SUBSCRIPTS e = electrolyte as a neutral species i = any arbitrary species 0 = solvent SUPERSCRIPTS = adjacent to high pressure side of membrane = adjacent to low pressure side of membrane b = bulk solution
i P
literature Cited
Bennion, D. N., Department of Engineering, University of California, Los Angeles, Rept. 66-17 (1966). FUND.\hi. 8, 36-49 Bennion, D. N., Ithee, B. W., IND.EXG.CHF:M., (1969). Glueckauf, E., First International Symposium on Water Desalination, Washington, I).C., Vol. 1, pp 143-50 1963. Govindan, T. S., Sourirajan, S., IND.ENG.CHEM.,PROC.DES. 5, 422-29 (1966). DEVELOP. Johnson, J. S., Department of Engineering, University of California, Los Angeles, Rept. 67-41 (1967). Johnson, J. S., Bennion, D. N., Chem. Eng. Progr. Symp. Ser. 64, NO. 90, 270-9 (1968). Kedem, O.,Katchalsky, A,, J . Gen. Physiol. 45, 143-79 (1961). Laidler, K. J., Schuler, K. E., J . Chem. Phys. 17, 851-65 (1949).
Loeb, Syndey, Sourirajan, S., Advan. Chem. Ser., No. 38, 117-32 (1962). Lonsdale, H. K., “Desalination by Reverse Osmosis,” U. Merten, Ed., pp 93-160, M.I.T. Press, Cambridge, Mass., 1966. Lonsdale, H. K., Merten, U., Riley, R. L., J . Appl. Polym. Sci. 9, 1341-62 (1965). hlanjikian, S., Department of Engineering, University of California, Los Angeles, Rept. 65-13 (1965). Michaels, A. S., Bixler, H. J., Hodges, R. AI., Department of Chemical Engineering, M.I.T., Cambridge, Mass., Rept. 315-1 DSR 9409 (1964). Osborn, J. C., RI. S. thesis, Department of Engineering, University of California, Los Angeles, December 1969. Reid, C. E., Breton, E. J., J . Appl. Polym. Sci. 1, 133-43 (1959). Reid, C. E., Kuppers, J. R., J . A d . Poiurn. Sci. 2.264-72 (1959). Reid; C. E., Spencer,’H. G’., J . Phys. C h n . 64, lh87-8 (1960). ’ Rosenfeld, Judy, Loeb, S., Department’of Engineering, University of California, Los Angeles, Rept. 66-62 (1966). Sherwood, T. K., Brian, P. L. T., Fisher, It. E., IND.ENG. CHEM., F U N D A h i . 6, 2-12 (1967). Shpall, R . T., AI. S. thesis, Department of Engineering, University of California, Los Angeles, 1969. Souriraian. S..J . A m l . Chem. 14. 306-13 11964). Spiegler, K.S:,Tm&. Faraday Sbc. 54, 1408-28 (1958). Wills, G. B., Lightfoot, E. N., IXD.ENG.CHEM.,FUNDAM. 5, 115-20 (1966). RECEIVED for review May 11, 1970 ACCEPTEDJanuary 19, 1971
Equilibrium Theory of Cycling Zone Adsorption Ramesh Gupta and Norman H. Sweed Department of Chemical Engineering, Princeton Cniversity, Princeton, N . J .
08540
Cycling zone adsorption is a separation process in which fluid i s passed through a series of adsorbent zones, the periodic temperature changes in adjacent zones being one half cycle out of phase with one another. This paper presents a method for computing the separation as a function of fluid displacement, cycle time, and adsorption equilibrium parameters. Criteria are developed for optimum separation. This analysis i s restricted to linear isotherms and instantaneous local equilibrium.
I n recent years much attention has been paid to periodic operation of fixed beds. Wilhelm, Sweed, and their co-workers (1966, 1968) have obtained large separation factors in parametric pumping. Such separation results from a coupling between the cycles of flow and of a thermodynamic potential, such as temperature, pH, or pressure. Recently, Pigford and his coworkers (1969) observed that the reversal of fluid flow is not essential to obtain separation. The new method of separation called “cycling zone adsorption” is illustrated in Figure 1. Fluid with a solute conceiitration of Co is passed through a series of packed beds of adsorbent (zones), the temperatures in adjacent zones being T Cand T H . After a half cycle, the temperatures in the zones are alternated, from T c to T Hand from T Hto Tc, respectively. The solute concentration in the product stream, in general, is different during the two half cycles. Separation factor is defined as [ c f H ] / [ c f , ] , where the brackets indicate the average solute concentration in the product stream during the indicated half cycle. Mathematical Model of Cycling Zone Adsorption
R i t h the assumptions of nondispersive flow, constant solid and fluid density, radial uniformity, and steady plug flow the dimensionless fluid phase material balance is 280
Ind. Eng. Chem. Fundom., Vol. 10, No. 2, 1971
where t = time, z = axial distance, CY = fluid velocity, = fluid phase coiicentration, and $s = solid phase concentration. For the special case of instantaneous local equilibrium and linear isotherms
$f
9s = K(T)9f
(2)
where K ( T ) is the temperature-dependent adsorption constant. Combining Equations 1and 2 QJf
b.?
CY -
Defining K c becomes
=
+ (I + K ( T ) )d9f at
K ( T c ) and KH
when a zone is cold, and
-=
=
0
(3)
K ( T H ) , then Equation 3
Figure 2a shows the concentration profile when zone 1 is cold, zone 2 is hot, and the cycle is about t'o start. Figure 2b represents the concentration profile after the fluid has flowed for a half cycle and hence has moved a distance a. Note t h a t the solute moves a distance a/(1 K C )in the cold zone and a distance a / ( l K H ) in the hot zone. The concentration "belts" crossing from zone 1 into zone 2 move faster in zone 2 by a factor (1 K c ) / ( l K H ) due to iiicreased velocity into the hot zone. Similarly, the concentration belts moving out of zone 2 and forming the product stream have their velocity iiicreased by a factor (1 KH). The product concentration is calculated by proper averaging. Figure 2c shows the concent>rationprofile after the temperature has been switched. The cold zone becomes hot and the hot zone becomes cold. In zone 1, the temperature rise results in desorption of solute, hence the fluid concentration increases by a factor (1 K c ) / (I K H ) .Similarly, temperature lowering of zone 2 results in the fluid concentration decreasing by a factor (1 KH)/ (I K C ) .Figure 2d shows t'he fluid concentration profile after the fluid has flowed for the second half of the cycle. Now the concentration waves move a distance a / ( l KH) in zone 1 and a distance a / ( l K C ) in zone 2. The concentration belts crossing from zone 1 to zone 2 move slower by a factor (1 K H ) / ( ~ K C )due to decreased velocity of concentration waves in the cold zone. The product concentration is calculated in the same may. It should be noted t h a t Figures 2a through 2d represent a full cycle of operat,ion. The concentrat'ion profile of Figure 2a will be obtained on cooling zone 1 and heating zone 2 of Figure 2d. A computer program using the above algorit'hm was written to keep track of different concentration belts in different zones and move them at' appropriate velocities depending upon the temperature of the zone. After start-up transients disappear, regular concentratmionprofiles are established in various zones. Separation factors have been computed for several values of CY, K C , and K H .It was found that for each set of equilibrium coefficients K C and K H there are regions of a where bhe separat'ion fact.or increases with the number of zones. There are also regions of CY where the separation factor does not increase with zone number but fluctuates, increasing some times and decreasing the other times. St'ill a third case is possible: the separation is independent of zone number and equals
when a zone is hot. The coefficient of the concentration gradient in Equation 4a or 4b is the velocity a t which a concentration wave moves in any zone.
+
+
Computational Algorithm
Figure 2 shows the concentration profiles i n the first two zones of a C Z 1 unit, calculated using Equations 4a and 4b. The profiles are shown a t four different times in the cycle, after start-up transients have disappeared and steady periodic separation has been established. For the sake of illustration, let Kc = 3, KH = 1, and CY = 1. For these values
CO
+
+
+
+ +
+ +
CO
+
+
+
Cfc
'fH
I S T H A L F CYCLE
2 N D H A L F CYCLE
Figure 1. Multiple-zone, direct mode operation of cycling zone adsorber
+
ZONE I
ZONE 2
TC
(2b1
co 0
1 0.25
C,/R
'H
1
TC
0.50
0.75
I
co 1.00
0
Co/R
0.25
TH 0.50
0.75
1.00
TH
I CO/ R2 TH
Figure 2. Concentration profiles in the first and second zones at four different stages in the cycle; KC = 3, K H = 1 , a = 1 Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
281
and the separation factor remains constant. Noting that a/(l K C )and a / ( l K H )are the lengths of two adjacent concentration belts in a zone, the criterion for case 1 to exist is
+
+
1
N = A
CY
I
l+Kc .40.44.50
.60 6 7 . 7 0
.00
.90
1.00
a Figure 3. Effect of fluid displacement a on separation; Kc = 3 , K x = 1
'
-
(6)
CY
l+Kn
where N is any positive integer. Case 2. Separation factor increases regularly with the Bone number and is given by Equation 5. T h e largest integral number, N , of pairs of concentration belts t h a t a zone can contain is
ZONE I
ZONE 2
TC
TH
0
TH 0.50
0.25
1.00
0.75
Figure 4. Concentration profiles in the first and second zones at four different stages in the cycle; KC = 5, K H = 2, a = 1
1.0, i.e., no separation. Figure 3 shows such regions for values of CY between 0.4 and 1.0, for Kc = 3 and KH = 1. Note that the separation factor is constant a t 1.0 (Le., no separation)] when CY equals z/3 and 4/9. In the regions of CY where separation is achieved] the separation factor is independent of CY and is
(5)
separation factor = 2
1
N < 01
1$KC
+
+-1 + K H CY
+
Then ( 1 - N [ c Y / ( ~K c ) f CY/(^ K H ) ] is ) the remainder of the zone. For the separation factor to increase regularly with the zone number, the remainder must be longer than the cold belt but shorter than the hot, that is
where m is the zone number. Criteria for Separation
(7 1
To determine the regions of CY where separation occurs, we consider the three following cases. ( 1 ) The separation factor is independent of the zone number and is constant a t a value of 1. ( 2 ) The separation factor increases regularly with the zone number. (3) The separation factor us. zone number exhibits an oscillatory behavior. Case 1. Separation factor is constant. Figure 4 illustrates this case, showing the concentration profiles in the first and the second zone for K C = 5, K H = 2, and CY = 1. Note that each zone contains an integral number of pairs of concentration belts. A pair of belts contains one belt of length a/(l K c ) and one belt of length a / ( l K H ) .During one half cycle, exactly one concentration belt crosses into the next zone. Concentration profiles of adjacent zones are identical a t a half cycle phase difference. Thus the effluent streams from adjacent zones a t a half cycle phase difference are identical
+
282
+
Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
For this condition, the concentration waves reinforce each other from zone to zone. Figure 2 is one illustration of this case. It is not immediately obvious that the separation factor should have the form given in Equation 5 for all values of CY satisfying Equation 7 . To show the origin of Equation 5 , we will derive it for the first zone. The same approach can be extended to subsequent zones. Assume for now that the remainder of the zone
{ l - N ( 1A++K c- r >~ } + K H
+
is equal to a / ( l K c ) , L e . , it satisfies the left-hand equality of Equation 7. This situation is shown in Figure 2. During the cold half cycle, all of the fluid leaving the first zone has
concentration Co/R. On the hot half cycle, the zone effluent is composed of parts of two belts: one belt has length a / R and has concentration COR;the other has length a [ ( R- 1)/R] and concentration Co. By proper concentration averaging of the hot effluent and dividing by the concentration of t h e cold product, one obtains Equation 5 for the separation factor. If the remainder of the zone exceeds a / ( l K c ) but is less than a / ( l K H ) ,the hot effluent contains parts of three belts. The first and third belts both have concentration Co and their combined length is a [ ( R- ])/I?]; the second belt has length a / R and concentration COR.Clearly the average concentration of the hot effluent is the same for all values of CY that satisfy Equation 7. I n a similar fashion the derivation of Equation 5 can be extended to m zones. Case 3. No regular pattern. T h e separation factor u s . zone number exhibits a n oscillatory behavior if t h e reK H ) ] ] is less than mainder { 1 - N [ a / ( l K C ) a / ( l a n entire cold belt or greater than an entire hot belt.
+
+
+
+
+
pumping is that resonance call be obtained a t all the frequencies (a’s). On the other hand, a system like cycling zone adsorption with only one periodic variable produces resonance for some frequencies only. Severtheless, cycling zone adsorption, which has an advantage of nonreversal of the fluid flow, is similar to parametric pumping in the sense that a periodic time coefficient has been replaced by a periodic position coefficient, Nomenclature
K = slope of adsorption isotherm Kc R = I 1 KH t
=
T = temperature, “C z = dimensionless axial position a = dimensionless velocity, fraction of bed volume dis-
placed 1half cycle
4f = dimensionless fluid concentration 4 8
Discussion
The criterion for separation developed in this paper assumes linear adsorption isotherms and instantaneous local equilibrium. For certain regions of a , the separation increases from zone to zone. h computer model based on finite differences was used t o compute separation in cycling zone adsorption. Finite differencing introduces some numerical errors which are similar to the axial concentration spreading due to finite mass transfer or dispersion. It was found that for proper values of a the separation factor leveled off after a few zones. This is similar to what has been observed in parametric pumping. Ritter and Douglas (1970) have defined parametric pumping as a situation where a periodic input signal causes state variables to fluctuate, and simultaneously a time variable coefficient in the system equation is forced to be periodic. The advantage of having two cycling variables in parametric
+ + dimensionless time
=
dimensionless solute concentration in the solid
SUBSCRIPTS C = cold €I = hot literature Cited
Pigford, R. L., Baker, B., 111, Blum, D. E., IND.ENG.CHI:M., FUNDAM. 8, 448 (1969). Ritter, A. B., Douglas, J. M., 1x0. ENG.CHEM., FUKDAX. 9, 22 ( 1970). Wilhelm, R. H., Rice, A. W., Bendelius, A. R., IND.EYG.CHEM., FIXDAM. 5 , 141 (1966). Wilhelm, R. H., Rice, A. W., Rolke, R. W.,Sweed, N. H., . IND.ENG.CHEM.,FUYDSM. 7, 337 (1968). RECEIVED for review July 13, 1970 ACCEPTED February 4, 1971 Computations were carried out at the Princeton University Computer Center, which is supported in part by National Science Foundation Grants GJ-34 and GU-3137. Financial support was provided by the National Science Foundation under Grant GK1427x1.
Cycling Zone Adsorption: Quantitative Theory and Experimental Results Burke Baker, 111,’ and Robert 1. Pigford2 Department of Chemical Engineering and Lawrence Radiation Laboratory, University of California, Berkeley, Calij. 94720
A theoretical explanation for cycling zone adsorption, a wave-propagational separation process given a qualitative explanation in an earlier article, i s presented. Experimental results confirming the theoretical predictions are also included. It i s found that in addition to accounting for the separation effect the theory predicts the effect can be amplified through the interaction of the concentration and progressing thermal waves. Possible process schemes utilizing the interaction are discussed.
I n a recent article by Pigford, Baker, and Blum (1969b) a qualitative description was presented of a cyclic fixed-bed separation process given the name “cycling zone adsorption.” The first objective of this article will be to explain the process theoretically. This leads to certain predictions of ways of Present address, Shell Oil Co., Deer Park, Texas. To whom correspondence should be sent.
introducing heat into the bed that will increase the concentration change. Finally, experimental evidence of the predictions is offered. Consider the process shown in Figure 1, in which a fluid having a constant solute concentration, yf, is passed through a bed of solid particles. The temperature of the bed is cyclically altered, either by heating and cooling the walls, as in Ind. Eng. Chern. Fundam., Vol. 10, No. 2, 1971
283
