Nonsteady State Diffusion

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Non-steady state implies a change in concentration with time. There is a first derivative in time, ∂/∂t, and a second derivative in location, ∂2/∂x2, which makes this a notoriously difficult problem for computer programs. Luckily, there are some exact solutions, one of them being penetration into a flat layer:

c = c0 {1 − erf ( x )} 
2√D¯¯t
where c0 is the concentration at x=0 where the layer starts, and erf() is the so-called error function. The solution is shown in the figure, for three values of t, and it looks like a filled portion gradually shifting towards higher x values. Therefore, this solution is also referred to as the moving front solution. It allows for an estimation when the end of a layer at x=L is reached, which turns out to be:
tL = L2
6D
 but that is better proven in a derivation of another type. An interesting thing about diffusion comes forward here: the time that it takes to overcome a certain distance increases much more rapidly than the distance itself. This can be checked from the formula: if for a certain distance L the tL is 1 second, then for 2L that is 4 seconds and for 10 L already 100 seconds. Or: diffusion may be fast on short distances, it will be very slow on large distances. Though shown here for a flat layer, that is generally true.

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