Question

A hollow aluminum sphere, 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 0.15 and 0.18 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 0.12 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/m²K1f65W are dissipated by the heater under steady-state conditions, what is the thermal conductivity of the insulation? W/mK

Answer 1

Ans

Thermal conductivity of aluminium is, k_al = 205 W /m K

Steady state heat dissipated from the sphere, Q = 65 W

20 C 0.3 m h=30 W/m2K kal 0.15 m 0.18 m 250 C

Therefore, we have

$Q = \frac{250 - 20}{\frac{0.18 - 0.15}{4\pi k_{al} *0.18*0.15} +\frac{0.3 - 0.18}{4\pi k_{i} *0.3*0.18} +\frac{1}{hA}}$

$\Rightarrow 65= \frac{250 - 20}{\frac{0.18 - 0.15}{4\pi *205 *0.18*0.15} +\frac{0.3 - 0.18}{4\pi k_{i} *0.3*0.18} +\frac{1}{30*4*\pi * 0.3^2}}$

$\Rightarrow 65= \frac{230}{0.0299 +\frac{0.1768}{ k_{i} } }$

$\Rightarrow k_{i} = 0.05 W/m K$ (ans)

where, 'k_i' is the thermal conductivity of the insulation