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... 618 Modeling of Combust
Trang 1Appendix 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 jth reactant, R j is the jth 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 kth product,
and P k is the kth 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)
r1R1+r2R2+ ↔$ p1 1P +p2 2P +$
j
m
k k k
n
R
∑ ↔∑
P
K
k
= ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦
⎡⎣ ⎤⎦
P
1 2
1 2
1 2
1 2
$
$
pp k n
j r j m
k
j
=
=
∏
1
1
R
RT
s s r r
S s S s r
=⎛
⎝⎜
⎞
⎠⎟
+ + − − −
1 2 1 2
1 1 2 2 1 1
$ $
$
R R
S
R
2 2
1
1
r
s r
s k n
r j m
P RT
y y
k
k j
$=⎛
⎝⎜
⎞
⎠⎟
−
∑
=
=
∏
∏
© 2006 by Taylor & Francis Group, LLC
Trang 2618 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
P
p k r k
−
∑
K
y y y
p k n
r j m
k k
k j
=
∏
∏
P
R 1
1
b T
© 2006 by Taylor & Francis Group, LLC