,

where is concentration, is time, is distance, is the dispersion coefficient, is the pore velocity, and is the reaction constant.

The initial and boundary conditions are

,

,

and

.

The analytical solution to this equation is [1]:

,

with .

The mixing-cell model is an explicit finite-difference approximation to the diffusion-advection equation. The time derivative is approximated with a forward difference, while a backward difference is used to approximate the space derivative. For the cell, this approximation is written as:

. The left-hand side and the first term on the right-hand side describe the transport according to the mixing-cell model. In order to eliminate the effect of the second right-hand side term, the following relationship must hold [2]:

.

Taking the dispersion to be proportional to , then , thus

and

, where because cannot be negative.

The results are shown as a three-dimensional plot of the analytic solution (blue curve) and the mixing-cell model (red markers) for several values of the variables , , , and , which are obtained with , where is the column length. The results of the two methods are in very close agreement.