The figure shows a system of two concentric shells of radii r 1 and r 2 and kept at temp. T 1 and T 2 , respectively. In between the shells, a substance of thermal conductivity K is filled. Th

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The figure shows a system of two concentric shells of radii $r_{1}$ and $r_{2}$ and kept at temp. $T_{1}$ and $T_{2}$ , respectively. In between the shells, a substance of thermal conductivity K is filled. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to :

TestHub · Accepted Answer

Consider a concentric spherical shell of radius r and thickness dr as shown in figure. The radial rate of flow of heat through this shell in steady state will be

$H = \frac{d Q}{d t} = - K A \frac{d T}{d t} = - K (4 π r^{2}) \frac{d T}{d r} \Rightarrow \int_{r_{1}}^{r_{2}} \frac{d r}{r ^{2}} = - \frac{4 π K}{H} \int_{T_{1}}^{T_{2}} d T \Rightarrow H = \frac{d Q}{d t} = \frac{4 π K r _{1} r _{2} ( T _{1} - T _{2} )}{r _{2} - r _{1}} \Rightarrow \frac{d Q}{d t} \propto \frac{r _{1} r _{2}}{( r _{2} - r _{1} )}$

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