On the modelling of turbulent reacting flows in furnaces and combustion chambers

Acta Astroaautica. Vol.6, pp. 449-465. PergamonPress 1979. Printedin Great Britain

On the modelling of turbulent reacting flows in furnaces and combustion chamberst E. E. K H A L I L Department of Mechanical Power Engineering, Faculty of Engineering, Cairo University, Cairo, Egypt (Received 22 August, 1977; revised 3 July 1978)

Abstract--A general computer programme was developed to calculate the local flow properties in turbulent reactive and non reactive flows with recirculation; these calculations were obtained by solving the appropriate conservation equations in finite difference form with the corresponding boundary conditions. The calculation procedure employs a two equation turbulence model, and embodies various combustion models appropriate to diffusion, premixed and arbitrary fuelled flames. The phenomenon of unmixedness caused by turbulent fluctuations, which lead to a situation where the instantaneous value of fuel and/or oxidant concentrations and, therefore, their corresponding chemical reaction rates vanish, is investigated. The combustion models considered here are characterized by, for instance, instant reaction with clipped Gaussian probability distribution of concentration, which corresponds to random variation of fuel concentration with time, finite reaction rate with an eddy break up formulation, and finite reaction rate with a second order closure which accounts for temperature and concentration fluctuations. The radiative heat flux, which appears in the energy conservation equation, is obtained using a coupled four flux representation and integrating the radiation intensity distribution over a solid angle of 2w. The validity of the computational procedure incorporating the proposed turbulence, combustion and radiation models was assessed by comparisons with the experimental data in reacting and non reacting flows, and indicated satisfactory agreement. The obtained agreement assesses the validity of the physical assumptions of the models and supports the use of such procedure for furnace design purposes.

Introduction SIGNIFICANT theoretical advance has been made since the late 1960s, and stems from the utilization of digital computers to allow the solution of simultaneous partial differential equations which represent the conservation of mass, momentum, energy and species. The computed results obtained in furnace flow configurations aid the design, as they replace the "rule of thumb" often used in practical design of furnaces and combustion chambers. Attempts to calculate furnace and combustion chamber performance have been reported* in a range of tPaper presented at the 6th International Colloquium on Gasdynamics Explosions and Reactive Systems, Stockholm, Sweden. :~See Anasoulis et al. (1973), El Ghobashi et al. (1974), Hutchinson et al. (1975, 1976 a,b, 1977), Khalil et al. (1975) and Khalil (1978). 449

450

E.E. Khalil

furnace flows; these authors reported reasonable agreement between measured and calculated local flow properties in turbulent recirculating confined flames, and suggested the use of such solution procedure to real furnace flows, (Khalil et al., 1977a,b). The solution of the time dependent three dimensional equations appropriate to turbulent reacting flows cannot be obtained, at present, due to limitations of computer storage and computing time and, as a consequence, equations based on time averaged values of velocities, densities, temperature and chemical species concentrations have been solved, (Khalil et al., 1975) and (Khalil, 1976). The exact equations for time averaged properties contain unknown terms that correlate velocity components, densities and any dependent variable, typically, u v , up', u~b and p'~. Various turbulence models were suggested to evaluate the second order correlations in the momentum equation.t The validity of these models was assessed by comparisons with experiments. The two equation turbulence model of Launder et al. (1974), is used in the present work. In addition to the turbulence model, a combustion model is required in reacting flows to provide information needed to determine the rate of fuel consumption and the correlation terms u~b and p'~b, which appear in the fuel conservation equation. In the present investigation, three different combustion models were incorporated, and their validity in various flame types was assessed by comparisons with the available experiments. The first model is characterized by fast chemical reaction with a random temporal distribution of fuel concentration (Naguib, 1975) and (Hutchinson et al., 1976), and is suitable for diffusion flames; the second by finite chemical reaction rate with an eddy break up formulation to account for the effect of turbulence on the reaction rate (Spalding, 1970). The third model postulates a finite reaction rate with a second order closure and accounts for temperature and concentration fluctuation (Borghi, 1974) and (Hutchinson et al., 1976, 1977). The radiative heat transfer contribution to the energy balance appears as a source term in the enthalpy conservation equation, and is represented by a four flux model (Lockwood et al., 1975 and Khalil et al., 1977a). The present calculation procedure is similar to that of Gosman et al. (1974) and solves the governing equations in an iterative procedure; this procedure incorporates, a two equation turbulence model, three combustion models and a four flux radiation model. The validity of the present procedure and the models embodied within is assessed by comparisons with the experimental data of Bilger et al. (1972) in hydrogen diffusion flames, Steward et al. (1972) and Spadaccini et al. (1976). The substance of the paper is contained in two main sections. In the first section, the calculation procedure is described together with the various modelling assumptions. In the following section, the calculated results are presented and compared with the corresponding measurements in three different furnace flames. The paper ends with a brief discussion and conclusion section.

tSee Launderet al. (1974), Launder et al. (1975) and Pope et al. (1976).

On the modelling of turbulent reacting ]lows in furnaces and combustion chambers

451

Calculation proeedure and modelling assumptions Governing equations and boundary conditions The present computational procedure solves the governing time-averaged conservation equations of mass, momentum, species and energy, expressed in the following elliptic form as O--x

r ~ t~("~ = ~-~ F®~

+

F®r Or

(1)

where • is a general dependent variable, which can represent the three velocity components U, V, W, turbulent kinetic energy (k), its dissipation rate (E), stagnation enthalpy (H), mixture fraction (f), fuel mass fraction (Mtu), the rms of its fluctuations ~ and the correlation mrumox. The exchange coefficient F~ and source/sink terms S® are of the form given by Hutchinson et ai. (1977). The specification of boundary and inlet conditions is necessary to render the governing elliptic equations solvable. These conditions are specified along the four boundaries of the solution domain, which conforms to a symmetrical half section of the furnace. Wall functions (Launder and Spalding, 1974), are used to link the wall values to the near wall node points and were based on the logarithmic law of the wall. At the solid walls velocities were presumed zero, and temperatures were specified from the measurements. Inlet velocity, turbulence intensity and temperature profiles were either specified from the measurements, or reasonably assumed according to the recommendations of Khalil (1976), where the influence of these assumptions on the obtained results was quantified for similar flows.

Physical modelling Various modelling assumptions are implied in eqn (1) and relate to turbulence, combustion and radiation characteristics of the flow. Turbulence model. The present turbulence model is based on the solution of the transport equations for turbulent kinetic energy (k) and its dissipation rate (e). The effective viscosity is calculated from: ~o~ = ~ + C,,#k21e.

(2)

The various Reynolds stresses which appear in the momentum equations were obtained from the product of the velocity gradients and an effective viscosity. The validity of the k-~ model was assessed in detailt and the model constants were consistent with those of Launder and Spalding (1974). Combustion model. Three combustion models are described and discussed in turn. Model 1: This model is appropriate for idealized diffusion flames and postulates an infinitely fast one step reaction, i.e. physically controlled, with fuel and

tSee Gosman et al. (1977), Hutchinson et al. (1975, 1976, 1977) and Khalil (1976).

4_52

E . E . Khalil

oxygen existing at the same location but at different times. Equations for the mixture fraction f and the corresponding fluctuations g are modelled in the form l, and solved with the aid of the present numerical procedure. The temporal distribution of f was assumed to conform to a clipped Gaussian distribution in the physical limits of 1 > f > 0 (Lockwood and Naguib, 1975), details of the probability distribution and its derivation can be found in Khalil (1976). Values of temperature, mass fractions of fuel and oxidant are calculated from the knowledge of the probability P ( f ) and the relation between these entities and f ~)(f) =

t

l

(1)(f)P(f) df + Ati)I. + B ~ ,

(3)

where A and B are constants which depend on the inlet streams and correspond to the areas of the clipped parts of the Gaussian distribution. The properties and the shape of P ( f ) are obtained from the values of T and g, the mean and variance of the distribution. The validity of this model was assessed in jet diffusion flames, (Naguib, 1975) and in confined diffusion flames,f and the obtained agreement suggested the adequacy of the model assumptions to represent diffusion flames. M o d e l 2: In turbulent premixed flames, the previous assumptions of model 1 are not appropriate and a model to account for the finite rate of reaction is required. An Arrhenius type rate of reaction is compared with an eddy break up rate (Spalding, 1970), and the smaller of the two rates controls the reaction. The corresponding rates are expressed as; Rfu = - lOl°~2Mf.Mox exp ( - 1.84 x 104/T)

(4)

and the eddy break up rate, /~ebu

= - p -~ m ~ ~u.

(5)

The inverse of these reaction rates are proportional to the time scales of chemical kinetics, ~,., and turbulent mixing, ~m, respectively and their ratio (Damkohler number No = rmh'~) determines the dependence of the reaction on turbulence and chemistry. The values of m~u are obtained from the solution of a modelled conservation equation expressed in the form 1, and additional terms to account for the influence of chemical reaction on the generation and destruction of ~ are included (Khalil, 1976). This model assumes a clipped Gaussian probability distribution of fuel mass fraction P(Mfu) and the average values of any scalar entity are obtained in a similar manner to that of model 1 (Hutchinson et al., 1976). The validity of this model was assessed by comparisons with experiments and the obtained agreement suggested its use in premixed flames (Hutchinson et al., 1976). fSee Elghobashi et al. (1974), Hutchinson et al. (1974) and Khalil (1976).

On the modelling of turbulent reacting flows in furnaces and combustion chambers

453

Model 3: In flow situations, where turbulent mixing and chemical kinetics are both of significant influence on the reaction, a model which accounts for the turbulent and chemical fluctuations effects is necessary. The approach of Borghi (1974) was developed and modified for furnace flows (Hutchinson et al., 1977), and the rate of fuel consumption can be written as; R~. = -~210'°ICdf.Mox exp ( - 1.84 x 104/T){1 + F}

(6)

where

F = F ( T , JVlfu, Mox, T-~, mfumox, m~,T', m o x T ' , . . . ). This expression was obtained by expanding the exponential term in the instantaneous rate in a Taylor series, and on time averaging, the above equation emerged. The values of the second order correlations are obtained from the modelled form of their conservation equation expressed in the form 1. The temporal distribution of Mru is assumed to conform to a clipped Gaussian form in the range of 1 > MI. > 0, and its value P(MI.) is obtained from the local values of ~tt. and m ~.. The use of this model involves the solution of many correlations and, hence, is restricted to regions where both models 1 and 2 fail to yield reasonable results, typically, in regions, where the time scale of turbulence and that of chemistry are of the same order i.e. (No---1). In shear flows, the modelled correlations equations can be simplified by neglecting the convection and diffusion of the correlations and, hence, obtaining simple algebraic expressions (Khalil, 1976). To overcome the modelling of many conservation equations for these correlations, described above, Pope (1976) suggested the calculation of the rate of reaction from the integration of the instantaneous rate and probability; 1

Rf. =

f0

R~.(Mf.)P(MI.) dMs.

(7)

and solves transport equation for P(MI,). Preliminary attempts to use this approach were reported by Khalil (1978), but work is in progress to compare the two approaches, i.e. eqns (6) and (7), in a wide range of flow configurations. The effect of the density fluctuation correlations on the local flow properties was considered and these correlations were obtained from algebraic expressions of their exact conservation equations .in a manner similar to Hutchinson et al. (1977). The local density of the flow was obtained from the perfect gas law and Dalton's law of partial pressures. The specific heat of a mixture of gases was obtained from a fitted third order polynomial (JANAF, 1971). The heat of reaction of the fuel and thermodynamic properties of the mixtures were obtained from JANAF (1971). Radiation model. Many models have been proposed to account for the radiation heat transfer contribution to the total energy transport in turbulent furnace flames, these models solve the transfer equation with various dis-

454

E.E. Khalil

tributions of the total radiation intensity. Flux methods are appropriate to the present types of solution procedure; the transport equations governing the net flux in the coordinate directions are expressed in a form similar to eqn (1). The four flux model of Lockwood and DeMarco (1975) is modified here for axisymmetric flows and the two transport equations of Rx and Rr are obtained in the following form (Khalil, 1976) -----4 0 1 0 Rx = ~ a(2Rx - Rr Ox a ax

-

-

o ' T 4)

4 10r0 Rr = ~ a ( 2 R r - Rx - ~rT 4) r Or a Or

(8) (9)

and the source term in the enthalpy equation is Sn = ~ a(Rx + Rr

-

2o'T4).

(10)

The value of the average absorption coefficient, a, can be found by matching a one grey gas representation to the multi-grey gas representation (Truelove, 1975), at a path length equal to the mean beam length of the enclosure and a temperature equal to the mean radiating temperature of the furnace gas. Numerical procedure

The present numerical procedure solves the finite difference form of the governing conservation equations simultaneously in an iterative procedure. The solution algorithm is that of Gosman et ai. (1974) embodied in the TEACH-T computer program, modified and tested for reacting flows by Hutchinson et al. (1975, 1976, 1977). Further details of the solution procedure were reported by Khalil (1976). The present computer program requires an available storage of 35k words of memory, and the approximate computing time, which depends on the flow situation, is nearly 10 min for moderately swirling flows S = 0.6 and decreases to 7.5 min for non swirling flows. A converged solution is obtained when the maximum value of the normalized residuals in all conservation equations is less than 10-4. The present grid arrangement comprises a 20 x 23 non-uniform nodes distribution. To aid the convergence of the solution, under-relaxation was used and the under-relaxation parameters were 0.5 for velocity components and 0.7 for all other scalars. The obtained results were independent of the grid size and arrangement, as a 40 x 40 grid arrangement did not influence the results by more than 2%. Calculated results and comparisons with experiments The present section describes the comparisons between calculated and measured flow properties and heat transfer in a wide range of furnace flow configurations. These test cases were selected to represent diffusion, premixed

On the modelling of turbulent reacting ]tows in furnaces and combustion chambers

455

and arbitrary fuelled flames; and the calculations were performed to assess the validity of the computational scheme and, particularly the various combustion models assumptions. In each test case, flow configuration, flame types are indicated and the inlet and boundary conditions were specified as in Table 1, and the corresponding computational grid is shown in Fig. 1 and shows the main dimensions of the furnaces considered. Three test cases have been selected and considered for the comparisons, as indicated in Table 1, and were reported by Bilger et al. (1972), Stewards et al. (1972) and Spadaccini et al. (1976). These data correspond to turbulent reacting flows in various geometries. The type of fuel used in each case was different, which allows the assessment of the model behaviour for air to fuel density ratios between 12 and 0.6. In each test case; flow patterns, mean gas temperature, reactants distribution and wall heat flux are discussed. The flow of Bilger et al. (1972) Detailed profiles of mean gas temperature, species concentrations and centreline velocity distribution were reported in a hydrogen flame issuing in a coflowing air stream (Bilger et al., 1972). The furnace and burner dimensions are shown in Table 1, and the present comparison conforms to fuel to air inlet velocity ratio of 10 and the flow rate of the air was 84 times the stoichiometric Table 1. Dimensions of furnaces considered

••rnace

Bilger

(1972)

flow

Steward et al (1972)

Spadaccini et al (1976)

Properties

Furnace diameter

0.305

0.254

0.122

i. 800

O. 720

1.OO

Df Furnace length

Flame type

Diffusion

Premixed

Arbitrary

fuelled

Dj

0.00762

0.0032

0.063

D

0.00802

0.0032

O~0637

O~305

O.250

O~O937

O~O

O~O

O~6

0,066

0.071

0.376

Hydrogen

Propane

Methane

Burner geometry

o

% Swirl number Thermal input

Fuel

S MW

456

E.E. Khalil

Fig. I. Furnace and grid arrangement.

requirements. Inlet profiles of mean velocity and turbulent kinetic energy were measured and were used in the present calculations as inlet conditions; the inlet profile of e was assumed according to a mixing length hypothesis (Khalil et al., 1975). An equivalent diameter was used and corresponds to nearly 40 Dj, it did not influence the calculated flame properties as the flame was narrow. The present results were obtained by model 1 which is appropriate to the present flow as the local Damkohler number was much greater than unity (Khalil, 1976). The calculated centreline distribution of mean velocity and kinetic energy of turbulence are shown in Fig. 2 together with the available measured data. Measured and predicted velocities were in good agreement. The calculated and measured distributions of mean gas temperature, fuel and oxygen concentrations along the flame centreline are shown in Fig. 3, and are in reasonable agreement. The good quality of the agreement for the results is aided by the boundary layer nature of this diffusion flame. Calculated and measured temperature profiles are shown in Fig. 4 at various distances downstream of the burner exit. Reasonable agreement is obtained with maximum discrepancies of the order of the experimental scatter at xlDj of 160 but generally within _-+100 K. It can be seen from the above comparisons that the assumptions of model 1 were adequate for this type of flame; and the model is more economical than model 3 for diffusion flames (Hutchinson et al., 1976b).

On the modelling of turbulent reacting ]lows in furnaces and combustion chambers

U U'~

~"

oeeee

457

Exp'. , BiFger et ol.

I0 •

Colculot ions

0,.5,

•

i

i

i

A

i

40

i

|

80

I

i

120

I

160

___X Dj

Fig. 2. Measured and calculated centerline distribution of mean axial velocity and turbulence intensity in the furnace of Bilger et al. (1972).

fraction

mole I

T K

Fu

~Ox

•

i

i

40

i

mA

80

A

120

160

x Di

Fig. 3. Measured and calculated centerline distribution of temperature, fuel and Oxygen in the furnace of Bilger et al. (1972).

The flow of Steward et al. (1972) This flow was different from that of Bilger et al., as the central fuel, premixed with air was issued in a co-flowing air stream in a cylindrical enclosure. The enclosure was oil cooled and was equipped to measure wall heat fluxes. Mean axial velocity and gas temperature profiles were also measured and are compared, in the present work, with the corresponding calculations. Inlet profiles were not measured and therefore, were assumed according to Khalil

458

E . E . Khalil

e o o o o E x p ! flilger el ol. - Colculoflons

2r D-'~"

-.-~,

ee~

8

°N

2r

2.~.r

Oj II

~

r

2

16 xioo T K

2r Dl 16. 11~~82

4 8

16 xlOO T K

12 16 xloo T K

Fig. 4. Measured and calculated profiles of mean gas temperature in the furnace of Bilger et al. 0972).

(1976). The associated uncertainties, did not influence the calculated results of 0 and T by more than 3.5%, and this influence diminished at distances greater than 0.17 D i downstream the burner exit. In obtaining the present calculations, the assumptions of model 2 were incorporated in the calculation procedure, and were justified on the grounds that the local Damkohler numbers were very small; typically ~ 0.1. The mean axial velocity profiles at two axial locations downstream the jet exit are shown in Fig. 5 where, general agreement was observed in regions of forward flow. Discrepancies near the edges of the wall recirculation zone are attributed to the overpredicted rate of spread of the jet; this wall recirculation zone formed due to the high jet momentum. Radial profiles of mean gas temperature for the same flow conditions are shown in Fig. 6; the lines represent the calculated profiles and the symbols denote the measurements. Near the base of the flame, the central core temperature is low and the peak temperature corresponds to the flame brush, and its location moves outwards, at locations of xID r of 1.64, then

On the modelling of turbulent reacting/Rows in furnaces and combustion chambers

eeeeeExp,

i

Steword

et ol.

459

C alculotlons

U m/s

U m/s

~ '0,588

"~'f- 0 , 9 3 8 15

I0

I0

h

50

i

50

r, m m

I00

-

•

• r,

I00

m m

Fig. 5. Measured and calculated profiles of mean axial velocity in the furnace of Steward et aL (1972).

e,o

T K

Exp,

, St•word

et ol

T K

I00

xlO0

20

, ~

Calculations T~

20

.>0

x

X.L Of

x t00

_..x= 2 . 6 5 Dt

o-;

• 0.238 o 0.588

15

\ \

o 1.29

15

10-

° "---2"" 5~

5 5o

I0O

r, m m

5o

ioo

I r,

L I

I

L

L

i

i

i

I

i

mm

r. mm

Fig. 6. Measured and calculated profiles of mean gas temperature in the furnace of Steward et al. (1972).

towards the flame centreline as shown at xlD I of 2.65. General agreement is apparent except at the outer edges of the flame and the maximum discrepancies are less than _+ 150 K. The radiative heat flux distributions along the furnace wall are shown in Fig. 7; the measured distribution was obtained by the aid of a wide angle radiometer, and the calculated distribution, was obtained from the solution of eqns (8)-(10) incorporated in the radiation model. Both measured and cal-

460

E . E . Khalil

(3W kW/mz O e e • o Exp. ~ Steward et ol. Calculations 15

,o

5

Io

'

~'

2'o

Df Fig. 7. M e a s u r e d and calculated distribution of wall heat flux in the f u r n a c e of Steward et al. (1972).

oj • Exp., Spadoccini et al., u m/=

Calculations

T K x I000

•

•

•

-if,

Of

0

.

I

4.47

D

•

.38

IO0

I

!

0 I00

0 .

0.13

~

~

=

~

-

0.6(?

0.06

T Itt

fuel

'

air

,.Oo fuel

air

LO 2_..C D~

Fig. 8. M e a s u r e d and calculated profiles of m e a n axial velocity and gas temperature in the f u r n a c e of Spadaccini et al. (1976), S = 0.6.

On the modelling of turbulent reacting flows in furnaces and combustion chambers

461

culated distributions are in good agreement and assess the validity of the radiation model assumptions.

The tlow of Spadaccini et al. (1976) An axisymmetric sudden expansion furnace with a central fuel injection and annular air was used by Spadaccini et al. (1976) to simulate furnace flows. The overall dimensions of the furnace are shown in Table 1. The air flow rate through the annulus was adjusted to give air to methane velocity ratio of 22 and to allow for 10% excess air. The air swirl number, S, was 0.6, and the tangential momentum was imparted to the flow with the aid of vanes. Measurements of mean velocities, gas temperature and species concentrations were reported (Spadaccini et al., 1976), at various locations downstream the burner exit and are used for comparisons in the present subsection. Inlet profiles were not measured but were assumed according to Khalil (1976) to conform to developed flows, and

O, • E x p ~. ,

Spacloccini

el a l . ,

CH4%I

Calculations

02

-67,

I0 1 0 I

• ......

4..25

IO

Or .....

I.

12.86

-~o Io 0 1 '

e.

1.47

20 IO

IO 0

:! i I I''

1.0 2.~_r Df

Fig. 9.

,

'

' '

~0.82

,o

Io 0

'e'°'

"'

0.08 , ,

I

I

ILo2_!Df

Measured and calculated profiles of fuel and Oxygen concentrations in the furnace of Spadaccini et al. (1976), S = 0.6.

462

E . E . Khalfl

made use of mixing length hypothesis (Khalil et al., 1975). The uncertainty limits on these assumptions result in less than 4% difference in the calculated downstream properties and this difference diminished away from the burner vicinity. Measured and calculated profiles of mean axial velocity and gas temperature are compared in Fig. 8 and are in good agreement. This agreement assesses the validity of the assumptions of model 3 which is appropriate for that type of flow as the local Damkohler number is of the order unity. Discrepancies between the measured and predicted axial velocity were less than 10%, and increased in regions of very small velocities. The maximum discrepancy between the calculated and measured temperature was less than 150K, and the agreement improved away from the burner exit. Radial profiles of measured and calculated species concentrations decreased rapidly away from the burner exit due to both chemical reaction and diffusion effects. The rate of flame spread was correctly predicted by this model and the observed discrepancies were of the order of 10%. The corresponding radial profiles of oxygen concentrations are shown in the same figure and indicate the coexistence of fuel and oxygen although not at the same time. The agreement between measured and calculated oxygen concentration is fair, and the observed discrepancies in the vicinity of the wall are attributed to the measuring precision; the calculations qualitatively represent the measurements in the wall and burner exit vicinity. No wall heat flux measurements were reported by Spadaccini et al. (1976), but these can be readily predicted with the aid of the present procedure; and the total wall heat flux corresponds to 20% of the input energy and consequently, a relatively high exhaust gas temperature, (1700 K), was calculated. Discussion and conclusion The various comparisons presented in the previous section show that trends are correctly predicted and quantitative agreement with measurements are obtained. The level of agreement shown in the previous section is sufficient to justify the use of calculation procedures for furnace design purposes, although it is clear, however, that improvement can be made to the various modelling assumptions. The limitations of the turbulence model, i.e. the eddy viscosity concept can, in principle, be reduced by increasing the number of equations used to characterize the turbulence model. Equations for the normal and shear stresses can be introduced to replace those calculated from effective viscosity concept (Pope et al., 1976). The uncertainties introduced by the two equation turbulence model are unlikely to affect the calculation of furnace flow properties as those attributed to the combustion model. A tentative conclusion which may be drawn from the calculated results in the three flows considered here is that model 1 is appropriate for diffusion flames with high Damkohler number and represents the available measurements with a precision of the same order as the measurements. The use of clipped Gaussian probability distribution yields better results than the square wave distribution as shown by Khalil (1976) for a wide range of diffusion flames. The use of model 2 for premixed flames yielded ealculations, which are in good agreement with the

On the modelling of turbulent reacting flows in furnaces and combustion chambers

463

corresponding measurements; however, the form of the eddy break up term is in need of further considerations. In the flame situations, where the local Damkohler number is of the order unity, models 1 and 2 are not appropriate and model 3, which accounts for turbulent mixing and chemical kinetics, is used to calculate the local flow properties. The agreement obtained with the aid of this model for the flow of Spadaccini et al. (1976) suggested the use of the model for arbitrary fuelled flames. Further development of model 3 is required to include higher order correlations and detailed chemical kinetics. A model that calculates the average rate of reaction from the instantaneous rate and local probability distribution is a favourable approach to follow (Pope, 1976). The present radiation model is adequate for furnace design purposes and the modelled equations, which govern the transport of the net fluxes, are easily coupled to the solution procedure. The present model can be improved to include the effects of non grey gases and soot radiation. It may be concluded that the present procedure with its two equation turbulence model, various combustion models, four flux radiation model is able to represent flows of the type described by Bilger et al. (1972), Steward et al. (1972) and Spadaccini et al. (1976) with a certainty which is of similar magnitude to that of the measurements. This suggests that the present calculation procedure can be used with confidence for industrial design purposes (Khalil et al., 1977b). There is an immediate need for more comprehensive comparisons, and for the improvement of the species temporal variation and the eddy break up assumptions embodied in the combustion model. The constants used in the turbulence model and combustion models are termed universal and, therefore, should not be changed from one flow to another. These constants were obtained from simple flow situations as explained in the relevant references. The calculations could be improved by changing the constants, but that would mean fitting the procedure to the flow and the concept of a general model will not hold.

References Anasoulis R. F., McDonald H. and Buggelin R. C. (1973) Development of a combustor flow analysis: Part 1: Theoretical Studies, U.S. Air Force Rep. APL-TR-73-98. Bilger R. W. and Kent J. H. (1972) Measurements in turbulent diffusion flames. Sydney Univ., Dept. of Mech. Engng Rep. F41. Borghi R. (1974) Computational studies of turbulent flow with reaction, Project SQUID on Turbulent Mixing, Purdue Univ., U.S.A. El Ghobashi S. and Pun W. M. (1974) A theoretical and experimental study of turbulent diffusion flames in cylindrical furnaces. Proc. 15th Syrup. (Int.) Combustion, 1353-1365, The Combustion Institute, U.S.A. Gosman A. D. and Pun W. M. (1974) Calculation of recirculating flows, Imperial College, Dept. of Mech. Engng Rep. HTS/74/2. Gosman A. D., Khalil E. E. and Whitelaw J. H. (1977) The calculation of two dimensional turbulent recirculating flows. Proc. Syrup. Turbulent Shear Flows, Penn. State Univ., U.S.A. Hutchinson P., Khalil E. E., Whitelaw J. H. and Wigley G. (1975) Influence of burner geometry on small scale furnaces performance. Proc. 2nd European Syrup. Combustion, 659-665.

464

E . E . Khalil

Hutchinson P., Khalil E. E., Whitelaw J. H. and Wigley G. (1976a) The calculation of furnace flow properties and their experimental verification. J. Heat Transfer 98, 276-283. Hutchinson P., Khalil E. E. and Whitelaw J. H. (1976b) The calculation of wall heat transfer rate and pollution formation in axisymmetric furnaces. Proc. 4th Members Cong. Int. Flame Research Foundation 1. Hutchinson P., Khalil E. E. and Whitelaw J. H. (1977) The measurement and calculation of furnace flow properties. Proc. 15th Aerospace Science Meeting, paper 77-50-FE. Energy 1, 212. J A N A F (1971) Thermochemical tables, National Bureau of Standards, NSRDS 37, COM-71-50363, U.S.A. Khalil E. E., Spalding D. B. and Whitelaw J. H. (1975) The calculation of local flow properties in two dimensional furnaces. Int. J. Heat. Mass. Transfer. 18, 775-791. Khalil E. E. (1976) Flow and combustion in axisymmetric furnaces. Ph.D. thesis, Univ. of London, U.K. Khalil E. E. (1978) Aerodynamic and heat transfer characteristics of axisymmetric confined gaseous flames. Proceedings of the 6th Int. Heat Transfer Conf., Toronto. Khalil E. E. and Truelove J. S. (1977a) Calculation of radiative heat transfer in furnaces. Atomic Energy Research Establishment, Rep. A.E.R.E. R/8747, U.K. Khalil E. E. and Whitelaw J. H. (1977b) Burner design criteria, Imperial College, Dept. of Mech. Engng Rep. Launder B. E. and Spalding D. B. (1974) The numerical computation of turbulent flows. Compur Methods in Appl. Mech. and Engng 3, 269-289. Launder B. E., Reece G. J. and Rodi W. (1975) Progress in the development of Reynolds stress turbulence closure. J. Fluid Mechanics 68, 537-566. Lockwood F. C. and Naguib A. S. (1975) The prediction of the fluctuations in the properties of free, fround-jet, turbulent diffusion flames. Combustion and Flame 24, 104--125. Lockwood F. C. and DeMarco A. G. (1975) A new flux model for the calculation of three dimensional radiative heat transfer. La revista du combustibile 5, 29, 184-196. Naguib A. S. (1975) The prediction of axisymmetrical free jet, turbulent reacting flows. Ph.D. Thesis, Univ. of London, U.K. Pope S. B. (1976) The probability approach to the modelling of turbulent reacting flows. Combustion and Flame 27, 299-312. Pope S. B. and Whitelaw J. H. (1976) The calculation of near wake flows. J. Fluid Mechanics 73, 9-32. Spadaccini L. J., Owen F. K. and Bowman C. T. (1976) Influence of aero-dynamic phenomena on pollutant formation in combustion. Environmental Protection Agency Rep. EPA-600/2-76-2470. Spalding D. B. (1970) Mixing and chemical reaction in steady confined turbulent flame. Proc. 13th Syrup. (Int.) Combustion, 649--657, The Combustion Institute, U.S.A. Steward F. R., Osuwan S. and Picot J. J. C. (1972) Heat transfer measurements in a cylindrical test furnace. Proc. 14th Symp. (Int.) Combustion, 651-660, The Combustion Institute, U.S.A. Truelove J. S. (1975) Mathematical modelling of radiant heat transfer in furnaces, Atomic Energy Research Establishment, Rep. A.E.R.E. R/7817.

Appendix Nomenclature a c~ D f g H i k

absorption coefficient, m -t constant of turbulence model Diameter, m mixture fraction m (MI~ - l(4odi) square of the mixture fraction fluctuation stagnation enthalpy, kJ/kg stoichiometric mass of oxygen per unit mass of fuel kinetic energy of turbulence ~(I/2)(~ 2+ t32+ ~'~), m2/s 2

L

Mo

length, m

mean mass fraction of species a, kg/kg fluctuating component of species mass fraction, kg/kg No Damkohler number-= ~'m/7". P(O) probability distribution of • Q heat flux W/m 2 r radius, m Rf~ rate of fuel consumption, kg/m3.s Rx, R, nett radiation fluxes in the x, r directions, W/m 2 S swirl number ----2fpUWr 2 dr/ (Do f pUUr dr) S . source of entity

mo

On the modelling of turbulent reacting flows in furnaces and combustion chambers

T 0

gas temperature, K mean axial velocity, m/s u fluctuating component of axial velocity. m/s mean radial velocity, m/s v fluctuating component of radial velocity, mJs if" mean tangential velocity, m/s w fluctuating component of tangential velocity, m/s x distance along the axis of the furnace from burner exit, m F . turbulent exchange coefficient, kg/ms p density, kg/m 3 general dependent variable ,b fluctuating component of co • rate of dissipation of turbulent kinetic energy, m2]s3 r time scale, s viscosity, kgJms tr Stefan Boltzmann constant W/m2K4

465

Subscripts a A b c eft f

fu j l m

ox t w o

species a air burner exit chemical reaction effective (including turbulence) furnace fuel jet laminar mixing oxygen turbulent wall outer

Superscripts [ --

fluctuating mean

the

effect

of