Week 5.1 - Compact Notation Derivation for a simple Mechanism
AIM- The aim of this exercise is to derive reaction rate ODE of a simple reaction mechanism using a compact notation.
REACTION MECHANISM:
CO + O2 ⇌ CO2 + O (r1)
O + H2o ⇌ OH + OH (r2)
CO + OH ⇌ CO2 + H (r3)
H + O2 ⇌ OH + O (r4)
RATE OF CHANGE OF CONCENTRATION:
1) CO
`(d[CO])/(dt) =-k_(f_1)[CO][O_2]-k_(f_3)[CO][OH]+k_(r_1)[CO_2][O]+k_(r_3)[CO_2][H]`
2) O2
`(d[O_2])/(dt) =-k_(f_1)[CO][O_2]-k_(f_4)[H][O_2]+k_(r_1)[CO_2][O]+k_(r_4)[OH][O]`
3) CO2
`(d[CO_2])/(dt) =k_(f_1)[CO][O_2]+k_(f_3)[CO][OH]-k_(r_1)[CO_2][O]-k_(r_3)[CO_2][H]`
4) O
`(d[O])/(dt) =k_(f_1)[CO][O_2]+k_(f_4)[H][O_2]-k_(f_2)[O][H_2O]+k_(r_2)[OH][OH]-k_(r_1)[CO_2][O]-k_(r_4)[OH][O]`
5) H_2O
`(d[H_2O])/(dt) =k_(f_2)[O][H_2O]-k_(r_2)[OH][OH]`
6) OH
`(d[OH])/(dt) =k_(f_2)[O][H_2O]-k_(r_2)[OH][OH] +k_(f_3)[CO][OH]-k_(r_3)[CO_2][H]+ k_(f_4)[H][O_2]-k_(r_3)[OH][O]`
7) H
`(d[H])/(dt) =k_(f_3)[CO][OH]-k_(r_3)[CO_2][H] +k_(f_4)[H][O_2]-k_(r_4)[OH][O]`
COMPACT NOTATION:
Labelling of Species
j Species
1 CO
2 O_2
3 CO_2
4 O
5 H_2O
6 OH
7 H
Labelling of reaction
i Reaction
1 r1
2 r2
3 r3
4 r4
STOICHIOMETRIC COEFFICIENT MATRIX FOR REACTANT SIDE:
`(nu_(ji))prime= [[1,1,0,0,0,0,0],[0,0,0,1,1,0,0],[1,0,0,0,0,1,0],[0,1,0,0,0,0,1]]`
STOICHIOMETRIC COEFFICIENT MATRIX FOR PRODUCT SIDE:
`(nu_(ji))primeprime= [[0,0,1,1,0,0,0],[0,0,0,0,0,2,0],[0,0,1,0,0,0,1],[0,0,0,1,0,1,0]]`
Difference of two matrix
`nu_(ji) =nu_(ji)primeprime -nu_(ji)prime`
`(nu_(ji))= [[-1,-1,1,1,0,0,0],[0,0,0,-1,-1,2,0],[-1,0,1,0,0,-1,1],[0,-1,0,1,0,1,-1]]`
NET REACTION RATE FOR EACH REACTION:
The net reaction rate is is computed as
`q_i= k_(fi)prod_(j=1)^N[X_j]^(nuprime_(ji))-k_(ri)[X_j]^(nuprimeprime_(ji))`
`Q_1 =k_(f1)[CO][O_2] -k_(r1)[CO_2][O]`
`Q_2 =k_(f2)[O][H_2O] -k_(r2)[OH][OH]`
`Q_3 =k_(f3)[CO][OH] -k_(r3)[CO_2][H]`
`Q_4 =k_(f4)[H][O_2] -k_(r4)[OH][O]`
NET PRODUCTION RATE FOR EACH SPECIES:
The net production rate for each species is given by
`dotomega_j= sum_(i=1)^L nu_(ji)q_(i)`
where 'nu_(ji)' is net stoichiometric reaction matrix.
1) For 1st species
`dotomega_1= nu_(11)q_1 + nu_(12)q_2+nu_(13)q_3+nu_(14)q_4`
`dotomega_1= -k_(f1)[CO][O_2] +k_(r1)[CO_2][O]-k_(f3)[CO][OH] +k_(r3)[CO_2][H]`
2) For 2nd species
`dotomega_2= nu_(21)q_1 + nu_(22)q_2+nu_(23)q_3+nu_(24)q_4`
`dotomega_2= -k_(f1)[CO][O_2] +k_(r1)[CO_2][O]-k_(f4)[H][O_2] +k_(r4)[OH][O]`
3) For 3rd species
`dotomega_3= nu_(31)q_1 + nu_(32)q_2+nu_(33)q_3+nu_(34)q_4`
`dotomega_3= k_(f1)[CO][O_2] -k_(r1)[CO_2][O] + k_(f3)[CO][OH] -k_(r3)[CO_2][H]`
4) For 4th species
`dotomega_4= nu_(41)q_1 + nu_(42)q_2+nu_(43)q_3+nu_(44)q_4`
`dotomega_4= k_(f1)[CO][O_2] -k_(r1)[CO_2][O] -k_(f2)[O][H_2O] +k_(r2)[OH][OH] + k_(f4)[H][O_2] -k_(r4)[OH][O]`
5) For 5th Species
`dotomega_5= nu_(51)q_1 + nu_(52)q_2+nu_(53)q_3+nu_(54)q_4`
`dotomega_5=- k_(f2)[O][H_2O] +k_(r2)[OH][OH]`
6) For 6th Species
`dotomega_6= nu_(61)q_1 + nu_(62)q_2+nu_(63)q_3+nu_(64)q_4`
`dotomega_6= 2k_(f2)[O][H_2O] -2k_(r2)[OH][OH] -k_(f3)[CO][OH] +k_(r3)[CO_2][H]+ k_(f4)[H][O_2] -k_(r4)[OH][O]`
7)For 7th Species
`dotomega_7= nu_(71)q_1 + nu_(72)q_2+nu_(73)q_3+nu_(74)q_4`
`dotomega_7= k_(f3)[CO][OH] -k_(r3)[CO_2][H]- k_(f4)[H][O_2] +k_(r4)[OH][O]`
ODE SYSTEM:
`[[dotomega_1],[dotomega_2],[dotomega_3],[dotomega_4],[dotomega_5],[dotomega_6],[dotomega_7]]=[[(d[CO])/dt],[(d[O_2])/dt],[(d[CO_2])/dt],[(d[O])/dt],[(d[H_2O])/dt],[(d[OH])/dt],[(d[H])/dt]]`