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 λ_m 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)λ_m/3(1−R) allows the representation of the full range of possible values of the particle reflection coefficient, 0<R0 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%.