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, $\mathrm{A}=\mathrm{\pi }\left({\mathrm{R}}_2^2-{\mathrm{R}}_1^2\right)$

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

$$\begin{gathered} =\frac{KA.∆T}{l} \\ =\frac{K\mathrm{\pi }\left({\mathrm{R}}_2^2-{\mathrm{R}}_1^2\right)\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)}{l} \end{gathered}$$

(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, q$=KA.\frac{d\mathrm{T}}{dr}$
$$\begin{gathered} q=KA.\frac{\mathrm{dT}}{\mathrm{d}r} \\ q=K\left(2\mathrm{\pi }rl\right)\frac{\mathrm{dT}}{\mathrm{d}r} \\ {\int }_{{\mathrm{R}}_1}^{{\mathrm{R}}_2}\frac{dr}{r}=\frac{2\mathrm{\pi }Kl}{q}{\int }_{{\mathrm{T}}_1}^{{\mathrm{T}}_2}d\mathrm{T} \\ {\left[\ln \left(r\right)\right]}_{{\mathrm{R}}_1}^{{\mathrm{R}}_2}=\frac{2\mathrm{\pi }Kl}{q}\left[{\mathrm{T}}_2-{\mathrm{T}}_1\right] \\ \ln \left(\frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}\right)=\frac{2\mathrm{\pi }Kl}{q}\left({\mathrm{T}}_2-{\mathrm{T}}_1\right) \\ q=\frac{2\mathrm{\pi }Kl\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)}{\ln \left({\displaystyle \frac{{\mathrm{R}}_2}{{\mathrm{R}}_1}}\right)} \end{gathered}$$