Question

A hollow tube has a length l, inner radius R₁ and outer radius R₂. 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 T₁ and T₂ (T₂ > T₁) (b) the inside of the tube is maintained at temperature T₁ and the outside is maintained at T₂.

Answer 1

(a)  When the flat ends are maintained at temperatures T₁ and T₂ (where T₂ > T₁):

Area of cross section through which heat is flowing, = ${\displaystyle A=\pi (R_2^2-R_1^2)}$

Rate of flow of heat = ${\displaystyle \frac{d\theta }{dt}}$

${\displaystyle =\frac{KA(R\pi -R_1^2)(T_2-T_1)}{l}}$

( b )

When the inside of the tube is maintained at temperature T₁ 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, ${\displaystyle q=KA.\frac{aT}{dr}}$`

${\displaystyle q=KA.\frac{dt}{dr}}$
${\displaystyle q=K\left(2\pi rl\right)\frac{dt}{dr}}$
\[\int\limits_{R1}^{R2}\] ${\displaystyle \frac{dr}{r}=2\pi Kl}$  \[\int\limits_{T1}^{T2}\]  ${\displaystyle dT}$ 
${\displaystyle {[In \left(r\right)]}_{R1}^{R2} \frac{dr}{r}=\frac{2\pi rL}{q} [T_2-T_1]}$

In ${\displaystyle \left(\frac{R_2}{R_1}\right)=\frac{\text{2piKl}}{q}[T_2-T_1]}$

${\displaystyle q=\frac{2\pi Kl(T_2-T_1)}{\text{in}}\left(\frac{R_2}{R^1}\right)}$