# Chemical kinetics

## Chemical kinetics

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

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### Reaction rates

Chemical reactions occur on highly variable durations. Some of them are almost instantaneous while others may last for several years. The reaction rate measures the change in the concentration of either a reactant or a product per unit of time.

### Arrhenius law

The determination of the rate constant $k$ of a chemical reaction has driven many experimental studies which have led to the description of empirical laws based on a temperature dependency. The Arrhenius equation is, among those, the most commonly used. It combines temperature and reactants concentration

 1$^{st}$ order 2$^{nd}$ order 3$^{rd}$ order $s^{-1}$ ... ...

${\mathcal {A}}_{j}$ is the pre-exponential factor,

${\mathcal {\beta }}_{j}$ is the temperature exponent and

$E_{a_{j}}$ the activation energy

${\frac {E_{a_{j}}}{{\mathcal {R}}T}}$ is a non-dimensional term

${\mathcal {R}}={\text{8.314}}\;J/(mol.K)$

$E_{a}$ is expressed in $J/mol$

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

### 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 ${\text{M}}$ 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.

$......$

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

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

• and the high pressure limit

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

The expression taken at any pressure is based on a combination of both low and high-pressure Arrhenius expressions. The term $P_{r}$ is here equivalent to a pressure and ${\text{M}}$ 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 concentration of the species involved within the studied reaction are considered for

${\mathcal {Q}}_{f,j}=k_{f,j}\sum ...$

$k_{r,j}={\frac {k_{f,j}}{EQ_{j}}}$

${\mathcal {Q}}_{r,j}=k_{r,j}.\sum ...$

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.

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

### Species production/consumption source terms

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

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