Week 5.1 - Compact Notation Derivation for a simple Mechanism
Compact Notation Derivation for a simple Mechanism:
Aim:
We will derive the reaction rate ODEs & production of each species for the simple reaction mechanism given below by using the compact notation (indices for reactions and species).
First we will find the net reaction rate (`dotq`) for all four reactions using reactant and product matrices. Then we will find production of each species (`(d[N])/(dt)`) using matrix notation w.r.t net reaction rate.
Description:
Any chemical reaction takes place through a series of several elementary reactions. The single reaction which is used to represent the overall reaction is called Global reaction. In order to simulate the actual reaction in computer we need to use the reaction mechanism. A reaction mechanism consists of set of elementary reactions, which are actually taking place in a reaction. The accurate we predict the elementary reactions in a mechanism, the more accurate our results are going to be in a simulation.
Reaction Mechanism
1. `CO + O_2 harr CO_2+ O`
2. `O+ H_2O harr OH + OH`
3. `CO + OH harr CO_2 + H`
4. `H +O_2 harrOH + O`
Above mentioned mechanism of a reaction consists of only 4 elemenetary reactions, 7 species and 3 elements.
But in many practically relevant mechanisms, we encounter arround 100 species and thousands of reactions. In these cases, in order to input these equations to the chemical kinetics solver, we need to use a notation (indexing) for each species and reactions, so that we can do calculations in a systematic manner for mechanisms of any size.
Now, we represent the above mentioned small reaction mechanism in index notation.
Procedure:
Firstly, we use index notation for the species and reactions, so that we can feed the information to the computer systematically and solve using a solver which is already available.
| j | species |
| 1 | CO |
| 2 | `O_2` |
| 3 | `CO_2` |
| 4 | O |
| 5 | `H_2O` |
| 6 | OH |
| 7 | H |
| i | Reaction |
| 1 | `CO + O_2 harr CO_2+ O` |
| 2 | `O+ H_2O harr OH + OH` |
| 3 | `CO + OH harr CO_2 + H` |
| 4 | `H +O_2 harrOH + O` |
Stoichiometric co-efficient matries:
Reactants side Stoichiometric co-efficient matrix
`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]]`
Products side Stoichiometric co-efficient matrix
`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]]`
Matrix useful in compact notation
`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]]`
rate of reaction progress is given by (relation was found experimentally)
`q_i = k_(fi) prod_(j=1)^N [X_j]^(v'_(ji)) - k_(ri) prod_(j=1)^N [X_j]^(v''_(ji))`
where N is the number of species
where,
`k_(fi)` is the rate co-efficient for the forward reaction of i th reaction
`k_(ri)` is the rate co-efficient for the reverse reaction of i th reaction
`X_j` is the concentration of the j th species.
Expanding above equation for each reaction we get
`q_1 = k_(f1)[CO]^1[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^0- k_(r1)[CO]^0[O_2]^0[CO_2]^1[O]^1[H_2O]^0[OH]^0[H]^0`
`q_2 = k_(f2)[CO]^0[O_2]^0[CO_2]^0[O]^1[H_2O]^1[OH]^0[H]^0- k_(r2)[CO]^0[O_2]^0[CO_2]^0[O]^0[H_2O]^0[OH]^2[H]^0`
`q_3 = k_(f3)[CO]^1[O_2]^0[CO_2]^0[O]^0[H_2O]^0[OH]^1[H]^0- k_(r3)[CO]^0[O_2]^0[CO_2]^1[O]^0[H_2O]^0[OH]^0[H]^1`
`q_4 = k_(f4)[CO]^0[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^1- k_(r4)[CO]^0[O_2]^0[CO_2]^0[O]^1[H_2O]^0[OH]^1[H]^0`
Net production rate of a particular species (j) in the reaction is given by
`omega_j = sum_(i = 1)^Lv_(ji)q_i`
where L is number of reactions.
we can write
`d([N_j])/dt = omega_j= sum_(i = 1)^Lv_(ji)q_i`
These are the ODEs to be solved numerically, for obtaining concentration as the time progresses.
As this notation is in index form, we can implement this procedure for any sized reaction mechanism.