Thermodynamics - Semester 1 2025 – Assignment
PROBLEM 1 Multi-Mode Heat Transfer (18 pts)
Consider heat transfer in two thermos flasks. Flask A has a vacuum layer sealed between the inner and outer shells. Flask B is same as Flask A except for several holes drilled near the top and bottom of the outer shell (as shown in the figure). The diameter of the inner shell is 8.5 cm and of the outer shell is 9.5 cm. Both shells are 17 cm tall and made of the same steel with negligible thickness. The steel surfaces are gray with an emissivity of 0.1. The flasks are well insulated at the top and the bottom. The inner surface is fully covered by the outer surface in terms of thermal radiation.
The flasks are placed in a room where the ambient air and wall temperatures are both at 20ºC. The natural convection coefficient from the ambient air, ho, is 6 W/(m2-K). The water temperature, Twater, is uniform in both flasks, and the inner surface temperature is always at the same as that of the water. The density and specific heat of water is 1000 kg/m3 , and 4200 J/(kg-K), respectively.
1. Analyze the heat transfer of Flask A and draw a schematic for the energy flows. Derive the equations to determine the outer surface temperature, Ts,o, and the water cooling rate, dTwater/dt. (2 pts)
2. Solve the equations derived in part 1 using a finite difference method. The initial water temperature is 95°C. Calculate and report Twater, Ts,o, and dTwater/dt from 0 to 60 minutes at a step of 5 minutes. Plot Twater, Ts,o, and dTwater/dt as a function of time. (2 pts)
Note: write a small computer routine to assist the calculation.
3. Analyze the heat transfer in Flask B and draw a schematic of energy flows. Derive the equations to determine Ts,o and dTwater/dt. In this case, the air layer temperature is assumed to be uniform. and constantly below the inner surface temperature by 20°C. The convection coefficient in the air layer hi = 20 W/(m2-K). (4 pts)
4. Solve the equations for Flask B using a finite difference method. The initial water temperature is 95°C. Calculate and report Twater, Ts,o, and dTwater/dt from 0 to 60 minutes at a step of 5 minutes. Plot Twater, Ts,o, and dTwater/dt as a function of time. (4 pts)
5. Discuss why the temperature drops differently in the two flasks. ( 1 pt)
6. Discuss what could be the actual temperature distribution in the water and in the air layer. Draw a schematic to show such temperature distribution in the two fluids. (2 pt)
7. Prove that air is flowing in the layer between the inner and outer surfaces. (3 pts)
PROBLEM 2 Heat Conduction (12 pts)
Heat transfer is crucial in designing nuclear power plants. A schematic of a nuclear reactor is shown below. The inner tubular rod is the nuclear fuel (thorium) which generates heat uniformly at 108 W/m3 . The fuel rod is surrounded by a graphite shell which is cooled by water. The inner surface of the thorium rod is well insulated. Thermal conductivity of all materials involved is invariant with temperature.
1. Determine the fuel rod’s outer surface temperature T2 . (2 pts)
2. Derive the equation for temperature distribution along the radius of the fuel rod. (3 pts)
3. Solve the temperature distribution using the conditions provided. (3 pts)
4. To prevent melting the fuel rod and the graphite layer, T1 and T2 must be kept below 1000 K. Are these conditions satisfied? (1 pt)
5. One engineer suggested that as long as the fuel rod is properly sealed from the cooling water, a thinner graphite layer should be used. From heat transfer’s point of view, do you agree? (3 pts)