The figure shows a system of two concentric spheres of radii r1 and r2 and kep

Discussion Forum : Heat Transfer

Question -

The figure shows a system of two concentric spheres of radii $r_{1}$ and $r_{2}$ and kept at temperatures $T_{1}$ and $T_{2}$ , respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to

Options:

A .

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

B .

(r_{2} - r_{1})

C .

(r_{2} - r_{1}) (r_{1} r_{2})

D .

I n (\frac{r_{2}}{r_{1}})

Answer: Option A
:
A
Consider a concentric spherical shell of radius r and thickness dr as shown in fig.

The Figure Shows A System Of Two Concentric Spheres Of Radii...

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 r} = - 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_{1}} d t

Which on integration and simplification gives

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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