CONTINUOUS THERMODYNAMICS FINITE DIFFUSION MODEL FOR MULTICOMPONENT FUEL SPRAY EVAPORATION

Transcription

1 CONTINUOUS THERMODYNAMICS FINITE DIFFUSION MODEL FOR MULTICOMPONENT FUEL SPRAY EVAPORATION Dongyao Wang 1 and Chia-fon F. Lee Departent of Theoretical and Applied Mechanics 1 Departent of Mechanical and Industrial Engineering University of Illinois at Urbana-Chapaign Urbana, IL INTRODUCTION The different volatility of the coponents of a fuel ixture will result in different vaporization rates of coponents at the droplet surface, and the fuel becoes non-unifor inside the liquid phase because of the ass diffusion resistance inside the droplet. For practical, ultidiensional siulations of IC engines, zero-diensional odels for the liquid phase are necessary because of their low coputational cost and the fact that thousands of coputational parcels are typically needed to represent the fuel spray in an engine. Zeng and Lee [1] developed a zero-diensional odel capable of including finite diffusion and preferential vaporization. This odel uses discrete coponent representation. Discrete ulticoponent fuel odels use a set of coponents to atch the distillation behavior of a real fuel. A large nuber of coponents are typically required in order to represent a coercial fuel adequately. However, the coputational cost increases quickly as the nuber of the coponents increases, since each coponent adds an additional transport equation. Coercial fuels usually contain hundreds of coponents, so it is usually not practical to represent every coponent using a discrete representation for spray siulation. An alternative approach is continuous representation, which uses a probability density function based on the principles of continuous therodynaics to represent the fuel. For engineering calculations, a typical coercial fuel consisting of hundreds of coponents can be reasonably represented by a continuous distribution function. As the ixture is assued of alkanes and properties of the species can be characterized by the variable of olecular weight, so only a few paraeters are required to describe the characteristics of the ixture, such as the ean and variance of the distribution. Thus, only the paraeters describing the ulticoponent fuel have to be deterined beforehand, rather than having to solve a transport equation for each coponent. The ethod of continuous therodynaics was previously used by Lippert and Reitz [], Yi et al.[3] and Zhu et al. [4] for ultidiensional engine odeling. These studies ade iportant contributions to the application of continuous therodynaics to IC engine siulations. However, these studies are based on the assuption of infinite diffusion in the liquid phase, so non-unifority inside the droplet (i.e., the effects of finite diffusion) is not considered. In this paper, a coprehensive odel considering preferential vaporization of a coplex fuel ixture using continuous distributions is presented. The odel consists of a gas phase sub-odel, which deterines evaporation fluxes, and a liquid phase sub-odel with finite diffusion. This odel was validated with experiental data for the vaporization of isolated JP-4 fuel droplet, and then it was used to analyze the vaporization behavior of single diesel fuel droplet. Finally, gasoline hollow-cone spray siulations were ade using this odel. This odel was ipleented into KIVA-3V for coputations. MATHEMATICAL MODEL Gas Phase Sub-Model A fuel ixture is described by a continuous distribution function, f(w), characterized by the olecular weight, w, of each coponent. In this study, f(w) is a Γ-distribution function, which is often used to represent petroleu fractions (Cotteran et al. [5]): α 1 ( w γ ) w γ f ( w) = exp( ), (1) α β Γ( α) β 1

2 where γ is the origin, α and β are paraeters controlling the shape of the distribution, with the ean olecular weight, θ=αβ+γ, and the variance, σ =αβ. In previous studies, using the second oent was found to be ore convenient for calculation than using the variance. Here, θ is the first oent of f(w), and =σ +θ is defined as the second oent. The evaporation fluxes are given by Equations ()-(4): Sh cd yfr yf N = ln(1 + B), B =, () R 1 yfr ~ D Nθ = yfr + (1 yfr ) θ RN D, (3) Dˆ N = yfr + (1 yfr ) R N, (4) D where the first and second oent fluxes, N θ and N, are associated with ass and variance fluxes. The effect of convection on vaporization is assued to be adequately expressed by a Sherwood nuber, Sh, and a Nusselt nuber, Nu, for olar flux and heat flux, respectively. The heat flux is obtained as: NC p ( Ts T ) q = H fg. (5) RNC p 1 exp( ) Nu k Liquid Phase Sub-Model The transient behavior of the liquid phase significantly affects the vaporization of a spray and thus, the vapor distribution in an IC engine. In the present study, an effort is ade to properly odel the nonunifority of the liquid phase due to preferential vaporization rather than using infinite diffusion odel (i.e., well ixed assuption) as in previous studies. It is assued that the effect of internal circulation can be expressed adequately by an effective diffusivity and conductivity [6,7]. It is reasonable to assue that every coponent in the liquid phase diffuses at the sae rate, though we consider the effect of olecular weight on the diffusion rate in the gas phase. As Abrazon and Sirignano [7] and Runge et al. [8] assued unifor liquid diffusivity for all the coponents of the liquid ixture, unifor liquid phase diffusivity was adopted in the present study. It is possible to find a relation between the surface and average properties of a droplet to approxiate the effect of finite diffusion rates in the liquid phase by using the sae approach as Zeng and Lee [1]. An expression for the difference between the surface and ean values can be obtained: dφ d λ1 E( Pev) Rφ Pev = 0. (1 + ) Φ d, (6) dt ts κ λ1 where, Φ d = Φ s Φ, λ 1 = , t s = R / κ, Pev = RR / κ and E(P ev ) is the coefficient of the effect of droplet surface regression, which is noralized to 1 at P ev = 0 and decreases as P ev increases; and ~ θ D Nθ R Nθ 1 Φ = = =, κ D ˆ, φ N R N. (7) c CpT k / ccp q / Cp The average quantities are deterined by: d dt R 3 g θ Nθ c = 3R N, (8) CpT q

3 where the fluxes are given by the gas phase sub-odel. Thus, the surface value, Φ s = Φ d + Φ can be obtained. The effective diffusivity and conductivity correlations of Abrazon and Sirignano [7] are used. The vapor-liquid equilibriu relations and fuel vapor property correlations given by Cotteran et al. [5] and Tai and Hallet [9] are used here. The present odel involves fuel liquid properties, so the proper correlations are introduced. The liquid viscosity correlation is based on Orrick and Erbar Method [10]. The Sastri Method [10] and the Hayduk and Minas Method [10] are extended in this odel to use for conductivity and diffusivity correlation in the liquid phase, respectively. The above odel was ipleented into a odified version of KIVA-3V [11]. With a continuous fuel representation, the vapor-liquid equilibriu and liquid phase quantities are olar based; however, original KIVA is ass based. Thus, olar and ass basis variables are converted in each tie step. RESULTS AND DISCUSSIONS Three cases of siulations were ade for validation of the present odel, analysis of single droplet vaporization behavior, and siulation of spray. The paraeters of distribution by Hallett [1] were used for our coputations. For diesel, gasoline and JP-4 fuels, the initial eans and variances are θ = 185 and σ = 43, θ = 85.5 and σ = 35.8 and θ = 113 and σ = 7.5, respectively, and with γ = 0. Coparisons With Experiental Data The present continuous distribution vaporization odel was used to siulate the evaporation of a JP-4 droplet. Runge et al. [8] easured the teperature of a JP-4 droplet in 94K air and oving at 3 /s. This experient showed that the droplet teperature initially dropped rapidly and then increased to about 93K. This behavior is due to the lighter coponents vaporizing rapidly and absorbing heat fro the liquid. Siulation using the present odel was ade at the sae conditions as those of the experient. Figure 1 shows that the coputational results agree well with the experiental data. Runge et al. [8] also showed coputational results with a discrete 33-coponent odel. Copared with their results, the present odel not only reduced the coputational cost draatically, but also iproved the agreeent with experient rearkably. Single Stagnant Droplet Vaporization Diesel droplet vaporization using continuous therodynaics was studied with a finite diffusion odel in the liquid phase. The coputations were ade using a single, 100 µ droplet with an initial teperature of 300 K, in abient air at 1 at and 1000 K. Figure shows the difference between the coposition at the droplet surface and the average coposition of the diesel fuel droplet when 50% of the fuel ass has evaporated. Since lighter coponents at the droplet surface vaporize faster than heavier coponents, the coposition at the surface is different fro that inside the droplet. If the infinite diffusion odel is used, the coposition of the droplet is considered unifor (i.e, the two curves will overlap). The infinite diffusion odel siplifies the proble enorously, but it neglects soe iportant physical processes. In the present odel, diffusion in the liquid phase is at a finite rate. The siulation predicts that the surface distribution has a higher ean olecular weight and a saller variance than the average distribution. Since the coposition of the droplet is varying, as the ore volatile coponents vaporize, the overall boiling point increases and the droplet never reaches an equilibriu teperature throughout its whole lifetie, which is an iportant feature of the distillation of practical fuels. Gasoline Spray The present odel was used to spray siulations under typical engine operating conditions suarized as follows. Abient conditions are 450 K and 4.4 bars, the total fuel ass injected is 0.01 gra, and the duration of injection is 1.3 s. A two-diensional cylindrical sector is used for coputational efficiency, with a grid of by 3 in the radial and axial directions, respectively. The initial spray cone angle is set as 45 degrees with a cone angle thickness of 10 degrees. Two cases 3

4 were investigated: one with iso-octane as the fuel with the vaporization odel using discrete representation [1], and the other with a ulti-coponent gasoline fuel using continuous representation and the present vaporization odel. The spray evaporation history is shown in Figure 3. Initially, the ulti-coponent fuel vaporizes slightly faster, but later on, it vaporizes uch slower than the single coponent case. This phenoenon is due to the lighter coponents vaporizing rapidly at first, while the heavier coponents with higher noral boiling points vaporize ore slowly. Since heavier coponents vaporize later, the overall fuel vapor ean olecular weight and average variance increase. The teporal variation of fuel vapor coposition is shown in Figure 4. The difference in vaporization behavior will naturally affect the global spray behavior. Droplet positions predicted in the two cases are shown in Figure 5. It is noticed that there are ore sall droplets in the gasoline spray than in the iso-octane spray, especially in the center area of the spray. In the ulticoponent case, a partially vaporized droplet consists of heavier coponents and thus can last longer. In the center of the spray, with a lower pressure, sall single coponent droplets can be depleted quickly, however, ulti-coponent droplets with heavier coponents ay persist. Another iportant difference is that the single coponent iso-octane spray tips see to collapse inward, whereas, the ulti-coponent spray tips stretch outward due to the heavier coposition of the droplets, which results in slower vaporization and higher penetration. SUMMARY AND CONCLUSIONS A preferential vaporization odel for coplex coercial fuels has been developed using continuous therodynaics. The fuel coposition is odeled using continuous distribution function, the effect of finite diffusion in liquid phase is odeled by the differences between surface and average values. Siulation of single JP-4 droplet vaporization agrees with the experient very well. Siulation of single diesel droplet shows that the present odel can capture the ajor features of coercial fuels. This odel is zero-diensional with a very low coputational cost, and so it could be eployed for siulations of a hollow-cone gasoline spray. Copared with an iso-octane spray, the spray structure of a ulti-coponent fuel is substantially different, which will lead to considerable effects on cobustion characteristics. ACKNOWLEDGEMENTS This work was supported in part by the National Science Foundation under grant No. CTS , with Farley Fisher as technical onitor, and by the Departent of Energy under grant No. 99- DOE SBC-L/P-0030DOE. We also thank Dar-Lon Chang and Yu-Yu Lee for their assistance in preparing the anuscript. REFERENCES [1] Zeng, Y. and Lee, C.F., ILASS Aericas 11th Annual Conference on Liquid Atoization and Spray Systes, Sacraento, CA, pp , (1998). [] Lippert, A.M. and Reitz, R.D., SAE Paper [3] Yi, Y., Zhu, G.S., and Reitz, R., SAE Paper [4] Zhu, G.S., Reitz, R., Xin, J., and Takabayashi, T., SAE Paper [5] Cotteran, R.L. and Prausnitz, J.M., Ind. Eng. Che. Process Des. Dev. 3:434 (1985). [6] Jin, D. and Boran, G.L., SAE Paper [7] Abrazon, B., and Sirignano, W.A., Int. J. Heat and Mass Transfer 3:1605 (1989). [8] Runge, T., Teske, M., and Polyeropoulos, C.E., Atoization & Sprays, 8:5-44(1998). [9] Tai, J., and Hallett, W.L.H., Che. Engng Sci. 50:933 (1995). [10] Reid, R.C., Prausnitz, J.M., and Poling, B.E., 1987, The Properties of Gases and Liquids (4 th edition), McGraw-Hill, New York. [11] Asden, A., KIVA-3V: A Block-Structured KIVA Progra for Engines with Vertical or Canted Valves, Los Alaos Natl. Lab. Rep. LA MS,

5 [1] Hallett, W.L.H., Cobust. Flaes 11:334 (000). NOMENCLATURE B Transfer nuber C p heat capacity D diffusion coefficient c Molar density f distribution function k theral conductivity H fg Enthalpy of vaporization, overall q Heat flux N olar vaporization flux, overall R droplet radius T teperature t Tie y Molar fraction w Molecular weight α,β,γ Paraeters of distribution function κ effective diffusivity θ Mean of distribution function Φ diffusion variable σ Variance of distribution function φ Boundary condition ψ nd oent of distribution function Subscript d difference between the surface and ean values F Fuel g gas phase liquid phase average value s surface value R Surface of droplet Abient Overhead -, ~, Average and weighted average over distribution function Average Surface 90 Teperature K 85 Coputation Experient Molar Fraction Tie s Figure 1. Coparison of droplet teperature for the coputation and the experient by Runge et al [8]. A JP-4 fuel droplet with initial diaeter of 646 µ oves at 3 /s in air at 94K and 1 at Molecular Weight Figure. Surface and average distributions of the droplet when 50% of the droplet ass has evaporated. 5

6 Gasoline Isooctane Fuel vapor ass (g) Coposition Mean Molecular Weight Variance Tie s Figure 3. Overall fuel vapor ass history of the spray for single coponent and continuous representation fuel odels Tie s Figure 4. Variation of fuel vapor coposition as the spray vaporizes using continuous fuel representation. Gasoline fuel with continuous representation 1 s 3 s 5 s Gasoline fuel with single coponent representation ( iso-octane) Figure 5. Positions of droplets of sprays predicted with single coponent and continuous representation fuel odels. 1