Week 5.1 - Compact Notation Derivation for a simple Mechanism

Objective: Deriving the reaction rate ODE of the given reaction mechaninsm using compact notation.

Solution:

Let the unique species in the set of reactions be identified by j index and the individual reaction be identified by i index.

The Species are:

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

The reaction index are:

i Index	Reactions
1	`CO + O_2 = CO_2 + O`
2	`O + H_2O = 2OH`
3	`CO + OH = CO_2 + H`
4	H + O_2 = OH + O

The reactants matrix (`v_{ji}'`):

`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]]`

The product matrix (`v_{ji}''`):

`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}'`

`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]]`

The net production rate is given by,

`omega_j = sum_{i = 1}^{L} v_{ji}q_i = {d[X_j]}/{dt}`,

Where, L is the no. of individual chemical reation equations

j is the species index

i is the chemical reaction index.

Net production rates in matric form:

`[[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]]`

where, individual chemical reactions are:

`q_1 = K_{f1}[CO][O_2] - K_{r1}[CO_2][O]`

`q_2 = K_{f2}[O][H_2O] - K_{r2}[OH]^2`

`q_3 = K_{f3}[CO][OH] - K_{r3}[CO_2][H]`

`q_4 = K_{f4}[H][O_2] - K_{r4}[OH][O]`

Reaction Rate ODE:

`{d[CO]}/{dt} = -K_{f1}[CO][O_2] + K_{r1}[CO_2][H] - K_{f3}[CO][OH] +K_{r3}[CO_2][H]`

`{d[O_2]}/{dt} = -K_{f1}[CO][O_2] + K_{r1}[CO_2][O] + K_{r4}[OH][O] - K_{f4}[H][O2]`

`{d[CO_2]}/{dt} = K_{f1}[CO][O_2] - K_{r1}[CO_2][O] - K_{r3}[CO_2][H] + K_{f3}[CO][OH]`

`{d[O]}/{dt} = K_{f1}[CO][O2] - K_{r1}[CO_2][O] - K_{f2}[O][H_2O] + K_{r2}[OH][OH]-K_{r4}[OH][O] + K_{f4}[H][O_2]`

`{d[H_2O]}/{dt} = K_{r2}[OH]^2 - K_{f2}[H_2O][O]`

`{d[OH]}/{dt} = -K_{f3}[CO][OH] + K_{r3}[CO_2][H] + K_{f2}[O][H_2O] - K_{r2}[OH][OH] - K_{r4}[OH][O] + K_{f4}[H][O_2]`

`{d[H]}/{dt} = K_{f3}[CO][OH] - K_{r3}[CO_2][H] + K_{r4}[OH][O] - K_{f4}[H][O_2]`