2. (30 pts) A 50 gallon residential electric hot water heater has outside dimensions: diameter D = 52 cm and height H = 1.5 m. The heater element supplies q =

5 kW of heat to the water in the tank. The tank is filled with 50 gal of water that is initially at To = 20°C. The tank is installed in a cold garage at temperature T = 10°C. The convective heat transfer coefficient between the tank and the air is h = 10 W m² K¹. 00 • Assume that the hot water tank is at a uniform temperature T(t), which is also the water temperature. (Real hot water tanks have insulation that we are neglecting here). • Assume convection heat transfer all around the outside of the tank. Neglect radiation or other heat loss. a. (8 pts) Draw a schematic of the hot water heater tank in the garage. Make sure to indicate the heat transfer processes and a control volume that you will study. b. (8 pts) Set up a transient energy balance for the hot water heater system. Reduce this to form a differential equation for the temperature of the tank as a function of time. c. (8 pts) Solve the differential equation to obtain a solution for the temperature of the tank as a function of time. How long will it take to warm the tank up to T = 55°C (a typical hot-water tank set point)? d. (6 pts) Make a graph of the water tank temperature as a function of time. Also show the tank temperature plot if the heat loss coefficient doubled to h=20 W m²² K-¹.