A simple formula for diffusion calculations involving wall reflection and low density

Abstract

Diffusion theory is often employed to calculate the effects of wall destruction on the local concentration of an active species immersed in a scattering gas. In many situations the spatial dependence of the concentration is given to a good approximation by the fundamental diffusion mode, and the local loss frequency can be calculated using the container’s fundamental mode diffusion length Λ. The additional assumption that the density of the active species may be taken to be zero at the container boundaries gives a value of Λ=Λ0 which depends only on the container dimensions, but use of Λ0 can be seriously in error if the diffusion mean free path λm is comparable to the dimensions, or if the particle reflection coefficient R becomes of significance. An improved boundary condition may be written simply in terms of the linear extrapolation length λ, whose inverse is the logarithmic gradient of the particle density at the boundary. The equation λ=2(1+R)λm/3(1−R) allows the representation of the full range of possible values of the particle reflection coefficient, 0<R<1, and extends the usefulness of the diffusion approximation to low scatterer densities. In the collisionless limit, the predicted particle loss frequency is identical to that predicted from the average chord length. Using this boundary condition, the dependence of Λ on λ/Λ0 has been computed for a range of simple container shapes, by solving the transcendental equations involved. This has allowed the identification of a dimensionless scaling variable, l0λ/Λ20, where l0 is the ratio of the container volume to its surface area. For all cases considered the simple empirical approximation Λ2=(Λ20+l0λ) is accurate when λ is very large or very small compared to Λ0, and disagrees most with the numerical solutions in the region where λ and Λ0 are comparable, with the worst case error being 11%.