and not that given in (7). It cannot be assumed, therefore, that c in equation (8) is constant, and in fact it is found to vary for different values of B/H2 and B/H1, and must be ascertained experimentally.
_Relation between the Expressions (5) and (8)._--For a rectangular orifice the area of the orifice is [omega] = B(H2 - H1), and the depth measured to its centre is ½(H2 + H1). Putting these values in (5),
Q2 = (2/3)cB [root](2g) [H2^(3/2) - H1^(3/2)].
Hence, for the same value of c in the two cases,
Q2/Q1 = (2/3)[H2^(3/2) - H1^(3/2)] / [(H2 - H1)[root]{ (H2 + H1)/2}].
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.
§ 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
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