An approximate expression is presented for the rate coefficient k to an embedded active site. The site is located at a depth (sR) in a hollow channel whose sides are located at the polar angle ϑ0, in a sphere of radius R. The result is k (ϑ0,s) = (2πDR)[1−cosϑ0][(1−s)/s +η (cosϑ0)]−1, where D is the diffusion coefficient and η (x) is given as a series of Legendre polynomials.

A simple exact solution for a point source above a reacting surface

The subject of this paper is competitive effects between multiple reaction sinks. A theory based on off-center monopoles is developed for the steady-state diffusion equation and for the convection-diffusion equation with a constant flow field. The dipolar approximation for the diffusion equation with two equal reaction centres is compared with the exact solution. The former turns out to be remarkably accurate, even for two touching spheres. Numerical evidence is presented to show that the same holds for larger clusters (with more than two spheres). The theory is extended to the convection-diffusion equation with a constant flow field. As one increases the convective velocity, the competitive effects between the reactive centres gradually become less significant. This is demonstrated for a number of cluster configurations. At high flow velocities, the current methodology breaks down. Fixing this problem will be the subject of future research. The current method is useful as an easy-to-use tool for the calibration of other more complicated models in mass and/or heat transfer.

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