Liquid oxygen at -183°C is stored in a thin walled spherical container with an outside diameter of 2 m. This container is surrounded by another sphere of 2.5 m inside diameter to reduce heat loss. The inner spherical surface has an emissivity of 0.05 and the outer sphere is black. Under normal operation the space between the spheres is evacuated but an accident resulted in a leak in the outer sphere and the space is filled with air at 100 kPa. If the outer sphere is at 25°C, compare the heat losses before and after the accident.
GIVEN
? A sphere filled with liquid oxygen surrounded by a larger sphere
? Sphere diameters ? Di = 2 m
? Do = 2.5 m
? Emissivity of inner sphere (?) = 0.05
? Outer sphere temperature (To) = 25°C = 298 K
? Liquid oxygen temperature (Ti) = –183° = 90 K
? Outer sphere is black
FIND
The rate of heat loss with
(a) A vacuum between the spheres
(b) Air at 100 kPa between the spheres
ASSUMPTIONS
? Steady state
? The internal convective resistance and the resistance of the inner sphere wall are negligible
SKETCH
PROPERTIES AND CONSTANTS
The thermal expansion coefficient (B) ? 1/T = 1/(194 K) = 0.0052 1/K
Extrapolating from Appendix 2, Table 27, for dry air at the mean temperature of –79°C from values at 0°C and 20°C
(a) With the space evacuated, there will only be radiative heat transfer as given by Equation (1.17)
The characteristic length for the problem is: b = (Do – Di)/2 = (2.5 m – 2.0 m)/2 = 0.25 m
The Rayleigh number is
(b) The following criteria must be satisfied to use Equation (8.34) for the convective heat transfer
Therefore, the condition is met.
The effective thermal conductivity of the air space is
The total rate of heat transfer will be the sum of the convective and radiative heat transfer
Where Reff is given by Equation (2.51)
The leak causes the rate of heat loss to increase 3.8 times
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