EFFECT OF AXIAL MIXING ON MASS TRANSFER N I EXTRACTION COLUMNS N. N. 11 E. N. ZlEGlER Neglect of the effect of axial mixing in the design and evaluation of packed spray towers for extract ion may lead to an overestimation of column performance of as much as 30% The authors review the problem of axial mixing and put it in perspective for two of,the most used types of extraction towers
a stream of material flows steadily through a W hen tank, a column, or a pipe in which it takes part in some process such as mass transfer, chemical reaction, or simple mixing, it is usually necessary to make one of the following assumptions for the purposes of calculation: The fluid in the vessel is completely mixed, so that its properties are uniform and identical with those of the outgoing stream; elements of fluid which enter the vessel at the same moment move through it with constant and equal velocity on parallel paths and leave at the same moment-Le., plug flow. There arc many cases in which neither type of flow corresponds exactly to the facts. For instance, in the case of a solvent extraction column, backmixing usually occurs in the solvent and raffinate phases (Figure 1). Early design techniques which did not take cognizance of backmixing were based on methods employed in distillation operations. The performance of extractors was compared to staged columns. An operating line (curve) could, for example, be drawn on a triangular (three-component, mutually soluble systems) diagram (97). Alternatively, the height equivalent of a theoretical stage could be calculated using log mean concentration difference at the tower ends as the driving force for diffusion (82). Morello and Poffenberger (68) and Geankoplis and Hixon (37) noted a concentration jump or discontinuity at the continuous phase inlet in spray towers. These observations shed doubts on the validity of using logarithmic mean concentrations for design purposes. Vertuzaev (94) pointed out recently that computation of the driving force for mass transfer in extractors without accounting for longitudinal mixing may lead to an overestimation of its value by 10 to 30% or more. Brenner (8) calculated a difference in the instantaneous and average solute concentration values. Vermeulen et al. (93) have also indicated that, depending upon the circumstances, the height of transfer unit values corrected for backmixing may be as low as 15y0 of the uncorrected values. This review is concerned with the theories of axial mixing and the specific research results on packed and spray towers. The choice of the above two types of extractors among other types mainly is based on their popularity. I n most instances the information on mixing phenomena in these extractors can readily be extended to cover other varieties. Also, a recent article on reactor design (93a), received too late for this review, may be of interest. Determination of Axial Dispersion
The result of the lowering of the driving potential for mass transfer due to longitudinal backmixing is that a longer column for a given mass transfer coefficient will be needed to effect the same overall separation. This is 30
INDUSTRIAL A N D ENGINEERING CHEMISTRY
illustrated schematically in Figure 2. The phenomenon of axial mixing or longitudinal dispersion may result from: Turbulence imparted by the rising drops Channeling flow due to the particular column or packing geometry Entrainment of field liquid in the wake of the drops Turbulent diffusion in the direction of the axis of the extractor Radial diffusion as a consequence of nonuniform velocity Forced backmixing due to pulsation True molecular diffusion A distribution of residence times for the dispersed phase flow Nonuniform velocity due to friction at the wall and subsequent transverse diffusion. This is also called Taylor diffusion, from G. I. Taylor’s analysis of molecular diffusion effects during laminar flow in pipes (88, 89) These mechanisms, whether acting separately or in combination, are known to decrease the concentration gradient for mass transfer. Because of the axial dispersion effect, there are three different definitions of the “number of transfer units, NTU” as noted by Miyauchi (63). 1, True value :
No, = Ko,ah/U,
(1)
where
KO, = overall mass transfer coefficient related to phase x a = surface area per unit volume of packed bed h = total height of a packed bed U, = superficial volumetric flow rate of x phase through unit cross section of column 2. Measured value. I n this case, the concentration
Figure 7 . Backmixing in an extraction column
DISPERSED
’PHASE
distributions for phases x and y in the column follow, respectively, curves ABDE and FGHK of Figure 2.
where Q = dimensionless intercept in the plot of equilibrium concentrations m = solute partition coefficient 3. Piston flow values, This is defined in terms of the logarithmic mean driving force computed from the exterior incoming and outgoing concentrations at both ends of a column. In general,
Two early flow models (20, 28) were concerned with molecular diffusion in flow reactors. Hulbert (42), Danckwerts (27), and Wehner and Wilhelm (97) solved the diffusive flow equations for first-order reacting systems. The differences between the models proposed were in the boundary conditions only. Alternatively, the diffusion coefficient could be viewed as a longitudinal dispersion coefficient to account for backmixing (4-6, 23, 44, 49, 50, 52, 55, 88-90, 707). The dispersion coefficient will not, however, account fully for the concentration jump at the tower inlet. Two models have been found useful for describing the axial mixing in solvent extraction columns. The first is the dispersion model which is an extension of the early flow models to two-phase flow. It assumes that the various factors causing axial mixing can be described by a diffusionaltype process superimposed on plug flow for both phases. The second model, the mixing cell model, characterizes the mixing in extractors as a number of completely mixed cells connected in series. The backmixing in an extraction column is, therefore, a function of only one parameter-the number of mixing cells in the column. The relative utility of these models is still uncertain, but the parameters of the dispersion model have been developed to a much greater extent. The dispersion model was proposed by Miyauchi (63) and also by Sleicher (83). The equations can be written in the dimensionless form (1/P,B)d2C,/dZ2 - dC,/dZ -
CONTINUOUS I PHASE
AUTHORS N. N . Li is wi t h the Central Basic Research Laboratory, Esso Research and Engineering Co. E. N . Ziegler is Assistant Professor in the Department of Chemical Engineering, Polytechnic Institute of Brooklyn. T f i e authors acknowledge the assistance of T . Vermeulen and R. B. Long in preparation of the manuscr2pt.
V O L . 5 9 NO. 3 M A R C H 1 9 6 7
31
Noz[Cz - (Q . %.
+ mC,)l
=0
cally by computer and detailed numerical results are available (62). A graphical solution has been presented recently for the Miyauchi-Sleicher equations by Rod (74). In addition, Rod (75) has discussed the influence of forward mixing on mass transfer illustrated by worked examples. The forward mixing in dispersed phase is a result of the distribution of drop slip velocities caused by their diameter distribution. Figure 3 from work by Miyauchi (63) shows the change in concentration profiles that would occur in different c o h n n s with the same mass transfer HTU, but with different amounts of axial dispersion. Here A is the extraction factor (ratio of continuous to discontinuous phase flow rate of solute under conditions of equilibapply if the rium). Values marked with a dagger raffinate and extract phases are interchanged. For illustrative purposes only, the P values in the two phases have been shown as equal. The curves for P,B = m show the gradients calculated in the absence of axial dispersion (the piston flow case), and those for P,B = 0 show the behavior in the extreme case of complete mixing. Thus, the less the Peclet-number values, the greater the extent of lengthwise mixing. The greatest change for each phase occurs at the point where it enters the column. The mass-transfer rate appears to be lower than it really is, and the apparent H T U is larger than the true mass-transfer HTU. The extent of longitudinal dispersion can be measured by unsteady-state experiments, usually in the absence of extraction; or by sampling the column contents during extraction to determine the actual gradients, plotting them, and fitting them with the profiles calculated for different amounts of axial dispersion. Miyauchi and Vermeulen (65) also proposed approximate empirical equations relating N,,, number of transfer units in the case of axial dispersion controlling, to P a ; these have proved useful for rapid estimation of N,.
(5)
boundary conditions (27, 22)
with the
a t Z = 0:
-dC,/dZ
=
PZE ( 1
- Cz), -dC,/dZ
=
0
(6)
at Z = 1:
-dC,/dZ
=
0, -dC,/dZ = P,B (Cvi - C,l)
(7)
in which:
E C,, C,
= dimensionless height, h/dp
= dimensionless continuous and discontinuous phase concentrations. Subscript 1 and superscript 1 refer, respectively, to concentrations inside and outside a column P, and Pv = packing Peclet numbers (Uodp/&) Z = fractional column height 4 = particle diameter E1 = axial dispersion coefficient
(t)
The direction of mass transfer is taken from phase X to phase Y , and a linear equilibrium distribution is assumed. These equations are based on some simplifying assumptions. The first states that the axial mixing can be described by Fick's law in which the axial flow rate and the mixing coefficient do not vary with either the height or cross-sectional area of the apparatus. If the ratio of tower diameter to packing size is greater than 8 to 1, the constant flow rate assumption is reasonable. Also, mixing coefficients have been shown not to vary with respect to column height by Liles and Geankoplis (57). Further, it is assumed that the dispersed phase can be treated as a second continuous phase. Finally, constancy of holdup throughout the region of interest is assumed. The above differential equations with all the terms expressed in dimensionless groups were solved numeri-
-
-It 1
----- - ,---
1.8
Q.6
0.4
0.2
'
0
---. ...
COLUMN HEIGHT
-
.
F i p e 2. Cornenfration disttibution in a typical exfration
-. -*
z
Figwe 3. Typicd concmfrution u' prajfbs th &$fusion predicted model by (63) ' O l 0
32
INDUSTRIAL A N D ENGINEERING CHEMISTRY
d 1 .o
As illustrated in Figure 2, the true NTU can be related to Nmd and N,, by a difference in reciprocals:
Some extensions of Miyauchi’s model have been made.
In many countercurrent operations, a stagnant section at the end of the extractor may exist. For example, in spray towers the inlet of the dispersed phase sometimes is placed well below the outlet of the continuous phase to avoid drop entrainment. As shown in Figure 4, Wilburn (99) divided the extractor into three sections and solved the mass transfer equation for each section. The concentration profiles in the extractor changed appreciably upon introduction of these modified boundary condiFigwe 4. Schematic repesentation of
an cxtrmtor
tions into Miyauchi’s model. Instead of assuming C, to be a linear function of C, and the volumetric fraction of each phase e to be constant as in Miyauchi’s model, Wdburn and Nicholson (IW)approximated the equilibrium distribution by a cosine function and the volumetric fraction by an exponentially damped polynomial function of axial distance and solved the nonlinear continuity equation by a matrix method. The nonlinear approximations are necessary to describe the system of uranyl nitrate transfer between a nitric acid-aqueous phase and a tributyl phosphatekerosine organic phase. As the authors pointed out, their method of obtaining a solution needs an initial rough approximation to the solution. Such an approximation can be obtained, however, from a complete linearization of the equations. Another model for longitudinal dispersion was considered by Miyauchi and Vermeulen (66) in which hackflow, superimposed on the net flows through a column, was simulated by a series of perfectly mixed stages in cascade. Exchange of materials between two adjacent stages is caused by net flows, Us and Uv, of main streams and a recycled fluid, denoted by U, which occurs in each direction and is the sum of individual phase backflows of Us and Uv as shown diagrammatically in Figure 5. The material balance equation around the jth stage for x is
Kmd [XI
- (b + m FA I
(9)
where the first term represen@.&w downward and the second term flow (backmixing) upward. The last term designates mass transfer from the Y stream to X stream with 6 designating the intercept value for a partition equilibrium line. The above equation can be simplified and written in terms of the number of transfer units, N,:
(1
+ a*)
@,A
where a is
- X,) - a d X , - Xj-1) = Nmlxj - (6 + mPJ1
(10)
U/O.
Figur8 5. Mixing cell moh6 for counfmwe%t operation (66)
*
2.0
2 2
1.0
ai 0.5
0.21 200
500
1000
2000
0.41 0.5
Figure 6. H. and H, for t h imbutanol-wdm system (93)
5
2
1
IO
20
N“
6. ILIS./HR-FI.’I
Figure 7.
No*p
(15
a function of
VOL 59
No. and PxB (63)
NO. 3 M A R C H 1 9 6 7
33
V."
0.4
0.1 0.05
Figure 9. Axial Ped86 number as a funclion of Reynolds numbn
104
z
5 d.U&
103
z
5
10-2
z
5
U. (HOURS)
Figwt. 8. Cmelnfion fm mkl p h m dispersion in continuous phase
The Peclet number based on column height for either phase can be related to a number of mixing cells n in that phase (93):
N,,
=
2
(. + +>- 1
The diffusion model of Equation 1 is demonstrated to be but a limiting case of the backflow model. Figure 6 shows the magnitude of the axial mixing correction calculated in the case of Colburn and Welsh's data (78). The corrected values of H T U for the continuous phase are as little as 15%, and as much as 50%, under the apparent experimental values (93). Column Peclet numbers were estimated from Figure 7 hased on Miyauchi's diffusion model solution. The single-phase plot of apparent us. true NTU's (Figure 8) (45)was then used to obtain the ratio between apparent (Hcp) and true (Hc) HTU. For design purposes, the reverse of this calculation is needed. For a selected packing, the true HTU is calculated first. A column height is then assumed, which gives the NTU. From this and the estimated Peclet number, the apparent NTU and the actual extent of extraction are read. Where dispersion occurs in both phases, the calculations are similar but more complicated. These effects become more pronounced in industrial-scale columns where larger packing sizes are generally used. Packed Column
Axial dispersion coefficients in single-phase flow were measured by several investigators over a wide range of Reynolds numbers. The experimental technique involves measuring the outlet response resulting from an inlet disturbance. The disturbance can be a step or pulse or continuous sine-wave function formulated by the injection of a tracer into the main flow stream. These methods were discussed in detail in several dissertations (24, 47, 46). 34
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
I n the past few years several papers (7, 5, 77) discussed another method of evaluating axial dispersion by measuring the spread of a tracer's concentrationtime curves. The essence of this method is to evaluate mixing by injecting a tracer and analyzing its behavior as it moves through the bed. A radioactive tracer can be used so that its concentrations can be measured from outside the bed without disturbing the flow patterns by using scintillation detectors. The advantage of this method is that there is no need for an exact tracer input and any desired pulse can be used because this method compares the concentration-time responses at two positions downstream. The difference in the variances of the two response curves is a measure of the mixing occurring between the two measurement positions. Viscosity has a large effect upon axial dispersion. Low dispersion is associated with high viscosity. Different types of packings give different values of axial Peclet numhers. But no effect of packing arrangements (regular and random) was observed. A general correlation has been obtained which fits reasonably well the data for different types of packings with porosity ranging from 26 to 73%. These data are summarized in Figure 9. The Peclet number is shown as a function of Reynolds number. The curves represent the data of Cairns and Prausnitz (77), Carberry and Bretton (72),Ebach and White (24), Jacques and Vermeulen (46), Moon (67), and Strang and Geankoplis (86). A marked transition is observed between the Peclet numbers in the laminar range and those in the turbulent range; the latter correspond more closely to the values observed for gas flow (67). Sater and Levenspiel (77) correlated their Peclet numbers as a function of flow rate and packing geometry by using the correlation developed for holdup (72). The dispersion should be a function of the same factors that determine the holdup.
P
=
A (Re)B(Ga)C(adp)E
where: (Re) = dL/p, Reynolds number (Ga) = dpag$/pa, Gallileo number
(12)
If the viscosity and surface area are constant, the above equation is reduced to a direct correlation of dispersion with Reynolds number. Because of the lack of data presented in the paper, no comparison can be made with other investigators’ results summarized in Figure 9. The effect of pulsation has been studied by Moon ( 6 7 ) , who found that Peclet numbers decrease as the amplitude increases and, to a lesser extent, as the frequency increases. Hayford (37) has correlated his data on the effect of pulsation by the equation
where s = pulse amplitude in tower and f = pulse frequency. For counterflowing liquid-liquid systems, measurements of Peclet number have been made for sphere and ring packings using kerosine and mineral oil with water (47, 45, 46, 67). For both pulsed and unpulsed columns, results show that the axial Peclet number of one phase decreases as the flow rate of the other phase increases. This effect is much greater if the dispersed phase wets the packing. In addition, P, increases with increasing Lye, and P , increases with either an increasing U d for a nonivetting dispersed phase or a decreased Ud for a wetting dispersed phase. As noted by Treybal (97), ho\vever, these data are insufficient as yet to assist the analysis of extraction data or to be used for design. T o characterize the effect of backmixing on extraction and to verify the diffusion model, work has been reported on measuring concentration distribution within the column. From this information the coefficient of mass transfer can be evaluated. As discussed previously, this approach for calculating coefficients should be more adequate than the alternative of using a logarithmic mean driving force computed only from the end concentrations of the incoming and outgoing extremes. Experimental concentration profiles for the extraction systems of water-crotonic acid-isododecane and water-acetic acid-diisobutyl ketone in both dispersed and continued phase with and without pulsation were measured by Moon (67). Comparison of the experimental and theoretical profiles shows that the two-phase, onedimensional diffusion model adequately represents the fluid behavior in packed extraction column. Spray Tower
The spray tower is a countercurrent contacting device for a continuous and a discontinuous (droplet) phase which, for convenience, will hereafter be designated CP and DP, respectively. Early work in this area was done by Geankoplis et al. (37, 32), who measured the concentration profiles for the systems of FeCla-isopropyl ether-HCl(aq.) and toluene-acetic acid-water. Mass transfer coefficients were calculated over small increments of the column. The large end effect found at the CP inlet was apparently not influenced by the direction of solute transfer. The
concentration jump at the CP inlet is accompanied by a high transfer efficiency (95). Kewman (69) deduced that backmixing was responsible for changing the effective concentrations at the tower ends. The system adipic acid-diethyl ether-water was later used by Gier and Hougen (33) to study end effects in packed and spray towers. The tower packing reduced the bulk mixing of the CP and thereby established more favorable concentration gradients than in the spray tower. If Newman’s deductions were correct, such changes in mixing would be expected to produce the corresponding changes in the concentration gradient. Patton (73) confirmed Newman’s explanation via dye tracer experiments. The increase in mass transfer at the DP inlet is possibly dependent to a greater extent on drop formation than on mixing of the DP (27, 56, 87, 98). Under certain conditions, the D P discontinuities were not observed (37, 32, 48, 57). Keith and Hixon (47) indicated that above the jetting point no unusual transport would occur at the DP inlet. The concentration level influenced extraction rates (96). Heertjes et al. (39) found a CP of constant concentration in the process of contacting a partially soluble binary system. Brutvan (9) recently has determined axial dispersion coefficients, El, of the continuous phase in a simulated spray tower with glass beads as the DP. Because the beads were rigid, they did not become distorted in shape or undulate as drops do in passing through a fluid. His values for Peclet number were considerably lower than those found for packed beds ( 7 7 , 72, 24, 45, 57, 67) and about half as large as those of Hazlebeck and Geankoplis (38). E1 increased slightly with an increase in the dispersed phase flow. Hazlebeck and Geankoplis measured axial dispersion coefficients using water as the continuous and methyl isobutyl ketone as the dispersed phase. E1 varied with the velocity of the continuous phase (Vu)in the following manner: E1 = 9.00
(U,)O*45
Peclet numbers were about l/10 of those reported for packed beds. The multiplate pulsed spray tower possesses the advantage of reproducing the drop formation process a number of times; in the jetting region this would enhance mass transfer. Van Dijk (92), Blaedel and Hyman ( 7 ) , and Groot (36) were among the pioneers of pulsed flow in extraction processes. The transfer efficiencies reported for the pulsed sieve plate towers were 2 to 3 times those of packed colurnns which are, in turn, more efficient than spray towers (70, 74, 77, 79, 26, 35, 54, 78).
The backmixing in pulsed streams was discussed by Swift and Burger (87), O’Brien (77), and Christensen (75). Axial mixing coefficients were determined by a number of experimenters (76, 25, 59, 60, 64, 70, 76, 79, 80, 84, 85, 87) by a tracer method suggested by Gilliland and Mason (34) and others (27, 55). The dynamic behavior of pulsed columns has also been investigated (2, 3, 73, 29) by testing various mathematical models and numerical solutions. VOL. 5 9
NO. 3
MARCH 1967
35
I n general, the process of backmixing in spray towers is believed to be initiated largely in the wakes of droplets. Magarvey and Bishop (58) photographically observed the wakes of single drops in the system of organic liquid drops-water using a water-soluble dye. The wake appeared to be independent of any gross oscillations of the drops. Garner and Tayeban (30) discussed the role of the wake as a solute reservoir attached to the drop. Hendrix, Dave, and Johnson (40) found via a dye technique that significant portions of the CP were conveyed in the wakes of droplets. When droplet oscillations became significant the wakes were continually shed and (presumably re-established). Conclusions
The prediction of the effects of axial mixing on mass transfer is not yet possible in many instances. In general, however, the simple dispersion models discussed above can give reasonable estimates of these effects if sufficient experimental data are available to determine the CP dispersion coefficient. The model will work best if radial diffusion is sufficiently high to equalize channeling effects or local inhomogeneities and if molecular diffusion is insignificant. Using an effective diffusivity to represent a drop phase by the same model probably is a poor approximation, A more exact approach, however, requires information on the joint effects of drop dynamics, size distribution, and drop-to-drop interaction in the presence of a continuous phase. In addition, more axial mixing data in the DP of spray towers and radial mixing data in the CP and DP are needed to further our understanding of the mixing phenomenon. The effects of tower height and diameter on backmixing have not been studied experimentally, particularly in towers of large diameter (over 1 foot) and of substantial height (over 10 feet). The dense phase regime in the tower is still to be investigated. In particular, it would be interesting to study the possible enhancement of extraction and the extent of backmixing at comparable flow rates in sparse and dense beds. REFERENCES ( 1 ) Ark, R., Amundson, N. R., A.1.Ch.E. J . 3, 280 (1957). ( l a ) Bell, R . L. Ph.D. Dissertation, L’niv. of Washington, Seattle, 1964. (Ib) Bell, R . L., Babb, A. L., Chcm. Eng. So. 20, 1001 (1965). (2) Biery, J. C., Ph.D. dissertation, Iowa State University, 1961. (3) Biery, J . C., Boylan, D. R., IND.ENG.CHEM.FUNDAMENTALS 2,44 (1963). (4) Bischoff, K . B., Chem. E n s . Sci. 1 6 , 128 (1961). (5) Bischoff, K. B., Levenspiel, O., Ibid., 17, 245 (1962). (6) Zbid., 17, 257 (1962). (7) Blaedel, W. J., Hyman, H . H., U. S. Atomic Energy Comm. Rept. CN-3525 (1946).
(8) Brenner, H., Chem. E n s . Sa. 17, 229 (1962). (9) Brutvan, D. R., Ph.D. dissertation, Rensselaer Polytechnic Institute, 1958. (10) hurns, W. C., Groot, C., Slansky, C . M., U. S. Atomic Energy Comm. Rep!. HW-14728 (1949). (11) Cairns, E . J., Praurnitz, J. M . , Chrm. E n g . Sri. 12, 20 (1960). (12) Carberry, J. J., Bretton, R. H., A.I.Ch.E. J . 4, 367 (1958). (13) Champagne, F., M.S. thesis, University of Washington, 1962. (14) Chantry, W. A,, Von Berg, R . L., Wiegandt, H . F., IND.Exo. CHEM.47, 1153 (1955). (15) Christensen, C. M . , Ph.D. dissertation, Cornel1 University, 1951. (16) Claybaugh, B. E., Ph.D. dissertation, Oklahoma State University, 1961. (17) Cohen, R . M., Beyer, G. H., Chem. E n g . Progr. 49, 279 (1953). (18) Colburn, A. P., Welsh, D. G., Trans. A m . Ins!. Chcm. Ens. 38, 179 (1942). (19) Cooper, V. B., Broot, C., U. S. Atomic Energy Comm. Rept. HW-20305 (1950). (20) Damkohler, D., Z . Electrochem. 42, 846 (1936). ( 2 1 ) Danckwerts, P. V., Chem. Eng. Sa. 2, 1 (1953). (22) Danckwerts, P. V., Trans. Faroday Soc. 40, 300 (1950). (23) Deisler, P. F., Wilhelm, R . H . , I N D .ENO.CHEH.45, 1219 (1953). (24) Ebach, E. 4 . , While, R . R., A.1.Ch.E. J . 4, 161 (1958). Nag;ita, S., Chrm. Ens. (Japan) 23, 146 (1959). (25) Eguchi, W.,
36
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
(26) Ellison, C. V., U.S. Atomic Energy Comm. Rept. ORNL-912 (1951). (27) Farmer, W. S., Ibid., ORNL-635 (1950). (28) Forster, V. T., Geib, K. H., Ann. Phyrik 5, 250 (1934). (29) Foster, H. R . , Babb, A. L., thesis, University of Washington, 1963. (30) Garner, F. H., Tayeban, M., Anolcr Fir. Quim. LVI-B, 479 (1960). (31) Geankoplis, C . J., Hixon, A. N., INn. ENO.CHEM.42, 1141 (1 950). (32) Geankoplis, C. J., Wells, P. L., Hawk, E. L., Ibid., 43, 1848 (1951). (33) Gier, T. E., Hougen, J. O., Zbid., 45, 1362 (1953). (34) Gilliland, E. R., Mason, E. A,, Zbid., 41, 1191 (1 949). (35) Griffith, W. L., Jasney, G. R., Tupper, H. T., U. S. At. Energy Gomm. ~ ~ p f . K-972 (1952). (36) Groot, C., Ibid., HW-11841 (1748). (37) Hayford, D . A,, Ph.D. dissertation, Virginia Polytechnic Institute, 1962. 2, 310 (38) Hazlebeck, D . E., Geankoplis, c. H., I N D .END. CHEM.FUNDAMENTALS (1 963). (39) Heertjes, P. M., Holve, W.A . , Tlsma, H., Chem. E n ? . Sci. 9, 122 (1954). (40) Hendrix, C. 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