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Question
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. `( 4pir^2 ). dt/dx `
Here, the negative sign indicates that the temperature decreases with increasing radius.
`int_{r1}^{r2}\ (dr)/r^2 = (-4piK)/q int_{T1}^{T2}\ dT`
`[(-1)/r]_{r1}^{r2} = -(4piK)/q (T_2 - T_1)`
`⇒ q = (4piK(T_1 - T_2) ( r_1 - r_2))/(r_2-r_1)`
`⇒ q = (4xx 22/7 xx K(50-10)(0.2)xx0.05)/ (0.2-0.05)`
⇒ K = 3W /m°c
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