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

Explanation:

(a) Flat ends at $\mathrm{T}_{1} \& \mathrm{T}_{2}, \mathrm{T}_{2}>\mathrm{T}_{1}$

Area of cross-section = $\pi\left(R_{2}^{2}-R_{1}^{2}\right)$

Rate of flow of heat    $\frac{d Q}{d t}=\frac{K A \Delta T}{1}$

$\frac{d Q}{d t}=K \pi\left(R_{2}^{2}-R_{1}^{2}\right)\left(T_{2}-T_{1}\right) / l$

(b) Inside temp.= T1

Outside temp. =T2

Consider a cylindrical shell of radius r and thickness dr

$q=\frac{K A d T}{d r}$

$q=\frac{K 2 \pi r l d T}{d r}$

Integrating both sides  

$\int_{r_{2}}^{r_{2}} \frac{d x}{x}=\frac{2 \pi k l \int_{T_{1}}^{T_{2}} d T}{q}$

$q=\frac{2 \pi K l\left(T_{2}-T_{1}\right)}{\ln \frac{r_{2}}{r_{1}}}$