Week 5.1 - Compact Notation Derivation for a simple Mechanism

Aim: To derive the reaction rate ODEs of a simple reaction mechanism using the compact notation.

Introduction:

Combustion is a process of burning the fuel inside a chamber and heat is extracted. In the combustion process, many reactions occur before the final product is formed and the reactions are neutralized. The reactions and species formed depend on the fuel, air, and temperature. in order to know the products from a simulation first, we need to know the primary components like fuel, air, and constraints like temperature. Then the species that are formed and the chain reactions. To accurately determine the products we need the data of the reaction rates of the intermediate species direction of a reaction. From all the data mentioned above, we can find the products accurately but the number of reactions and species formed are numerous. It is difficult to calculate the reaction rates. Here the ODE's help to get the reaction rates. calculating thousands of ODE's manually is difficult so the coefficients of species should be input into a computer understandable format. i.e in the matrix form and the odes are solved by the program. The ODE's are derived in the following.

Reaction Mechanisms

`CO + O2 ⇌ CO2 + O`

`O + H2O ⇌ OH + OH`

`CO + OH ⇌ CO2 + H`

`H + O2  ⇌ OH + O`

 Species numbering

j	Species
1	O2
2	CO
3	H2O
4	CO2
5	O
6	H
7	OH

Reaction numbering

i	Reaction
1	r1
2	r2
3	r3
4	r4

Stochiometric coefficients for reaction side

`V_(ji)' = [[1,1,0,0,0,0,0],[0,0,1,0,1,0,0],[0,1,0,0,0,0,1],[1,0,0,0,0,1,0]]`

Stochiometric coefficients for product side

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

Net stoichiometric coefficients

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

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

`q_i = kf_iprod_(j = 1)^N[X_j]^(vj' )- kr_iprod_(j = 1)^N[X_j]^(vj'')`

q_i -Gives the direction of the reaction.If it is positive forward reaction and vice versa

`q_1 = kf_1[O_2][CO] - kr_1[CO_2][O]`

`q_2 = kf_2[O][H_2O] - kr_2[OH]^2`

`q_3 = kf_3[CO][OH] - kr_3[CO_2][H]`

`q_4 = kf_4[O_2][H] - kr_4[O][OH]`

Production rate for the species

`omega_j = sum_(i=1)^Lv_(ji)q_i`

`omega_1 = -(kf_1[O_2][CO] - kr_1[CO_2][O])-(kf_4[O_2][H] - kr_4[O][OH])`

`omega_2 = -(kf_1[O_2][CO] - kr_1[CO_2][O])-(kf_3[CO][OH] - kr_3[CO_2][H])`

`omega_3 = -(kf_2[O][H_2O] - kr_2[OH]^2)`

`omega_4 = kf_1[O_2][CO] - kr_1[CO_2][O] + kf_3[CO][OH] - kr_3[CO_2][H]`

`omega_5 = kf_1[O_2][CO] - kr_1[CO_2][O] -(kf_2[O][H_2O] - kr_2[OH]^2) + kf_4[O_2][H] - kr_4[O][OH]`

`omega_6 = kf_3[CO][OH] - kr_3[CO_2][H] - (kf_4[O_2][H] - kr_4[O][OH])`

omega_7 = 2(kf_2[O][H_2O] - kr_2[OH]^2) - (kf_3[CO][OH] - kr_3[CO_2][H]) + kf_4[O_2][H] - kr_4[O][OH]

`[[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]]`

Reaction rates of the species:

`(d[O_2])/(dt) = kr_1[CO_2][O] -kf_1[O_2][CO] + kr_4[O][OH]-kf_4[O_2][H]`

`(d[CO])/(dt) = kr_1[CO_2][O] - kf_1[O_2][CO] + kr_3[CO_2][H] - kf_3[CO][OH]`

`(d[H_2O])/(dt) = kr_2[OH]^2 -kf_2[O][H_2O]`

`(d[CO_2])/(dt) =kf_1[O_2][CO] - kr_1[CO_2][O] + kf_3[CO][OH] - kr_3[CO_2][H]`

`(d[O])/(dt) = kf_1[O_2][CO] - kr_1[CO_2][O] +kr_2[OH]^2-kf_2[O][H_2O]+ kr_4[O][OH]-kf_4[O_2][H]`

`(d[H])/(dt) = kf_3[O_2][Co] - kr_1[CO_2][O] + kr_3[CO_2][H] - kf_3[CO][OH]`

`(d[OH])/(dt) = 2kf_2[O_2][H_2O] - kr_2[OH]^2 + kr_3[CO_2][H] - kf_3[CO][OH] + kr_4[O][OH] - kf_4[O_2][H]`