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Encyclopedia Britannica



DISCHARGE WITH VARYING

This article appears in Volume V14, Page 53 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: DIO-DRO
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.


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