Modeling Chemical Reactions (2024)

Modeling Chemical Reactions in Cantera¶

Here, we describe how Cantera calculates chemical reaction rates for variousreaction types.

Elementary Reactions¶

The basic reaction type is a hom*ogeneous reaction with a pressure-independentrate coefficient and mass action kinetics. For example:

\begin{equation*}\mathrm{A + B \rightleftharpoons C + D}\end{equation*}

with a forward rate constant \(k_f\) defined as a modified Arrhenius function:

\begin{equation*}k_f = A T^b e^{-E_a / RT}\end{equation*}

where \(A\) is the pre-exponential factor, \(T\) is the temperature,\(b\) is the temperature exponent, \(E_a\) is the activation energy,and \(R\) is the gas constant. The forward reaction rate is then calculatedas:

\begin{equation*}R_f = [\mathrm{A}] [\mathrm{B}] k_f\end{equation*}

An elementary reaction can be defined in the YAML format using theelementary reaction type.

In YAML, the reaction type entry can be omitted, as it represents the default. Incase the type entry is omitted and a species occurs on both sides, Canterainfers that the reaction type is three-body.

Three-Body Reactions¶

A three-body reaction is a gas-phase reaction of the form:

\begin{equation*}\mathrm{A + B + M \rightleftharpoons AB + M}\end{equation*}

Here \(\mathrm{M}\) is an unspecified collision partner that carries away excess energy tostabilize the \(\mathrm{AB}\) molecule (forward direction) or supplies energy to break the\(\mathrm{AB}\) bond (reverse direction). In addition to the generic collision partner\(\mathrm{M}\), it is also possible to explicitly specify a colliding species. In this case,the reaction type is automatically inferred by Cantera.

Different species may be more or less effective in acting as the collision partner. A species thatis much lighter than \(\mathrm{A}\) and \(\mathrm{B}\) may not be able to transfer much ofits kinetic energy, and so would be inefficient as a collision partner. On the other hand, a specieswith a transition from its ground state that is nearly resonant with one in the\(\mathrm{AB^*}\) activated complex may be much more effective at exchanging energy than wouldotherwise be expected.

These effects can be accounted for by defining a collision efficiency\(\epsilon\) for each species, defined such that the forward reaction rate is

\begin{equation*}R_f = [\mathrm{A}][\mathrm{B}][\mathrm{M}]k_f(T)\end{equation*}

where

\begin{equation*}[\mathrm{M}] = \sum_{k} \epsilon_k C_k\end{equation*}

where \(C_k\) is the concentration of species \(k\). Since any constantcollision efficiency can be absorbed into the rate coefficient \(k_f(T)\), the default collisionefficiency is 1.0.

New in Cantera 3.0: The rate coefficient \(k_f(T)\) may be implemented using anyrate parameterization supported by Cantera, not just the modified Arrhenius form.

Sometimes, accounting for a particular third body's collision efficiency may require analternate set of rate parameters entirely. In this case, two reactions are written:

\begin{equation*}\mathrm{A + B + M \rightleftharpoons AB + M \quad (R1)}\end{equation*}

\begin{equation*}\mathrm{A + B + C \rightleftharpoons AB + C \quad (R2)}\end{equation*}

where the third-body efficiency for C in the first reaction is set to zero.

A three-body reaction may be defined in the YAML format using thethree-body reaction type.

Changed in Cantera 3.0: The type field of the YAML entry may be omitted. Reactionscontaining the generic third body M (as in R1) are automatically identified asthree-body reactions. Reactions are also identified as three-body reactions if all ofthe following conditions are met:

• There is exactly one species appearing as both a reactant and product (as in reaction R2)

• All reactants and products have integral stoichiometric coefficients

• The sum of the stoichiometric coefficients for either the reactants or products is 3.

If the third body efficiency of C in R1 is not set to zero, these are consideredduplicate reactions and must be marked as such.

Falloff Reactions¶

A falloff reaction is one that has a rate that is first-order in \([\mathrm{M}]\) at lowpressure, like a three-body reaction, but becomes zero-order in \([\mathrm{M}]\) as \([\mathrm{M}]\)increases. Dissociation/association reactions of polyatomic molecules oftenexhibit this behavior.

The simplest expression for the rate coefficient for a falloff reaction is theLindemann form 1:

\begin{equation*}k_f(T, [{\mathrm{M}}]) = \frac{k_0[{ \mathrm{M}}]}{1 + \frac{k_0{ [\mathrm{M}]}}{k_\infty}}\end{equation*}

In the low-pressure limit, this approaches \(k_0{[\mathrm{M}]}\), and in thehigh-pressure limit it approaches \(k_\infty\).

Defining the non-dimensional reduced pressure:

\begin{equation*}P_r = \frac{k_0 [\mathrm{M}]}{k_\infty}\end{equation*}

The rate constant may be written as

\begin{equation*}k_f(T, P_r) = k_\infty \left(\frac{P_r}{1 + P_r}\right)\end{equation*}

More accurate models for unimolecular processes lead to other, more complex,forms for the dependence on reduced pressure. These can be accounted for bymultiplying the Lindemann expression by a function \(F(T, P_r)\):

\begin{equation*}k_f(T, P_r) = k_\infty \left(\frac{P_r}{1 + P_r}\right) F(T, P_r)\end{equation*}

This expression is used to compute the rate coefficient for falloffreactions. The function \(F(T, P_r)\) is the falloff function.

A falloff reaction may be defined in the YAML format using thefalloff reaction type.

Chemically-Activated Reactions¶

For these reactions, the rate falls off as the pressure increases, due tocollisional stabilization of a reaction intermediate. Example:

\begin{equation*}\mathrm{Si + SiH_4 (+M) \leftrightarrow Si_2H_2 + H_2 (+M)}\end{equation*}

which competes with:

\begin{equation*}\mathrm{Si + SiH_4 (+M) \leftrightarrow Si_2H_4 (+M)}\end{equation*}

Like falloff reactions, chemically-activated reactions are described byblending between a low-pressure and a high-pressure rate expression. Thedifference is that the forward rate constant is written as being proportionalto the low-pressure rate constant:

\begin{equation*}k_f(T, P_r) = k_0 \left(\frac{1}{1 + P_r}\right) F(T, P_r)\end{equation*}

and the optional blending function \(F\) may described by any of theparameterizations allowed for falloff reactions.

Chemically-activated reactions can be defined in the YAML format usingthe chemically-activated reaction type.

Pressure-Dependent Arrhenius Rate Expressions (P-Log)¶

This parameterization represents pressure-dependent reaction ratesby logarithmically interpolating between Arrhenius rate expressions at variouspressures. Given two rate expressions at two specific pressures:

\begin{equation*}P_1: k_1(T) = A_1 T^{b_1} e^{-E_1 / RT}\end{equation*}

\begin{equation*}P_2: k_2(T) = A_2 T^{b_2} e^{-E_2 / RT}\end{equation*}

The rate at an intermediate pressure \(P_1 < P < P_2\) is computed as

\begin{equation*}\log k(T,P) = \log k_1(T) + \bigl(\log k_2(T) - \log k_1(T)\bigr) \frac{\log P - \log P_1}{\log P_2 - \log P_1}\end{equation*}

Multiple rate expressions may be given at the same pressure, in which case therate used in the interpolation formula is the sum of all the rates given at thatpressure. For pressures outside the given range, the rate expression at the nearestpressure is used.

Negative A-factors can be used for any of the rate expressions at a given pressure.However, the sum of all of the rates at a given pressure must be positive, dueto the logarithmic interpolation of the rate for intermediate pressures. When aP-log type reaction is initialized, Cantera does a validation check for a range oftemperatures that the sum of the reaction rates at each pressure is positive. Unfortunately, ifthese checks fail, the only options are to remove the reaction or contact the authorof the reaction/mechanism in question, because the reaction is mathematically unsound.

P-log reactions can be defined in the YAML format using thepressure-dependent-Arrheniusreaction type.

Chebyshev Reaction Rate Expressions¶

Chebyshev rate expressions represent a phenomenological rate coefficient\(k(T,P)\) in terms of a bivariate Chebyshev polynomial. The rate constantcan be written as:

\begin{equation*}\log k(T,P) = \sum_{t=1}^{N_T} \sum_{p=1}^{N_P} \alpha_{tp} \phi_t(\tilde{T}) \phi_p(\tilde{P})\end{equation*}

where \(\alpha_{tp}\) are the constants defining the rate, \(\phi_n(x)\)is the Chebyshev polynomial of the first kind of degree \(n\) evaluated at\(x\), and

\begin{equation*}\tilde{T} \equiv \frac{2T^{-1} - T_\mathrm{min}^{-1} - T_\mathrm{max}^{-1}} {T_\mathrm{max}^{-1} - T_\mathrm{min}^{-1}}\end{equation*}

\begin{equation*}\tilde{P} \equiv \frac{2 \log P - \log P_\mathrm{min} - \log P_\mathrm{max}} {\log P_\mathrm{max} - \log P_\mathrm{min}}\end{equation*}

are reduced temperatures and reduced pressures which map the ranges\((T_\mathrm{min}, T_\mathrm{max})\) and \((P_\mathrm{min},P_\mathrm{max})\) to \((-1, 1)\).

A Chebyshev rate expression is specified in terms of the coefficient matrix\(\alpha\) and the temperature and pressure ranges.

Note that the Chebyshev polynomials are not defined outside the interval\((-1,1)\), and therefore extrapolation of rates outside the range oftemperatures and pressure for which they are defined is strongly discouraged.

Chebyshev reactions can be defined in the YAML format using theChebyshev reaction type.

Blowers-Masel Reactions¶

In some circ*mstances like thermodynamic sensitivity analysis, ormodeling heterogeneous reactions from one catalyst surface to another,the enthalpy change of a reaction (\(\Delta H\)) can be modified. Due to the change in \(\Delta H\),the activation energy of the reaction must be adjusted accordingly to provide accurate simulation results. Toadjust the activation energy due to changes in the reaction enthalpy, the Blowers-Masel rate expression isavailable. This approximation was proposed by Blowers and Masel 6 to automaticallyscale activation energy as the reaction enthalpy is changed.The activation energy estimation can be written as:

\begin{equation*}E_a = \begin{cases} 0 & \text{if } \Delta H \leq -4 E_a^0 \\ \Delta H & \text{if } \Delta H \geq 4 E_a^0 \\ \frac{\left( w + \frac{\Delta H }{2} \right) (V_P - 2 w + \Delta H) ^2} {V_P^2 - 4 w^2 + \Delta H^2} & \text{Otherwise} \end{cases}\end{equation*}

where

\begin{equation*}V_P = 2 w \frac{w + E_a^0}{w - E_a^0},\end{equation*}

\(w\) is the average of the bond dissociation energy of the bond breaking and that being formed,\(E_a^0\) is the intrinsic activation energy, and \(\Delta H\) is the enthalpy change of the reaction.Note that the expression is insensitive to \(w\) as long as \(w \ge 2 E_a^0\), so we can usean arbitrarily high value of \(w = 1000\text{ kJ/mol}\).

After \(E_a\) is evaluated, the reaction rate can be calculated using the modified Arrhenius expression

\begin{equation*}k_f = A T^b e^{-E_a / RT}.\end{equation*}

Blowers Masel reactions can be defined in the YAML format using theBlowers-Masel reaction type.(New in Cantera 2.6)

Surface Reactions¶

Heterogeneous reactions on surfaces are represented by an extended Arrhenius-like rate expression, which combines the modified Arrhenius rate expression withfurther corrections dependent on the fractional surface coverages\(\theta_{k}\) of one or more surface species. The forward rate constant for areaction of this type is:

\begin{equation*}k_f = A T^b \exp \left( - \frac{E_a}{RT} \right) \prod_k 10^{a_k \theta_k} \theta_k^{m_k} \exp \left( \frac{- E_k \theta_k}{RT} \right)\end{equation*}

where \(A\), \(b\), and \(E_a\) are the modified Arrheniusparameters and \(a_k\), \(m_k\), and \(E_k\) are the coveragedependencies from species \(k\).

In the YAML format, surface reactions are identified by the presenceof surface species and support severaladditional options.

In YAML, the surface reaction type defaults to interface-Arrhenius, wherethe rate expression uses the Arrhenius parameterization (seeYAML documentation).As an alternative, Cantera also supports the interface-Blowers-Masel surfacereaction type, which uses the Blowers-Maselparameterization (seeYAML documentation;New in Cantera 2.6).

Sticking Reactions¶

Sticking reactions represent a special case of surface reactions, where collisionsbetween gas-phase molecules and surfaces result in the gas-phase molecule sticking tothe surface. This process can be described as a reaction which is parameterized by asticking coefficient:

\begin{equation*}\gamma = a T^b e^{-c/RT}\end{equation*}

where \(a\), \(b\), and \(c\) are constants specific to thereaction. The values of these constants must be specified so that the stickingcoefficient \(\gamma\) is between 0 and 1 for all temperatures.

The sticking coefficient is related to the forward rate constant by theformula:

\begin{equation*}k_f = \frac{\gamma}{\Gamma_\mathrm{tot}^m} \sqrt{\frac{RT}{2 \pi W}}\end{equation*}

where \(\Gamma_\mathrm{tot}\) is the total molar site density, \(m\) isthe sum of all the surface reactant stoichiometric coefficients, and \(W\)is the molecular weight of the gas phase species.

Sticking reactions can be defined in the YAML format by specifying the rate constantin the reaction'ssticking-coefficient field.

The sticking reaction type defaults to sticking-Arrhenius, wherethe rate expression uses the Arrhenius parameterization (seeYAML documentation).As an alternative, Cantera also supports the sticking-Blowers-Masel surfacereaction type, which uses the Blowers-Maselparameterization (seeYAML documentation;New in Cantera 2.6).

Two-Temperature-Plasma Reactions¶

The two-temperature-plasma reaction is commonly used for non-equilibrium plasmas. Thereaction rate of a two-temperature-plasma reaction depends on both gas and electrontemperature 8, and can be expressed as:

\begin{equation*}k_f = A T_e^b \exp \left( - \frac{E_{a,g}}{RT} \right) \exp \left(\frac{E_{a,e}(T_e - T)}{R T T_e}\right),\end{equation*}

where \(A\) is the pre-exponential factor, \(T\) is the temperature, \(T_e\)is the electron temperature, \(b\) is the electron temperature exponent,\(E_{a,g}\) is the activation energy for gas, \(E_{a,e}\) is the activationenergy for electron and \(R\) is the gas constant. (New in Cantera 2.6)

Additional Options¶