Week 5 - Compact Notation Derivation for a simple Mechanism

COMPACT NOTATION DERIVATION OF SIMPLE MECHANISM

Objective: To derive the reaction rate ODEs of a simple reaction mechanism.

Reaction mechanism: Four reactions are giving for which we have to derive the matrix compact notation. Reactions are fairly simple.

`CO+ O_2⇔ CO_2+ O`

`O + H_2O ⇔ OH + OH`

`CO + OH ⇔ CO_2 + H`

`H + O_2 ⇔ OH + O`

Compact notation: It is a very useful way or tool to describe the properties of each species involoved in all reactions, rather than specifying the value or properties every time the species appear in every single reaction.

Species compact notation and reactions:

species	j
CO	1
O_2	2
CO_2	3
O	4
H_2O	5
OH	6
H	7

i	Reactions
1	R1
2	R2
3	R3
4	R4

Writing the above species in the matrix form for the ease of computation.

`v_(ji)` = stochiometric coefficient that describes the behaviour of the reaction i.e.., whether the forward or backward reaction take place.

`v'_(ji)` = stochimetric coefficient of the reactants

`v''_(ji)` = stochimetric coefficient of the products

`v'_(ji)` =`[[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]]`

`v''_(ji)` =`[[0,0,1,1,0,0,0],[0,0,0,0,0,2,0],[0,0,1,0,0,0,1],[0,0,0,1,0,1,0]]`

`v_(ji)` =`v''_(ji)`-`v'_(ji)` =`[[-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]]`

The net production rate of each species in the reactions with respect to time`omega_j`. L = Reaction number

`omega_j` = `sum_(i=1)^Lv_(ji).q_i` = `(d[X_j])/dt`

Here `q_i`= net rate of each reaction.

`q_i` = `k_(f1) oversetNunderset(j=1)Pi[X_j]^(v'_(ji)) - k_(ri) oversetNunderset(j=1)Pi[X_j]^(v''_(ji))`

where kfi = forward reaction constant , kri = reverse reaction constant.

`q_1`=`k_(f1)[CO]^1[O_2]^1 - k_(r1)[CO_2]^1[O]^1`

`q_2`=`k_(f2)[O]^1[H_2O]^1 - k_(r2)[OH]^2`

`q_3`=`k_(f3)[CO]^1[OH]^1 - k_(r3)[CO_2]^1[H]^1`

`q_4`=`k_(f4)[H]^1[O_2]^1 - k_(r4)[OH]^1[O]^1`

where

`omega_1`=`(d[CO])/dt`=`sum_(i=1)^Lv_(1i).q_i`= `v_(11)q_1+v_(12)q_2+v_(13)q_3+v_(14)q_4` = `-q_1-q3`

similarly,

`omega_2`=`(d[O_2])/dt`=`-q_1-q4`

`omega_3`=`(d[CO_2])/dt`=`q_1+q3`

`omega_4`=`(d[O])/dt`=`q_1-q2+q4`

`omega_5`=`(d[HO_2])/dt`=`-q_2`

`omega_6`=`(d[OH])/dt`=`2q_2-q_3+q_4`

`omega_7`=`(d[H])/dt`=`q_3-q_4`

writing above equation in matrix form

`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]] =[[-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]][[q_1],[q_2],[q_3],[q_4]]`

Compact notation of all the species ODEs are derived. Results can be derived with help of any numerical iteration technique.