MASS TRANSFER C. R. WILKE
AND
J. M. PRAUSNITZ
Universify of California, Berkeley 4, Calif.
R
ESEARCII and publication in the field of mass transfer has continued actively during the past year. This article summarizes the most interesting contributions of a fundamental nature. Some articles, including those on general performance of packings and other mass transfer equipment, have been omitted, or treated more briefly than otherwise, assuming that more complete coverage of practical applications will be given in the various Unit Operations rcviews.
Molecular Diffusion in Gases and Liquids
A valuable revicw of fundamentai material transport equations i s presented by Opfell and Sage (69). The various types of diffusion coefficients are defined and equations for their interrelationship are presented. Particular attention is given to the effect of hydrodynamic velocity resulting from the diffusion process on the value of the various diffusion coefficients. General relations employing fugacity as the driving force for diffusion are presented along with limiting forms applicable in the case of ideal gas behavior. Application of these relations is employed by Carmichael, Sage, and Lacey (19) in the measurement of diffusion coefficientsin some hydrocarbon systems. Experimental R ork on diffusion in gases a t high pressures has continued during the past year. O’Hern and Martin (68) measured the diffusion of CIAO2 and C’202 over a range of pressures from near atmospheric up t o 200 atni. and temperatures from 0” to 100” C. A quasi steady state method was employed involving diffusion between two chambcrs connected by a porous metal plug. Over the conditions studied, the product Dp ( D = diffusion coefficient; p = gas density) was essentially independent of pressure a t constant tempmature. This result is in agreement with data of Becker, Vogcll, and Zigan (11) for diffusion of NZ15in N214 a t 20 and 90 atm. The latter investigators found that Dp for C1302 in C l 2 0 ~ increased 30% with pressures up t o 52 atm. and attributcd the behavior to formation of a dimer in the carbon dioxide sgstcm. All these results cast doubt on the general validity of the Enskog dense gas correction (26). In the case of O’Hern and Martin’s system Dp would decrease a t 200 atm. to about onc half the value a t atmospheric pressure if the data were in agreement with the Enskog theory. On the other hand, Jcffries and Drickanier (44) obtained satisfactory agreement mith the Ensltog theory for carbon dioxidemethane mixtures a t pressures up to 200 atm. as reported in the previous issue of this review. These conflirting results indicate need for further study and clarification of this important problem. Lee and Wilke (4g)reviewed existing methods for estimating diffusion coefficients of gases and vapors. Comparison of various methods with a body of selected experimental data indicated the procedures outlined by Hirschfelder, Bird, and Spotz (37) crith force constants based on viscosity to be most satisfactory, although the Arnold method ( 9 ) gave nearly comparable results. An empirical modification of the Hirschfelder, Bird, and Spotz method is presented. Li and Chang ( 5 2 )suggest a modification for the Eyring theory March 1956
for viscosity to give a proper meaning to the relative velocity involved. This result leads t o a new expression relating the coeficient of self-diffusion and viscosity:
where D = diffusion coefficient 7 = viscosity k = Boltzmann’s constant T = absolute temperature N = Avogadro’s number V = molar volume of the liquid u = number of closexst neighbor to the diffusing or flowing molecule in all directions = number of closest neighbor to the diffusing or flowing molecule in one layer Comparison of Equation 1 with available sclf-diffusion data gives 2u of about 2~ for most systems corresponding t o
a value of
u - r
cubical packing of the rnoleculcs. New data on the effect of pressure on diffusion in liquids were obtained by Watts, Alder, and Hildebrand (77). The selfdiffusion coefficient of carbon tetrachloride was determined over a temperature range of 25’ to 50” C. a t 1 and 200 atm. The group Dq/T was constant within 7% over the range of exprrimental conditions. Longsworth (65)studied the diffusion of hydrogen deuterium oxide, urea, glycine, alanine, dextrose, cycloheptamylose, and bovine plasma albumin in dilute aqueous solutions over a temperature range from 1’ to 37’ C. The Stokes-Einstein radius increased by as much as 8% and decreased by as much as 4% for various substances over the temperature range. This behavior is equivalent to variation in Dq/T. The temperature coefficient for diffusion was less than that predicted by the Stokes-Einstein relation only for solute molecules so small that the solvent (water) cannot be considered a continuum. Diffusion coefficients among a wide variety of organic substances and for various materials in aqueous solutions were measured by Lewis (61). The data were compared with some of the methods for estimation of diffusion coefficients. Effect of stirring in the diaphragm-cell constant is interpreted on the basis of forced convection-mass transfer theory. Stirring is believed to have been inadequate in some experimental studieR reportcd in the literaturc. An empirical correlation of diffusion constants for nonionic substanccs in dilute solutions was developed by Wilke and Chang (’78). The diffusion factor, l’/Dv, is expressed as a function of the molar volume of the solute and the molecular weight of the solvent. -4distinction is made between behavior in nonassociated solvents, such as hydrocarbons, and associatcd solvente, such as water and alcohols, by means of an “association parameter” which defines an effective molecular weight for the solvent. Reasonably satisfactory correlation of data for a wide variety of systems is obtained.
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FUNDAMENTALS REVIEW
...
For the first time a general scheme for engineering design of columns for gas-liquid contacting operations
cu
'Q X cu
t
P
0 0
Figure 1.
N= I
0.0500WT. %o S.A.M.
N=3 N=5
+
#=IO
x
Nf14
Gas absorption by columns of spheres (55)
Solid line refers to best line for single spheres; dotted line refers to theoretical equation ( 5 5 ) for single spheres based on penetration theory; N, number of spheres stacked vertically
Turbulent Diffusion Turbulent transport of momentum, heat, and mass in ducted coaxial streams was studied theoretically and experimentally by Alexander and others (6-8). Semi-two-diinensional solutions of the equations of transport involving two empirical coefficients were obtained by application of Reichardt's hypothesis. These solutions were used to interpret the experimental radia 1 and axial temperature and velocity profiles obtained when a heated jet of air from an 0,898-inch nozzle (shandard ASME) was discharged into a Cinch steel duct, well insulated over its entire 10-foot length. The second empirical coefficient, which is a measure of the limitation imposed by the duct on the growth of the scale of turbulence, was unimportant. The experimental momentum- and energy-flux profiles were, in effect, correlated by a single spreading coefficient, which was 0.0480 for momentum transport and 0.0545 for energy transport. Forstall and Gaylord investigated momentum and mass transfer in a submerged water jet which was axially symmetric (28). Momentum and material diffusion were measured for a round water jet issuing into stationary water; the measurements were
610
made with an electrical conductivity cell using sodium chloride as a tracer. The bchavior of the watcr jet was the same as that found by others for an air jet issuing into air; therefore, constants obtained from measurements in air can be applied to water. The turbulent Schmidt number was in the range 0.75-0.85 which is in agreement with earlier results of air jets ( 1 ) . Radial velocity and concentration profiles were closely approximated by error curves. A theoretical study of turbulent jet mixing of two gases a t constant temperature was presented by Pai (60). Using mixing length theory, Pai derives the fundamental equations of both twodimensional and axially symmetrical, turbulent jet mixing and suggests a general method of solution. For an axially symmetric jot from a small opening the radial distributions of velocity and density can be approximated by error functions. The ratio of the turbulent cxchange coefficient for density distribution to that for velocity distribution is larger than 1.0 and may be as high as 2.0. This ratio rises as the difference in density of the two gases incremeil. Turbulent mass transfer in gases flowing through a fixed bed was studied by Fahien and Smith ( 2 7 ) and by Plautz and Johnstone (61). Fahien and Smith measured thc eddy diffusivity of carbon dioxide in air for various pipe and packing sizes and considered its variation with radial position. The modified Pcclet number DpulE, where Dp = pal tick diameter, u = velocity, and E = eddy diffusivity, inrreases from the center toward the pipe wall; the increase is probably due to the radial variation of the fraction voids. However, if the ratio of packing diameter to tube diameter is less than 0.05 the Perlet number is essentially invariant rvith radial position. Plautz and Johnstone made measurements for sulfur dioxide diffusing into air in an 8-inch tube and found that the Peclet nuinher is constant a t a value of about 12 when the modified Reynolds number excceds 500. Heat transfer measurements made in the mme apparatus indicated that the modified Pcclet number for heat transfer is about 25% less than that for mass transfer. Croockewit, IIonig, and Kramers ( 2 0 ) used the technique of frequency response analysis t o measure longitudinal diffusion in liquid flow through an annulus between a stationary outer cylinder and a rotating inner cylinder. A dilute aqueous solution of ammonium chloride was fed a t one end in such a manner that the salt concentration varied sinusoidally with time. From continuously recorded clectrical conductivity measurenients made a t the entrance and a t a point down-stream, D (the apparent longitudinal diffusivity) was calculated. The dimensional quantity, Dn-'Rl-'(RZ - Rl)-l, where n = angular velocity of rotor, Rl = rotor radius, R, = stator radius, is equal to about 0.16 for moderate rotor velocities. .4t high rotor speeds the regular vortex pattern changes to a more random turhulence and the dimensionless group decreases, probably because of the diminishing scale of turbulence. The authors suggest that if U L / 2 D >> 1, where u = mean velocity and T, = reactor length, the influenre of back-mixing is negligible. Diffusion and Convection
A comprehensive mathematical review of the cquations of change of fluid mechanics as a general framework for the description of heat and mass transfer processes is presented by Bird, Curtiss, and Ilirschfelder (16). The fluxes of mam, momentum, cnergy, and chemical kinetics are expressed in terms of differential equations for the flux vectors. Expressions for the flux
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MASS TRANSFER
where & = rate of absorption for the column C* = concentration of solute a t the liquid-gas interface C , = initial concentration of solution in bulk liquid A = area of liquid-film surface tc = time during which an element of surface is exposed t o the gas D = diffusion coefficient of solute in the liquid
To check the validity of Equation 3 values of the diffusion cocfficient of sulfur dioxide in water nere calculated from observed rates of absorption. The relatively good agreement of D so obtained with other reported measurenients of D is comidered to be “persuasive” evidence in favor of the penetration theory. A companion study ( 5 6 ) with short columns 1 to 5 em. in length revealed evidence of apparent stagnation of the surface in the lower 1 cm. of length above the point of liquid discharge. Such stagnation causes low rates of absorption and could result in unreliable measurements .cr-ith short columns. Stirba and Hurt ( 7 4 ) measured the absorption of pure carbon dioxide into nater in wetted wall toners and the dissolution of organic solids coated on a tube wall into liquid films flowing dot7 n the tube. The data were interpreted in each case on the basis of the appropriate diffusion-convection equationq, and values of the apparent diffusivity were calculttl ed from thc mass transfer rate?. Values of the apparent diffusion coefficients were several times larger than the niolecular diffusion coefficients at Reynolds numbers as lorn as 300. The authors suggest that the results indicate existence of some turbulence a t Reynolds numbers where the flow would normally be considerrd completely laminar. Because of the law of molecular diffusivity, mass transfer rates can be affected by a small amount of eddy diffusion, an cffect M hich might not show up in the corresponding heat transfer experiment where the thermal diffusivity is usually much larger than the mass diffusivity. Further, the results cast some doubt on the usurtl assumption tliat the molerular and eddy diffusivity are additive in defining the total diffusion flux. X further study of the penetration theory was made by Lynn, Straatemeier, and Kramers ( 5 7 ) for absorption of pure sulfur dioxide into water in laminar flow over single and vertical rows of spheres. On the basis of a derived expression for the thickness and surface velocity for liquid flow downuard over a sphcrc the penetration theory permitted the prediction of absorption rates with less than 10% error for single spherrs over a range of diameters from 1.00 to 2.94 cm. Results obtained for vertical columns of spheres placed one upon the other are extremely interesting. A s shown in Figure 1 the derivation from the single sphere equation is not large for a colunin containing as many as 14 spheres. The dotted and solid lines in Figure 1 are for single spheres. It may be concluded from these data that nearly ncgligible mixing of fluid streamlines occurs a t points of contact between the spheres. Mass transfer and psychromctrie studirs with various gaFes were made by Lynch and W l k e ( 5 4 ) in an effort t o determine the effectof the Schmidt number in turbulent flow processes. Water was vaporized into air, helium, and Freon-12 in turbulent gas flow over a wet-bulb thermometer and through a tower (1 foot in diameter) packed with 1-inch Raschig rings. The wet- and drybulb measurements indicate that t hc gas-film mass transfer coeficient varies with the Schmidt number t o thr - l/z power for flow perpendicular to single cylinders.
C. R. WILKE received a B.S. in 1940 from the University of Dayton, M.S.in 1942 from State College of Washington, and Ph.D. (1944) from the University of Wisconsin. Since 1946 he has been a t the University of California where he is professor of chemical engineering and chairman of the division of chemical engineering. H e is a member of the AIChE, whose junior award he received in 1951, the ACS, Electrochemical Society, and ASEE.
JOHN M. PRAUSNITZ, assistant professor of chemical engineering at the University of California (Berkeley) attended Cornell and the University of Rochester. His Ph.D was obtained from Princeton University, where he served as instructor for two years. Prausnitz’s main interests are in rate processes in chemical reactors. H e is a member of the American Institute of Chemical Engineers.
vectors are given in ternis of the tlansport coefficients and the latter are related to intermolecular forces. A review of mathematical relations for the dk namic, teniperature, and diffusion boundary layers in laminar flow of gases along a flat plate is given by Berman ( l a ) . Observations of flow patterns and mass transfer around single spheres suspended in a flowing fluid are described by Garner and Grafton (30’). Local Sheruood numbcrs for dissolution of benzoic acid into water were determined by follouing the change in size of t h e spheres photographically. hssuniing the limiting condition at zero flou to be that for free convection the following equation is proposed for average niass transfer coefficient? over the sphere as a whole:
+
k d = 0.50 ( G ‘ S C ) ~ . ~0.48 ~ Reo 5Sc0.33
Dwhere 12 d
D G’
Sc
(2)
= mass transfcr coefficient = diameter of sphere =
diffusion coefficient
= Grashof number for
inass transfer
Schmidt number Re = Reynolds number =
This equation differs someuhat from that recommended by R a m and LIarshall for gas-phase transfer (63). Jlowever, this is not unexpected since the Schmidt number-Grashof number product docs not enter as a simple 0.25-power function for Schmidt nunibers near unity ( 1 4 ) . Available data on liquid-phase free convection from vertical plates aIe presented by Ibl ( 4 2 ) in a general review of free and forced convection -mass tranwfer correlations. Good agreement n.ith earlier free convection correlations ( 7 9 ) was obtained over a range of G’ from 11 to 2 X 102 and Sc X G’ from l o pto 10l2. A nunibcr of studies of mass transfer in wetted wall towers have been reportrd ( I ? , 29, 55-67, 7 4 ) . Interest in this type of apparatus has bcen stimulated in efforts to test various theoretical developments. One of the most significant investigations appears to be that of Lynn, Straatcmeier, and Kramers ( 5 6 ) who measured the absorption of pure sulfur dioxide into water and aqueous solutions of hydrochloric acid, sodium bisulfite, and sodium chloride. Laminar liquid flow was assumed in all rases and small concentrations of Teepol, an alkyl sulfatr wetting agent, were used to prevent rippling. Results for relatively long columns were interpreted on the basis of the unsteady state-penetration theory of Higbie ( 3 4 ) according to which the rate of absorption is espresscd by the equation:
(3)
March 1956
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671
FUNDAMENTALS REVIEW Heights of atransfer unit, H.T.U., were measured in the packed tower for flow of the gases countercurrent to water over a range of gas and liquid rates. When compared at equal-gas-flow Reynolds numbers a t constant liquid rate, H.T.U. varied as the 0.9 power of the Schmidt group. When compared a t equal values of pu2 ( p = gas density, u = velocity), H.T.U. varied as thc 0.47 power of the Schmidt group. With reference to the psychrometric study, the latter method of comparison of H.T.U.’s seems preferable and indicates that further study of criteria for dimensional similitude in packed columns may be needed. A similar investigation was conducted by Yoshida (80) who vaporized water into air, and helium and carbon dioxide in a 4inch diameter tower packed with 1-inch Raschig rings. At conditions corresponding t o equal-gas-flow Reynolds numbers, H.T.U. was found to vary as the Schmidt number t o the 0.77 power. However, if the Schmidt number for helium-water vapor is taken as 1.33, corresponding to the value used by Lynch and Wilke, the Schmidt number exponent becomes 0.9 on B Reynolds number basis. A cross plot of 1I.T.U. us. Schmidt number a t u l / p = 1000, based on Figure 5 of Yoshida’s paper, shows 1I.T.U. t o vary as the Schmidt group to about the 0.44 power. Therefore, it may be concluded that tho results of Yoshida and of Lynch and Wilke show essentially the same Schiiiidt number effect, although the most suitable method for correlation of gas-film data may not yet be established. The inside resistance to mass transfer in drops of water falling through carbon dioxide gas was studied by Hughes and Gilliland ( 4 1 ) . Mass transfer data for carbon dioxide absorption were obtained a t sevcral temperatures with several different dropforming tips. The results can be separated into two contributions: absorption during drop formation and absorption during fall of the drop. The results for absorption during fall were correlated by a straight line on a semilogarithmic plot of Sherrvood number us. reduced time of fall. The authors propose the following mechanism: Surface effects cause a relatively stagnant film which effectively “insulatcs” the drop. Oscillations of the drop cause internal motion reducing the stagnant film thickness to less than 1% of the drop diameter. At any time the Sherwood number is approximately proportional to the amplitude of the oscillations which are causcd by the ‘[snapping back” of the tail of the drop just as separation from the tip occurs. These oscillations decay exponentially with time. Interfacial Resistance Evidence in support of the existence of a considerable resistance to mass transfer at a liquid-liquid interface is given by Sinfelt and Drickamer (7.9) in a study of the effect of molecular properties. The diffusion of molecular sulfur across a saturated liquidliquid interface for sevcral organic liquids was studicd using a radioactive tracer technique. The interfacial resistance does not depend on the polarity of the liquids nor on the interfacial tension. The results show that high interfacial resistances are encountercd with liquids which show a high degree of hydrogen bonding. Free energies of activation for transfer across the interface were computed and found to be in the range 11 t o 16 kcal. per gram mol. with the higher energies corresponding to the cases where hydrogen bonding was considerable. The authors suggest that interfacial transfer in systems involving negligible resistance is analogous to van der Waal’s adsorption while for systems involving high resistance the process is comparable to chemisorption. Transfer between Two Fluid Phases An extensive study of liquid holdup and mass transfer in ring and saddle packings is reported by Shulnian and associates (70-7$). By mcasurement of rates of sublimation of paclrings fabricated from naphthalene with and without liquid flowing,
672
effective wetted areas for liquid vaporization were determined. A general gas-film correlation for dry packings is proposed.
where li,
= gas-film coefficient
M , = molecular weight of the gas ~ U B M =
log mean partial pressure of inert gas
G = superficial mass velocity of the gas SCG= Schmidt number for the gas ’
D, p E
= diameter of a sphere possessing same area
as a
piece of packing = gas viscosity = void fraction including effects of the packing and liquid lloldup
,4pplication of Equation 4 to Fellinger’s data on ammonia absorption was made to calculate effective wetted areas for gas absorption. These areas are tabulated for different packings for use in design. Effective areas for absorption are generally less than for vaporization bccause of holdup of essentially stagnant liquid a t points of packing contact which is active only in vaporization. The authors believe this explains much of the disagreement among such data in the literature. On the basis of the foregoing results for effective areas, available liquid-film data were corrected for variations and a general dimensionless correlation was obtained as expressed by Equation 5. k d = 25.1 DL
p-
where k~
DL
D;L)0’45
(ScL)0.5
= liquid-film mass transfer coefficient
= = 7 = SCL=
L
(-
liquid-diffusion coefficient superficial mass velocity of the liquid liquid viscosity Schmidt number for the liquid
Studies with various liquids were made to determine the effect of physical properties on holdup and hence on effective interfacial areas with nonaqueous systems. In previously publishcd studies, confusion with respect to the role of the Schmidt number as measured by vaporization of various pure liquids into air is believed due to variations of effective area produced by the different liquids. Although the broad generalizations offered may bear further confirmation in some cases, the study appears t o be a significant contribution t o the general understanding of packed tower performance. Additional performance data for tower packings are reportod by Leva (60)for absorption of carbon dioxide in sodium hydroxide solutions and by Hikita and associates (36) for absorption of pule sulfur dioxide and carbon dioxide into water. The latter obtained good agreement with the ShertTood-Holloway equation for liquid-film coefficientswith 15- and 25-mm. Raschig rings. Quigley, Johnson, and Harris (6.9) measured bubble size and gas holdup in the flow of air through single orifices into various liquids in a 2.5-inch square column. Equations 6, 7 , and 8 are the empirical relations for bubble diameter and gas-liquid interfacial area and gas holdup. Dg =
0.222 De”33Q G O . ’ * ~
v0.O2
+ 3.02 X lo-‘
Qo’.09
(6)
(7)
N
==
2.44 X 10 - 4 Q$J4
(8)
where DB = equivalent sphere diameter of the bubble, feet QG = volumetric air flow rate/orifice, cubic feet per hour 7 = liquid-kinematic viscosity, squarc feet per hour A = interfacial bubble area per nozzle, cubic feet per foot of liquid seal H = holdup of gas in foam, cubic feet per cubic foot of foam From measurements of oxygen absorption by water in copjunction with previous data of West and associates on carbon dioxide absorption and desorption a general liquid-film mass
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MASS TRANSFER
Figure 2.
Comparison of j factor data for countercurrent spray tower continuous phase coefficients with available data for fixed and fluidized beds (64) Key
AA BB
cc
DD E€
FF GG HH
Type of Contacting
Fluid
SCL
Countercurrent spray towers Fixed beds Fixed beds Fluidized beds Fixed beds Fluidized beds Fixed beds Fixed beds
liquid liquid liquid liquid Liquid Gas Gas Gas
170-300,000 160-1 2,000 1300-1400 1200-1 300 776-866 2.565 0.61 5 0.61 5
transfer correlation was obtaincd in the form of a modified j number as a function of Reynolds number for the bubble. Application of the relations to the calculation of plate efficiency for liquid film-controlled systems is suggested and equations are given for the over-all Murphree point efficiency based on liquid conccntrations. The term over-all efficiency secms inappropriate in this cam, however, since only the liquid-film resistance is included with no allowance for gas-film resistance. Extension of the data for mass transfer between air and water to other liquids and application of the single bubble results to niultiplehole sieve trays is proposed tentatively. In absence of more general experimental confirmation, however, considerable caution seems indicated in making such extension. Liquid-film results for desorption of oxygen, carbon monoxide, and hydrogen on a perforated plate are described by Kuzminykh and Koval (47). Effects of variables including air velocity, liquid rate, and viscosity of the solution on the volumetric mass transfer coefficient, k ~ u ,are expressed by graphs and empirical equations. At a given air and water rate kLa was observed t o vary as the diffusion coefficient to the 0.7 power in substantial agreement with the assumption of Quigley and associates (61) cited above. Mass transfer between liquid drops and a continuous phase was investigated by Ruby and Elgin (64) for liquid-liquid extraction in a spray tower. Measurements of drop s h e and holdup permitted calculation of interfacial areas. Through use of suitable binary liquids of low mutual solubility the mass transfer coefficient, k,, between the drop surface and the continuous phase was isolated. The important conclusion was reached that k, is essentially independent of the flow rates of either phase for a given liquid-liquid system and drop size. The flow variable which
March 1956
affects IC, is the slip velocity which is assumed equal to the freefall velocity of the drop in this case. Variations of H.T.U. with flow rates in such cases are due primarily to change in holdup, and hence interfacial area. Mass transfer data for four different systems were correlated in terms of a modified j numberReynolds numbcr plot. These parameters are defined by Equations 9 and 10.
where k, pe
LZ.
=
mass transfer coefficient for continuous phase
= continuous phase density = liquid phase mms velocity corresponding to slip
velocity
Dp = drop diameter U , = slip velocity ve = continuous phase viscosity General agreement was found between other j number correlations for packed and fluidized beds and the j number defined above as illustrated in Figure 2. A mathematical treatment of countercurrent mass transfer operations in the unsteady state is developed by Jaswon and Smith ( 4 9 ) . The approach differs from the common method of the Laplace transforms. For systems which can be described by constant gas-film and liquid-film mass transfer coefficients and a linear-equilibrium isotherm the transient behavior can be calculated in terms of the phase flows and column holdup. A numerical example of the solution to a batch distillation problem is presented.
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FUNDAMENTALS REVIEW Mass Transfer to and within Porous Solids Mass transfer in ion exchange was treated in a variety of experimental investigations. lIiggins and Roberts (35) describe a new type of continuous countercurrent ion exchange contactor which retains the high throughput characteristic of the conventional fixed bed. Hiester, Phillips, and Cohen ( 3 3 ) present a design method for the exchange of gross components in a continuous countercurrent ion exchange column. The number of theoretical equilibrium contacts 1s found graphically and the height equivalent to a theoretical stage is obtained from a modified mass trnnsfcr correlation which takes into account the operating variables of the column and the characteristics of the ionresin system. Continuous countercurrent ion exchange was also treated by Koenig and others ( 4 6 ) who report mass transfer data in spinner columns. By means of a previously obtained correlation to evaluate particle settling rates in spinner columns, the results were presented in terms of ma% transfer coefficients. Biebcr, Steidler, and Selke (IS) investigated ion exchange rate mechanisms by using a shallow bed technique for silver-hydrogen exchange with -kmberlite IR-120 cation exchange resin. The solid- and liquid-diffusion steps were important rate factors while the chemical exchange reaction offered negligible rebietance The diffusion of ions in the solid phase of an ion exchanger was the subject of a brief paper by Holm (38). The diffusivities of Naf, Ca++, and Sn+3in thc resin Don-ex 50 were measured; a plot of diffusion coefficient 2)s valence gave a straight line of negative slope. Self-diffusion of cations, nonexchange anions, and solvent in a cation exchange-resin system was invevtigated by Tetenbaum and Gregor ( 7 5 ) . The self-diffusion of potassium, the exchange cation, chlorine, the nonexchange anion, and q-ater, the solvent, were measured in a styrene-sulfonic acid-resin system using an isotopic tracer technique. I n the range 0,0001 to 0.lM the diffusion coefficient for potassium was 21% of the value in free solution, that for chlorine 37%, and that for deuterated water 85% of the free solution value. The kinetics of sorption by porous solids was investigated by Carman and Raal (18) who measured the diffusion of normal butane a t 10’ C. into porous silica plUg? using one nonsteady flow and two steado flow methods. I n the monolayer region and also over most of the multilayer and capillary condensation regions &ad? flow and nonsteady flow methods shoxed no consistent differences. At high butane concentrations diffusion of capillary condensate, but gradually gave way to viscous f l o ~ the apparent diffusion coefficients obtained by different methods were in good agreement. The contribution of surface diffusion was large ; in thesc experiments the apparent surface diffusivities were of the order of 10-4 sq. cm. per sccond. Additional work on diffusion in silica is reported by Haul ( S I ) who m e a w r d the separation of oxygen isotopcs b y diffusion
-
through a compressed silica plug of 60% porosity at 76.8”C. The diffusion coefficient for Knudsen flow m-as 0.36 X sq. crn. per second and that for surface diffusion was 1.46 X sq. cm. per second. Of the flow, 37% was surface flow. The ratio of the effective diffusion coefficient for Oz’8 and O P was 0.84 and the limiting separation factor 1.2. Mass transfer data from a gas stream to fluidized pororls solids are reported by Hsu and Slolstad ( 4 0 ) who investigated the isothermal adsorption of carbon tetrachloride from anitrogencarbon tetrachloride mixture on fliiidiied activated charcoal. At high relative saturation the over-all mass transfer coefficient decreased as the relative saturation of the charcoal increased. At low and moderate relative saturation, hon ever, mass transfer through the gas film surrounding the solid was the rate-determining step in the adsorption, and the ovcr-all mass transfer coefficient was almost independent of the relative saturation. A study of diffusion in porous catalysts and adsorbents by Hoogschagen (39) is concerned with the diffusional and flow tramfer of reactants and products in and out of the pore structure of a catalyst particle when the reaction takes place at interior surfacer. The effect of restricted internal diff wion is to retlnce the catalyst activity; this effect can be detected either by a difference in activity between large and small particles of the same catalytic material or by a nonlinear relationship between the logarithm of the reaction rate constant and the reciprocal temperatiire. The author discusses methods for meawring diffusibilities and labyrinth factors and relates them t o the effectiveness factor of porous catalysts. The results are applied to ammonia synthesis a t atmospheric pressure and t o the watergas shift reaction. Another study of diffusion in catalysts is reported b y Bockhoven and van Raayen (16) v h o measured the activity of ammonia synthesis catalyst particles of different mesh size a t I and 30 atm. and at different temperatures. h plot of the logarithm of the rate coiistant against the reciprocal temperature showed a decreasing slope a t increasing temperature except for the small particle size at 30 atin. Thc results agree with the assumption that the reaction rate was retarded by restricted diffusion through the pores of the ratalyst particles. Procedures for the numerical solution of the steady state and unsteady state diflusion equations applicable to throngh-flow drying are given by Van Arsdel (7’6). Ratch and continuous operation are considered for the special case in \\ hich the ratecontrolling step ir determined b y the internal diffusional resistance in the material. Diffusive Separation Processes An extensive series of theoretical and experiniental studies of thermal diffusion in liquids wa.9 published by Drirkamer and aqsociates (21-23, 65-68), The status of these developments including modifications of previous theories is summarized by
“Deve/opment of the field of mass transfer has made it possible to condense and unify knowledge regarding a variety of chemical engineering processes. The quantitative correlation of momentum, mass, and heaf transfer, both in torbolent flow and on a molecular scale, has pointed the way to the development of the three as a single subject. “More research is badly needed on the physics of transfer across a phase boundary in order to provide an adequate basis for a sound theory of the basic mechanism. Development of the subject is seriously hampered by this lack, especially in the irnporfant case o f mass transfer with chemical reaction. ” THOMAS K. SHERWOOD Massachusetts Institute of Technology 674
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Dougherty and Drickamer ( 2 2 ) . Proceeding on the basis of an interpretation of Denbigh’s “net heat of transport,” a theory is developed which permits prediction of the separation using only molecular weights, molar volumes, activation energies for viscous flow, and excess thermodynaniic properties. Excellent qualitative agreemmt, and in many cases essentially quantitative agreement, is obtained in comparison of the theory with a wide variety of data on binary mixtures. Agreement on the effect of pressure for several systems is particularly striking over a range of 10,000 atm. Although the authors indicate need for refinement of the theory in several aspects the work should be very valuable for the preliminary evaluation of thermal diffusion processe? and the interpretation of molecular behavior in liquids. Some data on separation of hydrocarbons in a microthermal diffusion column consisting of two coaxial cylinders are reported by van Schooten and van Nes (69). Additional theoretical discussions appear in the papers of Alexander (2-6) and Bierlein (14). The m e of crystal sieves for the separation of molecules is discustsed by Barrer ( I O ) who studied the srlectivc occluding or sorbing properties of zeolites. Separations of gaseous and liquid mixtures can be achieved by a percolation method in which the mixture is allowed to flow through a column of the zeolite. Typical separations studied were the separations of mixtures of ethyl aleohol-mater, nitrogen-argon, and niethaneethane derivatives. Two paper$ by Kammermeyer and TIagerbaumer ( S I , 46) are concerned with the operating characteristics and uses of membranes for effecting separation? in the gaseous and liquid statee. Equilibrium time, rates of permeation, and ratios of the permeability rates which indicate the separation to be expected were determined for five different porous-glass membranes using oxygen, nitrogen, and carbon dioxide. Separation of binary organic liquid mixtures was achieved by straight forward pressurr permeation through microporous membranes.
Bibliography Albertson, RI. L., Dai, Y. B., others, Proc. A m . SOC. Civil Engrs. 74, 1571 (1948). Alexander, K. F., Z. physik. Chem. 203, 181 (1954). Ibid., p. 203. Ibid., p. 213. Ibid., p. 228. Alexander, I,. G., Kivnick, Arnold, others, A.I.Ch.E. Journal 1, 55 (1955). Ibid., p. 61. Ibid., p. 69. Arnold, J. I€., IND. ENG.CHEN.22, 1091 (1930). Barrer, R. >I., Rrennstof-Chem. 35, 325 (1954). Becker, E. W., Vogell, W., Zigan, F., 8. Naturjorsch. 8a, 686 (1953). Berman, Kurt, J. A p p l . Mechanics, Paper X o . 54-A-29,Jan. 21, 1954. Bieber, Herman, St,eidler, F. E., Selke, U’. A, Chem. Eng. Progr. S y m p o s i u m Ser. 50, No. 14, 17 (1954). Bierlein, J . A , , J . Chem. Phys. 23, 10-14 (1955). Bird, R. B., Curtiss, C. F., Hirschfelder, J. O., Chem. Eng. Progr. S y m p o s i u m Ser. 51, No. 16, 69 (1955). Bockhoven, C., van Raayen, W., J . Phys. Chem. 58, 471 (1954). Brotz, Walter, Chem.-Ing.-Tech,. 26, 470 (1954). Carman. P.C.. Raal. F. A.. Trans. Faradau SOC.50. 842 (1954). Carmichael, 1,. T., Sage, B. H., I.accy, W.%., A.I.Ch.h’. journal 1 , 385 (1955). Croockewit, P., Honig, C. C., Kramrrs, H., Chem. Eng. Sci. 4, 111 (1955). Dougherty, E. I+ Jr., Drickamer, H. G., J . Chem. Phys. 23, 295-309 (1955). Dougherty, E. I,., Jr., Drickamer, H. G., J . Phys. Chem. 59,443 (1955). Drickamer, H. G., Rutherford, W. AI., Chem. Eng. Progr. Symposium Ser. 51, KO.16, 87 (1955). Eckert, E . R. G., “Introduction t o the Transfer of Heat and Mass,” pp. 158-63, RicGraw-Hill. New York, 1950. Eisenberg, RI., Tobias, C. W., Wilke, C. R., Chem. Eng. Progr. Symposium Ser. 51, No. 16, 1 (1955).
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(26) (27) (28) (29)
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