SCATTERING KERNELS FOR THE BOUNDARY CONDITIONS OF THE BOLTZMANN EQUATION ON THE MOVING BOUNDARY OF TWO-PHASE SYSTEMS* Nicola Bellomo**, Roberto Monaco**

SOMMARIO: 1l presente lavoro studia il problema della condizione al contorno per l'equazione di Boltzmann sulla frontiera mobile fra la fase liquida e quella di vapore di un sistema bifasico in condizioni di disequilibrio termodinamico. St propone, sulla base dei metodi matematici della teoria cinetica e della teoria degli escattering kernels~ e sulla base dei risultati precedenti di una teoria statistica di transizione di fase, una teoria fisico-matematica per lo studio del problema. In termini matematici si ottiene un operatore che correla la funzione di distribuztone delle molecole che lasciano la superficie liquida alia funzione di distribuzione delle molecole che arrivano sulla superficie. Tale operatore rappresenta la formulaztone della condizione al contorno. SUMMARY: This paper deals with the problem o f the boundary conditions for the Boltzmann equation on the moving boundary between a liquid and a vapour phase with condensation and vaporization in non-eq'uilibrium conditions. A physical theory is proposed on the basis o f the mathematical theory o f the scattering kernels, o f the kinetic theory and o f the statistical theory o f phase-transition. The mathematical result o f this theory consists in a set o f equations correlating, by means o f integral operators, the distribution function o f the molecules leaving the liquid surface to the one o f the molecules arriving on the surface o f the liquid phase.

INTRODUCTION. This paper deals with the problem of the boundary conditions (for short b.c.) at the free-boundary in a one-dimensional liquid-vapour system. In fact, as is known [1, 2], in kinetic theory, a correct statement of the b.c. is necessary in order to apply the Boltzmann Equation (for short B.E.) for the study o f the diffusion into the vapour medium, see the discussion in ref. [31. In particular, we should like to emphasize that even if the known analytical and numerical mathematical methods dealing with the B.E., see ref. [2, 4, 5], have reached a high

* This work has been realized within the sctivities of the Italian Council for the Research, Gmppo Nazionale per la Fisica Matematica, and presented at the Euromech Colloquium 86 on the <
standard of sophistication, the particular problem of the b.c. for moving boundaries with phase transition still needs to be correctly treated. In fact, because of the phase-transition occurring at the interface, the complete physical-mathematical analysis must include both the kinetics of the interaction between molecules and liquid surface and the mechanics of the vaporization/condensation process. The above-described problem has been studied by Lang and Kuscer [6] with a formulation based on the theory of the scattering kernels, their work is however limited to a phenomenological description of the phenomenon without quantitative results, and by Bellomo [7, 3] with a formulation based on a phase-transition model which assumes that the distribution function of the molecules leaving the. surface is maxwellian with drift velocity and correlates the said drift velocity to the vaporization-condensation coefficients evaluated by means of the model. More recently Bellomo et alli [8] have proposed a rigorous theory for the liquid-vapour phase transition, which gives, as main result, the vaporization and condensation functions (the ratio between vaporizing/condensing and impinging unit number fluxes; these ratios are, according to the theory, functions of the temperature of the liquid surface). Let us comment, with regard to this point that some authors define these ratios with the term <~and assume, a priori, them to be constant quantities for a given liquid-vapour system. On the other hand, the experimental results are close, qualitatively and quantitatively, with the theory [8], which states that the said ratios are functions, depending also on the properties of the liquid-vapour system, which increase with the temperature of the surface of the liquid and assume different values for equilbrium and non-equilbrium conditions. In this paper we shall propose a theory based on the results of the quoted paper [8] and on the theory of the scattering kernels proposed by Cercignani and Lampis.[9, 10] for half-space systems with solid boundary. The said theory will be modified in order to include the more general problem of the b.c. at moving interfaces with phase-transitiun. This work is based on the mathematical methods of kinetic theory [1 ] and of stochastic calculus [ 11 ] as well as on the known physical theories of two-phase systems [12].

MATHEMATICAL DESCRIPTION OF THE PROBLEM. Let us take into account a one-dimensional liquid-vapour 127

In order to deal with the problem mathematically described in the preceding section, we shall propose a general physical theory based to some axioms, which express, by means of the equations of the statistical mechanics, some phenomenologieally known physical results and which lead to a set of algebrical and integral equations (on fi and fr) describing the mechanics of the molecular interaction and of the phase transition. a) The mechanics o f the interaction o f the vapour molecule with the surface o f the liquid is described by the scattering kernel proposed by Cercignani and Lampis [9, 10]. Namely:

z

~37

IUI

U2: w

It)

Ut=(U22+ u3j'2'I/2

R(U -~ V; %; a t, k]m, T) =

2[~n at(2 _ at)l- 1 lr[2(k/m)T]ll2 V1 •

(3)

exp

2(k/m)T

/

Fig. 1

[V,--(l --at)Ut ]2

expl system, see fig. II (z zw: vapour phase) with given profile of the temperature in the liquid phase. In particular: i) T = Tt ( t ; z ) temperature of the liquid surface. ii) g(t)=OT[az)z= z normal gradient o f T, at the surface, on the liquid-siaffe, The determination of the boundary conditions at the interface on the vapour side consists in finding: a) The distribution function o f the vapour molecules f(V, t; z w) leaving the surface o f the liquid. b) The time evolution of the displacement of the moving boundary: t

zw(t) = zw(0) +

i

Zw dt

(1)

f3 is a continuous, measurable in the sense of Lebesgue, application from R3®R+®R+ into R+;V = V I + V 2+ + V3 (see fig. 1). If the b.c. are correctly stated and the initial condition f ( V , z ; t = 0) is given, then the B.E., or a model equation of the B.E., can be applied to evaluated, for Vt > 0 and Vz > z w, the distribution function f.

I

I"

at(2--at)(k/m)T 1"

.1012"(I ;.2(k~ l-~")~12u~v~ b) The molecules which after the ~nteraction have a velocity V > Vc are scattered with a distribution function fs deduced by the scattering kernel o f eq. (3), and the molecules with V ~ Vc condense. c) The distribution function o f the molecules which vaporize from the liquid phase is maxwellian at the temperature Tv (in non-equilibrium conditions Tv 4: T). d) The phase transition liquid-vapour is described by the axioms o f the physical theory proposed in [8] and in particular: the mean energy o f the condensing molecules equals the energy o f the liquid phase, and in non-equilibrium conditions the vaporizing condensing functions, respectively qo~ and ~o~ have a symmetrical behaviour with respect to the equilibrium function ~oe. That is:

~e(T) = [exp (y2) _ l --y2]/exp (y2);

(4)

y ( T ) = VJT)/[2(g/m)T] v2

_3 __6 LCT)-2Ck-m)TdfH(IT)I 2 d

--

=

--I

THEORY.

Let us indicate separately the contribute, to the distribution function f, of the molecules arriving and leaving the surface of the liquid phase: f = fl(U) + fr(V) fl

128

(5)

(2)

= distribution function of the molecules which will interact with the surface (U 1 > 0). ft includes the part of the vapour molecules which after the interaction will condense, = distribution function of the molecules leaving the surface (Vl > 0) after interaction or vaporization.

=

/:

r/4 exp (-- r/2) dr/

= V/[2(k/m)T] 1/2.

Ifo"

~/2 exp (-- r/2) dr/ (6)

In eq. (3) a n and a t have the physical meaning of accomodation coefficients respectively of the energy normal to the surface and of the tangential momentum for each molecule and I 0 is the modif. Bessel function of the first kind, see [9, I0]. In eqs. (4, 5, 6) d is the number of degreesof freedom of each molecule, L is the latent heat of vaporization per MECCANICA

unit mass, N + and N - are, respectively, the vaporizing and condensing unit number fluxes and N l is the unit number flux of the molecules impinging on the surface of the liquid. In equilibrium conditions: N ÷ = N-;

,g; = ~aa- = 9.

and according to axioms (a, b):

4 =f(I uz I/I ~1)" e .

(l/(2k/mT)3/2)4 • dU

(13)

(7)

The discussion of the uniqueness of the solutions of eq. (5) and of the qualitative behaviour of the functions appearing in the set of eqs. (4, 5, 6) can be found in paper [8] and in particular: - re is a strictly monotone application from B(T)= = ]Ts, T ] into 10, I ] - y is a strictly monotone application from B(T) into

10, 04 T = Tc ~* ~oe = 1, y = ,o(gas-like situation)

where:

V> Vc~, P= R(U-* V;%,~ t) on the other hand f * , according to axiom (c), can be written as follows:

f+ = (n+fir3/2)exp {-- V 2/(2k/m Tv)}

(14)

n+(2k/mTv)1t2 2v~

(15)

= (2%-~-).

-

Tc is the critical temperature and T is the solidification temperature.

fu (U1/(2k/mT)312)4dU ~>o

ANALYSIS. The physical theory proposed in the preceding section can be utilized in order to write an equation, correlating fr to 4 ' of the type:

f = F(4;%;o~,, b)

=~

{vzl(2klmT)}(IU1I1[ VII)"

(9)

Jv<% •R ' 4 d U d V l f V

(17)

In particular eq. (15) express the equality of mass flux according to eq. (14) and axiom (d), and equations (16, 17) are known results of kinetic theory. Consequently, after some manipulations, and setting eqs. ( 1 3 , . . . , 17) into eq. (12), the integral operator, deemed by eq. (8), can be written as follows:

f, = F(4; %, % b)

(18)

=

J

(10)

and consequently the non-equilibrium condensation function:

fv

{ VIt(2k/mT) 3/2}-

(11)

o --I

"('Ull/lgl')RfiidUdV l : U

UI~dU 1

t>0 The distribution function ~ .of the molecules leaving the surface can be written as superposition of the distribution function fs of the scattered molecules and o f the distribution function/* of the vaporized ones:

SEPTEMBER 1978

~fcud(2k/mT,)3n)"4 dU.

(16)

qUI'/[VI')R'4dUdV 1

= ~(T;f/, %, a i, b)

I, = 4 +:+



f_:(U -- (U1))2 (1/(2k/rnTv)312)4 • dU

=l(I u~I/I ~l)(ll(2k/mT)3t2) "P "4dU +

--1

~pd-(~ ; f/, %, ott, b) =

,

(8)

where F is an integral operator on 4 and b is a set of function of T. In the next section we shall write, according to the physical axioms proposed in the preceding section, two functional equations which let the evaluation of the coefficients % and a t according to the physical conditions of the whole system. As a first step, let us evaluate, according to axiom (d), the condensation velocity Vc by means o f the following equation:

H(T)

2 (k/m)Tv =

(12)

,2klmT~2

2~,

U~ "4 dU -l>0

-- fv, > 0 ( V J ( 2 k / m T ) 3 / 2 ) ( I U I I / I V I I ) ' R ' 4 d U d V I x x exp {--

V2/(2k/mTv)}

Tv being given by eq. (16). Moreover knowing ~a- and r e allows the determination of the law of evolution of the moving boundary. In fact, according to axiom (d), the difference ( ~ - - ¢ - ) can be written as follows:

(¢2-¢;) = 2(~,-~;)

(19)

see also [6]. Consequently:

zw(t ) = zw(O) +

(20)

129

]

aT JO

"

dzw

z = Zw(t) ~, h ~ = L(T) "~t aZ zw

~>0

+ q(t)

df Therefore, if the set (c~n, ctt) is given, the problem of the boundary conditions is def'med by eqs. (16, 17, 18). Such a set depends on the physical state of the whole system and can be detemfined, according to our theory, in the sense which will be defined in the next section together with the mathematical description of the general problem of the process of heat and mass transfer in the two-phase system, the equations of continuum mechanics being applied in the liquid phase and the ones of kinetic theory in the vapour phase. Limiting a preliminar analysis to the physical theory proposed in the preceding section, let us point out the following points: 1) The formulation o f the boundary conditions on two-phase systems has been proposed, by means o f the theory o f the scattering kernels, for the first time by Lang and Kuscher in the above-quoted paper [6]. In our work we indicate how the theory o f the scattering kernel must match the theory o f phase transition. 2) The scattering kernel proposed by Cercignani and Lampis [9] can be suitably utilized for the formulation o f the boundary.conditions i f the limitation o f the condensation velocity Vc is applied to the scattered molecules. V can be evaluated with the kinetic theory o f paper [8]. 3) The vaporizating molecules are not maxwellian at the temperature T o f the surface o f the liquid phase (also according to Lang and Kuscher [6]). On the other hand a maxwellian distribution function at the temperature Tv ~ T is assumed in first approximation. This point should however, in our opinion, furtherly investigated. HEAT TRANSFER PROCESS AND EVALUATION OF THE ACCOMODATION COEFFICIENTS. A mathematical description of the heat transfer process in the two-phase system when the equations of the continuum mechanics are applied in the liquid phase and the ones of kinetic theory in the vapour phase is necessary both to def'me the. general problem of the boundary conditions and to propose a correct methodology to evaluate the accomodation coefficients otn and ~ . In this sense, the formulation of the classical Stefan problem can be proposed as consisting in:

finding two functions." T(z, t) : t > 0, z E [0, z w ]

and

f ( z , V, t) : t > 0, z > z w

Z > zw(t)

~

(21)

= Q(f x f)

dt

see also refs. [13, 14]. In eq. (21), h is the diffusion coefficient, L the latent heat of vaporization, q the heat flux from the surface of the liquid towards the vapour phase, and Q the integral operator of the Boltzmann equation [ 1]. Therefore, considering that the solution of the Stefan's problem in the liquid phase, z E[0, zw] , gives T(Zw, t ) and g(t), the problem of the boundary conditions can be formulated, still in agreement with the statements (i, ii) and (a, b) of the second section of this paper, as consisting in:

finding the function f ( z w, V, t) and the location Zw(t) o f the moving boundary at given zw(O) and given set T(z w, t), g(t) o f functions corresponding to a given set U. The said function f can be determined by means of eqs. (16, 17, 18) of the preceding section andz w by eqs. (16, 17, t9, 20). In these equations the accomodation coefficients can be evaluated by the eq. of the energy flux and by the eq. stating that the mean energy of the scattered molecules equals the energy corresponding to the Tv-temperature. Respectively:

q(t)=hg(t)--L(T)

dzW =

~I

- ~ m V 2.

(22)

~>0 ( d / 3 ) • U]

( 2 - k l m - Tv)3t2 [i dU--

--

fv, 1 ~

+ ( d . k12)

(dl3)E(IU~IIIV~I)R

m V 2

>o 2

.f/dUdV+

(2k/mT)312(2k/mTv) 3/2

fu (U~l(2klmTv)312)(~o, --~T) ~ "fi Tv

dU

~>0

3

2(klm)T~ =

L I ->V

2

V2(I U~ III ~1)"

(23)

¢

• (ll(2klmT)m)(ll(2klmT~)312)

• R. f~ dU dV.

Eqs. (22, 23) let the evaluation of the accomodation coefficients a n and ~ . And in particular, the equilibrium conditions are verified, namely:

which at given set o f functions U = (To], T02,f0): Tol = T(z;t =O),

To2= T ( z = O , t ) : z E [ O , Zw];

fo = f ( z , V; t = 0) : z > z w ;

°÷1

= g(t)

Z =Z w

satisfy the following set o f differential equations: z E [ O , Zw] =* ~ =h Ot 130

" az 2 '

h = h T,

g=O,~o~=so~=~o e , T =

T=~n=et t= 1,(dzJdt)=O

(24)

CONCLUSIONS. In this paper we have proposed a physical theory describing with the equations of the statistical mechanics the heat and mass transfer process, at a molecular scale, on a moving boundary between a liquid phase and a vapour phase. The

MECCANICA

mathematical elaboration of the said theory gives the complete formulation of the boundary conditions, in kinetic theory, without the uncertainess of undetermined parameters. Thus our work shows (see the last section) that for a liquid-vapour system the solution of a problem, still unsolved for solid boundaries, can be achieved. The afore-mentioned theory consists in a set of axioms deduced by the theory of the scattering kernels and by a kinetic theory of phase-transition proposed in a preceding paper and compared in that paper with the experimental results of other authors. If the approach is considered too heavy on a mathematical-numerical point of view, then simplified models, like the one of ref. [7], or simplifications of this theory can be applied.

On the other hand we would like to emphasize that the assumption of maxwellian molecules from the liquid surface, at the temperature of the liquid phase, already rejected by Lang and Kuscher, should be removed in the study of practical problems of condensation-vaporization by kinetic theory (see in particular the recent work by Matsushita [15], who is extremely accurate in the analysis of the diffusion process in spite of the fact that maxwellian molecules are assumed for the boundary conditions). From this point of view, it should be interesting to investigate the practical mathematical implications and the difference of quantitative results involved by a more correct statement of the boundary conditions.

Received: 5 May 1977.

REFERENCES. It] CERCIGNANI C., Mathematical Methods in Kinetic Theory, Hanum, New York, 1969. [2] CERCIGNANIC., Theory and Application of the Boltzmann Equation, Scottish Press, Edimburgh,London, 1975. [3] BELLOMO N., Physical Mathematical Problems in the kinetic Theory of Vaporization and Condensation, l'Aerotecnica, J. of AIDAA, 1-2, 13, 1976. [4] LOYALKAS. et alii, Some NumericalResultsfor the BGK Model: Thermal Creep and Viscous Slip Problems with Arbitrary Accomodation at the Surface, The Physics of Fluids, 18, 1094, 1975. [5] GAJESKllP. et alii, The VapourPhase Phenomena in Nonsteady Condensation Process, in Rar. Gas Dynamics, Ed. M. Becket and M. Fiebig,DFVLR,Porz-Wahn,FI3, 1974. [6] LANG H. and KUSCHER1., Reciprocity and Phenomenological Coefficients in Liquid-Vapour Phase-change, in Rar. Gas Dynamics, Ed. M. Becket and M. Fiebig, DFVLR, Porz-Wahn, FI2, 1974. [7] BELLOMO N., A physical-mathematical model describing the interface and transport properties in a liquid-vapour system in nonequilibrium conditions, Meccanica, J. of AIMETA, 2, 71, 1976.

[8] BELLOMO14. et afii, A new statistical kinetic theory of phase-transition in liquid-vapour systems, Mech. Research Comm., 3, 417, 1976. [9] CERCIGNANIC. and LAMPISM., Kinetic models for gas-surface interactions, Transp. Theory and StatisticalPhys., 1,101, 1971. [10] CERCIGNANIC., Scattering kernels for Gas-surface interaction, Transp. Theory and StatisticalPhys., 2, 27, 1972. [11 ] MdSHANEE., Stochastic Calculus and Stochastic Models, Academic Press, N. Y., 1974. [12] LEVlCH B., Theoretical Physics, North Holland, Amsterdam, 1971. [13] DUVAUTG., The solution era two-phase Stephan problem by a variational inequality, in Moving Boundary Problems in Heat Flow and Diffusion, Edited by J. Ockendon and W. Hodgkins, ClarendonPress, Oxford, 173, 1975. [14] PRIMICERIOM., Problemi al contorno libero per l'equazione della diffusione, Rend. Matematico Univ. e Politecnicodi Torino, 32, 183, 1974. [15] MATSUSHITA T., Study of Evaporation and Condensation Problems by Kinetic Theory, Report No. 541 of the Inst. of Space and Aeron. Science,Tokio, 1976.

DISCUSSION.

f

I. KUSCHER*. The theory of evaporation and condensation presented by the authors should not be called a physical theory. Rather, it represents a purely mathematical model, containing several arbitrary assumptions With no physical foundation. It is true that such models can still be very useful as long as a physical theory is lacking. (The Cercignani-Lampis model of surface scattering is a good example.) However, one usually requests that the model must not contradict those theoretical aspects that are already known with certainty. The proposed model fails in this respect since it violates the law of reciprocity (or the principle of detailed balance), which follows directly from the basic laws of physics. Let me explain this in detail, by aid of the notation from a paper by H. Lang and myself (RGD, 1974). Tile relation of the outgoing and ingoing distributions of vapour molecules near the liquid surface has the form

* Univerzav Ljubljan,Fakulteta za naravoslovjein tehnologijo. SEPTEMBER 1978

f - ( v ' ) P(v' -* v) d3v '.

1

According to reciprocity, the scattering kernel P (proportional to the R of the authors) and the distribution re(V) of the evaporating molecules must obey the following two relations:

I v~l exp

(-mv'2/2gro) e(v'-+v) =

(1)

= vxexp ( - - m v 2 / 2 k T o ) P ( - - v - + - - v ' ) ,

~(v) = no(r o)

(2)

exp(--mv2/2kTo)[1--fP(--v-+v')d3v

t.

,,+

The authors take P(v'-+ v) proportionale to the Cercignani-Lampis kernel, however with the modification that P = 0 whenever v < vc. By this additional assumption relation (1) is violated, unless one would postulate that P 131

also vanishes for v' ~ ve. After the last amendment is incorporated in the proposed model, another discrepancy shows up. According to relation (2), fe should have a discontinuity at v = i-. Contrary to this, the authors assume that fe is smooth, namely equal to some Maxwellian. Thus I see no way by which one could save the model from contradictions. Violation o f reciprocity is almost tantamount to violation

of the second law of thermodynamics. Whether or not a model with such a deficiency is fit for publication is, of course, a matter of taste. My opinion in this respect is negative, so that I cannot reccomend publication of the paper. (I also refrain from analyzing further details) I feel sorry that I have to say so, since the problem tackled by the authors is important and unsolved so far, so that any real advance should have great merit.

AUTHORS' CLOSURE. The reciprocity law for gas-surface interaction has been derived, independently, by Cercignani & Lampis [I] and by Kuscer [2] under the following basic axioms: -Time-reversal i n v a r i a n c e - (see ref. [2], page 229, line 7) - Validity of Liouville's theorem - (see ref. [2], page 229, line 12) (]). Afterwords that law has been extended by Lang & Kuscher [3 ] to gas-liquid systems under these assumptions: microscopic reversibility and thermal equilibrium of the liquid - (see ref. [3], page F. 12 - 4 , line 3). Further studies are due to Kuscher [4], still under the same -

kind of assumptions. Our paper deals with the classical Stefan problem with time-dependent, non-equilibrium conditions, where all the above mentiofied assumptions are no-longer valid. Let us quote the following, very simple, counter example: - a particle which, after interaction from the vapour phase with the liquid surface, condense into the liquid phase, does not vaporize unless heat subministration occurs (latent heat Q) The validity of the Liouville's theorem is limited to equilibrium reversible processes.

o f vaporization). Moreover, the particle loses its identificability. The authors' opinion can be, consequently summarized as follows:

a) Reciprocity can be applied only to particles which caming from the vapour impinge the surface o f the liquid and are scattered without condensation. b) Reciprocity cannot be applied, in the known form, to the vaporizing flux. In our paper point (a) has been considered. In fact P(U -~ V) obey the reciprocity law and P(-- V -~ -- U) is certainly equal to zero for V ~ Vc, in fact the path U ~ V (V < Vc) is not a reversible one. With regard to point (b), see the discussion proposed on point (3) of the section analysis of our paper. In conclusion, the authors wish to declare that they have been very interested by this discussion and are gratefull to professor Kuscher for this taking part to it. On the other hand, they f'md surprising the extention of the reciprocity law to non-equilibrium problems with moving boundary, as it appears from Kuscher's comment, with his mathematical formulation which does not take into account some macroscopic quantities such as the latent heat of vaporization.

REFERENCES. [1] CERCIGNANI C., LAMPIS M., Transp. Theory and Stat. Phys., 1, 1971, 101. [2] KUSCHERI., Surface Science, 25, 1971,225. 132

[3] LANG H. & KUSCHERI., in Rarefied Gas Dynamics, Proc. of the IX R.G.D. Symposium, 2, 1974, F. 12. [4] KUSCHERI., Int. Summer School in Statistical Mechanics, Jadwisin, Poland 1977. MECCANICA