HYDRAULICS.] HYDROMECHANICS 471 and the velocity of discharge will be (4). We may express this result by saying that differences of pressure at the free surface and at the orifice are to be reckoned as part of the effective head. Hence in all cases thus far treated the velocity of the jet is the velocity due to the effective head, and the discharge, allowing for con traction of the jet, is Q = c u> v = c<a / 2gh (5), where o> is the area of the orifice, co> the area of the contracted sec tion of the jet, and h the effective head measured to the centre of the orifice. If h and 01 are taken in feet, Q is in cubic feet per second. It is obvious, however, that this formula assumes that all the filaments have sensibly the same velocity. That will be true for horizontal orifices, and very approximately true in other cases, if the dimensions of the orifice are not large compared with the head h. In large orifices in say a vertical surface, the value of h is different for different filaments, and then the velocity of different filaments is not sensibly the same. SIMPLE ORIFICES HEAD CONSTANT. 35. Large Rectangular Jets from Orifices in Vertical Plane Sur faces. Let an orifice in a vertical plane surface be so formed that it produces a jet having a rect angular contract ed section with vertical and hori zontal sides. Let ~> T I _ t. b (fig. 46) be the breadth of the jet, Aj and 7t. 2 the depths below the free surface of its upper and lower surfaces. Con sider a lamina of the jet between the depths h and Fi S- 46. li + dh. Its normal section is bdh, and the velocity of discharge /Zgh. The discharge per second in this lamina is therefore b/ l 2cjh dh, and that of the whole jet is therefore 7 = h. 2 - (6), where the first factor on the right is a coefficient depending on the form of the orifice. Now an orifice producing a rectangular jet must itself be very approximately rectangular. Let B be the breadth, H 1( H 9 , the depths to the upper and lower edges of the orifice. Put (7). Then the discharge, in terms of the dimensions of the orifice, instead of those of the jet, is Q=|cBV 2^(H 2 3 -H 1 -) (8), the formula commonly given for the discharge of rectangular orifices. The coefficient c is not, however, simply the coefficient of contraction, the value of which is B7H 2 ^H7) 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 and , and must be ascertained experimentally. Edation between the Expressions (5) and (8). For a rectangular orifice the area of the orifice is = B(H 2 - Hj), and the depth mea sured to its centre is $ (H 2 + Hj). Putting these values in (5), From (8) the discharge is Q 2 = fcBVity(H, ? -H, 5 ) . Hence, for the same value of c in the two cases, 1 - -0-9427 If Hj varies from to oo , *L varies from to 1. The following table gives values of the two estimates of the discharge for different H. Q? H, Q 2 H 2 Qi H 2 Qi o-o 943 8 999 0-2 979 0-9 999 0-5 995 1 -o i-ooo 07 998 Hence it is obvious that, except for very small values of l , the H 2 simpler equation (5) gives values sensibly identical with those of (8). When 1 < 5 it is better to use equation (8) with values of W 2 c determined experimentally for the particular proportions of orifice which are in question. 36. Large Jets having a Circular Section from Orifices in a Ver tical Plane Surface. Let fig. 47 represent the section of the jet; O O Fig. 47. 00 being the free surface level in the reservoir. The discharge through the horizontal strip aabb, of breadth aa = b, between the A! + y and depths h^ + y + dy, is The whole discharge of the jet is But b = d sin ; y = ^d(l - cos = d sin $ d(f>. Let then From eq. (5), putting = d?, k = h 1 + , c = l when d is the dia meter of the jet and not that of the orifice, r / 2f j( k i +4 ) /" si A ^ J >-/o For 7^ = oo , 6 = and ~ = 1 ; and for 7i 1= , e = l and ?_ = 96. So that in this case also the difference between the simple formula (5) and the formula above, in which the variation of head at differ ent parts of the orifice is taken into account, is very small. NOTCHES AND WEIRS. 37. Notches, Weirs, and Eyewashes. A notch is an orifice extend ing up to the free surface level in the reservoir from which the dis charge takes place. A weir is a structure over which the water flows, the discharge being in the same conditions as for a notch. The for mula of discharge for an orifice of this kind is ordinarily deduced
by putting H t = in the formula for the corresponding orifice, ol>
