Two concentric spheres 0.2 m and 0.3-m in diameter, with the space between them evacuated, are to be used to store liquid air (133 K). If the surfaces of the spheres have been flashed with aluminum and the liquid air has a talent heat of vaporization of 209 kJ/kg, determine the number of kilograms of liquid air evaporated per hour.

Question

GIVEN

• Two concentric spheres with the space between them evacuated and liquid air in the inner sphere

• Diameters ? D1 = 0.2 m ? D2 = 0.3 m

• Liquid air temperature (Ta) = 133 K

• Room temperature (T?) = 293 K

• Surfaces of the spheres have been flashed with aluminum

• Heat of vaporization of liquid air (hfg) = 209 kJ/kg = 209,000 J/kg

FIND

• The number of kilograms of liquid air evaporated per hour ( )m

ASSUMPTIONS

• Steady state

• Convective thermal resistance between the liquid air and interior sphere is negligible

• Thermal resistance of the sphere walls is negligible

• Natural convection on the exterior is negligible

• The room behaves as a blackbody enclosure

• The thickness of the sphere walls in negligible

SKETCH



PROPERTIES AND CONSTANTS

the Stephan-Boltzmann constant (?) = 5.67 × 10–8 W/(m2 K4)

the hemispherical emissivity of the spheres will be approximated by that for oxidized aluminum at 310 K: ? = ?1 ? 2 = 0.11

Answer 1

Since A2 completely surrounds A1 and A1 cannot see itself, F12 = 1.0 and F11 = 0



The shape factors from a given surface must sum to unity



Also



The net rate of heat transfer from A1 to A2 must equal the rate of heat transfer from the exterior sphere to the surroundings



The rate of heat transfer between the spheres is



where f12 is given for concentric spheres.



Solving for T2



The rate of evaporation of the liquid air is



COMMENT

The rate of evaporation would be reduced to 0.024 kg/h if the two aluminum surfaces in the evacuated space could remain polished so that ?1 = ? 2 = 0.04.