The kinetic mechanisms of NOx formation and destruction described in the preceding sections have all been obtained from laboratory experiments using either a laminar premixed flame or shock-tube studies where molecular diffusion conditions are well defined. In any practical combustion system, however, the flow is highly turbulent. The turbulent mixing process results in temporal fluctuations in temperature and species concentration that will influence the characteristics of the flame.
The relationships among NOx formation rate, temperature, and species concentration are highly nonlinear. Hence, if time-averaged composition and temperature are employed in any model to predict the mean NOx formation rate, significant errors will occur. Temperature and composition fluctuations must be taken into account by considering the probability density functions that describe the time variation.
In turbulent combustion calculations, Ansys Fluent solves the density-weighted time-averaged Navier-Stokes equations for temperature, velocity, and species concentrations or mean mixture fraction and variance. To calculate NO concentration, a time-averaged NO formation rate must be computed at each point in the domain using the averaged flow-field information.
Methods of modeling the mean turbulent reaction rate can be based on either moment methods [714] or probability density function (PDF) techniques [279]. Ansys Fluent uses the PDF approach.
Important: The PDF method described here applies to the NOx transport equations only. The preceding combustion simulation can use either the generalized finite-rate chemistry model by Magnussen and Hjertager, the non-premixed or partially premixed combustion model. For details on these models, refer to Species Transport and Finite-Rate Chemistry, Non-Premixed Combustion, and Partially Premixed Combustion.
The PDF method has proven very useful in the theoretical description of turbulent flow [280]. In the Ansys Fluent NOx model, a single- or joint-variable PDF in terms of a normalized temperature, species mass fraction, or the combination of both is used to predict the NOx emission. If the non-premixed or partially premixed combustion model is used to model combustion, then a one- or two-variable PDF in terms of mixture fraction(s) is also available. The mean values of the independent variables needed for the PDF construction are obtained from the solution of the transport equations.
The mean turbulent reaction rate 
can be described in terms of the instantaneous
rate 
and a single or joint PDF of various variables.
In general,
 | (9–104) |
where 
,... are temperature
and/or the various species concentrations present. 
is the probability density
function (PDF).
The PDF is used for weighting against the instantaneous rates of production of NO (for example, Equation 9–15) and subsequent integration over suitable ranges to obtain the mean turbulent reaction rate. Hence we have
 | (9–105) |
or, for two variables
 | (9–106) |
where 
is the mean turbulent rate of production of NO, 
is the instantaneous molar rate of NO production, 
is the instantaneous density, and 
and 
are the PDFs of the variables 
and, if relevant, 
. The same treatment applies for the HCN or NH3 source
terms.
Equation 9–105 or Equation 9–106 must be integrated at every node and at every iteration. For a PDF in terms of temperature, the limits of integration are determined from the minimum and maximum values of temperature in the combustion solution (note that you have several options for how the maximum temperature is calculated, as described in Setting Turbulence Parameters in the User's Guide). For a PDF in terms of mixture fraction, the limits of the integrations in Equation 9–105 or Equation 9–106 are determined from the values stored in the look-up tables.
In the case of the two-variable PDF, it is further assumed that
the variables 
and 
are statistically independent, so that 
can be expressed as
 | (9–107) |
Ansys Fluent can assume 
to be a two-moment beta function that is appropriate
for combustion calculations [231], [444]. The equation for the beta function is
 | (9–108) |
where 
is the Gamma function,
and 
and 
depend on the mean value of the quantity
in question, 
, and its variance, 
:
 | (9–109) |
 | (9–110) |
The beta function requires that the independent variable 
assumes values between 0
and 1. Thus, field variables such as temperature must be normalized.
See Setting Turbulence Parameters in the User's
Guide for information on using the beta PDF when using
single-mixture fraction models and two-mixture fraction models.
Ansys Fluent can also assume 
to exhibit a clipped Gaussian
form with delta functions at the tails.
The cumulative density function for a Gaussian PDF (
) may be expressed in terms of the error function
as follows:
 | (9–111) |
where 
is the error function, 
is the quantity of interest, and 
and 
are the mean and variance values of 
, respectively. The error function may be expressed in terms of the incomplete
gamma function (
):
 | (9–112) |
The variance of the field variables (temperature and species mass fraction) 
can be computed by solving the following transport equation during the
combustion calculation or pollutant postprocessing stage:
 | (9–113) |
where the constants 
, 
, and 
take the values
0.85, 2.86, and 2.0, respectively.
Note that the above variance transport equation is solved only for the temperature variance. Solving an additional equation may be computationally intensive, and therefore may not always be a preferred option for a postprocessing treatment of NOx prediction.
In addition to the variance transport equation, Ansys Fluent provides an option to calculate
algebraically. This is an approximate method which assumes equal production
and dissipation of variance, and is as follows:
 | (9–114) |
The term in brackets is the dissipation rate of the independent variable. The algebraic option is a preferred choice when PDF is constructed with more than one variable (like temperature-species PDF in Equation 9–107). When the turbulence chemistry interaction mode is temperature-species, the species variance is computed algebraically, while the temperature variance can be computed either algebraically or by using the transport equation.
For a PDF in terms of mixture fraction, the mixture fraction
variance has already been solved as part of the basic combustion calculation,
so no additional calculation for 
is required.