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14.1.1 Theory



Species Transport Equations


When you choose to solve conservation equations for chemical species, FLUENT predicts the local mass fraction of each species, $Y_i$, through the solution of a convection-diffusion equation for the $i$th species. This conservation equation takes the following general form:


 \frac{\partial}{\partial t} (\rho Y_i) + \nabla \cdot (\rho \vec{v} Y_i) = -\nabla \cdot \vec{J}_{i} + R_{i} + S_{i} (14.1-1)

where $R_{i}$ is the net rate of production of species $i$ by chemical reaction (described later in this section) and $S_{i}$ is the rate of creation by addition from the dispersed phase plus any user-defined sources. An equation of this form will be solved for $N-1$ species where $N$ is the total number of fluid phase chemical species present in the system. Since the mass fraction of the species must sum to unity, the $N$th mass fraction is determined as one minus the sum of the $N-1$ solved mass fractions. To minimize numerical error, the $N$th species should be selected as that species with the overall largest mass fraction, such as N $_2$ when the oxidizer is air.

Mass Diffusion in Laminar Flows

In Equation  14.1-1, $\vec{J}_{i}$ is the diffusion flux of species $i$, which arises due to concentration gradients. By default, FLUENT uses the dilute approximation, under which the diffusion flux can be written as


 \vec{J}_{i} = - \rho D_{i,m} \nabla Y_i (14.1-2)

Here $D_{i,m}$ is the diffusion coefficient for species $i$ in the mixture.

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 Section  8.9.2 for details.

Mass Diffusion in Turbulent Flows

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


 \vec{J}_{i} = - \left ( \rho D_{i,m} + \frac{\mu_t}{{\rm Sc}_t} \right ) \nabla Y_i (14.1-3)

where ${\rm Sc}_t$ is the turbulent Schmidt number ( $\frac{\mu_t}{\rho D_t}$ where $\mu_t$ is the turbulent viscosity and $D_t$ is the turbulent diffusivity). The default ${\rm Sc}_t$ 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 warranted.

Treatment of Species Transport in the Energy Equation

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


{\nabla} \cdot \left [ \sum^{n}_{i=1} \;\; h_{i} \vec{J}_{i} \right ]

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


 {\rm Le}_i = \frac{k} {\rho c_p D_{i,m}} (14.1-4)

for any species is far from unity, neglecting this term can lead to significant errors. FLUENT will include this term by default. In Equation  14.1-4, $k$ is the thermal conductivity.

Diffusion at Inlets

For the pressure-based solver in 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 inlet species mass fraction specified by you. 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. See Section  14.1.5 for information about specifying the net inlet transport of species.



The Generalized Finite-Rate Formulation for Reaction Modeling


The reaction rates that appear as source terms in Equation  14.1-1 are computed in FLUENT by one of three models:

The generalized finite-rate formulation is suitable for a wide range of applications including laminar or turbulent reaction systems, and combustion systems with premixed, non-premixed, or partially-premixed flames.

The Laminar Finite-Rate Model

The laminar finite-rate model computes the chemical source terms using Arrhenius expressions, and ignores the effects of turbulent fluctuations. The model is exact for laminar flames, but is generally inaccurate for turbulent flames due to highly non-linear Arrhenius chemical kinetics. The laminar model may, however, be acceptable for combustion with relatively slow chemistry and small turbulent fluctuations, such as supersonic flames.

The net source of chemical species $i$ due to reaction is computed as the sum of the Arrhenius reaction sources over the $N_R$ reactions that the species participate in:


 R_{i} = M_{w,i} \sum_{r=1}^{N_R} \; \hat{R}_{i,r} (14.1-5)

where $M_{w,i}$ is the molecular weight of species $i$ and $\hat{R}_{i,r}$ is the Arrhenius molar rate of creation/destruction of species $i$ in reaction $r$. Reaction may occur in the continuous phase between continuous-phase species only, or at wall surfaces resulting in the surface deposition or evolution of a continuous-phase species.

Consider the $r$th reaction written in general form as follows:


 \sum_{i=1}^{N} \nu'_{i,r} {\cal M}_{i} \stackrel{k_{f,r}}{\s... ...eftharpoons}{k_{b,r}}} \sum_{i=1}^{N} \nu''_{i,r} {\cal M}_{i} (14.1-6)


where      
  $N$ = number of chemical species in the system
  $\nu'_{i,r}$ = stoichiometric coefficient for reactant $i$ in reaction $r$
  $\nu''_{i,r}$ = stoichiometric coefficient for product $i$ in reaction $r$
  ${\cal M}_{i}$ = symbol denoting species $i$
  $k_{f,r}$ = forward rate constant for reaction $r$
  $k_{b,r}$ = backward rate constant for reaction $r$

Equation  14.1-6 is valid for both reversible and non-reversible reactions. (Reactions in FLUENT are non-reversible by default.) For non-reversible reactions, the backward rate constant, $k_{b,r}$, is simply omitted.

The summations in Equation  14.1-6 are for all chemical species in the system, but only species that appear as reactants or products will have non-zero stoichiometric coefficients. Hence, species that are not involved will drop out of the equation.

For a non-reversible reaction (that is, the Include Backward Reaction button is disabled), the molar rate of creation/destruction of species $i$ in reaction $r$ ( $\hat{R}_{i,r}$ in Equation  14.1-5) is given by


 \hat{R}_{i,r} = {\Gamma} \left( \nu''_{i,r} - \nu'_{i,r} \ri... ... \left[C_{j,r} \right]^{( \eta'_{j,r} + \eta''_{j,r})} \right) (14.1-7)


where      
  $C_{j,r}$ = molar concentration of species $j$ in reaction $r$ (kgmol/m $^3$)
  $\eta'_{j,r}$ = rate exponent for reactant species $j$ in reaction $r$
  $\eta''_{j,r}$ = rate exponent for product species $j$ in reaction $r$

For a reversible reaction, the molar rate of creation/destruction of species $i$ in reaction $r$ is given by


 \hat{R}_{i,r} = {\Gamma} \left( \nu''_{i,r} - \nu'_{i,r} \ri... ...r} \prod_{j=1}^{N} \left[C_{j,r} \right]^{\nu''_{j,r}} \right) (14.1-8)

Note that the rate exponent for the reverse reaction part in Equation  14.1-8 is always the product species stoichiometric coefficient ( $\nu''_{j,r}$).

See Section  14.1.4 for information about inputting the stoichiometric coefficients and rate exponents for both global forward (non-reversible) reactions and elementary (reversible) reactions.

${\Gamma}$ represents the net effect of third bodies on the reaction rate. This term is given by


 {\Gamma} = \sum_{j}^{N} { \gamma_{j,r} C_{j} } (14.1-9)

where $\gamma_{j,r}$ is the third-body efficiency of the $j$th species in the $r$th reaction. By default, 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 $r$, $k_{f,r}$, is computed using the Arrhenius expression


 k_{f,r} = A_r T^{\beta_r} e^{-E_r/RT} (14.1-10)


where      
  $A_r$ = pre-exponential factor (consistent units)
  $\beta_r$ = temperature exponent (dimensionless)
  $E_r$ = activation energy for the reaction (J/kgmol)
  $R$ = universal gas constant (J/kgmol-K)

You (or the database) will provide values for $\nu'_{i,r}$, $\nu''_{i,r}$, $\eta'_{j,r}$, $\eta''_{j,r}$, $\beta_{r}$, $A_{r}$, $E_r$, and, optionally, $\gamma_{j,r}$ during the problem definition in FLUENT.

If the reaction is reversible, the backward rate constant for reaction $r$, $k_{b,r}$, is computed from the forward rate constant using the following relation:


 k_{b,r} = \frac{k_{f,r}}{K_{r}} (14.1-11)

where $K_{r}$ is the equilibrium constant for the $r$th reaction, computed from


 K_{r} = \exp \left( \frac{\Delta S_r^0}{R}- \frac{\Delta H_r... ...)^ {\displaystyle{\sum_{i=1}^{N} (\nu''_{i,r} - \nu'_{i,r}) }} (14.1-12)

where $p_{\rm atm}$ 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:


 \frac{\Delta S_r^0}{R} = \sum_{i=1}^N \left( \nu''_{i,r} - \nu'_{i,r} \right) \frac{S_{i}^0}{R} (14.1-13)


 \frac{\Delta H_r^0}{RT} = \sum_{i=1}^N \left( \nu''_{i,r} - \nu'_{i,r} \right) \frac{h_{i}^0}{RT} (14.1-14)

where $S_{i}^0$ and $h_{i}^0$ are the standard-state entropy and standard-state enthalpy (heat of formation). These values are specified in FLUENT as properties of the mixture material.

Pressure-Dependent Reactions

FLUENT can use one of three methods to represent the rate expression in pressure-dependent (or pressure fall-off) reactions. 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 thus 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 [ 212] form. There are also two other related methods, the Troe method [ 122] and the SRI method [ 357], 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 ( $k$) and the low-pressure limit ( $k_{\rm low}$) are as follows:


$\displaystyle k$ $\textstyle =$ $\displaystyle AT^{\beta} e^{-E/RT}$ (14.1-15)
$\displaystyle k_{\rm low}$ $\textstyle =$ $\displaystyle A_{\rm low}T^{\beta_{\rm low}} e^{-E_{\rm low}/RT}$ (14.1-16)

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


 k_{\rm net} = k\left(\frac{p_r}{1+p_r}\right) F (14.1-17)

where $p_r$ is defined as


 p_r = \frac{k_{\rm low}[M]}{k} (14.1-18)

and $[M]$ is the concentration of the bath gas, which can include third-body efficiencies. If the function $F$ in Equation  14.1-17 is unity, then this is the Lindemann form. FLUENT provides two other forms to describe $F$, namely the Troe method and the SRI method.

In the Troe method, $F$ is given by


 \log F = \left\{1 + \left[\frac{\log p_r + c}{n - d(\log p_r +c)}\right]^2\right\}^{-1} \log F_{\rm cent} (14.1-19)

where


$\displaystyle c$ $\textstyle =$ $\displaystyle -0.4-0.67\log F_{\rm cent}$ (14.1-20)
$\displaystyle n$ $\textstyle =$ $\displaystyle \phantom{-}0.75-1.27\log F_{\rm cent}$ (14.1-21)
$\displaystyle d$ $\textstyle =$ $\displaystyle \phantom{-}0.14$ (14.1-22)

and


 F_{\rm cent} = (1-\alpha)e^{-T/T_3} + \alpha e^{-T/T_1} + e^{-T_2/T} (14.1-23)

The parameters $\alpha$, $T_3$, $T_2$, and $T_1$ are specified as inputs.

In the SRI method, the blending function $F$ is approximated as


 F = d\left[a \exp\left(\frac{-b}{T}\right) + \exp\left(\frac{-T}{c}\right)\right]^X T^e (14.1-24)

where


 X = \frac{1}{1 + \log^2 p_r} (14.1-25)

In addition to the three Arrhenius parameters for the low-pressure limit ( $k_{\rm low}$) expression, you must also supply the parameters $a$, $b$, $c$, $d$, and $e$ in the $F$ expression.

figure   

Chemical kinetic mechanisms are usually highly non-linear and form a set of stiff coupled equations. See Section  14.1.7 for solution procedure guidelines. Also, if you have a chemical mechanism in CHEMKIN [ 178] format, you can import this mechanism into FLUENT as described in Section  14.1.9.

The Eddy-Dissipation Model

Most fuels are fast burning, and the overall rate of reaction is controlled by turbulent mixing. In non-premixed flames, turbulence slowly convects/mixes fuel and oxidizer into the reaction zones where they burn quickly. In premixed flames, the turbulence slowly convects/mixes cold reactants and hot products into the reaction zones, where reaction occurs rapidly. In such cases, the combustion is said to be mixing-limited, and the complex, and often unknown, chemical kinetic rates can be safely neglected.

FLUENT provides a turbulence-chemistry interaction model, based on the work of Magnussen and Hjertager [ 229], called the eddy-dissipation model. The net rate of production of species $i$ due to reaction $r$, $R_{i,r}$, is given by the smaller (i.e., limiting value) of the two expressions below:


 R_{i,r} = \nu'_{i,r} M_{w,i} A \rho \frac{\epsilon}{k} \min_... ...left(\frac{Y_{\cal R}}{\nu'_{{\cal R},r} M_{w,\cal R}} \right) (14.1-26)


 R_{i,r} = \nu'_{i,r} M_{w,i} AB \rho \frac{\epsilon}{k} \frac{ \sum_{P} Y_{P} }{\sum_{j}^N \nu''_{j,r} M_{w,j}} (14.1-27)


where $Y_P$ is the mass fraction of any product species, $P$  
  $Y_{\cal R}$ is the mass fraction of a particular reactant, $\cal{R}$  
  $A$ is an empirical constant equal to 4.0  
  $B$ is an empirical constant equal to 0.5  

In Equations  14.1-26 and   14.1-27, the chemical reaction rate is governed by the large-eddy mixing time scale, ${k}/{\epsilon}$, as in the eddy-breakup model of Spalding [ 350]. Combustion proceeds whenever turbulence is present ( ${k}/{\epsilon}>0$), 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, FLUENT provides the finite-rate/eddy-dissipation model, where both the Arrhenius (Equation  14.1-8), and eddy-dissipation (Equations  14.1-26 and  14.1-27) reaction rates are calculated. The net reaction rate is taken as the minimum of these two rates. In practice, the Arrhenius rate 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.

figure   

Although FLUENT allows multi-step reaction mechanisms (number of reactions $> 2$) with the eddy-dissipation and finite-rate/eddy-dissipation models, these will likely produce incorrect solutions. The reason is that multi-step chemical mechanisms are based on Arrhenius rates, which differ for each reaction. In the eddy-dissipation model, every reaction has the same, turbulent rate, and therefore the model should be used only for one-step (reactant $\rightarrow$ product), or two-step (reactant $\rightarrow$ intermediate, intermediate $\rightarrow$ product) global reactions. The model cannot predict kinetically controlled species such as radicals. To incorporate multi-step chemical kinetic mechanisms in turbulent flows, use the EDC model (described below).

figure   

The eddy-dissipation model requires products to initiate reaction (see Equation  14.1-27). When you initialize the solution for steady flows, FLUENT sets all species mass fractions to a maximum of the user specified initial value and 0.01. This is usually sufficient to start the reaction. However, if you converge a mixing solution first, where all product mass fractions are zero, you may then have to patch products into the reaction zone to ignite the flame. See Section  14.1.7 for details.

The Eddy-Dissipation Model for LES

When the LES turbulence model is used, the turbulent mixing rate, ${\epsilon}/k$ in Equations  14.1-26 and   14.1-27, is replaced by the subgrid-scale mixing rate. This is calculated as


 \tau^{-1}_{sgs} = \sqrt{2 S_{ij} S_{ij} } (14.1-28)


where      
  $\tau^{-1}_{sgs}$ = subgrid-scale mixing rate (s $^{-1}$)
  $S_{ij}$ = $\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) $ = strain rate tensor (s $^{-1}$)

The Eddy-Dissipation-Concept (EDC) Model

The eddy-dissipation-concept (EDC) model is an extension of the eddy-dissipation model to include detailed chemical mechanisms in turbulent flows [ 228]. It assumes that reaction occurs in small turbulent structures, called the fine scales. The length fraction of the fine scales is modeled as [ 127]


 \xi^{*} = C_{\xi} \left( \frac{\nu \epsilon}{k^2} \right)^{1/4} (14.1-29)

where $^*$ denotes fine-scale quantities and


  $C_{\xi}$ = volume fraction constant = 2.1377
  $\nu$ = kinematic viscosity

The volume fraction of the fine scales is calculated as ${\xi^{*}}^3$. Species are assumed to react in the fine structures over a time scale


 \tau^{*} = C_{\tau} \left( \frac{\nu}{\epsilon} \right)^{1/2} (14.1-30)

where $C_{\tau}$ is a time scale constant equal to 0.4082.

In FLUENT, 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 $\tau^{*}$, governed by the Arrhenius rates of Equation  14.1-8, and are integrated numerically using the ISAT algorithm [ 290]. ISAT can accelerate the chemistry calculations by two to three orders of magnitude, offering substantial reductions in run-times. Details about the ISAT algorithm may be found in Sections  18.2.4 and 18.2.5. ISAT is very powerful, but requires some care. See Section  18.3.9 for details on using ISAT efficiently.

The source term in the conservation equation for the mean species $i$, Equation  14.1-1, is modeled as


 R_i = \frac{{\rho (\xi^{*})}^2}{\tau^{*}[1-{(\xi^{*})}^3]}(Y^{*}_i - Y_i) (14.1-31)

where $Y^{*}_i$ is the fine-scale species mass fraction after reacting over the time $\tau^{*}$.

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 the slow CO burnout in rapidly quenched flames, or the NO conversion in selective non-catalytic reduction (SNCR).

See Section  14.1.7 for guidelines on obtaining a solution using the EDC model.


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