The figure shows a system of two concentric spheres of radii r₁ and r₂ kept at temperatures T₁ and T₂ respectively. The radial rate of flow of heat in substance between the concentric spheres is proportional to

 

Consider an element of radial thickness dr at a distance ‘r’ from the centre of shells.
Let dT be the temperature difference across the shells. The rate of flow of heat through the element
$\mathrm{H}=\mathrm{Ka}\left(-\frac{\mathrm{dT}}{\mathrm{dr}}\right)$

$\Rightarrow \mathrm{H}=-\mathrm{K}4{\mathrm{\pi r}}^2\frac{\mathrm{dT}}{\mathrm{dr}}$

$\Rightarrow \frac{\mathrm{dr}}{{\mathrm{r}}^2}=\frac{-4\mathrm{\pi K}}{\mathrm{H}}\mathrm{dt}$

Integrating on both sides,

 ${\int }_{{\mathrm{T}}_1}^{{\mathrm{r}}_2} \frac{\mathrm{dr}}{{\mathrm{r}}^2}=\frac{-4\mathrm{\pi K}}{\mathrm{H}}{\int }_{{\mathrm{T}}_1}^{{\mathrm{T}}_2} \mathrm{dT}$

$\Rightarrow \frac{1}{{\mathrm{r}}_1}-\frac{1}{{\mathrm{r}}_2}=\frac{4\mathrm{\pi K}}{\mathrm{H}}\left({\mathrm{T}}_1-{\mathrm{T}}_2\right)$

$\Rightarrow \frac{{\mathrm{r}}_2-{\mathrm{r}}_1}{{\mathrm{r}}_1{\mathrm{r}}_2}=\frac{4\mathrm{\pi K}}{\mathrm{H}}\left({\mathrm{T}}_1-{\mathrm{T}}_2\right)$

$\therefore \mathrm{H}\propto \left(\frac{{\mathrm{r}}_1{\mathrm{r}}_2}{{\mathrm{r}}_2-{\mathrm{r}}_1}\right)$