A semi-infinite bar with a constant source (i.e. constant concentration) of solute material diffusing in from one end.
In this case, the solution is obtained by stacking a series of “thin sources” at one end of the bar, and summing the effects of all of the sources over the whole bar.
The initial concentration of solute in the bar is \(C_0\), therefore
\[C\left( {x,t = 0} \right ) = {C_0}\]
The concentration of solute at the end of the bar is a constant, \(C_s\), therefore
\[C\left( {x = 0,t} \right ) = {C_s}\]
These boundary conditions give the following solution:
The integral can only be solved numerically with a computer, so erf tables are used to solve the diffusion equation where necessary.
This animation shows the applications of Fick’s 2nd law and its solutions.
3. Non-Analytical Solutions
For more complicated situations we cannot obtain an analytical solution for Fick’s 2nd law. In these cases numerical analysis is used. Solutions obtained in this way are approximations, however, they can be made as precise as needed.
The following demonstration shows how numerical analysis can be used to approximate solutions for various conditions.
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.