Week 5 - Compact Notation Derivation for a simple Mechanism
Aim:- Derivation of simple reaction mechanism using compact notation.
Introduction:-
In the assignment, I have derive reaction mechanism for elementary reaction in chemical process. The motive behind deriving this reaction mechanism is to find out the chemical kinetics for elementary reaction which will help us in knowing the different physical concept running behind a chemical process or combustion process like ignition delay, heat release rate and emissions(amount of pollutants ejected out of combustion).
Theory:-
As We know When a combustion process takes place then hydrocarbon and water molecule are formed as a product. During this conversion of reactant species to product species, there is multistep elementary reaction that take place as no reaction is there that place at single step. This elementary reaction is taken into accountability because this reactions are responsible for determining thermodynamic properties of the product. Hence it becomes necessary for us to determine the reaction pathway which depends on how fast this reaction takes place. In order to predict how fast this reaction takes place, We need two things :-
1. Operating condition (Pressure and temperature).
2. Rate of a reaction (Forward reaction and backward reaction)
If we are successfully able to predict this two things, then we are able to get good combustion result and if it happens, then we are also able to compute thermodynamic properties very accurately.
So keeping in my mind this concept , We basically write elementary reaction in terms of reaction mechanism which is nothing but set of intermeditary species and elementary reactions which is subjected to different operating conditions.
Elementary reaction types :-
(i) Bimolecular reaction :- those reaction which involve two molecular species.
example, `A + B harr C +D`
(ii) Unimolecular reaction :- those reaction which involve only one molecular specie.
example, `A harr B` (Isomerization)
`A harr B + C` (Decomposition)
(iii) trimolecular reaction:- those reaction which involve three molecular species.
example, `A + B + C harr C + M`
We have reaction mechanism :-
O + H2O `harr` 2OH
CO + OH `harr` CO2 + H
H + O2 `harr` OH + O
CO + O2 `harr` CO2 + O
So Ordinary differential equation for above given reaction mechanism will be as ,
`(d[O2])/(dt)= -k_(f1)[CO][O2] + k_(r1) [CO2][O] - k_(f4)[H][O2] + k_r4[OH][O].`
`(d[CO])/dt = -k_(f1)[CO][O2] + k_(r1)[CO2][O] - k_(f3)[CO][OH] + k(r3)[CO2][H]`
`(d[O])/dt = -k_(f2)[O][H2O] + k_(r2)[OH]^2 - k_(f4)[H][O2] + k_(r4)[OH][O]`
`(d[H20])/dt = -k_(f2)[O][H2O] + k_(r2)[OH]^2`
`(d[OH])/dt = -k_(f3)[CO][OH] + k_(r3)[CO2][H] -k_(f4)[H][O2] + k_(r4)[OH][O]`
`(d[H])/dt = -k_(f4)[H][O2] +k_(r4)[OH][O]`
`(d[O2])/dt = -k_(f4)[H][O2] + k_(r4)[OH][O] -k_(f1)[CO][O2] + k_(r1)[CO2][O]`
Now, We will derive the chemical equation in terms of stiochiometric coefficients matrices :-
COMPACT NOTATION :-
| J | SPECIES |
| 1 | O2 |
| 2 | H2 |
| 3 | H2O |
| 4 | HO2 |
| 5 | O |
| 6 | H |
| 7 | OH |
MATRIX FOR REACTANT SPECIES :-
1 1 0 0 0 0 0 0 0 1 1 0 0 0
0 0 0 1 1 0 0 0 0 0 0 0 2 0
`nu_(ji)prime=` 1 0 0 0 0 1 0 `nu_(ji)primeprime=` 0 0 1 0 0 0 1
0 1 0 0 0 0 1 0 0 0 1 0 1 0
-1 -1 1 1 0 0 0
0 0 0 -1 0 0 -1
`nu_(ji)^(n et)`= -1 0 1 0 -1 0 1
0 -1 0 1 0 1 -1
here, `nu_(ji)prime` stands for stiochiometric coefficients for reactants and `nu_(ji)primeprime` stands for stiochiometric coefficients for products.
General equation for reaction rate for reaction mechanism is defined as:-
`q_i = k_(fi)prod_(j=1)^N[X_j]^(nuprimeij) - k_(ri)prod_(j=1)^N[X_j]^(nuprimeprimeij)`.
for example,
`q_1 = k_(f1)[X_1]^(nuprime_11)[X_2]^(nuprime_21)[X_3]^(nuprime_31)....[X_N]^(nuprime_(N1))`
Rection rate is denoted by `'q_i'`
Now, using this above reaction equation, we get our net reaction rate for given reaction mechanism as:-
`q_1 = k_(f1)[CO]^(1)[O2]^(1)[CO2]^(0)[O]^(0)[H20]^(0)[OH]^(0)[H]^(0)-k_(r1)[CO]^(0)[O2]^(0)[CO2]^(1)[O]^(1)[H20]^(0)[OH]^(0)[H]^(0)`
or, `q_1=k_(f1)[CO][O2]-k_(r1)[CO2][O]`
`q_2 = k_(f2)[CO]^(0)[O2]^(0)[CO2]^(0)[O]^(0)[H20]^(1)[OH]^(0)[H]^(0)-k_(r2)[CO]^(0)[O2]^(0)[CO2]^(0)[O]^(0)[H20]^(0)[OH]^(2)[H]^(0)`
or, `q_2= k_(f2)[O][H2O] - k_(r2)[OH]^2`
`q_3 = k_(f3)[CO]^(1)[O2]^(0)[CO2]^(0)[O]^(0)[H20]^(0)[OH]^(1)[H]^(0)-k_(r3)[CO]^(0)[O2]^(0)[CO2]^(1)[O]^(0)[H20]^(0)[OH]^(0)[H]^(1)`
or, `q_3=k_(f3)[CO][OH] - k_(r3)[CO2][H]`
`q_4 = k_(f4)[CO]^(0)[O2]^(1)[CO2]^(0)[O]^(0)[H20]^(0)[OH]^(0)[H]^(1)-k_(r4)[CO]^(0)[O2]^(0)[CO2]^(0)[O]^(1)[H20]^(0)[OH]^(1)[H]^(0)`
or, `q_4= k_(f4)[O2][H] -k_(r4)[O][OH]`