In Equation 7–1, is the diffusion flux of species , which arises due to gradients of concentration and temperature. By default, Ansys Fluent uses the dilute approximation (also called Fick’s law) to model mass diffusion due to concentration gradients, under which the diffusion flux can be written as

(7–2)

where

= mass diffusion coefficient for species in the mixture

= mass fraction of species

= thermal (Soret) diffusion coefficient (see Thermal Diffusion Coefficients)

= temperature

For certain laminar flows, the dilute approximation may not be acceptable, and full multicomponent diffusion is required. In such cases, the Maxwell-Stefan equations can be solved; see Full Multicomponent Diffusion in the User's Guide for details.

Equation 7–2 is strictly valid when the mixture composition is not changing, or when is independent of composition. This is an acceptable approximation in dilute mixtures when , for all except the carrier gas. Ansys Fluent can also compute the transport of non-dilute mixtures in laminar flows by treating such mixtures as multicomponent system. Within Ansys Fluent, can be specified in a variety of ways, including by specifying , the binary mass diffusion coefficient of species in species . is not used directly, however; instead, the diffusion coefficient in the mixture, , is computed as [299]:

(7–3)

where and are the mole fraction and mass fraction of species , respectively, and is the porosity. You can input or for each chemical species, as described in Mass Diffusion Coefficient Inputs in the Fluent User's Guide.

In turbulent flows, Ansys Fluent computes the mass diffusion in the following form:

(7–4)

where is the turbulent viscosity, and is the turbulent Schmidt number calculated as:

(7–5)

where is the turbulent diffusivity. The default value of is 0.7. Note that turbulent diffusion generally overwhelms laminar diffusion, and the specification of detailed laminar diffusion properties in turbulent flows is generally not necessary.

In turbulent flows, the total diffusion is the sum of the laminar diffusion given by Equation 7–3 and the turbulent diffusion given by Equation 7–5

A careful treatment of chemical species diffusion in the species transport and energy equations is important when details of the molecular transport processes are significant (for example, in diffusion-dominated laminar flows). As one of the laminar-flow diffusion models, Ansys Fluent has the ability to model full multicomponent species transport.

 

7.1.1.1.3.2. Maxwell-Stefan Equations

From Merk  [436], the Maxwell-Stefan equations can be written as

(7–6)

where,
	= the mole fraction
	= the diffusion velocity
	= the binary mass diffusion coefficient of species in species
	= the thermal diffusion coefficient.

For an ideal gas the Maxwell diffusion coefficients are equal to the binary diffusion coefficients. If the external force is assumed to be the same on all species and that pressure diffusion is negligible, then . Since the diffusive mass flux vector is , the above equation can be written as

(7–7)

After some mathematical manipulations, the diffusive mass flux vector, , can be obtained from

(7–8)

where is the mass fraction of species . Other terms are defined as follows:

(7–9)

where,
	and = matrices
	= matrix of the generalized Fick’s law diffusion coefficients   [649]
	= the molecular weight
	= a subscript used to define a mean molecular weight in the mixture

The solver will use a modification of the Chapman-Enskog formula  [421] to compute the diffusion coefficient (cm²/s) using kinetic theory:

(7–10)

where,
	= molar mass (g/mol)
	= temperature (K)
	= absolute pressure (atm)
	= non-dimensional diffusion collision integral, which is a measure of the interaction of the molecules in the system

is a function of the dimensionless temperature , which is calculated as:

(7–11)

where is the temperature (in Kelvin).

is the Boltzmann constant, which is defined as the gas constant, , divided by Avogadro’s number. in Kelvin for the mixture is the geometric average:

(7–12)

The relation between and is obtained from the tabulation values given in [248], which is approximated in Ansys Fluent as:

(7–13)

For a binary mixture, (in angstroms, where 1 angstrom = 1.0*10^-10 m) is calculated as the arithmetic average of the individual s:

(7–14)

The laminar mass diffusivity of all species is calculated as:

where denotes the mass diffusivity of species in the mixture, is the thermal conductivity, is the mixture density, and is the mixture specific heat.

Anisotropic species diffusion in porous media is modeled as follows:

The thermal diffusion coefficients can be defined as constants, polynomial functions, user-defined functions, or using the following empirically-based composition-dependent expression derived from  [327]:

(7–17)

This form of the Soret diffusion coefficient will cause heavy molecules to diffuse less rapidly, and light molecules to diffuse more rapidly, towards heated surfaces.

For many multicomponent mixing flows, the transport of enthalpy due to species diffusion

can have a significant effect on the enthalpy field and should not be neglected. In particular, when the Lewis number

(7–18)

for any species is far from unity, neglecting this term can lead to significant errors. Ansys Fluent will include this term by default. In Equation 7–18, is the thermal conductivity.

For the pressure-based solver in Ansys Fluent, the net transport of species at inlets consists of both convection and diffusion components. For the density-based solvers, only the convection component is included. The convection component is fixed by the user-specified inlet species mass fraction. The diffusion component, however, depends on the gradient of the computed species field at the inlet. Thus the diffusion component (and therefore the net inlet transport) is not specified a priori. For information about specifying the net inlet transport of species, see Defining Cell Zone and Boundary Conditions for Species in the User’s Guide.

When no turbulence-chemistry interaction (TCI) model is used, finite-rate kinetics are incorporated by computing the chemical source terms using general reaction-rate expressions, without attempting to account explicitly for the effects of turbulent fluctuations on the source-term calculations. This approach is recommended for laminar flows, where the formulation is exact, or for turbulent flows using complex chemistry where either the turbulence time-scales are expected to be fast relative to the chemistry time scales or where the chemistry is sufficiently complex that the chemistry timescales of importance are highly disparate.

The net source of chemical species due to reaction is computed as the sum of the reaction sources over the reactions that the species participate in. For reactions in a porous medium, the net source due to reactions is obtained by multiplying the reaction rates with porosity:

(7–19)

where is the molecular weight of species , is the porosity, and is the molar rate of creation/destruction of species in reaction . Reaction may occur in the continuous phase at wall surfaces.

Consider the ^th reaction written in general form as follows:

(7–20)

Equation 7–20 is valid for both reversible and irreversible reactions. For irreversible reactions, the backward rate constant, , is zero.

The summations in Equation 7–20 are for all chemical species in the system, but only species that appear as reactants or products will have nonzero stoichiometric coefficients.

The molar rate of creation/destruction of species in reaction is given by

(7–21)

Note that the rate exponent for the reverse reaction part in Equation 7–21 is the product species stoichiometric coefficient ().

For information on more general reaction mechanisms in CHEMKIN format, see Getting Started with Ansys Chemkin. For information about inputting the stoichiometric coefficients and rate exponents within the Ansys Fluent user interface, for both global forward (irreversible) reactions and elementary (reversible) reactions, see Inputs for Reaction Definition in the Fluent User's Guide. The information below refers to mechanisms manually entered in Ansys Fluent.

represents the net effect of third bodies on the reaction rate. This term is given by

(7–22)

where is the third-body efficiency of the ^th species in the ^th reaction. By default, Ansys Fluent does not include third-body effects in the reaction rate calculation. You can, however, opt to include the effect of third-body efficiencies if you have data for them.

The forward rate constant for reaction , , is computed using the Arrhenius expression

(7–23)

You (or the database) will provide values for , , , , , , , and, optionally, during the problem definition in Ansys Fluent.

If the reaction is reversible, the backward rate constant for reaction , , is computed, by default, from the forward rate constant using the following relation:

(7–24)

where is the equilibrium constant for the ^th reaction, computed from

(7–25)

where denotes atmospheric pressure (101325 Pa). The term within the exponential function represents the change in Gibbs free energy, and its components are computed as follows:

(7–26)

(7–27)

where and are the entropy and enthalpy of the ^th species evaluated at temperature and atmospheric pressure. These values are specified in Ansys Fluent as properties of the mixture material.

Ansys Fluent also provides the option to explicitly specify the reversible reaction rate parameters (pre-exponential factor, temperature exponent, and activation energy for the reaction) if desired. In this case, the backward rate constant of the reversible reaction is computed using the relationship similar to Equation 7–23:

(7–28)

For more information about specifying the backward reaction rate parameters, see Inputs for Reaction Definition in the Fluent User's Guide.

General formulations for pressure-dependent reactions are available through use of CHEMKIN imported mechanisms and detailed descriptions of these formulations can be found in Getting Started with Ansys Chemkin. If you choose to manually enter reaction expressions through the Ansys Fluent interface, Ansys Fluent can use one of three methods to represent the rate expression in pressure-dependent (or pressure fall-off) reactions. These are a subset of the formulations available through use of CHEMKIN imported mechanisms. A “fall-off” reaction is one in which the temperature and pressure are such that the reaction occurs between Arrhenius high-pressure and low-pressure limits, and therefore is no longer solely dependent on temperature.

There are three methods of representing the rate expressions in this fall-off region. The simplest one is the Lindemann  [377] form. There are also two other related methods, the Troe method  [206] and the SRI method  [634], that provide a more accurate description of the fall-off region.

Arrhenius rate parameters are required for both the high- and low-pressure limits. The rate coefficients for these two limits are then blended to produce a smooth pressure-dependent rate expression. In Arrhenius form, the parameters for the high-pressure limit () and the low-pressure limit () are as follows:

(7–29)

(7–30)

The net rate constant at any pressure is then taken to be

(7–31)

where is defined as

(7–32)

and is the concentration of the bath gas, which can include third-body efficiencies. If the function in Equation 7–31 is unity, then this is the Lindemann form. Ansys Fluent provides two other forms to describe , namely the Troe method and the SRI method.

In the Troe method, is given by

(7–33)

where

(7–34)

and

(7–35)

The parameters , , , and are specified as inputs.

In the SRI method, the blending function is approximated as

(7–36)

where

(7–37)

In addition to the three Arrhenius parameters for the low-pressure limit () expression, you must also supply the parameters , , , , and in the expression.

For Chemically Activated Bimolecular pressure dependent reactions, the net rate constant at any pressure is

(7–38)

Under some combustion conditions, fuels burn quickly and the overall rate of reaction is controlled by turbulent mixing. In high-temperature non-premixed flames, for example, turbulence slowly convects/mixes fuel and oxidizer into the reaction zones where they burn quickly. In certain premixed flames, the turbulence slowly convects/mixes cold reactants and hot products into the reaction zones, where reaction occurs rapidly. In such cases, one approximation is to assume the combustion is mixing-limited, allowing neglect of the complex chemical kinetic rates and instead assuming instantaneous burn upon mixing.

For mixed-is-burned approximation, Ansys Fluent provides a turbulence-chemistry interaction model, based on the work of Magnussen and Hjertager  [410], called the eddy-dissipation model. With this model, the net rate of production of species due to reaction , , is given by the smaller (that is, limiting value) of the two expressions below:

(7–39)

(7–40)

In Equation 7–39 and   Equation 7–40, the chemical reaction rate is governed by the large-eddy mixing time scale, , as in the eddy-breakup model of Spalding  [624]. Combustion proceeds whenever turbulence is present (), and an ignition source is not required to initiate combustion. This is usually acceptable for non-premixed flames, but in premixed flames, the reactants will burn as soon as they enter the computational domain, upstream of the flame stabilizer. To remedy this, Ansys Fluent provides the finite-rate/eddy-dissipation model, where both the finite-rate reaction rates (Equation 7–21), and eddy-dissipation (Equation 7–39 and  Equation 7–40) rates are calculated. The net reaction rate is taken as the minimum of these two rates. In practice, the finite-rate kinetics acts as a kinetic “switch”, preventing reaction before the flame holder. Once the flame is ignited, the eddy-dissipation rate is generally smaller than the Arrhenius rate, and reactions are mixing-limited.

When the LES turbulence model is used, the turbulent mixing rate, in Equation 7–39 and   Equation 7–40, is replaced by the subgrid-scale mixing rate. This is calculated as

(7–41)

The eddy-dissipation-concept (EDC) model is an extension of the eddy-dissipation model that allows inclusion of detailed chemical mechanisms in turbulent flows  [409]. It assumes that reactions occur within small turbulent structures, called the fine scales. These fine scales occupy a fraction of the computational cell, characterized by a volume fraction . Combustion at the fine scales is assumed to occur as a constant pressure reactor, with initial conditions taken as the current species and temperature in the cell. Reactions proceed over the time scale , governed by the reaction rates of Equation 7–21, and are integrated numerically using an ODE solver. The source term in the conservation equation for the mean species , Equation 7–1, is modeled as

(7–42)

where is the fine-scale species mass fraction after reacting over the time scale .

The EDC model can incorporate detailed chemical mechanisms into turbulent reacting flows. However, typical mechanisms are invariably stiff and their numerical integration is computationally costly. Hence, the model should be used only when the assumption of fast chemistry is invalid, such as modeling low-temperature or high-pressure combustion, the slow CO burnout in rapidly quenched flames, or the NO conversion in selective non-catalytic reduction (SNCR).

For guidelines on obtaining a solution using the EDC model, see Solution of Stiff Chemistry Systems in the User's Guide.

 

7.1.1.2.5.1. The Standard EDC Model

In the standard EDC model [217], the volume fraction of the reactive fine scales is modeled as:

(7–43)

where the length scale is computed as:

(7–44)

where

	= length scale constant = 2.1377
	= kinematic viscosity
	= turbulent kinetic energy
	= rate of dissipation of turbulent kinetic energy

The integration time scale is modeled as:

(7–45)

where is a time scale constant equal to 0.4082.

 

7.1.1.2.5.2. The Partially Stirred Reactor EDC Model

The standard EDC model (see The Standard EDC Model for details), has been successfully applied in modeling turbulence-chemistry coupling by incorporating detailed chemical mechanisms. However, the main shortcoming of the standard EDC model is that neither the length scale constant nor the time scale constant depends on local flow and chemistry variables. Studies have shown limitations of the standard EDC model. For example, when applied to the simulation of Moderate and Intense Low-oxygen Dilution (MILD) systems (slow chemistry due to strong oxygen dilution), the EDC model gives a significant overestimation of temperature levels [361], [362].

Several authors have offered the generalization of the EDC model for applications to a wider flame regime. The Partially Stirred Reactor (PaSR) model [176], [720] has been applied to both mild and supersonic combustions. It is conceptually similar to the standard EDC model, but is characterized by a different definition of the reacting volume fraction and time scale:

(7–46)

(7–47)

In Equation 7–46, is defined as:

(7–48)

In Ansys Fluent, the chemistry time scale is computed based upon the practice outlined in [166]:

(7–49)

where are the reaction rates (in kg/m³/s) of CH₄, H₂, O₂, H₂O, and CO₂ as these slower major species dominate the fine-scale time-scales and larger flame structure.

The mixing time scale is taken as a fraction of the integral time scale of turbulence :

(7–50)

where can be a constant value or a function of the local turbulent Reynolds number ():

(7–51)

where is the constant of the k-ε turbulence model.

When Equation 7–51 is used to compute , the model is referred to as the fractal method [176]. The exponent coefficient is expressed as:

(7–52)

where is the fractal dimension. A value of = 3 corresponds to the Kolmogorov time-scale, whereas a value of = 5 results in the integral time.

Note that when the LES turbulence model is used, is computed from the geometric mean of the Kolmogorov time scale and the turbulence time scale [366].

Premixed flames have typical laminar flame thicknesses of the order of a millimeter. Since the laminar flame propagation speed is determined by species diffusion, heat conduction and chemical reaction within the flame, sufficient grid resolution inside the flame is necessary to predict the correct laminar flame velocity. In contrast, the combustor dimensions are usually much larger than the laminar flame thickness and the flame cannot be affordably resolved, even with unstructured and solution-adaptive grids.

The premixed laminar flame speed, denoted , is proportional to where is a diffusivity and is a reaction rate. The laminar flame thickness is proportional to . Hence, the laminar flame can be artificially thickened, without altering the laminar flame speed, by increasing the diffusivity and decreasing the reaction rate proportionally. The thickened flame can then be feasibly resolved on a coarse mesh while still capturing the correct laminar flame speed.

The thickening factor is calculated in Ansys Fluent as:

(7–53)

where is the grid size, is the laminar flame thickness, and is the user-specified number of points in the flame (default of 8). The grid-size, , is determined as where is the cell volume and is the spatial dimension (2 or 3). The laminar flame thickness, , is a user-input, and may be specified as a constant, a User-Defined Function, or calculated as , where is the thermal diffusivity evaluated as ( is the thermal conductivity, is the density and is the specific heat).

All species diffusion coefficients, as well as the thermal conductivity, are multiplied by the thickening factor, , and all reaction rates are divided by . However, away from the flame, these enhanced diffusivities can cause erroneous mixing and heat-transfer, so the flame is dynamically thickened only in a narrow band around the reaction front. This band is calculated by multiplying with, a factor calculated as

(7–54)

In Equation 7–54, is the spatially filtered absolute value of the reaction rate, and is a constant with a default value of 10. The absolute reaction rate is filtered several times, and is the maximum value of in the domain. ranges from unity in the band around the flame to zero outside this band.

The Thickened Flame Model (TFM) [497] is most often used with a single step chemical mechanism where the global reaction has been tuned to provide the correct laminar flame speed. The TFM can, in principal, be used with multi-step reaction mechanisms, however all composition profiles should be adequately resolved within the flame. The TFM is available in both the pressure-based as well as density-based solvers.

While the TFM can be used to model laminar flames, its most common application is as an LES combustion model for turbulent premixed and partially-premixed flames. The turbulent flame speed of a premixed flame is determined principally by the flame wrinkling, which increases the flame surface area.

The thickening of the flame alters its interaction with the turbulence. This should be considered in the calculations of the species formation and destruction rates. As the flame thickens, the ability of turbulence to wrinkle the flame diminishes, which impacts the turbulent flame speed propagation. The decrease in the turbulence chemistry interaction is compensated by multiplying the reaction rates by the efficiency function . In Ansys Fluent, the Charlette and Colin efficiency functions are available for modeling this phenomenon.

The Colin efficiency function is computed by:

(7–55)

(7–56)

The turbulent Reynolds number in Equation 7–56 is based on the integral length scales and the RMS velocity fluctuations at integral length scales. The integral length scale is the length of the largest eddies present in the domain. It depends on the geometry, and a value of 1/4 to 1/2 of a characteristics dimension (such as a burner diameter, an inlet diameter, or a size of a bluff body) is typically used. The RMS velocity fluctuations at the integral length scale is computed as:

(7–57)

where is the RMS velocity at the integral length scale .

(7–58)

Here, the superscript 0 refers to the flame without thickening, and the superscript 1 refers to the thickened flame.

When the flow is laminar, equals 0, and the Colin efficiency function is reduced to 1. When the turbulent velocity fluctuations vary, such that

(7–59)

the Colin efficiency function approaches a theoretical maximum limit:

(7–60)

The Charlette efficiency function [100], [687] is computed as follows:

(7–61)

where

= filter width that corresponds to the thickened flame, which is usually an order of magnitude higher than the physical grid length scale. The default value is 10 times the local cell grid spacing.

= laminar flame thickness.

= sub grid scale turbulent velocity.

= laminar flame speed.

= sub grid Reynolds number at the filter scale.

= exponent set to a constant value of 0.5.

is calculated as:

(7–62)

with = 1.4, and

(7–63)

(7–64)

(7–65)

(7–66)

In the above equations, is the Kolmogorov constant with a value of 1.5.

All species reaction rates are multiplied by . The effective species diffusivities (and thermal conductivity) in each cell are dynamically determined as:

(7–67)

where is the molecular (laminar) diffusivity and is the turbulent diffusivity. may be computed with any of the available methods in Ansys Fluent, including kinetic theory and User Defined Functions (UDFs). Since the kinetic coefficients of 1-step reactions are invariably adjusted to capture the actual laminar flame speed, the same transport properties that are used in this tuning simulation should be used in the TFM simulation. In the narrow band around the flame where is one, the turbulent diffusivities are switched off and the molecular diffusivities are enhanced by a factor . Away from the flame where is zero, the effective diffusivity is the non-thickened value of .

When using the finite-rate (FR) chemistry, or when using eddy-dissipation (ED), finite-rate/eddy-dissipation (FR/ED) or EDC model, as described above, chemical species evolve according to the prescribed kinetic mechanism. In the Relaxation to Chemical Equilibrium model, the species composition is driven to its equilibrium state. The reaction source term in the ^th mean species conservation equation (Equation 7–1), is modeled as

(7–68)

where,
	= the density
	= the mean mass fraction of species
	superscript denotes chemical equilibrium
	= a characteristic time-scale

Equation 7–68 implies that species react toward their chemical equilibrium state over the characteristic time, , as in the Characteristic Time model [320]. Since chemical equilibrium does not depend on reactions or reaction rates, for a given , the reaction source term in Equation 7–68 is independent of the reaction mechanism.

The Relaxation to Chemical Equilibrium model is an option available for any of the turbulence-chemistry interaction options. When no turbulence-chemistry model is used, the characteristic time is calculated as

(7–69)

where is a convection/diffusion time-scale in a cell. is a cell chemical time-scale modeled as

(7–70)

In Equation 7–70, is a model constant (default of 1), index denotes the user-specified fuel species, is the fuel species mass fraction, and is the rate of change of the fuel species mass fraction, while denotes the minimum over all specified fuel species. Hence, represents a chemical ignition time-scale for the fuel species, and is used to prevent a premixed flame from auto-igniting, similar to the FR/ED model. Note that a kinetic mechanism is required to evaluate .

For the ED model, the characteristic time-scale is evaluated as , where the turbulent time-scale is , and the default turbulent rate constant is . For the FR/ED model, the characteristic time-scale is calculated as . Typically, the ED and FR/ED models employ a 1-step reaction, or a 2-step reaction where hydrocarbons pyrolize to CO, which then oxidizes to CO₂. The Relaxation to Chemical Equilibrium model can be considered as an extension of the ED type models where species react towards chemical equilibrium instead of complete reaction. The model should provide more accurate predictions of intermediate species such as CO and radicals required for NOx modeling such as O and OH.

Since chemical equilibrium calculations are typically less computationally expensive than detailed chemistry simulations, the Relaxation to Chemical equilibrium can be used to provide a good initial condition for full-kinetic steady-state simulations. The model can also be used in lieu of the Eddy-Dissipation model where the solution tends to chemical equilibrium instead of complete reaction.

The assumption of chemical equilibrium can lead to large errors in rich zones for hydrocarbon fuels. Ansys Fluent offers the option of reducing the reaction rate in Equation 7–68 by increasing the characteristic time-scale in rich zones. With this option, the local equivalence ratio in a cell is calculated as

(7–71)

where , , and denote the atomic molar fraction of oxygen, hydrogen and carbon. The time-scale, , is then multiplied by a factor when the local equivalence ratio exceeds the specified rich equivalence ratio . The default rich equivalence ratio is 1, and the default exponential factor, , is 2. The option to enable the slow reaction rate in rich mixtures is available in the text interface.

Since chemical equilibrium calculations can consume computational resources, ISAT tabulation is the recommended (and default) solution method. The initial Ansys Fluent iterations speed up significantly as the table is built up with sufficient entries and the majority of queries to the table are interpolated. Since chemical equilibrium compositions are uniquely determined by the initial temperature, pressure and elemental compositions of the mixture, the ISAT table dimensions are for isobaric systems. In most combustion cases, the number of elements is much smaller than the number of chemical species, and the ISAT table is built up relatively quickly. The Direct Integration option may also be selected, in which case equilibrium calculations will always be performed at all cells.