You are asked to design a micro hydropower system, which is simplified as the figure below. 1-ton of water at 25C is stored in a cylindrical tank, and water is being drained through a pipe, whose inner diameter is D, horizontally placed at the bottom of the tank. It is assumed that the efficiency of the turbine is 75% and the velocity of water after the turbine is zero. H(t) duz z Pat AT turbine The dynamic equations for the flow through the capillary are given as follows: drain a 10 10 r. (r du ) = 9H (0) + = 0 (rou) L (1) If the flow is slow enough, quasi-steady flow assumption is applied, u = - [-(2)]. Find the expression of the wall shear stress at the pipe wall. (2) Find the expression of H(t) as a function of time. (3) How long does it take to 99% empty the tank? (4) Find the expression of the rate of the electricity generation (J/s). (5) Find the expression of the overall electricity generation (J) until the tank is 99% empty. C(t)R 4 (6) To maximize the overall electricity generation (J), how the cross-sectional area of the tank Ay should be chosen? (7) If the pipe is vertically attached at the bottom of the tank, repeat the calculations from (1) to (6).