Thermal Conductivity of Insulation Calculation

What is the thermal conductivity of the insulation material in the given scenario?

If 80W are dissipated by the heater under steady-state conditions, what is the thermal conductivity of the insulation?

Answer:

Thermal conductivity of the insulation is k_ins = 0.062 W/m.k

A hollow aluminum sphere (kAl = 234 W/mK), with an electrical heater in the center, is used in tests to determine the thermal conductivity of insulating materials. The inner and outer radii of the sphere are 0.18 and 0.21 m, respectively, and testing is done under steady-state conditions with the inner surface of the aluminum maintained at 250°C. In a particular test, a spherical shell of insulation is cast on the outer surface of the sphere to a thickness of 0.15 m. The system is in a room for which the air temperature is 20°C and the convection coefficient at the outer surface of the insulation is 30 W/m2K.

Given values:

Inner radius of sphere (r_i) = 0.15 m

Outer radius of sphere (r_o) = 0.18 m

Thickness of insulation (r) = 0.12 m

Outer radius of insulation (r) = 0.30 m

Convection coefficient (h) = 30 W/m².K

Inner surface temperature (T_s) = 250°C = 523 K

Room temperature (T(∞)) = 20°C = 293 K

Thermal conductivity of aluminum (k_Al) = 234 W/m.K

The formula for the rate of heat transfer is q = [T_s - T(∞)]/R, where R is the total thermal resistance.

By calculating the thermal conductivity of insulation, we find that k_ins = 0.062 W/m.k