402
12. Special Topics
there are many situations that are not close to either limit; they are called
partially premixed. For a complete treatment of the theory of combustion,
the reader is advised to consult the well-known work by Williams (1985).
The key parameter in reacting flows is the ratio of the flow time scale to
the chemical time scale; it is known as the Damkohler number, Da. When
the Damkohler number is very large, chemical reaction is so fast that it takes
place almost instantaneously after the reactants have mixed. In this limit,
flames are very thin and the flow is said to be mixing-dominated. Indeed,
if the effects of heat release can be ignored, it is possible to treat the limit
Da + cc as one involving a passive scalar and the methods discussed at the
beginning of this chapter can be used.
For the calculation of practical combustors, in which the flow is almost
always turbulent, it is necessary to rely on solution of the Reynolds-averaged
Navier-Stokes (RANS) equations. This approach and the turbulence models
that need to be used in conjunction with it for non-reacting flows were described in Chapter 9. When combustion is present, it is necessary to solve
additional equations that describe the concentrations of the reacting species
and to include models that allow one to compute the reaction rate. We shall
describe some of these models in the remainder of this section.
The most obvious approach, that of Reynolds-averaging the equations for
a reacting flow does not work. The reason is that chemical reaction rates are
very strong functions of temperature. For example, the reaction rate between
species A and species B might be given by:
where E, is called the activation energy, R is the gas constant, and YA and
YB are the concentrations of the two species. The presence of the Aarhenius
factor e(-EaIRT) is what makes the problem difficult. It varies so rapidly with
temperature that replacing T with its Reynolds-averaged value produces large
errors.
In a high Damkohler number non-premixed turbulent flame, the reaction
takes place in thin wrinkled flame zones. For this case, several approaches
have been used, of which we will briefly describe two. Despite the significant
difference in philosophy and appearance between them, they are more alike
than it would seem.
In the first approach, one takes the point of view that, since mixing is the
slower process, the reaction rate is determined by how fast it takes place. In
that case, the rate of reaction between two species A and B is given by an
expression of the form:
where T is the time scale for mixing. For example, if the k-E model is used,
T = k / ~ .
In the k-w model, T = l/w. A number of models of this kind have
been proposed, perhaps the best known of which is the eddy break-up model
12. Special Topics
there are many situations that are not close to either limit; they are called
partially premixed. For a complete treatment of the theory of combustion,
the reader is advised to consult the well-known work by Williams (1985).
The key parameter in reacting flows is the ratio of the flow time scale to
the chemical time scale; it is known as the Damkohler number, Da. When
the Damkohler number is very large, chemical reaction is so fast that it takes
place almost instantaneously after the reactants have mixed. In this limit,
flames are very thin and the flow is said to be mixing-dominated. Indeed,
if the effects of heat release can be ignored, it is possible to treat the limit
Da + cc as one involving a passive scalar and the methods discussed at the
beginning of this chapter can be used.
For the calculation of practical combustors, in which the flow is almost
always turbulent, it is necessary to rely on solution of the Reynolds-averaged
Navier-Stokes (RANS) equations. This approach and the turbulence models
that need to be used in conjunction with it for non-reacting flows were described in Chapter 9. When combustion is present, it is necessary to solve
additional equations that describe the concentrations of the reacting species
and to include models that allow one to compute the reaction rate. We shall
describe some of these models in the remainder of this section.
The most obvious approach, that of Reynolds-averaging the equations for
a reacting flow does not work. The reason is that chemical reaction rates are
very strong functions of temperature. For example, the reaction rate between
species A and species B might be given by:
where E, is called the activation energy, R is the gas constant, and YA and
YB are the concentrations of the two species. The presence of the Aarhenius
factor e(-EaIRT) is what makes the problem difficult. It varies so rapidly with
temperature that replacing T with its Reynolds-averaged value produces large
errors.
In a high Damkohler number non-premixed turbulent flame, the reaction
takes place in thin wrinkled flame zones. For this case, several approaches
have been used, of which we will briefly describe two. Despite the significant
difference in philosophy and appearance between them, they are more alike
than it would seem.
In the first approach, one takes the point of view that, since mixing is the
slower process, the reaction rate is determined by how fast it takes place. In
that case, the rate of reaction between two species A and B is given by an
expression of the form:
where T is the time scale for mixing. For example, if the k-E model is used,
T = k / ~ .
In the k-w model, T = l/w. A number of models of this kind have
been proposed, perhaps the best known of which is the eddy break-up model