= B(H - h) [root](2gh) dh/H;
and total discharge of notch
_
/H
= Q = B[root](2g) | (H - h)h^(½) dh/H
_/0
= (4/15) B[root](2g)H^(3/2).
or, introducing a coefficient to allow for contraction,
Q = (4/15)cB [root](2g) H^(½),
[Illustration: 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 ½BH is the area of
section of the stream through the notch, the formula is again of the
form
Q = c × ½BH × k[root](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._
+----------+-------------+---------------------------------+-----------------------------------------+
| | | Planks 2 in. thick, | |
| Heads in | Sharp Edge. | square on Crest. | Crests 3 ft. wide. |
| inches +------+------+-----+-----+-------+-------------+------+------+------+------+------+------+
| measured | | | | | |10 ft. long, | 3 ft.| 3 ft.| 3 ft.| 6 ft.|10 ft.|10 ft.|
|from still| 3 ft.|10 ft.|3 ft.|6 ft.| 10 ft.| wing-boards | long,| long,| long,| long,| long,| long,|
| Water in | long.| long.|long.|long.| long. | making an |level.|fall 1|fall 1|level.|level.|fall 1|
|Reservoir.| | | | | |angle of 60°.| |in 18.|in 12.| | |in 18.|
+----------+------+------+-----+-----+-------+-------------+------+------+------+------+------+------+
| 1 | .677 | .809 |.467 |.459 |.435[4]| .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*| .. | .. | .. | .. | .. | .. | .. | .. |
+----------+------+------+-----+-----+-------+-------------+------+------+------+------+------+------+
[Illustration: 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 [root]{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 = (2/3)h, and,
inserting this value,
Q = 0.385 bh [root](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 sluice
openings than the ordinary weir formula for sharp-edged weirs. It
should be remembered, however, that the friction on the sides and
crest of the weir has been neglected, and that this tends to reduce a
little the discharge. The formula is equivalent to the ordinary weir
formula with c = 0.577.