Week 5.1 - Compact Notation Derivation for a simple Mechanism

COMPACT NOTATION DERIVATION FOR A SIMPLE MECHANISM

AIM: to derive the reaction rate ODEs of a simple reaction mechanism using compact notation.

OBJECTIVE:

Derive the reaction rate ODEs of a simple reaction mechanism using compact notation for the following equations:

Reaction Mechanism

• CO + O2 ⇌ CO2 + O
• O + H2o ⇌ OH + OH
• CO + OH ⇌ CO2 + H
• H + O2 ⇌ OH + O

SOLUTION AND RESULTS:

The reaction rate ODEs for the reaction mechanism are as follows:

`(d[CO])/(dt)=-k_(f1)[CO][O2]+k_(r1)[CO2][O]+k_(r3)[CO2][H]-k_(f3)[CO][OH]`

`(d[O_2])/(dt)=-k_(f1)[CO][O2]+k_(r1)[CO2][O]+k_(r4)[OH][O]-k_(f4)[H][O2]`

`(d[CO_2])/(dt)=k_(f1)[CO][O2]-k_(r1)[CO2][O]-k_(r3)[CO2][H]+k_(f3)[CO][OH]`

`(d[O])/(dt)=k_(f1)[CO][O2]-k_(r1)[CO2][O]-k_(f2)[O][H2O]+k_(r2)[OH][OH]-k_(r4)[OH][O]+k_(f4)[H][O2]`

`(d[H_2O])/(dt)=k_(r2)[OH][OH]-k_(f2)[H2O][O]`

`(d[OH])/(dt)=-k_(f3)[CO][OH]+k_(r3)[CO2][H]+k_(f2)[O][H2O]-k_(r2)[OH][OH]+k_(f2)[O][H2O]-k_(r2)[OH][OH]-k_(r4)[OH][O]+k_(f4)[H][O2]`

`(d[H])/(dt)=k_(f3)[CO][OH]-k_(r3)[CO2][H]+k_(r4)[OH][O]-k_(f4)[H][O2]`

Compact Notation:

The species Index is given as follows: The species index is denoted by j

Compact notation:

`[[j,Species],[1, CO],[2,Ounderset(2)()],[3,COunderset(2)()],[4,O],[5,Hunderset(2)() O],[6,OH],[7,H]]`

`[[j,Reactions],[1, R1],[2,R2],[3,R3],[4,R4]]`

Reactants 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 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]]`

`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:

`q_i=k_(fi)prod_(j=1)^N[X_j]^(v_(ji)')-k_(ri)prod_(j=1)^N[X_j]^(v_(ji)'')`

`q_1=k_(f1)[CO]^1[O2]^1[CO2]^0[O]^0[H2O]^0[OH]^0[H]^0-k_(r1)[CO]^0[O2]^0[CO2]^1[O]^1[H2O]^0[OH]^0[H]^0`

`q_1=k_(f1)[CO]^1[O2]^1-k_(r1)[CO2]^1[O]^1`

`q_2=k_(f2)[CO]^0[O2]^0[CO2]^0[O]^1[H2O]^1[OH]^0[H]^0-k_(r2)[CO]^0[O2]^0[CO2]^0[O]^0[H2O]^0[OH]^2[H]^0`

`q_2=k_(f2)[O]^1[H2O]^1-k_(r2)[OH]^2`

`q_3=k_(f3)[CO]^1[O2]^0[CO2]^0[O]^0[H2O]^0[OH]^1[H]^0-k_(r3)[CO]^0[O2]^0[CO2]^1[O]^0[H2O]^0[OH]^0[H]^1`

`q_3=k_(f3)[CO]^1[OH]^1-k_(r3)[CO2]^1[H]^1`

`q_4=k_(f4)[CO]^0[O2]^1[CO2]^0[O]^0[H2O]^0[OH]^0[H]^1-k_(r4)[CO]^0[O2]^0[CO2]^0[O]^1[H2O]^0[OH]^1[H]^0`

`q_4=k_(f4)[H]^1[O2]^1-k_(r4)[OH]^1[O]^1`

The Net production rates of a species:

`omega_j=sum_1^Lv_(ji)q_i=(d[X_j])/(dt)` where L is number of reactions

the Matrix representation:

`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]]=[[-1,0,-1,0],[-1,0,0,-1],[1,0,1,0],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]][[q_1],[q_2],[q_3],[q_4]]`

As shown above we can derive the compact notations of ODE for all the species.