Simulating Condensation in a Supercritical Gas Jet

Transcription

1 Simulating Condensation in a Supercritical Gas Jet L. Qiu, Y. Wang, H. Wang, Q. Jiao and R. D. Reitz * Engine Research Center University of Wisconsin Madison, WI USA Abstract The KIVA code was modified to account for real gas effects with the Peng-Robinson equation of state. The updated code, built upon the homogeneous phase equilibrium assumption, is capable of dealing with two-phase compressible flow problems in a fully Eulerian scheme. Investigations were conducted to examine the condensation process in a supercritical gas jet as verification of the algorithm s implementation. The simulation results were found to be consistent with the experimental results in two aspects. The first observation is that condensation is predicted to occur if and only if the temperature difference is large enough to promote strong heat transfer. Secondly, a condensed liquid phase is found to form at the jet boundary where the energy and mixing interactions between the hot gas jet and the cold surrounding gas (nitrogen) are strong. The intensive heat exchange finally brings mixture into two-phase region by condensation. * Corresponding author: reitz@engr.wisc.edu

2 Introduction Phase transition is a common phenomenon in spray and atomization applications, such as in diesel engines and gasoline-direct-injection (GDI) engines. Investigations are complicated by the underlying complex physical processes of the two-phase flows, such as turbulence effects, mechanisms of jet breakup and atomization, droplet evaporation and distortion in convective environments, etc. The essential issue of two-phase simulation lies in modeling phase transition processes-the interactions between the liquid and gas phase. From the standpoint of numerical simulation, there are generally two approaches. One is the Eulerian treatment of both the liquid and gas phases. Depending on the details of modeling, there are 5 or 7 equation models. While more information can be resolved, these computations generally place a high requirement on computer resources. The other treatment is to solve the two phases with the continuous Eulerian phase and a dispersed Lagrangian phase. This methodology has been widely used in stateof-the-art multi-dimensional engine simulations at high injection pressures [1]. Nonetheless, due to the lack of information about internal nozzle flow, cavitation is usually not modeled so its effects on the boundary conditions are generally neglected in typical engine simulations, such as those using the open source KIVA-3V Release 2 code [2]. At the same time, because the grid size near the nozzle is on the same order as the nozzle diameter, other detailed information at the nozzle exit is not available. Supercritical injection has gained attention over the years, but has been mainly limited to the aerospace engineering area. For hypersonic flights, regenerative endothermic fuel is used to cool aircraft frames and combustor components to meet thermal requirements. During the cracking process, the fuel is heated up to its supercritical state readily. Therefore, future aircraft are expected to operate with supercritical fuels [3]. Besides, there are also some potential benefits of injecting supercritical diesel fuels in engines. When a fuel is injected beyond its thermodynamic critical point, there is no defined boundary between the fuel and the ambient air; and the mixing between them will be more effective when the surface tension disappears but diffusion overwhelms. Large Eddy Simulation (LES) modeling shows that, under high injection pressure conditions, a n-heptane liquid jet maybe approximated as a dense fluid, for which the surface tension effects diminish [4]. Experiments [5] show that, when a supercritical jet is issued into a nitrogen environment, condensation occurs only when the temperature difference between the reservoir and the cylinder is large. The simulation work here focuses on the underlying physics associated with the phase transition process. The paper is organized as follows. The equation of state is introduced first, followed by discussions of the thermodynamics with its applications. Afterwards, the generalized thermodynamic relations considering real gas effects, as well as the coupling between thermodynamics and fluid dynamics are analyzed. The updated code is then used to study a supercritical jet in which condensation occurs. Equation of State To consider the non-ideality of thermodynamic properties at high pressures, a proper equation of state is needed. While there are no general guidelines for selecting a specific equation of state, the Peng- Robinson equation of state [6] is chosen for several reasons. First, it is a simple form of the cubic equation of state so it is easy to implement. Second, it has better performance for the prediction of vapor-liquid phase equilibrium properties over others, including the Soave- Redlich-Kwong equation of state [6]. It is also the most popular equation of state used in the natural gas and petroleum industry. Third, it has been successfully applied to model droplet evaporation at high-pressure engine-like conditions [7, 8]. The Peng-Robinson equation of state is of the form: ( ) ( ) (1) Here, is pressure, is temperature, is the molar volume. and are two parameters determined from: where { [ ( )] (2) (3) Here,, and are the critical temperature, pressure and acentric factor, respectively. In case of a mixture, the classical Van der Waals mixing rule is used: { (4) ( ) (5) Here, is the mole fraction of species i in the mixture. is the interaction parameter between component i and j, and it is generally independent of pressure or temperature. More advanced mixing rules are not pursued in the current work. The fugacity coefficient is determined according to 1

3 [ ( ) ( ) ( ) ] ( ( ) ( ) ) (6) Here, is the compressibility factor; and are the dimensionless parameters of and : maximum entropy. Mathematically, this equilibrium criterion may be interpreted as: { ( ) (8) { (7) Phase equilibrium Considerations on the dynamics of phase transition, which requires analyses of non-equilibrium thermodynamics such as nucleation, are superficial if the equilibrium state ( destination state ) is not known. Phase equilibrium, hence, is a starting point of further nonequilibrium analyses and it is of primary importance from the point view of thermodynamics. In the current work, the homogeneous phase equilibrium assumption is adopted. The main underlying assumptions are: (1) No interface exists between the liquid and vapor phases. This means that capillary pressure effects are not considered. Hence, tracking the interface, as generally in two-fluid models, is not needed and the pressure is universal. (2) Thermodynamic equilibrium is attained instantaneously. From the point of thermodynamics, this assumption means that the mixture in each computation cell is at a global thermodynamically stable state. (3) The two phases have the same velocity and temperature, so no thermal gradients or relative motion exists. The current equilibrium solver, built upon equilibrium thermodynamics, is composed of two main parts, depending on the specified thermodynamic conditions. The first one is the TPn phase equilibrium calculation, in which the temperature, pressure and composition are fixed. The second is the UVn phase equilibrium calculation, in which the internal energy, volume and composition are fixed. The nomenclature here is adopted from the Chemical Engineering literature. Theoretically, the UVn calculation is more challenging since all the intensive variables are unknown, but the thermodynamic properties, such as the Gibbs free energy, are generally expressed as a function of temperature, pressure, and composition. However, as will be shown later, the UVn calculation is needed as it plays a major role in the full determination of the thermodynamic state. The UVn solver was developed based on the entropy maximization method suggested by Castier [9], which is based on the fundamental thermodynamics postulate [10]: for a mixture at fixed internal energy, volume and composition, the equilibrium state has the 2 Here, S denotes the entropy of a defined thermodynamic system; stands for internal energy; is the volume; is the mole number of species. is the index for phase and is for species. Note that a multicomponent mixture and a potentially multi-phase situation is considered. Castier has already documented a detailed discussion on the numerical algorithms used for solving the maximization problem in [9] so the details are not repeated here. One special noteworthy feature is that pressure can be negative during iterations but it will be positive upon numerical convergence and all the phases should have same temperature and pressure. Phase stability and splitting Since an essential part of the current Eulerian implementation is the thermodynamic treatment, it is necessary to highlight its distinguishing features. In both the TPn and UVn calculations mentioned above, the phase stability and splitting are called routinely to add or remove phases as necessary during the course of searching for equilibrium state. It is hence guaranteed that a new phase is only added or removed to minimize Gibbs free energy or increase entropy, depending on the specifications. For a single phase, it is proved by Castier [9] that the Gibbs free energy minimum state also corresponds to an entropy maximum state, so the general well-established TPn stability test can be used. For a mixture at a given temperature, pressure and composition, the equilibrium state has the minimum Gibbs free energy. This is the necessary and sufficient condition of phase equilibrium. However, the equality of fugacity, widely used in the literature [7, 8], is not a sufficient condition since it only represents a local stationary point, but not necessarily the global minimum point in the Gibbs free energy phase space [11]. The tangent plane distance (TPD) method suggested by Michelsen [12, 13] is adopted here for rigorous phase stability test. The TPD function is defined as: ( ) [ ( ( )) ( ) ( ( )) ( ) ] (9) Here, and are the molar compositions of the trial and test phases; ( ) and ( ) are the fugacity coefficients of component i. The test phase is thermodynamically stable if and only if

4 ( ) (10) for all feasible. For a multi-component mixture, geometrically, the Gibbs free energy is a hyper-surface and the tangent plane is a hyper-plane [11]. The quasi- Newton s Broyden Fletcher Golfarb Shanno (BFGS) algorithm for unconstrained local minimization [14] is implemented to locate minima for its fastness and effectiveness over the full Newton s method. The general initialization strategies for stability testing suggested by Li and Firoozabadi [15] were adopted and they were found to be reliable and robust. Phase splitting occurs whenever a phase stability test yields a negative TPD value, indicating that the original mixture is thermodynamically unstable (i.e., phases separate). It is used to determine the equilibrium composition in each phase. The traditional phase splitting method is the flash calculation, which is mainly based on the Rachford-Rice algorithm. While it is straightforward for the implementation of this algorithm in traditional two-phase vapor-liquid equilibrium computations, determining the multi-phase flash is not a trivial task. The multi-phase equilibrium, however, occurs quite often in real situations, such as in the carbon dioxide (CO2) utilization for the enhanced oil recovery at low temperatures, as well as when organic species and water coexist. The Rachford-Rice equation for multi-phase flash [13, 16] is: with { [ ( ) ] ( ) (11) [( ) ] ( ) (12) Here, and are the numbers of components and phases; is the equilibrium ratio of component i in phase j relative to phase ; is the mole fraction of phase j. In the current application the feasible region algorithm suggested by Okuno et al. [16], which is based on minimization of a non-monotonic convex function with linear constrains, was applied for multi-phase flash computation (number of phases is greater than or equal to 3). The method guarantees convergence, independent of the number of phases for both positive and negative flash. Another feature of this method is that it limits the solution in a small feasible region constrained by: (13) where { }, { }, and { { }} for ( ). This feasible region does not contain any poles where the Hessian matrix is ill conditioned, so a minimization scheme with the aid of line search using the Hessian matrix is applicable. In addition, the constraint { } creates a much smaller feasible region compared to the traditional constraint { }. For positive flash, another constraint { } is needed to limit positive composition. All the above-mentioned numerical methods on phase stability and splitting have been applied to construct a general phase equilibrium solver called: Applied Phase-related Equilibrium (APPLE) solver in previous work. Currently, the solver uses the Peng- Robinson equation of state. More information and verifications can be found in [17]. The APPLE solver is robust and efficient; it can be used for multi-phase (including three-phase equilibrium) equilibrium calculations with rigorous thermodynamic stability tests and fast phase splitting calculations. Generalized equation of state in KIVA The standard KIVA code solves the conservation equations at each time step in three consecutive phases: In the Lagrangian Phase A, the source terms due to spray and combustion/chemistry are solved. In Phase B the cell is moved with the local fluid velocity, and all the physical properties (pressure, temperature, velocity, diffusion, etc.), except for convection are solved. This phase is the main part of the fluid dynamics solver. Finally, in Phase C (Rezoning stage) the cell boundaries are mapped back to the position where they should be, as determined by the specified mesh motion. In this study, Phase A is irrelevant because a non-reactive Eulerian flow is considered; Phase B and C are of interest and were modified to consider the real gas effects. Discussions on generalizing the PvT relationship with a real gas equation of state are thoroughly documented in [7] and [17], so only a brief discussion is pursued here. In Phase B, the main parts are the internal energy equation and the SIMPLE loop for implicit-pressure iteration. Some related equations are now updated to consider real gas effects, such as the linearized isentropic relation. Another important aspect to be mentioned, which is incorrectly discussed in Ref. [18], is the simultaneous determination of temperature and pressure using the equation of state, once the internal energy from solving the Eulerian conservation equations (including the convection at the end of phase C) is available. Because the internal energy is a function of temperature and pressure, a robust and effective UVn solver is needed. Similarly, in the case of explicit pres- 3

5 sure-based conservation equations, the corresponding thermodynamic solver would be the HPn solver, in which the enthalpy and pressure are directly available from the fluid solver. Fluid solver KIVA with real equation of state End of Phase C U, V and spd Determine T and P from one phase configuration T >0 and P >0 Stability test Thermo. solver T <0 and P <0 2-phase mixture Figure 1. Schematic diagram of the coupling between fluid dynamics and thermodynamics. Another important thing to note is that the current implementation, built upon homogeneous phase equilibrium, enables a full separation between thermodynamics and fluid dynamics: the thermodynamic state is determined fully from the outcome of the fluid dynamics and the thermodynamic is fixed during the fluid dynamic calculation. To promote the treatment more clearly, a coupling scheme between the fluid solver and thermodynamic (UVn) solver is shown schematically in Figure 1. It is remarked that a dynamic update of the thermodynamics during the iterations on fluid dynamics is not practical. On the other hand, as long as the time step is small enough, this treatment is suitable. As such, theoretically, any equation of state can be used. Depending on the objectives and purpose, the formula could be different as long as it is thermodynamically valid and consistent with the phase equilibrium calculations. For instance, Giljarhus et al. [19] used the Span- Wagner equation of state, a density-energy based equation of state, to solve for a one dimensional phase transition problem for carbon dioxide (CO2), since this equation is accurate in predicting properties of pure CO2 for a very wide range of temperatures and pressures. Simulation setup Details about the planar laser induced fluorescence (PLIF) experiments done using fluoroketone (C6F12O) as injectant can be found in Ref. [5]. Fluoroketone was selected because it has a favorable lower critical point (442 K, 18.7 bar); hence it is easier to attain the desired experimental conditions. At the same time, it is safe to use since it is not flammable. Actually, it is referred as Novec 1230 and is used as a flame suppressor by the 3M company [20]. For the simulation, however, the thermodynamic properties (i.e., viscosity, conductivity, etc) of this fuel are rather limited. Tuma [21] noticed that the perfluorohexane (C6F14) has similar properties to those of (a) (b) Figure 2. Density contour plot of pure species from Peng-Robinson equation of state. (a): Perfluorohexane (C6F14). (b): Fluoroketone (C6F12O). 4

6 Case Chamber conditions Injection conditions U_inj Tr Pr T[k] P[bar] Tr Pr T [k] P[bar] [m/s] Table 1. Simulation cases for the supercritical gas injection. fluoroketone s critical point, latent heat and surface tension, etc. at normal conditions, and its properties can be found from the DIPPR database [22]. Density contours of pure fluoroketone and perfluorohexane are shown in Figure 2 using the Peng-Robinson equation of state. For a wide range of conditions, the two species have similar P-T-ρ relationships. Therefore, in the current simulation, perfluorohexane is used. A 2D axisymmetric mesh is used with gradually increasing grids in the radial and axial directions. Five nonreacting cases of supercritical injection into sub-critical environments were simulated, as tabulated in Table 1. The first four cases are from the experiments of Roy et al. [5]. Case 5, as will be discussed later, was added to show phase changes. Taking Case 1 as an example, the Reynolds number is around 2.4e+06 using the dynamic viscosity data from DIPPR. For pure species, its supercrtical state is achieved if its pressure or temperature is above the critical point [23]. Similarly, determining the supercritical state of a mixture needs the knowledge of the mixture s critical point, which is generally not a linear combination of all the species, as will be shown later. Besides, the critical point of a mixture depends on the composition. Here, the reduced properties, such as reduced pressure and reduced temperature were used to specify the boundary conditions, and were all based on the critical point of pure perfluorohexane. For all the cases, the injection pressure and temperature were kept supercritical; the chamber temperature was kept subcritical but its pressure was kept supercritical. Thus, emphasize was placed on the effects of temperature [5]. Results and Discussion Phase equilibrium calculations for the binary mixture (perfluorohexane and nitrogen) of interest are shown in Figs. 3 and 4 using the APPLE solver. Figure 3(a) is the global phase diagram of the mixture at two feeding compositions (10% and 30% nitrogen) with the true critical locus. The critical points are found to be (447 K, 25 bar) for the 10:90 mixture, and (440 K, 45 bar) for the 30:70 mixture. For each feeding composition, the bubble point and dew point line are used to construct the phase envelop. The region between these two lines is for two-phase mixtures. Figure 3(b) shows the phase transition during a constant pressure heating process of the 30% nitrogen mixture. Close to the bubble point, the heat addition increases vapor phase amount. Nonetheless, after the minimum liquid phase point at around 85% at 300 K, further heating does not (a) (b) Figure 3. Global phase diagrams of the binary mixture (perfluorohexane and nitrogen). (a): Pressure-temperature diagram at two feed compositions. (b): Liquid phase mole fraction at 60 bar. 5

7 (a) (b) Figure 4. Equilibrium phase diagram of the binary mixture (perfluorohexane and nitrogen) at three temperatures. (a): Pressure-composition diagram. (b): Equilibrium ratio-pressure diagram. enhance the evaporation process as would expected but condensation occurs. This phenomenon is generally referred as retrograde condensation [25]. Correct capture of this peculiar behavior of multicomponent mixture shows the robustness of the present APPLE solver. Some more calculations are shown in Figure 4 at three temperatures using the APPLE solver. As shown in Figure 4(a), when pressure is increased, the liquid phase composition of perfluorohexane decreases while its vapor phase composition increases. Another feature to be noted is that the mixture critical pressure increases while the critical temperature reduces (also shown at Figure 3(a) more directly). A simple linear function of the critical points of the pure species (denoted as stars in Figure 3(a)) can only give the pseudo-critical point but not the thermodynamically true critical point. Actually, there is a non-linear relationship for both critical pressure and temperature, which is a general feature of mixtures of long chain hydrocarbons with nitrogen. Figure 4 (b) shows the corresponding equilibrium ratios of the two species. As the equilibrium ratio approaches unity, the mixture approaches its critical point, where the two-phase compositions become identical, so there are not obvious phase differences but instead a homogeneous mixture exists with similar thermodynamic properties (i.e., density, compressibility). Thermodynamic properties show great departures from the ideal gas state at high-pressure conditions. For instance, the heat capacity increases substantially near the critical point but then decreases. Pressure effects on enthalpy and entropy become important. All these nonidealities need to be considered for a complete treatment of trans-critical and supercritical problems. Figure 5 shows the constant pressure specific heat of nitrogen predicted from the Peng-Robinson equation of state. Also included is the data from NIST [24]. It is obvious that the sudden jump and drop near the critical temperature cannot be captured with the ideal gas relationship. On the other hand, the Peng-Robinson equation of state does a much better job of predicting non-linear transcritical behavior. Figure 5. Comparison of properties of pure nitrogen at supercritical conditions (P=35 bar). PR: Peng-Robinson equation of state. IG: Ideal gas law. Comparison with the experimental data for the supercritical gas jet is shown in Figure 6, along with simulation results using the real gas equation of state and the ideal gas law (standard KIVA). For simplification, only Case 1 and Case 4 are shown. The physical dimensions have been set to be the same as experimental images. It is seen that the results from the ideal gas law underestimate the density, and the prediction is improved when real gas effects are considered. This is seen in two aspects. First, the jet core is captured well and it is comparable with the experiments. Second, the prediction of the density difference between the jet and 6

8 Experimental data Simulation results Using real gas equation of state Ideal gas Case 1 Case 1 Case 1 Case 1 Case 4 Case 4 Case 4 Case 4 Figure 6. Comparison of predicted mixture density with experimental images from Roy et al. [5]. Also shown are the compressibility factor (z_comp1), temperature and C6F14 mass fraction contours. 7 ambient gas is improved. It is hence reasonable to expect that when the non-ideality becomes more important at high pressures (such as Pr>3) and low temperatures, compressibility effects will become more important. The current calculations do not predict any phase change in Case 1, as opposed to the experiment observation in [5]. This could be because a different fuel is used in simulation. The underlying physics, however, should be very close. In the experiments with fluoroketone, condensation is observed to occur when the temperature difference between the injection reservoir and the chamber is large. Similarly here with perfluorohexane, when the chamber temperature is further lowered from Tr=0.69 to Tr=0.6, condensation starts. This is because the larger temperature difference leads to a strong heat transfer process that finally alters the local conditions inside the phase envelope and a liquid phase forms. What s more, the condensation happens with a pressure increase due to the heating effects.

9 In addition, liquid phase is found near the jet outer boundary in the experiments. This is also consistent with the current simulation results seen in Figure 7, which show that an incipient liquid phase forming at the jet periphery. The jet boundary, defined by the distinctive species concentration but not surface tension, is where the turbulent kinetic energy is large due to mixing effects and is the place of the development of the turbulent mixing layer. At the same time, the large temperature gradient enhances the local heat transfer process. Therefore, a strong interaction between turbulent mixing and convective heat transfer between injectant (with high injection velocity and high temperature) and nitrogen (with zero velocity and low temperature) is expected. If there is any liquid phase formation, it has a good chance to be located near the jet periphery. Based on these analyses, it seems reasonable that the liquid phase will formed close to the injector, where the interfacial shear force is large due to the large relative velocity. This is also shown in the current simulation. Downstream of the nozzle exit, due to compressibility effects and entrainment and spreading, there is a loss of jet kinetic energy. However, the experiment shows the liquid drops form at locations beyond 10 injector diameters downstream. The authors attribute this to the heat transfer effects that finally lead to local subcritical conditions where the surface tension asserts its importance [5]. Based on the current thermodynamic treatment, the phase separation, in terms of condensation, is a direct result of entropy maximization. Consequently, it is an entropy-driven physical process. In other words, the condensation is not occurring with a release of latent heat, which is generally accompanied by a discontinuous change of the corresponding first derivative properties, such as entropy or volume. This is generally the case for condensation of a pure species under subcritical conditions. It can be shown that there is no latent heat associated with the phase change process across phase boundaries (i.e., the bubble or dew point lines). The continuous change of entropy indicates the phase separation takes place without discontinuity of the thermodynamic potentials (i.e., Gibbs free energy). As a result, the phase change can be categorized as a second order phase transition according to the canonical classification of Ehrenfest [26]. Finally, it is remarked that the liquid phase mole fraction, as an output scalar, is averaged. Because the KIVA code uses the staggered grid, it saves scalar properties at the cell vertices. The post-processing then averages the scalars around its neighbors to get cellcentered values. Since not all cells have a second (liquid) phase, different from the fact that they all start as single gas phase, the damping effect of the averaging is more severe than for the other scalars. For Case 5, calculation shows that the actual maximum liquid phase mole fraction is around 2%. Conclusions In this work, the standard KIVA 3v code was generalized with the Peng-Robinson equation of state to account for real gas effects. APPLE, a robust phase equilibrium (TPn) solver developed at the ERC, which is composed of phase stability and splitting routines based on rigorous equilibrium thermodynamic analyses, is applied to determine the true equilibrium state. Furthermore, a UVn solver is also applied for the coupling between the fluid dynamics solver and the thermodynamic solver. Finally, the updated solver is used to study supercritical gas jet behaviors. The simulations show agreement with experimental observations on two aspects. First, a liquid phase is found to form when the temperature difference between the injectant and the chamber gas is large enough. Second, the liquid phase is found to occur at the jet outer boundary, where turbulent mixing and convective heat transfer effects are very intensive. These strong interactions finally send the mixture into the two-phase region on the phase diagram and a liquid phase forms. Acknowledgements Financial support of this research by the DOE Sandia laboratories is greatly acknowledged. One of the authors, Qiu, L., would like to express his gratitude to Dr. Li (now with ExxonMobil) and Prof. Firoozabadi at the Reservoir Engineering Research Institute (RERI) for helpful discussions of phase equilibrium calculations. Helpful discussions and communications with Prof. Castier from the Chemical Engineering department at TAMU-Qatar on the implementation of entropy maximization algorithm are also greatly appreciated. Nomenclature A dimensionless parameter in Eq. (6) a energy parameter in Eq. (1) B dimensionless parameter in Eq. (6) b volume parameter in Eq. (1) h variable used in Eq. (12) k equilibrium ratio L constraint n mole numbers nc number of components np number of phases P pressure Pr reduced pressure R universal gas constant S Entropy constraint spd species density T temperature Tr reduced temperature 8

10 Figure 7. Liquid phase mole fraction (xf_phase2) and pressure field in Case 5 at different times. Arrows shows velocity vectors. Liquid phase starts to form around 0.5ms. 9

11 U V v X x Y Z z internal energy volume molar volume constraint mole fraction array constraint compressibility factor feed composition Greek Letters factor in Eq. (2) molar phase fraction binary interaction parameter acentric factor constant in Eq. (3) fugacity coefficient Subscripts c critical properties i species index j phase index Abbreviations APPLE applied phase related equilibrium DIPPR Design Institute of Physical Properties ERC Engine Research Center GDI gasoline-direct-injection HPn specified enthalpy, pressure and mole numbers KIVA open source fluid solver LES large eddy simulation PLIF planar laser induced fluorescence TPD tangent plane distance TPn specified temperature, pressure and mole numbers UVn specified internal energy, volume and mole numbers References 1. Reitz, R.D., Atomization and Spray Technology, 3: (1988). 2. Amsden, A. A., Los Alamos National Laboratory Report LA MS, Edwards, 31st Aerospace Sciences Meeting & Exhibit, Reno, NV, USA, January Oefelein, J., Dahms, R. and Lacaze, G., SAE Int. J. Engines 5(3):(2012) 5. Roy, A., Joly, C., Segal, C., Journal of Fluid Mechanics, (2013). 6. Peng, D.-Y., Robinson, D.B., Industrial & Engineering Chemistry fundamentals 15(1):59-64 (1976). 7. Trujillo,M. F., Torres, D. J., O Rourke, P J., International Journal of Engine Research, 5: (2004) 8. Zhu, G.-S.; Reitz, R.D., ASME Journal of Gas Turbines and Power, 123: (2001). 9. Castier, M. Fluid Phase Equilibria, 276:7-17 (2009). 10. Callen H.B., Thermodynamics and an Introduction to Thermostatistics, Wiley, 1985, p Baker, L. E., Pierce, A. C., Luks, K. D., Transactions of AIME, 273 (1982). 12. Michelsen, M. L., Fluid Phase Equilibria, 9: 1-19 (1982). 13. Michelsen, M. L., Fluid Phase Equilibria, 9: (1982). 14. Hoteit, H., Firoozabadi, A., AIChE Journal 52(8): (2006). 15. Li, Z., Firoozabadi, A., SPE Journal (2012). 16. Okuno R., Johns R., Sephernoori K., SPE Journal (2010). 17. Qiu, L, Wang, Y., Wang, H, Jiao, Q., Reitz, R., AIChE journal, to be submitted. 18. Trujillo M. F., P.J. O Rourke and D. Torres, "Generalizing the Thermodynamics State Relationships in KIVA-3V," Los Alamos Technical Report, LA-13981, October Giljarhus, K.E.T, Munkejord, S.T., Skaugen, G, Industrial & Engineering Chemistry Research, 51: (2011) aaaaaakiumpavebaoebaab21fmyaaaa_-. Last visited March 20th, Tuma, P. E., Proc. 24th IEEE Semi-Therm Symposium, San Jose, CA, USA, pp , March 16-20, Klein. S., Nellis, G., Thermodynamics, Cambridge University, 2012, p Last visited March 20th, Firoozabadi, A., Thermodynamics of Hydrocarbon Reservoirs, McGraw-Hill, 1999, p Callen H.B., Thermodynamics and an Introduction to Thermostatistics, Wiley, 1985, p. 218 and Ch