A hollow sphere with inside diameter (ID) of
*r*_{o} has an outside diameter (OD) of
*r*_{o}. The inside wall is held at
*T*_{i} and the outside is kept at
*T*_{o} (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

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

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