Week 5.1 - Compact Notation Derivation for a simple Mechanism

Objective: To derive the reaction rate ODEs & production of each species for the simple reaction mechanism and represent it in the form of compact notations. and write desired matrices.

`R_1 text( ) CO+O_2underset(k_(r_1))(overset(k_(f_1))↔)CO_2+O`

`R_2 text( ) CO+OHunderset(k_(r_2))(overset(k_(f_2))↔)CO_2+H`

`R_3 text( ) H+O_2underset(k_(r_3))(overset(k_(f_3))↔)OH+H`

`R_4 text( ) O+H_2 Ounderset(k_(r_4))(overset(k_(f_4))↔)OH+OH`

where, `k_(f_i)` is forward reaction rate constant

`k_(r_i)` is reverse reaction rate constant

Species: `CO ,O_2,CO_2,O,H_2O,OH,H`

Species indexing

j	Species
1	`O_2`
2	`CO`
3	`O`
4	`CO_2`
5	`H_2O`
6	`OH`
7	`H`

Reaction indexing

i	reactions
1	`R_1`
2	`R_2`
3	`R_3`
4	`R_4`

Rate equations for each species

`[d[O_2 ]]/[dt]=-k_(f_1 ) [O_2 ][CO]+k_(r_1 ) [CO_2 ][O]-k_(f_4 ) [H][O_2 ]+k_(r_4 ) [OH][O]`

`[d[CO]]/[dt]=-k_(f_1 ) [O_2 ][CO]+k_(r_1 ) [CO_2 ][O]-k_(f_3 ) [CO][OH]+k_(r_3 ) [CO_2][H]`

`[d[O]]/[dt]=k_(f_1 ) [O_2 ][CO]-k_(r_1 ) [CO_2 ][O]-k_(f_2 ) [O][H_2 O]+k_(r_2 ) [OH][OH]+K_(f_4 ) [H][O_2 ]-k_(r_4 ) [OH][O]`

`[d[CO_2 ]]/[dt]=k_(f_1 ) [O_2 ][CO]-k_(r_1 ) [CO_2 ][O]+k_(f_3 ) [[CO][OH]-k_(r_3 ) [CO_2][H]`

`[d[H_2 O]]/[dt]=-k_(f_2 ) [O][H_2 O]+k_(r_2 ) [OH][OH]`

`[d[OH]]/[dt]=k_(f_2 ) [O][H_2 O]-k_(r_2 ) [OH][OH]-k_(f_3 ) [CO][OH]+k_(r_3 ) [CO_2][H]+K_(f_4 ) [H][O_2 ]-k_(r_4 ) [OH][O]`

`[d[H]]/[dt]=-k_(f_4 ) [O_2 ][H]+k_(r_4 ) [OH][O]+k_(f_3 ) [[CO][OH]-k_(r_3 ) [CO_2][H]`

Matrix for reactant side

`v_(ji)^'=[[1,1,0,0,0,0,0],[0,0,1,0,1,0,0],[0,1,0,0,0,1,0],[1,0,0,0,0,0,1]]`

Matrix for product side

`v_(ji)^('') = [[0,0,1,1,0,0,0],[0,0,0,0,0,2,0],[0,1,0,0,0,1,0],[0,0,1,0,0,1,0]]`

`v_(ji) = v_(ji)^('')-v_(ji)^'=[[-1,-1,1,1,0,0,0],[0,0,-1,0,-1,2,0],[0,-1,0,1,0,-1,1],[-1,0,1,0,0,1,-1]]`

Net reaction rate `q_i`

`q_i=k_(f_i) underset(j=1)(overset(N)∏)[X_j ]^(v_(ji)^' )-k_(r_i ) underset(j=1)(overset(N)∏)∏[X_j ]^(v_(ji)^('') )`

`q_1 = k_(f_1 ) [O_2 ][CO]-k_(r_1 ) [CO_2 ][O]`

`q_2=k_(f_2 ) [O][H_2 O]-k_(r_2 ) [OH][OH]`

`q_3 = k_(f_3 ) [CO][OH]-k_(r_3 ) [CO_2][H]`

`q_4 = K_(f_4 ) [H][O_2 ]-k_(r_4 ) [OH][O]`

Now,

`ω_j=underset(i=2)(overset(L)∑)v_(ji)*q_i`

where `omega_j` is product rate of each species and L is number of reactions

`omega_1=v_11*q_1+v_12*q_2+v_13*q_3+v_14*q_4`

`omega_2 = v_21*q_1+v_22*q_2+v_23*q_3+v_24*q_4`

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`[(omega_1),(omega_2),(omega_3),(omega_4),(omega_5),(omega_6),(omega_7)]=[[-1,0,-1,0],[-1,0,0,-1],[1,0,1,0],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]]xx[[q_1],[q_2],[q_3],[q_4]]`