Compact Notation Derivation for a simple Mechanism
Aim
To Derive compact Notation Derivation For below Reaction
`CO+O2 to CO2+O`
`O+H2O to OH+OH`
`CO+OH to CO2+H`
`H+O2 to OH +O`
Solution
Give Problem has 7 species & 4 Reaction mechanism
Species
| j | Species |
| 1 | CO |
| 2 | O2 |
| 3 | CO2 |
| 4 | O |
| 5 | H2O |
| 6 | OH |
| 7 | H |
Reaction
`CO+O2 overset(kf1) (harr) underset(kr1) CO2+O`
`O+H2O overset(kf2) (harr) underset(kr2) OH+OH`
`CO+OH overset(kf3) (harr) underset(kr3) CO2+H`
`H+O2 overset(kf4) (harr) underset(kr4) OH +O`
`Q_i=k_(fi)prod_(j=0)^N[X_(j)]^(v_(ji))-k_(ri)prod_(j=0)^N[X_(j)]^(v_(ji))`
`Q_1=kf1[CO][O2]-kr1[CO2][O]`
`Q_2=kf2[O][H2O]+kr2[OH][OH]`
`Q_3=kf3[CO][OH] +kr3[CO2][H]`
`Q_4=kf4[H][O2]+kr4[OH][O]`
ODE's
`(d(co))/dt=k_(f1)[CO][O2]-k_(r1)[CO2][O]+k_(f3)[CO][OH]-k_(r3)[CO2][H]`
`(d(O2))/dt=k_(f1)[CO][O2]-k_(r1)[CO2][O]+k_(f4)[H][O2]-k_(r4)[OH][O]`
`(d(O))/dt=k_(f2)[O][H2O]-k_(r2)[OH][OH]+k_(f4)[H][O2]-k_(r4)[OH][O]`
`(d(H2O))/dt=k_(f2)[O][H2O]-k_(r2)[OH][OH]`
`(d(OH))/dt=k_(f2)[O][H2O]-k_(r2)[OH][OH]+k_(f3)[CO][OH]-k_(r3)[CO2][H]+k_(f4)[H][O2]-k_(r4)[OH][O]`
`(d[H])/dt=k_(f3)[CO][OH]-k_(r3)[CO2][H]`
`(d(co2))/dt=k_(f1)[CO][O2]-k_(r1)[CO2][O]+k_(f3)[CO][OH]-k_(r3)[CO2][H]`
which is written as
`[(R1),(R2),(R3),(R4)]`
which can be formed by
Reactant
`vji'=[(1,1,0,0,0,0,0),(0,0,1,1,0,0,0),(1,0,0,0,0,1,0),(0,1,0,0,0,0,1)]`
Product
`vji''=[(0,0,1,1,0,0,0),(0,0,0,0,0,2,0),(0,0,1,0,0,1,0),(0,0,0,1,0,1,0)]`
Reaction
`vji=vji''-vji'=[(-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)]`
where
column=Species
Row=Reaction
Now,
`omega_j=sum_(i=1)^Lv_(ij).q_i`
`omega_j=Productrate,L=Number of reaction`
`omega_1=v_11.Q_1+v_12.Q_2+v_13.Q_3+v_14.Q_4=-Q_1-Q_3`
`omega_2=v_21.Q_1+v_22.Q_2+v_23.Q_3+v_24.Q_4=-Q_1-Q_4`
`((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)]X[(Q_1),(Q_2),(Q_3),(Q_4)]`
Result
Hence Compact notation for simple mechanics is derived & Form matrix for Program calculation