474 HYDROMECHANICS [HYDRAULICS, 43. Separating Weirs. Many towns derive their water supply from streams in high moorland districts, in which the flow is ex tremely 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 sepa rate 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. 54 shows one of these sepa rating weirs in the form in Fig. 54. which they were first introduced on the Manchester Waterworks; fig 55 a more modern weir of the same kind designed by Mr Binnie for the Bradford Waterworks. When the quantity of water coming Fig, 55. down the stream is not excessive, it drops over the weir into a trans verse 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, prolmbly with accuracy enough for practical purposes, that the particles describe the parabolas due to the mean Telocity of the water passing over the weir, that is, to a velocity 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. 56). Then, if a particle passes from a to & in t seconds, 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 verti- cally_and equal to g on any scale; af horizontally and equal to 1 ygh . Divide af, fc 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 /give the parabolic path of the jet. MOUTHPIECES HEAD CONSTANT. 44. Cylindrical Mouthpieces. When water issues from a short cylindrical pipe or mouthpiece of a length at least equal to 1J times its smallest transverse dimension, thestream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without contraction, at the point of discharge. The discharge is found to be about one-third greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the action being something like that which occurs at an abrupt change of section. __ :L q, e Fig. 56. Let fig. 57 represent a vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the axis of the jet XX be taken as the datum with reference to which the head is esti mated. Let n be the area of the mouthpiece, u the area of the stream at the contracted section EF. Let v, p be the velocity and pressure at EF, and v 1 ,j7 l the same quantities at GH. If the discharge is into the air, p 1 is equal to the atmo- spheric pressure p a . The total head of any filament which goes to form the jet, taken at a point where its velocity is sensibly zero, is Ji + ^; at EF the total head is ^1 + 2-; at GH it is ^- + Q- . 2g G 2g G Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), 32, may have the value Adding this head lost to the head at GH, before equating it to the heads at EF and at the point where the filaments start into motion, G 2*7 G 2<7 G 2g But ( I >v = nv 1 , and w = c e n > if c c is the coefficient of contraction
within the mouthpiece. Hence
