- 6.5.1 The main ideas
- 6.5.2 Application to the well-stirred reactor
- 6.5.3 Application to the paddle-stirred reactor
- 6.5.4 Application to a 3D combustor
- 6.5.5 Implementation in PHOENICS

The fundamental ideas of MFM, which have been decribed in section 5.3, remain the same whether or not chemical reaction is present.

It needs merely to be emphasised that each fluid of the population, as well as having the attribute which distinguishes it from other members of the population, can have its own temperature and a full range of chemical-species concentrations.

Often, two-dimensional populations are appropriate. In combustion applications, these will usually be:

- the fuel/air ratio; and
- some measure of the reactedness.

An important idealisation, partly realised experimentally by way of
the "Longwell bomb", is the steady-flow chemical reactor that is so
well stirred by mechanical means that variations of the **time-average**
temperature and concentration from place to place are absent.

Turbulent fluctuations remain in it, however; and the multi-fluid turbulence model can simulate their influence on reactor performance.

Some 25-fluid results from one such study will be shown.

It will be seen that the shape of the population distribution depends on the dimensionless micro-mixing and chemical-rate parameters.

The influence of the "population grid", ie the number of fluids, will also be demonstrated.

- The micro-mixing constant is varied over the range:
500 > chsoA > 6 ,
for fixed reaction-rate constant, chsoB = 5 .

- It will be seen that, as the micro-mixing constant diminishes:
- the average reactedness diminishes,
- the breadth of the population distribution increases,
- the proportions of both unburned and fully-burned gas increase.
The results are, for chsoA = : 500 ; 200; 100; 50; 20; 10; 7; 6

- Reduction of chsoA to 5.0 causes the reaction to be totally extinguished, ie the population consisted entirely of fluid 1, the unburned gas.

Similar calculations have been performed for other numbers of fluids, with the following results.

Conclusion: If the 100-fluid results are taken as correct, the 10-fluid solution may be regarded as good enough for many practical purposes. So the use of MFM need not be very expensive.no of fluids | chsoA = 500 200 100 50 20 10 7 6 -------------|----------------------------------------------- 100 | .980 .964 .940 .897 .780 .591 .416 .313 25 | .980 .964 .941 .898 .782 .593 .417 .302 10 | .965 .965 .942 .901 .789 .604 .431 .315 5 | .976 .964 .944 .908 .805 .638 .492 .409 4 | .970 .960 .943 .910 .815 .668 .548 .484 3 | .948 .941 .931 .910 .847 .742 .652 .601

In order to illustrate the use of MFM with a two-dimensional fluid population, two further pictures will be shown.

They come from a study of a reactor into which enter steadily:

- a cold fuel-lean gas stream, and
- a hot fuel-rich gas stream.

The two-dimensional histogram has fuel/air ratio as the horizontal axis and reactedness as the vertical axis.

The extent of filling of the grid of boxes indicates how much of each fluid is present.

Case for which the mixing constant is halved

Industrial reactors, unlike the ideal well-stirred one, are far from having uniform time-mean concentrations and temperatures.

They are also three-dimensional; and unsteady analysis may be needed in order to represent properly the effect of the stirring paddle.

PHOENICS may be used for simulating such reactors; and the use of the multi-fluid model reveals the importance of being able to simulate the micro-mixing process.

These points will be illustrated by the following extract from a recent study.

The geometry and computational domain are shown below.

The impeller speed is 500 rpm, the dynamic laminar viscosity is 1.0cP and the water density is 1000 kg/m

The grid is divided into two parts, namely an inner part which rotates at the same speed as the impeller, and an outer part which is at rest.

The total number of cells was 31365 (45 vertical, 41 radial and 17 circumferential).

A view of the 3-dimensional body-fitted grid

The sketch below illustrates the apparatus and the initial state of the two liquids.

They are both at rest, and are separated by a horizontal interface

The paddle is supposed to be suddenly set in motion.

The computational task is to predict both the macro-mixing, represented by the subsequent distributions of velocity, pressure and time-average concentration, but also the extent to which the two liquids are mixed together at any point.

An 11-fluid version has been employed to simulate the mixing of an acid upper liquid and an alkaline lower liquid in the tank.__________|.|__________ | ||| | | upper |.| | | liquid ||| | | |.| acid | |..........|||..........| | lower |.| alkali | | liquid ||| | | --------- | |paddle ///////// | | . | ------------|----------- .< axis of rotation The stirred mixing tank

The salt-concentration distribution is calculated on two different assumptions, namely:

- that the multi-fluid model is valid, so that each fluid reacts
at its own rate, according to its own acid-base ratio; and
- that fluctuations can be neglected, so that the salt-production rate depends only on the total-population acid-base ratio.

Assumption (2) is the conventional, ie single-fluid, model.

The salt concentrations predicted by the multi-fluid model.

The salt concentrations predicted by the single-fluid model.

The single-fluid model predicts appreciably higher salt yields than the multi-fluid model.

They are larger than the multi-fluid values, because micro-mixing is presumed (wrongly) to be perfect.

Some fluid-population histograms are now shown, for points located on a radius near the top of the tank. Their shapes would be hard to "presume" correctly.

Radius index =: 1; 2; 3; 4; 5; 6

- It is possible, and not very expensive, to compute the
probability-density functions describing the extent of micro-
mixing of initally-separated materials in a stirred tank.
- The 11 fluids used in the calculations were probably too few; but
any number can be used (100 is common) so that (fluid-) grid-
independence can be tested.
- When a chemical reaction can take place between the materials,
the yield can be computed accurately only by way of a multi-fluid model.
- Even without further refinement, MFM is probably more reliable than any presumed-pdf method, because it PREDICTS instead of PRESUMING.

Smoke is produced in combustors in regions of high temperature and excess fuel. Its rate may be computed, given:

(1) suitable chemical-kinetic rate formula; and (2) the distributions of temperature and fuel concentration.

Of course, it is not the time-mean temperature and concentration that are relevant, because of the turbulent fluctuations.

This will be demonstrated by attachment of a smoke model to PHOENICS library case 492, together with an 11-fluid MFM, with fuel-air ratio as the distinguishing attribute.

The conclusions are similar to those for the paddle-stirred reactor: only when the multi-fluid nature of turbulence is accounted for can chemical-reaction-rate predictions be regarded as credible.

* Smoke-production rate is taken as: const * (f - f_stoich) * T**5* The equilibrium SCRS scheme is | * <-temperature used, which entails that all | * * fluid properties are functions | * * of the mixture fraction, as 1.0 - * + * +<-fuel indicated. |+ * + + *+ | + * + + + * * Here "fuel" signifies the | +* + + + fuel-rich air fuel mixture | *+ + + + which is injected into this | * + + particular combustion chamber. |* + +<-oxid + + | + + + + * The micro-mixing rate is taken | + + + +<-product as 10 * epsilon/k * the |+ + + + product of the mass fractions. 0.0 |_________+_________| 1.0 mixture fraction ->

In the following contour diagrams, the flow is from right to left.

Only one sector of the combustion chamber is shown, because the pattern of injection ports is repeated 6 times around the circumference.

Longitudinal velocity, w1, contours

These and other contoured values are the averages over all fluids

Unburned-fuel contours. It can be seen that a small amount of unburned fuel escapes from the chamber near the combustor wall.

Now follow a set of fluid-population distributions for 10 points located on a radius in the middle of the exit plane.

Radius index = 1; 2; 3; 4; 5; 6; 7; 8; 9; 10

Each of the fluids produces smoke at the rate which corresponds to its own fuel-air ratio and temperature. This leads to an all-fluid-average smoke concentration as follows.

Smoke concentration according to the multi-fluid model.

Next will be shown computations based on the neglect of the mixture-fraction fluctuations.

Smoke concentration according to the single-fluid model

The maximum value is 25 % greater; and the distribution is different.

- The difference between the single-fluid and multi-fluid smoke-
production rates is, as expected, significant.
- Neglect of the influence of fluctuations, or basing calculations
on unlikely-to-be-correct pdf presumptions, is therefore dangerous
for combustor designers seeking to reduce smoke.
- A similar demonstration could easily be made for NOX or other
pollutant.
- The sufficiency of 11 fluids can be tested by "fluid-grid-
refinement" studies; but these are easy to do, as was shown for
the well-stired reactor.
- Greater realism can of course be attained by the use of a two- dimensional population, with fuel-air ratio as one dimension and reactedness (or an equivalent) as another.

- Two-dimensional-population calculations have already been made
for turbulent diffusion flames (Spalding, 1995). They appear to
present no problems of convergence, no matter how many fluids
are used.
- Of course, the greater the number of fluids, the larger is the
computer time. Therefore, for large 3D (and especially
time-dependent) calculations, the number of fluids will be made
no larger than necessary.
- There is no need for the number to be the same over the whole
field; or to remain constant throughout the time span. Population-
grid-adaptation strategies have already been devised.
- Computational expense will therefore present no serious barrier to
the use of MFM for combustor simulation.
- The main need is for acceptance and use of the model, folllowed by experimental validation.

Examples of chemically-reacting flows, simulated by way of the MFM, are to be found in the MFM-option Library of PHOENICS.

In all the examples provided so far, the number of fluids is uniform and constant, the use of computer-time-economising devices being still a matter for research and development.

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