Modeling of Combustion Systems A Practical Approach 8 doc

617
Appendix H
Equilibrium Primer
Consider a general reaction:
or equivalently
(H.1)
Here, j is an index from 1 to m reactants, r
j
refers to the number of moles
of the j
th
reactant, R
j
is the j
th
reactant, the double-headed arrow (↔) means
that both the forward and reverse reactions occur (typically at different rates),
k is an index from 1 to n products, p
k
is the number of moles of the k
th
product,
and P
k
is the k
th
product.
For Reaction H.1, we may define an equilibrium constant with the follow-
ing relation:
(H.2)
The brackets denote the volume concentrations of the enclosed species. If

we wish to express K
eq
in terms of mole fraction rather than concentration
for gases, then, using the ideal gas law, we obtain
(H.3)
rr pp
11 22 11 22
RR PP++↔++$$
rp
jj
j
m
kk
k
n
R
==
∑∑
↔
11
P
K
pp
rr
k
=
⎡
⎣
⎤
⎦

⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
PP
RR
P
12
12
12
12
$
$
pp
k
n
j

r
j
m
k
j
=
=
∏
∏
⎡
⎣
⎤
⎦
1
1
R
K
P
RT
yy
yy
ss rr
S
s
S
s
r
=
⎛
⎝

⎜
⎞
⎠
⎟
++−−−
12 12
1
1
2
2
1
1
$$
$
R RR
S
R
2
2
1
1
r
sr
s
k
n
r
j
m
P

RT
y
y
kk
k
k
k
j
$
=
⎛
⎝
⎜
⎞
⎠
⎟
−
∑
=
=
∏
∏
© 2006 by Taylor & Francis Group, LLC
618 Modeling of Combustion Systems: A Practical Approach
Please note that P and R (italicized) are the pressure and universal gas
constant, while P and R (nonitalicized) refer to products and reactants,
respectively.
For constant-pressure systems such as combustion, we may also incorpo-
rate the pressure dependence as part of the equilibrium constant, as
:

(H.4)
In either case, the equilibrium constant is defined as products over reactants.
K (or K
y
) is a function of temperature according to an Arrhenius relation:
(H.5)
Thus, K (or K
y
) is only constant if the reaction temperature is constant. Since
temperature is in the exponential, it usually overwhelms the temperature
effects shown in Equation H.3. In particular, note that K
y
is neither necessarily
dimensionless nor independent of pressure, except in the case that the num-
ber of moles remains invariant.
KK
RT
P
yy
pr
kk
=
⎛
⎝
⎜
⎞
⎠
⎟
−
∑

K
y
y
y
p
k
n
r
j
m
k
k
k
j
=
=
=
∏
∏
P
R
1
1
KAe
b
T
=
−
© 2006 by Taylor & Francis Group, LLC