Transmission Of Heat Question 131
Question: The figure shows a system of two concentric spheres of radii r1 and r2 and 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
[AIEEE 2005]
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) In $ ( \frac{r _{2}}{r _{1}} ) $
Show Answer
Answer:
Correct Answer: A
Solution:
Consider a concentric spherical shell of radius r and thickness dr as shown in fig. The radial rate of flow of heat through this shell in steady state will be $ H=\frac{dQ}{dt}=-KA\frac{dT}{dr}=-K,(4\pi r^{2})\frac{dT}{dr} $
Therefore $ \int _{,r _{1}}^{,r _{2}}{\frac{dr}{r^{2}}=-\frac{4\pi K}{H}\int _{,T _{1}}^{T _{1}}{dT}} $ Which on integration and simplification gives $ H=\frac{dQ}{dt}=\frac{4\pi Kr _{1}r _{2}(T _{1}-T _{2})}{r _{2}-r _{1}} $
Therefore $ \frac{dQ}{dt}\propto \frac{r _{1}r _{2}}{(r _{2}-r _{1})} $
BETA
