First slide

Conduction

Question

Fig. shows a system of two concentric spheres of radii r₁and r₂ kept at temperatures T₁ and T₂ respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to

Difficult

A

$\frac{r_{1} r_{2}}{(r_{2} - r_{1})}$
B

$r_{2} - r_{1}$
C

$\frac{(r_{2} - r_{1})}{r_{1} r_{2}}$
D

$\log_{e} (\frac{r_{2}}{r_{1}})$

Solution

We know that $Q = - KA (\frac{dT}{dr})$
or    $Q = - K 4 {πr}^{2} (\frac{dT}{dr}) (∵ A = 4 {πr}^{2})$
$∴ \frac{dr}{r^{2}} = - \frac{4 πK}{Q} dT$
Integrating this expression within proper limits, we get
   $\int_{r_{1}}^{I_{2}} \frac{dr}{r^{2}} = - \frac{4 πK}{Q} \int_{T_{1}}^{T_{2}} dT$
or
   $\begin{aligned} \frac{1}{r_{1}} - \frac{1}{r_{2}} = \frac{4 πK}{Q} (T_{1} - T_{2}) \\ \frac{r_{2} - r_{1}}{r_{1} r_{2}} = \frac{4 πK}{Q} (T_{1} - T_{2}) \end{aligned}$
or
$\begin{array}{l} Q = \frac{4 πK (T_{1} - T_{2}) r_{1} r_{2}}{(r_{2} - r_{1})} \\ \propto \frac{r_{1} r_{2}}{(r_{2} - r_{1})} . \end{array}$

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