Liquid oxygen LOX is being shipped through interstellar spac

Liquid oxygen (LOX) is being shipped through interstellar space in a spherical tank, constructed of several layers of material. The tank has an inner diameter of 5m, and the LOX is contained in a 5cm-thick shell of polycrystalline fused silicon dioxide, which is sheathed in a 2cm-thick pressure-bearing shell of 2024-T6 aluminum. The tank is illuminated faintly by sunlight, with a flux of 10 W/m^2. The outer shell is coated with magnesium oxide, and has alpha = 0.09 and epsilon = 0.90. Space may be assumed to be at 3K. Data for the solid materials may be evaluated at 100K, due to the table limitations. You may assume the shell is isothermal in the phi and theta directions, and all is at steady-state. To avoid expansion stresses on the shell, the LOX (melting point 54.9K) is kept from freezing by a small heater internal to the tank. How much heat must be supplied by this heater?

Solution

r₁=2.5m, r₂=2.5+.05=2.55m, r₃=2.5+.05+.02=2.57m

k₁(silica dioxide)= 1.38 W/m-K, k₂(Aluminium)= 151 W/m-K

T_i(LOX)= 54.9K, T_o(space)= 3K

Heat Transfer through the spherical shell , Q₁= ( T_o - T_i ) / [ { ( r₂ - r₁ ) / 4*pi*k₁*r₁*r₂ } + { (r₃ - r₂ ) / 4*pi*k₂*r₂*r₃ } ]

So, by substituting the values, Q₁ = 114.338 kW

Q₂ , Heat irradiated by outer surface = ( 1 - 0.09 )*10 + 0.90*5.67*10^-8*3⁴ = 9.1 W/m² = 9.1 * 4*pi*2.57² = 0.755 kW

Net heat that must be supplied = Heat transfer from the spherical shell = Q₁ + Q₂ = 115.09 kW