Modeling Chemical Reactions in Cantera

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

Elementary Reactions

The basic reaction type is a homogeneous reaction with a pressure-independent rate 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 calculated as:

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

An elementary reaction can be defined in the CTI format using the reaction() entry, or in the YAML format using the elementary reaction type.

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 to stabilize the \(\mathrm{AB}\) molecule (forward direction) or supplies energy to break the \(\mathrm{AB}\) bond (reverse direction).

Different species may be more or less effective in acting as the collision partner. A species that is much lighter than \(\mathrm{A}\) and \(\mathrm{B}\) may not be able to transfer much of its kinetic energy, and so would be inefficient as a collision partner. On the other hand, a species with 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 would otherwise 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 constant collision efficiency can be absorbed into the rate coefficient \(k_f(T)\), the default collision efficiency is 1.0.

A three-body reaction may be defined in the CTI format using the three_body_reaction() entry, or in the YAML format using the three-body reaction type.

Falloff Reactions

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

The simplest expression for the rate coefficient for a falloff reaction is the Lindemann form 2:

\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 \(k0{[\mathrm{M}]}\), and in the high-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 by multiplying 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 falloff reactions. The function \(F(T, P_r)\) is the falloff function, and is specified by assigning an embedded entry to the falloff field.

A falloff reaction may be defined in the CTI format using the falloff_reaction() entry, or in the YAML format using the falloff reaction type.

The Troe Falloff Function

A widely-used falloff function is the one proposed by Gilbert et al. 1:

\begin{equation*} \log_{10} F(T, P_r) = \frac{\log_{10} F_{cent}(T)}{1 + f_1^2} \end{equation*}
\begin{equation*} F_{cent}(T) = (1-A) \exp(-T/T_3) + A \exp (-T/T_1) + \exp(-T_2/T) \end{equation*}
\begin{equation*} f_1 = (\log_{10} P_r + C) / (N - 0.14 (\log_{10} P_r + C)) \end{equation*}
\begin{equation*} C = -0.4 - 0.67\; \log_{10} F_{cent} \end{equation*}
\begin{equation*} N = 0.75 - 1.27\; \log_{10} F_{cent} \end{equation*}

A Troe falloff function may be specified in the CTI format using the Troe() directive, or in the YAML format using the Troe field in the reaction entry. The first three parameters, \((A, T_3, T_1)\), are required. The fourth parameter, \(T_2\), is optional; if omitted, the last term of the falloff function is not used.

The SRI Falloff Function

This falloff function is based on the one originally due to Stewart et al. 4, which required three parameters \(a\), \(b\), and \(c\). Kee et al. 5 generalized this function slightly by adding two more parameters \(d\) and \(e\). (The original form corresponds to \(d = 1\) and \(e = 0\).) Cantera supports the extended 5-parameter form, given by:

\begin{equation*} F(T, P_r) = d \bigl[a \exp(-b/T) + \exp(-T/c)\bigr]^{1/(1+\log_{10}^2 P_r )} T^e \end{equation*}

In keeping with the nomenclature of Kee et al. 5, we will refer to this as the SRI falloff function.

An SRI falloff function may be specified in the CTI format using the SRI() directive, or in the YAML format using the SRI field in the entry.

Chemically-Activated Reactions

For these reactions, the rate falls off as the pressure increases, due to collisional 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 by blending between a low-pressure and a high-pressure rate expression. The difference is that the forward rate constant is written as being proportional to 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 the parameterizations allowed for falloff reactions.

Chemically-activated reactions can be defined in the CTI format using the chemically_activated_reaction() entry, or in the YAML format using the chemically-activated reaction type.

Pressure-Dependent Arrhenius Rate Expressions (P-Log)

This parameterization represents pressure-dependent reaction rates by logarithmically interpolating between Arrhenius rate expressions at various pressures. 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 the rate used in the interpolation formula is the sum of all the rates given at that pressure. For pressures outside the given range, the rate expression at the nearest pressure 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, due to the logarithmic interpolation of the rate for intermediate pressures. When a P-log type reaction is initialized, Cantera does a validation check for a range of temperatures that the sum of the reaction rates at each pressure is positive. Unfortunately, if these checks fail, the only options are to remove the reaction or contact the author of the reaction/mechanism in question, because the reaction is mathematically unsound.

P-log reactions can be defined in the CTI format using the pdep_arrhenius() entry, or in the YAML format using the pressure-dependent-Arrhenius reaction 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 constant can 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 of temperatures and pressure for which they are defined is strongly discouraged.

Chebyshev reactions can be defined in the CTI format using the chebyshev_reaction() entry, or in the YAML format using the Chebyshev reaction type.

Surface Reactions

Heterogeneous reactions on surfaces are represented by an extended Arrhenius- like rate expression, which combines the modified Arrhenius rate expression with further corrections dependent on the fractional surface coverages \(\theta_{k}\) of one or more surface species. The forward rate constant for a reaction 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 Arrhenius parameters and \(a_k\), \(m_k\), and \(E_k\) are the coverage dependencies from species \(k\).

Surface reactions can be defined in the CTI format using the surface_reaction() entry, with coverage information provided using the coverage keyword argument supplied to the Arrhenius() directive. In the YAML format, surface reactions are identified by the presence of surface species and support several additional options.

Sticking Coefficients

Collisions between gas-phase molecules and surfaces which result in the gas- phase molecule sticking to the surface can be described as a reaction which is parameterized by a sticking coefficient:

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

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

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

\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, \(m\) is the 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 CTI format using the stick entry, or in the YAML format by specifying the rate constant in the reaction’s sticking-coefficient field.

Additional Options

Reaction Orders

Explicit reaction orders different from the stoichiometric coefficients are sometimes used for non-elementary reactions. For example, consider the global reaction:

\begin{equation*} \mathrm{C_8H_{18} + 12.5 O_2 \rightarrow 8 CO_2 + 9 H_2O} \end{equation*}

the forward rate constant might be given as 6:

\begin{equation*} k_f = 4.6 \times 10^{11} [\mathrm{C_8H_{18}}]^{0.25} [\mathrm{O_2}]^{1.5} \exp\left(\frac{30.0\,\mathrm{kcal/mol}}{RT}\right) \end{equation*}

Special care is required in this case since the units of the pre-exponential factor depend on the sum of the reaction orders, which may not be an integer.

Note that you can change reaction orders only for irreversible reactions.

Normally, reaction orders are required to be positive. However, in some cases negative reaction orders are found to be better fits for experimental data. In these cases, the default behavior may be overridden in the input file.

References

1

R. G. Gilbert, K. Luther, and J. Troe. Ber. Bunsenges. Phys. Chem., 87:169, 1983.

2
  1. Lindemann. Trans. Faraday Soc., 17:598, 1922.

3

Gregory P. Smith, David M. Golden, Michael Frenklach, Nigel W. Moriarty, Boris Eiteneer, Mikhail Goldenberg, C. Thomas Bowman, Ronald K. Hanson, Soonho Song, William C. Gardiner, Jr., Vitali V. Lissianski, , and Zhiwei Qin. GRI-Mech version 3.0, 1997. see http://combustion.berkeley.edu/gri-mech/version30/text30.html.

4

P. H. Stewart, C. W. Larson, and D. Golden. Combustion and Flame, 75:25, 1989.

5(1,2)

R. J. Kee, F. M. Rupley, and J. A. Miller. Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics. Technical Report SAND89-8009, Sandia National Laboratories, 1989.

6

C. K. Westbrook and F. L. Dryer. Simplified reaction mechanisms for the oxidation of hydrocarbon fuels in flames. Combustion Science and Technology 27, pp. 31—43. 1981.