Sonderforschungsbereich/Transregio 40 Annual Report 2016 193 Construction of Libraries for Non-Premixed Tabulated Chemistry Combustion Models including Non-Adiabatic Behaviour due to Wall Heat Losses By G. Frank, J. Zips AND M. Pfitzner Institut für Thermodynamik, Universität der Bundeswehr München Werner-Heisenberg-Weg 39, 85577 Neubiberg Non- Flamelet Libraries in the literature are often associated with radiative heat losses, which is oftentimes surpassed by the heat losses due to cooled walls. Different methods of generating non- flamelets are compared with special emphasis on their applicability for flamelets subjected to wall heat losses. 1. Introduction Simulating combustion processes is still a very challenging task for Computational Fluid Dynamics (CFD). A combustion process is described by hundreds of elementary reactions, forming a stiff complex set of partial differential equations to be solved numerically. To get a result in a reasonable time, models simplifying the computational complexity are used for combustion processes. Tabulated chemistry models decouple the time-consuming solution of the complex chemistry from the main fluid flow simulation. For this the thermodynamic state (temperature T and composition Y i ) is parametrized by a set of certain parameters and saved in a table. Transport equations for those parameters are solved during runtime in the flow simulation and the thermodynamic state is then retrieved via table look-up. One of the most often used tabulated chemistry models is the flamelet concept, based on the theory by Peters [1]. Here, the turbulent flame is assumed to consist of a collection of laminar flames, called flamelets. Each thermodynamic state is defined by the mixture fraction Z and the scalar dissipation rate χ. Turbulence-chemistry interaction is oftentimes modelled using assumed probability density functions (PDFs). The thermodynamic state then additionally depends on the variance of the mixture fraction Z 2. Most tabulated chemistry models for non-premixed combustion are based on laminar flamelets, which are generated by solving a one-dimensional laminar counterflow diffusion flame (see Fig.1). Different numerical tools exist with the capability of using detailed chemical reaction mechanisms. In this work the FlameMaster code by Pitsch [2] was used, which solves the flamelet equations in mixture fraction space. For this, a coordinate transformation of the species- and energy conservation equations is done from physical space into mixture fraction space, leading to following equation: ρ Φ i t = ρ χ 2 2 Φ i Z 2 +ω i (1.1)
194 G. Frank, J. Zips & M. Pfitzner S-Curve 2000 T max [K] 1000 500 0.1 1 10 100 χ st [1/s] FIGURE 1. Schematic counterflow diffusion flame configuration. FIGURE 2. S-Curve of a counterflow diffusion flame. χ = 2D Z 2, (1.2) with Φ i being a reactive scalar (Y i, T ) and ω i being the source term of the scalar. For those equations the Lewis number of each species is assumed to be unity. For details of the calculation of the diffusion coefficient D and the source terms for species mass fractions and temperature the reader is referred to [2]. For steady flamelets, the unsteady term on the left hand side of equation (1.1) vanishes. Those equations can have three solutions, which are shown by the classical S-Curve Fig. 2: An unburnt solution, where no chemical reactions occur (lower branch of the curve), a fully reacted solution (top branch of the curve), and an unstable burning solution (middle branch of the curve). The classic flamelet model by Peters [1] only uses the top branch of the S-Curve with the parameters Z, χ. A Flamelet-Progress Variable (FPV) combustion model can also represent the two other branches of the S-Curve. Combustion processes near cooled walls are subjected to heat loss. Thus the reaction rates of the elementary reactions drop, slow reaction with lower activation energy become more prominent and recombination processes can occur, consequently changing the thermodynamic states compared to the flame. Including heat loss effects into tabulated chemistry models has been studied by many researchers (e.g. [3, 4]) in the scope of radiation heat losses. As additional parameter for the tabulation of the thermodynamic states the enthalpy of the mixture, or a derivative of it, is used. A Direct Numerical Simulation (DNS) by Wang et al. [5] investigating flame-wall interaction validates the use of the enthalpy for wall heat losses. Another near wall effect noticed by [6] and [7] is that burned products can be trapped between the wall and the flame, affecting diffusion and mixing in the near wall region. In the scope of premixed flames, Fiorina et al. [8] extended the intrinsic low-dimensional manifold (ILDM) by an enthalpy coordinate. The ILDM library was built using isothermal burner-stabilized flames. Their model has been verified by two dimensional CFD calculations of premixed and partially premixed burner stabilized flames. In the scope of this work, we will concentrate on the construction of flamelet libraries for non-premixed flames subjected to wall heat losses. First, we will discuss a chemistry table built with the assumption of chemical equilibrium and with the chemistry code Cantera [9]. Afterwards, methods to include non- effects into one dimensional counterflow diffusion flames are presented. The method of frozen chemistry is often found in commercial solvers. Flamelets with a convective sink term as proposed by Lee et al. [10]
1 Non-Adiabatic Libraries for Tabulated Chemistry Models 195 0.75 hnorm [-] 0.5 0.25 T/T max [-]: 0 0.5 1 0 0 0.25 0.5 0.75 1 FIGURE 3. Equilibrium chemistry look-up table for temperature at a specific pressure parametrized by mixture fraction Z and normalized enthalpy h norm. The temperature ranges from T min = 270 K to T max = 3500 K. are discussed as the third option. The last two methods both change the boundary conditions of the flamelets and represent the methods proposed by Hossain et al. [3] for radiative heat losses and by Wu and Ihme [11] for wall heat losses. Those five methods are of increasing complexity, their advantages, disadvantages and their applicability are discussed. 2. Chemical Equilibrium A very quick and easy way to generate a table is to assume chemical equilibrium. Chemistry is assumed to be infinitely fast, i.e.τ chem. Consequently, the equilibrium state can be calculated by imposing an enthalpy level and a constant operating pressure. In order to compute the thermochemical equilibrium, it is common to use the Gibb s free energy given by G = H TS, (2.1) with H being the enthalpy, T the absolute temperature and S the entropy, respectively. The resulting database can be parametrized in mixture fraction and enthalpy space for a specific pressure level. Turbulence-chemistry interaction can be modeled using a presumed PDF approach which increases the look-up table by one dimension. Fig. 3 shows a generic temperature table in mixture fraction and normalized enthalpy space. The equilibrium approach has two main advantages: first, the table generation is fast and simple to implement. Second, the table generation allows to represent all possible thermochemical states and no interpolation during the look-up process is necessary. However, the assumption of chemical equilibrium is very strict and mostly not fulfilled in a real flame. Obviously the effect of strain on a flame is neglected and close to walls, heat losses cause non-equilibrium effects that cannot be reproduced.
196 G. Frank, J. Zips & M. Pfitzner 2400 2100 χ st = 0.3 s -1 1800 T [K] 1200 900 600 300 0 0.2 0.4 0.6 0.8 1 FIGURE 4. S-Curve for the frozen chemistry. FIGURE 5. Temperature profiles for frozen chemistry. 3. Counterflow Diffusion Flames Most tabulated chemistry models for non-premixed combustion are based on counterflow diffusion flames. In the following section four methods to include heat loss effects in counterflow diffusion flames are discussed. The reference is a counter flow diffusion flame with methane as fuel and air as oxidizer. The boundary temperatures were chosen as T = 300K. Extinction scalar dissipation rate for this flame is χ 0.3s 1. The detailed chemistry mechanism GRI-3.0 [12] was used. 3.1. Frozen Chemistry The frozen chemistry approach can be found often in commercial solvers (e.g. FLU- ENT [13]). Here, the flamelets are calculated solving steady, counter flow diffusion flames. The table is parameterized in terms of mixture fraction, mixture fraction variance and scalar dissipation rate, for which transport equations are solved during runtime. The composition of the mixture is read from the tabulated flamelets. An additional transport equation for enthalpy is solved to account for heat losses and the temperature is then calculated from the composition and enthalpy. Physically, it can be interpreted as a cooling down of burned products while omitting any recombination effects that may occur. The range of tabulated states for this model is shown in Fig. 4 as grey area. All states below the stable branch of the S-Curve can be calculated. Typical temperature profiles are plotted in Fig. 5. The main disadvantage of this method is, that the influence of the heat loss on the chemistry and the reaction rates is omitted and the expected influence of the radical formation near cooled walls cannot be mapped in the table. 3.2. Mechanisms of Heat Loss Flame-wall interaction can be characterised by the angle between flame front and wall, with the two extrema being either the flame front perpendicular to the wall (head on configuration Fig. 6) or the flame front parallel to the wall (side wall configuration Fig. 7). The heat loss mechanisms for both cases will be investigated in the following sections. 3.2.1. Flamelets with convective sink term When we assume the very thin flame front being perpendicular to the wall (Fig. 6), then the heat loss vector is perpendicular to the mixture fraction gradient of the flame as well. A one dimensional flamelet at a certain distance from the wall is therefore subjected
Non-Adiabatic Libraries for Tabulated Chemistry Models 197 FIGURE 6. Head-on configuration. FIGURE 7. Side wall configuration. 2400 2100 χ st = 0.3 s -1 1800 T [K] 1200 900 600 300 0 0.2 0.4 0.6 0.8 1 FIGURE 8. S-Curve for flamelets with a convective sink term FIGURE 9. Temperature profiles for flamelets with a convective sink term. to a heat loss of q = α C (T wall T) (3.1) with α C being the convective heat transfer coefficient. Please note, that a configuration as shown in Fig.6 would in reality lead to a triple flame [15]. For this problem it should be seen as simplified thought model. The convective heat transfer coefficient is antiproportional to the convective timescale α C 1/τ conv. Lee et al. [10] propose using a Nusselt correlation to estimate this heat transfer coefficient. With this heat sink term in the unsteady flamelet equations (1.1) the physical temperature dependence of the reaction rates of the elementary reactions is considered. In Fig. 8 a comparison of the S-curve and the S-curve with a convective heat loss is shown. It can be seen that at high scalar dissipation rates the heat loss source terms have little effect on the flame. Since the scalar dissipation rate is antiproportional to the flamelet lifetime, the heat loss is a process on a slow timescale that does little effect on short living flames. Additionally, the flame thickness reduces with higher dissipation rate, and thinner flames can exist nearer to the wall, which was found in [5]. The range of tabulated states with this method is indicated by the grey area in Fig. 8. The area between the stable branch with heat loss and the unburned state needs to be interpolated in the table. For a low scalar dissipation rate of of χ st = 0.3s 1, where the flame extinguishes after a while, the temperature curves are shown in Fig. 9. The highest enthalpy losses
198 G. Frank, J. Zips & M. Pfitzner T max [K] 2250 2000 1750 0.1 1 10 100 χ st [1/s] FIGURE 10. S-Curve for flamelets with adjusted boundaries. T [K] 2400 2100 1800 1200 900 600 χ st = 0.3 s -1 300 0 0.2 0.4 0.6 0.8 1 FIGURE 11. Temperature profiles for flamelets with adjusted boundaries. ( h = c p T ) occur at stoichiometric mixture fraction, where the temperature difference to the wall is the highest. After extinction of the flame, the hot combustion products are cooled down to the boundary temperature. In the instationary simulation also the slow recombination of radicals at low temperatures can be tabulated. A drawback of this method is the huge dependency on the unknown convective time scale τ conv. This time scale firstly determines how much of the area below the S-Curve can be tabulated with actual flamelets, and how much data needs to be interpolated. Moreover, this time scale also influences the production rates of species. 3.2.2. Flamelets with adjusted boundaries When the flame front is parallel to the wall, the heat loss vector is parallel to the mixture fraction gradient as well. Here heat conduction will occur through the flamelet, a process in physical space, which can be hardly transformed into mixture fraction space, since the flame thickness would be an influencing factor. If a diffusion flame brushes on a surface as shown in Fig. 7, the boundary conditions change. In Fig. 7 the rich fuel side is trapped between flame and wall, it could be also possible that the lean side is between flame and wall. For flamelets with radiation heat losses Hossain et al. [3] propose to feed slightly lean fuel and slightly rich air into the counterflow diffusion flame. This reduces the mixture fraction range on both sides as can be seen in Fig. 11. The maximum flame temperature of the reference flamelet can be reduced by around 300 K, which is sufficient for radiation losses, but for wall heat losses a much higher enthalpy losses are expected. This method seems unsuitable for heat loss effects due to cooled walls. 3.2.3. Flamelets with permeable wall A fairly new model to incorporate wall heat losses has been proposed by Wu and Ihme [11]. In the counterflow diffusion flame an isothermic permeable wall is introduced that subjects the flame to heat losses (see Fig. 12). In mixture fraction space this translates to an additional thermal boundary condition at a certain mixture fraction Z wall. For Z > Z wall the species simply diffuse to the fuel side. Preliminary results for the resulting temperature profiles are shown in Fig. 13. This method is quite promising, because it seems to allow to tabulate all states below the stable branch in the S-Curve while incorporating heat loss effects on the production rates of the species.
Non-Adiabatic Libraries for Tabulated Chemistry Models 199 2100 1800 T [K] 1200 900 600 300 0 0.2 0.4 0.6 0.8 1 FIGURE 12. Schematic counterflow diffusion flame with permeable wall. FIGURE 13. Temperature profiles for flamelets with permeable wall. 4. Conclusions Heat loss effects in tabulated chemistry models have been investigated mainly in scope of thermal radiation heat losses. But heat loss also occurs near cooled walls, where it reduces the reaction speed and possibly quenches a flame. When a flame can be represented by its mixture fraction Z and scalar dissipation rate χ, its composition Y i and temperature T changes according to the enthalpy of the system subjected to wall heat losses. It would be desirable, if models based on tabulated chemistry can reproduce these changes. This work presents a short comparison of five methods to build chemistry tables with heat losses. The first section presented a tabulation method based on chemical equilibrium. This method is very quick and does not need any interpolation in the table. It represents the effect of the reduced enthalpy level on chemical kinetics. However, effects due to strain of the flames are not considered. Additionally, reactions near the wall are probably not in chemical equilibrium. In the next section a simple extention of the original flamelet model, here dubbed frozen chemistry model, as used in many commercial solvers is presented. In this method the composition of an counterflow diffusion flame is assumed to be constant and the temperature is calculated according to the total enthalpy in the system. The flamelet table is easily generated and any state below the stable branch of the S- Curve can be represented. The effects of strain are incorporated. But the effects of the heat losses on the production rates of species is omitted, therefore the composition near the wall is predicted incorrectly. A flamelet with a convective sink term is used to generate the table. Here again strain effects are incorporated in the model. Additionally the effect of heat loss on the chemistry is incorporated for flamelets at low strain. Even extinction can be tabulated. But the table cannot be filled completely with this method and states between the stable burning branch with heat loss and the unburned solution needs to be interpolated. Also species composition and production rates are dependent on an unknown timescale τ conv. For radiation heat losses an adjustment to the boundaries of the counterflow diffusion flame can be used to generate different enthalpy levels in the flame. But the enthalpy losses that the flames are subjected to, are not high enough to sufficiently model heat loss near cooled walls. This model is not suited to include wall heat loss effects. A fairly new model introducing a cooled permeable wall into the counterflow diffusion flame have been proposed by Wu and Ihme [11]. Only preliminary results are available until now, but this models is quite promising. The effort for the table generation is slightly
200 G. Frank, J. Zips & M. Pfitzner larger, than for the frozen chemistry model, however it should be possible to tabulate all states below the stable burning branch of the S-Curve while still incoporating the effects of heat losses on the composition. Ongoing work in the project will use four of the presented methods on a generic test case of a reacting flow over a wall and evaluate their capability of predicting the correct heat flux to the wall. Acknowledgments Financial support has been provided by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) in the framework of the Sonderforschungsbereich Transregio 40. References [1] BRAY, K. AND PETERS, N. (1994) Laminar Flamelets in Turbulent Flames. Turbulent Reacting Flows, Academic Press, London 63 113. [2] PITSCH, H. (1993). Entwicklung eines Programmpaketes zur Berechnung eindimensionaler Flammen am Beispiel einer Gegenstromdiffusionsflamme. Diploma Thesis, RWTH Aachen [3] HOSSAIN, M., JONES, J. AND MALALASEKERA, W. (2001) Modelling of a Bluff- Body Nonpremixed Flame Using a Coupled Radiation/Flamelet Combustion Model Flow. Turbulence and Combustion 67, 217 234. [4] COELHO, P.J., TEERLING, O.J. AND ROEKARTS D. (2003) Spectral radiative effects and turbulence/radiation interaction in a non-luminous turbulent jet diffusion flame. Combustion and Flame 133, 75 91. [5] WANG, Y. AND TROUVÉ, A. (2006) Direct numerical simulation of nonpremixed flame-wall interactions. Combustion and Flame 144, 461 475. [6] DABIREAU, F., CUENOT, B.. VERMOREL, O. AND POINSOT, T. (2003) Interaction of flames of H 2 + O 2 with inert walls. Combustion and Flame 135, 123 133. [7] FRANK, G., FERRARO, F. AND PFITZNER, M. (2013). RANS Simulations of Chemical Reactions in Cooling Films. 5 th European Conference for Aeronautics and Space Sciences (EUCASS), Munich [8] FIORINA, B. BARON, R., GICQUEL, O., THEVENIN, D., CARPENTIER, S. AND DARABIHA, N. (2003) Modelling non- partially premixed flames using a flame-prolongation of ILDM. Combustion Theory and Modelling 7(3), 449 470. [9] CANTERA DEVELOPERS. URL: http://cantera.github.io/docs/sphinx/html/index.html. [10] LEE, D.J., THAKUR, S., WRIGHT, J., IHME, M. AND SHYY, W (2001) Characterization of Flow Field Structure and Species Composition in a Shear Coaxial Rocket GH2/GO2 Injector: Modelling of Wall Heat Losses. 47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit San Diego, California, paper AIAA 2011-6125. [11] WU, H. AND IHME, M. (2015) Modeling of Wall Heat Transfer and Flame/Wall Interaction A Flamelet Model with Heat-Loss Effects. 9th U.S. National Combustion Meeting Cincinnati, Ohio, May 17-20. [12] SMITH, G. ET AL.. URL: http://www.me.berkeley.edu/gri_mech. [13] ANSYS, INC. (2012). ANSYS FLUENT Theory Guide. Version 14.5.
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