FUNDAMENTALS REVIEW (2F) Graaf, J. G. A. de, and Held, E. F. M. van der, A p p l . Sci. Research (A), 3, 393 (1953). (3F) Hahnemann, H. W., Allgem. Wdrmetech., 4,240 (1953). (4F) Hyman, S. C., Bonilla, C. F., and Ehrlich, S. W., Chem. Eng. Progr. Symposium Ser., 5 , p. 21 (1953). (5F) Ostrach, S.,Natl. Advisory Comm. Aeronaut., Rept. 1111 (1953). (6F) Ostrach, S., Trans. Am. Soc. Mech. Engrs., 75, 1191 (1953). (7F) Poppendiek, H. F., Ibid., 75, 1191 (1953). (8F) Rich, B. R.,Ibid., 75,1191 (1953). (9F) Schmidt, E., and Leidenfrost, W., Forsch. Gebiete Ingenieurw., 19, 65 (1953).
Film a n d T r a n s p i r a t i o n Cooling (1G) Eckert, E. R. G., Heat Transfer Symposium, Univ. of Michigan, Ann Arbor, p. 195 (1953). (2G) Eckert, E. R. G., and Livingood, J. N. B., Natl. Advisory Comm. Aeronaut., Tech. Note 3010 (1953). (3G) Schneider, P. J., J . AppZ. Phys., 24, 271 (1953).
Change of Phase (1H) Baker, M., Touloukian, Y. S., and Hawkins, G. A,, Refrig. Eng., 61, 986 (1953). (2H) Bromley, L. A., Leroy, N. R., Robbers, J. A,, IND.ENG. CHEM.,45, 2639 (1953). (3H) Chu, J. Chin, Lange, A. M., Conklin, D., Ibid., 45,1586 (1953). (4H) Harvey, B. F., and Foust, A. S., Chem. Eng. Progr. Symposium Ser., 5, 91 (1953). (5H) Myers, J. E., and Katr, D. L., Ibid., p. 107. (6H) Rohsenow. W. M., Heat Transfer Symposium, Univ. of Michigan, Ann Arbor, p. 101 (1953). (7H) Schweppe, J. S., and Foust, A. S., Chem. Eng. Progr. Symposium Ser., 5, 77 (1953). (SH) Traupel, W., Allgem. Wdrmetechnik, 4, 105 (1953). 1
.
Radiation (1J) Hand, I. F., Heating and VentiZating, 50,73 (1953). (25) Held, E. F. M.van der, Allgem. Warmetechnilc, 4, 236 (1953). (35) Jordan, R. C., and Threlkeld, J. L., Heating, Piping, Air Conditioning, 25,111 (1953). M e a s u r e m e n t Techniques (1K) Benderskey, D., Mech. Eng., 75, 117 (1953). (2K) Coulbert, C. D., Ibid., 74, 1005 (1952).
(3K) (4K) (5X) (6K)
Kawata, S., and Omori, Y., b. Phys. Sac. Japan, 8, 768 (1953). Kraus, W., A1Zgem. Wdrmetechnik,4,113 (1953). Lowell, H. H., J . Appl. Phys., 24, 1473 (1953). Werner, F. D., and Keppel, R. E., Proo. Third Midwestern Conference on Fluid Mechanics, University of Minnesota, Minneapolis, p. 463 (1953).
H e a t Transfer Applications Brooks, R. D., and Rosenblatt, A. L., Mech. Eng., 75, 363 (1953). Callaghan, E. E., and Serafini, J. S., Natl. Advisory Comm. Aeronaut., Tech. Note 2861 (1953). Callaghan, E. E., and Serafini, J. S., Ihid.. 2914 (1953). Coppage, J. E., and London, A. L., Trans. Am. SOC.Mech. Engrs., 75,1191 (1953). Dunlap, I. R., Jr., and Rushton, J. H., Chem. Eng. Progr. Symposium Ser., 5 , p. 137 (1953). Ellerbrock, H. H., Jr., Schum, E. F., and Nachtigall, A. J., Natl. Advisory Comm. Aeronaut., Tech. Note 3060 (1953). Foust, A. S.,Heat Transfer Symposium, Univ. of Michigan, Ann Arbor, p, 1 (1953). Gelder, T. F., Lewis, J. P., and Koutz, S. L., Natl. Advisory Comm. Aeronaut.. Tech. Note 2866 (1953). Gray, V. H., and 'Bowden, D. T., 'Ibid.', Research Mem. E53C27 (1953). Hahnemann, H. W.,Forsch. Gebiete Ingenieurw., 19, 81. 105 (1953). Huth, J. H., J . AeTonaut. Sei., 20, 613 (1953). Juhasz, I. S., and Hooper, F. C., IND.ENO.CHEW,45, 1359 (1953). Linden. A. J.. Alloem. Warmetechnik. 4. 107 (1953). . , Livingood, J,' N."B., and Eckert, E. R. G., Trans. Am. Sac. Mech. Engrs., 75, 1271 (1953). Lynch, E. P., Chem. Eng. Prom. Symposium Ser., 5. p. 121 (1953). Manson, S. V., Natl. Advisory Comm. Aeronaut., Tech. Note 2988 (1953). Manson, S. V., Heat Transfer Symposium, Univ. of Michigan, Ann Arbor, p. 9 (1953). h'lessinger, B. L., J . Aeronaut. Sci., 20,29 (1953). Parker, H. M., Natl, Advisory Comm. Aeronaut., Tech. Note 3058 (1953). Paschkis, V., and Heisler, M. P., Chem. Eng. Progr. Symposium Ser., 5 , p. 65 (1953). Trocki, T., and Nelson, D. B., Mech. Eng., 75,472 (1953).
MASS TRANSFER f@! R O B E R T L. P I G F O R D University of Delaware, Newark, Del.
The most interesting developments during the past year have been In the field of ion exchange kinetics, including both new experiments and calculation methods. Other important developments have taken place in calculational methods for diffusion and in experimental studies of thermal and molecular diffusion rates.
LTHOUGH a large volume of information is already avail-
A
able on the resistance to mass transfer from a geometrically simple surface to a fluid that flonv past it there is still a need for additional data of this kind, especially if the experiments are carried out carefully. By comparing various hypotheses about mass transfer mechanism with such data more may be learned about the processes by which turbulent exchange and other mass transfer phenomena take place.
Mass Transfer Rate Measurements a t Surfaces Evaporation of water into a turbulent air jet from a porous flat surface parallel to the direction of air flow was investigated experimentally by Spielman and Jakob (8.5). The jet was formed by a rectangular nozzle located about 3 inches from the first May 1954
of 11 porous plates, each 3 inches long in the direction of flow. The best of the data obtained on local mass transfer coefficients was found to be correlated by the empirical equation
(hcx/Du)(pBlv/l') = 0.031 ( z u l p / p ) o . * [ l
-
(2s1/2)o.*]-o.11
(1)
where k, = local mass transfer coefficient, pound-mole per (hour) (square foot) (Pol1nd-mole Per cubic foot), based on the d r ~ i t = i g ~ ~evaporating ~ n ~ strip ~ o from ~ cnozzle ~ ~ outlet, ~ ~ ~ ~ feet x8t E distance of leading edge of first active mass transfer area from nozzle, feet D o= diffusion coefficient in gas, square feet per hour P B M = logarithmic mean partial pressure of air, atmospheres
z
~ ~ ~ ~ $ in air ? jet~passing~ Over~evapo~ rating strip, feetper hour p / p = kinematic viscosity of gas, square feet per hour
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
937
;
~
~
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
:I
THEORETICAL
-
0.2
0.1
I
I
p , = vapor pressure of water, atmospheres p , = observed smallest partial pressure of vapor measured in gas jet above an evaporating strip located a t distance 2, atmospheres p , = uniform partial pressure of vapor in jet issuing from nozzle, atmospheres vo = uniform velocity of jet a t nozzle, feet per hour The jet velocity decreased as the jet expanded in floning unrestrained over the evaporating surface, the decrease in local maxima following the empirical equation (ZJ]/DO)
= 1/1.97 ~
o
.
~
~
~
(2:
The local minimum partial pressure also varied according to
where pa = partial pressure of vapor in room air, atmospheres, and a = vertical width of nozzle, feet. The last tTTo equations permit the local jet conditions above the surface to be related to orifice conditions and, consequently, for evaporation rates into jets of specified original conditions to be estimated from Equation 1.
The new results obtained by Spielman and Jalrob are in good agreement with data published previously b y Maisel and Sherwood (63)from their similar experiments carried out inside a Kind tunnel. The effect of the unwetted starting length, z,~, is that proposed by Maisel and Sherwood and is needed to account for differences in the points of origin of the velocity and concentration boundary layers. As pointed out by Naisel and Shervood in a written discussion of the Spielman and Jakob paper, it is very likely that had the boundary layer been laminar over part of the surface and turbulent over the remainder the distance zerfrom the point of the flow transition in the boundary layer to the nozzle would also have been needed as an additional correlating factor.
938
Apparently 6he boundary layer in Spielman and Jakob’s work was turbulent very nearly throughout it,s length. Licht and Pansing (69) measured rates of mass transport of acetic acid between a continuous water phase and isolated freely falling drops of methyl isobutyl ketone, perchloroethylene, and carbon tetrachloride-oil mixtures. The data seemed to indicate that variations in the time of formation of a drop did not influence the amount of mass transfer during the formation period or that the total mass exchange during this period was much smaller than that occurring during t,he period of free fall. During the eecond period the controlling diffusional resistance appeared to be in the drops when, as in the case of perchloroethylene and carbon Mrachloride, the drop surfaces did not oscillate, and in the continuous phase of water around the drops when, as in t’he case of methyl isobutyl ketone, the drops were distorted and flattened during their fall. Internal movement within the drops appeared to originate during their formation and to persist for about 12 seconds during fall of the 0.24-em. to 0.42-em. diamet,er drops, causing the apparent diffusion coefficient to exceed the true molecular value until the internal currents had dkappeared. The most important factors affect,ing mass exchange to drops seemed to be those influencing “the existence and magnitude of oscillations in the falling drop.” Ingebo (61, 62) continued his studies of rates of evaporation of liquid spheres into air, investigating the influence of gas pressure in the range from 450 to 1500 mm. of mercury. I n correlating the rate data he expressed the heat transfer coefficient, as a function of several variables, including the Schmidt number rather than the commonly employed Prandtl number, and including a nevi group, g L L 2 / C 2 , where gL is the local aceeleration owing to gravity, L is t’he mean free path of the gas molecules, and C is their root-mean-square velocity. It is not clear why thcse quantities should affect the heat transfer or the mass transfer rates, for the lowest pressure mas too high for single-molecule diffusion t o be important, and in order for local gravitational acceleration bo affect the rate natural convection mould have to have been present, in which case other variable3 should also have had an influence. R a k s of mass transfer a t drop surfaces have been thoroughly studied by previous authors as well as Ingebo. The result is especially simple when the fluid around the sphere is stagnant. I n this case the mass transfer coefficient, k,, in gram-moles per (second) (square em.) (gram-mole per cc.), is given by
where T = drop radius, em., and D , = diffusion coefficient, square em. per second. This result is in accord with the well knovn diffusion theory. Bradley (10) has shown how the ordinaiy theory can be modified to allow for van der Waals attraction of the diffusing gas molecules by the molecules forming the liquid drop. The resulting correction is insignificant unless the drop size is of the order of magnitude of molecular dimensions, as it would be in the formation of molecular clusters in a gas. Wilke, Eisenberg, and Tobias (98) measured limiting currents for the deposition of copper on plane, vertical cathodes from unstirred solbtions of copper sulfate, sulfuric acid, and glycerol. Since the cathodic reaction w-as diffusion controlled, the current measurements were, in effect, measurements of the rate of mass transfer of copper ions to the electrode by diffusion through a laminar, natural convection boundary layer. The resulting mass transfer data were found to be expressed b y Sh = 0.673 (Gr.Sc)’j4
(5)
in good agreement a i t h other free convection data. Sh is the Sherwood number, k,z/DL, Gr is the Grashof number, and Sc is the Schmidt number, M / ~ D Lmrhere , z is the height of the surface. The density differences in this case causing natural convection currents are principally the result of differences in concen-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46. No. 5
FUNDAMENTALS REVIEW 1 .o
Vielstich (94) discussed the well-known relationships between equivalent laminar film thickness for momentum, heat, and mass transfer in laminar boundary layers and in turbulent streams inside tubes, pointing out that the ratio of any two layer thicknesses depends on the value of the Prandtl or the Schmidt numbers. Kohler (55) showed how the fluid mechanical methods of Prandtl, von KBrm h , and Sutton could be applied to the evaporation of snow in the open atmosphere.
0.8
0.6 0
5 0.4
0.9
0
x/iuot
Figure 2.
Concentration i n Mixing Zone between Two Liquids Plowing through Tube i n Laminar Motion, According to Taylor (89)
Turbulent Mass Transfer Measurements
Abscissa is ratio of distance measured from original interface between liquids to distance original interface has moved at the average fluid velocity. The symbols are characterized by the following values of Dt/R*:
0 = 0.06
A = 0.02
+
-
Q.02
tration of copper near to and far from the electrode. quently the Grashof number is computed from Gr =
gL(po
-- p i ) p 2 Z s / p i f i 2
Conse-
(6)
pi being taken as a linear function with the density difference of the concentration difference causing diffusion.
Boundary Layer Calculations Paralleling the development of improved experimental data concerning mass transfer from surfaces into fluids theoretical calculation of expected rates has advanced, especially through a paper by Lin, Moulton, and Putman (61), who revised the previously available calculations of turbulent mass exchange inside a cylindrical tube by allowing for the gradual development of weak eddies within the laminar sublayer near the tube wall. The calculated effect of these eddies on the distribution of mean velocity in this region was small because the kinematic viscosity of fluids is large, being about 24.5 times the eddy viscosity according to their estimates, even a t the boundary of the laminar sublayer farthest from the wall. The ratio of eddy viscosity to molecular viscosity was taken proportional t o the cube of the distance from the mall. Although the calculated equation for velocity distribution in the laminar zone is complex the resulting values of u+ and y f do not deviate noticeably from the previously used equation, u+ = y+, which agrees with the best data. Owing to the large Schmidt numbers of liquid systems (very low molecular diff usivity compared with kinematic viscosity) these weak eddies have a pronounced effect on concentration profiles in the laminar sublayer, causing the drop in solute concentration from the turbulent core value to the wall value to occur within a very thin region, thinner than that in which the velocity drops to zero. (The boundary of the laminar zone for mass transfer is no longer equal to y + = 5 but may be much nearer the wall.) A small amount of turbulence near the wall thus has a very marked influence on liquid-phase mass transfer. I n a second part of their paper Lin, Moulton, and Putnam (61) reported very interesting measurements of time-mean concentration profiles in the laminar layer adjacent to electrodes at which difl'usion-controlled reactions were taking place. Figure 1 shows their results, the line representing the expected profile as calculated from the theory just described. The concentration profiles were measured with a Zehnder-Mach interferometer. Since for the system chosen the kinematic viscosity was 900 times the molecular diffusivity (Sc = 900) the agreement between measured and computed concentrations is a good confirmation of the magnitudes of eddy diffusivity used in the laminar zone. May 1954
Radial transport of small liquid spray droplets across a turbulent stream in a cylindrical tube was investigated by Longwell and Weiss (68). The transport phenomenon was found to be described by an ordinary diffusion equation with an eddy diffusivity that was uniform over any cross section, but varied somewhat in the direction of flow. Diffusivity values for a volatile liquid were about 1.1 to 2.0 times those for a nonvolatile liquid, probably due to the inherently lower mobility of liquid droplets than vapor molecules. The authors suggest that this should be expected in view of the failure of droplets, because of their inertia, to follow the rapid random motion of eddies. Concentration distributions from distributed sources could be predicted by super-position of solutions for a point source. The former are useful as an approximation when the source is a spray nozzle. Schlinger and Sage (79) studied the turbulent mixing in a straight tube of a central cylindrical jet of natural gas and an annular air jet. By assuming negligible eddy transport in the direction of flow and uniform eddy diffusivity in the radial direction across the channel they were able to use the conventional diffusion equation to calculate concentration profiles, finding the best values of eddy diffusivity by adjusting the calculated profile to fit the measured concentrations. The ratio of eddy diffusivity to average stream velocity varied from 2.3 X feet a t U = 25 feet per second to 1.9 X feet a t U = 50 feet per second. Use of mass transfer relationships to calculate conditions in a homogeneous reaction space has for some time been a goal of those interested in turbulence problems. A beginning attempt to relate compositions in a turbulent combustion space t o eddy transport equations has been made by Berry, Mason, and Sage (O),who studied local time-mean gas compositions, temperatures, and velocities in a cylindrical combustion chamber used for burning natural gas with air. Their data will need to be interpreted more completely to relate the combustion process to turbulent mass exchange, but they were able to show that nitrogen was distributed radially across the reaction zone approximately as would be expected if the eddy diffusivity were uniform over any cross section. e = 160
D=3
Operating Characteristics of Equipment Rates of mass transfer of diethylamine from a continuous water phase to a dispersed toluene phase were studied by Liebson and Beckmann (60),using 3- to 6-inch diameter towers filled with I/*to 1-inch diameter Raschig rings. Individual values of the height of a transfer unit for the separate phases were computed from the data and are shown by Figure 2. The increased mass transfer resistance of the larger packings was apparently caused by the smaller dispersed-phase holdup in these packings, as found by Gaylor and Pratt (36),for Liebson and Beckmann's
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
939
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT photographs of column internal behavior showed that the drops of toluene were about the same size for all packings larger than 1/2 inch. The smallest, l/a-inch packings seemed to give an entirely different behavior, the drops being large and comparable in size with the space between packing pieces. As a result the height of transfer unit (H.T.U.) values for these packings were larger. Decreasing the ratio of column diameter to packing size had a slightly unfavorable effect, as shown by the figure. Fleming and Johnson (38) found that flow conditions near the inlet to a spray-type, liquid-liquid extractor had a strong influence on total rates of mass transfer when operation vas carried out near the loading point. Flow conditions near the limiting flow rate were the same whether a solute was being transferred between the phases or not. Dobratz, Moore, Barnard, and Meyer ( 2 7 ) measured rate3 of absorption of hydrogen chloride from concentrated feed gas in mater, using a cooled, wetted-wall absorber. The mass transfer coefficients agreed on the average with values expected from previous studies of wetted-mil column performance, but the effect of gas velocity was more pronounced. This may have been caused by the formation of fog in the vapor phase or to tmophase flow conditions inside the tube a t the higher gas velocities, the liquid layer being torn into spray by the gas. Rates of batch absorption of carbon dioxide in sodium carbonate-bicarbonate solutions were measured by Bedaliar (6) using a baffled, stirred vessel. The mass transfer coefficients based on a unit volume of the tank contents varied shaiply with the degree of conversion of the reagent. The rates were greatest at zero conversion, decreased rapidly as the first carbon dioxide mas dissolved, and then increased to maximum values near 29% conversion to bicarbonate. The average coefficients between 0 and 90% conversion increased in proportion to the 0.G power of the power input by the agitator. No explanation vas offered for the extreme variations of the coefficient with degree of conversion. Transverse diffusion of mass and heat in a water-fluidized bed of closely sized glass spheres was studied evperimentally by TVicke and Trawinski (96) using a point source to inject hydrochloric acid or warm water into the main water stream. Diffusion coefficients increased approximately in proportion to the mean fluid velocity, decreased with increasing distance from the source, and increased with increasing particle diameter. The coefficients for the two different processes agreed approximately, although the heat transfer values were slightly higher. Presumably this difference resulted from the fact that the impervious particles were able to transport enthalpy but not mass. Wilhelm (97) gave a general discussion of rate processes, including mass transfer, that are of importance in designing catalytic reactors. Rates of radial diffusion in fixed beds were shoarn t o vary with fluid flow conditions around the solid particles. .4t high Reynolds numbers both air and water gave equal values of the Peclet number, Dpu/Et = 12, where Et is the eddy diffusion coefficient calculated from the lateral spread of solute from a point source. The observed value is about what would be evpected from a side-stepping action of well mixed fluid streams as they f l o ~ around successive solid obstacles. At Reynolds numbers below about 100, a gaseous system seemed to remain n-ell mixed by molecular diffusion, but Et values for a liquid system vere low. I n the latter case filaments of fluid were seen t o flow around solid particles without mixing appreciably with stream. coming from neighboring regions. I n both eases, measured valucs of Et seemed ultimately to approach the molecular diffusion coefficients. Further data on rates of solution of solid paiticles in fixed and liquid-fluidized beds were reported by Evans and Gerald ($1). Chu, Kalil, and Tetteroth ( 1 6 ) measured Fimilar rates of evaporation of naphthalene particles into air. Both groups found that mass transfer and fluid fiiction effects were related.
940
Diffusive mixing and simultaneous longitudinal transport of mass down a tube with the fluid in streamline motion (parabolic distribution of velocity) was analyzed mathematically and some experimental data were taken by Taylor (89). The same transport and mixing phenomena had been observed experimentally by Griffiths ( S 7 ) in 1911 in experiments that he designed to measure very low velocities during flow of a liquid along a glass tube. He injected a drop of fluorescent solution as a marker in the fluid. Griffiths called att'ention to the surprising observation that the color spread out symmetrically from the central small source region that moved along the tube a t the average fluid velocity, even though the local fluid velocities were not uniform through the cross section occupied by the marker liquid. Except for molecular diffusion, this should have caused concentric layers of fluid to telescope over each other; the flow tends to produce a very broad distribution of color along the tube's length. Griffiths correctly pointed out, however, that if the rat,e of radial diffusion in the tube mere fast enough compared with the rate of transport' of mass along the tube the radial variation of velocity ehould have had no effect, as he observed. Taylor (89) indicated that two extreme cases arise in the dispersion of soluble material in a laminar stream; that in which the mean velocity is great enough for radial diffusion between the telescoping layers of fluid to be insignificant, and that in which the flow is so slow that radial diffusion is able t o keep the concentration nearly uniform a t all points in a cross section, in spite of the velocity distribution. I n the latter case he showed that the dispersion of concentration along the tube is governed by a virtual diffusivity, D',given by
B' = R 2 2 / 4 8 D
(9)
where R = radius of tube, ZT. = mean fluid velocity, D = molecular diffusion coefficient. Note that the virtual diffusion coefficient varies inversely as the true local diffusion coefficient. Taylor found mathematical expressions for the following cases involving slow fluid motion; ( a ) dissolved material initially concentrated at a point and ( b )tube initially filled with solvent, which is displaced by a n infinite supply of fluid of concentration Ca. He found that in case (a) the length of tube, I,,, that contains 90% of the injected material is given by
L1 = 4.55
"Jm
(8)
I n case ( b ) the length of the tube, Lz, that contains the rnised region (0.1CO < C < 0.9C0) is given by
L2 = 3.62 "JEt
(9)
where t = time of floiv during which dispersion has taken place. Each of the cases analyzed experimentally-both loiv speed and high speed flow--was confirmed experimentally using potassium permanganate solution and clear wtter in a glass capillary tube. Some of the results are illustrated by Figure 3, which shows the concentration, averaged over the tube's c r o s section, in the mixing zone between clear mater and solution after this zone had moved down the tube for a time, t. Whether one extreme case or the other applies is shown by Taylor to depend on the magnitude of tD/R2, which governs the extent of radial diffusion. The figure brings out the fact that molecular diffusion cause8 the mixed zone to be much narrower than if the flow were fast enough for diffusion t o be insignificant. Khen the flow rate is slow enough for diffusion to have an importmt influence measurements of the concentration distribution in the niixiny zone can be used to find the molecular diffusion coefficient. Values of D determined in this may for poLassium permanganate in Rater were found by Taylor to agree with literature values. Lauwerier (3) solved the mathematical problem of diffusion from a continuously acting line source stretched across a flowing
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 5
FUNDAMENTALS REVIEW I
I
I
I
i l l 1
I EFFECT
OF
PACKING
PACKING SIZE
I
SIZE
COLUMN DIAMETER 3-in. 4-in. 6-in,
V 4 - h RINGS
0
318-in.RlNGS
Q
RINGS
0
0
4 0
0 D
o
S I @ -In. RINGS 3I4-ln.RINGS
Figure 3.
Q
-
ary of zero concentration moves grad ually into the medium. The method can be applied with some difficulty to the cylindrically symmetric diffusion problem, and the results computed bv Fujita show that the rate of movement of the zone of reaction into a cylinder is faster than its rate of progress into a semi-infinite slab. Diffusion coefficients computed by Fujita from Hermans’ data on copper ion absorption by cellulose xanthate are closer to values independently measured than Hermans’ tentative values were. I n addition to reviewing the wellknown basis for calculating rates of mass transfer between gas and liquid phases without chemical reaction, van Xrevelen (56) also described the effect of chemical reaction in the liquid phase on the rate of absorption of a gas treating the case of the kinetically second-order homogeneous reaction
Heights of Transfer Units for Packed Column Solvent Extraction from Liebson and Beckmann (60)
L4SB-C
Data on towera of various dinmeters filled with Rarchig rings used to transfer diethylamine from water to toluene (dispersed). a p = superficial packing area, sq. it., per cu. ft. of packed volume; F = fraction free apace in packing
fluid stream within which the velocity varies linearly with distance perpendicular to the direction of flow u = uo(l
+ ay)
(10)
where u = local velocity and y = distance perpendicular to flow direction. When an absorbing wall is placed at y = - I / a the concentration distribution downstream from the source is given approximately by
7 = av
4
= aaDx/uo
which is a slightly slteived distribution of the error curve-type. Note that q is the line source strength and that 5 refers to distance from the source measured in the direction of flow. The gradual penetration of a diffufiing substance into a material with which it reacts is a mathematical problem of considerable interest to those studying kinetics of gas absorption with simultaneous chemical reaction of the dissolved gas with the solvent (84). If the fluid can be regarded as a semi-infinite stagnant region, and if the chemical reaction can be assumed extremely fast, the mathematical calculations can be carried out in a simple way following a procedure used originally in a heat transfer problem by Neumann and described in standard texts. Several years ago Hermans (43)discussed an analogous problem of diffusion in a polymeric material with which the diffusing molecules react, becoming fixed rigidly to the polymer structure and consequently being removed from the diffusing stream. An example of such a situation is the absorption of copper ions into a cylindrical thread of cellulosc xanthate. The mathematical difficulties inherent in the cylindrically symmetrical problem prevented Hermans from obtaining more than a tentative analysis of his experimental data. Fujita (85) has now shown how the Ranie type of computations used by von Xarman and Pohlhausen for their approximate theories of the laminar boundary layer in flow across a flat plate can also be employed for the diffusion problem in which a boundMay 1954
in the liquid near the interface. His calculations were made on the assumption of the steady state diffusion theory, which assumes that diffusion and reaction occur in a hypothetical film of constant thickness. This film is supposed to have no capacity for retaining molecules that enter it and to lie between the gas phase and a well stirred main liquid mass. In spite of these physically unrealistic assumptions and certain mathematical simplifications, the computed influence of tho reactant concentration of the rate of absorption relative to that without reaction is in agreement with the more exact calculations carried out by Perry and Pigford ( 7 7 ) . Diffusion of dissolved molecules through a dilute solution toward the growing face of a crystal on which they condense was studied mathematically by Seeger (83). Assuming that the plain faces of a polyhedral crystal advance parallel to themselves as they grow and, consequently, that the rate of diffusion of molecules toward any crystal face is uniform over the face, it must follow from diffusion calculations that the actual concentration of dissolved solid molecules is greater near edges and corners of a crystal than near the center of a face. Thus, the degree of supersaturation of the solution is greatestnear the edges and corners, presumably leading to dendritic rather than polyhedral growth in certain circumstances. Seeger estimates that in the case of a cube or a regular octahedron the extreme variation of solution concentration across a crystal face is 25 to 40% of the difference between the concentration in the bulk of the solution and the concentration near the surface a t an edge. Acrivos and Amundson ( 1 ) showed how a Reeves analog computer could be used for computing transient composition changes in a four-stage gas absorber, using a nonlinear and time dependent phase equilibrium relationship. Computations were carried out both for a step function upset in feed composition and for a square wave variation. Agreement within 1% was obtained with values of exit stream compositions computed previously from an analytical solution of a similar simpler problem. Lewis and White (68) showed how the graphical method of Mickley ( 6 7 ) for computation of changes in gas composition and temperature in a vaporizer or condenser handling an inert gas containing small amounts of water vapor could be extended to mixtures for which the psychrometric ratio is not equal to the specific heat. Lewis and White used a modified enthalpy function, as suggested by Mizushina and Kotoo ( 7 0 ) and made
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
94 1
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table I .
Summary of Equations (40) for I o n Exchange, Adsorption, and Heat Transfer i n Fixed Beds
Symbols not defined in t e s t : G = niass velocity of fluid; a = interfacial area per unit volume oi packed space; k:, ic, a n d k’ = rate constants; K” = eauilihriuin constants. Gqriilihriuni Ilefinitions Rate Expression Condition u u c Y r
IC, I 1-ixcd Iled
44 =
at
h a t ‘l’iaiisfor t o I’ixcd Bcd ( A Linear I’ruhlotu)
Adsorption by Fixed Bed, Langntuir Equilibrium Isotherm
step-by-step calculations graphically on gas enthalpy-gas ternperature coordinates. Schumacher (80) continued his st:t.ic. of artic’les on separation by thermal diffusion. discussing the steady of mult,icoinponerit mixture. in a Clusius-Di diffusion column. Ahin distillat’ion, t,he compounds of iiitermediate volatility tend to conceiit,rate in t,he middle of the column. This makes it, possible to introduce an estra component in a mixture fed to a thermal-diffusion apparat,u? in order t o increase the separation between two or more original components. ISxamples demonstrated by Srhuniacher’* calculations include air-carbon dioxide-krypton, krypton-silicon tet~u~luoride-xenon, aiid air-carbon diozI‘de-krpptoii~silicon.tel,nfli~oridP-xerion, in vhich the added components are italicized. In each of thesc cases the original coinporient,s were calculated to be separat,ed very completely. Similarly complete separations ~ c r also c obtained in the case of the argon isotopes of atomic weight? 36. 38. aiid 40 using hydrogen chloride and deuterium chloride as added component’... David ( 2 3 ) cori*idcml tlie powihlc variations in diincnsionil of ;LClueius-type thermal diffueioa us used for the continuous he optimum tliainctere of separation of isot,olm. He dis inner and outer (hot and cold rical surfaces, as ne11 as t,he powibility of varying thc diaineters along the axis of the tuhc. By using tapered tubes in a design he was able tsorrducc t,he calcxlated energy conmmplion and the calculated time required Eo], the production of enrichcd product, as compared nith a tubc operated batchrvise. Rceornincnded diniensioiis \ for columns designed t,o enrich C13Hain natural methane and t o separate t’hehelium ieotopes. Stuke ( 8 8 ) discussed mass exchange clieinical reactions, such as
+
(1iC12N)gab
(C13N--jiiquirl =
(IIC13N)aaa
+
(C125-)liqu,d
including the phase equilibrium constants antl the calculittion of concentration changes in continuous countercurrent equipment.
942
Gillis arid Kedcm ( 3 9 ) gave a aerier wlution for the diffusive mixing of two large fluid i n a s w having initially uniform concentrations and a linear variation of diffusivity wit,h conccntration. Cra,riB ( 1 8 ) publishrtl a, mathematical study of tliffusion in als n-ith particular reference to t,he variations cient that occur during vapor ahsorption owing to swelling of the polymer resulting from t’he internal effect of the eolverit arid stresse? induced by the local swelling and sloustructural clianyc~in the polymeric material. Comparison ol’ calculated and measured rates of absorption of nic:thylcne chloride by polyst,J-reneshoTved t’hat t’he structural changes t,ook placc: i n a time comparablc irith the time required for absorption it,sclf and t,hat the internal stresse. increased the diffusion coeffioit:iit hy a fact80rof two or three. Espcrinieiital measurements of rates of ahsorptiori of incthyleric chloride in polyst’yrene anti cellulose acetate were presented b y Park ( 7 5 ) . These data also showed t,he strong influence of the variation in D during absorption, t,he instantaneous rate frcqumtly increasing after an initial slow absorption during xhicli ahout a third of the final maximum ab3orption had occurred. T h e changes in rate were not obeervcd whcii thc polyrnc.rie materktl initially containrid some of the solvent. In agreement with the analysis of the sanie problem by Crank ( l a ) , t’he cff observed could not be accounted for by a dependence of D oil concentration alone: I ) must have varied with both time arid eoncent,ration. Drechsei, Hoard, arid Long (28) found similar variatioits i n ahsorption rates of acetone in cellulose nitrate and concluded that D must he a function of both time and conccntrat~ion. Rateof absorption were much greater; after sevcral cyclcs of rcpcat’ed abPorpt,ioii arid demrption then thcy wrre initially in the solventfree polynicr. The orientat,ion of the polymer I t,o the surfacc of tlic riitratc films used wax iucr vent, absorption.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 46,No. 5
FUNDAMENTALS REVlEW
Ion Exchange
where
JCxchnnge of ions between solutions and resin particles containing acidic or basic groups fixed in the resin structure has been a recognized procedure for purifying liquids for many years. It, i R only within the last few years, however, that anything like an adequate quantitative treatment of ion exchange as a mass transfer rate-controlled process has become available. Several of the most useful developments have taken place during the last year. Two very valuable papers on the mathematical theory of isothermal fixed-bed adsorbers and ion exchangers were published by Goldstein (40, 41)as a result of work that he did in this country while he was a visiting lecturer a t Harvard University. The first paper dealt with the influence of a finite rate of interphase mass exchange on the curves of exit-fluid concentration versus timefrequently referred to as “break-through” curves; the second was concerned with the limiting case of the general mathematical theory when the interphase mass t’ransfer resistance is so small that phase equilibrium is maintained locally a t all times. The appropriate theory for the simplest linear case, in which the local rate of transfer is proportional to a difference in concentrations between the phases has been known for some time (46,46); it is mathematically equivalent to the problem of the regenerative-type heat exchanger (4,81). The rate expression for this simplest case is linear in the fluid and solid concentrations and so can be solved by simple mathematical procedures. I n actual adsorption or ion exchange units, however, the rate is not a linear function of concentrations, and the equilibrium concentrations are not linearly related, even if the operation is isothermal. Goldstein (40-41 ) has succeeded in showing that solutions for these cases are related to the solution already known for the linear case. Goldstein solves the nonlinear system of equations
( a ~ / a ~+)(av/ay) (dv/dy) = u
- Tli
+
(T
=o
- 1)uu
(12)
F
=
fJ =
exp[(l -
1
T)(Z
N/F,v = l M / F
- y ) I l l - J(z,ry)j
- (1 - r ) ( l - u,) J(z,y) = 1 - e - ”
May 1954
(14)
+
c = concentration of adsorbatr in fluid, equivalents per unit volume of fluid co = total concentration of ions in liquid for ion exchange Q = total concentration of ions in solid phase for ion exchange, maximum capacity of solid phase for one-component adsorption V = superficial linear velocity of fluid e = fractional free space in bed ha = heat transfer coefficient,based on a unit volume of packed space cP = specific heat, subscripts f and s denoting fluid and solid, respectivdy T = temperature, subscriptsf, s, 1, and 2 denoting fluid, solid, entrance fluid. and initial solid. resuectivelv X = distance through bed in t h i d i r e c t i b of fluid flow. I
~~
The function J(z,y)is that found in previous papers on the linearized problem; numerical values can be found in Brinkley and Brinkley ( I d ) and on charts published by Hougen and Watson (48), Hougen and Marshall ( 4 7 ) , and Marshall and Pigford (66). A study of the concentration patterns calculated from these new equations shows that a t least in the equilibrium case ( k or k‘ very large) the wave of concentration in the solid bed is more diffuse and less abrupt than it would be for the simple, linear approximation. Rates of ion exchange between an aqueous solution of sodium ions and beds of Dowex 50 sulfonated polystyrene in the acid form were measured by Gilliland and Baddour (Sa). The rate seemed to be expressed correctly by the bilinear expression (Table I), which reduces a t equilibrium to the mass-action expression for univalent ions. Gilliland and Baddour pointed out that rate constant k in the bilinear expression was affected partly by diffusional resistance in the solid phase and partly by that in the liquid. The former may predominate for large resin particles; the latter depends on the fluid velocity around the particles. The data were expressed by the dimensional equation
(13)
where u and v refer to local concentrations in the fluid and in the bed, respectively, as defined by Table I ; x and y are the coordinates of position and time, respectively; and T depends on the phase equilibrium constant. The first equation is a statement of the material balance on a differential segment of the bed; the second expresses the experimentally determined rate of mass t,ransfer as a function of the concentrations in the phases. Goldstein solved two problems: ( a ) in which the input concentration, U O ,is const>antfor all times after the operation begins and (6) in which the input concentration is constant after the operation begins and until a time, T,when it falls to zero and is maintained there for all later times. The first problem corresponds to the gradual consumptiori of absorptive capacity by a fixed bed initially free of absorbed material; the second, to adsorption followed by elution from the partially spent bed. In ( u ) the Bolutions for all cases listed in Table I can be expressed by u =
q = concentration of adsorbate on solid, equivalents pcr unit
volume
l/kD2 =
+
’010 Reo.*4 0.049
+ 720
where k = rate constant, as given in Table I, in cc./(meq.) (second); D = particle diameter, Re = Dvp/p; V = superficial fluid velocity; and ~ / p= fluid kinematic viscosity, square cm./second. Rates of exchange were measured for both directions of the exchange reaction. Increasing the value of co decreased the rate constant approximately in accord with predictions. The expression adopted for the solid-phase diffusional resistance, unlike that of Vermeulen (92),is mathematically correct only if there is rapidapproach to uniform composition throughout each resin bead. Consequently, the empirical equations for k may be expected to hold best when the diameter is small and the solid-phase resistance is small. Nevertheless, the empirical equation correlated all the data very well for particles ranging in diameter from 0.022 to 0.100 em. The total transfer resistance ranged from about eight times to about 1.1 times that in the solid phase alone. Further theoretical consideration of performance curves for fixed-bed ion exchangers was given by Vermeulen (Qd), who discussed especially operations in which slow diffusion of ions inside the resin particles controls the rate of mass exchange. By introducing a simple mathematical expression that closely approximates the true series equation for the rate of change of the bulk concentration of a sphere that is exposed to a concentration a t its surface, Vermeulen showed how parameters z: and 8, equivalent to x and y, respectively, in Goldstein’s nomenclature, could be defined. According to Vermeulen Z = (4x2D,/dp2)Dc(eX/V)
A”
and 0-9
In (2 4.G)ds
where DG is the distribution coefficient, qmps/cD.
INDUSTRIAL AND ENGINEERING CHEMISTRY
943
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT iilthough the surface concentvat,ion of the resin particles will not be constarit, !Then there is a finite liquid concentration in equilibrium with the solid phase (I< ti infinit,y), Vermeulen usetl the same solid-diff usion equation. Extending his calculations, he presented dimensionless plots of effluent concentrations with the phase equilibrium constant as a parameter. His results show that because of the relatively rapid diffusion during the initial penetration of solute just under the surface of the solid particles and before the diffusing ions reach the center of the particles in appreciable numbers, the previously used "linear" approximations, such as that, of Gilliland and Baddour (58)give too high effluent concentrations near the beginning of break-through. I n a companion experimental study Vermeulen and Huffman (98)determined break-through curves for nonaqueous solvents containing organic bases, which displaced hydrogen ions from a sulfonated polystyrene resin (Dovrex-50). Using t8heequations previously derived by Vermeulen (Qd), values of the solid-phase diffusion coefficient, D,, were computed by comparing t,he sharpness of experimental curves with those characterized by different values of 2. The apparent values of D , varied more than 1000fold vihen the solvent used to feed an organic amine to the bed was changed from water to acetone, There was no apparent correlation of D, with the extent of swelling of the resin or with any common physical property of the pure solvents. The equilibrium absorptive capacity of the resin, as determined from the rate measurements, was nearly independent of the nature of the solvent but was only about half the equilibrium capacity measured after prolonged contact between liquid and solid. Thus, a slow transfer process appeared to follow the fast one that determined the behavior of the bed in the flow experiments, not all tho sulfonic acid groups being quickly aceespible. Wetting the resin particles with water before using them in an immiscible solvent increased the rate, An interesting illustrative example showed how the generalized curves could be used for design. Rates of mass transfer between aqueous solutions containing sodium ions and cation exchange resins mere considered briefly by Pepper and Reichenberg ( 7 6 ) . For lo^ sodium ion concentrations the rat,e of exchange vias governed by the liquid-phase resishnce, but for concentrations greater than 1-normal diffusion inside the particles was rate controlling. Diffusion coefficients were sixfold smaller in a strongly cross-linked sulfonated polystyrene resin than in one less completely cross linked, but the activat,ion energies were about equal for the two resins and were approximately the same as for diffusion of the ions in waterabout 6 kcal./gram mole. The lower diffusion coefficients in all the resins than in water were explained by assuming that not all the viater of hydration in the resin is available for internal diffusion of the ions, about, five molecules of water being tied to each sulfonic acid group. Studies of rates of diffusion of sodium and bromine ions through membranes of ion exchange resins pvere also reported by Schlagl (78). He found, as had Pepper and Reichenberg, that diffusion coefficients of both sodium and bromine ions were lower in the resins than in water. I n a cation exchange resin containing sulfonic acid groups the sodium ions diffused more slowly than the bromine a t low concentrations, apparently because the cations were held tightly by the resin while the bromine was not adsorbed. At higher concentrations the more numerous bromine ions competed with sulfonic acid groups for sodium, and the sodium diffusion coefficient was larger. With an anion exchanger the concentration effects Trere opposite, an increasing concentration of salt, increasing the mobility of the anion and decreasing t'hat of the cation. Dickel and Meyer ( 2 6 ) also measured rates of exchange of hydrogen and alkali metal ions between water solutions and films of resin materials. They found that the rates of exchange were limited by the rate of diffusion of ions t,hrough the fluid to the resin surface. Dickel and Bohm (66) discussed the use of ion exchange resin
944
in countercurrent, equipment, referring to a previously pu1)liskiotl derivation ( 2 4 ) of equations for change9 in the relative proport ion of ions along the column. ApplicaiioriH to t,ht monium and potasqium ions and oC prascoilynium and rieotlyniuiii ions were discussed. Weiss ( 9 6 ) pointed out t'he advant'age in adsorption operations of using solid pellets having porous out,er shells. I n moving-bed operations t'he particles must l x laiger than ahout, 0.3 mni. in diameter, and the diffusional resistance inside these partidei: may limit the efficient use of the adsorbent, unless the Furface of the particles is treated chemically to increase their porosit LIinty, Ross, and Weiss (69) discussed the technique of surf preparation of sulfonated polystyrene beads and the protiuctii of surface-porous particles of diatomaceous earth coated on quartz particles as centers. The finely divided surface coating was lield t o the quartz by a thin layer of wat'er and an immiscible solvent such as amyl acetate. Minty, LIcSeil, Ross, Sninton, antl R c (68) discussed the use of these composite adsorbent particles or separation of penicillin from aqueous solut'ion hy partition chromatography, displacing the adsorbed penicillin ions hy a pho+ phoric acid-amyl acetate solut'ion. 7 .
Measurements of Ordinary and Thermal Diffusion Coefficients Self-diffusion coefficients of liquid carbon disulfide were measured by Koeller and Driclramer (63, 64)using pressures up to 10,000 atm. At these conditions t,he liquid \!-as compressed by an amount approximately equal to the free volume that is present at, low pressure. As a result the range of free motion available to the diffusing molecules is very much restricted and the diffusion coefficient drops rapidly after the density exceeds values about, 30% greater than t'he normal value. The enthalpy of activation also increases, as does the act'ivation volume. The latter increaw means that in t'he very dense fluid the local expansion needivl to permit a molecule to diffuse is felt by a greater number of surrounding molecules than a t low pressures. Koeller and Driclramer (68, 54)also measured liquid diffusion coefficients of tagged carbon disulfide through several organic liquids a t high pressures. The enthalpy, entropy, and volunir of activation computed from the data show some irregularities not observed in the one-component' measurements and are intrrpreted by the authors in ternis of increased order in the solution as the density increased. In every case, however, t,he diffusion coefficient became smaller the higher the pressure. Similar measurements Trere also made by Cuddeback, Koeller, and Drickamer (10)and by Cuddeback and Drickamer (19) using tritiated water in natural water, as well as solutions of various tagged sulfur compounds in wat'er and in alcohol. In these aqueous solutions the effects of pressure on D are more complicated than those observed in t,he same laboratory for nonaqueous systems, probably owing t,othe existence of icelike, tetrahedral structure in the aqueous liquid. The same general downward trend of D with increased density was observed, but inaxima antl minima in D were found, possibly because the tetrahedral structure of water molecules collapses when one water molecule diffuses. The st'ructure of aqueous salt solutions appeared to be even less stable toward diffusion under pressure t,han did water. Cummings and Gbbelohde (22) used the Stefan evaporating liquid technique t o measure diffusion coefficients in hydrogen, nitrogen, oxygen, and argon of several aliphatic and cyclic hydrocarbons ranging from Cg to C I ~ . Effective collision diameter$ of the hydrocarbon molecules were computed from thc r c s u h by subtracting the known collision radii of the srveep-gas molccules from the diameters of the gas pairs. The yesult,s showed that normal and ipoparaffins of the same molecular weight had the same radii and thus must have been mhstantially crumpled in the vapor p h a x For each of the heavier normal parafins, but, not for the cyclic compounds or the isomeric paraffins, the rolli-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol, 46, No. 5
FUNDAMENTALS REVIEW Pion diameter of the hydrocarbon molecule was 4 to 8% greater for collisions with hydrogen than for collisions with nitrogen, oxygen, or argon. Thus, encounters with hydrogen appeared t o involve inelastic collisions which transferred translational energy into internal degreees of freedom. Simihr inelastic collisions appear to occur between ethylene and hydrogen, but not between ethane and hydrogen. Strehlow (87) measured diffusion coefficients as a function of temperature for four binary systems involving hydrogen, helium, argon, and n-butane in order t o compute Lennard-Jones molecular force constants. Brandstaetter (11) discussed the possibility that traces of materials dissolved in water may hinder the passage of gas through a bubble-liquid interface and thus influence the rate of diffusion of gas from a bubble into a liquid. Andrussow ( 3 ) presented another in his series of articles concerning transport processes in pure gases and gas mixtures, pointing out that if diffusion, coefficient, thermal conductivity, and viscosity are taken t o be proportional to a power of the absolute temperature the three different exponents are related. He list’ed the empirically determined variation of each exponent with temperature for common gases and showed how one exponent can be calculated from another if the effect of temperature on the ratio of specific heats is also known. Moyle and Tyner (72) measured diffusion coefficients of 2naphthol in water, obtaining values about 10% higher than those given b y McCune and Wilhelm ( 6 5 ) ,who uaed this substance t o study turbulent diffusion in packed beds and fluidized systems. The Schmidt number of 2-naphthol in water varies from 1605 to 142 as the temperature decreases from 70 O to 10 C. The viscosities of mixtures of hydrogen or helium with other gases were measured a t 18’ C. by Heath (48). The variation of viscosity with composition is not linear, a small amount of the smaller molecules producing a sharp rise in p. The effect was found to be in accord with a formula of Chapman, based on the assumption of an inverse nth power attraction between molecular pairs. One of the const,ants in the formula is proportional to the value of D for the g i s pair, which consequently can be determined indirectly by fitting the formula t’o the mixture data. Heath found that the values of D thus obtained agreed with the values measured directly. The same conclusions had previously been reached by Wilke (Yg), who preferred to use the more exact formulas of Hirschfelder and his associates. Heath concluded that the simple formula for mixture viscosities is probably as good as the more complex one. Franck (34) showed how an old formula due t o Maxwell for calculations of the electrical conductivity of binary, heterogeneous alloys from conduct,ivities of the constituents can be used for estimating viscosities and self-diffusion coefficients in suspensions and dense gases. Results computed for dense carbon dioxide from rather arbitrary assumptions agreed rather well with the measurements of Drickamer and Timmerhaus (30). Othmer and Thakar (74) correlated literature data on diffusion coefficients in dilute liquid systems, finding that D L was related to liquid viscosity and molecular volume and that the temperature variation of D L can be expressed as a straight line on log (DL) versus log (pL) coordinates. Thermal diffusion coefficients( D T )in pure carbon dioxide and pure methane a t pressures up to 160 atm. were measured by Caskey and Drickamer (15) using radioactively tagged isotopes of slightly different molecular weight. The results show that for each substance the coefficient changes sign twice as the density is increased along an isotherm passing near the critical point. Predictions based on the kinetic theory of gases with the most advanced form of molecular interaction forces, modified to allow for the influence of gas density, failed to account for the effects observed. However, the thermodynamic approach usually referred to as the “thermodynamics of irreversible processes” correctly predicted the effect seen. Hirota and Xobayshi ( 4 6 ) measured thermal diffusion coeffiMay 1954
cients for mixtures of hydrogen and nitrogen and found that, especially a t high prcssurcss, it way necessary to allon- for turliulcnce in the laminar flow inside their column, its suggestd hy Ilrickamer, Mellow, and Tung (29). Srivastava and IIadan (86) re-examined literature data on gaseous thermal diff usion coefficients for argon and found that the concltants in the Lennard-Jones formula varied with the temperature. Banigan ( 5 )showed that solvent extraction of resin from Guayule rubber followed the well-known, uneteadg-state diffusion relationships, the diffusion coefficients being about of the magnitude expected for liquid systems.
Diffusive Separation Processes Clusius and Ramirez (17) demonstrated the separation of rare earths in aqueous solution by taking advantage of differences in the mobility of their ions. The apparatus consisted of a horizontal tube filled with the aqueous solution and with a porous packing material to reduce convection currents. An electric current flowed through the aqueous solution in the tube between a cathode chamber a t one end and a n anode chamber a t the other. There was also a slow mass flow of the electrolyte in the direction from cathode to anode and those elements having greater ionic mobilities were found to be concentrated a t the cathode. Lanthanum was quantitatively separated from neodymium and samarium; praseodymium was separated from neodymium and samarium; and samarium was partially concentrated. Separation of gas mixtures into fractions that diffuse rapidly and slowly through another gas was described by Schwertz ( 8 2 ) . According to Schwertz, “the gas mixture to be separated flows along one side of a porous barrier and the condensable vapor along the other. If the pressure drop across the barrier is zero the process is referred to as free double diffusion; if the pressure drop is different from zero the process is referred to as forced double diffusion.” After solving the mathematical problem of molecular diffusion of a single gaseous component from a central cylindrical stream of uniform velocity into an annular stream having the same velocity, Schwertz applied the result to the free double diffusion process by assuming that the presence of two simultaneously diffusing components could be treated as though each had no effect on the other and that the presence of a finely perforated barrier could be allowed for b y multiplying each of the gaseous diffusion coefficients by a constant factor. Separation factors as large as 3.4 were obtained in a laboratory apparatus used for separating hydrogen from coke oven gas with steam as a sweep vapor. The variation of the separation factor with the fraction of the feed gas allowed to pass through the porous membrane was similar to that expected from the diffusion theory. Further studies of rates of permeation of pure gases through porous glass were reported by Huckins and Kammermeyer (49). The data indicated that a t average pressures greater than about 0.1 of the,vapor pressure a small fraction of the total movement of gas occurred by adsorption and two-dimensional movement along thc surface of the pores. By extrapolating to zero pressure, rates of molecular effusive flow were obtained; these varied with temperature and molecular weight a[: expected for ideal gas kinetic theory. I n a second paper Huckins and Kammermeyer ( 6 0 ) showed that permeability values measured for pure gases could be used with appropriate material balances to compute the product gas composition streaming through a porous plug as a function of the fraction of the feed gas that is permitted to flow through. Equations were given for binary and ternary mixtures and the first of these wap confirmed experimentally using mixtures of hydrogen with nitrogen, oxygen, and carbon dioxide. Brubaker and Xammermeyer (13) measured rates of permeation of five pure gases through various plastic membranes. -4lthough
INDUSTRIAL AND ENGINEERING CHEMISTRY
945
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT the otisorved effect8 of membrane thickness, pressure drop, and tjcmperaturr: wure expected, there r a s no apparent, way of (~rr(:latirig t>hcI :d)solute rates with plnst,ic propeities.
Interface Phenomena Maximum rates of masp transfer at a liquid-gas interface were studied experimentally by measuring the rates of evaporation of pure glycerol into an evacuated space (91). The fat'es were found t o correspond rather closely w-ith t,he values computed from the Knudsen equation, which assumes that the rate is equal t o the t'otal rate of impact of molecules from a vapor at a pressure equal to the equilibrium vapor pressure of the liquid. In other words, in a situation involving true liquid-vapor equilibrium every vapor molecule that strikes the liquid surface is asmmed to condense. The rate of evaporation of a liquid is therefore assumed to be equal t'o the maximum rate of condensation of its equilibrium vapor. The ratio of the measured evaporation rate to the expected rate is called the evaporation coefficient, which Trevoy's measurements show is equal t o unity a t least for freshly formed surfaces of pure liquids. This is in contradiction t,o data previously published by Wyllie (99) and by Alty ( d ) , Tvhich show that the rates of evaporation of pure polar liquids from stagnant pools correspond to evaporation coefficients as low as 0.01. Previous observations by Hickman and Trevoy (44,46) have also shown that torpid liquid surfaces-Le., those that are old or contaminated-may evaporate slowly. The apparent lack of agreement between experiments on freshly formed surfaces and on stagnant ones suggests that there may be some kind of gradual change in the structure or composition of the surface of a polar liquid after it is first formed; the interface molecules graduaily become lese active. The subject needs further careful study. Hickman and Trevoy (44,46) discussed some of their previously reported experiments and presented new data on rates of cvnporafrom torpid and working liquid surfaces.
References Acrivos, A , and Amundson, S . R., IXD.Exo. CIIRX.,45,467-71 (1953). Alty, T., Phil. Mag,, 15, 82 (1933). Andrussow, L., 2. Elektrochem., 57, 124-30 (1953). Anzelius, h.,2.angew. Math. M e c h . , 6, 291-4 (1926). Banigan, T. F., Jr., IND.ESG.CHEY.,45, 577-81 (1953). Bedakar, S. G., J . - 4 p p l . Ciiern. (London), 3, 524 (1953). Behr, E. A , , Briggs. D . It., and Kaufert, F H., J . Phus. C'lre~iz., 57, 47G (1953) Berman, A . I., and Harris, J. S.,ReE. Sci. Pnstr., 25, 21-9 (1954). Berry, V. J., Mason, D. M ~and , Sage, B. H., IKD.ENQ.C1imfe, 45, 1596-1602 (1953). Bradley, K.S.,J . Phgs. Chein., 57, 307-9 (1953). Urandstaetter, F.. Oster. A k a d . TViss. Matlb.-natuiw. R I . Sitzber., IIa 161, 4-6, 107-130 (1952); A p g I . Alech. Rea., 6 , 578 (1953). I3rinkley, S. K., Jr., and Brinkley, K. F., Math. Tables and Other Aids to Computation, 2, 221 (1947); cf. also Brinkley, S.K., J . A p p l . Phys., 18, 582-5 (1947). Brubalier, D. W.,and Kammermeyer, I
