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6.5 The multi-fluid model for chemically-reacting flows


6.5.1 The main ideas

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:

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

6.5.2 Application to the well-stirred reactor

(a) The well-stirred reactor, with a single entering stream

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.

(b) The 25-fluid results

(c) Summary of population-grid-dependence results

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

    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

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.

(d) The well-stirred reactor with 2 streams of differing compositon

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:

  1. a cold fuel-lean gas stream, and
  2. 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.

Reference case

Case for which the mixing constant is halved

6.5.3 Application to the paddle-stirred reactor

(a) The problem

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.

(b) The geometry

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).

The general arrangement

A view of the 3-dimensional body-fitted grid

(c) The starting condition

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.

        |          |||          |
        | upper    |.|          |
        | liquid   |||          |
        |          |.|  acid    |
        | lower    |.| alkali   |
        | liquid   |||          |
        |       ---------       |
        |paddle /////////       |
        |           .           |
                    .< axis of
         The stirred mixing tank

An 11-fluid version has been employed to simulate the mixing of an acid upper liquid and an alkaline lower liquid in the tank.

(d) The distributions after 10 paddle rotations

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

  1. that the multi-fluid model is valid, so that each fluid reacts at its own rate, according to its own acid-base ratio; and

  2. 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

(e) Conclusions

6.5.4 Application to a 3D combustor

(a) The problem of predicting smoke production

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.

(b) The thermodynamics, stoichiometry and smoke kinetics

    * 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 ->

* Smoke-production rate is taken as: const * (f - f_stoich) * T**5

(c) Computed contours of some all-fluid-average quantities

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

Temperature contours

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

Oxidant contours

(d) Computed fluid-population distributions, with mixture fraction as population-distinguishing attribute

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

(e) Corresponding smoke concentration contours

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.

(f) Conclusions

(d) Future developments

6.5.5 Implementation in PHOENICS

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.