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VII.4 Combustion modelling - direct approach

Since combustion is basically a chemical reaction process combining reactants to form products, a direct approach to modelling it would be to work with the individual components. We can write down balance equations for the various reactants and the energy equation, together as usual with the continuity and momentum equations, and solve the entire system to calculate the flow. The quantity of chemical species i present at any point in the flow is indexed by the mass fraction Y i - the mass of the species per unit mass of the mixture (eg. kg of species i per kg of gas). A transport equation can be written for each Y i in the system

@rYi- @t + \~/ .rYiu-= \~/ .rDi\ ~/ Yi + rSi
(VII.1)
where Di is some diffusivity, and of course
sum Yi = 1 i
The source term Si accounts for the addition or removal of species i due to the combustion processes. At the same time an energy equation must be solved. This can be an equation for the internal energy e, or equivalently for the enthalphy h (automatically taking into account the pdV work being done in expansion) or the temperature T . For example, a balance equation for T can be formulated
@rT-- @t + \~/ .rT u-= \~/ .rD\ ~/ T + rST
(VII.2)
The source term here includes such effects as radiation loss or gain, pressure work as well as the chemical energy release due to the combustion. All the equations describing the reacting components are very similar in form, and for notational simplicity are sometimes grouped into a vector denoting “reactive scalars”
yi = {Y1,Y2,...,Yn, T}
(VII.3)
(for n species in the reaction), with a governing transport equation of the form
@ryi- @t + \~/ .ryiu-= \~/ .rDi\~ / yi + rSyi
(VII.4)
There are a number of problems with doing this however. Firstly, in a complete system n is probably very large : to make this a realistic model we need to reduce its complexity somehow. Also we have not yet discussed the effects of turbulence on the flow.

A great deal of information has been collected about the chemical reactions occurring in various important combustion processes. Significant data includes the rates of reaction kf and kb for the forward and backward reaction processes and the stoichiometric coefficients for the reactions. From this information the heat release from the reactions, and ultimately the source terms Syi, can be determined. This information can be combined into tables called elementary reaction mechanisms which describe in detail the chemical reactions proceeding during combustion. These are extensively used when the detail of the chemical processes is important, but as they often take into account hundreds of individual reactions, and are frequently numerically ‘stiff’ (ie. difficult to solve), they are difficult to incorporate into a full fluid flow/combustion model. To develop such a model it is necessary to simplify the processes involved whilst at the same time without losing the more important details.

Hydrocarbon combustion consists of chain reactions of the general form

A + B --> C
C + D --> E
E + F --> G
...
N + O --> P
The intermediate steps of the process A + B --> P often involve short-lived radical species which remain in quasi-steady state throughout the process : these are fast reactions which proceed rapidly. If these can be eliminated (the Quasi-Steady-State Assumption, QSSA) then the reaction can be effectively represented by a reduced set of slower, rate-determining reactions, called a global reaction mechanism. For example, an elementary reaction mechanism for methane oxidation might detail 277 reactions involving 49 chemical species but is frequently reduced down to a global reaction mechanism involving just 4 steps
I CH4 + 2H + H2O = CO + 4H2
II CO + H2O = CO2 + H2
III H + H + M = H2 + M
IV O2 + 3H2 = 2H + 2H2O

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