by NSF grant GK-1275 and a n unrestricted research grant from the Petroleum Research Fund of the American Chemical Society. They thank A. H. P. Skelland for reviewing the manuscript and offering critical comments.
X
Nomenclature
+= dimensionless velocity profiles for 5 _< X and f Case I1 Q = dimensionless volume rate of flow = dimensionless quantity given in Figure 2 Q’ S/Dt = Jaumann derivative
B F, Q
= gap between parallel surfaces = force of fluid on moving surface = gravitational acceleration
= distance above gravity reference plane = length m,n = rheological constants associated with q m’,n’ = rheological constants associated with 0 6 = defined by p pgh = pressure = volume rate of flow 8 = defined by l / n Ti = uniform velocity of moving surface
h L
Ashare, E., Bird, R. B., Lescarboura, J. A., A.I.Ch.E. J . 11, 910-16 (19$Ei).
Bird, R. B., Lectures in Transport Phenomena,” Chap. 1 of “Today Series,” R. B. Bird, W. E. Stewart, E. N. Lightfoot, and T. W. Chapman, A.1.Ch.E. Meeting, March 1969, New Orleans, La. Bird, R. B., Carreau, P. J., Chem. Eng. Sci. 23, 427-34 (1968) (Equation 22). Oldrovd. J. G.. Proc. Rou. Soe. London A200.523-41 (1950):A283.
= local velocity in z-direction
W*
= width s, y, z = rectangular coordinates
aoundary Layer Theory,” pp. 60-2, McGraw.ark, 1955. I. P., “Non-Newtonian Flow and Heat Transfer,” k, 1967. 12, 890-3 (1966).
GREEKLETTERS
4
= normal stress function associated with = shear rate
7w
- ryu
7
= shear viscosity
6
= normal stress function associated with rZr rz2 = dimensionless group defined in Equation 7
A
2 X,
literature Cited
+
uz
= dimensionless integration constant = dimensionless position in s-direction P = density 1 = stress tensor rZE = component of stress tensor = dimensionless velocity in z-direction @ (
-
RECEIVED for review January 21, 1969 February 17, 1969 ACCEPTED
COMMUNICATIONS ZONE REFINING CONSIDERED AS A MULTISTAGE SEPARATION METHOD Some aims are suggested for a science of separations which seeks to unify the understanding of diverse multistage separation methods. Zone refining, examined in this context, is found to be an unusual type of equilibrium separation method in which multistage separation is achieved without conventional countercurrent flow and reflux.
W E examine a typical multistage separation method, zone refining, and consider how a concentration difference is produced and how it is multiplied. Examining such matters is one task of a science of separations, the purpose of which is to establish a common basis for understanding the many and diverse methods of purification and separation in use today. A few steps by way of classification have been taken toward such a unifying science. For example, Benedict and Pigford (1957) distinguished between separation methods based on the irreversible flow of heat or matter, such as diffusion methods, and those using potentially reversible processes, such as distillation and liquid-liquid extraction. Pfann (1966a) in a cursory survey suggested kinetic for the former group and change-of-phase (which is not broad enough) for the latter. Pratt (1967) suggested nonequilibrium for the former, and equilibrium for the latter. At this point we suggest the terms “kinetic” and “equilibrium” for these two classes of separation methods. Examples of methods of the two classes are shown in Table I (after Pratt). [We question Pratt’s classifying the gas centrifuge as an equilibrium process. Surely it is akin to electrodiffusion (electrostatic potential gradient), thermal diffusion (temperature gradient), and mass diffusion (concentration gradient).]
Suggested goals of a science of separation include: to learn how to perform single-stage separations more effectively, to seek new methods of producing a concentration difference, to seek new methods of utilizing countercurrent flow and reflux, to seek and define the underlying unity of the various disciplines involved, and to express this underlying unity in basic theoretical form (Pfann 1966a). I n a typical equilibrium process involving two phases, such as multistage distillation, separation is achieved by producing the two phases, bringing them into contact so that equilibration can occur, and moving the phases in countercurrent fashion. Equilibrium is approached by means of diffusional processes in both phases, with transfer of components across the interface. Separation methods based on crystallization Table 1.
Classification of Multistage Separation Processes
Equilibrium Distillation Absorption Liquid-liquid extraction Adsorption Chemical exchange
Kinetic Mass diffusion Sweep diffusion Thermal diffusion Electrodiff usion Molecular distillation Irreversible electrolysis
Ion exchange
Countercurrent gas centrifuge VOL.
B
NO.
2
MAY
1969
357
differ from this scheme in one important respect, and those based on zone refining differ in a second important respect, which we now examine. Although the separation achieved in crystallization is based on the equilibrium difference in concentrations of a component in solid and liquid phases, this equilibrium is seldom if ever achieved for the solid and liquid phases as a whole. Diffusivity in the solid is almost always too small to permit equilibration. How then does a macroscopic concentration difference arise? Equilibrium is realized or is usually closely approached in a narrow region coniprised of a few atom layers on either side of the solid-liquid interface. As the crystal grows, the interface advances into the liquid and, on the strength of this equilibrium, rejects one or another component preferentially. This advancing interface has been called, with some accuracy, a “thermodynamic broom” ; its sweeping action produces a macroscopic segregation. Thus, crystallization is a n equilibrium separation method, but it differs from the usual. Consider a cylinder of liquid binary solution freezing slowly from one end to the other. As the interface advances, components in the solid, once they are more than a few atom distances behind the interface, are for all intents and purposes removed from the system. The remaining liquid, in which mixing is fairly complete, becomes enriched in one component. This process was designated “normal freezing” for purposes of discussion (Pfann, 1952). K e see that it is exactly analogous to differential or Rayleigh distillation, and in fact the Rayleigh equation describes the distributions of components in the two processes. Given the concentration difference realizable by a moving solid-liquid interface, how can it be enhanced by countercurrent procedures? One approach is to use normal freezing as the basic step in complex pyramid or diamond schemes involving separation of solid and liquid fractions, remelting, refreezing, and recombining (Pratt, 1966, p. 254). These schemes are inefficient and time-consuming. Another is to imitate the countercurrent flow and reflux of a distillation column, and indeed, various procedures have been demonstrated for mechanically moving crystals downward and liquid upward through a column hot at the bottom and cool a t the t o p f o r example, the column crystallizer (Schildknecht, 1961). The imitation, though good, is incomplete, however, for although liquid and vapor can equilibrate by interdiffusion of components, a crystal can interact with a liquid only by growing or melting. This approach, despite mechanical complexity, has shown promise, in the laboratory at least. A third approach to multistage separation using the solid-liquid transformation is zone refining (Pfann 1952, 1966b). I n a batch zone refiner molten zones one after another traverse a cylindrical charge of solid. The moving zones carry toward the end of the charge solutes that lower the freezing point of the major component. Eventually a steady state of maximum separation is reached. I n considering how this steady state is reached we encounter the second way in which zone refining differs from conventional countercurrent methods: the achievement of “multistage” separation without any readily evident countercurrent flow or reflux. Clearly, in batch zone refining, a multistage separation is achieved in the steady state. [An excellent approximation to it is: C(z) = A@”
where C(z) denotes solute concentration at distance 358
I&EC
FUNDAMENTALS
1:
along
a charge of unit cross section, and A and B are constants given by k=-
B1 and A eBz- 1
=
CoBL eBL - 1
u~here1 denotes zone length, L charge length, k the distribution coefficient defined as the ratio of solute concentration in the freezing solid to that in the bulk liquid, and CO the mean solute concentration. B is large, and the separation is large, when k and 1 are small.] One is tempted to seek a good analogy between a batch zone refiner and a batch distillation column at total reflux. I think such a good analogy does not exist. True, the molten zone, as it traverses the charge, freezes out a cmcentration in equilibrium with that of the bulk liquid. True, the melting interface of the zone senses the solute concentration in the charge by mixing the melted increments with the liquid of the zone. But countercurrent flow and reflux, in the usual meaning of these terms, simply do not exist in a batch zone refiner. How, then, is a large “multistage” separation achieved? To find a n answer, it is helpful to think of the steady-state separation in a batch zone refiner as a balance of two fluxes of solute as defined by Reiss (1954). The first is a convective flux representing the sweeping action of the freezing interface of the zone, proportional to the concentration and to [1(1 - k ) / k ] . The second is an opposed diffusive flux representing the back-mixing of the material entering the zone a t the meltihg interface, proportional to the concentration gradient and to (12/2). The convective flux piles solute against one or the other end of the charge; the diffusive flux permits solute to slide backward. For the greatest separation, 1 should be small, to minimize the diffusive flux. On the other hand, small 1 connotes a small convective flux, and hence a long time to reach the steady state. I n a continuous zone refiner there are readily evident flows of materials through the refiner governed by the material balance feed equals product plus waste. But the separation effect, just as in a batch zone refiner, is perhaps best visualized in terms of the above fluxes. The foregoing brief and incomplete discussion is submitted as exemplary of the kinds of questions to be answered by a developing science of separations. ,4s new separation methods, such as the parametric pumping methods of Wilhelm and coworkers, are introduced, similar questions arise, and should be answered. Acknowledgment
I thank R. L. Pigford and R. P. Andres for helpful discussions. Over the years, I have been indebted greatly to the late R. H . Wilhelm for stimulating discussions an matters of separation science. Literature Cited
Benedict, &I.,Pigford, T. H., “Nuclear Chemical Engineering,” p. 472, McGraw-Hill, New York, 1957. Pfann, W. G., Ann. N . Y . Acad. S a . 137, 5 (1966a). Pfann, W. G., Trans. A.I:M.E. 194, 747 (1952). Pfann, W. G., “Zone Meltlng, 2nd ed., Wiley, New York, 1966b. Pratt. H. R. C.. “Countercurrent Separation Processes,” Elsevier, New York, 1967. Reiss, H., Trans. A.Z.M.E. 200, 1053 (1954). Schildknecht, H., 2.Anal. Chem. 181, 254 (1961). W. G. PFANX Bell Telephone Laboratories, Inc. Murray Hill, N . J. 07974 RECEIVED for review December 26, 1968 ACCEPTEDFebruary 11, 1969
