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SAE TECHNICALPAPER SERIES 1999-01-0528

Scaling Liquid-Phase Fuel Penetration in DieselSprays Based on Mixing-Limited Vaporization

Dennis L. SiebersSandia National Laboratories

Reprinted From: Technology for Diesel Fuel Injection and Sprays(SP-1415)

International Congress and ExpositionDetroit, Michigan

March 1-4, 1999

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ISSN 0148-7191Copyright 1999 Society of Automotive Engineers, Inc.

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1

1999-01-0528

Scaling Liquid-Phase Fuel Penetration in Diesel Sprays Basedon Mixing-Limited Vaporization

Dennis L. SiebersSandia National Laboratories

Copyright © 1999 Society of Automotive Engineers, Inc.

ABSTRACT

A scaling law for the maximum penetration distance ofliquid-phase fuel in a diesel spray (defined as the liquidlength) was developed by applying jet theory to a simpli-fied model of a spray. The scaling law accounts for injec-tor, fuel, and in-cylinder thermodynamic conditions onliquid length, and provides significant insight into the fuelvaporization process. As developed, the scaling law isvalid for single-component fuels, but can be used tomodel multi-component fuels through use of single-com-ponent surrogate fuels.

Close agreement between the scaling law and measuredliquid length data over a very wide range of conditions isdemonstrated. The agreement suggests that vaporiza-tion in sprays from current-technology, direct-injection(DI) diesel injectors is limited by mixing processes in thespray. The mixing processes include entrainment ofhigh-temperature air and the overall transport and mixingof fuel and air throughout the spray cross-section. Animplication of mixing limited vaporization is that the pro-cesses of atomization and the ensuing interphase trans-port of mass and energy at droplet surfaces are notlimiting steps with respect to fuel vaporization in DI dieselsprays.

The scaling law provides a fundamental baseline on liq-uid fuel penetration and vaporization in diesel sprays thatcan be compared with the vaporization aspects of themulti-dimensional diesel spray models under develop-ment. The scaling law can also provide design guidanceon the expected maximum extent of liquid-phase fuelpenetration in engines. Application of the scaling law to aheavy-duty and a light-duty DI diesel shows that liquid-phase fuel impingement on piston bowl walls is not aserious concern in heavy-duty engines using a typicaldiesel fuel, as observed previously through experiments,but may be an issue in light-duty engines.

INTRODUCTION

Improvement of the diesel engine combustion processcontinues to be an important facet of the efforts to meetstringent new emissions regulations, and at the same

time, enhance the performance of diesel engines. Oneimportant issue with respect to optimizing in-cylinder pro-cesses in direct-injection (DI) diesel engines, especiallyin small-bore, automotive DI diesels, is the penetrationand vaporization of liquid-phase fuel. Penetration of theoverall spray in a DI diesel is needed to promote fuel-airmixing, but impingement and collection of liquid-phasefuel on piston bowl and other in-cylinder walls can lead togreater emissions. As a result, understanding how vari-ous parameters affect the penetration of the liquid-phasefuel and what processes control fuel vaporization in a die-sel spray are important, both to the engine designer andto those developing multi-dimensional computationalmodels for use as engine design tools.

This paper presents a scaling law for the maximum pene-tration distance of liquid-phase fuel in a diesel spray,defined as the liquid length. The scaling law accounts forthe effects of injector, fuel, and in-cylinder thermody-namic conditions on liquid length. The assumptions usedin developing the scaling law, the form of the scaling law,and the close agreement achieved with experimentaldata provide significant insight into the processes control-ling fuel vaporization and liquid-phase fuel penetration.

The development of the scaling law is based on recentexperimental results and conclusions presented in Ref. 1.The results and conclusions from Ref. 1 are summarizedin the next section. They indicate that vaporization in aDI diesel spray is limited by mixing processes in thespray. The mixing processes include entrainment ofhigh-temperature air and the overall transport and mixingof fuel and air throughout the spray cross-section.

Injector and in-cylinder thermodynamic conditions overwhich the scaling law applies include conditions thatoccur in current and proposed DI diesels. With respect tofuels, the scaling law as developed applies strictly for sin-gle-component fuels, but can be used to model multi-component fuels through use of surrogate single-compo-nent fuels, as will be shown for a standard #2 diesel fuel.

The remainder of the paper is divided in to seven sec-tions and two appendices. In the first section, previousresearch on liquid-phase fuel penetration in diesel spraysis summarized. An emphasis in this section is placed on

2

recent experimental work presented in Ref. 1, since aconclusion drawn from that work is the basis for the liquidlength scaling law developed in this paper. The next fivesections discuss: the derivation of the liquid length scal-ing law, a comparison with data, the implications of thescaling law with respect to fuel vaporization, small sys-tematic trends in the data relative to the scaling law, andfurther insight drawn from the scaling law, respectively.The latter of these sections includes a discussion ofexpected liquid lengths in typical heavy-duty truck andproposed light-duty automotive DI diesel engines. Thefinal section presents a summary and conclusions.

In the appendices, the spray spreading angle dataacquired simultaneously with the liquid length data inRef. 1, and the orifice area-contraction coefficients for theorifices used to acquire the data in Ref. 1 are presented.Both of these data sets are required in the liquid lengthscaling law, but have not been previously presented.

BACKGROUND

Research on liquid-phase fuel penetration in vaporizingsprays is beginning to provide significant improvement inour understanding of the penetration and vaporization ofliquid-phase fuel in diesel sprays. Early research on liq-uid-phase fuel penetration primarily revealed the evolu-tion of the penetrating tip of the liquid region in a spray,and identified several parameters important to liquid-phase fuel penetration. This research is discussed indetail in Ref. 1. In summary, it showed that liquid fuel ini-tially defines the penetrating tip of a diesel spray. Thiscontinues until the liquid fuel penetrates to a point wherethe total fuel evaporation rate in the spray equals the fuelinjection rate. When this condition occurs, the tip of theliquid region stops penetrating and begins fluctuatingabout a mean axial location [2-5], with only vapor-phasefuel penetrating beyond this location. Parameters thatwere found to affect the location of the tip of the liquid-phase fuel included the injector orifice diameter [2], ambi-ent gas conditions [2], and fuel volatility [2], while injec-tion pressure seemed to have little effect [3,6].

More recently, three other papers have reported resultson liquid-phase fuel penetration that provide both quanti-tative data and an improved understanding of the pro-cesses controlling fuel vaporization. Two papers by Decand coworkers provided data on liquid-phase fuel pene-tration for typical heavy-duty diesel engineconditions [7,8]. Their results on liquid length (a) con-firmed the general nature of liquid-phase fuel penetrationnoted earlier, (b) demonstrated the strong effects of in-cylinder gas temperature and density on liquid length, (c)showed that liquid lengths were relatively short for typicalheavy-duty engine conditions, and (d) showed that theenergy for vaporizing the fuel comes mainly from high-temperature air entrained into the spray. Their last obser-vation was based on the facts that the liquid length wasestablished before ignition occurred, that liquid length didnot shorten significantly after ignition, and that vaporiza-tion occurred largely upstream of the combustion zone

formed after ignition. The third paper, Ref. 1, presented acomprehensive investigation of the effects of injector,fuel, and in-cylinder ambient gas conditions (i.e., pres-sure, temperature and density) on liquid length in a dieselspray conducted over a wide range of well characterizedconditions. This latter work is summarized and dis-cussed in more detail next, since a main conclusionderived from it forms the basis for the scaling law devel-oped in this paper, and since the extensive data pre-sented in it are compared with the scaling law. Thefollowing discussion on Ref. 1 only covers the pointsimportant to the development of the scaling law and/orapplication of the scaling law in this paper. For moreextensive discussions of the experimental apparatus andtechniques, the data analysis, and the results, consultRef. 1.

REVIEW OF REFERENCE 1 – The investigation ofliquid-phase fuel penetration presented in Ref. 1 wasconducted in an optically accessible, constant-volumecombustion vessel designed for conducting diesel com-bustion experiments at pressures up to 35 MPa. Fuelwas injected with an electronically controlled, common-rail diesel fuel injector capable of injection pressures upto 200 MPa. The ambient gas composition in the com-bustion vessel can be varied. The ambient gas for theexperiments reported in Ref. 1 was inert with a composi-tion of 89.7% N2, 6.5% CO2, and 3.8% H2O and a molec-ular weight of 28.67.

Inert conditions were used to focus the research on theprimary mechanism for fuel vaporization, i.e., entrain-ment of high-temperature ambient gases [7], and to pro-vide data for comparison with multidimensional modelsunder simplified, well characterized conditions. The inertenvironment resulted in liquid lengths 4% shorter thanwould have been obtained in air, as a result of heatcapacity effects [1]. (The liquid length scaling law devel-oped in this paper will be used to show the inert ambientgas composition effects relative to air.)

Parameters varied in the investigation in Ref. 1 included:injection pressure, orifice diameter, orifice aspect ratio,ambient gas temperature, ambient gas density, fuel tem-perature, and fuel volatility. The ranges considered forthe engine related parameters included those in currentand proposed advanced diesel engine technologies.Table 1 lists the ranges considered for the various param-eters.

The fuels considered were heptamethylnonane (HMN),n-hexadecane (cetane), and a standard diesel fuel (DF2).Table 2 gives some of the basic properties of the DF2.HMN and cetane were used because they are well char-acterized single-component fuels that span the distillationtemperature range for the DF2. HMN and cetane haveboiling points equivalent to the 30% and 80% tempera-ture points on the DF2 distillation curve, or 520 K and560 K, respectively. Properties sources for HMN and cet-ane are discussed later.

3

Important characteristics of the orifices used in the exper-iments are listed in Table 3 (see Ref. 1 for more details).The values given in the table for each orifice are thediameter (d), the discharge coefficient (Cd), the area-con-traction coefficient (Ca) for two injection pressure differ-ences across the orifice (72 and 138 MPa), and thelength-to-diameter ratio (l/d). All the orifices had sharpedged inlets and outlets.

The discharge coefficients given in Table 3 were mea-sured with standard techniques described in Ref. [9].The area-contraction coefficients, not previously given inRef. 1, appear in the scaling law and are discussed indetail in Appendix B of this paper. The area-contractioncoefficient accounts for flow area loss in the orifice due tocavitation or other orifice flow phenomena. In the follow-ing discussions, area-contraction coefficients for injectionpressures other than listed in Table 3 were linearly inter-polated from the values in the table as needed.

The liquid lengths reported in Ref. 1 were time-averagedliquid lengths determined from time-averaged Mie-scat-tered light images of the liquid-phase fuel in a dieselspray. The liquid lengths determined from the imageswere repeatable to ±4%, and were estimated to accountfor more than 97% of the liquid-phase fuel. It was alsoshown that the quasisteady liquid lengths reported hadinstantaneous turbulent fluctuations of about ±11%.

Table 1. Parameter ranges covered in Ref. 1.

Table 2. Phillips research grade DF2 fuel properties.

The dominant trends observed in the liquid length data inRef. 1 with respect to injector and in-cylinder thermody-namic conditions were that liquid length (a) decreaseslinearly with decreasing orifice diameter, approachingzero as the diameter approaches zero, (b) is insignifi-cantly affected by injection pressure, and (c) decreaseswith increasing the ambient gas density or temperature,but in a non-linear manner. With respect to fuel parame-ters, liquid length was found to increase with decreasingfuel volatility, and decrease linearly with increasing fueltemperature. In addition, the results for different fuelssuggested that the liquid length of a multi-component fuelis controlled by its lower volatility fractions, suggesting abatch distillation-type process with higher volatility com-ponents evaporating first and lower volatility componentscontrolling the liquid length.

Two of the most revealing trends in the previous para-graph relative to fuel vaporization and the liquid lengthscaling law developed in this paper are the first two.These trends can be observed in the data in Figs. 1and 2 (reproduced from Ref. 1). Figure 1 shows the lin-ear dependence of liquid length on orifice diameter overa wide range of conditions, while Fig. 2 shows that liquidlength is insignificantly affected by injection pressure overa similar wide range of conditions. As discussed inRef. 1, these trends strongly suggest that vaporization ina diesel spray is controlled by mixing processes in thespray, as opposed to interphase transport rates of mass,momentum, and energy at droplet surfaces. Mixing pro-cesses, as used here, refer to the entrainment of high-temperature air into the spray and the overall transportand mixing of fuel and air throughout the spray cross-sec-tion. These mixing processes are determined by the tur-bulence generated by the spray.

Figure 1. Liquid length versus orifice diameter for a wide range of conditions. The terms in the legend are the ambient gas temperature (Ta) and density (ρa), the orifice pressure drop (∆Pf), and the fuel type. The lines in the figures are linear least squares fits to the data for each set of conditions given in the legend. The aspect ratio of the orifices were nominally 4.2 and 438 K, respectively.

Table 3. Injector tip parameters for the seven tips used in the experiments in Ref. 1. (The Ca for the 363 µm orifice was not measured at 72 MPa.)

Orifice Discharge Area-Contraction Length-to-Diameter Coefficient Coefficient Diameter

D Cd Ca Ca l/d(µm) (72 MPa) (138 MPa)100 0.80 0.91 0.86 4.0180 0.77 0.85 0.82 4.2251 0.79 0.88 0.79 2.2246 0.78 0.89 0.81 4.2267 0.77 0.89 0.82 8.0363 0.81 - 0.85 4.1498 0.84 0.94 0.88 4.3

0 100 200 300 400 500 6000

25

50

75

100

125

Liqu

id L

engt

h [m

m]

Orifice Diameter [µm]

Ta ρa ∆Pf Fuel(K) (kg/m3) (MPa)

696 14.8 136 HMN1000 7.3 136 HMN1007 14.7 137 HMN1300 14.8 135 HMN 996 30.3 133 HMN1000 14.8 56 Cetane1300 30.2 131 Cetane1000 14.8 54 DF21296 30.2 163 DF2

4

Figure 2. Liquid length versus the pressure drop across the injector orifice for a wide range of conditions. The term d in the legend is the orifice diameter. See Fig. 2 for definitions of the other terms. The lines in the figures are linear least squares fits to the data for each set of conditions given in the legend.

The mixing limited conclusion was arrived at by examin-ing the changes in liquid length that would be expectedwith a change in orifice diameter or injection pressure fortwo limiting cases of vaporization: control by mixing pro-cesses or control by interphase transport processes atdroplet surfaces. Only control of vaporization by mixingexplained the trends in Figs. 1 and 2.

Vaporization limited by mixing processes was examinedusing jet theory. From jet theory, the mass of ambientgas entrained into a spray up to any axial distance fromthe orifice ( a(x)) is linearly proportional to the orificediameter (d), the axial distance from the orifice (x), thefuel injection velocity (Uf), and the spray spreading angle(θ) [1], or:

(1)

The terms ρa and ρf in Eq. (1) are the ambient gas andinjected fuel densities, and the velocity Uf is set by theinjection pressure through Bernoulli’s equation. Alsofrom jet theory, fuel mass is conserved in the spray, andtherefore, the fuel mass flow rate at any axial location ( f (x)) is the injected fuel flow rate ( f ) or:

(2)

Based on Eqs (1) and (2), it was argued in Ref. 1 that anincrease in injection velocity (i.e., injection pressure) willresult in equal increases in the mass flow rates of the fueland the entrained ambient gas at any axial location. As aresult, fuel injection pressure should have no effect onthe length of spray needed to entrain enough high-tem-perature ambient gas to vaporize the fuel. The net result

for vaporization limited by turbulent mixing processesshould be no change in liquid length with an increase ininjection pressure, as observed in Fig. 2.

With respect to orifice diameter, Eqs. (1) and (2) showthat an increase in the diameter will result in an increasein the fuel mass flow rate at any axial location propor-tional to the square of the orifice diameter change, butonly a linear increase in the total entrained ambient gasflow rate. Thus, a linear increase in the axial length of thespray needed to entrain enough high-temperature ambi-ent gas to vaporize the fuel is required. The net result formixing limited vaporization should be a linear depen-dence of liquid length on orifice diameter, as observed inFig. 1.

Droplet vaporization theory combined with the experi-mentally observed effects of orifice diameter and injec-tion pressure on droplet size were used to examinevaporization limited by local interphase transport pro-cesses at droplet surfaces. The trends in liquid lengthestimated with respect to orifice diameter and injectionpressure, however, did not agree with Figs. 1 and 2. Inthis limit, liquid length should decrease with a decrease inorifice diameter, but with a less than linear dependenceon orifice diameter; and liquid length should increasewith an increase in injection pressure. (This limit is dis-cussed in more detail in Ref. 1).

In the following section, a scaling law for liquid length in adiesel spray is developed using jet theory based on theobservation that fuel vaporization in a diesel sprayappears limited by mixing processes in a spray. The scal-ing law will help explain all of the liquid length trendsobserved in Ref. 1 with respect to various injector, fueland in-cylinder thermodynamic conditions. The observa-tions will strengthen the concept that vaporization is lim-ited by mixing processes, as well as provide more insighton vaporization in a diesel spray.

THE SCALING LAW DEVELOPMENT

A scaling law for liquid length can be developed from anidealized diesel spray model using principles of conser-vation of mass, momentum and energy. This section dis-cusses the spray model and assumptions used inderiving the scaling law and the scaling law derivation. Italso discusses several details relevant to the scaling law.Many of the assumptions made in the derivation are verysimplistic; however, they have proven successful in thepast in allowing basic features of gas jets and sprays tobe understood [e.g., 9-14].

The idealized spray model used is similar to thatemployed by Naber and Siebers [9] to develop a compre-hensive transient diesel spray penetration correlationvalid over a range of conditions similar to that in Table 1.The application of the model by Naber and Siebers, inturn, built on earlier applications of the model to dieselsprays by Wakuri [12] and Hays [13]. The differencebetween the current use of the spray model and the pre-

25 50 75 100 125 150 175 2000

25

50

75

100

125Ta ρa d Fuel(K) (kg/m3) (µm)

1000 7.2 246 HMN1000 14.8 246 HMN1000 30.2 246 HMN 700 14.8 246 HMN1300 14.8 246 HMN

Liqu

id L

engt

h [m

m]

Orifice Pressure Drop [MPa]

Ta ρa d Fuel(K) (kg/m3) (µm)

700 7.3 100 Cetane1300 30.1 498 Cetane1300 30.2 100 DF21000 14.8 246 DF21000 14.8 498 DF2

),2/()( θ⋅⋅⋅⋅ρ⋅ρ∝ tanUxdxm ffaa�

.)( 2ffff Udmxm ⋅⋅ρ∝= ��

5

vious application by Naber and Siebers is the inclusion ofconservation of energy considerations needed to accountfor fuel vaporization processes.

SPRAY MODEL AND ASSUMPTIONS – The liquid lengthscaling law for diesel sprays was developed using integralcontrol surface techniques applied to an idealized modelof a diesel spray. Figure 3 shows a schematic of the ide-alized spray model and the control surface used for themass, momentum, and energy balances. The idealizedspray is assumed to have the following physical charac-teristics: (a) quasisteady flow with a uniform growth rate(i.e., a constant spreading angle α), (b) uniform velocity,fuel concentration, and temperature profiles (i.e., perfectmixing inside the spray boundaries), and (c) no velocityslip between the injected fuel and the entrained ambientgas.

The first assumption is supported by the axially uniformspray angles measured in the non-head portion of aspray [9]. In addition, the first assumption has beenestablished for the non-head region of transient gasjets [15,16]. The assumption of uniform velocity, fuel con-centration, and temperature profiles is clearly a simplifi-cation. More realistic profiles could be employed, ashave been used by others for analyzing turbulent jets andsprays [e.g., 14,17]; however, the form of the scaling lawthat is derived in this paper would be the same. The novelocity slip assumption is doubtful very near the injectororifice, but should begin applying well before the axiallocation of complete vaporization. This latter assumptionhas also been successfully applied in developing an over-all transient spray penetration correlation for dieselsprays [9].

Figure 3. Schematic of the “idealized” spray model used to develop the liquid length scaling law.

In Fig. 3, the downstream side of the control surface(x=L) is defined to be at the axial location in the spraywhere the fuel has just completely vaporized (i.e., the liq-uid length location in the idealized spray). The thermody-namic assumptions up to this location in the spray are:(a) the vapor phase fuel is at a saturated condition, (b)

the vapor phase fuel is in thermodynamic equilibrium withthe entrained ambient gas and the liquid phase fuel (i.e.,all three are at the same temperature), (c) idealizedphase equilibrium assumptions (i.e., Raoult’s and Dal-ton’s rules) apply, (d) gas absorption in the liquid phase isneglected, and (e) the recovery of kinetic energy in thefuel vaporization region of the spray is neglected.

The assumption of thermodynamic equilibrium betweenthe saturated fuel vapor and entrained ambient gas isequivalent to the “adiabatic saturation1” condition dis-cussed by El Wakil et al. [18, also see 19] for the denseinner region of a diesel spray. Whether “adiabatic satura-tion” applies in the inner region of a spray, however, hasnot been shown. This assumption will be examined andtested by the comparison of the scaling law to the data.The other thermodynamic assumptions will also be dis-cussed more later, since the scaling law can be used toassess their impact as well.

Finally, no consideration is given in the spray model toatomization processes or droplets. The idealized modeltreats the spray more as a locally homogeneous flowwhere the transport rates between phases at droplet sur-faces are fast relative to the transport rates as a result ofthe mixing processes in the spray [20].

SPRAY SPREADING ANGLE – The spray spreadingangle α in Fig. 3 will appear in the liquid length scalinglaw to be derived. The spreading angle is a measure ofthe growth rate of the spray caused by entrainment ofambient gas. The spreading angle of a diesel spray isknown to be a function of orifice geometry parameters(i.e., sharp versus smooth orifice edges, the aspect ratio,the orifice orientation, etc.) and the ratio of the fuel andthe ambient gas densities [e.g., 9, 21-24]. In the scalinglaw, the spreading angle will account for the effects theseparameters on the turbulent transport of momentum fromthe injected fuel to the entrained gas [9].

Unfortunately, because turbulence is a major factor indetermining the spray spreading angle, and because thedetails concerning flow through an orifice are not wellunderstood, no comprehensive theory exists for deter-mining the spreading angle of a spray in terms of theindependent parameters involved. As a result, experi-mentally measured spray spreading angles will be usedin the scaling law. The spray spreading angle used isone commonly measured for a diesel spray, the angledefining the spray outer boundary. These spray outerboundary angles are presented in Appendix A. Theywere measured simultaneously with each liquid lengthpresented in Ref. 1, but have not been reported previ-ously. Also discussed in Appendix A is the schlierentechnique used to measure the angles.

Entrained gas (Pa , Ta , ρa)

(Pa , Tf , ρf)

Vaporizationcomplete (x=L)

x

1. Adiabatic saturation here refers to a thermo-dynamic equilibrium state that occurs when a liquid is adiabatically mixed with a gas in a closed volume and the vapor phase of the liq-uid reaches a saturated condition.

6

The measured angles presented in Appendix A arerelated to the idealized spray angle α in Fig. 3 by a con-stant. This was shown in Ref. 9 by assuming self pre-serving flow and an “equivalence” between the mass andmomentum flow rates of a “real” spray and the idealizedspray in Fig. 3 at any axial distance x. These assump-tions lead to the relationship:

(3)

where θ is the measured spray angle for a real spray andthe constant a has a value of 0.66.

The value used for the constant a in Eq. (3) was derivedfrom a best fit of transient spray penetration data to thespray penetration correlation developed by Naber andSiebers [9]. However, they also estimated a value for theconstant a within a few percent of 0.66 using the self pre-serving flow assumption, the mass and momentum“equivalence” assumption, and typical radial velocity pro-files in sprays, with no reference to any measured spraypenetration data or spray angle data.

DERIVATION – The liquid length scaling law is derivedby solving mass, momentum, and energy balances forthe axial location at which fuel vaporization is complete inthe idealized spray model (i.e., the downstream end ofthe control surface in Fig. 3, x=L). The derivationinvolves three steps. First, a relationship is derived fromconservation of mass and energy for the ratio of the fueland the ambient gas mass flow rates that results in com-plete fuel vaporization when the two are mixed. Second,a relationship is derived from conservation of mass andmomentum for the axial variation of the ratio of the fueland the ambient gas mass flow rates in the spray. Third,the liquid length scaling law is then derived by substitut-ing the mass flow rate ratio determined in the first stepinto the relationship for the mass flow rate ratio as a func-tion of axial distance from the second step, and solvingfor the axial location at which vaporization is complete,x=L. The following three sections present each of thesteps, respectively.

Requirement for Complete Fuel Vaporization – Based onthe assumptions made, there is one mixture of fuel andambient gas that will result in complete vaporization ofthe fuel for any set of fuel and ambient gas propertiesand initial conditions. This mixture ratio can be deter-mined from the mass and energy balances for the controlsurface shown in Fig. 3, and a real gas equation of state(to be discussed later). The mass and energy balancesare:

(4)

(5)

(6)

The term f in the preceding equations is the mass flowrate of the injected fuel. From conservation of fuel mass,

f is also the fuel mass flow rate at the liquid length, or

f (L). The term a(L) is the total entrained ambientgas mass flow rate up to L, which is equal to the ambientgas mass flow rate through the downstream control sur-face at L. The terms ρf , Uf , and Af are the density, theaxial velocity and the cross-sectional area of the fuel atthe orifice exit, respectively; and the terms ρf (L), ρa(L),U(L), and A(L) are the partial fuel density, the partialambient gas density, the spray velocity, and the spraycross-sectional area at L, respectively. By definition atx=L, the fuel is completely vaporized and the partial den-sities of the fuel and the ambient gas are dependent ontheir respective partial pressures.

The terms hf and ha in Eq. (6) refer to the specific enthal-pies of the fuel and ambient gas, respectively. The leftside of Eq. (6) is the sum of the initial enthalpy of the fuelat the orifice exit and the enthalpy of the ambient gasentrained into the spray. The fuel at the orifice exit isassumed to be at the measured injector tip temperature,Tf , and the ambient pressure, Pa. The ambient gas spe-cific enthalpy is determined at the measured ambient gaspressure and temperature, Ta. The right side of Eq. (6) isthe sum of the enthalpies of the fuel and ambient gas atL, after mixing and complete vaporization of the fuel. Byprevious assumption, at this location the vaporized fuel isat a saturated condition in thermodynamic equilibriumwith the ambient gas, both with a temperature Ts. Sincethe fuel is saturated, the saturation temperature Ts alsodefines the partial pressure of the vapor fuel, Ps, throughthe saturation pressure-temperature relationship for thefuel. The partial pressure of the entrained ambient gas inthe spray is given by the difference between the ambientpressure and the partial pressure of the vaporized fuel inthe spray, or Pa-Ps.

The ratio of the fuel and the ambient gas mass flow ratesrequired to vaporize the fuel is obtained by rearrangingEq. (6), and substituting in Eqs. (4-5) and the real gasequation of state of the form P=ZρRT/M. First, rearrang-ing Eq. (6) yields the desired mass flow rate ratio in termsof the specific enthalpies:

(7)

The enthalpy difference in the numerator is the specificenthalpy transferred from the entrained ambient gas toheat and vaporize the fuel. The enthalpy difference in thedenominator is the specific enthalpy required to heat andvaporize the liquid fuel.

Next, substitution of Eq. (4-5) and the equation of stateinto the left side of Eq. (7) gives the fuel and the ambientgas mass flow rate ratio, now defined as B, in terms ofthe fuel and ambient gas properties, the initial fuel andambient gas conditions (Ta, Pa, and Tf), and oneunknown, Ts:

),2/()2/( θ⋅=α tanatan

)()()()( LULALLmUAm ffffff ⋅⋅ρ==⋅⋅ρ= ��

)()()()( LULALLm aa ⋅⋅ρ=�

).,()()()(

),()(),()(

sasaasff

aaaaafff

PPThLmThLm

PThLmPThLm

−⋅+⋅=⋅+⋅

��

��

),()(),(),(

)()(

affsf

sasaaaa

a

f

PThThPPThPTh

LmLm

−−−

=�

�

7

(8)

The terms Zf and Za are the vaporized fuel and ambientgas compressibilities, respectively, and Mf and Ma arefuel and ambient gas molecular weights, respectively.

The unknown, Ts, in Eq. (8) can be solved for iterativelygiven the fuel and ambient gas properties and initial fueland ambient gas conditions. Once determined, Tsdefines B, as well as the pressures, temperatures, andenthalpies of the fuel and ambient gas at the liquid lengthlocation.

The term B given by Eq (8) is analogous to the mass andthermal transfer numbers used in droplet vaporizationstudies [25]. Furthermore, the method of solving for Ts isanalogous to that used in determining the surface tem-perature of a vaporizing liquid droplet [25].

Axial Variation of the Fuel/Ambient-Gas Ratio – The fol-lowing discusses the derivation of the relationshipneeded for the axial variation of the ratio of the fuel andambient gas mass flow rates in a spray. Several defini-tions used in the derivation are also introduced and dis-cussed.

To derive a relationship for the axial variation of the ratioof the fuel and ambient gas mass flow rates, an additionalsimplifying assumption is made. The additional assump-tion is that the axial variation of this ratio in a vaporizingspray is similar to that in a non-vaporizing, isothermalspray. This assumption essentially implies that the drop-lets in the spray (non-vaporizing or vaporizing) are smallenough that they follow the gas flow, or that the spray haslocally homogeneous flow, as discussed in Kuo [20]. Theassumption also implies that temperature effects do notsignificantly alter the mean fuel/ambient-gas ratio at anyaxial location in the spray.

This additional assumption is supported by the observa-tions made by Naber and Siebers [9] on the scaling andcorrelation of transient diesel spray penetration data forboth non-vaporizing and vaporizing sprays. They show-ed that characteristic time and length scales developedfrom an isothermal, non-vaporizing spray analysis corre-lated both non-vaporizing and vaporizing transient spraypenetration data. In addition, for transient spray penetra-tions lengths corresponding to the liquid lengths mea-sured in Ref. 1, the non-vaporizing and vaporizing spraypenetration data agreed to within a constant factor. Bothobservations indicate that the various effects of tempera-ture on fuel penetration cancel to within a constant factor,and can be accounted for with a constant that will appearin the scaling law (to be discussed).

The definitions employed in the derivation are:

(9)

(10)

(11)

(12)

(13)

Equation (9) gives the area of the spray cross-sectionderived from the geometry of the spray. The diameter dfin the orifice area in Eq. (10) is the effective orifice diame-ter given by Eq. (11). In Eq. (11), the diameter d is thephysical diameter of the orifice and Ca is the orifice coef-ficient of area-contraction. Both d and Ca are given inTable 3 for each orifice from Ref. [1]. When Ca’s at injec-tion pressures other than listed in Table 3 are needed, alinear interpolation is used. Appendix B presents furtherdiscussion of Ca and its measurement.

The term x+ defined in Eq. (12) is a spray penetrationlength scale introduced in Ref. [9], and the term inEq. (13) is the axial distance in the spray normalized byx+. The product of df and the density ratio term in x+ haslong been used to account for the effects of an injectedfluid density different from the ambient gas on themomentum of steady jets [11] and transient sprays[e.g., 26]. More recently, however, Naber and Siebers [9]showed that this product alone did not account for all thedensity effects on a spray for the range of conditions rele-vant to a diesel. They found that tan(α/2) term had to beincluded in the length scale to account for an additionaleffect of the density difference between the ambient gasand the injected fluid on the turbulent transport betweenthe two fluids.

With the additional isothermal, non-vaporizing assump-tion, the following mass and momentum balances can bewritten for the spray:

(14)

(15)

(16)

As opposed to Eq. (5), the density ρa in Eq. (15) is a con-stant. Also, as is conventionally assumed for non-vapor-izing sprays [9], the area occupied by the fuel isneglected in the term A(x) in Eq. (15)

Employing Eq. (3) and the definitions in Eqs. (9-13), thespray mass and momentum balances given by Eqs. (14-16) can be rearranged to give the desired relationship forthe axial variation of the fuel and ambient gas mass flowrates in the spray (in dimensionless from):

),()(

),(),(

][),(),(

affsf

sasaaaa

asassf

fssasa

PThThPPThPTh

MPPPTZMPPPTZ

B

−−−

=⋅−⋅

⋅⋅−= [ ]2)2/()( α⋅⋅π= tanxxA

2

4 ff dA ⋅π=

dCd af ⋅=

)2/(α⋅

ρρ

=+

tand

x f

a

f

+= xxx /~

)(xmUAm fffff�� =⋅⋅ρ=

)()()( xUxAxm aa ⋅⋅ρ=�

).()()()( xUxmxUxmUm afff ⋅+⋅=⋅ ���

8

(17)

Liquid Length Scaling Law – The scaling law for liquidlength can now be developed in either a non-dimensionalor dimensional form. The dimensionless form is derivedby substituting B (determined with Eq. (8)) into Eq. (17),and solving for (i.e., the corresponding to the liquidlength):

(18a)

The dimensional form is arrived at substituting Eq. (3)and the definitions given by Eqs. (9-13) into Eq. (18a):

(18b)

Equation (18a) or (18b) coupled with Eq. (8) (rewrittenbelow for completeness) form the liquid length scalinglaw.

(8)

The constant a in Eq. (18b) is from Eq. (3) and has avalue of 0.66. A value for the constant b in Eqs. (18a)and (18b) of 0.25 results from the analysis. However,given all the simplifying assumptions made in the devel-opment of the scaling law, there is no reason to expectthat this value for b will be work. Instead, the value rec-ommended for b is 0.41. The value of 0.41 is derivedfrom a best fit of the liquid length scaling law to the cet-ane and HMN liquid length data. The fit will be presentedin the Comparison with Data section. This value of bappears to apply for a very wide range of fuels, since it isalso valid for the methanol liquid length data presented inRef. [27].

Examining the source of the terms in Eq. (18b) showsthat the square-root term containing B accounts for theeffects of parameters involved in the energy equation onthe liquid length. The terms in front of the square-rootterm with B account for the effects of various parameterson the liquid length through their effects on the spraymass and momentum transport.

The method for determining a liquid length from the scal-ing law is to iteratively solve Eq. (8) for Ts, which definesB for a given set of conditions. The value for B is thensubstituted into Eq. (18b).

FUEL AND AMBIENT GAS PROPERTIES – To comparethe liquid length scaling law with measured liquid lengths,

extensive property data for the ambient gas and the fuelsused in the experiment are needed. These propertieswere determined by using computer software providedwith the API Technical Data Book [28]. The API propertysubroutines were incorporated into a computer programwritten to perform the iterative solution of Eq. (8) for Band to compute the liquid fuel density at the orifice exit foreach set of conditions. The software uses the three-parameter corresponding states correlation of Lee andKesler [29] based on an equation of state of the formP=ZρRT/M to compute thermodynamic properties suchas enthalpy, entropy, and heat of vaporization.

The properties for an extremely large number of hydro-carbon fuels (up to C30), as well as other compounds andcommon gases are available in the API data base. Anexception is the detailed property data for HMN. Prop-erty data for HMN only very recently became available inthe DIPPR data base [30] and had to be merged into thecomputer software provided with the API data base.

COMPARISON WITH DATA

The trends in the measured liquid length data are com-pared with those predicted by the scaling law in this sec-tion. The first subsection compares the dimensionlessform of the scaling law given by Eq. (18a) to the liquidlength data acquired for all conditions with cetane andHMN. This comparison to Eq. (18a) was used to definethe value recommended for the constant b in Eqs. (18a)and (18b). The remaining subsections make detailedcomparisons between the predicted and measuredeffects of the various parameters on liquid length.

In many of the figures in the following subsections, aswell as the remainder of the paper, continuous curves areplotted for liquid lengths predicted with the scaling law asa function of various parameters. The spreading anglesneeded in Eq. (18b) to compute these continuous curvesare interpolated as required from measured spreadingangle data. Equation A2 in Appendix A is used to inter-polate ambient gas and fuel density effects on spreadingangles for each orifice. Spray spreading angle variationsfor each orifice with respect to other parameters are verysmall (as discussed in Appendix A), and are interpolatedlinearly with respect to each parameter. The orifice area-contraction coefficients in Eq. (18b) for injection pres-sures other than the two listed in Table 3 are linearlyinterpolated (and in a few cases extrapolated) from thevalues in the table.

OVERALL COMPARISON – Figure 4 shows the compar-ison between Eq. (18a) and the measured liquid lengthdata for cetane and HMN from Ref. 1. The figure is a plotof the measured liquid lengths normalized by x+ (sym-bols) versus the value for B determined from Eq. (8) foreach condition and fuel. A total of approximately 75 datapoints are shown for each fuel that include data for condi-tions covering the ranges given in Table 1 and data foreach of the orifices given in Table 3. Many data points liedirectly on top of each other near B values of 0.25, 0.35,

1~161

2)()(

2 −⋅+=

xxmxm

a

f�

�

11),,(

2~2

−

+⋅=

faa TPTBbL

11),,(

2)2/(

2

−

+

θ⋅

⋅ρρ⋅=

faa

a

a

f

TPTBtan

dC

ab

L

),()(),(),(

][),(),(

affsf

sasaaaa

asassf

fssasa

PThThPPThPTh

MPPPTZMPPPTZ

B

−−−

=⋅−⋅

⋅⋅−=

9

0.65, 0.85, and 1.2. These values for B correspond to thethermodynamic conditions repeated for each of the vari-ous orifices and injection pressures considered. Thesolid curve in the figure is Eq. (18a) with the constant bset at 0.41. The value selected for the constant b waschosen to give the best fit to the liquid length data overthe largest range of conditions in this figure.

Figure 4 shows that there is close agreement betweenthe scaling law and the measured data over the range ofconditions considered. The standard deviation (σ)between the scaling law and all the measured liquidlengths is 4%. The dashed lines in Fig. 4 represent aband of ±8% (or ±2σ). The data points falling outside ofthe ±8% band (generally on the low side) are for theextreme low temperature-density conditions and thehighest density conditions of the experiment, as will beshown.

Figure 4. Dimensionless liquid length ( =L/x+) versus B determined with Eq. (8) for all the conditions covered in the experiment in Ref. 1 using cetane and HMN.

GAS TEMPERATURE AND DENSITY TRENDS – Thetemperature and density trends predicted by the scalinglaw for cetane are compared in Fig. 5 with the measuredliquid length data from Ref. [1]. Figure 6 shows the samecomparison for HMN. The figures are plots of measuredliquid lengths (symbols) and liquid lengths estimatedusing the scaling law given by Eqs. (8) and (18b) (curves)versus ambient gas temperature for five different ambientdensities. The orifice diameter, the injection pressuredrop across the orifice, and the fuel temperature for bothfigures are nominally 246 µm, 136 MPa, and 438 K,respectively.

The figures show that the scaling law closely predicts thestrong effects of temperature and density on liquid length.This is especially true for typical small-bore automotiveand heavy-duty DI diesel top-dead-center (TDC) temper-atures and densities, which are represented by theshaded area centrally located in each figure. The mostsignificant disagreements in trend between the data and

the scaling law are at the low-temperature, low-densityextreme of the data and at the high-density extreme ofthe data. At the low-temperature, low-density extreme,the scaling law tends to over predict the measured liquidlength. At the high density extreme, an over predictionalso occurs, although to a lesser extent. (The potentialsources of the disagreement are discussed later.)

Figure 5. Cetane liquid length as a function of gas temperature for five gas densities (ρa). The symbols are measured data from Ref. 1 and the curves are the scaling law predictions for each gas density. The orifice pressure drop, the orifice diameter, the ambient gas temperature, and the fuel temperature were 136 MPa, 246 µm, and 438 K, respectively. The light gray region in the figure represents typical TDC gas temperatures and densities in light- and heavy-duty DI diesels.

Figure 6. HMN liquid length as a function of gas temperature for five gas densities. The symbols are measured data from Ref. 1 and the curves are the scaling law predictions for each gas density. See Fig. 5 for the experimental conditions. The light gray region in the figure represents typical TDC gas temperatures and densities in light- and heavy-duty DI diesels.

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

6

~L

B

CetaneHMN

600 700 800 900 1000 1100 1200 1300 14000

20

40

60

80

100

Liqu

id L

engt

h [m

m]

Gas Temperature [K]

ρa (kg/m3)

3.67.3

14.830.259.0

600 700 800 900 1000 1100 1200 1300 14000

20

40

60

80

100

Liqu

id L

engt

h [m

m]

Gas Temperature [K]

ρa (kg/m3)

3.67.3

14.830.259.0

10

Figure 7. HMN liquid length as a function of gas density for five gas temperatures (Ta). The symbols are measured data from Ref. 1 and the curves are the scaling law predictions for each gas temperature. See Fig. 5 for the experimental conditions.

Figure 7 is a plot of the same data as shown in Fig. 6 forHMN, except as a function of density for five tempera-tures. This figure more clearly shows the non-linearity inthe effect of density on liquid length and that the scalinglaw models this variation very well for most conditions.

The disagreement at the low density extreme is less visi-ble in this figure because the extreme sensitivity of liquidlength to gas density at the lower densities tends to maskthe disagreement.

FUEL VOLATILITY TRENDS – Figures 5 and 6 togetheralso show that the scaling law accurately models therange of fuel volatilities represented by HMN and cetane,a range that spans a significant portion of the volatilityrange of standard diesel fuel. The HMN and cetaneatmospheric pressure boiling points correspond to the30% and 80% points on a standard diesel fuel distillationcurve. The overall agreement between the data and thescaling law for the two fuels is the same in each figure.Moreover, when using the same value for b in Eq. (18),similar agreement occurs between the scaling law andmethanol liquid length data. These results suggest thatthe value of 0.41 for b applies for a wide range of single-component fuels. (Methanol liquid length data is pre-sented in a concurrent paper by Higgins et al. [30] on theeffects of a wide range of fuel physical properties on liq-uid length. Fuels covered in the liquid length investiga-tion in Higgins et al. include Fischer-Tropsch diesel, bio-diesel, gasolines, and M-85.)

FUEL TEMPERATURE TRENDS – Figure 8 comparesthe effect of fuel temperature on liquid length measuredfor cetane with that predicted by the scaling law. The fig-ure is a plot of liquid length versus the liquid fuel temper-ature at the orifice exit. The solid symbols in the figure

are the measured data for cetane at three different ambi-ent gas conditions. The ambient gas conditions shownwere selected to provide a range of liquid lengths span-ning those measured in the experiments in Ref. 1. Theorifice diameter and injection pressure drop across theorifice were 246 µm and 135 MPa, respectively.

Figure 8. Liquid length versus the injected fuel temperature for cetane at three different ambient gas conditions given in the legend. The symbols are measured data from Ref. 1 and the curves are the scaling law predictions for each gas temperature and density condition. The orifice diameter and injection pressure drop across the orifice were 246 µm and 135 MPa, respectively

The lines in Fig. 8 are the scaling law predictions as afunction of fuel temperature for the three ambient gasconditions. These predictions have been normalized toeliminate differences between the scaling law and thedata caused by ambient gas temperature and densityeffects already discussed in conjunction with Figs. 5and 6. This normalization allows the predicted trendswith respect to the injected fuel temperature to be moreclearly compared with the measured trends.

To normalize the predictions in Fig. 8 for an ambient gascondition, the predictions are multiplied by the ratio of themeasured and the predicted liquid lengths for the corre-sponding ambient gas condition shown in Fig. 5. Exami-nation of Fig. 5 shows that the normalization is onlysignificant for the 700 K, 7.3 kg/m3 condition in Fig. 8.(Note: This normalization technique will be used again infollowing figures, as noted, to examine predicted liquidlength trends with respect to other parameters.)

Comparison of the measured data to the computed linesin Fig. 8 shows that the effects of the fuel temperature onliquid length are accurately accounted for by the scalinglaw. The data and the scaling law show that liquid lengthdecreases linearly with increasing fuel temperature, aspreviously discussed in Ref. 1. The decrease in liquid

0 10 20 30 40 50 60 700

20

40

60

80

100Li

quid

Len

gth

[mm

]

Gas Density [kg/m3]

Ta (K)

700850

100011501300

360 380 400 420 440 4600

20

40

60

80

100

Liqu

id L

engt

h [m

m]

Fuel Temperature [K]

Ta (K) ρa (kg/m3)

700 7.3 995 14.8 1295 30.3

11

length is about 0.2% per Kelvin increase in fuel tempera-ture for conditions relevant to DI diesels.

ORIFICE DIAMETER TRENDS – By inspection ofEq. (18b), it is clear that the linear dependence of liquidlength on orifice diameter observed in the experimentaldata in Fig. 1 is modeled by the scaling law. A compari-son of the scaling law to the measured data for variousorifice diameters is shown in Fig. 9 to highlight this obser-vation. In addition, the comparison in Fig. 9 allows thepredicted effects of tip to tip variations in the indepen-dently measured orifice area-contraction coefficient (seeAppendix B and Table 3) and the spray spreading angleto be examined (see Appendix A).

Figure 9. Liquid length versus orifice diameter for a wide range of conditions. The measured data are given by the solid gray symbols, and the scaling law predictions for each condition by the open symbols. The lines in the figures are linear least squares fits to the scaling law predictions for each set of conditions given in the legend. The terms in the legend are defined in Fig. 1. The aspect ratio of the orifices were nominally 4.2 and 438 K, respectively.

Figure 9 is a plot of liquid length versus orifice diameterfor a diameter range from 100 to 500 µm. (For reference,typical light- to heavy-duty DI diesel engine injector ori-fice diameters fall in the 160 to 250 µm size range.) Eachsolid gray symbol-type in Fig. 9 represents liquid lengthdata measured for the different orifice diameters for onefixed set of conditions. The set of conditions for eachsymbol is given in the legend. These measured data area subset of the data previously shown in Fig. 1. Theexperimental conditions selected for Fig. 9 were chosento include a variety of conditions and a wide range of liq-uid lengths. The remaining conditions previously shownin Fig. 1 were left out for clarity or because the fuel usedwas DF2, a multi-component fuel. The aspect ratio of theorifices and the fuel temperature for the data shown were4.2 and 438 K, respectively.

The open symbols in Fig. 9 are the liquid lengths pre-dicted with the scaling law for each set of conditions andorifice. These scaling law predictions were normalized inthe same manner as described for the scaling law predic-tions in Fig. 8. (i.e., The predictions were multiplied bymeasured/predicted liquid length ratios from Figs. 5 and6 for each corresponding ambient gas condition andfuel.) As stated before, the normalization eliminates dif-ferences between the scaling law and the data caused byambient gas temperature and density effects already dis-cussed in conjunction with Figs. 5 and 6. With respect toFig. 9, the normalization allows any trends with respect tothe orifice diameter, the orifice area-contraction coeffi-cient, and the spreading angle effects to be more clearlycompared.

The lines shown in Fig. 9 are least-squares fits to thescaling law predictions. (Lines representing continuouspredictions of liquid length as a function of orifice diame-ter are not possible since the orifice area-contractioncoefficients and the spray spreading angle do not vary ina systematic manner with orifice diameter, i.e., from ori-fice to orifice.)

Figure 9 shows excellent agreement between the scalinglaw predictions and the measured liquid length data overa wide range of conditions with respect to orifice diame-ter. Comparison of the individual points in the figureshows that the scaling law predictions (open symbols) fallvery nearly on top of the measured data (solid gray sym-bols). In addition, the linear dependence on orifice diam-eter is clearly evident from the least squares fits. Thisdegree of agreement strongly supports the assumptionsused to derive the scaling law.

The point to point agreement shown in Fig. 9 also pro-vides a check on the independently measured sprayspreading angles and orifice coefficients that appear inthe scaling law. Attempts to use spray angles and area-contraction coefficients averaged over all the orifices,instead of the values measured for each orifice, resultedin systematic differences between the measured and pre-dicted liquid lengths for each orifice.

INJECTION PRESSURE TRENDS – Inspection of thescaling law given by Eqs. (8) and (18b) shows that thescaling law does not contain a direct injection pressure(i.e., injection velocity) dependence. This is in agreementwith the general trend observed in the data in Fig. 2which shows that injection pressure has virtually no effecton liquid length. Injection velocity appears in the massand momentum equations used to derive the scaling law,but cancels out in the derivation, thus eliminating themajor parameter affected by injection pressure.

The small systematic changes in liquid length withrespect to injection pressure that can be observed inFig. 2 are the result of small systematic effects of injec-tion pressure on the spray spreading angle and the ori-fice area-contraction coefficient. This is shown in Fig. 10,a plot of liquid length versus the injection pressure dropacross the orifice. (For reference, typical light- to heavy-

0 100 200 300 400 500 6000

25

50

75

100

125

Liqu

id L

engt

h [m

m]

Orifice Diameter [µm]

Ta ρa ∆Pf Fuel(K) (kg/m3) (MPa)

696 14.8 136 HMN1000 7.3 136 HMN1300 14.8 135 HMN 996 30.3 133 HMN1000 14.8 56 Cetane1300 30.2 131 Cetane

12

duty DI diesel engine orifice injection pressure drops fallin the 70 MPa to 140 MPa range.)

Figure 10. Liquid length versus the pressure drop across the injector orifice for a wide range of conditions. The measured data are given by the solid gray symbols, and the scaling law predictions for each condition by the open symbols. The lines in the figures are linear least squares fits to the scaling law predictions for each set of conditions given in the legend. The terms in the legend are defined in Figs. 1 and 2. The aspect ratio of the orifices were nominally 4.2 and 438 K, respectively.

Each solid gray symbol-type in Fig. 10 represents liquidlength data measured for different injection pressures forone fixed set of conditions. The set of conditions for eachsymbol is given in the legend. These measured data area subset of the data previously shown in Fig. 2. Theexperimental conditions selected for Fig. 10 were chosento include a variety of conditions, a wide range of liquidlengths, and the most systematic variations of liquidlength with injection pressure. The remaining conditionsshown in Fig. 2 were left out for clarity or because thefuel used was DF2, a multi-component fuel.

The open symbols in Fig. 10 are the predictions of liquidlength made with the scaling law for each set of condi-tions, and the lines are least squares fits to the scalinglaw predictions. Like Figs. 8 and 9, the scaling law esti-mations in Fig. 10 have been normalized to eliminate thedeviations of the scaling law from the measured data as aresult of gas temperature-density effects previously dis-cussed in conjunction with Figs. 5 and 6.

Figure 10 shows that there is excellent agreementbetween the scaling law and the experimentally observedtrends with respect to injection pressure. In every case,the liquid lengths determined with the scaling law fall veryclose to the measured values. The scaling law indicatesthat the small systematic variations in liquid length withinjection pressure appear because of variations in the ori-

fice area-contraction coefficient and the spray spreadingangle with injection pressure. The measured spreadingangle variations with injection pressure for the specificconditions shown in Fig. 10 are shown in Fig. A3 inAppendix A. The measured orifice area-contraction coef-ficient variations with injection pressure are shown inTable 3 and Appendix B.

IMPLICATIONS OF THE MAJOR TRENDS

The comparisons in Figs. 5-9 demonstrate that the liquidlength scaling law is in close agreement with the experi-mental data over a very wide range of fuel, injector, andambient gas conditions. The trends in the data withrespect to all major parameters are reproduced by thescaling law. This close agreement between the scalinglaw and liquid length data strongly supports the originalpremise that mixing processes in a DI diesel spray con-trol fuel vaporization (as they have been shown to controlcombustion once the initial premixed burn phase is com-pleted [19]).

Moreover, the close agreement suggests that many of theassumptions applied in developing the scaling law arereasonable with respect to the overall fuel vaporizationprocess. One critical assumption is that the fuel reachesa saturated condition in thermodynamic equilibrium withthe entrained ambient gas. Achieving this condition inthe dense inner region of a real spray, or at least closelyapproaching it, suggests a mechanism for control ofvaporization by mixing. When such a saturated condi-tions exists, the only means to further vaporize fuel is toheat the liquid fuel to a higher temperature and/or trans-port fuel vapor away from the inner region of the spray. Ahigher liquid fuel temperature would allow further vapor-ization by raising the fuel vapor pressure. Transport offuel away from the inner region of the spray (or equiva-lently, dilution of the spray by entrained ambient gas)would allow further fuel vaporization by lowering thevapor fuel partial pressure below the fuel saturation pres-sure. Since both the energy (i.e., high-temperature ambi-ent gas) entrainment needed to heat the fuel and thetransport of fuel and ambient gas in the spray are con-trolled by spray mixing processes, vaporization will becontrolled by mixing processes.

Thermodynamic equilibrium between saturated fuelvapor and entrained ambient gas in the dense region ofthe spray was first discussed by El Wakil et al. [18, alsosee 19]. They analytically explored the possible range offuel/air ratios and equilibrium temperatures that couldexist in the dense core of a spray for adiabatic mixing, butthey had no evidence or measurements that indicated“adiabatic saturation” occurred. The scaling law andmeasurements presented in this paper and Ref. 1 offerstrong indirect evidence that an adiabatic saturation con-dition does exist in the inner region of a DI diesel spray.Direct measurement of fuel vapor concentrations andtemperature, however, are needed to conclusively provethe occurrence of adiabatic saturation.

25 50 75 100 125 150 175 2000

25

50

75

100

125

Liqu

id L

engt

h [m

m]

Orifice Pressure Drop [MPa]

Ta ρa d Fuel(K) (kg/m3) (µm)

1000 7.2 246 HMN 600 14.8 246 HMN1000 30.2 246 HMN

Ta ρa d Fuel(K) (kg/m3) (µm)

700 14.8 246 HMN 700 7.3 100 Cetane1300 30.1 498 Cetane

13

The discussion in the previous paragraphs has an impor-tant further implication. Namely, for current-technologyDI diesel conditions, local mass, momentum, and energytransport processes at droplet surfaces do not control theoverall rate of vaporization. The atomization processmust be producing small enough droplets with enoughsurface area, such that the droplet surface transport pro-cesses are fast relative to the spray mixing processes. Ifthis were not true, the scaling law, which contains nophysics regarding droplet vaporization, should not agreeso closely with the data over such a wide range of condi-tions.

The above discussion also means that better atomization(i.e., smaller droplets) alone will not promote increasedfuel vaporization in a DI diesel spray. The close fit of thescaling law indicates that parameters such as injectionpressure and orifice diameter affect fuel vaporizationthrough their affect on mixing processes, not throughtheir known affects on droplet size through atomization.

DISCUSSION OF THE SECOND-ORDER TRENDS

Although very close agreement between the liquid lengthscaling law and the experimental data was shown, smallsystematic differences did appear in the trends withrespect to ambient gas density and temperature.Figures 5 and 6 showed that the scaling law over esti-mated liquid length at the very highest high ambient gasdensity conditions and the very lowest ambient gas tem-perature and density conditions. The most likely sourcefor the differences at the highest gas density (i.e., highestgas pressure) conditions is the simplified phase-equilib-rium analysis used in the scaling law. However, thephase-equilibrium considerations may also be contribut-ing to the deviations at the lowest temperature and den-sity conditions through the constant b in Eqs. (18a) and(18b). This constant was set to give the best overallagreement, and therefore, may be accounting for some ofnon-ideal phase-equilibrium effects and distributing therest throughout the range of ambient gas conditions con-sidered.

The phase-equilibrium assumptions used to develop thescaling law were that Raoult’s and Dalton’s rules apply,and that no gas absorption by the liquid-phase occurs.Among other simplifications, these “ideal” phase-equilib-rium assumptions result in neglecting two non-idealphase-equilibrium effects on fuel vapor pressure [31].The two neglected effects on fuel vapor pressure are off-setting to some degree. The first is the increase in fuelvapor caused by ambient gas pressures greater than thefuel vapor pressure for a given temperature (i.e., thePoynting effect [31]). The second is the decrease in thefuel vapor pressure caused by gas absorption into the liq-uid phase at higher pressures.

Figure 11 demonstrates a possible impact the “ideal”phase-equilibrium assumptions. This figure is a plot ofthe percent deviation between the predicted and themeasured liquid lengths versus the fuel vapor partial

pressure at the liquid length location, L. The fuel vaporpressures were derived from the solution of Eq. (8). Datafor all conditions examined for cetane and HMN areincluded in the figure.

Figure 11. Percent deviation between the scaling law and the measured liquid length data versus the fuel vapor partial pressure at the liquid length location. Data for all conditions for cetane and HMN are shown. The fuel vapor partial pressures were determined from the solution of Eq. (8).

Figure 11 shows that there is a strong correlationbetween fuel vapor partial pressure (derived from the“ideal” phase-equilibrium analysis) and the deviationbetween the data and the scaling law. The trend inFig. 11 is the strongest correlation found between anyparameter and the small systematic deviation of the scal-ing law from the data noted in Figs. 5 and 6. This trendsuggests that a more realistic phase-equilibrium analysis,such as those used in multidimensional diesel spraymodels [32-34], may decrease some of the remaining dif-ferences between the scaling law and the data. This wasexamined by correcting the fuel vapor pressure used inideal phase-equilibrium analysis by including estimatesfor the Poynting effect and the effect of gas absorption onvapor pressure. The corrections resulted in generallybetter agreement in Figs. 5 and 6.

The use of a more realistic phase-equilibrium analysis,however, should have very little impact on the observa-tions and conclusions of this paper. The primary effectshould be better agreement in Figs. 5 and 6, and slightlylower predicted temperatures at the liquid length location(to be discussed).

Another likely contributor to the differences at the low gastemperature and density conditions in Figs. 5 and 6 is thebeginning of a shift in the processes controlling fuelvaporization in a spray. As lower and lower gas tempera-ture and density conditions are considered, the localtransport processes at droplet surfaces must ultimatelybegin controlling fuel vaporization. Consider moving

0.03 0.10 0.30 1.00-40

-20

0

20

40

60

Dev

iatio

n fr

om D

ata

[%]

Vapor Pressure [MPa]

CetaneHMN

14

toward lower gas temperatures along the lowest ambientgas density curve in Fig. 5 with all other conditions fixed.As the gas temperature is lowered, the mass and energytransport rates at droplet surfaces will decrease, but theturbulent transport of ambient gas to the inner region ofthe spray and fuel transport away from the inner regiondo not2. Therefore, as the ambient temperature isdecreased, the overall droplet surface transport ratesshould ultimately become slow relative to the spray mix-ing processes. When this transition occurs, the droplettransport processes would be unable to maintain a satu-rated condition in the spray inner region, and control ofvaporization should shift to the droplet transport pro-cesses. (A similar shift in processes controlling vaporiza-tion probably occurs at the very tip and periphery of theliquid-phase fuel region in a spray. This shift would occurbecause of the rapid decline noted in the quantity of liq-uid phase fuel at these locations [1,7], and the in abilityfor the remaining liquid fuel to provide enough fuel vaporto maintain a saturated condition3.)

A third factor that should be mentioned as a possible con-tributor to the small systematic disagreements is therecovery of the injected fuel kinetic energy as thermalenergy as the spray decelerates. The specific kineticenergy of a diesel spray at an injection pressure of140 MPa (or an injection velocity of about 600 m/s) is ofthe same magnitude as the specific latent heat of vapor-ization of diesel fuel (~200 kJ/kg). However, if kineticenergy recovery were an important factor for fuel vapor-ization, liquid length should show a significant decreasewith increasing injection pressure (i.e., kinetic energy).The lack of an injection pressure effect suggests that theultimate recovery of the injected fuel kinetic energythrough turbulent dissipation must occur largely down-stream of the vaporization regions.

A factor that does not appear to contribute to the low gastemperature-density disagreement shown in Figs. 5and 6 is wall effect. This appears true at least for liquidlengths up to a few millimeters from the wall, which islocated at 98 mm [1]. If wall effects were important, adeviation from the dominate trends observed in the fueltemperature plot (Fig. 8) and the orifice diameter plot(Fig. 9) should be noted for the longer liquid lengths thatapproach the wall location. The longer liquid length datain Figs. 8 and 9 show no such wall effect.

OBSERVATIONS BASED ON SCALING LAW

Additional observations regarding diesel spray vaporiza-tion processes that can be made based on the liquidlength scaling law are discussed in this section.

MAXIMUM LIQUID-PHASE FUEL TEMPERATURE – Oneparameter resulting from the solution of Eq. (8) is theequilibrium temperature of the saturated fuel vapor andentrained ambient gas (Ts) at the liquid length location.This temperature is the maximum temperature that anyliquid fuel reaches. Figure 12 is a plot of the equilibriumtemperatures derived from Eq. (8) for cetane (solidcurves) and HMN (dashed curves) for conditions corre-sponding to those in Figs. 5 and 6. They are plotted ver-sus the ambient gas temperature for five different gasdensities. Also included in the figure is the critical tem-perature for each pure fuel, represented by the horizontalarrows. The figure applies for a fuel temperature of440 K. Parameters such as injection pressure and orificediameter have no effect on the information presented inFig. 12, since they do not appear in Eq. (8).

Figure 12 indicates that as either the ambient gas tem-perature or ambient gas density increase, the final tem-perature of the liquid-phase fuel increases. In addition,the lower volatility fuel, cetane, reaches the higher finaltemperature at each condition. The trends in the figureresult from the non-linear interaction of the various ther-modynamic properties of each fuel with the ambient gasthermodynamic properties in Eq. (8).

Figure 12. Equilibrium temperature at the liquid length location in the spray determined from Eq. (8) for cetane and HMN versus gas temperature for five different gas densities. The conditions in the figures correspond to those in Figs. 5 and 6.

As important however, Fig. 12 shows that neither fuelreaches its critical temperature for any condition, even atthe most extreme ambient gas temperature and densityconditions of the experiment. These extreme conditionsare well beyond any conditions expected in current or

2. Appendix A shows that the spray spreading angle, i.e., the entrainment of ambient gas into the spray, is not affected by ambient gas tem-perature for a fixed ambient gas density.

3. The vaporization process as described in this paragraph is more representative of a time-averaged picture. A description of an instan-taneous picture of vaporization would proba-bly have the same processes and transitions occurring, but on a length scale comparable with the larger scale turbulent structures in the spray.

600 700 800 900 1000 1100 1200 1300 1400400

500

600

700

800

Max

imum

Liq

uid

Fue

l Tem

pera

ture

[K]

Gas Temperature [K]

CetaneHMN

Gas Density (kg/m3)

59.0

30.2

14.8

7.3

3.6

Critical Temperatures

Cetane (723 K)

HMN (692 K)

15

proposed advanced DI diesel engines. The same wasfound for many fuels explored as surrogates for dieselfuel using the scaling law (to be discussed later). Thisobservation suggests that standard diesel fuel vaporizesthrough subcritical vaporization processes in a DI diesel,and never reaches supercritical conditions. This trueeven though the in-cylinder gas temperature and pres-sure are typically well above the fuel critical temperatureand pressure. (Note: Estimates of the effects of the useof a more realistic phase-equilibrium analysis in the scal-ing law suggest that the temperatures in Fig. 12 shouldbe slightly less than shown. Also, the critical temperaturefor the saturated fuel/ambient-gas mixture will be slightlyless that the pure fuel critical temperature shown [31].The general observations made, however, will notchange.)

ENERGY REQUIRED TO VAPORIZE FUEL – The spe-cific enthalpy (i.e., energy) required to vaporize the fuel,determined from the solution of Eq. (8), is shown inFig. 13 for cetane (solid curves) and HMN (dashedcurves). The conditions in Fig. 13 correspond to those inFig. 12. The figure shows that as either the gas tempera-ture or density increase, more energy is required tovaporize a given mass of fuel. This increasing specificenergy requirement is a consequence of the increase inthe final equilibrium temperature of saturated fuel vaporshown in Fig. 12 and the nature of the thermodynamicproperties of a fuel. For any fuel (any liquid for that mat-ter), the specific energy required to vaporize the fuel fromany initial liquid state is greater for a higher final saturatedvapor temperature, almost independent of the initial liq-uid-phase fuel pressure. This can be observed byinspection of a pressure-enthalpy diagram for any single-component fuel.

Figure 13. The total specific energy required to vaporize cetane or HMN determined from Eq. (8) versus gas temperature for five different gas densities. The conditions in the figures correspond to those in Figs. 5, 6 and 12.

With respect to fuel volatility, Fig. 13 shows that thehigher volatility fuel, HMN, requires less energy to vapor-ize for any set of conditions. This is in part due to thehigher volatility of HMN (i.e., higher vapor pressure)which makes HMN easier to vaporize. But the form of thescaling law also indicates that the lower latent heat ofvaporization and liquid-phase heat capacity of HMN aresignificant factors as well. This is dramatized by recentexperimental liquid length measurements for methanol[30] which show that methanol, a very high volatility fuelrelative to DF2, has liquid lengths very close to those forDF2 as a result of methanol’s high heat of vaporization.This implies that the use of atmospheric pressure boilingpoint (a measure of fuel volatility) to correlate liquidlengths, as was done in Refs. [1,8], only applies for nar-row fuel classes with similar thermodynamic properties.

GAS TEMPERATURE AND DENSITY EFFECTS – Clearexplanations for the effects of orifice diameter and injec-tion pressure on fuel vaporization and liquid length weresummarized in the Background Section. These explana-tions were built around Eqs. (1) and (2) and conservationof mass and momentum considerations. With the scalinglaw and Figs. 12 and 13, a clearer understanding of themechanisms by which ambient gas temperature and den-sity affect liquid length can also be provided.

One obvious effect of a higher ambient gas temperature(for a constant density) is that the entrained ambient gashas a higher specific energy. This translates to the needfor a shorter spray entrainment length to entrain enoughenergy to vaporize the fuel (i.e., a shorter liquid length).However, Fig. 12 shows that the equilibrium temperatureof the vaporized fuel also increases as ambient gas tem-perature increases, resulting in a greater vaporizationenergy requirement (Fig. 13) or the need for a longerspray entrainment length. Thus, the effect of an increasein ambient gas temperature on liquid length is the netresult of the higher specific energy content of the ambientgas and competing fuel property effects that lead to agreater fuel vaporization energy requirement. The bal-ance results in the decreasing liquid length with increas-ing gas temperature observed in Figs. 5 and 6.

Ambient gas density (for a constant temperature) alsoaffects liquid length through two competing mechanisms.First, increasing ambient gas density affects the spraydevelopment. Equation (1) shows that as ambient gasdensity increases, the mass of ambient gas entrainedinto the spray increases. The increase in mass entrain-ment occurs directly as a result of the higher gas density,and also as a result of the increased spray spreadingangle that occurs with increased gas density (see Appen-dix A). The greater mass entrainment translates to ashorter spray entrainment length requirement to entrainenough energy to vaporize the fuel. The competingmechanism is the increased equilibrium temperature(Fig. 12) and concurrent greater energy requirement forfuel vaporization dictated by the fuel properties (Fig. 13).

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16

This greater energy requirement translates to a longerspray entrainment length requirement. As with ambientgas temperature, the balance of these mechanismsresults in the decreasing liquid length with increasingambient gas density most clearly observed in Fig. 7.

AIR VERSUS INERT AMBIENT GAS EFFECTS – Theliquid length experiments in Ref. 1 were conducted in aninert environment composed of 89.7% N2, 6.5% CO2,and 3.8% H2O with a molecular weight of 28.67. Thisenvironment has a slightly different heat capacity andmolecular weight than air, which have an effect on liquidlength. The scaling law can be used to assess this effect.Figure 14 compares the liquid lengths for cetane deter-mined with the scaling law for both the inert environment(solid curves - previously shown in Fig. 5) and air(dashed curves). The figure shows that air results inabout a 4% increase in liquid length for all conditions, pri-marily as a result of its the lower heat capacity relative tothe inert gas used in the experiments in Ref. [1]. In anengine, this means that exhaust gas recirculation willhave small effects on the liquid length, not only becauseof potential in-cylinder gas temperature effects, but alsobecause of heat capacity changes due to in-cylinder gascomposition changes.

Figure 14. Liquid lengths predicted for cetane injected into the inert ambient gas used in Ref. 1 and into air versus gas temperature for five gas densities. The solid curves are for the inert ambient gas and the dashed curves for air. The conditions correspond to those in Fig. 5 for cetane.

VAPORIZATION SURROGATE FOR DIESEL FUEL –Because single-component fuels are generally very wellcharacterized relative to #2 diesel fuel (DF2), single-com-ponent fuels are often used as surrogates for DF2 instudies of diesel sprays. In this section, the understand-ing of fuel vaporization derived through the liquid lengthscaling law is used to select a single-component fuel thatclosely matches the vaporization characteristics of DF2.Such a surrogate is useful for estimating DF2 liquid

lengths under various conditions, as will be done in thenext section, and for fundamental experimental and mod-eling studies of diesel sprays requiring a reasonable sim-ulation of the diesel spray evaporation process.

For a single-component fuel to be a good vaporizationsurrogate for DF2, it must: (a) generate a spray with thesame injected mass and momentum as a DF2 spray,(b) have fuel volatility characteristics similar to DF2, and(c) have energy requirements for fuel vaporization similarto DF2. Matching the injected fuel mass and momentummatches the overall spray development for both fuels(i.e., the transient penetration, the entrainment, the turbu-lent mixing, etc.,). Matching the fuel volatility and vapor-ization energy requirements matches the overallvaporization characteristics. For a fixed set on injectorand ambient gas conditions, the three criteria can be metby matching fuel densities, volatilities (e.g., atmosphericpressure boiling points), and liquid lengths.

A number of single-component fuels were considered inthe search for a DF2 vaporization surrogate. Theyincluded fuels that have already been used in the litera-ture as DF2 surrogates for various reasons (e.g., n-dode-cane, cetane, α-methylnaphthalene, and HMN) and sev-eral other fuels whose thermodynamic property data indi-cated that they might make good surrogates for DF2(e.g., n-tetradecane, α-tetradecene, n-heptadecane, andn-octadecane). Detailed property data for all fuels con-sidered are available in either Ref. [28] or [30].

Figure 15 presents the liquid lengths predicted with thescaling law for the three single-component fuels investi-gated (solid and dashed curves) that most closely match-ing those measured for DF2 (symbols). The three fuelsare n-heptadecane, cetane, and α-methylnaphthalene.The DF2 data was reproduced from Ref. 1, and the con-ditions in the figure correspond to the those in Figs. 5and 6.

Figure 15 shows that n-heptadecane provides the bestmatch to the liquid lengths measured for DF2. Theagreement shown between n-heptadecane and DF2 is asgood as that observed between the scaling law and cet-ane or HMN in Figs. 5 and 6, respectively. Moreover, thedensity of n-heptadecane is only 7% less than the densityof DF2, and its the boiling point (575 K) corresponds tothe 90% point on the DF2 distillation curve. Based on thescaling of diesel spray processes presented in this paperand in Ref. [9], the 7% density difference will have lessthan a 4% effect on various critical spray developmentprocesses (such as ambient gas entrainment and thetransient spray penetration) relative to diesel fuel. Over-all, the results indicate that n-heptadecane is the bestvaporization surrogate for DF2 of the fuels investigated.

Figure 15 also provides an additional interesting observa-tion. The fact that atmospheric boiling point of n-hep-tadecane corresponds to the temperature at the 90%point on the DF2 distillation suggests that the lower vola-tility fractions in DF2 control liquid length. This observa-tion supports the suggestion in Ref. 1 that vaporization in

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Inert GasAir

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17

a complex fuel, such as DF2, occurs in a batch distillationmode in a diesel spray, with the less volatile, higher boil-ing point fractions vaporizing last, and therefore, control-ling liquid length.

Figure 15. A comparison of liquid lengths predicted for various single-component fuels with those measured for DF2 as a function of gas temperature for five gas densities. The symbols are the DF2 data from Ref. 1. The single-component fuels are represented by the curves as defined in the figure legend. The conditions correspond to those in Figs. 5 and 6.

The liquid lengths shown in Fig. 15 for cetane and α-methylnaphthalene are also in reasonable agreementwith those for DF2. Considering fuel density and fuel vol-atility in conjunction with Fig. 15, indicates that cetane isthe next best vaporization surrogate. The density of cet-ane, like n-heptadecane, is also only 7% less than thedensity of the DF2, and its atmospheric pressure boilingpoint lies at the 80% point on the DF2 distillation curve.In the case of α-methylnaphthalene, however, the fueldensity consideration indicates that it is not a good vapor-ization surrogate. The density of α-methylnaphthalene is20% greater than the density of DF2. This large densitydifference will begin to result in other changes in thespray that will make the overall spray development andvaporization unrepresentative of DF2, in spite of thefavorable liquid length comparison with DF2.

Other single-component fuels considered, includingthose often used in place of DF2 in the literature, werefound to have liquid lengths significantly different than theDF2 liquid lengths. None are recommended as single-component fuel vaporization surrogates for DF2.

LIQUID LENGTH IN LARGE-BORE AND SMALL-BOREENGINES – In this section, the scaling law is used toaddress the questions: Does liquid fuel impinge on pistonbowl walls in a DI diesel when using a typical diesel fuel,and under what conditions? The scaling law and the

vaporization surrogate for DF2 (n-heptadecane)described in the previous section are used to examinethe range of liquid lengths to be expected in representa-tive large-bore, heavy-duty and small-bore, light-duty DIdiesel engines. The goal is to provide a general pictureof liquid fuel penetration for each of the two classes ofengines over a range of operating conditions based onwhat is presently known about fuel vaporization. Forheavy-duty DI diesels, experimental research hasalready shown that for a typical diesel fuel and operatingconditions, liquid-phase fuel is not likely to impinge onpiston bowl walls [1,7,8]. The following discussion willconfirm this observation. However, for light-duty DI dieselengines, there is a concern that significant fuel impinge-ment on piston bowl walls will occur as a result of thesmaller piston bowl geometries.

Table 4 gives the values of the parameters important toliquid–phase fuel penetration that were selected for the“hypothetical” heavy-duty and light-duty DI dieselengines. The values chosen are typical of each engineclass. The orifice area-contraction coefficient (Ca)selected is an average of those presented in Ref. [9] andTable 3 of this paper. An intake pressure of less than oneatmosphere is intended to represent light load operationat high altitude. A fuel temperature of 100 °C is used torepresent a warm running engine condition; but for coldstart conditions, the fuel is assumed to be at the intake airtemperature. Since the cold start liquid lengths predictedare based on significant extrapolations of the scaling lawbeyond the fuel temperature range considered in theexperiments, they should be viewed as approximationsonly. (Injection pressure is not included in the Table 4because liquid lengths are nearly independent of injec-tion pressure as shown in Figs. 2 and 10). The sprayspreading angles needed in the scaling law were interpo-lated from the data in Appendix A.

Figure 16 is a plot of the liquid lengths expected in theheavy-duty engine defined in Table 4 as a function ofTDC temperature for the parameter ranges given in thetable. The polytropic coefficient used to convert intaketemperatures and pressures in the table to TDC condi-

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id L

engt

h [m

m]

Gas Temperature [K]

α-methylnapthalenen-heptadecanecetane

Gas Density (kg/m3)

3.6

7.3

14.8

30.259.0

Table 4. Parameters used in scaling law for heavy- and light-duty DI diesel engines.

Heavy-Duty Light-Duty

Compression Ratio 15 19

Orifice diameter 194 160 µm

Orifice Ca 0.84 0.84

Warm Running:

Intake Temperature 60-160 45-120 °C Intake Pressure 0.7-3.0 0.7-2.3 atm

Fuel temperature 100 100 °CCold Start:

Intake Temperature 0-40 0-40 °C Intake Pressure 0.7-1 0.7-1 atm

Fuel temperature 0-40 0-40 °C

18

tions was 1.35. The expected liquid lengths for warmrunning engine conditions are represented by the lowergray region centered on 1000 K in the figure. The uppergray region represents the range of liquid lengthsexpected for cold start engine conditions. The cross-hatched horizontal band running through the center ofthe figure between 50 and 60 mm represents typical dis-tances between the injector and the far wall of the pistonbowl in heavy-duty engines.

The upper and lower edges of the region for warm run-ning conditions represent operation at boost pressures of0.7 atm and 3.0 atm, respectively. The left and rightsides of this region correspond to operation at the lowestand the highest intake temperatures, respectively. Thedata point in the middle of the region for warm engineconditions (the symbol) is the liquid length measured forDF2 by Canaan et al. [8] in a diesel very close to the onedefined in Table 4. The TDC gas temperature and den-sity for the Canaan et al. data point are 992 K and16.6 kg/m3, respectively. For reference, the liquid lengthspredicted with the scaling law for lines of constant density(the light gray curves) are also included. (The dashedcurves are explained later.)

Figure 16. Liquid lengths expected in the heavy-duty diesel engine defined in Table 4. The lower gray region represents the range of liquid lengths expected for warm running engine conditions. The upper gray region represents the range of liquid lengths expected for cold start engine conditions. The cross-hatched horizontal band across the middle of the figure represents typical maximum distances from the injector to the piston bowl wall. The light gray curves are liquid lengths for lines of constant density and are included for reference.

Figure 16 shows that even for the extreme range of oper-ating conditions represented in the figure, liquid fuel isunlikely to impinge on piston bowl walls in a heavy-duty

diesel other than for cold start conditions. The warm run-ning operating conditions for which liquid is the closest toimpinging on the piston bowl are the lightest load condi-tions, which correspond to the upper left corner of thelower gray region. Overall, the nominal liquid penetrationdistance is about half-way across the piston bowl. Theseobservations are in agreement with previous experimen-tal results [1,7,8].

There are several caveats that must be considered withrespect to Fig. 16, however. The first is spray targetingand the piston bowl shape. The figure assumes thesprays are targeted in the direction with the maximumdistance to the opposing piston bowl wall. A different tar-geting could easily shorten the distance to the pistonbowl wall and result in wall impingement of liquid-phasefuel. Second, piston movement can also shorten the dis-tance to the wall. Third, the scaling law gives the timeaveraged liquid length, but turbulence causes approxi-mately ±11% fluctuations in the instantaneous liquidlength [1]. The dashed lines above and below the upperborder of the warm running conditions region in Fig. 16represent the effect of turbulent fluctuations in the liquidlength on the upper boarder. Factoring in turbulent fluctu-ations, the liquid length approaches closer to the wall atlighter load conditions, but still does not reach it. A finalconsideration is the effect of combustion. Experimentalresults in an engine indicate combustion will potentialshorten the liquid length slightly, but that most vaporiza-tion is complete before the combustion zone is reached[7,8]. The liquid length predictions made with the scalinglaw developed for vaporizing, non-combusting sprays aretherefore slightly conservative (i.e., long with respect tothose expected in combusting sprays).

Figure 17 shows the same information presented inFig. 16, only for the small-bore engine defined in Table 4.The various features in the figure have the same mean-ing as discussed with Fig. 16. The piston bowl wall isnow shown in a range typical of the smaller bore engines,22 to 28 mm. The results indicate that liquid fuelapproaches the wall more closely than in the heavy-dutyengine for all conditions, and even impinges for the lighterload conditions. However, the impingement is not assevere as might have been anticipated, mainly becauseof the strong effect of orifice diameter on liquid length andthe higher densities resulting from the higher compres-sion ratio.

Based on the scaling law, the most direct way to shortenthe liquid length in the small-bore engine for all conditionsis to reduce the orifice diameter. An orifice diameter of100 µm would give the same picture of liquid length rela-tive to the piston bowl wall as observed in for the heavy-duty engine in Fig. 16. An additional important point isthat increased injection pressure will not make the wallimpingement problem more severe than is indicated inFig. 17 for small-bore engines.

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Piston Bowl Wall

Cold Start Conditions

Warm Engine Conditions

Liqu

id L

engt

h [m

m]

Gas Temperature [K]

19

Figure 17. Liquid lengths expected in the light-duty diesel engine defined in Table 4. The various features are defined with Fig. 16.

The caveat beyond those already discussed with Fig. 16that needs to be addressed in conjunction with small-bore engines is the effect of high swirl flows on liquidlength. High swirl is typically used in small-bore enginesto compensate for the mixing the fuel jet cannot provideas a result of orifice diameter and injection pressure limi-tations. Although the effect of swirl on liquid length needsto be investigated, it is presently believed that since mostfuel vaporization occurs in the early very high velocityregion of the spray, the high swirl typical of small-boreengines will have very little effect on the general picturepresented in Fig. 17.

SUMMARY AND CONCLUSIONS

Recent experimental research on liquid-phase fuel pene-tration and vaporization in sprays suggests that vaporiza-tion in a spray from a high-pressure, direct-injection (DI)diesel injector approaches a limit controlled by spray mix-ing processes [1]. These mixing processes includeentrainment of high-temperature air and the overall trans-port and mixing of fuel and air throughout the spraycross-section. Based on this finding, a scaling law for themaximum penetration distance of liquid-phase fuel in anevaporating diesel spray (defined as the liquid length)was derived using jet theory applied to a simplified modelof a diesel spray. The scaling law accounts for injector,fuel, and in-cylinder gas temperature, density and pres-sure effects on liquid length. As developed, the scalinglaw is valid for single-component fuels, but can be usedto model multi-component fuels through use of single-component surrogate fuels.

A comparison of the scaling law was made with mea-sured liquid length data previously presented in Ref. 1.The comparison covered a very wide range of conditions,including those in current-technology and proposedadvanced DI diesel engines. Fuels considered in thecomparison included two single-component fuels,

n-hexadecane (cetane) and heptamethylnonane (HMN),and a standard #2 diesel fuel (DF2). The volatilities ofthe single-component fuels, cetane and HMN, span asignificant fraction of the DF2 volatility range.

The comparison showed that scaling law reproduced allmajor trends in the experimental data with respect to var-ious parameters shown in Ref. 1. The liquid lengthtrends reproduced were:

1. The linear dependence of liquid length on orificediameter

2. The decrease of liquid length toward zero as orificediameter approaches zero.

3. The insignificant effect of injection pressure on liquidlength.

4. The strong, non-linear effects of ambient gas temper-ature and density on liquid length.

5. The increase in liquid length with decreasing fuel vol-atility.

6. The linear decrease in liquid length with increasingfuel temperature.

In addition, the scaling law provided new insight and/orclarification on the effects of some of the parametersobserved in the experiments, and several new observa-tions on diesel spray vaporization:

1. Fuels with properties comparable to typical dieselfuels do not appear to reach supercritical tempera-tures before they are completely vaporized. As aresult, vaporization in a diesel spray occurs throughsubcritical vaporization processes for conditions rele-vant to present and proposed DI diesels.

2. The total energy required to vaporize a fuel is impor-tant in determining liquid length, in addition to the fuelvolatility. Recent liquid length measurements formethanol [30] dramatize this observation. The recentmeasurements show that methanol, a very high vola-tility fuel relative to DF2, has liquid lengths very closeto those for DF2 as a result of methanol’s high heat ofvaporization. This observation implies that the use ofatmospheric pressure boiling point (a measure of fuelvolatility) to correlate liquid lengths, as was done inRefs. [1,8], only applies for fuels with similar thermo-dynamic properties.

3. The fuel n-heptadecane, a single-component fuelwith a boiling point near the 90% temperature on theDF2 distillation curve, models the vaporization char-acteristics of DF2. This observation also supportsthe conclusion in Ref. 1 that vaporization in a multi-component fuel, such as DF2, occurs in a batch dis-tillation mode. Therefore, the less volatile, higherboiling point fractions control liquid length in a multi-component fuel.

The close agreement shown between the liquid lengthscaling law and the measured data supports the conceptthat vaporization in a diesel spray from current-technol-ogy DI diesel injectors approaches a limit controlled by

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spray mixing processes. Furthermore, the thermody-namic equilibrium assumptions used to develop the scal-ing law suggest a mechanism for the limitation. Fuelvapor in the inner region of a spray, where a high numberdensity of small droplets exist, reaches a saturated condi-tion in thermodynamic equilibrium with the entrainedambient gas. For vaporization to proceed, ambient gasentrainment and mixing within the spray are needed todilute and/or heat the inner region of a spray.

An implication of mixing limited vaporization is that atom-ization and local interphase transport processes at drop-let surfaces are not limiting factors for fuel vaporization, atleast for current-technology DI diesel injectors. This fur-ther implies that better atomization (i.e., smaller droplets)alone will not promote increased fuel vaporization in a DIdiesel spray. Parameters such as injection pressure andorifice diameter appear to affect fuel vaporization throughtheir affect on mixing processes, not through their knownaffects on droplet size (i.e., atomization.)

The scaling law provides a fundamental baseline on liq-uid fuel penetration and vaporization in diesel sprays thatcan be compared with the vaporization aspects of themulti-dimensional diesel spray models under develop-ment. The scaling law can also provide design guidanceon the expected maximum extent of liquid-phase fuelpenetration in engines. Application of the scaling law to aheavy-duty and a light-duty DI diesel shows that liquid-phase fuel impingement on piston bowl walls is not aserious concern in heavy-duty engines using a typicaldiesel fuel, as observed previously through experiments,but may be an issue in light-duty engines.

ACKNOWLEDGMENTS

Support for this research was provided by the U.S.Department of Energy, Office of Heavy-Duty VehicleTechnologies and Office of Advanced Automotive Tech-nologies. The research was performed at the Combus-tion Research Facility, Sandia National Laboratories,Livermore, CA.

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2. Browne, K. R., Partridge, I. M., and Greeves, G., “FuelProperty Effects on Fuel/Air Mixing in an Experimental Die-sel Engine,” SAE Paper 860223, 1986.

3. Kamimoto, T., Yokota, H., and Kobayashi, H., “Effect of HighPressure Injection on Soot Formation Processes in a RapidCompression Machine to Simulate Diesel Flames,” Trans-actions of the SAE, Vol. 96, Sect. 4, pp. 4.783-4.791, 1987.

4. Hodges, J. T., Baritaud, T. A., and Heinze, T. A., “PlanarLiquid and Gas Fuel and Droplet Size Visualization in a DIDiesel Engine,” Transactions of the SAE, Vol. 100, Sect. 3,pp. 1284-1302, 1991.

5. Bower, G. R. and Foster, D. E., “The Effect of Split Injection

on Fuel Distribution in an Engine-Fed Combustion Cham-ber,” Transactions of the SAE, Vol. 102, Sect. 3, pp. 1187-1202, 1993.

6. Yeh, C.-N., Kamimoto, T., Kobori, S., and Kosaka, H., “2-DImaging of Fuel Vapor Concentration in a Diesel Spray viaExciplex-Based Fluorescence Technique,” SAE Paper932652, 1993.

7. Espey, C. and Dec, J. E., “The Effect of TDC Temperatureand Density on the Liquid-Phase Fuel Penetration in a D.I.Diesel Engine,” Transactions of the SAE, Vol. 104, Sect. 4,pp. 1400-1414, 1995.

8. Canaan, R. E., Dec, J. E., Green, G. M., and Daly, D. T.,“The Influence of Fuel Volatility on the Liquid-Phase FuelPenetration in a Heavy-Duty D.I. Diesel Engine,” SAEPaper 980510, 1998.

9. Naber, J. D. and Siebers, D. L., “Effects of Gas Density andVaporization on Penetration and Dispersion of DieselSprays,” Transactions of the SAE, Vol. 105, Sect. 3, pp. 82-111, 1996.

10. Abramovich, R. G. N., The Theory of Turbulent Jets, MITPress, pp. 586-600, 1963.

11. Thring, M. W. and Newby, M. P., “Combustion Length ofEnclosed Turbulent Jets,” 4th Symposium on Combustion,pp. 789-796, 1953.

12. Wakuri, Y., Fujii, M., Amitani, T., and Tsuneya, R., “Studiesof the Penetration of Fuel Spray in a Diesel Engine,” Bulle-tin of JSME, Vol. 3, No. 9, 1960.

13. Hays, W. J., Personal Communication, March, 1995.

14. Adler, D. and Lyn, W-T., “The Evaporation and Mixing of aLiquid Fuel Spray in a Diesel Air Swirl,” Proc. Instn. Mech.Engrs., Vol. 184, pp. 171-180, 1970.

15. Witze, P. O., “Hot-Film Anemometer Measurements in aStarting Turbulent Jet,” AIAA Journal, Vol. 21, No. 2,pp. 308-309, 1983.

16. Tomita, E., Hamamoto, Y., Tsutsumi, H., and Yoshiyama,S., “Measurement of Ambient Air Entrainment into Tran-sient Free Gas Jet by Means of Flow Visualization,” SAEPaper 950056, 1995.

17. Newman, J. A. and Brzustowski, T. A., “Behavior of a LiquidJet Near the Thermodynamic Critical Region,” AIAA Jour-nal, Vol. 9, No. 8, pp. 1595-1602, 1971.

18. El Wakil, M. M., Myers, P. S., and Uyehara, O. A., “FuelVaporization and Ignition Lag in Diesel Combustion,” Trans-actions of the SAE, Vol. 64, pp. 712-729, 1956.

19. Heywood, J. B., Internal Combustion Engine Fundamen-tals, McGraw-Hill Publishing, New York, NY, 1988.

20. Kuo, K. K., Principles of Combustion, John Wiley and Sons,New York, NY, 1986.

21. Hiroyasu, H. and Arai, M., “Structure of Fuel Sprays in Die-sel Engines,” Transactions of the SAE, Vol. 99, Sect. 3,pp. 1050-1061, 1990.

22. Varde, K., Popa, D., and Varde, L., “Spray Angle and Atom-ization in Diesel Sprays,” Transactions of the SAE, Vol. 93,Sect. 4, pp. 779-787, 1984.

21

23. Reitz, R. D. and Bracco, F., “On the Dependence of SprayAngle and Other Spray Parameters on Nozzle Design andOperating Conditions,” SAE Paper 790494, 1979.

24. Wu, K.-J., Su, C.-C., Steinberger, R. L., Santavicca, D. A.and Bracco, F. V., “Measurements of the Spray Angle ofAtomizing Jets,” Journal of Fluids Engineering, Vol. 105,pp. 406-413, 1983.

25. Lefebvre, A., Atomization and Sprays, Hemisphere Publish-ing Company, New York, 1989.

26. Bracco, F. V., “Modeling of Engine Sprays,” Transactions ofthe SAE, Vol. 94, Sect. 7, pp. 144-167, 1985.

27. Higgins, B. S., Mueller, C. J., and Siebers, D. L., “Measure-ments of Fuels Effects on Liquid-Phase Penetration in DISprays,” SAE Paper 1999-01-0519, 1999.

28. Technical Data Book-Petroleum Refining, Twelfth Revision,American Petroleum Institute (API), Washington D. C.,1997.

29. Lee, B. I. and Kesler, M. G., “A Generalized Thermody-namic Correlation Based on Three-Parameter Correspond-ing States,” AIChE Journal, Vol. 21, No. 3, pp. 510-527,1975.

30. Physical and Thermodynamic Properties of Pure Com-pounds: Data Compilation, Eds., Daubert, T. E., Danner,R. P., Design Institute for Physical Property Data (DIPPR),American Institute of Chemical Engineers, Taylor and Fran-cis, Washington D. C., 1998.

31. Prausnitz, J. M., Lichtenthaler, R. N., and Azevedo, E. G.,Molecular Thermodynamics of Fluid-Phase Equilibria, 2ndEd., Prentice-Hall Publishing, Englewood Cliffs, NJ, 1986.

32. Curtis, E. W., Uludogan, A., and Reitz, R. D., “A New HighPressure Droplet Vaporization Model for Diesel EngineModeling,” SAE Paper 952431, 1995.

33. Varnavas, C. and Assanis, D., “A High Temperature andHigh Pressure Evaporation Model for the KIVA-3 Code,”SAE Paper 960629, 1996.

34. Hohmann, S., Klingsporn, M., and Renz, U., “An ImprovedModel to Describe Spray Evaporation Under Diesel-LikeConditions,” SAE Paper 960630, 1996.

35. Chavez, H., Knapp, M., Kubitzek, A., Obermeier, F., andSchneider, T., “Experimental Study of Cavitation in the Noz-zle Hole of Diesel Injectors Using Transparent Nozzles,”SAE Paper 950290, 1995.

36. Soteriou, C., Andrews, R., and Smith, M., “Direct InjectionDiesel Sprays and the Effect of Cavitation and HydraulicFlip on Atomization,” SAE Paper 950080, 1995.

NOMENCLATURE

Greek:

Subscripts:

a a constant with a value of 0.66 in Eq. (3)A cross-sectional area of spray in the radial direction Ax cross-sectional area of spray in the axial directionb a constant with a value of 0.41 in Eq. (18)B the ratio of the fuel and ambient gas mass flow

rates resulting in complete vaporization of the fuel, defined by Eq. (8)

Ca orifice area-contraction coefficientCd orifice discharge coefficientCv orifice velocity coefficientd orifice diameterh enthalpyl length of the orificeL liquid length (i.e., the maximum penetration

distance of liquid-phase fuel)dimensionless liquid length (=L/x+)mass flow ratemomentum flow rate

M molecular weightP pressureR universal gas constantT temperatureU axial velocityUb fuel velocity at orifice exit based on Bernoulli’s

equationx axial coordinate of the sprayx+ characteristic length scale defined by Eq. (12)Z compressibility factor

α full cone angle of the model spray∆ incremental changeθ full cone angle of the real sprayρ densityσ standard deviation

a ambient gasf fuels saturated fuel vapor condition at the liquid length

22

APPENDIX A: SPRAY SPREADING ANGLE

The spray spreading angle data used in the liquid lengthscaling law are presented in this appendix. Both themeasurement technique and the spray spreading angledata are discussed.

MEASUREMENT – The spray spreading angles for thevaporizing sprays were determined from time-averagedschlieren images of the sprays. The schlieren imageswere acquired simultaneously with the time-averagedMie-scattered light images of the liquid-phase fuel usedto determine the liquid lengths in Ref. 1. The conditionscovered in the experiments are listed in Table 1 of thepaper. The optical setup for the simultaneous schlierenand Mie-scattered light image acquisition is described inRef. 1.

Figure A1.Time-averaged schlieren images of three sprays injected from left to right into the ambient gas density (ρa) given in each image. The orifice pressure drop, the orifice diameter, the ambient gas temperature, the fuel temperature, and the fuel were 135 MPa, 246 µm, 1000 K, 438 K and DF2, respectively. (The horizontal width of each image corresponds to 58 mm.)

Figure A1 shows three example time-averaged schlierenimages used for determining the spreading angle of aspray. The parameter varied from image to image inFig. A1 is the ambient gas density, which is given in theupper left corner of the image. The injector tip in theimages is located at the left edge, and the fuel is injectedfrom left to right. The spray is the dark, cone shaped

region emanating from the injector tip located at the leftside. (The white lines in Fig. A1 are discussed later.)(The location of liquid phase fuel inside the sprays inFig. A1 can be found in Fig. 8 of Ref. 1.)

Spray spreading angles were determined from theschlieren images, such as shown in Fig. A1, by a thresh-old intensity method. The threshold selected was half-way between the average background intensity (the lightgray region in an image in Fig. A1) and the spray center-line intensity in the axial region of the spray used fordetermining the spray spreading angle. Any location inthe region of the image used for determining the sprayspreading angle with an intensity below the threshold isconsidered part of the spray.

The region of the spray used for determining a spreadingangle included the region between an axial distance2.5 mm from the injector tip and a distance 20% beyondthe measured liquid length location. The region of thespray considered was restricted axially for signal-to noisereasons. The near injector region of the spray was notused because of the noise generated in the schlierenimages by the proximity to the relatively cold injector tip.The region of the spray significantly beyond the liquidlength location was not considered because the schliereneffect defining the spray begins to fade out in the down-stream axial direction due to ambient gas entrainment.This latter problem was mainly an issue for the shortestliquid length conditions for which the region downstreamof the liquid length location was the majority of the imagefield. In these images, the schlieren effect induced by thespray almost completely disappeared by the right side ofthe image. This resulted in significant noise in the mea-sured spreading angle when it was included in the evalu-ation.

With the spray identified in the image through applicationof the threshold, the spreading angle is determined fromthe following expression derived from geometric consid-erations:

(01)

The terms x1 and x2 are the axial distances correspond-ing to 2.5 mm from the injector tip and a distance 20%longer than the liquid length, respectively. The area Ax isthe area in the image between the distances x1 and x2with an intensity below the threshold. The white lines inFig. A1 represent the full spray angle (θ) determined fromeach image using Eq. (A1). The values for the half angle(θ/2) are given in the upper left corner of each image.Spray angles identified in this manner lie in the very outerintermittent region of the spray [9].

SPREADING ANGLE DATA – Figures A2 through A5show the effects of ambient gas temperature, fuel type,injection pressure, ambient gas density, and orifice

ρa= 7.3 kg/m3

θ/2 = 5.8°

14.8 kg/m3

7.0°

30.2 kg/m3

8.0°

−=θ −

21

22

1

2 xx

Atan x

23

details on the spray spreading angle. The effect of ambi-ent gas temperature and fuel type are most clearly visiblein Fig. A2. Figure A2 is a plot of the spreading anglesdetermined for cetane, HMN, and DF2 as a function ofambient gas temperature for five ambient gas densities.Each density condition is presented as a “sub-plot” in thefigure. The spray angle for each fuel and each conditionin Fig. A2 was acquired simultaneously with the liquidlength at the corresponding condition in Figs. 5, 6, and 15for each respective fuel.

Figure A2. Spray spreading angle versus ambient gas temperature for five gas densities and three fuels. The orifice pressure drop, the orifice diameter, the ambient gas temperature, and the fuel temperature were 136 MPa, 246 µm, 1000 K, and 438 K, respectively.

Figure A2 shows that neither the fuel type nor the ambi-ent gas temperature have any significant effect on thespray spreading angle for the range of temperature andfuels considered. The insignificant temperature effectagrees with previous observations for vaporizing spraysby Naber and Siebers [9]. The lack of effect of fuel typewas extended to include fuels such as bio-diesel, Fis-cher-Tropsch diesel, methanol, and gasoline in a recentinvestigation of alternative fuel effects on liquid length[30].

The effect of injection pressure on spreading angle isshown in Fig. A3. The conditions for which data areshown correspond to those used in Fig. 10. Some of theconditions presented in the figure were selected to showthe maximum observed effect of injection pressure. Thefigure indicates that injection pressure also has very littleeffect on the spray spreading angle as previously notedfor vaporizing sprays [9]. The small effect noted caneither be an increase or a decrease with increasing injec-tion pressure. The small effects observed in the figureare believed to be caused by details of the flow throughthe orifice.

Figure A3. Spray spreading angle versus the pressure drop across the injector orifice for conditions given in the figure legend. (The conditions correspond to those used in Fig. 10.) The terms in the legend are the ambient gas temperature (Ta) and density (ρa), the orifice diameter (d), and the fuel type. The lines in the figures are linear least squares fits to the data for each set of conditions given in the legend. The aspect ratio of the orifice and the fuel temperature were nominally 4.2 and 438 K, respectively.

Figure A4. The spray spreading angle acquired with the 246 µm diameter orifice versus the ambient gas to fuel density ratio for all conditions for the three fuels. The curve is given by Eq. (A2). The dashed curves are Eq. (A2) shifted by ±7%, which corresponds to ±2σ in the measured angles at any density condition.

The effects of ambient gas density on the spray spread-ing angle are shown in Fig. A4. All the spray spreadingangle data acquired with 246 µm diameter orifice listed in

' ' ' ' ' ' ' ' '79

11

' ' ' ' ' ' ' ' '68

10

' ' ' ' ' ' ' ' '579

' ' ' ' ' ' ' ' '468

600 700 800 900 1000 1100 1200 1300 1400357S

pray

Spr

eadi

ng A

ngle

(θ /

2)

Gas Temperature [K]

Cetane HMN DF2ρa

(kg/m3)

59.0

30.2

14.8

7.3

3.6

25 50 75 100 125 150 175 2000

4

8

12

16

Spr

ay S

prea

ding

Ang

le (

θ /2)

Orifice Pressure Drop [MPa]

Ta ρa d Fuel(K) (kg/m3) (µm)

1000 7.2 246 HMN 600 14.8 246 HMN1000 30.2 246 HMN

Ta ρa d Fuel(K) (kg/m3) (µm)

700 14.8 246 HMN 700 7.3 100 Cetane1300 30.1 498 Cetane

24

Table 3 for all three fuels (cetane, HMN, and DF2) areshown in Fig. A4 as a function of the ambient gas/fueldensity ratio. This figure includes data spanning therange of conditions given in Table 1. The figure empha-sizes that for a given orifice, the primary parameteraffecting the spreading angle is the ambient gas/fuel den-sity ratio [e.g., 9, 21-24].

The curve in the Fig. A4 is:

(A2)

where the constant c is 0.260. Equation (A2) is an empir-ically derived equation that fits the vaporizing sprayspreading angle data for the 246 µm orifice. The firstdensity ratio term in the equation with an exponent of0.19 agrees with the density dependence observed previ-ously for non-vaporizing sprays over a larger density ratiorange [9]. The second density ratio with an exponent of0.5 was included to account for the observed vaporiza-tion effects on the spreading angle, also previously noted[9]. The dashed lines are ±7% shifts in Eq. (A2), whichcorrespond to ±2σ in the angles measured at each gasdensity for all other conditions.

Figure A5. The deviation of the measured spray angle data acquired for four of the orifices in Table 3 versus the ambient gas/fuel density ratio. The orifice diameters and the values used for c in Eq. (A2) for each orifice are given in the figure. The dashed curves represent ±7% shifts in the constant c, which corresponds to ±2σ in the measured angles at any density condition.

Equation (A2) is presented for two reasons. First, it wasused to interpolate the spreading angle data as a func-tion of gas and fuel densities and allow the continuouscurves in various liquid length figures in the paper to begenerated with the scaling law. Second, it is used in thenext figure to quantify orifice dependent effects on sprayspreading angle for the various orifices used in the exper-iment.

Figure A5 is plot of the deviation of the measured spread-ing angles from Eq. (A2), where the constant c has beenoptimized to provide the best fit to the data for each of theother four orifices listed in Table 3. The value for c foreach orifice is given in the figure. The data for these ori-fices is not as extensive as for the 246 µm orifice shownin Fig. A4, but does include a number of ambient temper-ature and injection pressure conditions, as well as datafor all three fuels for two of the orifices. The dashed linesin each figure represent ±7% deviations from Eq. (A2).

Figure A5 shows two important trends. First, the spread-ing angle for a given density ratio varies from orifice toorifice. Second, the dependence of the spreading angleon the ambient-gas/fuel density ratio is the same for eachorifice. The first trend is indicated by the differing valuesfor c given in the figure. The second trend is indicated bythe fact that there is no trend in the deviation between thedata and Eq. (A2) as a function of the density ratio.These trends indicate that there are orifice details thateffect the flow through an orifice and the ensuing spraydevelopment, but not the relative dependence of thespray spreading angle on the ambient gas and fuel densi-ties. The importance of orifice effects on the sprayspreading angle has been noted by many others previ-ously [e.g., 9, 21-24], however, the physical explanationof the effect remains unknown.

-0.3

0.0

0.3

-0.3

0.0

0.3

0.010 0.100-0.3

0.0

0.3Dev

iatio

n fr

om E

q. (

A2)

ρa / ρf

Cetane HMN DF2

-0.3

0.0

0.3

c - Eq. (A2)0.255

0.271

0.258

0.276

d (µm)100

180

363

498

25

APPENDIX B: ORIFICE AREA-CONTRACTION COEFFICIENT

The technique used to measure the area-contractioncoefficients (Ca) in Table 3 is presented in this appendix.The technique gives an absolute measure of Ca, and isan improvement over the technique described in Ref. [9]based on a reference orifice. Before presenting the area-contraction coefficient measurement technique, however,a brief discussion is presented on the relationship of thedischarge coefficient and the area-contraction coefficient,and the need for two coefficients to characterize an ori-fice.

ORIFICE COEFFICIENTS – The development of a sprayis dependent on both the mass and momentum flow ratesfrom an orifice. To characterize both flow rates, two ori-fice coefficients are needed. The discharge coefficient isthe most commonly measured orifice coefficient. Twoother orifice coefficients are the area-contraction coeffi-cient and the velocity coefficient (Cv). They are definedsuch that their product equals the discharge coefficient:

(B1)

Any two of the three coefficients (Ca, Cv, and Cd) can beused to characterize the mass and momentum flowsthrough an orifice. Using Cv and Ca, the mass flow rate( f ) and momentum flow rate ( f ) from an orifice aregiven by the following:

(B2)

(B3)

where Ub is given by Bernoulli’s equation:

(B4)

The term ρf is the fuel density, Af is the orifice exit area,and Pf and Pa are the fuel and ambient gas pressures,respectively. The velocity Ub is the maximum potentialfluid velocity at the orifice exit, while the product Cv ⋅Ub isthe average velocity at the orifice exit over the areaCa ⋅Af.

The area-contraction coefficient in the above equationsaccounts for loss of flow area in an orifice as a result ofvapor bubbles generated by cavitation reaching the ori-fice exit [35], “hydraulic flip” [36], and/or non-uniformvelocity profiles at the orifice exit. Based on Eq. (B1),any change in Ca must result in change in Cv and thevelocity at the orifice exit.

The equations demonstrate that two coefficients areneeded to characterize the mass an momentum flowrates. For the liquid length investigation presented in thispaper, the area-contraction coefficient and the dischargecoefficient were measured for each orifice used. Themore common situation in the literature, however, is thatonly a discharge coefficient is available. In this case,either Ca or Cv are effectively assumed to be one.

Depending on which assumption is made, errors up to20% in the momentum flow rate can occur for typical die-sel injector conditions even though the mass flow rate iscorrect, as will be shown in the next section.

AREA-CONTRACTION COEFFICIENT – The orificearea-contraction coefficients in Table 3 were determinedfrom the spray momentum measured with a force trans-ducer and the following relationship derived fromEqs. (B1-B4):

(B5)

The spray momentum was measured with a Kistler 6121piezoelectric pressure transducer calibrated to measurethe force (i.e., the momentum) induced by a sprayimpinging on the transducer diaphragm. The transducerwas placed approximately 3 mm in front of the orifice.This distance was close enough to the orifice that theentire spray impinged on the central region of the trans-ducer diaphragm, but far enough away that the flowthrough the orifice was not restricted. The momentum( f in Eq. (B5)) was measured once steady flow throughthe orifice was established, which typically occurs in lessthan 100 µs for the injector used [9]. (Several othertransducers were tried, but the Kistler 6121 was the onlyone with no significant zero drift during an injection as aresult of the spray impingement.)

Figure B1 is a plot of the area-contraction coefficientmeasured as a function of injection pressure for the267 µm diameter orifice listed in Table 3 with an l/d of 8.The ambient pressure for these experiments was atmo-spheric (Pa = 0.101 MPa), and the fuel temperature wasroom temperature. Also shown in the figure are the inde-pendently measured orifice discharge coefficient and thevelocity coefficient determined using Eq. (B1). The figureshows that the area-contraction coefficient decreaseswith increasing injection pressure, while the dischargecoefficient remains nearly constant. This means the flowarea is effectively decreasing as injection pressureincreases, most likely as a result of increasing cavitation.In addition, the corresponding increase in Cv withincreasing injection pressure indicates that the orifice exitvelocity more closely approaches the Bernoulli velocitygiven by Eq. (B3) as injection pressure increases. Thislatter trend is in a agreement with the velocity measure-ments made by Chavez et al. [35] in an injector orifice.

The area-contraction coefficient for the other orifices inTable 3 were measured at two pressures and are given inthe table. Each shows a decrease with increasing injec-tion pressure as observed in Fig. B1. The effect of fueltemperature on Ca was also examined and found to beinsignificant for temperatures up to 440 K.

Equations (B2) and (B3) coupled with the results inFig. B1 and Table 3 show that at the lower injection pres-sures considered (~80 MPa), if only a Cd is available and

vad CCC ⋅=

bvffaf UCACm ⋅⋅ρ⋅⋅=�

bvff UCmM ⋅⋅= �

�

fafb PPU ρ−⋅= /][2

fafdfa MPPCAC �/][2 2 −⋅⋅⋅=

26

Cv is assumed to be one, a 15% over estimate of thespray momentum would occur. Assuming Ca is onewould result in a 15% under estimate of the spraymomentum. At the higher injection pressures(~140 MPa), assuming a Cv of one would only result in afew percent over estimated of the momentum flux, butassuming a Ca of one would result in a 20% under esti-mated. These observations indicate that the details ofcavitation and other orifice processes have significanteffects on the entire evolution of a spray and need furtherinvestigation.

Figure B1. The orifice discharge, area-contraction, and velocity coefficients versus injection pressure for the 267 µm diameter orifice listed in Table 3 with an l/d of 8.0.

40 80 120 160 2000.4

0.6

0.8

1.0

1.2

1.4

Orif

ice

Coe

ffici

ents

Injection Pressure [MPa]

CdCaCv