Thermodynamics filled in class notes_Part_118

Thermodynamics filled in class notes_Part_118 - 243 7.1....

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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. (7.103-7.104) can be written compactly as dρO = 2r13 , dt dρO2 = −r13 . dt (7.105) (7.106) CC BY-NC-ND. 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 BY-NC-ND. 18 November 2011, J. M. Powers. (7.116) ...
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