Heat conduction through hollow cylinder with constant internal and external temperature

Question

Consider a hollow cylinder of inner radius $r_1$ and outer radius $r_2$. The entire surface of the cylinder is kept at constant (but distinct) temperatures along the interior and exterior.

I found that the steady state temperature function is (in cylindrical coordinates) $$T = -kr_2 \log \left(\frac{r}{r_1}\right) + T_0.$$

I am then asked to find the heat flux through a Gaussian cylinder of radius $r\in (r_1,r_2)$. The answer given is that the flux per unit length $\phi$ is $$\phi = 2\pi r h_r,$$ where $h_r$ is the $r$ component of $\nabla \mathbf{h}$ (in cylindrical coordinates). My question is, how is this consistent with Gauss' Theorem (Divergence Theorem)? Because I found that $\nabla \cdot \mathbf{h} = 0$, so shouldn't the flux be $$\iint_S \mathbf{h}\cdot\mathrm{d}\mathbf{S} = \iiint_V \nabla\cdot \mathbf{h}\, \mathrm{d}V = \iiint_V 0\, \mathrm{d}V = 0?$$

Any help would be appreciated.

Answer 1

Here is the correct development: $$\phi=2\pi r q_r$$where $q_r$ is the rate of heat flow per unit area in the radial direction: $$q_r=-k\frac{dT}{dr}$$where k is the thermal conductivity of the annular cylinder. These combine to give $$\frac{dT}{dr}=-\frac{\phi}{2\pi rk}$$integrating between $r_1$ and r gives:$$T-T_1=\frac{\phi}{2\pi k}\ln{(r_1/r)}$$The parameter $\phi$ can be obtained by applying the boundary condition at $r_2$: $$\phi=2\pi k\frac{(T_1-T_2)}{\ln{(r_2/r_1)}}$$