Compact Notation Derivation for a simple Mechanism   (Cantera, Skill Lync, Medium)

Aim: To write a derivation of compact notation for a simple mechanism to form expressions for production rates of different species

Introduction:

Each reaction has several intermediary reactions which take place between initial and final steps of reactions. These all make up a reaction mechanism. The rates of production/depletion of any species in the reaction can be modeled as an ordinary differential equation (ODE). The rate of production of a species depends on the concentration of that species as well as on concentrations of other species. In reality, there are thousands of reactions included in one reaction mechanism. Solving these reactions is only practical with computer calculations, but the computer doesn't understand the reactions in their native representations. So, a compact notation was developed to make represent the reactions in a form readble by a computer. This report contains the derivation of the compact notation.

Derivation:

The reactions used for this report are:

`CO + O_2 harr_(kr_1)^(kf_1) CO_2 + O`

`O + H_2O harr_(kr_2)^(kf_2) OH + OH`

`CO + OH harr_(kr_3)^(kf_3) CO_2 + H`

`H + O_2 harr_(kr_4)^(kf_4) OH + O`

These reactions can be represented with their reaction rates as:

kf --> Rate of forward reaction

kr --> Rate of reverse reaction

The rate of formation or depletion of any species is directly proportional to concentration of species in the reaction. So for our set of reactions we can write.

`(d[CO])/dt` = `− kf_1 [CO][O_2] + kr_1 [CO_2][O] − kf_3 [CO][OH] + kr_3 [CO_2][H]`

`(d[O_2])/dt` = `− kf_1 [CO][O_2] + kr_1 [CO_2][O] − kf_4 [H][O_2] + kr_4 [OH][O]`

`(d[CO_2])/dt` = `kf_1 [CO][O_2] − kr_1 [CO_2][O] + kf_3 [CO][OH] − kr_3 [CO_2][H]`

`(d[O])/dt` = `kf_1 [CO][O_2] − kr_1 [CO_2][O] − kf_2 [O][H_2O] + kr_2 [OH][OH] + kf_4 [H][O_2] − kr_4 [OH][O]`

`(d[H_2O])/dt` = `− kf_2 [O][H_2O] + kr_2 [OH][OH]`

`(d[OH])/dt` = `2kf_2 [O][H_2O] − 2kr_2 [OH][OH] − kf_3 [CO][OH] + kr_3 [CO_2][H] + kf_4 [H][O_2] − kr_4 [OH][O]`

`(d[H])/dt` = `kf_3 [CO][OH] − kr_3 [CO_2][H] − kf_4 [H][O_2] + kr_4 [OH][O]`

(The plus and minus signs in front of each term tells us about whether the forward or backward reaction would increase or decrease the concentration. When due to a reaction the concentration of species increases, a plus sign is used and if concentration decreases a minus sign is used.)

The species and reactions have to be defined with indices for a computer to reference them accordingly:

For species we define indices as 'j' and for reactions we define it as 'i'

Stoichiometric coefficient matrix:

`v_(ji)' ->` Stoichiometric coefficient matrix of reactant

`v_(ji)'' ->` Stoichiometric coefficient matrix of product

`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)'`

`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]]`

Reactions rates:

`q_i = kf_i prod_(j=1)^N [X_j]^(v_(ji)') - kr_i prod_(j=1)^N [X_j]^(v_(ji)'')`

Accordingly,

`q_1 = kf_1[CO]^1[O_2]^1 − kr_1[CO_2]^1[O]^1`

`q_2 = kf_2[O]^1[H_2O]^1 − kr_2[OH]^2`

`q_3 = kf_3[CO]^1[OH]^1 − kr_3[CO_2]^1[H]^1`

`q_4 = kf_4[H]^1[O_2]^1 − kr_4[OH]^1[O]^1`

The production rates of a species:

`ω_j = sum_(i=1)^L v_(ji)q_i = (d[X_j])/dt`

here, L is number of reactions

`[[ω_1],[ω_2],[ω_3],[ω_4],[ω_5],[ω_6],[ω_7]] =[[−1,0,−1,0],[-1,0,0,-1],[1,0,1,0],[1,-1,0,1],[0,-1,0,0],[0,2,-1,1],[0,0,1,-1]] [[q_1],[q_2],[q_3],[q_4]]`

For CO:

`(d[CO])/dt = ω_1 = -q_1 - q_3 = − (kf_1[CO]^1[O_2]^1 − kr_1[CO_2]^1[O]^1) − (kf_3[CO]^1[OH]^1 − kr_3[CO_2]^1[H]^1)`

For O2:

`(d[O_2])/dt = ω_2 = -q_1 - q_4 = − (kf_1[CO]^1[O_2]^1 − kr_1[CO_2]^1[O]^1) − (kf_4[H]^1[O_2]^1 − kr_4[OH]^1[O]^1)`

For CO2:

`(d[CO_2])/dt = ω_3 = q_1 + q_3 = (kf_1[CO]^1[O_2]^1 − kr_1[CO_2]^1[O]^1) + (kf_3[CO]^1[OH]^1 − kr_3[CO_2]^1[H]^1)`

For O:

`(d[O])/dt = ω_4 =q_1 - q_2 +q_4 = (kf_1[CO]^1[O_2]^1 − kr_1[CO_2]^1[O]^1) − (kf_2[O]^1[H_2O]^1 − kr_2[OH]^2) + (kf_4[H]^1[O_2]^1 − kr_4[OH]^1[O]^1)`

For H20:

`(d[H_2O])/dt = ω_5 = -q_2 = − (kf_2[O]^1[H_2O]^1 − kr_2[OH]^2)`

For OH:

`(d[OH])/dt = ω_6 =2q_2 - q_3 + q_4 = 2(kf_2[O]^1[H_2O]^1 − kr_2[OH]^2) − (kf_3[CO]^1[OH]^1 − kr_3[CO_2]^1[H]^1) + (kf_4[H]^1[O_2]^1 − kr_4[OH]^1[O]^1)`

For H:

`(d[H])/dt = ω_7 = q_3 - q_4 = (kf_3[CO]^1[OH]^1 − kr_3[CO_2]^1[H]^1) − (kf_4[H]^1[O_2]^1 − kr_4[OH]^1[O]^1)`

These ODEs are solved to get concentration of species. The only unknowns in these equations are rates of reaction. These are found using Arrhenius equation:

`k_(fi) = A_iT^(beta_i)exp((-E_i)/(R_cT))`

`k_(ri) = (k_(fi))/K_(ci)`

`K_(ci) = K_p(P_(atm)/(RT))^(sum_(k=1)^Kv_(ki)`

`K_(p) = exp((DeltaS_i^o)/R - (DeltaH_i^o)/(RT))`

In these equations entropy(S) and enthalpy(H) are found using NASA polynomials. In this way, for every timestep of combustion, these calculations are done along with calculations of governing equations of other physics models and turbulence models

Conclusion:

The derivation of compact notation of simple mechanism is studied. Also, an extra part of finding reaction rates is studied.