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.
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
From Appendix 1, Table 5, the Stephan-Boltzmann constant
From Table 11.3, the hemispherical emissivity of the spheres will be approximated by that for
oxidized aluminum at 310 K: ? ? = ??1 ???2 = 0.11
Since A2 completely surrounds A1 and A1 cannot see itself, F12 = 1.0 and F11 = 0
From Equation (11.46)
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 given by Equation (11.75)
where f12 is given in Equation (11.76) for concentric spheres.
Solving Equations [1] and [2] for T2
The rate of evaporation of the liquid air is
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