WOLFRAM|DEMONSTRATIONS PROJECT

Packed Bed Reactor with Three Chemical Species: Alternative Computational Methods

​
Péclet number
5
Damköhler number
3.1
equilibrium constant
1.
Consider the reversible reaction
A+B
k
f
⇌
k
r
C
. This reaction involves three chemical species,
A
,
B
, and
C
, and takes place in a packed bed reactor (PBR) with axial mixing. Our objective is to compare alternative solutions of this problem using Mathematica's NDSolve and using Chebyshev's orthogonal collocation method.
Introduce the dimensionless parameters
τ=
L
u
,
P
e
=
uL
D
(Péclet number), and
D
a
=kτ
(Damköhler number). Then the governing equations are:
∂
C
A
∂t
=
1
P
e
2
∂
C
A
∂
2
x
-
∂
C
A
∂x
-
D
a
 r
,
∂
C
B
∂t
=
1
P
e
2
∂
C
B
∂
2
x
-
∂
C
B
∂x
-
D
a
 r
,
and
∂
C
C
∂t
=
1
P
e
2
∂
C
C
∂
2
x
-
∂
C
C
∂x
+
D
a
 r
,
where
r=
C
A
C
B
-
C
C
K
e
and
K
e
=
k
f
/
k
r
.
Initially, the reactor contains only species
A
and
B
such that
C
A
(x,t=0)=0.6mol/l
and
C
B
(x,t=0)=0.4mol/l
.
The feed to the reactor is also composed of only
A
and
B
such that
C
A
(x=
-
0
,t)=0.6mol/l
and
C
B
(x=
-
0
,t)=0.4mol/l
.
The Demonstration solves this system of partial differential equations for user-set values of the Péclet and Damköhler numbers and the equilibrium constant
K
e
. The Mathematica NDSolve results are shown as colored solid curves, while the Chebyshev orthogonal collocation values are shown as colored dots. As you can see, the two approaches are in excellent agreement.