Environmental Engineering Reference
In-Depth Information
Please note that this definition of the mole-based conversion differs from the relative
degree of conversion of a reactant Equation (3.10), which is mass based.
Consider the following reaction:
a
A+
b
B
c
C+
d
D. If the rate law depends on
several species, we must relate the concentrations of the different species to each
other. This can be done using the stoichiometric coefficients
a
,
b
,
c
, and
d
. The
stoichiometric coefficients also give the relationship between the reaction rate
r
and the net production rates of the components,
R
A
,
R
B
,
R
C
,
R
D
:
!
r
=
−
R
ð Þ
a
=
−
R
ð Þ
b
=
R
C
c
=
R
D
d
ð
Eq
:
6
:
2
Þ
Note that (
R
B
),
R
C
and
R
D
are positive. A and B are reactants that disappear
during the reaction, so the net rates of production
R
A
,
R
B
are negative. In this chapter,
the analysis of the reactor will be based on the key component A.
When there are gaseous species involved in the reaction, the volume of the system
might be a function of the conversion. For instance, for an ideal gas, V = nR
u
T/
p
. If the
pressure
p
and temperature T are constant and the number of moles n increases due to
the reaction (in case
c
+
d
>
a
+
b
), the gas will expand. Since the concentration is the
ratio between the number of moles and the volume that these moles occupy, a change
in the volume will affect the concentration.
To account for changes in the volume of the system, we define the expansion
factor
−
R
A
), (
−
ε
A
as
ε
A
=
V
X
A
=1
−
V
X
A
=0
V
X
A
=0
ð
Eq
:
6
:
3
Þ
Here,
ε
A
refers to reactant A, but it can be defined for any component. In
those reactions where a change in the number of moles is associated with a change
in the reaction volume,
ε
A
can be calculated from the stoichiometric coefficients.
For instance, for the hypothetical reaction 2 A
!
5 B starting from pure A, the
expansion factor would be
2)/2 = 1.5. Starting from a mixture
consisting of 50 vol.% A and 50 vol.% inerts that do not contribute to the reaction,
ε
A
=(7
ε
A
=(5
−
−
4)/4 = 0.75. For most liquids,
ε
= 0 (no change in volume), while for many
gases
0.
The relation between the concentration and the conversion is
ε
6¼
c
A0
−
c
A
ε
ð Þ
c
A0
+
c
A0
1+
X
A
=
ε
A
c
A
;
X
A
=
−
2
dc
A
ð
Eq
:
6
:
4
Þ
c
A0
+
ð
ε
A
c
A
Þ
Together with the conversion, it is common to define the extent of the reaction
ξ
.
For closed systems (BR), it is defined as
=
n
i
−
n
i
0
ν
i
ξ
ð
Eq
:
6
:
5
Þ
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