A spherical tank with an inner radius 1 m is filled with liquid hydrogen at 1 atm and 20 K. The surface temperature along the inner surface of the tank is also at 20 K. Assume the thickness of the tank wall is 0.1 m and the outer surface temperature of the tank wall (on the outside) is 300 K. When the tank is not insulated, some of the liquid hydrogen from the tank will evaporate as a result of the heat gain from the surroundings. The heat of vaporization of liquid hydrogen is 225 kJ/kg. This means if 225 kJ is gained then 1 kg of liquid hydrogen will evaporate. The density of liquid hydrogen in the tank when it is stored at 500 bar is: 600 kg/m³. The thermal conductivity of the tank wall is 0.01 W/m-K. How much of the liquid hydrogen will evaporate?

1. 1. Choose the appropriate transport equation for energy conservation and reduce the equation by eliminating all the zero terms. 
2. 2. Solve the differential equation that you obtained in part (a) to get a general solution for the temperature variation within the walls of the tank as a function of two arbitrary constants C₁ and C_{2 .}
3. 3. Employing your boundary conditions associated with this problem, evaluate the values of the constants C₁ and C_{2 .}
4. 4. Using the temperature profile that you obtained in Part (c), determine the heat gained by the liquid hydrogen assuming steady state conditions exist at the conditions defined.  
5. 5. Determine the mass of liquid hydrogen that will evaporate. What fraction of the tank gets empty because of this?