Heat and Thermodynamics 3 Question 32
32. A point source of heat of power $P$ is placed at the centre of a spherical shell of mean radius $R$. The material of the shell has thermal conductivity $K$. If the temperature difference between the outer and inner surface of the shell is not to exceed $T$, the thickness of the shell should not be less than …… .
(1991, 1M)
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Solution:
- Thermal resistance $=\frac{l}{K A}=\frac{t}{K\left(4 \pi R^{2}\right)}(t=$ thickness $)$ Now, rate of heat transfer $=\frac{\text { Temperature difference }}{\text { Thermal resistance }}$
$$ =\frac{T}{t / 4 \pi K R^{2}}=\frac{4 \pi K T R^{2}}{t} $$
Equating this rate with the power of the source.
$\therefore P=\frac{4 \pi K T R^{2}}{t}$ or $t=\frac{4 \pi K T R^{2}}{P}$
or thickness $t$ should not exceed $\frac{4 \pi K T R^{2}}{P}$.
