6.1 The special difficulty about turbulent chemical reaction

(a) The practical background

Chemically-reacting laminar fluids are, in principle, easy to simulate by numerical means; for the equations governing them are known, and chemical-kineticists have provided data on the source rates of many of the practically-interesting chemical species.

To simulate a laminar flame, for example, all that is needed is a powerful computer, equipped with the appropriate software.

However, most flames of industrial and environmental importance, whether in furnaces, engines, oil-platform explosions, or forest conflagrations, are turbulent.

The same is true of the paddle- or jet-stirred reactors of the chemical, pharmaceutical and petroleum industries

(b) The special difficulty of modelling turbulent chemical reaction

Turbulent systems, by contrast with laminar ones, present difficulties of principle which have not so far been adequately surmounted

They result from the facts that:

1. Chemical-kineticists' data apply to INSTANTANEOUS conditions.

2. Turbulent fluctuations of temperature and concentration are of too small a scale to be resolved by any current computer; so only TIME-MEAN VALUES OF TEMPERATURE AND CONCENTRATION are computed;

3. the relationship of the TIME-MEAN REACTION RATES to the TIME-MEAN TEMPERATURES AND CONCENTRATIONS IS NOT THE SAME AS that of the INSTANTANEOUS RATES to the INSTANTANEOUS TEMPERATURES AND CONCENTRATIONS.

The need for simultaneous presence

In order to understand this mathematically, suppose that a reaction between species A and B proceeds at a rate proportional to their instantaneous concentrations [A] and [B].

In the absence of fluctuations, the rate is thus:

constant * [A] * [B]

However, if (to take an extreme view) fluctuations cause [A] to be finite only when [B] is zero, and vice versa, the time-average reaction rate, which is proportional to the time-average product of [A] and [B], is zero.

The non-linearities stemming from the temperature influence

The time-average reaction rate is even harder to predict when the effects of temperature and ITS fluctuations are brought into consideration; for reaction rates are non-linearly dependent on temperature. Thus, if the rate is proportional to:

T ** n,

where T stands for absolute temperature and n is an exponent between 6 and 10, it is easy to work out that

0.5 * (T1 ** n + T2 ** n)

has a very different magnitude from (0.5 * (T1 + T2) ) ** n.

The relation between reaction rate and temperature is in fact more more complex than a simple power law. The following diagram shows the typical shape. Its rise and fall result from the fact that the increase of temperature is accompanied by decreases in the amounts of available reactants; and, without them, reaction cannot proceed.

Variation of reaction rate, oxygen content and unburned-fuel content for a sub-stoichiometric fuel-air mixture

          ^
          |*      oxygen---> x               # #
          |    *                x           #   #
          |        *               x       #     #
          |           *               x   #       #
     rate |               *              x         #<--reaction
          | unburned fuel---> *        #    x       #  rate
          |                       *   #        x     #
          |                         # *           x
          |                       #       *          x#
          |                     #             *         x
          |                  #                    *     #
          |           #                               *  #
          ------------------------------------------------*
          0    reactedness ( = (T - Tu)/(Tb - Tu)  )      1

(c) A homely illustration

The point can also be explained and understood as follows:

when the man is a night-worker and his wife a day-worker, their contribution to the population explosion is hard to estimate.

Being present at the same place AND at the same time is as important to interactions between chemical species as it is to those between humans.

wbs