Week 5.1 - Compact Notation Derivation for a simple Mechanism

AIM:

To derive the reaction rate ODEs of a simple reaction mechanism by using compact notation and to write down the desired matrices.

REACTION MECHANISM:

CO+O2⇌CO2+OCO+O2⇌CO2+O

O+H2o⇌OH+OHO+H2o⇌OH+OH

CO+OH⇌CO2+HCO+OH⇌CO2+H

H+O2⇌OH+OH+O2⇌OH+O

DERIVATION TO FIND RATE OF CHANGE OF CONCENTRATION:

RATE OF CHANGE OF CONCENTRATION FOR CO:

d[CO]dt=−kf1[CO][O2]−kf3[CO][OH]+kr1[CO2][O]+kr3[CO2][H]d[CO]dt=-kf1[CO][O2]-kf3[CO][OH]+kr1[CO2][O]+kr3[CO2][H]

RATE OF CHANGE OF CONCENTRATION FOR O2:

d[O2]dt=−kf1[CO][O2]−kf4[H][O2]+kr1[CO2][O]+kr4[OH][O]d[O2]dt=-kf1[CO][O2]-kf4[H][O2]+kr1[CO2][O]+kr4[OH][O]

RATE OF CHANGE OF CONCENTRATION FOR CO2:

d[CO2]dt=kf1[CO][O2]−kr1[CO2][O]+kf3[CO][OH]−kr3[CO2][H]d[CO2]dt=kf1[CO][O2]-kr1[CO2][O]+kf3[CO][OH]-kr3[CO2][H]

RATE OF CHANGE OF CONCENTRATION FOR O:

d[O]dt=kf1[CO][O2]−kr1[CO2][O]−kf2[O][H2O]+kr2[OH][OH]+kf4[H][O2]−kr4[OH][O]d[O]dt=kf1[CO][O2]-kr1[CO2][O]-kf2[O][H2O]+kr2[OH][OH]+kf4[H][O2]-kr4[OH][O]

RATE OF CHANGE OF CONCENTRATION FOR H2O:

d[H2O]dt=−kf2[O][H2O]+kr2[OH][OH]d[H2O]dt=-kf2[O][H2O]+kr2[OH][OH]

RATE OF CHANGE OF CONCENTRATION FOR OH:

d[OH]dt=kf2[O][H2O]−kr2[OH][OH]+kf4[H][O2]−kr4[OH][O]−kf3[CO][OH]+kr3[CO2][H]d[OH]dt=kf2[O][H2O]-kr2[OH][OH]+kf4[H][O2]-kr4[OH][O]-kf3[CO][OH]+kr3[CO2][H]

RATE OF CHANGE OF CONCENTRATION FOR H:

d[H]dt=kf3[CO][OH]−kr3[CO2][H]−kf4[H][O2]+kr4[OH][O]d[H]dt=kf3[CO][OH]-kr3[CO2][H]-kf4[H][O2]+kr4[OH][O]

COMPACT NOTATION:

we have 7 species and 4 reactions,

j	species
1	CO
2	O2
3	CO2
4	O
5	H20
6	OH
7	H

i	reaction
1	r1
2	r2
3	r3
4	r4

STOICHIOMETRIC COEFFICIENT MATRIX OF REACTANT SIDE:

Vji'=⎡⎢ ⎢ ⎢ ⎢⎣1100000000110010000100100001⎤⎥ ⎥ ⎥ ⎥⎦Vji′=[1100000000110010000100100001]

STOICHIOMETRIC COEFFICIENT MATRIX OF PRODUCT SIDE:

Vji''=⎡⎢ ⎢ ⎢ ⎢⎣0011000000002000100010001010⎤⎥ ⎥ ⎥ ⎥⎦Vji′′=[0011000000002000100010001010]

difference of the matrix,

Vji⇒Vji''−Vji'⇒⎡⎢ ⎢ ⎢ ⎢⎣−1−111000000−1−120−10100−110−10101−1⎤⎥ ⎥ ⎥ ⎥⎦Vji⇒Vji′′-Vji′⇒[-1-111000000-1-120-10100-110-10101-1]

NET REACTION RATE FOR EACH REACTION:

Q1=kf1[CO][O2]−kr1[CO2][O]Q1=kf1[CO][O2]-kr1[CO2][O]

Q2=kf2[O][H2O]−kr2[OH][OH]Q2=kf2[O][H2O]-kr2[OH][OH]

Q3=kf3[CO][OH]−kr3[CO2][H]Q3=kf3[CO][OH]-kr3[CO2][H]

Q4=kf4[H][O2]−kr4[OH][O]Q4=kf4[H][O2]-kr4[OH][O]

NET PRODUCTION RATE FOR EACH SPECIES:

ω1=−kf1[CO][O2]+kr1[CO2][O]−kf3[CO][OH]+kr3[CO2][H]ω1=-kf1[CO][O2]+kr1[CO2][O]-kf3[CO][OH]+kr3[CO2][H]

ω2=−kf1[CO][O2]+kr1[CO2][O]−kf4[HO][O2]+kr4[OH][O]ω2=-kf1[CO][O2]+kr1[CO2][O]-kf4[HO][O2]+kr4[OH][O]

ω3=kf1[CO][O2]−kr1[CO2][O]+kf3[CO][OH]−kr3[CO2][H]ω3=kf1[CO][O2]-kr1[CO2][O]+kf3[CO][OH]-kr3[CO2][H]

ω4=kf1[CO][O2]−kr1[CO2][O]−kf2[O][H2O]+kr2[OH][OH]+kf4[H][O2]−kr4[OH][O]ω4=kf1[CO][O2]-kr1[CO2][O]-kf2[O][H2O]+kr2[OH][OH]+kf4[H][O2]-kr4[OH][O]

ω5=−kf2[O][H2O]+kr2[OH][OH]ω5=-kf2[O][H2O]+kr2[OH][OH]

ω6=2kf2[O][H2O]−2kr2[OH][OH]−kf3[CO][OH]+kf3[CO2][H]+kf4[H][O2]−kr4[OH][O]ω6=2kf2[O][H2O]-2kr2[OH][OH]-kf3[CO][OH]+kf3[CO2][H]+kf4[H][O2]-kr4[OH][O]

ω7=kf3[CO][OH]−kr3[CO2][H]−kf4[H][O2]+kr4[OH][O]ω7=kf3[CO][OH]-kr3[CO2][H]-kf4[H][O2]+kr4[OH][O]

THE FINAL ODE SYSTEM BECOM

⎣ω1ω2ω3ω4ω5ω6ω7⎤ = ⎣d[CO]dtd[O2]dtd[CO2]dtd[O]dtd[H2O]dtd[OH]dtd[H]dt

CONCLUSION:

Thus the derivation of ODEs of simple reaction mechanism by using compact notation is carried out.