Week 5 - Compact Notation Derivation for a simple Mechanism

Aim: Derive the reaction rate ODEs of simple mechanism using compact notations.

Objective:

• Form reaction rate ODEs using compact notations.

Theory:

Reaction rate or rate of reaction is the speed at which a chemical reaction takes place. We define reaction rate as the speed at which reactants are converted into products. Reaction rates can vary dramatically. Chemical Kinetics is the part of physical chemistry that studies reaction rates.

These rates can be modelled as an ODE as a function of the concentration of the species involved in the reaction. The rate of product formation is actually the rate of depletion of the reacting species. Hence, we arrived at a system of coupled ODEs which can be solved by various integration techniques. This can feasible only if there are a few reaction rate equations, but in reality reactions are complex in nature because of many intermediate poroducts, and hence, multiple ways to complete a given reaction. Thus, the number of ODEs can span upto thousands, hence we use computational methods for this. In this project we'll use compact notations for the given reactions:

Compact Notations:

We need to define indices of the reacting and product species and the reactions.

where, `j` denotes the index of species and i denotes the index of the reactions.

For the mechanism, we have 7 species and 4 reactions. We define `(nu_{ji})^'` which is the coefficient matrix where each element denotes the coefficient of the species on reactant side in the reactions.

`(nu_{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)]`

Similarly, `nu''_(ji)` is defined which has coefficients of species on the product side in the reactions.

`(nu_{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)]`

Next, we form `(nu_(ji)) = (nu_{ji})^('') - (nu_{ji})^'`

`(nu_{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 rate of reaction `(q)` for each reaction is given as:

`q_i = k_{fi}prod_(j=1)^N [X_j]^(nu'_{ji}) - k_{ri}prod_(j=1)^N [X_j]^(nu''_{ji})`

where, `N` is the number of species in reaction mechanism

`q_1 = k_{f1}[CO]^1[O_2]^1[CO_2]^0[O]^0[H_2O]^0[OH]^0[H]^0 - k_{r1}[CO]^0[O_2]^0[CO_2]^1[O]^1[H_2O]^0[OH]^0[H]^0`

hence, similarly we have,

`q_1 = k_{f1}[CO][O_2] - k_{r1}[CO_2][O]`

`q_2 = k_{f2}[O][H_2O] - k_{r2}[OH]^2`

`q_3 = k_{f3}[CO][OH] - k_{r3}[CO_2][H]`

`q_4 = k_{f4}[O_2][H] - k_{r4}[O][OH]`

Now, as we have reaction rates, the production rates can be determined as:

`omega_j = sum_(i=1)^L nu_(ji) q_i`

where, `L` is the number of reactions in the mechanism. Hence,

`omega_1 = - q_1 - q_3`

`omega_2 = - q_1 - q_4`

`omega_3 = q_1 + q_3`

`omega_4 = q_1 - q_2 + q_4`

`omega_5 = -q_2`

`omega_6 = 2q2 - q_3 + q_4`

`omega_7 = q_3 - q_4`

The production rate is also the rate of change of species with time,

`omega_j = (d[X_j])/(dt)`

Hence, we can form the rate of change of each specie as a matrix

`[(d[CO]"/"dt),(d[O_2]"/"dt),(d[CO_2]"/"dt),(d[O]"/"dt),(d[H_2O]"/"dt),(d[OH]"/"dt),(d[H]"/"dt)]=[(omega_1),(omega_2),(omega_3),(omega_4),(omega_5),(omega_6),(omega_7)]`

`{d[CO]}/{dt} = k_{r1}[CO_2][O] - k_{f1}[CO][O_2] + k_{r3}[CO_2][H] - k_{f3}[CO][OH]`

`{d[O_2]}/{dt} = k_{r1}[CO_2][O] - k_{f1}[CO][O_2] + k_{r4}[O][OH] - k_{f4}[O_2][H]`

`{d[CO_2]}/{dt} = k_{f1}[CO][O_2] - k_{r1}[CO_2][O] + k_{f3}[CO][OH] - k_{r3}[CO_2][H]`

`{d[O]}/{dt} = k_{f1}[CO][O_2]-k_{r1}[CO_2][O] + k_{r2}[OH]^2 - k_{f2}[O][H_2O] + k_{f4}[O_2][H] - k_{r4}[O][OH]`

`{d[H_2O]}/{dt} = k_{r2}[OH]^2 - k_{f2}[O][H_2O]`

`{d[OH]}/{dt} = 2k_{f2}[O][H_2O] - 2k_{r2}[OH]^2 + k_{r3}[CO_2][H] - k_{f3}[CO][OH] + k_{f4}[O_2][H] - k_{r4}[O][OH]`

`{d[H]}/{dt} = k_{f3}[CO][OH] - k_{r3}[CO_2][H] + k_{r4}[O][OH] - k_{f4}[O_2][H]`

Hence, a system of coupled ODEs are formed which can be solved by coding any numerical integration technique.