Week 5.1 - Compact Notation Derivation for a simple Mechanism
Aim: To derive the reaction rate ODE's and production rate ode for the given reaction with a compact notation so that it can be fed into the computer program easily.
The four reaction are given as
`CO + O2 iff CO2 + O` -------------------------------------------------------reaction 1
`O + H2O iff OH + OH`-------------------------------------------------------reaction 2
`CO + OH iff CO2 + H`-------------------------------------------------------reaction 3
`H + O2 + M iff OH + O + M`----------------------------------------------------reaction 4
Here in Table 1 we have given notation for each species
| j | Species |
| 1 | O2 |
| 2 | CO |
| 3 | CO2 |
| 4 | O |
| 5 | H2O |
| 6 | OH |
| 7 | H |
Table: 1
Notation for reaction
| i | Reaction |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
`ddotVji`=stochiometric coefficient for the products
`dotVji`=stochiometric coefficient for the reactants
`dotVji=([1, 1, 0, 0, 0, 0, 0, 0 ],[0, 0 ,0 ,1 ,1 ,0 ,0, 0],[0, 1, 0, 0, 0, 1, 0, 0],[1, 0 ,0 ,0 ,0 ,0, 1 ,1])`
`ddotVji=([0 ,0, 1, 1, 0, 0, 0, 0],[0,0,0,0,0,2,0,0],[0 ,0, 1, 0, 0, 0, 1, 0],[0, 0, 0, 1, 0, 1, 0 ,1])`
now,
`Vji=ddot Vji -dot Vji`
`Vji=([-1 ,-1 ,1 ,1 ,0 ,0 ,0, 0],[0 ,0 ,0 ,-1 ,-1 ,2 ,0 ,0],[0 ,-1 ,1 ,0 ,0 ,-1 ,1 ,0],[-1 ,0 ,0 ,1 ,0 ,1 ,-1, 0])`
REACTION RATE FOR EACH REACTION
now for reaction no 1
`qi=(Kfi) underset(i=1) overset(N)(Pi) *[Xj]^(dotVji) - (Kri) underset(i=1) overset(N)(Pi) *[Xj]^(ddotVji)`
now for i=1 that means for the first reaction
`q1=(Kf1){[X1]^(dotV11)*[X2]^(dotV21)*[X3]^(dotV31)*[X4]^(dotV41)*[X5]^(dotV51)*[X6]^(dotV61)*[X7]^(dotV71)}-(Kr1){[X1]^(ddotV11)*[X2]^(ddotV21)*[X3]^(ddotV31)*[X4]^(ddotV41)*[X5]^(ddotV51)*[X6]^(ddotV61)*[X7]^(ddotV71)}`
`q1=(Kf1){[O2]^(dotV11)*[CO]^(dotV21)*[CO2]^(dotV31)*[O]^(dotV41)*[H2O]^(dotV51)*[OH]^(dotV61)*[H]^(dotV71)}-(Kr1){[O2]^(ddotV11)*[CO]^(ddotV21)*[CO2]^(ddotV31)*[O]^(ddotV41)*[H2O]^(ddotV51)*[OH]^(ddotV61)*[H]^(ddotV71)}`
Similarly we can find out q2 q3 and q4
picking coefficient from the matrices
`q1=Kf1[O2][CO] - Kr1 [CO2][O]`
`q2=Kf2 [O][H2O] - Kr2 [OH]^2`
`q3=Kf3[OH][CO] - Kr3 [CO2][H]`
`q4=Kf4[O2][H] - Kr4 [OH][O]`
PRODUCTION RATE FOR EACH SPECIES
Now production rate is given by
`omegaj=underset(i=1)overset(L)sum *Vji * qi`
for species number 1
`omega1=underset(i=1)overset(L)sum *V1i * qi`
`omega1=V11q1+V12q2+V13q3+V14q4`
`omega1=(-1)q1+(0)q2+(0)q3+(-1)q4`
`omega1=-q1-q4`
`omega1 =-(Kf1[O2][CO] - Kr1 [CO2][O])-(Kf4[O2][H] - Kr4 [OH][O])`
`omega2=underset(i=1)overset(L)sum *V2i * qi`
`omega2=V21q1+V22q2+V23q3+V24q4`
`omega2=(-1)q1+(0)q2+(-1)q3+(0)q4`
`omega2=-q1-q3`
`omega2 =-(Kf1[O2][CO] - Kr1 [CO2][O])-(Kf3[CO][OH] - Kr3 [CO2][H])`
similarly we can find out for the 7 species
`omega3 =Kf1[O2][CO] - Kr1 [CO2][O])+Kf3[CO][OH] - Kr3 [CO2][H]`
`omega4 =Kf1[O2][CO] - Kr1 [CO2][O]-Kf2[O][H2O] + Kr2 [OH][OH]+Kf4[O2][H] - Kr4 [OH][O]`
`omega5 = -Kf2[H2O][O] + Kr2 [OH][OH]`
`omega6 =Kf2[H2O][O] - Kr2 [OH][OH]-Kf3[CO][OH] +Kr3 [CO2][H]+Kf4[O2][H] - Kr4 [OH][O]`
`omega6 =Kf3[CO][OH] -Kr3 [CO2][H]-Kf4[O2][H] + Kr4 [OH][O]`
Omega can also be find out using following
`[omega]=[Vji]*[q]`
`([omega1],[omega2],[omega3],[omega4],[omega5],[omega6],[omega7])=([-1 ,-1 ,1 ,1 ,0 ,0 ,0, 0],[0 ,0 ,0 ,-1 ,-1 ,2 ,0 ,0],[0 ,-1 ,1 ,0 ,0 ,-1 ,1 ,0],[-1 ,0 ,0 ,1 ,0 ,1 ,-1, 0])*([q1],[q2],[q3],[q4])``