Analytical solution for the 1D convection-diffusion equation

Question

The Ogata and Banks analytical solution of the convection-diffusion equation for a continuous source of infinite duration and a 1D domain:

where C [mol/L] is the concentration, x [m] is the distance, R is the retardation factor, D [m2/day] is the effective dispersion/diffusion, v [m/day] is the flow velocity, Ci [mol/L] is the initial concentration in the column, and Co [mol/L] is the influent (or injected) concentration.

The time-evolution plot below shows two cases of specie production and degradation obtained from the numerical solution of equation (1) with the following parameters:

Production (red line): Ci = 0 mol/L, Co = 1.2 mol/L, R = 1, D = 0.00048 m2/day, v = 0.24 m/day.

Degradation (green line): Ci = 1 mol/L, Co = 0 mol/L, R = 1, D = 0.00048 m2/day, v = 0.24 m/day.

The analytical equation above matches (this is not shown in the plot) the production but not the degradation, whose curve is clearly an inversion of the production curve. So, I have been looking for the equivalent analytical expression that describes the degradation process (meaning that Ci would be contained therein). I am thus hoping that there is someone who may know this equation or could point me in the right direction. Thank you in advance.

Answer 1

It looks like equation (2) above assumes $C_i=0$. The more general equation is just $$C(x,t)=C_i+\frac{C_0-C_i}{2}\left[\mathrm{erfc}\left(\frac{x-vt}{2\sqrt{Dt/R}}\right) +\exp\left(\frac{vx}{D/R}\right)\mathrm{erfc}\left(\frac{x+vt}{2\sqrt{Dt/R}}\right)\right].$$

The essential form is the same; I've just created a new reference and scaled the forcing function. Convection-diffusion equations are linear, and so many can be manipulated in this way:

$$C~\mathrm{(conc.~or~temp.,~for~example)}=C_\mathrm{final}+(C_\mathrm{initial}-C_\mathrm{final})\times f(x,y,z,t),$$

where $f=1$ at $t=0$ and $f\to 0$ as $t\to\infty$.