CGPET 2013 Solved Paper

Consider a shell of thickness (dr) and of radii ( r ) and the temperature of inner and outer surfaces of this shell be T, (T−dT).
‌

dQ

dt

= rate of flow of heat through it
‌=‌

kA[(T−dT)−T]

dr

=‌

−KAdT

dr

‌=−4πkr2‌

dT

dT

‌‌[∵A=4πr2]
To measure the radial rate of heat flow, integration technique is used, since the area of the surface through which heat will flow is not constant.
Then,
‌(‌

dQ

dt

)‌

r2

∫

r1

‌

1

r2

‌dr=−4πK‌

T2

∫

T1

dT
‌‌

dQ

dt

(‌

1

r1

−‌

1

r2

)=−4πK(T2−T1)
‌‌

dQ

dt

=‌

−4πk1r2(T2−T1)

(r2−r1)

‌⇒‌‌‌

dQ

dt

∝‌

(r1r2)

(r2−r1)

‌