Compact Notation Derivation for a simple Mechanism
Objective: Derive the reaction rate ODEs of a simple reaction mechanism using compact notation.
Reaction mechanism
`CO+O_2=>CO_2+O` R1
`O+H_2O=>OH+OH` R2
`CO+OH=>CO_2+H` R3
`H+O_2=>OH+O` R4
For solving the reaction rate of different species we need to solve a system of coupled ODEs. The compact notation is used for describing ODE to the computer program.
Compact notation
Species representation
|
j |
Species |
|
1 |
CO |
|
2 |
O2 |
|
3 |
CO2 |
|
4 |
O |
|
5 |
H20 |
|
6 |
OH |
|
7 |
H |
Reaction representation
|
i |
Reaction |
|
1 |
R1 |
|
2 |
R2 |
|
3 |
R3 |
|
4 |
R4 |
We need to create the stoichiometric coefficient matrix of reactants and the stoichiometric coefficient matrix of products.
Stoichiometric coefficient matrix of reactants
`v_(ji)'=`
Number of rows = Number of reactions
Number of columns= Number of species
Stoichiometric coefficient matrix of products
`v_(ji)''=`
`v_(ji)=v_(ji)''-v_(ji)'`
`v_(ji)=`
j=Species index
i=Reaction index
Net reaction rate (`q_i`) for each reaction,
`q_i=kf_i*Pi_(i=1)^N [X_j]^(v_(ji)' )-kr_i*Pi_(i=1)^N [X_j]^(v_(ji)'' )`
`kf_i=`Forward reaction rate
`kr_i=`Reverse reaction rate
`[X_j]=`Concentration of species
Net reaction rate for reaction R1
`q_1=kf_1*[CO]^1*[O_2]^1*[CO_2]^0*[O]^0*[H20]^0*[OH]^0*[H]^0-kr_1*[CO]^0*[O_2]^0*[CO_2]^1*[O]^1*[H20]^0*[OH]^0*[H]^0`
`=kf_1*[CO]^1*[O_2]^1-kr_1*[CO_2]^1*[O]^1`
Net reaction rate for reaction R2
`q_2=kf_2*[CO]^0*[O_2]^0*[CO_2]^0*[O]^1*[H20]^1*[OH]^0*[H]^0-kr_2*[CO]^0*[O_2]^0*[CO_2]^0*[O]^0*[H20]^0*[OH]^2*[H]^0`
`=kf_2*[O]^1*[H_2O]^1-kr_2*[OH]^2`
Net reaction rate for reaction R3
`q_3=kf_3*[CO]^1*[O_2]^0*[CO_2]^0*[O]^0*[H20]^0*[OH]^1*[H]^0-kr_3*[CO]^0*[O_2]^0*[CO_2]^1*[O]^0*[H20]^0*[OH]^0*[H]^1`
`=kf_3*[CO]^1*[OH]^1-kr_3*[CO_2]^1*[H]^1`
Net reaction rate for reaction R4
`q_4=kf_4*[CO]^0*[O_2]^1*[CO_2]^0*[O]^0*[H20]^0*[OH]^0*[H]^1-kr_4*[CO]^0*[O_2]^0*[CO_2]^0*[O]^1*[H20]^0*[OH]^1*[H]^0`
`=kf_4*[O_2]^1*[H]^1-kr_4*[O]^1*[OH]^1`
Net production rate for each species,
`w_i=Sigma_(i=1)^L v_(ji) * q_i`
Where `w_i=(del[X_i])/(delt`
`w1=v_(11)*q_1+v_(12)*q_2+v_(13)*q_3+v_(14)*q_4`
In this mechanism w1 is the net production rate for CO
`w1=-kf_1*[CO]^1*[O_2]^1+kr_1*[CO_2]^1*[O]^1+kf_3*[CO]^1*[OH]^1-kr_3*[CO_2]^1*[H]^1`
Similary
`w2=v_(21)*q_1+v_(22)*q_2+v_(23)*q_3+v_(24)*q_4`
`=-kf_1*[CO]^1*[O_2]^1+kr_1*[CO_2]^1*[O]^1-kf_4*[O_2]^1*[H]^1+kr_4*[O]^1*[OH]^1`
`w3=v_(31)*q_1+v_(32)*q_2+v_(33)*q_3+v_(34)*q_4`
`=kf_1*[CO]^1*[O_2]^1-kr_1*[CO_2]^1*[O]^1+kf_3*[CO]^1*[OH]^1-kr_3*[CO_2]^1*[H]^1`
`w4=v_(41)*q_1+v_(42)*q_2+v_(43)*q_3+v_(44)*q_4`
`=kf_1*[CO]^1*[O_2]^1-kr_1*[CO_2]^1*[O]^1-kf_2*[O]^1*[H_2O]^1+kr_2*[OH]^2+kf_4*[O_2]^1*[H]^1-kr_4*[O]^1*[OH]^1`
`w5=v_(51)*q_1+v_(52)*q_2+v_(53)*q_3+v_(54)*q_4`
`=-kf_2*[O]^1*[H_2O]^1+kr_2*[OH]^2`
`w6=v_(61)*q_1+v_(62)*q_2+v_(63)*q_3+v_(64)*q_4`
`=2*kf_2*[O]^1*[H_2O]^1-2*kr_2*[OH]^2-kf_3*[CO]^1*[OH]^1+kr_3*[CO_2]^1*[H]^1+kf_4*[O_2]^1*[H]^1-kr_4*[O]^1*[OH]^1`
`w7=v_(71)*q_1+v_(72)*q_2+v_(73)*q_3+v_(74)*q_4`
`=kf_3*[CO]^1*[OH]^1-kr_3*[CO_2]^1*[H]^1-kf_4*[O_2]^1*[H]^1+kr_4*[O]^1*[OH]^1`