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REESE LIBRARY
UNIVERSITY OF CALIFORNIA.
Received
Accessions No. ^/ 5<- Shelf No.
Jb
HYDRAULIC
TABLES, COEFFICIENTS, AND lOBMULffi,
FOR
FINDING THE DISCHARGE OF WATER FKOM ORIFICES,
NOTCHES, WEIRS, PIPES, AND RIVERS.
BY
JOHN NEVILLE, CIVIL ENGINEEB, M.E.I.A.,
COUNTY 6UBVEYOB OF LOUTH AND OF THE COUNTY OF THE TOWN OF DBOOHEDA.
WITH EXTENSIVE ADDITIONS, NEW FORMULA, TABLES, AND GENERAL
INFORMATION ON RAIN-FALL CATCHMENT-BASINS, DRAINAGE, SEWERAGE,
WATER SUPPLY FOR TOWNS AND MILL POWER,
" It ought to be more generally known, that theory is nothing more than the conclusions
of reason from numerous and accurately observed phenomena, and the deductions of the
laws which connect causes with effects ; that practice is the application of those general
truths and principles to the common affairs and purposes of life ; and that science is the
recorded experience and discoveries of mankind, or, as it has been well defined, ' the know-
ledge of many, orderly and methodically digested, and arranged, so as to become attainable
by one.'" AMERICAN QUABTEBLY
LONDON:
JOHN WEALE, 59, HIGH HOLBORN.
1860-1.
TO
MAJOR-GENERAL SIR THOMAS AISKEW LARCOM, K.C.B.,
LL.D., F.R.S., M.B.I.A., ETC.,
OF THE
ROYAL ENGINEERS,
UNDER SECRETARY OF STATE FOR IRELAND,
THIS WORK IS INSCRIBED
BY THE
AUTHOR.
CONTENTS.
Introduction to the Second Edition * *; &.* ;'-;..**< .-*>
Introduction to the First Edition -*. ' * ~ 4.-; - #T- *1
SECTION I. Application and Use of the Tables, Formulae,
&c. Extra Horse-power required in Pumping Engines
from Friction in the Pipes Table of Heads due to
Friction and of Discharges f . ~ w
SECTION II. ' Formulae for the Velocity and Discharge
from Orifices, Weirs, and Notches Coefficients of Ve-
locity, Contraction, and Discharge Practical Remarks
on the Use of the Formulae ; ^. . -.# i f*-j.
SECTION III. Experimental Results and Formulse Co-
efficients of Discharge for Orifices, Notches, and Weirs
SECTION IV. Variations in the Coefficients from the Posi-
tion of the Orifice General and Partial Contraction
Velocity of Approach Various Practical Formulae for
the Discharge over Weirs and Notches Central and
Mean Velocities . . . . ' . ". '" .' *~"l
SECTION V. Submerged Orifices and Weirs Contracted
River Channels . j 4 * . ,, . ..
SECTION VI. Short Tubes, Mouth-pieces, and Approaches
Alteration in the Coefficients from Friction by in-
creasing the Length Coefficients of Discharge for
Simple and Compound Short Tubes Shoots
SECTION VII. Lateral Contact of the Water and Tube
Atmospheric Pressure Head measured to the Dis-
charging Orifice Coefficient of Resistance Formula
for the Discharge from a Short Tube Diaphragms
Oblique Junctions Formula for the Time of the Sur-
face smking a given depth Lock Chambers *' : '
SECTION VIII. Flow of Water in Uniform Channels
Mean Velocity Mean Radii and Hydraulic Mean
Depths Border Train Hydraulic 1 nclination Ef-
fects of Friction Formulse for calculating the Mean
Velocity Application of the Formulae and Tables to
the Solutions of three useful Problems
PAGES
v. to xii.
1 to!4
1643
4463
6395
96139
140150
150165
166181
162234
11
CONTENTS.
SECTION IX. Best Forms of the Channel Regimen
Velocity Equally Discharging Channels . . . 235253
SECTION X. Effects of Enlargements and Contractions
Backwater Weir Case Long and Short Weirs . 254 271
SECTION XI. Bends and Curves Branch Pipes Dif-
ferent Losses of Head General Equation for finding
the Velocity Hydrostatic and Hydraulic Pressure
Piezometer Catchment Basins Rain-fall per annum 271 289
SECTION XII. Rain-fall Catchment Basins Discharge
into Channels Discharge from Sewers Loss from
Evaporation, &p. . . . ; , . . 289315
SECTION XIII. Water Supply for Towns -Strength of
Pipes Sewerage Estimates and Cost Thorough Drain-
ageArterial Drainage ^ > . . + . , p. 315364
SECTION XIV. Water Power Dynamometer, or Friction
Brake Overshot, Breast, and Undershot Vertical
Wheels Turbines and Horizontal Wheels Hydraulic
$am Water-pressure Engine Work of Flour Mills . 364399
TABLES NOT EMBODIED IN THE TEXT.
TABLE I. Coefficients of Discharge from Square and Dif-
ferently proportioned Rectangular Lateral Orifices in
Thin Vertical Plates . . ... . . 400401
TABLE II. For finding the Velocities from the Altitudes
and the Altitudes from the Velocities .... 402 411
TABLE III. Square Roots for finding the Effects of the
Velocity of Approach, when the Orifice is small in pro-
portion to the Head. Also, for finding the Increase in
the Discharge from an Increase of Head. (See pp. 101
and 107.) ......... 412413
TABLE IV. For finding the discharge through Rectan-
gular Orifices j in which n =-^. Also, for finding the
Effects of the Velocity of Approach to Weirs, and the
Depression on the Crest. (See pp.101 and 107.) . . 414 417
TABLE V. Coefficients of Discharge for different Ratios
of the Channel to the Orifice . . 418423
CONTENTS.
Ill
TABLE VI. The Discharge over Weirs or Notches of One
Foot in Length, in Cubic Feet per Minute. (See pp.
IQQ and 111 to 127.) 424429
TABLE VII. For finding the Mean Velocity from the
MaximumVelocity'at the surface, in Mill-races, Streams,
and Rivers with Uniform Channels ; and the Maximum
Velocity from the Mean Velocity. (See p. 184.) y 430
TABLE VIII. For finding the Mean Velocities of Water
flowing in Pipes, Drains, Streams, and Rivers. (See
p. 195) ..."...,'' 431443
TABLE IX. For finding the Discharge in Cubic Feet, per
Minute, when the Diameter of a Pipe or Orifice, and
the Velocity of Discharge, are known, and vice versd . 444445
TABLE X. For finding the Depths on Weirs of different
Lengths, the Quantity discharged over each being sup-
posed constant. (See p. 271.) 446
TABLE XI. Relative Dimensions of Equally-Discharging
Trapezoidal Channels, with Slopes from to 1, up to 2
to 1. (See pp. 249 and 252.) ..... 447
TABLE XII. Discharges from the Primary Channel in
the first column of Table XI. (See p. 253.) ; . 448449
TABLE XIII. The Square Roots of the Fifth Powers of
Numbers for finding the Diameter of a Pipe, or Dimen-
sions of a Channel from the Discharge, or the reverse ;
showing the relative Discharging Powers of Pipes of
different Diameters, and of any similar Channels what-
ever, closed or open. (See pp, 31, 42, 228, &c.) . . |j 450
TABLE XIV. English and French Measures . . . 451454
TABLE XV. Weight, Specific Gravity, Ultimate Strength
and Elasticity of Various Materials . . . 655
COBRECTION.
2 2
For o = 3 c dN / 2# h, p. 120, line 18, read o =g-<W 2$r.
((UNIVERSITY jj
INTRODUCTION
TO THE SECOND EDITION.
IN order to render this edition more valuable to
the hydraulic engineer it has been considerably ex-
tended by the insertion of several new tables, formulae,
experimental coefficients, examples, and general esti-
mates of cost. It is hoped that the great extent and
practical nature of the additions will render this
work far more useful than the first in the ever
varying requirements of the profession. The six
pages on catchment basins, rain-fall, and water power
in the first edition have been extended into three new
sections of one hundred and ten pages, embodying
the subjects of water supply, sewerage discharge,
drainage, and the method of determining the useful
effect of water employed in turning the various kinds
of water wheels. Considerable additions have also
been made in Sections I, III, IV, VIII, and IX;
the practical formula for gauging by weirs have been
extended through twenty-one extra pages of new
matter ; the portion treating of the conveying powers
of pipes and rivers has been also extended by the
insertion of new tables at pages 42, 152, 191, 220,
and 252, and of several new formulae, amongst which
we believe that at page 215 (119A), will be found, in
practice, the simplest and most accurate yet dis-
VI INTRODUCTION TO THE SECOND EDITION.
covered for ranges of velocity varying between one
inch and twenty feet per second, and for all descrip-
tions of channels, pipes, and rivers with which the
engineer has to deal.
We have noticed at page 95 the erroneous notation
of Morin and other engineers in giving only two-
thirds of the co-efficient of discharge for weirs.
This notation assumes that the theoretical discharge
from a notch is the same as if all the particles of
water had the same velocity as those undermost,
which, being too large by one-third, the experimental
coefficient has to be reduced in the same proportion.
Mr. Blackwell and Mr. Hughes, in this country, lend
themselves to this nomenclature ; the latter gentle-
man says, page 328 of his useful treatise on Water
Works*, " Mr. Neville, in his tables of the discharge
over weirs, termed the theoretical discharge 321A^
instead of 481/^ in cubic feet per minute." We
were correct in doing so, for a weir one foot long ;
and those who adopt the latter formula are in error :
this is a matter of demonstration, not of opinion.
We can see no reason for sanctioning a different
notation for notches, or orifices at the surface, and
sunk orifices. The coefficients when in thin plates,
with large cisterns, have nearly the same general
value, -615 to -628, and it tends to confusion to adopt
in one place a coefficient for a correct formula, and
in another a coefficient for an incorrect one ; although
the final result by an equality of contrary errors
may be the same in both. We may here observe
* Weale's Series.
INTRODUCTION TO THE SECOND EDITION. Vll
how very general the coefficient of two-thirds, and
thereabouts, is for all orifices, notches, and also for
the useful effect derived from the application of
water power, as well as the relation of the velocity
due to the fall and the velocity of water wheels to
give a maximum result. The modifications of coeffi-
cients dependent on the position, thickness, form,
and approaches of an orifice are seldom understood.
The defects in the ordinary formula when the velo-
city of approach has to be considered are pointed
out at pages 100, 115, and 116, and it is to be re-
gretted that the authority of D'Aubuisson and others
has misled many as to the correct form. Before the
effective power of a water wheel, or water engine,
can be determined we must know how to gauge the
water supplied to it correctly. This can be done
only by the application of formulae varied to suit the
circumstances of the case under consideration. From
causes, which it is not necessary to enter into here,
this has seldom been done, and very little dependence
can be placed on results obtained by the formula in
common use when applied generally. It is pleasing
to follow Francis and Thomson through the steps by
which they get the effective power of their wheels,
and we have accordingly made considerable use of
their labors in Section XIV.
We have modified some of the old partial equations
for the velocity in pipes, beginning at (86), from the
(rs . 2 H a . , a
form v = - + - 2 - , mto v = c^rs ;
principally for the purpose of giving the numerical
values of c in the form v = c \/rs. Of course these
Vlll INTEODUCTION TO THE SECOND EDITION.
modifications are only applicable when the value of
O-T is small compared with that of the velocity.
Some remarks will be found at page 224 respecting
M. Darcy's new formula for the velocity of water in
iron pipes, as given by Morin in his Hydraulique.
We have, only since those remarks were written, seen
the original Memoire, printed in Tome XY, Memoires
presentes par divers savants a I'Academie des Sciences
de rinstitut Imperial de France, Paris, 1858. This
Memoire extends through 263 quarto pages, of which
34 contain tables calculated from the formula. We
have not had time to do more than glance through it,
but the deduced formula appears to be entirely de-
rived from the author's experiments. This we con-
sider, to some extent, objectionable ; but, however
this may be, there can be no doubt that the value of
the coefficient c, in the formula v c \/ r s, increases
with the inclination, $, as well as with the diameter,
4 r, of the pipe ; and as M. Darcy's formula makes
the value of the multiplier, c, depend alone on the
value of r, or 4 r, there appears an omission, in making
the coefficient of friction entirely independent of the
hydraulic inclination, and dependent only on the size
of the channel. We shall give a few examples, taken
at discretion, to show how limited this formula must
be in its application.
1. Couplet's experiment, No. 43, p. 103, reduced
to feet, gives r = -3997 feet,* zz -0035, r s = -001339,
and the observed velocity v =2 3-478 feet =. 95 \/r~s
nearly. Darcy's formula would give v = 110-8
\/f r s, our formula 106 \/r~s nearly, and Weisbach's
INTRODUCTION TO THE SECOND EDITION. IX
105 \/r^ nearly. The pipe was probably an old
one, and a deduction of about 10 per cent, might be
made for the state of the bore. We have here, how-
ever, no means of judging the effect of a change of
inclination on the multiplier, c.
2. From Du Buat's experiments with an inch
pipe, nearly, Nos. 50 and 51, p. 103, we get, after
reducing them to feet, in experiment 50, r zz -0222,
s zz -228 and v zz 6*33 feet zz 89-2 \/rs ; or, after
making the necessary deductions in the head for the
velocity and the orifice of entry with the coefficient
815, s zz -147 and v zz 6*33 feet = 111*4 \/r~s. In
experiment 51, we also get in feet r zz -0222, s zz
3074, and v zz 7-54 zz 92 \/r~s ; or, by making
allowance for the head due to the velocity and the
orifice of entry, as before, s zz *179, and v zz 7*54
feet zz 119*7 \/rs. Here we see how the velocity
or value of the inclination, s, affects the value of
the multiplier, the diameter remaining constant.
M. Darcy's formula, in each case, would only make
v zz 80-3 \/r~s.
3. In the excerpt proceedings of the Institution
of Civil Engineers, p. 4, 6th February, 1855, James
Simpson, president, in the chair, there is given for
the "Colinton pipe" 16 inches diameter, eight or
nine years in use, three observations. First, 29,580
feet long, a head of 420 feet and a discharge of 571
cubic feet per minute : these give v zz 6-816 feet zz
99*2 </7s nearly. Secondly, a length of 25,765 feet
a head of 184 feet, and a discharge of 440 cubic feet
per minute : these give v zz 5*252 feet zz 96*3 \/rs.
And thirdly, a length of 3,815 feet a head of 184
X INTRODUCTION TO THE SECOND EDITION.
feet, and a discharge of 1,215 cubic feet per minute :
these give v zi 14-5 feet z= 115 \/r s nearly. In these
three examples, the diameter, castings, and age of
the pipes are the same. Yet we see, clearly, that the
inclination affects the multiplier of \/r s, which in*
creases with the inclination, s, although M. Darcy's
formula would make the multiplier the same in each
case, and for all inclinations, viz, v = 110 \/rs<
Making those allowances inseparable from the state
of the pipe, and all experimental observations, these
results, as well as those from Du Buat's experiments,
confirm the accuracy of our general formula (119 A)
page 215*, and those others we have given following
it, as well also as that of Weisbach.
Dr. Young's formula, page 207, bears a resem-
blance to that of M. Darcy, in making the multi-
plier of \/r s depend only on the diameter; but it
works in a contrary manner : for the high velocities
being derived from pipes, with small diameters in the
experiments at his command, the value of c in v
c \/r s, reduced from his formula, becomes larger in
general for small than for larger diameters. No
doubt an allowance should be made in small pipes
for a thin film of water adjoining the pipe with little
or no velocity ; but within the limits which the en-
gineer has to deal this may be neglected. Its effect,
* The form in which we first discovered this formula was
v = 1 140 7 rj [ X vTrs. For measures in metres it becomes
. ( (rs))
v = 77-3 (rs)* 4-9 (rs)s; in which r is half the radius of a cy-
lindrical pipe, or the hydraulic mean depth of any channel.
INTRODUCTION TO THE SECOND EDITION. XI
as well as that of all the other resistances, junctions,
contractions, deposits, &c., is greater in small than
in large pipes. We must refer to the body of the
work for further remarks on this subject, but from
lately appearing at such length in the Memoires of
the Imperial Academy of Sciences, M. Darcy's for-
mula called for especial notice here.
The Statistics of rain-fall and catchment-basins
have not yet received the attention which the subject
deserves. The distribution of rain gauges with re-
ference to elevation, contour, temperature, and iso-
thermal lines has not been sufficiently attended to.
The connexion of the rain-fall with the discharge
generally, for the whole catchment, for the tributary
catchments, and their sub-catchments, at the sea in
the middle districts and at the sources, noting the
geology, must be" observed for several years before
the questions of supply, discharge, absorption, and
evaporation in any climate can be answered. The
maximum and minimum discharges in each year and
series of years must be observed, as well as the
average mean discharges, and the maximums and
minimums of these also, before the physical con-
nexion of climate and catchment can be correctly
ascertained, and the engineer furnished with reliable
data. Heretofore observations, even when best, have
been partial or limited, and a wide field is here yet
open to competent physicists in connexion with our
drainage works.
The general items of cost given in SECTION XIII.
will be found of use ; they are intended, how-
ever, more as guides than as standards for other
Xll INTRODUCTION TO THE SECOND EDITION.
works, the cost of which must depend on their own
circumstances. Those who have practical experience
of the differences between estimates, cost, and value,
and how they are affected by time, locality, quality,
and quantity, will estimate for each case in detail ;
but the discrepancies between estimates and cost,
even under the same circumstances, are too well
known to call for any remark here.
A few words about our publisher. Mr. Weale
having purchased our interest in this edition, at once
decided on adopting larger type and better paper,
at a heavy extra outlay to himself. We had reason
to be satisfied with the manner in which the first
edition was brought out, in this, however, he has
excelled, and we hope his enterprise will receive a fair
return.
Jocelyn Street^ Dundalk,
October, 1860.
it f r inc r
f C NIVERSITY
X^UFORN!^
INTKODUCTION
TO THE FIEST EDITION.
IN preparing the following work, we had three ob-
jects in view : first, a collection of useful hydraulic
formulae ; secondly, a collection of experimental re-
sults, and coefficients ; and, thirdly, a collection of
useful practical tables, some calculated entirely from
the formulae and experiments, and others for the
purpose of rendering the calculations more easy.
The TABLES at the end of the volume are all
original, with the exception of TABLE I., which con-
tains the well-known coefficients of PONCELET and
LESBROS ; but those are newly arranged, the heads
reduced to English inches, and the coefficients for
heads measured over and back from the orifice,
placed side by side, for more ready comparison.
The coefficients in the small Tables throughout the
work have been all calculated by us from the original
experiments ; the formulae have been carefully ex-
amined, and the continental ones reduced to English
measures some of them, as will be seen, for the
first time. No labour has been spared in preparing
the TABLES, and they are all purely hydraulic, though
some of them are capable of being otherwise applied.
We have filled no gap by the introduction of Tables
applicable to other subjects, and in e very-day use.
2 INTRODUCTION TO THE FIRST EDITION.
The correction of some of the experimental for-
mulae, particularly the continental ones, as printed
in some English books, cost us some labour. Even
Du Buat's well-known formula is frequently mis-
printed; and in a late hydraulic work, \/d -1, one
of the factors, is printed \A/--1 in every page where
it is quoted. It is not always that such mistakes
can be avoided, but experimental formula are so
often copied from one work into another without
sufficient examination, that an error of this kind
frequently becomes fixed ; and when applied to prac-
tical purposes erroneous formula get the correct ones
into disrepute. See note to formula (91).
The TABLES of velocities and discharges over weirs
and notches have been calculated for a great number
of coefficients to meet different circumstances of ap-
proach and overfall, and for various heads from ^th
of an inch up to 6 feet. TABLE II. embodies the
velocities acquired by falling bodies under the head
of " theoretical velocity," and the velocities, suited
to various coefficients, for heads up to 40 feet.
The formulae for calculating the effects of the ve-
locity of approach to orifices and weirs, and the
necessary corrections for the ratio of the channel to
the orifice, as well as TABLE V., we believe to be
original. They will be found of much value in de-
termining the proper coefficients suited to various
ratios. The remarks throughout SECTION IV. are par-
ticularly applicable to the proper use of this TABLE.
TABLE VII. of surface and mean velocities will be
found to vary from those generally in use, and to be
much more correct, and better suited for practical
INTRODUCTION TO THE FIRST EDITION. 3
purposes, particularly as applied to finding the mean
velocities in rivers.
We have extended TABLE VIII. so as to make it
directly available for hydraulic mean depths, from
jVth of an inch to 12 feet, and for various hydraulic
inclinations, even up to vertical, for pipes. The fall
in rivers seldom exceeds 2 or 3 feet per mile, or the
velocity 5 or 6 feet per second. The extension of the
Table for great inclinations, and consequently great
velocities, was made for purposes of calculation, and
to include pipes. It must be understood through-
out this TABLE that the velocities are those which
continue unchanged for any length of channel, viz.,
when the resistance of friction is equal to the accele-
ration of gravity, the moving water and channel
being then in train. Several of Du Buat's experi-
ments were made with small vertical pipes. This
TABLE is equally applicable to pipes and rivers, and
gives directly either the hydraulic inclination, the
hydraulic mean depth, or the velocity when any two
of them are known.
Hydraulic formulae have been frequently rendered
unnecessarily complex, and unsuited for practical
application, by combining them with those of mere
mensuration in order to find the discharge. We have
therefore given formulae for finding the mean velocity
principally, unless in a few instances, as in orifices
near the surface, where the discharge itself is first
necessary to find the mean volocity ; this once deter-
mined, the calculation of the discharge becomes one
of simple mensuration.
We have preferred giving the mean velocity to the
B3
4 INTRODUCTION TO THE FIRST EDITION.
discharge itself in TABLE VIII., because, while an
infinite number of channels having the same hydraulic
inclination (s) and the same hydraulic mean depth (r)
must have the same velocity (v), yet the sectional
areas, and consequently the discharges, may vary
upwards from 6 -2 83 2 r 2 , the area of a semicircular
channel, to any extent ; and the operation of multi-
plying the area by the mean velocity, to find the
discharge, is so very simple that any tabulation for
that purpose is unnecessary. Besides this, the banks
of rivers, unless artificially protected, remain very
seldom at a constant slope, and therefore any TABLES
of discharge for particular side slopes are only of use
so far as they apply to hypothetical cases. Indeed
we have seen, in new river cuts, the banks, cut first
to a given slope, alter very considerably in a few
months ; while the necessary regimen between the
velocity of the water and the channel was in the
course of being established. The velocity suited to
the permanency of any proposed river channel,
though too often entirely neglected, is the very first
element to be considered.
For circular pipes, however, TABLE IX. gives the
discharge in cubic feet per minute when the velocity
in inches per second is known, or found from TABLE
VIII., and is calculated for pipes from Jth of an inch
up to 12 inches in diameter. TABLE XII. gives also
the discharges in cubic feet per minute from the
different equivalent river channels in TABLE XI.
TABLE X., for finding the heads on weirs of different
lengths, TABLE XI., of equally discharging river
channels, and TABLE XII., of the actual discharges
INTRODUCTION TO THE FIEST EDITION. 5
from the equivalents in TABLE XI., will be found of
great practical value when new weirs and water-cuts
have to be made. TABLES XI. and XII. are equally
applicable to channels having side slopes, the widths
being then the mean or central widths.
When the discharge and fall are known, and the
hydraulic mean depth and the dimensions of any
channel have to be determined, Problem III., section
8, as illustrated in EXAMPLE 17, section 1, gives a
new and perhaps the most practically useful solution
yet published. Tables XI., XII., and XIII. are par-
ticularly applicable to this problem.
A uniform notation is preserved throughout the
work, so that the different experimental formulae can
be compared without any further reduction. The
letter h is used in every instance for the head, c for
the coefficient, r for the mean radius or hydraulic
mean depth, and s for the sine of the hydraulic
inclination, unless it be otherwise stated. In order
to designate particular values, the primary letters
have deponent or initial letters below to explain them.
Thus h t is the head to the fop of an orifice, h^ the
head at the bottom, A w the head on a weir, h t the head
due to/riction, c d the coefficient of discharge, c v the
coefficient of velocity, c c the coefficient of contraction,
&c. When the whole head is made up of different
elements, such as the portions due to friction, velocity,
contractions, bends, &c., it is expressed by the capital
letter H.
Some writers and engineers appear to confound
the inclination of a pipe, simply so called, or the head
divided by the length, with the hydraulic inclination ;
6 INTRODUCTION TO THE FIRST EDITION.
and consequently have fallen into error in applying
such of the known formulae as take into consideration
only the head due to the resistance of friction. When
pipes are of considerable length, and the water is
supplied from a reservoir at one end, the inclination,
found as above, and the hydraulic inclination, may be
taken equal to each other without sensible error ; but
for shorter pipes, of say up to 100 feet long, or even
longer, the greater number of formulae, as Du Buat's
and others, do not directly apply ; and it is necessary
to take into consideration the head due to the orifice
of entry, the velocity in the tube, and also to the
impulse of supply when there are junctions. These
separate elements, and their effects, will be considered
in the following pages ; but it will be of use to refer
here to some late experiments, and the imperfect
application of formulae to them, first premising that
a pipe may be horizontal, or even turn upwards, and
yet have a considerable hydraulic inclination.
Mr. Provisos valuable experiments* with li-inch
pipes, from 20 to 100 feet long, have been used in a
recent work for the purpose of testing the accuracy
of Du Buat's and some other formulae ; but the head
divided by the length is assumed to be the hydraulic
inclination throughout, and no allowance is made for
the head due to the orifice of entry and velocity in
the pipe. Of course the writer's conclusions are
erroneous. We have shown, SECTION I., page 30,
how very nearly the formulae and experiments agree.
The formulae appear to have been also misunder-
* Transactions of the Institution of Civil Engineers, Vol. II.,
pp. 201210.
INTRODUCTION TO THE FIRST EDITION.
stood by the surveyor who experimented for the
General Board of Health ; for the inclination of the
pipe in itself is assumed to be the hydraulic inclina-
tion, and no allowance is made for the head due to
the impulse of supply. We quote from the CIVIL
ENGINEER AND ARCHITECT'S JOURNAL, Vol. XV.,
page 366, in which it is stated that "the chief results
as respect the house drains are thus described in the
examination of the surveyor appointed to make the
trials."*
" What quantity of water would be discharged through a 3-inch
pipe on an inclination of 1 in 120 ? Full at the head, it would
discharge 100 gallons in three minutes, the pipe being 50 feet in
length. This is with stone-ware pipe manufactured at Lambeth.
This applies to a pipe receiving water only at the inlet, the water
not being higher than the head of the pipe.
" What water was this? Sewage-water of the full consistency,
and it was discharged so completely that the pipe was perfectly
clean.
"At the same inclination what would a 4-inch pipe discharge
with the same distances ? Twice the amount (that I found from
experiment) ; or, in other words, 100 gallons would be discharged
in half the time. This likewise applies to a pipe receiving water
only at the inlet, and of not greater height than the head. In
these cases the section of the stream is diminished at the outlet
to about half the area of the pipe.
"Before these experiments were made, were there not various
hypothetical formulae f proposed for general use? Yes.
* Minutes of Information with reference to Works for the
removal of Soil, Water, or Drainage, &c., &c. Presented to both
Houses of Parliament, 1852.
f It is a mistake to call those formulae hypothetical, unless so
far as the hypothesis is founded on facts. Every formula with
which we are acquainted is founded on experiments, and has
been deduce^ from them, but those formulae are too often hypo-
thetically applied to short tubes without the necessary corrections.
It will be seen from SECTION VIII. that the experiments from
8 INTRODUCTION TO THE FIEST EDITION.
" What would these formulae have given with a 3-inch pipe, and
at an inclination of 1 in 100 ? and what was the result of your
experiments with the 8-inch pipe ? The formulae would give 7
cuhic feet, the actual experiment gave 11 J cubic feet ; converting
it into time, the discharge, according to the formulae, compared
with the discharge found by actual practice, would be as 2 to 3.
" How would it be with a 4-inch pipe ? The formulae would
give about 14-7 cubic feet per minute, whereas practice gave 23
cubic feet per minute.
" Take the case of a 6-inch pipe of the same inclination? The
results, according to Mr. Hawkesley's formula, would be 40^ cubic
feet per minute ; from experiment it was found to be 63 cubic
feet per minute.
" Then with respect to mains and drainage over a flat surface,
the result of course becomes of much more value, as the difference
proved by actual practice increases with the diminution of the
inclination ? Certainly, to a very great extent. For example, the
tables give only 14-2 cubic feet per minute as the discharge from
a pipe 6 inches diameter, with a fall of 1 in 800 ; practice shows
that, under the same conditions, 47*2 cubic feet will be discharged.
" Will you give an example of the practical value of this when it
is required to carry out drainage works over a very flat surface ?
An inclination of 1 in 800 gives only 14 cubic feet per minute,
according to theory, while, according to actual experiment, and
with the same inclination, 47 cubic feet are given.
" Then this difference may be converted either into a saving of
water to effect the same object, or into power of water to remove
feculent matter from beneath the site of any houses or town?
It may be so.
" And also the power of small inclinations properly managed ?
Yes ; for example, if it was required to construct a water course
that should discharge, say 200 feet per minute, the formula would
require an inclination of 1 in 60=2 inches in 10 feet ; whereas,
experiment has shown that the same would be discharged at an
inclination of 1 in 200 =$ inch in 10 feet, thus effecting a con-
siderable saving in excavation, or a smaller drain would suffice at
the greater inclination."
which the formulae there given were derived, were in every way
greatly more extensive than those made by the directions of
the Board of Health. The formula named is, substantially,
Eytelwein's algebraically transformed.
INTRODUCTION TO THE FIEST EDITION.
We have extracted and tabulated the results given
above, in the following Table, and also eight of the
experiments made for the Metropolitan Commissioners
of Sewers'* ; and assuming for the present, with the
surveyor, examined by the Commissioners, that the
inclinations of the pipes and hydraulic inclinations of
the formulae are the same, which is incorrect, we give
the calculated discharges, found by nieans of TABLES
VIII. and IX., in the last column of the Table.
Diameter of
pipe
in inches.
Inclination of
pipe.
Discharge in
cubic feet
per minute
by experiment.
Hypothetical
discharge by
Du Buat's
formula.
3
1 in 120
5-3
6-6
4
1 in 120
10-7
14
3
1 in 100
11-2
7-5
4
1 in 100
23
15-6
6
1 in 100
63-5
43-8
6
1 in 800
47-2
13-3
6
Iin60
75
59-3
6
1 in 100
63
43-.S
6
1 in 160
54
33-4
6
1 in 200
52
29-2
6
1 in 320
49
21-8
6
1 in 400
48-5
19-6
6
1 in 800
47-2
13-3
6
Level
46
00
Du Buat's formula, therefore, gives larger results
than the experiments in the two first cases, because
the water received at one end only barely filled it,
and the pipe was not full at the lower end ; but less
in the others. If in these the head due to the im-
* Adcock's Engineer's Pocket Book, 1852, pp. 261 and 262.
10 INTRODUCTION TO THE FIRST EDITION.
pulse of entry, at the upper end, and at the side
junctions, were known, and the proper hydraulic
inclination determined by the experiments, the for-
mulae would be found to give larger approximate
results in every case, as might have been expected
from the sewage-water used. In the last eight ex-
periments it is stated*, that " the water was admitted
at the head of the pipe, and at five junctions or tri-
butary pipes on each side, so regulated as to keep the
main pipe full," and that " without the addition of
junctions the transverse sectional area of the stream
of water near the discharging end was reduced to
one-fifth of the corresponding area of the pipe, and
that it required a simple head of water of about 22
inches to give the same result as that accruing under
the circumstances of the junctions." It is also stated,
that " in the case of the 6-inch pipe, which discharged
75 cubic feet per minute, the lateral streams had a
velocity of a few feet per minute."
Now, the head of " about 22 inches" is wholly
neglected in the foregoing calculations, though in a
pipe 100 feet long it would be equal to an inclination
of 1 in 55 ! It however includes three elements at
least, viz. the portion due to the orifice of entry,
the portion due to the velocity in the pipe, and
the portion due to friction. Let us assume the
case of the horizontal pipe, which discharged 46 cubic
feet per minute^. This is equal to a mean velocity of
* Adcock's Engineer's Pocket Book, 1853, pp. 261 and 262.
| The horizontal pipe would discharge equally at both ends,
unless there was a head of water at either, or an equivalent in
INTRODUCTION TO THE FIRST EDITION. 11
46-9 inches per second; with this velocity, we find
from TABLE VIII. the hydraulic inclination of a 6-inch
pipe to be 1 in 94, and, therefore, the head due to
friction in a pipe 100 feet long is 12-7 inches. As-
suming the coefficient for the orifice of entry and
velocity to be -815, we also find from TABLE II. a
head of 4^ inches due to these. We then have,
Head due to the velocity and orifice of
entry. - lJ . ' Mi . . . . 4-25 inches
Head due to the resistance of friction 12-70
Radius of pipe . . ^ ', 3-00
Total ^ "^ 19-95
which is about 2 inches less than the observed head :
this, however, is not stated definitely. It is therefore
evident, that the formula gives, if anything, larger
results than these experiments*, as might have been
expected, instead of less in the ratio of 2 to 3, as is
stated in the Report.
Wherever junctions are applied, as in the examples
above referred to, the formulae in general use require
correction; for the quantity of water then flowing
below each junction is increased. A certain amount
of error is, perhaps, inseparable from every calcula-
tion of this kind ; but before we condemn formulas
deduced from experiment by men every way quali-
the velocity of approach. Of course, a smaller pipe with a fall,
must be better than the larger one with none at all, in preventing
deposits.
* This is also true of the other formulae, for finding the dis-
charge from pipes, given in this work.
12 INTRODUCTION TO THE FIRST EDITION.
fied for the task, it would be well that we should
learn to understand and properly apply them.
The diameter of a short pipe gives in itself the
means of increasing very considerably the surface
inclination of the fluid stream, by reducing the sec-
tion at the lower end. If we assume a horizontal
pipe 50 feet long and 6 inches in diameter, we per-
ceive, that if the receiving end be full, and the
discharging end one-third full, this inclination
6-2 1
will be 5Q x 12 = ^I5Q ; an( ^ ^ a ^ ^ e discharging en( i
cannot be kept full unless a head of several inches
be maintained at the receiving end, or an equivalent
from a lateral supply. When the pipe is about two
diameters long it becomes a short tube ; and when
the length vanishes, the transverse section becomes,
simply, a discharging orifice.
We have been led into the foregoing remarks, not
from any desire to find fault with a Report contain-
ing so much valuable information as the one referred
to, but for the purpose of defending from unmerited
reproach, in a Blue Book, the researches in this de-
partment, of
" Those dead but sceptred sovereigns who still rule
Our science from their urns "
Du Buat, Young, Eytelwein, Prony, and others.
We do not pretend to any particular accuracy in
the sketches scattered throughout the work ; they are
only intended to illustrate the text, and were sketched
while writing it, without further aim ; neither do we
pretend to have entered fully into the principles or
INTEODUCTION TO THE FIEST EDITION. 13
practice of hydraulics, our object being to select,
construct, and arrange useful hydraulic formulae, ex-
periments, and Tables for the use of all classes of
engineers. We make, however, no apology for pre-
ferring formulae, in their simplicity, to any written
rules which may be deduced from them, as being
in every way more general, concise, and elegant.
In conclusion, it is hoped that any errors of con-
sequence in the work, will be found corrected in the
errata.
Roden Place, Dundalk,
January, 1853=
ON THE
DISCHARGE OF WATER
FEOM
OEIFICES, WEIRS, PIPES, AND RIVERS.
SECTION I.
APPLICATION AND USE OF THE TABLES, FORMULA, &C.
To find the velocity of a falling body from the height
fallen, or the height fallen from the velocity.
RULE. MULTIPLY THE SQUARE ROOT OF THE
HEIGHT IN INCHES BY 27 '8, AND THE PRODUCT WILL
BE THE VELOCITY IN INCHES.* To FIND THE HEIGHT
FROM THE TELOCITY, SQUARE THE VELOCITY IN INCHES
AND DIVIDE THE SQUARE BY 772*84, THE QUOTIENT
WILL BE THE HEIGHT IN INCHES. See equation (1).
TABLE II., column 1, will give the velocity from the
height, found in the column of " altitudes," or the
height from the velocity, directly.
EXAMPLE 1. What is the velocity acquired by a
heavy body falling th of an inch ? In the Table
opposite to ith of an inch, found in the column
headed " altitudes /*," we find 9-829 in column 1, for
the required velocity, in inches per second.
EXAMPLE 2. What is the velocity acquired by a
* The square root of the height in feet multiplied by 8-OS5
gives the velocity per second in feet ; and the square of the
velocity in feet divided by 64-4 will give the height in feet.
16 THE DISCHARGE OF WATER FROM
fall of II feet 3 inches? Opposite to 11 feet 3
inches, as before, we find 323-007 inches, for the
velocity required.
EXAMPLE 3. What height must a heavy body fall
through to acquire a velocity of 40J feet per second ?
Here 40J feet is equal 486 inches, opposite to the
nearest number to which, found in column 1, we find
25 feet 6 inches for the required fall. In this ex-
ample, the nearest number to 486 found in the Table
is 486*301. The difference *301 corresponds, very
nearly, to f ths of an inch in altitude, and, therefore,
the true head according to the rule would be 25' 6f";
but for all practical purposes the difference is im-
material.
By means of TABLE II. we can find, directly, or
by simple interpolation, the velocity due to all
heights from TTO part of an inch up to 40 feet, and
the heights from the velocities. For a greater height
than 40 feet it may be divided by 4, 9, or some
square number s~, and the velocity found for the
quotient, from the Table, multiplied by 2, 3, or s,
the square root of the divisor, will give the velocity
required.
EXAMPLE 4. What is the velocity acquired by a
45
fall of 45 feet ? -j- = 11' 3", the velocity corre-
sponding to which, found from the Table, is 323-007.
Hence, 323-007 x</4 = 323-007 2 = 646"-014 =
53' 10"-014 is the velocity per second required. The
reverse of this example is equally simple.
Columns 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 in the
Table, give the values of \/Zgh multiplied by the
ORIFICES, WEIRS, PIPES, AND RIVERS. 17
coefficients therein stated. These columns will be
found of great practical use in finding the mean
velocities in the vena-contracta, in the orifice, and in
short tubes ; and consequently also in finding the
mechanical force, as well as the discharge. An
examination of the coefficients in the small Tables
in SECTION III., and also of those in TABLES I. and
V., at the end of the work, will show how much they
vary ; but those most generally useful, and their
products by the theoretical velocity due to different
heads, up to 40 feet, are given in the columns
referred to.
EXAMPLE 5. What is the discharge from an
orifice 4 inches by 8 inches, the centre sunk 20 feet
below the surface of a reservoir ? From TABLE II.,
we find 430-676 inches equal 35-89 feet for the
x A.
theoretical velocity of discharge : hence,
TT
2
35-89 = ^ x 35-89 = 7-976 cubic feet per second is
the theoretical discharge. If the discharge takes
place through a thin plate, or if the inner arrises
next the water in the reservoir be perfectly square,
and the water in flowing out does not fill the passage
so as to convert the orifice into a short tube, the
coefficient will be found from TABLE I. to be -603.
The true discharge then is 7-976 x -603 = 4-809
cubic feet per second.
For,, the determination of the coefficient suited to
any particular orifice, and the circumstances of its
position, we must refer generally to the following
pages. If in the example just given, the arrises
c
18 THE DISCHARGE OF WATER FROM
next the reservoir were rounded into the form of the
contracted vein, see Fig. 4, the coefficient would
increase from -603 to -974 or -956, for a passage not
exceeding a couple of feet in length. With the
former the discharge would be 7-976 x -974 = 7-769
cubic feet, and with the latter 7-976 x -956 = 7-625
cubic feet. We may find from Table II. the latter
results otherwise. With a head of 20 feet and the
coefficient -974, the velocity is 419-48 inches = 34-957
'2
feet ; hence, the discharge is g x 34-957 = 7-768
cubic feet. With a coefficient of -956, the velocity
is 411-73 inches = 34-31 feet, and | x 34-31 = 7-624
cubic feet. These results are the same, practically,
as those previously found.
If the inner arrises be square, and the passage out
be from 18 inches to 2 feet long, the orifice will be
converted into a short tube, the coefficient for which
is -815. With this coefficient, and a head of 20 feet,
we find as before, from TABLE II., the mean velocity
of discharge equals 351 inches = 29-25 feet; hence,
2
the discharge now is ^ x 29-25 = 6-5 cubic feet per
second.
The velocities in inches per second, given in TABLES
II. and VIII., or elsewhere in the following pages, may
be converted into velocities in feet per minute, by mul-
7 60
tiplying by 5, equal j^
EXAMPLE 6. The discharge from a small orifice
having its centre placed 1 feet below the surface of a
OEIFICES, WEIRS, PIPES, AND RIVERS. 19
reservoir is 18 feet per minute, what will be the dis-
charge from the same orifice at a depth of 17 feet?
The discharges will be to each other as \/10 : \/W 9
or as 1 : N/F7; or, from TABLE III., as 1 : 1-3038,
whence we get the discharge sought equal 1-3038 x
18 = 23-4684 cubic feet.
EXAMPLE 7. What is the value of the expression
( C z }*
c d \ 1 + 2 d g [ in equation (45), when c d = '617, and
m c a/
m = 2 ? Here we have
* -2807
1052 ;
c 2 d -617 2 -3807
m z -c\ 4--617 2 3-6193
whence the first expression becomes equal to '617
(M052)* equal, from TABLE III, -617 x 1-0513 =
649, the value sought. TABLE V. contains the values
of this expression for various values of c d and m,
which latter, m, stands for the ratio of the channel
to an orifice ; and we can immediately find from it,
opposite 2 in the first column, and under the coeffi-
cient -617 in the sixth column, -649 the value sought.
When the head due to the pressure, and to the velocity
of approach, are both known, we can determine the
new coefficient of discharge by the above expression,
and thence the discharge itself. The coefficient suited
to the velocity of approach may however be found
directly in TABLE V. The usual methods for finding
the effects of the velocity of approach, given by
d'Aubuisson and others, are incorrect in principle,
see SECTION IV.
EXAMPLE 8. What is the discharge from an
orifice 17 inches long and 9 inches deep, having the
upper edge placed 4 inclies below the surface, and the
c3
20 THE DISCHARGE OF WATER FROM
lower edge 13 inches? The expression for the dis-
charge is^x Av / 272xc d |(l +~)^-(-^) 2 [equa-
tion (43), in which we must take d = 9 inches; h t = 9
inches ;A = 17x9 = 153 square inches ; and\/2^^,
found from TABLE II. = 83-4 inches. We have, also,
= - = -444^ and hence the value of
(1-444)4 - (-444)2 = (from TABLE IY.) 1-44.
Assuming the coefficient of discharge to be -617, we
then have the discharge in cubic inches per second
equal to
| x 153 x 83-4 x -617 x 1-44 =
o
I x 12760-2 x -88848 = 7558.
Consequently, TI^ = 4-374 is the discharge in cubic
feet per second. Prom equation (6), we get the dis-
charge equal to
| x -617 x 27-8 x 17 x {13* 4*}.
But 13t - 4t = 46-872 - 8, from TABLE IY., equal
to 38-872, whence the discharge is
| x -617 x 27-8 x 17 x 38-872 = 114351 x 17 x
38-872 = 194-3967 x 38-872 = 7557 cubic inches
= 4*374 cubic feet, the same as before.
It is shown in equation (31), that by using the
mean depth for orifices near the surface, the discharge
/"'*'
will approximate very closely to the true discharge,
and that even for weirs the error will not exceed
6 per cent. The discharge is then expressed by
ORIFICES, WEIRS, PIPES, AND RIVERS. 21
x 8i x 9 x 17 = (from TABLE II.) 50-01 x
153 = 7651-53 cubic inches = 4*427 cubic feet per
second. The head to the centre of the orifice is
here 8i inches, and the depth of the orifice 9 inches,
therefore, in equation (31), h = d very nearly ; and,
therefore, this result must be multiplied by *989, as
shown in that equation ; then -989 x 4-427 = 4-378
cubic feet, which gives a result differing from those
otherwise found, by a very small quantity, which,
practically, is of no value. By means of TABLE VI.
the discharge from rectangular orifices near the sur-
face can be found with very great facility.
We may always find the discharge from an orifice
near the surface with sufficient accuracy ', for practical
purposes, by taking the head to the centre, in the same
manner as if the orifice were sunk to a considerable
depth ; then by applying the corrections given in equa-
tion (31) ; or if the orifice be circular, those given in
equation (28) ; extreme accuracy ', according to the cor-
rect formula, is obtainable.
EXAMPLE 9. What is the discharge from a circular
orifice 4 inches in diameter ', having its centre placed 4
inches below the surface, when the coefficient of dis-
charge is *6L7 ? The area of the orifice is 4 x 4 x
7854 = 12-566 square inches. The velocity in the
orifice at the mean depth of 4 inches, with a co-
efficient of *617, is 34-31 inches, whence the discharge
is 12-566 x 34-31 = 431-139 cubic inches = -2496
cubic feet per second, or 14-97 cubic feet per minute.
By means of TABLE IX. the discharge in cubic feet
per minute can be found very readily when the velo-
city, 34-31 inches per second, is known. Thus,
22 THE DISCHAKGE OF WATEK FEOM
Inches. Cubic feet.
For a velocity of 30-00 the discharge is 13-089
4-00 1-745
30 0-131
-01 0-004
./ 34-31 ^ 14-969
By applying the coefficient found from equation
(28), which is -992, when the depth at the centre is
twice the radius, as it is in this example, we get
992 x 14-97 = 14*85 for the correct discharge in
cubic feet per minute. Here the difference in the
results is only 1 in 125.
The application of TABLE VI. will enable us to
find the discharge from rectangular orifices near the
surface very quickly. Resuming " EXAMPLE 8," the
discharge may be found from this Table for each foot
in length of the orifice, as follows. The discharge in
cubic feet per minute, when the coefficient is -617 for
a notch 1 foot long and 13 inches deep, is 223-323 ;
and for a notch of 4 inches deep, 38-116 ; therefore,
the discharge from an orifice 9 inches deep, with the
upper edge 4 inches below the surface, is 223-323
38-116 = 185-207 cubic feet per minute. But as the
length of the orifice is 17 inches, this must be multi-
17
plied by ^ an(i the product 262-377 is the dis-
charge in cubic feet per minute; this is equal to a
discharge of 4-373 cubic feet per second, and agrees
with that before found. This is the simplest way of
finding the discharge from rectangular orifices near
the surface.
EXAMPLE 10. What is the discharge in cubic feet
OEIFICES, WEIES, PIPES, AND EIVEKS. 23
per minute, from an orifice 2 feet 6 inches long and 7
inches deep, the upper edge being 3 inches below the
surface, and the coefficient of discharge -628 ? From
TABLE VI. we find the discharge from a notch 1 foot
long and 10 inches deep to be 153-353, and for a
notch 3 inches deep, 25199. The difference, or
128-154, multiplied by 2i, will be the discharge re-
quired ; viz. 2 x 128-154 = 320-385 cubic feet per
minute.
EXAMPLE 11. The size of a channel is 2*75 times
the size of an orifice, what is the coefficient of dis-
charge when that for a very large channel in pro-
portion to the orifice is -628 ? We find from TABLE V.
the coefficient to be -645, when the approaching
water suffers full contraction. By attending to the
auxiliary Tables in the text, we find, for this case,
/> i
-r T = cTfTK = *36. We must, therefore, multiply
channel 2-75
2-75 by -857, which gives 2*36 for the ratio of the mean
velocities in the orifice and in the channel approaching
it. With this new value of the ratio of the channel to
the orifice, we find, as before, the value of the co-
efficient from TABLE V. to be -651. The remarks
throughout the work, with the auxiliary tables, will
be found of much use in determining the coefficients
for different ratios of the channel to the orifice,
notch, or weir, and the corrections suited to each.
If in this example we were considering, other
things being the same, the alteration in the coeffi-
cient for a notch, or weir, it would be found from
the Table, column 4, to be *672 instead of *645
found in column 3, for an orifice sunk some depth
24 THE DISCHARGE OF WATER FROM
below the surface. For the corrections suited to mean
and central velocity, and to the nature of the approaches,
we must refer to the body of this work and to the
auxiliary tables therein at the end of SECTION IY.
EXAMPLE 12. What is the discharge aver a weir
50 feet long ; the circumstances of the overfall, crest,
and approaches, being such that the coefficient of
discharge is '617, when the head measured from the
water in the weir basin, 6 feet above the crest, is
17i inches? TABLE VI. will give the discharge in
cubic feet per minute, over each foot in length of
weir, for various depths up to 6 feet. It is divided
into two parts ; the first for " greater coefficients,"
viz. -667 to -617; and the second for "lesser co-
efficients," viz. -606 to -518. The coefficient assumed
being -617, we find the discharge over 1 foot in
length, with a head of 17J inches, to be 348-799
cubic feet per minute ; hence the required discharge
is 50 x 348-799 = 17439-95 cubic feet.
The determination of the coefficient suited to the
circumstances of the overfall, crest, approaches, and
approaching section, will be found discussed else-
where through this work. The valuable Table de-
rived from Mr. BlackwelPs experiments will also be
of use ; but the heads being taken at a much greater
distance back from the crest than is generally usual,
the coefficients taken from it for heads greater than
5 or 6 inches, will be found under the true ones for
heads measured immediately at or about 6 feet, above
the crest. For heads measured on the crest, the
small Table of coefficients in SECTION III., applicable
to the purpose, will be of use.
OEIFICES, WEIRS, PIPES, AND RIVERS. 25
EXAMPLE 13. What is the mean velocity in a large
channel, when the maximum velocity along the central
line of the surface is 31 inches per second? TABLE
VII. gives 25-89 inches for the required velocity, and
for smaller channels 24*86 inches. In order to find
the mean velocity at the surface from the maximum
central velocity, the latter must be multiplied by
914.
The velocity at the surface is best found by means
of a floating hollow ball, which just rises out of the
water, The velocity at a given depth is best found
by means of two hollow balls connected with a link,
the lower being made heavier than the upper, and
both so weighted by the admission of a certain quan-
tity of water that they shall float along the current,
the upper one being in advance but nearly vertical
over the other. The velocity of both will then be
the velocity at half the depth between them. The
velocity at the surface, found by means of a single
ball, being also found, the velocity lost at the half
depth is had by subtracting the common velocity due
to the linked balls from that of the single ball at the
surface. The velocity at any given depth is then
easily found by a simple proportion ; but the result
will be most accurate when the given depth is nearly
half the distance between the balls, which distance
can never exceed the depth of the channel. Pitot's
tube, Woltmann's tachometer, the hydrometric pendu-
lum, the rheometer, and several other hydrometers,
have been used for finding the velocity ; but these
instruments require certain corrections suited to each
separate instrument, as well as kind of instrument,
26 THE DISCHAKGE OF WATEK FROM
and are not so correct or simple, for measuring the
velocity in open channels, as a ball and linked balls.
EXAMPLE 14. What is the discharge from a river
having a surface inclination of 18 inches per mile, or
1 in 3520, 40 feet wide, with nearly vertical banks,
and 3 feet deep ? The area is 40 x 3 = 120 feet,
and the border 40 + 2 x 3 = 46 feet ; therefore the
120
hydraulic mean depth is -TTT = 2-61 feet zz 2 feet 7-3
inches*. With this and the inclination we find from
TABLE VIII. 28-27 + 2-75 X ^ = 28-87 inches per
second =: 28-87 x 5 =z 144-35 feet per minute for
the mean velocity ; hence we get 144-35 x 120 =
17,322 cubic feet per minute for the required dis-
charge. For channels with sloping banks we have
only to divide the border, which is always known,
into the area for the hydraulic mean depth, with
which, and the surface inclination, we can always
find the velocity by TABLE VIII., and thence the
discharge. Unless the banks of rivers be protected
by stone pavement or otherwise, the slopes will not
continue permanent ; it is therefore almost useless to
give the discharges for channels of particular widths
and side slopes, When the mean velocity is once
known, the remaining calculations are those of mere
mensuration, and they should be made separately.
This example may also be solved, practically, by
means of TABLES XI. and XII. A channel 40 x 3
* For greater hydraulic depths than 144 inches, the extent of
the TABLE, divide hy 9, and find the corresponding velocity.
This multiplied by 3 will be the velocity sought.
ORIFICES, WEIKS, PIPES, AND RITE
has the same conveying power as one 70 x
XL, which latter, TABLE XII. discharges with a fall
of 18 inches in the mile, 17,157 feet; or about one
per cent, less than that previously found.
EXAMPLE 15. The diameter of a very long pipe is
li inch, and the rate of inclination , or ivhole length
of the pipe divided by the whole fall, is 1 in 71^ ;
what is the discharge in cubic feet per minute ? The
1*5
hydraulic mean depth, or mean radius, is -j- = -375
3
inches = ^ inch. Consequently we find from TABLE
VIII. the velocity in inches per second equal to
25-09 - 1-92 x ^ = 25-09 - -29 = 24-80. The
discharge in cubic feet per minute for a IJ-inch pipe
is now found most readily by means of TABLE IX.,
as follows :
Inches. Cubic feet.
For a velocity of 20-0 the discharge is 1-227
4-0 -245
8 -049
,, 34-8 1-521
Whence the discharge in cubic feet per minute is
1-521.
For short pipes, of 100 or 200 feet in length, and
under, the height due to the velocity and orifice of
entry must be deducted from the whole height to find
the proper hydraulic inclination, and also the height
due to bends, curves, cocks, slides, and erogation.
The neglect of these corrections has led some writers
into mistakes in applying certain formulae, and in test-
28 THE DISCHAEGE OF WATER FEOM
ing them by experimental results obtained with short
pipes. We shall now apply the TABLES to the deter-
mination of the discharge from short pipes, and
compare the results with experiment, referring gener-
ally to equation (153) and the remarks preceding it
for a correct and direct solution.
EXAMPLE 16. What is the discharge in cubic feet
per minute from a pipe WO feet long, with a fall or
head of 35 inches to the lower end, when the diameter
is 1 J inch ? Find also the discharge from pipes 80
feet t 60 feet, 40 feet, and 20 feet, of the same diameter
and having the same head. If the water be admitted
by a stop-cock at the upper end, the coefficient
due to the orifice of entry will probably be about -75
or less, -815 being that for a clear entry to a short
cylindrical tube. The approximate inclination is
100 x 12
or i= 1 in 34*3 ; but as a portion of the fall
must be absorbed by the velocity and orifice of entry,
we may assume for the present that the inclination
is 1 in 35. With this inclination and the mean radius
IL- 3
-jr zz g inches, we find the mean velocity from TABLE
VIII. to be 38-06 inches. Now when the coefficient
due to the orifice of entry and velocity is -75, we
find from TABLE II. the head due to this velocity to
be 3| inches nearly, whence 35 - 3f = 31f = 31-625
100 x 12
inches is the height due to friction, and ^^
o J.*D^jO
equals 1 in 37-9, the inclination, very nearly. With
this new inclination we find, as before, from TABLE
VIII. the mean velocity of discharge to be now 36-35
ORIFICES, WEIRS, PIPES, AND RIVERS. 29
inches ; and by repeating the operation we shall find
the velocity to any degree of accuracy in accord-
ance with the table, and the shorter the pipe is, the
oftener must it be repeated. The height due to 36-35
inches taken from TABLE II. as before, with a coeffi-
cient of -750, is 3^ = 3-125 inches. The corrected
fall due to the friction is now 35 - 3-125 == 31-875,
and 01.07- equal 1 in 37-6, the corrected inclina-
tion. With this inclination we find the corrected
velocity to be now 36-53 inches per second. It is
not necessary to repeat the operation again. The
discharge determined from TABLE IX. is as follows :
Inches. Cubic feet.
For a velocity of 30-00 the discharge is 1-841
6-00 -368
50 -031
03 -002
36-53
The experimental discharge found by Mr. Provis was
2-264 cubic feet per minute in one experiment, and
2-285 in another. The discharge from the shorter
pipes may be found in a similar manner, and we
place the results alongside the experimental ones
given in the work referred to below* in the following
short table :
* " Transactions of the Institution of Civil Engineers," vol. ii.
p. 203. " Experiments on the Flow of Water through small
Pipes." By W. A. Provis. The small Tables in SECTIONS VI.
and VIII. of this edition give at once the coefficient to be mul-
tiplied by^^n, or 8-v/lf, to find the velocity when the ratio of
the diameter to the length of the pipe is known. They will be
found of great advantage in calculating directly the velocity from
short pipes. For long pipes, see the TABLE pp. 42 and 43.
30 THE DISCHARGE OF WATER FROM
EXPERIMENTAL AND CALCULATED DISCHARGES FROM SHORT PIPES.
1,
4
1
.a
!*
A
o
^o
fa
1
.a
1
III
II
jji
II
11
w
I'l
if
If
3
w
w
100
35
2-275
37-082
3*
31|
37-6
36-53
2-242
80
35
2.500
40-750
3f
3H
30-8
41-18
2-521
60
35
2-874
46-846
5
30
24-0
48-02
2-946
40
35
3-504
57-115
n
27i
17-5
58-50
3-590
20
35
4.528
73-801
12*
221
10-7
78-61
4-824
The velocities in the fourth column have been cal-
culated by the writer from the observed quantities
discharged, from which the height due to the orifice
of entry and velocity in column 5 is determined, and
thence the quantities in the other columns as above
shown. The differences between the experimental
and calculated results are not large, and had we used
a lesser coefficient than 8 750 for calculating the re-
duction of head due to the velocity, stop-cock, and
orifice of entry, say -715, the calculated results, and
those in all of Mr. Provis's experiments in the work
referred to, would be nearly identical.*
EXAMPLE 17. It is proposed to supply a reservoir
near the town of Drogheda with water by a long pipe,
having an inclination of 1 in 480, the daily supply to
be 80,000 cubic feet ; what must the diameter of
* In a late work, " Researches in Hydraulics," the author is
led into a series of mistakes as to the accuracy of Dtf Buat's and
several other formulae, from neglecting to take into consideration
the head due to the velocity and orifice of entry when testing
them by the experiments above referred to.
ORIFICES, WEIRS, PIPES, AND RIVERS. 31
the pipe be ? The discharge per minute must be
* =56* cubic feet, nearly. Assume a pipe
whose "mean radius" is 1 inch, or diameter 4
inches, and the velocity per second found from TABLE
VIII. will be 14*41 inches. We then have from
TABLE IX.,
Inches. Cubic feet.
For a velocity of 10-00 a discharge of 4-363
4-00 1-745
40 -175
01 -004
14-41 6-287
The discharge from a pipe 4 inches in diameter would
be therefore 6-287 cubic feet per minute. We then
have
4* : d$ : : 6'287 : 56, or 1 : rf* : : -196 : 56 : : 1 : 286;
therefore d^ =. 286, and d =. 9-61 inches, nearly, as
may be found from TABLE XIII., &c. This is nearly
the required diameter. It is to be observed that the
diameters thus found will not always agree exactly
with those found from Du Buat's or other formulas,
nor with each other, because the discharges are not
strictly as d^ ; but in practice the difference is imma-
terial, and the approximative value thus found can
be easily corrected. If we assumed a pipe whose
diameter is 1, the operation would have been more
simple ; for the velocity would then be, TABLE VIII.,
at the given inclination, 6-4 inches ; and the discharge
175 cubic feet, TABLE IX. Hence we get d%
* Hydraulic Tables, Weale, 1854, give at once this discharge
for a pipe between 9 and 10 inches diameter, also the TABLE, p. 42.
32 THE DISCHARGE OF WATER FROM
= 320, and, therefore, TABLE XIII., d 10
inches nearly, which differs about half-an-inch from
the former value, 9-6 inches, found by assuming a
pipe of 4 inches to calculate from. It is necessary
to understand that different results must be expected,
in working from practical formulae, for different
operations. When once an approximative value is
obtained, it can be easily corrected to any required
degree of accuracy.
Again the velocity in inches per second, from a cylin-
drical pipe 6 inches in diameter, is nearly equal to the
discharge in cubic feet per minute ; and as 6^ m 88*2,
we have 88*2^ : d^ : : the velocity in inches per
second from a 6-inch pipe : the discharge per minute
from a pipe whose diameter is d. Hence this pro-
portion would enable us to find, very nearly, the
discharge from the diameter and fall ; or the diameter
from the discharge and fall by finding the velocity
only, due to a 6-inch pipe. See TABLE pp. 42 and 43.
EXAMPLE 18. The area of a channel is 50 square
feet, and the border 20*6 feet ; the surface has an in-
clination of 4 inches in a mile ; what is the mean
50
velocity of discharge? ^g = 2*427 feet = 29-124
inches is the hydraulic mean depth ; and we get
from Table VIII. 12-03 - -- - = 12-03 -
19 = 11*84 inches per second for the required velo-
city. Though this velocity will be found under the
true value for straight clear channels, it will yet be
more correct for ordinary river courses, with bends
OKIFICES, WEIES, PIPES, AND KIVERS. 33
and turns, of the dimensions given, than the velocity
found from equation (114). For a straight clear
channel of these dimensions, Watt found the mean
velocity to be 13-5 to 14 inches; that is to say,
17 at top, 10 at bottom, and 14 in the middle, Our
formula v = 140 (r sfi 11 (r sft gives v = 1*143
feet, or nearly a mean of these two.
EXAMPLE 19. A pipe 5 inches in diameter, 14,637
feet in length, has a fall of 44 feet ; what is the dis-
charge in cubic feet per minute ? The inclination is
1 i r* Q IT K
-jj = 332*7, and mean radius j = 1J. We then
find from TABLE VIII. the velocity equal to 19-81 +
L \*. 5 4 - = 19-81 + -16 = 19-97, or 20 inches per
second very nearly ; and by TABLE IX. the discharge
in cubic feet per minute is, as before found to be,
13-635. The TABLE, p. 42, gives, by inspection, 13-6
feet.
EXAMPLE 20. What is the velocity of discharge
from a pipe or culvert 4 feet in diameter, having a
fall of 1 foot to a mile ? Here s =: goftQ' and r =
1 foot. We then find the velocity of discharge from
TABLE VIII. to be 14-09 inches, equal to 1-174 feet per
second. By calculating from the different formulae
referred to below, we shall find the velocities, when
r s = -0001894, and \/Ts = -01376, as follows.
Velocity
in feet.
Eeduction of Du Buat's formula .... equation (81) 1-174
Girard's do. (Canals with aquatic
plants and very slow velocities) (86) -621
Prony's do. (Canals) .... (88) 1*201
D
34 THE DISCHAKGE OF WATER FROM
Velocity
in feet.
Seduction of Prony's formula (Pipes) . . equation (90) 1-257
Prony's do. (Pipes and Canals) . (92) 1-229
Eytelwein's do. (Kivers) ... (94) 1-200
Eytelwein's do. (Kivers) ... (96) 1-285
Eytelwein's do. (Pipes) ... (98) 1-364
Eytelwein's do. (Pipes) ... (99) 1-350
Dr. Young's do. . . .>. . (104)1-120
*D'Aubuisson's do. (Pipes) . . (109)1-259
*D'Aubuisson's do. (Kivers) . . (111) 1-199
,, The writer's do. (Clear straight
Channels with small velocities) (114) 1-268
Weishaeh's do. (Pipes) ... (119) 1-285
The author's, for Pipes and Kivers 1-295
We have calculated this example from the several
formulae above referred to, whether for pipes or rivers
in order that the results may be more readily com-
pared. The formula from which the velocities and
tables for the discharges of rivers are usually cal-
culated is, for measures in feet, v 94-17 </ r s.
This gives the mean velocity, for the foregoing ex-
ample, equal to 1-295 feet per second. This is the
same as is found from my general formula for all
velocities; but the particular expression, v = 99-17
\/ r s, is only suited for velocities of about 15 inches
per second; the results found from it for lesser
velocities are too much, and for higher velocities too
little, if bends and curves be allowed for separately-
For ordinary practical purposes the result of Du
Buat's general formula, equation (81), may be safely
adopted ; and we have, accordingly, preferred retain-
ing the results in TABLE VIII. calculated for our first
edition from it, notwithstanding the greater accuracy
and simplicity of our own general equation (119 A)
* These two formulae of D'Aubuisson's are, simply, adoptions of Eytel
vein's and Prony's.
ORIFICES, WEIRS, PIPES, AND RIVERS. 35
for the velocity in pipes and rivers, viz., v = 140 (rs)*
ll(r*)i.
Dr. Young's formula gives lesser results for rivers
and large pipes than Du Buat's, but they are too
small unless when the curves and bends are numerous
and sudden. Girard's formula (86) is only suited for
small velocities in canals containing aquatic plants,
and it is entirely inapplicable to rivers or regular
channels for conveyance of water. A knowledge of
various formulae, and their comparative results, ap-
plied to any particular case, will be found of great
value to the hydraulic engineer, and the differences
in the results show only an amount of error that may
be expected in all practical operations, and which be-
comes of less importance when we consider that by
increasing the dimensions of a channel every way,
by only one-third, we shall more than double its dis-
charging power. See TABLE XIII.
EXAMPLE 21. Water flowing down a river rises to
a height of 10 J inches on a weir 62 feet long ; to what
height will the same quantity of water rise^ on a weir
62
sim ilarly circumstanced, 120 feet long ?
nearly. In TABLE X. we find, by inspection, opposite
to -517, the ratio of the lengths, the coefficient *644,
rejecting the fourth place of decimals ; whence 10|
X *644 ~ 6-76 inches, the height required. When the
height is given in inches it is not necessary to take out
the coefficient to further than two places of decimals.
EXAMPLE 22. The head on a weir 220 feet long is
6 inches ; what will the head be an a weir 60 feet long,
similarly circumstanced ' f the same quantity of water
D 3
36 THE DISCHAEGE OF WATER FROM
fiO
flowing over each ? o 273. As this lies between
27 and '28, we find from TABLE X. the coefficient
/*
4208; hence .TZZ 14*2 6 inches, the head required.
TABLE X. will be found equally applicable in
finding the head above the pass into weir basins, and
above contracted water channels. See SECTION X.
EXAMPLE 23. A river channel 40 feet wide and
4*5 feet deep is to be altered and widened to 70 feet ;
what must the depth of the new channel be so that the
surface inclination and discharge shall remain un-
altered? In "TABLE XL, OF EQUALLY DISCHARGING
RECTANGULAR CHANNELS," W6 find Opposite to 4*54,
in the column of 40 feet widths, 3 in the column of
70 feet widths, which is the depth required in feet.
EXAMPLE 24. It is necessary to unwater a river
channel 70 feet wide and 1 foot deep, by a rectangular
side cut Wfeet wide ; what must the depth of the side
cut be, the surface inclination remaining the same as
in the old channel? In TABLE XI. we find 4 -5 feet
for the required depth. When the width of a channel
remains constant, the discharge varies as \/f~s X d,
in which d is the depth ; and when the width is very
large compared with the depth, the hydraulic mean
depth r approximates very closely to the depth d, and
therefore d r; consequently the discharge then
varies as d^ x **, and when the discharge is given d%
must vary inversely as /; or more generally dr must
vary inversely, as /, when the width and discharge
remain constant.
In narrow cuts for unwatering, it is prudent to
make the depth of the water half the width of the
ORIFICES, WEIRS, PIPES, AND RIVERS. 37
cut very nearly, when local circumstances will admit
of these proportions ; for then a maximum effect will
be obtained with the least possible quantity of exca-
vation ; but for rivers and permanent channels the
proper relation of the depth to the width must be
regulated by the principles referred to in SECTION IX.
TABLE XI. is equally applicable, whether the mea-
sures be taken in feet, yards, or any other standards
whatever.
EXAMPLE 25. A new river channel is to have a fall
of eighteen inches in a mile, and must discharge 18,700
cubic feet per minute, what shall the dimensions be ?
In TABLE XII., in the column of 18 inches per mile,
we shall find opposite to 18,766, that a primary chan-
nel 70 x 2*125 will be sufficient ; and opposite to 2*125
in TABLE XI. we shall find the equivalent rectangular
channels 60x2*37; 50x2*70; 40x3*19; 35x3*52;
30x3*96; 25x4*61; 20x5*58; 15x7*29; and 10x
11 *37, to select from. If the sides shall have any given
slopes, the discharge will not be practically affected
as long as the depth and area of the rectangular
channel and the one with sloping banks remain the
same. See SECTION IX.
EXAMPLE 26. A pipe WO feet long and 1 inch in
diameter has a head of 150 feet over the lower end, what
will be the discharging velocity? Here r rz: '020833
in feet, and s = 1*5, therefore r s = *03125. Hence
by formulse(119A) v = 140 x ('03125/ 11 X (-03125)*
= 140 x '1766 - 11 X '315 = 24*724 - 3*465 =
21*259 feet per second. If allowance is required
for the orifice of entry, the velocity is corrected as
follows. A square orifice of entry has a coefficient
of -815. The head due to this coefficient for a
38 THE DISCHAKGE OF WATEK FKOM
velocity of about 20| feet, or 246 inches, is about
10 feet, TABLE II. The head due to friction is
therefore 150 - 10 zz 140 feet, and s =
r s now becomes 1-4 X '020833 zz -02917. Hence
v zz 140 \/~r~s 11 \/ rs now becomes 140 x *171
-11 X '308 nearly, equal to 23-940- 3-388 zz 20'552
feet, the velocity for a square junction.
EXAMPLE 27. A sewer 9 feet in diameter has a
fall of 2 feet per mile^ what will be the velocity and
discharge of water flowing through it when full ?
Hererzz 2-25 and s= 777^7^ therefore rs= -0008523,
Jo40
(r*)* = -02919 and (r*)* = '0948 ; and by formula
(119A) we have v zz 140 (r *)* - 11 (r *)* zz 140 x
02919 - 11 x '0948 zz 4-0866 - 1-0428 zz 3-0438
feet per second. Hence the discharge per minute is
9 2 x '7854 x 3-0438 x 60 zz 63-62 X 182-6 =11,617
cubic feet nearly. The velocity from a circular pipe
or sewer is however greatest when the circumference
is open for about 78 \ degrees at the top, but the
velocity of sewage matter would not be equal to that
of water. It would vary according to the dilution
in the sewer, and 50 per cent, should be allowed,
at least, in deduction, unless the dilution be very
considerable.
The TABLE for the values of r s and v, calculated
from the formula (119 A) SEC. VIII., will give the
velocity at once when r s is known, and r s when the
velocity is known, from the latter of which a definite
value of r or s can be fixed upon, when the other
may be found, by an operation of simple division.
EXAMPLE 28. Water is to be pumped through a
ORIFICES, WEIRS, PIPES, AND RIVERS. 39
pipe 3000 feet long and 2 feet in diameter, with a
velocity not exceeding 4 feet per second, what head
must be allowed extra for friction in the pipe in
calculating horse power ? We shall find from our
TABLE of the values of the velocity and product of
the hydraulic mean depth and hydraulic inclination,
given near the conclusion of SECTION VIII. , that for
a velocity of 4 feet per second r szz -00142. The dia-
meter of the pipe is 2 feet, therefore r -5, whence
,9 = -p zz -00284. and as the length of the pipe is
*o
3000 feet we get 3000 x '00284 zz 8*52 feet, the head
required. The TABLE, p. 43, would give 9-6 feet
nearly, which corresponds with Du Buat's formula.
If the velocity in the pipe was 10 feet instead of
4 feet per second, then, from our table, rs zz -007576,
arid zz s zz - = = -015152, and, therefore.
r -5
h = ls = 3000 x -015152 zz 45-456 feet, or about
six times as much as when the velocity was only
4 feet per second. The great loss of head arising
from pumping at high velocities, from friction alone,
is therefore apparent. Were the velocity double, or
8 feet per second, the head would be 30 feet nearly,
or from the TABLE, p. 43, 31-6 feet.
For velocities of about 2-1 feet per second, v is
equal to 100 \/ r s, and for velocities of about 5| feet
per second, v zz 110 \/~r~s. If / be the length of a pipe,
we would find in the former case the head h in feet due
Iv
to friction from the formula h zz -. Q zz Is ; and
Iv '
in the latter h zz zz Is.
In questions of this kind, however, the diameter of
40 THE DISCHARGE OF WATEE FKOM
a pipe, d should be used in preference to the
hydraulic mean depth, and as d = 4 r we shall find in
lv
the first case h = 2509.7 ^ s > an( ^ i n the second
lv*
If we wish to substitute the fall per mile for the
hydraulic inclination, the first of these will again
become h = -* = Is for the loss per mile ; and
in the second case, h = Y^. = i s f OT the loss per
mile in feet.
If the velocity were so low as about 1 foot per
second, then v = 90 \/ r s, and we should find
, lv j
--
If for the inclination we substitute the fall per
mile, this will become h = -* = Is for the loss
per mile in feet.
The loss of head varies in the same pipe with the
velocity, and must be calculated differently, for small
and for high velocities, when using the common for-
mulae. The TABLE near the end of SECTION VIII. will
?* ^
always give the correct value of rs, and thence *=~'
In addition to the loss of head arising from fric 1
tion, losses also o'ccur from straight or curved bends,
from diaphragms, from junctions, and from the ori-
fices of entry and discharge ; these must be deter-
mined separately for each case, as is shown hereafter,
and added together and to the loss arising from fric-
tion, and the sum to the height the water is to be
raised, before the full or total head for determining
the power of an engine can be accurately known.
OEIFICES, WEIES, PIPES, AND KIVEES. 41
The TABLE on the next two pages will be found of
great practical utility in solving all questions con-
nected with water-pipes and sewers discharging fully-
diluted sewage. In using it we can interpolate, by
inspection, for intermediate diameters or inclinations.
For greater diameters, divide those given by 4, and
multiply the corresponding velocity found in the
table by 2, and the corresponding discharge in the
table by 32. If the object be to find the size
of the channel, divide greater given velocities by 2,
and multiply the diameters or inclinations found from
the table by 4 ; also divide greater discharges by 32,
and multiply the diameters found from the table by
4. The small auxiliary table, p. 43, embodied in the
larger one, is of great use in making allowance for
the velocity and orifice of entry in short pipes, before
finding the head due to friction. The table also gives
the different diameters and inclinations which, taken
together, give the same velocity or discharge ; and it
enables us, from inspection, to select that relation of
diameter to declivity which is best suited for other
engineering aspects of the question. Taken in con-
nexion with TABLES VIII., XL, XII., and XIII., this
table completes the means of finding, by inspection,
the dimensions, inclinations, velocities, and discharges
of every class of water-channel or sewage-conduit
required in engineering practice.
TABLE XIV. gives the comparative values of Eng-
lish and French measures ; and TABLE XV. gives the
weight, specific gravity, and ultimate strength and
elasticity of various materials with which the engineer
has to operate.
TABLE for finding, very nearly, the velocity and discharge from Cylindrical Water Pipes or Sewers, when the
diameter and fall are given. Any two of the four quantities, the velocity, discharge, diameter, and fall or
inclination, being given, the others can be found in THE TABLE from inspection.
sift
-w o 5 a
A * p,.
flj*
Mean by- 1
draulic incli-l
nation of 1
pipe or sewerj
The VELOCITY IN INCHES PER SECOND is given in the first horizontal line for each inclination
or fall ; and the DISCHARGE IN CUBIC FEET PER MINDTE in the next following one.
lin. 2 in. 1 3 in.
diameter diameter | diameter
4 in.
diameter
5 in.
diameter
6 in.
diameter
7 in.
diameter
8 in.
diameter
9 in.
diameter
10 in.
diameter
One in
1
5280
1-7
2-5
3-2
3-8
4-2
4.7
5-1
5-5
5-9
6-2
05
27
79
1-6
29
4-6
6-8
9-6
129
169
2
2640
2-5
3-8
4-7
5-6
6-3
6-9
7-5
8-1
8-6
9-1
07
41
1-2
2-4
4-3
68
101
14-2
191
24-9
3
1760
3-1
4-7
5-9
7-0
7-8
8-7
9'4
10-2
10-8
11-5
08
51
1-4
30
53
85
126
17-7
239
31-2
4
1320
3-6
5-5
6-9
8-2
9-2
10-2
ii-l
11-9
12-7
13-4
10
60
1-7
3-6
63
100
148
20-8
280
36-7
5
1056
4-1
6-2
7-9
9'3
10-4
11-6
12-5
135
14-4
15-2
11
68
19
4-0
7-1
11-3
16-8
236
31-7
41-5
6
880
4-6
6-9
8*7
10-3
11-5
12-8
13-9
15-0
15-9
16-9
12
76
2-2
45
79
126
186
261
351
460
7
754
5-0
7-5
9-5
11-2
1-2-6
14-0
15-1
16-3
17-4
18-4
14
82
2-3
49
86
13-7
20-2
285
383
502
8
660
5-4
8-1
10-2
12-0
13-5
15-0
16-3
17-6
18-7
20-0
15
89
2-5
53
92
14-8
21-8
306
41-3
541
9
587
5-7
8-7
11-0
12-9
14-5
16-1
17-4
18-8
20-0
21-2
16
95
27
56
99
15-8
233
328
441
57-7
10
528
6-1
9-2
116
13-7
15-4
17-1
18-5
19-9
21-2
22-5
17
1-00
29
60
92
167
24-7
348
46-8
61-3
11
480
6-4
9-7
12-3
14-4
16-2
18-0
19-5
21-0
22-4
23-7
17
11
3-0
6-3
111
17-7
261
36-7
49-4
64-7
12
440
6-7
10-2
12-9
15-2
17-1
18-9
20-5
22-1
23-5
24-9
18
11
32
6-6
11-6
186
27-4
386
519
679
13-2
400
7-1
10-8
13*6
16-0
18-0
20-0
21-7
23-3
24-8
26-3
19
1-2
33
69
123
196
28-9
40-7
548
71-7
15-1
350
7-7
11-6
14-7
17-2
19-4
21-6
23-4
25-2
26-8
28-4
21
1-3
36
75
33-2
21-2
31-2
439
591
77-4
17-6
300
8-4
12-7
16-0
18-8
21-2
23-5
25-5
27-5
29-2
31-0
23
1-4
3-9
82
14-4
231
34-1
40-8
646
845
21-1
250
9-4
14-1
17-8
20-9
23-5
26-1
28-3
30-5
32-5
34-4
26
15
4-4
92
160
25-7
378
533
71-7
93-8
26-4
200
10-6
16-0
20-2
23-8
26-8
29-7
32-2
34-7
36-9
39-1
29
1-7
5-0
104
18-2
29-2
431
60-6
816
1067
35-2
150
12-5
19-0
23-9
28-1
31-6
35-1
38-1
41-0
43-7
46-3
34
21
59
123
21-6
345
509
71-6
96-4
126-2
52-8
100
15-9
24-1
30-4
35-7
40-1
44-6
48-3
52-1
55-4
58-7
43
2-6
7-4
15-6
27-3
438
64-6
90-9
122-3
1601
587
90
16-9
25-6
32-3
38-0
42-7
47-4
51-4
55-4
58-9
62-5
46
28
79
166
29-1
46-6
687
96-7
130-2
170-3
C6-
80
18-1
27-5
34-6
40-7
45-8
509
55-2
59-4
63-2
67'0
49
30
8-5
17-8
312
499
73-7
103-7
1396
1827
754
70
19-6
30-0
37-5
44-1
49-6
55-1
59-7
64-4
68-4
72-5
53
32
9-2
192
338
54-1
79-8
112-3
1512
197-7
88-
60
21-5
32-6
41-1
48-3
54-4
60-4
65-5
70-6
75-1
79-5
59
36
101
211
371
59-3
87-5
123-2
165-8
2169
105-6
50
24-0
36-4
45-8
53-9
60-7
67-4
73-1
78-7
83-7
88-7
65
4-0
112
235
413
662
97-6
137-4
1849
242-0
132-
40
27-4
41-6
52-5
61-7
69-4
77-1
83-6
90-1
95-8
101-5
75
4-5
128
26-9
47-3
75-7
111-7
157-2
211-6
2768
176
30
32-6
49-5
62-5
73-4
82-6
91-7
99-5
107-2
1140
120-8
89
5-4
153
320
563
90-0
1329
1871
251-8
3294
212-2
25
36-4
55-3
69-8
8-2-0
92-2
102-4
lii-l
119-7
127-3
134-9
99
6-0
171
35-8
62-9
ICO 6
148-4
2089
2812
367-9
264-1
20
41-7
63-3
79-9
93-8
105-6
117-3
127-2
137-0
145-7
154-4
114
69
196
409
720
1151
1699
2392
3219
421-2
352- 15
49-6
75-3
95-0
111-7
125-6
1396
151-3
163-1
173-4
183-8
1-35
8-2
23-3
48-3
856
1370
202-2
284-6
383-1 1501-2
528
10
63-3
96-0
1212
142-4
160-2
178-0
192-9
207-9
221-1 1234-3
173
105
29-7
621
109-2
174-7
257-8
3629
4885 6390
TABLE for finding, very nearly, the velocity and discharge from Cylindrical Water Pipes or Sewers, when the
diameter and fall are given. Any two of the four quantities, the velocity, discharge, diameter, or inclination
being given, the others can fee found in THE TABLE from inspection.
il
Mean hy- 1
draulic incli-l
nation of 1
pipe or sewer|
The VELOCITT IN INCHES PER SECOND is given in the first horizontal line for each inclination
or fall ; and the DISCHARGE IN COBIC FEET PER MINUTE in the next following one.
12 in. 1 14 in.
diameter) diameter
16 in.
diameter
18 in.
diameter
20 in. i 22 in.
diameter diameter
24 in. 26 in.
diameter| diameter
28 in.
diameter
30 in.
diameter
One in
i
5280
6-8
7-4
8-0
8-5
8-9
9-4
9-8
10-3
10-7
ll'l
26-9
40-
56-
75-
98-
124-
155-
189-
228-
271-
2
2640
10-1
10-9
11-7
12-5
13-2
13-9
14-5
15
1
15-7
16-3
40
58
82
110
144
183
228
278
336
400
3
1760
13
14
15
16
17
17
18
19
20
20
50
73
103
138
180
229
285
349
421
500
4
1320
15
16
17
18
19
20
21
22
23
24
58
86
120
163
212
269
335
410
494
588
5
1056
17
18
20
21
22
23
24
25
26
27
66
97
136
184
240
305
380
464
560
665
6
880
19
20
22
23
24
26
27
28
29
30
73
108
151
203
265
337
420
514
620
737
7
754
20
2-2
24
25
27
28
29
30
32
33
80
118
165
222
289
368
458
560
676
804
8
660
22
24
25
27
29
30
31
33
34
35
86
127
177
239
312
397
493
604
728
866
9
587
23
25
28
29
30
32
33
35
36
38
92
135
192
255
333
424
527
645
778
925
10
528
25
27
29
31
32
34
36
37
39
40
97
144
201
271
354
450
560
685
826
982
11
480
26
28
30
32
34
36
38
39
41
42
103
152
213
286
373
475
590
723
871
1036
12
440
27
30
32
34
36
38
39
41
43
44
108
159
223
300
392
498
620
759
916
1089
13-2
400
29
31
34
36
38
40
42
43
45
47
114
168
236
317
414
526
655
801
9i
36
1149
15-1
350
31
34
36
39
41
43
45
47
49
50
123
181
254
342
446
568
707
865
1043
1240
17-6
300
34
37
40
42
45
47
49
51
53
55
134
198
278
374
487
620
772
944
1139
1354
21-1
250
38
41
44
47
50
52
55
57
59
61
149
219
308
415
541
688
857
1048
1264
1504
26-4
200
43
47
50
53
56
59
62
65
67
70
169
249
350
472
616
783
975
1192
1438
1710
35-2
150
51
55
59
63
67
70
73
76
79
82
200
294
414
558
728
925
1152
1409
1700 2021
52-8
100
65
70
75
80
85
89
93
97
101
105
254
374
526
708
923
1174
1462
1788
2157 2565
58*7
90
69
74
80
85
90
95
99
Sop
Velocit
ies in inches, oer
270
398
559
753
982
1250
1556
c >t>
second, due to heads.
<D bo
66
80
74
80
86
91
96
101
106
-gg
'-*-*
290
427
600
808
1054
1340
1669
f~s
SEi
W 3 C
||
jllsj
75-4
70
80
86
93
99
105
110
115
rii^
?
5**
314
463
650
875
1141
1451
1807
i
27-8
22-7
20
88
60
88
95
102
109
115
120
126
2
39
32
28
344
506
712
959
1251
1591
1981
3
48
39
34
105-6
50
98
106
114
121
128
134
141
4
56
45
40
384
565
795
1070
1396
1775
2211
6
68
56
49
132
40
119
124
130
138
146
153
161
8
79
64
56
467
665
908
1224
1597
2031
2529
10
88
72
63
176
30
133
144
155
165
174
183
192
12
96
78
69
523
770
1082
1456
1900
2417
3010
24
136
111
97
212-2
25
149
161
173
184
195
204
214
36
167
136
119
584
860
1208
1627
2122
2699
3362
48
193
157
138
264-1
20
170
184
198
211
223
234
245
60
215
176
154
668
984
1383
1862
2430
3090
3848
72
236
192
169
352
15
202
219
236
251
265
278
292
84
255
208
182
795
1171
1646
2216
2891
3677
4579
96
272
222
194
528
10
258
279
301
320
338
355
372
108
289
235
207
1014
1496 2098 2826
3686 4688
5839
120
305
248
218
44 THE DISCHARGE OF WATER FROM
SECTION II.
FORMULA FOR THE VELOCITY, AND DISCHARGE, PROM
ORIFICES, WEIRS, AND NOTCHES. COEFFICIENTS OF
VELOCITY, CONTRACTION, AND DISCHARGE. PRACTICAL
REMARKS ON THE USE OF THE FORMULAE.
The quantity of water discharged in a given time
through an aperture of a given area in the side or
bottom of a vessel, is modified by different circum-
stances, and varies more or less with the form, posi-
tion, and depth of the orifice ; but the discharge may
be easily found, when we have determined the velocity
and the contraction of the fluid vein.
VELOCITY.
If g be the velocity acquired by a heavy body
falling from a state of rest for one second, in vacuo,
then it has been shown by writers on mechanics, that
the velocity v per second acquired by falling from a
height k, will be
(1.) v \/ 2gh.
The numerical value of g varies with the latitude ; we
shall assume 2 g = 772-84 inches zz 64*403 feet.
These will give for measures in inches,
v = 27-8 v/V and h = jf ~ -001293t> 2 ,
'
and for measures in feet,
* The velocities for different heights are given in the column
number 1, TABLE II.
ORIFICES, WEIES, PIPES, AND RIVERS.
45
v = 8-025 v/ h, and h =
64-403
If v be in feet, and h in inches, then
v = 2-317
h =
= -01553 v*.
= -1864 v\*
COEFFICIENT OF VELOCITY.
Let the vessel A B c D, Fig. 1, be filled with water
to the level E F : then it has been found, by experi-
ment, that the velocity of discharge through a small
orifice o, in a thin plate, at the distance of half the
diameter outside it, in the vena-contracta, will be very
nearly that due to a heavy body falling freely from
the height A, of the surface of the water E F, above
the centre of the orifice. The velocity of discharge
* The force of gravity increases with the latitude, and decreases
with the altitude above the level of the sea, but not to any con-
siderable extent. If A be the latitude, and h the altitude, in feet,
above the mean sea level, then we may, generally, take
g = 32-17 (1 -0029 cos 2A) x (I ),
in which E, the radius of the earth at the given latitude is equal
to
20887600 ( 1 + -0016 cos 2X).
46 THE DISCHARGE OF WATER FROM
determined by the equation v v/2 g h, for falling
bodies, is, therefore, called the " theoretical velocity''
If we now put v d for the actual mean velocity of dis-
charge in the vena-contracta, and c v for its ratio to
the theoretical velocity v, we shall get v d zz c v v ; and
by substituting for v, its value
(2.) v,
c v is termed "the coefficient of velocity ;" its numerical
value, at about half the diameter from the orifice, is
about *974 ; and, consequently,
t? d = -974 v/2 g h.
This for measures in inches becomes
v d = 27-077
and for measures in feet
v* = 7-816
The orifice o, is termed an horizontal orifice in Fig.
1, and in Pig 2 a vertical or lateral orifice. When
* The velocities for different heights calculated from this
formula, are given in the column numbered 2, TABLE II. It has
been latterly asserted in a Blue Book that theoretically v d =
f V 2 g h. It is not necessary here to combat this error, which
confounds the discharge with its velocity, and a single practical
fact, applicable only to a thin plate, with a theoretical principle.
The experimental^ discharge approximates to f */ % g h multiplied
by the area of the orifice ; but the theoretical velocity \/ 2 g h
always approximates to the experimental velocity, or *974 */ 2 gh,
obtained immediately outside the orifice in the vena-contracta. It
would be unnecessary to allude to this theory here if it were not
supported and put forward by three eminent engineers whose
authority may mislead others. Vide p. 4. Brief observations of
Messrs. Bidder, Hawksley, and Bazalgette on the answers of the
Government Referees on the METROPOLITAN MAIN DEAINAGE, or-
dered by the House of Commons to be printed 13th July, 1858.
OEIFICES, WEIRS, PIPES, AND RIVERS. 47
small, each is found to have practically the same
velocity of discharge, when the centres of the con-
tracted sections are at the same depth, h, below the
surface ; but when lateral orifices are large, or rather
deep, the velocity at the centre is not, even prac-
tically, the mean velocity ; and in thick plates and
modified forms of adjutage, the mean velocities are
found to vary.
VENA-CONTRACTA AND CONTRACTION.
It has been found that the diameter of a column
issuing from a circular orifice in a thin plate, is con-
tracted to very nearly eight-tenths of the whole
diameter at the distance of the radius from it, and
that at this distance the contraction is greatest. The
ratio of the diameter of the orifice to that of the
contracted vein, vena-contracta, is not always found
constant by the same or different experimentalists.
Newton makes it 1 : -841, J md ' Before, that of the j. 707
I areas as 1 : )
(7156
(622
1 : -6432
1 : -64
656
Poleni
1 :
f -846
\ -788
Borda
1 :
802
Michellotti
1 :
8
Bossut
1 :
f -81
{818
Du Buat
1 :
816
Venturi
1 :
798
Eytelwein
1 :
8
Bayer
1 :
7854
,,1: '667
1: '637
1: -64
1 : '617
Bayer's value for the contraction has been deter-
mined on the hypothesis, that the velocities of the
particles of water as they approach the orifice from
all sides, are inversely as the squares of their
48
THE DISCHARGE OF WATER FROM
distances from its centre ; and the calculations made
of the discharge from circular, square, and rectan-
gular orifices, on this hypothesis, coincide pretty
closely with experiments.
FORM OF THE CONTRACTED VEIN.
Let o R zz d, Fig. 3, be the diameter of an orifice ;
then at the distance R s zz -^ the contraction is found
to be greatest ; we shall assume the contracted di-
ameter o r zz '7854 d. If we suppose the fluid
column between o R and o r to be so reduced, that
the curve lines R r and o o shall become arcs of circles,
then it is easy to show
from the properties of
the circle, that the ra-
dius c r must be equal
to 1-22^. The mean
velocity in the orifice,
OR, is to that in the
vena-contracta, o r, as *617 : 1 ; and the mouth piece,
R r o o, Fig. 4, in which o p zz \ o R, and or zz -7854 x
o R, will give for the velocity of discharge at o r, the
vena-contracta,
v, zz -974
c--
Fig. 4
zz 7-816
in feet very nearly. In
speaking of the velo-
city of discharge from
orifices in thin plates,
we always assume it
to be the velocity in
the vena-contracta, and not that in the orifice itself,
ORIFICES, WEIRS, PIPES, AND RIVERS. 49
which varies with the coefficient of discharge, unless
in TABLE II., where the mean velocity in the latter,
as representing c d v/ 2 g h, is also given.
COEFFICIENTS OF CONTRACTION AND DISCHARGE.
If we put A for the area of the orifice o R, Pig. 3,
and c c x A for that of the contracted section at o r,
then c c is called the "coefficient of contraction." The
velocity of discharge v^ is equal to c v \/2 gh, equa-
tion (2). If we multiply this by the area of the
contracted section c c x A, we shall get for the dis-
charge
D =i c v x c c x A N/ 2 g h*
It is evident A \/ 2 g h would be the discharge if
there were no contraction and no change of velocity
due to the height h; c v x c c is therefore equal to the
coefficient of discharge. If we call the latter c d , we
shall have the equation
(3.) Ct = c v xc c ,
and hence we perceive that the "coefficient of discharge"
is equal to the product of the coefficients of velocity
and contraction. In the foregoing expression for the
discharge D, h must be so taken, that the velocity at
that depth shall be the mean velocity in the orifice A.
In full prismatic tubes the coefficients of velocity and
discharge are equal to each other.
* The expression c v c c */2gh=:c d */2gh is the coefficient
of the area A, and, consequently, represents the mean velocity in
the orifice ; the coefficient of which is, therefore, equal to c d .
The values of the velocity c d \/2 g h t for different heights and
coefficients, are given in TABLE II.
50 THE DISCHARGE OF WATER FROM
MEAN AND CENTRAL VELOCITY.
In order to find the mean velocity of discharge
from an orifice, it is, in the first instance, necessary
to determine the velocity due to each point in its
surface, and the discharge itself; after which, the
mean velocity is found by simply dividing the area
of the orifice into the discharge. The velocity due
to the height of water at the centre of a circular,
square, or rectangular orifice, is not strictly the
mean velocity, nor is the latter in these, or other
figures, that at the centre of gravity. When, how-
ever, an orifice is small in proportion to its depth in
the water, the velocity of efflux determined for the
centre approaches very closely to the mean velocity ;
and, indeed, at depths exceeding four times the depth
of the orifice, the error in assuming the mean velocity
to be that at the centre of the orifice is so small as to
be of little or no practical consequence, and for
lesser depths it never exceeds 6 per cent. It is, there-
fore, for greater simplicity, the practice to determine
the velocity from the depth h of the centre of the
orifice, unless in weirs or notches ; and the coeffi-
cients of discharge and velocity in the following
pages have been calculated from experiments on this
assumption, unless it shall be otherwise stated.
DISCHARGES THROUGH ORIFICES OF DIFFERENT FORMS IN
THIN PLATES.
The orifices which we have to deal with in practice
are square, rectangular, or circular; and sometimes,
perhaps, triangular or quadrangular in form. It will
ORIFICES, WEIKS, PIPES, AND RIVERS.
51
be necessary to give here only the theoretical ex-
pressions for the discharge and velocity for each kind
of form, but as the demonstrations are unsuited to
our present purposes we shall omit them.
TRAPEZOIDAL ORIFICES WITH TWO HORIZONTAL SIDES.
Put d for the vertical depth of an orifice, h t for the
altitude of pressure at top, above the upper side, and
h b for the altitude at bottom, above the lower side,
we then get
^b ^t = d.
Let us also represent the top or upper side of the
orifice A or c, Fig. 5, by / t , and the lower or bottom
lt + /b
side by / b , and put
= /.
Now, when l t = / b , the trapezoid becomes a paral-
lelogram whose length is I and depth d ; and putting
h for the depth to the centre of gravity, we get the
equation
*t + 2 = ^ 2 = *
The general expression for the discharge, D, through
a trapezoidal orifice, A, is then
B3
52 THE DISCHARGE OF WATER FROM
(4.) D^v2 X
in which c d is the coefficient of discharge ; and when
the smaller side is uppermost as at c,
(5.) D=
PARALLELOGRAMIC AND RECTANGULAR ORIFICES.
When / t z= 4 = I, the orifice becomes a parallelo-
gram, or a rectangle, B, and we have for the discharge
(6.) D=
NOTCHES.
When the upper sides of the orifices A, B, and c,
rise to the surface as at A O , B O , and c , h t becomes
nothing, and we get, as h b = d, for the trapezoidal
notch A with the larger side up,
(7.) D = c d N/2^x 3"" 2
for the trapezoidal notch, c , with the smaller side up,
(8.) D = C
the same in form, but not in value, as the preceding
equation ; and for a parallelogramic or rectangular
notch B ,
ORIFICES, WEIRS, PIPES, AND RIVEES.
53
(9.) D = c, v/^7 X -Id = - cj N/ 2 ff .
o o
It is easy to perceive that the forms of equations (4)
and (5), and also of equations (7) and (8), are iden-
tical. The values for the discharge in equations (6)
and (9) are equally applicable, whether the form of
the orifice be a parallelogram or a rectangle, the only
difference being in the value of the coefficient of
discharge, c d , which becomes slightly modified for
each form of orifice.
TRIANGULAR ORIFICES WITH HORIZONTAL BASES, AND
RECTILINEAL ORIFICES IN GENERAL.
When the length of the lower side, 4 = 0, the
orifice becomes a triangle, D, Fig. 6, with the base
upwards.
Fig. 6
In this case, equation (4) becomes
(10.) ..
which gives the discharge through the triangular
orifice, D.
When / t = 0, in equation (5), the orifice becomes a
triangle, F, with the base downwards ; in this case,
we find for the value of the discharge,
54 THE DISCHARGE OF WATER FROM
(11.) r, = c dN /2^xf />(*!-! X**-**
As any triangular orifice whatever can be divided
into two others by a line of division through one of
the angles parallel to the horizon ; and as the dis-
charge from the triangular orifice D or F is the same
as for any other on the same base and between the
same parallels, we can easily find, by such a divi-
sion, the discharge from any triangle not having one
side parallel to the horizon, and thence the discharge
from any rectilineal figure whatever by dividing it
into triangles.
If the triangle F be raised so that the base shall be
on the same level with the upper side of the trian-
gular orifice D ; if, also, the bases be equal, and also
the depths, we shall find, by adding equations (10)
and (11), and making the necessary changes indicated
by the diagram,
(12.) D = c d s2 X {+- 2 X
for the discharge from a parallelogram E with one
diagonal horizontal. Now this is the same as the
discharge from any quadrilateral figure whatever,
having the same horizontal diagonal, and also having
the upper and lower angles on the same parallels, or at
the same depths, as those of the parallelogram. If
the orifices D, F, and E rise to the surface of the
water, as at D O , E O , F O , we shall then have for the dis-
charge from the notch D O ,
which for a right angled triangle becomes
ORIFICES, WEIRS, PIPES, AND RIVERS. 55
: D = C d N X <.
For the discharge from the notch F O ,
(14.) D = c dV /2^ x ^^:
and for the discharge through the notch E O ,
(15.) D=c d v/2^x ] ^{4~2^} = c d v/2^x -9752/^1
When the parallelogram E O becomes a square / = 2 d,
and hence,
(16.) D=c dV / 2^X-9752/*x v/J = c d v/2^X'34478 l\
The foregoing equations will enable us to find an
expression for the discharge from any rectilineal
orifice whatever, as it can be divided into triangles,
the discharge from each of which can be determined
as already shown in the remark following equation
(11.) The examples which we have given will be
found to comprehend every form of rectilineal orifice
which occurs in practice ; but for the greater number
of orifices, sunk to any depth below the surface, the
* In the Civil Engineer and Architect's Journal, 1858, p. 370, it
is stated that Professor Thompson, Belfast College, gave at the
British Association in Leeds for a right angled triangle, for
discharges of from 2 to 10 cubic feet per minute, the expression
Q = '317 H"ST, in which Q is the quantity in cuhic feet per minute,
and H the head in inches. Now the ahove equation for a coeffi-
cient of -617 becomes, for inch measures, D = 17-153 x T % d 2 =
_5
9-15 d2; or by multiplying by 60, and dividing by 1728, to
reduce the discharge to feet per minute, we get D = -317 d^,
identically the same as Professor Thompson derived from his
experiments. All sections of a triangular notch are similar tri-
angles, and hence the advantage of a triangular-notch-gauge,
where it can be used, as, probably, the coefficient remains constant.
Professor Thompson, I believe, first drew attention to this.
56
THE DISCHAEGE OF WATEE FROM
discharge will be found with sufficient accuracy by
multiplying the area by the velocity due to the
centre.
CIRCULAR AND SEMICIRCULAR ORIFICES.
The discharge through circular and semicircular
orifices in thin plates can only be represented by
means of infinite series. Let us represent by s 1 the
sum of the series
Fig.7
-\ \x / v Y \ f 1 \
= H
e^^-gjF^=^
m
_f\ 1357 9\/l 1 3 5\/^
^2 4 6 o 10 12 2 4 6 o /
Let us also represent by s 2 the sum of the series
_M(i.IV . (I.I.*
3*1416( ^2 * 3^ A ^~ V2 ' 4 * 6
2*4*6*8 1(K ^3 5
then the discharge from the semicircle G, Fig. 7, with
the diameter upwards and horizontal, is
(17.) D = c d \/%g~h x 3-1416 r 8 (^ + s 2 ).
And the discharge from the semicircle i, with the
diameter downwards and horizontal, is
(18.) D = c d \/2gh x 3-1416 7-2 (s 1 .s 2 ).
If we put A for the area, we shall also have for the
discharge from a circle H,
OKIFICES, WEIES, PIPES, AND EIVEKS. 57
(19.) v = c (i \2g/i X
In each of these three equations (17), (18), and (19),
h is the depth of the centre of the circumference
below the surface, and r the radius.
When the orifices rise to the surface, we have for
the discharge from a semicircular notch G O , with the
diameter horizontal and at the surface,
(20.) D = c d v2#^ X '9586 r 2 = c d v2#r X '6103 A ;
when the circumference of the semicircle is at the
surface, and the diameter horizontal, as at I ,
when the horizontal diameter of the semicircle is
uppermost, and at the depth r below the surface,
(22.) D = c d v/27rx 1-8667 r^^v/^^x 1-1884 A;
and when the circumference of the entire circle is at
the surface, as at H O ,
(23.) D = c d v/2<7>x 3-0171 r 2 = c d \/2^r x '9604A.
If we desire to reduce equations (20), (21), and
(22), to others in which the depth h of the centre
of gravity from the surface is contained, we have only
to substitute ^ for r in equation (20), and we
shall get, for the discharge from a semicircle with the
diameter at the surface,
(24.) D = c d \/2gh X '0367 A :
also, by substituting 7/yrinr for r in equation (21), we
get, for the discharge from a semicircle when the
circumference is at the surface and the diameter
horizontal,
(25.) D = c d V Zgh X -9653 A ;
58 THE DISCHARGE OF WATER FROM
and when the horizontal diameter is uppermost, and
at the depth r below the surface r = i .4044 an ^
(26.) D = c d v/2#A x -9957 A.
As A stands for the area of the particular orifice
in each of the preceding expressions for the dis-
charge, it must be taken of double the value, in
equation (23) for instance where it stands for the
area of a circle, that it is in equations (20), (21), or
(23), where it represents only the area of a semicircle.
MEAN VELOCITY.
The mean velocity is easily found by dividing the
area into the discharge per second given in the pre-
ceding equations. For instance, the mean velocity
in the example represented in equation (9), is equal
, which is had by dividing the area Id
into the discharge ; and in like manner the mean
velocity in equation (23) is -9604 c d \/ 2g r.
PRACTICAL REMARKS ON THE DISCHARGE FROM CIRCULAR
ORIFICES.
It has been shown, equation (19), that, for the
discharge from a circle, we have
x 2 A*!
in which h is the depth of the centre, A the area,
and s l the sum of the series
I 1 -/' 1 l \f l -r 1 l 3 5 \( l l 3 V-
II Vl'TAi'lJF UTi'lJUTOF
ORIFICES, WEIRS, PIPES, AND RIVERS.
59
and it has also been shown, equation (23), that, when
the circumference touches the surface, this value
becomes
D = c d v/2^r x '9604 A.
Now when h is very large compared with r, it is easy
to perceive that 2 ^ = 1, and hence
(27.) D = c d \/%gh x A.
As this is the formula commonly used for finding the
discharge, it is clear, if the coefficient c d remain con-
stant, that the result obtained from it for D would be
too large. The differences, however, for depths
greater than three times the diameter, or 6 r, are
practically of no importance ; for, by calculating the
values of the discharge at different depths, we shall
find, when
h = r, that D = c d
(28.)
;?
3 r
2'
7 r
X '960 A ;
X '978 A;
X -985 A;
X '989 A;
X '992 A ;
X '996 A;
X -998 A;
X '9987 A ;
X '9991 A.
These results show very clearly that, for circular
orifices, the common expression for the discharge
c d N/2^^ x A is abundantly correct for all depths
h
h
h
=
3r,
4r,
5r,
)>
D
D
D
= Cd
= C d
= C d
60 THE DISCHARGE OF WATER FROM
exceeding three times the diameter, and that for
lesser depths the extreme error cannot exceed four
per cent, in reduction of the quantity found by this
formula. We shall show, hereafter, when discussing
the value of c d , that from the sinking of the surface,
and perhaps other causes, the discharge at lesser
depths is even larger than that exhibited by the
expression c d \/2 g h, x A, the value of the coefficient
of discharge, c d , being found to increase as the depths
h decrease. In fact, the sides of the orifice, the
rounding of the arrises, and the depth and position
with reference to the sides of the vessel, and surface
of the water, are of far greater practical importance
than extreme accuracy in the mathematical formula.
PRACTICAL REMARKS ON THE DISCHARGE FROM RECTANGULAR
ORIFICES.
It has been shown, equation (6), that the discharge
from rectangular orifices, with two sides parallel to
the horizon or surface of the water, is expressed by
the equation
D = c d x \ v/2~> x I (h\ - hj },
o
in which / is the horizontal length of the orifice, A b
the depth of water on the lower, and h t the depth on
the upper, side. As it is desirable in practice to
change this form into a more simple one, in which
the height h of the centre and depth d of the orifice
only shall be included, we then have h b == h + and
ORIFICES, WEIRS, PIPES, AND RIVERS. 61
h t = h --- By substituting these values of A b and
2
k t in the foregoing equations, and developing the
result into a series, the terms of which, after the
third, may be neglected, and putting A. for the area
I d, we shall find,
_ ( d* ^
(29.) D = c d v/2 g h x A|l ^g-^ [very nearly.
We have therefore for the accurate theoretical dis-
charge
(30.) D = d --
for the approximate discharge
D = c v2 X A
,
jl
and for the discharge by the common formula
D = c d v2 X A.
When the head (h) is large compared with (d) the
height of the orifice, each of the three last equations
gives the same value for the discharge ; but as the
common expression c d \/2 g h X A is the most simple ;
and as the greatest possible error in using it for lesser
depths does not exceed six per cent., viz. when the
orifice rises to the surface and becomes a notch, it is
evidently that formula best suited for practical pur-
poses. The following table and equations will show
more clearly the differences in the results as obtained
from the true, the approximate, and the common for-
mulce, applied to " lesser " heads ; and they will also
explain, to some extent, why " coefficients " deter-
mined from the common formula, and that used by
Poncelet and Lesbros, should decrease as the orifice
approaches the surface.
62
THE DISCHARGE OF WATER FROM
(31.)
d
2>
5d
D=<V
x '9693 A. X '9733 A.
*=
36?
?9 99
99 99
99 99
99 99
99 99
99 99
9? 99
X-9796A.
X-9854A.
X-9890A.
x '9953 A.
X-9974A.
x '9815 A.
x '9983 A.
X-9988A.
x -9864 A.
X-9896A.
x '9954 A.
x '9974 A.
x '9983 A-
X-9988A.
99 99
X'999lA.
X '9994 A.
X -9991 A.
,, x *yyy'A. ,, ,,
x '9999 A. X '9999 A.
In the foregoing Table the first column contains
the head at the centre of the orifice expressed in
parts of its height d; the second contains the values
of the discharges according to equation (30) ; and
the third column contains 'the approximate values
determined from equation (29), the results in which
are something larger than those in column 2, derived
from the correct formula. The numerical coefficients
of A, at every depth, are less in both than one, the
constant coefficient according to the common formula.
The latter, therefore (as in circular orifices), gives re-
sults exceeding the true ones, but the excess is in-
OKIFICES, WEIES, PIPES, AND EI\ T EKS. 63
appreciable at greater depths than h =z 3d, and for
lesser depths than this the error cannot exceed six
per cent. It may be useful to remark here, that
when the orifice rises to the surface and becomes
a notch, the "centre of mean velocity" is at four-ninths
of the depth, and the centre of gravity at two-thirds
of the depth from the surface. The former fraction
is the square of the latter.
SECTION III.
EXPERIMENTAL RESULTS AND FORMULAS. COEFFICIENTS
OF DISCHARGE.
We have heretofore dwelt but very partially on the
numerical values of the general coefficient of discharge
c d . In order to determine its value under different
circumstances more particularly, it will be now ne-
cessary to consider some of the experiments which
have been made from time to time. These do not
always give the same results, even when conducted
under the same circumstances and by the same par-
ties, and there appears to exist a certain amount of
error, more or less, inseparable from the subject. The
experiments with orifices in thin plates afford the
most consistent results ; but even here the differences
are sometimes greater than might be expected. In
many of the earlier experiments the value of the co-
efficient c d comes out too large, which arises, very
probably, from the orifices experimented with not
being in thin plates, and partaking, more or less, of
the nature of short tubes or mouth-pieces with rounded
64
THE DISCHARGE OF WATER FROM
arrises, which, we shall see, give larger coefficients than
simple orifices. When an orifice is in the bottom of
a vessel, it would appear more correct to measure the
head from the surface to the vena-contracta than from
the surface to the orifice itself ; and as any error in
measuring the head in any experiment must affect the
value of the coefficient derived from such experiment,
so as to increase it when the error is to make the head
less, and vice versa, it appears that heads measured to
an orifice in the bottom of a vessel, and not to the
vena-contracta, must give larger coefficients from the
experimental results than, perhaps, the true ones.
The coefficients in the following pages have been al-
most all arranged and calculated, by the writer, from
the original experiments.
In 1739 Dr. Bryan Robinson made some experi-
ments on the discharge through small circular ori-
fices, from one-tenth to eight-tenths of an inch in
diameter, with heads of two and four feet*, which
give the following coefficients.
COEFFICIENTS FKOM DR. B. ROBINSON'S EXPERIMENTS.
Heads.
10 inch
diameter.
^jinch
diameter.
30 inch
diameter.
To inch
diameter.
2 feet head
768
767
761
728
4 feet head
768
774
765
742
These results are pretty uniform, and the values
from which they are derived are said to be " means
taken from five or six experiments ;" as values of c d
they are, however, too high. The apparatus made
use of is not described ; but it is probable, from the
* Helsham's Lectures, p. 390. Dublin, 1739.
ORIFICES, WEIRS, PIPES, AND RIVERS. 65
results, that the plate containing the hole or orifice
was of some thickness, and that the inner arris was
slightly rounded. There is here, however, a very
perceptible increase in the coefficients for the smaller
orifices, but none for the smaller depth.
In a paper in the Transactions of the Royal Irish
Academy, vol. ii. p. 81, read March 1st, 1788, Dr.
Mathew Young determines the value of the coefficient
for an orifice ^ inch in diameter, with a mean head of
14 inches, to be *623. The manner in which this
value is determined is very elegant, viz. by comparing
the observed with the theoretical time of the water,
in the vessel^ sinking from 16 inches to 12 inches.
The following experiments by Michelotti, with
circular orifices from 1 to about 3 inches diameter,
and with from 6 to 23 feet heads, give for the mean
value c d 613 ; and for square orifices of from 1 to
9 square inches in area, at like depths, the mean
value of c d zz-628. The experiments are given in
French feet and inches, according to which standard
we have, in feet, D:z:7-77 A \/hx t, t being the time
in seconds/ As the time of discharge in these ex-
* The value of \/ % g h, equation (1), for measures in French
feet, is 7-77 ^/ h, and for measures in French inches, 36-9 \/ h, g
being equal to 30-2 feet, or 362-4 inches, French measure. One
French foot is equal to 1-06578 English feet, and the inches
preserve the same proportion. The resulting coefficients must be
the same, whatever standards we make our calculations from.
Many of the most valuable formulae and experiments in hydraulics
are given in French measures of the old style. As our object,
however, in the present section, is simply to determine from ex-
periment the relation of the experimental to the theoretical dis-
charge, it is not necessary to reduce the experiments to other
measures than those in the original ; but the value of the force of
P
66 THE DISCHAKGE OF WATER FltOM
periments varies from ten minutes to an hour, and as
the depths are considerable, the results must be
looked upon as pretty accurate ; and it is worthy
of note that here the coefficients are larger for
square than for circular orifices.
We may remark here in passing how universal the
coefficients -613 to *6'28 are for all forms of orifices
in thin plates, or with the outside arrises chamfered.
Indeed, we may always use the coefficient '62 with
certainty, for practical purposes, for every orifice of
this kind, whether at the surface in the form of a
notch, or at the sides or bottom of a vessel, if the
section of the approaching water be large in propor-
tion to the area of the discharging orifice or notch.
By coefficient we of course here mean that decimal
which, multiplied by the theoretical value, gives the
practical result ; and this is substantially the same
for notches and orifices sunk below the surface, as
will appear farther on. There appears to us, how-
ever, an utter want of accuracy in using the coeffi-
cient '62 or thereabouts in gauging for all orifices,
weirs included, no matter what the thickness or form
of the orifice or crest of a weir may be, or area
of the approaching channel. These will cause the
coefficient to vary from '5 to 1 or more, and hence
the necessity for endeavouring to reduce this portion
of our subject to rule.
gravity, g, must, of course, be taken in those measures with which
the experiments were made. In the French decimal, or modern
style, the metre is equal to 3-2809 English feet, or 39-371 inches.
The tenth part of a metre is the decimetre, and the tenth part of
the decimetre the centimetre, as the names imply. TABLE
XIV. contains the weights and measures in general use in Great
Britain and France, with their general ratios to each other.
OEIFICES, WEIKS, PIPES, AND RIVERS.
67
COEFFICIENTS FROM MICHELOTTl's EXPERIMENTS.
Description and
size of orifice, in
French inches.
Depth of the
centre of the
orifices in
French feet.
Quantity
discharged
in cubic feet.
Time of
discharge
in seconds.
Theoretical
time,
calculated
from
t= 5_
7-77 A \A.
Resulting
coefficients
of discharge.
Square orifice,
3"X3"
6-613
6-852
11-676
11-818
21.691
21-715
463-604
566-458
516-785
612-118
415-437
499-222
600
720
510
600
300
360
371-3
445-6
311-4
366-6
183-7
220-6
619
619
610
611
612
613
Mean value of the coefficient ; square orifice 3" )
<3"
614
Square orifice,
2"X2"
6-625
11-426
21-442
329-806
423-465
385-333
900
900
600
594-
580-4
385-7
660
646
643
Mean value of the coefficient ; square orifice 2" I
< 2"
649
Square orifice,
l" X l"
6-757
11-889
21-507
158-549
163-792
562-944
1800
1440
3600
1585-
880-6
2249-9
628
612
625
Mean value of the coefficient ; square orifice l" >
<l"
621
Circular orifice,
3" diameter
6-694
11-590
21-611
542-85
570-972
521-299
900
720
480
550-1
439-6
293-8
611
610
612
Mean value of the coefficient ; circular orifice 8" diameter . . -611
Circular orifice,
2" diameter
6-785
11-722
21-903
488-687
589-535
575-486
1800
1680
1200
1108-1
1016-4
725-9
616
605
605
Mean value of the coefficient ; circular orifice 2" diameter . . '609
Circular orifice,
1" diameter
6-875
11-743
22-014
247-354
324-11
444-535
3600
3600
3600
2227-
2233-
2237-2
619
620
621
Mean value of the coefficient ; circular orifice 1" diameter . . -620
F 3
68
THE DISCHARGE OF WATER FROM
The experiments made by the Abbe Bossut, con-
tained in the following table, give the mean value of
c d , for both circular and square orifices, equal to -616
nearly ; and it may be perceived that, for the small
depth in the last experiment, the coefficient rises so
high as -649. These and other experiments led the
COEFFICIENTS FROM BOSSUT'S EXPERIMENTS.
Description, position, and size of
orifice, in French inches.
Depth of
the centre
of the
orifice in
French
inches.
Number
of French
cubical
inches
discharged
per minute.
Theoretical
discharge
per minute,
Eesulting
coefficients.
Horizontal and circular, ~" diameter
140-832
2311
3760-8
614
Horizontal and circular, 1" diameter
140-832
9281
15043-3
617
Horizontal and circular, 2" diameter
140-832
37203
60173-1
618
Horizontal and rectangular, 1" X k"
140-832
2933
4788-4
613
Horizontal and square, 1" X 1"
140-832
11817
19153-7
617
Horizontal and square, 2" X 2"
140-832
47361
76614-6
617
Lateral and circular, J" diameter . .
108-
2018
3293-3
613
Lateral and circular, 1" diameter . .
108-
8135
13173-3
617
Lateral and circular, " diameter . .
48-
1353
2195-5
616
Lateral and circular, 1" diameter . .
48-
6436
8782-2
616
Lateral and circular, 1" diameter . .
0-5833
628
968-
649
Abbe to construct a table of the discharges, at dif-
ferent depths, from a circular orifice 1 inch in dia-
meter, from which we have determined the following
table of coefficients. These increase, as the orifice
COEFFICIENTS DEDUCED FROM BOSSUT'S EXPERIMENTS.
Heads,
in feet.
Coefficients.
Heads,
in feet.
Coefficients.
Heads,
in feet.
Coefficients.
1
621
6
620
11
619
2
621
7
620
12
618
3
621
8
619
13
618
4
620 ,
9
619
14
618
5
620
10
619
15
617
OEIFICES, WEIRS, PIPES, AND EIVERS.
69
approaches the surface, from *617 to *621 ; and at
lesser depths than L foot other experiments show an
increase in the coefficient np to -650. The experi-
ments of Poncelet and Lesbros show, however, a
reduction in the coefficients for square orifices 8"x 8"
as they approach the surface from -601 to *572.
Brindley and Smeaton's experiments, with an orifice
1 inch square placed at different depths, give a mean
COEFFICIENTS CALCULATED FEOM BRINDLEY AND SMEATON'S EXPERIMENTS.
tean -637.
value for c d of -637. The last experiment, with an
orifice only \ inch by \ inch, gives so small a coeffi-
cient as *557 placed at a depth of 6 feet !
For notches 6 inches wide and from 1 to 6J inches
deep, Brindley and Smeaton's experiments give the
mean value of c d 637. The coefficients of discharge
COEFFICIENTS FOR NOTCHES, CALCULATED FROM BRINDLEY AND SMEATON'S
EXPERIMENTS.
1 foot head :
2 feet head :
3 feet head :
4 feet head :
5 feet head :
6 feet head :
orifice 1" x 1"
orifice 1" x 1*
orifice 1" x 1"
orifice 1" x 1"
orifice 1" x 1"
orifice J" x J"
coefficient -639
coefficient *635
coefficient -648
coefficient -632
coefficient -632'
coefficient -557
Ratio of the
length
to the depth.
Size of
notches
in inches.
Coefficients.
Ratio of the
length
to the depth.
Size of
notches
in inches.
Coefficients.
92 to 1
6 X6
633
3-7 to 1
6X If
638
1-07 to 1
6 X 5|
571
4-4 to 1
6X1
654
1-2 to 1
6X5
609
4-8 to 1
exit
681
1-92 to 1
6 X 3*
602
6- tol
6X1
713
2-4 tol
6 X 31*
636
Mean value. '637
* The depth is misprinted 2^ inches in the Encyclopaedias,
the resulting coefficient for which would be *568 instead of '636
as above, for a depth of 2~ inches.
70
THE DISCHAEGE OF WATEK FROM
for notches and orifices appear to differ as little from
each other as those for either do in themselves. The
results also show a general though not uniform in-
crease in the coefficients for smaller depths.
Du Buat's experiments with notches 18 '4 inches
long, give the mean value of c d =z-632, which differs
very little from the mean value determined from
Brindley and Smeaton's experiments.
COEFFICIENTS FOB NOTCHES, CALCULATED FKOM DU BUAT's EXPERIMENTS.
Ratio of the
length
to the depth.
Size of
notches
in inches.
Coefficients.
Ratio of the
length
to the depth.
Size of
notches
in inches.
Coefficients.
2-72 to 1
3-94 to 1
18-4 X 6-753
18-4 X 4-665
630
627
5-75 to 1
10-3 to 1
18-4 X 3.199
18-4 X 1'778
624
648
Poncelet and Lesbros' experiments give the coeffi-
cients in the following table, for notches 8 inches
COEFFICIENTS FOR NOTCHES, BY PONCELET AND LESBROS.
Ratio of the
length
to the depth.
Size of
notches
in inches.
Coefficients.
Ratio of the
length
to the depth.
Size of
notches
in inches.
Coefficients.
9tol
8X9
577
3-33 to 1
8 X2-4
601
1 tol
8X8
585
5 tol
8X1-6
611
1-3 to 1
8X6
590
6-7 tol
8X1-2
618
2 tol
8X4
592
10 tol
8 X 0-8
625
2-5 tol
8 X 3-2
595
20 tol
8X0-4
636
wide; the mean value of all the coefficients in these
experiments is *603. Here the coefficients increase
in every instance as the depths decrease, or as the
ratio of the length of the notch to its depth increases.
We shall have to refer to the valuable experiments
made at Metz, on the discharge from differently-pro-
portioned orifices immediately.
Rennie's experiments for circular orifices at depths
ORIFICES, WEIRS, PIPES, AND RIVERS.
COEFFICIENTS FOR CIRCULAR ORIFICES, FROM RENNIE's EXPERIMEN
Heads at the
centre of each
orifice in feet.
J inch
diameter.
inch
diameter.
finch
diaiheter.
linch
diameter.
Mean
values.
1
671
634
644
633
645
2
653
621
652
619
636
3
660
636
632
628
639
4
662
626
614
584
621
Means
661
629
635
616
635
from 1 foot to 4 feet, and of diameters from i inch
to 1 inch, give the following coefficients. Here the
increase in the coefficients for lesser orifices and at
lesser depths exhibits itself very clearly, notwithstand-
ing a few instances to the contrary. The mean value
of the coefficient c d derived from the whole, is -635.
For small rectilineal orifices the coefficients were as
follows :
COEFFICIENTS FOR RECTANGULAR ORIFICES, FROM RENNIE's EXPERIMENTS.
Heads at the
centre of gravity,
in feet.
Square orifice,
1 inch X 1 inch.
Rectangular
orifice, longer side
horizontal,
2" X 4".
Rectangular
orifice, longer side
horizontal,
li" X |".
a 1!
iflj
Equilateral
triangle
of 1 square inch,
with base up.
1
617
617
663
}>
596
2
635
635
668
M
577
3
606
606
606
572
4
593
593
593
593
593
Means
613
613
632
593
585
The most valuable series of experiments of which
we are possessed are those made at Metz, by Poncelet
and Lesbros. They were made with orifices 8
inches wide, nearly, and of different vertical dimen-
sions placed at various depths down to 10 feet. The
discrepancies as to any general law in the relation of
the different values of the coefficient of discharge c d
72 THE DISCHARGE OF WATER FROM
to the size and depth of the orifice in the preceding
experiments, have been remedied 'to a great extent by
these. They give an increase of the coefficients for
the smaller and very oblong orifices as they approach
the surface, and a decrease under the same circum-
stances in those for the larger square and oblong
orifices. There are a few depths where maximum
and minimum values are obtained : we use the terms
"maximum and minimum values" for those which
are greater in the one case and less in the other than
the coefficients immediately before and after them,
and not as being numerically the greatest or least
values in the column. We have marked with a *, in
the arrangement of the coefficients, TABLE I,, these
maximum and minimum values. The heads given in
this table were measured to the upper side of the
orifices, and by adding half the depth (d) to any
particular head, we shall obtain the head at the
centre.
As a perceptible sinking of the surface takes
place in heads less than from five to three times the
depth of the orifice, the coefficients are arranged in
pairs, the first column containing the coefficients for
heads measured from the still water surface some
distance back from the orifice, and the second ob-
tained when the lesser heads, measured directly at
the orifice, were used. A very considerable increase
in the value of the coefficients for very oblong and
shallow small orifices, may be perceived as they ap-
proach the surface, and the mean value for all
rectilinear orifices at considerable depths, seems to
approach to *605 or *606.
OEIFICES, WEIKS, PIPES, AND RIVERS. 73
We have shown, equation (29), that the discharge is
approximately, in which expression d is the depth of
the orifice, and h the head at its centre. Now it is
to be observed, that it is not the value of c d simply,
which is given in TABLE I., but the value of c d x
2 g h, equation (29).
The coefficients in the table are, therefore, less than
the coefficients of discharge, strictly so called, by a
c d?
quantity equal to Q ^ , g . The value of this expression
is in general very small, and it is easy to perceive
from the first of the expressions in equation (31), p. 62,
that it can never exceed 4-2 per cent., or more correctly
0417 in unity. If we wish to know the discharge
from an orifice 4 inches square = 4" x 4", with its
centre 4 feet below the surface, which is equivalent
to a head of 3 feet 10 inches at the upper side, we
find from the table the value of c d jl ,A =
601 ; hence we shall get
D = -601 x A v/2# h = -601 x - x 8'025 x 2 =
601 x i x 16-05 = - x 9-646 = 1-072
9 9
cubic feet per second. In the absence of any expe-
riments with larger orifices, we must, when they
occur, use the coefficients given in this table ; and,
in order to do so with judgment, it is only necessary
to observe the relations of the sides and heads. For
example, if the size of an orifice be 16"x4", we must
74
THE DISCHARGE OF WATER FROM
seek for the coefficient in that column where the ratio
of sides is as four to one, and if the head at the
upper side be five times the length of the orifice, we
shall find the coefficient *626, which in this case is
the same for depths measured behind, or at the
orifice. For lesser orifices, the results obtained
from the experiments of Michelotti and Bossut,
pages 67 and 68, are most applicable; and also
the coefficients of Rennie, page 71. It is almost
needless to observe, that all these coefficients are
only applicable to orifices in thin plates, or those
having the outside ar-
rises chamfered as
in Fig. 8. Very lit-
tle dependence can be
placed on calculations
of the quantities of
water discharged from
other orifices, unless where the coefficients have been
already obtained by experiment or correct inference
for them. If the inner arris next the water be
rounded, the coefficient will be increased.
NOTCHES AND WEIRS.
We have already given some coefficients, pages 69
and 70, derived from the experiments of Du Buat,
Brindley and Smeaton, and Poncelet and Lesbros, for
finding the discharge over notches in the sides of
large vessels ; and it does not appear that there is
any difference of importance between these and those
for orifices sunk some depth below the surface, when
the proper formula for finding the discharge for each
ORIFICES, WEIRS, PIPES, AND RIVERS. 75
is used. If we compare Poncelet and Lesbros' co-
efficients for notches, page 70, with those for an
orifice at the surface, TABLE L, we perceive little
practical difference in the results, the head being
measured back from the orifice, unless in the very
shallow depths, and where the ratio of the length to
the depth exceeds five to one. The depths being in
these examples less than an inch, it is probable that
the larger coefficients found for the orifice at the
surface, arise from the upper edge attracting the
fluid to it and lessening the effects of vertical con-
traction, as well as from less lateral contraction. In-
deed, the results obtained from experiments with
very shallow weirs, or notches, have not been at
all uniform, and at small depths the discharge must
proportionably be more affected by movements of
the air and external circumstances than when the
depths are considerable. We shall see that in Mr.
Blackwell's experiments the coefficient obtained for
depths of 1 and 2 inches was -676 for a thin plate 3
feet long, while for a thin plate 10 feet long it in-
creased up to -805.
The experiments of Castel, with weirs up to about
30 inches long, and with variable heads of from 1 to
8 inches, lead to the coefficient -497 for notches ex-
tending over one-fourth of the side of a reservoir ;
and to the coefficient -664 when they extend for the
whole width. For lesser widths than one-fourth, the
coefficients decrease down to -584 ; and for those
extending between one-third of, and the whole width,
they increase from -600 to -665 and -680. Bidone
finds c d = -620, and Eytelwein c d = -635. It will be
76 THE DISCHARGE OF WATER FROM
perceived from these and the foregoing results, that
the third place of decimals in the value of c d , and
even sometimes the second, is very uncertain ; that
the coefficient varies with the head and ratio of the
notch to the side in which it is placed ; and we shall
soon show that the form and size of the weir, weir-
basin, and approaches, still further modify its value.
When the sides and edge of a notch increase in
thickness, or are extended into a shoot, the coeffi-
cients are found to reduce very considerably ; and
for small heads, to an extent beyond what the in-
crease of resistance, from friction alone, indicates.
Poncelet and Lesbros found, for orifices, that the
addition of a horizontal shoot, 21 inches long, re-
duced the coefficient from *604 to '601, with a head
of 4 feet ; but for a head of only 4J inches, the
coefficient fell from -572 to -483, the orifice being
8" x 8". For notches 8 inches wide, with a hori-
zontal shoot 9 feet 10 inches long, the coefficient fell
from -582 to -479, for a head of 8 inches ; and from
622 to -340, for a head of only 1 inch. Castel found
also, for a notch 8 inches wide with a shoot 8 inches
long attached and inclined at an angle 4 18', that the
mean coefficient for heads from 2 to 4J inches was
only -527. Little dependence can be placed on ex-
perimental results obtained for shoots which partake
of the nature of short pipes, and should be treated
in like manner to find the discharge.*
We have obtained the following table of coefficients
from some experiments made by Mr. Ballard, on the
river Severn, near Worcester, " with a weir 2 feet
* Trait6 Hydraulique, par D'Aubuisson, pp. 46, 94 et 95.
OKIFICES, WEIRS, PIPES, AND RIVERS.
77
COEFFICIENTS FOR SHORT WEIRS OVER BOARDS.
Heads measured on the crest.
Depths
in inches.
Coefficients.
Depths
in inches.
Coefficients.
Depths
in inches.
Coefficients.
1
762
3
801
5
733
i|
662
8J
765
H
713
ii
673
3*
748
i
735
u
692
31
740
5|
729
2
684
4
759
6
727
|
702
4*
731
7
716
^
756
4i
744
8
726
8*
786
4f
745
Mean
732
long, formed by a board standing perpendicularly
across a trough."* The heads or depths were here
measured on the weir, and hence the coefficients are
larger than those found from heads measured back
to the surface of still water.
Experiments made at Chew-Magna, in Somerset-
shire, by Messrs. Blackwell and Simpson, in 1850f,
give the following coefficients.
COEFFICIENTS DERIVED FROM THE EXPERIMENTS OF BLACKWELL AND SIMPSON.
Heads
in inches.
Coefficients.
Heads
in inches.
Coefficients.
Heads
in inches.
Coefficients.
Ito I
591
44
743
6
749
Ito 11
626
41
760
6 3
748
16
16
16
23 to 21
16
682
4 I
741
6fg to 6i
747
24
665
41
16
750
61 5
16
772
2 32
670
725
7 21
717
35
32
21
665
5
780
8
802
g29
653
51
781
8 to 8 13
737
32
16
16
215
654
5L 3
749
8 16
750
16
32
16
725
itoS
751
9
781
4
745
^
728
Mean
723
* Civil Engineer and Architect's Journal for 1851, p. 647.
f Civil Engineer and Architect's Journal for 1851, pp. 642
and 645.
78
THE DISCHARGE OF WATER FROM
" The overfall bar was a cast-iron plate 2 inches
thick, with a square top." The length of the over-
fall was 10 feet. The heads were measured from
still water at the side of the reservoir, and at
some distance up in it. The area of the reservoir
was 21 statute perches, of an irregular figure, and
nearly 4 feet deep on an average. It was supplied
from an upper reservoir, by a pipe 2 feet in diameter
and of 19 feet fall ; the distance between the supply
and the weir was about 100 feet. The width of the
reservoir as it approached the overfall was about 50
feet, and the plan and section, Fig. 9, of the weir
and overfall in connection with it, will give a fair
idea of the circumstances attending the experiments.
For heads over 5 inches the velocity of approach to
the weir was "perceptible to the eye," though its
amount was not determined. We perceive that the
coefficient (derived from two experiments) for a depth
of 8 inches is *802, while the coefficient (derived
from three experiments) for a depth of 7J inches is
ORIFICES, WEIRS, PIPES, AND RIVERS. 79
717, and for depths from 8 to 8H inches the mean
coefficient is -743 : as all the attendant circumstances
appear the same, these discrepancies and others must
arise from the circumstances of the case : perhaps
the supply, and, consequently, the velocity of ap-
proach, was increased while making one set of ex-
periments, without affecting the still water near the
side where the heads appear to have been taken.
By comparing the results with those obtained by one
of the same experimenters, Mr. Black well, on the
Kennet and Avon Canal, we shall immediately per-
ceive that the velocity of approach, and every
circumstance which tends to alter and modify it,
has a very important effect on the amount of the
discharge, and, consequently, on the coefficient.
The experiments made by Mr. Blackwell, on the
Kennet and Avon Canal, in 1850*, afford very valuable
instruction, as the form and width of the crest were
varied, and brought to agree more closely with actual
weirs in rivers than the thin plates or boards of
earlier experimenters. We have calculated and ar-
ranged the coefficients in the following table from
these experiments. The variations in the values for
different widths of crest, other circumstances being
the same, are very considerable ; and the differences
in the coefficients, at depths of 5 inches and under,
for thin plates and crests 2 inches wide, are greater
than mere friction can account for ; and greater also
than the differences at the same depths between the
coefficients for crests 2 inches thick, and 3 feet long.
* Civil Engineer and Architect's Journal, 1851, p. 642.
80
THE DISCHARGE OF WATER FROM
When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients
are marked with a *. The effect of the converging wing-boards is very strongly marked.
NOTE. Francis' experiments give a coefficient of '565 for a level crest 3 feet wide, and a head slope of 3| to 1, see p. 121.
Crests 3 feet wide.
00
||5
t- O . O . CO b-
CO OS .rH .'I* O 1 ', '. *
in
i-H OS 00 CO 00 CO O
00 fr- * rH rH \ CO OO O
co-^ oo ^i ^ ^p
{
!i
* * # * *
Gx t?* t^ t>- O O b*
CS OS m O OS * 00 CO CO
^ Tj< O ^ ^ ^ -^
ll
l^-COOSO ,rHt .00
CO CO CO O . CO CT 1 ^ * OS
TtlJOOT^ OO * ^
fe
15
CO <M
OCOl>rHCO .COrHCJ
T^H -rH CO CO ^^ * rH OS OS
OOOrflO O^-rH
bo
Si
CO
C7<?C?rHOSCSrHOOOCO
OOOrJ<rHtOOOt~l.-~ * * ',
g
o
a
o
1
1
X
II 1
* O CO rH
It
o o
llliillllil
Planks 2 inches tl
I!
*-***** #
OSr-it^OrHOOOOOOSOSOOS
ocoost^ooocscocooi^
rjiOOOCOCOCOOiOOOO
II
t-OSCOOSOOCO?.^ r
COOCO"+ODCSrHOO! 1
TjlOOOOOCOCOCOCO
Thin plates.
II
0,2
# * *
8 g*
t" O O l>* O5 CO ^
-IIOAJ989I 9qj Ut
'saqout ut spB9H
rHdCO^OCOJ^OOCSOcMT*
rH rH rH
OEIFICES, WEIRS, PIPES, AND RIVERS.
81
The plan and section, Fig. 10, will give a fair idea
of the approach to, and nature of the overfall made
The dotted lines on
Plan show the sub-
merged masonry ap-
pearing at C in Section.
use of in these experiments. The area of the
reservoir was 2 A. In. 3 OP., and the head was mea-
sured from the surface of the still water in it, which
remained unchanged between the beginning and end
of each experiment. The width of the approach A B
from the reservoir was about 32 feet ; the width at
a b about 13 feet, below which the waterway widened
suddenly, and again narrowed to the length of the
overfall. The depth in front of the dam appears to
have been about 3 feet ; the depth on the dam, next
the overfall, about 2 feet ; and the depth on the sunk
masonry in the channel of approach, about 18 inches.
Altogether, the circumstances were such as to in-
crease the amount of resistances between the reservoir,
from which the head was measured, and the overfall,
particularly for the larger heads, and we accordingly
82 THE DISCHARGE OF WATER FROM
see that the coefficients become less for heads over
six inches, with a few exceptions. The measure-
ments of the quantities discharged appear to have
been made very accurately, yet the discharges per
second, with the same head and same length of over-
fall, sometimes vary ; for instance, with the plank
2 inches thick and 10 feet long, the discharge per
second for 4 inches head varies from 6-098 cubic
feet to 6-491 cubic feet, or by about one-sixteenth of
the whole quantity. Most of the results, however,
are means from several experiments. The quantities
discharged varied from one-tenth of a cubic foot to
22 cubic feet per second, and the duration of the
experiments from 24 to 420 seconds. If we compare
the coefficients for a plank 10 feet long and 2 inches
thick in the foregoing table with those for the same
overfall at Chew-Magna, we shall immediately per-
ceive how much the form of the approaches affects
the discharge. Indeed, were the area of the reser-
voir at Chew-Magna even larger than that for the
Kennet and Avon experiments, it would be found,
notwithstanding, that the coefficients in the former
would still continue the larger, though not fully
as large as those found under the particular cir-
cumstances.*
* There is a very important omission in all the preceding
experiments on weirs and notches. In Fig. 10, for instance, it
would have been necessary to obtain the heads at A B and a b in
each experiment, above the crest, and also the head on and a few
feet above the crest itself. These are, perhaps, best calculated by
means of the observed velocity of approach. They would indicate
the resistances at the different passages of approach, and enable us
to calculate the coefficients correctly, and thereby render them
more generally applicable to practical purposes. The coefficients
ORIFICES, WEIRS, PIPES, AND RIVERS.
83
The following table gives the mean results of 88
experiments made by Francis, at the Lower Lock,
Lowell, Massachusetts, in 1852. The duration of
each of these experiments varied from 180 to 822
seconds. The coefficients in column 10 have been
1
2
3
4
5
6
7
8
9
10
1
it
il
>*
gi
fr*
11
iff
.s|
^ 03
multiplier
ormula in
i
PH
. -4j
2 a
o
o -*^* ^
r<
** T 1 S
Q) tM
.2 p<
Si
\
Observed n
over weir i
Observed di
cubical fe
cond.
Observed ve!
proach in
second.
5-^ A
Values of
formula it
| + o
Values of th
c in the
columnS.
Correspondi
the coefflc
charrge c<
1
9-997
1-55
62-6
78
1-56
1-56
62-6
3-32
621
2
9-997
1-24
45-6
59
1-25
1-25
45-4
3-33
623
3
9-997
1-00
33-4
44
1-00
1-00
32-5
8-32
621
4
7-997
1-01
26-8
36
1-02
1-02
26-3
3-36
628
5
9-997
1-05
36-
97
1-06
1-06
35-8
3-35
626
6
9-995
0-98
32-6
54
0-99
.98
32-4
3-34
624
7
9-995
1-00
33-5
55
1-01
1-00
33-3
3-33
623
8
9-997
0-80
23-5
33
80
80
23-4
3-32
621
9
9-997
0-82
25-
75
83
83
24-8
3-34
624
10
9-995
0-80
23-9
40
80
80
23-8
3-34
624
11
9-997
0-62
16-2
23
62
62
16-0
3-33
623
12
9-997
0-65
17-5
53
65
65
17-2
3-33
623
13
7-997
0-68
14-6
45
68
68
14-5
3-34
623
calculated by ourselves, and the other results con-
densed from the large table given in Francis' Book.*
in the two previous tables are not as valuable as they otherwise
would be from this omission. The level of still water near the
banks is below that of the moving water in the current, therefore,
heads measured from still water must give larger coefficients than
if taken from the centre of the current. This may account, to
some extent, for the larger coefficients in the first table, but apart
from this, the short contracted channel immediately above the
water-fall, Fig. 9, must increase the coefficients.
.* Lowell Hydraulic Experiments. New York, 1855.
G3
84 THE DISCHAEGE OF WATEE FEOM
The heads given in the 6th column are those which
would give the observed discharge from the formula
o
As we have also equation (39)
we must, therefore, have
the values of which are given in column 6. The
values of h" in column 8 are those which would be
found by resolving the equation
D=O(/+ -Ink")*!*
n being the number of end contractions, and c a
multiplier varying from 3*32 to 3-36.
2
In this table the theoretical head ~ zz -0155 vjdue to
*&
the velocity of approach has been used and does not
exceed -02 of a foot. We are of opinion, however,
that the head is much greater, and should be taken
v 2
"2 ^o~~ = '04 vl or thereabouts. This would reduce
c d x &g
the values of the coefficient of discharge c d in the 10th
column. The differences between A, ti, and h" in
columns 3, 6, and 7 are here, practically, of little
moment, and the value of c d in column 10 would be
nearly the same derived from either. The crest of
the weir experimented upon was 1 inch thick. The
weir measuring 10 feet x 13 inches x 1 inch, the top
was rounded off at both arrises, leaving the central
horizontal portion one quarter of an inch wide.
The general result of these experiments verifies the
OEIFICES, WEIKS, PIPES, AND KIVEES.
85
ordinary coefficient for notches in thin plates from
617 to -628 for the value of c d .
Professor Thomson's experiments with right-
angled triangular notches, in thin plates, give a mean
coefficient of -617. Vide Note p. 55.
HEAD, AND FROM WHENCE MEASURED.
By referring to TABLE I., we shall see that there
is a difference in the coefficients as obtained from
heads measured on or above the orifice. This dif-
ference is greater in notches, or weirs, than in orifices
sunk below the surface ; and when the crest of a
weir is of some width, the depths upon it vary. In
the Kennet and Avon experiments, the heads mea-
sured from the surface of the water in the reservoir,
and the depths at the "outer edge" (by which we
understand the lower edge) of the crest were as
follows :
DIFFEEENCE OF HEADS MEASUEED ON AND ABOVE WEIES.
I
Heads on crests
Heads on
II
2 inches thick.
crests 3 feet wide.
a -9
I
I
ao'3
l!
Jfc
!fe
|1
fi
bJDOJ
V
1
eo
1
D
11
w
11-
03 W
ll
||
"S-2.9
1
1
..
7
16
..
i
4
r
16
5
16
2
..
i
1
16
3
. .
ll?
U to 1J
4
3to2g
3J
U
If
u
..
..
li
5
3*
3 r
2 i
14
..
14
6
41
4
2|
..
'i j'*:;;
24
2|
2i
7
..
ft
2
..
8
61
'J ;>".
; : r
31
34
9
grV
..
..
.......
..
4i
N
..
10
4
I
86 THE DISCHARGE OF WATER FROM
No intermediate heads are given, but those registered
point out very clearly the great differences which
often exist between the heads measured on a weir, or
notch, and those measured from the still water above
it ; and how the form of the weir itself, as well as
the nature of the approaches, alters the depth pass-
ing over. On a crest 2 feet wide, with 14| inches
depth on the upper edge, we have found that the depth
on the lower edge is reduced to 11J inches, or as 1*26 to
1. The head taken from 3 to 20 feet above the crest,
where the plane of the approaching water surface
becomes curved, is that in general which is best
suited for finding the discharge by means of the
common coefficients, but a correct section of the
channel and water-line, showing the different depths
upon and for some distance above the crest, is neces-
sary in all experiments for determining accurately
by calculation the value of the coefficient of dis-
charge c d .
Du Buat, finding the theoretical expression for the
discharge through an orifice of half the depth h,
equation (6)
to agree pretty closely with his experiments, seems
to have assumed that the head h is reduced to ^ in
passing over. This is a reduction, however, which
never takes place unless with a wide crest and at its
lower edge, or where the head h is measured at a
ORIFICES, WEIRS, PIPES, AND RIVERS. 87
considerable distance above the weir, and when a
loss of head due to the distance and obstructions in
channel takes place. When there is a clear weir
basin immediately above the weir, we have found
that, putting h for the head measured from the sur-
face in the weir basin, and h w for the depth on the
upper edge of the weir, that
(32.) h A w
for measures in feet, and
(33.) A A w = -
for measures in inches. The comparative values of
h and h w depend, however, a good deal on the par-
ticular circumstances of the case. Dr. Robinson
found* Am 1-111 A w , when h was about 5 inches.
The expressions we have given are founded on the
hypothesis, that h A w is as the velocity of discharge,
or as the \/A nearly. For small depths, there is a
practical difficulty in measuring with sufficient accu-
racy the relative values of h and A w . Unless for
very small heads the sinking will be found in general
to vary from to -, and in practice it will always
be useful to observe the depths on the weir as well
as the heads for some distances (and particularly
where the widths contract) above it.
In order to convey to our readers a more definite
idea of the differences between the coefficients for
heads measured at the weir, or notch, and at some
distance above it, we shall assume the difference of
A w A w .,1 A A w
the heads A A w zz ; then , , =r, and-zz r ?
* Proceedings of the Eoyal Irish Academy, vol. iv. p. 212.
00 THE DISCHARGE OF WATER FROM
hence h = r *~ h w and A ff = ^_ k.
r r + J-
Now the discharge may be considered as that
which would take place through an orifice whose
depth is h w with a head over the upper edge equal to
A ^ w zz-^; hence from equation (6) the discharge
is equal to
and substituting for h% its value ( r ' A W V, we shall
\ r '
find the value of
(34.) D =
As the value of the discharge would be expressed by
2
X
s
if the head h h^ were neglected, it is evident the
coefficient is increased, under the circumstances,
from c d to
or, more correctly, the common formula has to be
multiplied by (l + i)* (!)*, to find the true dis-
charge, and the value of this expression for different
values of - = n w m be found in TABLE IV. If we
suppose that
* z. ^w ., 1 1
h-hv = JQ, then - =j-g = n ;
and we find from the table(l + i) l ~( -)* = 1-1221.
ORIFICES, WEIRS, PIPES, AND RIVERS. 89
Now if we take the value of c d for the full head h to
be -628, we shall find 1-1221 x '628 =-705, rejecting
the latter figures, for the coefficient when the head is
1 2
measured at the orifice ; and if - = JQ n, we should
find in the same manner the new coefficient to be
1-2251 x '628 = -769 nearly. The increase of the
coefficients determined, page 77, from Mr. Ballard's
experiments is, therefore, evident from principle, as
the heads were taken at the notch ; and it is also
pretty clear that, in order to determine the true dis-
charge^ the heads both on, at, and above a weir should
be taken. Most of the discrepancies in the coefficients
determined from experiment have arisen from imper-
fect and limited observations of the facts. Amongst
these the velocity of approach should never be
neglected by observers, as its effect on the discharge
is often considerable in increasing the quantity. The
effect of the form of the weir and approaches is
scarcely ever sufficiently considered by professional
men. Most of the discussions which arose with
reference to the gaugings on the Metropolitan MAIN
DRAINAGE QUESTION would have been obviated if the
calculators, or engineers, had taken into account the
different circumstances attendant on it, instead of
applying generally a formula suited to a particular
case, namely, a thin crest, a small notch, and a large
body of water immediately above it ; and applied a
correct formula for finding the effects of the velocity
of approach.
The two following tables have been reduced to
English measures of feet, from Boileau's experiments ;
90
THE DISCHARGE OF WATER FROM
they show the relation of the head to the depth on
the crest at the upper arris. The coefficient for the
head h being known, we may, from our equation (34),
calculate that due to h & on the weir.
TABLE showing the ratio of the head, h, to the depth., h^, on a Plank
Weir of the full width of the Channel, immediately at the upper
edge, or j, see equation (33), when the sheet of water is free
after passing over, with air under it.
Head h in
feet.
Values of the head h divided by the thickness of the sheet of water
passing over the weir immediately at the upper edge ; average r- = F
= 1-2 between heads of 3 and 14 inches.
Height of weir
in feet,
8&.
Height of weir
in feet,
1-07'.
Height, of weir
in feet,
1-33'.
Height of weir
in feet,
Ml*.
1
1-339
1-285
13
1-282
1-320
1-250
16
1-260
1-285
1-228
20
1-234
1-243
1-249
1-214
23
1-223
1-232
1-231
1-205
26
1-216
1-232
1-223
1-200
3
1-212
1-228
1-218
1-199
33
1-210
1-225
1-217
1-199
39
1-206
1-221
1-112
1-197
46
1-202
1-216
1-206
53
1-199
1-201
. .
59
1-196
1-195
..
66
1-192
1-191
82
1-186
..
99
1-184
. .
*''"*. '"
1-15
1-182
*W' C
If we were to use the head h w instead of h, to cal-
culate the discharge, when j- zi 1*2, then a coefficient
/i w
of -628 for the head h would become -769 for the
head A w in equation (34) : for - = '2, and, therefore,
TABLElV.,-628x(l'2)*-(-2)* = -628x1-2251 =-769.
ORIFICES, WEIRS, PIPES, AND RIVERS.
91
TABLE showing the ratio j , equation (33), when the sheet of water
' l Vf
passing over is in contact with the crest and with the water im-
mediately below a Plank Weir.
h
Values of i for different heights of weirs and for different
"w
Head h
in feet.
heads : mean value for heads between 3 and 14 inches, equal
1=1-25.
Height of weir
in feet,
1-07'.
Height of weir
in feet,
1-1'.
Height of weir
in feet,
1-38'.
43
1-283
46
1-275
1-291
49
1-256
1-266
1-281
53
1-250
1-258
1-271
59
1-236
1-245
1-254
66
1-225
1-232
1-241
73
1-216
1-223
79
1-208
1-216
86
1-202
1-208
92
1-198
1-203
99
,{!;* -
1-198
.:.
If we were to use the head h w instead of h to cal-
culate the discharge, when y- = 1-25, then a coeffi-
cient of -628 for the head h would become -799 for
the head h w in equation (34) : for - = -25 ; and,
therefore, the value of c d {(l + -)* - (-)*J,TABLE IV.,
is -628 x (1-25)* - (-25)*-= -628 x 1'2725 = -799:
and so on we may calculate the value of the coeffi-
cient to be applied to the depth h w on the weir, for
any other ratios between h and h w by means of
equation (34).
Boileau made some valuable experiments at Metz,
92 THE DISCHARGE OF WATER FROM
which were published in 1854. They give the fol-
lowing results for vertical plank weirs extending from
side to side of the channel, when the water passed
over without adhering to the crest :
Height of weir over bot-
torn of channel in feet. Head above Mean coefficient.
3- -2 to 1-6 -645
1-3 -16 to -5 -622
6 -15 to -25 -625
When the water passing over was joined to the crest,
and no air between the sheet passing over and the
water below the weir, the experiments gave
Height of weir over bot- -^ , ,
torn of channel in feet. Mean coefficient.
2- 1- to 1-6 -694
1-3 -6 to 1-8 -690
6 -36 to 1-3 -675
When the plank weir leant up-stream 4 inches to a
foot, the mean value of c d was -620, the height of
weir being 1-5 foot, and with heads from -23 to *5
foot. When its crest was rounded to a semi-cylin-
der, the coefficient was, with a head of % 26 foot, '696,
and with a head of -52 foot, *843 ; the water adhering
to the crest. With a head of -6 foot the coefficient
was -867, and with a head of -85 foot, '840, when the
water passed over without air between it and the
water below the crest. The following tables give the
experimental and reduced coefficients for vertical
plank weirs of different heights, and with different
heads, when the water passes over in a full sheet, and
also when it is joined to the crest and lower water. Also
for plank weirs suitable for sluices, leaning up-stream
with a slope of one-third horizontal to one vertical.
ORIFICES, WEIRS, PIPES, AND RIVERS.
93
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94
THE DISCHARGE OF WATER FROM
COEFFICIENTS of Vertical Plank Weirs at right angles to the Channel,
when the edge is chamfered at the lower arris, and when the head
passing over is in contact with the water at and below the Weir ; or
when the water immediately below the Weir rises to the crest. The
maximum coefficient -733 appears to obtain when the height of
the Weir is double the depth passing over the crest.
_g
Heights of weirs, in feet, over the bottom of the channel, and
a
+5
corresponding values of the coefficient of discharge c d in the
*;.J
13
formula v = c d X f A/2 g h.
1$
w
66'
82'
99'
1-15'
1-32'
1-48'
1-65'
1-81'
1-98'
S
30
727
30
33
724
33
36
721
36
39
718
39
43
714
43
46
709
46
49
702
708
715
724
..
49
53
694
699
708
718
. .
53
56
687
693
700
712
729
56
59
679
687
694
705
721
59
63
676
682
689
700
717
63
66
672
678
684
696
714
66
73
667
672
678
690
708
733
73
79
661
666
673
685
705
729
79
86
655
660
669
681
700
724
86
92
648
655
666
678
699
720
92
99
640
652
666
678
693
703
712
720
729
99
1-05
631
645
657
669
*681
691
702
"11
720
1-05
1-12
627
636
646
657
667
679
690
"00
711
1-12
1-19
625
636
646
657
666
675
685
694
703
1-19
1-25
625
636
646
657
666
675
682
690
696
1-25
1-32
625
636
646
657
666
673
679
685
691
1-32
1-39
666
672
678
682
684
1-39
1-45
t
i
664
670
675
679
684
1-45
1-52
(
,
661
667
672
676
681
1-52
1-58
.
.
658
663
669
672
675
1-58
1-65
655
658
663
666
667
1-65
The following table gives the result of experiments
on chamfered plank weirs, for gauging, extending
across a channel at right angles to it, when the back-
water below was joined to the head- water at passing
over, and when there was no air between :
Height of weir over the bot- ") '
torn of the channel below. . . J
Heads passing over the weir in }
each case, when absorbed 'at >'23
the crest into the back-water J
feet feet feet feet feet feet feet feet
66 -83 1-00 1-16 1-32 1-48 1'65 2-00
31 -38 -45 -51 -59 -66
92
which shows that the head was drowned (noyee) when
the depth of the lower channel below the crest of the
weir was less than 24 times the head passing over,
taking a general average.
ORIFICES, WEIRS, PIPES, AND RIVERS.
95
TABLE of Experimental Coefficients for Plank Weirs leaning up-
stream, when the crest has the down-stream arris rounded to a
quadrant; and when the crest is cylindrical and projecting up-
stream in the form of a knob.
Head h in
Plank weir leaning np-sti earns
one-third to oni- ; the lower arris
of crest rounded off to a quadrant
of a circle with a radius the full
thickness of the plank.
Plank weir leaning upwards one-
third to one, the crest rounded and
projecting in front beyond the
plank, so as to be thicker than it.
feet.
Water free from
curve of crest
13 foot thick.
Water in contact
with curve of
crest -17 foot
thick.
Water in contact
with curve of
crest -3 foot
thick.
Water in contact
with curve of
crest -33 foot
thick.
16
589
651
20
589
672
23
594
697
26
612
697
30
633
721
670
33
642
747
604
686
36
649
766
625
700
39
655
768
648
714
43
661
795
669
727
46
667
802
687
741
49
675
702
753
53
679
715
765
56
685
729
775
59
741
786
63
753
795
66
762
802
69
808
72
813
The effect of the form of the crest in increasing
the coefficients is distinctly observable in this Table,
although the weirs experimented on overhung the
water above, between the crest and the bottom of
the channel.
We must protest against the notation adopted by
Boileau and Morin, of giving only two-thirds of the
coefficient of discharge, c d , for weirs, instead of the
full and true value. The correct formula for the dis-
charge from a weir, is D zz - lh \/2gh. Now they
3
assume a coefficient due to an incorrect formula D zz
lh \/2gh, which reduces c d to - c d to give the same
3
final results. This leads also to an unnecessary dis-
tinction between the coefficients of orifices at the
surface, or notches, and orifices sunk to some depth,
which, practically, have the same, or nearly the same,
general value.
96 THE DISCHARGE OF WATER FROM
SECTION IY.
VARIATIONS IN THE COEFFICIENTS FROM THE POSITION OF
THE ORIFICE. GENERAL AND PARTIAL CONTRACTION.
VELOCITY OF APPROACH. PRACTICAL FORMULAE FOR THE
DISCHARGE OVER WEIRS AND NOTCHES. CENTRAL AND
MEAN VELOCITIES.
A glance at TABLE I. will show us that the coeffi-
cients increase as the orifices approach the surface,
to a certain depth dependent on the ratio of the sides,
and that this increase increases with the ratio of the
length to the depth : some experimenters have found
the increase to continue uninterrupted for all orifices
up to the surface, but this seems to hold only for
depths taken at or near the orifice when it is square
or nearly so : it has also been found that the co-
efficient increases as the orifice approaches to the
sides or bottom of a vessel: as the contraction
becomes imperfect the coefficient increases. These
facts probably arise from the velocity of approach
being more direct and concentrated under the re-
spective circumstances. The lateral orifices A, B, c,
D, E, F, G, H, i, and K, Fig. 11, have coefficients dif-
fering more or less from each other. The coefficient
for A is found to be larger than either of those for
B, c, E, or D ; that for G or K larger than that
for H or i ; that for H larger than that for i ; and
that for F, where the contraction is general, least of
all. The contraction of the fluid on entering the
orifice F removed from the bottom and sides is com-
plete; it is termed, therefore, " general contraction;"
that at the orifices A, E, G, H, i, K, and D, is inter-
fered with by the sides ; it is therefore incomplete,
ORIFICES, WEIRS, PIPES, AND RIVERS.
97
and termed "partial contraction'' The increase in
the coefficients for the same-sized orifices at the same
Fig-.ll
mean depths may be assumed as proportionate to
the length of the perimeter at which the contraction
is partial, or from which the lateral flow is shut off ;
for example, the increase for the orifice G is to that
for H as cd -{-de : de; and in the same manner the
increase for G is to that for E as cd + de : c d. If
we put n for the ratio of the contracted portion c d e
to the entire perimeter, and, as before, c d for the
coefficient of general contraction, we shall find the
coefficient of partial contraction to be equal to
(35.) c d + *09 n = c d -f- *1 n nearly,
for rectangular orifices. The value of the second
term *09 n is derived from various experiments. If
we assume *617 for the mean value of c d , we may
change the expression into the form (1 + *146 n) c d .
When n=%, this becomes 1-036 c d ; when n = |, it
becomes T073 c d ; and when n f , contraction is
prevented for three-fourths of the perimeter, and the
coefficient for partial contraction becomes l'L09c d .
The form which we have given equation (35) is,
however, the simplest ; but the value of n must not
exceed f . If in this case c d =. -617, the coefficient
H
98 THE DISCHARGE OF WATER FROM
for partial contraction becomes '617 + -09 x f =
617 + -067 =z -684, Bidone's experiments give for
the coefficient of partial contraction (1 + -152n)c d ;
and Weisbach's (1 + -132 w) c d .
VARIATION IN THE COEFFICIENTS FROM THE EFFECTS OF
THE VELOCITY OF APPROACH.
Heretofore we have generally supposed the water in
the vessel to be almost still, its surface level un-
changed, and the vessel consequently large compared
with the area of the orifice. When the water flows to
the orifice with a perceptible velocity, the contracted
vein and the discharge are both found to be in-
creased, other circumstances being the same. If the
area of the vessel or channel in front exceed thirty
times that of the orifice, the discharge will not be
perceptibly increased by the induced velocity in the
conduit ; but for lesser areas of the approaching
channel corrections due to the velocity of approach
become necessary. It is clear that this velocity
may arise from either a surface inclination in the
channel, an increase of head, or a small channel of
approach.
We get equation (6) for the discharge from a
rectangular orifice A, Fig. 12, of the length /, with a
head measured from still water
in which A b and k t are measured to the surface at some
distance back from the orifice, as shown in the section.
The water here, however, must move along the channel
towards the orifice with considerable velocity. If A
be the area of the orifice, and c the area of the
OEIFICES, WEIRS, PIPES, AND RIVERS. 99
channel, we may suppose with tolerable accuracy that
Fig. 12
this velocity is equal to -v , in which v represents
c
the mean velocity in the orifice. If we also represent
by v & the velocity of approach, we get the equation
(36.) '; : ,f;':'; : * = x...
and consequently the theoretical height (h a ) due to
it is
(37.) h & = ^ x - = '0155
o 2 c
in feet measures.*
The height A a may be considered as an increase of
head, converting A b into h b + h & , and h t into h t + h & .
The discharge therefore now becomes
(38.) D =
which, for notches or weirs, is reduced to
* When the approaching velocity passes through the orifice without
contraction, it is evident that the head h & required to produce that
A s v 2
velocity, in the orifice with contraction, must be h & = ^ x <> - 2 -^
case equation (40) becomes
instead of h & = ^2 x cf~>
In like manner we must have h & =
^ = c"a x
= '04 vl in feet measures when v & is the velocity of approach
and c d = -617.
H3
100 THE DISCHARGE OF WATER FROM
(39.) D =1 c^^Tg { (Ab + Aj'r _ At} *
as h t then vanishes. As D is also equa to A x v ,
equation (37) may be changed into
D 2 1 D 2
(40.) h a X - - -0155 -gin feet measures.
zg
If this value for h & be substituted in equations (38)
and (39), the resulting equations will be of a high
order and do not admit of a direct solution ; and in
(38) and (39), as they stand, h & involves implicitly
the value of D, which we are seeking for. By find-
ing at first an approximate value for the velocity of
approach, the height A a due to it can be easily found,
equation (37) ; this height, substituted in equation
(38) or (39), will give a closer value of D, from
which again a more correct value of h a can be de-
termined ; and by repeating the operation the values
of D and h & can be had to any degree of accuracy.
In general the values found at the second operation
will be sufficiently correct for all practical purposes.
It has been already observed that, for orifices, it is
advisable to find the discharge from a formula in
which only one head, that at the centre, is made use
of ; and though TABLE IV., as we shall show, enables
us to calculate the discharge with facility from either
formula, it will be of use to reduce equation (38) to
* The formula for the discharge over weirs, taking into account
the velocity of approach, D = 2-95 c d l*J h -f -115 vj, given by
D'Aubuisson, Traite Hydraulique, seconde Edition, pp. 78 et 95,
and adopted by some English writers and engineers, is incor-
rect in principle. In feet measures it becomes D = 5-35c d ZfeX
V h + -03494 t?J, which form, with alterations in the numerals
and measures, was used for calculating discharges of sewers
during the METROPOLITAN MAIN DRAINAGE discussion.
ORIFICES, WEIES, PIPES, AND EIVERS. 101
a form in which only the head (h) at the centre
is used. The error in so doing can never exceed six
per cent., even at small depths, equation (31), and
this is more than balanced by the observed increase
in the coefficients for smaller heads.
The formula for the discharge from an orifice, h,
being the head at the centre, is
and when the additional head h a due to the velocity
of approach is considered,
which may be changed into
(41.) D=
Equation (39), for notches, may be also changed to
the form
(42.) D=
this is similar in every way to the equation
(43.) ^
for the discharge from a rectangular orifice whose
depth is d, with the head h t , at the upper edge.
TABLE III. contains the values of j 1 + v- a | in equation
(41), and TABLE IV. the values of
in equation (42), or the similar expression in (43),
T- or -> being put equal to n; and we perceive that
the effect of the velocity of approach is such as to in-
crease the coefficient from c d to c d { 1 + j- & } for orifices
ft i
102 THE DISCHARGE OF WATEE FROM
sunk some distance below the surface, and into
for weirs when h & is the height due to the velocity of
approach, h the depth of the centre of the orifice,
and A b the head on the weir. A few examples, show-
ing the application of the formulas (41), (42), and
(43), and the application of TABLES I., II., IIL, and
IV. to them, will be of use. We shall suppose, for
the present, the velocity of approach v & to be given,
and no extra head be required to maintain it through
v 2
the orifice : in other words when h=
= '017^ in feet measures nearly.
EXAMPLE I. A rectangular orifice, 12 inches wide
by 4 inches deep, has its centre placed 4 feet below the
surface, and the water approaches the head with a
velocity of 28 inches per second ; what is the dis-
charge ? For an orifice of the given proportions,
and sunk to a depth nearly four times its length, we
shall find from TABLE I.
*616 +'627 /jn-i i
c d -T 621 nearly.
As the coefficient of velocity, equation (2), for water
flowing in a channel is about -956, we shall find,
column No. 3, TABLE II. the height A a =l|izl-125
inch nearly, corresponding to the velocity 28 inches.
Equation (41),
now becomes
D=12 X4>/2^X -621 f
OEIFICES, WEIES, PIPES, AND KIVEES. 103
We also find v/2# h = 192 -6 inches when h = 48
inches, TABLE II. ; therefore
D=12x4xl92-6X'62lfl+ }*
* 48 '
=9244-8x'621{l + -0234p=9244-8x -621x1-0116,
(as {1-0234}>=1-0116 from TABLE III.) = 9244-8 x
628 nearly = 5 805 -1 cubic inches=3-36 cubic feet
per second. Or thus: The value of -621 x (1-0234)
being found equal -628, D=AX-628v/2#x48. Now
for the coefficient -628, and h = 48 inches, TABLE II.
gives us -628v/2^ x 48 =120-96 inches; hence we
get D=12 x 4x120-96=5806-08 cubic inches=3'36
cubic feet, the same as before, the difference -38 in
the cubic inches being of no practical value. If we
v 2
find h & from the formula h & = n 2 =2*6 inches. then
r
we shall get i>=3-41 cubic feet nearly.
If the centre of the orifice were within 1 foot of
the surface, the effect of the velocity of approach
would be much greater ; for then
Cd x x + * = (from TABLE L) . 623
= (from TABLE III.) -623 x 1-047 = -652 instead of
628. In this case the discharge is D = 12 x 4 x -652
X 12 = 12 x 4 x -652x96-3 (from TABLE II.)
= 12 X 4 x 62-8 = 3014-4 cubic inches = 1-744
cubic feet per second. Or we may find the value of
652 v/2# h directly from TABLE II. thus :
The value of -628 y/2^ x 12 = 60-48 -628
The value of -666 \/1g x 12 = 64-14 -652
38 ~T~ 3-66 ::~24: 2-31.
Hence -652 \/2^h = 60-48 + 2-31 = 62-79, and the
104 THE DISCHAEGE OF WATER FROM
discharge zz 12 x 4 x 62-79 x 3013-92 cubic inches
zz 1-744 cubic feet per second, the same as before.
v 2
If we take A a zz f~ 2=: 2 -6 inches, we shall find DZZ
Agc<L
1-833 cubic feet nearly.
EXAMPLE II. A rectangular notch, 7 feet long, has
a head of 8 inches measured at about 4 feet above the
orifice, and the water approaches the head with a
velocity of 16 J inches per second ; what is the dis-
charge? For a still head we shall assume c d zz -628
in this case, and we have from equation (42)
As in the last example, we shall find from TABLE II.
(h & ) the height due to the velocity of approach (16J
inches) to be - zz 0-375 inch, assuming the coefficient
8
of velocity to be -956. We have, therefore, h & zz
375, h b = 8, c d zz -628, and A zz 7 x 12 x 8, or for
measures in feet - zz -047, h b zz -> and A zz 7 x -
n^ 3
hence
D zz - x 7 x ~^g X - X 628{(l-047) l -(-047) f }.
3 8 3 v /
The value of (1-047)* - (-047)* will be found from
TABLE IV. equal to 1-0612 ; the value of v 2# x ^
will be found from TABLE II. equal to 6-552, viz. by
dividing the velocity 78-630, to be found opposite 8
inches, by 12 ; hence
Dzz-x7x-X 6-552 x '628 X 1'0612
3 3
zz x 7 X 4-368 x -628 x 1'0612
OKIFICES, WEIKS, PIPES, AND RIVERS. 105
= - x 7 x 4-368 x -666 nearly
3
= - x 7 x 2-909 -1 X 1-939
3
= 13-573 cubic feet per second = 814-38 cubic feet
per minute. Or thus: From TABLE VI, we find,
when the coefficient is -628, the discharge from a
weir 1 foot long, with a head of 8 inches, to be
109-731 cubic feet per minute. The discharge for
a weir 7 feet long, when -r = *047 is therefore
109-731 x 7 x 1-0612 = 815-12 cubic feet per
minute. The difference between this value and that
before found, 814-38 cubic feet is immaterial, and has
arisen from not continuing all the products to a suffi-
v *
cientnumberof places of decimals. If A a :n 2 =. -87
&g c&
inch, then D zz 14-51 cubic feet per second nearly.
We have, in equations (36) and (37), pointed out
the relations between the channel, orifice, velocity of
approach, and velocity in the orifice, viz.
. = X t^andA. = X =, in which A. =
(neglecting, for the present, the coefficient of velocity
in passing through the orifice). As v is the actual
v
velocity in the orifice, must be the theoretical
velocity due to the head h + h^ and therefore
v* vl v\
;, and h =. -3 ; hence
** 4A**=?
C C d A V & A
106 THE DISCHARGE OF WATER FROM
We have hence
(44.) i a - C ' A "
\ / j - Q *2 Q ?
fl/ C "" Cft A
substituting this value in equations (41) and (42),
there results
(45.)
D = A v^AXCajl +^T
or
D = A < X
in which m = -, for the discharge from an orifice at
some depth, and for the discharge from a weir,
(46.) ^A
The two last equations give the discharge when the
o
ratio of the channel to the orifice - = m is known,
and also when the whole quantity of water passing
through the orifice, that due to the velocity of approach
as well as that due to the pressure, suffers a contrac-
tion whose coefficient is c d . When h & zz ^~
"9 ^ C
is when the velocity of approach v & passes through
the orifice without contraction, we shall get
h. vl A 2 1
h ~ vl - vl -" c 2 - A 2 ^ m 2 - 1'
consequently, in this case, equation (45) becomes
(46o.) D rz A x/2^1 X c d x (l + i^if > ' T ^
and equation (46) in like manner changes into
OEIFICES, WEIES, PIPES, AND E1VEES. 107
The last members of these two equations are the
same as the like members in (45) and (46), when
c d? within the brackets = 1 ; consequently we shall
easily find their values for the coefficient 1 in the
last page of TABLE Y., for the respective values of
c 'h \
m == - and also for those of ^ = 2 _ , When
c d zz 1, equation (45) may be changed into
Igh \\
This is the equation of Daniel Bernoulli, and only a
particular case of the one we have given.
2 2 1
If we put n = gfc&^ the values of (l + gjtn^l)*
no o *|
and of {l + m ^ } - { m 2 C l_ ^2}^ respectively, can
be easily had from TABLES III. and IV. We have,
however, calculated TABLE V. for different ratios of
the channel to the orifice, and for different values
of the coefficient of discharge. This table gives at
once the values of
as new coefficients, and the corresponding value of
T~* or y- zz a o4f
h> h* nP cl
h i A S
* When T 5 = -5 T- sa ^ - -3 we shall have in EXAMPLE II.
/& Wl * X O^ - * A
-* = -11 and (l + ~* - ( 1-133, TABLE IV., (or
108 THE DISCHARGE OF WATER FROM
It is equally applicable, therefore, to equations (41)
and (42) as to equations (45) and (46). For in-
stance, we find here at once the value of 628
{(1-047)1 --(-047)*} in EXAMPLE II., p. 104, equal
to -666, as T- a i= *047, and the next value to it for
the coefficient -628, in the table, is *046, opposite
to which we find -666, the new coefficient sought.
The sectional area of the channel in this case, as
appears from the first column, must be about three
times that of the weir or notch.
TABLE V. is calculated from coefficients, c d , in still
water, which vary from -550 to 1. Those from '606
to -650, and the mean value -628 are most suited for
application in practice. When the channel is equal
to the orifice, the supply must equal the discharge,
and for open channels, with the mean coefficient
628, we find, accordingly, from the table, the new
coefficient 1*002 for weirs ; or 1 very nearly as it
should be. We also find, in the same case, viz. when
A =z c, and c d zz -628, that for
short tubes, Fig. 13, the re-
sulting new coefficient be-
comes *807. This, as we
shall afterwards see, agrees
very closely with the experimental results. When
the coefficients in still water are less than -628, or
more correctly -62725, the orifice, according to our
TABLE V. for the coefficient 1.) Hence in this case *628 x
1-133 = -712 the new coefficient suited to the velocity of
v 2
approach. Here of course h & = ^^ (see Note p. 99).
OEIFICES, WEIRS, PIPES, AND EIVEES. 109
formula, cannot equal the channel unless other re-
sistances take place as from friction in tubes longer
than one and a half or two diameters, or in wide
crested weirs ; and for greater coefficients the junc-
tion of the short tube with the vessel must be
rounded, Fig. 14, on one or
Fig. 14
more sides ; and in weirs or
notches the approaches must
slope from the crest and ends
to the bottom and sides, and
the overfall be sudden. The
converging form of the approaches must, however,
increase the velocity of approach ; and therefore
v a is greater than - x V Q when c is measured between
c
r o and R o, Fig. 14, to find the discharge, or new
coefficient of an orifice placed at r o.
As the coefficients in TABLE V. are suited for
orifices at the end of short cylindrical or prismatic
tubes at right angles to the sides or bottom of a
cistern, a correction is required when the junction is
rounded off as at nor o, Fig. 14. When the chan-
nel is equal to the orifice, the new coefficient in
equation (45) becomes
The velocity in the short tube Fig. 14 is to that in
( 1 )
the short tube Fig. 13 as 1 to c d y^ 2 nearly, or
vl C d ;
for the mean value c d zz -628, as 1 to -807. Now, as
- is assumed equal to - in the cylindrical or pris-
J10 THE DISCHARGE OF WATER FROM
matic tube, Fig. 13, - - z=- in the tube Fig. 14
A V &
with the rounded junction, for v & becomes TOTI hence,
in order to find the discharge from orifices at the end of
the short tube, Fig. 14, we have only to multiply the
/t
numbers representing the ratio - in the first column,
^L
TABLE V., by -807, or more generally by c d jj-^ 2} ,
and find the coefficient opposite to the product.
Thus if c d =; *628, we find, when - zi l,c d j-j -- 2}
A. N JL c/d )
zz -807 in the table. If, again, we suppose - z= 3,
A.
then 3 x '807 2-421, the value of - for the tube
Fig. 14, and opposite this value of -, taken in column
1, we shall find -651 for the new coefficient. For the
cylindrical or prismatic tube, Fig. 13, the new coeffi-
cient would be only -642. When the head h & is how-
v 2
ever equal to ^ * c z the results must be modified
accordingly (see Note p. 99).*
* Professor Rankine gives the value of the coefficient of dis-
charge, or contraction, for varying values of A and c at a diaphragm
in a pipe by the formula
618
When- = o, c d = 1; and when - = 1, c d = -618 ; as it should be
c c
very nearly for an orifice in a thin plate, to which only, and to A, in
the short tube, Fig. 14, the formula is suited (see SECTION X).
ORIFICES, WEIRS, PIPES, AND RIVERS. Ill
PRACTICAL FORMULA FOR THE DISCHARGE OVER WEIRS.
In order to reduce the preceding formulae for
weirs and notches to some of the forms in common
use, with definite combined numerical coefficients,
by substituting 8*025 for v/2^, equation (39) be*
comes for feet measures
(A.) D a =5-35 c d / {&+ A>-A a *},
and for inch measures, as \f%g = 27*8, the dis-
charge, taken also in cubic feet, becomes
(B.) D a = -01072 c d I {(h b - A a )t - A>* }
When the length I is taken in feet and the depth in
inches, we shall have
(0.) D a =-1287c d /{(A,^AJ*-A>*}-
The three last equations being for seconds of time,
we shall get, when the time is taken in minutes for
feet measures, the discharge in cubic feet
; (D.) D a -321 c d
for inch measures
(E.) D. = -6433c
and for lengths (I) in feet and depths in inches
, : (F.) * = W*Cil{(h, + h$-hJ}. (j If
The latter equation, when the coefficient of discharge,
c d , is taken at -614 becomes
,D a = 4-74 / {(A b + A a )-A fc *}, and
(G.) JDn:4-74/^, when the velocity of ap-
( proach vanishes.
For a coefficient of -617
112 THE DISCHARGE OF WATER FROM
-^}' and
(H.) -p = 4*76// when the velocity of ap-
( proach vanishes.
For a coefficient of -623
a
' ' JD=:4'81/A with no perceptible approach.
For a coefficient of -628
- A.*}' and
a
^ ' JDzr4-85 /A with no perceptible approach.
For a coefficient of -648
\ '* 1 D zz 5 I ti% with no perceptible approach.
For a coefficient of f or -667
D a iz5-14 / {(A b + A a )^ - Aj}' and
' *' JDzz5*14 /At with no perceptible approach.
For a coefficient of -712
= 5-5 I ((Ah-1-A.Fr-A.IV' and
' ' JDz=5.5 / 7^ with no perceptible approach.
And finally for a coefficient of -81
/D a = 6-3 /{(AH+AJ* A^Vand
(0.)
D 6-3 / A when the velocity of approach
vanishes.
The theoretical value of k & in each of the foregoing
equations is in terms of the velocity of approach v a
* *
h * - 27
in which 2 g must be taken equal to 64-403 for heads
in feet, and equal to 772-84 for heads in inches. But
it is evident that in order to produce the velocity
per second v & passing through the notch with a
ORIFICES, WEIRS, PIPES, AND RIVERS. 113
nearly still-water basin above it, that A a must be
v l v l
increased from its theoretical value ^- to 2 -?in
* 9 C ^ L 9
which expression c d is the coefficient of discharge
due to the particular notch, or weir, and its attendant
circumstances ; whence we must take
/P v * Theoretical head
Now, unquestionably, the most general coefficient
both for notches and submerged orifices, in thin
plates, for gauging, whether triangular, rectangular,
or circular, is -617, when the orifice or notch is small
compared with the approaching channel ; whence for
measures in feet
A a zz -0408 v% and v a zz4'95
For measures in inches,
A a = -0034 t and v & = 17-2
And for measures in which v & is expressed in feet per
second, and 7 a in inches
h & zz '49 vl 9 and v & zz 1*43 v/A a .
By substituting these values of A a , found in terms of
the approaching velocity, according to the standards
used in the equations from (A) to (p) inclusive, and
also in equation (H), we shall be enabled to find the
proper discharge from a notch in a thin plate. The
values of A a , equation (p), can be found at once in
inches from the observed values of v & , to be also taken
in inches/for coefficients varying from -584 to -974, by
means of TABLE II. Thus, with a coefficient of -617,
we shall find, for an approaching velocity of 36
inches per second, that h & becomes 4f = 4-4 inches
nearly, while for a coefficient of -666, it is only 3|zz
i
114 THE DISCHARGE OF WATER FROM
3-8 inches ; and for a coefficient of 1, the theoretical
head is but If zzl-7 inches nearly.
From the very nature of the case the approaching
velocity must continue nearly unimpaired through
the notch with but a very slight reduction arising
from the viscidity of the water when it enters the
aperture, and separates from the lateral fluid. But
in order to give this unimpaired velocity by means of
an extra head h & , it is evident that h & must be in-
creased above the theoretical value by the amount
due to the coefficient of discharge; or, as before
v 2 v*
stated. h a must be increased from ~ to -%- This
* 9 c &^9
value of h a is, perhaps, something too large, owing
to the reduction of v & at the moment it enters the notch
and is acted upon by the overfall, drawing it away,
as it were, from the lateral water above the crest.
The numerical results of the respective formula
from (A) to (o), inclusive, can be obtained by modify-
ing the form as in equation (42) into
(Q.) J
in which D is the discharge found, when there is no
velocity of approach, by the common form D = 5*35 x
c d / h*, for which separate values are given in equa-
tions from (H) to (o) inclusive ; and numerical values
in TABLE VI. ; and l+ "" a multi l )lier
suited to the velocity of approach, the values of
which can be found from TABLE IV. Suppose, for
OEIFICES, WEIKS, PIPES, AND KIVEES. 115
example, D =: 158-1 cubic feet per minute, A b 10
inches, and h a = 4 inches, which is that due to an
approaching velocity of 3 feet per second with a
coefficient of -648 ; then the multiplier becomes
(l + -4)l_ -4! 1-4035, TABLE IV. Hence the dis-
charge due to an approaching velocity of 3 feet is
158-1 x 1*4035 = 221-9 cubic feet, or an increase of
about 40 per cent. Also, if the common formula
were used, it is plain that the coefficient -648 should
be increased to -648 x 1-4035, or to -909 nearly,
which approximates within 10 per cent, of the
theoretical value. Nothing can show more clearly
the necessity for varying the coefficients when the
ordinary formulae are used, even for a notch in a thin
plate : for other notches the coefficients, even for
still water above the crest, vary considerably.
The form of the equation used by D'Aubuisson
and several other writers is
(R.) D a =
in which c and c are numerical coefficients, and v a the
velocity of approach. This form is incorrect in
principle, although the values of c and c can be so
taken as to give resulting values for D a approximately
correct. For feet measures, and time in seconds,
Professor Downing makes, after D'Aubuisson, p. 37
of his translation,
D a = c d X 5-35 / \hl + -03494^.
Doctor Robinson* gives for like measures and time,
values varying from
D a = 3-55 / <hl + -1395^/ to
* Proceedings Royal Irish Academy, vol. iv. p. 212. -1395 vl is
nine times the theoretical head, and too much.
i3
116
THE DISCHARGE OF WATER FROM
Mr, Taylor finds (for the Government Beferees, see
Report on the Main Drainage of the Metropolis,
13th July, 1858, p. 32) the discharge in cubic feet
per minute, when the depth is taken in inches, and
the length in feet to be,
and the Messrs. Hawksley, Bidder, and Bazalgette
assume, (p. 33 ibid,) for like measures,
D a = 5
which they consider is in " excess." The following
table, copied and extended from the report just
referred to, shows the results of the last two
formulae, and of our equations (L) and (N), in which
the depth, A b , must be taken equal to 10 inches,
and the length, /, equal to 1 foot,
Formulae.
Mean velocities approaching the notch in feet per second,
and discharges in cubic feet per minute.
5
1
1-5
2
2-5
. 3
158-1
158-5
159-5
161-4
164
167
171
D a = 5 <v // i 3 + -1875 v Jfe*
Equation(L)when the '
head, 7& a , due to the ve-
locity of approach is
158-1
159-2
162-1
166-8
173
180
189
taken at only its theo-
retical value .
Equation (L) when the '
head, h & , due to the ve-
locity of approach is in-
158-1
160
167
177
190
205
222
creased for the coeffi-
cient of velocity -648
Equation(N)when the '
head, h a , due to the ve-
locity of approach is
173-9
175-1
178-3
183-5
190-1
198-3
207-5
taken at only its theo-
retical value .
Equation (N; when the '
head due to the velocity
of approach is increased .
173-9
176
182
192
204
218
234
for the coefficient of ve-
locity -712 .
. = 6V*i + *..'*J
173-9
175-7
180-8
188-9
199-8
213
228
OEIFICES, WEIKS, PIPES, AND EIVERS. 117
In equations (L) and (N) we can get, TABLE II., the
values of the head, A a , due to velocity of approach v & ,
as follows :
^a = '5, !, 1-5, -2, 2-5, 3-0 ; in feet per second.
7i a = '047, -186, -419, -745, 1-16, 1-68 ; theoretical head in inches.
Then
7^ = 'HI, -447, -997, 1-77, 2-76, 4- ; for a coefficient of -648.
and
7t a = -093, -366, -833, 1-47, 2-29, 3-31 ; for a coefficient of -712.
Whence as A b = 10 inches, we shall have in equa-
tion (Q),
~ = -Oil, -045, -1, -18, -28, -4 ; for a coefficient of -648,
and
5l = -009, -037, -083, -15, -23, -33 ; for a coefficient of -712 ;
%
-m o 73
and hence, by means of TABLE IY. (l + ^f (jj^
becomes of the following respective values suited to
the above velocities,
1-015, 1-059, 1-122, 1-205, 1-3, 1-403 ; for a coefficient of -648,
and
1-013, 1-049, 1-104, 1-175, 1-254, 1-344 ; for a coefficient of -712.
These latter values multiplied in order by the initial
values of the discharges, 158-1 and 173-9, in the
above table, give the discharges in the third and
fifth lines corresponding, due to the respective velo-
cities of approach.
The accordance between the results in the last
two lines of the table is remarkable. TABLE Y.
shows that if the coefficient be '617 when the water
above the crest is still, it will be increased to -712
when the approaching channel is about 1-83 times
the section of the water in the notch. If the arrises
of the two-inch thick waste board be rounded,
the coefficient must also be considerable, although
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ig the Metropolitan Sewage D
TROACH IN FEET PER SECOND.
nanner of obtaining the expe f
to -737 nearly.
scriptions and Formulae.
g
1
o
1
1 velocities of stream in \
t per second . . . j
OJ O ^>_, O)
5 v 5 y fl
5 ^ 4 f
fJLii
: 4-8 \/A 3 H- -1875 Ji v* & )
lal quantities measured .
2S of the coefficients, #, \
the formula
= x \/h* + -8 A 2 v V
obtained from the actual
an titles discharged . ./
, ' *
A
w
t ?|j 8
r2 .S
fi S c^
H
c
< < o <
o ^
ORIFICES, WEIRS, PIPES, AND RIVERS. 119
uncertain ; but as the equation D a = 5'5\/hl +'81% hi
appears to have been framed by Mr. Taylor, to
express special experiments made for Mr. Simpson,
in which the quantities varied from 5 to 152 cubic
feet per minute, and for heads on a four-foot weir
varying from 1 inch to 8 inches,* we must conclude
the coefficient for heads measured from still water
above the crest in those experiments suited to the
form of the weir used, and its attendant circum-
stances, is -712.
The equations (39) and those from (A) to (o) may
be easily changed into forms in which only the
depth h b) the velocity of approach, and the coefficient
of velocity (in this case equal to that of discharge) c d
are introduced. It is, however, only necessary here to
reduce the general form (A) p. Ill, for feet measures,
which becomes, after, substituting for h & its value
c 2 x* 2 tf an( * ma ki n g some reductions,
and for time in minutes the discharge is
R91
(T.) D a = -- l{ (64-4 C\h, + tfl* - }
in which v & still continues the velocity in feet per
second, as determined from observation. These for-
* Vide p. 22, Letter dated 16th August, 1858, from the
Government Referees to the Eight Hon. Lord John Manners,
on the subject of the Metropolitan Main Drainage.
120 THE DISCHARGE OF WATER FROM
mulge may be again reduced to many others. If we
take h b in inches (T) becomes
(U.) D a =
Mr. Pole, in a letter to Mr. Simpson and Captain
Gallon, already referred to, gives the special value,
D a = 1-06 / {(3 ^+^)t _<;*},
which corresponds very closely with the experiments
made for Mr. Simpson. If we assume c d = -712,
which also closely corresponds with those experi-
ments, our equation (U) becomes for them
D a = 1-225 / {(2-72 h, + !)* J} ;
but the amount of the discharge must always depend
on the coefficient c d , equation (U) suited to the special
circumstances of the case under consideration.
The form of equation for the discharge proposed
by Mr. Boy den * includes the effects of the end con-
tractions : it is
D =z c {/ bnh^h^
in which c zz f c d \/%gh, n the number of end con-
tractions, / the length of the weir, h^ the head
measured from the surface of the water above the
curvature of approach, and b a coefficient due to the
nature of the end contractions. The mean nume-
rical exponent of this formula, derived by Francis
from his experiments, is for feet measures, per second,
Dzz3-33 (I -
* Francis's Lowell Hydraulic Experiments, p. 74.
t Ibid, p. 119.
OKIFICES, WEIRS, PIPES, AND RIVERS. 121
but the value of c varied from 3-303 to 3-3617.
These results give corresponding values of c d = -617
to -628, and when c = 3-33, c d = -623. The experi-
mental results compared with this formula have been
referred to at p. 83.
Francis's Lowell experiments on a wooden dam
10 feet long, level and 3 feet wide at the crest, with
a head slope of 3i to 1 in a channel 10 feet wide,
give, for heads between 6 and 20 inches, a mean
coefficient of -563 or -565. This for feet measures
would give for the discharge per second
Dm 3-02Ai
For greater depths, on this width of crest, the dis-
charge would probably rise as high as 3-lA^or 3-3^.
The section of the dam was the same as that erected
by the Essex Company across the Merrimack Kiver,
at Lawrence, Massachusetts. See, also, table of
coefficients, p. 80.
In equation (13), pp. 54 and 55, we have given a
general expression for the value of D through a tri-
angular notch. Professor Thomson, of the Queen's
College, Belfast, in a paper read at the British Asso-
ciation at Leeds in 1858, says :
" The ordinary rectangular notches, accurately ex-
perimented on as they have been, at great cost and
with high scientific skill, in various countries, with
the view of determining the necessary formulas and
coefficients for their application in practice, are for
many purposes suitable and convenient. They are,
however, but ill adapted for the measurement of very
variable quantities of water, such as commonly occur
to the engineer to be gauged in rivers and streams.
122 THE DISCHARGE OF WATER FROM
If the rectangular notch is to be made wide enough
to allow the water to pass in flood times, it must be
so wide that for long periods, in moderately dry
weather, the water flows so shallow over its crest,
that its indications cannot be relied on. To remove,
in some degree, this objection, gauges for rivers or
streams are sometimes formed, in the best engineer-
ing practice, with a small rectangular notch cut down
below the general level of the crest of a large rectan*
gular notch. If now, instead of one depression being
made for dry weather, we use a crest wide enough for
use in floods, we conceive of a large number of de-
pressions extending so as to give the crest the
appearance of a set of steps of stairs, and if we
conceive the number of such steps to become in-
finitely great, we are led at once to the conception of
the triangular instead of the rectangular notch. The
principle of the triangular notch being thus arrived
at, it becomes evident there is no necessity for
having one side of the notch vertical, and the other
slanting; but that, as may in many cases prove
more convenient, both sides may be made slanting,
and their slopes may be alike. It is then to be
observed, that by the use of the triangular notch,
with proper formulas and coefficients derivable by
due union of theory and experiments, quantities of
running water from the smallest to the largest may
be accurately gauged by their flow through the same
notch. The reason of this is obvious, from consider-
ing that in the triangular notch, when the quantity
flowing is very small, the flow is confined to a small
space admitting of accurate measurement ; and that
ORIFICES, WEIRS, PIPES, AND RIVERS. 123
the space for the flow of water increases as the
quantity to be measured increases, but still continues
such as to admit of accurate measurement.
" Further, the ordinary rectangular notch, when ap-
plied for the gauging of rivers, is subject to a serious
objection from the difficulty or impossibility of pro-
perly taking into account the influence of the bottom
of the river on the flow of the water to the notch*
If it were practicable to dam up the river so deep
that the water would flow through the notch as
if coming from a reservoir of still water, the diffi-
culty would not arise. This, however, can seldom be
done in practice, and although the bottom of the
river may be so far below the crest as to produce
but little effect on the flow of the water when the
quantity flowing is small, yet when the quantity
becomes great, the velocity of approach comes to
have a very material influence on the flow of the
water, but an influence which is usually difficult, if
not impracticable to ascertain with satisfactory ac-
curacy. In the notches now proposed of a trian-
gular form, the influence of the bottom may be
rendered definite, and such as to affect alike (or at
least by some law that may be readily determined
by experiment) the flow of the water when very
small, or when very great, in the same notch.
The method by which I propose that this may be
effected consists in carrying out a floor, starting
exactly from the vertex of the notch, and extending
both up-stream and latterally, so as to form a bottom
to the channel of approach, which will both be
smooth and will serve as the lower bounding surface
124 THE DISCHARGE OF WATER FROM
of a passage of approach unchanging in form while
increasing in magnitude, at the places at least which
are adjacent to the vertex of the notch. The floor
may be either perfectly level, or may consist of
two planes, whose intersection would start from the
vertex of the notch> and would pass up-stream per-
pendicularly to the direction of the weir board ; the
two planes slanting upwards from their intersection
more gently than the sides of the notch. The level
floor, although theoretically not quite so perfect as
the floor of two planes, would probably for most
practical purposes prove the more convenient ar-
rangement.
" With reference to the use of the floor it may be
said, in short, that by a due arrangement of the
notch and the floor a discharge orifice and channel
of approach may be produced, of which (the upper
surface of the water being considered as the top
of the channel and orifice) the form will be un-
changed or but little changed, with variations of
the quantity flowing ; very much less certainly than
is the case with rectangular notches.
" Whatever may be the result in this respect, the
main object must be to obtain, for a moderate
number of triangular notches of different forms,
and both with and without floors at the passage
of approach, the necessary coefficients for the va-
rious forms of notches and approaches selected,
and for various depths in any one of them, so as
to allow of water being gauged for practical pur-
poses, when in future convenient, by means of
similarly formed notches and approaches. The util-
ORIFICES, WEIRS, PIPES, AND RIVERS. 125
ity of the proposed system of gauging it is to be
particularly observed, will not depend upon a per-
fectly close agreement of the theory described with
the experiments, because a table of experimental
coefficients for various depths, or an empirical for-
mula slightly modified from the theoretical one, will
serve all purposes.
"To one evident simplification in the proposed
system of gauging, as compared with that by rect-
angular notches, I would here advert, namely, that
in the proposed system the quantity flowing comes
to be a function of only one variable namely, the
measured head of water while in the rectangular
notches it is a function of at least two variables,
namely, the head of water, and the horizontal width
of the notch ; and is commonly also a function of a
third variable very difficult to be taken into account,
namely, the depth from the crest of the notch down
to the bottom 'of the channel of approach, which
depth must vary in its influence with all the varying
ratios between it and the other two quantities of
which the flow is a function.
" The proposed system of gauging also gives facil-
ities for taking another element into account which
often arises in practice namely, the influence of
back water on the flow of the water in the gauge,
when, as frequently occurs in rivers, it is found
impracticable to dam the river up sufficiently to give
it a clear overfall free from the back or tail water.
For any given ratio of the height of the tail water
above the vertex of the notch to the height of head
water above the vertex of the notch, I would an-
126 THE DISCHARGE OF WATER FROM
ticipate that the quantities flowing would still be
approximately at least, proportional to the \ power of
the head, as before ; and a set of coefficients would
have to be determined experimentally for different
ratios of the height of the head water to the height
of the tail water above the vertex of the notch.
" I have got some preliminary experiments made on
a right-angled notch in a vertical plane surface, the
sides of the notch making angles of 45 with the
horizon, and the flow being from a deep and wide
pool of quiet water, and the water thus approaching
the notch uninfluenced by any floor or bottom. The
principal set of experiments as yet made were on
quantities of water varying from about 2 to 10 cubic
feet per minute ; and the depths or heads of the water
varied from 2 inches to 4 inches in the right-angled
notch. From these experiments I derive the formula
Q = 0-317 H*
where Q is the quantity of water in cubic feet per
minute, and H the head as measured vertically in
inches from the still water level of the pool down to
the vertex of the notch. This formula is submitted
at present temporarily as being accurate enough for
use for ordinary practical purposes for the measure-
ment of water by notches similar to the one experi-
mented on, and for quantities of water limited to
nearly the same range as those in the experiments ;
but as being, of course, subject to amendment by
more perfect experiments extending through a wider
range of quantities of water."
In the first edition of this book we gave the gene-
ral form of the equation for the discharge through
ORIFICES, WEIRS, PIPES, AND RIVERS. 127
triangular notches, and also showed the general appli-
cation of the coefficients -617 to -628 for all forms
of orifices and notches in thin plates. '617, as shown
in note p. 55, gives a result identical with the prac-
tical results of Professor Thomson's experiments.
The great advantage of the triangular notch for
gauging is, that the sections for all depths flowing
over are similar triangles, and therefore the coeffi-
cient probably remains constant, or nearly so, not
only for one but for all species of triangles, when the
depth at the open is not very little indeed in propor-
tion to the width flowing over at the surface.
The disadvantage of the proposed triangular form
of depression, if permanent in a dam, would be that
the angular point should be at a lower level than the
top of a horizontal crest to maintain the same level,
above, of the water during floods ; and therefore the
power of the water and head would be reduced at
the period when most required for mill-power or
navigation purposes ; that is, during dry weather.
For drainage purposes the winter level or that du-
ring floods, must evidently be kept down, unless
when the banks are steep, and along rapids ; but
these remarks do not apply to dams erected across
inillraces or streams where the banks are, generally,
considerably above floods. These remarks refer to
occasions for permanent gauging to find the relations
of evaporation, absorption, and discharge in given
catchment areas. In notch gauging to determine the
useful effect of water engines, rectangular forms in
thin plates have the coefficients already well deter-
mined, and the calculations are easy.
128 THE DISCHAKGE OF WATEK FKOM
DIFFERENT EFFECTS OF CENTRAL AND MEAN VELOCITIES.
There is, however, another element to be taken
into consideration, and which we shall have to refer
to more particularly hereafter ; it is this, that the
central velocity, directly facing the orifice, is also
the maximum velocity in the tube, and not the mean
velocity. The ratio of these velocities is 1 : -835
/-i
nearly; hence, in the example, p. 110, where - 3,
A
we get 3 x *835 = 2-505 for the value of - in column
A
1, TABLE Y., opposite to which we shall find -649,
the coefficient for an orifice of one-third of the
section of the tube when cylindrical or prismatic,
Fig. 13; and 3 x '835 x '807 = 2-02 nearly, oppo-
site to which we shall get -661 for the coefficient
when the orifice is at the end of the short tube,
Fig. 14, with a rounded junction. We have, there-
c c*
fore, - x *835 equal to the new value of for
A A
finding the discharge from orifices at the end of
/i
cylindrical or prismatic tubes, and - x '835 x '807
A
C C
z= - x '67 nearly for the new value of J - when
A A
finding the discharge from orifices at the end of a
short tube with a rounded junction.
The ratio of a mean velocity in the tube to that
facing the orifice cannot be less than *835 to 1, and
varies up to 1 to 1 ; the first ratio obtaining when
the orifice is pretty small compared with the sec-
ORIFICES, WEIRS, PIPES, AND RIVERS.
129
Fig 15
D
tion of the tube, and the other when they are
equal. If we suppose
the curve D c, whose
abscissae (A b) repre-
sent the ratio of the
orifice to the section
of the tube, and whose
ordinates (be) repre-
sent the ratio of the mean velocity in the tube to
that facing the orifice, to be a parabola, we shall
find the following values :
Katio of the orifice
to the channel, or
values of
AB
1
2
3
4
5
6
7
8
9
1-0
Values of
dc.
165
163
158
150
139
124
106
084
059
031
000
Ratio of the mean velocity
of approach in a tube or
channel to that
directly opposite the
orifice, or values of b c
835
837
842
850
861
876
894
916
941
969
1-000
These values of b c are to be multiplied by the cor-
/-
responding ratio - in order to find a new value,
opposite to which will be found, in the table, the
coefficient for orifices at the ends of short prismatic
K
130
THE DISCHARGE OF WATER FROM
or cylindrical tubes ; and this new value again mul-
tiplied by -807, or more generally by c d (T 3}*, will
/~i
give another new value of -, opposite to which, in
A
the table, will be found the coefficient for orifices
at the ends of short tubes with rounded junctions.
EXAMPLE III. What
shall be the discharge
from an orifice A, Fig.
16, 2 feet long by 1
foot deep, when the
117 '945
12
value of ' is 3, and
A
the depth of the centre
of A I foot 6 inches
below the surface? We have D t zz 2 x 1 X
(TABLE II.) = 2 x 9-829 x 19-658 cubic feet per
second for the theoretical discharge. From the table
on last page the coefficient for the mean velocity,
/i
facing the ori&ce, is about -86 ; hence - x *86 = 3 x
A.
86 zz 2-58. If we take the coefficient from TABLE L,
we shall find it (opposite to 2, the ratio of the length
of the orifice to its depth) to be -617 ; and, for this
coefficient, opposite to 2-58, in TABLE V., or the next
number to it, we find the required coefficient -636 ;
hence the discharge is -636 x 19-658 = 12-502 cubic
feet per second. If we assume the coefficient in still
water to be -628, then we shall obtain the new co-
* See p. 106, with reference to the modifications of equations
*
(45) and (46) into (45 a) and (46a) suited to & a
ORIFICES, WEIRS, PIPES, AND RIVERS. 131
efficient '647, and the discharge would be '647 x
19-658 = 12-719 cubic feet. If the junction of the
tube with the cistern be rounded, as shown by the
dotted lines, we have to multiply 2*58 by *807, which
/
gives 2-08 for the new value of -, opposite which we
A.
shall find, in TABLE V., when the first coefficient is
628, the new coefficient -659; and the discharge in
this case would be -659 x 19-658 zz 12-955 cubic
feet per second.
It is not necessary to
take out the coefficient
of mean velocity facing
the orifice to more than
two places of decimals.
For gauge notches in
thin plates placed in
streams and millraces, Fig. 17, the mean coefficient
628, for still water, may be assumed ; thence the new
/-!
coefficient suited to the ratio - may be found, as in
A
the first portion of EXAMPLE III. We shall leave
the working out of the results when h a is taken equal
to the student.
EXAMPLE IV. What shall be the discharge through
the aperture A, equal ^feet by I foot, when the channel
is to the orifice as 3-375 to 1, and the depth of the
centre is 1 *2 5 foot below the surface, taken at about 3
feet above the orifice? Here the coefficient of the
approaching velocity is -85 nearly, whence the new
/-(
value of - is 3-375 x -85 = 2-87 ; and as C A = -628,
K3
132 THE DISCHARGE OF WATER FROM
we shall get from TABLE V. the new coefficient '644.
Hence
D = 2 x 1 x x -644 (TABLE !!.)= 2 x 8*972 x -644
12 V
= 17-944 x -644 = 11*556 cubic feet per second.
Weisbach finds the discharge, by an empirical
formula, to be 11*31 cubic feet. If the coefficient
be sought in TABLE I., we shall find it -617 nearly,
from which, in TABLE V., we shall find the new
coefficient to be *632 : hence 17-944 x *632 = 11-341
cubic feet per second. If the coefficient *6225 were
used, we should find the new coefficient equals *638,
and the discharge 11-468 cubic feet. Or thus: The
ratio of the head at the upper edge to the depth of
q
the orifice is = *75, and from TABLE IY. we find
12
(1*75)* (-75)* = 1-6655. Assuming the coefficient
to be *644, we find from TABLE VI. the discharge per
minute over a weir 12 inches deep and 1 foot long to
208-650 + 205'119 on/^oo^ i i j
be - - = 206*884 cubic feet nearly; and
2
as the length of the orifice is 2 feet, we have
2 X 206*884 X 1-6655 nn AC . n , . , ,. ,
- = 11*482 cubic feet per second, which
60
is the correct theoretical discharge for the coefficient
644, and less than the approximate result, 11*556
cubic feet above found, by only a very small dif-
ference. The velocity of approach in this example
must be derived from the surface inclination of the
stream. The working out of this example and the
v 2
increase of the discharge when h & = a 2 will afford
practice to the student.
ORIFICES, WEIRS, PIPES, AND RIVERS.'
For notches or Poncelet weirs the approachin
velocity is a maximum at or near the surface. If the
central velocity at the surface facing the notch be 1,
the mean velocity from side to side will be -914. We
may therefore assume the variation of the central to
the mean velocity to be from 1 to '914; and hence
the ratio of the mean velocity at the surface of the
channel to that facing the notch or weir cannot be
less than -914 to 1, and varies up to 1 to 1 ; the first
ratio obtaining when the notch or weir occupies a
very small portion of the side or width of the channel,
and the other when the weir extends for the whole
width. Following the same mode of calculation as
at p. 129, Fig. 15, we shall find as follows :
Ratio of the Values of Values of
width of the notch J l. ~
to the width of _. a 6 > _.*' C >
thechaunel. ]?lg. 15. Fig. 15.
-086 -914
1 -085 -915
2 -083 -917
3 -078 -922
4 -072 -928
5 -064 -936
6 -055 -945
7 -044 -956
8 -031 -969
9 -016 -984
1-0 -000 1-000
These values of b c are to be used as before in
/i
order to find the value of -, opposite to which in the
At
134 THE DISCHARGE OF WATER FROM
tables, and under the heading for weirs, will be found
the new coefficient.
EXAMPLE V. The length of a weir is 10 feet; the
width of the approaching channel is 20 feet; the head,
measured about 6 feet above the weir, is 9 inches ; and
the depth of the channel 3 feet: what is the discharge ?
Assuming the circumstances of the overfall to be
such that the coefficient of discharge for heads,
measured from still water in a deep weir basin or
reservoir, will be -61 ?, we shall find from TABLE VI.
the discharge to be 128-642 x 10 = 128642 cubic
feet per minute; but from the smallness of the
channel the water approaches the weir with some
velocity, and - = = 8. We have also the width
A 10 X ^
of the channel equal to twice the width of the weir,
and hence (small table, p. 133,) 8 x -936 = 7-488 for
P
the new value of -. From TABLE V. we now find the
A.
*fi24
new coefficient - = -623, and hence the dis-
2
charge is - - = 1298-93 cubic feet per minute.
Or thus : As the theoretical discharge, TABLE VI., is
2084-96 cubic feet, we get 2054-96 x -623 = 1298-93,
the same as before. In this example, however, the
mean velocity approaching the overfall bears to the
mean velocity in the channel a greater ratio than
1 : -936, as, though the head is pretty large in pro-
portion to the depth of the channel, the ratio of the
sections - = - is small. We shall therefore be more
C 8
correct by finding the multiplier from the small table,
ORIFICES, WEIRS, PIPES, AND RIVERS. 135
/-i
p. 129. By doing so the new value of - is 8 x -838
A
= 6*704. From this and the coefficient *617 we shall
find, as before from TABLE V., the new coefficient to
be -627 ; hence we get 2084-96 x -627 = 1307-27
cubic feet per minute for the discharge.
The foregoing solution takes for granted that the
velocity of approach is subject to contraction before
arriving at the overfall or in passing through it ;
now, as this reduces the mean velocity of approach
from 1 to -784, TABLE V., when the coefficient for
heads in still water is '617, we have to multiply the
ri
value of- = 6-704, last found, by -784, and we get
A.
6-704 x -784 = 5-26 for the value - due to this cor-
A.
rection, from which we find the corresponding co-
efficient in TABLE V. to be -629, and hence the cor-
rected discharge is 2084-96 x -629 = 1311-44 cubic
/-i
feet. It is to be borne in mind that the value of - in
A.
TABLE Y. is simply an approximate value for the ratio
-of the velocity in the channel facing the orifice to the
velocity in the orifice itself ; and the corrections
applied in the foregoing examples were for the pur-
pose of finding this ratio of velocity more correctly
r\
than the simple expression -gives it. The following
auxiliary table will enable us to find the correction,
and thence the new coefficient, with facility. Thus,
if the channel be five times the size of the orifice,
and a loss in the approaching velocity takes place
equal to that in a short cylindrical tube, we get
136
THE DISCHAKGE OF WATEE FKOM
AUXILIARY TABLE, TO BE USED WITH TABLE V. FOR MORE NEARLY FINDING
THE COEFFICIENT OF DISCHARGE NEARLY SUITED TO EQUATIONS (45 a)
AND (46 a).
Eatio of the orifice 1
to the channel, or J
l
Multipliers due to I
the difference of I
the central and 1
mean velocity only.|
Multipliers for finding the new values of-^ in TABLE V., when
the water approaches and passes through the orifice, without
contraction or loss of velocity.
Coefflc*'
639
Coeffic*'
628
Coefflc 1
617
Coeffic*'
606
Coeffic* 1
595
Coeffic*'
584
Coeffic 1
573
835
69
67
65
64
62
60
58
1
837
70
68
66
64
62
60
59
2
842
70
68
66
64
62
61
59
3
850
71
69
67
65
63
61
59
4
861
72
69
68
66
64
62
60
5
876
73
71
69
67
65
63
61
6
894
74
72
70
68
66
64
62
7
916
76
74
72
70
68
66
64
8
941
78
76
74
72
70
68
66
9
969
81
78
76
74
72
70
68
1-0
1-000
831
807
784
762
740
719
699
5 x '842 zz 4-210 for the new value of -, opposite to
which, in TABLE V., will be found the coefficient
sought. If the coefficient for still water be '606, we
shall find it to be *6 12 for orifices and '623 for weirs.
But when the water approaches without loss of
velocity, we find from the auxiliary table '64 for the
multiplier instead of -842, and consequently the new
value of -becomes
A
5 x '64 zz 3-2, from which we
shall find *617 to be the new coefficient for orifices
and -636 for weirs. The auxiliary table is calculated
by multiplying the numbers in the second column
(see third column, table, p. 129) by the value of c d x
J
- 3-2! , which will be found from TABLE V., for
the different values of c d in the table, viz.
ORIFICES, WEIRS, PIPES, AND RIVERS. 137
639, -628, -617, -606, -595, -584., and -573, to be
831, -807, -784, -762, -740, -719, and -699 respec-
tively, as given in the top and bottom lines of figures.
c 2
When -3-^2- in equations (45) and (46) is equal to
, 2 _^j in equations (45a) and (46), then c d zz 1,
and c d (l + ^ - 2 } in equation (45) is equal to
^ vn c d j
1 1 H -- 2 1 I i n equation (45a) ; and c d \(l +
in e( l uatio11 ( 46 ) is e( l ual to
l +^^=Tl""^=r ^ equation (460); and
therefore the coefficient found from TABLE V. for
c d = 1 will give the multiplier for d , outside the
brackets, in (45o) and (46a), to find the new coeffi-
cients. Thus in the last example m = 5, and hence
TABLE V. for c d = 1, we find { 1 +-^r l f = 1-021
and {(1 + ^f - fcj^r-i)*} - 1-OB5. Hence
1-021 x '606 = -619 nearly ; and 1-055 x '606 = -639
nearly, the new coefficients found from the other
method being *617 and -636, the difference by both
methods being of no great practical importance.
It is necessary to observe, that in equations (45),
(46), (45a), and (46o), the head due to the velocity of
supply or approach, A a , must be extra to the head k,
and no part of it, and that as is indicated by the
equations m can never be so small as unity. These
equations are not, therefore, strictly applicable to
orifices in the short tubes, Fig. 15 and Fig. 16, al-
138
THE DISCHARGE OF WATER FROM
though they can be made practically so within definite
limits. The initial value of c d itself varies consider-
ably with the position and form of the orifice ; for a
mean value of *707 it changes according to the rela-
707
tion of c and A into -7271 an( ^ f r a value of
618 for an orifice, central in a thin plate, Professor
Bankine's formula,, p. 110, is applicable.
In weirs at right
angles to channels
with parallel sides,
the sectional area can
never equal that of
the channel unless it be measured at or above the
point A, where the sinking of the overfall commences ;
and unless also the bed c D and surface A B have
the same inclination. In all open channels, as mill-
races, streams, rivers, the supply is derived from
the surface inclination of A B, and this inclination
regulates itself to the discharging power of the over-
fall. When the overfall and channel have the same
width, and it is considerable, we have, as shall
appear hereafter, 91 \/ h s for the mean velocity in
the channel, where h is the depth in feet and s the
rate of inclination of the surface A B. We have also
- \/1gh for the theoretical velocity of discharge at
3
the overfall, of equal depth with the channel, and,
when both velocities are equal,
- \/!Tgh zz 5-35 \/~h = 91 \HTs ;
8
from which we find
ORIFICES, WEIRS, PIPES, AND RIVERS.
139
,= -=00348,
the inclination of B A when the supply is equal to
the theoretical discharge at the overfall. If the co-
efficient at the overfall were *628, or, which is nearly
the same thing, if a large and deep weir basin inter-
vene between the weir and channel, Fig. 19, A a
would be level, the velocity of approach would be
destroyed, and we should have
5-35 X '628 \/~h = 3-36 %/T= 91 \Ths ;
and thence the inclination of A B
s = =-00136
734
very nearly. When we come to discuss the surface
inclination of rivers, we shall see that the conditions
here assumed and the resulting surface inclinations
would involve a considerable loss of head. If the
quantity discharged under both circumstances be the
same, and h be the depth in the first case, Fig. 18,
we shall then have the head in the latter case, Fig.
19, equal ( ) T h = 1-36 h very nearly, from which
vJ'SG''
and the surface inclination the extent of the back-
water may be found with sufficient accuracy. When,
in Fig. 19, the inclination of A B exceeds , the head
734
at a must exceed the depth of the river above A.
We must refer to pages further on, SECTION X, for
some remarks on the backwater curve.
140
THE DISCHARGE OF WATER FROM
SECTION V.
SUBMERGED ORIFICES AND WEIRS. CONTRACTED
RIVER CHANNELS.
The available pres-
sure at any point in
the depth of the ori-
fice A, Fig. 20, is equal
to the difference of the
pressures on each side.
This difference is equal to the pressure due to the
height k, between the water surfaces on each side of
the orifice ; in this case, the velocity is
(47.) ^
and the discharge
(48.) D = / c? c d \/2 g h ;
in which, as before, / is the length, and d the depth
of the rectangular orifice A.
When the orifice is
partly submerged, as
in Fig. 21, we may
put ^ b h = d<2 for
the submerged depth,
and k~h t = d l} the remaining portion of the depth ;
whence d l + d 2 = d is the entire depth. The dis-
charge through the submerged depth d 2 iscjd 2 ^x
\/ 2 g h, and the discharge through the upper portion
d l is
ORIFICES, WEIRS, PIPES, AND RIVERS. 141
whence the whole discharge assuming the coeffi-
cient of discharge c d is the same for the upper and
lower depths is
(49.) D = c d /\/2^m 2 \/A-)- - (h? A t 2") L
We may, however, equation (31), assume that
very nearly, and hence
(50.) D =
Ash i + -=h -^ this equation may be changed
into
(51.) = c,ld. 2
In either of these forms the values of
c d v 2 ff A,
can be had from TABLE II., and the value of the dis-
charge D thence easily found.
When the water approaches the orifice with a
determinate velocity, the height h & due to that velo-
city can be found from TABLE II., and the discharge
is then found by substituting h + A a and h t + A a for
h and h t in the above equations.
In the submerged
weir, Fig. 22, A be-
comes equal to d^ and
h t =. ; the discharge,
equation (49), then be-
comes
142 THE DISCHARGE OF WATER FROM
( 52
When the water approaches with a velocity due
to the height h & , then h becomes h + A a , h t = h &) and
equation (49) becomes
(53.) D=
In the improvement of the navigation of rivers, it
is sometimes necessary to construct weirs so as to
raise the upper waters by a given depth, d^ The
discharge D is in such cases previously known, or
easily determined, and from the values of d^ and D
we' can easily determine, equation (52), the value of
(54.)
3
or, by taking the velocity of approach into account,
(55.) d 2 =- *
This value of d 2 must be the depth of the top of the
weir below the original surface of the water, in order
that this surface should be raised by a given depth,
d lt When h & is small compared with d 2 , we may take
= 2 _ x -in equation (55).
3 3 \/d l + A a
EXAMPLE VI. A river whose width at the surface
is 7 Q feet, whose hydraulic mean depth is k&feet, and
whose cross sectional area is 325 feet, has a surface
inclination of 1 foot per mile to what depth below,
or height above the surface must a weir at right angles
\i
xv^V.
rv^v^
ORIFICES, WEIRS, PIPES, AND RIVERS. 143
to the channel be raised, so that the depth of water
immediately above it shall be increased by %\feet ?
When the hydraulic mean depth is 4-4 feet, and
the fall per mile 1 foot, we find from TABLE VIII.
that the mean velocity of the river is 29-98 or 30
inches very nearly per second. The discharge is,
therefore, 325x2izz812-5 cubic feet per second,
or 48750 cubic feet per minute. Hence, =
696*4 cubic feet, must pass over each foot in length
of the weir per minute. Assuming the coefficient
c d -628 in the first instance, we find from TABLE
VI. the head passing over a weir corresponding to
this discharge to be 27-4 inches ; but as the head is
to be increased by 3J feet, or 42 inches, it is clear
that the weir must be perfect ; that is, have a clear
overfall, and rise 42 27-4 = 14-6 inches over the
original water surface. In order that the weir may
be submerged, or imperfect, the head could not be
increased by more than 27-4 inches. Let us, there-
fore, assume in the example, that the increase shall
be only 18 instead of 42 inches; the weir then
becomes submerged, and we have, from equation (54),
d 2 = u " u * _ -i x 18" (as 7 = 1 foot).
628 \/18"x2# 3
The value of the first part of this expression is
found from TABLE VI. or TABLE II. equal to
696-4 696-4
= 07,1. ! = 1*88 feet = 22-56 m.;
- X-X 370-341 370 ' 341
18 2
net
hence 22-56 yzz 10-56 inches is the value of c? 2 ;
144 THE DISCHARGE OF WATER FROM
that is, the submerged weir must be built within
10-56 inches of the surface to raise the head 18
inches above the former level. If, however, the
velocity of approach be taken into account, we shall
find this velocity equals ,o = 2 feet per second
very nearly ; and the height, or value of A a , due to
o
this velocity, taken from TABLE II., is - = -75 inches
nearly ; therefore, from equation (55),
d = 696-4 2 (18-75)* -(-76)*
628 v/2^ x 18-75" s X \Sl8-T5
The value of x 696 ' 4 = (from TABLE VI.)
628 x/2#x 18-75"
696-4 696-4 = 1 . 84fflet = 22H)8ilL5
393-75 378 ' 81
1875
2 (18-75)* -(-75)* 2 2 (-75)*
x ^- -i x 18 ' 75 -3 -
o fiK
= 12-5 --x ^ - 12-5- -1 =z 12-4.
3 4*33
Hence d 2 = 22-08 12-4 ~ 9*68 inches, or about 1
inch less than the value previously found from equa-
tion (54). The mean coefficient of discharge was
here assumed to be -628. Experiments on submerged
weirs show that the value of c d varies from -5 up
to *8, but as this coefficient would reduce the value
of d<i, or the depth of the top of the weir below the
surface, it is safer (where a given depth above a weir
* This is found from TABLE II. more readily.
ORIFICES, WEIRS, PIPES, AND RIVERS. 145
must be obtained) to use the lesser and ordinary
coefficients of perfect weirs, with a clear overfall, for
finding the crest levels of submerged weirs, when it is
necessary to construct them. If the coefficient -8 were
used in the previous calculation, we should have found
^_ -628x^22-08 _ 124 _ 17 , 33 _ 12 . 4 = 4 . 93 in>j
or not much more than half the previous value ;
but this would only increase the whole height of the
weir by 9-68 - 4-93 = 4-75 inches.
As D = cJ\7Tg {(d, + h h] for a perfect
weir with a free overfall, it is clear that when D is
greater than -c^l \/'Yg {(^ + h^ hf}, the weir
3
is imperfect or submerged. For backwater curve
see SECTION X.
In the following table of coefficients from Lesbros*
d 2 is measured from that point below the weir
where its value is a minimum. On examining equa-
tion (52), it will be seen that the equation D zz c d /
(d l + d 2 ) v/2 g d l adopted by Lesbros is incorrect,
and can only be safely used within the limits of his
experiments.
* Vide p. 84, deuxieme Edition, Hydraulique, par Arthur
Morin. Paris, 1858.
146
THE DISCHARGE OF WATER FROM
Values of
d l
pqy
Values of the
coefficient c , in the
d
formula
D = c 1 (d + d\
d V i a/
X-V/20^.
Values of
d i
Values of the
coefficient c, in the
d
Oin ula
D =- c I (d + d \
d \ l */
X*/20d~.
d i+ d *
001
227
060
519
002
295
080
517
003
363
100
516
.-
,
150
512
004
430
200
507
005
496
250
502
006
556
300
497
007
597
350
492
008
605
400
487
009
600
450
480
010
596
^___
_^__
015
580
500
474
020
570
550
466
025
557
600
'459
030
546
700
444
035
537
800
427
040
531
900
409
045
526
1-000
390
050
522
>
The experimental values are those shown between the horizontal lines, the others above
the upper ones, and below the lower ones, were deduced from calculations bj Lesbros.
The true value of the discharge is expressed by
the equation D zz c d / j - di + d 2 r X v/2 g d 1} and
the values of c d in the above table are, therefore,
too small, applied to the correct formula. When
d^ = d 2 the table gives c d = -474. Now for weirs
in which the sheet passing over is " drowned," the
general value of the coefficient is about -67 ; this
would give the coefficient for the lower portion d 2 ,
in the true formula, equal to -503, and a mean co-
efficient c d in the correct formula (52) equal to -569
nearly. When d 2 = 200 d l9 the apparent limits of
the experiments on the other side, then the mean
value of c d zz *496 nearly in equation (52). These
results would show that the coefficient due to the
submerged depth d 2 , in the first and last experiments,
ORIFICES, WEIRS, PIPES, AND RIVERS.
147
is equal to about *5 nearly, (but varies to *6 nearly
in some of the middle experiments,) or thereabouts,
and, therefore, equation (52) for submerged weirs,
as the coefficient for the upper part d l is -67, would
become
(52A.) D = I X { '445 d, + -5 d 2 } x
which for feet measures would become again
(52B.) D = /X v/^ X {3-56 ^ + 44}*
for the discharge in cubic feet per second over a
submerged weir, Fig. 22.
CONTRACTED RIVER CHANNELS.
When the banks of a river, whose bed has a
uniform inclination, approach each other, and con-
tract the width of the channel in any way, as in
Fig. 23, the water will rise in the channel above the
contracted portion A, until the increased velocity of
discharge compensates for the reduced cross section.
If we put, as before, d for the increase of depth
immediately above the contracted width, and d 2 for
the previous depth of the channel, we shall find the
quantity of water passing through the lower depth,
d 2 , equal to cjd 2 \/2gd ly in which / is the width of
the contracted channel at A, and the quantity of
water overflowing through d l equal to - c d / d l \/2gd l ;
3
L3
148 THE DISCHARGE OF WATER FROM
and hence the whole discharge through A is
(56.) D = c d / v/^M (d 2 + * d\
\ O '
When our object is to find the width I of the con-
tracted channel, so that the depth of water in the
upper stretch shall be increased by a given depth d l9
we shall find
(57.) /=
When the velocity of approach is considerable, or
when the height h & due to it becomes a large portion
of d 1} its effect must not be neglected. In this case,
as before, we find the discharge through the depth
d 2 equal to c^ld 2 v2#(<^i + A a ) ; and the discharge
through the depth d l equal to - c d / v/2^ {(d l + A a ) 7
Ay } ; and hence the whole discharge is
(58,) = cj^{d,(
from which we shall find
(59.) /=
If the projecting spur or jetty at A be itself sub-
merged, these formulae must be extended ; the man-
ner of doing so, however, presents no difficulty, as it
is only necessary to find the discharges of the
different sections according to the preceding formulae,
and then add them together ; but the resulting for-
mula so found is too complicated to be of much
practical value.
ORIFICES, WEIRS, PIPES, AND RIVERS. 149
HEADS ARISING FROM PIERS AND BACKWATER ABOVE BRIDGES.
Equations (56), (57), (58), and (59), are applicable
to cases of contraction of river channels caused by
the construction of bridge-piers and abutments, when
the width / is put for the sum of the openings between
them. The value of the coefficient c d will depend on
the peculiar circumstances of each case; we have
seen that it rises from -5 to -7 in some cases of
submerged weirs, and for cases of contracted chan-
nels it rises sometimes as high as -8, particularly
when they are analogous to those for the dis-
charge through mouth-pieces and short tubes. When
the heads of the piers are square to the chan-
nel, the coefficient may be taken at about *6 ; when
the angles of the cut-waters or sterlings are ob-
tuse, it may be taken at about '7 ; and when curved
and acute, at -8. With this coefficient, a head of 2|
inches will give a velocity of very nearly 36 inches, or
3 feet per second ; but as a certain amount of loss
takes place from the velocity of the tail-water being
in general less than that through the arch, also from
obstructions in the passage, and from square-headed
and very short piers, the coefficient may be so small
in some cases as -5, which would require a head of
6f inches to obtain the same velocity. This head is
to the former as 54 to 21. The selection of the proper
coefficient suited to any particular case is, therefore,
a matter of the first importance in determining the
effect of obstructions in river channels : we shall have
to recur to this subject again, but it is necessary to
observe here, that the form of the approaches, the
150
THE DISCHARGE OF WATER FROM
length of the piers compared with the distance between
them, or span, and the length and form of the obstruc-
tion compared with the width of the channel, must be
duly considered before the coefficient suited to the
particular case can be fixed upon. Indeed, the coeffi-
cients will always approximate towards those, given
in the next section, for mouth-pieces, shoots, and short
tubes similarly circumstanced. For some further
remarks on contracted channels, see SECTION X.
SECTION VI.
SHORT TUBES, MOUTH-PIECES, AND APPROACHES. ALTERATION
IN THE COEFFICIENTS FROM FRICTION BY INCREASING THE
LENGTH. COEFFICIENTS OF DISCHARGE FOR SIMPLE AND
COMPOUND SHORT TUBES. SHOOTS.
The only orifices we have heretofore referred to
were those in thin plates or planks, with a few inci-
dental exceptions. It has been shown, page 48, Fig. 4,
that a rounding off, next the water, of the mouth-
piece increases the coefficient ; and when the curving
Fig. 24.
assumes the form of the vena-contracta, the coefficient
increases to -986, or nearly unity. The discharge
OKIFICES, WEIES, PIPES, AND EIVEKS. 151
from a short cylindrical tube A, Fig. 24, whose length
is from one and a half to three times the diameter, is
found to be very 'nearly an arithmetical mean between
the theoretical discharge and the discharge through a
circular orifice in a thin plate of the same diameter
as the tube, or -814 nearly. If, however, the inner
arris be rounded, or chamfered off in any way, the
coefficient will increase until, in the tube B, Fig. 24,
with a properly-rounded junction, it becomes unity
very nearly. In the conical short tubes c and D the
coefficients are found to vary according to some func-
tion of the converging or diverging angles o, o, and
according as we take the lesser or greater diameter to
calculate from. When the length of the tube exceeds
twice the diameter, the friction of the water against
the sides may be taken into account.
The following table, calculated by us, for a coeffi-
cient of friction -00699, due to a discharging velocity
of about eighteen inches per second, see SECTION VIII.,
shows the resistance arising from friction in pipes of
different lengths in relation to the diameter, and will
be found of considerable practical value. It will be
perceived that the calculations are made for three
different orifices of entry. First, when the arrises
are rounded, as in B, Fig. 24, with a coefficient of
986 ; secondly, when the arrises are square, as in A,
with a coefficient of -815 ; and, thirdly, when the pipe
projects into the vessel, when the coefficient of entry
becomes reduced to -715. The velocity is
v = c d v/2#A,
h being measured to the lower end of the tube.
152
THE DISCHAEGE OF WATER FEOM
COEFFICIENTS FOE SHORT AND LONG TUBES.
Number of diame-
ters in the length
of the pipe.
Corresponding coefficients
of discharge, showing
the effects of friction.
Number of diame-
ters in the length
of the pipe.
Corresponding coefficients
of discharge, showing
the effects of friction.
2 diameters
986
814
715
650 diameters
228
225
223
5
936
779
690
700
220
217
215
10
884
747
668
750
213
211
209
15
840
720
649
800
206
205
203
20
801
695
630
850
201
199
197
25
767
673
615
900
195
193
192
30
737
653
598
950
190
189
187
35
711
634
584
1000
186
184
183
40
693
617
570
1100
177
'176
175
45
665
601
558
1200
170
169
168
50
646
586
546
1400
158
157
156
100
513
480
458
1600
148
147
146
150
439
418
403
1800
139
139
138
200
389
375
364
2000
132
132
131
250
354
345
334
2200
126
126
125
300
327
318
311
2400
120
120
120
350
304
297
292
2600
116
116
116
400
287
280
276
2800
112
112
112
450
271
266
262
3000 ,,
108
108
108
500
258
254
250
3200
105
105
104
550
247
243
240
3400
102
102
101
600
237
234
231
3600
099
099
099
We see from this table, that the effect of adding
to the length of the pipe is greatest next the orifice of
entry. The effect of a few diameters added to the
length in long pipes is, practically, immaterial ; but
in short pipes it is considerable.
As for orifices in thin plates, so also for short
tubes, the coefficients are found to vary according to
the depth of the centre below the surface of the
water, and to increase as the depths and diameter of
the tube decrease. Poleni first remarked that the
discharge through a short tube was greater than that
OEIFICES, WEIRS, PIPES, AND RIVERS.
153
through a simple orifice, of the same diameter, in the
proportion of 133 to 100, or as -617 to -821.
CYLINDRICAL SHORT TUBES, A, FIG. 24.
The experiments of Bossut, as reduced by Prony,
give the following coefficients, at the corresponding
depths, for a cylindrical tube A, Fig. 24, 1 inch in
diameter and 2 inches long. The depths are given in
COEFFICIENTS FOE SHORT TUBES, FROM
Heads
in feet.
Coefficients.
Heads
in feet.
Coefficients.
Heads
in feet.
Coefficients.
1
818
6
806
11
805
2
807
7
806
12
804
3
807
8
805
13
804
4
807
9
805
14
804
5
806
10
805
15
-.803
Paris feet in the original, but the coefficients remain
the same, practically, for depths in English feet.
Yenturi's experiments give a coefficient *823 for a
short tube A, 1 J inch in diameter and 4j inches long,
at a depth of 2 feet 8J inches, the coefficient through
an orifice in a thin plate of the same diameter and at
the same depth being -622. We have calculated these
coefficients from the original experiments. The mea-
sures were in Paris feet and inches, from which
the calculations were directly made ; and as the
difference in the coefficient for small changes of
depth or dimensions is immaterial or vanishes, as
may be seen by the foregoing small table, and as 1
Paris inch or foot is equal to 1*0658 English inches
or feet, the former measures exceed the latter by only
154
THE DISCHARGE OF WATER FROM
about iVth. We may therefore assume that the coeffi-
cient for any orifice, at any depth, is the same, whether
the dimensions be in Paris or English feet or inches.
This remark will be found generally useful in the con-
sideration of the older continental experiments, and
will prevent unnecessary reductions from one stand-
ard to another where the coefficients only have to be
considered.
The mean value derived from the experiments of
Michelotti, at depths from 3 to 20 feet, and with short
tubes A fromi inch to 3 inches in width, isc d zz -814.
Buff's experiments* give the following results for a
tube A of an inch wide and A of an inch long, nearly.
BUFF'S COEFFICIENTS FOE SMALL SHORT TUBES.
Head
in inches.
Coefficient.
Head
in inches.
Coefficient.
Head
in inches.
Coefficient.
1*
855
6
840
23
829.
^
861
14
840
32
826
The increase for smaller tubes and for lesser depths
appears by comparing these results with the foregoing,
and from the results in themselves, generally. Weis-
bach's experiments give a mean value for c d zz -815,
and for depths of from 9 to 24 inches the coefficients
843, -832, -821, -810 respectively, for tubes -, -, -,
10' 10' 10'
and - of an inch wide, the length of each tube being
three times the diameter. D'Aubuisson and CastePs
* Annalen der Physik und Chemie von Poggendorff, 1839,
Band 46, p. 243.
ORIFICES, WEIRS, PIPES, AND RIVERS.
155
experiments with a tube -61 inch diameter and 1-57
inch long, give -829 for the coefficient at a depth of
10 feet. When a pipe projects into a cistern and has
a sharp edge, the coefficient falls so low as -715.
We have calculated the coefficients in the two fol-
lowing short tables, from Rennie's experiments with
glass orifices and tubes, Table 7, p. 435, Philosophical
Transactions for 1831. The form of the orifices, or
length of the shorter tubes is not stated, but it is
probable from the result, that the arrises of the ends
were in some way rounded off; it is stated they were
" enlarged." Indeed, the discharges from the short
tube or orifice of J inch diameter exceed the theo-
retical ones in the proportion of 1-261 to l,and 1*346
to 1. These results could not have been derived from
a simple cylindrical tube, but might have arisen from
the arrises being more or less rounded at both ends,
and the orifice partaking of the nature of a compound
tube, which may be constructed, as we shall hereafter
show, so as to increase the theoretical discharge from
1 up to 1-553. The resulting coefficients for the |
COEFFICIENTS FOE SHORT TUBES, THE ENDS ENLARGED.
Head
in feet
4 inch
diameter.
inch
diameter.
finch
diameter.
linch
diameter.
1
1-231
831
766
912
2
1-261
839
820
920
3
1-346
838
821
8SO
4
1-261
831
829
991
and | inch tubes, approach very closely to those
obtained by other experimenters, but those for the
inch tube are too high, unless the arris at the ends
was also rounded. The coefficients derived from the
156
THE DISCHAEGE OF WATEE FEOM
experiments with a cylindrical glass tube 1 foot long,
as here given, are very variable ; like the others they
COEFFICIENTS DERIVED FROM EXPERIMENTS WITH A GLASS TUBE ONE FOOT LONG.
Heads
in feet.
iinch
diameter.
| inch
diameter.
1 inch
diameter.
1 inch
diameter.
1
892
703
691
760
2
914
734
718
749
3
931
723
709
777
4
914
725
677
815
are, however, valuable, as exhibiting the uncertainty
attending "experiments of this nature," and the ne-
cessity for minutely observing and recording every
circumstance which tends to alter and modify them.
Indeed, for small tubes, a very slight difference in the
measurement of the diameter must alter the result a
good deal, particularly when it is recollected that
measurements are seldom taken more closely than
the sixteenth of an inch, unless in special cases. As
the author, however, states, p. 433 of the work re-
ferred to, that the " diameters of the tubes at their
extremities were carefully enlarged to prevent wire
edges from diminishing the sections;" this circum-
stance alone must have modified the discharges, and
would account for most of the differences.
The coefficient for rectangular short tubes differs
in no way materially from those given for cylindrical
ones, and maybe taken on an average at -814 or -815.
SHOKT TUBES WITH A ROUNDED MOUTH-PIECE, B, FIG. 24.
When the junction of a short tube with a vessel
takes the form of the contracted vein, Figs. 3 and 4,
page 48, the mean value of the coefficient c d 956,
OKIFICES, WEIRS, PIPES, AND RIVERS.
157
and the actual discharge is found to be from 93 to 99
per cent, of the theoretical discharge. Weisbach, for
a tube 1| inch long and T 9 o inch diameter, rounded at
the junction, found at 1 foot deep c d = -958, at 5 feet
deep Q969, and at 10 feet deep c d 975. These
experiments show an increase in the coefficients, in
this particular case, for an increase of depth. Any
other form of junction than that of the contracted
vein, will reduce the discharge, and the coefficients
will vary from -715 to -814, and to -986, according to
the change in the junction from the cylindrical, pro-
jecting into the vessel, to the square and properly
curved forms. The coefficients derived from Ventures
experiments will be given hereafter.
SHORT CONICAL CONVERGENT TUBES, C, FIG. 24.
The experiments of D'Aubuisson and Castel lead to
the following coefficients of discharge and velocity*
from a conically convergent tube c at a depth of 10
COEFFICIENTS FOB CONICAL CONVEEGENT TUBES.
Converging
angle o.
Coefficient
of discharge.
Coefficient
of velocity.
Converging
angle o.
Coefficient
of discharge.
Coefficient
of velocity.
1
858
858
14
943
964
2
873
873
16
937
970
3
908
908
18
931
971
4
910
909
20
922
971
5
920
916
22
917
973
6
925
923
26
904
975
80
931
933
30
895
976
lOo
937
950
40
869
980
12
942
955
50
844
985
* Traite d'Hydraulique, Paris, p. 60.
158 THE DISCHAEGE OF WATEK FROM
feet. We have interpolated the original angles and
coefficients so as to render the table more convenient
to refer to, for practical purposes, than the original.
The diameter of the tube at the smaller or discharging
orifice in the experiments was *61 inches, and the
length of the axis T57 inch; that is, the length was
2-6 times the smaller diameter of the tube. The
coefficient became *829 for the cylindrical tube, i.e.
when the angle at o was nothing. The angle of con-
vergence o determines, from the proportions, the
length of the inner and longer diameter of the tube.
The coefficients of discharge increase up to '943 for
an angle of 13 J or 14 degrees, after which they again
decrease ; but the coefficients of velocity increase as
the angle of convergence, o, increases from -829, when
the angle is zero up to -985 for an angle of 50
degrees.
When D is the discharge and A the area of the sec-
tion, we have, as before shown, D =C A A \/^gh; but
as, in conically convergent or divergent tubes, the
inner and outer areas (or, as they may be called, the
receiving and discharging sections) vary, it is clear
that, the discharge being the same, and also the theo-
retical velocity \/2 g h, the coefficient c d must vary
inversely with the sectional area A, and that c d XA
must be constant. For the coefficients tabulated, the
sectional area to be used is that at the smaller or
outside end of a convergent tube c, Fig. 24.
For a short tube c, whose length is -92 inch, lesser
diameter 1-21 inch, and greater diameter 1-5 inch,
we have found, from Venturi's experiments, that
c d 607 if the larger diameter be used in the calcu-
OEIFICES, WEIES, PIPES, AND RIVERS. 159
lation, and c d =-934 when the lesser diameter is made
use of, the discharge taking place under a pressure
of 2 feet 8J inches.
The earlier experiments of Poleni, when reduced,
furnish us with the following coefficients : A tube 7-67
inches long, 2167 inches diameter at each end, gave
c d zz-854; the like tube with the inner or receiving
orifice increased to 2| inches, c d 903 ; increased to
3-5 inches, c d 898 ; increased to 5 inches, c d 888 ;
and increased to 9*83 inches, c d 864. The depth
or head was 21-33 inches, the discharging orifice
2-167 inches diameter, and the length 7-67 inches, in
each case.
In the conically divergent tube D, Fig. 24, the co-
efficient of discharge is larger than for the same tube
c, convergent, when the water fills both tubes, and
the smaller sections, or those at the same distances
from the centres o o, are made use of in the calcu-
lations. A tube whose angle of convergence, o, is 5
nearly, with a head of from 1 to 10 feet, whose axial
length is 3 J inches, smaller diameter 1 inch, and lar-
ger diameter 1-3 inch, gives, when placed as at c, -921
for the coefficient ; but when placed as at D, the co-
efficient increases to -948. In the first case the
smaller area, used in both calculations, being the re-
ceiving, and in the other the discharging, orifice.
The coefficient of velocity is, however, larger for the
tube c than for the tube D, and the discharging jet
of water has a greater amplitude in falling. The
effects of conically diverging tubes will, however, be
better perceived from the experiments on compound
short tubes.
160
THE DISCHARGE OF WATER FROM
EFFECTS OF COMPOUND ADJUTAGES AND ADMISSION OF AIR
INTO SHORT TUBES.
If the tube A, Fig. 24, be pierced all round with
small holes at the distance of about half its diameter
from the reservoir, the discharge will be immediately
reduced in the proportion of '814 to *617. Venturi
found the reduction for a tube li inch diameter and
4i inches long, at a depth of 2 feet 10i inches, as 41
to 31, or as *823 to -622. As long as one hole re-
mained open, the discharge continued at the same
reduced rate ; but when the last hole was stopped,
the discharge again increased to the original quantity.
If a small hole be pierced in a tube 4 diameters long,
at the distance of li or 2 diameters at farthest from
the junction, the discharge will remain unaffected.
This shows that the contraction in the cylindrical
tube extends only a short distance from the junction,
probably li or li diameter, including the whole cur-
vature of the contraction.
The contraction at the entrance into a tube from a
reservoir accounts for the coefficients for a short tube
A, Fig. 24, and the short tubes, diagrams 1 and 2,
Fig. 25, being each the same decimal nearly, when
Fig. 25.
OR : or : : I : '8, or when or is not less than ORX '79,
ORIFICES/WEIRS, PIPES, AND RIVERS. 161
O Tl
and is at the distance of nearly - - from o R. The
2
form of the junction o o r R remaining as we have de-
scribed it, the following coefficients will enable us to
judge of the discharging powers of differently formed
short mouth-pieces. They have been deduced and
calculated by us, principally, from Venturi's ex-
periments.*
These coefficients show very clearly that any cal-
culations from the mere head of water and size of the
orifice, without taking into consideration the form of
the discharging tube and its connection with the re-
servoir, are very uncertain ; and that the discharge
can only be correctly obtained when all the circum-
stances of the case, including the form of the dis-
charging orifice and its approaches, have been duly
considered. .oil
JfcfM 0%0 89MCfg 6rft prated ajj I {I
When a tube similar to diagram 5, Fig. 25, has the
junction o o r R rounded, as in Fig. 4, page 48, the
outer extremity s t s T, such that s t or, ss = 9 st,
and the diameter s T = 1-8 times the diameter ,9 , with
a short central cylindrical piece o r s t between, the
coefficient of discharge corresponding to the diameter
or=:rs will increase to 1-493 or 1-555 ; that is, the
1-493 1.555
discharge is =: 2-4, or zz 2-5 times as much
*622 *622
as through an orifice (whose diameter is o r) in a thin
plate, and - - : srjK9 times as much as through a
i * 822 ;sifj ."asifour T of s a bur, . ,.'i fl {
B*6* j.,.. . .0 .o/C ni KB sniff ift/i
\ * See Nicholson's translation of Venturi's Experimental IB
quiries, published in the Tracts on Hydraulics, London, 1836.
The coefficients in the table, next page, have been all calculated
for the first time by us.
M
162
THE DISCHARGE OF WATER FROM
TABLE OF COEFFICIENTS FOR SHORT TUBES AND MOUTH-PIECES.
Description of orifice, mouth-piece, or short tube.
Coefficients
for the
diameter
OR.
Coefficients
for the
diameter
or.
1. An orifice 1 inch diameter in a thin plate . .
2. A cylindrical tube 1 inch diameter and 4 ;
inches long, A, Fig. 24
622
823
974
823
3. A short tube with a sharp end projecting intc
the cistern
715
715
4. A cylindrical tube, B, Fig. 24, having the June
tion rounded, as in Fig. 4, page 48
611
956
5. A short conical convergent mouth-piece, c, Fig
24, of the proportions of o o r R, Fig. 25
607
934
6. The like tube divergent, with the smaller dia
meter at the junction with the reservoir
length 3 inches, lesser diameter 1 inch, anc
greater diameter 1*3 inch .
561
948
7. The tube, o o u v r R, diagram 2, Fig. 25, when
o R = 1 inch, o r = 1-21 inch, u v 1-21 inch
and ouc=rv= 2 inches, the cylindrical por-
tion bein 01 shown bv dotted lines
600
923
8. The same tube when o u 11 inches
567
873
The same tube when o u 23 inches
531
817
9. The tube, oossitrn, diagram 2, Fig. 25, in
which OR=S=ST = I inch, from o to s
If inch, and ss = 3 inches, gives the same co-
efficient as the cylindrical tube, result No. 2
(see No. 19) viz
823
1-266
0. The tube, diagram 1, Fig. $5, o R = l inch. . .
1. The same tube, having the spaces oso and
r t R between the mouth-piece o o r n and the
cylindrical tube o s T R open to the influx of
804
785
1-237
1-209
12. The double conical tube, o o s TTR, diagram 3,
Fig. 25, when OR = ST = ! inch, or = l-21
inch, o o= '92 inches, and o s = 4-l inches . .
3. The like tube when, as in diagram 4, Fig. 25,
o o T R o s T T and o o s = 1*84 inch . . . .
928
823
1-428
1-266
4. The like tube when, sT = l-46 inch, and os
2'17 inches
823
1-2C6
15. The like tube when ST = 3 inches, and os =
911
1-400
16. The like tube when os = 6| inches, and ST
enlarged to 1*92 inch . .
1-020
1-569
17. The like tube when ST = 2| inches, and os
12i inches
1-215
1-855
18. A tube, diagram 5, Fig. 25, when o s = r f=3
inches, or=s=l-21 inch, and the tube
o s T r the same as described in No. 12, viz.
g T 1 1 inch and s s = 4'1 inches
895
1-377
9. The tube, diagram 2, Fig. 25, when s T is en-
larged to 1-97 inch, and s s to 7 inches, the
other dimensions remainin ' as in No 9.
945
1-454
20. When the junction of o s r t with s s T t, dia-
gram 2, Fig. 25, is improved, the other parts
850
1-309
2 ^ Another experiment gives
847
1'303
ORIFICES, WEIRS, PIPES, AND RIVERS. 163
short cylindrical tube A, Fig. 24, whose diameter is also
o r. Venturi was of opinion that this discharge con-
tinued even when the central cylindrical portion orst
was of considerable length ; but this was a mistake,
as the maximum discharge is obtained when it is
reduced so that o o r a and s s t T shall join, as in
diagram 3, Fig. 25. We see from No. 16 of the fore-
, 1-569 . 1-569
going coefficients that - = 2-52 and 7 =1-91 are,
'622 "822
perhaps, nearer to the maximum results obtainable
by comparing the discharge from a compound tube
ooSTrR, diagram 3, Fig. 25, with those through an
orifice in a thin plate, and through a short cylin-
drical tube. When the form of
Fig\26
the tube becomes curvilineal
throughout, as in Fig. 26,
s T = 1/8 o r and o s = 9 o r,
the coefficient suited to the diameter o r will be 1*57
nearly, and the discharge will be - - =2 '5 2 times as
*622
much as through an orifice o r in a thin plate.
The whole of the preceding coefficients have been
determined from circumstances in which the co-
efficient for an orifice in a thin plate was -622, and
for a short cylindrical tube *822 or -823. When the
circumstances of head and approaches in the reser-
voir are such as to increase or decrease those
primary coefficients, the other coefficients for com-
pound adjutages will have to be increased or de-
creased proportionately.
After examining the foregoing results, it appears
sufficiently clear that the utmost effect produced by
M3
164 THE DISCHABGE OF WATER FROM
the formation of the compound mouth -piece o o s T r R,
with the exception of No. 17, is simply a restoration
of the loss effected by contraction in passing through
the orifice o R in a thin plate, and that the coefficient
2 *5 applied to the contracted section at o r is simply
equal to the theoretical discharge* or the coefficient
unity, applied to the primary orifice o R ; for, as
orifice o R : orifice o r : : 1 : '64, very nearly,
when o o r R takes the form of the vena-contracta, and
the coefficient of discharge for an orifice o r in a
thin plate is -622, we get the theoretical discharge
through the orifice o R, to the actual discharge
through an orifice o r, so is 1 to -622 x '64, so is
1 : -39808 :: 1 : '4 very nearly ; and as -4 x 2'5 = 1,
it is clear that the form of the tube oosflrR^ when
it produces the foregoing effect, simply restores the
loss caused by contraction in the vena-contracta*
Venituri's sixteenth experiment, from which we have
derived the coefficients in No. 17, gives the coefficient
f%i5 m ior ine orifice d W E?^ 1i fMl f Hnmcai > ^SF thaf J
greater discharge than the theoretical, through the
receiving orifice, may be obtained. It is, however,
observable that Venturi, in his seventh proposition,
does not rely on this result, and Eytelwein's expe-
riments do not give a larger coefficient than 2-5
applied to the contracted orifice o r, which, we have
above shown, is equal to the theoretical discharge
through o R.
1 SHOOTS.
When the sides and under edge of an orifice or
notch increasie in thickness, so as to be converted
into a shoot or small channel, open at the top, the
OEIFICES, WEIKS, PIPES, AND KIVERS. 165
coefficients reduce very considerably, and to some
extent beyond what the increased resistance from
friction, particularly for small depths, indicates.
Poncelet and Lesbros* found for orifices 8" x 8", that
the addition of a horizontal shoot 21 inches long
reduced the coefficient from -604 to *601, with a head
of about 4 feet ; but for a head of 4 J inches the
coefficient fell from -572 to -483. )!i 'S)W' notches
8" wide, with the addition of a horizontal shoot
P' 1,0" long, the coefficient fell from -582 to -479 for
a head of 8"; and from -622 to -340 for a head of I".
Castel also found for a notch 8" wide, with the addi-
tion of a shoot 8" long, inclined 4 18', the mean co-
efficient for heads from 2" to 4i", to be -527 nearly.
The effects arising from ' friction alone will be per-
ceived from the short table at the beginning of this
section, p. 1521' ( TIJ > r]y ^
The orifice of entry into a shoot and its position
with reference to the sides and bottom modify the
discharge, the head remaining constant. Lesbrosf
has given the coefficients suited to different positions
: df "shootis both within and without a cistern, and
from notches and submerged Orifices ; but, however
valuable these ate in some respects, they are of little
practical use to the engineer. The general principles
which are involved in the modification of these coeffi-
cients have, however, been already pointed out by us
when discussing the effects 1 ' of ' ; fchfc 'position of ; toe
orifice, and the addition of short tubes, on the dis-
charge. Equation (74s. ), . 1 8$, ! -is here &ppllfcabte j
* Trait6 d'Hydraulique, pp. 46 et 94.
Morin's Hydrauliqne, deuxieme Edition, pp. 29 et 40.
166
THE DISCHAKGE OF WATEE FKOM
SECTION YIL
LATERAL CONTACT OF THE WATER AND TUBE. ATMOSPHERIC
PRESSURE. HEAD MEASURED TO THE DISCHARGING ORI-
FICE. COEFFICIENT OF RESISTANCE. FORMULA FOR THE
DISCHARGE FROM A SHORT TUBE. DIAPHRAGMS. OB-
LIQUE JUNCTIONS. FORMULA FOR THE TIME OF THE
SURFACE SINKING A GIVEN DEPTH. LOCK CHAMBERS.
The contracted vein
o r is about 8 times
the diameter o R; but
it is found, notwith-
standing, that water,
in passing through a
short tube of not less
than 14 diameter in length, fills the whole of the
discharging orifice s T. This is partly effected by
the outflowing column of water carrying forward and
exhausting the air between it and the tube, and by
the external air then pressing on the column so as to
enlarge its diameter and fill the whole .tube. When
once the water approaches closely to the tube, or is
caused to approach, it is attracted and adheres with
some force to it. The water between the tube and
the vena-contracta is, however, rather in a state of
eddy than of forward motion, as appears from the
experiments, with the tube, diagram 2, Fig. 25,
giving the same discharge as the simple cylindrical
tube. If the entrance be contracted by a diaphragm,
as at o R, Fig. 27, the water will also generally fill
the tube, if it be only sufficiently long. Short cylin-
drical tubes do not fill when the discharge takes place
in an exhausted receiver ; but even diverging tubes,
OKIFICES, WEIRS, PIPES, AND RIVERS. 167
V, Fig. 24, will be filled, under atmospheric pressure,
when the angle of divergence, o, does not exceed 7 or 8
degrees, and the length be not very great nor very short.
When a tube is fitted to the bottom or side of a
vessel, it is found that the discharge is that due to
the head measured from the surface of the water to
the lower or discharging extremity of the tube. It
must, however, be sufficiently long, and not too long,
to get filled throughout. G-uiglielmini first referred
this effect to atmospheric pressure, but the first
simple explanation is that given by Dr. Mathew
Young, in the Transactions of the Royal Irish
Academy, vol. vii., p. 56. Yenturi, also, in his fourth
proposition, gives a demonstration.
The values of the coefficients for short cylindrical
tubes, which we have given p. 162, have been derived
from experiment. Coefficients which agree pretty
closely with them, and which are derived from the
coefficients for the discharge through an orifice in a
thin plate, may, however, be calculated as follows :
Let c be the area of the approaching section, Fig. 27,
A the area of the discharging short tube, and a the
area of the orifice o R which admits the water from
the vessel into the tube : also put, as before, h for
the head measured from the surface of the water to
the centre of the tube, and diaphragm o R ; v for the
actual velocity of discharge at s T ; v & for the velocity
of approach in the section c towards the diaphragm
o R; and c c for the coefficient of contraction in
passing from o R to o r ; then we have c X v & = A X v,
the contracted section o r z= C Q x a, and consequently
the velocity at the contracted section is equal to
168 THE DISCHAKGE OF WATEE FKOM
= - Now a theoretical head equal to
ac c ac c
,2
: i? - -
is necessary to change the velocity v & into v by the
action of gravity ; but as the water at the contracted
A v
section o r, moving with a velocity , strikes against
ac c
the water between it and T s, moving, from the nature
of the case, with a slower velocity,* a certain loss of
effect takes place from the impact. If this be, sup-
posed, suddeD, then writers on mechanics have shown
that a loss of head, equal to that due to the difference
of the velocities,- - v, before and after the impact
must take place. This loss of head is therefore equal to
V^Au'.ii- \) y 2
-^2 * jnorli fttiw
whence we must have the whole head, ' J()1
: ewoflo] 8J3 i-)li>h
,noi)oyay[Mj^i-'g^n:jfojlJ ad o Js
(bO.| ^:-r j-jo) -ofifo^oftib
from which we find for the velocity from a short tube,
'-f< ;< J \\ f ' j-
orlt Q _ ;i(M]iJ)fu^n 9jo o^inoo oil]
Now, as v2 ^ h would be the velocity of discharge
were there no resistances^ jss sustained, it^s
( _ l _ -||
evident that J ^ _ ^ , /_A_ Tx 2 [ becomes as it
( C 2 ' \ac c ) } f'K)il gniaajsq
Sir EolJert Kane's translation of Riinhnan's book on
Horizontal Water Wheels, p, 49frjjnoo Olli Jfi Ydioolyv OilJ
ORIFICES, WEIRS, PIPES, AND RIVERS. 169
were a coefficient of velocity. When the diameter of
the diaphragm o R becomes equal to the diameter s T
of the tube, A zz a, and as the coefficient of velocity be-
comes equal to the coefficient of discharge when there
is no contraction, we get in such case this coefficient,
which we shall also call c of, expressed by the formula
oj fcaoji>i-- [ - g 1 s 1
J ' " 1 1 ^ + ( -- 1) ( '*
i t - 111 J-ilOiJOiJj' i ! Mlkll 1 ) Y'UiHHODQJJ
and when the approaching section c is very large
compared with the area A,
'.oitfAHoaM uo ^,oa-^.ao>:ATj4ifc3a uo TKaioi'a'iaoo
(63.) '" '6'-of=\,,,i ,A'
* "T ( -- 1) i 1 ',)lTAM8iaM
If c c -z -64, we shall find from the last equation
c of= -872 ; if C G -601, c o/z= -833 ; if c c = -617,
c of= -847 ; and if ^ -62l;-^/ i !> 856. These
results are in excess of those derived from experi-
ment with cylindrical short tubes, perfectly square
at the ends and of uniform bore. As some loss, how-
ever, takes place in the eddy between o r and the
tube, and from the friction at the sides, not taken
into account in the above calculation, they will ac-
count for the differences of not more than from 4 to
* When the diaphragm is placed in a> tube of uniform bore,
then c= A, and we shall get }l
0} oi/b Bestir io4i3i<mJ fdi c B/J dud ; L-
ac c a
and the loss of head, in passing the diaphragm, becomes
/A V v* aow <ro
^- 1 *'
i
It is evident from the equations that and c c depend mutually on
each other, and that they cannot be assumed arbitrarily. See equations
(66), (67)*..(1&3), (124), and (125), with the corresponding remarks.
170 THE DISCHARGE OF WATER FROM
6 per cent, between the calculation and experiment.
If C G be assumed for calculation equal *590, then
cof-=. '821 ; and as this result agrees very closely
with the experimental one, c c should be taken of this
value in using the foregoing formulae, from (60) to
(63), for practical purposes. The thickness of the
diaphragm itself and the relation of that thickness to
the diameter, as well as the form of the orifice a, are
necessary elements in the consideration of this question.
COEFFICIENT OF RESISTANCE. LOSS OF MECHANICAL POWER
IN THE PASSAGE OF WATER THROUGH THIN PLATES AND
PRISMATIC TUBES.
The coefficients of contraction, velocity, and dis-
charge have been already denned. The coefficient of
resistance is the ratio of the head due to the resistance,
to the theoretical head due to the actual or final
velocity. If v be this latter velocity, the theoretical
head due to it is ^- ; and if c r be the coefficient of
resistance, then the head due to the resistance itself
is, from our definition, c r x s-. Now if c v be the
coefficient of velocity, the theoretical velocity of dis-
charge must be , and the head due to it is equal
-a 2"-; k u t as the theoretical head due to v is
s , we shall have
for the head due to the resistance ; and, therefore,
from our definition, the coefficient of resistance
ORIFICES, WEIRS, PIPES, AND RIVERS.
171
(64)
from which we shall find the coefficient of velocity
These equations enable us to calculate the coefficient
of resistance from the coefficient of velocity, and
vice versa. If c v = 1, c r = 0, as it should be. The
following short table, calculated from equation (65),
will be of use. In short tubes, the coefficient of
velocity c v is equal to the coefficient of discharge c d .
COEFFICIENTS OF VELOCITY AND EESISTANCE.
Coefficient
of velocity.
Coefficient
of resistance.
Coefficient
of velocity.
Coefficient
of resistance.
Coefficient
of velocity.
Coefficient
of resistance.
990
020
910
208
830
452
970
063
890
263
820
488
950
109
870
320
814
508
930
156
850
383
810
525
The coefficient of velocity for an orifice in a thin
plate, or for a mouth-piece, Fig. 4, is -974 ; while
that for a short prismatic tube, A, Fig. 24, is -814
nearly. The coefficient of resistance in the former
case is -054, and in the latter -508 ; there is, there-
fore, 9-4 times as great a loss of mechanical power in
the passage through short prismatic tubes, as through
orifices in thin plates or tubes with a rounded junc-
tion, as in Fig. 4, the quantities of water discharged
and the discharging velocities being the same.
If the quantities discharged and the heads be the
same in both cases, then we shall have
2 2
equal the head ;
* See the tables of resistances, discharge, and contraction,
pp. 174 and 176.
172 THE DISCHARGE OF WATER FROM
that is > -663 x 2 ff = -949 1 2g> r " 949 * = ' 663 v '
whence we get v\ = -698 v\ and v* =. I -43 1 v* for the re-
lation of the discharging velocities, v , from an orifice,
and, Vty from a short tube. The height due to the re-
1 t' 2
sistance is therefore, ('.oi 4 2 ~~ l) o 1 for short pris-
matic tubes, and (79742 ~~l) ~o - ' for orifices in
thin plates. These are to each other as -508 to
054 x 1-431, or as 5-08 to -773, that is to .say,
the loss of mechanical power arising from the re-
sistance in passing through short tubes is 6 '5 7 times
as great as when the water passes through thin plates
or mouth-pieces, as in Fig. 4 ; and the discharging
mechanical power in plates, is to that in tubes as
1*431 to 1, or as 1 : *698, the heads and quantities
discharged being the same^ ol9V ^ j
The whole loss of mechanical power in the passage
is 5*4 per cent, for the plates, and about 51 per cent.
for short tubes. If the loss compared with the whole
head be sought, we get, when v is the discharging
velocity, ^j for the theoretical velocity due to the
mi& oilBflighq ho/Ia irgiJO'iJ) ouc8S./$q 9|[J
head in short tubes, and its square Tgrp = TgTjo
is as the whole head ; therefore, the whole head is to
.9iHa oil,t gniod ssiiioolsv guigidBdoaibi^dJ bn^j
the head due to the discharging velocity as ^ to v\
or as 1 to *663 ; and as -508 is the coefficient of re-
sistance* for the discharging velocity, '508 x '663
zz '337 is the coefficient of resistance due to the
:injJaiao'i i
* Table, p. 171. ,jTl brm iTJ .qq
ORIFICES, WEIRS, PIPES, AND RIVERS. 173
whole head ; this is equal to a loss of 34 per cent,
nearly, or about one-third. In like manner, we find
974 2 x '054 = -0512 for the coefficient when the dis-
charge takes place through thin plates, or 5| per
cent, of the whole head.
DIAPHRAGMS! '^> ^
When a diaphragm, o R, Fig. 27, is placed at the
entrance of a short tube, we have shown, page 168,
A 2
janiwollirt orfKIwijsIiiofBo r (^ c ~ 1 )^ A ui _
that a loss of head equal - -^ - takes place
when v is the discharging velocity, whence the co-
efficient of resistance is equal to ( l) ,* according
.31/roAar-fiAirt ao't aD/tATaraaa ax A ,sfo;. ^ ^c
to our definition. The coefficient of contraction c c ,
as we have before shown, page 170, should be taken
equal to -590 in the application of formula (63) ; and,
as it must also be taken equal to about *621 when the
area of the tube i'ls very large compared with the
area a of the orifice o R in the diaphragm, we may
assume that when is equal to
A
0, .!. -2, -3* -4, -5, -6, -7, -8, -9, and 1
O.aoU^jTLiilO;! 10 JiTQi . , , , , , , ,
successively, the coefficient c c must be taken equal to
621, -TO^HO? 48, '609, -606, -603, -600, -597, -593, and -590,
in the same order. As the approaching section c
' * tor the sudden alteration in the velocity passing through a
diaphragm, we must reject the hypothesis of D'Aubuisson,
Traite d'Hydraulique, p. 238, and adopt that of Navier, taking
the loss of head to correspond to the square of the difference and
not to the difference of the squares of the velocities in and after
passing the orifice. The coefficient of contraction must, however,
be varied to suit the ratio of the channels, as it is in this and the
following pages.
174
THE DISCHARGE OF WATER FROM
may be considered exceedingly large, the value of
the coefficient of discharge or velocity, as the tube
o R s T is supposed full, in equation (61), becomes
(66.) *
and the coefficient of resistance
( 67 -)
from which equations and the above values of c c , cor-
responding to -, we have calculated the following
A.
values of the coefficients of discharge and resistance
through the tube o K s T, Fig. 27.
COEFFICIENTS OF CONTRACTION, DISCHARGE, AND RESISTANCE FOR DIAPHRAGMS.
Ratio
1
1
1
Ratio
1
"S
.2
1
a
6 ^
s j 3
I*
a
3 fe
I*
ss ***
A *
1
1
1
A *
1
1
1
o-o
621
000
infinite.
0-6
603
493
3-115
0-1
618
066
231-
0-7
600
587
1-907
0-2
615
139
50-8
0-8
597
675
1-198
0-3
612
219
. 19-8
0-9
593
753
762
0-4
609
307
9-6
1-0
590
821
483
0-5
606
399
5-3
-
In this table c c is the coefficient of contraction, c d
the coefficient of discharge, suited to the larger section
of the pipe A, at s T ; and c r the coefficient of re-
sistance. The discharge is found from equation (61),
as c is here very large compared with A, to be
1
(67A.) D = A
ORIFICES, WEIRS, PIPES, AND RIVERS. 175
A ^
The coefficient of resistance c r is here equal ( 1) ,
and the coefficient of discharge c 6 zz ^r-- -TI-*
v* T c *)
The tube must be so placed, that the water, after
passing the diaphragm, shall fill it ; for instance,
between two cisterns, when the height h must be
measured between the water surfaces, or when the
tube is sufficiently long to be filled ; in this case,
however, the height must be determined from the dis-
charging velocity , as a portion of the head is required
to overcome the friction, which we shall have im-
mediately to refer to more particularly.
The table shows that the head due to the resistance
is 5*3 times that due to the discharging velocity,
when the area of the diaphragm is half the area of
the tube ; that is, the whole head required is 6-3
times that due to the velocity, and that the coeffi-
cient of discharge is reduced to -399. In order to
find the coefficients suited to the smaller area of the
orifice in the diaphragm o R, when it is to be used in
calculations of the discharge, we have only to divide
the numbers corresponding to into those of c d , op-
A
posite to them in the table. Thus, when - zz -8, we
A
have the coefficient of discharge suited to the area a,
* For the loss sustained by contraction in the bore of a pipe
by a diaphragm, see equations (123), (124), and (125). The
actual value of c c in equation (67A) depends on the thickness of
the diaphragm as well as on the relation of a and A. The form
of the orifice a also affects the value of c c .
178
THE DISCHARGE OF WATER FROM
67
Fig-,28
od i
equal -^- = -844, and so of other values of the
ratio -. The cc efficients in the table, page 174, are
A
for the larger orifice A in the formula D = A c d \/2 g h.
^onuiaflKHOWi TUBES i SOBLIQUE AT THE JUNCTION.
When a tube is at- fp
tached obliquely, as in
Fig. 28, we have found
that if the number of
degrees in the angle
TO s, formed by the
direction of the tube o s, with the perpendicular o T,
be represented by <, then -814 -0016 <j> will give the
coefficient of discharge corresponding to the obliquely
attached short tube in the Figure. This formula is,
however, empirical, but it is simple, and agrees pretty
closely with experimental results. As the coefficient
[ . .UGG- . 1 '(. at as'fGf/Dsib lo irr^b
of resistance is equal -, 1, equation (64), we have
odi To -co isucma 9dKc3 Dan
here ^r ^i , __ -6oi6 V> 2 ~" ^ ; ^ rom ^ ese equations
we Have calculated the following table for heads
measured to the middle of the outside orifice ; :-rm(f
COEFFICIENTS OF DISCHARGE AND RESISTANCE FOR OBLIQUE JUNCTIONS.
in degrees.
Coefficient
of discharge.
Coefficient
of resistance.
in degrees.
Coefficient
of discharge.
Coefficient
of resistance.
814
508
35
758
740
5
806
539
40
750
778
10
15
20
"E
569
603 sr
635 (A
45
i B i ,-'742' -
734
81<6 [
3,^87
25
. h7T.4 E '!
ingJWth-i
60
718
940;;ili
30
766
704
ilV tidj gfO
710
984
ORIFICES, WEIRS, PIPES, AND RIVERS.
177
The coefficient of resistance for a tube at right angles
to the side, is to the like coefficient when it makes
an angle of 45 degrees as -508 to -816, or as 1 to 1-6
nearly ; and the loss of head is greater in the same
proportion. If the short tube be more than three or
four diameters in length, friction will have to be taken
into account. The head h is measured to the outside
orifice.
FORMULA FOR FINDING THE TIME THE SURFACE OF WATER
IN A CISTERN TAKES TO SINK A GIVEN DEPTH. DIS-
CHARGE FROM ONE VESSEL OR CHAMBER INTO ANOTHER.
LOCK CHAMBERS.
In experiments for find-
ing the value of the coeffi-
cients of discharge, one of
the best methods is to ob-
serve the time the water
discharged from the orifice
takes to sink the surface in a prismatic cistern a given
depth; the ratio of the observed to the theoretical
time will then give the coefficient sought. A formula
for finding the theoretical time is, therefore, of much
practical value. In Fig. 29, the time of falling from
s t to s T, in seconds, is
1
F 1 i *F
Fi.29
_ _
---i
1
.
'
'
R
...^
in which a is the area of the orifice o R, and A the
area of the prismatic vessel at st or ST; this formula is
for measures in feet. For measures in inches, we have
EXAMPLE VII. A cylindrical vessel 5 *74 inches in
diameter has an orifice *2 inch in diameter at a depth
178
THE DISCHARGE OF WATER FROM
0/"16 inches below the surface, measured to the centre ;
it is found that the water sinks 4 inches in 51 seconds ;
what is the coefficient of discharge ?
The theoretical time t is found from equation (69),
equal
5-74*X-785 6> _ ^3^
JL ij y X *^j X io04: "OOO
1 T.fJKfJf? Q~f .Q
= -556 ~ X '5359 =31-8 seconds; hence,-gpzz-624
is the coefficient sought. When the orifice o R and
the horizontal section of the vessel are similar figures,
A S T 2
- is equal ^ ; and therefore, for circular cisterns
tt OR
and orifices, it is unnecessary to introduce the mul-
tiplier -7854.
We have given above, formulae for -the time water in
a prismatic vessel takes to fall a given depth, when dis-
Fig. 29<*.
charged from an orifice at the side or bottom. The time
' the surface s T, diagram 1, Fig. 29#, takes to rise
to s t, when supplied through an orifice or tube o R,
from an upper large chamber or canal, whose surface
2 A/*
s 1 tf remains always at the same level, is "TTo^'*
* The time of rising from s to s is exactly double the time it
would take, if the pressure/ remained uniform, to fill the same
depth below R.
ORIFICES, WEIRS, PIPES, AND RIVERS. 179
and we thence get the time of rising from R to s for
measures in feet
and for measures in inches
( 69B -) '=
in which A is the area of the horizontal section at s T ;
a the sectional area of the communicating channel or
orifice o R ; c d the coefficient of discharge suited to it,
and ^ and/, as shown in the diagram.
In order to find the time of filling the lower vessel
to the level s T, supposing it at first empty, we have
the contents of the portion below o R equal to A^ 2 >
and the time of filling it equal to
(69c.)
8-025 c d ah\
then the time of filling up to any level s T, for mea-
sures in feet, is equal to the sum of (A) and (c) ;
that is,
8-025 c d ah$
and for measures in inches
27-8 c d a
"When s T coincides with * t
A (2 A, + fl)
T = 8-025 c,ahi>
for measures in feet, and
N3
180 THE DISCHARGE OF WATER FROM
A (2 fr + ft,)
27-8 c,ah\ >
for measures in inches. These equations are ex-
actly suited to the case of a closed lock-chamber
filled from an adjacent canal.
When the upper level s' T' is also variable, as in
Diagram 2, the time which the water in both vessels
takes to come to the same uniform level s' t' s t, is
2AQ(A 1 +/ 1 -j)* = 2AO(/+/ 1 )*_
' c d a(A+c)v/2^ c d tf(A+c)v/2#'
in which h + jfj h = f -f f^ is the difference of
levels at the beginning of the flow ; c the horizontal
section of the upper chamber ; and the other quanti-
ties as in Diagram 1. As c^ =: Af, we find
Now, in order to find the time of falling a given
depth d below the first level s' T', we have the head
above s't's t equal to^ d in the upper vessel, and
the depth below it in the lower vessel equal to
^ -- ; whence the difference of levels in the two
A
vessels at the end of the fall d, is
The time of falling through d is, therefore, from
equation (69 H),
2AC
ORIFICES, WEIRS, PIPES, AND RIVERS. 181
in which \^g = 8*025 for measures in feet, and
equal 27*8 for measures in inches. The whole time
of filling to a level the lower empty vessel, is found
by adding the time of filling the portion below R, de-
termined in a manner similar to equations (68) and
(69) to be
to the time of filling above R, given in equation (H),
when h is taken equal to zero. Equations (H), (i),
and (K) are applicable to the case of the upper and
lower chambers of a double lock, after making the
necessary change in the diagrams.
The above equations require further extensions
when water flows into the upper vessel while also
flowing from it into the lower ; such extensions are,
however, of little practical value, and we therefore
omit them. For sluices in flood-gates with square
arrises, c d may be taken at about -545, but with
rounded arrises, the coefficient will rise much higher.
See SECTIONS III. and VI.
182 THE DISCHARGE OF WATER FROM
SECTION VIII.
FLOW OF WATER IN UNIFORM CHANNELS. MEAN VELOCITY.
MEAN RADII AND HYDRAULIC MEAN DEPTHS. BORDER.
TRAIN. HYDRAULIC INCLINATION. EFFECTS OF FRIC-
TION. FORMULA FOR CALCULATING THE MEAN VELOCITY.
APPLICATION OF THE FORMULA AND TABLES TO THE
SOLUTIONS OF THREE USEFUL PROBLEMS.
In rivers the velocity is a maximum along the
central line of the surface, or, more correctly, over
the deepest part of the channel ; and it decreases
thence to the sides and bottom : but when backwater
arises from any obstruction, either a submerged
weir, Fig. 22, or a contracted channel, Fig. 23, the
velocity in the channel approaching the obstruction
is a maximum at the depth of the backwater below
the surface, and it decreases thence to the surface,
sides, and bottom. When water flows in a pipe of
any length, the velocity at the centre is greatest, and
it decreases thence to the sides or circumference of
the pipe. If the pipe be supposed divided into two
portions in the direction of its length, the lower por-
tion or channel will be analogous to a small river
or stream, in which the velocity is greatest at the
central line of the surface, and the upper portion will
be simply the lower reversed. A pipe flowing full
may, therefore, be looked upon as a double stream,
and we shall soon see that the formulae for the dis-
charge from each kind are all but identical, though a
pipe may discharge full at all inclinations, while the
inclinations in rivers or streams, having uniform
motion, never exceed a few feet per mile.
ORIFICES, WEIRS, PIPES, AND RIVERS. 183
MEAN VELOCITY.
It is found, by experiment, that the mean velocity
is nearly independent of the depth or width of the
channel, the central or maximum velocity being the
same. From a number of experiments, Du Buat
derived empirical formulae equivalent to
v= V f=v^ v*+i, v b =(v* - I) 2 , and v=(v|+l) 2 ;
in these equations v is the mean velocity, v the max-
imum surface velocity, and v b the velocity at the
sides, or bottom, expressed in French inches. Tables
calculated from these formulae do not give correct
results for measures in English inches, though they
are those generally adopted. Disregarding the dif-
ference in the measures, which are as 1 to 1*0678, it
will be found that, in the generality of channels, the
mean velocity is not an arithmetical mean between
the velocity at the central surface line and that at
the bottom, though nearly so between the mean
bottom and mean surface velocities. Dr. Young,*
modifying Du Buat's formula, assumes for English
inches that v + v* zz v, and hence vzzv + i (v + 1)*
This gives results very nearly the same as the other
formula for v, but something less, particularly for
small surface velocities. For instance, Du Buat's
formula gives 5 inch for the mean velocity when
the central surface velocity is 1 inch, whereas Dr.
Young's makes it -38 inch. For large velocities both
formulas agree very closely, disregarding the differ-
ence between the measures, which is only seven per
* Philosophical Transactions, 1808, p. 487.
184 THE DISCHARGE OF WATER FROM
cent. They are best suited to very small channels or
pipes, but unless at mean velocities of about 3 feet
per second, they are wholly inapplicable to rivers.
Prony found, from Du Buat's experiments, that
/2-37187 + vx
for measures in metres v (Q.I^QTO ! ) v > m which
v is also the maximum surface velocity. This, re-
duced for measures in English feet, becomes
and for measures in English inches,
93*39
For medium velocities v = -81 v. The experiments
from which these formulae were derived were made
with small channels. We have calculated the values
of v from that of v, equation (71), and given the
results in columns 3, 6, and 9, in TABLE VII.
Ximenes, Funk, and Briinning's experiments in
larger channels give the mean velocity at the centre
of the depth equal '914 v, when the central or
maximum surface velocity is v ; but as the velocity
also decreases in nearly the same ratio at the surface
from the centre to the sides of the channel, we shall
get the mean velocity in the whole section equal
* Francis, Lowell Experiments/p. 150, finds this formula to
give 1 5 per cent, less than the result found hy weir measurement
from the formula D = 3 '33 (I ! n h) hi, the quantity discharged
being about 250 cubic feet per second, and the velocity about 3 '2
feet. It appears, however, that Francis uses the mean surface
velocity, and not the maximum surface velocity required by the
formula : if the latter were used, the difference would be reduced
to 6 per cent., or thereabouts, in equation (72).
^ , V*X^. fc^X^oL V^UStAjfc-^jj.
JU
2.
v v
^
"^<\\\,
.
0.
.
* C .
Q
O
ORIFICES, WEIRS, PIPES, AND RIVERS. 185
914 x '914 v z= '835 v ; and hence we have, for
large channels,
(72.) v = -835 v,
in which equation v is the maximum velocity at the
surface. We have also calculated the values of v
from this formula, and given the results in columns
2, 5, and 8 of TABLE VII. This table will be found
to vary considerably from those calculated from Du
Buat's formula in French inches, hitherto generally
used in this country, and much more applicable for
all practical purposes.
MEAN KADITJS. HYDRAULIC MEAN DEPTH. BORDER.
COEFFICIENT OF FRICTION.
If, in the diagrams 1 and
2, Fig. 30, exhibiting the
sections of cylindrical and
rectangular tubes filled with
flowing water, the areas be
divided respectively by the perimeters A c B D A and
A B D c A, the quotients are termed " the mean radii"
of the tubes, diagrams 1 and 2 ; and the perimeters
in contact with the flowing water are termed "the
borders." In the diagrams 3 and 4, the surface A B
is not in contact with the channel, and the width of
the bed and sides, taken together, A c D B, becomes
" the border." " The mean radius " is equal to the
area A B D c A divided by the length of the border
A c D B. " The hydraulic mean depth" is the same
as " the mean radius" this latter term being perhaps
most applicable to pipes flowing full, as in diagrams
1 and 2 ; and the former to streams and rivers which
186 THE DISCHARGE OF WATER FROM
have a surface line A B, diagrams 3 and 4. We shall,
throughout the following equations, designate the
value of the " mean radius," " hydraulic mean depth,"
. area A B D c A Jt
or quotient, border BDCA ? b J the letter r > remarking
here that for cylindrical pipes flowing full, or rivers
with semicircular beds, it is always equal to half the
radius, or one-fourth of the diameter.
Du Buat was the first to observe that the head
due to the resistance of friction for water flowing in
a uniform channel increased directly as the length of
the channel /, directly as the border, and inversely
as the area of the cross-flowing section,! very nearly ;
that is, as -. It also increases as the square of the
velocity, nearly; therefore the head due to the re-
v* I v z I
sistance must be proportionate to - If c t X ^
zz h { , then c f is the coefficient for the head due to the
resistance of friction, as h f is the head necessary to
overcome the friction ; c is therefore termed " the co-
efficient of friction"
* M. Girard has conceived it necessary to introduce the coeffi-
cient of correction 1-7 as a multiplier to the border for finding r,
to allow for the increased resistance from aquatic plants ; so that,
according to his reduction,
area
A* .... . ,
1*7 border*
See Eennie's First Keport on Hydraulics as a Branch of En-
gineering ; Third Eeport of the British Association, p. 167 ; also,
equation (85), p. 201.
f Pitot had previously, in 1796, remarked that the diminution
arising from friction in pipes is, cateris paribus, inversely as the
diameters.
ORIFICES, WEIRS, PIPES, AND RIVERS.
187
HYDRAULIC INCLINATION. TRAIN.
If / be the length of a pipe or channel, and h { the
height due to the resistance of friction of water
flowing in it, then -/ is the hydraulic inclination. In
i
Fig. 31 the tubes A B, c D, of the same length /, and
Fig.31
whose discharging extremities B and D are on the
same horizontal plane B D, will have the same hy-
draulic inclination and the same discharge, no matter
what the actual inclinations or the depth of the en-
trances at A and c may be, so they be of the same
kind and bore ; and as the velocities in A B and c D
are the same, the height h due to them must be the
same when the circumstances of the orifices of entry
A and c are alike. We have the whole head H h +
h f (see pp. 166 and 167). The hydraulic inclination is
not therefore the whole head H, divided by the length
/ of the pipe, as it is sometimes mistaken, but the
height h f (found by subtracting the height h, due to
the entrance at A or c, and the velocity in the pipe,
from the whole height) divided by the length /.
When the height h is very small compared with the
whole height H, as it is in very long tubes with
moderate heads, j may be substituted for j without
error ; but for short pipes up to 100 feet in length
188
THE DISCHARGE OF WATER FROM
the latter only should be used in applying Du Buat's
and some other formulae ; otherwise the results will
be too large, and only fit to be used approximately
in order to determine the height h from the velocity
of discharge thus found. When the horizontal pipe
c D, Fig. 32, is equal in every way to the inclined
pipe A B, and the head at A is that due to the velocity
in c D, the discharge from the pipe A B will be equal
to that from c D ; but a peculiar property belongs
to the pipe A B in the position in which it is here
placed ; for if we cut it short at any point e, or
lengthen it to any extent, to E, the discharge will re-
main the same and equal to that through the hori-
zontal pipe c D. The velocity in A B at the angle of
inclination ABC, when A c = h { , and A B zi c D, is
therefore such that it remains unaffected by the
length A E or A e, to which it may be extended or
cut short ; and at this inclination the water in the
pipe A B is said to be " in train." In like manner a
river or stream is said to be "in train" when the
inclination of its surface bears such a relation to
the cross section that the mean velocity is neither
decreased nor increased by the length of the chan-
nel ; and we perceive from this that the acceleration
caused by the inclination is exactly counterbalanced
ORIFICES, WEIRS, PIPES, AND RIVERS. 189
by the resistances to the motion when the moving water
in a pipe or river channel is in train.
o
As h = (1 + r ) o where c r is the coefficient of
the height due to the resistance at the orifice of entry
V 2 l
A or c, and h { = c { * , we therefore get
(73.) H = (l+c t ) + c f X r
and hence we find the mean velocity of discharge
as c\ zz j-y, equation (65). We have also
C T -f- 1
(74B.) v z
the values of the second member on the right hand
side of this equation, or of
fcM?feb$ 58S
UiP r]
are given, for different values of c , c d , and -, in the
small table at p. 152, and below at p. 191.
When h is small compared with A f , or, which comes
to the same thing, 1 + c t small compared with c t x - f
190 THE DISCHARGE OF WATER FROM
Iv*
(75.) v = c t
and
TT
If, in the last equation, we substitute s for y, equal
the .sine of the angle of inclination A B c, we then
have
(77.) || . =
The average value of c f for all pipes with straight
channels, with velocities of about 1*5 foot per second,
is -0069914, from which we find equation (77) be-
comes, for measures in feet,
(78.) v = 96 \/rs.
As the mean value of the coefficient of resistance c y
for the entrance into a tube is '508, and as
2# z= 64-403, and c f = -0069914, equation (74), for
measures in feet, becomes
f 64 -403 H U
v= ~~T , or
1-508 + -0069914-
-0234r+-OOOi085/J ~(234-hl-085/
(79.)
* \\
58c/+l-085/J *
This, multiplied by the section, gives the discharge.
For velocities between 2 and 2J feet per second,
c t zz -0064403, and therefore
ORIFICES, WEIRS, PIPES, AND RIVERS,
Hr ^1
191
UI 50
0234 r + -0001 /
in which d = 4 r i= diameter of a pipe.
The following table is calculated from equation
VALUES or -
3 + * X -
C 7*
N umber of diame-
cers in the length
of the pipe.
Corresponding coeffici-
ents of discharge.
Number of diame-
ters in the length
of the pipe.
Corresponding coeffici-
ents of discharge.
2 diameters
986
814
715
900 diameter
239
236
233
5
957
791
698
950 " +
234
230
227
10
919
769
683
1000
228
225
222
15
886
749
669
noo
218
215
213
20
855
731
656
1200
209
207
205
25
828
713
643
1400
194
192
191
30
804
698
632
1600
182
180
179
35
781
683
620
1800
173
171
170
40
760
668
610
2000
163
162
161
45
741
655
600
2200
156
155
154
50
723
643
590
2400
149
149
148
100
595
548
514
2600
144
143
142
150
518
485
462
2800
139
138
137
200
464
440
422
3000
134
133
133
250
424
405
391
3200
130
129
129
300
392
378
366
3400
126
125
125
350
367
356
345
3600
122
121
121
400
346
336
329
3800
119
119
118
450
329
319
314
4000
116
116
115
500
314
307
300
4200
113
113
113
550
301
295
289
4400
111
111
111
600
289
283
278
4600
108
108
108
650
279
273
269
4800
106
106
106
700
269
265
261
5000
104
104
104
750 ;-;,
261
257
253
5200
102
102
102
800
253
249
246
5400
100
100
100
850
246
242
239
5600
098
098
098
192 THE DISCHARGE OF WATER FROM
(74B) for a velocity of about 20 feet per second,
when c f zr -004556, and for different orifices of entry,
in which c d varies from -986 for a rounded orifice,
to '715 when the pipe projects into the vessel. It
gives directly the coefficient, which, multiplied by
H, gives the velocity in the pipe, taking friction
into account.
The small table, SECTION VI., p. 152, gives the like
coefficients of v/2#H in equation (74B), when c f =
00699 suited to a velocity of about 18 inches per
second, and can be applied in like manner. The
value of \/2#H is given, in inches, in column 2,
TABLE II. For feet it is equal 8v/i nearly.
DU BUAT'S FORMULA.
The coefficient of friction c t is not, however, con-
stant, as it varies with the velocity. That which
we have just given answers for pipes when the
velocity is 20 feet per second. For pipes and
rivers it is found to increase as the velocity de-
creases ; that is, the loss of head is proportionately
greater for small than for large velocities. Du Buat
found the loss of head to be also greater for small
than large channels, and applied a correction accord-
ingly in his formula. This, expressed in French
inches, is
(80 .) = - - , - -3 (H - -I),
(^-hyp-iog^-M-e/
maintaining the preceding notation, in which s j.
In this formula -1, in the numerator of the first term,
is deducted as a correction due to the hydraulic
ORIFICES, WEIRS, PIPES, AND RIVERS. 193'*ITY
\s^U ff
mean depth, as it was found that 297 (ri 0-1) agreed
more exactly with experiment than 297 rf simply. The
f \ \*
second term hyp. log. (^ + I'^J, of the denominator
is also deducted to compensate for the observed loss
of head being greater for less velocities, and the last
term -3 (rl _ -1) is a deduction for a general loss of
velocity sustained from the unequal motions of the
particles of water in the cross section as they move
along the channel. These corrections are empirical ;
they were, however, determined separately, and after
being tested by experiment, applied, as above, to the
radical formula v = 297 \/r~s.
Du Buat's formula was published in his Principes
d'Hydraulique, in 1786. It is, as we have seen,
partly empirical, but deduced by an ingenious train
of reasoning and with considerable penetration from
about 125 experiments, made with pipes from the
19th part of an inch to 18 inches in diameter, laid
horizontally, inclined at various inclinations, and
vertical ; and also from experiments on open chan-
nels with sectional areas from 19 to 40,000 square
inches, and inclinations of from 1 in 112 to 1 in
36,000. The lengths of the pipes experimented with
varied from 1 to 3, and from 3 to 3600 feet.
In several experiments by which we have tested
this formula, the resulting velocities found from it
were from 1 to 5 per cent, too large for small pipes,
and too small for straight rivers in nearly the same
proportion. As the experiments from which it was
derived were made with great care, those with pipes
particularly so, this was to be expected. Expe-
194 THE DISCHARGE OF WATEK FROM
riments with pipes of moderate or short lengths
should have the circumstances of the orifice of entry
from the reservoir duly noted ; for the close agree-
ment of this formula with them must depend a great
deal, in such pipes, on the coefficient due to the
height h, which must be deducted from the whole
head H before the hydraulic inclination, -j- zz s, can
be obtained ; but for very long pipes and uniform
channels this is not necessary.
The experiments from which Du Buat's formula
was constructed are given in full by the late Dr.
Eobinson in his able article on "rivers" in the
Encyclopaedia Britannica, pp. 268, 269, and 270,
where the calculated and observed velocities are
placed side by side in French inches per second. In
all these experiments Du Buat carefully deducted
the head due to the velocity and orifice of entry before
finding the hydraulic inclination s, and those who
attempt to calculate the velocity from the head and
length of the channel only, without making this
deduction, will find their calculated results very dif-
ferent from those there given. If there were bends,
curves, or contractions, deductions would have to be
made for these in like manner before finding s.
Under all the circumstances, and after comparing
the results obtained from various other formulae, we
have preferred calculating tables for the values of v
from this formula reduced for measures in English
inches, which is
=
OEIFICES, WEIKS, PIPES, AND RIVERS. 195
or more simply,
, Q1 x 307 (r 1)
(81.) v = --- - - -_. 3r *_.i.
This gives the value of v a little larger than the
original formula, but the difference is immaterial.
For measures in English feet it becomes
(82.) = t 88-51 (r*- -03) _ _ . 084 (fi _ . 03>
(-)-hyp.log.(- + l-6)'
The results of equation (81) are calculated for
different values of s and r, and tabulated in TABLE
VIII. , the first eight pages of which contain the
velocities for values of r varying from Tsth inch to
6 inches ; or if pipes, diameters from | inch to
2 feet, and of various inclinations from horizontal to
vertical. The last five pages contain the velocities
for values of r from 6 inches to 12 feet, and with
falls from 6 inches to 12 feet per mile.
EXAMPLE VIII. A pipe, 1J inch diameter and
100 feet long, lias a constant head of 2 feet over the
discharging extremity ; what is the velocity of dis-
charge per second ?
II 3. , .,100 1
The mean radius r=- = - inches, and zz50n:-,
48 2 S
is the approximate hydraulic inclination. At page 2
of TABLE VIII., in the column under the mean radius
o
-, and opposite to the inclination 1 in 50, we find
8 f
30*69 inches for the velocity sought. This, however,
is but approximative, as the head due to the velocity
should be subtracted from the whole head of 2 feet,
before finding the true hydraulic inclination. This
o3
196 THE DISCHAKGE OF WATEE FEOM
head depends on the coefficient of resistance at the
entrance orifice, or the coefficient of discharge for a
short tube. In all Du Buat's experiments this latter
was taken at -8125, but it will depend on the nature
of the junction, as, if the tube runs into the cistern,
it will become as small as -715 ; and, if the junction
be rounded into the form of the contracted vein, it
will rise to -974, or 1 nearly. In this case, the co-
efficient of discharge may be assumed -815,* from
which, in TABLE II., we find the head due to a velo-
city of 30-69 inches to be 1- = 1-87 inch nearly,
which is the value of h ; and hence, H h h t = 24
- 1-87 = 22-13 inches ; and| = ^^ = 54 ' 2 =p
the hydraulic inclination, more correctly. With this
O i '
new inclination and the mean radius -, we find the
8
velocity by interpolating between the inclinations
1 in 50 and 1 in 60, given in the table to be
30-69 _ 1-34 ~ 29-35 inches per second. This
operation may be repeated until v is found to any
degree of accuracy according to the formula ; but it
is, practically, unnecessary to do so. If we now wish
to find the discharge per minute in cubic feet, we can
easily do so from TABLE IX., in which, for an inch
and a half pipe, we get
Inches. Cubic Feet.
For a velocity of 20-00 per second, 1-22718 per minute.
9-00 -55223
30 -01841
04 -00245
29-34 .1-80027
* See EXAMPLE 16, pp. 28, 30.
ORIFICES, WEIRS, PIPES, AND RIVERS. 197
The discharge found experimentally by Mr. Provis,
for a tube of the same length, bore, and head, was
1-745 cubic foot per minute.
If we suppose the coefficient of discharge due to
the orifice of entry and stop-cock in Mr. Provisos
208 experiments* with li inch lead pipes of 20, 40,
60, 80, and 100 feet lengths, to be '715, the results
calculated by the tables will agree with the experi-
mental results with very great accuracy, and it is
very probable, from the circumstances described,
that the ordinary coefficient -815 due to the entry
was reduced by the circumstances of the stop-cock
and fixing to about -715 ; but even with -815 for the
coefficient, the difference between calculation and
experiment is not much, the calculation being then
in excess in every experiment, the average being
about 5 per cent., and not so much in the example
we have given.
TABLE VIII. will give the velocity, and thence the
discharge, immediately, for long pipes, and TABLE X.
enables us to calculate the cubic feet discharged per
minute, with great facility. For rivers, the mean
velocity, and thence the discharge, is also found with
quickness. See also TABLES XI., XII., and XIII.,
and the table at pp. 42 and 43.
EXAMPLE IX. A watercourse is 7 feet wide at the
bottom, the length of each sloping side is 6*8 feet, the
width at the surface is 18 feet, the depth 4 feet, and
the inclination of the surface 4 inches in a mile ; what
is the quantity flowing down per minute ?
* Transactions of the Institution of Civil Engineers, pp. 201,
210, vol. ii.
198 THE DISCHAEGE OF WATEK FROM
/TO I I7\ v> _
4
_
?. =2-4272 feet=29-126 inches
7 + 2 x 6-8 ""20-6
zzr, is the hydraulic mean depth ; and as the fall is
4 inches per mile, we find at the llth page of TABLE
VIII., the velocity v - 12-03 - -16 = 11-87 inches
per second ; the discharge in cubic feet per minute
is, therefore,
50 x X 60 = 2967-5.
12
If 94-17\/ r s = v, we have v = 94-17-Y/2 -427 +
15840
= 94 ' 17 X = =M7 feet = 14-04 inches.
Watt, in a canal of the fall and dimensions here given,
found the mean velocity about 13 inches per second.
This corresponds to a fall of 5 inches in the mile,
according to the formula. Du Buat's formula is less
by 12 J per cent, or ^th ; the common formula too
much by 5 per cent.
In one of the original experiments with which the
formula was tested on the canal of Jard, the mea-
surements accorded very nearly with those in this
example, viz. - = 15360, and r = 29*1 French
s
inches ; the observed velocity at the surface was
15*74, and the calculated mean velocity, from the
formula, 11-61 French inches.* TABLE VII. will
give 12-29 inches for the mean velocity, corre-
* These measures reduced to inches, give r = 31-014, v =
12-374; and the surface velocity 16-775 inches; reduced for mean
velocity 13-101 inches.
ORIFICES, WEIES, PIPES, AND RIVERS. 199
spending to a superficial velocity of 15*74 inches.
This shows that the formula also gives too small
a value for v in this case, by about rrth of the
result, it being about ; part in the other. The
8'3
probable error in the formula applied to straight
clear rivers of about 2 feet 6 inches hydraulic mean
depth is nearly Ath or 8 per cent, of the tabu-
lated velocity, and this must be added for the more
correct result ; the watercourse being supposed
nearly straight and free from aquatic plants.
Notwithstanding the differences above remarked
on, we are of opinion that the results of this formula,
which we have calculated and tabulated, may be
more safely relied on as applied to general prac-
tical purposes than most of those others which we
shall proceed to lay before our readers. Eivers or
watercourses are seldom straight or clear from weeds,
and even if the sections, during any improvements,
be made uniform, they will seldom continue so, as
" the regimen" or adaptation of the velocity to the
tenacity of the banks, must vary with the soil and
bends of the channel, and can seldom continue per-
manent for any length of time unless protected.
From these causes a loss of velocity takes place,
difficult, if not impossible, to estimate accurately, but
which may be taken at from 10 to 15 per cent, of
that in the clear unobstructed direct channel ; but be
this as it may, it is safer to calculate the drainage or
mechanical results obtainable from a given fall and
river channel, from formula which give lesser, than
200 THE DISCHARGE OF WATER FROM
from those which give larger velocities. This is a
principle engineers cannot too much observe.
We have before remarked, that for both pipes and
rivers the coefficient of resistance increases as the
velocity decreases. This is as much as to say, in the
simple formula for the velocity, v zz m \/r s, that m
must increase with v, and as some function of it.
This is the case in TABLE VIII., throughout which the
velocities increase faster than \/r, the \/s, or the
v/rsT In all formulae with which we are acquainted
but Du Buat's and Young's, the velocity found is con-
stant when \/r s'or r X s is constant. In Du Buat's
formula for r x s constant, v obtains maximum
values between r zz f inch and r zz 1 inch ; the dif-
ferences of the velocities for different values of r
above 1 inch, r x s being constant, are not much.
We may always find the maximum value, or nearly
so, by assuming r zz J inch, and finding the corre-
4- 7* c
spending inclination from the formula ~o~~j which is
equal to it. For example, if r 12 inches, and s zz
, the velocity is found equal 9*52 inches ; but
when r s is constant, the inclination s corresponding
4 X 12" 1
to r zz j- inch is zz , from which we find
3 X 10560 660'
from the table vzzlO-25 inches for the maximum
velocity, making a difference of fully 7 per cent.
When r zz -01 of an inch, or a pipe is Ath part
of an inch in diameter, Du Buat's formula fails, but
it gives correct results for pipes ith of an inch in
diameter, and two of the experiments from which it
OEIFICES, WEIES, PIPES, AND EIVERS. 201
was derived were made with pipes 12 inches long and
only -rsth part of an inch in diameter.
COULOMB having shown that the resistance opposed
to a disc revolving in water increases as the function
a v + b v* of the velocity v, we may assume that the
height due to the resistance of friction in pipes and
rivers is also of this form ; and that
(83.) k t = (av + bv*) 1 -,
and consequently^
(84.) r s av + bv*, and v=
G-IRARD first gave values to the coefficients a and b.
He assumed them equal, and each equal to -0003104
for measures in metres, and thence the velocity in
canals,
(85.) v = (3221-016 rs + -25^ -5;*
which reduced for measures in English feet becomes
(v = (10567-8 r s + 2-67)* 1-64, or
(v = 103 \/Ts 1-64, nearly.
The value of a = b = -0003104 was obtained by
means of twelve experiments by Du Buat and Chezy.
Of course the value is four times this in the original,
as we use the mean radius in all the formulae instead
of the diameter. This formula is only suited for
very small velocities in canals, between locks, con-
taining aquatic plants ; it is inapplicable to rivers
and channels in which the velocity exceeds an inch
per second.
PRONY found from thirty experiments on canals,
* See Brewster's Encyclopedia, Article Hydrodynamics, p. 259.
202 THE DISCHAEGE OF WATER FROM
that a = -000044450 and b = 000309314,* for mea-
sures in metres, from which we find
(87.) v = (3232-96 r s + -005 16)* -0719 ;
this reduced for measures in English feet is,
rt> = (10607-02 r* + -0556)* -236 ;f or
(v = 103 \/rs -24 nearly :
the velocities did not exceed 3 feet per second in the
experiments from which this was derived.
For pipes, Prony found,J from fifty-one experi-
ments made by Du Buat, Bossut, and Couplet, with
pipes from 1 to 5 inches diameter, from 30 to 7,000
feet in length, and one pipe 19 inches diameter and
nearly 4,000 feet long, that a = -00001733, and b
zz -0003483, from which values
(89.) v = (2871-09 rs + -0006192)* -0249,
for measures in metres, and for measures in English
feet,
(v = (9419-75 rs + -00665)* '0816; or
( ' (v = 97 v/ '08 nearly.
Prony also gives the following formula applicable
to pipes and rivers. It is derived from fifty-one
selected experiments with pipes, and thirty-one with
open channels :
(91.) v = (3041-47 rs + -0022065)* -0469734,g
* Reeherches Physico-Mathe'matiques sur la ThSorie des Eaux
Courantes.
j- For canals containing aquatic plants, reeds, &c., we must sub-
stitute jTy for r. See note, p. 186.
J Keeherches Physico-Mathematiques sur la The"orie du Mouve-
ment des Eaux Courantes, 1804.
Keeherches Physico-Mathe'inatiques sur la Theorie des Eaux
Oourantes. A reduction of this formula into English feet is given
at page 6, Article Hydrodynamics, Encyclopedia Britannica ; at
ORIFICES, WEIRS, PIPES, AND RIVERS.
203
TABLE of the fifty-one Experiments referred to in Equation (89), the
value of g in the sixth being taken at 9-8088 metres.
It will be perceived that Prony did not take into calculation, in framing his formula, the head
due to the velocity in the pipe and to the orifice of entry.
Number of
selected expe-
riments.
Names of
Experimenters.
Heads measured
to the lower ori-
fice in metres.
n!
w
Length of the
pipes in metres.
Values of
9 r *
V
in metres.
Experimental
values of the
velocity v in
metres.
Calculated
velocity from
formula (89) in
metres.
1
Du Buat
0041
0271
19-95
000314
0430
0427
2
Couplet
1511
1353
2280-37
000404
0544
0591
3
Couplet
3068
1353
2280-37
000523
0854
0921
4
Du Buat
0135
02707
19-95
000459
0980
0926
5
Couplet
4534
1333
2280-37
000590
1117
1263
6
Couplet
6105
1333
2280-37
000638
1301
1330
7
Couplet
6497
1333
2280-37
000670
1411
1433
8
Couplet
6767
1333
2280-37
000683
1441
1467
9
Du Buat
0189
0271
3-75
001426
2352
2895
10
Du Buat
1137
0271
3-75
'001138
2826
3088
11
Du Buat
1137
0271
375
001309
2888
3088
12
Bossut
1083
0271
16-24
001337
3308
3359
13
Bossut
3248
0361
58-47
001446
3400
3553
14
Du Buat
1605
0271
19-95
001482
3604
3713
15
Bossut
3248
0361
48-75
001549
3807
3915
16
Du Buat
2106
0271
19-95
001713
4091
4287
17
Bossut
3248
0361
38-98
001687
4366
4402
18
Du Buat
2425
0271
19-95
001830
4408
4618
19
Bossut
3248
0544
58-47
001672
4433
4416
20
Du Buat
2425
0271
1995
001793
4500
4618
21
Bossut
3248
0544
48-73
001795
4955
4860
22
Bossut
6497
0361
58-47
'001922
5115
5122
23
Bossut
3248
0361
29-23
001918
5128
5122
24
Du Buat
3335
0271
19-95
002050
5411
5450
25
Bossut
3248
0544
38-98
'001981
5605
5458
26
Du But
3709
0271
19-95
'002174
5676
5766
27
Bossut
6497
0361
48-73
002073
5693
5634
28
Du Buat
3952
0271
19-95
002223
5916
5961
29
Bossut
3248
0271
26-24
002201
6032
5990
30
Bossut
3248
0361
19-49
002333
6323
6327
31
Bossut
3248
0544
29-23
002300
6444
6344
32
Bossut
6497
0361
38-98
002267
6498
6323
33
Bossut
6497
0544
58-47
002214
6695
6344
34
Bossut
6497
0544
48-73
002392
7436
6972
35
Bossut
6497
0361
29'23
002588
74
7343
36
Du Bu^t
6416
0271
19-95
002750
7761
7660
37
Bossut
3248
0544
19-49
002812
7908
7823
38
Du Butt
1624
0271
3-75
003620
7943
8930
39
Bossut
6497
0544
38-98
002656
8363
7819
40
Bossut
3248
0361
9-74
003287
8976
9048
41
Bossut
65
0361
1949
003161
9332
9048
42
Bossut
65
0544
29-23
003062
9681
9071
43
Couplet
3-9274
4873
1169-42
003785
1-0600
1-0592
44
Bossut
3248
0544
9'74
004073
1'0915
1-1164
45
Bossut
6497
0544
19-49
003821
1-1640
1-1164
46
Bossut
6497
0361
9-74
004491
1-3138
1-2896
47
Du Buat
4873
0271
3-17
006470
1-5784
1-7043
48
Du Buat
5671
0271
3-75
006307
1-5919
1-6898
49
Bossut
6497
0544
9-74
005578
1-5945
1-5890
50
Du Buat
7219
0271
3-17
007838
1-9301
2-0798
51
Du Buat
9745
0271
3-17
008882
2-2994
2-4205
204 THE DISCHARGE OF WATER FROM
for measures in metres, which, reduced for measures
in English feet, is
v = (9978-76 r* + -02375)* -15412; or
k ' v = 100 vr s -15 nearly.
EYTELWEIN, following the method of investigation
pursued previously by Prony, found from a large
number of experiments, a = -0000242651, and b =
000365543 in rivers, for measures in metres ; and,
therefore,
(93.) v = (2735-66 rs + -001102)* -0332.*
This reduced for measures in English feet, is
v = (897543 r s + -0118858)* '1089 ; or
(94.) v = 94-5 v/rs -11 nearly = 1-3 v//> -11
= v/f-fyv -11
when f is the fall in feet per mile. He also shows,f
that irths of a mean proportional between the fall in
two English miles and the hydraulic mean depth,
gives the mean velocity very nearly. This rule for
measures in inches is equivalent to
(95.) v = 324 >/ ;
and for measures in feet
(96.) t? = 93-4\/r*.
For the velocity of water in pipes, he found,J from
the fifty-one experiments of Du Buat, Bossut, and
page 164, Third Report, British Association, by Rennie ; and at
pages 427 and 533, Article Hydrodynamics, Brewster's Encyclo-
pedia. This reduction, v = 0-1541 + (-02375 + 32806*6 rs)*
is entirely incorrect ; and. being the same in each of those works,
appears to have been copied one from the other.
* Memoires de 1'Academie de Berlin, 1814 et 1815. See
equation (110).
f Handbuch der Mechanik und der Hydraulik, Berlin, 1801.
J Memoires de I'Acad&nie des Sciences de Berlin, 1814 et 1815.
ORIFICES, WEIRS, PIPES, AND RIVERS. 205
Couplet, that a = -0000223, and b = -0002803, from
which we get for measures in metres,
(97.) v = (3567-29 r s + -00157,* -0397 ;
which reduced for measures in English feet becomes
r v = (11703-95 rs + -01698)* -1303 ; or
(98>) \v = 108 \/rs -13 nearly.
Another formula given by Eytelwein for pipes, which
includes the head due to the velocity for the orifice
of entry, is
in which H is the head, I the length, and d the dia-
meter of the pipe, all expressed in English feet. This
is a particular value of equation (74) suited to velo-
cities of about 2J feet per second. It must be here
mentioned, that much of the valuable information
presented by Eytelwein is but a modification of
what Du Buat had previously given, to whom only
for much that is attributed to the former we are
primarily indebted.
In the foregoing as well as in the following equa-
tions for the velocity, we have, unless otherwise
stated, maintained one class of standards. It is
evident if we change these standards in part, or in
whole, that apparently different forms of the equa-
tions will arise ; thus if for s, the hydraulic incli-
ryyi
nation, we substitute T^on? we shall have the fall
m in feet per mile, in place of the inclination s; so
that equation (94), for instance, would become
v = (l-7nr+-012)* -11 = (l*7mr) i -11 nearly,
in which v is the velocity in feet per second, m the
206 THE DISCHARGE OF WATER FROM
fall in feet per mile, and r the "hydraulic mean
depth " in feet. In like manner equation (98) would
become
v = (2-2 m r + -02)* -13 = (2 -2 m r)* -13.
The first of these reductions, viz. :
v = (1-7 m r + -0119)* -109,
is given in a book of tables calculated for river
channels for the Commissioners of Public Works, Ire-
land, the original equation being Eytelwein's, and not
D'Aubuisson's, who merely copied it, and is suited
for velocities averaging about 1*3 feet per second.
Mr. Hawksley gives for pipes the formula
itJiMi
l,+ 1-5
in which / is the length in yards, H the head in
inches, d the diameter in inches, and v the velocity
in yards per second. For uniform feet measures,
for, v, d, and H, this becomes
which is only Eytelwein's equation (99) slightly
modified. Eytelwein's equation expressed in the
measures used by Mr. Hawksley would be very
nearly
d { ^ )i
which is far the simplest of the two ; both, however,
are but particular cases of the general equation 74),
and .only suited for velocities of about 2j feet per
second.
DR. THOMAS YOUNG* also derives his formula from
* Philosophical Transactions for 1808.
OBIFICES, WEIRS, PIPES, AND RIVERS. 207
the supposition, that the head due to the resistance
of friction assumes the form of equation (83) ; calling
the diameter of a pipe d, he takes
and the whole height H = h { + j^, expressed in
inches. He found from some experiments of his
own, those collected by Du Buat, and some of
G-erstner's, that
(100.) a = -0000002
and
then as zz -00171, we get
586
/ al vM a I
\2/ + -00341 dJ I '" 2
b I + -00341 3/--J 2 b I + -00341 d'
When the length / of the pipe is very great compared
with the head due to the orifice of entrance and
velocity, -00171 v*, we have
n end a 2 * a
TT
or by substituting for ^ its value s, equal the sine of
the inclination,
n (sd a 2 J a
(104.) v= -- +
208 THE DISCHARGE OF WATER FROM
The values of a and b are for measures in inches.
For most rivers, in which d must be taken equal 4 r,
he finds for French inch measures, v = v/20000^ ;
this reduced for English inches is
(105.) ^ = 292x7^;
which again reduced for feet measures, becomes
(106.) v = 84-3v/.
These latter values, for rivers, are even smaller than
those found from Du Buat's formula ; less than the
observed velocities, and less than those found from
any other formula, with the exception of Girard's.
The values of the coefficients a and b vary in this
formula with the value of d = r; they are expressed
generally in equations (101) and (102), from which
we have calculated the following table for different
values of d and r.
An examination of this table will show that a
obtains a minimum value when d is between
10 and 11 inches ; and b when the diameter is
between J and f of an inch. Now, it appears from
equation (102), that v increases with <\ nearly,
or, which is the same thing, as b decreases, there
must, cceteris paribus^ be a maximum value of v for a
H d
given value of -j-, or r s, when d is between J and }
inch ; but as -~-. has a minimum value when d is
nearly 12 inches, the maximum value of v referred
to will be found between values of d from I inch to
12 inches ; in fact, when d 10 inches nearly. We
have already pointed out a similar peculiarity in Du
ORIFICES, WEIRS, PIPES, AND RIVERS. 209
Buat's general theorem, at page 195. It will not be
a a*
necessary to take out the values* of -^-j and ^ to more
than one place of decimals.
The values of 77-7- are also given in the following
table, and may be used in equation (104) for finding
the discharge from long pipes. It is, however, neces-
sary to remark, that this equation is sometimes mis-
applied in finding the velocity from short pipes, and
those of moderate lengths. It is necessary to use
equation (102), which takes into consideration the
head due to the velocity and orifice of entry for such
pipes.
For a pipe 11 inches in diameter, the expression
for the velocity, equation (104), becomes for.jnch
measures,
and for feet measures, also substituting 4 r for d,
(
very nearly. For a pipe -7 inch in diameter we should
find in a like manner for feet measures,
(106s.) v = 118(r*)i-5,
which is only suitable for pipes with very high velo-
cities.
SIR JOHN LESLIE states,* that the mean velocity
of a river in miles per hour, is Ifths of the mean
proportional between the hydraulic mean depth and
* Natural Philosophy, p. 428.
210
THE DISCHAKGE OF WATEE FKOM
ft
U5 > CO
co co co
10 10 o us o
t>-<Nib-O
b- <N Oi CO O
M CM <M (M
O O
O O O
OJ O l>
OOOOOOiOOO
c?ot^o<Mot-oo
(MCCOO<t>OOOC?
OKIFICES, WEIKS, PIPES, AND EIVEKS. 211
the fall in two miles in feet. This rule is equivalent,
for measures in feet, to
(107.) v zz 100 v/777
and is applicable to rivers with velocities of about 2i
feet per second.
D'AuBuissoN, from an examination of the results
obtained by Prony and Eytelwein, assumes* for
measures in metres that a zz -0000189, and b zz
0003425 for pipes, substituting these in equation (84)
and resolving the quadratic
(108.) v zz (2919-71 rs + -00074)^ _ -027 ;
which reduced for measures in English feet becomes
(v zz (9579 rs + -00813)* - -0902, or
\v = 98 v/_-l nearly.
For rivers he assumes with Eytelwein,f a zz
000024123 and b zz -0003655, for measures in
metres, and hence
(110.) v zz (2735-98 rs + -0011)* -033 ;
which for measures in English feet is
v zz (8976-5 rs + -012)' _ -109, or
v zz 94-5 N/^S '11 nearly.
When the velocity exceeds two feet per second, he
assumes, from the experiments of Couplet, a zz 0, and
b zz -00035875 ; these values give
(112.) v zz v/ 2787-46 rs,
for measures in metres, and
(113.) v zz 95-6 \/7s zz v/ 9145 rs
for measures in English feet. Equations (110) and
(111) are the same as (93) and (94), found from Eytel-
* Traite d'Hydraulique, p. 224.
t Traite d'Hydraulique, p. 133. See Equation (93).
p 3
212 THE DISCHARGE OF WATER FROM
wein's values of a and b, and it may be remarked that
D'Aubuisson's equations for the velocity generally,
are simply those of Prony and Eytelwein.
The values which we have found to agree best
with experiments on clear straight rivers are a z=
0000035, and b = -0001150 for measures in English
feet, from which we find
v = (8695 -6 r s + -00023)*- '0152, or
which for an average velocity of 1J foot per second
will give v =. 92'3\A*s nearly, and for large velo-
cities vzz93'3v/rA- ; for smaller velocities than 1J foot
per second, the coefficients of \/r s decrease pretty
rapidly. This formula will be found to agree more
accurately with observation and experiment than any
other we know of this form.
WEISBACH is perhaps the only writer who has
modified the form of the equation r s a v + b v*.
In Dr. Young's formula, a and j b vary with r, but
7 j 2
Weisbach assumes that A f (a + -i)~ x ~o~ > and
finds from the fifty-one experiments of Couplet,
Bossut, and Du Buat, before referred to, one experi-
ment by Guemard, and eleven by himself, all with
pipes varying from an inch to five and a half inches
in diameter, and with velocities varying from 1J inch
to 15 feet per second, that a -01439, and b z=
0094711 for measures in metres ; hence we have for
the metrical standard
This reduced for the mean radius r is
ORIFICES, WEIRS, PIPES, AND RIVERS. 213
(116.)
from which we find for measures in English feet
and thence
(118.) ,. = (, + 5!") *;
and by substituting for 2^7, its value 64*403,
(119.) ,. = (-00005585 +
/.
In equation (117), (-003597 + - -4~ -) = c t is the
coefficient of the head due to friction. The equation
does not admit of a direct solution, but the coeffi-
cient should be first determined for different values
of the velocity v and tabulated, after which the true
value of v can be determined by finding an approx-
imate value, and thence taking out the corresponding
coefficient from the table, which does not vary to any
considerable extent for small changes of velocity.
In the following small table we have calculated the
coefficients of friction, and also those of t; 2 , in equa-
tion (119), for different values of the velocity v.
214
THE DISCHARGE OF WATER FROM
TABLE OF THE COEFFICIENTS OF FRICTION IN PIPES.
Velocity
in feet.
Cf
Cf
64-4
644
Cf
v/^ 4
Cf
Velocity
in feet.
Cf
Cf
64-4
644
c r
^/m
1
017159
0002664
3078-07
55-5
2-4
006365
0000988
10121-5
100-5
2
013186
0002047
4885-2
69-9
2-5
006309
0000979
10214-5
101-0
3
011427
0001774
5636-9
75-08
2-6
006257
0000972
10288-1
101-4
4
010378
0001611
6270-3
78-8
2-7
006207
0000964
10373-4
101-8
5
009662
0001500
6666-6
81-6
2-8
006160
0000956
10460-2
102-2
6
009133
0001418
7052-2
84-0
2-9
006115
0000949
10537-4
102-6
7
008723
0001354
7385-5
85-9
3-
006073
0000943
10604-4
102-9
8
008391
0001303
7674-6
87-6
3-5
005890
0000914
10940-9
104-6
9
008117
0001260
7936-5
89-1
4-
005741
0000891
11223-3
105-9
1-0
007886
0001224
8169-9
90-4
5-
005514
0000856
11682-2
108-0
1-1
007686
0001193
8382-2
91-5
6-
005348
0000830
12048-2
109-7
1-2
007512
0001166
8576-3
92-6
7-
005218
0000810
12345-6
111-1
1-25
007433
0001154
8665-5
93-1
8-
005113
0000794
12632-2
112-4
1*8
007358
0001142
8756-5
93-5
9-
005026
0000780
12820-5
113-3
1-4
007221
0001121
8920-6
94-4
10-
004953
0000769
13003-9
114-0
1-5
007098
0001102
9074-4
95-2
15-
004704
0000730
13698-6
117-0
1-6
006987
0001085
9216-5
96-0
16-
004669
0000725
13793-1
117-4
1-7
006886
0001069
9354-5
96-7
20-
004556
0000707
14144-2
118-9
1-75
006839
0001062
9416-2
97-03
25-
004455
0000691
14471-7
120-3
1-8
006794
0001054
9487-6
97-4
30-
004380
0000680
14705-9
121-2
1-9
006715
0001042
9596-9
97-9
35-
004322
0000671
14903-1
122-0
2-
006629
0001029
9718-2
98-5
40-
004275
0000664
15060-2
122-7
2-1
006556
0001018
9823-2
99-1
45-
004236
0000658
15197-5
123-3
2-2
006488
0001007
9930-5
99-6
50-
084203
0000653
15313-8
123-7
2-3
006424
0000997
10003-
100-
100-
004208
0000625
16000-0
126-4
If the value of
the equation v =
644
here found, be substituted in
c f
rs. we shall have the value
of v. According to this table the coefficient of fric-
tion for a velocity of six inches is more than twice
that for a velocity of twenty feet, and the velocity is
less in the proportion of 81*6 to 118*9, or of 81*6 (rs)*
ORIFICES, WEIRS, PIPES, AND RIVERS. 215
to 118*9 (rs)*. On comparing these coefficients and
those for pipes in the preceding formulae, with those
for rivers of the same hydraulic depth, we perceive
that the loss from friction is greatest in the latter,
as might have been anticipated ; but this evidently
arises from lesser velocities.
It has been already remarked that the coefficient
of friction decreases as the velocity increases. The
only general formula which properly meets this de-
fect in the common formulae is Weisbach's, but it
does not give the velocity v directly, as this quantity
is involved in both sides of his equation. As for
several hydraulic works it is necessary to convey
water through pipes to work machines under high
heads, and for which the common formula would
give results considerably under the true ones, it
appeared to me desirable to obtain some simple ex-
pression for the velocity which might be easily
remembered and applied, which would be equally
correct with other formulae for medium velocities of
from one to two and a half feet, and which at the
same time would give practically correct results for
lesser and greater velocities within the limits of
experiment. By reducing the velocity found from
experiment to the form v = m ^/^~ s for every case,
and afterwards applying a correction of the form
n \/^rs to meet the increasing value of m as v in-
creased, I discovered that the expression
(119 A) v= 140 (rs)* - 11 (r*)*
gave results not differing more from experiments than
these frequently do from each other. The following
table exhibits the velocities compared with those
216 THE DISCHARGE OF WATER FROM
obtained from the experiments made by Du Buat,
Couplet, Watt, Mr. Provis, and Mr. Leslie, in the
Minutes of the Institution of Civil Engineers for
February 1855. The last experiment was furnished
to me by Mr. Hodson of Lincoln. Numbers 34
and 35 were made by myself, and give the mean
results of several experiments made with great care ;
the coefficient of the orifice of entry was found to
be -860.* The measures have been all reduced to
English feet. The results found by the same experi-
menters, at the same time, with the same apparatus,
sometimes differ by three or four per cent., as may be
seen by referring to Mr. Provis' experiments, (Trans-
actions of the Institution of Civil Engineers, vol. n.,
p. 203,) and the difference in the experiments shown
in the table are apparent. The difference in the
velocities found from the experiments, do not exceed
those inseparable from practical investigations, and
they differ as much in themselves as from the formula,
which for cylindrical pipes of diameter d may be
thus expressed,
v zz 70 (d 5)4 6-93 (d )*, or
(119B.)
1 v = 70 (d *)4 - 7 (d s)* nearly.
The expression fails when 70 (d sfi is equal to or
less than 6-93 (d s$, but as this only happens when
f\\ \8
r s =1^40) = '000000235, and for velocities below
one inch per second, its practical value is not thereby
affected. The expression of Du Buat fails with a
* The coefficient for the orifice of entry was found by cutting
off the pipe at two diameters from the cistern at the conclusion of
the experiments, and finding the time of emptying. Vide p. 177.
ORIFICES, WEIRS, PIPES, AND RIVERS.
217
TABLE showing the Experimental Results of observed Velocities in Water
Channels, with the Author's general formula for Pipes and Rivers, viz.
v = 140 (rsj*-Il (r s)*.
c
K
Heads in
feet (H).
Lengths in
feet (I).
Values
of r.
Values
of s.
Values
of r 8.
Velocities 1
from
experiment. I
Velocities I
from the 1
formula. 1
Velocities
expressed in
the form v =
m \/ r s.
Experimen-
ters' Names.
1
08333
1086-
052083
000076
00000396
100
105
52-6 ^/T, Mr. Leslie
2
01332
65-37
022204
000116
00000434
140
113
54-0 Couwlet
8
14583
1086-
052083
000133
00000693
118
157
60-0
Mr. Leslie
4
49566
7482-
111000
000066
00000134
178
167
61-5
Couplet
6
20833 1086-
052083
000190
00000989
217
206
65-0
Mr. Leslie
6
45833
1086-
052083
000417
00002170
361
345
74-1
7
1-4876H
7482-
111000
000198
00002220
366
348
74-1
Couplet
8
g
1-448001086-
2-781251086-
052083
052083
001321
002538
0000688
0001322
715
1-085
711
1-050
85-7
91-3
Mr. Leslie
10
2-427200
000063
0001532
1-166
1-143
92-4
Watt"
11| -50000| 100-
031250
004741
0001482
1-023
1-122
92-2
Mr. Provis
12
2-78125
1086-
052083
004348
0002265
1-461
1-438
95-5
Mr. Leslie
L3
4-76042
1086-
052083
006410
0003340
1-725
1-796
98-3
14
1-06580
127-9
044630
007748
0003458
1-839
1-840
98-9
Bossut
15
50000
40-
031250
010810
0003378
1-711
1-816
98-7
Mr. Frovis
in
1-06580
95-92
044630
010050
0004485
2-111
2-124
100-3
Bossut
17
1-5
100-
031250
014156
0004422
2-005
2-103
100-5
Mr. Provis
18
9-9896
1086-
052083
009174
0004779
2-095
2-185
100-6
Mr. Leslie
1!)
8575
40-
031250
018042
0005638
2-380
2-414
101-7
Mr. Provis
'20
2-1316
191-9
044630
010548
0004708
2-463
2-183
100-6
Bossut
21
2-1316
159-9
044630
012524
0005589
2-440
2-404
101-7 ,
22
ft
052083
014286
0007440
2-800
2-823
103-5 ,
Mr. Leslie
2:)
2-1316
127-9
044630
015350
0006851
2-744
2-696
103-0 ,
Bossut
24
n
n
044630
027921
0012465
3-819
3-760
106-5 ,
25
n
J}
052083
025000
0013021
3-783
3-852
106-7 ,
Mr. Leslie
36
3-27416
40-
031250
018040
002093
5-054
5-006
109-3 ,
Mr. Provis
27
2-3684
10-39155
022204
133689
0029679
6-322
6-048
111-0 ,
Du Buat
28
3-27416
20-
031250
111200
0034750
6-723
6-572
111-5 ,
Mr. Provis
2!)
3-4525
20-
031250
113900
0035594
7-086
6-668
111-9 ,
30
7-135
62-8822
029605
098861
0029268
6-157
5-999
110-9 ,
Couplet
31
14-270
125-7644
029605
106151
0031426
6-151
6-239
111-3
:S2
21-405
188-6466
029605
108579
00321455
6-145
6-316
111-4
33
3-1974
10-39155
022204
176991
0039292
7-544
7-039
112-3
Du Buat
34
11-125
9-292
021250
713000
01515125
14-583
14-513
117-9
Mr. Neville
35
20-8
19-2
021250
814000
01729750
15-667
15-617
118-4
36
ISO-
100-
020833
1-400000
0291667
21-7
20-6
120-3^/71
Mr.Hodson
tube of one twenty-fifth part of an inch in diameter,
no matter what the head may be, as it then makes
the velocity equal to nothing, although some of the
experiments from which it was derived were made
with tubes but the eighteenth part of an inch dia-
218 THE DISCHARGE OF WATER FROM
meter. The following expression is free from this
defect :
(119o.) v = 60 (rsfi + 120 (rs)$,
and will give results approximating very closely to
those found from Du Buat's formula, and, therefore,
with those experiments with which it most nearly
coincides, but agreeing much more closely with
Watt's and other experiments, on rivers. It gives
higher results than the previous formula for velo-
cities below six inches, but the results found by
different experimenters differ very much in those.
For higher velocities it appears to differ occasionally
only about one-twentieth from observation, being in
general less, as far as twenty feet per second, where
it coincides very closely with Mr. Hodson's expe-
riment. As the errors appear to be of an opposite
kind generally, in the two last expressions, we may
get by combining them
(119D.) v = 100 (rs)* + 60 (r s)* 5 -5 (r*)*,
an expression which, however, wants simplicity for
ready practical application. When the length of the
pipe does not exceed from 1000 to 2000 diameters, a
correction is due to the velocity in it, and to the
orifice of entry before finding the "hydraulic incli-
nation" (s). The coefficient used in reducing the
foregoing experiments for the orifice of entry was
v 2
815, which gives 1*508 -^ for the height due to the
joint effects of velocity and orifice. This must be
deducted from the head (H) before dividing it by the
length (/) to find the inclination (s) in our table.
ORIFICES, WEIRS, PIPES, AND RIVERS. 219
The following table, calculated from the formula
(119A), v = 140 (rs)? 11 (r s)?, gives the corre-
sponding values of r s and v, so that when one is
known the other is immediately found from inspec-
tion. Thus, if r s = -03125, we find
v 20-6 when rs= -029167
v 24-7 when r s = -041666
Difference 4-1 corresponds to -012499
03125
02917
Difference -00208
Whence -0125 : 4-1 : : -00208 : -7 nearly, and
20-6 4 7 z= 21-3 is the velocity sought; the same
practically as found in EXAMPLE 26, p. 37. If
allowance is to be made for the head due to the
orifice of entry and velocity, this head can be de-
termined from the velocity due to the value of r s
in the table next less than the given value with
sufficient accuracy. In this case, this velocity is
20-6 feet per second zz 247 inches nearly. If the
orifice of entry be square, the coefficient is -815, and
the head due to the velocity and this coefficient is,
TABLE II., 10 feet nearly. If r be known separately,
and also ,9, as well as the head H, and the length of
the pipe /, we had at first
H H 10 h
j zz s, and, therefore, * ~ 7 "" s '
In EXAMPLE 26, p. 37, H = 150, and 1= 100 feet,
therefore, the new value of j z= =r^ is 14 ; and as
r must be equal '020833, rs = -02917 : the value
220
THE DISCHARGE OF WATER FROM
TABLE for finding the Velocity in feet per second, from the product of
the hydraulic mean depths and hydraulic inclinations, and the reverse
calculated from the Authors formula v = 140 (rs) 4 11 (rs)*, in
which r, s, and v, are feet measures.
Values of r s.
Velo-
city V.
Values of rs.
Velo-
city V.
Values of r
Velo-
city V
Values of rs
Velo-
city V.
00000296
083
0001302
1-04
000689
270
003559
6-67
00000332
091
0001322
1-05
000710
2-75
003599
6-71
00000395
104
0001420
1-09
000744
2-83
003630
6-74
00000427
111
0001482
1-12
000758
2-85
003788
6-90
00000543
133
0001532
1-14
000789
2-91
003929
7-04
00000592
142
0001578
1-16
000805
2'94
003946
7-05
00000690
158
0001610
1-17
000833
3-00
003977
7-08
00000734
167
0001657
1-19
000852
304
004104
7-20
00000947
198
0001736
1-21
000900
3-13
004167
7-27
00000989
206
0001776
1-24
000947
3'22
004356
7-44
00001184
231
0001815
1-26
001042
340
004546
762
00001263
241
0001894
1-30
001105
3-51
004630
7-69
00001420
261
0002052
1 35
001136
3-57
004735
7-78
00001578
280
0002131
1-38
001231
3-73
005556
8-49
00001677
292
0002265
1-43
001246
3'76
006944
9-61
00001894
316
0002367
1-47
001263
3'78
007576
10-0
00001973
325
0002552
1-50
001302
3-85
008333
10'5
00002170
345
0002604
1-55
001326
3-89
009259
11 1
00002367
365
0002652
1-57
001420
4-04
010417
118
00002565
385
0002778
1-61
001515
4'18
011905
12-7
00002841
411
0002841
1-63
001576
4-28
013889
13-8
00003255
448
0003030
1-69
001610
4-32
015151
14-5
00003354
457
0003157
1-73
001667
4-41
016667
15-3
00003551
473
0003220
1 75
001705
4'46
017297
156
00003748
489
0003314
1-79
001735
4'51
020833
17-1
00003946
505
0003378
1-80
001799
4'60
027778
20-2
OOC04143
521
0003409
1-81
001894
4-73
029167
20-6
00004340
536
0003551
1-85
001989
4-87
041666
24-7
00004632
558
0003630
1-89
002052
4-94
055556
28-8
00005130
594
0003706
1-90
002083
4-98
062500
30-6
00005327
608
0003788
1-92
002093
5-00
072916
33-2
00005524
622
0003946
1-98
002178
5-10
083333
35-6
00005919
648
0004022
1-10
002210
5-14
104167
40-0
00006314
674
0004103
2-02
002273
5-22
125
43-9
00006708
699
0004261
36
002375
5*35
145583
47-6
0000688
711
0004419
2-10
002462
5-46
166667
51-1
00007102
724
0004485
2-12
002533
553
208333
57-3
00007694
760
0004546
2-14
002652
56-8
229167
60-2
00008049
781
0004708
2-18
002683
5-72
250000
63-0
00008523
808
0004735
2-18
002841
5'90
270833
65-7
00008681
828
0004893
2-23
002968
6-05
312500
70-7
00009270
849
0005051
2-27
002999
6-08
333333
73-2
00009470
861
0005208
2-31
003030
6-11
354167
75-5
00010259
903
0005303
2-33
003143
6-23
375000
77-7
00010654
923
0005638
2-41
003157
6-25
395833
80-0
00011048
945
0006061
2-52
003214
6-31
416667
62-1
00011364
960
0006155
2-54
003220
6-32
437500
84-2
00011837
983
0006313
2-57
003314
6-42
458333
86-2
00012232
1-00
0006440
2-60
003409
6-51
479166
88-3
00012627
1-02
0006629
2-64
003475
6-58
500000
90-2
ORIFICES, WEIRS, PIPES, AND RIVERS. 221
corresponding to which, in our table, is 20*6, the
velocity when allowance is made for the head due
to the velocity and orifice of entry. *+
In general, by taking the value of v for the next
less value of r s in the table, we shall find the
velocity with sufficient accuracy, and also the value
of r s from that of v by taking it as the next
greater. If we had taken r s -0008523, the table
would give v = 3-04 feet, the same practically as
already found in EXAMPLE 27, p. 38.
The value of r s, when known, determines the value
of v. If r be assumed of any convenient dimen-
sions, s is determined ; and, in like manner, any
suitable value of s determines r ; thus :
r s _ r s
=s, and z= r.
r s
It is well to remark, here again, that for pipes the
value of r is the fourth part of the diameter J, and
that
r zz j, and 4 r = d.
In 1857, M. Darcy, inspecteur des ponts et
chausees, published his Eecherches experimentales
relatives au Mouvement de 1'Eau dans les Tuyaux,*
the result of 198 experiments, in which the velocities
varied from '03 to 5 or 6 metres per second, or from H
inch to 16 or 19 feet, and with pipes varying from
J inch to 20 inches diameter. The formula by which
he presents the results is in metres,
(a.) R j = x u 2 ,
in which R is the radius of the pipe, j the hydraulic
* Morin's Hydraulique, deuxieme edition, Paris, p. 164.
222
THE DISCHARGE OF WATER FROM
inclination, b-^ a variable coefficient dependent on the
circumstances, and u the velocity per second. For
wrought and cast iron pipes of the same state of
bore, the value of b is expressed by M. Darcy, by
the equation
(ft.) Z, = -000507 + 0000 647 , '
R
the agreement between which and experiment is
shown in the following table. ;.
Diameters
in English
inches.
Diameters in
metres.
Values of ^
from experi-
ments.
Values of b^
by the
formula.
Kemarks.
5
0122
001'673
001568
1-
0266
000918
000993
1-5
0395
000785
000835
3-2
0819
000695
000665
5-4
1370
000553
000601
Well polished bore.
7.4
11-7
1880
2970
000584
000612
000576
000551
f Pipe already in use,
\ but the bore cleaned.
19-7
5000
000509
000532
For iron coated with bitumen, the value of b in
a pipe *196 metres in diameter was -0004334 ; for
a newly cast pipe of -188 metres, b was -000584;
and for a pipe -2432 metres in diameter, ^ was
001168 ; the relative proportions of b in these
three instances, being as
1-1 to 1-5 and to 3 ;
and, therefore, the velocities, or discharges, would
be inversely as the square roots of these, or as
95 to -82 and to -58.
By substituting our notation for that of M. Darcy,
we shall have in metres, from equations (a) and (b),
ORIFICES, WEIRS, PIPES, AND RIVERS, - 223
which for feet measures becomes (as 1 metre z=
3-281 feet)
r " 3-281 x '0000016175! v 2
rs= {'0002535 + - ^- ~| x 3^281 :
hence we get
rs
**= ^-00007726 + miQ ' 2
and, therefore,
-00007726
For all half-inch pipes this becomes
-00023278
for all inch pipes,
t f* C \ TV
v = 1-000155021 =
for all two-inch pipes,
for all four-inch pipes,
c =1-0000967) =
for all six-inch pipes,
v = 1-000090221 = 105 ' 3
224 THE DISCHARGE OF WATER FROM
for all nine-inch pipes,
= 107-8 v
for all twelve-inch pipes,
*= 1-
-00008374
for all eighteen-inch pipes,
* =1-
00008158)
for all twenty-four-inch pipes,
rs
={:
0000805)
= 111-5
and when r is large, as for very large pipes and
channels, we get the velocity
f
v
-00007726J
= 113-8 vW
There is evidently, on an examination of these
results, a great error in the formula of M. Darcy.
As long as the diameter of a long pipe continues
constant, the velocity is always represented by a
given fixed multiple of \/~rs, or of the square root
of the product of the hydraulic inclination and
hydraulic mean depth, no matter how small or great
the velocity In the pipe may be. For an inch pipe
this multiplier for feet measures is 80-3. Now with
a lead pipe I have found, from several experiments,
that for a velocity of about 15 feet per second, the
multiplier to be 117 or 118 ; and for a velocity of
about 22 feet per second, Mr. Hodson's experiment
gives a multiplier of about 120. Taking the other
ORIFICES, WEIRS, PIPES, AND RIVERS, 225
extreme for large pipes, the multiplier derived from
M. Darcy' s formula is 113*8, no matter how small
the velocity may be. Now we have experiments in
abundance to prove that for velocities of about 12 or
13 inches per second, the multiplier cannot exceed
95. We, therefore, look upon these researches of
M. Darcy as partial and defective, and his formula
as a representation, at best, of a limited range of
velocities, in which those at either side are omitted
or not perceived.
For small pipes, any obstruction arising from de-
fective bore, decomposition, encrustation, or from
diminished bore, affects the discharge much more
considerably than the same obstructions in a large
pipe. In order to compare correctly the effects of
the state of the bore on the discharge, we must use
pipes of exactly the same diameter, and determine
the value of b : from experiments in which the velo-
city is the same, otherwise the results, as deduced
by M. Darcy and given by Morin, cannot be de-
pended upon.
COEFFICIENTS DUE TO THE ORIFICE OF ENTRY. PROBLEMS.
Unless where otherwise expressed, the head due
to the velocity and orifice of entry is , not considered
in the preceding equations. In equation (74), where
it is taken into calculation generally,
Cf X -
/1\ 2
in which 1 + c r is equal to ( ) , c r being the coefficient
226 THE DISCHARGE OF WATER FROM
of resistance clue to the orifice of entry, and c v the
coefficient of velocity or discharge from a short tube.
If the tube project into the reservoir, and be of small
thickness, c v will be equal -715 nearly, and therefore
c r zz -956 ; if the tube be square at the junction, the
mean value of c v will be -814, and therefore c r = -508;
and if the junction be rounded in the form of the
contracted vein, c v is equal to unity very nearly, and
C T z= 0. For other forms of junction the coefficients
of discharge and resistance will vary between these
limits, and particular attention must be paid to their
values in finding the discharge from shorter tubes
and those of moderate lengths ; but in very long tubes
1 + c r becomes very small compared with c f x ,
and maybe neglected without practical error. These
remarks are necessary to prevent the misapplication
of the tables and formulae, as the height due to the
velocity and orifice of entry is an important element
in all calculations for short tubes.
We have considered it unnecessary to give any
formulae for finding the discharge itself, because, the
mean velocity once determined, the calculation of the
discharge from -the area of the section is one of
simple mensuration ; and the introduction of this
element into the three problems to which this por-
tion of hydraulic engineering applies itself, renders
the equations of solution complex, though easily
derived ; and presents them with an appearance of
difficulty and want of simplicity which excludes
them, nearly altogether, from practical application.
The three problems are as follows :
ORIFICES, WEIRS, PIPES, AND RIVERS. 227
I. Given the fall, length, and diameter of a pipe
or hydraulic mean depth of any channel, to find the
discharge.
Here all that is necessary is to find the mean
velocity of discharge, which, multiplied by the area
of the section (equal d 2 x *7854 in a cylindrical
pipe), gives the discharge sought. TABLE VIII.
gives the velocity at once for long channels, accord-
ing to Du Buat, or it can be found from equation
(119A) by calculation. TABLE IX. gives the dis-
charge in cubic feet per minute for different diameters
of pipes, and velocities in inches per second, when
found from TABLE VIII. or formula (119A). See
also TABLES XI. and XII. For a pipe 6 inches in
diameter, the velocity per second is practically equal
to the discharge in cubic feet per minute. See
also the tables, pp. 42, 43, 252, and 253.
II. Given the discharge and cross section of a
channel, to find the fall or hydraulic inclination.
If the cross section be circular, as in most pipes,
the hydraulic mean depth is one-fourth of the dia-
meter ; in other channels it is found by dividing the
water and channel line of the section, wetted peri-
meter, or border, into the area. The velocity is
found by dividing the area into the discharge, and
reducing it to inches per second ; then in TABLE
VIII., under the hydraulic mean depth, find the
velocity, corresponding to which the fall per mile
will be found in the first column, and the hydraulic
inclination in the second. This result can be cor-
rected by trial and error to accord with formula
(11 BA), and the table for the values of r s and v, p. 220,
Q3
228 THE DISCHARGE OF WATER FROM
calculated from it. See also the tables, pp. 42, 43,
252, and 253.
III. Given the discharge, length, and fall, to find
the diameter of a pipe, or hydraulic mean depth and
dimensions of a channel.
This is the most useful problem of the three.
Assume any mean radius r a , and find the discharge D a
by Problem I. We shall then have for cylindrical
pipes
rf : 7*5 : : D a : D : : 1 : ;
and as r a , D, and D a are known, r% becomes also
known, and thence r. TABLE XIII. will enable us to
find r with great facility. Thus, if we had assumed
r & =. 1 and found D a = 15, D being 33, we then have
1 : r*:: 1 : - :: 1 : 2*2, therefore r* = 22 ;
15
and thence by TABLE XIII., r = 1*37, the mean radius
required, four times which is the diameter of the
pipe. For other channels, the quantity thus found
must be the hydraulic mean depth ; and all channels,
however varied in the cross section, will have the
same velocity of discharge, when the fall, length, and
hydraulic mean depth are constant. If r a be as-
sumed equal to 1J inch, the velocity found from
TABLE VIII. will then be the discharge in cubic feet
per minute nearly, and this "mean radius" can
always be assumed for the first term of the pro-
portion. See also the tables, pp. 42, 43, 252, and 253.
In order to find the dimensions of any polygonal
channel whatever, which will give a discharge equal
to D, we may assume any channel similar to that
OEIFICES, WEIES, PIPES, AND RIVERS. 229
proposed, one of whose known sides is s a , and find
the corresponding discharge, D a , by Problem L, or
from TABLES XI. and XII. ; then, if we call the like
side of the required channel, s, we shall have
sn:s a ( j, and thence the numerical value from
TABLE XIII. The result, as before, can be corrected
to accord with any of our formulae by the method of
trial and error.
As it frequently happens that deposits in and en-
crustations on a pipe take place from time to time,
which diminish the flowing section considerably, it
is always prudent, when calculating the necessary
diameter, to take the largest coefficient of friction, c f ,or
to double its mean value, particularly for small pipes,
when calculating the diameter from any of the for-
mulae. Some engineers, as D'Aubuisson, increase
the quantity of water by one-half to find the dia-
meter ; but much must depend on the peculiar cir-
cumstances of each case, as sometimes less may be
sufficient, or more necessary. Tne discharge increases
in similar figures, nearly as r% or as d? 9 that is, as the
square root of the fifth power of the diameter, and
the corresponding increase in the diameter for any
given or allowed increase in the discharge can be
easily found by means of TABLE XIII., as shown
above. If we increase the dimensions by one-sixth,
the discharge will be increased by one-half nearly,
and by doubling them the discharge is increased in
the proportion of 5 1 to 1.
For shorter pipes, we have to take into considera-
tion the head due to the velocity and orifice of entry.
230 THE DISCHARGE OF WATER FROM
Taking the mean coefficient of velocity or discharge,
we find from TABLE II. the head due to the velocity,
if it be known ; this subtracted from the whole head,
H, leaves the head, h iy due to the hydraulic inclina-
tion, which is that we must make use of in the table.
If the velocity be not given, we can find it approxi-
mately ; the head found for this velocity, due to the
orifice of entry, when deducted, as before, will give a
close value of h t , from which the velocity may be
determined with greater accuracy, and so on to any
degree of approximation. In general, one approxi-
mation to h t will be sufficient, unless the pipes be
very short, in which case it is best to use equation
(74). EXAMPLE VIII., p. 195, and the explanation
of the use of the tables, SECTION I., may be usefully
referred to.
TABLES XI., XII., and XIII. enable us to solve
with considerable facility all questions connected
with discharge, dimensions of channel, and the ordi-
nary surface inclinations of rivers. The discharge
corresponding to any intermediate channels or falls
to those given in TABLES XI. or XII., will be found
with abundant accuracy, by inspection and simple
interpolation ; and in the same manner the channels
from the discharges. Rivers have seldom greater
falls than those given in TABLE XII., but in such an
event we have only to divide the fall by 4, then
twice the corresponding discharge will be that re-
quired. TABLE XIII. gives the comparative dis-
charging powers of all similar channels, whether
pipes or rivers, and the comparative dimensions from
the discharges. We perceive from it, that an increase
OEIFICES, WEIKS, PIPES, AND RIVERS. 231
of one-third in the dimensions doubles, and a de-
crease of one-fourth reduces the discharge to one-
half. By means of this table, we can determine by
a simple proportion, the dimensions of any given
form of channel when the discharge is known. See
EXAMPLE 17, p. 30. See also the tables pp. 42,
43, 252 and 253.
The mean widths in TABLES XI. and XII. are cal-
culated for rectangular channels, and those having
side slopes of 1J- to 1. Both these tables are, how-
ever, practically, equally applicable to any side
slopes from to 1 up to 2 to 1, or even higher, when
the mean widths are taken and not those at top or
bottom. A semihexagon of all trapezoidal channels
of equal area has the greatest discharging power, and
the semisquare and all rectangles exactly the same
as channels of equal areas and depths with side
slopes of H to 1. The maximum discharge is ob-
tained between these for the semihexagon with side
slopes, of nearly \ to 1, but for equal areas and
depths the discharge decreases afterwards as the slope
flattens. The question of "HOW MUCH?" is here,
however, a very important one ; for, as we have
already pointed out in equations (28) and (31), the
differences for any practical purposes may be imma-
terial. This is particularly so in the case of chan-
nels with different side slopes, if, instead of the top
or bottom, we make use of the mean width to calcu-
late from. We then have only to subtract the ratio
of the slope multiplied by the depth to find the
bottom, and add it to find the top. If the mean
width be 50 feet, the depth 5 feet, and the side slopes
232 THE DISCHARGE OF WATER FROM
2 to 1, we get 50 (2 x 5) z= 40 for the bottom, and
50 + (2 x 5) = 60 for the top width.
Side slopes of 2 to 1 present a greater difference
from the mean slope of H to 1, than any others in
general practice when new cuts are to be made. A
triangular channel having slopes of 2 to 1, and bot-
tom equal to zero, differs more in its discharging
power from the half square, equal to it in depth and
area, than if the bottom in each was equally in-
creased, yet even here it is easy to show that this
maximum difference is only 5 per cent. If the
bottom be increased so as to equal the depth, it is
only 4J per cent. ; when equal to twice the depth, 3 -8
per cent. ; and when equal to four times the depth, to
2 per cent. ; while the differences in the dimensions
taken in the same order are only 2-2, 1-8, 1-5, and
0-8 per cent. For greater bottoms in proportion to
the depth the differences become of no comparative
value. It therefore appears pretty evident, that
TABLES XL and XII. will be found equally applicable
to all side slopes from to\upto1to\,by taking the
mean widths. When new cuts are to be made, we see
no reason whatever in starting from bottom rather
than mean widths, to calculate the other dimensions ;
indeed, the necessary extra tables and calculations
involved ought entirely to preclude us from doing
so. Besides, the formulas for finding the discharge
vary in themselves, and for different velocities the
coefficient of friction also varies.* Added to which
* The coefficient m in the formula v = m (r s)* in rivers for
velocities from 3 inches to 3 feet per second, varies from about
72 to 103 ; yet, strange to say, most tables are calculated from
ORIFICES, WEIRS, PIPES, AND RIVERS. 233
the inequalities in every river channel, caused by
bends and unequal regimen, preclude altogether any
regularity in the working slopes and bottom, though
the mean width would continue pretty uniform under
all circumstances.
The quantities in TABLE XII. are calculated, from
the velocities found from TABLE VIII., to correspond
to a channel 70 feet wide and of different depths, the
equivalents to which are given in TABLE XI. In
order to apply these tables generally to all open
channels, the latter are to be reduced to rectangular
ones of the same depth and mean width, or the re-
verse, as already pointed out. If the dimensions of
the given channel be not within the limits of TABLE
XI., divide the dimensions of the larger channels by
4, and multiply the corresponding discharge found
in TABLE XII. by 32 ; for smaller channels, multiply
the dimensions by 4, and divide by 32. In like
manner, if the discharge be given and exceed any to
be found in TABLE XIII., divide by 32, and multiply
the dimensions of the suitable equivalent channel
found in TABLE XI. by 4. If we wish to find equiva-
lent channels of less widths than 10 feet for small
discharges, multiply the discharge by 32, and divide
the dimensions of the corresponding equivalent by 4,
Many other multipliers and divisors as well as 4 and
32 may be found from TABLE XIII., such as 3 and
one coefficient alone ; or, rather, from a formula equivalent to
94-17 (r s)$, which gives results suited only to a velocity of 16
inches. Dimensions of channels calculated by means of this
formula are too small in one case, and too large in the other. In
pipes the variation of the coefficients is shown in the small tables,
pp. 214 and 217.
234 THE DISCHARGE OF WATER FROM
15-6, 6 and 88-2, 7 and 130, 9 and 243, 10 and 316,
12 and 499, &c. The differences indicated at pages
198 and 199, must be expected in the application of
these rules, which will give, however, dimensions for
new channels which can be depended on for doing
duty.
It will be seen from TABLE XIII. that a very
small increase in the dimensions increases the dis-
charging power very considerably. TABLE XII.
also shows that a small increase in the depth alone
adds very much to the discharge. If we express in
this latter case a small increase in the depth, d, by - ,
7i
then it is easy to prove that the corresponding in-
crease in the velocity, v, will be ; and that in the
A 1
discharge D, o~, if the surface inclination continue
unchanged ; but as it is always observable in rivers
that the surface inclinations increase with floods, the
differences in practice will be found greater than
these expressions make it. As in a large river the
surface inclination must be very small, four times
the fall will add very little to the sectional area ; yet
this increase of fall will double the discharge, and
we thence perceive how tributaries can be absorbed
into the main channel without any great increase to
its depth.
OKIFICES, WEIRS, PIPES, AND EIVERS.
235
SECTION IX.
BEST FORMS OF THE CHANNEL. REGIMEN. VELOCITY.
EQUALLY DISCHARGING CHANNELS.
We have seen above, that the determination of the
hydraulic mean depth does not necessarily determine
the section of the channel. If the form be a circle,
the diameter is four times the mean radius ; but,
though this form be almost always adopted for pipes,
the beds of rivers take almost every curvilineal and
trapezoidal shape. Other things being the same,
that form of a river channel, in which the area
divided by the border is a maximum, is the best.
This is a semicircle having the diameter for the sur-
face line, and in the same manner, half the regular
figures, an octagon, hex-
agon, and square, in Fig.
33, are better forms for
the channel, the areas
and side slopes being
constant, than any others
of the same number of
sides. Of all rectangular channels, Diagram 4, in
which A B c D is half a square, is the best cross section ;
and in Diagram 3, A c D B, half a hexagon, is the best
trapezoidal form of cross section. When the width
of the bottom, c D, Diagram 3, is given, and the slope
Aa
= n, then, in order that the discharge may be the
greatest possible, we must have
236 THE DISCHARGE OF WATER FROM
A **
-12
in which A is the given area of the channel. As,
however, we have never known a river in which the
slope of the natural banks continued uniform, even
though made so for any improvements, we consider it
almost unnecessary to give tables for different values
of n. If, notwithstanding, we put $ for the inclina-
tion of the slope A c, equal angle c A a, we shall find,
1
as cot. < = n, and vri 2 + 1 = - - , that the fore-
sin. <f>
going equations become
( A sin. <fr | * _ CJD
12 cos. *! = 2 {(n* + l)*-n}'
and
(121.) CD = ca x cot. <,*
C cl
which will give the best dimensions for the channel
when the angle of the slope for the banks is known.
When the discharge from a channel of a given
area, with given side slopes, is a maximum, it is easy
to show that THE HYDRAULIC MEAN DEPTH MUST BE HALF
OF THE CENTRAL OR GREATEST DEPTH. This Simple
principle enables us to construct the best form of
channel with great facility. Describe any circle on
the drawing-board ; draw the diameter and produce
* When c D = o. The channel is triangular ; we get A = c a 2
OKIFICES, WEIES, PIPES, AND RIVERS. 237
it on both sides, outside the circle ; draw a tangent to
the lower circumference parallel to this diameter, and
draw the side slopes at the given inclinations, touching
the circumference also on each side and terminating
on the parallel lines : the trapezoid thus formed will
be the best form of channel, and the width at the
surface will be equal to the sum of the two side
slopes. It is easy to perceive that this construction
may be, simply, extended for finding the best form
of a channel having any polygonal border whatever
of more sides than three and of given inclinations.
Commencing with the best discharging form of chan-
nel, which in practice will have the mean width, about
double the depth ; an equally discharging section of
double the width of the first will have the contents one-
eleventh greater, and the depth less in the proportion
of 1 to 1*85. A channel of double the mean width
of the second must have the sectional area further
increased by about one-fifth, and a further decrease
in the depth from 1*67 to 1 nearly. The greater
expense of the excavation at greater depths will,
in general, more than counterbalance these differences
in the contents of the channel. When the banks rise
above the flood line, and are unequal in their section,
the wider channel involves further upper extra cut-
ting, but there is greater capacity to discharge extra
and extraordinary flooding, the banks are less liable
to slip or give way, the slopes may be less, and the
velocity being also less, the regimen will, in general,
be better preserved. The table of equally discharg-
ing channels, p. 2 5 2, will afford the means of calculating
the difference of cubical contents.
238
THE DISCHARGE OF WATER FROM
If"
% % % ^ %
uo i i <?q T-I o
O <M - t- 00
rH CN? T^ O J>-
C{ CH Cq rH
lf|
333333333 2 "
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OKIFICES, WEIES, PIPES, AND RIVERS. 239
When the sectional area is given, the above table
shows that the semicircle is the best discharging
channel, and the complete circle the worst; the
latter is so, however, only compared with the open
channels given in the table, it being the very best
form for an enclosed channel flowing full. T/ie best
form of channel is particularly suited for new cuts in
flat, marsh, callow, and fen lands, in which it is also
often advisable to cut them with a level bed, up from
the discharging point, in order to increase the hy-
draulic mean depth, and consequently the velocity
and discharge.
As the quantity of water coming down a river
channel in a season varies very considerably, we
have observed it in one case to vary from one to
thirty, and occasionally in the same channel from
one to seventy-five, the proportion of the water
section to the channel itself must also vary, and
those relations of the depth, sides, and width to each
other, above referred to, cease to hold good and be
the best under such circumstances. If the object be
to construct a mill-race, temporary drain for un-
watering a river, or other small channel, in which
the depth remains nearly constant, channels of the
form of a half hexagon, diagram 3, Fig. 33, will be,
perhaps, the best, if the tenacity of the banks per-
mit the slope ; but rivers, in which the quantity of
water varies considerably, require wider channels in
proportion to the depth ; and also, that the velocity
be so proportioned to the tenacity of the soil, or as
it is termed " the regimen" that the banks and bed
Fig. 34
240 THE DISCHAKGE OF WATER FROM
shall not vary from time to time to any injurious
extent, and that any deposits made during their
summer state, and during light freshes, shall be
carried off periodically by floods. Another circum-
stance, also, modifies the effects of the water on the
banks. It is this, that at curves, and turns, the
current acts wit,h greatest effect against the bank,
concave to the di-
rection in which
it is moving; deep-
ening the channel
there ; undermin-
ing also the bank,
as at A, Fig. 34 ; and raising the bed towards the
opposite side B. The reflexion of the current to
the opposite bank from A acts also in a similar man-
ner, lower down, upon it ; and this natural operation
proceeds, until the number of turns, increased length
of channel, and loss of head from reflexion and
unequal depths, bring the currents into regimen with
the bed and banks. At all bends it is, therefore,
prudent to widen the channel on the convex side B,
Fig. 34, in order to reduce the velocity of approach ;
and if the bed be here also sunk below its natural
inclination, as we see it in most rivers at bends, the
velocity will be farther reduced, and the permanence
of the bed better established.
The circumstances to be considered in deciding on
the dimensions and fall of a new river course, after
the depth to which the surface of the water is to be
brought has been decided on, are the following :
ORIFICES, WEIRS, PIPES, AND RIVERS. 241
The mean velocity must not be too slow, or
aquatic plants will grow, and deposits take place,
reducing the sectional area until a new and smaller
channel is formed within the first with just sufficient
velocity to keep itself clear. This velocity should
not in general be less than from ten to fourteen
inches per second. The velocity in a canal or river
is increased very considerably by cutting or re-
moving reeds and aquatic plants growing on the
sides or bottom.*
The mean velocity must not be too quick, and
should be so determined as to suit the tenacity
and resistance of the channel, otherwise the bed and
banks will change continually, unless artificially
protected ; it should not exceed
25 feet per minute in soft alluvial deposits.
40 clayey beds.
60 sandy and silty beds.
120 gravelly.
180 strong gravelly shingle.
* " M. Girard a fait observer, avec raison, que les plantes aqua-
tiques, qui croissent toujours sur le fond et sur les berges des
canaux, augmentent conside"rablement le perimetre mouille, et par
suite la resistance ; il a rapelle* que Du Buat, ayant mesure la
vitesse de 1'eau dans le canal du Jard, avant et apres la coupe des
roseaux dont il e"tait garni, avait trouve un resultat bien moindre
avant qu'apres. En consequence, il a presque double la pente
donne"e par le calcul . . ." Traite d'Hydraulique, p. 135.
When the fall does not exceed a few inches per mile, the velocity,
as determined from the inclination, is very uncertain, and for this
reason it is always prudent to increase the depths and sectional
areas of channels in flat lands, as far as the regimen will permit.
In such cases the section of the channel should approximate
towards the best form. See p. 238.
R
242
THE DISCHARGE OF WATER FROM
240 feet per minute in shingly.
300 ., shingly and rocky.
400 and upwards in rocky and shingly.*
A velocity of 180 feet per minute will remove an-
gular stones the size of an egg. Mr. Phillips, under
the Metropolitan Commissioners of Sewers, states
that 2| feet per second, or 150 feet per minute,
is sufficient to prevent soil depositing in sewers.
The fall per mile should decrease as the hydraulic
mean depth increases, and both be so proportioned
that floods may have sufficient power to carry off the
deposits, if any, periodically. The proportion of the
width to the depth of the channel should not be
derived, for new cuts or river courses, from any
formula, but taken from such portions of the old
channel as approximate in depth and in the inclina-
tion of the surface to that proposed. When the
depth is nearly half the width, the formula shows,
cceteris paribus, that the discharge will be a maxi-
mum ; but as (altogether apart from the question
* TABLE OF VELOCITIES OP SOME MOVING BODIES COMPARED WITH THOSE OF RIVERS.
Objects in motion.
Miles
per
hour.
Feet
per
second.
Objects in motion.
Miles
per
hour.
Feet
per
second.
Current of slow rivers . .
Currents of ordinary rivers,
up to
A
ii
1
Railway trains, French . .
German .
Sound when atmosphere
is at 32 Fahr
27
24
743
89*
53i
1 090
Currents of rapid rivers .
7
lof
Ditto 60 Fahr
765
1,122
Man walking
3
4?
850
1 247
Horse trotting
7
10i
TVtt 'h fh h t '
Swiftest race-horse . . .
Moderate winds ....
60
7
36
88
101
52$
stands at 30 inches . .
Common musket-ball . .
Rifle-ball
917
850
1000
1,344
1,247
1/67
80
117t
1 091
1 600
Swift English steamboats
navigating the channels .
Swift American Kiver
steamers
14
18
**t
20|
26^
Bullet discharged from air-
gun, air being compress-
ed into the hundredth
477
700
Fast sailing vessels . . .
Railway trains, English .
American .
Belgian .
12
32
18
25
Hi
47
26*
36f
A point on earth's surface
at the equator moving
round the axis ....
Earth moving round sun .
1,040
68,182
1,525
100,000
ORIFICES, WEIRS, PIPES, AND RIVERS. 243
of expense) the quantity of water discharged daily,
at different seasons, may vary from one to seventy,
or more, and " the regimen" has to be maintained,
the best proportion between the width and depth
of a new cut should be obtained, as we have stated,
from some selected portion of the old channel, whose
general circumstances and surface inclination ap-
proximate to those of the one proposed ; and the
side slopes of the banks must be such as are best
suited to the soil. The resistance of the banks to
the current being in general less than that of the
beds, which get covered with gravel, and the neces-
sary provision required for floods, appears to be
the principal reason why rivers are in general so
very much wider than about twice the depth, the
relation which gives the minimum of friction.
The following table is given by Bennie, as an
approximation, generally, to the actual state of
rivers.* The surface inclinations, however, given
in this table for the first and second classes, are
very considerable for large rivers, and would give
velocities which would effectually scour them. For
a hydraulic mean depth of 12 feet, the velocity, with
a fall of 10000' would be 2 feet 8 inches per second
by Du Buat's formula ; and 3-3 feet per second by
our formula. The description, therefore, can only
apply to small channels. In fact, 4 inches to a
mile, or , is a considerable inclination for a
large river.
* Eeport to the British Association 1834.
R3
244
THE DISCHARGE OF WATER PHOM
i* 2
2 a S
tjj
|||
*8J
i?l
'S'o
DISTINCTIVE ATTBEBUTE8
**f
- 'o "3
6
| **
*
00 *H rfl '>
09 THE
f
"o > &o
i.'||
o"
a! 2
1 .&
I-H
VABIOUS KINDS OF BIVEBS.
g | fl
'S'wo
fi "3 *
o-g
B& 13
1*1
u - a
"! a
ill
III
t$l
-||o
Channels wherein the resist-
ance from the bed, and other
obstacles, equal the quantity of
current acquired from the de-
clivity ; so that the waters would
1st.
0"
12000
14
stagnate therein, were it not for
the compression and impulsion
of the upper and back waters .
Artificial canals in the Dutch i
and Austrian Netherlands . . J
2nd.
1
180
1000
8
Rivers in low flat countries, "]
full of turns and windings, and
of a very slow current, subject
to frequent and lasting inunda-
- 3rd.
1
120
10
5200
6
Rivers in most countries that \
are a mean between flat and
hilly, which have good currents,
but are subject to overflow ; also
4th.
li
80
15
Wtro
4|
the upper parts of rivers in flat
Rivers in hilly countries with\
a strong current, and seldom
subject to inundations ; also all 1
rivers near their sources have (
5th.
*i
55
21*
T2"0(7
3f
this declivity and velocity, and
often much more J
Rivers in mountainous coun- \
tries having a rapid current and (
straight course, and very rarely [
6th.
3
40
30
2600
3
Rivers in their descent from\
among mountains down into the 1
plains below, in which plains [
7th.
5
24
50
"2000
2J
they run torrent- wise. . . . j
Absolute torrents among i
8th.
8
15
80
x
2
mountains . . i
1 7
ORIFICES, WEIRS, PIPES, AND RIVERS.
245
The following information with reference to the sur-
face inclinations of the Thames, is from Rennie's Report
on Hydraulics,* as a branch of engineering science.
Names of places.
Length.
Fall
Fall in feet
per mile.
Ratio of
inclinations.
From Lechdale at St. John's
Bridge to Oxford at FoUy
Miles. Fur.
28
Feet. In.
47
1-68
TTTTS
From Oxford to Abingdon
9
13 11
1-73
1
From Abingdon to Walling-
ford Bridge ....
14
27 4
1-95
From Wallingford to Read-
in s Bridsre
18
24 1
1-81
From Reading to Henley
Bridge
9
19 3
2-14
TT A & 7
From Henley to Mario w
Bridge
9
12 2
1-35
From Marlow to Maiden-
head Bridge
8
15 1
1-86
From Maidenhead Bridge to
Windsor Bridge ....
From Windsor 'to Staines
Bridge
7
8
13 6
15 8
1-93
1-96
2 1*8" 3 ^
ff,T3TF
From Staines to Chertsey
Bridge ....
4 6
6 6
1-44
From Chertsey to Tedding-
ton-Lock .
13 6
19 8
1-45
s-T^rr
From Teddington-Lock to
London Bridge ....
From London to Yanlet
Creek
19
40
2 9
2 1
145
052
3.64 1
TnTi4'14
TO 1637
From Lechdale to Yanlet
Creek
186 4
218
Deduct . . .
40
.-
From Lechdale to London .
146 4
For enclosed channels, the circular form of sewer
will have the largest scouring power, at a given
hydraulic inclination. For the area of the sections
being the same, the velocity in the circular channel
will be a maximum. When the supply is intermit-
tent, and the channel too large, the egg-shaped form
* Report, for 1834, of the British Association.
246 THE DISCHARGE OF WATER FROM
with the smaller end for the bottom, or the sides
vertical with an inverted ridge-tile or V bottom for
drains, will have a hydrostatic flushing power to
remove soil and obstructions, which a cylindrical
channel, only partly fall, does not possess ; because
a given quantity of water rises higher against the
same obstruction, or obstacle, to the flow in the
pipe. It must be confessed, however, that for small
drains and house-sewage, this gain is immaterial, and
is at best but effected by a sacrifice of space, mate-
rial, and friction in the upper part of drains, from
6 to 12 inches in diameter. Besides this, the mere
hydrostatic pressure is only intermittent, and during
an ordinary, or heavy, fall of rain, the hydrodynamic
power is always more efficient in scouring properly-
proportioned cylindrical drains ; and the workman-
ship in the form and joints is less imperfect than for
more compound forms, as those with egg-shaped and
inverted tile bottoms. The moulds and joints of
cylindrical stone- ware drains, exceeding 12 inches
in diameter, are seldom, however, in large quantities
perfect ; and the expense will exceed that of brick,
stone, or other sufficient drains in most localities.
As to the increased discharging power which it is
asserted by some, stone-ware cylindrical drains pos-
sess over other ordinary drains, no doubt it is true
for small sizes, because the form, jointing, and sur-
face are in general more smooth and circular ; and
for sewage matter* the friction and adherence to the
sides and bottom is less ; any advantage from these
causes becomes, however, immaterial for the larger
* Weisbach found the coefficient of resistance 1'75 times as
great for small wooden as for metallic pipes. All permeable pipes
ORIFICES, WEIRS, PIPES, AND RIVERS. 247
sizes, as these can be constructed of brick or stone
abundantly perfect to any form, and sufficiently
smooth for all practical purposes, for in the larger
properly-proportioned sizes the same amount of sur-
face roughness opposed to the sewage matter is, com-
paratively, of no effect. The judicious inclination
and form of the bottom, and properly curved junc-
tions, are the principal points to be attended to.
Smaller drains tile-bottomed, with brick or stone
sides, and flat-covered, have one great advantage
over circular pipes,* They can be opened up, for
examination and repairs at any time with facility,
and at the smallest expense ; but greater certainty
must be attached to the working of small stone-ware
drains than to equally-sized small brick or stone
drains, and they will be found, in general, also
cheaper. This, however will depend on the locality.
It may be observed in numerous experiments,
that water flowing from a pipe does not entirely fill
the orifice of exit, when the velocities are not con-
siderable, and yet the results are found to be but
slightly affected if a little more than three-fourths of
the circumference be full. It is easy to demonstrate
that the full circle does not give the maximum dis-
charging velocity as has been generally believed, but
present greater resistance than impermeable ones ; hence the prin-
cipal advantage derived from glazing.
* Half-socket joints at bottom would remedy this imperfection
in small pipes, and they could be better laid and cemented. A
semicircular flange laid on at top would effectually protect the
joint on the upper side. Latterly Doulton has cut off an upper
segment from the pipe, which can be removed for cleaning. And
it may be demonstrated, that when this is a segment of 78J
degrees, the lower portion will discharge more than a full pipe at
the same inclination.
248
THE DISCHARGE OF WATER FROM
when filled to the height
of the chord ac of arc
a e c of 784 degrees, and
where the velocity is 9i
per cent, over that due
to the full circle, for then
,1 area ad c
the =- is a maxi-
arc aae
mum, and the length of
the arc adc is equal to the tangent of the supple-
mental arc a e c, as may be without difficulty demon-
strated. The hydraulic mean depths of the circle
and larger segment are to each other as -5 to -6, and
their square roots, which are as the velocities or
scouring powers, are as 1 to 1-095. The discharging
powers are to each other as 1x3*1416 to 1-095
X 2*946, or as 1 to 1-026, which shews that the seg-
ment adc has also a greater discharging power than
the whole circle of nearly three per cent. These facts,
which were first pointed out by the author, are not
unimportant in matters connected with pipe-drains
and sewerage. The effects of greater velocity and
discharge here pointed out, are sometimes increased,
in short pipes, from the fall between the surface a c, and
the surface from which the head is measured, being
greater than the fall to the top of the pipe at e, or from
the inclination of the surface of the water in the pipe
being greater than the inclination of the pipe itself.
EQUALLY DISCHARGING CHANNELS.
In order that different channels should have the
same discharging power, the inclination of the sur-
face being the same, the areas must be inversely as
ORIFICES, WEIRS, PIPES, AND RIVERS.
249
the square roots of the hydraulic mean depths. The
channel a dcv, Fig. 35, will have the same discharge
as the channel A D c B if they be to each other
as f ADCB ,i to ,
I A D 4- DC 4- C B J (
AD + DC + CB
ad + dc + CB
and hence the square root of the cube of the channel
area, divided by the border, must be constant. With
a fall of one or more feet to a mile, two channels, one
70 feet wide and 1 foot deep, and the other 20 feet
wide and 2i feet deep, will have the same dis-
charge. If we put w for the width and d for
the depth of any rectangular channel, then
w
w
y
+ 2 d)
m ; we therefore have the cubic equation
(122.) d--d =
w 3 w 2
for finding the depth, d of any other rectangular
channel whose width is w, of the same discharging
power. We have calculated the depths d for different
widths of channel from this equation, assuming a
width of 70 feet and different depths to find m from.
The results are given in TABLE XI., which will be
found sufficiently accurate for all practical purposes,
when the banks are sloped, by taking the mean width.
This table is equally applicable to any measures
whatever, to their multiples, and sub-multiples.
If the hydraulic inclinations vary, then the \/rs
must be inversely as the areas of the channels when
\/rs x channel or the discharge is constant ; and if
250 THE DISCHARGE OF WATER FROM
the area of the channel and discharge be each con-
stant, r must vary inversely as s; and r s be also
constant. For instance, a channel which has a fall
of four feet per mile, and a hydraulic mean depth of
one foot, will have the same discharge as another chan-
nel of equal area, having a hydraulic mean depth of
four feet, and a fall per mile of only one foot. If in
TABLE XII. we take the same discharge from the
columns for different inclinations, we shall get the
mean rectangular dimensions corresponding to them
in the first column, and thereby be enabled to select
an equally discharging channel from TABLE XL, suited
to an increase or decrease of the hydraulic inclina-
tion.*
We have, however, calculated for this edition the
table at p. 252, of equally discharging river channels,
with a primary channel having a mean width of 100,
instead of 70, as in TABLE XI. ; and in the table at
p. 253 we have given the discharges at different in-
clinations from this new primary channel, to find
those from its equivalents. The tables at pp. 42, 43,
253, and TABLE VIII., have been calculated from
Du Buat's formula. For slow velocity of only a few
inches per second, the dimensions should be increased
by about one-sixth, and the discharges by about one-
half.
With reference to pipes, it is apparent that a given
depth of roughness or contraction arising from any
* Tables similar to numbers XL, XII., and XIII., but on a
much more extended scale, have been printed and published by
MB. WEALE, on a separate sheet for office use, and may be had
from him.
ORIFICES, WEIRS, PIPES, AND RIVERS. 251
cause will have a greater effect the smaller the dia-
meter becomes. Now in practice, it is necessary to
increase the diameter beyond what is found by calcu-
lation. For small service pipes half-an-inch is the
smallest diameter in general use. For mains and
sub-mains the value of c t in equation (74B), or at p.
214, should at least be doubled, or the discharge taken
at one and a half times its amount to find the dia-
meter. By enlarging the diameter by one-seventh, one-
half the amount will be added to the discharge, very
nearly; and by increasing the diameter by one-third, the;
discharge will be doubled. In a broad and practical
sense, and considering the losses arising from depo-
sitions,* pipes under two inches should have one-
third or more added to their calculated dimensions,
and larger pipes from one-third to one- seven th even
after making allowance for junctions, bends, and con-
tractions. For large conduits or channels the allow-
ance need not be so large, if the maximum quantity
to be conveyed be duly estimated.
* Mr. Bateman lately in giving evidence says :^ " He wished
to mention a circumstance which might he useful with regard to
the spongillse found in the Dublin water pipes. At Manchester,
before the introduction of soft water, the city was supplied with
hard water, which favoured the growth of a small fresh- water
mussel, which thickly line the reservoirs and pipes. There were
myriads of them, and they lay in the pipes as thick as paving
stones. These were caused by the large quantity of lime in the
water. He was curious to see what would be the effect of passing
water without lime. This was done ten or eleven years ago, and
the result was that these mussels had entirely disappeared. There
was no longer anything from which they could make their shells,
and for years, on their discharge, the small pipes were found
choked with them. If soft water were supplied to Dublin in
place of the present hard water, which probably favoured the
growth of spongillaD, they would probably disappear."
TABLE of mean widtJis and depths of equally discharging trapezoidal River-channels, or
Sewers, with side slopes up to 2 J to 1. Practically all river-channels may be reduced to
rectangular sections of equal areas and depths to find the discharge.
Primary
Channel
Mean rectangular dimensions of equally discharging water-channels or sewers, in any
measures whatever, inches, feet, yards, fathoms, or their aliquot parts, or multiples.
Primary
Channel
Mean
width
100
Mean
width
90
Mean
width
80
Mean
width
70
Mean
width
60
Mean
width
50
Mean
width
40
Mean
width
30
Mean
width
20
Mean
width
15
Mean
width
10
Mean
width
100
1
11
12
13
14
16
18
22
29
35
47
1
125
13
14
16
17
20
23
28
37
45
60
125
2
21
23
25
28
32
37
45
60
73
98
2
25
27
29
32
35
40
46
56
75
92
1-25
25
3
32
35
38
42
48
56
68
90
1-11
1-52
3
375
40
44
.48
53
60
70
85
1-13
1-40
1-94
375
4
43
46
51
56
64
74
91
1-21
1-50
2-08
4
5
54
58
64
71
80
93
14
1-53
1-90
2-67
5
6
64
70
76
85
96
1-12
37
1-85
2-31
3-28
6
625
67
73
79
88
1-00
1-16
43
1-93
2-42
3-44
625
7
75
81
89
99
1-12
1-31
61
2-17
2-73
3-92
7
75
80
87
95
06
1-20
1-41
73
2-34
2-95
425
75
8
86
93
1-02
13
1-29
1-50
85
251
3-17
4-59
8
875
94
02
1-12
24
1-40
1-64
2-02
2-76
3-50
5-10*
875
9
97
05
1-15
27
1-45
1-69
2-08
2-84
3-61
5-28
9
1-0
1-07
16
1-27
42
1-61
1-88
2-32
3-18
4-07
5-99
1-0
1-125
1-21
31
1-43
60
1-81
2-13
2-63
3-62
4-64
6-92
1-25
1-2
1-29
40
1-53
70
1-94
2-27
2-81
3-88
5-00
7-50
1-2
1-25
1-35
46
1-60
78
2-02
2-37
2-94
4-06
5-24
7-89
1-125
1-3
1-40
51
1-66
85
2-10
2-47
3-06
4-24
5-48
8-29
1-3
1-375
1-48
60
1-76
1-96
2-23
2-62
3-25
4-51
5-85
8-89
1-375
1-4
1-50
63
1-79
1-99
2-27
2-66
3-31
4-60
5-97
9-10
1-4
1-5
1-61
75
1-92
2-14
2-43
2-86
3-56
4-97
6-47
9-92
1-5
1-6
1-72
1-86
2-05
2-28
2-60
3-06
3-81
5-34
6-U8
10-78
16
1-625
1-75
1-89
2-08
2-32
2-64
3-11
3-87
5-43
7-11
11-00
1-625
1-7
1-83
1-98
2-17
2-43
2-76
3-26
4-06
5-72
7-50*
11-66
1-7
1-75
1-88
2-04
2-24
2-50
2-85
3-36
4-19
5-91
7-77
12-10
175
1-8
1"93
2-10
2-30
2-57
2-93
3-45
4-32
6-09
8-03
12-54
1-8
1-875
2-02
2-19
2-40
2-68
3-05
3-60
4-51
6-38
8-43
13-23
. 1-875
1-9
2-04
2-22
2-43
2-71
3-10
3-65
4-57
6-48
8-57
13-46
1-9
2-0
2-15
2-33
2-56
2-86
3-26
3-86
4-83
6-87
9-11
14-39
2-0
2-1
2-26
2-45
2-69
3-01
3-43
4-06
5-09
7-27
9-67
15-35
2-1
2-2
2-37
2-57
2-82
3-15
3-60
4-26
5-36
7-66
10-23
1632
2-2
2-3
2-47
2-69
2-95
3-30
3-77
4-4'J
5-62
8-07
10-80
17-31
2-3
2-4
2-58
2-80
3-08
3-44
3-94
4-67
5-89
8-48
11-38
18-33
2-4
2-5
2-69
2-92
3-21
3-59
4-11
4-87
6-16
8-79
11-97
19-35
2-5
2-6
2-80
3-04
3-34
3-74
4-28
5-08
6-42
9-31
12-57
20-40
2-6
2-7
2-91
3-16
3-47
3-88
4-55
5-28
6-69
9-73
13-17
21-46
2-7
2-8
3-01
3-27
3-60
4-03
4-62
5-49
6-97
10-16*
13-78
22-52
2-8
2-9
3-12
3-39
3-73
4-18
4-79
5-70
7-24
10-59
14-40
23-63
2-9
3-0
3-23
3-51
3-86
4-42
4-96
5-91
7-52
11-02
15-03
24-75
3-0
3-1
3-34
3-63
3-99
4-47
5-13
6-12
7-79
11-46
15-68
3-1
3-2
3-45
3-75
4-13
4-62
5-30
6-33
8-07
11-90
16-32
el
3-2
3-3
3-55
3-86
4-26
4-77
5-48
6-54
8-35
12-35
16-97
,.
3-3
3-4
3-66
3-98
4-39
4-92
5-65
6-75
8-64
12-80
17-63
||
3-4
3-5
3-77
4-10
4-52
5-06
5-82
6-96
8-92
13-26
18-29
PL
3-5
3-6
3-88
4-22
4-65
5-21
6-00
7-18
9-21
13-71
18-96
in
3-6
3-7
3-99
4-34
4-78
5-36
6-17
7-39
9-49
14-18
1965
gs
3-7
3-8
4-09
4-46
4-91
5-51
6-35
7-60
9-78
14-65
20-34
^
3-8
3-9
4-20
4-58
5-05
5-66
6-52
7-82
10-07
15-12
*j "5-^
3-9
4-0
4-31
4-69
5-18
5-81
6-70
8-04
10-36
15-59
ilif-si
4-0
4-1
4-42
4-81
5-31
5-96
6-87
8-25
10-66
16-07
"t* o3 H^'n fl>
Ss^g-s^
4-1
4-2
4-53
4-93
5-44
6-11
7-05
8-47
10-95
16-55
^llt = l
4-2
4-3
4-64
5-05
5-57
6-26
7-23
8-69
11-25
17-04
2**lid
4-3
4-4
4-74
5-17
5-71
6-41
7-40
8-91
11-55
17-53
KfS-&8
4-4
4-5
4-85
5-29
5-84
6-56
7-58
9-13
11-85
18-02
IHi^s
4-5
4-6
4-96
5-47
5-97
6-72
7-76
935
12-15
18-52
!*uHi
4-6
4-7
5-07
5-53
6-10
6-87
7-94
9-57
12-45
19-02
ftSI&:J
4-7
4-8
5-18
5-64
6-24
7-02
8-12
9-79
12-75
19-53
li*tgS1
4-8
4-9
5-29
5-76
6-37
7-17
8-29
10-02
13-06
20-04
S|.sB*-8
4-9
5-0
5-40
5-88
6-50
7-32
8-47
10-24
13-37
?iliii*
5-0
TABLE of the Discharges in cubic feet per minute from the primary Channel in the opposite
page, taken in feet; and from the corresponding equivalent Channels, also taken in feet.
Depths of the prln.a- 1
ry channel in oppo- 1
site table, having a 1
mean width of 100 ; 1
In feet.
Discharges In cubic feet per minute. Interpolate for Intermediate falls or Inclinations: divide
greater falls or Inclinations by 4, and donble the corresponding discharges. If the dimensions be
in Inches, divide the discharges here given by 500 ; if in tenths, by 316 ; If in eighths, by ibl ; if in
sixths, by 88; if in fifths, by 56; if in quarters, l>y 32; if in thirds, by 15-6; and if in halves, dy
5-66. Reverse the operation and multiply for like multiples.
Depths of the prima-
ry channel In oppo-
site table, having a
mean width of lOo;
in feet.
4 inches
per mile
lin
15840.
1 6 inches
per mile
lin
10560.
9 inches
per mile
lin
7040.
;12inches
per mile
lin
5280.
15inchef
per mile
lin
4224.
18inche
per mile
lin
3520.
21 inches
per mile,
lin
3017-1.
24 inches
per mile,
lin
2640.
80 inches
per mile,
lin
2112.
36inche
per mile
lin
1760.
1
IK
140
176
207
235
260
284
305
345
383
1
125
157
198
250
294
332
373
402
433
490
543
125
2
325
409
515
606
6*6
760
828
b91
1,009
1,117
2
25
455
675
725
853
966
1,070
1,166
1,256
1,422
1,574
25
3
601
760
957
1,125
1,275
1,412
1,539
1,658
1,876
2,076
3
375
844
1,068
1,344
1,582
1,793
1,985
2,162
2.329
2,637
2,919
375
4
931
1,178
1,484
1,745
1,977
2,189
2,386
2,569
2,900
3,220
4
5
1,308
1,653
2,()81
2,447
2,775
3.071
3,347
3.6<6
4,0*3
4,513
5
6
1,721
2,178
2.743
3,227
3,657
4,047
4,410
4,752
5.401
5,956
6
625
1,830
2,316
2,917
3,431
3,887
4,303
4,690
5,053
5,795
6,332
625
7
2,177
2,750
3,463
4,0'2
4,614
5,109
5.568
5,999
6,936
7,516
7
75
2,414
3,029
3,844
4,520
5,123
5.674
6,180
6.660
7,567
8,342
75
8
2,660
3,363
4,236
4,982
5,646
6,253
6,811
7,340
8,309
9,194
8
875
3,044
3,850
4,846
5,703
6,463
7,157
7,770
8,401
9,513
10,527
875
9
3,175
4,017
5,060
5,951
6,743
7,467
8,082
8,765
9,926
10,984
9
1-0
3,731
4,711
5,933
6,973
7,903
8.750
9,513
10,273
11,634
12,877
1-0
1-125
4,441
5,614
7,071
8,313
9,421
10,430
11,369
12,216
13,867
15,347
1-125
1-2
4,88!)
6,186
7,791
9,163
10,381
11,494
12,521
13,494
15,280
16.914
1-2
1-25
5,207
6,582
8,291
9,752
11,048
12,232
13,336
14,361
16,261
18,000
1-25
1-3
5,529
6,981
8,793
10,357
11,718
12,974
14,146
15,234
17,246
19,091
1-3
1-375
6,004
7,591
9,561
11,245
12,734
14,107
15,386
16,576
18,752
20,756
1-375
1-4
6,167
7,797
9,821
11,544
13,087
14,491
15,794
17,031
19,262
21,318
1-4
1-5
6,844
8,653
10,^98
12,818
14,524
16,081
17,523
18,917
21,376
23,658
1-5
1-6
7,538
9,520
12,002
14,115
15,994
17,709
19,296
20,829
23,539
26,053
1-6
1-625
7,705
9,741
12,272
14.428
16,348
18,102
19,724
21,286
24,061
26,631
1-625
1-7
8.252
10,432
3,139
15,432
17,509
19,360
21,126
22,780
25,769
28,523
1-7
1-75
8,617
10,893
3,719
16,134
18,282 20,241
22,060
23,776
26,907
29,784
1-75
1-8
1-875
8,993
9,561
11,369
12,088
4,318
15,226
16,851 19,079
17,905 |20,287
21,124
22,463
23,024
24,476
24,821
26,372
28,081
29,860
31,085
33,052
1-8
1-875
1-9
9,741
12,316)15,515
18,245 120,672
22.890
24,946
26,872
30,426
33,682
1-9
2-0
10,515
13,297
16,753
19,702
22.320
24.718
26,935
29,019
32,852
36,358
2-0
2-1
11,307
14,300
18,020 21,192
23,991
26,561
29,074
31,213
35,334
39,106
2-1
2-2
12,110 15,314 19,297 22,689
25,708
28,467
31,024
33,424
37,838
41,878
2-2
2-3
12,935 16,357 ! 20,608 24,235
27,456
30,407
33,134
35,694
40,410
44,724
2'3
2-4
13,781
17,425 121,954 25,816
29,250
32,32
35,299
38,022
43,048
47,643
2-4
2-5
14,647
18,520 123,332 ! 27,436 31,087
34,425
37,516
40,407
45,750
50,634
2-5
2-6
15,538
19,645 '24,747 129,100 132,974
36,514
39,794
42,856
48,526
53,706
2-6
2-7
16,430
20,773 26,167 30,770 34,867
38,610
42,078
45,316
51.311
56,789
2-7
2-8
17,333
21,915
27,605 32,462 36,784
40,733
44.390
47,809
54,131
59,913
2-8
2-9
18,257
23,084
29,076 ;34,193 38,744
42,905
46,755
50,359
57,017
63,110
2-9
3-0
19,203
24,280 30,581 35,963 (40,750
45,127
49,175
52,968
59,968
66,379
3-0
3-1
20,167 25,498 32,120
37,767 (42.794
47,392
51,640
55,634
62,986
69,709
3-1
3-2
21,146 26,737 33,673
39,600 144,871
49,692
54,148
58,327
66,033
73,097
3-2
3-3
22,118
27,969 35,225
41,425
46,939
51,978
56,640
61,017
69,077
76,465
3-3
3-4
23,106
29,220 36,798
13,275
49,036
54,302
59,171
63,745
72,164
79,879
3-4
3-5
24,115
30,497
38,407
45,166
51,1*0
56,675
61,758
66,534
75.322
83,371
3-5
3-6
25,139|31,795 40,040
47,086
53,356
59,084
64,384
69,366
78,526
86,915
3-6
3-7
26,18233.116
41,702
49,041
55,572
61,532
67,058
72,249
81,789
90,524
3-7
3-8
27,23:334,446
43,379
51,013
57,807
64,009
6:>,753
75,158
85,078
94,162
3-8
3-9
28-287
35,777
45,060
52,989
50,046
66.489
72,455
78.061
88.371
97,810
3-9
4-0
29,356
37,128
46,766
54.9M
52,318
69,006
75,197
81,012
91,710
101,512
4*0
4-1
30,438
38,495
48,492
57,024 64,616
n,553
77,973
83, '.*99
95,093
105,259
4-1
4-2
31,538
39.H84
50,246 59,0*6 66.960
74.141
80,7^3
87,033
98,535
109,065
4-2
4-3
32,654
41,294
"2,027 161,180 6.327
76,769
83,655
90,116
102,025
112,93"
4-3
4-4
33,776
+2,712
53,816 63,283
71,709
79,406
86,529
93.209
105,531
116,811
4-4
4-5
34,908
44,138 55,613 !6r>,394 '
"4.100
32,052
89.413
96,318
L09.054
I2<),709
4-5
4-6
36,041
45,579 ;57,429 67,527 76,500 J84,725
92,327
99,460
Ll26i4
124,647
4-6
4-7
37,193
47,034 59,2(i2 6.*, 682 78,955 (87.426
95,271
102,632
116,209 ;
128,625
4-7
4-8
38,36:3
18,514 61,128 71,874 |81,438 |90,173
98,266
105,860
119,866 132,672
4-8
4-9
39,544
50,009
^3,011 74.087 83,944 <
)2,946
01,289 :
L09.119
123,559 ]
L36.758
4-9
5-0
40,725
51,507
54,895 7 6,298 |86,450 <
>5,720
04,313 :
112,376
127,248 ]
L40,841
5-0
254 THE DISCHARGE OF WATER FROM
SECTION X.
EFFECTS OF ENLARGEMENTS AND CONTRACTIONS. BACKWATER
WEIR CASE, LONG AND SHORT WEIRS.
When the flowing section in pipes or rivers expands
or contracts suddenly, a loss of head always ensues ;
this is probably expended in forming eddies at the
sides, or in giving the water its new section. A side
current, moving slowly upwards, may be frequently
observed in the wide parts of rivers, when the channel
is unequal, though the downward current, at the
centre, be pretty rapid ; and though we may assume
generally that the velocities are inversely as the sec-
tions, when the channels are uniform, we cannot
properly do so when they are not, and the motions
so uncertain as those referred to. When a pipe is
contracted by a diaphragm at
Fig. 3 6
the orifice of entry, Fig. 27,
we have seen (equation 60),
that the loss of head is,
(123.) h =
When the diaphragm is placed in a uniform pipe,
Fig. 36, then A zz c, and we get the loss of head
(124.) h= - 27 >
and the coefficient of resistance
as in equation (67). The coefficient of discharge c d is
ORIFICES, WEIRS, PIPES, AND RIVERS.
255
here equal to the coefficient of contraction c c , o * very
nearly. Now we have shown in equation (45), and
the remarks following it, that the value of the co*
efficient of discharge, c d , varies according to the ratio
of the sections, A * and in TABLE V. we have calcu-
a
lated the new coefficients for different values of the
ratios, and different values of the primary coefficient
c d . If we assume c d , when A is very large compared
with a, to be '628, we then find by attending to the
remarks at pp. 109 and 128, that the different values of
c d corresponding to -807 x^, taken from TABLE V.,
a
are those in columns Nos. 2 and 5 of the next small
TABLE OF COEFFICIENTS FOR CONTRACTION, BY A DIAPHRAGM IN A PIPE,
a
A
Pd
C r
a
A
CA
c r
628
infinite
6
713
1-790
1
630
221-2
7
753
807
2
636
47-1
8
807
301
3
647
17-2
85
845
154
4
661
7-7
9
890
062
5
683
3-7
1
1-000
000
* The general value of c c , as given by Professor Eankine, is
618
c c = ~ ^-7, which is equal to unity when = A, as it
should be ; and equal to -618, when a is very small, compared with
A, as it also should be when the diaphragm is a thin plate, but
not otherwise. If the thickness of the diaphragm be twice the dia-
meter of the orifice a, the coefficient of discharge would be -815 ;
and if the higher arris be rounded, this would be increased to 1, in
which cases the expression would clearly fail ; the thickness of the
diaphragm and the form of the aperture a must also be considered.
256 THE DISCHARGE OF WATER FROM
table, the values of the coefficient of resistance, in
columns 3 and 6, being calculated from equation (125)
for the respective new values of the coefficient of
discharge thus found. The table shows that when
the aperture in a diaphragm is Aths of the section of
the pipe, that 47 times the head due to the velocity
is lost thereby. If the aperture in the diaphragm be
rounded at the arrises, the loss will not be so great, as
the primary coefficient c d will then be greater than
that due to an orifice in a thin plate : see coefficients,
p. 174
When there are a number of diaphragms in a tube,
the loss of head for each must be found separately,
and all added together for the total loss. If the
diaphragms, however, approach each other, so that
the water issuing from one of the orifices a, Fig. 36,
shall pass into the next before it again takes the
velocity due to the diameter of the pipe, the loss will
not be so great as when the distance is sufficient to
allow this change to take place. This view is fully
borne out by the experiments of Eytelwein with tubes
1-03 inch in diameter, having apertures in the diaph-
ragms of -51 inch in diameter.
Venturi's twenty-fourth experiment, with tubes
varying from -75 inch to -934 inch in diameter at the
junction with the cistern, so as to take the form of
the contracted vein, and expanding and contracting
along the length from -75 to 2 inches and from 2
inches to *75 inch alternately, shows the great loss of
head sustained by successive enlargements and con-
tractions of a channel, even when the junction of the
parts is gradual. Calling the coefficient for the short
ORIFICES, WEIRS, PIPES, AND RIVERS. 257
tube, with a junction of nearly the form of the con-
tracted vein, 1, then the following coefficients are
derivable from the experiment :
Short tube with rounded junction . . , . 1-
One enlargement ......... "741
Three enlargements ........ -569
Five enlargements , -454
Simple tube with a rounded junction of the
same length, 36 inches, as the tube with
the five enlarged parts . . . . , , . ? 786
The head, in the experiment, was 32 J inches. Venturi
states that no observable differences occurred in the
times of discharge when the enlarged portions were
lengthened from 3i to 6^ inches. See tables, pp. 152
and 191.
With reference to this experiment, so often quoted,
it is necessary to remark that the diameters of the
enlarged portions were 2 inches each, while the lengths
varied only from 3 to 6-J- inches, and consequently
were at most only 3 times the diameter. Now with
such a large ratio of the
width to the length of the H
enlarged portions, a A B b,
Fig. 37, it is pretty clear
that a good deal of the head
is lost by the impact of the moving water on the
shoulders A and B. That this is so is evident from
the fact, stated by the experimenter, of the time of
discharge remaining the same when a A, in five dif-
ferent enlargements, was increased from 3i to 6i
inches ; though this must have lengthened the whole
258 THE DISCHARGE OF WATER FROM
tube from 36 to 50 inches,* thereby increasing the
loss from friction proportionately, but which happened
to be compensated for by the reduction in the resist-
ances from impact at A and B, and in the eddies, by
doubling the lengths from a to A.
If, however, the length from a to A be very large
compared with the diameter, and the junctions at a,
A, B, and by be well grafted, less loss will arise from
the enlargement than if the smaller diameter con-
tinued all along uniform. The explanation is clear,
as the resistance from friction is inversely as the
square roots of the mean radii ; and the length being
the same, the loss must be less with a large than a
small diameter,
These remarks, mutatis mutandis, apply equally to
rivers as to pipes. We have already, pp. 140 and 147,
pointed out the effects of submerged weirs and con-
tracted river channels, and given formulae for calcu-
lating them.
BACKWATER FROM CONTRACTIONS IN RIVERS.
A river may be contracted in width or depth, by
jetties or by weirs ; and when the quantity to be dis-
charged is known, we have given, in formulae (9), (55),
and (57), equations from which the increase of head
may be determined. The effect of a weir, jetty, or
contracted channel of any kind, is to increase the
depth of water above ; and this is sometimes neces-
sary for navigation purposes, or to obtain a head for
mill power. When a weir is to rise over the surface,
we can easily find, from the discharge and length, the
discharge per minute over each foot of length, with
which, and the coefficient due to the ratio of the
* The dimensions throughout this experiment are given as in
the original, viz. in French inches.
ORIFICES, WEIRS, PIPES, AND RIVERS. 259
sections, on and above the weir, found from TABLE V.,
we can find the head from TABLE VI. For submerged
weirs and contracted widths of channel, the head can
be best calculated, by approximation, from the equa-
tions above referred to.
The head once determined, the extent of the back-
water is a question of some importance. If F c o D,
Fig. 38, be the original surface of a river, and a A B p
the raised surface by backwater from the weir at a, then
the extent a F of this backwater, in a regular channel,
will be from 1-5 to 1*9 times ac drawn parallel to the
horizon to meet the original surface in c. This rule
Fig-. 38
will be found useful for practical purposes ; but in
order to determine more accurately the rise for a
given length, B : B 2 or B! B, of the channel, it is neces-
sary to commence at the weir and calculate the
heights from A to B, B to B I? and from BJ to B 2 sepa-
rately, the distance from A to B 2 being supposed
divided into some convenient number of equal parts,
so that the lengths A B, BB^ &c., may be considered
free from curvature. Now, as the head A D is known,
or may be calculated by some of the preceding for-
mulae, the section of the channel at the head of the
weir also becomes known, and thence the mean velo-
city in it, by means of the discharge over the weir.
Putting A for the area of the channel at A n, d for its
depth A H, and v for the mean velocity ; also A X for
s3
260 THE DISCHARGE OF WATER FROM
the area of the channel at B i, d 1 for its depth, and v l
for its mean velocity ; b m the mean border between
the sections at AH and BI; r m the mean hydraulic
V -j V-i
depth ; ' the mean velocity ; AD A; B o zz ^ ;
the sine of angle o D E zz s ; and the length A B zz D o
nearly zz /; we get A x v = A X x v i and r m zz ' , -;
^ ^m
but as, in passing from B to A, the velocity changes
from v l to v, there is a loss of head equal -^ ?
and if c t be the coefficient of friction, there is a loss
of head from this cause equal c f X -- X
^*m
hence the whole change of head in passing from B to A
is equal to c { x X Q -^ . But this
*
change of head is equal to BE ADZZBO + OE AD
zz h^ + Is h, whence we get
(126.)
AV A + A! , ,
or as v l = . and r m zz ( ', . we get, by a few re-
A! ZO m
ductions and change of signs,
(127.) A- A! = *- * x ft m X X
and therefore we get
ORIFICES, WEIES, PIPES, AND RIVERS. 261
_A 2 -A*
, 6 m x (A + A,) v ^
~ Ct ' 2 A? v 2#
from which we can calculate the length / corre-
sponding to any assumed change of level between A
and B. Then, by a simple proportion we can find
the change of level for any smaller length. To find
the change of level directly from a given length does
not admit of a direct solution, for the value of h k
in equation (127) involves A 1? which depends again
on h h ly and further reduction leads to an equation
of a higher order ; but the length corresponding to
a given rise, k ly is found directly by equation (128).
When the width of the channel, w, is constant, and
the section equal to w x d nearly, the above equa-
tions admit of a further reduction for A X = d l w and
A = dw, by substituting these values in equation
(127) it becomes, after a few reductions,
(129.) h h l =dd 1
dxd l t
or, as it may be further reduced,
J^
s ~~ Ct X X
(130.) A-A, zz
d\ f *g
Now, we may take in this equation for all practical
purposes,
approximately, b being the border of the section at
262 THE DISCHARGE OF WATER FROM
AH; and also, , a l = -?, approximately; there-
! a
fore we shall have
f dw 2 a
(131.) h-h 1 = - -3 ^r--X/5
and
f 2 -
(132.) /= b d j" 9
Now, as -T = -, 2# r 64-4, and the mean value of
the coefficient of friction for small velocities c f -z
0078, we shall get
64 -4 ds- -0078 ^ 2
(133.) *, = A 64-4 rf- 2 t/ X /5 " /
and
64-4^5- -0078 - r v*
very nearly* Having by means of these equations
found A B from B o or B E, and B o from A B, we can in
the same manner proceed up the channel and calcu-
late B! c, B 2 G!, &c>, until the points B, B!, Ba in the
curve of the backwater shall have been determined,
and until the last nearly coincides with the original
surface of the river. When A x = 0, we shall have
64-4 tfs -0078- v 2
r
ORIFICES, WEIRS, PIPES, AND RIVE
^
If we examine equation (134) it appears that w
64-4 d -=L 2 v 2 , /must be equal to zero; or. when
d v 2
2 == gr-ij equal the height due to the velocity v.
When / is infinite, 64 -4 d must exceed 2# 2 , and 64
equal to -0078 ^ 2 ;
64*4 7*5
> .Q078
This is the velocity clue to friction in a channel of
the depth d, hydraulic mean depth r, and inclination
5 ; and, as in wide rivers r d nearly, v m 90'9
\/d~s, but when the numerator was zero we had from
it v m \/32-2 6/ ; equating these values of v, we get
5 ='0039 n --nearly : see p. 139. Now, the larger
the fraction s is, the larger will the velocity v become ;
and the larger v becomes, the more nearly, in all
practical cases, will the terms
64-4 d -2 v 2 and 64-4tfs - -0078 ^ 2 ,
in the numerator and denominator of equation (134),
approach zero ; when 64-4 d 2 v 2 becomes zero first,
/ = ; when 64'4 ds -0078 - v 2 becomes zero first,
/ equals infinity ; and when they both become zero
at the same time, l = h h l9 and 5 m , see p. 139 ;
if 5 be larger than this fraction, the numerator in
equation (134) will generally become zero before the
denominator, or negative, in which cases / will also
be zero, or negative ; and the backwater will take the
264 THE DISCHARGE OF WATEE FROM
form F c 2 b 2 j b a 1 a, Fig. 38, with a hollow at C*
Bidone first observed a hollow, as F c 2 b 2 , when the
inclination s was -. When the inclination of a river
30
channel changes from greater to less, the velocity is
obstructed, and a hollow similar to F c 2 b 2 sometimes
occurs ; when the difference of velocity is consider-
able, the upper water at b 2 falls backwards towards
c 2 and F, and forms a bore, a splendid instance of
which is the pororoca, on the Amazon, which takes
place where the inclination of the surface changes
from 6 inches to ^th of an inch per mile, and the
velocity from about 22 feet to 4J feet per second.
WEIR CASE, LONG AND SHORT WEIRS.
When a channel is of very unequal widths, above
a weir, we have found the following simple method of
calculating the backwater sufficiently accurate, and
the results to agree with observation. Having as-
certained the surf ace fall due to friction in the channel
at a uniform mean section, add to this fall the height
which the whole quantity of water flowing down would
rise on a weir having its crest on the same level as the
lower weir, and of the same length as the width of the
channel in the contracted pass. The sum will be the
head of water at some distance above such pass very
nearly. A weir was recently constructed on the
river Blackwater, at the bounds of the counties
Armagh and Tyrone, half a mile below certain mills,
which, it was asserted, were injuriously affected by
backwater thrown into the wheel-pits. The crest of
the weir, 220 feet long, was 2 feet 6 inches below the
ORIFICES, WEIRS* PIPES, AND RIVERS. 265
pit; the river channel between varied from 50 and
57 feet to 123 feet in width, from 1 foot to 14 feet
deep ; and the fall of the surface, with 3 inches of
water passing over the weir and the sluices down>
was nearly 4 inches in the length of half a mile.
Having seen the river in this state in summer, the
writer had to calculate the backwater produced by
different depths passing over the weir in autumn and
winter, which in some cases of extraordinary floods
were known to rise to 3 feet. The width of the
channel about 60 feet above the weir averaged 120
feet; The width, 2050 feet above the weir and 550
feet below the mills, was narrowed by a slip in an
adjacent canal bank, to 45 feet at the level of the top
of the weir, the average width at this place as the
water rose being 55 feet. The channel above and
below the slip widened to 80 and 123 feet. Between
the mills and the weir there were, therefore^ two
passes ; one at the slip, averaging 55 feet wide ;
another above the weir, about 120 feet wide. As-
suming as above, that the water rises to the heights
due to weirs 55 and 120 feet long, at these passes,
we get, by an easy calculation, or by means of TABLE
X., the heads in columns two and four of the follow-
ing table, corresponding to the assumed ones on the
weir, given in the first column.
As the length of the river was short, and the
hydraulic mean depth pretty large, the fall due to
friction for 60 feet above the weir was very small,
and therefore no allowance was made for it ; even the
distance to the slip was comparatively short, being
less than half a mile, and as the water approached it
266
THE DISCHARGE 0V WATER FROM
TABLE OF CALCULATED AND OBSERVED HEIGHTS ABOVE M'KEAN's
WEIR ON THE RIVER BLACKWATER.
Heights at
M'Kean's
weir 220 feet
long, in
inches.
Heights 60 feet above the
Weir channel 120 feet wide.
Heights 2050 fef>t above the
weir channel 55 feet wide ;
average.
Calculated
inches.
Observed
inches.
Calculated
inches.
Observed
inches.
1*
2J
2J
4J
5*
2
:.
^ii ?'!<
3
4*
i .
7i
7
4
6
..
10
9
5
7 *
. 12*
llj
6
9
9
15
!%
7
10*
10*
17*
18|
8
12
'" t?::v
20
20*
9
13*
12*
22*
20|
10
15
. .
24J
20
11
16*
. .
27|
24
12
18
17
30
31
13
19*
18*
32|
33
15
22*
21
37|
40
18
27
25
45J
46
21
31*
29|
53
54
24
36
34
604
62
with considerable velocity, this was conceived, as the
observations afterwards showed, to be a sufficient
compensation for the loss of head below by friction.
The observations were made by a separate party,
over whom the writer had no control, and it is neces-
sary to remark, that with the same head of water on
the weir, they often differed more from each other
than from the calculation. This, probably, arose from
the different directions of the wind, and the water
rising during one observation, and falling during
another.
ORIFICES, WEIRS, PIPES, AND RIVERS. 267
The true principle for determining the head at #,
Fig. 39, apart from that due to friction, is that
pointed out at pages 142 and 147 ; but when the passes
are very near each other, or the depth d 2 , Fig. 23, is
small, the effect of the discharge through d 2 is incon-
siderable in reducing the head, as the contraction and
loss of vis-viva are then large, and the head d^ be-
comes, that due to a weir of the width of the con-
tracted channel at A, nearly. The reduction in the
extent of the backwater, by lowering the head on
a longer weir, is found by taking the difference of the
amplitudes due to the heads at g, Fig. 39, in both
cases, as determined from equations (56), (128), etseq.
This will seldom exceed a mile up the river, as the
surface inclination is found to be considerably greater
than that due to mere friction and velocity, and hence
the general failure of drainage works designed on the
assumption that the lowering of the head below, by
means of long weirs, extends its effects all the way
up a channel. We must nearly treble the length of
a weir before the head passing over can be reduced
by one-half, TABLE X., even supposing the circum-
stances of approach to be the same : surely several
engineering appliances for shorter weirs, during
periods of flood, would be found more effective and
far less expensive than this alternative, with its extra
sinking and weir basin for drainage purposes.
The advocates for the necessity of weirs longer
than the width of the channel, for drainage purposes,
must show that the reduction of the head and extent
of backwater above ^, Fig. 39, is not small, and
that the effects extend .the whole way up the channel,
268
THE DISCHAKCiE OF WATEK FBOM
or at least as far as the district to be benefited.
Practice has heretofore shown, that long weirs have
failed (unless after the introduction of sluices or
other appliances) in producing the expected arterial
drainage results, notwithstanding the increased leak-
age from increased length, which must accompany
their construction.
The deepening in the weir basin a b B E A is mostly
of use in reducing the surface inclination between
a b and A B by increasing the hydraulic mean depth ;
but, thereby, the velocity of approach is lessened,
and therefore the head at E increased. When the
length of a weir basin a E exceeds that point where
these two opposite effects balance each other, there
will be a gain by the difference of the surface in-
clinations in favor of the long weir : but unless a E
exceeds half a mile, this difference cannot amount to
more than 3 or 4 inches, unless the river be very
small indeed ; and if the channel be sunk for the long
weirs B A or b a ly it should also be sunk to at least the
same depth and extent for the short weirs B e, b a,
otherwise there is no fair comparison of their separate
merits. The effect of the widening between a b and
A B, the depth being the same, is also to reduce the
OKIFICES, WEIES, PIPES, AND EIVEES. 269
surface inclination from a toE ; but, as before, unless a D
be of considerable length, this gain will also be small.
Now A B, at best, is but a weir the direct width of
the new channel at A B, and if the length a E be
considerable, we have an entirely new river channel
with a direct weir at the lower end, and the saving of
head effected arises entirely from the larger channel,
with a direct transverse weir at the lower end.
By referring to TABLE VIII., it will be found that
for a hydraulic mean depth of 5 feet a fall of 74
inches per mile will give a velocity of 2 feet per
second ; if we double the depth, a fall of 4 inches
will give the same velocity ; and for a depth of only
2 feet 6 inches, a fall of 121 inches is necessary,
This is a velocity much larger than we have ever
observed in a weir basin, yet we easily perceive
that the difference in the inclinations for a short
distance, E a of a few hundred feet, must be small.
If one section be double the other, the hydraulic
mean depth remaining constant, the velocity must be
one-half, and the fall per mile one-four-th, nearly.
This would leave 7| 2 = 5| inches per mile, or 1
inch per 1000 feet nearly, as the gain with a hydrau-
lic mean depth of 5 feet for a double water channel.
For greater depths the gain would be less, and the
contrary for lesser depths.
Is the saving of head and amplitude of backwater
we have calculated worth the increased cost of long
weirs and the consequent necessity and expense of
sinking and widening the channels for such long
distances ? We think not ; indeed, the sinking in the
basin immediately at the weir is absolutely injurious,
270 THE DISCHAEGE OF WATER FROM
by destroying the velocity of approach, and increasing
the contraction. The gradual approach of the bottom
towards the crest, shown by the upper dotted line
b E in the section, Fig. 39, and a sudden overfall, will
be found more effective in reducing the head, unless
so far as leakage takes place, than any depth of
sinking for nearly 80 or 100 feet above long weirs.
In most instances, the extra head will be only per-
ceived by an increased surface inclination, which
may extend for a mile or more up the channel,
according to the sinking and widening.
It is a general rule that, for shorter weirs, the co-
efficients of discharge decrease ; this arises from the
greater amount of lateral contraction, and is more
marked in notches or Poncelet weirs, than for weirs
extending from side to side of the channel ; but for
weirs exceeding 10 feet in length the decrease in the
coefficients from this cause is immaterial, unless the
head passing over bear a large ratio to the length ;
and we , even see from the coefficients, page 80, de-
rived from Mr. Blackwell's experiments, that with
10 inches head passing over a 2-inch plank, the co-
efficient for a length of 3 feet is *614 ; for a length of
6 feet *539 ; and for a length of 10 feet -534 ; show-
ing a decrease as the weir lengthens, but which may,
in the particular cases, be accounted for. We have
before referred to other circumstances which modify
the coefficients, yet we may assume generally, without
any error of practical value, that the coefficients are
the same for different weirs extending from side to
side of a river. If, then, we put w and w l for the
lengths of two such weirs, we shall have the relation
ORIFICES, WEIRS, PIPES, AND RIVERS. 271
of the heads d and d l for the same quantity of water
passing over, as in the following proportion :
d : d l ; : w^ : w 3 ;
and therefore
(135.) #=(-)* X d,
M0i'
By means of this equation we have calculated
TABLE X,, the ratio being given in columns 1, 3,
w l
l
5 and 7, and the value of ( IL Y, or the coefficient by
\Wif
which d is to be multiplied, to find d l in columns 2,
4, 6 and 8. It appears also, that if we take the heads
passing over any weir in a river in an arithmetical
progression, the heads then passing over any other weir
in the same river must also be in arithmetical progres-
sion, unless the quantity flowing down varies from
erogation or supply, such as drawing off by millraces,
&;c. If c d be the coefficient for a direct weir, *94 d
will answer for an obliquity of 45, and -91 c d for an
angle of 65.
SECTION XL
BENDS AND CURVES. BRANCH PIPES. DIFFERENT LOSSES OF
HEAD. GENERAL EQUATION FOR FINDING THE VELOCITY.
HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZOMETER.
CATCHMENT BASINS. RAIN-FALL PER ANNUM.
The resistance or loss of head due to bends and
curves has now to be considered. If we fix a bent
pipe, F B c D E G, Fig. 40, between two cisterns, so as
272
THE DISCHARGE OF WATER FROM
to be capable of re-
volving round in col-
lars at F and G, we
shall find the time the
water takes to sink a
given distance from/
to F in the upper cistern the same, whether the tube
occupy the position shown in the figure or the hori-
zontal position shown by the dotted line YbcdeG.
I This shows that the resistances due to friction and to
\ bends are independent of the pressure. If the tube
were straight, the discharge would depend on the
length, diameter, and difference of level between
f and a, and may be determined from the mean
velocity of discharge, found from TABLE VIII. or
equation (79). Here, however, we h^ve to take into
consideration the logs sustained at the bends and
curves, and our illustration shows that it is unaffected
by the pressure.
/ The experiments of Bossut, Du Buat, and others,
show that the loss of head from bends and curves
like that from friction increases as the square of the
velocity ; but when the curves have large radii, and
the bends are very obtuse, the loss is very small.
With a head of nearly 3 feet, Venturi's twenty- third
experiment, when reduced, gives for a short straight
tube 15 inches long, and 1^ inch in diameter, having
the junction of the form of the contracted vein, very
nearly *873 for the coefficient of discharge. When of
the same length and diameter, but bent as in Diagram
I, Fig. 40, the coefficient is reduced to -785 ; and
when bent at a right angle as at H, Fig. 40, the co-
ORIFICES, WEIRS, PIPES, AND RIVERS.
273
efficient is further reduced to -560. In these respec-
tive cases we have therefore*
1.
2.
, _
v = -873 \/^g~h, and h = 1-312 x 3-;
-2
vzz-785
, and k = 1-623 x
_
3. w = -560 v/2 # A, and A = 3-188 x j- ;
showing that the loss of head in the tube H, Fig. 40,
v 2
from the bend, is 1-876 x o~> or nearly double the
theoretical head due to the velocity in the tube. The
v 2
loss of head by the circular bend is only -3 11s,
or not quite one-sixth of the other.
Du Buat deduced, from about twenty-five experi-
ments, that the head due to the resistance in any
bent tube ABCDEFGH, diagram 1, Fig. 41, depends
IT O
on the number of deflections between the entrance at
A and the departure at H ; that it increases at each
* It is stated that the time necessary for the discharge of a
given quantity of water through a straight pipe being 1, the time
for an equal quantity through a curve of 90 would be 1-11, with
a right angle 1-57; two right angles would increase the time to
2'464, and two curved junctions to only 1*23. Vide REPORT ON
THE SUPPLY OF WATER TO THE METROPOLIS, p. 337, APPENDIX
No. 3.
274 THE DISCHARGE OF WATER FROM
reflection as the square of the sine of the deflected
angle, A B R for instance, and as the square of the
velocity; and that if <, < 1? < 2 , < 3 , &c., be the number
of degrees in the angles of deflection at B, c, D, E, &c.,
then for measures in French inches the height A b , due
to the resistances from curves, is
v 2 (sin. 2 4> + sin. 2 ^ + sin. 2 4> 2 + sin. 2 4> 3 + &c.)
*t>.) /**= 3000
which for measures in English inches becomes
^ 2 (sin. 2 < + sin. 2 ^ + sin. 2 < 2 + sin. 2
(137.) h*- 3197
and for measures in English feet,
58.) K- 266-4
or, as it may be more generally expressed for all
measures,
(139.) ^ = (sin/^+sin. 2 ^ + sin. 2 < 2 + sin. 2 4> 3 + &c.) X
2
, in which 8>27 zz ^grj = '00375 v 2 in feet.
The angle of deflection, in the experiments from
which equation (136) was derived, did not exceed
36. We have already shown the loss of head from
the circular bend in diagram L Fig. 40, where the
angle of deflection is nearly 45, to be *311 K =
00483 v\ but as the sin. 45 = -707 ; sin. 2 45 = -5
we get -00483 v* = -00966 v* x sin. 2 45, or more than
two and a half times as much as Du Buat's formula
would give; and if we compare it with Rennie's
experiments,* with a pipe 15 feet long, J inch dia-
meter, bent into fifteen curves, each 3 inches radius,
* Philosophical Transactions for 1831, p. 438.
ORIFICES, WEIRS, PIPES, AND RIVERS. 275
we should find the formula gives a loss of head not
much more than one half of that which may be
derived from the observed change, -419 to -370 cubic
feet per minute in the discharge. See p. 278.
Dr. Young* first perceived the necessity of taking
into consideration the length of the curve and the
radius of curvature. In the twenty-five experiments
made by Du Buat, he rejected ten in framing his
formula, and the remaining fifteen agreed with it very
closely. Dr. Young finds
(140.) ^OOOOM^x*.
where </> is the number of degrees in the curve N P,
diagram 2, Fig. 41, equal the angle N o P ; P = o N
the radius of curvature of the axis ; h b the head due
to the resistance of the curve, and v the velocity, all
expressed in French inches. This formula reduced
for measures in English inches is
(141.) ^ = -000004*^ X*.
P
and for measures in English feet,
(U2 .) , b
Equation (140) agrees to ^ n of the whole with
twenty of Du Buat's experiments, his own formula
agreeing so closely with only fifteen of them. The
resistance must evidently increase with the number
of bends or curves ; but when they come close upon,
and are grafted into each other, as in diagram 1,
Fig. 41, and in the tube F B c D E G, Fig. 40, the
motion in one bend or curve immediately affects those
* Philosophical Transactions for 1808, pp. 173 175.
T3
276 THE DISCHARGE OF WATER FROM
in the adjacent bends or curves, and this law does
not hold.
Neither Du Buat nor Young took any notice of
the relation that must exist between the resistance
and the ratio of the radius of curvature to the radius
of the pipe. Weisbach does, and combining Du Buat's
experiments with some of his own, finds for circular
tubes,
(143.) Ab = Tf0 X {-131 + 1-847 (^J) x
$*;. s 2^
and for quadrangular tubes,
d_ 7
p-
(144) A b=T f X ("124 4 3-104(^7) x ;
in which < is equal the angle N o P = N i R, diagram 2,
Fig. 41 ; d the mean diameter of the tube, and p the
radius N o of the axis. When - exceeds -2, the value
of -131 + 1-847 * exceeds -124 + 3-104
and the resistance due to the quadrangular tube ex-
ceeds that due to the circular one. We have ar-
ranged and calculated the following table of the
numerical values of these two expressions for the
more easy application of equations (143) and (144).
This table will be found of considerable use in cal-
culating the values of equations (143) and (144), as
the second and fifth columns contain the values of
131 + 1-847 o~ 5 and the third and sixth columns
the values of -124 + 3-104 (:;)> corresponding to
ORIFICES, WEIRS, PIPES, AND RIVERS.
277
different values of ~- ; and it is carried to twice the
& P
extent of those given by Weisbach.
TABLE OF THE VALUES OF THE EXPEESSIONS
131 + 1-847 (-\* and -124 + 3-104 ( -
d
2p '
Circular
tubes.
Quadrangular
tubes.
d
V
Circular
tubes.
Quadrangular
tubes.
1
131
124
6
440
643
15
133
128
65
540
811
2
1S8
135
7
661
1-015
25
145
148
75
806
1-258
3
158
170
8
977
1-545
35
178
203
85
1-177
1-881
4
206
250
9
1-408
2-271
45
244
314
95
1-674
2-718
5*
294
308
1-00
1-978
3-228
For bent tubes, diagrams 3, 4, and 5, Fig. 41, the
loss of head is considerably greater than for rounded
tubes. If, as before, we put the angle N i R = <, i R
being at right angles to i o the line bisecting the
angle or bend, we shall find, by decomposing the
V* V 2
motion, that the head ^ becomes ^ x cos. 2 $ from
the change of direction ; and that a loss of head
(145.) A b = (1 - cos. 2 2 $) ^- = sin. 2 2 < f-
must take place. When the angle is a right angle,
v 2
as in diagram 4, cos. 2 <#> = 0, and k b z= ^- ; that
^9
is to say, the loss of head is exactly equal to the
* The values corresponding to Q =-55 are -350 and *507 for
circular and quadrangular tubes.
278 THE DISCHARGE OF WATER FROM
theoretical head. When the angle or bend is acute,
v 2
as in diagram 5, the loss of head is (1 + cos. 2 2 <j> ~- 9
for then cos. 2 < becomes negative. Weisbach does
not find the loss of head in a right angular bend
v 2
greater than -984 - ; while Venturi's twenty-third
experiment, made with extreme care, p. 273, shows
the loss to be 1-876 s When the pipes are long,
however, the value of o - is in general small, and
this correction does not affect the final results in any
material degree.
Rennie's experiments,* with a pipe 15 feet long, J
inch in diameter, and with 4 feet head, give the dis-
charge per second
Cubic feet.
1. Straight, see table, p. 152 . . -00699
2. Fifteen semicircular bends . . -00617
3. One bend, a right angle 8J inches
from the end of the pipe . . -00556
4. Twenty-four right angles . . -00253
From these data we find consecutively, the theoreti-
cal discharge being -021885 cubic feet per second,
Q
and the theoretical head H s , that
1. v = -319 \/2#H, and therefore H = 9-82 x j- ;
2. v = -282 \X2^H, H = 12-58 x^-;
* Philosophical Transactions for 1831, p. 438.
ORIFICES, WEIRS, PIPES, AND RIVERS. 279
^2
3. v n -254 \/2#H,and therefore H = 15*50 x
^2
4. v = -116 v/2#H, HIZ 74-34 x g--
The loss of head, therefore, by the introduction of
15 semicircular bends, is 2*76 ; by the intro-
2
duction of one right angle, 5-68 n ; and by the
2
introduction of 24 right angles, 64-52 -, or about
12 times the loss due to one right angle. This
shows that the resistance does not increase as the
number of bends, as we before remarked, p. 256,
when they are close to each other. The loss of head
from one right angle, 5*68 o , is more than double
the loss from 15 semicircular bends, or 2-76 ^~.
The loss of head for a right angular bend, determined
v 2
from Venturi's experiment, is 1-876 s-; formula
v 2
(145) makes it ^- ; and Weisbach's empirical for-
v 2
mula, (-9457 sin. < + 2-047 sin. 4 <) H-, makes it only
v 2
984 s . The formulae now in use give, therefore,
results considerably under the truth. It appears
to us, that the velocity of the water moving
directly towards the bend must be taken into
consideration, and also the loss of mechanical ef-
fect from contraction, and eddies at the bend, as
280
THE DISCHAEGE OF WATEE FEOM
well as the loss arising from the mere change of
direction.
BRANCH PIPES.
When a pipe is joined to another, the quantity of
water flowing below the junction B, diagram 1,
Fig. 42, must be equal to the sum of the quantities
Fig*. 42
flowing in the upper branches in the case of supply ;
and when the branch pipe draws off a portion of the
water, as in diagram 2, the quantity flowing above
the junction must be equal to the quantities flowing
in the lower branches. Both cases differ only in the
motion being from or to the branches, which, in
pipes, are generally grafted at right angles to the
main, for practical convenience, as shown at bb, and
then carried on in any given direction. The loss of
head arising from change of direction, equation (145),
2
is sin. 2 2 < n~, in which 2 < zz angle ABO; but as in
general 2 <#> is a right angle for branches to mains,
v 2
this source of loss becomes then simply s . I n
addition to this, a loss of head is sustained at the
junction, from a certain amount of force required to
unite or separate the water in the new channel. In
the case of drawing off, diagram 2, this loss was
estimated by D'Aubuisson, from experiments by
Genieys, to be about twice the theoretical head due
ORIFICES, WEIRS, PIPES, AND RIVERS. 281
2 v 2
to the velocity in the branch, or o , so that the
whole loss of head arising from the junction is
v 2 2v 2 3 v 2
o + ~ - =: ^ , or three times the theoretical head
2# n 1g 2g>
due to the velocity. In the case of supply, the loss
is probably nearly the same. The actual loss is,
however, very uncertain; but, as was before ob-
served when discussing the loss of head occasioned
v 2
by bends, two or three times ^ is in general so
comparatively small, that its omission does not ma-
terially affect the final results. A loss also arises
from contraction, &c. See pp. 175, 176.
The calculations for mains and branches become
often very troublesome, but they may always be
simplified by rejecting at first any minor corrections
for contraction at orifice of entry, bends, junctions,
or curves. If, in diagram 2, Fig. 42, we put h for
the head at B, or height of the surface of the
reservoir over it ; h & for the fall from B to A ; A d for
the fall from B to D ; / equal the length of pipe from
B to the reservoir ; / a equal the length B A ; 4 equal
the length B D ; r equal the mean radius of the pipe
B c ; r a the mean radius of the pipe B A ; r d the mean
radius of B D ; v the mean velocity in B c ; v a the
velocity in B A ; and v d the velocity in B D, we then
find, by means of equation (73), the fall from the
reservoir to A equal to
(146.) A+A>
the fall from the reservoir to D equal to
282 THE DISCHARGE OF WATER FROM
(147.) k + ht
and, as the quantity of water passing from c to B is
equal to the sum of the quantities passing from B to
A and from B to D,
(148.) vr 2 = v a rl + v d rl
By means of these three equations we can find any
three of the quantities A, A a , A d , r, r a , r d , b, b a , b A ,
the others being given. Equations (146) and (147)
may be simplified by neglecting c r , the coefficient due
to the orifice of entry from the reservoir, and 1, the
coefficient of velocity. They will then become
(148A.) h + A. = c t x - X + X
and
(149.) A + A d = c f
The mean value of c { for a velocity of 4 feet per
second is -005741. and of ^, -0000891. The values
*9
for any other velocities may be had from the table of
coefficients of friction given at p. 214. When /, h,
and r are given, the velocity v can be had from
the equation, v = ( x -y) , or more immediately
from TABLE VIII.
GENERAL EQUATION FOR MEAN VELOCITY.
We are now enabled to give a general equation for
finding the whole head H, and the mean velocity v, in
any channel ; and to extend equations (73) and (74)
so as to comprehend the corrections due to bends,
ORIFICES, WEIKS, PIPES, AND EIVERS. 283
curves, &c. Designating, as before, the height due
to the resistance at the orifice of entry by
h r , and the corresponding coefficient by c r ;
h { the head due to friction, and c ( the coefficient of friction ;
7i b the head due to hends, and c b the coefficient of hends ;
h c the head due to curves, and c c the coefficient of curves ;
h e the head due to erogation, and c e the coefficient of erogation ;
/i x the head due to other resistances, and c x their mean coefficient ;
then we get
(150.) n = h t + h f + h l) + h c +h e + ^ + ~;
that is to say, by substituting for h r , h f , &c., their
values as previously found,
v* I v 2 v 2
V 2 V 2 V 2
X + Ce X + Cx X
or, more briefly,
7
(151.) H =
from which we find
(152.) v=
c t x -
It is to be observed here, that for very long
uniform channels, the value of the mean velocity
will be found in general equal to I ^ rH | a s the
\ Cf I )
other resistances and the head due to the velocity
are all trifling compared with the friction, and may
be rejected without error ; but, as we before stated,
it is advisable in practice, when determining the
diameter of pipes, p. 229, to increase the value of c
284 THE DISCHARGE OF WATER FROM
table, p. 214, or to increase the diameter found
from the formula by one-sixth, which will increase
the discharging power by one half. (See TABLE XIII.)
In equations (74) and (151), the coefficient of fric-
tion c { depends on the velocity v, and its value can
be found from an approximate value of that velocity
from the small table, p. 214. If, however, we use
both powers of the velocity, as in equation (83), we
shall get, when H is the whole head, and h the head
from the surface to the orifice of entry
I v 2
(av + bv 2 )- + (I + ;)*- + h=n,
a quadratic equation from which we find
f (H-fe)2<7r / gal
1(1 + c T )r + 2gbl
(1 + c r )r
for a more general value of the velocity than that
given in equation (74). If now we put c s = c r + c b
+ c c + c e + c x in equation (151) we shall find
gal
Zgbl
for a more general expression of equation (152), when
the simple power of the velocity, as in equation (83),
is taken into consideration. For measures in English
feet, we may take a -0000223 and b = -0000854,
which correspond to those of Eytelwein, in equation
(97). The value of a is the same in English as in
French measures, but the value of b in equation (83),
for measures in metres, must be divided by 3-2809
to find its corresponding value for measures in
v 2
English feet. In considering the head ^- c n due to
contraction at the orifice of entry as not implicitly
comprised in the primary values of a and b, equation
ORIFICES, WEIRS, PIPES, AND RIVERS.
285
(83), Eytelwein is certainly more correct than D'Au-
buisson, Traite d'Hydraulique, pp. 223 et 224, as
this head varies with the nature of the junction, and
should be considered in connection with the head
due to the velocity, or separately. It can never be
correctly considered as a portion of the head due to
friction. In all Du Buat's experiments, this head
was considered as a portion of that due to the velo-
v 2
city, and the whole head, (1 + c r ) 9 ? deducted to
find the head due to friction and thence the hydraulic
inclination.
VALUES OF a AND I FOR MEASURES IN ENGLISH FEET.
Equation (88.)
(90.)
(94.)
(98.;
(109.)
(HI.)
(114.)
Mean values for all
straight channels,
pipes, or rivers
These mean values of a and b give the equation
r s = -0001040 v 2 + -0000221 v,
from which we find
9615 r s =v 2 + -21 v,
and thence
(153.) v =(9615r* + -Oil)* -105 = 98 v/~ 1,
very nearly, suited to velocities of about 2 feet, p. 217.
a.
b.
)
0000445
0000944
)
0000173
0001061
)
0000243
0001114
)
0000223
0000854
)
0000189
0001044
)
0000241
0001114
)
0000035
0001150
0000221
0001040
286
THE DISCHARGE OF WATER FROM
HYDROSTATIC AND HYDRAULIC PRESSURE. PIEZOMETER.
When water is at rest in any vessel or channel, the
pressure on a unit of surface is proportionate to the
head at its centre,* measured to the surface, and is
expressed in Ibs. for measures in feet, by 624 H s, in
which H is the head, and s the surface exposed to the
pressure, both in feet measures. This is the hydro-
static pressure. In the pipe A B c D F E, Fig. 43, the
pressure at the points B, c, D, F, and E, on the sides of
the tube will be respectively as the heads B b, c c, D d,
F/J and E e, if all motion in the tube be prevented by
stopping the discharging orifice at E. In this case the
pressure is a maximum and hydrostatic ; but if the
discharging orifice at E be partially or entirely open,
a portion of each pressure at B, c, D, F, &c., is ab-
sorbed in overcoming the different resistances of
friction, bends, &c., between it and the orifice of entry
at A, and also by the velocity in the tube, and the
difference is the hydraulic pressure.
* This is only correct when the surface is small in depth com-
pared with the head. If H he the depth of a rectangular surface
in feet, and also the head of water measured to the lower hori-
zontal edge, then the pressure in Ibs. is expressed by 31 H 2 ; and
the centre of pressure is at f rds- of the depth.
OEIFICES, WEIES, PIPES, AND EIVEES. 287
Bernoulli first showed that the head due to the
pressure at any point, in any tube, is equal to the
effective head at that point, minus the head due to the
velocity. When the resistances in a tube vanish, the
effective head becomes the hydrostatic head, and by
representing the former byA ef we shall have, adopting
the notation in equation (150),
and consequently the head due to the hydraulic pres-
sure equal
v * X
If small tubes be inserted, as shown in Fig. 43, at
the points B, c, D, and F, the heights B b 1 , c c 1 , D d 1 , F/ 1 ,
which the water rises to, will be represented by the
corresponding values of A p in the preceding equation;
and the difference between the heights c c 1 , F/ 1 , at
c and F, for instance, added to the fall from c to F
will, evidently, express the head due to all the re-
sistances between c and F. When H = E e, and the
orifice at E is open, we have, from equation (150),
, and therefore Ap=0,
that is, the pressure at the discharging orifice is
nothing.
The vertical tubes at B, c, D, F, when properly
graduated, are termed piezometers or pressure gauges;
they not only show the actual pressure at the points
where placed, but also the difference between any
two ; D d 1 B b 1 , for instance, added to the difference
of head between D and B, or D d 2 will give D d 1 sb l
+ D d 2 for the head or pressure due to the resistances
288 THE DISCHARGE OF WATER FROM
between B and D. This instrument affords, perhaps,
the very best means of determining the loss of head
due to bends, curves, diaphragms, &c. The loss of
head due to friction, bend, diaphragms, &c., between
K and L, Fig. 43, is equal to K k L / + K v. If M
be the same distance from L as K is, L / M m will be
the height due to the friction (L and M being on the
same level) ; therefore K k L / + KV L / + M m m
K k + K v + M m 2 L / is the head due to the
diaphragm and bend both together. If the diaphragm
be absent, we get the head due to the bend, and if
the bend be absent, the head due to the diaphragm in
like manner.
When the discharging orifice, as at E, is quite open,
we have seen that the pressure there is zero; but
when, as at G, it is only partly open, this is no longer
the case, and the hydraulic pressure increases from
zero to hydrostatic pressure, as the orifice decreases
from the full section to one indefinitely small com-
pared with it. A piezometer, placed a short distance
inside G, will give this pressure ; and the difference
between it and the whole head will be the head due to
the resistances and velocity in the pipe : from which,
and also the length and diameter, the discharge may
be calculated as before shown. Again, by means of
the head M m 1 , and that due to the velocity of ap-
proach, we can also find the discharge through the
diaphragm G ; see equation (45) and the remarks fol-
lowing it. This result must be equal to the other,
and we may in this way test the formulae anew or
correct them by the practical results.
The velocity of discharge of the tube A c D E, may
ORIFICES, WEIRS, PIPES, AND RIVERS. 289
be calculated by means of any piezometric height
c c 1 ; for by putting the whole fall from c 1 to E equal
(2flrrH c lH
to H C I, we get, disregarding bends, v = \ - \ , j n
1 Me 1 J
which I C \G E. This is evident from equation (152),
as we have supposed that no part of the head is ab-
sorbed in generating velocity, or in overcoming the
resistance of bends. If the bend at D were taken
into consideration, then v = '
-fX^r +
SECTION XII.
RAIN-FALL CATCHMENT BASINS. DISCHARGE INTO CHANNELS.
DISCHARGE FROM SEWERS. LOSS FROM EVAPORATION, ETC.
A catchment basin is a district which drains itself
into a river and its tributaries. It is bounded gene-
rally by the summits of the neighbouring hills, ridges,
or high lands forming the water-shed boundary ; and
may vary in extent from a few square miles to many
thousands ; that of the Shannon is 4,544 square
miles. The average quantity of water which dis-
charges itself into a river will, cceteris paribus, depend
on the extent of its catchment basin, and the whole
quantity of rain discharged on the area of the catch-
ment basin, including lakes and rivers.
The quantity of rain which falls annually varies
with the district and the year ; and it also varies
at different parts of the same district. The average
quantity in Ireland may be taken at about 34 inches
290
TABLE of some Catchment Basins in Ireland.
Names of Drainage-districts,
or Rivers.
Counties or Towns.
Area of
Catchment
in acres.
Area of
Catchment in
square miles.
Avonmore . .
Wicklow and Wexford
128 000
200-
Wicklow . .
179,840
281-
Ballinasloe .
Mayo
70000
110-
Barrow, Nore, and Suir . . .
Waterford
2,176,000
3400-
Blackwater and Boyne ....
Meath, &c
695,040
1086-
Blackwater ....
Waterford Youghal
780,160
1219-
Blackwater
Armagh
336,640
526-
Blackwater
Meath and Kildare
50,000
78-1
Bandon River
Cork
145,920
228-
Bann, Upper and Lower, and
the Main
810,240
1266-
Boyne
Meath, Westmeath Kildare
and King's
304,139
478-2
Brusna (Ferbane)
King's . . .
389,120
608-
Wexford
26,752
41-8
Ballinamore and Bally connel
Cavan, Fermanagh, Leitrim,
and Roscommon
101,455
158-5
Breeogue
Sligo . .
180,408
282-
Cork
23,500
36-7
Cappagh
34,856
54-4
Sliffo
90744
141-8
Camoge
Limerick . . .
61,184
95-6
Galway, Mayo, and Roscom-
mon . . .
96,161
150-2
Dodder
Dublin
35,200
55-
Deel
Meath and Westmeath
64,000
100-
Dee
Louth and Meath
78,000
121-9
Erne
Belturbet Enniskillen
1,014,400
1585-
Fovle
Londonderry
944,640
1476-
Fergus
Clare and Galway
134,400
210-
Fane
Louth
87,400
136-6
Glyde
Louth, Meath Monaghan
and Cavan
176,813
276-3
Inny . .
Meath Westmeath Long-
ford, and Cavan
231,116
361-1
Kilbeggan . .
Westmeath and King's
88,030
137-5
lifi'ey and Tolka
Dublin, &c
328,320
513-0
Lee ...
Cork
470,400
735-
Lough Gara and Mantua . .
Loughs Oughter and Gowna
and River Erne
Lough Neagh
Roscommon, Mayo, and Sligo
Cavan, Leitrim, and Longford
Londonderry Antrim Down
128,000
260,480
200-
407-
and Armagh
1,411,320
2205-2
Lough Mask and River Robe
Mayo and Galway
225,000
351-5
Loughs Corrib, Mask, and
Carra .
Galwav and Mayo
780,000
1218-7
Lonford
Longford .
72320
113-0
Moy .
Mayo, Ballina
661,120
1033-
r ?
Main . ,
Antrim
37,600
90-
54,000
84-4
Ma^hera .
Down
19000
29-7
Nobber
Meath
40,000
62-3
Quoile
Down
57000
89-1
Rinn and Black River ....
Strokestown
Leitrim and Longford ....
Roscommon
74,000
70000
115-6
109-4
Different Counties, Towns of
Athlone, Limerick
2,908 160
4544-
Slaney . .
Wexford
521 600
815-
ORIFICES, WEIRS, PIPES, AND RIVERS. 291
deep, that which falls in Dublin being 27 inches, and
that in Cork 41 inches nearly. The average yearly
fall in Dublin for seven years, ending with 1849, was
26*407 inches ; and the maximum fall in any month
took place in April 1846, being 5-082 inches. "The
average fall in inches per month for seven years,
ending with 1849, was as follows : October, 3 -060 ;
August, 2-936 ; January, 2-544 ; April, 2-503 ; No-
vember, 2-300 ; July, 2-116 ; June, 2-005 ; Decem-
ber, 1*938; September, 1-860 ; May, 1-814; March,
1-739 ; February, 1-534."* A gauge at London-
derry, 1795 to 1801, gives 31 inches average ; one
at Belfast, from 1836 to 1841, gives 35 inches; at
Mountjoy, Phoenix Park, 182 feet above low water,
1839 and 1840, there is an average of 33 inches ;
and at the College of Surgeons, 52 feet over low
water, the average is 30 inches for the same two
years. Sir Robert Kane assumes that 36 inches is
the average fall in Ireland, and that out of that depth
12 inches, or one-third, passes on to the sea, two-
thirds being evaporated and taken up by plants.
The quantity varies a good deal with the altitude of
the district. In parts of Westmoreland it rises
sometimes to 140 inches ; in London, an average
of 20 years' observations, gives a fall of nearly 25
inches.
Forty years' observation at Greenwich, Kent, at
155 feet above the level of the sea, gives the following
results :
* Proceedings of the Koyal Irish Academy, vol. v., p. 18.
TJ3
292
THE DISCHAEGE OF WATER FROM
Description of fall.
Winter.
Spring.
Summer
Entire
Years.
Mean annual fall . . .
inches.
7-86
inches.
7-25
inches.
10-47
inches,
f 25-48
Maximum fall; being a mean of
five of the wettest years during
forty years
11-05
10-86
14-96
{ 25-58
( 34-00
Minimum fall ; being a mean of
five of the driest years during
forty years . .
5-22
4-05
6-80
{ 36-87
( 18-40
(16-07
In this table Winter comprises November, De-
cember, January, and February ; Spring, the next
four months ; and Summer, the months of July,
August, September, and October. The last column
contains means of two classes of years : the first
figures showing the ordinary years from January to
December, and the second, under the first, years
from November to October.* We see here that the
mean maximum is fully double of the mean minimum,
and about one-and-a-half times the mean annual fall,
and therefore the necessity for calculating from the
minimum fall for all water works in which it is
an element, and from the maximum for sewerage
works where it is not intended to pass off a portion
on the surface or through other available channels.
In the district surrounding the Bann reservoirs in
the County Down, the average fall has been so high
as 72 inches. In Keswick, the average fall is said to
be 67| inches, and in Upminster, Essex, only 19|
inches. Indeed, it is requisite to obtain the fall
from observation for any particular district, when it
* See Mr. James Simpson in the Metropolitan Main Drainage
Report, 1857, p. 115.
ORIFICES, WEIRS, PIPES, AND RIVERS. 293
is necessary to apply the results to scientific pur-
poses ; and not the mean average fall alone, but also
the maximums and minimums in a series of years and
months in each year.
Mr. Symons gives (see Builder for I860, p. 230)
the following heavy falls of rain during 1859 :
Wandsworth, June 12th, 217 inches in two hours ;
Manchester, August 7th ? 1*849 inches in twenty-four
hours ; Southampton, September 26th, 2-05 inches
in two-and-a-quarter hours ; Truro, October 25th,
during the day, 2-4 inches. The mean falls in the
South Western Counties were 39-1 inches ; in the
South Eastern Counties, 30-2 inches ; in the West
Midland Counties, 28 inches ; in the Eastern Coun-
ties, 2 5 *4 inches ; in the North Midland Counties,
24 inches ; in the North Western Counties, 39
inches ; in the Northern Counties, 55 inches ; and
the average of all England, 31-857 inches.
As an instance of extraordinary rain-fall, in
connexion with the sewage question, it is stated that
4 inches of rain fell in one hour in the Holborn and
Finsbury sewers' district, on the 1st of August,
1846 ; at Highgate, 3-5 to 3 -3 inches ; and at
Greenwich, 0-95 inches.*
In the upland districts about Manchester, Mr.
Homershamf gives the result of observations at
Fairfield, Bolton, Rocksdale, Marple, Comlis re-
servoir, Belmont, Chapel-en-le-Frith, and Whiteholme
reservoir, for four years. These give a maximum
fall of 61-4 inches at Belmont Sharpies in 1847, and
* Metropolitan Main Drainage Keport, p. 16.
f Report on the Supply of Water to Manchester. WEALE.
294 THE DISCHARGE OF WATER FROM
a minimum of 24-8 at Whiteholme reservoir in 1844.
The general average for the four years being 42-49
inches.
April is the driest month, and October, or about
it, the wettest month, and the average fall during
the year varies sometimes as two to one.
The proportion between the quantity which falls,
and that which passes from a catchment basin into
its river, also varies very considerably. When the
sides of a catchment basin are steep, and the water
passes off rapidly into the adjacent river or tri-
butaries, there is less loss by evaporation and perco-
lation than when they are nearly flat. The soil,
subsoil, and stratification, have also considerable
effect on the proportion. Reservoirs being generally
constructed adjacent to steep side falls, give a much
larger proportion of the quantity fallen than can be
obtained from rivers in flatter districts ; besides, the
quantity of rain which falls on the high summits,
near reservoirs, almost always considerably exceeds
the average fall. As 640 acres is equal to one
square mile, and one acre is equal to 43,560 square
feet, a fall of one inch of rain is equal to 3,630 cubic
feet per acre, and to 3,630 x 640 = 2,323,200 cubic
feet per square mile : the proportion of this fall, for
each acre, or square mile of the catchment basin,
which enters the river, must depend entirely on the
district and local circumstances, the full or maximum
quantity being retained on lakes. A stream de-
livering 53 cubic feet per minute supplies an equiva-
lent to 12 inches of rain-fall collected per square
mile per annum.
ORIFICES, WEIRS, PIPES, AND RIVERS.
295
It is too often taken for granted that the discharge
from a catchment basin takes place, into the con-
veying channels, in nearly the same time that a
given quantity of rain falls. Perhaps the largest
registry on record in Great Britain is a fall of four
inches in an hour. The maximum fall in any hour of
any year seldom exceeds half of this amount, and
then perhaps only once in several years. The quan-
tity which falls will not be discharged into the
channels in the same time. The quantity discharged,
QUANTITY PER ACRE FOR A GIVEN DEPTH OF FALL.
Fall in
inches.
Cubic feet
per acre.
Fall in
inches.
Cubic feet
per acre.
Fall in
inches.
Cubic feet
per acre.
Fall in
inches.
Cubic feet
per acre.
2
7260
i
1815
1
454
JL_
181
If
6352
-i
1361
|
403
TO-
121
ii
5445
907
1
363
1
91
2
4
TO
4:
U
4537
i
726
A
302
50
73
i
3630
i
605
yV
259
"g"0"
61
I
2723
i
T
519
TV
227
TV
52
and time, will depend a good deal on the season and
district. The arterial channel receives the supply at
different places and from different distances, and the
water in passing into and from it does not encounter
the same amount of resistance as if it all passed
first into the upper end. Less sectional area is
therefore necessary than if the whole discharge had
to pass through the whole length of the channel and
during the time of fall. The relation of the quantity
of rain -fall to the portion which flows into the main
channel, as well as the time which it takes to arrive
at it, and the places of arrival, must be known
296 THE DISCHARGE OF WATER FROM
before the proper size of a new channel can be
determined, particularly sewers in urban districts.
A pipe sufficient to discharge the water from 200
acres need not be 20 times the discharging power of
one exactly suited to 10 acres of the same district,
for the discharge from the outlying 190 acres will
not arrive at the main in the same time as that from
the adjacent 10 acres.
The following table of rain-fall, at Athlone, central
in Ireland, was furnished to the Royal Irish Academy
by General Sir H. D. Jones, and is printed in the
Proceedings.* The average for four years, gives 29
inches, and the effect on the Upper and Lower Sills
of the Lock as affecting the rise and fall of the
Shannon, affords valuable data, although not analysed.
The rise and fall on the sills is the sum of the
monthly risings and fallings for each year, and must
be divided by 12 to get the average monthly rise and
fall. In 1845 the greatest rise was in January,
2 feet 9 inches at the upper sill, and 3 feet 1 1| inches
at the lower sill. In 1846 the greatest rise was
2 feet 5 inches in October, at the upper sill ; and
5 feet 6J inches on the lower sill, in August.
Upper SiU. Lower Sill.
Maximum rise in Maximum rise in
one month. one month.
1845 .... 2 ft. 9 in. January . . 3 ft. !! in. January.
1846 . . . . 2 ft. 5 in. October . . 5 ft. 6 in. January.
1847 .... 3ft. 1 in. November. .4ft. 6 in. May.
1848 .... 3 ft. 3 in. February . . 4ft. 11 in. February.
The sum of the risings and fallings for each month,
taken as a mean of four years, is nearly the same
* Vol. iv.
ORIFICES, WEIRS, PIPES, AND RIVERS.
297
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298
THE DISCHARGE OF WATER FROM
at either sill. The general average of the rise and
fall for the upper sill, is about 1 foot 3i inches each
way, and 1 foot 101 inches at the lower sill. These
would give 2 feet 7 inches for the average difference
of level in the Shannon above, and 3 feet 9 1 inches
for that in the Shannon below. In Lough Allen
catchment of 146 square miles, the maximum rise
was sometimes 6 inches in 24 hours, calculated at
568 inch of depth of rain, over the catchment area.
Above Killaloe, the catchment is 3,611 square miles,
and the floods about once a year rose 6 inches in
24 hours, or *296 inch in depth of rain over the
catchment. Once, in 1840, it is reported to have
risen 12 inches, or *6 inch of rain over the catchment
in one day.
MAXIMUM DISCHARGES OF THE SHANNON AND ERNE, AND A TRIBUTARY OF THE LATTER,
THE WOODFORD RIVER.
BIVERS IN IRELAND.
Extent of
catchment,
statute acres.
Square
miles.
Maximum
discharge
per minute
in cubic feet.
Cubic feet
per minute
from each
acre.
Cubic feet
per minute
from each
square mile
Shannon, at Killaloe, measured
previous to the commence-
ment of Shannon Works,
3,000,000
4687'5
1,000,000
0-33
211
Lower Erne, measured during
the very high floods of Jan.
1851, at BeUeek
974,000
1521-9
657,511
0-67
429-
Upper Erne, measured during
the very high floods of Jan.
1851, at Belturhet
309,000
482-8
257,771
0-83
531
Woodford River, Counties of
Lei trim and Cavan, measured
during the very high floods
of Jan. 1851, at Ballyconnell
90,000
140-6
101,035
1-12
717
Yellow River, or upper portion
of the Woodford River, mea-
sured during the very high
floods of Jan. 1851, Co. Leitrim
5,000
7-8
52,125
10-43
6675-
ORIFICES, WEIRS, PIPES, AND RIVERS. 299
These results show how difficult it is to draw any
inference from discharge and area of catchment
alone, as the discharge, per minute per acre, must
vary with the contour and elevation of the district
in the same course ; and with the climate also, in
different countries. We have observed ourselves
the maximum discharges to vary up to 6 cubic feet
per minute per acre, the lesser maximums being due
to broad flat districts, and the greater maximums to
higher and steeper districts, near the sources. In
the Proceedings of the Institution of Civil Engi-
neers, Ireland, vol. iv., from which we have collected
and arranged some of the foregoing information, it
is stated, p. 96, that the ratio of the discharge to the
rain-fall, on a catchment on the Glyde, of 79,433
acres, for three months, ending March 13th, 1851,
was 1*49 to 1 up to January 13th ; 1*39 to 1 up to
February 13th ; and 3-86 to 1 up to March 13th,
making a general average of 1*59 to 1 ; the whole
rain-fall for the three months being only 5 '89 inches,
while the discharge was 9-35 inches ! We fancy
there is a mistake here. The whole catchment of
the Glyde is 176,813 acres, and there is no data to
show the discharge previous to or after the rain-fall
from which to calculate the difference due to it per
se for the three months; nor is the place or method
of gauging stated. The supply from springs and
the actual discharge before and after rain-fall must
be correctly gauged before the proportion passing
into the main channel in a given time, can be pro-
perly estimated ; the results just stated clearly con-
tradict themselves. The following anomalous results
300
THE DISCHARGE OF WATER FROM
from p. 47 of the same work are also worthy of
note. In five different districts the discharge is
gauged, or estimated, greater than the fall, as shown
in the following table. It is not stated, however, if
District.
River.
Catchment
in Acres.
DECEMBER 1850.
JANUARY 1851.
Total fall
of Bain by
gauge in
inches.
Total
depth of
discharge
off catch-
ment in
inches.
Total fall
of Rain by
Castlebar
gauge in
inches.
Total
depth of
discharge
off catch-
ment in
inches.
Saleen . . .
Saleen
2,625
20,640
33,500
70,000
3,200
32,000
. 3-55 ,
J4-00
6-26
5-46
5-46
6527
5-705
. 6-33 .
9-20
$55
8-18
7-39
Ijannagh . . .
Castlebar . .
Manulla . . .
Robe
Balla
Mask and Robe .
DaUa . . . . I
Dalla
Owenmore .
I
the depths passed off, estimated over the catchments,
include the flow before the commencement of the
rain. If so the results are so far useless ; and if
they do not include it, there must be an error
somewhere. Indeed, in the Eobe we have evidence
that not more than 58 per cent, passed from the
catchment to the river, from Mr. Betagh's paper,
the results of which are arranged below. Also, in
July 1850, it is shown that in the Lannagh district
only -53 inch in depth passed off the catchment
from a fall of 1'83 inches, or about one-third of the
depth. The method of determining this was un-
objectionable. Where such discrepancies as above
exhibited exist, it is important that the method of
gauging, and the whole calculation, should be shown,
in order that other engineers should be able to judge
ORIFICES, WEIRS, PIPES, AND RIVERS. 301
of their accuracy ; otherwise the results should be
rejected, no matter under whose authority they may
be published.
The following information has been collected and
arranged by us from a paper by Mr. Betagh, in the
Proceedings of the Institution of Civil Engineers,
Ireland, vol. iv. In January 1851, 3*41 inches of
rain fell in seven days, producing the maximum dis-
charge of 85,836 cubic feet; while in December
1852, 3-17 inches, also falling in seven days, pro-
duced 115,656 for the maximum. At the beginning
of the first fall there was flowing 2 6, 640 feet, leaving
the effects of the seven days' rain 85,836 - 26,640
zz 5 9,1 96 cubic feet, while in the second year the
quantity flowing at first was 75,360 cubic feet, leaving
the effects of the seven days' rain-fall equal to
115,656 - 75,360 zz 40,296 cubic feet. The effect
of the previous state of the weather on the catch-
ment must always modify, to a considerable extent,
the discharge from a given rain-fall, and this has
more to do with the results than the effect of arterial
drainage itself, unless so far as one is a result of the
other. Taking the mean of 1851 and 1852, it ap-
pears that the evaporation in the Ballinrobe catch-
ment was to the rain-fall as 41 '6 to 98*7, or about 42
per cent. This is certainly, from the nature of the
catchment, less than the average through Ireland,
which cannot be less than 60 per cent. In high,
steep districts, fully three-fourths or 75 per cent, of
the rain-fall can be collected, and at times, when the
catchment is saturated, nearly the whole ; even in
some few limited cases, when springs or hidden
302
THE DISCHARGE OF WATER FROM
TABLES showing in detail, for the years 1851 and 1852, the Monthly Fall of Rain and
the corresponding Discharge of the River Robe, at Ballinrobe, County Mayo ; the
catchment basin being 70,000 acres, or 110 square miles ; the lower end 100 feet,
the upper end 336 feet ; and the average height of the surface about 180 feet above
the level of the sea. The average fall of the river, not including the rapids, is
from one to two feet per mile ; the catchment is about 20 miles long, about one-
tenth of the area bog or low marsh, and nine-tenths clayey and gravelly. The
river is about 33 miles long.
OBSERVATIONS IN 1851.
MONTHS.
Eain-fall each
Month
in inches.
Discharge each
Month
of rain-fall in
inches.
Discharge in cubic feet per mi-
nute, from a catchment of 70,000
acres, for each month.
Discharge in cubic feet per
minute, per acre, for each
month.
Maximum
minimum.
Average.
Maximum.
Minium m .
Average.
January
9.2
7.4
85,836
20,133
43,373
1-158
287
620
February .
6-8
4.7
72,448
18,420
30,410
1-034
263
434
March . .
4.4
3-6
49,137
10,860
20,945
702
155
300
April . .
3-4
2-5
24,200
5,760
14,355
345
082
205
May . . .
1-0
8
5,820
4,125
5,001
083
059
071
June . . .
3-8
8
7,040
1,114
4,230
100
016
060
July. . .
3-8
5
4,920
1,500
2,558
070
021
036
August . .
2-4
0-9
17,055
1,240
4,866
243
017
069
September .
1-9
0-5
4,746
1,200
2,854
067
017
040
October . .
5-0
1-6
23,980
6,940
12,588
342
099
179
November .
1-3
1-2
12,852
6,000
7,827
183
085
111
December .
Total
2-6
2-5
44,715
6,210
14,373
638
088
205
45-6
27-
352,749
83,502
163,380
4-965
1-189
2-33
supplies are re-tapped, a larger discharge may take
place than that due to the catchment and rain-fall ;
but these do not affect the general question.
" The future population of the suburbs of London
is calculated at 30,000 inhabitants per square mile.
According to the following data, some of the densest
portions of our large towns have a population of
220 persons to an acre. The population on the
ORIFICES, WEIRS, PIPES, AND RIVERS.
303
RIVER ROBE OBSERVATIONS IN 1852. Continued from last page.
MONTHS.
Eain-f all each
Month
in inches.
Discharge each
Month
of rain-fall in
inches.
Discharge in cubic feet per mi-
nute, from a catchment of 70,000
acres, for each month.
Discharge in cubic feet per
minute, per acre, for each
month.
Maximum.
Minimum .
Average.
Maximum.
Minimum.
Average.
January. .
7-5
54
41,600
12,852
28,730
594
183
410
February .
4-8
4-3
56,400
8,190
25,296
805
117
361
March . .
1-0
0-7
9,600
2,737
6,702
137
039
095
April . ' *.
1-1
0-5
3,931
1,468
2,477
056
020
035
May. .' ; .
1-9
0-4
3,931
1,050
1,861
056
015
026
June . . .
6-6
1-2
22,764
1,400
6,547
325
020
093
July. . ..
2-5
1-0
15,439
3,172
6,057
220
045
087
August . .
4-5
0-6
3,856
2,236
3,070
055
032
043
September .
1-8
0-5
3,427
2,642
2,874
048
037
041
October . .
3-9
1-0
32,040
1,114
5,932
457
016
084
November .
5-5
5-2
45,360
17,000
30,742
648
242
439
December .
Total
12-0
9-5
115,656
23,232
54,846
1-657
331
783
53-1
30-1
354,004
77,093
175,134
5-058
1-097
2-497
north side of the Thames is about 75 persons per
acre, and on the south side 28 persons per acre.
Taking the average density of population in our
twenty-one principal towns, there appear to be 5045
inhabitants to the square mile ; but, from the fol-
lowing table, extracted from Dr. Duncan's report
on Liverpool, it will be seen that if we select five
of our most populous cities, the average in these is
much greater, while in others, it is equally certain
that the crowding is far less than the general stand-
ard to which we have referred :
304 THE DISCHARGE OF WATER FROM
Inhabitants to a Square Mile.
Towns. Total Area. Builded Area.
Leeds 20,892 . . 87,256
London .... 27,423 . . 50,000
Birmingham . . 33,669 . . 40,000
Manchester . . . 83,224 . . 100,000
Liverpool . . . 100,899 . . 138,224
Dr. Duncan, however, states that there is a district
in Liverpool containing 12,000 inhabitants crowded
together on a surface of only 105,000 square yards,
which gives a ratio of 460,000 inhabitants to the
geographical square mile. In the East and West
London Unions, Mr. Farr has estimated that there
are nearly 243,000 inhabitants to a geographical
square mile ; but, great as this overcrowding is, the
maximum density of Liverpool exceeds that of the
metropolis by nearly double."*
The amount of sewage from each person is cal-
culated about FIVE CUBIC FEET PER PERSON, including
the supply from manufactories, breweries, distilleries,
&c. SEVEN FEET PER HEAD has been recommended
as data to calculate from by Captain Galton, Messrs.
Simpson and Blackwell, in their Eeport on the
Main Drainage, and it has been found that about
half of the estimated quantity of sewage would pass
off in six or eight hours.
In calculating the size of sewers, however, the
rain-fall must be provided for, in addition to the
sewage matter from houses and public establish-
ments. Mr. Bazalgette calculated this for the Lon-
don sewerage at ith of an inch fall in 24 hours in
the urban districts, and ith of an inch for the
suburban districts. Captain Galton and the Messrs.
* Illustrated News, September 8th, 1855.
ORIFICES, WEIRS, PIPES, AND RIVERS. 305
Simpson and Blackwell assumed fths of an inch fall
during eight hours' maximum flow. This would be
]452 feet per acre. Assuming the highest data, we
shall have to provide sewers to discharge in eight
hours
1,452 cubic feet of rain water per acre,
3J cubic feet of sewage nearly per person.
Assuming a population of 80 persons per acre, then
these figures would become
in eight hours, or
1,452 cubic feet for ram, '
OOA i,- f 4. f \ at)OUt 3 i cubic f eet
280 cubic feet for sewage.
per minute, per acre,
which shows that the sewage is not more than -Hh
of the rain water ; and that, in calculations for the
size of sewers, the surface water is the most im-
portant element to be considered. If we had as-
sumed a larger fall of rain, the difference between
sewage and rain would be greater. On the 20th
June, 1857, the day after heavy rain, the referees on
the Metropolitan Drainage question found the Nor-
folk-street sewer to discharge 3 feet ; the Essex-
street sewer 5| feet; the Northumberland-street
sewer 3J feet ; and the Savoy-street sewer 20J feet
per minute per acre ; but the last result has been
controverted.
It appears that the daily amount of sewage
varies from 4*8 cubic feet per head in the more
thickly inhabited portions of London, occupied by
a larger portion of the poorer classes, to 8 cubic feet
per head in the western districts, where the value of
water is more appreciated, and the cost less a matter
of consideration; and the average of the whole
306 THE DISCHARGE OF WATER FROM
metropolitan districts appears to be 5-8 cubic feet per
head per diem. If the day be divided into three periods
of eight hours each, the amount of the maximum flow
is between nine A.M. and five P.M. and 49 per cent.
of the whole, whilst only 18 per cent, flows during
the eight hours of minimum flow, which occur be-
tween eleven P.M. and seven A.M.* The advantage
of storm flows in flushing is shown by the heavy
rain which occurred on the 20th of June, causing a
flow in the Savoy-street sewer .which was equivalent
to 20 times the ordinary flow at the time. This was
six times the maximum flow, and although the sewer
had been scoured, to a considerable extent, by a heavy
fall of rain on the previous night, the sample con-
tained more than double the amount of total impurity
contained in specimens of ordinary sewage.
In a town district, such as that drained by the
Savoy and Northumberland-street sewers, the quan-
tity running off into sewers, within six hours after
the fall, varies from 10 to 60 per cent, of the quan-
tity fallen. Of the rain during the storm of the
20th June, 1857, nearly one inch-and-a-quarter in an
hour, 65 per cent., ran off within 15 hours of the
fall, viz. :
46 per cent, in 45 minutes after the rain ceased,
14 in the next 6| hours,
5 in the next 7i hours.
In a suburban locality, such as the Counter Creek
sewer drain, the quantity reaching the sewers would
* Metropolitan Main Drainage Keport, pp. 15, 17.
OEIFICES, WEIKS, PIPES, AND KIVEES. 307
vary from to 30 or 40 per cent, in 24 hours after
the rain.*
In the Holborn and Finsbury divisions Mr. Roe
calculated that an 18-inch cylindrical pipe, laid at
an inclination of 1 in 80, is sufficient for 20 acres of
house-sewage, while a 5-inch pipe, laid at an inclina-
tion of 1 in 20, is necessary for 1 acre, and a 3-inch
pipe, laid also at I in 20, for j acre. A pipe 30"
in diameter, laid with an inclination of 1 in 200,
would discharge 1700 cubic feet per minute, and per-
fectly drain 200 acres of urban land covered with
houses to the extent of 4000 or upwards, and each
house having a water supply of 150 gallons per
diem. In each of these cases, however, the dis-
charge must depend on the head and length of the
pipe as well as the inclination at which it is laid.
Assuming the inclination of those pipes to corre-
spond with the hydraulic inclination, we have calcu-
lated their discharging powers with water to be
respectively 807, 72, 20, and 1700 cubic feet per
minute, the areas to be drained being 20, 1, J, and
200 acres. In all calculations of this kind it is
necessary , for accuracy, to ascertain not only the max-
imum rain-fall per hour, but also the proportions dis-
charged per hour, according to the season and district,
into the main channel, as well as the junctions or
places of arrival. In urban districts, 1500, 2100,
and sometimes 3600 cubic feet per hour per acre,
have to be discharged after extraordinary rain-falls.
These may be taken as maximum results. The
* Metropolitan Main Drainage Keport, pp. 75, 76.
x3
308 THE DISCHARGE OF WATER FROM
gaugings of the Westminster sewers in summer
give 53 cubic feet per hour for the urban, and 17
feet for the suburban, according to Mr. Hawkins.
In urban districts, however, a much larger quantity
of water is conveyed more rapidly, cceteris paribus,
to the mains, than in suburban districts and catch-
ment basins generally, in which the maximum dis-
charge per acre per hour, even in the steeper and higher
districts, seldom exceeds 700 cubic feet, and varies
from about 20 cubic feet for the larger and flatter
districts upwards. This arises from the impervious
nature of the surfaces it falls upon in towns, and the
lesser waste in passing to the drains, as well as a
large portion of the supply being often artificial.
From 70 to 90 cubic feet* per acre per hour, is
generally taken for the maximum discharge from the
average number of catchment basins ; this is nearly
equal to a supply of one-fiftieth part of an inch in
depth from the whole area. Thorough-drainage in-
creases the supply and discharge. Every catchment
basin has, however, its own peculiar data, and a
knowledge of these is necessary before we can draw
any correct conclusions for new waterworks in connec-
tion with it. It may be remarked, however, that any
conclusions drawn from experiments on the supply
of tributaries, particularly in high districts, are
wholly inapplicable to the main channel into which
they flow. The flow into tributaries and mountain
streams, or rivers, is always more rapid than into
* Some interesting observations on rain-fall and flood dis-
charges are given in the Transactions of the Institution of Civil
Engineers, Ireland, for 1851, pp. 19-33, and pp. 44-52.
OBIFICES, WEIES, PIPES, AND RIVERS.
309
main channels and rivers in flat districts, and the
supply from springs often forms a large portion of
the water flowing in them.
TABLE showing Summer Discharges of some English Rivers, as collected from
various authorities, re-arranged, showing to some extent the effect of
Springs in supplying Channels in different places.
NAMES OF RIVEKS.
1
Valley. Hill.
Catchment in 1
square miles.
1
B
11
Discharge per 1
square mile in I
cubic feet per 1
minute.
Representing
inches of rain-fall I
per annum.
Total average 1
rain-fall in inches 1
per annum.
Gade, at Hunton
Bridge, chalk . .
150 to 500
69-5
2,500
36-2
8-19
Lea, at Lea Bridge,
chalk. (Rennie,
April 1796) . . .
30 to 600
570-0
8,880
15-58
3-53
-
Loddon, (Feb. 1850,)
green sand . . .
110 to 700
221-8
3,000
13-53
3-01
25-4
Medway, driest sea-
sons,(Rennie!787,)
clay
481-5
2,209
4-59
1-04
Mimram, atPanshan-
ger, chalk . . .
200 to 500
29-2
1,500
51-4
11-58
26-6
Medway, ordinary
summer run, (Ren-
nie, 1787,) clay . .
481-5
2,520
5-23
2-19
Nene, at Peterbo-
rough, oolites, Ox-
ford clay, and lias .
10 to 600
620-0
5,000
8-45
1-88
23-1
Plym, at Sheepstor,
granite ....
800 to 1,500
7-6
500
71-4
15-10
45-0
Severn, at Stone-
bench, silurian . .
400 to 2,600
3,900
33,111
8-49
1-98
Thames, at Staines,
chalk, green sand,
Oxford clay, oolites,
&c
40 to 700
3,086
40,000
12-98
2-93
24-5
Verulam, at Bushey
Hall, chalk . . .
150 to 500
120-8
1,800
14-9
3-37
Wandle, below Car-
shalton, chalk .;-..
70 to 350
41-0
1,800
43-9
9-93
24-0
Trent, at its mouth,
oolites and Oxford
100 to 600
3,921
310 THE DISCHARGE OF WATER FROM
The above information has- been obtained from
Mr. Beardmore and Mr. Hughes' books, and from
Rennie's reports. The effect of the geology and
fissures in the chalk and mountain limestone forma-
tions on the springs of a catchment basin, and on
maintaining the summer discharge, should be carefully
noted as one of the elements entering into catchment
basin statistics. Indeed, the maximum and minimum
discharges from catchments are of as much impor-
tance to the engineer as the averages, and, for many
purposes, more important. There were abundant
opportunities of acquiring this information for all
our Irish rivers, but we are not aware if these were
turned to account.
The effects of evaporation are very variable ; some-
times 58 or 60 per cent, of the annual fall is carried
off in this way from ordinary flat tillage soils, and
other estimates are much higher ; much, however,
depends on the soil, subsoil, inclination, stratification,
and season. The evaporation from water surfaces
exceeds the annual fall in these countries by
about one-third ; and that from flat, marsh, and
callow lands exceeds the evaporation from ordinary
tillage, porous, and high lands. When the flat lands
along the banks of rivers extend considerably on
both sides, an extra fall is necessary into the main
channel, along the normal drains, otherwise such
lands must suffer from excessive evaporation as well
as floods. Evaporation also varies with the climate,
and in this country we may assume that one-third of
the whole rain-fall passes on to the sea.
In a paper in the Journal of the Royal Agricultural
OKIF1CES, WEIRS, PIPES, AND RIVERS.
311
Society of England, vol. v, part 1, 1844, Mr. Josiah
Parkes shows, that 42 1 per cent, of the whole annual
rain of England filters through the soil, and 57| per
cent, evaporated, being the mean results of eight
years' observations, from 1836 to 1843, both included.
The mean evaporation and filtration for each month
during this period is shown and arranged by us in
the following table :
MONTHS.
Total
falling.
Evaporated.
Remaining.
Deposited in Tons
and Cubic feet per
acre.
Inches.
Inches.
per cent.
Inches.
per cent.
Cubic feet
Tons.
January ....
1-847
540
29-3
1-307
70-7
4,744
132
February . . .
1-971
424
21-6
1-547
78-4
5,616
156
March ....
April
1-617
1-456
1-856
2-213
540
1-150
1-748
2-174
33-4
79-0
94-2
98-3
1-077
0-306
0-108
0-039
66-6
21-0
5-8
1-7
3,910
1,111
392
142
109
39
11
4
]\lav .
June
July
2-287
2-245
982
0-024
1-8
87
2-4
August ....
2-427
2-391
98-6
0-036
1-4
131
3-6
September . . .
2-639
2-270
80-1
0-369
13-9
1,339
37
October ....
2-823
1-423
50-5
1-400
49-5
5,082
141
November . . .
3-837
0-579
15-1
3-258
84-9
11,826
328
December . . .
1-641
0-164
00-0
1-805
ico-o
6,552
182
Yearly averages .
26-614
15-320
57-6
11-294
42-4
40,932
1145
The maximum quantity, 32-10 inches, fell in 184.1,
and the minimum in 1837, 21-10 inches. The maxi-
mum quantity which fell in January was 3-95 inches,
and the minimum -31 inch; in February 2*85 and
1-02 inches; in March 3-65 and 0*34 inches; in
April, 2-57 and -34 inches; in May 5-00 and -70
312
THE DISCHARGE OF WATER FROM
inches ; in June 3-31 and 1*33 inches ; in July 4-36
and 1-30 inches; in August 3-65 and 0-95 inches; in
September 4-50 and 0*63 inches ; in October 4-82
and 1*41 inches ; in November 5*77 and 2*05 inches;
and in December 3-02 and *40 inches. The greatest
quantities fall in September, October, and November;
and the least in February, March, and April. The
general mean fall for England is said to be 3 H inches,
and near London 25 inches.
The amount of rain varies, not only at different
places and different elevations, but also at different
elevations in the same place. The following table
shows the amount of rain collected in each month in
1855 at Greenwich observatory, at different elevations:
MONTH IN 1855.
Osier's anemo-
meter gauge,
inches.
On the roof
of the
library.
Cylinder
partly sunk in
the ground.
January
0-2
1*0
1 5
February
0'2
1-4
1-0
0*5
1'3
2-0
AprU...
0-1
0-1
0-1
May .,
0-5
1-5
1-8
0'5
0*7
0'9
July
3*1
4'8
5.3
August
0-6
0*8
1-4
September
0'8
1-1
2-0
2'6
4*5
6'2
0'5
ri
1*5
0*4
0-9
1-1
Totals
10-0
19-2
23-8
The cylinder gauge was placed 155 feet above the
OKIFICES, WEIES, PIPES, AND EIVERS. 313
level of the sea; the gauge on the roof of the
library 22 feet over the cylinder gauge, and Osier's
anemometer gauge 28 feet higher than the gauge on
the roof of the library. In the valleys in the lake
districts, Westmoreland and Cumberland, the annual
fall varies occasionally from 50 to 100 inches, and
the maximum fall is said to obtain at about 2000 feet
above the level of the sea on high catchments.
At Ballinrobe, a gauge placed on the church
tower, 60 feet above the ground, indicated 42 per cent,
less rain than one on the ground ; and another experi-
ment with a change of gauges, gave 68 per cent, less
at the greater elevation.
At Kinfauns Castle, Scotland, a gauge 600 feet
high on a hill, gave 41 J inches, while one at the base,
580 feet lower, gave only 25 \ inches. In Keswick,
the fall is 65! inches, and in Carlisle only 30 inches.
At Kendall the fall is 60 inches ; at Manchester 33
inches ; at Lancaster 45 inches ; at Liverpool 34 inches.
From the 23rd of February to the 6th of June, 1 860,
the rain at Dublin was 8 inches. At the Leefin Moun-
tain, which is 2000 feet high, the rain was 13-1 inches.
From the 23rd of February to the 9th of July, the
rain at Dublin was 10-674 inches ; and at the same
time, on the Leefin Mountains (over Ballysmutten),
181 inches ; that is, an increase of nearly 80 per cent,
in that time. From the 23rd February to the 21st
August, inclusive, the rain-fall at Dublin was 17 inches ;
at Blessington 21 inches ; at Ballysmutten, on the site
of a proposed reservoir, 27 inches. This showed an
increase over Dublin of 10 inches. It would appear
that from 50 to nearly 80 per cent, more rain fell at
Ballysmutten than at Dublin.
314 THE DISCHARGE OF WATER FROM
Experiments were made at York in 1832, 1833, and
1834, for the British Association, with three gauges,
the first placed on a large grass plot in the grounds
of the Yorkshire Museum ; the second at a higher
elevation, 43 feet 8 inches, on the roof of the Mu-
seum; and the third on a pole 9 feet above the
battlements of the great tower of the Minster, at an
elevation over the gauge on the ground of 212 feet
10i inches. The quantities received were as follows :
Average depth for
Depth for three years. one year.
First gauge . . 64-430 inches . . 91-477 inches
Second gauge. . 52-169 \.-.. 17-389
Third gauge . . 38-972 . . 12-991
Professor Phillips gives the following formula for
calculating the difference between the ratios of rain
falling on the ground and at any height h in the same
place 1 the temperature of the season, and c a co-
efficient dependent upon it ; then the difference d is
> - ' d==ch Tw'
The mean height at which rain begins to be formed
by this formula is 1,747 feet over the ground; and at
356 feet high, the depth which falls is one-half of
what falls on the ground.*
A discussion of the mean temperature in connexion
with the fall of rain, has been made at Greenwich for
the years 1852, 1853, and 1854 ; and at Oxford for
the years 1855, 1856, and 1857. The result shows an
average of 160-3 rainy days at Greenwich for each
year, and 146-6 at Oxford. The difference of the mean
temperatures of the day of rain and the day before
is less than that of the day of rain and the day after.
* Vide Civil Engineer and Architect's Journal for 1860, p. 167.
ORIFICES, WEIRS, PIPES, AND RIVERS. 315
Mean tempera- Mean tempera- Mean tempera-
ture, day ture, day of ture, day
before rain. rain. after rain.
Greenwich observations 49-25 . 49-27 . 48-98
Oxford do. 49-50 . 49-63 -V 49-44
Dividing the winds into two groups, northerly
and southerly, the Oxford observations give the di-
rection for 218'5 days' fair weather. The wind was
northerly for 131-5 days, and southerly for 87 days.
For the remaining 146-5 rainy days, the wind was
northerly for 64-5 days, and southerly for 82 days.
SECTION XIII.
WATER SUPPLY FOR TOWNS. STRENGTH OF PIPES. SEWER-
AGE ESTIMATES AND COST. THOROUGH-DRAINAGE.
ARTERIAL DRAINAGE.
SUPPLY. QUALITY.
The supply of water to towns has become latterly
a subject of considerable importance. Three points
have to be considered, first, a sufficient supply
at high pressure, when it can be obtained within a
reasonable expenditure ; secondly, the quality ; and,
thirdly, the cost. The advantages in towns of high
pressure are now apparent to all in overcoming fire ;
fronts of houses and pavements may also be cleaned,
and streets watered if the supply be abundant. The
highest apartments ean be supplied, and even mecha-
nical power can be obtained for many purposes, as
grinding coffee, at a reasonable cost. Mr. Glynn
says,* " In many parts of London water is supplied
at 4< for 1000 gallons, at a pressure of 150 feet :
a gallon of water weighs 10 Ibs., so that 1000
* Power of Water. WEALE.
316 THE DISCHARGE OF WATER FROM
gallons of water falling 150 feet, are equal to
1,500,000 Ibs. falling one foot ; and if 1500 gal-
lons of water be used in one hour, they are equal
to 37,500 Ibs. falling one foot in one minute, or
somewhat more than a horse's power, which is 33,000;
therefore, it may be assumed, that the cost of a horse's
power for an hour in such cases, is only 6<"
The number of gallons of water required for the
supply of each person, including all collateral uses,
has been differently estimated, and varies in almost
every town, and even in the same city London, for
instance, when supplied by different companies and
under different systems. 44 gallons per head, per
diem, were supplied by the several companies of
London in 1853, while evidence has been given to
show that the actual average consumption for all
purposes did not exceed 10 gallons per head, per
diem ; the remainder having been wasted under an
imperfect system of distribution. It is asserted that
when the supply is 25 gallons per head, per diem,
that 5 gallons of it are used for purposes requiring
nitration, 10 gallons for purposes not requiring fil-
tration, and 10 gallons wasted, or two-fifths of the
supply. As there must be a considerable, loss under
even the best system of supply, we may assume,
with the Board of Health, that a minimum supply of
75 gallons per house, per diem, or 15 gallons per
person, per diem, is necessary.
The following is an abstract of the average num-
ber of gallons of water furnished per diem, by
different water companies in London, during the
year 1853, to each house, including manufactories
and public establishments as houses :
ORIFICES, WEIRS, PIPES, AND RIVERS.
317
New River Company
Gallons.
Per House.
Per Person.
193
38-3-5
East London Water Works . . , .
187
37-4-5
West Middlesex Water Works
204
40-5-5
Grand Junction Water Works
f319
(336
f 63-4-5
1 67-1-5
Southwark and Vauxhall Companies' Houses
175
35
Ditto average houses, manufactories, public
209
227
41-4-5
45-2-5
Chelsea Water Works . . . . ..-
Hampstead Water Works ....
111
22-1-5
Kent Water Works
Mean Values .
270
55
2233
446-3-5
223-3-10
44-3-5
These quantities have been calculated from the par-
liamentary returns made in 1854; and if there be
any truth in the calculations and returns of the
quantities actually consumed per person said to be
10 gallons we get the proportion, as 10 is to 34 so is
the quantity consumed to the quantity wasted. But,
even assuming the quantity consumed to be 20 gal-
lons per head, what an immense loss is here exhibited
from want of a suitable system of distribution.
For large towns it is safe to provide for many
purposes, besides mere personal or house wants ; and
it is safer, where it can be done without much cost,
to provide for a supply of 40 gallons to each inha-
bitant, even if this quantity shall not be used or
318 THE DISCHARGE OF WATER FROM
raised. For high pressure, the supply required will
generally vary from 15 to 42 gallons, or from 3 to
7 cubic feet to each inhabitant, or an average of
about 30 gallons, including the supply to stables,
offices, manufactories, and breweries.
The quality of water for drinking, washing, or
cooking, is also an important element in selecting
a source of supply. Hardness is measured by the
number of grains of chalk or carbonate of lime to
a gallon of water, each called a degree. The average
hardness of spring water is about 26, that is, 26
grains of carbonate of lime to one gallon of water.
Eivers and brooks have an average hardness of 13,
and water derived from surface drainage 5 ; hence
the great advantage of the latter kinds of water in
washing. The average hardness of the London pipe
waters is from 10 to 16. The following report
and analyses furnished to me, in 1855, by Professor
Sullivan, of the Museum of Irish Industry, Dublin,
will show what is generally required on this head :
" On the annexed page you will find the numerical
results of my analyses of the four samples of water
which you left with me for examination. From the
table you will perceive that the water of the Mattock
Eiver appears to be the purest, so far as the nature
and the amount of the foreign substances held dis-
solved in it is concerned. The water of the Boyne
comes next in quality to that of the Mattock River,
the pump water being in every sense the worst, so far as
amount of ingredients can be taken as a test of the
quality of a water ; in this respect, indeed, it resembles
the water of the deep wells of London and elsewhere.
ORIFICES, WEIRS, PIPES, AND RIVERS.
319
" As the ordinary mode in which the quality of
a water, for drinking and for culinary and like pur-
poses, is judged of is, by the comparative amount of
organic matter, the total amount of dissolved matter,
and its hardness, according to the 'soap test,' I
shall give in the following table the numbers repre-
senting each of these qualities :
TABLE showing the number of grains of Organic Matter, and the number of
grains of Solid Matter, in an imperial gallon of
Water from
Number of
Grains of
Organic Matter,
per
Imperial Gal.
Number of
Grains of
Solid Matter,
per
Imperial Gal.
Degree of
Hardness
according to
the
Soap Test.
No. 1. Tullyescar . . .
8-975 grs.
31,175
15 8-10ths.
2. Eiver Mattock .
2- (about)
15,360
9 l-10th.
3. Kiver Boyne . .
3-250
22,700
14 9-10ths.
4. Burn's Pump . .
7-100
76,850
34 4-10ths.
" In order to render this table more instructive, it
may be well to subjoin a few of the results obtained
from the analyses of the waters of other localities.
TABLE showing the number of grains of Solid Matter contained in one gal-
lon of the following Water :
Thames, at Greenwich . *' i*-> . V*? 27-9 grains.
London .,.*,, . .. . : . .... , . 5 ., .28-0
Westminster ...._;"..;* ..= * f . 24-4
Twickenham . . . 22-4
Teddington .... 17-4
New River (London) . . . . 19-2
Lea ... ;'- 23-7
Trafalgar Square Fountain, Deep Well . 68-9
Well in St. Giles', Holborn . . . 105-0
Artesian Well at Grenelle (Paris) . T"; 9-86
" The following are some of the results obtained
from an examination of the waters in the neighbour-
320
THE DISCHARGE OF WATER FROM
hood of Dublin, or which have been proposed as a
source of supply :*
Locality from whence Water
was obtained.
Total Number
of Grains
per Imperial
Gallon.
Total Number
of Grains
of Organic
Matter.
Degree of Hard-
ness according
to the Soap
Test.
Royal Canal (12th Lock) . .
21-0
2-80
degs.
14-0
Grand Canal (7th Lock) . .
16-300
2-30
10 3-4ths.
River Liffey, at Kippure . .
3-525
1-90
2-10ths.
at Phoulaphouca
5-125
1-50
2-10ths.
Lough Dan, Co. Wicklow . .
2*800
1-225
8-10ths.
River Dodder, at City Weir .
Lough Owel ,
8-350
10-225
1-625
1-550
1 8-10ths.
6 7-10ths.
" The quality of a water for drinking purposes
depends in a great degree upon the condition in
which the organic matter is found, much more than
upon its quantity. This is, however, a question
outside of the domain of chemistry, and can only
be solved by the aid of the microscope. I may,
however, venture to remark that the organic matter
contained in the water of the Boyne and the Mat-
* While these pages were passing through the press, Dr. Apjohn
gave the following analyses :
* Total matter Organic
dissolved, matter. Hardness.
Grand Canal mean of seven analyses . 20-78 -95 15-9
Royal Canal mean of five analyses . 20-76 1-64 14-1
Liffey mean of eleven analyses . . 8*62 1-77 6*1
Analysis of the deposition on pipes from the Portobello basin :
Water . . . "..;.' ... ' . 2-20
Organic Matter '; . ' . ; . , . 9-71
Sand . . . . 10-20
Per Oxide of Iron and Alumina . 3-50
Carbonate of Lime . .... . 74-20
Carbonate of Magnesia . '..' . -19
100
ORIFICES, WEIRS, PIPES, AND RIVERS. 321
tock River is of vegetable origin, and would not,
so far as I believe, be injurious to health.
" As a general rule, I believe that the water of
clear flowing rivers, even though it may contain a
large amount of solid matter, and even of organic
matter, will be found wholesomer than well water,
especially in towns.
u For certain manufacturing purposes, and for
culinary purposes, too large an amount of lime is
injurious, but I believe that a certain quantity pre-
sent in water, is not only not injurious, but in my
opinion is of the greatest utility, and renders the
waters wholesome. I think the rage for extracting
pure water containing only one grain of solid matter
to the gallon, or thereabouts, for supplying towns, is
carried too far. Such water is, no doubt, the best
on a hill side ; but, I question whether it is equally
well adapted for resting in basins, tanks, pipes, &c.,
with that containing some lime. The Eiver Dodder
and Lough Owel waters appear to me the best
adapted for city and town supplies. The River
Mattock contains rather more than either, but it
is decidedly better than the water of either of the
canals from which our Dublin supply is drawn.
" Drogheda is rather badly situated for a supply
of very soft water, as almost the whole drainage
basin of the Boyne is either situated upon limestone,
or the feeders of that river rise through the cal-
careous drift gravel which covers so much of the
country. The water of the Boyne appears to be
an excellent water for most purposes, and perhaps
the difference between it and the Mattock River
322
THE DISCHARGE OF WATER FROM
Tabular Results of the Special Analyses of Four Samples of Water
from the neighbourhood of Drogheda.
Nature of dissolved matter.
No. 1.
Tullyescar.
No. 2.
Mattock
River.
No. 8.
Boyne
River.
No. 4.
Burns's
pump water.
Observations.
Carbonate of lime . . .
Carbonate of magnesia .
9-350
0-429
7-302
0-510
11-648
0-888
21-475^
0-585J
(^Inclusive of a very
small quantity of
-j phosphate of lime
and iron not sepa-
t^rated from the lime.
Sulphate of lime . . .
9-043
2-514
4-459
4-568
Chloride of magnesium .
0-743
1-258
1-685
8-445
Chloride of calcium . .
..
9-524
Chloride of soldium . .
..
0-991
..
Magnesia existing as cre-
nate, &c., in the water .
0*464
..
..
..
Lime do. do. . .
0-548
Silica do. do. . .
0-627
0-322
2-212
Potash and eoda existing
in water, as nitrates,
crenates, and other or-
ganic salts .....
1-644
2-785
0-448
22-393
Organic matter . . . .
Total number of grains
per Imperial gallon . .
8-975
8-250
7-100
31-175
15-360
22-700
76-850
may in part be accounted for by its being taken
near the banks, or more probably, perhaps, because
it was above and close to where some small stream
entered.
" The quantity of solid matter in it, however, was
not more than I would expect considering the nature
of the locality. I did not draw attention in my
Report to a point of some importance namely, the
proportion of lime and magnesia existing as car-
bonates, and as sulphates, and chlorides. The whole
ORIFICES, WEIRS, PIPES, AND RIVERS. 323
of the lime and magnesia existing as carbonates, and
as sulphates, and chlorides, is precipitated by boil-
ing, the water being thus proportionably rendered
less hard ; lime and magnesia existing as sulphates
or chlorides, on the other hand, are not precipitated.
This difference is of great consequence in culinary
operations, as where boiled water is used, the car-
bonates of lime and magnesia are not injurious, and
if no sulphates or chlorides be present, the water
may be soft after boiling. The same observation
applies to water applied to washing clothes when
boiled. And lastly, sulphate of lime forms one of
the worst elements of fur or deposits upon steam
boilers."
The saving in soap effected by a reduction of 10
degrees in hardness, is found to be over 50 per cent.
Some of the metropolitan waters analyzed by Dr.
Eobert Dundas Thomson, F.R.S., were found, in
May 1860, much more impure than others, the sam-
ples of which had been taken at the beginning of
the month, before the impurities conveyed by the
rains had contaminated them. The supply afforded
by large and small rivers, as in London, in this
table, contrasts most unfavourably with that afforded
by the drainage of mountain ridges, as at Glasgow
and Manchester. The specimens of water from the
two latter cities were taken by the instructions of
Mr. Bateman, F.R.S., the engineer, from the main
pipes during the month. It should be the object of
the London Companies to avoid pumping the water
in its most impure state, and to store it when in the
condition of the greatest purity.
Y3
324
THE DISCHARGE OP WATER FROM
Total
Impurity
per gallon.
Organic
Impurity
per gallon.
Grs., or .
o-o
Grs., or .
o-o
Loch Katrine water, new supply to Glasgow
3-16
0-96
Manchester water supply ....
4-32
0-64
THAMES COMPANIES : Chelsea .
17-84
17-08
1-48
1-64
Grand Junction
20-72
2-00
West Middlesex
20-08
20-80
2-08
2'40
OTHER COMPANIES": New Kiver
18-52
23-64
1-56
3-20
21-68
2-96
The table is read thus : Loch Katrine water con-
tains in the gallon 3-16 degrees or grains of foreign
matter in solution, of which *96 degrees or grains
are of vegetable or animal origin.
Professor Apjohn gives the following analyses of
waters furnished to the city of Dublin in 1860. It
shows how necessary it is to distinguish the time of
taking specimens for analysis, and the previous state
of the weather as affecting the foreign matters in the
water. The specimens were collected on the 5th and
19th of May, 1860. The quantity operated upon in
each instance was an imperial gallon, or 277-273
cubic inches :
ORIFICES, WEIRS, PIPES, AND RIVERS.
325
Carbonate of lime . . .
Carbonate of magnesia . . j
Sulphate of lime and chlo-
rides of sodium and mag-
CITY WATER COURSE, DODDER.
5th May.
1 4-056
19th May.
7-308^1
2-269
nesium
Silex 0-166
Organic matter 1*811
8-302
0-700
2-171
0-526
Specific gravity of
specimen (5th May)
1-00011.
Specific gravity of
specimen (19th
May) 1-00014.
11-086
PORTOBELLO BASIN.
7-687
Carbonate of lime
Carbonate of magnesia
Sulphate of lime and chlo- )
rides of sodium and mag- j- 4-058
nesium j
Silex 0-073
Organic matter 3-308
15-126
11-660-
0-764
3-751
0-194
2-289
18-658
Specific gravity of
specimen (5th May)
1-00023.
Specific gravity of
specimen (19th
May) 1-00031.
It will be observed that the quantities of saline
and other ingredients found in specimens of same
water collected at the two separate periods above
mentioned are materially different; those obtained
at the later date (May 19) containing the larger
portion of foreign matters. The extent of this
variation is very considerable, and it appears to Dr.
Apjohn to have been the consequence of a very con-
siderable fall of rain, which took place in the in-
terval between the periods at which the specimens
were taken up for analysis.
When the means of the preceding analyses are
taken, we obtain the following results :
City Water Course. Portobello Basin.
Mean amount of saline matter . 8-598 14-094
organic matter . 1-456 2*798
326 THE DISCHARGE OF WATER FROM
SOURCES AND GATHERING GROUNDS.
The sources from which a water supply for towns
may be derived are lakes, rivers, and streams,
springs, wells, and gathering grounds. Of the latter
it may be said that, however ably put forward under
the auspices of the Board of Health, it is far safer
to resort to good river waters than trust to what
has been termed, with some satirical truth, "new
fangled schemes of pot-piped gathering grounds."
Springs and wells afford, at best, but a partial sup-
ply unless for villages or manufactories; and we
must almost always trust to lakes, rivers, or streams,
with sometimes reservoirs, for stowage, for a suffi-
cient supply for large towns. The Grot on aqueduct,
conveying water with an average of three degrees of
hardness, to New York, is perhaps the noblest work
for water supply of modern times. The length of
the aqueduct is about 44 miles, with a channel in-
clination of about 15 inches per mile. The receiving
reservoir is about two miles higher up the channel
than the distributing reservoir, which latter is 115
feet over the level of the sea, and commands the
highest buildings of the city. In the driest weather
the supply is equal to 28,000,000 gallons.* The cost
of the work, including the purchase of land and
water rights, was 8,575,000 dollars, or 8 per lineal
foot nearly. The cost of distributing pipes was
1,800,000 dollars. Latterly we have had the Loch
Katrine and Glasgow aqueduct, also a noble work,
* Schrainke's Croton Aqueduct, New York.
ORIFICES, WEIRS, PIPES, AND RIVERS.
327
constructed after this model by Mr. Bateman, not-
withstanding the previous supply of that city, or a
portion of it, the G-orbals, from gathering grounds at
a high level. It is, however, sometimes necessary
to make use of such grounds, particularly when
TABLE showing the Quantities of Gathering Ground and Reservoir Room to
supply a given population with 15, 30, and 40 gallons of water per head
per diem. The reservoir room is calculated to hold 12 inches in depth
of rain-fall per mile as a guide for lesser depths. For 4 inches the
results are to be divided by 3; and for 6 inches by 2.
ttd
*4
SB
S.g-1
1
*|^* S
f*J
JH
si
it
*i
11 a
li
Ifftl
|P|
<B
If
ft
|4*
i
flijfj
flf|
II
It
il
||f|l
If
||^||
l||!-'
9
o m
-w S i
^"2
tc'3 it, *<
^*" <H
It
si
ti
ft
k!l
||
f|f||
||*t
2a
(SS
&$
SJflt
1"
llsls
iiii
2,500
1,250
937
4-179
0375
0789
2-196
5,000
2,500
1,875
8-358
075
1577
4-393
7,500
3,750
2,812
12-536
1125
2366
6-589
10.000
5,000
3,750
16-715
16
3154
8-786
12,500
6,250
4,687
20-894
1875
3942
10-982
15,000
7,500
5,625
25-072
225
4731
13-179
17,500
8,750
6,562
29-251
2625
5519
15-375
20,000
10,000
7,500
33-430
800
6308
17-572
25,000
12,500
9,375
41-788
375
7885
21-965
30,000
15,000
11,250
60-145
45
9462
26-358
35,000
17,500
13,125
58-5
525
1-1039
30-75
40,000
20,000
15,000
66-9
6
1-2616
35-144
45,000
22,500
16,875
75-217
675
1-4193
39-537
50,000
25,000
18,750
83-57
75
1-577
43-93
55,000
27,500
20,625
91-932
825
1-734
48-32
60,000
30,000
22,500
100-29
9
1-8924
52-716
65,000
32,500
24,375
108-65
975
2-0501
57-109
70,000
35,000
26,250
117-
1-05
2-2078
61-502
75,000
37,500
28,125
125-36
1-125
2-3655
65-895
80,000
40,000
30,000
133-72
1-2
2-5232
70-288
85,000
42,500
31,875
142-1
1-275
2-6809
74-681
90,000
45,000
33,750
150-435
1-36
2-8386
79-074
95,000
47,500
35,625
158-8
1-425
2-970
83-467
100,000
50,000
37,500
167-16
1-5
3-154
87-86
105,000
52,500
39,375
175-5
1-57
3-311
92-25
110,000
55,000
41,250
183-86
1-65
3-469
96-64
115,000
57,500
43,125
192-22
1-72
3-62
01-10
120,000
60,000
46,000
200-58
1-8
3-785
05-43
328 THE DISCHARGE OF WATER FROM
flanking or lying above glens where an embankment
may be easily thrown across, and the supply stored
for use, which would otherwise pass quickly off.
The table, page 327, gives the areas of reservoirs
and gathering grounds according to a collection of
one foot in depth from the catchment ; it can be
easily modified when the storeage or required supply
exceeds or falls short of this depth. One acre of
gathering ground with a collection of twelve inches of
rain-fall from it annually will give a daily supply of
Jive cubic feet per head to twenty-four inhabitants.
The next table will be of use in showing the
actual quantities which have been collected, or could
have been collected, for storeage. Hbmersham,
Hughes, and Beardmore's books have been consulted
in arranging it.
The various methods employed for purification may
be classed under three heads : mechanical, by filtering
or straining ; chemical, or antiseptic media, such as
peat and animal charcoal, and precipitation by the
use of lime water ; and the natural precipitation of
impurities when the water is at rest, as well as the
purification which takes place from oxidation and
neutralization on thorough exposure by the ozone of
the atmosphere. This latter plan has, however, been
tried, and signally failed. Filter beds may be con-
structed to have a surface area of one square yard
to every 1,000 gallons to be filtered in twenty-four
hours. For executed works the proportions vary
from 1 in 460 to 1 in 1140. The cost of filtering,
in capital, may be said to vary from 30 to 70 for
each million of gallons, or 30*. to 70s. annually.
ORIFICES, WEIRS, PIPES, AND RIVERS.
329
TABLE showing information with reference to size of Reservoirs, Catchment Areas,
(&c., collected and arranged from various authorities. The first, fifth, and sixth
columns contain information with reference to reservoirs and the collecting areas;
the second, third, and fourth, show for different districts the whole rain-fall, and
the portions or per centage flowing off and available.
Names of Drainage Areas
and names of Reservoirs.
a
'*
1
If
Rain-fall in
inches per
annum.
IP
*ss
M
rS g S
|||
P.Sto
!
933 .
tJD > CJ
1a!l
Si 1
Reservoir room
per square mile
in millions of cu-
bic feet of water.
Contents of
Reservoir in mil- 1
lions of cubic 1
feet.
Ashton
59
40-0
15-5
39
21-0
12
Albany Works, U.S
29
1-1
32
Ballinrobe, Ireland
11-0
49-3
28-5
58
Belmont (moorland, mean
of four years)
2-81
54-5
39-6
72
26-8
75
Bolton
80
25-6
20
Bute (low country)
45-4
23-9
53
Bateman's Evidence on the
drainage area of Long-
dendale :
First half of 1845, very dry
. .
21-2
13-5
64
..
..
Second half of 1845
38-6
27-25
71
First half of 1846
22-5
17-5
78
Oct., Nov., and Dec., 1846
10-2
8-67
85
Bann Reservoir (moorland
..
72-
48-0
66
Drainage areas on south
side of Longridge Fell,
near Preston, May 1852,
i ::
54-'
15-5
18-0
oo.ft
29
33
40
to April 1853
/
A/Si U
4bO
r
DilworthReservoir of Pres
ton Works, Lancashire
092
..
..
54-0
5
6-00
37-0
22-3
60
7-66
46
Grreenock
7-88
60-0
41-0
68
38*
300
Homersham's estimate of
24,000 cubic feet of Re-
servoir to each acre oi
drainage
1
15-36
15-36
Longdendale
23-8
12'3
292
Proposed Reservoir for
Wolverhampton Works
22-
..
7
16
Rivington Pike
16-25
55-5
24'25
44
29-6
481
Sheffield
1-42
36"5
52
Turton and Entwistle .
3-18
46-2
41-0
89
31-43
100
330
THE DISCHARGE OF WATER FROM
COST.
With reference to cost, the following tables, ar-
ranged by us from various sources, will afford in-
formation from works executed.
The actual cost of all works for house service
varies very much in different towns, and with the
quantities supplied, from a general average of Id.
per house per week, to 2d., and from an annual
rate of 9d. in the pound to Is. 6<, and higher.
The cost of raising and supplying 1000 gallons
from a height of 135 feet in Nottingham, is said to
be 3d., and the charge for house service to vary
from 5s. to 60s. annually. In Eugby, the average
cost per house is 19s. per year, 4&d. per week, or an
annual charge of 3s. 3d. per year, or ^d. per week
per head of the population, and for a bare supply
of 13 gallons. In Croydon, for a supply of only
14 gallons per head, the cost of works varied from
lid. to 2%d. per house per week. The parliamentary
returns, showing the number of houses supplied,
and cost of supply, by different water companies of
London, in 1834, give the following results :
COMPANIES.
Number
of
Houses.
Daily
average
Supply in
Gallons.
Height of
Supply
over
Thames.
Amount of
charge per
Company.
73,212
241
145
s. d.
166
Chelsea .
13891
168
135
1 13 3
West Middlesex
16000
185
155
2 16 10
Grand. Junction
11,140
350
152
286
46,421
120
107
129
South London
12,046
100
80
15
Lambeth
16,682
124
185
17
7,100
156
60
113
ORIFICES, WEIRS, PIPES, AND RIVERS.
331
Cost of house apparatus for private supply from
street mains, as averaged by the Board of Health,
for first-rate houses, is 3 13s. Id. ; second-rate
houses, 2 18s. 6d. ; third-rate, 2 3s. 3d. ; fourth-
rate and cottages, 17s. 5d. ; average cost for houses
and cottages, 2 8s. Id.
The actual cost of private works to take water
from mains for the supply of cottages is shown in
the following table :
Work
executed in
NAME OF PLACE.
Mean Expense
of Private
Works for
each Cottage.
Annual Value
of
each Cottage.
Jan. 1852
Mar.1852
1852
Aug.1852
Mean valu<
Rugby, mean of 6 Cottages
s. d.
1 12 11
200
1 18 li
2 11 10
s. d.
5 10
400
326
10
Barnard Castle 11
Tottenham .... 6 ....
3s for each Cottage
209
5 13 1J
The water rate charged by the Local Board at
Tottenham, is given as follows :
In the Special District Rate
Assessment.
Water Bate
per week.
Water Rate
per annum.
Above
And not
exceeding
On Premises
assessed.
s. d.
8. d.
10
8. d.
8. d.
026
10
15
n
039
15
20
H
050
n
20
25
n
063
tt
25
30
n
080
n
30
40
H
11
40
50
14
and 3s. for every additional rate of 10.
332
THE DISCHARGE OF WATER FROM
p ^
O h
H
* I
papuadxg ^
-idsouo pred
^ ^
rH
CO
O CO CO O O
t- i> CSJ O CO
O* rH CO O .-H
CO
0*
'^T
1 :
:
;
O O
O ' ' '.
i I t I
CO
saiqB, S
03
'
:
O O O O
t> O O O
i-H rH rH
CO
Oi
r^ CO O
CO
co CO O
^
898UOH
1
rH
CO
00*
ssBio qs
^
CO
00
rH
I 1
sasnoH
o
* :
0?
CO
o*
rH
o
rH
rH
O
o
rH
rH
89SUOH
i>4 ^^
s : S
o
rH
O
2
o?
rH
rH
CO
rH
i r
sasnoH
BSBIQ pug
^
<* *
"^ CO
00
CO
CM
o
o
rH
CO
O
CO
rH
rH
o*
rH
53%
o
:
0?
rH
CO
g
"*
O
: : : :
s
sraooa 91
Suture^uoo
898HOH
- : S
rH
;
/. O
ii : : : :
00
;
SUTUIB^UOO
B98UOH
^
oj 00
H? CO
O
CO
CO
!!:=::
^ <*
CO
sraoog OT
Smure^uoo
sWnoH
r>* O O
^00 CO
^ rH CO
o
o
T-H
9
8 ' I
CO
00
<M
sraoojj g
sasnoH
1 2 :
' '
i?
o
^g, o
o
o
o
rH
rH
Surarajuoo
898UOH
l>i O O
-^ c?
rH
^ rH rH
O
a
rH
CO O O
0? O
rH
rH 1-1 rH
rH
Sutura;uoo
B98UOH
r>4 O O
. o O
^ rH rH
O
;
O O O O
0* O
rH rH
rH TH rH
O
00
1-1
sraooa e
CO
rH
'
'
00
l-H t- O
co
Smuro^uoo
ns o
-s
CO
fc. CO ' '
rH
O
m
1
' j
VauxhaU
M
d
o ;
lal Charge
COMPAQ
New River . .
East London
Southwark &
1
1
ill J I
rl 6 M W
Average Annt
ON |
rH (M
CO
2
O CO t~ 00 OS
ORIFICES, WEIRS, PIPES, AND RIVERS.
333
PUBLIC WORKS OF WATER SUPPLY, PRESTON.
Yards. Cost of Pipes. s. d.
44 of 2-in. iron pipes, including valves,
fire-plugs, outlet-pipes, and all
appurtenances, at Is. 7^. ... 398
1,496 of 3-in. ditto, at 3s. 4<Z 249 6 8
321 of 4-in. ditto, at 4s. 9d 76 4 9
625 of 5-in. ditto, at 6s 187 10
30 of 9-in. ditto, at
14 5
2,516
530 16 1
Water Supply and its Cost for some Cities and Towns, from a Paper read to
the British Association at Leeds, in 1858, by Dr. Strang, of Glasgow.
Vide Builder, for 1858, p. 653.
TOWNS.
Population
within
bounds of
Supply.
Daily Supply.
Daily
Supply
'or each
inhabi-
tant.
Cost of
undertaking.
Daily
Supply
for every
1 ex-
pended.
Prospective
Supply daily
in addition.
London ....
Paris
2,667,917
1,100,000
Gallons.
81,025,842
26,350,000
Gallons.
30-3
24-
7,102,823
800,000
Gallons.
11-4
33-
Gallons.
20,000,000
Hamburgh .
160,000
5,000,000
31-25
170,000
29-50
New York . .
713,000
28,000,000
39-27
1,800,000
15-5
..
Manchester .
500,000
11,000,000
22-
1,300,000
8-5
14,000,000
Liverpool . .
Leeds
500,000
153,000
11,000,000
1,850,000
22-
12-
1,640,000
283,871
7-
7-
Edinburgh .
215,000
4,800,000
22-3
456,000
10-5
2,000,000
Aberdeen . .
65,000
1,200,000
18-4
50,000
24-
..
Dundee ....
96,000
1,750,000
18-2
139,000
12-5
..
Greenock . .
40,000
2,112,500
52-8
90,000
23-4
..
Paisley
48,000
1,021,452
21-
60,000
17-
Glasgow
420,000
16,710,000
39-8
651,199
26-
20,000,000
The cost of pumping varies with circumstances ;
we believe that pumping engines cannot be put down at
less than from 60 to 100 per horse power, dependent
334
THE DISCHARGE OF WATER FROM
on the size of the engine, although the Board of
Health adopted a standard of .50 per horse power.
For the town of Drogheda we estimated for two
engines at 75 per horse power. The following in-
formation has been furnished to us respecting the
cost of the Waterworks, Cork, by Sir John Benson
the engineer, who designed and carried out the works.
CORK WATER WORKS.
Steam engine 100-horse
power.
Iwo 50-horse power Tur-
bines.
Reservoirs
One of 3,500,000 gallons.
One of 563,000 gallons.
Cost per head.
Valuation standard per
pound on the valuation.
Yearly cost per five in-
habitants.
Water supplied.
Direct acting Cornish Engine with
three cylindrical flue boilers, includ-
ing engine and boiler house, setting
boilers, chimnies, &c., &c., per horse
power
Two Turbines completed with four 11
in. ram pumps on each, including
buildings, cisterns, sluices, gates,
screens, per horse power
One reservoir on a levelof 186 feet over
weir
One reservoir on a level of 360 feet over
weir
The inhabitants in 1851, 86,000. ,
The inhabitants in 1861, 100,000.
City valuation, 112,000 ,
Distribution per house of every five
persons
Quantity supplied, including manufac-
tories, to one person per day
s. d.
55
44
4,900
15 3
13
11 7
050
30 gallons.
The total estimated cost of engines, including
pumps, engine houses, wells, &c., for raising the
London sewage, is 70 per horse power, and the
annual cost 20 per horse power. *
When coals are 10*. per ton, the cost of an engine
* Main Drainage Eeport, 1857, p. 29.
ORIFICES, WEIRS, PIPES, AND RIVERS. 335
exceeding 100-horse power, single acting Cornish,
working night and day, will be 10 per horse power;
when coals are 15s. per ton, the cost would be 13
per horse power ; when coals are 20s. per ton, the
cost would be 16 per horse power; when coals are
25,9. per ton, the cost would be 19 per horse power.
These estimates have been given by Mr. Hughes, and
include every expense of coals, wages, oil, tallow,
materials for packing, cleaning, &c., but none for
interest of capital or depreciation of machinery.*
At Ely the cost of pumping is stated by a writer
in the Builder to be as follows :
To pump one million gallons 140 feet high, the old engine
consumes s. d.
Four tons of coal, at 16s. per ton 340
Oil, tallow, and packing 12
090
Total cost of pumping one million gallons 4 5
which gives Id. per 1,000 gallons pumped 140 feet
high (not a very high price).
The new engine requires
Five and a half tons of coal at 16s 480
Oil, tallow, and packing 1 10
Wages 120
Total cost of pumping one million gallons 140 feet high 700
which is 65 per cent, more money than the old
engine requires.
While another writer in the same periodical states,
that the cost of pumping 1,000,000 gallons with the
old engine was 4 18s. 8|^., and with the new
engine, 4 10s. Id. In the following table, arranged
from information in Mr. Hughes' book,f the estimated
cost of pumping engines for various works, English
and American, is given :
* Main Drainage Keport, 1857, p. 447. f WEALE, London.
336
THE DISCHARGE OF WATER FROM
ESTIMATED COST OF ENGINES FOR PUMPING, AND OTHER INFORMATION.
Description of Works, and Engineers' estimates,
with short description of engines.
Millions of Gal- 1
Ions daily.
Height to be
raised in feet. 1
"o <u -u
cc- ^
J3.a
11-
Estimated cost in
s.
Work to be done 1
in units of 33,000 Ibs.j
raised 1 ft. high perl
minute, or horse I
power.
Cost for each unit
of 33,000 Ibs. raised
1 foot high per mi-
nute, or horse
power, in a.
Daglish's estimate for Liverpool Works, exclusive
of buildings one engine and pumps
2
800
600
8470
126-26
67-1
Ditto Two engines and pumps
2
300
600
9850
126-26
78-0
Ditto For one engine and pump
1
75
75
3400
15-78
215-4
Homersham, for Watford supply
8
174
1392
85,357
292-92
120-7
Forester & Co.'s estimate for Liverpool engines,
one or two, exclusive of buildings
2
300
600
9700
126-26
76-8
Ditto ditto ditto
1
75
75
3200
15-78
202-8
Hurry and Bateman's estimate, Wolverhampton
Water Works
w
Hit
314\
264)
471)
1188 J
85,400
349-11
101-4
Harvey and West's for ditto, exclusive of buildings,
84 inch cylinder engine
11
306
459
11000
96-59
113-9
Ditto ditto 64 inch cylinder engine
Ditto ditto 58 inch cylinder engine
Messrs. Hawthorne for ditto, double power expan-
sive condensing beam engines, exclusive of
8
14
n
180
140
814
270
210
471
7000
6500
6055
56-82
44-19
99-11
123-2
147-1
61-1
Ditto
if
163
244
2520
51-45
48-9
Messrs. Hawthorne's high-pressure double acting
horizontal engines, to be raised in 6 hours
Ditto to be raised in 9 hours
It
22
22
132
108
1170
780
27-78
22-73
42-1
34-3
is
22
66
570
13-89
41-0
Mr. Hocking's estimate for Wolverhampton, single
14
500
750
20800
157-83
131-8
5J
500
750
13200
157-83
83-6
Ditto to be raised in 12 hours, by two 60 inch ,
two 45 inch, and one 36 inch cylinders
Ditto exclusive of buildings
14
Is
500
500
1500
1500
32,600
21800
315-65
315-65
103-3
69-0
Mr. Quick's estimate, for Grand Junction Works,
one 64, or two 45 inch cylinder, single acting
Cornish
5
46
230
7000
48-40
144-6
Ditto for Southwark and Vauxhall
8
40
320
7000
67-34
103-9
Ditto for West Middlesex
5
46
230
10000
48*40
206-6
Sandys, Vivian & Co., two engines for Liverpool,
2
300
600
8000
126-26
63-4
Ditto one engine, exclusive of buildings
Seaward and Capel's, for Wolverhampton, exclusive
1
11
75
566
75
849
1800
r 22,000^1
1 * <
15-78
178-65
114-1
f 123-2 \
West's estimate, 65 inch cylinder, exclusive of
Is
814
471
I 25,000 J
6000
99-11
1 139-9 j
60-5
Ditto 50 inch cylinder, including buildings . .
Ditto exclusive of buildings
U
1
163
163
244
244
7100
5000
51-45
51-45
138-0
97-2
Ditto raised in 6 hours, by a 35 inch cylinder
22
132
4600
27-78
165-5
Ditto exclusive of buildings
1A
22
132
8100
27-78
111-6
{i
566
849
38200
158-66
240-8
Ditto 65 inch cylinder engine, including
buildings (?)
814
471
8400
99-11
84-8
AMERICAN ESTIMATES AND WORKS.
Mr. McAlpine's, for Brooklyn, double acting, expan-
sive, high-pressure, condensing engine, 72 inch
cylinder engine, to be raised in 12 hours, ex-
5
190
1950
18,000
399-83
45-0
Ditto non-condensing 30 inch cylinder engine ,
raised in 24 hours, exclusive of buildings
Ditto in 12 hours, exclusive of buildings
Ditto exclusive of buildings
5
10
20
190
190
190
1035
8800
7600
8000
39,000
53,000
199-92
799-66
1599-23
15-0
48-7
83-1
Ditto ditto
30
190
11,400
85,000
2398-99
35-4
Ditto Albany Water Works, 58 inch condens-
ing beam engine, and one duplicate not con-
densing, working for day only
Ditto Chicago Works, as in last, including
buildings, 46 inch cylinder condensing engine
}*{
3
1561
238J
107
624"!
476J
642
13,320
11,258
231-48
135-10
67'5
83-3
3
116
696
11,642
146-46
79-5
OEIFICES, WEIKS, PIPES, AND EIVEES. 337
In EXAMPLE 28, pages 39 and 40, we have pointed
out the method of calculating the increase of horse
power required in raising water through pipes from
friction, and also the great increase of this extra
head if the velocity increases ; the increase being
nearly as the square of the velocity. In addition to -
this, an allowance of horse power must be made for
bends, curves, junctions, and other obstructions, for
the effects of which see SECTION XI. The more
slowly the water is pumped, the less will the loss be
from these causes through the same pipe. It is
therefore, so far, advisable to give as large a diameter
to the pipes supplying a reservoir from a pumping
engine as other aspects of the question, cost, and
engine power, will admit.
THICKNESS OF PIPES FOR WATER WORKS.
It is evident that the thickness of a pipe should be
at least sufficient to bear the pressure of the atmos-
phere, and therefore the whole pressure in a pipe is
best expressed by a determinate number of pressures,
each equal to that of a column of water 33 feet high.
If n be the number of such pressures, or the number
of units each equal 33 feet high, d the diameter of
the pipe in inches, and t the thickness, also in inches,
we shall have for
(A).
( 1. Iron pipes, plate t = -0009 nd + -13.
2. Iron pipes cast horizontally . t = -0024 nd + -33.
3. Iron pipes cast vertically . . t = 'OOlQnd+ -32.
4. Copper pipes, plate .... Z = '0015nd + -16.
5. Lead pipes * = -0024nd+ -19.
6. Zinc pipes t-='005lnd+ -16.
7. Artificial stone * = -0054 nd -f 1-60.
z
338 THE DISCHARGE OF WATER FROM
For cast-iron pipes the engineer of the Paris water
works, M. Dupuis, adopted in his practice a formula
which is equivalent to
(B.) * 0016rad + -32 + -013d
in the foregoing measures. This formula may also be
expressed as follows :
(0.) t = (-0016/1 + -013) d + -32.
If d be 12 inches, and n = 9, corresponding to a pres-
sure of 297 feet, we shall find from the last equation,
All pipes should however be proved with ten atmos-
pheres, or 330 feet, and in practically applying the
above formulae in equation (A), for finding the
thickness of pipes, the value of n should always have
10 added to it. Hence, applying formula (A), No. 3,
to our example, we get Jzz-OOlGx 19 x 12 + -32=:
6848 inch, which is the same practically as found
from equation (C).
SEWERAGE COST.
As for water works, the minimum rain-fall of a
district should be calculated upon, so the maximum
fall must be considered in sewerage and drainage.
We have already shown, page 305, that for a popu-
lation of 80 persons per statute acre, and a dis-
charge of two-fifths of an inch in eight hours,
sewers should be calculated to discharge about 31
cubic feet per minute, the rain supply being about
seven times the house supply, or sewage, including
house water supply. Instances are quoted in which
the discharge, after a heavy rain-fall, amounted to
ORIFICES, WEIRS, PIPES, AND RIVERS, 339
201 cubic feet per minute per acre, as in the Savoy-
street sewer, which of course was principally sur-
face water, as the sewage of 80 persons at 7 cubic
feet per person, one-half of which, if discharged in
80 x 7
eight hours, would only be g x o = ^ cubic feet
35
per hour, or TTQ = *59 feet nearly per minute, which
is only about the thirty-third part of 201 feet. In
other words, the storm waters were thirty-three times
the amount of house sewage. It would be waste to
provide drainage for so much surface water con-
sidered in itself, where it can be passed off from
the surface channels. But sewage is not water,
and it is essential, in the greater number of cases,
that sewers should be flushed occasionally. It is
absurd to calculate the size of sewers, as if the
sewage matter were thoroughly diluted or passed
off like water. In fact, the sewage in part lies at
the bottom of the sewer, or is deposited there in
nine cases out of ten, while the house supply of
water passes on and escapes over it, removing only
diluted and detached portions. It is, therefore, of
importance, where artificial flushing and cleansing
out are not provided, that storm waters should oc-
casionally pass through and flush a system of sewers,
particularly the main or arterial lines. An engineer
must be guided, in calculating the dimensions, &c.,
of main sewers, by the circumstances of each case.
The inclinations to be obtained, the form of the
bottom or invert, the rain-fall, the amount of sewage
which will not affect the size to any considerable
z 3
340 THE DISCHARGE OF WATER FROM
extent, the material and the cost consistent with
permanency.
The discharging power of a water channel is more
than doubled by increasing its dimensions by one-
third; and it is increased in the proportion of 5*7 to
1 by doubling the dimensions. By giving four times
the fall, the same channel will only double the
discharge. Now a pipe 2 feet in diameter with a
fall of 1 in 200, would discharge fully 1000 cubic
feet of water flowing full with a velocity of 5*4 feet
per second : at 3 j cubic feet per minute per acre, for
a population of 80 to the acre, the thoroughly diluted
sewage of 280 acres would be passed off by one such
pipe ; that is, the sewage from 20,400 persons, on
280 acres, and also two-fifths of an inch of rain
falling for eight hours, can be conveyed by a 2
feet pipe, with a fall of 1 in 200. But as this rain
supply is about seven times the house supply, passing
2^ feet off per person in eight hours, made up of
faeces and used-up water supply. It is apparent
that such a pipe would convey about eight times the
sewage alone of the district, if flowing as water ; but,
under any circumstances, would be abundantly large
for the duty, even when assuming the whole quantity
to pass in at the upper end. For a fall of 1 in 800,
two such pipes would be required, or one pipe 32
inches in diameter ; for a fall of 1 in 3200, four 2
feet pipes would be required, or one pipe 3 feet 6
inches.
' House drains should not be less than 6 inches in
diameter, and should have facilities for being cleaned,
either by using half-flange joints, or by having a
OKIFICES, WEIES, PIPES, AND KIVEKS.
341
moveable upper segment. The inclination for these
drains should be uniform, but the amount is not so
important as some appear to think, if proper provi-
sion be made for cleaning. Where flushing is used,
cast-iron pipes are the best, but they are also the
most expensive. House drains of brick with a v
tile bottom covered with flags or bricks are perhaps
the best, as the capacity can be considerably aug-
mented by adding to the height of the sides, and they
can be at all times easily opened and cleaned. If
inclinations from 1 in 50 to 1 in 20 can be had, so
mucn the better. The following items as to cost
have been selected from the Builder :
COST OF SEWERS, NEWPORT, MONMOUTHSHIRE.
Total
lengths.
Average
depths.
Sizes of sewers.
Thick-
nesses.
Cost
per foot
lineaL
1,322 ft.
15 ft. 6 in.
ft. in. ft. in.
4 6 by 3 6
9 in.
8. d.
11 8
2,217 ft.
13 ft. in.
4 6 by 3
9 in.
10 1
6,1 10 ft.
12 ft. in.
3 by 2 2
9 in.
7 7i
12,354 ft.
11 ft. 8 in.
3 by 2 2
6 in.
5 3|
1,953 ft.
9 ft. 3 in.
2 6 by 1 10
6 in.
4 7
9,663 ft.
10 ft. in.
2 6 by 1 10
4in.
3 8J
690 ft.
10 ft. 2 in.
2 3 by 1 9
4in.
3 6J
3,264 ft.
8 ft. 6 in.
1 2 diameter
4^ in.
2 4|
COST OF SEWERS AND PIPES IN PRESTON.
The following extract from the recently published
summary of public works executed during the year
ending April 30th, 1859, contains some useful infor-
mation :
342
THE DISCHARGE OF WATER FROM
Yards. s. d. 8. d.
60 of Brick Sewers, 2ft. 6in. dia-
meter, at 7s 21
538 3ft. by 2ft., at 17s. 6d 470 14 3
294 3ft. 6in. by 2ft. 4in., at 28s 412 12 11
372 3ft. 9in. by 2ft. 6in., at 28s.... 520 16
250 4ft. 3in. by 2ft. lOin., at 41s. U. . 521 16 8
56 4ft. 6in. by 3ft., at 75s. Id 211 13 4
66 4ft. 6in. diameter, at 40s. 9d 134 9 6
2,293 2 8
1,678
42 of Cast-iron Sewer, 2ft. diameter,
at 36s 75 12
22 of Earthenware Pipe Sewer, 6in.
diameter, at 4s 480
1,129 9in. diameter, at 7s. 5d ... 418 13 5
565 12in. diameter, at 8s. 9d 247 2 9
88 15in. diameter, at 11s. 3d 49 10
98 1 Sin. diameter, at 13s 63 14
145 21in. diameter, at 18s. Qd 134 2 6
917 10 8
2,047
Total, including superintendence, also man-boles,
street gullies, and all appurtenances 3,286 5 4
TABLE showing the prices of Tubular Drains as made by the Board of Health
in 1852, fifty per cent, being added for profit, &c. ; and the sale prices in
the market.
Diame-
ter in
inches.
Lengths.
For 1,000 feet
For 1,000 feet
For 1,000 feet
Red earthen-
ware pipes
made by the
Board.
s. d.
6 15
9 14
15 1 6
Red pipes
at
Sale prices.
S. d.
20 16 8
25
37 10
Stoneware
glazed at
Sale prices.
S. d.
25
29 3 4
50
Assumed gain.
On
red ware
pipes.
S. d.
14 1 8
15 6
22 8 6
Over glazed
stoneware
pipes.
s. d.
18 5
19 9 4
34 18 6
Did the Board of Health here add the cost of their own
establishment and staff to the cost of production ? The manufac-
turer must surely live.
ORIFICES, WEIRS, PIPES, AND RIVERS.
343
1
8i
R
H rt
73' OOOOOi-lrHOtM
!
,3 *00 >0 CO *<N<M
2||j
73' 00001>t>COCOCO
i
73* *O <M -* <M rH
co <N <N rH rH rH
"ft^
iW
73' OOCOOlOrHOO
rH
73 CO CO O <M rH
oi CO CO <M rH O
f
>
If
Si
I
H rt
78 rH
ST. THOMAS'S, EXETEE.
Pipes laid 8 to 7 feet deep.
73 CO CD CO 00 CO ~ O O 00
rH
<* *CO " O 00 10 CO * <M !N ,H rH
i 1 rH
Illl
73 OCOCO^COdd
oi lOdddfMdd
Ts'COCO CO CO O CO CO O O O
CO CO " d d <N rH ' rH rH rH rH
ol^l
-14
73 CSCO'OCOOCOrH
73' _ O OOooco ^OOOO
,5 O 00 CO CO (M -HrHOO
1 SOUTHAMPTON.
Pipes laid 9 feet deep.
1 s *
' Oi fOlO^ '
1 TOTTENHAM.
Pipes laid 5 to 13 feet :deep.
rH rH
co- CO J> Tfl CO O rH
nil
73' t- 00
^ co co co d
73 10 00 O O rH CO
drHrHrHOO*
iW
73* .0 co co *
oj CO CO d rH
73 <* IO O rH 00 OJ
=5 ' t~ IQ CO 0? rH rH
|3
.s.
j!h
73* O O O t-H t-
co ' CO CO d d
f
73
O {O
73 00 d rH t> IO
co O -^ d d <M
III!
73 O O 00 00 00
73* 05 CO CO CO CO
* *
--a?.
rH rH rH rH rH
il4
73* O <N <M CO rH
4 *U5 CON rH O
73 rH O 5 rH rH
tS CO d rH I 1 O
p.
1
g 3
s
H - 1
>eco co d J> -i os o
rfCO ' rH CO CO <*< CO CO
1
1
73 d rH rH CO O d 00 CO
aj rHO5tO^ COdd
rH
if
73'CO CO d ^ CO d
73 COOJCOCOO OO5OS
iW
73O O O CO d CO O
aSCO OOCOCOdrHO
rH
^ * "oOtU5COd rHOO
sadid |o ja^eureid
gddrHrHiH <N d d rH rH rH
344
THE DISCHARGE OF WATER FROM
ESTIMATE FOR SEWERS AT BRIGHTON.
DESCRIPTION OF SEWEBS.
Length in yards.
.Price
per
yard.
Amount.
Brick Sewers:
Diameter.
6ft
4,850
350
3,990
1,890
2,820
8,580
22,480
53,896
2,010
8.
3 10
2 10
2 8
2 2
1 16
18
8.
13 6
10
8.
7
3
20
4
3 10
8.
16,975
875
9,600
3,969
5,076
7,722
6,389 11
22,215
5,250
3,780
12,000
2,400
10,500
5,000
11,178 9
122,930
4ft. Gin
4ft. 6in. by 3ft
3ft. 9in. by 2ft. Gin
3ft. by 2ft
2ft. 3in. by 1ft. Gin. .
Total brick sewers .
Earthenware Pipe Sewers :
15 inches diameter .
9,466
44,430
750
1,260
12
Total earthenware pipe sewers .
Cast-iron Pipe Sewers :
1ft. Gin.
Total cast-iron pipe sewers ....
Total length of sewers
78,386
Number
600
600
3,000
Or 44 miles 956 yards.
Man-holes and ventilating shafts
Gullies
Outlet works, overflows, and ex-
tra work on steep gradients,
&c
Contingencies, in eluding repairs,
&c. of existing sewers, 10 per
cent
Total.
The following estimates have been made for lay-
ing pipes at Tottenham, not including their cost :
Depth
6 feet.
Depth
8 feet.
lid.
Depth
10 feet.
Diameter
of pipe
in inches.
6
12
The cost of laying alone at St. Thomas's, Exeter,
was
OEIFICES, WEIRS, PIPES, AND RIVERS. 345
6 inch pipes 5d. per foot lineal 3 to 4 feet deep.
9 5d. 3 to 4 feet deep.
12 Sd. 5 feet deep.
15 9d. 5 feet deep.
18 lid. 5 feet deep.
2d. per foot lineal for relaying pitching ; 4d. for macadamized
roads ; and Qd. for pavements.
The author has constructed a large quantity
of main sewers, from 18 inches to 2 feet and 2
feet 6 inches wide, and 4 feet 6 inches high; the
side walls built with rubble masonry, 9-inch seg-
ment inverts laid with 4i-inch courses in cement ;
the top sometimes flagged, when flags of sufficient
length could be procured, and sometimes arched
with rough rubble arches. The invert was laid
on, well bedded, well rammed, rubble to prevent
subsidence, and preserve the bottom inclination
uniform. The cost, at an average depth of about
9 feet, was 9s. per running foot, the side walls
being about 18 inches thick. Upright side walls,
where rubble is cheap, have many advantages
in giving a considerable increase of capacity for a
small outlay. The tenement and house drains were
of earthenware pipes. Cast-iron gully grates and
traps, weighing 3 cwt., cost 30s. each; the grate
fastened by a wrought-iron chain.
The following regulations have been laid down
for Cambridge and Carlisle :
STIPULATIONS FOR CAMBRIDGE DRAINAGE.
" Water from the rear of premises should not be conveyed to
the front under the hasement floor.
" Rain-water from the roofs should not be conveyed into the
basement, but conducted into the sewer by shallow drains.
346 THE DISCHARGE OF WATER FROM
" Cast-iron pipes may be used for basement drains in some
instances.
" The scullery sink should be kept as high as possible, and
approached by a step. A flap trap should be fixed between the
sink and sewer.
" There should be no water-closet on the basement floor ; if it
cannot be arranged elsewhere, the soil-pipe should have a flap
trap, or similar contrivance, to prevent the influx of sewage
water."
FOE CARLISLE DRAINAGE.
" STIPULATION 1. If water-closets are to be generally used,
the description of such to be sanctioned by the Board, the same
to be fixed to the satisfaction of the Surveyor.
" 2. All down-spouts to be connected with the sewers where
it may be proper to connect the same ; in all cases where they
are not connected with the sewer they are to be connected with
the channel.
" 3. All stench traps to be similar to samples furnished by
the Surveyor, or others approved by him, and properly fixed to
his satisfaction.
" 4. All sewers to water-closets not to be less than six inches
diameter.
" 5. All sewers to yards, stables, kitchens, and sculleries, not
to be less than four inches diameter.
" 6. In every case the whole of the fall to be made available
from the junction with the main sewer to the end of the private
drain, that is to say, only one inclination to be used from the
junction with the public sewer to the end of the private drain ;
and all branches from the private drain to sinks, water-closets,
&c., to have one inclination from the junction of such drain.
None of the above instructions to be departed from without the
express sanction of the Surveyor.
" 7. In no case must a private drain be put in with a less fall
than one in fifty, without the sanction of the Surveyor.
" 8. No pipes, water-closets, stench traps, gullies, kitchen
sinks, bends, junction or tapering pipes, to be used without
being approved by the Surveyor.
" 9. All ash pits and dung depots to be raised to the level of
the adjoining ground, to be properly paved and drained as the
Surveyor may direct.
ORIFICES, WEIRS, PIPES, AND RIVERS. 347
" 10. All buildings, outhouses, &c., to be properly spouted,
and the water conveyed into the sewers where approved of by the
Surveyor."
THOROUGH LAND DRAINAGE.
The following instructions and general specifica-
tions, have been prepared by the Commissioners of
Public Works in Ireland, for the use of the District
Inspectors, and persons reporting on thorough-
drainage. The drains are made in general parallel,
and to suit the fall of the ground. The depths must
alter in order that the bottoms should have an un-
interrupted fall, and may vary from 2 feet to 4 feet
6 inches in practice, averaging, say about 3 feet
6 inches, but dependent on circumstances. The
portions printed in italics are from specifications
prepared by officers of the Board, and are varied
according to each particular case :
GENERAL OBSERVATIONS.
" No drainage works should be undertaken until it has been
clearly ascertained that the surface level of the maximum floods
in the main drain can be discharged at a level that will admit of
the submain drains venting the waters from the lowest point of the
lands proposed to be thorough-drained, at a level sufficiently below
the surface of such land, that the highest floods shall not prevent
the free discharge of such submain.
" When sufficient out-fall can be obtained, no open main drain
should be of a less depth than five feet, and in all cases a greater
depth is desirable, in order to insure a permanent and efficient
drainage, and at the same time to prevent cattle, &c., from
crossing.
"As it has been found by practical experiments on different
varieties of soils, that deep drains, say from four to five feet deep,
are more effective than shallow ones : they should always be
estimated for, when the open main drains admit of their being
348 THE DISCHARGE OF WATER FROM
cut to that depth, or when, hy a moderate outlay per acre, the
main drains can be cut to a sufficient depth ; the distance be-
tween the parallel drains must necessarily vary with the texture
of the soil, forty feet may be taken as a general rule.
OPEN MAIN DRAINS.
" Main drains should have gradients of such inclination, and
be sunk to a depth that will admit of the above stipulations, as
to the discharge of the submain drains being carried out. They
should have such width at bottom and side slopes as may be
necessary ; and be free of sharp angles, projecting stones, and
other impediments to the quick discharge of the waters.
" The spoil or material raised in sinking and improving the
drains, where not available for filling up useless holes or drains,
should be removed to a proper distance from the edge of the
main drains, and dressed off in a workmanlike manner.
" The abutments and piers of such bridges as have sufficient
breadth of water-way, should, if necessary, be carefully under-
pinned; and those bridges which are insufficient to discharge
floods, should be taken down and rebuilt of suitable dimensions.
COVERED MAIN DRAINS.
" Whenever, from the nature of the lands, the extent of the
district under drainage, and the quantity of water to be voided,
it may be necessary to form covered main drains to receive the
water discharged from the submains, their dimensions must be
proportional to the amount of water to be voided, well flagged or
paved at bottom, the sides built of stone or brick, and covered
with a flag or arch at top.
SUBMAINS.
" The submains to be of such depth and width at top and bot-
tom as may be necessary. The fall in each to be as great as the
above-described main drainage of the district will allow, and not
to be allowed to run beyond a suitable length without discharging
itself into a covered or open main drain.
OEIFICES, WEIES, PIPES, AND KIVERS. 349
THE MINOR DRAINS
" To be of such depth, width at top and bottom, and at such
distance apart, as will secure the perfect drainage of the land, to
be run in a straight direction parallel to each other, directly up
and down the declivity, unless where the declivity happens to be
very steep, and then to be carried across the fall at such an angle
as to secure a free discharge for the water. The fall in each
minor drain to be as great as the main drainage and submain
drainage, previously described, will admit.
" In filling in the stones, great care should be taken that the
bottom of the drain be clean, and that no clay or dirt be put in
with them ; a sod, grass side down, or a few inches of tough clay,
to be placed on the surface of the stones, and trodden firmly.
The drain should then be filled up with the stuff previously sho-
velled out, observing to keep the active soil for the top. The
putting in of the stones to be commenced at the highest part or
head of the drain.
" In using draining pipes or other tiles, care should be taken
that they be laid firmly on the bottom for their entire length, so
as to prevent them being deranged by the filling of the drain, and
that the points be fitted as closely together as possible.
" In cases of unfavourable ground, caused by running sand or
otherwise, whereby the level of the conduit might be deranged,
collared pipe tiles offer considerable advantages in the way of
remedy.
" When gripes may be necessary on the sides of farm roads,
they should be on the field side of the fences."
SPECIFICATION FOR MAIN DRAINAGE.
OPEN MAIN DRAINS.
" The deepening and improving of the main drain, No. ,
is to be commenced at the point on the accompanying map,
and from thence a gradient carried up to the point , having
an inclination of at least feet per statute mile, and sunk to
the depth of feet. It shall be feet wide at bottom, and
the side slopes shall average at least, unless in rock cutting,
when the side slopes may be diminished to six inches to one foot;
all sharp angles, projecting stones, and other impediments to the
350 THE DISCHAEGE OF WATER FROM
free discharge of the water, must be carefully removed. The
spoil or material raised in sinking and improving the drain, when
not immediately used for top-dressing the adjoining lands, or for
filling useless holes or drains, is to he removed to a distance of
feet from the edge of the main drain, and dressed off in a
workmanlike manner.
" The bridge marked at the point on the accompanying
map to be .
" The whole to be executed in a proper and workmanlike manner,
and the works to be maintained in good .order for so long as any
interest shall be payable for the money advanced on account of its
execution."
SPECIFICATION FOR THOROUGH-DRAINAGE
(WITH TILES).
COVERED MAIN DRAINS.
" These shall be cut jiffy-four inches deep, thirty-six inches wide
at top, twenty-four inches wide at bottom ; the materials used in
them shall be double row of three-inch pipe tiles.
" The side walls shall be inches in height, inches
thick, and well at bottom. They shall be covered with a
flag not less than in thickness.
SUBMAINS.
" These shall be cut fifty inches deep, thirty inches wide at top,
eighteen inches wide at bottom. They shall be carried along the
low side of the fields, or portions of land to be drained, at a dis-
tance from the fence of fifteen feet, and through natural hollows
where necessary. No submain to be allowed to run beyond a
length of two hundred yards without discharging itself into a
covered or open main drain.
MINOR DRAINS.
" These shall be cut forty-eight inches deep, sixteen inches wide at
top, five inches wide at bottom, and at a distance of forty feet
apart. They shall be run in a straight direction, parallel to each
other, directly up and down the declivity (when possible). No
minor drain to be allowed to run beyond a length of two hundred
yards without discharging itself into a submain.
OKIFICES, WEIES, PIPES, AND RIVERS. 351
FILLING IN.
"All the drains (or a large number of them) having been opened
and cut in a workmanlike manner, and it being ascertained that
no water is standing in any of them, the filling in may be com-
menced.
MINOR DRAINS.
" Into each minor drain shall be put pipe tiles twelve inches in
length, one-and-a-half inch in the ope, for one hundred yards, com-
mencing from the upper end of the drain, and pipe tiles twelve
inches in length, one-and-three-quarter inch in the ope, in con-
tinuation from thence to the submains.
SUBMAINS.
" Into each submain shall be put pipe tiles twelve inches in length,
two inches in the ope, for one hundred yards, commencing from the
upper end of the drain, and pipe tiles twelve inches in length, three
inches in the ope, in continuation to the end or point where they
discharge themselves.
GENERAL RULES.
" All tiles to be of good sound material, and well burned. The
tiles shall be laid firmly on the bottoms of the drains for their
entire length ; the joints fitted as closely as possible, they shall
be carefully covered with a thin grassy sod or screen. The stuff pre-
viously taken out of the drains shall then be returned, observing
to keep the active soil uppermost.
" The mouths of the covered main or submain drains shall be
built about with solid masonry set in mortar, carried up with the
same slope as the sides of the open main drain, into which they
discharge themselves.
" Before laying the tiles, great care must be taken that the
bottom of the drains be clean. The putting in of the tiles to be
commenced at the highest point or head of the drains.
" In case of an entire field being thorough-drained, a drain
shall be cut at the top of it, parallel to the fence, and running at
a distance from it equal to one-half of the distance between each
of the minor drains, into one or more of which (as may be neces-
sary) it shall discharge itself. The remainder of the minor
drains to be discontinued at a distance from this drain equal to
352 THE DISCHARGE OF WATER FROM
one-half the entire distance between each of the minor drains ;
this drain to be of the same dimensions, and filled with the same
materials, and in like manner, as the above described.
" No open drain shall run into a closed one.
" In passing through unfavourable ground, caused by running
sand or otherwise, whereby the level of the conduit might be de-
ranged, and where pipe tiles are the materials used for forming
the conduit, collars must be used, so as to connect the ends of the
tiles, and they must be fitted as closely as possible.
" Soles must, in all cases, be used when laying single D tiles,
and they must be so laid that the ends of the tiles shall rest
equally on them ; when inverted D tiles are used, they shall also
be connected from end to end by placing one-half of the upper
tiles on one-half of the adjoining tiles below them.
" The whole to be executed in a proper and workmanlike
manner ; and the work to be maintained in like good order as
when approved of at its completion, for so long as any interest
shall be payable for the money advanced on account of its
execution." [Collars for up to k-inch pipes can be had at the Florence
Court Tilery.]
SPECIFICATION FOR THOROUGH-DRAINAGE (WITH
BROKEN STONES).
COVERED MAIN DRAINS.
" These shall be cut forty >-tivo inches deep, thirty inches wide
at top, twenty-four inches wide at bottom ; the materials used in
them shall be .
" The side walls in them shall be twelve inches in height, six
inches thick, and well at bottom. They shall be covered
with .
SUBMAINS.
" These shall be cut forty-two inches deep, eighteen inches
wide &t top, fourteen inches wide at bottom. They shall be car-
ried along the low side of the fields, or portions of land to be
drained, at a distance from the fences of thirteen feet, and
through natural hollows, where necessary. No submain to be
allowed to run beyond the length of one hundred and fifty yards,
without discharging itself into a covered or open main drain.
/ x<s iegs.
ORIFICES, WEIRS, PIPES, AND RIVERS. 353
ft VNJYE115S1TY |
MINOE DRAINS.
" These shall be cut thirty-six inches deep, fifteen inches wide
at top, four inches wide at bottom, and at a distance of twenty-
six feet apart. They shall be run in a straight direction, parallel
to each other, directly up and down the declivity (when possible).
No minor drain to be allowed to run beyond a length of two
hundred yards without discharging itself into a submain.
FILLING IN.
" All the drains (or a large number of them) having been
opened and cut in a workmanlike manner, and it being ascertained
that no water is standing in any of them, the filling in may be
commenced.
MINOR DRAINS.
" Into each minor drain shall be put ten inches of broken
stones in depth, the stones having been broken to a size not ex-
ceeding two-and-a-half inches in diameter. Great care should be
taken that the bottom of the drain be clean, and that no clay or
dirt be put in along with the stones ; a sod (or clay, as may be
convenient) three inches thick shall be placed carefully on top,
and the whole trampled upon or rammed hard. The drain shall
then be filled up with the stuff previously shovelled out, ob-
serving to keep the active soil for covering the top. The putting
in of the stones shall invariably be commenced at the highest
part or head of the drain.
FILLING IN SUBMAINS.
" In each submain a conduit shall be formed of six inches in
height, four inches wide, and the filling in completed as above
described.
GENERAL RULES.
" The mouths of the covered main or submain drains shall be
built about with solid masonry set in mortar, carried up with the
same slope as the sides of the open main drain into which they
discharge themselves.
" Before filling in the stones, great care must be taken that the
bottom of the drains be clean, and that no clay or dirt be put in
A A
354 THE DISCHARGE OF WATER FROM
along with them. The putting in of the stones to be commenced
at the highest part or head of the "drains.
" In case of an entire field being thorough-drained, a drain
shall be cut at the top of it, parallel to the fence, and running
at a distance from it equal to one-half the distance between each
of the minor drains, into one or more of which (as may be
necessary) it shall discharge itself. The remainder of the minor
drains to be discontinued at a distance from this drain, equal to
one-half the entire distance between each of the minor drains ;
this drain to be of the same dimensions, to be filled with the
same material, and in like manner, as the above described.
" No open drain shall run into a closed one.
" The whole to be executed in a proper and workmanlike man-
ner, and the work to be maintained in like good order as when
approved of at its completion, for so long as any interest shall be
payable for the money advanced on account of its execution."
One of the officers of the Commissioners of Public
Works, Ireland, the Inspector of Drainage for Ros-
common, a gentleman residing in that county, writes
to me as follows, with reference to tile and broken
stone drains on the carboniferous formation :
" With respect to tile drainage my experience has
not been very extensive, as the proprietors of the dis-
trict, with scarcely any exception, give a decided
preference to broken stones ; but from what I have
seen, I am very much inclined to prefer good well-
burnt pipes to any other draining material, provided
that collars be used, but not otherwise. As to the
best diameters, I have found the li" collared pipes
of the Clonbrock Tile Works (now closed) very satis-
factory ; but when the length of minor drains ex-
ceeded 100 yards, I should like an increase to li or
1|. For submains (say 150 or 180 yards long) I
have recommended pipes of 2 inches, 2i, and 3
inches in succession, all of which were to be had
OKIFICES, WEIKS, PIPES, AND KIVEES. 355
with collars: if 4-inch pipes were to be had with
collars, I should have recommended longer submains.
The larger-sized pipes are not provided with collars
in our present tileries, and on this account I generally
put a note on the margin of the printed form, sug-
gesting that a stone duct of the ordinary size of
submain, say 6 inches in height and 4 inches wide,
be substituted for the tile filling.
" I decidedly prefer an open duct to broken stone
filling ; and in nine-tenths of my own drainage I
have made the minor drains on the same plan as the
submain, with an open stone conduit ; the only differ-
ence being, that the minor drains are a few inches
shallower, with a smaller duct. The increase of
expense is a mere trifle, and when the substratum (as
very frequently occurs here) is a fine calcareous gravel,
containing 40 to 60 per cent, carbonate of lime, the
additional spoil is a very cheap fertilizer for the land.
" With respect to depths and distances apart, the
two most commonly used in my specification are
3i feet deep, 33 feet apart, and 4 feet deep, 42
feet apart. These arrangements will not suit all
cases, and I vary accordingly. Thus, in one case
of exceedingly retentive land of peculiar texture,
4 feet drains, 27 feet apart, produced the required
result, while in another, 3J feet drains, 66 feet apart,
effected all that was required. In the latter case
there was a mixed soil, which might be described
as " half wet ;" yet the water lingered sufficiently
long to make the land unsound for sheep, and greatly
to injure the crops in quality as well as quantity."
Mr. Josiah Parkes says, in 1843 : " Experiment
AA3
356 THE DISCHARGE OF WATER FROM
and experience have rapidly induced the adoption
of a system of parallel drains, considerably deeper,
and less frequent, than those commonly advocated
by professed drainers, or in general use. I gave
several instances of this practice in Kent in the
Report of last year, 1843, already alluded to, and
it is rapidly extending. Mr. Hammond stated to
you that he drained < stiff clays 2 feet deep, and 24
feet between the drains, at 3 4s. 3d. per acre,
and porous soils 3 feet deep, 33J feet asunder, at
2 5s. 2d. per acre.' I now find him continuing
his drainage at 4 feet deep, wherever he can obtain
the outfall, from a conviction founded on the ex-
perience of a cautious progressive practice as to the
depth and distance, that depth consists with economy
of outlay as well as with superior effect. He has
found 4-feet drains to be efficient, at 50 feet asunder,
in soils of varied texture not uniform clays and
executes them at a cost of about 2 5s. per acre,
being 185. 4d. for 871 pipes, and 1 6s. 6d. for
53 rods of digging. Communications have been
recently made to me by several respectable Kentish
farmers, of the satisfactory performance of drains
deeply laid in the Weald clays, at distances ranging
from 30 to 40 feet, but I have not had the oppor-
tunity of personally inspecting these drainages.
" The following little table shows the actual and
respective cost of the above three cases of under-
draining, calculated on the effects really produced,
that is, on the masses of earth effectively relieved
of their surplus water at an equal expense. I con-
ceive this to be the true expression of the work
ORIFICES, WEIRS, PIPES, AND RIVERS.
357
done, as a mere statement of the cost of drainage
per acre of surface conveys but an imperfect, in-
deed a very erroneous, idea of the substantive and
useful expenditure on any particular system. This
will be apparent on reference to the two last columns
of the table, which give the cost in cubic yards and
square yards of soil drained for one penny, at the
above mentioned prices, depths, and distances.
Depth of drains,
in feet.
Distance
between the
drains, in feet.
Mass of soil
drained per acre,
in cubic yards.
Mass of soil
drained for one
penny, in cubic
yards.
Surface of soil
drained for one
penny, in square
. f yards.
2
24
3226 *
4-1
6'27
3
33*
4840
8-93
8-93
4
50
6453
12-00
8-96
" I may here observe, that Mr. Hammond, when
draining tenacious clays, chooses the month of
February for the work, when he lays his pipes
(just covering them with clay to prevent crumbs
from getting in), and leaves the trenches open
through March, if it be drying weather, by which
means he finds the cracking of the soil much accele-
rated, and the complete action of the drains ad-
vanced a full season. The process of cracking may,
doubtless, be hastened both by a choice of the
period of the year in which drains are made, and
by such a management of the surface as to expose
it to the full force of atmospheric evaporation."
With reference to drains, we have known a case
in the Queen's County in which inch pipes had to
be taken up, and pipes of 2J-inch bore substituted.
358 THE DISCHARGE OF WATEK FROM
The drains were 40 feet apart, and 4 feet deep, and
the pipes had collars. The minor drains should
discharge into submains at convenient distances,
say 100 yards, on flat grounds. Small pipes will
choke unless the velocity in them be sufficient to
carry off deposits, and the diameters should vary
according to the inclinations of the ground, and
distance apart of the drains.
Mr. Mechi, in 1844, lays down the following
rules :
" 1st. That it is not the size or form of the drains
that regulate perfect drainage ; but the depth at
which they are placed. The depth aiso governs the
distances at which the drains should be cut accord-
ing to the quality of the soil.
" 2nd. The pipes of 1-inch bore, without stones,
are amply sufficient, placed at 4 feet deep and 30
feet wide in dense soils, and the same depth and 50
feet wide in mixed soils.
" 3rd. The deep drains receive more water than
shallow ones, and consequently lay dry a greater
extent of ground.
"4th. The deep drains begin and end running
sooner than shallow ones, and carry off more water
in a given time.
"5th. That where shallow drains are made and
deep ones cut below them, the shallow ones no longer
act, all the water passing to the deeper drains.
" 6th. That when round stones are used as well as
pipes, the latter should always be placed at the
bottom, as I find, practically, water flows more
quickly through pipes than amongst stones.
ORIFICES, WEIRS, PIPES, AND RIVERS.
359
" Before persons begin draining. I would recommend
their perusing attentively the facts developed by
Mr. Parkes, at pages 39 and 40, and my remarks at
page 36 of Letters on Agricultural Improvements.
u Pipes made to socket into each other (by Ford's
TABLE showing a Return of the number of Acres thorough-drained
in the years 1843 and 1844, by the different Competitors for
Sir Richard O'DonnelVs Gold Medal, together with the Average
Prices per Perch, and Cost per Acre respectively. (Given in
Reports to the Royal Agricultural Improvement Society of Ireland.)
Competitors.
Number
of statute
acres.
Number
of
statute
perches.
Average
price per
statute
perch.
Average
price per
statute
acre.
Cost to the
tenants.
Kate of wages
per day and
by task work.
Marquess of Waterford
A. R. P.
501 2 15
66*900
9d. in
1842, re-
duced to
5 id. in
1845.
5 in
1842, re-
duced to
3 2s. 8d.
in 1845.
5 per cent,
charged to
the rent.
Viscount Templetown
564 25
54,351
4R
1 14 4
500
..
Sir R. O'Donnell, Bart.
551
53,478
U.
1 12 4
408 6
Sd. a day.
The Earl of Caledon .
321 19
47,183
d.
295
396 9 9
Wd. to 1. a
day.
J. L. W. Naper, Esq. .
254 1 29
34,433
9fd.
583
700 7 4
Id. a day
in Winter,
andlOd.in
Summer.
Lord Blayney . . .
201 2 30
34,634
5d.
3 12
427 18
lOd. to Is. &
day.
TABLE showing a Return of the number of Acres thorough -drained
by Proprietors, for the Society's Gold Medal, and the Average
Prices per Perch and per Acre respectively.
The Earl of Erne . .
A. R. P.
110 37
p.
..
8. d.
3 12 3
..
..
Lord Dufferin and
Clanboye . . . .
203 1
29,478
lOfd.
6 11 9
, .
Is. per day.
Messrs. Andrews . .
117 1 4
16,614
9d.
5 15
.
13d. per day.
Dr. O'Neill . . . .
115 12
2 16 3
7d. to 8d. a
day.
360 THE DISCHAEGE OF WATER FROM
Patent Socketing Machine) are best adapted to loose
or mixed soils.'*
Pipes laid, however, too near the surface, are
frequently choked with the roots of plants. The
principal advantage of submains alongside open
mains is, that the mouths of the minor drains
should not be choked from vegetation, and that the
water from them, flowing into and taken up by this
submain, may be discharged by a few apertures
only, and thereby keep themselves open, or as much
so as the nature of the case will admit. The fore-
going tables show the cost per statute acre, in
Ireland, of thorough-drainage, which must vary with
circumstances, locality, and the value of labour.
The average cost per statute acre for Sir Eichard
O'Donnell's Gold Medal, was 3 5*. Id., and
4 13s. IQd. for the Society's Gold Medal ; average
of both, 4 per statute acre nearly.
The average number of acres now annually im-
proved in Ireland, is about 5530, at an average cost
per acre of 4 17 s.
In Ireland, thorough-drainage is almost generally
carried out by loan, under the Commissioners of
Public Works, and there is no branch of the public
service has given more satisfaction to owners of pro-
perty. The works are, we believe, always executed
within the estimates, and the owner having the ex-
penditure in his own hands, can satisfy himself of
its proper application. No loans are made unless
where immediately, or prospectively, a return of
6i per cent, is estimated on the expenditure, a rent
charge for this amount being made for 22 years.
ORIFICES, WEIRS, PIPES, AND RIVERS.
361
TABLE showing Estimates of the Quantities and General Cost for
the Thorough- Drain age of a Statute Acre of Land, with Broken
Stones or Tiles, with the distances apart for different class soils.
Description of land, section
of parallel drain, and cost.
Distance between!
the
parallel drains. 1
Lineal
perches
of
drains
per
statute
acre.
Cubic yards per
acre.
Expense
per statute
acre.
Number
of feet
of tiles
per
statute
acre.
Of Clay
excav.
Of
broken
stones.
/
Feet.
Perches.
Cu. yds.
Cu. yds.
8. d.
Number
12
220
277
38-5
768
3,630
13
203
255|
35-5
6 15 4
3,351
14
188
237
33-0
658
3,111
Hard subsoil stiff and
15
176
221
30-8
5 17 4
2,904
sandy clay drains, from
16
165
207
28-9
5 10
2,722
12 to 20 feet apart, at
17
155J
iftj
27-2
5 3 6
2,562
Sd. per lineal perch.
18
146
184
25-9
4 17 6
2,420
19
139
175
24-3
4 12
2,293
20
132
166J
23-1
480
2,178
/
21
125J
182
29-3
4 14 6
2,074
22
120
174f
28-1
4 10
1,971
23
115
167
23-9
463
1,894
Freestone bottom drains,
24
110
159i
25-7
426
1,815
from 20 to 30 feet
25
105
153J
24-7
3 19 1J
1,742
apart, at 9d. per
26
101
147*
23-7
3 16 1J
1,675
27
97f
142
22-9
3 13 4
1,613
28
94
136J
22-0
3 10 6
1,556
29
91
132
21-3
383
1,502
30
88
127}
20-6
360
1,452
j
31
85
151J
24-7
3 10 10
1,405
32
82
147
24-0
389
1,361
33
80
142|
23-2
368
1,320
Beds of gravel, sand, and
34
77f
188}
22-6
3 4 8
1,280
rocky stratification,
35
75|
134
21-9
3 2 9J
1,245
from 30 to 100 feet <
36
73*
1301
21-3
3 1 1J
1,210
apart, at IQd. per
37
71
126
20-6
2 19 2
1,177
lineal perch . .
38
69
123
20-0
2 17 6
1,146
39
67*
119|
19-6
2 16 1J
1,117
40
65|
H6
19-1
2 14 8
1,089
100
27
47
8-0
126
436
362 THE DISCHARGE OF WATER FROM
ARTERIAL DRAINAGE.
The effect of thorough-drainage on the arterial
channels of a district, is to discharge the rain-fall
into the main channels in a shorter time than before,
particularly during wet seasons. This frequently
causes floods to rise higher as well as more rapidly.
During dry seasons the supply is less, and so far,
when it is limited, an injury is done to the adjacent
districts requiring it for use. The effect of ob-
structions in the main channel is to impound the
upland water, sometimes made available for water
power or navigation purposes, but in general, to the
injury of the drainage of adjacent lands, and the
regimen of the river, particularly in flat districts.
The arterial drainage in Ireland has effected a vast
amount of good, but up to 1853 the estimates appear
to have been usually doubled ; the estimates for
eleven of these works being 124,647, and the ex-
penditure, 293,532. The average cost per acre, on
the land improved by these projects, varied from
1 19s. Sd. to 5 17s. Id., the average of the
eleven districts being 4 3s., which is about the
average for thorough-drainage.*
The following table affords valuable information of
the cost of arterial drainage works in Ireland : it
is extracted from the Report of the Commissioners
of Inquiry, presented to the House of Commons
June 16th, 1853.
* See Parliamentary Eeport, by Sir Richard Griffith, Sir
W. Cubitt, and Jas. M. Eendel, June 16th, 1853.
ORIFICES, WEIRS, PIPES, AND RIVERS.
363
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si
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364
THE DISCHARGE OF WATEK FROM
At
The abstract of 84 arterial drainage awards, made
by the Commissioners of Public Works in Ireland,
in 1854, gives for different years, 1849 to 1854
Number of
districts
drained.
Total
combined
catchment
acreage of
districts.
Area of
flooded lands.
Average cost of
arterial drainage
per acre improved
by drainage.
s. d.
12 districts
90,332
9,453
3 14 2
27
95,582
11,579
3 16 7
19
237,466
13,707
4 17 3
16
374,427
29,452
3 13 4
2
49,840
3,275
500
8
266,420
21,033
394
84
1,114,067
88,501
3 17 7
The last line gives the general average, and shows
that in these 84 districts, about 1 acre in 13 is the
average of flooded lands to^ the catchment area, or
8 per cent, nearly.
l*.
SE
f
CTION
XIV.
WATER AND HORSE POWER. FRICTION BRAKE, OR DYNAMO-
METER, CALCULATION OF THE EFFECTIVE POWER OF
WATER WHEELS. OVERSHOT, UNDERSHOT, AND BREAST
VERTICAL WHEELS. HORIZONTAL WHEELS AND TURBINES.
HYDRAULIC RAM. WATER ENGINE.
Taking the representative of a horse's power at
33,000 foot-pounds* or 33,000 Ibs. raised one foot
* 16,500 foot-pounds, or one half of the above, is much nearer
the average power of a horse, working for 10 hours only, as the
work is ordinarily done through the country ; 33,000 Ibs. raised
one foot per minute, is equivalent to 884 tons, nearly, raised one
foot in an hour. Therefore, a river discharging 884 tons, over a
fall one foot high in an hour, or 884 tons, over a fall 24 feet high
ORIFICES, WEIRS, PIPES, AND RIVERS.
365
high in one minute, the theoretical horse-power of an
overfall is expressed by the fall in feet, multiplied by
the discharge in cubic feet per minute, the product
multiplied by 62 (the weight in Ibs., nearly, of a
cubic foot of water), and divided by 33,000. The
following table gives the weight in air of a cubic foot
of pure water at different temperatures, Fahrenheit's
thermometer.
WEIGHT OF A CUBIC FOOT OF WATER.
The weight of 36 cubic feet of water is one ton, nearly.
Tempe-
rature, in
degrees.
Weight of a
cubic foot of
water. Pounds
Avoirdupois.
Tempe-
rature, in
degrees.
Weight of a
cubic foot of
water. Pounds
Avoirdupois.
Tempe-
rature, in
degrees.
Weight of a
cubic foot of
water. Pounds
Avoirdupois.
32
62 ' 375
51
62-365
69
62-278
33
62-377
52
62-363
70
62-272
34
62-378
53
62-359
71
62 264
35
62-379
54
C2-356
72
62 257
36
62-380
55
62-352
73
62 249
37
62 381
56
62 349
74
62 242
38
62-381
57
62 345
75
62-234
39
62 382
58
62-340
76
62-225
40
62-382
59
62 ' 336
77
62-217
41
62-381
60
62-331
78
62-208
42
62-381
61
62-326
79
62-199
43
62-380
62
62-321
80
62-190
44
62*379
63
62-316
81
62-181
45
62-378
64
62-310
82
62 172
46
62-376
65
62-304
83
62-162
47
62 '375
66
62'298
84
62-152
48
62 '373
67
62-292
85
62-142
49
62-371
68
62-285
86
62-132
50
62-368
87
62-122
The effective power of a fall depends on the nature,
in 24 hours, has also a horse power. The drainage of 10 square
miles, with an average collection of 12 inches annually of rain in
depth, will give an annual unceasing one-horse power for each
foot of fall in a receiving channel ; or five square miles will give
the same result, if the collection amounts to 24 inches in depth.
The collection of 10 square miles, one foot deep, yearly, is nearly
equal to the delivery of 530 cubic feet per minute, for the same
period.
366 THE DISCHARGE OF WATER FROM
proportions, and construction of the wheel or
machine, and also upon the manner in which the
theoretical power is applied. When the velocity of
a stream acting on a wheel only is known, the theo-
retical head, A, due to it is found in feet from the
formula h -0155 v 2 , v being the volocity in feet per
second.
In order to gauge the quantity of water applied to
a wheel, and thereby determine with accuracy its
effective power, the water used must be passed
through a notch, orifice, or over a weir, the coefficients
for which had been previously ascertained from expe-
riment. Greater accuracy can be obtained from
gaugings through thin plates, or planks having the
down-stream arrises chamfered, than with any other
form of orifice or notch ; and when it can be effected,
the channel above should be sufficiently enlarged to
prevent the effects of an approaching current. We
have already in the body of this work dwelt in detail
on the various formulae required for gauging under
different circumstances. The accuracy of the results
showing the effective powers of wheels depends in
the first place, on the accuracy of the gaugings or
estimates of the quantity of water used, and next on
the fall employed.
FRICTION BRAKE, OE DYNAMOMETER.
The power applied to a revolving shaft through a
water wheel of any construction, is the weight of
water multiplied by the fall. It is evident that the
portion of this power available to turn a shaft and
machinery, or the effective power, must depend on
ORIFICES, WEIRS, PIPES, AND RIVERS.
367
the construction of the wheel, as a portion of the
theoretical power is lost mechanically, in applying it;
in changes of direction, friction, eddies, and discharg-
ing currents. The greater the effective power con-
veyed to a shaft, the greater becomes the power of
the wheel, or medium through which the original
power is transmitted. The mechanical effect produced
by a revolving shaft is best measured by a friction
brake, the principle of which is as follows. In
Fig. 44, let the friction pulley AA be
diagram 1,
FIG. 44.
FRICTION BRAKES, OR DYNAMOMETERS.
firmly fixed to the revolving shaft or axis of the wheel ;
E and E, two wooden clamps grasping the friction
pully by means of the screw bolts, delineated, which
can be tightened on the axis, and also to the arm F,
by means of suitable nuts. The more tightly the
bolts are screwed, the greater will be the friction
368 THE DISCHARGE OF WATEE FROM
between the friction pulley AA, and the clamping
pieces EE. If, while the axis and friction pulley A A,
are revolving in the direction indicated by the arrow,
a weight be applied in the scale at i, so that the arm
F shall not be carried round, but remain fixed ; it is
clear that the work done by the revolving shaft in
one revolution, will be measured by the circumference
of the friction pulley, multiplied by the friction due
to the pressure on it, or by its equivalent, the weight
in the scale i, multiplied by the circumference of a
circle whose radius is L, or by 2 L x 3*14159 x w, in
which expression w is the weight in Ibs. in the
scale i. If n be the number of revolutions in a
given time, say one minute, we shall therefore have
the useful effect of the wheel on the shaft in foot-
pounds per minute, equal to
2LX 314159 Xwxn.
We have also the power of the water acting on the
wheel, equal to
hx DX 62-37,
in which h is the head and D the discharge in cubic
feet per minute ; therefore, we shall have for the ratio
of the effect to the power the expression
2LX 3-14159 x w xn
hXVX 62-37
If the revolving shaft be horizontal, the weight of
the arm F, acting at its centre of gravity and reduced
to the length L, where the weight w is suspended, will
have to be included in the weight w. If the weight
w be suspended at the end of a connexion of levers,
or other mechanical powers, the length L will have to
be determined accordingly. Diagram 2, Fig. 44,
ORIFICES, WEIRS, PIPES, AND RIVERS.
369
shows the Armstrong-brake ; Diagram 3, the common
form ; and Diagram 4, Egen's brake.
Fig. 45, is a general representation of the brake
FIG. 45.
DYNAMOMETER FOE DETERMINING THE USEFUL EFFECT OF THE
TREMONT TURBINE.
used by Francis, in the Lowell experiments. The
length of the arm of the brake L, was 9-745 feet;
the length of the vertical arm I of the bell crank 4-5
BB
370 THE DISCHARGE OF WATER FROM
feet ; and the length of the horizontal arm /' 5 feet.
The following detailed description is by Francis :
" The Friction Pulley A is of cast-iron, 5 -5 feet in
diameter, two feet wide on the face, and three inches
thick. It is attached to the vertical shaft by the
spider B, the hub of which occupies the place on the
shaft intended for the bevel gear.
" The friction pulley has, cast on its interior circum-
ference, six lugs, c c, corresponding to the six arms
of the spider. The bolt holes in the ends of the arms
are slightly elongated in the direction of the radius,
for the purpose of allowing the friction pulley to
expand a little as it becomes heated, without throwing
much strain upon the spider. When the spider and
friction pulley are at the same temperature, the ends
of the arms are in contact with the friction pulley.
The friction pulley was made of great thickness for
two reasons. When the pulley is heated, the arms
cease to be in contact with the interior circumference
of the pulley, consequently they would not prevent
the pressure of the brake from altering the form of
the pulley. This renders great stiffness necessary in
the pulley itself. Again, it is found that a heavy
friction pulley insures more regularity in the motion,
operating, in fact, as a fly-wheel, in equalizing small
irregularities.
" The brakes E and F are of maple wood ; the two
parts are drawn together by the wrought-iron bolts
c c, which are two inches square.
" The bell crank F', carries at one end the scale i, and
at the other the piston of the hydraulic regulator K ;
this end carries also the pointer L, which indicates
ORIFICES, WEIRS, PIPES, AND RIVERS, 371
the level of the horizontal arm. The vertical arm is
connected with the brake F, by the link M.
" The hydraulic regulator K, shown in the figures, is a
very important addition to the Prony dynamometer,
first suggested to the author by Mr. Boy den, in 1844.
Its office is to control and modify the violent shocks
and irregularities, which usually occur in the action
of this valuable instrument, and are the cause of some
uncertainty in its indications.
"The hydraulic regulator used in these experiments,
consisted of the cast-iron cylinder K, about 1*5 feet
in diameter, with a bottom of plank, which was
strongly bolted to the capping stone of the wheel pit,
as represented in figure 1. In this cylinder, moves
the piston N, formed of plate-iron 0-5 inches thick,
which is connected with the horizontal arm of the bell
crank by the piston rod o. The circumference of the
piston is rounded off, and its diameter is about TV inch
less than the diameter of the interior of the cylinder.
The action of the hydraulic regulator is as follows.
The cylinder should be nearly filled with water, or
other heavy inelastic fluid. In case of any irregularity
in the force of the wheel, or in the friction of the
brake, the tendency will be, either to raise or lower
the weight, in either case the weight cannot move,
except with a corresponding movement of the piston.
In consequence of the inelasticity of the fluid, the
piston can move only by the displacement of a portion
of the fluid, which must evidently pass between the
edge of the piston and the cylinder ; and the area of
this space being very small, compared to the area of
the piston, the motion of the latter must be slow,
BBS
372 THE DISCHARGE OF WATER FROM
giving time to alter the tension of the brake screws
before the piston has moved far. It is plain that
this arrangement must arrest all violent shocks, but,
however violent and irregular they may be, it is
evident, that, if the mean force of them is greater in
one direction than in the other, the piston must move
in the direction of the preponderating force, the
resistance to a slow movement being very slight. A
small portion of the useful effect of the turbine must
be expended in this instrument, probably less, how-
ever, than in the rude shocks the brake would be
subject to without its use.
" For the purpose of ascertaining the velocity of
the wheel, a counter was attached to the top of the
vertical shaft, so arranged, that a bell was struck at
the end of every fifty revolutions of the wheel.
" To lubricate the friction pulley, and at the same
time to keep it cool, water was let on to its surface
in four jets, two of which are shown. These jets were
supplied from a large cistern, in the attic of the
neighbouring cotton mill, kept full, during the work-
ing hours of the mill, by force pumps. The quantity
of water discharged by the four jets was, by a mean
of two trials, 0-0288 cubic feet per second.
" In many of the experiments with heavy weights,
and consequently slow velocities, oil was used to
lubricate the brake, the water, during the experiment,
being shut off. It is found, that, with a small
quantity of oil, the friction between the brake and
the pulley is much greater than when the usual
quantity of water is applied; consequently, the
requisite tension of the brake screws is much less
OKIFICES, WEIES, PIPES, AND RIVERS. 373
with the oil, as a lubricator, than with water. This
may not be the whole cause of the phenomenon, but,
whatever it may be, the ease of regulating in slow
velocities is incomparably greater with oil as a
lubricator, than with water applied in a quantity
sufficient to keep the pulley cool. The oil was
allowed to flow on in two fine continuous streams ;
it did not, however, prevent the pulley from becoming
heated sufficiently to decompose the oil, after running
some time, which was distinctly indicated by the
smoke and peculiar odour. When these indications
became very apparent, the experiment was stopped,
and water let on by the jets, until the pulley was
cooled. As the pulley became heated, the brake
screws required to be gradually slackened. Water,
linseed oil, and resin oil, were each used for lubrica-
tion."
Fig. 46 is a representation of a brake used by
Professor Thomson, at Crawford and Lindsay's mill,
to determine the power of a turbine put up there,
by Mr. Gardner, of Armagh. One of the common
causes of the swinging or vibratory jumps of the arms
F, in Figs. 44, 45, and 46 is, that very often the
friction pulley, or drum A, A, must be made in two
parts, so as to be fixed to its place on the shaft.
This fixing is liable to give an oval shape, and causes
an irregular action with the clamps E, E. 0S gives
greater regularity of motion than water, but without
the use of the latter abundantly, the friction pulley
would usually get too much heated. The following
calculations from practical operations will point out
pretty clearly the use of the brake, and the manner
374
THE DISCHARGE OF WATER FROM
of determining the useful effect in the tables of expe-
riments, by Francis and Thomson, pp. 383 and 392 :
FIG. 46.
FRICTION BRAKE, USED AT CRAWFORD AND LINDSAY'S MILL, BY PROFESSOR
JAMES THOMSON, TO DETERMINE THE POWER OF THE TURBINE.
Length of the brake, L, Fig. 45, adjusted . . . 9-745 feet.
Effective length of vertical arm I 4-5
Effective length of horizontal arm I 1 .... 5'0
The effective length of the brake was therefore
9-745 x 5
4-5
= 2 x 10 -827778 feet; and the circumference
of a circle of this radius = 10-827778 x 3-14159 =
68-0329 feet.
In the first experiment on the Tremont turbine,
ORIFICES, WEIRS, PIPES, AND RIVERS. 375
page 393, the number of revolutions of the wheel per
second was '89374, and the weight in the scale
1443-34 Ibs. The useful effect of the brake was
therefore in foot-pounds per second 68-0329 x
89374 x 1443-34 = 87680-3 Ibs. raised one foot per
second. The quantity of water which passed the
gauge-weir in cubic feet per second was 139-4206,
and the total fall acting on the wheel 12-864 feet;
therefore, the total power of the water acting on the
wheel was 12-864x1394206x62-375=111870 feet-
pounds per second, 62-375 being taken as the weight
in Ibs. of a cubic foot of water at 32 Fahrenheit.
The ratio of the useful effect, at the given velocity of
the wheel (viz. 450 revolutions in 503*5 seconds), to
Qrr/^or\ .0
the power expended, is therefore i 11070 = '784, or
about 78j per cent. The effect in the experiments
generally appears to have been a maximum, when
the velocity of the interior circumference of the wheel
was about 66 per cent, of the velocity due to the fall ;
and this was about half of the maximum velocity,
which was 1-333 times that due to the fall alone,
when the turbine was doing no work.
In Thomson's brake for determining the useful
effect of the vortex turbine, erected from his designs
at Bally sillan, Ireland, L = 4 ft. 2 in., and the cir-
cumference of a circle that would be described by the
arm 3-14159 x 8 feet 4 inches. = 26-18 feet. In the
first experiment, taken from the tabulated results,
page 383, we get 26-18, the circumference multiplied
by 46-31, the weight in Ibs., and the product by
323*3, the number of revolutions per minute, equal
376 THE DISCHARGE OF WATER FROM
to 391,967 foot-pounds, for the effect transmitted
from the turbine or work done. We have also 3544,
the number of cubic feet of water passed to the
wheel per minute, multiplied by 62*37, the weight of
a cubic foot of water in Ibs., multiplied by 23*73 feet,
the available fall, equal to 524,526 foot-pounds :
391 967
therefore Swa '747 is the useful effect, that in
the table being -7481, which probably arose from
taking a different weight per cubic foot for the water.
Of course the difference is immaterial. The drum
attached to the vortex wheel shaft for fixing the
brake to, was in two parts, bolted together, and
firmly enclosing the shaft. It was of cast-iron, 20
inches diameter, and 8 inches wide ; the shaft to
which it was attached was 2f inches diameter. The
arm of the brake was 5''4" x 6" x 4", of timber, and
extending 1 foot 2 inches beyond the centre of the
shaft and drum. The clamping pieces were about
2 feet 5 inches long externally.
FOR OVERSHOT WHEELS the ratio of the power to
the effect may be taken as 3 to 2, and therefore the
effective horse-power, taking 33,000 foot-pounds
per minute as a standard, will be 49,500 Ibs. of
water falling one foot in one minute. The maximum
effect varies with the construction of the wheel.
Smeaton found it '76 times the theoretical power ;
Weisbach -78 times for the wheel of a Stamp Mill
at Frieberg, which was 23 feet high, 3 feet wide,
carrying 48 buckets.* To find the effective horse-
* Some valuable experiments on the power of water wheels are
given by Rennie, in Weale's Quarterly Papers of Engineering,
vol. vi. They however require reduction.
ORIFICES, WEIRS, PIPES, AND RIVERS.
377
power, the theoretical horse-power must be here di-
vided by the coefficient of effect -76 or -78, which will
give 43,600 foot-pounds, or 43,300 foot-pounds per
minute. The following experimental results from a
model wheel are by Smeaton.
TABLE containing the Result of Sixteen Experiments, on a Model
Overshot Wheel, by Smeaton.
.9
S .
S
-1
4J
o
iS ^
If .
4
'SJ!
'p_g
|.J
1 .
J
"f o
<g
"a .9
S a
a 1
jl
"o"
-s
I.S
oS^
1
No
I.r
S
*< r
^ *"* "S
o.9
*" eS
t'B
|p
N
oo
Ifl
i
o
|1
!ji
g
ft
I? 1
S
P
s
^
ft
fl
B
1
27
30
19
9|
810
720
556
69
77
2
27
56f
161
14J
1530
1360
1060
69
78
3
27
56|
20|
12^
1530
1360
1167
76
84
rH
00
4
27
63|
20*
13f
1710
1524
1245
73
82
5
27
78|
21*
15i
2070
1840
1500
73
82
6
28*
73*
18f
17J
2090
1764
1476
70
84
S3
7
28*
96|
20J
2755
2320
1868
68
80
8
30
90
20
1H
2700
2160
1755
65
81
9
30
96|
aof
20^
2900
2320
1914
66
82
00
10
30
113*
21
23*
3400
2720
2221
65
82
11
33
56|
20*
13*
1870
1360
1230
66
90
12
13
33
33
106f
146f
23
dil
27*
3520
4840
2560
3520
2153
2846
61
59
84
81
*
14
35
65
19f
16*
2275
1560
1466
65
95
15
35
120
9if
25*
4200
2880
2467
59
86
00
16
35
163*
25
5728
3924
2981
52
76
In this table the effective power of the water
must be reckoned upon the whole descent, because it
must be raised that height, in order to be in a
condition of producing the same effect a second time.
378 THE DISCHARGE OF WATER FROM
The ratios between the powers so estimated, and
the effects at the maximum, deduced from the several
sets of experiments, are exhibited at one view, in
column 9, of Table II. ; and from hence it appears,
that those ratios differ from that of 10 to 7-6 to that
of 10 to 5-2, that is, nearly from 4 to 3 to 4 to 2. In
those experiments where the heads of water and
quantities expended are least, the proportion is nearly
as 4 to 3, but where the heads and quantities are
greatest, it approaches nearer to that of 4 to 2 ; and
by a medium of the whole, the ratio is that of 3 to 2,
nearly. We have seen before, in our observations
upon the effects of undershot wheels, that the general
ratio of the power to the effect, when greatest, was 3
to 1 ; the effect, therefore, of overshot wheels, under the
same circumstances of quantity and fall, is at a
medium double to that of the undershot ; and, as a
consequence thereof y that non-elastic bodies, when acting
by their impulse or collision, communicate only a part
of their original power ; the other part being spent
in changing their figure, in consequence of the
stroke.
The powers of water, computed from the height of
the wheel only, compared with the effects as in
column 10, appear to observe a more constant ratio :
for, if we take the medium of each class, which is set
down in column 11, we shall find the extremes to
differ no more than from the ratio of 10 to 8*1 to that
of 10 to 8'5 ; and as the second term of the ratio
gradually increases from 8-1 to 8-5, by an increase of
head from 3 inches to 11, the excess of 8-5 above 8-1
is to be imputed to the superior impulse of the water
ORIFICES, WEIRS, PIPES, AND RIVERS. 379
at the head of 11 inches, above that of 3 inches : so
that if we reduce 8-1 to 8, on account of the impulse
of the 3-inch head, we shall have the ratio of the power,
computed upon the height of the wheel only, to the
effect at a maximum, as 10 to 8 or as 5 to 4, nearly ;
and from the equality of the ratio between power and
effect subsisting, where the constructions are similar,
we must infer, that the effects, as well as the powers,
are as the quantities of water and perpendicular
heights, multiplied together respectively.
FOR BREAST WHEELS, the ratio of the theoretical
power to the effective power must vary considerably,
the mean value being about 1 to *5 and, therefore, the
effective horse-power would be 66,000 foot-pounds
in one minute. Morin gives an efficiency of from *52
to -7. Egen, with a wheel 23 feet in diameter, 4i feet
wide, having 69 ventilated buckets, very well con-
structed, found at best an efficiency of only -52,
under ordinary circumstances '48, the mean amount
being -5. Very wide wheels give a larger effect,
sometimes as high as '7 ; but a great deal depends on
the manner of bringing on the water and the con-
struction of the wheel and buckets.
FOR UNDERSHOT WHEELS the mean effect may be
taken at one-third or -33, or 100,000 foot-pounds in
one minute for an effective horse's power; a maximum
effect of -5 is sometimes approached, and a minimum
of -26 or less. The following results, obtained from
a model, are given by Smeaton. The virtual or
effective head is here termed the theoretical head
due to the velocity of the wheel, at the circum-
ference, which was 75 inches girth.
380
THE DISCHARGE OF WATER FROM
TABLE containing the Eesult of Twenty-seven Experiments, on a
Model Undershot Wheel, by Smeaton.
1
'S rf
c!
1
SI
1
g S
Srfl
TJj
.
&
"5 ,2
'"SI!
ill
fi
ti
j|
||
l&
*l 8
1|
|
|
. <
m
J2
<B
"o g
11
11
" a
02
||
^21
l|
i
H
jl o
||
?|S
i)
p
p3
Q
fl
+* -3
4->
o f*
o'S *
.SP.fH
H
P*
H
*
2 I
H
ri
1
33
88
15-85
so-
13 10
10 9
275-
4358
1411
324
340
775
2
30
86
15-0
so-
12 10
9 6
264-7
3970
1266
320
350
740
3
27
82
13-7
28-
11 2
8 6
243-
3329
1044
315
340
750
4
24
78
12-3
27-7
9 10
7 5
235-
2890
901-4
312
355
753
5
21
76
11-4
25-9
8 10
6 5
214-
2439
735-7
302
345
732
6
18
70
9-95
23-5
6 10
5 5
199-
1970
561-8
285
336
802
7
15
65
8-54
23-4
5 2
4 4
178-5
1524
442-5
290
360
830
8
12
60
7-29
22-
3 10
3 5
161-
1173
328
280
377
910
9
9
52
5-47
19-
2 12
2 8
134-
733
213-7
290
365
910
10
6
42
3-55
16-
1 12
1 10
114-
404-7
117-0
282
380
930
11
24
84
14-2
30-75
13 10
10 14
342-
4890
1505
307
366
790
12
21
81
13-5
29-
11 10
9 6
297-
4009
1223
301
362
805
13
18
72
10-5
26-
9 10
8 7
285-
2993
975
325
360
875
14
15
69
9-6
25-
7 10
6 14
277-
2659
774
292
362
900
15
12
63
8-0
25-
5 10
4 14
234-
1872
549
294
397
870
16
9
56
6-37
23-
4
3 13
201-
1280
390
305
410
950
17
6
46
4-25
21-
2 8
2 4
167-5
712
212
298
455
900
18
15
72
10-5
29-
11 10
9 6
357-
3748
1210
323
402
805 1
19
12
66
8-75
26-75
8 10
7 6
330-
2887
878
305
405
810
20
9
58
6-8
24-5
5 8
5
255-
1734
541
301
422
910
21
6
48
4-7
23-5
3 2
3
228-
1064
317
299
490
960
22
12
68
9-3
27-
9 2
8 6
359-
3338
1006
302
397
917. 1
23
9
58
6-8
26-25
6 2
5 13
332-
2257
686
904
452
950
24
6
48
4-7
24-5
3 12
3 8
262-
1231
385
313
510
935
25
9
60
7-29
27-3
6 12
6 6
355-
2588
785
303
455
945 1
26
6
50
5-03
24-6
4 6
4 1
307-
1544
450
292
490
930 I
27
6
50
5-03
26-
4 15
4 9
360-
1811
534
295
520
925 1
ORIFICES, WEIRS, PIPES, AND RIVERS. 381
Smeaton derived the following " maxims " from
the foregoing experiments. Their truth, independent
of any experiment, will be apparent :
/. That the virtual or effective head being the same, the effect will
be nearly as the quantity of water expended.
II. That the eoopense of water being the same, the effect will be
nearly as the height of the virtual or effective head.
III. That the quantity of water expended being the same, the effect
is nearly as the square of the velocity.
IV. The aperture being the same, the effect will be nearly as the cube
of the velocity of the water.
FOR TURBINES OR HORIZONTAL WHEELS, a USeM
effect of two-thirds or '67 may be assumed, or
49,500 foot-pounds in a minute for a horse-power,
and the efficiency varies from -5 to -8, or less.* Pon-
celet's turbine gives an efficiency of *5 to *6. Float-
ing wheels -38, impact wheels from -16 to *4, and
Barker's mill from -16 to -35. We believe that the
efficiency of the turbine has been too often over-
estimated, and that the great advantage of this
wheel, as a medium of power, is derived from its
capability of employment for all falls, whether large
or small, without any considerable loss of effect.
In Ireland, Mr. Gardner, of Armagh, was amongst
the first, if not the first, to apply this wheel to prac-
tical purposes ; and Professor Thomson has, in his
* In our first edition we gave an efficiency of -821, on the
authority of a paper by Dr. Robinson, Armagh, in the Proceed-
ings of the Royal Irish Academy, vol. iv., p. 914. On again
glancing over this paper, we believe there are mistakes, which
vitiate the results there given; first, in the formula for calcu-
lating the discharge over the weir, and next, in the formula for
finding the effect of the brake. Francis gives an efficiency of -88,
p. 3, his book.
382 THE DISCHARGE OF WATER FROM
vortex wheels, produced, we believe, the highest effi-
ciencies which have yet been obtained in practice.
In the experiments on the Ballysillan wheel, higher
efficiencies would probably have been attained with a
supply pipe of larger diameter. It will be seen from
the remarks, at pp. 171 and 172, and the tables, at
pp. 152 and 191, that quite apart from bends, &c.,
a loss of mechanical power always results from the
passage through orifices and pipes ; and that it
is necessary to take this loss into account, before
the head acting on the wheel can be accurately used
to determine its effective power. The table, next page,
contains the experiments on the Ballysillan turbine.
The following remarks on the vortex turbine, read
at the meeting of the British Association at Belfast,
in 1852, are also by Professor Thomson :
"Numberless are the varieties, both of principle
and of construction, in the mechanisms by which
motive power may be obtained from falls of water.
The chief modes of action of the water are, however,
reducible to three, as follows : First, the water may
act directly by its weight on a part of the mechanism
which descends while loaded with water, and ascends
while free from load. The most prominent example
of the application of this mode is afforded by the
ordinary bucket water wheel. Secondly, the water
may act by fluid pressure, and drive before it some
yielding part of a vessel by which it is confined.
This is the mode in which the water acts in the water
pressure engine, analogous to the ordinary high-pres-
sure steam-engine. Thirdly, the water, having been
brought to its place of action subject to the pressure
ORIFICES, WEIRS, PIPES, AND RIVERS.
383
the
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atin
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384 THE DISCHAEGE OF WATEE FEOM
due to the height of fall, may be allowed -to issue
through small orifices with a high velocity, its inertia
being one of the forces essentially involved in the
communication of the power to the moving part of
the mechanism. Throughout the general class of
water wheels called turbines, which is of wide extent,
the water acts according to some of the variations of
which this third mode is susceptible. In our own
country, and more especially on the Continent, turbines
have attracted much attention, and many forms of
them have been made known by published descrip-
tions. The subject of the present communication is a
new water wheel, which belongs to the same general
class, and which has recently been invented and
brought successfully into use by the author.
"In this machine the moving wheel is placed within
a chamber of a nearly circular form. The water is
injected into the chamber tangentially at the circum-
ference, and thus it receives a rapid motion of rotation.
Retaining this motion it passes onwards towards the
centre, where alone it is free to make its exit. The
wheel, which is placed within the chamber, and which
almost entirely fills it, is divided by thin partitions
into a great number of radiating passages. Through
these passages the water must flow on its course
towards the centre ; and in doing so it imparts its own
rotatory motion to the wheel. The whirlpool of water
acting within the wheel chamber, being one principal
feature of this turbine, leads to the name Vortex as
a suitable designation for the machine as a whole.
" The vortex admits of several modes of construc-
tion, but the two principal forms are the one adapted
OKITICES, WEIKS, PIPES, AND RIVERS. 385
for high falls and the one for low falls. The former
may be called the High-pressure Vortex, and the
latter the Low-pressure Vortex. Examples of these
two kinds are in operation at two mills near Belfast.
" The height of the fall for the first vortex is about
37 feet, and the standard or medium quantity of water,
for which the dimensions of the various parts of the
wheel and case are calculated, is 540 cubic feet per
minute. With this fall and water-supply the esti-
mated power is 28 horse-power, the efficiency being
taken at 75 per cent. The proper speed of the wheel,
calculated in accordance with its diameter and the
velocity of the water entering its chamber, is 355
revolutions per minute. The diameter of the wheel
is 22* inches, and the extreme diameter of the case is
4 feet 8 inches.
"In the second vortex, the fall being taken at 7
feet, the calculated quantity of water admitted, at the
standard opening of the [guide-blades, is 2,460 cubic
feet per minute. Then, the efficiency of the wheel
being taken at 75 per cent., its power will be 24
horse-power. Also, the speed at which the wheel is
calculated to revolve is 48 revolutions per minute.
"The two examples which have now been described
of vortex water wheels, adapted for very distinct cir-
cumstances, will serve to indicate the principal features
in the structural arrangements of these new machines
in general. Respecting their principles of action some
further explanations will next be given. In these
machines the velocity of the circumference is made
the same as the velocity of the entering water, and
thus there is no impact between the water and the
c c
386 THE DISCHARGE OF WATER FROM
wheel ; but, on the contrary, the water enters the
radiating conduits of the wheel gently, that is to say,
with scarcely any motion in relation to their mouths.
In order to attain the equalization of these velocities,
it is necessary that the circumference of the wheel should
move with the velocity which a heavy body would attain,
in falling through a vertical space equal to half the
vertical fall of the water, or in other words, with the
velocity due to half the fall ; and that the orifices
through which the water is injected into the wheel-
chamber should be conjointly of such area that when
all the water required is flowing through them, it also
may have a velocity due to half the fall. Thus one-half
only of the fall is employed in producing velocity in
the water ; and, therefore, the other half still remains
acting on the water within the wheel-chamber at the
circumference of the wheel, in the condition of fluid
pressure. Now, with the velocity already assigned
to the wheel, it is found that this fluid pressure is
exactly that which is requisite to overcome the cen-
trifugal force of the water in the wheel, and to bring
the water to a state of rest at its exit ; the mechanical
work due to both halves of the fall being transferred
to the wheel during the combined action of the moving
water and the moving wheel. In the foregoing state-
ments, the effects of fluid friction, and of some other
modifying influences, are, for simplicity, left out of
consideration ; but in the practical application of the
principle, the skill and judgment of the designer
must be exercised in taking all such elements, as far
as possible, into account. To aid in this, some prac-
tical rules, to which the author as yet closely adheres,
ORIFICES, WEIRS, PIPES, AND RIVERS. 387
were made out by him previously to the date of his
patent. These are to be found in the specification of
the patent, published in the Mechanics' Magazine for
January 18 and January 25, 1851 (London).
" In respect to the numerous modifications of con-
struction and arrangement which are admissible in
the vortex, while the leading principles of action are
retained, it may be sufficient here merely to advert,
first, to the use of straight instead of curved radiating
passages in the wheel ; secondly, to the employment,
for simplicity, of invariable entrance orifices, or of
fixed instead of moveable guide-blades ; and lastly, to
the placing of the wheel at any height, less than about
thirty feet, above the water in the tail-race, combined
with the employment of suction pipes descending
from the central discharge orifices, and terminating
in the water of the tail-race, so as to render available
the part of the fall below the wheel.
" In relation to the action of turbines in general,
the chief and most commonly recognized conditions,
of which the accomplishment is to be aimed at, are
that the water should flow through the whole machine
with the least possible resistance, and that it should
enter the moving wheel without shock, and be dis-
charged from it with only a very inconsiderable
velocity. The vortex is in a remarkable degree
adapted for the fulfilment of these conditions. The
water moving centripetally (instead of centrifugally,
which is more usual in turbines), enters at the period
of its greatest velocity (that is, just after passing the
injection orifices) into the most rapidly moving part
of the wheel, the circumference ; and, at the period
cc3
388 THE DISCHARGE OF WATER FROM
when it ought to be as far as possible deprived of
velocity, it passes away by the central part of the
wheel, the part which has the least motion. Thus, in
each case, that of the entrance and that of the dis-
charge, there is an accordance between the velocities
of the moving mechanism and the proper velocities of
the water.
" The principle of injection from without inwards,
adopted in the vortex, affords another important
advantage in comparison with turbines having the
contrary motion of the water; as it allows ample
room, in the space outside of the wheel, for large and
well-formed injection channels, in which the water
can be made very gradually and regularly to converge
to the most contracted parts, where it is to have its
greatest velocity. It is as a concomitant also of the
same principle, that the very simple and advantageous
mode of regulating the power of the wheel, by the
jnoveable guide-blades already described, can be intro-
duced. This mode, it is to be observed; while giving
great variation to the areas of the entrance orifices^
retains at all times very suitable forms for the con-
verging water channels.
u Another adaptation in the vortex is to be remarked
as being highly beneficial, that, namely, according to
which, by the balancing of the contrary fluid pressures
due to half the head of water and to the centrifugal
force of the water in the wheel, combined with the
pressure due to the ejection of the water backwards
from the inner ends of the vanes of the wheel when
.they are curved, only one-half of the work due to the
fall is spent in communicating vis viva to the watery
ORIFICES, WEIRS, PIPES, AND RIVERS. 389
to be afterwards taken from it during its passage
through the wheel ; the remainder of the work being
communicated through the fluid pressure to the wheel,
without any intermediate generation of vis viva.
Thus the velocity of the water, where it moves fastest
in the machine, is kept comparatively low ; not ex-
ceeding that due to half the height of the fall, while
in other turbines the water usually requires to act at
much higher velocities. In many of them it attains
at two successive times the velocity due to the whole
fall. The much smaller amount of action, or agita-
tion, with which the water in the vortex performs
its work, causes a material saving of power by
diminishing the loss necessarily occasioned by fluid
friction.
" In the vortex, further, a very favourable influence
on the regularity of the motion proceeds from the
centrifugal force of the water, which, on any increase
of the velocity of the wheel, increases, and so checks
the water supply ; and on any diminution of the
velocity of the wheel, diminishes, and so admits the
water more freely; thus counteracting, in a great
degree, the irregularities of speed arising from varia-
tions in the work to be performed. When the work
is subject to great variations, as for instance in saw-
mills, in bleaching works, or in forges, great incon-
venience often arises with the ordinary bucket water-
wheels and with turbines which discharge at the
circumference, from their running too quickly when
any considerable diminution occurs in the resistance
to their motion.
"The first vortex which was constructed on the
390 THE DISCHARGE OF WATER FROM
large scale was made in Glasgow, to drive a new
beetling-mill of Messrs. C. Hunter and Co., of
Dunadry, in County Antrim. It was the only one in
action at the time of the meeting of the British
Association in Belfast ; but the two which have been
particularly described in the present article, and one for
an unusually high fall, 100 feet, have since been com-
pleted, and brought into operation. There are also
several others in progress ; of which it may be suffi-
cient to particularize one of great dimensions and
power, for a new flax-mill at Ballyshannon in the
West of Ireland. It is calculated for working at 150
horse-power, on a faU of 14 feet, and it is to be
impelled by the water of the Eiver Erne. This great
river has an ample reservoir in the Lough of the
same name ; so that the water of wet weather is long
retained, and continues to supply the river abun-
dantly even in the dryest weather. The lake has
also the effect of causing the floods to be of long
duration, and the vortex will consequently be,
through a considerable part of the year, and for long
periods at a time, deeply submerged under back-
water. The water of the tail-race will frequently be
seven feet above its ordinary summer level ; but as
the water of the head-race will also rise to such a
height as to maintain a sufficient difference of levels,
the action of the wheel will not be deranged or
impeded by the floods. These circumstances have
had a material influence in leading to the adoption in
the present case of this new wheel in preference to
the old breast or undershot wheels."
The next tables have been arranged by us from
ORIFICES, WEIRS, PIPES, AND RIVERS. 391
Mr. Francis' valuable experiments. They show the
ratio of the effect to the power in two wheels, the
first a centre-vent wheel, erected at the Boott Cotton
Mills, and the second a turbine, erected at the
Tremont Mills, Lowell, Massachusetts.
The maximum effect -794 was obtained from the
Tremont turbine experiments, when the velocity of
the interior circumference of the wheel was to that
due to the whole fall as -63 to 1 ; and an effect of 78
per cent, was obtained when these velocities were as
51 to 1. In the Boott centre-vent wheel the max-
imum effect -797 was obtained when the velocity of
the exterior circumference was to that due to the fall
as *64 to 1 ; and a like effect was produced when
this ratio was -708 to 1. Indeed, between these
ratios the useful effect was nearly the same; an
effect of *78 to '79 was obtained for all such ratios
between limits of *59 and -71 to 1, averaging a ratio
of -65 to 1. If a turbine have a variable fall, say
from 2 to 1, and be of sufficient capacity to give the
required power always, the dimensions should be
determined from the lesser fall, and if correctly so
determined, it will not have sufficient velocity for
the greater fall. When the fall is greatest the
quantity in the same place is generally least, giving
thereby a lessened effect when most is required. For
such cases two turbines may be used with advantage.
392
THE DISCHARGE OF WATER FROM
TABLE showing the Results of Experiments upon a Model of a Centre-vent Water Wheel, and
also upon a Centre-vent Water Wheel at the Boott Cotton Mills, Lowell, Massachusetts,
arranged from Mr. Francis 1 valuable Experiments. Diameter of Wheel to the outside of
the Buckets, about 9'3 feet. Depths of External Guide Curves about -75 foot. External
height of Wheel about 1-5 feet. Number of Buckets, 40. Mean height of the Orifices be-
tween the Guides, I foot. Diameter of Supply Pipe, 8 feet. The first Seven Experiments
were made on a Model, the Exterior Diameter of the Wheel being 22| inches, Interior
Diameter 19 inches, height between the Crowns 2^| inches, and the number of Buckets 36.
The Construction of the Wheel is shown in Mr. Francis' Book, and all necessary Details,
lent Wheel of Francis' and Thomson's Vortex Wheel
Blades being fewer in the latter, and capable of
The general principle of the Centre-vent Wheel of Francis' and Thomson's Vortex Wheel
appears to be the same; the Guide Bla
adjustment.
(lumbers of the 1
Experiments, j
Falls acting on 1
the Wheels.
i Depth on the 1
Weirs in feet. 1
Cubic feet of
Water acting on 1
the Wheel per 1
second.
Number of
pounds avoir- 1
dupois, if raised 3
one foot per
second.
Revolutions of 1
the Wheel per 1
second.
Weight in the I
scale in pounds I
avoirdupois. 1
Ratios of the ef-
fect to the power,
calculated by
means of Prony's
Brake.
Velocity due to
the fall acting on
the Wheel, in
feet per second.
Velocity of the
outside circum-
ference of the
"Wheel, in feet
per second.
1
2-52
365
2-15
337-7
1-14
16-
679
12-73
6-83
2
2-46
366
2-16
331-5
1-34
14-
711
12-58
8-03
3
2-50
367
2-17
338-0
1-54
12-5
716
12-68
9-23
4
2-60
372
2-21
358-2
1-70
12-
705
12-93
10-03
5
2-60
373
2-22
361-3
1-73
11-5
692
12-95
10-36
6
2-60
373
2-22
559-6
1-71
11-5
689
12-93
10-26
7
2-60
374
2-23
360-8
1-90
10-0
648
12-93
11-15
8
14-60
1-296
67-53
61493-4
1-48
390-9
377
30-65
17-39
9
14-67
1-262
64-89
59364-4
1-26
775-6
203
30-72
25-77
10
14-57
1-282
66-43
60347-0
1-14
963-3
332
30-61
21-21
11
14-16
1-284
66-61
58821-6
1-08
1069-
382
30-18
15-97
12
14-20
1-290
67-03
59351-7
1-02
1150-8
381
30-22
14-65
13
14-14
1-288
66-89
59002-2
96
1243-0
373
30-16
13-18
14
14-24
1-294
67-37
59858-1
93
1293-6
269
30-27
7-46
15
14-gO
1-211
61-08
54486-4
90
1345-5
. .
30-33
29-75
16
14-29
1-514
85-
75732-8
86
1396-1
303
30-32
29-63
17
14-23
1-531
86-35
76608-0
83
1444-0
376
30-25
28-16
18
14-20
1-539
87-08
77093-2
80
1494-7
413
30-22
27-35
19
14-19
1-547
87'68
77607-2
77
1548-7
444
30-21
26-43
20
14-19
1-554
88'28
78143 8
71
1657-0
489
30-22
25-12
21
13-78
1-576
90'17
77480-4
1-55
582
29-77
20-37
22
13-61
1-594
9170
77812-1
1-35
316-0
596
29-58
17-54
23
13-94
1-418
77-11
67076-7
1-22
5197
29-9
34-51
24
13-52
1-642
95-76
80736-3
1-10
720-2
245
29-49
32-44
25
13-37
1*673
98-49
82145-2
1-00
832'3
437
29-33
29-14
26
13-37
1-695
100-42
83728-2
92
934-7
576
2932
26-43
27
13-40
1-718
102-42
85571-4
84
1033-3
657
29-35
24-20
28
13-38
1-723
102-82
85800-0
77
1115-0
690
29-34
23-07
29
13-34
1-731
103-52
86138'0
77
1115-0
710
29 -30
21-89
30
13-32
1-734
103-77
86218'S
69
1204-8
720
29-27
21-26
31
13-33
1-733
103-70
86229-3
62
1278-0
728
29-29
20-86
32
13-30
1-739
104-23
86483*7
46
1482-6
731
29-25
20-50
33
13-70
1-598
92*02
78648-0
85
1482-6
29-70
3678
34
13-40
1-832
112-52
94057-5
82
1544-9
797
29-36
20-81
35
13-43
1-837
112-99
94662-2
79
1604-8
796
29-39
20-51
36
13-33
1-832
112-56
93603-9
1-25
..
797
29-28
19-93
37
13-38
1-837
113-00
94296-4
1-15
118-6
797
29-33
19-71
88
13-39
1-838
113-07
94415-2
96
325-4
796
29-34
19-36
39
13-38
1-839
113-16
94471-1
77
519-9
796
29-34
19-03
40
13-36
1-838
113-09
94219-0
67
612-2
797
29-31
18-67
41
13-38
L-844
113-67
94881-9
56
704-4
791
29-34
18-32
42
13-40
1-851
11429
95571-2
46
777-6
787
29-36
18-00
43
1332
1-848
113-97
94703-1
30
882-1
781
29-27
17-38
44
13-54
1-809
110-45
93270-1
62
118-6
29-51
45
13-57
1-807
110-32
93422-4
69
73-1
29-55
46
13-60
1-088
99-79
84635-0
39
296-4
29-57
37-70
ORIFICES, WEIRS, PIPES, AND EIVERS.
393
TABLE showing the Results of Experiments upon the Turbine at Tremont Mills, Lowell,
Massachusetts, arranged from Mr. Francis' valuable Experiments. Diameter, measured
to the Exterior Circumference of Crowns of the Wheel, 8'333 feet. Height of Buckets
from top of the Disc to the bottom of the Garniture, '97 feet. Number of Buckets, 44.
Width, of the Buckets '8 foot, nearly. Width of Guide Curves, fr'dfect, nearly. Number of
Ditto, 33. A Double Weir with 4 end constructions and 16'98 feet long, used for gauging
the Water the Crest being &5feet above the floor of the Wheel Pit. The Falls show the
difference of heads in the Forebay and Wheel Pit. For further details, see Francis'
Lowell Hydraulic Experiments, pp. 1 to 43. Tlie supply pipe is fully a, quadrant, and
varies from 6 to 9 feet in diameter.
1 Numbers of the 1
Experiments. I
Falls actinp: on I
the Wheel.
Depths on the I
Weir in feet. 1
Cubic feet of
Water acting on
the Wheel per
second.
Number of pounds!
avoirdupois, if 1
raised one foot perl
second.
Number of revolu-S
tions of the Wheelj
per second.
1 1
ill
C o "p
pl
*
Ratio of the effect!
to the power, cal- 1
dilated by meansl
ofProny's brake!
or dynamometer. 1
Velocity due to 1
the fall acting on 1
the Wheel, in feet I
per second.
Velocity of the in-
terior circumfer-
ence of the Wheel,
in feet per second.
1
12-864
1-88
139-42
111870-0
894
1443-34
784
28-76
18-95
2
12-869
1-88
139-47
111951-2
892
1443-34
784
28-77
18-96
3
12-611
2-02
154-40
121444-2
893
1443-34
353
28-48
32-49
4
12-696
1-97
149-46
118363-5
1-596
307-03
507
28-58
29-32
5
12-777
1-94
146-02
116373-2
1-532
411-48
622
28-67
26-40
6
12-819
1-92
143-91
115067-3
1-461
519-77
703
28-71
23-86
7
12-856
1-91
142-52
114284-2
1-382
638-36
735
28-76
22-63
8
12-888
1-90
142-04
114187-1
1-309
750-42
750
28-79
21-71
9
12-896
1-90
141-28
113640-9
1-245
854-87
766
28-80
20-57
10
12-883
1-89
140-08
112568-7
1-184
957-35
779
28-79
19-57
11
12-899
1-88
139-90
112563-3
1-125
1057-49
781
28-80
19-13
12
12-905
1-88
139-01
111893-5
1-784
789
28-81
18-92
13
12-899
1-88
139-03
111859-4
1-784
788
28-80
18-77
14
12-902
1-87
138-85
111740-3
1-067
1156*27
790
28-81
18-66
15
12-906
1-87
138-51
111504-9
1-02J:
1229-41
792
28-81
18-37
16
12-915
1-87
138-27
111384-0
999
1269-42
794
28-82
17-73
17
12-934
1-87
138-23
111521-1
970
1319-22
792
28-84
17-25
18
12-939
1-86
137-71
111139-7
945
1359-23
783
28-85
15-74
19
12-940
1-84
135-14
109077-1
923
1397-12
770
28-85
13-69
20
12-963
1-84
135-34
109433-4
911
1416-70
769
28-88
13-72
21
12-977
1-83
133-75
108265-8
902
1433-43
725
28-89
11-29
22
12-948
1-S2
133-43
107764-7
897
1443-06
679
28-86
9-63
23
12-954
1-87
138-62
112009-3
1-781
791
28-86
17-64
24
12-932
1-87
138-50
111720-6
892
1454-24
791
28-84
17-32
25
12-951
1-87
138-37
111777-2
885
1464-80
788
28-86
16-74
26
12-758
1-92
143-33
114060-7
875
1474-37
346
28-65
31-45
27
12-909
1-86
137-75
110917-6
880
1474-37
675
28-82
24-21
28
12-950
1-86
137-00
110664-7
873
1485-63
725
28-86
21-73
29
12-965
1-84
135-10
109252-2
866
1498-66
750
28-88
19-74
30
12-999
1-82
133-30
108082-5
851
1524-67
760
28-92
18-35
31
13-026
1-81
131-99
107246-3
1-790
763
38-95
17-05
32
13-028
1-80
130-89
106366-6
836
1552'-44
750
28-95
15-01
33
13-077
1-68
118-55
96699-5
813
1597-08
300
29-00
28-66
34
13-134
1-66
116-10
95112-0
784
1648-87
453
29-06
25-84
35
13-215
1-63
113-24
93346-0
742
1724-49
609
29-15
21-28
36
13-282
1-59
109-71
90893-3
695
1816-71
652
29-23
17-88
37
13-310
1-58
107-95
89620-7
646
1911-45
656
29-26
16-44
38
13-362
1-54
103-85
86555-6
1-786
;;
619
29-32
13-06
39
12-883
1-86
137-36
110380-8
647
1911-45
.-781
28-79
18-12
40
12-896
1-86
136-97
110176-6
600
2011-52
784
28-80
17-42
41
12-912
1-85
136-55
109973-0
532
2167-38
783
28-82
16-73
42
13-369
1-27
78-84
65746-8
454
2367-88
141
29-32
24-30
43
13-395
1-25
76-62
64018-1
4213-38
,332
29-35
20-36
44
13-435
1-22
74-05
62062-0
3946-38
440
29-40
16-37
45
13-478
1-19
71-87
60424-6
1-783
.
463
29-44
14-25
46
13-513
1-17
70-01
59006-4
832
156V-21
455
29-48
11-87
47
13-559
1-11
64-50
54554-9
817
1590-50
330
29-53
6-36
48
13-985
' -78
38-22
33340-0
803
1614-79
-150
29-99
13-14
49
14-001
78'
38-57
33683v
789
1641-34
102
30-01
14-61
50
14-020
76 :
37-17
32508-Q
769
1679-62
240
30-03
8-32
394
THE DISCHAKGE OF WATER FROM
TABLE for Turbines of different Diameters, modified from Francis,
operating with different Falls; assuming the useful effect is
seventy-five per cent, of the power expended, that the Velocity of the
Interior Circumference is fifty-six per cent, of the Velocity due
to the Fall ; and that also, the Height between the Crowns is one-
tenth of the Outside Diameter.
Outside diameter 2 ft.
Outside diameter 3 ft.
Outside diameter 4 ft.
3utside diameter 5 ft.
Inside 1-56
Number of buckets 36.
Inside ,. 2-38
Number of buckets 39.
tnslde 3-24 ,.
Number of buckets 42.
Inside 4-11
Number of buckets 45.
1
11
jj
1
||
i
1
ft
j.
ft
11
^
fe
P.
i
Hi
i
1!
|1|
i
oj
ll
||,3
1
i!
hi
1
li
*j g
-23*
I
I!
fll
|g n
1
33
o a
m
III
S
|
P
W
1
li
o n
1*
B
)
a
II
B
s
K
B
PH
5
4-5
1-9
123
10-1
4-3
80
17-88
7-6
59
27-9
11-9
47
6
4-9
2-5
135
11-0
5-6
88
19-6
10-0
65
30-6
15-6
51
7
5-3
3-1
146
11-9
7-1
95
21-17
12-6
70
33-1
19-7
55
8
5-7
3-8
156
12-7
8-7
102
22-63
15-4
75
35-3
24-0
59
9
6-0
4-6
165
13-5
10-3
108
24-00
18-4
79
37-5
28-7
63
10
6-3
5-4
174
14-2
12-1
114
25-30
21-5
84
39-5
33-6
66
11
6-6
6-2
183
14-9
13-9
119
26-53
24-8
88
41-5
38-8
69
12
6-9
7-1
191
15-6
15-9
125
27-71
28-3
92
43-3
44-2
72
13
7-2
8-0
199
16-2
17-9
130
28-84
31-9
95
45-1
49-8
75
14
7-5
8-9
206
168
20-0
135
29-93
35-6
99
46-8
55-7
78
15
7-7
9-9
213
17-4
22-2
139
30-98
39-5
103
48-4
61-7
81
16
8-0
10-9
220
18-0
24-5
144
32-00
43-5
106
50-0
68-0
83
17
8-2
11-9
227
18-5
26-8
148
32-99
47-7
109
51-5
74-5
86
18
8-5
13-0
234
19-1
29-2
153
33-94
51-9
112
53-0
81-1
88
19
8-7
14-1
240
19-6
31-7
157
34-87
56-3
115
54-5
88-0
91
20
8-9
15-2
247
20-1
34-2
161
35-78
60-8
118
55-9
95-0
93
21
9-2
16-4
253
20-6
36-8
165
36-66
65-4
121
57-3
102-2
96
22
9-4
17-5
259
21-1
39-5
169
37-52
70-2
124
58-6
109-6
98
23
9-6
18-7
264
21-6
42-2
172
38-37
75-0
127
59-9
117-2
100
24
9-8
20-0
270
220
45-0
176
39-19
79-9
130
61-2
124-9
102
25
10-0
21-2
276
22-5
47-8
180
40-00
85-0
132
62-5
132-8
104
26
10-2
22-5
281
22-9
50-7
183
40-79
90-1
135
63-7
140-9
106
27
10-4
23-8
286
23-4
53-7
187
41-57
95-4
138
65-0
149-1
108
28
10-6
25-2
292
23-8
56-7
190
42-33
100-7
140
66-1
157-4
110
29
10-8
26-5
297
24-2
59-7
194
43-08
106-2
143
67-3
165-9
112
30
10-9
27-9
302
24-6
62-8
197
43-82
111-7
145
68-5
174-6
114
31
11-1
29-3
307
25-0
66-0
200
44-54
117-4
147
89-6
183-4
116
32
11-3
30-8
312
25-5
69-2
203
45-25
123-1
150
70-7
192-3
118
33
11-5
32-2
317
25-8
72-5
207
45-96
128-9
152
71-8
201-4
120
34
11-7
33-7
321
26-2
75-8
210
46-65
134-8
154
72-9
210-6
122
35
11-8
35-2
326
26-6
79-2
213
47-33
140-8
157
73-9
220-0
123
36
12-0
36-7
331
27-0
82-6
216
48-00
146-9
159
75-0
229-5
125
37
12-2
38-3
335
27-4
86-1
219
48-66
153-0
161
76-0
239-1
127
38
12-3
39-8
340
27-7
89-6
222
49-32
159-3
163
77-0
248-9
129
39
12-5
41-4
344
28-1
93-2
225
49-96
165-6
165
78-1
258-8
130
40
12-6
43-0
349
28-5
96-8
227
50-60
172-0
167
79-1
268-8
132
ORIFICES, WEIRS, PIPES, AND EIVEES.
395
TABLE of Turbines of different Diameters, modified from Francis,
operating with different Falls; assuming the useful effect is
seventy-Jive per cent, of the power expended, that the Velocity of
the Interior Circumference is fifty-six per cent, of the Velocity due
to the Fall, and that also, the Height between the Crowns is one-
tenth of the Outside Diameter.
Outside diameter 6 ft.
Outside diameter 7 ft.
Outside diameter 8 ft.
Outside diameter 10 ft.
Inside 5
Number of buckets 48.
Inside 5-90 ,,
Number of buckets 51.
Inside 6-81
Number of buckets 54.
Inside 8-67 ,.
Number of buckets 60.
I
||
O
1
IK
i
1
fj
*3
1
Is.
j
1
1
||l
1
T
Jf
llH
lal
o
r
V
K V
53
l|l
p
II
^3
|
||
|f
w
fs
If
1
I
"3 a
Water
in cub)
se
1
11
1
m
I
1
5
40-2
17-1
38
54-8
23-3
32-5
71-5
30-4
28-1
111-8
47-5
22-1
6
44-1
22-5
42
60-0
30-6
35-6
78-4
40-0
30-8
122-5
62-5
24-2
7
47-6
28-3
45
64-8
38-6
38-4
84-7
50-4
33-3
132-3
78-7
26-2
8
50-9
34-6
48
69-3
47-1
41-1
90-5
61-5
35-6
141-4
96-2
28-0
9
54-0
41-3
51
73-5
56-2
43-6
96-0
73-4
37-8
150-0
114-7
29-7
10
56-9
48-4
54
77-5
65-9
46-0
101-2
86-0
39-8
158-1
134-4
31-3
11
59-7
55-8
57
81-3
76-0
48-2
106-1
99-2
41-7
165-8
1550
32-8
12
62-4
63-6
59
84-9
86-6
50-3
110-8
113-1
43-6
173-2
176-7
34-3
13
64-9
71-7
62
88-3
97-6
52-4
115-4
127-5
45-4
180-3
199-2
35-7
14
67-3
80-1
64
91-7
109-1
54-4
119-7J142-5
47-1
187-1
222-6
37-0
15
69-7
88-9
66
94-9
121-0
56-3
123-9 158-0
48-7
193-6
246-9
38-3
16
72-0
97-9
69
98-0
133-3
58-1
128-0 174-1
50-3
200-0
272-0
39-6
17
74*2
107-2
71
101-0
146-0
59-9
131-9190-6
51-9
206-2
297-9
40-8
18
76-4
116-8
73
103-9
159-0
61-7
135-8207-7
53-4
212-1
324-6
42-0
19
78-5
126-7
75
106-8
172-5
63-3
139-5225-3
54-9
217-9
352-0
43-1
20
80-5
136-8
77
109-6
186-3
65-D
143-1 243-3
56-3
223-6
380-1
44-3
21
82-5
147-2
79
112-3
200-4
66-6
146-6261-7 57-7
229-1
4090
45-4
22
84-4
157-9
80
114-9
214-9
68*2
150-1:280-7
59-0
234-5
438-5
46-4
23
86-3
168*8
82
117-5
229-7
69-7
153-5 300*0
60-4
239-8
468-8
47-5
24
88-2
179*9
84
120-0
244-8
71-2
156-8 319*8
61-7
244-9
499-7
48-5
25
90-0
191*2
86
122'5 260-3
72-7
160'0 340'0
62-9
250-0
531 '2
49-5
26
91-8
202-8
87
124-9
276-1
74-1
163'2 360'6
64-2
254-9
563*4
50-5
27
93-5
214-6
89
127-3
292-2
75*5
166*3
381-6
65-4
259-8
596*3
51-4
28
95-2
226-7
91
129-6
308-5
76'9
169*3
403*0
66-6
264-6
629*7
52-4
29
96-9
238-9
92
131-9
325-2
78*3
172-3
424-8
67-8
269-3
663*7
53-3
30
98-6
251-4
94
134-2
342-2
79-6
175-3
446-9
68-9
273-9
698*3
54'2
31
100-2
264-1
95
136-4
359-4
80-9
178-2
469-5
70-1
278-4
733'5
55-1
32
101-8
277-0
97
138-6
377-0
82-2
181-0
492-4
71-2
282-8
769*3
56-0
33
103-4
290-0
98
140-7
394-8
83-5
183-8
515-6
72-3
287-2
805*7
56-9
34
105-0
303-3
100
142-9
412-9
84-7
186-6
539-2
73-4
291-5
842'6
57-7
35
106-5
316-8
101
144-9 431-2
86-0
189*3
563-2
74-5
295-8
880*0
58-5
36
108-0
330-5
103
147-0
449-8
87-2
192-0
587-5
75-5
300-0
918'0
59-4
37
109-5
344-3
104
149-0
468-7
88-4
194-6
612-2
766
304-1
956'5
60-2
38
39
111-0
112-4
358-4
372-6
106
107
151-0487-8
153-OJ507-2
89-6
90-8
197-3
199-8
637-1
662-5
77-6
78-6
308-2
312-2
995-5
1035-1
61-0
61.8
40
113-8
387-1
108
154-9
526-8
91-9
202-4
688-1
79-6
316-2
1075-2
62-6
396 ,THE DISCHARGE OF WATER FROM
THE HYDRAULIC RAM has been applied with advan-
tage in raising water to a considerable height by the
momentum of a larger quantity at a lower level.
The shock of the valves, and vibration of the ma-
chine, require heavy and strong setting, and con-
siderable strength in all the parts. This limits its
application, and prevents its use for raising large
quantities of water. The work done by the ram, in
over one thousand experiments by Eytelwein, did
not exceed in any of them 1480 Ibs. raised one
foot in one minute ; and in France, the ram put
up by the younger Montgolfier, said to be the
largest constructed, raised only 7400 Ibs. one foot
high per minute, and had a useful effect, it is re-
ported, of -65. This ram was put up at Mello, near
Clermont-sur-Oise. Its dimensions were
Length of the body pipe or injection pipe 108 feet.
Diameter . ; ... . 4-3 inches.
Weight of body pipe . . v . 3190 Ibs.
Weight of head . . |'^- '^'V ^ ; 440 Ibs.
Contents of air-chamber . ->v 1 J gallons.
This ram worked under a head of 37 feet, discharg-
ing in use 31i gallons each minute, and raising
3-85 gallons a height of 195 feet.
The largest ram employed by Eytelwein in his ex-
periments had the following dimensions
Length of the body pipe or injection pipe 43 feet 9 inches.
Diameter of ditto . . . .0 feet 2 inches.
Contents of air-chamber . . ; 1-94 gallons.
Area of tail or escape valve . -^ ,' T ' . 3-74 square inches ;
and his experiments led to the following practical
formula by D'Aubuisson
OKIFICES, WEIKS, PIPES, AND EIVEKS.
397
in which D is the water used per minute in gallons,
d the quantity raised in gallons, h the head used,
and h' the lift of the quantity d. By a slight re-
duction we get
dh' = 1-42 D (h - -28 v/AAO
for the effect produced, which is reduced nearly one-
sixth for practical application, giving the formula
dA'=l-2 D (A - -2 v/A"A')
for the work done.
EXPERIMENTAL RESULTS. HYDEAULIC RAM.
umber
of
strokes
minute.
Height in feet
of
Ratio of
leights.
Gallons of water per
minute.
**
Ratio
D
d~
Fall
h
Elevation
h'
h 1
h
Expended
D
Raised
d
Experi-
ments.
Formula.
Ft. Iu.
Ft. In.
66
icy o"
26' 4"
2-63
10-65
3-39
97
2-92
04
10 2
32 4
3-18
13-97
383
873
92
3-67
00
9 11
38 8
3-9
12-01
2-622
85
87
4-58
52
8
32 4
4-
8-16
1-687
847
85
4-72
45
8 9
38 8
4-4
10-85
2-09
845
84
5-2
42
7 5
38 8
5-21
9-92
1-5
787
78
6-62
36
6
38 8
6-5
8-89
1-05
754
71
8-62
26
4 6
32 4
7-2
5-23
495
672
67
10-7
31
5
38 7
7-7
8-05
704
667
65
11-54
23
4 1
38 8
9-47
11-11
649
548
56
17-2
17
3
32 2
10-7
10-8
479
473
51
22-6
15
3 3
38 8
11-9
12-34
363
352
45
33-8
14
2 6
38 8
15-5
11-95
22
284
32
54-6
10
\,
MIJ
38 8
19-3
9-81
088
181
18
106-6
39
THE DISCHARGE OF WATER FROM
Eytelwein recommends, that the length of the
body pipe should not be less than three-fourths of
the height to which the water is to be raised ; its
diameter in inches equal -58 V/D"; the diameter of
the rising pipe -3 \/:D ; and the contents of the air-
chamber equal to that of the rising pipe.
The following table gives the result of experiments
made by Montgolfier and his son :
TABLE OF EXPEEIMENTAL KESULTS HYDEAULIC HAM.
Height.
Water per Minute.
dh
Dh
Mean Batio
dlif
Dh
Fall
h
Elevation
h
Expended
D
Delivered
d
Ft. In.
Ft. In.
Gallons.
Gallons.
8' 6"
52' 8"
15
1-37
57
, .
37 2
195
31
3-85
653
34 9
111 11
18-5
3-74
651
65
3 3
14 11
437
59-18
629
..
22 10
196 10
2-86
0-22
671
'"
Latterly, the Messrs. Easton and Amos have
patented improvements in this machine, and have
raised water to a height of 330 feet. The injection
pipe is laid by them at an inclination of about one
in four for high falls, and varies down to one in
eighteen for smaller falls. The quantities raised in
their practice vary up to six gallons per minute.
WATER PRESSURE ENGINES give a useful effect
varying up to 70 per cent, for the best constructed.
An immense amount of mechanical skill and inven-
tion has been brought to bear on their construction,
ORIFICES, WEIRS, PIPES, AND RIVERS. 399
and in Weisbach's book* a useful effect of 83 per
cent, has been calculated ; this is, however, a result
seldom obtained in practice, where two-thirds, or 66
per cent., is nearer to the general efficiency. Jordan
got a maximum efficiency of '66 from one of the
Clausthal engines, making four strokes per minute,
and -71 making three strokes per minute. These re-
sults were for the combined engine and pumps, from
which it was calculated that the efficiency of the
engine alone, was in the first case -83, and in the
second -85. It would be a great mistake to calculate
on such high efficiencies.
CORN MILLS will grind about a bushel of corn per
horse-power per hour, but much depends on the state
of the stones and of the grain. The value of the work
done in an hour being once known, the value of the
standard horse-power can be determined accordingly.
* Vol. ii., p. 342.
400
THE DISCHARGE OF WATER FROM
TABLE I. Coefficients of Discharge from Square and differently
proportioned Rectangular Lateral Orifices in thin Vertical Plates,
arranged from Poncelet and Lesbros.
,
Square orifice
Rectangular
Rectangular
8" X 8".
orifice 8" X 4".
orifice 8" X 2",
J" p "
*i (
Ratio of the sides
Ratio of the sides
Ratio of the sides
. _--
S
Itol.
2tol.
4 to 1.
|!|i
iff
|J
Jfj
Li
Sjj
1 |
|l ^
1 |
ill
1 1
IM!
i
ill
'rt
i
~
J "o
fi
If
w a
11
w s
o-ooo
619
667
713
0-197
025
597
630
668
0-394
050
595
618
607
642
0-591
075
594
593
615
612
639
0-787
100
572
594
596
614
615
638
1-181
150
578
593
600
613
-620
637
1-575
200
582
593*
603
612
623
636
1969
250
585
593
605
612*
625
636
2-362
300
587
594
607
613
627
635
2-756
350
588
594
609
613
628
635
3-150
400
589
594
610
613
629
635
3-545
450
591
595
610
614
629
634
3-937
500
592
595
611
614
630
634
4-724
600
593
596
612
614
630
633
5-512
700
595
597
613
614
630
632
6-299
800
596
597
614
615
631*
631
7-087
900
597
598
615
615
630
631
7-874
1-000
598
599
615
615
630
630
9-843
1-250
599
600
616
616
630
630
11-811
1-500
600
601
616 -
616
629
629
15-748
2-000
602
602
617
617
628
629
19-685
2-500
603
603
617*
617*
628
628
23-622
3-000
604
604
617
617
627
627
27-560
3-500
604
604
616
616
627
627
31-497
4-000
605
605
616
616
627
627
35-434
4-500
605*
605*
615
615
526
626
39-371
5-000
605
605
615
615
626
626
43-307
5-500
604
604
614
614
625
625
47-245
6-000
604
604
614
614
624
624
51-182
6-500
603
603
613
613
622
622
55-119
7-000
603
603
612
612
621
621
59-056
7-500
602
602
611
611
620
620
62-993
8-000
602
602
611
611
618
618
66-930
8-500
602
602
610
610
617
617
70-867
9-000
601
601
609
609
615
615
74-805
9-500
601
601
608
608
614
614
78-742
10-000
601
601
607
607
613
614
118-112
15-000
601
601
603
603
606
606
See pages 71, 72, and 73.
ORIFICES, WEIRS, PIPES, AND RIVERS.
401
TABLE J. -Coefficients of Discharge from Square and differently
proportioned Rectangular Lateral Orifices in thin Vertical Plates t
arranged from Poncelet and Lesbros.
Rectangular
Rectangular
Rectangular
o ,0
orifice 8" Xl-18".
orifice 8" X 0-8".
orifice 8"X 0-4".
3
j *o
Ratio of the sides
Ratio of the sides
Ratio of the sides
r-j
03 60
7 to 1 nearly.
10 to 1.
20 to 1.
J^ J
S-I W .
si
If
|l
3s
1
sl
Has
a
1 1
Jll
ll'!l
li 1
%'%
1 1
Sell
IF
-I]
W
Heads 1
back fro
orifk
~m~a o
-H ^
*o <o *fl
3$
CS <0
1*3
766
783
795
725
750
705
778
025
0-197
630
687
660
720
701
762
050
0-394
632
674
660
707
697
745
075
0-591
634
668
659
697
694
729
100
0-787
638
659
659
685
688
708
150
1-181
640
654
658
678
683
695
200
1-575
640*
651
658
672
679
686
250
1-969
640
647
657
668
676
681
300
2-362
639
645
656
665
673
677
350
2-756
638
643
656
662
670
675
400
3-150
637
641
655
659
668
672
450
3-543
637
640
654
657
666
669
500
3-937
636
637
653
655
663
665
600
4-724
635
636
651
653
660
661
700
5-512
634
635
650
651
658
659
800
6-299
634
634
649
650
657
657
900
7-087
633
633
648
649
655
656
1-000
7-874
632
632
646
646
653
653
1-250
9'843
632
632
644
644
650
651
1-500
11-811
631
631
642
642
647
647
2-000
15-748
630
630
640
640
644
645
2-500
19-685
630
630
638
638
642
643
3-000
23-622
629
629
637
637
640
640
3-500
27-560
629
629
636
636
637
637
4-000
31-497
628
628
634
634
635
635
4-500
35-434
628
628
633
633
632
632
5-000
39-371
627
627
631
631
629
629
5-500
43-307
626
626
628
628
626
626
6-000
47-245
624
624
625
625
622
622
6-500
51-182
622
622
622
622
618
618
7-000
55-119
620
620
619
619
615
615
7-500
59-056
618
618
617
617
613
613
8-000
62-993
616
616
615
615
612
612
8-500
66-930
615
615
614
614
612
612
9-000
70-867
613
613
612
612
611
611
9-500
74-805
612
612
612
612
611
611
10-000
78-742
608
608
610
610
609
609
15-000
118-112
See pages 71, 72, and 73.
402
THE DISCHARGE OF WATER FROM
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes feet
inch to feet 3|- inches.
1
Coefficients of velocity, and the corresponding velocities of
discharge in inches per second.
r
o ,
.S 2
^ "a jf
"s^ si
^5*5.
"o"*^
^"*3 in
"o
*J
l?|l
jii
ill
111
ill
ill
:j3
k S| 1
^1
^|
**gi
If 1
* B!
5
T-i H | "3
<* *j
rf !
Tj5 5> g
*>J
5 *!
6 '6^
2-78
2-71
2-66
2-39
2-27
2-22
0^
3-48
3-38
3-32
2-99
2-83
278
0^-g.
6-95
6-77
6-64
5-98
5-66
5-56
Oi
9-829
957
9-40
8-45
8-01
7-86
-JL
12038
11-72
11-51
1035
9-81
9-63
Oi
13-900
13-54
1329
11 95
11-33
o o-j?
15541
15-14
14-86
13-36
12-67
12-43
Of
17024
16-58
16-27
1464
1387
13-62
S of
18-388
19-658
17-91
19 15
17-58
1879
15-81
16-91
14-99
16-02
14-71
1573
T 9
20850
20-31
19-93
1793
1699
16-68
Of
21-978
21-41
21-01
18-90
17-91
17-58
o ott
23-051
22-45
22-04
19-82
18-79
1844
o of
24-076
23-45
23-02
20-70
19-62
1926
o olf
25-059
24-41
24-00
21-55
20-4-2
20-05
Oi
26-005
25-33
24-86
22-36
21-19
2080
0^-f
26-917
26-22
25-73
23-15
21-94
21-53
1
27-800
27-08
26-58
23-91
22-66
22-24
1
29-486
28'72
28-19
25-36
24-03
23-59
o u
31-081
30-27
29-71
26-73
2533
24-87
l|
32-598
3175
31-16
28-03
26-57
26-08
1|
34-048
33-19
32-58
29-30
27-75
2726
l|
35-438
3452
33-88
30-48
28-88
28-35
1|
36-776
3582
35 16
31-63
29-97
29-4-2
13
38-067
3708
3639
32-74
31 02
30-45
2 8
39-315
38-29
37-59
33-81
32-04
31-45
2i
40525
39-47
38-74
34-85
33-03
3242
2|
41-700
40*62
3987
35-86
33-99
33-36
2s
42-843
4173
40-96
36-84
34-92
3427
2|
43-956
42-81
42-02
37-80
35-82
3516
2|
45-041
43-87
43-06
38-74
36-71
3603
2f
46-101
44-90
44-07
39-65
3757
36-88
2
47-137
45-90
45-06
40-54
38-42
37-71
3
48-151
46-90
46-03
41-41
39-24
38-52
3$.
49-144
47-87
4698
42-26
40-05
39-32
3|
50-117
4881
47-91
43-10
40-85
40-09
3f
51072
49-74
4882
4392
4162
4086
3|
52-009
50-66
49-72
44-73
4239
41 61
3|
52-930
5155
5060
45-52
43 14
42-34
3f
53-834
52-43
51-47
46-30
43-88
43-07
3|
54725
53-30
52-3-2
47-06
44-60
4378
ORIFICES, WEIRS, PIPES, AND RIVERS.
403
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes feet T g- inch to feet 3| inches.
Coefficients of velocity, and the corresponding velocities of
1
discharge in inches per second.
1
Hi
t*g
*sS
"o^
*ag
**.$
,M
^ ^*
ffi r * fc ^P
re "
f/3 -*
so r* ^
o> 10
** 2
ill
S3
a QO .2
fS
li?l
J oo "a
g <b .2
1 f 1
1
"1
ii
*71
k 7 e
1 1
7 1
3
^ *<
' & O
* & 3
s *!
s * J
9 *J
3
195
1-85
1-75
1-72
1-68
1-62
0/ <Uo
2-43
2-31
2-l
2-15
2-11
2-03
0^
487
4-63
4-36
4-29
4-21
4-06
i
6-88
655
6-17
6-06
5-96
5-74
o o{
8-43
8-02
7-56
7-43
7-29
7-03
03
9-73
9-26
8-73
8-58
8-42
8 12
0|
10-88
10-35
9-76
9-59
9'42
908
3
1192
11-24
10-69
10-50
10-32
994
o ojr
1-2-87
1225
11-55
11-35
11-14
10-74
o oX
13-76
12-97
1234
12 13
11-91
11-48
o o| 6
1460
13-89
1309
12'86
12-64
1218
0_9
15-38
1464
13-80
1356
13*32
12-84
(15^
16-14
15-35
1448
1422
13-97
13-46
oli
16-85
1603
15-12
1485
14-59
14-06
0|
17-54
16-69
15-74
1546
15-19
14-63
o 0*1
18-20
17-32
16-33
16-04
1576
15-09
Oi
18-84
17-93
16-90
16-61
16-31
15-72
o olf
19-46
1851
1746
17-15
16-85
1624
1
20-64
19-64
18-52
18'i9
17-87
17-22
li
21-76
20-70
19-52
19 18
18-84
1815
o 4
22-82
21-71
20-47
20*11
19-75
19-04
If
23-85
22-69
21-38
21-01
20-63
19-88
]
24-81
23-60
22-26
21-87
21-48
20-70
If
25-74
24-49
23-10
22-69
22-29
21-48
If
26-65
25-35
23-91
23-49
23-07
22-23
If
2752
26-18
24-69
24-26
23-82
22-96
2
28-37
2699
2545
25-00
24-50
23-67
2|
29-19
27-77
26-19
25-73
25-27
24-35
2|
29-99
28-53
26-91
2643
25-96
25-02
2i
30-77
29-27
27-60
27-12
26-64
25-67
2
3153
3000
28-29
2779
27-29
26-30
2|
32-27
30-70
28-95
28-44
27-94
26-92
2|
3300
31-39
29-60
29-08
28-57
27-53
2|
3371
32-07
30-24
29-71
29 18
28-12
3
34-40
32-73
3086
30-32
29-78
28-70
3|
3508
33-38
31-47
30-92
30-37
29-27
3}
35-75
34-01
32-07
31-51
30-95
29-83
3|
3641
34-64
32-66
32-09
31-52
30-37
3j
37-05
35-25
33-24
32-66
32-08
30-91
85
3768
35-85
3381
3322
32-62
31-44
3f
38-31
36-45
34-37
33-77
33-16
31-96
3|
DD3
404
THE DISCHARGE OF WATEE FROM
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes feet 4 inches to 1 foot.
T3
Coefficients of velocity, and the corresponding velocities of
"8
discharge in inches per second.
1 i
sill
r^
r .|
TO 00 *
i*i
-o-s |
-^
s^^ "3
o "S
,2 > "S
5 S 9
to "S
c5 "3
J
?||'.a
^ 85
S -1
a 05 .2
f> cq jS
gal
1
e ^ Q
-'ij
-i|
1 1
ii|
-i|
6//
4
55-600
54-15
53-15
47-82
45-31
44-48
4ft
56-462
54-99
53-98
48-56
46-02
45-17
4j
57-311
55-82
54-79
49-29
46-71
45-85
4f
58-148
56-64
55-59
50-01
47-39
46-52
4J
58-973
57-44
56-38
50-72
48-06
47-18
4f
59-786
58-23
57-16
51-42
48-73
47-83
4f
60-589
59-01
57-92
52-11
49-38
48-47
4|-
61-368
59-77
58-67
52-78
50-02
49-09
5
62-163
60-55
69*43
53-46
50-66
49-73
5J
62-935
61-30
60-17
54-12
51-29
50-35
51
63-698
62-04
60-90
54-78
51-91
50-96
5|
64-452
62-78
61-62
55-43
52-53
51-56
5
65-197
63-50
62-33
56-07
53-14
52-16
5f
65-933
64-22
63-03
56-70
53-74
52-75
5f
66-662
64-93
63-73
57-33
54-33
53-33
5
67-383
65-63
64-42
57-95
54-92
53-91
6
68-096
66-33
65-10
58-56
55-50
54-48
61
69-500
67-69
66-44
59-77
56-64
55-60
6|
70-876
69-03
67-76
60-95
57-24
56-70
6f
72-227
70-35
69-05
62-11
58-86
57-78
7
73-552
71-64
70-32
63-25
59-95
58-84
7i
74-854
72-91
71-56
64-37
61-01
59-88
7$
76-133
74-15
72-78
65-47
62-05
60-91
7|
77-392
75-38
73-99
66-56
63-07
61-91
0> 8
78-630
76-59
75-17
67-62
64-08
62-90
8J
79-849
77-77
76-34
68-67
65-08
63-88
8
81-050
78-94
77-48
69-70
66-06
64-84
8|
82-234
80-10
78-62
70-72
67-02
65-79
9
83-40
81-23
79-73
71-72
67-97
66-72
9
84-550
82-35
80-83
72-71
68-91
67-64
9J
85*685
83-46
81-92
73-69
69-83
68-55
Of
86-805
84-55
82-99
74-65
70-75
69-44
10
87-911
85-63
84-04
75-60
71-65
70-33
10i
89-004
86-69
85-09
76-54
72-54
71-20
10
90-082
87-74
86-12
77-47
73-42
72-07
lOf
91-148
88-79
87-14
78-39
74-29
72-92
11
92-202
89-80
88-15
79-29
75-14
73-76
111
93-244
90-82
89-14
80-19
75-99
74-59
Hi
94-274
91-82
90-13
81-08
76-83
75-42
11|
95-294
92-82
91.10
81-95
77-66
76-23
1
96-302
93-80
92-06
82-82
78-49
77-04
ORIFICES, WEIRS, PIPES, AND RIVERS.
405
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes feet 4 inches to 1 foot.
Coefficients of velocity, and the corresponding velocities of
ti
discharge in inches per second.
' |
S~
w .g
**|
H*
**
*ftg
J J
1 S 13
Hi
|I|
Hj
||j
I1J
jl
-1*1
ij
OS J
SfjJ
rfi|
**j
1
38-92
37-03
34-92
34-31
33-69
32-47
6 I
39-52
37-60
35-46
34-84
34-22
32-97
4*
40-12
38-17
35-99
35-36
34-73
33-47
4J-
40-70
38-73
36-52
35-88
35-24
33-96
4f
41-28
39-28
37-03
36-39
35-74
34-44
4
41-85
39-82
37-55
36-89
36-23
34-92
4f
42-41
40-35
38-05
37-38
36-72
35-38
4|
42-96
40-87
38-54
37-86
37-19
35-84
4|
43-51
41-40
39-04
38-35
37-67
36-30
5
44-05
41-91
39-52
38-83
38-14
36-75
6k
44-59
42-42
40-00
39-30
38-60
37-20
5i
45-12
42-92
40-48
39-77
39-06
37-64
51
45-64
43-42
40-94
40-23
39-51
38-07
5|
46-15
43-91
41-41
40-68
39-96
38-51
5f
46-66
44-40
41-86
41-13
40-40
38-93
5|
47-17
44-88
42-32
41-58
40-83
39-35
5
47-67
45-35
42-76
42-02
41-27
39-77
6
48-65
46-29
43-65
42-88
42-12
40-59
6J
49-61
47-20
44-51
43-73
42-95
41-39
6
50-56
48-10
45-36
44-56
43-77
42-18
6|
51-49
48-99
46-19
45-38
44-57
42-95
7
52-40
49-85
47-01
46-18
45-36
43-71
7J
^3-29
50-70
47-81
46-97
46-14
44-46
7f
54-17
51-54
48-60
47-75
46-90
45-20
7f
55-04
52-37
49-38
48-51
47.65
45-92
8
55-89
53-18
50-15
49-27
48-39
46-63
8J
56-74
53-98
50-90
50-01
49-12
47-33
8|
57-56
54-77
51-64
50-74
49-83
48-02
8f
58-38
55-54
52-38
51-46
50-54
48-71
9
59-19
56-31
53-10
52-17
51-24
49-38
9|
59-98
57-07
53-81
52-87
51-93
50-04
9|
60-76
57-81
54-51
53-56
52-60
50-69
9|
61-54
58-55
55-22
54-24
53-27
51-34
10
62-30
59-28
55-89
54-92
53-94
51-98
10
63-06
60-00
56-57
55-58
54-59
52-61
10J
63-80
60-70
57-24
56-24
55-24
53-23
10f
64-54
61-41
57-90
56-89
55-87
53-85
11
65-27
62-10
58-56
57-53
56-51
54-45
Hi
65-99
62-79
59-70
58-17
57-13
55-06
HJ
66-71
63-47
59-84
58-80
57-75
55-65
llf
67-41
64-14
60-48
59-42
58-36
56-24
1
406
THE DISCHABGE OF WATEK FEOM
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes 1 foot O inch to 5 feet 3 inches.
*2
Coefficients of velocity, and the corresponding velocities of
1
discharge in inches per second.
i|
*8 *" "i 8
r^
Is
* -U
|.s?
!~;i
,5 ^ .3
"3 - H
a
" 5
* *? g
i H 9
a
k | -3
^ ^i
^ i
> S3 -
8 -3
> * 1
52
*
ei & J
rfij
4>|
>!
rfV|
0}
98-288
95-73
93-96
84-53
80-10
78-63
1
100-234
97-63
95-82
86-20
81-69
80-19
1
102-144
99-49
97-65
87-84
83-25
81-71
2
104-018
101-31
99-44
89-46
8477
83-21
2}
105-859
103-11
101-20
91-04
86-28
8469
3
107-669
10487
102-93
9-260
87-75
86-14
3}
109 449
10660
104-63
94-13
89-20
87-56
4
111-200
108-31
106-31
95-63
90-63
88-96
4}
112-924
10999
107 96
97 11
9203
90-34
5
114-6-22
111-42
109-58
9858
93-42
91-70
5}
116-296
113-27
111-18
100-01
94-78
93-04
6
117-945
114-78
11276
10143
96-13
94-36
7
1-21177
118-03
115-85
10421
98-76
96-94
8
124 325
121 -09
11886
106-92
101-33
99-46
9
127-896
124-08
121-79
109-56
103-83
101-92
10
130-394
127-00
12466
112 14
106-27
10431
11
183-324
129-86
.127-46
114-66
108-66
106-66
2
136-192
132-65
130-20
117-12
111-00
108-95
2 H
140-383
13673
13421
120-73
11441
112-31
2 3
144-453
140-70
138-10
1-24-23
117-73
115-56
2 4}
148-411
144-55
141-88
127-64
120-96
118-73
2 6
152-267
148-31
14557
130-95
124-10
12181
2 7}
156027
151-97
149-16
134-18
127-16
124-82
2 9
159-699
155-55
152-67
137-34
130-15
127-76
2 10}
163-288
159-04
156-10
140-43
133-80
130-63
3
166-800
162-46
159-46
143-45
13594
133-44
3 1}
170-240
165-81
162-75
14641
138-75
136-19
3 3
173-611
169-10
165-97
149*31
141-49
13889
3 4}
176-918
172-32
169-13
152-15
144-19
141-53
3 6
180-165
175-48
172-24
154-94
146-83
144-13
3 7}
183-354
178-59
175-29
157-68
149-43
146-68
3 9
186-488
181-64
178-28
1' 0'38
151-99
149-19
3 10}
189-571
184-64
181-23
163-03
154-50
151-66
4
192604
187-60
184-13
165-64
156-97
154-08
4 2
196576
191-46
187-93
169-06
16021
157-26
4 4
200-469
195-26
191-65
172-40
16338
160-37
4 6
204287
198-98
195-30
17569
16I5-49
163-43
4 8
208-036
202-63
198-88
178-91
169-55
166-43
4 10
211-718
206-21
202-40
182-08
172-55
16937
5
215338
209-74
205-86
185-19
17550
172-27
5 3
220-656
214-92
210-95
189-76
179-83
176-52
OEIFICES, WEIES, PIPES, AND EIVEES.
407
TABLE II. For finding the Velocities from the Altitudes, and tlie
Altitudes from the Velocities.
Altitudes 1 foot OJ inch to 5 feet 3 inches.
Coefficients of velocity, and the corresponding velocities of
discharge in inches per second.
9
s
*j
o^i
**g
oa ** S
M
1*1
s
o to .^
1 is
! 2 ^
QJ CO "
|S1
H ? s
fj
> 7 i
3 *
> 5 -3
s|
s|
II fi
*J1
<!!
r-l & O
ri'll
*ij
9
/ //
68-80
65-46
61-72
60-64
59-56
67-40
1
70-16
66-76
62-95
61-84
60-74
58-54
1 1
71-50
68-03
64-15
63-02
61-90
59-65
1 1J
72-81
6928
65 32
64-18
6303
60-75
1 2
74-10
70-50
6648
65-32
64-15
61-82
1 2
75-37
71-71
67-62
66-43
6525
C2-88
1 3
76-61
72-89
68-73
67-53
66-33
63-92
3
77-84
74-06
69-83
68-61
67-34
64-94
4
79-05
75-21
70-92
6967
68-43
65-95
41
80-24
76-34
71-98
70-72
69-46
66-94
5
81-41
77-45
73-03
71-75
70-48
67-92
5J
82-56
78*55
74-07
72-77
71-47
68-88
6
84-82
80-70
76-10
7477
73-43
70-77
1 7
87-03
82-80
78-08
76-71
7534
72-61
1 8
8918
84-85
80-00
78-60
77-20
74-40
1 9
9128
86-84
81 89
80-45
79-02
76-15
1 10
9333
88-79
83-73
82-26
80 79
77-86
1 11
9533
90-70
8553
84-03
82-53
7954
2
98-27
93-50
88-16
86-62
85-07
81-98
2 1
101 12
96-21
90 72
89-13
87-54
84-36
2 3
103-89
98-84
93-20
91-57
8994
8667
2 4
106-59
101-41
95-62
9395
92-27
88-92
2 6
10922
103-91
97-99
96-27
94-55
91-12
2 7
111-79
10636
100-29
98-53
96-78
9326
2 9
114-30
108-75
102-54
100-75
98-95
9V36
2 10
11676
111-09
104-75
102-92
101-03
97-41
3
119-17
11338
106-91
105-04
103-17
99-42
3 1
121-53
115-62
109-03
107-12
105-21
101-39
3 3
123-84
117-83
111-10
109-16
10721
103-32
3 4
126-12
119-99
113-14
111-16
109-18
105-22
3 6
128-35
122*11
115-15
11313
111-11
107-08
3 1\
130-54
124-20
117-11
115-06
11301
108-91
3 9
13270
126-25
119*5
11697
114-88
110-71
3 10
134-82
12827
120-96
118-84
11672
112-48
4
137-60
130-92
123-45
12129
119-12
114-80
4 2
14033
133-51
125-89
12369
121-48
117-07
4 4
H300
13606
128-29
12605
123 80
11930
4 6
145-63
138-55
130-65
128-36
126-07
121-49
4 8
148-20
141-00
132-96
130-63
128 30
123-64
4 10
150-74
143 42
135-23
132-86
130-49
12576
6
15446
146-96
138-57
136-14
133-72
128-86
5 3
408
THE DISCHAEGE OF WATEE FKOM
TABLE IL For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes 5 feet 6 inches to 17 feet.
.
Coefficients of velocity, and the corresponding velocities of
!
discharge in inches per second.
lj
*jAj
"s^S
<3^S
**!
^rf
<3~g
TO
SsT *" *o
3 t^ "
5 r- ^
S oo ^
3 >~- ^
3 ^< ^
I
*!*
|j
5iS
Jij
P S |
10 ?> J
<o 8
o
5 6
225-848
219-98
215-91
194-23
184-07
180-68
5 9
230-924
224-92
220-76
198-59
188-20
184-74
6
235-891
229-76
225*51
202-87
192-25
188-71
6 3
240-755
234-50
230-16
207-05
196-22
192-60
6 6
245-524
239-14
234-72
211-15
200-10
196-42
6 9
250-200
243-69
239-19
215-17
203-91
200-16
7
254-791
248-17
243-58
219-12
207-65
203-83
7 3
259-301
252-56
247-89
222-99
211-33
207-44
7 6
263-734
256-88
252-13
226-81
214-94
210-99
7 9
268-093
261-12
256-30
230-56
218-50
214-47
8
272-383
265-30
260-40
234-25
221-99
217-91
8 3
276-607
269-41
264-44
237-88
225-43
221-29
8 6
280-766
273-47
268-41
241-46
228-82
224-61
8 9
284- 65
277-46
272-33
244-98
232-17
227-89
9
288-906
281-39
276-19
248-46
235-46
231-12
9 3
292-891
285-28
280-00
251-89
238-71
234-31
9 6
296-823
289-11
283-76
255-27
241-91
237-46
9 9
300-703
292-88
287-47
258-60
245-07
240-56
10
304-534
296-62
291-13
261-90
248-19
243 '63
10 3
308-317
300-30
294-75
265-15
251-28
245-65
10 6
312-054
303-94
297-32
268-37
254-32
249-64
10 9
315-747
307-54
301-85
271-54
257-33
252-60
11
319-398
311-09
305-34
274-68
260-31
255-52
11 3
323-007
314-61
308-79
277-79
262-25
258-41
11 6
326-576
318-09
312-21
280-86
266-16
261-26
11 9
330-107
321-52
315-58
283-89
269-04
264-09
12
333-600
324-93
318-92
286-90
271-88
266-88
12 3
337-057
328-29
322-23
289-87
274-70
269-65
12 6
340-479
331-63
325-50
292-81
277-49
272-38
12 9
343-867
334-93
328*74
295-73
280-25
275-09
13
347-222
338-19
331-94
298-61
282-99
277-78
13 3
350-545
341-43
335-12
301-47
285-69
280-44
13 6
353-836
344-64
338-27
304-30
288-38
283-07
13 9
357-097
347-81
341-39
307-10
291-03
285-68
14
360-329
350-96
344-47
309-88
293-67
288-26
14 6
366-707
357-17
350-57
315-37
298-87
293-37
15
372-976
363-28
356-57
320-76
303-98
298-38
15 6
379-141
369-28
362-46
326-06
309-00
303-31
16
385-208
375-19
368-26
331-28
313-94
308-17
16 6
391-181
381-01
373-97
336-42
318-81
312-94
17
397-063
386-74
379-50
341-47
323-61
317-65
ORIFICES, WEIRS, PIPES, AND EIVEES.
409
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes 5 feet 6 inches to 17 feet.
Coefficients of velocity, and the corresponding velocities of
discharge in inches per second.
a
a
**
**!
^i
*^g
**i
***
f|
Is 2
I'g j
ill
S co ^
a s
igi
JgS
jl
3|
l& I;
?;1
ii SB
^fi
Is I
1
*;|
00 J
. ii %
it!
d&i
CQ " 1
1
158-09
150-41
141-83
139-35
136-86
131-90
t 8
161-65
153-80
145-02
142-48
139-94
134-86
5 9
165-12
157-10
148-14
145-55
142-95
137-76
6
168-53
160-34
151-19
14855
145-90
140-60
6 3
171-87
163-52
154-19
151-49
148-79
143-39
6 6
175-14
166-63
157-13
154-37
151-62
146-12
6 9
178-35
169-69
160-01
157-21
154-40
148-80
7
181-51
172-69
162-84
159-99
157-14
151-43
7 3
184-61
175-65
165-62
162-72
159-82
154-02
7 6
187-67
178-55
168-36
165-41
162-46
156-57
7 9
190-67
181-41
171-06
168-06
165-06
159-07
8
193-62
184-22
17371
170-67
167-62
161-54
8 3
196-54
186-99
176-32
173-23
170-14
163-97
8 6
199-41
189-72
178-90
175-76
172-63
166-36
8 9
202-23
192-41
181-43
178-26
175-08
168-72
9
205-02
195-07
183-94
180-71
177-49
171-05
9 3
207-78
197-68
186-40
183-14
179-87
173-34
9 6
210-49
200-27
188-84
185-53
182-23
175-61
9 9
213-17
202-82
191-25
187-90
184-55
177*85
10
215-82
205-34
193-62
190-23
186-84
180-06
10 3
218-44
207-83
195-97
192-54
189-10
182-24
10 6
221-02
210-29
198-29
194-82
191-34
184-40
10 9
223-58
212-72
200-58
197-07
193-55
186-53
11
226-10
215-12
202*85
199-30
195-74
188-64
11 3
228-60
217-50
205-09
201-50
197-91
190-72
11 6
231-07
219-85
207-31
203-68
200-04
192-78
11 9
233-52
222-18
209-50
205-83
202-16
194-82
12
235-94
224-48
211-67
. 207-96
204-26
196-84
12 3
238-34
226-76
213-82
210-08
206-33
198-84
12 6
240-71
229-02
215-95
212-17
208-38
200-82
12 9
243-06
231-25
218-06
214-24
210-42
202-78
13
245-38
233-46
220*14
216-29
212-43
204-72
13 3
247-69
235-65
222-21
218-32
214-42
206*64
13 6
249-97
237-83
224-26
220-33
216-40
208-54
13 9
252-23
239-98
226-29
22-2-32
218'36
210-43
14
256-70
244-23
230-29
226-26
222-22
214-16
14 6
261-08
248-40
234-23
230-13
226-02
217-82
15
265-40
252-51
238-10
233-93
229-76
221-42
15 6
269-65
256-55
241-91
237'67
233-44
224-96
16
273-83
260-53
245-66
241-36
237-06
228-45
16 6
277-94
264-44
249-36
244-99
240-62
231-89
17
410
THE DISCHARGE OF WATER FROM
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes 17 feet 6 inches to 40 feet.
>CJ
Coefficient of velocity, and the corresponding velocities of
1
S*"
discharge in inches per second.
2|
**ij
**f
**S
"* CO
"3^5; 00
**|
JI
l^'rt "I
Is!
1 -*>
3s
J9 j
I
> * 1 '*
Ill
u I
III
i 1 . W
jj
Jijt
ci s> 8
*M
II 1
. <u
IO c*
<IJ
17 6
402-860
392-39
385-13
346-46
328-33
322-29
18
408-575
397-95
390-60
351-37
332-99
326-86
18 6
414-211
403-44
395-99
356-22
337-58
331-37
19
419-772
408-86
401-30
361-00
34-2 11
33582
19 6
425-258
414-20
406-55
365-72
346-59
340-21
20
430-676
419-48
411-73
370-38
351-00
344-54
20 6
436-026
424-69
416-84
374-98
35536
348-82
21
441-311
429-84
421-89
379-53
359-59
353-05
21 6
446-534
43492
426-89
384-02
363-93
357-23
22
451-697
439-95
431-82
388-46
368-13
361-36
22 6
456-801
444-92
436-70
392-85
372-29
365-44
23
461-848
449-84
441-53
397-19
376-41
369-48
23 6
466-841
450-70
446-30
401-48
380-48
373-47
24
471-782
459-52
451-02
405-73
384-50
377-43
24 6
476671
464-28
455-70
409-94
388-49
38134
25
481-510
468-99
460-32
414-10
392-43
385-21
25 6
486-301
473-66
464-90
418-22
396-34
389-04
26
491-046
478-28
469-44
422*30
40020
392-84
26 6
495745
482-86
473-93
426*34
40403
396-60
27
500-400
487-39
478-38
430-34
40783
400-32
27 6
505-012
491-88
482-79
434-31
411-58
404-01
28
509 682
49H'33
487-16
438-24
41531
40767
28 6
514-112
500-75
491 49
442-14
419-00
411-29
29
518-602
505-12
49578
446-00
4-22-66
414-88
29 6
5-23-054
509-45
500-04
449-83
42629
418-44
30
527-468
513-75
504-26
453-62
429-89
421-97
30 6
531-845
518-02
508-44
457-31
433-45
425-48
31
536-187
522-25
512-59
461-12
436-99
428-95
31 6
540-494
526-44
516-71
464-8-2
440-50
432-40
32
544-767
530-00
520-80
468-50
443-98
435-81
3-2 6
549006
534-73
5-24-85
472-15
447-44
43920
33
553-213
53883
5-28-87
475*76
450-87
442-57
33 6
557-388
54-2-90
532-86
479 35
454-27
445-91
34
561-532
54'r93
536-83
48-2-92
407-65
449-23
34 6
565-646
550-94
540-76
186-46
461-00
45-2-5-2
35
569730
554-92
544-66
489-97
464-33
455-78
36
577-812
56279
552-39
496-92
4709-2
46-2-25
37
585 782
570-55
560-01
503-77
477-41
468-63
38
593-646
578-21
567-53
510-54
483 82
4749-2
39
601 406
585-77
574-94
517-21
49015
481 12
40
609-067
593-23
58227
523-80
496-39
487-25
ORIFICES, WEIRS, PIPES, AND RIVERS.
411
TABLE II. For finding the Velocities from the Altitudes, and the
Altitudes from the Velocities.
Altitudes 17 feet 6 inches to 40 feet.
Coefficients of Telocity, and the corresponding velocities of
ri
discharge in inches per second.
1
Hg
^|
**i
^g
*%i
* % : 4
.a |
iij
^|
Hi
III
ycd
II S
ill
01 C
*!
00
e I o
V
r4 H
N o
1
sJ
u
a
^ o
/ //
282-00
268-30
253-00
248-56
244-13
235-27
17 6
286-00
272-11
256-59
252-09
247-60
238-61
18
28995
275-86
260 12
255-57
251-01
241-90
18 6
29384
279-57
263-32
259-00
254-38
245-14
19
297-68
283-22
267-06
26238
257*71
248-35
19 6
301-47
28683
270-46
265-73
260-99
251-51
20
305-22
290-39
273-82
269-03
264-23
254-64
20 6
30892
293-91
277-08
272-23
26737
257-67
21
312-57
297-39
2SO-42
275-51
27060
260-78
21 6
316-19
300-83
283-67
27870
27373
263-79
22
319-76
304-23
286-87
281-85
276-82
266-77
22 6
323-29
307-59
290-04
284-96
279-88
269-72
23
326-79
310-92
293-18
288-04
28291
272-64
23 6
330-25
314-21
296-28
291-09
285-90
275-52
24
333 67
317-46
299-35
294-11
288-86
278-38
24 6
33706
320-69
302-39
297-09
291-80
281-20
25
340-41
323-88
305-40
300-05
294-70
284-00
25 6
34373
32704
308-38
302-98
297-57
286-77
26
347-02
330-17
311-33
305-87
30042
289-52
26 6
350-28
333-13
314-25
308-75
303-94
292-23
27
353-51
336-34
317-15
311-59
30604
29493
27 6
35671
339 38
32002
314-41
308-81
297-60
28
359-88
342-40
322-86
317-20
311-55
300-24
28 6
363-02
34539
325-68
319-98
314-27
302 86
29
366-14
34835
328-48
322-72
316-97
305-46
29 6
369-23
351-29
331-25
325-45
319-65
3<'8-04
30
372-29
354-21
334-00
3-28-15
32230
310-60
30 6
375-33
357 10
386-73
330-83
324-93
313-13
31
37835
359-97
339-43
333-48
327-54
315-60
31 6
381-34
362-81
342-11
336-12
330-13
31814
32
384-30
36564
344-78
338-74
332-70
320-62
32 6
387 25
368-44
347-42
341-33
335-25
323-08
33
390-17
371-22
350-04
34301
337-78
325-51
33 6
393-07
37398
352-64
346-47
340-29
327-93
34
395-95
376-72
355-23
349-00
34-2- / 8
330-34
34 6
39881
379-44
35779
351-52
345-26
332-72
35
404-47
384-82
36287
356-51
350-15
337-44
36
410-05
390-13
367-87
361-43
35498
342 10
37
41555
395-37
37-2-81
366-28
35975
346-69
38
420-98
40054
377-68
371 11
36445
35122
39
42635
405-64
382-49
375-79
369-09
355-70
40
412
THE DISCHARGE OF WATER FROM
TABLE III. Square Roots for finding the effects of the Velocity of
Approach ivhen the Orifice is small in proportion to the Head.
Also for finding the Increase in the Discharge from an Increase
of Head. (See p. 101.)
No.
Square
root.
No.
Square
root.
No.
Square
root.
No.
Square
root.
1-000
1-0000
1-115
1-0559
1-475
1-2141
1-975
4053
1-001
1-0005
1-120
1-0583
1-49
1-2207
1-99
4107
1-002
1-0010
1-125
1-0607
1-5
1-2247
2-00
4142
1-004
1-0020
1-13
1-0630
1-51
1-2288
2-01
4177
1-005
1-0025
1-135
1-0654
1-525
1-2349
2-025
4230
1-006
1-0030
1-14
1-0677
1-54
1-2410
2-04
4283
1-008
1-0040
1-145
1-0700
1-55
1-2450
2-05
4318
1-009
1-0044
1-15
1-0723
1-56
1-2490
2-06
4353
1-010
1-0050
1-155
1-0747
1-575
1-2550
2-075
1-4405
1-011
1-0055
1-16
1-0770
1-58
1-2570
2-09
1-4457
1-012
1-0060
165
1-0794
1-59
1-2610
2-10
1-4491
1-014
1-0070
17
1-0817
1-6
1-2649
2-11
1-4526
1-015
1-0075
175
1-0840
1-61
1-2689
2-125
1-4577
1-016
1-0080
18
1-0863
1-625
1-2748
2-14
1-4629
1-018
1-0090
185
0886
1-64
1-2806
2-15
1-4663
1-019
1-0095
19
0909
1-65
1-2845
2-16
1-4697
1-020
1-0100
195
0932
1-66
1-2884
2-175
1-4748
1-0225
1-0112
2
0954
675
1-2942
2-19
1-4799
1-025
1-0124
21
1000
69
1-3000
2-2
1-4832
1-0275
1-0137
22
1045
7
1-3038
2-21
1-4866
1-03
1-0149
23
1091
71
1-3077
2-225
1-4916
1-0325
1-0161
24
1136
725
1-3134
2-24
1-4967
1-035
1-0174
25
1-1180
74
1-3191
2-25
1-5000
1-0375
1-0186
26
1-1225
75
1-3229
2-26
1-5033
1-04
1-0198
27
1-1269
76
1-3267
2275
1-5083
1-0425
1-0210
28
1-1314
775
1-3323
2-29
1-5133
1-045
1-0223
29
1-1358
79
1-3379
2-3
1-5166
1-0475
1-0235
30
1402
80
1-3416
2-31
1-5199
1-05
1-0247
31
1446
81
1-3454
2-325
1-5248
1-055
1-0271
325
1511
825
1-3509
2-34
1-5297
1-06
1-0296
34
1576
84
1-3565
2-35
1-5330
1-065
1-0320
35
1619
85
1-3601
2-36
1-5362
1-07
1-0344
36
1662
86
1-3638
2-375
1-5411
1-075
1-0368
375
1726
875
1-3693
2-39
1-5460
1-08
1-0392
39
1-1790
89
1-3748
2-4
1-5492
1-085
1-0416
40
1-1832
9
1-3784
2'41
1-5524
1-09
1-0440
41
1-1874
91
1-3820
2'425
1-5572
1-095
1-0464
1-425
1-1937
925
1-3875
2-44
1-5621
1-1
1-0488
1-44
1-2000
94
1-3928
2-45
1-5652
1-105
1-0512
1-45
1-2042
1-95
1-3964
2-46
1-5684
1-110
1-0536
1-46
1-2083
1-96
1-4000
2-475
1-5732
ORIFICES, WEIRS, PIPES, AND RIVERS.
413
TABLE III. Square Eoots for finding the effects of the Velocity of
Approach when the Orifice is small in proportion to the Head.
Also for finding the Increase in the Discharge from an Increase
of Head. (See p. 101.)
No.
Square
root.
No.
Square
root.
No.
Square
root.
No.
Square
root.
2-49
1-5780
3-0000
1-7321
4-5
2-1213
26
5-0990
2-5
1-5811
3-025
1-7393
5'0
2-2361
27
5-1962
2-51
1-5843
3-05
1-7464
5-5
2-3452
28
5-2915
2-525
1-5890
3-075
1-7536
6-0
2-4495
29
5-3852
254
1-5937
3-1
1-7607
6-5
2-5495
30
5-4772
2-55
1-5969
3-125
1-7678
7-0
2-6458
31
5-5678
2-56
1-6000
3-15
1-7748
7-5
2-7386
32
5-6569
2-575
1-6047
3-175
7819
8'0
2-8284
33
5-7446
2-59
1-6093
3-2
7889
8'5
2-9155
34
5-8310
2-6
1-6125
3*225
7958
9-0
3-0000
35
5-9161
2-61
1-6155
3-25
8028
9-5
3-0822
36
6-0000
2-625
1-6202
3-275
8097
10-0
3-1623
37
6-0828
2-64
1-6248
3-3
8166
10-5
2-2404
38
6-1644
2-65
1-6279
3-325
8235
11-0
3-3166
39
6-2450
2'66
1-6310
3-35
8303
11-5
3-3912
40
6-3246
2-675
1-6355
3-375
8371
12'0
3-4641
41
6-4031
2-69
1-6401
3-4
8439
12-5
3'5355
42
6-4807
2-7
1-6432
3-425
8507
13'0
3-6056
43
6-5574
2-71
1-6462
3-45
8574
13-5
3-6742
44
6-6332
2-725
1-6508
3-475
8641
14-0
3-7417
45
6-7082
2-74
1-6553
3-5
8708
14-5
3-8079
46
6-7823
2-75
1-6583
3-525
8775
15-0
3-8730
47
6-8557
2-76
1-6613
3-55
8841
15-5
3-9370
48
6-9282
2-775
1-6658
3-575
8908
16-0
4-0000
49
7-0000
2-79
1-6703
3-6
8974
16-5
4-0620
50
7-0711
2-8
1-6733
3-625
9039
17-0
4-1231
51
7-1414
2-81
1-6763
3-65
9105
17-5
4-1833
52
7-2111
2-825
1-6808
3-675
9170
18-0
4-2426
53
7-2810
2-84
1-6852
3-7
1-9235
18-5
4-3012
54
7-3485
2-85
6882
3-725
1-9300
19-0
4-3589
55
7-4162
2-86
6912
3-75
1-9365
19'5
4-4159
56
7-4833
2-875
6956
3-775
1-9429
20-0
4-4721
57
7-5498
2-89
7000
3-8
1-9494
20-5
4-5277
58
7-6158
2-9
7029
3-825
1-9558
21-0
4-5826
59
7-6811
2-91
7059
3-85
1-9621
21-5
4-6368
60
7-7460
2-925
7103
3-875
1-9685
22-0
4-6904
61
7-8102
2-94
7146
3-9
1-9748
22-5
4-7434
62
7-8740
2-95
7176
3-925
1-9812
230
4-7958
63
7-9373
2-96
7205
3-95
1-9875
23-5
4-8477
64
8-0000
2-975
7248
3-975
1-9938
24-0
4-8990
65
8-0623
2-99
7292
4-0
2-0000
25-0
5-0000
66
8-1240
414
THE DISCHARGE OF WATER FROM
TABLE IV. For finding the Discharge through Eectangular Ori-
fices ; in which n = -. Also for finding the effects of the
ct
Velocity of Approach to Weirs, and the Depression on the Crest.
(See p. 101.)
1+ n
.
(1 + n)*
(H) f -n f
l +
.*
(1 4 n)*
(1+71)2 n f
1-000
0000
1-0000
i-oooo
115
0390
1-1774
1-1384
1-<01
0000
1-0015
1-0015
120
0416
1-1853
1-1437
1 -002
0001
1-0030
1-0029
125
0442
1-1932
1-1491
1004
0003
1-0060
1-0058
13
0469
1-2012
1-1543
1-005
0004
1-0075
1-0072
135
0496
1-2092
1-1596
1-006
0005
1-0090
1-0086
14
0524
1 2172
1 1648
1 008
0007
1-0120
1-0113
145
0552
1-2251
1-1700
1-009
0009
1-0135
1 0127
15
0581
1-2332
1-1751
1-010
0010
1-0150
1-0140
155
0610
1-2413
1-1803
1-011
0012
T0165
1-0154
16
0^40
1-2494
1-1854
1-012
0013
1-0181
1-0167
165
0670
1-2574
1-1904
1-014
0017
1-0211
1-0194
17
0701
1-2655
1-1955
1-015
0018
T0226
1-0207
175
0732
1-2737
1-2005
1-016
0020
1 0241
1-0221
18
0764
1-2818
1 -2054
1-018
0024
1-0271
1-0247
185
0796
1-2900
1-2104
1-019
0026
1-0286
1-0260
19
0828
1 2981
1-2153
1-020
0028
1-0301
1-0273
1-195
0861
1 -3063
1-2^02
1-0225
0034
1-0339
1 0306
1-2
0894
1-3145
1-2251
1-025
0040
1 0377
1-0338
1-21
0962
1-3310
1-2348
1-0275
0046
1-0415
1-0370
1-22
1032
1-3475
1-2443
1-03
0052
1-0453
1-0401
123
1103
1-3641
1-2538
1-0325
0059
1-0491
1-0433
1-24
1176
1 3808
1-2632
1-035
0065
1-0530
1-0464
1 25
1250
1-3975
2725
1-0375
0073
1-0568
1-0495
1-26
1326
1 4143
2818
1-04
0080
1-0606
T0526
1-27
1403
1 4312
2909
1-0425
0088
1-0644
1-0557
1-28
1482
1-4482
3000
1-045
'0095
1-0683
1-0587
-29
1562
T4652
3090
1-0475
0104
1-0721
1-0617
30
1643
1-4822
3179
1-05
0112
1-0759
1-0648
31
1726
1 4994
3268
1-055
0129
1-0836
1-0707
325
18 3
1 5252
3399
1-06
0147
T0913
1-0766
34
1983
1-5512
3529
1-065
0166
1-0991
1-0825
35
2071
1-5686
3615
P07
0185
1 1068
1-0883
36
2160
1-5860
3700
1-075
0205
1 1146
1-0940
375
-2296
1-6123
3827
1-08
0226
1 1-224
1-0997
1-39
2436
1-6388
3952
1 085
0248
1 1302
1-1054
1 40
2530
1 -6565
4035
l-i <9
0270
1-13NO
rino
T41
2625
1-6743
4118
1 -095
0-293
1-1458
1-1166
1-425
2771
1 7011
4240
1-1
03 1 6
1-1537
1 1221
1-44
2919
1-7-J80
4361
1-105
0340
1-1616
1 1275
1-45
3019
1-7460
4442
1-110
0365
1-1695
1-1330
1-4G
3120
1-7641
4521
i
Values Of n from to '46. ^Continued on next page.
ORIFICES, WEIRS, PIPES, AND RIVERS.
415
TABLE IV. For finding the Discharge through 'Rectangular Ori-
fices ; in which n = - . Also for finding the effects of the
Velocity of Approach to Weirs, <&c. f See p. 101.)
l+tt
.<
(1 + n)^
(1+nf-J
1 +n
.*
(l + ) f .
(l-fn)*-n*
1*475
3274
17914
4640
1-975
9627
2-7756
1-8128
1-49
3430
8188
4758
1-99
9850
2-8072
18222
1-5
3536
8371
4836
2-
1-0000
2-8284
1-8284
1-51
3642
8555
4913
2-01
1-0150
2-8497
18346
1525
3804
8832
5028
2-025
1-0377
2-8816
8439
1-54
3968
9111
5143
2-04
1-0606
2-9137
8531
155
4079
9297
5218
2-05
1-0759
2-9352
8592
1-56
4191
9484
5294
2-06
1-0913
2-9567
8653
1-575
4360
9766
5406
2'075
1-1146
2-9890
8744
1-58
4417
9860
5443
2-09
1-1380
30215
8835
1-59
4532
2-0049
5517
2-10
1 -1537
3-0432
1-8895
16
4648
2-0239
1-5591
211
1-1695
30650
1-8955
1-61
4764
2-0429
1-5664
2-125
1-1932
3-0977
1-9045
1 -625
4941
2-0715
1-5774
2-14
1-2172
3-1306
19134
164
5120
2-1002
1-5882
2-15
1-2332
3-1525
1-9193
1 65
5240
2-1195
1-5954
2-16
1-2494
3-1745
1-9252
1-66
5362
2-1388
1-6026
2-175
1-2737
3-2077
1-9340
1-675
5546
2-1678
1-6132
2-19
1'2981
3-2409
1-9428
1-69
5732
2-1970
1-6238
2-2
1-3145
3-2631
1-9-186
1-7
5857
22165
1-6309
2-21
1-3310
3-2854
1-9544
1-71
5983
2-2361
1-6379
2225
1-3558
3-3189
T9631
1-725
6173
2-2656
1-6483
0-04
1-3808
33525
T9717
1-74
6366
2-2952
1-6586
2-25
13975
3-3750
1-9775
1-75
6495
2-3150
1-6655
2-26
1-4143
3-3975
1-9832
1-76
6626
2-3349
1-6724
2"275
1-4397
3-4314
1-9917
1-775
6823
2-3648
1-6826
2'29
1-4652
3-4654
2-0002
1-79
7322
2-3949
1 6927
2-3
1-4822
3-4881
2-0059
1-80
7155
2-4150
6994
2-31
1-4994
3-5109
2-0115
1-81
7290
2-4351
7061
2-325
T5252
3-5451
2-0200
1-825
7493
2-4654
7161
2-34
1-5512
3-5795
2-0284
1-84
7699
2-4959
7260
2-35
1-5686
36025
2-0339
1-85
7837
2-5163
7326
2-36
1-5860
36255
2-0395
1-80
7975
2-5367
7392
2-375
1-6123
3-6601
2-0478
1-875
8185
2-5674
7490
2-39
T6388
3-6948
2-0561
1-89
8396
2-5983
7587
2'4
V6565
37181
2-0616
1-9
8538
2-6190
7652
2-41
1-6743
3-7U3
2-0670
1-91
8681
2-6397
7716
2425
17011
3-7763
2-0752
1-925
8896
2-6709
7813
244
1-7280
38114
2-0834
194
9114
2-7021
7907
245
1-7460
38349
2-0888
1-95
9259
2-7230
7971
2-46
1-7641
3-8584
2-0942
1-96
9406
2-7440
8034
2475
1-7914
3-8937
2-1023
Values Of n from -475 to 1'475. [Continued on next page.
416
THE DISCHARGE OF WATER FROM
TABLE IV. For finding the Discharge through Eectangular Ori-
fices ; in which n = -,. Also for finding the effects of the
Velocity of Approach to Weirs, dc. (See p. 101.)
14-*
**.
(l + n) f .
(!+)*_*.
l + n.
nl
d + nf.
(l+n) f -n f .
2-49
1-8188
3-9292
2-1104
3-
2-8284
5-1962
2-3677
2-5
1-8371
3-9528
2-1157
3-025
2-8816
5-2612
2-3796
2-51
1-8555
3-9766
2-1211
3-05
2-9352
5-3626
2-3914
2-525
1-8832
4-0123
2-1291
3-075
2-9890
5-3922
2-4032
2-54
1-9111
4-0481
2-1370
3-1
3-0432
5-4581
2-4149
2-55
1-9297
4-0720
2-1423
3-125
3-0977
5-5243
2-4266
2-56
1-9484
4-0960
2-1476
3-15
3-1525
5-5907
2-4382
2-575
1-9766
4-1321
2-1554
3-175
3-2077
5-6574
2-4497
2-59
2-0049
4-1682
2-1633
3-2
3-2631
5-7243
2-4612
2'6
2-0239
4-1924
2-1685
3-225
3-3189
5-7915
2-4726
2-61
2-0429
4-2166
2-1737
3-25
3-3750
5-8590
2-4840
2-625
2-0715
4-2530
2-1815
3-275
3-4314
,5-9268
2-4953
2-64
2-1002
4-2895
2-1893
33
3-4881
5-9947
2-5066
2-65
2-1195
4-3139
2-1944
3-325
35451
6-0630
2-5179
2'66
2-1388
4-3383
2-1996
3-35
3-6025
6-1315
2-5290
2-675
2-1678
4-3751
2-2073
3-375
3-6601
6-2003
2-5401
2-69
2-1970
4-4119
2-2149
3-4
3-7181
6-2693
2-5512
2-7
2-2165
4-4366
2-2200
3-425
3-7763
6-3386
2-5623
2-71
2-2361
4-4612
2-2251
3-45
3-8349
6-4081
2-5732
2-725
2-2656
4-4983
22327
3-475
3-8937
6-4779
2-5842
2-74
2-2952
4-5355
2-2403
3-5
3-9528
6-5479
2-5951
2-75
2-3150
4-5604
2-2453
3-525
4-0123
6-6182
2-6059
2-76
2-3349
4-5853
2-2504
3-55
4-0720
6-6887
2-6167
2-775
2-3648
4-6227
2-2579
3-575
4-1321
6-7595
2-6274
2-79
2-3949
4-6602
2-2654
36
4-1924
6-8305
2-6381
2'8
2-4150
4-6853
2-2703
3-625
4-2530
6-9018
2-6488
2-81
2-4351
4-7104
2-2753
3-65
4-3139
69733
2-6594
2'825
2-4654
4-7482
2-2827
3-675
4-3751
7-0451
2-6700
2-84
2-4959
4-7861
2-2902
3-7
4-4366
7-1171
2-6805
2-85
2-5163
4-8114
2-2951
3725
4-4983
7-1893
2-6910
2-86
2-5367
4-8367
2-3000
3-75
4-5604
7-2618
2-7015
2-875
2-5674
4-8748
2-3074
3-775
4-6227
7-3346
2-7119
2-89
2-5983
4-9130
2-3147
3-8
4-6853
7-4076
2-7223
2'9
2-6190
4-9385
2-3196
3-825
4-7482
7-4808
2-7326
2-91
2-6397
4-9641
2-3244
3-85
4-8114
7-5542
2-7429
2-925
2-6708
5-0025
2-3317
3875
4-8748
7-6279
2-7531
2'94
2-7021
5-0411
2-3389
3-9
4-9385
7-7019
27634
2-95
2-7230
5-0668
2-3438
3925
5-0025
77761
2-7735
2-96
2-7440
5-0926
2-3486
3-95
50668
7-8505
2-7837
2-975
2-7756
5-1313
2-3558
3-975
5-1313
7-9251
2-7838
2-99
2-8072
5' 1702
2-3630
4-
5-1962
8-
2-8038
Values of n from *475 to 1-475.
[ Continued on next page.
ORIFICES, WEIRS, PIPES, AND RIVERS.
417
TABLE IV. For finding the Discharge through Eectangular Ori-
fices ; in which n = T. Also for finding the effects of the
Velocity of Approach to Weirs, &c. (See p. 101.)
l-t-n.
4
<l + n)*.
i*M
1 + n.
f.
d + ) f .
(l+n) 2 -7i^
4-5
6-5479
9-5459
2-9980
26-
125-0000
132-5745
7-5745
5-0
8-0000
11-1803
3-1803
27-
132-5745
140-2961
77216
5-5
9-5459
128986
3-3527
28-
140-2961
148-1621
7-8660
60
11-1803
14-6969
3-5166
29-
148-1621
156-1698
8-0077
6-5
12-8986
16-5718
3-6732
30-
156-1698
164-3168
8-1470
7-0
146969
18-5203
3-8234
31-
164-3168
172-6007
8-2839
7-5
16-5718
20-5396
3-9678
32-
172-6007
181-0193
8-4186
8-0
18-5203
226274
4-1071
33-
181-0193
189-5706
8-5513
8-5
20-5396
24-7815
42419
34-
189-5706
198-2524
8-6818
9-0 22-6-274
27-0000
4-3726
35-
198-2524
207-0628
8-8104
95
24-7815
29-2810
4-4995
36-
207-0628
216-0000
8-9372
10-0
27-0000
31-6-2-28
4-6-228
37-
216-0000
225-0622
9-0622
10 5
29-2810
34-0239
4-7429
38-
225-0622
234'2477
9-1855
11-0
31-62-28
36-4829
4-8601
39-
234-2477
243-5549
9-3072
11-5
34-0239
38-9984
4-9745
40-
2435549
252-9822
9-4273
12-0
36-4829
41-5692
5-0863
41-
252-9822
262-5281
9-5459
12-5
38-9984
44-1942
5-1958
42-
262-5281
272-1911
9-6630
13-0
41-5692
46-8722
5-3030
43-
272-1911
281-9699
9-7788
13-5
44-1942 I 49-6022
5-4080
44-
281-9699
291-8630
9 8931
14-0
46-8722 | 52-3832
5-5110
45-
291-8630
301-8692
10-0062
14-5
49-6022
55-2144
5-6122
46-
301-8692
311-9872
10-1180
15-0
52-3832
58-0947
5-7115
47-
311-9872
322-2158
10-2286
15-5
55-2144
61-0236
5-8092
48-
322-2158
332-5538
10-3380
16-0
58-0947
64-
5-9053
49-
332-5538
343-0000
10-4462
16-5
61-0236
67-0247
6-0011
50-
343-0000
353-5534
10-5534
17-0
64-
70-0928
6-0928
51-
353-5534
364-2128
10-6594
17-5
67-0247
73-2078
6-1831
52-
364-2128
374-9773
10-7645
18-0
70-0928
76-3675
6-2747
53-
374-9773
385 8458
10-8685
18-5
73*2078
795715
6-3637
54-
385-8458
396-8173
10-9715
19-0
76-3675
82.8191
6-4516
55-
396-8173
407-8909
11-0736
19-5
79-5715
86-1097
6-5382
56-
407-8909
419-0656
11-1747
20-0
82-8191
89-4427
6-6236
57-
419-0656
430-3406
11-2750
205
86-1097
92-8)77
6-7080
58-
430-3406
441-7148
11-3742
21-0
894427
96-2341
6-7914
59-
441-7148
453-1876
1 1-4728
21-5
92-8177
99-6914
6-8737
60-
453-1876
464-7580
11-5704
2-2-0
96-2341
103-1892
6-9551
61-
464-7580
4764252
11-6672
22-5
99-6914
1067269
7 0355
62-
476-4252
488-1885
11-7633
23-
103-1892
110-3041
7-1149
63-
488-1885
500-0470
11-8585
23-5
106-7269
1139205
7-1936
64-
500-0470
512-0000
11-9530
24-
110-3041
117-5755
7-2714
65-
512-0000
524-0468
12-0468
25-
117-5755
125-
7-4245
66-
524-0468
5361865
12-1397
Values of n from 3-5 to
E E
418
THE DISCHARGE OF WATER FROM
TABLE V. Coefficients of Discharge for Different Eatios of the
Channel to the Orifice.
Coefficients for Heads in still water '550 and -573.
Coefficient -550 for heads in still
Coefficient -573 for heads in still
water.
water.
Ratio
of the
Ratio of
Coefficients
Coefficients
Ratio of
Coefficients
Coefficients
channel
the height
for orifices :
for weirs :
the height
for orifices :
for weirs :
to the
due to the
the heads
the heads
due to the
: the heads
the heads
orifice.
velocity of
measured
measured
velocity of
measured
measured
approach
to the
the full
approach
to the
the full
to the head.
centres.
depth.
to the head.
centres.
depth.
so-
000
550
550
000
573
573
so-
001
550
551
001
573
574
15-
001
550
551
001
573
574
lo-
003
551
552
003
574
576
o-
004
551
553
004
574
576
s'
005
551
554
005
574
577
7-
006
552
5,55
007
575
578
6-
008
552
557
009
576
580
5-5
010
553
558
Oil
576
582
5-0
012
553
559
013
577
584
4-5
015
554
562
016
578
586
4-0
019
555
565
021
579
589
3-75
022
556
566
024
580
592
3-50
025
557
569
028
581
594
3-25
029
558
572
032
582
598
3-0
035
559
575
038
584
602
2-75
042
561
580
045
586
607
2-50
051
564
586
055
589
614
2'25
064
567
594
069
593
623
2'00
082
572
606
089
598
636
1'95
086
573
609
094
599
639
1-90
091
575
612
100
601
643
1-85
097
576
615
106
603
647
1-80
103
578
619
113
604
651
1-75
110
579
623
120
606
655
1-70
117
581
627
128
609
660
1-65
125
583
632
137
611
666
1-60
134
586
637
147
614
671
1-55
144
588
643
158
617
678
1-50
155
591
649
171
620
6H5
1-45
168
594
656
185
624
694
1-40
183
598
664
201
628
703
1-35
199
602
673
220
633
713
1-30
218
607
683
241
638
724
1-25
240
612
695
266
645
737
1-20
265
619
707
295
652
753
1-15
297
626
723
330
661
770
MO
333
635
741
372
671
791
1-05
378
646
762
424
684
816
1-00
434
659
787
489
699
845
See the auxiliary table, p. 136.
ORIFICES, WEIRS, PIPES, AND RIVERS.
419
TABLE V. Coefficients of Discharge for different Ratios of the
Channel to the Ori/ice.
Coefficients for heads in still water *584 and *595.
Coefficient -584 for heads in still
Coefficient -595 for heads in still
water.
water.
Ratio of
the
Ratio of
Coefficients
Coefficients
Ratio of
Coefficients
Coefficients
channel
the h eight
for orifices:
for weirs :
the height
for orifices :
for weirs :
to the
due to the
the heads
the heads
due to the
the heads
the heads
orifice.
velocity of
measured
measured
velocity of
measured
measured
approach
to the
the full
approach
to the
the full
to the head.
centres.
depth.
to the head.
centres.
depth.
30-
000
584
584
000
595
595
20-
001
584
585
001
595
596
15-
002
584
585
002
595
596
10-
003
585
587
004
596
598
9-0
004
585
588
004
596
599
1-0
005
586
588
006
597
600
7-0
007
586
590
007
597
601
6-0
010
587
592
010
598
603
5-5
Oil
587
593
012
599
605
5-0
014
588
595
014
599
607
4-5
017
589
598
018
600
610
4-0
022
590
601
023
602
613
3-75
025
591
604
026
603
616
3-50
029
592
606
030
604
619
3-25
033
594
610
035
605
622
3-0
039
595
614
041
607
627
275
047
598
620
049
609
633
2-50
058
601
627
060
613
641
2-25
072
605
637
075
617
651
20
093
611
651
097
623
666
1-95
099
612
654
103
625
669
1-90
104
614
660
109
627
673
1-85
111
615
662
115
628
678
1-80
118
617
666
123
630
682
1-75
125
620
671
131
633
687
T70
134
'622
676
140
635
693
1-65
143
624
682
149
638
699
1-60
154
627
689
160
641
706
155
166
631
696
173
644
713
1-50
179
634
703
187
648
721
1-45
194
638
712
202
652
730
1-40
211
643
722
220
657
741
1-35
230
648
732
241
663
752
T30
253
654
745
265
669
765
1-25
279
661
759
293
677
780
1-20
310
669
775
325
685
797
1-15
.348
678
794
366
695
818
4-10
393
689
816
414
707
842
1-05
448
703
842
473
722
870
1-00
518
719
874
548
740
905
See the auxiliary table, p. 136.
EE 3
420
THE DISCHARGE OF WATER FROM
TABLE V. Coefficients of Discharge for different Eatios of the
Channel to the Orifice.
Coefficients for heads in still water -606 and -617.
Coefficient -606 for heads in
Coefficient -617 for heads in
still water.
still water.
Ratio
of the
Ratio of
Coefficients
Coefficients
Ratio of
Coefficients
Coefficients
channel
the height
for orifices :
for weirs :
the height
for orifices :
for weirs :
to the
due to the
the heads
the heads
due to the
the heads
the heads
orifice.
velocity of
measured
measured
velocity of
measured
measured
approach
to the
the full
approach
to the
the full
to the head.
centres.
depth.
to the head.
centres.
depth.
30-
000
606
606
000
617
617
20-
001
606
607
001
617
618
15-
002
607
607
002
618
619
1O
004
607
609
004
618
620
9-0
005
607
610
005
618
621
8-0
006
608
611
006
619
622
7-0
008
608
612
008
619
624
6'0
010
609
615
on
620
626
5-5
012
610
616
013
621
628
5-0
015
611
619
015
622
630
4'5
018
612
621
019
623
633
4-0
023
613
625
024
624
637
375
027
614
628
028
626
640
3'50
031
615
631
032
627
643
3'25
036
617
635
037
628
647
3'00
043
619
640
044
630
653
275
051
621
646
053
633
660
2'50
062
625
654
065
637
668
2'25
078
629
665
081
642
679
2-00
101
636
681
105
649
696
1-95
107
638
685
111
650
700
1'90
113
639
689
118
652
704
1-85
119
641
693
125
654
709
T80
128
644
698
133
657
714
1-75
136
646
703
142
659
720
1-70
146
649
709
152
662
726
1-65
156
652
716
163
665
733
1-60
167
655
723
175
669
741
T55
180
658
731
188
673
749
1-50
195
662
739
204
677
759
1-45
212
667
749
221
681
768
1'40
231
672
760
241
687
780
1-35
252
678
772
264
694
793
1-30
278
685
-786
291
701
808
125
307
693
803
322
709
825
1-20
342
702
821
359
719
845
1-15
384
713
843
404
731
868
1-10
436
726
868
459
745
895
1-05
499
742
898
527
763
928
1-00
580
762
936
615
784
969
See the auxiliary table, p. 136.
ORIFICES, WEIRS, PIPES, AND RIVERS.
421
TABLE V. Coefficients of Discharge for different Ratios
of the Channel to the Orifice.
Mean Coefficient -628.
Coefficients for heads in still water -628 and -639.
Coefficient -628 for heads in
Coefficient -639 for heads in
still water.
still water.
Ratio
of the
Ratio of
Coefficients
Coefficients
Ratio of
Coefficients
Coefficients
channel
the height
for orifices
for weirs:
the height
for orifices
for weirs :
to the
due to the
the heads
the heads
due to the
the heads
the heads
orifice.
velocity of
measured
measured
Telocity of
measured
measured
approach
to the
the full
approach
to the
the full
to the head
centres.
depth.
to the head
centres.
depth.
30-
000
628
628
000
639
639
20-
001
628
629
001
639
640
15-
002
629
630
002
640
641
10-
004
629
632
004
640
643
9-0
005
630
632
005
641
644
8-0
006
630
634
006
641
645
7-0
008
631
635
008
642
647
6-0
on
631
638
Oil
643
649
5-5
013
632
640
014
643
651
5-0
016
633
642
017
644
654
4-5
020
634
645
021
646
657
4-0
025
636
649
026
647
662
375
029
637
652
030
648
665
3-50
033
638
656
034
650
668
3-25
039
639
659
040
652
673
3-0
046
642
666
048
654
678
2-75
055
645
672
057
657
686
2-50
067
649
682
070
661
695
2-25
084
654
694
088
666
708
2-0
109
661
711
114
674
727
1-95
116
663
715
120
676
731
1-90
123
665
720
128
679
736
1-85
130
668
725
135
681
741
1-80
139
670
731
144
684
747
75
148
673
737
154
686
753
70
158
676
743
165
690
760
65
169
679
750
176
693
768
60
182
683
758
190
797
776
55
196
687
767
205
701
786
50
213
692
777
222
706
796
45
231
697
788
241
712
808
40
252
703
800
262
718
820
35
276
709
814
289
725
836
30
304
717
830
319
734
853
25
338
726
846
354
743
872
20
377
734
866
396
755
895
15
425
750
894
447
769
921
10
484
765
924
509
785
953
05
557
784
"959
588
805
991
00
651
807
1-OC2
690
831
1-038
See the auxiliary table, p. 136.
422
THE DISCHARGE OF WATER FROM
TABLE V. Coefficients of Discharge for Different Eatios of the
Channel to the Orifice.
Coefficients for heads in still water *650 and '667.
Coefficient -650 for heads in
Coefficient -667 for heads in
still water.
still water.
Ratio
of the
Ratio of
Coefficients
Coefficients
Ratio of
Coefficients
Coefficients
channel
to the
the height
due to the
for orifices :
the heads
for weirs :
the heads
the height
due to the
for orifices :
the heads
for weirs :
the heads
orifice.
velocity of
measured
measured
velocity of
measured
measured
approach
to the
the full
approach
to the
the full
to the head.
centres.
depth.
to the head.
centres.
depth.
so-
000
650
650
000
667
667
so-
001
650
651
001
667
668
15-
002
651
652
002
667
669
lo-
004
651
654
004
668
671
g-
005
652
655
006
669
67-2
s'
007
652
656
007
669
673
7-0
009
653
658
009
670
675
6-0
012
654
661
012
671
678
5-5
014
655
663
015
672
680
5-0
017
656
665
018
673
682
4-5
021
657
669
022
674
687
4-0
027
659
674
029
676
692
3-75
031
660
677
033
678
6'c>6
3-50
036
662
681
038
679
700
3-25
042
663
686
044
681
705
3-0
049
666
692
052
684
711
275
059
669
699
062
687
720
2-50
073
673
709
077
692
731
2-25
091
679
723
096
698
745
2-0
118
687
742
125
707
766
1-95
125
689
747
132
709
771
1-90
133
692
752
140
712
-777
1-85
141
694
758
149
715
783
1-80
150
697
764
159
718
790
1-75
160
700
771
170
721
797
1-70
172
704
779
182
725
805
1-65
184
707
786
195
729
814
1-60
198
711
795
210
733
823
1-55
213
716
805
227
738
833
50
231
721
816
'246
744
846
45
251
727
828
268
751
859
40
275
734
842
293
758
874
35
302
742
858
322
764
888
30
333
751
876
356
776
911
1-25
371
761
896
398
788
934
1-20
415
773
920
446
802
961
1-15
469
788
949
506
818
992
1-10
537
806
983
580
838
1-030
1-05
621
828
T024
675
863
1-076
i-oo
732
855
1-074
800
894
1-133
See the auxiliary table, p. 136.
ORIFICES, WEIES, PIPES, AND RIVERS.
423
TABLE V. Coefficients of Discharge for different Ratios of the
Channel to the Orifice.
Coefficients for heads in still water N /~ 7 5~ = -7071 and 1.
Coefficient '7071 for heads in still
Coefficient 1-000 for heads in still
water.
water.
Ratio of
the
Ratio of
Coefficients
Coefficient
Ratio of
Coefficients
Coefficients
channel
the height
for orifices
for weirs :
the height
for orifices
for weirs :
to the
due to the
the heads
the heads
due to the
the heads
the heads
orifice.
velocity of
measured
measured
velocity of
measured
measured
approach
to the
the full
approach
to the
the full
to the head
centres.
depth.
to the, head
centres.
depth.
30-
001
707
708
001
1-001
1-002
20-
001
708
708
003
1-001
1-004
15-
001
708
709
005
1-002
1-006
lo-
005
709
712
010
1-005
1-014
o-
006
709
713
013
1-006
1-017
s'
008
710
714
016
1-008
1-021
7-
010
711
717
021
1-010
1-028
6-
014
712
721
029
1-014
1-038
5-5
017
713
723
034
1017
1-045
5-0
020
714
727
041
1-021
1-055
4-5
025
716
731
052
1-026
1-067
4-0
032
718
737
067
1-033
1-084
3-75
037
720
742
077
1-038
1-096
3-50
043
722
747
089
1-044
rno
3-25
050
724
753
105
1-051
1-127
300
059
728
760
125
1-061
1-149
2-75
071
732
770
152
1-073
1-178
2-50
087
737
783
190
1-091
1-216
2-25
110
745
801
246
1-116
1-269
200
143
756
826
333
1-155
1-347
T95
151
759
832
356
1-165
1-367
1-90
161
762
839
383
1-176
T389
T85
171
765
846
412
1-188
1-413
1-80
182
769
854
446
1-203
1-441
1-75
195
773
863
484
1-218
1-471
1-70
209
778
873
529
1-237
1-505
1-65
225
783
883
579
1-257
1-543
1-60
243
788
895
641
1-281
1-589
1-55
263
795
908
711
1-308
1-638
1-50
286
802
923
800
1-342
1-699
1-45
312
810
939
903
1-379
1-767
1-40
342
819
958
1-042
1-429
1-854
1-35
378
830
980
1 216
1-489
1-958
1-30
421
842
1'003
1-449
1-565
2-088
T25
471
857
1-033
1-778
1-667
2-259
1-20
532
875
1-066
2-273
T810
2-499
1-15
608
897
1-107
3-100
2-025
2-844
1-10
704
923
1-155
4-762
2-400
3-440
1-05
830
957
1-216
9756
3-280
4-803
1-00
'1-000
1-000
1-293
infinite.
infinite.
infinite.
See the auxiliary table, p. 136, also p. 138,
424
THE DISCHARGE OF WATER FROM
TABLE VI. The Discharge over Weirs or Notches of one foot in length,
in Cubic feet per minute.
Depths I inch to 10 inches. Coefficients -667 to -617.
GREATER COEFFICIENTS.
The Formulce at the heads of the Columns give the Value of the Discharge,
D, in Cubic feet per minute, when I, the length of the Weir, is taken
feet, and the head, h, in inches. For I \fh* we may substitute I h
retaining tJie same standards..
Heads
in
inches.
Theoretical
discharge,
D =
7-72 l\/h3.
Coefficient
667,
D =
5-15 l\fh*.
Coefficient
650.
D =
5-02 J\A3.
Coefficient
639.
D =
4-93 l\/hf.
Coefficient
628.
D =
4-85 Z\A 3 -
Coefficient
617.
D =
4-76 l\/h*.
25
965
644
627
617
606
596
5
2730
1-821
1-775
1-744
1-714
1-684
75
5-016
3-345
3-260
3-205
3-150
3-095
1-
7-722
5-151
5-019
4-934
4-849
4-764
1-25
10-792
7-198
7-015
6-896
6*777
6-659
1-5
14186
9-462
9221
9-065
8-909
8-753
1-75
17-877
11-924
11-620
11-423
11-227
11-030
2-
21-842
14-569
14-197
13957
13-717
13-477
2-25
26-062
17-383
16-940
16-654
16-367
16-080
2-5
30-524
20-360
19-841
19-505
19-169
18-833
2-75
35-215
23-489
22-890
22-503
22'115
21-728
3-
40-125
26-763
26-081
25-640
25-199
24-757
3-25
45-244
30-178
29-408
28-911
28-413
27-915
3-5
50563
33-726
32-866
32310
31-754
31-197
3-75
56-077
37-403
36-450
35-833
35-216
34-599
4'
61-777
41-205
40-155
39-476
38-796
38-116
4-25
67-658
45-128
43978
43-233
42-489
41-745
4-5
73-714
49-167
47-914
47-103
46*292
45-482
4-75
79-942
53-321
51-962
51-083
50-203
49324
5'
86-335
57-585
56-118
55-168
54-218
63269
5-25
92-891
61-958
60-379
59-357
58-335
57-314
5-5
99-604
66-436
64-743
63-647
62-551
61-456
5-75
106-472
71-017
69-207
68-036
66-864
65-693
6-
113-491
75698
73-769
72521
71-272
70024
625
120 657
80-478
78-427
77-100
75-772
74-445
6-5
127-969
85355
83-180
81-772
80-365
78-957
6-75
135-422
90-326
88-024
86-535
85-045
83-555
7*
143-015
95-391
9-2-960
91-387
89-813
88-240
7-25
150-744
100-546
97'983
96-325
94-667
93-009
7-5
158-608
105-792
103-095
101-350
99 606
97-861
7-75
166-604
111-125
108-292
106-460
1046-^7
102-795
8'
174-731
116-546
113-575
111-653
109731
107-809
8-25
182-984
122-05]
118-940
116927
114-914
112-901
8'5
191-365
127-640
124-387
122-282
120-177
118-072
875
199-869
133-313
129915
127716
1-25-518
123-319
9-
208-496
139-067
135-52-2
133-229
130935
128-642
9-25
217243
144-901
141-207
138-818
136-428
134-039
9'5
226-111
150-816
146972
144-485
141-997
139-510
9-75
235-093
156-807
152-810
150225
147-639
145-053
10-
244-193
162-877
158-725
156-039
153-353
150-666
See pp. Ill to 127.
ORIFICES, WEIRS, PIPES, AND RIVERS.
425
TABLE VI. The Discharge over Weirs or Notches of one foot in length,
in Cubic feet per minute.
Depths 10-25 inches to 32 inches. Coefficients -667 to -617.
GEEATER COEFFICIENTS.
The Formula at the heads of the Columns give the Value of the Discharge
D, in Cubic feet per minute, when I, the length of the Weir, is taken in
feet, and the head, h, in inches. For I \fh? we may substitute I h \/Ji,
retaining the same standards.
Heads
in
inches.
Theoretical
discharge
D-
7-72 I \A 3 -
Coefficient
667.
D =
5-15 l\/lfl.
Coefficient
650.
D =
5-02 I \/W.
Coefficient
639.
D =
4-93 I </h?.
Coefficient
628.
D =
4'85 I v/P.
Coefficient
617.
D =
4-76 I \/h~3.
10-25
253-407
169.023
164-715
161-927
159-140
156-352
10-5
262-734
175-244
170-777
167-887
164-997
162-107
10-75
272-173
181-540
176-913
173-919
170-925
167-931
IV
281-723
187909
183-120
180-021
176-92-2
173-823
11-25
291-382
194-352
189-398
186-193
182-988
179 782
11-5
301-148
200-866
195-746
192-434
189' 121
185-808
11-75
311-024
207-451
202-164
198-743
195-321
191-900
12-
321-
214-107
208-650
205-119
201-5&8
198-057
12-5
341-275
227-628
221-826
218-072
214-318
210-564
13-
361-950
241-421
235-268
231-286
227-305
223-323
13-5
383-031
255-482
248-970
244-757
240-543
236-330
14-
404-507
269806
262-930
258-480
254-030
249-581
14-5
426-368
284-387
277-139
272-449
267-759
263-069
15-
448-611
299-223
291-597
286-662
281-728
276-793
15-5
471-228
314-309
306-298
301-115
295-931
290-748
16-
494-212
329-639
321-238
315801
310-365
304-929
165 i 517-558
345-211
336-413
330-720
325-026
319-333
17-
541-261
361 021
351-820
345-866
339-912
333-958
17-5
565-315
377 065
367-455
361-236
355-018
348-799
18-
589715
393340
383'315
376-828
370-341
363-854
18-5
614-443
409-833
399'388
392-629
385-870
379-111
19-
639*533
426-569
415*696
408-662
401 627
394-592
19'5
664-944
443-518
432'214
424-899
417-585
410-270
20-
690-682
460-685
448*943
441-346
433-748
426-151
20-5
716737
478-064
465-879
457*995
450-111
442-227
21-
743 1-25
495 664
483-031
474-857
466-683
458-508
21-5
769-823
513 472
500-385
491-917
483-449
474 981
22-
796 832
531 487
517*941
509-176
500-410
491-645
22-5
824-151
549-709
535*698
526-632
517-567
508-501
23-
851 775
568-134
553-654
544284
534-915
525-545
235
879 700
586.760
571-805
562-128
552-452
542-775
24-
907-925
605-586
590-151
580-164
570-177
560-190
25'
965-253
643 824
627-414
616797
606-179
595-561
26-
1023-748
682-840
665*436
654-175
642-914
631-653
27-
1083-375
722611
704-194
692-277
680-360
668-442
28-
1144-116
763 125
743*675
731-090
718-505
705-920
29'
1205 950
804-369
783*868
770-602
757-337
744-071
30
1268-864
846-332
824-762
810-804
796-847
782-889
31-
1332-833
889 000
866-341
851-680
837019
822-358
32-
1397-842
982-361
908-597
893-221
877-845
862-469
See pp. Ill to 127.
426
THE DISCHARGE OF "WATER FROM
TABLE VI. The Discharge over Weirs or Notches of one foot in length,
in Cubic feet per minute.
Depths 33 inches to 72 inches. Coefficients '607 to -617.
GEEATEE COEFFICIENTS.
The Formula at the heads of the Columns give the Value of the Discharge,
D, in Cubic feet per minute, when I, the length of the Weir, is taken in
feet, and the head, h f in inches. For I \/h 3 we may substitute
retaining the same standards.
Heads
in
inches.
Theoretical
discharge
D =
7'72 l\/M.
Coefficient
667.
D =
5-15 I -V//A
Coefficient
650.
D =
5-02 I \A 3 .
Coefficient
639.
D =
4-93 I \/~ti.
Coefficient
628.
D =
4-85 I \/W.
Coefficient
617.
D =
4-76 l\/M.
33-
1463-875
976-405
951-519
935-416
919314
903-211
34-
1530-917
1021-122
995-096
978-256
961-416
944-576
35-
1598-951
1066-500
1039-318
1021-730
1004-141
986-553
36-
1667-964
1112-532
1084-177
1065-829
1047-481
1029-134
37-
1737-943
1159208
1129-663
1110-546
1091-428
1072-311
38-
1808-875
1206-520
1175-769
1155-871
1135-974
1116-076
39-
1880-746
1254-458
1222-485
1201-797
1181-108
1160-420
40-
1953-544
1303-014
1269-804
1248-315
1226-826
1205337
41-
^027-258
1352181
1317-718
1295-418
1273-118
1250*818
42-
2101-876
1401-951
1366-219
1343-099
,1319-978
1296-857
43-
2177-387
1452'317
1415-302
1391-350
1367-399
1343-448
44-
2253-783
1503-273
1464-959
1440-167
1415376
1390-584
45-
2331-052
1554'812
1515-184
1489-542
1463-901
1438259
46-
2409-183
1606-925
1565-969
1539*468
1512-967
1486466
47-
2488-170
1659-609
1617-311
1589-941
1562-571
1535-201
48-
2568-
1712856
1669-200
1640-952
1612-704
1584-456
49-
2648*666
1766660
1721-633
1692-498
1663-362
1634-227
50-
2730-160
1821-021
1774-604
1744-572
1714-540
1684-509
51-
2812-474
1875'920
1828-108
1797-171
1766-234
1735-296
52-
2895-597
1931-363
1882-138
1850-286
1818-435
1786-583
53-
2979'525
1987-313
1936-691
1903*916
1871-142
1838-367
54-
3064-253
2043-857
1991-764
1958-058
1924-351
1890-644
55-
3149755
2100-887
2047-341
2012-693
1978-046
1943-399
56-
3236050
2158-445
2103433
2067-836
2032-239
1996-643
57-
3323-117
2216-519
2160-026
2123-472
2086-917
2050-363
58-
3410-946
2275-101
2217-115
2179-594
2142-074
2104-554
59-
3499-542
2334-195
2274-702
2236-207
2197712
2159-217
60-
3588'889
2393-789
2332-778
2293-300
22o3-822
2214-344
61-
3678 984
2453-882
2391-340
2350-871
2310-402
2269-933
6*2-
3769-825
2514-473
2450-386
2408-918
2367-450
2325-982
63-
3861-393
2575-549
2509-905
2467-430
2424-955
2382-479
64-
3953-694
2637-114
2569-901
2526-410
2482-920
2439*429
65-
4046720
2699-162
2630-368
2585-854
2541340
2496-826
06-
4140-465
2761-690
2691-302
2645-757
2600-212
2554-667
(37-
4234-922
2824-693
2752-699
2706-115
2659 531
2612-947
68-
4330-086
2888-167
2814-556
2766-925
2719-294
2671-663
69-
4425-954
2952-111
2876-870
2828-185
2779-499
2730-814
70-
4522-516
3016-518
2939-635
2889-888
2840-140
2790-392
71-
4619-774
3081-389
3002-853
2952-036
2901-218
2850-401
7-2-
4717-718
3146-718
3066 518
3014622
3962-727
2910-832
See pp, 111 to
ORIFICES, WEIRS, PIPES, AND RIVEES.
427
TABLE VI. The Discharge over Weirs or Notches of one foot in length,
in Cubic feet per minute.
Depths J inch to 10 inches. Coefficients -606 to '518.
LESSER COEFFICIENTS.
The Formula at the heads of the Columns give the Value of the Discharge,
D, in Cubic feet per minute, when I, the length of the Weir, is taken in
feet, and the head, h, in inches. For l^/ h* we may substitute
retaining the same standards.
Heads
in
inches.
Coefficient
606
D =
44B {*/*?
Coefficient
595
D =
4-59 I \A 3 -
Coefficent
584
D =
4-51 1</M.
Coefficient
562
D =
4-34 I \/W.
Coefficient
540
D =
4-17 l*/h*.
Coefficient
518
D =
l\/ti*.
25
585
574
564
542
521
500
5
1-654
1-624
1-504
1-534
1-474
1-414
75
3-039
2-985
2-929
2-819
2-708
2-598
1-
4-680
4-595
4-510
4-340
4-170
4-000
i-25
6-540
6-421
6-303
6-065
5-828
5-590
To
8-597
8-441
8-284
7-973
7-660
7-348
1-75
10-833
10-637
10-440
10-047
9-653
9-260
2-
13-236
12-996
12-756
12-275
11-795
11-314
225
15-794
15-507
15-220
14-647
14-073
13-500
2-5
18-498
18-162
17-826
17-155
16-483
15-811
2-75
21-340
20-953
20-566
19-791
19-016
18-241
3-
24-316
23-874
23-433
22-550
21-668
20-785
3-25
27-418
26-920
26-422
25-427
24M32
23-436
3'5
30-641
30-085
29-529
28-416
27-304
26-192
3'75
33-982
33-366
32-749
31-515
30-281
29-048
4-
37-437
36-757
36-078
34-719
33-i60
32-000
4-25
41-001
40-256
39-512
38-024
36-535
35-047
4-5
44-671
43-860
43-049
41-427
39-806
38-184
4-75
48-445
47-565
46-686
44-927
43-169
41-410
5-
52-319
51-369
50-420
48-520
46-621
44-722
5'25
56-292
55-270
54-248
52-205
50-161
48-117
5'5
60-360
59-264
58-169
55-977
53-786
51-595
575
64-52-2
63-351
02-180
59-837
57-495
55-153
6-
68-776
67-527
66-279
63-782
61-285
58-788
6-25
73-118
71-791
70-464
67-a09
65-155
62-500
6'5
77-549
76-142
74-734
71-919
69-103
66-288
6'75
82-066
80-576
79-086
76-107
73-128
70-149
7-
86-667
85-094
83-521
80-374
77-228
74-082
7-25
91-351
89*693
88-034
84-718
81-402
78-085
75
96-116
94-372
92*627
89-138
85-648
82-159
7-75
100-962
99-129
97-297
93-631
89-966
86-301
8-
105-887
103-965
102-043
98-199
94-355
90511
8-25
110-889
108-876
106-863
102-837
98-812
94-786
8-5
115-967
113-862
111-757
107-547
103-337
99-127
8-75
121-121
118-922
116-723
112-326
107-929
103-532
9-
126-349
124-055
121-762
117-175
112-588
108-001
9-25
131-649
129-259
126-870
122-090
117-311
112-532
95
137-023
134-535
132-048
127-074
122-100
117-125
975
142-467
139-881
137-294
132122
126-950
121-778
10-
147-991
145-295
142-609
137-237
131-864
126-492
See pp. Ill to
428
THE DISCHARGE OF WATER FROM
TABLE VI. The Discharge over Weirs or Notches of one foot in length,
in Cubic feet per minute.
Depths 10-25 inches to 32 inches. Coefficients -606 to -518.
LESSEE COEFFICIENTS.
The Formula at the heads of the Columns give the Value of the Discharge,
D, in Cubic Jeet per minute, when I, the length of the Weir, is taken in
feet, and the head, h, in inches. For l*J h 3 we may substitute I h <J h>
retaining the same standards.
Heads
in
inches.
Coefficient
606.
D =
^eSZ-v/k 3 -
Coefficient
595.
D =
4-59 l\/W.
Coefficient
584.
D =
4-51 1 \/l#.
Coefficient
562.
D =
4-34 1 -vA.
Coefficient
540.
D =
4-17 l\/h*.
Coefficient
518.
D =
4 I */h\
10-25
153-565
150-777
147-990
142-415
136-840
131-265
10-5
159-217
156-327
153-437
147-657
141-876
136-096
1075
164-937
161-943
158-949
152961
146-974
140-986
11-
170-724
167625
164-526
158-328
152-130
145-933
11-25
176-577
173372
170 167
163-756
157-346
150-936
11-5
182-496
179-183
175-870
169-245
162-620
155-995
11-75
188-479
185-059
181-636
174-794
167-9,52
161-109
12-
194-526
190-995
187-464
180-402
173-340
166-278
12-5
206-810
203-056
199 302
191-794
184-286
176-778
13-
219-342
215-360
211-379
203-415
195-453
187-490
13-5
232-117
227903
223690
215263
206-837
198-410
14-
245-131
240-682
236-232
227333
218-434
209-535
145
258-379
253-689
248-999
239619
230-239
220-859
15-
271-858
266-924
261 989
252-119
242250
232-380
15-5
285-564
280-381
275-197
264.830
254-463
244-096
16-
299-492
294*056
288-6-20
277-747
266875
256001
16-5
313-640
307*947
302-253
290-868
279-481
268-095
17-
328-004
322-050
316-096
304-189
29-2-281
280-373
17-5
342-581
336-362
330-144
317-707
305-270
292-833
18-
357-367
350-880
344-394
381-4-0
318-446
305-472
18-5
372-352
365 594
358-835
345-317
331-799
318-241
19
387-557
380'o22
373-487
359-418
345-348
331-278
19-5
402-956
395-642
388-3-27
373-699
359-070
344-441
20-
418*553
410-959
403-358
388-163
372-968
357-773
20'5
434-343
426-458
418-574
402-806
387-038
371-270
21-
450-334
442-159
433-985
417-636
401-288
384-939
21'5
466-513
458-045
449-577
432-641
415-704
398-768
22-
482-880
474-115
465-350
447-819
430-289
41-2759
22-5
499-436
490-370
481 -304
463-173
445-042
426-910
23-
517-176
506-806
497-437
478-698
459*959
441-219
23'5
533-098
523-421
513-745
494-391
475-038
455 685
24-
550-203
540-215
530-228
510254
490280
470-305
25-
584943
574-326
563*708
542-472
521-237
500-001
26-
620-391
609-130
597 869
575-346
552'824
530-301
27-
656-525
644-608
632-691
608-857
585-023
561-188
28-
693334
680-749
668-164
642-993
61 7-828
592-652
29'
730-806
717-540
734-275
677-744
651-213
624-682
30-
768-932
754-974
711-017
713-102
6^5-187
657-272
31-
807-697
793-036
778-374
749-052
739730
690-407
32-
847-092
831-716
8A6-340
785-587
7b4-835
724-082
See pp. Ill to 127.
OKIFICES, WEIES, PIPES, AND EIVEES.
429
TABLE VI. The. Discharge over Weirs or Notches of one foot in length,
in Cubic feet per minute.
Depths 33 inches to 72 inches. Coefficients -606 to '518.
LESSER COEFFICIENTS.
The Formula at the heads of the Columns give the Value of the Discharge,
D, in Cubic feet per minute, whence I, the length of the Weir, is taken in
feet, and the head, h, in inches. For l^/ h 3 we may substitute I h^/ h,
retaining the same standards.
TT__ J_
Coefficient
Coefficient
Coefficient
Coefficient
Coefficient
Coefficient
Heads
606.
595.
584.
562.
540.
518.
in
D=
D =
D =
D =
D =
D =
inches.
4-88 I\A*'
4-59 I \/W.
4-51 1 \/h*.
4-34 14/h*.
4-17/ <sA a .
4^/13.
33-
887-108
871-006
854-903
822-698
790-493
758-287
34-
927736
910-896
894-056
860-375
8^6-695
793-015
35-
968-964
951-376
933-787
898-610
863-434
828-257
36-
1010-786
992-439
974-091
937-396
900-701
864-005
37-
1053-193
1034-076
1014-959
976-724
938 489
900-254
38-
1096-178
1076-281
1056-383
1016-588
976793
936-997
39-
1139-732
1119-044
1098-356
1056-979
1015-603
974-226
40-
1183-848
116-2-359
1140-870
1097-892
1054-914
1011-936
41-
1228-518
1206-219
1183919
1139-319
1094-719
1050-120
42-
1273-737
1250-616
1227-496
1181-254
1135-013
1088-772
43-
1319-497 1295-545
1271-594
1223-691
1175-789
1127-886
44-
1365792
1341-001
1316-209
1266-626
1217-043
1167-460
45-
1412-618
1386-976
1361334
1310-051
1258-768
1207-485
46-
1459-965
1433-464
1406-963
1353-961
1300-959
1247-957
47-
1507831
1480-461
1453-091
1398-352
1343-612
1288-872
48'
1556-208
1527-960
1499-712
1443216
1386720
1330-224
49'
1605-092
1575-956
1546-821
1488-550
1430-280
1372-009
50-
1654-477
1624-445
1594-413
1534-350
1474-286
1414-223
51-
1704-359
1 673-422
1642-485
1580-610
1518-736
1456-862
52-
1754732
1722-880
1691-029
1627-326
1563-622
1499-919
53-
1805-592
1772-817
1740-043
1674-493
1608-944
1543-394
54-
1856-937
1823-231
1789-524
1722-110
1654-697
1587-283
55-
1908-751
1874-104
1839-457
1770-162
1700-868
1631-573
56-
1961-046 1925-450
1889-853
1818-660
1747-467
1676-274
57-
2013-809 1977-255
1940-700
1867592
1794-483
1721-375
58-
2067-033
2029-513
1991992
1916-952
1841-911
1766870
59-
2120-722
2082-227
2043-733
1966-743
1889-753
1812-763
60-
2174-867
2135-389
2095-911
2016-956
1938-000
1859045
61-
2229-464
2188-995
2148-527
2067-589
1986-651
1905-714
62-
2284-514
2243-046
2201-578
2118-642
2035-706
1952-769
63-
2340-004
2297-529
2255-054
2170-103
2085-152
2000-202
64-
2395-939
2352-448
2308-957
2221-976
2134995
2048-013
65-
2452 312
2407-798
2363-284
2274 257
2185-229
2096-201
66-
2509-122
2463-577
2418-032
2326-941
2235-851
2144-761
67-
2566-363
2519-779
2473194
2380-026
2286-858
2193-690
68-
2624-032
2576-401
2528-770
2433-508
2338-246
2242-985
69-
2682-128
2633-443
2584-757
2487-386
2390-015
2292-644
70-
2740-645
2690-897
2641-149
2541 654
2442-159
2342-663
71-
2799-583
2748-766
2697-948
2596-313
2494-678
2393-043
72-
2858-937
2807-042
2755-147
2651-358
2547*568
2443-778
See pp. Ill to 127.
430
THE DISCHARGE OF WATER FROM
TABLE VII. For finding the Mean Velocity from the Maximum
Velocity at the Surface, in Mill Races, Streams, and Rivers with
uniform Channels ; and the Maximum Velocity from the Mean
Velocity. (See p. 184.)
For the Velocity in feet per minute, multiply by 5.
frfl^
.S
> *
S
* -d
M "
ri .
"S
"3,
"3 1
*O >F * ^
.M "-" fl
'111
'i
','>
'^"3
Ml
B g
3S
> a
>" p,
ill
'3 C 2o
m
d QQ
11$
P.
?ll
Sis,
1
"is &
G S
lls
w a
ss
'5 * J
gal
||I
Pi
00
Sf &>J3
III
Pi-
|&|
III
3*1
m
".9
l-i
333
g P
3*3
s ^l
S P
i
84
75
41
34-24
33-37
81
67-64
68-86
2
1-67
1-51
42
35-07
34-23
82
68-47
69-77
3
2-51
2-27
43
35-91
35-09
83
69-31
7068
4
3-34
3-04
44
36-74
35-95
84
70-14
71-59
5
4-18
3-81
45
37-58
36-82
85
70-98
72-50
6
5-01
4-58
46
38-41
37-69
86
71-81
73-42
7
5-85
5-36
47
3925
38-56
87
72-65
74-33
8
6-68
6-14
48
40-08
39-43
88
73-48
75-24
9
7-52
6-92
49
40-92
40-30
89
74-32
76-16
10
8-35
7-71
50
41-75
41-17
90
75-15
77-08
11
9-19
8-50
51
42-59
42-05
91
75-99
77-99
12
10-02
9-29
52
43-42
42-92
92
76-82
78-91
13
10-86
10-09
53
44-26
43-80
93
77-66
79-83
14
11-69
10-88
54
45-09
44-68
94
78-49
80-75
15
12-53
11-69
55
45-93
45-56
95
79-33
81-67
16
13-36
12-49
56
46-76
46-44
96
80-16
82-59
17
14-20
13-30
57
47-60
47*32
97
81-00
83-51
18
15-03
14-11
58
48-43
48'2t
98
81-83
84-43
19
15-87
14-92
59
49-27
49-09
99
82-67
85-36
20
16-70
15-73
60
50-10
49-98
100
83-50
86-28
21
17-54
16-55
61
50-94
50-87
101
84-34
87-20
22
18-37
17-37
62
51-77
51'76
102
85-17
88-13
23
19-21
18-19
63
52-61
52-65
103
86-01
89-06
24
20-04
19-02
64
53-44
53*54
104
86-84
89-98
25
20-88
19-85
65
54-28
54-43
105
87-68
90-91
26
21-71
20-68
66
55-11
55-33
106
88-51
91-84
27
22-55
21-51
67
55-95
56-22
107
89-35
92-77
28
23-38
22-34
68
56-78
57-12
108
90-18
93-69
29
24-22
23-18
69
57-62
58-02
109
91-02
94-62
30
25-05
24-02
70
58-45
58-91
110
91-85
9555
31
25-89
24-86
71
59-29
59-81
111
92-69
96-49
32
26-72
25-70
72
60-12
60-71
112
93-52
97-42
33
27-56
26-54
73
60-96
61-61
113
94-36
98-35
34
28-39
27-39
74
61-79
62-52
114
95-19
99-28
35
29-23
28-24
75
62-63
63-42
115
96-03
100-21
36
30-06
29-09
76
63-46
64-32
116
96-86
101-15
37
30-90
29-94
77
64-30
65-23
117
97-70
102-08
38
31-73
30-79
78
65-13
66-13
118
98-53
103-02
39
32-57
31-65
79
65-97
67-04
119
99-37
103-95
40
33-40
32-51
80
66-80
67-95
120
100-20
104-89
ORIFICES, WEIRS, PIPES, AND RIVERS.
431
TABLE VIII, For finding the Mean Velocities of Water flowing
in Pipes, Drains, Streams, and Rivers.
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
Diameters of pipes \ inch to 2 inches. Falls per mile 1 inch to
12 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii," and
velocities in inches per second.
Falls.
Inclinations
one in
^inch.
J inch.
J inch.
inch.
J inch.
F. I.
1
63360
14
24
38
49
57
2
31680
22
37
59
76
90
3
21120
28
48
75
97
1-15
4
15840
34
57
89
1-15
1-36
5
12672
38
65
1-02
1-30
1-55
6
10560
42
72
1-13
45
1-72
7
9051
46
78
1-24
58
1-88
8
7920
50
85
1-33
71
202
9
7040
53
'90
1-43
83
2'16
10
6336
57
96
1-51
94
2-30
11
5760
60
I'Ol
1-60
96
2'42
1
5280
63
1-06
1-68
2'15
2'54
1 3
4224
71
1'20
1-90
2-43
2'88
1 6
3520
79
1'33
2'10
2'69
3-19
1 9
3017
87
1-45
2-29 2'94
3'48
2
2640
93
1'56
2-47
3-16
375
2 3
[Interpolated.
99
T67
2'63
3-37
399
2 6
2112
1-05
1-77
279
3-58
4-24
2 9
Interpolated.
1 11
1-87
294
3'77
4-47
3
1760
1-16
1-96
3'09
3'96
4-69
3 3
[nterpolated.
1-21
2-05
3-23
4-14
4-91
3 6
1508
1-26
2-14
3'37
4-32
5-12
3 9
[nterpolated.
1-31
2-22
3'50
4-48
5'31
4
1320
1-36
2-30
3-63
4-65
5-51
4 6
Interpolated.
45
2-45
3'87
4-96
5'88
5
1056
54
2'6t
4-11
5-27
6-24
5 6
Interpolated.
62
2-75
4-33
5'55
6'58
6
880
71
2-89
4-55
5'83
6-91
6 6
Interpolated.
78
3-02
4'76
6'10
7-22
7
754
1-86
3-15
4-97
6-36
7'54
7 6
Interpolated.
193
3-27
5-16
6'61
7'83
8
660
2-01
3-39
5'35
6-86
8-12
8 6
Interpolated.
2-07
3-51
5-53
7-09
8'40
9
587
2-14
3-62
5-72
7-32
8-68
9 6
Interpolated.
2-20
3'74
5-89
7-55
8'94
10
528
2-28
3-85
6-07
I'll
9-21
10 6
Interpolated
2-33
3-95
6-24
7'99
9-47
11
480
2-40
4-06
6-40
8'20
972
11 6
Interpolated.
2-46
4*16
657
8-41
9-97
12
440
2-52
4-27
6-73
8-62
10-21
See p. 195.
432
THE DISCHAEGE OF WATEE FEOM
TABLE VIII. For finding the Mean Velocities of Water flowing
in Pipes, Drains, Streams, and Rivers.
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
Diameters of pipes J inch to 2 inches. Falls per mile 13 feet to
5280 feet.
Falls per mile in feet, and the
hydraulic inclination.
" Hydraulic mean depths," or " mean radii," and
velocities in inches per second.
Falls.
Inclinations
one in
T V inch.
J- inch.
| inch.
inch.
\ inch.
13-2
400
2-66
4-50
7-10
9-10
10-78
13-6
Interpolated.
2-71
4-59
7-24
9-27
1098
14-1
375
2-76
4-67
7-37
9-44
11-18
14-6
Interpolated.
2-82
4-76
7-52
9-63
1141
15-1
350
2-87
4-85
7-66
9-82
11-63
15-6
Interpolated.
2-94
4-96
7'83
10-03
11-88
16'2
325
3-00
5-07
7-99
10-24
12-13
17'6
300
3-14
5-30
8'37
10-72
12-70
19'2
275
3-30
5-58
8-80
11-27
13-35
21-1
250
3-48
5-89
9-39
11-90
14-10
23-5
225
3-70
6-26
9-87
12-65
14-99
26-4
200
3'96
670
10-57
13*54
16-04
30-2
175
4-28
7-24
11-42
14-63
17-33
35'2
150
4-68
7-92
12-49
1600
18-96
37-7
140
4-88
8-24
13-00
16-66
19-74
42-2
125
5'21
8-81
13-90
1780
2r09
48-
110
5-62
9-50
14-98
19-19
22-74
52'8
100
5-94
10-05
15-85
20-30
24-06
587
90
6-33
10-69
16-87
21-61
25-60
66'
80
6-78
11-47
18-10
23-17
27'46
75'4
70
7'35
12-42
19-59
25-09
29*73
88'
60
8'05
13-61
21'48
27-51
32-60
105'6
50
8-99
15-19
23-96
30-69
36-37
117-3
45
9'57
16-18
25-53
32-70
38-75
132-0
40
10'28
17-37
27'41
35-11
41-60
150-8
35
iri4
18-84
29-71
38-06
45-10
176-
30
12-23
20-68
32'62
41-78
49-51
212-2
25
13-66
23-09
36-43
4667
55-30
264-
20
15-64
26-44
41-71
53-43
63-30
352'
15
18-61
31-46
4963
63-57
7533
528-
10
23-73
40-11
63-28
81-06
96-05
586-7
9
25-26
42-70
67-37
86-29
102-25
660'
8
27-08
4578
72-22
92-51
109-61
754-3
7
29-29
49-51
78-10
100-04
118-54
880-0
6
32-05
54-15
85-43
109-43
129-66
1056-
5
35-08
60-15
94-89
121-54
144-02
1320-
4
40-40
68-29
107-73
137-99
163-51
1760-
3
47-48
80-25
126-61
162-17
192-16
2640-
2
59-47
100-53
158-59
203-14
240-70
5280-
1
88-13
148-97
235-02
301-04
356-70
See p. 195.
ORIFICES, WEIRS, PIPES, AND RIVERS.
433
TABLE VIII. For finding the Mean Velocities of Water flowing in
Pipes, Drains, Streams, and Rivers.
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
Diameters of pipes 2J inches to 5 inches. Falls per mile 1 inch to
12 feet..
Falls pei 1 mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii," and
velocities in inches per second.
Falls.
Inclinations
4 inch.
if inch.
inch.
1 inch.
1 J in. in-
one in
O
terpolated.
F. I.
1
63360
65
73
79
85
96
2
31680
1-02
1-13
1-23
1-33
1-49
3
21120
1-30
1-45
1-58
1-70
1-91
4
15840
1-54
1-71
1-87
2-01
2-26
5
12672
1-76
1-95
2-13
2-29
2-58
6
10560
1-95
2-17
2-36
2-55
2-86
7
9051
2-13
2-37
2-58
2-78
313
8
7920
2-30
2-55
2-78
3-00
3-37
9
7040
2-46
2-73
2-98
3-21
361
10
6336
2-61
2-90
3-16
3-40
3'83
11
5760
2-76
3-06
3-33
3-59
4-04
1
5280
2-89
3-21
3-50
3-77
4'24
1 3
4224
3-28
3-64
3-97
4-27
4-81
1 6
3520
363
4-03
4-39
4-73
5'32
1 9
3017
3-96
4'39
4-79
5-16
5'80
2
2640
4-26
4-73
5-16
5-55
6-25
2 3
Interpolated.
4-55
5-04
5-50
5-92
6-66
2 6
2112
4-83
5'35
5-84
6-29
7-07
2 9
Interpolated.
5-09
5-64
6-15
6-12
7-46
3
1760
5-34
5'92
6-46
6-96
7-83
3 3
Interpolated.
5-58
6-19
6-75
7-27
8-18
3 6
1508
5-82
6-46
7-04
7-59
8-53
3 9
Interpolated.
6-05
6-71
7-31
7-88
8-86
4
1320
6-27
6-95
7-58
8-17
9-19
4 6
Interpolated.
6-69
7-42
8-09
8-71
9-80
5
1056
7-10
7-88
8-59
9-25
10-41
5 6
Interpolated.
7-48
8-30
9-05
9-76
10-97
6
880
7-86
8-72
9-51
10-25
11-53
6 6
Interpolated.
8-22
9-12
9-94
10-71
12-05
7
754
8-57
9-51
10-37
11-17
12-57
7 6
Interpolated.
8-92
9-89
10-78
11-62
13-06
8
660
9-24
10-25
11-18
12-04
13-54
8 6
Interpolated.
9-55
10-60
11-56
12-45
14-01
9
587
9-87
10-95
11-94
12-86
14--17
9 6
Interpolated.
10-18
11-28
12-31
13-26
14-91
10
528
10-48
11-62
1267
13-65
15-36
10 6
Interpolated.
10-77
11-95
13-03
14-03
15-78
11
480
11-00
12-27
1338
14-41
16-21
11 6
Interpolated.
1134
12-58
13-72
14-82
16-64
12
440
11-62
12-89
14-05
15-22
17-07
434
THE DISCHARGE OF WATER FROM
TABLE VIII. For finding the Mean Velocities of Water flowing in
Pipes, Drains, Streams, and Rivers.
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
Diameters of pipes 2J inches to 5 inches. Falls per mile 13 feet to
5280 feet.
Falls per mile in feet, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii,"
and velocities in inches per second;
Falls.
Inclinations
one in
g inch.
| inch.
I inch.
1 inch.
1 in. in-
terpolated.
F.
13-2
400
12-26
13-60
14-83
15-98
17-98
13-6
Interpolated.
12-49
13-86
15-11
16-28
18-31
14-1
375
12-72
14-11
15-39
16-58
1865
14-6
Interpolated.
12-98
14-39
15-70
16-91
19-02
15-1
350
13-23
14-68
16-00
17-24
19-40
15'6
Interpolated.
13-52
14-99
16-35
17-62
19-81
16-2
325
1380
1531
16-79
17-99
20-23
17-6
300
14-45
16-02
17-48
18-83
21-18
19'2
275
15-19
16-85
1837
19-79
22-26
21'1
250
16-04
17-80
1940
20-91
23-52
23-5
225
1705
18-91
20-62
22-21
24-99
26-4
200
18-25
20-24
22-07
23-78
26-75
30-2
175
19-71
21-87
23-85
25-69
28-90
35-2
150
21-57
23-92
26-09
28-11
31-62
37-7
140
22-45
24-91
27-16
29-26
32-92
42-2
125
23-99
26-62
29-03
31-27
35-18
48-
110
25-87
28-69
31-29
33-71
37-92
62'8
100
27-36
30-35
33-10
35-66
40-11
587
90
29-12
32-31
35-23
37-96
42-69
66'
80
31-23
34-64
37-78
40-70
45-79
75'4
70
33-82
37-51
40-91
44-07
49-58
88-0
60
37-08
41-13
44-86
48-33
54-36
105'6
50
41-37
45-78
50-04
5391
60-65
117-3
45
44-08
4889
53-32
57-44
64-62
132-
40
47-32
52-49
57-25
61-67
69-37
150'8
35
51-30
56-90
62-06
66-86
75-20
176-
30
56-32
62-47
68-13
73-40
82-56
211-2
25
62-90
69-77
76-09
81-97
92-21
264-
20
72-01
79-87
87-11
93-84
105-56
352-
15
85-68
95-05
103-66
111-67
125-61
528-
10
109-26
121-19
132-17
142-39
160-17
586-7
9
116-31
129-01
140-70
151-58
170-50
660-
8
124-68
138-30
150-83
162-49
182-78
754-3
7
134-84
149-57
163-12
17573
197-67
880-
6
147-69
163-60
. 178-42
192-22
21622
1056-
5
163-82
181-71
198-17
213-50
240-15
1320-
4
185-99
206-31
225-00
242-39
27266
1760-
3
218-58
242-46
264-42
284-86
320-43
2640-
2
273-79
303-70
331 22
356-82
401-37
5280-
1
405-74
450-07
490-84
528-79
594-82
ORIFICES, WEIRS, PIPES, AND RIVERS.
435
TABLE VIII. For finding the Mean Velocities of Water flowing in
Pipes, Drains, Streams, and Rivers.
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
OPEN DEAINS AND PIPES,
Diameters of pipes 6 indies to 13 inches. Falls per mile 1 inch
to 12 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii," and
velocities iu inches per second.
Falls.
Inclinations,
IT? inch.
If- in. in-
2 inches.
2 J inches
3 inches.
one in
terpolated.
F. I.
1
63360
MW
1-15
1-24
1-40
1-55
2
31680
1-66
1-80
1-94
2-19
2-41
3
21120
2-12
2-30
2-48
2-80
3-08
4
15840
2-52
2-73
2-94
334
3-65
5
12672
2-86
3-11
3-35
3-77
4-16
6
10560
3-18
3-45
3-72
4-19
4-62
7
9051
3-47
3-77
4-06
4-58
5-04
8
7920
3-75
4-06
4-38
4-94
5-44
9
7040
4-01
4-34
4-68
5-28
5-81
10
6336
4-25 4-61
4-97
5-60
617
11
5760
4-49
4-86
5-24
5-91
6-51
1
5280
4-71
5-11
551
6-21
6-84
1 3
4224
5-34
5-79
6-24
7-03
775
1 6
3520
5-91
6-41
6-91
7-79
8-58
1 9
3017
6-44
6-99
7-53
8-49
9-35
2
2640
6-94
7-53
8-11
9-14
10-07
2 3
[nterpolated.
7-40
8-03
8'65
9-74
10-74
2 6
2112
7-86
8-52
9-18
10-35
11-40
2 9
Interpolated.
8-28
8-98
9-67
10-90
12-01
3
1760
8-70
9-43
10-16
11-45
12-62
3 3
[nterpolated.
9-09
9-85
10-62
11-97
13-19
3 6
1508
9-48
10-28
11-08
12'48
13-76
3 9
[nterpolated.
9-84
10-67
11-50
12-96
14-29
4
1320
10-21
11-07
11-93
13-44
14-81
4 6
interpolated.
10-89
11-80
12-72
14-34
15-80
5
1056
11-56
12-54
13-51
15-23
16-78
5 6
Interpolated.
12-18
13-21
14-24
1604
17-68
6
880
1280
13-88
14-96
16-86
18-58
6 6
Interpolated.
13-38
14-51
15-64
17-62
19-42
7
754
13-96
15-14
16-32
18-39
20-26
7 6
Interpolated.
14-51
15-73
16-95
19-10
21-05
8
660
15-05
16-32
17-58
19-82
21-84
8 6
Interpolated.
15-56
16-87
18-18
20-49
2-2-58
9
587
16-07
17-4*3
18-78
21-17
23-32
9 6
Interpolated.
16-57
17-97
1936
21-82
24-04
10
528
17-06
18-50
1994
22-47
24-76
10 6
Interpolated,
17-54
19-01
20-49
23-09
25-45
11
480
18-01
1953
21-04
23-72
26-13
11 6
Interpolated.
18-47
20-02
21-57
24-32
26-79
12
440
18-92
20-51
22-11
24-91
27-45
PF2
436
THE DISCHARGE OF WATER FROM
TABLE VIII. For finding the Mean Velocities of Water flowing in
Pipes, Drains, Streams, and Rivers,
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
Diameters of pipes 6 inches to 14 inches. Falls per mile 13 feet
to 5%8Qfeet.
Falls per mile in feet, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii," and
velocities in inches per second.
Tails.
Inclinations,
one in
1 J inch.
2 inches.
2 J inches.
3 inches.
i^- inches.
13-2
400
19-97
23-34
26-30
28-98
31-44
13-6
20-34
23-77
26-79
29-52
32-03
14-1
375
20-72
24-21
27-28
30-06
32-62
14-6
21-13
24-69
27-83
30-67
33-27
15-1
350
21-55
25-18
28-38
31-27
33-93
15-6
22-01
25-72
28-99
31-94
34-66
16-2
325
22-48
26-27
29-60
32-62
35-39
17-6
300
23-53
27-50
30-99
34-15
37-05
19-2
275
24-74
28-90
32-57
35-89
38-94
21-1
250
26-13
30-53
34-41
37-91
41-14
23-5
225
27-76
32-44
36-56
40-28
43-71
26-4
200
29-72
34-72
39-13
43-12
46-79
30-2
175
32-11
37-52
42-28
46-59
5055
35-2
350
35-13
41-04
46-26
50-97
55-30
37-7
140
36-57
42-73
48-16
53-07
57-58
42-2
125
39-08
45-66
51-46
56-71
61-53
48*
110
42-13
49-23
55-48
61-13
6633
52'8
100
44-57
52-07
58-69
64-67
70-17
587
90
47-43
55-42
62-46
68-83
74-68
66*
80
50-87
59-44
66-99
73-81
80-09
75-4
70
55-08
64-36
72-50
79-92
86-72
88-
60
60-39
70-57
79-53
87-63
95-09
105*6
50
67-38
78-73
88-73
97-77
106-08
117-3
45
71-79
83-88
94-54
104-17
113-03
132'
40
77-07
90-06
101-50
118-84
121-35
150-8
35
83-55
97-63
110-03
121-24
131-55
176-
30
91-72
107-18
120-79
133 10
144-41
211-2
25
102-44
11970
134-90
148-65
161-29
264-
20
117-28
137-03
154-44
170-18
184-65
352-
15
139-56
163-06
183-78
202-50
219-72
528-
10
177-95
207-92
234-33
258-21
280-16
586-7
9
189-43
221-34
249-45
274-87
298-24
660-
8
203-07
237-28
267-42
294-67
31972
754-3
7
219-61
256-61
289-20
318-67
345-77
880-
6
240-22
281-36
316-33
348-57
378-20
1056-
5
266-81
311-75
351-35
387-15
420-07
1320-
4
302-92
353-95
398-91
43955
47693
1760-
3
356-00
415-96
468-80
516-57
560-49
2640-
2
44593
521-04
587-22
647-06
702-08
5280-
1
660-84
772-16
87023
958-91
1040-44
ORIFICES, WEIRS, PIPES, AND RIVERS.
437
TABLE VIII. For finding the Mean Velocities of Water flowing
in Pipes, Drains, Streams, and Rivers.
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
Diameters of pipes 14 inches to 22 inches. Falls per mile 1 inch
to 12 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii,"
and velocities in inches per second.
Falls.
Inclinations,
one in
3 \ inches.
4 inches.
4J inches.
5 inches.
5 \ inches.
F. I.
1
63360
1-68
1-80
1-91
2-02
2-13
2
31680
2-61
2-81
2-98
3-15
3-32
3
21120
3-34
3-59
3-82
4-03
4-24
4
15840
3-96
4-25
4-52
4-78
5-02
5
12672
4-51
4-84
5-15
5-44
5-72
6
10560
5-01
5-37
5-72
604
6-35
7
9051
5-47
5-87
6-24
6-60
6-94
8
7920
5-90
6-33
6-74
7-12
7-48
9
7040
631
6-77
7-20
7-61
8-00
10
6336
6-70
7-18
7-64
8-08
8-49
11
5760
7-06
7-58
8-06
8-52
8-96
1
5280
7-42
7-96
8-47
895
9-41
1 3
4224
8-41
9-02
9-60
10-14
10-66
1 6
3520
9-31
9-99
10-63
11-23
11-80
1 9
3017
10-15
10-89
11-58
12-24
12-86
2
2640
1093
11-73
12-47
13-18
13-86
2 3
Interpolated.
11-65
12-50
13-30
1405
14-77
2 6
2112
12-37
13-28
14-12
14-93
15-69
2 9
Interpolated.
13-03
13-68
14-88
15-72
16-53
3
1760
13-69
14-69
15-63
16-52
17-36
3 3
Interpolated.
14-31
15-35
16-33
17-26
18-14
3 6
1508
14-92
16-01
17-03
18-00
18-92
3 9
Interpolated.
15-50
16-63
17-69
18-70
19-65
4
1320
16-07
17-25
18-35
1939
20-38
4 6
Interpolated.
17-14
18-39
19-56
20-68
21-73
5
1056
18-21
19-53
20-78
2196
23-08
5 6
Interpolated.
19-18
20-58
21-90
23-14
24-32
6
880
20-16
21-63
23-01
24-32
25-56
6 6
Interpolated.
21-07
22-61
24-05
25-42
26-72
7
754
21-98
23-59
25-09
26-52
27-87
7 6
Interpolated.
22-84
24-50
26-07
27-55
28-96
8
660
23-69
25-42
27-04
28-58
30-04
8 6
Interpolated.
24-50
26-29
27-97
29-55
31-06
9
587
25-31
27-54
28-89
30-53
32-09
9 6
Interpolated.
26-09
27-99
29-78
31-47
3308
10
528
26-87
28-83
30-67
32-41
34-06
10 6
Interpolated.
27-61
29-62
31-52
3331
35-01
11
480
28-35
30-42
32-37
34-20
35-95
11 6
Interpolated.
29-07
31-19
33-18
35-07
36'86
12
440
29-79
31-96
34-00
35-93
37-77
438
THE DISCHARGE OF WATER FROM
TABLE VIII. For finding the Mean Velocities of Water flowing
in Pipes, Drains, Streams, and Rivers.
For a full cylindrical pipe, divide the diameter by 4 to find the
hydraulic mean depth.
Diameters of pipes 16 inches to 2 feet. Falls per mile 13 feet to
SSSOfeet.
Falls per mile in feet
and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii,"
and velocities in inches per second.
Falls.
Inclinations,
one in.
4: inches.
4:3% inches.
5 inches.
5i-r inches.
6 inches.
13-2
400
33-74
35-89
37-93
3987
41-72
13-6
Interpolated.
34-37
36-56
38-64
40-61
42-50
14-1
375
35-00
3723
39-35
41-36
43-28
14-6
Interpolated.
35-70
37-98
40-14
42-19
44-15
15'1
350
36-40
38-73
40-92
43-02
45-01
15-6
Interpolated.
37-19
39-56
41-81
43-94
45-99
10-2
325
37-97
40-40
42-69
44-87
46-96
176
300
39-75
4229
44-69
46-97
49-16
19-2
275
41-78
44-45
46-97
49-38
51-67
21-1
250
4414
46-95
49-62
52-16
54-58
23-5
225
46'90
49-90
52-72
55-42
58-00
26-4
200
50-20
53-41
56-44
59-32
62-08
30-2
175
54-24
57-71
60-98
64-10
67-07
35-2
150
59-34
63-13
66-71
7012
73-37
37-7
140
61-78
65-72
69-45
73-00
76-39
42-2
125
66-02
70-23
74-22
78'Gl
81-64
48-
110
71-17
75-72
80-01
84-10
8800
52-8
100
75-29
80-09
84-64
sa-97
93-10
58-7
90
80 13
85-25
90-08
94*69
99-09
66-
80
85-93
91-42
96-61
101-54
106-26
75-4
70
93-04
98-98
104-60
109-95
115-05
88-
60
102-02
108-54
114-70
120-56
126-16
105-6
50
113-82
121-09
l-27'96
13450
140-74
117-3
45
121'27
129-01
136-34
143-30
14996
132-
40
130-20
138-51
146-38
153-86
161-00
150-8
35
141 14
150-16
158-68
166-79
17453
176-
30
154-95
164-84
174*20
183-10
191-61
211-2
25
173-05
184-10
194-56
204-50
214-UO
264-
20
198-12
210-77
222-73
234-11
244-98
352-
15
235-75
25080
265-04
278-58
291-52
528-
10
300-60
319-80
337-95
355-22
371-71
586-7
9
320-00
34043
359-76
378-14
395-70
660-
8
343-04
35965
385-67
40537
424-20
754-3
7
370-99
394-68
417-08
438-39
458-76
880-
6
405-79
431-70
456-21
47952
501-79
1056-
5
450-71
479-49
506-71
532-60
557-34
1320-
4
511-72
54439
575-30
604-69
63278
1760-
3
601-38
639-78
676-10
71064
743-65
2640-
2
753-29
80139
846 '89
890-16
931-50
5280-
1
1116-35
1187-62
1255-04
1319-17
1380 44
ORIFICES, WEIRS, PIPES, AND RIVERS.
439
TABLE VIII. For finding the Mean Velocities of Water flowing
in Pipes t Drains, Streams, and Rivers.
The hydraulic mean depth is found for all channels, by dividing
the wetted perimeter into the area.
Hydraulic mean depths 6 inches to 10 inches. Falls per mile 1 inch
to 12 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii,"
and velocities in inches per second.
Falls.
Inclinations,
one in
6 inches
7 inches.
8 inches
9 inches
10 inches.
F. I.
1
63360
2-23
2-41
2-58
2-75
2-90
2
31680
3'47
3-76
4-03
4-28
4-52
3
21120
4-43
4-80
5-15
5-47
5'78
4
15840
5-26
5-69
6-10
6-49
6-85
5
12672
5-98
6-48
6-95
7-39
7-80
6
10560
665
7-20
7-72
8-20
8-66
7
9051
7-26
7-86
8-43
8-96
9-46
8
7920
7-83
8-48
909
9-67
10-21
9
7040
837
9-07
9-72
10-33
10-91
10
6336
8-88
9-63
10-32
10-97
11-58
11
5760
9-37
10-16
10-89
11-57
12-22
1
5280
9-84
1067
11-43
12-15
12-83
1 3
4224
11-16
1209
12'95
13-77
14-54
1 6
3520
12-35
13-38
14'34
15-25
16-10
1 9
3017
13-46
14-58
15'63
16-61
17-54
2
2640
14-50
1571
16'84
17-90
18-90
2 3
Interpolated.
15-45
16-75
18'24
19-08
20-15
2 6
2112
16-42
17-79
19'64
20-26
21-40
2 9
Interpolated.
17-29
18-74
20*37
2134
22-54
3
1760
18-17
19-69
21-10
22-42
23-fi8
3 3
Interpolated.
18-99
20-57
22-05
23-43
24-75
3 6
. 1508
19-80
21-46
23'00
24-44
2581
3 9
Interpolated.
20-56
22-28
23'88
2o'38
26-80
4
1320
21-33
2311
24-77
26-32
27-80
4 6
interpolated.
22-74
2464
26-41
28-07
29-64
5
1056
24-16
26-17
28-05
2981
31-48
5 6
interpolated.
25-45
27-58
29-56
31-42
33-17
6
880
26-75
28-98
31-06
33-02
34-86
6 6
nterpolated.
27-96
3029
32'47
34-51
36-44
7
754
29-17
31-60
3387
36-00
3802
7 6
Interpolated.
3030
32-83
35-19
37-40
39-50
8
660
31-43
34-06
3650
38-80
40-97
8 6
Interpolated.
32-51
35-22
37-75
4012
42-37
9
587
33-58
3639
38'99
41-45
43-77
9 6
nterpolated.
34-61
3750
40'20
4272
45-11
10
528
35-65
38-63
41'40
4400
46-46
10 6
'nterpolated.
36-63
39-69
42'54
45-22
47-75
11
480
37'62
40-76
43 '69
4644
49-03
11 6
nterpolated.
38-57
41-79
4479
47-61
50-27
12
440
39-52
42-82
45-90
48-78
51-51
440
THE DISCHARGE OF WATER FROM
TABLE VIII. For finding the Mean Velocities of Water flowing
in Pipes, Drains, Streams, and Rivers.
The hydraulic mean depth is found for all channels by dividing
the wetted perimeter into the area.
Hydraulic mean depths 11 inches to 91 inches. Falls per mile 1 inch
to 12 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii," and
velocities in inches per second.
Falls.
Inclinations,
one in
11 inches.
12 inches.
15 inches.
18 inches.
21 inches.
F. I.
1
63360
3-05
3-19
3-57
3-92
425
2
31680
4-75
4-97
5-57
6-12
6-62
3
21120
6-07
6-35
7-12
7-82
8-46
4
15840
7-19
7-53
8-44
9-27
10-03
6
12672
8-19
8-57
9-61
10-55
11-42
6
10560
9 10
9-52
10-67
11-72
12-68
7
9051
9-94
10-39
11-66
12-80
13-85
8
7920
10-72
11-21
12-57
13-81
1494
9
7041
11-46
11-99
13'44
14-76
15-97
10
6336
12-16
12-72
14-27
15-66
16-95
11
5760
12-83
13-42
15-05
16-53
17*88
1
5280
13-48
14-09
15-81
17-36
18-78
1 3
4224
15-27
15-97
17-91
19-67
21-28
1 6
3520
1691
17-68
19-83
21-78
23-56
1 9
3017
18-23
19-27
21-62
23-73
25-68
2
2640
19-85
20-76
23-28
25-63
27-66
2 3
Interpolated.
21-16
22'13
24'82
27-29
2949
2 6
2112
22 -4-8
23-51
26'36
28-95
31-32
2 9
Interpolated.
23-68
24-76
2777
30-49
3299
3
1760
24-88
26-02
29*18
32-04
34-67
3 3
Interpolated.
25-99
27-18
30-47
33-48
36-22
3 6
1508
27-11
28-35
31-77
34-92
37-78
3 9
Interpolated.
28-15
29'45
33-01
36-26
39-23
4
1320
29-20
30-54
34-25
37-60
40-69
4 6
Interpolated.
31-13
32-56
36-52
40-10
43-39
5
1056
33-07
34-59
38-79
42-59
46*09
6 6
Interpolated.
34-85
36-44
40-87
44-88
48-56
6
880
36'62
38-30
42-95
47-16
51-03
6 6
Interpolated.
38-28
40-03
44'90
49-30
5334
7
754
3993
41-76
46"84
51-43
55-65
7 6
Interpolated.
41-48
43-39
46"66
53-43
57-81
8
660
43-04
45-01
50'48
55-42
59-97
8 6
Interpolated.
44-50
46-54
52'20
57-32
62-02
9
587
45-97
48-08
53 "92
59-21
64-06
9 6
Interpolated.
47-39
49-56
55'58
61-03
66-04
10
528
48-80
51-04
57'24
62-85
68-01
10 6
Interpolated.
50-15
52-45
58'83
64-59
69-89
11
480
51-51
53-87
60'41
66-33
71-78
11 6
Interpolated.
52'81
55-23
61'94
6&-01
73-59
12
440
54-11
56-59
63'47
69-68
75-40
OEIFICES, WEIRS, PIPES, AND RIVERS.
441
TABLE VIII. For finding the Mean Velocities of Water flowing
in Pipes, Drains, Streams, and Rivers.
The hydraulic mean depth is found for all channels by dividing
the wetted perimeter into the area.
Hydraulic mean depths 24 inches to 4 feet. Falls per mile 1 inch
to 12 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations,
" Hydraulic mean depths," or " mean radii,"
and velocities in inches per second.
Falls.
Inclinations,
one in
24 inches.
30 inches.
36 inches.
42 inches.
48 inches.
F. I.
1
63360
4-54
5-09
5-59
6-04
6-47
2
31680
7-09
7-94
8-71
9-42
10-08
3
21120
9-06
10-15
11-14
12-04
12-89
4
15840
10-73
12-03
13-20
14-27
15-27
5
12672
1-2-22
13-69
15-03
16-25
17-39
6
10560
13-57
15-21
16-69
18-05
19-31
7
9051
14-83
16-61
1823
19-71
21-09
8
79-20
15-99
17-92
19-66
21-27
22-76
9
7041
17-10
19-16
21-02
22-73
24-33
10
6336
18-15
20-33
22-31
2413
25-82
11
5760
19-15
21-45
23-54
25-46
27-24
1
5280
20-11
22-53
24-72
2673
28-61
1 3
4224
22-78
25-53
28-01
3029
32-42
1 6
3520
25-23
28-27
31-02
33-54
35-90
1 9
3017
27-49
3081
3380
36-55
39-12
2
2640
29-62
33-18
36-41
3938
42-14
2 3
Interpolated.
31-57
35-38
38-82
41-98
44-92
2 6
2112
33-53
37-57
41-22
44-58
47-71
2 9
Interpolated.
3532
39-58
43-43
46-96
50-26
3
1760
37-11
41-58
45-63
49-34
5281
3 3
InterDolated.
38-78
43'45
47-68
51-56
55-18
3 6
1508
40-45
45-32
49-73
53*78
57-55
3 9 Interpolated.
4 13-20
42-00
4356
47-07
48-81
51-64
53-56
5585
57-92
5977
61-98
4 6
Interpolated.
46-45
5205
57-11
61 76
66-09
5
1056
4934
5528
60-66
65-60
70-20
5 6
Interpolated.
51-99
58-25
63-91
69-12
73-97
6
880
54-63
61-22
67-17
72-64
7774*
6 6
Interpolated.
57-11
6399
70-21
75-93*
81-25
7
754
59-58
66-76
73-25
7921
84-77
7 6
Interpolated.
6l-9
69-35
76-09*
87-29
88-06
8
660
64-21
71-94
78-94
85-37
91-35
8 6
Interpolated.
66-40
74-40
81-63
88-26
94-47
9
587
68-59
76-85*
84-32
91-19
97-59
9 6
Interpolated.
70-60
79-22
86-92
94'00
100-59
10
528
72-81
81-58
89-52
96-81
103-60
10 6
Interpolated.
74-83
83-84
91-99
99-49
106-47
11
480
76-84*
86-10
94-47
102-17
109-33
11 6
Interpolated.
78-78
88-28
96-86
104-75
112 10
12
440
80-72
90-45
99-25
107-33
114-86
442
THE DISCHAEGE OP WATER FROM
TABLE VIII. For finding the Mean Velocities of Water flowing
in Pipes, Drains, Streams, and Rivers.
The hydraulic mean depth is found for all channels hy dividing
the wetted perimeter into the area.
Hydraulic mean depths 4 feet 6 inches to 7 feet. Falls per milel inch
to 12 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or " mean radii,"
and velocities in inches per second.
Falls.
Inclinations,
one in
54 inches.
50 inches.
66 inches.
72 inches.
34 inches.
F. I.
1
63360
6-86
7-24
7-60
7-94
8-58
2
31680
10-70
11-29
11-85
12-38
13-39
3
21120
13-68
14-62
15-14
15-83
17-11
4
15840
16-21
17-10
17-95
18-76
20-28
5
12672
18-46
19-47
20-43
21-35
23-13
6
10560
2050
21-63
22-70
23-72
25-64
7
9051
22-39
23-62
24-79
25-90
28-00
8
7920
24-16
25-48
26-74
27-95
30-21
9
7041
25-83
2724
28-59
29-88
32-30
10
6336
27-41
28-91
30-34
31-71
34-28
11
5760
28-92
30-51
32-01
3346
36-17
1
5280
30-37
32-03
33-62
35-13
37-98
1 3
4224
34-41
36-30
3-iO
39-81
43-04
1 6
3520
38-10
40-19
42-18
44-08
47-65
1 9
3017
41-5-2
43-80
45-97
48-04
51-93
2
2640
44-73
47-18
49-52
51-75
55-94
2 3
interpolated.
47-69
50-30
2-79
55-17
59-64
2 6
2112
50-65
53-42
56-07
58-59
63-34
2 9
interpolated.
53-35
56-28
59-06
61-72
66-72
3
1760
56-06
59-13
62-05
64-85
70-10
3 3
interpolated.
58-57
61-79
64-84
67-76
73-25
3 6
1508
61-09
64-44
67-63
7067
76-40*
3 9
[interpolated.
63-44
6692
70-^3
73-39
79-35
4
1320
6580
69-41
72-84
76-11*
8229
4 6
[interpolated.
70-16
7401
77-67*
81-16
87-74
5
1056
74-52
78-61*
82-50
86-21
93-20
5 6
Interpolated.
78-52*
82-83
86-92
90-84
98-20
6
880
82-5-2
87-05
91-35
95-46
103-20
6 6
Interpolated.
86-25
90-98
95-58
99-78
107-87
7
754
89-99
9492
99-62
104-10
11254
7 6
Interpolated.
93-48
98-61
103-48
108-14
116-91
8
660
96-98
102-30
107-35 112-19
121-28
8 6
Interpolated
100-29
105-79
111-02
116-01
125-42
9
587
103-59
109-27
114-68
119-84
129-56
9 6
Interpolated
10678
112-64
118-21
123-53
133-55
10
528
109-97
116-01
121-74
127-22
137-54
10 6
Interpolated
113-02
119-22
12511
130-74
141-34
11
480
116-06
12-2-43
128-48
134-27
145-15
11 6
Interpolated
119-00
125-52
131-73
137-66
148-82
12
440
121-93
128-61
134-97
141-05
152-49
ORIFICES, WEIKS, PIPES, AND EIVEES.
443
TABLE VIII. For finding the Mean Velocities of Water flowing in
Pipes, Drains, Streams, and Rivers.
The hydraulic mean depth is found for all channels by dividing
the wetted perimeter into the area.
Hydraulic mean depths 8 feet to 12 feet. Falls per mile 1 inch to
19 feet.
Falls per mile in feet and
inches, and the
hydraulic inclinations.
" Hydraulic mean depths," or "mean radii," and
velocities in inches per second.
Falls.
Inclinations,
one in
96 inches.
108
inches.
120
inches.
132
inches.
14.4
inches.
F. I.
1
63360
9-18
9-75
10-28
10-79
11-27
2
31680
14-32
15-20
16-03
16-82
17-57
3
21120
18-30
19-43
20-49
21-50
22-46
4
15840
2169
23-02
24-28
25-47
26-62
5
12672
24-70
26-21
27-64
29-00
30-31
6
10560
27-43
29-11
30-70
32-21
33-66
7
9051
29-96
31-80
33-53
35-18
36-76
8
7920
32-32
34-30
36-18
37-96
39-66
9
7041
34-55
36-67
38-67
40-58
42-40
10
6336
36-67
38-92
41-04
43-07
45-00
11
5760
38-69
41-06
43-31
45-44
47-48
1
5280
40-63
43-12
45-48
47-72
4986
1 3
4224
46-04
48-87
51-54
54-07
56-50
1 6
3520
50-98
54-11
57-06
59-87
62-56
1 9
3017
55-60
58-96
62-18
65-25
68-17
2
2640
59-85
63-52
66-98
70-28
73'44*
2 3
Interpolated.
63-80
67-72
71-41
74-93*
78-29
2 6
2112
67-76
71-91.
75-84*
79-58
83-15
2 9
Interpolated.
71-38
75-75*
79-89
83-83
87-59
3
1760
75-00*
7959
83-94
88-08
92-03
3 3
Interpolated.
7837
83-17
87-71
92-03
06-16
3 6
1508
81-74
86-75
91-48
95-99
100-30
3 9
Interpolated.
84-88
90-09
9501
9969
104-16
4
1320
88-03
93-43
98-53
103-38
108-02
4 6
Interpolated.
93-87
99-62
105-06
110-24
115-18
5
1056
99-70
105-82
111-59
117-09
122-34
o 6
Interpolated.
105-06
111-49
11758
123-38
128-91
6
880
110-41
117-17
123-57
129-66
135-48
6 6
Interpolated.
115-40
122-47
129 16
135-53
141-61
7 .
754
12D -40
127-76
134-75
141-39
147-73
7 6
Interpolated.
125-07
132-74
139-99
146-88
153-47
8
660
129-75
137-70
145-22
152-38
15921
8 6
Interpolated.
134-18
142-40
15U-18
157-57
164-64
9
587
138-00
147-10
155-13
162-77
170-07
9 6
Interpolated.
14287
151-63
159-91
167-78
175-31
10
528
147-14
156-16
164-68
172-80
180-55
10 6
Interpolated.
151-21
160-48
169-24
177-58
18555
11
480
155-29
164-80
173-80
182-36
190-54
11 6
Interpolated.
15921
168-97
178-19
186-97
195-38
12
440
163-13
173-13
182-59
191-58
200-17
444
THE DISCHARGE OF WATER FEOM
TABLE IX. For finding the Discharge in Cubic Feet per Minute,
when the Diameter of a Pipe, or Orifice, and the Velocity of
Discharge are known ; and vice versa.
sss
$ .,3
e
5.s
Discharge in cubic feet per minute, for different velocities.
Velocity of
100 inches
per second.
Velocity of
200 inches
per second.
Velocity of
300 inches
per second.
Velocity of
400 inches
per second.
Velocity of
500 inches
per second.
i
170442
3409
5113
6818
8522
1
68177
1-3635
2-0453
2-7271
3-4089
1
1-53398
3-0679
4-6019
6-1359
7-6699
2-727077
5-4541
8-1812
10-9083
13-6354
it
4-26106
8-5221
12-7832
17-0442
21-3053
1}
6-13593
12-2718
18-4080
24-5437
30-6797
if
8-35167
16-7033
25-0550
33-4067
41-7584
2
10-90831
21-1817
32-7249
43-6332
54-5415
2J
13-80583
27-6117
41-4175
55-2233
69-0291
&t
17-04423
34-0885
51-1327
68-1769
85-2212
2f
2062352
41 2470
61-8706
824941
103-1176
3
24-54369
49-0874
73-6311
98-1748
121-7185
3i
28-80475
57-6095
864143
115-2190
144-0238
3
33-40669
66-8134
100-2201
133-6268
167-0335
3|
38-34952
76-6990
115-0486
153 3981
191-7476
4
43-63323
87-2665
130-8997
174-5329
218-1662
4J
49-25783
98-5157
147 7735
197-0313
2462892
if
5522331
110-4466
165-6699
2208932
276-1166
4|
61*52968
123-0594
184-5890
246-1187
307-6484
5
68-17692
136-3539
204-53U8
2727077
340-8846
6*
75-16506
150-3301
225-4952
300 6603
375-8253
6*
82-49408
1649882
247-4822
329-9763
412-4704
6f
90-16399
180-3280
270-49-20
360-6560
450-8200
6
98-17478
196-3495
294-5243
392-6991
490-8739
6*
106-5-2645
213-0529
319-5794
426-1058
532-6323
6J
115-2190
230-4380
345-6570
460-8760
576-0950
6|
124-25245
248-5049
3727574
497-0098
621-2623
133-6268
267-2536
400-8804
534-5072
668-1340
7*
143-34199
286-6840
430-0260
573 3680
716-7100
7*
153-39809
306-7962
460-1943
613-5924
766-9905
1
163-79507
327-5901
491-3852
655-1803
818-9753
8
174-53293
349-0659
523-5988
698-1317
872-6647
&
197-03132
394-0626
591-0940
788-1253
985-1566
9
22089325
441-7865
6626798
883-5730
1104-4663
9*
246-11871
492-2374
738-3561
984-4748
1230-5936
10
272-70771
545-4154
818-1231
1090 8308
1363-5386
IQi
300-66025
601-3205
901-9808
1202-6410
1503-3013
11
329-97633
6599527
989-9290
13199053
1649-8817
MJ
360-65595
721-3119
1081-9679
1442-6238
1803-2798
12
392-6991
785-3982
11780973
1570-7964
1963-4955
OEIFICES, WEIES, PIPES, AND EIVEES.
445
TABLE IX. For finding the Discharge in Cubic Feet, per Minute,
when the Diameter of a Pipe, or Orifice, and the Velocity of dis-
charge are known ; and vice versa.
Discharge in cubic feet per minute, for different velocities.
5s
1-S1
sXs
Velocity of
600 inches
per second.
Velocity of
700 inches
per second.
Velocity of
800 inches
per second.
Velocity of
900 inches
per second.
Velocity of
1000 inches
per second.
1-0227
1-1931
1-3635
1-5340
1-7044
i
4-0906
4-7724
5-4542
6-1359
6-8177
i
9-2039
10-7379
12-2718
13-8058
15-3398
1
16-3625
19-0895
21-8166
24-5437
27-2708
25-5664
29-8274
34-0885
38-3495
42-6106
H
36-8155
429515
49-0874
55-2234
61-3593
*!
50-1100
58-4617
66-8134
75-1650
83-5167
if
65-4499
76-3582
87-2665
98 1748
109-0831
2
828350
96-6408
110-4466
124-2525
138-0583
*!
102-2654
119-3096
136-3538
153-3981
170-4423
2
1237411
144-3646
164-9882
185-6117
206-2352
2f
147-2621
171-8059
196-3496
220-8933
245*4369
3
172-8285
201-6333
230-4380
259-2428
288-0475
31
200-4401
233-8468
267-2535
300-6602
334-0669
3$
230-0971
268-4467
306-7962
345-1457
383-4952
8|
261-7994
305-4326
349-0659
392-6991
436-3323
4
295-5470
344-8048
394-0626
443-3205
492-5783
J
331-3399
386-5632
441-7865
497-0098
552-2331
4
369-1781
430-7077
492 2374
553-7671
615-2968
4f
409-0615
477-2384
545-4154
6135923
681-7692
5
450-9904
526-1554
601-3205
676-4855
751-6506
fij
494-9645
577-4586
659-9526
742-4467
824-9408
ty
540-9839
631-1479
721-3119
811-4759
901-6399
fif
589-0486
687-2235
7853982
883-5730
981*7478
6
639-1587
745-6852
852-2116
958-7381
1065*2645
*i
691-3141
806-5330
921-7520
10369710
1152-1900
6*
745-5147
869-7672
994-0196
11182721
1242-5245
6f
801-7608
935-3876
1069-0144
1202-6412
1336-2680
7
860-0519
1003-3939
1146-7359
1290-0779
1433-4199
7i
920-3885
1073-7866
1227-1847
1380-5828
1533-9809
7
982-7704
1146-5655
1310 3605
1474 1556
1637-9507
7f
1047-1976
1221-7305
1396-2634
1570-7964
1745-3293
8
1182-1879
1379-2192
1576-2506
1773-2819
1970-3132
8
1325-3595
1546-2528
1767-1460
1988-0393
2208-9325
9
1476-7123
1722-8310
1968-9497
2215-0684
2461-1871
$$
1636-2463
1908-9540
2181-6617
2454-3694
2727-0771
10
1803-9615
2104-6218
2405-2820
2705-9423
3006 6025
10,}
1979-8580
2309-8343
2639-8106
2969-7870
3299-7633
11
2163-9357
2524-5917
2885 2476
3245-9936
3606-5595
Hi
2356-1946
2748-8937
3141-5928
3534-2919
3926-9910
12
446
THE DISCHARGE OF "WATER FROM
TABLE X. For finding the depths of Weirs of different lengths,
the quantity discharged over each being supposed constant.
See pages 270 and 371.
Ratios of
lengths.
Coeffi.
cients.
Ratios of
lengths.
Coeffi-
cients.
Ratios of
lengths.
Coeffi-
cients.
Ratios of
lengths.
Coeffi-
cients.
01
0464
405
5474
605
7153
805
8654
02
0737
410
5519
610
7193
810
8689
03
0965
415
5564
615
7232
815
8725
04
1170
420
5608
620
7271
820
8761
05
1357
425
5653
625
7310
825
8796
06
1533
430
5697
630
7349
830
.8832
07
1699
435
5741
635
7388
835
8867
08
1857
440
5785
640
7427
840
8903
09
2008
445
5829
645
7465
845
8938
10
2154
450
5872
650
7504
850
8973
11
2296
455
5916
655
7542
855
9008
12
2433
460
5959
660
7580
860
9043
13
2566
465
6002
665
7619
865
9078
14
2696
470
6045
670
7657
870
9113
15
2823
475
6088
675
7695
875
9148
16
2947
480
6130
680
7733
880
9183
17
3069
485
6173
685
7771
885
9218
18
3188
490
6215
690
7808
890
9253
19
3305
495
6258
795
7846
895
9287
20
3420
500
6300
700
7884
900
9322
21
3533
505
6342
705
7921
905
9356
22
3644
510
6383
710
7959
910
9391
23
3754
5J5
6425
715
7996
915
9425
24
3862
520
6466
720
8033
920
9459
25
3969
525
6508
725
8070
925
9494
26
4074
530
6549
730
8107
930
9528
27
4177
535
6590
735
8144
935
9562
2.8
4280
540
6631
740
8181
940
9596
29
4381
545
-6672
745
8218
945
9630
30
4481
550
6713
750
8255
950
9664
31
4580
555
6754
755
8291
955
8698
32
4678
500
6794
760
8328
960
9732
33
4775
565
6834
765
8365
965
9762
34
4871
570
6875
770
8401
970
9799
35
4966
575
6915
775
8437
975
9833
36
5061
580
6955
780
8474
980
9866
37
5154
585
6995
785
8510
985
9900
38
5246
590
7035
790
9546
990
9933
39
5338
595
7074
795
8582
9D5
9967
40
5429
600
7114
SCO
8618
1-000
1-0000
ORIFICES, TVEIRS, PIPES, AND RIVERS.
447
TABLE XL Mean relative Dimensions of equally Discharging Trape-
zoidal Channels, with Side Slopes varying from to 1, up to % to 1.
Half sum of the top and bottom is the mean width. The ratio of
the slope, multiplied by the depth, subtracted from the mean
width, will give the bottom ; and if added, will give the top.
TABLE XII. gives the discharge in cubic feet per minute from the
primary channel, 70 wide, and the corresponding depths taken
in feet. For lesser or greater channels and discharges, see
Eules, pp. 237, 229, 232, 249, and 252.
The mean widths are given in the top horizontal line, and the corresponding depths in
the other horizontal lines. They may be taken in inches, feet, yards, fathoms, or
any other measures whatever.
70
60
50
40
35
30
25
20
15
10
125
13
15
17
20
23
26
29
35
-48
25
27
30
35
40
45
52
58
71
98
375
41
46
54
60
67
76
88
1-09
1-51
5
55
62
73
80
89
1-02
1-19
1-48
2-04
625
68
78
91
1-00
1-12
1-29
150
1-88
2-62
75
82
94
1-10
1-20
1-35
1-56
1-82
2-28
3-22
875
96
MO
1-29
1-41
1-58
1-83
2-14
2-69
3-86
1-
1-10
1-26
1-48
1-62
1-81
2-10
2-46
3-11
4-50
1-125
1-24
1-42
1-67
1-83
2-04
2-37
2-79
3-54
5-19*
1-25
1-39
1'58
1-86
2-04
2-28
2-65
3-12
3-98
5'89
1-375
1-53
1-74
2-05
2-25
2-51
2-92
3-46
4-43
6-60
1-5
1'67
1-90
2-24
2-46
2-75
3-20
3-80
4-88
7-31
1-625
1-81
2-06
2-43
2-67
2-99
3-47
4-15
5-34
8-08
Kfi
1-95
2-22
2-62
2-88
3-23
3-75
4-50
5-80
8-86
1-875
2-09
2-38
2-81
3-09
3-47
4-03
4-86
6-29
9-68
2-
2-23
2-54
3-00
3-31
3-72
4-32
5-22
6-78
10-50
2-125
2-37
2-70
3-19
3-52
3-96
4-61
5-58
7-29
11-37
2-25
2-51
2-86
3-38
3-73
4-21
4-91
5-95
7-81*
12-25
2-375
2-65
3-02
357
3-94
4-45
5-20
6-31
8-32
13-12
2-5
2-79
3-18
3-76
4-16
4-70
5-50
6-68
8-84
14-00
2-625
2-93
3-34
3-95
4-38
4-95
5-79
7-06
9-38
14-92
2-75
3-07
3-51
4-15
4-60
5-21
6-09
7-45
9-93
15-84
2-875
3-21
3-67
4-34
4-82
5-46
6'39
7-83
10-48
16-76
3-
3-35
3-84
4-54
5-04
5-72
6-69
8-22
11-03
17-68
3-125
3-49
4-00
4-73
5'26
5-97
7-00
8-62
11-60
18-68
3-25
3-63
4-17
4-93
5-49
6-23
7-31
9-02
12-17
19-68
3-375
3-77
4-33
5-13
5-72
6-49
7-62
9-42
12-74
20-68
3-5
3<91
4-50
5.33
5-95
6-75
7-93
9-82
1332
21-68
3-625
4-05
4-66
5-53
6-17
7-01
S-25
1023*
13-92
2276
3-75
4-19
4-82
5-73
6-40
7-28
8-57
1065
14-53
23-84
3-875
4-33
4-98
5-93
6-62
7-54
8-89
ii-or.
15-14
24-92
4-
4-48
5-14
6-13
6-85
7-81
9-21
11-48
15-75
26-00
4-25
4-76
5-46
6-54
7-30
8-35
9-85
12-33
16-98
28-18
4-5
5-05
5-79
6-95
7-75
8-90
10-50
1319
18-22
30-36
4-75
533
6-12
7-35
8-20
9-45
11-14
14-07
19-50
32-68
5-
5-62
6-45
7*75
8-66
10-00
11-79
14-96
20-80
35-00
5-25
5-90
6'78
8-16
9-14
10-55
12-51*
15-86
22-13
37-40
5-5
6-18
7-12
8-57
9-62
11-10
13-24
16-77
23-47
39-81
5-75
6-46
7-46
8-98
10-11
11-66
13-94
17-71
24-86
42-33
6-
6 f 75
7-80
9-40
10-60
12-22
14-65
18-65
26-25
44-86
448
THE DISCHARGE OF WATER FROM
TABLE XII. Discharges from the Primary Channel in the first
column of Table XI.
If the dimensions of the primary channel be in inches, divide
the discharges in this table by 500 ; if in yards, multiply by
15'6 ; if in quarters, multiply by 32 ; and if in fathoms, by
88-2, &c. : see pp. 233, 234. The final figures in the dis-
charges may be rejected when they do not exceed one-half
per cent., or 0-5 in 100. See pages 226 to 234.
Depths of a
channel
whose mean
width is
70: in feet.
Falls, inclinations, and discharges in cubic feet per minute.
Interpolate for intermediate falls ; divide greater falls by 4,
and double the corresponding discharges.
linch
per mile,
2 inches
per mile,
1 in 31680.
3 inches
per mile,
1 in 21120.
6 inches
per mile,
1 in 10560.
9 inches
per mile,
1 in 7040.
12 inches
per mile,
1 in 5280.
15 inches
per mile.
1 in 4224.
125
47
72
93
139
175
205
233
25
136
210
268
403
506
596
675
375
249
389
498
746
940
1105
1252
50
387
603
770
1155
1454
1709
1935
625
541
849
1078
1617
2036
2395
2714
75
714
1112
1420
2128
2681
3153
3573
875
900
1401
1791
2685
3382
3978
4507
1-
1100
1714
2190
3283
4134
4862
5507
1-125
1310
2042
2614
3909
4927
5792
6577
1-25
1534
2384
3058
4581
5766
6780
7690
1-375
1767
2757
3521
5279
6661
7823
8863
1-50
2013
3142
4006
6016
7588
8915
10099
1-625
2268
3540
4525
6781
8541
10044
11381
1-75
2534
3950
5053
7570
9537
11210
12703
1-875
2812
4384
5599
8386
10570
12429
14083
2-
3090
4821
6161
9230
11628
13675
15513
2-125
3377
5273
6738
10092
12718
14956
16943
2-25
3674
5736
7331
10981
13833
16281
18435
2-375
3977
6210
7937
11889
14981
17645
19960
2-50
4293
6699
8563
12829
16161
19045
21534
2-625
4616
7203
9204
13800
17380
20434
23135
2-75
4947
7716
9865
14782
18624
21886
24800
2-875
5280
8233
10525
15773
19887
23360
26473
3-
5621
8762
11204
16788
21165
24833
28176
3-125
5972
9310
11900
17b'30
22454
26410
29925
3-25
6329
9862
12614
18897
23780
27994
31714
3-375
6689
10420
13320
19963
25145
29570
33507
3-50
7049
10995
14048
21052
26509
31262
35329
3-625
7418
11574
14785
22153
27906
32860
37186
3-75
7794
12163
15526
23284
29321
34479
39080
3-875
8178
12753
16283
24416
30756
36170
41013
4-
8566
13354
17070
25592
32225
37898
42954
4-25
9355
14582
18643
27936
35191
41368
46916
4-50
10173
15849
20267
30366
38254
44982
50973
4-75
11001
17140
21908
32818
41356
48630
55102
5-
11833
18454
23595
35355
44546
52378
59346
5-25
12696
19802
25362
37939
47795
56209
63688
5-50
13576
21172
27248
40564
51097
60079
68U97
5-75
14478
22580
29160
43253
54478
64058
72591
6-
15393
23995
31122
45969
57897
68082
77154
OftlFICES, WEIRS, PIPES, AND RIVERS.
449
TABLE XII. Discharges from the Primary Channel in the first
column of Table XI.
If the dimensions of the primary channel be in inches, divide the
discharges in this table by 500 ; if in yards, multiply by 15*6,
if in quarters, multiply by 32, and if in fathoms, by 88'2 etc. :
see pp. 233 and 234. The final figures in the discharges may
be rejected when they do not exceed one-half per cent., or 0*5
in 100. See pages 226 to 234.
Falls, inclinations, and discharges in cubic feet per minute.
Interpolate for intermediate falls; divide greater falls by 4, and
double the corresponding discharges.
Depths of a
channel
whose mean
width is
0: in feet.
18 inches
per mile,
1 in 3520.
21 inches
per mile,
1 in 3017.
24 inches
per mile,
1 in 2640.
27 inches
per mile,
1 in 2347,
30 inches
per mile,
1 in 2112.
33 inches
per mile,
1 in 1920.
36 inches
per mile,
1 in 1760.
258
281
303
323
343
362
380
125
748
815
877
936
993
1049
1100
25
1387
1511
1627
1736
1843
1952
2037
375
2145
2336
2515
2684
2852
3023
3155
50
3004
3274
3527
3753
4021
4207
4414
625
3957
4311
4645
4966
5287
5553
5817
75
4991
5422
5859
6274
6650
6992
7342
875
6097
6622
7159
7631
8107
8540
8974
1-
7266
7920
8531
9124
9660
10200
10693
1-125
8514
9284
9995
10658
11318
11923
12520
1-25
9816
10697
11539
12307
13045
13741
14479
1-375
11182
12185
13152
14007
14862
15656
16448
1-50
12601
13730
14821
15786
16750
17657
18552
1-625
14069
15331
16525
17616
18700
19698
20696
1-75
15593
16997
18306
19517
20728
21840
22944
1-875
17157
18697
20141
21469
22803
24017
25242
2-
18766
20446
22030
23480
24938
26269
27601
2-125
20410
22247
23965
25547
27129
28578
30027
2-25
22104
24087
25947
27662
29395
30934
32512
2-375
23848
25988
27992
29841
31701
33381
35096
2-50
25669
27953
30100
32069
34086
35910
37725
2-625
27479
29933
32247
34384
36512
38471
40415
2-75
29318
31947
34408
36697
38958
41055
43135
2-875
31206
34002
36624
39050
41464
43680
45896
3-
33141
36112
38897
41482
44048
46398
48747
3-125
35126
38266
41223
43954
46672
49174
51664
3-25
37109
40438
43556
46438
49330
51951
54586
3-375
39140
42631
45925
48963
51993
54775
57550
3-50
41184
44872
48343
51537
54728
57659
60580
3-625
43273
47158
50807
54162
57514
60585
63656
3-75
45407
49468
53300
56840
60341
63560
66784
3-875
47551
51818
55832
59514
63200
66576
69951
4-
51911
56586
60973
64974
69013
72694
76383
4-25
56448
61508
66176
70623
75017
79017
82994
4-50
61014
66500
71625
76408
81097
85426
89767
4-75
65713
71628
77140
82250
87351
92015
96653
5-
70509
76863
82779
88200
93731
98729
103745
5-25
75383
82159
88434
94344
100200
105550
110905
5-50
80379
87590
94348
100616
106823
112540
118254
5-75
85407
93093
100275
106911
113505
119616
125664
6-
450
THE DISCHARGE OF WATER FROM
TABLE XIII. The Square Roots of the fifth powers of numbers for
finding the Diameter of a Pipe, or dimensions of a Channel from
the Discharge, or the Reverse; showing the relative Discharging
Powers of pipes of different Diameters, and of any similar Channels
whatever, closed or open. See pages 81, 230, 233, etc.
If d be the diameter of a pipe, in feet, and D the discharge in
cubic feet per minute, then for long straight pipes we shall ha,ve
for velocities of nearly 3 feet per second, D = 2400 (d 5 s) J , and
d = *044( \ ; or if D be the discharge per second, D = 40 (d 5 s)*,
/D 2 \i
228 I - ) . See pages 190 to 224, and pages 42 and 43.
V s '
and d
Relative
dimen-
sions or
Relative
discharg-
Relative
dimen-
sions or
Relative
discharg-
Relative
dimen-
sions or
Relative
discharg-
Relative
dimen-
sions or
Relative
discharg-
diameters
of pipes.
ing
powers.
diameters
of pipes.
ing
powers.
diameters
of pipes.
ing
powers.
diameters
of pipes.
ing
powers.
25
031
10-5
357-2
30-5
5138-
61-
29062-
5
177
11-
401-3
31-
5351-
02-
30268-
75
485
11-5
448-5
31-5
5569-
63-
315U3-
1-
1-
12-
498-8
32-
5793-
64-
32768-
1-25
1-747
12-5
552-4
32-5
0022-
65-
34063-
1-5
2-756
13-
609-3
33-
(>256-
66-
35388-
1-75
4-051
13-5
669-6
33-5
6496-
67-
36744-
2-
5-657
14-
733-4
34-
6741-
68-
38131-
2-25
7-594
14-5
800-6
34-5
6991-
69-
39548-
2-5
9-882
15-
871-4
35-
7247-
70-
40996-
2-75
12-541
15-5
9459
35-5
7509-
71-
42476-
3-
15-588
16-
1024-
30-
7776-
72-
43988-
3-25
19-042
16-5
1105-9
36-5
8049-
73-
45531-
3-5
22-918
17-
1191-6
37-
8:327-
74-
47106-
3-75
27-232
17-5
1281-1
37-5
8611-
75-
48714-
4-
32-
18-
1374-6
38-
8901-
76-
50354-
4-25
37-24
18-5
1472-1
38-5
9197-
77-
52027-
4-5
42-96
19-
1573-6
39-
9498-
78-
53732-
4-75
49-17
19-5
1679-1
39-5
9806-
79-
55471-
5-
55-90
20-
1788-9
40-
10119-
80-
57243-
5-25
63-15
20-5
1902-8
41-
10764-
81-
59049-
5-5
70-94
21-
2020-9
42-
11432-
82-
60888-
5-75
79-28
21-5
2143-4
43-
12 125-
83-
62762-
6-
88-18
22-
2270-2
44-
12842-
84-
64669-
6-25
97-66
22-5
2401-4
45-
13584-
85-
66611-
6-5
107-72
23-
2537-
46-
14351-
86-
68588-
6-75
118-38
23-5
2677-1
47-
15144-
87-
70599-
7*
129-64
24-
2821-8
48-
15963-
88-
72645-
7-25
141-53
24-5
2971-1
49-
16807-
89-
74727-
7-5
154-05
25-
3125-
50-
17678-
90-
76843-
7-75
167-21
25-5
3283-6
51-
18575-
91-
78996-
8-
181-02
26-
3446-9
52-
19499-
92-
81184-
8-25
195-50
26-5
3615-1
53-
20450-
93-
83408-
8-5
210-64
27-
3788-
54-
21428-
94-
85668-
8-75
220-48
27-5
3965-8
55-
22434-
95-
87965-
9-
243-
28-
4148-5
56-
23468-
96-
90298-
9-25
260-23
28-5
4336-2
57-
24529-
97-
92668-
9-5
278-17
29-
4528-9
58-
25020-
98-
95075-
9-75
296-83
29-5
4726-7
59-
2(5738-
99-
97519-
10-
316-23
30-
4929-5
60-
27886-
100-
100000-
ORIFICES, WEIRS, PIPES, AND RIVERS. 451
TABLE XIV. Weights and Measures, English and French, with
their relative values.
MEASUEES OF LENGTH.
12 inches 1 foot.
7-92 inches 1 link.
3 feet 1 yard.
5 \ yards = 16^ feet 1 pole or perch.
100 links = 22 yards 1 chain = 4 perches.
40 perches = 220 yards 1 furlong.
8 furlongs =17 60 yards 1 mile.
6 feet 1 fathom.
120 fathoms 1 cable's length.
1 Nautical mile 6082-7 feet.
69-12 miles 1 Geographical deg.
3 miles 1 league.
The Irish perch is 21 feet, or seven yards. Three inches make
a palm ; 4 inches a hand ; 5 feet a pace. In cloth measure
2 \ inches = 1 nail ; 4 nails = 1 quarter ; 4 quarters 1 yard.
11 Irish miles = 14 English.
MEASUEES OP SUEFACE.
144 square inches 1 square foot.
62-7264 1 square link.
9 square feet 1 square yard.
30 \ square yards = 272 J square feet ... 1 square perch.
10,000 square links =4,356 ... 1 square chain.
10 square chains = 160 square perches... 1 acre.
1 rood = 210 square yards ... 40 perches.
4 roods =4,840 ... 1 acre.
640 acres =3,097,600 ... 1 square mile.
The Irish perch is 49 square yards, or 441 square feet ; 1 Irish
acre=la. 2r. 19-17p. statute; and 1 statute acre=0a. 2r. 18-77p.
Irish. The Irish acre is to the English acre as 196 is to 121.
100 square feet, is a square of roofing, slating, or flooring. The
Cunningham acre is = la. Ir. 6-61p. English; and 1 English
acre is = Oa 3r. 3'904p. Cunningham measure.
452 THE DISCHARGE OF WATER FROM
CUBIC MEASURES, AND MEASURES OF CAPACITY AND WEIGHT.
1728 cubic inches 1 cubic foot,
27 cubic feet 1 cubic yard.
16Jx lx 1 =24-75 cubic feet ... 1 perch of masonry
16Jxl6|x lf= 306 cubic feet ... 1 rod of brickwork,
21 x IJx 1 = 30 J cubic feet ... 1 Irish perch of masonry
The standard gallon, imperial measure, contains 10 Ibs.
avoirdupois, of distilled water at 62 Fahrenheit, the barometer
standing at 30 inches.
6-232 gallons 1 cubic foot.
8*665 cubic inches ... ... ... 1 gill.
4 gills 34-659 cubic inches 1 pint.
2 pints 69*318 cubic inches... ... 1 quart.
2 quarts 138-637 cubic inches ... 1 pottle.
2 pottles 277-274 cubic inches ... 1 gallon.
2 gallons 554*548 cubic inches ... 1 peck.
4 pecks 2218- 191 cubic inches ... 1 bushel.
The old Irish gallon contained 217-6 cubic inches, nearly, and
1 Irish gallon is therefore = -7850 imperial gallon. The Irish
barrel of lime still measures 40 Irish gallons, or 31*321 imperial
gallons, or 4 bushels, very nearly. It is measured by a cylindrical
measure 12 inches high, and about 21 J inches in diameter, con-
taining half the Irish barrel. In the old English liquid measures
for ale and beer, 36 gallons = 1 barrel = 36 gallons, 3J quarts,
imperial measure, nearly.
For old dry measures, 82 bushels = 1 chaldron = 31 bushels,
1 pint, imperial measure, nearly.
And 36 bushels of coal = 1 chaldron of coal = 34 bushels 3
pecks, and 1 gallon, imperial measure. The Irish barrel of
wheat is 20 stone ; barley 16 stone ; oats 14 stone.
TROY WEIGHT,
24 grains 1 pennyweight.
20 pennyweights ... ... ... 1 ounce
12 ounces ... 1 pound.
One pound Troy = 22-816 cubic inches of distilled water, baro-
meter 30 inches ; thermometer 62,
ORIFICES, WEffiS, PIPES, AND RIVERS.
453
APOTHECARY'S WEIGHT.
90 Troy grains 1 scruple.
3 scruples 1 drachm.
8 drachms ... ... ... ... 1 ounce.
12 ounces .< 1 pound.
The ounce weighs 480 grains, and the pound 5760 grains, both
in Troy and Apothecary's weight.
AVOIRDUPOIS OR COMMERCIAL WEIGHT.
One pound Avoirdupois = 27*7274 cubic inches, when the
barometer stands at 30 inches, and Fahrenheit's thermometer
at 62.
16 drachms = 437-5 Troy grains 1 ounce
16 ounces = 7,000 Troy grains 1 pound
14 pounds =98,000 Troy grains 1 stone
8 stone = 112 pounds ... ... 1 cwt.
20 cwt. = 2,240 pounds 1 ton
One pound Troy= -82286 pounds Avoirdupois, and one pound
Avoirdupois, is equal to 1-2153 pounds Troy. One ton of water
contains about 36 cubic feet, equal to 224 imperial gallons,
nearly. Ten pounds of distilled water is equal to one gallon,
the Barometer and Thermometer being as above stated.
FRENCH MEASURES AND WEIGHTS COMPARED
WITH ENGLISH.
MEASURES OF LENGTH.
1 metre .. .. 3-2808992 feet
1 decimetre . . . . -3280899
1 centimetre . . . . -0328090
1 millimetre . . . . -0032809
1 Jdlometre(orlOOOmetres)-621383mile
1 foot English . .
1 inch
1 yard
1 perch, 5 yds.
1 mile
3047945 metre
0253995
9143835
5-0291092
1-60932 kilometre
1000 metres = 100 decametres = 10 hectometres = 1 kilometre =3280-849
feet. The metre is the 10,000,000th part of a quadrental arc of the meridian
or 39-3708 inches English.
454
THE DISCHARGE OF WATER FROM
MEASURES OF SURFACE
1 centiare (one square ) 1n 7ftlq
metre) . . j U
1 declare.. .. 107-6430
1 are 1076-430
f .
*
119-6033 sq. yds.
11-9603
1-1960
1 are
1 declare
( 1 centiare or sq.
\ metre.
17
100 ares = 10 declares = 1 hectare = 2-471143 English acres, and
hectares are nearly equal to 42 English acres.
The old Paris foot is equal 1-06578 English feet ; the French inch =
1-06578 English inches ; the French line -08882 of an English inch ; thetoise
is equal to 6 French feet = 76-736 English inches = 6-39468 feet. The
perches is 18 French feet ; and the perch royal 22 French feet. The French
square foot or inch = 1-13581 English square feet or inches, and the cubic
foot or inch = 1-21061 English.
MEASURES OF SOLIDITY AND CAPACITY.
1 millistere
.. -0353166 cubic ft. T ^.^.^
0610279
1 centistere
.., -353166
1 decistere
.. 3-53166
1 centilitre
610279
1 stere (one cubic > Q n.oi cc
metre J 3
1 decilitre
1 litre
. . 6-10279
.. 61-0279
1 decastere
353-166
1 decalitre
..610-279
1 hectostere
3531-66
1 hectolitre
6102-79
1 kilostere
35316-6
1 kilolitre
61027-9
The stere and kilolitre are each a cubic m^tre, and the litre is a cubic deci-
mStre ; 50 litres are nearly 11 English gallons, and 1 hectolitre 2-751207
English bushels.
MEASURES OF WEIGHT.
0648 gramme = 1 grain, and 7000 grains = 1 Ib. Avoirdupois.
1 milligramme
1 centigramme
1 decigramme
1 gramme
015432 grains
15432
1-5432
15-432
1 gramme . .
1 decagramme
1 hectogramme
1 killogramme
15-432 grains
02205 lb.avoir.
2205
2-2046
1-01605 tonnes = 1 ton ; and 1 tonne = -984206 ton.
A gramme is the weight of a cubic centimetre of water and its maxim,
density in vacuo, 1 kilogramme = 2-6795 Ibs. Troy=2-2046 Ibs. Avoirdupois.
1 metrical quintal 220-46 Ibs. Avoirdupois, and 10 quintals is equal to the
weight of a cubic metre of water.
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ORIFICES, WEIRS, PIPES, AND RIVERS,
455
TABLE XV. Shewing the Weight, Specific Gravity, strength and elasticity
of various materials employed by the Physicist and Engineer. When
used bi/ the Engineer only about one-fourth of the ultimate strengths
here given should be calculated from.
MATERIALS.
Moduli
of
Rupture.
Moduli
of
Elasticity.
Crushing
forces per
sq. inch,
inlbs.
Tenacities
persq. in.
in Ibs.
Weights
of a cubic
foot in
Ibs.
specific
gravi-
ties.
Acacia, English Growth . .
Ash
11,200
12,000
1,150,000
1,600,000
9,000
16,000
17,000
44-3
48-0
71
77
8,900,000
10,300
18,000
525-0
8-40
Beech
9,300
1,350,000
8,500
16,000
48-0
77
Brick Red
800
280
135-5
2-20
112-5
1-80
Do. Pale Red
550
300
130-3
2-08
Cedar, American, Fresh . .
Do. do. Seasoned
490,000
5,600
4,900
11,400
19,000
56-8
47-0
538-0
0-91
0-75
8-61
Do. Sheet
30,000
549-0
8-80
Do. Wire-drawn
9,900
1,670,000
60,000
12,400
560-0
43-6
8-88
0-70
Do. Memel ....
10,400
1 530 000
37-0
0-60
Do. Norway Spruce
Elm Seasoned
6100
700000
10,300
17,600
13500
21-2
36-8
0-34
0-59
Fir, New England
6 600
2 190000
10,000
34-5
0-55
Do. Riga
7,600
1,100,000
6,100
12,000
47-0
0-75
Glass
8,000,000
33,000
2,400
153-3
2-45
Iron, Wrought, English . .
57,000
57,000
481-2
487-0
7-70
7-80
Do. rolled in Sheets and
Rivetted
Cast Iron Carron, cold blast
Do. Hot blast
38*500
37,500
17,270,000
16,080,000
106,000
108,000
31,000
16,700
13,500
487
441
440-0
7-8
7-07
7-04
Do. Buffery .
37 500
14 000 000
90,000
17,500
441-0
7-06
Larch, green
5,000
900,000
3,200
10,200
36-6
0-52
6,900
1,050,000
5,500
8,900
35-0
0-56
Lead, cast English
Do. milled sheet
Marble, white Italian ....
Do. black Gal way ....
Mortar, old, good
Oak, English
1,100
2,700
10,000
720,000
2,520,000
1,450,000
250
6,600
1,800
3,300
"so
17,300
717-4
712-9
165-0
168-4
107-1
58-3
11-44
11-40
2-64
2-70
1-75
0-93
Do. Canadian
10500
2 150 000
6 500
10200
54-5
0-87
8700
1 190 000
12 700
47-4
76
Do. African . . .
13600
2 280 000
60-7
0-97
Do. Adriatic
8,300
970000
62-0
0-99
9,800
1,230,000
7,800
41-2
0'66
Do. red
8900
1 840 000
5300
41-2
0-66
Silver, Standard
Slate, Welsh
11,800
15,800,000
40,900
12,800
644-5
180-5
10-31
2-89
Do. Westmoreland
Do. Valentia
5200
1,290,000
174-4
180-0
2-70
2-88
Steel, soft
120.000
486-2
7-80
Do, razor tempered ....
Stone, granite average ....
Do. Rochdale
5,500
2400
29,000,000
8,000
150,000
490-0
168-0
161-0
7-84
2-70
2-58
Teak, dry
14800
2 400 000
12101
15000
41-1
0-66
Tin, cast
4 600 000
5,300
455-7
7-30
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