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WOLFRAM NOTEBOOK
Mixing-Cell Model for the Diffusion-Advection Equation
Mixing-Cell Model for the Diffusion-Advection Equation
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
∂
∂
2
x
∂C
∂x
where is concentration, is time, is distance, is the dispersion coefficient, is the pore velocity, and is the reaction constant.
C
t
x
v
k
The initial and boundary conditions are
C(x,0)=0
C(0,t)==1
C
0
and
∂C(∞,t)
∂x
The analytical solution to this equation is [1]:
C(x,t)=experfc+experfc
C
0
1
2
(v-u)x
2
x-ut
2
t
1
2
(v+u)x
2
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 cell, this approximation is written as:
th
i
C
i,j+1
C
i,j
Δt
C
i-1,j
C
i,j
Δx
2
v
2
vΔx
2
2
∂
∂
2
x
C
i,j+1
2
v
2
vΔx
2
Taking the dispersion to be proportional to , then , thus
v
=αv
vΔt
2
Δx
2
Δt=
Δx-2α
v
(Δx-2α)>0
Δt
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.
v
L
k
Δx=
L
10
L