Week 5.1 - Compact Notation Derivation for a simple Mechanism

Given:

`CO + O_2 harr CO_2 + O`

`O + H_2O harr OH + OH`

`CO + OH harr CO_2 + H`

`H + O_2 harr OH + O`

Here there are 4 reactions(i) and 7 species(j).

Compact notation: This helps to solve the reaction mechanism which includes a large number of species.

`[[j,Species],[1,CO],[2,O_2],[3,CO_2],[4,O],[5,H_2O],[6,OH],[7,H]]` and `[[i,Reactions],[1,R_1],[2,R_2],[3,R_3],[4,R_4]]`

Matrix form for `V_(ji)^'` for reactants and `V_(ji)^('')` for products 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))`

`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))`

`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))`

Reaction rate(`q_i`):

`q_i = K_(fi)Pi_(j=1)^N*(X_j)^(V_(ji)^') - K_(ri)*Pi_(j=1)^N * (X_j)^(V_(ji)^(''))`

`K_(fi)` = forward reaction

`K_(ri)` = reverse reaction

`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]`

The net production rate for particular species(`omega`):

`omega_j = Sigma_(i=1)^L V_(ji)*q_i`

j = species number = 7

L = number of reactions = 4

`q_i` = reaction rate

`omega_1 (CO) = V_(11)*q_1 + V_(12)*q_2 + V_(13)*q_3 + V_(14)*q_4` = `-q_1 - q_3`

`omega_2 (O_2) = V_(21)*q_1 + V_(22)*q_2 + V_(23)*q_3 + V_(24)*q_4` = `-q_1 - q_4`

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`omega_7 (H) = V_(71)*q_1 + V_(72)*q_2 + V_(73)*q_3 + V_(74)*q_4` = `q_3 - q_4`

`((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))*((q_1),(q_2),(q_3),(q_4))`