The thermal NO mechanism is a predominant source of NOx in gas flames at temperatures above 1800 K. The NO is formed from the combination of free radical O and N species, which are in abundance at high temperatures. The two-step mechanism, referred to as the Zeldovich mechanism, is thought to dominate the process:

(7–109)

(7–110)

In sub or near stoichiometric conditions, a third reaction may be important:

(7–111)

When this step is included with the first two, it is referred to as the extended Zeldovich mechanism.

The rates of each of these three reactions (using the subscripts 1,2,3 to describe the three reactions) are expressed as [43]:

(7–112)

(7–113)

(7–114)

When multiplied by the concentrations of the reactants, they yield rates in terms of , which can be converted to a volumetric mass source term.

The first step tends to be rate limiting, producing both an NO and N radical species. The N radical is assumed to be oxidized by reaction Equation 7–110 in the Zeldovich mechanism and also by reaction Equation 7–111 in the extended Zeldovich mechanism. Either way, these second oxidation reactions are assumed to be fast and if Reaction Equation 7–109 occurs, then two NO molecules will be formed. The thermal NO formation in , , is therefore related to the rate of reaction Equation 7–109:

(7–115)

(7–116)

Here, denotes the molecular mass of NO. Thus, if the molar concentrations [O] and [N₂] of O radicals and N₂ are known, the thermal NO mechanism can be calculated.

When using the Laminar Flamelet model, almost always the O radical concentration can be taken without further assumptions from the solution because the model predicts it directly. However, when using the Eddy Dissipation model (EDM) and/or the Finite Rate Chemistry model (FRC), O radical concentrations usually are not known directly but must be derived from other quantities. Here, the O radical concentration is estimated from the molecular oxygen dissociation,

(7–117)

(Westenberg, 1975):

(7–118)

By substitution, the effective source term for NO then reads:

(7–119)

(7–120)

At temperatures lower than 1800 K, hydrocarbon flames tend to have an NO concentration that is too high to be explained with the Zeldovich mechanisms. Hydrocarbon radicals can react with molecular nitrogen to form HCN, which may be oxidized to NO under lean flame conditions.

(7–121)

(7–122)

The complete mechanism is very complicated. However, De Soete (see also Peters and Weber, 1991) proposed a single reaction rate to describe the NO source by the Fenimore mechanism,

(7–123)

(7–124)

and denote molar mass of NO and the mean molar mass of the mixture, respectively. The Arrhenius coefficients depend on the fuel. (De Soete, 1974) proposed the following values:

Methane fuel

(7–125)

Acetylene fuel

(7–126)

The fuel nitrogen model assumes that nitrogen is present in the fuel by means of HCN. HCN is modeled to either form or destroy NO depending on the local conditions in the mixture, with HCO acting as an intermediate species. The mechanism consists of three reaction steps:

The reaction rates in [mol/s] for the reactions are, respectively:

(7–127)

where

(7–128)

(7–129)

where denotes the molar fraction of and denotes the mean molar mass of the mixture.

Under fuel rich conditions, when the amount of oxygen available is not sufficient to oxidize all of the fuel, the excess fuel may lead to reduction of NO. This process can be described by a global reaction:

(7–130)

Note that this is only a global representation; the physical process actually occurring is much more complicated. The real process involves many intermediate components appearing during combustion of the fuel, such as CHx radicals, which attack the NO.

The stoichiometric coefficients for fuel, carbon dioxide and water vapor are fuel-dependent. For a given fuel they can easily be derived from the element balance. For methane, the global NO reburn reaction is:

(7–131)

The reaction rate will be fuel dependent. For coal volatiles, the reaction rate defaults to:

(7–132)

The same reaction rate is applied for the predefined NO reburn reaction for methane.

The above reaction rates are applicable to laminar flow, premixed chemistry. In turbulence systems, fluctuations can have a dominant impact on the NO formation rate. For both the thermal and prompt NO mechanisms, there is a strong dependence of the rates on the temperature due to their high activation energy. Thus, temperature fluctuations, particularly positive fluctuations, can dramatically increase the NO formed in flames. These temperature fluctuations are included in CFX using a statistical approach.

In order to determine the mean rate for NO formation, a presumed probability density function (presumed PDF) method is used to compute the weighted average of the reaction rate:

(7–133)

This integration is carried out separately for each reaction step. For simplicity, the subscripts (thermal or prompt) have been omitted. The integration range for temperature, , is the range of temperatures occurring. The default for NO reactions is the range [300 K; 2300 K], but this may be modified by you on a per reaction scope.

The probability density function (PDF) is computed from mean temperature, , and the variance of temperature, . The shape of is presumed to be that of a normalized beta function (-function):

(7–134)

Where:

(7–135)

and:

(7–136)

For vanishing temperature variance (vanishing temperature fluctuations), the beta function is approaching to a single Dirac peak (delta-function). In the limit, the integrated reaction rate is the same as for that for the standard Arrhenius rate. For very large fluctuation, the beta function goes towards a double Dirac peak, and for small to medium temperature variance the shape of the PDF is similar to that of a Gaussian distribution.

Arrhenius reaction rates integrated over a PDF for temperature is not limited to NO formation but may be used for any reaction.

For the temperature variance, , that is needed from constructing the probability density function (PDF) used for the temperature integration, the following transport equation is solved:

(7–137)

Default values for the model coefficients are and .

The above equation is missing some physical aspects namely the production of temperature fluctuations due to heat release by chemical reaction. Heat release is fluctuating, too, because of turbulent fluctuations of the reactants. However, in the current model temperature variance is only needed as input to another model: The construction of a probability density function with presumed shape. For this purpose, the above equation provides sufficient accuracy.

For convenience of setting up a case, the temperature variance equation can be run without specifying BC or IC data. In the first release, these are not offered by the user interface (CFX-Pre). If absent, zero fluctuations are assumed at inlets, openings and walls with specified temperature. At walls with specified heat flux or transfer coefficient, the default BC for temperature variance is zero flux.

Because the model for NO formation is implemented by means of REACTION objects, you have full control of all aspects of the model. CFX provides the same flexibility for the NO model as for the generic combustion and reaction system.