Week 5 - Compact Notation Derivation for a simple Mechanism
FINDING THE REACTION RATE ODEs AND PRODUCTION RATE
Notation
|
j |
Species |
|
1 |
CO |
|
2 |
O2 |
|
3 |
CO2 |
|
4 |
O |
|
5 |
H2O |
|
6 |
OH |
|
7 |
H |
Reactant mole concentration matrix(vji’)(matrix A)
The first column is for species 1 , the second column for second species and so on, row are no. as perreaction no.1,2,3,4. Here the vji notation does not mean that j is row and i is column , it is actually just a way of saying that the 'j' species in the reaction no. 'i' and the actual order of matrix is i x j.
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
Product mole concentration matrix(vji’’)
0 0 1 1 0 0 0
0 0 0 0 0 1 1
0 0 1 0 0 0 1
0 0 0 1 0 1 0
Both matrix are of size I x j, where i=no. of reactions and j=no. of species
Here the matrix size is 4 x 8
Vji’-vji’’=vji
Transpose of the subtraction of matrix A and B gives the matrix with notation vji.(to get order j x i now).
Vji=Net mole concentration matrix
1 0 1 0
1 0 0 1
-1 0 -1 0
-1 1 0 -1
0 1 0 0
0 -1 1 -1
0 -1 -1 1
This is an 8 x4 matrix with notation as vji.
q=kf∏[x](for vji')-kr∏[x](for vji'')
[x] =concentration of species
[x] for vji'=concentration of species at the reactant side(no. of moles)
[x] for vji''=concentration of species at the product side(no. of moles)
kf=forward reaction constant
kr=reverse reaction constant
eg-q1=kf1(product of concentration of reactants in reaction1(i.e first row)^vji')-kr1(product of concentration of products in reaction1(i.e. first row in product side)^vji'')
q1=kf1[CO]^1[O2]^1-kr1[CO2]^1[O]^1
q2=kf2[O][h2o]-kr2[OH][H]
q3=kf3[CO][OH]-kr3[CO2][H]
q4=kf4[O2][H]-kr4[OH][O]
q matrix is 4 x1 matrix,i.e
q1
q2
q3
q4
ωj=production rate of species
ω matrix is, an 8 x1 matrix
ω1 = d/dt[CO]
ω2 = d/dt[O2]
ω3 = d/dt[CO2]
ω4 = d/dt[O]
ω5 = d/dt[h2O]
ω6 = d/dt[OH]
ω7 = d/dt[H]
ωj=∑qi x vji, i.e summation of product of reaction rate and mole concentration matrix
CO is species 1, in vji matrix, v11 means v for species1 in reaction 1
ω for CO is q1(v11)+q2(v12)+q3(v13)+q4(v14) i.e.
ω for CO is (kf1[CO][O2]-kr1[CO2][O])(1) +(kf3[CO][OH]-kr3[CO2][H])(1)
similarly for all species, i.e the production matrix is ,
ω=8 x 1 matrix
Vji=8 x 4 matrix
q= 4 x1 matrix
i.e. ω=vji x q is the final matrix.