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The simulation of smoke generation in a 3-D combustor, by means of the multi-fluid model of turbulent chemical reaction


Brian Spalding, of CHAM Ltd

Paper presented at the "Leading-Edge-Technologies Seminar"


"Turbulent combustion of Gases and Liquids",

organised by the Energy-Transfer and Thermofluid-Mechanics Groups of the Institution of Mechanical Engineers

at Lincoln, England, December 15-16, 1998


It is well known that, in order to predict correctly the rate of any chemical reaction which takes place in a turbulent fluid, proper account must be taken of the fluctuations of concentration and temperature.

Such account can be taken when, and only when, these fluctuations are expressed in the form of "probability-density functions" (ie PDFs) of the appropriate quantities; but properly-calculated PDFs are rarely used because of the computational expense of the Monte-Carlo-type methods which until recently, have been the only ones available.

Since 1995 however, the Multi-Fluid Model (MFM) of turbulence, which computes PDFs in discretised form, has greatly reduced the computational burden; and its embodiment in the PHOENICS computer code makes it accessible to design engineers.

The present paper illustrates the application of MFM to the calculation of smoke-generation rates in:

It shows that the effect of the fluctuations is very significant; and also that it can be computed with sufficient accuracy with a rather small number of fluids, and so with acceptably small computer times.

The method is recommended as being ready for experimental use by combustor designers.


  1. The problem of predicting the rate of generation of smoke
  2. Solution by way of the multi-fluid model of turbulence
  3. Application to an idealised 1-D combustor
  4. Application to a particular 3-D combustor
  5. Conclusions
  6. References


  1. 3D-combustor calculations; convergence and computer times
  2. The influence of the number of fluids
  3. Contours for the 50-fluid model
  4. Conclusions

Note: Figures, the titles of which are indicated by underlining, are not provided in the printed paper. They may be inspected, together with the text, on the web-site: www.cham.co.uk

1. The problem of predicting the rate of generation of smoke

Smoke is produced in gas-turbine combustors as a consequence of the thermally-induced breakdown of hydrocarbon molecules in fuel-rich regions within the flame. Since smoke production is undesirable, designers strive to reduce the extent of these regions by controlling the fuel-air-mixing process.

Computational fluid dynamics (CFD) is used for predicting the sizes and locations of such regions. However, whereas CFD can predict reasonably well the locations at which the time-average mixture ratio will be fuel-rich, that is not all that is required.

The reason is that the fluctuations of concentration and temperature which characterise turbulence entail that fuel-rich mixture is present, for part of the time, even in locations where the time-average mixture ratio is fuel-lean; and vice versa.

What is needed therefore is knowledge of the proportions of time in which, at each point in the combustor, the mixture is in the smoke-generating states; and this knowledge is obtainable only from what is called the "probability-density function" (PDF) of the mixture ratio.

Methods of calculating PDFs have been available for many years. They were conceived by Dopazo and O'Brien (1974); and Pope (1982, 1985, 1990) and co-workers have been implementing them by the use of Monte-Carlo techniques; these, however, are regarded by most combustor designers as being too expensive for regular use.

Recently, the present author has been showing, in a series of papers (Spalding, 1995, 1996, 1997, 1998), that a different mathematical method can lead to the desired information about the PDFs rather economically; and moreover that this new method has several further advantages which the Monte-Carlo technique does not.

The present paper continues the demonstrations, by considering the generation of smoke in:

  1. an idealised one-dimensional combustor, and
  2. a particular three-dimensional gas-turbine-type combustor,
in which the local and instantaneous smoke-generation rate is taken to be a particular function of temperature and unburned-fuel concentration.

The study shows that:-

There is no reason, in the author's view, for not immediately incorporating the multi-fluid model into the computer codes used by combustor designers.

2. Solution by way of the multi-fluid model of turbulence

2.1 The nature of MFM

The multi-fluid model (MFM) of turbulence is best regarded by those who are familiar with the Dopazo-O'Brien-Pope "PDF-transport" ideas as simply providing an alternative mathematical technique.

Specifically, it employs discretization in place of stochasticism. There is nothing especially strange about this: Monte Carlo methods can be employed for solving heat-conduction problems, even though most practitioners nowadays prefer the discretized finite-difference/element/volume methods. The same choice is available now for PDF calculations.

To those who are unfamiliar with the Monte-Carlo approach, MFM can be best regarded as applying to fluid properties, which are often thought of as "dependent variables", the same discretization techniques as are commonly applied to the "independent variables", x, y, z and time.

There is nothing new about this either. The "six-flux model" of radiation (Spalding, 1980) is of this kind; and so is the "discrete-ordinates" method (Zhang, Soufiani and Taine, 1988).

Further, particle-size distributions in multi-phase flows have long been handled by similar dependent-variable-discretization techniques (Sala and Spalding, 1973; Fueyo, 1992).

2.2 MFM applied to smoke generation

The central idea of MFM is that of a "population of fluids", the members of which are distinguished by one or more "population- distinguishing attributes" (PDAs), which are then arbitrarily discretised by the analyst.

However, each of the fluids can also possess an unlimited number of "continuously-varying attributes" (CVAs).

As far as smoke-generation modelling is concerned, the most relevant PDA appears to be the fuel/air ratio; it is therefore the one used in the present study. Other variables, such as temperature, free-fuel mass fraction, and smoke content, thus become CVAs.

It may be remarked that the distinction between PDAs and CVAs is not emphasised in the Monte-Carlo-method literature, and is perhaps not present.

3. Application to an idealised 1-D combustor


  1. The idealised combustor
  2. The idealised smoke-generation reaction
  3. How the calculations are conducted
  4. The results for the base case
  5. The influence of the micro-mixing constant
  6. The influence of the temperature-dependence of the reaction
  7. The influence of the fluid-population discretization
  8. Conclusions

3.1 . The idealised combustor

The sketch (click here) illustrates the situation which is under consideration. It shows that:-

It should be understood that the one-dimensional model is employed only because it is economical and sufficient. The simulation procedure is applicable without difficulty to two and three-dimensional circumstances, and to time-dependent conditions.

3.2 . The idealised smoke-generation reaction

Whereas the main combustion reaction is supposed to proceed in accordance with the "mixed-is-burned" rule, the smoke-generation reaction is supposed to be kinetically controlled.

Specifically, the rate is taken as being governed by the formula:

Of the two constants, the first is arbitrary, because it is comparisons which are in question, not absolute values; but the second is given values between 1 and 10, in order that the effect of the steepness of temperature dependence can be explored.

Subsequent oxidation of the smoke is not considered, because it is not relevant to the main purpose of the paper.

3.3 . How the calculations are conducted

The calculations to be reported have been carried out with the aid of the PHOENICS (version 3.1) computer code, for which the multi-fluid model is an optional, but standard, attachment.

A grid of 50 uniform intervals was chosen for the flow direction.

Runs took between 30 seconds and 4 minutes to achieve convergence on a Pentium-200 personal computer, depending on the number of fluids employed.

No non-standard settings or additions were needed.

A typical screen dump at the end of a base-case run, shown below, reveals the smooth approach to convergence. This run lasted 30 seconds.

Convergence display

3.4 . The results for the base case

The results of the calculation will be presented, in the first instance, by way of plots with various local gas conditions as ordinates and with distance along the pipe as abscissa, as follows:

Further insight into the way in which the fluid population changes with position in the combustor is provided by the following PDF print-outs, in which the figure on the left represents the PDF, in histogram style, while the figure on the right serves simply as a reminder of the random-mixture concept which underlies MFM.

In the following list, cell 5 is near the inlet to the combustor, and cell 50 is adjacent to the outlet. indicate which computational cell is in question, out of the 50 which are provided for the whole combustor.

cell 5 cell 10 cell 15 cell 20 cell 30 cell 40 cell 50

It is interesting to observe that the shapes of these PDFs are rather unlike those which are customarily presumed by those who seek to replace calculation by presumption.

Numerical data for the smoke-production rate are contained in the following table, which confirms that accounting for fluctuations predicts significantly less smoke production

SFM stands for "single-fluid model", and MFM for multi-fluid model.

15.0 7.0 20 6.09E43.32E4

3.5 . The influence of the micro-mixing constant

Calculations have also been carried out for a range of values of the micro-mixing constant CONMIX. The numerical results are shown in the following table.

21.0 7.0 20 6.09E48.59E5
15.0 7.0 20 6.09E43.32E4
310.07.0 20 6.09E44.91E4
4100.0 7.0 20 6.09E45.56E4

Increasing the micro-mixing constant, CONMIX, evidently increases smoke production. This is understandable, becaause it brings the mixture closer to the no-fluctuations state postulated by the single-fluid model.

The following pictures explain how this occurs by showing how the population-average temperature rises with CONMIX.

Population-average temperature distributions according to the single-fluid and multi-fluid models, for CONMIX = 1.0 , 5.0 , 10.0 , 100.0

Changes in other aspects of the solution are illustrated in the following series of pictures

The smoke-concentration distributions: 1.0 , 5.0 , 10.0 , 100.0

The mass fractions of individual fluids: 1.0 , 5.0 , 10.0 , 100.0

The root-mean-square fluctuations of mixture fraction: 1.0 , 5.0 , 10.0 , 100.0

The population-average free-fuel mass fraction: 1.0 , 5.0 , 10.0 , 100.0

3.6 . The influence of the temperature-dependence of the reaction

Calculations have also been made with the base-case values of CONMIX and NFLUIDS, but with greater and smaller values of the temperature-dependence constant, SMOEXP. The results are shown in the next table.

55.0 1.0 20 1.95E31.65E3
65.0 3.0 20 1.19E37.76E4
15.0 7.0 20 6.09E43.32E4
75.0 10.0 20 4.17E42.16E4

Evidently, increasing the temperature exponent makes the effect of fluctuations more pronounced.

The following pictures show the corresponding smoke distributions along the length of the combustor.

Mass fractions of smoke according to the single- (upper curve) and twenty-fluid (lower-curve) models, for the temperature exponent: SMOEXP = 1.0 3.0 7.0 10.0

3.7 . The influence of the fluid-population discretization

CONMIX and SMOEXP have physical significances; but, as in all CFD calculations, some purely numerical parameters may also influence the results.

In MFM calculations, the most doubtful such parameter is NFLUIDS, which measures the fineness of discretization of the fluid population.

Accordingly, some calculations have been carried out to explore its effect, with the result shown in the following table.

85.0 7.0 3 6.09E41.22E4
95.0 7.0 5 6.09E43.65E4
105.0 7.0 10 6.09E43.99E4
15.0 7.0 20 6.09E43.32E4
115.0 7.0 1006.09E43.23E4

Inspection of this table, with the presumption that the 100-fluid value is the correct one, shows that:

Changes in other aspects of the solution are illustrated in the following series of pictures

The smoke-concentration distributions: 3.0 , 5.0 , 10.0 , 20.0 , 100.0

The root-mean-square fluctuations of mixture fraction: 3.0 , 5.0 , 10.0 , 20.0 , 100.0

Population-average temperature distributions: 3.0 , 5.0 , 10.0 , 20.0 , 100.0

The population-average free-fuel mass fraction: 3.0 , 5.0 , 10.0 , 20.0 , 100.0

3.8 . Conclusions

The above-reported study appears to the present author to justify the following conclusions:-

4. Application to a particular 3-D combustor

The case considered

The same smoke-generation model has been activated for case 492 of the PHOENICS input-file library, which represents a (rather simple) three-dimensional gas-turbine-like combustor, which is fed by a rich-fuel-air vapour mixture, and primary-, secondary- and dilution-air streams.

The grid is coarse, namely 6*10*13 for a 30-degree sector; but it suffices for the present purposes, which are:-

The calculations performed

The calculations performed have all had the same values of CONMIX and SMOEXP as in the one-dimensional study; but the number of fluids has been varied, from 10 to 50.

As will be seen, even the 10-fluid model gives a prediction which may be accurate enough for many purposes.

The graphical convergence monitor for the 40-fluid run shown here gives proof of a satisfactorily converging calculation.

The results

The results are tabulated in a similar manner to before; but approximate computer times are here added, as follows.

Run NFLUIDSSFM rateMFM rateseconds
12 10 7.4 E-42.38E-3 139
1320 ditto 2.28E-3217
14 30 ditto 2.31E-3267
15 40 ditto 2.26E-3485
16 50 ditto 2.27E-3599

Two points may be remarked upon, namely:

The following figures show the computed PDFs for a location in the middle of the outlet plane of the combustor, for 10 fluids, 40 fluids, 50 fluids.

The shapes are all similar; and the root-mean-square and population-average values do not differ much.

The following contour plots show various aspects of the 50-fluid calculation:

  1. The smoke distribution at the outlet, computed by the multi-fluid model,

  2. The smoke distribution on an axial plane according to:
    (a) the single-fluid (no fluctuations) model and
    (b) the multi-fluid model
    The flow is from right to left.

  3. The population-average mixture-fraction distribution according to:
    (a) the single-fluid model and
    (b) the multi-fluid model
    These last two are in close agreement, as they should be.

  4. The distribution of population-average unburned-fuel concentration according to:
    (a) the single-fluid model and
    (b) the multi-fluid model
    The last two pictures should be, and are different. Specifically, the fluctuations allow unburned fuel to be present in regions where, according to the single-fluid model, it could not exist.

  5. The distribution of population-average temperature according to:
    (a) the single-fluid model and
    (b) the multi-fluid model
    The highest temperature encountered is greater for the single-fluid than for the multi-fluid model.

  6. The concentrations of fluids: fluid 1 (pure air) , fluid 6 , and fluid 11.


It is interesting to speculate as to why the one-dimensional study showed MFM to predict less smoke than SFM, while the three-dimensional study showed the opposite. At least two reasons may have had an influence, namely:

  1. It was pure fuel vapour which entered the 1D combustor, but an air-fuel mixture which entered the 3D one; so the latter had less smoke-generation propensity.
  2. Whereas the 50-cell geometrical gid was certainly sufficiently fine for the 1D analysis, the 780-cell grid was certainly not fine enough for the 3D one. In particular, the number of cells in the fuel-rich region near the injector was rather small.

The second observation represents a contrast with typical engineering practice, in which very large numbers of geometrical cells are employed, in the hope of procuring high accuracy; but this hope is rendered vain by the conventional employment of far too coarse a population grid.

Thus, in practice the fluctuations are ignored, as when a single-fluid model is is used; or some version is utilised of the eddy-break-up model, which in MFM terminology, is just a two-fluid model.

What is desirable, of course, is to maintain a proper balance between the two kinds of discretization. MFM allows this.

5. Conclusions

  1. It has been shown that the Multi-Fluid Model of Turbulence is capable of simulating, in a plausible and convincing way, the influence of turbulent fluctuations of concentration and temperature on the rate of smoke generation.

  2. A similar conclusion could undoubtedly have been drawn if the reaction product considered had been NOX rather than smoke.

  3. The computed PDFs are very varied in shape. It therefore appears to be dangerous to employ methods of prediction which entail presuming that they have a particular mathematical form.

  4. The computer times present no serious deterrent, particularly because sufficient accuracy can often be obtained by the use of a coarse "population grid", which may consist of as few as ten distinct fluids.

  5. There also exists the possibility, not illustrated in the present paper, of using different numbers of fluids in different parts of the geometrical space. Great economy can result from this.

  6. The computer times are much smaller for MFM than those reported by users of the Monte-Carlo-based PDF-transport method.

  7. The Monte-Carlo method also appears to lack the ability to perform the population-grid-refinement studies which have been illustrated in the present paper.

  8. Another difference is the distinction made only by MFM between the two classes of scalar variable, namely those which are discretized (the PDAs) and those which are not.

6. References

C Dopazo and EE O'Brien (1974)
Acta Astronautica vol 1, p1239
N Fueyo (1992)
"Two-fluid models of turbulence for axi-symmetrical jets and sprays"; PhD Thesis, London University
FC Lockwood & AS Naguib (1975)
"The prediction of the fluctuations in the properties of free, round-jet, turbulent, diffusion flames", Comb & Flame, vol 24 p 109
SB Pope (1982)
Combustion Science and Technology vol 28, p131
SB Pope (1985)
Progr Energy Combust Sci vol 11, pp119-192
SB Pope (1990)
"Computations of turbulent combustion; progress and challenges" Twenty-Third International Symposium on Combustion, The Combustion Institute, pp 591-612
DB Spalding (1971)
"Concentration fluctuations in a round turbulent free jet"; J Chem Eng Sci, vol 26, p 95
DB Spalding (1980)
"Mathematical Modelling of Fluid-Mechanics, Heat-Transfer and Chemical-Reaction Processes: A lecture course". Imperial College of Science and Technology, London, Mechanical Engineering Dept. Report, HTS/80/1
R Sala and DB Spalding (1973)
"A mathematical model for an axi-symmetrical diffusion flame in a furnace". La Rivista di Combustibili, vol 27, pp 180-186
DB Spalding (1995a)
"Models of turbulent combustion" Proc. 2nd Colloquium on Process Simulation, pp 1-15 Helsinki University of Technology, Espoo, Finland
DB Spalding (1995b)
"Multi-fluid models of turbulent combustion"; CTAC95 Conference, Melbourne, Australia
DB Spalding (1995c)
"Multi-fluid models of Turbulence", European PHOENICS User Conference, Trento, Italy
DB Spalding (1996a)
"Older and newer approaches to the numerical modelling of turbulent combustion". Keynote address at 3rd International Conference on COMPUTERS IN RECIPROCATING ENGINES AND GAS TURBINES, 9-10 January, 1996, IMechE, London
DB Spalding (1996b)
"Multi-fluid models of Turbulence; Progress and Prospects; lecture to be presented at CFD 96, the Fourth Annual Conference of the CFD Society of Canada, June 2 - 6, 1996, Ottawa, Ontario, Canada
DB Spalding (1996c)
"Progress report on the development of a multi- fluid model of turbulence and its application to the paddle-stirred mixer/reactor", invited lecture at 3rd Colloquium on Process Simulation, Espoo, Finland, June 12-14
DB Spalding (1997)
"Boiling, condensation, multi-phase flow, chemical reaction and turbulence; the multi-fluid approach" Lecture at International Symposium on The Physics of Heat Transfer in Boiling and Condensation 21-24 May, 1997, Moscow
DB Spalding (1998a)
"CAD to SFT, with Aeronautical Applications" Proceedings of the 38th Israel Annual Conference on Aerospace Systems, Israel, Feb 25-26; pp s7 1-32
DB Spalding (1998b)
"Turbulent mixing and chemical reaction; the multi-fluid approach ". Lecture at the University of Delft, Netherlands
DB Spalding (1998c)
"The modelling of turbulent combusting systems via MFM". Invited Lecture at Eurotherm 56, Heat Transfer in Radiating and Combusting Systems, 1-3 April 1998, Delphi, Greece
L Zhang, A Souffiani & J Taine (1988)
Int J Heat & Mass Transfer, vol 31, no 11, pp 2261-2272