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WOLFRAM|DEMONSTRATIONS PROJECT

Absorption with Chemical Reaction in a Semi-Infinite Medium

​
collocation points
35
k
0.5
D
0.2
t
0.5
concentration profile
rate of absorption
Consider the unsteady-state absorption with a chemical reaction in a semi-infinite medium. The governing equation is:
∂c
∂t
=D
2
∂
c
∂
2
x
-kc
,
where
D
and
k
are the diffusion coefficient and first-order reaction rate constant, respectively.
The initial and boundary conditions are:
t=0
,
c(x,0)=0
,
x=0
,
c(0,t)=
c
A0
,
x=∞
,
c(∞,t)=0
,
where
c
A0
is the saturation concentration and
x
is the position.
This problem admits an analytical solution [4] given by:
c(z,t)/
c
A0
=
1
2
-
k
2
x
D
e
erfc
x
4Dt
-
kt
+
1
2
k
2
x
D
e
erfc
x
4Dt
+
kt
.
The rate of absorption is given by [4]:
-D
∂c
∂x
x=0
=
kD
erf(
kt
)+
-kt
e
πkt
.
This Demonstration plots the solution
c(x,t)
, as well as the rate of absorption versus time. The numerical solution obtained using the Chebyshev orthogonal collocation is given by the red dots. The analytical solution is given by the blue curve. The numerical rate of absorption is shown with a red curve. The analytical rate of absorption is given by the blue dashed curve. Excellent agreement between both solutions is observed.
You can vary the values of
t
,
D
, and
k
as well as the number of Chebyshev collocation points,
N+1
.
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