5. (Balances-40 pts) We want to drain a cylindrical tank filled with water (density p = 1000kg/m, viscosity u = 1 CP) through a cylindrical outlet as shown below. The system is in thermal equilibrium with surroundings at 25C Open air (Pa = 1 atm) Water (p = 1 g/cm = 1 cp) H = 10 m D = 10 m Valve 1 (a) Derive analytical formula for the time to fully drain the tank after opening the valve 1 (5 pts) (b) Calculate full evacuation times when internal diameter (I.D.) of outlet is 1 mm and 3 mm, respectively. Other dimensions are given in the figure. Assume coefficient of contraction to be 0.63. (5 pts) (c) What are Reynolds numbers of the flow after valve 1 a) at the beginning and b) when the height of the liquid left in the tank becomes 1 cm? The I.D. of the outlet (R) is 3 mm. Will they be laminar or turbulent? (5 pts) Now the top of the tank is closed and a valve is introduced on the top part of the tank (valve 2) to let in controlled air flow in as shown below. Controlled Air inflow Barometer Barometer Valve 2 Hvalve 2 Water (p = 1 g/cm? 1 = 10P) 15 V2 =const. H = 10 m Water (p = 1 g/cm", u=1 cp) t = 0 H = 10 m t=t D = 10 m D = 10 m Valve 1 Valve 1 R = 3 mm R = 3 mm (d) Can the tank be drained if the valve 1 is opened and valve 2 is closed? (5 pts) (e) We want to keep the discharge velocity constant at all times during draining. Derive an analytical formular for the required rate of the air pressure change inside the tank (10 pts). Calculate the final pressure of gas inside the tank right after full evacuation of water. (5 pts) (1) What is Reynolds number of the flow at valve 1 if we change pressure inside the tank by 50 u Pa/s? Assume coefficient of contraction to be 0.63. Will it be turbulent? [5 pts]