CL: 1-D Conduction through a Sphere

A hollow sphere with inside diameter (ID) of ro has an outside diameter (OD) of ro. The inside wall is held at Ti and the outside is kept at To (these are our boundary conditions!). We know k. We would like to determine what the rate of heat flow is.

Again, we start with Fourier's Law only now in spherical coordinates:

$\displaystyle{\frac{q}{A} = -k\frac{dT}{dr}}$

Writing the surface area in spherical coordinates (recall that the surface area is our area perpendicular to the heat flow) gives us

$\displaystyle{\frac{q}{4\pi r^2} = -k\frac{dT}{dr}}$

Rearranging so that we can integrate,

$\displaystyle{\frac{q}{4\pi} \int_{r_i}^{r_o} \frac{dr}{r^2} = -k \int_{T_i}^{T_o} dT}$

Integrating and rearranging again gives:

$\displaystyle{q = \frac{4\pi k}{\frac{1}{r_i}-\frac{1}{r_o}} \left (T_i - T_o \right)}$

Writing this in terms of a resistor gives

$\displaystyle{q = \frac{\Delta T}{R}}$

where R is given as

$\displaystyle{R_{sph} = \frac{\frac{1}{r_i}-\frac{1}{r_o}}{4\pi k}}$

OUTCOME:

Calculate the thermal resistance and magnitude of conductive heat flow/flux through a spherical shell