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Unformatted text preview: 243 7.1. ISOTHERMAL, ISOCHORIC KINETICS
7.1.1.2 Single reversible reaction The two irreversible reactions studied in the previous section are of a class that is common
in combustion modeling. However, the model suﬀers a defect in that its link to classical
equilibrium thermodynamics is missing. A better way to model essentially the same physics
and guarantee consistency with classical equilibrium thermodynamics is to model the process
as a single reversible reaction, with a suitably modiﬁed reaction rate term.
7.1.1.2.1 Mathematical model 7.1.1.2.1.1 Kinetics For the reversible O − O2 reaction, let us only consider reaction
13 from Table 7.2 for which
13 : O2 + M ⇌ O + O + M. (7.100) For this system, we have N = 2 molecular species in L = 1 elements reacting in J = 1
reaction. Here
a13 = 1.85 × 10 11 mole
cm3 −1 (K )−0.5 , β13 = 0.5, E 13 = 95560 cal
.
mole (7.101) Units of cal are common in chemistry, but we need to convert to erg , which is achieved via
E 13 = 95560 cal
mole 107 erg
J 4.186 J
cal = 4.00014 × 1012 erg
.
mole (7.102) For this reversible reaction, we slightly modify the kinetics equations to
dρO
= 2 a13 T β13 exp
dt −E 13
RT ρO2 ρM − 1
ρρρ
,
Kc,13 O O M (7.103) =k13 (T )
=r13 dρO2
dt = − a13 T β13 exp −E 13
RT ρO2 ρM − 1
ρρρ
Kc,13 O O M . (7.104) =k13 (T )
=r13 Here we have used equivalent deﬁnitions for k13 (T ) and r13 , so that Eqs. (7.1037.104) can
be written compactly as
dρO
= 2r13 ,
dt
dρO2
= −r13 .
dt (7.105)
(7.106) CC BYNCND. 18 November 2011, J. M. Powers. 244 CHAPTER 7. KINETICS IN SOME MORE DETAIL In matrix form, we can simplify to
d
dt ρO
ρO2 2
(r13 ).
−1 = (7.107) =ν Here the N × J or 2 × 1 matrix ν is
2
.
−1 ν= (7.108) Performing row operations, the matrix form reduces to
d
dt ρO
ρO + 2ρO2 2
(r13 ),
0 = (7.109) or
10
12 d
dt ρO
ρO2 = 2
(r13 ).
0 (7.110) So here the N × N or 2 × 2 matrix L−1 is
10
.
12 L−1 = (7.111) The N × N or 2 × 2 permutation matrix P is the identity matrix. And the N × J or 2 × 1
upper triangular matrix U is
2
.
(7.112)
U=
0
Note that ν = L · U or equivalently L−1 · ν = U:
10
2
·
12
−1
= L −1 = =ν 2
.
0 (7.113) =U Once again the stoichiometric matrix φ is
φ= 1 2 . (7.114) And we see that φ · ν = 0 is satisﬁed:
12· 2
−1 =0. (7.115) As for the irreversible reactions, the reversible reaction rates are constructed to conserve
O atoms. We have
d
ρ + 2ρO2 = 0.
dt O
CC BYNCND. 18 November 2011, J. M. Powers. (7.116) ...
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 Fall '11
 ParkSou
 Dynamics, Combustion

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