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

enter image description here

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.

enter image description here

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.


1 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 $t\to\infty$.

  • $\begingroup$ Thank you @Chemomechanics. I appreciate. If possible, can I get the reference for this? $\endgroup$
    – gary105
    Jun 30, 2021 at 5:53
  • $\begingroup$ Incropera & DeWitt’s Fundamentals of Heat and Mass Transfer and Crank’s Mathematics of Diffusion are good general references. I hadn’t seen this particular equation before you posted it. $\endgroup$ Jun 30, 2021 at 14:14

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