A hollow tube possesses a length l , inner radius R_1 , and outer...

Consider an elemental cylindrical shell of radius $r$ and thickness $dr$.

The surface area of this shell is $A = 2\pi r l$.

According to Fourier's law of heat conduction, the rate of heat flow $H$ is given by:

$H = -K A \dfrac{dT}{dr} = -K (2\pi r l) \dfrac{dT}{dr}$

Rearranging the terms for integration:

$\dfrac{dr}{r} = -\dfrac{2\pi K l}{H} dT$

Integrating both sides within the limits $r = R_1$ to $R_2$ and $T = T_1$ to $T_2$:

$\int_{R_1}^{R_2} \dfrac{dr}{r} = -\dfrac{2\pi K l}{H} \int_{T_1}^{T_2} dT$

$\ln\left(\dfrac{R_2}{R_1}\right) = \dfrac{2\pi K l}{H} (T_1 - T_2)$

Taking the magnitude of heat flow, we get:

$H = \dfrac{2\pi K l (T_2 - T_1)}{\ln(R_2 / R_1)}$