a single combustor as experimentally studied by Sommerfeld and Qui [17] is chosen ...... [26] Lain S. and Sommerfeld Martin, â Study of Turbulence Modulation in.
Modeling and Simulation of Effects of Turbulence on Vaporization,Mixing and Combustion of Liquid-Fuel Sprays A. Sadiki, M. Chrigui and J. Janicka Institute of Energy and Power Plant Technology, Dept.of Mechanical Engineering,Technical University of Darmstadt Germany
M.R. Maneshkarimi Institute of Mechanics, Dept. of Mechanics, Darmstadt University of Technology, Germany Abstract. The objective of this work is twofold. Firstly, the effects of turbulence intensity variations on the turbulent droplet dispersion, vaporization and mixing for non-reacting sprays (with and without swirl) are pointed out. Secondly, the effects of the coupling of the turbulence modulation with external parameters, such as swirl intensity, on turbulent spray combustion are analyzed in configurations of engineering importance. This is achieved by using advanced models for turbulence, evaporation and turbulence modulation implemented into FASTEST-LAG3D-codes. 1) To highlight the influence of turbulence modulation on some spray properties, a thermodynamically consistent modulation model has been considered besides the standard assumption and the well known Crowe’s model. For turbulent droplet dispersion, we rely on the Markov-sequence formulation. 2) In order to characterize phase transition processes ongoing on droplets surfaces, a non-equilibrium evaporation model shows better agreement with experiments in comparison with the quasi-equilibrium-based evaporation models often used. 3) The results of turbulence intensity variations reveal the existence of a limited range out of which the increase or decrease of the turbulence intensity affects no more the efficiency of the heat and mass transfer. A derived characteristic number, a vaporization Damkh¨oler number, possesses a critical value which separates two different behaviour regimes with respect to the turbulence/droplet vaporization interactions. 4) Under reacting conditions, it is shown how the evaporation characteristics, mixing rate and combustion process are strongly influenced by swirl intensity and turbulence modulation. In particular, the turbulence modulation modifies the evaporation rate, which in turn influences the mixing and the species concentration distribution. In the case under investigation, it is demonstrated that this effect can not be neglected for low swirl intensities (Sw.Nu. ) in the region far from the nozzle, and close to the nozzle for high swirl number intensities. In providing these particular characteristics, a reliable control of the mixing of gaseous fuel and air in evaporating and reacting sprays, and a possible optimization of the mixing process can tentatively be achieved.
1. Introduction In many technical combustion devices, such as automotive engines, gas turbine combustors, the fuel is supplied as a liquid with varying physical and chemical properties to form a combustible mixture of fuel vapor and air. Focused on the fuel preparation in gas turbine combustors, the performance, the stability and emissions of the combustors strongly depend on the fuel-air mixing processes, which in turn are determined by the turbulent dispersion, the evaporation of fuel droplets and the related interaction phenomena within © 2005 Kluwer Academic Publishers. Printed in the Netherlands.
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3 the turbulent carrier phase. In order to provide reliable design criteria along with appropriate tools to handle with modern combustion technology requirements, early detailed information of different design variants and parametric studies is necessary. For systems that are extremely costly to build and to test, only a CFD-based analysis is capable of delivering deeper insight into the transport processes encountered. However, an accurate description of the flow properties using CFD decisively relies on the availability of adequate and efficient physics-preserving mathematical models. These may consist of a set of sub-models suitable for each specific phenomena and associated interactions. The interactions between gas and spray droplets depend on many factors (e.g. gas and droplet physical properties, phase velocities, concentrations, temperatures and turbulent fluctuations, etc). They could be classified in three main categories including the turbulent dispersion of droplets, the so-called turbulence modulation, i.e. the modification of the continuous phase turbulence characteristics by droplets or by interface transport and, the modification of interface transport by turbulence [1, 2, 3 and 4]. A considerable amount of works has been done in a large diversity of mono-dispersed gas-liquid flows, and various parameter studies have been carried out [1, 3, 4, 8, 9, 16 and 17]. However, the study of the effect of the turbulence on the modification of the droplet interface transport is rather rare, although the turbulence is one of the major factors controlling droplet dispersion, formation of droplet high concentration regions and thus sprays flame structure [3, 4, 16, 17, 19, 20]. In internal combustion engines, available results are usually limited to those of the experiments of turbulence augmentation by additional injections of air or gaseous fuel or by adjusting the swirl intensity without investigating deeply the modification of interface transport and the influence of the turbulence modulation process on it (e.g. [44]). The effect of turbulence on vaporization and mixing of liquid-fuel sprays has been experimentally investigated in [3, 4]. Non-reacting and reacting spray jets in a duct with square cross-section have been chosen. They found that the turbulence strongly influences the evaporation rate and suggested that the turbulence modifies the mass transfer from liquid droplets in dependence on a characteristic, defined vaporization Damkh¨oler number [3]. Furthermore with the increase of the evaporation rate, the mixing of gaseous fuel and air becomes controlling process of combustion [4]. Although in many recent works some numerical calculations have been performed in which the effects of the turbulence on droplets have been pointed out (e.g. [1, 6, 9, 25, 26]), they deal mostly with mono-dispersed sprays using 2-D numerical codes in which the standard K-Epsilon model was coupled to quasi-equilibrium evaporation models [1, 8-10, 16, 19, 20, 25] without a fully consideration of turbulence modulation processes. Thereby no reliable prediction of the turbulent kinetic energy could be expected. This is also the case in LES (e.g. [12, 13, 27]). From these studies, it is also well-
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4 known that the swirl intensity can strongly influence the turbulent mixing process and therefore the spray combustion (see [5, 24] and therein cited references). An accurate capture of the turbulent kinetic energy is therefore decisive for better prediction of these effects, and in particular of the spray combustion characteristics. In this connection, the modification of the continuous phase turbulence characteristics by droplets or by interface transport, and in turn the modification of interface transport by turbulence can no longer be neglected. In general, there are three main approaches for simulating turbulent combusting sprays: The Eulerian-Lagrangian approach [e.g. 12, 25, 27], the Eulerian-Eulerian method [e.g. 5] and the PDF approach [e.g. 30, 31]. Comprehensive reviews of the spray modeling and simulation as well as of combustion models can be found for instance in [32, 33, 37-46]. In this paper, interest is focused on the Eulerian-Lagrangian approach to analyze the effects of the turbulence on mass and heat transfer in dilute sprays, and in particular how the coupling effects of turbulence and swirl variations can influence droplet dispersion, evaporation, mixing and combustion. For this purpose, two spray configurations are numerically analyzed based on Reynods averaging based numerical simulations (here, RANS). The first configuration focuses on the effects of turbulence properties on droplet dispersion, vaporization and mixing of a non-reacting spray. For this case a single combustor as experimentally studied by Sommerfeld and Qui [17] is chosen. Kohnen et al. [25] used the standard k-Epsilon model coupled to the quasi-equilibrium evaporation model. Liu et al. [30] applied the PDF approach and the quasi-equilibrium evaporation model. Both compared the numerical results to experimental data only. No additional analysis could be provided. The second configuration aims at the coupling of conjugate effects of the turbulence, turbulence modulation and the swirl intensity on the gas phase along with the spray combustion. The used configuration is a swirl stabilized combustor similar to that investigated by Wittig et al. [5] where we vary swirl intensities in the simulation. In [5] an Eulerian two-phase model in conjunction with an Eddy dissipation and a presumed-shape PDF combustion model has been used. For the turbulence, a modified k-Epsilon model has been applied without considering the droplet/particle source terms. Furthermore, turbulence intensity and swirl number variation effects have not been addressed there. In contrast to existing works, advanced models for turbulence (RANS second order model [15], algebraic Reynolds stress model (ARSM) [21] and explicit algebraic Reynolds stress model (EARSM) [34]), for evaporation (equilibrium and non-equilibrium models) [11, 32, 33, 7, 8] are considered in the present work, while the Markov-sequence model based on the calculation of Lagrangian and Eulerian correlations factors is applied to simulate the turbulent droplets dispersion [1, 2, 25]. A fully consistent two-way coupling
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5 [2, 14, 28] along with the modification of the interphase transport by the turbulence is taken into account by means of a thermodynamically consistent model by Sadiki and Ahmadi [2, 14]. To illustrate the performance of the presented, coupled models, numerical and available experimental results will be compared for both configurations. In the following section, a brief description of the governing equations and models used is provided followed by the numerical procedure. Then, the configurations and the applied boundary conditions are summarized. Before concluding the paper, numerical analysis results in the two investigated configurations are presented and discussed.
2. Modelling and Numerical Procedure To account for the instantaneous flow properties encountered by the droplets, involving each droplet history starting from the injection into the flow, an Euler-Langrangian approach is adopted. The droplets are described by a Lagrangian transport through a continuous carrier gas flow which is captured by an Eulerian approach. 2.1. D ROPLET L AGRANGIAN D ESCRIPTION To compute the properties of droplets moving in turbulent flow a Lagrangian approach is employed. The trajectories of individual particles are obtained from motion equations, where all external effects except drag, buoyancy and gravity forces are neglected due to the large density ratio between droplets and carrier phase in the configurations under study [1, 2, 19, 25, 28]. Collisions between droplets were neglected, as the sprays are considered as diluted. To obtain the velocity fluctuations of gas occurred in the motion equation of droplets at droplet locations, a dispersion model following the MarkovSequenz-formulation (see in [2, 25, 26]) is used. This model is based on the calculation of Lagrangian and Eulerian correlations and exhibits a satisfactory ability in calculating the gas fluctuations. For evaporating droplets, two additional equations which give the rate of change of droplet diameter and temperature with respect to time are also needed. 2.2. E VAPORATION
MODELS AND T URBULENCE /D ROPLET VAPORIZATION INTERACTION
To characterize the deviation of phase transition on the droplet surface from the equilibrium, a characteristic parameter has been introduced in [7]. It depends among other on the ratio of the non-equilibrium and quasi-equilibrium
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6 Peclet numbers. The value of this parameter is zero for the case of equilibrium, and it grows remaining positive with the increase of the deviation from the equilibrium. When this parameter is greater than one, it was then shown that the use of non-equilibrium models is needed in order to have adequate estimates of the evaporation rate of small droplets. This is the case for the configuration under study in this work. To account for the 3D-evaporation of droplets, equilibrium and nonequilibrium models are therefore considered. For a detailed theoretical analysis and a comparative evaluation of different evaporation models, the reader may also refer to [8] and therein cited references. According to the film theory, the vaporization rate is calculated in both models by considering the mass transfer around the droplet as
"!$#&%
!$#')( *,+-(. (1) +-( * is the mass transfer Spalding number. The quantity ( * is the surface mass 0 * %21 where 1 is the ratio of molecfraction determined by (/* 0*+ + 0* ular weights. ( . is the mass fraction in the gas phase, while 43 and 3 where
are the averaged values of the mixture density and binary diffusion coefficient throughout the film. For the equilibrium models usually used, the molar mass fraction is related to the saturation pressure through the ClausiusClapeyron equation [1, 11, 20, 30, 32, 33]. In order to incorporate the effects of Stefan flow on heat and mass transfer, Abramzon and Sirignano [11] in, and Sherwood, , numbers. In the case troduced modified Nusselt, of non-equilibrium evaporation model [7, 8], the molar mass fraction, , is determined by the following relation:
0*
8
576
0 *
@LK M ONQPSR (2) JG I + E M represents the half of the blowing Peclet number. N is the Prandl number A B for the gas phase, represents the Knudsen length, and POR is the parti0 *:9 =; 0 *:9 +
@A B C/D FEHGJI
where
cle/droplet relaxation time. The Lagrangian equation describing the transient temperature of a single droplet is given by:
CUT R K M N PSR d D @[Z T T \` d Y 5 6 X W N ] R U % _ ^ CV K M N PSR\E + a 9 b c e W + 5Y6 % !ghHf i (3) ! h f represents the heat transfer Spalding number. It is related to ! # , and
calculated iteratively with the help of the following system equations in the equilibrium evaporation case:
! h f j k!# 9 % l + e
with
a ` 8 m a 9 Y 5 6 A3n e
(4)
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7
5 6 + h h ! h %j "! h % "! h f % (5) 5Y6 7 ! hf h e ! #f 9 % , # + e #' # !$# 9 %8 -!$# 9 % " ! #f 9 (6) Z a ` In Eq. (3)-(6), represents the ratio of the gas heat capacity, 9 , to that of A n is the Lewis Number and ` is the latent heat of the liquid phase a 9 b , evaporation. In the case of non-equilibrium evaporation model, the function according to: W in Eq. (3) is calculated @ K M N PSR d D @ n (7) + E i W + E Because the influence of the turbulence on the mass and heat transfer rate is investigated, let us define the time scales characterizing the turbulence and the thermodynamics of vaporization. To define the adequate turbulence time scale, we follow [3] by considering the fraction of turbulent energy that effectively contributes to the turbulent mass and heat transfer. By doing so, we implicitly eliminate the so-called Fr¨ossling effect which accounts for the influence of the mean relative velocity. Thus, the relevant turbulence time scale appears to be that of eddies having a length scale equal to, or smaller than the instantaneous droplet diameter. As time scale characterizing the thermodynamics of vaporization, a residence time of the fuel vapor within the film thickness around the droplet can be used. As a first approximation, this can be estimated as the ratio between the film thickness and the Stefan velocity or the radial blowing velocity
$
V R " ! A %$#&%(' < 6)
*
6f
V` A
+
(8)
with being the droplet diameter. In Eq.(8) is the turbulent length scale and the velocity fluctuation. The radial blowing velocity in (8) is determined using the evaporation rate also defined as
d -,U R! + e
(9)
while the film thickness is calculated according to [3]. To derive a measure of the relative importance of these time scales, a vaporization Damk¨ohler number, defined as the ratio between the turbulence time scale and the vaporization time scale in eq. (8),
V ". ` V < ` R e
(10)
is introduced. It characterizes the turbulence-droplet vaporization interaction regimes.
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8 2.3. G OVERNING
EQUATIONS FOR TURBULENT GASEOUS FLOW
The turbulent fluid phase is described following a RANS-modeling approach. The stationary, general form of the transport equation needed emerges as an elliptic differential equation:
63 %
3 % 3 % % % + + ( ( ( + (11) + % 3 9 3 9 9 ` 3 in which may represent the mean value of mass density, , velocity T components, (6 e e ) , enthalpy/temperature, , turbulent kinetic energy,
, turbulent dissipation rate, , components of the deviatoric part of the Reynolds stress tensor, (anisotropy tensor components) and chemical species mass fraction ( ! , ! , vapor fuel ), respectively. represents an effective diffusion coefficient and the well known turbulence source terms in monophase flows. The latter have been already documented in many literatures [15, 21, 24, 34]. In the last years, efforts to better capture streamline curvature effects and swirled flows phenomena have been achieved in the RANS framework by means of second order models or by their corresponding algebraic formulations, besides non-linear turbulence and mixing models [15, 21, 24]. We therefore apply for the Reynolds stress tensor turbulence models of second order (RSM) by Jones and Musonge [15] and the derived, explicit algebraic Reynolds stress model by Gatski and Speziale [34] as well as implicit algebraic model by Launder, Reece & Rodi [21]. The standard k-Epsilon model is also used for comparison. All these models have been modified for two phase flow description by including source terms for phase exchange and phase ). For simplicity, the scalar flux vector has transition processes, ( been postulated by means of a gradient assumptions. The additional source terms in (11) characterize the direct interaction of mass, momentum, energy and turbulent quantities between the two phases and account for the twoway coupling between the fluid turbulence and the evaporating droplets. The is due to the inter-phase transport without phase transition, and quantity express the classical two-way coupling term in absence of evaporation and combustion. takes into consideration the transfer caused by the phase transition processes, like evaporation described above. Details about these terms and their respective relationships can be found in [2]. For the selfconsistency of the paper they are summarized in Table 1. Thereby , and are the mean gas phase (axial, tangential and transversal) velocity components. , and represent the three velocities components of the droplet, respectively. represents the number of real particles in one computational droplet, is the cell volume. is the momentum transferred to the
3 9 3 9 9 `
3
9 9 9 9 `
63 3 3 5 99
63 3
9
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9 9 ` h 9
9
gas phase in the absence of phase transition, is the gas momentum flux ejected by the droplet during its vaporization. represents the heat repabsorbed or released by the droplet during a thermal interaction. resents the heat which is released by the droplet into the fluid because of vaporization. represents the heat flux into the particle/droplet and is the gravitation. represents the value of the evaporation rate in a control volume (CV), where is the droplet mass flow cross a CV per second. is the Lagrangian integration time step. It is given by
H b 9 `
h 9 9 `
V
V i min P T T T % e e e P represents the droplet relaxation time, T is the turbulent time scale, where T represents T is the time to cross
the time to cross a turbulent eddy, and
a CV. 2.4. D ROPLET /G ASEOUS
TURBULENCE INTERACTION :
T WO - WAY
COUPLING
Focused on phase interactions with regard to turbulent quantities such as turbulent kinetic energy, the presence of small particles/droplets may attenuate the turbulence of the gas phase while big particles/droplets can augment it. In fact, an overbalancing of the particle-induced turbulence attenuation and production is observed which cannot be well captured by the state-of-the art approaches (e.g [1, 6, 10, 12, 25, 27]), as described in Crowe [6] who used the energy balance to attempt a first physically consistent description (see also [26]). Because all the phenomena involved are thermodynamical processes, we use in this work, besides the standard expression for two-way coupling, a model compatible with the second law of thermodynamics to better account for the direction of the process evolution. The droplet/particle source term for the turbulent kinetic energy is given in this model by [2, 14]
O9 G 6 9 + G f +
(12) 6 9 % 6 9 + 6 9 U% e where f % 6 6 9 % + 9 (13) 6 9 + 6 9 U% is a model parameter. The parameter f in Eq.(13) depends on droplets properties, as shown in [2, 14]. The second term in Eq.(12) represents the usual . The first term in Eq.(12) dissipative standard contribution. In this case G accounts for the production of turbulent kinetic energy. This corresponds with G to the model by Crowe [6, 26] resulting from the energy balance consideration.
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10
-
#+$
: 5 # $
; # $
"CB @ 6D ,
#+$ , /. 0132%4 ! " #%$'&()# *
" 5 # $ &()# 65 7 # $ , /.98 132%4
! " ; # $ &?132%4 " A # ,
5 ; @ E
N
0
M
G G
F
G H G G G
?OP P B P Q ? SR
P BUP TW QV YX T V
G G IHIKJ " K L K L , G G H G G H " L L , YOP P P BZP TX QV ?TW V B P Q Y R
Table 1 Source terms due to the presence of evaporating droplets. Unlike the model in [6] that regularly overpredicts the turbulent kinetic energy [26, 28], the thermodynamically consistent model (12) captures well both the enhancement and the diminution of the turbulence of the gas phase due to the presence of both big and small droplets in polydispersed sprays, as it accounts for the irreversibilty requirements of any thermodynamical process. It was successfully validated in different monodispersed two-phase flow configurations in [28, 29]. To improve the prediction of mass and heat transfer processes involving evaporation and combustion, which in turn affect the turbulence, such a consistent approach is now applied to complex polydispersed sprays. Thus, transport equations of turbulent quantities in Eq. (11) have been modified by including this physically consistent consideration of turbulence modulation phenomena, besides the other special cases (standard and Crowes’ model). 2.5. C OMBUSTION
AND
R ADIATION C ONSIDERATIONS
3
in equations (11) For the combusting spray case, chemical source terms for ! , ! and vapor fuel must be simultaneously provided. Although many efforts have been done in the last decade in combustion modeling [38, 39], we chose for the complex swirled, reacting two-phase flow case a simple turbulence-based model, the eddy dissipation model (EDM) [5, 10, 22] to calculate the mean source terms. It is known that this model does not necessarily predict correctly the mean temperature field for simple flame configurations
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11 [38]. But its release in complex configurations [5, 10], as it is the case here, has been proven in [5, 10], for instance. Radiation affects the temperature field of the reacting flow and thus many other processes, like droplet evaporation, chemical reaction and pollution formation. But in modern low-emission gas turbine combustors characterized by high bulk flow velocities and short residence times as it is the case here [5], radiation plays only a minor role on the temperature inside the combustor [36], although it still may influence the emission. A comparative study of efficient numerical methods for radiation in gas turbine has been carried out in [37]. To save computational time and because we do not focus on , an approximate prediction of radiation is satisfying here. The radiation contribution to the enthalpy balance is therefore evaluated by simply solving differential equations for the radiant fluxes following the Four-Flux method [23] in which the black body emissive power is assumed. A critical review of advanced methods for predicting radiative transfer in participating media, such as discrete ordinate method can be found in [33, 37]. The equations system of gas-phase is further closed by the equation of state which determines the distribution of density for ideal gas.
5
5
3. Numerical procedure The computational method is based on an Euler-Lagrangian coupling approach. For the Lagrangian phase a 3-D Lagrangian code (LAG-3D) for particle tracking has been extended. The Lagrangian equations for droplets are discretized using first order scheme and solved explicitely. Source terms for the gas phase (see Table 1) are computed in each cell with the contributions of all the relevant droplets. For the Eulerian description of the turbulent gas phase far from the droplet, the simulation is performed using the three dimensional CFD-code FASTEST in which the equations are solved by finite volume method. The diffusion terms are discretized with central schemes on a non orthogonal block structured grid. The velocity-pressure coupling is accomplished by a SIMPLE algorithm. The whole system is solved by the SIP-solver. The droplet injection is based on a stochastic approach by considering the droplet mass flux and the droplet size distributions obtained at the inlet far off the nozzle exit from experimental measurements [5, 17]. The droplet sizes are sampled from the droplet distribution functions obtained experimentally. The droplets are tracked until they reach the exit boundary or are completed vaporized. Average values and variances of droplet characteristic variables (velocity, temperature, etc.) are evaluated in each cell. Numerically, the interaction between the continuous and the dispersed phases is taken into account by means of several couplings between two mod-
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12 ules involved. Following a steady method, after several iterations of gas phase alone, the gas variables are kept frozen and all the droplets representing the entire spray are injected in the computational domain. The computed droplets source terms are inserted in the calculations of gas phase and kept frozen till the next coupling takes place in which the old particle sources are replaced by newly calculated sources. High levels of under-relaxation technique were used in order to obtain successful convergence following [2, 25, 28].
4. Configurations and Boundary Conditions Two configurations are used to outline the effects of turbulence on mass and heat transfer in evaporating and reacting sprays. The first configuration allows to study the effects of turbulence properties on droplet dispersion, vaporization and mixing of a non-reacting spray. It corresponds to an isopropyl spray issuing into a co-flowing heated air-stream as experimentally investigated in [17], see figure 1. The inner diameter of the main section is 200 mm. The annulus has an outer diameter of 64 mm and an inner diameter of 40 mm. This cells in axial, is represented by a computational domain with radial and tangential direction, respectively. This was found to be sufficient. The inlet conditions used for the carrier gas correspond to the measured values from the experiment at mm downstream. The hollow-cone pressure atomizer in the cylindrical center-body of the inlet tube injects 8 types of droplet classes. These classes are distinguished by the droplet diameter, start velocities, start location and the PDF. These were used as inlet conditions for droplets. The initial temperatures of the droplets and gas are set to 32°Cand 28°C, respectively . The co-flow has a temperature of 80°C. The maximum . The total droplet flow air velocity in the considered test case was 18 rate was 0.44 . The second configuration permits to investigate the conjugate effects of the turbulence modulation and the swirl intensity variation effects on the gas phase turbulence, and focuses on combusting sprays. The configuration is a swirl stabilized combustor similar to that investigated by Wittig et al. [5]. To account for the swirl effects, we vary swirl intensities in the simulation between zero and 1. The configuration consists in a cylindrical combustor where the swirled air stream enters the chamber after flowing through swirler and, the liquid fuel (dodecane) is injected into the combustion chamber from were axially located nozzles. In accordance with [5] the properties of used for the fuel vapor. According to experimental data the particle size distribution is determined from Gaussian distribution. The geometry of chamber and details of operating conditions (used as boundary conditions) at the combustor inlet are summarized in figure 2. Due to the symmetry assumption, half the domain is presented in computed results. These are obtained on a
K
K
D
D
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13
Figure 1. Spray issuing into a co-flowing heated air-stream
computational domain with respectively.
,
cells in axial and radial directions,
5. Results and Discussions To evaluate the performance of the approach used, numerical and available experimental results are compared in both configurations. The results shown have been achieved by using different model combinations as summarized in Table 2. This allowed numerical analysis of the ongoing interaction processes, and enabled to judge the ability of different combination models employed in retrieving some essential spray properties, such as droplet mass flux, evaporation rate, species concentration distribution as well as spatial mixing efficiency.
Figure 2. Cylindrical Combustion Chamber
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14 Configuration
Turbulence model
Evaporation model
Modulation model
Dispersion model
Chemistry
1 (Figure 1)
Second order [15]
Equilibrium [11] Non-equilibrium [8]
Standard [25]
Markov-Sequenz [1]
-
2 (Figure 2)
EARSM [34]
Equilibrium [11] Non-equilibrium [8]
Statndard [25] Sadiki [2]
MarkovSequenz [1]
EDM [22]
ARSM [21]
Equilibrium [11] Non-eqiulibrium [8]
Standard [25] Sadiki [2]
MarkovSequenz [1]
EDM [22]
Equilibrium [11] non-equilibrium [8]
Standard [25] Sadiki
MarkovSequenz [11]
EMD [22]
Standard [9]
F
Table 2: Different model combination used for the simulations
5.1. F IRST
CONFIGURATION : I SOPROPYL
S PRAY
To verify the code and to validate the approach used, different calculations have already been carried out [e.g. 2, 28]. In particular, a comparison of the streamlines, isotherms and cross-sectional profiles of velocity for both phases with experimental data have been reported in [35]. We focus here on the analysis of the mass and heat transfer process. Figure 3 represents the radial distribution of the droplet mass flux at different axial positions downstream. In accordance with the experiment, the concentration of droplets decreases while moving away from the nozzle due to the evaporation. The results demonstrate that the dispersion model used does a satisfactory job. A comparison between equilibrium and non equilibrium evaporation models reveals that the latter delivers results closer to experimentalresults, even . though it seems to underestimate the droplet mass flux when The equilibrium model following Abramzon and Sirignano allows almost a constant evaporation rate, which is not enough to give a droplet mass fraction properly. Figure 4 confirms that the non-equilibrium evaporation model agrees most favorably with the experimental measurement of the normalized evaporation rate along the central axis of the chamber (the evaporation rate at has been considered as reference value for the normalisation).
&
i
5.1.1. Some parameter studies After this comparison, a numerical analysis of the ongoing processes due to turbulence intensity variations is now presented. Figure 5(left) shows that with the increasing of the turbulence intensity, the efficiency of the mass transfer increases, approximately during the first half of the droplet lifetime. This influence decreases with decreasing droplet diameter. It must be mentioned that the turbulence effect manifests itself well before the droplet dimensions become comparable to the smallest turbulence eddies, Figure 5(right). Only turbulence eddies with scales equal to or smaller than the instantaneous droplet diameter will contribute. As pointed out in [3], a pure
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15 0.025 m 0.1
eq nep exp.
0.08 Radius [m]
0.05 m 0.1
0.08
0.08
0.06
0.06
0.06
0.04
0.04
0.04
0.02
0.02
0.02
0
0 0 0.1 0.2 0.3 0.4 0.5
0 0 0.1 0.2 0.3 0.4 0.5
0.2 m
Radius [m]
0.1 m
0.1
0 0.1 0.2 0.3 0.4 0.5
0.3 m
0.4 m
0.1
0.1
0.1
0.08
0.08
0.08
0.06
0.06
0.06
0.04
0.04
0.04
0.02
0.02
0.02
0
0 0 0.1 0.2 0.3 0.4 0.5
0 0 0.1 0.2 0.3 0.4 0.5
0 0.1 0.2 0.3 0.4 0.5
Figure 3. Comparison between results with equilibrium non-equilibrium evaporation and models. -axis represents value of droplet mass flux and -axis the radial position, respectively.
Normalized evaporation rate [-]
1
neq eq exp.
0.8
0.6
0.4
0.2
0 0
0.05
0.1
0.15 0.2 0.25 Axial position [m]
0.3
0.35
0.4
Figure 4. Normalized evaporation rate: Comparison between results with equilibrium and non-equilibrium evaporation models.
turbulence effect is expected for smaller droplets that are transported by the carrier phase without mean relative velocity while the influence of mean relative velocity may be significant for larger droplets. The reference value for the normalisation is set to the value of the evaporation rate resulting from a turbulent kinetic energy boundary condition of 60% of experimental data.
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16 1.03 X/D=0.5 X/D=1.3 X/D=2.3
1.03
normalized evaporation rate [-]
normalized evaporation rate [-]
1.035
1.025 1.02 1.015 1.01 1.005
X/D=3.3 X/D=4.6 X/D=6.0
1.02 1.01 1 0.99 0.98 0.97 0.96
1 60
70
80
90
100 110 120 130 140 150
60
70
80
90
100 110 120 130 140 150 k [%]
k [%]
Figure 5. Influence of turbulence variation on the evaporation rate within: (left) the first half part; (right) the second half part of the configuration
To evaluate the relative importance of the evaporation process and the gaseous turbulence within the system, let us retrieve and compare the characteristic time scales of the processes involved as introduced in Eq. (8). The first task is achieved through Fig. 6(left), while the second is accomplished by means of Fig. 6 (right), 7 and 8. The axial profiles of the process characteristic time scales are plotted in Fig. 6 where the reference value for the normalisation is set to the value of . One can observe that a process inversion occurs each time scale at after the first half of the droplet lifetime at corresponding to a ". critical value of , see Eq. (10). This is confirmed in Fig. 7 in which the ratio of these times is depicted (see profile k1:mp). Because this behavior may be affected by the turbulence modulation through changes in the turbulent kinetic energy, it is wisely indicated to detect its persistency or sensitivity with respect to the turbulent kinetic energy. We then varied turbulent kinetic energy intensities from 60% to 140%, and recorded their effect on the behavior of the evaporation rate along the axial direction in Fig.7. This figure confirms the findings from Fig. 6 (left), and reveals the existence of a limited range for the effects of the turbulence intensity variations (around 40%) out of which the increase or decrease of the turbulence intensity affects no more the efficiency of the mass transfer. Within this sensitive range, . it is easy to distinguish two regions separted by the critical value ". In the first region ( . ), when the turbulence intensity increases, the efficiency of the mass transfer increases, approximately during the first half of the droplet lifetime according to the findings in Fig. 6(left). This influence decreases with decreasing droplet diameter. Based on the fuel vapor concentration gradient near the droplet surface, which is the driving force for evaporation, the beneficial effect of a high turbulence intensity near the droplet surface could contribute to the buildup of the fuel vapor concentration gradient by removing the fuel vapor from the surface. Consequently in the case of the residence time of the fuel vapor in the vicinity of the droplet
i `
/`
i =
`d
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17 3
1.4
ted tv normalized heat rate [-]
2.5 normalized time [-]
neq 0.6*k neq 1.0*k neq 1.4*k
1.3
2 1.5 1
1.2 1.1 1 0.9 0.8 0.7
0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Axial position [m]
0.6 0.04
0.06
0.08
0.1 0.12 0.14 Axial position [m]
0.16
Figure 6. (Left): Turbulent time scale and time scale characterizing the thermodynamics of vaporization at different axial positions. (Right): Influence of the turbulence intensity through the turbulent kinetic energy on the normalized heat flux rate along the axial direction.
surface is large for low turbulence intensity, any acceleration of its removal will strongly contribute to an increase in the vaporization rate. Below the critical value, for ". the turbulence energy is able to increase the mass ) an inverse phenomenon is observed: the transfer. For higher value (". increase of the turbulent intensity even decreases the vaporization rate. These results which distinguish two characteristic interaction regimes based on a vaporization Damk¨ohler number agree well with the experimental results in [3]. A similar characteristic behaviour can be well observed for the heat flux rate as shown in Fig. 6(bottom).
`
\`
For the mean (cell) droplet size used for calculations, this behavior was observed independently from evaporation models. It also could be pointed out that the droplets clearly modify the turbulent kinetic energy mainly near the centerline where the droplet density is high. This was achieved by using different modulation models as droplet source terms in Eq. (11) for the turbuin axial direction, lent quantities. The influence was observed up to afterwards it is negligible in the remaining positions because most of droplets are either already evaporated or become very small. Coupled to swirling phenomena, this effect is more strengthened, as it will be shown in the second configuration (see, Fig.14-15). As pointed out in the introduction, the fuel preparation in gas turbine combustors and in many internal combustion engines strongly depends on fuel-air mixing processes. The question is now how to characterize the influence of the turbulence intensity variation on the mixing degree of the gaseous fuel and air. This can be achieved by means of the so-called spatial mixing deficiency (SMD). According to [18], it is defined as:
&
3 % b :; < 3 % e SMD b ;=<
with
3 %8 b:; < 3 e #
(14)
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0.18
0.2
18
Figure 7. Influence of the turbulent intensity through the turbulent kinetic energy (k) on the evaporation rate ( ) and the vaporization Damk¨ohler number ( 5 ) along the axial direction.
% 3 + b:; < 3 % % ! (15) b ;=< 3 + # In (14)-(15) b: ; < is the average of the mean vapor mass fraction over a plane section having cells, and b: ; < is the corresponding root mean square. The SMD parameter describes the heterogeneity of the distribution of the fuel vapor mass fraction, , in the combustion chamber. A zero value indicates perfect mixing at a given plane. Fig. 8 displays the influence of the turbulence intensity on the mixing process in the combustion chamber at different axial planes located at the positions between and 0.4 nozzle downstream. A significant modification of about 6.2% is observed between and in accordance to the results in Fig. 7. The effects of swirl intensities appear to be more significant, as it will be highlighted in the second configuration.
i =
i =
5.2. S ECOND CONFIGURATION : C OMBUSTION
CHAMBER
With regard to design efforts, by adjusting the swirl intensity it is possible to improve the mixing quality of the flow and to influence or to control physicochemical processes. Because of convergence problems experienced
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19
1
k=0.6 k=1.0 k=1.4
0.9 0.8 0.7 SMD [-]
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.05
0.1
0.15 0.2 0.25 Axial position [m]
0.3
0.35
0.4
Figure 8. Influence of the turbulence intensity through the turbulent kinetic energy on the spatial mixing deficiency (SMD) along the axial direction.
with the second order model in this complex two-phase flow configuration, we use the robustness of the two-equation models in the frame of algebraic formulations to predict the influences or effects of swirl intensity coupled to internally induced turbulence due to evaporating droplets on combustion. 5.2.1. First Case: Non-reacting swirled Spray The computed cross-sectional profiles of gas axial velocity component along the chamber using different turbulence models are shown in Fig. 9 and compared with available experimental data. x = 0.1 m
x = 0.02 m 10
20
30
-10 40 0.05 0.05
0
10
x = 0.3 m 20
30 0.05 0.05
1
2
3
x = 0.4 m 4 0.05 0.05
5
10
15
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0.04 0.04
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0.03 0.03
0.03
y
0
y
-10
y
y [m]
0.05
0.02
0.02 0.02
0.02 0.02
0.02 0.02
0.02
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Measured Calculated k,e 0.01 0.01 Calculated EARSM Calculated LRR
0.01 0.01
0.01
0.01
0
-10
0
10
u [m/s]
20
30
0 0 -10 40
0
10
u [m/s]
20
30
0 0
1
2
u [m/s]
3
4
0 0
5
10
15
0
u [m/s]
Figure 9. Cross-sectional profiles of gas axial velocity component along the chamber
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20 It turns out that in the vicinity of inlet port all turbulence models deliver good agreement with experimental data without any considerable deviation from each other. In contrast, the second section where the experimental data are still available reveals a high performance of EARSM [34] in capturing well the velocity field in a turbulent swirling flow. Based on this result, further investigations in the remaining part of this paper have been carried out by using this model. In order to characterize phase transition processes ongoing on the droplet surface, and then to qualify the suitable evaporation model, the calculated radial distribution of the droplet mass flux is shown in Fig. 10 at different axial positions using equilibrium and non-equilibrium evaporation models. Generally, the concentration of droplets decreases while moving away from the nozzle due to the evaporation. A comparison between equilibrium and non-equilibrium evaporation models does not reveal a prediction difference as ). As expected, with decreasing long as big particles are involved ( ; droplet diameter the non-equilibrium effects become predominant ( 40 ) and lead to an enhancement of droplet evaporation rate, a behavior similar to figure 3. This is also confirmed in Fig. 11 (with combustion) which shows the droplet concentration zones inside the chamber obtained by using equilibrium and non-equilibrium models in comparison with experimental visualization. A qualitatively better agreement with experiment is achieved
=&
&
X=20 mm
X=5 mm 0.5
Radius [m]
0.04
1
1.5
Equilibrium model Non-equilibrium model
0
0.03
0.02
0.02
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0.01
0
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1
1.5
Droplet Mass Flux [kg/m2s]
0.1
X=40 mm 0.15
0.04
0.04
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0
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0
0.04
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0
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0
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0.15 2
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0
Radius [m]
0
Radius [m]
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0.002
0.004
0.006 0.05
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0.03
0.02
0.02
0.01
0.01
0
0
0.002
0.004
0 0.006 2
Droplet Mass Flux [kg/m s]
Figure 10. Droplet mass flux along the chamber: Comparison between results with equilibrium and non equilibrium evaporation models
by using a non-equilibrium model delivering the correct spray penetration length. Regarding the influence of the turbulent kinetic energy on the vaporization rate, a similar behavior to that in figure 7 has been observed along the axial direction. It appeared clearly that there exists a critical number ) below which the turbulence energy is able to increase the mass ( . transfer. For higher value of this number an inverse behavior is even observed. For the mean droplet size used in calculations, this behavior is also observed independently from evaporation models. The influence of the swirl intensity on the mixing process is depicted in Fig. 12 in which variations of the spatial mixing deficiency parameter (SMD)
`&
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21
are shown at different axial planes located at the positions between and 0.4 nozzle downstream. The results show that an enhancement in swirl intensity leads to a rapid and efficient mixing process providing a homogeneous mixture very close to the inlet port. This fact may be related to the presence of two (internal and corner) recirculation zones formed in the flow for high degrees of swirl (Sw.Nu.¿0.6). As pointed out in [35], an increase in swirl number obviously enhances the size of the internal recirculation zones.
5.2.2. Second Case: Swirled Combusting Spray In this section, some results of scalar distribution obtained under consideration of the conjugate effects of the swirl intensity, the turbulence and the combustion are presented. In comparison to Fig. 12 for non-reacting case, the influence of the swirl intensity on the mixing in reacting case is shown in Fig. 13(Left). One can see that the mixing process parameters are quantitatively different. The heterogeneity seems to be more persistent due to density and viscosity modifications of the flow. While the mixing is already homogeneous in non-reacting case for swirl number Sw.Nu=1, it reachs an at homogeneous state far away at in the reacting case for the same swirl number. In Fig. 13 (right) the effects of swirl intensity on combustion process are shown. An increase in swirl number enhances the temperature and the size of the hot zone in chamber. The hot zone appeared at the outlet area (Fig. 14 (right-a)) is shifted inside the combustor which leads to an homogenization of the temperature distribution in the combustion chamber (Fig. 13(Right-b). This is due to the modification of the flow and mixing dynamics. To investigate the influence of turbulence modulation on some spray properties, let us use different modulation models deduced from Eq. (12) allowing different predicted values of the source terms in the turbulent kinetic energy along with different degrees of turbulence modification prediction.
i =
i
Figure 11. Droplets concentration zones using non-equilibrium (left) and equilibrium (right)
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22
0
0.1
0.2
0.3
0.4
0.4
SMD
Sw. Nu.= 0.0 Sw. Nu.= 0.5 Sw. Nu.= 1.0 0.3
0.3
0.2
0.2
0.1
0.1
0
0 0
0.1
0.2
0.3
X [m]
Figure 12. Spatial Mixing Deficiency (SMD) along the chamber (Non-reacting case)
0.1
0.2
0.3
0.4
0.9
0.9
0.8
0.8
Tt: 918.625
1216.52
1864.58
1899.83
2016.34
2048.51
Sw. Nu.= 0.6
0.6
0.6
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Sw. Nu.= 0.6 0.05 0.04
Y [m]
SMD
0.04
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Sw. Nu.=0.0 Sw. Nu.=0.6 Sw. Nu.=1.0
0.7
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0.03 0.02 0.01
0
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0.3
0 0.4
0 0
0.05
0.1
0.15
0.2 X [m]
0.25
0.3
0.35
0.4
Figure 13. (Left): Spatial Mixing Deficiency (SMD) along the chamber (reacting case). (Right): Effects of swirl intensity on temperature [K] distribution (reacting case). Top (a), Bottom (b).
Focused on the evaporation rate under reacting conditions, Fig. 14 displays effects of the modulation modelling on the calculation of the droplet evaporation rate along the spray core. The influence of the swirl intensity can also be seen. For a given modulation model, a comparison between results for different swirl numbers shows that in contrast to the non-reacting case, the evaporation rate is arising at the beginning and reaches different maximum values while moving along the spray core. These values decrease with decreasing swirl numbers. That is due to the fact that in the combusting case the ambient temperature is much higher than that in non-reacting case and, this might accelerate the vaporization rate. It can be also seen that the
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23 penetration length of spray increases with decreasing the swirl intensity due to the absence of the internal recirculation zone and the relative reduction of as shown in Fig. 13(Left). This gives rise gas temperature for Sw.Nu on one hand to a longer droplet residence time so that the modulation effects become considerable and, on the other hand to a delay in the mixing process which can strongly affect the combustion (see also Fig.12(Left)). For high degree of swirl number (Sw.Nu.=1), due to the high ambient temperature in the combustor, a fast evaporation of droplets in the vicinity of inlet port occurs, so that the droplets/turbulence interaction becomes negligible far away from the nozzle ( ) where droplets are already evaporated. Only in the region close to the nozzle where droplets are still present, the droplet/turbulence interaction is considerable, and an accurate prediction is relevant for a reliable prediction and control of the mixing preparation process. In general, for the same swirl number, it obviously appears in the reacting case that modulation models affect the prediction of the evaporation rate. Thus, besides the role of the temperature, the effect of the turbulence modulation is not negligible. Figure 15 demonstrates clearly this statement. It reveals the influence of the turbulence modulation modelling on the prediction of species concentration in swirled combusting sprays. Comparing the results of different modulation modeling it appears that the model by Sadiki & Ahmadi causes a considerable reduction in predicting the chemical reactions approximately in the middle of combustion chamber exclusively for Sw.Nu. . Due to the well-known reliability of the thermodynamically consistent modulation model by Sadiki and Ahmadi [2, 28, 29], it turns out that the prediction of turbulence interactions in spray achieved by this model may be considered as physically consistent. Figure 15 shows also the influence of swirl intensity on the concentration . of oxygen and carbon dioxide along the centerline at the position It can be observed that increasing the swirl intensity enhances the speed of chemical reaction in ! -consumption and ! -creation in combustion chamber. It is clear that for low swirl intensities (Sw.Nu. ) the oxidator is not consumpted entirely and can be observed at the outlet.
i
i K
i
&
i
6. Conclusions In this work, a complete model has been formulated in order to investigate the influence of turbulence on the global mass transfer rates from fuel droplets. For small droplets, it could be shown that non-equilibrium effects are no longer negligible in the dynamic vaporization process, so that a nonequilibrium evaporation model becomes suitable. The overbalancing of the droplet-induced attenuation (due to small droplets) and production (due to big droplets) in polydispersed sprays has been accounted for by means of
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24 Droplets evaporation rate along the spray core Standard Model (Sw.Nu.=1) Sadiki & Ahmadi (Sw.Nu.=1) Standard Model (Sw.Nu.=0.6) Sadiki & Ahmadi (Sw.Nu.=0.6) Standard model (Sw.Nu.=0.0) Sadiki & Ahmadi (Sw.Nu.=0.0)
1.6E-08 1.4E-08
. mp [ kg/s ]
1.2E-08 1E-08 8E-09 6E-09 4E-09 2E-09 0
0.02
0.04
0.06
0.08
X [m]
Figure 14. Effects of swirl intensity on evaporation rate along the spray core along with comparison between two different modulation models. (Standard model and model by Sadiki & Ahmadi) along the line Y=0.01 (m)
0.15
along the line Y=0.01 (m)
0.3
Sadiki & Ahmadi (Sw.Nu.=0.0) Sadiki & Ahmadi (Sw.Nu.=0.6) Sadiki & Ahmadi (Sw.Nu.=1) Standard Model (Sw.Nu.=0.0) Standard Model (Sw.Nu.=0.6) Standard Model (Sw.Nu.=1)
0.2
O2 mass fraction
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Sadiki & Ahmadi (Sw.Nu.=0.0) Sadiki & Ahmadi (Sw.Nu.=0.6) Sadiki & Ahmadi (Sw.Nu.=1) Standard Model (Sw.Nu.=0.0) Standard Model (Sw.Nu.=0.6) Standard Model (Sw.Nu.=1)
0.1
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X [m]
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0.3 0.25
0.05
0
0.2
0.25
0
0.1
0.2
0.3
0.1
0.05
0
X [m]
Figure 15. Effects of swirl intensity on concentration of and alongthe combustion chamber along with comparing two different modulation models. ( Standard model and model by Sadiki & Ahmadi )
the thermodynamically consistent (turbulence) modulation model by Sadiki and Ahmadi. Coupled to appropriate turbulence models and suitable combustion/radiation models, a CFD-analysis could be performed to gain deeper insight into the processes ongoing in spray systems. To characterize the turbulence-droplet vaporization interaction regimes, a vaporization Damk¨ohler number appeared to be useful in non-reacting cases. To demonstrate the ability of the model combinations used in predicting evaporating spray and droplet properties under turbulent conditions, some representative results calculated by coupling the advanced turbulence model
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25 and the equilibrium and non equilibrium evaporation models have been presented in comparison to available experimental data. Furthermore, the use of the spatial mixing deficiency parameter confirms in the second configuration that the increasing tangential motion of gas causes a more mixing rate of the fuel vapor and the ambient air which provides appropriate conditions for earlier ignition and complete combustion of the gas-fuel mixture. In nonreacting case the effect of the turbulence intensity variations on this parameter seems to be restricted to the region in which the turbulence affects positively the evaporation and heat flux rate. In order to provide more detailed characteristics of these effects, and then to be capable of optimizing the mixing process in a combustion chamber, a temporal mixing deficiency needs to be considered, as well. This task, left for future work, can be well performed by using URANS or LES. Dealing with combustion, the coupling between modulation and swirl number effects lead to following observations: 1) For low levels of swirl intensity (Sw.Nu. ) the modulation effects are dominant. The modulation models affect significantly the calculated vaporization rates. This, in turn, affects the prediction of the fuel vapor distribution, the mixing process and consequently the chemical reactions. Due to the well-known reliability of the thermodynamically consistent modulation model by Sadiki and Ahmadi, the prediction of turbulence interactions in spray achieved by this model may be considered as reasonable. 2) For high degree of swirl number (Sw.Nu.=1), due to the fast evaporation of droplets in the vicinity of inlet port, the droplets/turbulence interaction becomes negligible so that the turbulence modulation does not play a significant role in the results for chemical species concentrations.
i
Acknowledgments The research reported in this paper is sponsored by the Deutsche Forschungsgemeinschaft (DFG) through the Sonderforschungsbereich SFB568 (TPA4). [1] Crowe, C.T, Sommerfeld, M., Tsuji, Y., ”Multiphase flow with droplets and particles”, Boca Raton, Boston, New York, Washington, London. CRC Press LCC, 1998. [2] Chrigui, M., Ahmadi, G., Sadiki, A., ”Study on Interaction in Spray between Evaporating Droplets and Turbulence Using Second Order Turbulence RANS Models and a Lagrangian Approach”, Progress in Computational Fluid Dynamics, pp. 162-174 Special issue, 2004. [3] G¨okalp, I., Chauveau, C., Simon, O., Chesneau, X., ”Mass transfer from Liquid fuel droplets in turbulent flow”, The combustion Institute 1992.
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26 [4] Sornek, R.J., Dobashi, R., Hirano, T., ”Effect of Turbulence on Vaporization, Mixing, and Combustion of Liquid-fuel Sprays”, Combustion and Flame 120:479-491, 2000. [5] Klose, G., Schmehl, R., Meier, R., Meier, G., Koch., R., Wittig, S., ”Evaluation of advanced two-phase flow and combustion models for predicting low emission combustors”, Proc. of ASME Turbo Expo 2000, May 8th-11th, Munich, Germany, 2000. [6] Crowe, T., ”On models for turbulence modulation in fluid-particle flows”, Int. J. of multiphase flow, 26 pp. 719.727 (2000). [7] Smirnov, N.N., Kulchitskiy, A.V., Dushin, V.R., Kazakova A.E., ”Droplet Size Effect in Non-Equilibrium Evaporation”, Proc. of Microgravity Sciences and Processes Symp., October 15, 2002. [8] Miller, R.S., Harstad, K., Bellan, J., ”Evaluation of equilibrium and non-equilibrium evaporation models for many-droplet gas-liquid flow simulations”, Int. Journal of Multiphase Flow 24 (1998) 1025-1055. [9] Berlemont, A., Grancher, M., Gouesbet, G., ”On the Lagrangian simulation of the turbulence influence on the drolpet evaporation”, Int. J. Heat Mass transfer. Vol. 34 No11 pp 2805-2812, 1991. [10] Krebs, W., Gruschka, U., Fielenbach, C., Hoffmann, S., ”CFD-analysis of reacting flow in an annular combustor” , Progress in Computational Fluid Dynamics, Vol.1, Nos 1/2/3, pp. 104-116, 2001. [11] Abramzon, B., Sirignano, W.A., ”droplet vaporization Model for Spray Combution Calculations”, Int. J. Heat Mass Transfer, Vol 32, pp. 16051618, 1989. [12] Sankaran, V., Menon, S., ” LES of spray combustion in compressible high Reynolds number swirling flows”, Proc. of the 2nd International Symp. on Turbulence and Shear Flow Phenomena, Stockholm, Sweden, 2001. [13] Bellan, J., ”Perspectives on Large Eddy Simulation for Sprays”, Issues and Solutions, Atomization and Sprays, Vol. 10, pp. 409-425, 2000. [14] Sadiki, A., Ahmadi, G., ”Modeling of chemically active multiphase turbulent flows on rational extended thermodynamics”, Advances in Turbulence IX, ETC9, U.K. 2002. [15] Jones, W.P., ”Turbulence modeling and numerical solution methods for variable density and combusting flows. In Libby P.A. und Williams F.A.
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27 Herausgeber, Turbulent Reacting Flows”, Pages 309-374. Academic Press, London, San Diego, New York, 1994. [16] Ochs, M., ”Verdunstung monodisperser, frei beweglicher BrennstoffTropfen in einer turbulenten Heissluftstr¨omung”, Ph. D. Thesis, Diss. ETH Nr. 12984, 1999 [17] Sommerfeld, M., Qiu, H.-H.,” Experimental studies of spray evaporation in turbulent flows International Journal of Heat and Fluid Flow ”, Vol. 19, 10-22, 1998. [18] Priere, C., Gicquel, L.Y.M., Kaufmann, P., Krebs, W. and Poinsot, T., ” Large Eddy Simulation predictions of mixing enhancement for jets in cross flows, Journal of Turbulence 5, 005, 1-24 (2004) [19] Zamuner, B., Gilbank, P., Bissieres, D., Berat, C., ”Numerical Simulation of the reactive two-phase flow in a kerosene/air tubular combustor”, Aerospace Sci. and Tech., Vol. 6, pp.521-529, 2002. [20] Hollmann, C., Gutheil, E., ” Flamelet-Modelling of Turbulent Spray Diffusion Flames based on Laminar Spray Flame Library” , Combust. Sci. and Tech., Vol. 135, pp. 175-192, 1998. [21] Rodi, W., ”A new algebraic relation for calculating the Reynolds stresses”, ZAMM, 56, P. 219, 1976. [22] Lilleheie, L.I., Byggstyl, S., Magnussen, B.F., ”Calculations of a Methane Turbulent Diffusion Flame Using Detailed Chemical Kinetics in the Eddy Dissipation Concept”, Technical reports, SINTEF/Norwegian Institute of Technology, Division, 1989. [23] Gosman, A.D., Lockwood, F.C., ”Incorporation of a flux model for radiation into a finite-difference procedure for furnace calculations”, 14th Symp (int.) on Combustion. The Combustion Institute. pp.661-671, 1973. [24] Sharif, M.A.R., Wang, Y.K.E., ”Evaluation of the Performance of the three turbulence closure models in the prediction of confind swirling flows”, ”Computers and Fluids”, 24, n1, pp. 81-100,1995. [25] Kohnen, G., R¨uger, M., Sommerfeld, M., ”Convergence behaviour for numerical calculations by the EULER/LAGRANGE Method for strongly coupled phases”, FED-Vol185, Numerical Methods in Multiphases Flows ASME 1994.
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