Water-Supply and Irrigation Paper No. 150

Series M, General Hydrographic Investigations. 16

DEPARTMENT OF THE INTERIOR

UNITED STATES GEOLOGICAL SURVEY .

CHARLES D, WALCOTT, Director

WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS

BY

ROBERT E. HORTON

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Water-Supply and Irrigation Paper No. 150

Series M, General Hydrographic Investigations, 16

DEPARTMENT OE THE INTERIOR

UNITED STATES GEOLOGICAL SURVEY

CHARLES D. WALCOTT, Director

WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS

BY

ROBERT LL HORTON

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CONTENTS AND ILLUSTRATIONS.

Page.

Introduction................................................

Definitions of terms.....................................

Notation................................................

Base formulas...........................................

Equivalent coefficients...................................

Approximate relative discharge over weirs................

References..............................................

Theory of weir measurements................................

Development of the weir................................

Theorem of Torricelli....................................

w.

Elementary deduction of the weir formula................

Application of the parabolic law of velocity to weirs ...

General formula for weirs and orifices....................

Vertical contraction.....................................

Velocity of approach........................................

Theoretical formulas.....................................

Distribution of velocity in channel of approach............

Distribution of energy in channel of approach.............

The thin-edged weir.........................................

Earlier experiments and formulas........................

Castel..............................................

Poncelet and Lesbros................................

Boileau............... .............................

East Indian engineers’ formula.......................

Experiments and formula of James B. Francis............

Experiments and formula of Fteley and Stearns............

Bazin’s experiments.....................................

Bazin’s formulas for thin-edged weirs.....................

Derived formulas for thin-edged rectangular weirs.........

Fteley and Stearns-Francis formula...................

Hamilton Smith’s formula...........................

Smith-Francis formula...............................

Parmley’s formula..................................

Extension of the weir formulas to higher heads...........

Comparison of weir formulas.............................

Comparison of various velocity of approach corrections

End contractions—incomplete contraction.................

Compound weir........................................

Triangular weir..........................................

General formula.....................................

Thomson’s experiments..............................

CO

The thin-edged weir—-Continued.

Trapezoidal weir.....................................................

The Cippoletti trapezoidal weir .................................

Cippoletti’s formula...........................................

Requirements and accuracy of weir gagings.................................

Precautions for standard weir gaging...................................

Plank and beam weirs of sensible crest width..........................

Reduction of the mean of several observations of head..................

Effect of error in determining the head on weirs........................

Error of the mean where the head varies..............................

Weir not level........................................................

Convexity of water surface in leading channel..........................

Results of experiments on various forms of weir cross sections...............

The use of weirs of irregular section...................................

Modifications of the nappe form .......................................

Experimental data for weirs of irregular cross section...................

Base formula for discharge over weirs of irregular cross section......

Bazin’s experiments on weirs of irregular cross section..............

Bazin’s correction for velocity of approach .....................

Recomputation of coefficients in Bazin’s experiments...........

Cornell University hydraulic laboratory...........................

Experiments of United States Board of Engineers on Deep Waterways. Experiments at Cornell University hydraulic laboratory on models of

old Croton dam................................................

Experiments of United States Geological Survey at Cornell University

hydraulic laboratory............................................

Experiments on model of Merrimac River dam at Lawrence, Mass...

Flow over weirs with broad crests.....................................

Theoretical formula of Unwin and Frizell..........................

Blackwell’s experiments on discharge over broad-crested weirs......

East Indian engineers’ formula for broad-crested weirs..............

Fteley and Stearns experiments on broad-crested weirs..............

Bazin’s formula and experiments on broad-crested weirs............

Experiments of the United States Geological Survey on broad-crested

weirs..........................................................

Table of discharge over broad-crested weirs with stable nappe.......

Effect of rounding upstream crest edge.................................

Experiments on weirs with downstream slope or apron of varying inclination ........:......................................................

95

107

110-

110

112

114

116

117

119

121

122

Triangular weirs with vertical upstream face and sloping aprons..... Triangular weirs with upstream batter 1:1 and varying slope of apron. Experiments on weirs of trapezoidal section with upstream slope of

J:l, horizontal crest, and varying downstream slopes..............

Combination of coefficients for weirs with compound slopes.............

Weirs with varying slope of upstream face.............................

Dams of ogee cross section, Plattsburg-Chambly type...................

Experiments on discharge over actual dams............................

Blackstone River at Albion, Mass.................................

Muskingum River, Ohio..........................................

Ottawa River dam, Canada........................................

Austin, Tex., dam................................................

Roughness of crest............. ......................................

Falls ,...............................................................

Weir curved in plan..................................................

§ is

Page.

Submerged weirs.........................................................

Theoretical formula ..........i.......................................

Fteley and Stearns submerged-weir formula............................

Clemens Herschel’s formula..........................................

The Chanoine and Mary formula......................................

R. H. Rhind’s formula...............................................

Bazin’s formulas.....................................................

Increase of head by submerged weirs..................................

Rankine’s formulas...............................................

Colonel Dyas’s formula............................................

Submerged weirs of irregular section..................................

Bazin’s experiments................................................

Data concerning East Indian weirs ................................

United States Deep Waterways experiments.......................

Weir discharge under varying head.......................................

Prismatic reservoir, no inflow.........................................

Approximate time of lowering prismatic or nonprismatic reservoir.......

Reservoir prismatic, with uniform inflow..............................

General formulas.................................................

Formulas for time of rise to any head H, prismatic reservoir with uniform inflow....................................................

Nonprismatic reservoir, uniform inflow................................

Variable inflow, nonprismatic reservoir..............................

Tables for calculations of weir discharge...................................

Table 1. Head due to various velocities................................

Table 2. Percentage increase in discharge by various rates of velocity of

approach..........................................................

Tables 3, 4. Discharge over a thin-edged w'eir by the Francis formula.___

Tables 5, 6. Three-halves powers......................................

Table 7. Flow over broad-crest weirs with stable nappe.................

Table 8. Backwater caused by a dam or weir...........................

137

137

138

139

140

141

141

142

142

143

143

143

144

146

146

147

147

148

148

149

153

154

156

157

159

162

171

177

180

Index

187

Plate I. Bazin’s coefficients.........................

II. Effect of errors in weir experiments.........

III. Modifications of nappe form................

IV-XII. Bazin’s experiments......................

XIII, XIV. Cornell hydraulic laboratory experiments.... XV-XVIII. United States Deep Waterways experiments .

XIX-XXII. Croton dam experiments....................

XXIII-XXXII. United States Geological Survey experiments

XXXIII. Merrimac River dam experiments...........

XXXIV-XXXV. Cross sections of ogee dams.................

XXXVI. Coefficient diagram for ogee dams...........

XXXVII. Experiments on actual dams................

XXXVIII. Diagram of variable discharge...............

Fig. 1. Torricellian theorem applied to a weir.................

2.    Rectangular orifice...................................

3.    Distribution of velocities..............................

4.    Triangular weir......................................

5.    Trapezoidal weir.....................................

32

54

60

66

86

90

94

106

108

130

130

132

150

11

12

16

46

Fig. 6. Sections of the Francis weir........

7.    Inclined weir.....................

8.    Broad-crested weir.................

9.    Coefficient curve for triangular weirs

10.    Fall.....................J........

11.    Weir curved or angular in plan......

12.    Submerged weir....................

13.    East Indian weir section............

14.    East Indian weir section............

15.    Concave backwater surface..........

16.    Convex backwater surface ..........

Page.

51

57

110

125

135

136

137

145

145

180

181

WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

By Robert E. Horton.

INTRODUCTION.

DEFINITIONS OF TERMS.

The word “weir” will be used to describe any structure used to determine the volume of flow of water from measurements of its depth on a crest or sill of known length and form. In this general sense timber and masonry dams having various shapes of section, reservoir overflows, and the like may be weirs. Terms, more or less synonymous, used to describe such weirs are “comb,” “wasteway,” “spillway,” “overwash,” “rollway,” and “overfall.”

The French term “nappe,” suggesting the curved surface of a cloth hanging over the edge of a table, has been fittingly used to designate the overfalling sheet of water.

The expression “wetted underneath” has been used to describe the condition of the nappe designated by Bazin as “noyees en dessous,” signifying that the water level between the nappe and the toe of the weir is raised by vacuum above the general water level below the weir.

“Thin-edged weir” and “sharp-crested weir” are used to designate a weir in which the nappe, or overfalling sheet, touches only the smooth, sharp upstream corner or edge of the crest, the thickness of which is probably immaterial so long as this condition is fulfilled.

A “suppressed weir” has a channel of approach whose width is the length of the weir crest.

A “contracted weir” has a crest length that is less than the width of the channel of approach.

The term “channel of approach,” or “leading channel,” defines the body of water immediately upstream from the weir, in which is located the gage by which the depth of overflow is measured.

“Section of approach” may refer to the cross section of the leading channel, if the depth and width of the leading channel are uniform; otherwise it will, in general, apply to the cross section of the channel of approach in w'hich the gage is located.

“ Weir section ” refers to the cross section of the overflowing stream in the plane of the weir crest.

“Crest contraction” refers to the diminished cross section of the overflowing stream resulting from the upward curvature of the lower water filaments in passing the crest edge. It does not include the downward curvature of the water surface near the weir crest.

The “vertical contraction of the nappe” includes both the crest contraction and the surface contraction.

“Incomplete contraction” may take place either at the crest or at the ends of a weir, and will occur when the bottom or side walls of the channel of approach are so near the weir as to prevent the complete curvature of the water filaments as they pass the contracting edge.

Dimensions are uniformly expressed in feet and decimals, velocities in feet per second, and quantities of flow in cubic feet per second, unless otherwise stated in the text.

In the preparation of this paper much computation has been involved and it is expected that errors will appear, which, if attention is called to them, may be corrected in the future. Information concerning such errors will be gratefully received.

NOTATION.

The symbols given below are used in the values indicated. The meaning qf additional symbols as used and special uses of those that follow are given in the text:

D=Measured or actual depth on the crest of weir, usually determined as the difference of elevation of the weir crest and the water level, taken at a point sufficiently far upstream from the weir to avoid the surface curve.

//=The head corrected for the effect of velocity of approach, or the observed head where there is no velocity of approach. As will he explained, D is applied in formulas like Bazin’s, in which the correction for velocity of approach is included in the coefficient, if is applied in formulas where it is eliminated. i’=Mean velocity of approach in the leading channel, usually taken in a cross section opposite which D is determined.

v*

h=Velocity head=<x <7=Acceleration by gravity. Value here used 32.16.

P=Ideight of weir crest above bottom of channel of approach, where channel is rectangular.

IT= Width of channel of approach where D is measured.

A=Area of cross section of channel of approach.

67=Area of channel section where D is measured, per unit length of crest. a=Area of weir section of discharge=Z> L. ~

Actual length of weir crest for a suppressed weir, or length corrected for end contractions, if any.

L'—Actual length of crest of a weir with end contractions.

Ar=Numher of complete end contractions.

J?=Breadth of crest of a broad-crested weir.

A= Batter or slope of crest, feet horizontal to one vertical.

d=Depth of crest submergence in a drowned or submerged weir. Q=Volume of discharge per unit of time.

C, M, m, ju, a, f, etc., empirical coefficients.

BASE FORMULAS.

The following formulas have been adopted by the engineers named:

Q=| MLII^gJI

—H'LH'ltyH. =mLI)ji2gD.

Hamilton Smith (theoretical).

Bazin, with no velocity of approach.

Bazin, with velocity of approach.

= CLIP.

= CLIP+fL.

Francis a (used here).

Ftelev and Stearns.

EQUIVALENT COEFFICIENTS.

The relations between the several coefficients, so far as they can be given here, are as follows:

M=

CM ICO

M.

M is a direct measure of the relation of the actual to the theoretical weir discharge.

C— M^2g=~^ M 72y=8.02 /t=5.35 M.

3

_M~—

= 0.1870 a

_ 3 C

2/'“ 2 JTg

>‘=8X2=⁰-1247 a

APPROXIMATE RELATIVE DISCHARGE OVER WEIRS.

10

3.33=^

For a thin-edged weir, the coefficient C in the Francis formula is Let C' be the coefficient for any other weir, and x the relative discharge as compared with the thin-edged weir, then

x—

3 O'

10

(1)

or, as a percentage,

^ = 100 «=30 o.

«The coefficient C of Francis includes all the constant or empirical factors appearing in the formula, which is thus thrown into the simplest form for computation.

This expression will be found convenient in comparing the effect on discharge of various modifications of the weir cross section. For a broad-crested weir with stable nappe, Cx — 2.64, see p. 121. The discharge over such a weir is thus seen to be 79.2 per cent of that for a thin-edged weir by the Francis formula.

REFERENCES.

The following authorities are referred to by page wherever cited in the text:

Bazin, H., Recent experiments on flow of water over weirs. Translated by Arthur Marichal and J. C. Trautwine, jr. Proc. Engineers’ Club Philadelphia, vol. 7, No. 5, January, 1890, pp. 259-310; vol. 9, No. 3, July, 1892, pp. 231-244; No. 4, October, 1892, pp. 287-319; vol. 10, No. 2, April, 1893, pp. 121-164.

Bazin, H., Experiences nouvelles sur l’ecoulement en deversoir, 6me art., Annales des Ponts et Chaussees, Memoires et Documents, 1898, 2me trimestre, pp. 121-264. This paper gives the results of experiments on weirs of irregular section.

. Bazin’s earlier papers, published in Annales des Ponts et Chaussees, 1888, 1890, 1891, 1894, and 1896, giving results of experiments chiefly relating to thin-edged weirs and velocity of approach, have been translated by Marichal and Trautwine. Bellasis, E. S., Hydraulics.

Bovey, H. T., Hydraulics.

Francis, James B., Lowell hydraulic experiments.

Frizell, James P., Water power.

Fteley, A., and Stearns, F. P., Experiments on the flow of water, etc. Trans. Am.

Soc. Civil Engineers, January, February, March, 1883, vol. 12, pp. 1-118. Merriman, Mansfield, Hydraulics.

Rafter, George W., On the flow of water over dams. Trans. Am. Soc. Civil Engineers, vol. 44, pp. 220-398, including discussion.

Smith, Hamilton, Hydraulics.

THEORY OF WEIR MEASUREMENTS.

DEVELOPMENT OF THE WEIR.

The weir as applied to stream gaging is a special adaptation of mill dam, to which the term weir, meaning a hindrance or obstruction, has been applied from early times. The knowledge of a definite relation between the length and depth of overflow and the quantity also probably antedates considerably the scientific determination of the relation between these elements.

In theory a weir or notch" is closely related to the orifice; in fact, an orifice becomes a notch when the water level falls below its upper boundary.

THEOREM OF TORRICELLI.

The theorem of Torricelli, enunciated in his JDe Motu Gravium Naturaliter Accelerate, 1643, states that the velocity of a fluid passing through an orifice in the side of a reservoir is the same as that which would he acquired by a heavy body falling freely through the vertical

a Commonly applied to a deep, narrow weir.

THEORY OF WEIR MEASUREMENTS.

11

height measured f rom the surface of the fluid in the reservoir to the center of the orifice.

This theorem forms the basis of hydrokinetics and renders the weir and orifice applicable to stream measurement. The truth of this proposition was confirmed by the experiments of Mariotte, published in 1685. It can also be demonstrated from the laws of dynamics and the principles of energy.a

ELEMENTARY DEDUCTION OF THE WEIR FORMULA.

In deducing a theoretical expression for flow over a weir it is assumed that each filament or horizontal lamina of the nappe is actuated by gravity acting through the head above it as if it were flowing through an independent orifice. In fig. 1 the head on the successive orifices being If, If, If, etc., and their respective areas Ax, A₂, A₃, etc., the total discharge would be

-0

Fig. 1.—Torricellian theorem applied to a weir.

If the small orifices A be considered as successive increments of head If the weir formula may be derived by the summation of the quantities in parentheses, ^comprises n elementary strips, the breadth of each

is The heads on successive strips are y, etc., and the total

becomes

.    . (3)

etc., for a rectangular weir.

series Vi+V^+V^-T to Hence the discharge is

where

The sum of the

col bo

q=l-h

n

= LIlfifcjH.

- Ifcl

CM CO

CM ICO

The above summation is more readily accomplished by calculus.

a See Wood, Elementary Mechanics, p. 167, also p. 291.

APPLICATION OF THE PARABOLIC LAW OF VELOCITY TO WEIRS.

The following elementary demonstration clearly illustrates the character of the weir:

According to Torricelli’s theorem (see fig. 1), the velocity (v) of a filament at'any depth (a?) below surface will be v—pgx. This is the equation of a parabola having its axis OX vertical and its origin O at water surface. Replacing the series of jets by a weir with crest at A, the mean velocity of all the filaments will be the average ordinate of the parabola OPQ. The average ordinate is the area divided by the height, but the area of a parabola is two-thirds that of the circumscribed rectangle; hence the mean velocity of flow through the weir is two-thirds the velocity at the crest, i. e., two-thirds the velocity due to the total head AT on the crest. The discharge for unit length of crest is the head AT, or area of opening per unit length, multiplied by the mean velocity. This quantity also represents the area of the parabolic velocity curve OPQX. The mean velocity of flow in the nappe occurs, theoretically, at two-thirds the depth on the crest.

The modification of the theoretical discharge by velocity of approach, the surface curve, the vertical contraction at the crest, and the various forms that the nappe may assume under different conditions of aeration, form of weir section, and head control the practical utility of the weir as a device for gaging streams.

GENERAL FORMULA FOR WEIRS AND ORIFICES.a

Consider first a rectangular opening in the side of a retaining vessel. The velocity of flow through an elementary layer whose area is Ldy will be from Torricelli’s theorem:

v=^2gy.

y h, h₂

Fig. 2.—Rectangular orifice.

The discharge through the entire opening will be, per unit of time, neglecting contractions,

Q=

pgy- Ldy

(4)

a The correlation of the weir and orifice has been given by Merriman. See Hydraulics, pp. 42-43.

This is a general equation for the flow through any weir or orifice, rectangular or otherwise, Q being expressed as a function of y. In the present instance L is constant. Integrating,

ICO

bi

n|cc

Si

ior a weir or notch, the upper edge will be at surface, Hx — O, and calling II^ — H in equation (5),

(6)

C=|zV2fflli

In the common formula for orifices, only the head on the center of gravity of the opening is considered.

Expressing Il₂ and IIX in terms of the depth II on the center of gravity of the opening and the height of opening^, Merriman obtains, after substituting these values in and expanding equation (5) by the binomial theorem, the equivalent formula,

Q—dL^lgh J^l —

1 JL _ ¹

96 IF 2048 IF 21845 IF\

etc.

]'

• (7)

The sum of the infinite series in brackets expresses the error of the ordinary formula for orifices as given by the remainder of the equation. This error varies from 1.1 per cent when h=d to 0.1 per cent when h=3d.

VERTICAL CONTRACTION.

Practical weir formulas differ from the theoretical formula (6) in that velocity of approach must be considered and the discharge must be modified by a contraction coefficient to allow for diminished section of the nappe as it passes over the crest lip. Velocity of approach is considered on pages 14 to 20. Experiments to determine the weir coefficient occupy most of the remainder of the paper. The nature of the contraction coefficient is here described.

Vertical contraction expresses the relation of the thickness of nappe, s, in the plane of the weir crest, to the depth on the crest, FI. If the ratio s II were unity, the discharge would conform closely with the expression

Q=2/3 LH V2gH.

The usual coefficient in the weir formula expresses nearly the ratio sill.

The vertical contraction comprises two factors, the surface curve or depression of the surface of the nappe and the contraction of the under surface of the nappe at the crest edge. The latter factor in

particular will vary with form of the weir cross section, and in general variation in the vertical contraction is the principal source of variation in the discharge coefficient for various forms of weirs.

The usual base weir formula, (>=2/3 LII^2gII, is elsewhere given for an orifice in which the upper edge is a free surface. If instead the depth on the upper edge of the orifice is d, the surface contraction, there results the formula

. . (8)

This is considered as the true weir formula by Merriman.® In this formula only the crest-lip contraction modifies the discharge, necessitating the introduction of the coefficient. The practical difficulties of measuring d prevent the use of this as a working formula.

Similarly a formula may be derived in which only "the effective cross section s is considered, but even this will require some correction of the velocity. Such formulas are complicated by the variation of s and d with velocity of approach.⁶ Hence, practical considerations included, it has commonly been preferred to adopt the convenient

2 _

base formula for weirs,    MLII^2g H, or an equivalent, and throw

all the burden of corrections for contraction into the coefficient M.

VELOCITY OF APPROACH.

THEORETICAL FORMULAS.

Before considering the various practical weir formulas in use some general considerations regarding velocity of approach and its effect on the head and discharge may be presented.

In the general formula (4) for the efflux of water when the water approaches the orifice or notch with a velocity v, then with free discharge, writing D-\-h in place of II, for a rectangular orifice, we have

Q=

'I)₂ T" A

4%gy . Ldy .    .

.    .    (9)

Dx and D₂ being the measured depth on upper and lower edges of the

qs2

orifice, and A=—, the velocity head.

To assume that D-\-h equals II is to assume that the water level is

a Hydraulics, p. 123.

b See Trautwine and Marichal’s translation of Bazin’s Experiments, pp. 231-307, where may also be found other data, including a r6sum6 of M. Boussinesq’s elaborate studies of the vertical contraction of the nappe, which appeared in CompteS Rendus de 1’Academic des Sciences for October 24,1887.

VELOCITY OF APPROACH.

15

increased by the amount A, or, as is often stated, that His “measured to the surface of still water.” This is not strictl}⁷ correct, however, because of friction and unequal velocities, which tend to make Er—D>h, as explained below.

For a weir, equals zero; integrating,

2    _ 3

Since Q —    we have

*«

t©|co

co|*o

.    .    (9a)

This is the velocity correction formula used by James B. Francis."

Since h appears in both the superior and inferior limits of integration, it is evident that h increases the velocity only, and not the section of discharge. The criticism is sometimes made that Francis’s equation has the form of an increase of the height of the section of discharge as well as the velocity.

The second general method of correcting for velocit}? of approach consists of adding directly to the measured head some function of the velocity head, making

- JI=D+ah

in the formula

Q= GLU^gH

or

9b

Q—CL(D-\-och)^2g(DJrah).

This is the method employed by Boileau, Ftele}^ and Stearns, and Bazin. No attempt is made to follow theory, but an empirical correction is applied, affecting both the velocity and area of section.

By either method v. must be determined by successive approximations unless it has been directly measured.

Boileau and Bazin modify (95) so as to include the area of section of channel of approach, and since the velocity of approach equals Q A, a separate determination of v is unnecessary. Bazin also combines the factor for velocity of approach with the weir coefficient.

The various modifications of the velocity correction formulas are given in conjunction with the weir formulas of-the several experimenters.

a Bovey gives similar proof of this formula for the additional cases of (1) an orifice with free discharge, (2) a submerged orifice, (3) a partially submerged orifice or drowned weir, thus establishing its generality.

DISTRIBUTION OF VELOCITY IN CHANNEL OF APPROACH.

The discharge oyer a weir takes place by virtue of the potential energy of the la}⁷er of water lying above the level of the weir crest, which is rendered kinetic by the act of falling over the weir. If the water approaches the weir with an initial velocity, it is evident that some part of the concurrent energy will facilitate the discharge.

The theoretical correction formulas may not truly represent the effect of velocity of approach for various reasons:

1.    The fall in the leading channel adjacent to the measuring section is the source of the velocity of approach, and this fall will alwaj's be greater than that required to produce the existing velocities, because some fall will be utilized in overcoming friction.

2.    The velocity is seldom uniform at all parts of the leading channel and the energy of the water varies accordingly⁷. This effect is discussed later (p. 17).

3.    It is not certain just what portion of the energy of the water in the section of the leading channel goes to increase the discharge.

u:

i___r

->■

AC

hr

Fig. 3.—Distribution of velocities.

In general the threads of the water in the cross section of the channel of approach to a weir have varying velocities. It follows that, as will be shown, the ratio of the actual energy of the approaching water to the energy due to the mean velocity will be greater than unity, and for this reason the correction for velocity of approach will be greater than if the energy were that due to a fall through a head produced by the mean velocity v. The more nearly uniform is the velocity of the water in the leading channel the smaller will be the necessary coefficient a in the velocity head formula. The velocity may be rendered very nearly uniform by the use of stilling racks or baffles. Where this was done in the experiments on which a formula was based (that of Francis, for example) a larger velocity of approach correction than that obtained by the author may be necessary in applying the formula to cases where there is wide variation in the velocity in the leading-channel. To avoid such a contingency it is desirable, when practicable, to measure head to surface of still water, because more accurate results can be obtained and wash against instruments prevented.

The vertical and horizontal velocity curves in an open channel usually closely resemble parabolas. A weir interposes an obstruction in the lower part of the channel, checking the bottom velocities. The velocity is not, however, confined to the filaments in line with the section of the discharge opening of the weir. As a result of viscosity of the liquid, the upper rapidly moving layers drag the filaments underneath, and the velocity may extend nearly or quite to the channel bottom. There will usually, however, be a line (A B C, fig. 3), rising as the weir is approached, below which there is no forward velocity.

The line ABC is the envelope of the curves of vertical velocity in the channel of approach.

There will be a similar area of low velocity at each side of the channel, for a contracted weir. The inequality of velocities for such weirs being usually greater than for suppressed weirs, it follows that a larger coefficient in the formula for velocity of approach may be required. This is confirmed by experiment.

Various assumptions have been made as to what portion of the energy of the approaching stream goes to increase the discharge, (a) that resulting from the mean velocity deduced from the discharge divided by the area of the entire section of the channel of approach; (~b) that of the mean velocity obtained by using the sectional area of the moving water, above the line A B C, fig. 3; (c) that of the filaments lying in line with or nearest to the section of the weir opening, determined approximately b}r the surface velocity.a

DISTRIBUTION OF ENERGY IN CHANNEL OF APPROACH.

Consider unit width of the channel of approach:

Let vs — Surface velocity. vm— Mean velocity. vb — Bottom velocity. v = Velocity at a height x above bottom. X— Depth of water in channel of approach. w = Weight of unit volume.

The general formula for kinetic energy is

-K.E.= ™

%/

(10)

where W— weight of the moving mass.

If the velocit}⁷ increases uniformly from bottom to surface, the velocity at height x will be

v=vb+~^ (v-vb).

a Smith, Hamilton, Hydraulics, p. 68.

-2

IKK 150—06-

Let dx be the thickness of a lamina one unit wide at height x. The total kinetic energy for the depth X will be

r*

tH

If the velocity is uniform, the total kinetic energy per unit width is found by integration to be

K. E. =

wXvn³

2g

(12)

Integrating for the simple case where vb=0 and the velocity increases uniformly from the bottom to the surface so that    we have

K. E. =

(13)

wXvm³

Comparing this with the expression for kinetic energy of a stream flowing with the uniform velocity v (formula 12), we find the mass energy of the stream with uniformly varying velocity to be twice as great as for the uniform velocity.

By a similar integration the ratio of the total kinetic energy to the kinetic energy corresponding to the mean velocity in the channel of approach can be obtained for any assumption as to the distribution of Arelocities in the leading channel. The resulting ratio will depend upon the relative areas of section with low and high velocities which go to make up the mean, and in practice it will generally exceed unity.

The lowering of the water surface from the level of a still pond will also be greater in the case of unequal velocities than in the case of a uniform velocity equal to their mean. The theoretical weir formula indicates the same discharge in case of a uniform velocity of approach v as in case of varying velocities whose mean is equal to v, although in the former case the actual drawing down of the head if it were measured would be found greater. If h were the velocity head corresponding to the mean velocity, and if v₂, v₃, etc., vn were the actual velocities in the n unit areas of cross section, the actual velocity head h' will be such that

Qy>

(V+V+eto. vn*)=w Qh' = Integral K. E.

Now,

/i?=wQh='K. E. of average velocity.

As shown above, the integral K. E. is the greater.

It follows that h">h.

If

Then

h' — ah.

Introducing velocity of approach in the discharge formula we substitute D-\-h for II, and integrate between the limits zero and D. Hence, for the same discharge, the area of weir section is greater without velocit}^ of approach by nearly the amount hi.

For a given measured head D, the effect of velocity of approach, whatever it may be, appears as an increase in the mean velocity of discharge in the plane of the weir. The relation of the mean velocity of discharge for a weir with velocity of approach to that for a weir without such velocity is shown by the following expression, the mean head being the same in both cases:

Mean velocity in the plane of the weir

then

: I)² : (D+hy-K*.

It will be seen that the discharge over a weir with velocity of approach is less than that for the same total head and greater than that for the same measured head without velocity of approach, and that with a given measured head the greater the velocity of approach the greater will be the discharge.

In a weir section opening out of still water there is always a considerable surface velocity, the parabolic law (see fig. 3) being modified by fluid friction, which tends to equalize the velocities. Velocity of approach, being usually greater at the surface, furthers this equalization. Some of the kinetic energy of the swifter-moving filaments is transferred to their slower-moving neighbors, the result being that while the kinetic energy of the whole mass Q passing the weir per second remains constant, yet the average velocity is accelerated and the discharge rate is increased as compared with the theoretical quantities. This will be clearer if we consider two contiguous filaments, each having unit section «, one with a velocity of 1, the other of 2 feet per second. The two will discharge 2+1 units flow per second, having the total kinetic energy indicated below:

2X 2s

K. E.=

lxl⁹

aw-f

cm

2g

am—9vr

If, now, the velocities are equalized, 9 units of kinetic energy will be equally divided between the two filaments, so that, the new velocity being -y,

2awv X v² _ 'daw ~2g

and .

v—

loi |<N

rH

Oi

i—

The average velocity before equalization was 1.5.

The discharge from two filaments having equal velocities will be 3.802 units, as compared with 3.00 for two filaments having unequal velocities.

THE THIN-EDGED WEIR.

EARLIER EXPERIMENTS AND FORMULAS.

Prior to 1850 the practice of weir measurement was in a somewhat chaotic condition, especially in England, Germany, and the United States. There were many experimental results, but the experiments were made on so small a scale that the various influences affecting the measurements and the lack of proper standards made the results erratic and untrustworthy in detail. Greater advancement had been made in France by.such savants as Dubuat, Eytelwein, ITAubuisson, Castel, Poncelet, Lesbros, and Boileau. Some of the work of the early French experimenters has proved, in the light of wider experience, to be of considerable value.

EXPERIMENTS OF CASTEL.

The first experiments deserving consideration are of those of M. Castel, conducted at the waterworks of Toulouse in 1835 and 1836/* Castel erected his apparatus on a terrace in conjunction with the water tower, which received a continuous supply of 1.32 cubic feet per second, capable of being increased to 1.77 cubic feet per second. The weir consisted of a wooden dam, surmounted byr a crest of copper 0.001 foot in thickness, situated in the lower end of a leading channel, 19.5 feet long, 2.428 feet wide, and 1.772 feet deep. Screens were placed across the upper end of the channel to reduce oscillations. The head was measured at a point 1.60 feet upstream from the weir by means of a point gage. The overflow was measured in a zinc-lined tank having a capacity of 113.024 cubic feet. The length of the crest for weirs with suppressed contractions varied from 2.393 to 2.438 feet. Heights of weirs varying from 0.105 to 0.7382 were used, and a Originally published in Memoires Acad. Sci. Toulouse, 1837. See D’Aubuisson’s Hydraulics, Bennett’s translation, pp. 74-77. Data recomputed by Hamilton Smith in his Hydraulics, pp. 80-82 and 138-145. The recomputed coefficients will be found valuable in calculating discharge for very small and very low weirs.

a similar series of experiments was performed on suppressed weirs 1.1844 feet long. The head varied for the longer weirs from about 0.1 to 0.25 foot. Additional experiments were made on contracted weirs having various lengths, from 0.0328 to 1.6483 feet, in a channel 2.428 feet wide, and for leng-ths from 0.0328 to 0.6542 foot in a channel 1.148 feet wide. The experiments on these narrow slit weirs included depths varying from 0.1 or 0.2 foot to a maximum of about 0.8 foot.

D’Aubuisson gives the following formula, derived from the experiments of Castel for a suppressed weir:

.    .    (14)

<?=3.4872ZA> V/i+0.035 TP .    .

where W is the measured central surface velocity of approach, ordinarily about 1.2-y.

EXPERIMENTS OF PONCELET AND LESBROS.

The experiments made by Poncelet and Lesbros, at Metz, in 1827 and 1828, under the auspices of the French Government, were continued by Lesbros in 1836. The final results were not published, however, until some years later.®

The experiments of Poncelet and Lesbros and of Lesbros were performed chiefly on a weir in a fixed copper plate, length 5.562 feet. The head was measured in all cases in a reservoir 11.48 feet upstream, beyond the influence of velocity of approach. The crest depth varied from about 0.05 to 0.60 or 0.80 foot. The experiments of Lesbros are notable from the fact that a large number of forms of channel of approach were employed, including those with contracted and convergent sides, elevated bottoms, etc. The experiments of Lesbros on these special forms of weirs have been carefully recomputed by Hamilton Smith, and nm^ be useful in determining the discharge through weirs having similar modifications.⁵

EXPERIMENTS OF BOILEAU.

The experiments of Boileau⁰ at Metz, in 1846, included 3 suppressed weirs, having lengths and heights as follows:

(1)    Length 5.30 feet, height 1.54 feet.

(2)    Length 2.94 feet, height 1.12 feet.

(3)    Length 2.94 feet, height 1.60 feet.

The depth of overflow varied from 0.19 to 0.72 foot. Boileau obtained the following formula for a suppressed weir:

0—3.3455

P+D

MP+Df-D

I ID* ...    .    (15)

a Experiences hydrauliques snr les lois de l’Ocoulement de l’eau, Paris, 1852. b Smith, Hamilton, Hydraulics, pp. 96 and 97 and 104-107. Also plates 1-2 and 8. cGaugeage de cours d’eau, etc., Paris, 1850.

This formula includes the correction for velocity of approach. The coefficient 0, it will be noticed, is given as a constant. Boileau afterwards gave a table of corrections varying with the depth, indicating a discharge from 96 to 107 per cent of that obtained with the constant coefficient. Additional experiments by Boileau on suppressed weirs having a crest length of about 0.95 foot have been recomputed by Hamilton Smith.® The heights of weirs were, respectively, 2.028, 2.690, 2.018, and 2.638 feet. In these experiments the discharge was determined by measurement through orifices.

EAST INDIAN ENGINEERS⁵ FORMULA.⁶

The East Indian engineers⁵ formula for thin-edged weirs is

where

Q=t₆ MI *JL2gH³= Clin C= I J/=5.35 M

• • (1«)

0.04 [34.6+77]

Reducing-,

M= 0.654 -0.01 if 67= 3.4989—0.0535 II

(IT)

This formula applies to a suppressed weir. Method of correction for velocity of approach is not stated. Coefficient M has a maximum value 0.654, and decreases slowly as the head increases. Limits of applicability of formula are not stated. Values of Care given below:

Coefficient C for thin-edged weirs, East Indian engineers’- formula.c

Velocity of approach correction, K, Parmley and Bazin formulas.

H—Ho+ft

11=1)+A.

Bazin's experiments on weirs of irregular section—Continued.

Q=C.LHK

United States Geological Survey experiments at Cornell hydraulic laboratory on model of flat-top weirs with vertical faces.

Q=LKH^2gR(l—K) = K^2g(l-K)Liri

EXPERIMENTS OF DWIGHT PORTER, BLACKSTONE RIVER, AL

(I)

Q=3.S3L&%.

Table 4.—Discharge over a thin-edged weir per foot of crest—Continued.

Column (2) in the following table gives the values

CLASSIFICATION OF THE PUBLICATIONS OF THE UNITED STATES GEOLOGICAL SURVEY.

Coefficient C for thin-edged weirs, East Indian engineers’- formula.c

II in | feet.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3.488 3.435 3.381 3.328 3.274 3.221 3.167 3.114 3.060 3.007

3.472 3.419 3.365 3.312 3. 258 3.205 3.151 3.098 3.044 2.991

3.028 3.023 2.975    2.969 j

a Hydraulics, pp. 133-135.

5 Given in J. Mullins’s Irrigation Manual, introduced in United States by G. W. Rafter and used in region of upper Hudson River. Not given in Bellasis’s recent East Indian work on hydraulics.

c For East Indian engineers’ broad-crested weir formula, using coefficients derived from the above, see p. 114.

EXPERIMENTS AND FORMULA OF JAMES B. FRANCIS.

The experiments on discharge over thin-edged weirs,® upon which the Francis formula is based, were made in October and November, 1852, at the lower locks of the Pawtucket canal, leading from Concord River past the Lowell dam to slack water of Merrimac River. Additional experiments were made by Francis in 1848b at the center vent water wheel at the Boott Cotton Mills in Lowell, with gates blocked open and with constant head. A uniform but unknown volume of water was thus passed through the turbine and over a weir having various numbers of end contractions, the effect of which was thus determined. Similar experiments were made in 1851 at the Tre-mont turbine,'⁰ where a constant volume of water was passed over weirs of lengths ranging from 3.5 to 16.98 feet and with from two to eight end contractions. These experiments were made to determine the exponent n in the weir formula

Q= CLHn.

Francis here found 71=1.47, but adopted the value 71=1.5=3/2, in the experiments of 1852.

The Pawtucket canal lock was not in use at the time of the Lowell experiments in 1852 and the miter gates at the upper lock chamber were removed and the weir was erected in the lower hollow quoin of the gate chamber. The middle gates at the foot of the upper chamber were replaced by a bulkhead having a sluice for drawing off the water. A timber flume in the lower chamber of the lock was used as a measuring basin to determine the flow over the weir. Its length was 102 feet and its width about 11.6 feet. A swinging apron gate was so arranged over the crest of the weir that, when opened, the water flowed freely into the measuring basin below, and when closed, with its upper edge against the weir, the overflow passed into a wooden diverting channel, placed across the top of the lock chamber, and flowed into Concord River. An electric sounder was attached to the gate framework, by which a signal was given when the edge of the swinging gate was at the center of the nappe, when either opening or closing. By this means the time of starting and stopping of each experimental period was observed on a marine chronometer. The depth on the weir was observed by hook gages. The readings were taken in wooden stilling boxes, 11 by 18 inches square, open at the top, and having a 1-inch round hole through the bottom, which was about 4 inches below the weir crest. The weir was in the lower quoin of the gate recess, and the hook gage boxes were in the upper quoin, projecting slightly beyond the main lock walls. In weirs with end

a Francis, J. B., Lowell Hydraulic Experiments, pp. 103-135.

6 Idem, pp. 96-102.

o Idem, pp. 76-95.

contractions the full width of the channel was used. For suppressed weirs, a leading channel having a width equal to the length of the weir crest was formed by constructing vertical timber walls within the main canal, extending 20 feet upstream from the weir and having their upper ends flaring about 1 foot toward the canal walls. Water was freely admitted on both sides of these timber walls. The hook gage boxes were outside of this channel. The holes in the bottom were plugged, and flush piezometer pipes were used to connect the hook-gage boxes with the inner face of the side walls of the channel of approach. Observations of the head by hook gage were taken at intervals of about 15 seconds. Each experimental period covered from 190 to 900 seconds. The hook-gage readings were reduced to weir crest level as a datum and arranged in groups of two or three, which agreed closely. The mean head was determined by the correction formula (48). In one period, 18 observations of heads ranged from 0.6310 to 0.6605 foot; their arithmetical mean was 0.6428; the computed correction was minus 0.0008.

The measured head was corrected for velocity of approach by using the theoretical formula given below. The range and character of the experiments, together with the general results, are shown in the following table:

Thin-edged weir experiments of J. B. Francis at the lower lochs, Lowell, Mass., 1852.

Serial numbers of

ments.

From—

Total number.

Width of channel at upstream side of weir.

Depth of channel below

top of weir at hook

Range of ve-Range of ob- j locity of served head, approach, in in feet.

feet per second.

03

.be

From—

To—

From—

To—

Length of weir crest in

feet.

Number end contractions.

Approximate mean corrected head H, in feet.

r:

~-

Maximum.

Minimum.

Mean.

4 j 13.96 6 : 13.96 23 ' 13.96 2 | 13.96 ; 8 I 13.96

7 | 9.992

5    i 9.992

6    | 13. 96

5    13.96

5    9.992

7    13.96 13.96

4 I 13.96

5.048    1.52430    1.56910    0.7682    0.7889 , 9.997

2.014    .66940    i    .68815    . 4382    . 4526 j 7.997

2    1.56    , 3.3318    3.3002    3.3181

2    1.26    3.3412    3.3159    3.3338

2    1.00    3.3333    3.3110    3.3223

4    1.02    3.3617    3.3586    3.-3601

2    1.06    3.3567    ■    3.3498    3.3527

0    0.98    3.3437    I    3.3366    3.3409

0    1.00    3.3349    3.3243    3.3270

2    0.80    3.3287    3.3188    3.3246

2    0.83    3.3435    3.3376    3.3403

0    0.80    3.3424    .    3.3341    3.3393

2    0.62    3.3306    3.3237    3.3275

2    0.65    3.3278    !    3.3244    3.3262

4    0.68    3.3382    j    3.3333 ! 3.3368

From a discussion of these experiments Francis presents the final formula—

0=3.33777*.

If there are end contractions,

• • (18)

Z=Z'-0.1iV77

If there is velocity of approach,

eota$

IP=[(Z>+A)

The mean velocity v was determined by successive approximations; h was determined by the usual formula—

The Francis formula for velocity of approach correction is cumbersome, and several substitutes have been devised, some of which are described in the following- paragraphs.

(1) Determine the approximate velocity of approach by a single trial computation of Q, using D—II.

Then use

‘2g

II— 7>+s¹ —D-\-h

to determine the final value of Q. For a given value of v this gives too large a value of 77, but the approximate value of vx is somewhat too small, partially counterbalancing the error and usually giving a final value of Q sufficiently precise.

(2) By developing into series and omitting the powers h ID above the first, h being always relatively small, the following closely approximate equivalent of the Francis correction formula, given by Emerson/4 is obtained:

77= D+h-

IV

(19)

(3) Hunking and Hart⁵ derive from the Francis correction formula the following equivalent expression:

. . (20)

• (21)

where G is the area of channel section in which D is measured, per unit length of crest.

WEIR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

For a suppressed weir,

G=l>+Z>.

For a contracted weir,

G=

Z'-0.1 A/A.

Hunking and Hart have computed values of K bv the solution of the above formula for each 0.005 increment in D G to 0.36. The results extended by formula (23) are given below.

Velocity of approach correction, factor K, Hunking and Hart formula, II I=KD I.

DIG

o.o

0.1

0.2

0.3

0.4

0.5

0.6

.000

.005

.010

.015

.020

I .025 I .030

.035

.040

: .045

.050

.055

.060

.065

.070 .075 !

.080

.085

.090

.095

1.00000 | 1.000006 1.000026 1.000058 1.000103 1.000161 1.000231 1.000314 1.000409 1.000518 1.000638 1.000772 1.000917 1.001075 1.001246 1.001429 1.001624 1.001832 1.002051 1.002284

1.002528    1.009980    j

1.002785    1.010480

1.003053    1.010994

1.003335 j 1.011519 1.003628 j 1.012057 ; 1.003933    1.012607

1.004251    1.013169

1.004581    1.013744

1.004923 j 1.014331 1.005278 • 1.014931 1.005644    1.015543

1. 006023    1.016167

1.006414    1.016805

1.006817    1.017455    J

1.007232    1.018107    j

1.007659    1.018792

1.008099    1.019480

1.008551    1.020180

1.009015    1.020893

1.009491    1.021620

1.022359 ! 1.023110 j 1.023875 1.024653 j 1.025444 1.026248 1.027065 1.027895 1.028739 1.029596 1.030467 1.031350 1.032248 1.033117 1.034113 1.035109 1.035856 1.036852 1.037848 j 1.038844

1.039840 1.040836 1.041832 1.042828 1.043824 1.045069 1.046065 1.047061 1.048306 1.049302 1.050298 '

1.051543 1.052788 .

1.053784 1.055029 1.056274 1.057270 1.058515 1.059760 1.061005    -1.088146

1.062250    1.08964

1.063495    1.091134

1.064740    1.092628

1.065985    1.094122

1.067230 | 1.095616 1.068724 | 1.097359 1.069969    1.098853

1.071214    1.100347

1.072708    1.102090

1.073953    1.103584

1.075198    1.105078

1.076692    1.106821

1.078186    1.108564

].079431    1.110058

1.080925    1.111801

1.082419    1.113544

1.083664    1.115038

1.085158    1.116781

1.086652    1.118524

1.120267

The general formula for K is too complex for common use. The expressions

(23)

(24)

K= 1+0.2489

and

K= 1+

are stated to give results correct within one-hundredth and one-tiftieth of L per cent, respectively, for values of K less than 0.36.

EXPERIMENTS AND FORMULAS OF FTELEY AND STEARNS.

The first series of experiments by Fteley and Stearns on thin-edged weir discharge* were made in March and April, 1877, on a suppressed weir, with crest 5 feet in length, erected in Sudbury conduit below Farm Pond, Metropolitan waterworks of Boston.

Water from Farm Pond was let into the leading channel through a Fteley, A., and Stearns, F. F., Experiments on the now of water, etc.: Trans. Am. Soc. C. E., vol. 12, Jan., Feb., Mar., 1883, pp. 1-118.

head-gates until the desired level for the experiment, as found by previous trial, was reached. A swinging gate was then raised from the crest of the weir and the water was allowed to flow over. The maintenance of a uniform regimen was facilitated by the large area and the consequent small variation of level in Farm Pond, so that the outflow from the gates was sensibly proportional to the height they were raised. The water flowed from the weir into the conduit channel below, and was measured volumetrically. For the smaller heads the length of the measuring basin was 22 feet, and for the larger heads 367 feet.

The crest depth was observed by hook gage in a pail below the weir, connected to the channel of approach by a rubber tube entering the top of the side wall, 6 feet upstream from the weir crest. Hook-gage readings of head were taken every half minute until uniform regimen was established, and every minute thereafter. The depths in the measuring basin were also taken by hook gage. The bottom of the conduit was concave, and was graded to a slope of 1 foot per mile. It was covered with water previous to each experiment, leaving a nearly rectangular section.

The experiments in 1877 included 31 depths on a suppressed weir of 5 feet crest length, 3.17 feet high. The observed heads varied from 0.0735 to 0.8198 foot.

In 1879 a suppressed weir, with a crest length of 19 feet, was erected in Farm Pond Gate House. Head-gates and screens were close to weir; otherwise the apparatus for measuring head and starting and stopping flow was similar to that used in previous experiments. The crest of the weir was an iron bar 3i inches wide and one-fourth inch thick, planed and filed and attached to the upper weir timber with screws. No variation in level of the weir crest occurred. As in the preceding experiments, no by-pass was provided, and the entire overflow entered Sudbury conduit below the weir. The conduit was partly filled with water at the start, leaving a nearly rectangular section, 11,300 feet in length and about 9 feet wide. A difference of 3 feet in water level was utilized in measuring discharge, the total capacity being 300,272 cubic feet. Semipartitions were provided to reduce oscillation of the water. Mamr observations, covering a considerable period of time, were required to determine the true water level. This series of experiments included 10 depths on a suppressed weir 19 feet long and 6.55 feet high, with measured heads varying from 0.4685 to 1.6038 feet and velocities of approach ranging from 0.151 to 0.840 foot per second.

From measurements on weirs 5 and 19 feet in length, respectively, and from a recalculation of the experiments of James B. Francis, Fteley and Stearns obtained the final formula

In the above, if there is velocity of approach,

H=D+ah.

oc =1.5 for suppressed weirs.

oc =2.05 for weirs with end contractions.

The value of the velocity head coefficient oc was determined from 94 additional experiments on the 5-foot weir in 1878. These involved measured heads ranging¹ from 0.1884 to 0.9443 foot, heights of weir ranging from 0.50 to 3.47 feet, and velocities of approach reaching a maximum of 2.35 feet per second. Also 17 experiments were made on weirs 3, 3.3, and 4 feet long respectively; the- first with two and the last two with one end contraction. These experiments included measured heads varying from 0.5574 to 0.8702 foot, and velocities of approach from 0.23 to 1.239 feet per second.

In all experiments on velocity of approach, the head was measured 6 feet upstream from crest. The width of channel was 5 feet.®

Fteley and Stearns found the following values of oc for suppressed weirs:

Fteley and Stearns’s value of oc for suppressed weirs.

Measured depth on weir, in feet.

Depth of channel of approach below weir crest, in feet.

0.50

1.00

1,70

2.60 b

0.2

1.70

1.87

1.66

1.51

.3

1.53

1.83

1.65

1.50

.4

1.53

1. 79

1.63

1.49

.5

1.53

1.75

1.62

1.48

.6

1.52

1.71

1.60

1.47

.8

1.51 cl. 50

1.68

1.59

1.46

«1. 65

1.57

1.45

.9

1.49

1.63

1.56

cl. 44

1.0

1.48

1.61

1.54

1.43

1.1

1.59

1.53

1.42

1.2

1.57

1.51

1.41

1.3

1.55

1.49

1.40

1.4

1.54

1.48

1.39

1.5

1.52

1.46

1.38

1.6

1.51

1.44

1.37

1.7

1.49

1.43

1.36

1.8

1.41

1.35

1.9

1.40

1.34

2.0

1.38

1.33

a Fteley and Stearns, idem, pp. 5-23.

& Applicable to greater heights of weir. c Limit of experiments.

Current-meter measurements showed a nearly uniform distribution of velocities in the channel of approach above the 19-foot weir, a fact to be taken account of when the formulas are applied to cases where the velocity of approach varies in different portions of the leading channel.

If there are end contractions, the net length-of weir should be determined by the Francis formula,

The head should be measured at the surface of the channel of approach, 6 feet upstream from the weir crest.

BAZIN’S EXPERIMENTS.

Bazin’s experiments on thin-edged weirs were performed in the side channel of the Canal de Bourgogne, near Dijon, France, and were begun in 1886. Their results were published in Annales des Fonts et Chaussees and have been translated by Marichal and TrautwineA

The standard weir consisted of horizontal timbers 1 inches square, with an iron crest plate 0.2T6 inch in thickness. Air chambers were placed at the ends of the weir on the downstream side, to insure full aeration of the nappe. End contractions wTere suppressed. The height of the first weir was 3.27 feet above channel bottom, and the head was measured in “Bazin pits,” one at each side of the channel 16.10 feet upstream from the weir crest. The pit consisted of a lateral chamber in the cement masonry forming the walls of the canal. The chamber was square, 1.61 feet on each side, and communicated with the channel of approach by a circular opening 1 inches in diameter, placed at the bottom of the side wall and having its mouth exactly flush with the face of the wall. The oscillations of the water surface in the lateral chamber were thus rendered much less prominent than in the channel of approach. The water level in the Bazin pit was observed by dial indicators attached to floats, the index magnifjdng the variations in water level four times, the datum for the indicators having been previously determined by means of hook gages placed above the crest of the weir and by needle-pointed slide gages in the leading channel.

A drop gate was constructed on the crest of the weir to shut off the discharge at will. In each experiment the head-gates through which the water entered the leading channel were first raised and the water was allowed to assume the desired level. The weir gate was then raised, and the head-gates were manipulated to maintain a nearly con-

a Bazin, H., Recent experiments on flow of water over weirs, translated from the French by Marichal and Trautwine: Proc. Engineers' Club Phila., vol.,7, Jan., 1890, pp. 259-310; vol. 9, pp. 231-244, 287-319; vol. 10, pp. 121-164.

stant inflow. The arithmetical mean of the observations during- each period of uniform regimen was used as the measured head for that experiment.

The overflow passed into a measuring channel, 656.17 feet in length, whose walls were made of smooth Portland cement concrete. The channel was 6.56 feet wide, its side walls were 3.937 feet high, and its lower end was closed by water-tight masonry. Its bottom was graded to a slope of about 1:1,000. The volume of inflow was determined by first covering the channel bottom with water, then noting the change of level during each experimental period, the capacit}^ of the channel at various heights having previously been carefully determined. A slight filtration occurred, necessitating a correction of about one-eighth of 1 per cent of the total volume. The observations for each regimen were continued through a period of 12 to 30 minutes.

Sixty-seven experiments were made on a weir 3.72 feet high, includ ing heads from the least up to 1.017 feet. Above this point the volumetric measuring channel filled so quickly as to require the use of a shorter weir. Thirty-eight experiments were made with a standard weir, 3.28 feet long and 3.72 feet high, with heads varying from the least up to 1.34 feet. For heads exceeding 1.34 feet it was necessary to reduce the height of the weir in order that the depth above the weir should not exceed that of the channel of approach. Forty-eight experiments were made on a weir 1.64 feet long and 3.297 feet high, with heads ranging from the least up to 1.780 feet. These experiments sufficed to calibrate the standard weir with a degree of accuracy stated by Bazin as less than 1 per cent of error.

In order to determine the effect of varying velocities of approach the following additional series of experiments were made on suppressed weirs 2 meters (6.56 feet) in length.

Experiments on suppressed weirs 2 meters in length.

Number of experiments.

Range of head in feet.

From—

To—

Height of experimental weir, in feet.

28+30

29+29

27+41

44

2. 46

1.64

1.15

0. 79

The standard weir was 3.72 feet high, and the experimental weirs were placed 46 to 199 meters downstream. The discharge was not measured volumetrically. A uniform regimen of flow was established and the depths on the two weirs were simultaneously observed during each period of flow.

These experiments afforded data for the determination of the relative effect of different velocities of approach, corresponding to the different depths of the leading channel.

From these experiments Bazin deduces coefficients for a thin-edged weir 3.72 feet high, for heads up to 1.97 feet, stated to give the true discharge within 1 per cent.⁰⁵

BAZIN’S FORMULAS FOR THIN-EDGED WEIRS.

Starting with the theoretical formula for a weir without velocity of approach, in the form

and substituting

Q=MLHj2gH

D+a

for II, in the case of a weir having velocity of approach, there results,

Bazin obtained, by mathematical transformation, the equivalent⁵

or

Bazin writes

Q=^LD 4%gl) (l+« y#)*, Q=M    jfy-D-

m—yu ( 1 + nr

(26)

for which equation he obtains, by mathematical transformation, the approximate equivalentc

7)1=yU

1+

-y²

■ • (27)

2gD

The calculation of the factor v appearing in this formula requires the discharge. Q to be known.

Assuming that the channel of approach has a constant depth P below the crest of the weir, and that its width is equa¹ to the length of the

« Bazin, H., Experiences nouvelles sur l’ecoulement en deversoir: Ann. Ponts et Chaussdes, Mem. et Doc., 1898, 2m« trimestre. See translation by Marichal and Trautwine in Proc. Eng. Club Phila., vol. 7, pp. 259-310; vol. 9, pp. 231-244.

6 The steps in the derivat!on of this formula are given by Trautwine and Marichal in their translation of Bazin’s report of hit experiments, in Proc. Eng. Club Phila., vol. 7, p. 280.

cThe steps in detail are gh n by Trautwine and Marichal in their translation of Bazin, in Proc. Eng. Club Phila., vol. 7, No. 5, p. 281.

weir, v may be expressed in terms of these factors, and of the discharge

(Q=mLDV 2 gD).

UP+D).

Using this value of v, Bazin obtains the expression

. . (28)

where go—

| oc m².

co is a nearly constant factor, varying only with

m². The value of go as well as that of a can be determined by comparative experiments on thin-edged weirs of different heights.a

From a discussion of his own experiments and those of Fteley and Stearns, Bazin finally obtained the formulas

Q—juLIfilSgll, no velocity of approach; Q=mZJJ^2gD, with velocity of approach.

0.00984b

•    •    (29)

M=0.405

0.003X3.281

= 0.405+

.    .    (30)

COi Ox

For a weir with velocity of approach a— and co=0.55. Substituting in equations (27) and (28),

m—

M ^1+    •

CO I CM

• • (31)

in—

“ [¹⁺⁰-⁶⁵ (pTpJ

• (32)

These formulas give values of m agreeing with the results of the experiments within 1 per cent for weirs exceeding about 1 foot in height within the experimental range of head.

Approximately, for heads from 4 inches to 1 foot,

.    .    (33)

m—0.425+0.21

correct within 2 to 3 per cent.

The following table gives Bazin’s experimental coefficients, the head and height of weir (originally meters) having been reduced to feet:

a For detailed analysis see Trautwine and Manchal, Proc. Eng. Club Phila., vol. 7, pp. 282-283. b Experimental tabular values of p. differing very slightly from the formula within the range of Bazin’s experiments are also given.

Values of the coefficient m.

8S

fe

is

BAZIN’S EXPERIMENTAL COEFFICIENT

.2

.3

.5

.6

.7

M?,

Ss

U. S. GEOLOGICAL SURVEY

m FOR THIN-EDGED WEIRS OF VARYING HEIGHT, FOR USE IN THE FORMULA Q=mLDV29 D.

-V

°\

Heads in feet.

.8

.9

1.0

1.1

1.2

8*1

VI

1.5

1.6

1.7

1.8

1.9

2.0 B

fc.

X-

51

rk

7%

4-

do.

ft

“V’S;

*T2

'?

0>-

Xy

7o;

is

150—06-3

r“

o5:

Sir

£_

c>

Vi

«!

“Itt

jy j-W

ill

WATER-SUPPLY PAPER NO. 150 PL. I

.

'

Values of the Bazin coefficient C in the formula Q—CLH² for a thin-edged weir, without

end contraction.

Measured head D.

Height of crest of weir above bed of channel of approach, in feet.

0.66    0.98

1.31

1.64

1.97

2.62

3.28    4.92

6.56

00

Measured head D.

Feet.

Meters.

0.164

.197

.230

.262

.295

3.665    3.585    , 3.545

3.609 3.569 3.553 3.537 3.529

3.601 3.569 3.545 3.529 3.513

3.497 3.497

3.593 3.553 3.529 3.505 3.489

3.593 3.553 3.521 3.505 3.481

3.594 3.550 3.522 3.499 3.481

0.05

.06

.07

.08

.09

3.681    3.585    j 3.545

.328

.459

.525

.591

3.521 3.513 3.513 3.513 3.521

3.505 3.497 3.489 3.489 3.489

3.489 3.473 3.465 3.457 3.457

3.481 3.465 3.449 3.440 3.432

3.473 3.449 3.432 3.424 3.416

3.473 3.449 3.432 3.416 3.408

3.466 3.441 3.422 3.405 3.392

.10

.12

.14'

.16

.18

.656

.722

.787

.853

.919

3.978 ¹ 3.785    3.665 | 3.593 | 3.537

3.513

3.457 3.457 3.465 3.465 3.473

3.432 3.432 3.432 3. 440 3.440

3.408 3.400 3.400 3.400 3.400

3.392 3.392 3.384 3.384 3.384

3.380 3.371 3.364 3.358 3.353

.20

.22

.24

.26

.28

.984 j 4.010    3.810

3.834 3.858 3.874 3.898

1.050

1.116

1.181

1.247

3.689 3.705 •3.721 3.745 3.761

3.609 3.625 3.641 3.657 3.673

3.553 3.561 3.577 3.593 3.601

3.481 3.497 3.505 3.513 3.521

3.449 3.449 3.457 3.465 3.465

3.400 3. 400 3.400 3.400 3.400

3.376 3.376 3.376 3.376 3.376

3.348 3.343 3.338 3.333 3.328

.30

.32

.34

.36

.38

1.312

1.378

1.444

1.509

1.575

3.922 3.938 3.962 3.978

3.785 3.801 3.818 3.834 3.850

3.681 3.697 3.713 3.729 3.745

3.617 3.625 3.641 3.657 3.665

3.529 3.537 3.545 3.553 3.561

3.473 3.481 3.489 3.489 3.497

3.400 3.408 3.408 3.408 3.408

3.376 3.376 3.376 3.376 3.376

3.323 3.319 3.316 3.311 3.306

.40

.42

.44

.46

.48

1.640

1.706

1.772

1.837

1.903

1.969

3.866 '3.874 3.890 3.906 3.922 3.930

3.753 3.769 3.785 3.793 3.810 3.818

3.681 3.689 3.697 3.713 3.721 3.737

3.569 3.577 3.585 3.593 3.601 3.617

3.505 3.513 3.513 3.521 3.529 3.537

3.416 3.416 3.416 3.424 3.424 3.424

3.376 3.376 3.376 3.376 3.376 3.376

3.303 3.298 3.294 3.289 3.285 3.282

.50

.52

.54

.56

.58

.60

Meters.

0.20

0.30

0.40

0.50

0.60

0.80

1.00

1.50

2.00

00

This table, unfortunately, is inconvenient for interpolation in English units. The values also differ slightly from those computed from the formulas. The table illustrates the difficulty of practical application of a weir formula in which the coefficient varies rapidly both with head and height of weir.

A table has been added giving values- of ft computed by formula (30) for a thin-edged weir without velocity of approach.

Values of u in the Bazin formula for weirs of infinite height, with no velocity of approach.

H.

Feet.

o.

0.01. 0.02.

0.03.    0.04.

0.05.

0.06.

0.07. | 0.08.

0.09.

0.1.

1.389    0.8970

1.9    . 4102    . 4102    . 4101

2.0    .4099 ................

0.0

.4

0.7331 | 0.6510    0.6018    0.5693    0.5457 | 0.5280    0.5142 J    0.5034

.4137 j .4136 .4130 .4124 .4119 .4114 | .4114 .4110 .4107 .4104 .4101

DERIVED FORMULAS FOR THIN-EDGED RECTANGULAR WEIRS.

A number of weir formulas have been derived from subsequent analysis or recomputation of the experiments of Francis, Fteley and Stearns, and Bazin, differing more or less from those given by the experimenters.

FTELEY AND STEARNS-FRANCIS FORMULA.a

#=3.33Z//*+0.007Z

.    (34)

Correction for end contractions is to be made by the Francis formula; velocity of approach correction by the Fteley and Stearns formulas

II—5A, H=D+2.05A,

for suppressed weir, for contracted weir.

HAMILTON smith’s FORMULA.h

The base formula adopted is

Q=| MLirhtjH

(35)

a Fteley and Steams, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, p. 82. 6Smith, Hamilton, Hydraulics, pp. 123-132.

The velocity of approach correction is made by the use of the formulas

II-— -D~\~ 1.4: h. II=D-\-\jsh,

for contracted weirs.a for suppressed weirs.

A diagram and tables of values of the coefficient Mare given by the author. The correction for partial or complete contraction is included in the coefficient, separate values of AT being given for suppressed and contracted weirs.

2 _

Making 0= g JHj2g, the Smith formula (35) may be written

to'co

Q — CL II ,

which is directly comparable with the Francis formula.

Smith’s coefficients in the above form are given in the following tables.

Hamilton Smith’s coefficients for weirs with contraction suppressed at both ends, for use in

the formula Q= CLH¹.

n=

Head, in feet.

19

16

10

J/=length of weir, m feet.

■2a

0.666

(OCCSIOOiilxOJtOlCM Oi    in

1.0

1.1

1.2

1.3

1.4

1.6

1.6

1.7

3.616    3.615    3.620    3.520    3.526 ........................

3.333 3.349 3.381 3.413

3.611 3.542 3.510 3.494 3.483 3.478 3.478 3.483 3.494 3.510

i, to determine D,

is also recommended. 6 Approximate.

Hamilton Smith’s coefficients for weirs with two complete end contractions, for use in the

formula Q=CLHJ.

11=

Z'=length of weir, in feet.

Head.

0.66

2.6

10

15

19

.0

.7

3.381 3.312 3. 269 3. 237 3.215 3.183 3.156 3.140 3.130

3.419 3.344 3.306 3.274 3.253 3.215 3.189 3.172 3.156

.8

.9

1.0

1.1

1.2

1.3

1.4

3.456 3.392 3.349 3.322 3.296 3.258 3.237 3.215 3.199 3.183 3.167 3.156 3.140 3.130 3.114 3.103

1.5

1.6

3.162

3.494 3.419 3.376 3.344 3.322 3.285 3.264 3.247 3.231 3.221 3.210 3.199 3.189 3.178 3.167 3.156 3.151 3.140

1.7

3.231 ! 3.247 3.226 j 3.242 3.215 J 3.231 3.205 j 3.226 3.194 I 3.215 3.199    3.205

.......    3.178

3.504 3.429 3.386 3.360 3.338 3.306 3.290 3.280 3.274 3.269 3. 258 3.253 3.242 3.237 3.231 3.221 3.215 3.210 3.205

3.295 | 3.301 3.285 ! 3.290 3.280 ! 3.285 3.274    3.280

2.0

(i Approximate.

Hamilton Smith’s coefficient C for long iveirs.

II

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

3.5096 3.4957 3.4818 3.4678 3.4539 3.4400 3.4314 3.4229 3.4143 8.4058

3.3972 3.3908 3.3844 3.3780 3.3716 3.3652 3.3537 3.3512 3.3488 3.3463

3.3438 3.3411 3.3384 3.3358 3.3331 3.3304 3.3277 3.3250 3.3224 3.3197

3.3170 j 3.3010 3.3154    3.3005

3.3138    3.2999

3.3122    3.2994

3.3106 | 3.2988 3.3090    3.2983

3.3074    3.2978

3.3058    3.2972

3.3042    3.2967

3.3026    3.2961

3.2956 3.2945 3.2935 3.2924 3.2913 3.2902 3.2892 3.2881 3.2870 3.2860

3.2849 3.2838 3.2828 3.2817 3.2806 3.2796 3.2785 3.2773 3.2762

3.2752

Hamilton Smith’s formula is based on a critical discussion of the experiments of Lesbros, Poncelet and Lesbros, James B. Francis, Fteley and Stearns, and Hamilton Smith; including series with and without contractions and having crest lengths from 0.66 to 19 feet.

SMITH-FRANCIS FORMULA.

The Smith-Franeis formula,“ based on Francis’s experiments, reduced to the basis of correction for contractions and velocity of approach used with Hamilton Smith’s formula, is, for a suppressed weir,

Q= 3.29

(36)

(37)

for weir of great length or with one contraction, Q=3.WLH^ . .    .    .

for weir with full contraction,

0=3.29

(38)

If there is velocity of approach,

for a contracted weir, for a suppressed weir.

H=D+1A 7i,

l-j h,

parmlfy’s FORMULA.⁶

Parmley’s formula is

0= CKLD*

.    (39)

If there are end contractions, the correction is to be made by the Francis formula,

L=Z'-0AArH

The factor iTrepresents the correction for velocity of approach. The factor has been derived by comparing the velocity correction factor in the Bazin formula (formula 32), written in the form

K—

[i+⁰-⁵<3>]’

with the approximate Francis correction as deduced by Hunking and Hart (formula 23), written in the form

K=

1+0.2489

where a is the area of the section of discharge, for either a suppressed or contracted weir, and A is the section of the leading channel. It is observed that there is an approximately constant relation between the two corrections, that of Bazin being 2.2 times that of Francis.

a Smith, Hamilton, Hydraulics, pp. 99 and 137.

& Rafter, G. W., On the flow of water over dams: Trans. Am. Soc. C. E., vol. 44, pp. 350-359, discussion by Walter C. Parmley.

Parmley adopts the Bazin correction and gives the following table, which may also conveniently be applied in computing discharge by Bazin’s formula.

The discharge coefficient C used by Parmley is that for a weir with no velocity of approach, as in the Francis formula. It is not, however, constant. Its values have been deduced from a mean curve representing the experiments of Francis, Fteley and Stearns, and Bazin.

Velocity of approach correction, K, Parmley and Bazin formulas.

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.1

0.2

0.3

0.4

0.5

........I    1.0055

1.0001 |    1.0006

1.0002 j    1.0079

1.0005 I    1.0093

1.0009 |    1.0108

1.0014    1.0124

1.0020    1.0141

1.0027 ' 1.0159 1.0035    1.0178

1.0044    1.0198

1.0220 1.0243 1.0266 1.0291 1.0317 1.0344 1.0372 1.0401 1.0431 1.0463

1.0495 1.0529 1.0563 1.0599 1.0636 1.0674 1.0713 1.0753 1.0794 1.0837

1.0880 1.0925 1.0970 1.1017 1.1065 1.1114 1.1164 1.1215 1.1267 1.1321

1.1375 1.1431 1.1487 1.1545 1.1604 1.1664 1.1725 1.1787 1.1850 1.1915

Parmley's weir formula, coefficient C.

o.oo

0.01 [ 0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

<M CO Tt<

I-¹ .

co oo cj o

co to

Id O 00 1    1-4 r-t I—t r-I 1—1 r-H

1.9

2.0

3.580 3.478 3.420 3.385 3.368 3.358 3.351 3.346 3.340 3.334 | 3.329 3.324 3.319 3.313 3.307 3.301 3.296 3.290 I 3.285 3.280

3.532 3.451 3.404 3.378 3.363 3.355 3.349 3.344 3.338 3.332 3.327 3.322 3.317 3.311 3.305 3.299 3.294 3.288 3.283

EXTENSION OF THE WEIR FORMULA TO HIGHER HEADS.

It will be noticed that all the accepted formulas for discharge over thin-edged rectangular weirs are based on experiments in which the head did not exceed 2 feet above crest. It is often desirable to utilize the weir for stream gagings where the head is greater, especially for the determination of maximum discharge of streams, the head frequently being as large as 6, 8, or even 10 or 12 feet.

In the experiments at Cornell University on weirs of irregular section it was often necessary to utilize depths on the standard weir exceeding the known limit of the formula. A series of experiments was accordingly carried out in which a depth on a standard thin-edged weir (16 feet long) not exceeding the limit of the formula was utilized to determine the discharge over a similar but shorter standard thin-edged weir (6.56 feet long) for depths up to approximately 5 feet.® The results of these experiments, as recomputed, eliminating slight errors in the original, are given below.

It will be noted that the weir was short and the velocity of approach relatively large, yet, according to the results when corrected by the Francis method, the average value of C for heads from 0.75 to 4.85 feet is 3.296, or 98.88 per cent of the Francis coefficient for a thin-edged weir. The average value of C for heads from 0.746 foot to 2 feet is 3.266, and for heads from 2 to 4.85 feet, 3.278.

United States Beep Waterways experiments at Cornell hydraulic laboratory for extension of

thin-edged weir formula.

Standard weir, 16 feet long, 13.13 feet high.

Lower thin-edged weir: P= 5.2, L=6.66.

Cor. D, longitudinal, piezometer, centimeters.

Q, Bazin formula, in cubic feet per second.

Observed P, flush, piezometer, centimeters.

in feet.

-P J Hunking P+D 1 and 1 Hart.

in

Q, cubic feet per second, per foot (corrected) .

II-

5)

3    3 3    @    3    a    3 3    3    $1 S’S S

N    Ifl’ 00    ri    T)i    N    6 d    h    05 'J' H

T—li—It-HC'IC'JC4'C0C0C0'^lO1>Q0

CC'Wg*4ifcv)0)MCC(0W^H

22.744 27.855 33.175 39.419 44.000 49. 699 55.213 55.128 68.238 80.566 105.639 130.286 142.557

0.7462 .9139 1.0885 1.2933 1.4436 1.6306 1.8115 1.8088 2.2389 2.6434 3.4660 4.2747 4.6773

0.1255    |    1.0041

.1495    j    1.0056

.1731    !    1.0075

.1992    1.0099

. 2173    1.0122

. 2387    1.0141

.2583    1.0166

. 2581    1.0166

. 3010    1.0225

. 3370    1.0283

.4000    1.0398

.4512    1.0504

.4735    I    1.0557

0.6469 .8787 1.1434 1.4849 1.7564 2.1116 2.4787 2.4730 3.4254 4.4193 6.6095 9.2867 10. 6789

If it is borne in mind that the influences which go to make up variation in the weir coefficient are more potent for low than for larger heads, it may be confidently asserted that the Francis formula is applicable within 2 per cent for heads as great as 5 feet, and by inference it is probably applicable for much greater heads as well.

COMPARISON OF WEIR FORMULAS.

The later weir formulas all give results agreeing, for the range of heads covered, within the limit of accuracy of ordinary stream measurements. Which of the several formulas to use will be determined by convenience and by the conditions attending the measurements.

The Francis formula is applicable for weirs with perfect bottom contraction and for an}^ head above 0.50 foot.

The-Hamilton Smith, Fteley and Stearns, and Bazin formulas are more accurate for very slight heads, or where bottom contraction is imperfect, this element, which tends to increase discharge, being-included in the larger velocity of approach correction. These formulas are, however, based on experiments none of which exceeded 2 feet head, and they have not been extended.

For suppressed weirs in rectangular channels having conditions closely duplicating Bazin’s experiments, his formula is probably most applicable. The head should preferably be measured in a Bazin pit, opening at the bottom of the channel, 16.4 feet upstream from the weir. In a suppressed weir, if the nappe is allowed to expand laterally after leaving the weir, the computed discharge by any of the formulas should be increased from one-fourth to one-half of 1 per cent.

Comparative discharge by various formulas over weirs of great height and length; no end, contractions nor velocity of approach.a

Formula.

Coefficient C, for heads ranging from 0.20 to 4 feet.

Per cent of discharge by Francis formula for heads ranging from 0.20 to 4 feet.

0.20

0.50

1.00

4.00

0.20

0.50

1.00

4.00

Castel.....................

Boileau...................

Weisbach.................

Francis...................

Fteley and Stearns........

Bazin.....................

Fteley-Steams-Francis....

Hamilton Smith..........

Smith-Francis............

Parmley..................

East Indian engineers....

8.4872    | 8.4872

3.3455    3.3455

.......... 3.3136

3.4872 3.3455 3.4025

3.33

3.5004 3.642684 3. 3800 3.3972

3.29

3.478

3.488

3.33

3.3269 3.406094 3.3300 3.3010

3.29

3.368 3.472

3.33

3.317 3.326696 3.319

3.284

3.29

3.334 3.445

3.4872 3.3455

3.33

3.3109 3. 26783 3. 31375

3.284

3. 29

3.285

104.616

100.365 102.075 100.00 105.012 109.281 101.400 101.916 98. 70 104.340 104.640

104. 616

100.365

100.0 99.807 102.183 99.90 99. 030 98.70 101.040 104.16

104.616

100.365 99.408 100.0 99.51 99.801 99.570 98.520 98.70 100. 020 103.35

104.616

100.365

100.0 99.327 98.035 99.412

98. 550

a Computed by H. R. Beebe, C. E.

Table showing comparative discharge per foot of crest for suppressed weirs of various lengths, heads, and velocities of approach.a

Length (L)............................

Height (P)............................

Head (D).............................

Approximate velocity of approach (v)

10

10

1.0

1.0

1.0

1.90

1.18

1.16

J

co

2.15

Castel ...................

Boiieau..................

Francis..................

Fteiey and Stearns.......

'Bazin....................

Fteley-Stearns-Francis ____

Hamilton Smith..........

Smith-Francis . _ _ _ „.......

Parmley.................

3.7822    3.6127

3.8630    3.5484

3.5373    3.4218

3.7268 ! 3.4729

3.    7845    3. 3766

3.7297    3.4752

3.9220    3.6392

4.    0581    3. 7109

3. 7924    3. 5337

3. 6217 3. 5484 3. 4218 3. 4730 3. 3766 3. 4752 3.4872 3.4847 3.5337

3. 5308 3.4144 3. 3632 3.3669 3,4002 3.3690 3.3878 3.3876 3.3347

30. 3037 30.9046 28.2983 29.7470 29.7555 29.7000

31.573

Average

3. 800

3.532    3.490

3. 395

30.040

a Computed by H. R. Beebe, C. E.

COMPARISON OF VARIOUS VELOCITY OF APPROACH CORRECTIONS.

The various inodes of correction for velocity of approach used by different investigators can be rendered nearly identical in form, vary-ing, however, in the value of the coefficient a adopted.

Comparative coefficients of correction for velocity of approach for thin-edged weirs with end

contractions suppressed.

Experimenter.

Value of a. in the for-vi

mula H=D+a

Values of to in the formula

K=l+to

(?)

<*=1.8 a—1.56 a—1.5

Boileau...........

Lesbros ..........

Fteley and Stearns

Francis

aa=1~5

Bazin

<r=1.69 or

wi tal sH

o go=0.2489

go—0.55

a Emerson.

6 Hunking and Hart.

The above values were all derived from experiments on thin-edged weirs. Bazin’s experiments covered the larger range of velocities and were most elaborate. It may be noted that the correction applied by Bazin is two and two-tenths times that of Francis for a given velocity

of approach. Bazin s correction is, in effect, an increase in the measured head of 1.69 times the velocity head, while Francis increases the

less than the velocity head accord-

Ol I CO

measured head by an amount

ing to Emerson’s formula.

Ratio of the various corrections for velocitiy of approach, for suppressed weirs.

Bazin.

Fteley and Stearns.

Hamilton

Smith.

Francis.

Bazin............

Fteley and Stearns Hamilton Smith Francis...........

1.000

.887

. 789

.454

1.127 1.000

.887

.511

1. 271

1. 128 1.000

. 576

2.2

1.957

1. 736 1.000

The factors in the above table are not strictly accurate, for the reason that the expressions used to deduce the equivalents from the different formulas are in some cases approximations. They serve to illustrate the relative magnitude of the different corrections for thin-edged weirs without end contractions. For thin-edged weirs with end contraction, Hamilton Smith uses the coefficient a= 1.4 and Fteley and Stearns give the coefficient o'=2.05.

There are no experiments available relative to the value of the velocity correction for other than thin-edged weirs. It is necessary, therefore, to utilize the values above given for weirs of irregular section. It will be seen that it matters little in what manner the correction for velocity of approach is applied, either by directly increasing the observed head, as in the formulas of Hamilton Smith and Fteley and Stearns, or by including the correction in the weir coefficient, as is done by Bazin, or by utilizing a special formula to derive the corrected head, after the manner of James B. Francis. The three methods can be rendered equivalent in their effect.

The important point is that the corrected result must be the same as that given by the author of the formula which is used to calculate the discharge. As to the relative value of the different modes of applying the correction, it ma}T be said of that of Francis, that in its original form it is cumbersome, but it renders the correction independent of dimensions of the leading channel, as do also the formulas for correction used by Hamilton Smith, and Fteley and Stearns. Inasmuch as the velocitj^ head is a function of the discharge, successive approximations are necessary to obtain the final corrected head by any one of these three formulas.

By using the Hunking and Hart formula the correction for the Francis weir formula becomes fairly simple, as it does not require the determination of the mean velocity of approach by successive approximations, but to apply this formula it is necessary to know the dimensions of the leading channel and of the weir section. The approximation given by Emerson is also much simpler than the original Francis formula.

Bazin’s method of including the velocity correction in the coefficient makes the weir coefficients obtained by the experiments comparable one with another only when both the head and velocity of approach are the same in both cases.a His correction also involves the dimen* sions of the leading channel as factors. Obviously, in the case of many broad-crested weirs utilized for measuring flow, the dimensions of the leading channel can not be ascertained accurately and there is great variation of velocity in different portions of the section of approach. It becomes necessary that the correction should be in such a form that it is a function of the velocity and not of the channel dimensions.

It is to be noticed that where an attempt has been made in the weir experiments to eliminate velocity of approach effect from the coefficient the velocity has been nearly equalized by screens and has been determined by successive approximations. It is suggested that where the velocities vary widely they be determined by current meter in several subdivisions of the section, the approximate integral kinetic energy estimated, and a value of a selected depending on the

ratio of -j~ so obtained, where h is the velocity head corresponding to

the mean velocity and ti is the velocity head which would result if the actual velocities were equalized. Inasmuch as the surface velocity usually exceeds the mean velocity in the channel of approach in about the same ratio that h' exceeds A, the suggestion is made by Hamilton Smith* that where the velocity of approach is unavoidably variable, or the boundaries of the current are uncertain, the surface velocity vs be measured by floats and applied directly in the determination of the quantity A.

The variations in discharge over a thin-edged weir, by the different formulas, are often less than the difference in the correction for velocity of approach would indicate. In the formula of Fteley and Stearns, as compared with Francis, for example, the larger velocity correction is in part compensated by a smaller weir coefficient, and the same is true of the formulas of Hamilton Smith and Bazin for cases where the head is large.

a See special discussion of the point, p. 63.

b Smith, Hamilton, Hydraulics, p. 84.

END CONTRACTIONS—INCOMPLETE CONTRACTION.

The formula for end contractions deduced by James B. Francis is very generally used. The correction is made to the length of weir, the result obtained being the length of a suppressed weir that will give the same discharge.

(40)

I —JJ—l) Nil

b = A coefficient, the value of which, deduced by Francis, is b=0.1.

L' ~ Actual length of weir crest.

L — Length of equivalent suppressed weir crest.

N— Number of end contractions.

H— Effective head, feet.

The experiments of Fteley and Stearns," while somewhat discordant, indicate an average value of b for heads from 0.3 to 1 foot, of about 0.1. The value of b apparently decreases as the head increases. It also decreases if the end contraction piece is so near the side of the channel as to render the contraction incomplete.

Hamilton Smith shows that side contractions and bottom or crest contraction are mutually related, and that the side width of the channel of approach should be fully three times the least dimension of the weir. Usually 7 is much greater than 77, and the side width may be made at least as great as 3H. The specification of Francis is, side width II.

Smith’s rule indicates that to provide complete contraction the area of leading section A must bear a relation to the area of weir section a depending upon the relative head and length of crest.

For three weirs of equal section a, the following values of A, the necessary channel-section area, are given:

77= 1 77= 3 77=12

A= 72=6a A = 26±=2.2a H = 105 = 8.7«

a=12 a=12 a=12

11 = 12 Z' = 4 Z' = 1

Hamilton Smith prefers to use separate coefficients for suppressed weirs from those for contracted weirs, the relation between the coefficients being expressed by the formula

(41)

Gr= a (l+e £)

Op—Coefficient for partially suppressed weir, as with complete suppression on sides and full contraction at bottom.

a Fteley and Stearns, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 108-113.

Cc— Coefficient for completely contracted weir.

' X — Least dimension of weir, whether L or H.

R = Wetted perimeter of weir=Z+2/Z

Y — Distance from any side of weir to the respective side of channel, where there is partial suppression.

& = Length of sides on which'there is partial suppression.

Smith’s values of contraction coefficient z in formula 31 are

Y/X    z

CO tH O

0.000

.005 .025 .06 .16

The ratio Y AT approximately measures the amount of contraction.a

Bazin does not give a formula for weirs with end contractions. The Bazin formula may be applied to weirs in which the height of weir is so small that the bottom contraction is partially suppressed. The Bazin coefficient then includes:

1.    Effect of contraction from surface curve.

2.    Effect of crest contraction and its modification by both velocity of approach and by partial suppression, if any.

3.    Effect of velocity of approach proper.

4.    Effect of distribution of velocities in channel of approach.

5.    Loss of head from friction and eddies.

As the Bazin weirs were very low, and these factors go to increase the correction necessary, it will be seen that the relatively large velocity of approach correction required by Bazin’s formula may be readily accounted for.

The experiments of Flinn and Dyer on the Cippoletti weir (see p. 48) indicate that the effect of end contraction may be somewhat greater than that indicated Hy the Francis formula. Any experiments in which similar volumes of water have been successively passed over weirs with and without end contractions may be utilized to determine the effect of such contractions.

It may be added that a more elaborate stud}⁷ of end contractions is desirable. It is to be borne in mind, however, that to secure greater accuracy in this regard a more complicated or variable correction than that of Francis must probably be used, and the result will be to greatly increase the labor of weir computations in the interest of what is usually a comparatively small matter, the better remedy being probably the use of weirs with end contractions suppressed, wherever practicable.

a Smith, Hamilton, Hydraulics, pp. 118-123. Smith’s critical discussion of this subject will be found of value in calculating discharge for weirs with partially suppressed contraction either at sides or bottom.

COMPOUND WEIR.

A weir with a low-water notch depressed below the general crest level may sometimes be used to advantage in gaging small, variable streams. The discharge over such a weir, constructed with end contractions on both sections, can be calculated as for two separate weirs, the lower short section having end contractions for all heads. The flow over the two upper sections is computed as for a suppressed weir.

Such a weir has been used for the determination of the low-water flow of very small streams, for which purpose it is well adapted, the entire stream when at low stages flowing in the central notch, in a stream relatively deep and narrow.

The measurement of very thin sheets of water on a broad weir is subject to peculiar difficulties, including uncertainty of fcoefficient, adhesion of nappe to weir face, dispersion by winds, and a large percentage error in the results if there is a small error in measuring the head.

TRIANGULAR WEIR.

GENERAL FORMULA.

Referring to fig. 4, we may write

i'x

Fig. 4.—Triangular weir.

Substituting, in equation (4),

Q=

II

r0

gy

II

dy

Thomson’s experiments.

The mean coefficient of contraction for a t’hin-edged triangular weir deduced experimentally by Prof. James Thomson, of Belfast, is 31= 0.617,a the formula being

• • (43)

Q=^MI H*=1.32LH% . .

a British Association Report, 1858 (original not consulted). Merriman gives the mean value of M for heads between 0.2 and 0.8 foot as 0.592.

For a right-angled notch,

(44)

Z=2ZTand $=2.64ZT^

The length of the contracting edges in a triangular notch being proportional to the depth, it is believed that the coefficient of discharge is somewhat more constant than for a rectangular weir.a

TRAPEZOIDAL WEIR.

The discharge in this case may be determined directly from the integral formula (4) as for a triangular weir, by integrating between the limits AD and CE, fig. 5. It may also be derived as follows:

if-r—f

&l

Fig. 5.—Trapezoidal weir.

z=slope of one side to the vertical.

yy—II

V2gy [L+2z(3—y)\ dy

y—o

By integration,

(45)

Q=y%g LIF+ypg 11%

in which coefficients of contraction for the horizontal crest and for the end slopes must be introduced.

THE CIPPOLETTI TRAPEZOIDAL WEIR.

The discharge over a trapezoidal without contraction would be the sum of that for a rectangular weir added to that for two triangular weirs forming the ends. From the experiments of James B. Francis6 it appears that each end contraction reduces the effective length of the weir 0.1II. The contraction decreases the discharge by the amount

Q=| Mx0.lHj2g    ^ V^ZT*

If the ends of the weir, instead of being vertical, are inclined outward in such manner that the discharge through the added area counterbalances the decrease from the end contraction, then the effective

a The coefficient 2.64 is the same as that deduced for broad crest weirs with stable nappe. A table of values of 2.6472® is given on page 177, which may be applied in calculating flow over triangular weirs. b Lowell Hydraulic Experiments.

ikb 150—06-

length of the weir will remain constant as the head increases, the same as in a suppressed weir. The discharge through the end triangle ABC will be, from equation (42)

Where z is the width or base of the end triangle. Equating the two expressions for Q, and solving for 2, we find, assuming M to have the same value in both cases, "

z—

This condition defines the Oippoletti weir.a

CIPPOLETTl’s FORMULA.

Cippoletti derived his formula from a discussion of the experiments of James B. Francis, selecting a coefficient 1 per cent greater, making

Q=l X0.629 ZJIj2g&=3.3G7LJIi .... (47)

L is the length of the crest or base of the trapezoid.

Flinn and Dyer& experimented at the testing flume of the Holyoke Water Power Company by passing the same volume of water successively over a trapezoidal experimental weir and over the gaging weir of the turbine testing flume 19.7 feet downstream. The latter, it is stated, complied in form with Francis’s specifications.

The depths were observed b}^ hook gage; eleven readings, as a rule, being taken and their arithmetical mean used for the determination of a head. The thirty-two series of valid experiments range from 0.3 foot depth on a weir with sill length of 3 feet to a head of 1.25 feet on a sill 9 feet long.

The discharge over the standard weir was calculated by the formulas of J. B. Francis and of Hamilton Smith. The correction for velocity of approach at the experimental weir was made by the formula of Hamilton Smith, for use with contracted rectangular weirs,

B=D+lAh.

Flinn and Dyer’s coefficients are as follows:

0= 3.283

Mean of 32 experiments,

Mean after rejecting 5 diminished weights, C=3.301 In general, the coefficient diminished as the head increased, suggesting that the end inclination should slightly exceed -IT in the Cippoletti weir, to provide complete compensation, and that the end contraction

“First described by Cesare Cippoletti in Canal Villoresi, Modulo per la Dispensa delle Acque, 1887. 6 Flinn, A. D., and Dyer, C. W. D., The Cippoletti trapezoidal weir: Trans. Am. Soe. C. E., vol. 32, 1894, pp. 9-33.

coefficient in the trapezoidal weir may be greater than 0.17/, as used by Francis.

The question is complicated b}^ velocity of approach. For example, had the Francis velocity-correction formula been used by Flinn and Dyer, their values of C would have been larger. As a tentative conclusion it is probable that the application of either the Francis formula with his velocity-head correction or the Flinn and Dyer coefficient with the Smith velocity correction will, when applied to a Cippoletti weir, give results as accurate as the precision of the coefficients will justify.

REQUIREMENTS AND ACCURACY OF WEIR GAGINGS.

PRECAUTIONS FOR STANDARD WEIR GAGING.

Certain specifications were laid down by James JB. Francis as guides in cases where the utmost precision is desired in weir measurements.a The limits of applicability of the weir have been greatly extended since 1852, and some of the uncertainties as to the effect of various modifications of weir construction have been removed.

In general, for standard thin-edged weirs—

1.    The upstream crest edge should be sharp and smooth.

2.    The overflowing sheet should touch only the upstream crest corner.

3.    The nappe should be perfectly aerated.

4.    The upstream face of the weir should be vertical.

5.    The crest should be level from end to end.

6.    The measurements of head should show the true actual elevation of water surface above the level of the weir crest.

7.    The depth of leading channel should be sufficient to provide complete crest contractions, and, if they are not suppressed, the width of channel should be sufficient to provide complete end contractions.

8.    A weir discharging from a quiet pond is to be preferred. If this is not available, the velocity of approach in the leading channel should be rendered as uniform as possible and correction made therefor by the method employed by the experimenter in deriving the formula.

In order to fulfill these requirements, certain secondary conditions are necessary. The depth on the weir should be measured at a point far enough upstream from the crest to be unaffected by the surface curvature, caused by the discharge.

« Francis, J. B., Lowell Hydraulic Experiments, pp. 133-135.

The distance upstream to the point of measuring the head has been as follows:

Distance upstream from weir to gage used by various experimenters.

Experimenter.

Date.

Distance upstream, in feet.

Poneelet and Lesbros

Lesbros..............

Francis..............

Hamilton Smith, jr... Fteley and Stearns ... Bazin...............

1828

1834

1852

; 1874-1876

1878

1886

11.48

11.48

6. 00

7. 60

6.00

16. 40

Six feet upstream from crest is a distance frequently used, but this may be insufficient for suppressed weirs, and also for those having irregular cross sections or upstream slopes. Boileau considered the origin of the surface curvature to be at a distance from the weir equal to about 2.5 times the height of crest above the bottom of the channel of approach, indicating that for a suppressed weir the head should be measured at least this distance from the crest." For a weir discharging from a still pond the head can be measured at any considerable distance from the weir. Hamilton Smithb states that, for weirs with full contraction, 7/can be measured at any convenient point from 4 feet to 10 feet from the crest.

The head may be measured directly by a graduated scale or hook gage, or by means of a piezometer tube having its orifice flush with the side wall of the leading channel, and at right angles to the direction of flow of the water.

The depth of the leading channel in Francis’s experiments was 4.6 feet below crest, and Francis lays down the rule that the depth of the leading canal should be at least three times the head on the weir. Hamilton Smith Axes the minimum depth of the leading channel below the crest at 27T.

Fteley and Stearnsc state that the depth of the leading channel below weir crest should be at least 0.5 foot, in order that correction for velocity of approach may be reliably made for depths occurring in their measurements, and that a greater depth of leading channel is to be preferred.

To provide complete end contractions, Francis states that the distance from the side of the channel of approach to the end of the weir overflow should be at least equal to the depth on the weir. Hamilton

<*Fteley and Steams, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, p. 47. b Smith, Hamilton, Hydraulics, pp. 129-131. o Ibid., pp 112-114.

Smith considers that the distance from the end of the weir to the side of the channel should be at least 2i7, and that the depth of channel below crest, also the side distance, should in no case be less than 1 foot. Francis further specifies. that the length of weir crest should be at least three times the depth of overflow. The nappe should not be allowed to expand laterally immediately⁷ below a suppressed weir.

In order that the nappe may be perfectly aerated, Francis considers that the fall below crest level on the downstream side should be not less than ^II, increasing for very long weirs or in cases where the downstream channel is shallow. He found, however, no perceptible difference in the discharge fora head of 0.85 foot, whether the water on the downstream side was 1.05 feet or 0.0255 foot below crest level. Fteley and Stearns and Hamilton Smith agree that, if the water is

3 i?

?    3-8

* &A

If!

•11 I

•I

-------~u!0

5-

si

fed

—

(D

8.

deep below, it may rise to crest level on downstream side of weir without sensible error, and Fteley and Stearns add that a weir may be submerged to a depth of 15 per cent of the head without an error exceeding 1 per cent.

The thickness of crest lip is immaterial so long as the edge is sharp and square and the nappe cuts free and is freely aerated. The latter conditions require, however, that the crest shall be thin, especially where the head is slight.

Fig. 6 shows cross sections of the crest of the weir used by James B. Francis at the Lower Merrimac locks at Lowell, in 1852, in deriving his formula. The crest consisted of a cast-iron plate 13 inches wide and 1 inch thick, planed true and smooth on all surfaces. Its upper

edge was chamfered on the downstream side at an angle of 45° to a thickness of 0.25 inch at the edge. As shown, the nappe cut clear from the top of the crest in an unbroken sheet. The lowest head used by Francis was over 0.5 foot. For very low heads the crest lip should be thinner. A wooden crest tends, by capillary attraction, to cause the nappe to adhere to the flat top surface under low heads. A wooden crest is cheap, easily adjusted, and convenient for temporary use, but it will, in time, tend to become somewhat rounded, reducing the vertical contraction of the nappe.

A cast-iron crest will usually have to be made to order. A large steel angle bar may often be obtained from stock sizes of the rolling mills more cheaply. Such a bar, with legs, say 3 and 6 inches, respectively, with the 6-inch flat face planed and its edge trued, will form a rigid and permanent crest. The 3-inch leg may be bolted to the top of the timbers forming the body of the weir.

It may be added that approximate corrections for rounding of upstream corner of the crest, inclination of the weir upstream or downstream, or incomplete contractions can be made from data now available. In constructing gaging weirs preference, however, should be given to those forms which render the determination of the discharge the most simple, and the extent to which the preceding specifications may be departed from judiciously will depend upon the exigencies of the case and the purposes for which the results are desired.

PLANK AND BEAM WEIRS OF SENSIBLE CREST WIDTH.

Experiments on weirs with crest boards 1, 2, or 4 inches in thickness were made by Blackwell, Fteley and Stearns, and Bazin. The results show that for depths exceeding 1.5 to 2 times the crest width the nappe will break free, and if properly aerated the coefficient will then be identical with that for a thin-edged weir.

When the nappe adheres to the crest the coefficients are very uncertain for such weirs, adhesion of nappe to downstream face of crest and modified aeration entering to give divergent values.

The precise stage at which the change from an adhering to a free nappe or the reverse occurs is not constant, but varies with velocity of approach and with rate of change of the head as the changing point is approached, being different for a sudden and for a gradual change, and also when the point of change is approached by an increasing as compared with a decreasing head.

REDUCTION OF THE MEAN OF SEVERAL OBSERVATIONS OF HEAD.

In measurihg a constant volume of water, several observations of the head on the weir are desirable, the accuracy of the result, according to the theory of least squares, being proportional to the square root of the number of observations.

In weir experiments it is often impossible to maintain a perfectly uniform head or regimen. If the variations are minute the arithmetical mean may be used directly. If the variations are of wider range, or if the utmost precision is required, the following correction formula of Francis may be applied: a

Let Dx, D%, D₃, etc., Dn represent the several successive observed heads.

Let t₁, 4, 4? etc., 4 represent the corresponding- intervals of time between the several observations.

Let Trepresent their sum, or the total time interval.

Q—the total volume of water flowing over the weir in the time T.

j9=the mean depth on the weir that would discharge the quantity Q in the time T.

L = the length of weir crest.

C— the weir coefficient.

We have, very nearly,

Q=h OLDCLD*f7ZAl+etc.+% CLL>J

ju    jj    A

Also,

Q=TCLD*

Equating, eliminating the common factor CL, and solving for L>, we

have

Di

(48)

EFFECT OF ERROR IN DETERMINING THE HEAD ON WEIRS. 6

Consider the formula

Q= CLE%,

Differentiating, we have

dQ=| CL^HdH.

The error of any gaging when H+dH is taken as the head instead of the true head H being used will be dQ, and the ratio of this quantity to the true discharge Q will be

3 dir 2 IL

dQ_3CL 4H

This formula will give nearly the correct value of the error if the increment dH approaches an infinitesimal.

a Francis, J. B., Lowell Hydraulic Experiments, p. 113.

6Rafter, G. W., On the flow of water over dams: Trans. Am. Soc. C. E., vol. 44, p. 686; data here given based on discussion by Walter C. Parmley.

In the following- table is shown the effect of errors of one thousandth, five thousandths, one hundredth, and five hundredths foot, respectively, for various heads. This clearly illustrates both the necessity of proper care and the folly of ultra precision in measuring the relatively large values of II with which we are mainly concerned. The curves of error on PI. II are equilateral l^perbolas, which have been reduced to straight lines by plotting on logarithmic scales.

Percentage error in discharge resulting from various errors in the measured head on weirs.

Head, in feet.

0.001

Error in measured head, in feet.

0.005

0.01

0.05

Per cent.

Per cent.

Per cent.

Per cent.

0.1

1.5

7.5

15

.5

.3

1.5

1.0

.15

.75

CO rH

5.0

.03

.15

10.0

.015

.075

An error of a half-tenth foot under 5 feet head causes the same error in the result as an error of one-half hundredth foot with a head of one-half foot.

In weir experiments it is important to know the effect of an error in head lion the resultant coefficient of discharge C. The error in C is evidently equivalent to the error in Q found above, where H is constant.

ERROR OF THE MEAN WHERE THE HEAD VARIES.

In determining the volume of flow over dams where gaging records are kept, the method usually pursued has been to have readings taken twice daily, as at morning and evening, showing the depth flowing over the crest of the dam. The average of the two readings for each day has been found and the volume of flow corresponding to this average head has been taken as the mean rate of flow over the dam for the day.

It is evident, however, that as the discharge varies more rapidly than the head (usually considered to be proportional to the three-halves power of the head), the volume of discharge obtained as above described will be somewhat less than the amount which actually passes over the dam. The following analysis has been made to show the magnitude of the error introduced by using the above method.

Assuming that the initial depth on the crest of the dam is zero, but increases at a uniform rate to IIX at the end of a time interval T\ the

to

05

ip.

Measured head In feet. .....r

os os -ts oo so ©

to

05

OS

S 06 00

U. S. GEOLOGICAL SURVEY

Per cent of error in discharge.

EFFECT OF ERRORS IN WEIR MEASUREMENTS.

M    O    O    O    O    O    O

O    to    Oo    -*    o>    Cn    W*.

ISS S S g s

2.00

3.00

4.00

5.00'

6.00

7.00

/'

7_

\y

'°x

Js

N;

PS

'O'

'Of

Ai

A'

!

I

■<

yl-y~

8.00

9.00

10.00

mean head deduced from observations at the beginning and end of the period would be i the head at any time t would be

IL—ft

where y is a constant.

We may write the usual formula for weir discharge Q=CLH^\ then, if the head varies from zero to 11the total volume of flow in the time T will be

*s|w

.s

Or

The total discharge

^av

corresponding to the average head ^ Hx is

¹

CL

T— CL

«)

(51)

The ratio of the discharge is

Volume bj average head _ Actual volume

rH id d HO

«|is>

Or

CD

It appears that where the initial or terminal head is zero the volume of flow determined by using the average head will be 11.6 per cent too small. This percentage of error is the same whatever may be the maximum head 11, and whether the stream is rising or falling. It is also independent of the rate of change in the head.

Conditions like those above discussed occur at milldams during the season of low water, when the pond is allowed to till up at night and the water is drawn down to crest level or below during the day when mills are running.

The following example will illustrate. Suppose a sharp-crested weir without end contractions, with crest 1 foot long, on which the water rises to a depth of 1 foot in a period of 10 seconds—

\

Mean depth on a weir with varying head.

Time, in seconds,......................

Head, in feet, at end of each second...

Average head for period...............

Average head, second to second.......

.1

.05

.05

to

at

C5

<i

00

CO

to

CO

vO

vO iC tH <M

CM

at

to

at at

CO C5

CO

i-⁴

at

at

C5 CO at at

05

Using the average head during each second the volume of flow may be approximately integrated by finite differences, as follows, the discharge being taken from Francis’s tables:

Discharge over a weir with varying head.

Time, in seconds.

Average head, in feet.

Discharge, in second-feet.

0 to 1 ] to 2

0. 05

0. 037

. 15

.25

.35

.45

.55

.65

.75

.85

.95

. 194

.416

.690

1.005

1.    358 1.745 2.163

2.    609

3.    083

13. 30

Total

The average head for the entire period, 0.5 foot, gives a discharge for 10 seconds of 11.773 second-feet, or 88.5 per cent of that given above, the numerical result agreeing closely with that obtained by analysis. The volume of flow from average head equals seven-eighths of the true integral volume of flow, approximately.

If there is an initial head H₀, then when the head varies uniformly,

H—Ho+ft

CLH*= CL (JL₀+ft)\

The total volume of flow in time T will be

SI

taw

.+

Si

.0x1 to

H,¹

CLH(>\

The average head during time T is

S„=II₀+\fT

The total volume of flow corresponding to this head is

Q&y— CL

The ratio of the actual or integral discharge to the discharge by the average head is

Volume by average head 5 Integral volume

³ \?Hc+fTy

-**]

• (S3)

The value of this ratio is independent of the coefficient or length of weir, but varies with the rate of change of head.

WEIR NOT LEVEL.

If the crest of a gaging weir is not truly horizontal, but is a little inclined, the discharg-e may be closely approximated by the use of the average crest depth Hin the ordinary formula, or more precisely by the formula below, applicable also to weirs of any inclination.

d!

'at-'

Fig. 7.—Inclined weir

The flow through the elementary width dl is dQ= ClP’dl

II-H

Total discharge — Q—

cin di— c

'L

UI-+

H-H,

dl

'0

Integrating,

201

5(11-11)

II - II?

.....(54)

In this formula either the mean coefficient deduced by Thomson

2

(see p. -16) for a triangular weir, in which - (7=1.32, or that of Francis, in which ~ O = 1.332, may be used. If there are end contractions, the net length,

(M>

L=H—0.2

should be used.

The discharge using the average head,

c\ 5

is

.    .    (55)

q=cl(Ms±h₁ y

The extent of variation from the true discharge resulting from the use of formula (55) in place of the integral formula (54) is illustrated by the following:

Let L—10 feet, Ha—1.0 foot,    C—1.332.

Discharge by (55) for average head = 33.30 cubic feet per second. If    —0.01 foot—true discharge, (>=33.30 cubic feet per second.

If H₂ — IIx = 0.10 foot—true discharge, Q—33.30 cubic feet per second. If H₂ — Hx = 0.50 foot—true discharge, (>=33.54 cubic feet per second.

In general, since the discharge varies more rapidly tfian the head, the effect of calculating the discharge from the average head will be to give too small discharge, the error increasing with the variation in crest level.

Hence the discharge obtained by using the average crest level for a weir having an inclined or uneven crest will be somewhat deficient. The magnitude of the variations in height of the crest will determine whether the average profile can be used or whether the crest should be subdivided into sections, each comprising portions having very nearly the same elevation (whether adjacent or not), and the discharge over each section computed as for a separate weir.

In general it may be stated that the error in the value of Qx increases directly in proportion as the ratio of the difference in the limiting heads to the average head is increased.

CONVEXITY OF WATER SURFACE IN LEADING CHANNEL.

If there are wide variations in velocity in the measuring section, the level of the water surface may be affected, since water in motion exerts less pressure than when at rest.

Conditions of equilibrium cause the swift-moving current to rise above the level of the slower-moving portions. If the head is measured near still water at the shore, the result may be slightly too small.

The difference in heighta may be expressed in the form

(56)

D-I\=r

v²—v²

The coefficient r is often assumed equal to unity, but evidently varies with the distribution of velocities whose resultant effect it measures.

a Humphreys and Abbot, Physics and Hydraulics of the Mississippi River, 1876, p. 320.

RESULTS OF EXPERIMENTS ON VARIOUS FORMS OF WEIR

CROSS SECTIONS.

THE USE OF WEIRS OF IRREGULAR SECTION.

Many cases arise where it is desired to estimate, approximately, at least, the flow over dams of peculiar cross section.

The construction of so-called standard or thin-edged weirs that shall be permanently useful to measure the flow of large and variable streams is so difficult and expensive as to be frequently impracticable. Existing milldams often afford a convenient substitute. In the following pages are presented the results of the leading experiments to determine proper coefficients for “irregular” weirs, followed by a grouping of experiments on similar models, whether all by one experimenter or not. The data are not always as complete or consistent as could be desired, but the need for fair working coefficients is very great, and, in the line of making use of all the available information, the several diagrams of comparison and the conclusions therefrom are presented, with the understanding that these are not final, although it is quite certain that the laws of coefficient variation are correctly outlined by the data at present available, and they form, therefore, a safe working hypothesis.

Weir models of irregular section are calibrated in order that existing dams of similar cross section may be used for stream gaging. It becomes necessary to calibrate the experimental models for a wider range of heads than has commonly been employed in experiments on standard thin-edged weirs, in order that the range of rise and fall of the stream from low water to high may be included.

While the recent experimental data include heads as great as from 4 to 6 feet, jmt it is often necessary to determine the discharge for still greater heads, and experiments on certain forms with heads up to 10 or 12 feet are needed.

In this connection the greater relative facility of securing accurate results with weirs for high than for low heads may be noted.

The proportional error resulting from variations in crest level, as well as uncertainties as to the nappe form and consequent value of the coefficient, largely disappear as the head increases. The effect of form of crest and friction is also relatively diminished. It is probably true that the coefficients for many ordinary forms of weir section would tend toward a common constant value if the head were indefinitely increased. The above facts render milldams especially useful for the determination of the maximum discharge of streams. Dams can be used for this purpose when the presence of logs and drift carried down by the flood preclude the use of current meters or other gaging instruments.

MODIFICATIONS OF THE NAPPE FORM.

The elaborate investigations of Bazin relative to the physics of weir discharge set forth clearly the importance of taking into consideration the particular form assumed by the nappe. This is especially true in weirs of irregular section in which there is usually more opportunity for change of form than for a thin-edged weir. In general the nappe may—

1.    Discharge freely, touching only the upstream crest edge.

2.    Adhere to top of crest.

3.    Adhere to downstream face of crest.

4.    Adhere to both top and downstream face.

5.    Remain detached, but become wetted underneath.

6.    Adhere to top, but remain detached from face and become wetted underneath.

7.    In any of the cases where the nappe is ‘‘wetted underneath” this condition may be replaced by a depressed nappe, having air imprisoned underneath at less than atmospheric pressure.

The nappe ma}'' undergo several of these modifications in succession as the head is varied. The successive forms that appear with an increasing stage may differ from those pertaining to similar stages with a decreasing head. The head at which the changes of nappe form occur vary with the rate of change of head, whether increasing or decreasing, and with other conditions.

The law of coefficients may be greatly modified or even reversed when a change of form takes place in the nappe.

The effect of modifications of nappe form on various irregular weir sections is shown in PI. III. The coefficients are those of Bazin and include velocity of approach. The coefficient curve for any form of weir having a stable nappe is a continuous, smooth line. When the nappe becomes depressed, detached, or wetted underneath during the progress of an experiment, the resulting coefficient curve may consist of a series of discontinuous or even disconnected arcs terminating abruptly in u points cP'arret” where the form of nappe changes. The modifications of nappe form are usually confined to comparatively low heads, the nappe sometimes undergoing several successive changes as the head increases from zero until a stable condition is reached bejmnd which further increase of head produces no change. The condition of the nappe when depressed or wetted underneath can usually be restored to that of free discharge by providing adequate aeration.

The weir sections shown in PI. Ill are unusually susceptible of changes of nappe form. Among weirs of irregular section there is a large class for which, from the nature of their section, the nappe can assume only one form unless drowned. Such weirs, it is suggested, may, if properly calibrated, equal or exceed the usefulness of the thin-edged

Observed depth on weir (feet).

3.0

.50

.75

1.00

1.25

0T'g

®S2/s?ua$

S:

OK'S

■1

3.30

-Jit

W*

Series 1290 *

-Hi

J.64 '

Cb

VARIATIONS IN WEIR COEFFICIENT WITH CHANGE OF NAPPE FORM.

3.40

3.80

Value of the coefficient C=m i'29 (m being the original Bazin coefficient).

13

c.

3.50

3.60

3.70

3.90

4.00

01‘t

08'?

OS'?

Of? •

s¹

ikk 150—06-5

4S>

V>

\l-

Bs-

tn

\m

S.eries143 0'

i-

/.64 '

/: c/i

/.64 '

JU

jSSSr

py*

il

f\>

CD

QO'-

-Q

a®

end

?

S

?

i

weir for purposes of stream gaging, because of their greater stability of section and because the thin-edged weir is not free from modification of nappe form for low heads.

As an example, Bazin gives the following coefficients applying to a thin-edged weir 2.46 feet high, with a head of 0.656 foot, under various conditions:

Condition of nappe.

'C=mV2g

Bazin coefficient m.

Per cent of the Francis coefficient.

Free discharge, full aeration .....................

Nappe depressed, partial vacuum underneath_____

Nappe wetted underneath, dowr stream water level, 0.42 foot below crest.....................

Nappe adhering to downstream face of weir, res-sault at a distance.............................

0. 433

3. 47

.460

3. 69

104. 1 110. 7

.497

.554

3. 99

4. 45

119. 7

133.5

These coefficients include velocity of approach effect, which tends to magnify their differences somewhat. There is, however, a range of 25 per cent variation in discharge between the extremes.®

The departure in the weir coefficient from that applying to a thin-edged weir, for most forms of weirs of irregular section, results from some permanent modification of the nappe form. Weirs with sloping upstream faces reduce the crest contraction, broad-crested weirs cause adherence of the nappe to the crest, aprons cause permanent adherence of the nappe to the downstream face.

EXPERIMENTAL DATA FOR WEIRS OF IRREGULAR CROSS

SECTION.

The only experiments on irregular or broad-crested weirs in which the discharge has been determined volumetrically are those of Black-well on weirs 3 feet broad, of Francis on the Merrimac dam, and of the United States Geological Survey for lower heads, on various forms of section. So far as the writer is aware, all other such experiments have been made by comparison with standard weirs.

In the following pages are included the results of the experiments of Bazin on 29 forms of cross section; also those of the United States Deep Waterwa}rs Board under the direction of George W. Rafter, and those of John R. Freeman at Cornell University hydraulic laboratory. The results of 20 series of experiments, chiefly on weirs with broad and ogee crest sections, made under the writer’s direction at Cornell University hydraulic laboratory, are here for the first time published.

a Bazin’s general discussion of the above and other modifications of the coefficient has been translated by the writer, and may he found in Rafter’s paper, On the flow of water over dams: Trans. Am. Soc. C. E., vol. 44, pp. 264-261.

As it lias been necessary to reduce the experimental data to a uniform basis for purposes of comparison the original data, together with the results obtained by recalculation, have been ^included for the Bazin, United States Deep Waterways, and Freeman experiments.

BASE FORMULA FOR DISCHARGE OVER WEIRS OF IRREGULAR CROSS

SECTION.

Precedent to the opening of the hydraulic laboratory of Cornell University the most elaborate experiments on weirs of irregular cross section were those of Bazin. His experiments were all reduced in such manner as to include the velocity of approach correction in the discharge coefficient.

In America the formula most commonly used is that adopted by James B. Francis, in which velocity of approach is eliminated from the coefficient by correcting the head, thus reducing the conditions as nearly as possible to the basis of no velocity of approach before applying the formula.

In order to render Bazin’s results comparable with the later experiments, it has been necessary to adopt a standard or base formula to which all the experiments should be reduced. The considerations leading to the adoption of the formula of Francis here used are given below.

In the process of gaging streams at dams the head is usually measured in comparatively still water in an open pond. This condition could not be duplicated in the Cornell experiments. As the formula of James B. Francis is most simple in form for the case of a weir with no velocity of approach, and as it is often convenient to compare the discharge over a dam with that for a thin-edged weir of standard form, a weir formula of the base form used by Francis has been adopted in reducing the experiments. In this formula,

Q= CLIP.

Z=Length of crest corrected for end contractions, if any.

ZT=Head on weir crest corrected for velocity of approach by the Francis correction formula or an equivalent method.

C—A coefficient determined from experiments on a model dam.

In this connection it maybe remarked that the formula of Bazin includes the correction for velocity of approach in the weir coefficient; hence the coefficient for a given weir is comparable only with that for another weir under the same head when the velocity of approach is the same in both cases. Bazin’s formula also expresses the velocity of approach implicitly by means of the depth and breadth of the leading channel. In actual gagings the leading channel is often of irregular form, hence it becomes necessary to eliminate the depth and breadth of the channel from the formula.

There is considerable variation in the magnitude of the correction for velocity of approach used by different experimenters. As a rule, the velocity of approach is negligible at gaging stations at dams. It became necessary, therefore, in reducing these experiments to determine from the measured discharge and observed head what the head would have been had the same discharge taken place over a weir in a still pond. To accomplish this the formula for correction for velocity of approach adopted by James B. Francis has been used. This being the case, it is to be noted that in applying the coefficients, which, as given, have been reduced as nearly as possible to the basis of no velocity of approach, the same method of velocity correction must be used, and if it is used no error will result where the actual velocity of approach is nearly the same as that which occurred in the experiments.

bazin’s experiments on weirs of irregular cross section.

These include a wide variety of forms, many of which will seldom be found in America, and the use of which for purposes of gaging would be ill advised.

The small size of the models used, high velocity of approach, and narrow range of heads covered, limit the application of these results. No effort has been made to present all the results in this paper.® Certain series, useful for comparison, have been recomputed as described below, and by grouping similar sections we may determine the general effect of various slope and crest modifications.

bazin’s correction for velocity of approach.

The base formula for weir discharge adopted by Bazin and the method of taking into account the velocity of approach are described in connection with his experiments on thin-edged weirs (p. 31).

The following discussion shows the complex character of the Bazin coefficients, and the fact that they do not express directly the relative discharging capacity of weirs of irregular section.

The effect of velocity of approach is to increase the discharge at a given observed head, Z>, over what it would be if the same head were measured in still water, as in a deep, broad pond.

Bazin’s coefficients in the form published are not readily applicable in practice to weirs of other heights, or to weirs in ponds, or otherwise to any but weirs in restricted channels of the depth and width of the weir.

a For complete original data, see Bazin, as translated by Mariclial and Trautwine in Proc. Engineers Club Phila., vol. 7, pp. 259-310; vol. 9, pp. 231-244, 287-319; vol. 10, pp. 121-164; also numerous experiments reduced to English units by Rafter and others, Trans. Am. Soc. C. E., vol. 44, pp. 220-398.

The Bazin coefficients as published may be considered as comprising- two principal factors. M being the Bazin coefficient, we may write

M=FG.

F—velocity of approach effect.

C — contraction effect.

Bazin uses a correction formula for velocity of approach, derived from the expression

v²

TT-—Z^+1.69

Consider a standard weir and experimental weir both of the same height, but of different form, the measured depth being the same, and the Bazin coefficients being M and m, the velocity of approach and discharge Vand v and Q and g, respectively, and G and Gx the coefficients in a formula in which the velocity of approach correction is eliminated from the coefficient and applied to the head; then the discharge for the standard weir would be,

using the Bazin coefficients,

Q—MLD V 2gD=M’Dz,

where M' — M4 2g and Z=1.0; using the coefficient C\

CJF = G(I)+ah)\

taking roots

to

Bazin does not give the quantities of flow in the tables of results of his experiments, hence to determine h it is necessary to calculate Q. -y, and h from the known values M and D and from F, the height of weir.

D being the same for both the standard and the experimental weirs, we have for the experimental weir

Gx being the coefficient for the experimental weir, and hx the velocity head.

Hence, by multiplication,

and

or

I)

D+ ah

1)7 ah

D+ah ■ X D+ah

1)

3.

C\ D+ah,

M~ CX D+ah J.

The velocity of approach for a given depth on a weir is proportional to C\ hence, since h is proportional to vz, we have

Hence,

1/ ('

D+ah

ah

(57)

qfYl,    #    #    s

The ratio used by Bazin is not, therefore, precisely a measure of

the relative discharging capacities of the two weirs under similar conditions of head and velocity of approach, for the reason that the velocity of approach will not be the same for both weirs if the Bazin coefficients are different. The ratio m\M is made up of two factors, one of which, C, Cexpresses the absolute relative discharging capacities of the two weirs under similar conditions of head and velocity of approach, and the other expresses the effect of the change in discharging capacity on the velocity of approach for a given depth on a weir of given height.

Thus the coefficient Mfor any weir has, by Bazin’s method of reduction, different values for every depth and for every height of weir that may occur.

only

Cx

For reasons elsewhere stated it is preferred to express by

the relative discharging capacities of the weirs where the velocit}^ of approach is the same in both. It is then practically a measure of the vertical contraction of the nappe, and is constant for a given head for any height of weir, and may be sensibly constant for various depths on the weir.

RECOMPUTATION OF COEFFICIENTS IN BAZIN’S EXPERIMENTS.

In reporting the results of his experiments on weirs of irregular 'section, Bazin gives the observed heads on the standard weir of comparison, the absolute coefficient m applying for each depth on the experimental weir and the ratio ml If of the experimental and standard weir coefficients.

The results give coefficients which strictly apply only to weirs having both the same form of section and the same heights as those of Bazin. Although weirs of sectional form geometrically similar to Bazin’s are common, yet few actual weirs have the same height as his. There appear to be two elements which may render inaccurate the application of Bazin’s absolute coefficients to weirs of varying height: (1) The difference in velocity of approach; (2) the difference in contraction of the nappe for a higher or lower weir. .

In order to render the results of Bazin’s experiments comparable one with another and with later experiments, a number of series have been recomputed, the velocity of approach being treated in the same manner as in the computation of experiments at Cornell hj^draulic laboratory.

The method is outlined below, the references being to the tables of Bazin’s experiments given on pages 68 to 81.

Column 2 gives the observed head reduced to feet for the experimental weir.

Column 4 the absolute coefficient C\ = m ^2g.

(These have been reduced from Bazin’s original tables.)

Column 5 gives the discharge per foot of crest over the experimental weir calculated by the formula

Q-mLD V2gD= CxLlft,

quantities in column 3 being taken directly from a table of three-halves powers.

In column 6 the actual velocity of approach, v=’ *s griverb

and in column 7 the velochy head, h

The discharge over the standard weir was calculated by Bazin by using his own formula and velocity of approach correction. He does not give the discharge, however, and we have been obliged to work back and obtain it from the data given for the experimental weir.

Having determined the actual discharge and the observed head, we are now at liberty to assume such a law of velocity of approach correction in deducing our new coefficients as we choose. We will therefore deduce the coefficients in such form that when applied to a weir

U. S. GEOLOGICAL SURVEY •Rad. 0.33'

WATER-SUPPLY PAPER NO. 160 PL. IV

ii

Round crest

math

Series

86 0.328 89 0.656

//3 / 3/5

//* 2.62'\

//S 6.56'

Coeffi

cient

C.

3.6

3.5

3.4

■g

3.3

O'

3.2

3.1

■y

3.0

2.9

.    -A"

z$-

<3

2.8

2.7

q/ SER >Ei //ft &~ei t %¥ Rounded

mf

^2.62^

corne

r»frearn

-■a-—ft— -

■E

Crest*

2.6

2.5

Ij²--

N7

-Crest

J.idth

5.5(5

LEGEND Series No. 86 A

2.1

!/3

" no- <$>

- //5 ©

2.2

.1

.2

.3

.1    .5

.6    .7    .8    .9    1.0    1.1

Corrected head in feet.

1.2    1.3    1.1    1.5

EXPERIMENTS OF BAZIN ON BROAD-CRESTED WEIRS.

\

/

.

*

xv

■>

T“

-i

I Slope JO: A IvSer/es /42

\SJope 5 ■'/, \Series/4/

±_

Symbol lJ

Sloped:/,

Series/38

Coeffi

cient

C.

SI ope 3'I Series 145

ss

I!

f!

Symbol Q

.1

Id

CO

bi

Corrected head in feet.

.6    .7    .8    .9    1.0

1.1    1.2    1.3    1.4    1.5

3.9

Downstre,

3.8

ci ujjjXu-

e-m s

ij-

3.7

Downstream s/ope [?■'/

SEIZES ]/J8

_^9j—p-UQ—ip-

G3

IS

H7j

lift—ng:

3.5

3}

j4j

3.4

3.7

3.fi

' 3.5

tT

Downstream s/ope c.

SERIES 137

CD-

©| ©

3.4

3.5

1 Dow/lstream slope 3 j'/

3.4

SERIFS

145

SHI

5D

■gS-Ej.

3.3

3.2

@1

3;

3.1

US

3.0

2.9

3.2

Downstream slope t .7

SERIES

141

3.1

37)

3.0

,2l

.3.

3.0

SERIES

Downstream slope /O'-(

2.9

2.8

6.

's=gP@='£^

.2

.3

.4

.5

.6    .7    .8    9.    1.0

Corrected head In feet.

1.1    1.2    1.3    1.4    1.5    1.6

EXPERIMENTS OF BAZIN ON WEIRS OF TRIANGULAR SECTION WITH VARYING DOWNSTREAM SLOPE.

I

■ ■/;

I

Co

cii

<\j

effi-

ent

r.

.1

3.8

3.7

3.6

3.5

3.4

3.3

3.2

3.1

3.0

2.9

2.8

2.7

3.6

3.5

3.4

3.3

3.2

3.1

3.0

2.9

2.8

2.7

0-66

KS

S$£gus*

fgllj

HHH;

Corrected head in feet.

.2

.3

.4

.5

.6

.7

.8

.9

1.0    1.1    1.8    1.3    1.4    1.5

SERIES

/47

WiA.

ffl

3D

t-

444

SERIES /5/

31

O^r

.2

.2

.4

.5

.6    .7    .8    .9    1.0    1.1    1.2    1.3    1.4    1.6

Corrected head in feet.

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING DOWNSTREAM SLOPE.

V

Coeffi

cient

C.

Corrected head In feet.

.2

.3

.4

.5

.6

.8

.9

1.0    1.1    1.2    1.3    1.4    1.5

3.4

SER/ES /S3

3.3

3.2

-s>4

21!

M-

3.1

-A

41 rig

Ji

3.0

2.9

2.8

©/

2.7

3.4

3.3

3.2

SER/ES /S4

4-'/

3.1

A°P- ,

3.0

-p£

2.9

2.8

2.7

2.6

2.5

2.4

2.3

2.2

SER/ES /56

3.1

3.0

rw

2.9

2.8

e>

.2

.3

.4

.5

.6

.8

.9

1.0    1.1    1.2    1.3    1.4    1.5

Corrected head in feet.

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING DOWNSTREAM SLOPE.

Velocity-of-approach correction by the Francis method. (For cross section see PI. VI.)

l

{

\

I

I

U. S. GEOLOGICAL SURVEY -y.3/7.'.>i

---

VO \ *

5/opeZ:/

SERIES

158

S/ope4-/

SERES

6S!

5/ope 6-/ SERIES

160

Co

ri

effl-

ent

C.

.1

.2

.3

.5

.6

Corrected head in feet.

.7

.8

9.

1.0    1.1    1.2    1.3    1.4    1.5

SERIES 158

3.0

₅r-f

fwTT~

-nzwr-----

^Tt

2.9

iWr

2.8

2.7

sr

3.2

3.1

SERIES 159

3.0

D°

vV/7

str

sarT,m

slop

±1

Z5

19

©-

2.9

gr

2.8

i!)

(3J

±EO

2.7

2.6

3.1

SERIES 160

3.0

:/

2.9

2.8

.1    .2    .3

.4

.5

.6

.8

.9

Corrected head in feet.

1.0    1.1    1.2    1.3    1.4    1.5    1.6

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING DOWNSTREAM SLOPE.

ikr 150—06-6

%

\

\

Coeffi

cient

.1

.2

1.3

1.2

1.1

4.0

3.9

3.8

3.7

3.9

3.8

3.7

3.6

3.5

3.1

K—Y.64-—>|

0-©

.2

ST I |

.3

.5

mass

TXE:

.3

.1

.5

S/opeEl,

No.161

Slope 2-1. Ro.163

Corrected head in feet.

.6

.7

.8

.9

'ES 161

bovjpsfregrp <7)rr<ifc®

SERIES

163

tar

.6

1

-<

S

S/opeS/,

RO./65

1.0    1.1    1.2    1.3    1A    1.5    1.6

(T

X DownsTrepni slope 2

SERIES /65

-Q

©.I®

JiL

CD

(7J

.8

.9

Corrected head in feet.

1.0

jfe;

1.1    1.2    1.3    1.1    1.5    1.6

U. S. GEOLOGICAL SURVEY effi-

Co

ci

ent

C.

3.9

3.8

3.7

3.6

3.5

Slope of upstream face. Run for unit rise.

MEAN CONSTANT COEFFICIENTS FOR VARYING SLOPE OF UPSTREAM FACE.

*1

Series

164

Coeffi

cient

0 .1

2.

.3

.4

.5

Corrected head in feet.

.6    .7    .8    .9    1.0

1.1    1.2    1.3    1.4    1.5

3.9

SET, /£-.; /,

___ Upstri '.arh s/ope

Z-J

3.8

EFUES /\63

3.9

A Upstream 3/

op

./

3.8

W.

83.

3.7

SEE

/£.> /

4-0

3.8

Up

sfr ea

s/cpe '/z

37

"ST

—(2iy^

72

3.6

S EE'EE 139

3.7

_©

@ (a yp

stt ea

M>

m slope i§)~@

'/3

~©—71)-

j3 58

3.6

s/on ? r ert/ca/

3.6

3.5

-JA

^-Ar-P~~©~4^

Upstream

58

.2

.3

.4

.5

.6

.7

.8

Corrected head in feet.

.9    1.0    1.1    1.2    1.3    1.4

1.5

1.6

EXPERIMENTS OF BAZIN ON WEIRS OF TRIANGULAR SECTION WITH VARYING UPSTREAM SLOPE.

V

I

m

effi-

Co

cient

ft

.1    .2    .3 A .5

Corrected head in feet.

.6    .7    .8    .9    1.0 LI 1.2    1.3    1.4    1.5

3.6

SERI

ES

178

3.5

Upstream slope 2'J

3.4

ay®’

3.3

m—

3.2

sc?

3.1

3.0

2.9

2.8

3.7

3.6

SERIES 176

91

3.5

Upstream $/ope /•'/

3.4

wM'

ny—-

3.3

3.2

3.1

3.0

2.9

2.8

2.7

3.6

SERIES

151

3.5

Upstream^ slope '/e -l

Ml

3.4

~w_

3.3

3.2

tfC

3.1

z.

3.0

2.9

2.8

2.7

rV'

.3

.4

.6

.8

Corrected head in feet.

.9    1.0    1.1    1.2    1.3    1.4    1.5

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING

UPSTREAM SLOPE.

Coe

cie

effi-

ent

C.

■ Corrected head in feet.

.1

.2

.3

.1

.5

.6

.7

.8

3.6

.9

1.0    1.1    1.2    1.3    1.4    1.5

3.5

SERIES ISO

20

3.4

Upstream slope 4?-'/

3.3

3.2

3.1

3.0

2.9

2.8

G>

-l

2.7

' 2.6

YL

2.5

3.5

SERIES

MS

3.4

Upstream slope, vertical

3.3

3.2

Jr

3.1

M-

3.0

M-

2.9

2.8

2.7

2.6

2.5

.1

.2

.3

.4

.5

.6

.8

.9

1.0    1.1    1.2    1.3    1.4    1.5

Corrected head in feet.

Si

Vertical. Ni>. 149^

2.46-

EXPERIMENTS OF BAZIN ON WEIRS OF TRAPEZOIDAL SECTION WITH VARYING UPSTREAM SLOPE.

in which there is velocity of approach we may apply the correction formula of Francis,

11=

A sufficient approximation to this formula ior our present purposes may be obtained if we simply make

11=

where v is the velocity of approach corresponding to the trial discharge for the head 1), no successive approximations being made, as would be necessary to determine the true head IIby the Francis correction formula.

For example, in an extreme case, using a thin-edged weir

O a S3

Z>=1.0, P= 1.0, v (approx.)j)^2)—~2~:= 1-665

rz

4=2^ = 0.0431 whence H=D-\--^=1.0431, and $=3,547.

By the Francis correction formula we hnd, using three successive approximations,

$₁ = 3.5183 giving v=1.7591 $₂ = 3.5387 giving v=1.7694 $3 = 3.541 as the final discharge,

that the difference is 0.11 of 1 per cent. We are therefore justified in using this method to determine values of C to two places decimals, or to within one-fourth to one-half per cent.

We have also used J%(/=8.02, as in the reduction of the Cornell experiments.

Column 8 gives the corrected head,

11=1)+A.

Column 10 gives the final coefficient C deduced by the formula

c=

II

3*

Pis. IV to XII show the resulting discharge coefficients.

Bazin’s experiments on weirs of irregular section.

Bazin’s Series, No. 86. Crest length, 6.55 feet. Crest height, 2.46 feet.

jicuT

Cross section.

Period.

Observed head, experimental weir D, in feet.

I)%

C.,

Q, flow per foot, experimental weir, in cubic feet per

second.

•to

to

fat

10

10

11

12

13

14

15

16

17

18

0.1820 .2119 .2509 .2781 .3067 .3392 .3678 .4016 .4251 .4527 .4770 .5075 .5360 . 5639 .5973 a. 5304 a. 6032 a. 6347

0.0777 .0976 . 1257 .1466 .1701 .1974 .2232 .2549 .2771 .3049 .3294 .3616 .3924 .4235 .4613 .3858 .4683 .5060

2.7829 2.8712 2.9674 2.9754 3.0957 3.0957 3.1839 3.2321 3.2641 3.3283 3.3844 3.4406 3.4807 3. 5368 3.5930 3.4566 3. 4486 3.4646

0.2160 .2801 .3733 .4369 .5273 .6119 .7098 .8233 .9033 1.0153 1.1137 1.2449 1.3656 1.4992 1.6561 1.3349 1.6156 1.7508

0.082

0.1820 .2119 . 2509 ! .2781 .190 | 0.0006    .3073

.105

.138 ....

.159

0.0777 .0976 .1257 .1466 .1701 .1983 .2241 . 2539 .2786 .3069 .3315 .3642 .3957 .4280 .4671 .3897 .4740 .5120

2.78

2.87

2.97

2.98

3.10

3.09

3.17

3.24

3.24

3.31

3.36

3.42

3.45

3.50

3.54

3.42

3.41

3.42

Bazin’s Series, No. 89. Crest length, 6. 55 feet. Crest height, 2.46 feet.

t—2.46

Cross section.

10

11

12

13

14

15

16

17

18

19

20

21

0.2079 .2873 . 3641' .4337 .4963 .5619 .6331 .6890 .7490 .7985 .8546 .9228 .9648 1.0236 1.0784 1.1312 1.1866 1.2375 1. 2959 a 1. 0807 a 1.1587

0.0948 .1538 .2196 .2859 .3494 .4213 .5036 .57195 .6482 .7135 .7906 .8867 .9479 1.0362 1.1193 1. 2028 1. 2932 1. 3767 1.4754 1.1239 1.2477

2.7669 j 2.7669 2.76C9 2.8230 j 2.8792 ; 2.9674 : 3.0476 3.0877 3.1679 3.2160 3.2962 3.3524 3.4326 3.4887 3.5288 3.5849 3.6331 3.6732 3.7052 3.5368 3.5448

0.2626 .4260 .6083 .8062 1.0063

1.2513 1.5360 1. 7673 2.0548 2.2975 2.6090 2.9704

3.2513 3.6163

3.    9511 4.3060

4.    6943 5.0525 5.4737 3.9786 4.4169

0.098

0.2079 .2873

.216 j    0.0008    . 3649

. 560    .0049    . 6939

. 640    . 0064    . 7554

. 705    . 0078    . 8063

. 785    . 0097    . 8643

. 879    .0120    .9348

. 949    .0140    .9788

1. 038    .0168    1.0404

1.118    .0195    1.0979

1.201    .0224 | 1.1536

1.292    . 0259    1.2125

1.364    .0288    1.2663

1.456    . 0331    1.3290

1.122    . 0195    1.1002

1.219    . 0231    1.1818

.155

0.0948 . 1538 .2205 .2869 . 3515 .4235 . 5084 .5782 .6561 .7236 . 8031 .9041 .9687 1.0606 1.1505 1.2396 1.3359 1.4245 1.5321 1.1537 1.2850

2.77

2.77

2.76

2. 81

2.86

2.95

3.02

3.06

3.13

3.18

3.25

3.28

3.36

3.41

3.43

3.47

3.51

3.54

3.57

3.45

3.44

a Nappe free from the crest.

Bazin's experiments on weirs of irregular section—Continued.

yj/Jt

Bazin’s Series, No. 113. Crest height, 2.463 feet.

Cross section.

Period.

Observed head, experimental weir D, in feet.

Ci

Q, flow per foot, experimental weir, in cubic feet per

second.

H*-

9    10

0.208

.289

.363

.443

.518

.592

.667

.736

.805

10

.863

11

.936

12

.989

1.055 1.076 1.114 1.159 1.197 18    |    1.252

1.320

13

14

15

16

17

19

0.0948 .1554 .2187 .2949 .3728 . 4555 . 5447 .6314 .7223 .8017 .9056 .9835 1.0530 1.1162 1.1758 1.2478 1.3096 1.4009 1. 5166

2.64

2.66

2.66

2.65

2.66

2.64

2.71

2.75

2.78

2.80

2.85

2.88

2.91

2.94

2.95

3.00

3.01

3.06

3.11

0.2503 .4134

l«—2.6Z -»

Bazin’s Series, No. 114. Crest height, 2.46 feet

Cross section.

10

11

12

13

14

15

16

17

1.034 1.0514 1.112 1.1727 1.171 1.2672 1.243 1.3858 1.301 1.4839 1.384 1.6282

2.47

2.54

2.59

2.60

2.59

2.60

2.62

2.63

2.63

2.65

2.66

2.68

2.69

2.70

2.73

2.73

2.76

0.2275 .3764 .5411 .7409 .9267 1.1424 1.3952 1.6511 1. 9173 2.1950 2.4943 2.8178 3.1546 3.4214 3.7832 4.0510 4.4938

0.056 |.....

0.204

.137

.280

. 193    0.0006

. 256    .0011

.313 I .0015 . 376    . 0022

.446 ! .0031 .517 ! .0042 .587 j .0054 .655 ¹ .0068 . 728    .0083

1.168 | .0213

.3526 .4341 . 5055 .5802 .6601 .7392 .8154 .8888 .9663 1.0442 1.1240 1.1847 1.2591 1. 3191 1.4053

0.0921 .1482 . 2097 .2860 . 3594 .4417 .5362 .6353 .7358 .8381 .9494 1.0667 1.1917 1.2899 1.4127 1.5149 1.6654

2.47

2.54

2.58

2.59

2.58

2.59

2.60

2.59

2.60

2.62

2.63

2.64

2.65

2.65

2.67

Bazin's experiments on weirs of irregular section—Continued.

—*    i*----6.56

Bazin’s Series, No. 115. Crest height, 2.46 feet.

Cross section.

Period.

Observed head, experimental weir D, in feet.

Ci

Q, flow per foot, experimental weir, in cubic feet

per

second.

10

2.25

10

11

12

13

14

15

16

17

18

19

1.023 1.0347 1.097 1.1490 1.178 1.2786 1.260 1.4144 1.330 1.5338 1.388 1.6353 1.424 1.6993 1.467 1.7768

2.41

2.45

2.51

2.50

2.55

2.54

2.60

2.60

2.60

2.61

2.63

2.64

2.65

2.68

2.69

2.70

2.70

0.1953    0.073    ........

.3270    .120    ........

.4902    .175    ¹    0.0005

Bazin’s Series, No. 116. Crest height, 2.46 feet.

Had.-o.33

0.33-

Cross section.

10

11

12

33

14

15

16

17

0.177 0.0745 .225    .1068 ;

1.053 1.0805 1.120 1.1853 1.185 1.28995 1.251 1.3992 1.317 !l. 5114

2.71

2.83

2.90

2.92

2.95

2.98

2.99

3.01

3.00

3.00

3.02

3.02

3.05

3.06

3.08

3.09

3.10

0.2019 .3022 .4672 .6494 .8467 1.0662 1.1776 1.5375 1.8063 2.0706 2.3625 2.6519 2.9863 3.3063 3.6507 3.9859 4.3375 4. 7156

0.076

.112

.169

.229

.292

.360

.392

.497

.569

.640

.713

.793

.864

.942

1.019

1.092

1.169

1.250

0.177

.225

.296

.3678

.4363

.5060

.5394

.6429

.7181

.7874

.8568

.9267

0.0008 .0013 .0020 .0024 .0039 .0051 .0064 .0078 .0097 .0116 ! .9975 .0137    1.0667

. 0162    1.1362

. 0185    1.2035

.0213

.0243

1.2723 1.3413

0.0745 .1068 .1611 .2232 .2879 .3599 .3957 .5156 .6084 .6982 .7933 .8925 .9963 1.1021 1.2108 1.3203 1.4346 1.5529

2.71

2.83

2.90

2.91

2.94

2.96

2.98

2.98

2. 97

2.96

2.98

2.97

3.00

3.00

3.02

3.02

3.02

Bazin’s experiments on weirs of irregular section—Continued.

-^ I*---- 6.S6

Bazin’s Series, No. 117. Crest height, 2.46 feet.

Rad-033

033

W/

Cross section.

Period.

Observed head, experimental weir J), in feet.

Ci

Q, flow per foot, experimental weir, in cubic feet per

second.

CO

9    10

0.158

.204

.289

.361

.426

.494

.562

.635

.708

10

.771

11

.834

12

.912

13

.989

14

15

16

17

18

1.064

1.129

1.197

1.267

1.336

0.06282 .09212 .1554 .21691 .27808 .34724 .42134 .5060 .59578 .67702 .76168 .87096 .98352 1.0975 1.1996 1.3096 1.4262 1.5442

2.19

2.64

2.57

2.65

2.73

2.77

2.83

2.82

2.86

2. 89

2.91

2.93

2.95

2.95

2.95

2.97

2.98

2.99

0.0005 .0011 .0016 . 0025 .0033 .0045 .0058 .0070 .0090 .0110 .0132 .0152 .0175 .0202 .0228

.204

.289

Bazin’s Series, No. 136. Crest length, 6.519 feet. Crest height, 2.46 feet.

Cross section.

0.183

.244

.304

.364

.424

.484

.542

.597

.658

10

.713

11

.776

12

.830

13

.887

14

.953

15

16

17

18

19

20

21

1.010

1.068

1.122

1.179

1.244

1.299

1.361

0. 0783 .1206 . 1676 .2196 . 2761 .3367 .3990 .4613 .5338 .6021 .6836 . 7562 .8354 .9303 1. 0150 1.1037 1.1885 1.2802 1.3875 1. 4805 1.5878

3.90

3.86

3.85

3.86

3.88

3.87

3.88

3.89

3. 91

3.92

3.93

3.97

3.96

3. 98

3.97

4.00

3.99

4.01

4.01

4.01

4.03

0.305 |    0.12    0.0002    0.1832

4.029    1.16    .0209    1.0309

4.417 ,    1.25    .0243    1.0923

4.748    1.33    . 0275    1.1495

5.133    1.41    .0309    1.2099

5.564    1.50    .0350    1.2790

5.935    1.58    .0388    1.3378

6.408 j 1.68    .0439    1.4049

.647 .849 i.07i : 1.304 1.548 1.793 ! 2.088 2.360 2.684 3.001

Bazin’s experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 137. Crest length, 6.523 feet. Crest height, 2.46 feet.

Cross section.

Period.

Observed head, experimental weir J), in feet.

if

Q, flow per foot, experi-r mental ¹ weir, in cubic feet

per

second.

10

li

12

13

14

15

16

17

18

0.482

.680

.856

1.051 1.295 1.540 1.783 2.080 2.382 2.652 2.973 3.334

3.    662 4.043 4.392

4.    778 5.140 5.533

0.18    0.0006    0.2686    0.1395

.1922 .2454 .3049 .3707 . 4440 .5144 . 5945 .6797 .7589 .8524 .9420

.24

.30

.36

.44

.51

.57

.66

.74

.81

.89

.98

1.05

1.14

1.22

1.30

1.38

1.46

.0009 .0014 .0020 .0030 .0040 . 0051 .0068 . 0085 .0102 .0123 .0149 .0171 .0202 .0231 .0263 .0296 .0331

.3329 .3924 .4530 .5160 .5820 . 6421 .7068 . 7735 . 8322 .8993 . 9609 1.0291    1.0438

1.0982    1.1506

1.1651    1.2575

1.2273    1.3591

1.2916    1.4686

1.3551    1.5773

3.46

3.54

3.49

3.45

3.49

3. 47

3.47

3.50

3.50

3.49

3.49

3.54

3.51

3.51

3.49

3.52

3.50

3. 51

Bazin’s Series, No. 138. Crest length, 6.532 feet. Crest height, 1.64 feet.

Cross section.

'OCOvJCtiOirfxCOlOH

0.194

.263

.327

.391

.447

.510

.571

.626

.685

.745

.807

.873

.927

.992

1.045

1.110

1.176

1.233

1.289

1.355

1.429

0.0854 .1349 .1870 .2445 .2989 .3642 .4314 .4953 .5670 .6431 .7250 .8157 .8926 .9880 1.0683 1.1695 1.2753 1.3691 1.4645 1.5773 1.7082

3.57

3.50

3.48

3.50

3.56

3.63

3.62

3.71

3.66

3.69

3.70

3. 72

3.72

3. 76

3.80

3.78

3.79

3.81

3.82

3.82

3.83

0.305

0.17

.473

.651

.858

1.064 1.321 1.560 1.838 2.075 2.373 2.683 3.036 3.318 3.715 4.060 4.422 4.851 5.220 5.577 6.036 6. 542

.25

.33

.42

.50

.61

.70

.81

.89

.99

1.09

1.21

1.29

1.41

1.51

1.61

1.72

1.82

1.90

2.01

2.13

0.0004 .0010 .0017 .0027 .0039 .0058 .0076 .0102 .0123 .0152 .0185 .0228 .0259 .0309 .0354 .0403 .0460 .0515 .0561 .0628 .0705

0.1944 .2640 .3287 .3937 .4519 .5158 .5786 .6362 .6973 .7602 .8255 .8958 .9529 1.0229 1.0804 1.1503 1.2220 1.2845 1.3451 1.4178 1.4995

.5820 , 3.56 .6826 ! 3.58 .7507 j 3.57 . 8481    3.58

1.8362 j 3.56

Bazin’s experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 145. Crest length, 6.541 feet. Crest height, 1.64 feet.

Cross section.

Period.

Observed head, experimental weir D, in feet.

IP

Ci

Q, flow per foot, experimental weir, in cubic feet per

second. ;

IP

!)

10

10

.11

12

13

14

15

16

17

18

19

. 835 j . 7631 .902 : .8567 .962 : .9435 1.032 j 1.0484 1.087    1.1333

1.152    1.2364

1.210    1.3310

1.274    1.4380

1.334    1.5408

1.396    1.6494

1.467    1.7768

1.9016 j 3.41

.856

1.053

1.316

1.553

1.805

3.53    3.707    1.39    ! .0300    1.0620

3.58    4.045    1.48    .0341    1.1211

3.58    4.403    1.58    . 0388    1.1908

3.61    4.801    1.68    .0439    1.2539

3.61    5.198    1.78    .0493    1.3233

3.62    5.575    1.88    . 0549    1.3889

3.64    6.006    1.98    .0609    1.4.569

3.64    6.479    2.09    . 0679    1.5349

Bazin’s Series, No. 141. Crest length, 6.520 feet. Crest height, 2.46 feet. •

Cross section.

10

11

12

13

14

15

10,

17

18

19

20

0.215    0.0997 ,

.281    .1490    ¹

.355 | .2116 .425    .2771

. 489 ' . 3420 .561 I .4202 .624 ; .4929 . 692    .5757

1.029 j 1.0438 1.113    1.1742

1.165    1. 2575

1.237 ! 1.3758 1.298    1.4788

1.369    1.6018

1.431    1.7118

1.463 | 1.7696

3.02

3.09

3.07

3. 04

3.08

3.08

3.17

3.11

3.12

3.15

3.17

3.19

3.17

3.20

3.21

3.20

3.22

3.22

3.24

3.25

0.301

.460

.650

.842

1.053 1.294 1. 562 1.791 2.059 2.347 2.653 2.983 3.309 3. 757 4.045 4.416 4.766

5.152

5.540 5.752

0.11

.17

.24

.29

.37

.43

.51

.57

.64

.72

.79

.87

.95 ,

1.04

1.12

1.19

1.27

1.34

1.42

1.47

0.0002    0.2152    0.0997

.0140    1.0430    1.0652

.0168    1.1298    1.2012

. 0195    1.1845    1.2884

.0220    1.2590    1.4127

.0251    1.3231    1.5218

.0279    1.3969    1.6511

.0313    1.4623    1.7677

.0336 | 1.4966 J 1.8307

3.02

3.09

3.06

3.02

3.06

3.06

3.14

3.08

3.08

3.10

3.12

3.13

3.12

3.13

3.14

3.13

3.13

3.12

3.13

3.14

irr 150—06-7

Bazin's experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 142. Crest length, 6.523 feet. Crest height, 2.46 feet.

fo: /

Cross section.

Period. |

Observed head, experimental j ₍ weir D, J [ in feet. !

Ci

Q, flow per foot, experimental weir, in cubic feet

per

second.

(5    7

II

10

0.300

.369

.447

.509

.591

.666

.727

.795

.861

10

.934

11

12

13

14

15

16

17

1.007

1.079

1.149

1.222

1.285

1.362

1.430

0.464 |

.643

.851

2.98

1.038 1.308 1.554 1.810 2.084 2.349

2.664 2.980 3.338

3.665 4.037 4.370 4.770 5.147

0.17 | 0.0004 j 0.3004 0.1643 I 2.82 .23    .0008    .3698    i    .2251 : 2.86

Bazin’s Series, No. 139. Crest length, 6.532 feet. Crest height, 1.64 feet.

Cross section.

10

r ii

12

13

14

15

16

17

18

19

20

0.190    0.0828

.1273 .1743 .2297 .2860 .3536 .4101 . 4823 .667 i .5447 .733 ! .6276 . 7128 j . 7865 .8753 .9538 1.023    1.0347

1.092    1.1411

1.151    1.2348

1.210    1.3310

1.258    1.4110

1.326    1.5269

1.394 i 1.6459 j

.253

.312

.375

.434

.500

.552

.615

.798

.852

.915

.969

3.66

3.68

3. 72

3.66

3.73

3.72

3.78

3.76

3.82

3.79

3.80

3.84 I

3.86

3.87

3.92

3.90

3. 90

3.94

3.95

3.93

0. 303

.467

.647

.841

1.067 1. 317 1.550 1.812 2.081 2.380 2.709 3.022 3.378 3.692 1 4.038 4.446 4.816 5.240 5.570 6. 013 6.484

0.17    0.0004    0.1904    0.0828 j 3. 66

Bazin's experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 140. Crest length, 6.532 feet. Crest height, 1.64 feet.

Cross section.

Period.

Observed head, experimental weir D, in feet.

ir-

Q, flow per foot, experi-,, mental ¹ weir, in cubic feet

per

second.

10

10

11

12

13

14

15

16

17

18

19

20

21

0.192    0.0835

1.023 1.0347 J 1.080    1.1224

1.143    1.2220

1.195    1.3063

1.254    1.4043

1.316    1.5097

1.375 | 1.6123

3. 77

3.74

3.75

3.71

3.77

3.74

3.81

3.82

3.83

3.84

3.88

3.88

3.92

3.90

3.95

3.93

3.97

3.96

3.97

3.99

4.01

0.315

.473

.641

.838

1.086 1.276 1.551 1.792 2.072 2.342 2.700 2.968 3.375 3.674 4.069 4.402 4.843 5.187 5.558 6.024 6.456

0.17 0.0004 0.1924    0.0841

.1273 .1726 .2278 .2919 .3472 .4157 .4811 .5570 .6289 .7209 .7961 .8998 .9880

.25

.33

.42

.52

.60

.71

.80

.90

.99

1.11

1.20

1.32

1.41

1.53

1.62

1.74

1.83

1.92

2.03

2.14

.0010 . 0017 .0027 .0042 .0056 .0078 .0099 .0126 .0152 .0192 .0224 .0271 .0309

.0364    1.0594    1.0898

. 0408    1.1208    1.1869

.0471    1.1901    1.2981

.0521    1.2471    1.3925

.0573    1.3113    1.5011

.0641    1.3801    1.6211

. 0712    1.4462 1.7388

3.74

.2530 .3097 .3727 . 4402 .4936 .5568 .6139 .6766 .7342 .8042 .8594 .9321 .9919

3.72

3.71

3.68

3.72

3. 66

3. 73

3.72

3.72

3. 72

3. 74

3.73

3.75

3.72

3. 73

3.71

3.73

3.72

3.70

3.72

3.71

Bazin’s Series, No. 147. Crest length, 6.536 feet. Crest height, 2.46 feet.

*-4.0.66

is

Cross section.

0.231

.308

.373

.438

.503

.569

.637

.681

.734

10

.797

11

.845

12

.898

13

.953

14

15

16

17

18

19

20

1.015

1.063

1.115

1.165

1.217

1.265

1.332

1.894

0.1110 .1709 . 2278 . 2899 .3568 .4292 .5084 . 5620 . 6289 .7115 .7768 .8510 ,9303 1.0226 1.0960 1.1774 1.2575 1.3426 1.4228 1.5373 1.6459

2.75

2.85

2.86

2.97

3.02

3.13

3.20

3.24

3.34

3.40

3.44

3.53

3.57

3.63

3.67

3. 73

3.79

3.83 |

3.85

3.89

0.305

0.11

.485

.652

.861

1.078 1.343 1.626 1.821 2.109 2.417 2.673 3.004 3.320 3.703 4.037 4.401 4..775 5.132 5.467 5.991 6.567

.18

.23

.30

.37

.44

.54

.58

.66

.74

.81

.89

.97

1.06

1.15

1.22

1.31

1.40

1.47

1.58

0.0002 .0005 .0008 .0014 .0021 .0030 .0045 .0052 .0068 .0083 . 0102 .0123 .0146 . 0175 .0206 . 0231 . 0267 . 0305 .0336 .0388 . 0439

0.2312 .3085 .3738 .4394 . 5051 . 5720 .6415 .6862 .7408 .8053 .8552 .9103 .9676 1.0325 1.0836 1.1381 1.1917 1.2475 1.2986 1.3708 1.4379

0.1110 .1709 .2287 .2909 .3589 .4326 .5132 , .5682 .6378 . 7223 .7906 .8681 .9523 1.0484 ! 1.1286 1.2140 1.3013 1.3925 1.4805 1.6053 1.7244

2.75

2.84

2.85

2.96

3.00

3.10

3.17

3.20

3.31

3. 35

3.38

3.46

3.49

3. 53

3.58

3.62

3.67

3.68

3.70

3.73

Bazin's experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 149. Crest length, 6.518 feet. Crest height, 2.46 feet.

■-2.46- Hi

» *0.66

Cross section.

o>

•§

Observed head, experimental weir ]), in feet.

1)*

Ci

Q, flow per foot, experimental weir, in 3ubic feet per

second.

if*

(i

10

0.248

.317

.390

.455

.521

.585

.653

.705

.766

©SooEcicnEcotoE©

.818

.882

.942

.999

1.051 1.103 1.165 1.209 1.281 1.330 1.385 1.446

0.1235 .1785 .2436 .3070 .3761 .4475 .5277 .5920 .6705 .7398 .8283 ; .9143 .9985 1.0774 1.1584 ; 1.2575 , 1.3294 1.4499 1.5338 1.6300 1.7388

2.55

2.58

2.67

2.73

2.82

2.89

2.97

3.00

3.08

3.16

3.23

3.30

3.36

3.39

3.45

3.49

3.62

3.57

3.60

3.63

3.67

0.316

.462

.651

.838

1.060 1.295 1.568 1.776 2.067 2.338 2.674 3.016 3.356 3. 627 4.002 4.397 4.682 5.177 5.508 5.917 6.386

0.12 0.0002 . 17    . 0004

. 23    . 0008

. 80    . 0099

.89 j .0123 .97 I .0146 1.03 I .0165 1.12 j ' . 0195 1.20 , . 0224 1.27    .0251

1.38    .0296

1.45    . 0327

1.54    .0369

1.64 ' .0418

0.2482 .3174 .3908 .4563 .5230 .5850 .6570 .7099 .7724 .8258 .8979 .9543 1.0156 1.0665 1.1225 1.1874 1.2341 1.3106 1.3627 1.4219 1.4878

0.1235 .1785 .2445 .3080 .3782 .4476 .5325 .5983 .6783 .7507 .8509 .9318 1.0241 1.1006 1.1885 1.2932 1.3708 1.5011 1.5912 1.6956 1.8151

2.56

2.69

2.66

2. 72

2.80

2.89

2.94

2.97

3.05

3.11

3.14

3.23

3.28

3.30

3.37

3.40

3.42

3.45

3.46

3.49

3.52

Bazin’s Series, No. 150. Crest length, 6.518 feet. Crest height, 2.46 feet.

0.66

Cross section.

(OCCviaoi*.coiOH

. 928    . 8940

. 982    . 9732

1.043    1.0652

1.095    1.1459

1.152    1.2364

1.215    1.3393

1.259    1.4127

1.315    1.5080

1.323    1.5218

1.380    1.6211

1.439    1.7262

2.63

2.65

2.78

2.82

2.91

2.96

3.07

3.12

3.18

3.26

3.31

3.37

3.42

. 3.47

3.51

3.56

3.58

3.63

3.65

3.65

3. 68

3.73

0.314

.488

.648

.877

1.065 1.329 1.560 1.819 2.070 2.393 2.681 3.013 3.328 3.713 4.037 4.414 4.797

5.118

5.512

5.548 5.962 ; 6.416 !

0.12    0.0002    0.2482    0.1235

1.06    . 0175    1.0605    1.0913

1.13 ' .0199    1.1149    1.1774

,1.22    . 0231 j 1.1751    1.2737

1.30 j .0263    1.2413    1.3825

1.38 | .0296    1.2856    1.4584

.0331    1.3481    1.5651

. 0336    1.3566    1.5807

.0374    1.4174    1.6868

.0418    1.4808    1.8023    |

1.46 !

1.47

1.55

1.64

2.54

2.66

2.77

2.81

2.89

2.94

3.04

3.09

3.14

3.21

3.25

3.31

3.34

3.40

3.43

3.46

3.47

S. 51

3. 52 3.51

3.54

3.56

Bazin's experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 151. Crest length, 6.550 feet. Crest height, 2.46 feet.

A,

Cross section.

Period.

Observed head, experimental weir D, in feet.

3

If¹

Ci

Q, flow per foot, experimental weir, in

per

second.

-1

ac

at

10

tOOO'JClOi^WWM

1.027    1.0408

1.090 1.1380 1.112    1.1727

1.140 1.2172 1.209    1.3294

1.248    1.3942

1.314 1.5063 1.352    1.5721

1.416    1.6850

2. 71

0.244

0.09

2.81

2.79

2.79

2.92

2.95

2.98 i

3.05

3.09

3.20

3.24

3.30

3.31

3.36

3.39

3.46

3.51

3.52

3.57

3.60

3.61

3.64

3.68

3.71

3.75

.329

.474

.684

.867

1.089 1.174 1.324 1.594 1.856 2.129 2.362 2.486 2.712 2.997 3.332 3.653 4.013 4.177 4.382 4.801 5.060 5.557 6.825 6.337

.12

.17

.24

.30

.37

.39

.43

.51

.59

.66

.72

.76

.81

.89

.97

1.04

1.13

1.17

1.22

1.31

1.36

1.47

1.52

1.64

0.0001    0.2011    0.0901

.0002    .2422    .1191

.0004 .0009 .0014

.0021 j

.0024 .0030 .0040 .0054 .0068 .0081 .0090 .0102 .0123 .0146 .0168 .0199 .0213 .0231 .0267 .0288 .0336

2.71

2.76

0359

.0418

.3074 .3919 .4464 .5161 .5394 .5760 .6470 .7004 .7628 .8081 .8350 .8772 .9333 .9896 1.0438 1.1099 1.1333 1.1631 1.2357 1.2768 1.3476 1.3879 1.4578

.1701 .2454 '. 2979 .3707 . 3957 .4371 .5204 .5857 .6665 .7263 .7631 .8213 .9013 .9850 1.0667 1.1695 1.2060 1.2543 1.3741 1.4431 1.5651 1.6352 1.7604

2.79

2.79

2.91

2.94

2.97

3.03

3.06

3.17

3.19

3.25

3.26

3.30

3.32

3.38

3.47

3.43

3.46

3.50

3.49

3.51

3.55

3.56

3.60

Bazin's experiments on weirs of irregular section■—Continued.

Bazin’s Series, No. 153. Crest length, 6.515 feet. Crest height, 2.46 feet.

» * 0 66

Cross section.

Period.

[Observed j head, experimental weir D, in feet.

I)i

Q, flow per foot, experimental weir, in cubic feet per

second.

<2

Hi

10

tOCCMOCnifcWlOHi

0.314

0.12

.457

.633

.645

.847

1.052 1.311 1.560 1.801 1.820 2.085 2.101 2.349 2.678 2.984 3.307 3.330 3.672 3.956 4.394 4.733 5.159 5.139 5.507 5.943 6.158

.16

.22

.23

.29

.35

.43

.50

.67

.58

.65

.65

.72

.80

.88

.96

.96

1.05

1.11

1.21

1.28

1.38

1.37

1.45

1.54

1.58

0.0002

.0004

.0008

.0008

.0013

.0019

.0030

.0039

.0051

.0052

.0066

.0066

.0081

.0099

.0120

.0143

.0143

.0171

.0192

.0228

.0255

.0296

.0292

.0327

.0369

.0388

.5069 .5790 .6409 .7011 .7062 .7666 .7686 .8221 .8889 .9490

.3610 .4406 . 5132 .5870 .5933 .6717 .6743 . 7452 .8381 .9245

2.91 !

2.98

3.04

3.07

3.07

3.10

3.12

3.15

3.20

3.23

Bazin’s experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 154. Crest length, 6.516 feet. Crest height, 2.46 feet.

Cross section.

Period. J

[Observed head, experimental weir D, in feet.

D®

Ci

Q, flow per foot, experimental weir, in cubic feet

if®

per

second.

10

0.236

.308

.373

.447

.508

.577

.643

.706

.760

10

.823

11

.888

12

.946

13

14

15

16

17

18

19

20

1.011 1.075 1.138 1.195 1.250 1.310 1.370 1.430

0.1147 .1709 .2278 .2989 .3621 .4382 .5156 .5933 .6626 .7466 .8368 .9201 1.0165 1.1146 1.2140 1.3063 1.3975 1.4994 1.6035 1.7100

2.70

2.74

2.83

2.85

2.95

2.97-

3.04

3.07

3.17

3.20

3.20

3.24

3.28

3.31

3.36

3.37

3.40

3.43

3.45

3.48

0.311

0.12

.469

.645

.852

1.068 1.301 1.569 1.821 2.102 2.389 2.678 2.981 3.334 3.674 4.066 4.415 4.760 5.145 5.520 5.951

.17

.23

.29

.36

.37

.51

.57

.65

.73

.80

.87

.96

1.03

1.13

1.20

1.28

1.36

1.44

1.53

0.0002    0.2632

.0004 .0008 .0013 .0020 .0021 .0040 .0051 .0066 .0083 .0099 .0118 .0143 .0165 .0199 .0224 .0255 .0288 .0322 .0364

.3084 .3738 .4483 .5100 .5791 .6470 .7111 .7666 .8313 .8979 .9578 1.0253 1.0915 1.1579 1.2174 1.2755 1.3388 1.4022 1.4664

0.1349 .1709 .2287 .2999 .3642 .4406 .5204 .5996 .6717 .7576 .8509 .9376 1.0377 1.1396 1.2461 1.3426 1.4397 1.5494 1.6601 1.7750

2.30

2.74

2.82

2.84

2.94

2.95

3.02

3.04

3.11

3.16

3.15

3.18

3.21

3.22

3.26

3.29

3.31

3.32

3.32

3.35

Bazin's experiments on weirs of irregular section—Continued.

Bazin's Series, No. 156.

Crest height, 2.46 feet.

Crest width, 0.66 foot.

Upstream slope, J to 1.

Downstream slope, 5 to 1.

Period.

Observed head, experimental weir J), in feet.

Ci

Q, flow per foot, experimental weir, in cubic feet per

second.

V-

Ml

h • in

TO

coviaioi^coioH

0.246    0.1220

I. 018    1.0272

J. 074    1.1130

1.139    1.2156

1.203    1.3194

1.267    1.4262

1.341    1.5529

1.394    1.6459

1.457    1.7587

2.76

2.80

2.84

2.90

2.91

2.95

3.01

3.06

3.10

3.13

3.17

3.24

3.22

3.30

3.29

3.31

3.34

3.36

3.36

3.39

0.337

.486

.671

.864

1.054 1.289 1.534 1.804 2.070 2.384 2.653 3.028 3.309 3.673 3.999 4.367 4.764 5.218 5.530 5.962

0.12    0.0002    0.2462

2.76

.42

.49

.57

.64

.72

.79

.88

.95

1.04

1.11

1.19

1.26

1.37

1.43

1.52

.0027 | .5787 .0037 .0051 .0064 .0081 .0097 .0120 .0140 .0168 .0192 • .0220 .0247 .0292 .0318 .0359

.6417 .7081 .7704 .8421 .8977 .9680 1.0320 1.0908 1.1582 1.2250 1.2917 1.3702 1.4258 1.4929

0.1220 .1734 .2370 .2989 .3642 .4406 .5144 .5958 ' 3.03 .6757 | 3.06 .7727 .8510 .9524 1.0484 1.1396 1.2462 1.3558 1.4686 1.6035 1.7028 1.8241

2.80

2.83

2.89

2.89

2.92

2.98

3.08

3.12

3.18

3.16

3.22

3.21

3.22

3.24

3.25

3.25

3.27

y.3/.

Basin’s Series, No. 158. Crest length, 6.520 feet. Crest height, 2.46 feet.

Cross section.

10

11

12

13

14

15

16 |

17

18

19

0.234    0.1132

.530 | .3858 .600 | .4648 .672 | .5509 | .733    .6276 j

. 799    .7142

. 860    .7975

.930 ! .8969 .984 I .9761 1.055    1.0836

1.125    1.1933 j

1.177    1.2769 j

1.243    1.3858

1.297    1.4771

1.361    1.5878

1.412    1.6779

1.457 1.7587

2.79 2.72

2.77

2.79

2.81

2.82 2.86 |

2.90

2.91

2.95

3.00

3.04 |

3.10

3.12 I

3.15

3.19

3.22

3.25

3.30

3.32

0.316

.474

.656

.862

1.085 1.311 1.576 1 1.821 2.078 2.354 2.691 2.967 3.348 , 3.713 j 4.022 i 4.434 4.766 5.168 5.544 5.839

0.12    0.0002    0.2342    0.1132 | 2.79

Bazin’s experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 159. Crest length, 6.511 feet. Crest height, 2.46 feet.

*-v

Cross section.

Period.

Observed head, expert- ni mental weir D, in feet.

Ci

Q, flow per foot, experimental weir, in cubic feet per

second.

H \ H*

10

0.234 .304 .379 j .387 ! .457 .516 .526 i

.599

.664

10

.670

11

.735

12

.797

13

.861

14

.876

15

.935

16

.994

17

18

19

20

21

22

23

24

25

26

1.068 1.126 1.145 1.198 1.261 1.320 1.332 1.389 1.445 1.456

0.1132 I .1676 .2333 .2408 .3090 .3707 .3815 .4636 .5411 .5484 .6302 .7115 .7989 .8199 .9042 .9910 1.1037 1.1948 1.2252 1.3112 1.4161 1.5166" 1.5373 1.6370 1.7370 1.7569

2.68

2.75

2.82

2.82

2.81

2. 91

2.84

2.82

2.87

2.83

2.88

2.94

2.99

2.94

2.93

3.01

3.03

3.10

3.05

3.08

3.11

3.15

3.13

3.14

3.19

3.19

0.303

.462

.657

.680

.868

1.079 1.085 1.308 1.553 1.552 1.813 2.092 2.389 2.411 2.649 2.983 3.333 3.704 3.751 4.035 4.416 4.777 4.820 5.150

5.551

5.614

0.11    |    0.0002    |    0.2342

.17    .0004    |    .3044

.23    j    .0008    !    .3798

.30

.36

•.36

.43

.50

.49

.56

.64

.72

.72

.78

.86

.94

1.03

1.04

1.10

1.19

1.27

1.27

1.33

1.42

1.43

.0009 .0014 .0020 . 0020 .0030 .0039 .0037 .0049 .0064 .0081 .0081 .0095 .0115 .0137 .0165 .0168 .0188 .0220 .0251 .0251 . 0275 .0313 .0348

0.1132 .1676 .2342 .3879    .2417

.4584    .3100

.5180    .3728

.5280 ¹ .3836 .6020 ¹ .4671 .6679 ! .5460 . 6737    . 5533

.7399    .6366

.8034 I .7196 .8693 I .8101 .8843 ;    .8311

.9445 ! .9172 1.0055    1. 009

1.0817    1.1255

1.1425    1.2204

1.1618    1.2526

1.2168    1.3425

1.2830    1.4533

1 3451    1.5599 ■

1.3571    1.5808

1.4165    1.6850

1.4763    1.7932

1.4908    1.8188

2.68

2.74

2.80

2.81

2.80

2.89

2.83

2.81

2.84

2.80

2.85

2.91

2.95

2.90

2.89

2.96

2. 96

3.04

3.00

3.00

3.03

3.06

3.05

3.06

3.09

3.09

Bazin’s experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 160.

Crest height, 2.46 feet.

Crest width, 1.31 feet.

Upstream slope, i to 1.

Downstream slope, 6 to 1.

Period.

Observed head, experimental weir D, in feet.

TP

Ci

Q, flow per foot, experimental weir, in cubic feet

per

second.

I <To

fcE

10

0.451 ! 0.3029 . 522    .3772

. 593    .4567

. 663 . 5399 . 735 . 6302

.798

.863

.930

.998

10

11

12

13

14

15

16

1.074 1.129 1.193 1.254 1.326 1.389 1.457

.7128 .8017 .8969 .9970 1.1130 1.1996 1.3030 1.4043 1.5269 1. 6370 1. 7587

2. SI    0. 8540    0.29

2.82    1.0637

2.84    1.2970

2.88    1.5549

2.89    1.8213 2.0742 2.3410 2.6638 2.9810 '

2.91

2.92

2.97

2.99

3.02

3.03

3.06

3.08

3.10

3.14

3.16

.36

.42

.50

.57

.64

.70

.78

.86

0.0013 .0020 .0027 .0039 .0051 .0064 .0076 .0095 .0115 .0140 .0159 .0185 .0209 . 0243 .0279 .0313

0.4523 0.3039    2.80

.5240    .3793    2.80

.5957 .6669 .7401 .8044 .8706 .9395 1.0095 1.0880 1.1449

1.2115    1.3334

1.2749    1.4397

1.3503    1.5686

1.4169    1.6867

1.4883    1.8151

.4601 .5447 .6366 .7209 .8129 .9100 1.0135 1.1349 1.2252

2.82

2.86

2.86

2.88

2.88

2.93

2.94

2.94

2.97

2.98

3.00

3.02

3.05

3.06

Bazin’s Series, No. 161. Crest length, 6.543 feet. Crest height, 1.64 feet.

Cross section.

0.701

0.36

.906

.45

1.131

1.371

1.625

1.883

2.167

2.458

2.782

3.107

3.461

3.806

4.073

4.248

4.434

4.665

4.825

5.003

5.171

5.380

5.378

5.808

6.001

6.242

6.446

.56

.65

.75

.85

.95

1.05

1.16

1.27

1.38

1.49

1.57

1.61

1.67

1.72

1.78

1.81

1.84

1.90

1.95

2.00

2.05

2.10

2.15

0.0020 .0031 .0049 .0066 .0087 .0112 .0140 .0171 .0209 .0251 .0296 . 0345 .0383 . 0403 .0434 .0460 .0493 .0509 .0526 .0561 .0591 .0622 .0653 .0686 .0719

.3571

.4179

.4786

.5377

.5922

.6530

.7101

.7709

.8291

.8936

.9535

.9983

Bazin's experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 163. Crest length, 6.535 feet. Crest height, 1.64 feet.

Cross section.

Period.

Observed ¹ head, experimental weir D, in feet.

Ci

Q, flow per foot, experimental weir, in cubic feet per

second.

i*

10

10

11

12

13

14

15

16

17

18

19

20

21

1.056    1.0851

1.114    1.1758

1.171    1.2672

1.231    1.3658

1.285 1.4567 1.339    1.5495

3.81

3. 83

3.84

3.83

3.83

3.82

3.86

3.94

3.91

4.04

3.98

4.02

4.02

4.04

4.06

4.06

4.05

4.07

4.07

4.12

4.17

0.301

0.17

.463

.641

.850

1.053 1.295 1.513 1.799 2.063 2.376 2.683 3.023 3.329 3. 735 4.048 4.425 4.779 5.169 - 5.576 6.014 6.464

.25

.33

.42

.51

.61

.69

.81

.90

1.01

1.11

1.22

1.32

1.44

1.53

1.64

1.74

1.84

1.94

2.06

2.17

0.0004 0.1844 .0009 . 001*7 .0027 .0040 .0058 .0074 ;    .5434

.0102    .6032

.0126 .0159

.0188 | .7878 .0231 . 0271 . 0322

.2449

.3047

.3687

.4270

.4915

. 6656 .7179

.8501 .9091 .9812 .0364    1.0344

.0418 | 1.0978 . 0471    1.1611

. 0526    1.2236

.0585    1.2895

. 0660    1.3510

. 0732    1.4122

.4001 . 3.78 .4683 | 3.84 .5435 | 3.80 .6084 ¹ 3.90 .6995 . 3.84 .7837 | 3.86 . 8667    3.84

Bazin’s Series, No. 164. Crest length, 6.534 feet. Crest height, 1.64 feet.

Cross section.

10

11

12

13

14

15

16

17

18

19

20

0.244    0.1206

1.052    1.0790

1.115    1.1774

1.162    1.2526

1.219    1.3459

1.277    1.4431

1. 330    1.5338

3.86

3.91

3.87

3.90

3.87

3.87

3.94

3.94

3.97

4.00

4.03

4.05

4.07

4.10

4.09

4.12

4.13

4.15

4.18

4.19

0.467

0.25

.659

.859

1.080 1.296 L 536 1.797 2.069 2.334 2.684 2.978 3.325 3.704 4.055 4.417 4.862 5.163 5.602 6.019 6.411

.34

.43

.52

.61

.70

.81

.90

.99

1.11

1.21

1.32

1.44

1.54

1.64

1.76

1.84

1.96

2.06

2.16

0.0009 .0018 .0030 .0042 .0058 .0076 .0102 .0126 . 0152 .0188 .0228 .0271 .0322 .0369 .0418 .0482 .0526 . 0597 .0660 .0725

0.2449 .3068 .3700 .4292 .4878 .5476 .6022 .6636 . 7172 .7848 .8398 .9041 .9722 1.0299 1.0938    1.1442

1.1632 ₍ 1.2543 1.2146    1.3392

1.2787    1.4465

1.3430    1.5564

1.4025    1.6601

3.86

3.88

3.86

3. 87

3.87

3.86

84 weir experiments, coefficients, and formulas.

Bazin's experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 165. Crest length, 6.544 feet. Crest height, 1.64 feet.

Cross section.

►0

G>

o'

Observed head, experimental weir D, in feet.

Ci

Q, flow per foot, experimental weir, in cubic feet per

second.

IP I

10

^GCvJoiOi^oilOH

0.337

.904 i .8595 .969 j .9538 1 1.029    1.0438

1.090 I 1.1380 1 1.153    1.2381 i

1.217    1.3426

1. 279    1.4465

1.341    1.5529

1.401    1.6583

1.448    1.7424

3.56

3.56

3.56

0. 698

.904

1.125 1.363 1.618 3.54    1.880

3.55

3.64

3.61

0.35    0.0019    |    0.3389

1.42    .0313 : 1.0603

1.52 | .0359    1.1259

1.62    .0408    1.1938

1.71    .0455    1.2625

1.83    .0521    1.3311

1.92    .0573    1.3983

2.02 I .0634    1.4644

2.10 I .0686    1.5166

0.1974 .2568 .3202 .3902 . .4660 .5435 .6276 .7142 .8003 .8926 .9940 1.0913 1.1948 1.3046 1.4178 1.5355 1.6530 1.7714 1.8680

3.54

3.52

3.51

3.49

3.47

3.46

3.44

3.45

3.46

3.47

3.48

3.47

3.47

3.47

3.46

3.48

3.45

3.46

3.47

Bazin’s Series, No, 176. Crest length, 6.519 feet. Crest height, 2.46 feet.

J*0.66

Cross section.

CD OC

0.317

.441

.645

.858

1.055 1.318 1.550 1.813 2.066 2.385 2.724 2.995 3.373 3.671 4.034 4.416 4.782 5.115 5.558 5.925 6.422

Bazin's experiments on weirs of irregular section—Continued.

Bazin’s Series, No. 178. Crest length, 6.518 feet. Crest height, 2.46 feet.

I * 0.66

Mil

Cross section.

Period.

Observed head experimental weir D, in feet.

Oi

®’    o»o'73^Ha

os :

GC I

SO

10

11

12

13

14

15

16

17

18

19

20

21

0.222    0.1046

.847 I .7796 .906    . 8624 j

.966 ! .9494 1.028    1.0423

1.083    1.1271

1.142    1.2204

1.195 | 1.3063 1.259    1.4127

1.314    1.5063

1.366 j 1.5965 1.424 I 1.6992

2.83

2.95

2.96

3.08

3.13

3.19

3.24

3.28

3.33

3.36

3.43

3.46

3.51

3.53

3.58

3.60

3.64

3.66

3.69

3.72

3.75

0.297 j 0.11    j    0.0002    0.2222    0.1046    | 2.84

1.323 1.558 1.794 2.085 2.355 2.675 2.983 3.331 3.671 4.045 4.392 4.755 5.170 5.572 5.952 6.375

CORNELL UNIVERSITY HYDRAULIC LABORATORY.

This laboratory, erected in 1898, includes a reservoir formed by a masonry dam on Fall Creek, at Ithaca, N. Y. An experimental channel is supplied with water from the pond and has, as its general dimensions, length, 400 feet; breadth, 16 feet; depth, 10 feet; bottom grade, 1:500. Fall Creek drains an area of 117 square miles, and affords a minimum water supply estimated at 12 second-feet. The hydraulic laboratory is located at Triphammer Falls, where a descent of 189 feet occurs. The weirs used in the experiments here described were erected in the concrete-lined experimental channel. The water supply was regulated by wooden head-gates, operated by lever, rack, and pinion, the outflow from the canal passing over the declivity below.

a In reducing the experiments at Cornell hydraulic laboratory the value of g for Ithaca, latitude 42° 27', altitude 500 feet, has been taken as 32.16, makingl 2p=8.02, .^=0.015547, |l2</=5.35.

EXPERIMENTS OF UNITED STATES BOARD OF ENGINEERS ON DEEP

WATERWAYS.

These experiments were performed at Cornell University hydraulic laboratorj^ in May and June, 1899, for the United States Board of Engineers on Deep Waterways, under the immediate direction of George W. Rafter, engineer for water supply, in conjunction with Prof. Gardner S. Williams. The results of the original computations were published in Trans. Am. Soc. C. E., vol. 44, together with an extended discussion. In the experiments a closely regulated volume of water was passed over a standard thin-edged weir which was placed near the upper end of the experimental canal and had a height of 13.13 feet and a crest length of 16 feet, end contractions suppressed. The nappe was aerated, but was not allowed to expand on downstream side. The water flowed down the experimental canal past a series of screens and baffles and over the experimental weir placed at the lower end of the channel.

The experimental weirs were about 4.5 feet high and 6.56 feet crest length. A leading channel of planed boards, 6.56 feet wide and 48 feet in length, extended upstream from the experimental weir, having at its upper end flaring sides extending 8.3 feet upstream and meeting the sides of the main channel.

The head on both weirs was read by means of open manometers connected to galvanized-iron piezometer pipes, placed horizontally across the bottom of the narrow leading channel, 37 feet upstream from the weir. At the standard weir two piezometers were used, one termed the middle piezometer, placed across the leading channel, 8 inches above the bottom and 10 feet upstream from the standard weir. A second or upstream piezometer was placed 25 feet upstream from the standard weir. Readings of both piezometers were taken. It was decided, however, to use the middle piezometer as the basis of calculation of discharge over the standard weir. Near the close of the experiments it was found that this did not give results agreeing with those which would have been obtained from a piezometer placed flush with the bottom of the channel, as is shown to be necessary from the experiments of H. F. Millsa and others. A correction curve was accordingly deduced from comparative experiments between the middle piezometer and the flush piezometer, and the readings of the middle piezometer thus corrected were applied in the Bazin formula to calculate the discharge over the standard weir for heads not exceeding the limit of Bazin’s experiments. For depths on the standard weir greater than 2 feet the discharge was computed by using coefficients deduced for higher heads on a shorter experimental weir, on the basis of the Francis formula. Owing to the uncertainty as to the piezometers and

WATER-SUPPLY PAPER NO. 150 PL. XIII

200 feet

150

<k

Is

Scale

re4kng~%—jT

Baffles*

0>

W.S.in res eryoh

Regulating gates

Baffles ' *

if)}

Experime weir Deflecting boar for admitting \ air under nappe|

nta/

II

I Experimental x weir Deflecting boards for admitting airundemappe

l

*1

Upper weir chamber

Walk

*•§ -»

'I

%

9

&

Fall CreeTc

II

Experimental canal

GENERAL PLAN

Lower we/r chamber

PLAN

Lower weir chamber

II

I!

ii

LONGITUDINAL SECTION

Baffles]^

Baffles

HI

■r^&pde

yvall-

.1

appe

I .fPl

| |

£>'-$ Standard1'_^

o^:    weir

6

a.?

Standard

weir

pl

S;l||

HYDRAULIC LABORATORY AT CORNELL UNIVERSITY, ITHACA, N. Y.

Lab

Si

U. S. GEOLOGICAL SURVEY

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CORNELL HYDRAULIC LABORATORY, ARRANGED FOR WEIR EXPERIMENTS.

• 1.

-r

X

other conditions, the original results of the experiments were credited with a possible error of 5 or H per cent.

In connection with the experiments on models of the Croton dam, a very thorough comparison of the so-called upstream piezometer with other methods of obtaining the head on a standard weir was made by Professor Williams. It was found that the upstream piezometer gave the actual head on the standard weir correct^. These "results were communicated to the writer, and a recomputation of the Deep Waterways experiments has been made, using readings of the upstream piezometer to calculate the standard weir discharge by Bazin’s formula. This method of calculation eliminates the necessity for correcting the piezometer readings at the standard weir, as was necessary in the previous reductions. The discharge over the experimental weir has been calculated from readings of a piezometer placed 38 feet upstream from the weir and 8 inches above channel bottom, corrected to the basis of a Hush piezometer.

The United States Deep Waterways experiments included, for each experimental model, a smaller number of heads or periods than either the Croton or United States Geological Survey experiments. They were also the first experiments of the kind conducted at the Cornell laboratory, and the experience gained has probably contributed to the securing of somewhat greater accuracy in the later experiments. It is believed, however, that, as recomputed, the United States Deep Waterways experiments do not differ much in accuracy from those made on models of the Croton dam, which are stated by John R. Freeman to be reliable within about 2 per cent. The coefficients obtained by recomputation, when compared with the original United States Deep Waterways coefficients, show few differences exceeding 2 per cent. The variations are plus and minus in about equal numbers, and it is believed that these experiments are entitled to greater weight than they have hitherto received.

In the accompanying tables a summary of the recomputation is given.

xkk 150—00-8

Recomputation of United States Beep Waterways Board experiments on flow of water over model dams, Cornell University hydraulic laboratory, 1899.

Weir model.

i* j Period.

Corrected depth D, experimental weir, centimeters.

D, in feet.

Q=

flow

per

foot, in cubic feet per second.,

h=

V- H=

2g D+h

Cl

Number of observations

of

head.

(i

9    10

Series 1.

4.972 | 39.73

4.972    3.584

P=4.91.    £=6.58.

Series 2.

«• 0.66

P=4.90.    £=6.58.

05 O' ^ W tO

5.05    42.13

4.15    31.17

3.35 ¹ 22.89 2.55 I 15.03

1.75

8.39

.923

3.02

.34

.82

41

23

15

21

15

18

Series 8.

5■>

5.23

P=4.91.    L—6.58.

CO

3.49

1.75

3.393 3.383 3. 382

tO H W W On h-L

Series 4.

5.11

0.66

■hr?

P= 4.91.    £=6.58.

05 O' w to

4.28

3.43

. 2.57

1.73

3.547 3.373

3.484

3.485 3.485

to to to to to

CO    ■<!    k.

Series 5.

3L

** 0.66

P=4.90.    £=6.58.

4.23    0.2782    5.0902

. 1959    4.2299

. 1092    3.3512

. 0504    2.5344

. 0239    1.6859

3.50

2.65

1.80

1.24

3.574 3.593 3.503 3.289 3. 751

Series 7. “0.66

£’=4.865.    £=6.58.

142. 75 119.20 96.72 74.50

4.682    41.40

3.969    30. 72

3.173    21.64

2.444 | 14.07

4.30 3.47 2.67 1.92

.2875

.1872

.1108

.0573

4.9695 4.1562 3.2838 2.5013

3.737 3.626 3. 637 3.557

Series 8. « “ 0.66

P=4.90.    £=6.58.

144.00 120.50 97. 27 74.35 49. 77

. 2848    5.007g

. 1850    4.1380

. 1067    3.299

. 0573    2.4963

.0213 | 1.6543

3. 678 3.623 3.591 3.595 3.585

Series 9.

..0,33

JV/

P=4.94.    £=6.58.

147.10 123.00 99.62 76.22 51.00

. 2742    5.0992

. 1776    4.2116

.1067    3.3927

. 0573    2.5573

.0220    1.6950

3.575 3.525 3. 437 3.482 3.557

(i Same as series 7, but upstream face covered with pinch mesh galvanized wire netting.

Recomputation of United States Deep Waterways Board experiments onflow of water over model dams, Cornell University hydraulic laboratory, 1899—Continued.

Weir model.

Period.

Corrected depth D, experimental weir, centimeters.

£>, in feet.

Q=

flow per foot, in cubic feet per second.

h—

Va

H=

D+h

Number of / i obser-vations

of

head.

(5

10

Series 10.

P=4.57.    £=6.58.

Cl CO ^ lO

153.82 132.94 110.98 87.99 61.70

5.046

4.361

3.641

2.887

2.024

42.01 31.27 22.33 14.77

8.12

4.37    0.2969

5.3429 4.5514 3.7560 2.9486 2.0479

3.402 3.220 3.06& 2.917 2.771

21

21

20

17

17

Series 11.

ftsd-0.33\

*!

P=4.57.    £=6.58.

M CO iC

149.15 127. 70 105.60 82.52 58.80

4.892 4.189 3.464 2.707 1.930

42.06 31.66 22.02 14.34

4.45

3.60

2.72

1.97

8.26

1.26

Series 12.

—6.56 - .

CO to

P=4.56.

£=6.58.

154.55 j 5.069 126.80    4.160

96.75    3.174

66.30 j 2.174

30.69 21.58 14.12

7.85

3.19    .1582    5.219

2.48    . 0956    4.2556

1.82    . 0515    3.2255

1.17    .0213    2.1953

2.574 .... 2.458 .... 2.438 .... 2.413 ....

Series IS.

Rad* 0331

-6.56

-P=4.56.    £=6.58.

■M CO Tf iO

1.010 i 2.89

30.80

.51

Series lk.

P'y

*'0.20

[0.7S

Yi-il-P.

P= 4.53.    L=6.58.

N CO ^ lO CO

41.16

30.64

21.68

14.07

4.23

3.45

2.72

1.97

7.78

2.75

1.24

.51

.2782 .1850 . 1150 .0603 .0239 .0040

Series 15.

► - 10.56 --•*0.30 /*>.7S

l 6 -1    ~ ~ ,z

». 55

O--

P=4.53.    £=6.58.

Series 16.

CN CO Tt*

79.52 53.65

4.250 3.444 2.608 1.760

C4 I' C4 iC !-< 00 O Oi H H T)I bi W N H

0.75-

P=4.57.    £=6.58.

H Cl CO ^

L26.75    4.157

L02.55    3.364

78.02 ' 2.559 52.00 ¹ 1.706

30.69 21.87 14.02

7.73

Series 17.

iy¹

P= 4.57.    L=6.58.

1    127.30

2    103.32

4.175 3.389 2.571 1.691 1.063

GO    CO

O    GO    CO    I>    00

r—1    T-H    rji    fT    CO

CO    C-I    H

3.55

2.75

2.02

1.24

.68

.1959

.1176

.0634

.0239

.0072

Series 18.

i

iJ

P=4.65    £=6.58.

1    38.523

2    149.362

3    125.693

4    100.766

5    75.427

.92

Recompute!tion of United States Deep Waterways Board experiments on flow of water orer model dams, Cornell University hydraulic laboratory, 1899—Continued.

Weir model.

ii Period.

Corrected depth D, experimental weir, centimeters.

Q=

flow

D, in feet.

per

foot, in cubic feet per second.

V-

H=

V+h

Cr

Number of observations

of

head.

Series 19.

WM.

P=5.28.    L=6.58.

27.04 51.36 142.128 119.442 97.858 77.246 53.42

0.8869 j 2.75 1.685 ! 7.46

4. 662 3.918 3.210 2.534 1.752

0.44 I    0.0030    0.8899

1.07    .0178    1.7028

4.20    .2742    4.9362

3.23    .1622    4.0802

2.45    .0933    3.3033

1.82 j    .0515    2.5855

1.09 !    .0185    1.7705

3.276 3.357 3.651 3.594 3.461 3.401 3.260

Column 5 shows the discharge over the experimental weir per foot of crest, deduced from the readings of the upstream piezometer at the standard weir, by Bazin’s formula, and corrected for slight leakage.

Column 3 shows the head on the experimental weir, in centimeters, taken by a piezometer 38 feet upstream and 8 inches above channel bottom, corrected to reduce it to the equivalent reading of the flush piezometer.

Column 4 shows the equivalent head in feet.

Column 6 shows the absolute velocity of approach.

Column 7 shows the velocity head.

Column 8 shows the head corrected for velocity of approach; tbe correction being made by the simple addition of the velocity head to the measured head, which is assumed to be a sufficiently precise equivalent to the Francis correction formula for this purpose.

Column 9 gives the coefficient 6⁷x, deduced from the foregoing.

The resulting coefficient diagrams are shown on Pis. XV to XVIII, inclusive.

EXPERIMENTS AT CORNELL UNIVERSITY HYDRAULIC LABORATORY ON MODELS OF OLD CROTON DAM.⁸

These experiments were made in November and December, 1899, by Prof. Gardner S. Williams, under the direction of John K. Freeman. The standard weir used was located near the head of the experimental canal, water being admitted and regulated by head-gates in the usual manner. The standard weir was 11.25 feet high and 16 feet long on the crest. The experimental weir was placed 232.5 feet farther downstream, and also occupied the full width of the experimental canal. The models of the Croton dam were constructed of framed timber and were 6 to 9 feet high.

a Report on New York’s water supply, Freeman, 1900, pp. 139-141.

U. S. GEOLOGICAL SURVEY

WATER-SUPPLY PAPER NO. 150 PL. XV

Coefli

cient

c.

3.40

Series /0

3.20

Series /<?

3.00

2.80

2.60

2.40

ZT

2.20

.8 1.2

1.6    2.0    2.4    2.8    3.2    3.6    4.0    4.4    4.8    5.2    5.6

Corrected head in feet.

6.0

a?

6.56'--*

Y-4.57'-^

Series/O

Series !2

effi-

Co

cient

3.60

3.40

Series II Ser/ei A3

JO

3.20

3.00

2.80

2.8/

A"

,8

1.2    1.6    2.0    2.4    2.8    3.2    3.6    4.0    4.4    4.8

Corrected head in feet.

5.2

5.6    6.0

K P.62’1

(*- - -6.56— -8

5;

'O

Series //

5 eries !3

3.74

3.58

3.49

3.39

WATER-SUPPLY PAPER NO. 150 PL. XVI

VL

6.1

Nt

Coefficient C. 3.701-1—

Cornell experiments> 1899

5.1

Vf

SER/ES /&

3.1

2.1

09*8

3.50

Of‘8

3.30

Batter upstream face.

AVERAGE CONSTANT COEFFICIENT FOR HEADS EXCEEDING 2 FEET.

U2.

5.0

£L

4.0

AT CORNELL UNIVERSITY, 1899.

:Q-

B)

JS-

Si

<5

CO

UJ

is

/.•/

^y-

I—

AT Z

u.

AT S

3

CL

A"#"    >

0--

/•T C jzjjbq 99 udg '/oqujAg

co

■n

Coeffi

cient

er

c.

3.80

3.60

3.40

3.20

1.0

\

co

§

■<

S'

k S'

I

£ r

O Series 7 P/aned t

boarc/s Wire chin covered

^0.67'

Coeffi

cient

C.

8.

3.80

SEIZES 7-8

3.60

=1—:

3.40

v-0.33

z\.

dry

SERiES 9

3.60

3 58

3.40

73 ' /S.O'

O Series /4

r~ja

•Mj

Series /S

1iHo./

*---J0.5Q — -4

SERIES 74. Sharp upstream angle

75. Upstream ang/e rounded; rad.- 0.33'

3.40

■A-

3.20

3.00

.4

.8    1.2    1.6    2.0    2.4    2.8    3.2    3.6    4.0    4.4    4.8

Corrected head H In feet.

5.2

5.6

*

V

!

Crest batter 6-7

Coeffi

cient

0.75 —

4.5'

r.

3.60

SERIES 16

K-4.75'-H

3.40

Series /6, Little Falls No. /

3:8

3.20

.4

.X

1.2    1.6    2.0    2.4    2.8    3.2    3.6    4.0    4.4    4.8

Corrected head H in feet.

5.2

Crest batter 6 -7

0.75

3.60

SERIES 17

JL

3.40

©■

Series 17, Little Falls No. 2

3.20

.4

.8

1.3

1.6    2.0    2.4    2.8    3.2    3.6    4.0    4.4    4.8    5.2

Corrected head H in feet.

3.80

KD

18

3.60

r—

₄.6

z o'--*r

:C/y

3.40

WF.

x

-k

_3±2

3.20

Series !8, Indian Lake

.4

.8 1.2 1.6

2.0    2.4    2.8    3.2    3.6    4.0    4.4    4.8    5.2    5.6

Corrected head EL in feet.

O'

30

»T

3.60

wMa i

CO

SERIFS

19

3.6

-<3>-

3.40

Series !9,

submerged weir section

3.20

TU

0    .4

.8 1.2

1.6

2.0    2.4    2.8    3.2    3.6    4.0    4.4    4.8    5.2    5.6    6.0

Corrected head H in feet.    *

.

*

/

«<

WEIRS OF IRREGULAR SECTION.

91

The head on the weirs was measured by means of open glass manometers connected to piezometer tubes in the channel above each weir. The piezometer tubes were made of 1-inch galvanized-iron pipe with small holes drilled along the sides, the ends being plugged. At the standard weir three piezometers were used, placed parallel to the current, at about mid-depth of the channel, one being near each side and one at mid-width of the channel, the mid-length of the pipes being 26.5 feet upstream from the standard weir. A hook gage in the same section was used to check the observed head.

Experiments on volume of flow over models of old Croton dam, Cornell University hydraulic laboratory, 1899.

Period

No.

Ob-

served

depth

on

model dam, in feet.

Mean velocity of approach, in feet per

pecond.

Correc

tion

for

velocity of approach, in feet.

Cor

rected

head

on

model dam, in feet.

Dis

charge

over

model

dam

per

foot of length, in cubic feet

Ci

per second.

Series 1—Model A.

Round crest, old Croton dam, smooth pine, crest and slope 16 feet long. Nov. 28-29, 1899.

0.0439 . 0259 .0087 . 0031 .0008 .0000

2.7668 2.2116 1.4475 .9861 .5915 . 1230 14.762 10.562 5.604 3.154 1.451 .147

3.208 3.211 3.218 3.222 3.190 3.408

Series la—Model A.

Round crest, old Croton dam, unplaned plank, crest 16 feet long, smooth slope. Nov. 6, 1899.

2.0897    . 978    . 0149

2.1046 1.8395 1. 5946 1.2596 .9947 . 6305 .4873

.991 ! 2.913

Series 2—Model A.

Round crest, old Croton dam, 16-foot smooth crest, rough slope formed of cleats and stone to simulate concrete and riprap. Dec. 4, 1899.

2. 9753 2.9091 2.5305 2.1661 1.6358 1.2657 1.1465 .7209 .4877

1.087 ! 3.192

Series 3—Model A.

Round crest, old Croton dam, 16-foot crest, covered with wire cloth of No. 18 wire, i-ineh mesh,a rough slope, as in series 2. Nov. 28, 1899.

2.0187

1.4129

.8656

.4251

.1144

9.037 j 3.148 5.308 i 3.161 2.527 j 3.138 .861 j 3.099 .124    3.205

Experiments on volume of floiv over models of old Croton dam, Cornell University hydraulic laboratory, 1899—Continued.

Period

No.

Mean velocity of .

Observed dePth . ap-model rFoach, dam, 1 m Ieet in feet.

Correc

tion

for

mi

velocity of ap-

per proach, second, in feet.

Cor

rected

head

on

model dam, in feet.

Discharge over model dam per foot of length, in cubic feet per second.

Series 1—Model B.

Angular crest, old Croton dam, 16-foot crest, all uhplaned plank. Nov. 15, 1899.

0.0147    1.8782

9.506 ! 3.693 3. 272    3.668

1.870    3. 630

. 741    3. 606

. 272    3.601

Series 2— Model B.

Angular crest, old Croton dam, 167foot unplaned plank, crest slope roughened with cleats and stone. Nov. 28, 1899.

.0262

.0084

.0024

.0004

.0001

.0000

2.4388 1.5335 .9635 .5161 .3052 .0890

13.478 6.945 3.466 1.369

.631

.094

3.539 3.657 3.665 3.692 3.742

3.540

Series 2—Continued model B.

Conditions as in preceding. Nov. 16, 1899.

.0151

.0024

.0010

.0002

.0000

1.9081 .9629 .7038 .3943 .1952

Series 3—Model B.

Angular crest, old Croton dam, wire cloth on crest, rough slope. Nov. 16, 1899.

10.385 3.463 2.241

.260

3.622 3.586 3.548 3.567

Series 1—Model C.

Round crest, old Croton dam, 12-inch timber on crest, 16 feet long, rough slope. Dec. 1, 1899.

.0187

.0040

.0017

.0008

.0003

.0001

.0000

.0000

2.0128 1.1857 .8849 .6881 .4989 .2993 .1177 .0846

9.904 4.211 2.594 1.722 1.065

.522

.139

.067

3.468 3.262 3.116 3.017 3.022 3.188 3.450 2. 723

Series 1—Continued model C.

Conditions as in preceding. Dec. 4,1899.

1    j 2.7146

2    I 2.4519

3    1.5566

. 4    1.1046

5    .1070

1.632

1.436

.774

.016

. 0165

. 0414 ! 2. 7560 . 0320    2.4839

. 0093    1.5659

.0000 ¹ .1046 .0000    .1070

15.917 13. 629 6.660

.112

.118

3.479 3.482 3.399 3.311 3.371

Series 1—Model D.

a In experiments with wire cloth over crest, 0.004 foot is deducted from observed depth to compensate for thickness of wire.

Angular crest, old Croton dam, 12-inch timber, on 16-foot crest, rough slope. Nov. 16, 1899.

WEIRS OF IRREGULAR SECTION.

Experiments on volume of flow over models of old Croton dam, Cornell University hydraulic laboratory, 1899—Continued.

Ob-

Period

served

depth

Mean velocity of

No.

on

ap-

model dam, in feet.

proacli, in feet

Correc

tion

for

velocity of

Cor

rected

head

on

per

second.

ap

proach, in feet.

model dam, in feet.

Discharge over model dam per foot of length, in cubic feet seco

Ci

Series 1—Model E.

16-foot angular crest, old Croton dam, without timber, but with obstructed channel, with sharp contraction. Nov. 18,1899.

1    2.3051

2    ! 1.8125

3    1.2278

4    j .8598

5    : .5745

6    j .3245

7    | .1120

' 8 | .1102

1.154

.845

.507

.313

.179

.078

.017

.017

0.0207 .0111 .0040 .0015 .0005 .0001 .0000 .0000

2.3258 1.8236 1.2318 .8613 .5750 .3246 .1120 .1102

11.778 8.214 4.632 2.744 1.516

.641

.133

.140

3.321 3.336 3.388 3.433 3.477 3.466 3.548 3.827

Series 1—Model E, repeated. Conditions as in preceding. Nov. 25, 1899.

1    1.4569

2    .9168

3    | .6866

4    : .4820

5    | .2811

6    .1416

.636

.338

.237

.148

.071

.028

.0063

.0018

.0009

.0003

.0001

.0000

1.4632

.9186

.6875

.4823

.2812

.1416

5.957 2.982 2.037 1.240

.579

.226

3.366 3.387 3.573 3.702 3.883 4.203

Series 2—Model E.

Angular crest, old Croton dam, 16 feet long without timber, and with slope instead of sharp edge to upstream end of obstruction. Nov. 27, 1899.

2.2927 2.2914 1.1395 1.1403 1.1006 1.1099 .4763 . 0233

1.139

1.138

.473

.453

.448

.457

.141

.031

.0202

.0202

.0034

.0031

.0031

.0032

.0003

.0000

2.3129 all. 613 2.3116 611.606 1.1429    4. 278

1.1434 a 4.097 1.1037 64.034 1.1131    4.119

.4766 j 1.180 . 0233    .025

3.302 3.302 3.501 3.351 3.479 3. 507 3.586 7.029

a Trap open.

6 Trap closed.

At the experimental weir two similar piezometers, each about one third of the width of the channel from the side, were used. Owing to the long back slope of some of the model dams, the head was measured 69.75 feet upstream from the crest of the experimental weirs. Headings of all the piezometers were taken at half-minute intervals, two and sometimes three observers working at each weir. The mean of ten to twenty observations was used to determine the head for each period in the experiment. Freeman states that he considers the results of these experiments for heads up to 2.5 feet, including all sources of errors, as certainty correct within 2 per cent, and probably much closer. In reducing the experiments, the head on the experimental weir is corrected by a method comparable with that of Francis. Freeman does not give the resulting coefficients for the weir formula, but presents the results in the form of diagrams showing the discharge per foot of crest for the various models. In the accompanying tables the computations have been carried out to

show the coefficients, some errors in the original data having been omitted.

Column 3 shows the observed head on the experimental dam, in feet.

Column 7 shows the computed discharge over the experimental dam, per foot of crest. This was determined bj calculating the discharge over the standard weir by means of both the Francis and Bazin formulas, the mean of the two having been used. The result corrected for slight leakage, divided by 16 (the length in feet of the experimental weir model), appears in column 7.

Columns 4 and 5 show the velocity of approach and the corresponding velocity head at the experimental weir. The velocity of approach correction was made by adding directly the velocity head as given to the observed depth on the model dam, this being considered a sufficiently close approximation to the Francis method of correction.

The

Columns 1 to 7 are taken from the original computations, coefficient C₁ has been computed from the data in columns 6 and 7 by the formula

Cl=

in

Pis. XIX to XXII show the resulting coefficients applicable in the formula here adopted,

Q=C.LHK

correction for velocity of approach being made by the Francis correction formula or an equivalent method.

These experiments were performed for the specific purpose of determining the discharge over the old Croton dam. They include two main groups: (1) Experiments on round-crested portion of the dam; (2) experiments on the angular-crested portion of the darn. Each group includes series of experiments on: (a) Model of smooth-planed pine; (b) model of unplaned plank; (c) model with cleats and fragments of stone on the upstream slope to simulate the natural back filling; (d) model with rough slope and with 4-inch-mesh wire cloth on crest to simulate cut stone; (e) model surmounted by 12-inch-square timber on crest. Experiments were added with a construction to simulate a natural rock ledge lying upstream from the angular portion .of the dam.

The experiments were abbreviated owing to lateness of season and trouble from air in the gage pipes.

The value of the results is limited by the narrow range of heads covered. The models were of unusual forms, and show some peculiar differences when an attempt is made to compare the results with those of other weirs of similar slopes. The data are of value as showing the effect of various degrees of roughness on the discharge.

Co

cii

effi-

ent

C.

3.60

.2

.4

.6

.8

Corrected head II in feet.

1.0    1.2    1.4    1.6    1.8    2.0    2.2    2.4    2.6    2.8    3.0

3.40

ER/ES /,M

OD

EL

3.20

TsF

©A

Smooth

3.ZI

3.00

Sf/T/i-S

3.20

ia,\MqpEL Unp/aned p/an'k

JTS

3.00

2.80

ER/ES 2

, MODEL

3.40

3.20

j:

Qr-®-

s/cpe

g)

3.22

2)

3.40

SERIES 3,

MODEL] A

3.20

/ugh s/ope

and wire c/oth on ere

st

3.15

3.00

.2

.4

.6

.8

1.0    1.2    1.4    1.6    1.8    2.0    2.2    2.4    2.6    2.8    3.0

Corrected head H in feet.

fe

- 6.75- -*hr

---/S.o'--

-»k— 6.

TT .T-"

—j>j<- 5.ZS —- Zo5^— -v

--jr*

'O

- M

lo'o

Cana/ bottom i

*/

___Cana/ bottom ->

SERIES /, Z,AND 3, MODEL ,A 7

SERIES /a, MODEL A    /"~

EXPERIMENTS ON ROUND-CRESTED MODELS OF OLD CROTON DAM.

ikr 150—06-9

\

>;

05

to

be

be

Coe

cie

effi-

ent

SERIES /, MODEL B

3.80

Unplaned

p/ank

3. 70

3.60

3.40

SERIES

2,

SERIES

2,

MODEL B MODEL B

cont/r'ue'd

3.80

©.

ZAy?/

?ned

p/j nk.

Roug'h s/obe

-3.66

3.60

3.40

SERIES 3

ODEL

3.80

tfOb gk

s/.OQ

e.

Wi

re

c/ptk

on ere

3.60

3 62

■©-

w.

3.40

.2

.4

.6 .8

1.0    1.2    1.4    1.6    1.8    2.0    2.2    2.4    2.6    2.8

Corrected head H in feet.

— 35.83'

— /3./‘ -

-n

--/o. o'

ft

tt

TT

-8.05-

Cans/ bottom >

SERIES f ADO 2, MODEL 3

SERIES 3, MODEL B, ROUGH SLOPE

EXPERIMENTS ON ANGULAR-CRESTED MODELS OF OLD CROTON DAM.

Coeffi

cient

a

3.60

O.

SERI ES

•si'/P/E'S

hoE!L

MQOE^L

C,

CGH7 IMllEh

32s

49

3.40

3.20

He

3.00

Nk

htt.

.2

.4

.6

.8

1.0    1.2    1.4    1.6    1.8    2.0    2.2    2.4    2.6    2.8    3.0

Corrected head H in feet.

l~_^kL'£*,aiyi-h-er

E---JC.o'--—6.0— ^t~5.Z5 V -5.£5——7.5

~-l--f

r c?eTtlJnT'~r ~

. o,

s

s>

Cana/ bottom

SERIES I, MODEL C

Coeffi

cient

C.

3.80

SERIES

i, \w'de\. d

3.60

:S:

3.40

(2

.2

.4

.6    .8    1.0    1.2    1.4    1.6    1.8

Corrected head H in feet.

|!l____—j—5.oL-*ft— 5.0’—>^—5.o‘-+fr----u.i' — JA iz"xiztimber*\

r'Tt'i    1

f I    \

Cana/ bottom

SERIES /, MODEL D

OLD CROTON DAM MODELS WITH CREST TIMBER.

Co

ci.

U. S. GEOLOGICAL SURVEY effi-

ent

C.

3.80

SERIES /,

MODEL

3.60

3.40

<4)

-®=

3.

3Z

3.20

0    .2    .4    .6

.8    1.0    1.2    1.4    1.6    1.8    2.0    2.2    2.4    2.6    2.8

Corrected head H In feet.

Coeffi

cient

C.

4.40

4.20

SERIES

/, MODEL

E,

COD T! AID ED

4.00

3.80

-o

o»

(4

3.60

<5*

3.40

PS.

3 36

3.20

.2

.4    .6    .8    1.0    1.2    1.4    1.6    1.8

Corrected head II in feet.

Coe

cie

effi-

SERIES

MODEL E

ent

C.

3.60

3.40

3^30 '

3.20

(a)

.2

Trap open'

la + Z

.4

.6

.8    1.0    1.2    1.4    1.6    1.8    2.0    2.2    2.4

Corrected head Hin feet.

r-

— 35.63 --/6'--

—13.O'— —

■h    /

---too

62 —

—6 06- —

___--------Cana/ bottom >_

SERIES /, MODEL E. End a open

SERIES Z. MODEL E End a with s/op/ng approach

ANGULAR CROTON DAM MODEL, WITH CONSTRUCTION TO SIMULATE ROCK LEDGE.

WATER-SUPPLY PAPER NO. 150 PL. XXIII

3°

ft*

0

Form of crest

0.75

5.4    5.6.    5.8

4.0    4.2    4.4    4.6    4.8    5.0    5.2

ONE END CONTRACTION.

Fr³-⁰'-

1.5

Form of crest

6.4    5.6    6.8

tm

LS£j/?/£:.Si Sf

Jt

1.4    1.6    1.8    2.0    2.2    2.4    2.6    2.8    3.0    3.2    3.4    3.6    3.8

Corrected head IIin feet.

MODEL OF OGEE DAM OF LOZIER MOTOR COMPANY, PLATTSBURG, N. Y. LENGTH OF CREST, 7.938 FEET.

3.6    3.8    4.0    4.2    4.4    4.6    4.8    5.0    5.2

|2

JiL

1.6    1.8    2.0    2.2    2.4    2.6    2.8    3.0    3.2    3.4

Corrected head Hin feet.

U. S. GEOLOGICAL SURVEY

1.2

.8 1.0

.6    .8    1.0    1.2    1.4

LENGTH OF CREST, 7.979 FEET.

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC LABORATORY, 1903.

ONE END CONTRACTION.

MODEL SIMILAR TO PLATTSBURG DAM, WITH BACK SLOPE MODIFIED.

Coefli r cient C.

3.80

3.60

ors

0Z’?.

3.00

9" f-

Tinn⁰

.2    .4

Coeffi

cient

C.

4.00

3.80

3.60

3.40

3.20

1

52

4.0

WATER-SUPPLY PAPER NO. 150 PL. XXIV

17-f

OT

3.8

- -3.o'

3.0

3.8

3.0

0 ft '

Form of crest

3.6

3.4

3.2

3.0

2.8

2.6

2.4

2.2

,OT—

Form of crest

3.6

3.4

0.75 -*<

3.8

3.0

313

2.0

Corrected head II in feet.

LENGTH OF CREST, 15.969 FEET. NO END CONTRACTION.

2.8

2.6

rz

2.2

1.8

in

1.4

OGEE MODEL AS IN SERIES 32.

S3

ISEW/F5

2.0

1.8

Corrected head II In feet.

1.6

1.4

1.2

1.2

1.0

1.0

U. S. GEOLOGICAL SURVEY

.8

.6

.4

Elk

.8

.6

Coeffi

cient.

#-<!3

15

.2

MODEL OF OGEE DAM OF LOZIER MOTOR COMPANY, PLATTSBURG, N. Y

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC

LABORATORY, 1903.

LENGTH OF CREST, 15.969 FEET.

3.80

3.60

3.40

3.20

3.00

Coeffi

cient

C.

3.60

3.40

3.20

3.00

2.80

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY AT CORNELL UNIVERSITY HYDRAULIC LABORATORY.

In April, 1903, the writer was instructed to plan and execute a series of experiments on models of dams similar to those in use at gaging stations of the Geological Survey in New York, Michigan, and elsewhere.

The experiments were performed at the hydraulic laboratory of Cornell Universit}r, mainly during the months of May and June, 1903, and were conducted, under the supervision of the writer, by Prof. Gardner S. Williams, director of the laboratory.

The various types of dams most commonly occurring were grouped as follows:

1.    Weirs with broad horizontal or slightly inclined crests.

2.    Weirs with vertical downstream faces and inclined upstream slopes.

3.    Weirs having compound slopes, including those with inclined upstream faces and with either broad crests or with sloping aprons.

4.    Completely or partially curved weir sections, including those of ogee profile.

It was found impossible to include in the experiments all the forms of section desired, and it was accordingly determined to limit the experiments to the thorough stud3r of two classes—weirs with broad crests and weirs with ogee sections—and to extend, if possible, the measurements to -include dams with vertical downstream faces and sloping upstream approaches. The order of operation used in previous experiments was transposed, the experimental models being built on a bulkhead forming the standard weir hitherto used and located near the head of the experimental canal.

The quantity of water passing over the experimental weir was meas- • ured on a standard weir below, 6.65 feet high and having a crest length of 15.93 feet. The head on the standard weir was measured in a Bazin pit, 3 by 4 feet in section, reaching to the depth of the bottom of the canal, and communicating therewith through a pipe 4 inches in diameter and about 3.5 feet long, opening at the bottom of the channel of approach, 29.88 feet upstream from the weir. The head on the standard weir was observed in the gage pit by means of a hook gage reading to millimeters and estimated to about one-fifth millimeter. The conditions at the standard weir were thus closely comparable to those obtained in Bazin’s experiments, and his formula for this height and length of weir was applied to determine the discharge. Observations to determine the leakage between the experimental and standard weirs were made, and corrections were applied for whatever leakage was indicated, the amount being usuall}T less than 0.01 cubic foot per second per foot of crest. The discharge over the standard weir was computed in cubic meters per second and has been reduced to cubic feet per second, the discharge table being as follows:

Discharge over standard weir at different heads.

Head, in meters.

Head, in meters.

Q in cubic meters, per second.

Q in cubic meters, per second.

0. 05

.10

. 15

.20

.25

.30

.40

.50

0.111863 . 296230 .53207 .81166 1.12871    |

1.48032 ! 2.27850 I 3.19350 |

0.60 j 4.21730 .70    5.34459

. 80    6.57096

. 90    7.89078

1. 00    9. 30650

1.10 10. 81066 1. 20    12.40420

The discharge curve for the standard weir has also been carefully checked by comparing the depth flowing over with that on a similar weir, using the formula and method of determining the head adopted by Fteley and Stearns; it has also been checked by float and current-meter measurements, and for lower heads by means of volumetric measurement of the discharge in the gaging channel, so that it is believed that the discharge in these experiments is known within 1 or 2! per cent of error as a maximum.

The work of calibrating the standard weir had been accomplished by Professor Williams and his assistants before the experiments of the United States Geological Survey were taken up, so that somewhat more certainty attaches to the results of these later experiments than to earlier experiments made before the standard-weir discharge had been thoroughly checked.

It was the wish of the Geological Survey that the conditions at the experimental weirs should conform to those actually existing at dams which are utilized as weirs, in connection with the stream-gaging operations. In such cases it is often impracticable to utilize gage pits of the form adopted by Bazin or to use piezometer or hook gages. The usual method is to read the depth directly on a graduated vertical scale or measure the distance to water surface from a suitable bench mark. The method adopted in the weir experiments consisted of reading directly the distance to water surface from bench marks located above the central line of the channel. The readings were taken by means of a needle-pointed plumb bob attached to a steel tape forming a point gage, readings being taken to thousandths of a foot.

In series XXXV and

Two gages were used, one located 10.3 feet upstream from the crest and another 16.059 feet upstream, following, for the higher heads, the readings of the upstream tape were used. For heads where no general difference was apparent the average of the readings of the two tapes was taken. In general, the

surface curve did not perceptibly affect the reading- of the gage nearest the weir for depths below 3 feet. The readings of the tapes were checked from time to time by observations with hook gages, thus practically eliminating the effect of temperature on the tapes. Observations of the head were usually taken at intervals of thirty seconds. Great care was used to maintain a uniform regimen of flow during each experimental period, and the variations of head were very slight. The character of the observations is illustrated by the following data taken from the experiments:

Headings of tapes to determine head at experimental weir.

Series XL. Period 10.

Date 6,22,03.

Series XLI. Period 5.

Date 6,23,03.

Time.

Readings.

Time.

Readings.

h. m. s.

42. 681

1 34 30

43. 633

.681

.632

.680

.633

.681

.630

h-i h-i to to

Cn CO to CC

co o O O

.682

.630

42.680

.633

.681

.635

.680

.635

12 54 30

.679 : .680 :

.634

.633

Mean

42.6805

.630

.635

Series XLIII. Period 3.

Date 6,26,03.

.630

.635

.635

Time.

Readings.

.633

.630

12 42 10

Mean

2. 305

.302

.302

.302

.301

.302

.301

.300

.300

2. 3015

1 51 20

.630

.630

.630

.630

.634

.630

.628

.630

.632

.628

.631

Mean

43. 6317

For the lower heads the discharge over the experimental weir was volumetrically determined by measuring the rise of water in the canal, as follows:

List of experimental periods for which the discharge was volumetrically determined.

Series.

Periods.

Series.

Periods.

Series.

Periods.

30

31

34

1,13,14,16 10,11 1, 9,10

39

40    ;    1,2

41

37

38

5,6

13,14    ’

-1,2,3 43 « !    1,2, 3, 4

43

47    1, 2, 3

United States Geological Survey experiments at Cornell University hydraulic laboratory on

model of Plattsburg dam.

[Series No. XXX. Height of weir=P, 11.25 feet; length of weir„crest=£, 15.969 feet; width of channel=6, 15.970 feet; height of upstream crest corner, 10.50 feet; crest width, 3 feet.]

Measured head on experimental weir, in feet.

v=

No.

Number of observations.

Maxi

mum.

Mini- Mean mum. | =D.

J mean area of: v<]}°c'

Sener0n of 4

footof Proach'

crest infeot crest- per second.

Head corrected for velocity of approach, j charge

in feet. Per r>is-footof charge crest, coeffi-in cu- i cient H Jbic. feet: C\.

¹¹    (per sec

ond.

Q-dis-

in

10

11

10

11

12

13

14

15

16

17

0.0993    0.0818    0.096    11.347

.7993    .7863    j    2.790    14.041

3.1993    3.1473    3.187    14.438

2.3993    2.3473    2.384    13.635

1.7973    1.7853    1.793    13.043

1.1803    1.1723    :    1.174 j 12.425

1.0133    1.0073    j    1.010    |    12.260

.8043    .8003    j    .802    j    12.053 j

.6143    ,6113    !    .613    !    11.864 j

.5083    .5073    .508    ,    11.758

.4283    .4263    .427    !    11.678

.2943    . 2903    . 291    11.542

.2933    .2903    .292    11.543

.1783    .1763    ,    .178    11.427

. 1793    .1733    .179    11.429

.0893    .0893    !    .089    11.340

1.5373    1.5303    1.532    12.783

2.1033 ■ 2.0843 I 2.094    13.345

. 007    .027    .089    . 076    2. 841

.517    1.904    1.535    6.607 3.471

.793    3.051    2.104 10.576 3.466

United States Geological Survey experiments at Cornell University hydraulic laboratory on model of Plattsburg dam—Continued.

[Series No. XXXI. Height of weir=P, 11.25 feet; length of weir crest=D, 7.538 feet; width of channels, 15.970 feet; height of upstream crest coiner, 10.50 feet; width of crest, 3 feet.]

Measured head on experimental weir, in feet.

v—

No.

Number of observations.

Maxi

mum.

Mini

mum.

Mean

=i>.

A=

area of section per foot of crest.

mean

veloc-

ity

of approach, in feet per

Second.

Head corrected for velocity of approach, in feet.

(^dis

charge

per

foot of

crest, in cubic feet

Discharge coefficient Cj.

per

second.

10

11

h=effective length of crest weir.

12

10

11

12

25    5.1003 15 | 4.2873 44 ; 3.6893 21 ¹ 2.8213

39    : 2.2713

40    1.3903

4.9273 4.1693 3.5253 2. 8003 2.2623 1.3813 . 9653 1.8723 .6083 .6063 .2993 .2993

5.0014 259.535 4.2214 247.078 3.6191 507.460 2.8185 224.674 2.2696 215.780 1.3861 201.998 .9663 195.094 1.8749 209.605 .6087 189.383 .6063 189.345 .3017 184.471 .3017 184.481

1.201    11.257

. 326 2.570

5.023 4.235 3.621 2.823 2.273 1.387 .966. 1.876

.065

.475

.609

.060

.020

.021

.472

.165

.166

.606

.301

.302

3.429 3.288 3.083 .469    2.838

. 490    2.956

8.814 1.562 1.456

7.436 7.514 7.580 7.656 7.711 7.799 7.841 7.750 7.877 7.877 7.908 7.908

[Series No. XXXII. Height of weir=P, 11.25 feet; length of weir crest=P, 7.979 feet; width of chan-nel=6, 15.970 feet; height of upstream crest corner, 9.75 feet; width of crest, 3 feet.]

10

11

10    2.1193    2.0943

23 | 4.0343 ! 3.9883

26    : 2.6693 | 2.6333

34    .9953    .9923

32    5.5543    5.2983

27    | 4.7343    4.6443

27    4.0173    3.9883

28    | 3.3053 | 3.2673

31    2.5863    2.5193

28 j 1.9913    1.9693

53    1.4113    1.4023

2.1173 j213.476 4.0053 243.627 2.6469 221.934 .9942 195.540 5.4712 267.038 4.6947 254.637 4.0007 240.808 3.2890 232.188 2.5596 220.539 1.9801 211.285 1.4015 202.044

2.117 4.018 2.651

.135

.992

.994

1.404

1.141

.925

.701

.506

.357

.221

12.855 10.235 8.040 5.984 4.104 2.790 1.661

5.487 4.714 4.013 3.296 2.564 1.982 1.402

11.029 29.846 15.398 3.342 50.440 38.704 29.395 21.278 14.460 9.686 5.704

3.580 3.706 3.567 3.370 3.924 3.782 3.656 3.556 3.524 3.471 3.434

7.767

7.577 7.714 7.880 7.430 7.508

7.578 7.649 7.723 7.781 7.839

irr 150—06-10

United States Geological Survey experiments at Cornell University hydraulic laboratory on model of Plattsburg dam—Continued.

[Series No. XXXIII. Height of weir=P, 11.25 feet; length of weir crest=i, 15.969 feet; width of channel= b, 15.970 feet; height of upstream crest corner, 9.79 feet; width of crest, 3 feet.l

Measured head on experimental weir, in feet.

No.

Number of obser

va

Maxi

mum.

Mini

mum.

Mean = V.

A = area of section per foot of crest.

mean

veloc-

ity

Head corrected for velocity of approach, in feet.

tions.

of approach, in feet per second.

II*

II

(^discharge per foot of crest, in cubic feet per second.

Dis

charge

coeffi

cient

Cl

10

11

0. 7563 .9973 1.3963

1.    7893 2.3733

2.    6683 .5593 .6603 .6804

0.7493 .9873 1.3903

1.    7803

2.    3363 2.6503 j

.5493 . 6493 .6783

0.752

.992

1.394 1.784 2.352 2.660

.553

.653

.679

12.003 12.243 12.644 13.034 13.603 13.910 11.803 11.904 11.930

0.183

.281

.468

.671

.990

1.170

1.183

.152

.156

3.363 3.473 3.580 3.651 3.698 3.709 3.232 3.422 3.330

[Series No. XXXIV. Height of weir=P, 11.25 feet; length of weir crest=i, 15.969 feet; width of channel=6, 15.970 feet; height of upstream crest corner, 8.37 feet; width of crest, 3 feet.]

CO*d050i)^WtCM

l J—¹ t—i 1—A I—l

CO tO h-¹ O CD

0.6453 0.6383    0.641 11.891

.6423    .6383    .640    11.891

2.0253 2.0043    2.015 13.2Q6

1.6303    1.6203    1.628    12.878

1.2333    1.2293    1.232    12.483

.9523    .9483    .950    12.201

.4013    .3983    i    .400    11.650

.6243    .6243    .624    11.875

.2243    .2203    .222    11.473

.1103    .1093    .110    11.360

5.0963    5.0793    3.089    14.340

2. 7763    2.7263    2.749    14.000

2.4383    2^3963    2.421    13.672

2.8013    2.7963    .800    12.050

0.137

0.513

.141

.513

.816

.592

.391

.261

.070

.136

.029

.009

1.509 1.258 1.051

.194

2.882 2.086 1.371

.927

.253

.493

.105

.036

5.515 4.615 3.805

.716

0.641

.640

2.025 1.633 1. 235

.951

.400

.625

.222

.110

.800

United States Geological Survey experiments at Cornell University hydraulic laboratory

on model of Chambly dam.

[Series No. XXXV. Height of weir=P, 11.25 feet; length of weir crest=7,, 15.969 feet; width of channel=6,15.970 feet; height of upstream crest corner, 10.25 feet; width of crest 4.5 feet.]

Measured head on experimental weir, in feet.

v=

No.

A =

area of section

mean

veloc

ity

Head corrected for velocity of approach, in feet.

Number of observations.

Maxi

mum.

Mini

mum.

Mean

=D.

per

foot of crest.

of approach, in feet

per

second.

Q=discharge per

foot of crest, in cubic feet per second.

Dis

charge

coeffi

cient

Ci.

11

12

13

14

15

16

17

18

19

20

21

22

23

24

2.5503    2.5463

1.0103    1.0063

1.5643    1.5523

2.0273    2.0133

1. 7513    1.7303

1.2673    1.2583

.7593    .7543

. 4453    .4373

.3043 i .3003 3.7303 : 3.7063 3.2193    3.1813

3.0023    2.9873

2.6743    2. 6643

2.3533    2.3283

1.4923    1.4873

0.549 1.008 1.559 2.021 1.739 1.262

.757

.441

.302

3.755 3.195 2.987 2.642 2.317 1.463

.184

.391

.331

. 253

.221

.183

.132

.083

M M P ti p M    05 K K S P M K K G K K' P

ri rl ii i *

0.110

.272

.511

.744

.602

.379

.176

.079

. .045 1.740 1.406 1.283 1.077

.901

.478 :

.023

.065

.053

.035

.028

.021

.012

.007

0.407

1.014

1.954

2.890

2.304

1.422

0.549 1.009 1.503 2.029 1.744 1.265

.660

.758

.293

.166

7.403 5.789 5.224 4.334 3.555 1.775

.079

.245

.191

.127

.104

.078

.048

.024

.441

.302

10

11

1.297 3.331 6.548 9.874 7.816 4.747 2.114

.919

.525

3.798    26.114

3.224    20.319

3.011 J 18.267 2.658 | 14.963 2.330 ! 12.221 1.466    6.072

.184

.258

.391

.758

.331

.253

.221

.183

.132

.083

.609

.404

.325

.238

.136

.079

3.189 3.285 3.352 3.416 3.392 3.337 3.205 3.132 3.159 3.528 3.510 3.496 3.452 3.438 3.420 3.260 3.096 3.194 3.258 3.120 3.039 2.826 3.321

[Series No. XXXVI. Height of weir=P, 11.25 feet; length of weir crest=£, 15.969 feet; width of chan-nel=6, 15.970 feet; height of upstream crest corner, 10.50 feet; width of crest, 4.5 feet, with 4 inches radius quarter round.]

18

18

3    30

23

21

19

22

2.7653 I 2.6913 2.3513 | 2.2913 2.9373 | 2.8923 1.5173    1.4993

1.1143    1.1013

.7553    .7493

. 4943    .4903

• 2. 741 2.316 1.915 1.507 1.111

.753

.492

13.991 13.567 13.166 12. 758 12.361 12.004 11. 743

1.185

.936

.702

.501

.320

.177

.095

.654

.753

.345

.492

16.586 12.698 9.243 6.390 3.950 2.123 1.113

3.610 3.569 3.467 3.441 3.367 3.248 3.221

United States Geological Survey experiments at Cornell hydraulic laboratory on model oj Dolgeville dam with injured apron.

[Series No. XXXVII. Height of weir=P, 11.25 feet; length of weir crest=P, 15.969 feet; width of channel=6, 15.970 feet; height of upstream crest corner, 10.25 feet; width of crest, 6 feet.]

Measured head on experimental weir, in feet.

v=

No.

Num-ber of | obser-va-

| tions.

Maxi

mum.

A=

area of section iper foot

Mini- Mean = I cr°l

mean velocity of

ap-

mum.

D.

proach, in feet j per sec-1 ond.

Head corrected for velocity of approach, in feet.

<2=discharge

per

ii§

foot of crest, in cubic feet

Dis

charge

coeffi

cient

per second.

Cl

10

11

0.916 I 12.166 j 0.251 j 0.878

30    0.9203 I 0.9113

35 ! 3.5613 ' 3.4233 I 3.565 I 14.816    1.535 ! 6.828

5a

28    |    2.9123    2.8573

52    |    2. 3293    j    2.2903

59    1.6413    1.6213

41    1.3973    ;    1.3923

2.927    14.178

2.324    13.574

1.635 | 12.885 .415    11.666

1.170 I 5.060 .906 ! 3.570 .585    2.100

. 069    .267

.415 !    .810    I 3.029

United States Geological Survey experiments at Cornell hydraulic laboratory on model oj

Dolgeville dam.

[Series No. XXXVIII. Height of weir=P, 11.25 feet; length of weir crest—P, 15.969 feet; width of channel=6, 15.970 feet; height of upstream crest corner, 10.25 feet; width of crest, 600 feet.]

on experimen-, in feet.

!'=

No.

A= area of section

mean velocity of

Head corrected for velocity of approach, in feet.

Number of obser

ap

va

Maxi

mum.

Mini

mum.

Mean=

D.

of

crest.

tions.

proach, in feet per second.

If%

II

Q=discharge per foot of crest, in cubic feet per second.

Dis

charge

coeffi

cient

Cl

(i

10 11

0.3973 1.6843 1.1143 .7513 . 5093 .2173 3.4726 3.1176 2. 6176 2.2096 1.9306 1.5286

0.3913 1.6713 1.1893 .7473 .5083 . 2113 3.4586 3.0806 2.5926 2.1806 1.8156 1.5106

0.395    11.646

1.689    12.940

1.112    12.362

.749 I 12.000 .504    11.754

.209 ; 1L 460 3.470    14.721

3.100    14.351

2.605 | 13.856 2.198 ! 13.449 1.920 ! 13.171 1.517 ! 12.768

0.396 1.695 1.114

.750

.504

.209

3.501 3.124 2. 621 2.209 1.928 1.519

0.790 7.826 4.113 2.223 1.186

.294

21.583 18.238 14.124 11.381 9.558 6.557

3.176 3.548 3.501 3.426 3.310 3.081 3.295 3.303 3.329 3.467 3.570 3.503

[Senes No. XXXIX. Height of weir=P, 11.25 feet; length of weir crest=P, 15.969 feet; width of channel--?), 15.700 feet; height of upstream crest corner, 10.25 feet; width of crest, 6 feet.]

26    0.6883

0.6793

10

22    |    .4183

33    3.9806

24    3.1506

23    2.5066

56    1.8986

32    1.2716

.8286 82 I .5626

39

f 0.683 { 11. 934 j 0.170 1 .683 j 11.934 .418 ! 11.668

.167

.081

.4153

1.8756 !    1.884    ;    13.135

1.2626 :    1.268    !    12.518

.8206 j    .826    |    12.077

.5606 I .561 11.811

.699

.401

.218

.125

United States Geological Survey experiments at Cornell hydraulic laboratory on model of flat-top weirs with vertical faces.

[Series No. XL. Height of weir=P, 11.25 feet; length of weir crest=L, 15.969 feet; width of channel= b, 15.970 feet; width of broad crest, 0.479 foot; nappe aerated.]

No.

COOOMCJCn^WtCM

Measured head on experimental weir, in feet.

A=

Number of observations.

area of section per foot

Maxi- { Mini- Mean= mum. j mum.

D.

of

crest.

v—

mean velocity of

ap

proach, in feet

per second.

Head corrected for velocity of approach, in feet.

u'i

(^discharge per foot of crest, in cubic feet per second.

Dis

charge

coeffi

cient

a-

10

11

0.6343 .2643 .2613 .1216 1.9916 1. 6256 1.2556 .9756 . 8213 .6513 .4503

0.502

0.632

1.580

.133

.136

.044

2.821 2.065 1. 412

.967

.743

.524

.301

.260

.264

.124

1.996 1.622 1.259

.978

.820

.650

.449

.343

.380

.129

9.401 6.819 4. 713 3.209 2.469 1.654

.867

3.148 2.584 2.794 2.943 3.332 3.301 3.838 3.318 3.325 3.154 2.881

[Series No. XLI. Height of weir = P, 11.25 feet; length of weir crest = L, 15.969 feet; width of channel = 6,15.970 feet; width of broad crest, 1.646 feet; nappe partly aerated.]

25

25

26

31

28

27

25

34

30

10

18

11

34

3.8606 3.1906 2.6906 2.0306 1.6043 1.2373 .9443 .6733 .4893 .3303 .2113

3.8256 3.1666 2. 6656 2.0116 1.5973 1. 2293 .9393 .6693 .4873 . 3293 .2093

12

13

22

14

20

15

44

.1253

.4206

.4206

.1213

.4136

.4126

3.842    15.092

3.177    14.428

2.674    13. 925

2.022    13.272

1.601    12.852

1.233    12.484

.942    12.192

. 671    11. 922

.488    11.739.

. 330    11.581

.210    11.461

.122    11.373 ¹

.788    12.038 I

.417    11.668 |

.417    11.668

1.692 1.317 1.050

.680

.462

.307

.203

.123

.078

.045

.024

.011

.153

.064

. 065

7. 651 5. 730 4.413 2.889 2.032 1.372

3.883 3.202 2.690 2.028 1.604 1.234

.915

.942

.575

.341

.190

.096

.043

.699

.270

.270 j

.692

.488

.330

25.531 18.995 14.624 9.021 5.936 3.835 2.476 1.472

.910

.520

.210 | .272 .122 j .130 .788    1.840

.417

.417

.750

.759

3.337 3.315 3.314 3.123 2.922 2.796 2.706 2.560 2.669 2.742 2.827 3.047 2.631 2.782 2.815

[Series No. XLII. Height of weir=P, 11.25 feet; length of weir crest=D, 15.969 feet; width of channel^, 15.970 feet; width of broad crest, 12.239 feet; nappe partly aerated.]

.639

0.1706 4.3706 3.8316 3. 0446 3.7356 3.5856 3.3966 2.2506 1.4706 1.0986 .6406 .6126

0.1626 4.3416 3.8006 3.0256 3.7186 3.5776 3.3806 2.2406 1.4526 1.0906 .6376 .6106 0.168 4.353 3.809 3.032 1.728 2.580 3.387 2.243 1.449 1.096

.611

11.418 15. 604 15.060 14.283 12.979 13.831 14. 638 13.494 12. 700 12.347 11.890 11.862 0.016 1.584 1.321

.971 .467

.787

*1.130

.657 .369 .253 .116 .109

0.069 9.196 7.510 5.317 2.278 4.166 6.285 3.373 1.748 1.149

.511 .478

0.168 4.389 3.835 3.046 1.732 2.589 3. 406 2.249 1.451 1.097

.639 .611

0.180 24.716 19.896 13.876 6.066 10.882 16.535 8.870 4.682 3.129 1.375 1.290

2.611 2.688 2.649 2.610 2.663 I 2.612 2. 631 2. 629 2.701 2.723 2.689 2.700

United States Geological Survey experiments at Cornell hydraulic laboratory on model of flat-top weirs with vertical faces—Continued.

[Series No. XLIII. Height of weir=P, 11.25 feet; length of weir crest- /., 15.969 feet; width of channel= 6, 15.970 feet; width of broad crest, 16.302 feet; surface somewhat rough; nappe partly aerated.]

Measured head on experimental weir, in feet.

v=

No.

Num-ber of obser-

A=

area of section per foot

mean veloc- j ity or

ap-

Head corrected! for velocity | Q=dis-of approach, ; charge in feet.

va-

Maxi

mum.

Mini

mum.

Mean=

D.

of

crest.

tions.

proach, in feet per second.

ni

per

- foot of crest, in cubic H Ifeet per second.

Dis

charge

coeffi

cient

Ci.

10

11

12

12

10

26

33

27

34

30

9-

23

10

25

11

26

0.8536 .4496 .3246 .6936 4.4506 3.6806 2. 9426 2.3626 1.8956 1.4826 1.2086

0.8506 .4426 .3106 .6856 4.4176 3.6306 2.9306 2.3586 1.8866 1.4756 1.2026

0.851    12.102

. 447    11.698

.312    11.563

‘ . 689    11.940

4.432    15. 683

3.661    14.912

2. 935    14.186

2.360    13.611

1.890    13.141

1.480    12. 731

1.206    12.457

2.627 ; 2.579 .811 | 2.713 .468 ¹ 2.684 1.544    2.696

25.011 | 2.647 18.657 j 2.637 13.200    2.608

9. 607    2. 637

6.841    2.624

4. 767    2. 531

3.520    2. 652

[Series No. XLflla. Height of weir=P, 11.25 feet; length of weir crest=P, 15.969 feet; width of channel^, 15.970 feet; width of broad crest, 16.302 feet; smooth planed surface.]

Oi O 1—I <M r-i rH rH

0.361    11.612 | 0.050 I 0.217

0.361    0.576    2. 653

[Series No. XLIV. Height of weir=P, 11.25 feet; length of weir crest=P, 15.969 feet;

nel=5, 15.970 feet; width of broad crest, 8.980 feet.]

10

11

12

0.3136 . 1596 .4196 3.0666 2.3306 2.8656 1.5166 1. 2556 1.0396 .8996 .7326 . 5046

0.3116 .1576 .4176 3.0506 2.3116 2.8486 1.5106 1.2506 1.0356 . 8916 . 7306 . 5006

0.312

.159

.419

3.058 2.319 1.856 1.513 1.252 1.038

.897

.732

.502

0.174

0.312

0.469

.064

.271

5.386 3.547 2.537 1.864 1.404 1.059

.850

.627

. 356

.159

.419

3.073 2.326 1.860 1.515 1.254 1.039

.897

.732

.502

.163

.679

14.105 9.307 6. 712 4.928 3.718 2.803 2.254 1.676

.971

2. 691 2.570 2.504 2.619 2.624 2.646 2.642 2.649 2.646 2. 652 2.676 2.727

United States Geological Survey experiments at Cornell hydraulic laboratory on model of flat-top weirs with vertical faces—Continued.

[Series No. XLV. Height of weir=P, 11.25 feet; length of weir crest—X, 15.969 feet; width of channels, 15.970 feet; width of broad crest, 5.875 feet; nappe partly aerated.]

Measured head on experimental weir, in feet.

v=

A=

area of

No.

mean velocity of

Number of obser-I va-tions.

per foot

ap

Maxi

mum.

Mini

mum.

Mean =

D.

of

crest.

proach, m feet

per

second.

Head corrected! for velocity ! Q₌dis-of approach,    charge

in feet.    per

foot of crest, in cubic 11 feet per econd.

Dis

charge

coeffi

cient

Ci-

10

11

O rH rH rH

0.1766 . 2566 .3906 .9906 1.2456 . 9126 .7346 1.0006 .5906 .5916 .5216

0.1726 . 2526 .3886 .9786 1.2396 .9066 .7306 .9916 .5806 .5896 .5116

0.174 ' 11.424 .253 ! 11.504 .390 j 11.640 .982 ! 12.233 1.242    12.492

. 908    12.159

. 733    11.983

1.996    13.247

1.585    12.836

.590    11.841

. 520    11. 771

0.018 | 0.072

.029 | .127 .055 | .243

.209

.293

.189

.139

.564

.410

.103

.087

.975

1.386

.867

.627

2.830 2.000

.454

.376

.521 I 1.022 I 2.722

[Series No. XLVI. Height of weir=P, 11.25 feet; length of weir crest=X, 15.969 feet; width of chan-nel=6, 15.970 feet; width Of broad crest, 3.174 feet; nappe partly aerated.]

26

41

34

23

30

32

33

32

32

10

31

11

38

12

33

0.2526 .1916 .4186 2.9686 2.4956 2.0376 1.6006 1.2326 .9726 .7866 .6026 .5056

0.2446 .1896 .4156 2.9416 2.4806 2.0126 1.5906 1.2286 .9706 .7816 .6006 .5026

0.250    11.501    0.029    | 0.125

[Series No. XLVII. Height of weir=P, 11.25 feet; length of weir crest=X, 15.969 feet; width of channel =5, 15.970 feet; width of broad crest, 0.927 foot; nappe partly aerated.]

27

29

31

29

29

26

29

30

27

10

34

11

27

12

30

0.1666 .2816 . 4156 2.9446 2.5306 2.0196 1.5786 1.2296 1.0096 .7786 .6296 . 4616

0.1636 .2726 .4106 2. 9206 2.5116 2.0106 1.5706 1.2236 1.0046 .7756 .6276 .4606

0.165

.278

.412

2.933

2.522

2.014

1.592

1.226

1.007

.777

.629

.461

11.416 11.529 11.663 14.184 13.772 13.264 12.842 12.477 12.258 12.027 11.879 11.712

0.016

0.067

0.165

.033

.060

1.187

.970

.722

.512

.345

.248

.163

.117

.073

.147

.265

5.076 4.037 2.874 2.015 1.361 1.012

.685

.499

.313

.278

.412

2.954

2.535

2.021

1.596

1.228

1.008

.777

.629

.461

.859 | 2.744

United States Geological Survey experiments at Cornell hydraulic laboratory on model of Merrimac River dam, at Lawrence, Mass.

[Height of weir=P, 6.65 feet; length of weir crest=L, 9.999 feet; width of channel=6,15.97 feet.]

No.

la

2a

3a

21

10

11

22

23

12

24

13

25

19

Measured head on experimental weir, in feet.

Number of obser-

va-

tions.

Maxi-

mum.

Mini

mum.

Mean

I).

4.001 3.930 3.630 3.630 3.166 2.815 2.510 2.223 2.130 2.041 1.850 1.746 1.645 1.496 1.322 1.268 1.089

.764

.584

. 583

.198

A= area of section per foot of crest.

10.651 10.580 10.280 10.280 9.816 9.465 9.160 8.873 8.780 8.691 8.500 8.396 8.295 8.146 7.972 7.918 7.739 7.414 7.234 7.233 6.848

v=

mean

veloc-

ity

of approach, in feet per second.

2.618 2.563 2.309 2.319 1.939 1.654 1.424 1.200 1.127 1.066

.932

.860

.802

.691

.600

.556

.462

.284

.195

.192

.039

Head corrected for velocity of approach, in feet.

II*

II

Q=discharge per foot of crest, in cubic feet per second.

Dis-

charge

coeffi

cient

Cx.

10

11

1.092

1.141

.669

.447

.446

.088

.765

.585

.584

.198

0.270    3. 067

In the accompanying tables (pp. 98-106), columns 2, 3, and 4 show, respectively, the number of observations of head and the maximum and minimum readings in each experimental period. In column 5 is given the mean head on the experimental weir deduced from the tape observations above described. Column 6 shows the area of cross section of the channel of approach per foot of crest. For suppressed weirs this quantit}⁷ equals the sum of the height of weir plus the measured depth on crest. For weirs with one end contraction the quantity A is obtained by dividing the total area of the water section, where D is measured, by the net length of the weir crest corrected for the end contraction. For those series where the depth on the experimental weir was increased by contracting the weir to about one-half of the channel width and introducing one end contraction, the net length of crest has been determined by the method of Francis, by deducting one tenth the head from the measured length of crest. The discharge per foot of crest of the experimental weir given in column 10 has been deduced from the discharge over the standard weir,

Plates XXIII and XXIV will be found immediately preceding page 95.

WATER-SUPPLY PAPER NO. 150 PL. XXV

ms-

O.

n

a.

Form of crest

4.0

- 3.8

3.6

3.4

3.2

3.0

1r*—89‘2-

2.8

2.6

LENGTH OF CREST, 15.969 FEET.

2.2

2.0

1.8

Corrected head II in feet.

9‘1

FI

£ri)

1.2

SER/ES\34

0‘I

.8

.6

OGEE MODEL SIMILAR TO PLATTSBURG DAM, WITH BACK SLOPE MODIFIED.

UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC LABORATORY, 1903.

EXPERIMENTS OF

U. S. GEOLOGICAL SURVEY

m.

.2

Coeffi

cient,

c.

4.00

3.80

09*8

3.40

3.20

3.00

2.80

2.60

1

PL. XXVI

Y?o

4.0

3.8

3. \55

3.8

WATER-SUPPLY PAPER NO. 150

l-°

e*‘

Cf/f^O.33

Form of crest

3.6

z-°

or

3.4

3.2

3.0

2.8

2.6

2.2

2.0

1.8

Corrected head H in feet.

UPSTREAM END EXTENDED WITH 4-INCH RADIUS QUARTER ROUND.

3\5

Form of crest

3.6

VS

3.2

8.0

2.8

2.6

VZ

2.2

2.0

1.6

1.4

|9C

1.2

SEF/ES

1.0

.8

.6

MODEL OF OGEE DAM AT CHAMBLY, QUEBEC.

Ff'FS

1.8

Corrected head H In feet.

16

JB

1.2

1.0

.8

.6

U. S. GEOLOGICAL SURVEY

.4

.2

.2

LENGTH OF CREST, 15.969 FEET.

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC

LABORATORY, 1903,

MODEL OF OGEE DAM AT CHAMBLY, QUEBEC.

Coeffi

cient

iez

C.

3.80

3.60

3.40

3.20

3.00

Coeffi

cient

C.

3.60

3.40

3.20

3.00

2.80

PL. XXVII

30

WATER-SUPPLY PAPER NO. 150

--so'---

©

Soz=u

Form of crest

•r

3.0    3.2    3.4    3.6    3.8    4.0

_u

Am

O'f

3.8

- 6.0' -

Form of crest

3.6

3.1

3.2

2.6 2.8

SERIES 38

1.8    2.0    2.2    2.4

Corrected head H In feet.

LENGTH OF CREST, 15.969 FEET.

KIT

2.8    3.0

LENGTH OF CREST, 15. 969 FEET.

2.6

1.6

1.4

1.2

MODEL OF OGEE DAM AT DOLGEVILLE, N. Y.

jdr

1.6    1.8    2.0    2.2    2.4

Corrected head Hin feet.

PART OF APRON CARRIED AWAY IN SECOND EXPERIMENT.

1.4

1.0

1.0 1.2

U. S. GEOLOGICAL SURVEY

.8

.6

.2

.8

.6

.4

MODEL OF OGEE DAM AT DOLGEVILLE, N. Y.

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC LABORATORY, 1903.

'Coeffi

cient

lei

C.

3.60

3.40

3.20

3.00

'Coeffi-|

cient

C.

3.60

.3.40

:3.20

3.00

S'

/

T0

To

4.0

PL. XXVIII

WATER-SUPPLY PAPER NO. 150

/v^ppe free ^

Form of Crest

8‘S

3.3

II

3.6

3.4

3.2

3.0

2.8

2.6

NAPPE PARTLY AERATED.

32

3.

je

e fre

14,0.4-79'

Form of crest

3.6

3.4

.3.2

3.0

2.o

-6.0

Form of crest

3.8

8.0

S.i

3.3

rz

Z‘7

3.0

LH

IT

TJ

SEF7E

1.0

CROAD-CREST WZI.T. WIDTH, 1.6<6 FEET; LENGTH, 13. 9G9 TFFT.

40

ISEF/FS

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

BROAD-CREST WEIT. WIDTH, 0.479 FOOT; LENGTH, 15.C69 FEET. NAPPE AERATED.

AO

3.0

z.a

3.0

rz

kl

ki

in

Corrected head 77 in feet.

"1.6--- 1.3'

FT F'I_

LENGTH OF CREST, 15.969 FEET.

STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS CORNELL UNIVERSITY HYDRAULIC LABORATORY, 1903.

NAPPE AERATED.

.8

U. S. GEOLOGICAL SURVEY

.6

#■

(n)

JL

JvT

.4    .6 "    ~.8--l!j

MODEL OF DOLGEVILLE CAM WITH APRON REMOVED, LEAVING TRAPEZOID.

EXPERIMENTS OF UNITED

Coeffl.

eient

r.

3.20

3.00

2.80

00‘S

3.40

3.20

00‘S

2.80

2.60

||«5 5 y v

3.20

T. ■

■

J .

/

i

. '

■

:

• •; '    .....-t-----■-!    ■

LJ

s?

■1

:

5.4

WATER-SUPPLY PAPER NO. 150 PL. XXIX

4.4    4.6    4.8    5.0

--/Z.Z39

/"or/77 of Crest

3.4    3.6    3.8    4.0    4.2

4.4    4.6    4.8    5.0    5.2

4.2

K---

42

SERIES]

1.6    1.8    2.0    2.2    2.1    2.6    2.8    3.0    3.3

~ Corrected head H in feet.

NAPPE PARTLY AERATED.

j---le.Soz

Form of crest

-v

3.4    3.6    3.8    4.0

,8    1.0    1.2    ' 1.4

BROAD-CREST WEIR. WIDTH, 12.239 FEET; LENGTH, 15.969 FEET.

K---

2.2    2.4    2.6    2.8    3.0    3.2

Corrected head II in feet.

U. S. GEOLOGICAL SURVEY

.6

.2

1.0    1.2    1.4    1.6    1.8    2.0

Id

.8

.6

Coeffi

cient

C.

2.80

2.60

OFZ

.2

EDGES OF TRANSVERSE BOARDS NOT FLUSH; NAPPE PARTLY AERATED.

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC

LABORATORY, 1903.

WIDTH, 16.302 'FEET; LENGTH, 15.969 FEET.

BROAD-CREST WEIR.

irr 150—06-11

Coeffi

cient

C.

2.80

2.60

OFZ

' •

2-

O'*

3.8

-H

a.

t' ■

/5.302'

Form of crest

3EF\eA 4\3a\

1.4

Corrected head H in feet.

t\§T

1.2

.8 1.0

.6

.4

CO

P> 1©

<D-

(3)1

.2

BROAD-CREST WEIR. WIDTH, 16.302 FEET; LENGTH, 15.969 FEET. SMOOTH CREST SURFACE; NAPPE PARTLY AERATED.

--6.980'----->|

3.6

k---

4-

SEFtES 4

Form of crest

rs

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

.8

.6

Corrected, head H in feet.

BROAD-CREST WEIR. WIDTH, 8.980 FEET; LENGTH, 15.969 FEET. NAPPE PARTLY AERATED.

is

3.00

2.80

2.60

ors

2.20

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC LABORATORY, 1903.

.3

Coefti

cient

cxei

C.

2.80

2.60

Ofg

/

'

8'8

WATER-SUPPLY PAPER NO. 150 PL. XXXI

.64

2.8

2.6

f'6

2.2

2.0

3.0

2.8

5.675

Form of crest

NAPPE PARTLY AERATED.

2.8

2.4

2.2

NAPPE PARTLY AERATED.

1.8

9*1

1.2

Corrected head H in feet.

1.0

W5\

.8

R/ES

U. 8. GEOLOGICAL SURVEY

SE

.6

.4

.2

BROAD-CREST WEIR. WIDTH, 5.875 FEET; LENGTH, 15. 969 FEET.

2.0

A^Z./7-f-

SERIES

Form of crest

1.8

91

ft

1.2

Corrected head H Jn feet.

jin—

1.0

.8

.6

.4 .

BROAD-CREST WEIR. WIDTH, 2.174 FEET; LENGTH, 15.969 FEET.

Coeffi

cient

iei

a

3.00

2.80

2.60

.2

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYQRAULIC LABORATORY, 1903.

Coeffi

cient

3.00

2.80

2.60

\‘

WATER-SUPPLY PAPER NO. 150 PL. XXXII

13/

3.2

3.0

TV²⁷'

Form of crest

2.8

2.6

2.4

2.2

2.0

8T

1.6

Corrected head H in feet.

1.2

<n

BROAD-CREST WEIR. WIDTH, 0.927 FOOT; LENGTH, 15.069 FEET. NAPPE PARTLY AERATED.

S\ER/E$ \f-7

.8

.6

VEY

U. S. GEOLOGICAL SUR

<sr

.2

EXPERIMENTS OF UNITED STATES GEOLOGICAL SURVEY ON VOLUME OF FLOW OVER MODEL DAMS, CORNELL UNIVERSITY HYDRAULIC

LABORATORY. 1903.

Coeffi

cient

3.40

3.20

3.00

2.80

2.60

OF!

obtained, as described above, by dividing the total discharge by the net length of the experimental weir. The mean velocity of approach -y, given in column 7, has been obtained by the formula

Q

V=A-

The correction for velocity of approach has been carefully computed by the Francis formula

T1S = (I) -M,)*-/<*,

where

h-Z-

The resulting values of 77³ are given in column 8. The corresponding values of //, given in column 9, have been obtained bv interpolation from a table of three-halves powers. The discharge coefficient C\ given in column 11 has been obtained by the formula

cx=

7/2

This coefficient represents the discharge per linear foot of crest, if the head is 1 foot, with no velocity-of approach, it being the coefficient in a weir formula of the same form as that used by J. B. Francis for a thin-edged weir.

Pis. XXIII to XXXII show the coefficient diagrams deduced from these experiments.

EXPERIMENTS ON MODEL OF DAM OF THE ESSEX COMPANY, MERRIMAC RIVER, AT LAWRENCE, MASS."

A series of experiments covering five different depths on crest was made by James B. Francis at lower locks, Lowell, Mass., November, 1852. The model had a crest length of 9.999 feet, with end contractions suppressed. Height of water was measured by hook gage in a chamber at one side of the channel, 6 feet upstream from crest, so arranged as to give substantially the height of the still-water surface above the crest without correction for velocity of approach. The discharge was volumetrically determined as in Francis’s thin-edged weir experiments.

The experiments of Francis covered depths on crest ranging from 0.5872 foot to 1.6338 feet. From these experiments he deduced tne formula for discharge,

Q=S. 01208Z77¹⁵³.

a Francis, J. B., Lowell Hydraulic Experiments, pp. 136-137.

108 WErR EXPERIMENTS, COEFFICIENTS, AND FORMULAS.

If the discharge were expressed in terms of the usual formula,

3

Q— CXLH%, with a varying coefficient C„ we should have a continuously increasing coefficient.

A series of experiments on a similar model dam, 6.65 feet high, with crest length of 15.932 feet, was made at Cornell University hydraulic laboratory in 1903. The model there used differed from that shown by Francis only in the substitution of a flatter upstream slope near the bottom of the canal, as shown in PI. XXXIII. The end contractions . were suppressed and the depth on crest was measured with steel tape and plumb bob suspended over center of channel at points 14.67 feet and 29.82 feet, respectively, upstream from crest of experimental weir. Discharge was previously measured over the standard weir, calibrated by Bazin’s and Fteley and Stearns’s formulas, located at head of experimental canal.

The experiments covered a range of heads varying from 0.198 foot to 4.94 feet. In the majority of the experiments the head was observed at both points. The upper point of measuring depth was at the upstream end of the inclined approach. The lower point was over the incline, where the area of the section of approach was smaller and the velocity larger than in the deeper channel above. The experiments have been reduced with reference to the heads measured 29.82 feet upstream from crest. By comparison of the depths simultaneously observed at the two points correction factors have been deduced for the reduction of the remaining experiments, in which the head was observed at the downstream point of observation only.

The observed head has been corrected for velocity of approach by the formula of Francis. The resulting mean coefficient curve, based on 19 valid observations, shows a larger coefficient of discharge in the

3

formula Q— C^LIl² than does that of Francis.

For a head of 1 foot the formula of Francis for the Merrimac dam gives a discharge of 90.3 per cent of that for a thin-edged weir. The Cornell experiments show 94.5 per cent of the discharge over a thin-edged weir under the same head.

4.2

WATER-SUPPLY PAPER NO. 150 PL. XXXIII

0'f

3.8

3.6

re

3.2

3.0

9.75---

2.8

Cana/ bottom ^

-,-tr'OS-----»j

-1-1-1- I 1-1-1-u

Cross section

---

2.6

rz

2.2

2.0

1.8

1.6

1.4

Corrected head H in feet.

V>

<0

1.2

1.0

.8

U. S. GEOLOGICAL SURVEY

.6

.4

.2

EXPERIMENTS ON MODEL SIMILAR TO MERRIMAC RIVER DAM AT LAWRENCE, MASS., MADE FOR UNITED STATES GEOLOGICAL SURVEY

AT CORNELL UNIVERSITY HYDRAULIC LABORATORY, 1903.

Coeffi

cient

iei

C.

re

3.2

3.0

.8

Discharge per foot of crest, Francis formula for Merrimac dam, compared with Cornell experiments on similar cross section.

Depth on crest, H.

Q per foot of crest, in cubic feet second, Francis.

per

Coefficient C\ in formula Q=CiLH%.

Francis’s

formula.

Cornell experiments.

Depth on crest, H.

'Q per foot of crest, in cubic feet per second, Francis.

Coefficient C\ in formula Q=CiLH%.

Francis’s I Cornell exformula. periments.

0.15

.20

.25

.30

.35

.40

.45

.50

.55

.60

.65

. 70

. 75

.80

0. 1653 .2567 .3611 .4774 .6043 . 7431 .8877 1.0430 1. 206 1.379 1.5581 1.7452 1.9395 2.1408

2.845 2. 871 2. 889 2. 905 2.913 2.937 2. 940 2. 940 2. 956 2. 966 2. 973 2. 980 2. 986

2. 992

3. 05

0. 85

3. 06

.90

„ 3.07

3. 08

3. 09

3.11

3. 12

3.13

3.135

3.14

3. 14

3.14

3.14

3.15

. 95

1.00

1.15 1.25 1.50 1.75 2. 00

2.    50 3.00

3.    50

4.00

2.3490 2.5636 2.7846 3. 0121

3.    7500

4.    2378 5.6012

8. 6975

16.1750

25.1200

2.997 3. 002

3. 007

3. 012 3.041 3. 033 3. 048

3.075

3.113

3.140

3.15

3.15

3. 15

3.13

3.12

3. 10

3. 12

3.14

3. 20

3. 26

3.31

3. 36

Aside from Blackwell's experiments the Francis formula for the Merrimac dam was until recently the only one available for a large dam of irregular section, and for want of more appropriate data it has been used for the calculation of discharge over many forms of weirs of irregular section, and in spite of Francis’s explicit caution, it has been applied where the heads differed widely from those used in the original experiments.

Considering the limited experiments on which it is based, Francis’s Merrimac dam formula gives good agreement with the much more extended experiments on a similar section made at Cornell hydraulic laboratory.

FLOW OVER WEIRS WITH BROAD CRESTS.

THEORETICAL FORMULA OF UNWIN AND FRIZELL.a

Consider a weir of such breadth that the nappe becomes of sensibly uniform depth in the portion BO, tig. 8, the upstream corner of the weir being rounded to prevent vertical contraction and the surface slightly inclined downstream so that it becomes parallel with the surface of the nappe BO.

Fig. 8.—Broad-crested weir.

The fall causing the velocity V in the section 7? C is II—d. It follows that if v is the mean velocity in B O

v=4%{/(II—d)

Q— Ldv—LdV 2(7(77— d)

In this equation Q is 0 when <7=0 or <7=77. There must, therefore, be an intermediate value of <7 for which Q will be a maximum. Differentiating we ffnd for the condition of a maximum,

¹⁻¹

ct    ^    t

Giving 77— <7=- and <7=- 77, or, for maximum discharge, one-third

the head would be expended in producing the velocity of Mow. With this value of d the expression for discharge becomes

2 LII

3 V3

V 2 gll— 0.384907)77/ 2</77

or if

4%g—8.02,

0=3.0877)77

.    .    58

idee

In this formula frictional resistance has been neglected. The discharge given is the maximum for the conditions, and would result only if the stream discharges itself in.accordance with the “principle of least energy.”

Blackwell’s experiments, given elsewhere, show a considerably larger coefficient for weirs 3 feet broad, slightly inclined downward, than for those with horizontal crests.

a Given by W. C. Unwin, in article 1 on Hydrodynamics in Ency. Brit. Independently derived by J. P. Frizell. See bis Water Power, pp. 198-200.

Let d—KH, then from the formula first given

Q=LKH^2gR(l—K) = K^2g(l-K)Liri

(59)

6^ = 8.02iF/l—K

The theoretical coefficient C\ can be computed from this equation if iThas been determined experimentally.

From profiles taken in connection with United States Deep Waterways experiments at Cornell University hydraulic laboratory in 1899 the following values of D and d for broad-crested weirs have been scaled and the ratio d. D computed. D was taken 1 feet upstream from the upper face of the weir, and does not include velocit}^ of approach correction; values of dx and d₂ were taken at the lower-crest lip and center of crest, respectively. The value of d₂ at center of crest has been used in the computations.

Values of I) and d for broad-crested weirs.

dx

d»

K=h

0. 90

0. 35

0. 52

0. 58

•3

--B —

Broad-crested weir.

D

_L_

-    6.39-

Broad-crested weir.

1.15

1.80

2. 60

3.55

5.15

1.00

1.32

1.98

2. 85

3.90

4. 65

.45

.68

.59

.75

1.14

.63

1.20

1. 72

2.20

.35

.53

.75

1.08

1.50

2.10

1.75

2. 52

3.15

.50

.70

.98

1. 70

2.50

3.10

.67

.71

.61

.50

.50

.50

.60

.64

.61

For low heads a sudden drop begins near the upstream crest corner and terminates at a distance 1.5 to 2 I) below the upstream corner. From this point to within a distance about equal to D from the downstream crest corner the surface is nearly parallel with the crest. If the width of crest is not greater than 2.5 to 3 D the nappe passes over the broad crest in a continuous surface curve, becoming more nearly convex as the ratio D B increases.

For low heads Cornell experiment 13, crest 6.56 feet wide, with rounded upstream corner, complies very well with the theory of dis-

charge in accordance of the principle of least energ}T. The coefficient computed as above is

Cx = 8.02 X 0.585- 0.585 = 8.02X0.585X0.6442 = 3.02

The experimental coefficient with head corrected for velocity of approach is 2.82.

The following additional data may be cited:

Trail twine" quotes data of Elwood Morris, C. E., for Clegg’s dam, Cape Fear River, North Carolina. Horizontal crest 8.42 feet wide, vertical faces. //=1.25 feet, d (throughout central portion of crest) = 0.50 foot, d H—QA0.

Thos. T. Johnston⁶ gives data of elaborate protiles of the nappe for Desplaines Iiiver dam, Illinois. Horizontal planed stone coping, vertical downstream face; upstream face batter, 1 2:1. H= 0.587 foot. c/=0.315 to 0.307 foot in central, nearly level portion at distances 1.5 to 4 feet from upstream edge of crest. Johnston and Cooley deduce the coefficient C=1.69 for this case;

Blackwell’s experiments on discharge of water over broad-

crested weirs.

Experiments made by Thomas E. Blackwell,c M. Inst. C. E., are of interest as being probably the first recorded for weirs with broad crests. The discharge was volumetrically measured, and the conditions were generally favorable to accuracy. The experiments were made on a side pond of the Kennet and Avon Canal, 106,200 square feet surface area, closed by a lock at each end, the water being admitted from time to time as required, the relation between area of reservoir and volume of discharge being such that there was no sensible variation in water level during an experiment.

The weir was constructed in a dock to which the water had access through an irregularly shaped channel 40 feet in width, cut off from the main pond by a submerged masonry wall 9 feet wide, situated 25 feet upstream from the weir, having its top 18 inches to 20 inches below water surface.

The water level in the pond being constant when outflow took place, the weir, which had a crest adjustable in a vertical plane, was set with its crest level at the depth below water surface desired for an experiment, by means of adjusting screws at the ends of the weir; the water

a Engineers’ Pocket Book.

& Johnston, T. T., and Cooley, E. L., New experimental data for flow over a broad-crest dam: Jour. Western Soc. Engrs., vol. 1, Jan., 1896, pp. 30-61.

c Original paper before Institution of Civil Engineers of London, reprinted in the Journal of the Franklin Institute, Philadelphia, March and April, 1852.

was then allowed to waste through the weir until a uniform regimen of flow was established.

A gaging tank having a floor of brick laid in cement, with plank sides, and 449.39 cubic feet capacity, was erected at the foot of the weir. At a given signal the lid of this tank was raised, the time noted, and the rate of filling of the tank recorded by several observers. Such leakage from the tank as occurred was separately measured and allowed for. There was no correction for velocity of approach or for end contractions. *

The wind was so slight as to be negligible, except during one series when there was a brisk wind blowing downstream. The experimenter states that parallel experiments on a quiet day indicated an increase of about 5 per cent in discharge due to this wind.

The crest of the thin-edged weir consisted of an iron plate barely one-sixteenth inch thick. A square-top plank 2 inches thick was attached to the weir, and an apron of deal boards, roughly planed so as to form an uninterrupted continuation downstream, constituted the wide-crested weir used in the experiments.

The coefficient 6⁷j from Blackwell’s experiments has been worked out and is given in the following table. The measured depths taken in inches have also been reduced to feet. irr 150—06-12

Blackwell’s experiments on broad-crested weirs, Rennet and Avon Canal, England, 1850.

Weir.

Measured head, in inches.

Head, in feet.

Q per foot, in cub. ft. per sec.

C\—M V2p

Weir.

jasured head, in inches.

Head, in feet.

Me

Q per foot, in cub. ft. per sec.

C₁=M\/2g

II.

1 Thin plate, 10 feet long.

10.083 0.104 0.539 4.32

2    .167 . 292; .535 4.29

3    .25

4    . 333 . 675 . 437 3.50

5    .417 .935 .433 3.47

8    1 667 1.691 .387 3.10

9    .7501.842 .353 2.83

.429! .428 3.43

X.

L— 3 feet. %

Ho

rizonta!

VII.

L=3 feet.

--j:o -

Slope hi?

7    .583 9 .750

.060 . 311 2.49 .194 .355 2.85 .360 . 359 2.88 .468 .303 2.43 1.005 .354 2.84 1.254 .351 2.82 1.729 .332 2.66

XI.

L=6 feet.

//////////////////

Series VII, average

2.71

VIII.

L=3 feet.

- 3'.0 —--

Slope me - - -

Y,

7    .5831.222 .342 2.74

8    .667 L 426 .327 2.62

9    .7501.709 .328 2.63

10.083 0.058 0.301 2.41

2    .167 .175 .321 2.57

3    . 250 . 295 . 294 2.36

4    .333 .431 .279 2.24

5    .417,' .689 . 319 2.56

6    . 500 . 947 . 334 2.68

7    .5831.162 .325 2.61

8    .6671.369 .313 2.51

9    .7501.642 .317 2.54

ltol| .093 .071 ...........

3    .250 .329 .328 2.63

4    . 333 . 511 .331 2.65

6    .500 .963 ...........

7    .583 9 .750

10 .883 121.0

1.191

1.670

1.895

2.495

.331 2.65

.310 2.49 .311 2.49

XII.

L=10 feet.

Hor

ni.iL

9    .750 10 .833

.049 .254 2.04 .174 .319 2.56 .745 . 345 2.77 .972 .342 2.74 1.362 .312 2.50 1.688 .324 2.60 1.847 .303 2.43

IX.

£=10 feet.

1_____ Slope /:/g

ltof .073 . 060 2 .167 .181 4 .333 .530 6 .5001.028 8 .6671.476

.311 2.49 .330 2.65 .343 2.75 .362 2.90

.338

2.71

Series X, XI, XII, average ..............

52.03

2.48

Series VIII-IX, average .............

2.71

EAST INDIAN ENGINEERS* FORMULA FOR BROAD-CRESTED WEIRS.®

This formula is

M'LH V 2 gH, or if V 2 ^=8.02,    CXLH* (60)

Where M' = coefficient for thin-edged weirs=0.654—0.01ZT,

M' = M-

0.025Jf(£+l)

H+l

aMuliins, Gen. Joseph, Irrigation Manual, Madras, 1890.

6See table giving values of M and equivalent values of C, p. 22.

Experimental data not given. This formula gives values of M' or Cx decreasing as breadth of crest B increases, and for low heads increasing to a maximum for a head of about 1 foot, then slowly decreasing.

The formula reduces to

1-0.025 (Z?+l)

//+1

—RG . . . (62)

For B=0

H+ 0.975

GX=G

11+1

— RG

which differs by the ratio R from the equivalent value of G for a thin-edged weir.

Values of coefficient 0, of discharge over a broad-crested weir and of R, the ratio of the former to C the coefficient of discharge over a thin-edged weir, by Mullins’s formula.

li

II

1 foot.

2 feet.

3 feet.

4 feet.

feet.

It

Ci

C\

Ci

Ci

10

-    .991

0.975

.983

.988

.990

.992

.993

.994

.994

.995

.995

3.359 3.335 3.296 3.253 3.204 3.155 3.104 3.054 3.003 2.950

0.962

.975

.981

.985

.988

.989

.991

.992

.992

.993

3.316 3.307 3.275 3.237 3.191 3.144 3.095 3.045 2.995 2.944

0.95

.967

.975

.98

.983

.986

.988

.989

.99

3.319 3.279 3.255 3.220 3.177 3.132 3.085 3; 037 2.988 2.937

0.938

.958

.969

.975

.979

.982

.984

.986

.988

.989

3.230 3.251 3.234 3.204 3.164 3.121 3.075 3.028 3.042 2.930

5 feet.

6 feet.

7 feet.

8 feet.

feet.

Ci

Ci

Ci R

Ci

0.925

.95

3.182

0.912

3.22

.942

. 962    3.213

.97 [ 3.193 .975 ! 3.149 .978 | 3.11 .981 ; 3.06 .983 I 3.017

.956

.965

. 971

.975

.978

.980

.985

2.97

.982

10

.986

2.92

.984

3.057 3.138 3.150 3.138 3.110 3.076 3.036 2.994 2.950 2.093

The values of Gx given in the above table have been deduced from the corresponding values of C for a thin-edged weir by Mullins’s formula. The ratio R may, if desired, be applied approximately to correct values of G derived from other standard weir formulas.

FTELEY AND STEARNS EXPERIMENTS ON BROAD-CRESTED WEIRS.®

The formula of Fteley and Stearns is based on five series of experiments made in the Sudbury River conduit, Boston, 1877, on weirs 2, 3, 4, 6, and 10 inches wide, respectively. Suppressed weirs 5 feet long were used, the depths being as follows:

Fteley and Stearns experiments.

Width of crest, in inches.

Number of experiments.

Range of depth observed on broad crests, in feet.

From—

To—

21

25

22

17

The results are given the authors in the form of a table of corrections to be added algebraically to the measured head for the broad-crested weir to obtain the head on a thin-edged weir that would give the same discharge.

Fteley and Stearns’s correction c may be found approximately from the formula

c=0.2016V[(0.807^—^T)²+0.2146 iT|-0.1876 A . . (63) or if £=0.2016, m—0.1876, ^=0.2146, O—0.807, then

Q= OL\H-mB+kJ{0£-HY+K&$ . . . (64) If the head on a broad-crested weir is H, the discharge will be

Q= CL{H+cY .

C being the coefficient of discharge for thin-edged weirs.

If C₁ is the coefficient for the broad weir, then we may also write

Q=CJAP-

Hence

(66)

CH+c i

C~ H

From formula (66) have been calculated Fteley and Stearns’s coefficients for weirs with nappe adhering to crest for use in the formula

CO

Q= OLE

ci Fteley and Stearns, Experiments on the now of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 86-96.

Values of the ratio of the coefficient of discharge for a broad-crested weir, by Fteley and Stearns’s experiments, to that for a thin-edged weir.

Ci

correction for velocity of approach being made by adding 1.5 measured head to obtain H.u

V¹

to the

It

Width of crest, in inches.

10

12

0.0

.1 |    0,7466 | 0.74798

. 9806 1.0030

.5

.6

.7

.8

.9

1.0

1.1

1.2

1.3

1.4

1.5

0. 7562 j 0.7589    0.7576

. 7740 j    . 7679    I    . 7644

.9508 .9732 .9948 1.0148 1.0317

1.0090

1.0317

1.0499

0.7576 . 7624 .7685 . 7768 . 7865 .7989 . 8150 . 8339 . 8552 .8824 . 9027 . 9252 .9465 . 9657 . 9850 I

bazin’s formula and experiments on broad-crested weirs.

These included series of about 20 periods each for depths not exceeding- 1.4 feet on weirs of 0.164, 0.328, 0.656, 1.315, 2.62, and

6.56 feet breadth of crest. The coefficient C[ in the formula Q— CxLB'i, deduced from a recomputation of the experiments on weirs 2.46 feet high, using the Francis velocity of approach correction, is given on PI. IV.

Other experiments were made for the four narrower weirs with heights 1.148 and 1.64 feet, to determine the comparative velocity of approach effect.

Bazin shows that if the nappe is free from the downstream face of the weir it may assume two forms: (1) It may adhere to the horizontal crest surface; (2) it may become detached at the upstream edge in such a manner as to flow over the crest without touching the downstream edge. In the second case the influence of the flat crest evidently disappears and the discharge is like that over a thin-edged weir. The nappe usually assumes this form when the depth I) exceeds twice the breadth of crest B, but it may occur whenever the depth exceeds §B. Between these limits the nappe is in a state of instability; it tends to detach itself from the crest, and may do so under the

a Fteley and Stearns’s formula for a thin-edged weir has been used to calculate Q in deriving these coefficients, the experiments having been made under conditions similar to those under which their formula was derived.

influence of any external disturbance, as, for example, the entrance of air or the passage of a floating object over the weir.

■    •    (67)

When the nappe adheres to the crest, the coefficient C\ depends chiefly on the ratio DIB and may be represented by the formula c\ = c (0.70+0.185 DIB) .    .

in which C is the coefficient for a thin-edged weir.

When DIB—1.50 to 2, Cj C— 0.98 to 1.07 if the nappe adheres to crest, or Cj (7=1.00 if nappe is detached, and for I) 7?>2, Cj (7=1.00. Between the limits 77=1.57? and 77=27? the value which the coefficient (7j will assume in a particular case is uncertain. Bazin considers that his formula gives accurate results for adhering nappes with breadth of crests up to 2 or 3 feet. For a crest 6.56 feet wide and 71=1.476 feet he finds the result by formula (67) 93.4 per cent of that given directly by the experiment.

Values of the ratio CJC, for a broad-crested weir, with adhering nappe, by Bazin’s

formula.a

C,/C=0.700+0.185 DIB.

.8

.9

1.0

1.1

1.2

1.3

1.4

1.5

0 I 0.01

0.709 j 0.7018 .7185 :    .7204

.7370 ; .7388 .7555 j .7574 .7740 ! .7758 .7925 j .7944 .8110 ' .8128 .8295    .8314

.8480 I .8498 .8665 j .8684 .8850 j .8868 . 9035    .9054

0.02

0.03

0.04    0.05 | 0.06

0.07

0.08

0.09

0. 7037 .7222 .7407 .7592 .7777 .7962 .8147 .8332 .8517 . 8702 .8887 .9072 . 9257 . 9442 .9627 .9812

0. 7056 .7240 .7426 .7610 .7796 .7980 .8166 .8350 .8536 . 8720 .8906 .9090 .9276 .9460 . 9646 .9830

0. 7074 .7259 .7444 .7629 .7814 .7999 .8184 .8369 .8554 .8739 .8924 .9109 .9294 .9479 .9664 .'9849

0.7092    0.7111

.8942 | .8961 .9128 ' .9146 . 9312 | . 9331 .9498    .9516

.9682    .9701

.9868 j .9886

0.7130 .7314 .7500 .7684 .7870 .8054 .8240 .8424 .8610 .8794 .8980 .9164 .9350 . 9534 .9719 .9904

. 9553    . 9572

.9738 | .9756 . 9923 ! . 9942

a If there is velocity of approach, the value of D/B, not H/B, should be used as an argument. The ratio Ci/6'may be applied in a formula which includes the velocity of approach correction, either in the head H or in the coefficient.

Bazin’s formula gives ratios which continually increase as II increases, B remaining constant, and which continually decrease as B increases, 77 remaining constant. It gives, however, a constant ratio for all widths or heads where the ratio 77 B is unchanged.

Compared with their respective standard weir formulas, Mullins’s formula gives for a broad-crested weir a continuously decreasing ratio of discharge as B increases from zero, II remaining constant, and a continuously increasing discharge as //increases from zero, B remaining constant; Fteley and Stearns’s experiments give a discharge ratio which is less than unity, but which varies in an irregular manner, depending on the head and breadth of weir.

On referring to PI. IV, in which the Bazin coefficients are given in a form comparable with the experiments of the United States Geological Survey, it will be noticed that, except for the lowest heads, the coefficient curves are simple linear functions of the head. The rate of increase of the coefficients as the head increases grows rapidly less as the breadth of the weir increases, indicating that for a very broad weir the coefficient would be sensibly constant throughout the range of stability of the nappe.

For the narrower weirs the coefficients tend to increase rapidly almost from the start toward the value for a thin-edged weir or detached nappe. For the weirs 2.62 and 6.56 feet breadth of crest the total variation in the coefficient for the range of heads covered by the experiments is comparatively small. The average coefficients are as follows:

Average Bazin coefficients, broad-crested weirs.

Bazin series No.

Crest width, in feet.

Range of head, in feet.

From—

To—

Average constant coefficient, C\.

113

1. 312

Lowest.

0. 60

2. 64

114

2. 624

0. 35

.85

2. 59

1.32

2. 62

115

6. 56

.55

Highest.

a 2. 58

a Coefficient increases slowly throughout.

The average coefficients show a fair agreement with the constant coefficient for broad-crested weirs with stable nappe deduced from the experiments of the United States Geological Survey (page 120).

EXPERIMENTS OF THE UNITED STATES GEOLOGICAL SURVEY ON BROAD-

CRESTED WEIRS.

The method of conducting these experiments and the detailed results are given on pages 95-107. The coefficient curves are presented on Pis. XXVIII to XXXII. It may be remarked here that the models were larger and the range of breadth of crest and depth of flow experimented upon was greater than in the earlier experiments described. In general, the laws of behavior of the nappe pointed out by Fteley and Stearns and Bazin were confirmed.

■k

The following table presents a resume of the results:

Resume of United Stales Geological Survey experiments on broad-crested weirs.

Series.

40

47

41

46

45

44

42

Nappe

un

stable

Nappe

de-

Breadth , of crest, ! in feet. ;

for

heads less than values below, in feet.

tached

from

crest

at

head, in feet.

0.479 I 0.3 .927    .3

1.646 3.174 5.875 8. 980

0.8

1.8

2.8

12.239

43    16.302

43a '    16.302

Coefficient C\ varies between the limits—

Head, in feet. I Coefficient. J

Above head,

From— To— From— | To— in feet.

Ci

CO M N lO lO 1C

to

CO

CC OO OO CO 05 o o © TH <M*

2.64

2.57

2.56

2.70

cd cd

t-¹

1.0

2.72

2. 73

2. 62

2.73

2.68 ;■

2.72

3.32

3.31

3.32

2.64

2.64

2.62

2.64

2,72

2. 63

2.64

OO OO O0 CO Oi o

© r—* cd rH    cd

3. 32

3.31    |

3.32

Increases| 2.64 | 2. 62

!-

I-¹    t-¹

a 2. 64

62. 63

2.64

2.634

a Coefficient shows tendency to increase slowly with head.

6 Edges of planed and matched boards not flush. Crest smoothed in series 43a.

The deductions that follow have been based on a consideration of earlier experiments as well as those here given for the first time.

1.    For depths below 0.3 to 0.5 foot the nappe is very unstable, owing probably to magnified effect of crest friction and to the varying aeration or adhesion of the nappe to the downstream weir face.

2.    For heads from 0.5 foot to 1 or 2 feet for very broad weirs, or from 0.5 foot to the point of detachment for narrower weirs, the coefficient is somewhat variable and changes in an uncertain manner. For the broader weirs, the range of variation of Cx between the depths indicated is narrow, from 2.73 to 2.62.

3.    When the nappe becomes detached the coefficient remains nearly identical with that for a thin-edged weir. For the narrower weirs the coefficient increases rapidly within the range of tendency to detachment indicated by Bazin, i. e., for heads between I) and 2D.

♦    4. On the broader weirs for depths exceeding 1 to 2 feet up to the

limit of the experiments (about 5 feet), the experiments indicate a sensibly constant coefficient for all depths. Where there is any tendency to variation within the range indicated there is a gradual increase in C\.

For weirs of 5 to 16 feet breadth the experiments show no conspicuous tendency for the coefficient Cx to change with variation in either H or B, the range of value of Cx being from 2.62 to 2.64.

The line of detachment of the nappe for a weir of 5 feet breadth would be 7.5 to 10 feet head or perhaps more, and a higher head for

broader crests. If this depth were ever reached it may be surmised that the coefficient Cx would increase to about 3.33 at the point of detachment. It would also appear, as is in fact indicated in Bazin’s formula, that the coefficient should very slowly increase with II and decrease as B increases, independent of the tendency to detachment of the nappe, and owing to the decreased relative effect of crest friction and contraction.

The United States Geological Survey experiments indicate that this effect is of relatively little significance for large heads and broad weirs, and hence a constant coefficient covering a wide range may be safely adopted.

The average coefficient, 2.64, which we have tentatively chosen for weirs exceeding 3 feet in breadth under heads exceeding 2 feet, may apparently be applied for considerably lower heads for weirs of 5 feet or more crest breadth with but small error.

TABLE OF DISCHARGE OVER BROAD-CRESTED WEIRS WITH STABLE NAPPE.

A table has been calculated, using U₁ = 2.64 and covering heads varying by 0.1 foot increment from zero to 10 feet (p. 177). It is considered applicable for weirs of 3 feet or more crest breadth when HiB lies between the general limits 0.25 to 1.5. The coefficient 2.64 gives a discharge 79.2 per cent of that for a thin-edged weir by the Francis formula. The relative discharge obtained by other formulas and experimenters is shown in the following¹ table:

Comparison of broad-crested, weir formulas and experiments giving percentage of discharge over a thin-edged weir.a

1 foot width.

2.62 feet width.

Formula or experiments.

HIB=0.5 11=0. 5

1.0

1.5 i 0.5

Mullins 6.......i...........

Fteley and Stearns.........

Bazin formula.............

U. S. Deep Waterways experiments.................

U. S. Geological Survey experiments .................

96.7

78.6

79.2

81.0

Ox tO Cn

98.0

1.0 1.5 0.25 [ 0.5 2.62 | 3.93 1.64 j 3.28

93.2 I 95.5

1.0

6.56

97.5

1.5

9.81

98.2

98.5

97.8

79.2

82.8

90.3

97.5

c 79.5

88.5 I 97.8    74.6    79.2

93.3    114.1    72.0    71.1

C81.3 e 86.7    79.2 d 79.2

88.5

72.3

d 79.2

97.8

73.2

d 79.2

a No velocity of approach.

ft East Indian engineers’ formula, given in Mullins’s Irrigation Manual, Madras Presidency. <~Weir 2.17 feet broad. cl Weir 5.88 feet broad.

Considering the low heads used, it may be noted that before Bazin’s experiments only those of Blackwell included a weir breadth sufficient to eliminate the early tendency to detachment and permit the existence of the stable period for which a constant coefficient applies.

Blackwell’s experiments on weirs 3 feet broad indicate a maximum coefficient Cx of 2.65 to 2.77 for a head of about 0.5 foot, decreasing as the head increased. •

The experiments of the United States Deep Waterways Board on models with 2.62 and 6.56 foot crest width are shown on PI. XV. For the narrower weir the coefficient increased uniformly with the head. The nappe did not leave the crest, although the experiments were continued to the limit H\B— 2, at which stage the coefficient exceeded that for a thin-edged weir. For the broader weir the coefficients are much less variable and the curves indicate that the coefficients approach a constant as the breadth of crest is increased.

It will be noted that considerable care must be exercised in determining the condition of the nappe for broad-crested weirs of inconsiderable width, while for those of greater breadth the wind may exert considerable influence on the nappe on the broad crest under lower heads. The constant coefficient 2.64 has been deduced from experiments on weirs with smooth, planed crests and sharp upstream crest angles. The effect of crest roughness on weir discharge is discussed on page 133.

EFFECT OF ROUNDING UPSTREAM CREST EDGE.

Experiments bj⁷ Fteley and Stearns'* indicate that the effect of rounding the upstream crest corner is to virtually lower the weir, b}7 allowing the water to pass over with less vertical contraction. To determine the discharge over a thin-edged weir, with upstream crest corner rounded to a radius i?, add to the measured head the quantit}⁷

(68)

A==0.70i?

The above formula was deduced by Fteley and Stearns from experiments on weirs with crest radii of one-fourth, one-half, and 1 inch. For heads nojt exceeding 0.17, 0.26, and 0.45 foot, respectively, the nappe adhered to the crest, and the formula does not appty.

The correction formula (68) is equivalent to increasing the discharge coefficient in the ratio

ZT+0.7 R

II

or nearly in the ratio

H+B

IT ‘

A second series of experiments was made with rounded upstream edges of similar radii applied to a crest 4 inches wide, giving the correction formula for this case,

K—0A1B <* Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 97-101.

EFFECT OF ROUNDING UPSTREAM CREST EDGE.

123

where K' is a correction to be added to the measured head before applying the formula for discharge over the broad-crested weir. This formula is applicable for depths of not less than 0.17 and 0.26 foot, respectively, on weirs with radii of one-fourth and one-half inch. Fteley and Stearns’s formulas show the effect to decrease with the breadth of crest. It also decreases, when expressed as a percentage, with the head. These formulas are probably applicable to weirs with smaller, though not to those with greatly larger, radii than those of the experimental weirs.

Bazin experimented upon two weirs, duplicated in the United States Deep Waterways experiments, having crest widths of 2.624 and 6.56 feet, respectively, with an upstream crest radius of 0.328 foot (PI. IV).

Broad-crested weirs with rounded upstream corner.

Coefficient C\, Bazin’s experiments.

Head, in feet.

Crest width, 2.62 feet.

Crest width, 6.66 feet.

With angle crest.

With rounded crest.

With angle crest.

With rounded crest.

0. 25

2.52

2.85

2. 40

2. 58

.50

2. 59

2.95

1.00

2. 64

3. 00

1.50

2. 69

3.04

2.515 2. 575 2.635

2. 76

2.89

2.92

Coefficient Ci, United States Deep Waterways experiments.

1.50

2. 67

2. 92

2. 39

2. 81

2. 00

2. 75

3.00

2. 41

2.81

3. 00

2. 93

3.17

2. 44

2. 81

4.00

3.11

3.34

2. 47

2. 81

5.00

6. 00

3.30 Nappe free.

3.51

3. 00

2.50

2. 53

2. 81

2. 81

United States Deep Waterways series 14 and 15, PI. XV, show the effect of rounding the upstream crest corner, radius 0.33 foot, on a model of the Kexford flats, New York, dam. In this case, with a weir 22 feet broad with 6:1 slope on each face, the effect of rounding becomes comparatively slight, the average increase being about 2 per cent.

United States Geological Survey experiments, series Nos. XXXV and XXXVI, PL XXVI, show the effect of the addition of a 4-inch radius (0.33 foot), quarter-round extension to the upstream face of the model of an ogee-section dam, having 4.5 feet crest width, 4.5 :1 slope.

Effect of rounded upstream crest corner on an ogee dam.

Head, in f6et.

Chambly model, series 35.

Same, with ⁽ Difference rounded up- per cent of stream crest ! Francis’s coefficient.

corner.

0.50

3. 18

3. 23

1.00

3.30

3. 34

2.00

3.42 |

3.51 *

3.00

3.49

3. 64

+ 1. 5 +1.2 +2.7 +4.5

EXPERIMENTS ON WEIRS WITH DOWNSTREAM SLOPE, OR APRON, OF VARYING INCLINATION.

Aside from the experiments of Blackwell on weirs with very slight inclination and a few series by other experimenters on weirs of irregular section involving aprons, the data on this subject are limited to those of Bazin’s experiments.

Bazin selected a number of weir types, each having a constant top width, height, and upstream inclination and applied to each a number of different downstream slopes.a

TRIANGULAR WEIRS WITH VERTICAL UPSTREAM FACE AND SLOPING

APRONS.

Such weirs are occasionally used, as where the apron slopes to the stream bed in log slides. A similar form in which the downstream slope terminates at a greater or less distance from the vertical upstream face is not uncommon, and to this form the Bazin experiments may probably be applied, provided the breadth of the sloping apron is considerable. The experiments are of special interest, however, as showing the effect of attaching a sloping apron to the downstream face of a thin-edged weir, and by inference affording an indication of the effect of a similar apron attached to any form of cross section. The results of Bazin’s experiments recomputed on the basis of the Francis formula are shown on PI. V.

Four series of experiments on weirs 2.46 feet high are included. For all these series the coefficient C tends to remain nearly constant for the range of heads covered, 0.2 foot to 1.5 feet, there being a slight increase in C with the lower heads only.

Two series on weirs 1.64 feet high are also given. In series 145, slope of apron 3 :1, there is a general increase in coefficient with head below 0.9 foot. Series 138, for a weir 1.64 feet high, is duplicated on a weir 2.46 feet high, and the latter series is given preference in the general curve. The lower weirs indicate in both cases slightly higher coefficients, possibly owing to the incomplete elimination of the effect of excessive velocity of approach.

a Bazin did not attempt to collate the results extensively. His general resume has been translated by the writer, and may be found in Rept. U. S. Board of Engineers on Deep Waterways, pt. 2, 1900, pp. 646-658.

The average constant coefficients for the several series are shown in the following table:

Mean coefficients, triangular weirs with varying apron slope.

Range of C.

Range of head.

Series.

Height.

Average 0.

From—

To—

From—

To—

136

2. 46

137

2. 46

CO co

1.40    3.84

3. 88

3.85

3. 48

3. 52

3.50

138

1.64

145

1.64

141

2.46

5:1

N Ol CO

1. 5    3. 56

3.58

3. 57

1.5

1.5

3.39

3.08

3. 41

3. 40

3.14

3.13

142

2. 46

10:1

1. 5    2. 90

2. 93

2. 91

3.9

3.8

A Weirs /. 64 ft. high.

1.46 »    "    *

3.7

3.5

-qr

Form of experimental weirs

Coefficient C.

3.4

3.3

3.2

3.1

c--

3.85

S°'²

3.0

2.9

2.8 __

JJL

%r

10

11

12

Slope or batter of downstream face of weirs. Fig. 9.—Coefficient curve for triangular weirs.

The mean coefficients have also been plotted on tig. 9 and a general curve drawn. This curve becomes approximately a straight line when plotted on logarithmic cross-section paper. Its equation expressed in logarithmic form is

3^85 £0.12

where 8 is the batter or slope of apron.

C=

If N=6, then, solving- by logarithms,

log 6=0.7781513 log 6⁰¹²=0.0933782

log 0o.i2 =9.9066218

log 3.85=0.5854607 log (7=0.4920825 (7=3.105

Fig. 9 gives C=3.07; the difference is 1 per cent.

The following conclusions deduced from the recomputed data conform in general with those of Bazin:

1.    For steep apron slopes where the nappe tends to break free, the apron materially increases the discharge by permitting- a partial vacuum to be formed underneath the nappe.

2.    For flat apron slopes the conditions approach those for a horizontal crest.

3.    For an apron slope of about 3 :1, the discharge is nearly the same as for a thin-edged weir.

4.    For slopes greater than 3 :1 the apron diminishes the discharge the amount of diminution increasing as the slope becomes flatter.

TRIANGULAR WEIRS WITH UPSTREAM BATTER 1:1 AN1) VARYING SLOPE

OF APRON.

Three series of experiments by Bazin are included (PI. IX), all made from weirs 1.64 feet high. The results are comparable among themselves, but owing to the high velocity of approach their general applicability is less certain.

Series No. 161, downstream slope 1:1, shows a generalty decreasing coefficient with an apparent tendency to become constant through a narrow range of heads, from 0.5 to 0.9 foot, with (7=about 4.11.

Series No. 163 and 165, with apron slopes of 2:1 and 5:1, give coefficient lines-, which ma}^ be fairly represented b}r the constants 3.82 and 3.47, respectively. These coefficients compare with those for vertical weirs with the same apron slopes as follows:

Comparative coefficients.

Batter of apron.

Vertical

face.

Face inclined 1:1.

('

Difference, per cent, Francis’s coefficient.

1:1

3. 85

4. 11

2:1

3. 53

3. 82

5:1

3.13

3. 47

+ 7.8 + 8.7 +10.2

EXPERIMENTS ON WEIRS OF TRAPEZOIDAL SECTION WITH UPSTREAM SLOPE OF -J-rl, HORIZONTAL CREST, AND VARYING DOWNSTREAM SLOPES.

Five series of Bazin’s experiments on weirs 2.46 feet high, with crest width of 0.66 foot, are shown on Pis. VI and VII. The curves indicate coefficients increasing- with the head, the rate of increase being-more rapid for the steeper apron slopes. There is a tendency to depression at about 0.4 foot head, representing-, possibly, the point at which the nappe changes from adhering to depressed condition on the downstream face. The curves are all convex, and apparently approach a constant, which was not, however, reached within the limit of experiments, except, perhaps, for the flattest slope of 5:1. The coefficients increase in value as the steepness of the apron slope increases.

Three series of experiments on weirs similar to those above described, but with flat crests 1.317 feet wide, are shown on PI. VIII. The coefficient curves are of uncertain form for heads below 0.6 foot. For greater heads they may be represented by inclined straight lines. The coefficients increase uniformly with the head, the initial values for 0.6 foot head being nearly the same for the several slopes, the increase being more rapid for the steeper downstream slopes.

It may be seen from the following table that increased width of the flat crest, as compared with that of the preceding weir, causes a decrease in the discharge.

Comparative coefficients at 1-foot head, weirs with flat crests and \:1 upstream slope.

Slope of apron.

Crest width, in feet.

0.66

1.317

1:1

3. 52

2:1

3. 38

2. 985

3:1

4:1

5:1

3. 265 3. 205 3.195

2. 94

6:1

2. 93

COMBINATION OF COEFFICIENTS FOR WEIRS WITH COMPOUND

SLOPES.

Series 163 for an apron slope 2:1 represents a weir form which would be produced by placing, vertical face to vertical face, a weir with back slope 1:1 and a weir'with apron slope 2:1. For the former, Bazin’s experiments indicate 10 per cent excess discharge over that for a thin-edged weir, and for the latter (from PI. V) (7=3.50, equivalent to 5

per cent excess over a thin-edged weir. If the discharge over the 1:1 upstream slope was similarly increased by the addition of an apron, C would be 3.66X1.05 = 3.84. PI. IX indicates C— 3.82.

The above method of determining the coefficient for weir of irregular cross section by combining the coefficients for two principal elements of which it is composed, as separate weirs, is restricted in its application and may lead to inconsistencies.

WEIRS WITH VARYING SLOPE OF UPSTREAM FACE.

Experiments were made by Bazin on thin-edged weirs inclined at various angles. Bazin found the ratio of the coefficient of discharge to that for a vertical thin-edged weir to be sensibly constant for all heads within the limits of his experiments, 0.0 to 1.5 feet. Bazin's results were expressed in the form of a modulus by which to multiply the coefficient for a vertical weir to obtain that for an inclined weir. Assuming the Francis coefficient 3.33 to apply to a vertical weir, the coefficients for weirs of various inclinations would be as follows:

Coefficients for inclined weirs, Bazin’s experiments.

Bazin’s

modulus.

1 horizontal to 1 vertical___

Upstream inclination of the weir.... 2 horizontal to 3 vertical...

1 horizontal to 3 vertical...

0. 93

.94

.96

Vertical weir

1.00

Downstream inclination of the weir..

1    horizontal to 3 vertical.

2    horizontal to 3 vertical... • 1 horizontal to 1 vertical...

2 horizontal to 1 vertical... 4 horizontal to 1 vertical...

1.04

1.07

1.10

1.12

1.09

3. 097 3.130 3.197 3.330 3. 463 3. 563 3. 663 3. 996 3. 630

On PI. XVI are shown the results of United States Deep Waterways experiments on weirs 4.9 feet high, having horizontal crests 0.67 foot broad, and with various inclinations of the upstream slope. The experiments cover heads from 1.75 to 5.2 feet, but only 3 or 4 points are given on each coefficient curve. The results indicate in a general way, however, nearly constant coefficients for each inclination of the upstream face. The values of the coefficients are considerably smaller than those obtained by Bazin, whose experiments were on weirs 2.46 feet high with sharp crests.

Pis. X, XI, and XII show the results of experiments of Bazin on weirs of irregular section, with various upstream slopes. PI. X includes 5 series of experiments on weirs 1.64 feet high, with sharp

crest angles, and 2 : 1 downstream slopes. The coefficient curves show a depression period at from 0.3 to 0.7 foot head, beyond which the coefficients may be fairly represented by constants up to 1.5 foot head (the limit of the experiment). A general curve showing the constant coefficient in terms of a downstream slope or batter has been added. This indicates a maximum coefficient of discharge for an upstream slope of about 2.6 : 1. Bazin found, for thin-edged weirs, with inclined downstream slopes, a maximum coefficient for an inclination of 30°, or If : 1.

Pis. XI and XII show coefficient curves for weirs having the same upstream slopes as in PI. X, but 2.16 feet high, and with Hat crests 0.67 foot wide. The coefficient curves are convex outward, indicating that they may approach constant values at some point beyond the limits of the experiments. The marked difference in character of these coefficient curves, as compared with those in the preceding group, is notable. For weirs with flat crests 0.67 foot wide the coefficients for a given head uniformly increase as the slope becomes flatter up to a batter of about If : 1. They are also greater for all heads within the limit of the experiments than the coefficients for weirs with sharp crest angles. The comparative values are indicated in the following table:

Comparative coefficients, weirs with varying upstream slope.

Pl.X,

sharp crest, IT | 2:1 downstream stream slope.

Pis. XI and XII, 0.67 feet crest width, 2 :1 downstream slope.

slope; average con- | stant coef- [ ficient.

Head, in feet.

0.5

1.0

1.5

Vert.

hi I

2. 87

3.58 i 2.78

3. 68

3. 26 3. 51 3. 34 3. 56

3. 72

2.92

3. 38

3. 62

1:1

3. 83

3. 03

3. 42

3. 65

2:1

3.87

3.13

3. 43

3. 61

It will be seen that the addition of the flat crest has an effect in this case similar to that observed in Pis. VI and VIII, showing the results of experiments by Bazin on weirs with various downstream slopes.

United States Deep Waterwaj^s series No. 7, PI. XVII, may be compared with Bazin’s series No. 178, shown on PL XI. The former gives a coefficient of 3.55 for a head of 2 feet on a wTeir 1.895 feet high, the coefficient slowly increasing with the head. The latter gives a coefficient of 3.6 for a head of 1.5 feet, decreasing rapidly as the head decreases.

JBE 150—06-13

United States Geological Survey series No. XXXIX, PI. XXVIII, and United States Deep Waterways series No. 18, PI. XVIII, represent weirs with vertical downstream faces and inclined crests. The upstream slope does not, however, extend back to the bottom of the channel of approach, but is cut off abruptly b}T a vertical upstream face. The average coefficients deduced from these series have been plotted on a general curve on PI. XVI, the coefficients agreeing closely with those of the United States Deep Waterways experiments on weirs of similar upstream slope, extending to the channel bottom. United States Geological Survey series No. XXX represents the Dolge-ville dam, with rounded crest removed, leaving a trapezoid with crest 6 feet broad and 1 foot lower at upstream than at downstream edge. The coefficient is not constant, but apparently approaches a constant value of about 3.25 for heads exceeding 3 feet. United States Deep Waterways series No. 18 represents a model of the spillwa}r of the Indian Lake dam, having a crest 7 feet wide, 1.5 feet lower at upstream than at downstream edge, which gives an average constant coefficient of 3.42.

It is suggested that if the upstream slope of an inclined weir is continued back 6 feet or more and terminates in a vertical upstream face, the discharge coefficient will not differ materially from that for an upstream slope extending to the channel bottom.

DAMS OF OGEE CROSS SECTION, PLATTSBURG-CHAMBLY TYPE.

The United States Geological Survey experiments on dams of this t}rpe are shown on Pis. XXIII to XXVII. Cross sections of the various dams, with lines indicating the comparative size of the models used in the United States Geological Survey experiments, are shown on PI. XXXIV. Cross sections of other ogee dams used as weirs are shown on PI. XXXV.

This class includes dams with downstream crest radius sufficiently large to retain the nappe always in contact, yet not so large as to simulate a broad flat crest. We thus exclude-the Dolgeville section on the one hand, in which the nappe as observed in the existing dam partially or completely breaks free near the crest for other than very low stages, and on the other hand, the Austin dam, with a crest radius of 20 feet, which appears, from the meager data available, to lie outside this class.

We have arranged the available data in order, advancing with decreased breadth and increased inclination of sloping upstream face.

The coefficients for various depths are as follows:

!.0~

A

-JP-6Pi

— 14' —

0.75

rf-3^

|0‘

P/

Base of 'spop'e!

Base of modeA

PLATTSBURG,N.Y.

DOLGEV/LLE, N. Y.

(B)

r--

---go' —

il

Crest Austin dam

*-4.M

i <\j

i"

I*

(E)

/8.0-

Bjsse_ofmo defy

Smooth stone —

R=8

nTT

h\⁴-'⁴'

CHAMBL Y, P. Q.

<_

- 60 '

~V?%rn

<f77,

AUSTIN\ TEX.

66'---

(C)

(D)

COMPARATIVE SIZE OF MODELS AND SECTIONS OF OGEE DAiyiS.

/s'

■<s"

I.ol

''r

----23.’a"---

BEA VER RIVER, N. Y.

-R

HANN AW A FALLS, N.Y.

EL 1019.o'

T/? - /.O' \ ,

!v>

1010 \

/s.o'-

Crest, Trenton Fa/is

/000\

20.5

990

^■26.5/ -

990 k-

O--

El. 968'.

ll

R^/.Oi

0.75

k--—

ss'

///

S''

TRENTON FALLS, N. Y.

t/onk Falls ,N. Y.

Crest section

CROSS SECTIONS OF OGEE DAMS.

'<

%:e±

b\\

Note — Crest rad/us must be such that nappe adheres to face of dam Crest uv/dth B must be at /east 3 ft

0/ 6

C = [ 3.62 -./6(S-D ] H Example S = 2 I H - 40 C =3.46 X 4yz°

=. 03/03

C = 3.708 computed C =from experiments 3.70

Zo

= / 07/6

Zzo

bog 4

•O

Experimental data

Crest s/ope

4:/

2:/

/.04:/

4.5:/

Head- feet

P/attsburg (Peg/

" (Spec/

//

Chamb/y

COEFFICIENT DIAGRAM FOR OGEE DAMS.

Symbol

•>

Series

U. S. G. S. 30-3/

32-33

rO

CO

r5

Coeffi

cient

ciei

C.

4.0\

I

\

/

Comparative coefficients, dams of ogee cross section.

Dam

Cham-

bly.

Platts- I burg.

Modified Plattsburg.

Approx, constant coefficient ...................

Breadth of slope, in feet.

Batter of slope...........

Crest radius, in feet......

Experiment..............

Series....................

3.48

3.48

3.48

3.70

3.70

3.70

3.435

4.5    3    13    3    13    3    3    13

4J:1    4:1    4:1    4:1    '    2:1    2:1    2:1    1.04:1

2.0    S 3.0    [    3.0    3.0    (    3.0    3.0    3.0    j 3.0

.......!U. S. G.S.jU. S. G.S. Mean of ]U. S. G.S. U. S. G.S.¹ Mean of |U. S. G.S.

a 33    | 32-33

a 30

6 31

::n '■ I

6 32

Head 0. 5 foot......

1.0 foot......

2.    0 feet......

3.    0 feet......

4.    0 feet......

5.    0 feet......

6.    0 feet......

3. 30

3. 42

3. 49

| 3.22 |......

3.43    3.29

3. 42

3. 47

3.53 ; 3.52

3. 35

3. 43

3.54

3. 62

3.36

3. 385

3. 45

3. 53

3.29 ! 3.22 J 3.255 3.37 | 3.44    3.405

3.51 | 3.67 ! 3.59

3.46

3.75

3.57 j 3.72 3.645 ; 3.87 3. 67 * 3. 74 3. 705    3. 88

3. 73

a 15.969 feet crest length, without end contraction. 6 7.979 feet crest length, one end contraction.

It appears that the rounded crest changes the character of the law of coefficient from a value tending toward a constant for each back slope to a slowly increasing function of the head. Compared with the constant coefficients for weirs with similar upstream slopes extending back to canal bottom, and with vertical faces, we find that the constant values deduced for these cases correspond with the values of the varying coefficients for ogee sections at a medium head of 2 to 4 feet.

slope,

By plotting the data for weirs of ogee section on logarithmic cross-section paper the following convenient approximate formula has been deduced, applicable for weirs with 2 or 3 feet crest radius and upstream slopes 3 to 4.5 feet broad. S indicates the batter ratio of the horizontal run vertical rise

If £=2:1

C— [3.62—0.lfi (/S'—1) | R <7=3.46x4*°'

.    . (71)

//= 4.0

log 4** = 0.030103 The experiments give (7=3.74.

(7=3.46 X 1.0716 = 3.70.

EXPERIMENTS ON DISCHARGE OVER ACTUAL DAMS.

On PI. XXXVII are shown the results of a number of experiments made b}^ measuring the discharge over existing dams b}⁷ means of floats or current meters. Aside from those for the Austin, Tex., dam, the data have been collected by Mr. George T. NellesA

a Discussion of paper by G. W. Rafter on the flow of water over dams: Trans. Am. Soc. C. E., vol. 44, pp. 359-362.

BLACKSTONE RIVER AT ALBION, MASS.

This is a timber dam 217 feet long, with horizontal crest 1 foot wide, vertical downstream face, and upstream slope covered with riprap. Discharge was measured by current meter 500 feet below dam, and the depth was measured by hook gage 20 feet upstream from crest. Coefficients have not been corrected to eliminate velocity of approach. They illustrate the uncertainty of discharge for broad-crested weirs of small width under low heads.

MUSKINGUM RIVER, OHIO.

Discharge was measured by rod floats in a cross section 500 feet above the dams, which are constructed of timber cribs tilled with stone. Data by Maj. W. H. Bixby, U. S. Army.

Discharge data for Muskingum River dams.

Number of dam.

Length on crest, j in feet.

I Area of Dis-Mean | discharge charge, height, injsection, in in cubic feet. square feet per I feet. ; second.

Mean velocity, n feet pei second.

I MS |

feet.

feet.

Coeffi- | cient C. i

CO -h t - CC

12.6: 7,765    18,118

15.9 ! 8,360    25,559

14.2    8,230    21,015

515    16.0    7,330    22,310

848

535

472

2.    333

3.    045 2. 553 3.044

8. 00

6. 70

7. 00

2. 86

4. 66

4. 40

5.16    5. 90

4. 419 4. 723 4. 812 3. 015

The depth on crest has not been corrected to eliminate velocity of approach.

OTTAWA RIVER DAM, CANADA.

Data by T. C. Clark, C. E. Dam 30 feet high, with upstream and downstream faces planked and sloping’ 3:1, forming sharp crest angle at junction.

Discharge data for Ottawa River dam.

Length of

Depth on

Discharge, in cubic feet per second.

Discharge coefficient C.

1,600

1,760

2.5

10.0

26,000 190, 000

4. 106 3. 408

These data are notable as giving the only authentic value of discharge over a dam under so great a head as 10 feet. The high coefficient found for a head of 2.5 feet renders the results somewhat doubtful.

U. Si. GEOLOGICAL SURVEY

WATER-SUPPLY PAPEfi NO. 150 PL. XXXVlf

Coeffi

cient

c.

3.40

\\'

V?

\°-

R = 20'

3.20

Section of crest

3.00

.2

.4

.6

.8

Corrected head H in feet.

1.0    1.2    1.4    1.6

TAYLOR-HOWARD EXPERIMENTS ON DAM

AT AUSTIN, TEX.

Coeffi

cient

er

4.00

AO'-4

3.80

V H.

3.60

&)

kr

-2

3.40

wv

.2

.4

.6

.8

1.0

1.2

Corrected head H in feet.

EXPERIMENTS OF DWIGHT PORTER, BLACKSTONE RIVER, ALBION, MASS.

IT %

V.

I'

ii

- - /B. 6- /6 -

Timber cr/bs . stone ri//eo

Coeffi

cient

ei

5.00

--0.’

4.50

15--

4.00

3.50

3.00

3.0

4.0

5.0

6.0

Corrected head H in feet.

MUSKINGUM RIVER DAM.

DATA BY MAJ. W. H. BIXBY, U. S. A.

EXPERIMENTS TO DETERMINE COEFFICIENT C FROM ACTUAL DAMS.

CROTON DAM, ROUND-CREST SECTION, MODEL A (PL. XIX).

Crest rounded, radius 10 feet. Upstream slope about 6:1.

Comparative coefficients with varying roughness, Croton round crest.

la

Series

Head, in feet.

Smooth-pine

crest.

Unplaned-plank crest and slope.«

Broken- ! Broken-stone slope, | stone slope, unplaned ) wire cloth crest.

on crest.

3.16

0. 25

3. 34

2. 84

3.18

.50

3.24

2. 91

3. 18

3.09

1.00

3. 21

3. 04

3.19

3.15

1.50

3. 21

3.12

3.20

3. 15

3. 21    3. 15

3. 22 !    3.15

2.00

2. 50

3. 21

3. 21

3. 15

3.15

3. 00

3. 21

3. 22

3.15

a This series appears doubtful.—R. E. H.

CROTON DAM, ANGULAR SECTION, MODEL B (PL. XX).

Apron slope 1.25:1, upstream slope 6.24:1 for 13 feet, then rough and slope about 4:1 to bottom.

Comparative coefficients, varying roughness, Croton angular crest.

Unplaned

plank.

Series.

Head.

Unplaned

plank,

rough-stone

approach.

Rough-stone approach, wire cloth on crest.

0. 25

3. 61

3,56

.50

3. 63

3. 66

3. 57

1.00

1.50

3. 67 3. 68

3. 66

3. 66

3. 58

3. 60

2. 00

3. 70    3. 66

3. 61

2. 50 |    3. 70

3. 66    3. 62

The data given above are somewhat discordant, but indicate that in general the decrease in discharge resulting from the roughness of the various materials forming the crests and approaches of dams will not exceed from 1 to 2 per cent for low heads, and usually decreases as the depth of overflow increases,

FALLS.

Bellasis® presents the following analysis for a fall in which there is neither a raised weir nor a lateral reduction in section. If v is the mean velocit}⁷ at CD, near to AB, then v is both the velocity of approach and the velocity in the weir formula

CM ICO

C7>

tc

ft

A,

<o

‘/////.sy/////////////////////////////////////.

Fig. 10.—Fall.

where c is a coefflcient of velocity.

V² —

^ cl 2yj9-f-y ac²v²

(^1 — I ac2) v²-(jl)

Wl tC

c^2gD

v= —

1-

ZD I

ft

Making a-=1.00 and c=0.79.

=4.74 V/>

(72)

_ 3.424 VX> ^“Vl-0.277

Q=vDL=4.74 LD

Tbe depth X>is to be measured so near ALA? that the water shall have acquired its velocity of efflux. The depth will, of course, be affected by the surface curve, the upstream extension of which will be longer according as the slope of the leading channel is flatter, being very great for a horizontal channel. The formula needs experimental verification, but affords a convenient basis of approximation of the flow through troughs and sluices and over aprons and falls.

Experimental data for c are needed.

a Hydraulics, p. 99.

WEIR CURVED IN PLAN.

Milldams of both wood and masonry are often constructed to bow upstream, sometimes to secure the added strength of arched form, or to secure additional spillway length, or to follow the crest of a favorable rock ledge, or to throw the ice-bearing current away from intake gates. The dam may follow the arc of a circle, or, as is common with timber dams, there may be an abrupt angle in the plan of the dam. Fig. 11 shows a graphical comparison of curved and angle dams with a straight dam across the same channel, the former being each 13.5 per cent longer than the straight crested dam.

If such an arched spillway opens out of a broad, deep pond, the discharge over it would be greater than for a straight overfall very nearly in proportion to the excess in length of the arc as compared with the length of its chord.

When the stream is confined in a restricted channel, the increased velocity of approach above the longer spillway will become a factor. Thus if two dams—one straight, the other arched—were placed in the same straight, uniform channel, and the depth on crest measured at the same distance upstream from each, then, with the same measured head on both, the velocit}^ of approach to the arched dam would be

Fig. 11.—Weir curved or angular in plan.

greater nearly in the same proportion that its length of crest and discharge are greater than for the straight crest. Properly corrected for velocity of approach, the arched dam will give a correct measurement of the discharge, the length of the arc being used as the crest length. When the length of the arc greatly exceeds the channel width, the velocity of approach may become excessive, introducing uncertainty as to the proper correction coefficient, difficulty in measuring the head, and an uplifting of the central swifter-flowing portion of the stream surface.

The circular overflow lip of a vertical artesian-well casing’ is sometimes used to approximate the flow, the measured depth of water above the lip of the pipe, together with its circumference, being used in the weir formula.a a Experiments showing the discharge over a circular weir to be proportional to the length of the arc were made by Simpson at Chew Magna, Somersetshire, England, 1850, not recorded in detail.

SUBMERGED WEIRS.

THEORETICAL FORMULA.

In a “submerged,” “drowned,” “incomplete,” or “partial” weir the water on the downstream side stands above the crest level.

The submerged weir is not extensively used as a device for stream gaging. A knowledge of the relations of head, rise, and discharge of such weirs is, however, of great importance in works of river improvement, canals, etc,, and the leading formulas are here presented.

It may be added that for situations where head can not be sacrificed, precluding the use of an ordinary weir, and where the velocity is not a continuous function of the depth, as in race ways, making a channel-rating curve inapplicable, the use of submerged weirs to measure or control the discharge merits consideration. Their use for such purposes as the equable division and distribution of water in power canals has hitherto been very restricted, owing to the lack of experimental coefficients.

Let Af=Head on upstream side, corrected for velocity of approach.

X>=Measured head, upstream side of weir.

d = Measured head, downstream side of weir, or the depth of drowning, taken below the ressault.

Z = Dift'erence of elevation, upstream and downstream sides

=71—d.

P = Height of weir above channel bottom.

L = Length of weir crest, feet.

^=Mean velocity of approach.

A = Head on a thin-edged weir that would give the same discharge.

M' and O' coefficients of discharge for a submerged weir.

I«N

I*"* * t)

Pig. 12.—Submerged weir.

The theoretical formula of Dubuat for discharge is obtained by regarding the overflow as composed of two portions, one through the upper part D—d, treated as free discharge, the other through the lower part d, treated as flow through a submerged orifice.

Combining the two discharges,

Q— (?i+ $2—3 4^gL{D—dy+Ld^]L2ff(D—d)

irk 150—06-14

By reducing⁻, including a coefficient, and using the head // corrected for velocit}" of approach, we have the general formula for a submerged weir.

The head due to the velocity of retreat should m strictness be subtracted from the depth of submergence d. This is not commonly done, however, in the experiments, where the usual method of producing the submergence is by damming and retarding the water below. In practice, if the velocity of retreat is large, the correction should be made.

The theory of formula (73) makes C —1.5 times the value of the coefficient C in the free portion of the discharge.® This value is adopted by Dubuat and Weisbach.

D’Aubuisson gives C = 1.43 C.

Francis’s early experiments make C = 1.38 C.

From gage records of large rock-tilled crib dams on Kentucky River, having planked upstream slope 3:1 and vertical steps below⁻crest—height of dams about 20 feet, heads 4 to 7.5 feet, mean 5.3 feet—Nelles found results as"follows:

6" = 1.5 C. 6⁷' = 1.53C.

Dam No. 3, water falling slowly 4 days,

Dam No. 2, water falling slowly 3 days,

Dam No. 1, water rising and falling slowly 5 days, C' —1.46 6*1

FTELEY AND STEARNS SUBMERGED-WEIR FORMULA. &

Fteley and Stearns use the base formula

.    .    (74)

q=gl(h+ . .

Coefficients for the above formula were derived from experiments on thin-edged weirs, by Fteley and Stearns and by J. B. Francis, and give correct results for weirs for which the free discharge would be correctly calculated by the Francis formula.

The head on upstream side varied from 0.3251 to 0.9704 foot, and

d

-jj varied from —0.063 to 0.081 with air under nappe, and from 0.077

to 0.975 with no air under nappe, and in appl}Ting the formula the same conditions should be complied writh. The authors comment that where sufficient head can not be obtained for a weir of the usual free-discharge type, a submerged weir may be used, provided that the head does not vary greatly.

a See valuable discussion of submerged weirs by Geo. T. Nelles in Trans. Am. Soc. C. E., vol. 44, pp. 359-383.

b Fteley and Stearns, Experiments on the flow of water, etc.: Trans. Am. Soc. C. E., vol. 12, pp. 101-108.

From a large-scale curve Fteley and Stearns derive the following table of coefficient C\ for formula (74):

Fteley and Stearns’s coefficients for submerged ~weirs.

0-00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

.1

.2

.3

.4

.5

.6

.8

.9

3. 365 3. 286 3.214 3.155 3.113 3.092 3. 092 3. 122 3.190

3. 330 3. 359 3. 278 3.2C7 3.150 3.110 3. 091 3.093 3.127 3. 200

3. 331 3. 352 3.271 3. 201 3. 145 3. 107 3.090 3. 095 3.131 3. 209

3. 335 3.343 3. 264 3.194 3. 140 3. 104 3.090 3. 097 3.137 3.221

3.343 : 3.360 3.335 i 3.327 3.256 | 3.249 3.188    3.182

d •

Where jj is less than 0.15 Q is not sensibly affected by submergence.

Where ~ is from 0.5 to 0.8 C may be taken at 3.10.

Correction for velocity of approach was made by the formula

H=D+

No correction was made for velocity of retreat.

The formula is probably applicable to larger dams and greater depths

by selecting proper values of C\ cjj being a relative quantity.

A number of empirical formulae for submerged-weir discharge are also used.

CLEMENS HERSCHEL’S FORMULA.a

Herschel’s formula, based on experiments of J. B. Francis, 1848, Fteley and Stearns, 1877, and J. B. Francis, 1883, is

.    .    (75)

$=3.33 L{NHf=3.33 Za¹ .    .

In this formula the measured head⁶ is reduced to an equivalent'head that would give the same discharge over a free overflow. The value of

/\

the coefficient W=j^ depends on the proportional submergence jj-

n Hershel, Clemen, The problem of the submerged weir: Trans. Am. Soc. C. E., vol. 14, May, 1885, pp. 190-196.

b Corrected for velocity of approach by method for Francis’s formula before applying in above formula.

The values of this ratio, together with their probable error, are given below.

Coefficient N, HerscheVs submerged-weir formula.a

tola.

o.o

iO.Ol

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0

SOOMOlOli^COtO

.574    .557    .539    .520    .498    ' .471

« Values for ~ exceeding 0.80 less accurately determined.

-^=0.02 to 0.14, variation of N— ±0.005 to 0.007.

=0.15 to 0.22, variation of JV= ±0.008 to 0.010.

= 0.24 to 0,32, variation of .2^=4=0.012 to 0.014.

= 0.33 to 0.41, variation of .N= ±0.015 to 0.017.

= 0.42 to 0.59, variation of 2V= ±0.018.

= 0.60 to 0.65, variation of N= ±0.017 to 0.015.

= 0.66 to .071, variation of JV= ±0.014 to 0.012.

= 0.72 to .084, variation of JV= ±0.011 to 0.009.

This table indicates that for depths of submergence not exceeding 20 per cent, the head will not ordinarily be increased more than 2 per cent.

The discharge over a submerged weir, according to Herschel’s formula, bears the ratio N ~ to that over an unsubmerged weir under the same head.

THE CHANOINE AND MARY FORMULA.

.    . (76)

Q=M'LH^gZ .    .

This expression has a form similar to that for the ordinary formula for submerged orifices. It is applicable only under conditions identical with those for which M' has been determined. a

a Van Nostrand’s Eng. Mag., vol. 34, p. 176.

R. H. RHIND’S FORMULA.«

Q—3\PL'J’&g d*]Z-\-0.01P+g ^^+0.035'?»²^j . . (77)

This may be reduced to the theoretical formula (73) by omitting the correction for velocity of approach.

BAZIN’S FORMULAS. &

By duplicating, with various depths of submergence, his experiments on thin-edged weirs Bazin deduced the following expressions for the coefficients for submerged weirs to be applied in the discharge formula

Q~m! LD ^]‘2gD.

Let Prepresent, as heretofore, the height of weir crest above channel bottom, the coefficient m being that which would apply to the same weir with free discharge.

(1)    Accurate formula with small values of d:

(78)

m!—m

1.06+0.0.05

(2)    Accurate formula with large values of d:

m!—m

1.08+0.18 y)    •    •

• ■ (79)

(3) Approximate formula for all cases:

m! = m

(^1.05 +0.21

³lD-d

~D

(80)

The above formulas are for weirs without end contractions.

The coefficient m contains the correction for velocity of approach of the free-discharge weir, and m! contains the necessary factor (if any) for the resulting modification of the velocity of approach effect, when the weir becomes drowned. Theyr are only strictly accurate, therefore, when m' is substituted for m in Bazin’s formula.

In Bazin’s formulas the height P of the weir enters as a controlling factor in (1), and is present less prominently in (2) and (3).

The modification by drowning is made to depend on-p in (2), and

D-d    ¹

jointly in formula (3).

on this ratio and that of the cube root of

It is often difficult to determine P or to apply these formulas to a weir fed by a large pond and having end contractions.

a Proc. Inst. Civil Engineers, 1886.

6 Bazin, H., Experiences nouvelles sur l’6coulement en deversoir, 6me art., Ann. Fonts et Chauss6es, Memoires et Documents, 1898.

WEIR experiments, coefficients and formulas.

142

Assume P—oc Then (2) becomes

a-mX 1.08

T)

=mX 1.08

• • (81)

and differs from (3) when similarly reduced only in the substitution of 1.05 for 1.08 as a coefficient.

Ex.

D=4'

d=2f

P— 00

If

m=0.425

m' = 0.425x1.08

.4-! tc

= 0.364,

the discharge being 89.4 per cent of that over an unsubmerged weir under the same head.

Comparison of submerged-weir formulas.a

.75

2.0

d, feet, if, feet

am

Percentage of unsubmerged-weir discharge.

Fteley-Stearns .. j 99.91 Herschel Bazin (3)

100.15 100.43

95.06 95.83 95.40

89. 29 90.56 89.78

95.01 95.83 95.40

82.61 84.24 83.34

64.02

64.95

66.15

a Weir assumed to be very high so that there is no velocity of approach or of retreat. The coefficient of discharge for a thin-edged weir with free discharge has been taken at 3.33 for the Fteley-Stearns and Herschel formulas.

INCREASE OF HEAD BY SUBMERGED WEIRS.

Any of the submerged-weir formulas may be transformed into expressions giving the rise in water level caused by the construction of a submerged weir in a channel or canal; in this form they are most useful in the design of slack-water navigation works.

rankine’s formulas."

Weir not drowned, with flat or slightly rounded crest:

//= A =

³/Q2 V D

-1

, approximate .    .

• • (82)

Weir drowned:

First approximation— //' = A+J

Second approximation-

H=JP-d(x-\ „

a Civil Engineering, p. CS9.

COLONEL DYAS’S FORMULA A

This is intended to determine the height of a weir on the crest of a fall in an irrigation or other canal to maintain a desired uniform depth and slope.

D—Depth on weir, feet.

X= Depth of uniform channel, feet. P— X— D—Height of weir necessary. A = Area uniform channel section, feet. R— Hydraulic radius, feet.

S— Slope or fall in feet, per foot.

L — Length of weir crest, feet.

to

Mjh-

K5

04

00

CM

1C

<>

4-

If A = 1000 X=10' R- 8.33 ff= 0.001 L=100,

7) =

900 X 1000² X 8.38 X 0.001

10000

—125.81 x 8.33x0.001

= 9.0856-1.0441 = 8.04

P= 10-8.04= 1.96 feet.

In this case length, of weir equals width of channel, and the velocity of approach would be the mean velocity, which by Kutter’s formula will vary, say, from 8 to 10 feet per second under the conditions, depending on the value of the coefficient of roughness n. This would make the flow in the channel 8,000 to 10,000 cubic feet per second.

As a check on the calculated depth Z>, it will be found that the flow over a weir 100 feet long under a head 8.04 feet (corrected for the large velocity of approach) will also be from, say, 8,000 to 10,000 cubic feet per second, depending upon the coefficient used in the weir formula.

SUBMERGED WEIRS OF IRREGULAR SECTION.

For certain forms of irregular weirs having vertical downstream faces, the discharge when subject to submergence may probably be * approximated by applying the ratio of drowned to free discharge for a thin-edged weir similarly submerged as a correction to the coefficient for free discharge over the weir in question. For broad-crested weirs or weirs with aprons this method probably will not be applicable.

bazin’s experiments.

For many of the model weirs of irregular section for which free-discharge coefficients were obtained by Bazin, duplicate series of coefficients with various degrees of submergence were also obtained.

a Wilson, H. M.,'Irrigation in India: Twelfth Ann. Rept. U. S. Geol. Survey, 1890-91, pt. 2, p. 482.

Many of these data have been reduced to English units by Nelles.3 Evidently each form of weir section will require a special formula or table of coefficients, and little more can be done than to refer to the original data for each specific case;

By way of general illustration of the character of submergence effect on weirs of irregular section, the writer has deduced the following roughly approximate formulas from Bazin’s experiments on triangular weirs with vertical upstream faces and sloping aprons. The weirs were 2.46 feet high and the end contractions were suppressed. Coefficient curves for free discharge are given on PI. V.

Three series are included:

Series 195, batter of face 1:1.

Series 196, batter of face 2:1.

Series 197, batter of face 5 :1.

Experiments in which the proportional submergence was nearly the same were grouped, and the average values of A, D, and d were

determined. From these the mean values of

and

were computed

and platted and a straight-line formula deduced.

(85)

(86)

£=0.08+0.17 B.

The initial effect occurs when

d 0.17^-0.20 D~ 17^+0.08

In the above formulas A is the measured head on a weir with free overflow, having the same form of cross section, that would give the same discharge. D is the depth on the submerged weir, d is the depth of submergence, a?hd B is the batter or slope of the apron.

DATA CONCERNING EAST INDIAN WEIRS.

The following data compiled by Nelles* are derived from observations on actual dams under heads unusually great. The calculated coefficients in the ordinary weir formula (a)

in the theoretical submerged-weir formula (£)

Q=M'L^gz(d +1Z^,

and in the Rhind formula (77) are given in columns 14, 13, and 12, respectively (p. 145), the observed head being corrected for velocity of approach.

i> Loc. cit.

a Trans. Am. Soc. C. E., vol. 44, pp. 359-383.

SUBMERGED WEIRS.

I—*

tO

Approximate section of weir.

15

Fig. 13.

Fig. 14.

>Same as Fig. 13.

•OCI

Do.

Do.

Coefficients of discharge in the various submerged-flow formulas.

Formula! Formula b, ob- a, observed served head. head.

Free

weir.

Sub

merged

weir.

Rhind’s

(77).

II

0.241

0.491

0.511

0.266 0.245

0.261 0.355

0. 265 0.274

0. 244 0. 260

13

12

0.806

0.940

0.970

0.790

0.786

0.764 0.763

0. 561 0.597

0. 570 0.677

0.881

0.953

0.979

0.878 0.876

0.893 0.891

0. 595 0.633

0. 594 0. 648

Data of submerged flow at certain large masonry dams in India.

of main

Calculated

Corre-

spond-

Mean

surface

veloc-

?!

ity of

Observed

feet.

second.

feet.

ap-

fall over

o>

114,000

%i<s>

96'0

Os

os

iO

i>*

10.65

6.0

10

Total

o d tN n

8.51

body alone

260,000

307,500

680,000

787,000

488,000

609,000

82,800

112,700

0. 76

2.10

" a;;

0.68

g'S

a-S

weir, feet.

per

second.

s>

1) — (l = Z

GO

7. si

0.86

2.81

1.13

6.99

11. 62

7.74

6. 07.

3.40

0.47

0.525

2.00

1.20

CG U 01 OJ

*H +-»

« 3

6o

&S

surface in—

Lower

o~-S

oS

ft'-M

CO

iC tO tO os l> CO CO CO CO <N CO O

00 l> rH iO rfi 00 tH    rH

15.20

1.7*

7.9 1

17.55

7.55

10.05

30.15

25.09

19.59

15.20

9.45

12.10

8.10

7.10

s.

feet.

HHHOX^ <M C<J (M H CO 00

OS 00 <n* 00    ©

rH    rH    rH

18.60

8.18

11.34

18.02

■ 02

0.52

30.675

25.615

20.115

17.45

17. 20

11.45

13.30

4.30

- : .i i

Make-up of weir.

s &s

OOlOH^tCO^^CDOOHO O „-H COI^OOOOOOO CO O r—t H (M O H H OJ CC tH !>• O OS Ol CO Tf OS CO OS tH >0 H rH OS o rH rH lO CO rH rH O CO rr 00 rH rH CO CO 04 rH

CO rf

COCO

lOZD

1,607

1,980

Description of

parts.

=3

eo

& : : >03

'C 0) O >

^ o

111! Ill

CC

Piers

Piers

Main body

I

03

1«

CC

Piers

Main body

Sluices

Breaches

'O a) d

in

Center sluices

Main body

Total overfall

cc

03

■s .s

r/2

Piers

Total overfall ..

Name of weir.

Pattia

Byturnee

Burrah

Brahmini ...

Kajooree.

Manhanuddy.

Beropa

Name of river.

Brahmini

Byturnee

Do,

Brahmini

Manhanuddy..

Do

Do

UNITED STATES DEEP WATERWAYS EXPERIMENTS.a

These experiments were made in 1899 at Cornell University hydraulic laboratory on a model having completely rounded profile, being a design for a submerged dam for regulation of Lake Erie.

The coefficient curve for free discharge is given on PI. XVI. The absolute coefficients and the relative discharge with various degrees of submergence are shown below. The Francis formula is used.

Q= CLIP.

Absolute coefficients.

Submer-

D gence from ! backwater.

Feet.

Feet.

0. 0 0. 00

3. 70

.66

3. 67

.2

1.32

3. 64

.3    1.98

.4    2.64

3. 60

3. 54

.5

3.30

3. 47

.6

3. 99

3. 36

4. 62

3.17

.8

5.28

2. 88

.9

5.94

2. 30

Relative coefficients, United States Deep Waterways submerged-weir model.

■eta d

Ciio

1. 000

0.5

0. 937

.991

.6

.907

.983

.972

.8

.856 :    '

.778 !

.956

.9

.621

1.0

C is the coefficient for free discharge over a similar weir under the same head.

WE lit DISCHARGE UNDER VARYING HEAD.

Problems of weir discharge under varying head occur in the design of storage reservoirs for river regulation, and in determining the maximum discharge of streams.

't Kept. IT. S. Board of Engineers on Deep Waterways, pt. 1, p. 291.

An effort has been made in the present chapter to record the various working¹ formulas resulting from the solution of this mathematically difficult portion of the theory of the weir, and to give numerical data to facilitate calculations.

It is assumed that there is no velocity- of approach, or, if any, that the head has been corrected therefor. The weir coefficient is also assumed to continue constant through the range of variation of the head. Notation:

T =Time in seconds required for the head to change between two assigned values.

77^= Initial depth on weir, feet. dlt — Depth on weir at the time t.

S = Reservoir surface area, square feet.

L — Length of overflow weir. feet.

7 = Rate of inflow to reservoir, cubic feet per second.

Q = Rate of outflow at time t.

PRISMATIC RESERVOIR, NO INFLOW, TIME REQUIRED TO LOWER WATER SURFACE FROM H₍₎ TO H,.<f

clQ— C’LdPdt= - Sdll dt=-

CL1I

to: co

did

nil

dt=

-~CL

to, CO

~ d ll

ir₀

T=

2N

cl V V/4 - UL • • •

2 _

C— g J/i/2<7=5.35 M T— oo , when ddt — 0.

If N= 1,000,000, 7^ = 4,    7/^=0.1, C= 3.33, and 7 = 100,

Where

T=

2    3,000,000

100 X~ 10

2 =15,972 seconds=4.44 hours.

To lower the reservoir from 77=4 to 77=1 would require 3,000 seconds.

APPROXIMATE TIME OF LOWERING PRISMATIC OR NONPRIS-

MATIC RESERVOIR.

Choosing small successive values of II₀—ddt, we naiy solve this problem approximately, as shown in the following table:

Time required to lower reservoir from II₀ to ddt =

Mean Q

(88)

<t Des Ingenieurs Taschenbuch, I, 1902, p. 230.

We may take the mean discharge between the narrow limits II₀ and

Qm

.    .    (89)

or, using⁻ the average head,

Qm— CL

• .    (90)

In the following- example we have used the latter value, and have made H₀—IIt—0.5 foot. A similar solution may be made for a non-

prismatic reservoir, using successive values of    as the reservoir

area, and determining the increments of T by formula (88).

-    Example of varying discharge.

Ho

Ht

Avurage ! Qpersecond.

1000

T for increment

Ho-Hf.

Total T, in seconds.

4.0

3.5

3. 75

3.5

3.0

3.0

2.5

3. 25 2. 75

2.5

2.0

2.25

2.0

1.5

1. 75 !

1.5

1.0

1.25 i

2,417.0 | 0.4137 : 1,951.0 1, 519. 0 1,124.0

771.0    1. 2970

465.4    2.1500

.5126 j . 6580 | . 8970 i

207

207

256

463

793

330

The total time required in seconds is 2,961, as compared with 3,000 b}^ formula (87).

The time required, using the average Q instead of the average IIin the calculation, that is, using formula (89) instead of (90), is 2,933.5 seconds.

The time T is directly proportional to the area of storage surface and inversely proportional to the length of spillway. It is also usually proportional to the value of C in the weir formula.

RESERVOIR PRISMATIC, WITH UNIFORM INFLOW.«

GENERAL FORMULAS.'

Starting with reservoir full to crest level, JI₀~0, to find the time required for the depth of overflow to reach a given stage, II,.

a Mullins, Lieut. Gen. J., Irrigation Manual, Madras Govt., 1890, App. V, pp. 214-223.

When individual values of the increment II₂—J1X are small, not over 0.5 foot each, if successive values are taken, we have approximately:

(91)

j-Qi+Q*

• • (92)

£=time required to rise through the increment

A summation of the successive values of t required for the water to rise each increment will give the total time of rise from II₀ to Ht. Formula (92) will give the maximum run-off from a catchment area tributary to a reservoir if two successive values of AT and the corresponding value of t are known.

Formula (92) may also be used to determine T for a nonprismatic reservoir with a variable rate of inflow by choosing such increments, I1₂—I1₁, that the average values of S, I, and Q will be nearly correct. Variations in the weir coefficient Gmay also be considered.

FORMULAS FOR TIME OF RISE TO ANY HEAD H, PRISMATIC RESERVOIR WITH UNIFORM INFLOW.

Several analytical solutions of this problem have been made. Starting at spillway level, let Ha equal the depth of overflow corresponding to the quantity of inflow 1. The problem is stated by the following differential equation whose primitive is required:

(Rate inflow—rate outflow) dt—d (increase in storage), or

(93)

(I— CLIP) dt— SdH

In the solution, mathematical substitutions are necessary in order to render the time-outflow equation integrable in known forms. A very clear demonstration for a special value of C has been given by Frizell.® By modifying Frizell’s formula to adapt it to the use of any value of Cin the weir formula, the following equation is obtained:

3 CLb

2/S

T=nat. log

— V 3 tan

-1

bj 3

(94)

where b=

CL

When H—II(n the second member becomes the sum of an infinite and two finite quantities, T is then infinite, and the outflow can never

“Water Power, pp. 200-203.

become equal to the inflow, or //can never equal Uin which quantity it approaches as a limit as T increases. Frizell places H—rHm r having any value less than unity, and, being very nearly unity, Jr will be more nearly so, and is taken as equal to unity, without great error, enabling the two inverse trigonometric constants to be evaluated in terms of arc, giving finally^:

t=

nat. log

-\f i+V/'+/’

3( 6⁷²Z²/)

1 ~~Jr

0.88625

.    .    (95)

Nat. log .2^=2.302585 log₁₀ N

E. Ludlow Gould" gives the following formula, identical with the above except in the form of the constant of integration:

T=

2N

3( C²Z?1)

nat. log

Jl+jr+r 1 — Jr .

oil a

.    .    (96)

11

as before. Gould does not consider Jr constant, but derives

the values of the function in brackets for various values of r\ from which the following table has been derived:

<^= nat. log

h+Jr+r 1 — Jr

—J3 j tan

VT (

cr.¹ 4

. (97)

Values of cf>, Gould’s formula.

Ha I °

0. 0919 0. 1072 0. 1226:0. 13788 .2483, .2646    .2810 .2973

.4137:    .4319    .4501 .4683

.59571 .6139    .6321 .6534

.8024| .8237    .8450 .8663

1.0444 1.0705 1.09661.1128 1.3380 1.3733 1.40861.4439 1.7073 1.7590 1.81071.8624 2.3035: 2.3808 2.45822.5355 3.5441 4.0096 4.47504.9405

0.0    0.00000.0153    0.0306 0.0459 0.0613 0.0766

.4

.5    .8876    .9137    .9399    .9660    .9921    1.0183

.6    1.14891.1750    1.2012'    1.2322    1.2674    1.3027

.7    1.47921.5145    1.5498    1.5851    1.6203    1.6556

.8    1. 9141 1. 9658    2. 0176    2. 0715,    2.1488j    2. 2262

.9 ! 2.6129 2.7681 i 2.9233    3.0785    3.2347:    3.3889

a Engineering News, Dec. 5,1901, pp. 430-431.

PL. XXXVIII

1.9

WATER-SUPPLY PAPER NO. 150

2 xf (x)

XCOHeOrHCOHt>» GOOO(MCO 0(NOOTt<NOOON«OCOOj

oooHdco^iocoi^aio

1.8

sec x.

/(*) = (sin x)³

f(x)x

1.7

x arc.

0.05

.10

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

1.8

II

in

WiCH0505I>®>0'^M05iOt-i

OOnHMCO^f'iOtOl'*l>*COX

1.5

(?)

/(*)

a:=sin_l

see x

sec x dx

(sin a-y

0.562 |    1.0167.    0.5708

.669 ,    1. 0480    I    .701

. 740 I    1.094    8095

1.203 1.423 1.772 2.545

CO CO H r4 rH <N

880

914

946

974

1.3

a)e

6= j (sin x)’

Jo

Angle.

Are.

(°E\

Ah) h

VJL

^ ooo ^ © © o

TT< Tf iO ^ rH <N OQ

o 2SS

o o o o co o o o

GO C© SO *“H iC 1C ^ CO

0.1786 .3037 .4174

0.1777

299

4053

.750

YU

.871 ' 1.008 1.178

Note.—Values plotted on

l.a

VI

1.0

6817

7653

8459

9240

.9

.8

.7

.6

Corrected head H In feet.

DIAGRAM OF VARIABLE DISCHARGE.

^he-

*3

|bf ■

Coeffi

cient

c.

0'X

.9

«.

j.

“3

■s

«3

'

We may write formula (9)

(98)

(I)

R. S. Woodward suggests the formula*

T=

±s

3 ‘4o‘L‘i I

(sin XY sec XdX

.    .    99)

where

X— sin⁻¹ y

/ cl in

= sin_1

Ha

1 his, like the preceding expressions, becomes infinity when the

•    •    •    •    7t

integral is carried over the entire range X=0 to X=~, conforming, with the physical conditions.

II

Ha

The writer has evaluated this function for finite values of

by mechanical quadrature, as shown in the diagram, PL XXXVIII. The diagram illustrates the rapid rise until a head closely approaching IIa is attained, occupying’ a comparatively short time interval, while for further increments of head the time interval is relatively very great.

E. Sherman Gouldb gives the same integral developed as an infinite series

where

If S'

7=

K=

T=

2X7/

2 SII

i-H aq

W|co

+^+

_|------1

Rn

2+3,

. (100)

oc|

I S

.8 CO

x+

bC

a Engineering News, December 5, 1901, p. 431.

& Engineering News, November 14, 1901, pp. 362-363.

ibb 150—06-15

If we write

F=

2$

3 tl CP II I'

then Frizell’s formula may be written

T—

Fxp(jD

E. L. Gould’s formula may be written

T=Fx <f>

II

Ha.

II

Woodward’s formula may be written

E. Sherman Gould’s formula may be written T—Fx S

The formulas are therefore identical, the transcendental factors bearing the relation.

T— 2 Ftp

Ha.

The E. L. Gould, Woodward, and E. S. Gould formulas are appli-

That of Frizell can be strictly

II

cable for any value of the ratio yj.

** a

applied only when

XII \

S' yjjJ converges very slowly as the argument approaches unity. For rough calculations E. S. Gould gives the rule

77- T(/I(pI/y=8II

II

Ha

is nearly unity. In the E. S. Gould formula

where yu is the coefficient in the weir formula for reducing final head to mean head.

T—

SIT

/- OZ(Mll)

•    •    (103)

The ratio yn of the constant mean head which would give the total discharge SIIin the time 7⁷he finds by trial.

E. S. Gould gives the values

yu = 0.67 for small values of II to ju = 0.75 for large values of II.

Comparing the formulas,

Let S= 1,000,000 square feet

<7=3.33 = ¹⁰

Z=100

7=10,000 cubic feet per second 30^—9.655 feet.

II —(l    3 =

° \CL

Required the time to rise to a height 7/a=0.977= 8-6895 feet.

F-

2$

Henri

= 643.5

(95)    7 =1677.6 seconds.

(96)    7 = 1681.5 seconds. (99) 7 = 1660.2 seconds.

Frizell E. L. Gould W oodward

E. S. Gould (approximate) (103) 7 — 1488.3 seconds.

The difference in the value of T by the first three formulas represents the difference in the values of the transcendental portions of the equations as evaluated by different methods.

The time required to rise from IIa to Hb will be the difference of the times 71 and T₂ by the above formulas.

NONPRISMATIC RESERVOIR, UNIFORM INFLOW.

P. P. L. O’CONNELL."

Representing the reservoir by a cone having its apex at distance A, below plane of the overflow,

= S₀ = 7r(aA)

(104)

Area at overflow level

Area at any other level = S—7*[ol(A-\- 1I)Y

where oc is the slope of the sides, or where there is a foot horizontal run to 1 foot vertical rise. From (104) with S₀ and a given, A may be determined.

Where the factor

/l-V OF

7=

Tta'¹

7,³    -H

CL

IC

(2A7ta+A²)

37,

nat. log

R+IJH+I*

2 A²—4+7,²

37,

a,/3 tan

-V3 H 27,+ J7J

. . (105)

a Mullins’s Irrigation Manual.

E. L. GOULD.a

Calling i the angle of inclination of the banks, P₀ the perimeter, at spillway level, exclusive of overflow,

S — S₀-\-BH-\- Bx II² where B~ P₀ cot i

H

. — —- (S₀( CL)³ + B (ICL)³ nat. log

3ltC*L*\_y    '

(so( CLp -B(LCL)^) V3

Bx — n cot B

Vl + Vr+r

1 —Vr

tan

1+2-Jr

BXL

3    3

nat. log (1—+)++

—B{ICLf 4t

. (106)

For i=90° and B— 0, the above formula reduces to (96), the equation for a prismatic reservoir.

VARIABLE INFLOW, NONPRISMATIC RESERVOIR.

This problem may be solved by dividing the reservoir into successive levels, and solving by the formulas previously given, as if each layer represented a portion of a reservoir with a constant inflow equal to the average rate, or if the formulas for prismatic reservoir are used, then each layer will be supposed to represent a portion of a prismatic reservoir of area equal to the average area of the layer.

Mullins’s formula may often be more conveniently used and a better solution be obtained than by attempting to average the area and inflow, as would be necessary to apply the analytical formulas given.

The general differential equation for rise in time T with a variable inflow and reservoir area is

..    .    (107)

( I-Q) (IT—Belli . .

If we can express I as a function of T, and S and Q as functions of //, and integrate between the limits 77=0, 77= 77,, we may obtain an equation between II and Tsimilar to those given for prismatic reservoirs with constant inflow.

We may write the ordinary weir formula,

0= CL II*.

«Loc. cit.

The area S can usually be readily expressed in terms of the area at crest level and slope of the reservoir sides (assumed constant within the narrow limits 0, H)\ the inflow /often increases nearly as a linear function of Twhile a stream is rising rapidly; we have, then,

q, = cl in

st=s₀+2 a4snii+ a* ip It=I₀+fT.

Substituting in (107)

( T₀+fT- Gini)dT= (8₀+%atjS₀//+ II)dll .    (108)

The complete primitive of this differential equation can be determined only as an infinite series.®

Rivers during flood usually rise rapidly and fall slowly. The time-inflow function can sometimes be approximated by a modified sinusoid.

where n— or >1

I=l₀+Im sin (it)i*

(109)

7=Total duration of flood.

Im= Maximum rate of inflow.

7^=Time elapsed from beginning of rise to maximum.

The constants are so chosen that the arc value of the duration of the flood from stage I₀ through to the same stage is n, or,

(bt)n—7t

7ln

1)— rji

(110)

For the maximum we will have, differentiating (109),

or e>=

n—log

cos (bTJ«=0, .

,IS

i-!

I-S

IS

81

rH

bO

3.32204 log (jr) •    •

Tm log 2

common logarithms being used.

If T-1000 and if T⁷ =200, then *i=3.322 log 5 = 2.322,

(112)

^2.322

1000

=0.0143, and -=0.43066 ’ n

uSeddon, James A., C. E. (Proc. Am. Soc. C. E., vol. 24, June, 1898, pp. 559-598), has solved equation (108) for the Great Lakes reservoir system, assuming an annual cycle following the law I=Im+A sin T, Im being the mean inflow and Tthe time arc on a circle whose circumference represents- one year. He also assumes Q=Q₀+bH, or a linear function of the height if.

Exaviple of variable food discharge computed, by formula (109).

j t, in seconds.

(M)

log (M)

log (M)"

Angle.

sin (bt)n

100

1.43

200

2. 86

300    4.29

400    5.72

500

7.15

600

8. 58

0.155336 . 456366 .632457 .757396 . 854306 . 933487 1.000434 1.058426 1.109578

0.06695 . 196694 . 27259 . 32644 . 36820 . 40233 .43100 .45618 .47790

1.1667 1.573 1.8732 2.1206 2.3346 2.5254

2.    6978 2.8588

3.    0054

66 55 90 00 107 21 121 08 133 49 144 42 155 00 163 52 172 12

0. 9199 1. 0000 . 9938 . 8560 . 7216 .5779 . 3907 .2779 .1320

Ihe form of the graph of the flood may be determined by plotting the quantities in the last column of this table in terms of t. The resulting curve rises rapidly to a maximum when £=200, after which it descends slowly.

TABLES FOR CALCULATIONS OF WEIR DISCHARGE.

The investigations at Cornell University have greatly extended the limit for which weir coefficients are definitely known. The experiments of Bazin did not reach beyond 1.8 feet head maximum. The tables of Francis for thin-edged weirs extended to a head of 3 feet.

The experiments at Cornell have furnished the coefficients for a variety of weir forms for heads up to 4, 5, and 6 feet. At such heads the nappe form has become stable for nearly all forms of weirs. We may now predict the probable extension of the coefficient curves for higher heads with more confidence than could be done by starting from a lower datum.

Owing to their usefulness in the approximate determination of flood discharges, the weir tables have been carried up to a head of 10 feet.

In the tables here given the head is uniformly expressed in feet. For computing the flow over irrigation modules and other small weirs where the head is measured in inches, weir tables expressed with the inch as the argument of head are convenient. Numerous tables of this character are available. The following may be referred to:

The Emerson weir tables, computed by Charla A. Adams, pages 251-285 of Emerson’s Hydrodynamics, published by J. and W. Jolly, Holyoke, Mass. These give discharge in cubic feet per minute for weirs with two end contractions having lengths of 2, 3, 4, 5, 6, 7, 8,10,12,16, and 20 feet. The discharge is computed by the Francis formula for heads from 0.001 foot to 2 feet, advancing by thousandths of a foot, with auxiliary table of decimal equivalents of fractional parts of inches.

The Measurement and Division of Water, Bulletin No. 27, Agricultural Experiment Station, Fort Collins, Colo. This publication gives tables of discharge in cubic feet per second, computed by the Francis formula, for a weir 1 foot long, for heads in inches and sixteenths, from inch to 30 inches, with auxiliary table for end contractions, and for velocity of approach correction by the Fteley and Stearns rule    A similar weir table for a weir 1 inch long is given. Also a table

of discharge for Cippoletti weirs (C=3.36§), for lengths of crest sill of 1, 1.5, 2, 3, 4, 5, and 10 feet. Head in inches and decimals with feet-equivalents.

Special Instructions to Watermasters as to Measurements of Water, State Engineer’s Office, Salt Lake City, Utah, 1896. Table of discharge, in cubic feet per second, for 1-foot crest, based on the Francis formula, with auxiliary table for end contractions and velocity of approach. The head is expressed in inches and thirty-seconds (with equivalents in feet) for fa inch to 36 inches. A similar table for heads in inches and sixteenths, from to 36 inches, gives the discharge in cubic feet per second by the Francis formula for weirs with two end contractions and for the crest lengths of 1,1J, 2, 2J, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 feet. A table for trapezoidal weirs (C=3.367) of various crest lengths is also given.

California Hydrography, by J. B. Lippincott, Water-Supply Paper No. 81, United States Geological Survey. This publication contains a table of weir discharge in cubic feet per second for heads, advancing by sixteenths, from ^ inch to 10 inches (with equivalent decimals of a foot), for weirs with two end contractions having crest lengths as follows: 4, 6, 9, 12, 15, and 18 inches, 2, 2.5, 3, 3.5, 4, 4.5, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, and 20 feet. Based on the Francis formula. Also published as a circular.

The tables that follow are all original computations, with exception of the “ Francis weir tables,” page 162, and the table of head due to various velocities, page 158.

TABLE i.—HEAD DUE TO VARIOUS VELOCITIES.®

This table gives values of the expression

v²

h=

based on the constant of gravity for the latitude and altitude of Lowell, Mass.,

ff=32.1618, 4=0.01554639.

a Francis, Lowell Hydraulic Experiments, extended.

Table 1.— Values of h=

I

, or heads due to velocities from 0 to 4-99 feet per second.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0 | 0.0000    0.0000    0.0000

.1 ! .0002 . 0002 . 0002

.2

.3

.4

.5

.6

.7

.8

.0006 . 0014    .0015

.0025 .0039 . 0056 .0076 .0099

0.0000 .0003 .0008 .0017 .0029 .0044 .0062

0.0000

.0083

.0107

.0134

.0003

.0009

.0018

.0030

.0045

.0064

.0085

0.0000 o.oooi o.oooi

.0004 .0011 .0021 .0034 .0051 .0070 .0092 .0118 .0146 !

.0137    . 0140    . 0143

0.0001 0.0001

1.0

.1

CO Tf iO CO

CO 00

0.0155 .0188 .0224 .0263 . 0305 .0350 .0398 .0449 .0504 .0561

0. 0159 .0192 .0228 .0267 .0309 .0354 .0403 .0455 .0509 . 0567

0.0162 .0195 .0231 .0271 .0313 .0359 .0408 I .0460 .0515 .0573

0.0165 .0199 .0235 .0275 .0318

0.0168 .0202 .0239 .0279 .0322

.0364

.0413

.0465

.0521

.0579

.0369

.0418

.0471

.0526

.0585

0.0171 .0206 .0243 .0283 .0327 .0374 .0423 .0476 .0532 .0591

0.0175 .0209 .0247 .0288 .0331 .0378 .0428 .0482 .0538 . 0597

0.0178    0.0181

. 0544    . 0549

. 0603    . 0609

0.0185 .0220 .0259 .0300 .0345 .0393 .0444 .0498 .0555 .0616

2.0

.1

0. 0622 .0686 .0752 .0822 .0895 .0972 .1051 .1133 .1219 .1307

0.0628 .0692 .0759 .0830 .0903 .0979 .1059 .1142 . 1228 .1316

0.0634 .0699 .0766 .0837 .0910 .0987 .1067 .1150 .1236 .1326

0.0641 .0705 .0773 .0844 .0918 .0995 .1075 .1159 .1245 . 1335

0.0647 .0712 .0780 .0851 .0926 .1003 .1084 .1167 .1254 .1344

0.0653 .0719 .0787 .0859 .0933 .1011 .1092 .1176 .1263 .1353

0.0660 .0725 .0794 .0866 .0941 .1019 .1100 .1184 .1272 .1362

0.0666 .0732 .0801 .0873 .0948 . 1027 * .1108 .1193 .1281 .1371

0.0673 .0739 .0808 .0881 .0956 .1035 .1117 .1201 .1289 .1381

0,0679 . 0746 . 0815 .0888 .0964 .1043 .1125 .1210 .1298 .1390

CO

.2

.3

.4

.5

.6

.7

.8

.9

0.1399 .1494 .1592 .1693 .1797 .1904 .2015 .2128 .2245 .2365

0.1409 .1504 .1602 .1703 .1808 .1915 .2026 .2140 .2257 .2377

0.1418    0.1427

0.1437 .1533 .1632 .1734 .1840 .1948 .2060 .2175 .2292 .2413

0.1446 .1543 .1642 .1745 .1850 .1959 .2071 .2186 .2304 .2426

0.1456    0.1465

0.1475 . 1572 .1673 .1776 .1883 .1992 .2105 .2221 .2340 .2463

0.1484 .1582 .1683 .1787 .1894 .2004 .2117 .2233 .2352 .2475

4.0

.1

.2

.3

.4

.5

.6

.7

.8

0.2487 .2613 .2742 .2875 .3010 .3148 .3290 .3434 .3582 .3733

0.2500 . 2626 .2755 .2888 .3023 .3162 .3304 .3449 . 3597 .3748

0.2512 .2639 .2769 .2901 .3037 .3176 .3318 .3463 i .3612 .3763

0.2525 .2652 .2782 .2915 .3051 .3190 .3333 .3478 .3627 .3779

0.2537 .2665 .2795 .2928 .3065 .3204 .3347 .3493 .3642 .3794

0.2550 .2677 .2808 .2942 .3079 .3218 .3362 .3508 .3657 .3809

0.2563 .2690 .2821 .2955 .3092 .3233 .3376 . 3522 .3672 .3825

0.2575 .2703 .2835 .2969 .3106 .3247 .3390 .3537 .3687 .3840

0.2588 .2716 .2848 .2982 .3120 .3261 .3405 .3552 .3702 .3856

0.2601 .2729 .2861 .2996 .3134 .3275 .3420 .3567 .3717 .3871

TABLES FOR CALCULATING WEIR DISCHARGE.

159

This value will suffice in ordinary corrections for velocity of approach for localities in the United States.

Velocity of approach correction.

Francis, and as used in portions of this paper (approximate)

Fteley and Stearns, contracted weir.......................

Hamilton Smith, suppressed weir.........................

Hamilton Smith, contracted weir..........................

H=D+h

J/=Z>+1.5/t

H=D+1

H—D-flAJt

TABLE 2.—PERCENTAGE INCREASE IN DISCHARGE BY VARIOUS RATES OF VELOCITY OF APPROACH.

This table has been calculated from the Francis correction formula,

The percentage increase in discharge over that at the same measured head with no velocity of approach is

Percentage=loO

= K ....    (113)

2.2

0.0752 0.0206

Table 2.—Percentage increase in discharge over weirs for various rates of mean velocity of approach. measured head in feet; 77=head corrected for velocity of approach; *=mean velocity of approach, feet per second;    11= [(7>+/,)8_/₍8J 3

2.1

2.0

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

X'X

1.0

0.9

0.8

0.7

0.6

0.5

0. 0686 0.0179

0.0622 0.0154

0.0561 0.0132

0.0504 0.0111

0.0449 0.0095

0.0398 0.0079

0.0350 0.0064

0.0305 0.0051

0.0263 0.0041

0.0224 0.0033

0.0188 0.0025

0.0155 0.0019

0.0126 0. 0014

| 0.0099 0.0010

0.0076 0.0007

0.0056 0.0005

0.0039 0.0003

Percentage increase in discharge=100 1-

33.55    36.30    39.06

18.11    19.72    21.34

12.57    13.72    14.86

9.66    10.56    14. 76

7.21 I 7.93    8.68    9.43

30.78 16. 53 11.49 7.97 I 8.88

5.45    5.96    6.49

3.78    4.17    4.57    4.98 j

2.75    3.04    3.36 3.68    3.95

2.31    2.56

1. 99    2.21

2.83    3.10    3.38    |'

2.48    2.67    2.93    I

1.94    2.14    2.36    2.57

1.56 |    1.73    1.92    2.10    2.30

1.57    1.64    1.90    2.07

1.43    1.57    1.73    1.90

1.45    1.59    1.74    |

1.34    1.47    1.61    I

1.50

20.80    22.51    | 25.39    27.74

10.86    12.16    13.54    14.94

7.49    8.40    9.33    10.42

5.67    6.45    7.14

4.66    5.24    5.86    6.54

3.22    3.50    4.01    4.48 |    4.94

2.42    2.72    3.06    3.42

1.94    2.19    2.45

1.63    1. 84    2.07

1.40    1.58    1.78

1.23    1.39    1.57    |    1.75

1.10    1.24    1.40

.98    1.12    1.26    I 1.42

. 90    1.00    1.15

1.28

1.05    1.18    1.31

.98 [    1.12    1.21

1.02    1.13    |    1.25 |    1.37

.95    1.05    !    1.17

.91

1.28    1.40

.99 |    1.10    1.21    1.32

• 93 ;    1.03    1.13    1.24

.98    1.07    1.17

.93    1.02    1.11

. 90    1.09

4.0

3.9

0.2487 0.1242

ssssisssssss

N N rl d l> !» rH (N O

OSlOrHCOC^rHrHrHrH

8.86

7.91

7.11

6.44

5.88 | 5.40 :

5.01

4.68

4.41 I

4.13

3.89 I 3.69 i 3.51 !

3.36

12.19    14.13    16.34    18.62

9.77

0.2365 0.1150

ssssssss

cdidairHCDCOrHrH CsiOCOCOC^tHt—I H

9.84

8.54

7.56

6.7

6.15

5.63

5.1

4.8

4.4

4.2

3.9

3.7

3.5

3.3

3.2

9. 62

8.39

7.20

6.16

5.11

8.16

4.20

6.56

4.98

4.10

2.79

2.13

1. 70

1.43

1,23

1.08

.96

.83

.78

5.69

4.33

3.56

2.43

1.84

1.47

1.24

90 ;x

.93

.83

.75

.68

4.88

4.14

3.44

2.86

3. 70

3.05

2.07

1.58

1.26

1.06

.90

.80

.71

.65

.58

3.12

2.66

2.11

2.59

2.15

1.76

1.75

1.34

90'X

.89

.76

.67

.60

.54

.49

1.45

1.11

.88

.74

.63

.55

.49

ff

.40

61‘X

.91

.71

.60

.47

.45

- .40

.35

.33

6.44

3.33

2.22

1.62

1.40

.94

.72

.56

.47

.40

.35

.31

.27

.27

5.02

2.58

1.74

1.26

1.07

.72

.59

.43

.36

.27

.24

.22

.20

.88

.84

.90

.80

.75    .84

.80

.71

• 67 |    .75    .84

.94

.87

.80

.75

.71

.72 I .84

.63

.53

.45

.37

.30

.23

.17

.76

.67

.58

.49

.41

.34

.28

.22

.17

.72

.66

.63

.59 i .67

.63

.60

.55

.53

.63

.54

.46

.39

.32

.26

.20

.15

.58

.50

.43

.36

.29

.24

.18

.14

.55

.47

.40

.34

.28

.23

.18

.14

.51

.44

.38

.31

.26

.21

.16

.13

.48

.46

.42

.36

.30

.25

.20

.12

.40

.34

.28

.23

61 •

.15

.11

.84

3.8

0.2128 0.1064

cc

53.07 38.12 29.97 24.86 17.48 13.55 11.07

9.38

8.14 7.20

6.45

5.86 5.35

4.93

4.52

4.15

4.00

3.76

3.55

3.36

3.20

3.04

.79

.71

.64

.57

.50

.44

.37

.31

.27

.23

61 •

.14

.10

N©SSSS^SSS§HHHHHHHHOOOO

CO rH rH

2.65

1.34

06

62

56

37

29

22

19

16

14

12

IT

10

60

08

08

07

07

06

90

90

.05

3.7

3.6

3.5

3.4

3.3

3.2

3.1

3.0

0.2128 0. 0982

0.2015 0.0904

0.1904 0.0830

0.1797 0.0761

0.1693 0.0696

0.1592 0.0634

0.1494 0.0576

0.1399 0.0524

2.9

2.8

2.7

2.6

2.5

2.4

0.1307 0.0472

0.1219 0.0426

0.1133 0.0381

0.1051 0.0340

0.0972 0.0303

0.0896 0.0268

'■D

CD

SS

50.79 36.47 28.64 23.74 16.67 12.92 10.54

SSoocoSScio tj? oo tj? i> c4' 10 e4 o

OOrHCODIC'IrHrHrH

46.50 33.26 26.06 21.58 15.12 11.69

9.53

'.'."I

8.93

7.75

6.85

6.14 5.*57

5.09

4.69

4.35

4.06

3.80

3.58

3.38

3.20

3.04

2.89

8.50

8.00

7.78

7.37

7.00

".63

6.52

5.84

5.30

4.84

4.46

4.13

3.86

3.61

3.40

3.20

3.04

2.88

2.74

6.24

5.5

5.0

4.59

4.2

3.9

3.6

3.4

3.2

3.0

2.9

2.7

03*

4.S6

5.24

4.75

1.3.4

I ""

3.71

3.46

3.2 1

3.li4

2.88

2. 72

2. 44

2.47

te?!?

Percentage increase in discharge=100

57.34 32.19 22.80 17.77 14.57 10.03

54.22 30.29 37.16 16.58 13.66

iC    03    03    rH    rH

8 S S 58 § SS 8 3 S 3

44.91 24.77 17.64 13.42 11.04

9.47

8.84

N.2-

7.65

8.30

7.78

7.28

6.79

6. 62

5.85

N..V,

7.63

7.17

6.74

6.32

5.90

5.50

5.11

4.73

7.24

6.27

5.54

4.96

1.4"

4.11

3.78

3.57

3.27

3 06

2.88

2.72

2.57

2.44

2.42

6.81

r.

6.07

5.70

5.33

4.98

4.64

4.31

3.99

93

6.04

4.93

4.61

4.31

4.01

3.73

3.44

5.23

4. 68

4.25

3.88

3.57

3.31

3.09

2.88

2. 72

2.56

2.43

2.30

2.19

4.41

4.00

3.36

8.17

2.96

2 73

If Z

2.17

2.05

4.63

I.

4.07

3.80

::.51

3.28

3.04

l. 14

3.89

3.64

3.40

3.16

2.93

2.71

3.76

3.53

3.30

3.09

2.87

2.66

2.46

3. 13

09

2.94

•2. C.7

2.40

2.03

1.94

8.21

2.74

2.55

■2.

2.25

2.12

c4

1.90

08 I

3.01

08 'Z

19 ‘Z

2.42

2.24

2.77

2.58

2.22

2.06

2.56

2.39

2.06

1.90

2.39

2.22

2.10

1.98

1.87-

1.78

89 T

2.23

81'Z

1.96

1.85

1.78

1.66

1.57

2.07

1.94

1.82

1.72

1.62

1.54

1.47

1.92

08 T

1.69

1.59

1.51

1.43

1.30

1.78

1.66

1.56

1.47

68 ’X

1.32

1.25

=a

0.4

0.0025 0.0002

2.3

0.0822 0.0235

8    8    9    3    S

03    cd    CD    03    O

rH    03    t-H    t—I    rH

; in

5. 11

4.36

3.68

3.18

2.80

2.26

2.10

1.90

1.75

89 T

1.53

1.44

1.35

1.28

1.22

1.15

<=!

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

O'OI

=5

03 rH © *

.6

.8

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

O’OX

To use this table the discharge corresponding to the measured head D may be taken directly from Table 3 or 4 and the quantity so obtained increased by the percentage indicated in Table 2. This table is especial^ useful where the velocity of approach is measured directly. If the velocity of approach is determined from the approximate discharge

by the formula    successive approximate corrections may be

required.

Table 2 shows directly the relative error introduced by various velocities of approach. The large error introduced by moderate velocities with low heads and the comparatively small error resulting from higher velocities under great heads are conspicuous.

TABLES 3 AND 4.—DISCHARGE OVER A THIN-EDGED WEIR BY THE FRANCIS FORMULA.

These tables give the discharge in cubic feet per second, for a crest length of 1 foot, without contractions, computed by the formula

Q=3.S3L&%.

Table 3.—Discharge over a thin-edged weir per foot of crest.

Head if, feet.    .000

.001

.002

.003

.004

.005

.006

.007 ! .008

.009

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0000 .0033 .0094 .0173 .0266 .0372 .0489 .0617 .0753 .0899

0.0001    0.0003    0.0005    0.0008    0.0012

0.0015 .0067 .0140 .0227 .0329 .0441 .0565 .0698 .0840 .0990

0.0020 .0074 .0148 .0237 .0339 .0453 .0578 .0712 .0855 .1006

0.0024

.0080

.0156

.0247

.0350

.0465

.0590

.0725

.0869

.1022

0.0028 .0087 .0164 .0256 .0361 .0477 .0604 .0739 .0884 .1037

0.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

0.1053 .1215 .1384 .1561 .1744 .1935 .2131 .2334 .2543 .2758

0.1069 .1231 .1402 .1579 .1763 .1954 .2151 .2355 . 2564 .2780

0.1085 .1248 .1419 .1597 .1782 .1973 .2171 .2375 .2586 .2802

0.1101 .1265 .1436 .1615 .1801 .1993 .2191 .2396 .2607 .2823

0.1117 . 1282 .1454 .1633 .1820

.2012 j .2032 .2212 ,2417 .2628 .2845

0.1133 .1299 .1472 .1652 .1839

.2232

.2438

.2650

.2867

0.1149 .1316 .1489 . 1670 .1858 .2052 .2252 .2459 .2671 .2890

0.1166 .1333 .1507 .1689 .1877 .2072 .2273 .2480 .2693 .2912

. 2934    . 2956

. 0.20 .21 .22 .23 .24 .25 .26 .27 .28

0.2978

.3205

.3436

.3673

.3915

.4162

.4415

.4672

.4934

.5200

0.3001 .3228 .3460 .3697 .3940 .4187 .4440 .4698 .4960 .5227 0.3023 .3250 .3483 .3721 .3964 .4213 .4466 .4724 .4987 .5254

0.3046 .3274 .3507

.3745

.3989

.4238

.4491

.4750

.5013

.5281

0.3068 .3297 .3530 .3769 .4014 .4263 .4517 .4776 .5040 .5308 0.3091 .3320 .3554 .3794 .4038 .4288 .4543 .4802 .5067 .5336 0.3113 .3343 .3578 .3818 .4063 .4313 .4568 .4828 .5093 .5363 0.3136 .3366 .3601 .3842 .4088 .4339 .4594 .4855 .5120 .5390 0.3159 .3389 .3625 .3866 .4113 .4364 .4620 .4881 .5147 .5417 0.3182 .3413 .3649 .3891 .4138 .4389 .4646 .4907 .5174 .5444

Table 3.—Discharge over a thin-edged weir per foot of crest—Continued.

Head H, feet.

.000

.001

.002    .003

.004

.005

.006

.007

.008

.000

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

0.5472 .5748 .6028 .6313 .6602 .6895 .7193 .7495 .7800 .8110

0.5499 .5775 .6056 .6341 .6631 .6925 .7223 .7525 .7831 .8142

0.5527    0.5554

.5803 .6085    .6113

.6370

.6660 ; .6689 .6954 . 7253 .7555 .7862 .8173

.5831

.6399

.6984 .7283 . 7586 .7893 . 8204

0.5582 .5859 .6141 .6428 .6719 .7014 .7313 .7616 .7924 .8235

0.5609 .5887 .6170 .6457 .6748 .7043 .7343 .7647 .7955 .8267

0.5637 .5915 .'6198 .6486 .6777 .7073 .7373 .7678 .7986 .8298

0.5664 .5943 .6227 .6515 .6807 .7103 .7404 .7708 .8017 .8330

0.5692 .5972 .6255 .6544 .6836 .7133 .7434 .7739 .8048 .8361

0.5720 .6000 .6284 .6573 .6866 .7163 .7464 .7770 .8079 .8393

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.8424 .8742 .9064 .9390 .9719 1.0052 1.0389 1.0730 1.1074 1.1422

0.8456 .8774 .9096 .9422 .9752 1.0086 1.0423 1.0764 1.1109 1.1457

0.8488 .8806 .9129 .9455 .9785 1.0119 1.0457 1.0798 1.1143 1.1492

0.8519 .8838 .9161 .9488 .9819 1.0153 1.0491 1.0833 1.1178 1.1527

0.8551 .8870 .9194 .9521 .9852 1.0187 1.0525 1.0867 1.1213 1.1562

0.8583 .8903 .9226 .9554 .9885 1.0220 1.0559 1.0901 1.1248 1.1597

0.8615 .8935 .9259 .9587 .9919

0.8646 .8967 .9292 .9620 .9952 1.0254    1.0288

1.0593 1.0627 1.0936    1.0970

1.1282    1.1317

1.1632    1.1668

0.8678 .8999 .9324 .9653 .9985 1.0321 1.0661 1.1005 1.1352 1.1703

0.8710 .9032 .9357 .9686 1.0019 1.0355 1.0696 1.1039 1.1387 1.1738

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

1.1773 1.2128 1.2487 1.2849 1.3214 1.3583 1.3955 1.4330 1.4709 1.5091

1.1809 1.2164 1.2523 1.2885 1.3251 1.3620 1.3992 1.4368 1.4747 1.5130

1.1844 1.2200 1.2559 1.2921 1.3287 1.3657 1.4030 1.4406 1.4785 1.5168

1.1879 1.2235 1.2595 1.2958 1.3324 1.3694 1.4067 1.4444 1.4823 1.5206

1.1915 1.2271 1.2631 1.2994 1.3361 1.3731 1.4105 1.4481 1.4862 1.5245

1.1950 1.2307 1.2667 1.3031 1.3398 1.3768 1.4142 1.4519 1.4900 1.5283

1.1986 1.2343 1.2703 1.3067 1.3435 1.3806 1.4180 1.4557 1.4938 1.5322

1.2021 1.2379 1.2740 1.3104 1.3472 1.3843 1.4217 1.4595 1.4976 1.5361

1.2057 1.2415 1.2776 1.3141 1.3509 1.3880 1.4255 1.4633 1.5014 1.5399

1.2093 1.2451 1.2812 1.3177 1.3546 1.3918 1.4293 1.4671 1. 5053 1.5438

0.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

1.5476 1.5865 1.6257 1.6652 1.7050 1. 7451 1.7855 1.8262 1.8673 1.9086

1.5515 1.5904 1.6296 1.6691 1.7090 1.7491 1.7896 1.8303 1.8tl4 1.9128

1.5554 1.5943 1.6335 1.6731 1.7130 1.7531 1.7936 1.8344 1.8755 1.9169

1.5593 1.5982 1.6375 1.6771 1.7170 1.7572 1.7977 1.8385 1.8796 1.9211

1.5631 1.6021 1.6414 1.6810 1.7210 1.7612 1.8018 1.8426 1.8838 1.9252

1.5670 1.6060 1.6454 1.6850 1.7250 1.7652 1.8058 1.8467 1.8879 1.9294

1.5709 1.6100 1.6493 1.6890 1. 7290 1.7693 1.8099 1.8508 1.8920 1.9336

1.5748 1.6139 1.6533 1.6930 1.7330 1.7733 1.8140 1.8549 1.8962 1.9377

1.5787 1.6178 1.6572 1.6970 1.7370 1.7774 1.8181 1.8590 1.9003 1.9419

1.5826 1.6217 1.6612 1.7010 1.7410 1.7814 1.8221 1.8632 • 1. 9045 1.9461

0.70

.71

.72

.73

.74

.75

.76

.77

.78

1.9503 1.9922 2.0344 2.0770 2.1198 2.1629 2.2063 2.2500 2.2940 2.3382

1.9544 1.9964 2.0387 2,0812 2.1241 2.1672 2.2107 2.2544 2.2984 2.3427

1.9586 2.0006 2.0429 2.0855 2.1284 2.1716 2.2150 2.2588 2.3028 2.3471

1.9628 2.0048 2.0472 2.0898 2.1327 2.1759 2.2194 2.2632 2.3072 2.3515

1.9670 2.0091 2.0514 2.0941 2.1370 2.1802 2.2237 2.2675 2.3116 2.3560

1.9712    1.9754

2.0133    2.0175

2.0557    2.0599

2.0983    2.1026

2.1413    2.1456

2.1846    2.1889

2.2281    2.2325

2.2719    2.2763

2.3161    2.3205

2.3604    2.3649

1.9796 2.0217 2. 0642 2.1069 2.1499 2.1932 2.2369 2.2807 2.3249 2.3694

1.9838 2.0260 2.0684 2.1112 2.1543 2.1976 2.2412 2.2851 2.3293 2.3738

1.9880 2.0302 2.0727 2.1155 2.1586 2.2019 2.2456 2.2896 2.3338 2.3783

Table 3.—Discharge over a thin-edged weir per foot of crest—Continued.

Head H, feet. .000

.001

.002

.003

.004

.005

.006

.007

.008

.009

0.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

2.3828 2.4276 2. 4727 2.5180 2. 5637 2.6096 2.6558 2. 7022 2.7490 2.7959

2.3872 2.4321 2.4772 2.5226 2.5683 2.6142 2. 6604 2.7069 2. 7536 2.8007

2.3917    2.3962

2.4366    2.4411

2.4817    2.4862

2.5271    2.5317

2.5728 2. 5774 2.6188 2.6234 • 2.6650 2.6697 2.7116    2.7162

2.7583    2. 7630

2.8054    2.8101

2.4006    2.4051    2.4096    2.4141

2.4456    2.4501    2.4546    2.4591

2.4908    2.4953    2.4999    2.5044

2.5363    2.5408    2.5454    2.5500

2.5820    2.5866    2.5912    2.5958

2.6280 j 2.6327    2.6373    2.6419

2.6743    2.6790    2.6836    2.6883

2.7209    2.7256    2.7302    2.7349

2.7677    2.7724    2.7771    2.7818

2.8148 ! 2.8195    2.8243    2.8290

2.4186    2.4231

2.4636    2.4681

2.5089    2.5135

2.5545    2.5591

2.6004    2.6050

2.6465    2.6511

2. 6929    2.6976

2.7396    2.7443

2. 7865    2. 7912

2.8337    2.8385

0.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

2.8432    2.8479

2.8907    2.8955

2.9385    2.9433

2.9865    2.9914

3.0348    3.0397

3.0834    3.0883

3.1322    3.1371

3.1813    3.1862

3.2306    3.2355

3.2802    3.2851

2.8527 2.9003 2.9481 2.9962 3.0445 3.0931 3.1420 3.1911 3.2405 3.2901

2.8574    2.8622    |    2.8669

2.9050    2.9098    2.9146

2.9529    2.9577    2.9625

3.0010    3.0058    3.0107

3.0494    |    3.0542    ;    3.0591

3.0980    j    3.1029    3.1078

3.1469    3.1518    3.1567

3.1960    !    3.2010    i    3.2059

3.2454    !    3.2504    |    3.2554

3.2951    3.3001    |    3.3051

2.8717 | 2.8764    2.8812

2.9194 j 2.9241

2.    9673- 2.9721

3.0155 ! 3.0203 3.0639    3.0688

3.1127 j 3.1175 3.1616 I 3.1665 3.2108 i 3.2158

3.    2603 ; 3.2653    3.2702

3.3100 ! 3.3150    3.3200

2.9289 2.9769 3. 0252 3.0737 3.1224 3.1714 3.2207

2.8860 2.9337 2.9817 3. 0300 3.0785 | 3.1273 j 3.1764 3.2257 ! 3.2752 3.3250

1.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

3.3300 3.3801 3.4304 3.4810 3.5318 3.5828 3.6342 3.6857 3.7375 3.7895

3.3350 3.3851 3.4354 3.4860 3.5369 3.5880 3. 6393 3.6909 3.7427 3. 7947

3.3400 3.3901 3.4405 3.4911 3.5420 3.5931 3.6444 3.6960

3.7479

3.8000

3.3450 3.3500 i 3.3550 3.3951    3.4002    3.4052

3.4455    3.4506    3.4557

3.4962    3.5013    3.5063

3.5471    3.5522    3.5573

3.5982 | 3.6033 j 3.6085 3.6496 | 3.6547 i 3.6599 3.7012    3.7064    3.7116

3.7531    3. 7583    3.7635

3.8052    3.8104    3.8156

3.3600 | 3.3650 3.4102    3.4153

3.4607 ! 3.4658 3.6114 3. 5165 3.5624 3.5675 3.6136 3.6187 3.6651    3.6702

3.7167 3.7219 3.7687 j 3.7739 3.8209 i 3.8261

3.3700 3.4203 3.4708 3.5216 3.5726 3.6239 3.6754 3.7271 3. 7791 3.8313

3.3751 j 3.4254 3.4759 3.5267 3.5777 3.6290 3.6805 3.7323 3.7843 3.8365

1.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

3.8418    3.8470

3.8943    3.8996

3. 9470    3. 9523

4.0000    4. 0053

4.0532    4.0586

4.1067    4.1120

4.1604    4.1657

4.2143    4.2197

4.2684    4. 2738

4.3228    4.3282

3.8523 3.9048 3.9576 4.0106 4.0639 4.1174 4.1711 4.2251 4.2793 4.3337

3.8575 3.9101 3.9629 4.0160 4.0692 4.1228 4.1765 4.2305 4.2847 4.3392

3.8628 3. 9154 3.9682 4.0213 4.0746 4.1281 4.1819 4.2359 4.2901 4.3446

3.8680 3.9206 3.9735 4.0266 4.0799 4.1335 4.1873 4.2413 4.2956 4.3501

3.8733 . 3.8785    3.8838

3.9259 3.9312    3. 9365

3.9788    3.9841    3.9894

4.0319    4.0372    4.0426

4.0853 4.0906 I 4.0960 4.1389    4.1442    4.1496

4.1927    4.1981    ' 4.2035

4.2467    4.2522    4.2576

4,3010 4.3065    4.3119

4.3555 4.3610 4.3665

3.8890 3.9418 3.9947 4.0479 4.1013 4.1550 4.2089 4.2630 4.3173 4.3719

1.20

.21

.22

.23

.24

.25

.26

.27

.28

4.3774 4.4322 4.4873 4.5426 4.5981 4.6538 4.7098 4.7660 4.8224 4.8790

4.3829 4.3883 4.4377 4.4432 4.4928    4.4983

4.5481    4.5537

4.6036    4.6092

4.6594 4.6650 4.7154 4.7210 4.7716 4.7772 4.8280 4.8337 4.8847 4.8903

4.3938 4.3993    4.4048

4.4487    4.4542    4.4597

4.5038    4.5094    4.5149

4.5592    4.5647    4.5703

4.6148    4.6203    4.6259

4.6706    4.6762    4.6818

4.7266 4.7322    4.7378

4.7829    4.7885    4.7941

4.8393 4.8450 4.8506 4.8960    4.9017    4.9074

4.4103    4.4158    4.4212    4.4267

4.4652    4.4707    4.4763    4.4818

4.5204    4.5260    !    4.5315    4.5370

4.5759    4.5814    I    4.5870    4.5925

4.6315    4.6371    i    4.6427    4.6482

4.6874    4.6930    4.6986    4.7042

4.7435    4.7491    4.7547    4.7603

4.7998    4.8054    4.8111    4.8167

4.8563    4.8620    4.8676    4.8733

4.9131    ,4.9187    4.9244    4.9301

Table 3.—Discharge over a thin-edged weir per foot of crest—Continued.

Head II, feet. . 000

.001 .002

.003

.004

.005

.006    .007

.008

.009

1.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

4.9358 4. 9929 5.0502 5.1077 5.1654 5.2233 5.2814 5.3398 5.3984 5.4572

4.9415 4.9986 5.0559 5.1134 5.1712 5.2291 5.2873 5.3456 5.4142 5.4630

4.9472 5.0043 5.0616 5.1192 5.1769 5.2349 5.2931 5.3515 5.4101 5.4689

4.9529    4.9586 4.9643

5.0158    5.0215

5.0789 5.1365 5.1827 I 5.1885    5.1943

5.2407 J 5.2465 j 5.2523 5.2989    5.3048    ! 5.3106

5.3573    5.3632    5.3691

5.4160    5.4219    5.4277

5.4748 5.4807    5.4866

5.0100 5.0674 5.1249

5.0731 5.1307

4.9700 I 4.9757    4.9814

5.0272 | 5.0330    5.0387

5.0846 | 5.0904    5.0961

5.1423    5.1480    5.1538

5.2001    5.0259    5.2117

5.2582    5.2640    5.2698

5.3164    5.3223    5.3281

5.3749 5.3808    5.3866

5.4336 5.4395 5.4454 5.4925 5.4984 5.5043

4.9872 5.0444 5.1019 5.1596 5.2175 5.2756 5.3340 5.3925 5.4513 5.5102

1.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

1.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

5.5280    j    5.5339    5.5398 j 5.5457

5.5872    j    5.5932    5.5991

5.6467    J    5.6526    5.6586

5.7064    !    5.7123    I    5.7183

5. 7662    ,    5.7722    j    5.7782

5.8263    |    5.8323    5.8384

5.8866 | 5.8926 : 5.8987 | 5.9047 j 5.9108 5.9471    5.9532 ! 5.9592 j 5.9653

6.0078 : 6.0139 I 6.0200 i 6.0261 6.0565 j 6.0626    6.0687    ,    6.0748    [    6.0809    j    6.0870

6.1237    6.1298    !    6.1360    !    6.1421    |    6.1482

6.1850    6.1912    6.1973    j    6.2034    6.2096

6.2465    6.2527    ¹    6.2588    ^    6.2650    6.2712

6.3082    6.3144 j    6.3206    ;    6.3268    6.3330

6.3701    6.3763    6.3825    ;    6.3887    6.3949

6.4322 !    6.4385 !    6.4447    ;    6.4509    6.4571

6.4945 i 6.5008 6.5070 6.5133 6.5195 6.5570 ! 6.5633 6.5696 6.5758 6.5821 6.6198 |    6.6260    6.6323    |    6.6386    6.6449

6.6827 j    6.6890    6.6953    I    6.7016    6.7079

5.5162    5.5221

5.5754    5.5813

5.6348    5. 6407

5.6944    5. 7004

5.7542    5.7602

5.8143    5.8203

5.8745    5.8806

5.9350    5.9410

5.9957 : 6.0017

5.5516 5.5576 5.6050    5.6110    5.6169

5.6646    5.6705    5.6765

5.7243    5.7303    5.7363

5.7842 j 5.7902 j 5.7962 5.8444 j 5.8504    5.8564

! 5.9168 5.9714 ; 5.9774 6.0322    6.0382

6.0931 ! 6.0993

5.5635 5.6229 5.6825 5.7423 5.8023 5.8625 5.9229 5.9835 6.0443 6.1054

5.5694 5.6288 5.6884 5.7482 5.8083 5.8685 5.9289 5.9896 6.0504 6.1115

6.1176 6.1789 6.2404 6.3020 6.3639 ! i 6.4260 6.4883 6.5508 6.6135 6.6764

6.1543    6.1605

6.2157    6.2219

6.2773    6.2835

6.3391    6.3453

6.4012    6.4074

6.4634    6.4696

6.5258 6.5320 6.5884    6.5946

6.6512    6.6575

6.7142    6.7205

6.1666 6.2280 6.2897 6.3515 6.4136 6.4758 6.5383 6.6009 6.6638 6.7268

6.1727 6.2342 6.2959 6.3577 6.4198 6.4821 6.5445 6.6072 6.6701 6. 7331

1.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

6.7394 j    6.7458    6.7521    6.7584

6.8027 '    6.8091    6.8154    6.8217

6.8662    6.8726    6.8789    6.8853

6.9299 i    6.9363    j    6.9426    6.9490

6.9937 ¹    7.0001    j    7.0065    7.0129

7.0578 |    7.0642    !    7.0706    7.0771

7.1221 |    7.1285    7.1349    ; 7.1414

7.1865    7.1930    7.1994    7.2059

7.2512    7.2576    j    7.2641 j 7.2706

I 7.3160    7.3225    ;    7.3290    7.3355

6.7647 | 6.7711 6.8281 | 6.8344 6.8916    6.8980

6.9554    6. 9618

7.0193    7.0258

7.0835    7.0899

7.1478    7.1543

7.2124    7.2188

7.2771    7. 2836

7.3420    7.3485

6.7774    6. 7837

6.8408    6.8471

6.9044    6.9108

6.9682    6.9746

7.0322    7.0386

7.0963 ! 7.1028 7.1607    7.1672

7.2253 7.2318 7.2901    7.2965

7.3550    7.3615

6.7901 6.8535 6.9171 6.9810 7.0450 7.1092 .7.1736 7.2382 7.3030 7.3680

6.    7964 6.8598 6.9235 6.9874 7.0514 7.1156 7.1801 7.2447 7.3095

7.    3745

1.70

.71

.72

.73

.74

.75

.76

.77

.78

j 7.3810    |    7.3876    7.3941'    7.4006    7.4071

7.4463    7.4528    7.4593    7.4659    7.4724

7.5117    !    7.5182    7.5248    1 5313    7.5379

7.5773    I    7.5839 : 7.5904    7.5970    7.6036

7.6431    :    7.6497 I 7.6563    7.6628    ¹ 7.6694

7.7091    i    7.7157    7.7223    7.7289    7.7355

7.7752    7.7819    7.7885    S    7.7951    7.8018

7.8416    7.8482    7.8549    |    7.8615    7.8682

7.9081    7.9148    7.9215    7.9281    7.9348

7.9749    7.9816    7.9882    j    7.9949 ; 8.0016

7. 4136 7.4789 7.5445 7. 6102 7.6760 7.7421 7. 8084 7.8748 7.9415 8.0083

7.4201 7.4855 7.5510 7.6167 7.6826 7.7487 7.8150 7.8815 7.9482 8.0150

7.4267 7. 4920 7.5576 7.6233 7.6892 7.7554 7.8217 7.8882 7.9648 8.0217

7.4332 7.4986 7.5641 7.6299 7.6958 7.7620 7.8283 7.8948 7.9615 8.0284

7.4397 7.5051 7.5707 7.6365 7.7024 7.7686 7.8349 7.9015 7.9682 8.0351

Table 3.—Discharge over a thin-edged weir per foot of crest—Continued.

Head H, feet.

.000

.001

.002

.003

.004

.005

.006

.007    .008

1.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

8.0418 8.1089 8.1762 8.2437 8.3113 8.3792 8.4472 8.5154 8.5838 8.6524

8.0485 8.0552 8.1156    8.1223

8.1829 8.1897 8.2504    8.2572

8.3181    8.3249

8.3860 8.3928 8.4540 8.4608 8.5223    8.5291

8.5907 8.5975 8.6593 8.6661

8.0619    8.0686

8.1291    8.1358

8.1964    8.2032

8.2640    8.2707

8.3317 i 8.3385 8.3996 ' 8.4064 8.4677 i 8.4745 8.5359 ' 8.5428 8.6044    8.6112

8.6730 : 8.6799

8.0753 8.1425 8.2099 8.2775 8.3452 8.4132 8.4813 8.5496 8.6181 8.6868

8.0820 8.1493 8.2167 8.2842 8.3520 8.4200 8.4881 8.5564 8.6250 8.6936

8.0888 j 8.0955 8.1560 I 8.1627 8.2234 : 8.2302 8.2910 I 8.2978 8.3588    8.3656

8.4268 | 8.4336 8.4949 I 8.5018 8.5633    8.5701

8.6318    8.6387

8. 7005 | 8.7074

8.1022 8.1695 8.2369 8.3046 8.3724 8.4404 8.5086 ! 8.5770 ! 8.6455 8.7143 !

1.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

8.7212    8. 7281

8.7901 j 8.7970 8.8592    8.8662

8.    9285    8.9355

8.9980    9.0050

9.0677    9.0747

9.1375    9.1445

9.    2075    9.2145

9.2777    9.2848

9.3481    9.3552

8.7349 8.8039 8.8731 8.9424 9.0119 9.0816 9.1515 9.2216 9.2918 9.3622

8.7418    8.7487

8.8108    8.8177

8.8800    8.8869

8.9494    8.9563

9.0189    9.0259

9.0886 j 9.0956 9.1585    9.1655

9.2286    9.2356

9.2988    9.3059

9.3693    9.3763

8.7556 8.8246 8.8939 8.9633 9.0328 9.1026 9.1725 9.2426 9.3129 9.3834

8. 7625    8.7694 j 8.7763

8.8316    8.8385    8.8454

8.9008    8.9077    8.9147

8.9702    8.9772    8.9841

9.0398    9.0468    9.0537

9.1096    9.1165    9.1235

9.1795 9.1865    9.1935

9.2496 9.2567    9.2637

9.3199 9.3270    9.3340

9.3904 9.3975    9.4045

8.7832 8.8523 8.9216 8.9911 9.0607 ; 9.1305 ’ 9.2005 1 9.2707 9.3411 | 9.4116

2.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

9.4187    9.4257    9.4328    9.4399

9.4894    9.4965    9.5036    9.5106

9.5603    9.5674    9.5745    9.5816

9.6314    9.6385    9.6456    9.6527

9.7026    9.7098    9.7169    9.7240

9.7741    9.7812    9.7884    9.7955

9.8457    9.8528    9.8600    9.8672

9.9174    9.9246    9.9318    9.9390

9.9894    9.9966 10.004    10.011

10.062    10.069    10.076    10.083

9.4469    9.4540    9.4611    9.4682

9.5177    9.5248    9.5319 9.5390

9.5887    9.5958    9.6029    9.6100

9.6599    9.6670    9.6741    9.6812

9.7312    9.7383    9. 7455    9. 7526

9.8027    9.8098    9.8170    9.8242

9.8744    9.8815    9.8887    9.8959

9.9462    9.9534    9.9606    9.9678

10.018    10.025    10.033    10.040

10.090    10.098    10.105 10.112

9.4752 9.5461 9.6171 9.6884 9.7598 9.8313 9.9031 9. 9750 10.047 10.119

9.4823 9.5532 9.6243 9.6955 9.7669 9.8385 j 9.9103 j 9. 9822 ;

10.054 J

10.127

2.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

10.134 10.206 10.279 10.352 10.425 10.498 10.571 10.645 10. 718 10.792

10.141    10.148    10.156

10.214    10.221    10.228

10.293    10.301

10.359    10.366    10.374

10.432    10.439    10.447

10.513    10.520

10.586    10.593

10.652    10.659    10.667

10.733    10.741

10.800    10.807    10.814

10.286

10.505 10.579

10.726

10.163    10.170

10.235    10.243

10.308    10.315

10.381    10.388

10. 454    10. 461

10.527    10.535

10.601 10.608 10. 674    10.682

10.748    10. 755

10.822    10.829

10.177    10.185

10.250 10.257 10.323    10.330

10.396    10.403

10.469    10.476

10.542    10.549

10.615    10.623

10.689    10.696

10.763    10.770

10.837    10.844

10.192 10. 264 10.337 10.410 10.483 10. 557 10.630 10.704 10. 777 10.851

10.189 10.272 10. 344 10.417 10.491 10.564 j 10.637 | 10.711 10.785 10.859

2. 20

.21

.22

.23

.24

.25

.26

.27

.28

10.866 10.940 11.015 11.089 11.164 11.239 11.314 11.389 11.464 11.540

10.874 10.948 11.022 11.097 11.171 11.246 11.321 11.396 11.472 11.547

10.881    10.888    10.896

10,955    10.963    10.970

11.030    11.037    11.045

11.104    11.112    11.119

11.179    11.186    11.194

11.254    11.261    11.269

11.329    11.336    11.344

11.404    11.412    11.419

11.479    11.487    11.494

11.555    11.562    11.570

10.903 10.978 11.052 11.127 11.201 11.276 11.351 11.427 11.502 11.578

10.911 40.918 10.985 ! 10.992 11.059 111. 067 11.134 111.141 11.209 11.216 11.284    11.291

11.359    11.366

11.434    11.442

11.510 11.517 11.585 ill. 593

10.926 11.000 11.074 11.149 11.224 11.299 11.374 11.449 11.525 11.600

10.933 11.007 11.082 11.156 11.231 11.306 11.381 11.457 11.532 11.608

Table 3.—Discharge over a thin-edged weir per foot of crest—Continued.

Head H, feet.

.000

.001

.002

.003

.004    .005

.006

.007

.008 .000

2.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

11. 615 11.691 11.767 11.843 11.920 11.996 12.073 12.150 12.227 12.304

11.623 11.699 11.775 11.851 11.927 12.004 12.081 12.157 12.234 12.312

11.631    11.638

11.706 ; 11.714 11.783    11.790

11.859    11.866

11.935    11.943

12.012    12.019

12.088    12.096

12.165    12.173

12.242    12.250

12.319    12.327

11.646    11.653

11.722 I 11.729 11.798    11.805

11.874    11.882

11.950    11.958

12.027 , 12.035 12.104    12. Ill

12.181 12.188 12.258    12.265

12.335 I 12.342

11.661    11.669

11.737    11.744

11.813    11.821

11.889    11.897

11.966    11.973

12.042    12.050

12.119    12.127

12.196    12.204

12.273    12.281

12.350 I 12.358

11.676 11.684 11.752    11.760

11.828    11.836

11.904    11.912

11.981    11.989

12.058 | 12.065 12.134 | 12.142 12.211 J 12.219 12.288 ; 12.296 12.366    12.373

2.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

12.381    12.389

12.459    12.466

12.536    12.544

12.614    12.622

12.692    12.700

12.770    12. 778

12.848 | 12.856 12.927    12.935

13.005    13.013

13.084    13.092

12.397    12.404    12.412    12.420

12.474    12.482    12.490    12.497

12.552 [ 12.560 12.567    12.575

12.630 12.637    12.645    12.653

12.708    12.715    12.723    12.731

12.786 ' 12.794    12.801    12.809

12.864    12.872    12.880    12.888

12.942    12.950    12.958    12.966

13.021    13.029    13.037

13.100 13.108 13.116    13.124

13.045

12.428    12.435

12.505 ! 12.513 12.583 j 12.591 12.661 ; 12.669 12.739 ; 12.747 12.817    12.825

12.895    12.903

12.974    12.982

13.053    13.060

13.131    13.139

12.443 12.521 12.598 12.676 12.754 12.833 12.911 12.990 13.068 13.147

12.451 12.528 12.606 12.684 12.672 12.840 12. 919 12.997 13.076 13.155

2.50

.51

.52

.53

.54

.55

.56'

.57

.58

.59

13.163 13.242 13.321 13.401 13.480 13.560 13.640 13.720 13.800 13.880

13.171 13.250 13.329 13.409 13.488 13. 568 13.648 13.728 13.808 13.888

13.179    13.187

13.258    13.266

13. 337    13.345

13.417    13.424

13.496    13.504

13.576    13.584

13.656 13.664 13.736    13.744

13.816    13.824

13.896    13.904

13.195 13.274 13.353 13.432 13.512 13.592 13.672 13.752 13.832 13.912

13.202 13.282 13.361 13.440 13.520 13.600 13.680 13.760 13.840 13.920

13.210 13.290 13.369 13.448 13.528 13.608 13.688 13.768 13.848 13.928

13.218 13.297 13.377 13.456 13.536 13.616 13. 696 13.776 13.856 13.936

13.226 13.305 13.385 13.464 13.544 13.624 13.704 13.784 13.864 13.944

13.234 13.313 13. 393 13.472 13.552 13.632 13.712 13. 792 13.872 13.953

2.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

13.961 14.041 14.122 14.203 14.284 14.356 14.447 14.528 14.610 14.692

13.969    13.977

14.049    14.057

14.130 I 14.138 14.211    14.219

14.292    14.300

14. 373    14.382

14.455    14.463

14.536 i 14.545 14.618 , 14.626 14.700 | 14.708

13.985 14.065 14.146 14.227 14.308 14.390 14.471 14.553 14.634 14.716

13.993 14.074 14.154 14.'235 14.316 14. 398 14.479 14.561 14.643 14.725

14.001 14.082 14.162 14.243 14.325 14.406 14.487 14.569 14.651 14.733

14.009 14.090 14.171 14.252 14.333 14.414 14.496 14.577 14.659 14.741

14.017 14.098 14.179 14.260 14.341 14.422 14.504 14.585 14.667 14.749

14.025 14.106 14.187 14.268 14.349 14.430 14.512 14.594 14.675 14.757

14.033 14.114 14.195 14.276 14.357 14.438 14.520 14.602 14.684 14.766

2. 70

.71

.72

.73

.74

.75

.76

.77

.78

.79

14.774    14.782    j 14.790

14.856    14.864    14.872

14.938    14.946    14.955

15.021    15.029    ■ 15.037

15.103    15.112    15.120

15.186    15.194    15.203

15.269    15.277    15.285

15.352    15.360    | 15.369

15.435    15.443    j 15.452

15.519    15.527    | 15.535

14.798 14.807 14.881    14.889

14.963    14.971

15.045 15.054 15.128 15.136 15.211    15.219

15.294    15.302

15.377    15.385

15.460 15.468 15.544 15.552

14.815 14.897 14.979 15.062 15.145 15.227 15.310 15.394 15.477 15.560

14.823 14. 905 14.988 15.070 15.153 15.236 15.319 15.402 15.485 15.569

14.831    14.839

14.913    14.922

14.996    15.004

15.078    15.087

15.161    15.169

15.244    15.252

15.327    15.335

15.410    15.419

15.494    15.502

15.577 15.585

14.848 14.930 15.012 15.095 15.178 15.261 15. 344 15.427 15.510 15.594

ibb 150—06-16

Table 3.—Discharge over a tlun-edged weir per foot of crest—Continued.

Head H, feet.

.000

.001

.002    .003

.004    .005    .006    .007

.008

2.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

15.602 15. 686 15.769 15.853 15.938 16.022 16.106 16.191 16.275 16.360

15.610 15.694 15.778 15.862 15.946 16.030 16.115 16.199 16.284 16.369

15.619 15.702 15.786 15.870 15.954 16.039 16.123 16. 208 16.292 16.377

15.627    15.635

15.711    15.719

15.795    15.803

15.879    15.887

15.963    15.971

16.047    16.056

16.132    16.140

16.216    16.225

16.301    16.309

16.386 I 16.394

15.644    15.652

15.728    15.736

15.811    15.820

15.895    15,904

15.980    15.988

16.064 I 16.072 16.148    16.157

16.233 j 16.242 16.318 ! 16.326 16.403 j 16.411

15.661    15.669    15.677

15.744    15.753    15.761

15.828    15.837    15.845

15. 912    15.921    15.929

15.997    16.005    16.013

16.081    16.089    16.098

16.165    16.174    16.182

16.250    16.258    16.267

16.335    16.343    16.352

16.420    16.428    16.437

2.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

3.00

16.445    16.454

16.530    16.539

16.616    16.624

16. 701    16.710

16.787    16.795

16.872    16.881

16.958    16.967

17.044    17.053

17.130    17.139

17.217    17.225

17.3033 ........

16.462 16.547 16.633 16. 718 16.804 16.890 16.975 17.062 17.148 17.234

16.471    j    16.479    16.488    16.496    16.505

16.556    16.565    16.573    16.582    16.590

16.641    |    16.650 ; 16.658    16.667    16.675

16.727    |    16. 735 I 16.744    16.752    16. 761

16.812    16.821    16.830    16.838    16.847

16.898    16.907    16.915    16.924    16.932

16.984    16.993    17.001    17.010    17.018

17.070    17.079    17.087    17.096    17.105

17.156    17.165    17.174    17.182    17.191

17.243    !    17.251    17.260    17.269    17.277

16.513    16.522

16.599    16.607

16.684    16.693

16.770    16.778

16.855    16.864

16.941    16.950

17.027    17.036

17.113    17.122

17.199    17.208

17.286    17.295

Table 4.—Discharge over a thin-edged weir per foot of crest.

Head H, feet.

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0    0.0000

.5    1.1773

1.5476 .7    1.9503

. 8    2.3828

. 9    2.8432

.6

0.0033    0.0094    0.0173

1.2128    1.2487    1.2849

1.5865    1.6257    j    1.6652

1.9922    2.0344    :    2.0770

2.4276 | 2.4727    |    2.5180

2.8907 I 2.9385    !    2.9865

0.0266    0.0372

.9719 ; 1.0052 1.3214    1.3583

1.7050 ; 1.7451 2.1198 ! 2.1629 2.5637 ; 2.6096 3.0348    3.0834

0.0489    0.0617

1.0389    1.0730

1.3955    1.4330

1.7855 j 1.8262 2.2063 | 2.2500 2.6558 | 2.7022 3.1322 | 3.1813

1.1074 j 1.1422 1.4709 I 1.5091 1.8673    1.9086

2.2940 I 2.3382 2.7490 ¹ 2.7959 3.2306 | 3.2802

1.0    3.3300

1.1    3.8418

1.2    4.3774

1.3    4.9358

1. 4    5. 5162

1.5 j 6.1176 1. 6 : 6.7394

1.7    ! 7.3810

1.8    j 8.0418

1.9    8.7212

3.3801 3.8943 4.4322

4.    9929

5.    5754 6.1789 6.8027 7.4463 8.1689 8. 7901

3.4304 j    3.4810    3.5318    3.5828    3.6342

3.9470    4.0000    4.0532    4.1067    4.1604

4.4873    4.5426    4.5981    4.6538    4.7098

5.0502    5.1077    5.1654    5.2233    5.2814

5.6348    5.6944    5.7542    5.8143    5.8745

6.2404 ,    6.3020    6.3633    6.4260    6.4883

6.8662    6.9299    6.9937    7.0578    7.1221

7.5117 ;    7.5773    7.6431    7.7091    7.7752

8.1762 |    8.2437 j 8.3113    8.3792 !    8.4472

8.8592    8.9285    j 8.9980 ! 9.0677'    9.1375

3.    6857    3.7375

4.    2143 4.2384

4.7660    4.8224

5.3398    5.3984

5.9350 i 5.9957 6.5508    6.6135

7.1865    7.2512

7.8416    7.9081

8.6154    8.5838

9.2075    8.2777

3.7895 4.3228 4.8790 5.4572 6.0565 6.6764 7.3160 7.9749 8.6524 9.3481

Table 4.—Discharge over a thin-edged weir per foot of crest—Continued.

Head H, feet.

.00

.01

.02

.03

.04

.05

.00

.07

.D8

.09

2.0 I 9.4187

2.1    ! 10.1340

2.2    10.8660

2.3    11.6150

2.4    12.3810

2.5    13.1630

2.6    13.9610

2.7    ¹ 14.7740

2.8    15. 6020

2.9    16.4450

9.4894 10.2060 10.9400

11.    6910

12.    4590 13.2480 14.0410 14.8560 15.6860 16.5300

9. 6314

9.5603 10.2790    10.3520

11.0150    11.0890

11.7670    11.8430

12.5360    12.6140

13.3210    13.4010

14.1220    14.2030

14. 9380    15.0210

15.7690    15.8530

16.6160    16.7010

9.7026    9.7741

10.4250    10.4980

11.1640    11.2390

11.9200 11.9960 12.6920    12. 7700

13.4800 13.5600 14.2840 14.3650 15.1030 15.1860 15.9380    16.0220

16.7870 16.8720

9.8457 10.5710 11.3140 12.0730 12.8480 12.6400 14.4470 15.2690 16.1060 16.9580

9.9174 10. §450 11.3890 12.1500 12.9270 13.7200 14.5280 15.3520 16.1910 17.0440

9.9894    10.0620

10.7180    10.7920

11.4640    11.5400

12.2270    12.3040

13.0050    13.0840

13.8000    13.8800

14.6100    14.6920

15.4350    15.5190

16.2760    16. 3600

17.1300    17. 2170

OHNMr}(iCt0GC005 CO* CO CO CO CO CO CO CO CO CO

65

tOtOtOlOtOtOHHMM Cn    4-    00    JO    H    O    JO    JO    CO    <1

05    05    Ol    *S|    00    00    CO    O    h-⁴

^05c0i^0<1050vl MMtOOi^MlOHOl C0 05 t0050t*4^t0t4

17.3899 18.2634 19.1515 20.0533 20.9690 21.8980 22.8405 23. 7962 24.7645 25.7459

17.4698 18.3516 19.2410 20.1442 21.0613 21.9917 22.9354 23.8924 24.8621 25.8748

17.5634 18.4399 19.3307 20.2354 21.1538 22.0856 23.0306 23. 9887 24.9600 25.9437

17. 6503 18.5285 19.4206 20.3267 21.2464 22.1795 23.1259 24.0852 25.0576 26.0429

17.7376 17.8248 18.6170    18. 7056

19.5105    19. 6007

20.4179    20.5C93

21.3390    21.4319

22.2734    22.3677

23.2211    23.3167

24.1818    24.2787

25.1555    25.2537

26.1422    26.2414

17.9124 18.7945 19.6910 20.6011 21.5248 22.4618 23.4122 24.3756 25.3520 26.3410

18.0000 18.8838 19.7812 20.6930 21.6180 22.5564 23.5081 24.4728 25.4502 26.4405

18.0876 18.9727 19.8718 20. 7849 21.7113 22. 6510 23.6040 24.5697 25.5488 26.5401

*.*.*.*. *.■ *. cO CO    C7I CO to o

^    TM    <0    Os    CO GO    lO

(£•    CO    to    to    G N    CO

S3 Si

.y:

26. 7399 27.7466 28.7652 29.7962 30.8391 31.8941 32.9607 34.0373 35.1288 36.2297

26.8401 27.8478 28.8678 29.9001 30. 9440 32.0003 33.0679 34.1475 35.2354 36.3406

26.9404 27.9494 28.9703 30.0040 31.0493 32.1065 33.1755 34.2560 35.3480 36.4515

27.0406 28.0509 29.0732 30.1079 31.1545 32.2128 33.2830 34.3646 35.4578 36.5624

27.1412    27.2417

28.1525    28.2544

29.1761    29.2790

30.2118    30.3163

31.2597    31.3649

32.3193 I 32.4259 33.3906    33.4985

34.4735    34.5824

35.5677    35.6780

36.6736    36. 7845

27.3423 28.3563 29.3823 30.4205 31.4705 32.5324 33.6064 34.6913 35.7882 36.8961

27.4432    27.5411

28.4582    28.5604

29.4855    29.5890

30.5251 ! 30.6297 31.5764 i 31.6820 32.6393    32.7462

33.7143    33. 8225

34.8005 | 34.9097 35.8984    36.0086

37.0073 37.1188

5.0 37.2304

5.1    38.3529

5.2    39.4865

5.3    40.6310

5.4    ! 41.7866

5.5    , 42.9523

5.6    44.1292

5.7    45.3166

6.8    46.5141

5.9    47.7226

37.3423 38.4658 39.6004 40. 7462 41.9024 43.0700 44.2474 45. 4359 46.6347 47.8438

37.4542 38.5787 39. 7146 40.5281 42.0186 43.1871 44.3659 45.5554 46.7552 47.9653

37.5661 38.6919 39.8288 40.9766 42.1352 43.3043 44.4845 45. 6746 46.8757 48.0869

37.6783 37.7905 j 37.9027 38.8052    38.9184    I 39.0319

39.9430    40.0576    j 40.1718

41.0919 41.2074    41.3230

42.2517 42.3683 j 42.4848 43.4219 42.5394 I 43.6573 44.6030 44.7216 ■ 44.8404 45.7945 45.9140    46.0339

46.9963 47.1172    47.2380

48.2084 j 48.3303    48.4522

38.0153 39.1455 40.2867 41.4386 42.6017 43.7752 44.9593 46.1538 47.3589 48.5744

38.1275 39.2591 40.4012 41.5544 42.7186 43.8931 45.0782 46.2740 47.4798 48.6963

38.2404

39.    3726

40.    5161 41.6703 42.8355 44.0109 45.1974 46.3939 47.6010 48.8185

6.0    48.9407

6.1    50.1694

6.2    51.4082

6. 3    52.6570

6.4    j 53.9157

6.5    ' 55.1832

6.6    j 56.4625 6. 7 I 57. 7505

6.8    I 59.0482

6.9    ? 60.3556

49.0632 50.2930 51.5324 52. 7822 54.0419 55.3116 56.5910 57.8801 59.1788 60.4868

49.1858 50.4162 51.6570 52.9077 54.1684 55.4392 56. 7192 58.0093 59. 3090 60.6183

49.3083 50.5401 51. 7818 53.0336 54.2950 55.5667 56.8478 58.1388 59.4428 60.7499

48.4312    49.5537    49.6766

50.6637    50.7875 50.9114

51.9034 52.0313    52.1531

53.1591    53.2850    53.4109

54.4219    54.5487    54.6756

55.6943    55.8221    55.9500

56.9766 57.1055 57.2340 58.2687 58.3982    58.5281

59. 5700    59. 7009 ! 59.8314

60.8814 I 61.0129 I 61.1445

49.7999 51.0356 52.2813 53.5371 54.8025 56.0779 57.3233 58.6580 59.9623 61.2763

49.9230    50.0462

51.1595    51.2837

52.4062    52.5314

53.6630 53. 7892 54. 9297    55.0569

56.2061    56.3343

57.4921    57. 6213

58. 7882    58.9180

60.0935    60.2244

61.4082 | 61.5404

Table 4.—Discharge over a thin-edged weir per foot of crest—Continued.

Head H, feet.

.00

.01

.02 .03 .04 .05

.00 .07 .08

8.0

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

p p p p

CO 10

9.4

9.5

9.6

9.7

9.8

9.9

10.0

61.6736 62.9986 64.3343 65.6793 67.0336 68.3969 69.7695 71.1508 72.5414 73.9410

61.8048 63.1318 64.4685 65.8145 67.1694 68.5337 69. 9070 71.2896 72. 6809 74.0815

61.9370 63.2650 64.6027 65. 9493 67.3053 68.6706 70.0449 71.4282 72.8208 74.2220

62.0692    62.2017

63.3992    63. 5317

64.7369    64.8711

66.0845    66.2197

67.4415    67.5777

68.8078    68. 9447

70.1827    70.3209

71.5670    71.7059

72.9603    73.1002

74.3626    74.5031

62.3343 63. 6653 65.0056 66.3552 67.7139 69.0818 70.4591 71.8451 73.2400 74.6439

62.    4671    62.6000

63.    7991    63.9327

65.1268    65.2750

66. 4908    66. 6263

67.8504    67.9869

69.2794 . 69.3566

70.    5973    70. 7355

71.    9843    72.1235

73.3802 73.5201 74.7848    74.9260

62. 7329 64.0665 65.4095 66. 7618 68.1235 69.4941 70.8737 72.2627 73. 6603 75.0669

62.8657 64.2004 65.5444 | 66.8977 i 68.2600 69.6316 71.0123 72.4743 73.8005 75.2081

75.3492 76. 7665 78.1924 79.6273 81.0705 82.5224 83.9833 85.4521 86.9297 88.4192

89.9100 91.4125 92.9237 94.4428 95.9703 97. 5057 99.0492 .00. 6010 .02.1607 .03.7282

75.4908    75.6320    75.7735

76.9087    77.0509    77.1934

78.3356    78.4788 78.6220

79. 7711    79.9153 80.0592

81.2154    81.3602    81.5054

82.6682    82.8141    82.9600

84.1298 | 84.2763 84.4228 85.5996    85.7472    85.8947

87.0778 87.2264 87.3745 88. 5647 88.7139 88.8630

75.9150    76.0569    76.1987

77.3360    77.4784    77.6210

78. 7655    78.9087    79.0522

80.2034    80.3479 80.4921

81.6503    81.7955    81.9406

83.1058    83.2517    ' 83.3979

84.5697    84.7165 84.8634

86.0455 86.1897    86.3376

87.5231    87.6716    87.8204

89.0126 89.1617    89.3113

76.3406    76.4824

77. 7638    77. 9067

79.1957    79.3396

80.6366    80. 7811

82.0862    82.2314

83.5440    83. 6902

85.0106    85.1578

86.4854    86.6336

87.9689    88.1178

89.4608    89.6103

76.6243 78.0496 79.4834 80.9260 82.3769 83.8367 85.3049 86.7815 88.2666 89.7602

90. 0599 91.5633 93.0782 94.5950 96.1234 97.6596 99.2040

100.7565 100. SI 23 101.0678 101.2237 102.3169 102.4734 102.6299 102.7868 103.8853 104.0429 104.2000 104.3575 105.4618 105. 6199 105.7781 |l05. 9363

90.2064 91. 7142 93.2267 94. 7475 96.2766 97.8138 99.3589

90.3599    90.5101

91.8650    92.0159

93.3785    93. 5304

94. 9000    95.0529

96.4298    96.5833

97.9679    98.1021

99.5141    99.6689

90.6602 92.1671 93.6822 95.2054 96. 7368 98.2763 99.8241 101.3799 102.9433 104.5121 106.0945

90.4778 92.3183 93.8341 95.3582 96.8903 98.4308 99.9793 101.5357 103.1001 104.6726 106.2530

90.9609 92.4694 93.9863 95.5111 97.0442 98. 5853 100.1344 101. 6919 103.2570 104.8304 106.4115

91.1115 92. 6206 94.1384 95. 6639 97.1977 98.7398 100.2899 101.8481 103.4141 104.9882 106.5700

91.2620 92. 7721 94. 2906 95.8171 97.3516 98.8943 100.4455 102.0042 103.5710 105.1461 106.7285

When applied to a weir with JV end contractions, the measured crest length ZI should be reduced by the formula

L^L'-O.IJSTH

When applied to a weir having appreciable velocity of approach, the measured head should be corrected by the correction formula of Francis (see p. 15), or by one of the simpler approximate equivalents; or the correction may be applied as a percentage to the discharge, by the use of Table 2.

Table 3, taken from Lowell Hydraulic Experiments, by James B. Francis, gives the discharge for heads from zero to 3 feet, advancing by thousandths.

Table 4 is original and gives the discharge for heads from zero to 10.09 feet, advancing by hundredths.

By increasing⁻ the quantities from either table 1 per cent, the discharge by the Cippoletti formula will be obtained,

Q= 3.36f Z/A

In calculating discharge by this formula, the head should be corrected for velocity of approach by the formula

II=D+1M.

TABLES 5 AND 6.—THREE-HALVES POWERS.

These tables of three-halves powers (cubes of the square roots) were prepared by the writer to facilitate the calculation of discharge over weirs of various forms, by the use of coefficients taken from the diagrams that accompany this paper and the base formula

Q= CLIP.

Table 5.—Three-halves powers for numbers 0 to 1.49.

Numbers.

.000 .001

.002

.003 .004 .005

.000 .007 .008

.000

.010

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0000j 0.0001 . 0010 .00118 .0028; . 00304 .0052    .00548

. 0080    . 00832

. 0112    .01155

. 0147    .01508

. 0185    . 01891

. 0226    .02304

. 0270    . 02746

0.0002 . 00136 . 00328 .00576 . 00864 .01190 .01546 .01932 . 02348 . 02792

0.0003 .00154 . 00352 . 00604 . 00896 . 01225 . 01584 . 01973 . 02392 .02838

0.0004 .00172 .00376 .00632 .00928 .01260 .01622 . 02014 . 02436 . 02884

0.0005 .00190 . 00400 .00660 .00960 . 01295 .01660 .02055 .02480 . 02930

0.0006 . 00208 . 00424 . 00688 .00992 .01330 .01698 .02096 . 02524 . 02976

0.0007 .00236 . 00448 .00716 . 01024 . 01365 .01736 . 02137 .02568 .03022

0.0008 .00244 .00472 .00744 . 01056 . 01400 . 01774 . 02178 .02612 .03068

0.0009 .00262 .00496 .00772 . 01088 . 01435 . 01812 . 02219 . 02656 . 03114

0.0010 .0028 .0052 .0080 .0112 .0147 .0185 .0226 .0270 .0316

0.10

.11

.12

.13

.14

.15

.16

.17

.18

.19

0.0316 .0365 .0416 .0469 .0524 .0581 .0640 .0701 .0764 .0828

0.20

.21

.22

.23

.24

.25

.26

.27

.28

0.0894 .0962 .1032 .1103 .1176 .1250 .1326 .1403 .1482 .1562

0.03356 0.03405 .03854    .03905

.04372    .04425

.04910    .04965

.05468    . 05525

.06046    .06105

. 06644    .06705

.07262    .07325

.07896    . 07960

.08544    .08610

0.03209 . 03701 . 04213 . 04745 .05297 . 05869 . 06451 . 07073 . 07704 .08346

0.09008 .09690 .10391 . 11103 .11834 .12576 . 13337 . 14109 .1490 .15701 0.03258 .03752 .04266 . 04800 .05354 .05928 . 06522 .07136 .07768 .08412 0.09076 .09760 .10462 .11176 . 11908 . 12652 . 13414 . 14188 .1498 .15782 0.03307 . 03803 . 04319 . 04855 .05411 . 05987 . 06583 . 07199 . 07832 . 08478 0.09144 .09830 . 10533 . 11249 . 11982 . 12728 . 13491 . 14267 .1506 .15863 0.09212 .09900 .10604 .11322 .12506 .12804 .13568 . 14346 .1514 ; 15944 0.09280 .09970 .10675 .11395 .12130 .12880 .13645 .14425 .1522 .16025 0.03454 . 03956 . 04478 . 05020 . 05582 . 06164 . 06766 . 07388 . 08024 .08676 0.09348 . 10040 . 10746 .11468 . 12204 .12956 .13722 .14504 1530 . 16106 0.03503 . 04007 . 04531 . 05075 .05639 . 06223 . 06827 . 07451 . 08088 . 08742 0.09416 .10110 . 10817 . 11541 .12278 . 13032 . 13799 . 14583 .1538 . 16187 00355.2 . 04058 . 04584 . 05130 .05696 . 06282 .06888 . 07514 . 08152 . 08808

0.09484 .1018 .10888 .11614 . 12352 .13108 .13876 . 14662 .1540 .16268

0.03601 0.0365 .04109 . 0416 .04637 .0469 .05185 .0524 .05753 .0581 .06341 .0640 .06949 . 0701 .07577 .0764 .08216 .0828 .08874 . 0894

009552 0.0962 .1025    .1032

.10959 .1103 .11687 .1176 .12426 .1250 .13184 .1326 .13953 .1403 .14741 .1482 . 1554    .1562

.16349 .1643

Numbers.

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

0.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

0.70

.71

.72

.73

.74

.75

.76

.77

.78

Table 5.—Three-halves powers for numbers 0 to 1.49—Continued.

.000 .001

0.1643 0.16513 . 1726    .17344

. 1810 .18186 . 1896    .19047

. 1983    .19918

. 2071    .20799

. 2160    .21691

. 2251    .22601

. 2342    . 23514

. 24361 .24454

0.2530 0.25395 . 2625i . 26347 . 2722    .27318

. 2820    .28299

.2919    .2929

. 3019    .30291

.3120    .31302

. 3222    .32323

. 3325    .33355

. 3430    . 34406

0.3536 .3642 .3750 .3858 .3968 .4079 .4191 .4303 .4417 .4532

0.4648 .4764 .4882 .5000 .5120 .5240 .5362 .5484 .5607 .5732

0.5857 .5983 .6109 .6237 .6366 .6495 .6626 .6757 .6889 . 7022

.002

.003

.004    .005

.000    .007

.008

.009

.010

0.16596 .17428 .18272 . 19134 . 20006 .20888 . 21782 . 22692 . 23608 . 24548

0.25490 .26444 .27416 . 28397 .2939 . 30392 .31404 . 32426 . 33460 .34512

0.16679 0.16762 0.16845 . 17512    .17596    .17680

. 18358    .18444    .18530

. 19221    .19308    .19395

. 20094    .20182    .20270

. 20977    .21066    .21155

. 21873    . 21964    . 22055

.22783    .22874!    .22965

. 23702    . 23796    . 23890

.24642    .24736;    .24830

0.16928 0.17011 .17764    .17848

.18616    .18702

. 19482    .19569

.20358    .20446

. 21244    . 21333

.22146    .22237

.23056    .2317

.23984    .24078

. 24924    .25018

0.17094 . 17932 .18788 . 19656 . 20534 . 21422 . 22328 .23238 . 24172 . 25112

0.17177 0.1726 .18016 .1810 .18874 .1896 .19743 .1983 .20622 .2071 .21511 .2160 .22419 .2251 .23329 .2342 .24266 .2436 .25206 .2530

0.25585 . 26541 . 27514 . 28497 .2949 . 30493 . 31506 . 32529 .33565 .34618

0.25680 0.25775 . 26638! . 26735 . 27612¹ .27710 . 28596! . 28695 . 2959 .30594 .31608    .31710

.32632    .32735

. 33670    .33775

. 34724    . 3483

.2969 .30695

0.25870 . 26832 . 27808 . 28794 .2979 . 30796 . 31812 . 32838 .33880 .34936

0.25965 .26929 . 27906 . 28893 .2989 . 30897 .31914 . 32941 . 33985 .35042

0.26060 . 27026 .28004 . 28992 .2999 . 30998 . 32016 .33044 .34090 . 35148

0.26155 0.2625 .2722 .2820 .2919 .3019 .3120 .3222 .3325 .3430 . 3536

.27123 .28102 .29091 .3009 .31099 .32118 .33147 .34195 .35254

0.35466 . 36528 . 37608 . 38690 . 39791 . 40902 . 42022 . 43144 .44285 .45436

0.46596 . 47758 . 48938 . 50120 .5132 . 52522 . 53742 . 54963 . 56195 .57445

0.58696 . 59956 . 61218 .62499 . 63789 . 65081 . 66391 . 67702 . 69023 . 70353

0.35572 . 36636 . 37716 . 38800 . 39902 . 41014 . 42134 . 43258 . 44400 . 45552

0.46712 . 47876 .49056 . 50240 .5144 . 52644 .53864 . 55086 . 56320 .5757

0.58822 .60082 .61346 . 62628 . 63918 . 65212 . 66522 . 67834 .69156 . 70486

0.35678 . 36744 . 37824 .38910 .40013 .41126 . 42246 . 43372 .44515 . 45668

0.35784 . 36852 . 37932 . 39020 . 40124 . 41238 . 42358 . 43486 .44630 .45784

0.35890 . 36960 . 38040 . 39130 . 40235 . 41350 . 42470 . 43600 . 44745 . 45900

0.35996 . 37068 . 38148 . 39240 . 40346 . 41462 . 42582 . 43714 . 44860 . 46016

0.36102 . 37176 .38256 . 39350 . 40457 .41574 .42694 .43828 .44975 . 46132

0.36208 . 37284 . 38364 . 39460 . 40568 . 41686 . 42806 . 43942 . 45090 . 46248

0.36314 . 37392 .38472 .3957 .40679 . 41798 .42918 .44056 .45205 . 46364

0.3642 .3750 .3858 .3968 .4079 .4191 .4303 .4417 .4532 .4648

0.46828 .47994 . 49174 .5036 .5156 . 52766 . 53986 . 55209 . 56445 . 57695

0.46944 .48112 .49292 .5048 .5168 .52888 .54108 . 55332 . 56570 .57820

0.47060 . 48230 . 49410 .5060 .5180 . 53010 . 54230 . 55455 . 56695 . 57945

0.47176 . 48348 . 49528 .5072 .5192 . 53132 .54352 . 55578 . 56820 . 58070

0.47292 . 48466 . 49646 .5084 .5204 .53254 .54474 . 55701 .56945 . 58195

0.47408 .48584 .49764 .5096 .5216 .53376 .54596 .55824 .57070 .58320

0.47524 0.4764 .48702 .49882 .5108 . 5228 .53498 .54718' .5484 .55947; .5607 .67195; .5732 .584451 .5857

.4882 .5000 . 5120 .5240 .5362

0.58948 . 60208 . 61474 . 62757 . 64047 . 65343 . 66653 . 67966 . 69289 . 70619

0.59074 . 60334 . 61602 .62886 . 64176 . 65474 .66784 .68098 .69422 .70752

0.59200 . 60460 . 61730 . 63015 . 64305 . 65605 . 66915 .68230 .69555 .70885

0.59326 .60586 . 61858 . 63144 . 64434 . 65736 . 67046 . 68362 . 69688 . 71018

0.59452 .60712 .61986 . 63273 . 64563 .65867 . 67177 .68494 . 69821 . 71151

0.59578 .60838 .62114 . 63402 . 64692 . 65998 .67308 .68626 .69954 . 71284

0.59704 0.5983 .60964 .62242 .63531 .64821 . 66129 . 67439 .68758 .70087 .71417

.6109 .6237 .6366 .6495 .6626 .6757 . 6889 .7022 . 7155

Numbers.

0.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

0.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

1.00

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

l' 09

1.10

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22

1.23

1.24

1.25

1.26

1.27

1.28

1.29

Table 5.—Three-halves powers for numbers 0 to 1.49—Continued.

.000 .001

0.7155 0.71685 . 7290    .73035

. 7425    .74387

. 7562    .75757

. 7699    .77128

. 7837    .78508

. 7975    . 79890

. 8115    .8129

.8255    .82691

.8396

0.8538 .8681 .8824 . 8969 .9114 .9259 .9406 .9553 .9702 .9850

1.0000 1.0150 1.0302 1.0453 1.0606 1.0759 1.0913 1.1068 1.1224 1.1380

1.1537 1.1695 1.1853 1.2012 1.2172 1.2332 1. 2494 1.2656 1.2818 1.2981

1.3145 1.3310 1.3475 1.3641 1.3808 1.3975 1.4144 1.4312 1.4482 1.4652

.002

.00!}

.004

.005

.000

.007    .008    .009

.010

0.71820 0.71955 0.72090 . 7317    . 73305    .73440

. 74524    . 74661    .74798

. 75894    . 76031    .76168

. 77266    . 774C4    .77542

.78646    . 78784. .78922

.80030    .80170    .8031

. 8143    .8157    .8171

.82832    .82973    .83114

.84102    .84244    .84386    .84528

0. 72225 .73575 . 74935 .76305 .77680 . 79060 . 8045 . 8185 .83255 .84670

0.85523 . 86953 . 88385 . 89835 . 91285 . 92737 . 94207 . 95679 . 97168 .9865

0.85666 . 87096 .8853 .89980 .9143 .92884 .94354 . 95828 . 97316 .9880

1.0015    1. 0030

1.01652 1.01804 1.03171 1.03322 1.04683 1.04836 1.06213 1.06366 1.07744 1.07898 1.09285 1.09440 1.10836 1.10992 1.12396 1.12552 1.13957 1.14114

0.85809 0.85952 0.86095 .87239    .87382    .87525

. 88675    . 88.820    . 88935

.90125    . 90270    . 90415    .    9056

. 91575    .91720    .91865

.93081    .93178    . 93325

.94501    .94648    .94795

.95977    .96126    .96275

.97464    . 97612    . 97760

.9895    .9910    .9925

0.86238 0.86381 .87668    .87811

. 89110    .89255

. 90705 .92010    .92155

. 93472!    .93619

. 94942;    .95089

. 96424    . 96573

. 97908    .98056

.9940 ¹ .9955

0.86524 .87954 .8940 .9085 .9230 .93766 . 95236 . 96722 . 98204 .9970

0.86667 .88097 .89545 .90995 .92445 .93913 .95383 .96871 . 98352 .9985

0.8681 .8824 .8969 .9114 .9259 . 9406 . 9553 . 9702 . 9850 1.0000

1.0045 1.01956 1. 03473 1.04989 1.06519 1.08052 1. 09595 1.11148 1.12708 1.14271

1.0060 1.02108 1.03624 1.05142 1.06672 1.08206 1.09750 1.11304 1.12864 1.14428

1.0075 1.02260 1. 03775 1.05295 1.06825 1.08360 1.09905 1.11460 1.13020 1.14585

1.15528 1.15686 1.15844 1.16002 1.17108 1.17266 1.17424 1.17582 1.18689 1.18848 1.19007 1.19166 1.20280 1.20440 1.20600 1.20760 1.21880 1.22040 1.22200 1.22360 1.23482 1.23644 1.23806 1.23968 1.25102 1.25264 1.25426 1.25588 1.26722 1.26884 1.27046 1.27208 1.28343 1.28506 1.28669 1.28832 1.2997l! 1.30138 1.30302 1.30466

1.31615' 1.31780 1.33265¹ 1.33430 1.34916¹ 1.35082 1.36577¹ 1.36744 1.38247¹ 1.38414 1.39919 1.40088 1.41608 1.41776 1.4329 i 1.4346 1.4499 j 1.4516 1.4669 i 1.4686

1.31945 1.32110 1.33595 1.33760 1.35248 1.35414 1.36911 1.37678 1.38581 1.38748 1.40257 1.40426 1.41944 1.42112 1.4363    1.4380

1.4533    1.4550

1.4703    1.4720

1.16160 1.17740 1.19325 1.20920 1.22520 1.24130 1.25750 1.27370 1.28995 1.30630

1.32275 1.33925 1.35580 1.37245 1.38915 1.40595 1.42280 1. 4397 1.4567 1.4737

1.0090 | 1.0105 1.02412; 1.02564 1.03926 1.04077 1.05448 1.05601 1.06978 1.07131 1.08514¹ 1.08668 1.10060 1.10215 1.11616 1.11772 1.13176' 1.13332 1.14742 1.14899

1.16318; 1.16476 1.17898 1.18056 1.19484 1.19643 1.21080 1.21240 1. 22680! 1.22840 1.24292' 1.24454 1.25912 1.26074 1.27532; 1.27694 1.29158 1.29321 1.30794| 1.30958

1.32440; 1.32605 1.34090 1.34255 1.35746 1.35912 1.37412 1.37579 1.39082 1.39249 1.40764 1.40933 1.42448 1.42616 1.4414 1.4584 1.4754

1.4431 1.4601 1. 4771

1.0120 1.02716 1.04282 1.05754 1. 07284 1.08822 1.10370 1.11928 1.13488 1.15056

1.16634 1.18214 1.19802 1.21400 1.23000 1.24616 1.26236 1.27856 1.29484 1.31122

1.32770 1.34420 1.36078 1.37746 1.39416 1.41102 1. 42784 1.4448 1.4618 1.4788

1.0135 1.0150 1.02868 1. 0302 1.04379 1. 0453 1.059071.0606 1.07437 1.0759 1.089761.0913 1.10525 1.1068 1.12084 1.1224 1.13644 1.1380 1.15213 1.1537

1.167921.1695 1.18372 1.1853 1.19961 1.21560 1.231601.2332 1. 24778 1.2494 1. 26398 1.2656 1.28018 1.2818 1.29647 1.2981 1.312861.3145

1. 32935 1.3310 1. 34585,1.3475 1.36244 1.3641 1.37913 1.3808 1.39583 1.3975 1. 412711.4144 1.42952 1.4465 1.4635 1.4805

1. 2012 1.2172

1.4312 1.4482 1.4652 1.4822

Numbers.

1.30

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

1.40

1.41

1.42

1.43

1.44

1.45

1.46

1.47

1.48

1.49

0.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.10

.11

.12

.13

.14

.15

.16

.17

.18

Table 5.—Tliree-halves powers for numbers 0 to 1.49—Continued.

.000 .001

.002

.003

.004 | .005

.006

.007

.008

.009

.010

1.4822 1.48392 1.48564 1.4994 1.50112 1.50284 1.5166 1.51832 1.52004 j 1.5338 1.53554 1.53728 1.5512 1.55294 1.55468 1.5686 1.57034 1.57208 j 1.5860 1.58775 1.58950 1.6035 1.60526 1.60702 1.6211 1.62287 1.62464 1.6388 1.64057 1.64234

1.6565 1.65828 1.66006 1.66184 1.6743 1.67608 1.67786 1.67964 1.6921 1.69389 1.69568 1.69747 1.7100 1.7118 1.7136    1.7154

1.7280 1.7298    1.7316    1.7334

1.7460 1.74781 1.74962 1.75143 1.7641 1.76592 1,76774 1.76956 1.7823 1.78412 1.78594 1.78776 1.8005 1.80233 1.80416 1.80599 1.8188........................

1.48736 1.48908 1.50456 1.50628 1.52176 1.52348 1.53902 1.54076 1.54250 1.55642 1.55816 1.55990 1.57382 1.57556 1.59125 1.59300 1.59475 1.60878 1.61054 1.61230 1.62641 1.62818 1.62995 1.64411 1.64588,' 1.64765

1.66362 1.66540 1.66718 1.66896 1.68142 1.68320 ' 1. 68498 1.68676 1.69926 1.70105 1.70284 1.70463 1.7172    1.7190    j 1.7208    1.7226

1.7352    1.7370    1.7388    1.7406

1.75324 1.75505 1.75686 1.75867 1.77138 1.773201 1.77502 1.77684 1.78958 1.79140 1.79322 1.79504 1.80782 1.80965 1.81148 1.81331

1.49080 1.49252 1.50800! 1.50972 1.52520 1.52692

1.49424 1.49596 1.49768 1.4994 1.51144 1.51316 1.51488 1.5166 j 1.52864 1.53036 1.532081.5338 1.54424 1.54598 1.54772 1.549461.5512 1.56164 1.56338 1.56512 1.566861.5686 1.57730 1.57904 1.58078 1.58252 1.584261.5860 1.59650 1.59825 1.60000 1.601751.6035 1.61406 1.61582 1.61758 1.619341.6211 1.63172 1.63349 1.63526 1.637031.6388 1.64942 1.65119 1.65296 1.654731.6565

1.67075 1.672521.6743 1.68854 1.690321.6921 1. 70642 1.708211.7100 1.7244    1.7262 1.7280

1.7424    1.7442 1.7460

1.76048 1.762291.7641 1.77866 1.780481.7823 1.79686 1.798681.8005 1.81514 1.816971.8188 j

Table 6.—Three-halves powers for numbers from 0 to 12.

10

11

0.0000,1. 0000:2.8284 5.1962 .00101.0150 2. .8497 5.2222 . 00281.0302 2.8710 5.2482 . 00521.0453 2.8923 5.2743 . 00801.0606 2.9137 5.3004 . 01121.0759 2. 9352 5.3266 . 01471.0913 2.9567 5.3528 . 01851.1068 2.9782 5.3791 . 02261.1224 2. 9998 5.4054 . 02701.1380^3.0215 5.4317

*0.03161.1537 3.0432 5.4581 j . 0365 1.1695 3.0650 5.4845 .04161.1853 3.0868 5.5110 ! .04691.2012 3.1086 5.5375 . 05241.2172,3.1306 5.5641 . 05811.2332i3.1525 5.5907 . 06401.2494 3.1745 5.6173 ! .07011.2656 3.1966 5.6440 . 07641.2818 3. 2187 5. 6708 . 08281. 2981,3.2409 5. 6975

8.000011.180314.696918.5203 8.030011.213914.7337 8.060111.2475 14. 7705 8. 0902 11.281114.8073 18.6394 8.1203 11.3148 14.8442 18.6792 8.1505 11. 348514.8810 18. 7190 8.1807 11. 3822 14.9179 18.7589 8.2109 11.4160 14.9549 18. 7988 22.9251 8.2412 11.4497 14.9919 18.8387 22.9677 8.271511.4836 15.0289 18.8786 23.0103

22.6274 27.0000j31.6228 18.5600 22.6699 27.0450 31.6702 18.599722. 712327.089031.7177 22.7548 27.135131. 7652 22. 7973 27.1802(31.8127 22.8399 27.2253,31.8602 22.8825 27.2705,31. 9078 27.3156 31.9554 27.3608 32.0030 27.4060 32.0506

8.3019 11.5174 15.065918.9185 8.3323 11. 5513 15.103018.9585 8.3627 11.5852 15.1400 18.9985 8.3932 11.6192 15.1772 19.0386 8.4237 11. 6532 15.2143 19.0786 8.454211.6872 15.2515 19.1187 8.484811. 7213 15.288719.1589 8.5154 11. 7554 15.326019.1990 8.546011. 7895 8.5767 11.8236

36.4829 36.5326 36.5824 36.6322 36. 6820 36. 7319 36. 7818 36. 8317 36. 8816 36. 9315

23.0530 27.4512 32.0983    36. 9815

23.0957 27.4965 32.1460 37.0315 23.1384 27.5418 32.1937    37.0815

23.1812 27.5871 32.2414    37.1315

23.2240 27.6324 32.2892    37.1816

23.2668 27. 6778 32. 3370 37.2317 23.3096 27. 7232 32.3848    37.2817

23.3525 27.7686 32.4326    37.3319

37.3820 37.4322

15.363219.2392 23.3954 27.8140 32. 4804 15.4005 19. 2794 23. 4383 27.8595,32.5283

0.20

.21

.22

.23

.24

.25

.26

.27

.28

.29

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

0.60

.61

.62

.63

.64

.65

.66

.67

.68

Table 6.—Three-halves powers for numbers from 0 to 12—Continued.

0.08941. 3145 . 0962 1.3310 . 10321. 3475 . 11031.3641 . 11761.3808 . 12501.3975 .13264.4144 .14031.4312 . 1482 1.4482 | .15621.4652

3.2631 5.7243 3.2854 5.7512 3.3077 5.7781 3.3301 5.8050 3.3525 5.8320 3.3750 5.8590 3.3975 5.8861 3.4201 5.9132 3.4427 5.9403 3.4654 5.9675

8.6074 11.8578 8.638211.8920 8.669011.9263 8.699811.9606 8. 7307 11.9949 8.7616 12.0293 8. 792512.0636 8.823512.0981 8.854512.1325 .8.8856;12.1670

15.437919.3196 15.475219. 3599 15.5126 19.4002 15.5501 19.4405 15.586619.4808 15.625019. 5212 15.6616 19.5576 15. 7001 19.6021 15.7376 19. 6425 15. 775219. 6830

23.4812 23.5242 23.5672 23.6102 23. 6533 23.6963 23.7394 23.7825 23.8257 23.8689

10

11

27.9050 32.5762 ! 37.4824 27.9514 32.6241j 37.5326 27.9960 32. 6720 , 37.5828 28.0416 32.7200    37.6331

28.0872 32.7680    37. 6833

28.1328 32.8160    37. 7336

28.1784 32.8640    37. 7840

28.2241 32.9121    37.8343

28.2698 32.9600    37.8847

28.3155133.0083 ' 37.9351

!0.16431.4822 >3.4881 5.9947 . 1726 1.4994 3.5109 6.0220 . 1810 1. 5166 3. 5337 6.0493 . 1896 1.5338 3.55666.0767 . 1983,1.5512 3.5795 6.1041 . 2071 1.5686 3.6025 6.1315 . 21601.5860 3.6255 6.1590 . 22511. 6035 3.6486 6.1865 . 23421.6211 3. 6717 6.2141 .24361.6388 3.6949 6.2417

0.2530|1.6565 .26251.6743 .27221.6921 . 28201.7100 .29191.7280 . 30191.7460 .31201.7641 .32221.7823 . 33251.8005 : .34301.8188

3.7181 6.2693 3.7413 6.2970 3.7646 6.3247 3.7880 6.3525 3.8114 6.3803 3.8349 6.4081 3.8584 6.4360 3.8819 6.4639 3.9055 6.4919 3.9292 6.5199

0.35361.8371 3.9529|6.5479 . 3642 1.8555 .37501.8740 . 38581.8925 .3968 .4079 .4191 .4303 .4417 .4532

3.9766:6.5760 4.0004 6.6041 4.0242 6.6323 1.9111 4.0481 6.6605 1.9297 4.0720 6.6887 1.9484 4.0960 6.7170 1.9672 4.1200 6. 7453 1.9860 4.1441 6.7737 2.0049 4.1682 6.8021

0.4648 2.0238 4.1924 6.8305 . 4764 2.0429 4.2166 6.8590 . 4882 2.0619 4.2408 6.8875 . 5000 2.0810 4.2651 6. 9161 . 5120 2.1002 4.2895 6.9447 . 5240 2. II95J4.3139 6.9733 . 5362 2.1388 4.3383 7.0020 . 5484 2.1581 4.3628 7.0307 .5607 2.1775 4.3874 7.0595 . 5732 2.1970 4.4119 7.0883

8.9167|l2.201515.812919. 7235 23.9121 28.3612 33.0564 8.9478|l2.2361115.8505 19.7641 23.9553 28.4069 33.1046 8. 979012.2706 15.888219.8046 23.9986 28.4527 33.1527 9.0102 12.3053115.926019. 8452 24. 0418 28.4985 33.2009 9.041442. 339915. 9637 19.8858 24.0851 28.5444 33.2492 9.0726 12.374616.001519.9265 24.1285 28.5902 33.2974 9.104012.4093 j 16.0393 19.9672 24.1718 28.6361 33.3457 9.1353jl2.4440 16.0772 20. 0079 24.2152 28.6820 33.3940 9.166712.478816.1150 20.0486 24.2586 28.7279 33.4423 9.198112.513616.1529 20.0894 24.302128.7739 33.4906

9.229512.5485 16.19.09 20.1302 24.3455 28.8199 33.5390 9.261ojl2.583316. 2288 20.1710 24. 3890|28.8659 33.5874 9.2925 12.618246.2668 20. 2118 24.4325 28. 9119 33.6358 9.324112.653216.3048 20.2527 24.476128.9579 33.6842 9.3557 12.688216.3429 20.2936 24.5196 29.004033.7327 9.387312. 723216.3810 20.3345 24.563229.050433.7811 9.418912. 758216.4191 9.450612.793316.4572 9.482412.828416.4954 9.514112.863516.5336

20.3755 24.6068J29.0962 33.8297 20.4165 24. 6505 29.1424 33.8782 20.4575 24.6941 29.1885 33.9267 20.4985 24.7378 29. 2347133.9753

37.9855 38.0359 38.0864 38.1369 38.1874 38.2379 38.2884 38.3390 38.3896 38.4402

38.49Q8 38.5415 38.5922 38. 6429 38.6936 38.7443 38.7951 38.8459 38.8967 38.9475

9.5459 12.8986 9.577812.9338 9.6097 12.9691 9.641613.0043 9.6735 13.0396 9. 705513.0749 9.737513.1103 9.769513.1457 9.801613.1811 9.833713.2165

9.8659 9.8981 9.9303 9.9626 9.9949 10.0272 10.0596 10.0920 10.1244 10.1569

13.2520 13.2875 13.3231 13.3587 13.3943 13.4299 13.4656 13.5013 13.5370 13.5728

16.5718 16.6101 16. 6484 16. 6867 16.7250 16.7634 16.8018 16.8402 16.8787 16. 9172

16.9557 16.9943 17.0328 17.0714 17.1101 17.1488 17.1874 17.2172 17.2649 17.3037

20.5396 24. 7815 29.2810|34.0239 20.5807 24.8253 29.3272 34.0725 20.6218 24. 8691 29.3735 34.1211 20.6630 24.9129 29.4198 34.1698 24.956729.466l|.34.2185 20. 7453 25.0005 29.512434.2672 20. 7866 20.8278 20.8691 20.9104

20.7041

25.0444 25.0883 25.1322 25.1762

20.9518 20.9931 21.0345 21.0759 21.1174 21.1589 21.2004 21.2419 21.2834 21.3250

29.5588 34. 3159 29.6052 34.3647 29.6516 34.4135 29.6980 34.4623

38.9984 39.0493 39.1002 39.1511 39.2020 39.2530 39.3040 39.3550 39.4060 39.4571

25.2202 29.7445 34.5111 25.2642 29.7910 34.5599 25.3082 29.8375 34.6088 25. 3522 29.884134.6577 25.3963 29.9306 34.7066 25.4404 29.9772 34.7557 25.4845 30.0238 34.8045 25.5287 30.0704 34.8535 25.5729 30.1171 34.9025 25.6171 30.1638 34.9516

39.5082 39.5593 39.6104 39.6615 39.7127 39.7639 39.8151 39.8663 39.9176 39.9689

Table 6.—Three-halves powers for numbers from 0 to 12—Continued.

10

it

0.70

.71

.72

.73

.74

0.5857 2.2165 . 5983 2.2361 . 6109 2.2558 .6237 2.2755 . 6366 2.2952

4.436617.1171 4.4612j7.1460 4.4859 7.1749 4.5107 7.2038 4.5355 7.2328

10.1894 13. 6086 10.2214 13.6444 10.254513.6803 10.287113. 7161 10.3197 13.7521

30.2105

(2

.75

.77

.78

.79

4202:21.4499J25.7499 30.3040 17.4591:21.4916 25. 7942 30. 3507 17.4981 21.5333 25.8395 30. 3975 17. 5370|21.5751 25.8828 30.4444 17. 5760,21.6169 25.9272 30.4912 17. 6150 21.6587'25.9716

. 6889 2.3748i4.6352!7.349210.4506 13.896117,6541 21.7005 26.0161 30.5850 35.3939 . 7022 2.3949 4.6602 7.3783 10.4834 13. 9321 17.693121.7423,26.0605 30.6319'35.4431

. 6495 2.3150 4.5604,7.261810.3524 13. 7880 . 6626 2.3349 4.5853 7.2909 10.385113.8240 . 6757 2.3548 4.6102:7.3200 10.4178 13.8600

30.5381 35.3446

40.0202 40.0715 40.1.228 40.1742 40.2256 40.2770 40.3284 40.3798 40.4313 40.4828

0.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

0.7155 2.4150 4.6853 7.4076 10.5163 13.9682 17.7322 21.7842 26.1050 30.6789 35.4924 . 7290 2.43514. 7104 7.4368,10.5492 14.0044 17.7714 21.826126.1495 30.7258 35.5417 . 7425 2.4553 4.7356 7.4661,10.5812 14.0406 17.8105 21.8681 26.1941 30. 7728 35.5911 . 7562 2.4756 4.7608 7.4955^10.6150 14. 0768 17.8507 21.9100 26.2386 30.8198 35. 6404 . 7699 2.4959 4.7861 7.5248 10.648014.113017.8889|21.9520 26.2832 30.8669 35. 6898 . 7837 2. 5163 4.8114 7.554210.6810 14.1493,17.9282|21.9940 26.3278 30.9139 35.7392 . 7975 2.5367 4.8367 7. 5837 10. 7141 14.1856 17.9674 22.0361 26.3725 30.9610 35.7886 . 8115 2.5572 4.8621 7.6132 10. 7472 14.2219 18. 0067 22.078126.4171 31.0081 35.8380 . 8255 2.5777 4.8875 7.6427 10. 7803 14.2582 18. 0461l22.1202 26.4618 31.0553 35.8875 . 8396 2.6983 4.9130 7.6723 10.8134 14.2946 18.0854 22.1623 26.5065 31.1024 35.9370

40.5343 40.5859 40.6374 40.6890 40.7406 40. 7922 40.8439 40.8955 40.9472 40.9989

0.90

.91

.92

.93

.94

.95

.96

.97

.98

.99

0.8538 .8681 .8824 .8969 .9114 .9259 .9406 .9553 .9702 .9850

1.00

2.6190 4.9385 2.6397 4.9641 2.6604 4.9897 2. 6812 5.0154 2. 7021 5.0411 2.7230 5.0668 2.7440 5.0926 2.7650 5.1184 2.7861 5.1443 2.8072 5.1702 2.8284 5.1962

7.701910.8466 14. 331118.1248j22.204526.552331.149635. 9865 7.7315 10.8798 14.3675 18.1642:22. 2467 26.5960 31.1968,36.0360 7. 7702 10.9131114.4040 18.2037 22.2889 26.6408 31.2441 36.0856 7. 7909 10.9464 14.4405 18. 2432 22.3311 26.6856 31.2913 36.1352 7.820710.9797 14.4770 18.2827 22.3733 26.7305!31.3386'36.1848 7.850511.013114. 5136 18.3222 22.4156 26.7753:31.3850 36. 2344 7.8803 11.0464 14. 5502 18.3617 22.4579 26.8202 31.4332 36.2841 7.910211.0799 14.5869 18.4013 22.5003 26.8651 31.480636.3337 7.940111.1133 14. 6235 18.4409 22.5426 7.970011.1468 14. 6602 18.4806 22.5850 8.000011.1803 14.6969 18. 5203 22.6274

26.9100 26.9550 27.0000

31.5280 36. 3834 31.5754 36.4331 31.6228 36.4829

41.0507 41.1024 41.1542 41.2060 41.2578 41.3097 41.3615 41.4134 41.4653 41.5173 41.5692

The tables of three-halves powers may conveniently be used in conjunction with Crelle’s Rechentafeln, or similar tables of the products of pairs of factors. C will usually be constant, or nearly so. Entering Crelle’s tables with C or CL as an argument, the discharge

corresponding to values of II'² read from the tables here given maj^ be taken out directly, and usually with sufficient precision at least for 1 foot length of crest, without any arithmetical computation. Table 5

gives II² for values of II from zero to 1.5 feet, advancing by thousandths. In Table 6 the increment is 0.1 foot, and the range zero to

12 feet. Should H² be required for larger values of H, it may be found from the three-halves power of ’ by the formula

sj

00

bM

lolcc

TABLE 7.—FLOW OVER BROAD-CRESTED WEIRS, WITH STABLE

NAPPE.

This table gives values of

where

Q= C.Zffi,

L=1

6'₁ = 2.64

The derivation of this coefficient is given in connection with discussion of broad-crested weirs (pp. 119-121). It may be applied to broad-crested weirs of any width of cross section exceeding 2 feet within such limiting heads that the nappe does not adhere to the downstream face of the weir for low heads nor tend to become detached with increased head. Under the latter condition the coefficient increases to a limit near the value which applies for a thin-edged weir, a point being finally reached where the nappe breaks entirely free from the broad crest and discharges in the same manner as for a thin-edged weir. The coefficient, 2.64, may often be applied for weirs exceeding 2-feet crest width and for heads from 0.5 foot up to 1.5 or 2 times the breadth of weir crest. If corrections for the velocity of approach are required the Francis correction formula, or its equivalent, should be used.

Table 7.—Weir discharge per foot of crest length.

[Coefficient Ci=2.64.]

Head H, feet.!

(i

cs

I—i

0.00

.01

.02

0.000 j 2.64 .003 i 2.68 .007 | 2.72

7.47

7.52

7.58

.03

.014

2.76

7.64

.04

.021

2.80

7.69

.05

.030

2.84

7.75

,06

.039

2.88

7.81

.07

.049

2.92

7.86

.08

.060

2.96

7.92

.09

.071

3.00

7.98

0.10

0.083

3.04

8.03

.11

.096

3.09

8.09

.12

.110

3.13

8.15

.13

.124

3.17

8.21

.14

.138

3.21

8.26

.15

.153

3.26

8.32

.16

.169

3.30

8.38

.17

.185

.18

.202

3.34 ; 3.38

8.44

8.50

.19

( . 218

3.43

8.56

S S E E K K K E EE EEEEEE55SS

OOtOCOOOSIOCnOi^    COWtOMMO©M(X)vi

29.5

29.8

29.7

29.8

29.9

30.0

30.0

30.1

30.2

30.3

30.4

30.5

30.6

30.7

30.8

30.8

30.9

31.0

31.1

31.2

ggggggggg&    &£)&£&&&&&&

cOOO^IOiOl^COIOOcO    COvlOlCJi^WWMOO

59.7

59.8

59.9

60.1

60.2

60.3

60.4

60.5

60.6

60.7

60.8

61.0

61.1

61.2

61.3

61.4

61.5

61.6

61.8

61.9

W W CO CO CO io to to !C to liioIofCHHMHMH O'^WtO HtOCCMOJO' WIOHOcDvlOlOi^CO

Table 7.—Weir discharge per foot of crest length—Continued.

Head II, feet.

0.20

0.236

3.47    8.61

15.1

22.7

31.3

40.8

.21

.254

3.51

8.67

15.2

22.8

31.4

40.9

.22

.272

3.56

8.73

15.2

22.9

31.5

41.0

.23

. 291    3.60

8.79

15.3

23.0

31.6

41.0

.24

.310

3.64

8.85

.25

.330

3.69

8.91

15.4    j 23.0

16.5    I 23.1

31.7

31.8

41.1

41.2

O'    O'    O'    in    Cn

r    r    r    r    r

03    (O    M    O

62.0

73.7

86.0

62.1

62.2

62 3

62.4

62.6

73.8

73.9

74.0

74.1

74.3

86.1

86.2

86.4

86.5    |

86.6

.26

.350

3. 73

8.97

15.5

23.2

31.8

41.3

51.6

62.7

74.4

86.8

.27

.370

3.78

9.03

.28

. §91

3.82

9.09

.29

.412

3.87

9.15

15.6    | 23.3

15.7    23.4

15.8    I 23.4

31.9

32.0

32.1

41.4

41.5

41.6

51.7

51.9

52.0

62.8

62.9

63.0

74.5

74.6

74.8

86.9

87.0

87.1

0.30

0.434

3.91

9.21

15.8    23.5

32.2

41.7

52.1

63.1

74.9

87.3

.31

.456

3.96

9.27

.32

.478

4.00

9.33

15.9 i 23.6 16.0 ! 23.7

32.3

41.8

52.2

63.2

75.0

87.4

32.4

41.9

52.3

63.4

75.1

87.5

.33

.500

4.05

9.39

16.0

23.8

32.5

42.0

52.4

63.5

75.2

87.6

.34

.524

4.10

9.45

16.2

23.9

32.6

42.1

52.5

63.6

75.4

87.8

.35

.547

4.14

9.51

16.2

24.0

32.7

42.2

52.6

63.7

75.5

87.9

.36

.570

4.19

9.57

16.3

24.1

32.8

42.3

52.7

63.8

75.6

88.0

.37

.594

4.23

9.63

16.3

24.1

32.8

42.4

52.8

63.9

75.7

88.2

.38

.618

4.28

9.69

16.4

24.2

32.9

42.5

52.9

64.0

75.8

88.3

.39

.643

4.33

9.75

16.5

24.3

33.0

42.6

53.0

64.2

76.0

88.4

0.40

0.668

4.37

9.82

.41

.693

4.42

9.88

.42

.719

4.47

9.94

33.1

42.7

53.1

64.3

76.1

88.5

33.2

42.8

53.2

64.4

76.2

88.7

33.3

42.9

53.4

64.5

76.3

88.8

.43

.744

4.51

10.0

16.8

24.6

33.4

43.0

53.5

64.6

76.4

88.9

.44

.771

4.56

10.1

16.8

24.7

33.5

43.1

53.6

64.7

76.6

89.0

.45

.797

4.61

10.1

16.9

24.8

.46

.824

4.66

10.2

17.0 J 24.9

33.6

33.7

43.2

53.7

64.8

76.7

89.2

43.3

53.8

65.0

76.8

89.3

.47

.851

4.70

10.2

17.1

24.9

33.8

43.4

53.9

65.1

76.9

89.4

.48

. 878    4. 75

10.3

17.1

25.0

33.9

43.6

54.0

65.2

77.0

89.6

.49

.905

4.80

10.4

17.2

25.1

34.0

43.6

54.1

65.3

77.2

89.7

0.50

0.934

4.85

10.4

17.3

25.2    34.0

43.7

54.2

65.4

77.3

89.8

.51

.961

4.90

.52

.990

4.95

.53

1.02

5.00

.54

1.05

5.04

.55

1.08

5.09

.56

1.11

5.14

.57

1.14

5.19

10.9    '    17.8 | 25.8

34.1

43.8

54.3

65.5

77.4

90.0

34.2

34.3

44.0

54.4

65.6

77.5

90.1

44.1

54.6

65.8

77.7

90.2

34.4    44.2

34.5

34.6

34.7

54.7

65.9

77.8

90.3

44.3

54.8

66.0

77.9

90.5

44.4

44.5

54.9

66.1

78.0

90.6

55.0

66.2

78.2

90.7

.58

1.17

5.24

10.9

17.9

25.9

34.8

44.6

55.1

66.3

78.3

90.8

.59

1.20

6.29

11.0

18.0

26.0

34.9

44.7

55.2

66. b

73.4

91.0

0.60

1.23

5.34

.61

1.26

5.39

.62

1.29

5.44

11.1 18.0 11.1 | 18.1 11.2 I 18.2

26.0

26.1

35.0

44.8

55.3

66.6

78.5

91.1

35.1

44.9

55.4

66.7

78.6

91.2

.63

1.32

5.49

11.2

18.2

26.2    35.2

^26.3    35.3

45.0

45.1

55.5

55.6

66.8

66.9

78.8

78.9

91.4

91.5

.64

1.35

5.54

11.3    18.3

26.4

35.4

45.2

55.7

67.0

79.0

91.6

.65

1.38

5.60

11.4

18.4

26.5

35.4

45.3

55.9

67.2

79.1

91.8

.66

1.42

5.65

11.4

18.5

26.6

.67

1.45

6.70

11.5

18.6

26.6

.68

1.48

5.75

11.6

18.6

26.7

35.5

35.6

35.7

45.4

45.5

45.6

66.0

56.1

56.2

67.3

67.4

67.5

79.3

79.4

79.5

91.9

92.0

92.1

.69

1.51

5.80 11.6

18.7

26.8

35.8

45.7

56.3

67.6

79.6

92.3

Table 7.— Weir discharge per foot of crest length—Continued.

Head H, feet.

rc

CO

10

J_

0.70

1.55

5.85

.71

1.58

5.90

.72

1.61

5.96

.73

1.65

6.01

.74

1.68

6.06

.75

1.71

6.11

.76

1.75

6.16

.77

1.78

6.22

.78

1.82

6.27

.79

1.85

6.32

0.80

1.89

6.38

.81

1.92

6.43

.82

1.96

6.48

.83

2.00

6.54

.84

2.03

6.59

.85

2.07

6.64

.86

2.10

6.70

.87

2.14

6.75

.88

2.18

6.80

.89

2.22

6.86

0.90

2.25

6. 91

.91

2.29

6.97

.92

2.33

7.02

.93

2.37

7.08

.94

2.41

7.13

.95

2.44

7.19

.96

2.48

7.24

.97

2.52

7.30

.98

2.56

7.36

.99

2.60

1.00

2.64

7.41 7.47

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ggggggggggg gggssg?, 33 3 33333S§S§§ •

rf^OOtOMOOO-lOlCn^

^TOSCH^COtOH-iOOO^JOS

67.7

67.9

68.0

68.1

68.2

68.3

68.4

68.6

68.7

68.8

68.9

69.0

69.2

69.3

69.4

69.5

69.6

69.7

69.9

70.0

70.1

70.2

70.3

70.4

70.6

70.7

70.8

70.9

71.0

71.2

71.3

79.8

79.9

80.0

80.1

80.2

80.4

80.5

80.6

80.7

80.9

81.0

81.1

81.2

81.4

81.5

81.6

81.7

81.9

82.0

82.1

82.2

82.4

82.5

82.6

92.4

92.5

92.7

92.8

92.9

93.0

93.2

93.3

93.4

93.6

93.7

93.8

94.0

94.1

94.2

94.4

94.5

94.6

94.7

94.9

95.0

95.1

95.3

95.4

82.7 ' .95.5

82.9

83.0

83.1

83.2

83.4

83.5

95.6

95.8

95.9

96.0

96.2

96.3

TABLE 8.—BACKWATER CAUSED BY A DAM OR WEIR.

In a channel of uniform depth, width, and slope, let I)—Original uniform depth.

4 = Depth at the dam or obstruction.

4— Depth at a point upstream.

1=Distance upstream to the point dv w=Width of channel.

4=Distance upstream to the “hydrostatic limit.”

S= Natural uniform slope or inclination of water surface and stream bed, assumed parallel. g=Acceleration of gravity.

0= Coefficient in the Chezy or slope formula v= CsjRS,

area of section wetted perimeter The value of G varies for rivers from about 50 to 140.

The distance upstream from the obstruction at which the depth will be 4 may be found by the formula

^2 4

where R is the hydraulic radius=

.    (115)

d

R is a function of whose value can be expressed mathematically

only as a transcendental equation. The numerical values of this function are given in Table 8. Fx will be found opposite the argument

and R\ opposite

4*

i<-

dr os te tic

Fig. 15.—Concave backwater surface.

The inverse problem of finding the depth at any given distance l upstream can be solved only by successive trials.

Using the above equation, a series of values of dx may be determined giving in tabular form the corresponding values of l. From this data the form of the surface curve may be graphically shown or the depth of back piling atain⁷ point may be interpolated.

IiD=5, 4 = 10,    0=75, £=0.0001,

= 0.5, and ^ = 0.1318,

G-f><

10000

75²

32.16

=49625.

Column (2) in the following table gives the values of l for various values of dx computed by means of formula (115).

Form of backwater curve above a dam.

di-8

Depth,

feet

(1).

6.5

6.4

6.3

6.2

6.1

5.5

5.3

5.1

Hydrostatic Depth of

Distance from dam depth at “ back to depth dlt distanced, piling,” feet

feet

feet

(2).

(3).

(4).

8.83

.17

7.57

6.15

5.30

5.13

a 5. 00 a 5. 00 a 5. 00 a 5. 00 a 5. 00 J a 5. 00 | a 5. 00 :

.43

.85

1.20

1.27

1.30

1.20

1.10

1.00

.50

.30

.10

a Above hydrostatic limit.

If the pond formed by the dam were level, the hydrostatic depth d at any distance upstream would be

(116)

S=d₂—l sin S

Column (3) in the above table shows this factor for the several values of l. The true “back piling¹” or rise due to the surface curvature is expressed by the difference dx — d, as given in column (4).

This quantity has a maximum value at the hydrostatic limit, or terminus of the level pond, where <$=D.

Its location is such that if lx is the distance upstream from the dam

(117)

k=

d^—D sin S

?!

difL

__

Fig. 16. —Convex backwater surface.

In the example given the hydrostatic limit occurs at a distance ^ = 50,000 feet above the dam, at which point the maximum back piling of about 1.31 feet occurs.

Above the hydrostatic limit the depth of back piling is dx—D.

the pond surface will not be concave, but a reniou or hydraulic jump will occur, having- a height

(H9)

When **

So

VII

d^hfg

where vl is the mean velocity corresponding to dv To tind the distance upstream to the point where the jump occurs, solve equation (115) for the value of d₂, found b}^ formula (119).

If £=0.004 C—100

£_

C‘l

= 0.003216.

500

If Vx = 5    w=100    £>=5    i?=j^=4.55.

*>i =

100^4.55. J =13.49 feet per second.

185

04.32

=7.48.

Let the depth at the dam be 10 feet; using d₂ as found above as the terminal depth in formula (115), we obtain

10-7.48

0.004

0.004

31l) (F,-FJ

I)

I)

= 0.669

= 0.5

F₂=0.1318 1=630+(250-311) X 0.126 = 622.3 feet.

<“7.48 Fx =0.2578

d₂

The hydrostatic limit in this case is

10-5

0.004

= 1,250 feet.

If the channel above an obstruction consists of successive reaches having different slopes or cross sections, the depth at the head of the first reach or level may be found by the method outlined, and using this as the initial depth Y₂, a similar solution may be made for the second and succeeding levels.a

a Table 8 has been extended from Bresse’s original table by interpolation. Demonstrations of the formulas here given may be found in Merriman’s or Bovey’s Hydraulics. In case of a fall a different function must be employed. Its values will be found in the works mentioned.

Table 8.—Backwater function (F) for a dam or obstruction.

[The column headings are hundredths for values of ^ from zero to 0.29, thousandths for values of ^ from 0.30 to 0.899, and ten-thousandths for values of ^ from 0.900 to 0.999.]

10

05

e<

es

ha

hi

hj

hj

hj

hj

CM

0.0000 .0050 . 0201

0.0001    0.0003    0.0005    0.0009    0.0013    0.0018

. 0061    .6072    .0085    .0098    .0113    .0128

. 0221    .0243    . 0266    . 0290    . 0314    . 0340

0.0026

.0045

.0367

0.0034 j 0.0042 .0162 , .0181 . 0395    .0425

0.30

.31

.32

.33

.34

.35

.36

.37

.38

.39

0.0455 .0486 .0519 .0553 .0587 .0623 .0660' .0699 .0738 .0779

0.0458 j 0.0461    0.0464    0.0467    0.0471

.0489 . 0522 .0556 .0591 .0627 .0664

.0703 | .0707 .0742 ! .0746 .0783

.0493

.0526

.0560

.0594

.0630

.0668

.0787

.0496

.0529

.0563

.0698

.0634

.0672

.0711

.0750

.0792

.0499

.0533

.0567

.0601

.0638

.0676

.0715

.0754

.0796

.0503 .0536 .0570 .0605 . 0642 .0680 .0718 .0758 .0800

0.0474 .0506 .0539 .0573 .0609 .0645 .0683 .0722 .0763 .0804

0.0477 . 0509 .0543 .0577 . 0612 .0649 .0687 . 0726 .0767 .0808

0.0480 0.0483 .0512 ■ . 0546 .0580 ; .0584 .0616 .0653 .0691 .0730 ! .0734 .0771 .0813

. 0516 . 0550

.0619

.0656

.0695

.0775

.0817

0.40

.41

.42

.43

.44

.45

.46

.47

.48

.49

0.0821 | .0865 | .0909 ¹ .0995 . 1003 : . 1052 ! . 1102 : . 1154 j .1207

j .1262

0.0825 .0869 .0918 .0996 .1008 .1057 .1107 .1159 .1212 . 1268

0.0830    j    0.0834    0.0839    0.0843    0.0847

0.0852 .0896 . 0969 .1001 .1037 .1087 .1138 .1191 .1246 .1301

0.0856    0.0861

.1307 I .1312

0.50

.51

.52

.53

.54

.55

.56

.57

.58

.59

0.1318 .1376 . 1435 .1497 .1560 .1625 .1692 .1761 .1832 .1905

0.1324 i 0.1330 0.1335 .1382 , .1388 .1394 .1441 .1447 .1454 .1503 .1510 ! .1516 .1566 .1573 | .1580 .1632 .1638 j .1645 .1699 .1706 I .1713 .1768 .1775 .1782 .1839 .1847 i .1854 . 1912 .1920 .1928

0.1341 0.1347 0.1353 .1400 .1406 i .1411 .1460 .1466 i .1472 .1522 .1528 ! .1535 .1586 .1592 .1599 .1652 .1658 j .1665 .1720 ,.1726 .1733 .1789 .1796 .1804 .1861 ! .1868 | .1876 .1935 .1942 j .1950

0.1359 .1417 .1478 .1541 .1606 .1672 .1740 .1811 .1883 .1958

0.1364 j 0.1370 .1423 j .1429 .1485

,1547| .1554 .1612 .1679

.17471 .1754 .1818 .1890 . 1965

.1491

. 1618 .1685

. 1825 .1898 .1972

0.60

.61

.62

.63

.64

.65

.66

.67

,t:s

0.1980 .2058 .2138 .2221 .2306 .2395 .2486 .2580 .2667 .2778

0.1988 .2066 .2146 .2230 .2315 .2404 .2495 . 2589 .2678 . 2788

0.1996 ¹ 0.2003 . 2074 .2155 .2238 .2324 .2413 .2505 .2597 .2689 .2799

.2082 .2163 . 2246 .2333 . 2422 .2514 .2606 .2700 .2810

0.2011 . 2090 . 2171 .2255 .2342 .2431 ,2524 .2615 .2711 .2820

0.2019 .2098 .2180 .2264 .2350 .2440 .2533 .2624 .2722 .2830

0.2027 .2106 .2188 .2272 . 2359 .2450 .2542 .2632 .2734 .2841

0.2035 .2114 . 2196 .2280 .2368 .2459 .2552 .2641 ,2745 .2852

0.2042 j 0.2050 . 2130

.2122

.2204 | .2213 .2289 .2377 .2468 .2551 .2650 . 2756 .2862

. 2298 .2386 .2477 .2571 .2658 .2767 .2872

ibb 150—06-17

Table 8.—Backwater function (F) for a dam or obstruction—Continued.

[The column heading's are hundredths for values of ~ from zero to 0.29, thousandths for values of from 0.30 to 0.899, and ten-thousandths for values of ® from 0.900 to 0.999.]

•>

it

0.70    I    0.2883 !    0.2894    0.2905    0.2915    ¹    0.2926    0.2937    0. 2948

0.2959 .3070 . 3186 . 3306 . 3432 . 3563 . 3699 . 3842 . 3993 .4149

0.80

.81

.82

.83

.84

.85

.86

.87

.88

.89

0.4198    0.4215 | 0.4231 j 0.4248    0.4264    0.4281    0.4298

. 6173    .6204    .6235    .6265    .6296    .6327    |    . 6359

. 5776

0.4315 .4490 . 4675 . 4871 . 5081 .5304 . 5544 . 5803 . 6084 . 6392

0.4333 j 0.4350 .4508 :    .4526

.4695 I .4714 .4892    .4912 }

.5102    .5124

.5328    . 5351

. 5569    . 5594

.5830 i .5857 .6114 ! .6143 .6424 ! .6457

0.900

.901

.902

.903

.904

.905

.906

.907

.908

.909

0.910

.911

.912

.913

.914

.915

.916

.917

.918

.919

0.6506 | 0.6509 .6539 1 .6542 .6573 i .6576 .6608 ; .6611 .6642 | .6646 .6678 j .6681 .6712 ; .6716 . 6748    .6752

. 6784    . 6788

.6820    .6824

0.6512 . 6546 . 6580 . 6614 . 6650 . 6684 . 6720

. 6755

. 6791 . 682,S

.6832 !    .6835 j

0.6858 .6895 . 6933 . 6971 .7010 .7049 .7089 .7129

.7170

. 7210

0.6861 . 6899 .6937 . 6975 . 7013 . 7053 .7093 . 7133 .7174 . 7215

0.6865 . 6903 . 6941 .6979 .7017

. 7057

.7097

.7137

.7178

.7219

0.6869 ] 0.6872 .6906 j .6910 .6944 | .6948 .6982 I .6986 . 7021    . 7025

0.920

.921

. 922

. 923 .924

. 925 . 926 . 927

0. 7260 . 7302 . 7345 . 7388 . 7432 . 7476

.928

0.7231 I 0.7235    0.7239    0.7244    0.7248    0.7252    0.7256

.7273 [    .7277    .7281    .7286    .7290    .7294    .7298

.7315 !    .7319    . 7324    . 7328    . 7332    . 7336    j    .7341

.7358 ]    .7362    .7367    .7371    .7375    .7380    j    . 7384

. 7401 !    . 7405 ] . 7410 ! . 7414    .7419 I . 7423    j    . 7427

. 7445 '    . 7450    I    . 7454    I    . 7458    . 7463    !    .    7468    !    . 7472

.7490    . 7494! .7499    . 7504    . 7508    . 7512 1 .7517    . 7522

. 7535 j    . 7540.!    . 7544    I    . 7549    . 7553    .    7558    ]    . 7563    .7567

. 7381    . 7586    ]    . 7590    . 7595 ,    .    7600    .    7604    ,    . 7609    j    . 7614

. 7628 :    . 7633    !    . 7637    :    . 7642    . 7647    j    .    7652    j    . 7656    I    . 7661

0.7265    0.7269 I

.7307    .7311

. 7349    .7354

. 7392    . 7397

. 7436    . 7441

.7481 j .7486 ; .7526 , .7530 i .7572 I .7576 .7619 ; .7623 .7666 I .7670

__

S'

Table 8.—Backwater function (F) for a dam or obstruction—Continued.

§■

25

Sj

S’

aita 1 3

I 1

-e o

I ⁵

I 1

<p

Cll-B

0.930

.931

.932

.933

.934

.935

.936

.937

.938

.939

0.7675 .7723 .7772 .7821 .7871 .7922 .7973 .8026 .8079 .8133 j .8138

0.7680 .7728

.7777

.7826

.7876

.7927

.7978

. 8031 .8084

0.7694 .7743 .7792 .7841 .7891 .7942 .7994 .8047 .8101 .8155

0. 7699 .7748 .7796 .7846 .7896 .7948 . 8000 .8052 .8106 .8160

0.7704 | 0.7709 .7752 j .7757 .7801 .7851 .7902 .7953 .8005 .8058 .8111 .8166

0. 7713    0.7718

. 7762    . 7767

. 7806 .7856 .7907 .7958 .8010 .8063 .8117 .8172

.7811 . 7861 .7912 .7963 . 8015 .8068 .8122 .8177

. 7816 .7866 .7917 .7968 .8021 .8074 .8128 .8182

0.940

.941

.942

.943

.944

.945

0.8188 [ 0.8194 .8244 .8301 .8359 j .8365 .8418 .8478

.8250

.8307

.8424

.8484

0.8199

. 8256

.946

.947

.948

.949

.8602

.8665

.8730

.8608

.8672

.8736

.8313 .8371 . 8430 .8490 . 8552 . 8615 .8678 . 8743

0.8205 . 8262 .8318 .8377 .8436 . 8496 . 8558 . 8621 . 8684 .8750

0.8222    0.8227    0- 8233    0- 8238

. 8280    . 8286    . 8292    . 8298

. 8769    . 8776    . 8782    . 8788

0.950

.951

.952

.953

.954

.955

0.8795 j 0.8802 .8863 j . 8870 .8931 | .8938 .9002 j .9009 .9073 | .9080 .9147 j .9154

.956

.957

.958

. 959

.9221 j .9229 .9298 ’ .9306 .9376 ! .9384 .9457 j .9465

0.8809 .8877 .8945 .9016 .9088 .9462 . 9236 .9314 .9392 .9473

0.960 , 0.9539    0.9548

.964 , .9890 | .9899 . 965 | . 9985 ¹ .9994 .966 | 1.0080    1.0090

. 967 j 1.0181    1.0191

.968 : 1.0282    1.0292

.969 | 1.0389    1.0399

0.970 j 1.0497    1.0508

.971    1.0610    1.0622

.972    1.0727    1.0739

.973 | 1.0848    1.0861

.974    1.0974    1.0987

.975    1.1105    1.1119

. 976    1.1241    1.1255

. 977    1.1383    1.1398

.978    1.1531    1.1546

.979    1.1686    | 1.1702 [

0.8829 .8897 .8966 .9038 .9110 .9184 .9260

. 8815    0.8822

.8890 .8959 .9030 .9103 .9177 .9252

.9329 ! .9337 .9408 .9490

0.9556    0.9564    0.9573    !    0.9582

. 9658,

.9745 ! .9754 . 9835 .9928

1.0023 | 1.0032

1.0100    1.0110    1.0120    j    1.0130 j

1.0201    1.0211    1.0221    !    1.0231 |

1.0303    1.0314    !    1.0324    ¹    1.0335 j

1.0410    1.0421    I    1.0432    1.0443    !

.8883 . 8952 . 9023 •9095 .9169 . 9244 .9321 .9400 .9482

.9416

.9498

0.8836    0.8843    0.8849    0.8856

.9506    . 9514    . 9523    . 9531

.9641

.9727

.9817

.9909

1.0004

. 9650 .9736 .9826 .9918 1.0013

.9666

.9844 ;

.9937

1.0519 1.0633 1.0751 1.0873 1.1000 1.1132 1.1269 1.1413 1.1562 1.1718 I 1.1735

1.0530 | 1.0542 1.0645 I 1.0657 1.0763 ; 1.0775 1.0886 j 1.0898 1.1013 1.1026 1.1146    1.1159

1.1284 1.1298 1.1427    1.1442

1.1578 1.1593 1.1751

1.0553 1.0668 : 1.0788 1.0911 1.1040 1.1173 1.1312 1.1457 1.1608 1.1767

0.9590    0.9598    0.9607    0.9616

. 9675    . 9684    . 9692    . 9700

. 9763    .9772    .9781    .9790

.9854    . 9863    . 9872    . 9881

. 9947    . 9956    . 9966    . 9975

1.0042    1.0051    1.0061    1.0070

1.0140    1.0150    1.0160    1.0170

1.0241    1.0251    1.0261    1.0271

1.0346    1.0356    1.0367    1.0378

1.0453    ;    1.0464    1.0475    1.0486

1.0565    j    1.0576    1.0587    1.0598

1.0680    ;    1.0692    1.0704    1.0715

1.0800    1.0812    1.0824    1.0836

1.0924    ;    1.0936    1.0949    1.0961

1.1053    1.1066    1.1079    1.1092

1.1187    1.1200    1.1214    1.1227

1.1326    1.1340    1.1355    1.1369

1.1472    1.1487    1.1501    1.1516

1.1624    1.1640    1.1655    1.1670

1.1783    1.1799    1.1816    1.1832

Table 8.—Backwater function (F) for a dam or obstruction—Continued.

[The column heading's are hundredths for values of P from zero to 0.29, thousandths for values of P from 0.30 to 0.899, and ten-thousandths for values of !>

— from 0.900 to 0.999.]

IS

4-

S’

GO

5C

0.980 j 1.1848    1.1865    1.1882    1.1899    1.1916    1.1934    1.1951    1.1968    1.1985    1.2002

.981    1.2019    1.2037    1.2055    1.2073    1.2091    1.2109    1.2127    1.2145    1.2163    1.2181

1.2275    1.2294    1.2314    1.2333    1.2352    1.2371

1.2471    1.2491    1.2511    : 1.2531    1.2552    1.2572

1.2678    1.2700    1.2721    1.2742    1.2764    1.2786

1.2899    1.2922    1.2945    1.2968    1.2991    1.3014

1.3136    1.3160    1.3185    1.3210    1.3235    1.3259

1.3391    1.3418    1.3444    1.3471

1.3667    1.3696    1.3725    1.3754    1.3783    1.3812

1.3968    1.4000    1.4032    j 1.4064    1.4095    J    1.4127

.982

.983

.984

.985

.986

.987

.988

.989

1.2199    1.2218    ,i    1.2237    1.2256

1.2390    1.2410    1.2430    1.2451

1.2592    1.2614    j    1.2635    1. 2656

1.2807    1.2830    |    1.2853    1.2876

1.3037    1.3062    !    1.3086    1.3111

1.3284    1.3311    j    1.3337    1.3364

1.3551    1.3580    1.3609    1.3638

1.3841    1.3873    |    1.3905    1.3936

1.3498

1.3524

0.990

.991

.992

.993

.994

.995

.996

.997

.998

.999

1.4159    1.4194    1.4229    1.4264

1.4510    1.4549    i    1.4588    1.4628

1.4902    1.4947    1.4991    1.5036

1.5348    1.5399    1.5451    1.5502

1.5861    1.5922    1.5983    1.6043

1.6469    1.6543    1.6618    1.6692

1.7213    1.7309    1.7405    1.7501

1.8172    1.8307    ,    1.8442    1.8577

1.9523    1.9754    1.9985    2.0216

2.1834 ........................

1.4299    1.4334    1.4370    1.4405    1.4440

1.4667    1.4706    1.4745    1.4784

1.5080    1.5125    1.5170    1.5214    1.5259

1.5553    1.5604    1.5656    1.5707    1.5758

1.6104    1.6165    1.6226    1.6287    1.6347

1.6767    1.6841    1.6915    1.6990    1.7064

1.7597    1.7692    1.7788    1.7884    1.7980

1.8712    1.8848    1.8982    1.9118    1.9253

2.0447    2.0678    2.0910 ! 2.1141    2.1378

1.4824

1.4475 1.4863 1.5303 1.5810 1.6408 1.7139 1.8076 1.9388 2.1603

1.000

^TSOT IK?

kr

INDEX.

Albion, Mass., dam at, flow over...........

Angular weirs. See Weirs, angular. Approach, channel of. See Channel of approach.

section of, definition of.................

velocity of. See Velocity of approach.

Austin, Tex., dam at, flow over............

Authorities cited, list of....................

Page.

132

133

10

Backwater, depth of..........

depth of, figures showing

table showing........

Bazin, H., base formula of....

coefficients of, for thin-edged weirs.... plate showing......................

, 180-182 . 180,181 . 183-186

61

32

correction of, for velocity of approach . 63-65 experiments of, on effect of rounding

upstream crest edge............

on submerged weirs of irregular section

123

Channel, leading, definition of_____

Channel of approach, definition of.

depth of, for weir gaging.......

energy in, distribution of......

velocity in, distribution of.....

Chanoine and Mary, formula of, for sub- '

merged weirs...................

Cippoletti, Cesare, formula of, for trapezoidal weirs...........

weir of, definition of..........

Clegg’s dam, flow over............

Coefficients, relations of...........

Compound weirs, flow over.......

Contracted weirs, definition of_____

Contraction, crest, definition of....

Contraction, end, formula for......

Contraction, vertical, definition of. effect of.........................

Page.

50

17-20

16-17

140

48-49

47-48

112

46

44-45

13-14

on thin-edged weirs. on trapezoidal weirs, on triangular weirs..

143-144 .. 29-31

127

124-126

on weirs of irregular cross section.. 63-85 on weirs with compound slopes.. 127-128 on weirs with varying upstream

slopes.....

plate showing

formula of, comparison of, with other formulas.

128-129

66

Cornell University, hydraulic laboratory of,

description of...................

hydraulic laboratory of, experiments

86-87

at.

39,85-107

for submerged weirs, for thin-edged weirs, for weirs with end contraction, use

.. 40-42 141-142 . 31-34

experiments at, plate .showing

Crest, character of.....................

roughness of, corrections for......

Crest, contraction, definition of.....,.

Croton dam, crest of, correction for...

model of, flow over, experiments on.... 90-94 flow over, experiments on, plate showing........................

86

52

133-134

134

94

of

45

formulas and experiments of, on broad-crested weirs on falls......

Bellasis,

Blackstone River, dam on, flow over.......

Blackwell, T. E., experiments of, on broad-crested weirs Boileau, P., experiments of, on thin-crested

weirs...........................

formula of, compared with other formulas ....

Broad-crested weirs.

crested.

117-119

135

132

112-114,122

21-22

40-41

See Weirs, broad-

Castel, M., experiments of, on thin-crested

weirs...........................

formula of, compared with other formulas ..........................

Chambly dam, model of, experiments on..

20-21

40-41

101

Dams, backwater caused by, depth of.... 180-182 backwater caused by, depth of, table showing

Dams, actual, flow over, experiments on.. 131-133 flow over, experiments on, plate showing..............................

Dams, model, crest of, correction for.......

flow over, experiments on.....

Dams, submerged, data concerning

D’Aubuisson, J. F.-, formula of.....

Deep Waterways, U. S. Board of Engineers on, experiments of, on submerged weirs...................

experiments of, on weirs of irregular

section.........................

on weirs with varying upstream

slopes.....

plate showing

formula of, for broad-crested weirs... 121-122

Definitions of terms................

Desplaines River dam, flow over...

Dimensions, methods of expressing

183-186

132

133

.. 88-90 144-145

21

146

85-90

128-130

90

7-8

112

ax'

Page.

102

143

53

135

135

28

40

51

37

130

106

154

45,48

114-110,121 ......22,40

144-145

17-20

107-109

40-42

23-26 j

formula of, compared with other formu-

26,29,34, 40-41 90-94

110-112

26,29,34,40-41

53-58

49-53

119-121

123,124 98-107

150-152

151-152

25-26

54

...... 39-40

.... 142-143 53-54,97-98

150

146-156

139-140

95-107

25-26

148-156

112

45,48 | Lawrence, Mass., dam at, model of, experiments on

dam at, model of, experiments on, plate

showing ........................

9 Leading channel, definition of.............

Lesbros, experiments of, on thin-edged

weirs...........................

formulas of, compared with other formulas...........................

discharge by, table showing...... 162-171 ! Lesbros and Poncelet, experiments of, on

44 j    thin-edged weirs .

62 [ Lowell, Mass., experiments at

Merrimac River, dam on, experiments on..

dam on, experiments on, plate showing. Metz, Germany, experiments on thin-edged

weirs at.........................

Morris, Elwood, on Clegg’s.dam............

Muskingum River, dam on, flow over......

60-61

60

131

8-9

Orifice, flow through..................

flow through, figure showing......

Ottawa River dam, Canada, flow over.

12

40-41

37-38

21

85-90

142

Dolgeville dam, model of, experiments

on

Dyas, —, formula of, for submerged weirs .. Dyer, C. W. D., and Flinn, A. D., experiments of........................

East Indies, engineers of, formulas of, for

broad-crested weirs......

formulas of, for thin-edged weirs, submerged dams in, data concerning

Energy, distribution of, in channel of approach..........................

Error, effect of, in determining head.......

Essex County,dam of, model of, experiments on

Falls, flow over.............................

flow over, figure showing..............

Farr/i Pond, Mass., experiments on thin-

edged weirs at..................

Flinn, A. D., and Dyer, C. W. D., experiments of........................

Flow, method of expressing................

Formulas, comparison of...................

list of...................................

Francis, J. B., base formula of.............

experiments and formulas of, for thin-crested weirs...................

las

for end contractions................

for weirs of irregular cross sections.

weir of, diagram showing..............

Francis, J. B., and Smith, H., formula of,

for thin-edged weirs............

Francis, J. B., Stearns, F. P., and Fteley, A., formula of, for thin-edged weirs

Freeman, J. R., experiments of............

Frizell, J. P., formula of, for broad-crested weirs

Fteley and Stearns. Sec Stearns and Fteley. Fteley, A., Stearns, F. I\, and Francis, J.

B., formula of, for thin-edged weirs

Gaging, accuracy of........................

requirements for.......................

Geological Survey, United States, experiments of, on broad - crested weirs

experiments of, on rounding upstream

edge.......................

on weirs of irregular section.. on weirs with varying slopes.

plate showing................

Gould, E. L-, formula of, for discharge from nonprismatic reservoir.... formula of, for discharge from prismatic reservoir Gould, E. S., formula of, for discharge from prismatic reservoir

Hart and Hunking, formula of, for thin-edged weirs....................

Page.

Head, determination of, error in, effect of. 53-57 determination of, error in, effect of, plate showing effect of velocities on, table showing. 157-159

increase in, effect of......

from submerged weir.

variation in..............

diagram showing.....

effect of...............

Herschel, C., formula of, for submerged

weirs..............

j Horton, R. E., experiments of.

| Hunting and Hart, formula of, for thin-edged weirs....................

Inclined weirs. See Weirs, inclined. India. See East Indies.

Inflow, effect of, on reservoir........

Johnston, T. T., on flow over Desplaines River dam......................

107-109 106

21 41 21

23-26,107-109 106 106 21-22 112 132

Nappe, definition of........................

form of, modifications of...............

modifications of, plate showing ....

Nelles, George T., data collected by........

Notation, explanation of ...................

O’Connell, P. P. L., formula of, for discharge

from nonprismatic reservoir....    153

12-13

12

132

Parabolic law of velocity, application of, to

weirs...........................

Parmley, W. C., formula of, compared with

other formulas.................

formula of, for thin-edged weirs.......

Plattsburg dam, model of, experiments on. 98-100 Poncelet and Lesbros, experiments of, on thin-edged weirs

Rafter, G. W., experiments of..............

•Rankine, W. J., formula of, for submerged

weirs...........................

Reservoirs, lowering of, time required for. 146-156 Rhind,R. H.,formulaof,forsubmergedweirs 141

Section of approach, definition of... Sharp-crested weirs, definition of... Smith, Hamilton, base formula of... formula of, for thin-edged weirs

37

116-117 122-124 . 26-29

las.

. 40-41 138-139

34

Submerged weirs. See Weirs, submerged. Suppressed weirs, definition of............

.. 46-47 171-176 .. 10-11

11

12

26

38

20-21

110-112

16-17

......... 14-20,58

14-16,41-43,63-66

159-162 .. 17-20 .. 14-20 157-159

180-182

7 ! Weirs, backwater caused by, depth of, table showing 9    definition of....

22,    discharge over, relative approximate.. 10-11

variation in.........

diagram showing

flow over, calculation of, tables for .. 156-186 measurement of, formulas for.

9-12,11-13,40-43 10-14

136

136

124

flow over..................

figure showing........

table showing..........

Weirs, compound, flow over....

7 Weirs, contracted, definition of

Weirs, curved, flow over.......

flow over, figure showing ..

I Weirs, Fast Indian, flow over ..

! flow over, figures showing .

I Weirs, flat-top, models of, experiments on. 103-105 Weirs, inclined, flow over....

flow over, figure showing Weirs, irregular, flow over, experiments on. 61-110

flow over, formulas for, basic ..........

use of...................................

Weirs, of sensible crest width, flow over...    52

Weirs, ogee cross-sectioned, flow over.... 130-131

|    flow over, plate showing.....

j Weirs, submerged, flow over.....

I    flow over, figure showing.....

j increase of head due to.......

Weirs, suppressed, definition of ..

Weirs, thin-edged, definition of .. discharge over, table showing flow over, measurement of, experiments on and formulas for measurement of, formulas for, comparison of...........................

formulas for, extension of..........

15 j Weirs, trapezoidal, aprons of, variation in,

effect of.........................

cross-section of.........................

flow over, formulas for, figures showing.    47

formulas for........................

Weirs, triangular, aprons of, variation in, effect of

coefficient curve for, figure showing ...    125

flow over, experiments on, figure showing

experiments on and formulas for .. 46-47

Weirs, uneven, flow over...................

Weisbach, formula of.......................

“ Wetted underneath,” definition of.......

8 Williams, G. S., experiments by...........

Woodman, R. S., formula of, for discharge from prismatic reservoir

Uaga

34-36,40-41,44

Smith, H., and Francis, J. B., formula of,

for thin-edged weirs...........

Stearns, F.P.,andFteley,A.,baseformulaof 9 experiments of, on broad-crested weirs

on rounding upper crest

on thin-edged weirs.....

formula of, compared with other formu-

for submerged weirs for thin-edged weirs Stearns, F. P., Fteley, A., and Francis, J. B.,

formula of, for thin-edged weirs. 26, 29,34,40-41

Thin-edged weirs. See Weirs, thin-edged. Thomson, James, experiments of, on coefficient of contraction for thin-

edged weirs......

Three-halves powers, table of Torricelli, G., theory of.......

theory of, application of, to weir, figure

showing........................

Toulouse, France, experiments at, on thin-

edged weirs....................

Trapezoidal weirs. See Weirs, trapezoidal. Triangular weirs. See Weirs, triangular.

United States Board of Engineers on Deep Waterways. See Deep Waterways.

United States Geological Survey. See Geological Survey.

Unwin, W. C., formula of, for broad-crested weirs

Velocities, method of expressing...........

Velocity, parabolic law of, application of..

Velocity of approach, distribution of.......

distribution of, figure showing.........

effect of, on weir discharge

correction for..........

table showing (Hunting and

Hart formula)................

table showing (Parmly and Bazin formula)...

table showing..........

energy of, distribution of ...

formulas for................

head due to, table showing Vertical contraction. See Contraction, vertical.

Weir section, definition of.

Weirs, aprons of, variation in, effect of .. 124-127 backwater caused by, depth of.

Page.

183-186

146-154 150 9.

theory of...........................

gaging at. See Gaging, head on. See Head.

Weirs, angular, flow over ...................

flow over, figure showing..............

Weirs, broad-crested, edge of, rounding of, effect of........................

110-122 110 177-179 46 136 136 144-145

127-130 57 62-63 59

130 137-146 137 142-143

162-171 20-29,31-46 40-41

39-40 127 47 47 124-126

46 57-58 40 90-107 151-152

/

CLASSIFICATION OF THE PUBLICATIONS OF THE UNITED STATES GEOLOGICAL SURVEY.

[Water-Supply Paper No. 150.]

The serial publications of the United States Geological Survey consist of (1) Annual Reports, (2) Monographs, (3) Professional Papers, (4) Bulletins, (5) Mineral Resources, (6) Water-Supply and Irrigation Papers, (7) Topographic Atlas of United States—folios and separate sheets thereof, (8) Geologic Atlas of the United States—folios thereof. The classes numbered 2, 7, and 8 are sold at cost of publication; the others are distributed free. A circular giving complete lists may be had on application.

Most of the above publications may be obtained or consulted in the following ways:

1.    A limited number are delivered to the Director of the Survey, from whom they may be obtained, free of charge (except classes 2, 7, and 8), on application.

2.    A certain number are delivered to Senators and Representatives in Congress, for distribution.

3.    Other copies are deposited with the Superintendent of Documents, Washington, D. C., from whom they may be had at prices slightly above cost.

4.    Copies of all Government publications are furnished to the principal public libraries in the large cities throughout the United States, where they may be consulted bv those interested.

The Professional Papers, Bulletins, and Water-Supply Papers treat of a variety of subjects, and the total number issued is large. They have therefore been classified into the following series: A, Economic geology; B, Descriptive geology; C, Systematic geology and paleontology; D, Petrography and mineralogy; E, Chemistry and physics; F, Geography; G, Miscellaneous; H, Forestry; I, Irrigation; J, Water storage; K, Pumping water; L, Quality of water; M, General hydrographic investigations; N, Waterpower; O, Underground waters; P, Hydrographic progress reports. This paper is the sixteenth in Series M, the complete list of which follows. (PP= Professional Paper; B=Bulletin; WS=Water-Supply Paper):

Series M—General Hydrographic Investigations.

WS 56. Methods of stream measurement. 1901. 51 pp., 12 pis.

WS 64. Accuracy of stream measurements, by E. C. Murphy. 1902. 99 pp., 4 pis.

WS 76. Observations on the flow of rivers in the vicinity of New York City, by H. A. Pressey. 1902. 108 pp., 13 pis.

WS 80. The relation of rainfall to run-off, by G. W. Rafter. 1903. 104 pp.

WS 81. California hydrography, by J. B. Lippincott. 1903. 488 pp.,1 pi.

WS 88. The Passaic flood of 1902, by G. B. Hollister and M. O. Leighton. 1903. 56 pp., 15 pis.

WS 91. Natural features and economic development of the Sandusky, Maumee, Muskingum, and Miami drainage areas in Ohio, by B. H. Flynn and M. S. Flynn. 1904. 130 pp.

WS 92. The Passaic flood of 1903, by M. O. Leighton. 1904. 48 pp., 7 pis.

WS 94. Hydrographic manual of the United States Geological Survey, prepared by E. C. Murphy J. C. Hoyt, and G. B. Hollister. 1904. 76 pp., 3 pis.

WS 95. Accuracy of stream measurements (second edition), by E. C. Murphy. 1904. 169 pp., 6 pis. WS 96 Destructive floods in the United States in 1903, by E. C. Murphy. 1904. 81 pp.,13 pis.

. WS 106. Water resources of the Philadelphia district, by Florence Bascom. 1904. 75 pp.,4 pis.

WS 109. Hydrography of the Susquehanna River dfainage basin, by J. C. Hoyt and R. H. Anderson. 1904. 215 pp., 28 pis.

WS 116. Water resources near Santa Barbara, California, by J. B. Lippincott. 1904. 99 pp., 8 pis.

WS 147. Destructive floods in the United States in 1904, by E. C. Murphy and others. 1905. 206 pp., 18 pis.

WS 150. Weir experiments, coefficients, and formulas, by R. E. Horton. 1906. 189 pp., 38 pis.

Correspondence should be addressed to

The Director,

United States Geological Survey,

Washington, D. C.

January, 1906.

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