Estimating the Diffusion Depth and Time to Approach Steady State

6 Estimating the Diffusion Depth and Time to Approach Steady State 

A rough estimate of the diffusion penetration distance from a point source is the location where the concentration has fallen off by as 1/e of the concentration at x = 0. This occurs when 

An estimate of the penetration distance for the error-function solution (Eq. 5.23) is the distance where c(x,t) = c0/8, or equivalently, erfjz/(2s/dI)] = -3/4, which corresponds to 

A reasonable estimate for the penetration depth is therefore again 2 y/Dt. 

To estimate the time at which steady-state conditions are expected, the required penetration distance is set equal to the largest characteristic length over which diffusion can take place in the system. If L is the characteristic linear dimension of a body, steady state may be expected to apply at times r L2/Dmm, where Dmin is the smallest value of the diffusivity in the body. Of course, there are many physical situations where steady-state conditions will never arise, such as when the boundary conditions are time dependent or the system is infinite or semi-infinite. 

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