A hollow metallic sphere of radius 20 cm surrounds a concentric metallic sphere of radius 5 cm. The space between the two spheres is filled with a nonmetallic material. The inner and outer spheres are maintained at 50°C and 10°C respectively and it is found that 100 J of heat passes from the inner sphere to the outer sphere per second. Find the thermal conductivity of the material between the spheres.

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Solution

A = 4πr²
Let:
Radius of the inner sphere = r₁
Radius of the outer sphere = r₂
Consider a shell of radii r and thickness dr.

For this shell,
$Rate of flow of heat, q = - K \cdot (4 π r^{2}) \cdot \frac{d T}{d r}$
Here, the negative sign indicates that the temperature decreases with increasing radius.
$\int_{r_{1}}^{r_{2}} \frac{d r}{r^{2}} = \frac{- 4 π K}{q} \int_{T_{1}}^{T_{2}} dT$
${[\frac{- 1}{r}]}_{r_{1}}^{r_{2}} = \frac{- 4 π K}{q} (T_{2} - T_{1}) (\frac{1}{r_{1}} - \frac{1}{r_{2}}) = \frac{4 π K (T_{1} - T_{2})}{q} \Rightarrow q = \frac{4 π K (T_{1} - T_{2}) (r_{1} r_{2})}{(r_{2} - r_{1})} \Rightarrow q = \frac{4 \times \frac{22}{7} \times K (50 - 10) (0.2) \times 0.05}{(0.2 - 0.05)} \Rightarrow K = 3 W / m ° C$

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A hollow metallic sphere of radius 20 cm surrounds a concentric metallic sphere of radius 5 cm. The space between the two spheres is filled with a nonmetallic material. The inner and outer spheres are maintained at $50^{\circ} C$ and $10^{\circ} C$ respectively and it is found that 100 J of heat second. Find the thermal conductivity of the material between the spheres.