NUMERICAL SIMULATION OF HEAT AND MASS TRANSFER, MIXTURE FORMATION IN COMBUSTION CHAMBER OF GAS TURBINE ENGINE

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1 MATHEMATICA MONTISNIGRI Vol XL (2017) MATHEMATICAL MODELING NUMERICAL SIMULATION OF HEAT AND MASS TRANSFER, MIXTURE FORMATION IN COMBUSTION CHAMBER OF GAS TURBINE ENGINE A.A. SVIRIDENKOV, P.D. TOKTALIEV AND V.V. TRETYAKOV Baranov Central Institute of Aviation Motors (CIAM) Moscow, Russia Summary. Results of numerical simulation of mixture formation an liqui fuel evaporation processes in combustion chamber of gas turbine engine are presente. Simple mathematical moel of non-equilibrium evaporation of liqui fuel is propose, moel postulates epenence of roplet evaporation rate from fuel vapor partial pressure above the roplet surface. To estimate significance of propose epenence numerical simulation of isothermal an non-isothermal turbulent flow in computational omain, which imitates geometry of combustion chamber, is performe. For isothermal case influence of interfacial impulse exchange, relate with iscrete phase propagation, on flow structure is estimate. In nonisothermal case it was shown, that taking into account non-equilibrium of evaporation in propose way is essential at temperatures of gas phase T< K. 1 INTRODUCTION Aircraft engine performance epens strongly of the efficiency of mixing an burning of the liqui fuel in the combustion chamber. Description of propagation an evaporation of the iscrete liqui or soli phase of the fuel in challenging conitions occurring in the combustion chambers is actual problem for the moern aircraft engines. This problem is irectly relate to the task of escribing the turbulent multi-component flow (i.e. bulk phase) in the combustion chamber. Goal of this paper is to evelop a numerical metho for couple solution of the problems of computational flui ynamics, the propagation an evaporation of the rop-liqui phase of fuel in the non-equilibrium regime in the combustion chamber of an aircraft engine. First attempts of escription of heating an evaporation of the rop phase of fuel for the following ignition have been mae in en of the 19th century. This activity has been intene for evelopment of the internal combustion engines for the liqui fuels. So in 1877 one of the first moels of evaporation of a single rop has been propose, this moel uses an assumption that the evaporation process epens only on the iffusion of vapors from the surface [1]. In the first half of the 20th century mathematical escription of the heating an evaporation of the fuel roplets has been evelope, however, most moels use in practice were base on experimental ata an ecouple escription of the rop-liqui phase an surrouning gas. These moels are presente in the survey papers [1, 2] an the references therein. Development of computational flui ynamics (CFD) in the secon half of the 20th century not only allowe us to consier the problem of the rop-liqui phase an the surrouning gas flow in couple formulation, but also raise the task of the computational complexity of the iscrete phase moels. This perio until the mi-90s is characterize by the use of methos of continuum mechanics an the refinement of the mechanisms an characteristics of mass, 2010 Mathematics Subject Classification: 70-04, 76T15, 80A20. Key wors an Phrases: Combustion Chamber, Heat an Mass transfer, Multiphase Flow 127

2 momentum, energy transfer in the liqui-liqui phase an the surrouning gas. Continuous meium mechanics an more accurate efinition of the mechanisms an characteristics of mass, momentum, energy transfer in the roplet-liqui phase an the surrouning gas have been use until the mi-90s. Mathematical moels of heating an evaporation of the ropliqui phase an their application are given in [3-8]. In the last years the multicomponent structures of the iscrete phase an the evaporation kinetics have been use for escription of the heating an evaporation of the rop-liqui phase. Extene survey of these moels can be foun in the monograph [9] an [10-12]. A number of particular problems such as evolution of roplet shape, computation of real properties of the bulk an iscrete phases an features of coalescence an ecay of iniviual rops are iscusse in [13-15]. Analytical solutions in the form of Stokes an Oseen for iniviual rops are often use at low Reynols numbers (Re <100) [16]. Now all the mathematical moels for the heating an evaporation of the iscrete ropliqui phase in a turbulent flow can be ivie into several groups [15] in orer of increasing complexity an the computational efforts require: equilibrium moels of stationary heat transfer of the rop with the surrouning meium (this case correspons to the same roplet temperature in time an over the entire volume of the roplet), moels using the approximation of the infinite molecular thermal conuctivity of the roplet liqui, moels with finite heat conuctivity taking into account the flow insie the rop an moel base on the solution of the full Navier-Stokes equations for two-phase flow. Application of moels of the last group for solving engineering problems nees impressive computational efforts; therefore, results of such computation are use for the verification of moels of the first groups. In aition, the most engineering applications assume that the sizes of iniviual roplets () are small in comparison with the characteristic linear imensions of the flow L (i.e. L), so the imensions of the roplets are neglecte. Thus, the problem of couple computation of the iscrete phase an the flow fiel reuces to solving the Navier-Stokes equations for the bulk phase an the trajectory problem for a set of material rop-points. Local heat an mass transfer problem can be solve separately for each of rop-point. The moels can be classifie by the criterion of equilibrium of the single rop evaporation. In the equilibrium approximation it is assume that the vapor near the interface is saturate an the evaporation rate of the liqui from the surface of the rop oes not epen on the vapor concentration near its surface. This approximation is vali for large heat fluxes to a rop [17] an with enough accuracy correspons to the typical conitions in the combustion chamber with the initial temperature of the iscrete phase T f ~300 K an the temperatures of the bulk flux T>1000 K. However, heat fluxes into the rop can be significantly less on the transitional moes of the combustion chamber or in the preheating of the fuel-air mixture. Goal of this paper is formulation of the mathematical moel of the heating an nonequilibrium evaporation of the rop-liqui phase of the fuel for real conitions of the combustion chamber of the gas turbine engine an also etermination of the temperature range of the bulk phase with account of non-equilibrium effects. Also the paper presents the results of mathematical moeling of the complex physicochemical process of heating an evaporation of the iscrete phase of fuel in the combustion chamber of an aircraft engine using the propose mathematical moel. 128

3 2 MODEL OF DROP-LIQUID FUEL EVAPORATION Couple solution of the problem of flow fiel an iniviual roplet characteristics etermination can be buil by sequential etermination of roplet parameters an then using this ata in solution proceure of Navier-Stokes equations for bulk phase. In this approach, the exchange of mass, momentum an energy between the bulk an iscrete phases is escribe by source terms in Navier-Stokes equations an heat-an-mass transfer of the rop equations. Usually these source terms are calculate locally for each noe (volume) of computational omain in case of using mesh methos for numerical solution. Thus, it is possible to use various numerical algorithms to solve aeroynamic, trajectory problems, the problem of heat-an-mass transfer of a rop. Matching of the problems can be accomplishe by exchanging for each cell in computational omain the values of the change in the mass of liqui an vapor, momentum, an energy. The calculation of the flow fiel in this paper is base on the solution of the unsteay Navier-Stokes equations in the framework of the etache vortex metho [18]. For the subgri viscosity, the Smagorinsky moel with the constant C=0.1 is use. The computational proceure is base on the SIMPLEC algorithm [19]. The solution of the trajectory problem is carrie out by the metho of [20] an is base on the solution of a system of orinary ifferential equations with initial conitions escribing the motion of roplets, each of equations has the following form: mi() t X i = Fr, i(, t Xi, X i), i = 1.. N t = 0: X = X, X = X i,0 i,0 here i - number of roplet, m i (t) - mass of i-th roplet, X - vector of roplet coorinates, F r,i - resistance force acting on the roplet when it moves. The moeling of the roplets heating an evaporation processes in this paper is base on integral approach [4]. Thus, the iterative process of heat an mass transfer calculation for each time step can be organize accoring to the following scheme: 1. Set the initial values of all variables - pressure p, velocity V, temperature T, mass fractions of vapor an air Y i for bulk phase, coorinates an velocities for all roplets X i, X i, iameters an surface temperatures for all roplets i,,t p,i ; 2. Solution of gas-ynamic problem (Navier-Stokes equations), ODE system for roplet motion, equations of heat an mass transfer for each of the roplets; 3. Upating the source terms in the Navier-Stokes equations associate with the interphase exchange of mass, momentum an energy; 4. Checking the convergence, avancement to the next time step. Let us consier more closely the problem of etermining the characteristics of a single roplet for known local values of bulk phase characteristics at the time t=t i. From the balance of heat fluxes on the surface of roplet of raius r, the following transcenental equation can be obtaine for etermining the rate of evaporation of a roplet [4]: m 4π r Cp( T T) λ = ln 1 + t C Q. (1) p H LV + m / t 129

4 Here m mass of the roplet, T roplet surface temperature, T temperature of bulk phase in computational cell with the roplet, C p bulk phase specific heat coefficient uner constant pressure, bulk phase heat conuction coefficient, L V latent heat of evaporation, Q H heat flux to roplet, leaing to roplet heating without evaporation: Q = 4 π rα( T T ), (2) H where - heat transfer coefficient between roplet an surrouning bulk phase. As 3 2 m πρ πρ = = ( ), where roplet iameter, liqui of the roplet t t 6 2 t mass ensity, then equation (1) takes the following form: t 4λ Cp( T T) = ln 1 + Q. (3) ρc p H LV + m / t In case of equilibrium evaporation, when Q H = 0, equations (1) an (3) are transforme into m 4π r Cp( T T) λ = ln 1 +, t C p LV 4λ Cp( T T) = ln 1 +. t ρc p LV Equating the mass fluxes through a spherical surface of raius r an the surface of a roplet of raius r an assuming that the roplet evaporation rate is etermine by the iffusion of vapors from its surface, one can obtain the following equation, which relates the roplet evaporation rate with the istribution of fuel vapor concentrations, see, for example [4] an [10]: 2 m 4π rd V ρ Y =, (4) t 1 Y r where Y imensionless vapor concentration, ensity of gas mixture (air an vapor), D V vapor iffusion coefficient. Integration of this equation uner the assumption that the values of an D V can be replace by their mean values gives the following expression for the istribution of the imensionless vapor concentration Y over the raius r: 1 Y 1 Y = 1 Y 1 Y where Y an Y imensionless vapor concentration at the roplet surface an in the surrouning space respectively. It can be note, that in equation (1) function uner the logarithm sign has a maximum value at Q H = 0, as in the equilibrium evaporation case. For normal liquis, for example water r / r, 130

5 an kerosene, C p 10 3 J/K, L V J/kg. Then, at low gas flow tempearatures of the orer 600 K an initial roplets temperatures of the orer of 300 K, T 300 K an x = C p T/L V 0.6, that is, less than 1. In this case, equation (1) can be represente in the m 4π r λ form: = ( T T ) or t QH LV + m / t m 4 π rλ( T T) = 1 + t m LV + Q H t 1. (5) Equation (5) allows to fin the rate of roplets evaporation at low temperatures of the gas flow (bulk phase). Let's fin the rate of heating of a single roplet assuming that its heating takes place so fast that the temperature of the roplet surface is approximately equal to the temperature of the liqui insie the roplet. The heat flux into the roplet Q is equal to the sum of heat fluxes associate with the heating of roplet Q H, roplet evaporation Q V an the loss of heat, relate with vapors convection (neglecting raiation heat flux). The equations of heat balance on the surface of a roplet of raius r an on the sphere of raius r have the following form: T m m Q = m + LV Cp,VT t t t 2 T m Q= 4 πr λ pt. r t In the last equation, the first term on the right-han sie represents the heat flux to the roplet 2 T ue to the thermal conuctivity of the gas: QH = 4πr λ, an the secon represents the r convective heat losses associate with vapor motion.; C specific heat of a roplet liqui, C p,v vapor specific heat coefficient at constant pressure. Equating the heat fluxes through the spherical surfaces of the roplet an in the surrouning space an integrating this equation over the raius from r to an from T to T in temperatures, assuming that the values of the quantities in the equation can be replace by their mean values, one can obtain: T 1 m = 4 π r λ( T T ) ( L C T + C T ) t mc t V p p, V. This equation escribes the heating of the roplet an iffers from the analogous equation of [21], where it is assume that C p,v T = C p T an where the heat Q H is calculate using the heat transfer coefficient of the roplet with the surrouning meium, see (2). The corresponing equation of [4] is written for the case m/t =0. To obtain the epenence of the roplet heating rate on the istribution of the vapor concentration in the space surrouning the roplet, it is necessary to solve together the heat transfer an roplet evaporation equations, which have the following form [4]: 131

6 T m r = L + C T C T + Q, r t m 2 YV m = 4π rdvρm + YV. t r t 2 4 π λm ( V p, m p, V ) H In the latter expression, the first term on the right-han sie is the vapor mass flux ue to iffusion, the secon is the vapor mass flux ue to convection (the quantities associate to the mixture are enote by the subscript "m"). Eliminating the raial coorinate from the system of equations, one can obtain after transformations: T YV κ( T α ) = 1 Y V, (6) DVρ mcp, m L C T Q where κ =, λm Cpm, C m pm, C t pm, from T to T an from Y to Y to Y V, assuming that an are inepenent from the raial α = V pv, + H. Integration of (6) with respect to T 1 Y, coorinate gives (1- ) = T - T, where ς V T =. Since QH = Cm, then after 1 Y t transformations: κ T 1,, ( ) m CpVT L C V pm Tς T = +. (7) t m t C C C(1 ς ) Equation (7) expresses epenence of the roplet heating rate on its evaporation rate, on its temperature an ambient temperature, on the istribution of the vapor concentration, an on the properties of the air-vapor mixture an roplet liqui. Note that integrating the equation (4) in the ranges (r, r ) an (Y, Y ) assuming that the values of an D V can be replace by their mean values, gives the following expression for the roplet evaporation rate, which can be use for solving equation (7): m = 4 πdvρmrlv(1 + Bm), (8) t Y Y where B m - mass transfer coefficient: Bm =. 1 Y Accoring to [22] mass fractions of fuel vapor near the roplet surface Y can be expresse in the following way: Y 1 =, p μg 1+ 1 pv μl where p pressure in the surrouning meium, p v saturation vapor pressure near the roplet surface, g - molecular weight of air-vapor mixture in the roplet surrouning meium, l - 132

7 molecular weight of roplet liqui. To calculate saturation vapor pressure, Van er Waals equation can be use [22]: p = exp( A B/ ( T 43)), v here A an B Van er Waals constants, T - roplet surface temperature. Note that if the saturating vapor pressure near the roplet surface, p v, is equal to the gas pressure in the surrouning meium, p, then T = T vap, where T vap - evaporation temperature of the roplet: T vap = B 43 + A ln p. The heating of the roplet occurs as long as the temperature of the roplet is less than its equilibrium value, e, such that all the heat absorbe by the roplet goes to its evaporation an is tranferre to the fuel vapor. This situation correspons to the zero value of Q H in formulas (1), (3), (6) an expression in square brackets in (5). The couple solution of the heat balance equation (5) an the mass transfer equation (8) gives in this case following expression, see, for example, [4]: ζ 1 Te = T LV. (9) Here =, = (1 - Y V )/(1 Y V 0 ), where Y - vapor concentration in the surrouning meium, Y - vapor concentration at the roplet surface, mixture coefficient = D V m Y m / m, Y m vapor mass concentration in mixture. Note that equation (9) is transcenental, since the specific heat of evaporation L V in it is taken at the equilibrium temperature T e. Thus, in the case of noneqiulibrium evaporation of a single roplet uner the consiere assumptions, the ifferential equations (7) an (8) have to be solve, in the case of equilibrium evaporation, the problem reuces to solving equation (7) an the transcenental equation (9). 3 NUMERICAL SIMULATION Let's consier the application of the propose proceure for calculating characteristics of the heating an evaporation processes of the iscrete phase for the problem of multiphase flow moeling in the combustion chamber of an aircraft engine. The scheme of computational omain is represente in fig. 1. Air flow enters fire tube 1 through axial 2 an raial 3 inlets. Liqui fuel enters the combustion chamber through exit nozzle 4 of atomizer 5. Fig. 1a represents the entire computational omain with aitional volume imitating a free space. Bounary conitions with fixe mass flow rates an temperatures of gas an liqui were expose on surfaces corresponing to inlets 2 an 3, 4, with fixe static pressure in the outlet section corresponing to the base of the cyliner imitating a free space. For soli impermeable surfaces, no-slip, aiabatic conitions for continuous meium an elastic reflection conitions for iscrete phase were set. The bounary conitions for turbulent characteristics were similar, with a number of simplifications use in [23]. All simulations were performe using FORTRAN program moules on structure curvilinear gri. pm, 133

8 a b Fig. 1. Computational omain scheme (a) an zoome view of volume near the inlet section (b), D - exit iameter of the nozzle, D=44.5 mm 3.1 Characteristics of bulk phase As a first step in the solution of the problem of heating an evaporation of the iscrete phase, a numerical solution of the unsteay Navier-Stokes equations in the consiere computational omain without iscrete phase was obtaine. For this step no-slip bounary conitions were set on the surface corresponing to inlet 5 (fig. 1). In the secon step, an isothermal problem with a iscrete phase was consiere, the temperatures of iscrete an bulk phases were the same T=298 K. It was assume that the break up of the fuel film occurs in the immeiate vicinity of the nozzle outlet section (inlet 5 fig. 1). It was assume also that the seconary break up of roplets is etermine by the Weber criterion [24], an their coalescence is escribe by the probability laws of roplets collision. The physical time step for bulk an iscrete phases are the same an equals to t=10-5 s. At this step the influence of injecte mass of the iscrete phase on the flow structure in primary volume of combustion chamber was estimate. The calculations are performe for the liqui phase - a conventional one-component liqui fuel with chemical formula C 12 H 23 at various mass flow rates G f = 0, 3.75, 5 an 7.5 g/s. In fig. 2 as an example of simulation of the velocity fiel for an isothermal problem, the fiels of the instantaneous longituinal an circumferential velocity components for fuel mass flow rate G f = 5 g/s at a time moment corresponing to statistically steay flow t=0.05 s (N steps =5000) are shown. It can be seen from the figure that reverse flow zone (RFZ) exists in the near-axis area of the combustion chamber an velocity magnitue in the entire computational omain oes not excee 80 m/s. Reverse flow zone is use in most moern aviation combustion chambers to stabilize the combustion process, moreover experimental an numerical stuies [23] show, that characteristics of RFZ in most cases etermine the overall performance of combustion chamber. Thus, it is important for practical reasons to investigate influence of injecte iscrete phase on geometry of the RFZ. 134

9 a b Fig. 2. Instantaneous fiels of axial Ux (a) an circumferential Uz (b) components of the velocity in the mile axial section of the computational omain, isothermal problem, t=0.05 s. Colors correspon to the velocity components values The average geometric characteristics of the RFZ for ifferent mass flow rates of the iscrete phase represent in Table 1. Here the spatial characteristic scale is R 0 = 20 mm an abscissa of a section corresponing to maximum with of RFZ are given in last column of the table. Thus, numerical simulations show that the length of the region with large negative values of axial velocity insie the RFZ ecreases with increasing of iscrete phase mass flow rate G f. G f, g/s L RFZ /R 0 R RFZ / R 0 X RFZ, mm Table 1. Average geometric sizes of RFZ for ifferent mass flow rates of iscrete phase 3.2 Non-isothermal problem with iscrete phase Estimations for low temperatures of the bulk phase an kerosene roplets (C p 10 3 J/K, L V J/kg) show, in particular, that at gas flow temperatures of the orer 600 K an initial roplet temperatures of the orer 300 K, T 300 K an x = C p T/L V 0.6. In this case, the equilibrium approach can lea to a significant error in etermining the mass of vaporize 135

10 liqui for the roplet an the temperature of its surface. At high temperatures T>1000 K, the heat that evaporates the roplets can be neglecte, an the calculation proceure is substantially simplifie. Below results of numerical simulation of non-isothermal flow with iscrete phase in computational omain from fig. 1 are represente. Fig. 3 shows mean istributions (average for 200 time steps without weight function) of the specific mass concentrations of the liqui phase C for pressure level in computational omain c =5 atm an gas inlet temperature c = 400 K. Bounary conitions on surface 4 (fig. 1) for injecte roplet temperature is T = 300 K. Fig. 3-a correspons to the calculation of roplet evaporation with equilibrium moel (without taking into account the influence of fuel vapor concentrations on evaporation rate), fig. 3-b correspons to propose non-equilibrium moel. a b Fig. 3. Fiels of average specific concentration of liqui phase C in the mile axial section of computational omain for equilibrium moel (a) an non-equilibrium moel (b), bulk phase temperatures at the inlet section c = 400 K, axis correspons to imensionless coorinates X=x/R 0,Y= y/r 0. Colors correspon to the concentration C values It can be seen from these figures that uner consiere conitions the istributions of iscrete phase obtaine with non-equilibrium moel, which takes into account influence of saturate vapor pressure on roplet evaporation rate, qualitatively iffer from those obtaine with equilibrium moel. Similar results also occurs at pressure levels in computational omain c = 1 15 atm. With increasing temperature in the combustion chamber, the effect of 136

11 the evaporation non-equilibrium are becoming less visible, an at = 500 K there are only quantitative ifferences in the istributions of the roplet concentrations, even closer these istributions at c = 600 K an at higher temperatures. The last observation is illustrate in fig. 4 for pressure levels in computational omain c = 5 atm. Fig. 4-a an 4-b correspon to the calculations with the equilibrium an non-equilibrium moels respectively for inlet bulk phase temperature c = 500 K. a b Fig. 4. Fiels of average specific concentration of liqui phase C in the mile axial section of computational omain for equilibrium moel (a) an non-equilibrium moel (b), bulk phase temperatures at the inlet section c = 500 K, axis correspons to imensionless coorinates X=x/R 0,Y= y/r 0. Colors correspon to the concentration C values 4 CONCLUSION Results of liqui fuel mixing an evaporation processes numerical simulation in gasturbine engine combustion chamber are presente. Due to wie variation of thermoynamic parameters in inustrial combustors at low temperatures of bulk an iscrete phases nonequilibrium evaporation effects coul be significant. Multicomponent turbulent flow with iscrete phase numerical simulation technique is consiere. Moel of non-equilibrium evaporation of liqui fuel is propose, it takes into account in integral form epenence of single roplet evaporation rate on vapor partial pressure near roplet surface. Numerical simulation of isothermal flow with a iscrete phase reveale, that interphase exchange of 137

12 momentum simulate by source terms in Navier-Stokes equations (at typical mass flow rate ratio G f /G air <10%) coul consierably change the structure of mean flow in computational omain. At mass flow rate ratio G f /G air =5.5% momentum source, generate by iscrete phase leas to increase of mean integral flow scale (mean length of RFZ) on 9% in comparison with single-phase flow. Numerical simulation of non-isothermal flow with a iscrete phase reveale that at initial temperatures T,0 =300 an T c,0 =400 K ifference between equilibrium an non-equilibrium moels for values of local liqui phase specific concentration can be as large as local mean concentration. Increasing of gas phase temperature to T c,0 =500 K with fixe T,0 value leas to consierably lower ifference between equilibrium an non-equilibrium moels results. Thus, increasing of the heat flux per roplet leas to vapor partial pressure increasing near the roplet surface an vapor saturation. Equilibrium roplet evaporation moel has lower computational cost, therefore application of this moel at moerate an high temperatures of bulk phase T> K coul lea to economy of computational resources while liqui phase evaporation process escription accuracy remains the same. I.e. ifference connecte with non-equilibrium of evaporation process at temperatures T> K is negligible in escribe statement. Acknowlegement: The work is supporte by the Ministry of Eucation an Science of the Russian Feeration. Grant agreement , title Development an valiation of numerical methos for moelling of operational processes in combustion chambers of perspective gas turbine engines" (project ID RFMEFI62815X0003). REFERENCES [1] N.A. Fuchs, Evaporation an roplet growth in gaseous meia, Pergamon Press (1959). [2] R.W. Schrage, A theoretical stuy of interphase mass transfer, Columbia University Press (1954). [3] D.B. Spaling, Convective mass transfer. An introuction, Ewar Arnol (1963). [4] B.V. Raushenbach, S.A. Belyi, I.V. Bespalov, V.Ya. Boroachev, M.S. Volynskii, A.G. Prunikov, Fizicheskiye osnovy rabochego protsessa v kamerakh sgoraniya vozushnoreaktivnykh vigateley, Mashinostroyeniye (1964). [5] V.A. Boroin, Yu.F. Dityakin, L.A. Klyachko, V.I. Yagokin, Raspylivaniye zhikostey, Mashinostroyeniye (1967). [6] G.M. Faeth, "Current status of roplet an liqui combustion", Prog. Energy Combust. Sci, 3, (1997). [7] W.A. Sirignamo, "Fuel roplet vaporization an spray combustion theory", Prog. Energy Combust. Sci, 9, (1983). [8] A.H. Lefebvre, Atomization an sprays, Taylor & Francis (1988). [9] S.S. Sazhin, Droplets an sprays, Springer (2014). [10] S.S. Sazhin, "Avance moels of fuel roplet heating an evaporation", Prog. Energy Combust. Sci, 32, (2006). [11] C. Yin, "Moeling of heating an evaporation of n-heptane roplets: towars a generic moel for fuel roplet/particle conversion", Fuel, 141, (2015). [12] E. Mahiques, S. Deerichs, C. Beck, P. Kaufmann, J. Kok, "Coupling multicomponent roplet evaporation an tabulate chemistry combustion moels for large-ey simulations", Int. J. Heat Mass Transfer, 104, (2017). 138

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