#### Question

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} > T_{1}) (b) the inside of the tube is maintained at temperature T_{1} and the outside is maintained at T_{2}.

#### Solution

(a) When the flat ends are maintained at temperatures T_{1} and T_{2} (where T_{2} > T_{1}):

Area of cross section through which heat is flowing, = `A = pi (R_2^2 - R_1^2)`

Rate of flow of heat = `(d theta)/dt`

`= (KA ( Rpi - R_1^2) (T_2 - T_1))/l`

( b )

When the inside of the tube is maintained at temperature T_{1} and the outside is maintained at T_{2}_{:}

Let us consider a cylindrical shell of radius *r* and thickness *dr*.

Rate of flow of heat, `q= KA. {aT}/{dr}``

`q = KA. (dt)/(dr)`

`q = K (2pirl)dt/(dr)`

\[\int\limits_{R1}^{R2}\] `(dr)/r = 2piKl` \[\int\limits_{T1}^{T2}\] `dT`

`[In (r)]_{R1 }^{R2} (dr)/(r) =( 2pirL)/q [T_2 - T_1]`

In `((R_2)/(R_1)) = "2piKl"/ q [T_2 - T_1]`

`q = (2piKl(T_2-T_1))/"in" (R_2/R^1)`