Formation of thermal NO is a slow but highly temperature sensitive process and hence it is influenced strongly by the flow field. To predict thermal NO formation in the secondary non- premixed flame zone of rich Bunsen flames, an accurate model of the flow field is essential as buoyancy plays an important role in laminar flames of low Froude number. Direct inclusion of detailed chemistry into numerical simulation of fluid mechanics requires substantial computing time. For example, Smooke  computed a two-dimensional non-premixed CH4-air flame with a "C1" mechanism containing 15 species and 42 reactions. The required computational time per calculation is in the range of 150 hours on an FPS-264 computer in 1989. Because of the severe demand in computing time, numerical modeling with detailed chemistry is seemly far from being practical to serve as a design tool for industry. In this study, the potential of the flamelet approach is explored for modeling laminar flames with the specific goal of predicting NO formation within few hours of computing time on a typical workstation.
The structure of fuel rich Bunsen flames is known to exhibit a dual-flame feature - a thin inner premixed flame followed by a secondary nonpremixed flame formed by the product species from the premixed flame and ambient air. Detailed modeling of both flames can be a challenging task due to large computing time demand. In this study, the thin premixed flame cone is modeled as a discontinuous interface with its inner structure described by a planar unstretched laminar premixed flame. The secondary nonpremixed flame is treated by a flamelet model with its "fuel" compositions specified by the product species of the premixed flame, which consists of CO and H2 as the main fuel components. Since NO formation is too slow in comparison with fluid mechanics to be treated by the flamelet approach, the conservation equation for NO is solved simultaneously with the Navier-Stokes equations. In order to assess the influence of radiative heat loss on the NO predictions, an optically-thin radiation model is included in the simulation. The predicted radial temperature and NO profiles at three downstream axial locations are compared with recent experimental data obtained by Nguyen et al. . With the present numerical model, the dependence of NOx and CO emissions on the equivalence ratio of Bunsen flames is explored. The computed results are compared with the measurements by Muss .
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The flamelet approach can be considered as an extension of the "flame sheet" model which assumes infinitely fast chemical reaction such that the reaction zone is an infinitely thin interface. With the equal diffusivity assumption under constant pressure combustion without heat loss, the thermo-chemical properties are determined completely by the local mixing state which is described by the mixture fraction. The flamelet approach relaxes the infinitely fast chemistry assumption by introducing the scalar dissipation rate as a parameter to describe the degree of departure from the equilibrium state. However, the flamelet approach still relies on the assumption that the time scales for chemical kinetics are much shorter than the time scales of convection and diffusion. Under this condition of widely separated time scales, the combustion chemistry reaches a quasi- steady state and adjusts immediately to local flow condition. Peters  presented an extensive review of the flamelet approach for modeling turbulent combustion. By transformation of the physical coordinate into one with the mixture fraction and under the thin flame assumption, it was shown by Peters  that in the steady state, temperature and species are determined by the balance between diffusion and chemical reaction as(1)
For each equivalence ratio, Eqs. (1) and (2) are solved numerically for a series of scalar dissipation rates. The results are used to generate a flamelet library in which thermo-chemical properties are expressed as functions of mixture fraction and the scalar dissipation rate. When c = 0, the gas mixture is in the equilibrium state. We calculate the flamelet properties at c = 0 using the equilibrium chemistry package STANJAN  so that the range of the scalar dissipation rate in the library is extended to zero. The CPU time used in generating a flamelet library takes less than two hours on a DEC- 3000 300LX machine.
Using a flow field solution obtained with the equilibrium chemistry, Seshadri et al.  used the flamelet model as a post- processing tool for estimating benzene formation in a laminar jet diffusion flame. They found good agreement between the predicted benzene levels and experimental data in the mixture fraction space. Instead of using the flamelet model as a post-processing tool for predicting pollutant species, we have coupled combustion with flow field directly by solving the conservation equation for the mixture fraction simultaneously with the Navier-Stokes equations. Once the mixture fraction field is obtained, the local scalar dissipation rate is evaluated by . Given the distributions of both mixture fraction and the scalar dissipation rate, one can obtain density, temperature, and species concentrations from the flamelet library by a linear interpolation scheme.
Flamelet Libraries :
(Fuel : Burnt Premixed F=1.4 CH4-Air Mixture)
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Combustion generated pollutants, such as NO, are formed near the reaction zone. Since the formation rates of pollutant species are relatively slow compared to combustion, the concentrations of pollutant species depend strongly on flow residence time. The steady state assumption made for combustion species is not appropriate for pollutant species; consequently, the pollutant concentrations can not be obtained directly from the steady-state flame library or from the equilibrium chemistry. The present numerical scheme solves for the species conservation equation of NO directly during the simulation. For a given (f, c) state, the source term for NO is evaluated by the corresponding chemistry states of major and intermediate species from the flamelet library. Since local NO concentration will be needed in evaluating the reverse rate, the NO source term is split into two parts(5)
The coefficients and td-NO in Eq. (5) are determined by using the extended Zeldovich mechanism and the CHi+NO reburning reactions [14, 15] with the steady state assumption for the atomic nitrogen. The CHi+NO reburning reactions are NO consuming reactions mainly involving intermediate species such as CH2, CH3, and HCCO. These species are formed in the premixed flame and transported beyond the inner cone by convection and diffusion. Inclusion of reburning in the present numerical model reduces the peak NO level roughly by 10%. Contributions from other pathways, such as the Fenimore mechanism, are assumed to be secondary as we explained in the previous section and they are not included in the present model. The first term, , represents the production rate of thermal NO and it is positive. The second coefficient, td-NO, is the time scale of NO destruction. Figire 1a and Figure 1b present the computed distributions of and 1/t d-NO in terms of mixture fraction and scalar dissipation rate for a fuel with equivalence ratio of 1.4. The source is seen to peak around the stoichiometric and at log(c) = 1.6 or equivalently 45 sec-1 as shown in Figure 1a.
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Due to the severe computational demands, it is impractical to compute two-dimensional Bunsen flames to establish a basis for a wide range of fuel mixtures for validating the flamelet approach. Instead, the flamelet approach is applied to simulating one-dimensional opposed-jet flames and the results are compared directly with those obtained by using the detailed chemistry model. Details of the mathematical model and solution algorithm for the opposed-jet flames can be found in Ref. . Several flames with different fuel composition and strain rate have been computed. Compared below are results obtained for a fuel mixture consisting of products from a premixed methane-air flame with f = 1.4. The distance between the fuel and air nozzles is 2 cm. The inlet velocities of fuel and air are 30.0 cm/sec and 7.5 cm/sec respectively with a strain rate of ~ 38 sec-1, according to Seshadri and Williams . Comparisons of other flames show similar trends.
The performance of the flamelet approach is assessed by comparisons of predicted flame properties as presented in Figures 2 to 6. The results obtained with the detailed chemistry are indicated by solid lines and those from the flamelet model are denoted by dashed lines.
Figures of 1-D Predictions :
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Figure 7 presents the predicted temperature field for the Bunsen flame with equivalence ratio of 1.4 investigated by Nguyen et al. . As presented in this figure, a secondary nonpremixed flame is formed by the product gases from the inner fuel-rich premixed flame and the air from the coflowing annulus. Due to the presence of the inner cone, the flow expands radially outward in the near field. Figure 8 shows the predicted NO concentration distribution indicating that NO increases gradually above the inner premixed flame cone, reaches its maximum at the secondary nonpremixed flame tip, and then decreases farther downstream. Detailed comparisons of the predicted radial profiles of temperature and NO mass fraction with the data obtained by Nguyen et al.  at three axial locations will be presented later. Figure 9 presents the predicted energy loss due to radiative heat transfer in term of heat released by combustion, i.e., the radiant fraction. The predicted maximum radiant fraction is about 15% which is reasonable for soot-free methane flames .
Contour Plots :
Figure 13a and Figure 13b show respectively the temperature and the NO profiles at z = 21 mm where the inner cone is present. Reasonably good agreement between the calculation and the experiment is seen at this location. The presence of the premixed inner cone is evident by the sharp increases in both temperature and NO concentration at about 4 mm from the centerline. The radial position of the secondary nonpremixed flame is indicated by the peak location in the temperature and NO profiles at 13 mm from centerline as shown in these two figures. Small difference in the predicted temperature and NO profiles between adiabatic and radiative heat loss calculations indicates the radiation heat loss at this location is negligible due to small flow residence time.
Comparison of radial temperature and NO profiles at z = 50 mm (10 mm above the inner flame tip) are presented in Figure 14a and Figure 14b respectively. At this location the predicted flame temperature agrees well with the measurement in the central region of the flame. However, the present model predicts a wider secondary flame profile than the data. The peak NO level is overpredicted by 22% with the adiabatic model. Inclusion of radiative heat loss brings the model prediction to a better agreement with the data. As a wider flame profile is predicted by the model, the predicted NO radial profile is also wider than the experimental data.
Similar comparisons at 90 mm downstream of the burner (20 mm below the secondary flame tip) are presented in Figure 15a and Figure 15b. The computed flame temperatures are seen in close agreement with the experimental data when radiative heat loss is included. The predicted radiative heat loss fraction is about 10% as indicated in Figure 9 which decreases the predicted peak flame temperature by 150K. A strong impact of radiative heat loss on the predicted NO levels is observed. With the radiative heat loss, the predicted peak NO value is about 25% higher than the experimental data.
To assess the validity of the flamelet assumption, comparisons of temperature, and mass fractions of O2, CO, and NO in the mixture fraction space are presented in Figures 17 to 20. Also plotted in these figures are the corresponding equilibrium values except NO, which is too high to be shown. Figure 17 shows good agreement between the predicted and the measured temperature profiles. The measured temperature profiles peak at a mixture fraction value between 0.76 and 0.8 which is higher than theoretical stoichiometric value of 0.713. The difference is attributed to the inaccuracy in the measured O2 profiles. As shown in Figure 18, the computed and measured O2 profiles differ by about 0.06 in the mixture fraction domain. Further inspection of the measured O2 mass fraction in the coflowing air (f = 0) reveals that the experimental data show a value of 0.250 for air which should be 0.233 theoretically. This difference in the O2 mass fraction for f = 0 between the numerical model and measurement leads to different mixture fraction values for the same mixture composition between model and measurement. A stoichiometric mixture fraction of 0.762 is obtained based on the " 0.25O2+0.75N2 air " composition. By correcting the mixture fraction values deduced from the measured data, the measured profiles in mixture fraction space will be slightly stretched and shifted toward left by about 0.05. If one shifts the measured O2 value to the left by f = 0.05 in the mixture fraction space, the measured O2 profiles would coincide with the computed profiles. Similarly, the predicted CO profiles in Figure 19 would be in good agreement with the experimental data if the measured profiles are shifted to the left by f = 0.05. The predicted NO profiles in Figure 20 also would agree better with the measurements after the shift. We observe that, unlike those profiles shown in Figures 17, 18, and 19, both the predicted and the measured NO profiles in Figure 20 show dependence on the axial location. This observation supports our approach of solving the NO species equation simultaneously with the Navier-Stokes equations instead of treating it as a "flamelet" species such as O2 and CO.
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