Versteeg H., Malalasekra W. An Introduction to Computational Fluid Dynamics: The Finite Volume Method

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12.25 GENERATION OF LAMINAR FLAMELET LIBRARIES 397

vature along the mixture fraction surface. Nevertheless, they have been used

widely (Mauss et al., 1990; Seshadri et al., 1990; Lentini, 1994) to generate

ﬂamelets in mixture fraction space. A more accurate formulation which does

not rely on the above-mentioned assumptions has been presented by Pitsch

and Peters (1998). This new formulation uses a conserved scalar which does

not depend on the two-stream formulation for mixture fraction and also

allows us to incorporate non-unity Lewis numbers.

An example of laminar flamelet relationships

In this example we consider an experimental dataset available in the literature

(and on the Internet, Barlow, 2000) and show how a laminar ﬂamelet library

can be generated for particular ﬂame conditions. The resulting state relation-

ships are compared with experimental data to demonstrate the accuracy and

usefulness of the laminar ﬂamelet relationships. The calculation of ﬁeld quanti-

ties using the laminar ﬂamelet model will be demonstrated in section 12.30.

We use two popular codes – FlameMaster (Pitsch, 1998) and RUN-1DL

(Rogg, 1993; Rogg and Wang, 1997) – to generate ﬂamelets. The Flame-

Master code uses Method 2 described above; we have used Method 1 with

the RUN-1DL code, i.e. the calculations are performed in physical space. In

RUN-1DL ﬂamelets are generated using physical co-ordinates it is necessary

to convert the results obtained in physical co-ordinates to mixture fraction

space using Bilger’s mixture fraction formula (12.184). The chemical mech-

anism used in the calculation is the detailed mechanism for fuels up to pro-

pane by Peters (1993). The same chemical mechanism was used in both codes.

The experimental dataset considered here is the turbulent ﬂame experi-

ment, Flame B, of Barlow et al. (2000). This experiment and other datasets

have been widely discussed in the ‘Turbulent Non-premixed Flames’ (TNF)

forums. The fuel used is CO/H

with a volume ratio of 40/30/30. Flow-

dependent effects are represented by two different parameters, strain rate

and scalar dissipation rate. The scalar dissipation rate

is used in mixture

fraction space and the strain rate a is used in physical space. The expression

which is used to link scalar dissipation rate

(in axisymmetric cases) to strain

rate a is (Peters, 2000)

= exp{−2[erfc

−1

)]

} (12.189)

Here

is the scalar dissipation rate at the location where the mixture frac-

tion is stoichiometric and a

is the corresponding strain rate in physical space.

An alternative expression used for small values of

= 4a

[erfc

−1

)]

(12.190)

where erfc

−1

is the inverse of the complementary error function (not the

reciprocal). Barlow et al. (2000) compared laminar ﬂamelet relationships

with their experimental results using two chosen strain rates, a

= 10 s

−1

and

100 s

−1

, corresponding to stoichiometric scalar dissipation rates

= 4.6 s

−1

and 46 s

−1

, respectively. We have repeated their calculations and show the

comparison between the laminar ﬂamelet relationships and experimental

data at two axial locations of the ﬂame, x/D = 20 and 30. The comparison is

made in mixture fraction space.

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Figures 12.14a and b show the laminar ﬂamelet relationships for tem-

perature and species mole fractions as functions of the mixture fraction.

Experimentally measured conditional mean and r.m.s. ﬂuctuation values of

temperature and species mole fractions are shown as symbols and error bars,

respectively. The solid lines represent the calculation results. Figure 12.14a

shows the results obtained using the FlameMaster code and Figure 12.14b

shows the results obtained using the RUN-1DL code. It can be seen that

the two methods give remarkably similar results. The agreement between

ﬂamelet proﬁles and experimental data is excellent at both locations and well

within the experimental error bars. The key inputs for the generation of a

ﬂamelet library are the fuel composition and scalar dissipation rates or strain

rates. The locations and strain rates used here are those quoted by Barlow

et al. (2000). No attempt was made to ﬁnd the most suitable strain rate for

a given dataset. The close match between the experimental and calculated

values clearly demonstrates the usefulness and accuracy of laminar ﬂamelets

as a means of providing state relationships for combustion calculations in a

turbulent ﬂame.

398 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.14 Comparison of laminar ﬂamelet relationships with experimental conditional means for temperature, CO

O, H

and CO for ﬂame B of Barlow et al. (2000): (a) FlameMaster calculations; (b) RUN-1DL calculations

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12.26 STATISTICS OF THE NON-EQUILIBRIUM PARAMETER 399

In the laminar ﬂamelet model, the thermochemical composition of the

turbulent ﬂame is completely determined by two parameters: (i) the mixture

fraction

and (ii) a non-equilibrium parameter, the scalar dissipation rate

. In turbulent ﬂow ﬁelds, these parameters are statistically distributed. To

evaluate mean temperature, density and species mass fractions, it is therefore

necessary to know the statistical distribution of the mixture fraction and the

scalar dissipation rate in the form of a joint pdf G(

). The mean scalar

variables are then evaluated from

É =

(

)G(

(12.191)

In the current laminar ﬂamelet model formulations, it is assumed that the

mixture fraction and the scalar dissipation rate are statistically independent.

This assumption simpliﬁes the formulations, because the average value of a

scalar variable

in a turbulent ﬂow ﬁeld is now given by

É =

(

)G(

(12.192)

The pdf for mixture fraction is assumed to be a beta function. The pdf of the

scalar dissipation rate is assumed to be a log-normal function, an assump-

tion that is justiﬁed by experimental evidence presented in Effelsberg

and Peters (1988). Thus, the two independent pdfs are the beta pdf for the

mixture fraction

) =

where parameters

and

are calculated using (12.157), and the log-normal

pdf for the scalar dissipation

) = exp − (ln

−

)

where the parameters

and

are related to the ﬁrst and second moment of

via

9 = exp

(12.193)

9″

= 9

[exp(

) − 1] (12.194)

The variance 9″

is not available in this model. Therefore to obtain

and

it is necessary to specify 9. The mean scalar dissipation rate 9 is usually

modelled as

9 = C

7″

(12.195)

χσ π

−1

(1 −

)

−1

Γ(

)

Γ(

)Γ(

)



∞



∞



Statistics of the

non-equilibrium

parameter

12.26

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400 CHAPTER 12 CFD MODELLING OF COMBUSTION

Here the model constant C

is assigned a value of C

= 2.0. The variance

in the log-normal distribution is set equal to

= 2.0. Further details of these

two model constants can be found in Peters (1984). By calculating 9 from

equation (12.195) and with the speciﬁed value of

, the pdf G(

) can be

evaluated.

A different expression proposed by Bray and Peters (1994) for the scalar

dissipation rate is

9 = C

(∆

) (12.196)

With C

= 2.0 and (∆

) as the range of mixture fraction values within which

the scalar dissipation process is concentrated, then (∆

) is an indication

of the thickness of the reaction zone. With the reaction zone centred at the

location

, the thickness of the reaction zone is approximated as

(∆

) ≈ 2

(12.197)

where

is the stoichiometric mixture fraction. This expression has proved

to be successful in predicting lift-off heights in jet ﬂames.

To calculate the mean of a variable it is required to evaluate the integral

(12.192). Integration of the

-pdf was discussed before in section 12.20. The

same procedure applies here. Integration with the log-normal pdf G(

) can

be carried out using the approximation of Lentini (1994). More details of

the origin and the theory of laminar ﬂamelet models can be found in Peters

(1984, 1986), Bray and Peters (1994) and Veynante and Vervisch (2002).

, SO

, heavy-metal compounds and particles are the main pollutants

formed in combustion. The amount and nature of pollutants formed depend

on the type of fuel, combustion equipment and temperatures used in the

combustion process.

Among the combustion gases:

•NO

and SO

contribute to acid rain and other environmental effects

•CO

contributes to the greenhouse effect

is of particular concern:

• It has an impact on the formation of ozone

• It is linked to the formation of smog

• It is directly responsible for acid rain

• It has a considerable impact on human health since it causes conditions

such as acute respiratory diseases and respiratory infections

The species NO, NO

and N

O are collectively known as NO

and result

from two sources:

• Molecular N

(air)

• Fuel-bound nitrogen

Formation mechanisms of NO

in the combustion process are further

classiﬁed as follows:

Pollutant

formation in

combustion

12.27

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12.28 MODELLING OF THERMAL NO FORMATION IN COMBUSTION 401

• Thermal NO

: This is formed by the direct reaction of molecular

nitrogen and oxygen at high ﬂame temperatures, the formation of

which is described by the Zel’dovich mechanism. This will be

discussed further below.

• Prompt NO

: This is formed by oxidation of atmospheric nitrogen

in the ﬂame front. Prompt NO

formation occurs by the reaction of

molecular nitrogen with hydrocarbon free radicals in oxygen-deﬁcient

regions of the ﬂame. The mechanism often used to describe prompt NO

formation is called the Fenimore (1970) mechanism, which considers

the rapid formation of NO long before the time required to form

thermal NO. More detailed description of the mechanism and further

details can be found in Turns (2000) and Kuo (2005).

• Fuel NO

: This arises due to the presence of nitrogen in the fuels

such as heavy oil and coal. Fuel NO

formation reactions are not well

understood. The principal reaction process appears to be devolatilisation

of fuel nitrogen, forming intermediate compounds such as NO, NO

and HCN. Subsequent N

or NO formation depends strongly on

the fuel/air ratio. In general, a fuel with lower nitrogen content typically

produces less total NO

than a fuel with higher nitrogen content.

For further details see Kuo (2005).

In addition to these, there is a further route known as the N

O intermediate

mechanism, which is only important in high pressures. The relative contribu-

tion of different pathways depends on fuel type, temperature, pressure and

residence time. Out of these, thermal NO appears to be the main contributor

in most practical combustion situations. For a more detailed discussion of

formation in combustion the reader is referred to Turns (1995, 2000),

Warnatz et al. (2001) and Kuo (2005).

Among the NO

components NO is the main pollutant. Thermal NO is

formed by N

in the air stream reacting with O

at high temperature. The

reaction mechanism of thermal NO formation is well known as the

Zel’dovich mechanism. The key reactions are

(1) O + N

bcg NO + N (12.198)

(2) N + O

bcg NO + O (12.199)

(3) N + OH bcg NO + H (12.200)

Forward and backward reaction rate constants for these three key reactions

are given in Table 12.4 (Turns, 2000).

Table 12.4

Forward reaction rate Backward reaction rate

Reaction

constant (m

/(kmol)(s)) constant (m

/(kmol)(s))

1 k

1, f

= 1.8 × 10

exp(−38370/T ) k

1,b

= 3.8 × 10

exp(−425/T )

2 k

2, f

= 1.8 × 10

T exp(−4680/T ) k

2,b

= 3.8 × 10

T exp(−20820/T )

3 k

1, f

= 7.1 × 10

exp(−450/T ) k

1,b

= 1.7 × 10

exp(−24560/T )

Modelling of

thermal NO

formation in

combustion

12.28

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Compare the reaction rate constant of reaction (1) with the reaction

rate constant of a typical intermediate combustion reaction H + O

bcg

OH + O (see section 12.10). This reaction has a forward reaction rate

constant of k

= 2.00 × 10

exp(−70.3/T )cm

/mol.s (see Kuo, 2005), requir-

ing an activation energy of −585 kJ/kmol. Reaction (1) of the above mech-

anism by contrast has a very high activation energy (E

=−319027 kJ/mol)

and is sufﬁciently fast only at high temperatures. It has the slowest rate of the

reaction in the Zel’dovich mechanism and therefore it is the rate-limiting

reaction in the formation of NO. Because of this high-temperature depend-

ence it is a common practice to assume that the thermal NO mechanism is

unimportant below 1800 K. Thermal NO reactions (1)–(3) are much slower

than other combustion reactions, so NO formation takes too long to achieve

equilibrium. Hence, NO concentrations cannot be calculated using equilib-

rium chemistry. The rate of conversion is governed by the gas temperature

(exponentially), residence time of the gas in high-temperature regions, and

excess air levels (affecting oxygen availability). The use of laminar ﬂamelet

concepts for combustion assumes relatively fast chemistry. Even in the

absence of other NO production mechanisms the inclusion of the above

NO equations in a laminar ﬂamelet library calculation does not produce

experimentally observed NO levels (see Drake and Blint, 1989; Sanders and

Gökalap, 1995; Vranos et al., 1992). Laminar ﬂamelet modelling of thermal

NO requires some special treatment, which is explained below. As noted

above, the other two principal mechanisms of NO formation are prompt

(or Fenimore) NO formation and fuel-bound NO formation. These are not

included in the method outlined below.

As explained earlier, attempts to obtain concentrations of thermal NO from

a ﬂamelet library can be inaccurate. One way around this problem is to

include an accurate account of the effects of residence time on NO formation

by using the Zel’dovich mechanism in conjunction with a transport equation

for the mass fraction of NO:

(4J

) + (4U

) =+N

(12.201)

where

is the turbulent Schmidt number, usually assigned a value of

0.7. The average source term N

of the transport equation represents

the mean rate of production of NO, which can be evaluated using the pdf

approach:

= 4M

(

)G(

(12.202)

where G(

) and G(

) are pdfs introduced in section 12.26. The NO source

term M

(

) can be generated using the rate constants of equations

(12.198)–(12.200) and (12.69) and (12.71). It is stored in the laminar ﬂamelet

library alongside the ﬂamelet relationships for temperature T(

) and

species mass fraction Y

(



∞



∂

eff

∂

402 CHAPTER 12 CFD MODELLING OF COMBUSTION

Flamelet-based

NO modelling

12.29

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12.30 AN EXAMPLE 403

We consider an experimental ﬂame conﬁguration for which a set of well-

documented experimental data is available and show the predictions of

temperature, major and minor species and NO using the laminar ﬂamelet

calculation methodology explained in the above sections. The problem

considered here is the bluff-body CH

ﬂame experimentally studied by

Dally et al. (1996, 1998b). This experimental ﬂame is designated as HM1 in the

dataset. The experimental data were provided by Masri (1996) via the Sydney

University website. The bluff body considered here has an outer diameter of

= 50 mm and a concentric fuel jet diameter of D

= 3.6 mm. The fuel com-

position is 50% by volume CH

and 50% by volume H

. The mean velocity

of the fuel jet is 118 m/s and the mean air velocity is 40 m/s. A schematic

diagram of the ﬂame and important zones is shown in Figure 12.15. Further

experimental details are available in the aforementioned references.

Figure 12.15 Schematic

drawing of the bluff body

stabilised ﬂame and measurement

locations

Computational details

The governing differential equations (continuity, momentum, turbulence

variables k and

, mixture fraction, mixture fraction variance) were solved in

Favre-averaged form by means of an in-house ﬁnite volume code based on

the SIMPLE algorithm. The k–

turbulence model was used for turbulence

closure. In section 3.7.2 it was noted that the standard k–

model overpredicts

the decay rate and the spreading rate of a round jet. Several modiﬁcations to

An example to

illustrate laminar

flamelet modelling and

NO modelling of a

turbulent flame

12.30

ANIN_C12.qxd 29/12/2006 04:44PM Page 403

the k–

model are available for correcting this problem (McGuirk and Rodi,

1979; Pope, 1978). In this study the value of constant C

in the

-transport

equation is modiﬁed from 1.44 to 1.60 following the work of Dally et al.

(1998b) and Hossain et al. (2001). Following the International Workshop

on Measurements and Computations of Turbulent Nonpremixed Flames

(TNF) this modiﬁcation is now recommended practice for modelling bluff-

body ﬂames.

The computational domain has a size of 170 mm in the radial direction

and 216 mm in the axial direction. A grid arrangement with 89 radial and 99

axial nodes was used in all the calculations presented here. Sensitivity tests

were carried out to establish that the grid was sufﬁciently ﬁne to give accur-

ate results. A fully developed velocity proﬁle was speciﬁed as the boundary

condition at the air and fuel inlets.

Prediction of flame structure

The laminar ﬂamelet library for different scalar dissipation rates was generated

using the RUN-1DL program mentioned earlier. The calculations presented

here have been performed using unity Lewis number. Figure 12.16 shows

examples of the laminar ﬂamelet relationships for temperature, CO

, H

and OH mass fractions. Turbulent combustion calculations are carried out

404 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.16 A sample of laminar ﬂamelet proﬁles

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12.30 AN EXAMPLE 405

by following the methodology explained in sections 12.25 and 12.26. Local

values of the scalar dissipation rate were obtained using the approximate

relationship (12.196)–(12.197). The library stores relationships at discrete

values of scalar dissipation rate

. Local values of

are likely to be inter-

mediate between those of the library. The appropriate ﬂamelet relationships

required in the calculation process are obtained by linear interpolation of

values from neighbouring library look-up tables.

To illustrate the effectiveness of the laminar ﬂamelet model we present in

Figures 12.17–12.26 a comparison of numerical predictions with a selection

of experimental data in the near ﬁeld of the burner (i.e. in the region closest

Figure 12.17 Mean axial velocity proﬁles:

•

, measurements; , calculations

ANIN_C12.qxd 29/12/2006 04:44PM Page 405

to the inlets). To start, we note that it is essential that the ﬂow ﬁeld is

adequately represented in the simulation, because the mixture fraction stat-

istics completely determine the thermochemical state of the ﬂame. It is even

more important for this CH

ﬂame, because it has a low stoichiometric

mixture fraction (

= 0.05). As a result, a small error in the ﬂow ﬁeld will

lead to a large error in temperature, species concentrations and location of

the ﬂame front. Figures 12.17 and 12.18 show a comparison between mea-

sured and computed radial proﬁles of mean axial and radial velocities plotted

at different near-ﬁeld locations. The mean axial velocity measurements are

406 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.18 Mean radial velocity proﬁles:

•

, measurements; , calculations

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