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6. Application of MFM to the well-stirred reactor
6.1 With uniform fuel-air ratio; the 1D population

The main implications of the multi-fuel model are best seen in studies of simple flow. The simplest is the stirred reactor in which effective macro-mixing precludes variations with position.

 In the first example, it will be supposed        |     |     |
 that stream 1 is fully unreacted, that      1 ====>    |     |
 stream 2 is fully reacted, that 98 fluids        |  stirred  |
 of intermediate reactedness are created by       |  ///|/// 3===>
 micro-mixing and reaction (non-linearly          |  reactor  |
 dependent on reactedness) and that the      2 ====>          |
 outflowing stream 3 consists of 100 fluids       |___________|
 altogether, in to-be-computed proportions.

The flow is steady. Three cases will be shown, with mixing and reaction constants respectively = 10 & 10, 50 & 10, and 50 & 20.

This PDF is not unlike the 2-spike EBU presumption

With increased mixing constant, the PDF is not unlike an unsymmetrical "clipped-Gaussian"

With increased reaction constant, the PDF is even more unsymmetrical

6.2 With non-uniform fuel-air ratio; the 2D population

In most combustors, the fuel and oxidant enter separately; so it is best to postulate a two-dimensional population, with (say) fuel/air ratio and reactedness as the PDAs (ie population-distinguishing attributes).

The necessary number of fluids may then become large; however, it is possible to determine the necessary number by grid-refinement studies.

Four results will be shown, for a well-stirred reactor in which the two entering streams are a fully-unreacted lean-mixture gas and a fully-reacted rich-mixture gas.

The population grids will be: 3 by 3 (too coarse) ; 5 by 5 (still too coarse) ; 7 by 7 (fairly good) ; 11 by 11 (more than fine enough). Inspection of the average reactedness reveals the solution quality.

The 3 by 3 grid. Note the value of average R (ie reactedness)

The 5 by 5 grid. Average R is much smaller

The 7 by 7 grid. Average R is a little smaller

The 11 by 11 grid. Average R is almost the same.

6.3 Concluding remarks about the stirred-reactor results

The 100 fluids employed in the 1D-population studies were surely too many. Yet it is interesting to observe that such a number can be easily handled.

The shapes of the PDFs for the pre-mixed case vary with the values of the mixing and reaction constants (and of other parameters too, of which time-shortage precludes mention). They are "clipped-Gaussian" or "beta-function" only by (rare) chance.

The use of the 2D population for the non-pre-mixed reactor goes beyond what any "presumed-PDF" practitioner has ever dared (or should dare) to presume. Once again, the population distribution varies greatly with the defining parameters.

Such calculations take very little time; but pondering their significance needs much more. More "ponderers" are needed.

The cases shown are standard items of the PHOENICS input library; so any interested researcher can access and exercise them.