Compact Notation Derivation for simple Mechanism

Problem Definition:

- To derive the Compact Notation for the chemical reaction

------------------------------------------------------------------------------------------------------------------

Reaction:

1. `CO + O_2 overset(kf_1)underset(kr_1)(=)CO_2 + O`
2. `O+H_2O overset(kf_2)underset(kr_2)(=)OH+OH`
3. `CO+OH overset(kf_3)underset(kr_3)(=)CO_2+H`
4. `H+O_2overset(kf_4)underset(kr_4)(=)OH+O`

Net Production Rates:

Net rate of production of an species is the sum of production and destruction of same species for all the reaction.

`(d[x_i](t))/(dt) = f_i([X_1](t),[X_2](t),....,[X_n](t))`

A set of Ordinary Differential Equation is formed for all species, these ODE's are stiff system, which is very unstable numerically.

Compact Notation:

`(d[O_2])/(dt) = k_"r1"[CO_2][O_2]+k_"r4"[OH][O]-k_"f1"[CO][O_2]-k_"f4"[H][O_2]`

`(d[CO])/dt= k_"r1"[CO_2][O]+k_"r3"[CO_2][H]-k_"f1"[CO][O_2]-k_"f3"[CO][OH]`

`(d[O])/dt= 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]`

`(d[H_2O])/dt=k_"r2"[OH][OH]-k_"f2"[O][H_2O]`

`(d[CO_2])/dt=k_"f1"[CO][O_2]-k_"r1"[CO_2][O]+k_"f3"[CO][OH]-k_"r3"[CO_2][H]`

`(d[H])/dt= k_"f3"[CO][OH]-k_"r3"[CO_2][H]-k_"f4"[H][O2]+k_"r4"[OH][O]`

`(d[OH])/dt = k_"f2"[O][H_2O]-k_"r2"[OH][OH]-k_"f3"[CO][OH]+k_"r3"[CO_2][H]+k_"f4"[H][O_2]+k_"r3"[CO_2][H]`

Compact Notation:

`v'_"ij" = ([1, 1, 0, 0, 0, 0,0 ],[0, 0, 1, 1, 0, 0, 0 ],[0 ,1 ,0 ,0 ,0 ,0 , 1],[1, 0, 0, 0, 0, 1, 0 ])`

`v''_"ij"=([0 ,0 ,1 ,0 ,1, 0, 0],[0, 0, 0, 0, 0, 0, 2],[0, 0, 0, 0, 1, 1, 0],[0, 0, 1, 0, 0, 0, 1])`

`v_"ji" = v''_"ji" - v'_"ji"`

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

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

where, j = 1,2,3,4 represents the reactions

i = 1 to 8 represents the species.

The net production rate of each species in a Multistep Mechanism:

`dotomega_j = sum_(i=1)^(L) v_"ji"q_"i"`. for j = 1,2,3,.......,N

where,

`q_i = k_"fi"prod_(j=1)^(N)[X_j]^(v'_"ji") - k_"ri"prod_(j=1)^(N)[X_j]^(v''_"ji")`