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### 6.4 The 4-fluid model

See also a differently-worded account, and a transient-flame example in a 1995 lecture.

### (a) The central idea of the four-fluid model

The eddy-break-up model, like the Eddy-Dissipation Concept which was derived from it, envisages a two-fluid population.

The central idea of the four-fluid model is to suppose that the turbulent reacting mixture consists of fluids of more than two kinds.

Four being the minimum number which, it appears, will explain all the qualitatively observed facts, it is the number used.

### (b) Definition of the four fluids

Specifically, the mixture is supposed to consist of the following fragment classes, namely:

A, comprising fully-unburned gas;

B, comprising a mixture of unburned gas and combustion products of too low a temperature for the chemical reaction rate to proceed;

C, comprising a mixture of unburned gas and combustion products of higher temperature, sufficient to allow reaction; and

D, comprising fully-burned gas.

The eddy-break-up model allows for only A and D; the new 4-fluid model allows for B and C in addition.

### (c) The rates of production of B & C, and diminution of A, B, C & D

By analogy with the formulation successfully used for the EBU, it is postulated for the 4-fluid model that, per unit mass of mixture:

• fluid B is produced from fluids A and D at the rate: 0.5 * MA * MD * MIXRATE

• fluid B is produced from fluids A and C at the rate: MA * MC * MIXRATE

• fluid C is produced from fluids A and D at the rate: 0.5 * MA * MD * MIXRATE

• fluid C is produced from fluids B and D at the rate: MB * MD * MIXRATE

Of course, for each of the production-rate terms, corresponding diminution rates are experienced by the colliding partners.

### (d) The chemical rate of production of D

The sources and sinks of the last panel were associated wih micro-mixing, ie with coupling and splitting.

In addition to these rates of production of one fluid (and corresponding diminution of the others):

* fluid D is produced from fluid C (only) at the rate: MC * CHEMRATE

CHEMRATE is, of course, the rate of chemical conversion of the unburned fuel in fluid C to combustion products.

It is the element of the model which allows (as EBU did not) chemical-kinetic effects to be introduced rationally.

### (e) The first application of the 4-fluid model

The first application of the 4-fluid model was to the same problem as that for which the EBU was invented, namely that of steady turbulent flame spread in a plane-walled duct.

Through the upstream end flows both a stream of unburned combustible mixture, and a separate but thinner stream of fully-burned products to serve as an igniter, shown below.

```
ENTRANCE ##################### CHANNEL WALL ############## EXIT
UNBURNED GAS->                         ::::::::::::          ->
"     --->               ::::::::::                      -->
"     --->      :::::::::  FLAME SPREADING TO WALL       --->
"     --->::::::                                         ---->
BURNED GAS -->::  - --- - --- - --SYMMETRY PLANE - --- - --- ----->
```

### (f) Remarkable facts about the experimental findings

Experiments of this kind by Hottel, Scurlock and Williams (1954) have shown that the rate of spread of the flame, as measured by its angle for a fixed inlet-stream velocity, is very little dependent on:-

• the fuel-air ratio of the incoming mixture;

• the flow velocity of the incoming mixture; or

• the temperature of the incoming mixture.

However, the flame can be suddenly extinguished, if variations of any of these quantities becomes too extreme.

[ Note: Strictly speaking, Hottel et al used a bluff-body flame holder rather that a co-flowing igniting stream; but the difference has little significance. ]

### (g) The numerical simulation

This process has been simulated by activating the 4-fluid model within the PHOENICS computer program operating in "parabolic" (ie marching-integration) mode.

The magnitude of the MIXRATE quantity was computed in the same manner as for the eddy-break-up model, ie as proportional to the square root of the turbulence energy divided by the length scale.

The findings were in accordance with experiments in that:

• for a given MIXRATE, the flame angle was almost independent of flow velocity or CHEMRATE;

• but, once CHEMRATE was made low enough, flame propagation ceased.

The following pictures illustrate the predictions for a particular pair of MIXRATE and CHEMRATE values.

Flow is from left to right. The top boundary of the diagram represents the duct wall. The lower boundary represents the symmetry axis.

The contours are those of fluid A, the unburned incoming fluid.

CHEMRATE (here called chemfact) is large. This implies that the flame is mixing-rate limited.

Flow is from left to right. The top boundary of the diagram represents te duct wall. The lower boundary represents the symmetry axis.

The contours are those of fluid D, the products of combustion.

Flow is from left to right. The top boundary of the diagram represents te duct wall. The lower boundary represents the symmetry axis.

The contours are those of fluid B, the lower-temperature mixture, which is still too cold to burn.

Flow is from left to right. The top boundary of the diagram represents the duct wall. The lower boundary represents the symmetry axis.

The contours are those of fluid C, the higher-temperature mixture, which reacts rapidly to form D, even though its concentration is low.

Here are seen the mean-velocity contours. The largest velocities are on the axis, and at the down stream end.

This agrees with experiment.

The pressure, shown here, varies almost one-dimensionally, because normal-to-wall velocities are small.

The fall of pressure towards the exit is necessary to accelerate the gas stream as it decreases in mean density.

This picture shows how the effective visosity increases, in accordance with the steepening velocity gradient.

### (h) Concluding remarks

The above example has been discussed at some length because it shows how "refinement of the composition grid" (from 2 fluids to 4) enables new insights to be gained and improved realism to be added to a numerical simulation.

It also perhaps has some historic importance, being the first 4-fluid calculation ever made.

Later studies have shown that, as any CFD expert would expect, grid refinement from 2 to 4 is not enough for grid-independent results to be attained.

However, depending on the presumption made for the chemical kinetics) as few as 10 fluids will give acceptable accuracy.

### (i) The four-fluid model in PHOENICS

The four-fluid model can be be activated in PHOENICS in either of two ways, namely:

(1) By inserting MODEL 4FL in the Q1 file, below MFTMBEGIN ; or

(2) By introducing the relevant coding directly into an empty GROUND file, by means of the PLANT utility.

Both methods are exemplified in the PHOENICS Input File Libraries.

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