Question

The inner and outer surfaces of a hollow spherical shell of inner radius 'a' and outer radius 'b' are maintained at temperature $T_1$ and $T_2$ $(<T_1)$. The thermal conductivity of material of the shell is k. If the rate of heat flow inner to outer surface is $\frac{dQ}{dt}=N\pi kab\left(\frac{T_1-T_2}{b-a}\right)$,then value of $N$ is:

Answer 1

Taking a spherical shell at a radius r and thickness dr, we can write

$I=\frac{dQ}{dt}=k\left(4\pi r^2\right)\frac{dT}{dr}$

Thus, $\int \frac{dr}{r^2}=\int \frac{4\pi k}{I}dT$

$\frac{1}{a}-\frac{1}{b}=\frac{4\pi k}{I}(T_2-T_1)$

$I=\frac{dQ}{dt}=4\pi kab\left(\frac{T_1-T_2}{b-a}\right)$

Thus $N=4$

Answer is 4.