\.-
I
KARL KAMMERMEYER State University of Iowa, Iowa City, Iowa
Vapor Transfer through Barriers This correlation goes a long way toward a generalized approach and is usable in a practical manner. It does not work in all cases
CARMAN
( 7 7 ) presented a concise summary of the development of ideas on surface diffusion. Volmer (26-28) was the first to report experiments on mobility of molecules at a solid surface, and Damkohler (75) introduced the concept of parallel flow of a gaseous phase through a microporous medium, as consisting of a gas flow through the open space in the pores, and a surface flow (condensed phase) along the walls of the pores. Later investigators (6, 8, 70, 72-74, 25) made good use of Damkohler’s concept. Emphasis was placed mainly on the evaluation and interpretation of diffusion coefficients in conjunction with the application of Fick’s law to the vapor flow. This law is usually written as: dc/dt = D,(@c/W), where D, is often considered as a constant. Babbitt (3) pointed out that if there is an interaction between the diffusing gas and the solid, as in adsorption, the use of Fick’s law as the fundamental equation of diffusion is not valid. His mathematical procedures showed clearly that surface Row is not only a function of a pressure gradient, and therefore of the coheresponding concentration gradient, but is also influenced by concentration. Hence, the amount of vapor adsorbed per unit of adsorbing solid should have a definite bearing on the degree of surface flow, Barriers are either mainly microporous in structure or of a plastic sheet material type. Microporous materials are essentially capillary systems and therefore will adsorb gases and vapors to varying degrees. Plastic sheets or films may possess a certain ampunt of
porosity, the pores will be very small, perhaps about 10 A. in diameter, and the flow through the pores will, in general, not be a major portion of the total flow. Most of the flow through plastic films, particularly with vapors, is considered to take place by successive steps of absorption and solution, diffusion in the solid phase, and desorption ( 7 , 4, 5, 7, 76, 23, 37). The dissolved vapor corresponds to the adsorbed phase in the microporous solid; at least, it behaves in the same manner in respect to surface flow ( 2 ) . There remains the matter of nomenclature. “Surface flow,” “adsorbed flow,” and “condensed flow” have been, and are being, used interchangeably. All three terms are entirely suitable to describe what is going on in a microporous barrier. The vapor transport in addition to that through the gas space in the pores is by flow on the surface of the pores, as a layer adsorbed on the walls of the pores, or as a condensed phase traveling along the walls of the pores. On the other hand, the flow of the (presumably) dissolved vapor in a plastic film is by diffusion. T o use the term “diffusional flow” would not be correct, as this expression includes Knudsen flow, or molecular streaming through the gas space of the pores. The term “condensed flow” is used here to cover both surface flow in microporous barriers and diffusive solubility flow in plastic materials.
by Carman (70, 72, 73)) Barrer (6, 8), Flood (77-79), and in particular Vollmer (25, 30) and their coworkers. Vollmer published a significant article in 1954 on transport of gases and vapors in paper (25), in which the basic ideas used to account for the multiple flow process are well organized. Analysis of Vollmer’s Method. Vollmer was primarily interested in water vapor transmission (WVT) through papers. Even though he carried out his flow experiments, and therefore permeability determinations, at rather low absolute pressures, he obtained some Poiseuille or laminar flow, in conjunction with the Knudsen flow through the open pores of the paper structure. Figure 1 shows a plot of total flow us. relative humidity; as the plot is for constant temperature, the abscissa scale is equivalent to pressure-either upstream pressure on the barrier, or the average of the upstream and downstream pressures. Vollmer’s experiments were carried out with a pressure drop of only 2 mm. of mercury, while the absolute pressure varied from3 = 1 to 17 mm. of mercury, and p / p ~from 6 to almost 100% relative saturation. Curves I and I11 in Figure 1 can be calculated from several sets of experimental data by using basic equations of flow: Total flow = laminar flow Knudsen flow
Interpretation of Flow Phenomena
or
The concept of combined gaseous phase and condensed flow was utilized
where a is a variable fraction. P values represent quantity flowing (in moles,
+
P =
VOL. 50,
PL
NO. 4
+
LYPK
APRIL I956
(1)
697
Figure 1. Vapor transport through a porous barrier (25)
Figure 2. Adsorption isotherm and concentration gradient (25)
The differences between curves I1 and I11 are a measure of condensed flow, and to relate this flow with a con-
698
Correlation of AP with E
ISOBUTYLEVE
grams, standard cc., etc.) divided by the pressure drop, Ap, so that P = m/Ap where m is the total quantity flowing per unit time. The proportions of Knudsen flow, PK, and of laminar flow, PL, can be determined from basic equations given in detail in the original papers (25, 30). The curves in Figure 1 are obtained as follows: Curve I. The permeability toward air (or better toward nitrogen) is measured a t several $ values; this will be independent of humidity, unless water (or other solvent) vapor swells the barrier, and so decreases the pore size of the openings which transmit vapor by Knudsen and Poiseuille flow. If some Poiseuille flow is present, the curve (straight line) will have a corresponding slope with increasing pressure. The average pore radius may be needed in later calculations. This radius has to be obtained by measurements and use of basic flow equations (25). PL is measured at high average pressure drop. PK a t a very low average pressure drop. Probably only one permeability has to be determined for plastic films, as Poiseuille flow should be absent or a t least negligible, even at reasonably high Ap values. Curve 11. This curve is plotted from experimental data and represents total vapor flow as a function of pressure. I t could be plotted as upstream pressure, p , as p/fisstn,or as average pressure, fi. Curve 111. This curve is calculated from curve I. If Poiseuille flow is present, the Poiseuille and the Knudsen components of flow for several pressures have to be determined. The slope of curve I11 will then not be the same as the slope of curve I. If only Knudsen flow is present (plastic film), curve I11 should be parallel to curve I a t a value proportional to the mol. wt. of test gas mol. wt. of vapor
Figure 3.
AE/AP (25)
0
*oo
r
400 HY
em
800
D
2-
409
600
BOO
P Y * He
HO
Figure 4. Methyl bromide and isobutylene through polyethylene at 0" C.
(24)
P
8".
HO
Figure 5. Hydrogen sulfide through ethylcellulose at -35" C. ( 2 1 )
POLWINYL ALCOHO~
NYLON
P/Pe
P/Pe
Figure 6. Water vapor through nylon and poly(viny1 alcohol) at 25" C. (20)
x 4
- ~
-
_
w 0
100
200
MO
4w
Figure 8. Freon through Carbolac plug at 0" C. ( 1 2 )
INDUSTRIAL AND ENGINEERING CHEMISTRY
P MH
HG
P HM
HG
Figure 9. Ethyl chloride and methanol through activated carbon rods at 35" C. (19)
V A P O R TRANSFER centration gradient, an adsorption isotherm must be obtained. Figure 2 shows an isotherm and a curve giving the concentration gradient, AE/Ap. The gradient can be established by drawing tangents to the isotherm at respective values of E . For the examples given in his articlethree paper membranes and one poly(vinyl chloride) film-Vollmer obtained a straight line when plotting the condensed flow AP against the prodmt E(AE/Ap) (Figure 3). It would appear that Vollmer obtained the basis for using this product from Babbitt ( 3 ) , Equation 32, who gave the following equation for diffusion of a mobile monolayer where the mass transfer, Nu, is:
This equation, in effect, states that condensed flow should be proportional to the product of the concentration and a function of the concentration gradient. The most general expression is given by Babbitt as:
Vollmer (3, Equation 6) arrives at the following equation for condensed flow: AP =
Do 2?rr E ( A E / A p )
(4)
where D O = condensed flow coefficient, AE/Ap = surface concentration gradient, and E = concentration of adsorbed vapor corresponding to AE/Ap at the pertinent value of E . To compare Babbitt's differential equation with Equation 4, N(bf ( N ) / b x ) would correspond to E(AE/AF) in Equation 4 . The transformation of bx into Ap can be shown as: aE
AE Ax
---=--
ax
A E Ap A p Ax
Now A P = DoA E ( A E / A x ) =
condensed flow of vapor, where A = surface area of pore = 27rr(Ax) and Ax = distance along pore.
ll0
P x
90
a
80
n
3
=
70
LL
9z
60
50
E 40 z
s
30
20 10 0
KK)
so that
meakrement of a single water vapor transmission. This is Vollmer's claim for Equation 4. Taken as it stands, this claim is incorrect. Equation 4 must be modified for the general case to read : bpi
- APz
where DS represents the slope of the curve, and subscripts 1 and 2 correspond to two points on the curve. Thus, at least two determinations must be made for inert gas permeability and for vapor transfer a t corresponding p or p / p S values. When the E(AE/Ap) us. AP curve is established, the complete relationship between AP and p or p / p , can be calculated. The method does not possess the universal applicability which Vollmer claims for it. On the other hand, when modified, it becomes a valuable tool in analyzing data and in predicting relationships between humidity and condensed flow, and therefore total vapor flow. Testing of Method. Recent publications by Szwarc and others (27, 24, 29) made it possible to analyze Vollmer's method on vapor-polymer systems other than the water vapor-poly(viny1 chloride) system presented by Vollmer.
m = AP
-
=
D O2 ~ Er -A E AP
(4)
Inspection of Figure 3 shows that Equation 4 will hold as such only if the ordinate goes through the intersection of the E ( A E / A p ) curve with the abscissa. When this is the case, the curve can be established by an inert gas permeability measurement, determination of at least a portion of the adsorption isotherm, and
The rather good success with these systems suggested that the flow through strictly microporous systems might be analyzed in a similar fashion. Of course, Babbitt's (3) conclusions were sufficient to suggest such an approach to microporous flow. Condensed flow was correlated with both the amount adsorbed (or dissolved) and the concentration gradient for 10 systems. The vapors and solid barriers used in the systems are listed in Table I. From an over-all viewpoint the basic idea of the correlation worked very well. However, all the systems could not be correlated smoothly by using a plot of E ( A E / A p ) us. AP. The sample systems which Vollmer (25)used happened to be of a type that can be correlated by the E(AE/AF) product. I t was fortuitous that condensed flow did not set in with the paper barriers until a fairly high p / p s value was reached. The initial convex upward curvature of the isotherm shown in Figure 2 would have caused complications if condensed flow had been present in that region. Undoubtedly, it was this set of circumstances which led Vollmer to believe that the E ( A E / A p ) product would give a general correlation. Figures 4 to 9 demonstrate the variations in behavior that will be encountered.
Systems Investigated for Condensed Flow Correlation Temp.,
System
Vapor Methyl bromide Isobutylene Hydrogen sulfide Water vapor Water vapor Sulfur dioxide CF Clz (Freon) CFL!Iz (Freon)
c.
Barrier Reference Polyethylene film (24) 2 Polyethylene film (2.4) 3 Ethylcellulose film (21) 4 Nylon film (20) 5 Poly(viny1 alcohol) film (20) 6 Linde silica plug (18 ) 7 Linde silica plug (12) ' 8 Carbolac (activated car(12) bon) 9 Ethyl chloride 35 Activated carbon red (1Q) 10 Methanol 35 Activated carbon rod (19) King's data (22) on water vapor permeability through keratin membranes also were analyzed successfully, but are not reported in detail. 1
AP
&$
-
Table 1. D O2 ~ r ( A x )X E X A E / A p X A p / A x
200 WO 400 500 600 700 800 900 1000 1100 1200 1300 1400 x IO6
Figure 10. Condensed flow correlation for sulfur dioxide through Linde silica plug at 10°C. (12)
Then m
100
8
O
0 0 - 35 25 25 - 10 -33.1 0
VOL. 50, NO. 4
APRIL 1958
699
Ob
TOTAL
VAPOR
F-OW
E. IN - 0 I IOGG
-.
..
B 01
0
P-
0
10
20
30
40
50
0
n
6
4
+;E
Figure 1 1 . Hydrogen sulfide through ethylcellulose
Figure 13. Water vapor through poly(viny1 alcohol) a t 25" C. (20)
135
80
MM HE
Figure 15. Methanol flow
x 10'
*P
AT76 P
90
45
Figure 14. Ethyl chloride through activated carbon rods at 35" C. (17, 79)
through activated carbon rods a t 35" C. (17)
(27)
E
30
20 IN MM HG E IN G / G IN C C I M I N P
AP
1
0
3
6 & X
I
s
1
~
2
04
AP
Figure 12. Methanol through activated carbon rods (77)
Proposed General Correlation The types and number of systems which could be used to attempt a more general correlation were strictly limited by the availability and apparent quality of reported data. The ten cases presented cover all the types that were found. It was necessary to have a reasonable number of readings, not just a few, for both the adsorption isotherms and the condensed flow us. pressure curves. More systems of the same types are described in the literature (not many more), but these are mostly reported in too limited a manner to warrant analysis. Systems 1, 2, 4, 6, 7, and 8 could be correlated as shown in Figure 10, by
plotting AP against E(AE/A$). The system hydrogen sulfide with ethylcellulose film (Figure I I ) , could also have been correlated, but a plot of AP against E / ( A E / A p ) gave a definitely better correlation. The methanol-activated carbon system (Figure 12) required the E/'(AE/Ap) function for satisfactory correlation. The use of the E/(AE,IAp)correlation can be justified as follows : Babbitt's ( 3 ) general expression is given by Equation 3. At steady-state conditions, where b.V/bt = 0, the expression becomes :
dvu=- - AkT CA
Systems
- dj(iV) dx
Correlated
butylene ond Polyethylene a t r r vapor and Polhinyl
alcohol)*
ter vapor and Pol&inyl chloride)
ter vapor ond Paper oter vapor ond Keratin lfur dioxide ond Silica
Freon and Carbon
Ethyl chloride and Carbon
Methanol and Carbon +
Figure 1 6.
700
(low total flow region)
Methanol and Carbon (high total flow region)
Scheme for selecting correlation of surface flow and adsorption characteristics
INDUSTRIAL AND ENGINEERING CHEMISTRY
V A P O R TRANSFER Now if
of the validity of the procedure can be established only as more reliable data become available. The correlation must be considered as tentative in nature. O n the other hand, with every system tried so far it works, and therein liesjts value.
fW= 1 / N we get
K
K N
--N-
m
E x
A
m
It is immaterial whether the correlation is worked on the basis of 1 / [ E / (AE/Ap)], or by using E/(AE/Ap) directly. The seemingly abnormal behavior of the water-poly(viny1 alcohol) system (Figure 13) and the ethyl chlorideactivated carbon rod system (Figure 14) is explained in the detailed discussions of the systems. To analyze the various factors entering into the proper choice of the correlation, a summary of iystem types (Figure 16) permits selection of the proper correlation as a function of the types of curves involved. If either the adsorption isotherm or the condensed flow curve is composed of more than one sectional type-for instance, an S-shaped curvethe complete curve is subdivided into the types shown in Figure 16, and the sections are treated individually by using the E(AE/Ap) or E/(AE/Ap) correlation. The methanol-carbon system (Figure 12) presents an example of this procedure. A generalized method of correlation is possible by using some characterization factors which will define the slopes and shapes of the type-curves shown in Figure 16. The signs, positive or negative, of the first and second derivatives of thq curves will accomplish this. Equation 5 can then be written as: API
- AP2
=
where the numerical values of i and n are 1, and their signs are determined on the following basis. DETERMINATION OF SIGNSO F i AND n. The sign of i is obtained as the product of the signs of first and second derivatives dE/dp and d2E/dp2. The sign of n is obtained as the product of the signs of dAP/dp and dZAP/dp2. The correlations are then determined automatically from Equation 6-that is, when the product ( i ) ( n ) is + I , the proper correlation becomes E(AE/Ap), and when ( i ) ( n ) is -1, the correlation to be used is E/(AE/Ap). This scheme may appear somewhat artificial, and perhaps it is. However, the first and second derivatives adequately characterize the curve sections and this is what is needed. The extent
Discussion of Individual Systems On the basis of Brunauer’s classification of isotherms (9), the cases shown in Figure 16 can be characterized as: Case I covers Brunauer Types 11, 111, IV, and V, with these limitations. Types I1 and IV only if the initial convex upward portion is not too prominent Types I V and V, if final leveling-off of isotherm is somewhat near saturation pressure-p/p, N 1.0 Cases 11, 111, and IV cover Brunauer Type I isotherm. When various portions of the isotherm curve are equally pronounced, the correlation curve will contain rather definite breaks, while each of its portions maintains the straight line shape fairly well. Breaks or sudden changes in direction in the curve also can be caused by extremes in the shape of the isotherm or of the AP us. p curve. In addition to the five cases listed in Figure 16 there are other possible combinations, but none has been found so far. It is of practical significance that most of the systems encountered belong to Case I. In a very general manner, it is likely that most plastic films, siliceous type of microporous barriers (perhaps most ceramic types), and porous metals, will be covered by Case I. Most, if not all, of the carbonaceous materials will probably belong to Cases I1 to V. It is unlikely, but not impossible, that the condensed flow behavior of plastic films will differ from that of Case I. To encounter a condensed flow as in Case 111, which decreases as the pressure is increased, is surprising. No explanation seems to be known. The behavior of condensed flow responsible for Cases IV and V is caused by a leveling off of the total flow at higher pressures. Because the “gas” flow continues to increase with increasing pressure, a point is reached where condensed flow is a maximum, and then begins to decline. The experimental data for declining condensed flow with increasing pressure are too well substantiated to leave any doubt regarding their validity. The data used in analyzing the various systems were taken, wherever possible. from tables of observed values. In many cases, the data were reported as graphs and carefully enlarged plots were prepared. In some instances (when a point was grossly out of line) individual
readings were smoothed by cross-plotting the original data. The behavior of methyl bromide and isobutylene with polyethylene, of water vapor with nylon, of sulfur dioxide and Freon 12 with Linde silica, and of Freon 12 on activated carbon is represented by Figure 10. All these systems belong to Case I. WATERVAPORTHROUGH POLY(VINYL ALCOHOL)(FIGCRE 13). This system shows extreme behavior in both the adsorption isotherm and the permeability us. p / p Bfunction. There is an almost abrupt change in both curves a t a relative humidity of about 0.6. This behavior is reflected in the AP us. E(AE/Ap) plot; this is not surprising. When such extremes in behavior occur, it is necessary to proceed with caution. Apparently the branches of the plot can be described reasonably well by straight lines, but the curved portion covers too much of a range to be neglected. ETHYLCHLORIDE ON CARBONRODS (FIGURE14). The break in the AP us. E(AE/Ap) curve corresponds to the portion of the isotherm where the initial high rate of adsorption begins to taper off. Again this is a rather sudden change and so it shows up in the correlation curve. Such a sharp break seems unusual. METHANOL ON CARBON RODS. This system at‘ first seems more complicated than any of the others. Figure 15 shows the total flow of vapor and the gaseous phase flow, so that the condensed flow AP is obtained by the differences in the two curves. The condensed flow is given in Figure 9. At the point where the total flow curve levels out-at about p = 100 mm.-the condensed flow becomes directly proportional to p , because the gaseous flow is directly proportional to p . Therefore, a AP us. f[E,(AE/Ab) correlation is required only for the curved portion of the AP us. p curve. Figure 12 shows that a rather good straight line holds for the plot of AP us. E(AE/Ap) u p to a point which corresponds to a pressure of about 45 mm., a little before AP reaches its peak. The correlation is not satisfactory for the very flat portion of the curve between the maximum AP flow and the leveling off point (Figure 12). If ,this portion is plotted as AP us. E/(AE/Ap), in accordance with the rules given in Figure 16, the correlation is much better. Another source of trouble with this system proved to be the rather flat shape of the isotherm (Figure 9) over almost the entire range. It was necessary to make a few very accurate determinations of AE/Ap, and to read additional values from a plot of AE/Ap us. p . HYDROGEN SULFIDEON ETHYLCELLULOSE. The permeability data reported VOL. 50,
NO. 4
APRIL 1958
701
for this system (27) appear to scatter at pressures below about 100 mm,; they do not give a fairly smooth curve over the whole pressure range. This is not surprising, as difficulties in reproducibility of permeabilities have been encountered in previous work with acidic or alkaline vapors. It was thus necessary to draw a smooth curve through the hydrogen sulfide data (Figure 5). The hydrogen sulfide-ethylcellulose system was the only one found which represents Case 11. Figure 11 shows the correlation of AP against both E(AE/Ap) and E/(AE/Ap). The latter correlation is definitely the better. The E(AE/Ap) correlation also must be considered as satisfactory. While this situation may somewhat weaken the argument for the more involved scheme proposed (Figure 16), it confirms the scheme. I t is to be expected that the shapes of the isotherm and of the AP curve will affect the overall flow mechanism. Furthermore, it is unlikely that a single combination of terms will give a satisfactory correlation, and the water vapor flow through poly(vinyl alcohol) shows that the proposed method will not handle all situations. Consequently, the proposed rules are felt to constitute as systematic an approach as is possible at present. Summary
The vapor transfer through barriers can be satisfactorily correlated by the condensed flow (expressed in flow per unit time and area, divided by the pressure drop, Ap) as a function of both the concentration of the adsorbed (or dissolved) vapor and the concentration gradient created by the adsorption process. The condensed flow is the difference between total flow and the flow in the gaseous phase through the open pores of the structure. The gas flow through microporous barriers may be composed of molecular flow (independent of pressure) and laminar flow. Plastic films normally can be considered as having pores of such fineness that only molecular flow occurs and the gas flow can therefore be determined by a single permeability determination with an inert gas-Le., nitrogen-at the respective temperature. A scheme is proposed for the ready determination of characterization factors which, in turn, will determine the type of correlation that should be used. This scheme must be considered as tentative, until more sets of data become available. Five cases have been encountered so far, which represent differences in the shapes of the isotherms and of the curves of condensed flow us. pressure. Ten system from the literature have been analyzed and used to set u p the correlating scheme. T h e greatest utility of the method will probably be in determination of the
702
vapor transmission as a function of humidity, after the general shape of t h i s function has been obtained on initial samples. For subsequent samples of the same barriers and the same vapors, the vapor flow us. humidity curve can be obtained by relatively few measurements. Needed are a portion of the adsorption isotherm at the desired humidity, so that its slope can be determined, two vapor transport readings, and one gas permeability reading. The method will be particularly useful when the adsorption isotherm is determined over most of its range, as it then can be used in subsequent correlations, for the same system of barrier and vapor-for instance, in water vapor transmission of plastic films-or any other vapor transmission work where repeatable determinations of vapor transfer are needed. In the case of barriers whose gas permeability is independent of pressure (molecular flow only), gas flow need not be determined. Correlations can be made on the basis of total vapor flow. This shortcut is not permissible with porous bodies such as papers, ceramics, and porous metals, where there is laminar flow through the pores. Acknowledgment
The support of the U. S. Atomic Energy Commission is gratefully acknowledged. Nomenclature
A = area of surface per unit volume C = coefficient (Equations 2 and 3) c = concentration Do = condensed flow coefficient (Equation 4) D , = coefficient (Equation 5) E = quantity of vapor adsorbed per unit of solid i = exponent (Equation 6) k = constant (Equation 2) m = quantity flowing per unit time N = number of atoms adsorbed per unit volume n = exponent (Equation 6) P = permeability: (std. cc.) (cm.) or as (sq. cm.)(sec.)(cm. Hg Ab) defined separately in figures; sq. cm., area term based on superficial area perpendicular to direction of flow AP = condensed flow only 16 = absolute pressure p = average pressure = ( p l p 2 ) / 2 Ap = pressure drop over barrier r = radius of capillary or pore T = absolute temperature t = time u = velocity of motion relative to surface = distance in direction of flow x
+
GREEKLETTERS CY
X T
INDUSTRIAL AND ENGINEERING CHEMISTRY
= constant in Equation 1;
value between 0.81 if X is less than T . and 1.OO if X is greater than r . = mean free path = 3.1416
SUBSCRIPTS K = Knudsenflow L = Laminar or Poiseuille flow 1,2 = different locations Literature Cited (1) Amerongen, G. J. van, Rubber Chem. and Technol. 20, 479-514 (1947). (2) Babbitt, J. D., Can. J . Phys. 29, 427-36.437-46 11951). (3) Babbitt, ‘J. D., Can. J . Research, Sec. A. 28,449-74 (1950). (4) Barrer, R. M., “Diffusion in and
.
,
through Solids,” Cambridge Press, London, 1951. (5) . . Barrer. R. M., Kolloid Z. 120. 177-90 (1951) (esp. discussion b; F. H. Muller, Marburg, p. 190). (6) Barrer, R. M., Barrie, J. A., Proc. Roy. Soc. (London) 213A, 250-65 (1952). (7) Barrer, R. M., Skirrow, G., J . Polymer Sci.3, 549-63 (1948). (8) Barrer, R . M., Strachan, E., Proc. Roy. Soc. (London) 231A, 52-74 (1955). (9) Brunauer, S.,”Adsorption of Gases
and Vapors,” Princeton University Press, Princeton, N. J., 194.5. (10) Carman, P. C., Proc. Roy. Soc. (London) 203A, 55 (1950). (11) Carman, P. C., S. African Ind. Chemist 9. 115-18 (1955). (12) Caiman, P. ‘C., Galherbe, P. le R.. Proc. Roy. SOC. (London) 203A, 165-78 (1950). (13) Carman, P. C., Raal, F. A , , Ibid., 209A. 38-58 11951). (14) Carman, P. C., Raal, F. A., Truns. Fafaday SOC.50, 842-51 (1954). (15) Damkohler, Gerhard, Z. phys. Chem. 174A, 222-38 (1935). (16) Doty, P. M., Aiken, W. H., Mark, H., IND. ENG. CHEM., ANAL. ED. 686-90 (1944). 117) Flood. E. A,. Tomlinson. R. H., Leger, A. E., Can. J . Chem. 30; 348-71 11952). * - --,. (18 ) Ibid. ;pp. 372-85. (19) Ibid.,pp. 389-410. (20) Hauser, P. M,, McLaren, A. D., IND.EKG.CHEM.40.112-17 119481. (21) Heilman, h‘illiam, T‘ammela,‘Viljo, Meyer, J. A , , Stannett, Vivian, Szwarc, Michael, Ibid., 48, 821-4 (1956). (22) King, G., Trans. Faraday Soc., 41, 479-87 (1945). (23) Levi, D. L., Thomas, A. Morris, Brit. Electrical and Allied Industries Research Assoc., Tech.Rept. Ref. A/T 103 (1948). (24) Sobolev, I., Meyer, J. A., Stannett, V., Szwarc, M., IKD.ENG.CHEM. 49,441-4 (1957). (25) Vollmer. Wilfried. Chem. Ine. Tech. 26. 90-4 f1954).’ (26) Volmer, M:, Adhikari, G., Z. phys. Chem. 119,46-58 (1926). (27) Volmer, M., Adhikari, G., 2. Physik., 35,170-6 (1925). (28) Volmer. M., Esterman., I.., Ibid., 7, 13-17 (1921). (29) Waack, Richard, Alex, N. H., Frisch, H. L., Stannett, Vivian, Szwarc, Michael, IND.END.CHEM. 47;2524-7 (1955). (30) Wicke, E., Vollmer, UT.,Chem. Eng. Sci. 1, 282-91 (1952). (31) Wroblewski, S., Ann. Physik 8, 29 (1879).
-
RECEIVED for review March 13, 1957 .L\CCEPTED August 21, 1957 Division of Industrial and Engineering Chemistry, 131st Meeting, ACS, Miami, Fla., April 1957.
