
The formation of thermal NOx is determined by a set of highly temperaturedependent chemical reactions known as the extended Zeldovich mechanism. The principal reactions governing the formation of thermal NOx from molecular nitrogen are as follows:
A third reaction has been shown to contribute to the formation of thermal NOx, particularly at nearstoichiometric conditions and in fuelrich mixtures:
Thermal NOx Reaction Rates
The rate constants for these reactions have been measured in numerous experimental studies [ 36, 112, 250], and the data obtained from these studies have been critically evaluated by Baulch et al. [ 25] and Hanson and Salimian [ 136]. The expressions for the rate coefficients for Equations 20.15 20.17 used in the NOx model are given below. These were selected based on the evaluation of Hanson and Salimian [ 136].
=  =  
=  =  
=  = 
In the above expressions, , , and are the rate constants for the forward reactions 20.15 20.17, respectively, and , , and are the corresponding reverse rate constants. All of these rate constants have units of m /gmols.
The net rate of formation of NO via Reactions 20.15 20.17 is given by
(20.18) 
where all concentrations have units of gmol/m .
To calculate the formation rates of NO and N, the concentrations of O, H, and OH are required.
The QuasiSteady Assumption for [N]
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 N triple bond (dissociation energy of 941 kJ/gmol). This effect is represented by the high activation energy of reaction 20.15, which makes it the ratelimiting step of the extended Zeldovich mechanism. However, the activation energy for oxidation of N atoms is small. When there is sufficient oxygen, as in a fuellean flame, the rate of consumption of free nitrogen atoms becomes equal to the rate of its formation and therefore a quasisteady state can be established. This assumption is valid for most combustion cases except in extremely fuelrich combustion conditions. Hence the NO formation rate becomes
Thermal NOx Temperature Sensitivity
From Equation 20.19 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.
Decoupled Thermal NOx Calculations
To solve Equation 20.19, the concentration of O atoms and the free radical OH will be required, in addition to the concentration of stable species (i.e., O , N ). 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 [ 246] during laminar methaneair combustion. The results of these investigations indicate that the level of NOx emission can be underpredicted by as much as 28% in the flame zone, when assuming equilibrium Oatom concentrations.
Approaches for Determining O Radical Concentration
There has been little detailed study of radical concentration in industrial turbulent flames, but work [ 89] 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 [ 282] 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 NOx formation rate. Bilger et al. [ 33] 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 [ 243]. However, long computer processing time has made the method economically unattractive and its extension to turbulent flows difficult.
To determine the O radical concentration, FLUENT uses one of three approachesthe equilibrium approach, the partial equilibrium approach, and the predicted concentration approachin recognition of the ongoing controversy discussed above.
Method 1: Equilibrium Approach
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 [ 88]. According to Westenberg [ 401], the equilibrium Oatom concentration can be obtained from the expression
(20.110) 
With included, this expression becomes
where is in Kelvin.
Method 2: Partial Equilibrium Approach
An improvement to method 1 can be made by accounting for thirdbody reactions in the O dissociationrecombination process:
(20.112) 
Equation 20.111 is then replaced by the following expression [ 390]:
which generally leads to a higher partial Oatom concentration.
Method 3: Predicted O Approach
When the Oatom concentration is well predicted using an advanced chemistry model (such as the flamelet submodel of the nonpremixed model), [O] can be taken simply from the local Ospecies mass fraction.
Approaches for Determining OH Radical Concentration
FLUENT uses one of three approaches to determine the OH radical concentration: the exclusion of OH from the thermal NOx calculation approach, the partial equilibrium approach, and the use of the predicted OH concentration approach.
Method 1: Exclusion of OH Approach
In this approach, the third reaction in the extended Zeldovich mechanism (Equation 20.17) 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.
Method 2: Partial Equilibrium Approach
In this approach, the concentration of OH in the third reaction in the extended Zeldovich mechanism (Equation 20.17) is given by [ 26, 400]
(20.114) 
Method 3: Predicted OH Approach
As in the predicted O approach, when the OH radical concentration is well predicted using an advanced chemistry model such as the flamelet model, [OH] can be taken directly from the local OH species mass fraction.
Summary
To summarize, thermal NOx formation rate is predicted by Equation 20.19. The Oatom concentration needed in Equation 20.19 is computed using Equation 20.111 for the equilibrium assumption, using Equation 20.113 for a partial equilibrium assumption, or using the local Ospecies mass fraction. You will make the choice during problem setup. In terms of the transport equation for NO (Equation 20.11), the NO source term due to thermal NOx mechanisms is
where is the molecular weight of NO (kg/gmol), and is computed from Equation 20.19.