WOLFRAM|DEMONSTRATIONS PROJECT

Mixing-Cell Model for the Diffusion-Advection Equation

​
diffusivity
4.
pore velocity
4.
column length
120.
reaction constant
0.
The unsteady-state advection-diffusion equation is solved with a mixing-cell model. The results obtained with this simple model are in excellent agreement with the analytical solution when the correct choice of time and space steps is made. The equation describing the one-dimensional transport of a reactive component in porous media is:
∂C
∂t
=
2
∂
C
∂
2
x
-v
∂C
∂x
-kC
,
where
C
is concentration,
t
is time,
x
is distance,

is the dispersion coefficient,
v
is the pore velocity, and
k
is the reaction constant.
The initial and boundary conditions are
C(x,0)=0
,
C(0,t)=
C
0
=1
,
and
∂C(∞,t)
∂x
=0
.
The analytical solution to this equation is [1]:
C(x,t)=
C
0
1
2
exp
(v-u)x
2
erfc
x-ut
2
t
+
1
2
exp
(v+u)x
2
erfc
x+ut
2
t

,
with
u=v
1+4k

2
v
.
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
th
i
cell, this approximation is written as:
C
i,j+1
-
C
i,j
Δt
=v
C
i-1,j
-
C
i,j
Δx
+
2
v
Δt
2
-
vΔx
2
+
2
∂
C
∂
2
x
-k
C
i,j+1
.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]:
2
v
Δt
2
-
vΔx
2
+=0
.
Taking the dispersion

to be proportional to
v
, then
=αv
, thus
vΔt
2
-
Δx
2
+α=0
​and
Δt=
Δx-2α
v
,where
(Δx-2α)>0
because
Δt
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

,
v
,
L
, and
k
, which are obtained with
Δx=
L
10
, where
L
is the column length. The results of the two methods are in very close agreement.