charge was 239 gallons per minute. This gives the formula

P'=.00000356Xq*XZ

The difference in the value of the coefficient for very low
and very high pressures is so slight as to be of no practical im-
portance. It will be understood that the above formula does
not apply to leather hose, nor to any other than 2% inch rubber
lined hose. The coefficient is in its direct form, and conse-
quently applies only to the diameter for which it was deter-
mined.

SULLIVAN'S NEW HYDRAULICS. 139

36 Pressure Required at Hydrant or Steamer to Force
the Discharge of a Given Quantity in Gallons per Min-
ute. As the hoee we are considering was partially throttled
by the one inch smooth nozzle at discharge, the total pressure
was not all neutralized by resistance nor converted into ve-
locity, but a large portion of it remained to balance the fric
tion in the nozzle and to generate the velocity through the
nozzle. Therefore, in order to ascertain the value of the co-
efficient of velocity m, we must taks P=P'+Pv only, for the
hose, (not the nozzle).

To do this, we must first find the value of Pv, or the amount
of pressure which generates the given velocity in the 2^
inch hose. The quantity passing through the hose was 239
gallons per minute.

This is equal .5311 cubic feet per second. The area of
the hose is =.0341 square feet. The velocity in feet per sec-
ond through the hose was therefore

cubic feet .5311
v= ra -- :034T =15 ' 57 feet '

The pressure causing this velocity was
Pv= o - =1.6337 Ibs per square inch.
Hence, P=rP'+Pv=61+1.6337=62.&34 Ibs.

p = -y- xq*X I ^'Xq'X l =Q00003656Xq 1 X l.

Therefore the total pressure at hydrant or steamer that
is required to force the discharge of a given number of gal-
lons per minute, (q) through any length in feet of 2J^ inch
rubber lined hose, will be

P=.000003G56Xq 8 X 1.

This does not include the pressure required to balance
the friction in the nozzle, nor to lift the weight of the water

140 SULLIVAN'S NEW HYDRAULICS.

when the nozzle end of the hoze is elevated. This value of
P is that which is required to balance the friction in the
hose (not the nozzle) and to generate the velocity of flow in
the hose. If the discharge end of the hose is elevated, then
sufficient additional pressure must be added to the above
value of P to raise the weight of the given quantity to the
given height.

The pressure lost by friction in 2^ inch leather hose is

P'=.0000067464Xq 2 XZ

q=gallons discharged per minute.

Z=length in feet of hose.

From this value of the coefficient as compared with the
value of the coefficient for rubber lined hose, it is seen that
the friction loss in leather hose is nearly double that in rub
ber hose. For this reason leather hose has fallen into disuse
and will therefore not be discussed further.

37. Loss By Friction In Brass Fire Nozzles.

In conical pipes or nozzles which converge from a larger
to a smaller diameter, the velocity is inversely as the con-
stantly changing area and the resistance is inversely as ^/d 3 .
The velocity and resistance are therefore different at each
successive point along the length of such convergent pipe or
nozzle. The velocity is greatest in the portion having the
least diameter and least in the greatest diameter. If we take
the mean of all the varying velocities in such convergent noz-
zle, it will be found that this mean is very much greater
than the mean velocity through a pipe of uniform diameter
which uniform diameter is equal to the mean or averag-e
diameter of the convergent nozzle. It is therefore evident
that the friction in the nozzle will greatly exceed that in the
uniform diameter.

From the results of many experiments with very small
nozzles and large nozzles of cast iron from 8 to 12 feet in
length, the writer has discovered that the friction in a nozzle
or convergent pipe is nine times as great as in a pipe of uni-
form diameter which uniform diameter equals the mean di-
ameter of the convergent pipe, both being of the same mate-

SULLIVAN'S NEW HYDRAULICS. 141

rial and same length, and discharging equal quantities of
water in equal times.

The coefficient of resistance n, for smooth brass in terms
of head and diameter in feet is n=.0003268, or nearly the
same as for asphaltum coated pipes. A smooth brass tire
nozzle 18 inches in length and converging from 2^ inches
inside diameter at the butt to a diameter of one inch at dis-
charge, discharged .17134566 cubic feet per second when the
guage pressure at the buit of the nozzle was 10 pounds per
square inch. As the velocity pressure is parallel to the walls
of the pipe, it is not shown by a pressure guage. In order to
find the friction loss in the nozzle we must find the total
pressure at the butt of the nozzle and then find the pressure
which causes the velocity of final discharge from the nozzle.
The difference between the total pressure at the butt of the
nozzle and the pressure due to the velocity of discharge from
the nozzle, is evidently equivalent to the pressure lost by
friction in the nozzle.

The pressure causing the velocity in the hose at the butt
of the nozzle is to be found and added to the guage pressure
at the butt of the nozzle. The velocity in the hose while

q _ .17135
discharging .17135 cubic feet per second was v=

5.0248. The pressure causing this velocity is Pv=

=.17 Ib. Add this to guage pressure at butt of nozzle and the
total pressure at the butt is P=10.17 Ibs. The area in square
feet of the one inch discharge of the nozzle is .0055. Conse-
quently the final velocity of discharge from this one inch

nozzle was v = =~~0055 =31.20 feet per second. The

pressure causing this final velocity of discharge from the noz-

v a ~y 4.34.
zlewasPv= 2 ' ' =6.55 Ibe. The pressure lost in the

nozzle b^ friction was therefore 10.17 6.55=362 Ibs, or 8.34
feet head while the discharge was .17135 cubic feet per sec-

142 SULLIVAN'S NEW HYDRAULICS.

ond. The average or mean diameter of this nozzle was .1458
foot, and the area of this mean diameter was .0167 square
foot. Hence the velocity through the mean diameter while

discharging ,17i35 cubic feet per second was v^ = nig7~

=10.26 feet. The coefficient of resistance of a smooth brass
pipe of uniform diameter is n=.0003268. Hence the loss of
head in feet in a smooth brass pipe of uniform diameter
equal to the mean diameter of this nozzle, and of equal length

.0003268 .0003268X1.50X105.2676

wouldbeh'=- 73f -XlXV= .0557

.9267 feet. This is equal to only one ninth part of the actual
loss in the convergent nozzle. Hence in a formula for friction
loss in a conical or convergent pipe or nozzle we must take
the square of three times the velocity through the mean di-
ameter (3Xv) a or (3Xq) 2 =9v 8 or 9q 8 , or we must find the co-
efficient of friction n in terms of quantity or velocity and
multiply by 9 for a convergent nozzle or pipe, or we must
consider the nozzle as a pipe of uniform diameter and as be-
ing 9 times as long as the nozzle. If we consider it as a uni-
form diameter then that diameter must be equal to the aver-
age diameter of the nozzle or conical pipe and nine times as
long.

Hence the general formula for loss of head in feet by
friction in conical pipes, reducers and nozzles will be

In which

d=mean or average diameter of the convergent pipe.
v=velocity in feet per second in the mean diameter.
n=coefficient of friction in same terms as d.
The above value of n is iu terms of head and diameter in
feet.

SULLIVAN'S NEW HYDRAULICS.

143

In a nozzle of given length and form the loss by friction
will vary directly as the head or pressure at the butt of the
nozzle, or directly as V s or q 8 . Hence a constant multiplier
may be determined for each form and length of nozzle, by
which the loss for any discharge, head or pressure may at
once be found. For example, if the formula is in terms of di-
ameter in feet, pressure in Ibs. per square inch at butt of noz-
zle (guage pressure +Pv.) and v 8 , then
.0001418X I X9 v s

P ':

-, for brass smooth (not ring) nozzles

From which the following table of constants was calcu-
lated:

TABLE No. 20.

Table of multipliers for finding pressure lost by friction
in brass smooth (not ring) fire nozzles. For any head or total
pressure.

Length of
Nozzle
Inches

Diameter
at Butt
Inches

Diameter
at Discharge
Inches

Lbs. pressure lost
equal total pres-
sure at butt mul-
tiplied by the dec-
imal below:

18.000
12.000
3.500
18.000
12.000
3.204
18.000
12.000
2.9125

2 1-2
2 1-2
2 1-2
2 1-2
2 1-2
2 1-2
2 1-2
21-2
2 1-2

!l-8
.1-8
1-8
.1-4
.1-4
.14

.356
.2373
.06922
.49
.325
.087
.474
.316
.0767

These multipliers exhibit the relative efficiency of fire
nozzles of different lengths and forms, and show the import-
ance of making nozzles and reducers of short length. For
the least loss and greatest efficiency the rate of convergence in
a reducer or nozzle should be one inch in a length of 2.33 in-
ches,which will conform to the shape of the contracted vein or
vena contracta. (See 80.)

If we wish to determine the direct coefficient for a given
length and form of nozzle in terms of gallons discharged per
minute and pressure in Ibs. per square inch, take the experi-

144 SULLIVAN'S NEW HYDRAULICS.

mental data already given, for example, and we have
1=1.5 feet=18 inches
d=.1458 feet=mean diameter^ 2 ' 5 + 1 =1.75 inches

P' =10.17 6.55=3.62 Ibs.

Discharge=:.17185 cubic feet per second=77 gallons per
minute

P'd_ 3.62X1.75 _

Z=feet, and demean diameter in inches
q=gallons per minute.

CAUTION. This last formula is in the direct form, and
will apply only to the given nozzle for which
it was determined. If the direct coefficient,
.000407, be multiplied by the constant length
in feet I of the given nozzle, then .000407X1.5
=0006105, and the loss of pressure by friction
in this given form and length of nozzle for
any discharge in gallons per minute is

P'=.0006105Xq 2 .

A direct constant may be found in the same manner for
each length and form of nozzle.

It is interesting to compare the values of n for different
materials when the unit values of n are all in the same terms.
Thus u=0001418 for smooth brass. n.=0000754 for rubber.
These are the unit values of n in terms of P' and diameter in
feet, showing that rubber offers less resistance to flow than
smooth brass or asphaltum coatings.

In a constant diameter of pipe, or in a constant length
and form of nozzle, the friction will increase or decrease di-
rectly as the pressure or head. Hence if a total pressure at
the butt of the nozzleiof 10.17 Ibs will cause a loss by friction
of 3.62 Ibs in the given nozzle, then a total pressure of one Ib.

at the butt would cause a friction loss= 3 ' 62 =.356 Ib., and

10.17

SULLIVAN'S NEW HYDRAULICS, 145

any other total presssure at the butt would cause a loss of
P'=PX-356, for the given nozzle.

If a slope S .0004 in a cast iron pipe one foot diameter
will cause a loss of .0003938 foot head per foot length of pipe,
then the loss of head for any other slope of a one foot pipe

would be=^. 3 ? 3 ?_=.9845XS. And so of any other con-

.0004:

stant diameter or form of pipe or nozzle.

As friction increases as the square of the quantity dis-
charged, if the loss by friction in the nozzle is 3.62 Ibs. while
it is discharging 17 gallons per minute, the loss for a dis-

o Q q o

charge of one gallon per minute would be -

(77) 2 5929

.0006105 Ibs., and for any other discharge in gallons per min-
ute it would be = .0006105X(gallons) 2 .

If the loss of head in feet by friction in each foot length
of a 12 inch diameter cast iron pipe is .0003938 foot while the
pipe is discharging .7854 cubic foot per second, thon the loss
fora discharge of one cubic foot per second in such diameter

will be = 00 g^ 8 8 =.0006384 foot head per foot length and

for any greater or less discharge in cubic feet per second the
loss of head per foot length will be

h" = .0006384X (cubic feet per second) 8 .
Hence it is a simple matter to find the proper constant in
terms of head, pressure, velocity, slope or quantity for any
given form of nozzle or for any given diameter.

38 Friction In Ring Fire Nozzles. On account of the
abrupt shoulder or offset caused by the sudden contraction
of the diameter by the ring in what is termed a ring fire noz-
zle, very serious reactions and eddy effects occur in such noz-
zles, and the loss of head or pressure thus caused is very
great. In an experiment with a ring nozzle of brass, 18 inches
in total length, with a butt diameter of 2} inches and a ring
one inch diameter, and a total pressure at the butt equal to
23.237 feet head, the nozzle discharged . 13333 cubic feet per
second. The velocity through the one inch ring was there-

146 SULLIVAN'S NEW HYDRAULICS.

fore v= q - 13333 =24.243 feet per second. The head
due to this final velocity of discharge from the nozzle was

Hv=-^-= 587>72 =9.126 feet head.

2g 64.4

Deducting this from the total head at the butt of the
nozzle, and the friction loss in the nozzle was 23,2379.126=
14.111 feet head or more than half the total pressure at the
butt of the nozzle.

39 Hydraulic Giants, Cast Iron Nozzles for Power
Mains, Reducers, and Conical Pipes In General. The

writer has made many experiments on cast iron giants or con-
vergent pipes of various dimensions and under heads of 20
to 600 feet at the base of the giant. The results of these ex-
periments confirm the correctness of the general formula
heretofore given for finding the loss by friction in nozzles,
reducers and convergent pipes that is to say, the friution
in a cast iron giant or convergent pipe, will be nine times as
great for the same discharge as it would be in a uniform di-
ameter equal to the mean diameter of the giant, reducer or
convergent pipe. Hence the general formula for head in
feet lost by friction in such giant or convergent pipe is

*xz .............. (95)

In this formula

d=the mean or average diameter of the giant.

v=velocity in the mean diameter in feet per second.

n=the usual coefficient of resistance for the class of cast
iron or other material.

I ^length in feet of giant.

If d is taken in inches then n must also be in the same
terms.

Cast iron giants for discharging water upon impulse
water wheels are required to be of the best metal and without
flaws. They are usually under high pressure and the veloci-
ties through them are terrific. Hence they are scoured and

SULLIVAN'S NEW HYDRAULICS. U7

kept clean BO the coefficiennt will not increase after long use,
unless the water contains sand or gritty matter which cuts
the pipe walls and roughens them.

For this very dense, smooth cast iron, as usually found
in such nozzles, n= .0003623 in terms of diameter in feet.
Using the value of the mean diameter in feet of the cast iron
nozzle and the velocity v, through the mean diameter, and
the general formula for friction in such cost iron giants is

The coefficient n may be determined in terms of quantity
discharged per second or per minute so that the discharge
will correspond with a given loss of head. In an experiment
with a cast iron giant 8 feet in length and converging from
a diameter of fifteen inches at the base to a diameter of 3
inches at discharge, the loss of head in feet by friction while
the discharge was 8.845 cubic feet per second was 16.10 feet
head.

The mean diameter = == 9 inches =.75 foot.

Area of mean diameter=(.75) a X-7854=.4418 square foot.

Velocity through mean diameter ^ = ^ g =20022 ft.

per second.

Velocity in 15-inch diameter =7.209 feet per second.

Velocity of discharge in 3-inch diameter =180.16 feet per
second.

The mean of the velocities in all diameters =93.68 feet
per second.

Using the value of n applicable to this class of dense
cast iron and n=.0003623 in terms of head and diameter in
feet. Then,

.0003623X9 .0032607

b" = ^3 Xv 8 1= b X(20.022) 2X8 =16,10

148

SULLIVAN'S NEW HYDRAULICS.

feet head lost by friction. This corresponds exactly with
the actual result.

In this given nozzle therefore, as the loss of head in feet
by friction was 16.10 feet while the discharge was 8.845 cubic
feet per second, the loss by friction for a discharge of one cu-
bic foot per second would be h' = 16 - 1 =.2058 foot, and for

(O.OiO) 8

any other discharge in cubic feet per second it would be h"=
.2058Xq*. Here q s =cubic feet per second discharged. A con-
stant for any other length and mean diameter of nozzle or
conical pipe may be readily found in the same manner in any
terms desired. (See Table No. 25, 56,)

40. Table of Multipliers for Determining the Loss of
Head in Feet by Friction in Clean Cast Iron Nozzles of
Given Dimensions.

TABLE No. 21, (See Table No. 26.)

Head in feet lost

Length
in Feet of
Nozzle

Greatest
Diameter
Inches

Least
Diameter
Inches

n nozzle equals ef-
'ective head at
aase of nozzle mul-
tiplied by the dec-

imal below

.0415

.02113

.00823

.0324

.012033

.003815

.031

.01768

.0086

:<>iix5

1*6

.0098

.0244

.002394

.0058654

42: -The Total Head fn Feet H, or the Slope Required
to Cause the Discharge of a Given Quantity in Cubic Feet
Per Second in Ordinary Cast Iron Pipes.

By formula (30) the slope required to cause the discharge

SULLIVAN'S NEW HYDRAULICS. 149

of a given number of cubic feet per second, is

m q 8 m q 2 q*

.616853-j/d 1 1 .616853 -j/o 11 ~ "''^y'd 11

The total head in feet in the length in feet I, required to
cause a given diameter of common cast iron pipe to discharge
a given quantity q, in cubic feet per second, is

H= ^g^ X-^nX I =.000648452X-^ n X I

In these formulas d is expressed in feet. In table No.
18 the values of y'd 11 are given.

43 The Slope or Total Head in Feet Being Given, to
Find the Diameter in Feet of Common Cast Iron Pipe
Required to Discharge a Given Quantity in Cubic Feet per
Second.

From the above formula, S=.000648452X-^ T -
Whence,

,000648452Xq s

d "= -- s ~

, qil _(.000648452)2Xq 4

i2632

= n/;jF

Or in terms of total head in feet, for the given value of
m.

44. Head in Feet Lost by Friction In Different Diame-
ters of Clean, Ordinary Cast Iron Pipe While Discharging
Given Quantities in Cubic Feet Per Second.

By formula (32) the head in feet lost by friction for a
given discharge is

h " == .616863 v /d" ............................... (32)

150 SULLIVAN'S NEW HYDRAULICS.

The value of n in terms of diameter in feet for ordinary
clean cast iron pipe is n = . 0003938. Hence. . . .

.0006384
h"= I/dTrXq^XJ- It is convenient to have the loss of

head per 100 feet length of pipe, and therefore we may make
1=100 feet as a constant. The loss of head in feet by friction
in each 100 feet length of pipe for any given discharge in
cubic feet per second will then be

h"= ' /Jn Xq* Now if we take the quotent of-' /^n"

for each diameter of pipe in feet, the result will be a constant
for that diameter in feet, and when such constant is multi-
plied by the square of the discharge in cubic feet per second,
the product will equal the loss of head in feet per 100 feet
length of that diameter for the given discharge.

To facilitate such calculations the following table of
such constants is given.

45 Table of Multipliers for Determining the Loss of
Head in Feet by Friction Per 100 Feet Length of Ordi-
nary Clean Cast Iron Pipe for a Given Discharge In Cubic
Feet Per Second.

TABLE No. 22.

Multiply the constant which corresponds with the given
diameter on same line in the table by the square of the dis-
charge in cubic feet per second (q 8 ). The result will be the

SULLIVAN'S NEW HYDRAULICS.

151

loss of head in feet per 100 feet length of pipe for that dis-
charge.

Diam-

Inch's

Diam.
Feet

.06384

Diam
Inch's

Diam.
Feet. v

.06384

7"^*^-

"~ Constant

V/d 11
Constant

.1667

1214.611000

1.917

0.00178125

.25

130.737000

2.000

0.00141083

.3333

26.880000

2.0-3

0.00112791

.4167

7.871700

2.166

O.Q0090979

2.900000

2.250

0.00073800

.5833

1.236000

2.333

0.00060483

.6667

0.594000

2.416

0.00050000

.75

0.310657

2.500

0.00041347

.8333

0.174045

2.584

0.000344708

.9167

0.10:3000

2.666

0.000290000

1.000

0.0o384

2.750

0.000244785

1.083

0.041187

2.834

0.000207407

1.167

0.027305

2.916

0.000177333

1.250

0.0187104

3.000

0.000151675

1.333

0.0131385

3.166

0.000112800

1.417

0.0093>>23

3.333

0.000085018

1.500

0.0(W8(i3-0

3.500

0.000064970

1.583

0.00510310

3.666

0.000050347

1.667

0. 003*41 l.V>

4.000

0.000*311714

1.750

0.00294000

4.500

0.0000163100

1.833

0.00227919

5.000

0.00000900415

For convenience in referring to the table No. 22, the di-
ameters are given first in inches and then in feet. The total
head per 100 feet length ia equal the loss of head per 100
feet divided by .9815, provided the diameter is constant and

the discharge is free, or H=h"Xl-01573= 9g45 .

46. Head in Feet Lost by Friction In Asphaltum
Coated Pipes While Discharging a Given Quantity in Cubic
Feet Per Second. it we refer to the table of values of m as
developed in Group No. 2 from the experiments of Hamilton
Smith Jr., and of D'Arey and Bazin, on this class of pipe, it
will be seon'that the value of m, the coefficient of velocity,
varies from m=.00028 to m=.0003432 in the experiments of
Smiti., and from m= 000271 to m=.000289 in the experiments
of D'Arcy. Smith's experiments were on lap seamed riveted

152 SULLIVAN'S NEW HYDRAULICS.

pipes with Blip joints like stove pipe joints. D'Arcy's experi-
ments were on cast iron coated pipes which were free of rivet
heads and seams, but which were in shorter lengths and re-
quired more joints. The coefficient of friction n, for any given
class of perimeter, is always equal .9845 per cent of the value
of m for the given class of perimeter. Hence where the value
of m is known for any class of perimeter, the value of n for

n
that class is n=mX-9845, and m= . As it is prudent

to allow for errors in the experimental data from which the
above values of m were deduced and also for inferior quality
of the coating, and future deterioration of the coating and
slight deposits, we will adopt the value of n=. 00032 in
terms of diameter and head in feet. This should be a safe
and reliable value of n for either riveted pipe, welded pipe, or
cast iron pipe, which has been coated with asphaltum. The
coating material usually covers the rivet heads and fills the
longitudinal offset made by the lap of the plate. Hence there
should not be a great difference in the value of n or m for
either class of pipe after it has been coated. D'Arcy's coeffi-
cients are usually too small (that is, m or n, which makes C
too high) and the length of pipe used in his experiments was
rather short. The experiments of Smith are considered more
reliable. They are safer to use in practice at any rats.

TABLE No. 23.

Table No. 23 is based on the same principle as table No.
22, and its use is fully explained in 44, 45.

To USE TABLE No. 23.

To find the loss of head in feet per 100 feet length of as-
phaltum coated pipe for any given discharge in cubic feet per
second, multiply the constant in 3d column opposite the giv-
en diameter by the square of the discharge in cubic feet per
second, q s .

SULLIVAN'S NEW HYDRAULICS. 153

.051884

Head in feet lost per 100 feet length, h'
d is in feet.

dll

Diam-

.051864

Diam

.051864

eter
In.

eter
Feet.

Constant

eter
In.

eter
Feet.

Constant

.1667

986.758

1.917

0.00144710

.25

106.213

2.000

0.001146165

.3333;

21 . 417

2.0^3

0.000916325

.4167

6.400

2.166

0.000739120

2.347

2.25

0.000599600

.5833

1.006

2.333

0.000491370

.6667

0.482460

2.416

0.000405200

.75

0.252380

2.5

0.000386000

.8333

0.141400

2.584

0.000280000

.9167

0.083680

2.666

0.000236852

1.000

0.051864

2.75

0.000199000

1.083

0.033460

2.834

0.000170000

.167

0.022183

2.916

0.000144070

.25

0.015200

3.000

0.000123222

.333

0.0106738

3.166

0.0000916325

.417

0.0076270

3.333

0.0000690690

0.00557617

3.500

O.OOOOV27820

.583

0.004145M)

3.666

0.0003409000

.667

0.00312060

4.000

0.0000253241

.75

0.00240000

4.500

0.0000132509

.833

0.00185160

5.000

0.0000074314

REMARK The value of n used in above taole will allow
for the reduction of area and diameter by the thickness of
the coating, BO that the actual diameter before it is coated
may be used without any allowance for thickness of the coat.

47. To Find the Quantity Discharged when the Loss
of Head and Diameter are Given. The quantity in cubic
feet per second which is being discharged by any diameter
may be found from the loss of head as indicated by pressure
guages. We have just seen that the loss of head per 100 feet
length of coated pipe for a given discharge in cubic feet per

second is h" =

051864

Xq 2 . By transposing in this equation

we have the formula for finding the quantity discharged in
cubic feet per second from the amount of head in feet lost by
friction h", thus,

'' whence ' q= AJS&r

154 SULLIVAN'S NEW HYDRAULICS.

d=feet, and h"=head in feet lost per 100 feet length of the
pipe.

A similar formula for cast iron pipe may be deduced
from the coefficient values given in 44. The values of ^/d 11
will be found in table No. 18. For ordinary cast iron pipe,

not coated, we have h"=' 06384 Xq*=loss per 100 feet length
j/d 1 1

of pipe. Hence the discharge in cubic feet per second cor-
responding with this loss of head in feet per 100 feet length
of pipe is

48- To Find the Quantity that a Given Slope will Cause
a Given Diameter to Discharge.

The slope required to cause the discharge of a given quan-
tity in cubic feet per second is S=-
.00064845^

By transposition we have

/ Si/d 11 /

q= I/ .0006^845 = 39.27 j/Syd" =39.27 Vd 11 X v/S,

H__ total head in feet

~ I total length of pipe in feet
d=diameter in feet.
q=cubic feet per second discharged.
Table No 18 gives the values of yd", and Table No. 15
gives yS.

49 To Find the Total Pressure in Pounds Per Square
Inch that must be Exerted by a Pump Piston, or by Other
Means, in Order to Cause a Given Diameter of Asphaltum
Coated Pipe to Discharge a Given Quantity in Cubic Feet
Per Second.

By formula (44)

SULLIVAN'S NEW HYDRAULICS. 155

As we have adopted n=. 00032 as the safe coefficient of
friction in terms of head and diameter in feet for asphaltum
coated pipes, the corresponding value of the coefficient of flow

would be m Q ^ =.00032503, in terms of head and diame-
ter in feet.

To reduce this value of m to terms of pressure in pounds
per square inch and diameter in feet, it is simply necessary to
divide by the number of feet head required to cause a pres-
sure of one pound per square inch. II=PX2.304, and P=

H
2.304 '

Therefore if P=l pound per square inch, then H 1X2.-
304=2.304 feet.

00032503

Hence, m =~2~304 = 0001il07) * n terms of ^ an( * d in
feet.

.00014107 q"
Then, from formula (44), P= t6168 5^X ^/d l ^^ l=

.0002287

1/cin-Xq'xJ.

The total pressure to be exerted by the pump is there-
fore,

.0002287

P =Vd^ Xq Xl

d = diameter of coated pipe in feet.

J=length of pipe in feet.

q=cubic feet per second discharged.

See Table No. 18 for values of ^d* 1 .

CAUTION: It is assumed in the above formula that the
pipe is laid level, or that there is no differ-
ence in level between its two ends. If the
pipeia laid on a declivity, then this declivity
would supply a portion of the head or pressure.

156 SULLIVAN'S NEW HYDRAULICS.

If the discharge end of the pipe is above the
pump, then additional pressure will be re
quired at the pump sufficient to raise the
weight of the given number of cubic feet per
second to a height in feet equal to the differ-
ence in level between the pump and th dis-
charge end of the pipe.

50. To Find the Quantity Discharged From the Pres-
sure.

By transposition in the above formulafor P, we have

coated pi? 6 -

P=toial pressure in pounds per square inch.

d-diameter in feet of pipe.

Z=length of pipe in feet.

See "Caution" above. For value of ^/d 11 see Table No
18.

For cast iron pipe, not coated, m=.0004 in terms of head
and diameter in feet. Hence in terms of P and d, it will be

m=^3Q|=.0001736Jl. Therefore P ' 616853 X

59.6

51. ~ Pounds Pressure Per Square Inch Lost by Friction
fora Given Discharge In Cubic Feet Per Second.

By formula (45) the pressure lost by friction for a given
discharge is

As we have just found the values of m in terms of diam-
eter in feet and pressure in Ibs per square inch, for coated

SULLIVAN'S NEW HYDRAULICS. 157

pipes and for uncoated cast iron pipes, the corresponding val-
ues of n will be n=mX-9845.

Hence for asphaltum coated pipe n=.00014107X.9845==
.000138883415.

For cast iron pipe not coated, n=. 00017361 lX-9845=
.00017092.

The pressure in Ibs per square inch lost by friction for a
given discharge in cubic feet per second will be, for coated
pipe,

.00017092 q* .000225

~ .616853 X 7d^~ >< /dH Xq X '

And for cast iron pipe not coated,

.000138883415 xx q .000277

P '= .616853 X 7gn-X*= ^TdTlXq'X*

The quantity discharged may be found from the loss of
pressure thus

/ P' -/d 11
q = V .000225X I ' f r C ated pipe *

/ PVd 11
q "\ OOQ977N/ / ' ' or ca8 * i ron PiP e no * coated.

See table No. 18 for value of -/d' 1 .

52. Table lor finding the Slope of a Cast Iron Pipe
or the Total Head, in Feet Required to Cause a Given
Discharge in Cubic Feet per Second.

The quantity discharged by a constant diameter will be
directly as the velocity of flow. The velocity of flow will be
as i/H or -j/S. Hence S or H must vary as v s or q*. If the
slope or total head required in any given diameter of pipe,
one foot in length, to cause a discharge of one cubic foot per
second, be found, then, as S or H must vary as q 2 for that
given diameter, it follows that the slope or total head re-
quired for any other discharge will be equal to the slope or
head which causes a discharge of one cubic foot per second

158 SULLIVAN'S NEW HYDRAULICS.

multiplied by the square of the desired discharge in cubic
feet per second, q*.

If the required slope S is found, then the total head in

feet for any given length in feet will be H=SX M r S= =

total head required per foot length. ' =~g~; H=SX I-
By formula (30),

tn Q ^ Q *

TT = - 00064845 X n"' 0004.

Hence the slope, or the total head in feet pei foot length
of any given diameter of ordinary cast iron pipe, not coated,
required to cause the discharge of one cubic foot per second
will be

/jii And the slope required to cause the dis-
charge of any greater or less quantity in cubic feet will be
. ,00061845

In which,

d=diameter of pipe in feet.
'q=cubicfeet per second.

And the total head in feet required in any given length
in feet of pipe will be H=SXf-

TABLE No. 24.

To find the slope required to cause any given diameter in
feet of uncoated cast iron pipe to discharge a given quantity
in cubic feet per second: Rule. Multiply the slope in the
following table (No. 24) which is opposite the .given diameter,
by the square of the desired discharge in cubic feet per sec-
ond,

H=S X l.

SULLIVAN'S NEW HYDRAULICS,

159

Diam
eter

r/dll
froat

Slope
.00064845

Diam-
eter

v/d* 1

Pftftt

Slope
.00064845

Feet.

r eel

v/d 11

Feet

c 661

^d"

.1(567

.00005256

12.33734

1.917

35.84

.0000180929

.25

.0004883

1.32800

2.000

45.25

.0000143300

.3333

.002375

.27303

2.083

56.60

.0000114567

.4167

.00811

.07996

2.166

70.17

.0000092410

.Oi21

.0293416

2.25

86.50

.0000074:3653

.5833

.05157

.0125740

2.333

105.55

.00000614353

.6667

.1075

.0060800

2.416

128.00

.00000506600

.75

2055

.0031555

2.5

154.40

.0000042000

.8333

13668

.0017680

2.584

185.20

.0000035000

.9167

.6198

.00104622

2.666

219.90

.0000029500

.000

1.000

.000o4845

2.75

260.80

.00000248638

.083

1.55

.0004184

2.834

307.80

.00000210672

.167

2.338

.00027735

2.916

360.00

.00000180120

.25

3.412

.00019000

3.000

420.90

.00000154060

.343

4.859

.0001334533

3.166

566.00

.00000114570

.417

6.800

.OOOoa-,:;--m

3.333

750.90

.000000863563

9.301

.0000097180

3.5

982.60

.000000660000

.583

12.51

.0000518345

3.666

1268.00

.000000511400

1.667

16.62

.0000390000

4.000

2048.00

.000000316621

1.75

21.71

.00002'.iNV"7

4.5

3914.00

.00000 '165674

1.833

28.01

.0000231500

5.000

6979.00

.000000092916

See Table No. 16. These tables apply to pipes flowing
full bore and with free discharge. q:q ::i/S:v/S, for a given
diameter.

S3. Wooden Stave Pipes. In the western states,where
irrigation is practiced on an extensive scale, and in localities
without railway facilities, wooden stave pipe, invented by Mr.
J. T. Fanning, and described in his "Treatise on Water Sup-
ply and Hydraulic Engineering" page 439. has been adopted
in many instances in recent years.

In the very dry atmosphere of the arid west these pipes
have not proven satisfactory in many cases where they were
laid on the surface or without sufficient covering. In such
cases it shrinks and warps and leaks badly. Where properly
covered and kept constantly full of water it has been quite
satisfactory. It has not been in general use for a sufficient
length of time to test its durability. That would, of course,
depend upon the kind of wood used in manufacturing the
staves, and upon whether it was perfectly seasoned and
sound. If perfectly seasoned and treated with tar oil or sul-
phate of copper, it should be very durable. The quantity of
this class of pipe which is being used of late years in the

160 SULLIVAN'S NEW HYDRAULICS.

West for irrigation purposes in caseB where there is only
Email pressure to be sustained, and the general belief that
this wooden pipe is smoother and will give a higher dis-
charge under like conditions than uncoated iron pipes, de-
mands that it be given some notice here.

54. Coefficients of Flow in Wooden Stave Pipes
Compared with the Coefficients of Pipes of other Material.

By referring to the coefficient values developed from the
data of D'Arcy and Bazin, (See group No 5), it will be seen
that the average value of m for wooden conduits made of
closely jointed, planed poplar lumber is m = . 000060 in terms of
hydraulic mean radius in feet and head in feet. The rectan-
gular wooden conduits used in these experiments did not
contain the great number of joints which are necessary in
forming a circular conduit of wooden staves. It is fair to as-
sume then that the circular wooden conduit built up of narrow
staves with its many joints would not present a more uniform
surface to the flow than the rectangular conduit or flume of
planed, well jointed hard wood.

The nature of the wood of which the staves are made as
to density and freedom from knots, will undoubtedly affect
the value of the coefficient. It appears from the great num-
ber of experiments by D'Arcy and Bazin on such conduits
(only a few of which were quoted in Group No. 5) that m=
.00006 is about the average value of the coefficient in terms of
r in feet.

If this value of m be reduced to terms of diameter in feet,
we have for well jointed, planed hard wood conduits, m=

.00006X8=.00048, or C=^~- =45.64, in terms of head and

diameter in feet. For average cast iron pipe, not coated, m
.00040 and 0=50.00. For asphaltum coated riveted pipes, 0=
56.CO. For pipes lined with mortar composed of two-thirds
cement to one-third sand, 0=48.50. It is therefore apparent
that the wooden pipe offers much greater resistance to flow
than either of the others, and will therefore require a greater
diameter for an equal discharge.

SULLIVAN'S NEW HYDRAULICS. 161

The slope required in a wooden pipe in order to cause it
to discharge a given quantity in cubic feet per second would
be

" v q' .MOM q* _ - 0007781 v ...

~ .616853 X v/d 1 * T .616853 x v/d 11 "~ i/d 11 xq '
and qiqii/Si^S.

And the diameter in feet required to discharge a given
quantity for a given slope will be

55. Earthenware Or Vitrified Pipe This class of
pipe is made in very short lengths and consequently requires
many joints. It is subject to unequal settlement and leaks
unless very great care is taken to secure a firm bearing or
foundation upon which to lay the pipe. It also requires care
and experience to make and properly cement the joints. If
the pipe is made of clay containing a high percentage of
aluminum and is thoroughly glazed and p roperly laid and
very carefully jointed, it develops a coefficient m=.00036 or C
52.70, in terms of diameter in feet and slope or head in feet.
It therefore offers less resistance to flow than very smooth,
dense, clean cast iron pipe, provided all the above conditions
as to laying and jointing are complied with. As these con-
ditions are scarcely ever fulfilled, it is not prudent to depend
upon a greater discharge from such pipe than from ordinary
clean cast iron pipe. Hence all the tables heretofore given
for cast iron pipe may be adopted as applying also to earth-
enware glazed pipe. This class of pipe is very extensively used
for house drains, small sewers, land drains and irrigation pur-
purposes, and in other rough work where great care and
thorough workmanship are not usually exercised. Hence it
is not safe to take the value of C greater than C=50, or m =
.0001 in terms of head or elope and diameter in feet. For
small sewers, not exceeding about 18 inches diameter, this
class of pipe serves well. The flow of sewage is probably not
eo great as that of clear water because of the suspended, solid

162

SULLIVAN'S NEW HYDRAULICS.

matter that it carries. The value of C for a sewer would
therefore rot be quite BO great as for pure water flowing in
the samo class of pipe or conduit. C=50, should be a safe
value for fairly well laid and jointed earthenware glazed
pipe. In order to prevent deposits the mean velocity of flow
in a sewer should never be less than two and half feet per
second for small depths of flow. In order to ascertain the
mean velocity of flow in such sewer pipe when flowing only
part full, the coefficient maybe reduced to terms of hydraul
icmean depth r, in feet by multiplying m in terms of d in
feet by 0.125, or by dividing by 8. Then in terms of r in feet

0004
m = ! -g .0000."), and C 141.42. The mean velocity of flow

in a circular conduit, or in a pipe, will be the same for just
half full as for full, because-p- is the same for half full as

for full.

56 Table of Elementary Dimensions of Pipes.

TABLE No. 25.

Diam.
In.

Diam.
Feet

Area
Sq.
Feet

U.S. Gal.
In one
Ft.Lgth

Diam.
In.

Diam.
Feet

Area
Sq.
Feet

U.S. Gal
In one
Ft. Lsth

.0208

.0003

.0025

4.V4

3750

.1104

.8263

.0313

.0008

.0057

4.%

.3958

.1231

.9206

.0417

.0014

.0102

.4167

.1364

1.020

.0521

.0021

.0159

.1963

1.469

.0625

.0031

.0230

.6667

.3491

2.611

.0129

.0042

.0312

.8333

.5454

4.080

.0833

.0055

.0408

.7854

5.875

i-k

.1042

.0085

.0638

!l67

1.069

7.997

l.V>

.125

.0123

.0918

.333

1.396

10.440

1.3

.1458

.0167

.1249

1.767

13.220

.1667

.0218

.1632

.667

2.182

16.320

2.M

'1875

.0276

.2066

.833

2.640

19.75

2-H

.2083

.0341

.2550

3.142

23.50

2.X

.2292

.0412

.3085

2.167

3.6b7

27.58

.25

.0*91

.3672

2.25

3.976

29.74

3.!4

.2708

.0576

.43C9

2.333

4.276

31.99

3 K

.2917

.0668

.4998

2.5

4.909

36.72

3.%

.3125

.0767

.5738

2.667

5.585

41.78

.3333

.0873

.6528

2.833

6.305

47.15

4.M

.3542

498B

.7369

7.069

52.88

231 cubic inchee=d U. S. gallon. 7.48052 U. S. gallons=
1 cubic foot.

SULLIVAN'S NEW HYDRAULICS.

163

The area in square feet of a pipe is the same as the contents
Df one foot in length of the pipe in cubic feet. Hence by an
inspection of table No. 25, the diameter and also the velocity
required to carry a given number of cubic feet or of U. S. gal-
lons may be determined at once. If the velocity is one foot
per second in any diameter, the discharge in cubic feet per
second will equal the area in square feet of that diameter, or
the discharge in gallons per second will equal the number of
gallons in one foot length of pipe. For a discharge of 2, 3, 4.
etc times that quantity, the velocity must be 2, 3, 4, etc feet
per second. Tables No. 16 and 17 and 19 will show the slope
or head required to generate the required velocity, and also
the amount of head that will be neutralized by friction for
that velocity. The dimensions of the very small pipes given
in table No. 25 will be "found convenient in designing hy-
draulic giants and nozzles, and in selecting small service pipes,
and discharge pipes for small pumpd. See also Tables 22 and
23, and 26 and 27. Square inches multiplied by .00695=
square feet.

57. Length in Feet of Small Pipes Required to Hold
one U.S. Gallon of 231 Cubic Inches, and Areas Given in
Square Inches.

TABLE No. 26.

(1 square inch=.0069444 square feet).

Diam.
In.

Area
Sq.
Inches.

Length In
Feet to hold
1 Gallon.

Diam.
In.

Area
Sq.
Inches.

Length In
Feet to hold
1 Gallon.

.0490875

407. 43567 ;~0

2.%

5.412

3.5570

l /

.1963500

98.0S92083

2.%

5.940

3.2540

.4417875

43.5729833

2.X

6.492

2.9651

.7854

24.5098000

7.069

2.7225

1 ii

1.2271875

15.6862001)

3.^

7.670

2.5097

1 H

1.7671500

10.893iOOO

3.ii

8.296

2.3200

1 K

2.4050

8.0)41650

3.%

8.946

2.1517

l.X

2.7610

6.9721000

3.V*

9.fi21

3.1416

6.1274100

3.X

10.320

1.8652

2.H

3.5470

5.4270000

8.V

11.040

1.7360

2.?4

3.9760

4.8415000

3.%

11.790

1.6166

2 K

4.9090

3.9214000

12.570

1.5310

Length in feet to hold one gallon equals velocity in feet
per second required to discharge one gallon per second. The

164

SULLIVAN'S NEW HYDRAULICS.

velocity must be 7.5 times as great to discharge one cubic
foot per second.

A 4 inch cast iron pipe cannot supply one fire hydrant
with the ordinary supply of 255 gallons per minute without a
loss of head in such pipe of nearly one foot in each 10 feet
length of 4 inch pipe. Add to this the friction loss in the
hydrant, the hose and the nozzle, and the resistance of the
atmosphere and wind, and it is apparent that a hydrant will
be of little service when attached to a four inch pipe of any
considerable length.

58. Decimal Equivalents to Fractional Parts of one
Lineal Inch.

TABLE No. 27.

1-32=. 0312 1
1-16=. 06250
3-32=. 09375
1-8 =.125
1-8 -I- 1-32=. 15625
1-8+1-16=. 1875

18 +3-32=. 21875
1-4= .25
1-4+1-32=. 28125
1-4 + 1-16=. 3125
1-4 +3-32=. 34375
38= .375

|3 8+1-32=. 40625
3 8+1-16=. 4375
1 3-8 +3-32=. 46875
1-2= .5
12+1-32=. 53125
1-2 + 1-16=. 5625

5-8 =.625
5-8+1-16=. 6875
3-4=. 75
3-4 + 1-16=. 8125
7-8= .875
7-8+3-32=. 96875

Fractional inches in equivalent decimals of a foot.

Frac.

Deci.

Equiv

Frac.

Deci.

Equiv

Frac.

Deci,

Equiv

Inch

dec ft.

Inch

dec ft.

Inch

dec ft.

1-32

.03125

.00?04

3-8

.375

.03125

23-32

.71875

.059895

1-16

.0625

.005208

1332

.40675

.033854

3-4

.75

0625

3-32

.09375

.007X12

7-16

.4375

.036458

25-32

.78125

.065104

1-8

.125

.010416

15-32

.46875

.039062

1316

.8125

.067708

5-32

.15625

.010420

1-2

.041666

27-32

.84375

.070312

3-16

.1875

.015625

17-32

.53125

.044i7

7-8

.875

.072916

732

.21875

.018229

916

.5625

.046875

2932

.90625

.07552

.25

020833

19-32

.59375

.049479

15-16

.9375

.078125

932

.28125

.023437

5-8

.62>

.0-2083

31-32

.96875

.080729

5-16

.3125

.026041

21-32

.65625

.054607

1.00

.083333

11-32

.34375

.028645

11-16

.6875

.057291

Tenths of one foot in equivalent inches.

Foot

Inches

Foot

Inches 1

Foot

Inches

0.10
0.20
0.30
0.40

1 3 16
2.3-8
3.1932
4.25-32 i

0.50
0.60
0.70
0.80

6.00
7.3-16 i
8.3-8 !
9.19-32 !

0.90

10.25-32
12.00

SULLIVAN'S NEW HYDRAULICS.

165

59 Tables for Converting Measures,

TABLE No. 28. Lineal Measure.

Inch's

Feet

Yards

Fath.

Rods

Miles

Metres

12
36
72

198
7920
63360

.083333
1
3
6
16 l / s
660.
52SO.

.02778
.33333
1.
2.

220."
1760.

.013889
.16666
.5
1.

no! 4

880.

.005051
.060606
.181818
.363636
1.
40.
320.

.000016
.000189
.000563
.001136
.003125
.125
1.0

.0254
.304797
.914392
1.82878
5.02915
201.166
1609.33

TABLE No. 29. Land Measure (Lineal).

Inch's

Links

Feet

Yards

Chains

Miles

Metres

7 23-25
12
36
792
63360

.1261261
1.
1 17-33
4 6-11

100.
8000.

.083333
.066666
1.
3.
66.
5280.

.0277778
.222222
.333333
1
22.
1760.

.0012626
.01
.0151515
.0454545
1.
80.

.0000158
.00015
.00"1894
i 01X15682
.0125
1.

.0254
.201166
.304797
.914392
20.1166
1609.33

TABLE No. 30.* Metrical Equivalents. Lineal Measure.

Inches

Feet

Yards

Rods

Chains

Miles

1 Millimeter=
1 Centimeter=
1 Meter
1 Kilometer=

.03937
.393704
39.370432

.003281
.0328(19
3.2*0869
3280.8693

.001094
.Olf'936
1.093623
1093.6231

.001988
.198841
198.84057

.04971
49.710141

.000621
.621377

Milli-
meters

Centi-
meters

Meters

Kilo-
meters

Inch =
Foot =
Yard =
Rod =
Chain =
Mile =

25.399772
304.79727
914.391795
5029.15487

2.539977
30.47973
91.43918
502.9ln49
2011.66195

.253998
.304797
.914392
5.029155
20.11R62
1609.32956

.0003048
.0009144
.00"02915
.02011662
1.60933

ine.

*See "Rules and Tables" page 92, by Prof. W. J. M. Rank-

SULLIVAN'S NEW HYDRAULICS.

167

TABLE No. 33. Cubic Measure.

Cubic Inches

1.0

1728.0
46656.0

Cubic Feet I Cubic Yards ICubic Meters

.0005788
1.0
27.

.00000214
.037037
1.0

.000016387

.0283161

.764534

231 cubic inches=l U.S. gallon. 7.48052 U. S. gallons^
1 cubic foot.

The actual weight of 1 U. S. gallon of water at its maxi-
mum density is 8.345008 pounds. The weight is, however,
adopted by law as 8.33888 pounds avoirdupois.

1 U. S. gallon=.13368 cubic foot. 1 cubic foot per second
=448.8312 gallons per minute, or 26929.872 gallons per hour,
or 646316.928 gallons per 24 hours. 1 cubic foot per second=
60 cubic feet per minute, or 3600 cubic feet per hour, or 86400
cubic feet per 24 hours. This will cover one acre of ground
to a depth of 1 98347 feet, or 1.98347 acres to a depth of one
foot iu 2i Lours, or supply 200 gallons per person per 24 hours
for 3,231 58 persons. An 8 inch pipe will carry it at a velocity
of 2.864 feet per second.

TABLE No. 34. Metrical Equivalents. Cubic Measure.

Cubic In. ;U. S. Gal-|CubicFt,'Cubic yd|Perches :

leu. centimtr.
1 cu. rlecin.etr.
1 cubic meter

.061025386 ! .000264179 .000035316
61.025386 , .264179 .035316
6 '025. 386 [ 264.179 [ 35.316

.001307986 | .001426893
1.307986 | 1.426893

Cubic Cent.

Cubic Deem.

Cubic Meters

1 cubic inch
1 U. 8. gallon
1 cubic foot
1 rubic yard
1 Perch

16.386623 .
3785.31
28316.0844

.016386623
3.78531
28.3160844
764.5343
700.82309

.00378531
.0283160844
.7J5343
.70082309

1 Perch=24.75 cubic feet.

TABLE No. 35. Pressure. (Thurston).

Pounds per sq.
Inch.

Kilograms per [Kilograms per
Sq. CentimeterjSq. Centimeter

Pounds per.
sq. Inch.

1.0

.07030S27 1 1.0

14.22308

REMARK. The foregoing conversion tables are given in
order that the formulas may be used and coefficients deter-
mined either in English or metrical terms.

SULLIVAN'S NEW HYDRAULICS.
TABLE No. 31. Square Measure.

Sq.
Inches

Square
Feet

Square
Yards

Square
Rods

Square
Roods

Square
Acres

Square
Metres

144.

1296.
39204.
1568160.
C272640.

.00(59444
1.
9.
272. k
lftS90.
435(10

.0007716
.1111111
1.

30. y t

1210.
4840.

.0000255
.0036731
.0330579
1.
40.
160.

.00000064
. .OIX;0918
.0008264
.025
1.
4.

.00000016
.00002:$
.0002050
.00620
.25
1.

.0006452
.0929013
.836112
25.292
1011.6P6
4046.782

AcresX-C015625 = square miles. 1 square mile=27,878,400
square feet, or 3C97600 square yardb, or 640 acres, or one sec-
tion. One acre=10 square chains. Tho length of one chain
is 66 feet, or four rods. This Gunters chain has fallen into
disuse, and a steel tape 100 feet length is used instead. Areas
are taken in square feet, and when divided by 43,560, are re-
duced to acres.

TABLE No. 32* Metrical Equivalents. Square Measure.

Square
Inches

Square
Feet

Square
Yards

Acres

Square
Miles

1 sq. Centime-

ter=

.165003

.00107641

.00012

1 eq. Decime-

ter=

15.500309

.107541

.011960115

1 sq. Meter=

1550.030916

10.7641

1.1960115

.00024711

1 eq. Dekame-

ter=l Are=

15r-003.0916

1076.41

119. C011 5

.02411

.00003861

1 sq. Hectome-

ter=l Hectare

1 Kilometer=

1076.41

11960.115 2.4711
1196011.5 247.11

.003861
.3861

Sq. Centi-
meters

Sq. Deci-
meters

Sq. Meters.

Sq. Dekame-
ters or
Ares

ISq
Inch=
1 Square
Foot=
1 Square
Yard=

6.451484
929.013728
8361.123554

.06451484
9.29013728
83.61123554

.0006451484
.0929013728
.8361123554

.000929013728
.008361123554

'See "Conversion Tables," page 40, by Prof. Thureton,
and Trautwine's "Civil Engineer's Pocket Book", page 78,
Rankine's "Rules and Tables," p. p. 110-114.

CHAPTER V

Of Water Powers, Power Mains and Pipe Lines.

Work is expressed in units of weight lifted through one
unit of height; as in pounds lifted one foot, called foot pounds.
Here there is no reference to the units of time consumed
Power is expressed in units of work done in one unit of time;
as in pounds lifted one foot in one second of time, called foot
pounds per second.

One horse power is a conventional quantity equal to 550
foot pounds per second, or to 550 pounds lifted one foot in
one second, or to one pound lifted 550 feet per second.

As there are 60 seconds in one minute of time, theexpres
ion of horse power in terms of foot pounds per minute would
be 550X60=33,000. or in foot pounds per hour it woud be 33,-
000X60=1,980,000.

One pound of water falling one foot does work equal to
that of raising one pound one foot high. Hence the number
of pounds of water falling in one second multiplied by the
distance fallen in feet will equal the number of foot pounds
per second, and as 550 foot pounds per second equal one horse
power, the totil number of foot pounds per second divided by
550 will equal the horse power of the water. Expressed as a
formula, we have

cubic f eet per Bec.X weight of one cubic footX Head or fall in feet.
H ' P - = - 550

The weight of one cubic foot of water at its maximum
density is 62.5 Ibs. This is the weight always assigned, in or-
dinary cases, to one cubic foot of water. The formula may
therefore be written

62 5
H. P.= --Xcubic feet per second X head or fall in feet.

62 5
If we take the quotient of -- =.3136363, we have,

SULLIVAN'S NEW HYDRAULICS. 169

- H. P.=.1136363Xhead in feetXcubic feet per second.. .97

61. Formula for Cubic Feet Per Second Required to
Generate a Given Horse Power.

When the net head or fall in feet is given, then the cubic
feet per second required to develop any required horse power
will be

Horse Power Desired

Cubicfeet per sec.= jll36ae3xHeBd in Feet < 98 >

62. Formula for Net Head or Fall in Feet Required
to Develop a Given Net Horse Power.

The efficiency of a water wheel or other machine is the
ratio of effective power recovered from it to the total power
applied to it. To find the efficiency, divide the effective pow-
er delivered by the machine, by the total power applied to it.
The quotient is the efficiency. If a water fall of 100 horse
power is applied to a turbine and the turbine develops 80

horse power, then the efficiency of the turbine is E= j^ = 80
per cent.

The efficiency of the motor being given, then the net head
or fall in feet required to develop a given net horse power,
will be

Desired net H. P.-nper cent efficiency of motor
.1136363XCubic Feet per Second,

(99)

63. Head of Water Defined. By the term head is
meant the difference of level between the surface of the water
in the reservoir or head race, and the water surface in the tail
race, to which must be added the head due to the mean vel-
ocity of flow in the head race or stream above the fall.

The head due to the velocity in the head race is H=

(See formula (7) Chap. I.) In a pipe or power main the total
head is equal to the difference of level between the water sur-
face at the intake end of the pipe and the upper surface of
the jet at discharge, and the net head is equal to the total

170 SULLIVAN'S NEW HYDRAULICS.

head lees the amount of head neutralized by friction or re-
sistance in the pipe. Hence in a pipe or power main the loss
of head by friction must be first deducted from the total
head in order to ascertain the effective head at discharge.

64. To Find the Diameter in Feet of Pipe Required
to Carry a Given Quantity of Water with a Given loss
of Head in Feet,

Where the total head is known, and it is desired to lay a
pipe of such diameter as will convey a given number of cubic
feet per second with a predetermined loss of head by friction,
so that a given net head will be secured at discharge, such di-
ameter in feet may be found as follows:

x ' ....................... (32)

For ordinary clean cast iron pipe n=.0003938 in terms of
diameter in feet.

Hence the formula reduces to

and,

'.. whence.

4X <1 S (100)

If it is an asphaltum coated pipe, then take n=00032 in
terms of diameter in feet, and the formula for finding the di-
ameter required to carry a given quantity in cubic feet per
second with a given total loss of head in feet by friction in
the entire length of pipe line, will be
11

In these formulas

d=diameter of required pipe in feet
q=cubic feet per second it is to discharge
Z=total length of pipe in feet
h"=head in feet lost by friction in total length, I,

SULLIVAN'S NEW HYDRAULICS. 171

EXAMPLE OF THE USE OF THESE FORMULAS.

There is a total fall of 100 feet in a distance of 3,000 faet.
A diameter of asphaltum coated pipe is desired, which will
convey one cubic foot of water per second with a loss of head
not exceeding 6 feet, so that there shall remain an effective
head of 94 feet, at discharge while one cubic foot per second
is being drawn from the pipe at its lower end. By the above
formula

d= ll/ .000000269X J X9000000 _ 11
V 36 V

feet,

This diameter has an area =d 2 X-7 p 54=.48078 square feet.
The mean velocity required to discharge one cubic foot per

second in this diameter would be, v= ~ 48078 ~ 2.08 feet

per second.

The resu't may therefore be tested by the formula for
loss of head,

And we have,

.00032
h"= 692 X3000X4.3264-6.00 feet head lost.

It will be understood that the area at discharge is such
that it will admit of no greater discharge under the given net
head than the quantity q. The manner of discharge may be
through other small pipes tapped into the main if it is a
water works system, or the discharge may be through a re-
ducer or nozzle if the pipe is used as a power main for driv-
ing water wheels, or the discharge maybe full and the total
head lost except the velocity head, as may be desired.

65. To Find the Area and Diameter of the Nozzle
Tip or Aperture Required to Discharge a Given Quantity.

If there is a simple tip on the end of the pipe made in
the form of the contracted vein which reduces the diameter

172 SULLIVAN'S NEW HYDRAULICS.

at discharge, there will be a very small loss of head by
friction in efflux from the tip. The area in square feet of the
required aperture in such tip will be found as follows: As-
sume the diameter of the pipe to be .7824 feet, and net head
at discharge to be 94 feet, as in the preceding section, and the
quantity to be disharged as one cubic foot per second.

The velocity that will be generated by this net head at
discharge will be

v = 1 /iTgTl=8.025 1 /94~~=77.8052 f ee t per second.

Now, q= areaX velocity ^ay^g H. Whence a =

q 10

1/2 H = 77 8052~ = ' Q128526 8< l uare feet - The diameter in feet

is then

=' 128 foot = 1 636 inchee diam '

= .7854

eter.

See table No. 27, 58.

If the discharge is to be through a nozzle or reducer of
several feet length, there will be considerable loss of head by
friction in such nozzle or reducer, for which allowance must
be made. This loss will depend upon the length of the con-
vergent reducer or nozzle and its mean or average diameter as
well as its smoothness of internal circumference, and the
square of the velocity through it. We have seen heretofore
(37, 39) that the loss by friction in a convergent or conical
pipe is nine times as great as the loss in a pipe of uniform
diameter equal to the mean diameter of the convergent pipe.
It is therefore evident that such convergent pipes, reducers
or nozzles should be as short as possible.provided they do not
converge more rapidly than one inch in a length 2.33 inches,
which would make them conform to the form of the con-
tracted vein. Assuming the diameter of the base of the noz-
zle to be the same as the diameter of the pipe it is to join, and
that the net head at the base of the nozzle is 94 feet, and
that the reducer or nozzle is to be 6 feet in length, and is re-
quired to discharge one cubic foot per second under this net

SULLIVAN'S NEW HYDRAULICS. 173

head at the base, the problem now is to determine the area
and diameter of the small, or discharge end of the nozzle so
that it shall discharge this given quantity per second under
the given head at its base.

This will require one or more approximations, for the
reason that as the mean diameter of the proposed nozzle is
yet unknown we have no means of knowing the friction loss
that will occur in the nozzle, and hence do not kuow the value
of the net effective head at the point of final discharge from
the nozzle.

For first approximation assume that there will be three
feet head lost in the nozzle, leaving an assumed effective
head at discharge of 943=91 feet. The velocity of dis-
charge under the net head of 91 feet will be ^8.025/91=
76.5535 feet per second. Then the area in square feet re-
quired to discharge the quantity q, in cubic feet per second,

will be a=^!=g=: 7^35 = .01306276 square feet. The di-
ameter in feet answering to this area in square feet is

eter in inches is therefore .129X12=1 548 inches. (See 58.
Table 27).

For first approximation we have then the following di-
mensions of the nozzle: Greatest, or butt diameterr=.7824
foot =9.388 inches. Smallest, or discharge diameter =.129
foot=l 548 inches. Total length of nozzle 6 feet. The
average or mean diameter of the nozzle is therefore,

Mean d=' " "*"' - = .4557 foot, or 5.4684 inches.

Now, two tests must be applied to this nozzle in order to
ascertain whether or not it will fulfill the required
conditions:

(1) It must be tested by the formula (39) for friction lose
in nozzles in order to ascertain the actual loss of head that
will occur while discharging the given quantity.

(2) It must then be tested to ascertain whether or not

174 SULLIVAN'S NEW HYDRAULICS.

it will discharge the given quantity under the conditions
actually existing. If it fails to meet the requirements, fur-
ther approximation must be made.

(1) TEST FOR LOSS BY FRICTION.

The formula for loss of head in feet by friction in conver-
gent pipes and nozzles is

h"= X/X9V. (See 37,39).

In which

h"=head in teet lost by friction
i=rlength in feet of convergent pipe or nozzle*
d average or mean diameter of nozzle
v=mean velocity in the mean diameter of the nozzle
In the nozzle we are now considering the mean diameter
is .4557 foot, and the nozzle is required to discharge one
cubic foot per second. Hence the required mean velocity in
feet per second through this mean diameter to cause the dis-
charge of one cubic foot per second will be

v=-9_= -- - = _ L_=6.1312 feet per second.
a (.4557) s X-7854 .1631

Assuming the nozzle to be made of very dense, solid
smooth cast iron, the friction coefficient will be n .0003623 in
terms of diameter in feet. Applying the above formula for loss
by friction in this nozzle while discharging one cubic foot per
second, the velocity in the mean diameter being 6.1312 feet
per second, and we find the actual loss of head in the nozzle
to be

Hence at the point of discharge the effective head would
be 942.40=91.60 feet, whereas we had assumed that it
would be probably 91.00 feet. But as the assumed loss of head
(3 feet) and the actual loss (2.40 feet) are BO nearly equal, we
will now apply the test for quantity discharged under the
actual conditions. For this purpose we have the following:

Area of smallest diameter at discharge=.01306276 square
feet.

SULLIVAN'S NEW HYDRAULICS. 175

Effective head at point of discharge from nozzle 91.60
feet.

Velocity due to this net head, v=!/2gH. = 8.025 v /91.b p =
76.807275 feet per second. Quantity discharged q, will be

q=a v^.01306276=76.807275=:1.003315 cubic feet per sec-
ond.

If a closer result is desired, the smallest diameter may be
reduced by 1-16 inch and all the foregoing tests be again ap-
plied to the new proportions of the nozzle thus changed.
Table No. 21, 40, will be of assistance in such calculations.

66. Pipe Lines of Irregular Diameter. Where the head
or pressure is due to the slope or inclination of a pipe line,
and not to a pump, there will be very little pressure within
the pipe in the upper portion of the line. In such cases
large diameters with thin shells may be adopted in the
upper part of the line where the pressure is small. As the line
proceeds down the slope and the pressure increases, the di-
ameter is diminished and the pipe shell increased in thickness
in proportion to the increase of pressure.

If a pipe line is of uniform diameter and is laid on a uni-
form grade and has a full and free discharge, there will be no
radial pressure in the pipe at any point except the
very small pressure due to the vertical depth of the diameter.
In this cas? there is no object in increasing the thickness of
pipe shell at its lower end, because the total head or pressure,
under these conditions, will be converted into velocity of flow,
with the exception of the amount of head lost by friction, and
as the velocity head or velocity pressure is always parallel
to the pipe walls, it does not tend to burst the pipe.

Where a given pressure or head is to be maintained at
the lower end of the pipe,or at any point along its length, while
a given supply of water is being drawn from it for domestic
purposes, or for driving water wheels, the capacity of the
pipe must be such that the mean velocity of flow in it while
delivering the given supply, will not cause a loss of head by
friction exceeding a predetermined amount. The discharge
permitted from such pipe must therefore bo regulated by

176 SULLIVAN'S NEW HYDRAULICS.

the area of discharge BO that it will not exceed the givea
quantity. If the lower end of a pipe line be entirely closed
so there can be no discharge from it and no velocity within
it.the pressure at any point along the line will be that due to
the total head up to that point, which will be equal to the
difference in level between the given point in the pipe and
the water surface in the reservoir or source of supply. The
pressure at the lower end will be that due to the total head
in the pipe line. If a small orifice be opened in the lower end
of the pipe, it will at first discharge with a velocity due to the
total head, but this discharge will cause a small velocity to
be generated throughout the length of the entire pipe, and
this velocity will cause a small friction with the pipe walls
which will reduce the head by the amount of the friction
thus generated, and thus slightly check the velocity of dis-
charge through the orifice. The smaller the orifice relatively
to the area and capacity of the pipe, the smaller will be the
velocity in the body of the pipe to supply the quantity being
discharged; and as the loss of head by friction is as the square
of the velocity, the smaller the velocity becomes, the smaller
the loss by friction will become. If the orifice is enlarged so
that it may discharge a greater quantity per second, then
the velocity in the body of the pipe must increase proportion-
ately and loss of head or pressure will also increase as the
square of this greater velocity. If the entire end of the pipe
be opened so that the discharge is entirely free, then the
total head will be lost in friction due to the consequent high
velocity, except the small portion of the total head which re-

mains to generate the velocity, and which is h v=g^|. It is

evident then that if head or pressure is to be preserved the di-
ameter and area of the pipe must be sufficient to convey the re-
quired quantity at a low velocity, and the pipe must not be
permitted to discharge at anything like its full capacity. As
loss of head or pressure is directly as the roughness of the
pipe, and directly as the length, and inversely as j/d 3 , it is
necessary to take into account not only the diameter and ve-

SULLIVAN'S NEW HYDRAULICS. 177

locity but also the length of pipe, and the nature of the pipe
walls with regard to smoothness or roughness, and probable
future deterioration. The chemical qualities of the water
which is to flow through a pipe, and the effect they have upon
different classes of pipe and pipe coatings should be care-
fully ascertained before the pipe is selected. Some waters,
apparently almost pure, will corrode a pipe in a very short
time to such an extent as to reduce its capacity by nearly one
half. A pipe line made up of different diameters, gradually
decreasing as the slope increases, designed to convey a given
quantity and to maintain a given pressure, is some-
times less expensive than a pipe line of uniform
diameter. The velocities in the different diametera
of such irregular pipe lines will be inversely as the
areas of the different diameters and the friction loss in
each section will be as the square of the velocity in that sec-
tion and inversely as ;/d 8 . The loss of head in such a line
must be calculated separately for each different diameter.
In case the line is divided into divisions of equal lengths, and
each division is of a constant diameter but of a different di-
ameter from the rest of the line, the mean diameter of the
whole line cannot be adopted for such calculations, because
the mean of all the velocities in the different diameters will
greatly exceed the mean velocity in a pipe of uniform diameter
equal to the mean diameter of the line composed of different
diameters. As the friction is as the square of the velocity, it
is evident that it will be much greater in the line of decreas-
ing diameters than in a pipe of uniform diameter equal to the
mean diameter of the former. Where the saving of head or
pressure is a principal object there are only a few cases in
which it is cheaper or advisable to adopt large diameters for
the upper portion of the line and smaller ones for the lower
portion. What is saved in the cost of constructing such line
is lost in head or pressure, which maybe of more value than
the difference in cost between the two kinds of pipe line. For
example a pipe line 5,000 feet in length, made of lap welded

Sipe and thoroughly coated with asphaltum, in which the
ret 1,000 feet length has a diameter of three feet, the second

178 SULLIVAN'S NEW HYDRAULICS.

1,000 feet has a diameter of 2.75 feet, the third 1,000 feet has a
diameter of 2.5 feet, the fourth a diameter of 2 feet, and the
fifth a diameter of 1.5 feet, while discharging 8 cubic feet per
second, will have velocities and losses of head in the different
diameters as follows:

In section No. 1, =_!.=: ^=1.1178 feet, h"= .072 feet
a 7.069

In section No. 2, v=-*L = *L_ =1.3470 feet, h"= .119"
a 594

In section No. 3, v=-3_=-JL=1.6300 feet, h"= .202 "
a, 4 909

In section No. 4, v=-5-=_ =2.5400 f eet, h"= .680 "
a 3.1416

In section No. 5, v=^L= 8 =4.5200 feet, h"= 3.350 "
a 1767

4.423

The loss of head for this small discharge will be 4.423 feet
in the line of different diameters, and the mean of all the
velocities in the different diameters will be 2 2771 feet per sec-
ond.

Now if the sum of these five different diameters is divided
by 5 we have the mean diameter 2.35 feet. The area
of this mean diameter=4.3374 square feet. Conse-
quently if the entire pipe line had been of the uniform diam-
eter of 2.35 feet, the necessary velocity through it to cause a
discharge of 8 cubic feet per second would be

1.84442 feet, and the total loss of head

_
4.3374

would have been h" =1.424 feet. (For this class of pipe n=
.0003). As the friction is inversely as i/d 8 , and also directly
as v 2 , it is apparent that a small increase of the discharge
would greatly increase the loss by friction in sections No. 4
and 5 of the irregular diameter.

67 A Power Main with Nozzle, and Water Wheel to
Run at a Given Speed and Develop a Given Power.

In mountainous regions are many small torrents, the

SULLIVAN'S NEW HYDRAULICS. 179

sources of which are at such great altitudes as to afford
almost any head desired when the stream is confined within
a pipe or power main so as to preserve the head or pressure
by regulating the velocity of flow. Where the quantity of
water is small and the head is great, an impulse and re-
action water wheel will be much more efficient and satisfac-
tory than a turbine. The loss of head in a power main de-
pends upon the velocity of flow through it and upon its
length, diameter and smoothness and freedom from bends.
The velocity is governed by the quantity of wate'r the main is
permitted to discharge, and the quantity discharged is gov-
erned by the area of discharge at the point of the nozzle and
by the effective head at discharge.

The greater the length of the pipe line, the smaller the
velocity must be, for the loss of head by friction is directly as
the length and as the square of the velocity. Such power
mains or pipelines are usually constructed of riveted pipe
made of steel or wrought iron plate. The pipe is made in any
convenient lengths for transportation, or is made on the
ground where it is to be laid. After it is riveted into lengths
it is thoroughly coated by being submerged in a tank of hot
coating compound composed of 80 per cent asphaltum and 20
per cent crude petroleum which is maintained at a tempera-
ture of about 300 degrees Fahr. The pipe is allowed to re-
main submerged in the hot bath until the pipe metal attains
the same temperature as the bath. It is then withdrawn
from the bath and allowed to cool. In some cases coal tar
45 per cent and asphaltum 55 per cent is used as a coating
with fair results.

The quality or purity of the asphaltum used will deter-
mine the best proportion of asphaitum to crude petroleum to
use in the compound. The per cent of petroleum required
varies from 15 to 20. After the compound has been heated
and thoroughly mixed and incorporated it should be tested
by dipping into it a small sheet of the pipe metal and allow-
ing it to remain for ten minutes in the hot bath. It is then
withdrawn and placed in a large vessel of cold water and al-

180 SULLIVAN'S NEW HYDRAULICS.

lowed to cool. If the coating is too soft after cooling and has
a tendency to run or wrinkle, there is too much oil in it, and
the quantity of asp laltuin should be increased. If it is a mix-
ture of coal tar and asphaltum the coating will be too brittle
and easily knocked off with a hammer if the proportion of tar
is too great to that of asphaltum.

In any case the coating should be tough and elastic and
should adhere to the metal similar to paint. If the bath is
too hot, the coating will wrinkle on the inside of the pipe
when it is withdrawn and laid aside to cool. The lengths of
pipe are put together like stove pipe, by wrapping a cloth
around the end of one length and driving it into the end of
the length below, the laying always being started at the lower
end of the pipe line. This is called a slip joint. In cases
where the pressure is considerable a sleeve joint is used. A
sleeve joint consists of slipping an iron sleeve over the ends
where two pipe lengths join or are butted, and running in
melted lead between the sleeve and the pipe, having first
packed the joint sufficiently to prevent the lead from running
into the pipe where the ends come together.

In rocky, mountainous localities where trenching would
be quite expensive, the pipe is usually laid on the surface
without any trenching except where it is necessary to secure
a substantial bearing or foundation for the pipe.

In very cold weather the pipe is allowed to discharge
constantly, which prevents freezing within the pipe, or the
water is prevented from entering the pipe and the line left
empty when not in use.

Suppose a stream affords 10 cubic feet per second and
has a fall of 400 feet per mile, and it is required to construct
a water power plant that will develop 200 net horee power,
using a water wheel of 85 per cent efficiency. What head
will be required and what diameter and length of pipe, and
what will be the proportions of the discharge nozzle required?

By formula (99) 62, the net head required will be,
200-=- 85

SULLIVAN'S NEW HYDRAULICS. 181

As there is a fall of 400 feet per mile, it is seen that the
line will be a little longer than oce-half mile. The fall per

H 400

foot length will be S= -j-= 528Q = .075757575, and the

length in which there is a fall of one foot is Z=-~-= - ,-_

=13.20 feet.

Hence the length of pipe required, not making allowance
for friction loss, will be 207.065X13.20=2733.258 feet of pipe.

But as there will be loss of head by friction in the pipe
line and also in the nozzle, and it is required to have 207.065
feet net head at discharge from the nozzle, we must lengthen
the pipe line until the total head will cover these losses and
still leave the net head of 207.065 feet at discharge. If the
nozzle is to be 8 feet long we will assume that the loss of
head in the nozzle will be 6 feet while discharging 10 cubic
feet per second, and we will design the pipe line so that the
loss of head in the line by friction will be 6 feet also. Hence
the line must be extended further down the hill until we
have a total head in the whole length of the line including
the nozzle=207 .065+12=210.065 feet. In order to gain this
additional 12 feet head the line will have to be extended in
length by 158.40 feet, including the nozzle. The nozzle is to
be 8 feet in length, and therefore the pipe line without the
nozzle will be (2733.258+158 40) 8=2883.66 feet in length.
It is to be double riveted, asphaltum coated, slip joiL.t pipe,
and the total loss of bead in the whole line without the noz-
zle is to be 6 feet while discharging 10 cubic feet per second.
What diameter will be required?

By formula (101) 64, the diameter required will be
d= U/ .000000269Xq^ X ^ =1 . 795 f ee t=21.54 inches di-

ameter.

We have a net head now at the junction of the pipe line
with the nozzle of 213.065 feet. The next step is to ascertain
the required dimensions of the nozzle to discharge 10 cubic
feet per second under these conditions with a loss of head

182 S'ULLI VAN'S NEW HYDRAULICS.

not to exceed six feet in the nozzle. The method of doing
this is explained in 65. For this calculation we have the
length of the nozzle and itB butt or greatest diameter, and
the effective head at the butt of the nozzle. Now if we as-
sume that there will be probably a loss of 6 feet head in the
nozzle itself, the net head at discharge would be equal to the
head at the butt, le-s the amount lost in the nozzle, or 2 ] 8.065
6=207.065 feet. Hence by the rules given heretofore (65)
the area in square feet of the least diameter at discharge of
the nozzle will be

q 10.

= - 0866 8quare ' 66t

The diameter answering to this area is A.-*/ 1

v .7854

.11026 foot=1.323 inches

1.795+.11026
The mean diameter of the nozzle is= ^

.95263 foot.

Now we must test this nozzle to ascertain what the ac-
tual loss of head will be in it while it is discharging 10 cubic
feet per second. If the loss is not so great as six feet, as we
have assumed in the nozzle, then we mav shorten the pipe
line to some extent, or we may reduce the diameter of the
pipe line very slightly, and still obtain the required head and
power at discharge.

The velocity through the mean diameter of this nozzle in
order to discharge ten cubic feet per second would be

q _ 10

v= ~a" (.95263)"X.7854 = 14 ' 03 feet per Becond '

The actual loss of head by friction in the nozzle under
these conditions would be ( 37, 39, 65).

n 1 9 v 2 _ .0003623X8X9X ' 96.841

.9298

=55224feet.

As the net head at the base of nozzle is 213.065 feet, and
the loss in the nozzle is 5.5224 feet, we have a net head at

SULLIVAN'S NEW HYDRAULICS. 183

point of discharge from the nozzle of 207.5426 feet. The
required net head was 207.065 feet. Hence we have
.4776 foot head in excess of exact requirements, which
Is near enough the desired result.

TEST FOR QUANTITY DISCHARGED.

The area of smallest diameter of nozzle at discharge is
.0866 square foot ae above found, and the net head at dis-
charge ia 207.5426 feet.

Hence the quantity that will be discharged is q=a v, or

q=.OS66X8.025 1 /207.54 =10.000568 cubic feet per second.

Now the velocity of discharge from the nozzle is v=
8.025/26T5T =115.48 feet per second or 115.48X60= 6,928.80
feet per minute.

It has been established by experiment and experience
that the velocity of greatest efficiency of the circumference
of an impulse and reaction water wheel is about one-half the
velocity of discharge upon the wheel. The number of revo-
lutions per minute of the water wheel will depend upon its
circumference from center to center of the buckets taken as
its diameter. The circumference equals the diameter in feet
from center to center of buckets multiplied by 3.1416.

The circumference of the wheel when the load is on
should travel at one- half the velocity of the discharging
water. Hence the diameter of the wheel may be so propor-
tioned to the velocity of discharge as to run any desired num-
ber f revolutions per minute. Where high speed is desired
under a low head, two or more water wheels of equal diam-
eter may be placed upon one shaft and have separate nozzles.
In this way very small diameters of the wheels may be used
to secure high speed, and the water divided so as to avoid
placing very large buckets on small wheels and to also pre-
vent flooding the wheel. The power developed does not de-
pend upon the diameter of the water wheel, but depends up-
on its speed with reference to its diameter.

The point of the nozzle should be firmly set beyond the
possibility of slipping against the wheel, and should be as
close to the buckets as possible not to strike them or to have

184

SULLIVAN'S NEW HYDRAULICS.

the jet re-acted upon from the buckets. The distance between
the point of the nozzle and the center of the bucket on the
wheel will depend upon the diameter at discharge of the noz-
zle and the velocity of discharge upon the wheel. It should
not be so close that the jet will react upon itself on striking
the buckets.

68. Table of Eleventh Roots to Facilitate Calcula-
tions of Diameter Required to Discharge Given Quan-
tities.

The following table covers diameters from one inch to
32 inches both inclusive, and will be convenient in conjunction
with formulas for ascertaining the diameter in feet required
to generate a given discharge (formulas 28, 43, 65, 81, 100, 101)
or to cause a given discharge with a given loss of head.
TABLE No. 36.

Number

llth. Root

Number

llth. Root

.OOOOOOOOOU01345
.0000*002762

.08333
.1667

46.24
86.50

.417
.5

.000000^384

.25

156.40

583

.000005638

.3333

276.20

.667

.00006578

.4167

471.50

.75

.0004883

784.80

.002659

.5833

1285.00

^917

.01157

.6667

2048.00

2.000

.04223

.75

3203. 00

2 083

.1345

.8333

4948.00

2.167

.3842

.9167

7482.00

2.25

1.0000

1.000

11150.00

2.404

1.083

16370.00

2^416

5.467

1.167

23840.00

2 5

11.64

1.250

31300. (0

2.584

23.62

1.333

48560.00

2.667

REMARK 1. Where the pipe is to be of uniform diameter
and to have free discharge, as in the case of a pipe conveying
water from one reservoir to another, there is no object in pre-
serving the head by throttling the discharge, and in such case
the total head is consumed in balancing the resistance to flow
except that part of the head which is converted into velocity.
The diameter of a pipe which is required to convey a given
quantity of water per second under such conditions will be

.3805 H 2

SULLIVAN'S NEW HYDRAULICS. /85

m=coefficient of velocity in terms of diameter in fe*-t.

q=cubic feet per second that pipe is to discharge.
total head in feet

S =total length in feet= Bine of elo P 6

H=total head in feet.

REMARK 2 Where the pipe must convey a given quantity
per second to a given point and must maintain a given head
or pressure at that point while the given quantity is being
drawn from it, then the diameter required will be found as
pointed out in G4, or by the following general formula.

.38u5Xh" 8

This diameter will convey a given quantity with a given
loss of head which is pre-determined according to require-
ments.

In which,

h"=head in feet to be lost in friction.

n=coefficient of resistance applicable to class of pipe.

q cubic feet per second pipe is to discharge.

I =length of pipe in feet.

REMARK 3. The results of experiments by the writer on
"Converse Patent Lock Joint Pipe" made of wrought iron in
lengths of from 15 to 20 feet and lap welded, and coated with
asphaltum gave an average value of n=.000299 in terms of
diameter in feet. The small value of the coefficient of re-
sistance n, in this pipe is to be attributed to its uniformity of
diameter, and to the fact that it is made in long lengths BO
there are fewer joints per mile of pipe, and the joints are so
arranged as to present a continuous and uniform surface to
the flow. For this class of pipe take n=.0003 and m=: .00030472
in terms of diameter in feet. These values of the coeffi-
cients do not allow for future deposits in the pipe, if such
should occur, nor for deteriorationinthe pipe coating. It is not
probable that a first class asphaltum coating will deteriorate
to any considerable extent for a great number of years. This
remark has no reference to coatings made of coal tar com-
pounds.

The diameter (inside) in feet of Converse pipe, asphaltum
coated, required to convey a given quantity with a given loss
of head, would be

186 SULLIVAN'S NEW HYDRAULICS.

f q 4

Or if the discharge is to be free and full bore, and no at-
tempt made to preserve the head or pressure, the diameter
required to carry a given quantity will be

'a* m 2 11/m 2 11

= V >< l

11/m 2 I 2 q 4 11 / I s q 4

d =T/ :s805^-=- 25055 i/ -IP-

ll/ q 4
For riveted asphaltum coated pipe d =.2o41i/ -

69. Head Lost by Friction at Bends in Water Pipes.

The amount of the loss of head produced by a bend in a
pipe will depend upon the velocity of flow and the radius of
the central arc of the bend, and also upon the number of de-
grees included in the arc- of the bend. Whether the addi-
tional head required to overcome the resistance of a bend will
be proportional to the square or to the cube of the velocity is
doubtful. Weisbach's formula, which is most generally used
for determining the resistance of bends, gives results un-
doubtedly too low in all cases except for a bend of 90 with a
radius of central arc of bend equal to one half the diameter.

The resistance at a bend in a pipe or in an open channel
is caused by the change of direction of the flow. The more
abrupt the change, and the greater the amount of the change
in direction, the greater will the resistance be. It is evident
therefore that the resistance will be directly as the number
of degrees included in the central arc of the bend and in-
versely as the radius of that arc.

SULLIVAN'S NEW HYDRAULICS.

187

FIG.B.

Fig. A Bhowe a bend of 90, the radius c a, of the central
arc of the bend being equal 6 times the radius a b, of the
pipe, or equal three diameters of the pipe. Fig. B shows a
bend of 90 with the radius c e of the central arc of the bend
equal the radius of the pipe, or equal 12 diameter of pipe.

When the radius c a of the central arc of the bend is
only equal to the radius of the pipe, or to one half the diam-
eter, then the resistance or amount of head lost at such bend
will equal the head in feet which generates the velocity of

V s

flow, or h"=gj. For example suppose the velocity to be 3

feet per second through the pipe, and the bend is as shown
in Fig. B, then the head in feet lost by resistance at the bend

_ (3) 2

will be h"=

= - U foot

For a bend of 90 or any other constant number of de-
grees, the amount of change in the direction of the flow will
be the same for any length of radius c a, of the arc of the
bend, but the distance in which this change is effected will
be directly as the len th c a, of the radius of the bend.
Hence the shorter the radius of the bend the more
abrupt will be the change in direction of flow, and con-
sequently the greater the resistance. The central arc, a ed,
of the bend increases in length or becomes more gradual
directly as the radius of the bend c a increases in length and
hence the longer this radius c a becomes the more gradual
will be the change effected in the direction of the flow. The

183 SULLIVAN'S NEW HYDRAULICS.

resistance at a bend will therefore be directly as the number
of degrees included by the central arc of the bend, and in-
versely as the length of the radius of the bend and will in-
crease as v 2 (or possibly as v 8 ). The formula will therefore be

, __ Ay, r v v a _ A .5 v v 8 A .5 y v 2
~ 90 A R A -2g 90 A R A 64 4 90 X 64 4 A R

Which reduces to

In this formula (102)
r=$4 diameter of pipe=.5
R= radius of central arc o f bend in diameters of

the pipe and is to be expressed as 1, 2, 3 etc di-

ameters.
A number of degrees of the arc of bend as 30, 90,

180, etc,
v=mean velocity of flow through the pipe.

EXAMPLE OF THE USE OP THE FORMULA.

It is required to find the resistance at a bend of 180 in
an eight inch pipe where the mean velocity is 3 feet per sec-
ond and the radius of the central arc of the bend is equal 3
diameters.

,.._ A v* .007764 _ 180X9X.OQ7764 _ n4fifiju fflflf head
90 R 90X3

REMARK 1. The resistance at a bend is in addition to the
ordinary frictional resistances of the pipe walls. Hence for a
pipe which contains a bend, first calculate the loss of head by
friction as for a straight pipe, and then add the loss of head
due to the bend.

REMARK 2 It is assumed in all formulas for resistance
at bends that the resistance is independent of the diameter of
the pipe or width of the open channel, and that the resis-
tance of a bend depends solely upon the velocity, the radius
of the bend and the number of degrees included in the
central arc of the bend. It is doubtful whether the diameter
of a pipe exerts an influence on the resistance at a bend or
not. It probably does.

SULLIVAN'S NEW HYDRAULICS. 189

REMARK 3 The force exerted by a column of water im-
pinging upon a fixed surface is as the product of the quantity
of water by its head. The quantity ip directly as the velocity
and the head is as the square of the velocity. Consequently
the product is vXv 8 =v 3 . It is therefore possible that the
force or head or energy absorbed at a bend will vary as v in-
stead of v 8 .

70 Formulas of Weisbach and of Rankine for Resis
tance at Bends in Pipes.

The formula for resistance at bends proposed by Weis-
bach is

h'=.l31+1.847 (-LVx-^-X

In which

r=radius of pipe in feet=^ diameter in feet.

R=radius of axis of bend in feet.

A=central angle of bend in degrees.

2g=effect of gravity=64.4.

Professor W. J. M. Rankine's formula is simply a change
in form of Weisbach's formula, and is as follows:

In which

A=angle of bend in degrees

d=diameter in feet of pipe

r=radius of central arc of bend
To simplify Weisbach's formula, place the coefficient,

.131+1.847(-^-)*=Z. Then
V R /

, ., A ^, v 8 Av a Z y / A v 2 \
=ZX T80 X 60 = -11592- = Z \TI592 )

Remembering that in Weisbach's formula, r= half the
diameter of the pipe in feet, and R= radius of the central arc
of the bend in feet, the following table of values of Z will be
readily understood and applied:-

190 SULLIVAN'S NEW HYDRAULICS.

Value of Z in Weisbach's Formula.

.15

.225

.25

.275

.325

.35

.375

.425

.131

.133

.138

.145

.15

.155

.16

.17

.18

.195

,20i ;

.225

.45

.475

.525

.55

.575

.625

.65

.675

.725

.244

.264

.294

.32

.35

.39

.44

.49

.54

.60

.661

.73

.75

.775

.80

.825

.85

.875

.925

.95

.975

l.(X)

.806

.880

.98

1.0-

Ujj

1. 29L 41

1.51

1.68ll.8:r2.00 l

USE OP ABOVE TABLE. The velocity iu feet per second
through an eight inch pipe is 3 feet. There is a bend of 90
with a radius of bend equal 4 inches or half the diameter.
What is the loss of head in feet caused by the bend?

We see that as the radius of the central arc of the bend

is equal to half the diameter of the pipe; that -D-=1.00. Re-

ferring to the above table, and it is seen that when^s- = 1.00,
then Z=2.00. Hence by the formula,

h " =z -

The radius in feet of an 8 inch pipe=.3333 foot.
The diameter in feet of an 8 inch pipe=.6666foot.
Suppose the radius of the above bend R=.66G6 foot or
equal the diameter, and the radius of the pipe is .3333 foot.

- .6666 ~- 5

From the above table it is seen that when-p~ = -5, then
Z=.294. And in this case Weisbach's formula would give the
loss for 3 feet velocity of flow as

=.^[ggf-]==.(*feet

h " =Z 111592

This latter result is altogether too small.

SULLIVAN'S NEW HYDRAULICS.

191

71. Comparison of the Results by Weisbach's Formula
and by the Formula Herein proposed, for Bends of 90 with
Radii Varying form /?= y z d t o R= 3d, and Different Veloci-
ties.

In the following table the lose of head by friction has been
computed by our formula (102) and also by Weisbach's for-
mula for various velocities of flow through a bend of 90 in
which the radius of the central arc of the bend varies from R
=^ d to R=3d. It is possible that the results by either for-
mula are too small for the reason suggested in remark 3, 69

TABLE No. 38.
Table of computed results for comparison.

Velocities

Feet

.062

.14

.248

388

.761

.994

( Formula (102)

.0621

.14

.248

.388

.56

.761

.994

1 Weisbach

fld <

.031
.009

.07
02

.124
.026

.198
.056

.28
081

.38
.11

.497
.144

i Formula (102)
1 Weisbach

R=2d \

.0155
.005

.035
.011

.072
.(19

.C97
.029

.14

.042

.19
.057

.248
.074

j Formula (102)
1 Weisbach

.0103

0233

.0413

.'646

.093

.1656

j Formula (102)

.001

.01

.017

.027

.038

.052

.069

1 Weisbach

It would appear from au inspection of the results by
Weisbach's formula that there is little to be gained by mak-
ing the radius of the bend greater Iban twice the diameter of
the pipe. This is not true, however, in practice. The radius
of a becd should be made as great as the circumstances will
permit unless the velocity of flow through the pipe is to be
very small. The velocity should be the controlling feature
in determining the radius of the bend.

Fanning says ''Our bends should have a radius, at axis,
equal at least to 4 diameters." Trautwine advises a radius of
bend equal to 5 diameters length, or as much longer as it can
be made. If the velocity does not exceed 5 feet per second,
then a radius of 3 diameters will reduce the loss of head to
.0646 foot at a 90 bend.

192 SULLIVAN'S NEW HYDRAULICS.

72. Resistance at Bends. Rennie's Fxperiments.

While the results of experiments by Bennie on leaden
pipe one half inch diameter are not of great value as estab-
lishing any law of resistance at bends, yet they indicate
very clearly that the results by Weisbach's formula are too
low.

Bennie experimented with a leaden pipe 15 feet in length
and half inch in diameter under a total head of 4 feet. He ob.
tained the following results;

The straight pipe before being bent discharged .00699
cubic feet per second.

With one bend at right angles near the end, 00556

cubic feet per second.

With 24 right angle bends 00253

cubic feet per second.

It will be noted that the bends are described as right
angled. This may have crushed the pipe out of form and re-
duced the area at the bends, This would materially affect the
velocity and the resistance through the bend. Whether this
occured or not is not stated, Prom the area in square feet
of this half inch pipe and the quantity in cubic feet per sec-
ond it discharged we find that the velocities of discharge were
as follows:

Before the pipe was bent, v = JL= 06 ^ 9 =5 feet per sec-
ond.

With one right angle bend, v =-S_= - 00556 = 3.971 feet
per second.

With 24 right angled bends, v=-i-=-^HL= 1.80 feet

per second.

In order to prevent confusing the resistance of the pipe
walls with that of the bends, we will first find the value of
the coefficient of resistance n, of the pipe before it was bent.

The total head was 4 feet, and while the pipe was straight
the velocity of discharge was 5 feet per eecond. The head in
feet lost by friction along the walls of the straight pipe under
this velocity was equal the total head minus the head due to
the velocity of discharge, or was

SULLIVAN'S NEW HYDRAULICS. 193

4 ^f=4. .3882=3.6118 feet.

After one bend was made in the pipe, the total head re-
maining 4: feet, the velocity of discharge was only 3.971 feet
per second. Now from the data of flow in the straight pipe
before the bend was introduced we find the value of D to be

= Jv* 15X25 =- 000082 -

After one bend had been introduced the velocity was re-
duced to 3.971 feet per second, so the friction of the pipe walls
exclusive of the resistance of the bend was now

nlv"_.000082Xl5X15.76884
h =-/oT- .0085

But the total loss of head due to pipe walls and one bend
combined was equal the total head of 4 feet minus the ve-
locity head, or equal
/o 971 \

4 644 =* .244842=3.755158 feet.

If we deduct from this total loss the loss due to pipe
walls we have 3.7551582.2818=1.473358 feet head lost by the
resistance at the bend; which is equal 6 times the head gen-
erating the velocity. This would indicate that the resistance
at a bend is more nearly proportional to v* than to v*. as inti-
mated in remark 3, 69. The resistance at a bend in a very
small pipe is probably greater than in large pipes,

The total head remaining 4 feet, after 24 right angled
bends were made in this 15 foot length of half inch lead pipe
the velocity was 1.80 feet per second as determined from the
quantity discharged. The loss of head due to friction of pipe
walls, exclusive of the bends, was, for this velocity.

h , = nZv = . 000082X15X3.24 = mB feet head>
v 'd 3 .0085

The total loss of head due both to the 24 bends and the

friction of pipe wall wae H_ lL= 4. .05031=3.94969 feet.

The loss due to the 24 bends alone was therefore equal

194 SULLIVAN'S NEW HYDRAULICS.

the total loss minus the loss due to pipe walls=3.94969 .4688
=3.48 feet.

If the loss was equal at each bend, then h= 3 J 48 =

.145 foot head lost at each bend for a velocity of 1.80 feet per
second. In this caee the head lost at each bend was only
equal 2.88 times the head generating the velocity. It must
be remembered that all these bends are described as right
angled bends, It is probable that serious contractions of the
area of the pipe were produced at each such bend and that
the velocity of flow through the contractions was greater
than 1.80 feet per second.

Because of the direct action and equal reaction of the
water impinging upon the pipe wall at a right angled bend
the lose of head at such bend could not be less than twice

v 2
the head producing velocity, or h" = 2-sr- According to the

above results of Ronnie's experiments with 24 right angled
bends it appears that the loss at each bend was equal nearly

three times the head producing the velocity or h"=2.88 -|

But it is doubtful what the actual velocity was in the bends
as the areas were probably contracted.

Right angled bends or shoulders are, however, never in-
troduced into a water pipe, but the bends are always circular.
As a true right angled bend cannot be made without cutting
and fitting, or casting, it is probable that Rennie's pipe was
bent like Fig. B, 69.

73. Relation of Thickness of Pipe Shell to Pressure,
Diameter and Tensile Strength of Pipe Metal

When a pipe is filled with water and is closed at dis-
charge end so there can be no flow in it, the radial pressure
within the pipe tending to burst it will vary as the head of
water above any given point along the pipe, and at any

SULLIVAN'S NEW HYDRAULICS.

195

given point will be equal HX.434=lbs. pressure on each
square inch of the internal circumference.
A E F

In the Figure let R represent a reservoir, the water level
in which is A, and a pipe C D G, is laid from it ov*r hills and
depressions. When the pipe is closed at G, the pressure
within the pipe which tends to burst it will vary as the ver-
tical distance C E, D F, between the given point in the pipe
and the level of the water A E F B, in the reservoir. Hence
the thickness and strength of the pipe shell must be pro-
portion according to the position it is to occupy in the pipe
line. If the vertical distance CE is J30 feet then the pressure
at C on each squara inch of the internal circumference of the
pipe will be 130X .434=56.42 Ibs. But the pipe passing over the
hill at D is only 80 feet below the level of the water in the
reservoir, and consequently the pressure within the pipe at D
is equal 80X.434=34.7'2 Ibs. per square inch. A profile of the
pipe line showing the distance at all rises and depressions
along the line between the pipe and the level A E F B should
always be made before the thickness of pipe shell is calcula-
ted for any portion of the line. With such profile the thick-
ness and strength of the pipe for each division of the line
may be calculated so as to conform to the pressure it must
sustain.

The inclined line A, G, is the hydraulic grade line, or line
which indicates the hydraulic or running pressure in the
pipe when the pipe is open at G and discharging freely.

The hydraulic or running pressure within the pipe at any
given point along the pipe line is equal to the distance in-
feet, measured vertically, from the given point in the pipe to

196 SULLIVAN'S NEW HYDRAULICS.

the hydraulic grade line, A G., multiplied by .431. Thus, the
running pressure at C in the pipe ia equal the vertical dis-
tance C H in feet multiplied by .434. The difference in feet
between C E and C H shows the loss of head in feet by fric-
tion between the reservoir and C. If the pipe were laid on
the hydraulic grade line A, G, there would be no pressure in
it at all when discharging freely except that due to the depth
of the diameter. The pipe must be so laid that no part of it
will rise above the hydraulic grade line. If the pipe at D
should rise above the line A G, to K, then the line would re-
quire to be divided into two divisions, A K, and K G, both as
to diameter of pipe and as to the hydraulic grade line. The
diameter KG, if the same as A K, would not run full, for the
reason that K G would have a greater fall per foot length
than A K. Assuming the pipe to be laid as shown by C D G,
and that it is closed at G so there is no discharge, then the
internal pressure on each square inch at any given point in
the pipe will equal the vertical head in feet between the
given point in the|pipe and the line A E P B, multiplied by .434,
and the number of square inches subject to this pressure will
be directly as the diameter in inches of the pipe, because the
circumference is equal dX3.1416.

The total pressure on the inner circumference will there-
fore be HX.434XdX3.U16.

The pressure of quiet water is equal in all directions. In
a circular pipe the pressure radiates from the axis of the pipe
to every point in the circumference. The resultant of the
pressure on one half the circumference acts through the
center of gravity of that half, and equals the products of the
pressure into the projection ofjthat half circumference. The
projection of half the circumference equals the diameter of
the pipe. An equal resultant acts in the opposite direction
through the center of gravity of the other half circumference.
The resulting strain on the pipe shell at any point in the cir-
cumference is equal to the sum of these opposing resultants.
If therefore, the thickness and strength of the pipe shell is to
be lound simply in terms of the pressure resultant of one half

SULLIVAN'oQ NEW HYDRAULICS. 197

the circumference, due to the total head, it is evident that the
thickness and strength must equal twice this resultant, or,
2tS=PXd (103)

t= thickness of pipe shell in inches.

S= tensile strength in Ibs. per square inch of pipe metals.

P= pressure in Ibs. per square inch - H.X-434.

d= inside diameter of pipe ininches.

This gives a thickness and strength just sufficient to
equal or balance the pressure of the quiet water, as

t ^g- OM)

To be sufficiently strong to withstand the violent shocks
and sudden strains caused by water ram, and to provide for
defects in casting or in riveting, and to prevent breakage in
handling and from unequal settlement of the pipe in the
trench, it is necessary to make cast iron pipe very much
thicker and heavier than theory would indicate, and wrought
iron and steel pipe from three to six times as thick as the
quiet pressure alone would actually require. For these rea-
sons the formula (104) must have added to it another factor
called the factor of safety, and it then becomes

* Pd -XF (105)

The factor of safety F, may be equal 2, 3, 4 etc. according
to the service the wrought iron or steel pipe is to be put to.

This formula is not used for cast iron pipe for the reason
that cast iron pipe is BO brittle that it is necessary to give it
heavy dimensions regardless of the pressure it is to with-
stand, Wrought iron and steel pipe being flexible and
tough, does not require high factors of safety, but if laid as
a permanent line, the shell should be sufficiently thick to
prevent pitting through in case the coating is knocked off.
The factor of safety of a pipe is found by the formula

(106).

The value of S depends on the net strength of a riveted
joint, (See 74)

198 SULLIVAN'S NEW HYDRAULICS.

Many steel pipes have been in successful use under high
pressure for many years with factors of safety as low as 2.
These small factors of safety were used, however, where the
pipe was not subject to water ram.

For the reason heretofore mentioned, the formulas for
the thickness of cast iron pipe are necessarily arbitrary and
empirical.

For thickness in inches'of cast iron pipe of diameters of
less than 60 inches

t=(P+100)X.OOOH2Xd+.33(l. .01 d)

For thickness in inches of cast iron pipe of 60 inches di-
ameter or greater,

t=(P-r-100)X-OOOU2Xd.

t=thickness of pipe shell in inches.

P=pressure in pounds per square inch.

d=diameter (inside) of pipe in inches.

The tensile strength of cast iron pipe is ordinarily taken
as equal to 18,000 pounds per square inch. If made of the
best quality of iron and remelted four times, and cast verti-
cally with bell end down, the pipe would have a tensile
strength as great as 30,000 pounds per square inch, and would
be tough, so that a large part of its superfluous weight might
be dispensed with, and the thickness of shell greatly reduced
thus reducing the cost of freight, hauling and laying.

74. Values of S in Water Pipe~ t Formulas.The value
of S to be used in the formula (105) for determining the re-
quired thickness and strength of pipe shell depends on the
the nature of the pipe, whether steel or iron, and whether
welded or riveted, and if riveted, then whether single or
double riveted. The net strength of a riveted joint depends
on the ratio of shearing strength of rivets to tensile strength
of the plate, and also upon whether the riveting is done by
hand or by hydraulic power. In hand riveting the work is done
with cold rivets and the rivet boles are made from 1-32 to 1-16
jnch larger than the diameter of the rivet, and the effect of the

SULLIVAN'S NEW HYDRAULICS. 199

hammer in upsetting the rivet is not sufficient to swell the
rivet to its full length so as to completely fill the rivet hole.
Hand riveting does not leave as substantial a head on the
rivet as machine riveting and is inferior to machine riveting
in many respects. A formula for fixing the pitch of rivets in
a joint is necessarily based on the ratio of the given shearing
strength per square inch of the rivet metal to the given ten
sile strength of the plate metal. The formula must be varied
at these factors vary. The tensile strength of wrought iron
plates varies from 44,000 to 57,000 Ibs per square inch. A
good average wrought iron plate should have a tensile
strength of 50,000 pounds per square inch before the rivet
holee are made in it. The tensile strength of solid steel plate
varies from 56,000 to 108,000 Ibs per square inch.

The best iron rivets have a shearing strength of only
45,000 Ibs. per square inch. The results of a great many ex-
periments made by the Research Committee of the Institu-
tion of Mechanical Engineers (London, 1881) showed that the
ultimate shearing resistance of steel rivets was 49,280 Ibs.
per square inch for single riveted joints, and 53,760 Ibs. per
square inch for double riveted lap joints. It is very proba-
ble that iron rivets would not have a greater ultimate shear-
ing resistance than 40,000 Ibs. per square inch of livet area in
a single riveted joint riveted by hand. Very high steel of
great shearing strength is too brittle for rivets, although riv-
eted hot. Hence there is no advantage in adopting plates of
greater tensile strength than rivets of suitable shearing
strength can be found for. A steel plate of about 66,000 to
70,000 Ibs. per square inch tensile strength is as high as suit-
able rivets can be obtained for, and plates of this class will
require steel rivets of best quality. The value of S to be
used in the formula (105) should be the net strength of the
joint or pipe shell. We will first give the formula for propor-
tions of riveted joints, and then for testing the strength of
such joints. By these 'means the value of S must he deter-
mined in each case. (See 75 80.)

200 SULLIVAN'S NEW HYDRAULICS.

75. Riveted Steel Pipe For riveting cold, the best
grade of steel plate is open hearth mild steel of about 60,000
Ibs. per square inch tensile strength to be riveted with best
quality swede iron rivets of 45,000 Ibs, per square inch shear-
resistance. We have then 4500 =75 per cent as the ratio
oOOOO

of shearing strength of rivets to tensile strength of plates. In
this case 1 =1.33, is the ratio of area of rivets to net plate

required to balance the tensile strength of the plate. When
the rivet holes are made in the plate it is weakened as a whole
by a percentage found thus:

Let S=Original tensile strength :>f plate, unperforated.

S'=tensile strength of plate after holes are made.

P=pitch, inches, center to center of rivets In one
row.

d=diameter in inches of rivet hole (not of rivet).

t= thickness of plate in inches.

Then the per cent strength of the punched plate S' , to
the original unpunched plate will be

S' = P ~ d =per centS

The numerical value of S' will

r p j -\
Q' "I * *-* I v/Qx/4

O = ?s X&XI

We have just seen that in order to make the shearing
strength of the rivets equal to the tensile strengh of the plate
; in this case, the combined area of the rivets must equal 1.33
times the net plate area between holes. The plate area be-
tween the rivets holes is
(P-d)t

The area of the rivets is d 8 X. 7854. Hence the equation
(P d)Xt X 1.33=d*X .7854
From which

SULLIVAN'S NEW HYDRAULICS. 201

And
p= d.7854x2 +d=L20 -^L+d, for double riveted

joint,

But eupose the rivets had been steel rivets of 50,000 plbs.
shearing strength, and the plates as above, that is, of 60,000
Ibs. per square inch tensile strength. Then the pitch formu-
la would be worked out as follows:

50,000 Ibs. shearing strength

60,000 Ibs. tensile strength = ' 833 P er

Hence,- 030= 1-20. That is, the combined area of rivets

must be 1.20 times the net plate area between holes.
Then,

(P_d)tXl.20=d* .7854
Prom which,

p= [ d t xi 8 2o ] + d =- 6545 -r~+ d > for Bin * le riveted

And

d=1 .3094-+d, for double riveted

Observe that d=diameter of rivet hole, which is always
from 1 32 to 1-16 inch larger than the rivet before the rivet
is upset.

We are restricted to the use of the market sizes of rivets,
and should select a diameter of rivet equal to from 1.70 to 2.33
thicknesses of the plate. When the diameter of rivet is select-
ed then add 1-32 (.03125 inch) for value of d in the pitch formula

If steel plate of 70,000 Ibs. per square inch tensile
strength is used, then the best quality of steel rivets of not
less than 53,000 Ibs. per square inch shearing resistance
should be adopted. In this case the combined area of the
rivets must exceed the area of the net plate metal between
rivet holes by .32075 per cent, as below shown.
70000

'53000~ = per

202 SULLIVAN'S NEW HYDRAULICS.

Then,

(P d)tX 1.32075=d 2 .7854=total area of rivets.

And,

itXl.320?5 ] +d=-5946-^+d, for single riveted

joints
And

P= + d=1 - 19 -+ d ' for double riveted

joints.

If the pipe is to sustain an extremely high pressure, or is
subject to frequent water ram, it should be triple riveted
with a ribbon of lead 1-32 inch thick placed between the lap
of the plates. Then for a triple riveted joint with rivets and
plates of the above strengths, the pitch formula would be

P= [tx 1.32075 J +d=1.784 +d, center to center, in

one row.

After many tests of riveted joints (steel plates and steel
rivets) the Research Committee of the Institution of Mechan-
ical Engineers (London, 1881) reported that: "To attain the
maximum strength of joint the breadth of lap must be such
as to prevent it from breaking zig-zag. It has been
found that the net metal measured zig-zag should be
from 30 to 35 per cent in excess of that measured straight
across, JQ order to insure a straight fracture. This corres-

2 d

ponds to a diagonal pitch of -5- P -f-~ o~ if P be the straight

pitch and d=diameter of rivet hole To find the proper
breadth of lap for a double riveted joint it is probably best
to proceed by first setting this pitch off, and then finding
from it the longitudinal pitch, or distance between the cen-
ters of the two rivet lines running parallel across the plate."
If the net metal between two rows of rivet holes is equal
to twice the diameter of the rivet hole, the joint will be safe.

SULLIVAN'S NEW HYDRAULICS. 203

The distance of the rivet holes from edge of plate should be
equal to two diameters of the rivet hole.

In the experiments of the Research Committee they
found that a single riveted joint, riveted by hand, (steel rivets
and plate) would begin to slip or give when the stress or load
per rivet amounted to 6,600 Ibs. The plates were 3-8 inch
thick and rivets one inch diameter. A similar hand riveted,
double riveted joint, began to slip or give when the load per
rivet reached 7,840 Ibs. whereas a machine riveted joint of
similar proportions did not begin to slip until the load per
rivet was double that at which the hand riveted joints began
to give.

The value of hydraulic riveting is in the fact that.it holds
the plates more tightly together, and thus doubles the load
at which the slip in a joint commences. The size of rivet
heads and ends was found of great importance in single
riveted joints. An increase of one-third in the weight of
the rivets (all the excess weight being in the rivet heads and
ends) was found to add 8 1-2 per cent to the resistance of the
joint, for the reason that the large heads and ends held the
plates firmly together and prevented them from cocking so as
to place a tensile strain on the rivets. The committee also
found that the effect of punching instead of drilling the rivet
holes was to weaken the plates from 5 to 10 per cent in soft
wrought iron, and 20 to 25 per cent in hard wrought iron
plates, and 20 to 28 per cent in steel plates. This weakening,
of coursrf, extended only to the metal immediately around the
hole. They also found that the metal between the rivet holes
in mild steel plate has a considerably greater tensile strength
per square inch than the unperforated metal. The excess
tensile strength amounted to from 8 to 20 per cent, being
largest where the distance between rivet holes was least.

"A riveted joint may yield in three ways after being
properly proportioned, namely, by the shearing of its rivets;
or by the pulling apart of the net plate between the rive*
holes; or by the crippling (a kind of compression, mashing or
crumpling) of the plates by the rivets when the two are too

204 SULLIVAN'S NEW HYDRAULICS.

forcibly pulled against each other. It also compresses the
rivets themselves transversely at a less strain than a shearing
one; and this partial yielding of both plates and rivets al-
lows the joint to stretch considerably before there is any
danger of actual fracture. Or in steam or water joints it may
cause leaks without further inconvenience or danger."
Trautwine.

In view of the results of the experiments as to the slip-
ping, or giving or "crippling" of joints, as shown by the re
port of the Research Committee, it is evident that if an ab
solutely water tight joint is to be made to stand high pres-
sure, the pitch of the rivets must be less than would be in-
dicated by the theory of simply equalizing the shearing
strength of rivets and the tensile strength of plates. The
crushing or mashing load, within elastic, limits, must be
observed.

76. Table ot Proportions of Single and Double Riv-
eted Joint, Mild Steel, Water Pipe Joints. The pitch of
the rivets iu the following table is for sheet steel of 60,000 Ibs.
per square inch tensile strength, and for Swede Iron rivets
of 45,000 Ibs. per square inch shearing strength. The lap for
any class or strength of plate in the straight seams should
equal 5 diameters of the rivet hole in single riveted joints,
aid 8 diameters of the rivet hole in double riveted joints.
This gives two diameters distance between edge of rivet hole
and edge of plate in both single and double riv-
eted joints, and in double riveted joints also gives
two diameters (straight distance) between the two rows
of li vets, or three diameters straight across from one pitch
line to the other. Such lap gives more friction between the
plates, is more rigid and less straining on the rivets, and may
be scarped down better than a smaller lap. The round seams
should have a lap of three times the diameter of the rivet
ho!e, and pitch as for single riveted joint.

SULLIVAN'S NEW HYDRAULICS.
TABLE No. 39.

205

.2
"3

S
o>
a

*o,2

"S-2

S-s

2xl

.2+J

W .

G.No.

In.

0.220

7-16

.46875

.21972656

1.06875

1.66725

2.11-32

3.34

.203

3-8

.40625

lt)r>03906

0.89405

1.38185

2.1-32

3.14

.180

3-8

.40625

.1650:i90t5

0.95638

1.50651

2.1-32

3.14

.165

3-8

.40325

.16503906

1.00000

1.64285

2.1-32

3.14

.148

.28125

.07910256

0.6019r>

0.92262

.13-32

2.1-4

10-

.134

1-4

.2^12')

.07910256

0.63545

0.98963

.13-32

2.1-4

11.

.120

1-4

.28125

.07910256

0.67676

1.07227

.13-32

2.114

12.

.109

3-16

.21875

.04785156

0.48215

0.74555

.3-32

.3-4

13.

.095

3-16

21871

.04785156

0.52097

0.82319

.332

.34

14.

.083

3-16

.21875

.04785156

0.5646}

0.91058

.3-32

.34

15.

.072

1-8

.1562^

.02441400

0.35970

O.F6315

0.25-32

.14

16.

.065

.15625

.02*41406

0.38161

0.60697

0.25-32

.14

77 Table of Decimal Equivalents to Fractional Parts
of an Inch.

The following table will greatly facilitate calculations of
of riveted joints.

TABLE No. 40.

1-32=. 03125
1-16=. 0625
3-32=. 09375
1-8=. 125
1-8 +1-32=. 15625
1-8+116=. 1875
1-8 +3-32=. 21875
1-4= 25
14+1-32=. 28125
14+1 16=. 3125

1-4+3-32=. 34375
3-8=. 375
3-8+1 32=. 40625
3-8 + 1-16=. 4375
38+332=46875
1-2=. 50
1-2 +1-32=. 53125
1-2 +1-16=. 5625
1-2+3:52=. 59375
5-8=. 625

5-8 +1-32=. 65625
5-8+1-16---. 6875
58+3-2=.71875
34=. 75
3-4 +1-32=. 78125
34+1-16=. 8125
34 +3-32=. 84375
7-8=. 875
7-8 +1-32=. 90625
7-8 +1-16=. 9375
7-8+3-82=. 96875

SULLIVAN'S NEW HYDRAULICS.

78. Weight of Each Thickness, Per Square Foot, of
Sheet Iron and Steel

TABLE No. 40 A .

CT 1

Thickness
Inches

0.300
.284
.259
.238
.220
.203
.180
.165
.148

11.48 lb.-
10. 47 Ibs
9.6191bt

8.8921b<
8.2051b t
7.2751bt

5.9811bs

11.591bt
10.571b'
9.7151b^

8.981 ]b
8.287)bt
7. 348 Ibs
6.736 Ibf
6.041 Jhi-

0.134
.120

.083
.072
.065
.058
.049

4.8501bs
4. 405 Ibs
3 840 Ibs
3. 3551 be
2.9!01bs
2. 627 Ibs
2. 344 Ibs
l

5. 470 Ibs
4. 899 Ibs
4. 449 Ibs
3. 878 Ibs
3.3881b8
2. 939 Ibs
2. 653 Ibs
2. 367 Ibs
1.

79.- Calculating Weight of Lap- Joint Riveted Pipe.

In measuring the length of a sheet of metal to make a circle
of given inside diameter, allowance must be made for the
contraction or compression of the metal in bending, This
contraction or shortening of the plate in bending equals the
thickness of the plate to be bent. Consequently the length
of plate required to make a lap riveted pipe of a given inside
diameter in inches must be equal to (d+t)X3.14164- required
lap In inches. d= required inside diameter in inches, and
t= thickness of plate to be bent, in inches. The weight of
the metal punched or drilled out in making the rivet
holes for straight and round seams is about equal to 25 per
cent of the weight of the rivets. Consequently take the
weight of the solid plate of required dimensions (Table No.
40A) and add 75 per cent of total weight of rivets required.
If the pipe is to be coated or flanged, this 1 must also be added
to the weight. Allow for lap of each round seam as much
loss of length of pipe off each sheet of metal as six times the
diameter in inches of the rivet hole, except for the two
sheets forming the ends of a length of pipe which will be

For the straight seams the lap should be
Lap=dX8. for double riveted pipe joints.

SULLIVAN'S NEW HYDRAULICS. 207

Lap=dx5, for single riveted pipe joints.

And for round seams dx3=lap at each end of each
sheet.

Observe that d= diameter of rivet hole in calculating
lap, and in calculating the pitch of the rivets.

80. Tests for the Strength of a Riveted Lap Joint-

To ascertain the actual net strength of a riveted lap joint pro-
ceed as follows:

Let S^tensile strength per square inch of plate before
punched.

S' tensile strength of plate per square inch after
punched.

t=thickness of plate in inches or decimals of an inch.

d=diameter in inches of rivet hole.

P=pitch, or distance from center to center of rivets in
one row.

Then the net tensile strength of the punched plate will

EXAMPLE.

The original unpunched plate had a tensile strength of
say 60,000 Ibs. per square inch, or S=60,000.

The plate was of No. llguage steel and .12 inch thick,
or t=.12.

The diameter of rivet hole was d=.28125.

It was double riveted and the pitch of the rivets in one
row was P=1.07227.

Then,

The strength before the rivet holes were made was
S=SSXt-60,000=.12X7,200 Ibs.

Then the actual value of S to be used in the formula for
thickness and strength of pipe shell (105) would be

=- 7377 P er cent of S > or 60,OOOX-7377.

208 SULLIVAN'S NEW HYDRAULICS.

The test for actual strength of plate between rivet holes
in one row being satisfactory, we then test the joint for its
resistance to shearing of rivets. The area of net plate be-
tween two holes in one row was

f P d 1
Net plate = [-p J Xt.

But as the shearing resistance per square inch of rivet
metal was only 75 per cent of the tensile strength of the plate

60,000
metal, we made the rivet area = ^ QQQ=I 33 times the

net plate area.

Then, if R= resistance to shear of rivets, we have

R= [ p J XtXl-333X45,000=5311.48 Ibs.

This shows the tensile and shearing strength to be equal.
As to test for "crippling strength of joint, we have Traut-
wine's rule. N=number of rivets in one inch length of joint.

1 2

N -p- for single riveted joint, and N=-p- for dou-

joint. In this double riveted joint N=
=1.86522.

Then,

Crippling strength=NX2 tXdX60,000=7,554 Ibs.

The value of S to be used in formula (105) should be the
smallest of the three values above found if the pipe is to be
absolutely water tight, which in this case was S=60,000 X
.7377 per cent.

81. Testing Plates for Internal Defects. The quality
of iron or steel as to density will of courpe be determined by
the weight per cubic unit of the metal. Light weight indi-
cates weakness and impurities in the metal. Internal lamin-
ations may be detected by standing the plate on edge and
tapping it all over with a light hammer. If the sound is dull.

SULLIVAN'S NEW HYDRAULICS. 209

the plate is laminated internally, but if the ring IB clear and
sharp the plate is sound. Another test is to place supports
under the four corners of the plate and throw a thin layer
of dry fine sand upon the plate, and tap it lightly with a
hammer. If the plate is defective, the sand will collect over
the defective places, but if the plate IB sound the vibrations
will throw the sand off the plate.

82. Different Methods of Joining Pipe Lengths.
Cast iron pipe is usually made in lengths of 12 feet, having
an enlargement at one end of each length called a bell or hub,
to receive the spigot end of the next length. After the spigot
is inserted into the bell and adjusted so as to fit up closely at
the end and bring the pipe into line, a piece of jute, old rope,
or gasket cut long enough to reach around the pipe with a
small lap, is forced into the joint to prevent the melted lead
from running into the pipe. A fire-clay roll with a rope
centar is now wrapped around the pipe cloae to the bell with
its two ends turned out along the top of the pipe to guide the
melted lead into the joint. The lead is made sufficiently hot
to flow freely, and is poured in until the joint is full. The
lead is then calked back into the joint all around the pipe
with a calking tool.

Lap welded pipe, such as the converse lock joint pipe
have hubs similar to cast iron pipe, and the lead is poured by
the use of a pouring clamp. Lap welded and riveted pipe are
sometimes joined by a butt sleeve joint, In this case the
ends of two pipe lengths are butted evenly against each other
and an iron or steel sleeve somewhat thicker than the pipe
shell, is drawn over the joint, leaving a epace of % inch be-
tween the sleeve and pipe. A little packing is then inserted
to prevent the lead from running into the pipe, and the space
between the sleeve and pipe is then run full of melted lead.

When there is too much water in the trench to permit of
pouring hot lead in pipe joints, several pipe lengths may be
joined together on the surface and afterward lowered into
the trench by the use of several derricks, and these compound
lengths may be jointed in the trench by forcing small lead

210 SULLIVAN'S NEW HYDRAULICS.

pipe into the joint and setting it up firmly with a calking tool.
The method of making a slip joint was described in K7.

83. Reducers for Joining Pipe Lengths of Different
Diameters.

Where a pipe line is made up of different diameters, or
where a small pipe is to be connected to a larger pipe, a re-
ducer should be used which is simply a short length of pipe
converging from the larger to the smaller diameter. In the
investigation of friction in nozzles and converging pipes it
was shown that the friction in a converging pipe is much
greater than in a uniform pipe whose diameter is equal to
the mean or average diameter of the converging pipe. The
friction in a converging pipe depends upon its length and
mean diameter. Its mean diameter should be as great as
possible and its length as short as possible provided it does
not converge more rapidly or at a greater angle than the form
of the vena contracta or contracted vein. If d is the inside
diameter of the larger pipe, then in a length of the reducer

equal-o~, the diameter should converge to d'=dX-7854. For
example a pipe of 20 inches diameter is to be joined to a pips
of 3 inches diameter, and it. is required to find the length of
the reducer in inches.

Let d = diameter in inches of the large pipe=20.

d'=diameter in inches of the small pipe=3.
d 20

Then, in a length=-^- =~2~ = 1 inches, the reducer

must converge to a diameter=dX.7854 = 20 X -7854 = 15.708
inches. Hence total amount of convergence is d d'=20
15.708=4.295 inches in a length of 10 inches, or the rate of

convergence per inch length of the reducer is ,Q* J =.4292 of
an inch per inch length. Or the diameter will converge 1
inch in a length of ^92 = 2 -33 inches. It shoutd there-
fore converge from 20 to 3 inches diameter in a length I =
(d d')X2.33=(20 3)X2.33=39.61 inches. If a diameter of

SULLIVAN'S NEW HYDRAULICS. 211

one foot is to be joined to a diameter of .7854 foot, then the
length in feet of the reducer should be I s= (d d') X 2.33=
(1. .7854)X2.33= .5 foot.

In this latter case d and d' are expressed in feet. All
reducers and all nozzles for fire streams or power mains, and
conical pipes in general should conform to the foregoing pro-
portions where the most effective delivery and smallest loss
by friction and contraction are desired.

The rate of convergence is one inch in 2.33 inches length
or one foot in 2.33 feet length of the converging pipe and
hence the length of the convergent pipe or reducer will be
found by the general formula

f=(d d')X2.33 (107)

If I is expressed in inches then d and d* must be in inches

If l=is in feet, then d and d' must be in feet.

d=largest diameter.

d'=smallest diameter.

CHAPTER VI.

Plow in Open Channels of Uniform Cross Section.

84. Permanent and Uniform Flow. Permanent flow
may .occur in a channel either of uniform or non-uniform
cross section. The flow is said to be permanent when an
equal quantity flows through each cross section in equal
timea. If the cross sections of the channel are of unequal
area the velocities will be inversely as the areas, in the case
of permanent flow. Uniform flow can only occur in a chan-
nel of uniform cross-section and grade. By uniform flow is
meant that both the mean velocity and the quantity are
equal at all places along the channel. In this case the slope
of the water surface and the slope of the bottom of the chan-
nel are necessarily the same, otherwise the velocities or quan-
tities passing different points would not be equal. In natural
streams with firm beds which are not undergoing scour and
fill, the flow will become permanent if the supply of water is
constant and uniform. These conditions can scarcely occur
in large streams of great length, but may occur in email riv-
ers or creeks. In artificial channels such as irrigation canals
and mill races where the area of cross section and grade are
uniform, and where the quantify admitted into the canal is
constant and uniform, both permanent and uniform flow will
occur after sufficient time has elapsed for equilibrium to be
established between the acceleration of gravity and the re-
sistances to flow, provided seepage and evaporation are not
appreciably great, as sometimes they are.

If different portions of a canal are all of uniform sectional
area but the slope is different in the different divisions, the
flow may become permanent, but cannot become uniform un-
less the roughness and resistances in the portions of greatest
slope happen to be just enough greater than in the other divis-
ions to equalize the velocity head in all. In such

SULLIVAN'S NEW HYDRAULICS. 213

case each division might be considered separately
and the flow might be called uniform in and for
any given division of the canal in which the
area and slope are uniform. With the exception of flumes
aqueducts, and canals lined with masonry, there are few open
channels in which uniform flow takes place. The variations
in grade, area of cross-section and roughness of perimeter
may each be slight and yet the effect is marked. In uni-
form flow the resistances and accelerations of gravity must
be constantly equal to each other. If the slope varies the
head will be greater in one division than in another. If the
sectional area varies the resistances will be inversely as y'r 8 ,
and will also be increased by cross currents and re actions of
the particles of water which impinge upon the irregularities
of the perimeter and react therefrom. The resistances due
to mere irregularities of perimeter are similar to the resist-
ances of a bend in a pipe or open chancel . They deflect the
particles of water impinging upon them and thus destroy an
amount of head depending upon the angle of deflection and
the velocity of the particles affected.

Where the width of a channel is alternately small and
then greater, the resistances are similar to these in a con-
vergent or divergent pipe, and will vary with the mean value
of i/r 3 for a given convergent length of channel and with the
mean velocity through the section having the mean value of
r for the given length considered. It is apparent, therefore,
that a coefficient which would apply at one station or to one
given short length of a non-uniform channel, will not apply
at another station or to another given length unless the same
conditions of roughness and convergence of banks obtain at
both.

In natural streams containing bende of varying grade,
depth and width, there will be what may be termed velocity
of approach in many of its divisions which will cause veloci-
ties in short straight reaches of the channel which are ap-
parently greater than the velocity due to the apparent slope.
A coefficient of velocity C, developed from the data of flow

214 SULLIVAN'S NEW HYDRAULICS.

observed at such places will be much too high to be appli-
cable at any other reach or to any other conditions of flow.
Such conditions are most common at low water stages, and
may not obtain at the same place during medium and high
stages of water. In natural and non-uniform channels the
areas for different depths of flow and the various angles
made by the banks at different heights, and the varying de-
grees of roughness of the banks above the usual depth of
flow, so complicate the conditions for different depths of flow
at any given station that it is necessary to find the value of C
for the given station under each separate set of conditions.
In Section 13 an approximate method of determining C under
such conditions has been pointed out. It will require a con-
sideration of the form of the channel above and below the
observation station as well as at the station. No one formula
without the aid of auxiliary formulas, such as suggested
in 13, supplemented by experience and sound judgment,
can be made to apply to the conditions of flow in rivers and
irregular channels. With all attainable aids, we can only
expect fairly approximate results in such cases. We shall
therefore consider the flow in artificial channels of uuiform
grade and sectional area, or channels in which, by courtesy,
these conditions are said to be approximated. It would be
closer the truth to say that the flow is permanent to a degree
approaching uniform flow in each division of uniform slope.

The closer the actual conditions approach to uniform
flow the closer will be the computed results by the formula
for flow.

85. Resistances and Net Mean Head In Open Chan-
nels. In channels of uniform grade and cross section the re-
sistances to flow consist in the friction of the liquid in contact
with the perimeter and the internal resistances among the
particles of water themselves. The internal resistances are
caused by the distortion of the onward course of some of the
particles of water causing them to collide with and distort the
course of other particles.

These distortions have their origin in the small inoquali-

SULLIVAN'S NEW HYDRAULICS. 215

ties or roughnesses along the sides and bottom of the channel
against which the moving particles flow, and from which
they are hurled off in eddies angling across the path of the
parallel flow. Difference in the temperature of different par-
ticles of water, which may be caused in part by impact and
velocity, also causes upward and downward movements
among the particles of water. If each particle of water
moved uniformly in a course parallel to the bottom and sides
the term resistance to flow would include no element of any
importance except what is called the friction of the liquid
with the solid perimeter. The results of experiments estab-
lish the fact, however, that the sum total of all the resist-
ances whether internal or of friction at the perimeter, are
proportional to the extent of wetted perimeter, in channels of
uniform cross section and slope. The internal resistances
among the particles of water are not caused by friction of one
particle with another, but by the collisions and reactions of
particles travelling in different directions. There can be no
friction as between the particles themselves for they have no
roughnesses to interlock or by which they can take hold on
each other.

The molecules are independent, free bodies which act
upon each other by impact only, and not by friction.

If the flow could occur without any resistance of any
nature the effect of gravity would accelerate the flow so that
the rate of velocity at any given point down a uniform grade
would equal the square root of the total fall in feet between
the origin of flow and *he given point. The velocity on a uni-
form grade would therefore constantly increase each second.
As this result does not actually occur, but on the contrary
the mean velocity becomes uniform throughout the length
of such grade, it is evident that the acceleration of gravity
has been balanced by and is equal to the combined resist
ances to flow. It is equally evident that the resistances are
as the square of the velocity or are equal to the total head in
each foot length. If this were not true there would be a gain
in unresisted head in each foot length of channel, and to this
extent the acceleration of gravity would cause the velocity to

216 SULLIVAN'S NEW HYDRAULICS.

increase in each foot length of channel, and there could be no
such thing as uniform flow under any conditions, and all for-
mulas based upon the theory of uniform flow would necessar-
ily fail .Attention was called to this in the discussion of coeffi-
cients and the law of variation of coefficients. It is mentioned
again here because some hydraalicians of eminent ability
contend that the coefficient of friction or rather of resistance
will decrease with an increase in the velocity, which means
that the acceleration of gravity is greater than the combined
resistances to flow. If that contention can be established it
must be admitted that uniform flow is an impossibility either
in pipes or in channels of uniform grade and cross section.
The writer is not yet ready to make that admission. The law
governing the flow in pipes of uniform diameter is the same
which governs the flow in all uniform channels. The theory
of flow and resistance to flow was discussed in general hereto-
fore (3 to 7 inclusive) and need not be repeated here.

It is evident that the velocity of any given film or parti-
cle of water will depend upon the net unresisted head of such
film or particle after the resistances to its flow have been
balanced. It is equally evident that the mean of all the dif-
ferent velocities in a cross section will depend upon the
mean net head of all the particles. If the mean net head in-
creases more rapidly than the resistances, it follows that the
rate of velocity will increase in every successive foot length
of channel; which we know is not the case. In channels of
uniform grade and cross section the sum of the resistances
per foot length of channel is equal to the head included in
each foot length, and tbus leave the net unresisted head, or
velocity head, a uniform and constant quantity, and the uni-
form mean velocity is as the square root of this constant net
mean head.

There is no friction between the molecules of the atmos-
phere and the molecules of water at the surface. Before fric-
tion can occur between two independent bodies it is neces-
sary that both of the bodies should have projections or rough-
nesses which would interlock, and require force to separate.

SULLIVAN'S NEW HYDRAULICS. 217

When winds occur, the molecules of air are hurled against
the molecules of water and thus create resistance by distort-
ing the course of the water from its direct path, if the direc-
tion of the wind if not the same as that of the flow,but if the
wind follows the direction of the flow of the water with a
downward sweep it does not resist, but assists the flow. The
small bombardment of the water surface by molecules of air
caused by difference in temperature of different air strata
does not cause any appreciable resistance to or distortion of
the flow. In truth, it may be said that none of the resist-
ances to flow are due to pure friction, but are all due to
changes in direction of the courses of different molecules
which produces internal collisions and reactions as well as
collisions with and reactions from the solid perimeter. The
projections and inequalities along the perimeter, however
small they may be, distort the course of the molecules of
water impinging upon them, and the reaction sends them ed-
dying across the path of the adjacent molecules causing fur-
ther distortions and reactions among the molecules them-
selves. Roughnesses along the bottom of a channel cause
whirls and boils and vertical currents which spend their en-
ergies in reaching the water surface and there spread out in-
ert and without direction or velocity. For this reason the ve-
locity at the surface is less than it is below the surface, which
fact has led some persons to believe that there is friction be-
tween the atmosphere and the water surface.

Such boils rise above the surface of the water on the
same principle that water rhes above the surf ace in a Pitot
tube, and when it reaches the height due to its velocity, its
energy is spent, and it spreads out in all directions upon the
surface. Abrupt bends or changes in the direction of flow
produce impact and reaction and cause the formation of
whirls and cross currents which are finally overcome by con-
tact with the onward flow at the expense of considerable head,
the amount of which will depend upon the angle and the
radius of the bend. These remarks in connection with the
laws of resistance given at -. and the discussion of the re-

218 SULLIVAN'S NEW HYDRAULICS.

lationsof area to wetted perimeter and the resulting relations
between acceleration and resistance discussed in 3 to 7
both inclusive, it is believed will cover all the important fea-
tures relating to flow and resistance to flow in channels of
uniform grade and sectional area. There are, however
certain ratios and relations of surface, to mean and bottom
velocities in open channels which demand a separate and
more special investigation, as the knowledge of these re-
lations has always been involved in much uncertainty. The
writer's theory of these relations is entirely original, and is
based upon his theory of coefficients of resistance and upon
observation and experiment.

86. There is no Constant Ratio Between the Surface,
the Mean and the Bottom Velocities.

It cannot be denied that the velocity of flow of any given
particle of water will depend wholly upon the net unresisted
head of such particle.

The conditions under which the motion of any given par-
ticle takes place will vary with the relative position of the
particle in the cross section with reference to the perimeter,
which is the original place of impact and reaction. The dis-
tance that a rebounding particle will be projected into and
across the flow will depend upon the difference in the velocity
along and near the perimeter and the velocity at the center
and surface of the cross section, or the difference in the
velocity of the rebounding particle and that of the particles
with which it comes in collision. The action of a particle of
water is similar to that of a billiard ball. When it impinges
upon a projection along the perimeter its course is changed
so that it travels diagonally toward the opposite bank or
surface, but instantly meets the opposition of the particles
having a direction of flow parallel to the perimeter.

The force and direction of the reaction is changed and
reduced with each successive collision as the rebounding
particle travels across the parallel flow, until its direction
also becomes parallel and the resistances and collision cease
as to that particle.

SULLIVAN'S NEW HYDRAULICS 219

These impingements and reactions along the sides and
bottom are in continual progress and are naturally stronger
at the place of their origin along the perimeter than else-
where and grow weaker and weaker as they approach the
center of the volume of flow. The number of these reactions
will be directly as the roughness of the perimeter. If the
bottom of the channel is corrugated transversely the entire
volume of water will rise and fall and reproduce the corru-
gations on the surface, thus agitating the entire volume of
flow.

In such case there will be only a small difference in the
surface velocity and that at mid-depth, but the bottom ve-
locity will be almost nothing. If the sides and bottom of the
channel are fairly uniform and smooth there will be very
little disturbance at the surface and a small number of re-
actions from the bottom and sides, and the bottom velocity
will be proportionately much greater, which will result in in-
creasing the mean velocity. It is well known that the mean
velocity will increase very rapidly in uniform channels or di-
ameters, simply by increasing the hydraulic mean radius
without increasing the slope. This is accounted for by the
fact that as diameter or hydraulic mean radius increases, the
area of cross section of the column of water gains very rap-
idly on solid perimeter aud there will be a very large rela-
tive quantity passed which, in smooth, uniform channels of
large radius, will not come in contact with the perimeter nor
any other retarding influence. The result ie to increase the
rate of mean velocity, not by increasing the bottom velocity
but by increasing the area or section of the unretarded por-
tion of the vein, or the number of particles of water having
an unresisted head. An increase in hydraulic mean radius or
of diameter can not affect the velocity of the water in con-
tact with the perim iter or affected thereby. It do '8 not re-
move the resistance nor add anything to the net head or
freedom of flow of these particles. An increase in hydraulic
radius or diameter cannot relieve the roughness ol the peri-
meter nor the reactions therefrom, nor does it ad I anything

220 SULLIVAN'S NEW HYDRAULICS.

to their head. There is no conceivable reaeon.therefore, why
the bottom velocity should increase or decrease with changes
in hydraulic mean depth or diameter, because it will be af-
fected by the same retarding influences and resistances re-
gardless of the value of the diameter or hydraulic radius.
The velocity along the sides and bottom of a channel will
therefore depend solely upon thd degree of roughness of the
wetted perimeter and the slope of the channel, and will in no
manner be affected by an increaee in the hydraulic radius or
size of the channel. It cannot be maintained that the rapid
movement of the upper central core of the liquid vein will
assist the flow at the sides and bottom, because the minute
globules of water are independent of each other anJ are
without friction among themselves. There are no rough-
nesses upon these globules of water by which they can take
the slightest hold on each other. If there were any rough-
nesses upon them they would interlock and the flow would
become uniform and as great at the perimeter as at the cen-
ter, or would be brought to rest entirely by friction with the
perimeter. There is nothing to affect the velocity of flow
of any particular portion of the vein except the constant
net head it has remaining after the resistances to its flow
have been balanced. As an increase in hydraulic mean rad-
ius cannot relieve the roughness and reaction at the peri-
meter and the consequent loss of head to the portion of the
vein thus affected, it cannot therefore increase its velocity
which must depend solely upon the inclination of the chan-
nel and roughness of perimeter The velocity of the water
affected by the perimeter will be the same for the same slope
and same degree of roughness regardless of the sizj of the
channel and regardless of the mean and surface velocity.
This is directly confirmed by the fact that very high mean
and surface velocities may be permitted in large canals with-
out damage by erosion of the bed, while such mean velocity
in a small canal would rapidly destroy its bed. The
reason is that the small canal would require a steep slope
to generate a high mean velocity because the whole volume

SULLIVAN'S NEW HYDRAULICS. 221

of water in a small canal is affected by the resistance of
and reactions from the perimeter, and consequently the bot-
tom velocity which is controlled by the slope, would be dis-
astrously high.

The smoother the perimeter, the fewer the reactions and
disturbances, and the greater the area of cross section un-
affected by retarding influences, and as the area of unresisted
section increases, the mean velocity will increase. In such
case the ratio of surface to mean velocity will be small but
the ratio of bottom to mean or surface velocity will be great.
The mean velocity is apparently largely controlled by the
ratio of area to perimeter as well as by smoothness of peri-
meter and slope of channel.

The bottom velocity is controlled entirely by the slope and
the roughness of perimeter. After the depth of flow is sufficient
to remove the water surface from the small reaction from the
bottom in a fairly smooth channel, the surface velocity de-
pends only upon the slope and nothing else.

It is evident that there is no fixed ratio between any two
of these three velocities. The different velocities are.'dependi
ent upon separate and distinctly different conditions. The
mean velocity gains as area gains over perimeter without any
increase of slope, not because the maximum velocity gains, but
because a greater number of particles are set free from the
retarding influences of the perimeter and thus increase the
sectional area of the vein having the higher velocity. This
does not affect the bottom velocity because there is no
change of slope. If the channel is comparatively deep and
has a smooth bottom, a further increase in hydraulic mean
depth would not affect the maximum surface velocity which,
under these circumstances would be removed from the
effects of reactions from the bottom and would therefore
only be increased by an increase of slope simply. It is evi-
dent that the relation of the maximum surface velocity to
the bottom velocity is more constant than the relation of
surface to mean or of mean to bottom velocity, and it is also
evident that there are so many different influences affecting
the one which does not affect the other to an appreciable
degree, that it cannot be said that there is any given ratio or
relation between any two of them.

222 SULLIVAN'S NEW HYDRAULICS.

The relation between them will be very different in a
shallow rough, stony channel from what it will be in a deep
smooth channel, and the relation will change in any given
channel with changes in depth of flow. It has been demon-
strated that the mean velocity will increase as /r 3 while all
other conditions remain constant. The increase in r does
not affect the bottom velocity at all. An increase in r may or
may not increase the maximum surface velocity. The
various empirical formulas for deducing the mean or the
bottom velocity from the surface velocity are therefore
totally unreliable, for such a formula can only apply to one
set of given conditions. If such formula would apply to a
wooden trough two feet wide and one foot deep, it would not
apply to a canal five feet wide and three feet deep. If it
would apply to a canal with smooth and uniform perimeter
it would not apply to a rough canal of like dimensions. Such
formulas are therefore not of sufficient importance to de-
mand discussion.

87. The Eroding Velocity in Unpaved Channels in
Earth.

In irrigation engineering there is no one feature of greater
importance than the proper adjustment of the eroding
velocity, or velocity adjacent to the sides and bottom, to the
character of the soil which must form the perimeter of the
canal. There is one particular bed velocity best adapted to
each different class of earth. From considerations of econ-
omy it is desirable to maintain as high a velocity as the
nature of the material forming the canal bed will stand
without damage by erosion.

The stability of the bed of a canal will depend upon (1)
the nature of the material forming the bed, (2) the alignment
of the canal. (3) the angle made by the side slopes, (4) the vel-
ocity of flow of that portion of the vein adjacent to the sides
and bottom, (5) the action of frost, or climatic influences.

The destruction of the side slopes depends as much or
more upon the angle made by them as upon the velocity of
flow in contact with and adjacent to them. In cold climates

SULLIVAN'S NEW HYDRAULICS. 223

where frost penetrates the earth to a depth of several feet
the side slopes should be much flatter for the same nature of
material than in climates not subject to frost.

The eroding velocity in a majority of cases is only the
partial agent of destruction of the bed. Bad alignment and
side slopes too steep to withstand the disintegrating action
of alternate freezings and thawings are the principal factors
in destroying the uniformity and efficiency of the canal.

In a canal of uniform section with direct alignment the
only velocity which tends to erode the perimeter is the vel-
ocity of the water which is in contact with it, which velocity
is governed entirely by the slope and roughness of peri-
meter and is not affected by the value of the hydraulic mean
depth.

On the contrary if the canal has bends and curves, then
the surface, mean and bottom, and all intermediate vel-
ocities, become eroding velocities at all places where the
direction of flow is changed. The outer bank of the curve
must form the resistance which forces the change in direc-
tion of flow. The amount of this resistance will depend upon
the amount of change in direction of flow and the time or
distance in which the change is finally effected. It requires
work and power, (see 60) The resistance will therefore be
distributed along the outer curves over a distance depending
upon the abruptness of the curve or upon the distance in
which the total curvature is effected. The power expended
upon each square unit of area of the outer curve will there-
fore be directly as the radius of the curve. This is the
measure of resistance which each unit of area must be
sufficiently stable to offer, otherwise it will be eroded and
removed.

A comparison of the coefficients for straight flumes with
the coefficient of the crooked Highlme flume (Group No. 5)
would indicate that the resistance of a bend of 90 with a
radius equal one-half the width of the channel would amount
to at least twice the head iu feet generating the mean velocity
of flow. If this ratio of resistance holds good in channels of

224 SULLIVAN'S NEW HYDRAULICS

all widths then the resistance (which is equal to the head
required to balance it) would be

A 2v*X. 007764 AX2v 2 X.OQ776*

^go^X ~~R~~ 9oxR (108)

In which

A=angle in degrees included in central arc of bend.
R=radius of central arc of bend in widths of the
channel, not feet.

For further discussion see 69 et seq., where the for-
mula is explained in detail.

In channels with converging banks the resistance, which
they must be sufficiently stable to offer and withstand is
similar to that in a conical or convergent pipe( 37,39), and
therefore will vary as (3Xv) s , when v= the mean velocity
through the section of the convergent length at the point
where the value of r is the mean or average value of r for the
whole length of the convergent channel. If the channel is
both curved and convergent at the same place, then the
banks must be able to withstand the resistances due to both
causes. The necessity of direct alignment and of uniformity
of cross-section is therefore apparent, if we would avoid
erosion and yet maintain a reasonably high mean velocity.
In large rivers which have small slope and frequent bends
with cross-sections alternately wide and shallow and then
deep and narrow, all the velocities become eroding velocities
and their forces vary inversely as !/r 8 . The work done by
the impinging water is in the direction of straightening the
bends and trimming the sides so the width will be uniform,
and in bringing the slope of the bottom to uniform grade.
Unfortunately the banks and bends cave in and form new
resistances which divert the energies and directions of the
water to new quarters, and thus its work is self destructive.
In artifical channels this work should be done in advance so
that the energies of the water may be employed in a profit-
able way, and not wasted in building and destroying bars and
bends.

SULLIVAN'S NEW HYDRAULICS. 225

88. Eroding Velocity in Straight Canals of Uniform

Section. Theory and observation both indicate that a depth
of flow of one foot upon the perimeter of a straight canal of
uniform section will cause as great erosion as a flow of ten
feet depth or any greater depth. The power of erosion in a
straight, uniform canal varies with the square of the bottom
velocity, or as the square of the velocity in contact with the
sides and bottom. It has been shown that the velocity along
the sides and bottom is controlled by the slope and degree of
roughness of perimeter, in straight uniform channels, and that
this velocity cannot be affected in such channels by any
change in hydraulic mean radius.

As this bed velocity is not affected by the size of the chan-
nel, but is the same for the same slope of channel bed and
roughness of perimeter without regard to hydraulic mean
radius of the channel, we may conceive, for the purpose of de-
termining the eroding velocity in such straight uniform chan-
nel, that the central portion of the liquid vein has been re-
moved so that there remains only one foot depth of water
upon the sides and bottom of the channel.

Then find the sectional area of this layer of water in
square feet, and the length in lineal feet of the wet girth or
perimeter.

Then,
area in square feet of the layer of w*ter__ rnr hydraul - c

Wet girth in lineal feet
depth, so far as this one foot layer of water is concerned.

Then the velocity of flow of this layer of water one foot
depth upon the sides and bottom will be

In channels where the actual depth of flow exceeds one
foot, no matter how greatly, the value of r determined as above
will be less than unity, but will approach unity. In order to
err on the safe side and as a matter of convenience, we as-

226 SULLIVAN'S NEW HYDRAULICS.

eurne that r is a constant equal unity in channels where the
depth of flow is one foot or greater; and under these con-
ditions the eroding velocity or velocity of this layer of water
is

< 109 >

If the channel is so small that the actual value of r for
the whole volume of flow is less than rr=1.00, then the mean
velocity and all other velocities may be considered as equal
and may be found by the formula for mean velocity in chan-
nels of the given degree of roughness. In either case the ac-
tual eroding velocity will not exceed the computed eroding
velocity, and the computed result will be a safe guide in de-
termining the grade of the canal.

89. Slope or Grade of Canal to Generate a Qiven
Bottom or Eroding Velocity. The stability of the material
which forms the perimeter of the canal must be the controll-
ing factor in determining the grade or elope of the canal.
Very light soil will not stand a bottom velocity greater than
one half foot per second without serious erosion, while other
classes of soil will stand much higher bottom velocities with-
out damage. When it has been determined what bottom ve-
locity is best adapted to the material forming the perimeter,
then the slope or grade of the canal (without reference to its
size) which will be required to generate that given bottom ve-
locity will be

S=m v* ........................................ (110)

In which,

v 2 =the square of the proposed bottom velocity in feet
per second.

m=coefficient of velocity applicable to roughness of peri-
meter.

S=Slope required to generate the given bottom velocity.

If the channel is so small that the value of r for the en-
tire volume of flow is less than r=i.OO, then

S = :/7F> and tlie mean and bottom velocities will be
practically the same.

SULLIVAN'S NEW HYDRAULICS. 227

If the bottom velocity =.J-L has been decid6dt then
the mean velocity for any value of r will equal the bottom ve-
locity multiplied by fr 8 , or v=f/r 3 X -/

V m

The value of m may be selected from the groups of data
of flow in open channels heretofore given.

90. Stability of Channel Bed Materials, According
to the observations of Du Buat a bottom velocity of 3 inches
per second will just begin to work upon fine clay fit for pot-
tery; a bottom velocity of 6 inches per second will lift fine
sand; 8 inches per second will lift sand coarse as linseed; 12
inches per second will sweep along fine gravel. 24 inches per
second will roll along rounded pebbles an inch in diameter;
a bottom velocity of 3 feet per second will sweep along shiv-
ery, angular stones as large as eggs. Professor Rankine give a
the following table of the greatest velocities close to the
bed which are consistant with the stability of the materials
mentioned :-

Soft clay 0.25 feet per second .

Fine sand 0.50 " "

Course sand, and gravel as large as peas. .0.70 ' " "

Gravel as large as French beans 1.00 " " "

Gravel one inch diameter 2 25 " " "

Pebbles 1} inches diameter ,3.33 " "

Heavy shingle 4.00 " " "

Soft rock, brick, earthenware ,450 " " "

Rock, various kinds 6.00 and upwards.

See also "Civil Engineer's Pocket Book" by Trautwine,
pp. 563, 570, and "Irrigation Engineering" by H. M.Wilson
page 86, and Fanning, page 622.

The experiments of Du Buat were in a small wooden
trough with a smooth bottom so there was little friction be-
tween the moving particles of the material and the bottom of
the trough. Loose material on a smooth uniform floor
would be moved by a smaller bottom velocity than if it were

228 . SULLIVAN'S NEW HYDRAULICS.

incorporated in the bed of an earthen channel. It is probable
that in ordinary earth the bottom velocity should be about
.70 foot per second, and the slope should be S=tn v* = .00031X
(.70)* =.0001519.

91 Adjustment of Slope Or Grade, Bottom Velocities
and Side Slopes of Canals, to the Material Forming the
Bed. In order to preserve the efficiency and delivery of a ca
nal, its cross-section must be uniform, sy metrical and free of
deposits and plant growth. Caving and sliding banks, due to
the action of frost upon side slopes steeper than the natural
angle of repose of the material forming the sides of the canal,
when such material is reduced to powder by frost in winter
when the canal is empty, not only causes the filling up of the
canal, but also leaves the banks rough, irregular and ragged,
and greatly reduces its area, while it increases and roughens
the perimeter. The efficiency or delivery of a canal may be
reduced fully one third during one winter from this one cause
alone. The extent of damage thus done will not be fully dis-
covered until the water has again been admitted to the canal.
All the loose, disintegrated material will then be washed off
the sides and deposited in the bottom in irregular heaps.
These heaps will be acted upon by the mean velocity in the
same manner that a bridge pile or pier is attacked by the
flow, and will thus be cut away and redeposited on one side
where the velocity is not sufficiently great to keep the mater-
ial in suspension and in transit. This will change the direc-
tion of the current to the deepest part of the cross section
next the opposite bank which produces an undercutting and
caving at that point and a further deposit on the side oppo-
site the cutting. The thread of the current is caused to cross
from one side to the other and thus the energy of the stream
is expended in destroying the banks and in transporting ma-
terial from one point to another. There are few instances
in which the bottom of a canal has been scoured and eroded
to a serious extent. The silt and deposits nearly always come
from the banks which clearly indicates that the side slopes

SULLIVAN'S NEW HYDRAULICS. 220

are too steep for the material and for the climate, or that the
alignment is bad, for if the alignment IB bad and the ve-
locity too high, all the velocities are eroding velocities at
the bends, and consequently a very low mean velocity must
be adopted or the banks must be protected by paving or
otherwise.else the annual expense of cleaning and repairs will
be excessive. The proper side slopes of a canal will depend
upon the nature of the material forming the perimeter. The
side slope should never be steeper, in climates subject to
frost, than the natural angle of repose of the material when
thrown up in considerable heaps, loose and dry. In climates
subject to frost the side slopes will be thoroughly pulverized
by alternate freezing and thawing when the canal is empty
in winter, or above the water level if the water is not turned
out in winter. Under these conditions, if the side slope is
steeper than the natural angle of repose of the material
when it is perfectly loose and dry, the result is that the ma-
terial thus pulverized by frost will roll down into the canal
at each thawing until the slope finally reaches its natural an-
gle of repose in a rough and irregular way. The method of
determining the angle of repose is not by reference to pub-
lished tables of such angles for different materials, but by
throwing up a large heap of the material to be dealt with and
allowing it to assume any angle it will. The angle thus as-
sumed by the sides of the heap is as steep as the side slopes
of the canal should be in that class of material. The angle
of repose will be found to vary widely for different classes of
earthy material, and for most kinds the angle will be much
steeper if the material is damp or moderately wet than if it is
either dry or saturated. Hence the angle should be found
when the material is perfectly dry and loose.

The side slopes having been made to conform to the
angle of repose thus found, and due attention having been
given to the alignment, it is then necessary to so adjust the
slope of the bottom of the canal as to cause a bottom velocity
of flow most suitable to the material of the perimeter. If the
canal is to be of considerable width and to have a depth of

230 SULLIVAN'S NEW HYDRAULICS.

flow exceeding one foot, then the grade or slope should be

S=mv 8 .

Here m is to be selected from the values of m developed
for canals in like condition and in like material, given in the
groups of data of flow in open channels.

The value of v will depend upon the bottom velocity
which the given material of the perimeter will stand without
erosion. The suggestions heretofore (87) given may assist in
determining what value should be assigned to v in the above
formula.

If the canal is to be comparatively deep and narrow, as
it should be where practicable, then the grade should be
mv 8 mv a

But in this formula the value of r is found not by taking
the quotient of the total cross-sectional area of the column of
water by the wetted perimeter, but by assuming that there
is one foot depth of water adhering to the sides and bottom,
the area of which is to be divided by the total wet girth in
lineal feet. The resulting value of r is that which is to be
used in determining the slope to generate the given bottom
velocity.

If the value of r is the true value for total area divided
by wet perimeter, and v represents the desired mean velocity,
then the last formula will give the required slope to gener-
ate the given mean velocity, without reference to bottom
velocity.

In very light soil mixed with fine sand the action of waves
will reduce the side slopes much flatter than the angle of re-
pose of the material when dry or only damp. If fluming, pud-
dling, or paving cannot be resorted to where the canal passes
through such material, then the canal should have a cross-
section elliptical in form, and the bottom or scouring velocity
should not exceed .45 foot per second, and great care must be
taken to avoid bad alignment.

The grade of the canal having been determined with
reference to the greatest bottom velocity the material of the

SULLIVAN'S NEW HYDRAULICS 231

bed will safely stand, it then becomes necessary to determine
the dimensions of the canal with that given grade which will
cause the discharge or carriage of the required quantity of
water.

92. Dimensions of Canals to Carry Given Quantities.

In the case of canals with side slopes of about 2 horizontal
to 1 vertical, and of considerable capacity, the value of the

hyd aulic mean depth , may be approximately found by
formula (64) which is

In this connection see 19 and 3. The required value of r
being thus found in terms of cubic feet per second q, then,

a=r 8 Xl2- 566 * and wet perimeter, P=_ a _. For reasons here-
tofore pointed out these formulas are not generally applicable
to all forms of cross-section and capacities of open channels,
and when the values of a, p, and r have been calculated in
this manner, the general formula for velocity should be ap-
plied as a check. When the mean velocity is thus found,
then q=aXv.

For example suppose the grade decided upon for a canal
is S=.0002754=l in 3631.08, and the value of m applicable to
the class of gravelly earth is m=.00034. What area in square
feet and what wet perimeter and what value of r would be
required to cause the canal to discharge 1,000 cubic feet per
second, the side slopes being 2 to 1? In the first place find the
required value of r by formula (64) which will be r=5.121.

Then required area in square feet, a=r a X12.5664=329.554.

The required wet perimeter =JL= 32 ^f =64.353.
r 5.121

Taking 33.1668 feet of the wet perimeter as the bottom
width of the canal, there will have to be a depth at center
sufficient to take up the remaining 31.1862 feet of wet peri-
meter which is to be divided equally between the two side

232 SULLIVAN'S NEW HYDRAULICS.

elopes. Then the wet perimeter of one side slope will be=
31.1862

As the side slopes are 2 horizontal to 1 vertical, a verti-
cal depth of water equal about one half the length of one
side slope, or about 7 feet in this case, will be required. So
making the depth of water at the center equal 7 feet, and the
bottom width as above, equal 33.1668 feet, and the side slopes
2 to 1, we have the length of one side slope =-/ 7* +14* =15.65
feet. Then total wet perimeter =15.65+15.65-(-33.1668=
64.466 feet.

The actual area will be 330.1676 feet. The actual value of

r will be = 330 - 1676 =5.121. Now as a check on this calcu-
04.460

lation we must apply the general formula for mean velocity
to the slope and dimensions above found, and we have

.064. And the

quantity in cubic feet per second which will be discharged
will be q=areaXvelocity=330.1 676X3.064=1011.63 cubic feet,

,-Q

Tho bottom velocity in this canal would be v=.J =

While it is seen that the dimensions of a canal of this
form of cross section and capacity ruaybe closely ascertained
by the formulas for r, a and p, as above shown, yet these
particular formulas do not apply to small canals nor to rec-
tangular canals, with any degree of accuracy. These parti-
cular formulas do apply, however, with exactness to pipes or
circular closed channels running full.

93. Allowance In Cross Section of Canals For Leak-
age and Evaporation. The amount of loss by leakage and
evaporation from a canal will depend upon the climate, th e
nature of the soil, the length of the canal, the depth of flow,
and above all the position of the canal with reference to the

SULLIVAN'S NEW HYDRAULICS 233

elevations and depressions of the surface of the surrounding
country.

If the canal is constructed upon the highest line of the
land through which it passes, the leakage from it will be
great, and because of its elevated position it can never regain
any part of this loss by return seepage. Such location also
exposes the water surface to the action of the sun and wind,
and thus large losses occur by evaporation, especially if the
canal is wide and shallow. In arid regions where irrigation
is not general and abundant, the sub-surface water level i<3 at
considerable depth below the surface, but after irrigation has
been practiced for several years, the earth becomes saturated
and the sub-surface water level rises near to the surface. Un-
til this occurs the loss from new canals in such regions will
be very great. After irrigation has been practiced for a num-
ber of years, and has become general in the given locality, the
canals situated along side hills and skirting the valleys will
gain vastly more by seepage into the canal than will be lost
by leakage and evapoiation combined. In some canals in
Colorado the gain by seepage into the canal is as great as
two thirds the total original quantity admitted into the canal
at its head. This occurs only in canals located where irri-
gation has been practiced for years, and in canals so situated
on side hills or along the foot of the hill, as to admit of the
seepage flowing into the canal.

The loss by leakage and evaporation from new canals in
arid regions varies from 20 to 75 per cent of the quantity ad-
mitted into the canal, according to the nature of the soil and
the length of the canal. As the canal becomes silted and the
sub-surface water level rises, the leakage will decrease, and if
the canal is so located as to admit of it, the gain by return
seepage will, in the course of a lew years, more than balance
the loss by leakage and evaporation.

In regions where the rainfall is great it is probable that
the seepage into a new canal will offset the leakage from the
first opening of the canal, because the sub-surface water level
is already very close to the surface of the ground.

234 SULLIVAN'S NEW HYDRAULICS.

In making allowance in cross-sectional area of a canal
to cover these losses, it should be by way of extra depth.

94. Where a Flume Forms Part of a Canal. Where

the course of a canal would pass around on a very steep side
hill, or through stretches of very porous earth, or across low
depressions, flumes are frequently adopted as portions of the
canal for such reaches. In this event the question arises as
to the proper ratio of flume cross section to that of the canal,
of which the flume forms a part. The determination of this
question involves a consideration of the relative degree of
roughness of the two classes of channel, and the difference in
slope or grade of the flume and the canal, as well as the length
of the flume and its alignment. If the flume is short and
upon the same grade as that of the canal, and has no vertical
fall at its lower end, the water will not acquire a velocity in
such short flumes much greater than that in the canal, and
therefore the area of the flume under such conditions cannot
be reduced much below that of the wetted area of the canal.
While the velocity of flow will usually be greater in a flume
than in a canal of equal slope, yet at the entry to the flume
the water has only the velocity of the canal, and the head due
to that velocity. It must flow a sufficient distance in the
flume to acquire the greater velocity due to the smoother peri-
meter before the depth and area of the flume can be materi-
ally reduced from that of the connecting canal, otherwise
there will be an overflow at the upper junction of the flume
with the canal. The flume should converge from the mean
width oi the canal at the junction, to the standard section
adopted for the flume, in a length varying from 50 to 200 feet
according to the difference in slope and in roughness of the
flume and the canal. The value of C might be 56 for the canal
and anywhere from 70 to 130 for the flume, according to
the method and materials adopted in its construction and
alignment.

A straight canal in firm, dense earth and in best condi-
tion develops C 75.00, while a rough, crooked flume with
battens on the inside develops C 70.00. In such cases as

SULLIVAN'S NEW HYDRAULICS, 235

this the flume would require an area slightly in excess of
that of the canal, or would require an equal area and steeper
grade. On the other hand the value of C for a rough canal
may be as low as 40, while the value of C for a very smooth
well jointed hard wood flume of good alignment might be as
high as 130.

The slopes being equal, the velocities will be as f/r* in
the one is to J/r 3 in the other, as modified by the respective
values of C, or viviiCJ/r^CJ/r 3 . If the slopes are different
then v:v:: C/rVS:Cf/rVS

The value of C may be taken from the data of like
flumes and channels given in the groups, Chapter 2.

95 Mean Velocity In Uniform Sections of Canals
Found by Floats.

In straight sections of canals of uniform cross-section
where the thread of the greatest velocity is midway between
banks and just beneath the water surface, the place of mean
velocity will be found at .50 of total depth at a point midway
between the center of the canal and the bank, unless the
depth of flow is less than two feet, in which case the place
of mean velocity will be at or just above mid-depth at a point
midway between the bank and the middle of the canal, as-
Burning that the sides and bottom of the canal are fairly
smooth. In shallow canals with gravel and pebbles along
the bottom the place of mean velocity is very near mid-depth,
aometimes slightly above, and at one-fourth the width of the
canal from the bank. A large tin bucket loaded with gravel
and covered, may be suspended by a fine wire at this depth
and connected to a flat circular float on the surface no larger
than is absolutely necessary to support the submerged bucket
at proper depth. This double float is to be placed at some
distance above the upper end of a measured length of the
canal, and adjusted to proper position with reference to the
bank or width of the canal, and with reference to depth, and
allowed to travel over the given course a number of
times. The average time required for its passage over the
given number of feet length of the canal will closely approxi-

236 SULLIVAN'S NEW HYDRAULICS.

mate the rate of mean velocity. The difficulty of ascertain-
ing the exact number of seconds which elapse between the
time the float crosses the line at the upper station and arrives
exactly at the line of the lower station, will probably cause a
slight error in the final determination of the mean velocity.
For this reason the measured course should be several hun-
dred feet in length. If the channel is rough and winding the
float will be cast either too near the bank or into mid-cur-
rent, and the result is without value. Float measurement of
mean velocity is practicable only in channels of uniform
width and depth. The surface velocity has no particular re-
lation to the mean velocity, and it is therefore impossible to
deduce the mean from the surface velocity. The ratio be-
tween surface and mean velocity varies with the form of
cross-section, roughness of perimeter, uniformity of cross-
section, variation in slope, depth of flow and hydraulic radius
and alignment of the channel.

The surface velocity depends mainly on the slope, while
the mean velocity depends upon the value of {/r 3 as well as up-
on the roughness and slope of the channel. In rough, stony
channels of varying cross-section and small depth of flow
there is scarcely any difference between surface and mean
velocity.

CORRECTION OF TEXT.

It is probable that no one ever turned his manuscript
over to the printer without a lively sense of its probable de-
merits when it shall stare one in the face from the printed
page.

The greater part of the book was written several years
ago. and portions of it were published in various journals in
1894 and 1895, While the ultimate conclusions reached and
formulas deduced, as appear in the text, are correct, yet some
of the reasoning is at fault, and not clear. The author would
be glad to stop the press and re -write the entire book after
having seen half the printed "proof," but it ie too late.

He must therefore resort to the alternative of writing
a criticism of his own work, and thus forstall the other
fellow.

The three important principles which are sought to be
established are:-

(I) That it is the effective value of the head or slope which
varies with some function of the diameter or hydraulic
mean radius, or mean depth, and not the coefficient
that varies.

(II) That for any given class of wet perimeter, or any given
degree of roughness, the coefficient is necessarily a con-
stant for all heads, slopes, velocities, diameters or mean
hydraulic radii.

(Ill) That the value of the coefficient is governed absolutely
by the roughness of wet perimeter, and by nothing else,
and is therefore an absolutely reliable index of the
roughness of perimeter.

FIRST PROPOSITION.

That the Effective Value of a Constant Head or Slope
Varies With Some Function d/d 8 , or /R 8 ) of the Diam-

238 SULLIVAN'S NEW HYDRAULICS.

eter, or of the Hydraulic Mean Radius, and that the Coe*
ficient does not Vary with the Diameter or Hydraulic Mean
Radius at all.

If a series of pipes or open channels of exactly equal
roughness of perimeter, but of different diameters, or differ-
ent hydraulic mean radii, have exactly the same head or
slope per foot length, it ia well known that the pipe having
the greatest diameter, or the open channel having the great-
est hydraulic mean depth (R), will generate the greatest
velocity of flow, and the pipe having the least diameter, or
the open channel having the least mean hydraulic depth, will
generate the least velocity of flow. As all these pipes, or all
these channels, are of equal roughness, and all have exactly
equal heads or slopes, it is evident that the velocity would be
the same in each of them if the constant head or slope were
not made more effective with an increase in diameter or hy-
draulic mean depth. This being true, the next inquiry is,
what is the ratio of increase in the effectiveness of the given
head or slope as diameter or hydraulic mean depth increases?

To solve this problem we must appeal both to the laws
of friction or resistance, and of gravity. The resistance, or
head lost by resistance, will be directly as the roughness of
perimeter, and directly as the extent of perimeter, and also
directly as the square of the velocity.

As demonstrated in the text the wet perimeter or extent
ot friction surface, varies exactly with d or r. (See pp. 3
36,39,40.)

But if there were no friction or resistance, then the velocity
would be the same for the same actual slope regardless of the
value of d or r. While the friction surface and consequently
the absolute loss of head by resistance, increases only as d or
r, the cross section of the column of water increases as d 8 or
r, or as the sectional area.

The absolute head or slope therefore increases as the area,
or as d 8 or r 2 , while the absolute loss of head increases only
as d or r. It is evident then, that the absolute head or slope,
which varies as d 8 or r*, must be modified by the absolute
loss of head or slope which varies as d orr. Then the mean.

SULLIVAN'S NEW HYDRAULICS. 239

head, or relative head, of all the^ particles of water in the
croBB section will vary with d* as modified byd, or with r 1 as
modified by r. As r* must not be increased by r, but must be
modified by i, we must reduce both d and d, or r and r 1 , in
the same ratio, in order to obtain a reducing or modifying
multiplier. To accomplish this result, we say that j/d bears
the same relation to d that d bears to d 2 , or that y/r bears
the same relation to r that r bears to r*. In other words to
maintain the ratio, of r to r*, or d to d, and at the same time
obtain a multiplier which will give the combined net effects
of d and d 2 . or r and r s , upon the value of H or S, it is neces-
sary to take the square root of both d and d 8 , or of both r
and r*. We then say that, relatively, the area or absolute
head (d" or r 2 ) varies with y'd*=J,or with ^/r*=r, while the
friction surface or absolute loss of head varies with -/d or y'r.
and consequently the relative mean head of all the particles in
the cross section will vary with the resultant of these two
effects, which will be as d^/d, or as R^/R. Thus we obtain
the modifying multiplier j/d, or y/r, while we maintain the
correct ratio of friction surface to area, or of loss of head
to gain in head as d or r varies for a constant head or
slope.

It is evident then that the constant head or slope be-
comes more effective or less effective as dy/d=>/d 8 , or \/r*,
increases or decreases.

SECOND PROPOSITION.

That for any Given Degree of Roughness of Wet Peri-
meter, the Coefficient is a Constant for all Heads, Slopes,
Velocities, Diameters or Hydraulic Mean Depths.

It was shown in the foregoing discussion that the effect-
ive value of the head or slope varies with y / d 3 ory / r 3 . By the
law of gravity the square of the velocity must always be pro
portional to the head or slope in any given pipe or channel,
or v a =2gH. As a necessary consequence of this law, it is
obvious that anything which affects the effective value of the
head or slope must at the same time equally affect the value
of v s .

240 SULLIVAN'S NEW HYDRAULICS.

When we write m= ^3 and remember that the

effective value of S increases with i/r*, and that any increase
in the effective value of S must a'so increase v in the same
ratio, it is evident that as both dividend and divisor increase
aliKb the quoti?nt, m, will continue a constant for all values
of r, Sand v*. Their relation is such that we cannot increase
the effective value of S without also increasing the value of
v 1 in the same ratio. Hence m is necessarily a constant.

THIRD PROPOSITION.

That the Value ot the Coefficient is Governed Abso-
lutely by the Roughness ot the Wet Perimeter, and by
Nothing Else, and is Consequently an Absolutely Corrrect
Index of the Roughness.

When we inspect the formula for the coefficient, m= y8

it is apparent that m is simply the expression for the ratio of
effective slope to the square of the velocity. If the pipe or
channel is rough it will require a large value of the effective
slope, Sy/r 8 , to generate a small value of v*. Consequently
the ratio, m, of effective slope to v*,will be large in rough
channels. But if the channel is uniform in area, and smooth
then a small effective value of slope, Sy'r 3 , will generate a
relatively large value of v s , and hence the ratio, m, will be
small for smooth perimeters. As m is simply the expression
for this ratio, and as this ratio depends exclusively on the
roughness of perimeter, it is obvious that m will vary only
with the roughness.

The coefficient, C= A I y2 , is simply the square root of

VSyr 8

the reciprocal of m, and will consequently be a constant, like
m, for any given degree of roughness. But being the square
root of the reciprocal of m, C will vary with the roughness in
the exact opposite way from m that is, C will be large for
smooth perimeters and small for rough perimeters, while m
will be large for rough perimeters and small for smooth peri-
meters.

SULLIVAN'S NEW HYDRAULICS. 241

As either of the coefficients vary only with the rough-
ness of wet perimeter, but is very sensitive to uny change in
roughness, it will be found that C will decrease as depth of
flow increases in all channels where the sides are rougher
than the bottom, and will increase with increase of depth of
flow in all channels where the sides are smoother and more
uniform than the bottom. In other words C will vary as the
mean of the roughness varies. See in this connection 13
page 58, and also p p. 27, 28, 29, 41, 42.

The best form of the formula for general use is,

This form of the formula also shows by mere inspection
that the effective value of S varies with ^/r*.

If the formula is written, v=C V r * l/S, the actual result
would be the same whether we say that C or \/S varies with
* V r 3 , but as C insists on being constant, it is evident that it
is the effective value of S that varies with v/r 8 , and the writer
desires to correct all statements to the contrary, It is some-
what absurd to insist that the coefficient is a constant and at
the same time to claim that it varies. The coefficient in our
formula can vary only as the average of roughness of the en-
tire wet perimeter. In the Chezy or Kutter form
of formula, the coefficient must vary as the roughness
and also as {/r. (See pp. 6, 7, 42, 44.) Hoping that this ab-
surdity is fully corrected in this explanatory note, and asking
pardon for having committed such a glaring fault, the author
commits the work to the hands of the profession with the
further hope that its merits may outweigh its faults.

MARVIN E. SULLIVAN.
Longmont, Colorado,

November, 1st, 1899.

APPENDIX 1

Suggestions Relating to Weir and Orifice Measurements of
Flowing Water.

96 Remarks in Relation to Weir Coefficients. In the

third remark under Group No. 2 14, a general form of Weir
formula was suggested. It is not here intended to discuss the
well known theory of flow over measuring weirs with sharp
crests and full or partial contraction, any further than to
point out what the writer believes would be an improved
method of application which is believed would reduce the
errors in such determinations. From the nature of a meas-
uring weir it is impossible that the head or depth upon the
weir should ever be great, and consequently the velocities are
never very high, even in the cases where there is velocity of
approach. The amount of resistance to flow (being as v)
offered by the edges or perimeter of the notch is therefore a
small factor in the sum total of the coefficient of discharge.
The important factor is the coefficient of contraction. It is
usual to combine the coefficient of resistance with the coeffi-
of contraction and their product forms the coefficient of dis-
charge, which is usually assigned a mean value of .62. For
the reason that these two independent coefficients which
combined form the usual weir coefficient of discharge, do not
vary in the same manner under similar conditions, it has
been found necessary to find their combined value for each
given depth upon the weir and for each given length of notch,
and for each form of notch. If the length of weir notch re-
mains constant, a small change in depth upon the weir will
greatly affect the value of the combined coefficient, or coeffi-
cient of discharge. This cannot be attributed, except in very
small part, to the resistance at the edges of the notch, for
a small change in depth upon the weir does not greatly affect
the ratio of area to perimeter of the notch, which may be

SULLIVAN'S NEW HYDRAULICS. 24:'.

regarded as a very small fractional length of open channel.
The effect upon the combined coefficient of varying the depth
upon the weir must therefore be accounted for in the factor
representing contraction of the discharge. It is evident from
the discussion of coefficients of flow in pipes and open chan-
nels ( 3 to 7) that the resistance to flow offered by the edges
of the notch will vary as H and \/r a . But the coefficient of
contraction which is the controlling and important factor has
no known relation to the value of r. The coefficient of contrac-
tion is affected greatly by the position of the weir, the depth
upon the weir, the distance from the crest to the bottom of the
channel, the distance between the shoulders of the notch and
the banks of the channel, and the velocity of flow through the
notch.

The experiments of Mr. J. B. Francis upon the same weir
of constant length, and where all conditions were constant ex-
cept the depth upon the weir, show that a change of depth
alone upon any given sharp crested weir of the usual form
will greatly affect the value of the coefficient of discharge,
and further show that the variations of the coefficient of con-
traction apparently follow no law. The coefficient will de-
crease as depth increases until a certain depth is reached
(depending upon the proportions of the notch) and then in-
creases with a further increase in depth up to a certain point
where it will again begin to decrease to a small extent until
it becomes nearly constant for great depths (if such were
practicable).

To make the usual weir coefficients apply with any de-
gree of accuracy is not a simple matter by any means, for the
conditions must be identical with those under which the
given coefficient was determined, The ratio of area of notch
to area of channel, the depth or height of overfall, the height
of crest above the bottom of the channel on the upstream side
of the weir, the position of the weir, whether at right angles
to the thread of the channel, and vertical, and rigidly straight
or allowed to bend under pressure, all affect the coefficient of
contraction, in addition to the influence of varying the depth
upon the weir. There are so many different influences bear-

244 SULLIVAN'S NEW HYDRAULICS.

ing upon the coefficient of contraction that we can never be
certain of its value except under given favorable conditions
which do not often occur in actual practice. It is therefore
suggested that it would be safer practice where careful de-
terminations are to be made to avoid all these uncertainties
by suppressing all contraction. When this is done there re-
mains only the coefficient of resistance of the edges of the
notch to be dealt with, and the law of its variation is known.
In order to suppress contraction it is suggested that the
notch, whether rectangular, triangular or trapezoidal, should
be chamfered on the upstream side of the notch to the form
of the vena contracta^instead of placing the chamfered side
downstream. As illustrating the desultory manner or vari-
ation of the coefficient of discharge of a sharp crested weir
the first three columns I, H, and q, quoted by Fanning from
Francis' experimental data (Table 68, page 288 Water Supply
Engineering) are given in the following table, and the column

v was computed by the formula v=-^L, and from these data

the resulting values of m were computed,

The fundamental formula for flow over weirs with sharp
crests may be written

v= %-\l - > or v m %V 2gH=5.35 m^/H.
V m

Whence

m= 28 - 622 5H ^ . f m is U8ed as adiviBor>

v*
Or

m= - =- / v , if m is used as a multiplier.

5.35^ V 28.6225H '

, or q=AreaX5.35 m^E

SULLIVAN'S NEW HYDRAULICS.
TABLE. No. 41 Table of Weir Data.

245

Feet

Cubic
Feet
Sec.

Feet

Feet
Sec.

V 2

Feet
Sec.

R
Feet

Coefficient
"5.35T/H

9.997
9.997
9.997
9.997

0.62

S:i

1.56

16.2148
23.4304
45 5654
626019

6.198
7.997
12.496
15595

2.610
2.929
3.648
4014

6.8121
8.5790
13.2875
161122

0.5515
0.6900
1.000

1.188U

.6195
.6121
.6046
.6007

L= length in feet of notch.
H= depth in feet upon the weir.
q= cubic feet per second actually discharged.
A=LXH=Area in square feet=depth of water upon the
irXlength of notch.

v=-S-=mean velocity in feet per second.
R= hydraulic radius in feet of notch=-2_

m=Coefficient of discharge:

v
5.35/H'

In these experiments the conditions all remained con*
stant except the depth H, upon the weir.

In the formula
v=5.35 m v/H

if we combine the value of m with the constant 5.35=%\/2g,
the following values of the coefficient C result:-

fl=.62, m=.6195, 5.35Xm=C=3.3143.

H=.80, m=.6121, 5.35Xm=C=3.2747.

H=1.25, m=.6096, 5.35Xm=C=3.2613.

H=1.56, m=.6007, 5.35Xm=C=3.2137.

Whence,

q=AreaX<V H=C(LX H )/ H

It is evident, even for different depths upon the same
weir, that if the constant value C=3.33 is used, the results
must be erroneous. Suppose the velocity of approach is consid-
erable, as on mountain streams, and that the weir notch (rec-
tangular) is nearly as long as the stream is wide, as often

246 SULLIVAN'S NEW HYDRAULICS.

becomes necessary, then the eloping banks will approach the
submerged corners of the notch and greatly affect the coeffi-
cient of contraction, but to what extent, is merely surmise.
It is frequently the case that in order to stop the leaks under
and around the weir, earth, straw and brush are banked
against its upper side, thus training the flow upon the notch
and also preventing full contraction. This affects both the
real value of H or v z and the contraction of the discharge.

The range of experimental coefficients as determined by
Francis was very small, being mostly for weirs about 10 feet
length with depth upon the weir varying from about six
inches to 1.60 feet. The variation of the coefficient of contrac-
tion was found so fitful and irregular as the ratio of length to
depth was changed and with different depths upon any given
length of weir, that Mr. Francis advised caution in the appli-
cation of his formula and coefficients in cases not falling di-
rectly within the experimental conditions. It is assumed that
the contraction of the discharge over a sharp crested weir in
full contraction is analogous to the contraction of the jet
from a sharp edged orifice in thin plate. If the numerous
tables of experimental orifice coefficients determined under
various heads above the center, and under various proportions
of height to width of orifice be investigated, it will be found
that each form of orifice, or each ratio of height to width,
develops a distinct series of values of the coefficient as the
head varies.

The coefficient for an orifice will either decrease or in-
crease with the head upon the center in an irregular and
alternating manner which apparently depends upon the ratio
of height to length of orifice, as indicated in the following
table of experimental coefficients for square edged orifices in
thin plate and with full contraction, which were determined
by Poncelet and Lesbros.

Coefficients of discharge for square edged orifices in thin
plate and with full contraction.

SULLIVAN'S NEW HYDRAULICS. 247

TABLE No. 42 Table from Ponoelot and Lesbros.

Dimensions of Orifice in Inches.

Head above

Center

8 X 8

6X8

4X8

3X8

2X8

1X8

0.4X8

In Inches

0.4

.70

0.8

.65

.69

1.0

.64

.68

1.5

.61

.64

.68

2.0

.60

.62

.64

.68

2.5

.59

.61

.62

.64

.67

3.0

.60

.61

.62

.64

.67

3.5

.57

.60

.61

.62

.64

.66

4.0

.58

.60

.61

.63

.64

.66

4.5

.56

.59

.60

.61

.63

.64

.66

5.0

.57

.59

.61

.62

.63

.64

.66

8.0

.59

.60

.61

.62

.63

.64

.65

12.0

.60

.61

.62

.63

.64

36.0

.60

.61

.62

.63

60.0

.60

.61

.62

120.0

.60

'.60

.61

TABLE No. 43 Table From George Rennie.

Head above

Center

Dimensions of Orifice.

Coefficient

In Feet

1.0

1 inch diameter, circular.

.633

1.0

1X1 inches, square.

617

1.0

2.0

1 square inch area, triangular.
1 inch diameter, circular.

!596
.619

2.0

1X1 inch, square.

.635

2.0
3.0

1 square inch area, triangular.
1 inch diameter, circular.

.577
.628

3.0

1X1 inch, square.

.606

3.0
4.0

1 square inch area, triangular.
1 square inch area, triangular.

.572
.53

4.0

1X1 inch, square.

SULLIVAN'S NEW HYDRAULICS.

TABLE No. 44 Table from Gen. Ellis.

Coefficients of discharge for square edged.circular Orifices
in iron plate one half inch thick.

Head above
Center
In Feet

Diameter of Orifice in Feet.

Coefficient

2.1516
9.0600
17.2650
1.1470
10.8819
17.7400
1.7677
5.8269
9.6381

0.50
0.50
0.50
1.00
1.00

2.00
2.00

.60025
.60191
.59626
.5T<73
.59431
.59994
.58829
.60915
.61530

TABLE No. 45.

Coefficients for square orifice 1X1 foot with curved en-
trance and discharge slightly submerged. (Gen. Ellis.)

Head above
Center
In Feet

Dimensions of Orifice.

Coefficient

3.0416
10.5398
18.2180

Square, 1X1 Feet
Square, 1X1 Feet
Square, 1X1 Feet

.95118
.9*246
.94364

In these last experiments if the orifice had been in the
'orm of the vena contracta, and the discharge had been en-
tirely free instead of being partially under water, it is proba-
ble that the coefficient would have reached .98, and would
not have been affected in any manner except by the slight re-
sistance of efflux offered by the perimeter of the orifice. The
curving entrance had almost suppressed all contraction of
the jet in the above experiments.

The object of these tables and suggestions is to point out
the fact that all these uncertainties in the application of weir
and orifice coefficients may be easily avoided by so chamfer-
ing the inner edges of the weir notch or orifice as to make
them conform as nearly as possible to the form of the vena
contracta.

In many cases of the practical application of the ordinary

SULLIVAN'S NEW HYDRAULICS. 249

weir and orifice coefficients the conditions are such that com-
plete contraction cannot be obtained. In almost any case it
is much more convenient to suppress all contraction than to
obtain complete contraction, and when contraction is sup-
pressed there is no limit to the range of the remaining coeffi-
cient which should be determined in the same manner as the
value of m for pipes or open channels. When contraction is
suppressed (as pointed out 83). Then for a submerged
orifice.

v =A /IZL=8.025 J3Z1 ; m- 6 ^ H

q =AreaX8.025j H * /r> =

v m

And for weirs,

28.6225 H r/r
=

And.

When the numerical value of m is ascertained for any
thickness of plate it will apply to any shape or size of orifice or

weir notch, and the square root of its reciprocal^/ , may

then be taken and combined with the constant 8.025 or 5.35 as
the case may be.

APPENDIX II.

Useful Data and Tables Relating to Water Works and the
Water Supply of Cities and Towns.

97. Purposes to Which City Water is Applied.

In planning a water works system for town or city sup-
ply, the nature of the chief occupation of the inhabitants
must be considered as well as the number of inhabitants at

250 SULLIVAN'S NEW HYDRAULICS.

present, and the probable increase in population within the
next fifteen or twenty years. The purposes to which city
water will be applied will depend upon the humidity of the
climate. In the arid portion of the West the city water is
demanded for all purposes to which water is applied, such as
irrigation of lawns and gardens, and shade trees, street
sprinkling, carriage washing, watering horses and cows,
water for steam boilers and hydraulic motors, hydraulic lifts
or elevators, steam laundries, drinking fountains, ornamental
fountains, manufacturing purposes, extinguishment of fires
and ordinary household uses. Where manufacturing is the
chief business of a town the demand for water will be two or
three hundred per cent greater than in towns of equal size and
in like climates which are not manufacturing centers. In
some manufacturing towns situated on rivers the factories
have their own private water supply, and in such cases the
city water works is called upon only for water for ordinary
purposes. The coast states, and the Eastern and Southern
states, have frequent and large rainfalls and except at manu
facturing centers, the city water works in these states will
not be called upon except for ordinary purposes. In the arid
portion of the West the demand on the city water supply is
from fifty to one hundred per cent greater than in towns of
like population in other parts of the United States.

In non-manufacturing towns in such climates as in Ar-
kansas, Mississipi and Louis&na, the demand for all purposes
will not exceed 60 gallons per capita per 24 hours, while in
Colorado and other arid states the demand in small non-
manufacturing towns is from 110 to 150 gallons per capita
per 24 hours, and in older and larger cities the demand is
from 150 to 200 gallons per capita. Should an essentially
manufacturing city spring up in the arid West, it is probable
that the demand for water would reach 400 gallons per capita
per 24 hours.

98. Quantity of Water per Capita Required The

quantity of water required per capita per 24 hours for the
.present given number of inhabitants, and for all purposes,

SULLIVAN'S NEW HYDRAULICS.

251

depends upon the bection of the country and the chief occu-
pation of the inhabitants, as just pointed out. But in plan-
ning a water supply, the very rapid increase in the popula-
tion of towns and cities in the United States must be amply
allowed for. The U. S. census of 1890 shows that our popu-
lation is f aet gathering into the towns and cities. The popu-
lation of towns and cities, taken collectively, throughout the
United States, increa sed by 61.10 per cent from 1880 to 1890,
while the total population of town and country increased
only 24.85 per cent. The following table is valuable in this
connection.

TABLE No. 46.

Growth of population in cities and in the United States.

Cen-
sus
Year

Total Pop.
U.S.

Population
in Cities

Increase
in total
pop. per
cent

Per cent
total pop
in citie

of the
living
s.

1800

5,308,483

210,873

1810

7,239,881

356,920

36.28

4.93

1820

9,633,822

475,135

33.66

4.93

1830

12,866020

1,864509

32.51

6.72

1840
1850

17,069,453
23191,876

1,453.994
2,897,586

32.52
35.83

8.52
12.49

1860

31,443,321

5,072,256

35.11

16.13

1870
1880

38,558,371
50,155,783

8,071,875
11,318,547

22.65
30.08

20.93
22.57

1890

62,622,250

18,238,672

24.85

29.12

99. Table Showing the Consumption of Water Per
Capita Per 24 Hours in Various Cities and Towns, and
the Cost to the Consumer Per 1,000 Gallons, and the in-
crease In Population in Each City in 20 Years.

The foregoing table shows that the general average in-
crease of population in the towns and cities of the United
States was 61.10 per cent from 1880 to 1890. But the rate of
increase varies in different sections of the country and
also in different classes of cities and towns. The railroad
and general manufacturing centers increase most rapidly in
all parts of the country, while tho general growth of all cli
of towns increases most rapidly in the Western states.

SULLIVAN'S NEW HYDRAULICS.

TABLE No. 47.

cCt

l-H

Alabama

Birmingham

400

26,241

155

8c to 30 c

Gala.

Los Angeles

5,758

11,183

50,394

175

20c

Colo.

Denver

4,759

35,629

106,670

200

Conn.

New Britain

9840

11,800

19,010

10 c

Conn.
Conn.

Norwich
Hartford

16,653
37,180

15.112
42,015

16,195
53,182

50
125

15 c ;o 30 c
7Hcto30c

Conn.

New Haven

50840

62882

85,981

130

fV4cto30o

Georgia
Georgia

Atlanta
Augusta

21,789
15,389

37,409
21,891

65.515
33,1' 2

164
106

10 c

Georgia

Macon

10,810

12,479

22,698

6 c to 30 c

Illinois

Aurora

11.162

11,873

19,634

Illinois
Illinois

as? 80

298,977
5441

503,185

8,787

1,098,576
17,429

131&
70

8 c to 10 c
3c to8c

Illinois

Streator

1,486

5,157

6.671

120

10 c to25c

Illinois
Illinois
Indiana

Freeport
LaSalle
Indianopolis

7,889
5,200
48,224

8,516

7,847
75,056

10,159
11,610
107,445

46
70
90

10 c to 50 c
8 c to \~> c
6 c to 30 o

Indiana
Iowa

Richmond
Cedar Rapid*

9,445
5,940

12,472
10,104

16,845
17,997

74
68

5 c to 25 c
lOctoSOc

Iowa

Sioux City

3,401

7,366

37,862

10 c to 25 c

Iowa

Des Moines

5,241

22408

50,067

20cto 40 o

Kansas

Atchison

15,105

14,122

10 c to 50 c

Kansas

Minneapolis

2,000*

200

35 c

Kansas

Arkansas Citj

3,347

lOcto 40c

Ky.

Louisville

100,752

123,758

161,005

6 c to 15 c

Ky.

Lexington

14.801

16,656

22,355

10 c to 25 c

Ky.

Frankfort

5,396

6.958

8,500

100

6 c to 15 c

Ky.

Fulton

4,500*

200

Md.

Hagerstown

5.779

6,627

11,698

115

8 c to 40c

Mass.

Adams

12,090

5,591

9,206

Mass.

Fall River

26.766

48,961

74,351

Mass.

Holyoke

10,733

21,915

35,528

5 c to 15 c

Mass.
S ass.

Lowell
New Bedford

40,928
21,320

59,475
26845

77,605
40,705

75
113

2 1/, ctolSc

ass.

Newton

12,825

16,995

24,357

12cto35o

Mass.
Mich.

Springfield
Battle Creek

26,703

5,838

33,340
7,063

44,164
13,190

87
31

30c

Mich.

Bay City

7,064

20,693

27,836

5 c to 10 c

Mich.

Detroit

79,577

116,340

205,669

140

3%c

Mich.

Miss.

Sagnaw
Vicksburg

7,460
12,443

10,525
11,814

46,215
13,298

100
43

6 c to 11 c
6 c to 35 c

Missouri

5,555

6.522

21.842

25 c

Missouri

sfEolis

310,864

350,518

460,357

10 c to 30 c

SULLIVAN'S NEW HYDRAULICS.
TABLE No. 47 CONTINUED.

253

p
'

s *

Missouri
N. Hamp

Butler
Nashua

10,543

13 397

4,000*
19,266

25
150

6cto60c
10 c to 20 o

N. Hamp

Manchester

23,536

32,630

43,983

20 c

N.' Y.

Bayonne
Portland

3.834
3066

9.372
4050

18,996
8,561

7?
90

13 & c-23*c

10 c

N . Y.

Elmira

15863

20541

28.070

1% c to 45 o

N . Y.

Kingston

6,315

18,344

21,181

8cto30c

N. Y.

Olean

7,3S8

10 c to 40 c

N. Y,

Syracuse

43051

51.792

87.877

300

6 c to 25 c

N. Y.
N. Y.
N. Y.

Brooklyn
\ewYorkCity

396,099
12.733

942,292

566,663

18,892
1,206,299

804,377
31.942
1,513.501

100
70
92

IVt C-11& o
4 c to 25 c

Ohio

Dayton

30,473

33678

58,868

Ohio

Findlay

3,315

4.633

18,672

6 c to 12 c

Ohio

Oberlin
Sanduskj

13,000

15,838

4,000
19,234

20
154

30 c
4 c to 15 c

Ohio
Ohio
Ohio

Springfield
Toledo
Cincinnati

12,652
31,584
216.239

20,730
50.137
255,139

32,135
82,652
296.309

90
70
124

10 c to 40 c
SotolOc
17 c

Oregon
Penn.

Salem
Ml City

2276

7,315

10,943

100
230

15 c to 25 c
6c to25c

Penn.
Penn.

tfcKeesport
Williamsport

2,523
16,030

8,212
18,934

20,711
27,107

110

200

4!4cto30c
SctolOc

Penn.

^larrisburg

23,104

30,762

40,164

130

2!4 c to lOc

Penn.

Philadelphia

674 022

847.170

1,046.252

143

R I.

iVoonsocket

11527

16,050

20,759

22 y*

10 c to 80 c

Texas

Pawtucket
Laredo

6,619
2,046

19,0
3,321

2>, 502
11,313

79
150

tic. to 30 c
60 c

Texas

?ort Worth

6,663

20.725 130

20 c to 65 o

Va.

Richmond

51.038

66,600

80,388 ' 151

7 c to 15 o

*Estimated Population.

The above table will be useful in determining the quan-
tity of water required per 24 hours per person, and in de-
termining what extia capacity of reservoirs and conduits
should be provided for the increase in population during the
coming 20 years. The capacity of a water supply system
should not be based on the present number of inhabitants,
but upon the probable number of inhabitants 20 years hence.

254 SULLIVAN'S NEW HYDRAULICS.

What the increase of population will be in any given town or
city within any given number of years is a matter which
must be considered in the light of the local conditions and
surroundings of each given town or city. There are very
few cities or towns in the United States which do not increase
by 50 per cent within 20 years, and some increase by from 300
to 600 per cent within ten years. The general average increase
of population in all cities and towns in the United States for
the 10 years, 1880-1890, was 61.10 per cent.

100.~Formulas and Tables for Determining the Diam-
eter of the Conduit or Pipe Required to Convey any
Given Number of Gallons Per 24 Hours. -When the total
supply of water in gallons per 24 hours has been decided
upon, then the required diameter in feet of the circular brick
conduit or pipe, or other circular water way, may be at once
found by the formula

In this formula the value of m varies with the class or
roughness of the internal circumference of the waterway,
and the value of m must be in terms of diameter in feet.
The value of m for any class of wet perimeter will be found
by referring to the different groups of pipes and channels. If
the value of m, when found, is in terms of R in feet, it may
be converted to terms of d in teet as shown at section 10.

In the above formula q=cubic feet per second, and S

=the sine of the inclination of the waterway= -j-.

For a constant degree of roughness of perimeter, the

11 /m 2
value of ~yf ~^^ is a constant, and the formula may be

simplified accordingly. Thus, if we are going to adopt a

SULLIVAN'S NEW HYDRAULICS. 255

double riveted asphaltum coated Bteel pipe, then m=.00033,

11 /m 8
and j/ ogQg =0.2541, and the formula for any pipe in this

11 / Q 4
class reduces to d=.2541-| / / -g^. If we adopt an ordinary

uncoated, cast iron pipe, then m=.0004, and the formula re-
duces to

d= l

If the pipe is to convey water from the distributing res-
ervoir to the street mains, its capacity or diameter should be
such as to enable the pipe to maintain a given pressure in Ibs.
per square inch at the point of juncture with the street sys-
tem while it is supplying the given quantity of water in cubic
feet per second. It is also well to remember that the total
supply per 24 hours is usually drawn between 6 o'clock a. m
and 9 o'clock p. m., and for this reason the city supply pipe
leading from the distributing reservoir to the city must have
such diameter as will pass the entire 24 hours' supply within
12 hours' time, and also maintain a given pressure while so
discharging. In other words this pipe must carry a given
quantity of water within a given time with a given loss of
pressure or head at a given point. The formula for finding
the required diameter to carry a given quantity with a given
or pre-determined loss of head has already been given and
discussed . (See 64, 63.) The following tables will greatly
facilitate all such calculations, and show at once the value of
q, or cubic feet per second, corresponding to any supply in
gallons per 24 hours. (See also 102) .

256 SULLIVAN'S NEW HYDRAULICS.

TABLE No. 48.*

Gallons pei
24 hours=

Cub feet
per sec-
ond q

Loga
rithm
of q

Value of q 4

.00000154667

6.189397

.000,000,000,000,000,000,0 0,005,722,563,8T)0,675

10.

.0000154667

5.189397

.000,000,000,000,000,000,057,225,638,506,75

100.

.000154667

4.189397

. 030,000,000,000,0 X),572,256,335.067 .5

1,000.

.00154667

3.189397

.000,000,000,005,722,563,850,675

10,000.

.0151667

2.189397

000,000,057,225,638,506,75

100,000

.154667

1.189397

.000,572,256,335,067,5

1,000,000.

1.54667

0.189397

5.722,568,850,675

10,000,000.

15.4667

1.189297

57,225.638,506,75

100,000,000.

ir,4.667

2.189397

572,256,385.067,5

1,000,000,000.

1546.67

1. 189397

5,722,563,850,675.034,2

10.000,000,000.

15466.70

4.189397

57,225,638,506,750,342.032,1

100,000,000000.

154667.00

5.189397

572,256,385,067,503,420,321 .00

*One cubic foot=7.48 gallons. One gallon=23l cubic
inches.

TABLE No. 49.

Gal. per
24 hours

Cub. feet
per sec-
ond q

Loga-
rithm of

Value of q 4

.0000154667

5.1893 r 8

.000,000,000,000,000,000,057,225,638,508.75

.0000309334

5.490427

.000,000,000,000,000,000,915,720,15

.00003866675

5.587337

.000,000,000,000,000,002,235,678,513

.0000464001

5.666518

.000,000.000,000,000,004,63)

.0000618668

5.791458

. 000,000,000,000,000,014,65

.0(00773835

5.888367

. 000,000,000,000,000,035,77

.0000928002

5.967549

. 000,000,000,000,000,074,26

SULLIVAN'S NEW HYDRAULICS.

TABLE No. 49 Continued.

257

Gal. per
24 houre

Cub. feet
per sec-
ond q

Loga-
rithm of

Value of q*.

.00 1082669

4.034495

.000,000,000,000,000.137,398,72

.0001237336

4.092488

.000,000,000,000,000,234,396,1

.0001392003

4.1436399

.000,000,000,000,000,375,457,278

100

.000154667

4.189398

.000,000,000,000,000.572,^56,385,067.5

200

.000309334

4.490427

.000,000,000,000,009,157,201,5

250

.0003866675

4.587337

.000,000,000,000,022,356,735.513

300

.000464001

4.666518

. 000,000,000,000,046,350

400

.000618668

4.791458

. 000,000,000,000,146,500

500

.000773335

4.888367

. 000,000,000,000,357,700

600

.000928002

4.967549

. 000,000,000,000,742,600

700

.001082669

3.034496

.000,000,000,001,373,987,200

800

.001237336

3.092488

.000,OOU,000,002,343,96l,00

900

.001392003

8.1436399

.000,000,000,003,754,572,780

1,000

.00154667

3.189398

.000,000,000,005,722,563,850,675

2,000

.00309334

3.493427

.000,000,000,091,572,015

2,500

.003866675

3.587337

.000,000,000,223,-567,351,3

3,000

.00464001

3.666518

.000,000,000,463,5

4,000

.00618668

3.791458

.000,000,001,465

5,000

.00773335

3.888367

.000,000,003,577

6,000

.00928002

3.967549

.000,000,007,426

7,000

.01082669

2.034495

.000,000,013,739,872

8,000

.01237336

2.092488

.000,000,023,439,610

9,000

.01392003

2.5745399

.000,000,037,545,727,800

10,000

.0154667

2.189398

.000,000,057,225,638,506,750

20,000

.0309334 J2.490427

.000,000,915,720,015

25.000

.03866675

2.587337

.000,002,223,567,351,3

30,000

.0464001

2.666518

.000,004,635

258 SULLIVAN'S NEW HYDRAULICS

TABLE No. 49 Continued.

Gal. per
24 hours

Cub. feet
per sec-
ond q

Loga-
rithm of

Value of q*

40,000

.0618668

2.791458

.000,014,650

50,000

.0773335

2.888367

.000,035,770

60.000

.0928002

2.967549

.000,074,260

70,000

.1082669

1.034495

.000,137,398,720

80,000

.1237336

1.092488

.000,234,396,100

90,000

.1392003

1.5745399

.000,375,457,278

100000

.154667

1.189398

.000,572,256,385,067,5

200,000

.309334

1.490427

.(09,157,201,5

250,000

.3866675

1.587337

.022,356,735,130

300,000

.464001

1.666518

.046,35

400,000

.618668

1.791458

.146,5

500,001

.773335

1.888367

.357,7

600,000

.928002

1.967549

.742,6

700000

1.082669

0.034495

1.373,987,2

800,000

1.237336

0.092488

2.343,961

900,000

1.392003

0.1436399

3.754.572,78

1,000,000

1.54667

0.189398

5.722,563,850,675

2,000,000 3.09334

0.490427

91.572,015

2,500,000

3.866675

0.587337

223.567,351,3

3,000,000

4.64001

0-666518

463.500

4,000,000

6.18668

0.791458

1465.00

5,000000

7.73335

0.888367

3577.00

6,000,000

9.28002

0.967549

7426.00

7,000,000

10.82669

1.034495

13939.872

8,000.000

12.37336

1.092488

23439.61

9,000,000

13.92003

1.1436399

37545.727,8

SULLIVAN'S NEW HYDRAULICS.

TABLE 49 Continued.

259

Gal. per
24 hours

Cub. feet
per sec-
ond q

Loga-
rithm of

Value of q*.

10.000,000

15.4667

1.189398

57225.638,506,75

20,000,000

30.9334

1.490427

915720.16

25,000,000

38.66675

1.587337

2235673.513,03

30,000,000

46.4001

1.666518

46350' 0.00

40,000000

61.8668

1.791458

14650000.00

50,000,000

77.3335

1.888367

35770000.00

60,000,000

92.8002

1.967549

74260000.00

70000,000

108.2669

2.034495

137398720.00

80,000,000

123.7336

2.092488

234396100.00

90.000,000

139.2003

2.1436399

375457278.00

100000,000

154.667

2.189398

572256385.067,5

200,000,000

309.334

2.490427

9157201500.000

250,000,000

386.6675

2.587337

22356735130.30

101. To Find the Diameter in Feet of a Circular Con-
duit or Pipe With Free Discharge, as From One Reservoir
Into Another, which is required to Discharge a given quan-
tity in Cubic Feet Per Second, the Total Head or ths Slope
Being Known:

The general formula for finding the required diameter in
feet will be

Simplifying the formula as pointed out heretofore (100)
and for the following classes or degrees of roughness of peri-
meter we have,

(1) For ancoated clean cast iron pipe,m=.0004, and

SULLIVAN'S NEW HYDRAULICS.

(2) For uncoated steel or wrought iron, m=.00038, and

(3) For uncoated wooden stave pipe, made of dressed
hard wood, and closely jointed, m=.00048, and

(4) For cement mortar lined pipe, one third sand and two
thirds cement, m-. 000424, and

(5) For riveted pipe, thoroughly dipped and coated with
asphaltum and crude petroleum, m=. 000325, and

(6) For cast iron or welded pipes thoroughly asphaltum
coated and carefully laid and jointed, m=r.000305, and

For brick perimeters see 24. Always make an extra al-
lowance in the diameter of pip > or conduit for future deteri-
oration and for deposits.

102. To Find the Diameter in Feet of a Circular Con-
duit or pipe which is Required to carry a Given Quantity in
Cubic Feet Per Second to a Given Point and Maintain a
Given Head or Pressure at That Point while Delivering the
Required Quantity:

This formula is very important in designing power mains
for water wheels.'in which it is required to maintain a given
pressure or head at the base of the nozzle which discharges
upon the wheel or motor It applies equally well to hydraulic
giants used in placer mining, and to fire hose with nozzle at-
tached, and to all other cases where the discharge is partially

SULLIVAN'S NEW HYDRAULICS. 26l

throttled, as in the case of a supply pipe leading from the
distributing reservoir of a water works system to the street
mains. In the latter case it is desirable to so proportion the
diameter that it will convey the required quantity of water
and at the some time maintain not less than a given head
pressure at the point of its juncture with the street mains.
The general formula will be

In which,

h" = total head in feet to be lost in friction in the length I.

n=coefficient of resistance, and varies with different
classes of wet perimeter.

Simplifying the formula for given classes of perimeters
as heretofore pointed out ( 64, 68) and

(1) For uncoated clean cast iron, n=.0003938. and

(2) For uncoated clean steel orwrought iron, n=.00037411
and

(3) For uncoated wooden stave pipe, made of dressed
hard wood and closely jointed, n=.00047256, and

(4) For cement mortar lined pipe, one-third sand and
two-thirds cement, n=.0004175, and

d= .2653 11

262 SULLIVAN'S NEW HYDRAULICS.

(5) For riveted pipe, thoroughly dipped and coated with
asphaltum and crude petroleum, n=.00032, and

(6) For cast iron and welded pipes, thoroughly coated
with asphaltum and oil, and carefully laid and jointed, n=
.00030, and

REMAKK. For any given class of perimeter n=mX-9845,
and m=-qgT^, and the difference in value between

H/' m* 11 /~~n*~
y ogQc and ./ qgnc for any given roughness is equal .0008.

That is to say,

y 1 OQQ5 is .0008 less than the corresponding ualue of

n / m
^ ^805-

11 / m*~ 11 /~n*~~

If ,/ -oon^- =-2608, then 1 / / ^ ? ^=.2600, and soon.

V .OoUD V .ooUO

While the difference in value of m and n is small, yet it
must be remembered that m=the head per foot length of
pipe to balance the resistance and generate the mean velocity
of flow, and n is equal the friction head only, per foot length
of pipe. In a pipe of considerable length the difference be-
comes very considerable. (See, in this connection, 4 and 5)

Formulae (43) and (45) given in 17 may be adopted in-
stead of the above but in that event the value of m or n must
be converted to terms of P as in 17.

SULLIVAN'S NEW HYDRAULICS. 263

103 Velocities, Discharge and Friction Heads for
Slopes and Diameters.

The slope required to generate a velocity of one foot per
second in any given diameter with full and free discharge is

The slope required to generate any other velocity, either
greater or less than one foot per second, is

In the latter formula v must equal the square of the de-
aired velocity in feet per second. Having found the value of
S for v^l.OO in any given diameter, then the required value
of S to generate any other velocity in the given diameter,
will equal the value of S for v = 1.00 multiplied by the square
of the proposed velocity . The distance or length in feet i, of
pipe, in which there is a fall of one foot is

As the value of S shows the total head per foot length of
pipe, the fall in feet per 100 feet length is found by moving
the decimal point in the value of S two places to the right.
The friction head per foot length in any given uniform diam-
eter with full and free discharge is .9845 per cent of the value
of S for that pipe. The friction head may therefore be easily
found from the value of S. The friction head per 100 feet
length of pipe will be

h*=SX98.45, or h ' =(SX 100) (SX100X -0155).

When the friction head per 100 feet is ascertained for a
given diameter with v=1.00, then the friction head per 100
feet in the given diameter for any other velocity will equal
that for v=1.00 multiplied by the square of the proposed vel-
ocity. The following table (No. 50) is based on m=.0004 for
all clean iron pipes.

264

SULLIVAN'S NEW HYDRAULICS.

TABLE No. 50.

Velocities, Discharge and friction heads for given slopes
and diameters.

d;/d

8,1

si^

>fc

|||

U|
all

Dischar
Cubic t
Second.

3 inches

.25 feet

0.500

.0008

0.50

.07876

11.01804

.02455

.0032

1.00

.31504

22.03608

.04910

.0072

1.50

.70*84

33.05412

.07365

.0128

2.00

1.26016

44.07216

.09820

.0200

2.50

1.96900

55.09020

.12275

.0288

3.00

2.83536

66.10824

.14730

.0392

3.50

3.85928

77.12628

.17185

.0512

4.00

5.04070

88 14432

.19640

.0648

4.50

6.37964

99.16236

.22095

.0800

. 5.00

7.87600

110.18040

.24550

.0968

5.50

9.53008

121.19844

.27005

.1152

6.00

11.34158

132.21648

.29460

.1352

6.50

13.31060

143.23452

.31915

.1568

7.00

15.43715

154.25256

.34370

.2048

8.00

20.16281

176.28864

.39280

.3200

10.00

31.50400

220.36080

.49100

4 inches

.3333 feet

0.579

.00207850

1.00

.204625

39.18024

.08730

.00466662

1.50

.459428

58.77036

.13095

.0083140

2.00

.8185133

78.36048

.17460

.0129906

2.50

1.278924

97.95060

.21825

.0187065

3.00

1.811655

117.54080

.26190

.0254616

3.50

2.506694

136.72096

.30555

.0332560

4.00

3.254053

155.90120

.34920

.0420896

4.50

4.143721

175.08144

.392S5

.0519625

5.00

5.115708

194.26168

.43650

.0628746

5.50

6.190004

213.44192

.48015

.0748260

6.00

7.368619

232.62216

.52380

6.50 8.645544

251.80240

.56745

.'1018465

7.00 10.026788

270.98264

.61110

SULLIVAN'S NEW HYDRAULICS.
TABLE 50. Continued.

Is*

lo-S

O ***

v> *^ ^

g.2 o

002
02

?!1

ill

.1-5.2

fiu S

5 inches

4167 feet

0.645

.00148120

1.00

.14651329

61.17144

.13630

.00334845

1.50

.32965490

91.757160

.20445

:()05952<SO

2.00

.58605316

122.342880

.27260

.00930125

2.50

.91570806

152.92860

.34075

! "1339380

3.00

1.31861961

183.51432

.40890

.01823045

8.50

1.79276780

214.10004

.47705

.02381120

4.00

2.34421264

244.68576

.54520

.03013605

.03720500

4.50
5.00

2.96989412
3.66283225

275.27148
305.85720

.61335
.68150

.04501805

5.50

4.43202702

336.44292

.74965

.05357520

6.00

5.27447844

367.02864

.81780

6 inches

.50 foot

0.7071

.001131156

1 00

.1113623

88.14432

.1964

.002543101

1 50

.2503683

132.21648

.2946

.004524624
.007%9725

2.00
2 50

.4454492
.6960144

176.28864

.3928
.4910

.010180404

3 00

1 002260

264 [43296

.5892

.013*56051
.DIM 19*49(5
.022905909
.028278900

3.50
4.00
4.50
5.00

1.364198
1.781796
2.255086
2.784057

308.50512
352.57728
396.64944
440.72160

.6874
.7856
8838
.9820

.03J217469

5 50

3.368709

484.79376

[.0802

.040721616

6 00

4.008903

5?8. 86592

[ 1784

7 inches

.047791341

6.50

4.705057

572.93808

[.2766

.5833 feet

0.764

.000897583

1 '0

.088367

119.96424

.26730

.002019561
.003590332

1.50
200

.198825
.&5346S

179.94636

239.92J4S

.40095
.53460

.Ol)560lHt3

2 50

.552294

299 91060

.66825

.00-078247

3.00

795303

359.89272

.80190

.010995391
.013961328
.018176055

3.50
4.00
4.50

1.081996
1.374493
1.789433

419.87484
479.85(596
539.83908

.93555
1.06920

1.20285

8 inches

.022439575
027151885

5^50

2.209176
2.673103

599.82120
659.80332

1.33650
1.47015

.6667 feet

0.817

.000734357

1 00

.0722975

156.67608

.3491

.002937428
.006609213
.011749712
.018358925
.026436852
.035983493
.046998848

2:00

3.00
4.00
5.00
6.00
7.00
8 00

! 287 1888
.6506771
1.1567592
1.8074362
2.6027091
3 5425749
4 6270366

313.35216

470.02824
626.70432
783.38040
940.05648
1096.73256

.6982
1.0473
1.3964
1.7455
2.0946
2.4437
2.7928

.059482917
.073435700

9 00
10.00

5.8560932
7.2297500

li"li> !(472
1566.76080

3.1419
3.4910

SULLIVAN'S NEW HYDRAULICS
TABLE No. 50 CONTINUED.

as "to
*

O Jo

<D "S

-PH O

lit

fcfcS

I 33

S3l

inches
.75 Foot

0.866

.00061585835

1.00

.060631255

198.27984

.4418

.00246343340

2.00

.242525020

396.55968

.8836

.00554272415

3.00

.545681193

594.83952

1.3254

.<HMS53733tiO

4.00

.970100080

793.11936

1.7672

.01529f.45.s75

5.00

1.505936364

991.39920

2.2090

.02217090060

6.00

2.182725170

1189.67904

2.6508

.030177U:>915

7.00

2.970931474

1387.95888

3.0926

.03941493440

8.00

3.880400300

1586.23872

3.5344

.0498S452635

9.00

4.911131620

1784.51856

3.9762

.06158583500

10.00

6.063125500

1982.79840

4.4180

.07451886035

11. fO

7.336381802

2181.07824

4.8598

.08868360240

12.00

8.730900660

2379.35808

5.3016

10 inches

.8333 Ft

0.913

.00052576237

1.00

.051761306

244.77552

.5454

.00210304948

2.00

.207045222

489.55104

1.0908

.00473186133

3.00

465851748

734.32656

1.6 62

.00841219792

4.00

.828180886

979.10208

2.1816

.01314405925

5.00

1.294 32634

1223.87760

2.7270

.01892744532

6 00

i.,s'->:uor,992

1468.65312

3.2724

.02576235613

7.00

2.5363039H1

1713.42864

3.8178

.03364879168

8.00

3.312723541

1958.20416

4.3632

.04258675197

9.00

4.192665732

2202.97968

4.9086

.05257623700

10.00

5.176130600

2447.75520

5.4540

.06361724677

11.00

6.263117945

2692.53072

5.9994

12 inches

1.00 Feot.

1.00

.00040000000

1.00

03938

352.48752

.7854

.0016

2.00

; 15752

704.97504

1.5708

.0036

3.00

.35442

1057.46256

2.3562

.0064

4.00

.63008

1409.95(X'8

3.1416

.0100

5.00

.98450

1762.43760

3.9270

.0144

6.00

1.41768

2114.92512

4.7124

.0196

7.00

1.92962

2467.41264

5.4978

.0256

8.00

2.52032

2819.90016

6.2832

.0324

9.00

3.18978

3172! 38768

7.0686

.0400

10.00

3.94800

3524.87520

7.8540

.0484

11.00

4.76498

3877.36272

8.6394

14 inches

1.167 Feet

1.080

.00031736964

1.00

.026817735

479.7672

1.069

00126947856
.00285632676

2.00
3.00

.1249*0165
.281205370

959.5344
1439.3016

2.138
3.207

.00507791424

4.00

.49! 1920,558

1919.UC.NS

4.276

.00793424100

5.00

.78U26000

2398.8360

5.345

.01142o30704

6.00

1.1248208115

2878.6032

6.414

.01555111236

7.00

1.531007012

3358.3704

7.483

.02031165696

8.00

1.999682628

3838.1376

8.552

.02570694084

9.00

2.569280203

4317.9048

9.621

.TO! 73696400

10.00

3.124504100

4797.6720

10.690

.03840172844

11.00

3.790649969

5277.4892

11.759

SULLIVAN'S NEW HYDRAULICS.
TABLE 50 Continued.

267

||.

I 4

Is
S-s

Jh.

Ill

111

C Q m

fcWfe

11!

Q OS

ill

QOra

16 Inches

1.333 feet

1.155

00025980521

1.00

.025577823

626.5248

1.396

00103922084

2.00

.103350513

1253.0496

2.792

(X 1233,^24(^9

3.00

.23020 407

1879.5744

4.188

'415688336

4.00

.409245167

2506.0992

5.584

00649513025

5.00

.639445574

3132.6240

6.980

00935298756

6.00

.920801626

3759.1488

8.376

01273045529

7.00

1.253313324

4385.6736

9.772

01662753344

8.10

1.636980668

5012.1984

11.168

02104422201

9.00

1.971803657

5638.7232

12.564

02598052100

10.00

2.5577823

6265.2480

13.960

03143643041

11.00

3.126353005

6891.7728

15.356

18 inches

1.50 feet

1.224

0002124183

1.00

.02091259

793.0296

1.767

0008496732

2.00

.08365033

1588.0592

3.534

0019117737

3.00

.18821413

2379 0888

5.301

0'>339;?6928

4.00

.33460131

3172.1184

7.068

0053104575

5.00

.52281455

3965.1480

8.835

0076470588

6.00

.75284194

4758.1776

10.602

01' 140-4967

7.00

1.02471651

5551.2072

12.369

0135947712

8.00

1.33840523

6344.2368

14.136

0172"5S823

9 00

1.69391912

7137.2664

15.903

0212418300

10.00

2.0912590

7930.2960

17.670

0257026143

11.00

2.53 42238

8723.3256

19.437

20 inches

1.667 feet

1.291

000185865228

1 00

.0182984317

979.2816

2.182

000743460912

2.00

.0731937268

1958.5632

4.364

001672787052

3 00

.1646858853

2937.8448

6.546

002973843648
004646630700

4.00
5.00

.2927749072
.4574610000

3917.1264
4896. 40SO

8.728
10.910

(X)tit.i9114.>20>
009107396172

6.00
7 01

.6587435411

>9(231532

5875.6896
6854.9712

13.092
15.274

011895374592
015055083468

8.00
9.00

1.17109962*6
1.4821729675

-7834.2528
8813.5344

17.456
19.638

10 00

1 82984317

21.820

022489692588

11.00

2.2141102353

10772:0976

24.002

24 inches

2.00 feet

1.4142

.0001414227124

000565690849*

1.00
2.00

013923055

055692220

1409.35008

2818.70016

3.1416
6.2832

! 0012728044116
.0022623633984
.0035355678100
.0050912176464

4^00
5.00
6.00
7 00

.125307495

! 5012299 -i

.t;>222'.t')95

4228.06024

5637.40032
7046.75040
8456.10048
9S65. 45046

9.4248
12.5664
15.7080
18.8496
21.9912

! 009061051899

.0114552397044

s'.oo

9.00

l' 127767455

11274.80084
12684.15072

25.128S

2-S.2744

0141422712400
.0171121482004

10.00
11.00

1.39230550
1.684689655

1-1 CM. 500SO
15502.85088

31.4160
34.5578

SULLIVAN'S NEW HYDRAULICS
TABLE No. 50 CONCLUDED.

5-31

27 inches
2.25 Feet

30 inches
2.50 Feet

1.581

.00047407 4C740

.01)29(529629625
.0042671671660
.OJ5807 4074065
.0375851851840

0118518518500
0143407407385

.0001012018
.0004048072

0016192288
0:)25 300450
0036432648

0064769152
0081973458
0101218000
0122454178

.0000769823

0005928407
.0012317168
0019245575
0027713628
.0037721327
0049268672

1.00
2.00
8.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00

1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00

1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00

.011721478

.046885912
.105493302
.187543648

.421973208
.574352422
.750174592
.949439718
1.17214780

.1L941312
.24908300
.35867952

.63765248

1.20556172

.00757891
.03131564

.18947275
.27284076

CO! 13148583

.48505024
.61389171
.75789100
.91704811

1724.4288
3448.8576
5173.2864
6897.7152
8622.1440
10346.5728
12071.0016
13795.4304
15519.8592
17244.2880

2203.1592
4406.3184
6609.4776

11015.7960
13218.9552
15422.1144
17625.2736

19828.4328
22031.5920
24234.7512

3172.5672
6345.1844
9517.7016

15862.8360
19035.4032
22207.9704
25380.5376
28553.1048
31725.6720
34898.2492

3.976
7.952
11.928
15.904
19.880

31 ! 808
35.784
39.760
43.736

14.727
19.636
24.545
29.454
34.363
39.272
44.181

54.000

7.069
14.138
21.201

35.345
42.414
49.483
56.552
63.621

77.759

104. Thickness and Weight of Cast Iron Pipe. There
is a great want of uniformity in regard to the thickness of
cast iron pipe for any given pressure. Every city seems to
have adopted different thicknesses of pipe. The leading for-
mulas for thickness give greatly differing results for the same
conditions.

SULLIVAN'S NEW HYDRAULICS.

t=(.000058hd) +.0152d+.312 (J.B.Francis

t=(.0016 n d) +.013 d+.32 (M. Dupuit

t=(.00238 n d) +.34 (Julius Weisbach

The following formula gives thickness of cast iron pipe
as adopted in recent practice,

t=(p+100) .000142 d+.33 (1 .01 d) (110)

In the above formulas,
t=thickness of pipe shell in inches
d=inside diameter in inches
h=head of water in feet
p=pressure of water=HX-434
n=number atmospheres pressure at 33 feet each.
Fannings formula for the weight per lineal foot of cast
iron pipe, including the weight of the bell or hub is, for 12
foot pipes,

W=12 (d+t) Xl-08 tX3.1416X.2604

By a 12 foot pipe is meant a pipe which will actually lay
12 feet, or is 12 feet from bottom of bell to end of spigot. The
bell or hub adds about 1% per cent to the weight of a length
of pipe. The above formula allows for the extra weight of
bell. For more on weight of pipes, see "Gregory's Practical
Mathematics."

105 Dimensions and Weight ot Cast Iron Pipe Made
by The Colorado Fuel and Iron Company of Denver, for
100 Ibs. Pressure.

TABLE No. 51. By W. F. McCue

1
I

Length over all.
Feet Inches.

a I
!l

Will Lay
Feet Inches

Thickness of
Shell Inches*

Inside Diameter
of Bell Inches

0*0

Outside Diam.
of Spigot.

A 3

*.s

3
4

6
8
10
12
16
20

12-4
124
324
124
124
124
124
124

3
3
3
3V4
3K
4
4

121
12-1
12-1
12 Vt

1I-*

12
12

1332
7-16
1-2
17-32
1932
5-8
34
2732

4 1-2
5 1-2

7 5-8
9 7-8
11 7-8
133-1
18
22 1-8

7
81-4
10 7-8
13 38
15 5-8
17 3-4
22 1-2
26 7-8

43-8
5 3-8
7 1-2
9 3-4
11 3-4
13 5-8
17 7-8
22

15 1-2

33
44

175

*Sce Table No. 27. 58 fo- fractional inches in equivalent
decimals.

270

SULLIVAN'S NEW HYDRAULICS.

Packing (Jute Hemp) and Lead Required Per Joint For
Above Pipe.

Diameters =*

10"

12"

16'

20"

Lead, Ibs. Per

Joint

4 1-2

5 1-2

12 1-2

20 1-2

Packing, ozs

3 1-2

91-2

106 Weight Per Foot Length of Cast Iron Pipe For 150
and 200 Ibs. Pressure, as Made by Colorado Fuel and Iron
Co. of Denver, Colorado.

Diameters

3" 1 4'

10"

12"

16"

20"

Wt.Pei.ft.

150 pounds
Pressure

17 Ibs 23V, Ibe

361b<

48Jbs

70 Ibs

85 Ibs

140 Ibs

210 Ibs

Wt. Per. ft.

200 pounds
Pressure

1
19 lbs|26 Ibs

421bi

55 Ibs

78 Ibs

94 Ibs

15b Ibs

222 Ibs

REMARK The market prices of pig iron, cast iron and
lead and other metals fluctuate so rapidly that tables for esti-
mating the cost of pipe and laying are of no great value ex
cept in so far as such tables furnish the data as to the quan-
tity and weight required. The price of pig iron May 12th,
1898, was $6.65, and on July 28th, 1899, the price was $15.25.

The present price of cast iron pipe (August 2nd, 1899) is
$33.00 per ton of 2,000 Ibs., and of lead, $5.00 per 100 Ibs. in
Denver.

107 Manufacturers' Standard Casllron Water Pipe
For 100 Ibs. Pressure Per Square Inch.

TABLE No. 52.

Diameters In.

18 20

Thickness, In*

1-2

9-lfi

9-16

8-4

7 8 15-16

11-813-8

11-2

Wt. per. ft. Ib.

117

125

167 200

251

350

475

775

Wt.per. 12ft.

2ti4

;*;

540

720

900

1500

200012400

4200

5700

B800

*See Table No. 27, 58 for fractional inches in equivalent
decimals.

108 Cost Per 100 Feet Length, For Labor and Ma-
terial in Laying Cast Iron Water Pipe in Denver, Colorado,
in 1890.

The conditions were: Top of pipe 5 feet below surface.
Depth of trench 5 feet, plus outside diameter of pipe. Easy

SULLIVAN'S NEW HYDRAULICS.

271

trenching in sandy loam, Wages, foreman $3.00, calkere 12.50
laborers $1.75 per day of 10 hours, teams 83.00 per day, pipe
$33.00 per ton of 2000 Ibs.. lead $4.15 per 100 Ibs., packing 6
cents per Ib. No pavements to tear up. Backfilling done by
teams and scrapers. Average water pressure 80 Ibs. Thick-
ness of pipe for 120 Ibs. hydraulic pressure. Hemp packing.
Hauling pipe 60 cents per ton.

TABLE No. 53* Cost per 100 feet.

22
82

4:,
80

7fi
117
125

170
'iJO

a-,u
r.oo
no

$86.30

52.80
74.25
99.00
123.75
193.05
206.25
280. EO
412.50
577.50
825.00
1155.00

10.20

.94
1.28
1.88
2.63
4.50
6.30

$ 0.66 $0.

1.05

1.35

1.80

2.25

3.50

3.75

5.10

7. tO
10. EO
IB. 00
21.00

$ 2.15

3.13

4.03

4.14

5.36

8.07

9.30

11.17

14.96

16.15

23.75

49.87

$0.20
.20

.20
.'JO
.20

$0.15
..15
.15
.20
.20
.25
.25

.60

.70

1.00

15.00
16.00
20.00
31.00
32.00
38.00
44.00
50.00
60.00
75.00

*Allow 440 joints per mile when ebtimating cost of lay-
ing cast iron pipe. Wrought iron and steel pipe is made in
lengths of 15 to 27 feet according to conditions to be met.
Cast iron pipe is in lengths of 12 feet. See Remark under
Table No. 55.

109 Cost of Pipe Per Foot Laid in Boston.

Axis of pipe is 5 feet below surface. Labor $2.00 per
day. Cost of pipe 1^ cents per Ib., or $30.00 per ton of 2000
Ibs. Special castings 3 cents, lead 5 cents per Ib. Cost of
excavating rock $3.50 to $5,50 per cubic yard, measured to
neat lines.

This table is transcribed from "Details of Water Works
etc.", by W, R. Billings.

272 SULLIVAN'S NEW HYDRAULICS

TABLE No. 54. Cast Iron Pipe.

1 11*

t 1

fc|

"1 r-i

Jj]

-J

M .

&gj

111

bi
a

2 o

J4^3
jjfl

CUD"
'S u.

T3 a
a P
go
Hft

I' 3 !

O*0 CO

M *-

UT3 I-

H.S

0.45
0.50

21.7
85.0

-0.70
1.00

$0.38
0.57

$0.02
.08

$0.25
0.27

$0.70
.93

0.55

50.0

1.85

0.83

.08

.05

0.30

1.26

0.60

68.0

1.70

1.10

.10

.06

0.34

1.60

12
12

0.58
0.65

S:S

2.00
2.00

1.27
1.42

.13
.13

.07

0.37
0.37

1.84
1.99

0.66

118.0

2.70

1.87

.17

.08

0.45

2.57

0.75

185.0

2.70

2.12

.17

.08

0.45

2.82

0.73

162.0

3.35

2.55

.21

.09

0.55

3.40

0.85

183.0

3.35

2.94

.21

.09

0.55

8.79

0.81

216.0

4.00

3.44

.25

.10

0.68

4.47

0.94

250.0

4.00

3.95

.25

.10

0.68

4.98

0.93

308.0

5.00

4.92

.29

.11

0.80

6.12

1.04

410.0

6.00

6.58

.34

.12

1.00

8.04

1.12

490.0

6.70

7.80

.40

.15

1.30

9.65

1.25

660.0

8.00

10.40

.48

.20

1.75

12.83

REMARK From the high cost of trenching and the ref-
erence to rock excavation, the ground must have been very
"hard digging." Compare cost of trenching with that at
Omaha for like diameters.

IIOCost of Trenching, Laying, Calking and Back-
filling in Omaha, 1889, With wages of Foreman $2.50,
Calkers $2.25, Laborers $1.75 Per Day of 10 Hours. (W.

F. McCue, C. E., of Colorado Fuel & Iron Co.)

TABLE No. 55. Cast Iron Pipe.

Diam.
of Pipe

Width
of
trench
feet

Depth of
trench,
feet

Cost of
trench,
lineal
foot

Laying,
calking,
backfilling,
lineal foot.

Cost of
labor per
lineal foot
complete.

6
8
10
12
16

1.75
1.75
1.75
2.00
2.00
2.33

5.666
6.000
6.000
6.083
6.250
7.333

$0.104
0.10.1
0.107
0.126.
0.126
0.175

$0.036
0.036
0.043
0.053
0.056
0.063

$0.140
0.141
0.150
0.179
0.182
0.238

REMARK Mr. McCue lias been in charge of the con-
struction of nearly 700 miles of pipe line in the Eastern and

SULLIVAN'S NEW HYDRAULICS. 273

Western states, In a letter to the writer he says: "We gen-
erally employed 60 to 70 men in a gang enough laborers to
excavate the trench ahead of the layers. In laying 4 to 12
inch pipe, we had one yarner aud one calker. In laying pipe
16 inches diameter or larger, we had two yarners and two
calkers. In laying pipe larger than 12 inches diameter it is
necessary to use a derrick for lowering the pipe into the
trench.

One yarner and one calker will make about CO joints per
day of 10 hours in laying 4 or 6 inch cast iron pipe, and about
50 joints of 8 inch, 45 joints of 10 inch and 40 joints of 12
inch pipe. In laying pipe larger than 12 inchs, a derrick is
required, and progress is much less. Most of the backfilling
is done by team and scraper. The largest days work I ever
had done was 80 joints of 8 inch pipe yarned by one man and
calked by one man. In 1893, I took one yarner and one
calker, the fastest I ever saw, and laid and calked 272 joints
of 6 inch pipe in 35 hours. The cost was 1^ cents per lineal
foot including foreman, kettlemen, and 3 to lay pipe in the
trench. We use Jute hemp for packing."

IIIWeston's Tables for Estimating Cost of Lay lag
Cast Iron Pipe.

The following tables by E. B. Weston, C. E., of Provi-
dence, Rhode Island, were published in Engineering News,
June 21, 1890, together with other valuable data of like char-
acter. The elements of cost entering these tables are:

Wages, foreman $3.00, calkers $2.25, laborers $1.50, per
day. Teams $2.25 per day. Carting $1.00 per ton of 2240
Ibs. Depth of trench 4.67 feet plus one-half the outside
diameter of pipe. Lead, 5 cents per pound. Tools, blocks
and wedges 7 2 10 to 16 1-10 per cent of cost of trenching,
laying and backfilling the trench. In the tables the word
"trenching" includes excavation and backfilling. "Medium"
digging is ground equivalent to gravel and sand. "Hard"
digging id ground equivalent to hard or moist clay. Cost of
engineering and inspection not included in tables.

274

SULLIVAN'S NEW HYDRAULICS.

g 8

1 1

I I

3 |

*H ei

111

SSSS

s i 1 i 3

-H o O O O O O

S8SSS

3 i

3 e

ifllilll

SULLIVAN'S NEW HYDRAULICS.

275

112 Cubic Yards of Excavation in Trench Per Lineal
Boot, Vertical Sides Bell Holes Mot Included.

TABLE No. 57.

Depth of Trench in Feet.

10 11

5.50
6.

8. JO
9.

77S

0.370
0.444
0.518

3.740
0.815
1.888
1.968

[.037
1.111

[.185
[.259
:.333
1.407

0.926
015
111

0.888
.000

.111

0.388 0.444

0.518 0.592

0.644 0.740

0.777 0.888
0.876

3.000
3.166

i. :,:,:,
0.740
a. 926
1.111
296
1.481
1.666
1.851
2.037
2.222
2.407
2.592
2.777
2.963
J.148
3.333
3.518

0.611
0.815
1.018
1.222
1.425
628
833
2.037
2.237
2.444
2.647
2.851
i. or,:,
3.258
3.462
3.666
3.870

0.666
0.888
1.111

1.333
1 . 555

1.777
2.000

2' 222

3.611
3.851
4.092
4.000 4.333
4.222 U.573

0.722

0.962

203

444

2.648

0.777
037

2.S8

r,r,r>

074

2.852

3.111

.370

4.148

406

4.666

4.924

REMARK. The foregoing table (No. 57) will be useful in
estimating the cost of sewer work as well as in estimates of
cost of pipe laying. It is also the custom of some engineers
to excavate irrigation canals with vertical sides and allow for
the caving and sliding of the banks until they assume the
natural angle of repose. There is nothing to commend this
practice, but still it is followed to a considerable extent. In
sewer work where the ground is firm, the trench is excavated
in alternate sections, and tunnels driven through the short
blocks of ground between the excavated sections. This re-
duces the amount of excavation by about 30 per cent, and
saves the cost of sheeting and bracing. In estimating cost
of excavation in earth or rock, see Trautwine's "Engineers
Pocket Book."

113. -Bell Holes in Trench for Cast Iron Pipes. in

order that the "yarner" and the calker may have room to get

276 SULLIVAN'S NEW HYDRAULICS.

at all parts of the joint, the trench should be dug out 8 inches
deeper for a distance of four feet in front of the bell or hub,
and 8 inches wider on either side for the same distance to
give shoulder and striking room. This adds materially to the
cost of excavation, especially where the ground has a tendency
to cave and slide, or is very wet.

114 Depth of Trenches for Pipe. Pipes in which there
is a constant flow of water are in little danger from freezing
even if laid on the surface of the ground, but in the distribu-
tion or street system the flow id almost if not entirely stopped
during certain hours of the night when little or no water is
being drawn by consumers.

Pipes supplying reservoirs and having a constant dis-
charge may be covered to any convenient depth simply for
the protection of the pipe from injury by wagons, falling trees
etc., and to prevent too great expansion by heat or con-
traction by cold, and to get the pipe out of the way.

The general rule in the New England States is to make
the trenches for street pipes of such depth as will place the
center or axis of th3 pipe five feet under cover. That is, the
trench is five feet plus one half the outside diameter of the
pipe. The depth that a street pipe should be covered depends
on the climate, the nature of the ground and the diameter of
the pipe. Where the temperature gets down to from 25 to 40
degrees below zero (Fahr.) for two or three days at a time, 4
and 6 inch pipes will freeze solid when five feet under cover
in sandy and gravelly loam. This occurred in many towns in
Colorado in February, 1899. If the earth id dense and free
from stones and gravel it is not probable that frost will pene-
trate to a depth exceeding four and a half feet. Small pipes
laid in open, gravelly soil, should have the top of the pipe at
least six feet under cover.

115. Amount of Trenching and Pipe Laying Per Day
Per Man. The number of cubic yards of excavation done
per man per day will be less in deep trenchee than in com-

SULLIVAN'S NEW HYDRAULICS. 277

paratively shallow ones because of the extra effort required
to throw the dirt out of deep trenches. The nature of the
earth or rock to be excavated will, of course, be a controlling
element in determining the amount of excavation that can be
accomplished per day, by an average laborer. Quicksand,
water and caving banks may also be large items of expense
and prevent rapid progress. There are so many elements of
uncertainty invol/ed in making an estimate of the work that
one man will accomplish in a given time that it is best to as-
certain what has been actually accomplished under like con-
ditions in the past. By analysis of statements of work
actually done in a given time by a given number of men, we
can approximate the time required and the cost of doing
similar work. Mr. W. R. Billings, superintendent of the
Taunton, Mass., Water Works (1887) says*:

"The following notes of actual work are offered, not in
any sense as instances of model performance, but as simple
illustrations: Time July 6th 1887; gang 60 men, 16 inch pipe,
2 yarners, 2 calkers, 4 to 10 men digging bell holes, 30 boll
holes per day, 400 feet of pipe laid and jointed in 10 hours."

These notes are somewhat incomplete in that they do not
disclose the following items: (1) nature of earth excavated;
(2) depth of trench; (3) width of trench; (4) what part of the
total 400 feet length of trench and bell holes made on July
6th. (5) Was the trench back-filled for 400 feet on July 6th.
(6) How many of the 60 men were in the derrick gang. (7)
Did the derrick gang assist in excavating a part of the 400
feet of trench before beginning to lay pipe, or was a part of
the trench and bell holes made on the day before. Mr. Bil-
lings statement shows that 4 to 10 men working 10 hours
made 30 bell holes for 16 inch pipe. 30 bell holes would ac-
commodate only 360 feet of 12 foot pipes. He states that 400
feet were laid. It is therefore evident that some part (at
least 40 feet) of trench and bell holes must have been made on
some other day.

*Details of Water Works Construction, p. 55. (Published by "Engi-
neering Record" N. Y.)

278 SULLIVAN'S NEW HYDRAULICS

In another chapter of Mr. Billings work we find some
"Notes on the construction of two miles of 16 inch water
main," in 1887. The date shows that it is the same pipe above
referred to. From these notes we gather the following facts:
The pipe was hauled an average distance of 1)^ miles over
good roads for 64 cents per ton of 2240 Ibs. The first division
of the pipe line was 2,927 feet in length. The trenching was
in good ground except a short stretch of quicksand and water
The total cost of labor for this division of the line was 32.30
cents per lineal foot, including all labor charged on the time
book from foreman to water boy, in a gang of 60 men. An-
other division of 2,100 feet length furnished sandy digging
with some tendency to caving. A brook had to be crossed
and a blow-off located which required the trench to be 10 or
12 feet deep for 100 feet length. An old 8 inch pipe had to
be removed, and 18 services furnished with a temporary sup-
ply. The cost of labor per lineal foot for this division was

34.7 cents. In the next division the digging was dry and
sandy, and caving of the trench was almost constant. An old
8 inch pipe had to be taken up, and a temporary supply main-
tained for 53 services. The cost of labor on this division was

41.8 cents per lineal foot.

In the next division the digging was wet and dirty. Old
pipe had to be taken up, and temporary supply maintained
for 30 services, and four connections made for a manufactur-
ing company. The cost of labor in this division was 47.4
cents per lineal foot. The mill connections being the princi-
pal cause of the increased expense.

Mr. Billings states that a detachment of the same gang
of men laid 2,000 feet of 8 inch pipe in new ground, good dig-
ging, at a cost of 17.3 cents per lineal foot for all labor, and
1060 feet of 4 inch pipe at a cost of 13.10 cents per foot, and
600 fee't of 6 inch pipe at 15.38 cents per lineal foot.

While the depth and width of trench and daily wages
paid are not stated, it will be near enough to assume that the
trenches were 5 feet plus one half the diameter (outside) of
the pipe to be laid, and the width of trench 2.333 to 3.00 feet,

SULLIVAN'S NEW HYDRAULICS. 279

according to size of pipe (4*, 6", 8" and 16" diameters). For
amount of excavation in bell holes, refer to paragraph
110, ante.

Assume wages as follows: Foreman $3, Calkers and
yarners $2.25, Derrick gang (6 to 10 men) $1.75, laborers $1.50
per day of 10 hours. A gang of six men is sufficient to
handle the 4, 6, and 8 inch pipe, together with one yarner and
one calker. For the 16 inch pipe it will require 2 yarners
and 2 calkers and 10 men in the derrick gang. The remain-
der of the gang of 60 men will be laborers digging trench
and bell-holes ahead of the derrick gang. Subtracting the
number of calkers and yarnere and derrick gang and the
foreman from 60, the remainder shows the number of men en-
gaged in trenching and digging bell-holes. The length of
trench and bell-holes completed in 10 hours gives a basis of
calculating the cubic yards excavated by each man per day,
and the wages paid him per day furnishes the data for find-
ing the cost per cubic yard of excavation. In fairly good
digging it will be found that one man will make from 5.60 to
6.25 cubic yards of excavation per day of 10 hours. In ex-
cavating rock the average will be from, .50 to 1.50 cubic yards
of excavation per man per day, depending on the nature of
the rock. The excavation of deep, narrow trenches is very
much more expensive per cubic yard than railroad and canal
work in like earth or rock. See "Remark" under table No. 55,
Section 107. With labor at $2 per day the cost of excavation
in rock was $3.50 to $5.50 per cubic yard, measured in place,
in the City of Boston. This was an average of from .3636 to
.57 cubic yard of excavation in rock per man per day. In
very wet trenches the digging of sumps, sheet piling and
bracing, and pumping out of water is a heavy expense in ad-
dition to the ordinary cost, and will amount to from 40 cents
to $1.00 per lineal foot.

By reference to table No. 56, it will be seen that the cost
of laying 4", 6% 8" and 16" pipe as given above by Mr. Billings,
is about the same as given in Weston'a Table for "medium"
digging, and also about the same as shown in table No. 53 for

280 SULLIVAN'S NEW HYDRAULICS.

cost of laying pipe in Denver, in sandy loam. Referring to
cost of trenching in Omaha (table No. 55) with wages of
laborers at $1.75 per day, and we find that a trench, in good
digging, 5.666 feet deep and 1.75 feet wide, cost .10^ cents per
lineal foot. In a lineal foot of this trench there were 5.666X

1.75-i-27=.36724 cubic yards of excavation, or l =2.723 lin-
eal feet of trench to the cubic yard of excavation. With the
cost at .104 cents per lineal foot of trench, and 2,723 lineal
feet to the cubic yard, the cost per cubic yard of excavation
was 2.723 X.104=.284 cents.

With wages at $1.75 per day of 10 hours, the average work
done by one man in one day was ' =6.162 cubic yards.

One man would therefore average 2.723x6.162=16.779 feet
length of trench of those dimensions and in that kind of
ground, per day. If wages were reduced to $1.50 per day, the
cost of trench would be \ part less, or .24343 cents per cubic
yard of excavation. In stiff clay or cemented gravel, one man
will average about 4.50 cubic yards of excavation per day, and
the cost at $1.50 per day wages, will be 33% cents per cubic
yard, or if wages are $1.75 per day, the cost will be .39 cents
per cubic yard of excavation. Hence in stiff clay or cemented
gravel the average progress per man per day would be
12.25 lineal feet of trench 5.666 feet deep by 1.75 feet wide.
Trenches for larger diameters than 8 inches would be both
deeper and wider, and the cubic yards of excavation per
lineal foot would be increased in proportion.

One yarner and one calker will joint cast iron pipe about
as follows, in an average days work: 720 feet of 4 inch pipe,
or 660 feet of 6 inch, or 600 feet of 8 inch, or 540 feet of 10 inch,
or 480 feet of 12 inch, or 360 feet of 14 inch, or 200 feet of 16
inch pipe.

SULLIVAN'S NEW HYDRAULICS. 281

The number of joints made will depend on whether the
trench is wet or dry or stands up well or caves. The wages of
calkers and yarners are usually from 50 to 75 cents per day
more than the wages paid to ordinary laborers.

In estimating the cost of completing a cast iron pipe sys-
tem add five per cent to cost of the pipe in order to allow for
breakage. Also add the cost of engineering and inspection.

116 Lead Required Per Joint For Cast Iron Pipe.

The quantity of lead required per joint for cast iron pipe
depends on the dimensions of the lead space between the
bell and spigot, and also upon the manner in which the joint
is yarned or packed. There is no uniform rule observed in
the manufacture of cast iron pipe as to the dimensions of the
lead space, and consequently no rule can be framed for de-
termining the quantity of lead required per joint. The in-
ner diameter of some bells is uniform while in others it con-
verges. Inside of some bells there is a groove, semi-circular
in form, extending around the inner circumference of the
bell. Others are plain without grooves. Different foundries
adopt different depths and slopes of the lead space, and some
yarners will put twice as much yarn into a joint as others.
Some engineers adopt the rule of estimating 2 Ibs. of lead
for each inch diameter, as being approximately the quantity
required per joint. The result of this rule is too much for
small diameters and not enough for large diameters. The
amount of lead per joint used in recent practice is from one.
third to one-half less than formerly, and the tendency is to
reduce the quantity still more.

In laying 6", 8" and 16" pipe in Taunton, Mass., in 1887,
Mr. Billings used 7.68 Ibs. per joint for 6 inch pipe, 9.12 Iba.,
per joint for 8 inch and 21 Ibs. per joint for 16 inch pipe.

282

SULLIVAN'S NEW HYDRAULICS

8
3

o
-d

_S
8

follows:

i?
g

! I

k
8

-3

- 1

;s
S

!. 1

-f

In *<
^j

S?
S

n ffi Q
-I -2

"on S

'o ^

vi 3

sl I

^ <D ^

S
1

be
a

^ j-

<c P -u

-0

Diam.
in inches

Diameter
Inches

If belle and lead spaces were designed and specified by
engineers with regard to flexibility of joint and economy in
the quantity of lead and packing required, one third the lead
now used would be saved and the joints would be equally
strong and much more flexible and satisfactory. It is gen-

SULLIVAN'S NEW HYDRAULICS. 283

orally agreed that the calking tool does not affect or set up
the lead to a greater depth than one inch, and the lead be-
yond this is worse than wasted, for it only stiffens the joint,
which is really an element of weakness.

AB the circumference of a pipe increases directly as the
diameter, and as there is no sufficient reason for increasing
the depth of bell as the diameter increases, there is no reason
why the quantity of lead required per joint should not vary
directly with the diameter if the bell and spigot were prop-
erly designed.

Three inches is an ample depth of bell for any diameter,
and greater than necessary. Unequal settlement in the
trench must be prevented by proper laying anyway.

SULLIVAN'S NEW HYDRAULICS.

INDEX.

PAGE.

Abbot, Gen. H. L., discussion of data 84, Bee also 23, 30, 54, 98, 97

Abbot, Humphreys and, River Formula, , 11

Abrasion of Banks by frost and flow gl, 121,189. 222, 229

Acceleration of gravity 21,22,31, 32,60, 61,62,66,99,215,216

Acre, area of, in square feet 166

Action and reaction at bends 194,217,223

Action of frost on canal banks 81, 121, 222, 223, 228, 229

Adjustment of side slopes of canals 228,231

Air, friction of, at water surface 216, 217

Alignment of canal or flume 69, 96, 97, 124, 125, 223, 224, 229, 234, 235

Altitude affect 3 g in the formula 99

Amount of pipe laid per day 276,277, 281

Angle of bend in canals 186, 187, 189, 213, 217, 223, 224, 229

Angleof bend in pipes 186, 187,188,189

Angle of repose of earth 229, 230

Appendix I. New Orifice and Weir formulas 242, 249

Appendix II . Water works general data and f 01 mulas 249, '

Application and limitation of formulas 6,36,37,101,105,232

Aqueducts 27, 28,29,09,70,96, 97,124,125

Area of canal, how found 119, 120

Area of canals, tables of 121,125

Area of egg-shaped sewers 112, 113

Area of nozzle discharge; how determined 171, 173, 182

Area, of one acre in square feet 166

Area of rivets to net pipe metal 200 201, 202

Area of channel varies with square of friction surface, 34, 36, 39, 40, 49

104,105, 238, 239

Ashlar masonry, coefficient 69, 70, 74. See 28, 29, 58, 240, 241

Asphaltum, density of 57

Asphaltum coated cast iron pipes, coefficient 63, 61, 67, 151, 15:5, 260, 262

Asphaltum coated riveted pipe, coefficient, 63, 64, 67, 110, 111, 151, 152, 153,

186, 197. 260, 262

Asphaltnm coated welded pipe, coefilcient 64, 185, 197, 260, 262

Asphaltum coating, how made and applied Ill, 179, 185

Average consumption of water in towns and cities 249, 252

Average of roughness of perimeter 28,58, 80,81, 240,241

Average value of C for irregular perimeter 28, 48, 58, 75, 76, 81, 210, 241

Banks of channel, density of, affects coefficient 1..54, 96, 97

convergent and irregular, 6, 28, 29, 58, 76, 81, 84, 212, 213

......I........ .....214, 224, 236

Bazin's formula 10

Bell holes in pipe, trench 275,276

Bends in open channels 186, 188, 189, 213, 217, 223, 224, 229

Bends in pipes 186, 187,188

INDEX. 285

PAGE.

BlackwelTs formula 9

Bodies, law of falling 31,32.60,61.62,66

Boston, cost of pipe laying in 271, 272

Bottom velocity, controlled by grade 220, 221, 223, 225

grade required for 226,228,230

formula for 225,226,228,230

which erodes and moves material 227

value of R does not affect 219, 220,223, 225

ratio to other velocities 218, 219. 220, 22(5, 236

Brahms & Eytelwein's formula 10

Brass fire nozzles 140 143

Brass pipes, coefficient .".'.' .'.'.'.'l4i,'l42, 144

Brick conduits 27,28,29,70, 71,72,73,111,112

Brick /ewers HI, 112,113,114

Bursting pressure 52,195,196

Calculating weight of riveted pipes 206, 207

Canal, area of cross-section, how found 119, 120

areas of. table 121,125

alignment 69, 96. 97, 124, 125, 223, 224, 229, 234, 235

banks, affected by frost 81, 121, 222, 223. 228, 229

banks, convergent and irregular . ..6, 12, 27, 28, 58, 213, 221, 240, 241

banks, density of, affects coefficient 54, 55, 56, 57, 67, 96, 97, 124

benda in 69, 96, 97, 124, 186, 188, 189, 213, 217, 223, 22 1, 229

bed, stability of 222,226,227,230

bottom velocity in, controlled by grade 220, 221, 223, 225

bottom velocity, eroding power 227

bottom velocity, formula for 225, 226, 228, 230

bottom velocity, grade required for given 226, 228, 230

bottom velocity, no ratio to others 218,219,220, 226,236

bottom velocity, size of canal does not affect 219, 220, 223, 225

Coefficients, tables of ....87,96

coefficients vary with mean roughness, 27, 28, 29, 80, 58, 75, 76. 81 240

dimensions of, how found . Ill " II" "II 1 1 II 1 1 II II I .'I . iJ9, i20J 121, 125

djmensions to carry given flow 231,232

dimensioi s, formulas for R, p and a. . 231,232

effective value of grade or sk pe 237, 238, 239, 241

efficiency of

erod ng velocity in 222,223,224,229

evaporation and leakage 232,233

excavation, table 275