Week 5 - Compact Notation Derivation for a simple Mechanism
INTRODUCTION
The understanding of the underlying chemical reactions is very important in the study of combustion since it dictates the rate of reaction controls the rate of combustion. The chemical rates also determine the rate of formation of the pollutants. The study of the chemical reactions and the rate of chemical reactions is a branch of physical chemistry known as chemical kinetics. With the knowledge of chemical kinetics computer models is generated which can simulate chemical reactions.
The global reactions for example the formation of water from the combustion of hydrogen:
`2H_2 + O_2 \rightarrow 2H_2O`
consists of a large number of elementary reactions, or in other words, we can say the formation of water which is written in a single step takes place in multiple steps.
The representation of a multistep reaction in a computer is achieved using a compact notation which is the matrix form of the reaction rates also known as the compact notation is used to represent standard engineering reactant systems which are shown for an elementary set of reactants.
THEORY
The reactions for which the compact notation is to be derived is shown below:
`CO + O_2-> CO_2 + O-----------------------(R1)` is the first reaction
`O + H_2O->OH + OH-------------------------------(R2)` is the second reaction,
`CO + OH ->CO_2 +H--------------------------------(R3)` is the third reaction and
`H + O_2 ->OH +O------------------------------(R4)` is the fourth reaction.
The total number of reactions is 4 and the number of species taking part in the reaction is 7. The reactions can be identified using an index `i` and the species can be identified using the index `j`. This is illustrated using a table as shown below:
| `i` | Reaction |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
The species can be numbered as illustrated in the table below:
| `j` | Species |
| 1 | `O_2` |
| 2 | `CO` |
| 3 | `O` |
| 4 | `H_2O` |
| 5 | `OH` |
| 6 | `H` |
| 7 | `CO_2` |
The rate of formation or consumption of each of the species can be represented using ordinary differential equations.
It is also important to note that the forward and backward reaction rates for each of the reactions are represented using `k_{fi}`and `k_{ri}` respectively. Each of the ODE is shown below:
`\frac{d[O_2]}{dt} =-k_{f1}[CO][O_2] +k_{r1}[CO_2][O] + k_{r4}[OH][O]-k_{f4}[H][O_2]`
`\frac{d[O_2]}{dt} =-k_{f1}[CO][O_2] +k_{r1}[CO_2][O] + k_{r3}[CO_2][H]-k_{f3}[CO][OH]`
`\frac{d[O]}{dt} =-k_{f2}[O][H_2O] +k_{r2}[OH]^2 + k_{f1}[CO][O_2]-k_{r1}[CO_2][O]+k_{f4}[H][O_2]-k_{r4}[OH][O]`
`\frac{d[H_2O]}{dt} =-k_{f2}[O][H_2O] +k_{r2}[OH]^2`
`\frac{d[OH]}{dt} =k_{f2}[O][H_2O] -k_{r2}[OH]^2 - k_{r4}[OH][O]+k_{f4}[H][O_2]`
`\frac{d[H]}{dt} =-k_{f4}[H][O_2] +k_{r4}[OH][O] + k_{r4}[OH][O]-k_{f4}[H][O_2]`
`\frac{d[CO_2]}{dt} =k_{f1}[CO][O_2] -k_{r1}[CO_2][O] + k_{f3}[OH][CO]-k_{r3}[H][CO_2]`
The reactants and the products can be represented in the matrix form, which has the number of columns = number of species and the number of reactants equal to the number of rows. The reactant matrix is `v_{ji}\prime` and the product matrix is represented using `v_{ji}\prime \prime`.
The reactant matrix `v_{ji}\prime` is shown below:
`\begin{bmatrix} 1 & 1 & 0 & 0& 0 & 0 & 0 \\0& 0 & 1 & 1& 0 & 0 & 0 \\ 0 & 1 & 0 & 0& 1 & 0 & 0 \\1 & 0 & 0 & 0& 0 & 1& 0 \end{bmatrix}`
The product matrix `v_{ji}\prime \prime` is shown below:
`\begin{bmatrix} 0 & 0 & 1 & 0& 0 & 0 & 1 \\0& 0 & 0 & 0& 2& 0 & 0 \\ 0 & 0 & 0 & 0& 0 & 1 & 1 \\0 & 0 & 1 & 0& 1 & 0& 0 \end{bmatrix}`
The matrix `v_{ji} = v_{ji)\prime \prime - v_{ji)\prime`
The matrix `v_{ji}` can be defined as
`\begin{bmatrix} -1 & -1 & 1 & 0& 0 & 0 & 1 \\0& 0 & -1 & -1& 2& 0 & 0 \\ 0 & -1& 0 & 0& -1 & 1 & 1 \\-1 & 0 & 1 & 0& 1 & -1& 0 \end{bmatrix}`
The net reaction rate can be written as :`q_i = k_{fi} \prod_{j=1}^N [X_j]^{v_{ji} \prime}-k_{ri} \prod_{j=1}^N [X_j]^{v_{ji} \prime \prime}`
So we can write,
`q_1 = k_{f1}[O_2]^1[CO]^1[O]^0[H_2O]^0[OH]^0[H]^0[CO_2]^0-k_{r1}[O_2]^0[CO]^0[O]^1[H_2O]^0[OH]^0[H]^0[CO_2]^1`
`q_2=k_{f2}[O_2]^0[CO]^0[O]^1[H_2O]^1[OH]^0[H]^0[CO_2]^0-k_{r1}[O_2]^0[CO]^0[O]^0[H_2O]^0[OH]^2[H]^0[CO_2]^0`
`q_3=k_{f3}[O_2]^0[CO]^1[O]^0[H_2O]^0[OH]^1[H]^0[CO_2]^0-k_{r3}[O_2]^0[CO]^0[O]^0[H_2O]^0[OH]^0[H]^1[CO_2]^1`
`q_4=k_{f4}[O_2]^1[CO]^0[O]^0[H_2O]^0[OH]^0[H]^1[CO_2]^0-k_{r1}[O_2]^0[CO]^0[O]^1[H_2O]^0[OH]^1[H]^0[CO_2]^0`
`\omega_j` is the net production rate of a particular species `j` index denotes species number.
`\omega_j = \sum_{i=1}^L v_{ji}q_i`
We can say that `\omega_1` is for `O_2` and `\omega_1 = v_{1i}q_i`
`\omega_1=v_{11}q_1+v_{12}q_2+v_{13}q_3+v_{14}q_4`
`\begin{equation*} \omega_{j} = \begin{pmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \\ \omega_4 \\ \omega_5 \\ \omega_6 \\ \omega_7 \end{pmatrix} \end{equation*}`
`\begin{equation*} v_{j,i} =\begin{pmatrix} -1 & -1 & 1 & 0& 0 & 0 & 1 \\0& 0 & -1 & -1& 2& 0 & 0 \\ 0 & -1& 0 & 0& -1 & 1 & 1 \\-1 & 0 & 1 & 0& 1 & -1& 0 \end{pmatrix} \end{equation*}`
`\begin{equation*} q_{i} =\begin{pmatrix}q_1\\ q_2 \\ q_3\\ q_4 \end{pmatrix} \end{equation*}`
So we can easily pont out that `\omega_j =v_{ji}q_i`