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:
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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