+----------------+--------------+--------+
| Head h metres. | Head h feet. | [mu] |
+----------------+--------------+--------+
| 0.05 | .164 | 0.4481 |
| 0.10 | .328 | 0.4322 |
| 0.20 | .656 | 0.4215 |
| 0.30 | .984 | 0.4174 |
| 0.40 | 1.312 | 0.4144 |
| 0.50 | 1.640 | 0.4118 |
| 0.60 | 1.968 | 0.4092 |
+----------------+--------------+--------+
An approximate formula for [mu] is:
[mu] = 0.405 + 0.003/h (h in metres)
[mu] = 0.405 + 0.01/h (h in feet).
_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.
Coefficient.
Inclination up stream 1 to 1 0.93
" " 3 to 2 0.94
" " 3 to 1 0.96
Vertical weir 1.00
Inclination down stream 3 to 1 1.04
" " 3 to 2 1.07
" " 1 to 1 1.10
" " 1 to 2 1.12
" " 1 to 4 1.09
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.
[Illustration: FIG. 53.]
[Illustration: FIG. 54.]
_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 1½ 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 ½ 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 + (1/5)h1/p] [root 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.
[Illustration: FIG. 55.]
[Illustration: FIG. 56.]
§ 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 to flow away down the
original stream channel, or is stored in separate reservoirs and used
as compensation water. To accomplish the separation of the flood and
ordinary water, advantage is taken of the different horizontal range
of the parabolic path of the water falling over a weir, as the depth
on the weir and, consequently, the velocity change. Fig. 55 shows one
of these separating weirs in the form in which they were first
introduced on the Manchester Waterworks; fig. 56 a more modern weir of
the same kind designed by Sir A. Binnie for the Bradford Waterworks.
When the quantity of water coming down the stream is not excessive, it
drops over the weir into a transverse channel leading to the
reservoirs. In flood, the water springs over the mouth of this channel
and is led into a waste channel.
It may be assumed, probably with accuracy enough for practical
purposes, that the particles describe the parabolas due to the mean
velocity of the water passing over the weir, that is, to a velocity
(2/3)[root](2gh),
where h is the head above the crest of the weir.
Let cb = x be the width of the orifice and ac = y the difference of
level of its edges (fig. 57). Then, if a particle passes from a to b
in t seconds,
y = ½gt², x = (2/3)[root](2gh) t;
.: y = (9/16)x²/h,
which gives the width x for any given difference of level y and head
h, which the jet will just pass over the orifice. Set off ad
vertically and equal to ½g on any scale; af horizontally and equal to
2/3 [root](gh). Divide af, fe into an equal number of equal parts.
Join a with the divisions on ef. The intersections of these lines with
verticals from the divisions on af give the parabolic path of the jet.
[Illustration: FIG. 57.]