Supersaturation in the spontaneous formation of nuclei in water vapor

k prr T-









NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL MEMORANDUM 1368


SUPERSATURATION IN THE SPONTANEOUS FORMATION

OF NUCLEI IN WATER VAPOR*1

By Adolf Sander and Gerhard Damkohler


I. STATEMENT OF THE PROBLEM


According to experience, a certain supersaturation is required for
condensation of water vapor in the homogeneous phase; that is, for incep-
tion of the condensation, at a prescribed temperature, the water-vapor
partial pressure must lie above the saturation pressure. The condensation
starts on so-called condensation nuclei. Solid or liquid suspended par-
ticles may serve as nuclei; these particles may either a priori be present
in the gas phase (dust, soot), or may spontaneously be formed from the
vapor molecules to be condensed themselves. Only the second case will be
considered below. Gas ions which facilitate the spontaneous formation of
nuclei may be present or absent. The supersaturations necessary for
spontaneous nucleus formation are in general considerably higher than those
in the presence of suspended particles.

The condensed, thermodynamically stable phase pertaining to water
vapor below 00 is the ice. According to all experiences so far, one must
nevertheless assume in this temperature range the spontaneously formed
primary particles to be predominantly liquid, in accordance with Ostwald's
law of stages. The question is whether this is still valid at arbitrarily
low temperatures, or whether not perhaps, after all, below a certain tem-
perature the nuclei themselves already originate as (minute) crystals.
This problem has so far been treated only theoretically2 but not experi-
mentally. Thus the supersaturation of water vapor required for sponta-
neous formation of nuclei was measured in a temperature range as wide as
possible, namely between +350 and -750.


*"Ubersattigung bei der spontanen Keimbildung in Wasserdampf." Die
Naturwissenschaften, vol. 31, nos. 39/h0, Sept. 24, 1943, pp. 460-465.

From the department for motor research of the Hermann Goring
Institute for Aviation Research.

2R. Becker and W. During, Ann. Physics 24, pp. 719-752, 1935. -
M. Volmer, Kinetik der Phasenbildung, p. 200 ff. Dresden-Leipzig 1939.






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II. METHOD OF INVESTIGATION AND APPARATUS


The operating principle was as follows: Especially purified air
of known pressure, known temperature, and water-vapor content (lying still
below saturation) is adiabatically expanded to a certain terminal pressure.
During this process cooling off and, also, for a corresponding high expan-
sion ratio, supersaturation occur. If the supersaturation is sufficiently
high, fog formation will be observed. The critical supersaturation
( P partial pressure of water vapor at the final temperature
p. saturation pressure of water vapor at the final temperature/
is attained when due to the expansion about one fog droplet per cubic
centimeter and second becomes visible.

The purification of the initial air and the preadjustment of the water-
vapor content were carried out according to the scheme represented in
figure 1. This scheme also contains the various methods of operation used.
The suspended particles of the initial air could be removed optionally
either by a Schott bacteria filter (G5 on 3, fictitious pore diameter
according to Bechhold = 1.48i) or by a layer of absorbent cotton of 100-cm
length (it is true that such a layer of only 10-cm length also had proved
to be sufficient). The drying was done in two exchangeable freezing traps
containing a cotton wool filter of 5-cm length and cooled by liquid oxygen.
The water-vapor saturator through which was sent the entire air or only
a partial flow line (for the latter case the two flow manometers) con-
sisted of two washing bottles connected in series with adjoining moist
cotton wool filter; the entire arrangement was kept at a certain temper-
ature by a Hoppler thermostat.

The final adjustment of the water-vapor content was made in the sepa-
rator as shown in figure 2 which was kept at the same temperature as the
observation sphere b proper. The gas flow leaving the separator a,
being fully saturated there, was no longer completely saturated later in
the expansion sphere b since a pressure drop of about 10-mm Hg appeared
in the capillary connecting tubing between a and b at the steady flow
velocity of 5-cubic cm/sec used for flushing through and filling.

A methanol bath was used as a cold thermostat which was disposed in
a large Dewar vessel (500-mm height, 250-mm inside diameter) with observa-
tion strips. The cooling agent was liquid oxygen which was from time to
time injected into an immersed glass tube g. The stirring was done
mechanically, by means of an electrically driven stirrer of propeller
type f.

The temperature measurement was performed with a Hg-thermometer cali-
brated at the PTR or with a self-manufactered NH -tension thermometer con-

nected to it which was moreover compared with a second model arrangement.






INACA TM 1368


For expansion of the gas under investigation, in the observation
sphere b (0.71) the glass stopcock c (boring 10-mm) was opened quickly
toward a large prevacuum vessel (121) (not shown in figure 2) in which
various pressures could be measurably adjusted. In special tests with an
expansion sphere of the same size as the observation sphere which contained,
however, a metal membrane manometer for mirror reading, it was possible to
determine with the aid of films that the expansion time lasted about 0.1
second and the subsequent time of constant pressure more than 0.3 second.
No gas vibrations were observed with the connecting tubing used (about
2-m length and 20-mm inside diameter) between expansion sphere (0.72) and
prevacuum vessel (12Z).

The observation sphere was coated on the outside with a black lacquer
(graphite + vinidur adhesive solution PC 20) in order to keep off scattered
light. The illuminating light ray came from an arc lamp through a lens
system, entered from below into the observation sphere b through the
observation strips of the Dewar vessel and was lost in the expansion cock c.
In the first tests, we had operated with a small film projector. However,
the intensity of light of that projector was found to be too slight to
recognize reliably the condensate particles which are extremely small just
at low temperatures. The observation was made obliquely from above through
the observation strips of the Dewar vessel.

The content of ions of the expansion gas was either the natural one
or it had been reduced to zero in the customary manner, by applying a field
of about 50 volt/cm. For this purpose, two opposite inner segments of the
observation sphere had been silver-plated and connected with four B-batteries
in series (a 500 v) by platinum fused through the wall.

The observation sphere as well as the entire remaining apparatus could
be pumped out with a low-absolute-pressure aggregate, and pould then, after
it had been left standing for a while, be examined as to density by means
of a Geissler tube.


III. TEST RESULTS


If figure 3, the critical supersaturations measured p /p are rep-
resented as a function of the absolute temperature T. Therein p. sig-
nifies the saturation pressure of the supercooled water as it was taken
from Robitzsch's tables3.. Only for the curve branch on the upper right,
with a jump at the onset, reference was made to the saturation pressure


3M. Robitzsch, AusfUhrliche Tafeln zur Berechnung der Luftfeuchtigkeit.
Leipzig 1941.






NACA TM 1368


of ice, again using Robitzsch's figures. In order to exclude systematic
errors as far as possible the measuring points were obtained by very dif-
ferent methods. There were three possibilities of variation:

(a) The type of air purification and preadjustment of the water-vapor
content according to the scheme in figure 1 (marked by capital Latin
letters).

(b) The type of final water-vapor content according to the scheme in
figure 3 (marked by Roman numerals).

(c) Selection of the initial temperature in the observation sphere
so that for a certain expansion end temperature various temperature dif-
ferences (from 240 to 350) could be adjusted between center and wall of
the sphere.

In figure 3, the measuring points are distinguished only with respect
to variation possibility (b). However, none of the methods used for adjust-
ment of the water-vapor content shows any systematic deviations. On the
contrary, all measuring points lie so satisfactorily about the solidly
drawn curve of mean values that one is quite justified in excluding a
falsification of the measured values by insufficient purification of the
air (variation possibility (a)) or by insufficiently adiabatic expansion
(variation possibility (c)). Only at the lowest temperatures the meas-
uring points show somewhat more scatter the cause of which is, however,
in the poor visibility of the condensate particles, reduced more and more
with decreasing temperature.

In the temperature region investigated, the critical supersaturations
measured pl/P. (speed of nucleus formation J = 1 particle/cubic cm/sec)
can be satisfactorily represented by the following interpolation formulas:


In P1 = 780. 1.521 above -620 without ions4 (1)
p T


In !-= 1.537 above -62o with ions4 (2)
p T


In Pl = 13 3.748 below -620 with or without ions5 (3)



p = saturation pressure of liquid water.

5p = saturation pressure of ice.







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From the curves in figure 3 one can read off:

1. The influence of the gas ions favoring condensation disappears
at -62.

2. At the same temperature, a break in the supersaturation temper-
ature curve appears, in such a manner that the supersaturation pressures
measured at lower temperatures may lie higher but certainly not lower than
one should expect on the basis of the curve branch valid at higher tem-
peratures. (Compare the dashed extrapolation curve.)

In addition to these two quantitative findings there is a qualitative
one:

3. At very low temperatures, one finds a scintillating of the con-
densate particles; at -620 it is observable with certainty, at higher
temperatures one sometimes imagines seeing it. A rigorous temperature
limit for the start of scintillating cannot be defined.


IV. DISCUSSION OF THE TEST RESULTS AND COMPARISON

WITH THE THEORY USED SO FAR


From the quantitative findings 1 and 2, one may conclude that at -620
there starts a more or less sudden change in the spontaneous process of
nucleus formation.

The disappearing of the ion influence below -620 would suggest that
the nucleus forming at lower temperatures is in a higher order state
requiring more space than the nucleus type originating at higher temper-
atures, for surely the ion influence favoring the condensation must be
understood to mean that the water dipoles in the inhomogeneous field of
the ion are attracted and tend to arrange themselves as closely as pos-
sible around the latter whereby part of the surface work to be expended
for nucleus formation is compensated by electrostatic attraction energy.
This molecule grouping of maximum density about a central ion will hardly
be the molecule arrangement which must take place in ice and thus also in
the crystal nucleus as is suggested by the difference in density between
water and ice at 00. It would therefore be understandable if the gas ions
would favor the spontaneous formation of crystal nuclei either not at all
or at least less than the formation of droplet nuclei.

The scintillating of the formed condensate particles, observed with
certainty at -620, also supports the theory of a primary crystal-nucleus
formation although the latter cannot be proved directly by that fact, in
our opinion, for a water droplet, too, could suddenly crystallize throughout






NACA TM 1368


after a certain time and be transformed into a scintillating minute crys-
tal. In what time this would be possible under our test conditions, we
are not able to tell.

The break in the supersaturation temperature curve found at -62
likewise points at a sudden variation in the process of nucleus formation.
However, the direction of this break is strange and in contradiction to
the theory used so far. According to Becker and D'ring as well as
to Volmer- there should always be favored that type of nucleus which
requires for its formation the lesser partial pressure in the vapor phase.
This conception has the advantage of representing a perfect analogy to
the selection of the condensed phase thermodynamically stable in the
respective case where, for a prescribed temperature, there always forms
the phase which possesses the lower saturation pressure. However, the
present report would indicate another process for the formation of the
nucleus because of the required partial pressures, for below -620 there
would have originated precisely that type of nucleus which requires for
its formation a higher water vapor partial pressure than the type of
nucleus stable at higher temperature, as one can recognize by comparing
the extrapolation curve plotted in dashed lines with the actual measuring
points. According to Becker and Doring as well as to Volmer, the break in
the supersaturation temperature curve in.figure 3, seen from below, should
not be convex, but concave; however, this precisely could not be observed
within the comparatively high measuring accuracy.

In the theoretical treatment of the spontaneous process of nucleus
formation (in absence of ions), Becker and Doring as well as Volmer start
out from the same fundamental physical concept: To a vapor molecule,
further vapor molecules attach themselves on the basis of the natural
fluctuations in successive single steps. Thus aggregates of a higher num-
ber of molecules originate each of which may go over into the next highest
aggregate by addition of another vapor molecule, into the next lowest
aggregate by subtraction of a vapor molecule. The process of nucleus for-
mation itself is interpreted as a stationary chain of reactions so that
every aggregate occurs with a certain frequency. Then an expression for
the speed of nucleus formation may be derived, in principle, in a simple
manner. An explicit evaluation requires, of course, certain simplifying
assumptions; they were made in a somewhat different manner by Becker-Doring
and by Volmer. We checked their calculations and arrived under the same
physical presuppositions of theory but on the basis of a somewhat more
accurate calculation at a new formula. It yields numerical values for the
speed of nucleus formation which lie between those of Becker-Doring and of
Volmer. In the absence of ions, we have therefore for the spontaneous
formation of droplet nuclei the following theoretical relations:



6Compare especially the figure on p. 202 of his book (cited in
footnote 2).






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Becker-DSring (= to Volmer II)

Z 01 AK kT (4)
J= V (4)
nK 3kI

Volmer I

IA1KAK K
Z1WlOK AK kT k~T
Se e (5)
nK 3ikT

Sander-Damkxhler

AK i 4 (6)
ZW1OK AK ekT K]
-- e -
2nK 13 k

Therein signify:

J number of nuclei formed per cubic cm per second (= to the number
of fog particles observed per cubic cm)

Z1 number of vapor molecules per cubic cm

W1 number of vapor molecules impinging per second on 1 square cm at

the partial pressure p 1 =


NL Loschmidt number (= 6.0224 x 1023)

R gas constant per g-mol (= 8.315 x 107 erg/deg)

k R/NL = Boltzmann constant (= 1.3807 X 10-16 erg/deg)

M molecular weight of the vapor to be condensed

T absolute temperature

01 surface of the vapor molecule assumed to be spherical

OK surface of the droplet nucleus assumed to be spherical which is in
equilibrium with the external water-vapor partial pressure pl






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number of vapor molecules in the drop nucleus (nK : 100)

vaporization heat per molecule (for water X kJ 7.4 x 10-13 erg)

surface tension

0OK/3 = work of nucleus formation

may, with the Thomson equation


S-- = kT In p
dnK P.


be traced back to
and p. signify:

pl partial pressure of

pO saturation pressure
ature T


the supersaturation


pl/P wherein pl


the vapor to be condensed at the temperature T

of the vapor to be condensed at the temper-


With


OK = CnK2/3


there follows


AK "OK 4 J ( 3C\ 1
T = 3 =T 27 kT

\ P%0

For spherical droplets there applies with the condensate density d


c3 = 36i-M 2 (10)
K;)


and therewith


AK l6NL M\2(, \3 1
3 = 3I n
M 3R3W\; /^


(11)






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According to equations (11) and (4) to (6), there pertains to a
certain supersaturation pJ/P a perfectly defined speed of nucleus
formation J. If the latter becomes 1 particle/cubic cm/sec, we obtain
the critical supersaturation observed in our measurements which is plotted
in figure 3.

In figures 4(a) and 4(b), the experimental supersaturations (in the
absence of ions) of the present report are compared to the theoretically
calculated curves. For the latter, the numerical values of Moser7 were
used for the surface tension of water above 00. They lie highest among
the known values of literature8 (compare fig. 5) and are probably, for
this reason, too, the most correct ones, all the more so because one can
very easily lower the surface tension by slight contaminations with
surface-active matter, but is hardly able to increase it. Below 00 the
surface tension values had to be extrapolated. As may be recognized from
figure 4, our new theoretical formula shows the best agreement with our
measuring points, at least at and above 009. Below 00, however, our first
extrapolation of the surface tension values performed at first arbitrarily
(curve branch b in fig. 5) yields supersaturations which are too high.
We employed therefore the inverse method. Under the assumption that our
new formula (6) correctly renders the experimental data in the entire
temperature range to -620, we calculated from them backward the surface
tension of the water and obtained thus the curve branch c in figure 5.
It is pronouncedly curved; however, in view of the still more pronouncedly
cambered curves d and c of Ramsay and Shields and of Weinsteinl0 for
water, and of the glycerin curve f and gll (glycerin is also strongly
associated) this would not be unthinkable. It is noteworthy that the curve
branch c has a maximum for the surface tension of the water at about -500,
that is, not far from the point where the break in the supersaturation
temperature curve (compare fig. 3) was found.

7Moser, L. B. Eg. IIa, 148.

8Compare also Ramsay and Shields, and Weinstein, L. B. I, 199.

9The slight differences between experiment and theory above 00 are
most probably real and probably based on the fact that in our stationary
chain of reactions for excessive water-vapor partial pressure the molecule
aggregates exceeding the nucleus size are overheated because the conden-
sation heat cannot be carried off with sufficient rapidity. This point, not
yet taken into consideration in the theory used so far, will be discussed
more thoroughly elsewhere.

10L. B. I, 199.

11L. B. I, 255 and L. B. IIa, 156.





NACA TM 1368


For the speed of crystal-nucleus formation Becker and Doring
also had derived a formula which is based on the same fundamental physical
concepts as the formula for the formation of droplet nuclei. It is true
that a considerably larger number of simplifying assumptions was necessary
in the derivation of the crystal-nucleus formula because in crystal forma-
tion three dimensions may grow independently of each other and an aggregate
of n molecules can therefore assume very different shapes, in contrast
to the sphere-shaped droplet. The Becker-Dbring formula for the speed of
crystal-nucleus formation in the absence of ions reads with our above
symbols
AK

J 3 11 e (12)


Therein AK = a0K/3 represents the work of formation of the solid
crystal nucleus. It depends on the interfacial tension between solid
and gaseous phase (still unknown at present) as well as on the surface
of the determinative crystal nucleus; the form of the latter must be as
compact as possible, according to Becker and Doring, but is not exactly
defined. For a cube-shaped nucleus (we, too, shall calculate below with
such a nucleus) there results from equation (8)



S= 6 M2/3 (13)



and hence from equation (9)


AK 32NL 2 3 1
= -- () (in I)2 (14)
kT 3 (in )2



One should not overrate the importance of single numerical values
obtained with the equations (12) and (14); however, a temperature vari-
ation of the critical supersaturation


(l for J = 1 cm-3 8-1)
P.






NACA TM 1368


is significant since equation (12) represents solely the general Arrhenius
expression for a reaction velocity with the activation heat A which
K
appears perfectly plausible and has been assumed by Volmer for the nucleus
formation even before the report of Becker and Doring appeared.

In figure 6, there are plotted as functions of the temperature the
saturation pressure of supercooled water (curve a) and of ice (curve b),
furthermore the critical supersaturation pressures measured in the present
report (curve c) which are required for spontaneous nucleus formation in
the absence of ions, and finally the supersaturation curves d60, d70,

and d80 for the spontaneous formation of ice nuclei calculated with the
equations (12) and (14). The arbitrarily assumed interfacial tensions
a = 60, 70, and 80 erg/square cm between solid and gaseous phase correspond
to those supersaturation curves. All curves were based on the saturation
pressures of the tables of Robitzsch (footnote 3). One can see that the
experimental curve for the supersaturation pressures of the droplet-nucleus
formation can be intersected by an ice-nucleus curve in the temperature
range investigated only when the interfacial tension of the ice crystals
lies approximately between 68 and 72 erg/square cm. If it (the inter-
facial tension) were independent of the temperature, a convex viewed from
the abscissa axis break in the supersaturation pressure temperature curve
would never occur but always only a concave one; however, such a concave
break is precisely what was not found in the experiment. To explain a
convex break, one would have to assume a slight dependence on temperature
of the interfacial tension approximately as it is represented in figure 5
as the curve h12.


12The temperature coefficient to be read from figure 5, curve h:
-di/dT = 0.062 erg/square cm degrees is, with respect to order of magnitude,
completely in accord with a relation indicated by R. Fricke (Zur physikatischen
Chemie, vol. 52, 1942, pp. 284-294)


d= nk Z n i (15)
dT Va


wherein n = number of molecules per square cm surface, vi and va'
respectively = fundamental frequencies of the centers of the molecules
vibrating in the interior of the crystal or on the crystal surface, and
the summation E is to be extended over all lattice vibrations. If one
assumes that only one distinguishable lattice vibration is decisive and
that the molecules situated on the surface are bound normal to it by about
half the spring force as the molecules in the interior of the crystal,
there applies


footnote continued on following page






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We cannot yet state reliably at present how the break at -620, indi-
cated by our measuring points in figure 3, is to be explained. Should it
be based on the transition of spontaneous droplet-nucleus formation to
spontaneous ice-nucleus formation and we have named indications for this
being the case we would have to give up the prevailing notion regarding
the cause of such a transition (that always the type of nucleus forms
which requires the least supersaturation pressure). One will give up
this concept at first only reluctantly, particularly because of the above-
mentioned analogy with the transition from the vapor-water to the vapor-
ice equilibrium. Nevertheless this notion entertained so far, regarding
the transition of one type of nucleus to the other, does not take into
consideration a point which seems to us essential: the mobility of the
molecules in the nucleus surface. A droplet nucleus of almost spherical
shape can form only if the molecules being newly acquired push in between
the surface molecules already present, that is, if they are absorbed by
the surface. In the case of a crystal nucleus, in contrast, such a
pushing-in need not take place since the molecule being newly acquired,
is only added on, that is, in principle, adsorbed. The first process
presupposes a considerable mobility of the surface particles, the latter
does not. If a sort of two-dimensional melting point existed, that is,
if the surface mobility of the particles would suddenly disappear at a
certain temperature, no droplet nuclei could form any longer below this




lIn In 1/r = 0.3464
a

since, furthermore, n water molecules


n .= 8 23) (.33 X 12)3 = 1.033 x 1015



fall to the share of 1 square cm of the crystal surface, one obtains
according to (15)


-da/dT w (1.033 x 1015) (1.3807 x 1016)0.3464 = 0.0494 erg/square cm deg

since the tangential frequencies in the crystal surface also will differ
somewhat from the corresponding frequencies in the crystal interior,
this theoretical value of 0.0494 erg/square cm degree would have to be
increased slightly and would then come surprisingly close to our value,
inferred experimentally, of 0.062 erg/square cm degree.






NACA TM 1368


temperature and the crystal nucleus would be left as the only primary
condensation form, regardless whether the vapor partial pressure neces-
sary for the formation of this crystal nucleus is higher or lower than
that of the droplet nucleus. This conception could explain the strange
break in the supersaturation temperature curve found by us. Also, this
explanation does not perhaps imply an invalidation of the nucleus forma-
tion theory used so far but merely limits in a special manner the temper-
ature range of the droplet-nucleus and of the crystal-nucleus formulas.
The two regions would not overlap, as was assumed a priori by Becker and
During as well as by Volmer; rather, the two temperature regions would
be separated by the melting point of the two-dimensional surface phase.
If we denote it in the absolute temperature scale by TB, corresponding
to the "baking temperature" known from fritting processes and if we denote
likewise by TS the standard three-dimensional melting point, there would
result from our measurements TB/TS = 211/273 = 0.77. This value can
probably be fitted into the sequence determined by Tammannl3.

TB/TS = 0.33 0.52 0.57 0.90

for metals oxides salts C-compounds


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics


13G. Tammann, Z. angew. Chem. 39, 869, 1926 Gottinger Nachr. Math.-
naturwiss. K1. 1930, 227.






NACA TM 1568


Figure 1.- Air purification and preadjustment of the water-vapor content.


Methods of operation

Partial flow Partial flow Position of the
stop cocks
line I line II s p
5 6 7
A Bacteria filter Not used ( G )
S100 cm absorbent
B cotton Not used e
C Bacteria filter 100cm absorbent
D 100 cm absorbent 100 cm absorbent ( (
S cotton cotton






NACA TM 1568


vision thermometer


External air


(a) Separator (b) Observation sphere (c)Expansion stop cock
(d) Silver plating (e)Feeler of the NH3-tension thermometer
(f) Stirrer (g) Cooling tube (h) Mirror


Figure 2.- Apparatus.






NACA T 15368


Method of
operation


Partial
flow lines


Air ahead
of separator


Condensate in
separator spheres


Symbol
without with
ions


I or 2 Too humid

I or 2 Too dry


Yes

Yes


I or 2 Separator not used

I Separator not used


2 Separtor


No

No

Yes

No


not used


0 0








9 t


J.51



3.5 4.0 45 50
103/T-
I i I I I I I I I I


300 280


260


240


220


200K


--ir -e Z'.- >"-.-.--,. S:r:Sa'..'.I rdirg to .. 4.


A-c


A





NACA TM 1568


First extrapolation
of surface tension b





o:Becker and DWring,
o c(Volmer ll)
b- Sander and Damkihler
-c:Volmer I
d: Experiment
Second extrapolation -l/ b
of surface tension C
I


3.5 4.0
103/T
1 1 1 1 1


4.5

s


5.0


320 300280 260 240 220
,---- T


2000K


Figure 4.- Theoretical and experimental supersaturation curves
for water vapor.


C ;


2.0


1.5


1.0


/,0


f


1.5





NACA TM 1568


Erg/cm 2
nbI


-I 'IN


-4 Y =-


h


Ice h Interfacial tension ice/vapor
from experiment and formula
of Becker and Doring fI
I I I I '


-80


I I I
a: Experiments Moser
b: Extrapolation I
Water d: Experiments Weinstein
e: Experiments Ramsay


SGlerf: Experiments
Glycerin
{gly 9: Experiments
-d
e


-


Weinstein
Muller


Figure 5.- Surface tension of water, glycerin, and ice.


OC-


4 I -


-60 -40 -20 0 20 40 60 80 oC 100


_


JZ~


- --' -






NACA TM 1368


I I I I I
280 270 260 250 240


4.4 4.6
103/T .
I I
230 220
B--o T


I I I
210 200 1900K


Figure 6.- Saturation pressures (p .) and supersaturation
pressures (pl) of water vapor.


NAC A-Langley 11-9-53 1000










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