Page:EB1911 - Volume 14.djvu/60

is measured towards the weir, (b) in consequence of the crest contraction, (c) in consequence of the end contractions. It may be pointed out that while the diminution of the section of the jet due to the surface fall and to the crest contraction is proportional to the length of the weir, the end contractions have nearly the same effect whether the weir is wide or narrow.

Fig. 45.

J. B. Francis’s experiments showed that a perfect end contraction, when the heads varied from 3 to 24 in., and the length of the weir was not less than three times the head, diminished the effective length of the weir by an amount approximately equal to one-tenth of the head. Hence, if l is the length of the notch or weir, and H the head measured behind the weir where the water is nearly still, then the width of the jet passing through the notch would be l − 0.2H, allowing for two end contractions. In a weir divided by posts there may be more than two end contractions. Hence, generally, the width of the jet is l − 0.1nH, where n is the number of end contractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if cH is the thickness of the stream over the weir, measured at the contracted section, the section of the jet will be c(l − 0.1nH)H and (§ 41) the mean velocity will be 2/3 √(2gH). Consequently the discharge will be given by an equation of the form

Q = 2/3 c (l − 0.1nH) H √2gH

= 5.35c (l − 0.1nH) H^3/2.

This is Francis’s formula, in which the coefficient of discharge c is much more nearly constant for different values of l and h than in the ordinary formula. Francis found for c the mean value 0.622, the weir being sharp-edged.

§ 43. Triangular Notch (fig. 46).—Consider a lamina issuing between the depths h and h + dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is √(2gh). Hence the discharge for this lamina is

b√2gh dh.

But

B/b = H / (H − h); b = B (H − h) / H.

Hence discharge of lamina

= B(H − h) √(2gh) dh/H;

and total discharge of notch

= Q = B √(2g) ∫H0 (H − h) h^1/2 dh/H

= 4/15 B √(2g) H^3/2.

or, introducing a coefficient to allow for contraction,

Q = 4/15 cB √(2g) H^1/2,

Fig. 46.

When a notch is used to gauge a stream of varying flow, the ratio B/H varies if the notch is rectangular, but is constant if the notch is triangular. This led Professor James Thomson to suspect that the coefficient of discharge, c, would be much more constant with different values of H in a triangular than in a rectangular notch, and this has been experimentally shown to be the case. Hence a triangular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notch Professor J. Thomson found c = 0.617. It will be seen, as in § 41, that since 1/2BH is the area of section of the stream through the notch, the formula is again of the form

Q = c × 1/2BH × k √(2gH),

where k = 8/15 is the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form.

Coefficients for the Discharge over Weirs, derived from the Experiments of T. E. Blackwell. When more than one experiment was made with the

same head, and the results were pretty uniform, the resulting coefficients are marked with an (*). The effect of the converging wing-boards is very strongly marked.

Heads in inches measured from still Water in Reservoir.	Sharp Edge.		Planks 2 in. thick, square on Crest.				Crests 3 ft. wide.
	3 ft. long.	10 ft. long.	3 ft. long.	6 ft. long.	10 ft. long.	10 ft. long, wing-boards making an angle of 60°.	3 ft. long. level.	3 ft. long, fall 1 in 18.	3 ft. long, fall 1 in 12.	6 ft. long. level.	10 ft. long. level.	10 ft. long, fall 1 in 18.
1	.677	.809	.467	.459	.435[1]	.754	.452	.545	.467	..	.381	.467
2	.675	.803	.509*	.561	.585*	.675	.482	.546	.533	..	.479*	.495*
3	.630	.642*	.563*	.597*	.569*	..	.441	.537	.539	.492*	..	..
4	.617	.656	.549	.575	.602*	.656	.419	.431	.455	.497*	..	.515
5	.602	.650*	.588	.601*	.609*	.671	.479	.516	..	..	.518	..
6	.593	..	.593*	.608*	.576*	..	.501*	..	.531	.507	.513	.543
7	..	..	.617*	.608*	.576*	..	.488	.513	.527	.497	..	..
8	..	.581	.606*	.590*	.548*	..	.470	.491	..	..	.468	.507
9	..	.530	.600	.569*	.558*	..	.476	.492*	.498	.480*	.486	..
10	..	..	.614*	.539	.534*	..	..	..	..	.465*	.455	..
12	..	..	..	.525	.534*	..	..	..	..	.467*	..	..
14	..	..	..	.549*	..	..	..	..	..	..	..	..

Fig. 47.

§ 44. Weir with a Broad Sloping Crest.—Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a movement sensibly rectilinear and uniform (fig. 47). Let the inner edge be so rounded as to prevent a crest contraction. Consider a filament aa′, the point a being so far back from the weir that the velocity of approach is negligible. Let OO be the surface level in the reservoir, and let a be at a height h″ below OO, and h′ above a′. Let h be the distance from OO to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is Gh″; at a′ it is Gz. If v be the velocity at a′,

v²/2g = h′ + h″ − z = h − e;

Q = be √2g (h − e).

Theory does not furnish a value for e, but Q = 0 for e = 0 and for e = h. Q has therefore a maximum for a value of e between 0 and h, obtained by equating dQ/de to zero. This gives e = 2/3h, and, inserting this value,

Q = 0.385 bh √2gh,

as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonry

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1. The discharge per second varied from .461 to .665 cub. ft. in two experiments. The coefficient .435 is derived from the mean value.