by
on
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.
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
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:
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.
2.1 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).
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.
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.
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.
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.
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:
The greater steepness of the curves in the down-stream quarter of the duct is the result of the increased air-injection rate there.
Evidently the amount of smoke which will be generated, according to the multi-fluid model, is only about one half of that which the fluctuation-neglecting single-fluid model predicts.
This is because the mixture at any location consists of a population of fluids, each having its own values of:
Their distributions will be shown in the next two figures.
Understandably, the concentration of the pure-fuel fluid diminishes, that of the pure-air fluid increases, and that of the stoichiometric- mixture fluid rises and falls, with distance down the combustor.
The next sketch shows more profiles.
Fig. 3.4: mass fractions of fluids:1, 3, 5, 7, 9, 11, 13, 15, 17, 19 and 20.
Evidently, MFM's picture of the conditions in the combustor is more sophisticated than that of the single-fluid model; and it is (qualitatively at least) certainly more realistic.
Statistical properties of the fluid population can be easily deduced; of which three now follow.
Fig. 3.5: Root-mean-square fluctuations of mixture fraction
The root-mean-square fluctuations are greatest at the left-hand end, where pure fuel meets pure air. They diminish as the pure fuel disappears and fluids of intermediate fuel-air ratio come into existence.
Fig. 3.6: Mixture-average temperature according to the single-fluid and multi-fluid models
MFM (blue curve) shows a much lower population-average temperature than the single-fluid model; for only near-stoichiometric-mixture material has the maximum temperature; and there is not much of this in the population.
This observation explains why the smoke-generation rate is so much lower for MFM.
Interestingly, there is more free fuel according to MFM; and this would lead to an enlarged smoke-generation rate, were it not for the over-whelming effect of the temperature dependence.
It is also noteworthy that MFM predicts that the finite-free-fuel region extends downstream beyond the point at which the population-average mixture ratio is stoichiometric, which accords with experimental observations.
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.
| Run | CONMIX | SMOEXP | NFLUIDS | SFM rate | MFM rate |
| 1 | 5.0 | 7.0 | 20 | 6.09E4 | 3.32E4 |
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.
| Run | CONMIX | SMOEXP | NFLUIDS | SFM rate | MFM rate |
| 2 | 1.0 | 7.0 | 20 | 6.09E4 | 8.59E5 |
| 1 | 5.0 | 7.0 | 20 | 6.09E4 | 3.32E4 |
| 3 | 10.0 | 7.0 | 20 | 6.09E4 | 4.91E4 |
| 4 | 100.0 | 7.0 | 20 | 6.09E4 | 5.56E4 |
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
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.
| Run | CONMIX | SMOEXP | NFLUIDS | SFM rate | MFM rate |
| 5 | 5.0 | 1.0 | 20 | 1.95E3 | 1.65E3 |
| 6 | 5.0 | 3.0 | 20 | 1.19E3 | 7.76E4 |
| 1 | 5.0 | 7.0 | 20 | 6.09E4 | 3.32E4 |
| 7 | 5.0 | 10.0 | 20 | 4.17E4 | 2.16E4 |
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
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.
| Run | CONMIX | SMOEXP | NFLUIDS | SFM rate | MFM rate |
| 8 | 5.0 | 7.0 | 3 | 6.09E4 | 1.22E4 |
| 9 | 5.0 | 7.0 | 5 | 6.09E4 | 3.65E4 |
| 10 | 5.0 | 7.0 | 10 | 6.09E4 | 3.99E4 |
| 1 | 5.0 | 7.0 | 20 | 6.09E4 | 3.32E4 |
| 11 | 5.0 | 7.0 | 100 | 6.09E4 | 3.23E4 |
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
The above-reported study appears to the present author to justify the following conclusions:-
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 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 are tabulated in a similar manner to before; but approximate computer times are here added, as follows.
| Run | NFLUIDS | SFM rate | MFM rate | seconds |
| 12 | 10 | 7.4 E-4 | 2.38E-3 | 139 |
| 13 | 20 | ditto | 2.28E-3 | 217 |
| 14 | 30 | ditto | 2.31E-3 | 267 |
| 15 | 40 | ditto | 2.26E-3 | 485 |
| 16 | 50 | ditto | 2.27E-3 | 599 |
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:
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:
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.
