I am reading reference about diffusion and not quite clear about how to get the diffusion rate from the geometry and boundary condition. For example, a sphere with boundary condition that concentration $c=C_1$ at surface and inside of sphere and $c=C_0$ at infinity will give the diffusion equation:
$$\frac{dn}{dt}=-[4\pi (Dr)](C_1/C_0 -1)$$
To bring this further, consider a cylinder with a radius $r$ and height $h$. Concentration at surface and inside of the geometry is $C_0$ and at infinity is $C_1$. What is the diffusion rate of this. How about cylinder with constraint that we can only diffuse at top and bottom of the cylinder?
This is not easy: the problem requires solving Laplace's equation in the exterior of the solid. The diffusion equation is $v_t = k\Delta v$ where $k$ is the diffusivity constant. The steady distribution $u$ (independent of time) satisfies $u=C_1$ on the surface, $u=C_0$ at infinity, and $\Delta u=0$ in the exterior of the solid. The rate of diffusion through the surface is obtained by integrating the flux vector $-k\nabla u$ over the surface.
The difficult step is to find $u$. The case of the sphere is special because we know all radially symmetric harmonic functions: they are constants and multiples of $1/\|x\|$. Hence, $$u(x) = C_0+\frac{C_1-C_0}{\|x\|}$$ which leads to the result you mentioned.
However, solving Laplace's equation in the exterior of a cylinder is more difficult. (You may be able to find something relevant by Googling the italicized words, but I doubt it will be an explicit solution). The axial symmetry should help by reducing the problem to two cylindrical variables $r,z$ (no $\theta$ dependency). At worst, the resulting two-dimensional boundary value problem can be solved numerically using things like PDE Toolbox in Matlab.
If the side surface is impermeable, we have a mixed boundary problem for Laplace's equation: $u=C_1$ on the top and bottom, $\dfrac{\partial u}{\partial n} = 0$ on the side surface.