Week 5 - Compact Notation Derivation for a simple Mechanism
AIM
* To derive reaction rate ODE's for a given reaction mechanism using notations.
INTRODUCTION
To solve combustion using any CFD solver it is necessary that the solver should have a good reaction mechanism. Reaction mechanism is nothing but different reaction pathways, any fuel when undergoes combustion reaction generally we state that the end products will be `CO_2, H_2O,N_2` (also the pollutants such as soot, `No_x` etc if considered) but practically the combustion reaction is not that straight forward, before the combustion end products are formed in between there will lots of intermediary species and lots of elementary reactions that takes place, so a detailed reaction mechanism is the one which covers all the intermediary species along with its elementary reaction. The study of reaction mechanism is called chemical kinetics. Different fuels have different reaction mechanism, while simulating combustion of `CH_4` if we use recation mechanism of petrol the result will be completley off.
For a given fuel if we are able to predict all the intermediary species along with elementary reactions correct it will be a good reaction mechanism. Inorder to achieve this we need to know the operating conditions such as pressure and temperature also the reaction rate.
Determining the reaction rate is more of a pen and paper work when it comes to reaction mechanism which consists only 4 or 5 reactions but actual engineering simulations consists of hundreds of reaction and hundreds of species and hence it is necessary that we solve this using a computer, so as to feed the reactions into the computer it is ncessary that we convert the given reactions into such a format that the computer understands and hence notations and matrices are used.
Notations are used to describe the species as well as the reactions, using these notations appropriate ODE matrices are derived which can be solved using a computer to determine the reaction rate.
Reactions considered as follows:
`CO + O_2 harr CO_2 + O` ---> R1
`O + H_2O harr OH + OH` ---> R2
`CO + OH harr CO_2 + H` ---> R3
`H + O_2 harr OH + O` ---> R4
There are total of 7 species and 4 reactions. Species are indexed as j and recations are indexed as i.
* Species indentation:
| j | Species |
| 1 | `O_2` |
| 2 | `CO` |
| 3 | `H_2O` |
| 4 | `CO_2` |
| 5 | `O` |
| 6 | `H` |
| 7 | `OH` |
* Reaction indentation:
| i | Reaction |
| 1 | R1 |
| 2 | R2 |
| 3 | R3 |
| 4 | R4 |
* First 2 stoichometric coefficient matrices are defined one for the reactants and other for the products. These 2 matrices have number of moles of each species on reactant side and product side. The matrix that represent the reactant side named `V'_(ji)` and that of product side is named as `V''_(ji)`, if we consider the reactant side matrix `V'_(ji)` this can be read as number of moles of j th species in i th reaction on reactant side.
From the above reactions we get the matrices as follow,
`V'_(ji) = [[1,1,0,0,0,0,0],[0,0,1,0,1,0,0],[0,1,0,0,0,0,1],[1,0,0,0,0,1,0]]`
`V''_(ji) = [[0,0,0,1,1,0,0],[0,0,0,0,0,0,2],[0,0,0,1,0,1,0],[0,0,0,0,1,0,1]]`
* Another matrix called `V_(ji)` is defined which is computed subtracting `V'_(ji)` from `V''_(ji)`
`V_(ji) = V''_(ji) - V'_(ji)`
therefore `V_(ji) = [[0,0,0,1,1,0,0],[0,0,0,0,0,0,2],[0,0,0,1,0,1,0],[0,0,0,0,1,0,1]] - [[1,1,0,0,0,0,0],[0,0,1,0,1,0,0],[0,1,0,0,0,0,1],[1,0,0,0,0,1,0]]`
= `[[-1,-1,0,1,1,0,0],[0,0,-1,0,-1,0,2],[0,-1,0,1,0,1,-1],[-1,0,0,0,1,-1,1]]`
* Next q which is net reaction rate for each reaction is computed. Net reaction rate q is computed using the following formula,
`q_i = Kf_(i) prod_(j = 1)^N [X_j]^(V'_(ji) ) - Kr_i prod_(j=1)^N[X_j]^(V''_(ji))`
* Computing the reaction rates:
Reaction 1:
`q_1 = Kf_1*[O_2]^1*[CO]^1*[H_2O]^0*[CO_2]^0*[O]^0*[H]^0*[OH]^0 - Kr_1*[O_2]^0*[CO]^0*[H_2O]^0*[CO_2]^1*[O]^1*[H]^0*[OH]^0`
= `Kf_1*[O_2]*[CO] - Kr_1*[CO_2]*[O]`
Reaction 2:
`q_2 = Kf_2*[O_2]^0*[CO]^0*[H_2O]^1*[CO_2]^0*[O]^1*[H]^0*[OH]^0 - Kr_2*[O_2]^0*[CO]^0*[H_2O]^0*[CO_2]^0*[O]^0*[H]^0*[OH]^2`
= `Kf_2*[H_2O]*[O] - Kr_2*[OH]^2`
Reaction 3:
`q_3 = Kf_3*[O2]^0*[CO]^1*[H_2O]^0*[CO_2]^0*[O]^0*[H]^0*[OH]^1 - Kr_3*[O2]^0*[CO]^0*[H_2O]^0*[CO_2]^1*[O]^0*[H]^1*[OH]^0`
= `Kf_3*[CO]*[OH] - Kr_3*[CO_2]*[H]`
Reaction 4:
`q_4 = Kf_4*[O_2]^1*[CO]^0*[H_2O]^0*[CO_2]^0*[O]^0*[H]^1*[OH]^0 - Kr_4*[O_2]^0*[CO]^0*[H_2O]^0*[CO_2]^0*[O]^1*[H]^0*[OH]^1`
= `Kf_4*[O_2]*[H]- Kr_4*[O]*[OH]`
* Once the reaction rate of each reaction is computed, next the production rate of each species is computed using the below formulation,
`omega_(i) = sum_(L = 1)^NV_(ji)*q_(i)`
* Computing the production rate of each species:
Species 1:
`omega_1 = V_(11)*q_1 + V_(12)*q_2 + V_(13)*q_3 + V_(14)*q_4`, Since `V_(12) , V_(13)` are zero we have
=`-1*(Kf_1*[O_2]*[CO]-Kr_1*[CO_2]*[O])-1*(Kf_4*[O_2]*[H]-Kr_4*[O]*[OH])`
Species 2:
`omega_2 = V_(21)*q_1 + V_(22)*q_2+V_(23)*q_3+V_(24)*q_4`, Since `V_(22), V_(24)` are zeo we have
= `-1*(Kf_1*[O_2]*[CO] - Kr_1*[CO_2]*[O])-1*(Kf_3*[CO]*[OH]-Kr_3*[CO_2]*[H])`
Species 3:
`omega_3 = V_(31)*q_1 + V_(32)*q_2+V_(33)*q_3+V_(34)*q_4`, Since `V_(31), V_(33),V_(34)` are zero we have
= `-1*(Kf_2*[CO]*[OH] - Kr_2*[OH]^2)`
Species 4:
`omega_4 = V_(41)*q_1 + V_(42)*q_2+V_(43)*q_3 + V_(44)*q_4`, Since `V_(42), V_(44)` are zero we have
= `1*(Kf_1*[O_2]*[CO] - Kr_1*[CO_2]*[O]) + 1*(Kf_3*[CO]*[OH] - Kr_3*[CO_2]*[H])`
Species 5:
`omega_5 = V_(51)*q_1 + V_(52)*q_2 + V_(53)*q_3 + V_(54)*q_4`, Since `V_(53)` is zero we have
= `1*(Kf_1*[O_2]*[CO]-Kr_1*[CO_2]*[O])-1*(Kf_2*[H_2O]*[O]-Kr_2*[OH]^2)+1*(Kf_4*[H]*[O_2]-Kr_4*[O]*[OH])`
Species 6:
`omega_6 = V_(61)*q_1+V_(62)*q_2+V_(63)*q_3+V_(64)*q_4`, Since `V_(61), V_(62)` are zero we have
= `1*(Kf_3*[CO]*[OH] - Kr_3*[CO_2]*[H]) -1 *(Kf_4*[O_2]*[H]-Kr_4[O]**[OH])`
Species 7:
`omega_7 = V_(71)*q_1+V_(72)*q_2+V_(73)*q_3+V_(74)*q_4`, Since `V_(71)` is zero we haved
= `2*(Kf_2*[H_2O]*[O]-Kr_2*[OH]^2) - 1*(Kf_3*[CO]*[OH]-Kr_3*[CO_2]*[H]) + 1*(Kf_4*[O_2]*[H]-Kr_4*[O]*[OH])`
Since the production rates of all the species is computed, `omega_j = d/dt[X_j]`this can be represnted in a matrix form,
`[[omega_1],[omega_2],[omega_3],[omega_4],[omega_5],[omega_6],[omega_7]] = [[d/dt[X_1]],[d/dt[X_2]],[d/dt[X_3]],[d/dt[X_4]],[d/dt[X_5]],[d/dt[X_6]],[d/dt[X_7]]]`
the production rate of each species can be solved using integration package of cantera or integration package of matlab.