EVAPORATIOS AND CONDENSATION OF SPHERICAL BODIESIN NOXCONTI~VUUM REGIMES
is indicated by theoretical calculations given in the text for 0.15em, 1.2-.cm, and 3.0-cm cells (paper 11). For thicker cells the corrections would be much greater. The effect of steepness of schlieren patterns is illustrated by the simulat,ed virus model. Here the effect on s is not great because the diffusion rate is so low that the gaussian inflection leg
2857
does not change much during a n ordinary sedimentation experiment; the 2-time plot, is not. significantly tilted but only displaced along the time axis. The effect on the diffusion coefficient D is very great, however. More detailed discussion of magnitudes, though desirable, would be too lengthy for inclusion here.
Evaporation a.nd Condensation of Spherical Bodies in Noncontinuurn Regimes
by James R. Brock Department of Chemical Engineering, University of T e x a s , Austin, Texas
(Receiced February 17, 196.4)
The theory of the evaporation and condensation of spherical bodies is discussed for the slip-flow and free-molecule regimes. The value of the slip-flow density difference coefficient is discussed and the method of calculation in the slip-flow regime of evaporation and condensation rates of spherical bodies is outlined and jllustrated with a simple example. Expressions for the evaporation and condensation rates in the free-molecule regime are given in detail for general dynamical states of the gas phase.
Introduction The existence of lioncontinuum effects in gas-liquid and gas-solid systems in which interphase mass transfer is occurring has been recognized for some It seems useful to present a discussion of these effects consistent with the present understanding of noncontinuum phenomena. It is the purpose here to present a description of noncontinuum effects for the evaporation and condensation of spherical bodies for the two dynamical regimes where exact calculation for the gas phase is possible: the slip-flow and free-molecule regimes. For the slip-flow regime we will confine our explication to the quasi-steady evaporation or condensation of a stationarv soherical liauid droD immersed in 2% ._ quiescent' gas mixture' The manner of extending the methods applied in this simple example to more general gas phase dynamical states is indicated. In the free-molecule regime, evaporation and condensation rates of spherical bodies are discussed for general gas phase dynamical States for mass average _
I
&
velocity magnitudes which are small relative to mean molecular velocity magnitudes. The case of large mass average velocity magnitudes is also examined, but no account is taken of temperature, concentration, or velocity gradients in the gas.
Evaporation and Condensation in the Slip-Flow Regime In noncontinuum problems it is convenient to divide, for computational purposes, the realm of definition of Knudsen number4 0 5 @'a) 5 into three regimes which are termed slip-flow, 0 < @ / a ) < -0.25jr6; transition, 0.25 < (Z/a) < 10; and free molecule, (Z/a) > 10. (1) N. A. Fuchs, P h y s . 2 . Sowjet, 6 , 225 (1934). (2) R. S. Bradley, M. G. Evans, and R W. Whytlaw-Gray, proc, R ~ Usot. . (London), A I M , 368 (1946). (3) P. G. Wright, Proc. R o y . SOC.(Edinburgh), A66, 65 (1962). (4) Here 1 is the molecular mean free path and a is the radius of the body' ( 5 ) J. R. Brock, J . P h y s . Chem., 6 6 , 1763 (1962). (6) D. Willis, KTH Aero T N 52, Royal Institute of Technology, Stockholm, Sweden, 1960.
Volume 68, Number 10 October, 1964
JAMES R. BROCK
2858
The slip-flow regime covers that region of Knudsen number where one may apply the continuum equations of mass, momentum, and energy and account approximately for the noncontinuum effects through the introduction of slip-flow boundary conditions. This procedure, however, is foundjj6to fail as (2/a) increases. Before examining the slip-flow boundary conditions for problems of eyaporation and condensation, let us regard the continuum treatment of a simple physical system already described above for the slip-flow regime. Consider the quasi-steady evaporation or condensation of a stationary spherical liquid drop of component 2 of low vapor pressure immersed in a quiescent dilute binary gas consisting of component 2 in dilution and an insoluble component 1. We have for this physical system the continuum description of the condensation or evaporation
where D12 is the mutual diffusion coefficient, a t h e total rate of evaporation or condensation, nz the number density of 2 , and r is the radial distance from the center of the sphere. The integrated expression in terms of the concentration of 2 at the sphere surface, nZ(a), and a t a large distance from the surface, n2(.o), is ((a/47ra2) =
Dn[nz(a) - n*(..)la-’
(2)
which obviously fails for a -+ 0. Let us examine eq. 1 and its limitation in the slip-flow regime. Kow well removed from the drop surface, we expect eq. 1 to apply, but as the drop surface is approached, the diffusional transport process deviates increasingly from the linear law. Indeed, at the drop surface one finds from a gas-kinetic analysis that the number density gradients become infinite.7 Previous students of this problem1-3 have recognized that eq. 1 was not valid near the surface. Fuchs, in discussing the evaporation of spherical drops, introduced a distance A from the sphere surface; for r > A, eq. 1 was taken to apply, and for r < A, the vacuum evaporation rate was taken and both descriptions were matched at r = A. N o limitation of the validity of this procedure has been realized, although we will find here that in its previous applications it is identical with a slip-flow analysis. The A concept, while it may be applied in the slip-flow regime, 0 < ( l / a ) < -0.25 for the present simple one-dimensional physical system under consideration, is not capable of systematic extension to more complex physical systems. Through use of the usual slip-flow procedure, however, physical systems of arbitrary complexity may be discussed. The Journal of Physical Chemistry
Now the slip-flow procedure states that we apply the continuum equation, eq. 1, to the problem of evaporation or condensation of the spherical drop and use a slip-flow boundary condition, which accounts for the fact that the value of the concentration a t the surface is not that given by eq. 1, nZc(a),but is a value determined by the solution of the Boltzniann equations for the two species near the surface. The state of a dilute binary gas system diffusing in the normal direction to a surface has been studied previously by the author7 for a mixture of approximately equal molecular masses. For the present physical system, the appropriate slipflow boundary condition is analogous to the familiar velocity-slip and temperature-jump conditions. We may term this condition, by analogy, the density difference condition. The following general form is found for the density difference condition
This equation expresses the difference between the density of i found at the surface under the assumption that the gradient (dnildr) retains its value at some distance from the surface on extrapolation to the surface and the continuum value of the wall density from eq. 1; c d i is the density difference coefficient. The coefficient c d i has been evaluated7 through an approximate procedure using a linearized Boltzmann equation. For a dilute binary gas system of approximately equal molecular masses, cd2 may be shown to be
(4) where it is taken thats D12 = 1.2 0 4 p / p , p = 0.499pd, which is exact for mechanically similar molecules. The coefficient u2has the definition
(5) where Nz-(a) and N 2 + ( a )are, respectively, the actual fluxes toward and away from the surface. Nseq+ represents the value of the outward flux if all molecules 2 left in equilibrium with the surface. Thus a value u2 = 1 means all molecules 2 leaving the surface are in equilibrium with the surface. A value 6 2 = 0 indicates no incident molecule is adsorbed by the surface but is specularly reflected. (7) J. R. Brock, J . Catalysis, 2 , 248 (1963). (8) S. Chapman and T. G. Cowling,“Mathematical Theory of NonUniform Gases,” Cambridge University Press, London and New York, 1951.
EVAPORATION AND CONDENSATION OF SPHERICAL BODIESIN NONCOXTIXUUM REGIMES
2859
A semimacroscopic derivation gives an expression for cdi with no restriction on the molecular masses for a binary system undergoing equimolar counter diffusion a t a surface
condensation. We assume there is no average velocity in the gas as before. At the quasi-steady condition we have for the conduction heat flux
where ci is the mean molecular velocity of i of the binary mixture and 2 is the over-all molecular mean free path for the mixture. The density differences for the two components are simply related for this case
where A t is the gas phase thermal conductivity, @qz is the total heat flow, and qz is the heat of vaporization or condensation of 2. We have in addition the following relation for slip-flow between the drop temperature, T,, and the gas temperature, Tt, a t the surface
n1
- lZIC
= -
-=
UZQ
u1c1
(n2 - n2O)
+
(Q1
- .2> Ulh
2D dn2 12 dr
Combining eq. 1 and 3 we obtain the total rate of evaporation or condensation in the slip-flow regime @ =
4aDiza[nzc(a)- n2( a)1 1
+
cd2
(a)
(7)
Under the assumption that for the continuum regime there is interfacial equilibrium, then nzC(a)= n2,,(a). For practical application of the density difference boundary condition, the problems of calculation of reliable values of cdi are such that at present one must regard cdi as an empirical constant to be determined from experiment. Note that even where exact calculation for the gas phase for a particular molecular model has been carried out, eq. 4, the surface adsorption coefficient uz has an empirical introduction. No a priori methods are at present available for the calculation of such coefficients, although evaporation and condensation rates are well known to be influenced strongly by the surface state. The rate of evaporation or condensation of a spherical body in a binary gaseous system with general dynamical states of the gas phasemay be described approximately in the slip-flow regime through application of the continuum mass, momentum, and energy equations together with the full set of slip-flow boundary conditions at the surface as given by eq. 3 for the density difference and as given previously by the authorB,l0for the temperature-jump and veloeity-slip. Thus, given the set of slip-flow boundary conditions, the slip-flow description of the evaporation and condensation of spherical bodies becomes, usually, a trivial extension of the continuum description. As a simple illustration of the application of additional slip-flow boundary conditions, let us examine here the question of the temperature of the drop (which will determine the value nzc)when cooling or heating of the drop occurs owing to evaporation or
(9) where ct is the temperature-jump coefficient. Combining eq. 8 and 9 and integrating, we obtain for the total heat flow
Hence T , is the temperature a t which nzC(a)is to be evaluated for the drop of pure 2. However, when the heat flow by radiation is important, this contribution must be accounted
Of course, the form of R must depend on the conditions surrounding the drop. In the simplest case of the effectively infinite body of gas with containment surfaces at Tr( a) in radiant equilibrium with and surrounding the droplet at T,, we have for R
R
= 4na2u[e8Ta4 - a,Tt4( a ) ]
(12)
where cs and as are the radiation emissivity and absorptivity coefficients for the drop and u is the StefanBoltzmann constant. Equations 7 and 11 give the total rate of condensation or evaporation in terms of the surface properties of the drop and the concentration of 2 well removed from the surface. With the noted restrictions on the state of the gas phase and known values of Cd2, et, E,, and a,,these two equations will be found to describe approximately the evaporation and condensation of the spherical drop for 0 < (Z/a) < -0.25. (9) J. R. Brock, J . Colloid Sci., 17, 768 (1962). (10) 3. R. Brock, ihid., 18, 489 (1963).
Volume 68. Number 10 Octoher, 1904
The extension of these considerations to multicomponent gas systems is easily performed. Inasmuch as no new factors are introduced for multi-component systems, they will not be considered in detail here.
Evaporation and Condensation in the Transition Regime For the transition regime 0.25 < @ / a )< 10, no simple description of evaporation or condensation is available. For all physical systems involving the transfer of mass, momentum, and energy, no general methods have been found as yet for the transition regime. Evaporation and Condensation in the Free-Molecule Regime I n the discussion of the slip-flow regime above, it is to be noted that the spherical body has a paramount influence on the state of the surrounding gas and it is, therefore, not possible to specify an arbitrary state for the gas. For the free-molecule regime, (I/a) > 10, the simplification appears that the spherical body is so small that it does not affect the state of the surrounding gas. Hence the effect of arbitrary states of the gas on the evaporation or condensation may be accounted quite easily in this regime. We shall present hcre, principally for completeness of the discussion, expressions for evaporation and condensation rates in the free-molecule regime. Some of the results are interesting in their implication in the problem of droplet growth in the atmosphere. Consider the region of a dilute binary gas mixture surrounding a rigid spherical body where, for generality, both components of the mixture are evaporating or condensing. It will be assumed that the velocity of advance or regression of the surface owing to evaporation or condensation is negligible relative to the mean molecular velocity. Also, it will be assumed that the surface properties of the spherical body have no angular dependence. I n the region of the body, the state of the gas is specified by assigning number densities, ?al-, temperature, T - , mass velocity, ij, and gradients, Vni, V T - , Vp. It mill be taken that the over-all gas density is such that all fluxes in the gas may be accounted by expressions linear in the gradients. A reference COordinate system is taken fixed in the spherical body. With the specification above, the distribution functions, SI-, for the gas surrounding the spherical body have the form
Si-
-
= f L @ -{ lA i . 7
In T -
- -
d
Di.dl2 - Bi:Vij} (13) The .Journal of Phpical Chemistry
wheref,(O)- = nl-(P1-/n)8’2exp[-pl-~(vi -qIZ}j3, pi- = ( m , / 2 k T - ) , and is the molecular-velocity vector of
- - -
species i. The functions A,, D,, B , are defined and discussed elsewhere8x1Oto the level of approximation chosen here. For the molecular flux calculations, the distribution functions, fl+,for the molecules leaving the surface are defined by SI+
=
+ (1 - u1)j1-
U$l(O)+
(14)
where ui is previously defined and
Pi+
=
(mi/2lcT+)
The total rate of evaporation or condensation of the spherical body is given by the expression
where
and n’ is the unit surface normal. It is apparent that the calculations which follow may be readily extended to convex bodies of otherwise arbitrary shape. A simplification in the calculation of @ is found with the restriction
2p;vi.p:
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).
