A comprehensive study of flamelet tabulation methods for pulverized coal combustion in a turbulent mixing layer—Part II: Strong heat losses and multi-mode combustion

Combustion and Flame 216 (2020) 453–467

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Combustion and Flame journal homepage: www.elsevier.com/locate/combustflame

A comprehensive study of flamelet tabulation methods for pulverized coal combustion in a turbulent mixing layer—Part II: Strong heat losses and multi-mode combustion Xu Wen a,b, Martin Rieth c, Arne Scholtissek a, Oliver T. Stein d, Haiou Wang b, Kun Luo b, Andreas Kronenburg d, Jianren Fan b,∗, Christian Hasse a a

Simulation of Reactive Thermo-Fluid Systems (STFS), TU Darmstadt, Darmstadt 64827, Germany State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, PR China c Institute for Combustion and Gasdynamics (IVG), Chair for Fluid Dynamics, University of Duisburg-Essen, Duisburg 47057, Germany d Institut für Technische Verbrennung (ITV), Universität Stuttgart, Herdweg 51, Stuttgart 70174, Germany b

a r t i c l e

i n f o

Article history: Received 30 May 2019 Revised 25 November 2019 Accepted 16 December 2019 Available online 6 January 2020 Keywords: Pulverized coal combustion Turbulent mixing layer Flamelet tabulation methods Strong heat losses Multi-mode combustion

a b s t r a c t This paper is a continuation of our work done in Part I, in which the a priori and budget analyses were conducted, Wen et al. (2019). In this work, we focus on addressing specific and recurring issues in flamelet modeling for pulverized coal combustion, including strong heat losses, multi-mode combustion and reaction progress variable definition. First, extended flamelet formulations are developed that can take into account strong heat loss effects in pulverized coal combustion systems. Then, to characterize multi-mode combustion in pulverized coal flames, a coupled premixed and non-premixed flamelet model is developed using the combustion mode index. Finally, the effects of reaction progress variable definition on the flamelet predictions are quantified. A state-of-the-art direct numerical simulation database is employed to challenge the newly developed flamelet models. The tabulated thermo-chemical quantities are compared with the reference direct numerical simulation results through a priori analyses. Comparisons show that the newly developed flamelet models which take into account strong heat loss effects can predict the gas temperature and species mass fractions correctly. The adiabatic flamelet models overpredict the corresponding thermo-chemical quantities in regions where the interphase heat transfer is significant. Coupled with a linear extrapolation method, the prediction of the gas temperature with the adiabatic flamelet models can be improved. The performance of the multi-mode flamelet model depends on whether the local combustion mode can be correctly identified. The conventional combustion mode index based on the gradients of fuel and oxidizer species mass fractions cannot correctly identify the combustion mode in the entire combustion field. © 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction The process of pulverized coal combustion can be roughly divided into four subprocesses: water evaporation, devolatilization, char surface reactions and gaseous combustion of volatiles and char off-gases. The last step, gaseous combustion, is very important in predicting the pulverized coal flame structure. Efforts have been made to develop high-fidelity gaseous combustion models for pulverized coal combustion [1–12]. Particularly, the flamelet model [13] has been extended to simulate pulverized coal combustion [3–12]. Since the pioneering work conducted by

∗

Corresponding author. E-mail addresses: [email protected] (H. Wang), [email protected] (J. Fan).

Vascellari et al. [14], significant progress has been made in recent years. Watanabe and Yamamoto [3] were the first to devise a flamelet model for pulverized coal combustion in the context of direct numerical simulation (DNS), in which both devolatilization and char surface reactions were taken into account. Afterwards, they further extended the flamelet model to include the context of large-eddy simulation (LES), in which the turbulence-chemistry interactions were taken into account with the presumed probability density function (PDF) [4]. Rieth et al. [7] and Wen et al. [5] independently developed a steady laminar flamelet (SLF) model [13] and a flamelet/progress variable (FPV) model [15] for turbulent pulverized coal combustion, in which devolatilization, char surface reactions and radiation were all covered. Recently, Rieth et al. [10] extended the SLF model to include the FPV formulation, representing the transient states. Wen et al. [11], on the

https://doi.org/10.1016/j.combustflame.2019.12.028 0010-2180/© 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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Fig. 1. (a) Schematic of the computational setup of the pulverized coal flame in a turbulent mixing layer, and (b) the instantaneous distribution of the flame index FI at t = 10 ms at the central x–y plane [18]. The white dots represent the pulverized coal particles. The sample lines investigated in Section 3 are indicated in (b) as SAM1, SAM2 and SAM3, respectively. The definition of FI is the same as that in Ref. [18], i.e., F I = ∇ YF · ∇ YO2 , where YF and YO2 are the mass fractions of fuel and oxygen, respectively.

other hand, further extended the two-mixture-fraction FPV model, enabling it to formulate three mixture fractions to characterize mixing among multiple fuel streams. Vascellari et al. [12] recently developed a two-mixture-fraction flamelet model, which was evaluated against the experimental data provided by Xia et al. [16]. Wan et al. [9] developed a flamelet-generated manifold (FGM) [17] model to characterize the alkali metal emissions at an early stage of pulverized coal combustion, neglecting char surface reactions. Knappstein et al. [8], on the other hand, independently developed a two-mixture-fraction FGM model to characterize both the devolatilization and char conversion processes. Although significant progress has been made in flamelet modeling for pulverized coal combustion, the following three recurring issues were highlighted in previous flamelet studies: (1) The heat losses in pulverized coal combustion systems can be extremely strong, and affect the flamelet structure significantly [18,19]. The strong heat losses cannot be properly represented by the conventional adiabatic flamelet model, which results in the over-prediction of the gas temperature (see Fig. 14a in Ref. [19]). (2) The complicated multi-mode combustion exists in pulverized coal flame [18–23], which cannot be described by the conventional flamelet model based on a single-combustion mode. In particular, the CO and H2 mass fractions can be significantly over-predicted if the local combustion mode is not correctly represented by the flamelet model (see Figs. 16c and 17c in Ref. [19]). (3) The species compositions in volatile matter are complicated and may contain products [24]. Thus, it is not straightforward to define progress variable in flamelet model to characterize the progress of pulverized coal combustion. The main focus of the present work is to address these specific recurring issues in flamelet modeling for pulverized coal combustion. In the following context, the above three issues will be described in more detail. Compared to LES, DNS has the advantage that all length and time scales of turbulence and chemistry are resolved, such that the modeling of sub-grid scale (SGS) effects can be avoided. The rapid development of powerful supercomputers has made it possible to study pulverized coal combustion using the DNS technique. To date, several DNS cases have been conducted. The reviewing of the DNS studies has been done in Part I [25], and will be not repeated here. As in Part I, the DNS dataset of a pulverized coal flame in a turbulent mixing layer from Rieth et al. [18] is adopted for model evaluation. The schematic of the computational setup of the mixing layer is shown in Fig. 1. The operating conditions of

the mixing layer were extracted from the LES results of a semiindustrial pulverized coal furnace [7]. The details are reported in the work by Rieth et al. [18]. This specific DNS database is considered useful for developing a model for pulverized coal combustion. The availability of the detailed thermo-chemical variables makes it possible to validate the above identified issues for flamelet modeling comprehensively. In particular, the input data of flamelet models, i.e., the so-called trajectory variables, can be calculated from the DNS solutions prior to the simulation (i.e., a priori analysis). By this means, the validity of the flamelet library and the selected trajectory variables for pulverized coal combustion can be evaluated independently. Rieth et al. [10] conducted a priori and a posteriori analyses of the FPV model using this DNS dataset as the reference results. In previous works on flamelet modeling for pulverized coal combustion, the heat transfer between the coal particle phase and gas phase was taken into account by introducing the total enthalpy as one of the trajectory variables [3–12]. However, the interphase heat transfer in the flamelet library was always represented in different ways. A straightforward method used to represent interphase heat transfer, here denoted as “M1”, is to vary (increase and decrease) the temperature boundary condition of the adiabatic flamelet equations [3–5,26]. With this method, a series of flamelets with different enthalpy levels can be obtained. However, this method has the limitation that the temperature on the boundaries cannot be lower than the ambient gas temperature, so that the heat losses may not be fully represented in the flamelet library. Figure 2(a) plots a non-premixed flamelet library generated with this method. This flamelet library corresponds to the aforementioned DNS database for a pulverized coal flame in a turbulent mixing layer [18], which will be studied in this work. The total enthalpy extracted from the DNS database at t = 10 ms is also plotted for reference (similar observations for other time steps). Obviously, the entire range of enthalpies occurring in the studied pulverized coal flame cannot be fully represented by the non-premixed flamelet library by only varying the temperature boundary conditions. The same problem arises for the premixed flamelet library, as shown in Fig. 2(b). For premixed combustion, Proch and Kempf [27] proposed a new method, referred to as “M2”, to take into account the heat losses to the cold wall in gaseous combustion systems. In this method, the lower enthalpy is achieved by scaling the energy equation source term in adiabatic flamelet equations by a factor of less than unity. Rieth et al. [6,7] extended this method to include non-premixed combustion to take into account the heat loss effects in pulverized coal combustion systems. Since there is a non-linear relation between the scaling factor and enthalpy,

X. Wen, M. Rieth and A. Scholtissek et al. / Combustion and Flame 216 (2020) 453–467

(a)

455

(b)

Fig. 2. Total enthalpy distribution of the adiabatic non-premixed and premixed flamelets in the mixture fraction space at different conditions, and the corresponding values extracted from the whole computational domain in DNS at t = 10 ms [18]. The purple background color in (a) and (b) indicates the minor heat loss that is represented by decreasing the temperature boundary conditions. The flamelets are generated at operating conditions in the DNS. In (b), the green symbols correspond to premixed flamelets at different equivalence ratios, while the lines are obtained from the non-premixed flamelets at Z = 0 an Z = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

accessing the “M2” flamelet library requires iteration and therefore higher computational cost than with “M1”. In an attempt to achieve a balance between computational accuracy and efficiency, Rieth et al. [10] further coupled “M2” with “M1”. If the enthalpy in the flow field is lower than the lowest enthalpy in the “M1” table, the “M2” flamelet table is accessed; otherwise, the “M1” table is used. However, it is still unclear whether the whole enthalpy range in the flow field can be covered by this coupled flamelet table. Furthermore, it is also unclear how much of an improvement can be made with “M2” compared to “M1”. On the other hand, for premixed flamelet modeling in pulverized coal combustion, Vascellari et al. [12] took into account the heat loss effects by lowering the mass flow rate for the burner-stabilized premixed flame, as initially proposed by Fiorina et al. [28] for gaseous combustion. In this method, the lowest enthalpy in the flamelet table depends on the flammability limits of the gaseous flame. It is expected that the lowest enthalpy in a pulverized coal combustion system cannot be fully represented by this method, since the heat transfer between the cold coal particles and hot surroundings can be much more significant than in pure gaseous combustion. Note that the aforementioned flamelet models only represent the interphase heat transfer based on the gaseous flamelet equations, in which the two-phase coupling source terms are totally neglected. In fact, in pulverized coal combustion systems, the heat loss effects are generated due to interactions between cold particles and hot surroundings through two-way coupling source terms (e.g., convection, radiation, etc.). Thus, to incorporate the full range of total enthalpy in the flamelet library, two-way coupling between the two phases should be taken into consideration. In pulverized coal combustion system, multi-combustion modes can be generally observed (see Fig. 1(b) for example). However, the multi-mode combustion cannot be properly represented by the conventional single-mode flamelet model, which results in the incorrect prediction of species such as CO and H2 [19]. In our previous work [29], a multi-mode flamelet model was developed for pulverized coal combustion in the context of LES. In this coupled flamelet model, different flamelet libraries are accessed using the combustion mode index, i.e., in the premixed flame reaction zone, the premixed flamelet library is accessed, while in the non-premixed flame reaction zone, the thermo-chemical quantities are extracted from the non-premixed flamelet library. However, the flow field investigated in that work was too complicated to quantify the extent to which the multi-mode flamelet model can

improve the prediction accuracy compared to the classical singlemode flamelet model. Furthermore, the multi-mode flamelet model developed in that work was coupled with SGS models, which introduced uncertainties to the flamelet model. It is still unclear whether the multi-combustion modes in a pulverized coal flame can be accurately predicted by the coupled flamelet model. In flamelet tabulation methods, the reaction progress variable YPV is generally introduced as one of the trajectory variables [13,17]. However, different studies employed different definitions for YPV , even for the same type of coal [7,18,30]. While some procedures for automatic determination of YPV were proposed for gaseous combustion [31–33], there is no commonly accepted strategy for pulverized coal combustion. Further, a systematic assessment of the impact of different definitions of YPV has not been published for a coal combustion case to the authors’ knowledge. A suitable YPV should be defined according to the following principles [31–33]: (i) The transport equation for YPV can be easily formulated and solved in the flow solver. (ii) The reactive scalars included in YPV should all evolve on comparable time scales. (iii) The defined YPV should have a monotonic relationship with the thermo-chemical variables to ensure that the flamelet solutions are mapped uniquely. (iv) The “progress of reactions” can be characterized by the defined YPV . For pulverized coal combustion, these principles are difficult to fulfill simultaneously since multiple fuel streams with complex compositions, usually comprising major reaction products in YPV [24], are released from coal particles during the combustion process. In this case, it is difficult to fulfill both principles (iii) and (iv). It is essential to investigate how the definition of the reaction progress variable affects the flamelet predictions. In Part I [25], the suitability of both adiabatic premixed-flamebased flamelet model and non-premixed-flame-based flamelet model was evaluated through a priori analyses. Besides, the influence of the trajectory variables in both premixed and nonpremixed flamelet tables was evaluated through budget analyses. This work is a continuation of Part I, and the purpose is to address the aforementioned issues in flamelet modeling for pulverized coal combustion. To this end, extended flamelet formulations are developed that can take into account strong heat losses in pulverized

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coal combustion. Particularly, the heat transfer between the coal particle phase and gas phase is explicitly considered by introducing the two-way coupling source terms in the flamelet equations. The state-of-the-art DNS database for a pulverized coal flame in a turbulent mixing layer [18] is employed to challenge the flamelet models developed. To characterize the multi-mode combustion in the studied pulverized coal flame, a coupled multi-mode flamelet model is proposed, which dynamically utilizes different flamelet models depending on the local value of the combustion mode index. To clarify the suitability of the YPV definition, different flamelet tables are generated with different formulations of YPV . The performance of the flamelet models which are developed is evaluated by comparing them with the adiabatic and/or nonadiabatic premixed and non-premixed flamelet models, and the reference DNS results in different solution spaces. 2. Modeling methods In DNS, the non-filtered and non-averaged governing equations of mass, momentum, species mass fractions and total enthalpy are solved. The flamelet equations developed in this work are derived from the species mass fractions and total enthalpy governing equations. For completeness, these equations are repeated here. Taking into account the interphase mass and heat transfers, the governing equations for species mass fraction Yk and total enthalpy He can be written as,

  ∂ρYk ∂ρ u jYk ∂ ∂ Yk ρ Dk + = + ω˙ k + S˙ C,m,k ∂t ∂xj ∂xj ∂xj   ∂ρ He ∂ρ u j He ∂ ∂ He + = + S˙ C,He ρα ∂t ∂xj ∂xj ∂xj

(1)

S˙ C,m,k =

1 N p

− ⎪ ⎪ V ⎪ ⎪ ⎩

p=1

dmchar,k dt

0

for

k = volatiles,

for

k = char,

for

k = others

p

(2)

(3)

where V is the volume of the grid cell in which the particle resides, the summation index p loops over all particles in the local cell, and Np is the number of particles present in the cell volume. mvol, k and mchar,k denote the mass of species k originating from the volatile matter and char off-gases, respectively. α is the thermal diffusion coefficient calculated under the unity Lewis number assumption [18]. S˙ C,He takes into account the interphase heat transfer, which includes convection, chemical energy associated with devolatilization, char oxidation and radiation [18],

S˙ C,He = S˙ C,conv + S˙ C,chem + S˙ C,rad



2.1. Non-premixed flamelet tabulation method Following the flamelet transformation rules [13], the governing equations for species and enthalpy (1) and (2) of pulverized coal combustion can be transformed from physical-time space (x, y, z, t) into composition-time space (Z, τ ) as [37,38],





∂ Yk ∂ Yk ρχ ∂ 2Yk ˙ ρ − = ω˙ k − fd1 SC,m,k Yk + (1 − Z ) ∂τ 2 ∂ Z2 ∂Z ∂ T ρ c p,g χ ∂ 2 T − ∂τ 2 ∂ Z2



∂T = ω˙ T − c p,g fd1 S˙ C,m,k T + (1 − Z ) + fd2 S˙ C,T ∂Z

(5)

ρ c p,g

where ρ is the gaseous phase density, uj the gaseous phase velocity, and Dk the mass diffusivity of species k, which is calculated under the unity Lewis number assumption [18]. ω˙ k is the chemical reaction rate of species k, which is calculated with the reduced CRECK chemical reaction mechanism containing 52 species and 452 reactions [34,35]. S˙ C,m,k is the two-way coupling mass source term, which takes into account the mass transfer between the coal particle phase and gas phase,

  ⎧ dmvol,k 1 N p ⎪ ⎪ − ⎪ ⎪ ⎨ V p=1  dt p

volatiles and char off-gases in an individual particle. ∇ · qrad is the radiative heat exchange between the coal particle phase and gas phase, which is calculated with the discrete ordinates method [36]. For gaseous combustion, the source term in Eq. (2) disappears and He becomes a conserved scalar. In this case, the total enthalpy varies linearly with the mixture fraction, corresponding to a single line in Fig. 2. For pulverized coal combustion, the lowest enthalpy values shown in Fig. 2 is thought to result from the twoway coupling between the coal particle phase and gas phase (see Eq. (4)). Therefore, to incorporate the full range of total enthalpy in the flamelet library for pulverized coal combustion, the two-way coupling source term should be taken into account when solving the flamelet equations, which is the motivation behind the newly developed flamelet model described in Sections 2.1 and 2.2. The coal particles are treated in a Lagrangian manner, and the corresponding governing equations can be found in Ref. [18], and will not be repeated here.



Np dmvol,p dmchar,p dTp 1 =− m p c p,p −hvol −hchar + ∇ · qrad V dt dt dt p=1

(4) where mp is the particle mass, cp,p the particle heat capacity, Tp the particle temperature, hvol and hchar the enthalpies of volatiles and char off-gases, respectively. mvol, p and mchar,p are the mass of

(6)

where τ is the Lagrangian flamelet time. The transient terms can be neglected with the assumption of a fast chemical reaction. Z is the total mixture fraction for coal particles, χ the scalar dissipation rate, cp,g the gas phase heat capacity, ω˙ T the heat release rate, S˙ C,m,k and S˙ C,T the two-way coupling source terms due to pulverized coal combustion, which are obtained by separately performing a one-dimensional (1D) simulation for a pulverized coal counterflow flame. In the 1D pulverized coal counterflow simulation, the coal particles with the same properties as in the DNS are injected from the fuel side, while pure air is injected from the oxidizer side. The 1D governing equations are solved with the in-house code ULF (Universal Laminar Flame) [39]. fd1 and fd2 are newly introduced scaling factors. Eqs. (5) and (6) become the classical adiabatic flamelet equations if fd1 and fd2 are set to 0. Different states of pulverized coal combustion can be realized by varying the values of fd1 and fd2 . In this work, the flamelet equations with the two-way coupling source terms introduced in Eqs. (5) and (6) are only employed to take strong heat loss effects into account. For minor heat loss and heat gain processes, the conventional adiabatic flamelet equations with varying temperature boundary conditions are solved, i.e. “M1”, as described in the introduction section. In the very first approach, only fd1 is varied while fd2 is set to be constant while solving the flamelet equations. In summary, the non-premixed flamelet tables with strong heat losses are generated as follows: (i) The conventional non-premixed adiabatic flamelet equations ( fd1 = fd2 = 0 in Eqs. (5) and (6)) are solved with varied temperature boundary conditions. To represent the heat gain process, the temperatures at both the fuel and oxidizer boundaries are elevated compared to the original coal particle temperature. In line with the findings reported in our previous work [20], the fuel temperature Tf is set equal

X. Wen, M. Rieth and A. Scholtissek et al. / Combustion and Flame 216 (2020) 453–467

to the oxidizer temperature Tox while solving the flamelet equations. To represent the (minor) heat loss process, both Tf and Tox are successively lowered to the ambient temperature of 300 K. (ii) To represent strong heat loss effects, the non-adiabatic flamelet equations Eqs. (5) and (6) are solved with varied values of fd1 , in which the two-way coupling source terms are obtained from the 1D pulverized coal counterflow flame. For pulverized coal combustion, the mixing of gaseous fuels from devolatilization and char surface reactions is described with two corresponding mixture fractions Zvol and Zpro , which are defined as, Zvol = ξvol /(ξvol + ξ pro + ξox ) and Z pro = ξ pro/(ξvol + ξ pro + ξox ), respectively. ξ vol , ξ pro and ξ ox are the mass of gas originating from the volatile matter, char off-gases and oxidizer stream, respectively. To avoid possible numerical error in the triangular solution space (Zvol , Zpro ) [40], a coordinate transformation technique is adopted. The flamelet solutions are finally mapped to the unit square solution space (X, Z), X = Z pro/(Zvol + Z pro ) and Z = Zvol + Z pro. The aforementioned adiabatic and non-adiabatic flamelet equations are solved in the Z space for different values of X to characterize the mixing process. To represent the strain rate on the flame structure, the scalar dissipation rate χ in Eqs. (5) and (6) is varied from equilibrium to the extinction limit. In an attempt to represent possible unsteady phenomena, χ is transformed into the coordinate of the reaction progress variable YPV [15]. Note that although the unsteady states, such as local-extinction and reignition, are included in the flamelet library, Ihme and See [41] reported that these unsteady states are rarely accessed using the steady flamelet/progress variable (FPV) approach [15]. This is attributed to the fact that the unsteady phenomena are slow chemical processes, which cannot be simply represented by the reaction progress variable defined based on the major reaction products. The evolution of the unsteady phenomena can only be characterized by the unsteady flamelet model with additional trajectory variables (e.g., the local flamelet time [42], the reaction progress parameter and scalar dissipation rate [41]) being introduced. Based on the above introduced trajectory variables, the thermo-chemical quantities in the integrated flamelet library can be first parameterized as



= adi X, Z, T f or Tox , YPV ∪ nad (X, Z, fd1 , YPV )

(7)

with

= adi (X, Z, 300, YPV ) = nad (X, Z, 0, YPV )

(8)

Note that Tf and Tox in the adiabatic flamelet library adi , and fd1 in the non-adiabatic flamelet library nad are difficult to obtain from the flow field. In this work, both parameters are transformed into the coordinate of normalized total enthalpy Hnorm , which is defined as

Hnorm =

He − He,min (X, Z, YPV ) He,max (X, Z, YPV ) − He,min (X, Z, YPV )

scaling factor fd1 . According to the maximum and minimum values of He , the normalized total enthalpy Hnorm can be calculated according to Eq. (9). The thermo-chemical quantities in the integrated non-premixed flamelet library with strong heat loss can be finally parameterized as

= D (X, Z, Hnorm , YPV )

(9)

where He,max (X, Z, YPV ) and He,min (X, Z, YPV ) are the maximum and minimum enthalpy in the flamelet tables for specific values of X, Z and YPV . Note that the adiabatic and non-adiabatic flamelets are stored in a single flamelet library according to Eq. (7), and the transformation from the original coordinates of Tf (or Tox ) and fd1 to the targeted coordinate of Hnorm follows the conventional transformation procedure, i.e., from the coordinate of Tf (or Tox ) to Hnorm . Specifically, for specific values of X, Z and YPV , the maximum and minimum values of He are obtained by looping all of the flamelets, including both adiabatic and non-adiabatic. Note that the adiabatic flamelet can be identified by the temperature boundaries Tf (or Tox ), while the non-adiabatic flamelet can be located by the

(10)

The resulting integrated non-premixed flamelet library is shown in Fig. 3. The corresponding values of total enthalpy and gas temperature in the DNS solutions are also superimposed to illustrate whether the thermo-chemical quantities in the DNS can be fully represented by the flamelet library. The adiabatic flamelet library with varied temperature boundary conditions is the same as that in Fig. 2(a). The heat loss considered with this method is represented by the purple zone. It is seen that the non-adiabatic flamelet library generated with different values of fd1 contains values that are below the lowest thermo-chemical quantities in the adiabatic flamelet library. As indicated by the green zone in Fig. 3(a), the total enthalpy in the DNS dataset can be covered by the integrated flamelet library. Similarly, the gas temperatures shown in Fig. 3(b) also indicate that the varying range of gas temperatures can be covered by the new flamelet library. 2.2. Premixed flamelet tabulation method In general, the premixed flamelet tabulation method can be divided into two categories: the unstrained flamelet model and the strained flamelet model. The strain rate effects should be considered when the flame falls in the regimes of the thin reaction zone and the broken reaction zone, i.e., the Karlovitz number is larger than unity [43]. For the pulverized coal flame studied in this work [18], the Karlovitz number calculated at the stoichiometric mixture 2 /η 2 = 0.84, where τ fraction condition is Ka = τF,st /τη = lF,st F,st and lF,st are the flame thickness and flame time at the stoichiometric mixture fraction condition, respectively, while τ η and η are the Kolmogorov time and length scale, respectively. Thus, it is expected that the strain rate effects are not important for the pulverized coal flame studied. The unstrained premixed flamelet equations for pulverized coal combustion can be expressed as,

ρu

dYk dY d + ρu sL,u k + dτ dx dx

  dY ρ Dk k = ω˙ i + f p1 S˙ C,m,k

d dT dT ρu c p,g + ρu sL,u c p,g − dτ dx dx



adi X, Z, T f or Tox , YPV ∩ nad (X, Z, fd1 , YPV )

457

dx

  dT = ω˙ T + f p2 S˙ C,T λ dx

(11)

(12)

where ρ u is the density of the unburned mixture, sL,u the laminar flame speed, x the physical space coordinate, and λ the heat conductivity coefficient. S˙ C,m,k and S˙ C,T are the two-way coupling source terms, which are obtained from the 1D counterflow pulverized coal flame using the ULF package [39]. fp1 and fp2 are the scaling factors that take into account different states of pulverized coal combustion. Eqs. (11) and (12) are reduced to the classical adiabatic premixed flamelet equations if fp1 and fp2 are equal to 0. As in the non-premixed flamelet model, the premixed flamelet library with strong heat loss is generated in two steps, (i) At first, the adiabatic premixed flamelet equations ( f p1 = f p2 = 0) are solved for various temperature boundary conditions. To represent the heat gain process, the mixture temperature Tmix is elevated compared to the original coal particle temperature (600 K). To represent the (minor) heat loss process, the mixture temperature is lowered. (ii) Then, to represent strong heat loss effects, the flamelet equations with the two-way coupling source terms are solved. The scaling factor fp1 is varied by changing the flamelets from adiabatic flamelets at the lowest inlet temperature (300 K) to the extinction limit due to strong heat

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(a)

(b)

Fig. 3. Non-premixed non-adiabatic flamelet data for (a) total enthalpy He , and (b) the gas temperature T at different conditions. The dots in (a) and (b) are the corresponding thermo-chemical quantities from the whole computational domain in DNS, and the vertical dashed line in (b) indicates the location of the stoichiometric mixture fraction. The background color in (a) indicates minor (purple) and strong (light green) heat losses. The conditions for the non-premixed adiabatic flamelets with minor heat losses being considered are the same as those shown in Fig. 2(a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Premixed non-adiabatic flamelet data for (a) total enthalpy He , and (b) gas temperature T at different conditions. The dots in (a) are total enthalpy extracted from the DNS, while the purple and light green background colors in (a) and (b) indicate minor and strong heat losses, respectively. The red circle in (a) indicates the non-adiabatic flamelets generated at f pl = 0.3 and different equivalence ratios. The conditions for the premixed adiabatic flamelets with minor heat losses being considered are the same as those shown in Fig. 2(b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

loss. The scaling factor fp2 is kept as a constant value when fp1 > 0. The mixture fraction Z of the inlet mixture is varied from the lean flammability limit to the rich flammability limit. Outside the flammability limit, a linear extrapolation method is employed [44]. Note that the flammability limit is a function of the mixture temperature Tmix and varies depending on the flamelet boundary conditions. The fraction of fuel gases in the mixture is determined by the value of Z, while the fraction of species k in the fuel gases is calculated as, Y f uel,k = (ξvol Yvol,k + ξ proYpro,k )/(ξvol + ξ pro ) = (1 − X )Yvol,k + XYpro,k , where Yvol, k and Ypro,k are mass fractions of species k in volatile matter and char off-gases, respectively. According to the boundary conditions for the premixed flamelet equations, the adiabatic and non-adiabatic premixed flamelet solutions can be integrated into one and parameterized as



= adi (X, Z, Tmix , YPV ) ∪ nad X, Z, f p1 , YPV

(13)

with



adi (X, Z, Tmix , YPV ) ∩ nad X, Z, f p1 , YPV = adi (X, Z, 300, YPV ) = nad (X, Z, 0, YPV )

(14)

where adi (X, Z, Tmix , YPV ) corresponds to the adiabatic flamelet solutions calculated with varied mixture temperature boundary conditions, while nad (X, Z, fp , YPV ) corresponds to the non-adiabatic flamelet solutions calculated with varied scaling factors. As noted in the last subsection on the non-premixed flamelet tabulation method, the temperature boundary Tmix and the scaling factor fp1 are difficult to obtain in the flow solver. These parameters are transformed to the normalized He coordinate. Similarly to the non-premixed flamelet tabulation method, for specific values of X, Z and YPV , the minimum and maximum values of He are identified by looping all of the premixed flamelets, including both adiabatic and non-adiabatic. The adiabatic flamelet can be found by the Tmix , while the non-adiabatic flamelet can be located by the scaling factor fp1 . The thermo-chemical quantities in the integrated premixed flamelet library can be finally parameterized as

= P (X, Z, Hnorm , YPV )

(15)

Figure 4(a) and (b) shows the calculated total enthalpy and gas temperature in the mixture fraction space and physical space, respectively. For comparison, the corresponding thermo-chemical quantities from the DNS datasets are superimposed. One initial observation is that, in contrast to the adiabatic premixed flamelet li-

X. Wen, M. Rieth and A. Scholtissek et al. / Combustion and Flame 216 (2020) 453–467

(a)

459

(b)

Fig. 5. Gas temperature against different reaction progress variables for the premixed flamelet tables. The red circle in (b) indicates that flamelets do not have a monotonic relationship between the gas temperature and defined reaction progress variable. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

(c)

(d)

Fig. 6. Mass fractions of CO2 and CO against different reaction progress variables for the premixed flamelet tables. The red circle in (b) indicates that flamelets overlap in the YPV2 space, and the inset figure shows the local profiles of YCO2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

brary shown in Fig. 2(b), the total enthalpy in the DNS database can be fully addressed by the newly generated flamelet library. Note that, unlike the adiabatic premixed flamelet, where the total enthalpy maintains a constant value within a single flamelet, the total enthalpy within a single non-adiabatic flamelet spans a wide range (see vertically distributed dots in Fig. 4(a)). As indicated by the purple zone in Fig. 4(b), the burned gas temperature decreases slightly when the inlet mixture temperature is lowered. On the other hand, when the scaling factor in the non-

adiabatic flamelet equations is varied slightly, the burned gas temperature decreases significantly, as indicated by the green zone in Fig. 4(b). 2.3. Flamelet tabulation method for multi-mode combustion For the studied pulverized coal flame in the turbulent mixing layer, premixed and non-premixed combustion modes coexist in the combustion system (see Fig. 1(b)). In an attempt to reproduce

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multi-mode combustion, the premixed and non-premixed flamelet models described above are coupled using the combustion mode index. Specifically, in the identified premixed flame reaction zone, the thermo-chemical quantities are extracted from the premixed flamelet library, while in the non-premixed flame reaction zone, the non-premixed flamelet library is accessed. If the premixed and non-premixed flame coexist in the local cell, the thermo-chemical quantities extracted from both flamelet tables are weighted using the combustion mode index. In this work, the combustion mode index, RI, is defined as the normalized flame index FI [45],



1 RI = 1+ 2

∇YF · ∇YO2 |∇YF ||∇YO2 | +



(16)

where YF and YO2 are the mass fractions of fuel and oxygen, respectively. is a small positive number that is employed to avoid dividing by zero. To maintain consistency with the DNS dataset, YF is defined as, YF = YCH4 + YC6 H6 + YC2 H4 + YCO + YH2 [18]. In the premixed flame reaction zone, RI equals 1 and the corresponding premixed flame library is accessed, while in the non-premixed flame reaction zone, RI equals 0 and the corresponding non-premixed flame library is accessed. In the partially premixed flame reaction zone, RI ranges between 0 and 1, and both premixed and non-premixed flamelet tables are accessed. Note that RI is defined based on the original gas fuels and the oxidizer in the pulverized coal flame studied, and the intermediate species are not explicitly considered. It is expected that the value of RI is inaccurate for predicting the intermediate species. Note also that at conditions where the local oxygen concentration is negligible but the overall reaction rate is non-negligible, the calculated gradient of oxy-

gen is not strictly equal to 0 for the pulverized coal flame studied. At these conditions, the thermo-chemical quantities extracted from the premixed and non-premixed flamelet libraries would be weighted according to the calculated combustion mode index RI. The thermo-chemical quantities in the partially premixed flame reaction zone are obtained by a simple linear weighting procedure following previous works [5,46–48],

= (1 − RI ) · D (X, Z, Hnorm , YPV ) + RI · P (X, Z, Hnorm , YPV ) (17) For pulverized coal combustion, a similar method in the context of LES was evaluated in our previous work [5] in an a posteriori analysis. However, in that work, the evaluation of the multimode flamelet model may have been biased, since the SGS models employed introduce uncertainties. Furthermore, compared to that work, both premixed and non-premixed flamelet models presented in this work have been reformulated in terms of the flamelet table generation, trajectory variables definition, coordinate transformation, etc. Thus, it is essential to evaluate the performance of the multi-mode flamelet model further by excluding the SGS effects. Particularly, the suitability of the combustion mode index definition and the combination method employed in Eq. (17) should be evaluated specifically for pulverized coal combustion through an a priori analysis. 2.4. Definition of reaction progress variable To evaluate the effects of YPV definition on the flamelet predictions, the following expressions of YPV that widely used in previous

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works are tested [49,50],

YPV 1 = YCO2 + YH2 O + YH2 [5,11],

YPV 2 = YCO2 + YH2 O ,

YPV 3 = YCO2 + YCO [10, 26], YPV 4 = YCO2 [4,49],

YPV 5 = YCO2 + YH2 O + YH2 + YCO [31,50]

(18)

As described in the introduction, the principles of the YPV definition are difficult to fulfill simultaneously for pulverized coal combustion. For the Saar hvBb type coal [51] studied in this work, it is assumed that volatile matter contains CO and H2 in DNS [18], thus, YPV1 , YPV3 and YPV5 do not describe the progress of reactions (principle (iv) is not fulfilled). The distribution of T in YPV3 space is shown in Fig. 5(a). It is seen that the principle (iii) of a monotonic relationship between thermo-chemical variables and YPV is fulfilled, which means that the thermo-chemical quantities in the flamelet library can be unambiguously identified in the YPV direction. However, it can be also observed that the minimum value of YPV3 is larger than 0, which means that an additional extrapolation method is required to cover the full range of YPV . On the other hand, the definition of YPV2 can fulfill the principle (iv) with only major reaction product being included. However, in this case, YPV does not necessarily have a monotonic relationship with T, especially in fuel-rich conditions, as indicated by the red circle in Fig. 5(b). Further, the profiles of T are close to each other in the YPV2 space for the fuel-rich flamelets, indicating that T is not sensitive to the change of equivalence ratio. Thus, it is essential to investigate how the definition of YPV affects the performance of the flamelet models. Note that for gaseous combustion, Ihme et al. [31], Niu et al. [32] and Prüfert et al. [33] adopted optimization methods to formulate the reaction progress variable with

the property of monotonicity (principle (iii)). However, it is difficult to extend these advanced methods to pulverized coal combustion to also ensure that the reaction progress variable still describes the “progress of reactions” (principle (iv)). To illustrate the relation between the reaction progress variable and the thermo-chemical quantities further, the profiles of YCO2 and YCO are shown in the different reaction progress variable spaces for the premixed flamelets, see Fig. 6. It is seen that YCO2 shows a monotonic relationship with YPV3 while the profiles of YCO2 overlap in the YPV2 space for fuel-rich flamelets, as indicated by the red circle and the inset figure. The results also show that YCO2 varies rapidly at large values of YPV3 for the fuel-lean flamelet, which indicates that a small deviation of YPV3 may result in large errors of YCO2 . Furthermore, the extrapolation of YPV3 towards the lower limit of 0 is required, which may induce additional interpolation errors. For YPV2 , the profiles of YCO2 overlap in the fuel-rich conditions. Note that for the pulverized coal flame studied in this work, the maximum equivalence ratio is only around 2.5. Thus, it is expected that the overlapping of YCO2 in YPV2 space does not have significant effects on the performance of the flamelet models. For YCO shown in the bottom row of Fig. 6, the one-to-one relation with YPV2 and YPV3 can be observed in most of the regions. Further, it is seen that the profiles of YCO spread in a large region, especially for the fuel-lean flamelets, which indicates that YCO is sensitive to the change of equivalence ratio. 3. Results and discussion This section first investigates the validity of the extended premixed and non-premixed flamelet models, taking into account

X. Wen, M. Rieth and A. Scholtissek et al. / Combustion and Flame 216 (2020) 453–467

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strong heat losses. Then, the performance of the multi-mode flamelet model is evaluated by comparison with the classical premixed and non-premixed flamelet models. Finally, the effects of the reaction progress variable formulation on the performance of the flamelet model are evaluated. Three sample lines with different fractions of premixed combustion mode, i.e., SAM1, SAM2 and SAM3 shown in Fig. 1(b), are investigated to evaluate the performance of the different flamelet models. 3.1. A priori analysis of the extended premixed and non-premixed flamelet models Figure 7 compares the gas temperature, T, from different flamelet tabulation methods with the reference DNS results in different solution spaces. The composition spaces of Z and YPV are selected to show the flame structures since they are the trajectory variables in the flamelet library. It will be shown below that a negligible difference/improvement in physical space may become significant in composition space. Thus, the flame structures in different solution spaces are compared to show the performance of the different flamelet models. The sample location SAM1 shown in Fig. 1(b) is investigated. The background color in the first row indicates the combustion mode, while the profiles of RI are also shown in the composition spaces to illustrate the effects of combustion mode on the performance of the different flamelet models. In the left column, the gas temperature is obtained with different non-premixed flamelet models, while in the right column, it is obtained with different premixed flamelet models. For both premixed and non-premixed flamelet models, different types of flamelet tables are evaluated. The original adiabatic flamelet tables (see Fig. 2)

are denoted as “DFTadi ” and “PFTadi ”, the original adiabatic flamelet tables coupled with a linear extrapolation method are denoted as “DFText ” and “PFText ”, and the coupled adiabatic and non-adiabatic flamelet tables presented in this work are denoted as “DFTnew ” and “PFTnew ”. Note that the linear extrapolation method is only applied to the gas temperature, while the species mass fractions are collapsed to the adiabatic flamelet at the enthalpy limits,

⎧ H − He (X, Z, 0, YPV ) ⎪ ⎨T (X, Z, 0, YPV ) + e for He < He (X, Z, 0, YPV ), c p,g (X, Z, 0, YPV ) T = H − H X, Z, 1 , Y ( ) e e PV ⎪ ⎩T (X, Z, 1, YPV ) + for He > He (X, Z, 1, YPV ). c p,g (X, Z, 1, YPV )

(19) This artificial treatment is introduced because extremely low enthalpy states do not exist in the adiabatic flamelet library. Note that the gas temperature and the species mass fractions have strong non-linear relations, and a linear extrapolation for all thermo-chemical quantities does not fulfill the perfect gas assumption, which may lead to stability problems in the simulations [52]. Note also that in the a posterori simulations, the density needs to be calculated according to the extrapolated gas temperature using the ideal gas law, which ensures the stability of the simulations. This method has been evaluated in previous works [26,52] and reasonable results have been obtained. For different non-premixed flamelet tabulation methods, it can be observed that in the physical space, the gas temperature is over-predicted by “DFTadi ” between y = 8 mm and y = 12 mm where the cold coal particles in the upper stream are ignited by the lean hot products in the lower stream. The gas temperature profiles predicted with “DFText ” and “DFTnew ” are indistinguishable in the physical space. How-

X. Wen, M. Rieth and A. Scholtissek et al. / Combustion and Flame 216 (2020) 453–467

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ever, in the mixture fraction and reaction progress variable (YPV = YPV 2 ) spaces, a slight improvement can be observed when comparing the gas temperature predicted with “DFTnew ” and “DFText ”, which means that the impact of strong heat losses on the thermochemical variables can be reasonably represented by the newly proposed flamelet model. For the different premixed flamelet tabulation methods shown in the right column of Fig. 7, similar results are obtained as for the non-premixed flamelet models, i.e., the gas temperature is over-predicted by the original adiabatic premixed flamelet model, while the predicted results for “PFText ” and “PFTnew ” are similar overall. To evaluate the different flamelet tabulation methods further, another sample line is investigated along the y direction at (x = 0.75Lx , z = 0.5Lz ) and t = 10 ms. At this location, the fraction of the premixed flame reaction is significant (see SAM2 in Fig. 1(b)). The gas temperature predicted with different flamelet models is shown in Fig. 8. Again, the profiles of RI are presented to show the effects of the combustion mode on the performance of the flamelet models. It can be observed that for both the premixed and non-premixed flamelet models, the gas temperature is significantly over-predicted by the adiabatic flamelet models of “DFTadi ” and “PFTadi ” between y = 9 mm and y = 12 mm, where the heat transfer between the coal particle phase and gas phase is significant. Again, in the physical space, the linear extrapolation method seems to work for the gas temperature prediction. However, in the composition space, shown in the last two rows of Fig. 8, the newly proposed flamelet models for “DFTnew ” and “PFTnew ” perform better than those simply employing an extrapolation method, especially in the mixture fraction space. Note that the small gas temperature deviations in the a priori analysis may add up enormously when running a posteriori.

The major reaction product H2 O predicted with different flamelet tabulation methods is compared with the reference DNS results in Fig. 9. The sampled data is again extracted at SAM2. As noted earlier, the species mass fractions are collapsed to the adiabatic flamelets with the lowest enthalpy if the enthalpy in the flow field is lower than the lowest enthalpy in the flamelet table. Thus, the profiles of species mass fractions predicted with “DFTadi ” and “DFText ”, and “PFTadi ” and “PFText ” overlap. From the left column of Fig. 9, it can be observed that YH2 O is slightly over-predicted by the “DFTadi ” and “DFText ” methods in certain regions where the interphase heat transfer is significant, which corresponds to the location where the gas temperature is over-predicted (see Fig. 8). On the other hand, it is seen that the profiles of YH2 O predicted with “DFTnew ” follow the reference results closely in all solution spaces. In the right column of Fig. 9, YH2 O predicted with different premixed flamelet tabulation methods is compared with the DNS results. It is seen that the profiles of the predicted YH2 O largely overlap. The over-prediction of gas temperature does not result in an obvious over-prediction of YH2 O . The observed slight underprediction of YH2 O in Z space is considered to be related to the inaccurate representation of the local combustion mode. As indicated by the superimposed profiles of RI, the value of RI is less than 0.5 in the region where YH2 O is under-predicted. Thus, it is expected that in this region the non-premixed combustion mode is dominant, and the H2 O mass fraction cannot be accurately calculated by the employed premixed-flame-based flamelet model. Finally, the intermediate species CO predicted with different flamelet tabulation methods is compared with the DNS results in Fig. 10. As reported in previous works [19,20,48,53], the prediction of YCO and YH2 is sensitive to the underlying flamelets, i.e. whether it is a 1D premixed flamelet or 1D non-premixed flamelet, for

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Fig. 11. Comparisons of the CO and H2 mass fraction profiles in different solution spaces, contrasting the DNS results and the a priori profiles for the DFT, PFT and MMFT models along the y direction at (x = 0.75Lx , z = 0.5Lz ) and t = 10 ms. The sample location corresponds to SAM2 in Fig. 1(b). The background color has the same meaning as in Fig. 7.

gaseous, spray and pulverized coal combustion. Thus, these species mass fractions are particularly suitable candidates to evaluate the performance of premixed, non-premixed, and multi-mode flamelet models, which is the focus of the next subsection. The background color and the profiles of RI are shown to indicate the local combustion mode in different solution spaces. In the left column of Fig. 10, the performance of different non-premixed flamelet models is evaluated when predicting YCO . It can be observed that compared to the adiabatic “DFTadi ” and “DFText ” models, the peak value of YCO can be better represented by the newly developed “DFTnew ” model in both premixed and non-premixed flame reaction zones. The over-prediction of YCO with “DFTadi ” and “DFText ” is considered to be due to the fact that the heat loss effects are not properly represented (see the gas temperature profile in Fig. 8). In the right column of Fig. 10, YCO predicted with different premixed flamelet models is compared with the DNS results. It is seen that the profiles of YCO basically overlap with each other, and only a slight improvement can be observed for the newly developed “PFTnew ” model. The over-prediction of YCO in the non-premixed flame reaction zone by all premixed flamelet models is attributed to the fact that the interactions between iso-equivalence ratio surfaces cannot be represented since the premixed flamelet library is generated based on the 1D freely propagating premixed flames for separate equivalence ratios. Note that the peak value of YCO in the premixed flame reaction zone is correctly predicted by all premixed flamelet models. In summary, the newly proposed “DFTnew ” and “PFTnew ” methods which take strong heat loss effects into account perform better than the adiabatic flamelet models with or without the extrapola-

tion method, especially when predicting the gas temperature and intermediate species. Comparisons in Z and YPV spaces show that small deviations/improvements in the flamelet predictions in the physical space can become significant in the composition space, especially for intermediate species. 3.2. A priori analysis of the multi-mode flamelet model In the previous subsection, it was demonstrated that the combustion-mode-sensitive species such as CO cannot be correctly predicted by standard premixed or non-premixed flamelet models, even if the interphase heat transfer is already correctly represented. It is interesting to investigate whether the multi-mode flamelet tabulation (“MMFTnew ”) method proposed in Section 2.3 can improve the predictions of the combustion-modesensitive species. In Fig. 11, CO and H2 mass fractions predicted with the premixed, non-premixed and multi-mode flamelet models are compared with the reference DNS results. The sample data is extracted along the y direction at (x = 0.75Lx , z = 0.5Lz ) where the fraction of premixed flame is significant (see SAM2 in Fig. 1(b)). The background color in the first row indicates the range of RI, while the exact value of RI is shown as the green lines in different solution spaces. For YCO , it can be observed that “MMFTnew ” and “DFTnew ” perform better overall than “PFTnew ”, especially in the non-premixed-flame-dominated reaction zone. The peak value of YCO is slightly over-predicted by “MMFTnew ” in the non-premixedflame-dominated reaction zone (RI < 0.5). This is considered mainly to be related to the definition of the combustion mode in-

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Fig. 12. Comparisons of the CO and H2 mass fraction profiles, contrasting the DNS results and the a priori profiles for the DFT, PFT and MMFT models along the y direction at (x = 0.25Lx , z = 0.5Lz ) and t = 10 ms. The sample location corresponds to SAM3 in Fig. 1(b). The background color has the same meaning as in Fig. 7.

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Fig. 13. Comparison of the profiles of (a) C2 H4 and O2 mass fractions YC2 H4 , YO2 ; and (b) CO2 and H2 O mass fractions YCO2 , YH2 O , contrasting the DNS results and the a priori profiles for non-premixed flamelet tabulation (DFT) method obtained from different reaction progress variables along the y direction at (x = 0.5Lx , z = 0.5Lz ) and t = 10 ms. The sample location corresponds to SAM1 in Fig. 1(b).

dex. In the range between y = 8 mm and y = 11 mm, it is seen that the pure non-premixed flamelet model “DFTnew ” performs better than “MMFTnew ”. Thus, the over-prediction of YCO is mainly related to the positive RI value, which quantifies the portion of each combustion mode. For a similar coupled flamelet model, Kundsen and Pitsch [48] noted that the effects of the linear weighting method on the flamelet predictions are relatively small compared to the definition of the combustion mode index. In fact, Fiorina et al. [47] noted that the combustion mode index defined depending on the gradients of fuel and oxidizer breaks down in regions where the gradients of fuel and oxidizer are aligned, but combustion is still diffusion-controlled. Furthermore, the combustion mode index is defined in the 1D limit, which may introduce errors for the 3D turbulent pulverized coal flame studied here. To improve the prediction capability of the “MMFTnew ” method, a more appropriate combustion mode index should be formulated in future works. In the right column of Fig. 11, YH2 is compared for the flamelet predictions and the DNS results. Similarly to the results for YCO , YH2 is slightly over-predicted by “MMFTnew ” in the non-premixed-flamedominated reaction zone at around y = 10 mm. At this location, YH2 extracted from the non-premixed flamelet library agree well with the DNS results. Thus, the combustion mode is expected to be closer to the pure non-premixed combustion mode at this location. The over-prediction of RI results in the over-prediction of YH2 when weighted with the significantly over-predicted “PFTnew ” method. Comparisons in Z and YPV spaces show that the over-prediction of YCO and YH2 with “PFTnew ” and “MMFTnew ” happens around the position of peak Z and YPV . To further evaluate the performance of the proposed “MMFTnew ” model, another sample line at SAM3 (see Fig. 1(b)) is investigated where the fraction of premixed flame is minor. YCO and YH2 obtained from different flamelet models are compared to the DNS results in the physical space, as shown in Fig. 12. It can

be observed that outside the middle non-premixed flame reaction zone at around y = 10 mm, YCO and YH2 from all flamelet tables agree well with the reference results. However, in the middle non-premixed flame reaction zone, only “DFTnew ” gives correct predictions, while “PFTnew ” and “MMFTnew ” over-predict YCO and YH2 . As explained earlier, the over-prediction of “MMFTnew ” is considered to be due to the incorrect calculation of the combustion mode index, whereas the over-prediction of “PFTnew ” is associated with the inaccurate representation of the interactions between iso-mixture-fraction surfaces. Note that in the partially premixed flame, the thermo-chemical and fluid dynamic interactions, such as the heat transfer and radical exchange, between iso-mixture-fraction surfaces are expected [54,55]. Considering that the premixed flamelet library is generated based on the 1D freely propagating premixed flame at separate equivalence ratios, the interactions between different iso-mixture-fraction surfaces cannot be described by the employed premixed-flame-based flamelet model. Thus, the inaccurate prediction of the CO and H2 mass fractions in the non-premixed flame dominated reaction zone can be attributed to the missing of the interactions between different iso-mixture-fraction surfaces in the premixed flamelet library. 3.3. Effects of reaction progress variable definition In this section, the effects of reaction progress variable definition on the flamelet predictions are evaluated. As described in Section 2.4, when YPV is solely formulated to describe the progress of reaction, it may not have the property of monotonicity (see Fig. 5(b)), while YPV with monotonicity may not necessarily describe the reaction progress (see Fig. 5(a)). It is thus interesting to investigate whether the thermo-chemical quantities can be correctly predicted using different definitions of YPV . Figure 13 com-

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pares the a priori flamelet predictions with the reference DNS results. Only the results from the non-premixed flamelet tables are presented; those from premixed flamelet tables are similar. The only difference between the flamelet tables is the formulation of YPV (see Eq. (18)). The sample data is extracted along the y direction at (x = 0.5Lx , z = 0.5Lz ) and t = 10 ms, which corresponds to SAM1 in Fig. 1(b). It can be observed that for the mass fractions of C2 H4 , O2 and H2 O, all flamelet predictions agree well with the DNS results, with only negligible differences. However, for the CO2 mass fraction, flamelet predictions using YPV3 significantly deviate from the reference results in the range between y = 4 mm and y = 12 mm, while those employing other YPV definitions are similar and agree well with the reference results. Since YCO2 makes a large contribution to YPV3 , the deviation in the prediction of YCO2 may be related to the extrapolation of YPV3 towards the YPV = 0 side. Overall, it can be concluded that most species mass fractions are not sensitive to the formulation of YPV , and similar predictions can be obtained. However, specific species mass fractions may be incorrectly predicted for specific formulations of YPV . Thus, in Sections 3.1 and 3.2, YPV2 (i.e., sum of YCO2 and YH2 O ) is utilized in both premixed and non-premixed flamelet tables. This definition can not only give correct predictions of species mass fractions, but can also be used to describe the progress of reactions. 4. Conclusions This work has addressed specific and recurring issues in flamelet modeling for pulverized coal combustion, including strong heat losses, multi-mode combustion and reaction progress variable definition. Particularly, extended flamelet formulations are developed that can take into account strong heat loss effects in pulverized coal combustion. For the coexistence of multicombustion modes in pulverized coal flames, a multi-mode flamelet model is proposed. The effects of reaction progress variable definition on the flamelet model’s performance are quantified. The tabulated thermo-chemical quantities obtained with the newly developed flamelet models are compared with those obtained with conventional adiabatic and single-mode flamelet models and the reference DNS results. Comparisons show that the newly developed non-adiabatic flamelet models perform better than the adiabatic flamelet models with or without an extrapolation method for the prediction of gas temperature and species mass fractions. The gas temperature and intermediate species mass fractions are largely over-predicted by the adiabatic flamelet models in regions where the interphase heat transfer is significant. The results also show that, compared to the classical premixed and non-premixed flamelet models, the proposed multi-mode flamelet model cannot improve the predictions in the entire combustion field, which is considered to be related to the adopted combustion mode index. The multi-mode flamelet model over-predicts the CO and H2 mass fractions in reaction zones that are predominantly non-premixed. The combustion mode index should be reformulated in future works to further improve the prediction capability of the coupled premixed and non-premixed flamelet models. The investigation of the effects of reaction progress variable definition shows that while the predictions of most of species mass fractions are similar overall for different formulations of the reaction progress variable, some species mass fractions (e.g. YCO2 ) can not be predicted accurately for specific formulations of the reaction progress variable. Declaration of Competing Interest The authors have no conflicts of interest to declare.

Acknowledgments The authors acknowledge the financial support by the German Research Foundation (DFG) for the collaborative project “MultiDimensional Flamelet Modeling for LES of Pulverized Coal Flames (project number 238057103).” Part of the work was funded by the DFG (project number 215035359 – TRR 129). Xu Wen and Christian Hasse are grateful for the fruitful discussions with Luc Vervisch during his stay as a Mercator Fellow at TU Darmstadt.

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