Multicomponent Vaporization Modeling of Petroleum-Biofuel Mixture at High-Pressure Conditions

Transcription

1 ILASS Amercas, 3 rd Annual Conference on Lqud Atomzaton and Spray Systems, Ventura, CA, May 011 Multcomponent Vaporzaton Modelng of Petroleum-Bofuel Mxture at Hgh-Pressure Condtons L. Zhang and Song-Charng Kong Department of Mechancal Engneerng Iowa State Unversty Ames, IA USA Abstract Numercal smulaton of the vaporzaton of mult-component lqud fuels under hgh-pressure condtons s conducted n ths study. A hgh-pressure drop vaporzaton model s developed by consderng the hgh-pressure phase equlbrum whch equates the fugacty of each component n both lqud and vapor phases. Peng-Robnson equaton of state s used for the calculaton of fugacty. To model the vaporzaton of desel fuel under hgh-pressure condtons, contnuous thermodynamcs based on a gamma dstrbuton s coupled wth phase equlbrum by correlatng the parameters of the equaton of state wth the molecular weghts of the contnuous components. The hgh-pressure vaporzaton model s valdated usng the expermental data of n-heptane drops under dfferent ambent pressures and temperatures. Good levels of agreement are obtaned n drop sze hstory. Predcted results of the vaporzaton of desel fuel drops show that ncreasng ambent pressure leads to a shorter drop lfetme under hgh temperature condtons (e.g., 900 K). On the other hand, at a slghtly lower temperature of 700 K, the drop lfetme ncreases as the ambent pressure ncreases. The model was further appled to bodesel and ts blends wth desel fuel. The fuel blend s modeled based on a method that contnuous thermodynamcs s used to model desel fuel and bodesel s modeled as a mxture of ts fve representatve components. Results of sngle drop vaporzaton hstory show that drop lfetme ncreases as the volume fracton of bodesel n the fuel blend ncreases. It s also observed that the volume fracton of bodesel n the fuel blend ncreases durng vaporzaton and ts vapor concentrates near the tp of the lqud spray whle desel fuel vapor s around the entre lqud spray. Correspondng author

2 Introducton Lqud drop vaporzaton modelng n multdmensonal engne smulaton s of great mportance due to ts mpact on the accuracy of spray combuston predcton. Snce the compostons of practcal engne fuels are very complcated, an mportant means to mprove vaporzaton modelng s to replace the snglecomponent assumpton wth a mult-component approach. In general, researchers have used two major types of mult-component modelng approaches,.e., dscrete component and contnuous thermodynamcs. Dscrete component modelng approaches are preferred for fuels composed of only a few components or surrogates for complex fuels. For example, bodesel, whch can be regarded as the mxture of several fatty acd methyl esters, s one of the fuels that can be modeled usng ths approach. On the other hand, mult-component approach based on contnuous thermodynamcs can be used to model complex fuels such as gasolne and desel fuel, whch are usually composed of hundreds of components [1]. Usng ths approach, fuel s assumed to be composed of an nfnte number of contnuously dstrbuted components, and a probablty dstrbuton functon (PDF) s used to model the dstrbuton of the molecular weghts of the components n the fuel. In addton to the composton of mult-component fuels, vaporzaton modelng also needs to consder the effects of hgh pressure snce lqud fuel drops frequently experence such condtons n practcal combuston devces such as desel engnes and gas turbnes. Tradtonally, for computatonal effcency, drop vaporzaton models are mostly based on the Rault s law whch s a low-pressure smplfcaton of the general vaporlqud phase equlbrum. The phase equlbrum calculaton based on the Rault s law nvolves the assumptons of deal gas law and no gas dssolvng n the lqud phase. Snce such assumptons are not entrely true under hgh-pressure condtons, errors wll be generated n the predcton of drop vaporzaton rate and vapor dstrbuton at hgh-pressure condtons. To mprove the accuracy of mult-dmensonal computatonal flud dynamcs smulaton, general phase equlbrum relatons, whch are characterzed by the equalty of fugactes of both phases, need to be used to replace the Rault s law n drop vaporzaton smulaton. Ths mprovement requres the consderaton of real gas behavors whch can be modeled usng two-parameter cubc equatons of state, e.g., van der Waals, Redlch-Kwong (RK), Redlch-Kwong-Soave (RKS), and Peng-Robnson (PR), are more frequently used. Researchers have performed the analyss on the hgh-pressure drop vaporzaton behavor wthout forced or natural convecton by solvng onedmensonal governng equatons. Hseh et al. [] conducted the analyss of hgh-pressure drop vaporzaton modelng usng conservaton equatons based on a movng coordnate system and solved equatons numercally n both lqud and vapor phases to obtan the drop vaporzaton rate. The RK equaton of state was used to model the real gas behavor n systems composed of sngle-component drops such as n-pentane and ntrogen. Ja and Gogos [3] appled smlar numercal approaches n the vaporzaton modelng of n-hexane n a hghpressure ntrogen envronment and showed the advantage of PR equaton of state compared to RK equaton of state. Curts and Farrell [4] smulated the hghpressure drop vaporzaton at desel engne condtons and dscussed the effects of coupled dffuson process and anomales on drop vaporzaton. Hgh-pressure drop vaporzaton modelng can also be performed wthout solvng the tme-dependent governng equatons n both phases. For smplcty, pseudosteady solutons of the governng equatons, whch can be expressed wth the Spaldng mass transfer number, are used to evaluate drop vaporzaton rate. Ths approach s sutable for the cases wth forced convecton and can be appled effcently to the smulaton of desel fuel spray. The pseudo-steady vaporzaton rate determned by Spaldng mass transfer number s derved from the one-dmensonal speces concentraton equaton wth the assumpton that ambent gas does not dffuse nto the lqud phase. Aggarwal and Monga [5] used such a pseudo-steady formula for hgh-pressure vaporzaton rate and PR equaton of state for phase equlbrum n modelng the vaporzaton of bnarycomponent drops. The vaporzaton results of bnary component drops were compared wth equvalent sngle-component drops n the gas turbne envronment to study the mult-component effect. Zeng and Lee [6] emphaszed on the thermal and mass dffuson models (e.g. nfnte dffusvty and fnte dffusvty models consderng the effect of drop recesson) for modelng the vaporzaton of mult-component fuels based on the pseudo-steady vaporzaton rate formula and used PR equaton of state to account for the non-dealty of ambent gas under hgh-pressure condtons. In addton to the above research usng dscrete component approach, research on the hgh-pressure vaporzaton modelng of mult-component fuels was extended to usng contnuous thermodynamcs to model the composton of petroleum fuels coupled wth general phase equlbrum n hgh-pressure envronment [7]. The formulas for calculatng the parameters of the equaton of state based on contnuous thermodynamcs were also derved. In addton to tradtonal petroleum fuels, whch are questonable n avalablty n the future, borenewable fuels are beng used n many knds of combuston devces. Especally n desel engnes, snce bodesel s close to desel fuel n terms of physcal and chemcal propertes, t can be used wthout sgnfcant modfca- 1

3 tons to the engne. Due to the hgh mscblty, bodesel s usually blended wth regular desel fuel n practcal applcatons. Successful applcatons of bodesel n desel engnes rely on the detaled knowledge of ts spray and combuston behavors. Research has shown that most of bodesel derved from vegetable ols are manly composed of fve C16 to C18 fatty acds [9, 10]. The vaporzaton of bodesel and ts blends wth desel fuel at varous ambent temperatures was smulated under atmospherc pressure by Zhang and Kong [11], who assumed bodesel as a mxture of fve components and appled a mxng rule to obtan the propertes of bodesel. However, at hgh-pressure condtons, the mpact of the composton of bodesel on the vaporzaton behavor of the fuel mxture has not been studed extensvely. The purpose of present study s to predct the vaporzaton behavors of desel fuel, bodesel, and the blend of both fuels under engne operatng condtons whch are characterzed by hgh pressures and temperatures. A mult-component drop vaporzaton model for hgh-pressure condtons based on contnuous thermodynamcs s developed. The model s frst valdated usng the hgh-pressure vaporzaton experments of sngle-component fuel (.e. n-heptane). Both the drop vaporzaton hstory and the vapor dstrbuton of lqud spray usng bodesel and ts blends wth desel fuel under hgh-pressure condtons wll also be shown. MODEL FORMULATION In ths secton, the phase equlbrum n a vaporlqud system, whch s composed of fuel components and the speces of the surroundng gases at the surface of the lqud drop, wll be dscussed. Such an equlbrum provdes boundary condtons for the governng equatons of both phases. In solvng the onedmensonal governng equatons n the gas phase, the pseudo-steady assumpton wll be appled. The methods for calculatng the physcal propertes of the fuels at hgh-pressure condtons wll also be presented. Gas and fuel speces n both phases at the phase nterface can be regarded as a system n thermodynamc equlbrum at a specfc pressure and temperature. Such an equlbrum requres the mnmum of the total Gbbs free energy, namely, dg = 0 [1]. It can be shown that ths requrement wll lead to the equalty of chemcal potental, whch s the partal molar Gbbs free energy of each ndvdual component at constant temperature and pressure, n both phases, namely, µ = µ. (1), l, v In the above equaton, subscrpts l and v denote the lqud and vapor phases, respectvely. Ths relaton s usually further expressed n a more conventonal form by ntroducng the defnton of fugacty, ( ln ˆ ) RTd f = dµ. () f ndcates the fugacty of component, and ^ denotes the property of a component n a mxture. Therefore, the general phase equlbrum can be descrbed as the equalty of fugacty n both phases as fˆ, l, v = fˆ. (3) However, n practce, the fugacty coeffcent s more frequently used n descrbng phase equlbrum replacng fugacty n Eq.(3) and s defned by ˆ ˆ f φ =. (4) y P In the above equaton, y s the mole fracton of component at the phase nterface, and P s the pressure of the mxture. By substtutng Eq. (4) nto Eq. (3), the mole fracton of component of the fuel n the vapor phase (y ) can be obtaned as ˆ, l y = x φ. (5) ˆ φ, v where x s the lqud phase mole fracton of component at the phase nterface. In modelng the vaporzaton of petroleum fuels composed of hundreds of components, contnuous thermodynamcs s preferable because of ts capablty n depctng the complcated composton. By usng contnuous thermodynamcs, the composton of the fuel s descrbed by a dstrbuton of component molecular weghts wth a probablty densty functon (PDF). Therefore, the phase equlbrum n Eq. (5) can be further wrtten as ( ) ˆ φ ( ) ( ) ˆ φ ( ) y f v I v I x f l I l I F F F F =. (6) In the above equlbrum formula, subscrpt F denotes petroleum fuel, and fugacty coeffcents are functons of molecular weght. f s the probablty densty functon determnng the shapes of the molecular weght (I) dstrbutons n both phases. In ths paper, a gamma dstrbuton defned by the followng formula wll be used as the PDF for petroleum fuel.

4 α 1 ( I γ ) α Γ( ) I γ f ( I ) = exp β α β. (7) In the above dstrbuton, α and β are the parameters determnng the shape of the dstrbuton orgnated from γ, and Γ(α) s a gamma functon. Eq. (6) descrbes the phase equlbrum of each component of the multcomponent fuel. It can be seen that such phase equlbrum wll lead to a dfferent molecular weght dstrbuton n the vapor phase. Rearrangng Eq. (6) and ntegratng from γ to, the overall vapor mole fracton of the petroleum fuel at drop surface can be obtaned as ( I ) ( I ) ˆl v φf l y = y f ( I ) di = x f ( I ) di F F F ˆv φ. (8) γ Multplyng the molecular weght I and I on both sdes of Eq. (6), the ntegraton wll lead to the expresson of frst moment (.e., mean molecular weght) θ v and second moment Ψ v n the vapor phase as and γ F ( I ) ( I ) ˆl v φf l y θ = y f ( I ) IdI = x f ( I ) IdI F v F F ˆv φ (9) γ γ F ˆ φ ˆ φ ( I ) ( I ) l v F l y Ψ = y f ( I ) I di = x f ( I ) I di F v F F v (10) γ Usng the above two relatons and assumng γ to be constant over the phase nterface, the dstrbuton parameters,.e., α and β, n the vapor phase can be calculated. As can be seen n the above equatons, to calculate the mole fracton of the fuel n the vapor phase, the fugacty coeffcents, whch are functons of component molecular weght, need to be determned. It can be shown that the fugacty coeffcent of a component n a mxture can be calculated by consderng the departure functon of Helmholtz energy [13] as ˆ P RT RT ln φ = dv RT ln Z n V V. (11) T, V, n In the above equaton, Z s the compressblty factor of the mxture defned as Z=PV/RT, and n s the number of moles for component n the mxture. To calculate the fugacty coeffcent as descrbed n Eq. (11), the pressure-volume-temperature (P-V-T) relaton of the mxture must be known. In ths paper, a two-parameter cubc equaton of state, whch s applcable for both γ F 3 lqud and vapor, s selected to provde the real flud behavor,.e., the P-V-T relatonshp. Such equaton of state s applcable for both pure components and ther mxtures and can be expressed n a general form as NRT N a P = V Nb V + ubnv + wn b. (1) Peng-Robnson, Redlch-Kwong, Redlch-Kwong- Soave, and van der Waals equatons of state are most wdely used two-parameter cubc equatons of state. In the above equaton, N s the total number of moles of the flud, u and w are the two parameters that determne the type of the cubc equatons of state, and for Peng- Robnson equaton of state, u= and w=-1. Prevous research shows that t provdes the best accuracy n predctng the compressblty factor compared to the other equatons of state [3,4]. Therefore, Peng- Robnson equaton of state wll be used n ths study. a and b n Eq. (1) are functons of the crtcal temperatures and pressures of the components and the composton of the mxture. By usng contnuous thermodynamcs to model the mult-component fuel, both parameters are correlated as functons of molecular weght, and the mean values of the fuel can be evaluated usng the dstrbuton functon as and 1/ = ( ) ( ) 0 a a I f I di (13) = ( ) ( ). (14) b b I f I di 0 Eqs. (13) and (14) serve as the mxng rule for applyng the equaton of state for a mult-component fuel usng contnuous thermodynamcs. Even though the cubc equatons of state shown n Eq. (1) and the calculaton of fugacty coeffcent as descrbed n Eq. (11) are applcable n both lqud and vapor phases, the calculaton of fugacty coeffcents s carred out separately for both phases due to the dfference n the overall composton. Snce ths paper s focused on the vaporzaton characterstcs of lqud drops blended by two mult-component fuels, the ambent gas at the drop surface s a mxture of both fuel vapors and the ambent gases. Therefore, t s mportant to notce that n Eq. (1), parameters ncludng compressblty factor, volume, and parameters a and b should be the overall values of the gas mxture. There are several methods to obtan the overall values of a and b for the

5 mxture based on the mole fracton of each speces. In ths paper, a mxng rule, whch s essentally the same as that descrbed n Eqs. (13) and (14), s used, namely, and ( ) a y y a a k (15) 1/ m = j ( j ) 1 j j b = y b. (16) m a and b are functons of the crtcal temperature and pressure for the vapor phase speces, ncludng fuel vapors and ambent gases such as oxygen and ntrogen. For sngle-component fuels and ambent ar, both a and b can be calculated usng ther crtcal propertes. For mult-component fuels descrbed by contnuous thermodynamcs, such as desel fuel, t can be shown that both parameters n the above two equatons are the mean values obtaned by Eqs. (13) and (14). k s the j bnary nteracton coeffcent between speces and j, and the values can be found for some speces pars. On the other hand, snce the mxture n the lqud phase s also composed of all the speces exstng n the vapor phase, the calculaton of a and b for the mxture (a m and b m ) should also be based on certan mxng rules to account for the devaton from the deal soluton of lqud fuels under hgh-pressure condtons. In the lqud phase, the fugacty coeffcent of each component n the mxture can be calculated n the same way wth that of the vapor phase descrbed n Eqs. (15) and (16). Substtutng both mxng rules dscussed above nto Eq. (11), the fugacty coeffcent of contnuously dstrbuted components as a functon of molecular weght can be calculated as ( ) ( ) ( ) ( ) ˆ v b I ln φ ( I ) = ln ln Z + Z 1 v b b 1/ ( ) ( ) 1/ b a a b I a I + RTb u 4w v + b u + u 4w ln v + b u u 4w. (17) In the above equaton, v s the molar volume (m 3 /mol) of the mxture. Ths equaton provdes a means to determne fugacty coeffcent as a functon of molecular weght based on contnuous thermodynamcs. It can be seen from the above equaton that the correlatons of a and b as functons of molecular weght I must be obtaned to calculate the fugacty coeffcent. In ths paper, based on the data for C1 to C0 alkanes, a(i) for Peng- Robnson equaton of state s correlated as ( ) ( ) a 1/ I = T 1/ I T 1/, (18) and, b(i) s a lnear functon of the molecular weght descrbed as ( ) I b I =. (19) Substtutng the above two correlatons nto Eq. (17), the logarthmc fugacty coeffcent as a functon of molecular weght and can be generalzed as a lnear functon of molecular weght as ln ˆ( φ I ) = AI + B. (0) It can be seen that A and B are determned by pressure, temperature, and the compostons of the lqud or vapor phases. Substtutng the above relaton nto Eqs. (8), (9), and (10), the mole fracton, mean molecular weght, and second moment can be calculated as and y F [ γ ( ) + ( )] l [ 1 β ( A A )] α x exp A A B B F l v l v = θ = γ v Ψ = θ γ γ + v v l l v β Γ ( α + 1) l l [ β ( A A ) 1] Γ ( α ) l l v l β Γ ( α + ) l l [ β ( A A ) 1] Γ( α ) l l v l, (1), (). (3) It can be seen that all the three equatons above are actually mplct, because parameters A and B n the vapor phase are ndeed determned by the mxture s overall propertes whch are affected by the mean molecular weght (θ) of the fuel vapor through the mxng rule. By performng the teratve calculatons descrbed n Eqs. (17) to (3), the molecular weght dstrbuton for the mult-component fuel vapor at the drop surface can be obtaned. Due to the fact that the vapor mean molecular weght s smaller than that of lqud phase, the contnuously dstrbuted components wll vaporze n an order accordng to ther molecular weghts. In general, lghter components tend to vaporze earler than the heaver components. Ths phenomenon wll cause the composton dstrbuton n the lqud phase to shft to the hghmolecular-weght end and cause the mean molecular weght of the fuel to ncrease. 4

6 The phase equlbrum dscussed n the above secton relates the mole fracton of speces n each phase at the phase nterface and provdes the boundary condtons for the speces conservaton equatons of both phases. By calculatng the phase equlbrum of an n- heptane-ntrogen bnary system, Fg. 1 shows the calculated mole fractons of n-heptane at the phase nterface wth respect to ambent pressure for dfferent drop surface temperatures. The upper-rght part of each curve wth sold symbols s for the lqud phase, and the lower-left part wth open symbols s for the vapor phase. For each temperature, both parts of the curves orgnate from the saturated pressure, where the mole fracton of n-heptane s equal to unty n both phases. As pressure ncreases, the lqud and the vapor parts of the curve start to devate and fnally meet at the crtcal mxng pont. At the crtcal mxng pont, the temperature, pressure, and composton of both phases are dentcal, and there s no phase dfference n the system. Thus, phase equlbrum no longer holds at or beyond the crtcal mxng pont. It can be seen from Fg. 1 that under hgher pressure condtons, the amount of ambent gas that dssolves nto the lqud phase cannot be gnored. Consequently, n contrast to the assumpton of no ambent ar dssolvng n the lqud phase at low pressure condtons, the phase equlbrum of hgh-pressure condtons wll change the speces concentraton feld n the lqud phase. Fgure 1. Phase equlbrum for n-heptane-ntrogen system at varous temperatures and pressures. The speces dffuson n the lqud phase can be smplfed as one-dmensonal wth the assumpton of a sphercal drop and the absence of forced convecton. Ths paper assumes the components n a multcomponent fuel have the same dffusvty n the lqud phase. Therefore, only the bnary dffuson between the fuels n the mxture s consdered. Snce there s no bulk moton n the lqud phase, the governng equaton for 5 speces conservaton n the lqud phase for a bnary fuel mxture can be further smplfed as x = D t r r ( r x ) 1. (4) Subscrpt n the above equaton stands for an ndvdual n the mxture. At the drop center, symmetry requres x = 0. (5) r At the surface of the drop, mass conservaton for each fuel leads to x. (6) r m x 4π R D ρ = m ε tot, s tot In the above equaton, ε s the fracton of mass vaporzaton rate for fuel. x,s s the mole fracton of fuel at drop surface n the lqud phase, and the analytcal soluton of the above equatons obtaned by Brenn [14] s used to calculate the mass fracton dstrbutons for the multple components. Due to the fact that the bnary dffusvty between the ambent gas speces and the lqud component s very small n the lqud phase, the characterstc dffuson tme (t*=r /D) wll be relatvely large compared to the drop lfetme. Therefore, for smplcty, ths paper assumes that the ambent gas speces wll be restrcted at the drop surface durng vaporzaton, and the dffusve flux of the ambent gas speces toward drop center s nfntesmal. In the vapor phase, snce the characterstc tme s much smaller than that of the lqud phase, t s reasonable to obtan a pseudo-steady soluton of the speces conservaton equaton. By applyng the assumpton that the dffusve flux of the ambent gas speces nto the lqud phase s zero, the soluton of the speces conservaton equaton n the vapor phase wll be a smple formula relatng the speces vaporzaton rate and the mass fracton of the fuel vapor at the drop surface, namely, s ( ) m = m ε = 4π Rρε D ln 1 + B. (7) tot, g M, B M, s the Spaldng mass transfer number for fuel and s defned as B M, = y, s, y ε y, s. (8)

7 Subscrpts s and denote locatons at the drop surface and nfnty, respectvely. The mass fractons of speces n lqud and vapor phases,.e., x,s and y,s n Eqs. (6) and (8), can be related by the phase equlbrum relatons. In the case of forced convecton, the followng classcal correlaton for Nusselt number s used to account for the effect of convectve heat transfer. and 1/ 1/3 = (9) Nu Re Pr 1/ 1/3 = (30) Sh Re Sc Sc n Eq. (30) s the Schmdt number for the fuel. The Sherwood number s used to correct the mass vaporzaton rate descrbed by Eq. (7) where a non-convecton value of two s used. The enthalpy of vaporzaton determnes the amount of heat requred for the vaporzng lqud fuel mass and therefore has a sgnfcant effect on the drop heatng rate. In a lqud-vapor system composed of multple speces, the enthalpy of vaporzaton for a speces s the dfference of partal molar enthalpy n both phases as Results and dscusson The hgh-pressure vaporzaton model descrbed above s frst appled to n-heptane drop vaporzaton at varous ambent pressures and temperatures. Numercal results of the drop vaporzaton hstory are compared wth expermental data by Nomura et al. [16] for valdaton. The comparsons between numercal and expermental results are shown n Fgs. to 4. It can be seen that the numercal results n varous ambent condtons agree well wth the expermental data n both the shape of the drop sze hstory and the total lfetme. The tradtonal low-pressure model based on the Rault s law s also appled to the same condtons and compared wth the hgh-pressure model results and expermental data. It can be seen that as the ambent pressure ncreases, the error of usng the Rault s law ncreases. The applcaton of the Rault s law leads to longer drop lfetme and wll subsequently affect the dstrbuton of fuel vapor n practcal combuston devces under hgh-pressures. Consequently, the applcaton of hgh-pressure drop vaporzaton model n mult-dmensonal engne smulaton wll be an mportant advancement n mprovng the modelng accuracy. (,, ) (,, ) h = h T P y h T P x. (31) vap,, v, l It can be shown that the partal molar enthalpy for speces n a mxture can be obtaned usng a departure functon as h h = RT ( ln ˆ φ ) T P, y, (3) Fgure. Vaporzaton hstory of a sngle n-heptane drop at P=1.0 MPa, T=669 K. where denotes an deal state of very low pressure at the same temperature. At ths deal state, the partal molar enthalpy s equal to the molar enthalpy of that pure speces. It can be seen from ths equaton that the enthalpy of vaporzaton of a speces wll be dependent on all the speces that make up the two-phase system. At the crtcal mxng pont, the enthalpy of vaporzaton wll be zero snce there s no phase dfference n the system at ths pont. Fgure 3. Vaporzaton hstory of a sngle n-heptane drop at P=.0 MPa, T=656 K. 6

8 hgh-pressure condtons. One reason leadng to the dfference n drop surface temperature s the relatonshp between pressure and enthalpy of vaporzaton. By usng Eq. (34), the enthalpy of vaporzaton of desel fuel wth the ntal composton (α=3, β=8.15, γ=0) as a functon of temperature under dfferent ambent pressures s shown n Fg. 6. It can be seen that the enthalpy of vaporzaton decreases steadly to zero as the temperature ncreases to the value at the crtcal mxng pont of a partcular pressure. At hgher pressures, the enthalpy of vaporzaton s lower, leadng to a hgher drop vaporzaton rate. Fgure 4. Vaporzaton hstory of a sngle n-heptane drop at P=5.0 MPa, T=453 K. The vaporzaton modelng of desel fuel drops under engne operatng condtons s of prmary nterest snce t stll prevals n fuellng compresson-gnton engnes. In ths paper, the composton of desel fuel s modeled usng contnuous thermodynamcs based on a gamma functon shown n Eq. (7). For desel fuel, the dstrbuton parameters are α=3, β=8.15, and γ=0. The contnuous thermodynamcs model s coupled wth hgh-pressure phase equlbrum as descrbed by Eqs. (17), (18), and (19). Thus, addtonal numercal teratons are needed for the mplct soluton of the composton and fugacty coeffcents. Note that the fugacty coeffcents n the vapor phase are functons of the mean molecular weght whch, n the meantme, are also determned by the fugacty coeffcents as shown from Eqs. (18) to (3). The vaporzaton of desel fuel drops at ambent temperature 900 K and varous ambent pressures wthn the range of engne n-cylnder condtons s smulated. In these cases, desel drops wth ntal dameter of 0.1 mm and ntal temperature of 300 K s exposed to a pure ntrogen envronment. Hstores of the drop surface temperature and the square of the reduced drop dameter for dfferent ambent pressures are shown n Fg. 5. It can be seen that as ambent pressure ncreases, the heatng perod, whch s characterzed by the ncreased drop sze, s prolonged. On the other hand, f the pressure s hgher, the vaporzaton rate (.e., slope of the curve) followng the heatng perod wll ncrease, causng the overall drop lfetme to decrease. For nstance, the drop at 0 bar starts to vaporze later than the drop at 1 bar. However, when the drop at 0 bar starts to vaporze, the vaporzaton rate s much hgher,.e., steeper slope of the drop sze curve. Ths phenomenon can be explaned by the fact that at a hgher pressure, the drop surface temperature s hgher, resultng n a hgher vaporzaton rate. As can also be seen n Fg. 5, the drop surface temperature s sgnfcantly hgher for 7 Fgure 5. Drop sze and surface temperature hstores for desel fuel drop at dfferent ambent pressures, T amb =900 K. Fgure 6. Enthalpy of vaporzaton as a functon of temperatures for desel fuel at varous pressures. On the other hand, ambent temperature also affects the way that the drop lfetme vares wth ambent pressure. To nvestgate such effects of ambent temperature, calculatons were also performed for ambent

9 temperature 700K, and results are shown n Fg. 7. Dfferent from the results of the 900 K case, the drop lfetme at 700 K ncreases wth ncreased ambent pressure. In fact, from the phase equlbrum, hgher pressure reduces the mass transfer number and thus decreases the vaporzaton rate. Ths behavor counteracts the effect of ncreasng drop surface temperature that enhances the vaporzaton rate. These two effects compete wth each other and the net result vares. From ths study, t s shown that only f the ambent temperature s hgh enough, the drop surface temperature wll ncrease sgnfcantly at hgh-pressure condtons, causng a hgher vaporzaton rate and shorter drop lfetme (e.g., 900 K for desel fuel). It s antcpated that ths threshold temperature wll be dfferent for dfferent fuels. Fgure 7. Drop sze and surface temperature hstores at dfferent ambent pressures, T amb =700 K. Fg. 8 shows the evoluton of the mean molecular weghts of desel fuel n both phases at the drop surface. It can be seen that the molecular weght of the fuel vapor s smaller than that of the lqud. Ths phenomenon ndcates that the vaporzng components are the relatvely lghter components. As a result of contnuous vaporzaton, the mean molecular weght ncreases slowly at the begnnng. As the lghter components contnue to vaporze, the mean molecular weght ncreases rapdly at the end when the drop sze s very small. For the cases of 0 and 50 bar, where vaporzaton ends at crtcal mxng pont (see Fg. 5), the molecular weghts of the lqud and the vapor are equal at the crtcal mxng pont. Ths s consstent wth the essence of the crtcal mxng pont that there s no dfference n composton between the lqud phase and vapor phase. 8 Fgure 8. Varaton of mean molecular weght n both lqud and vapor phases at the surface of desel drop at varous ambent pressures, T amb =900 K. Bodesel s a renewable fuel whch can be produced from vegetable ols, anmal fat, or other bomass feedstock. Due to ts close physcal and chemcal characterstcs to regular desel fuel, t s a feasble alternatve fuel for compresson-gnton engnes. The composton of bodesel can be complcated due to the varaton n the feedstock. However, as dscussed above, the major components of bodesel are fve methyl esters, namely, palmtc (C 16 H 3 O ), stearc (C 18 H 36 O ), olec (C 18 H 34 O ), lnolec (C 18 H 3 O ), and lnolenc (C 18 H 30 O ). Therefore, the present paper assumes that bodesel s a mxture of the above fve components. The composton of bodesel,.e., the mass fractons of these components, derved from typcal feedstock can be found n lterature [11]. To perform the phase equlbrum calculaton for bodesel under hgh-pressure condtons, the mean values for a and b n the cubc equaton of state can be obtaned usng the mxng rule shown n Eq. (15) based on the composton of bodesel and crtcal temperatures and pressures of ts fve major components [11]. The present model was used to smulate the vaporzaton of bodesel drops under hgh pressure and hgh temperature condtons smlar to those n desel fuel vaporzaton study. The drop sze hstores of bodesel drops at 900 K and varous pressures are shown n Fg. 9. It can be seen that bodesel drops generally have longer lfetme than desel drops n the same ambent condton. The lfetme of the bodesel drop vares n the same way as the desel fuel drop when pressure changes,.e., hgher pressure reduces drop lfetme. On the other hand, n a desel engne, consderng engne performance and emssons, a practcal way of usng bodesel s to blend t wth desel fuel n dfferent volumetrc ratos due to ther hgh mscblty. In ths study, the vaporzaton of B0 drops, whch are blends of 0% bodesel and 80% desel on volume bass, under dffer-

10 ent ambent pressures are smulated. The hstores of drop sze under dfferent ambent pressures are shown n Fg. 10. Results show that the vaporzaton hstores of B0 drops are n between those of desel fuel and bodesel at the same ambent condtons. It s expected that the composton of B0 drop wll keep changng durng the vaporzaton process due to the dfference n the propertes of both fuels n the mxture. Fg. 11 shows the evoluton of the bodesel volume fracton n a B0 drop. It can be seen that the concentraton of bodesel components ncreases n all the cases durng the vaporzaton process except at the end, and for hgher ambent pressure cases, the volume fracton of bodesel ncreases at a slower rate. As shown n Fg. 8, at the end of vaporzaton when the drop s very small, the mean molecular weght of desel fuel becomes very large. These heavy desel fuel components vaporze more slowly than bodesel components. As a result, the volume fracton of bodesel decreases at the end of vaporzaton as shown n Fg. 11. Fgure 9. Drop sze hstores for bodesel at dfferent ambent pressures, T amb =900 K. Fgure 10. Drop sze hstores for B0 at dfferent ambent pressures, T amb =900 K. 9 Fgure 11. Hstores of volume fracton of bodesel for B0 at dfferent ambent pressures, T amb =900 K. In addton, to study the effect of the composton of desel-bodesel blend on vapor dstrbuton under hgh pressure condtons, the model was further appled to desel-bodesel sprays wth dfferent ntal compostons,.e., B0, B0, and B100, n an envronment of pure ntrogen at 38 bar and 900 K. Smulatons were performed at the same ambent condtons as n the experments by Pckett and Sebers [17]. The CFD code s based on an updated KIVA-3V [18] wth mprovements n varous physcal models, e.g., turbulence and spray atomzaton [19][0]. The computatonal doman has a heght of 10 cm and a dameter of 1.5 cm and s dscretzed wth a two-dmensonal mesh whch has 3 cells n the radal drecton and 50 cells n the axal drecton. Only non-combustng condtons are smulated here snce the emphass of ths work s to study the vaporzaton behavor. For each spray, the njected fuel mass for each fuel spray s 8 mg wth an njecton duraton of 3 ms. The hstores of the total njected fuel mass and total vapor mass for each spray n the doman are shown n Fg. 1. It can be seen that the vaporzed fuel mass decreases as the ntal volume fracton of bodesel n the blend ncreases due to the slower vaporzaton rate of bodesel components. The mass fracton dstrbutons of both desel fuel vapor and bodesel vapor for dfferent sprays at 1.4 ms after the start of njecton are shown n Fg. 13. For the B0 spray, desel fuel vapor exsts n a large regon smlar to that n desel fuel (B0) spray, whle there s a small amount of bodesel concentratng n a small regon near the tp of the spray. Determned by the crtcal propertes, bodesel (B100) spray produces much smaller amount of vapor n comparson wth desel fuel spray, and the vapor concentrates n a regon dstant from the njector.

11 Fgure 1. Hstores of total njected fuel mass and vapor mass n the doman for sprays of B0, B0, and B100, P amb =38 bar, T amb =900 K. Desel fuel vapor for B0 spray Desel fuel vapor for B0 spray Bodesel vapor for B0 spray Bodesel vapor for B100 spray Fgure 13. Desel fuel and bodesel vapor dstrbutons for B0, B0, and B100 sprays at t=1.4 ms, T amb =900 K, P=38 bar. Concluson Numercal modelng was performed to smulate the vaporzaton of mult-component fuel drops under engne operatng condtons whch are characterzed by hgh pressures and hgh temperatures. The phase equlbrum calculaton under hgh-pressure condtons s based on the equalty of fugacty of each component n both phases. To couple contnuous thermodynamcs wth hgh-pressure phase equlbrum, the parameters of Peng-Robnson equaton of state are correlated wth the contnuously dstrbuted molecular weghts of the components based on a gamma dstrbuton. Methods of estmatng propertes such as densty, dffusvty, and enthalpy of vaporzaton under hgh-pressure condtons are also developed. The present hgh-pressure vaporzaton model was valdated usng the expermental data of n-heptane 10 drops. The predcted hstory of drop sze durng vaporzaton agrees well wth the expermental data. The present model was further appled to predct the vaporzaton of desel fuel drop under dfferent ambent temperatures and pressures. Results of the 900 K ambent temperature case show that the drop lfetme decreases wth ncreased ambent pressure. At 700 K ambent temperature, the drop lfetme ncreases wth ncreased ambent pressure, opposte to the 900 K ambent temperature case. Ths phenomenon reveals that the effect of ambent pressure on the drop lfetme can depend on ambent temperature. It s also found that two competng factors, namely, reduced mass transfer number (decreasng vaporzaton rate) and reduced enthalpy of vaporzaton (ncreasng vaporzaton rate) for hgher pressure condtons determnes the way n whch the drop lfetme changes wth ambent pressure. In any case, model results show that the molecular weght of the vapor s smaller than that of the lqud and both molecular weghts keep ncreasng durng the process, ndcatng that lghter components vaporze earler than heaver components. The vaporzaton of bodesel and ts blend wth desel fuel was also smulated. Bodesel s modeled as the mxture of fve representatve components wth compostons determned by the feedstock. Results of bodesel drops show a smlar trend to desel fuel n drop lfetme wth respect to ambent pressure. In the vaporzaton modelng of the blend of desel fuel and bodesel, contnuous thermodynamcs s used to model desel fuel and bodesel s assumed to be a mxture of the fve components. The predcted vaporzaton hstory of the fuel blend les n between those of desel fuel and bodesel. The volume fracton of bodesel ncreases n most of the vaporzaton process except at the end of the process when the desel fuel components n the drop become very heavy. Such a behavor s also observed from the dstrbutons of desel fuel vapor and bodesel vapor n the sprays usng the fuel blends. Results of the sprays show that at a certan pont n tme, the vapor of bodesel only concentrates n a small regon near the tp of the spray, whle desel fuel vapor exsts n a much wder regon around the entre lqud spray. References 1. Tamm, J., and Hallett, W., Chemcal Engneerng Scence 18: (1995).. Hseh, K. C., Shuen, J. S., and Yang, V., Combuston Scence and Technology 76: (1991). 3. Ja, H., and Gogos, G., Internatonal Journal of Heat and Mass Transfer 30: (1993). 4. Curts, E. W., and Farrell, P. V., Combuston and Flame 90: (199).

12 5. Aggarwal, S. K., and Monga, H. C., Journal of Engneerng for Gas Turbnes and Power 14: (00). 6. Zeng, Y., and Lee, C. F., Journal of Engneerng for Gas Turbnes and Power 14: (00). 7. Zhu, G., and Retz, R. D., Internatonal Journal of Heat and Mass Transfer 45: (00). 8. Fang, T., Ln, Y. C., Foong, T. M., and Lee, C. F., Fuel 88: (009). 9. Ramos, M., Fernandez, C., Casas, A., Rodrguez, L., and Perez, A., Boresource Technology 100: (009). 10. Ramadhas, A. S., Muraleedharan, C., and Jayaraj, S., Renewable Energy 30: (005). 11. Zhang, L. and Kong, S.-C., Combuston and Flame 157: (010). 1. Dodge, B. F., Chemcal Engneerng Thermodynamcs, frst edton, McGraw-Hll, New York, N. Y., Red, R. C., Prausntz, J. M., and Polng, B. E., The Propertes of Gases & Lquds, fourth edton, McGraw-Hll, New York, N. Y., Brenn, G., Internatonal Journal of Heat and Mass Transfer 48: (005). 15. Takahash, S., Journal of Chemcal Engneerng of Japan 6: (1974). 16. Nomura, H., Uje, Y., Rath, H. J., Sato, J., and Kono, M., 6th Symposum (Internatonal) on Combuston/The Combuston Insttute, Japan, 1996, pp Pckett, L. M. and Sebers, D. L., Combuston and Flame 138: (004). 18. Amsden, A., KIVA-3V, Release Improvements to KIVA-3, LAMS Report (1999). 19. L, Y., and Kong, S.-C., Combuston Theory and Modelng 1: (008). 0. Zhang, L., and Kong, S.-C., Chemcal Engneerng Scence 64: (009). 11