The formation of thermal NOx is determined by a set of highly temperature-dependent chemical reactions known as the extended Zeldovich mechanism. The principal reactions governing the formation of thermal NOx from molecular nitrogen are as follows:
 | (9–5) |
 | (9–6) |
A third reaction has been shown to contribute to the formation of thermal NOx, particularly at near-stoichiometric conditions and in fuel-rich mixtures:
 | (9–7) |
The rate constants for these reactions have been measured in numerous experimental studies [68], [186], [450], and the data obtained from these studies have been critically evaluated by Baulch et al. [48] and Hanson and Salimian [234]. The expressions for the rate coefficients for Equation 9–5 – Equation 9–7 used in the NOx model are given below. These were selected based on the evaluation of Hanson and Salimian [234].
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In the above expressions, 
, 
, and 
are the rate constants for the forward reactions Equation 9–5 – Equation 9–7, respectively,
and 
, 
, and 
are the corresponding reverse rate constants. All
of these rate constants have units of m3/mol-s.
The net rate of formation of NO via the reactions in Equation 9–5 – Equation 9–7 is given by
 | (9–8) |
where all concentrations have units of mol/m3.
To calculate the formation rates of NO and N, the concentrations of O, H, and OH are required.
The rate of formation of NOx is significant only at high temperatures
(greater than 1800 K) because fixation of nitrogen requires the breaking of the strong
N2 triple bond (providing dissociation energy of 941 kJ/mol). This
effect is represented by the high activation energy of reaction Equation 9–5, which makes it the rate-limiting step of the extended Zeldovich mechanism. However, the
activation energy for oxidation of 
atoms is small. When there is sufficient oxygen, as in a fuel-lean flame, the
rate of consumption of free nitrogen atoms becomes equal to the rate of its formation, and
therefore a quasi-steady state can be established. This assumption is valid for most combustion
cases, except in extremely fuel-rich combustion conditions. Hence the NO formation rate
becomes
 | (9–9) |
From Equation 9–9 it is clear that the rate of formation of NO will
increase with increasing oxygen concentration. It also appears that thermal NO formation should
be highly dependent on temperature but independent of fuel type. In fact, based on the limiting
rate described by 
, the thermal NOx production rate doubles for every
90 K temperature increase beyond 2200 K.
To solve Equation 9–9, the concentration of O atoms and the free radical OH will be required, in addition to the concentration of stable species (that is, O2, N2). Following the suggestion by Zeldovich, the thermal NOx formation mechanism can be decoupled from the main combustion process by assuming equilibrium values of temperature, stable species, O atoms, and OH radicals. However, radical concentrations (O atoms in particular) are observed to be more abundant than their equilibrium levels. The effect of partial equilibrium O atoms on NOx formation rate has been investigated [445] during laminar methane-air combustion. The results of these investigations indicate that the level of NOx emission can be under-predicted by as much as 28% in the flame zone, when assuming equilibrium O-atom concentrations.
There has been little detailed study of radicals concentration in industrial turbulent flames, but work [151] has demonstrated the existence of this phenomenon in turbulent diffusion flames. Presently, there is no definitive conclusion as to the effect of partial equilibrium on NOx formation rates in turbulent flames. Peters and Donnerhack [515] suggest that partial equilibrium radicals can account for no more than a 25% increase in thermal NOx and that fluid dynamics has the dominant effect on the NOx formation rate. Bilger and Beck [60] suggest that in turbulent diffusion flames, the effect of O atom overshoot on the NOx formation rate is very important.
To overcome this possible inaccuracy, one approach would be to couple the extended Zeldovich mechanism with a detailed hydrocarbon combustion mechanism involving many reactions, species, and steps. This approach has been used previously for research purposes [440]. However, high computational cost of such approaches limits their use in turbulent reacting flows.
To determine the O radical concentration, Ansys Fluent provides the equilibrium, partial equilibrium, and the predicted concentration approaches described below.
The kinetics of the thermal NOx formation rate is much slower than the main hydrocarbon oxidation rate, and so most of the thermal NOx is formed after completion of combustion. Therefore, the thermal NOx formation process can often be decoupled from the main combustion reaction mechanism and the NOx formation rate can be calculated by assuming equilibration of the combustion reactions. Using this approach, the calculation of the thermal NOx formation rate is considerably simplified. The assumption of equilibrium can be justified by a reduction in the importance of radical overshoots at higher flame temperature [150]. According to Westenberg [704], the equilibrium O-atom concentration can be obtained from the expression
 | (9–10) |
With 
included, this expression becomes
 | (9–11) |
where 
is in Kelvin.
An improvement to method 1 can be made by accounting for third-body reactions in the O2 dissociation-recombination process:
 | (9–12) |
Equation 9–11 is then replaced by the following expression [693]:
 | (9–13) |
which generally leads to a higher partial O-atom concentration.
Ansys Fluent uses one of three approaches to determine the OH radical concentration: the exclusion of OH from the thermal NOx calculation, the partial equilibrium, and the use of the predicted OH concentration.
In this approach, the third reaction in the extended Zeldovich mechanism (Equation 9–7) is assumed to be negligible through the following observation:
This assumption is justified for lean fuel conditions and is a reasonable assumption for most cases.
In this approach, the concentration of OH in the third reaction in the extended Zeldovich mechanism (Equation 9–7) is given by [49], [703]:
 | (9–14) |
To summarize, the thermal NOx formation rate is predicted by Equation 9–9. The O-atom concentration needed in Equation 9–9 is computed using Equation 9–11 for the equilibrium assumption, using Equation 9–13 for a partial equilibrium assumption, or using the local O-species mass fraction. You will make the choice during problem setup. In terms of the transport equation for NO (Equation 9–1), the NO source term due to thermal NOx mechanisms is
 | (9–15) |
where 
is the molecular weight of NO (kg/mol), and 
is computed from Equation 9–9.