Week 5.1 - Compact Notation Derivation for a simple Mechanism

Abstract: To derive the net reaction rate for the simple mechanism

Reaction Mechanism:

CO + O2 <=> CO2 + O

O + H2O <=> OH + OH

CO + OH <=> CO2 + H

H + O2 <=> OH + O

Species j involved in this reaction:

1. CO

2. O2

3. CO2

4. O

5. H2O

6. H

7. OH

vji' is the stoichiometric coefficient of the reactant matrix.

`underset(ji)(v')=[[1,1,0,0,0,0,0],[0,0,0,1,1,0,0],[1,0,0,0,0,0,1],[0,1,0,0,0,1,0]]`

vji'' is the stoichiometric coefficient of the productt matrix.

`underset(ji)(v'')=[[0,0,1,1,0,0,0],[0,0,0,0,0,0,2],[0,0,1,0,0,1,0],[0,0,0,1,0,0,1]]`

`vji=vji''-vji'`

`underset(ji)(v)=[[-1,-1,1,1,0,0,0],[0,0,0,-1,-1,0,2],[-1,0,1,0,0,1,-1],[0,-1,0,1,0,-1,1]]`

The net rate of reaction qi is calculated using the below formula.

`underset(i)(q)=underset(fi)(k)underset(j=0)(overset(N)(Pi))[underset(j)(x)]overset(underset(ji)(v)') -underset(ri)(k)underset(j=0)(overset(N)(Pi))[underset(j)(x)]overset(underset(ji)(v)'')`

where N is the number of species.

`q1=kunderset(f1)()[CO][Ounderset(2)()]-kunderset(r1)()[COunderset(2)()][O]`

`q2=kunderset(f2)()[O][Hunderset(2)()O]-kunderset(r2)()[OH]overset(2)()`

`q3=kunderset(f3)()[CO][OH]-kunderset(r3)()[COunderset(2)()][H]`

`q4=kunderset(f4)()[Ounderset(2)()][H]-kunderset(r4)()[O][OH]`

The production rate `omega`for species j is calculated using the below steps.

`omegaunderset(j)()=sumoverset(L)()underset(i=1)()vunderset(ji)()qunderset(i)`

`omegaunderset(j)()=d/dt[Xunderset(i)()]`

`omegaunderset(j)()=vunderset(j1)()qunderset(1)()+vunderset(j2)()qunderset(2)()+vunderset(j3)()qunderset(3)()+vunderset(j4)()qunderset(4)()`

`(d[COunderset()()])/dt=-(kunderset(f1)()[CO][Ounderset(2)()]-kunderset(r1)()[COunderset(2)()][O])-(kunderset(f3)()[CO][OH]-kunderset(r3)()[COunderset(2)()][H])`

`(d[Ounderset(2)()])/dt=-(kunderset(f1)()[Ounderset(2)()][CO]-kunderset(r1)()[COunderset(2)()][O])-(kunderset(f4)()[Ounderset(2)()][H]-kunderset(r4)()[O][OH])`

`(d[COunderset(2)()])/dt=(kunderset(f1)()[Ounderset(2)()][CO]-kunderset(r1)()[COunderset(2)()][O])+(kunderset(f3)()[CO][OH]-kunderset(r3)()[COunderset(2)()][H])`

`(d[O])/dt=(kunderset(f1)()[CO][Ounderset(2)()]-kunderset(r1)()[COunderset(2)()][O])-(kunderset(f2)()[O][Hunderset(2)()O]-kunderset(r2)()[OH]overset(2)())+(kunderset(f4)()[Ounderset(2)()][H]-kunderset(r4)()[O][OH])`

`(d[Hunderset(2)()O])/dt=-(kunderset(f2)()[O][Hunderset(2)()O]-kunderset(r2)()[OH]overset(2)())`

`(d[H])/dt=(kunderset(f3)()[CO][OH]-kunderset(r3)()[COunderset(2)()][H])-(kunderset(f4)()[Ounderset(2)()][H]-kunderset(r4)()[O][OH])`

`(d[OH])/dt=2(kunderset(f2)()[O][Hunderset(2)()O]-kunderset(r2)()[OH]overset(2)())-(kunderset(f3)()[CO][OH]-kunderset(r3)()[COunderset(2)()][H])-(kunderset(f4)()[Ounderset(2)()][H]+kunderset(r4)()[O][OH])`

The above ODE's for all the species j can be solved using numerical techniques to evaluate their production rates.