which was obtained from Equation 2 by substitution for C B and t and setting Z equal to 1/V. From the material balance for reacted A, the molar flow rate of C is given by
A‘*,
=
.Y*
+
S B
+ A’c
(31)
and the fraction of A , appearing as C is given by _ -- 1
h-Ao
- 2[E3(a) + a E z ( a ) ]IC1 ,
= IC2
(32)
f
=
k
=
(33) Plots of &(a) and &(a) are given in Figures 2 and 4. Nomenclature
a
=
a’
=
C* CAO
= =
C B ,Cc
= = =
&(a)
&(a)
dimensionless woun for first-order reaction. kiL - kiVr C,,, 2Qj dimensionless group for first-order reaction, kzL _ _kzV,- ~ Urnax 2Qf concentration of reactant in react.or, moles/cm3 concentration of reactant a t inlet of reactor, moles/cm3 concentration of products in reactor, moles/cm3 second-order exponential integral third-order exponential integral
=
1
-
NA/NA,
L = NA = ATA, = ATB,Sc =
R
Qf
= =
RW
=
1
= = =
U U,,, V
fraction of A reacted, dimensionless, f
=
V, Z
= =
P
=
(moles/cm3j1-~ sec reactor length, cm molar flow rate of reactant A at exit, moles/sec molar flow rate of reactant a t inlet, moles/sec molar flow rate of products a t exit, moles/sec bulk flow rate, cm3/sec radius, cm radius of reactor, cm time, sec velocity, cm/sec center line velocity, cm/sec dimensionless group, 1 rate constants,
(a
reactor volume, rR,*.L, cm3 1/v k(n - ljCa,”-’L for nth-order reaction =
Urn,,
Literature Cited
Bosworth, R. C. L., Phd. Mag. 39, 847 (1948). Cleland, F. A., Rilhelm, R. H., A.Z.Ch.E. J . 2 , 4 6 9 (1956). Denbigh, K. G., J . A p p l . Cheni. 1, 227 (1951). “Handbook of Mathematical Functions,” Milton Abramowitz and I. A. Stegun, eds., National Bureau of Standards Applied hlathematics Series, 55 (1964). MARVIN RI. JOHYSON Phillips Petroleum Co. Bartlesville, Okla. 74003 RECEIVED for review February 24, 1970 ACCEPTED *July27, 1970
CORRESPONDENCE Parametric Pumping and Cycling Zone Adsorption SIR: I have what I believe are important points regarding the separation techniques of “parametric pumping,” “cycling zone adsorption” (Pigford et al., 1969b), and the like. Mechanism of Separation
The cause for separation in such devices is not nearly as complicated as has been implied by most of the papers in the area-e.g., “coupling of oscillatory thermal and mass fields with alternating flow displacements” (Wilhelm et al., 1966), “periodic, synchronous, coupled transport system” (filhelni et al., 1968), “The origin of the separation effect lies in the ability of the column to store solute temporarily on the solid adsorbent particles, withdrawing it from the lean bottom product and subsequently adding it to the top product as each of these penetrates portions of the column in turn” (Pigford et al., 1969a). Much of the confusion has arisen, I think, through efforts to contrast parapumping ideas with equilibrium stage concepts (Sweed and Rilhelm, 1969) rather than to show how they fit into equilibrium stage concepts, which are fundamental idealizations of all separation processes involving more than one phase. First of all, the cause of the separation is what has been for years called “regeneration”-Le., the sorption of a solute out of a volume of fluid by a volume of sorbent, followed by the desorption of the solute from the volume of sorbent into a second volume of fluid. It is most common in molecular sieve, chemisorption, and ion exchange systems where in practice the sorption equilibrium is easy to shift by large amounts, so that relatively large volumes of fluid are in contact n i t h relatively small volumes of sorbent with a 684 Ind. Eng. Chcm. Fundam., Vol. 9, No. 4, 1970
relatively large amount of solute transfer before each shift of the equilibrium. The equilibrium may be shifted in any number of ways -e.g., by changing temperature or some property of the two fluid phases such as p H or concentration of a third component. To effect a separation no column or columnar shape is required. For example, a n apparatus shaped as in Figure 1 could be used to effect a separation (however small) between the two reservoirs. T o effect a separation no reversal of velocity is required; even in a closed system, only recycle or regeneration is required. An apparatus as shom-n in Figure 2 could be used to effect a separation (however small) by proper altering of the temperature. Such devices are parametric pumps. However, they will not give very large separations, because they are essentially “onestage” devices. The use of a long column “stages” or “sequences” the parametric pumping operation in an efficient way, just as the sequencing of stages in series in a column greatly increases the efficiency of distillation, extraction, and chromatography. By “staging” here I mean staging in the usual sense of staging in space and not in time. Staging in time has been implied as the difference between parametric pumping and other types of separation (Sweed and Wilhelm, 1969; Wilhelm et al., 1968). This is not true. The increase in separation with time of a batch parametric pump (to some limiting value) is analogous to the increase in separation with time of a total reflux batch distillation column (to some limiting value). Thus the separation is caused by “regeneration” and is enhanced in the usual way by staging the cause of separation in a long column. Neither regeneration nor staging in a column is a new concept. Their use together is new and this is the contribution of Wilhelm.
F
Impossibility of Perfect Separation
Most of the major contributions in this area have either stated or implied the possibility of perfect separation ( h i s , 1969; Baker, 1970; Pigford et al., 1969a; Rhee and Amundson, 1969; Sn-eed and Rilhelin, 1969; Wilhelm et al., 1968) if there are no dissipative effects. This would be exciting if true. The primary arguments for this view are based upon the paper of Pigford et al. (1969a). .\gain the situation can best be explained by use of equilibrium stage concepts. Consider a closed, direct mode system like Pigford’s, except t h a t the column of length Z is divided into Jf equal-sized sections. Rather than force the liquid through the column at a velocity V in a t’ime T = Z/V per half cycle, as Pigford, Baker, and H u m (1969a) did, force t’he liquid through the column in JI flow increments of infinitesimal duration (Dirac function), leaving a time of duration T.w = Z / V X bet’ween flow increments for each section or stage within the column to reach equilibrium. Assume no mixing between the X increments (This is equivalent to Pigford’s assumpt’ion of no dispersion.) and assume that’ there are perfect mixing and equilibrium lvithin each increment a t the end of the time, T M .(Such a device is the discrete analog of Pigford’s [‘local equilibrium” continuum model.) For a n y finite equilibrium (nonperfect adsorption and nonperfect desorpt’ion), including linear equilibrium, such a system will reach a finite pair of concentrations in the bottom and top of the column as the number of cycles, n, increases without bound. If the number of sections or stages, M, into which the column is divided, is increased to a n infinite number, the above system becomes indistinguishable from Pigford’s local equilibrium model. Pigford et al. have in essence solved the case of a column with a n infinite number of stages. I n effect, their continuuin model [and the model of Sweed and Wilhelm (1969) ] says “divide a finite-sized column into a n infinite number of infinitesimally sized equilibrium st’ages.” This is fundamentally different from the usual continuum approach t,o stagewise processes (the so-called transfer unit concept) which says, “divide a finite-sized column into a n infinite number of infiriitesimally short lengths with a finite number of stages per unit length.” For t’his reason, Pigford’s model (including the Aris and Rhee and iimundson extensions) predicts separations independent of column size, length to diameter ratio, and flow volume. They a r e all models with a n infinite number of equilibrium stages. They are useful in determining the maxiniuni possible separation in a finite number of cycles, but they are misleading when trying to determine the actual separation in a real, finite-sized column, even when no dispersive effects exist. The equilibrium limit of a n y real separation process involving more than one phase depends upon the equilibrium relationship and the number of stages (a stage meaning here a n equilibrium contact and a separation). As long as the amount of product desired is finite (not infinitesimal), the only way to obtain a perfect separation is to have either a perfect equilibrium (infinite ratio of component in one phase to the other a t equilibrium) or a n infinite number of stages. Saying bhat the continuum, local equilibrium parametric pump model will approach a cyclic steady state of perfect separation is analogous to saying t h a t a totally refluxed distillation column will approach perfect separation between the overhead and bottoms at steady state if the column contains a n infinite number of plates. The reason that the theory of Pigford, Baker, and Blum (1969a) appears to agree so well with the results of Wilhelm et al. (1968) is that their system probably had a large number of stages. B u t the number of stages was finite. It would be impossible accurately to predict the limiting bottoms con-
\ / RESERVOIRS
Figure
. Parametric pump without
columnar shape
SORBING a DESORBING SECTION HEATING MEDIUM
HEATING MEDIUM AND COOLANT IN
COOLANT OUT
/
-
PUMP
Figure 2. Parametric pump without reversal in velocity
centration using the local equilibrium models of Pigford et al., .iris, and Rhee and Amundson, even without dissipative effects under any form of (nonperfect) equilibrium relationship, whether it be linear, nonlinear, Langmuir, or other&ise. The deviations of the data of Wilhelm et al. from the theory could be due primarily to dissipative effects (finite mass transfer resistance, axial dispersion), nonlinear equilibrium, or the finite size of the column. There is no reason a t present to favor one explanation over the other. Design of large, Industrial Scale Devices
Design of industrial scale parametric pumps, cycling zone absorbers, and the like will require a model which takes into account the effect of column length (number of stages) on the degree of separation. I n large scale devices the effect of axial dispersion will become important, just as it is in large scale chromotography. Thus, the model will require the inclusion of this effect. Probably the most convenient way to do this would be to use the idea of a “stage efficiency” or the “height equivalent of a theoretical plate” concept. Some means such as the patented baffling device of Baddour (1966) will be necessary in large scale columns to prevent axial dispersion. literature Cited
Aris, R., IND.ENG.CHEY.FUNDAM. 8, 603 (1969). Baddour, R. F., U. S. Patent 3,250,058 (1966). Baker, B. 111, IND.ENG.CHI:M.FVSDIM. 9, 304 (1970). Pigford, R. L., Baker, B. 111, Blum, D. E., IND.ENG.CHEM. FUNDAM. 8, 144 (1969a). Pigford, R. L. Baker, B. 111, Blum, D. E., I K DENG. . CHEY. FUNDAM. 8, 848 (1969b). Rhee. H. K.. Amundson. N. R.. IXD.EXG.Cmx. FUNDAM. 8. 144 (1969). !. CHEM.FVXDAM. 8, R., I N DESG. . CHEM.
W., Sweed, K. H.,
IND. ENG.CHEY.FUNDAX. 7, 337
Paul R. Harris S t a u f e r Chemical Co., Dobbs Ferry, N . Y . 10522 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
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