Week 5.1 - Compact Notation Derivation for a simple Mechanism
- Aim :
- To derive the reaction rate ODEs of a simple reaction mechanism.
- Use compact notation and write down desired matrices.
- Theory :
The given reactions are as follows:
CO + O2 ⇌ CO2 + O
O + H2o ⇌ OH + OH
CO + OH ⇌ CO2 + H
H + O2 ⇌ OH + O
There are total 4 reactions so let us name them R1, R2, R3 and R4 sequencially.
Let us assign reaction index to these reactions.
| Index (i) | Reaction |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
Also there are 7 different species that are involved in these 4 reactions namely CO, O2, CO2, O, H2O, OH, H.
Let's assign species index to different species.
| Index (j) | Species |
| 1 | CO |
| 2 | O2 |
| 3 | CO2 |
| 4 | O |
| 5 | H2O |
| 6 | OH |
| 7 | H |
Now reaction,
R1 = `CO + O_2 underset(K_(r_1))(overset(K_(f_1))(harr)) CO_2 + O`
where `K_(f_1)` and `K_(r_1)` are forward and reverse reaction rates for reaction 1 respectively
R2 = `O + H_2O underset(K_(r_2))(overset(K_(f_2))(harr)) OH + OH`
where `K_(f_2)` and `K_(r_2)` are forward and reverse reaction rates for reaction 2 respectively
R3 = `CO + OH underset(K_(r_3))(overset(K_(f_3))(harr)) CO_2 + H`
where `K_(f_3)` and `K_(r_3)` are forward and reverse reaction rates for reaction 3 respectively
R4 = `H + O_2 underset(K_(r_4))(overset(K_(f_4))(harr)) OH + O`
where `K_(f_4)` and `K_(r_4)` are forward and reverse reaction rates for reaction 4 respectively
`V_(ji)'` is stoichometric coefficients of reactants, which is given as
`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)''` is stoichometric coefficients of products, which is given as
`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] ]`
`V_(ji)`is the net stoichometric coefficients, which isgiven by
`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] ]`
Now, calculating net reaction rate `q_i`
`q_i=k_(fi)prod_(j=1)^N[X_j]^(V_(ji)') - k_(ri)prod_(j=1)^N[X_j]^(V_(ji)'')`
where N = number of species
for i = 1
`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)`
= `K_(f1)*([CO][O_2]) - K_(r1)*([CO_2][O])`
for i = 2
`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)`
= `K_(f2)*([O][H_2O]) - K_(r2)*([OH]^2)`
for i =3
`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)`
= `K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H])`
for i = 4
`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)`
= `K_(f4)*([[O_2][H]) - K_(r4)*([O][OH])`
Calculating production rate `omega_j`
`omega_j = sum_(i=1)^L V_(ji)q_i`
where L = number of reactions
for j = 1
`omega_1 = v_(11)q_1 + v_(12)q_2 + v_(13)q_3 + v_(14)q_4`
= `(-1)q_1 + (0)q_2 + (-1)q_3 + (0)q_4`
= `-q_1 - q_3`
= `-K_(f1)*([CO][O_2]) + K_(r1)*([CO_2][O]) - K_(f3)*([CO][OH]) + K_(r3)*([CO_2][H])`
Now `omega_1 = (d[CO])/dt` = `-K_(f1)*([CO][O_2]) + K_(r1)*([CO_2][O]) - K_(f3)*([CO][OH]) + K_(r3)*([CO_2][H])`
for j = 2
`omega_2 = v_(21)q_1 + v_(22)q_2 + v_(23)q_3 + v_(24)q_4`
= `(-1)q_1 + (0)q_2 + (0)q_3 + (-1)q_4`
= `-q_1 - q_4`
= `-K_(f1)*([CO][O_2]) + K_(r1)*([CO_2][O]) - K_(f4)*([O_2][H]) + K_(r4)*([O][OH])`
Now `omega_2 = (d[O_2])/d` = `-K_(f1)*([CO][O_2]) + K_(r1)*([CO_2][O]) - K_(f4)*([O_2][H]) + K_(r4)*([O][OH])`
for j = 3
`omega_3 = v_(31)q_1 + v_(32)q_2 + v_(33)q_3 + v_(34)q_4`
= `(1)q_1 + (0)q_2 + (1)q_3 + (0)q_4`
= `q_1 + q_3`
= `K_(f1)*([CO][O_2]) - K_(r1)*([CO_2][O] + K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H])`
Now `omega_3 = (d[CO_2])/dt =`K_(f1)*([CO][O_2]) - K_(r1)*([CO_2][O] + K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H])`
for j = 4
`omega_4 = v_(41)q_1 + v_(42)q_2 + v_(43)q_3 + v_(44)q_4`
= `(1)q_1 + (-1)q_2 + (0)q_3 + (1)q_4`
= `q_1 - q_2 + q_4`
= `K_(f1)*([CO][O_2]) - K_(r1)*([CO_2][O]) - K_(f2)*([O][H_2O]) + K_(r2)*([OH]^2 + K_(f4)*([[O_2][H]) - K_(r4)*([O][OH])`
Now `omega_4 = d[O])/dt=-k_(f1)[CO][O_2]+k_(r1)[CO_2][O]-k_(f2)[O][H_2O]+k_(r2)[OH]^2+k_(f4)[H][O_2]-k_(r4)[OH][O]`
for j = 5
`omega_5 = v_(51)q_1 + v_(52)q_2 + v_(53)q_3 + v_(54)q_4`
= `(0)q_1 + (-1)q_2 + (0)q_3 + (0)q_4`
= `-q_2`
= `-K_(f2)*([O][H_2O]) + K_(r2)*([OH]^2)`
Now `omega_5 = (d[H_2O])/dt ` = `-K_(f2)*([O][H_2O]) + K_(r2)*([OH]^2)`
for j = 6
`omega_6 = v_(61)q_1 + v_(62)q_2 + v_(63)q_3 + v_(64)q_4`
= `(0)q_1 + (2)q_2 + (-1)q_3 + (1)q_4`
= `(2)q_2 -q_3 + q_4`
= `2K_(f2)*([O][H_2O]) - 2K_(r2)*([OH]^2) - K_(f3)*([CO][OH]) + K_(r3)*([CO_2][H]) + K_(f4)*([[O_2][H]) - K_(r4)*([O][OH])`
Now `omega_6 = (d[OH])/dt = 2K_(f2)*([O][H_2O]) - 2K_(r2)*([OH]^2) - K_(f3)*([CO][OH]) + K_(r3)*([CO_2][H]) + K_(f4)*([[O_2][H]) - K_(r4)*([O][OH])`
for j = 7
`omega_7 = v_(71)q_1 + v_(72)q_2 + v_(73)q_3 + v_(74)q_4`
= `(0)q_1 + (0)q_2 + (1)q_3 + (-1)q_4`
= `q_3 - q_4`
= `K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H]) - K_(f4)*([[O_2][H]) + K_(r4)*([O][OH])`
Now `omega_7 = (d[H])/dt` = `K_(f3)*([CO][OH]) - K_(r3)*([CO_2][H]) - K_(f4)*([[O_2][H]) + K_(r4)*([O][OH])`
Thus, compact notations are derived. These can be used for solving ODEs and calculating production rates.