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Dalton J. E. Harvie

David F. Fletcher

Department of Chemical Engineering,

University of Sydney, NSW, 2006

A Simple Kinetic TheoryTreatment of Volatile Liquid-GasInterfacesKeywords: Boiling, Evaporation, Heat Transfer, Non-Equilibrium, Vaporization

1 Introduction

The Knudsen number  Kn  is defined as the ratio of the meanfree path of a gas,  , to a dimension of an object which interactswith the gas,  x,

Kn

 x . (1)

According to Rohsenow and Choi   1, when fluid velocities aresmall relative to the speed of sound, the dynamics of a gas can beloosely classified into four regimes, the boundaries of which are

determined by characteristic Knudsen numbers.When Kn0.01, the mean free path of a gas is small comparedwith the dimensions of the system, and the gas may be approxi-mated as a continuous medium. This is known as the Continuumregime. At slightly higher Knudsen numbers (0.01Kn0.1),the gas may be regarded as a continuum at several mean free pathsdistance from any adjoining medium, but near any interface, akinetic theory treatment must be used to describe the interactionbetween individual gas molecules and the adjoining surface. Thisregime is known as the Slip regime.

The Transition regime exists at higher Knudsen number levels(0.1Kn3). Here the mean free path of a gas has a lengthcomparable with the dimensions of the system, and collisions be-tween molecules and other molecules and collisions between mol-ecules and the system boundaries assume equal importance indetermining gas behavior. In the Free-Molecule regime (Kn3), the mean free path of the gas is large compared with thedimensions of the system, and collisions between molecules andother molecules occur only infrequently. Flow solutions in thefree-molecule regime are performed using rarefied gas kinetictheory  2– 4.

This study is concerned with determining the behavior of a gasat a volatile liquid-gas interface, within the slip flow regime. Morespecifically, the objective of this note is to provide simple mass,momentum, and thermal energy boundary conditions, which whenused in conjunction with continuum transport equations, are ableto model the molecular behavior of a gas up to a liquid-gas inter-face. For the transfer of tangential momentum and thermal energy,these boundary conditions take the form of effective velocity andtemperature discontinuities, respectively, which occur at a liquid-gas interface.

Mass transfer at a volatile liquid-gas interface has been previ-ously examined by researchers such as Schrage 5, Nabavian andBromley   6, and Collier   7. Heat and momentum transfer at aliquid-solid interface within the Slip flow regime has also beenstudied Loeb 2, Knudsen 4, Patterson 8, Rohsenow and Choi1, and Chapman and Cowling  9, amongst others.

Studies of heat and momentum transfer at a liquid-gas interfaceare less numerous, with the previous works falling into three maincategories. The simplest theories employ the Langmuir assump-

tion   10, whereby the liquid and continuum vapor phases areassumed to be separated by a discrete molecular layer. More com-plex theories involve solving the Boltzmann equation in a regionadjacent to the liquid phase, either by the Maxwell momentmethod  11, or by more direct methods  12–15. More recently,the direct-simulation Monte-Carlo  DSMC  technique of Bird 16has been used to determine the growth rates of droplets in super-saturated gas mixtures  17.

The approach chosen in the present study is essentially that dueto Langmuir   10, but includes the Schrage correction  5   to ac-count for molecular flow towards or away from the liquid surface.Heat and mass transfer using these methods has been previously

studied to good effect by Kang  18 and Young  19 at the surfaceof a droplet, however, similar studies involving flat surfaces donot appear to exist. A tangential momentum transfer analysis us-ing the Langmuir technique at a liquid-gas interface has not pre-viously been presented.

This note is divided into four sections. In the first, a kinetictheory mass balance at the interface is performed. This analysis islargely a summary of the work accomplished Schrage   5, butrepeated here for completeness. This section is followed by anexamination of the tangential momentum and thermal energytransfer that occurs at a liquid-gas interface. Finally, a short dis-cussion of the results and some applications for the theory aregiven.

2 Interface Mass TransferThe theory of condensation and evaporation rates existing at a

liquid-gas interface presented here is largely a summary of work performed by Schrage 5, this work being later reviewed by Col-lier  7.

Figure 1a   illustrates the variables used in this analysis. Themass flux of molecules impacting the liquid surface is   J i  , how-ever, only a proportion of these molecules,  J c , actually condenseinto the liquid. The remainder of molecules,  J r  , rebound from thesurface without entering the liquid. The proportion of moleculeswhich condense upon impact is specified by the AccommodationCoefficient    c  , which is here defined as the probability that animpacting molecule will condense into the liquid upon contactwith the surface. The mass flux of molecules evaporating from thesurface is  J e  , so the net mass flux of liquid evaporation is given

by

 J t  J e J c J e c J i . (2)

By assuming that the molecules which impact the liquidsurface initially have a Maxwellian velocity distribution, andthat the gas velocity directed away from the liquid surface isug  , the flux of molecules impacting the liquid surface is givenby  5,

 J ia  M 2  RP

T o, (3)

where

Contributed by the Heat Transfer Division for publication in the J OURNAL OF

HEAT TRANSFER. Manuscript received by the Heat Transfer Division May 23, 2000;

revision received November 13, 2000. Associate Editor: D. Poulikakos.

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aug   M 2 RT o, (4)and for small values of  a,

a1a . (5)

Note that Eq.  5   is an approximation to the exact solution, validwhen  a0.1  7. Also, the temperature used in these equations,T o , is the average gas temperature in the vicinity of the interface,

which may not be equal to the temperature of the liquid at theinterface,  T  f  .

The velocity of gas evaporating from the liquid surface is re-lated to the net evaporation rate via

u g J t 

 g

 J t  RT o

 M P  , (6)

where   g  is the density of the gas at the interface. Combining Eqs.2,  3, 4,  5, and  6  gives the net evaporation flux rate at thesurface as

 J t 2

2 c J e c  M 2  R PT o . (7)

In order to determine   J e  , we consider a liquid-gas interface

which is in dynamic equilibrium. Under these conditions, the netevaporating mass flux  J t   is zero, the temperature of the gas  T o   isequal to the temperature of the liquid  T  f  , and the pressure at theinterface is equal to the saturation pressure of the liquid corre-sponding to the temperature at the interface. Thus, Eq.  7  can berearranged to yield the dynamic equilibrium evaporation rate as

 J e c  M 2  RPsatT  f 

T  f . (8)

A commonly used assumption in molecular condensation theo-ries, which we will employ here, is that the evaporation mass fluxis determined solely by conditions existing within the liquid at theliquid-gas interface 5,7. Thus, the liquid evaporation mass flux isgiven at all times by Eq.  8, irrespective of whether the interface

is in equilibrium or not, and is a function solely of the temperatureof the liquid at the liquid-gas interface.Note that the pressure within the liquid is not necessarily equal

to the saturation pressure corresponding to the liquidtemperature—the saturation pressure in Eq.  8   has been empiri-cally determined to equate evaporation and condensation massfluxes when the interface is in equilibrium. Indeed, generallywhen condensation or evaporation are occurring at an interface,the interface velocities are much smaller than the speed of soundin either medium, and there is no mass flux induced pressurediscontinuity occurring across the interface.

Combining Eqs.   7   and   8   gives the total evaporating massflux at a liquid-gas interface as

 J t 2 c

2 c  M 

2  R P satT  f T  f  P

T o . (9)

3 Interface Tangential Momentum Transfer

The objective of this section is to determine an effective tan-gential velocity discontinuity, which when applied using con-tinuum equations within the body of a gas, will model the mo-lecular behavior of the gas at a liquid-gas interface. Themagnitude of this discontinuity is determined by equating the

stress imposed on a molecular gas by a volatile liquid surface withthe stress existing at an interface between a volatile liquid and acontinuum gas.

Figure 1b   shows variables used in this analysis. In the dia-gram, the  x  coordinate is directed normal to the interface, and the

 y coordinate tangential to the interface. Velocity in the  y   directionis represented by   v. The tangential velocity of the liquid at theinterface is   v f   , the average tangential velocity of molecules im-pacting the liquid is   v i , and the average tangential velocity of allgas molecules at the interface is   vo  . Thus, that the magnitude of the slip velocity at the interface is   vov f  .

In general, the tangential velocity of molecules impacting theliquid,   v i , is not equal to the average tangential molecular veloc-ity at the interface,   vo , because molecules impacting the liquidexperienced their last collisions at some distance to the right of the interface, and it is from these collisions that they have gained

their tangential velocity. Thus, the tangential velocity of mol-ecules impacting the liquid can be represented as,

v ivoU d v

dx , (10)

where the distance U   is the characteristic distance to the right of the interface from which the impacting molecules gained theirtangential velocity.

In a simple gas, molecules experience their last collision beforeimpacting the interface at an average distance of 2/3   from theliquid. The value of   U   in Eq.   10   is greater than 2/3 however,because more than a single collision is generally required to bringa molecule into equilibrium with its surroundings. A Chapman-Enskog analysis for the magnitude of  U  is given in Chapman andCowling  9, where it is found that for spherical rigid molecules,U 0.998, or approximately unity. For polyatomic molecules, thisvalue represents a reasonable approximation for momentum cal-culations, as the average momentum of molecules tangential to aninterface is not dependent on the internal molecular energy of individual molecules  9.

The average tangential velocity of molecules moving awayfrom the interface is not in general equal to the velocity of theliquid,   v f   , either. This is because molecules which are movingaway from the interface may be evaporating molecules, or alter-natively molecules which are rebounding from the interface with-out condensing. To further complicate the situation, moleculeswhich rebound from the interface may do so in a manner varyingfrom specularly to diffusely, depending on the molecular geom-etry of the interface.

Before examining further the conditions existing at a liquid-gasinterface, it is necessary to examine the assumptions used by pre-vious researchers regarding the tangential stresses existing at asolid-gas interface.

At a solid-gas interface, the stress acting on the solid due to thegas is a result of molecules which impact with the surface, and areaccelerated towards the velocity of that surface as a result of theimpact. Chapman and Cowling   9   and Rohsenow and Choi   1showed that following Newton’s Second Law, the magnitude of this stress is equal to

  xy , s J i v*v iv f , (11)

Fig. 1 Variable definition at the liquid-gas interface. Case   „a …shows variables used in the mass transfer analysis, case   „b …shows those used in the momentum transfer analysis, andcase „c …  shows those used in the heat transfer analysis.

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where they defined the Specular Reflection Coefficient for a solid-

gas interface,   v* , as the average proportion of the initial tangen-

tial momentum of a molecule which is transferred to the solidduring an impact with the interface.

Chapman and Cowling   9   and Rohsenow and Choi   1   bothassumed that the stress acting on a gas at a solid-gas interfacewould be equal in magnitude to the stress acting on the solid atthat interface. This assumption was reasonable as no net masstransfer occurs at a solid-gas interface. Employing this assump-tion, and noting that the mass flux of molecules impacting a solidinterface is equal to the mass flux of molecules rebounding from a

solid interface, Eq.  11   is equivalent to

  xy ,g J i v*v iv f , (12)

where     xy ,g   is the magnitude of the stress acting on a gas at asolid-gas interface.

Equation  12  is interesting in that it can be interpreted as mol-ecules that move away from a solid-gas interface come into equi-librium with their surroundings at a location where the averagetangential velocity of the gas is   v i  . We will apply this concept,which is a direct result of assuming that no stress discontinuityoccurs at a solid-gas interface, to molecules which move awayfrom a liquid-gas interface in our present analysis.

The stress acting on a gas near a liquid-gas interface,    xy ,v , iscomposed of a component due to molecules evaporating from theliquid, and a component due to molecules rebounding from the

interface without condensing, but after having transferred a pro-portion of their velocity to the liquid. Using the concepts of theprevious discussion, the magnitude of this stress can be repre-sented by

  xy ,g J ev iv f  J r  vv iv f . (13)

This stress is directed against the motion of the gas. Note that wehave now defined the Specular Reflection Coefficient for a vola-tile liquid-gas interface,   

v  , as the average proportion of the ini-

tial tangential momentum of a molecule which is transferred to theliquid, when the molecule rebounds from the interface. This coef-ficient is similar to the Specular Reflection Coefficient for non-volatile interfaces in that it represents the manner in which mol-ecules rebound from an interface.

The stress on a liquid at a liquid-gas interface,     xy , f   , is also

composed of two components. One component results from mol-ecules which condense into the liquid upon impact, and anotherfrom molecules which rebound from the interface without con-densing. In a similar fashion to the above, this stress can be rep-resented by

  xy , f  J cv iv f  J r  vv iv f , (14)

where we have again used the Specular Reflection Coefficient fora volatile liquid-gas interface,   

v .

Equation  14  will not be used further in this analysis, howevera comparison against Eq.   10   highlights some peculiarities of aliquid-gas interface within the slip regime. Firstly, if no conden-sation or evaporation occurs at the interface, then the tangentialstress on the liquid is equal to the tangential stress on the gas. Thisis as expected at a solid-gas interface. Also, if the rates of evapo-

ration and condensation are equal, as in an interface which is indynamic equilibrium, again the stress applied to the gas phase isequal to the stress applied to the liquid phase. However, if thecondensation and evaporation rates are different, then a stress dis-continuity occurs at the interface, and the stresses applied to eachphase are different. This phenomenon results from forces that aregenerated by molecules which must be accelerated to the velocityof their surroundings, and highlights one of the interesting differ-ences between volatile and non-volatile interfaces.

In this study we are only concerned with the stress imposed onthe gas at the liquid-gas interface. Using equations  2,  10, and13, noting that the rate of molecules rebounding from the inter-face is given by

 J r  J i J c , (15)

and defining the ratio of evaporation and condensation massfluxes as

  J e

 J c, (16)

the shear stress imposed on the gas is given by

  xy ,g J i  c v1 c voU  d vdxv f   . (17)To determine the magnitude of the slip velocity at the interface,

we equate the value for the actual shear stress on the gas at theinterface, given by Eq.  17, to the shear stress existing at a planein a Newtonian fluid,

  xy d v

dx . (18)

This yields

 d v

dx J i  c v1 c voU  d vdxv f   . (19)

The mass flux of molecules impacting the interface is given byEq.  3   of the previous section. Using Eqs.  4,  5, and  6, andexpressing the flux in terms of kinetic theory variables 9, we find

 J i1

4  c̄ 

 J t 

2 

 c¯ 

42 c 1  , (20)

where  c̄   is the mean molecular speed of the gas. Also, the coeffi-cient of viscosity can be expressed in terms of kinetic theoryvariables  9,

   1

2U  c̄ . (21)

Substituting Eqs.   20   and   21   into Eq.   19   and rearranginggives the interface slip velocity as

vov f U 11 v1 c  c v1 c  d v

dx  . (22)

As previously discussed, the Chapman-Enskog value of   U 1.0,

can be employed for the majority of gases. Note that if   c0, Eq.22 reduces to the velocity discontinuity existing at a non-volatilesolid-gas interface  9.

4 Interface Heat Transfer

In a similar manner to the above momentum analysis, to modelthe heat transfer occurring at a liquid-gas interface using con-tinuum equations, we assume that there exists an effective tem-perature discontinuity across the interface. To determine the mag-nitude of this discontinuity, we perform an energy balance on athin control volume which surrounds the liquid interface, usingboth kinetic theory and continuum methods.

Figure 1c  shows temperatures at a liquid-gas interface. In an

analogous manner to the velocity slip calculation, we define thetemperature of the liquid at the interface as  T  f   , the average tem-perature of molecules impacting the liquid as  T i , and the averagetemperature of all gas molecules at the interface as   T o . The ve-locity of the gas moving away from the interface is as previouslydefined, u g , the velocity of liquid moving towards the interface isu f  , and the net flux of heat into the liquid is denoted by  q  f  .

The average temperature of molecules impacting the solid isdetermined using an analogous equation to the impacting tangen-tial velocity Eq.  10. Thus,

T iT oU dT 

dx , (23)

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T oT  f U    2 c1  t  t  c1  1

 d T dx

  . (38)

As previously discussed, for most polyatomic molecules the Eu-

cken approximation of  U 9 5/4 can be employed. Note thatif there is no net mass transfer, then   1, and Eq.  38  yields thesame magnitude of temperature discontinuity that exists at a non-volatile solid-gas interface  9.

5 Model Discussion and Applications

Equations  9, 22, and 38  completely specify conditions at a

volatile liquid-gas interface. Following the restrictions placed onEq.  5, these equations are valid when

 J t 0.12 M  RT o

P. (39)

Equations  9   and  38   are coupled through the variables   ,   T o ,and  T  f  , so must be solved simultaneously. Equation  22 requires , so relies on a solution to Eq.  9.

Figure 2 shows the non-dimensional velocity and temperaturediscontinuities specified by Eqs.  22  and  38, plotted as a func-tion of the evaporation to condensation ratio,  . For this figure wehave assumed   c v t 1 and   5/3. As shown, the magni-tudes of the velocity and temperature discontinuities move in op-posite directions as      is varied. The velocity discontinuity de-creases with increasing   as an increased movement of moleculesfrom the liquid into the gas causes the tangential stress resistingthe movement of the gas to increase. The temperature discontinu-ity increases with increasing   as a higher    requires a higher heatflux, and consequently a steeper temperature gradient within thegas adjacent to the interface. Thus, the higher the  , the higher thetemperature of molecules impacting the liquid-gas interface is,and the higher the temperature discontinuity.

We have used a variation of this model in the analysis of vola-tile liquid droplet impacts with heated surfaces and shown thatnon-continuum effects become important under certain conditions20,21. There are other areas of heat transfer where these effectsare know to be important. One of these is in the collapse of vaporlayer films surrounding melt droplets in the triggering stage of vapor explosions 22. Analysis dating back to the 1980’s in threedifferent laboratories around the world showed that in determining

whether the vapor film collapsed or not following a triggeringevent, it was essential that the molecular behavior of the volatileinterface was modeled 23–25. However, when the mathematicaltreatments used in these studies were examined in detail it wasnoted that velocity slip was not included in some cases, and thatthere were differences between the thermal energy treatmentsused in all of them. This was the primary motivation for the firstprinciples treatment presented here. Another application area for

this theory is in the cooling of micro-chips. Although only pre-liminary studies have been conducted in this field, modeling cool-ant boiling in the narrow passages contained within these deviceswill require the examined non-continuum interface effects to beconsidered.

Finally, we note that there is considerable debate as to the mag-nitude of the Accommodation and Specular Reflection coefficientswhich have been used in this analysis. Rohsenow and Choi 1 listthe Specular Reflection coefficients for a number of non-volatilegas and surface combinations. Generally these coefficients areclose to unity, so   

v1 is probably an acceptable assumption for

the majority of liquid-gas combinations.There is some experimental data available on various volatileAccommodation coefficients  6,25–27, but consistent results be-tween studies are yet to be established. Also, the definitions usedfor these coefficients are not always the same between publica-tions. As previously noted,   t  should be larger than   c , a result of energy transfer between the liquid and gas phases from molecularrebounds as well as from molecular condensation. Until more ex-perimental investigations are conducted however, an assumptionof unity for both coefficients serves as a first approximation.

Nomenclature

c̄    mean molecular speed  m/sc

v   constant volume specific heat  J/kg.K

c p    constant pressure specific heat  J/kg.K E     energy  J/m2 f    ratio of molecular distance coefficients

 H     enthalpy  J/kg J     mass flux  kg/m2sk    thermal conductivity  W/mK

Kn    Knudsen number M    molecular weight  kg/kmol

P     pressure  Paq     energy flux  J/m2s

 R    Universal Gas Constant  J/kmol.KT     temperature  Ku    velocity normal to the interface  m/s

U     molecular tangential momentum distance coefficient

U     molecular thermal energy distance coefficientU     internal energy  J/kgv    velocity tangential to the interface  m/s x    coordinate normal to the interface  m y    coordinate tangential to the interface  m     ratio of specific heats     mean free path  m     dynamic viscosity  kg/ms      density  kg/m3

 c     Accommodation Coefficient r     Thermal Accommodation Coefficient for rebounding

molecules t     Thermal Accommodation Coefficient v    Specular Reflection Coefficient xy     tangential stress at interface  N/m

2

    ratio of evaporating to condensing fluxesSubscripts

c     condensinge     evaporating f     liquidg     gasi     impacting

o     average for gas at interfacer     reboundings     solid

sat     saturationt    net evaporating

Fig. 2 Non-dimensional velocity and temperature discontinui-ties at a volatile liquid-gas interface, as a function of the non-dimensional evaporation rate, as given by Eqs.   „22…   and   „38…,respectively

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