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

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12.22 EDDY BREAK-UP MODEL OF COMBUSTION 387

Figure 12.9 Comparison of predictions and experimental data in a furnace simulation (Case 6): radial temperature and

oxygen concentration proﬁles

Source: Gosman et al. (1978)

Now the reaction rate of fuel is given by

=−min 4 C

, 4 C

, 4 C ′

, −Ä

fu,kinetic

(12.167)

Nickjooy et al. (1988) used the above approach by considering a two-step

global reaction model which included CO formation to predict combusting

ﬂows in axisymmetric combustors. The model constants C

, C ′

, a, b and

c used in their study for the different cases considered can be found in the

above reference. Figure 12.10 shows the predictions reported by Nikjooy

et al. (1988).

The eddy break-up model makes reasonably good predictions and is

fairly straightforward to implement in CFD procedures. However, due to

the dependence of the fuel dissipation rate on the turbulence time scale k/

the quality of the predictions depends on the performance of the turbulence

model. Where a turbulence model fails to make accurate ﬂow predictions the

quality of combustion simulations will, of course, also be limited.

1 + s

ANIN_C12.qxd 29/12/2006 04:44PM Page 387

A modiﬁed version of the eddy break-up (EBU) model is known as the

eddy dissipation concept (EDC) of Ertesvag and Magnussen (2000). The

EDC model attempts to incorporate the signiﬁcance of ﬁne structures in a

turbulent reacting ﬂow in which combustion chemistry is important. A com-

prehensive review of the EDC model can be found in Magnussen (2005).

Without giving the conceptual details the model may be summarised as

follows.

In the EDC model the mass fraction occupied by the ﬁne structures is

deﬁned as

* = 4.6

1/2

(12.168)

Here

is kinematic viscosity, k and

are turbulent kinetic energy and dis-

sipation, and 4.6 is a model constant. The reacting fraction of the ﬁne

structure is deﬁned as

= (12.169)

where J

is the product mass fraction and, as in the EBU model,

min

= min(J

, J

/s). Then the reaction rate for fuel is obtained from

=−4 C

EDC

min J

, . (12.170)

Here C

EDC

is a model constant with a recommended value of 11.2.

All the above models except the equilibrium model either use single-step

combustion or accommodate only a limited degree of combustion chemistry

and do not predict intermediate and minor species. We now discuss the

1 −

/(1 + s)

min

+ J

/(1 + s)

νε

388 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.10 A comparison of the calculated and measured mixture fraction (above centre line), fuel and CO (below

centre line) for the Lewis and Smoot (1981) experiment

Source: Nikjooy et al. (1988)

Eddy dissipation

concept

12.23

Laminar flamelet

model

12.24

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12.24 LAMINAR FLAMELET MODEL 389

laminar ﬂamelet model, which is a compromise between simplicity and the

need to account for detailed chemistry. The model views the turbulent ﬂame

as consisting of an ensemble of stretched laminar ﬂamelets. Figure 12.11

shows a schematic sketch of the laminar ﬂamelet concept. Turbulent ﬂames

are described as wrinkled, moving laminar sheets of reaction, and major heat

release is considered to occur in narrow regions in the vicinity of stoichio-

metric surfaces. These are called ﬂamelets, which are considered to be

embedded within the turbulent ﬂow. The approach is based on the notion

that, if the chemical time scales are much shorter than the characteristic

turbulence time scales, the fuel and oxidant react in locally thin one-

dimensional structures normal to the stoichiometric contour, as illustrated

in the inset of Figure 12.11. These structures are assumed to resemble the

ﬂame sheets responsible for combustion in a laminar ﬂame: hence the name

laminar ﬂamelets. In a turbulent ﬂow the ﬂamelets are considered to

be stretched and strained by ﬂow and turbulence. These effects are incor-

porated in the modelling by including appropriate parameters, as we will

see below.

Figure 12.11 Schematic

representation of the laminar

ﬂamelet concept

The properties of laminar ﬂamelets, e.g. the temperature, density and

species mass fractions as a function of the mixture fraction

, are evaluated

once and for all outside the ﬂow ﬁeld calculation to yield a so-called lami-

nar ﬂamelet library. A library of such relationships used in a calculation is

shown later in an example (see Figure 12.16). The library comprises a set of

relationships

(

) between scalar ﬂow properties

and the mixture fraction

. Their form is similar to Figure 12.2, but turbulence causes the ﬂames to

be stretched, which alters the details of the relationships

(

). To accommo-

date the effect of ﬂame stretching due to the ﬂow ﬁeld in an actual turbulent

ﬂame, it is common practice to incorporate as a parameter either the strain

rate or the scalar dissipation rate, both of which are ﬂow properties, into

the laminar ﬂamelet library calculations.

A key advantage of the laminar ﬂamelet methodology is that detailed

chemistry can be incorporated relatively economically within the calculations,

because the library contains information relating to major species, minor

species, density and temperature. This enables us to evaluate important

aspects of the combustion process such as the formation of pollutants.

Without detailed chemistry it is only possible to calculate ﬂow and approx-

imate energy release due to combustion.

In section 12.25 we give an overview of the methods used to generate

a laminar ﬂamelet library. In section 12.26 we show how Favre-averaged

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values of a scalar variable in a turbulent ﬂow ﬁeld can be calculated from a

knowledge of the values of the mixture fraction

and the scalar dissipation

rate

and the pdfs G(

) and G(

). In sections 12.27 to 12.29 we discuss how

the formation of pollutants such as NO can be modelled within the laminar

ﬂamelet framework. Finally, in section 12.30 we give an illustrative example

to demonstrate the strengths and weaknesses of the laminar ﬂamelet model.

We summarise the main aspects of the laminar ﬂamelet method from the

work of Peters (1984, 1986), Bray and Peters (1994), Pitsch and Peters

(1998), Williams (2000) and Veynante and Vervisch (2002). The interested

reader is directed to these publications for further details of laminar ﬂamelet

theory for non-premixed combustion.

Laminar ﬂamelets are generated by solving the one-dimensional governing

equations of non-premixed combustion. The simplest model for counter-ﬂow

diffusion ﬂames assumes a steady state, laminar, stagnation-point ﬂow. The

governing equations for continuity, momentum, chemical species and energy

can be written and solved without much effort. A simple one-dimensional

non-premixed combustion situation is shown in Figure 12.12a, where fuel

and oxidant streams are opposed jets. There are two main methods of pro-

ducing laminar ﬂamelet libraries:

• Method 1: the physical (x–y) co-ordinate system is used to solve

governing equations for opposed ﬂow diffusion ﬂames

• Method 2: the governing equations of opposed ﬂow diffusion ﬂames are

transformed into mixture fraction space and solved

Method 1: Opposed flow diffusion flame configuration

Consider the opposed ﬂow diffusion ﬂame situation shown in Figure 12.12a.

Such a ﬂow might be created using two concentric axisymmetric opposing

nozzles, one delivering fuel and the other delivering oxidant. The conﬁgura-

tion produces an axisymmetric ﬂow ﬁeld with a stagnation plane between

the nozzles as shown. The location of the stagnation plane depends on the

velocities of the two streams, and can be adjusted by varying the velocity of

one stream and keeping the other constant. A diffusion ﬂame is established

between the two nozzles close to the stagnation plane. The position of this

thin ﬂat ﬂame depends on the velocities of the two streams and the proper-

ties of the fuel and oxidant. Since most fuels require more air than fuel by

mass the diffusion ﬂame usually sits on the oxidiser side of the stagnation

plane as shown. For calculation purposes the conﬁguration is considered to

be inﬁnitely wide and axisymmetric. The co-ordinate system used in the

Method 1 formulation is shown in Figure 12.12a. The oxidiser jet is located

at y =∞and the fuel jet at y =−∞. The model assumes the ﬂow to be

laminar, stagnation-point ﬂow in cylindrical co-ordinates.

The structure of the ﬂame is obtained by the solution of steady boundary

layer equations written along the stagnation streamline (see Smooke et al.,

1986; Lutz et al., 1997). The method looks for self-similar solutions that

are functions of the co-ordinate perpendicular to the ﬂame, i.e. y only (note

390 CHAPTER 12 CFD MODELLING OF COMBUSTION

Generation of

laminar flamelet

libraries

12.25

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12.25 GENERATION OF LAMINAR FLAMELET LIBRARIES 391

that y is the axial co-ordinate in this case). In the usual notation the vari-

ables are

(y)

T = T (y)

= Y

(y), k = 1,..., N

Figure 12.12 (a) An opposed

ﬂow ﬂame; (b) Tsuji burner

conﬁguration (after Tsuji and

Yamaoka, 1967)

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If y denotes the axial direction and x denotes the radial direction the steady

state conservation of mass is as follows:

+=0 (12.171)

The radial momentum equation is

u +

v =

− (12.172)

The species conservation equation is

u +

v + (

) − M

= 0 k = 1, 2,..., N (12.173)

The energy equation is

u +

v − k + c

+ h

= 0 k = 1, 2,...,N (12.174)

The system is closed with the ideal gas law

= (12.175)

In equations (12.173) and (12.174) the parameter V

is known as the

diffusion velocity, which considers the diffusion of species. For a multi-

component system the diffusion velocities are given by (see Lutz et al., 1997;

Turns, 2000)

=− (MW )

− (12.176)

where the X

denote mole fractions, (MW ) molecular weights and D

the

multi-component diffusion coefﬁcient, and D

is an equivalent diffusion

coefﬁcient of species k into the rest of the mixture. Since calculations of

laminar ﬂames attempt to consider detailed chemistry, these ﬁne details are

retained in the equations, and detailed transport properties and thermo-

dynamic data calculation routines are also employed in the calculations (see

Rogg and Wang, 1997).

In an opposing ﬂow the free stream radial and axial velocities at the edge

of the boundary layer are given by u

∞

= ax and v

∞

=−2ay, where a is the

strain rate. We may now introduce

U( y) == (12.177)

V( y) =

v (12.178)

∞

∂

∑

j =1

(MW )

mix

p(MW )

mix

∑

k=1

∂

∑

k=1

∂

(

vx)

∂

(

ux)

∂

392 CHAPTER 12 CFD MODELLING OF COMBUSTION

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12.25 GENERATION OF LAMINAR FLAMELET LIBRARIES 393

Introduction of equations (12.177) and (12.178) transforms the set of equa-

tions (12.171)–(12.175) into a system of ordinary differential equations valid

along the stagnation-point streamline x = 0:

+ 2a

U = 0 (12.179)

− V + a(

∞

−

) = 0 (12.180)

− (

) − V + M

= 0 k = 1, 2,..., N (12.181)

k − c

V −

− h

= 0 (12.182)

Smooke et al. (1986) stated that the appropriate boundary conditions for the

fuel ( fu) and oxidizer (ox) streams are as follows:

Fuel stream, y =−∞: V = V

U =

T = T

= Y

k, fu

Oxidiser stream, y =∞: U = 1 T = T

= Y

k,∞

Symmetry plane, y = 0: V = 0

The distances y =−∞and y =+∞refer to fuel and oxidiser steams. In

practice the inﬁnite interval is truncated and boundary data are supplied at

y =−L

(the fuel jet) and y =+L

(the oxidiser jet), where the separation

distance is 2L = L

+ L

(see Smooke et al., 1986).

Inspection of the above equations shows that the velocity gradient or

strain rate a is a parameter of the system of ordinary differential equations

(12.179)–(12.182) – the other quantities are the dependent and independent

variables and thermodynamic or transport properties. To generate laminar

ﬂamelet proﬁles in the physical space the system is solved for a series of

prescribed values of a corresponding to stretch conditions in practical tur-

bulent ﬂows (a = 0.1 to 5000 s

−1

). Normally, ﬁnite difference techniques

and Newton’s iteration method are used to solve the set of equations.

Another commonly used experimental situation for creating one-

dimensional non-premixed ﬂames is the Tsuji burner shown in Figure

12.12b (after Tsuji and Yamaoka, 1967). An alternative formulation of the

ﬂamelet equations based on Tsuji’s burner conﬁguration, which yields a

slightly different set of ordinary differential equations, is also a popular

method for laminar ﬂamelet calculations.

There are many programs available for laminar ﬂamelet calculations.

RUN1DL (Rogg, 1993; Rogg and Wang, 1997) is widely used for the gener-

ation of laminar ﬂamelet proﬁles in physical space. Another computer

program for solving counter-ﬂow diffusion ﬂames is the OPPDIF code of

Lutz et al. (1997). By solving the opposed ﬂow laminar ﬂame problem in

physical space the results obtained will give temperature, density, major and

minor species as a function of distance from the stagnation point. A typical

set of calculation results is given in Figure 12.13, which shows the tempera-

ture, density and all species concentrations for an opposed ﬂow laminar ﬂame

with a 1:1 mixture of CH

and H

as fuel for a strain rate of a

= 20 s

−1

∑

k=1

∑

k=1

∂

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In combustion procedures we solve the transport equations for mixture

fraction and mixture fraction variance. Flame property relationships, such

as those in Figure 12.2, lack detailed chemistry. The laminar ﬂamelet model

uses the laminar ﬂamelet library as a series of relationships Y

= Y

(

T = T (

(

) etc. in the form of a look-up table. This requires us

to convert the solutions of (12.179)–(12.182) in physical space Y

= Y

(y),

T = T(y),

(y) into the mixture fraction co-ordinate. We note that if we

go in Figure 12.13 from the fuel stream (y =−15 mm) to the oxidiser stream

( y =+15 mm), the mixture fraction will vary from 1 to 0. In between these

extremes the mixture fraction will gradually change. For each ﬂame and

each speciﬁc value of the strain rate there will be a one-to-one correspond-

ence between the values of the y-co-ordinate and the local mixture fraction

(y). For example, the ﬂame is located in the high-temperature region

around y =+5 mm, where the mixture fraction will be close to its stoichio-

metric value, so

(y = 5 mm). Thus, the results in Figure 12.13 can in

principle be replotted using the mixture fraction as the abscissa.

The most common method of evaluating the mixture fraction

in com-

busting ﬂows with an arbitrary number of fuel and oxidiser inlet streams is

known as Bilger’s mixture fraction formula (Bilger, 1988). It is based on

the notion that species are consumed or produced during chemical reactions,

but chemical elements are conserved during reactions. Mass fractions Z of

chemical elements (e.g. C, H, O) can be obtained from the mass fractions Y

of species containing these elements:

= Y

(12.183)

∑

i =1

394 CHAPTER 12 CFD MODELLING OF COMBUSTION

Figure 12.13 Laminar ﬂamelet

structure of an opposed ﬂow

(1:1) laminar ﬂame, air

and fuel temperatures 300 K,

strain rate a = 20 s

−1

, fuel left

hand side, air right hand side,

domain size −15 mm to +15 mm

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12.25 GENERATION OF LAMINAR FLAMELET LIBRARIES 395

where MW

is the molecular weight of species i, W

is the atomic weight of

element j, and a

is the number of atoms of element j in a molecule of

species i. The summation is carried out over all N molecular species. Using

these element mass fractions, Bilger’s formula for the mixture fraction for

the case of a fuel stream that contains only fuel and an oxidiser stream that

contains only element O is as follows:

= (12.184)

Here fu and ox refer to fuel and oxidant streams.

If both the fuel and oxygen streams contain the elements C, H, O this can

be written in a slightly modiﬁed form as

= (12.185)

Equation (12.185) can be used at every point of the one-dimensional calcula-

tion domain to obtain the mixture fraction from proﬁles of species mass

fraction such as those in Figure 12.13. In turn, this enables us to plot tem-

perature, density and species mass fractions as a function of

These deﬁnitions have also been widely used for the interpretation of

experimental data in terms of mixture fraction (Masri et al., 1996; Dally et al.,

1998b; Barlow et al., 2001) as well as in computational studies (Smooke et al.,

1990; Barlow et al., 2000; Hossain and Malalasekera, 2003) and offer a

consistent way of deﬁning mixture fractions, especially for obtaining mixture

fractions from experimental data. Smooke et al. (1986) and Sick et al. (1991)

have used experimental data for strained laminar ﬂame conﬁgurations, e.g.

using the Tsuji-type burner or similar to compare with numerical results and

validate the methodology for generating laminar ﬂamelet libraries.

Method 2: Flamelet equations in mixture fraction space

The ﬂamelet equations for temperature and species mass fraction of a

one-dimensional ﬂame can also be derived by using co-ordinate transforma-

tion (of the Crocco type) using mixture fraction

as the independent

co-ordinate. An opposed-ﬂow diffusion ﬂame is an example of a ﬂame that

can be entirely mapped into mixture fraction space. For details of the devel-

opment of these equations in mixture fraction space the reader is referred to

Peters (1984, 1986), who reviewed laminar ﬂamelet concepts for turbulent

combustion. Without proof we quote from these references the ﬂamelet

equations in mixture fraction space:

Species

−

− M

= 0 (12.186)

Temperature

−

−+M

= 0 (12.187)

The boundary conditions are

Fuel stream,

= 1.0: T = T

= Y

k,fu

k = 1,..., N

Oxidiser stream,

= 0.0: T = T

= Y

k,ox

k = 1,..., N

∑

k=1

∂

∂ξ

∂

∂ξ

∂

2(Z

− Z

C,ox

)/W

+ (Z

− Z

H,ox

)/2W

− 2(Z

− Z

O,ox

)/W

2(Z

C, fu

− Z

C,ox

)/W

+ (Z

H, fu

− Z

H,ox

)/2W

− 2(Z

O, fu

− Z

O,ox

)/W

+ Z

/2W

− 2(Z

− Z

O,ox

)/W

C, fu

+ Z

H, fu

/2W

+ 2Z

O, ox

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The initial conditions are known temperatures and mass fraction of the

species at t = 0, i.e. T = T

ambient

, Y

= 0.233, Y

= 0.767, Y

mass fractions

(e.g. Y

, Y

etc.) are speciﬁed using the given fuel composition and mass

fractions for all other species are set to zero.

In equations (12.186) and (12.187)

is the so-called scalar dissipation rate

(units s

−1

) and is a parameter of this set of ﬂamelet equations. The equations

are solved in pseudo-transient form, i.e. the transient term is retained, but

we are only interested in the ﬁnal steady state solution; this model is known

as the steady laminar ﬂamelet model (SLFM).

Equations (12.186)–(12.187) use a mixture fraction deﬁned on the basis

of two streams, fuel and oxidiser, as a co-ordinate whose direction is normal

to the stoichiometric surface

, as shown in Figure 12.12a. In turbulent

non-premixed combustion, the reaction zone is considered to be in the im-

mediate vicinity of the high-temperature region close to the stoichiometric

mixture and convected and diffused with the mixture fraction ﬁeld (Bray and

Peters, 1994).

When the method is used in a CFD computation, inﬂuence of the ﬂow

ﬁeld on the ﬂamelet structure is represented in the above equations by the

local scalar dissipation rate, deﬁned by

= 2D

(12.188)

where x, y, z are the co-ordinate directions and D

=Γ

is the diffusion

coefﬁcient for the mixture fraction. The scalar dissipation is the variable

which controls mixing, and represents the gradient of mixture fraction,

which is related to strain. When the ﬂame strain increases, the scalar dissi-

pation rate increases. The scalar dissipation rate implicitly incorporates

convection–diffusion effects normal to the surface of the stoichiometric

mixture (Peters, 1984), and it can be considered as the parameter which

describes the departure from equilibrium chemistry. The reciprocal of scalar

dissipation

−1

is a measure of diffusive time

. As this time decreases (i.e.

as the value of

increases) the heat and mass transfer through the stoichio-

metric surface are enhanced (Veynante and Vervisch, 2002). Furthermore, if

exceeds a critical value a ﬂame extinguishes as heat losses becomes larger

than chemical heat release.

To generate laminar ﬂamelet proﬁles in mixture fraction space the set

of governing partial differential equations (12.186) and (12.187) is solved

for given initial and boundary conditions for fuel concentration and tem-

perature for a series of prescribed values of the scalar dissipation rate

Different scalar dissipation levels give different ﬂame structures. A very low

scalar dissipation rate (i.e. long diffusion time) means that the combustion

takes place in conditions that are close to equilibrium, whereas a very high

rate means a highly strained ﬂame (close to extinction). Figure 12.16 below

shows a set of ﬂamelet relationships produced by this type of calculation.

The arrows drawn indicate the direction of increasing scalar dissipation rate.

The resulting ﬂamelet library is a collection of temperature, species and

density proﬁles in the mixture fraction space for different scalar dissipation

rates T(

), Y

(

). The computer program known as FlameMaster by

Pitsch (1998) can be used for this.

Finally, it should be noted that equations (12.186)–(12.187) have been

derived neglecting many higher-order terms involving convection and cur-

∂ξ

∂

∂ξ

∂

∂ξ

∂

396 CHAPTER 12 CFD MODELLING OF COMBUSTION

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