Q2/Q1 = 0.9427(1 - [sigma]^(3/2)) /
{1 - [sigma] [root]{(1 + [sigma])}}. (9)
If H1 varies from 0 to [infinity], [sigma]( = H1/H2) varies from 0 to
1. The following table gives values of the two estimates of the
discharge for different values of [sigma]:--
+------------------+--------+------------------+--------+
| H1/H2 = [sigma]. | Q2/Q1. | H1/H2 = [sigma]. | Q2/Q1. |
+------------------+--------+------------------+--------+
| 0.0 | .943 | 0.8 | .999 |
| 0.2 | .979 | 0.9 | .999 |
| 0.5 | .995 | 1.0 | 1.000 |
| 0.7 | .998 | | |
+------------------+--------+------------------+--------+
Hence it is obvious that, except for very small values of [sigma], the
simpler equation (5) gives values sensibly identical with those of
(8). When [sigma]<0.5 it is better to use equation (8) with values of
c determined experimentally for the particular proportions of orifice
which are in question.
[Illustration: FIG. 44.]
§ 40. _Large Jets having a Circular Section from Orifices in a
Vertical Plane Surface._--Let fig. 44 represent the section of the
jet, OO being the free surface level in the reservoir. The discharge
through the horizontal strip aabb, of breadth aa = b, between the
depths h1 + y and h1 + y + dy, is
dQ = b [root]{2g(h1 + y)} dy.
The whole discharge of the jet is
_
/d
Q = | b [root]{2g(h1 + y)} dy.
_/0
But b = d sin [phi]; y = ½d(1 - cos [phi]); dy = ½d sin [phi] d[phi].
Let [epsilon] = d/(2h1 + d), then
_
/[pi]
Q = ½d² [root]{2g(h1 + d/2)} | sin² [phi][root]{1 - [epsilon] cos [phi]} d[phi].
_/0
From eq. (5), putting [omega] = [pi]d²/4, h = h1 + d/2, c = 1 when d
is the diameter of the jet and not that of the orifice,
Q1 = ¼[pi]d² [root]{2g (h1 + d/2)},
_
/[pi]
Q/Q1 = 2/[pi] | sin² [phi] [root]{1 - [epsilon] cos [phi]} d[phi].
_/0