Diffusion to a Spherical Reactor
This Demonstration shows the steady-state chemical concentration around and within a spherical chemical reactor that consumes the chemical reaching the sphere via diffusion. The "cross-section view" shows the concentration on a plane through the center of the sphere, with the sphere's surface indicated by the black circle. The "radial view" compares the concentration with that of a fully absorbing sphere of the same size, with the gray region indicating the sphere's interior. The total reaction rate percentage is the rate at which the reactor consumes the chemical relative to the rate for a fully absorbing sphere.
The concentration at distance from the center of the spherical reactor satisfies the diffusion equation if and 0 if , in spherical coordinates, where is the diffusion coefficient, is the radius of the spherical reactor, and is the reaction rate coefficient. The sphere is in an unbounded fluid. In the units used here, the concentration approaches far from the sphere.
The reaction inside the sphere is first-order kinetics, meaning the reaction rate is proportional to the concentration: the number of molecules removed from the fluid per unit time and per unit volume is at a position with distance from the center of the sphere. The removal of the chemical in the reactor creates a concentration gradient and hence a diffusion of the chemical from the surrounding fluid into the sphere.
For comparison, the fully absorbing sphere satisfies for with boundary condition .
The fully absorbing sphere gives the maximum possible rate a sphere can absorb the diffusing chemical. The reactor gets close to this maximum by consuming arriving molecules before they have a chance to diffuse back out of the reactor. Achieving this requires the typical distance a molecule in the reactor diffuses before being consumed, , to be small compared to the radius of the sphere, . In this case, most of the consumption occurs close to the surface of the sphere.
H. C. Berg, Random Walks in Biology, ed., Princeton: Princeton University Press, 1993.