Predictions of the flow field and local gas composition in gas turbine combustors

PREDICTIONS OF T H E FLOW F I E L D AND LOCAL GAS COMPOSITION IN GAS TURBINE COMBUSTORS

W. P. JONES" AND C. H. PRIDDIN'*

A mathematical model for calculating the three-dimensional flow, gas composition and temperature fields in gas-turbine combustion chambers is described. The model makes use of the density-weighted averaged forms of the governing conservation equations with the two equation k-~ model for turbulent transport. The chemical reactions associated with heat release are assumed to be fast and fluctuations in scalar properties accounted for by use of a/~-probability density function (p.d.f.). For liquid spray fuelled combustors the droplet concentration field is described via an equation for the p.d.f, for droplet size. For a propane fuelled model gas turbine combustor the method yields calculated fields of gas composition and pollutants (NO, CO and UHC) which are in good overall agreement with measured values.

Introduction

The design of present day gas-turbine combustion systems is invariably based on the use of empirical models which attempt to correlate overall perform- ance in terms of a number of simple global parame- ters such as combustor volume, air inlet temperature and pressure, mass flow rate and fuel/air ratio. However, current interest in the reduction of combustion generated pollutant emissions requires a more fundamental approach and has led to an in- creased demand for prediction procedures for calculating the flow, gas composition and temperature fields in practical combustion systems. The work to be described here concerns the development of such a calculation method for gas-turbine combustors based on solution of conservation equations in differential form via a numerical procedure. The equations embody assumptions which will be re- ferred to as a turbulence model and a combustion model. The latter is capable of describing fuels which may be injected in gaseous form or in the form of liquid droplets. In the following sections the forms of these models and the solution procedure

"Department of Chemical Engineering and Chemical Technology, Imperial College, London SW7 2BY.

"'Rolls-Royce Ltd, Aero Division, P.O. Box 31, Derby DE2 8BJ.

is discussed. The application of the procedure to a model propane-burning combustor is described and the accuracy assessed through comparison with experimental measurements. Also included are some typical calculations of a liquid kerosene fuelled combustor, but here the absence of reliable experimental data precludes an assessment of the droplet model. Finally a summary of the more important conclusions is presented.

Formulation

A closed set of conservation equations in differential form describing the instantaneous velocity, gas composition and temperature fields can he written for both liquid and gaseous fuelled combustion systems. However, the flow in any practical combustion chamber is invariably turbulent and the variations in the dependent variables cover such a wide range of time and length scales as to preclude the direct numerical solution of the equations. As a consequence the dependent variables are decomposed into mean and fluctuating components and the resulting equations averaged to convert them into statistical equations describing the evolution of mean quantities. (In general the averaging process should involve ensemble averaging but for the stationary flows considered here this is indistin- guishable from time averaging.) As a result of the non-linearity of the equations averaging results in

399

400 COLLOQUIUM ON TURBULENT-COMBUSTION INTERACTIONS

a loss of information so that the equations are no longer closed and closure assumptions are necessary before solution is possible.

In the present work the dependent variables are decomposed using the density-weighted . de- composition and averaging procedure suggested by Favre (1). For high turbulence Reynolds number stationary turbulent flows to which we here restrict attention, the averaged forms of the governing equations can now be written:

( ~ ' ) , , = 0 (l)

(if0' u'), , = -(Pg" +~ u 'u ' ) , (2) - u ' (~0'4,o),, = -(p 6o ),, + t;ff,~ (3)

The scalar set d ) may represent enthalpy and a sufficient number of chemical species concentrations to close the system. Lower case letters represent fluctuating components and overbars averaged quantities.* To eqs. (1) to (3) must be added the equation of state:

P P = --RT ~ n (4)

For liquid fuel spray flames an equation repre- senting droplet number density conservation may be derived.

With the following assumptions:

i) droplets are taken to be spherical and of uni- form fluid density

2) the net formation rate through such processes as nucleation, particle breakup and coagulation is zero

3) fuel droplets form a suspended cloud, i.e. gravitational and other body forces are negligible

4) droplets occupy a negligible part of the total volume

5) the relative velocity between droplets and the surrounding fluid is zero

the averaged form of the spray equation may be written:

~ ' I ~ ) , , = - s ~ - t~ +~ as 7

- (t~u'f),, (5)

where ~" ds represents the mass fraction of droplets in the size range s to s + ds and s -=- ds/dt.

The unknown terms in eqs. (1) to (5) may be considered as falling within three categories:

a) correlations involving the fluctuating components of velocity which represent diffusive transport by turbulent motions,

b) the average (or mean) net formation rate of the scalar @o, and

c) a correlation involving the fluctuating compo- nent of rate of change of droplet size.

Unknowns in the first of these categories may be approximated by turbulence models which provide relationships between unknown correlations and 'known' quantities. For reacting flows where ~ may represent concentration of chemical species the net formation rates are invariably highly non-linear functions of temperature and concentration and a combustion model is needed for the evaluation of mean formation rate and mean fluid density. Un- knowns of category (c) are of significance only for liquid fuel spray flames. How these terms should be modelled is unclear and so, although under some circumstances they may be large, for the present they will be neglected.

Turbulence Model

The turbulence model used here is a form of the k-e two-equation model introduced by Jones (2). The Reynolds stress is assumed to be linearly related to the rate of strain:

2 f i U i U i i i - - m = - - g ( P k + # T U ,m)

3

-- IZT(g U ,t + g U ,,)

- 2 where/z, r = C pk /~ and k (=- u 'u , /2) and E are the turbulence energy and dissipation rate; their values are obtained from the solution of the transport equations:

(fiUik),~ = [ ~ tzr gltk,~} \ -fiu~u~ l~lJ,,-t~ (6) \ Ork /.l

(t;O' ~),, ~ " ) =

e l - i - j - p e a j U , - C2t~e2/k (7)

where a~ = u ' u , / k - 2 /3 8~. The equation for the conservation of a general

scalar and the droplet equation both contain a turbulent flux term which is approximated via a turbulence 'Prandtl-Schmidt' number

~ It = p u i / p ; U' = instantaneous velocity; u~ = U ~ - U' and similarly for ~ , ~ , .

$ s t i - i.e. ~ u J ~ = - - - g cb~., o r T

GAS TURBINE COMBUSTORS 401

- - - g ,,

t r T

The turbulence model contains a number of empirical constants which were assigned the values:

C~ = 0.09; C~ = 1.44; C2 = 1.92;crk = 1.0;

t r =1.3 and ~r+=0.9.

Combustion Model

The provision of an accurate combustion model for finite rate reacting turbulent flows in general presents many unresolved problems, see e.g. Jones and Whitelaw (3). Fortunately, the major features of many practical systems can be described with the assumption of 'fast' chemical reaction and the problems then become less severe. If reaction rates are sufficiently fast, then chemical equilibrium can be assumed to prevail everywhere at every instant. Furthermore, if the assumptions are made that all species and heat diffuse at the same rate and that the heat loss to the surrounds is negligible compared with the heat release, then the instantaneous chemical composition and temperature is determinable in terms of a single strictly conserved scalar quantity. All the relevant conserved quantities are linearly related and any one can be chosen.

In the present work the hydrogen element con- eentration, n H, (moles/unit mass) is chosen as the appropriate conserved scalar and the dependence of composition, temperature and density on n H obtained from equilibrium calculations performed with the computer programme of reference (4). Now the relationship between nH and mixture composition, temperature and density is highly non-linear and for the determination of mean values it is necessary to know the probability density function (p.d.f.) for n H or some equivalent information. Here the form of the p.d.f, chosen is the 13-probability density function suggested by Richardson et al (5).

a--I x )b - - I x ( 1 - i.e. p(x) 0 < x < l .

1

I x ~-' (1 - x) +-' dx o

where x = nn/nnMAx* and in which the exponents a and b can be obtained explicitly in terms of the density weighted mean and variance of n n, the values of which are obtained from the solution of the equations:

/I 'L x # -

( J ~ ( J i N H ) ' t : ~ T g NH") ',

= + ~ , ,H , i t , {~'I"

2 ~ - -

k

2 - - - - n H

(8)

(9)

Hence p (x) is a density-weighted function and thus any density-weighted mean value can be obtained from:

I = �9 (nH)p(x) dx (10) o

Unweighted mean values are calculated via:

I ~ p(x) dx ~ ( n , )

(~) = P J o p ( n . )

and the mean density is given by:

- - ' = [ ' p(x) dx v J oP(n.)

A double delta form for p has also been tried, with very little effect on the results.

Liquid Fuels

In the present work it is envisaged for liquid fuelled flames that droplets evaporate to form gaseous fuel which on mixing with air burns to form products. As for gaseous fuels the reaction rates are assumed to be sufficiently fast for the instantaneous state of the gas phase to be in chemical equilibrium. For liquid fuelled flames the hydrogen element concentration, for which the values of the mean and the variance are obtained from eqs. (8) and (9), is the total liquid plus gaseous hydrogen, and it is necessary to know the proportion of hydrogen element present in the gaseous phase. It is here assumed that the ratio of the mass of gaseous hydrogen per unit mass of gas to the total hydrogen (liquid + gas phase) mass fraction does not fluctuate with time. If the hydrocarbon fuel is taken to have hydrogen to carbon ratio tr then:

�9 r i l l

*nHMa• is the maximum possible value of nil, i.e. where when the mixture consists entirely of fuel.

MH Ii - - - F ds

M OH,


There is no experimental evidence in support of the above assumption, but it is correct in the limiting eases of hydrogen element being present entirely in one or other of the two possible phases. To complete the model, following ref (6), the instantaneous rate of change of droplet size through evapo- ration is assumed to be:

= _ ~ # In 1 + s -1 O'gpL

The specific enthalpy of the gas phase may be expressed as a function of n H a:nd eq. (10) then used to evaluate the mean value, S.

Nitric Oxide

Nitric oxide is normally present only in trace quantities and its calculation can be considered separately�9 It is presumed that the formation of NO can be described by the extended Zeldovieh mecha- nism:

O + N 2 ~ N O + N

N + O 2 ~ N O + O

N 2 + OH ~ N O + H

With a steady state approximation for N, neglect of the reverse reactions and oxygen atom concentration presumed to be in equilibrium, the instantaneous formation rate of NO is given by:

qtNo = 4.0917 • 1013 T -~

�9 exp {-67915/T}p t/~

no2 nN2 kg moles/(kg see)

The rate constants have been taken from ref (7). The temperature, oxygen and nitrogen concentrations and density can all be expressed in terms of n m and the mean NO formation rate (required in the NO conservation equation obtained from:

I' r = 't',,o (n . ) p(x) clx

o

Method of Solution

To obtain the results to be presented here, the full three-dimensional forms of the governing equations have been solved in cylindrical polar coordi- nates with the aid of a computer programme called PACEand developed by the authors for Rolls-Royce Ltd. The programme embodies a finite difference technique which utilises a guess and correct procedure similar to that of Chorin (8), Harlow and Welch

(9) and Patankar and Spalding (10). The calculations were carried out on an IBM 370-168 computer and typically needed 12 minutes run time and required 512 K bytes storage�9 Details of the computation grid and boundary conditions are given in ref (11). The program solves equations 1-7 for liquid-fuelled combustors; equation 5 is omitted for the gaseous fuelled case.

Discussion of Results

The Test Problems

The predictions reported in this paper are for a small scale research combustor built and tested at NGTE (12). The combustor details are shown in Fig. 1. The chamber is operated at 2.1 bar, with air inlet temperature of 570 K and fuelled with propane at 300 K. In addition, some sample calculations are shown for a combustor of the same size, operated at 10 bar and fuelled with liquid kerosene�9 This eombustor is shown in outline in Fig. 2.

Predictions of the Propane Combustor

Streakline Plots In order to render the three-dimensional velocity

field more easily understandable, several output analysis routines have been developed�9 The streakline plots shown in this paper represent the path of particles moved within a specified plane by the local velocity vector in that plane, for a specified tracking time. They provide a picture of the flow similar to that obtained in a water-flow perspex model illuminated by a plane of light (a technique often used in combustor testing).

The main features of the eombustor flow can be clearly seen on such a plot (Fig. 3). Up to the dilution jet plane, the flow is for practical purposes axi- symmetric, and Fig. 3a is sufficient to define the flow. The size and position of the recirculating flow is clearly displayed, as are the position and path of the dilution iets. Fig. 3b shows that these jets are in fact deflected circumferentially by the swirl which still persists, and this is the reason for the lack of radial penetration of the dilution air.

Axial velocity measurements only extend up to .047 m from the baseplate, but this is sufficient to show that the predicted recirculation length is too long--in fact about twice as long as measured. There is considerable doubt about the accuracy of the measurements as they were taken with a 7 hole total pressure probe which in highly fluctuating flow will give inaccurate results; it is believed, however, that the position of zero axial velocity can be located to a fair accuracy.


SWIRLER AIR DILUTION/COOLING AIR IGNITER

,0 . . . . 5,0 PLENUM LINERS

SCALE mm

FIC. I. NGTE propane fuelled research combustor.

Profile Comparisons The profiles of fuel/air ratio are plotted on Fig.

4 and compared with the measured values at each measurement plane. The general behaviour of the profile shapes is good, the largest error being in the profile at the first measurement plane, This exhibits a peak value higher than that measured,

a result consistent with the long recirculation length �9 since both imply too little mixing (turbulent transport too low) in the recirculating region. Although downstream the profiles are in quite good agreement, this over-rich peak is felt in the profiles of other quantities.

Carbon monoxide and unburned hydrocarbon

/ A,B,C,D AIR ENTRY FUEL + AIR ENTERS AT B

6 OFF 8 mm DIAMETER ~ H O L E S EQUISPACED

jSOLID WALL

I|RESONATOR CAP F _ L ~ ~ O F ATOMISER

" C B.....L~J~B C A ~ ACOUSTICATOMISER

Fie. 2. Kerosene fuelled acoustic atomiser eombustor.


FIc. 3. Streakline plot for the propane fuelled combustor.

~ - - I I I

I !

distributions are shown as plane averaged plots in Fig. 5. In both cases the model has overestimated the CO and HC levels and this results in the combustion efficiency being too low (Fig. 6). This is in spite of the infinitely fast reaction rate chemical equilibrium assumption. The presence of significant quantities of CO and HC is due in part to the overestimation of the maximum fuel /a i r ratio at the first measurement plane, but the modelling of concentration fluctuations may also be in error, since even those planes where the predicted and measured fuel /a ir ratio are in good agreement, exhibit excess CO and UHC levels, although here the discrepancies are considerably smaller. These results do illustrate the need for measurements of concentration fluctuations in combusting flows (in addition to mean values) as this is really the only way of assessing the validity of the chemistry/turbulence interaction modelling. Of course such quantities are extremely difficult to measure.

0.120

FUEL/AIR RATIO

0.100- /

/ o.o8~- . . . ~ " c

0 . 0 6 0 - - I- I- L

0.040- ~ " ~ L.

0 .020 - - -

0.00~ I v

~I ~ ~I~1 ~ 0'~2~ ~ ' ~ ~ I ~ RADIUS(m)

The predictions of NO concentration are shown on Fig. 7 as a contour plot. One half of this plot shows the measured data, its reflection being the predictions. While there is some discrepancy in the predicted levels, many features of the measured data are well predicted. In particular, the peak NO formation actually occurs downstream of the recir- eulation zone, and is confined to a plane upstream of the dilution jets. The dilution of the existing NO by the injected air can be seen in both measured and predicted contour plots.

Some examples are also given of the predictions of a liquid-fuelled combustor. For this case five size ranges of droplets are used. Unfortunately there are no detailed and reliable measurements available for the combustion chamber considered, and the predictions are included primarily to demonstrate the droplet model described in the preceding sections. The geometry of the combustor is shown in Fig. 2, and the corresponding streakline plot in Fig.

'X m x ~.0170 + 0.0470 Y 0.1070

0.1650

FIG. 4. Predicted and measured radial profiles of fuel /a i r ratio for the NGTE combustor.


C0% 6

4

3

2

I

g.05 g. 10 Xim)

0.15 g.2g

25"

UHC % AS PROPANE

2O

15

10"

5 "

- - p red ic t ion �9 da ta

!

0105 g " ] 0 0 . 1 5 0.20

FIG. 5. Axial variation of plane averaged CO and UHC levels.

8. It is apparent that the lack of swirl in this chamber allows the dilution jets to penetrate further and to impinge on the chamber centreline, causing some upstream flow which, because of the axial posit ion of the jets, does not merge with the upstream recirculating region, giving a very different flow pat- tern to the propane-fuelled geometry.

A final plot (Fig. 9) for this combustor shows the evolution of vapour phase hydrocarbons from the fuel droplets, and the subsequent combustion of this vapour. A plot of the mass fraction of unburned fuel at each axial plane shows that for this geometry no unburned fuel remains at the combustor exit for the conditions tested.

COMBUSTION EFFICIENCY %

100

9 0 -

80J

70

60 '

5s

w

I

0.15 0.2 g.05 0.1

X I m )

Flc.. 6. Axial variation of combustion efficiency.

pred ic t ion

�9 da ta


Prediction

Measurement

Fzc. 7. Contours of NO concentration.

NO:z: (p.p.m.v.) 0-100

liiiiiii~ii~ 100-200 GREATER THAN 200

FIG. 8. Streakline plot for the acoustic atomiser combustor.

% OF TOTAL FUEL UNBURNED

70

50=

30=

l~ L

C .02 .04 .06 .08 .10 .12 .14 .16

AXIAL DISTANCE (m) Fzc. 9. Axial variation of plane averaged unburnt hydrocarbons in liquid phase and gaseous phase.


Conclusions

A prediction method has been developed for the calculation of the flow, gas composition and temperature fields within gas turbine combustors. The flow patterns predicted show the correct overall features, but the present turbulence model yields a recirculation length longer than that observed. The reasons for this discrepancy are very difficult to ascertain in the absence of detailed velocity measurements, particularly in the vicinity of the swirler. Neverthe- less quite good predictions can be obtained of fuel/air ratio distribution, though CO and HC levels are overestimated, as is the NO concentration. In spite of these discrepancies the model provides useful indications of the regions of maximum poilu- tant formation and consumption.

Further complete sets of test data for liquid fuelled flames would test the validity of the model assumptions and give greater confidence in its use for new eombustor configurations. Visualisation techniques developed for use with the prediction method proved very useful in interpreting the flow patterns predicted. A further phase of the project will extend the programme to handle boundaries of arbitrary shape, and thus enable existing engine combustors to be calculated.

Nomenclature

F(s) Probability density function for droplet size g Metric tensor for co-ordinate system k Density weighted turbulence kinetic energy M~ Molecular weight of species ct n~ Concentration of species r moles /uni t mass P Static pressure p(x) Probability density function for a conserved

scalar R Gas constant s Droplet size T Temperature U, Velocity vector y Mass fraction of total (liquid + gas)

Density weighted dissipation rate of turbulence energy

I~ Fluid viscosity p Density cr Prandtl /Schmidt number r General scalar quantity ~Q Net formation rate per unit volume for ~

Subscripts

g Gaseous phase L Liquid phase T Turbulent

R E F E R E N C E S

1. FAVRE, A: Problems of Hydrodynamics and Continuum Mechanics, p. 231, Society of In- dustrial and Applied Mathematics, 1969.

2. JONES, W. P: Laminarisation in Strongly Acceler- ated Boundary Layers, Ph.D. Thesis, University of London, 1971.

3. JONES, W. P. AND WmTELAW, J. H: Coupling of Turbulence and Chemical Reaction. Proceed- ings of Workshop on Modelling of Combustion and Practical Systems, Los Angeles, 1978.

4. GOROON, S. AND McBRIDE, B. J: Computer Pro- gramme for Calculation of Complex Chemical Equilibrium Compositions, NASA SP 273, 1971.

5. RICHARDSON, J. N., HOWARD, H. C. AND SMITH, R. W: Fourth Symposium (International) on Com- bustion, 1953.

6. WILLIAMS, A: Combustion of Sprays of Liquid Fuels, Elek Science, 1976.

7. BAULCH, D. L., DRYSDALE, D. D., HORNE, D. G. AND LLOYD, A. C: Evaluated Kinetic Data for High Temperature Reactions, 2, Butterworth, 1973.

8. CHOmN, A. J: Numerical Solution of the Navier- Stokes Equations, Math. Comp. 22, 745, 1968.

9. HARLOW, F. H. AND WELCH, J. E: Numerical Cal- culations of Time-Dependent Viscous Incom- pressible Flow of Fluid with Free Surfaces, Phys. Fluids, 10, 314, 1967.

I0. PATANKAR, S. V. AND SPALDING, D. B: A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows, Int. J. Heat Mass Transfer, 10, 1787, 1972.

11. JONES, W. P., CLIFFORD, W. C., PRIDDIN, C. H. AND DE CHAIR, R: A Comparison between Predicted and Measured Species Concentrations and Ve- locities in a Research Combustor, Proc. AGARD Propulsion and Energetics Panel, 50th Meeting, Ankara, Turkey, 1977.

12. HOLDERNESS, F. H. ANt, MACFARLANE, J. J.: Soot Formation in Rich Kerosene Flames at High Pressure, AGARD Propulsion and Energctics Panel, 41st Meeting, 1971.

COMMENTS

H. G. Semer~tan, National Bureau of Standards, USA. If I understood you correctly, you assumed equilibrium concentration for all species except for

NO. Does this also apply to CO, or did you use a finite rate for the CO oxidation reaction? If you did not, could you comment on the implications

408 C O L L O Q U I U M ON TURBULENT-CO MBU STIO N INTERACTIONS

of this assumpt ion on predicted CO levels, since the quenching effect of dilution jets is being ig- nored? Because of the scale of the graph, it was difficult to see the difference between predicted and measured CO levels at the combustor exhaust. Could you please give us some numerical values?

Most researchers have predicted NO~ values which are usually lower than measured values, even when they account for superequi l ibr ium oxygen radical concentrations. Would you please comment on the fact that your predict ions give higher values than measured, despite the fact that you have neglected nonequi l ibr ium effects? Also, could you again give us some numerical values of predicted and measured NO x levels at the combustor exhaust?

Authors" Reply. It is true that CO concentrations are calculated via a chemical equil ibrium assumption. However we are considering a highly turbulent flow field in which fue l / a i r ratio (f.a.r.) and the equil ibrium composit ion fluctuate with time (and which is taken to be described by our turbulence- combust ion model). Thus the situation can arise where, though the t ime averaged f.a.r, is fuel lean, the instantaneous f.a.r, may take values richer than stoichiometric. In this situation the (fluctuating) gas composition will, for some fraction of the time, contain significant quanti t ies of CO (and U H C if the fluctuations are extreme enough). Hence appre- ciable mean CO concentrations may arise even when the mean f.a.r, is well below its stoichiometric value. Nevertheless, as Dr. Semerjian correctly implies, quenching of the CO oxidation reaction by dilution air in gas turbine combustors is well established. The accurate prediction of CO emissions particularly under idle condit ions will thus certainly require the inclusion in our model of a finite rate for CO oxidation. However this is by no means a simple matter: a tested and satisfactory model for finite-rate reacting turbulent flows and which properly ac- counts for the influence of turbulent f luctuations on mean reaction rate remains to be devised.

The assumpt ion of (instantaneous) chemical equi- l ibrium together with our assumed form of the probability density funct ion allows us to take full account of f luctuations in temperature and gas composit ion on the mean NO formation rate. For the test combustor, which was operated at 570 K inlet temperature and a pressure of 2.1 bar, the model leads to predicted NO levels higher than those measured. The reasons why we, in contrast to many other researchers, have not found it necessary to take account of super equi l ibr ium oxygen atom concentrations are at present difficult to ascertain. However, it is most likely that they lie either in our treatment of temperature and concentration fluctuations via the 13-p.d.f. or from the assumpt ion of chemical equi l ibr ium for all species except NO.

In answer to the other part o f Dr. Semerjian's

question, the following table gives the combustor exit concentrations, of CO, UHC and NO in p.p.m.

CO UHC NO

Measured 596 6 84

Predicted 11 0 122

M. M. M. Abou-Ellail, Cairo University, Egypt. You have chosen 5 size groups for your droplet size distribution. Are five size groups enough to describe the actual droplet size distribution for your calculation procedure?

Authors "Reply. At present we do not have reliable experimental data for our liquid fuelled combustor and so we have not yet carried out very detailed computat ions in which the spray equations are solved. I suspect though, that five droplet size increments will provide only a coarse solution to the droplet concentration probability density function. However it is a very simple matter to increase the number of droplet size increments and this we plan to do when suitable experimental data become available.

D. B. Spalding, Imperial College, U.K. and Purdue University, USA. It should be recognized that, when the flow is steady and the droplet sizes only decrease, as in the author 's situation, there is no need to have more than one 3D storage array for droplet concentration. The reason is that the concentration of droplets in a small-size range is inf luenced by the concentration in a larger-size range, but not vice versa. The consequence is that one may perform a "marching integration in size space," moving from the larger sizes to the small. Of course, iteration is needed, because the vaporization and burn ing rates influence somewhat the flow field. The technique is explained in the Imperial College Heat Transfer Section Report: D. B. Spalding, "Combus- tion Theory Applied to Engineer ing," 1977.

Authors" Reply. Professor Spalding's comments on the method of solution of the equation for the droplet concentration probability density function are of course correct. However, whether one retains in store all of the concentrations of droplets in each individual size range or overwrites them, thus re- taining at any instant only one concentration, is s imply a matter of computational convenience. The solutions obtained by either method will be identical,

GAS T U R B I N E C O M B U S T O R S 409

t h o u g h it m u s t be a d d e d tha t the compu t e r t ime r equ i r ed by Professor S p a l d i n g ' s m e t h o d will be greater because the concen t r a t ions ob ta ined at the p rev ious i teration are not re ta ined. I n our case we are no t ser ious ly l imi ted by c o m p u t e r s torage re- q u i r e m e n t s and it was more c o n v e n i e n t to retain

all d rop le t concen t ra t ions in store. However , in the even t o f a large n u m b e r o f d rop le t size i nc r emen t s b e i n g necessa ry c o m p u t e r s torage avai labi l i ty is l ikely to be a l imita t ion. I n th is s i tua t ion the so lu t ion t e c h n i q u e ou t l ined by Professor Spa ld ing wil l be usefu l .

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Predictions of the flow field and local gas composition in gas turbine combustors