Page:EB1911 - Volume 14.djvu/62

Rh Bazin arrives at the following values of m:—

Coefficients of Discharge of Standard Weir.

Bazin compares his results with those of Fteley and Stearns in 1877 and 1879, correcting for a different velocity of approach, and finds a close agreement.

Influence of Velocity of Approach.—To take account of the velocity of approach u it is usual to replace h in the formula by h + au2/2g where is a coefficient not very well ascertained. Then

Q = l (h + u2/2g) √ { 2g (h + u2/2g) }

= lh √(2gh) (1 + u2/2gh)undefined. (2)

The original simple equation can be used if

m = (1 + u2/2gh)undefined

or very approximately, since u2/2gh is small,

m = (1 + u2/2gh). (3)

Now if p is the height of the weir crest above the bottom of the canal (fig. 52), u = Q/l(p + h). Replacing Q by its value in (1)

u2/2gh = Q2 / {2ghl2(p + h)2} = m2 {h/(p + h) }2, (4)

so that (3) may be written

m = [1 + k {h/(p + h)}2 ]. (5)

Gaugings were made with weirs of 0.75, 0.50, 0.35, and 0.24 metres height above the canal bottom and the results compared with those of the standard weir taken at the same time. The discussion of the results leads to the following values of m in the general equation (1):—

m = (1 + 2.5u2/2gh)

= [1 + 0.55 {h/(p + h)}2 ].

Values of —

An approximate formula for is:

= 0.405 + 0.003/h (h in metres) = 0.405 + 0.01/h (h in feet).&emsp;&ensp;

Inclined Weirs.—-Experiments were made in which the plank weir was inclined up or down stream, the crest being sharp and the end contraction suppressed. The following are coefficients by which the discharge of a vertical weir should be multiplied to obtain the discharge of the inclined weir.

The coefficient varies appreciably, if h/p approaches unity, which case should be avoided.

In all the preceding cases the sheet passing over the weir is detached completely from the weir and its under-surface is subject to atmospheric pressure. These conditions permit the most exact determination of the coefficient of discharge. If the sides of the canal below the weir are not so arranged as to permit the access of air under the sheet, the phenomena are more complicated. So long as the head does not exceed a certain limit the sheet is detached from the weir, but encloses a volume of air which is at less than atmospheric pressure, and the tail water rises under the sheet. The discharge is a little greater than for free overfall. At greater head the air disappears from below the sheet and the sheet is said to be “drowned.” The drowned sheet may be independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a standing wave. In the latter case when the foot of the sheet is drowned the level of the tail water influences the discharge even if it is below the weir crest.

Weirs with Flat Crests.—The water sheet may spring clear from the upstream edge or may adhere to the flat crest falling free beyond the down-stream edge. In the former case the condition is that of a sharp-edged weir and it is realized when the head is at least double the width of crest. It may arise if the head is at least 1 the width of crest. Between these limits the condition of the sheet is unstable. When the sheet is adherent the coefficient m depends on the ratio of the head h to the width of crest c (fig. 53), and is given by the equation m = m1 [0.70 + 0.185h/c], where m1 is the coefficient for a sharp-edged weir in similar conditions. Rounding the upstream edge even to a small extent modifies the discharge. If R is the radius of the rounding the coefficient m is increased in the ratio 1 to 1 + R/h nearly. The results are limited to R less than in.

Drowned Weirs.—Let h (fig. 54) be the height of head water and h1 that of tail water above the weir crest. Then Bazin obtains as the approximate formula for the coefficient of discharge

m = 1.05m1 [1 + h1/p] 3√ { (h − h1) / h },

where as before m1 is the coefficient for a sharp-edged weir in similar conditions, that is, when the sheet is free and the weir of the same height.

§ 48. Separating Weirs.—Many towns derive their water-supply from streams in high moorland districts, in which the flow is extremely variable. The water is collected in large storage reservoirs, from which an uniform supply can be sent to the town. In such cases it is desirable to separate the coloured water which comes down the streams in high floods from the purer water of ordinary flow. The latter is sent into the reservoirs; the former is allowed