The water tank in Figure E-2 is filled by an inflow x(t) and is emptied by an outflow y(t). The outflow is controlled by a valve which offers resistance R to the flow of water out of the tank. The water depth in the tank is d(t) and the surface area of the water is A, independent of depth (cylindrical tank). The outflow is related to the water depth (head) by y(t) = d(t)/R.
The tank is 1.5 m high with a 1m diameter and the valve resistance is 10 s/m².
(a) Write the differential equation for the water depth in terms of the tank dimensions and valve resistance.
(b) If the inflow is 0.05m³ /s , at what water depth will the inflow and outflow rates be equal, making the water depth constant?
(c) Find an expression for the depth of water versus time after 1m³ of water is dumped into an empty tank.
(d) If the tank is initially empty at time t=0 and the inflow is a constant 0.2 m³/s after time t = 0 at what time will the tank start to overflow?
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