Week 5.1 - Compact Notation Derivation for a simple Mechanism
Aim:
To derive the reaction rate ODEs & production of each species for the simple reaction mechanism and represent it in compact notation form. The compact notation form for the reactions given.
Before arriving to the compact notation, we have to define the species and reactions.Here j is the number of species and i is the number of reactions.
Species Reactions
1. CO 1. R1
2. O2 2.R2
3. CO2 3.R3
4. O 4.R4
5. H2O
6. OH
7.H
So totally there are 7 species and 4 reactions.
Now we have to create stoichiometic coefficient matrix form for reactants and the products.These are represented as v'ji and v''ji for reactants and products.
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]]`
vji= v''ji - v'ji
vji= `[[-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]]`
As we have found stoichiometric coefficient of matrix for reactants and products.
Now we have to find the rate of reaction represented by q.
`q_i = K_(fi) pi_(j=1)^N [X_j]^(v'ji) - K_(ri)pi_(j=1)^N[X_j]^(v''ji)`
for reaction 1:
`q_1 = K_(f1) [CO].[O2] - K_(r1)[CO2].[O]`
for reaction 2:
`q_2 = K_(f2) [O].[H2O] - K_(r2)[OH].[OH]`
for reaction 3:
`q_3= K_(f3) [CO].[OH] - K_(r3)[CO2].[H]`
for reaction 4:
`q_4 = K_(f4) [H].[O2] - K_(r4)[OH].[O]`
The species corresponding to different reactions which has a stoichiometric coefficient matrix value above 1 is considered ,the other species whose power is zero is considered 1.
The reaction rate for different reactions has been found.So now that we have to find the production rate for each species Wj.
`omega_i = sum_(i=1)^L v_(ji).q_i`
The final compact notation form is represented as
`((omega_1),(omega_2),(omega_3),(omega_4),(omega_5),(omega_6),(omega_7))`= `[[-1,0,-1,0],[-1,0,0,-1],[1,-1,0,1],[0,2,-1,1],[0,0,1,-1],[1,0,1,0],[0,-1,0,0]] *[[q1],[q2],[q3],[q4]]`
https://drive.google.com/file/d/1TaXihOt_qpJQq5Ti8DIiI7qkGv3ExRqL/view?usp=sharing