First slide

Conduction

Question

A point source of heat of power P is placed at the centre of a spherical shell of mean radius fi. 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 L the thickness of the shell should not be less than

Difficult

A

$4 {πKTR}^{2} / P$
B

$4 {πK}^{2} TR / P$
C

$4 {πKT}^{2} R / P$
D

$4 πKTR / P^{2}$

Solution

$P = - KA \frac{dθ}{dr} = - K 4 {πr}^{2} (\frac{dθ}{dr})$
or $dθ = - \frac{Pdr}{4 {πKr}^{2}}$
Integrating, we have
$\int_{θ_{1}}^{θ_{2}} dθ = - \frac{P}{4 πK} \int_{r_{1}}^{T_{2}} (\frac{dr}{r^{2}})$
$∴ (θ_{1} - θ_{2}) = \frac{P}{4 πK} (\frac{1}{I_{1}} - \frac{1}{r_{2}}) = \frac{P}{4 πK} \frac{(I_{2} - r_{1})}{r_{1} I_{2}}$
So thickness
$I_{2} - I_{1} = \frac{4 πK (θ_{1} - θ_{2}) r_{1} r_{2}}{P} = \frac{4 {πKTR}^{2}}{P}$

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