the flow of water over sharp-edged notches and weirs

388 GOURLEY AND CRIMP ON THE FLOW OF WATER [Selected

(Paper No. 4 1 2 9.)

The Flow of Water over Sharp-Edged Notches and Weirs.” By HAROLD JOHN FREDERICE GOURLEY, M.Eng., Aesoc. M. Inst. C.E.,

and BERXARD SANTO CRIMP, Stud. Inst. C.E.

THE classic work of Francis, Fteley and Steams, Hamilton Smith, Blackwell, Castel, Bazin, Poncelet, Lesbros, and others, on the flow of water over sharp-edged rectangular weirs, has led to the general acceptance and use of the standard formula Q = cLHC, now familiar to a11 waterworks engineers. In this formula the coefficient c varied with both the length of the weir and the head of water discharged by the weir. In no ease known to the Authors can it be said that a formula has been framed which is applicable to fully-contracted rectangular weirs up to 20 feet (at least) in length, and for a wide range of head.

In a Paper presented t,o The Institution in 1910,l one of the Authors suggested that if some index other than 1 a 5 were used in formulas expressing the law of flow over weirs, the difficulty of variable coefficients might be overcome, and with this in mind, and in the hope of determining exactly how the ends affected the law of discharge, arrangements were made to determine experimentally the discharge over sharp-edged triangular notches and trapezoidal weirs of crest lengths 3, 6, 12, 18, 24, and 36 inches, and having ends inclined at 5 (vertically) to 1 (horizontally), 3 to 1, 2 to 1, and 1 to 1. The flow was measured by causing the water to pass over a rectangular’ weir, for which the coefficients had recently been determined.

The results of the experiments show (i) that for triangular notches the flow varies as H 2 ’ 4 7 , H being the head, (ii) for trapezoidal weirs the flow is obtained by adding the flow over a triangular notch of corresponding angle to that of a rectangular

H. J. F. Gourley, “Experiments on the Flow of Water over Sharp-Edged Circular Weirs,” Minutes of Proceedings Inst. C.E., vol. clxxxiv, p. 297.

Downloaded by [ York University] on [17/09/16]. Copyright © ICE Publishing, all rights reserved.

Papers.] OVER SHARP-EDGED NOTCHES AND WEIRS. 359

weir with end contractions and of the same crest length, and (iii) that the flow over rectangular weirs with end contractions varies as L1.02 and H1.47, being given by Q = 3 * 10 L1”JZ H1.47, where Q is the discharge in cubic feet per second, L the length in feet, and H the head in feet. This formula and others given later are believed to give the flow within f 1 per cent.

Menszcrenmzt qf Water.-An extensive series of experiments on fully-contracted rectangular and Cippolettil (i.e., end slopes of 4 to 1) weirs of lengths 6 inches to 36 inches was carried out by Messrs. Steward and Longwell at the experimental station a t the Boise Project of the United States Reclamation Service during 1911.2 The measurements were absolute, and after making slight corrections (not exceeding 1 per cent.) to compensate for the proximity of the bottom of the measuring channel to the sill of the weir, Table I (Appendix) was drawn up, using the coefficients given by these investigators as a basis, and this Table of coefficients has been used throughout the present experiments.

General Arrangement (Figs. l).-A &-foot gauging-weir on the River Alwen was raised teplporarily, and the water diverted to flow through a channel 3 feet wide cut down to rock, and with a gradient of 1 in 125 to a controlling penstock, thence to the upper or “ meter ” pool with a gradient of 1 in 25 ; after passing over the rectangular weir in this pool, the water flowed over the weir under investigation in the “ experimental” pool, and was led back to the river by a channel blasted out in the rock to a 1 in 15 gradient. The backing up of the river formed a pool some 45 feet wide and about 100 feet long. This comparatively large area compensated in a very satisfactory manner any small variations in the stream-flow. After a few trials it was found possible to fix on the penstock R

scale which enabled any desired rate of flow to be readily obtained. Pools (Figs. 2 and 3).-The pools were practically alike in size

and shape. In plan the width at 12 feet upstream was 10 feet, and a t 10 feet upstream the width had become equal to that measured at the weir, namely, 12 feet. Below the sill of the weir there was 3 feet of water, and this obtained to a line parallel to the weirs and 8 feet upstream, beyond which the bottom rose gradually to meet the discharge channel from the upper or the inlet channel, as the case might be.

~ ~ ~~ . ~ .

“Weir with Free Overfall end Constant Coefficient of Contraction,” Minutes

p W. a. Steward and J. S. Longwell, “Experiments on Weir Discharge,” of Proceedings Inst. C.E., vol. lxxxv, p. 454.

Trans. Am. Soc. C.E., vol. lxxvi, p. 1045 et s q .


Papers.] OVEIt SBAI1P-EDGED NOTCHES AND WElIiS. 391

It mill be seen that full provision for cwmplete contraction was

Pigs, 2. ' W W N01233S

made in these pools. A certain amount of clay puddle was


392 GOURLEY AND CRIMP ON THE FLOW OF WATER [~electec~

introduced in places where leakage might possibly have occurred.

Figs. 3. m

S E C T I O

S E C t l O N B B .

H A A .

This was protected from scour by light boarding or pitching.


papers.] OVER SHARP-EDGED NOTCHES AkND WEIRS. 393

Occasionally small slips of the sides of the intake channel took place, and at intervals the upper pool was c1e:tred of any material which may have been carried in, so that its full depth was maintained. Considerable protection from wind was afforded by banks of spoil deposited round the pools,

Weirs.-The sharp edges of the weirs were formed of :!+inch by d-inch steel plates having bevelled edges which projected 4 inch beyond the bevelled edge of the 3-inch planks to which the plates were screwed (Figs. 3). The metal sill of the rectangular or “ meter ” weir 3 feet 6 inches long WRS attached to the middle of a 9-inch by 3-inch plank 12 feet long, the top edge of which was planed and set truly horizontal. Two pieces of 3-inch plank held together by stout battens carried the end plates of the weirs, and the bottom edge of the lower piece was planed so that the combination might slide on the 12-foot plank, previously referred to, thus allowing a weir of any desired length (not exceeding 3 feet) to be readily obtained. The joint between the fixed and sliding pieces was made by lightly caulking with red lead and oakum, and leakage behind the sliding pieces was prevented by a packing of clay.

The arrangements for the triangular and trapezoidal or “ experimental ” weirs were of a similar nature. Care was taken during construction to ensure the plane of the weirs being vertical, and in setting either of the weirs to a definite opening every precau- tion was observed that the length should be exact to the nearest & inch, and that the inclination of the ends was exactly tts desired.

Neasurement of Head.-In each pool a t 9 feet back from the weir and 4 feet off the axial line of the pool, the lower portion of a pine ranging rod was driven into a hole bored in the rock, and a dome- headed screw was screwed into it, EO that the top of the dome was exactly level with the sill of the weir. From time to time the top of the dome was checked against the sill with a surveyor’s level, and occasionally the horizontality of the sill was also checked, though this latter operation was really unnecessary, as the 3-inch planking forming the end of the pool was carried down to allcl tied into the rock.

The rod was surrounded by two 9-inch stoneware pipes set vertically one above the other; the lower one rested on bricks placed at the bottom of the pool, and was filled with broken stone for 12 inches up; the upper pipe was kept 4 inch from contact with the lower one by packing pieces placed at the bottom of the socket, and the joint was made up with clay except for l inch. These stilling boxes answered quite well. There was no appreciable


394 COURLEY AND CRIMP ox THE FLOW OF WATER elected

oscillation of the water-surface in them, and the water readily responded to changes of level in the pools.

The heads were measured by a vernier hook-gauge capable of being read directly to foot and having a spirit-level attach- ment to ensure verticality. The gauge was carefully examined before use and found to be accurate.

Bafles.-The gradual changes from the channels t o the pools rendered baffles almost unnecessary, but two sets were introduced in each pool to ensure uniform distribution of flow over the section of the pool.

velocity of Approach.-There was no sensible velocity of approach, except with flows exceeding 10 cubic feet per second, and from the results of float-rod observations, in these few cases it was not considered necessary to make any allowance on this account.

Method of Malsing Obsewa,tions.-Prior to starting experiments, the lips of the weirs were cleaned with a slightly oiled rag. Having adjusted the penstock to give the required flow, a period varying between 10 to 15 minutes where the qunntit#y w : ~ small and 3 minutes for the largest flows was allowed to elapse, during which the water in the pools found its appropriate level. The head over the rectangular weir was read first-this took about 4 minute --and after the head in the lower pool had been observed a %minute interval was allowed t o elapse before the observations were repeated for the first time, then another %minute interval, and the second repetition followed. These readings generally agreed to 0 001 foot, but in a few cases more repetitions were necessary to obtain a satisfactory result.

To allow for the possibility of peculiar conditions prevailing during the experiments made on any one day, a second series of experiments was always made on the following day, and the two sets of observations were used to obtain the discharge curves. To render the conditions in the two pools as nearly alike as possible, the length of the rectangular weir used in a particular series was made appropriate to the weir under investigation in the lower pool. For example, the 3 to 1 triangular notch and trapezoidal weir of crest length 3 inches were " metered " with the &inch rectangular weir ; the 3 to 1 at 6 inches with the 12-inch; the 3 to 1 a t 12, 18 and 24 inches with the 24-inch, a.nd the 3 to 1 at 36 inches with the 36-inch rectangular weir. Maximum and Minimum Head.--The maximum head observed was,

as a rule, not far from 1 foot, and the lower limit was fixed at about 0.15 foot, as it was felt that the probable error of observation for small flows would exceed the & per cent. limit which it is thought


Papers.] OVEI1 SHARP-EDGED NOTCHES AND 1VEII:S. 395

applies to each series of experiments made if much lower heads were observed. After logarithmically plotting to a large scale the actuaI observations on the various weirs, the discharges for regular increments in head given in Tables I1 to VI1 (Appendix) were calculated.

Analysis of Results.-The first point of interest is the fact that, the experiments clearly show that the flow over any triangular notch can be represented by the law Q = b H2.47, in which Q is the discharge in cubic feet per second, 2, a coefficient varying with the angle of the notch, and H the observed head in feet.

The following values of b were obtained : 5 to 1 triangular notch . . . . . . . . . 0 = 0.52 3 , I 1 ,, ,, . . . . . . . . . 1,=0.86 1 ,, 1 ,, ), . . . . . . . . . b = 2 . 4 6

Messrs. Steward and Longwell give a discharge curve for a 4 to 1 triangular notch (facing p. 1090 of their Paper, previously quoted), and it is found that b = 0.61 represents the flow for head up to 3 feet within 1 per cent.

These four values of 2, give a straight line law from which the following corrected values are taken :

5 to 1 triangular notch (included angle 22" 38' ) . . h = 0'50 4 I , 1 ?, 1, ( ,, ,, 280 4') . . b = 0 '62 3 ,, 1 ,, I > ( 1, ,, 36'52'). . b = 0.83 2 3 , 1 I , 3 , > > ,, 63" S') . . h = 1.24 1 ,> 1 I , 1, ( >, .. 90'). . . b = 2.48

And, generally, if ?a is the tangent of half the included angle Q = 2.48.nHZ.47.

For comparison with Mr. B a d s results,l tjhe flow over a 90" V notch has been calculated from the formula Q = 2.48 H2'47 for heads between 2 inches and 10 inches. The corresponding quantities, which agree within about 1 per cent., are given in the Appendix, Table VIII.

I n some experiments on triangular notches carried out prior to commencing the work on trapezoidal weirs, the Authors obtained the following results :

0 197 t o 1 * 0 triangular notch (included

0.485 to 1.0 triangular notch (included angle 157" 42') . . . . . . Q = 12.80 H2.47

angle 128" 15') . . . . . . Q = 5-28 H 2 . 4 7

J. Barr, "Experiments on the Flaw of Water over Triangular Notches," Er~giwering, vol. lxxxix (1910), p. 473, Table W.



It was not possible to get more than about 3 inches head on these flat notches, and the experimental error is therefore likely to be comparatively high. The values of b given by thc above formula would be 12.60 and 5.12 respectively.

It is therefore evident that the true index for a V notch is 2.47, and from the above experimental results it would appear t’hnt the coeEcients increase with the ratio of the width to t>he height, thus conforming to the theoretical rule.

The fact that the index of H for a V notch was 2.47 suggested that the true index for a weir was 1 * 47 inste2.d of 1 5O.I A basic formula was accordingly assumed for a trapezoidal weir o f the form-

Q = aLH1.47 + bH2.4;

in which all t,he units are feet and L is the length of the weir-crest. This formula assumes that possibly the total flow over such a weir might be regarded as the sum of two end flows, which together form a triangular notch, with a middle part discharging as a weir. Of the constants, it was anticipated that a would not be affected by variation in the end slopes, but that b would depend upon these slopes.

For convenience in plotting, the formula was cast in the form-

Here a is determined as the tangent of the inclination of the line to the horizontal axis, and b the intercept on the vertical axis.

Q and -- for the L

From Tables I1 to VI1 the values of H2.47 H trapezoidat weirs were calculated and plotted to a large scale. The result was interesting in two respects ; first, there seemed to be a general parallelism in the “fair ” lines, which confirmed the idea that a would be constant for ,211 weirs, and, second, it was noted that the points corresponding to the longer weirs lay considerably to the left of the fair line for the points corresponding t o the shorter weirs, and doubtless in consequence of these departures the intercept on the vertical axis gave in every case a somewhat smaller value for 6 than that given earlier. This second point may be said to suggest that the flow over a weir with contractions increases more rapidly than the length, or what appears to amount

Francis appears, in 1851, t o have found that the index 1.47 wag applicable, but in 1852 he reverted t o 1.50, “ Lowell Hydraulic Experiments,” pp. 76-95.


Pq’ers.] OVER SHARP-EDGED NOTCHES AND WEIRS. 397

to the %me thing, that the contractive eff’ect of the ends reduces the flow by an amount which is proportionately greater for shorter weirs. The first point of view seemed more promising, and the general equation was put into the form-

Q = f l L x H l , 4 7 + b H 2 . 4 7 ,

and re-cast for plotting as-

Trials were made with values of X greater than unity, and finally it was found that when J: mas 1 - 0 2 the resulting curves closely agreed with the rectangular weirs’ flows, a.nd the departures for the trapezoidal weirs were within the limits of experimental error, whilst in every case the value of h agreed with the value for the corresponding V notch. The final plotting is reproduced in Fig. 4, and the following laws are derived from it :-

0 to 1 weir (rectangular) . . Q=3* 10 L1.02 H1‘47 l

5 ’, l ,, (trapezoidal) . . ~ ~ 3 . 1 0 ~ 1 . 0 2 ~ 1 . 4 7 + 0 .50 ~ 2 . 4 7

4 7 7 1 7 ) 1 7 . . & = 3 . 1 O L 1 ‘ 0 ” 1 ’ 4 7 + 0.62H2.47 3 3 , 1 > > ’7 . . 2 7, I ’> 3 7 Q=3.10 Ll.02 H 1 ‘ 4 7 + 1 24 H 2 . 4 7

1 1 , 1 7,

Q = 3-10 L1.02 H1’47 + 0 - 83 H 2 . 4 7

. . 9 , . . Q=3 10 L1.02 H 1 ’ 4 7 + 2-48 H 2 . 4 7

In the above formula Q is the discharge in cubic feet per second, L the crest-length in feet, and H the head in feet. It is thought that these formulas are correct to within 2 1 per cent.

The general law for a trapezoidal weir is

Q = 3 10 L1.02 H 1 . 4 7 + 2 * 48 . n . H2.47

Rectangular Weim-For weirs up to 3 feet in length the results from the suggested formula will be first compared with the quantities obtained in the Boise experiments. The results are given in Table IX.

Comparisons for weirs up to 19 feet long are made in Tables X and XI (Appendix). The generally close agreement between the various experimental results and the quantities calculated by the proposed formula will be noted.

From the preceding results and analysis it will be seen that the

‘ Mr. P. ?L Morley Parker has expressed the flow over rectangular weirs in formulas of B similar character.-“ Control of Water,” p. 112, 1913.



Fig. 4

1 1

J


Pitpers.1 OVER SIIARP-EDGED NOTCHES AND WEIBS. 399

flow over any trapezoidal weir is obtained by adding the flow of the V notch of the same inclination as the ends of the trapezium to the quantity discharged by a fully contracted rectangular weir of the same length as the crest of the trapezoidal weir.1 This being so, there is no object in using trapezoidal weirs a t all.

This combination of flows is somewhat remarkable, and i t a explanation not very clear. It would appear that the contractive effects of the sides of a notch are more intense than those of the ends of a rectangular weir, and as the notch is expanded to form a trapezoidal weir, the contractive effects of the sloping sides, which formerly balanced about the vertical plane containing the apex of the notch, now intrude into the flow over the crest ; the area of influence of the sloping sides is probably defined by lines starting from the ends of the crest, and equally inclined towards the middle of the weir, though not necessarily straight. It is suggested that the encroachment due to the ends of the trapezoidal weir correspond more or less to the end effects i n a rectangular weir with contrac- . tions, and possibly accounts for what, a t first sight, might appear an anomalous combination.

The Cippoletti formula was proposed to meet the difficulty of end contractions, and seems to have been based on the Francis formula :-

Q = 3 * 33 (L - ng) H1.50, where n is the number of end contractions.

If the expression is expanded it becomes Q = 3 33 LH1.50- 0.66 H 2 . 6 0

for a fully contracted weir. Approximately 0.66 is the coefficient for a 4 to 1 V notch, and presumably the contention was that by increasing the rectangle to a trapezium with 4 to 1 end slopes the second term would vanish. The present experiments indicate that there is no particular merit in the 4 to 1 slope, and by suitably altering the indices of H and L a formula is obtained which gives results in practical agreement with actual values over a wide range of flows.

Sunzmary.-(l) The flow over a triangular notch is proportional to H2.47, and varies in direct proportion to the ratio of width to height. The general law is Q = 2.48 . n . 212'47, in which Q is the discharge in cubic feet per second, 7) the tangent of half the included angle of the notch, and H the head in feet.

(2) The fiow over any trapezoidal notch is equal to the flow over n rectangular weir of equal length with two end contractions plus the flow through a triangular notch of corresponding angle,

' For R 4 t o 1 trapezoidal weir Messrs. Steward ancl Longwell arrived at the same conclusion.



(3) The flow over a rectangular weir with end contractions varies as H1.47 and increases rather more rapidly than the length, i.e., as L1.02, and is given by the formula Q = 3.10 L1.02 H1.47,

which applies to all weirs up t o a t least 19 feet in length, and, to judge from the comparisons made for short weirs, for heads up to half the length of the weir, provided the depth of pool below the sill of the weir is not less than twice the head. In the formula, Q is cubic feet per second, L the length in feet, and H the head in feet.

Conclusion,--The scheme of experimental work was drawn up in March, 1913; the constructional work took 2 months, and the experiments 5 months.

This Paper is presented by the kind permission of Messrs. Sir Alex. Binnie, Son and Deacon, MM. Inst . C.E., who defrayed a large proportion of the cost of construction. It is accompanied by three plans, two plotted curves, several photographs, and an Appendix containing thirty-nine Tables,l from which the Figures in the text and the following Tables have been selected.

~_____ __ ~

The Tables containing the results of the actual observations are filed for reference in the Institution Library.

[APPENDIX.


Papers.] OVER SHARP-EDGED NOTCHES AND \VVEIRS. 40 1

APPENDIX.

TABLE COEFFICIENTS FOR FULLY-CONTRACTED RECTANQULAR WEIRS USED IN

THIS RESEARCH.

~~~ ~~ ~~. ~ .

6 Inches.

3.601

3.177

3-120

3.092

3.091

3.116

3.119

3.128

3.145

3.162

1'00 3.168

Length of Crest.

12 Inches.

3.236

3.210

3.184

3.171

3.168

3.142

3.139

3.124

3.121

3.125

3.125

~~~~ _. .

24 Inches.

3.370

3.330

3.320

3.286

3.248

3.204

3.196

3.180

3.156

3.149

3.133

36 Inches.

3-442

3.393

3.360

3.347

3.328

3.289

3.255

3.242

3.237

3'224

3.210

~~~~ ~~~~ -

Formula :-Q = cLH"".

Q = discharge in cubic feet per second. L = length of weir in feet. H = head in feet. G = coefficient given in above Table.

NoTE.-The above Table is based upon the Boise experiments (Trans. Am. Soc. C.E., vol. lxxvi, p. 1077, Table XVIII). Slight corrections have been made to compensate for the proximity of the bottom of the pool to the sill of the weir in the above experiments.

[THE INST. C.E. VOL. cc.] 2 D


402 GOLTRLEY AND CRIMP ON THE FLOW OF WATER [Selected

( 7 LABLE IL-DISCHAIGE IN CUBIC FEET PE& SECOND OVER TIIAPEZOIDAL \vl<:lR

HATING END SLOPES OF 1 TO 1.

Values derived from lwge scale plotting of Authors’ experimental results.

Head.

E’oot. 0.2

0 .3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

~ ~ -.___ -. -

3 Inches. 1 6 Inches.

0.249 ~ 0.382

0.452 0.650

0.716

3.992 3.207

3.221 2.585

2.541 1.995

1.936 1.486

1-421 1.057

0.991

..

Length of Crest. ____

1% Inches.

.. 0.663

1.054

1.592

2.203

2.887

3.655

4.519

5 * 475

18 Inches. ~ 21 Inclles.

0.494 0’645

0.944 1.217

1.510

4.729

3.790 4.768

3.686 2.940

2.734 2.168

1.912

6.960 ~ 8.760

5.782 i 7.283

5.971

‘ 1

TABLE I I I . - ~ I S C I € A R G E IN CUBIC FEET PER SECOND OVER TRAPEZOIDAL IVEIR HAVING END SLOPES OF 2 (VERTICALLY) TO 1 (HORIZOSTALLY).

Values derived from large scale plotting of Authors’ experimental results.

Head.

Foot. 3 Inches.

0.2 0.096

0.3 0.197

0 .4

0.972 0 . 7

0.713 0 .6

0.501 0 .5

0.330

1.281 0.8

~

Leugth of Crest.

G Inches. l 2 Inches.

0.163 , 0.315

0.322 0.599

0.525 0.952

0.776 1.364

1.076 1.836

1.431 2.373

. ~ __ _______~__.

18 Inches. 24 Inches. , 36 Inches.

0.468 , 0.6381 0.966

0.878 , 1.170 ’ 1.752

1.368 1.813 , 2.687

1.934 2.544 3.761

2.586 , 3.373 4.958

3.310 I 4.284 6.281

~

,

0.9 1.629 I 2.292 3‘632 ~ 4.992 i 6.374 ’ 9.277 1

1.0 2.040 1 2.805 4.360 5.930 1 7.570 , 10.92


Papers.] OVER SHARP-EDGED NOTCHES AND WEIRS. 403

TADLI~ IV.-DISCHARGI.: IN CUBIC FEET PER SECOND OVER TRAPEZOIDAL WEIR HAVIPI'G END SLOPES 08 3 (VERTICALLY) TO 1 (HORIZONTALLY).

VaIues derived from large scale plottings of Authors' experimental results.

liead.

F O I L

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

B Inches. 0'159

0.295

0 . 4 7 7

0.697

0.955

1.252

I,engti1 of I'l,t!St,.

19 Inahes. 0.305

0.568

0.892

1.273

1.707

2.188

1.968

2.355

3.288

3.929

1s Illcllcs. 0.462

0.855

1.326

1.863

2.466

3.132

3.874

4.635

5.475

~ ~ -~ 24 Inches.

0.605

1.135

1.751

2.433

. ~~

3.201

4.064

5.000

5,986

7.050

TABLE V.---DISCHARQE IPI CUBIC FEET PER SECOND OVER TRAPEZOIDAL \vlCIR

HAVING END S L O ~ ~ E ~ OF 4 (VERTICALLY) TO 1 (HORIZONTALLY). These are Boise results (Trans. Am. Soc. C.E., vol. lx-xvi, p. 1064, Table VIII),

to which slight corrections hare been made.

Head. 1 Length of Crest. - .. ~~~

Foot. i ---6 Inches. ' 12 Inches. 0 '2

0.3

0.151

0.7 j 1.186

0.6 ' 0.906

0.669 0.5

0.463 0.4

0,288

0 . S ' 1.492

1.832

1.0

0.301

0'553

0.852

1.200

1.603

2,066

2.565

3.105

3.686

____-__~ 24 Inches. '

0.600

1.110

1.698

2.374

3.132

3'964

4.846 1 5'801 i 6.860

36 Inches. 0.910

1.670

2.566

3.588

4 740

5.992

7.303

8.692

10.19



TABLE VI.-DISCHARGE IN CUBIC FEET PER SECOND OVER TRAPEXOIDAL WEIR HAVING END SLOFES OF 5 (VERTICALLY) TO 1 (HORIZONTALLY).

Values derived from large scale plotting of Authors' experimental results.

Head.

Foot.

0 - 2

0 . 3

0 . 4

0.5

0.6

0.7

0.8

l T l

1 i

3 Inches.

0-0926

0'164

0.259

0.379

0'520

0.680

0'861

G Inches.

.. 0.286

0.448

0'642

0.867

1.126

1'414

Length of Crest.

12 Inches. 18 Inches.

. ' .. 0-553

1.305 0.861

0.848

3'446 2.532

2.981 2.054

! 2.365 1.619

1.816 1.221

____ 24 Inches.

0.600

1.095

1-678

2.353

3.089

3.888

4.748

0.9 1.060 1.724 3.037 4.357 5.657

1.0 , 1'284 ~ 2.067 1 3.580 1 5'100 ~ 6'620

1

-~ 30 Inches.

0-932

1. 695

2'595

3'604

4.727

5'940

7.233

8.592

10'04

TABLE VII.-DISCHARGIC IN CUBIC FEET PER SECOND OVER RECTANGULAR W E I R WITH END CONTRACTIONS.

Based upon Boise results (Tram. Am. Soc. C.E., vol. lxxvi, p. 1076, Table XVII).

Length of Crest. - __

G Inches. ' 12 Inches. 24 Inches. 1 30 Inches. -~

0.142

0.254

0.391

0-551

0.725

0.916

1.125

1.350

0.287

0.521

0.801

1-111

1.459

1. 830

2.223

2.668

0-580

1.080

~ 1.648

~ 2.270

2.965

3.730

4.515

5.390

0.910

1.650 1 2.525

~ 3.492

1 4.550

5.695

6.960

8.264

1.584 3.125 6.270 9.630


Pspers.1 OVER SHARP-EDGED NOTCHES AND WEIRS. 405

TABLE VIIL-COMPARISON WITH BARR'S RESULTS.

Discharge in Cubic Beet per Second. Head iu Inches. -

Actual (Ban's). i Formula (Authors').

2 , 0.0293 ! 0.0294

3

4

5

6

7

8

9

10

0.0797

0.1625

0-2825

0-4438

0.6508

0.9069

1.215

1.578

0.0789

0.1643

0.2852

0'4473

0-6550

0.9110

1.219

1 1.581

i

i

- 1 _ -

Percentage Difference.

_______- + 0.34

- 1.00

+ 1.09

+ 0.96

+ 0.79

+ 0.64

+ 0.45

+ 0.33

+ 0.19

Nom-Barr's results are hken from Engincering, vol. lxxxix (1910), p. 473, Table IV.

The formula used for the V notch is Q = 2.48 H*'", in which Q is the discharge in cubic feet per second and H is the head in feet.


TULE IX.-COXPBRISON BETWEEN ACTUAL DISCHARGES OBTAISED AT ROISE A S D THOSE GIVEN BY AUTHORS' FORXULA FOR RECTANGULAR W E I R S WITH CONTIIACTIONS

Boise.

:nbic Feet e r Second.

0.051 0.142 0.254

0.551 0.391

..

..

..

0.58 1 .os 1.65 2.27 2.98 3.74 4.54 5.41 6.31

Born~lla ~ 1 (Authors'), 1 Cubic Feet

0.0518 0.1435 0.2602 0.3975 0.5516

-- - ~: Cubic Feet,

per Second. 4-1.57 0,103 +1.06 0.287 +2.44 0.521 +1.66 0.801

.. ~' 1.459

~ 2,333

.. 1 2.668

+ O . l l ' 1 1.111

.. , 1.830

.. ! ! 3.125

- . ... ~~

Formnla (Anthors').

3nbic Feet Crlhic Feet ,er Second. per Second.

__--____

0.21 0.218 0.590 1.070 1.635

I 2.269 1 2.966 ~ 3.721

4.528 5.382 6.285

11.45 , 11.41 2.0 , ..

-l-

Percentage Difference. ___ _.___

1

+1.43 0.33 + l . i 2 0.97 -0.92 1 .66 -0.9-1 2 . 5 3 -0 .04 3.50 -0.47 , 4.57

-0.26 , 7.00 -0.51 l 5 . 7 3

-0.52 8 - 3 3 -0'40 ' 9.73 -0.35

~ 17.53

Cubic Feet per Second.

.. i 26.80

.- ~~

Formula [Authors').

Cnbic Feet ,er Second. 0.1050 0.2910 0,5280 0.8065 1.119 1.463 1.835 2.233 2.655 3.100

--

-0.49 -0.80

.foot Weir.

(Authors'). Formula

Cubic Beet )er Second.

0.3220 0.8927 1.6190 2 .473

4.513 3.432

5.628 6.848 8.142 9.507

___

17.26 26.33


- 2 . 4 1 -1.90 -1.88 -2.25 -1.94

-1.78 - 1 '81

-2.1; -2.26 -2.29 -1.54 -1.75

Nom.-Boise results taken from Trans. Am. Soc. C.E., vol. lxxvi, p. 1076. The Boise discharges for the 2-foot and %foot weirs are considered somewhat

large owing to proximity of bottom of weir pool. The correction required is probably about yZ per cent. for the %foot and 1 per cent. for the %foot weir a t l 2 inches head, and proportionately more for greater heads. These corrections would operate in redueins the differences in discharge given abore.


Pay~ers.] OVER SHARP-EDGED NOTCHES AND WEIRS. 407

TABLE X-COMPARISON BETWEEN ACTUAL DISCHARGES OBTAINED BY FRAKCIS AND THOSE GIYEN BY AUTHORS’ FORMULA FOR RECTANGULAR WEIRS WITH CONTRACTIOXS.’

~ Discharge in Cubic Feet per Second.1 ~ ~~~~~

i IIead. ________ Percentage

Difference. ’ Actnal Formula ~ (Francis). (Authors’).

_.__- - Foot.

0.5085 , 1-0627 1 0.7580 1 0.6049 I

1.0656 1 0.4936 1.1336 I 0.3745 0.4681 0.6554 I 0.9554 0.3662 , 0.8361 ~

6‘229 4.096 5.291 7.347

12.087 6-229

21.096 4.096 5.291 7.347

12.087 21.096

21.096 6.229

6.273 4.119 5.314 7.406

12.21 6.228

21.150 4.144 5.316 7.406

12.10 21.06

21.120 6.273

-I--- I

$0.70 t 0 . 5 6 +0.46

$1.02 +0.78

I $0.44 +0.47 $0.78 +0‘01 -0.17

’ +0.70 +O.ll

1 $0.25

COXPARISON BETWEEX ACTUAL DISCHARGES OBTAINED BY FTELEY AND STEARNS AND ‘THOSE GIVEN BY AUTHORS’ FORMULA FOR ItECTANGULAR WEIRS WITH COXTRACTIONS.~

Length.

2.313 Feet.

2.313 2.313 2.313 3.008 3-008 3.008 3.007 3-007 3.010 3.010

Head.

Foot.

0’749 0.583

0.957 0.892

0.330 0,216

0.485 0.623 0.742 0.874 0.873

)ischarge in Cubic Feet per Second ___--

Actual Tteley &Stearns: ____-

3.303 4.764 6’172 6.840 1.013

3.303 1.883

4.764 6.172

7.856 7.871

l., _ -

___~. -

(Authors’). Formula

3.300 4.770 6.166

1.002 6.837

3.290 1.868

4.753 6.146 7 828 7‘814


-_____

+0.12 -0.09

-0’04 -0.09

-0 .80 -1.09

-0.39 -0.23 -0 .42 -0.54 -0.53

1 Francis’s results are from p. 127, Hamilton Smith’s “Hydraulics,” 1886

Fteley and Stearns’ results are given on pp. 102 and 103, Hamilton Smith‘s edition.

“Hydraulics,” 1586 edition.


408 GOURLEY AND CHIMP ON THE FLOW OF WATER. [Selected

TABLE XI.-COMPARISOX BETWEEN DISCHARGES OBTAINED BY USING HAMILTON SMITH'S COEFFICIEXTTS AAD THOSE GIVEN BY AUTEORS' FORMULA FOR

RECTANGULAR WEIRS WITH CONTRACTIONS.

Eead. i %foot Weir. I/ 10-foot Weir.

' Formula 1 1 (Authors'). 1

--___ :ubic Feet Cubic Fcet er secontl. I per Secout!. 0.3310 0.3220 0-9045 0,8927

2.490 : 2.473 1.633 1.619

3.452 ~ 3.432

4.513 1 4.486 5.669 5.628 6.892 1 6.848 8.196 8.142 9.552 ' 9.507

17.26 ' 17.26 ~~ ~ ~~~~ -

Smith. ___.

Cubic Feet )er Second.

1.662 4.551 8.223

15.70

22.91 17.48

35-14 28.81

41.86 48.96 89.22


-2.72 -1.30 -0.86 -0.68 -0'82 -0.60 -0'72 -0.64 -0.66 -0'47

.. ~ .

~~ ~~~

(Authors'). Forn~ula

per Second. Cubic Feet

1.100 3.048 5.526 8 .443

11.71 15.32

23.38 19.22

27'80 32.46 58'92

~ ~ -~~

16-foot Weir. l 19-foot Weir .- ~~ .~~

Formula :Anthors'). I :ubic Feet )er Second.

1.663 4'609 8.355

12.77 17.72 23'16 29.05 35.36 42.01 49.08 89.08

l -~

Difference. ---ll___-

Cubic Feet 1~ per Second. +0.06 i ' 2.110 4-1.23 ! 5.765 +1.60 ' 1 10.44 +1.59 j 15.94 +1.37 22.18 4-1'09 ~ 29.06 +0*83 +0.62 +0,43 +0.24 - 0.16

44-60 36.54

53.11 62.12

113'6

~ ~~

Formula (Authors').

Cubic Feet )er Second.

2.118 5.870

___

10'64 16.26 22.56 29'50 37'00 45'03 53'52 62.50

113'4

~~~ -~~

Percentage Difference. -~

-0'90 +0.59 +0.73 +0.90 +0.60 +0'46 t 0 . 2 1 -0.09

-0 .25 ..

-0 '27 ~

~ ~- ~


~~~ ~ ~~ ~.

+0.38 t-1.73 +1.91 1-2.00 +1.71 +1.51 + 1 '26 +0.96 +0*77 +0.61 -0'16

NoTe.-Hamilton Smith's coefticients ' I For Weirs with Full Contraction " as given on p. 132 of his g ' Hydraulics " have been used. It should be noted that his coefficients for weirs up to 10 feet long are deduced from experiments on weirs with contractions, whereas for longer weirs he made use of experiments on weirs wit.11 suppressed contractions, and on this account there may be slight errors in his coefficients for longer weirs.


the flow of water over sharp-edged notches and weirs

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