A sphere 20 cm in diameter containing liquid air (–140°C) is covered with 5-cm-thick glass wool (50 kg/m3 density) with an emissivity of 0.8. Estimate the rate of heat transfer to the liquid air from the surrounding air at 20°C by convection and radiation. How would you reduce the heat transfer?

Question

GIVEN

• A sphere containing liquid air covered with glass wool

• Sphere diameter (Ds) = 20 cm = 0.2 m

• Liquid air temperature (Ta) = – 140°C = 133 K

• Surrounding air temperature (T?) = 20°C = 293 K

• Insulation thickness (s) = 5 cm = 0.05 m

• Insulation emissivity (?) = 0.8

FIND

• Rate of heat transfer from liquid air to surrounding air

(q)

• How can this be reduced?

ASSUMPTIONS

• Steady state conditions

• The surroundings behave as a black body enclosure at T?

• Surrounding air is still

• Thermal resistance of the convection inside the sphere and of the container wall are negligible

SKETCH



PROPERTIES AND CONSTANTS

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

the thermal conductivity of glass wool (ki) = 0.037 W/(m K)

Answer 1

The natural convection heat transfer coefficient on the exterior of the insulation depends on the exterior temperature of the insulation (Ts), an iterative procedure is therefore required. For the first iteration, let Ts = – 20°C (253 K)

for dry air at the mean temperature of 0°C

Thermal expansion coefficient (?) = 0.00366 1/K

Thermal conductivity (k) = 0.0237 W/(m K)

Kinematic viscosity (?) = 13.9 × 10–6 m2/s

Prandtl number (Pr) = 0.71

The characteristic length for the sphere is



The Grahsof and Rayleigh numbers based on this length are



Although the empirical relation extends only to Ra+ = 1.5 × 109, it will be extrapolated here to estimate the Nusselt number



The thermal circuit for the sphere is shown below



where Rci = interior convective resistance (negligible)

Rks = conductive resistance of the container (negligible)

Rki = conductive resistance of the insulation

Rco = exterior convective resistance

Rro = exterior radiative resistance





The exterior radiative resistance is





The net resistance for the thermal network is Rt = Rki + Ro where



The rate of heat transfer is given by



The accuracy of the insulation surface temperature guess can be checked from



Therefore, we need to reduce Tso. However, notice that nearly 94% of the total thermal resistance is due to the insulation. This means that adjusting Tso has little effect on the total rate of heat transfer. It also means that the heat gain by the liquid air can be most easily reduced by increasing the thickness of insulation, selecting an insulation with lower thermal conductivity, or both.