§ 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.
| Fig. 64. |
Time of Emptying or Filling a Vertical-sided Lock Chamber.—Suppose the lock chamber, which has a water surface of Ω square ft., is emptied through a sluice in the tail gates, of area ω, 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 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 −Ωdh, and the outflow from the sluice in the same time is cω √(2gh) dt. Hence the differential equation connecting h and t is
cω √(2gh) dt + Ωh = 0.
For the time t, during which the initial head H diminishes to any other value h,
−{Ω/(cω √2g) } ∫hH dh/√h = ∫0t dt.
∴ t = 2Ω (√H − √h) / {cω √(2g)}
= (Ω/cω) {√(2H/g) − √(2h/g) }.
For the whole time of emptying, during which h diminishes from H to 0,