A hollow tube has a length l, inner radius `R_(1)` and outer radius `R_(2)` . The material has a thermal conductivity K. Find the heat flowing through the walls of the tube if (a) the flat ends are maintained at temperature `T_(1)` and `T_(2)``(T_(2)gtT_(1))` (b) the inside of the tube is maintained at temperature `T_(1)` and the outside is maintained at `T_(2)` .

Answer

A, B

Explanation

`T_2gtT_1`
`A=pi(R_2^2-R_1^2)`
`so, (dQ)/(dt)=(KA(T_2-T_1))/l`
`=(K pi(R_2^2-R_1^2)(T_2-T_1))/l`
considering a concentric cylindtical cell of radius R and thickness DR.The radial heat flow through the cell.
H=(dQ)/(dt)=-KA(d theta)/(dr)`
`[-ve because as r increases Q decreases]`
`A=2 pi rl,`
`H=-2 pi rl K(d theta)/(dr)`
`or, int_(r_1)^(r_2)(dr)/r=-(2 pi rlK)/H int_(T_1)^(T_2)d theta`
Integrating and simplifying we get
`H=(dQ)/(dt)=(2 pi Kl(T_2-T_1))/(In(R_2//R_1))`.