Week 5.1 - Compact Notation Derivation for a simple Mechanism
AIM: To derive the reaction rate ODEs and production of each species for the simple mechanism given below.
Reactions: 1. `CO+O_2=CO_2+O`........R1
2. `O+H_2O=OH+OH`.........R2
3. `CO+OH=CO_2+H`..........R3
4. `H+O_2=OH+O` ..........R4
J Species:
- CO
- CO2
- O2
- O
- H2O
- OH
- H
Stoichiometry coefficient matrices for reactants:
`v_(ji)^'=``[[1,0,1,0,0,0,0],[0,0,0,1,1,0,0],[1,0,0,0,0,1,0],[0,0,1,0,0,0,1]]`
Stoichiometry coefficient matrices for products :
`v_(ji)^('')=[[0,1,0,1,0,0,0],[0,0,0,0,0,2,0],[0,1,0,0,0,0,1],[0,0,0,1,0,1,0]]`
now, `v_(ji)=v_(ji)^('')-v_(ji)^(')`
`v_(ji)=[[-1,0,-1,0],[1,0,1,0],[-1,0,0,-1],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]]`
`q_i=K_(fi)prod_(j=1)^N[X_j]^(v_(ji)^')-K_(ri)prod_(j=1)^(N)[X_j]^(v_(ji)^(''))`
`q_1=k_(f1)[CO]^1[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^0-kr_1[CO_2]^1[O]^1[CO]^0[O_2]^0[H_2O]^0[OH]^0[H]^0`
`q_1=k_(f1)[CO][O_2]-k_(r1)[CO_2][O]`
`q_2=k_(f2)[CO]^0[CO_2]^0[O_2]^0[O]^1[H_2O]^1[OH]^0[H]^0-k_(r2)[CO]^0[CO_2]^0[O_2]^0[O]^0[H_2O]^0[OH]^2[H]^0`
`q_2=k_(f2)[O][H_2O]-k_(r2)[OH]^2`
`q_3=k_(f3)[CO]^1[CO_2]^0[O_2]^0[O]^0[H_2O]^0[OH]^1[H]^0-k_(r3)[CO]^0[CO_2]^1[O_2]^0[O]^0[H_2O]^0[OH]^0[H]^1`
`q_3=k_(f3)[CO][OH]-k_(r3)[CO_2][H]`
`q_4=K_(f4)[CO]^0[CO_2]^0[O_2]^1[O]^0[H_2O]^0[OH]^0[H]^1-k_(r4)[CO]^0[CO_2]^0[O_2]^0[O]^1[H_2O]^0[OH]^1[H]^0`
`q_4=k_(f4)[O_2][H]-k_(r4)[O][OH]`
Now, for each species thee is production rate given by:
`w_j=sum_(i=1)^Lv_(ji)q_(i)`
where `w_j` denotes Net production rate of particular species ,`w=d/dt[X_j]`
' j ' denotes the species number
`w_1=v_11q_1+v_12q_2+v_13q_3+v_14q_4`
`w_2=v_21q_1+v_22q_2+v_23q_3+v_24q_4`
`w_3=v_31q_1+v_32q_2+v_33q_3+v_34q_4`
`w_4=v_41q_1+v_42q_2+v_43q_3+v_44q_4`
`w_5=v_51q_1+v_52q_2+v_53q_3+v_54q_4`
`w_6=v_61q_1+v_62q_2+v_63q_3+v_64q_4`
`w_7=v_71q_1+v_72q_2+v_73q_3+v_74q_4`
`[[w_1],[w_2],[w_3],[w_4],[w_5],[w_6],[w_7]]=[[-1,0,-1,0],[1,0,1,0],[-1,0,0,-1],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]][[q_1],[q_2],[q_3],[q_4]]`
further simplifying we get production rate for the species:
`w_1=-q_1-q_3`
`w_2=q_1+q_3`
`w_3=-q_1-q_4`
`w_4=q_1-q_2`
`w_5=-q_2`
`w_6=2q_2-q_3+q_4`
`w_7=q_3-q_4`
`[[w_1],[w_2],[w_3],[w_4],[w_5],[w_6],[w_7]]=[[d/dt[CO]],[d/dt[CO_2]],[d/dt[O_2]],[d/dt[O]],[d/dt[H_2O]],[d/dt[OH]],[d/dt[H]]]`