Week 5.1 - Compact Notation Derivation for a simple Mechanism

AIM:

To derive the reaction rate ODEs of a simple reaction mechanism by using compact notation and to write down the desired matrices.

REACTION MECHANISM:

`CO + O2 ⇌ CO2 + O`

`O + H2o ⇌ OH + OH`

`CO + OH ⇌ CO2 + H`

`H + O2 ⇌ OH + O`

DERIVATION TO FIND RATE OF CHANGE OF CONCENTRATION:

RATE OF CHANGE OF CONCENTRATION FOR CO:

`(d[CO])/dt = -k_(f1)[CO][O2] - k_(f3)[CO][OH] + k_(r1)[CO2][O] + k_(r3)[CO2][H]`

RATE OF CHANGE OF CONCENTRATION FOR O2:

`(d[O2])/dt = -k_(f1)[CO][O2] - k_(f4)[H][O2] + k_(r1)[CO2][O] + k_(r4)[OH][O]`

RATE OF CHANGE OF CONCENTRATION FOR CO2:

`(d[CO2])/dt = k_(f1)[CO][O2] - k_(r1)[CO2][O] + k_(f3)[CO][OH] - k_(r3)[CO2][H]`

RATE OF CHANGE OF CONCENTRATION FOR O:

`(d[O])/dt = k_(f1)[CO][O2] - k_(r1)[CO2][O] - k_(f2)[O][H2O] + k_(r2)[OH][OH] + k_(f4)[H][O2] - k_(r4)[OH][O]`

RATE OF CHANGE OF CONCENTRATION FOR H2O:

`(d[H2O])/dt = - k_(f2)[O][H2O] + k_(r2)[OH][OH]`

RATE OF CHANGE OF CONCENTRATION FOR OH:

`(d[OH])/dt = k_(f2)[O][H2O] - k_(r2)[OH][OH] + k_(f4)[H][O2] - k_(r4)[OH][O] - k_(f3)[CO][OH] + k_(r3)[CO2][H]`

RATE OF CHANGE OF CONCENTRATION FOR H:

`(d[H])/dt = k_(f3)[CO][OH] - k_(r3)[CO2][H] - k_(f4)[H][O2] + k_(r4)[OH][O]`

COMPACT NOTATION:

we have 7 species and 4 reactions,

j	species
1	CO
2	O2
3	CO2
4	O
5	H20
6	OH
7	H

i	reaction
1	r1
2	r2
3	r3
4	r4

STOICHIOMETRIC COEFFICIENT MATRIX OF REACTANT SIDE:

`V_(ji)' = [[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 OF PRODUCT SIDE:

`V_(ji)'' = [[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 the matrix,

`V_(ji) => V_(ji)'' - V_(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:

`Q_1 = k_(f1)[CO][O2] - k_(r1)[CO2][O]`

`Q_2 = k_(f2)[O][H2O] - k_(r2)[OH][OH]`

`Q_3 = k_(f3)[CO][OH] - k_(r3)[CO2][H]`

`Q_4 = k_(f4)[H][O2] - k_(r4)[OH][O]`

NET PRODUCTION RATE FOR EACH SPECIES:

`omega_1 = -k_(f1)[CO][O2] + k_(r1)[CO2][O] - k_(f3)[CO][OH] + k_(r3)[CO2][H]`

`omega_2 = -k_(f1)[CO][O2] + k_(r1)[CO2][O] - k_(f4)[HO][O2] + k_(r4)[OH][O]`

`omega_3 = k_(f1)[CO][O2] - k_(r1)[CO2][O] + k_(f3)[CO][OH] - k_(r3)[CO2][H]`

`omega_4 = k_(f1)[CO][O2] - k_(r1)[CO2][O] - k_(f2)[O][H2O] + k_(r2)[OH][OH] + k_(f4)[H][O2] - k_(r4)[OH][O]`

`omega_5 = - k_(f2)[O][H2O] + k_(r2)[OH][OH]`

`omega_6 = 2k_(f2)[O][H2O] - 2k_(r2)[OH][OH] - k_(f3)[CO][OH] + k_(f3)[CO2][H] + k_(f4)[H][O2] - k_(r4)[OH][O]`

`omega_7 = k_(f3)[CO][OH] - k_(r3)[CO2][H] - k_(f4)[H][O2] + k_(r4)[OH][O]`

THE FINAL ODE SYSTEM BECOMES,

`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]] = [[(d[CO])/dt],[(d[O2])/dt],[(d[CO2])/dt],[(d[O])/dt],[(d[H2O])/dt],[(d[OH])/dt],[(d[H])/dt]]`

CONCLUSION:

Thus the derivation of ODEs of simple reaction mechanism by using compact notation is carried out.