yang v. modelling

8/14/2019 Yang v. Modelling

1/18

925

Proceedings of the Combustion Institute, Volume 28, 2000/pp. 925942

INVITED TOPICAL REVIEW

MODELING OF SUPERCRITICAL VAPORIZATION, MIXING, AND COMBUS-TION PROCESSES IN LIQUID-FUELED PROPULSION SYSTEMS

VIGOR YANGThe Pennsylvania State University

104 Research Building EastUniversity Park, PA 16802 USA

This paper addresses the physiochemical mechanisms involved in transcritical and supercritical vapori-zation, mixing, and combustion processes in contemporary liquid-fueled propulsion and power-generationsystems. Fundamental investigation into these phenomena poses an array of challengesdue to the obstaclesin conducting experimental measurements and numerical simulations at scales sufcient to resolve theunderlying processes. In addition to all of the classical problems for multiphase chemically reacting ows,

a unique set of problems arises from the introduction of thermodynamic nonidealities and transportanom-alies. The situation becomes even more complex with increasing pressure because of an inherent increasein the ow Reynolds number and difculties that arise when uid states approach the critical mixingcondition. The paper attempts to provide an overview of recent advances in theoretical modeling andnumerical simulation of this subject. A variety of liquid propellants, including hydrocarbon and cryogenicuids, under both steady and oscillatory conditions, are treated systematically. Emphasis is placed on thedevelopment of a hierarchical approach and its associated difculties. Results from representative studiesare presented to lend insight into the intricate nature of the problem.

Introduction

Liquid droplet vaporization and spray combustionin supercritical environments have long been mat-ters of serious practical concern in combustion sci-ence and technology, mainly due to the necessity of developing high-pressure combustion devices suchas liquid-propellant rocket, gas-turbine, diesel, andpulse-detonation engines. Liquid fuels and/or oxi-dizers are usually delivered to combustionchambersas a spray of droplets, which then undergo a se-quence of vaporization, mixing, ignition, and com-bustion processes at pressure levels well above thethermodynamic critical points of the uids. Underthese conditions, liquids initially injected at subcrit-

ical temperature may heat up and experience a ther-modynamic state transition into the supercritical re-gime during their lifetimes. The process exhibitsmany characteristics distinct from those in an at-mospherical environment, thereby rendering conventional approaches developed for low-pressureapplications invalid.

Modeling supercritical mixing and combustionprocesses numerically poses a variety of challengesthat include all of the classical closure problems anda unique set of problems imposed by the introduc-tion of thermodynamic nonidealities and transportanomalies. From the classical point of view, reacting,multiphase ows introduce the complicating factorsof chemical kinetics, highly nonlinear source terms,and a variety of sub-grid scale (SGS) velocity and

scalar-mixing interactions. Floweld evolution is af-fected by compressibility effects (volumetricchanges induced by changes in pressure) and vari-able inertia effects (volumetric changes induced by variable composition and/or heat addition). The re-sultant coupling dynamics yield an array of physi-ochemical processes that are dominated by widely disparate time and length scales, manybeingsmallerthan can be resolved in a numerically feasible man-ner. The situation becomes more complex with in-creasing pressure because of an inherent increase inthe ow Reynolds number, which causes a furtherdecrease in the scales associated with SGS interac-tions and difculties that arise when uid states ap-proach the critical conditions. Near thecriticalpoint,

propellant mixture properties exhibit liquid-likedensities, gas-like diffusivities, and pressure-depen-dent solubilities. Surface tension and enthalpy of va-porization approach zero, and the isothermal com-pressibility and specic heat increase signicantly.These phenomena, coupled with extreme localprop-erty variations, have a signicant impact on the evo-lutionary dynamics exhibited by a given system.

This paper attempts to provide an overview of re-cent advances in theoreticalmodeling and numericalsimulation of supercritical vaporization, mixing, andcombustion processes in liquid-fueled propulsionsystems. Three subject areas are considered: (1)droplet gasication and combustion, (2) spray elddynamics, and (3) multiphase mixing and combus-tion processes. A variety of liquid propellants and


2/18

926 INVITED TOPICAL REVIEW

propellant simulants, including hydrocarbon andcryogenic uids, at both steady and oscillatory con-ditions, are treated systematically.

Supercritical Droplet Vaporization andCombustion

We rst consider the vaporization and combustionof an isolated liquid droplet when suddenly placedin a combustion chamber. This conguration allowsus to focus on the effects of pressure on the ther-modynamic and transport processes involved. As aresult of heat transfer from the surrounding gases,the droplet starts to heat up and evaporate due tothe vapor concentration gradient near the surface.Two scenarios, subcritical and supercritical condi-tions, must be considered in order to provide a com-plete description of various situations encountered.If the chamber condition is in the thermodynamicsubcritical regime of the injected liquid, the dropletsurface provides a well-dened interfacial boundary.To facilitate analysis, physical processes in the drop-let interior and ambient gases are treated separately and then matched at the interface by requiring liq-uid-vapor phase equilibrium and continuities of mass and energy uxes. This procedure eventually determines the droplet surface conditions and evap-oration rate. The situation, however, becomes qual-itatively different in the supercritical regime. Then,the droplet may continuously heat up, with its sur-face reaching the critical mixing point prior to theend of the droplet lifetime. When this occurs, thesharp distinction between gas and liquid disappears.The enthalpy of vaporization reduces to zero, and noabrupt phase change is involved in the vaporizationprocess. The density and temperature of the entireeld as well as their gradients vary continuously across the droplet surface. The droplet interior, how-ever, remains at the liquid state with a subcriticaltemperature distribution. For convenience of dis-cussion, the droplet regression is best characterizedby the motion of the surface which attains thecritical

mixing temperature of the system. The process be-comes totally diffusion controlled.Attempts to study supercritical droplet vaporiza-

tion and combustion have been made for more thanfour decades. The earliest development of a predic-tive model was initiated by Spalding [1] and subse-quently rened by Rosner [2]. Both studies consid-ered an isolated fuel droplet in a stagnantenvironment by approximating the droplet as a pointsource of dense gas with constant physical proper-ties. The same problem was re-examined by otherresearchers [35] in order to investigate the inu-ence of convection, density variation, and nite ratechemical kinetics.

Systematic treatment of droplet vaporization atnear-critical conditions was initiated by Manrique

and Borman [6], based on a quasi-steady model.They concluded that the effects of thermodynamicnonidealities, property variations, and high-pressurecorrections for phase equilibrium modify the vapor-ization mechanisms signicantly. In light of these

ndings, Lazar and Faeth [7] and Canada and Faeth[8] conducted a series of experimental and theoreti-cal studies on droplet combustion in both stagnantand forced convective environments, with special at-tention focused on the high-pressure phenomena of phase equilibrium. The effects of forced convectionin the gas phase were treated by conventional mul-tiplicative corrections. The assumptions of quasi-steadiness and uniform property distributionsadopted in Refs. [68] were later relaxed. In worksby Rosner and Chang [9] and Kadota and Hiroyasu[10], the effects of transient processes, natural convection, and the conditions under which a dropletmay be driven to its critical point were examined.More recently, several researchers [1117] haveemployed numerical techniques to simulate high-pressure droplet vaporization and combustion withconsiderable success. All of these models, however,adopted certain rudimentary assumptions and em-pirical formulas for uid properties that were ex-trapolated from low-pressure cases, with their ac-curacy for high-pressure applications subject toquestion. Furthermore, no effort was made to treatthe thermodynamic phase transition through thecritical point. In order to remedy these deciencies,a series of fundamental studies [1828] were con-ducted using state-of-the-art treatment of thermo-dynamic and transport phenomena. Of particularimportance is the unied analyses of thermophyscialproperties employed in Refs. [1920] and [2528]based on fundamental thermodynamic theories.These approaches allow for a self-consistent solutionfrom rst principles, thereby enabling a systematicinvestigation into underlying mechanisms involvedin supercritical droplet gasication and combustion.The effect of non-equilibrium phase transition ondroplet behavior was further addressed by Harstadand Bellan [2528] using Keizers uctuation theory.

In addition, Umemura and Shimada [2931] con-structed approximate analyses to elucidate many in-triguing characteristics of supercritical droplet gasi-cation. Extensive reviews of this subject wererecently given Givler and Abraham [32] and Bellan[33].

Thermodynamic and Transport Properties

Owing to the continuous variations of uid prop-erties in supercritical environments, classical tech-niques dealing with liquids and gases individuallyof-ten lead to erroneous results of droplet dynamics.The problem becomes even more exacerbatedwhenthe droplet surface approaches the criticalcondition.


3/18


4/18


F ig . 1. Comparison of oxygen density predicted by theBWR equation of state and measured by Sychev et al. [39].

F ig . 2. Relative errors of density predictions by threedifferent equations of state. Experimental data fromSychev et al. [39].

and SoaveRedlichKwong (SRK). The ECS prin-ciple is embedded into the evaluation procedure of the BWR equation of state and shows its superiorperformance with the maximum relative error of 1.5% for the pressure and temperature ranges underconsideration. On the other hand, the SRK and PRequations of state yieldmaximum errorsaround13%

and 17%, respectively.Thermodynamic Properties

Thermodynamic properties such as enthalpy, in-ternal energy, and specic heat can be expressed asthe sum of ideal-gas properties at the same tem-perature and departure functions which take into ac-count the dense-uid correction. Thus,

v p0h h T p dv T v RT (Z 1) (6)

v p0e e T p dv (7) T v

2 v p0C C T dvp p 2 T v2 p p

T R (8) T v v T where superscripts 0 refer to ideal-gas properties.The second terms on the right sides of equations 68 denote the thermodynamic departure functionsand can be obtained from the equation of state de-scribed previously.

Transport PropertiesEstimation of viscosity and thermal conductivity

can be made by means of the ECS principle. Thecorresponding-state argument for the viscosity of amixture can be written in its most general form as

l ( q, T ) l ( q , T )F v (9)m 0 0 0 l l where F l is the scaling factor. The correction factor v l accounts for the effect of non-correspondenceand has the magnitude always close to unity basedon the modied Enskog theory [40].

Because of the lack of a complete molecular the-ory for describing transport properties over a broadregime of uid phases, it is generally accepted that viscosity and thermal conductivity can be dividedinto three contributions and correlated in terms of density and temperature [41]. For instance, the vis-cosity of the reference uid is written as follows:

0 exc l ( q , T ) l (T ) Dl ( q , T )0 0 0 0 0 0 0 0crit Dl ( q , T ) (10)0 0 0

The rst term on the right-hand side represents the value at the dilute-gas limit, which is independent of density and can be accurately predicted by kinetic-theory equations. The second term is the excess vis-cosity which, with the exclusion of unusualvariationsnear the critical point, characterizes the deviationfrom l 0 for a dense uid. The third term refers tothe critical enhancement which accounts for the

anomalous increase above the background viscosity (i.e., the sum of and ) as the critical point is0 exc l Dl0 0approached. However, the theory of nonclassicalcritical behavior predicts that, in general, propertiesthat diverge strongly in pure uids near the criticalpoints diverge only weakly in mixtures due to thedifferent physical criteria for criticality in a pure uidand a mixture [42]. Because the effect of critical en-hancement is not well-dened for a mixture and islikely to be small, the third term is usually notcritDl 0considered in the existing analyses of supercriticaldroplet gasication.

Evaluation of thermal conductivity must be care-fully conducted for two reasons: (1) the one-uidmodel must ignore the contribution of diffusion toconductivity, and (2) the effect of internal degrees


5/18

SUPERCRITICAL VAPORIZATION, MIXING, AND COMBUSTION 929

F ig . 3. Vaporliquid phase equilibrium compositions forO2/H 2 system at various pressures.

F ig . 4. Pressuretemperature diagram for phase behav-ior of O2/H 2 system in equilibrium.

of freedom on thermal conductivity cannot be cor-rectly taken into account by the corresponding-stateargument. As a result, thermal conductivityof a puresubstance or mixture is generally divided into twocontributions [38]:

k ( q, T ) k (T ) k ( q, T ) (10)m m mThe former, , arises from transfer of energy viak (T )mthe internal degrees of freedom, while the latter,

, is due to the effects of molecular collisionk ( q, T )mor translation which can be evaluated by means of the ECS method. For a mixture, can be eval-k (T )muated by a semi-empirical mixing rule.

Estimation of the binary mass diffusivity for a mix-ture gas at high density represents a more challeng-ing task than evaluating the other transport proper-ties, due to the lack of a formal theory or even atheoretically based correlation. Takahashi [43] sug-gested a simple scheme for predicting the binary mass diffusivity of a dense uid by means of a cor-responding-state approach. The approach appears tobe the most complete to date and has demonstrated

moderate success in the limited number of testscon-ducted. The scheme proceeds in two steps. First, thebinary mass diffusivity of a dilute gas is obtained us-ing the Chapman-Enskog theory in conjunctionwiththe intermolecular potential function. The calcu-

lated data is then corrected in accordance with ageneralized chart in terms of reduced temperatureand pressure.

Vapor-Liquid Phase Equilibrium

The result of vapor-liquid phase equilibrium is re-quired to specify the droplet surface behavior priorto the occurrence of critical conditions. The analysisusually consists of two steps. First, an appropriateequation of state as described in the section on ther-modynamic and transport properties is employed to

calculate fugacities of each constituent species inboth gas and liquid phases. The second step lies inthe determination of the phase equilibrium condi-tions by requiring equal fugacities for both phases of each species. Specic outputs from this analysis in-clude (1) enthalpy of vaporization, (2) solubility of ambient gases in the liquid phase, (3) species con-centrations at the droplet surface, and (4) conditionsfor criticality. The analysis can be further used inconjunction with the property evaluation scheme todetermine other important properties such as sur-face tension.

As an example, the equilibrium compositions fora binary mixture of oxygen and hydrogen is shownin Fig. 3, where pr represents the reduced pressureof oxygen. In the subcritical regime, the amount of hydrogen dissolved in the liquid oxygen is quite lim-ited, decreasing progressively with increasing tem-perature and reducing to zero at the boiling point of oxygen. At supercritical pressures, however, the hy-drogen gas solubility becomes substantial and in-creases with temperature. Because of the distinctdifferences in thermophysical properties betweenthe two species, the dissolved hydrogen may appre-ciably modify the liquid properties and, subse-

quently, the vaporization behavior. The phase equi-librium results also indicate that the critical mixingtemperature decreases with pressure.

The overall phase behavior in equilibrium is bestsummarized by the pressure-temperature diagramin Fig. 4, which shows how the phase transitionoco-curs under different thermodynamic conditions.Theboiling line is made up of boiling points for subcrit-ical pressure. As the temperature increases, an equi-librium vapor-liquid mixture may transit to super-heated vapor across this line. The critical mixing lineregisters the variation of the critical mixing tem-perature with pressure. It intersects the boiling lineat the critical point of pure oxygen, the highest tem-perature at which the vapor and liquid phases of anO2/H 2 binary system can coexist in equilibrium.


6/18


F ig . 5. Time variations of droplet surface temperatureat various pressures. T 1000 K, T 0 90 K, D0 100 l m.

F ig . 6. Instantaneous distributions of mixture specicheat in the entire eld at various times. T 0 90 K, T 1000 K, p 100 atm, D0 100 l m.

Droplet Vaporization in Quiescent EnvironmentsSeveral theoretical works have recently been de-

voted to the understanding of droplet vaporizationand combustion under high-pressure conditions.

Both hydrocarbon droplets in air [1113,18,27,28]and liquid oxygen (LOX) droplets in hydrogen[14,17,1926,44] were treated comprehensively, with emphasis placed on the effects of transient dif-fusion and interfacial thermodynamics. The worksof Lafon et al. [22,44], Hsiao and Yang [20,45], andHarstad and Bellan [2528] appear to be the mostcomplete to date because of the employment of aunied property evaluation scheme, as outlined previously.

We rst consider the vaporization of LOX dropletin either pure hydrogen or mixed hydrogen/waterenvironments [22,44] due to its broad applicationsin cryogenic rocket engines using hydrogen and oxygen as propellants [46]. Fig. 5 shows the time var-iations of droplet surface temperature at various

pressures. The ambient hydrogen temperature is1000 K. Three different scenarios are noted. First,at low pressures (i.e., p 10 atm), the surface tem-perature rises suddenly and levels off at the pseudo wet-bulb state, which is slightly lower than the oxy-

gen boiling temperature because of the presence of hydrogen on the gaseous side of the interface. Forhigher pressures (i.e., p 50 atm), the surface tem-perature rises continuously. The pseudo wet-bulbstate disappears, and the vaporization process be-comes transient in nature during the entire dropletlifetime. For p 100 atm, the droplet surface evenreaches its critical state at 1 ms. Fig. 6 shows thedistributions of the mixture specic heat at varioustimes. The weak divergence of the specic heat nearthe droplet surface (i.e., dened as the surface at-taining the critical-mixing temperature) is clearly ob-served.

An extensive series of numerical simulations wereconducted for a broad range of ambient tempera-tures (500 T 2500 K) and pressures up to 300atm. The calculated LOX droplet lifetime in purehydrogen can be well correlated using an approxi-mate analysis that takes into account the effect of transient heat diffusion in terms of the reduced criti-cal temperature, . TheT * (T T )/(T T )c c 0resultant expression of the droplet lifetime takes theform

2R0 ls [0.0115 0.542 (1 T *)] f ( / ) (11)c 0l0

where the correction factor is chosen asl f ( / ) 0

l1 3.9[1 exp( 0.035( / 1))] (12) 0

The expression of may be related to the SpaldingT *ctransfer number, which has more physical signi-cance, as follows:

T T T * c cB (13)T T T 1 T *c 0 c

This correlation clearly shows that pressure affectsthe droplet lifetime through its inuence on the mix-ture critical temperature and ambient thermal dif-fusivity.

The behavior of hydrocarbon fuel droplets basi-cally follow the same trend as cryogenic droplets interms of their vaporization characteristics. In gen-eral, the droplet lifetime decreases smoothly withincreasing pressure, and no discernible variation oc-curs across the critical transition. In the subcriticalregime, the decrease in enthalpy of vaporizationwithpressure facilitates vaporization process and conse-quently leads to a decrease in droplet lifetime. Thesituation becomes different at supercritical condi-tions, at which the interfacial boundary disappears


7/18


F ig . 7. Time variations of droplet surface temperatureat various pressures, n-pentane/air system.

F ig . 8. Effect of pressure on milestone times associated with droplet gasicationand burningprocesses, n-pentane/air system. T 0 300 K, T 1000 K.

with the enthalpy of vaporization being zero. In thiscase, the decrease in droplet lifetime is mainly at-

tributed to the increase in thermal diffusivity nearthe droplet surface. As critical conditions are ap-proached, the divergence of the specic heat com-pensates for the effect of vaporization enthalpy re-duction, so that no abrupt change occurs during thecritical transition.

Droplet Combustion in Quiescent Environments

Much effort was devoted to the study of super-critical droplet combustion (e.g., Refs. [17,18,22]).In spite of the presence of chemical reactions in thegas phase, the general characteristics of a burningdroplet are similar to those involving only vaporiza-tion. We consider here the combustion of a hydro-carbon (e.g., n-pentane) fuel droplet in air [18]. The

droplet initial temperature is 300 K, and theambientair temperature is 1000 K. Fig. 7 shows the timehistory of the droplet surface temperature at variouspressures for D0 100 l m. Once ignition isachieved in the gas phase, energy feedback from the

ame causes a rapid increase in droplet surface tem-perature. At low pressures ( p 20 atm), the surfacetemperature varies very slowly following onset of ame development and almost levels off at thepseudo wet-bulb state. As the ambient pressure in-creases, the high concentrations of oxygen in the gasphase and the fuel vapor issued from the dropletsurface result in a high chemical reaction rate, con-sequently causing a progressive decrease in ignitiontime. Furthermore, the surface temperature jumpduring the ame-development stage increases withincreasing pressure. Since the critical mixing tem-perature decreases with pressure, the dropletreaches its critical condition more easily at higherpressures, almost immediately following establish-ment of the diffusion ame in the gas phase for p110 atm.

Figure 8 presents the effect of pressure on variousmilestone times associated with droplet gasicationand burning process for D0 100 l m [18]. Here,gasication lifetime is the time required for com-plete gasication; droplet burning lifetime is the gas-ication lifetime minus ignition time; single-phasecombustion lifetime is the time duration from com-plete gasication to burnout of all fuel vapor; com-bustion lifetime is the sum of single-phase combus-tion lifetime and droplet burning lifetime. Thegasication lifetime decreases continuously withpressure, whereas the single-phase combustion life-time increases progressively with pressure due to itsadverse effect on mass diffusion. More importantly,the pressure dependence of combustion lifetime ex-hibits irregular behavior. This phenomenon may beattributed to the overlapping effect of reduced en-thalpy of vaporization and mass diffusion with in-creasing pressure.

Since the time scales for diffusion processes areinversely proportional to the droplet diameter

squared, it is important to examine the effect of droplet size on the burning characteristics. In thisregard, calculations were conducted for large drop-lets having an initial diameter of 1000 l m, which iscomparable to the sizes considered in most experi-mental studies of supercritical droplet combustion[4749]. Furthermore, the ignition transient occu-pies only a small fraction of the entire droplet life-time for large droplets. The uncertainties associated with the ignition procedure in determining the char-acteristics of droplet gasication can be minimized.Consequently, a more meaningful comparison withexperimental data can be made. The combustion be-havior of a large droplet reveals several characteris-tics distinct from those of a small droplet. First, ig-nition for large droplets occurs in the veryearlystage


8/18


F ig . 9. LOX droplet gasication in supercritical hydrogen ow. p 100 atm, U 2.5 m/s.

of the entire droplet lifetime. The inuence of gas-ication prior to ignition on the overall burning

mechanisms appears to be quite limited. Second, thecombustion lifetime decreases with increasing pres-sure, reaching a minimum near the critical pressureof the liquid fuel. As the pressure further increases,the combustion time increases due to reduced massdiffusivity at high pressures. The gasication lifetimedecreases continuously with pressure, whereas thesingle-phase combustion lifetime increases progres-sively with pressure. At low pressures, the gasica-tion of liquid fuel primarily controls the combustionprocess, whereas in a supercritical environment, thetransient gas-phase diffusion plays a more importantrole.

Droplet Vaporization in Convective Environments

When a droplet is introduced into a cross-ow, theforced convection results in increases of heat andmass transfer between the droplet and surroundinggases, which consequently intensies the gasicationprocess. Although many studies have been con-ducted to examine droplet vaporization in forced-convective environments, effects of pressure andfreestream velocities on droplet dynamics, especially for rocket engine applications which involve super-critical conditions, have not yet been addressed indetail. Hsiao et al. [20,45] developed a comprehen-sive analysis of LOX droplet vaporization in a super-critical hydrogen stream, covering a pressure range

of 100400 atm. The model takes into account mul-tidimensional ow motions and enables a thorough

examination of droplet behavior during its entirelifetime, including dynamic deformation, viscousstripping, and secondary breakup. Detailed ow structures and thermodynamic properties are ob-tained to reveal mechanisms underlying droplet gas-ication as well as deformation and breakup dynam-ics.

Figure 9 shows six frames of isotherms and iso-pleths of oxygen concentration at a convective ve-locity of 2.5 m/s and an ambient pressure of 100 atm.The freestream Reynolds number Re is 31, based onthe initial droplet diameter. Soon after the introduc-tion of the droplet into the hydrogen stream, theow adjusts to form a boundary layer near the sur-face. The gasied oxygen is carried downstreamthrough convection and mass diffusion. The evolu-tion of the temperature eld exhibits features dis-tinct from that of the concentration eld due to thedisparate time scales associated with thermal andmass diffusion processes (i.e., Lewis number 1).The thermal wave penetrates into the droplet inte-rior with a pace faster than does the surroundinghydrogen species. Since the liquid core possesseslarge momentum inertia and moves slower com-pared with the gasied oxygen, at t 0.79 ms, thedroplet (delineated by the dark region in the tem-perature contours) reveals an olive shape, while theoxygen concentration contours deform into a cres-cent shape with the edge bent to the streamwise di-rection. At t 1.08 ms, the subcritical liquid core


9/18


F ig . 10. LOX droplet gasication in supercritical hydrogen ow at p 100 atm. (a) Spherical mode. U 0.2 m/s, t 0.61 ms. (b) Deformation mode. U 1.5 m/s, t 0.61 ms. (c) Stripping mode. U 5 m/s, t 0.17 ms. (d)Breakup mode. U 15 m/s, t 0.17 ms.

disappears, leaving behind a puff of dense oxygenuid which is convected further downstream withincreasing velocity until it reaches the momentumequilibrium with the ambient hydrogen ow.

Figure 10 summarizes the streamline patterns andoxygen concentration contours of the four differentmodes commonly observed in supercritical dropletgasication. The droplet may either remain in aspherical conguration, deform to an olive shape, or

even break up into fragments, depending on the lo-cal ow conditions. Unlike low-pressure cases in which the large shear stress at the gasliquid inter-face induces internal ow circulation in the liquidcore [50], no discernible recirculation takes place inthe droplet interior, regardless of the Reynolds num-ber and deformation mode. This may be attributedto the diminishment of surface tension at supercrit-ical conditions. In addition, the droplet regresses sofast that a uid element in the interphase region may not acquire the time sufcient for establishing aninternal vortical ow before it gasies. The rapid de-formation of the droplet conguration further pre-cludes the existence of stable shear stress in the liq-uid core and consequently obstructs the formationof recirculation.

The spherical mode shown in Fig. 10a typically occurs at very low Reynolds numbers. Although ow separation is encouraged by LOX gasication, no re-circulating eddy is found in the wake behind thedroplet. The vorticity generated is too weak to formany conned eddy. When the ambient velocity in-creases to 1.5 m/s, the droplet deforms into an oliveshape with a recirculating ring attached behind it, asshown in Fig. 10b. Owing to the droplet deformationand gasication, the threshold Reynolds numberabove which the recirculating eddy forms is consid-erably lower than that for a hard sphere. Figure 10cdepicts the ow structure with viscous stripping atan ambient velocity of 5 m/s, showing an oblatedroplet with a stretched vortex ring. The attenededge of the droplet enhances the strength of the re-circulating eddies and as such increases the viscousshear stress dramatically. Consequently, a thin sheetof uid is stripped off from the edge of the dropletand swept toward the outer boundary of the recir-culating eddy. At a very high ambient velocity of 15m/s, droplet breakup takes place, as clearly shownin Fig. 10d. The hydrogen ow penetrates throughthe liquid phase and divides the droplet into two


10/18


parts: the core disk and surrounding ring. The vor-tical structure in the wake region expands substan-tially as a result of the strong shear stress.

The effect of ambient pressure and Reynoldsnumber on droplet lifetime can be correlated with

respect to the reference value at zero Reynoldsnum-ber. The result takes the form

s 1f (14)1.1 0.88s 1 0.17Re ( p )f Re 0 r,O2 where Re and refer to the Reynolds number pr,O 2based on the initial droplet diameter and the re-duced pressure of oxygen, respectively. This corre-lation bears a resemblance to the popular Ranz-and-Marshall correlation [51] for droplet heat-transfercorrection due to convective effect. The Ranz-and-Marshall correlation applies only to low pressure

ows and has a weaker Reynolds number depen-dency.Drag coefcient has been generally adopted as a

dimensionless parameter to measure the drag forceacting on a droplet. Chen and Yuen [52] found thatthe drag coefcient of an evaporating droplet issmaller than that of a non-vaporizing solid sphere atthe same Reynolds number. Several researchers[53,54] numerically analyzed evaporating dropletmotion by solving the NavierStokes equations andproposed the following correlation:

0CDC

(15)D b(1 B)

where denotes the drag coefcient for a hard0CDsphere, and b is a constant which has a value of 0.2for Renksizbuluts model and 0.32 for Chiangs cor-relation. A transfer number B is adopted to accountfor the effect of blowing on momentum transfer tothe droplet. For droplet vaporization at low to mod-erate pressures ( pr 0.5), the Spalding transfernumber is widely used to characterize the vapori-zation rate:

C (T T )p sB (16)Dh v

The enthalpy of vaporization Dh v becomes zero atthe critical point, rendering an innite value for thetransfer number. This deciency may be remediedby introducing a transfer number suited for super-critical droplet vaporization [20,45].

T T cB (17)D T T c l where T l is the instantaneous average temperatureof droplet, and T c the critical mixing temperature.Since BD diverges at T c T l at the end of dropletlifetime, the calculation of drag force was terminated when T c T l becomes less than 1 K, at which the

droplet residual mass is usually less than one thou-sandth of the initial mass. The inuence on the ac-curacy of data reduction is quite limited. Followingthe procedure leading to equation 15, a correlationfor LOX droplet drag coefcient is obtained:

0CDC (18)D b(1 aB )D where a and b are selected to be 0.05 and 1.592( ) 0.7, respectively. The data clusters along the pr,O 2classical drag curve (equation 15) in the low Rey-nolds-number region, but deviates considerably athigh Reynolds numbers (i.e., Re 10). Although ashape factor may be employed to account for thisphenomenon arising from the increased form dragdue to droplet deformation, the difculty of calcu-lating this factor and conducting the associated data

analysis precludes its use in correlating the drag co-efcient herein. Instead, a simple correction factorRe0.3 is incorporated into equation 18 to provide thecompensation. The nal result is given below:

0 0.3C ReDC 0 Re 300 (19)D 0.71.592(P )r,O 2(1 aB )D

Droplet Response to Ambient Flow Oscillation

Although unsteady droplet vaporization and com-bustion have long been recognized as a crucialmechanism for driving combustion instabilities inliquid-fueled propulsion systems [55], they are ex-tremely difcult to measure experimentally. In par-ticular, the measurement of the effect of transverseoscillations on instantaneous evaporation and/orburning rates is quite formidable. The droplet vol-ume dilatation arising from rapid temperature in-crease may overshadow the surface regression as-sociated with vaporization and thereby obscure thedata analysis, especially in the early stage of thedroplet lifetime. Furthermore, conventional sus-pended-droplet experiments may not be feasible inthe presence of gravity due to reduced or diminished

surface tension. In view of these difculties, it is ad- vantageous to rely on theoretical analyses to study the responses of droplet vaporization and combus-tion to ambient ow oscillations. The model is basedon the general approach described in Refs. [20] and[44], but with a periodic pressure oscillation super-imposed in the gas phase. Both cases involving hy-drocarbon droplets in nitrogen [20] and LOX drop-lets in hydrogen [22] are examined carefully. Thepurpose of these studies is to assess the effect of ow oscillation on vaporization process as a function of frequency and amplitude of the imposed oscillationas well as its type.

The most signicant result is the enhanceddroplet vaporization response with increasing ambient meanpressure. Among the various factors contributing to


11/18


this phenomenon, the effect of pressure on enthalpy of vaporization plays a decisive role. At high pres-sures, the enthalpy of vaporization decreases sub-stantially and becomes sensitive to variations of am-bient pressure and temperature. Any small

uctuations in the surrounding gases may consider-ably modify the interfacial thermodynamics andcon-sequently enhance the droplet vaporization re-sponse. This phenomenon is most profound whenthe droplet surface reaches its critical condition at which a rapid amplication of vaporization responsefunction is observed. The enthalpy of vaporizationand related thermophysical properties exhibit ab-normal variations in the vicinity of the critical point,thereby causing a sudden increase in the vaporiza-tion response. On the other hand, the effect of meanpressure on the phase angle of response functionappears quite limited. The phase angle decreasesfrom zero at the low-frequency limit to 180 athigh frequency, a phenomenon which can be easily explained by comparing various time scales associ-ated with uid transport and ambient disturbance.

Supercritical Spray Field Dynamics

Modeling high-pressure, multiphase combustiondynamics in a fully coupled manner poses a variety of challenges. In addition to the classical closureproblems associated with turbulent reacting ows,the situation is compounded with increasing pres-sure due to the introduction of thermodynamicnon-idealities and transport anomalies. A comprehensivediscussion of this subject is given by Oefelein andYang [5658].

Experimental efforts to characterize propellant in- jection, mixing, and combustion processes at near-and super-critical conditions have only recently ledto a better qualitative understanding of the mecha-nisms involved [5965]. The current database, how-ever, is not adequate with respect to quantitativeas-sessments, and theoretical efforts are similarly decient because of a lack of validated theories, dif-

culties associated with numerical robustness, andlimited computational capacities. Depending on theinjector type, uid properties, and ow characteris-tics, two limiting extremes may be deduced [58]. Atsubcritical chamber pressures, injected liquid jetsundergo the classical cascade of processes associated with atomization. For this situation, dynamic forcesand surface tension promote the formation of a het-erogeneous spray that evolves continuously over a wide range of thermophysical regimes. As a conse-quence, spray ames form and are lifted away fromthe injector face in a manner consistent with thecombustion mechanisms exhibited by local dropletclusters. When chamber pressures approach or ex-ceed the critical pressure of a particular propellant,however, injected liquid jets undergo a transcritical

change of state as interfacial uid temperatures riseabove the saturation or critical temperature of thelocal mixture. For this situation, diminished inter-molecular forces promote diffusion-dominated pro-cesses prior to atomization and respective jets va-

porize, forming a continuous uid in the presenceof exceedingly large gradients. Well-mixed diffusionames evolve and are anchored by small but inten-sive recirculation zones generatedby theshear layersimposed by adjacent propellant streams. Theseames produce wakes that extend far downstream.

Figure 11, excerpted from Ref. 58, illustrates thebasic phenomena just described for the case of aLOX/gaseous hydrogen shear-coaxial injector ele-ment. The optical diagnostic studies conducted by Mayer et al. [59,60] and Candel et al. [63,64] dem-onstrated the dramatic effect of pressure on mixingand combustion processes within this type of injec-tor. When the liquid oxygen is injected at low-sub-critical pressures, jet atomization occurs, forming adistinct spray similar to that depicted in Fig. 11a.Ligaments are detached from the jet surface, form-ing spherical droplets, which subsequently break upand vaporize. As the chamber pressure approachesthe thermodynamic critical pressure of oxygen, thenumber of droplets present diminishes, and the sit-uation depicted in Fig. 11b dominates. For this sit-uation, thread-like structures evolve from the liquidcore and diffuse rapidly within the shear layer in-duced by the co-owing jets. At a downstream dis-

tance of approximately 50 diameters, the dense uidcore breaks into lumps, which are of the same or-der of magnitude as the diameter of the LOX jet.Results of optical measurements [5964] reveal thatame attachment occurs instantaneously after igni-tion in the small but intensive recirculation zone thatforms just downstream of the annular post. A well-mixed diffusion ame forms within this region, pro-ducing a wake that separates the oxygen stream fromthe hydrogen-rich outer ow. This wake persists atleast 15 jet diameters downstream. Fig. 12 shows theresultant ame structure and corresponding ow-eld. Here, the injected jet exhibits a pure diffusionmechanism at a pressure of 4.5 MPa, which isslightly below the thermodynamic critical pressureof oxygen, but signicantly above that of hydrogen.Hutt and Cramer [66] have reported similarndingsusing a swirl-coaxial conguration.

The trends outlined in the preceding text coincide with the results of high-pressure liquid nitrogen/he-lium cold-ow tests reported in Refs. [60] and [65].From a qualitative standpoint, these experimentshave demonstrated the effect of mixture propertieson injected liquid jets and the prevalence of onelimit over the other, as illustrated in Fig. 11. Thecutoff associated with these two limits, however, isnot distinct and does not necessarily coincide withthe critical point properties of either propellant.The


12/18


13/18


F ig . 12. Near injector region of a LOX/gaseous hydro-gen shear-coaxial injector. (a) Flame. (b) Correspondingoweld. Oxygen and hydrogen velocities are 30 and 300m/s, respectively. Oxygen and hydrogen injectiontempera-tures are 100 and 300 K. Oxygen jet diameter is 1 mm, andthe chamber pressure is 4.5 MPa [59].

and physicochemical processes within planar mixinglayers. The hydrogen/oxygen system is employed inall cases. The approach follows four fundamentalsteps: (1) development of a general theoreticalframework, (2) specication of detailed property evaluation schemes and consistent closure method-ologies, (3) implementation of an efcient and time-accurate numerical framework, and (4) simulationand analysis of a systematic series of casestudies thatfocus on model performance and accuracy require-

ments.The theoretical framework developed by Oefeleinaccommodates dense multicomponent uids in theEulerian frame and dilute particle dynamics in theLagrangian frame over a wide range of scales in afully coupled manner. This system is obtained by l-tering the small-scale dynamics from the resolvedscales over a dened set of spatial and temporal in-tervals. Beginning with the instantaneous system, l-tering is performed in two stages. First, the inu-ences of SGS turbulence and chemical interactionsare taken into account, yielding the well-known clo-sure problem for turbulent reacting gases. The sec-ond ltering operation incorporates SGS particle in-teractions. The resultant system of Favre-lteredconservation equations of mass, momentum, total

energy, and species concentration for a compressi-ble, chemically reacting uid composed of N speciesare derived.

Modeling SGS phenomena poses stringent nu-merical demands, and robust models are currently

beyond the state of the art. Because of the uncer-tainties associated with the current models and theintensive numerical demands that these modelsplace on computational resources, the current workneglects the effects of SGS scalar-mixing processesand focuses on detailed treatments of thermody-namic nonidealities and transport anomalies. To ac-count for these effects over a wide range of pressureand temperature, the extended corresponding statesprinciple outlined in the section on thermodynamicand transport properties is employed using two dif-ferent equations of state. The 32-term BWR equa-tion of state is used to predict the uid pressure

volumetemperature (PVT) behavior in the vicinity of the critical point. The SRK equation of state isused elsewhere. Having established an analyticalrepresentation for the real mixture PVT behavior,explicit expressions for the enthalpy, Gibbs energy,and constant pressure specic heat are obtained asa function of temperature and pressure using Max- wells relations to derive thermodynamic departurefunctions. Viscosity, thermal conductivity, and effec-tive mass diffusion coefcients are obtained in asimilar manner.

Turbulence quantities are modeled using thelarge-eddy simulation (LES) technique and the SGS

models proposed in Ref. [67]. Spray dynamics aretreated by solving a set of Lagrangian equations of motion and transport for the life histories of a sta-tistically signicant sample of particles. Here, thestochastic separated ow (SSF) methodology devel-oped by Faeth [68] is employed, with transcriticaldroplet dynamics modeled using a set of correlationssummarized previously.

The analyses developed were applied to study transcritical spray dynamics in a manner consistent with the phenomena depicted in Fig. 11a. Effort wasrst devoted to assessing the effect of pressure onthermophysical properties such as kinematic viscos-ity. This quantity is particularly signicant and has adirect impact on the characteristic scales associated with the turbulence eld. For both oxygen and hy-drogen, an increase in pressure from 1 to 100 atmresults in a corresponding reduction in the kinematic viscosity of up to three orders of magnitude. Thisimplies a three order of magnitude increase in thecharacteristic Reynolds number. Based on Kolmo-gorovs universal equilibrium theory, the order of magnitude of the Kolmogorov microscale, denotedhere as mt, and the Taylor microscale, kt, are relatedto the Reynolds number by equations of the form

g kt t3/4

1/2 Re , Re (20)t tl lt tHere, the Reynolds number is dened as Ret


14/18


15/18


F ig . 13. Physical model employed for the analysis of high-pressure hydrogen/oxygen mixing and combustionprocesses.

F ig . 14. Contours of density and temperature in the near-eld region for transcritical mixing [58].

resultant interactions cause disturbances that grow and coalesce immediately downstream of the splitterplate on a scale that is of the same order of magni-tudeas the splitter plate thickness. Combinedeffectsinduce increased unsteadiness with respect to theame-holding mechanism and produce signicantoscillations in the production rates of H2O and OHradicals. The temperature within the recirculationzone also uctuates about the stoichiometric value, with relatively cooler temperatures observed im-mediately downstream.

Conclusions

Fundamentals of high-pressure transport andcombustion processes in contemporary liquid-fueled

propulsion and power-generation systems were dis-cussed. Emphasis was placed on the development of a systematic approach to enhance basic understand-ing of the underlying physiochemical mechanisms.Results from representative studies of droplet va-porization, spray-eld dynamics, and mixing andcombustion processes were presented to lend insightinto the intricate nature of the various phenomenaobserved. In addition to all of the classical issues formultiphase chemically reacting ows, a unique setof challenges arises at high pressures from the intro-duction of thermodynamic nonidealities and trans-port anomalies near the critical point. The situationbecomes even more complex with increasing pres-sure because of an inherent increase in the ow Rey-nolds number. The resulting ow dynamics andtransport processes exhibit characteristics distinct


16/18


F ig . 15. Contours of density and temperature in the near-eld region for supercritical mixing [58].

from those observed at low pressures. In spite of theencouraging, but limited, results obtained to date,the current database is not adequate with respect toquantitative assessments due to the difculties inconducting experimental diagnostics. The theoreti-cal efforts are similarly decient because of a lack of validated theories and limited computational re-sources. Much effort needs to be expended to over-come these obstacles.

Acknowledgments

The author owes a large debt of gratitude to his col-

leagues and former students, especially Joe Oefelein,George Hsiao, Patric Lafon, Norman Lin, and J. S. Shuen.The paper would not have been completed without theirhard work, support, and intellectual stimulation. The work was sponsored partly by the Pennsylvania State University,partly by the U.S. Air Force Ofce of Scientic Research,and partly by the NASA Marshall Space Flight Center.

REFERENCES

1. Spalding, D. B., ARS J. 29:828835 (1959).2. Rosner, D. E., AIAA J. 5:163166 (1967).3. Brzustowski, T. A., Can. J. Chem. Eng. 43:3036

(1965).4. Chervinsky, A., AIAA J. 7:18151817 (1969).

5. Polymeropoulos, C. E., and Peskin, R. L., Combust.Sci. Technol. 5:165174 (1972).6. Manrique, J. A., and Borman, G. L., Int. J. Heat Mass

Transfer 12:10811094 (1969).7. Lazar, R. S., and Faeth, G. M., Proc. Combust. Inst.

13:801811 (1970).8. Canada, G. S., and Faeth, G. M., Proc. Combust. Inst.

15:418428 (1974).9. Rosner, D. E., and Chang, W. S., Combust. Sci. Tech-

nol. 7:145158 (1973).10. Kadota, T., and Hiroyasu, H., Bull. Jpn. Soc. Mech.

Eng. 15151521 (1976).11. Hsieh, K. C., Shuen, J. S., and Yang, V., Combust. Sci.

Technol. 76:111132 (1991).12. Curtis, E. W., and Farrel, P. V., Combust. Flame90:85102 (1992).

13. Jia, H., and Gogos, G., J. Thermophys. Heat Transfer 6:739745 (1992).

14. Delpanque, J. P., and Sirignano, W. A., Int. J. HeatMass Transfer 36:303314 (1993).

15. Jiang, T. L., and Chiang, W. T., Combust. Flame97:1734 (1994).

16. Jiang, T. L., and Chiang, W. T., Combust. Flame97:355362 (1994).

17. Daou, J., Haldenwang, P., and Nicoli, C., Combust.Flame 153169 (1995).

18. Shuen, J. S., Yang, V., and Hsiao, G. C., Combust.Flame 89:299319 (1992).

19. Yang, V., Lin, N. N., and Shuen, J. S., Combust. Sci.Technol. 97:247270 (1994).


17/18


20. Hsiao, G. C., Supercritical Droplet Vaporization andCombustion in Quiescent and Forced-Convective En- vironments, Ph.D. thesis, The Pennsylvania StateUniversity, University Park, PA, 1995.

21. Lafon, P., Modelisation et Simulation Nume rique de

LEvaporation et de la Combustion de Gouttes a HautePression, Ph.D. thesis, a LUniversite DOrleans,1995.

22. Lafon, P., Yang, V., and Habiballah, M., AIAA paper95-2432.

23. Haldenwang, P., Nicoli, C., and Daou, J., Int. J. HeatMass Transfer 39:34533464 (1996).

24. Meng, H., and Yang, V., AIAA paper 98-3537.25. Harstad, K., and Bellan, J., Int. J. Heat Mass Transfer

41:35373550 (1998).26. Harstad, K., and Bellan, J., Int. J. Heat Mass Transfer

41:35513558 (1998).27. Harstad, K., and Bellan, J., Int. J. Heat Mass Transfer

42:961970 (1999).28. Harswtad, K., and Bellan, J., Int. J. Multiphase Flow

26:16751706 (2000).29. Umemura, A., Proc. Combust. Inst. 26:463471

(1986).30. Umemura, A., and Shimada, Y., Proc. Combust. Inst.

26:16211628 (1996).31. Umemura, A., and Shimada, Y., Proc. Combust. Inst.

27:26592665 (1998).32. Givler, S. D., and Abraham, J., Prog. Energy Combust.

Sci. 22:128 (1996).33. Bellan, J., Prog. Energy Combust. Sci. 26:329366

(2000).34. Jacobsen, R. T., and Stewart, R. B. J., J. Phys. Chem.

Ref. Data 2:757922 (1973).35. Prausnitz, J. M., Lichtenthaler, R. N., and de Azevedo,

E. G., Molecular Thermodynamics of Fluid-PhaseEquilibria, Prentice-Hall, Englewood Cliffs, NJ, 1986.

36. Perry, R. H., Gree, D. W., and Maloney, J. O., PerrysChem. Eng. Handbook, McGraw-Hill, New York,1984.

37. Ely, J. F., and Hanley, H. J. M., Ind. Eng. Chem. Fun-damentals 20:323332 (1981).

38. Ely, J. F., and Hanley, H. J. M., Ind. Eng. Chem. Fun-damentals 22:9097 (1983).

39. Sychev, V. V., Vasserman, A. A., Kozlov, A. D., and Spir-idonov, G. A., Property Data Update 1:529536(1987).

40. Ely, J. F., J. Res. Nat. Bur. Stand. 6:597604 (1981).41. Vesovic, V., and Wakeham, W. A., Transport Proper-

ties of Supercritical Fluids and Fluid Mixtures, in Su- percritical Fluid Technology (T. J. Bruno and J. F. Ely,eds.), CRC Press, Boca Raton, FL, 1991, p. 253.

42. Levelt Sengers, J. M. H., Thermodynamics of Solu-tions Near the Solvents CriticalPoint, inSupercriticalFluid Technology (T. J. Bruno and J. F. Ely, eds.),CRCPress, Boca Raton, FL, 1991, p. 25.

43. Takahashi, S., J. Chem. Eng. (Japan)7:417420 (1974).

44. Lafon, P., Yang, V., and Habiballah, M., J. Fluid Mech.,in press (2001).

45. Hsiao, G. C., and Yang, V., AIAA paper 95-0383.46. Yang, V., Habiballah, M., Hulka, J., and Popp, M.

(eds.), Liquid-Propellant Rocket Combustion Devices,

Progress in Aeronautics and Astronautics, AIAA, Washington, DC, 2001.47. Faeth, G. M., Dominics, D. P., Tulpinsky, J. F., and

Olson, D. R., Proc. Combust. Inst. 12:918 (1960).48. Sato, J., Tsue, M., Niwa, M., and Kono, M., Combust.

Flame 82:142150 (1990).49. Sato, J., AIAA paper 93-0813.50. Prakash, S., and Sirignano, W. A., Int. J. Heat Mass

Transfer 23:253268 (1995).51. Ranz, W. E., and Marshall, W. R., Chem. Eng. Proc.

48:141146 (1952).52. Chen, L. W., and Yuen, M. C., Combust. Sci. Technol.

14:147154 (1976).

53. Renksizbulut, M., and Haywood, R. J., Int. J. Multi- phase Flow 14:189202 (1988).

54. Chiang, C. H., Raju, M. S., and Sirignano, W. A., Int. J. Heat Mass Transfer 35:13071324 (1992).

55. Yang, V., and Anderson, W. E. (eds.), Liquid RocketEngine Combustion Instability, Vol. 169, Progress in Aeronautics and Astronautics,AIAA,Washington,DC,1995.

56. Oefelein, J. C., and Yang, V., Prog. Aeronaut. Astro- naut. 171:263304 (1996).

57. Oefelein, J. C., Simulation and Analysis of TurbulentMultiphase Combustion Processes at High Pressures,Ph.D. thesis, The Pennsylvania State University, Uni- versity Park, PA, 1997.

58. Oefelein, J. C., and Yang, V., J. Prop. Power 14:843857 (1998).

59. Mayer, W., and Tamura, H., J. Prop. Power 12:11371147 (1996).

60. Mayer, W. O. H., Schik, A. H. A., Vielle, B., Chauveau,C., Gokalp, I., Talley, D. G., and Woodward, R. D., J.Prop. Power 14:835842 (1998).

61. Mayer, W., Ivancic, B., Schik, A., and Hornung, U.,AIAA paper 98-3685.

62. Ivancic, B., Mayer, W., Kruelle, G., and Brueggemann,D., AIAA paper 99-2211.

63. Candel, S., Herding, G., Snyder, R., Scouaire, P., Ro-lon, C., Vingert, L., Habiballah, M., Grisch, F., Pealat,M., Bouchardy, P., Stepowski, D., Cessou, A., andColin, P., J. Prop. Power 14:826834 (1998).

64. Kendrick, D., Herding, G., Scouaire, P., Rolon, C.,and Candel, S., Combust. Flame 118:327339 (1999).

65. Woodward, R. D., and Talley, D. G., AIAA paper 96-0468.

66. Hutt, J. J., and Cramer, J. M., AIAA paper 96-4266.67. Erlebacher, G., Hussaini, M. Y., Speziale, C. G., and

Zang, T. A., J. Fluid Mech. 238:155185 (1992).68. Faeth, G. M., Prog. Energy Combust. Sci. 13:293345

(1987).


18/18


COMMENTS

Josette Bellan, Jet Propulsion Laboratory, USA. My rstquestion regards your terminology of supercritical vaporization. Since, under supercritical conditions, the latentheat is null, there cannot be vaporization. Please explain.My second question regards the calculation of transportproperties such as the diffusivity and the thermal diffusionfactors for multicomponent mixtures at high pressures. Are you aware of established methodologies for the reliablecal-culation of these transport properties for multicomponentmixtures at high pressures? We should note that a singlespecies fuel burning in air constitutes already at least a ve-component mixture under the conservative assumptionthat combustion is completeand thusits productsarewaterand carbon dioxide.

Authors Reply. For simplicity, supercritical vaporizationin the present paper refers to vaporization of liquid drop-lets in supercritical environments. Calculations of ther-modynamic and transport properties for multicomponentmixtures at high pressures are discussed in great detail in

the paper. In general, thermodynamic and transport prop-erties except mass diffusivity can be treated reasonably wellby means of the corresponding-state principles. Estimationof mass diffusivity remains a challenging task due to thelack of a formal theory or even a theoretically based cor-relation. Only moderate success was achieved by Takahashi(Ref. [43] in this paper) for the limited number of casesstudied.

Yaakov Timnat, Techion, Israel. Could you comment onthe type of equation of state you recommend to use forobtaining the best results?

Authors Reply. As discussed in the paper, the Benedict- Webb-Rubin equation of state in conjunction with the cor-responding-state principles provides the best result of uid p-V -T properties over all the uid thermodynamic states,from compressed liquid to dilute gas.

yang v. modelling

Documents