DISCHARGE WITH VARYING  HEAD § 53 . Flow from a Vessel when the Effective Head varies with 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 ti square ft., is emptied through a sluice in the tail gates, of area w, 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 . 64) be the initial difference of level, h the difference Head water 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 —tdh, and the outflow from the sluice in the same time is cwJ (2gh)dt . Hence the differential equation connecting h and t is cal (2gh)dt-i-ih =o . B 1==~I B is the reservoir of water to be pumped ; C is the reservoir into which the water is pumped . ~1K N\ FIG .

59 . 4r . E For the time t, during which the initial head H diminishes to any other value h, {S2/(cw',/2g)} f hdh/,lh= ftdt . H 0 .'.t =2t2(V H —^( h)/{cwy' (2g) } =(S2/cw){J (2H/g) —I (2h/g)1 . For the whole time of emptying, during which h diminishes from H to o, T=(S2/cw) (2H/g) . 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 .