476 HYDROMECHANICS [HYDRAULICS. whence we find that ^ will become zero or negative if Or I (~ / u == h / Gh or, putting ^ = 34 feet, if In practice there will be an interruption of the full bore flow with a less ratio of , because of the disengagement of air from the water. u But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind is Q-../ that is, the discharge is the same as for a well-bellmouthed mouth piece of area <a, and without the expanding part, discharging into a vacuum. 47. Jet Pump. A divergent mouthpiece may be arranged to act as a pump, as shown in fig. 61. The water which supplies the energy Fig. 61. required for pumping enters at A. The water to be pumped enters at B. The streams combine at DD where the velocity is greatest and the pressure least. Beyond DD the stream enlarges in section, and its pressure increases, till it is sufficient to balance the head due to the height of the lift, and the water flows away by the discharge pipe C. Fig. 62 shows the whole arrangement in a diagrammatic way. A is the reservoir which supplies the water that eflects the pump ing ; B is the reservoir of water to be pumped ; C is the reservoir into which the water is pumped. DISCHARGE WITH VARYING HEAD. 48. Flow from a Vessel when the Effective Head varies ivith the Time. Various useful problems arise relating to the time of emptying and filling vessels, reservoirs, lock chambers, &c. , where the flow is dependent on a head -which increases or diminishes during the operation. The simplest of these problems is the case of filling or emptying a vessel of constant horizontal section. Time of Emptying or Filling a Vertical-sided Lock Chamber. Suppose the lock chamber, which has a water surface of n square feet, is emptied through a sluice in the tail gates, of area u, placed below the tail-water level. Then the effective head producing flow through the sluice is the difference of level in the chamber and tail bay. Let H (fig. 63) be the initial difference of level, h the difference Head, water level ^----fc-g ~~ _r^L|^r: H Tail wafer level of level after t seconds. Let - dh be the fall of level in the chamber during an interval dt. Then in the time dt the volume in the chamber is altered by the amount - Cldh, and the outflow from the sluice in the same time is cu^2ghdt. Hence the differential equation connecting /( and t is For the time t, during which the initial head H diminishes to any other value h, n rh dh^ft Cw^StyJB. ^h Jo n
- -^|=2(VB-VA)
_ n | /2H /-Ik ) ~CIVT v ?y For the whole time of emptying, during which /( diminishes from H to 0, Comparing this with the equation for flow under a constant head, it will be seen that the time is double that required for the discharge of an equal volume under a constant head. The time of filling the lock through a sluice in the head gates is exactly the same, if the sluice is below the tail-water level. But if the sluice is above the tail-water level, then the head is constant till the level of the sluice is reached, and afterwards it diminishes with the time. PRACTICAL USE OF ORIFICES IN GAUGING WATER. 49. If the water to be measured is passed through a known orifice under an arrangement by which the constancy of the head is ensured, the amount which passes in a given time can be ascertained by the formulae already given. It will obviously be best to make the ori fices of the forms for which the coefficients are most accurately de termined ; hence sharp-edged orifices or notches are most commonly used. Water Inch. For measuring small quantities of water circular sharp-edged orifices have been used. The discharge from a circular orifice one French inch in diameter, with a head of one line above the top edge, was termed by the older hydraulic writers a water- inch. A common estimate of its value was 14 pints per minute, or 677 English cubic feet in 24 hours. An experiment by Bossutgave 634 cubic feet in 24 hours (see Navier s edition of Belidor s Arch. Hi/dr., p. 212). Weisbach points out that measurements of this kind would be made more accurately with a greater head over the orifice, and he proposes that the head should be equal to the diameter of the orifice. Several equal orifice s may be used for larger discharges. 50. Pin Ferrules or Measuring Cocks. To give a tolerably definite supply of water to houses, without the expense of a meter, a ferrule with an orifice of a definite size, or a cock, is introduced in the service- pipe. If the head in the water main is constant, then a definite quantity of water would be delivered in a given time. The arrange ment is not a very satisfactory one, and acts chiefly as a check on extravagant use of water. It is interesting here chiefly as an ex ample of regulation of discharge by means of an orifice. Fig. 64 shows a cock of this kind used at Zurich. It consists of three cocks, the middle one having the orifice of the predetermined size in a small circular plate, protected by wire gauze from stoppage by im purities in the water. The cock on the right hand can be used )y the consumer for emptying the pipes. The one on the left and the measuring cock are connected by a key which can be locked b}
a padlock, which is under the control of the water company.
