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### (a) Discretized and non-discretized fluid attributes

An element of fluid can possess innumerable attributes, for example: temperature, vertical-direction velocity, moisture content, reactedness, and so on.

Each of these may be used as a population-distinguishing attribute; indeed, as will be seen, more than one can be used simultaneously.

Let moisture content be chosen as the discretized population marker, so that only a finite number of differently-moist fluids are regarded as constituting the population.

It is important to recognise that each of these fluids can still vary in temperature, in velocity, in carbon-dioxide content and in any other physically-meaningful attribute.

Moreover, these other attributes can vary continuously, in contrast to that which is (or those which are) used to define the population.

### (b) Differential transport equations for fluid attributes

If fm(i), now shortened to fmi, stands for the mass fraction of fluid of the i'th moisture range, then moisture transport within the population is fully described by the set of equations of the form:

D fmi/DT = div (gamma. grad (fmi) ) + net source of fluid i

There are as many equations as there are fluids (ie moisture bands).

Let now hi stand for the enthalpy of each fluid. Then the equations describing the enthalpies within the population have the new form:

```
D fmi.hi/DT  =  div (gamma. grad (fmi.hi )  +
+ fmi * net source of energy per unit mass of fluid i
+ fluid-mixing sources
```

The different treatments of fmi and hi arise because the moisture content is a discretized variable, whereas the enthalpy is not.

### (c) The source terms for non-discretized variables

There is nothing to say about the first of these source terms except that they are treated in the conventional manner.

What is meant by "fluid-mixing sources" is the following:

• Fluids distinguished by differing moisture contents are supposed to make random contact, as a consequence of which material of intermediate moisture content comes into existence.

• The contact between these differently-moist fluids is accompanied also by (for example) thermal exchanges.

• These exchanges affect the enthalpies of the fluids which have entered the contact, and of those which result from it.

• The effects have the form of sources in the transport equations for enthalpy.

### (d) The source terms for discretized variables

Let it be supposed that moisture is being created within the turbulent mixture, for example by the burning of a hydrocarbon.

This source has the effect of increasing the mass fractions of the more-moist fluids, and decreasing those of the less-moist ones.

Expressed mathematically, this entails that there is a source in the equation for fmi, ie fm(i), which is proportional to:

[ fm(i-1) - fm(i) ] * rate of moisture production

because fluid from the less moist (ie i-1) band enters, and fluid form the current (ie i ) band leaves (to enter the moister band, ie i+1 )

The moisture production thus has the effect of a "velocity in moisture space"; and its effect is taken account of by a kind of upwind differencing.

### (e) Coupling and splitting

There is a second source of fluid 1, which has already been alluded to, namely the "fluid-mixing" or "fluid-contact" source.

It is at this point that a physical hypothesis must be made, based upon intuition rather that exact analysis. It corresponds to the hypotheses made by Prandtl in 1925, with regard to turbulent transport, and to Kolmogorov in 1942 with regard to energy dissipation; but it is different from both.

The assumption has two parts, namely:

1. that all the fluids may collide indiscriminately, in pairs, at rates which are proportional to their individual mass fractions and to a measure, mdot (say), having the dimension of 1/time, of the fluctuating motion of the turbulence;
2. that as a result of each collision some material leaves the population elements of the colliding fluids and enters the population elements which are intermediate in attribute space.

### (f) Diagrammatic representation: "Mendelian coupling and splitting"

The two "parent fluids", from different attribute-space locations, F and M, lose mass; but this is gained by their offspring at O1 etc.

```
^ frequency in
| population         father                       mother
|/          ______ /                                   /
|          |      |          promiscuous             ******
|        __|______|__ <------- coupling --------->  ********
|           | .. |                                  *| .. |*
|           |*  *|                                  *| -- |*
|          /--**--/      ____  Mendelian  ______    /------/
|         /|//////|/   |      splitting        |   /|______|/
|          |//////|     v                       v   /      /
|          |  ||  |    _     _    _    _    _    _ /________/
|          |  ||  |   | |   | |  | |  | |  | |  | |  |  |  |
|          |__||__|   | |   | |  | |  | |  | |  | |  |__|__|
--------------F--------O1----O2---O3---O4---O5---O6------M---------
fluid attribute -------------------------->
```

### (g) The measure of the turbulent motion, mdot

The above-mentioned quantity of dimension 1/time, ie mdot, is of course deducible directly from the breadth of the population, if the distinguishing attribute involves velocity, and if a length-scale measure is chosen.

It may be deduced indirectly if a different attribute is chosen.

Moreover, there is nothing to prevent more-conventional means being chosen.

For example, if the modelling study is focussed on chemical reaction rather than on hydrodynamics, the latter may be handled by the k- epsilon model, and mdot can be taken as proportional to KE/EP.

In what follows, mdot is referred to as the micro-mixing rate.

### Illustration of the influence of the micro-mixing rate

The strong influence of the micro-mixing constant on the shape of the fluid-population distribution is shown in the next picture, which relates to the same stirred reactor as before, but with double the value of mdot.

Evidently, the FPD has a sharper peak.

This study is based on a Q1 file to be found in the PHOENICS library.

A very wide variety of fpd shapes can be generated by changing the values of mdot, of the reaction-rate constant, and of the reactedness ratio of the incoming fluid streams.

"Presuming" the shape of these appears to be a hopeless enterprise.

### (h) The gamma (turbulent-diffusion) terms

The exchange coefficient gamma in the transport equation can be taken as proportional to the square of the length scale multiplied by the mdot term, these quantities being either direct products of the MFM, or obtained in more conventional ways, according to what population-distinguishing attributes have been selected.

In the next section, a stand-alone MFM will be illustrated, which makes no use of any conventional model.

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