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 Verticalsided 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 tailwater 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)dtiih =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 tailwater level. But if the sluice is above the tailwater level, then the head is constant till the level of the sluice is reached, and afterwards it diminishes with the time.
End of Article: DISCHARGE WITH VARYING 

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