Rate Constant Parameterizations#
This page describes the different parameterizations available in Cantera for
calculating the forward rate constant
Arrhenius Rate Expressions#
An Arrhenius rate is described by the modified Arrhenius function:
where
YAML Usage
An Arrhenius rate can be specified for a reaction in the YAML format by providing an
Arrhenius rate expression for the reaction’s rate-constant
field.
Falloff Reactions#
A falloff reaction is one that has a rate that is first-order in the total concentration
of third-body colliders
The simplest expression for the rate coefficient for a falloff reaction is the Lindemann form [Lindemann, 1922]:
In the low-pressure limit, this approaches
Defining the non-dimensional reduced pressure:
The rate constant may be written as
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
This expression is used to compute the rate coefficient for falloff reactions. The
function
YAML Usage
A falloff reaction may be defined 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. [1983]:
YAML Usage
A Troe falloff function may be specified in the YAML format using the
Troe
field in the reaction entry. The first three parameters,
Tsang’s Approximation to #
Wing Tsang presented approximations for the value of
where
YAML Usage
A Tsang falloff function may be specified in the YAML format using the
Tsang
field in the reaction entry.
Added in version 2.6.
The SRI Falloff Function#
This falloff function is based on the one originally due to Stewart et al. [1989], which
required three parameters
In keeping with the nomenclature of Kee et al. [1989], we will refer to this as the SRI falloff function.
YAML Usage
An SRI falloff function may be specified 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:
which competes with:
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 proportional to the low-pressure rate constant:
and the optional blending function
YAML Usage
Chemically-activated reactions can be defined 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 [Gou et al., 2011]. Given two rate expressions at two specific pressures:
The rate at an intermediate pressure
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.
Caution
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.
YAML Usage
P-log reactions can be defined in the YAML format using the
pressure-dependent-Arrhenius
reaction type
.
Chebyshev Reaction Rate Expressions#
Chebyshev rate expressions represent a phenomenological rate coefficient
where
are reduced temperatures and reduced pressures which map the ranges
A Chebyshev rate expression is specified in terms of the coefficient matrix
Caution
The Chebyshev polynomials are not defined outside the interval
YAML Usage
Chebyshev reactions can be defined in the YAML format using the
Chebyshev
reaction type
.
Linear Burke Rate Expressions#
Linear Burke rate expressions employ the reduced-pressure linear mixture rule (LMR-R).
This mixture rule is used to evaluate the rate constants of complex-forming reactions,
and is a weighted sum of the bath gas rate constants (when pure) evaluated
at the reduced pressure (
where the reduced pressure,
and the fractional contribution of each component to the reduced pressure,
can be cast in terms of the absolute value of the least negative chemically significant
eigenvalue of the master equation for the
Evaluating all rate constants at the reduced pressure (
such that an alternate version of the generalized LMR-R equation can be written as
It is worth noting two convenient implications of this change in basis. First, when
LMR-R is implemented with the above equation, it is not necessary to specify
While full implementation of LMR-R via the above equation would require
where the sum over
While not required if unique
YAML Usage
Linear Burke rate expressions can be defined in the YAML format using the
linear-Burke
reaction type
.
Added in version 3.1.
Blowers-Masel Reactions#
In some circumstances like thermodynamic sensitivity analysis, or modeling heterogeneous
reactions from one catalyst surface to another, the enthalpy change of a reaction
(
where
and
After
Added in version 2.6.
YAML Usage
Blowers Masel reactions can be defined in the YAML format using the
Blowers-Masel 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
where
YAML Usage
In the YAML format, surface reactions are identified by the presence of surface species and support several additional options.
The surface reaction type
defaults to interface-Arrhenius
, where the rate
expression uses the Arrhenius
parameterization (see YAML
documentation).
Added in version 2.6: As an alternative, Cantera also supports the interface-Blowers-Masel
surface reaction
type
, which uses the Blowers-Masel
parameterization (see YAML
documentation).
Sticking Reactions#
Sticking reactions represent a special case of surface reactions, where collisions between gas-phase molecules and surfaces result in the gas-phase molecule sticking to the surface. This process can be described as a reaction which is parameterized by a sticking coefficient:
where
The sticking coefficient is related to the forward rate constant by the formula:
where
YAML Usage
Sticking reactions can be defined in the YAML format by specifying the rate constant in the reaction’s sticking-coefficient field.
The sticking reaction type
defaults to sticking-Arrhenius
, where the rate expression
uses the Arrhenius
parameterization (see YAML
documentation).
Added in version 2.6: As an alternative, Cantera also supports the sticking-Blowers-Masel
surface reaction
type
, which uses the Blowers-Masel
parameterization (see
YAML documentation).
Two-Temperature-Plasma Reactions#
The two-temperature-plasma reaction is commonly used for non-equilibrium plasmas. The reaction rate of a two-temperature-plasma reaction depends on both gas and electron temperature [Kossyi et al., 1992], and can be expressed as:
where
Added in version 2.6.
YAML Usage
Two-temperature plasma reactions can be defined in the YAML format by specifying
two-temperature-plasma
as the reaction type
and
providing the two activation energies as part of the rate-constant
.
Electron Collision Plasma Reactions#
The electron collision plasma reaction rate uses the electron collision data and the electron energy distribution to calculate the reaction rate. Hagelaar and Pitchford Hagelaar and Pitchford [2005] define the reaction rate coefficient (Eqn. 63) as,
where
To recast this in terms of moles rather than molecules, and letting
where
Added in version 3.1.
YAML Usage
Electron collision reactions can be defined in the YAML format by specifying
electron-collision-plasma
as the reaction type
and providing lists with the cross-sections
and corresponding energy-levels
.