Difference between revisions of "orch:Solvers"

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Chemical kinetics

This chapter reports the principles that drive the computation of combustion chemistry in most CFD softwares.

Chemkin (.scheme .therm .trans), Cantera (xml)

...

Arrhenius law

${\displaystyle {\mathcal {A}}_{j}}$ is the pre-exponential factor, ${\displaystyle {\mathcal {\beta }}_{j}}$ is the temperature exponent and ${\displaystyle E_{a_{j}}}$ the activation energy

${\displaystyle k_{j}={\mathcal {A}}_{j}T^{{\mathcal {\beta }}_{j}}\exp \left(-{\frac {E_{a_{j}}}{RT}}\right)}$

Three-body reactions

In the forward direction, three-body reactions involve two species A and B as reactants and yield a single product AB. In that case, the third body ${\displaystyle {\text{M}}}$ is used to stabilize the excited product AB*. On the contrary, in the reverse direction, heat provides the energy necessary to break the link between A and B.

${\displaystyle ......}$

The third body M can be any inert molecule.

Falloff reactions

Under specific conditions, some reaction rate expressions are dependent on pressure and temperature. This is especially true for the rate associated to unimolecular/recombination fall-off reactions which increases with pressure. In such cases, if the chemical process takes place in a high or low pressure limit typical Arrhenius laws are applicable to the reactions that are described. However, if the pressure is in between, an accurate description of the phenomenon requires a more complicated rate expression. In such a case, the reaction is said to be in the ”fall-off” region.

Lindemann

Several formulas (derived from the Lindemann description) are available to smoothly relate the limiting low and high-pressure rate expressions. With the Lindemann approach, Arrhenius parameters need to be given for both

• the low pressure limit

${\displaystyle k_{0}={\mathcal {A}}_{0}T^{{\mathcal {\beta }}_{0}}\exp \left(-{\frac {E_{a_{0}}}{RT}}\right)}$

• and the high pressure limit

${\displaystyle k_{\infty }={\mathcal {A}}_{\infty }T^{{\mathcal {\beta }}_{\infty }}\exp \left(-{\frac {E_{a_{\infty }}}{RT}}\right)}$.

The expression taken at any pressure is based on a combination of both low and high-pressure Arrhenius expressions. The term ${\displaystyle P_{r}}$ is here equivalent to a pressure and ${\displaystyle {\text{M}}}$ represents the concentration of the mixture, possibly estimated from third-body efficiencies.

Troe

   <reaction reversible="yes" type="falloff" id="0012">
     <equation>O + CO (+ M) [=] CO2 (+ M)</equation>
     <rateCoeff>
       <Arrhenius>
          <A>1.800000E+07</A>
          <b>0</b>
          <E units="cal/mol">2385.000000</E>
       </Arrhenius>
       <Arrhenius name="k0">
          <A>6.020000E+08</A>
          <b>0</b>
          <E units="cal/mol">3000.000000</E>
       </Arrhenius>
       <efficiencies default="1.0">AR:0.5  C2H6:3  CH4:2  CO:1.5  CO2:3.5  H2:2  H2O:6  O2:6 </efficiencies>
       <falloff type="Lindemann"/>
     </rateCoeff>
     <reactants>CO:1 O:1.0</reactants>
     <products>CO2:1.0</products>
   </reaction>

Reaction rates

The global rate of a reaction j (evolution in concentration per unit of time) varies depending on the proportion of the rates associated to the forward and backward directions.

${\displaystyle {\mathcal {Q}}_{j}={\mathcal {Q}}_{f,j}-{\mathcal {Q}}_{r,j}}$

Species production/consumption source terms

Species ${\displaystyle Y_{k}}$ source terms are deduced from

${\displaystyle {\dot {\omega }}_{k}=W_{k}\sum _{j=1}^{N_{R}}\nu _{k,j}{\mathcal {Q}}_{j}}$

Solver to build reference trajectories

DRGEP solver for species reduction

• Compute species direct inter-relations

• Compute species relations through indirect paths

• Compute relations between targets and

DRGEP solver for reactions reduction

QSS solver

• Solve for thermodynamic

${\displaystyle h_{k}=\Delta h_{f,k}^{0}+h_{sk}}$

${\displaystyle h_{sk}=\int _{T_{0}}^{T}Cp_{k}dT}$

Get Gibbs Free Energy

Get Equilibrium constants