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Assigned 11/02/09, due 11/09/09
(1)
In class, we derived a set of partial differential equations that, with some assumptions,
govern the spatial and temporal behavior of species concentrations in a chemically
reacting system.
dC
dt
v
C
D
C
r
k
k
k
k
ik i
i
M
= − ⋅∇
+
∇
+
=
∑
2
1
α
, for
k
from 1 to
N
, the number of species.
If the reactor to be modeled has turbulent flow or has complex flow patterns through a
bed of catalyst, then the diffusion coefficients,
D
k
are not molecular diffusion
coefficients.
Instead, they are effective diffusion coefficients that are related to mixing
in turbulent eddies and in complex flow patterns past catalyst particles.
In that case, the
diffusion coefficients are the same for all of the species.
We can define extents of
reaction such that the species concentrations are related to the extents of reaction by
C
C
k
k o
ik
i
i
M
=
+
=
∑
,
αξ
1
where the
C
k,o
are some reference concentrations (usually the feed concentrations) and
the
ξ
i
are the extents of reaction (with units of concentration).
Starting from the above
PDE’s for the concentrations (with the same effective diffusion coefficient for all species)
and this definition of the extents of reaction, derive a set of
M
equations in terms of the
extents of reaction that are equivalent to the
N
equations in terms of the concentrations.
(2)
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This document was uploaded on 07/08/2011.
 Fall '09

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