A methodology for soot prediction including thermal radiation in complex industrial burners Guillaume Lecocqa , Damien Poitoua , Ignacio Hern´andeza , Florent Duchainea , Eleonore Ribera , B´en´edicte Cuenota a
CERFACS, CFD Team, 42 Avenue G. Coriolis, 31057 Toulouse Cedex 01, France
Abstract This paper proposes a method for modeling soot when performing Large Eddy Simulation of complex geometries. To obtain a good trade-off between CPU cost and accuracy, soot chemistry is included via a tabulated flamelet approach, combined to a turbulent combustion model for Large Eddy Simulation based on a simplified description of chemistry. A semi-empirical soot model is chosen and validated on laminar premixed and counterflow diffusion flames. A proposed procedure enables to calculate radiation with a Discrete Ordinates Method approach and optimized spectral models. The developed soot model is applied to a real configuration, being the combustion chamber of a helicopter engine. To evaluate the importance of radiative heat losses, two cases are studied, using either adiabatic conditions or accounting for radiative heat gains/loss. Keywords: Turbulent combustion, Large Eddy Simulation, soot modeling, mixed reduced-tabulated chemistry, radiative transfer, coupling
Email address:
[email protected] (Guillaume Lecocq)
Preprint submitted to Combustion and Flame
January 7, 2014
1. Introduction Soot prediction remains nowadays both a necessity and a challenge. It is a necessity because soot particles have a non negligible impact on environment at several scales. They are known to be carcinogen for human beings [1]. They are also suspected to trigger the formation of contrails [2], whose long term presence impacts the local climate. Soot also has an impact, either positive or negative, on combustion and burners. If soot enhances heat fluxes in furnaces [3], improving the industrial process, their presence in the combustion chamber of an aircraft engine affects the burnt gas temperature, changing the heat balance of the system. This may on the one hand decrease the engine life expectancy, and on the other hand increase pollutant emissions such as NOx, both key issues for engine manufacturers. Soot production is the result of a complex process including homogeneous gaseous chemistry and heterogeneous chemistry on the soot particle surface. These chemical processes are very sensitive to temperature, and in order to predict soot with accuracy, one should account for the following effects: • flame temperature: the maximum soot volume fraction does not linearly increase with the burnt gas temperature, but follows a bell-shaped curve [4]. The explanation comes from the chemistry of the Polycyclic Aromatic Hydrocarbons (PAHs), which is promoted by an increasing temperature until it reaches a threshold above which PAHs are thermally decomposed. • thermal radiation: burnt gases and soot particles are responsible for radiative heat losses, which induces temperature changes and in turn modifications of the chemical and soot kinetics. In addition pressure and turbulence also play both a role in soot formation. The soot volume fraction increases with pressure, which is partially explained by an increase in 2
C2 H2 concentration [5]. Intense turbulence, characterized by short time scales, may alter the relatively slow formation of PAHs [6], limiting the subsquent nucleation phase. Finally, the chemical and thermodynamical processes of soot formation require the knowledge of some key species: C2 H2 (growth), PAHs (nucleation, condensation), plus O2 and OH for oxidation. Such species are however not present in the reduced chemistries often used in Large Eddy Simulation (LES) to compute the flame propagation in complex systems [7]. Soot formation was first numerically investigated in laminar premixed and counterflow diffusion flames [8, 9, 10, 11]. In most premixed flame studies, the thermal problem was not solved and the experimental temperature profile was prescribed. Diffusion flame studies mostly relied on a simplified radiation modeling [9], but dynamic coupling with thermal radiation was also used [12]. The production of soot in turbulent flames has been mainly simulated with a Reynolds-Averaged Navier-Stokes (RANS) approach. Recent works report LES of sooting flames [13, 14], even for real engineering devices [15]. Methods were proposed to couple LES with detailed radiation modeling. With such method, the thermal impact of soot was quantified in a complex geometry with intense turbulence by Amaya et al. [16]. The goal of this paper is to propose a method for modeling turbulent sooting flames in engineering devices. For such systems, LES appears a more and more attractive approach as illustrated in [17, 18]. Such computations are associated to a high CPU cost, and to minimize the added cost of combustion, LES usually uses reduced or tabulated chemistry for the gas. This also holds for soot chemistry, still keeping access to the key species mentioned above. In the present work, it is proposed to combine the reactive flow computation based on reduced chemistry [7] and a classical turbulent combustion model [19], with the tabulation of a detailed 3
chemistry including soot chemical paths. The tabulation is used to recover the necessary key species concentrations for the soot production, calculated with a semiempirical model [20] which offers a good trade-off between CPU cost, robustness and accuracy. The present document is organised in four sections. Section 2 presents the equations describing sooting flames with radiative loss in laminar and turbulent flows. Section 3 details numerical aspects and the coupling methodologies. Results are presented in Sections 4 and 5. Laminar cases are first shown, with the objective of validating the proposed model. Then the model is applied to the turbulent flame developing in the combustion chamber of a helicopter engine, to predict soot emission in both adiabatic and non-adiabatic cases. 2. Soot modeling in non-adiabatic turbulent flames 2.1. Model equations The conservation equations describing a reactive flow may be written in a matrix form: ∂w +∇·F=s ∂t
(1)
where w = (ρv1 , ρv2 , ρv3 , ρE, ρYk )T is the vector of conservative variables that is solved at each location x and time t. In this vector, ρ is the mixture density, v1 , v2 , v3 are the components of the velocity vector v, E is the total energy and Yk is the mass fraction of species k (1 ≤ k ≤ Nspecies ). The flux tensor F can be decomposed into an inviscid (with superscript I) and a viscous component (with superscript V ):
4
FI =
v : ρvT + P
(ρE + P )v ρ k Yk v
(2)
and
−τ
FV = −vT · τ + q Jk
(3)
using the hydrodynamic pressure P defined by the equation of state of perfect gas. In Eq. (3), q is the heat flux and Jk is the diffusive flux of species k. The stress tensor for a Newtonian fluid τ = [τij ] is: 1 τij = 2µ Sij − δij Sll with 3
1 Sij = 2
∂vj ∂vi + ∂xi ∂xj
(4)
where µ is the dynamic viscosity. For a reacting flow including radiation, the source term s is written: s = (0, 0, 0, ω˙ T + Sr , ω˙ k )T
(5)
where ω˙ T is the chemical heat release, Sr is the thermal radiative heat source and ω˙ k is the reaction rate for species k. 2.2. Description of chemistry Combustion chemistry can be described through detailed chemical mechanisms, involving hundreds of species and thousands of elementary reactions, or reduced chemistry, characterized by a few species and reactions [7] and optimally tuned to reproduce correct laminar flame speed and burnt gas temperature, for a given range of equivalence ratios and thermodynamic conditions. A reduced reaction scheme is not necessarily designed to both reproduce these trends and accurately predict intermediate species. To keep realistic chemical behaviors in LES computations, the technique of tabulated flamelets [21, 22] may be used, like for example the Flame Prolongation of Intrisic Low-Dimensional Manifold (FPI) approach [23] approach [24]. Such method 5
consists in computing a priori a set of laminar premixed flames with detailed chemistry, for a range of equivalence ratios (or mixture fraction Z). A table is then generated in which all physical quantities ϕ, being mass fractions, temperature, reaction rates, etc. are reported as functions of ϕ(Z, c) where c is the progress variable. A variety of models exists in the literature, depending on the definition and computation of c. Any monotonic variable varying from 0 in the fresh gases to 1 in the burnt gases may be used as a progress variable. Usually, c is a combination of species, for which a conservation equation can be built and solved with appropriate closure models in turbulent flows.
The originality of this work is to reconstruct c from solutions obtained with reduced chemistry, in such a way that c matches the progress variable calculated with complex chemistry. To this end, two different approaches are proposed and explained in the following. Definition of progress variable A progress variable has to be defined to use the tabluated flamelet technique to predict intermediate species. This variable may be defined from the fuel (lean combustion) or oxidizer (rich combustion) mass fractions as: clean = 1 −
YF YFo
and
crich = 1 −
YO2 YOo2
(6)
where YFo , YOo2 are the mass fractions of respectively fuel and the O2 in the fresh unburnt gases. By definition, clean and crich cannot describe non-premixed combustion. Both variables are well suited for reduced chemistry, as they will keep a monotonic behavior. This however is not guaranteed with detailed chemistry, and another definition is preferred, based on CO and CO2 mass fractions. This definition is often
6
used in tabulation methods [25], and writes: ctab =
YCO + YCO2 Yc = tab−eq tab−eq (YCO + YCO2 ) Yc
(7)
where the superscript tab − eq denotes tabulated thermochemical equilibrium values. As ctab is based on product species, it is 0 in the fresh gases and reaches 1 at equilibrium in the burnt gases.
Alternatively, the progress variable may be defined as the solution of a transport equation [25]: ∂ρctr ∂ ∂ + ρvi ctr = ∂t ∂xi ∂xi
∂ctr ρD + ρω˙ ctr ∂xi
(8)
associated to boundary conditions of 0 in the fresh gas and 1 in the burnt gas. The reaction rate appearing in Eq. (8) defines the progress variable. For example, taking is the progress variable source term from tabulated ω˙ ctr = ω˙ ctab /(Yc )tab−eq where ω˙ Ytab c chemistry, leads to the progress variable ctab defined in Eq. (7). In the present work, combustion chemistry of the LES is modeled with a reduced scheme. Using Eq. (6), a progress variable can be defined. Nevertheless, it is expected the thickness of the post-flame zone, where realistic kinetics slows down, would be underestimated. In order to localize the flame front with reduced chemistry variables and to reconstruct a realistic gradient of the progress variable accross the flame front, it is chosen to build the reaction rate ω˙ ctr as: • if ctr < c∗ then ω˙ ctr = ω˙ cred (ctr ), • else ω˙ ctr = ω˙ ctab (ctr ) where ω˙ cred is the progress variable source term from reduced chemistry and ω˙ Ytab is c the progress variable source term from tabulated chemistry, both taken at the value ctr . 7
The value of c∗ indicates the value of ctr at which the chemical model switches from reduced to tabulated chemistry, ie. at which the definition of the progress variable switches from clean(rich) to ctab . As tabulated chemistry is introduced here to capture soot production in the post-flame zone, c∗ is chosen so as to coincide with the flame / post-flame transition zone. In practice, any value between 0.1 and 0.8 can be chosen and a value of 0.5 is taken for the present simulations. Finally, as clean(rich) is known from the fuel (oxidizer) mass fractions, the quantity ω˙ cred is reconstructed as: ω˙ cred = (clean(rich) − ctr )/τ ∗ , τ ∗ being of the order of few simulation time steps. Such relaxation formulation has already proved to be usefull [26] and guarantees that ctr tends smoothly to clean(rich) in the reaction zone, without introducing significant additional time scales. All previous definitions of the progress variable assume adiabatic flames. In case of heat losses (through thermal radiation for example), non-adiabatic flames should be considered for the tabulation, introducing a new parameter in the table. Previous works have used such approach, tabulating a set of flames with different total enthalpies or different fresh gas temperatures to compute one-dimensional laminar burner-stabilized flames [27, 28]. In this work, the second technique is used, tabulating laminar adiabatic premixed flames with various fresh gas temperatures. The fresh gas temperature field can be retrieved from the solution energy field. At each location, the total enthalpy is linked to energy through the relation: 1 P ht = et − vi2 + 2 ρ
(9)
Conservation of enthalpy through the flame (pressure variation and differential diffusion effects being negligible) then allows to calculate the corresponding fresh gas temperature.
8
2.3. Soot modeling Modeling soot often relies on complex chemical mechanisms for gaseous chemistry coupled to a specific heterogeneous chemistry model for soot. Semi-empirical models, that are simplified but contain phenomenological source terms [29, 20, 30] mainly differ by the description of the inception phase, and in particular the choice of the precursor species. Moss et al. [31] for example choose the fuel as soot precursor, while Leung et al. [20] use C2 H2 . A more sophisticated approach is found in Di Domenico et al. [30], where a sectional method is used to model both the PAHs and the collision of the largest size PAHs leading to nucleation. Increasing model complexity then leads to sectional methods [32], stochastic approaches [33] or methods of moments [34]. The latter was already used in a LES framework for an academic case [13]. The semi-empirical model of Leung et al. is retained here. It requires the resolution of two equations only for the local soot mass fraction, Ys , and the number density of soot particles, ns , while the method of moments needs 4 to 6 equations and sectional approaches between 20 and 100. Despite their key role described previously, PAHs are neglected as no convincing simple model is available to include them. In the laminar regime, the soot model equations read: ∂ρns ∂ ρνns ∂T ∂ + (ρvi ns ) = kT ∂t ∂xi ∂xi T ∂xi 2 NA k1 (T )[C2 H2 ] + Cmin 1/6 1/2 1/6 6Ms 6κT ρYs 11/6 − 2Ca (ρn) (10) πρs ρs Ms ∂ρYs ∂ ∂ ρνYs ∂T + (ρvi Ys ) = kT ∂t ∂xi ∂xi T ∂xi + k1 (T )[C2 H2 ]Ms + k2 (T )[C2 H2 ]f (S)Ms − k3 (T )[O2 ] + k4 (T )XOH SMs (11)
9
where [X] is the concentration of species X in kmol.m−3 , Ms and ρs are the soot molar mass and density. NA and κ are Avogadro and Boltzmann constants. The rate constants Ai , ni and Ti are given in Table 2 for all source terms. The first term appearing on the right-hand side (RHS) of Eqs. (10) and (11) models the thermophoretic transport, where the constant kT is set to 0.54. The second RHS term in both equations accounts for soot particles nucleation. Soot growth rate is modeled by the third RHS term in Eq. (11). It involves a function f (S) where S is the soot volume surface per unit volume of gas: S = ns πd2s 1/3 6 ρYs with ds = π ρ s ns
(12) (13)
where ds is the mean soot particle size. The function was basically proposed to integrate aging effects that lead to a decay of soot surface reactivity. As proposed by Leung et al., f (S) = S 1/2 . Oxidation is accounted for with the last term of Eq. (11). The contribution of OH radicals was added to the original model, with corresponding constants coming from the work of Guo et al. [9]. The last term in Eq. (10) models coagulation, ie. the formation of larger soot particles. The soot surface growth rate pre-exponential factor of the original model of Leung et al. has been adjusted in this work and is different for the diffusion flames and the premixed flames to match experimental results. The whole parametrization is supplied in Table 2, and Ca = 9.0, ρs =2000 kg.m−3 and Ms =12.011 kg.kmol−1 . Note that the contribution of molecular diffusion to the mass flux was implicitly neglected in the transport equations of the soot moments. This is commonly assumed when modeling soot, due to the high Schmidt numbers of the soot particles.
It was recently shown this hypothesis 10
increased differential effects between soot and gaseous species, promoting soot spreading throughout mixture fraction space for nonpremixed flames [6]. 2.4. Thermal Radiation Modeling The radiative source term Sr in the energy balance equation (Eq. (5)) represents thermal radiative exchanges in the infrared spectrum. The physics of radiation imposes to solve non-local exchanges due to the propagation of photons at the speed of light. In gaseous media, the spectral properties are very complex and must be correctly modeled. Previous works on combustion have considered simplified models for radiation [35], considering absorption only: Sr ' 4σκp T 4 , or the optically thin approxima4 ). In both cases the spectral properties of gases are described tion: Sr ' 4σκp (T 4 −TW
with an absorption coefficient κp . To better account for the absorption of gases, it is necessary to solve the Radiative Transfer Equation (RTE) with a spectral model for absorption properties. Such radiation calculation has been recently coupled to LES of turbulent combustion [36, 37, 38]. These studies showed the importance of using a sufficiently accurate radiation model, based on the RTE and optimized spectral model. The RTE is given in its differential form (Eq. (14)) in the direction of propagation Ω, for a non scattering medium, with the associated boundary conditions (Eq (15)): Ω · ∇Lν (x, u) = κν L0ν (x) − Lν (x, u)
(14)
Lν (xw , u) = ν (xw )L0ν (xw ) + ρν (xw )Lν,incident (xw , u) | {z } | {z } Emitted part
(15)
Reflected part
where ν is the wavenumber, Lν (x, u) is the radiation intensity at point x in the direction u, and κν is the absorption coefficient, ν (xw ) is the wall emissivity and 11
ρν (xw ) is the wall reflectivity with ρν (xw ) = 1 − ν (xw ). L0ν is the equilibrium Planck function. The absorption coefficient κν is the sum of the contributions of the gaseous mixture and soot: κν = κgas,ν + κsoot,ν
(16)
Approaches to model the absorption coefficient of gases may be listed from the highest complexity (and CPU cost) to the simplest model: line-by-line [39], narrow bands [40, 41] or global (WSGG [42], SNB-FSK [43], SNB-FSCK [44]). In this work, global models are used. For academic configurations, the absorption coefficient is modeled with the detailed spectral model SNB-CK. When coupled with LES, the radiative source term computation has to respect a trade-off between accuracy and CPU cost. The tabulated SNB-FSK approach is chosen. The absorption of soot is calculated using the correlation of Liu et al. [45]: κν,soot = 5.5fv ν
(17)
where fv is the volume fraction of soot. The source term Sr results from a double integration of the RTE over the solid angle and the gas spectrum frequencies, and depends only on the position x: Z ∞ Z 0 Sr (x) = κν 4πLν (x) − Lν (x, u)dΩ dν 0
(18)
4π
2.5. Turbulence and turbulent combustion modeling A filtering is applied to Eqs. (1), (8), (10) and (11) to derive the filtered balance equations for LES (Eq. (19)), assuming a commutation between the filter and derivative operator [46]: ∂w ¯ ¯ =¯ +∇·F s ∂t 12
(19)
where the vector of conservative variables w¯ now includes ctr , ns and Ys . The conservative variables being weighted by density, the Favre average is introduced as ˜ = ρΦ/¯ ¯ ρ for all flow primitive variables. The filtered flux tensor F ¯ contains a reΦ solved part, expressed by Eqs. (2) and (3) using filtered variables, and an unresolved part which is modeled in the form of a subgrid flux tensor Ft :
t
vi vj − vei vej ) (21) τijt = − ρ(g
−τ qt t Jk t F = Jtc t Jns t JY s
(20) where
qit
=
e ρ(vg ei E) iE − v
t Ji,k
=
ρ(vg ei Yek ) (23) i Yk − v
t Ji,c
=
tr tr ) (24) ρ(vg ei cf ic − v
t Ji,ns
=
ρ(vg ei nes ) (25) i ns − v
t Ji,Y s
=
ρ(vg ei Yes ) (26) i Ys − v
(22)
Note that unresolved diffusive and thermophoretic transports are neglected. The subgrid turbulent stress tensor (Eq. (21)) is modeled with the turbulent viscosity concept using the Smagorinsky model [47]. Turbulent fluxes for thermal, species, progress variable and soot-related quantities diffusion (Eqs. (2226)) are modeled by classical gradient laws with turbulent Schmidt and Prandtl numbers. This is also applied to the soot model equations, where the nonresolved transport terms are closed with the same turbulent Schmidt number. The turbulent combustion source term is described using the dynamic Thickened Flame Model (TFLES), which artificially thickens the flame front in order to solve stiff gradients on the grid without altering global laminar flame characteristics. To account for subgrid flame wrinkling, an efficiency function is also introduced that leads to the correct turbulent flame speed. This model is detailed in [19] and has
13
been extensively used and validated in numerous configurations [48, 49, 50]. All other flow source terms, ie. Sr and ω˙ c , and all source terms for the soot model equations, are directly computed using the resolved filtered variables. In other words subgrid effects on these source terms are neglected. It has been shown in [51], also confirmed in [52] that this is reasonable for radiation. Concerning the progress variable source term, used in the post-flame region, it is justified by the low reaction rates there. It is much more critical to neglect subgrid effects in the soot model equations, as small scale turbulence certainly interacts with the soot formation process. It was shown by DNS [6] that turbulence could limit PAH formation and soot nucleation phase. Thus, choosing a PAH as a precursor species instead of C2 H2 would not be sufficient to account for this effect. Furthemore, small scale turbulence promotes soot spreading among mixture fraction space [53]. SGS modeling for soot was only very recently accounted for, with the proposition of a delta-type presumed probability density function by Mueller et al [14]. 3. Numerical aspects / methodology 3.1. LES of turbulent combustion The code AVBP1 is used for LES calculations [54, 26, 28, 17, 18]. It is a massively parallel code that solves the compressible Navier-Stokes equations together with conservation equations for energy and chemical species on unstructured or hybrid meshes. Momentum, energy and species conservation equations are solved using 1
http://www.cerfacs.fr/4-26334-The-AVBP-code.php
14
realistic thermochemistry, ie. real values for all thermodynamic properties taken from reference databases for each chemical species. The discretization schemes rely on a 3rd order in space and time Taylor-Galerkin scheme [55] and NSCBC boundary conditions [56] are applied. 3.2. Thermal radiation The RTE is discretized with a finite volume method on unstructured hybrid meshes and solved with the Discrete Ordinates Method (DOM) [57, 58, 59] in the solver PRISSMA2 . The RTE is solved for a set of Ndir directions (ordinates) by using a S4 quadrature (24 directions) [60]. The Diamond Mean Flux Scheme (DMFS) is used for the spatial integration [57]. The spectral properties are based on the spectroscopic data for CO, CO2 and H2 O over the wavelengths in the range ν = [150; 9300] cm−1 . In the SNB-CK model [40, 41] 367 narrow bands of width ∆νi = 25 cm−1 [61] are used and four additional bands ν = [9300; 20000] cm−1 are added to the visible spectrum to evaluate soot radiation. The SNB-FSK model [44] is similar but uses a tabulation to accelerate integration. It offers a good trade-off between accuracy and CPU cost and was used for coupling radiation and LES. The SNB-FSK model [44] was proposed to reduce spectral integration, only 5 to 15 spectral points are reconstructed from the 371 spectral bands. It offers a good trade-off between accuracy and CPU cost and was used for coupling radiation and LES. 3.3. Coupling methodology In a coupled combustion/radiation calculation, the radiative source term Sr must be known in the fluid solver while the radiative solver needs the local temperature, 2
PRISSMA, http://www.cerfacs.fr/prissma
15
pressure and composition of radiating species (H2 O, CO2 , CO and soot). Because of the double integration over directions and frequencies, the CPU cost of the radiative solver is much higher than the CFD solver. 3.3.1. Laminar flames To compute laminar sooting flames, the open-source software CANTERA [62] is used. CANTERA only solves the gaseous chemistry, and the soot model is integrated separately with a tool named CAN2SOOT, using the output of CANTERA for the mixture composition and temperature. At convergence, this procedure does not allow to take into account the feedback of the soot model on the flame, mainly the consumption of soot precursors within the flame. The impact on the result should be limited as the tabulated species are major species, and according to Xu et al. [63], their concentration is weakly impacted by soot formation. CAN2SOOT solves the transport equations for Ys and ns in onedimensional premixed or counterflow diffusion flames on non-uniform meshes [64]. A first-order finite difference spatial discretization is used and the steady-state solution is obtained by integrating the equivalent pseudo-transient problem with a first order (Euler’s method) temporal scheme. Finally, accounting for thermal radiation is performed in two ways. For laminar premixed flames, the experimental temperature profile is prescribed. For counterflow diffusion flames, the CANTERA and CAN2SOOT calculations are sequentially coupled to PRISSMA in an iterative loop until convergence is reached, in this case when the relative change in soot volume fraction peak is lower than 10−5 , as proposed by Liu et al. [45]. The coupling procedure is shown in Fig. 1. The calculated radiative source term is fed back to CANTERA, which plugs it in the energy equation. This procedure leads to a fully gas phase/soot/radiation coupled solution. For the calculation of radiation, perfectly 16
absorbing boundary conditions at T=300 K are imposed at the fuel and oxidizer inlet planes whereas pseudo-periodic (reflecting) boundary conditions are set on the four planes corresponding to the burned gas outflow, as sketched in Fig. 2. For these academic configurations, the computational cost is not an issue and the detailed spectral model SNB-CK is used. 3.3.2. Turbulent flames The coupling methodology between AVBP and PRISSMA has been presented and validated in a preceding paper [37], where both solvers run simultaneously and use the data obtained at the previous coupling iteration [36, 65]. Because of the different time scales in both solvers, synchronization in physical time and in restitution time is required: • Synchronization in physical time: the LES time step ∆tLES is fixed by the Courant-Friedrichs-Lewy (CFL) criterion due to the study of compressible flows: ∆tLES = CFL ×
∆xmin ∆xmin ≈ 0.7 ¯ + cs v cs
(27)
where ∆xmin is the smallest mesh characteristic width and cs is the local speed ¯. of sound, much larger than the norm of the local fluid velocity v Radiative fields change with a convection characteristic time [66]: τf =
∆xmin ubulk
(28)
where ubulk is the bulk flow velocity. Previous studies [67, 37] have shown that for low-Mach number flows the coupling frequency, ie. the number of LES iterations between two radiation calculations must be typically Nit ∼ 100. 17
• Synchronization in restitution time: the radiation calculation must take the same time necessary to perform Nit LES iterations: R Nit × tR LES = tRad
(29)
R where tR LES and tRad are respectively the restitution time required for one fluid
iteration and one radiative calculation. This load balancing is obtained by adjusting the number of cores allocated to each solver. The data exchange, communications and resource distribution between PRISSMA and AVBP is handled by the coupler Open-PALM3 , following the above constraints [68, 38]. 4. Results I: Laminar flames 4.1. Progress variable A one-dimensional laminar rich premixed kerosene/air flame is modeled with AVBP, under thermodynamic conditions representative of a helicopter engine. The reduced chemistry for kersoene-air flames 2S KERO BFER [7] is used and the transport equation for ctr is added. A reference is provided for the same flames (equivalence ratio, fresh gases temperature, pressure) computed with CANTERA [62] using a complex chemistry described by the kinetics of Luche (91 chemical species, 991 reactions) [69]. The composition of the kerosene surrogate is the one proposed by Luche [69] and is given in Table 1. A flamelet table is generated with CANTERA solutions, storing ϕ(c) where ϕ = YC2 H2 , YOH , T . 100 points discretize the progress variable space. 3
Open-PALM: http://www.cerfacs.fr/globc/PALM WEB
18
The progress variables crich , ctab and ctr are compared in Fig. 3. Note that the position of the two flame fronts, computed with AVBP and CANTERA, have been shifted to ease the comparison. The profiles are similar until a value of about 0.8. Above this value, while the crich progress variable evolves quickly towards equilibrium, as the result of reduced chemistry, ctab evolves more slowly, due to the presence of the post-flame zone with slow chemical kinetics. The progress variable ctr recovers this behaviour and shows the same profile shape than the reference progress variable ctab . This demonstrates the good correspondance between the transported progress variable ctr with the progress variables from reduced and detailed chemistries, and allows to use ctr to recover intermediate species from the tabulated flames for soot modeling. This is shown in Fig. 3 where the profiles YC2 H2 (ctr ), YOH (ctr ) and T (ctr ), normalized by the maximum value for mass fractions and the inlet value for temperature, show a very good agreement with the profiles computed with CANTERA. This strategy is therefore sufficiently accurate to supply the mass fractions of O2 , OH, C2 H2 as well as temperature to the soot model. Note that the profiles of YC2 H2 (crich ), YOH (crich ) and T (crich ) reconstructed with crich recover the temperature evolution and the peak of C2 H2 , but not the slow decay on the burned gas side due to the absence of the post-flame region. The consequence in terms of soot prediction would be a significant underestimation. 4.2. Laminar premixed flames As combustion in the application considered occurs mainly under a premixed flame mode, the proposed model is first tested against the measurements of Xu et al. [63], carried out for three rich burner-stabilized premixed ethylene-air flames. The structure of the three C2 H4 /air flames is first computed with the CAN19
TERA [62] code, using the UDEL chemical mechanism (70 species, 911 chemical reactions) [70] and prescribing the experimental temperature profile as 1D computations cannot reproduce it. Soot is computed with CAN2SOOT as explained in section 3.3.1. Figure 4(top) shows the predicted levels of C2 H2 , compared to experiments. Correct levels for the three cases are obtained, with a more pronounced difference for the intermediate case, and a generally too fast increase. Nevertheless, these results confirm that neglecting the feedback of soot production on the flame chemistry gives satisfying results. Note also that richer flames produce more C2 H2 , which promotes both nucleation and surface growth. This is confirmed in Fig. 4(bottom) where more soot is found for richer flames. The Leung et al. soot model enables to recover this trend, but the preexponential constant of the growth rate term was tuned to recover good orders of magnitude for the soot volume fraction (see Table 2). The agreement between the predicted levels and the experiments is overall good, however, far from the burner, the experimental soot profiles tend to level off, while the predicted soot profiles continue to increase. This can be explained [63] by HACA theory: the decay of H atoms concentration, induced by the temperature drop, prevents the soot surface activation from growth by C2 H2 addition. This physical feature is not integrated in the present model, but consequences in the complex turbulent configuration of Section 5 should be a priori limited as the premixed flame considered in the application is short and unsteady. To the authors’ knowledge, no soot profiles have been measured for kerosene or n-decane premixed flames. It is assumed that the calibrated soot model leads to correct order of magnitudes for any fuel, provided that the input data of the soot model (levels of temperature and mass fractions of chemical species) correspond to 20
the chosen fuel. 4.2.1. Laminar counterflow diffusion flames Three of the counterflow diffusion flames measured by Hwang and Chung [71] are then computed. The fuel is still C2 H4 , whereas the oxidizer stream is composed of air diluted with nitrogen. The dilution rate changes for the three flames, according to the following mole fractions: XoO2 =0.20, 0.24 and 0.28. The flames are named SF x where x is the oxygen mole fraction. Results are shown in Fig. 5. Good predictions for the soot volume fraction fv is found for case SF 0.24 flame. Soot is however overpredicted in the SF 0.2 flame. This disagreement was previously reported [72], where it was suggested that the experimental values corresponding to the SF 0.2 flame should be taken with caution. The radiative source term Sr for the SF 0.28 flame is represented in Fig. 6 for a case with uncoupled thermal radiation of gases only, uncoupled gas and soot radiation and finally coupled gas and soot radiation. In the first case, the single bump corresponds to the net energy exchange (emission-absorption) between the hot gases and the cold surroundings. Positive values of Sr correspond to a power loss (net emission). In the second case, a second bump appears, due to the presence of soot. Finally in the coupled case, the soot-driven radiation bump is decreased. The effect of radiation on soot is therefore to decrease the amount of soot produced, due to the temperature drop induced by the radiative heat loss (which is here about 5 %). Figure 7 shows the computed soot volume fraction profiles in the SF flames, with and without radiation. As mentioned above, it is observed that when gas and soot radiative losses are introduced, decrease up to about 6% at the peak is observed. Overall the present model allows a good representation of soot emissions 21
for academic configurations. However it should be kept in mind that the soot surface growth constant was adjusted for both premixed and diffusion flames. This was already mentioned by Mehta et al. [11], where several variations of a soot model were tested on eigth laminar premixed and diffusion flames. More complex soot models are then required, but the adjusted simplified models used in this section are sufficient for a first step towards soot prediction in complex systems. 5. Results II: Application to a helicopter combustion chamber 5.1. Case The chosen application is a sector of a helicopter combustion chamber of Turbomeca, recently studied by Amaya et al. [73] and Staffelbach et al. [17]. The computational domain is illustrated in Fig. 8 where the geometry has been rescaled for confidentiality reasons. The operating point considered corresponds to full thrust. Mesh refinement in the flame stabilization zone leads to a discretization of the domain through 11.9 106 tetrahedra. Wall laws are used and the sub-grid scale turbulent viscosity is closed with the Smagorinsky model [47]. A chemical table is generated for this application, discretized over 100 points in progress variable space, 100 points in mixture fraction space, as the equivalence ratio is non uniform in the combustion chamber and 20 points for fresh gas temperatures, ranging between 400 and 900 K. In this table the mixture fraction Z˜ is based on carbon atoms. This procedure is rigorous because no differential diffusion effects are accounted for in the species transport equations. The domain and grid for radiation calculation are extracted from the fluid domain, resulting in a mesh of only 2.6M of cells. The radiative calculation is performed with the S4 (24 directions) angular quadrature and the tabulated SNB-FSK approach
22
with 5 spectral quadrature points. Wall emissivity is 0.8 and wall temperature was provided by Turbomeca. The periodic boundaries of the sector are considered as purely reflecting walls. For the coupled combustion/radiation calculation the processor distribution between AVBP and PRISSMA to ensure the synchronization in CPU time with an acceptable accuracy is: PAVBP 111 = ∼ 14 PPRISSMA 8
(30)
where PAVBP and PPRISSMA are the numbers of processors allocated to AVBP and PRISSMA respectively. It must be pointed out that the mesh, initial solution and numerical set-up have been recovered from a previous work [16]. An other key point is in this present work there is no way to link results with experimental data, as to the authors’ knowledge, no measurements were performed inside aeronautical engines. For such systems, soot particles are described with a smoke number measured at the outlet of the engine [74]. 5.2. Results Adiabatic case Aeronautical engines use staged combustion: the kerosene/air mixture is first partially burned in a premixed flame zone (primary zone), and a secondary combustion occurs in a diffusion flame establishing between the burned gases and the fresh dilution air. Normalized fields reconstructed from the tabulated flamelets and an instantaneous solution are shown in Fig. 9. Note that the injection system has been removed for confidentiality reasons. It appears that C2 H2 exists only in the rich swirled premixed flame, while the maximum of OH is encountered in the highest temperature 23
zone, where the hot gases coming from the premixed flame burn with the dilution air in a diffusion flame. As a consequence soot remains confined between the swirling premixed flame, where it is produced via C2 H2 , and the diffusion flame, where it is oxidized by O2 and OH. Results also show the limit of the proposed approach linking reduced and complex chemistries. The progress variable, built for premixed flames, keeps the value 1 in the secondary combustion zone, filled with burnt gases issued from the premixed flame. As a consequence no acetylene is found in this zone, ie. no soot is produced, although a sooting non-premixed combustion occurs there. Future work is needed to extend the model to non-premixed combustion, using steady or unsteady diffusion flame tables.
Non-adiabatic case A second simulation is performed, coupling LES with thermal radiation. Figure 10 shows the total radiative source term, and the contribution of soot to it. The total radiative source term reaches maximum values in the diffusion flame zone, where the temperature is maximum. It is there 100 to 1000 times higher than in the premixed flame burnt gases, where however the soot contribution is highest. Neverthesless this contribution appears to be negligible. This counter-intuitive result may be explained by the staged combustion, with a highest temperature reached in the diffusion flame where soot is oxidized. Soot remain located in the premixed flame burnt gases, colder than the diffusion flame and their radiative contribution stays weak. The impact of the radiation on the mean flow is given in Figs. 11 and 12. The normalized difference for any quantity ϕ is calculated as: dϕ =
ϕUncoupled − ϕCoupled max(ϕUncoupled )
(31)
Radiation impacts the temperature maxima from -3.2% to 6.5%, corresponding to a 24
maximum absolute difference around 160K. The effect on major species H2 O, CO2 and kerosene is limited with a maximum normalized difference from 2 to 6%, and is slightly more important on CO with a maximal difference of 10%. As the activation energy of the secondary reactions involving such species is relatively low, they are more sensitive to temperature changes. Differences on soot volume fraction and particle number density are highlighted in Fig. 13. Due to lower temperature, the non-adiabatic case is lower, up to around 35 % and 21% respectively for fv and ns . The impact of radiation then appears greater for soot than for gaseous chemistry, probably due to different kinetics. Nevertheless, orders of magnitude are kept for soot related quantities in this configuration as soot does not strongly impact the radiative budget. It could be more important in other configurations with the interacting loop between temperature and soot emission. For previous 1D computations and LES, peak soot volume fraction decreases when including radiation. Moreover, the same order of magnitude is obtained for the relative drop (35% for the LES vs 10% for the 1D cases), proving the consistance of the soot model. In both cases, the order of magnitude of soot volume fraction is kept. This similarity between the two modeling ways can be qualitatively explained. The ratio between the radiative loss and the heat released is different for LES and 1D flames. It mainly results from the ratio between the volume of burned radiating gases and the surface of reacting flame that strongly depends on many parameters such as the geometry and the turbulence. It follows that the thermal effect on soot could not have been estimated a priori from 1D analysis. The proposed methods allows to combine all the effects of turbulence, thermal radiative loss to understand the key point in the mechanisms of soot formation in a complex geometry. 25
6. Conclusions The purpose was to check the behavior of a simple soot model from well-documented laminar flames and to use it for modeling soot emitted by turbulent reactive flows in complex geometries. The choice of the soot model is constrained by both the necessity to include key physical ingredients and an easy use for modeling sooting flows in complex geometries through low CPU cost and robustness. The model of Leung et al. appears as a good trade-off, as it implies the resolution of two transport equations only but accounts for acetylene concentration for soot surface growth and nucleation. When modeling turbulent reactive flows in realistic geometries, reduced chemical schemes are used. Acetylene is not counted among the species of such schemes. Strategies bridging reduced chemistries and tabulation techniques of complex chemistry effects were proposed. A rich premixed flame was modeled with reduced chemistry. Both strategies were used with this computation. With strategy II, it was possible to recover the detailed structure of the same flame computed with CANTERA, with detailed chemistry effects. The behavior of the model of Leung et al. was then regarded, first through welldocumented rich premixed flames. Their structure was computed with the CANTERA code and a detailed chemistry. The resulting profiles were used by an inhouse code to compute soot profiles, with the model of Leung et al. Good orders of magnitude were recovered. The counterflow diffusion flames were considered as well. The procedure mentioned above was completed by the computation of the radiative source term with the PRISSMA code. Iterations were performed in order to get the converged solution of the sequential computations of the flame structure, then the soot and finally the radiative source term profiles. Again, satisfying results were
26
obtained. The impact of radiation on temperature and soot was recovered: heat loss due to burned gases and soot emission induced decreasing peak values for soot. Finally, soot were computed in the combustion chamber of a helicopter engine. A LES from this case, relying on reduced chemistry was re-used. Strategy II and the Leung et al. model were employed to this end, first for a soot computation based on an adiabaticity assumption. Soot were produced and could grow in the rich zones of the premixed flame, and were then consumed in the diffusion flame. Very negligible amounts of soot were present in the burned gases exiting the combustion chamber. A second computation was carried out, with a coupling between LES, the radiative source term computation and soot modeling. As soot were located in a region colder than the diffusion flame zone, soot radiation was shown to be well lower than the the gas phase one. A maximum relative difference between mean temperature fields obtained for adiabatic/non-adiabatic computations was of the order of a few percents. Whereas the radiation effect on gaseous phase was observed to be weak, it impacted much more the soot volume fraction, for which the maximum relative difference reached a significant value of 35 %. Soot were probably more influenced due slower kinetics for reactions responsible of their evolution. This invites to model turbulent sooting flows in complex geometries through multi-physics computations. The present work could be nevertheless completed. To the author’s knowledge, experiments representative of the combustion encountered in the helicopter engine (kerosene/air turbulent swirling flames), for which soot were measured do not exist. The data provided could help to validate the methodology used to model soot in the combustion chamber. The modular aspect of the whole methodology used in the paper, makes easy its improvement. First, the soot model could be modified in order to integrate the effect of PAHs on soot. It can be done in several ways. For example, naphthalene is included in the chemical scheme of Luche. This species could be 27
tabulated to model nucleation instead of C2 H2 and condensation on soot particles. An other possibility could be to replace the model of Leung et al. by the model of Di Domenico et al., that accounts for PAHs through a sectional approach. Second, the tabulation method can be modified to describe more accurately the diffusion flame zone. Finally, turbulence effects at the sub-grid scale should be quantified and a closure derived. The tabulation method can be modified to describe more accurately the diffusion flame zone. Acknowledgements The authors wish to thank Dr. Arnaud Trouv´e for his advices concerning soot modeling. They also acknowledge Dr. Laurent Gicquel and Elena Collado for providing mesh, initial solution and numerical parameters for the LES. This work was partially funded by the STAE foundation through the ITAAC project and FNRAE within the STRASS project. Furthermore, this work was granted access to the HPC resources of CINES under the allocation 2010-025031 made by GENCI (Grand Equipement National de Calcul Intensif). References [1] H. Jung, B. Guo, C. Anastasio, and I. M. Kennedy. Quantitative measurements of the generation of hydroxyl radicals by soot particles in a surrogate lung fluid. Atmospheric Environment, 40(6):1043–1052, 2006. [2] B. K¨archer, O. M¨ohler, P. J. DeMott, S. Pechtl, and F. Yu. Insights into the role of soot aerosols in cirrus clouds formation. Atmospheric Chemistry and Physics, 7:4203–4227, 2007.
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Initialize
CANTERA
Compute flame
CAN2SOOT
Compute soot
PRISSMA Compute radiation
No. Feed back radiative source term
Convergence ?
Yes Stop Figure 1: Flow chart of the iterative coupled solution procedure.
39
Figure 2: Sketch of the 3D solution of the flame including soot distribution given to PRISSMA and the boundary conditions used for the calculation of radiation, where refers to the emissivity at the boundaries.
40
1 3
Normalized temperature [-]
Progress variable [-]
0.8
0.6
0.4
0.18
0.19
0.2 x [m]
0.21
0.22
1 0.17
0.23
1
1
0.8
0.8
OH normalized mass fraction [-]
C2H2 normalized mass fraction [-]
2
1.5
0.2
0 0.17
2.5
0.6
0.4
0.2
0 0.17
0.18
0.19
0.2 x [m]
0.21
0.22
0.23
0.18
0.19
0.2 x [m]
0.21
0.22
0.23
0.18
0.19
0.2 x [m]
0.21
0.22
0.23
0.6
0.4
0.2
0 0.17
Figure 3: From top to bottom: profiles of progress variable, normalized temperature, C2 H2 and OH mass fractions. Dashed line: CANTERA computation (ctab ), solid line: AVBP computation using crich to retrieve tabulated values of C2 H2 and OH, dot-dashed line: AVBP computation using ctr to retrieve tabulated values of C2 H2 and OH. For the sake of clarity, all the dashed line profiles have been slighlty shifted to the right.
41
0.04
C2H2 mole fraction [-]
0.03
0.02
0.01
0
0
0.005
0.01
0.015 x [m]
0.02
0.025
0.03
0
0.005
0.01
0.015 x [m]
0.02
0.025
0.03
7e-07 6e-07
fv [-]
5e-07 4e-07 3e-07 2e-07 1e-07 0
Figure 4: Laminar premixed flames of Xu et al. [63]. Profiles of C2 H2 mole fraction (top) and soot volume fraction (bottom). Symbols: experiment (squares: φ = 2.34, diamonds: φ = 2.64, circles: φ = 2.94). Lines: computations (solid: φ = 2.34, dashed: φ = 2.64, dot-dashed: φ = 2.94.
42
Figure 5: Laminar diffusion flames of Hwang & Chung [71]. Profiles of soot volume fraction from experiment (symbols) and computation (lines).
43
Figure 6: Laminar diffusion flames of Hwang & Chung [71]. Computed radiative source term for three levels of radiative coupling.
44
Figure 7: Laminar diffusion flames of Hwang & Chung [71]. Profiles of soot volume fraction with and without radiation.
45
Figure 8: Computational domain of a sector of an helicopter engine annular combustion chamber. For confidentiality reasons, the geometry has been rescaled.
46
Figure 9: Instantaneous fields in the vertical cut plane of the burner configuration. From top to bottom: normalized mass fraction of C2 H2 , OH, temperature and soot volume fraction. For confidentiality, the injection system has been removed.
47
Figure 10: Fields of the total (left) and soot contributed (right) radiative source terms. Both have been normalized by the maximum value of the total radiative source term.
Figure 11: Fields of normalized difference of temperature with and without radiation.
48
Figure 12: Normalized difference between the major species fields obtained with and without radiation.
Figure 13: Normalized difference between the mean soot fields obtained with and without the radiation.
49
Species
Mass fraction (%)
n-decane
76.7
n-propylbenzene
13.2
n-propylcyclohexane
10.1
Table 1: Composition of the kerosene surrogate.
Step
Ai
ni
Ti
Nucleation
0.1 · 105
0
21100
Surface growth
0.1 · 104 (premixed)
0
12100
1.4 · 104 (diffusion) O2 oxidation
0.1 · 105
0.5
19680
OH oxidation
106.0
-0.5
0
Table 2: Constants of the Leung et al. model with modified A2 values. The functions of temperature ki (T ) of Eqs. (10) and (11) are explicited as: ki (T ) = Ai T ni e−Ti /T [s−1 ].
50
