Diffusion through a Hollow Cylinder

We consider two problems. The first, and more important one, is the determination of fhe diffusional flux that results under these conditions. The second problem is the derivation of the concentration profile and is of mainly academic inferesf. Bofh problems involve the solution of a simple ordinary differential equation by the technique of separation of variables. 

Diffusional flow from (a) a hoUow cylinder (b) a spherical cavity. 

Separating variables and formally integrating between the limits of internal and external concentrations C, and C we obtain 

By multiplying numerator and denominator by (r - r, this expression can be cast into the frequently used alternate form  

The introduction of the resistance carries the advantage that one can, in the case of a composite cylinder made up of different materials, describe the system by simply adding resistances. This principle of adding resistances in series is routinely applied in electrical circuits. For a cylinder composed of three different materials, for example, one obtains 

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