chp14

Chapter 14. Modeling Non-Premixed

Combustion

In non-premixed combustion, fuel and oxidizer enter the reaction zonein distinct streams. This is in contrast to premixed systems, in whichreactants are mixed at the molecular level before burning. Examples ofnon-premixed combustion include methane combustion, pulverized coalfurnaces, and diesel (compression) internal-combustion engines.

Under certain assumptions, the thermochemistry can be reduced to asingle parameter: the mixture fraction. The mixture fraction, denotedby f , is the mass fraction that originated from the fuel stream. In otherwords, it is the local mass fraction of burnt and unburnt fuel streamelements (C, H, etc.) in all the species (CO2, H2O, O2, etc.). Theapproach is elegant because atomic elements are conserved in chemicalreactions. In turn, the mixture fraction is a conserved scalar quantity,and therefore its governing transport equation does not have a sourceterm. Combustion is simplified to a mixing problem, and the difficultiesassociated with closing non-linear mean reaction rates are avoided. Oncemixed, the chemistry can be modeled as in chemical equilibrium, or nearchemical equilibrium with the laminar flamelet model.

These models are presented in the following sections:

• Section 14.1: Description of the Equilibrium Mixture Fraction/PDFModel

• Section 14.2: Modeling Approaches for Non-Premixed EquilibriumChemistry

• Section 14.3: User Inputs for the Non-Premixed Equilibrium Model

• Section 14.4: The Laminar Flamelet Model

• Section 14.5: Adding New Species to the prePDF Database

c© Fluent Inc. November 28, 2001 14-1

Modeling Non-Premixed Combustion

14.1 Description of the Equilibrium Mixture Fraction/PDF Model

The non-premixed modeling approach involves the solution of transportequations for one or two conserved scalars (the mixture fractions). Equa-tions for individual species are not solved. Instead, species concentra-tions are derived from the predicted mixture fraction fields. The ther-mochemistry calculations are preprocessed in prePDF and tabulated forlook-up in FLUENT. Interaction of turbulence and chemistry is accountedfor with a probability density function (PDF).

Information about the non-premixed mixture fraction/PDF model is pre-sented in the following subsections:

• Section 14.1.1: Benefits and Limitations of the Non-Premixed Ap-proach

• Section 14.1.2: Details of the Non-Premixed Approach

• Section 14.1.3: Restrictions and Special Cases for Non-PremixedModeling

See Section 14.2 for an overview of modeling and solution procedures,and Section 14.3 for instructions on using the model.

14.1.1 Benefits and Limitations of the Non-Premixed Approach

Advantages of the Non-Premixed Approach

The non-premixed modeling approach has been specifically developed forthe simulation of turbulent diffusion flames with fast chemistry. For suchsystems, the method offers many benefits over the finite rate formulationdescribed in Chapter 13. The non-premixed model allows intermediate(radical) species prediction, dissociation effects, and rigorous turbulence-chemistry coupling. The method is computationally efficient in that itdoes not require the solution of a large number of species transport equa-tions. When the underlying assumptions are valid, the non-premixedapproach is preferred over the finite rate formulation.

14-2 c© Fluent Inc. November 28, 2001

14.1 Description of the Equilibrium Mixture Fraction/ PDF Model

Limitations of the Non-Premixed Approach

As detailed in Section 14.1.2, the non-premixed approach can be usedonly when your reacting flow system meets several requirements. First,the FLUENT implementation requires that the flow be turbulent. Sec-ond, the reacting system includes a fuel stream, an oxidant stream, and,optionally, a secondary stream (another fuel or oxidant, or a non-reactingstream). Finally, the chemical kinetics must be rapid so that the flow isnear chemical equilibrium. These issues are detailed in Sections 14.1.2and 14.1.3.

Note that the non-premixed model can be used only with the segregated!solver; it is not available with the coupled solvers.

14.1.2 Details of the Non-Premixed Approach

Definition of the Mixture Fraction

The basis of the non-premixed modeling approach is that under a certainset of simplifying assumptions, the instantaneous thermochemical stateof the fluid is related to a conserved scalar quantity known as the mixturefraction f . The mixture fraction can be written in terms of the atomicmass fraction as [213]

f =Zi − Zi,ox

Zi,fuel − Zi,ox(14.1-1)

where Zi is the elemental mass fraction for some element, i. The sub-script ox denotes the value at the oxidizer stream inlet and the subscriptfuel denotes the value at the fuel stream inlet. If the diffusion coefficientsfor all species are equal, then Equation 14.1-1 is identical for all elements,and the mixture fraction definition is unique. The mixture fraction isthus the elemental mass fraction that originated from the fuel stream.Note that this mass fraction includes all elements from the fuel stream,including inert species such as N2, and any oxidizing species mixed withthe fuel, such as O2.

If a secondary stream (another fuel or oxidant, or a non-reacting stream)is included, the fuel and secondary mixture fractions are simply the mass



fractions of the fuel and secondary streams. The sum of all three mixturefractions in the system (fuel, secondary stream, and oxidizer) is alwaysequal to 1:

ffuel + fsec + fox = 1 (14.1-2)

This indicates that only points on the plane ABC (shown in Figure 14.1.1)in the mixture fraction space are valid. Consequently, the two mixturefractions, ffuel and fsec, cannot vary independently; their values are validonly if they are both within the triangle OBC shown in Figure 14.1.2.

fox

B

C

A

O

ffuel

fsec1

1

1

0

Figure 14.1.1: Relationship of ffuel, fsec, and fox

FLUENT discretizes the triangle OBC as shown in Figure 14.1.2. Es-sentially, the primary mixture fraction, ffuel, is allowed to vary betweenzero and one, as for the single mixture fraction case, while the secondarymixture fraction lies on lines with the following equation:

fsec = psec × (1 − ffuel) (14.1-3)

where psec is the normalized secondary mixture fraction and is the valueat the intersection of a line with the secondary mixture fraction axis.



psec

f

f

fuel

sec

1

1

0O

C

B

Figure 14.1.2: Relationship of ffuel, fsec, and psec

Note that unlike fsec, psec is bounded between zero and one, regardlessof the ffuel value.

An important characteristic of the normalized secondary mixture frac-tion, psec, is its assumed statistical independence from the fuel mixturefraction, ffuel. Note that unlike fsec, psec is not a conserved scalar. Thenormalized mixture fraction definition for the second scalar variable isused everywhere except when defining the rich limit for a secondary fuelstream, which is defined in terms of fsec.

Transport Equations for the Mixture Fraction

Under the assumption of equal diffusivities, the species equations canbe reduced to a single equation for the mixture fraction, f . The re-action source terms in the species equations cancel, and thus f is aconserved quantity. While the assumption of equal diffusivities is prob-lematic for laminar flows, it is generally acceptable for turbulent flowswhere turbulent convection overwhelms molecular diffusion. The mean(time-averaged) mixture fraction equation is



∂

∂t(ρf) + ∇ · (ρ~vf) = ∇ ·

(µt

σt∇f)

+ Sm + Suser (14.1-4)

The source term Sm is due solely to transfer of mass into the gas phasefrom liquid fuel droplets or reacting particles (e.g., coal). Suser is anyuser-defined source term.

In addition to solving for the mean mixture fraction, FLUENT solves aconservation equation for the mean mixture fraction variance, f ′2 [105]:

∂

∂t

(ρf ′2

)+∇·

(ρ~vf ′2

)= ∇·

(µt

σt∇f ′2

)+Cgµt

(∇2f

)−Cdρ

ε

kf ′2 +Suser

(14.1-5)

where f′= f − f . The constants σt, Cg, and Cd take the values 0.85,

2.86, and 2.0, respectively, and Suser is any user-defined source term.

The mixture fraction variance is used in the closure model describingturbulence-chemistry interactions (see below).

For a two-mixture-fraction problem, ffuel and f′2fuel are obtained from

Equations 14.1-4 and 14.1-5 by substituting ffuel for f and f′2fuel for f ′2.

fsec is obtained from Equation 14.1-4 by substituting fsec for f . psec isthen calculated using Equation 14.1-3, and p′2

sec is obtained by solvingEquation 14.1-5 with psec substituted for f . Solution for p′2

sec instead off

′2sec is justified by the fact that the amount of the secondary stream is

relatively small compared with the total mass flow rate. To a first-orderapproximation, the variances in psec and fsec are relatively insensitive toffuel, and therefore p′2

sec is essentially the same as f ′2sec.

The Non-Premixed Model for LES

For large eddy simulations (LES), an equation for the mean mixturefraction is solved, which is identical in form to Equation 14.1-4 exceptthat µt is the subgrid-scale viscosity.

A transport equation is not solved for the mixture fraction variance.Instead, it is modeled as



f ′2 = CvarL2sgs|∇f |2 (14.1-6)

whereCvar = user-adjustable constantLsgs = subgrid length scale

Mixture Fraction vs. Equivalence Ratio

The mixture fraction definition can be understood in relation to commonmeasures of reacting systems. Consider a simple combustion systeminvolving a fuel stream (F), an oxidant stream (O), and a product stream(P) symbolically represented at stoichiometric conditions as

F + r O → (1 + r) P (14.1-7)

where r is the air-to-fuel ratio on a mass basis. Denoting the equivalenceratio as φ, where

φ =(air/fuel)actual

(air/fuel)stoichiometric

(14.1-8)

the reaction in Equation 14.1-7, under more general mixture conditions,can then be written as

φ F + r O → (φ+ r) P (14.1-9)

Looking at the left side of this equation, the mixture fraction for thesystem as a whole can then be deduced to be

f =φ

φ+ r(14.1-10)

Equation 14.1-10 is an important result, allowing the computation ofthe mixture fraction at stoichiometric conditions (φ = 1) or at fuel-richconditions (e.g., φ > 2).



Relationship of f to Species Mass Fraction, Density, and Temperature

The power of the mixture fraction modeling approach is that the chem-istry is reduced to one or two conserved mixture fractions. All thermo-chemical scalars (species mass fraction, density, and temperature) areuniquely related to the mixture fraction(s). Given a description of thereacting system chemistry, and certain other restrictions on the system(see Section 14.1.3), the instantaneous mixture fraction value at eachpoint in the flow field can be used to compute the instantaneous valuesof individual species mole fractions, density, and temperature.

If, in addition, the reacting system is adiabatic, the instantaneous val-ues of mass fractions, density, and temperature depend solely on theinstantaneous mixture fraction, f :

φi = φi(f) (14.1-11)

for a single fuel-oxidizer system. If a secondary stream is included, theinstantaneous values will depend on the instantaneous fuel mixture frac-tion, ffuel, and the secondary partial fraction, psec:

φi = φi(ffuel, psec) (14.1-12)

In Equations 14.1-11 and 14.1-12, φi represents the instantaneous speciesmass fraction, density, or temperature. In the case of non-adiabaticsystems, this relationship generalizes to

φi = φi(f,H∗) (14.1-13)

for a single mixture fraction system, where H∗ is the instantaneous en-thalpy (identical to H as defined in Equation 11.2-7):

H∗ =∑j

mjHj =∑j

mj

[∫ T

Tref,j

cp,jdT + h0j (Tref,j)

](14.1-14)

If a secondary stream is included,



φi = φi(ffuel, psec,H∗) (14.1-15)

Examples of non-adiabatic flows include systems with radiation, heattransfer through walls, heat transfer to/from discrete phase particles ordroplets, and multiple inlets at different temperatures. Additional detailabout the mixture fraction approach in such non-adiabatic systems isprovided on page 14-18.

The details of the functional relationship between φi (species mass frac-tion, density, and temperature) and mixture fraction (Equations 14.1-11through 14.1-15) depend on the description of the system chemistry. Youcan choose to describe this relationship using the flame sheet (mixed-is-burned), equilibrium chemistry, or non-equilibrium chemistry (flamelet)model, as described below.

Models Describing the System Chemistry

FLUENT provides three options for description of the system chemistrywhen you use the non-premixed modeling approach. These options are:

• The Flame Sheet Approximation (Mixed-is-Burned): The simplestreaction scheme is the flame sheet or “mixed-is-burned” approx-imation. This approach assumes that the chemistry is infinitelyfast and irreversible, with fuel and oxidant species never coexist-ing in space and complete one-step conversion to final products.This description allows species mass fractions to be determined di-rectly from the given reaction stoichiometry, with no reaction rateor chemical equilibrium information required. This simple systemdescription yields straight line relationships between the speciesmass fractions and the mixture fraction, as shown in Figure 14.1.3.

Because no reaction rate or equilibrium calculations are required,the flame sheet approximation is easily computed and yields a rapidcalculation. However, the flame sheet model is limited to the pre-diction of single-step reactions and cannot predict intermediatespecies formation or dissociation effects. This often results in a se-rious overprediction of peak flame temperature, especially in those



systems that involve very high temperature (e.g., systems usingpre-heat or oxygen-enrichment).

• Equilibrium Assumption: The equilibrium model assumes that thechemistry is rapid enough for chemical equilibrium to always existat the molecular level. An algorithm based on the minimizationof Gibbs free energy [120] is used to compute species mole frac-tions from f . Figure 14.1.4 shows the resulting mole fractions fora reacting system that includes 10 species in the combustion ofmethane in air.

The equilibrium model is powerful since it can predict the forma-tion of intermediate species and it does not require a knowledge ofdetailed chemical kinetic rate data. Instead of defining a specificmulti-step reaction mechanism (see Chapter 13), you simply definethe important chemical species that will be present in the system.FLUENT then predicts the mole fraction of each species based onchemical equilibrium.

FLUENT allows you to restrict the full equilibrium calculation tothose situations in which the instantaneous mixture fraction is be-low a specified rich limit, frich. In fuel-rich regions (e.g., equivalenceratio greater than 1.5 ) when the instantaneous mixture fractionexceeds frich, FLUENT assumes that the combustion reaction isextinguished and that unburned fuel coexists with reacted mate-rial. In such fuel-rich regions the composition at a given value ofmixture fraction is computed from the composition of the limitingmixture (f=frich) and that of the fuel inlet stream (f=1) based ona known stoichiometry. The stoichiometry is either supplied by youor determined automatically from chemical equilibrium at the richlimit (f=frich). This approach, known as the partial equilibriumapproach, allows you to bypass complex equilibrium calculationsin the rich flame region. The latter are time-consuming to com-pute and may not be representative of the real combustion process.When a full equilibrium approach is required, you can simply definethe rich limit as frich = 1.0.

Guidelines on the choice of which species to include in the equi-librium calculation are provided in Section 14.3. The species youinclude must exist in the chemical database accessed by prePDF.



Note that the species included in the equilibrium calculation shouldprobably not include NOx species, as the NOx reaction rates areslow and should not be treated using an equilibrium assumption.Instead, NOx concentration is predicted most accurately using theFLUENT NOx postprocessor where finite rate chemical kinetics areincorporated.

• Non-Equilibrium Chemistry (Flamelet Model): In combustion mod-els where non-equilibrium effects are important, the assumption oflocal chemical equilibrium can lead to unrealistic results. Typ-ical cases in which the equilibrium assumption breaks down aremodeling the rich side of hydrocarbon flames, predicting the inter-mediate species that govern NOx formation, and modeling lift-offand blow-off phenomena in jet flames.

Several approaches are available to overcome these modeling diffi-culties on a case-by-case basis; in FLUENT the partial-equilibrium/rich-limit approximation (described above) can be used to modelthe fuel-rich side of the hydrocarbon flames. Flamelet models havebeen proposed as a more general solution to the problem of mod-erate non-equilibrium flame chemistry. See Section 14.4 for detailsabout the laminar flamelet model in FLUENT.

PDF Modeling of Turbulence-Chemistry Interaction

Equations 14.1-11 through 14.1-15 describe the instantaneous relation-ships between mixture fraction and species mass fraction, density, andtemperature as given by the equilibrium, flamelet, or mixed-is-burnedchemistry model. The FLUENT prediction of the turbulent reacting flow,however, is concerned with prediction of the time-averaged values of thesefluctuating scalars. How these time-averaged values are related to theinstantaneous values depends on the turbulence-chemistry interactionmodel. FLUENT applies the assumed shape probability density function(PDF) approach as its closure model when the non-premixed modelingapproach is used. The PDF closure model is described in this section.



Mas

s F

ract

ion,

Y a

nd E

ntha

lpy,

h

Mixture Fraction, f0 1f st

h

YF

Y O Y

P

Figure 14.1.3: Species Mass Fractions and Enthalpy Derived Using theFlame Sheet Approximation

1.00E+00

8.00E-01

6.00E-01

4.00E-01

2.00E-01

0.00E+00

1.00E+008.00E-016.00E-014.00E-012.00E-010.00E+00

prePDF V4.00

KEY

Fluent Inc.

Mixture Fraction F

Mole Fraction

INSTANTANEOUS SPECIES COMPOSITION

CHEMICAL EQUILIBRIUM

H2CON2H2OCO2O2CH4

Figure 14.1.4: Species Mole Fractions Computed Based on ChemicalEquilibrium



Description of the Probability Density Function

The probability density function, written as p(f), can be thought of asthe fraction of time that the fluid spends at the state f . Figure 14.1.5illustrates this concept. The fluctuating value of f , plotted on the rightside of the figure, spends some fraction of time in the range denoted as∆f . p(f), plotted on the left side of the figure, takes on values such thatthe area under its curve in the band denoted, ∆f , is equal to the fractionof time that f spends in this range. Written mathematically,

p(f) ∆f = limT→∞

1T

∑i

τi (14.1-16)

where T is the time scale and τi is the amount of time that f spendsin the ∆f band. The shape of the function p(f) depends on the natureof the turbulent fluctuations in f . In practice, p(f) is expressed as amathematical function that approximates the PDF shapes that havebeen observed experimentally.

∆

τi

τi

ƒ

ƒ

ƒ

ƒp( )

Figure 14.1.5: Graphical Description of the Probability Density Func-tion, p(f)



Derivation of Mean Scalar Values from the Instantaneous Mixture Fraction

The probability density function p(f), describing the temporal fluctua-tions of f in the turbulent flow, has the very beneficial property that itcan be used to compute time-averaged values of variables that dependon f . Time-averaged values of species mole fractions and temperaturecan be computed (in adiabatic systems) as

φi =∫ 1

0p(f)φi(f)df (14.1-17)

for a single mixture fraction system. When a secondary stream exists,the average values are calculated as

φi =∫ 1

0

∫ 1

0p1(ffuel)p2(psec)φi(ffuel, psec)dffueldpsec (14.1-18)

where p1 is the PDF of ffuel and p2 is the PDF of psec. Here, statis-tical independence of ffuel and psec is assumed, so that p(ffuel, psec) =p1(ffuel)p2(psec).

Similarly, the true time-averaged fluid density, ρ, can be computed as

1ρ

=∫ 1

0

p(f)ρ(f)

df (14.1-19)

for a single mixture fraction system, and

1ρ

=∫ 1

0

∫ 1

0

p1(ffuel)p2(psec)ρ(ffuel, psec)

dffueldpsec (14.1-20)

when a secondary stream exists. ρ(f) or ρ(ffuel, psec) is the instantaneousdensity obtained using the instantaneous species mole fractions and tem-perature in the gas law equation. Equations 14.1-19 and 14.1-20 providea more accurate description of the time-averaged density than the alter-nate approach of applying the gas law using time-averaged species andtemperature.



Using Equations 14.1-17 and 14.1-19 (or Equations 14.1-18 and 14.1-20),it remains only to specify the shape of the function p(f) (or p1(ffuel) andp2(psec)) in order to determine the local time-averaged state of the fluidat all points in the flow field.

The PDF Shape

The shape of the assumed PDF, p(f), is described in FLUENT by one oftwo mathematical functions:

• the double delta function

• the β-function

The double delta function is the most easily computed, while the β-function most closely represents experimentally observed PDFs. Theshape produced by these functions depends solely on the mean mixturefraction, f , and its variance, f ′2. The choice of these functions (andothers, such as the clipped Gaussian) have their basis in experimentalmeasurements of concentration fluctuations [17, 105]. A detailed descrip-tion of each function follows.

The Double Delta Function PDF

The double delta function is given by

p(f) =

0.5, f = f −√f ′2

0.5, f = f +√f ′2

0, elsewhere

(14.1-21)

with suitable bounding near f = 1 and f = 0. One example of thedouble delta function is illustrated in Figure 14.1.6. As noted above, thedouble delta function PDF is very easy to compute but is invariably lessaccurate than the alternate β-function PDF. For this reason, it shouldbe employed only in special circumstances.



0 ff

p(f)

0.5

0

Figure 14.1.6: Example of the Double Delta Function PDF Shape

The β-Function PDF

The β-function PDF shape is given by the following function of f andf ′2:

p(f) =fα−1(1 − f)β−1∫fα−1(1 − f)β−1df

(14.1-22)

where

α = f

[f(1 − f)

f ′2− 1

](14.1-23)

and

β = (1 − f)

[f(1 − f)

f ′2− 1

](14.1-24)

Figures 14.1.7 and 14.1.8 show the form of the β function for two condi-tions of f and f ′2.



Probability Density p

5.53E+00

4.43E+00

3.32E+00

2.21E+00

1.11E+00

0.00E+00

1.00E+008.00E-016.00E-014.00E-012.00E-010.00E+00

prePDF V4.00

KEY

Fluent Inc.

Mixture Fraction F

F-Bar = 3.00E-01, F-Fluc = 5.00E-03.

BETA PROBABILITY DENSITY FUNCTION

Figure 14.1.7: β-Function PDF Shapes for f = 0.3 and f ′2 = 0.005

Probability Density p

1.63E+01

1.30E+01

9.78E+00

6.52E+00

3.26E+00

0.00E+00

1.00E+008.00E-016.00E-014.00E-012.00E-010.00E+00

prePDF V4.00

KEY

Fluent Inc.

Mixture Fraction F

F-Bar = 1.00E-01, F-Fluc = 1.00E-02.

BETA PROBABILITY DENSITY FUNCTION

Figure 14.1.8: β-Function PDF Shapes for f = 0.1 and f ′2 = 0.01



Importantly, the PDF shape p(f) can be computed at all points in theflow in terms of its first two moments, namely mean, f , and variance,f

′2. Thus, given FLUENT’s prediction of f and f ′2 at each point in theflow field (Equations 14.1-4 and 14.1-5), the known PDF shape can becomputed and used as the weighting function to determine the time-averaged mean values of species mass fraction, density, and temperatureusing, Equations 14.1-17 and 14.1-19 (or, for a system with a secondarystream, Equations 14.1-18 and 14.1-20). This logical dependence is de-picted visually in Figure 14.1.9 for a single mixture fraction. (When asecondary stream is included, the PDF shape will be computed for thefuel mixture fraction, ffuel, and the secondary partial fraction, psec, andthe order of the calculations is different, as shown in Figure 14.2.2.)

PDF Shape

Chemistry Model

(ƒ)φ

p(ƒ) = p (ƒ, ƒ¯

Look-up Table φ = φ (ƒ, ƒ )

)2´

2´i i

φ = ∫o

1

p(ƒ) φ (ƒ) dƒii

i

Figure 14.1.9: Logical Dependence of Averaged Scalars φi on f , f ′2, andthe Chemistry Model (Adiabatic, Single-Mixture-Fraction Systems)

Non-Adiabatic Extensions of the Non-Premixed Model

Many reacting systems involve heat transfer to wall boundaries, droplets,and/or particles by convective and radiative heat transfer. In such flowsthe local thermochemical state is no longer related only to f , but also



to the enthalpy H∗. The system enthalpy impacts the chemical equilib-rium calculation and the temperature of the reacted flow. Consequently,changes in enthalpy due to heat loss must be considered when computingscalars from the mixture fraction. Thus, the scalar dependence becomes

φi = φi(f,H∗) (14.1-25)

where H∗ is given by Equation 14.1-14. In such non-adiabatic systems,turbulent fluctuations should be accounted for by means of a joint PDFp(f,H∗). The computation of p(f,H∗) is not practical for most engineer-ing applications, however. The problem can be simplified significantly byassuming that the enthalpy fluctuations are independent of the enthalpylevel (i.e., heat losses do not significantly impact the turbulent enthalpyfluctuations). When this is assumed, we again have p = p(f) and

φi =∫ 1

0φi(f,H∗)p(f)df (14.1-26)

Determination of φi in the non-adiabatic system thus requires solutionof the modeled transport equation for time-averaged enthalpy:

∂

∂t(ρH∗) + ∇ · (ρ~vH∗) = ∇ ·

(kt

cp∇H∗

)+ Sh (14.1-27)

where Sh accounts for source terms due to radiation, heat transfer to wallboundaries, and heat exchange with the second phase. Figure 14.1.10depicts the logical dependence of mean scalar values (species mass frac-tion, density, and temperature) on FLUENT’s prediction of f , f ′2, andH∗ in non-adiabatic single-mixture-fraction systems.

When a secondary stream is included, the scalar dependence becomes

φi = φi(ffuel, psec,H∗) (14.1-28)

and the mean values are calculated from



PDF Shape

Chemistry Model

(ƒ, H )φ

p(ƒ) = p (ƒ, ƒ¯

Look-up Table φ = φ (ƒ, ƒ )

)2´

2´i i

φ = ∫o

1

p(ƒ) φ (ƒ,H )dƒii

i *

*

,H *

Figure 14.1.10: Logical Dependence of Averaged Scalars φi on f , f ′2,H∗, and the Chemistry Model (Non-Adiabatic, Single-Mixture-FractionSystems)

φi =∫ 1

0

∫ 1

0φi(ffuel, psec,H∗)p1(ffuel)p2(psec)dffueldpsec (14.1-29)

As noted above, the non-adiabatic extensions to the PDF model are re-quired in systems involving heat transfer to walls and in systems withradiation included. In addition, the non-adiabatic model is required insystems that include multiple fuel or oxidizer inlets with different inlettemperatures or that include flue gas recycle. Finally, the non-adiabaticmodel is required in particle-laden flows (e.g., liquid fuel systems orcoal combustion systems) since such flows include heat transfer to thedispersed phase. Figure 14.1.11 illustrates several systems that mustinclude the non-adiabatic form of the PDF model. Note that even ifyour system is non-adiabatic, you may want to perform the much sim-pler adiabatic calculation as an initial exercise. This will allow you tobound the non-adiabatic analysis in an efficient manner, as described inSection 14.3.



f = 1

f = 0

Fuel

Oxidant

Q or Qwall radiation

(c) Dispersed Phase Heat or Mass Transfer (e.g.,Liquid Fuel or Coal Combustion)

(b) Multiple Fuel or Oxidant Inlets at Different Temperatures

(a) Heat Transfer to Domain Boundaries and/orRadiation Heat Transfer

Oxidant

Oxidant

Fuel

T = T

T = T1

2

Oxidant Liquid Fuel orPulverized Coal

Figure 14.1.11: Reacting Systems Requiring Non-Adiabatic Non-Premixed Model Approach



14.1.3 Restrictions and Special Cases for the Non-PremixedModel

Restrictions on the Mixture Fraction Approach

The unique dependence of φi (species mass fractions, density, or tem-perature) on f (Equation 14.1-11 or 14.1-13) requires that the reactingsystem meet the following conditions:

• The chemical system must be of the diffusion type with discrete fueland oxidizer inlets (spray combustion and pulverized fuel flamesmay also fall into this category).

• The Lewis number must be unity. (This implies that the diffusioncoefficients for all species and enthalpy are equal, a good approxi-mation in turbulent flow).

• When a single mixture fraction is used, the following conditionsmust be met:

– Only one type of fuel is involved. The fuel may be made up ofa burnt mixture of reacting species (e.g., 90% CH4 and 10%CO) and you may include multiple fuel inlets. The multiplefuel inlets must have the same composition, however. Two ormore fuel inlets with different fuel composition are not allowed(e.g., one inlet of CH4 and one inlet of CO). Similarly, in spraycombustion systems or in systems involving reacting particles,only one off-gas is permitted.

– Only one type of oxidizer is involved. The oxidizer may consistof a mixture of species (e.g., 21% O2 and 79% N2) and youmay have multiple oxidizer inlets. The multiple oxidizer inletsmust, however, have the same composition. Two or moreoxidizer inlets with different composition are not allowed (e.g.,one inlet of air and a second inlet of pure oxygen).

• When two mixture fractions are used, three streams can be involvedin the system. Valid systems are as follows:



– Two fuel streams with different compositions and one oxidizerstream. Each fuel stream may be made up of a mixture of re-acting species (e.g., 90% CH4 and 10% CO). You may includemultiple inlets of each fuel stream, but each fuel inlet musthave one of the two defined compositions (e.g., one inlet ofCH4 and one inlet of CO).

– Mixed fuel systems including gas-liquid, gas-coal, orliquid-coal fuel mixtures with a single oxidizer. In systemswith a gas-coal or liquid-coal fuel mixture, the coal volatilesand char are treated as a single composite fuel stream.

– Coal combustion in which volatiles and char are tracked sep-arately.

– Two oxidizer streams with different compositions and one fuelstream. Each oxidizer stream may consist of a mixture ofspecies (e.g. 21% O2 and 79% N2). You may have multipleinlets of each oxidizer stream, but each oxidizer inlet musthave one of the two defined compositions (e.g., one inlet of airand a second inlet of pure oxygen).

– A fuel stream, an oxidizer stream, and a non-reacting sec-ondary stream.

• The flow must be turbulent.

It is important to emphasize that these restrictions eliminate the useof the non-premixed approach for directly modeling premixed combus-tion. This is because the unburned premixed stream is far from chemicalequilibrium. Note, however, that an extended mixture fraction formula-tion, described in Chapter 16, can be applied to premixed and partiallypremixed flames.

Figures 14.1.12 and 14.1.13 illustrate typical reacting system configu-rations that can be handled by the non-premixed model in FLUENT.Figure 14.1.14 shows a premixed configuration that cannot be modeledusing the non-premixed model.



60% CH4

40% COf = 1

f = 021% O 79% N 2

2

60% CH4

40% COf = 1

f = 02

2

35% O65% N

2

35% O

65% N

2

f = 0

f = 1

f = 0

f = 1

21% O79% N

2

2

60% CH420% CO

3 810% C H10% CO2

60% CH 420% CO

3 810% C H10% CO2

(a) Simple Fuel/Oxidant Diffusion Flame

(b) Diffusion System Using Multiple Oxidant Inlets

(c) System Using Multiple Fuel Inlets

Figure 14.1.12: Chemical Systems That Can Be Modeled Using a SingleMixture Fraction



CH /CO/C H4 3 8

Oxidant

CH /C H4 3 8

21% O2

Fuel

35% O2

(a) System Containing Two Dissimilar Fuel Inlets

(b) System Containing Two Dissimilar Oxidant Inlets

Figure 14.1.13: Chemical System Configurations That Can Be ModeledUsing Two Mixture Fractions



O2

N2

CH4

Figure 14.1.14: Premixed Systems CANNOT Be Modeled Using theNon-Premixed Model

Using the Non-Premixed Model for Liquid Fuel or CoalCombustion

You can use the non-premixed model if your FLUENT simulation includesliquid droplets and/or coal particles. In this case, fuel enters the gasphase within the computational domain at a rate determined by theevaporation, devolatilization, and char combustion laws governing thedispersed phase. In the case of coal, the volatiles and the products ofchar can be defined as two different types of fuel (using two mixturefractions) or as a single composite off-gas (using one mixture fraction),as described in Section 14.3.5.

Using the Non-Premixed Model with Flue Gas Recycle

While most problems you solve using the non-premixed model will in-volve inlets that contain either pure oxidant or pure fuel (f = 0 or 1),you can include an inlet that has an intermediate value of mixture frac-tion (0 < f < 1) provided that this inlet represents a completely reactedmixture. Such cases arise when there is flue gas recirculation, as de-picted schematically in Figure 14.1.15. Since f is a conserved quantity,the mixture fraction at the flue gas recycle inlet can be computed as

mfuel + mrecycfexit = (mfuel + mox + mrecyc)fexit (14.1-30)

or



fexit =mfuel

mfuel + mox(14.1-31)

where fexit is the exit mixture fraction (and the mixture fraction at theflue gas recycle inlet), mox is the mass flow rate of the oxidizer inlet,mfuel is the mass flow rate of the fuel inlet, mrecyc is the mass flow rateof the recycle inlet.

If a secondary stream is included,

ffuel,exit =mfuel

mfuel + msec + mox(14.1-32)

and

psec,exit =msec

msec + mox(14.1-33)

f = 1

f = 0m.

m.

F

O

m.R

fexit

fexit

Figure 14.1.15: Using the Non-Premixed Model with Flue Gas Recycle



14.2 Modeling Approaches for Non-Premixed EquilibriumChemistry

The FLUENT software package offers two different ways to model non-premixed equilibrium chemistry. You can choose either a single- or two-mixture-fraction approach depending on how many streams you have.prePDF stores information about the streams in “look-up tables”, whichare then used by FLUENT to solve for the mixture fraction, enthalpy, andscalar quantities. For more information about prePDF, see Section 14.3.

14.2.1 Single-Mixture-Fraction Approach

To keep computation time to a minimum, much of the calculation re-quired for the non-premixed model is performed outside of the FLUENTsimulation by preprocessing the chemistry calculations and PDF integra-tions in a separate code, called prePDF. Figure 14.2.1 illustrates how thecomputational effort is divided between the preprocessor (prePDF) andthe solver (FLUENT). In prePDF, the chemistry model (mixed-is-burned,equilibrium chemistry, or laminar flamelet) is used in conjunction withthe assumed shape of the PDF to perform the integrations given in Equa-tions 14.1-17, 14.1-19, and/or 14.1-26. These integrations are performedwithin prePDF and stored in look-up tables that relate the mean thermo-chemical variables φi (temperature, density and species mass fractions)to the values of f , f ′2

s , and H∗. Note that the scaled mixture fractionvariance is used for tabulation, where f ′2

s is defined as

f ′2s =

f ′2

0.25f (1 − f)(14.2-1)

Equations 14.1-4, 14.1-5, and 14.1-27 (for non-adiabatic systems) aresolved in FLUENT to obtain local values of f , f ′2, and H∗.

14.2.2 Two-Mixture-Fraction Approach

For the two-mixture-fraction (secondary stream) case, the preprocessorprePDF calculates the instantaneous values for the temperature, density,and species mass fractions (Equation 14.1-12 or 14.1-15) and stores themin the look-up tables. For the adiabatic case with two mixture fractions,


14.2 Modeling Approaches for Non-Premixed Equilibrium Chemistry

prePDF:

Integration: φi o

= ∫1

φ ( , H*)dƒ

Look-up Table

i i

FLUENT:

φi2. Look up Scalars

p(ƒ )

1. Solve , H*,

ƒ

ƒ ƒ

i

φ = φ (ƒ ƒ H*)s, ,

′2

2′

Figure 14.2.1: Separation of Computational Tasks Between FLUENT andprePDF for a Single-Mixture-Fraction Case



the look-up tables contain ρ, T , and Yi as functions of the fuel mixturefraction and the secondary partial fraction. For the non-adiabatic casewith two mixture fractions, the 3D look-up table contains the physicalproperties as functions of the fuel mixture fraction, the secondary partialfraction, and the instantaneous enthalpy.

The PDFs p1 and p2 of the fuel mixture fraction and the secondary par-tial fraction, respectively, are calculated inside FLUENT from the valuesof the solved mixture fractions and their variances. The PDF integra-tions for calculating the mean values for the properties are also performedinside FLUENT (using Equation 14.1-18 or 14.1-29, together with Equa-tion 14.1-20 or its non-adiabatic equivalent). The instantaneous valuesrequired in the integrations are obtained from the look-up tables.

Note that the computation time in FLUENT for a two-mixture-fraction!case will be much greater than for a single-mixture-fraction problem sincethe PDF integrations are being performed in FLUENT rather than inprePDF. This expense should be carefully considered before choosing thetwo-mixture-fraction model. Also, it is usually expedient to start a two-mixture-fraction simulation from a converged single-mixture-fraction so-lution.

Figure 14.2.2 illustrates the division of labor between prePDF and FLU-ENT for the two-mixture-fraction case.

14.2.3 The Look-Up Table Concept

Look-Up Tables for Adiabatic Systems

Figure 14.2.3 illustrates the concept of the look-up tables generated byprePDF for a single-mixture-fraction system. Given FLUENT’s predictedvalue for f and f ′2 at a point in the flow domain, the time-averagedmean value of mass fractions, density, or temperature (φi) at that pointcan be obtained from the table. FLUENT first uses Equation 14.2-1to compute the scaled mixture fraction variance f ′2

s because the single-mixture-fraction look-up tables contain property data as a function of fand f ′2

s , rather than f and f ′2.

The table, Figure 14.2.3, is the mathematical result of the integration ofEquation 14.1-17. There is one look-up table of this type for each scalar



prePDF:

Calculation:

Look-up Table

FLUENT:

1. Solve ƒ ,ƒ ,p ,p ,H*

φi

= φ (ƒ ,p ,H )i fuel sec

*

φi

= φ (ƒ ,p ,H )i fuel sec

*

´2

fuel secfuel sec´2

i2. Look up scalars φ3. Compute p (ƒ ) fuel1

4. Compute p (p ) sec2

5. Compute φ i

Figure 14.2.2: Separation of Computational Tasks Between FLUENT andprePDF for a Two-Mixture-Fraction Case



of interest (species mass fractions, density, temperature). In adiabaticsystems, where the instantaneous enthalpy is a function only of the in-stantaneous mixture fraction, a two-dimensional look-up table, like thatin Figure 14.2.3, is all that is required.

Scalar Value

Mean Mixture Fraction

Scaled Variance

Figure 14.2.3: Visual Representation of a Look-Up Table for the Scalar φi

as a Function of f and f ′2 in Adiabatic Single-Mixture-Fraction Systems

For a system with two mixture fractions, there will be a look-up table foreach instantaneous scalar property φi as a function of the fuel mixturefraction ffuel and the secondary partial fraction psec (Equation 14.1-12),as shown in Figure 14.2.4.

The look-up table structure is summarized in Table 14.2.1.

3D Look-Up Tables for Non-Adiabatic Systems

In non-adiabatic systems, where the enthalpy is not linearly related tothe mixture fraction, but depends also on wall heat transfer and/or radi-ation, a look-up table is required for each possible enthalpy value in thesystem. The result is a three-dimensional look-up table, as illustratedin Figure 14.2.5, which consists of layers of two-dimensional tables, eachone corresponding to the normalized heat loss or gain. The first layer orslice corresponds to the maximum heat loss for the system, where all the



Instantaneous Scalar

Value

FuelMixtureFraction

SecondaryPartial Fraction

Figure 14.2.4: Visual Representation of a Look-Up Table for the Scalarφi as a Function of ffuel and psec in Adiabatic Two-Mixture-FractionSystems

points in the look-up table are at the minimum temperature defined inthe problem setup. The maximum slice corresponds to the heat gain thatoccurs when all points have reached the maximum temperature defined.The zero heat loss/gain slice corresponds to adiabatic operation. Slicesinterpolated between the adiabatic and maximum slices correspond toheat gain, and those interpolated between the adiabatic and minimumslices correspond to heat loss.

The three-dimensional look-up table allows FLUENT to determine thevalue of each mass fraction, density, and temperature from calculatedvalues of f , f ′2, and H∗. This three-dimensional table is the visualrepresentation of the integral in Equation 14.1-26.

For two-mixture-fraction problems, the 3D look-up table allows FLUENTto determine the instantaneous values for the scalar properties from in-stantaneous values of ffuel, psec, and H∗. The three-dimensional tableis the visual representation of Equation 14.1-28. These instantaneousvalues are used to perform the integration of Equation 14.1-29.

See Table 14.2.1 for a summary of the look-up table structure.



Scalar Value

Mean Mixture Fraction

Scaled Variance

n+1

n-1heat loss/gain

heat loss/gain

heat loss/gain

n

normalized

normalized

normalized

Figure 14.2.5: Visual Representation of a Look-Up Table for the Scalarφi as a Function of f and f ′2 and Normalized Heat Loss/Gain in Non-Adiabatic Single-Mixture-Fraction Systems



Instantaneous Scalar Value

Fuel MixtureFraction

SecondaryPartialFraction

normalizedheat loss/gain

n-1

normalizedheat loss/gainn

normalizedheat loss/gain

n+1

Figure 14.2.6: Visual Representation of a Look-Up Table for the Scalarφi as a Function of ffuel, psec, and Normalized Heat Loss/Gain in Non-Adiabatic Two-Mixture-Fraction Systems



Summary of Look-Up Table Formats

Table 14.2.1 summarizes the look-up table format for different types ofnon-premixed models.

Table 14.2.1: Look-Up Table FormatsType of Model Adiabatic Non-Adiabaticsingle mixture fraction f , f ′2

s f , f ′2s , H∗

two mixture fractions ffuel, psec ffuel, psec, H∗

14.3 User Inputs for the Non-Premixed EquilibriumModel

A description of the user inputs (in prePDF and FLUENT) for the non-premixed equilibrium model is provided in the following sections:

• Section 14.3.1: Problem Definition Procedure in prePDF

• Section 14.3.2: Informational Messages and Errors Reported byprePDF

• Section 14.3.3: Non-Premixed Model Input and Solution Proce-dures in FLUENT

• Section 14.3.4: Modeling Liquid Fuel Combustion

• Section 14.3.5: Modeling Coal Combustion

14.3.1 Problem Definition Procedure in prePDF

As illustrated in Figures 14.2.1 and 14.2.2, the solution of chemicallyreacting flows using the non-premixed equilibrium approach begins withthe problem definition in prePDF.

For a single-mixture-fraction problem, you will perform the followingsteps in prePDF:


14.3 User Inputs for the Non-Premixed Equilibrium Model

1. Define the chemical species to be considered in the reacting sys-tem model and choose the chemical description of the system. Thedefault equilibrium chemistry option should invariably be used. Ifyou are modeling a mixing problem without reaction, or if a solu-tion cannot be obtained with the equilibrium option and alternatespecies components are not acceptable, you may want to use theinfinitely fast chemistry option.

2. Indicate whether or not the problem is adiabatic.

3. Choose the PDF (probability density function) shape that will beused to describe the turbulent fluctuations in the mixture fraction.The default Beta PDF should be selected unless you have a specialreason to use the double-delta PDF.

4. Compute the look-up table, containing mean (time-averaged) val-ues of species mass fractions, density, and temperature as a func-tion of mean mixture fraction, mixture fraction variance, and en-thalpy. The contents of this look-up table will reflect your preced-ing inputs describing the turbulent reacting system.

The look-up table is the output of prePDF. It is the stored result of theintegration of Equations 14.1-17 (or 14.1-26) and 14.1-19. The look-uptable will be used in FLUENT to determine mean species mass fractions,density, and temperature from the values of mixture fraction (f), mix-ture fraction variance (f ′2), and enthalpy (H∗) as they are computedduring the FLUENT calculation of the reacting flow. See Section 14.2and Figures 14.2.3 and 14.2.5.

For a problem that includes a secondary stream (and, therefore, a secondmixture fraction), you will perform the first three steps listed above forthe single-mixture-fraction approach, and then prepare a look-up tableof instantaneous properties using Equation 14.1-12 or 14.1-15.

The following step-by-step procedure explains how to use prePDF, takingyou through the problem definition procedure and explaining how yourinputs are used.



Step 1: Start prePDF

The way you start prePDF will be different for UNIX and Windows sys-tems. The installation process (described in the separate installation in-structions for your computer type) is designed to ensure that the prePDFprogram is launched when you follow the appropriate instructions. If itis not, consult your computer systems manager or your Fluent supportengineer.

Starting prePDF on a UNIX System

On a UNIX machine, type

prepdf

at the command prompt.

Starting prePDF on a Windows System

For a Windows system, there are two ways to start prePDF:

• Click on the Start button, select the Programs menu, select theFluent.Incmenu, and then select the prePDF program item. (Notethat if the default Fluent.Inc program group name was changedwhen prePDF was installed, you will find the prePDF menu item inthe program group with the new name that was assigned, ratherthan in the Fluent.Inc program group.)

• Start from an MS-DOS Command Prompt window by typing prepdfat the prompt. Before doing so, however, you must first modifyyour user environment so that the MS-DOS Command utility willfind prepdf. You can do this by selecting the program item “SetEnvironment”, which is also found in the Fluent.Inc programgroup. This program will add the Fluent.Inc directory to yourcommand search path.



The Thermodynamic Database

prePDF uses a thermodynamic database [112] and must be able to accessthe database file, THERMO.DB. This file must be present in the directorywhere you run prePDF, or it must be accessed through an environmentvariable, THERMODB, which points to the location of this file. In mostinstallations, you will be running prePDF using procedures supplied byFluent Inc. and these procedures will set this environment variable foryou.

Step 2: Allocate Memory

The amount of memory used by prePDF cannot be changed once it hasbeen allocated, without restarting the application. You must, therefore,take care to allocate enough memory for the maximum number of pointsin the PDF table, species, and flamelets that you are planning to use.

The parameters for which you can modify memory allocation are asfollows:

Maximum Number of Species is the maximum number of species in thePDF table. The default value is 20, and this parameter can beincreased up to a value of 65.

Maximum Number of f-mean Points is the maximum number of mixturefraction points in the PDF table. The default value is 45, and thisparameter can be increased up to a value of 100.

Maximum Number of f-var Points is the maximum number of mixture frac-tion variance points in the PDF table. The default value is 22, andthis parameter can be increased up to a value of 30.

Maximum Number of Enthalpy Points is the maximum number of enthalpypoints in the PDF table. The default value is 45, and this param-eter can be increased up to a value of 100.

Maximum Number of Scalar Dissipation Points in Adiabatic Flamelet PDFTable has a default value of 45, and can be increased up to a valueof 100.



Maximum Number of Flamelets is the maximum number of flamelets inthe flamelet model. The default value is 20, and this parametercan be increased up to a value of 30.

You can set these parameters in the Memory Allocation panel (Figure 14.3.1).

Setup −→Memory Allocation...

Figure 14.3.1: The Memory Allocation Panel in prePDF

When you click Apply, memory will be allocated for these parameters.If you need to allocate more memory later in the setup process, youwill need to save an input file, exit prePDF, and restart. Then, allocatethe appropriate amount of memory, read the input file into prePDF, andcontinue the problem setup.

Note that, if you read an input file or PDF file without first allocatingmemory, prePDF will allocate memory based on the number of speciesand points specified in the file. If the number is less than the defaultallocation, the default memory will be allocated. If it is more, memoryadequate for the number of species and points in the file will be allocated.



Step 3: Initialize the Problem Definition

Once you have allocated memory, your first task in prePDF is to definethe type of reaction system and reaction model that you intend to use.This includes selection of the following options:

• Addition of a secondary stream

• Partially premixed model option (see Chapter 16)

• Adiabatic or non-adiabatic modeling options (see Section 14.1.2)

• Equilibrium chemistry model or stoichiometric reaction (mixed-is-burned) model (see Section 14.1.2)

Procedures for setting up flamelet PDF models are described in!Section 14.4.6.

• Beta PDF or Double-Delta PDF (see Section 14.1.2)

• Empirically defined fuel and/or secondary stream composition

Your subsequent inputs and the inputs that prePDF will expect from youdepend on these choices.

You can make these model selections using the Define Case panel (Fig-ure 14.3.2).

Setup −→ Case...

Each of these modeling choices is described in detail below. Be sure toclick Apply after completing your inputs.

Enabling a Secondary Inlet Stream

If you are modeling a system consisting of a single fuel and a singleoxidizer stream, you do not need to enable a secondary stream in yourPDF calculation. As discussed in Section 14.1.2, a secondary streamshould be enabled if your PDF reaction model will include any of thefollowing:



Figure 14.3.2: The Define Case Panel in prePDF



• two dissimilar gaseous fuel streams: In these simulations, the fuelstream defines one of the fuels and the secondary stream definesthe second fuel.

• mixed fuel systems of dissimilar gaseous and liquid fuel: In thesesimulations, the fuel stream defines the gaseous fuel and the sec-ondary stream defines the liquid fuel (or vice versa).

• mixed fuel systems of dissimilar gaseous and coal fuels: In thesesimulations, the fuel stream must define the coal and the secondarystream must define the gaseous fuel. See Section 14.3.5 regardingcoal combustion simulations with the non-premixed combustionmodel.

• mixed fuel systems of coal and liquid fuel: In these simulations, thefuel stream must define the coal and the secondary stream mustdefine the liquid fuel. See Section 14.3.5 regarding coal combustionsimulations with the non-premixed combustion model.

• coal combustion: Coal combustion can be more accurately modeledby using a secondary stream. The fuel stream must define the charand the secondary stream must define the volatile components ofthe coal. See Section 14.3.5 regarding coal combustion simulationswith the non-premixed combustion model.

• a single fuel with two dissimilar oxidizer streams: In these simula-tions, the fuel stream defines the fuel, the oxidizer stream definesone of the oxidizers, and the secondary stream defines the secondoxidizer.

Using a secondary stream substantially increases the calculation time!for your simulation since the multi-dimensional PDF integrations areperformed in FLUENT at run-time.

Choosing Adiabatic or Non-Adiabatic Options

You should use the non-adiabatic modeling option if your problem defi-nition in FLUENT will include one or more of the following:



• radiation or wall heat transfer

• multiple fuel inlets at different temperatures

• multiple oxidant inlets at different temperatures

• flue gas recycle

• liquid fuel, coal particles, and/or heat transfer to inert particles

Note that the adiabatic model is a simpler model involving a two-dimen-sional look-up table in which scalars depend only on f and f ′2 (or onffuel and psec). If your model is defined as adiabatic, you will not needto solve the enthalpy equation in FLUENT and the system temperaturewill be determined directly from the mixture fraction and the fuel andoxidant inlet temperatures. The non-adiabatic case will be more complexand more time-consuming to compute, requiring the generation of three-dimensional look-up tables in prePDF. However, the non-adiabatic modeloption allows you to include the types of reacting systems describedabove.

Select Adiabatic or Non-Adiabatic under Heat transfer options in the DefineCase panel.

Using the Adiabatic Calculation to Determine Inputs to the Non-AdiabaticModel

Even if your FLUENT model will ultimately need to include non-adiabaticeffects, you may benefit from starting the analysis with an adiabaticcalculation in prePDF. This simpler calculation can be used to fine-tunethe inputs to the non-adiabatic model, identifying the peak temperaturethat needs to be considered in the non-adiabatic case and identifyingwhich chemical species are significant and need to be considered. Thusthe adiabatic prePDF calculation provides you with an understanding ofthe system that can help you to develop an efficient non-adiabatic model.

Choosing the Chemistry Model

In general, the equilibrium chemistry model is recommended over the sto-ichiometric reaction model. In this approach the concentration of species



of interest are determined from the mixture fraction using the assump-tion of chemical equilibrium (see Section 14.1.2). With this model, youcan include the effects of intermediate species and dissociation reactions,producing more realistic predictions of flame temperatures in combus-tion models. In contrast, the stoichiometric reaction (mixed-is-burned)modeling option provides a less accurate single-step description of thesystem chemistry. When you choose the equilibrium chemistry option,you will have the opportunity, in Step 8, to use a “partial equilibrium”model.

Select Equilibrium Chemistry or Stoichiometric Reaction under Chemistrymodels in the Define Case panel.

If you want to model non-equilibrium chemistry, you should use the!flamelet modeling approach described in Section 14.4. Procedures forusing this model is presented in Section 14.4.6.

Choosing the Chemistry Model for Non-Reacting Systems

If you are using the non-premixed combustion model to consider a non-reacting system, choose the Stoichiometric Reaction option in the DefineCase panel. When you are prompted to input the stoichiometric coeffi-cients for each species (Step 7, below), simply input zeros for all species.

Choosing the PDF Shape

The shape of the PDF you select will have some impact on the resultsyou obtain. In general, the default β-function PDF shape matches ex-perimental observations of f fluctuations much better than the double-delta function, and should be the one used. The double-delta function,on the other hand, is more efficient computationally during the gener-ation of look-up tables in prePDF. Since the look-up table generationis a pre-processing step, the double-delta PDF should be used only inspecial circumstances. When a secondary stream is included, you willnot choose the PDF type in prePDF. This step will occur in FLUENTinstead.

Select Delta PDF or Beta PDF under PDF models in the Define Casepanel.



Empirical Definition of the Fuel Stream(s)

The empirical definition option provides an alternative method for defin-ing the composition of the fuel or secondary stream. When you do notselect this option, you will define which chemical species are present ineach stream and the mass or mole fraction of each species, as describedbelow, in Step 5. When an empirically defined fuel or secondary streamis used, you will follow a different procedure for determining the chemicalcomposition of the stream:

1. Define the list of chemical species present in the stream (followingthe suggestions below, in Step 4), using the Define Species panel.The choice of species is not altered by your use of the empiricaldefinition option, except that you must also include atomic ele-ments (C, H, N, S, and O) and combustion products (CO2 andH2O) which are used for the calculation of the lower heating valueof the fuel.

2. In the Composition panel (Step 5, below), you will not input mass ormole fractions for each species. Instead, you will input the atomiccomposition of the stream (atom mole fractions of C, H, N, S, andO), its lower heating value, and its mean specific heat. prePDFwill compute the mole fraction of each chemical species from theseinputs.

The heat of formation of an empirical fuel stream is calculated from theheating value and the atomic composition. The fuel inlet temperatureand fuel specific heat are used to calculate the sensible enthalpy. prePDFperforms an equilibrium calculation, using the atomic composition andenthalpy, and returns the equilibrium molar species composition andtemperature of the fuel.

Note that the empirical definition option is available only with the fullequilibrium chemistry model. It cannot be used with the stoichiometricor partial equilibrium models, since equilibrium calculations are requiredfor the determination of the fuel composition. If your empirically definedfuel is a gaseous fuel, you should be aware of the modeling issues related



to gas-phase fuel inlet temperature in full equilibrium systems (see Step8, below).

The option for defining an empirical fuel stream is particularly usefulfor coal combustion simulations (see Section 14.3.5) or for simulationsinvolving other complex hydrocarbon mixtures. Because empirical fu-els use the full equilibrium model, which impacts your flow boundaryconditions at gas-phase fuel inlets, the empirical definition option is notgenerally recommended for gaseous fuels.

Turn on Fuel stream or Secondary stream under Empirically Defined Streamsin the Define Case panel to define a fuel or secondary stream empirically.

Step 4: Define the Chemical Species to be Considered

One of your most important modeling inputs will be the selection ofspecies to be included in the description of the reacting system. Allspecies that you include must exist in the chemical database and youmust enter their names in the same format used in the database. Youcan include a maximum of the number of species you entered in the Mem-ory Allocation panel (see Step 2 above) when you use the non-premixedmodel. (Note the additional requirements mentioned below for definingspecies when you have an empirically defined fuel stream.)

The number of species and their names are entered using the DefineSpecies panel (Figure 14.3.3).

Setup −→ Species −→ Define...

The steps for defining your species are as follows:

1. Specify the number of species you will define in the Maximum # ofSpecies field. (You can change the maximum number of species atany time by incrementing the counter.)

2. Define your first species by selecting it in the Database Species drop-down list. This list contains a complete listing of the species in thedatabase. The species name will appear in the Defined Species list.

3. Increase the Species # field (either use the counter arrow or typein the new value and press <RETURN>) and select the next species



Figure 14.3.3: The Define Species Panel in prePDF

from the Database Species list. Continue in this manner until allof the species you want to include are shown in the Defined Specieslist.

4. When you are satisfied with your selections, click Apply and closethe panel.

If you need to alter a species selection, click on the species name inthe Defined Species list and then select a new species from the DatabaseSpecies drop-down list.

Choosing Species for Empirically Defined Streams

For an empirically defined fuel or secondary stream, you must choosethe constituent elements in addition to the species that make up thechemical system. The elements allowed are C, H, N, S, and O. Also, the



species CO2 and H2O must be selected since they are the combustionproducts for the calculation of the lower caloric (heating) value of thefuel. If you are considering S (sulfur), you will also need to add SO2 inthe species list.

Including Solid and Liquid Species

Solid and liquid species can be included in the thermodynamic calcula-tions as well as gases. They are indicated by an L or an S in parenthesesafter the species name. If you select a solid or liquid species, you mustdefine the species density in the Condensed Species Densities panel (Fig-ure 14.3.4).

Setup −→ Species −→Density...

Figure 14.3.4: The Condensed Species Densities Panel in prePDF

In the panel, choose each solid or liquid species in the Defined Species listand enter its Density. Note that this density should be the density of thecondensed phase species and not the apparent density of the particles asdefined in FLUENT. For example, in coal combustion, you should enterthe density of C(s) and not the apparent density of the coal. When youhave set the density for all solid and liquid species, click Apply and close



the panel.

Guidelines on Choosing the Species to Include

In the combustion of simple hydrocarbons, many species and radicalshave been identified. Although you could, in principle, define an equi-librium system that includes a large number of species, you should limityour system description to those species that are of the most significance.The following suggestions may be helpful in the definition of the systemchemistry:

• In high-temperature flames (i.e., T > 2000 K) include radicals suchas H, O, and OH. These radicals are produced in dissociation re-actions at high temperatures and can have a significant impact onpeak flame temperatures.

• For heavy hydrocarbon fuels (e.g., fuel oils), lighter hydrocarbons(CH4, C2H4, etc.) will be formed in the rich mixture as a result ofpyrolysis and gasification reactions.

• For coal combustion, volatiles may be represented by a mixture ofCH4 (or a heavier hydrocarbon) and CO. Char in the coal should berepresented by C(s). Follow the other general guidelines outlinedhere to determine the other species that should be included in thecombustion system.

• If you want to consider the water content in a coal fuel, includeH2O(l). Include the liquid water content as part of the fuel compo-sition. Alternately, the water content can be included using vaporphase H2O. (See Section 14.3.5 for additional information.)

• If soot formation is of interest, C(s) can be included in the fuelstream definition. However, you should note that the equilibriummodel will not represent the complex finite rate chemistry whichis usually associated with soot formation.

Care should be taken to distinguish atomic carbon, C, from solid!carbon, C(s). Atomic carbon should be selected only if you areusing the empirically-defined input method.



• Combustion products should always include CO2 and H2O. In ad-dition, you may want to include CO and H2. Note that H2 shouldnot be included alone as it is produced in the water-gas shift reac-tion, CO + H2O −→ CO2 + H2.

• If your fuel composition is known empirically (e.g., C0.9H3O0.2),use the option for an empirically-defined stream (see Step 3).

• For hydrocarbon combustion systems, it is recommended that youinclude C(s) and H2O(l).

• If you wish to include the sulfur that may be present in a hydrocar-bon fuel, note that this may hinder the convergence of the equilib-rium solver, especially if the concentration of sulfur is small. It istherefore recommended that you include sulfur in the calculationonly if it is present in considerable quantities.

The simplest way to model sulfur is to represent it as SO2 andS(l), where SO2 will be formed in oxygen-rich mixtures and S(l)will be formed in fuel-rich mixtures. A more elaborate descriptionof a sulfur-containing fuel-oxidizer system could include SO2, H2S,COS, S(l), CS2, and S2.

It is extremely important that your choice of species provide a sensibledescription of the system chemistry. If this is not the case, the equilib-rium calculation may fail to converge or may produce incorrect results.

The species included in the equilibrium calculation should probably notinclude NOx species, as the NOx reaction rates are slow and should notbe treated using an equilibrium assumption. Instead, the NOx concentra-tion is predicted most accurately using the FLUENT NOx postprocessor,where finite-rate kinetics are included. (See Section 17.1.) Note that itis not important to include NOx predictions with the combustion simu-lation since the NOx species are present at low concentrations and havelittle impact on the combustion process.

Modifying the Database

If you want to include a new species in your reacting system that is notavailable in the chemical database, you can add it to the database files,



THERMO.DB (used by prePDF) and thermodb.scm (used by FLUENT).The format for THERMO.DB is detailed in [112]. You can generate thethermodb.scm file in prePDF using the File/Write/Thermodb... menuitem.

File −→ Write −→ Thermodb...

FLUENT will recognize the new species if the thermodb.scm file is inthe directory where FLUENT is started. To permanently store the newspecies in the standard database, copy the new species data from thegenerated thermodb.scm file to the default thermodb.scm file, as de-scribed in Section 14.5. If you choose to modify the standard databasefiles, you should create copies of the original files.

Step 5: Define the Fuel and Oxidizer (and Secondary-Stream)Compositions

After defining the species that will be considered in the reaction system,you must define their mole or mass fractions at the fuel and oxidizerinlets and at the secondary inlet, if one exists. (If you choose to definethe fuel or secondary stream composition empirically, you will insteadenter the parameters described at the end of this step.) For the exampleshown in Figure 14.1.12c, for example, the fuel inlet consists of 60% CH4,20% CO, 10% CO2, and 10% C3H8. This information is input using theComposition panel (Figure 14.3.5).

Setup −→ Species −→ Composition...

The procedure for defining mole or mass fractions is as follows:

1. Under Stream, select the Fuel, Oxidiser, or Secondary option.

2. Under Specify Composition In..., indicate whether you want to enterMole Fractions or Mass Fractions.

3. Select a species from the Defined Species list and then enter its moleor mass fraction in the selected stream (fuel, oxidizer, or secondary)by typing in the Species Fraction field. Repeat this process for allspecies in the Defined Species list until you have set all mole ormass fractions for the selected stream.



Figure 14.3.5: The Composition Panel in prePDF



4. Input the mole or mass fractions for each of the other streams byselecting the appropriate option (i.e., one that you did not choosein step 1) and repeating step 3.

5. When you are satisfied with all the settings, click the Apply buttonand close the panel.

You can check the current setting for a species in a particular streamby selecting the stream and choosing the species name in the DefinedSpecies list.

Un-Normalized Mole or Mass Fraction Inputs

If you input un-normalized mole or mass fractions when you are definingthe compositions, prePDF will scale your inputs so that they sum tounity, and inform you (in an Information dialog box) that the mole ormass fractions will be normalized.

Defining the Fuel Composition for Liquid Fuels

If your FLUENT model considers combustion of fuel that is evaporatedfrom liquid droplets, the composition of the vaporized fuel should bedefined in prePDF.

Defining the Fuel Composition for Coal Combustion

If your FLUENT model involves coal combustion, the fuel and secondarystream compositions can be input in one of several ways. You can use asingle mixture fraction (fuel stream) to represent the coal, defining thefuel composition as a mixture of volatiles and char (solid carbon). Alter-nately, you can use two mixture fractions (fuel and secondary streams),defining the volatiles and char separately. In two-mixture-fraction mod-els for coal combustion, the fuel stream represents the char and thesecondary stream represents volatiles. See Section 14.3.5 for more de-tailed descriptions of modeling options and input procedures for coalcombustion.



Composition Inputs for Empirically-Defined Fuel Streams

As mentioned in Step 3, you can define the composition of a fuel stream(i.e., the standard fuel or a secondary fuel) empirically instead of byspecifying mole or mass fractions. For an empirically-defined stream,you will enter atom fractions, the lower caloric (heating) value of the fuel(with the combustion product assumed to be water vapor and CO2), andthe mean specific heat of the fuel. The steps are as follows:

1. Enable the Fuel (or Secondary) option under Stream.

2. Select each element in the Defined Species list and enter its AtomFraction.

3. Enter the Lower Caloric Value and Specific Heat of the Fuel (orSecondary) stream.

4. When you are satisfied with all the settings, click the Apply buttonand close the panel.

Equilibrium Corrections to Fuel Composition

Note that for the full equilibrium option (i.e., when the fuel rich limitset in Step 8 is 1), prePDF will also perform an equilibrium calculationfor the fuel stream inputs (e.g., at f = 1). As a result, you may find thatprePDF will modify your inputs of fuel composition and temperature ifthe fuel system, as defined by your inputs, is not at equilibrium. Forexample, if you define the fuel as 0.5 CO and 0.5 CH4 at 300 K, prePDFwill correct the fuel to 0.35 C(S), 0.14 CH4, 0.009 CO, 0.1552 CO2, and0.3596 H2 at 751 K, which is the corresponding equilibrium composition.

Equilibrium calculations will always be used when you define your fuelempirically, since only the full equilibrium method is available for suchcases. If you define the fuel using species mole or mass fractions, thecorrection will occur only when the full equilibrium option is used. Ifyour fuel is a liquid or solid (coal) fuel, the equilibrium correction willhave no impact on your model setup.

For gas-phase fuels, the effect of the equilibrium calculation on the fuel!



composition and temperature is an important modeling issue that im-pacts your flow boundary conditions at gas-phase fuel inlets in FLUENT.If you are modeling a gaseous fuel, and you are using the full equilibriummodel or empirical definition of the fuel, you should review the additionalinformation on this topic that is included under Step 8, below.

Step 6: Define Operating Conditions

The thermodynamic operating conditions of your reacting system arerequired for construction of the look-up table and computation of theequilibrium chemistry model. These conditions are input using the Op-erating Conditions panel (Figure 14.3.6).

Setup −→ Operating Conditions...

Each of these inputs is described below. Remember to click the Applybutton when you are done.

Absolute Pressure is used to extract appropriate property data from thedatabase and in the calculation of chemical equilibrium via theminimization of the Gibbs free energy.

Min. Temperature is used to determine the lowest temperature for whichthe look-up tables are generated (see Figure 14.2.5). Your inputshould correspond to the minimum temperature expected in thedomain (e.g., an inlet or wall temperature). The minimum tem-perature should be set 10–20 K below the minimum system temper-ature. This option is available only when the Equilibrium Chemistrymodel is selected in the Define Case panel.

Max. Temperature is used to determine the highest temperature for whichthe look-up tables are generated (see Figure 14.2.5). It should beset to a value about 100 K above the peak temperature predictedby prePDF for the adiabatic system calculation. Note that if yourinput of the peak temperature is too low, prePDF’s calculation ofthe look-up tables will fail. This option is available only when theEquilibrium Chemistry model is selected in the Define Case panel.

Inlet Temperature contains temperature inputs for the fuel, oxidant, andsecondary stream inlets:



Figure 14.3.6: The Operating Conditions Panel in prePDF

Fuel is the temperature of the fuel inlet in your model. In adia-batic simulations, this input (together with the oxidizer inlettemperature) determines the inlet stream temperatures thatwill be used by FLUENT. In non-adiabatic systems, this in-put should match the inlet thermal boundary condition thatyou will use in FLUENT (although you will enter this bound-ary condition again in the FLUENT session). If your FLU-ENT model will use liquid fuel or coal combustion, definethe inlet fuel temperature as the temperature at which va-



porization/devolatilization begins (i.e., the Vaporization Tem-perature specified for the discrete-phase material—see Sec-tion 19.11). For such non-adiabatic systems, the inlet tem-perature will be used in prePDF only to adjust the look-uptable grid (e.g., the discrete enthalpy values for which thelook-up table is computed). Note that if you have more thanone fuel inlet, and these inlets are not at the same tempera-ture, you must define your system as non-adiabatic. In thiscase, you should enter the fuel inlet temperature as the valueat the dominant fuel inlet.

prePDF uses your input of fuel and oxidizer inlet tempera-!tures to determine the fuel and oxidizer enthalpies. When thefull equilibrium model is used (rich limit of 1.0), the equilib-rium calculation performed at f = 1 may result in a modifiedfuel composition and temperature. If you are using the fullequilibrium model for a gaseous fuel (or if you are using anempirically defined gaseous fuel), you should be aware of howthis equilibrium adjustment of the fuel temperature impactsyour fuel inlet boundary conditions in FLUENT. See Step 8,below.

Oxidiser is the temperature of the oxidizer inlet in your model. Theissues raised in the discussion of the input of the fuel inlettemperature (directly above) pertain to this input as well.

Secondary is the temperature of the secondary stream inlet in yourmodel. (This item will appear only when you have defineda secondary inlet.) The issues raised in the discussion of theinput of the fuel inlet temperature (directly above) pertain tothis input as well.

Nonadiabatic Flamelet Temperature Limits contains inputs for the tem-perature limits of a non-adiabatic flamelet system. These inputsare required only if you are using the laminar flamelet model andyou have defined your case as non-adiabatic. See Section 14.4.6 formore information on flamelet inputs.



Step 7: Define the Reaction Stoichiometry

The input of the reaction stoichiometry is required when you choose touse the stoichiometric reaction (mixed-is-burned) chemistry model. Ifyou are using the partial equilibrium approach (rich limit defined) youmay also choose to define the system stoichiometry at the rich limit (seebelow). In either case, your input of reaction stoichiometry defines a sim-ple one-step reaction between fuel species and oxidant species. Consider,for example, the following very simple system:

CH4 + 2O2 → CO2 + 2H2O

prePDF requires that you input the molar stoichiometric coefficients asfollows for this simple system: 1 for CH4, 2 for O2, −1 for CO2, and −2for H2O. Note the convention that the product stoichiometry is enteredwith a negative number.

You can input the coefficients using the Stoichiometric Coefficients panel(Figure 14.3.7).

Setup −→ Species −→ Stoichiometry...

Figure 14.3.7: The Stoichiometric Coefficients Panel in prePDF



Select each species in the Defined Species list and enter its stoichiometriccoefficient in the Coefficient field. When you have set the coefficients forall species, click Apply and close the panel.

Input of Stoichiometry for Fuel Mixtures

If your fuel stream consists of more than one species, you will need toinput the stoichiometry for the composite reaction. Suppose, for examplethat your fuel contains 40% CH4 and 60% CO by volume. Two molesof O2 are required for each CH4 and 0.5 moles of O2 are required foreach mole of CO. The molar stoichiometric coefficient for O2 would thusbe input as (0.4 × 2) + (0.6 × 0.5) = 1.1. The molar stoichiometry foreach product species would be determined in a similar fashion. The finalstoichiometry would then be

(0.4CH4 + 0.6CO) + 1.1O2 → CO2 + 0.8H2O (14.3-1)

Input of Stoichiometry for Two Fuel or Oxidant Streams

Defining stoichiometry for a two-mixture-fraction problem is similar tothat for a single-mixture-fraction problem with a single, mixed fuel stream(described above). Consider, for example, the following two-fuel system:one inlet of CO, one inlet of CH4, and one inlet of O2. A single reactionwill be defined:

CH4 + CO + 2.5O2 → 2CO2 + 2H2O (14.3-2)

Input of Stoichiometry for Partial Equilibrium Calculations

As described in Section 14.1.2, you can define a rich limit on the mixturefraction when the equilibrium chemistry option is used. Input of the richlimit is accomplished using the Solution Parameters panel, described be-low. For mixture fraction values above this limit, prePDF will suspendthe equilibrium chemistry calculation and will compute the compositionbased on mixing of the fuel with the composition at the rich limit. Whenyou choose this partial equilibrium approach, you can let prePDF com-pute the composition at the rich limit using equilibrium or you can input



the stoichiometry to be assumed at the rich limit. Generally, you shouldchoose to use the equilibrium calculation of the composition at the richlimit unless you have experimental data (e.g., laminar flame data) thatyou want to represent through the input of stoichiometric coefficients.

Step 8: Define Parameters Used in Creation of the Look-UpTable

prePDF requires several inputs that are used in the creation of the look-up tables. Several of these inputs control the number and distributionof discrete values for which the look-up tables will be computed. Theseparameters are input using the Solution Parameters panel (Figure 14.3.8).

Setup −→ Solution Parameters...

Figure 14.3.8: The Solution Parameters Panel in prePDF

The solution parameters are as follows:

Non-Adiabatic Model contains parameters related to non-adiabatic mod-els



Enthalpy Points is the number of discrete values of enthalpy atwhich the three-dimensional look-up tables will be computed.This input is required only if you are modeling a non-adiabaticsystem. In general, you should choose the number of enthalpypoints to be one and a half to two times the number of meanmixture fraction points considered. The default value of 31enthalpy points may be sufficient for your model, or you maywant to increase this number (up to a maximum of 45). Thenumber of points required will depend on the chemical systemthat you are considering, with more points required in highheat release systems (e.g., hydrogen/oxygen flames).

Fuel Mixture Fraction contains parameters related to the fuel mixturefraction:

Fuel Mixture Fraction Points is the number of discrete values of fat which the look-up tables will be computed. For a two-mixture-fraction model, this value will also be the number ofpoints used by FLUENT to compute the PDF if you selectbeta for the Probability Density Function in the Species Modelpanel (see Section 14.3.3). Increasing the number of pointswill yield a more accurate PDF shape, but the calculationwill take longer.

Automatic Distribution enables the automatic discretization of thefuel mixture fraction and its variance. This feature optimizesthe distribution of the discrete mixture fraction values by clus-tering them around the peak temperature value. The auto-matic discretization is recommended in most cases.

Distribution Center Point (available only when Automatic Distribu-tion is disabled) determines the distribution of the requestednumber of discrete values of f . The requested number ofpoints will be distributed on either side of the center pointwith more points concentrated near the center point and fewerat the endpoints. If the center point is defined as 0.5 (the de-fault), the values will be distributed uniformly over the rangebetween 0 and 1. Generally, you should choose this value tobe on the rich side of the stoichiometric value of f . This will



create more points (and hence better resolution and accuracy)in the stoichiometric range and below—where the calculationis most critical. Determination of the stoichiometric value off is discussed below. Note that you should not set the centerpoint above 0.8 or below 0.2.

Mixture Fraction Variance Points is the number of discrete values off

′2s at which the look-up tables will be computed. The number

of mixture fraction variance points should be roughly one-halfthe number of mean mixture fraction points requested. Lowerresolution is required because the variation along the f ′2

s axisis, in general, slower than the variation along the f axis of thelook-up tables.

Secondary Partial Fraction contains parameters related to the (optional)secondary partial fraction:

Secondary Partial Fraction Points is the number of discrete values ofpsec at which the look-up tables will be computed. Like FuelMixture Fraction Points, FLUENT will use the Secondary PartialFraction Points to compute the PDF if you choose the betaPDF option (see Section 14.3.3) for a two-mixture-fractionmodel. A larger number of points will give a more accurateshape for the PDF, but with a longer calculation time.

Automatic Distribution enables the automatic discretization of thesecondary partial fraction and its variance. The automaticdiscretization is recommended in most cases.

Distribution Center Point (available only when Automatic Distribu-tion is disabled) determines the distribution of the requestednumber of discrete values of psec. The requested number ofpoints will be distributed on either side of the center pointwith more points concentrated near the center point and fewerat the endpoints. If the center point is defined as 0.5 (the de-fault), the values will be distributed uniformly over the rangebetween 0 and 1. For an oxidant or non-reacting secondarystream, you should keep this default value. For a secondaryfuel stream, you should generally choose this value to be onthe rich side of the stoichiometric value of psec. This will cre-



ate more points (and hence better resolution and accuracy)in the stoichiometric range and below—where the calculationis most critical. Determination of the stoichiometric value offsec is discussed below. You can then use Equation 14.1-3to determine the corresponding value for psec. Note that youshould not set the center point above 0.8 or below 0.2.

Equilibrium Chemistry Model contains parameters related to the equilib-rium chemistry model (see Section 14.1.2). You will not set these ifyou have chosen the stoichiometry chemistry model or if you haveused the empirical definition option for fuel composition.

Fuel Rich Flamability Limit controls the equilibrium calculation forthe fuel mixture fraction. A value of 1.0 for the rich limitimplies that equilibrium calculations will be performed overthe full range of mixture fraction. When you use a rich limitthat is less than 1.0, equilibrium calculations are suspendedwhenever f or ffuel exceeds the limit. This “partial equilib-rium” model is a useful approach in hydrocarbon combus-tion calculations, allowing you to bypass complex equilibriumcalculations in the fuel-rich region. The efficiency of partialequilibrium will be especially important when your model isnon-adiabatic, speeding up the preparation of the look-up ta-bles.

If you use a rich limit below 1.0, prePDF will ask if you wantto define the reaction stoichiometry at the rich limit or if youwould like the program to compute the rich limit compositionusing equilibrium chemistry:

If you choose the automatic calculation, prePDF will deter-mine the composition at the rich limit using the equilibrium



calculation. If you do not choose the automatic calculation,you must input the molar stoichiometry at the rich limit usingthe Stoichiometric Coefficients panel (see Step 7, above).

Secondary Rich Flamability Limit controls the equilibrium calcula-tion for the secondary mixture fraction. If your secondarystream is not a fuel, you should keep the default value of 1.For a secondary fuel stream, you can consider modifying thevalue to use the “partial equilibrium model.” A value of 1.0for the rich limit implies that equilibrium calculations will beperformed over the full range of mixture fraction. When youinput a rich limit that is less than 1.0, equilibrium calcula-tions are suspended whenever fsec exceeds the limit. (Notethat it is the secondary mixture fraction fsec and not the par-tial fraction psec that is used here.) See the description of theFuel Rich Flamability Limit above for details.

Equilibrium Calculations in Fuel Rich Mixtures

Experimental studies and reviews [23, 213] have shown that although thefuel lean flame region approximates thermodynamic equilibrium, chemi-cal kinetics will prevail under fuel rich conditions. Therefore, when usingprePDF for non-empirically defined fuels, the partial equilibrium modelis strongly recommended. As described above, this approach suspendsthe equilibrium calculations in the rich mixture. Guidelines on how toset the rich limit values are given below.

If you are using the full equilibrium approach (rich limit of 1 or em-pirically defined fuel), you should be aware that prePDF will performan equilibrium calculation for the fuel (e.g., at f = 1). The resultingequilibrium fuel composition and temperature will, in most cases, differfrom your original inputs defining the fuel. This indicates that the fuelcomposition and temperature, as defined by you, were not at equilibriumconditions. When prePDF adjusts the fuel composition and temperatureto new equilibrium values, you will receive a warning message:



The purpose of this warning is to alert you that the fuel inlet temperatureand composition have been modified to new equilibrium values. Thisinformation is important because it impacts how you will define gaseousfuel inlet boundary conditions in FLUENT, as follows.

The new equilibrium fuel temperature and composition define the fueldensity at gas-phase fuel inlet boundaries in FLUENT. This equilibriumdensity should be used to compute the appropriate inlet velocity, pre-serving the desired mass flow rate of the fuel. You can determine theequilibrium fuel density using the VIEW-ALPHA/DENSITY text commandin prePDF at the final discrete F-MEAN point (f = 1). In non-adiabaticsystems, the density you should use is that on the enthalpy slice corre-sponding to your fuel inlet temperature. If your fuel inlet temperature isequal to the temperature you input in the Operating Conditions panel inprePDF, you should examine the density on the adiabatic enthalpy slice.If you have multiple fuel inlets at different inlet temperatures, you canperform an adiabatic calculation at each temperature to determine theequilibrium density.

Although prePDF will compute a new equilibrium temperature for the!fuel, you should use your original prePDF input of fuel inlet tempera-ture when you define a gas-phase fuel inlet in FLUENT. FLUENT usesthis original, non-equilibrium fuel temperature to compute the inlet fuelenthalpy. (This enthalpy is the same as that used in the prePDF equilib-rium calculation.) Based on this inlet enthalpy, FLUENT will determinethe equilibrium temperature, composition, and density at the fuel inlet.

If you are modeling a liquid or coal fuel using the discrete phase model,the modified equilibrium fuel temperature and composition do not im-pact your inputs in FLUENT.



Determining the Stoichiometric and Rich Limit Values of Mixture Fraction

Determination of the rich limit mixture fraction is an important partof your inputs in the Solution Parameters panel. Generally, you shouldchoose the rich limit to be equal to 1.5 to 2 times the stoichiometricmixture fraction:

frich ≈ 2.0fs (14.3-3)

The stoichiometric mixture fraction, in turn, can be computed from theair-to-fuel mass ratio, r, as described in Section 14.1.2 (Equation 14.1-10).Alternately, you can estimate the stoichiometric mixture fraction by ex-amining the instantaneous temperature vs. mixture fraction predictedby prePDF for your adiabatic system. The maximum temperature willoccur near the stoichiometric mixture fraction.

The combustion of methanol in air provides an example of how you cancompute the stoichiometric mixture fraction. Written in terms of molarstoichiometry, the reaction is

CH3OH + 1.5(O2 + 3.76N2) → CO2 + 2H2O + 5.64N2 (14.3-4)

To compute the stoichiometric mixture fraction, first write the reactionon a mass basis, in terms of the stoichiometric air-to-fuel ratio, r, andthe equivalence ratio, φ. The reaction becomes

φ CH3OH + r(O2 + 3.76N2) → (φ+ r) Products (14.3-5)

where r = 6.435. Using Equation 14.1-10, the stoichiometric mixturefraction (φ = 1.0) is then

fs =φ

φ+ r=

17.435

= 0.134 (14.3-6)

and the fuel rich mixture fraction, taken at φ = 2.0, is



frich =2

8.435= 0.237 (14.3-7)

Extension of this exercise to a fuel consisting of a mixture of hydrocar-bons is straightforward, with Equation 14.3-4 simply taking on a moregeneral form. Consider, for example, a fuel-air system in which the fuelis comprised of 60% CH4 and 40% CO:

(0.6CH4+0.4CO)+z(O2+3.76N2) → xCO2+yH2O+3.76zN2 (14.3-8)

After balancing this equation and solving for z, you can compute the air-to-fuel mass ratio and then compute the stoichiometric mixture fractionas described above.

Rich Limit Values for Secondary Streams

If the secondary stream is an oxidizer or an inert, the rich limit for thesecondary stream should be set to 1. If it is a second fuel, the single fuelsystem analysis above applies, since the secondary rich limit is defined interms of secondary mixture fraction, fsec (not secondary partial fraction,psec).

Step 9: Save Your Inputs

When all of the preceding procedures are complete, you should save yourinputs to an “input” file:

File −→ Write −→ Input...

This file contains all of your inputs defining the reaction system inprePDF. You will have the option to save either a binary (unformat-ted) file or a formatted (ASCII, or text) file. You can read and edit aformatted file, but it will require more storage space than the same filein binary format. Binary files take up less space and can be read andwritten by prePDF more quickly, but they cannot be transferred betweenall machine types.



Step 10: Compute the Look-Up Tables

After saving your inputs, you can initiate the computation of the look-uptables by prePDF:

Calculate −→PDF Table

The computations performed in prePDF for a single-mixture-fraction cal-culation culminate in the discrete integration of Equation 14.1-17 (or14.1-26) as represented in Figure 14.1.9 (or Figure 14.1.10). For a two-mixture-fraction calculation, prePDF will calculate the physical proper-ties using Equation 14.1-28 or its adiabatic equivalent. These compu-tations may take only a few moments for simple systems or they mayrequire up to an hour for a complex system (e.g., non-adiabatic systemswith 10 or more species). prePDF reports progress as the calculation pro-ceeds. Below, sample output is shown for an adiabatic, single-mixture-fraction calculation:



(*)- INITIALIZING AT ADIABATIC ENTHALPY LINE

.................ADIABATIC CALCULATION......................POINTS TO-GO EQUILIBRIUM DELTA-PDF

0 37516 359 T(K) = 2004. F-MEAN = 0.04 F-VAR = .00017 358 T(K) = 1386. F-MEAN = 0.04 F-VAR = .01818 357 T(K) = 1108. F-MEAN = 0.04 F-VAR = .03619 356 T(K) = 1053. F-MEAN = 0.04 F-VAR = .04020 355 T(K) = 1053. F-MEAN = 0.04 F-VAR = .04021 354 T(K) = 1053. F-MEAN = 0.04 F-VAR = .040...

356 19 T(K) = 467. F-MEAN = 0.96 F-VAR = .040357 18 T(K) = 467. F-MEAN = 0.96 F-VAR = .040358 17 T(K) = 467. F-MEAN = 0.96 F-VAR = .040359 16 T(K) = 467. F-MEAN = 0.96 F-VAR = .040360 15 T(K) = 467. F-MEAN = 0.96 F-VAR = .040...

(*)- CALCULATION SUCCEEDED

After completing the equilibrium calculation at the specified number ofmixture fraction points, prePDF reports that the calculation succeeded.The resulting look-up tables take the form illustrated in Figure 14.2.3 (orFigure 14.2.5, for non-adiabatic systems). These look-up tables can beplotted using the graphics tools available in prePDF, as described belowin Step 12.

Note that in non-adiabatic calculations, the report includes informationon the enthalpy point currently considered:



(*)- INITIALIZING ENTHALPY AT TEMPERATURE LIMITS

.................NON-ADIABATIC CALCULATION......................POINTS TO-GO H-POINT EQUILIBRIUM DELTA-PDF

0 153758 15367 8 T(K) = 298. F-MEAN = 0.00 F-VAR = .0009 15366 9 T(K) = 334. F-MEAN = 0.00 F-VAR = .000

10 15365 10 T(K) = 888. F-MEAN = 0.00 F-VAR = .00011 15364 11 T(K) = 1391. F-MEAN = 0.00 F-VAR = .00012 15363 12 T(K) = 1869. F-MEAN = 0.00 F-VAR = .00013 15362 13 T(K) = 2334. F-MEAN = 0.00 F-VAR = .00014 15361 14 T(K) = 2792. F-MEAN = 0.00 F-VAR = .00015 15360 15 T(K) = 3243. F-MEAN = 0.00 F-VAR = .000

You may notice that non-adiabatic calculations terminate before the tab-ulated number of points under TO-GO is zero. This is simply because thefinal calculations, at mixture fraction equal to 1.0, do not include multi-ple variance points.

For a two-mixture-fraction calculation, prePDF will print the followinginformation during the calculation:

(*)- INITIALIZING AT ADIABATIC ENTHALPY

...........ADIABATIC CALCULATION / SECONDARY STREAM.............POINTS TO-GO EQUILIBRIUM PROGRESS-VARIABLES

0 7251 724 T(K) = 600. F-FUEL = 0.000 P-SECND = 0.0002 723 T(K) = 1134. F-FUEL = 0.000 P-SECND = 0.0143 722 T(K) = 1622. F-FUEL = 0.000 P-SECND = 0.0294 721 T(K) = 2064. F-FUEL = 0.000 P-SECND = 0.0455 720 T(K) = 2357. F-FUEL = 0.000 P-SECND = 0.0626 719 T(K) = 2216. F-FUEL = 0.000 P-SECND = 0.0817 718 T(K) = 1954. F-FUEL = 0.000 P-SECND = 0.1008 717 T(K) = 1690. F-FUEL = 0.000 P-SECND = 0.1219 716 T(K) = 1432. F-FUEL = 0.000 P-SECND = 0.143

The resulting look-up tables take the form illustrated in Figure 14.2.4 (or



Figure 14.2.6, for non-adiabatic systems). These look-up tables can beplotted using the graphics tools available in prePDF, as described below.

For a non-adiabatic calculation, the current enthalpy point will be shownas in the example output listed above for the non-adiabatic single-mixture-fraction calculation.

Stability Issues in prePDF

Complex chemistry and non-adiabatic effects may make the equilibriumcalculation more time-consuming and difficult. In some instances theequilibrium calculation may even fail. You may be able to eliminate anydifficulties that you encounter using one of the following techniques:

• Try the calculation as an adiabatic system. Adiabatic system cal-culations are generally quite straightforward and can provide valu-able insight into the optimal inputs to the non-adiabatic calcula-tion. Using the adiabatic results, you can determine the maximumtemperature expected and correct this important input to the non-adiabatic case. You can determine which species are important tothe reacting system, and eliminate those that are unimportant.This information can be obtained by a simple review of the look-up tables generated by the adiabatic calculation. Selecting an ap-propriate temperature range and an appropriate list of chemicalspecies to include will greatly simplify the non-adiabatic calcula-tion. You might also simplify the non-adiabatic calculation by abetter choice of the rich cut-off limit, as described in Step 8, andadjust the mixture fraction center point to capture the adiabatictemperature curve more adequately.

• Try reducing the number of species considered. Simplify yourmodel of the reacting system. If your system includes heavy hydro-carbons, be sure that you are including basic hydrocarbons suchas CH4 in the system.

Additional stability issues may arise for solid or heavy liquid fuels thathave been defined using the empirical fuel approach. You may find that



for rich fuel mixtures the equilibrium calculation produces very low tem-peratures and eventually fails. This indicates that strong endothermicreactions are taking place and the mixture is not able to sustain them.In this situation, you may need to raise the heating value of the fuel un-til prePDF produces acceptable results. Provided that your fuel will betreated as a liquid or solid (coal) fuel, you can maintain the desired heat-ing value in your FLUENT simulation. This is accomplished by definingthe difference between the desired and the adjusted heating values aslatent heat (in the case of combusting solid fuel) or heat of pyrolysis (inthe case of liquid fuel).

Informational Messages and Errors Reported During the prePDFCalculation

prePDF may report a variety of error messages or informational reports asit computes the equilibrium chemistry and generates the look-up tables.These messages are detailed in Section 14.3.2.

Step 11: Save the Look-Up Tables

The look-up tables computed by prePDF are stored in a file that you willread into FLUENT. FLUENT will use the tables to extract the species,density, and temperature fields from the mixture fraction field that itpredicts as part of the flow-field calculation. The look-up tables must besaved before you exit from prePDF.

File −→ Write −→ PDF...

The file can be saved as formatted (ASCII, or text) or binary (unformat-ted), for either FLUENT 4 or FLUENT 6.

Be sure to save a PDF file for the appropriate solver.!

In addition to reading the PDF file into FLUENT for the flow analysis,you can read it back into prePDF at a later date if you want to examinethe look-up tables using the graphics tools described below, in Step 12.(All types of PDF files can be read back into prePDF.)



Step 12: Graphics and Alphanumeric Reports in prePDF

prePDF supplies a number of utilities that allow you to examine the resultof the look-up table computations.

Reviewing the Beta PDF Shape

You can plot the shape of the beta PDF using the Beta-Pdf panel (Fig-ure 14.3.9).

Display −→Beta PDF...

Figure 14.3.9: The Beta-Pdf Panel in prePDF

This utility simply plots the function, Equation 14.1-22, for any value off (Mean Mixture Fraction) and f ′2 (Mixture Fraction Variance) that youdefine in the panel. Figure 14.1.7 illustrates two of the many forms thatthe beta PDF shape may take. Note that none of your inputs in prePDFwill change the beta PDF shape for a given pair of f and f ′2. (Since theβ PDF plots are just for general informational purposes, you can plotthem even when you are working on a two-mixture-fraction problem, forwhich the PDFs will be calculated in FLUENT.)



Reviewing Instantaneous Values in Adiabatic Single-Mixture-FractionSystems

You can plot the variation of instantaneous species concentration, den-sity, or temperature with the instantaneous mixture fraction using theProperty Curves panel (Figure 14.3.10).

Display −→Property Curves...

Figure 14.3.10: The Adiabatic Property Curves Panel in prePDF

You can select temperature, density, species, or enthalpy as the variableto be plotted using the Plot Variable drop-down list. The resulting dis-plays show how these quantities vary with mixture fraction and can beused to determine the stoichiometric value of mixture fraction, the peaktemperature expected, and the most important species in the system.Figures 14.3.11 and 14.3.12 show instantaneous values derived for a verysimple hydrocarbon system.

For adiabatic systems, you can also write property data to a file in theformat used by the XY plotter in FLUENT. To write an XY plot file con-taining property data, use the WRITE-XY-FILE text command in prePDF:

VIEW-GRAPHICS −→ PROPERTY-CURVES −→WRITE-XY-FILE

When you select this command, you will be asked to select the propertyto be written and specify a name for the file:



1.00E+00

8.00E-01

6.00E-01

4.00E-01

2.00E-01

0.00E+00

1.00E+008.00E-016.00E-014.00E-012.00E-010.00E+00

prePDF V4.00

KEY

Fluent Inc.

Mixture Fraction F

Mole Fraction

INSTANTANEOUS SPECIES COMPOSITION


H2CON2H2OCO2O2CH4

Figure 14.3.11: Instantaneous Species Mole Fractions Derived From theEquilibrium Chemistry Calculation

3.28E+03

2.62E+03

1.97E+03

1.31E+03

6.56E+02

0.00E+00

1.00E+008.00E-016.00E-014.00E-012.00E-010.00E+00

prePDF V4.00

KEY

Fluent Inc.

Mixture Fraction F

Temperature

INSTANTANEOUS TEMPERATURE (KELVIN)


Figure 14.3.12: Instantaneous Temperature Derived From the Equilib-rium Chemistry Calculation



COMMANDS AVAILABLE FROM PROPERTY-CURVES:PLOT WRITE-XY-FILE QUIT

(PROPERTY-CURVES)-wxy

COMMANDS AVAILABLE FROM WRITE-XY-FILE:TEMPERATURE DENSITY SPECIES ENTHALPYQUIT

(WRITE-XY-FILE)-teENTER NAME OF XY PLOT FILE(S)- DEFAULT- sample.xy

Later, during your FLUENT session, you can read and plot this datausing the File XY Plot panel.

Plot −→File...

Reviewing Instantaneous Values in Adiabatic Two-Mixture-FractionSystems

If your problem includes a secondary stream, in addition to choosingthe variable to be plotted, you must also specify a constant value ofthe fuel mixture fraction or the secondary partial fraction at which theproperty curve should be drawn. Choose either fraction under ConstantValue of and specify its Value. The selected variable will be plotted as afunction of the fraction that is not held constant. In Figure 14.3.13, thesecondary partial fraction is held constant at 0.05, and the plot showshow temperature varies with fuel mixture fraction.

Reviewing Instantaneous Values in Non-Adiabatic Systems

If your single-mixture-fraction system is non-adiabatic, you can still re-view the variation of instantaneous scalar values with the instantaneousmixture fraction. In the non-adiabatic case, because the instantaneousresults depend on the mean enthalpy value, you will specify the meanmixture fraction, its variance, and the mean enthalpy value at which thevariable will be displayed. The Property Curves panel (Figure 14.3.14)



3.28E+03

2.62E+03

1.97E+03

1.31E+03

6.56E+02

0.00E+00

1.00E+008.00E-016.00E-014.00E-012.00E-010.00E+00

prePDF V4.00

KEY

Fluent Inc.

Temperature

F Fuel

CONSTANT SECONDARY PARTIAL FRACTION 0.050

INSTANTANEOUS TEMPERATURE (KELVIN)

Figure 14.3.13: Instantaneous Temperature Plotted for a Two-Mixture-Fraction Case

contains the fields required to input these parameters in non-adiabaticsystems.


Select temperature, density, species mass fraction, or enthalpy as thevariable to be plotted using the Plot Variable drop-down list. Then enterthe Mean Mixture Fraction, Mixture Fraction Variance, and Mean Enthalpyat which to display the selected variable. Click Display to generate thegraphical display.

Note that you cannot plot instantaneous property curves for a non-!adiabatic two-mixture-fraction case. You will instead plot the look-uptables of instantaneous values using the Nonadiabatic-Table panel.

Note also the following important difference in the Property Curves dis-!play for adiabatic and non-adiabatic cases. For adiabatic cases, the in-stantaneous property curves correspond to the values of the look-up ta-bles at f ′2

s = 0. In non-adiabatic systems, however, the property curves



Figure 14.3.14: The Non-Adiabatic Property Curves Panel in prePDF

represent only the intermediate values of the properties prior to the PDFintegrations, and therefore do not correspond to any of the values in thePDF look-up tables. In order to examine the values stored in the look-up tables for a non-adiabatic case you should use the Nonadiabatic-Tablepanel.

Reviewing the 2D Look-Up Tables Computed by prePDF for a SingleMixture Fraction

You can use either graphics or alphanumerics to display the 2D PDFlook-up tables generated for single-mixture-fraction adiabatic systems.To plot the tables, use the Pdf-Table panel (Figure 14.3.15).

Display −→PDF Table...

Using the Plot Variable drop-down list, you can display the look-up tablefor temperature, density, or any individual species fraction (defined inthe Species Selection panel that appears when you choose SPECIES).Figure 14.3.16 illustrates the look-up table generated for temperaturein a simple hydrocarbon combustion model. Similarly, you can use theVIEW-ALPHA command, available in the text interface, to display the



Figure 14.3.15: The Pdf-Table Panel in prePDF

look-up table in tabular form at each point in the discrete mean/variancematrix:

MAIN −→ VIEW-ALPHA

VIEW ALPHA: TEMPERATURE (K)

F-VAR F-MEAN= 1 2 3 4 5

15 1.5E+03 1.5E+03 1.5E+03 1.5E+03 1.5E+03

14 1.5E+03 1.5E+03 1.5E+03 1.5E+03 1.5E+03

13 1.5E+03 1.5E+03 1.5E+03 1.5E+03 1.5E+03

12 1.5E+03 1.5E+03 1.5E+03 1.5E+03 1.5E+03

11 1.5E+03 1.5E+03 1.5E+03 1.5E+03 1.5E+03

.

.

4 1.5E+03 1.6E+03 1.7E+03 1.7E+03 1.8E+03

3 1.5E+03 1.6E+03 1.7E+03 1.8E+03 1.9E+03

2 1.5E+03 1.7E+03 1.8E+03 2.0E+03 2.1E+03

1 1.5E+03 1.8E+03 2.1E+03 2.4E+03 2.7E+03

Reviewing the 2D Look-Up Tables Computed by prePDF for Two MixtureFractions

It is important for you to view your temperature and species tables in!prePDF to ensure that they are adequately but not excessively resolved.Inadequate resolution will lead to inaccuracies, and excessive resolutionwill lead to unnecessarily slow calculation times in FLUENT.



2.50E-01

2.00E-01

1.50E-01

1.00E-01

5.00E-02

0.00E+00

SCALED-F-VARIANCE

1.00E+00 8.00E-01 6.00E-01 4.00E-01 2.00E-01 0.00E+00

TEMPERATURE K

prePDF V4.00

2.3E+03

1.9E+03

1.5E+03

1.1E+03

7.0E+02

3.0E+02

Fluent Inc.

F-MEAN

MEAN FLAME TEMPERATURE

PDF TABLE - CHEMICAL EQUILIBRIUM

Figure 14.3.16: Two-Dimensional Look-Up Table for Temperature Gen-erated by prePDF for a Simple Hydrocarbon System (Single-Mixture-Fraction, Adiabatic System)



You can use either graphics or alphanumerics to display the 2D look-up tables of instantaneous properties generated for two-mixture-fractionadiabatic systems. To plot the tables, use the Property-Table panel:

Display −→Property Table...

Figure 14.3.17: The Property-Table Panel

You will use this panel exactly as described above for the Pdf-Table panel,but the resulting plot will show the selected variable as a function of in-stantaneous fuel mixture fraction and secondary partial fraction (insteadof as a function of mean fuel mixture fraction and variance).

Alphanumeric reports are generated in the same way as for single-mixture-fraction calculations, but the report will list the ffuel, psec points insteadof the mean/variance matrix.

VIEW ALPHA: TEMPERATURE (K)

P-SEC F-FUEL= 1 2 3 4 5

7 1.954E+03 1.773E+03 1.593E+03 1.420E+03 1.273E+03

6 2.216E+03 2.020E+03 1.824E+03 1.631E+03 1.442E+03

5 2.357E+03 2.263E+03 2.055E+03 1.845E+03 1.638E+03

4 2.064E+03 2.339E+03 2.279E+03 2.059E+03 1.836E+03

3 1.622E+03 2.043E+03 2.339E+03 2.267E+03 2.032E+03

2 1.134E+03 1.630E+03 2.071E+03 2.357E+03 2.225E+03

1 6.000E+02 1.180E+03 1.696E+03 2.145E+03 2.366E+03



Reviewing the 3D Look-Up Tables for Single-Mixture-FractionNon-Adiabatic Systems

The look-up tables generated for single-mixture-fraction non-adiabaticsystems contain the mean temperature, density, and species concentra-tions as a function of three quantities: mean mixture fraction, mixturefraction variance, and mean enthalpy. Consequently, when you ask todisplay the look-up tables in alphanumerics or graphics, you will bedisplaying them slice-by-slice. The graphical display begins with theNonadiabatic-Table panel (Figure 14.3.18).

Display −→ Nonadiabatic-Table...

Figure 14.3.18: The Nonadiabatic-Table Panel in prePDF

In this panel, you can select the variable to be plotted in the Plot Variabledrop-down list. Next, you must define how the three-dimensional arrayof data points available in the look-up table are to be sliced: whichdiscrete independent variable (either f or H∗) is to be held constant andwhether the constant value is to be selected as a numerical value (chooseValue as the Plot type) or by discretization index (choose Slice as thePlot type). If you choose the latter approach, click the Slice... button toselect the discretization index slice that you want.



Figure 14.3.19: The Slice Panel in prePDF

In the Slice panel (Figure 14.3.19), select which discretized variable is tobe constant (Enthalpy or F-Mean) and then pick the Slice # (discretiza-tion index). For example, in the panel in Figure 14.3.19, a look-up tablegenerated at the tenth discrete value of mean enthalpy has been re-quested. As discussed in Section 14.2, each slice actually corresponds toa normalized heat loss or gain. The enthalpy slice index correspondingto the adiabatic case is displayed in the Adiabatic Slice # field.

To generate the plot, click Apply and close the Slice panel, and thenclick Display in the Nonadiabatic-Table panel. A sample plot is shown inFigure 14.3.20.

Alternately, you may want to define a slice of the 3D look-up table basedon a specific value of one of the independent quantities. When this is thecase, select the Value option under Plot type in the Nonadiabatic-Tablepanel. To set the slice, click the Value... button to open the LookupPoints panel (Figure 14.3.21).

In this panel, you can select a slice of the 3D table that corresponds to:

• constant value of mean enthalpy (Enthalpy Values and ConstantEnthalpy options)



2.50E-01

2.00E-01

1.50E-01

1.00E-01

5.00E-02

0.00E+00

SCALED-F-VARIANCE

1.00E+00 8.00E-01 6.00E-01 4.00E-01 2.00E-01 0.00E+00

TEMPERATURE K

prePDF V4.00

2.4E+03

2.0E+03

1.6E+03

1.1E+03

7.2E+02

3.0E+02

Fluent Inc.

F-MEAN

MEAN FLAME TEMPERATURE FROM 3D-PDF-TABLE

MEAN ENTHALPY SLICE NUMBER 23

Figure 14.3.20: Display of a Single Slice of the Three-Dimensional Look-Up Table in a Non-Adiabatic System (Single Mixture Fraction)

• constant value of mean mixture fraction (Constant F-Mean Value)

• adiabatic enthalpy (Enthalpy Values and Adiabatic Relationship op-tions)

In addition, you supply the physical value of the selected quantity in theValue field. When the adiabatic enthalpy option is selected, you mustsupply the Fuel and Oxidiser Inlet Temperature instead of the fixed value.prePDF uses this information to construct the adiabatic relationship be-tween enthalpy and mixture fraction that is used to slice the table. Theadiabatic enthalpy option is very useful, as it allows you to generateadiabatic (2D) look-up tables for various combinations of fuel and oxi-dizer inlet temperatures from your 3D look-up table generated for thenon-adiabatic system.

Finally, you can set the Refinement Factor, which determines the res-olution of the plotted curve. A refinement factor of 1.0 (the default)implies that the plot will use the same number of discrete points that



Figure 14.3.21: The Lookup Points Panel in prePDF

you requested in the Solution Parameters panel. Increasing this factor willcause prePDF to compute and display additional data points, yielding asmoother plot but requiring some time to compute.

To generate the plot, click Apply and close the Lookup Points panel, andthen click Display in the Nonadiabatic-Table panel.

Reviewing the 3D Look-Up Tables for Two-Mixture-FractionNon-Adiabatic Systems

The look-up tables generated for two-mixture-fraction non-adiabatic sys-tems contain the instantaneous temperature, density, and species con-centrations as a function of three instantaneous quantities: fuel mixture



fraction, secondary partial fraction, and enthalpy. As for the single-mixture-fraction case described above, you will use the Nonadiabatic-Table panel to display the look-up table. The procedure is exactly thesame as that described above, except that you can choose only enthalpyslices or values on which to display the look-up table.

14.3.2 Informational Messages and Errors Reported by prePDF

The following messages may be issued by prePDF during the case setup,the calculation of the look-up tables, or postprocessing. The source ofthe message and the action required by you, if any, are detailed here.

BOTTOM TEMPERATURE TOO HIGH

From: Solver

Cause: The minimum temperature defined for the non-adiabatic calcu-lation is higher than the inlet temperatures.

Action: Correct the minimum temperature value.

CALCULATIONS INTERRUPTED

CONTINUE ? (ELSE RETURN TO MAIN MENU)

From: Solver

Cause: Ctrl-C has been pressed.

Action: Typing N will cause the solver to abort to the main menu andall previous equilibrium iterations will be lost. Typing Y or RETURNwill cause the calculations to continue.

ENTHALPY CURVE GENERATION FAILED

From: Graphics



Cause: For the non-adiabatic single-mixture-fraction case, prePDF wasunable to construct the instantaneous enthalpy curve required forthe calculation of the instantaneous property curves.

Action: Modify your input of mixture fraction, variance, or enthalpyfor the property curve plot.

ENTHALPY HIGHER THAN ENTHALPY CEILING

From: Graphics

Cause: For the non-adiabatic single-mixture-fraction case the enthalpyinput for the calculation of the instantaneous property curves istoo high.

Action: Decrease the enthalpy value for the property curve plot.

ENTHALPY LOWER THAN ENTHALPY FLOOR

From: Graphics

Cause: For the non-adiabatic single-mixture-fraction case the enthalpyinput for the calculation of the instantaneous property curves istoo low.

Action: Increase the enthalpy value for the property curve plot.

ERROR: NO ATOMS EXIST TO DEFINE THE EMPIRICAL STREAM

From: Setup

Cause: The empirical fuel stream option has been selected but no el-ements have been defined from which to construct the fuel. Ele-ments allowed are C, H, O, S, and N.

Action: Add the elements C, H, O, S, and N in the species list.

ERROR: NO CARBON DIOXIDE SPECIES EXISTS



From: Setup

Cause: The empirical fuel stream option has been selected but CO2

species has not been defined. CO2 is needed to calculate the heatof formation of the empirical fuel from its heating value

Action: Add CO2 in the species list.

ERROR: NO WATER VAPOR SPECIES EXISTS

From: Setup

Cause: The empirical fuel stream option has been selected but H2Ospecies has not been defined. H2O is needed to calculate the heatof formation of the empirical fuel from its heating value.

Action: Add H2O in the species list.

ERROR OPENING TEMPORARY LINKING FILE

From: Files or Setup

Cause: prePDF was unable to open the file DBLINK used for temporarystorage of thermodynamic data.

Action: Make sure you have write permission in the working directory.

ERROR WRITING TEMPORARY LINKING FILE


Cause: prePDF was unable to write the file DBLINK used for temporarystorage of thermodynamic data.

Action: Make sure there is space on the disk.

FAULT AT STOICHIOMETRIC REACTION CALCULATION FOR F-MEAN = xx



From: Solver

Cause: This message may appear for one of the following reasons:

• the temperature limits are not sufficient

• the stoichiometry defined is incorrect.

• the rich flammability limit with the automatic stoichiometrycalculation has been used and the value of rich flammabilitylimit has been set too low.

Action: Check inputs of temperature limits. Check inputs of stoichiom-etry. Check the setting of rich flammability limit.

INCORRECT STOICHIOMETRY DEFINED

From: Solver

Cause: The stoichiometry defined does not satisfy the element balance.

Action: Check inputs of stoichiometry.

SETTING UP DATA BASE LINK FILE FAILED


Cause: prePDF was unable to access the thermodynamic propertiesdatabase.

Action: Make sure the instructions for installation of prePDF have beenfollowed and all the environment variables are set correctly.

TOP TEMPERATURE TOO LOW

From: Solver

Cause: The maximum temperature defined for the non-adiabatic cal-culation is lower than the adiabatic flame temperature for thismixture.



Action: Increase the maximum temperature limit. It is recommendedthat this is set at Tadiabatic + 100 K. The adiabatic flame tem-perature Tadiabatic can be calculated by performing an adiabaticcalculation and reviewing the instantaneous temperature vs. mix-ture fraction curves predicted by prePDF.

UNABLE TO CALCULATE COMPOSITION AT F-MEAN xx AND TEMPERATURE xxxx K

From: Solver

Cause: The equilibrium calculation has failed. This may happen forthe following reasons:

• The species list defined is not adequate.

• The mixture is liquid for the conditions the equilibrium solverhas entered.

Action: Try using better temperature limits. Experiment with thespecies list, using an adiabatic calculation, adding or removingspecies based on the amount formed over the mixture fractionrange. Try moving the rich flammability limit closer to the sto-ichiometric mixture fraction.

UNABLE TO CALCULATE ENTHALPY CURVE AT I-FMEAN = xx J-FVAR = xx K-ENTH = xx

From: Solver

Cause: For the non-adiabatic calculation, prePDF was unable to con-struct the instantaneous enthalpy curve required for the PDF cal-culation.

Action: This message should not appear. Contact Fluent customersupport.



14.3.3 Non-Premixed Model Input and Solution Procedures inFLUENT

The non-premixed model setup and solution procedure in FLUENT dif-fers slightly for single- and two-mixture-fraction problems. Below, anoverview of each approach is provided. Note that your FLUENT casefile must always meet the restrictions listed for the non-premixed mod-eling approach in Section 14.1.3. In this section, details are providedregarding the problem definition and calculation procedures you followin FLUENT.

Single-Mixture-Fraction Approach

For a single-mixture-fraction system, when you have completed the cal-culation of the mixture fraction/PDF look-up tables in prePDF, you areready to begin your reacting flow simulation in FLUENT. In FLUENT,you will solve the flow field and predict the spatial distribution of f andf ′2 (and H∗ if the system is non-adiabatic or χst,d if the system is basedon laminar flamelets). FLUENT will obtain the implied values of tem-perature and individual chemical species mass fractions from the look-uptables.

Two-Mixture-Fraction Approach

When a secondary stream is included, FLUENT will solve transport equa-tions for the mean secondary partial fraction (psec) and its variance inaddition to the mean fuel mixture fraction and its variance. FLUENTwill then look up the instantaneous values for temperature, density, andindividual chemical species in the look-up tables, compute the PDFsfor the fuel and secondary streams, and calculate the mean values fortemperature, density, and species.

Note that in order to avoid both inaccuracies and unnecessarily slowcalculation times, it is important for you to view your temperature andspecies tables in prePDF to ensure that they are adequately but notexcessively resolved.



Step 1: Start FLUENT and Read a Grid File

Start your FLUENT session in the usual way, as described in Section 1.5,and then read in a grid file. The number and types of inlets in your modelmust meet the constraints of the non-premixed modeling approach, asdiscussed in Section 14.1.3 and illustrated in Figures 14.1.12, 14.1.13,and 14.1.14.

Starting a Non-Premixed Calculation From a Previous Case File

You can read a previously defined FLUENT case file as a starting pointfor your non-premixed combustion modeling. If this case file containsinputs that are incompatible with the current non-premixed combustionmodel, FLUENT will alert you when the non-premixed model is turnedon and it will turn off those incompatible models. For example, if thecase file includes species that differ from those included in the PDF filecreated by prePDF, these species will be disabled. If the case file containsproperty descriptions that conflict with the property data in the chemicaldatabase, these property inputs will be ignored.

See Step 2, below, for important information about PDF files created by!previous versions of prePDF.

Step 2: Activate the Non-Premixed Combustion Model

Preliminaries

Before turning on the non-premixed combustion model, you must enableturbulence calculations in the Viscous Model panel.

Define −→ Models −→Viscous...

If your model is non-adiabatic, you should also enable heat transfer (andradiation, if required).

Define −→ Models −→Energy...

Define −→ Models −→Radiation...

Figure 14.1.11 illustrates the types of problems that must be treated asnon-adiabatic. Note, however, that the decision to include non-adiabatic



effects is made in prePDF. FLUENT will turn off the energy equation ifyour prePDF inputs are for an adiabatic system.

Enabling Non-Premixed Combustion

Before any other modeling inputs (e.g., setting of boundary conditionsor properties) you should turn on the non-premixed combustion model,because activating this model will impact how other inputs are requestedduring your subsequent work. The non-premixed combustion model isenabled in the Species Model panel.

Define −→ Models −→Species...

Figure 14.3.22: The Species Model Panel in FLUENT

Select Non-Premixed Combustion under the Model heading. When youclick OK in the Species Model panel, a Select File dialog box will imme-diately appear, prompting you for the name of the PDF file containingthe look-up tables created in prePDF. (The PDF file is the file you savedusing the File/Write/PDF... menu item in prePDF after computing thelook-up tables.) FLUENT will indicate that it has successfully read thespecified PDF file:



Reading "/home/mydirectory/adiabatic.pdf"...read 5 species (binary c, adiabatic prepdf)pdf file successfully read.Done.

After you read in the PDF file, FLUENT will inform you that somematerial properties have changed. You can accept this information; youwill be updating properties later on.

You can read in an altered PDF file at any time by using the File/Read/Pdf...menu item.

Recall that the non-premixed combustion model is available only when!you used the segregated solver; it cannot be used with the coupledsolvers. Also, the non-premixed combustion model is available only whenturbulence modeling is active.

If you are modeling a non-adiabatic system and you wish to include theeffects of compressibility, re-open the Species Model panel (Figure 14.3.23)and turn on Compressibility Effects under PDF Options. This option tellsFLUENT to update the density, temperature, species mass fraction, andenthalpy from the PDF tables to account for the varying pressure of thesystem. When the non-premixed combustion model is active, you canenable compressibility effects only in the Species Model panel. For othermodels, you will specify compressible flow (ideal-gas, boussinesq, etc.) inthe Materials panel.

Using PDF Files Created by Previous Releases of prePDF

PDF files created by prePDF 1 cannot be read into FLUENT or intoprePDF 4 (the current version). If you have prePDF 1 files, read the inputfile into prePDF 4, recalculate the look-up tables, and save a new PDFfile to be read into FLUENT. As noted in Section 14.3.5, prePDF 1 inputfiles that were created for coal combustion systems must be modifiedbefore computing the PDF look-up tables in prePDF 4.

PDF files created by prePDF 2 can be read into FLUENT, but it is rec-ommended that you recalculate the look-up tables in prePDF 4. In theseversions of prePDF, the mixture fraction variance was not scaled to its



Figure 14.3.23: The Species Model Panel With Available PDF Options

maximum value while building the PDF table. This resulted in a lower-resolution table, especially for lower mixture fraction values, and at mix-ture fraction values close to 0 and 1. To take advantage of a more ad-vanced PDF table look-up scheme, you can read a PDF or input filecreated by prePDF 2 into prePDF 4 and recalculate the look-up table.

Two-mixture-fraction PDF files created in prePDF 2 should be read intoprePDF 4 and written out in FLUENT 6 format. (Two-mixture-fractionPDF files written by prePDF 2 cannot be read directly into FLUENT.)

Table 14.3.1 summarizes the recommended procedures for using old PDFfiles in FLUENT.

Retrieving the PDF File During Case File Reads

The PDF filename is specified to FLUENT only once. Thereafter, thefilename is stored in your FLUENT case file and the PDF file will beautomatically read into FLUENT whenever the case file is read. FLUENTwill remind you that it is reading the PDF file after it finishes readingthe rest of the case file by reporting its progress in the text (console)window.



Table 14.3.1: Using Old PDF Files in FLUENT

PDF file type Readable inprePDF 4?

Readablein FLUENT?

RecommendedProcedure

prePDF 1 input files only no Read input file intoprePDF 4, recalcu-late PDF table, andwrite a PDF file inFLUENT 6 format.

prePDF 2 (singlemixture fraction)

yes yes Read input or PDFfile into prePDF 4,recalculate PDF ta-ble, and write aPDF file in FLU-ENT 6 format.

prePDF 2 (twomixture fractions)

yes no Read PDF file intoprePDF 4 and writea PDF file in FLU-ENT 6 format.

prePDF 3 yes yes prePDF 3 files arecompatible withboth prePDF 4 andFLUENT 6.



Note that the PDF filename stored in your case file may not contain thefull name of the directory in which the PDF file exists. The full directoryname will be stored in the case file only if you initially read the PDFfile through the GUI (or if you typed in the directory name along withthe filename when using the text interface). In the event that the fulldirectory name is absent, the automatic reading of the PDF file may fail(since FLUENT does not know which directory to look in for the file), andyou will need to manually specify the PDF file. The safest approachesare to use the GUI when you first read the PDF file or to supply the fulldirectory name when using the text interface.

Step 3: Define Boundary Conditions

Input of Mixture Fraction Boundary Conditions

When the non-premixed combustion model is used, flow boundary con-ditions at inlets and exits (i.e., velocity or pressure, turbulence intensity)are defined in the usual way. Species mass fractions at inlets are not re-quired. Instead, you define values for the mean mixture fraction, f , andthe mixture fraction variance, f ′2, at inlet boundaries. (For problemsthat include a secondary stream, you will define boundary conditions forthe mean secondary partial fraction and its variance as well as the meanfuel mixture fraction and its variance.) These inputs provide boundaryconditions for the conservation equations you will solve for these quanti-ties. The inlet values are supplied in the boundary conditions panel forthe selected inlet boundary (e.g., Figure 14.3.24).

Define −→ Boundary Conditions...

Input the Mean Mixture Fraction and Mixture Fraction Variance (and theSecondary Mean Mixture Fraction and Secondary Mixture Fraction Vari-ance, if you are using two mixture fractions). In general, the inlet valueof the mean fractions will be 1.0 or 0.0 at flow inlets: the mean fuelmixture fraction will be 1.0 at fuel stream inlets and 0.0 at oxidizer orsecondary stream inlets; the mean secondary mixture fraction will be 1.0at secondary stream inlets and 0.0 at fuel or oxidizer inlets. The fuel orsecondary mixture fraction will lie between 0.0 and 1.0 only if you aremodeling flue gas recycle, as illustrated in Figure 14.1.15 and discussed



Figure 14.3.24: The Velocity Inlet Panel Showing Mixture FractionBoundary Conditions



in Section 14.1.2. The fuel or secondary mixture fraction variance canusually be taken as zero at inlet boundaries.

Input of Thermal Boundary Conditions and Fuel Inlet Velocities

If your model is non-adiabatic, you should input the Temperature at theflow inlets. While the inlet temperatures were requested in prePDF (asthe Fuel Inlet Temperature, Oxidiser Inlet Temperature, and (if applicable)Secondary Inlet Temperature in the Operating Conditions panel), theseinputs were used only in the construction of the look-up tables. The inlettemperatures for each fuel, oxidizer, and secondary inlet in your non-adiabatic model should be defined, in addition, as boundary conditions inFLUENT. It is acceptable for the inlet temperature boundary conditionsdefined in FLUENT to differ slightly from those you input in prePDF. Ifthe inlet temperatures differ significantly from those in prePDF, however,your look-up tables may provide less accurate interpolation. This isbecause the discrete points in the look-up tables were selected based onthe inlet temperatures as defined in prePDF.

When you are using the full equilibrium model (rich limit of 1.0), prePDFwill in most cases predict a modified equilibrium fuel temperature andcomposition. As detailed in Section 14.3.1, your inlet velocity at gas-phase fuel inlets should be based on the density corresponding to thisadjusted temperature and composition. The temperature at the gas-phase fuel inlet, however, should be retained at the value you used todefine the fuel inlet in prePDF. Similar equilibrium adjustments mayoccur, under unusual circumstances, at oxidizer inlets and your inputsshould be determined in the same way.

Wall thermal boundary conditions should also be defined for non-adiabaticnon-premixed combustion calculations. You can use any of the standardconditions available in FLUENT, including specified wall temperature,heat flux, external heat transfer coefficient, or external radiation. If ra-diation is to be included within the domain, the wall emissivity shouldbe defined as well. See Section 6.13.1 for details about thermal boundaryconditions at walls.



Step 4: Define Physical Properties

When you use the non-premixed combustion model, the material usedfor all fluid zones is automatically set to pdf-mixture. This material is aspecial case of the mixture material concept discussed in Section 13.1.2.The constituent species of this mixture are the species that you definedin prePDF; you cannot change them in FLUENT. When the non-premixedmodel is used, heat capacities, molecular weights, and enthalpies offormation for each species considered are extracted from the chemicaldatabase, so you will not modify any properties for the constituentspecies in the PDF mixture. For the PDF mixture itself, the densityis determined from the look-up tables and the specific heat is deter-mined via the mixing law discussed in Section 7.5.4, using specific heatvalues for the constituent species obtained from the chemical database(thermodb.scm).

The physical property inputs for a non-premixed combustion problem aretherefore only the transport properties (viscosity, thermal conductivity,etc.) for the PDF mixture. To set these in the Materials panel, choosemixture as the Material Type, pdf-mixture (the default, and only choice)in the Mixture Materials list, and set the desired values for the transportproperties.

Define −→Materials...

See Chapter 7 for details about setting physical properties. The trans-port properties in a non-premixed combustion problem can be definedas functions of temperature, if desired, but not as functions of composi-tion. In practice, since turbulence effects will dominate, it will be of littlebenefit to include even the temperature dependence of these transportproperties.

If you are modeling radiation heat transfer, you will also input radiationproperties, as described in Section 7.6. Composition-dependent absorp-tion coefficients (using the WSGGM) are allowed.

Step 5: Solve the Flow Problem

The next step in the non-premixed combustion modeling process in FLU-ENT is the solution of the mixture fraction and flow equations. First,



initialize the flow. By default, the mixture fraction and its variance haveinitial values of zero, which is the recommended value; you should gener-ally not set non-zero initial values for these variables. See Section 22.13for details about solution initialization.

Solve −→ Initialize −→Initialize...

Next, begin calculations in the usual manner.

Solve −→ Iterate...

During the calculation process, FLUENT reports residuals for the mixturefraction and its variance in the fmean and fvar columns of the residualreport:

iter cont x-vel y-vel k epsilon fmean fvar28 1.57e-3 4.92e-4 4.80e-4 2.68e-2 2.59e-3 9.09e-1 1.17e+029 1.42e-3 4.43e-4 4.23e-4 2.48e-2 2.30e-3 8.89e-1 1.15e+030 1.28e-3 3.98e-4 3.75e-4 2.29e-2 2.04e-3 8.88e-1 1.14e+0

(For two-mixture-fraction calculations, columns for psec and pvar willalso appear.)

Under-Relaxation Factors for PDF Equations

The transport equations for the mean mixture fraction and mixture frac-tion variance are quite stable and high under-relaxation can be used whensolving them. By default, an under-relaxation factor of 1 is used for themean mixture fraction (and secondary partial fraction) and 0.9 for themixture fraction variance (and secondary partial fraction variance). Ifthe residuals for these equations are increasing, you should consider de-creasing these under-relaxation factors, as discussed in Section 22.9.

Density Under-Relaxation

One of the main reasons a combustion calculation can have difficultyconverging is that large changes in temperature cause large changes indensity, which can, in turn, cause instabilities in the flow solution. FLU-ENT allows you to under-relax the change in density to alleviate this



difficulty. The default value for density under-relaxation is 1, but if youencounter convergence trouble you may wish to reduce this to a valuebetween 0.5 and 1 (in the Solution Controls panel).

Tuning the PDF Parameters for Two-Mixture-Fraction Calculations

For cases that include a secondary stream, the PDF integrations areperformed inside FLUENT. The parameters for these integrations aredefined in the Species Model panel (Figure 14.3.25).

Define −→ Models −→Species...

Figure 14.3.25: The Species Model Panel for a Two-Mixture-Fraction Calcu-lation

The parameters are as follows:

Compressibility Effects (non-adiabatic systems only) tells FLUENT to up-



date the density, temperature, species mass fraction, and enthalpyfrom the PDF tables to account for the varying pressure of thesystem.

Probability Density Function specifies which type of PDF should be used.You can pick either double delta (the default) or beta in the drop-down list. These choices are the same as what you saw in prePDFfor the single-mixture-fraction case. (See Section 14.1.2.) Thedouble delta PDF has the advantage of being faster than the betaPDF, and it is the default. The beta function, however, may be amore accurate representation of the PDF.

Number of Flow Iterations Per Property Update specifies how often thedensity, temperature, and specific heats are updated from the look-up table. Remember that when you are calculating two mixturefractions, the updating of properties includes computation of thePDFs and can be quite CPU-intensive. You should generally notreduce the Number of Flow Iterations Per Property Update belowthe default value of 10, unless you are experiencing convergencedifficulties.

For simulations involving non-adiabatic multiple strained flamelets,looking up the four-dimensional PDF tables can be CPU-intensiveif a large number of species exist in the flamelet files. In suchcases, the Number of Flow Iterations Per Property Update controlsthe updating of the mean molecular weight, which involves lookingup the PDF tables for the species mass fractions.

Step 6: Postprocessing the Non-Premixed Model Results inFLUENT

The final step in the non-premixed combustion modeling process is thepostprocessing of species concentrations and temperature data from themixture fraction and flow-field solution data. The following variables areof particular interest:

• Mean Mixture Fraction (in the Pdf... category)

• Secondary Mean Mixture Fraction (in the Pdf... category)



• Mixture Fraction Variance (in the Pdf... category)

• Secondary Mixture Fraction Variance (in the Pdf... category)

• Fvar Prod (in the Pdf... category)

• Fvar2 Prod (in the Pdf... category)

• Mass fraction of (species-n) (in the Species... category)

• Mole fraction of (species-n) (in the Species... category)

• Concentration of (species-n) (in the Species... category)

• Static Temperature (in the Temperature... category)

• Enthalpy (in the Temperature... category)

These quantities can be selected for display in the indicated categoryof the variable-selection drop-down list that appears in postprocessingpanels. See Chapter 27 for their definitions.

In all cases, the species concentrations are derived from the mixture frac-tion/variance field using the look-up tables. Note that temperature andenthalpy can be postprocessed even when your FLUENT model is an adi-abatic non-premixed combustion simulation in which you have not solvedthe energy equation. In both the adiabatic and non-adiabatic cases, thetemperature is derived from the look-up table created in prePDF.

Figures 14.3.26 and 14.3.27 illustrate typical results for a methane diffu-sion flame modeled using the non-premixed approach.

14.3.4 Modeling Liquid Fuel Combustion Using theNon-Premixed Model

Liquid fuel combustion can be modeled with the non-premixed model.In prePDF, the fuel vapor, which is produced by evaporation of the liq-uid fuel, is defined as the fuel stream, and the oxidizer (e.g., air) inletcomposition is defined as the oxidizer stream. (See Section 14.3.1.) Theliquid fuel that evaporates within the domain appears as a source of thefuel mixture fraction, f , when the non-premixed model is used.



Contours of Mean Mixture Fraction

1.00e+00

9.00e-01

8.00e-01

7.00e-01

6.00e-01

5.00e-01

4.00e-01

3.00e-01

2.00e-01

1.00e-01

0.00e+00

Figure 14.3.26: Predicted Contours of Mixture Fraction in a MethaneDiffusion Flame

Contours of Mass fraction of co2

1.35e-01

1.22e-01

1.08e-01

9.48e-02

8.12e-02

6.77e-02

5.42e-02

4.06e-02

2.71e-02

1.35e-02

0.00e+00

Figure 14.3.27: Predicted Contours of CO2 Mass Fraction Using theNon-Premixed Combustion Model



Within FLUENT, you define the liquid fuel model in the usual way. Thegas phase (oxidizer) flow inlet is modeled using an inlet mixture fractionof zero and the fuel droplets are introduced as discrete-phase injections(see Section 19.9). The property inputs for the liquid fuel droplets areunaltered by the non-premixed model, and should be input as usual(see Section 19.11). Note that when you are requested to input the gasphase species destination for the evaporating liquid, you should inputthe species that comprises the fuel stream as defined in prePDF.

Note that if the fuel stream was defined as a mixture of componentsin prePDF, you should simply select one of these components as the“evaporating species”. FLUENT will ensure that the mass evaporatedfrom the liquid droplet enters the gas phase as a source of the fuel mixturethat you defined in prePDF. The evaporating species you select hereis used only to compute the diffusion controlled driving force in theevaporation rate.

14.3.5 Modeling Coal Combustion Using the Non-PremixedModel

Coal combustion can be modeled with the non-premixed approach, us-ing either a single mixture fraction (fuel stream) or using two mixturefractions (fuel stream and secondary stream). This section describesthe modeling options and special input procedures for coal combustionmodels using the non-premixed approach.

Note that prePDF 4 uses a different formulation for coal combustion than!that used in prePDF 1. If you have a coal combustion system definedin a prePDF 1 input file, you can read that input file into prePDF 4 butyou will have to modify the definition of the coal composition beforecomputing the new prePDF 4 PDF look-up tables. Using a prePDF 1input file in prePDF 4, without making the required modifications tothe definition of coal composition, will result in an incorrect PDF look-up table. The correct prePDF 4 procedures for definition of the fuelcomposition are described in this section.

PDF files created by prePDF 2 or 3 will have the correct inputs for coalcomposition, so you need not change them. You may however, want toconsider recomputing the look-up tables in prePDF 4, as described in



Section 14.3.3. If you have a two-mixture-fraction PDF file from prePDF2 or 3, you will need to read it into prePDF 4 and write it out in FLUENT6 format.

Non-Premixed Modeling Options for Coal Combustion

There are three basic non-premixed modeling options for coal combus-tion:

• When coal is the only fuel in the system, you can model the coal us-ing two mixture fractions. When this approach is used, one streamis used to represent the char and the other stream is used to rep-resent volatiles. Generally, the char stream composition is repre-sented as 100% C(s). The volatile stream composition is definedby selecting appropriate species and setting their mole or massfractions. Alternately, you can use the empirical method (input ofatom fractions) for defining these compositions.

Using two mixture fractions to model coal combustion is more ac-!curate than using one mixture fraction as the volatile and charstreams are modeled separately. However, the two-mixture-fractionmodel incurs significant additional computational expense since themulti-dimensional PDF integrations are performed at run-time.

• When coal is the only fuel in the system, you can choose to modelthe coal using a single mixture fraction (the fuel stream). Whenthis approach is adopted, the fuel composition you define includesboth volatiles species and char. Char is typically represented byincluding C(s) in the species list. You can define the fuel streamcomposition by selecting appropriate species and setting their molefractions, or by using the empirical method (input of atom frac-tions). Definition of the composition is described in detail below.

Using a single mixture fraction for coal combustion is less accurate!than using two mixture fractions. However, convergence in FLU-ENT should be substantially faster than the two-mixture-fractionmodel.

• When coal is used with another (gaseous or liquid) fuel, you mustmodel the coal with one mixture fraction and use a second mix-



ture fraction to represent the second (gaseous or liquid) fuel. Thestream associated with the coal composition is defined as detailedbelow for single-mixture-fraction models.

Defining the Coal Composition in prePDF:Single-Mixture-Fraction Models

When coal is modeled using a single mixture fraction (the fuel stream),the fuel stream composition can be input using one of the following twomethods:

• Conventional approach: Define the mixture of species in the coaland their mole or mass fractions in the fuel stream:

1. Use the Define Species panel to select a list of species presentin the coal combustion system (e.g., C3H8, CH4, CO, CO2,H2O(l), H2O, H2, OH, H, O, C(s), O2, and N2).

2. Using the Composition panel, select the fuel stream and thendefine the mole or mass fractions of each of the species that arepresent in the coal fuel. Note that C(s) is used to representthe char content of the coal. For example, consider a coalthat has a molar composition of 40% volatiles and 60% charon a dry ash free (DAF) basis. Assuming the volatiles canbe represented by an equimolar mixture of C3H8 and CO,the fuel stream composition defined in the Composition panelwould be C3H8=0.2, CO = 0.2, and C(s)=0.60. Note that thecoal composition should always be defined in prePDF on anash free basis, even if ash will be considered in the FLUENTcalculation.

The following table illustrates the conversion from a typicalmass-based proximate analysis to the species fraction inputsrequired by prePDF. Note that the conversion requires thatyou make an assumption regarding the species representingthe volatiles. Here, the volatiles are assumed to exist as anequimolar mix of propane and carbon monoxide.



Proximate Analysis Weight % kg Moles Mole(DAF) (DAF) Fraction

(DAF)Volatiles 30

C3H8 0.1833 .00417 0.0715CO 0.1167 .00417 0.0715

Fixed Carbon (C(s)) 60 0.6 0.050 0.8570Ash 10 - - -(Total) .05834 1.0

Moisture in the coal can be considered by adding it in the fuelcomposition as liquid water, H2O(l). The moisture can also bedefined as water vapor, H2O, provided that the correspondinglatent heat is included in the discrete phase material inputsin FLUENT.

• Empirical fuel approach: Use the Empirically Defined Streams op-tion for the fuel stream. This method is ideal if you have an ele-mental analysis of the coal.

1. Use the Define Species panel to select a list of species presentin the coal combustion system (e.g., C3H8, CH4, CO, CO2,H2O(l), H2O, H2, OH, C(s), O2, and N2). In addition, youmust select atomic C, H, N, S, and O.

2. In the Composition panel, select the fuel stream and then de-fine the molar atom fractions of C, H, N, S, and O in the fuelstream. In addition, you will input the lower heating valueand mean specific heat of the coal. prePDF will use these in-puts to determine the mole fractions of the chemical speciesyou have included in the system.

Note that for both of these composition input methods, you should takecare to distinguish atomic carbon, C, from solid carbon, C(s). Atomiccarbon should only be selected if you are using the empirical fuel inputmethod.



Defining the Coal Composition in prePDF: Two-Mixture-FractionModels

If your FLUENT model will represent the coal using both the fuel streamand the secondary stream, one stream is used to represent char and theother stream is used to represent volatiles. (Typically the fuel stream isused to represent the char and the secondary stream is used to representvolatiles, but the reverse is also possible. In the procedures below, itis assumed that the fuel stream represents the char and the secondarystream represents the volatiles.)

The fuel stream and secondary stream compositions can be input usingone of the following two methods:

• Conventional approach: Define the mixture of species in the coaland their mole or mass fractions in the fuel and secondary streams:

1. Use the Define Species panel to select a list of species presentin the coal combustion system (e.g., C3H8, CH4, CO, CO2,H2O(l), H2O, H2, OH, H, O, C(s), O2, and N2).

2. Using the Composition panel, select the fuel stream and definethe mole or mass fractions of species used to represent thechar. Generally, you will input 100% C(s) for the fuel stream.

3. Using the Composition panel, select the secondary stream anddefine the mole or mass fractions of species used to representthe volatiles.

• Empirical fuel approach: Use the Empirically Defined Streams op-tion for the volatile (in this case, secondary) stream. This methodis ideal if you have an elemental analysis of the coal.

1. Use the Define Species panel to select a list of species presentin the coal combustion system (e.g., C3H8, CH4, CO, CO2,H2O(l), H2O, H2, OH, C(s), O2, and N2). In addition, youmust select atomic C, H, N, S, and O.

2. Using the Composition panel, select the fuel stream and definethe mole or mass fractions of species used to represent thechar. Generally, you will input 100% C(s) for the fuel stream.



This procedure assumes that you are not using an empiricalinput for the fuel stream (char).

3. Using the Composition panel, select the secondary stream anddefine the atom fractions of C, H, N, S, and O in the volatiles.In addition, you will input the lower heating value and meanspecific heat of the coal. prePDF will use these inputs todetermine the mole fractions of the chemical species you haveincluded in the system. For example, consider coal with thefollowing DAF (dry ash free) data and elemental analysis:

Proximate Analysis Wt % Wt %(dry) (DAF)

Volatiles 28 30.4Char (C(s)) 64 69.6Ash 8 -

Element Wt % (DAF) Wt % (DAF)C 89.3 89.3H 5.0 5.0O 3.4 3.4N 1.5 2.3S 0.8 -

(Note that in the final column, for modeling simplicity, thesulfur content of the coal has been combined into the nitrogenmass fraction.)

You can combine the proximate and ultimate analysis datato yield the following elemental composition of the volatilestream:Element Mass Wt % Moles Mole FractionC (89.3 - 69.6) 0.65 5.4 0.24H 5.0 0.16 16 0.70O 3.4 0.11 0.7 0.03N 2.3 0.08 0.6 0.03Total 30.4 22.7

This adjusted composition is used to define the secondarystream (volatile) composition.



The lower heating value of the volatiles can be computed fromthe known heating value of the coal and the char (DAF):

– LCVcoal,DAF = 35.3 MJ/kg

– LCVchar,DAF = 32.9 MJ/kg

You can compute the heating value of the volatiles as

LCVvol =35.3 MJ/kg − 0.696 × 32.9 MJ/kg

0.304

or

LCVvol = 40.795 MJ/kg

Note that for both of these composition input methods, you should takecare to distinguish atomic carbon, C, from solid carbon, C(s). Atomiccarbon should only be selected if you are using the empirical fuel inputmethod.

Coal Modeling Inputs in FLUENT

Within FLUENT, the coal combustion simulation is defined as usual whenthe non-premixed combustion model is selected. The air (oxidizer) inletsare defined as having a mixture fraction value of zero. No gas phase fuelinlets will be included and the sole source of fuel will come from thecoal devolatilization and char burnout. The coal particles are defined asinjections using the Set Injection Properties panel in the usual way, andphysical properties for the coal material are specified as described in Sec-tion 19.11. You should keep in mind the following issues when defininginjections and discrete-phase material properties for coal materials:

• In the Set Injection Properties panel, you will specify for the Ox-idizing Species one of the components of the oxidizer stream, asdefined in prePDF. This species concentration field will be used tocalculate the diffusion-controlled driving force in the char burnoutlaw (if applicable).

The specification of the char and volatile streams differs dependingon the type of model you are defining:



– If the coal is modeled using a single mixture fraction, thegas phase species representing the volatiles and the char com-bustion are represented by the mixture fraction used by thenon-premixed combustion model.

– If the coal is modeled using two mixture fractions, rather thanspecifying a destination species for the volatiles and char, youwill instead specify the Devolatilizing Stream and Char Stream.

– If the coal is modeled using one mixture fraction, and an-other fuel is modeled using a second mixture fraction, youshould specify the stream representing the coal as both theDevolatilizing Stream and the Char Stream.

• In the Materials panel, Vaporization Temperature should be set equalto the fuel inlet temperature used in prePDF. This temperaturecontrols the onset of the devolatilization process. Stated inversely,the fuel inlet temperature that you define in prePDF should be setto the temperature at which you want to initiate devolatilization.This way, the look-up tables produced by prePDF will include theappropriate temperature range for your process.

• In the Materials panel, Volatile Component Fraction and CombustibleFraction should be set to values that are consistent with the coalcomposition used to define the fuel stream (and secondary stream)composition in prePDF.

• Also in the Materials panel, you will be prompted for the BurnoutStoichiometric Ratio and for the Latent Heat. The Burnout Stoichio-metric Ratio is used in the calculation of the diffusion controlledburnout rate but has no other impact on the system chemistrywhen the non-premixed combustion model is used. The BurnoutStoichiometric Ratio is the mass of oxidant required per mass ofchar. The default value of 1.33 assumes that C(s) is oxidized byO2 to yield CO. The Latent Heat input determines the heat requiredto generate the vapor phase volatiles defined in the non-premixedsystem chemistry. You can usually set this value to zero whenthe non-premixed model is used, since your definition of volatilespecies will have been based on the overall heating value of the



coal. However, if the coal composition defined in prePDF includeswater content, the latent heat should be set as follows:

– Set latent heat to zero if the water content of the coal has beendefined as H2O(L). In this case, the prePDF system chemistrywill include the latent heat required to vaporize the liquidwater.

– Set latent heat to the value for water (2.25 × 106 J/kg), ad-justed by the mass loading of water in the volatiles, if thewater content of the coal has been defined using water va-por, H2O. In this case, the water content you defined will beevolved along with the other species in the coal but the prePDFsystem chemistry does not include the latent heat effect.

• The Density you define for the coal in the Materials panel shouldbe the apparent density, including ash content. This is to be dis-tinguished from the input of C(s) density in prePDF, where thedensity of dry ash free char should be used.

• You will not be asked to define the Heat of Reaction for Burnoutfor the char combustion. This quantity is computed based on yourinputs to prePDF.

Postprocessing Non-Premixed Models of Coal Combustion

FLUENT reports the rate of volatile release from the coal using the DPMEvaporation/Devolatilization postprocessing variable. The rate of charburnout is reported in the DPM Burnout variable.



14.4 The Laminar Flamelet Model

The laminar flamelet approach models a turbulent flame brush as anensemble of discrete, steady laminar flames, called flamelets. The in-dividual flamelets are assumed to have the same structure as laminarflames in simple configurations, and are obtained by experiments or cal-culations. Using detailed chemical mechanisms, prePDF can calculatelaminar opposed-flow diffusion flamelets for non-premixed combustion.The laminar flamelets are then embedded in a turbulent flame usingstatistical PDF methods.

The advantage of the laminar flamelet approach is that realistic chemi-cal kinetic effects can be incorporated into turbulent flames. As in theequilibrium approach of Section 14.3, the chemistry can be preprocessedand tabulated, offering tremendous computational savings. However, thelaminar flamelet model is limited to flames with relatively fast chemistry.The flame is assumed to respond instantaneously to the aerodynamicstrain, and thus the model cannot capture deep non-equilibrium effectssuch as ignition, extinction, and slow chemistry (like NOx).

Information about the flamelet model is presented in the following sec-tions:

• Section 14.4.1: Introduction

• Section 14.4.2: Restrictions and Assumptions

• Section 14.4.3: The Flamelet Concept

• Section 14.4.4: Flamelet Generation

• Section 14.4.5: Flamelet Import

• Section 14.4.6: User Inputs for the Laminar Flamelet Model

For general information about the mixture fraction model, see Section 14.1.



14.4.1 Introduction

In a diffusion flame, at the molecular level, fuel and oxidizer diffuseinto the reaction zone—where they encounter high temperatures andradical species—and ignite. More heat and radicals are generated inthe reaction zone, and some diffuse out. In near-equilibrium flames, thereaction rate is much faster than the diffusion rate. However, as the flameis stretched and strained by the turbulence, species and temperaturegradients increase, and radicals and heat more quickly diffuse out of theflame. The species have less time to reach chemical equilibrium, and thelocal non-equilibrium increases.

The laminar flamelet model is suited to predict moderate chemical non-equilibrium in turbulent flames due to aerodynamic straining by theturbulence. The chemistry, however, is assumed to respond rapidly tothis strain, so as the strain relaxes, the chemistry relaxes to equilibrium.

When the chemical time-scale is comparable to the fluid convectiontime-scale, the species can be considered to be in global chemical non-equilibrium. Such cases include NOx formation and low-temperatureCO oxidation. The laminar flamelet model is not suitable for such slow-chemistry flames. Instead, you can model slow chemistry using the tracespecies assumption (as in the NOx model), or using the EDC model (seeSection 13.1.1).

14.4.2 Restrictions and Assumptions

The following restrictions apply to all flamelet models in FLUENT:

• Only a single mixture fraction can be modeled; two-mixture-fractionflamelet models are not allowed.

• The mixture fraction is assumed to follow the β-function PDF,and the scalar dissipation is assumed to follow the double-delta-function PDF.

• Empirically-based streams cannot be used with the flamelet model.



14.4.3 The Flamelet Concept

Overview

The flamelet concept views the turbulent flame as an ensemble of thin,laminar, locally one-dimensional flamelet structures embedded withinthe turbulent flow field [27, 176, 177] (see Figure 14.4.1).

A common laminar flame used to represent a flamelet in a turbulent flowis the counterflow diffusion flame. This geometry consists of opposed,axisymmetric fuel and oxidizer jets. As the distance between the jets isdecreased and/or the velocity of the jets increased, the flame is strainedand increasingly departs from chemical equilibrium until it is eventuallyextinguished. The species mass fraction and temperature fields can bemeasured in laminar counterflow diffusion flame experiments, or, mostcommonly, calculated. For the latter, a self-similar solution exists, andthe governing equations can be simplified to one dimension, where com-plex chemistry calculations can be affordably performed.

In the laminar counterflow flame, the mixture fraction, f , (see Sec-tion 14.1.2 for definition) decreases monotonically from unity at the fueljet to zero at the oxidizer jet. If the species mass fraction and tempera-ture along the axis are mapped from physical space to mixture fractionspace, they can be uniquely described by two parameters: the mixturefraction and the strain rate (or, equivalently, the scalar dissipation, χ,defined in Equation 14.4-2). Hence, the chemistry is reduced and com-pletely described by the two quantities, f and χ.

This reduction of the complex chemistry to two variables allows theflamelet calculations to be preprocessed, and stored in look-up tables.By preprocessing the chemistry, computational costs are reduced consid-erably.

The balance equations, solution methods, and sample calculations of thecounterflow laminar diffusion flame can be found in several references.Comprehensive reviews and analyses are presented in [27, 51].



turbulent flamelaminar flamelet structure(see detail below)

x

fuel oxidizer

fuel-oxidizer distance

flame

velocity (u )

velocitygradient (a )

temperature (T )

oxidizer composition

velocity (u )

velocitygradient (a )

temperature (T )

fuel composition

ox

ox

ox

fuel

fuel

fuel

Figure 14.4.1: Laminar Opposed-Flow Diffusion Flamelet



Strain Rate and Scalar Dissipation

A characteristic strain rate for an opposed-flow diffusion flamelet can bedefined as as = v/2d, where v is the speed of the fuel and oxidizer jets,and d is the distance between the jet nozzles.

Instead of using the strain rate to quantify the departure from equilib-rium, it is expedient to use the scalar dissipation, denoted by χ. Thescalar dissipation is defined as

χ = 2D|∇f |2 (14.4-1)

where D is a representative diffusion coefficient.

Note that the scalar dissipation, χ, varies along the axis of the flamelet.For the counterflow geometry, the flamelet strain rate as can be related tothe scalar dissipation at the position where f is stoichiometric by [176]:

χst =as exp

(−2[erfc−1(2fst)]2

)π

(14.4-2)

whereχst = scalar dissipation at f = fst

as = characteristic strain ratefst = stoichiometric mixture fractionerfc−1 = inverse complementary error function

Physically, as the flame is strained, the width of the reaction zone di-minishes, and the gradient of f at the stoichiometric position f = fst

increases. The instantaneous stoichiometric scalar dissipation, χst, isused as the essential non-equilibrium parameter. It has the dimensionss−1 and may be interpreted as the inverse of a characteristic diffusiontime. In the limit χst → 0 the chemistry tends to equilibrium, and as χst

increases due to aerodynamic straining, the non-equilibrium increases.Local quenching of the flamelet occurs when χst exceeds a critical value.



Embedding Laminar Flamelets in Turbulent Flames

A turbulent flame brush is modeled as an ensemble of discrete laminarflamelets. Since, for adiabatic systems, the species mass fraction andtemperature in the laminar flamelets are completely parameterized by fand χst, mean species mass fraction and temperature in the turbulentflame can be determined from the PDF of f and χst as

φ =∫ ∫

φ(f, χst)p(f, χst) df dχst (14.4-3)

where φ is a representative scalar, such as a species mass fraction, tem-perature, or density.

In prePDF, f and χst are assumed to be statistically independent, so thejoint PDF p(f, χst) can be simplified as pf (f)pχ(χst). A β PDF shapeis assumed for pf , and transport equations for f and f ′2 are solved inFLUENT to specify pf . A double-delta PDF is assumed for pχ, which,like the β PDF, is specified by its first two moments. The first moment,namely the mean scalar dissipation, χst, is modeled in FLUENT as

χst =Cχεf

′2

k(14.4-4)

where Cχ is a constant with a default value of 2. The scalar dissipationvariance is assumed constant and specified by you in prePDF. Accordingto [27], it has become common practice to ignore the scalar dissipationfluctuations. Note, however, that a non-zero scalar dissipation variancewill result in a smoother variation of the physical properties along thescalar dissipation coordinate.

To avoid the PDF convolutions at FLUENT run-time, the integrations inEquation 14.4-3 are preprocessed in prePDF and stored in look-up tables.For adiabatic flows, single-flamelet tables have two dimensions: f andf ′2. The multiple-flamelet tables have the additional dimension χst.

For non-adiabatic laminar flamelets, the additional parameter of en-thalpy is required. However, the computational cost of modeling flameletsover a range of enthalpies is prohibitive, so some approximations are



made. Heat gain/loss to the system is assumed to have a negligible ef-fect on the species mass fractions, and the flamelet mass fractions atpredefined enthalpy levels are used [20, 164]. The temperature is thencalculated from Equation 14.1-14 for a range of mean enthalpy gain/loss,H∗. Accordingly, mean temperature and density PDF tables have an ex-tra dimension of mean enthalpy.

In prePDF, you can either generate your own flamelets, or import themas flamelet files calculated with other stand-alone packages. Such stand-alone codes include OPPDIF [147], RIF [8, 9, 181] and RUN-1DL [179].prePDF can import flamelet files in OPPDIF format or standard flameletfile format.

Instructions for generating and importing flamelets are provided in Sec-tion 14.4.4 and Section 14.4.5.

14.4.4 Flamelet Generation

The laminar counterflow diffusion flame equations can be transformedfrom physical space (with x as the independent variable) to mixturefraction space (with f as the independent variable) [182]. In prePDF, asimplified set of the mixture fraction space equations are solved [181].Here, N equations are solved for the species mass fractions, Yi,

ρ∂Yi

∂t=

12ρχ

1Lei

∂2Yi

∂f2+ Si

−12∂Yi

∂f

[ρχ

1Le2

i

∂Lei

∂f

]

−12∂Yi

∂f

[12

(1 − 1

Lei

)(∂ρχ

∂f+ ρχ

cpk

∂(k/cp)∂f

)](14.4-5)

and one equation for temperature:

ρ∂T

∂t=

12ρχ∂2T

∂f2− 1cp

∑i

H∗i Si



+1

2cpρχ

[∂cp∂f

+∑

i

1Lei

cp,i∂Yi

∂f

]∂T

∂f

− 1cp

[4σp

∑i

Xiai(T 4 − T 4b )

](14.4-6)

The notation in Equations 14.4-5 and 14.4-6 is as follows: Yi, T , ρ, andf are the ith species mass fraction, temperature, density, and mixturefraction, respectively. Lei is the ith species Lewis number, as definedin Equation 13.1-4. k, cp,i, and cp are the thermal conductivity, ithspecies specific heat, and mixture-averaged specific heat, respectively.Si is the ith species reaction rate, and H∗

i is the specific enthalpy of theith species.

The scalar dissipation, χ, must be modeled across the flamelet. Anextension of Equation 14.4-2 to variable density is used [114]:

χ(f) =as

4π3(√ρ∞/ρ+ 1)2

2√ρ∞/ρ+ 1

exp(−2[erfc−1(2f)]2

)(14.4-7)

The last term in Equation 14.4-6 is an optically thin model for radiativeenergy loss from the flamelet. Here, σ is the Stefan-Boltzmann constant,p is the pressure, Xi is the ith species mole fraction, ai are polynomialcoefficients for the Planck mean absorption coefficients (taken from [83]),and Tb is the far-field (background) temperature. Including the radiationterm offers the capability of slightly increased accuracy, but may causeflamelets to be extinguished at low strain rates. Hence, the radiationterm should be enabled with caution.

The default setting in FLUENT is Lei = 1, although prePDF offers the!capability to include differential diffusion effects. When activated, theLewis numbers are automatically calculated from Equation 13.1-4. How-ever, the mixture fraction space equations contain considerable simpli-fication, and most often, better results are obtained with the defaultLei = 1 setting. This default is recommended, especially for highly dif-fusive species such as H2. The differential diffusion option should beused only to “tweak” a Lei = 1 solution.



You can adjust the number of grid points used to discretize the mixturefraction space. Since the species mass fractions and temperature aresolved coupled and implicit in f space, the memory and time require-ments can increase drastically with the number of f grid points, andmoderate values are recommended.

prePDF also provides parameters to control the stability of the solution ofEquations 14.4-5 and 14.4-6. You can adjust two multiplication factorsfor the solution’s time steps if the solution diverges.

Non-Adiabatic Laminar Flamelets

For non-adiabatic flamelets, prePDF follows the approach of [20, 164] andassumes that flamelet species profiles are unaffected by heat loss/gainfrom the flamelet. This implementation treats the heat losses accuratelyand consistently. Furthermore, no special non-adiabatic flamelet profilesneed to be generated, avoiding a very cumbersome preprocessing step.In addition, the compatibility of prePDF and FLUENT with externalflamelet generation packages (e.g., OPPDIF, RIF, RUN-1DL) is retained.The disadvantage to this model is that the effect of the heat losses onthe species mass fraction is not taken into account. Also, the effect ofthe heat loss on the extinction limits is not taken into account.

In the equilibrium non-premixed model, the temperature limits are con-stant values Tmin and Tmax. For the non-adiabatic flamelet model, thesetemperature limits are surfaces or functions of mixture fraction andscalar dissipation to more closely bound the enthalpy domain.

The bottom temperature surface, Tmin(f, χ), is calculated as the mini-mum of the temperature from the flamelet calculation at point (f ,χ) andTad(f)−∆T−, but cannot be lower than the globally lowest temperaturein the flamelet calculation, TMIN:

Tmin(f, χ) = max(TMIN,min[(Tad(f) − ∆T−), Tfl(f, χ)]) (14.4-8)

The top temperature curve, Tmax(f, χ), is calculated as the maximumof the maximum environment temperature defined by you, TMAX, thetemperature from the flamelet calculation at point f , and Tad(f)+∆T+:



Tmax(f, χ) = max(TMAX, Tad(f) + ∆T+, Tfl(f, χ)) (14.4-9)

wheref = mixture fractionχ = scalar dissipationTMIN = globally lowest temperatureTMAX = maximum boundary (e.g., hot wall or inlet)

temperature in domain∆T− = maximum temperature drop expected

because of heat losses∆T+ = maximum temperature rise over the

adiabatic temperature curveTfl = temperature in flamelet profileTad(f) = adiabatic (equilibrium) flame temperature

After flamelet generation, the flamelet profiles are convoluted with theassumed-shape PDFs as in Equation 14.4-3, and then tabulated for look-up in FLUENT. You can set the resolution of the look-up table. Theassumption is made that both the enthalpy loss/gain and the scalar dis-sipation do not fluctuate. The PDF tables have the following dimensions:

Tmean(fmean, fvar,H∗, χ)

Yi,mean(fmean, fvar,H∗) for χ = 0 (i.e., equilibrium solution)

Yi,mean(fmean, fvar, χ) for χ 6= 0ρmean(fmean, fvar,H

∗, χ)

During the FLUENT solution, the equations for the mean mixture frac-tion, mixture fraction variance, and mean enthalpy are solved. Thescalar dissipation field is calculated from the turbulence field and themixture fraction variance. The mean values of cell temperature, density,and species mass fraction are obtained from the PDF look-up table



14.4.5 Flamelet Import

prePDF can import one or more flamelet files, convolute these flameletswith the assumed-shape PDFs (see Equation 14.4-3), and construct look-up tables for use in FLUENT. The flamelet files can be generated inprePDF, or with separate, stand-alone computer codes.

Two types of flamelet files can be imported into prePDF: binary filesgenerated by the OPPDIF code [147], and standard format files describedin Section 14.4.6 and in Peters and Rogg [179].

When flamelets are generated in physical space (such as with OPPDIF),the species and temperature vary in one spatial dimension. The speciesand temperature must then be mapped from physical space to mixturefraction space. If the diffusion coefficients of all species are equal, aunique definition of the mixture fraction exists. However, with differen-tial diffusion, the mixture fraction can be defined in a number of ways.

prePDF provides four methods of computing the mixture fraction profilealong the laminar flamelet:

• Average of C and H: Following Drake and Blint [54], the mixturefraction is calculated as the mean value of fC and fH, where fC

and fH are the mixture fraction values based on the carbon andhydrogen elements.

• Hydrocarbon formula: Following Bilger et al. [19], the mixturefraction is calculated as

f =b− boxbfuel − box

(14.4-10)

where

b = 2YC

Mw,C+ 0.5

YH

Mw,H− YO

Mw,O(14.4-11)

YC, YH, and YO are the mass fractions of carbon, hydrogen, andoxygen atoms, and Mw,C, Mw,H, and Mw,O are the molecular



weights. box and bfuel are the values of b at the oxidizer and fuelinlets.

• Nitrogen method: The mixture fraction is computed in terms ofthe mass fraction of the nitrogen species:

f =YN − YN,ox

YN,fuel − YN,ox(14.4-12)

where YN is the elemental mass fraction of nitrogen along theflamelet, YN,ox is the mass fraction of nitrogen at the oxidizer inlet,and YN,fuel is the mass fraction of nitrogen at the fuel inlet.

• Read from a file (standard format files only): This option is forflamelets solved in mixture fraction space. If you choose this method,prePDF will search for the mixture fraction keyword Z, as specifiedin [179], and retrieve the data. If prePDF does not find mixturefraction data in the flamelet file, it will instead use the hydrocarbonformula method described above.

The flamelet profiles in the multiple-flamelet data set should vary only inthe strain rate imposed; the species and the boundary conditions shouldbe the same. The formats for multiple flamelets are as follows:

• OPPDIF format: The multiple-flamelet OPPDIF files should beproduced using the CNTN keyword in the OPPDIF script. Alter-natively, you can use prePDF to merge a number of single-flameletOPPDIF files into a multiple-flamelet file.

• Standard format: If you have a set of standard format flameletfiles, you should manually merge the files into a multiple-flameletfile. (You can use a text editor or the UNIX cat command tomerge the files.) When you import the merged file into prePDF,prePDF will search for and count the occurrences of the HEADERkeyword to determine the number of flamelets in the file.

For either type of file, prePDF will determine the number of flamelet pro-files and sort them in ascending strain-rate order. For flamelets generated



in physical space, you can select one of the four methods available for thecalculation of mixture fraction. The scalar dissipation will be calculatedfrom the strain rate using Equation 14.4-2.

14.4.6 User Inputs for the Laminar Flamelet Model

Instructions for using prePDF to create PDF tables from generated andimported flamelets are provided here, along with information about re-lated files and formats.

Creating a PDF Table from Generated Laminar Flamelets

The procedures for the flamelet calculation approach described in Sec-tion 14.4.4 are presented in this section. See Section 14.3.1 for instruc-tions about starting prePDF.

Step 1: Activate Laminar Flamelet Generation

To specify the generation of a laminar flamelet library, you will use theDefine Case panel (Figure 14.4.2).

Setup −→Case...

In the Define Case panel, select the Laminar Flamelets option under Chem-istry models. Then choose Adiabatic or Non-Adiabatic under Heat transferoptions. Finally, select Generate under Flamelet options.

Step 2: Define the Flamelet

Once you have enabled the flamelet generation option, you can use theFlamelet Generation panel (Figure 14.4.3) to define the flamelet.

Setup −→Flamelet Generation...

Step 2a: Import the Chemical System Data

The first step in defining the flamelet is to read the species and reactiondefinitions for the chemical system. The species thermodynamic, trans-port and reaction data must be in CHEMKIN format [112]. Informationabout the format of these files is detailed later in this section.



Figure 14.4.2: The Define Case Panel



Figure 14.4.3: The Flamelet Generation Panel



To read the chemistry file into prePDF, click on the Chemistry Files...button in the Flamelet Generation panel. When you click this button, aSelect File dialog box will open, in which you can specify the chemistry fileto be read. Up to 100 species can be included in the flamelet calculation.If the number of species used in the flamelet calculation is greater thann, where n is the maximum number of species specified in the MemoryAllocation panel (see Section 14.3.1), prePDF will automatically selectthe n species with the largest concentrations for inclusion in the PDFfile.

After the chemistry file has been read, the species it contains will appearin the Defined Species list in the Flamelet Generation panel (as shown inFigure 14.4.3).

Step 2b: Define the Fuel and Oxidizer Compositions

To define the fuel and oxidizer compositions, you will specify the follow-ing parameters under Composition in the Flamelet Generation panel:

1. Select Fuel under Stream.

2. Select a species in the Defined Species list. Select either Mole Frac-tions or Mass Fractions under Species Composition In... and enterthe desired value in the Species Fraction field.

3. When you are satisfied with your entry, select the next species andrepeat the process until you have set all mole or mass fractions forthe fuel stream.

4. Input the mole or mass fractions for the oxidizer stream by selectingOxidiser under Stream and repeating steps 1–3.

You can check the current setting for a species in a particular streamby selecting the stream and choosing the species name in the DefinedSpecies list.



Step 2c: Set the Flamelet Parameters

In the Flamelet Generation panel, enter values for the following parame-ters under Laminar Flamelet Options.

Under Mixture Fraction, set the following parameter:

# Grid Points specifies the number of mixture fraction grid points dis-tributed between the oxidizer (f = 0) and the fuel (f = 1).Increased resolution will provide greater accuracy, but since theflamelet species and temperature are solved coupled and implicitin f space, the solution time and memory requirements increaselinearly with the number of f grid points. The default value of 32is enough to resolve the temperature distribution in most cases.

Under Scalar Dissipation, set the following parameters:

Start (χst in Equation 14.4-2) is the scalar dissipation for the first flameletin the library. When only one flamelet is generated, this will beits selected scalar dissipation. Note that prePDF will generate anequilibrium flamelet corresponding to χst = 0, so the Start scalardissipation should be set at a value near 0; the default value of 1s−1 is usually sufficient.

End is the scalar dissipation of the last flamelet if more than one lam-inar flamelet is being generated (i.e., # Grid Points for the ScalarDissipation is greater than 1). See below for details.

# Grid Points specifies the number of laminar flamelets to be calculated.

Grid Center Point is a non-dimensional parameter between 0.1 and 0.9that clusters the flamelets closer to the Start or End scalar dissi-pation values. A value of 0.5 will generate a flamelet library withequi-spaced scalar dissipation. In general, when using a large Endscalar dissipation, you may want to cluster near the Start scalardissipation and use a value between 0.1 and 0.5.

The maximum scalar dissipation in the flamelet file (End χst) should bethe scalar dissipation at the extinction limit; i.e., the maximum scalar



dissipation at which reaction can be sustained. The value of the scalardissipation at the extinction limit depends on the fuel and oxidizer com-position, operating pressure, and chemistry model, and thus determiningthe value may be sensitive to numerical parameters such as mixture frac-tion grid resolution in the flamelet computations.

The scalar dissipation at the extinction limit can be estimated by per-forming a series of flamelet computations at increasing scalar dissipationvalues and examining the resulting flamelet profiles

Do not include an extinguished (i.e., non-combusting) flamelet in the!multiple flamelet file. The last flamelet in the file should be the flameletat the extinction limit, and not an extinguished flamelet.

Note that the maximum scalar dissipation value in the reacting flowfield may be higher or lower than the scalar dissipation at the extinctionlimit. If the maximum scalar dissipation in the reacting flow field issignificantly lower than the extinction limit value, then the maximumscalar dissipation in the flamelet file may be reduced to a value slightlyhigher than the maximum flow-field scalar dissipation.

You may estimate the reacting flow-field scalar dissipation by followingthis approach:

1. Solve the combustion problem in FLUENT using the equilibriummodel. Later, you can use this solution as the starting point foryour laminar flamelet model by simply reading in the new laminarflamelet PDF file.

2. Create a custom field function called mean-scalar-dissipationas defined by the RHS of Equation 14.4-4, and determine the max-imum value of the function.

You can also choose whether or not to use the Differential Diffusion andInclude radiation options. See Section 14.4.4 for details.

After the flamelet parameters have been defined, click Apply in theFlamelet Generation panel to register the changes. prePDF automaticallycalculates an approximate value of the stoichiometric mixture fraction,



Stoichiometric f. Note that Stoichiometric f is calculated at the mixturefraction location with the maximum equilibrium temperature, and isthus an approximation of the actual stoichiometric mixture fraction.

After calculating the laminar flamelet (see Step 5, below), prePDF willautomatically write out the flamelet to a standard flamelet file, then im-port the flamelet and generate a PDF table. If the solution diverges,you can adjust the values of Initial CFL and Multiply factor under Solu-tion Controls to control the solution algorithm. The first time step iscalculated as the explicit diffusion stability-limited time step multipliedby the Initial CFL value. If the solution diverges before the first time stepis complete, the Initial CFL value should be lowered. Subsequent timesteps are continually multiplied by the Multiply Factor. If the solutiondiverges after the first time step, this factor should be reduced.

Step 2d: Define the Operating Conditions for the Flamelet

You can specify the flamelet operating pressure and the temperatures ofthe inlet streams using the Operating Conditions panel.

Setup −→Operating Conditions...

If you are creating a non-adiabatic flamelet, you will need to input theNonadiabatic Flamelet Temperature Limits, as described below:

Min. Temperature is the globally lowest temperature expected, TMIN inEquation 14.4-8.

Max. Temperature is the maximum boundary (hot wall or inlet stream)temperature in the domain, TMAX in Equation 14.4-9.

Temperature Drop is the maximum expected temperature drop due toheat loss from the domain (∆T− in Equation 14.4-8).

Temperature Increase is the maximum expected temperature increase dueto heat gain into the domain (∆T+ in Equation 14.4-9).

The default value of 1500 K for Temperature Drop can be used for mostcases involving hydrocarbon combustion. This value can be decreased



if the boundary conditions suggest moderate heat losses. In problemsinvolving extreme temperature conditions, it might be necessary to in-crease both the Temperature Drop and Temperature Increase values. Youcan check whether the FLUENT solution is within the PDF table tem-perature bounds using the non-premixed-combustion-parameters textcommand and setting the Enable checking of PDF table temperaturelimits? option to yes.

define −→ models −→ species −→non-premixed-combustion-parameters

See Section 14.3.1 for more information about setting operating condi-tions.

Step 3: Set the PDF Table Parameters

Once you have defined the flamelet, you will next define the PDF tableparameters using the Solution Parameters panel.

Setup −→Solution Parameters...

Under Fuel Mixture Fraction, set the number of Fuel Mixture FractionPoints and Fuel Mixture Fraction Variance Points. The default Auto-matic Distribution clustering in the mixture fraction coordinate is rec-ommended. See Section 14.3.1 for details about these parameters.

For adiabatic multiple-flamelet problems, you will next set the scalardissipation parameters in the Flamelet Parameters panel (Figure 14.4.4).

Setup −→Flamelet Parameters...

Under Scalar Dissipation, set the number of Scalar Dissipation Points toapproximately the same number of mixture fraction points in your PDFtable (Fuel Mixture Fraction Points in the Solution Parameters panel). Youcan cluster the mean scalar dissipation distribution in the PDF table bymodifying the Distribution Center Point. Finally, if you wish to use thedouble delta assumed shape PDF for the scalar dissipation, set the ScalarDissipation Variance to a value greater than zero. Please note that theassumed PDF option for scalar dissipation is available only for adiabaticmultiple-flamelet cases.



Figure 14.4.4: The Flamelet Model Parameters Panel

If you set the Scalar Dissipation Variance to 0, you will be effectively!ignoring turbulent fluctuations in the scalar dissipation, i.e., the meanand instantaneous scalar dissipation will be equal.

Step 4: Save an Input File

Next you can save an input file containing the flamelet definition andother specifications, using the File/Write/Input... menu item.

File −→ Write −→Input...

This can be important if you want to revisit your laminar flamelet setupbecause this setup is not stored in the PDF file.

Step 5: Calculate the Flamelet

To calculate the flamelet data, you will use the Calculate/Flamelet menuitem.



Calculate −→Flamelet

prePDF will immediately prompt you for the name of the flamelet file.The flamelet profiles will be written to this file automatically when thecalculation is complete. The flamelet file will be in the standard fileformat, and can be imported or merged with other flamelet files, asdescribed in later in this section.

After you specify the file name, prePDF will begin the flamelet calcula-tion, reporting its progress in the text window. Just before the flameletcalculation, prePDF will write a monitor file called FLAMELET.MON inyour working directory. This file contains the thermodynamic and chem-istry information, and can be useful for monitoring and troubleshootingflamelet generation calculations. After the flamelet calculation is com-plete, prePDF will write the flamelet to disk, and then construct the PDFfile.

If the flamelet calculation does not converge, decrease the Initial CFL (to-wards zero) and the Multiply factor (towards 1) in the Flamelet Generationpanel, and then recalculate the flamelet data.

Step 6: Save the PDF File

When prePDF has completed the PDF calculation, you can save the PDFfile as usual, with the File/Write/PDF... menu item. See Section 14.3.1.

File −→ Write −→PDF...

You can read the PDF file back into prePDF at a later time for post-!processing. You cannot, however, modify the solution parameters andrecalculate the PDF table unless you also import the original flameletfile.

Since the PDF table for multiple flamelets is three-dimensional, there areviewing options for both the instantaneous physical properties and thePDF integrated physical properties. These options are described below.

When you are satisfied with the PDF file and you have saved it, you cancontinue the non-premixed combustion modeling procedure in FLUENT.The procedures for setting up and solving a flamelet-based model in



FLUENT are the same as for other non-premixed combustion models.See Section 14.3.3 for details.

Step 7a: Postprocessing of Adiabatic Multiple-Flamelet Files

Reviewing Instantaneous Values

When you are plotting instantaneous values using the Property Curvespanel (Figure 14.4.5), you can plot the specified variable for a constantvalue of Dimensionless Scalar Dissipation. (You can also plot the variablefor a constant value of mixture fraction.)


Figure 14.4.5: The Property Curves Panel

Plotting the Scalar Dissipation Distribution

You can also plot the distribution of scalar dissipation, using the Dis-play/Scalar Dissipation menu item.

Display −→Scalar Dissipation

A scalar dissipation distribution plot is shown in Figure 14.4.6.



Scalar Dissipation (1/s)

1.00E+01

8.00E+00

6.00E+00

4.00E+00

2.00E+00

0.00E+00

6.00E+005.00E+004.00E+003.00E+002.00E+001.00E+00

prePDF V4.00

KEY

Fluent Inc.

Flamelet Number

SCALAR DISSIPATION DISTRIBUTION

Figure 14.4.6: Scalar Dissipation Distribution

Reviewing 3D Flamelet-PDF Tables

For multiple-flamelet models, you can view slices of the 3D flamelet-PDFtable using the Flamelet-PDF-Table panel (Figure 14.4.7).

Display −→Flamelet PDF Table...

The look-up tables generated for multiple-flamelet systems contain themean temperature, density, and species mass fractions as a functionof three quantities: mean mixture fraction, mixture fraction variance,and mean dimensionless scalar dissipation. Consequently, when you askprePDF to display the look-up tables, you will be displaying them slice-by-slice.

In the Flamelet-PDF-Table panel, you can select the variable to be plottedin the Plot Variable drop-down list. Next, you must define how the three-dimensional array of data points available in the look-up table is to besliced; i.e., which discrete independent variable (either f or χst,d) is tobe held constant and whether the constant value is to be selected as anumerical Value or by discretization index (Slice number).



Figure 14.4.7: The Flamelet-PDF-Table Panel

If you specify Slice as the Plot type, you will select which discretizedvariable is to be constant (Scalar Dissipation or F-Mean) and then specifythe Slice # (discretization index). For example, the panel shown inFigure 14.4.7 requests a look-up table generated at the tenth discretevalue of scalar dissipation. To generate the plot (see Figure 14.4.8), clickDisplay.

Alternately, you may want to define a slice of the 3D look-up table basedon a specific value of one of the independent quantities. When this is thecase, select Value as the Plot type. You can then select a slice of the 3Dtable that corresponds to a constant Scalar Dissipation Value or F-MeanValue, and supply the physical value of the selected quantity in the Valuefield.

Next, you can set the Refinement Factor, which determines the resolutionof the plotted curve. A refinement factor of 1.0 (the default) implies that



2.50E-01

2.00E-01

1.50E-01

1.00E-01

5.00E-02

0.00E+00

SCALED-F-VARIANCE

1.00E+00 8.00E-01 6.00E-01 4.00E-01 2.00E-01 0.00E+00

TEMPERATURE K

prePDF V4.00

1.8E+03

1.5E+03

1.2E+03

8.8E+02

5.9E+02

3.0E+02

Fluent Inc.MEAN FLAME TEMPERATURE FROM FLAMELET-PDF-TABLE

F-MEAN

SCALAR DISSIPATION SLICE NUMBER 10

Figure 14.4.8: Display of a Single Slice of a 3D Flamelet-PDF Table

the plot will use the same number of discrete points that you requestedin the Solution Parameters panel. Increasing this factor will cause prePDFto compute and display additional data points, yielding a smoother plotbut requiring some time to compute.

Finally, click the Lookup button to access the data to be plotted andthen click Display to display the plot.

Step 7b: Postprocessing of Non-Adiabatic Multiple Flamelet Files

Reviewing Non-Adiabatic Flamelet PDF Tables

For non-adiabatic flamelets, you can view slices of the four-dimensionalnon-adiabatic flamelet tables using the Nonadiabatic-Table panel (Fig-ure 14.4.9).

Display −→Nonadiabatic Table...

The Nonadiabatic-Table panel is also used for viewing the non-adiabaticPDF tables for equilibrium chemistry models. See Section 14.3.1 for



Figure 14.4.9: The Nonadiabatic-Table Panel for Flamelets

more information about using this panel.

In the case of non-adiabatic flamelets, there is the additional parameterof scalar dissipation. In addition to varying the mean enthalpy and meanmixture fraction, you can vary the display of the PDF table by changingthe value of Slice # under Scalar dissipation, which gives the table afourth “dimension”.

For the non-adiabatic multiple-flamelet PDF tables, you can plot thescalar dissipation distribution using the Display/Scalar Dissipation menuitem in the same way as for the adiabatic multiple-flamelet PDF tables.

Display −→Scalar Dissipation

Creating a PDF Table from Imported Laminar Flamelets

The procedure for importing flamelets is very similar to the flamelet gen-eration procedure described above, except that the flamelet generationstep is skipped.



Step 1: Activate Flamelet Import

In the Define Case panel (Figure 14.4.2), select the Laminar Flameletsoption under Chemistry models. Then choose Adiabatic or Non-Adiabaticunder Heat transfer options. Finally, select Import under Flamelet Options.

Setup −→Case...

Once you have enabled flamelet import, you will next define the flameletmodel parameters and solution parameters.

The parameters for the flamelet model and the solution process must!be defined before you import the flamelet file, because prePDF performsthe mixture-fraction calculations and generates the PDF look-up tableautomatically after it reads the data from the flamelet file.

Step 2: Set the PDF Table Parameters

Step 2a: Non-Adiabatic Table Parameters

If you are modeling a non-adiabatic combustor, you will need to setthe Nonadiabatic Flamelet Temperature Limits in the Operating Conditionspanel, as described in Step 2d on page 14-134.

Setup −→Operating Conditions...

Step 2b: Mixture Fraction Table Parameters

You will next set the resolution of the mixture fraction mean and variancein the Solution Parameters panel, as described in Step 3 on page 14-135.

Setup −→Solution Parameters...

Step 2c: Scalar Dissipation and Flamelet Conversion Parameters

If you are importing flamelets generated in physical space, such as withOPPDIF [147], prePDF will need to construct a mixture fraction profilefrom the species field (see Section 14.4.5). You can access the param-eters for this conversion in the Flamelet Model Parameters panel (Fig-ure 14.4.10).



Setup −→Flamelet Parameters...

Figure 14.4.10: The Flamelet Model Parameters Panel

The parameters for the flamelet import are as follows:

Pressure Conversion Factor specifies a factor for converting pressure datato SI units. If you are reading OPPDIF data, this parameter shouldbe set to 1, since OPPDIF files report the pressure in Pa. If youare importing standard flamelet files, which are formatted files, youshould check for the units of pressure reported in the file and inputthe appropriate conversion factor for the pressure to be in Pa.

Mixture Fraction Calculation specifies which of the methods described inSection 14.4.5 should be used to compute the mixture fraction.Read From File is the default and recommended option. If theflamelet file does not contain mixture fraction data, prePDF willreport this and use the Hydrocarbon Formula option instead.



Finally, set the scalar dissipation parameters as described in Step 3 onpage 14-135.

Step 3: Import the Flamelet File

After setting the parameters, you can import the flamelet file. ForOPPDIF flamelet files, use the File/Import/Oppdif Flamelets... menuitem:

File −→ Import −→Oppdif Flamelets...

For standard format flamelet files, use the File/Import/Standard Flame-lets... menu item:

File −→ Import −→Standard Flamelets...

After you specify the name of a standard format file to be imported,prePDF will ask you if the file was written for mixture fraction values inascending (starting from the oxidizer inlet), or descending (starting fromthe fuel inlet) order. By default, it will assume descending order.

After reading the flamelet file, prePDF will report the species data andthen integrate with the β PDF to generate the PDF look-up table.

If the number of species in the flamelet file is larger than the specieslimit in prePDF, the species with the lowest mass fractions are filteredout. The temperature curves are then constructed and the enthalpydomain is discretized. After this, the PDF integrations are performed.Note that this process may take some time, as four-dimensional tablesare being generated.



Step 4: Save and Postprocess the PDF File

When prePDF has completed the PDF calculation, you can save the PDFfile as usual, with the File/Write/PDF... menu item. See Section 14.3.1for details.

File −→ Write −→PDF...

You can read the PDF file back into prePDF at a later time for post-!processing. You cannot, however, modify the solution parameters andrecalculate the PDF table unless you also import the original flameletfile.

Postprocessing for the PDF look-up table data is the same as for thePDF table data resulting from flamelet generation (see Step 6 on page14-137).

Merging Single Flamelet Files into a Multiple Flamelet Library

If you have a number of single-flamelet files, you can use prePDF to mergethem and write out a single file containing the multiple flamelets. Youcan then read this file back into prePDF.

To perform the merging, you will use the MERGE-FLAMELETS text com-mand.

FLAMELET-MODEL −→MERGE-FLAMELETS

First, you will specify a name for the merged file and the number ofsingle flamelet files. Then you will specify the name of each of the singleflamelet files. After all names are entered, prePDF will merge the flameletdata and write out the merged file.

If your flamelet files have lowercase names, be sure to enclose the file!names in quotation marks.

If you have a number of single-flamelet files in the standard format, youwill need to merge them outside of prePDF, using your text editor or theUNIX cat command.



Files for Flamelet Modeling

In this section, information is provided about the standard flamelet filesused for the flamelet generation and import, and also the CHEMKIN [112]chemistry files used in the flamelet generation. The flamelet files gen-erated by OPPDIF software [147] or in OPPDIF format are binary, soformatting information is not relevant for them.

The Thermodynamic and Transport Databases

prePDF uses a thermodynamic database and must be able to access thedatabase file, THERMO.DB. This file must be present in the directory whereyou run prePDF, or it must be accessed through an environment variable,THERMODB, which points to the location of this file.

When the flamelet generation approach is used, prePDF also uses a trans-port database (for differential diffusion) and must be able to access thedatabase file, TRANSPORT.DB. This file must be present in the directorywhere you run prePDF or it must be accessed through an environmentvariable, TRANSPORTDB, which points to the location of this file.

In most installations, you will be running prePDF using procedures sup-plied by Fluent Inc. and these procedures will set the environment vari-ables for you.

Standard Flamelet Files

The data structure for the standard flamelet file format is based onkeywords that precede each data section. If any of the keywords in yourflamelet data file do not match the supported keywords, you will have tomanually edit the file and change the keywords to one of the supportedtypes. (The prePDF flamelet filter is case-insensitive, so you need notworry about capitalization within the keywords.)

The following keywords are supported by the prePDF filter:

• Header section: HEADER

• Main body section: BODY



• Number of species: NUMOFSPECIES

• Number of grid points: GRIDPOINTS

• Pressure: PRESSURE

• Strain rate: STRAINRATE

• Scalar dissipation: CHI

• Temperature: TEMPERATURE and TEMP

• Mass fraction: MASSFRACTION-

• Mixture fraction: Z

Sample File

A sample flamelet file in the standard format is provided below. Notethat not all species are listed in this file.

HEADERSTRAINRATE 100.NUMOFSPECIES 12GRIDPOINTS 39PRESSURE 1.BODYZ

0.0000E+00 4.3000E-07 2.1780E-06 1.2651E-05 7.8456E-052.1876E-04 5.9030E-04 9.4701E-04 1.4700E-03 1.8061E-032.1967E-03 2.6424E-03 3.1435E-03 4.3038E-03 5.6637E-038.9401E-03 1.2800E-02 1.7114E-02 2.1698E-02 2.6304E-022.8522E-02 3.0647E-02 3.2680E-02 3.4655E-02 4.2784E-025.2655E-02 6.5420E-02 8.2531E-02 1.0637E-01 1.4122E-011.9518E-01 2.8473E-01 4.4175E-01 6.6643E-01 8.6222E-019.5897E-01 9.9025E-01 9.9819E-01 1.0000E+00

TEMPERATURE3.0000E+02 3.0013E+02 3.0085E+02 3.0475E+02 3.2382E+023.5644E+02 4.3055E+02 4.9469E+02 5.8260E+02 6.3634E+026.9655E+02 7.6268E+02 8.3393E+02 9.8775E+02 1.1493E+031.4702E+03 1.7516E+03 1.9767E+03 2.1403E+03 2.2444E+032.2766E+03 2.2962E+03 2.3044E+03 2.3027E+03 2.2164E+03



2.0671E+03 1.8792E+03 1.6655E+03 1.4355E+03 1.1986E+039.6530E+02 7.5025E+02 5.7496E+02 4.4805E+02 3.6847E+023.2730E+02 3.0939E+02 3.0248E+02 3.0000E+02

MASSFRACTION-H23.2354E-07 7.4290E-07 1.6979E-06 3.8179E-06 8.3038E-061.2219E-05 1.7873E-05 2.1556E-05 2.5872E-05 2.8290E-053.0888E-05 3.3684E-05 3.6720E-05 4.3768E-05 5.4359E-051.0484E-04 2.6807E-04 6.1906E-04 1.2615E-03 2.3555E-033.1422E-03 4.1281E-03 5.3302E-03 6.7434E-03 1.4244E-022.4296E-02 3.7472E-02 5.5159E-02 7.9788E-02 1.1573E-011.7135E-01 2.6359E-01 4.2527E-01 6.5658E-01 8.5814E-019.5775E-01 9.8996E-01 9.9814E-01 1.0000E+00

MASSFRACTION-CH4. . . . .. . . . .. . . . .. . . . .

MASSFRACTION-O6.8919E-10 2.8720E-09 1.1905E-08 4.8669E-08 2.0370E-075.5281E-07 1.7418E-06 3.6996E-06 8.3107E-06 1.3525E-052.2484E-05 3.8312E-05 6.6385E-05 1.8269E-04 4.4320E-041.4284E-03 2.7564E-03 3.9063E-03 4.3237E-03 3.7141E-033.0916E-03 2.3917E-03 1.7345E-03 1.2016E-03 2.4323E-045.2235E-05 1.1469E-05 2.3011E-06 3.7414E-07 4.2445E-082.7470E-09 8.7551E-11 2.9341E-12 7.0471E-13 0.0000E+007.2143E-14 0.0000E+00 0.0000E+00 0.0000E+00

Missing Species

prePDF will check whether all species in the flamelet data file exist inthe thermodynamic properties databases THERMO.DB and thermodb.scm.If any of the species in the flamelet file do not exist, prePDF will issuean error message and halt the calculation. If this occurs, you can eitheradd the missing species to the databases (as described in Section 14.5),or remove the species from the flamelet file.

You should not remove a species from the flamelet data file unless itsspecies concentration is very small (10−3 or less) throughout the flameletprofile. If you remove a low-concentration species, you will not have thespecies concentrations available for viewing in the FLUENT calculation,but the accuracy of the FLUENT calculation will otherwise be unaffected.



If you choose to remove any species, be sure to also update the number!of species (keyword NUMOFSPECIES) in the flamelet data file, to reflectthe loss of any species you have removed from the file.

If a species with relatively large concentration is missing from the prePDFthermodynamic databases, you will have to add it (as described in Sec-tion 14.5). Removing a high-concentration species from the flamelet fileis not recommended.

Ordering of the Flamelet Data

The flamelet files are written in either ascending (starting from the ox-idizer inlet, where f = 0), or descending (starting from the fuel inlet,where f = 1) mixture fraction values. When you import the file, prePDFwill ask you to specify the order in which the file has been written.

Chemistry Files

Several chemistry files accompany prePDF, to be used as a basis to per-form the single-flamelet calculations. The chemistry files are written inthe standard CHEMKIN format, and are provided on the distributionCD in the following directory:

path/Fluent.Inc/prepdf4.⇓x/db/

where path is the directory where you have installed your Fluent Inc.software and the variable x corresponds to your release version, e.g., 0for prePDF 4.0.

Up to 100 species can be included in the flamelet calculation. If thenumber of species used in the flamelet calculation is greater than n,where n is the maximum number of species specified in the MemoryAllocation panel (see Section 14.3.1), prePDF will automatically selectthe n species with the largest concentration for inclusion in the PDFfile.



Description of Chemistry Files

Short descriptions of and references for the available chemistry files arepresented below. The files are named by the main fuel or mechanismname and the number of reactions, and end with the extension .che.

skeletal25.che: Skeletal mechanism for methane oxidation. 17 speciesand 25 reversible reactions (35 reactions if the forward and back-ward steps are counted separately). From Table B.1 on p. 262in [174].

kee58.che: Kee mechanism for methane oxidation. 18 species and 58reversible reactions. From Table B.2 on p. 263 in [174].

glarborg152.che: Glarborg mechanism for hydrocarbon oxidation upto C2. 33 species and 152 reversible reactions. From Table B.4 onp. 270 in [174].

methanol40.che: Mechanism for methanol combustion. The mecha-nism consists of H2-O2 chain reactions, HO2 formation and con-sumption, H2O2 formation and consumption, recombination reac-tions, CO-CO2 mechanism, CHO consumption, CH2O consump-tion, CH2OH consumption, and CH3OH consumption. 16 speciesand 40 reactions. From Chapter 16 and Table 1.1 in [179].

reduced25.che: Skeletal mechanism for methane oxidation. 17 speciesand 25 reversible reactions (35 reactions if the forward and back-ward steps are counted separately). From the Table on p. 161in [220].

drake67.che: Mechanism for the CO-H2-N2 system. NOx chemistry isalso included. 22 species and 67 reversible reactions. From Table1 on p. 154 in [53].

smooke46.che: Methane combustion. 17 species and 46 reversible re-actions. From Table 7 on p. 1787 in [221].

heptane42.che: A chemical kinetic mechanism for the oxidation of hep-tane. 20 species and 42 reversible reactions. From Table 1 on p. 296in [29].



hydrogen18.che: Mechanism for H2 combustion. 9 species and 18 re-actions. From Chapter 10 and Table 1.1 in [179].

hydrogen37.che: Mechanism for H2 combustion. NOx chemistry is alsoincluded. 13 species and 37 reactions. From Table 11.1 on p. 179in [179].

14.5 Adding New Species to the prePDF Database

prePDF uses the CHEMKIN database [112], THERMO.DB, for species ther-modynamic properties (see [112] for information on the parameters andthe format required for the THERMO.DB file). If you wish to add a newspecies, you will need to add the thermodynamic data to the THERMO.DBfile, as well as to the corresponding FLUENT database file, namedthermodb.scm.

You can use prePDF to generate the required thermodb.scm file:

1. Calculate the PDF look-up table in prePDF with the new speciesand the new THERMO.DB database file.

2. Generate a FLUENT property file with a default name prepdf.scm.

File −→ Write −→Thermodb...

3. You now have two choices:

• Rename prepdf.scm to thermodb.scm, and run FLUENT fromthe local directory where thermodb.scm is.

• If you would like to store the new species permanently, editthe file thermodb.scm in the

path/Fluent.Inc/fluent6.x/cortex/lib/

installation directory (where path is the directory in whichyou have installed FLUENT and the variable x correspondsto your release version, e.g., 0 for fluent6.0) and add inthe new species from prepdf.scm. Be careful to keep theScheme lists, enclosed within round parentheses, intact, andalso ensure that your editor does not insert carriage-returnsto break up the inserted Scheme lists. It is recommended that


14.5 Adding New Species to the prePDF Database

you save a backup copy of thermodb.scm before making anychanges.




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