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Unformatted text preview: 207 6.1. SUMMARY OF MULTIPLE REACTION EXTENSIONS
or
N Kc,j = ρi νij , j = 1, . . . , J. (6.13) i=1 For isochoric reaction, the evolution of species concentration i due to the combined eﬀect
of J reactions is given by
≡ ωi
˙ dρi
=
dt J νij aj T βj N −E j
RT exp j =1 ν′
ρk kj k =1 ≡kj (T ) forward reaction 1 1 − Kc,j N k =1 ν
ρk kj , reverse reaction ≡rj =(1/V )dζj /dt i = 1, . . . , N. (6.14)
The extension to isobaric reactions is straightforward, and follows the same analysis as for
a single reaction. Again, three intermediate variables which are in common usage have been
deﬁned. First one takes the reaction rate of the j th reaction to be rj ≡ aj T βj N −E j
RT exp ν′
ρk kj k =1 ≡kj (T ) forward reaction 1 1 − Kc,j N , νkj ρk
k =1 reverse reaction j = 1, . . . , J, (6.15) or rj = aj T βj exp −E j
RT ≡kj (T ), Arrhenius rate N ν′
ρk kj k =1 N 1
−
Kc,j forward reaction k =1 reverse reaction law of mass action ν ′′ ρk kj , j = 1, . . . , J, (6.16) = 1 dζj
(6.17)
.
V dt Here ζj is the reaction progress variable for the j th reaction.
Each reaction has a temperaturedependent rate function kj (T ), which is
kj (T ) ≡ aj T βj exp −E j
RT , j = 1, . . . , J. (6.18) CC BYNCND. 18 November 2011, J. M. Powers. 208 CHAPTER 6. THERMOCHEMISTRY OF MULTIPLE REACTIONS The evolution rate of each species is given by ωi , deﬁned now as
˙
J ωi ≡
˙ νij rj , i = 1, . . . , N. (6.19) j =1 The multireaction extension for mole change in terms of progress variables is
J dni = νij dζj , i = 1, . . . , N. (6.20) j =1 One also has
N dGT,P = µi dni ,
i=1
N J i=1 ∂G
∂ζj = ∂ζk
,
∂ζj (6.23) νik µi
i=1 k =1
J N = (6.22) k =1
J N
ζp νik dζk , µi = (6.21) νik δkj , µi
i=1 (6.24) j =1 N = µi νij , (6.25) i=1 = −αj , j = 1, . . . , J. (6.26) In a very similar fashion to that shown for a single reaction, one can further sum over all
reactions and prove that mixture mass is conserved, element mass and number are conserved.
A similar expression is obtained for temperature changes. Example 6.1
Show that element mass and number are conserved for the multireaction formulation.
CC BYNCND. 18 November 2011, J. M. Powers. ...
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This note was uploaded on 11/26/2011 for the course EGN 3381 taught by Professor Parksou during the Fall '11 term at FSU.
 Fall '11
 ParkSou
 Dynamics

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