JAMES R. BROCK
2862
set for a. For an inert component, 1, for example, the additional condition, = 0, together with eq. 22 and 23 completely determine a. The extension of these results to multicomponent systenis is readily performed. No new basic results are introduced. Accounting for an angular dependence of the surface properties of the spherical body presents no difficulty if this dependence can be specified. So long as the shape of the body remains convex, for a given time-dependent change in shape whose rate is small relative to the mean molecular velocity eq. 18 and 19 or 22 and 23 are easily modified, but no new basic considerations are introduced. The results presented here may be considered as adequate to describe evaporation or condensation of a collection of spherical bodies in the free-molecule regime so long as the distance, d, between the surfaces of the various bodies is such that (2/d) 0.
-
Conclusions If the condensed surface properties are known, evaporation and condensation rates for a spherical
body for completely general gas phase dynamical states may be calculated in the slip-flow, 0 < (Z/a) < -0.25, and free-molecule, @/a)> 10, regimes with the methods outlined here. Methods for calculation of evaporation and condensation rates are under study a t present for the transition regime, 0.26 < (Z/a) < 10. The major uncertainty in these various evaporation and condensation calculations, as has long been recognized in continuum calculations, is the prediction of the condensed surface properties, such as ui and ~ i . These surface properties are of greater importance in noncontinuum than in continuum considerations and hence are perhaps more easily studied experimentally for the free-molecule regime. Further experimental determinations of surface properties are needed, but equally important is the development of a priori methods for prediction of surface properties.
Acknowledgment. The author wishes to thank Prof. P. G. Wright, Queen's College, Dundee, for calling the author's attention to this problem, and the National Science Foundation for support through Grant G19432.
Free-Molecule Drag on Evaporating or Condensing Spheres
by James R. Brock Department of Chemical Engineering, The University of Texaa, Austin, Texas (Received February 17, 196'4)
The free-molecule drag on evaporating or condensing spheres is discussed. A gas-surface interaction parameter specifying the fraction of impinging molecules adsorbing on a surface but not undergoing condensation is introduced. The evaluation of this parameter from experimental free-molecule drag measurements is proposed.
Introduction The problem of calculating the free-molecule drag on various bodies has received attention by several investigators.'-3 The theory seems to be in fair agreement with the existing experimental data, although one finds here the ubiquitous difficulty of the specification of the gas-surface interaction. In connection with a part of an accompanying paper The Journal of Physical Chemistry
in which the free-molecule evaporation or condensation of spherical bodies is investigated, a question has arisen concerning the free-molecule drag on such spherical bodies. Accordingly, we consider here the (1) L. Waldmann, 2 . Naturforsch., 14a, 589 (1959). (2) M. Heineman, Commun. Pure AgpZ. Math., 1, 259 (1948). (3) P. S. Epstein, Phya. Rev., 2 3 , 710 (1924).
FREE-MOLECULE DRAGON EVAPORATING OR CONDENSING SPHERES
2863
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free-molecule drag on an evaporating or condensing sphere in a gas mixture whose state is specified by number densities, ni-, temperature, T-, and mass velocity, 0. It will be noted in the discussion that appropriate experimental free-molecule drag measurements can be used t,o obtain fundamental information on the condensation process a t a surface.
Free-Molecule Drag Description For calculation of the free-molecule drag, we require the velocity distribution functions at the sphere surface for impinging molecules, fi-, and (emitted molecules,
Vi’
- - 2n(n*Vi*) * - -
= Vi
and n is the outwardly directed surface normal. Further, we require that ni+ = fii+
fi-
for the physical system specified
where Pi- = mi/kT-. The function f i f it3 determined from a knowledge of the gas molecule-surface interaction. The simple one-parameter specification of this interaction given in the accompanying paper is not, however, sufficient for the drag calculation. For the drag calculation we introduce a parameter Si which is the total fraction of incoming molecules of species i adsorbed by the surface. By adsorbed molecule we mean here that the molecule achieves the temperature of the surface. We could introduce an additional parameter to distinguish between momentum and energy accommodation; such a procedure will receive later, brief consideration. The fraction (1 - Si) is assumed specularly reflected from the surface. Of the fraction Si we further distinguish fractions ai adsorbed but not undergoing condensation and (1 - ai) adsorbed and condensing. Thus an absolute condensation coefficient, yi, is, in terms of Si and ai yi
= Si(1
=z
&aij*(o)+
+ &(I -
ai)ji(o)++ (1 - Si)fi-(vi’)
where
&i+
=
j = 1, 2, . . .
&if(T+,~:)
Y
- 1 (6)
which is taken as the appropriate thermodynamic relation for the surface composition. T + may be calculated as described in the accompanying paper from the over-all evaporation or condensation equations. ni+ is calculated from the requirement
a,
The free-molecule drag, on the evaporating or condensing sphere is given by the expression
- ai)
and we have no possibility of learning anything experlmentally concerning Si and ai individually from measurements of condensation and evaporation rates alone. We shall see here, however, that drag measurements make possible the determination of 6i and ai when only one component of a gas mixture is undergoing condensation a t a surface. fi+, then, has the form fi+
+ &+
where
fi+.
The function above is simply
(5)
(2)
which becomes for 1 >> Pi-Vi’Q =
8 - a2ni-(2nmikT-)”‘
X
i 3
[I
+
Giffi
i8
(97 T-
ij
(10)
Discussion It may be seen then that appropriate experimental measurements on evaporating or condensing spheres Volume 68. Sumber 10
October, 1064
2864
can be used to determine the two parameters 6, and a,. The experiments would be of the following form. For convenience we consider a drop of a pure substance j in a gas mixture of none or more noncondensing species and j . The condensation coefficient, u3 = 6,(1 - a d ) ,is determined from the over-all evaporation or condensation rate as given in the accompanying paper. From drag measurements without j in the gas, in the instance of noncondensing gases, the 6, (a, = l ) , i # j, are determined from eq. 9 or 10. Finally, drag measurements with j present would then determine 6,a,. Thus from these measurements 6, and a , are separately determinable. Aside from the experimental difficulty, there is sonie uncertainty in this described method. We have not distinguished momentum and energy accommodation. I n the previous paper in calculating the total heat flow to an evaporating or condensing sphere, the molecules were assumed to have only translational communicable energy, aside from the latent condensation or evaporation energy. For such molecules it does not appear necessary to distinguish between momentum and energy accommodation. However, for polyatomic molecules with other forms of communicable energy, such a division may be in order. We might then say that of the fraction 6,a1 a fraction S l a i ~ , achieves momentum accommodation alone and a fraction 6,a,(l - E,) momentum and energy accommodation. On introduction of the parameter E , , eq. 9 would have the form
The Journal of Phusical Chemistry
JAMES R.BROCK
“I
+ 3- SiCri(1 - 4 €i)
g (11)
I n view of the empirical nature of 6 i , ai, e i and also of even the question of the correctness of describing the gas-surface interaction by such parameters, the validity of this further division is in need of examination. However, it does seem of interest to have some measure, even empirically, of the fraction of those molecules striking the surface which are adsorbed but do not undergo condensation. Careful free-molecule measurements on evaporating or condensing bodies, as suggested here, could be made to yield such information. We make the additional observation that analogous considerations could be applied, for example, to a sphere on whose surface a catalytic reaction occurs. From free-molecule drag measurements on such a sphere one could determine some measure of the fraction of impinging reactant molecules which are adsorbed but do not undergo reaction.
Acknowledgment. The author wishes to thank the National Science Foundation for support through Grant G 19432.
