GROWTH OF CLOUD DROPS BY CONDENSATION 
stances in high vacuum, introduced a compen- 
sated diffusion coefficient of the form 
ope see ec eat fa Peo 
2 | Dir + ax) | or R.T 
to take account of these effects, where a, the Cun- 
ningham constant, is about 0.7, and \ is the mean 
free path. 
Frisch and Collins [1952], in their investigation 
of the growth of aerosol particles, examined the 
boundary condition at the surface of the droplet 
appropriate for the solution of the diffusion equa- 
tion. A modified boundary condition was intro- 
duced which, when applied to the diffusion co- 
efficient, gives 
Dre = Dir/(r + y)] 
The parameter y is defined as 
y = (1/a)(¥/d) 
where \? is the mean square free path of the diffus- 
ing molecules, and @ is the probability that a 
molecule striking the droplet is absorbed. 
Rooth [1957] obtained a modified diffusion co- 
efficient of the form 
Dr = Dr/(r + 8) 
The thickness s of the layer through which mo- 
lecular diffusion acts is given by the relation 
8 = (D/a)V/2r/R,T 
When this expression for s is substituted into 
Langmui’s compensated diffusion coefficient, 
D, takes a form similar to Dg or Dre: 
2) | 
The actual value of y or s is subject to con- 
siderable uncertainty. Anderson [1957], based 
on some experimental cloud chamber data by 
Barrett and Germain [1947], suggests a value of y 
of the order of two microns. The value of s is 
taken by Rooth to be five microns at 10°C and 
1000 mb, based on the value of a = 0.036 ob- 
tained by Alty and Mackay [1935] by measure- 
ments of the rate of evaporation of drops. As 
Rooth points out, if the true value of a were larger, 
then y or s would be proportionally smaller and 
its effect would then soon approach the limit of 
meteorological insignificance. Because the correct 
values will remain uncertain until accurate data 
193 
regarding the exchange of water molecules across 
a liquid-vapor interface are available, modifica- 
tion of the diffusion coefficient as well as similar 
modification of the thermal conductivity has not 
been included in this study. 
The heat which must be transferred from the 
drop surface consists of the latent heat released 
by the condensation and the heat produced by 
friction of the drop falling through the air, less 
the heat stored in the drop and the energy re- 
quired for increasing the surface area of the drop 
lH We 8 rrég?ps(ps — pa) 
_ CH penne 2p sea 7’ps\Ps — p 
dt dt 27 mn 
(5) 
4 e al, 3 dr 
3 TT’ ps6 al omor at 
It is readily shown that all the other terms are 
very small compared to the first term on the right 
in (5). Neglecting them and combining (4) and (5) 
we obtain 
3 eeran/gt 
20 == CI =46) (6) 
K 
where 
6 = (psLr dr/dt)/KT, . 
The vapor pressure of a small drop containing 
a soluble nucleus is given by 
20 Tpnro® 
p>, = €s(T',) | ex 1 iC 
er = s( (esp tz) ( sis ) (7) 
where the second factor expresses the increase of 
vapor pressure due to the curvature, and the 
third factor the reduction due to the effect of the 
solute. 
It is convenient to replace the drop tempera- 
ture T,, by the ambient temperature 7’, in (7). 
This is accomplished using (6) and the Clausius- 
Clapeyron relationship. The result is 
Meer pi 
ee als ereciees) 
’ 20 1 Tpnto? 
ee rR Tans) per’ 
Substituting this expression and M = 4zr%p;/3 
in (3) and dropping the subscript » we have 
Rypsr dr e e(T) Lé 
= exp = 
D dt T TQ-+6) R,TQ + 4) 
: 20 1 Tpnro? 
&xP arRaT( + 8) par 
(8) 
(9) 
