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The growth of small droplets from a gas mixture; Results and estimates for very small radii
Physica A 173 (1991) 125-140 North-Holland
THE GROWTH OF SMALL DROPLETS FROM A GAS MIXTURE; RESULTS AND ESTIMATES FOR VERY SMALL RADII M.E. WIDDER and U.M. T I T U L A E R Institut fiir Theoretische Physik, Johannes Kepler Universitiit Linz, A-4040 Linz, Austria
Received 8 October 1990
To test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture, we develop a solution method especially suitable for radii smaller or not much larger than a mean free path. The method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere. For the simplest kinetic equation, a variant of the linear BGK-equation, and for radii below two mean free paths, this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent. For radii around two mean free paths, the new results agree with those of a variant of the moment method; for lower radii discrepancies of up to a few percent are found. Our new method does not work for the Klein-Kramers equation, a kinetic equation for Brownian particles. For the latter equation we obtain an estimate for the current at very low radii from the free space solution by Uhlenbeck and Ornstein. Its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the BGK-case.
I. Introduction and survey Understanding processes such as the growth of small liquid droplets from a supersaturated vapor, or from a gas mixture, requires the solution of kinetic equations in the space outside of a sphere. One method for obtaining such solutions, a variant of the moment method, was developed by Hubmer and Titulaer [1, 2] to treat a simple but realistic model for droplet growth from a pure vapor. To assess the quality of this method, we applied it in an earlier paper [3] to two highly simplified models for droplet growth from a gas mixture. F~r these models some exact results are known, especially in the limit of high droplet radii; some of these results were reproduced by the moment method up to five significant figures. For small radii the success of the method was less easy to estimate. With decreasing droplet radius, the moment approximations break down for numerical reasons as one includes more moments, the earlier the higher the number of moments. Though the values obtained for the growth rate at the point of 0378-4371/91/$03.50 ~
1991 - Elsevier Science Publishers B.V. (North-Holland)
126
M.E. Widder, U.M. Titulaer ! Growth o f small droplets from a gas mixture
breakdown are not too far removed from the known limit of vanishing radius (unlike the results obtained by earlier versions of the moment method [3, 4]), the results for radii smaller than the: mean free path of a vapor molecule were certainly no more accurate than a few percent, and the error was not easy to estimate. In the present paper we develop solution methods complementary to the ones in refs. [1-4]. They are designed to function especially well for small radii, though as yet they can only be applied to a few simple models, namely the ones treated in ref. [3]. For the first of these models, a variant of the BGK-model, our method uses a variant of Cercignani's integral equation approach [5]. A straightforward expansion in orders of the droplet radius breaks down: beyond linear order all terms diverge, for reasons similar to those causing the breakdown of the virial expansion for transport coefficients [6]. However, a rearrangement of the expansion leads to finite results in each order. The resulting series of approximations does not converge very well, but at least for the most important physical quantity to be derived from the solution, the growth rate of the droplet, a reliable extrapolation procedure can be devised. For the second of the models considered in ref. [3], the Klein-Kramers equation for Brownian particles, the method just sketched does not work: it is basically an expansion in the number of collisions experienced by the vapor molecule, a concept not well defined for a Brownian particle. Mathematically, this difficulty manifests itself by the appearance of ever higher derivatives of g-functions in the terms of the expansion. However, the behavior of the growth rate for very small radii can be extracted from the solution of the Klein-Kramers equation in free space given by Uhlenbeck and Ornstein [7]. We were not able to determine its behavior for small radii analytically, but our numerical results strongly indicate an analytic behavior different from the one found for the BGK-model. In section 2 we define the models treated, and outline the methods used to obtain solutions and approximations valid for small radii. Section 3 contains further analytical details for the BGK-case. Most of the material in these sections is available in principle in the literature; we provided it mainly to make the paper r_easo..a...: , hh, o,.,x-,.v,,tai,,~u, ~.~ ..~..., .. "-. .. -, d--~ llU to m.... a k,e our modifications comprehensible. In section 4 we present our numerical results (including a discussion of extrapolation procedures) and a comparison of our new results for the BGK-case with those of the moment method and of a variational method due to Loyaika [8]. We see that our new results agree with those of the moment method, and are of comparable accuracy, for radii around two mean free paths. The discrepancies with Loyalka's results are small, but outside of the margins of error of our method. The final section contains a few concluding remarks.
M.E. Widder, U.M. Titulaer / Growth of small droplets .from a gas mixture
127
2. Basic equations and solution procedures for small radii The equations treated in this paper have the structure
o)
0 + v" ~r P(v, r, t) = cap(v, r, t ) ,
(2.1)
where P is the distribution function for the velocity v and the position r of the vapor molecules and ca is an operator that describes the effect of the collisions of the vapor molecules with a carrier gas. We have assumed that the vapor admixture is so dilute that the vapor does not influence the carrier gas, and that collisions between vapor molecules may be neglected. For the Klein-Kramers equation, which describes "Brownian" vapor molecules, ca has the form CaK=3'~V" V+ rnfl OV " where m is the mass of the vapor molecule, y the friction coefficient (or velocity relaxation rate), and T = (k/3) -1 the carrier gas temperature. For the BGK-case, which corresponds to vapor molecules light compared to the carrier gas, ca has the form
~B= I(T~I--fM(V)
do'
,
(2.3)
where f ~ ( r ) is the Maxwell distribution at temperature T and y is the inverse relaxation time. As in ref. [3] we are looking for stationary, spherically symmetric solutions of (2.1) in the space outside of an absorbing sphere of radius R, which represents the liquid droplet. This means that there are no vapor particles at [r I = R with outwardly directed velocities. For simplicity, we shall measure velocities in units of the thermal velocity V,h ---(mfl) -1/2 and lengths in units of the mean free path l= y-~vth (for the Brownian case this quantity is the velocity persistence length). Due to the spherical symmetry, P depends only on the magnitudes of v and r and on ~ - - ~ " ~, the cosine of the angle between them. if we, moreover, introduce the scaled radius p = r/R, the equation to be solved in the BGK-case reads [2, 3] 5PP(t,, .~,., p)=- v /.t O'p +
1--/z 2 O) p
oix
e ( v , ft, p)
= -RP(v, la., p) + Rq~o(v) f dr' P(,J", tx', p) with the boundary condition
(2.4a)
128
M.E. Widder, U.M. Titulaer I Growth of small droplets from a gas mixture
P(v, It, 1)--0
(2.4b)
for It > 0 .
Here ~Oo(V) = (2'11") - 3 / 2 exp(-v2/2) is the Maxwell distribution in dimensionless units. In addition to (2.4b) we require that P approaches q~0(v) for r--->oo. In a first attempt one might try to solve (2.4) by expanding around the solution of the free flow equation, obtained by putting R = 0 in (2.4a). This leads to the ansatz
e(o,
p) =
-
o(O,
p) - Z
R ' P,(v, p~, p ) , -
(2.5)
i=l
where /30(v, It, p) is a solution of the free flow equation with the boundary condition (2.6)
/3o(V, It, 1)= q~o(v) O ( i t ) , and the P~ obey the recurrence relations b°Pi(v, It, p ) = -/3i_l(v, It, p) + q~o(v) f dr'/3i_,(v',/.t', p)
(2.7a)
with the boundary conditions Pi(v, ~, 1) = 0
for It >0, i > 0 .
(2.7b)
The most important physical quantity to be extracted from our solution is the reaction rate k ( R ) , defined as the ratio between the current density at the sphere and the density at infinity. From (2.5) one would obtain for this quantity an expansion of the type k ( R ) = Jo + R~ + R ~
(2.~)
+"..
The first term, j., is given by
i J' f
j,, = 2rr
dv
0
dp. V2VIt qg,,(V) = ( 2 7 r ) - ' j 2
(2.9)
11
which is clearly independent of the collision operator ~. For the higher terms one has
7, = 2"u
dv o
dp. .... |
v2v
1).
(2.10)
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
129
As we shall see in section 3, j, can be evaluated analytically and has the value 11 = --T~; all higher terms, however, diverge. This means that (2.5) and (2.8) are meaningless beyond the term linear in R. The divergencies can be attributed to collision sequences in which a vapor particle carries out free flights over distances very large compared to a mean free path. The contributions from such events can be suppressed by rearranging the expansion (2.5)- we replace t30 by a solution P0 of "•
!
5PPo + RP o = 0
(2.11)
with the unchanged boundary condition (2.6), and the /3 by Pi obeying the recurrence relations
S°Pi + RP i = ~,o(v) f d r ' Pi_,(v', la,', p)
(2.12)
with unchanged boundary conditions (2.7b). We see from (2.12) that Pi-, enters into the equation for P~ only via its density in position space ni_t(p). As we shall see in section 3, these densities,
hi(p) = 2rr
dv 0
d # v 2 Pi(v, ~ , p ) ,
(2.13)
-!
obey the recurrence relations ,,,( p) =
dp' n , _ , ( p ' ) p ' { T _ , ( R I p -
p'])
1
- T _,( R[~/p 2 - 1 + ~ / p ' ~ -
11)},
(2.14)
where Tl(x ) is the lth Abramowitz function [9],
--
-''2 {
i02 - x !o). do I_,t exp(- ~,
(2,!5)
0
As will also be shown in section 3, the full distributions Pi, and hence the contributions j, to the reaction rate, can be constructed from the n i_l(p). Hence, our problem has been reduced to an evaluation of the integrals (2.14). In section 4 we shall introduce our numerical procedure for accomplishing this task and show that it leads to a sufficiently rapidly converging series of type (2.8) for the reaction rate. (Note, however, that this series is no longer a
130
M.E. Widder, U.M. Titulaer I Growth o f small droplets from a gas mbcture
power series in R, since each of the Ji depends on R due to the additional term on the left-hand side in (2.12), compared to (2.7).) The methods described thus far in this section cannot be applied to the Klein-Kramers case: as we shall see, the distribution /30(o,/z, p) contains discontinuities in velocity space, which cause difficulties when the differential operator (2.2) is applied to them. As a preliminary to our alternative treatment for the I~dein-Kramers case we note that the combination (cf. (2.5))
Pa(v, p,, p ) = q~o(v)- P(v, Ix, p)
(2.16)
is the solution to a special albedo problem [10, 11] in which particles are injected into the system at p = 1 with a Maxwellian distribution. To calculate k(R) we must calculate how many of these particles eventually return to the sphere r = R (for the remainder of this section we return from the scaled variable p to the original variable r, measured in units of 1). In their classical p a p e r [8] Uhlenbeck and Ornstein determined the probability W(r, v, t; r o, Vo) that a particle with velocity v0 and position r 0 at t = 0 will have velocity v and position r at time t in the absence o f boundaries. Their expression can be written as e 3'
(-(a~72 + 2 h ~ . F + bF2)) ~ ,
W(r, v, t; ro, vo) =
8,.ff"~3 / 2 e x p
17= e ' ( v - Vo),
7 = r + v - r o - Vo,
(2.17)
with
(2.18) a =2t,
b = e 2 t - 1,
h =2-2e'
A=ab-
h2
From W one calculates the incoming normal current at any point of the surface; for the point Riz one has +x
+o~
0
j(R~ z, t; ro, vo) = ez
, v, t; r0, v0). ~
""Jc
--
~,¢
(2.19)
--"de
The total incoming current into the sphere is then
F(r,,, v,,; R)= - f dt RZ f f d2O . j ( R ~ ,
t; r,,, o,,).
(2.20)
0
Note, however, that this quantity is not the quantity we need: for the aibedo
M . E . Widder, U.M. Titulaer I Growth o f small droplets from a gas mixture
131
problem with an absorbing sphere only first passages back into the sphere should be counted, whereas (2.20) includes all subsequent passages as well. One may hope, however, that for a small enough sphere, multiple passages are highly improbable. However, this argument does not hold for particles emitted with very small velocities; in fact for v o ~ 0 one finds a divergence in ~'(ro, Vo). To eliminate the worst errors of this type we introduce the regularized return probability
r(r o, Vo; R ) = min[1, F(ro, Vo; R)].
(2.21)
To estimate the total return current we now integrate over all r 0 and v0; since our expression is reasonable only for very small r 0, we replace r o by 0, which eliminates the dependence of r on the direction of v0, and obtain as our final estimate for the return current density
j_(R) = ~
dv o
dp, v2vp, e -~2/2 r(O, v; R ) .
(2.22)
o
(For more detailed arguments concerning the weight function used in (2.22) see ref. [11].) As we shall see in section 4, a numerical evaluation of ]_ (R) for small R leads to a result that can be described by
kK(R ) "~ (2,rr)-'/2 - 0.28R °67 ,
(2.23)
in marked contrast to the linear R-dependence of the first correction term for the BGK-case.
3. Analytic derivations of results for the BGK-case
The aim of the present section is to provide derivations for some of the results of section 2, in particular the value -~2 for the quantity Ii defined in (2.8) and the recursion relation (2.14). To derive the first results we begin by decomposing the distribution/30(o,/z, p) into parts with given values of Iv[ and the angular momentum. Thus the boundary condition (2.6) is temporarily replaced by /3o(v, ~, 1; u, c) =
1 2"n'v
a(v
-
u) a(X/1
-
2 - c) O(p,).
(3.1)
132
M.E. Widder, U.M. Tindaer I Growth of small droplets from a gas mixture
The corresponding solution of the free flow equation (constant along the flight path) is
.) n(o~/1 -
to(O, ~, o ; . , c) = --!-1 ~(~ 2,wV2
~ - c) o ( . )
o ( o - 1)
(3.2)
The associated density profile is C
fro(O; u, c ) = O~pz _ cZ O(p - 1) ;
(3.3)
the outwardly directed current density equals u c p - 2 0 ( 0 - 1). The corresponding contribution to P~, calculated from (2.7) and evaluated for P = 1, reads /51(v,/.t, 1; u, c) = ~o(v) vc f d k -Itt
V k2 + 1 -
1.6 2
"V/k 2 q_ 1 - / . / , 2
for/.t < 0 ; _
(3.4)
¢2
the resulting contribution to 7, is given by 0
f,(u,c)=2~;do f d~% f',(o,~,a;.,¢). (}
(3.5)
- I
The quantity ]'1 itself now follows from
-ff
I, = 2"n" du 0
dc u2%(u) ~(u, c),
(3.6)
0
which is easily proved by checking that the corresponding integral over the boundary condition (3.1) reproduces (2.6). The fivefold integral obtained by substituting (3.5) and (3.4) into (3.6) can be evaluated analytically and yields 1 the result]! = 12To derive the recursion relation (2.14) we first rewrite (2.12), with fie defined in (2.4a), in terms of the variables P --- P ~ ,
q -- PV 1 - / x 2 ,
(3.7)
and the transformed distributioi, fun,-~ions
tO,tv, p, q)=-e t#'' ~, r, ( v, ~
P+ q2, Vip.,+q2)
(3.8)
M . E . Widder, U.M. Titulaer I Growth o f small droplets from a gas mixture
133
These functions obey the recursion relations
O qji(v, P, q) = 1 eRp,o tPo(V) ni_,(~ [ p2 + Op v
q2),
(3.9)
as one proves by substitution of (3.8) into (2.7a) and use of the definition (2.13). The solution of (3.9) that obeys the boundary condition (2.7b) can be written as P
f
~;(v, p, q ) =
ds e Rs/° v-'qo(V ) n i _ , ( ~ s 2 + q2),
(3.10)
$0
with so = ~/1 q2 for q < 1, p > O, and so = - ~ ni(p) to the density are given by -
otherwise. The corrections
n;(p)=2"rr f dv f dit v2 e-RP~"vo~(v, pit, p'~/1- it2),
(3.11)
as follows from (3.8) and (2.13). Substitution of (3.10) gives V~-p
pp.
-2
-1
-~
I
PP"
V~-p-2
ds)
~1 -p2(1-0.-')
x { T , [ R ( p i t - s)]ni_,(~s 2 + p 2 ( l - It2))} ,
(3.12)
where we used the definition (2.15). The transformation of variables
s' = pit - s,
i t , = ~/s 2 + p2(1 _ it2)
(3.13)
and use of the recurrence relation for the Abramowitz functions [9] finally lead to (2.14). The contributions j;(R) to the reaction rate, defined in analogy to (2.10), are given by 0
ji(R) = 27r f dv 0
fori>0,
f
-1
tx
V21.t e-gix/V f ds egS/vq~O(V) n,_,(X/s + --
oc
(3.14)
as one sees by subsitution of (3.8) and (3.10). The function ~b0, related to the P0 defined in (2.11) by the definition (3.8), is
134
M,E. Widder, U.M. Titulaer I Growth of small droplets from a gas mixture
constructed in analogy with /50, but with an exponential decay with rate R along the free flight trajectory. The result is
~o(V, p, q)=exp(R ~/1- q2) %(v) O(p) O(1- q) ;
(3.15)
the corresponding particle density is given by
R d/.t v2q:,o(v)exp(- v [pp' - ~ / 1 _ 02(1 _ t~2)]].'~
no(p) = 2"rr dv 0
ivri__p-2
(3.16) This function is used to start the recurrence scheme (2.14).
4. Numerical results and comparison with some earlier methods We solved the recursion relation (2.14), starting from (3.16), by numerical integration for a number of values of R between 0.1l and 2/, and for values of i up to 20 (to avoid infinite integration ranges we first transformed the variable in (2.14) from p to p-~). Subsequently we evaluated the contributions ji(R) to the current density at the droplet surface, defined in (3.14). To give an impression about the rate of convergence, the first six j~(R) are given in tables I and II for R - 0 . 1 / a n d R = l, respectively. We see that they decrease rather slowly; closer analysis shows a decrease roughly like i -3/2. This is not too surprising: for p and p' large compared to unity, the recursion relation (2.14) reduces to a smearing out of the function ni_~(p) over a distance of order l (or unity in scaled units). Thus, for p >> 1, the sequence n~(p) should approach a
Table I The successive back flow contributions j~ . . . . . j~, ( s e e ( 3 . I 4 ) ) for our variant of the BGK-model, together with the back flov~ corrections k ~ . . . . . k~s ( s e e ( 4 . 4 ) ) and the extrapolated k~ (see (4.5)) for a sphere with radius (k 1/.
L(o.l)
k~(O.l)
kyO.1)
-0.6492
x I0 2
-0.8803
-0.1128
x I 0 -'
- 0.8804 x 10 "
-0.8810×
-0.3773
x I0
- 0 . 8 8 0 5 x 10 z
-0.8810 × 10 '
-0.1794
x I0
-0.1042
x 10
-0,8805 -0.8806
x 10 : x 10 2
- 0 . 8 8 1 0 x 10 ~ - ( ) . 8 8 1 0 x 10 z
-0.6851
x 10 ~
- 0.8~
x 10 "
-0.8810
~
x 10 2
- 0 . 8 8 ( 0 x 10 : 10 :
x
10 2
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
135
T a b l e II T h e quantities listed in table I for a s p h e r e with radius l. j,(1)
k~(1)
-0.3143 -0.1325 -0.7366 -0.4735 -0.3327 -0.2483
x 10 -1 x 10 -!
-0.8660 -0.8664 -0.8668 -0.8670 -0.8673 -0.8675
x 10 -2
x 10 -2 x 10 -2
x 10 -2
k~(1) x x x x x x
10 -! 10 -I 10-' 10 -i 10- ! 10-1
-0.8707 -0.8706 -0.8705 -0.8703 -0.8702 -0.8702
x x x x x x
10 -1 10' 1010 -i 1010 -1
Gaussian, with a width increasing like V ~ - io, where i0 is an as yet unknown parameter of order unity that originates from initial slip effects [12] (the diffusion argument does not hold for small i, where hi(p) is still concentrated at small p). As long as the loss of probability caused by the absorbing sphere is small, the current density contributions j~ should decrease like the maximum of this spreading Gaussian, i.e. y,--- (i - io) -3/2 .
(4.1a)
A reasonable guess for the first correction is that it should be proportional to the accumulated loss of probability caused by the Jk with k < i, hence be asymptotically of relative order (i - i 0)- 1/2. j , - - - ( i - i0)-3/2[1 - c o n s t x ( i - io)-i'21
.
(4.1b)
We therefore fitted the ji(R) according to the prescription
j, = a ( i -
i0) -3/2 +
b(i-
io) -2 .
(4.2)
The values of a, b, and i i obtained in this way from Jn, J,,+l and Jn+2 are denoted by a n, b~., and i0n, respectively. In this notation we suppressed the dependence on R; in case of multiple solutions we chose the ones that fitted Jn-i, etc., the best. Using these parameters, we approximated the "back flow" correction (4.3)
kC(R) = k ( R ) - (2"tr)-''2 by the sequence of expressions (cf. (2.8), (2.9) and (3.14)) n+2
k,,(R) = c
N i--'l
R)',(R)+
N i=n+3
. -3/2 . - 2 ]. R ' [an(i- ton) + b,,(i- lot,)
(4.4)
136
M.E. Widder, U.M. Titulaer / Growth o f small droplets from a gas mixture
These quantities for n = 13 to n = 18 are given in the second columns of tables I and II. The third column contains the results from a second fit k~(R) of five successive values of k~(R) according to the formula
k~(R) = k ~ ( R ) + d n - " ,
(4.5)
which can be interpreted as an attempt to take further terms in (4.1), and corrections to the diffusion picture, into account. The results in the two last columns of the tables appear to approach a common limit; for R >0.41 the corresponding quantities approach the limit from different sides. Since, moreover, the fit parameters io, a, b, d and a depend only slightly on n, we feel confident in postulating k¢(0.1) = - ( 0 . 8 8 1 -+ 0.001) x 10 -2 ,
(4.6a)
k¢(1) = - ( 0 . 8 6 9 - + 0.002) x 10 -~ .
(4.6b)
The discretization errors in the numerical integrations are about one order of magnitude smaller than the ones quoted in (4.6); moreover they decrease rather quickly with i due to the smoothing of the density profiles ni(P) with increasing i. In table III we present the results obtained from estimates of type (4.6) for the scaled rate
k(R)-k(R)/k(O)
(4.7)
for several R between 0.1l and 21. The errors in the contributions k¢(R) are at most +-0.5%, except for R = 2, where it is about +-1%. Since k¢(R) is only part of /~(R), the percentual error in the latter quantity is reduced by a further factor of approximately [ ~ / 1 2 ] R / l . Also given in table !!I are the corresponding results obtained from the moment approximations D6, D 7 and D 8, described in refs. [1-3]. We also determined the highest D N for which the moment method did not yet break down (e.g. N = 12 for R = 2l). The latter results do not differ too much from the D 8, but only from R = 1.75/on do the higher D N all lie within the error bounds resulting from the procedure in the present paper. Thus, R ~ 1.751 is the point where the moment method becomes comparable in accuracy with the present one. In fig. 1 we show our results for /~(R) together with the D 6 and D 8 approximations of ref. [3] and the result of the straightforward R-expansion
[¢(R) = 1 - Rv'-~/12 .
(4.8)
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
137
Table III The scaled reaction rate k(R) (see (~t.7)) for our variant of the BGK-model, as calculated with the present method and with the moment method in the D 6, D 7 and D s approximations, for several values of R.
R
k(R)
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.25 1.50 1.75 2.00
present method
D~
D7
Ds
0.9779 0.9549 0.9318 0. 9089 0.8862 0.8644 0.8428 0.8220 0.8017 0.7822 0.7363 0.6942 0.6558 0.6202
0.953551 0.919731 0.895256 0.875193 0.857146 0.839967 0.823133 0.806435 0.789823 0.773317 0.732854 0.694165 0.657772 0.623892
0.965732 0.927860 0.901145 0. 879676 0.860641 0.842707 0.825265 0.808068 0.791040 0.774186 0.733091 0.694016 0.657400 0.623402
0.954481 0.915121 0.890376 0. 870690 0.853111 0.836329 0.819790 0.803302 0.786835 0.770433 0.730147 0.691611 0. 655386 0.621688
1.00 "'-.. ,,
0.85 =
-"-:~"~"~~'~:~
A
"~'~'%.+
: o.8o-
<'~'0.75-
-,~'~~ ~
1 0.60|
0.0
I
I
I
I
I
,
,
t
,
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
"'l
2.0
R-->
Fig. 1. The scaled reaction rate/~(R) (see (4.7)) for our variant of the BGK-model, as calculated with the present method (crosses) and with the moment method in the D, (dotted line) and Ds (dashed line) approximations, together with the analytic linear low-R approximation (solid line) and variational results due to Loyalka (stars). The radius R is measured in units of the mean free path/.
138
M.E. Widder, U.M. l~tulaer ! Growth o f small droplets from a gas mixture
Also indicated are the results of a simple variational calculation by Loyalka [8]. The two simple approximations show surprisingly good results for the values of R shown in the figure. The moment method, which works best for high R, shows deviations of up to 5% for D s in the region below R = 0.51; moreover it shows some spurious structure at very low R. The low-R estimates j_(R), defined in (2.22), for kC(R) in the KleinKramers case are given in table IV, together with the fit parameters a and a determined from two successive values and the fitting formula (4.9)
j _ ( R ) = a R '~ .
The results shown in the table clearly exclude a linear R-dependence, as found in the BGK-case, and suggest an exponent 2/3. The D 6 and D 7 results from ref. [3], when plotted on a linear scale, show a small-R singularity of roughly similar exponent and amplitude, but are shifted by a constant (R-~.ndependent) amount. However, they level off beyond R =0.11. At R = 1, where the approximations entering into the derivation of j _ ( R ) are certainly much too crude, but where the moment results should be reliable at least up to a few percent, the estimate ]_ (R) is off by at least a factor of 4. Since the low-R structure of the D,-approximations for the BGK-case turned out to be spurious, not too much significance should be attributed to the agreement of this structure in the present case. The precise analytic structure of k ( R ) for the Klein-Kramers case in the region R ~< 1 remains an open problem, but our results at least indicate that the qualitative difference between the two models found in ref. [3] is no mere artifact of the moment method.
T a b l e IV T h e low-R estimates j (R) (see (2.22)) for the K l e i n K r a m e r s equation, together with the fit p a r a m e t e r s a ---.-11 ~'--• A t%%% ~,,u a ~,~cc ~,~.~,)~ for several vamcs' ~,f K" ~'in units of the velocity persistence length l).
R I 10 I0 10 lO lO
j(R) ' : :~ 4 ~
0.3393 0.6644 0.1319 0.2783 0.5967 0.1284
× × x × ×
I0' i0' 10 " I0 ~ 10 :'
a
a
0.339 0.335 0.297 0.283 0.278
0.708 0.702 0.676 0.669 0.667
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
139
5. Concluding remarks The method developed in section 2 for the BGK-case is complementary to the moment method as developed in refs. [1-3]. The older method runs into difficulties for small radii #1", the new method loses accuracy as the radius increases. The most trivial reason for this is that the new method calculates the back flow kc(R ). For large spheres this quantity becomes almost equal to - k 0 ; the reaction rate k(R) is equal to a small difference (of order R -I) between two almost equal quantities, hence small errors in one of them may have large effects. In addition, the procedure used in section 4 for "predicting" the higher jn(R) becomes questionable for large spheres. In (4.1) we treated the loss of probability due to the central absorbing sphere as a small correction on the general diffusive motion; this is certainly incorrect for large spheres, which reabsorb virtually all the particles released near their surface. The observation of an initial i - 3 / 2 decay in the partial return currents ji(R) does not necessarily mean that effects of a finite radius are still unimportant, since the partial return currents of particles released from a plane absorbing wall also exhibit an .-3/ t E-dependence [11]. For a large but finite sphere one would therefore expect such a dependence both for low and for very high i (where the arguments of section 4 should become correct), but it is not clear whether the coefficients are the same, or whether thcrc is a transitional regime in between. The difficulties we encountered for R = 2/, which led us to assign a slightly larger error to our results for that R-value, may be indicative of trouble of this kind. Our method for the BGK-case should be extendible to collision operators that can be written as a finite linear combination of orthogonal projection operators, all but one of them of finite dimension. The number of equations to be solved in each iteration, however, increases linearly with the number of eigenfunctions with eigenvalues different from the one for the infinite-dimensional projector. For all such models, and for collision operators well approximated by such models, one expects a linear initial decrease of/c(R) at small R. The different qualitative behavior for the Klein-Kramers equation is related to the fact that such approximations should not work well for the collision operator c¢K, which has an infinite number of different eigenvalues, without an upper bound [10]. The models discussed in this paper are of course highly idealized ones; some of their shortcomings for describing actual droplet condensation were pointed out in ref. [3]. Since the present paper is mainly concerned with testing and ~'lThe practical importance of those difficulties should not be exaggerated. For radii of the order of one tenth of a mean free path, the treatment of the droplet as a macroscopic sphere becomes questionable, and errors of a few per cent in the treatment of the kinetic problem might become unimportant in view of errors caused by this inadequacy of the model.
140
M.E. Widder, U.M. Titulaer ! Growth o f small
droplets from
a gas mixture
comparing methods of solution we shall not repeat them here. We also refer to refs. [1-3] for a discussion of some other methods that have been used to treat problems of the type discussed here. Further references can be found in a recent review by Sone [13]. Taken together, the methods of the present paper and those of refs. [1-3] allow one to determine the growth rate k(R) with an error of at most about one tenth of a percent, at least for linear kinetic equations and for collision operators not too different from the BGK-operator (in the sense discussed earlier in this section). At least for the BGK-case, comparable accuracies can be obtained with less numerical effort by means of variational methods [14, 5]; unlike the methods discussed here, however, such methods do not easily provide reliable error estimates.
Acknowledgement The work reported in this paper was supported by the Austrian "Fonds zur F6rderung der Wissenschaftlichen Forschung".
References [1] G.F. Hubmer and U.M. Titulaer, J. Stat. Phys. 59 (1990) 441. [2] G.F. Hubmec and U.M. Titulaer, J. Stat. Phys., to be published; in: Proc. 17th Int. Symp. on karefied Gas Dynamics, to be published. [3] M.E. Widder and U.M. Titulaer, Physica A 167 (1990) 663. [4] V. Kumar and S.V.G. Menon, J. Chem. Phys. 82 (1985) 917. [5] C. Cercignani, Theory and Application of the Boitzmann Equation (Scottish Academic Press, Edinburgh, 1975); The Boltzmann Equation and Its Applications (Springer, New York, 1988) p. 222. [6] J.R. Dorfman and H. van Beijeren, in: Statistical Mechanics, part B, B.J. Berne, ed. (Plenum, New York, 1977). J.R. Dorfman, in: Fundamental Problems in Statistical Mechanics III, E.G.D. Cohen, ed. (North-Holland, Amsterdam, 1975). [7] G.E. Uhlenbeck and L.S. Ornstein, Phys. Rev. 36 (1930) 823. [8] S.K. Loyalka, J. Chem. Phys. 58 (1973) 354. [9] M. Abramowitz, J. Math. Phys. 32 (1953) 188. D. Welander, Ark. Fys. 7 (1954) 507. !!01 M.A. Burschka and U.M. Titulaer, J. Stat. Phys. 25 (1981) 569. !111 J.V. Selinger and U.M. Titulaer, J. Stat. Phys. 36 (1984) 293. 1121 U.M. Titulaer, J. Chem. Phys. 78 (1983) 1~4. 1131 Y. Sone, in: Proc. 17th Int. Syrup. on Rarefield Gas Dynamics. to be published. !141 S.K. Loyaika and J.W. Cipoila. Nuci. Sci. Eng. 99 (1988) !18. B.I.M. Ten Bosch, J.J.M. Beenakker and 1. K~,,~er, Physica A 134 (1986) 522.
THE GROWTH OF SMALL DROPLETS FROM A GAS MIXTURE; RESULTS AND ESTIMATES FOR VERY SMALL RADII M.E. WIDDER and U.M. T I T U L A E R Institut fiir Theoretische Physik, Johannes Kepler Universitiit Linz, A-4040 Linz, Austria
Received 8 October 1990
To test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture, we develop a solution method especially suitable for radii smaller or not much larger than a mean free path. The method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere. For the simplest kinetic equation, a variant of the linear BGK-equation, and for radii below two mean free paths, this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent. For radii around two mean free paths, the new results agree with those of a variant of the moment method; for lower radii discrepancies of up to a few percent are found. Our new method does not work for the Klein-Kramers equation, a kinetic equation for Brownian particles. For the latter equation we obtain an estimate for the current at very low radii from the free space solution by Uhlenbeck and Ornstein. Its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the BGK-case.
I. Introduction and survey Understanding processes such as the growth of small liquid droplets from a supersaturated vapor, or from a gas mixture, requires the solution of kinetic equations in the space outside of a sphere. One method for obtaining such solutions, a variant of the moment method, was developed by Hubmer and Titulaer [1, 2] to treat a simple but realistic model for droplet growth from a pure vapor. To assess the quality of this method, we applied it in an earlier paper [3] to two highly simplified models for droplet growth from a gas mixture. F~r these models some exact results are known, especially in the limit of high droplet radii; some of these results were reproduced by the moment method up to five significant figures. For small radii the success of the method was less easy to estimate. With decreasing droplet radius, the moment approximations break down for numerical reasons as one includes more moments, the earlier the higher the number of moments. Though the values obtained for the growth rate at the point of 0378-4371/91/$03.50 ~
1991 - Elsevier Science Publishers B.V. (North-Holland)
126
M.E. Widder, U.M. Titulaer ! Growth o f small droplets from a gas mixture
breakdown are not too far removed from the known limit of vanishing radius (unlike the results obtained by earlier versions of the moment method [3, 4]), the results for radii smaller than the: mean free path of a vapor molecule were certainly no more accurate than a few percent, and the error was not easy to estimate. In the present paper we develop solution methods complementary to the ones in refs. [1-4]. They are designed to function especially well for small radii, though as yet they can only be applied to a few simple models, namely the ones treated in ref. [3]. For the first of these models, a variant of the BGK-model, our method uses a variant of Cercignani's integral equation approach [5]. A straightforward expansion in orders of the droplet radius breaks down: beyond linear order all terms diverge, for reasons similar to those causing the breakdown of the virial expansion for transport coefficients [6]. However, a rearrangement of the expansion leads to finite results in each order. The resulting series of approximations does not converge very well, but at least for the most important physical quantity to be derived from the solution, the growth rate of the droplet, a reliable extrapolation procedure can be devised. For the second of the models considered in ref. [3], the Klein-Kramers equation for Brownian particles, the method just sketched does not work: it is basically an expansion in the number of collisions experienced by the vapor molecule, a concept not well defined for a Brownian particle. Mathematically, this difficulty manifests itself by the appearance of ever higher derivatives of g-functions in the terms of the expansion. However, the behavior of the growth rate for very small radii can be extracted from the solution of the Klein-Kramers equation in free space given by Uhlenbeck and Ornstein [7]. We were not able to determine its behavior for small radii analytically, but our numerical results strongly indicate an analytic behavior different from the one found for the BGK-model. In section 2 we define the models treated, and outline the methods used to obtain solutions and approximations valid for small radii. Section 3 contains further analytical details for the BGK-case. Most of the material in these sections is available in principle in the literature; we provided it mainly to make the paper r_easo..a...: , hh, o,.,x-,.v,,tai,,~u, ~.~ ..~..., .. "-. .. -, d--~ llU to m.... a k,e our modifications comprehensible. In section 4 we present our numerical results (including a discussion of extrapolation procedures) and a comparison of our new results for the BGK-case with those of the moment method and of a variational method due to Loyaika [8]. We see that our new results agree with those of the moment method, and are of comparable accuracy, for radii around two mean free paths. The discrepancies with Loyalka's results are small, but outside of the margins of error of our method. The final section contains a few concluding remarks.
M.E. Widder, U.M. Titulaer / Growth of small droplets .from a gas mixture
127
2. Basic equations and solution procedures for small radii The equations treated in this paper have the structure
o)
0 + v" ~r P(v, r, t) = cap(v, r, t ) ,
(2.1)
where P is the distribution function for the velocity v and the position r of the vapor molecules and ca is an operator that describes the effect of the collisions of the vapor molecules with a carrier gas. We have assumed that the vapor admixture is so dilute that the vapor does not influence the carrier gas, and that collisions between vapor molecules may be neglected. For the Klein-Kramers equation, which describes "Brownian" vapor molecules, ca has the form CaK=3'~V" V+ rnfl OV " where m is the mass of the vapor molecule, y the friction coefficient (or velocity relaxation rate), and T = (k/3) -1 the carrier gas temperature. For the BGK-case, which corresponds to vapor molecules light compared to the carrier gas, ca has the form
~B= I(T~I--fM(V)
do'
,
(2.3)
where f ~ ( r ) is the Maxwell distribution at temperature T and y is the inverse relaxation time. As in ref. [3] we are looking for stationary, spherically symmetric solutions of (2.1) in the space outside of an absorbing sphere of radius R, which represents the liquid droplet. This means that there are no vapor particles at [r I = R with outwardly directed velocities. For simplicity, we shall measure velocities in units of the thermal velocity V,h ---(mfl) -1/2 and lengths in units of the mean free path l= y-~vth (for the Brownian case this quantity is the velocity persistence length). Due to the spherical symmetry, P depends only on the magnitudes of v and r and on ~ - - ~ " ~, the cosine of the angle between them. if we, moreover, introduce the scaled radius p = r/R, the equation to be solved in the BGK-case reads [2, 3] 5PP(t,, .~,., p)=- v /.t O'p +
1--/z 2 O) p
oix
e ( v , ft, p)
= -RP(v, la., p) + Rq~o(v) f dr' P(,J", tx', p) with the boundary condition
(2.4a)
128
M.E. Widder, U.M. Titulaer I Growth of small droplets from a gas mixture
P(v, It, 1)--0
(2.4b)
for It > 0 .
Here ~Oo(V) = (2'11") - 3 / 2 exp(-v2/2) is the Maxwell distribution in dimensionless units. In addition to (2.4b) we require that P approaches q~0(v) for r--->oo. In a first attempt one might try to solve (2.4) by expanding around the solution of the free flow equation, obtained by putting R = 0 in (2.4a). This leads to the ansatz
e(o,
p) =
-
o(O,
p) - Z
R ' P,(v, p~, p ) , -
(2.5)
i=l
where /30(v, It, p) is a solution of the free flow equation with the boundary condition (2.6)
/3o(V, It, 1)= q~o(v) O ( i t ) , and the P~ obey the recurrence relations b°Pi(v, It, p ) = -/3i_l(v, It, p) + q~o(v) f dr'/3i_,(v',/.t', p)
(2.7a)
with the boundary conditions Pi(v, ~, 1) = 0
for It >0, i > 0 .
(2.7b)
The most important physical quantity to be extracted from our solution is the reaction rate k ( R ) , defined as the ratio between the current density at the sphere and the density at infinity. From (2.5) one would obtain for this quantity an expansion of the type k ( R ) = Jo + R~ + R ~
(2.~)
+"..
The first term, j., is given by
i J' f
j,, = 2rr
dv
0
dp. V2VIt qg,,(V) = ( 2 7 r ) - ' j 2
(2.9)
11
which is clearly independent of the collision operator ~. For the higher terms one has
7, = 2"u
dv o
dp. .... |
v2v
1).
(2.10)
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
129
As we shall see in section 3, j, can be evaluated analytically and has the value 11 = --T~; all higher terms, however, diverge. This means that (2.5) and (2.8) are meaningless beyond the term linear in R. The divergencies can be attributed to collision sequences in which a vapor particle carries out free flights over distances very large compared to a mean free path. The contributions from such events can be suppressed by rearranging the expansion (2.5)- we replace t30 by a solution P0 of "•
!
5PPo + RP o = 0
(2.11)
with the unchanged boundary condition (2.6), and the /3 by Pi obeying the recurrence relations
S°Pi + RP i = ~,o(v) f d r ' Pi_,(v', la,', p)
(2.12)
with unchanged boundary conditions (2.7b). We see from (2.12) that Pi-, enters into the equation for P~ only via its density in position space ni_t(p). As we shall see in section 3, these densities,
hi(p) = 2rr
dv 0
d # v 2 Pi(v, ~ , p ) ,
(2.13)
-!
obey the recurrence relations ,,,( p) =
dp' n , _ , ( p ' ) p ' { T _ , ( R I p -
p'])
1
- T _,( R[~/p 2 - 1 + ~ / p ' ~ -
11)},
(2.14)
where Tl(x ) is the lth Abramowitz function [9],
--
-''2 {
i02 - x !o). do I_,t exp(- ~,
(2,!5)
0
As will also be shown in section 3, the full distributions Pi, and hence the contributions j, to the reaction rate, can be constructed from the n i_l(p). Hence, our problem has been reduced to an evaluation of the integrals (2.14). In section 4 we shall introduce our numerical procedure for accomplishing this task and show that it leads to a sufficiently rapidly converging series of type (2.8) for the reaction rate. (Note, however, that this series is no longer a
130
M.E. Widder, U.M. Titulaer I Growth o f small droplets from a gas mbcture
power series in R, since each of the Ji depends on R due to the additional term on the left-hand side in (2.12), compared to (2.7).) The methods described thus far in this section cannot be applied to the Klein-Kramers case: as we shall see, the distribution /30(o,/z, p) contains discontinuities in velocity space, which cause difficulties when the differential operator (2.2) is applied to them. As a preliminary to our alternative treatment for the I~dein-Kramers case we note that the combination (cf. (2.5))
Pa(v, p,, p ) = q~o(v)- P(v, Ix, p)
(2.16)
is the solution to a special albedo problem [10, 11] in which particles are injected into the system at p = 1 with a Maxwellian distribution. To calculate k(R) we must calculate how many of these particles eventually return to the sphere r = R (for the remainder of this section we return from the scaled variable p to the original variable r, measured in units of 1). In their classical p a p e r [8] Uhlenbeck and Ornstein determined the probability W(r, v, t; r o, Vo) that a particle with velocity v0 and position r 0 at t = 0 will have velocity v and position r at time t in the absence o f boundaries. Their expression can be written as e 3'
(-(a~72 + 2 h ~ . F + bF2)) ~ ,
W(r, v, t; ro, vo) =
8,.ff"~3 / 2 e x p
17= e ' ( v - Vo),
7 = r + v - r o - Vo,
(2.17)
with
(2.18) a =2t,
b = e 2 t - 1,
h =2-2e'
A=ab-
h2
From W one calculates the incoming normal current at any point of the surface; for the point Riz one has +x
+o~
0
j(R~ z, t; ro, vo) = ez
, v, t; r0, v0). ~
""Jc
--
~,¢
(2.19)
--"de
The total incoming current into the sphere is then
F(r,,, v,,; R)= - f dt RZ f f d2O . j ( R ~ ,
t; r,,, o,,).
(2.20)
0
Note, however, that this quantity is not the quantity we need: for the aibedo
M . E . Widder, U.M. Titulaer I Growth o f small droplets from a gas mixture
131
problem with an absorbing sphere only first passages back into the sphere should be counted, whereas (2.20) includes all subsequent passages as well. One may hope, however, that for a small enough sphere, multiple passages are highly improbable. However, this argument does not hold for particles emitted with very small velocities; in fact for v o ~ 0 one finds a divergence in ~'(ro, Vo). To eliminate the worst errors of this type we introduce the regularized return probability
r(r o, Vo; R ) = min[1, F(ro, Vo; R)].
(2.21)
To estimate the total return current we now integrate over all r 0 and v0; since our expression is reasonable only for very small r 0, we replace r o by 0, which eliminates the dependence of r on the direction of v0, and obtain as our final estimate for the return current density
j_(R) = ~
dv o
dp, v2vp, e -~2/2 r(O, v; R ) .
(2.22)
o
(For more detailed arguments concerning the weight function used in (2.22) see ref. [11].) As we shall see in section 4, a numerical evaluation of ]_ (R) for small R leads to a result that can be described by
kK(R ) "~ (2,rr)-'/2 - 0.28R °67 ,
(2.23)
in marked contrast to the linear R-dependence of the first correction term for the BGK-case.
3. Analytic derivations of results for the BGK-case
The aim of the present section is to provide derivations for some of the results of section 2, in particular the value -~2 for the quantity Ii defined in (2.8) and the recursion relation (2.14). To derive the first results we begin by decomposing the distribution/30(o,/z, p) into parts with given values of Iv[ and the angular momentum. Thus the boundary condition (2.6) is temporarily replaced by /3o(v, ~, 1; u, c) =
1 2"n'v
a(v
-
u) a(X/1
-
2 - c) O(p,).
(3.1)
132
M.E. Widder, U.M. Tindaer I Growth of small droplets from a gas mixture
The corresponding solution of the free flow equation (constant along the flight path) is
.) n(o~/1 -
to(O, ~, o ; . , c) = --!-1 ~(~ 2,wV2
~ - c) o ( . )
o ( o - 1)
(3.2)
The associated density profile is C
fro(O; u, c ) = O~pz _ cZ O(p - 1) ;
(3.3)
the outwardly directed current density equals u c p - 2 0 ( 0 - 1). The corresponding contribution to P~, calculated from (2.7) and evaluated for P = 1, reads /51(v,/.t, 1; u, c) = ~o(v) vc f d k -Itt
V k2 + 1 -
1.6 2
"V/k 2 q_ 1 - / . / , 2
for/.t < 0 ; _
(3.4)
¢2
the resulting contribution to 7, is given by 0
f,(u,c)=2~;do f d~% f',(o,~,a;.,¢). (}
(3.5)
- I
The quantity ]'1 itself now follows from
-ff
I, = 2"n" du 0
dc u2%(u) ~(u, c),
(3.6)
0
which is easily proved by checking that the corresponding integral over the boundary condition (3.1) reproduces (2.6). The fivefold integral obtained by substituting (3.5) and (3.4) into (3.6) can be evaluated analytically and yields 1 the result]! = 12To derive the recursion relation (2.14) we first rewrite (2.12), with fie defined in (2.4a), in terms of the variables P --- P ~ ,
q -- PV 1 - / x 2 ,
(3.7)
and the transformed distributioi, fun,-~ions
tO,tv, p, q)=-e t#'' ~, r, ( v, ~
P+ q2, Vip.,+q2)
(3.8)
M . E . Widder, U.M. Titulaer I Growth o f small droplets from a gas mixture
133
These functions obey the recursion relations
O qji(v, P, q) = 1 eRp,o tPo(V) ni_,(~ [ p2 + Op v
q2),
(3.9)
as one proves by substitution of (3.8) into (2.7a) and use of the definition (2.13). The solution of (3.9) that obeys the boundary condition (2.7b) can be written as P
f
~;(v, p, q ) =
ds e Rs/° v-'qo(V ) n i _ , ( ~ s 2 + q2),
(3.10)
$0
with so = ~/1 q2 for q < 1, p > O, and so = - ~ ni(p) to the density are given by -
otherwise. The corrections
n;(p)=2"rr f dv f dit v2 e-RP~"vo~(v, pit, p'~/1- it2),
(3.11)
as follows from (3.8) and (2.13). Substitution of (3.10) gives V~-p
pp.
-2
-1
-~
I
PP"
V~-p-2
ds)
~1 -p2(1-0.-')
x { T , [ R ( p i t - s)]ni_,(~s 2 + p 2 ( l - It2))} ,
(3.12)
where we used the definition (2.15). The transformation of variables
s' = pit - s,
i t , = ~/s 2 + p2(1 _ it2)
(3.13)
and use of the recurrence relation for the Abramowitz functions [9] finally lead to (2.14). The contributions j;(R) to the reaction rate, defined in analogy to (2.10), are given by 0
ji(R) = 27r f dv 0
fori>0,
f
-1
tx
V21.t e-gix/V f ds egS/vq~O(V) n,_,(X/s + --
oc
(3.14)
as one sees by subsitution of (3.8) and (3.10). The function ~b0, related to the P0 defined in (2.11) by the definition (3.8), is
134
M,E. Widder, U.M. Titulaer I Growth of small droplets from a gas mixture
constructed in analogy with /50, but with an exponential decay with rate R along the free flight trajectory. The result is
~o(V, p, q)=exp(R ~/1- q2) %(v) O(p) O(1- q) ;
(3.15)
the corresponding particle density is given by
R d/.t v2q:,o(v)exp(- v [pp' - ~ / 1 _ 02(1 _ t~2)]].'~
no(p) = 2"rr dv 0
ivri__p-2
(3.16) This function is used to start the recurrence scheme (2.14).
4. Numerical results and comparison with some earlier methods We solved the recursion relation (2.14), starting from (3.16), by numerical integration for a number of values of R between 0.1l and 2/, and for values of i up to 20 (to avoid infinite integration ranges we first transformed the variable in (2.14) from p to p-~). Subsequently we evaluated the contributions ji(R) to the current density at the droplet surface, defined in (3.14). To give an impression about the rate of convergence, the first six j~(R) are given in tables I and II for R - 0 . 1 / a n d R = l, respectively. We see that they decrease rather slowly; closer analysis shows a decrease roughly like i -3/2. This is not too surprising: for p and p' large compared to unity, the recursion relation (2.14) reduces to a smearing out of the function ni_~(p) over a distance of order l (or unity in scaled units). Thus, for p >> 1, the sequence n~(p) should approach a
Table I The successive back flow contributions j~ . . . . . j~, ( s e e ( 3 . I 4 ) ) for our variant of the BGK-model, together with the back flov~ corrections k ~ . . . . . k~s ( s e e ( 4 . 4 ) ) and the extrapolated k~ (see (4.5)) for a sphere with radius (k 1/.
L(o.l)
k~(O.l)
kyO.1)
-0.6492
x I0 2
-0.8803
-0.1128
x I 0 -'
- 0.8804 x 10 "
-0.8810×
-0.3773
x I0
- 0 . 8 8 0 5 x 10 z
-0.8810 × 10 '
-0.1794
x I0
-0.1042
x 10
-0,8805 -0.8806
x 10 : x 10 2
- 0 . 8 8 1 0 x 10 ~ - ( ) . 8 8 1 0 x 10 z
-0.6851
x 10 ~
- 0.8~
x 10 "
-0.8810
~
x 10 2
- 0 . 8 8 ( 0 x 10 : 10 :
x
10 2
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
135
T a b l e II T h e quantities listed in table I for a s p h e r e with radius l. j,(1)
k~(1)
-0.3143 -0.1325 -0.7366 -0.4735 -0.3327 -0.2483
x 10 -1 x 10 -!
-0.8660 -0.8664 -0.8668 -0.8670 -0.8673 -0.8675
x 10 -2
x 10 -2 x 10 -2
x 10 -2
k~(1) x x x x x x
10 -! 10 -I 10-' 10 -i 10- ! 10-1
-0.8707 -0.8706 -0.8705 -0.8703 -0.8702 -0.8702
x x x x x x
10 -1 10' 1010 -i 1010 -1
Gaussian, with a width increasing like V ~ - io, where i0 is an as yet unknown parameter of order unity that originates from initial slip effects [12] (the diffusion argument does not hold for small i, where hi(p) is still concentrated at small p). As long as the loss of probability caused by the absorbing sphere is small, the current density contributions j~ should decrease like the maximum of this spreading Gaussian, i.e. y,--- (i - io) -3/2 .
(4.1a)
A reasonable guess for the first correction is that it should be proportional to the accumulated loss of probability caused by the Jk with k < i, hence be asymptotically of relative order (i - i 0)- 1/2. j , - - - ( i - i0)-3/2[1 - c o n s t x ( i - io)-i'21
.
(4.1b)
We therefore fitted the ji(R) according to the prescription
j, = a ( i -
i0) -3/2 +
b(i-
io) -2 .
(4.2)
The values of a, b, and i i obtained in this way from Jn, J,,+l and Jn+2 are denoted by a n, b~., and i0n, respectively. In this notation we suppressed the dependence on R; in case of multiple solutions we chose the ones that fitted Jn-i, etc., the best. Using these parameters, we approximated the "back flow" correction (4.3)
kC(R) = k ( R ) - (2"tr)-''2 by the sequence of expressions (cf. (2.8), (2.9) and (3.14)) n+2
k,,(R) = c
N i--'l
R)',(R)+
N i=n+3
. -3/2 . - 2 ]. R ' [an(i- ton) + b,,(i- lot,)
(4.4)
136
M.E. Widder, U.M. Titulaer / Growth o f small droplets from a gas mixture
These quantities for n = 13 to n = 18 are given in the second columns of tables I and II. The third column contains the results from a second fit k~(R) of five successive values of k~(R) according to the formula
k~(R) = k ~ ( R ) + d n - " ,
(4.5)
which can be interpreted as an attempt to take further terms in (4.1), and corrections to the diffusion picture, into account. The results in the two last columns of the tables appear to approach a common limit; for R >0.41 the corresponding quantities approach the limit from different sides. Since, moreover, the fit parameters io, a, b, d and a depend only slightly on n, we feel confident in postulating k¢(0.1) = - ( 0 . 8 8 1 -+ 0.001) x 10 -2 ,
(4.6a)
k¢(1) = - ( 0 . 8 6 9 - + 0.002) x 10 -~ .
(4.6b)
The discretization errors in the numerical integrations are about one order of magnitude smaller than the ones quoted in (4.6); moreover they decrease rather quickly with i due to the smoothing of the density profiles ni(P) with increasing i. In table III we present the results obtained from estimates of type (4.6) for the scaled rate
k(R)-k(R)/k(O)
(4.7)
for several R between 0.1l and 21. The errors in the contributions k¢(R) are at most +-0.5%, except for R = 2, where it is about +-1%. Since k¢(R) is only part of /~(R), the percentual error in the latter quantity is reduced by a further factor of approximately [ ~ / 1 2 ] R / l . Also given in table !!I are the corresponding results obtained from the moment approximations D6, D 7 and D 8, described in refs. [1-3]. We also determined the highest D N for which the moment method did not yet break down (e.g. N = 12 for R = 2l). The latter results do not differ too much from the D 8, but only from R = 1.75/on do the higher D N all lie within the error bounds resulting from the procedure in the present paper. Thus, R ~ 1.751 is the point where the moment method becomes comparable in accuracy with the present one. In fig. 1 we show our results for /~(R) together with the D 6 and D 8 approximations of ref. [3] and the result of the straightforward R-expansion
[¢(R) = 1 - Rv'-~/12 .
(4.8)
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
137
Table III The scaled reaction rate k(R) (see (~t.7)) for our variant of the BGK-model, as calculated with the present method and with the moment method in the D 6, D 7 and D s approximations, for several values of R.
R
k(R)
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.25 1.50 1.75 2.00
present method
D~
D7
Ds
0.9779 0.9549 0.9318 0. 9089 0.8862 0.8644 0.8428 0.8220 0.8017 0.7822 0.7363 0.6942 0.6558 0.6202
0.953551 0.919731 0.895256 0.875193 0.857146 0.839967 0.823133 0.806435 0.789823 0.773317 0.732854 0.694165 0.657772 0.623892
0.965732 0.927860 0.901145 0. 879676 0.860641 0.842707 0.825265 0.808068 0.791040 0.774186 0.733091 0.694016 0.657400 0.623402
0.954481 0.915121 0.890376 0. 870690 0.853111 0.836329 0.819790 0.803302 0.786835 0.770433 0.730147 0.691611 0. 655386 0.621688
1.00 "'-.. ,,
0.85 =
-"-:~"~"~~'~:~
A
"~'~'%.+
: o.8o-
<'~'0.75-
-,~'~~ ~
1 0.60|
0.0
I
I
I
I
I
,
,
t
,
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
"'l
2.0
R-->
Fig. 1. The scaled reaction rate/~(R) (see (4.7)) for our variant of the BGK-model, as calculated with the present method (crosses) and with the moment method in the D, (dotted line) and Ds (dashed line) approximations, together with the analytic linear low-R approximation (solid line) and variational results due to Loyalka (stars). The radius R is measured in units of the mean free path/.
138
M.E. Widder, U.M. l~tulaer ! Growth o f small droplets from a gas mixture
Also indicated are the results of a simple variational calculation by Loyalka [8]. The two simple approximations show surprisingly good results for the values of R shown in the figure. The moment method, which works best for high R, shows deviations of up to 5% for D s in the region below R = 0.51; moreover it shows some spurious structure at very low R. The low-R estimates j_(R), defined in (2.22), for kC(R) in the KleinKramers case are given in table IV, together with the fit parameters a and a determined from two successive values and the fitting formula (4.9)
j _ ( R ) = a R '~ .
The results shown in the table clearly exclude a linear R-dependence, as found in the BGK-case, and suggest an exponent 2/3. The D 6 and D 7 results from ref. [3], when plotted on a linear scale, show a small-R singularity of roughly similar exponent and amplitude, but are shifted by a constant (R-~.ndependent) amount. However, they level off beyond R =0.11. At R = 1, where the approximations entering into the derivation of j _ ( R ) are certainly much too crude, but where the moment results should be reliable at least up to a few percent, the estimate ]_ (R) is off by at least a factor of 4. Since the low-R structure of the D,-approximations for the BGK-case turned out to be spurious, not too much significance should be attributed to the agreement of this structure in the present case. The precise analytic structure of k ( R ) for the Klein-Kramers case in the region R ~< 1 remains an open problem, but our results at least indicate that the qualitative difference between the two models found in ref. [3] is no mere artifact of the moment method.
T a b l e IV T h e low-R estimates j (R) (see (2.22)) for the K l e i n K r a m e r s equation, together with the fit p a r a m e t e r s a ---.-11 ~'--• A t%%% ~,,u a ~,~cc ~,~.~,)~ for several vamcs' ~,f K" ~'in units of the velocity persistence length l).
R I 10 I0 10 lO lO
j(R) ' : :~ 4 ~
0.3393 0.6644 0.1319 0.2783 0.5967 0.1284
× × x × ×
I0' i0' 10 " I0 ~ 10 :'
a
a
0.339 0.335 0.297 0.283 0.278
0.708 0.702 0.676 0.669 0.667
M.E. Widder, U.M. Titulaer / Growth of small droplets from a gas mixture
139
5. Concluding remarks The method developed in section 2 for the BGK-case is complementary to the moment method as developed in refs. [1-3]. The older method runs into difficulties for small radii #1", the new method loses accuracy as the radius increases. The most trivial reason for this is that the new method calculates the back flow kc(R ). For large spheres this quantity becomes almost equal to - k 0 ; the reaction rate k(R) is equal to a small difference (of order R -I) between two almost equal quantities, hence small errors in one of them may have large effects. In addition, the procedure used in section 4 for "predicting" the higher jn(R) becomes questionable for large spheres. In (4.1) we treated the loss of probability due to the central absorbing sphere as a small correction on the general diffusive motion; this is certainly incorrect for large spheres, which reabsorb virtually all the particles released near their surface. The observation of an initial i - 3 / 2 decay in the partial return currents ji(R) does not necessarily mean that effects of a finite radius are still unimportant, since the partial return currents of particles released from a plane absorbing wall also exhibit an .-3/ t E-dependence [11]. For a large but finite sphere one would therefore expect such a dependence both for low and for very high i (where the arguments of section 4 should become correct), but it is not clear whether the coefficients are the same, or whether thcrc is a transitional regime in between. The difficulties we encountered for R = 2/, which led us to assign a slightly larger error to our results for that R-value, may be indicative of trouble of this kind. Our method for the BGK-case should be extendible to collision operators that can be written as a finite linear combination of orthogonal projection operators, all but one of them of finite dimension. The number of equations to be solved in each iteration, however, increases linearly with the number of eigenfunctions with eigenvalues different from the one for the infinite-dimensional projector. For all such models, and for collision operators well approximated by such models, one expects a linear initial decrease of/c(R) at small R. The different qualitative behavior for the Klein-Kramers equation is related to the fact that such approximations should not work well for the collision operator c¢K, which has an infinite number of different eigenvalues, without an upper bound [10]. The models discussed in this paper are of course highly idealized ones; some of their shortcomings for describing actual droplet condensation were pointed out in ref. [3]. Since the present paper is mainly concerned with testing and ~'lThe practical importance of those difficulties should not be exaggerated. For radii of the order of one tenth of a mean free path, the treatment of the droplet as a macroscopic sphere becomes questionable, and errors of a few per cent in the treatment of the kinetic problem might become unimportant in view of errors caused by this inadequacy of the model.
140
M.E. Widder, U.M. Titulaer ! Growth o f small
droplets from
a gas mixture
comparing methods of solution we shall not repeat them here. We also refer to refs. [1-3] for a discussion of some other methods that have been used to treat problems of the type discussed here. Further references can be found in a recent review by Sone [13]. Taken together, the methods of the present paper and those of refs. [1-3] allow one to determine the growth rate k(R) with an error of at most about one tenth of a percent, at least for linear kinetic equations and for collision operators not too different from the BGK-operator (in the sense discussed earlier in this section). At least for the BGK-case, comparable accuracies can be obtained with less numerical effort by means of variational methods [14, 5]; unlike the methods discussed here, however, such methods do not easily provide reliable error estimates.
Acknowledgement The work reported in this paper was supported by the Austrian "Fonds zur F6rderung der Wissenschaftlichen Forschung".
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