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20.1.3 Thermal NOx Formation

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:


$\displaystyle \mbox{O} + \mbox{N}_{2} \rightleftharpoons \mbox{N} + \mbox{NO}$     (20.1-5)
$\displaystyle \mbox{N} + \mbox{O}_{2} \rightleftharpoons \mbox{O} + \mbox{NO}$     (20.1-6)

A third reaction has been shown to contribute to the formation of thermal NOx, particularly at near-stoichiometric conditions and in fuel-rich mixtures:


 \mbox{N} + \mbox{OH} \rightleftharpoons \mbox{H} + \mbox{NO} (20.1-7)



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  20.1-5- 20.1-7 used in the NOx model are given below. These were selected based on the evaluation of Hanson and Salimian [ 136].


$k_{f,1}$ = $1.8 \times 10^{8} e^{-38370/T}$ $\phantom{X}$ $k_{r,1}$ = $3.8 \times 10^{7} e^{-425/T}$
$k_{f,2}$ = $1.8 \times 10^{4} T e^{-4680/T}$ $\phantom{X}$ $k_{r,2}$ = $3.81 \times 10^{3} T e^{-20820/T}$
$k_{f,3}$ = $7.1 \times 10^{7} e^{-450/T}$ $\phantom{X}$ $k_{r,3}$ = $1.7 \times 10^{8} e^{-24560/T}$

In the above expressions, $k_{f,1}$, $k_{f,2}$, and $k_{f,3}$ are the rate constants for the forward reactions 20.1-5- 20.1-7, respectively, and $k_{r,1}$, $k_{r,2}$, and $k_{r,3}$ are the corresponding reverse rate constants. All of these rate constants have units of m $^3$/gmol-s.

The net rate of formation of NO via Reactions 20.1-5- 20.1-7 is given by


 \frac{d[\mbox{NO}]}{dt} = k_{f,1} [\mbox{O}] [\mbox{N}_{2}] ... ..._{r,2} [\mbox{NO}] [\mbox{O}] - k_{r,3} [\mbox{NO}] [\mbox{H}] (20.1-8)

where all concentrations have units of gmol/m $^3$.

To calculate the formation rates of NO and N, the concentrations of O, H, and OH are required.



The Quasi-Steady 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 $_{2}$ triple bond (dissociation energy of 941 kJ/gmol). This effect is represented by the high activation energy of reaction  20.1-5, which makes it the rate-limiting 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 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


 \frac{d[\mbox{NO}]}{dt} = 2k_{f,1} [\mbox{O}] [\mbox{N}_{2}]... ...k_{f,3} [{\rm OH}]} \right) } \quad (\mbox{gmol/m}^3\mbox{-s}) (20.1-9)



Thermal NOx Temperature Sensitivity


From Equation  20.1-9 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 $k_{f,1}$, the thermal NOx production rate doubles for every 90 K temperature increase beyond 2200 K.



Decoupled Thermal NOx Calculations


To solve Equation  20.1-9, the concentration of O atoms and the free radical OH will be required, in addition to the concentration of stable species (i.e., O $_{2}$, N $_{2}$). 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 methane-air 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 O-atom 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 approaches--the equilibrium approach, the partial equilibrium approach, and the predicted concentration approach--in 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 O-atom concentration can be obtained from the expression


[\mbox{O}]= k_{p} [\mbox{O}_{2}]^{1/2} (20.1-10)

With $k_p$ included, this expression becomes


[\mbox{O}]= 3.97 \times 10^{5} T^{-1/2} [\mbox{O}_2]^{1/2} e^{-31090/T} \qquad\mbox{gmol/m$^{3}$} (20.1-11)

where $T$ is in Kelvin.

Method 2: Partial Equilibrium Approach

An improvement to method 1 can be made by accounting for third-body reactions in the O $_2$ dissociation-recombination process:


 \mbox{O}_2 + \mbox{M} \rightleftharpoons \mbox{O} + \mbox{O} + \mbox{M} (20.1-12)

Equation  20.1-11 is then replaced by the following expression [ 390]:


[\mbox{O}]= 36.64 T^{1/2} [\mbox{O}_2]^{1/2} e^{-27123/T} \qquad\mbox{gmol/m$^{3}$} (20.1-13)

which generally leads to a higher partial O-atom concentration.

Method 3: Predicted O Approach

When the O-atom 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 O-species 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  20.1-7) is assumed to be negligible through the following observation:


k_2[{\rm O}_2]_{\rm eq} \gg k_3[{\rm OH}]_{\rm eq}

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  20.1-7) is given by [ 26, 400]


[\mbox{OH}]= 2.129 \times 10^2 T^{-0.57} e^{-4595/T} [\mbox{O}]^{1/2} [\mbox{H}_2\mbox{O}]^{1/2} \; \; \; \mbox{gmol/m}^3 (20.1-14)

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 20.1-9. The O-atom concentration needed in Equation  20.1-9 is computed using Equation  20.1-11 for the equilibrium assumption, using Equation  20.1-13 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  20.1-1), the NO source term due to thermal NOx mechanisms is


 S_{\rm thermal, NO} = M_{w,{\rm NO}} \frac{d[{\rm NO}]}{dt} (20.1-15)

where $M_{w,{\rm NO}}$ is the molecular weight of NO (kg/gmol), and $d[{\rm NO}]/dt$ is computed from Equation  20.1-9.


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