PARAMETRIC PUMPING A Dyanzic P h c i p l e for Separatiiig Fluid Mixtures R I C H A R D H. W I L H E L M , ALAN W. R I C E , I ROGER W. ROLKE,? AND N O R M A N H . SWEED
Department of Chemical Engineering, Princeton Unicersity, Princeton, ,V. J . 08540
Parametric pumping i s an adsorptive separation technique based on periodic, synchronous, coupled transport actions. The net consequence of the coupling of alternating adsorbent-fluid position displacements with the cycling of a thermodynamic intensive parametric variable (temperature) i s a buildup of separation from cycle to cycle. Thereby the powerful separational effects o f countercurrent action are released in the uncommon circumstance of a continuously regenerating separation column having only a single fluid phase. Thus cycle-to-cycle time-staging as well as position-staging occurs within a column. Parametric pumping also i s a compound, macroscopic, active transport system in which a species i s moved at the expense of some form of energy (thermal, chemical potential) from a region of low concentration to high. The large separation capability o f the direct mode of parapumping i s demonstrated experimentally for a toluene-n-heptane mixture on silica gel. A very important theoretical phase-angle relationship between oscillatory parts of the system has been verified experimentally for this mode. New results are presented for the recuperative mode in which heat is exchanged internally; NaCI-Hz0 was separated in a continuous open system on ionexchange resins. Preliminary theoretical efficiencies have been computed.
of parametric pumping as a separation techA nique . is presented based on recent experience in laboratory ti OVERVIEW '
experimentation. mathematical model formulation, and system exploration. \filhelm et al. (1966) presented an initial description of parametric pumping. IVilhelm (1966) identified the principle as a form of active transport, a subject of general interest in biology; and recently IVilhelm and h e e d (1968) have presented experimental data sholving the system's great separative capability. Parametric pumping is a dynamic technique 1.vhich is rooted in this sequence of ideas and actions: An adsorptive system is driven through cyclic! relative displacements between adsorbent particles and interstitial fluid. The equilibrium solute content of the particles is caused to vary in a periodic fashion by cyclically varying the adsorbent temperature or other intensive thermodynamic variable which affects the equilibrium. This variation in turn forces a cyclic. diffusive solute exchange to occur between the solid and fluid phases. The above system components are coupled dynamically, causing the system states to vary periodically. The coupling involves bringing into synchronism the frequencies of the solidfluid displacements with those of the temperature cycles of the adsorptive solids. I t involves as well the establishment of effective phase angle and wave form relationships bet\veen these periodic actions. The resulting strategic interplay bet\veen the periodically varying interphase diffusive transport of solute and the periodically varying axial flux of solute gives rise to a nonzero: timeaveraged axial compositional gradient. Thus, briefly stated, parametric pumping is a separation technique based on a periodic: synchronous, coupled transport system. I n the separation of fluid mixtures by means of the "parapump" principle, each process variable, such as composition and temperature, alternates periodically about mean values a t all points in the system. By contrast, in the more common Present address, U. S. Army, Langley Research Center, Ilampton, Va. 23365. * Present address, Shell Development Co., Emeryville, Calif.
94608.
separation techniques-continuous distillation and solvent extraction, for example-these variables normally exhibit a steady-state behavior. Further, the following distinctive characteristics of parametric pumping arise as a natural consequence of its being dynamic. Considerinq single-column arrangements only, two limits of operation are discernjble, both of which may be described by the same set of basic parapumping differential equations. An open system provides a continuous, self-regeneratingcountercurrent separation. These elements are encountered as a matter of course in separation techniques involving two phases; they are unusual in the present circumstance of a fixed bed with only one mobile phase. Closed system operation, in turn, highlights an important new element-namely, separation may build on itself from cycle to cycle to augment separation capability as normally built u p from position to position. In other words, parapumping is staged in time as well as in position. As in single-column separations generally, transitions between the limits of open and closed (total reflux) operation lead to corresponding transitions in separation capability. Additionally, separations may be enhanced through staging in multicolumn arrays. Ideas regarding dynamic coupling were introduced in the nineteenth century by Faraday and Lord Rayleigh in connection \vith variable electrical reactance, a general rise in interest occurring some 40 years ago. Parametric amplifiers and optical lasers are recent examples of the application of parametric pumping principles in science and technology. In the laser, for example, the dynamic coupling of electromagnetic-light energy with molecular electron fields causes a "pumping up" of electrons to form metastable electron population densities on certain energy levels. Examples (Louisell, 1960) of parametric action in physics and electrical engineering commonly relate to self-sustaining oscillatory systems. These may, for example, be acted upon parametrically by electrical or mechanical sources having one or more frequencies different from the primary signal frequency; resonances and voltage amplification are evoked as the result of dynamic couplings. VOL.
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Such systems may or may not be synchronous. By contrast, the parametric separation action here described is non-selfsustaining in the usual sense, and it involves synchronous couplings.
V/A
I
ADSORBENT PHASE,(s) -
Parametric Pumping Elements
A parapump system is a dynamic, nonequilibrium system that nevertheless is governed by the limits set by the relevant equilibria. Equilibrium therefore is the first subject discussed. Next, the elements of parapump separation are considered in terms of generalized models at two levels of mathematical structure. These provide the basis for parametric pumping separation. Equilibria. Interphase equilibria are described symbolically by
4r*
=
$f*($JS,OS)
(1)
the variables being expressed in dimensionless form. 9f*represents the fluid-phase solute concentration in equilibrium with the solid-phase concentration, +s. Os represents any intensive thermodynamic variable which parametrically affects the compositional equilibrium and which can be experimentally varied in a cyclic manner. Of the many intensive variables available, those which most commonly affect heterogeneous equilibria to a significant extent are temperature, pressure, and other material components, such as hydrogen ion concentration-i.e., pH. Electrical and magnetic fields are others that suggest themselves. Skarstrom (1959) has used pressure as the intensive variable in a two-column parapump type of system to remove moisture from air. The parametric separation principle is not limited to fluidsolid adsorptive systems; it encompasses any combination of separable phases which can be cyclically displaced, one past the other. For example, a partitioning liquid phase such as liquid resin may be held in a porous matrix while a second fluid 338
I&EC FUNDAMENTALS
Figure 2. Differential section of column depicting primary periodic actions
(liquid or gas) shuttles back and forth. Further, more than one parametric variable-temperature and pH, or temperature and pressure, for example-may be brought to bear simultaneously. The number of equilibrium equations required in any given instance depends upon the number of components, separable phases, and thermodynamic variables, defined by the Gibbs phase rule. By way of illustration, the NaCl-H20 equilibrium on a mixture of weak cation and anion exchange resins is presented in Figure 1. The substantial effect of changing the system temperature is evident. These data underlie experimental and theoretical parapump separation results discussed below. Comparing the equilibria of different systems: the larger the effect of temperature or other intensive thermodynamic variable in changing the equilibrium solute content of the adsorbent, the larger will be the separation in a parapump system for any given process arrangement. Further, nonlinearity in the equilibrium, such as is evident in Figure 1, may greatly enhance parapump separation. For purposes of parametric pump separation, the need is apparent for adsorbents having
significantly large effects on equilibria due to changes in intensive parametric variables. Generalized Models. Consider the differential axial column element, dz, depicted in Figure 2. T h e fluid phase, f. is an "open" system because it exchanges solute mass with its surroundings-Le., the adsorptive phase, s: and the fluid in the adjacent column sections; heat also is exchanged. The t\vo primary periodic actions involved in parametric pumping are indicated by the vector arro\v pairs representing the alternating axial fluid velocity relative to the adsorbent phase and the sum of the alternating fluid and solid phase transients. For present purposes we omit details of how the transients occur and represent them by the function - q ( w l t ) and the velocity by V ( W ~ ) In , general, frequencies W I and up might be different but for parametric pumping to occur q and V are synchronous. Althey must be equal-Le., though synchronous, q and V may be displaced by a phase angle, e-that is, q = q(ut e) and V = V ( u t ) . Consider now the material balance over the fluid phase :
lations among the compositions in the t\vo phases, pf and as, and the independent variables. time. position. anti the parametric driver, 0. A material balance on the fluid phase
(7) an interphase transfer rate
an equilibrium expression
and the continuity equation
+
V ( u t ) dP, -= bZ
q(wt
+
e)
!e' call this the "Tinkerwhere 0, is the fluid composition. + toy" model. Solving for the concentration gradient (rvhich is actually the local separation over the differential element).
comprise, \vith appropriate initial and boundary conditions, the parapump system. The separational gradient in Equation 7, To, arises from the coupling of the first and the third terms. The second term is simply the fluid phase capacity, and the fourth is a dissipative loss. Equation 10 describes the effect of variable fluid density and flow geometry on the velocity. V. The above equations encompass liquid and gas cases, the latter having nonconstant density. Direct Mode
\$-e no\v ask Xvhether the local gradienr. when averaged over a cycle, is nonzero. indicating a useful separation, or zero, indicating that no net separation exists. In the former case we might expect the possibility that the second cycle would build on the first akin to a staged operation in time instead of position Therefore we examine 2r:w
q(wt
+
e)
dt
(4)
If V i s zero for at most a finite number of points, a new function may be defined
rvhere h is piecewise continuous if q is piecewise continuous. Therefore the integral
12='w
h(wt,e)dt = H(w,e)
(6)
exists. Is H(w,e) zero or finite? By choosing a simple case [q = cos (ut e), V = cos ( w t ) ] , it can be shown using Cauchy's method that H(u,e) = cos E, which clearly is not usually zero. Therefore, the time-averaged value (over a cycle) of the ratio of two synchronous periodic functions of any wave form is not generally zero. I n terms of parametric pumping, the time-average local separation [a q,, a z = H(u>E)] therefore generally is nonzero. Thus, ultimately, separation depends upon the mathematical properties of the particular cyclic, synchronous functions that describe the elementary actions of parametric pumping. The actual separation over a column will, of course, be the integral sum of the local contributions here described. The following more complete and general model expands upon the ideas above and is written to display the coupling re-
+
The direct mode of parametrically separating mixtures concerns apparatus arrangements whereby the velocity alternations. V ( t ) . and the cyclings of the parametric variable, O ( t ) , are independently controllable. The experimenter acts externally c n the system to couple these periodic actions by bringing them into synchronism and by establishing a desirable phase angle between them. The degree of freedom afforded by this independent, direct controllability permits placement of the system in operational domains in which large separations may occur, but at the expense of primary energy (heat) recovery. For many purposes, such recovery may not be important compared to the achievement of separation. By contrast, in the recuperative mode discussed in the next section, the periodic actions in question no longer are independent and smaller separations generally may be expected. These are counterbalanced by a recuperative, internal recovery of heat. Figure 3 presents the apparatus essentials for the direct thermal mode of separation for a single column. Figure 3a shows the relations among three elements: the bed of adsorbent particles, the thermal jacket, and the means for alternately displacing the fluid mixture, the pistons. Figure 3 b indicates the phase angle relationship (here a t zero) between the vertical fluid flow vectors and the synchronow lateral heat flow vectors ( Q ) : the latter serving to cycle the temperature of the vessel contents. -4lternatives to these system elements are evident: Seed for significant thermal penetration will determine bed diameter for any given cycle time and in the limit might require the use of an adsorbent wall film; the jacket heat source-sink may include spatial and temporal variants; and the axial fluid displacements may be impressed in different ways that affect the degree of end mixing. Experimental. Let us consider now the specific case of the direct-coupled thermal parapump separation of a liquid-phase toluene-heptane mixture on a silica gel adsorbent, a system previously reported by !t'ilhelm and Sweed (1968) and here VOL. 7
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DRIVEN PISTON
f
-0
Q
Q
Q
Q
Q
Q
Q
Q
(b) HEATING HALF-CYCLE
DRIVING PISTON
Figure 3. Apparatus direct mode
I
1
arrangement
10-3
COOLING HALF-CYCLE
for
separation
by
0
0
E;PERlMiNTAL
n/2
7
mi 3n/2
2n
c , P H A S E ANGLE, ( RADIANS )
1
I
0.59 FRACT. T O L .
Figure 5. Experimental effect of phase angle between flow and heating cycles on toluene separation factor Each run starts with 20% (volume) toluene in toluene-heptane mixture and continues f o r 2 3 cycles, each of 8.5 minutes' duration
/
CYCLING LIMITS BETWEEN 70'C B 4'C
0.48 FRACT. T O L ,
7 o o c B I5.C
0.49 F R A C T . T O L . 2.5 x I O - ) F R A C T ~ T O L .
CYCLE T I M E 8.5 MIN. 3 0 MIN. 140 MIN.
0
I IO
I
20
I 30
I 40
I
50
I 60
N U M B E R OF CYCLES
Figure 4. Experimental parametric separation of tolueneheptane liquid mixture with silica gel adsorbent Parametric cycling temperature limits indicated. Each experimental point represents a fresh start from equilibrium at ambient temperature with the fluid phase composition at 2070 (volume) toluene
presented with additional data and expanded theoretical analysis. The equipment consists of a jacketed glass column (100 X 1-cm. i.d.) packed with particles (30- to 60-mesh) of chromatographic grade silica gel (Matheson Coleman Bell), a constant rate, positive displacement, dual-syringe infusionwithdrawal pump (Harvard Apparatus Co.), sources of hot and cold water for the jacket (not shown in Figure 3), and a 340
I&EC FUNDAMENTALS
programmed cycle timer. The timer is adjusted to reverse the direction of the fluid stream periodically, and also to cycle the jacket temperature by connection to hot or cold sources. Both alternations have the same frequency and are in phase. Pump and timer adjustments establish the alternating velocities to be uniform at all times and displacements to be equal in each half cycle. T o compare experiments, each run was started under identical conditions: The interstitial fluid was a mixture of 20% (volume) toluene in n-heptane and was in equilibrium with the silica gel at ambient temperature. The bottom syringe was filled initially with 30 ml. of the same 207, solution, this volume being approximately equal to the interstitial volume in the column. The top syringe initially \vas empty. Temperature limits between hot and cold sources are shown in Figure 4. At the beginning of a run, the syringe pump is started and hot water is circulated through the jacket. \Vhen 30 ml. have been displaced upward, the pump reverses direction and the jacket is switched to the cold water source. This process continues until 30 ml. are pumped back downward through the column, completing one cycle. Since no product is removed, we call this method "total reflux." To secure for analysis a sample of mixed product in each piston, the final cycle of each run is carried to about 90% completion. All products containing more than about 1% toluene are analyzed by means of refractive index mith a sodium D line source; those containing less, by ultraviolet absorption (Cary 14 recording spectrophotometer, 2685-A. wave length). Separation factor (SF) is defined as the ratio of mixed average toluene volume concentrations in the syringes, compared top to bottom. Figure 4 shows the results of three experimental separation series, each performed with a different cycle time. Experimental points represent fresh starts from initial conditions to the number of cycles noted on the abscissa. The separation factor is observed first to increase approximately exponentially with number of cycles and finally to lean toward a limiting separation. The potential for still further separation by increasing the number of cycles is evident in each sequence. The actual toluene composition in each piston from which the sepa-
ration factor is calculated is indicated on the graph for selected experiments. The larger value is the mixed average concentration of a full 30 ml. of solution in the upper syringe a t the end of the upstroke; the smaller concentration is that of the lolver syringe a t the end of the downstroke. The largest separation attempted yielded a n average volumetric toluene fraction of 0.59 in the upper piston cylinder and 4.7 X in the lower (separation factor > lo6) after 5 2 cycles. Additional experimental results, shown in Figure 5, depict the effect of phase angle between flow and heating cycles. Each experiment lasted 23 cycles, each being 8.5 minutes. Qualitatively these phase-angle separation responses over the experimental column reflect the corresponding characteristics a t the local, differential level, as discussed earlier for the Tinkertoy model-Le., the separation gradient equals cos e. ,4s predicted, the direction of separation changes sign experimentally, higher concentration being a t the top of the column at one phase angle and at the bottom a t a 180' phase shift. Further, as predicted, there is no separation (SF = 1.0) a t two valc,es of C > 180' apart. However, the separation maxima are displaced slightly from the predicted 0 ' and 180' locations. This small shift of the wave is due to thermal and mass diffusive lags inherent in the experimental system. Model. A mathematical model next is presented in terms of material balances, rate equations, and equilibria for the experimental arrangement here described. T h e differential equations comprise an expansion of the vector Equations 7 > 8, 9, and 10 for the case of incompressible, one-dimensional flow (liquids) in a constant cross-section vessel. Material conservation for the ith component in the fluid and solid phases contained in a differential, axial section of the column, dz, is expressed through this equation :
INITIAL CONDITIONS
$f, (bottom reservoir, 0) is fixed; upper reservoir empty,
BOUNDARY CONDITIONS FOR TOTAL REFLUX
1. A t z = 0
2.
Atz = 1
1. i l t z = 1
J nr 2. A t 2
Dependent and independent variables are dimensionless, having been normalized through reference to initial fluid concentration. total column length, and frequency of displacement alternation. An approximate rate equation, convenient in its simplicity for expressing interphase transport and suitable for first-order explorations, is
Model formulation is completed by expressions for interphase equilibrium as follows :
4f3*=
If I
@j,*[m), 9sn
48*> , , , > @ S i > ,
[g
.
., 43"-11
(1 3)
I
There are n components taking part in the separation and hence ( n - 1) material balance equations are required, the final degree of freedom being utilized in the total continuity equation. Further. the constant density assumption for a liquid assures through the latter equation the position independence of the fluid velocity, which is the coefficient, af(t), in term [ a ] . The initial and boundary conditions which complete the statement of the model are these:
=
0
The set of boundary conditions used (A or B) depends on the direction of flow as indicated by the time in the cycle. Boundary condition 1 is that for total reflux where the effluent from a column end on one half cycle is mixed and returned on the next half cycle. Boundary condition 2 is the familiar Danckwerts condition. The initial conditions may be any which are physically realizable. I n the case of total reflux, here described, the magnitude of separation depends not only on the coupled actions expressed in Equations 11, 12, and 13 but also on the total solute content of the system. The latter quantity depends upon initial conditions. Dependent variables in the conservation, rate, and equilibrium equations are identified as +f, +j*, and +3 being, respectively, the dimensionless fluid composition, fluid composition in equilibrium with the adsorbent phase, and the adsorbent phase composition. Independent variables are the dimensionless time, t, axial position, z , and the parametric system temperature, 0. The terms in Equations 11 and 12 express mass conservation contributions attributable to these mechanisms: [ a ] axial convection, [ b ] interstitial, fluid phase transient, [c] adsorptive phase transient, [ d ] axial fluid-phase diffusive loss, and [ e ] lumped-parameter interphase diffusive transport rate. Parametric action resides in bringing into synchronism and into suitable phase relationship the alternating flow velocity in VOL. 7
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341
and that the system coefficients are physically attainable in workable combinations. Coefficients are identified thus : cy, velocity amplitude; K , ratio of volume of adsorptive phase to that of the interstitial fluid phase; 7, dissipative axial diffusivity of soluteD 1
= --
R2 w
0
IO
20 30 N U M B E R OF C Y C L E S
40
Le.. reciprocal characteristic diffusion time, D/R2, relative to the frequency, w , of the axial fluid convective displacements. (Dis a lumped interphase diffusion constant and R is the adsorbent particle radius.) The properties of the adsorptive equilibrium function, +f*[ 1, complete identification of the essential parameters. System Properties. Figure 6 presents two sets of computations using the mathematical model above for initial qualitative exploration within the limits of reasonable expenditures for computer time. Figure 6 shows the SF-cycle number dependence on A ; it is consistent with the experimental dependence on cycle time (Figure 4). Separation is retarded ultimately by two counteractions: dispersion due to noninstantaneous interphase transfer, represented by A, and axial, fluidphase diffusive losses, 7, caused by fluid mixing processes in the interstices between particles. I t is clear from the experimental results in Figure 4 that these dissipative actions, which tend to limit separation, fortunately are light, permitting large separations to occur. It is instructive next to display details of parametric pump action through computations of compositional profiles with position (Figure 7) and with time (Figure B), using Equations 11 and 12 as mcdels and based on experimental equilibrium values for the toluene-heptane-silica gel system of Robins (1967). The profiles represent limiting conditions-i.e., all system parameters have become steady periodic. The steep wave front of Figure 7 is that of an adsorption wave and its shape reflects the effect of the nonlinear, Langmuir-type adsorption isotherm. [However (Wilhelm et al.,
50
Figure 6. Computed dependence of separation factor on cycle number for different values of A, dimensionless ratio of cycle time to characteristic particle diffusion time [ a ] , c t f ( t ) , with the periodically cycling interphase transport, [ c ] . The latter action is imposed experimentally on the system by causing the temperature, e, in [g] to cycle periodically, the result of this action penetrating the system of equations successively through terms [f]and [ e ] and ultimately to The magnitude of separation by means of parapumping depends on the numerical values of the equation coefficients, cy, K , 7, A, the function df* [ 1, and the balance among them. The experimental results presented in this paper and by Bendelius (1961), Rice (1966), and Rolke (1967) demonstrate that the parametric pumping principle applied to separation is operable
IC].
z
2
3.:
k
I
I
1
I
I
I
I
7 I -
v)
0
a
Eu 3.0 0 3 -I LL
5
2.5
2
E
m
d2.0 3 0 W
AFTER DOWNFLOW
*+
* 1.5
I
z
0 5 1.0 0 a I 0 0
- 0.5 0
2 . l LL
2 00 BOTTOM
0.I
0.2
0.3
0.4 0.5 0.6 2 , AXIAL POSITION
0.7
0.8
Q.9
I .o TOP
Figure 7. Computed dimensionless interstitial fluid composition, +/, and adsorptive phase equilibrium composition, +I*, vs. axial position in bed a t two instants in cycle, after completion of upflow and downflow 342
l&EC FUNDAMENTALS
6, and desorption occurs. Abruptly at the step change of var-
.. .
EBOTTOM E N O O F COLUMN I
“5
9
‘t
ia -
I
w
I
LL
a
C~MEAN
I I
~ T C O L D
I
l
0
l 0.2
1
I 0.4
I
I
I
0.6
I
0.8
I
I I.o
F R A C T I O N OF CYCLE
Figure 8. Computed fluid compositions (interstitial fluid phase, 4, and equilibrium, $,*) vs. time (fraction of cycle) Symmetric square-wave displacements in phase with square w a v e thermal cycles. Steady periodic conditions prevail. In tent+j, +t* shaped curves,
-
---
1966), the fact of separation in parametric pumping does not depend upon nonlinearity in the equilibrium relationship.] The wave represents an instantaneous profile a t the end of the do\vnflow (cooling) half cycle. The desorption wave a t the end of the upflow half cycle is relatively diffuse, as is characteristic of such equilibria. The direction of the interphase diffusion gradient, (q,* - q,), alternates over each cycle. Parametric action thus serves to build up through successive adsorption-desorption steps a steep solid-fluid adsorption wave which constitutes an effective barrier to the passage of fluid-phase solute through the (bottom) end of the column. Time (in terms of fraction of a cycle) is the base against Lrhich fluid velocity, system temperature, and fluid compositions are related in Figure 8. Velocity is assumed to have a square Lvave form, symmetric about zero velocity, as noted. Temperature of both phases in turn is caused to cycle symmetrically about a mean, also with a square wave shape and in phase with velocity. The effects of these actions on fluid compositions, 0,.and q,*, are first noted in terms of Eulerian behavior (vertically shaded curve pair)-i.e., occurrences are examined a t a fixed point in space (here referred to the solid adsorptive phase) as time passes. The first of these locations is the top of the column; the second, one third from the top end. Those at the bottom, dilute end of the column are so small as not to be visible on the scale of the figure. Consider as typical the composition history a t the top end of the column over acycle. During the left lobe-Le., upstroke, heating half cycle, #,* >
iables, there is a reversal in the right lobe in which @*, < 4, and adsorption occurs. Finally, the cycle is closed by a step function return to the initial state. I n the desorption cycle, for example, the step change in temperature causes a rapid desorption. Under nonflow circumstances, the two curved compositional trajectories at the far left would continue to approach each other toward a state of equilibrium. I n parametric pumping, however, less saturated fluid is caused to flow past the reference location from elsewhere, maintaining thereby a n almost constant local driving force-i.e., interphase transferover the entire half cycle, the same being true of the adsorption half cycle. This tendency toward constancy of the interphase diffusive driving force over time (a half-cycle “stage”) as a result of parapumping dynamics is suggested to be a qualitative counterpart of a corresponding constancy over position as the result of countercurrent operation. The sequence of events above may also be viewed as a Lagrangian odyssey of a fluid element as it courses back and forth among the solid particles. Such composition curves are developed from intersections with the Eulerian results and are noted to be the tent-like curves in Figure 8. As before, the left lobe represents desorption, and the right, adsorption. I t is perhaps instructive to compare the Lagrangian behavior of parametric pumping, above, with elements of common gaseous molecular diffusion (Hirschfelder et al., 1954). For example, the convective displacement, cij(t), of a packet of fluid is considered analogous to the movement of a diffusing molecule over one mean free path; the contact of fluid and solid in parapumping is suggestive of gaseous molecular collision ; the release (or uptake) of solute by the adsorbent is similar to the adjustments of the gaseous system to the addition (or loss) of a molecule coming from another compositional domain. The above steps of transport, contact, and equilibration are: for the gaseous molecule, discretely separable, sequential actions; for the fluid packet in parametric pumping they are more nearly simultaneous and parallel, thus rendering the analog only partially valid. Severtheless, parapumping may be viewed as a compound, macroscopic diffusive transport mechanism. An alternating change in direction of the elementary displacement act is central to any diffusive process, microscopic or macroscopic. Thus the gaseous analog has molecular motion in both positive and negative directions, the result of which is a net transport of a species from a region of high concentration to low. In parapumping. the alternating direction of flow is a direct counterpart to molecular motion, but in this case the net transport of a species is from a region of low concentration to high. Parametric pumping therefore is a form of “active diffusion” or “active transport,” as biologists call the process of diffusion uphill, so to speak. Recuperative Mode
In this mode, fixed thermal boundary conditions a t the column ends cause the fluid to carry not only the mixture components but also heat into and out of the column as the flow direction alternates. The heat flux alternations establish the parametric temperature variations throughout the column. The combination of flow alternation, the direct thermal contact between phases. and the establishment of heat sourcesink positions at the column ends forms an internal recuperative heat exchange system as an integral part of parapumping. Such a system has a thermal advantage over the direct mode in which heat is introduced laterally to the flow direction. Figures 9 and 10 present the apparatus elements for the reVOL. 7
NO. 3
AUGUST 1968
343
PRODUCT
f
COOLER
DRIVING PISTON
74 I
& , T E M P E R A T U R E OF PART IC L E S #s, AVERAGE S O L U T E COMPOSITION OF PART I C L E S @ + . T E M P E R A T U R E OF INTERSTITIAL FLUID IN I N T E R S T I T I A L F L U I D
I
Id
Figure 9. Experimental arrangements for closed system recuperati*ve operation
cuperative mode, depicting, respectively, the closed and open operational limits. The diagram for the total reflux, closed-system variant (Figure 9) presents the locations of solid and fluid phase composition and temperature symbols used in the differential equations. The fluid mixture to be separated is caused to shuttle back and forth axially among the particles by an arrangement of pistons. High and low temperature boundary conditions in the fluid phase, respectively, are maintained a t the two column ends by means of exchangers. As in the case of the direct mode, there is coupled interaction of fluid displacement alternations, with periodic changes in temperatures and compositions. These events cause the development of an axial flux of adsorbable material and, in turn, formation of an axial separation. Interactions are described qualitatively as follows: A fluid volume is displaced downward from a warmer region, raising the temperature of the adjacent adsorbent. As a result of the temperature change, the adsorbent transfers solute to the fluid. The enriched (and cooled) fluid next is displaced axially upward to come in contact with warmer adsorbent. The cycle is completed as the fluid cools the adsorbent now adjacent and loses solute to it. The difference bet\veen the fluxes of solute in the two directions of motion is the net flux due to parametric pumping. The net flux serves to build up an axial concentration gradient because the counterflow actions of the hotter and colder fluid behave like countercurrent actions in two-fluid-phase separations. Additionally, separation builds u p from cycle to cycle. Ultimately, after running for a long time at total reflux, a limiting condition of separation is reached in which the net axial flux has become zero and the time-average values of compositions (and temperatures) a t all locations have become constant. Figure 10 illustrates a completely open-system operation; here none of the product issuing from the column is turned back as reflux. A variety of wave forms may be used in any 344
I&EC FUNDAMENTALS
1
PRODUCT HEATING CYCLE
WAIT
WAIT COOLING CYCLE
Figure 10. Operation for an open recuperative system with trapezoidal displacement
parapumping process but here, for convenience, a trapezoidal displacement wave is illustrated. Hot fluid feed displaces fluid previously in the column, followed by a period of waiting. Next, in a reverse flow direction, feed of the same composition but colder is introduced. Another waiting period closes the cycle. All features described above for closed-system operation, such as displacement-thermal-compositional coupling, axial fluxes, and limiting separations in the product streams are similar in the open system. Importantly, also, countercurrent action is primary to the development of a separation. However, the lack of reflux feedback diminishes the relative contribution of cycle-to-cycle accumulation to the over-all separation processes. Although open and closed systems are depicted in separate drawings. it is clear that a single column may be arranged to operate either way or in a partial reflux fashion. Variants are possible also in the manner of heating and cooling the column ends and in the types of displacement pumps employed. Because small quantities of heat are involved, particular care must be taken to insulate the column walls well, so that the ends are the only ports for heat. Experimental. LVe present next the results of experimental separations with the open-system arrangement depicted in Figure 10. A continuous NaCl-H90 separation was achieved for 90 cycles-i.e., a duration of 60 hours. Compositions of the time-average products drawn, respectively, from the top and bottom ends of the column are presented in Figure 11, as is the composition of the feed stream. The product streams from a feed composition of 0.0406~VNaCl averaged 0.0446N (top) and 0.0367'V (bottom), yielding a separation factor of 1.22. The separation is considered to be significant but certainly very much smaller then those measured with the direct thermal mode. In part this is due to the much lower rate of interphase transfer for the resins compared to the silica gel. ,41so presented in Figure 11 are the results of numerical calculation of the separation, based on the mathematical model presented in the next section. System constants for placement in the model were determined in separate experiments. Although the agreement between theory and experiment is seemingly close, this test of the model is not highly discriminating.
The results presented are therefore not to be taken as definitive, but they have encouraged undertaking a program of further model development for purposes of design, using detailed experimental spatial and temporal variations in temperatures and concentrations rather than over-all separation. Initial results (Rolke, 1967) of this program are presented below under the topic of process efficiency. Nevertheless, the present experiment-model agreement is deemed sufficiently convincing to justify using the model in a set of computer explorations (Figures 13, 14, and 15) to enhance understanding of recuperative parapumping. Returning to experiments, consider now apparatus essentials and operation details underlying the results of Figure 11 (see Rice, 1966, for complete details). The vertically mounted column was made of standard borosilicate glass tubing, 5.7'cm. i.d. and 55 cm. long, and \vas tightly packed with an equal-volume mixture of Rohm Rr Haas Amberlite IR-45 (anionic) and IRC-50 (cationic) resin. Provided were degassing equipment for the feed stream? heat source and sink heat exchangers, pumps, valves, and cycle timers. I n the heating cycle (Figure l o ) , the free liquid content of the column (495 ml.) is displaced downward during 2 minutes by feed having a temperature of 90' C. S e x t follo\v a waiting period of 6 minutes and then an upflow displacement of the column contents (2 minutes) by a feed stream a t 37' to 40' C. The final xvaiting-equilibration period is 30 minutes, longer than the previous period. The dissymmetry permitted an approximately equimolar exchange of solute to take place at each temperature. Neither total cycle time nor velocity wave shape \vas optimized. NaCl \.vas determined by the Mohr method for chloride ion. The p H values of the t\vo product streams \\.ere within 0.1 pH unit of each other and both values drifted from about 7.3 to 6.8 during the course of the run. The mathematical model for the recuperative mode comprises the material balance: rate, and equilibrium Equations 11, 12, and 13, previously written for the direct mode, Ivith a n additional set of heat balances defining the thermal fields. Consider now the heat balances (neglecting the enthalpies of mixing and adsorption) on the axial column section, dz. for fluid and solid phases:
0.06
c
0.05 W
t
J
\
0.04 J
0
-z
I
z* 0.03
0 c
J 3
0 v)
z
0.02
-I
0
z
0.01
I
C
40
20
60 C Y C L E NUMBER
as, + ae, 328, -~ + p as, --+-at
at
d.9
at
(14)
and
25 + r(0, - e,) at
=
0
8, and 0, represent the dimensionless temperatures in fluid and solid phases. INITIAL COKDITIOSS e,(z.o) =
e,(t,o)
=
eamblent.o I z I 1
BOVSDARYCONDITIOSS
1. At
2
=
0 O,(O:t)
2.
Atz = 1
=
0
I 100
1
Figure 1 1 . Experimental and calculated concentrations of products and feed for open recuperative system of NaCI-HzO-ion exchange resin
B. ( n
+ +)
T
< t < ( n + 1) T
1. A t z = 1
e,(i,t) = 1
Model.
cy/(if(t) -
I 80
2.
Atz
=
0 ae,(o,t) ~at
The terms in Equations 14 and 15 are identified: heat conservation contributions by [ h ] axial convection, [;] interstitial, fluid phase transient, [j]adsorptive phase transient, [ k ] axial fluid phase diffusive loss, and [ I ] lumped-parameter interphase diffusive transport rate. The system constants are: cy, velocity amplitude; 6 , ratio of average volumetric heat capacity of adsorbent phase to that of the fluid phase; $, dissipative axial diffusivity of heat; and y, characteristic time of displacement relative to time constant for thermal response of adsorptive particles. The latter quantity deserves further identification ; thus
=
+
[(A) 4.
\\here k;pC, is the thermal diffusivity of the adsorptive phase, R is the adsorptive particle diameter, and (J is the displacement frequency. I n the recuperative mode of parapump separation, thermal and compositional fields are driven synchronously by the alternating displacement velocity, oj(t), as represented in both terms [ u ] and [ h ] in Equations 11 and 14. In other words, in this mode the displacements simultaneously transfer both heat and mass? whereas in the direct mode the fields \\'ere independently adjustable. Furthermore, the parametric temperature variable, e,, which acts ultimately on the compositional fields through term [ g ] in Equation 13 is now defined in its spatial and temporal properties through the set of differential Equations 14 and 15 instead of being imposed directly, as in the direct mode. VOL. 7
NO. 3
AUGUST
1968
345
0.25 v)
W J
z
Q
2 0.20 W
r
n
5
0.10
W
I
Figure 14. Calculated separation factors for open, recuperative operation with sine velocity, for NaCI-HzO-ion exchange resin system
0.0 5
I
0 10-3
IO-*
IO-'
I
I IO
102
lo3
Y
Figure 12. Effect of heat transfer rate constant, duty per cycle for sine and Dirac velocities
y,on heat
5 EPAR AT ION
Figure 13. Calculated separation factors for closed, recuperative operation with sine velocity, for NaCI-H20ion exchange resin system
System Properties. Next we explore certain functional characteristics of the recuperative mode based on the mathematical model. Consider first the effect of the important thermal time constant ratio, y, on the relative heat requirements per cycle between column ends under steady periodic conditions for two velocity wave forms. The relationships in Figure 12 were computed with the thermal Equations 14 and 15 for a sine velocity wave and for an extreme contrast, a square 346
I&EC FUNDAMENTALS
wave displacement-Le., Dirac velocity. The same fullcolumn displacement and cycle time are provided in both cases. The value of the solid-fluid volumetric heat capacity ratio, @, used in the equation is characteristic of the waterresin system. Both curves in Figure 12 recede toward a zero heat duty as y -+ 0, because the particles become unresponsive to thermal changes and thus do not contribute to the complex series of events causing the heat dissipation. As y increases, the Dirac velocity case reaches a high limiting value of heat duty. With this wave form there is no interphase thermal exchange during the displacement act, and for values of y > lo2 there is substantially full thermal equilibration during each half cycle. The Dirac velocity produces the largest of all possible heat duties of any wave form. By contrast, the heat duty for the sinusoidal wave rises to a maximum value for y about 1 and then decreases, approaching zero duty as y + m , except for residual axial diffusive losses in term [k] in Equation 14, characterized by $. For very large values of y there is little lag between the fluid displacement and interphase heat transfer flux; in the limit the interphase A0 approaches zero and the system approaches reversibility. By contrast, the Dirac wave provides the largest possible A0 for interphase exchange and hence has the largest possible irreversibility. All wave forms which retain a finite portion of the cycle in flow condition-Le., term [ h ] in Equation 14 is operativewill display a maximum in the heat duty-y relationship. Figures 13, 14, and 15, all with the same displacements, display the inherent system properties of the recuperative mode. Specifically, the effect of different wave forms and degree of reflux on separation were calculated for wideranging values of the characteristic time constants of the system, y for interphase heat transfer and X for mass transfer. The full model (Equations 11 to 15) was employed in the numerical computations. [The Deans and Lapidus (1 960) model was employed with 50 axial stages. A more representative model will be presented in a separate publication.] The equilibrium data for the model were those of SaCl-HZO in Figure 1; all other system constants were characteristic of the mixed resin bed (see Table I). All separation factors are for steady-periodic conditions.
1040-2
Figure 15. Calculated separation factors for closed recuperative operation with Dirac velocity, for NaCIHzO-ion exchange resin system
magnitude of separation also to vary-for example, were the upper temperature increased by 100 "C., the theoretical calculated separation factor on the ridge (y = 1.0, X = lo3) would increase threefold. Different displacements (varying a) would affect both thermal and compositional fields: Large displacements aid the thermal alternations but lead effectively to greater mixing between the ends; short displacements have the opposite effect. We have found that about one column displacement is optimal. The shape and positioning of the surface with its three regions would also be affected by changing the equilibrium function, or the relative thermal and mass capacities ( p and K ) . Sensitivity to changes in the parametric variable, 8, and nonlinearities in the equilibrium are of particular importance. Let us consider further the vicinity of the ridge. I t might be interesting, although not necessarily efficient, to operate in this region because the separation is almost twice as great as on the plateau. However, on the ridge, X is greater than y-that is, the mass-transfer time constant must be greater than that for heat transfer. Symbolically, X > Y
that is, Table I. System Constants for Calculating Separation Factors and Heat Duties of Figures 12, 13, 14, and 15 0 . 5 sin t or a ( t ) 1.1 P 1.38 h' Initial fluid concentration 0.04,t.I NaCl 90' C. Top temperature Bottom temperature 25' C.
Figure 13 presents the y-X-SF surface for a total reflux, recuperative parapump system employing a sine wave velocity. Three regions of the surface draw our attention: First, for small values of y and A, there is no separation (SF = 1.0). Such a response is expected because with small time constants, very little heat or mass is exchanged between the phases. The packing is, in effect: inert. Second, for y 'u 1.0 and X > 1.0, there is a high ridge in the surface, resulting from the optimal actions of the thermal field a t this particular value of y . I n Figure 12 this y corresponds to a maximum in the heat duty too. The separational optimality is presumed to be due, in part, to phase angle relationships similar to those discussed earlier regarding the direct mode. However, in the recuperative mode we no longer can consider the existence of only a single system phase angle; the phase lag here is distributed in position, Apparently the net phase angle distribution is best at y = 1.0. I n addition to phase angles, the amplitudes of the temperature oscillations, which also are spatially distributed, contribute to the magnitude of the ridge separation. The third region of particular interest is that in which both y and X are large (y > lo*, X > lo2). Here the separation factor is about 1.5; it does not change as either y or h is increased. This uniformity occurs because for all practical purposes thermal and compositional equilibrium exists betlveen the phases throughout this cycle. Figure 12 indicates that for high y the heat duty is small (near zero) ; yet Figure 13 shows a large separation in the same region, implying high efficiency. Actually the heat duty curves neglect the enthalpy of mixing, which would be appreciable for large separations but small in the separation range presented. If the system constants Irere varied, we would expect the
or A Y
-
Dpcp
> 1.0
k
This means also that the Schmidt number must be less than the Prandtl number. For the SaC1-H20 system, X/y ^V assuming the mass diffusivity of the adsorbent particle a to be that of NaCl in H20. The locus of X/y = heavy broken line in Figure 13, represents a lower limit for the ratio in the system in question. The mass diffusivity in an adsorbent particle is orders of magnitude smaller than in water alone; the thermal diffusivity is about the same. Therefore, the area behind the demarcation line is completely inaccessible experimentally. Fortunately, the third region (large y, A) is accessible and has significant separation capability. The model system used to compute Figure 14 is identical to that of Figure 13, except that it is an open system having no reflux. The two response surfaces have similar shapes, each exhibiting the three predominant regions. The open system, however, has smaller separations because it does not take as much advantage of buildup from cycle to cycle as the closed parapump system does. The thermal duties, per cycle, are the same as for the closed system (Figure 12). The total heat duty for any separation is the duty per cycle multiplied by the number of cycles. Whereas product is removed after only one cycle in the open system, it is not withdrawn from the closed system until many cycles have elapsed, Thus, the efficiency of the closed system depends on not only separation but also the number of cycles needed to achieve it. The final separation surface (Figure 15) was calculated using a Dirac velocity with closed (total reflux) system operation. As above, the system constants are those of Table I. Using the Dirac function as a velocity means that the entire fluid displacement occurs instantaneously, followed by a period of no-flolr, followed by instantaneous displacement in the reverse direction and no-flow. Because the flow is so fast, there is no time for interphase heat or mass transfer to VOL. 7
NO. 3
AUGUST 1968
347
i 0
1
Table II.
System Constants Used in Efficiency Calculation
Temperatures, " C . TOP
Bottom Ambient Feed concentration Cycle time, min. Upflow Waiting Downflow Waiting Volume of hot feed, ml. Volume of cold feed, ml. Volume (void) in column, ml.
V
-I W
>
-v I
0
I
I
0.2
I
I
0.4
I
I
I
0.6 F R A C T I O N OF C Y C L E
I
0.8
I
I
I.o
Figure 1 6. Calculated velocity, temperature, and fluid composition a t a point in b e d as a function of time (fraction of cycle)
occur during flow. I t must take place during the no-flow period. Separation still occurs, however, as the result of the coupling of the displacement and the interphase transfer, in this case the two being in series rather than parallel. Noteworthy is the absence of the separational ridge which was present in both previous surfaces. I t is apparent that formation of the ridge (and the sharp front in Figure 7) is a direct consequence of the simultaneous flow and transfer. In the Dirac system there are no axial gradients in temperature or composition (no wave fronts). This is analogous to single stirred-tank apparatus. The Dirac velocity produces a separation of about 1.5 for large y and A. An important difference between operation with finite flow velocities, such as the sinusoidal, and infinite velocity such as the Dirac is the relative heat duty for identical separations. Thus, comparison of Figures 12, 13, and 15 shows that for large y and X both systems yield separation of 1.5, yet the sine wave case has a near-zero heat requirement while the Dirac has the greatest possible heat duty. These two systems then encompass limits of reversibility and irreversibility, respectively. The characteristic parapumping alternations in system states were illustrated for the direct mode in Figure 8. Comparison is afforded for the recuperative mode in Figure 16 for the case of a sine velocity wave. Computed driving forces a t a given column location for both thermal and mass interphase transfer mechanisms, (e, - 0,) and (+/* - $J~), are shown to alternate in each cycle. Efficiency. To predict accurately the experimental compositions withdrawn from the column ends a still more detailed model is required (Rolke, 1967). This model replaces the lumped parameter interphase transfer equations (15 and 12) by equations which include the intraparticle variations of 348
I&EC F U N D A M E N T A L S
90 25
31 0 . 0 4 M NaCl 38 2
38 580 450 420
temperature and composition. For the particular NaC1HnO-resin system investigated, the lumped heat transfer rate was sufficient (Little, 1967) to describe the thermal field, whereas the intraparticle rate was required for mass transfer. The computation time for the detailed or distributed parameter model is slower than the lumped model by a factor of about 10. Using the model with intraparticle mass transfer, a thermodynamic efficiency has been calculated. This efficiency is the thermodynamic minimum work required to achieve the separation in question divided by the work equivalent to the net axial heat flux calculated for the process in question. The thermodynamic minimum work is computed for a n isothermal, reversible process at temperature TA which produces the required product separation. The thermodynamic activities required for calculating minimum work (Dodge and Eshaya, 1960) have been estimated for dilute sodium chloride solutions (Robinson and Stokes, 1955). Axial internal heat flux or heat duty is based on calculations involving the thermal components of the parapumping model, assuming perfect external feed-product heat exchange. T o convert heat flux to its work equivalent, we have assumed that all heat must ultimately be received and rejected a t temperature TA. Using the system constants in Table I1 and the distributed parameter rate expression, we calculate a separation factor of 1.18 and a thermodynamic efficiency of about 0.35y0 for total feed, recuperative operation. This efficiency figure is only approximate and is only for the particular separation in question. These initial computed results give encouragement for further experimental and theoretical studies. An order of magnitude increase in the mass transferred per cycle due either to an increase in cycle time or a change to an adsorbent either with higher interior diffusivity or smaller particle size (none of which requires additional energy input) can increase the efficiency by about sixfold. The use of reflux does not affect the energy input per cycle, but it does reduce the product volume per cycle at the same time separation is increased. I t was computed that the efficiency of a total feed operation is higher than that of the corresponding half reflux-half feed operation, by a factor of 1.2 in a specific case. Multiple column operation requires ,V times the heat input compared to a single column for a cascade of N columns. Therefore, to be more efficient, the theoretical work of producing a separation for the cascade must be a t least N times greater than that for a single column. Acknowledgment
T h e authors thank Murray A. McAndrew for valuable discussions and J. E. Sabadell for help in analyses and other
experimental matters. Robert Kunin, Rohm and Haas Co., provided ion exchange resins and helpful information about
+s
them.
v v2
Nomenclature
L-nless otherwise noted, all quantities are dimensionless. = particle heat capacity, cal.:‘g. O C . D = particle mass diffusivity, sq. cm.//sec. / = velocity function, varies between + I and -1 F = interphase transfer rate expression (Equation 8) h = defined by Equation 5 H = defined by Equation 6 k = particle thermal conductivity, cal./’sec. cm. OC. n = number of cycles s = number of columns in a cascade ¶ = sum of solid and fluid composition transients R = particle radius. cm. SF = separation factor t = time Ta = ambient temperature, OC. velocity z = axial position in bed
c,
v =
0,
= = = = = =
8,
=
a /3 y E
7
=
X
=
p
= =
7
4,
=
$,*
=
velocity coefficient particle heat capacitylfluid heat capacity lumped interphase heat transfer rate constant phase angle axial mass diffusivity fluid phase temperature or other thermodynamic intensive variable solid phase temperature or other thermodynamic intensive variable solid fractionlfluid fraction in bed lumped interphase mass transfer rate constant particle density: g. k c . period of one cycle fluicl phase concentration fluid concentration in equilibrium ivith solid phase
u
= = = = =
solid phase concentration averaged over particle heat diffusivity cycle frequency gradient operator Laplacian operator
SUBSCRIPTS i = relating to component i f = relating to fluid phase s
= relating to solid (adsorbent) phase
literature Cited
Bendelitis, A . R . , B.S.E. thesis, Department of Chemical Engineering, Princton, Cniversity, 1961. Deans, H. .4., Lapidus, L., A.Z.Ch.E. J . 6, 656 (1960). Dodge, B. F., Eshaya, .4. M., Adcan. Chem. Ser., S o . 27, 7 (1960). Hirschfelder, J . O., Curtiss, C. F., Bird, R . B., “Molecular Theory of Gases and Liquids,‘’\Viley, New York, 1954. Little, D. S., B.S.E. thesis, Department of Chemical Engineering, Princeton University, 1967. Louisell, \V. H., “Coupled Mode and Parametric Electronics,” \$ley, New York, 1960. Rice, A . \V.,Ph.D. dissertation, Princeton University, 1966. Robins, G. S., B.S.E. thesis, Department of Chemical Engineering, Princeton University, 1967. Robinson, R . A , , Stokes, R . H., “Electrolyte Solutions,” Butterworths, London, 1955. Rolke, R . \V., Ph.D. dissertation, Princeton University, 1967. Skarstrom, C. \V., Ann. ,V. I’.Acad. Sei. 72, 751 (1959). \Vilhelm, R . H., “Parametric Pumping. A Model for Active Transport,” in “Intracellular Transport,“ p. 199, Academic Press. New York. 1966. \Vilhelk, R. H., Rice, A . I$’,, Bendelius, A . R . , I N D ENG. . CHEbf. FUNDAMENTALS 5 , 141 (1966). \Vilhelm, R. H., Sweed, N. H., Science 159, 522 (1968). RECEIVED for review March 18, 1968 ACCEPTED June 17, 1968 Part I1 of a series on parametric pumping. Studies supported by NSF grants GP-2286 and GK-1427X. Additional support through NSF graduate fellowships provided to three of the authors. Computations performed in the Princeton University Computer Center, which was aided by NSF grant GP-579.
EFFECT OF MIXING ON PERIODIC COUNTERCURRENT PROCESSES F. J . M . H O R N AND R . A. MAY
Chemical Engineering Department, Rice Universily, Houston, Tex.
7700 1
The effect of mixing on the stage efficiency of a periodically operated countercurrent process for the case of difficult separations is investigated by modeling each stage, during the time of liquid transport, by a series of rn well stirred tanks. The asymptotic value for the stage efficiency of such a column is found analytically. In a column with a finite number of stages the difference between the asymptotic and the real value is inversely proportional to the number of stages. Columns with a finite number of stages are calculated numerically. The transport number, 7 , which is defined as the ratio of the quantity of liquid transported per cycle to the liquid holdup of a stage, is an important parameter, since in practice it can be controlled easily. While for piston flow (rn = m ) the maximum stage efficiency is obtained for 77 = 1 , for the more realistic case where rn is finite the transport number maximizing the stage efficiency may be larger than 1 ,
(1964, 1967), Sommerfeld, Schrodt, Parisot, and (1966): and Robinson and Engel (1967) have shown that the stage efficiency of a countercurrent process can be improved by operating the process periodically in the foll o ~ i n gway. O n e of the phases is transported uniformly in one direction; the other phase is transported batchwise in the other direction. I n this paper only cases are considered where ORK
H Chien
the mass holdup of the uniformly transported phase in a stage is negligible-for instance, in distillation or absorption if the vapor phase is transported uniformly. BatchLsise and uniformly transported phases are called liquid and vapor, respectively, in what follows. I n previous papers, the case was studied in which piston flow can be assumed for the liquid phase during the time of do\vn transport-that is, no mixing VOL. 7
NO. 3
AUGUST 1968
349
