2. (20 pts) A 50 gallon residential electric hot water heater has outside dimensions: diameter D...

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2. (20 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 = 20C. The tank is installed in a cold garage at temperature T = 10C. The convective heat transfer coefficient between the tank and the air is h = 10 W m K-. 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. (4 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. (5 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. (6 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 = 55C (a typical hot-water tank set point)? d. (5 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 m K-. 2. (20 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 = 20C. The tank is installed in a cold garage at temperature T = 10C. The convective heat transfer coefficient between the tank and the air is h = 10 W m K-. 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. (4 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. (5 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. (6 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 = 55C (a typical hot-water tank set point)? d. (5 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 m K-.