Fuel Droplet Evaporation Modeling using Continuous Thermodynamics Method with Multi-Distribution Functions

International Multidimensional Engine Modeling User s Group Meeting at the SAE Congress, April 23, 2012, Detroit, MI Fuel Droplet Evaporation Modeling using Continuous Thermodynamics Method with Multi-Distribution Functions Cai Shen 1, Way Lee Cheng 1, Chia-fon F. Lee 1,2 1 Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, IL, 61801, USA 2 Center for Combustion Energy and State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China Abstract A finite diffusion droplet evaporation model for complex liquid mixture composed of different homogeneous groups is presented in this paper. Separate distribution functions are used to describe the composition of each homogeneous group in the mixture. Only a few parameters are required to describe the mixture. Quasi-steady assumption is applied in the determination of evaporation rates and heat flux to the droplet, and the effects of surface regression, finite diffusion and preferential vaporization of the mixture are included in the liquid phase equations using an effective properties approach. A novel approach was used to reduce the transport equations for the liquid phase to a set of ordinary differential equations. The proposed model is capable in capturing the vaporization characteristics of complex liquid mixtures. Introduction Commercial fuels are mixtures of hundreds of chemically different hydrocarbons with vastly different boiling points that could range from 340 K to over 700 K. Law [1] pointed out that liquid motion within the droplet, miscibility among the liquid components and relative volatilities of the components are the three controlling factors in the understanding of the multi-component fuel behaviors. The immensely different volatilities among the components imply significant differences in the evaporation rates. Moreover, the liquid constituents can evaporate only if it reaches the surface. As the more volatile species evaporates, less volatile constituents become dominating within the liquid phase. As a result, the species mass fractions and temperature are no longer uniform within the droplet. This process is known as preferential evaporation. In most numerical simulations, the fuel is usually represented by a single component, for example, tetradecane (C 14 H 30 ) is usually used to represent commercially available diesel. A major deficiency with this approach is that the influence of fuel composition is not accounted for, and, only the average evaporation behavior can be obtained. A possible solution to this is to use a set of fuel constituents to reproduce the distillation curve. An accurate representation of the fuel is essential for acquiring insightful information out of a simulation. The volatility of the fuel maintains a dominant position on spray penetration, and ignition is controlled by the most volatile species in the mixture [2, 3]. However, to represent each component in a commercial fuel, which consists of hundreds of component, using a discrete representation is impractical. Not only every component requires a separate transport equation, the exact composition is in generally unknown. As an alternative, continuous thermodynamics provides a more effective solution. The mixture is characterized by a probability distribution function (PDF) with respect to some characterizing variables, for examples, molecular weight or boiling points. Only a few parameters are required to describe the mixture, namely, the mean and variance of the probability distribution function. This approach was first developed in chemical engineering and has been applied in a variety of calculations including vapor-liquid equilibrium, liquid-liquid equilibrium, flash boiling and characterization of hydrocarbon mass fraction. It was first used for investigating the vaporization of isolated droplets by Tamim and Hallett [4]. The gas phase transport equations were solved with fine grid spacing close to the droplet surface. But this is infeasible for multi-dimensional spray simulations, which involve thousands of droplet parcels and complex engine geometry. Hallett [5] presented a quasi-steady analytical solution for droplet vaporization and applied his model to computations of diesel and gasoline, which captured important features, such as the distillation curves of commercial fuels, quite well. This model was extended to fuel droplet vaporization in a high-pressure environment [6]. The method of continuous thermodynamics has been applied in multi-dimensional engine modeling for spray calculations [7-11]. These studies made important contributions to the application of continuous thermodynamics to internal combustion engine simulations. However, most of the previous studies [9-18] are based on the assumption of infinite diffusion in

the liquid phase, so non-uniformity inside the droplet is not considered. Mathematical Formulations B j = y j surf y j Ṅ'' Ψ j Ṅ'' y j surf. (4) In this paper, a comprehensive model considering preferential vaporization of a complex fuel mixture using continuous thermodynamics without the infinite diffusion assumption in the liquid phase is presented. Gas Phase Model A fuel mixture is described by a continuous probability distribution function, f(ω), characterized by the molecular weight, ω, of each component. The fraction of a component is y F f(ω i )ω, where y F is the total molar fraction of fuel in gas phase, and in this study, f(ω) is a Γ- distribution function, which is often used to represent petroleum fractions [20], exp, (1) where γ is the origin, α and β are parameters controlling the shape of the probability distribution function, with the mean molecular weight, θ 1 = αβ + γ, and the variance, σ 2 = αβ 2. In previous studies, using the second moment, θ 2 = σ 2 2 + θ 1, was found to be more convenient for calculations than using the variance [12]. Here, θ j is the j th moment of f(w). The transport equations for energy and molar fuel fraction, y F, can be obtained by integrating energy and species conservation equations over the probability distribution function and two additional transport equations for θ 1 and θ 2 are achieved by performing proper weightings over the equation for fuel fraction. The gas phase transport equations for droplet vaporization, using continuous fuel representation, were first completed by Tamim and Hallett [12]. The methodology is similar to previously studies within the frame work of continuous thermodynamics. The vapor-phase quasi-steady solutions are given by: Ṅ'' = Sh C j vap D 0j 2 R ln (1 + B j ), (2) Ṅ'' Ψ j = Ṅ'' Φ j R Φ j Φ j R D 0 j 1 ( 1 + B ) / D Ψ j, (3) j where Ṅ'' is the total vapor flux, and Ṅ'' Ψ j is the flux for j th moment; D yf and D θj is the average diffusivity for fuel composition and j th moment, respectively, and B j is the transfer number given by, Note that the Sherwood number is evaluated using fuel fraction, since the diffusivities for composition and the moments are almost the same. The heat flux to the droplet, q '', is given by, q '' = Ṅ'' C p ( T T R ) 2 Ṅ'' C p R exp k vap Nu 1 ΔH fg, (5) where C p, ΔH fg, k vap and Nu are the vapor phase specific heat capacity, latent heat for vaporization, thermal conductivity and Nusselt number, respectively. Liquid Phase Model The transient behavior of the liquid phase significantly affects the evaporation of a spray and thus, the vapor distribution. In the present study, an effort is made to properly model the non-uniformity of liquid phase due to finite diffusion vaporization rather than using the infinite diffusion model as in the study by Tamim and Hallett [4]. The transport equations for the liquid phase are obtained with the same approach as used in the gas phase. The complete liquid phase equations are too complex to be solved analytically. Previous studies on the liquid phase during droplet vaporization have provided a lot of knowledge and inspiration on the simplifications to the problem while retaining reasonable accuracy. Internal circulation enhances diffusion in the liquid phase and this effect can be accounted for by an effective diffusivity and conductivity [21, 22]. In addition, assuming that in a short time interval, molar concentration and specific heat are constant, and that the flow is symmetric since the effects of internal vortices are included in the effective properties [23], the liquid phase equations can then be simplified and written as: C liq Φ t = 1 r 2 r r 2 C liq D Φ Φ r, (6) where the subscript liq represents liquid phase, Φ is a N vector of { Φ j } j=1. Note that the diffusivities in equation (6) are multi-component diffusivities, which are in general given by, D Φ = B 1, (7) for an ideal mixture. The elements of B are calculated 2

from the binary diffusion coefficients and molar composition of the mixture as, N B ii = y i Đ in + k=1 i k y i Đ, B ij = y i 1 ik Đ 1 ij Đ. (8) in The boundary conditions for liquid phase are given by, Φ r r=0 Φ = 0, C liq D Φ r r=r = Ṅ''Φ Ṅ Φ '' j = Ω Φ. (9) The conductivity and diffusivities in the equations are all effective properties. Symmetry imposes the boundary conditions at the droplet center, and conservation conditions are applied at the droplet surface, Φ r r=0 = 0, C D Φ liq Φ r r=r = N Yf Φ N Φ = Ω Φ. (10) By assuming C liq and D Φ constant, as in Toor [24] and Stewart and Prober [25], equation (6) can be written as: C liq Φ t = C liq D 1 Φ r 2 r r 2 Φ r, (11) the system of equations can then be linearized and decoupled by decomposing the diffusivity matrix using eigenvalue decomposition. It has been shown that eigenvalue decomposition always exists for the diffusivity matrix [26], Λ = Σ 1 D Φ Σ, (12) where Λ is the diagonal eigenvalue matrix, and Σ is the associated eigenvector matrix. Substitute equation (12) into equations (10) and (11) and multiply each of the equations by Σ 1, yielding equations (13) and (14), C liq Φ eig t = C liq Λ 2 Φ eig r 2. (13) where Φ eig = Σ 1 Φ and Ω Φ, eig = Σ 1 Ω Φ. The boundary can be transformed in a similar manner yielding: Φ eig r = 0, C liq Λ r=0 Φ eig r = r=r Ω Φ, eig. (14) These equations are in the same form as those for a discrete multi-component fuel representation by Zeng and Lee [27] if the first and second moments of the probability distribution function are considered as two discrete components. It is possible to find a relation between the surface and average properties of a droplet to approximate the effect of finite diffusion rate in the liquid phase by using the same approach as Zeng and Lee [7]: first, apply the quasi-steady state assumption and obtain an analytical solution of Equation (5). Then, obtain the transient part of the solution by separation of variables and retain only the first order term in the series solution. Finally, an approximate solution is derived using the initial condition. With this approximate solution, an expression for the difference between the surface and mean values can be obtained. The time differential form of this expression is dφ dt R 2 D R eff = λ2 1 5 D 1 eff Ω Φ I + Pe v Φ, (15) where Φ Δ = Φ r=r Φ mean (the difference between the surface and mean values), λ 1 = 4.4934, D eff = Ξ 1 (Pe v ) D, Ṙ is the regression rate of the droplet and Ξ 1 (Pe v ) is the enhancement coefficient for droplet surface regression, which is defined by Zeng and Lee [6] as: where ε(r,pe v ) = and Ξ(Pe v ) = [ ε 1 (1, Pe v )][ ε(r, Pe v )], (16) r 2 exp ( r 2 ) 2 Pe v 0 r λ 2 1 χ 2 exp ( χ2 ) 2 Pe v dχ (17) Pe v = R dr dt D 1 (18) The average values of the moments and temperature in the liquid phase are determined by: d dt ( C liq R 3 Φ ) mean = 3R 2 Ṅ''Φ, (19) d dt ( C liq R 3 C ) p liq T liq mean = 3R 2 q ''. (20) where the fluxes are given by the gas phase sub-model. Thus, the surface value can be obtained. Results and Discussion Model Verification Numerical calculations were carried out for heptanedecane droplets evaporating in moving air. Heptanedecane droplets evaporating in ambient air of 101 kpa and 348 K were simulated. The free stream velocity of the 3

ambient air was 3.1 m/s. Droplets of two different compositions, one composed of 75% heptane and 25% decane with initial diameter of 1484 μm and another one composed 25% heptane and 75% decane with initial diameter of 1360 μm, were simulated and compared with experimental data adopted from Daïf et al. [29]. Figure 1 shows the variation of normalized surface area both droplets. The proposed results predictedd the surface regression process well. For the 75% decane droplet, the rate of surface regression was almost constant and resembled that of pure decane [29]. Note that the liquid phase was dominated by decane rapidly after evaporation started. A two-stage processs was observed for the 75% heptane droplet. 30 atm. Droplet size was measured in the experiment usingg a video recording of a droplet suspended from a quartz wire. The regression of droplet surface area is shown in Figure 2. The numerical model predicted the initial increase off droplet size during heating-up and surface regression of the droplet was also correctly estimated, albeit with some discrepancies. Thermal radiative heat transfer was an important phenomenon and its effect wass included in the calculation by the correlations in Lippert et al. [31]. The discrepancies between numerical resultss and experimental measurements were very likely a consequence of uncertainties and/or fluctuations in measurements and cyclee variations in the measurements. The droplet surface area reduces in a non-lineaprocess. The maximum percentage manner after the initial heat-up increase in droplet surface during initial heat-up was about 6%. Thermal conduction during the heat-up stage caused thermal expansion in the droplet and the droplet size was the net difference between the thermal expansion and evaporation. Moreover, both high pressure real gas effects and multi-component effects might also contribute to this observation. Multi-components Droplet Evaporation Figure 1. Variation of droplet surface area for heptane- decane droplets evaporating at ambient air of 101 kpa and 348 K. Air velocity is 3.1 m/s. Experimental data are adopted from Daïf et al. [29]. The evaporation off 50 μm gasoline-kerosene and diesel- air of kerosene droplets vaporizing in quiescent ambient 10000 K and 405 kpa were simulated. Each component constitutes 50% by mole in the mixture. The probability distributions for the three fuels are depicted in Figure 3. The distributions for kerosene and diesel overlap with each other, except the spread of the kerosene distribution was smaller. The distribution of gasoline did not overlap with the other fuelss except the tail part. Figure 2. Variation of droplet surface areaa of a kerosenee droplet evaporating in ambient air of 573 K and 30 atm. Experimental data are adoptedd from Hiroyasu et al. [30]. The evaporation of a kerosene droplet vaporizing at ambient condition identical to those in Hiroyasu et al. [ 30] was simulated with the proposed model with the data compared to experimental measurements. Initial droplet size was 725 μm and ambient air is at 573 K and Figure 3. The probability distributions of diesel, gasoline and kerosene. Figure 4 shows that the mean temperature was lower than the surface temperature, especially during initial heat-up of the droplet, but the two values converged as the droplets 4

evaporate. The predicted lifetime for the droplet was 11.4 ms. The variations of the mean of the probabi ility distribution functions gasoline and kerosenee are shown in Figures 5 and 6, respectively. As in the other calculations shown earlier, the liquid phase mean of the distribution functions increased as the droplets evaporated. The vapor phase means also increased during the evaporation process and converged towards the initial liquid values at complete evaporation of the droplet. evaporation, the gasoline distribution overlapped with the kerosene distribution. The mean of the gasoline distribution surpassed that of kerosene right before complete evaporation of the droplet. The fuel vapor remained almost purely kerosene due to the liquid composition at thee times. This showed the interactions and effects of overlapping distribution functions on the evaporation process. Figure 4. Temperature variation of a 50 μmm 50% gasoline 50% kerosene (by mole) droplet evaporating in quiescent ambient air at 1000 K and 405 kpa. Figure 6. Mean of probability distribution function for kerosene in the 50% gasoline 50% kerosenee (by mole) droplet evaporating in quiescent ambient air at 1000 K and 405 kpa. (a) (b) Figure 5. Mean of probability distribution function for gasoline in the 50% gasoline 50% kerosene (by mole) droplet evaporating in quiescent ambient air at 1000 K and 405 kpa. From Figure 7, showing the liquid phase distributionss of gasoline and kerosene, showed that the gasoline distribution shifting much faster than its counterparts. The liquid phase gasoline composition at droplet surface steadily decreased for the duration of droplet life. By the time when 30% of droplet masss evaporation, most of gasoline has evaporated and the droplet consisted 90% of kerosene. Gasoline was effectively depleted at about 50% of droplet masss evaporation. Note that at 75% of droplet masss (c) (d) Figure 7. Liquid phase probability distributions for gasoline-kerosene droplet at (a) 25%, (b) 50% %, (c) 75% and (d) 98% of droplet mass evaporation. Conclusion A new, comprehensive and computationally efficient model for preferential vaporization for droplets using a continuous thermodynamics formulation was developed in this study, which is capable of accommodating the application of multiple probability distribution functions. The non-uniformity in the liquid phase due to 5

a finite diffusion effect was represented by the difference between the surface and mean values. The model was applied to study the evaporation of a gasoline-kerosene mixture. The distributions of gasoline and kerosene only overlapped in the tail regions. Initially, gasoline contained the most volatile components in the mixture. The mean and standard deviation of the distribution with broader spread shifted much faster than the one with narrow spread, in this case, gasoline and diesel. Upon complete droplet evaporation, the kerosene became the most volatile component in the mixture. The shifting of the distribution functions affected the rate of evaporation and also the vapor composition. The gasoline-kerosene droplet evaporated in a two-stage manner, with gasoline evaporated first, followed by kerosene. Although kerosene became the most volatile (lightest) components in the mixture by the end of the droplet life, the fuel vapor consisted mainly of gasoline or diesel because of the liquid composition at that instant. Acknowledgements This work was supported in part by the Department of Energy Grant No. DE-FC26-05NT42634, and by Department of Energy GATE Centers of Excellence Grant No. DE-FG26-05NT42622. References 1. C. K. Law, Prog. Energ. Combust., 8 (1982) 171. 2. C. K. Law, Prog. Energ. Combust., 8 (1982) 171-201. 3. W. L. H. Hallett and M. A. Ricard, Fuel, 71 (1992) 225-229. 4. E. C. Robert, E. D. John, M. G. Robert and T. D. Dan, SAE paper 980510 (1998). 5. J. Tamim and W. L. H. Hallett, Chem. Engng. Sci., 50 (1995) 2933-2942. 6. W. L. H. Hallett, Combust. Flames, 121 (2000) 334-344. 7. S. Zhu and R. D. Reitz, J. Eng. Gas Turbines Pwer, 123 (2001) 412-418. 8. A. M. Lippert and R. D. Reitz, SAE Paper 972882 (1997). 9. G. S. Zhu and R. D. Reitz, Int. J. Heat Mass Transf., 45 (2002) 495-507. 10. L. Zhang and S. C. Kong, Chem. Eng. Sci., 64 (2009) 3688-3696. 11. Y. Ra and R. D. Reitz, Int. J. Engine Res., 4 (2003) 193-218. 12. C. F. Lee, W. L. Cheng and D. Wang, Proc. Combust. Inst., 32 (2009) 2801-2808. 13. C. Fieberg,,L. Reichelt, L., Martin, D., Renz, U. and (2009), International Journal of Heat and Mass Transfer, 52, 3738 3746. 14. H. Ghassemi, S. W. Baek, and Q. Khan,. (2006a), Combustion Science and Technology, 178 (6)1031 1053. 15. H. Ghassemi, S. W. Baek, and Q. Khan, Q. S. Combustion Science and Technology, 178 (9) (2006b), 1669 1684. 16. S. D. Givler, and J. Abraham, Progress of Energy and Combustion Science, 22 (1996) 1 28., 17. T. B. Gradinger, an K d Boulouchos,. International Journal of Heat and Mass Transfer, 41, 41 (1998), 2947 2959. 18. G. S. Zhu, and S. K. Aggarwal,, International Journal of Heat and Mass Transfer, 43 (7) (2000), 1157 1171. 19. S. Zhu, and S. K. Aggarwal, (2000), Journal of Engineering for Gas Turbines and Power, 124 (2), 248 255. 20. S. Tonini, M. Gavaises, and A. Theodorakakos, A, International Journal of Thermal Sciences, 48 (3) (2009), 554 572. 21. D. Wang and C. F. Lee, (2002), Proceedings of 15 th ILASS Americas 2002. 22. B. Abramzon and W. A. Sirignano, 32 (1989) 1605-1618, Int. J. Heat Mass Transf., 23. D. Jin and G. L. Borman, SAE Paper 850264 (1985). 24. A. Y. Tong and W. A. Sirignano, Combust. Flame, 66 (1986) 221-235. 25. H. L. Toor, (1964) AIChE Journal, 10 (1964), 448-455 and 460 465. 26. W. E. Stewart, and R. Prober, 3 (1964), Industrial and Engineering Chemistry Fundamentals, 224 235. 27. Cullinan, 4 Industrial & Engineering Chemistry Fundamentals, (1965), 33 139. 28. Y, Zeng and C. F. Lee, ASME Journal of Engineering for Gas Turbines and Power, 124 (2002), 717 724. 29. S. C. Wong, and A. C. Lin, 237 (1992 Journal of Fluid Mechanics, 237, 671 687. 30. A. Daïf, M. Bouaziz, X. Chesneau, X. and A. Ali Chérif, A. 18 (1999), Experimental Thermal and Fluid Science, 18, 282 290. 31. H. Hiroyasu, T. Kadota, T., Senda and Imamoto, T. (1976), Heat Transfer - Japanese Research, 5(1), 50 68. 6