192 
air parcel: (1) the transport of water-vapor mole- 
cules towards the surface of the drop, and (2) at 
the same time the transport away from the sur- 
face of the drop of the latent heat released during 
condensation. 
The problem has been considered successively 
by various investigators who took into account 
more and more nearly the actual conditions which 
prevail in the atmosphere. Thus Houghton [1933] 
derived the equation for drop growth neglecting 
curvature, hygroscopicity, and heating caused by 
release of latent heat; Langmuir [1944] took cur- 
vature and heat transport into account but neg- 
lected hygroscopicity; while Howell [1949] and 
Best [1951] took all three factors into account. 
Even Howell neglected some terms and treated 
the problem as an equilibrium problem. In addi- 
tion, as pointed out by McDonald [1953], Howell 
made an error in his treatment of the hygroscopic 
effect. 
The differential equation for drop growth is too 
complicated to integrate by analytical means. 
Howell resorted to numerical and graphical tech- 
niques for its solution. He treated three cases 
involving various nucleus distributions and rates 
of cooling. With the availability of the electronic 
digital computer it is possible to deal with the 
equation in a somewhat more rigorous form than 
that used by Howell, in addition to using the cor- 
rect value for the hygroscopic factor. In addition 
we thought it would be desirable to try other as- 
sumptions regarding nuclei distributions and 
rates of cooling. As will be discussed subsequently, 
it turned out that the equation which Howell 
used is an adequate approximation, but that even 
with the electronic digital computer, the integra- 
tion of this equation is extremely time consuming. 
The equation of drop growth—The solution of 
transport processes in a system with nonuniform 
distribution of concentration of molecules and 
temperature must come from the kinetic theory. 
The theory is fully treated by Chapman and Cow- 
ling [1952]. The main development is based on 
the knowledge of the function giving the distri- 
bution of molecular velocities and of the effects 
of molecular encounters on this function, and ul- 
timately on the solution of Boltzmann’s equation 
for the velocity distribution. From this solution 
the transport processes in the system can be de- 
termined. 
In applying the theory it was necessary to make 
the following assumptions to reduce the com- 
plexity and thus decrease the mathematical diffi- 
culties. 
NEIBURGER AND CHIEN 
(1) A binary mixture of gases is considered, 
water vapor and dry air, and both constituents 
are treated as perfect gases. 
(2) The air parcel is considered a closed system 
with respect to mass. No exchange of matter is 
allowed between the parcel and its surroundings 
(that is, no entrainment), while exchange of en- 
ergy is permitted. 
(3) Drops in the air parcel do not affect each 
other, so that the field around each drop is con- 
sidered to have radial symmetry. 
(4) Condensation takes place only on nuclei, 
and these nuclei behave as though all are com- 
posed of the same hygroscopic substance, taken 
to be NaCl. 
(5) The gas that is immediately in contact with 
each drop is in equilibrium with the drop, so that 
the temperature and vapor pressure over the drop 
are completely determined by the properties of 
the drop. 
In Chapman and Cowling’s treatment it is 
shown that the flux of one substance, in our case 
water vapor, through another, that is, the ‘dry’ 
air, is given by 
Fur = —DVp» — mynD7V T/T (1) 
and the flux of heat is given by 
Fy = —KVT + nkT(C, — ©) Dr/D (2) 
The second term on the right in each of these 
equations is small compared with the first for the 
case of dilute mixtures such as water vapor in air. 
For the case of spherical symmetry, the trans- 
fers of mass and heat to the surface of a drop of 
radius r are obtained by integrating the above 
equations from the drop surface to infinity, giving 
dM 4rDr [fe er 
— = 4rDr(pr, — por) = a 3 
7 Dror per) R, (= =) @) 
= 4rKr(T,, — T+) (4) 
The law of diffusion treats diffusion as a con- 
tinuous process, implying that the individual dis- 
placements of the molecules are of infinitesimal 
length. When the mean free path is longer than 
the radius of the droplet, the question arises 
whether the same diffusion law can also be ap- 
plied. The problem is further complicated by the 
fact that at the iquid-vapor boundary molecules 
are evaporating as well as being condensed. 
Langmuir [1944], based on an equation used 
for calculating the rate of evaporation of sub- 
