A process fluid with a constant density, kg/m3 , enters a vertical cylindrical tank through an inlet pipe at its top. The tank is D m in diameter and 2.4 m tall. The fluid leaves the tank at the flat bottom through an outlet pipe with an internal diameter d m. When operated at a steady state with 1.2 m liquid level inside the tank, the outlet velocity of the fluid through the bottom pipe, v(t) m/s, can be estimated from Bernoulli equation as: v(t) = 4h(t) where h(t) m is the liquid level inside the tank.

(a) Obtain the transfer function between the liquid level inside the tank and the inlet volumetric flow rate when the tank is half full of the liquid. [10 marks]

(b) Determine how long it will take for the liquid level inside the tank to increase by 0.1 m when the inlet volumetric flow rate is suddenly increased by 0.01 m3 /s from the steady state value. Data: = 1000 kg/m3 , D = 1 m and d = 0.1 m