24 Analytical solution to the diffusion reaction equation

General form of the transport equation

The general form of the radionuclide transport equation is:
[latex]\frac{\partial}{\partial t}(\epsilon K N)+ \nabla (\epsilon v N) - \nabla^2 (\epsilon D N)+\epsilon K \lambda N =0.[/latex]

Simplifying assumptions

Assuming v = 0, we have the time-dependent diffusion reaction equation.

The transport equation reduces to:
[latex]\epsilon K \frac{\partial N}{\partial t}-\epsilon D \frac{\partial^2 N}{\partial x^2}+\epsilon K \lambda N=0.[/latex]

For a semi-infinite medium, the following side conditions are applied:
[latex]\displaylines{ N(x,0)=0,~~ 0 \lt x \lt \infty, \\ N(0,t)=N^*,~~ t \gt \infty, \\ N(\infty,t)=0,~~ t \gt \infty. }[/latex]

General procedure

1. Apply the Laplace transformation as before.
2. Obtain the general solution in the Laplace space.
3. Apply the side conditions to the general solution in the Laplace space.
4. Use the ‘sigma transformation’:
   [latex]\frac{\sqrt{\pi}}{2}e^{-2\sigma}=\int_{0}^{\infty}e^{-\xi^2-\frac{\sigma^2}{\xi^2}}d\xi[/latex]
5. Rearrange the integral similar to the general transport equation to obtain the solution in the real space, N(x,t).
6. Apply the exponential/error function transformation:
   [latex]\int e^{-a^2\xi^2-\frac{b^2}{\xi^2}}d\xi=\frac{\sqrt{\pi}}{4a}[e^{2ab}erf(a\xi+\frac{b}{\xi})+e^{-2ab}erf(a\xi-\frac{b}{\xi})][/latex]
7. Evaluate the expression using the limits of integration, rearrange, and apply the erfc.

The final solution is:
[latex]N(x,t)=\frac{N^*}{2}[e^{x\sqrt{\frac{K\lambda}{D}}}erfc(x\sqrt{\frac{K}{4Dt}}+2\sqrt{\lambda t}+e^{-x\sqrt{\frac{K\lambda}{D}}}erfc(x\sqrt{\frac{K}{4Dt}}-2\sqrt{\lambda t})][/latex]