troller gains. The improvements attainable by the use of cascade control can be seen for values of 711 greater than 1 and for dead times greater than 0.5. The parallel system with 711 = 1 produced a n interesting result (see Figure 8). The closed loop load transfer functions are identical with and R ithout cascade control. Two different values of K\f are used, but the presence or absence of the slave loop makes the ( z b f / L ) relationship reduced to eyactly the same ratio of polynomials i n s A time domain load respoiise comparison betmeeii parallel cascade and no cascade for the system with a dead time of 2 is shoa i i in Figure 10. Without cascade, the manipulative variable M does not move until time reaches 2 because of the dead time In the parallel cascade y s t e m , the slave loop begins to mol e the manipulative T. ariable immediately. The higher allowable gain with cascade control iewlts in less steady-state error in zhf. Nomenclature Bhf
=
Bs
=
CC
= = = =
CT D FC
feedback controller transfer function iii primary or master loop feedback controller transfer fuiiction in secondary or slave loop compositioii controller composition transmitter dead time flow controller
FT = flow transmitter GL = load transfer function GM = process t'ransfer function in master loop Gs
= =
Ks
= =
1, JI s TC
= =
TT
=
zM
=
K,
= =
zpt = zs = zpt =
process transfer function in slave loop gain of mast'er controller gain of slave controller ultimate gain input load disturbance manipulative variable Laplace transform variable temperature controller temperature transmitter process out,put variable of master loop set point of master controller process output variable of slave loop set point of slave controller
GREEKLETTERS 7b1
=
'TS
=
{
=
time constant of process transfer function in master loop time constant of process transfer fuiiction in slave loop closed loop damping coefficient
literature Cited
Franks, R. G., Worley, C. W., Ind. Eng.Chem. 48,1074 (1956). Jauffret, J. P., 113.Thesis, Lehigh University, Bethlehem, Pa., 1973.
RECEIVED for review April 13, 1973 ACCEPTEDJuly 24, 1973
Batch Fractionation of Ionic Mixtures by Parametric Pumping Timothy J. ButtsI1 Norman H. Sweed," and Arthur A. Camero Department of Chemical Engineering, Princeton Cniuersity, Princeton, .V. J . 08540
Direct, thermal parametric pumping has been used to fractionate experimentally K+-H+ and K+-Naf-H+ mixtures using Dowex 50x8 as the ion exchanger. The binary exchange equilibrium i s influenced b y temperature so that desorption of K+ and adsorption of H+ occur simultaneously on heating. K+ accumulates in the top reservoir with separation factors exceeding 2000:1, while H+ accumulates in the bottom with separation factors exceeding 2000: 1 in the opposite direction. In a ternary exchange experiment, the K f separation factor was 52,000: 1 , accumulating in the top, H+ was 97,OOO:l in the opposite direction, and No+ almost completely disappeared from both reservoirs. The influence of resin swelling due to temperature i s investigated with an equilibrium theory model.
P a r a m e t r i c pumping is a cyclic process which theory and experiment have shoivii can produce substantial separations of liquid mixtures. The misture to be separated is placed in a chromatographic bed wherein both the temperature aiid fluid flow direction are chaiiged periodically and synchroiiously. This cyclic operatioil interacts constructively with the natural depeiideiice of sorption equilibria on temperature to give separations which build on themselves cycle after cycle. All previous experiments and most theory (Sweed, 1971) have dealt with binary mixtures. However, Butts, et al. 1
Presently at Exxon Chemical Company, Florham, Park, S . J.
(1972) , did esainine theoretically the separation of multisolut'e mixtures. Their analysis was based on the equilibrium theory of the parametric pump, originally proposed by Pigford, et al. (1969), wherein adsorption equilibria are assumed linear aiid noncompetit'ive. 13utts, et nl. (1972), showed that parametric pumps can fractionate multisolute mixtures for such noncompetitive equilibria. However, in most real systems, adsorption is riot a linear phenomenon. Except a t low concentrations, adsorbates typically compete for adsorbent sites, giving rise to nonlinear, Langmiur type isotherms even when only a single adsorbing species is present. I n ion exchange the sorbing (i.e.) eschangInd. Eng. Chem. Fundam., Vol. 12, No. 4, 1973
467
Table 1. Selectivities and Heats of Ion Exchange
Table II. Column Void Fraction
Potassium-Hydrogen
0.05 M KCI Temp, OC
M o l e fraction H' on resin
0.0 0.2 0.4 0.6 0.8 1. o
cal/mole,
25OC
DH,'~
DH +K
+
(3 CIb
1,61 -1380 2.14 - 1580 2.68 - 1700 - 1800 2.98 3.06 - 1740 3.17 - 1690 Sodium-H ydrogen AH',"
M o l e fraction
cal/mole,
HC on resin
25'C
+
(55'CIb
1.35 1.75 2.15 2.35 2.45 2.55
1.09 1.37 1.65 1.78 1.87 1.96
D H + ~ ~ + DE +Ka
(3'CIb
DH + K t
(25"C)c
+
(25"C)c
1.37 1.20 0.0 - 980 1.59 1.35 0.2 - 1100 1.78 1.50 0.4 - 1200 1.81 1.53 0.6 - 1200 0.8 -1140 1.83 1,57 - 950 1.84 1.60 1.0 a From Boyd. b Calculated. Fram Bonner.
DH +Ke. + (55°C)b
1.04 1.13 1.25 1.27 1.32 1.37
ing) species compete for sites even a t very low concentrations due to the requirement for electroneutrality in the exchanger. Hence ion-exchange equilibria are nonlinear even a t low concentrations. I n this paper we demonstrate experimentally how parametric pumping takes advantage of the nonlinear. competitive nature of ion evchange to fractionate binary and ternary mixtures of cations H + and K+, and H+, E(+, and S a + , respectively. We first examine how temperature influences ion exchange in a sulfonated polystyrene resin, Dowex 50x8. Then, after describing the evperimental apparatus, we consider in detail batch binary separations including the influence of column fraction displaced per half-cycle, cycle time, and temperature range. We then examine the behavior of the ternary system. Effects of Temperature on Ion Exchange
Ion-exchange resins are characterized by their chemical composition, their equilibrium properties (selectivity, moisture content, capacity, extent of swelling, etc.), and their kinetics (i.e., rates of exchange), Several excellent monographs exist on this subject (especially Helfferich (1962)) so that we shall not dwell on ion-exchange properties except as they are affected by temperature. Ion-exchange selectivity is characterized by a selectivity coefficient, DB*, the selectivity of A compared to B.For the exchange reaction
A+
+ B + R - S A+R- + B +
where A+ and 13+ represent ions in solution and A+R- and B+R- represent ions on the resin phase
D B =~
[A+R-][B+] [A+][B+R-]
The brackets represent concentrations in moles per liter for the fluid phase, and moles per kilogram of dry resin in the H + f~ormfor the resin phase. Bonner, et al. (1953, 1954), obtained selectivity coefficient data for the binary mixtures H+-Kf and H+-Ka+ (as chlorides) on Dowes 50x8, a strong acid, sulfonated polystyrene resin. cross-linked with 8% divinylbenzene. These data, shown 468
Ind. Eng. Chem. Fundam., Vol. 1 2 , No.
4, 1973
0.1 M KCI
0.1 M HCI
+
0.05 M HCI
3 0.336 f 0.002. 0.234 & 0,001 ... 25 0.324 + 0.001 ... 55 0.301 i 0.001 0.228 & 0.005 a 95'33 confidence intervals shown.
in Table I, were determined at 25'C in solutions whose overall concentration was 0.10 N . Bonner's data show the strong influence of resin phase composition on selectivity, with D H ~ varying from a value of 1.35 for resin entirely in the K f form to 2.55 for resin in the H + form. S o explicit data are available in the literature for these selectivities a t temperatures different from 25"C, but Boyd, et al. (1956, 1964), have studied extensively the heats of ion exchange for these two mixtures. Table I summarizes Bonner's and Boyd's data. The data of Bonner and Boyd were combined using the van't Hoff equation to calculate selectivity coefficients a t 3 and 55OC, the temperatures t o be used in parametric pumping experiments. These data are also shown in Table I. Note that as temperature increases, the selectivity decreases, approaching 1.0. This dependence of selectivit'y on temperature is a t the heart of the parametric pumping effect. It is well known that ion-exchange resins swell or shrink when their ionic composition changes. The degree of swelling increases as the resin t'akes up the larger, more hydrated ions, such as H+, compared to the smaller K+. However, literature data are not available on the effects of temperature on swelling. To estimate this effect we measured column void fractions using a dye pulse technique a t several temperatures and compositions. particles swell they imbibe interparticle fluid, and the void fract'ion decreases. (At concentrations on t h e order of 0.1 M, the imbibed Auid is almost entirely water; the ions are not sorbed to any significant extent because of Donnan exclusion.) Table I1 shows the experimenbal void fractions in a column 60 em X 1.1 ern i.d. packed with 50-100 mesh Dowes 50x8.The same column was left intact throughout all measurements to eliminate variations due to repacking. The effect of concentration on void fraction is as expected with H + form resin being more swollen. The data show too that the resin particles swell as temperature rises, the effect being stronger for the less swollen K+. Because the resins were in single ion form, this swelling change is due not t o selectivity effects but to st'retching and contracting of the polymer matrix of the resin. Since in the H + resin the polymer is already stretched to near its maximum even a t cold temperatures, heating has little effect'. With Kf the polymer is in a more contracted state (like a coiled spring), and heating relaxes tension in the polymer. It will be evident later that the swelling of resins due to temperature change has a marked effect on parametric pumping separat,ions, while swelling from concentration change is of only secondary importance. Binary Solute Separations
X11 parametric pump separation experiments were carried out in the apparatus depicted in Figure 1. jacketed glass column, 60 em x 1.1 em i.d., was connected a t each end to a disposable, plastic 50-cm3 syringe. To ensure that syringe contents were kept well mised, a small magnetic stirring bar
Table 111.
System Parameters and Results for Selected Binary and Ternary Experiments
Initial fluid concn, meq uiv/m I
K No
Cycle time, rnin
H
Run
CY
0.0533 4
5
i'
0.0546 0.0554 ... 0.0544 0.0539
7
20.5
1.03
51.6 44.6
20.5
1.03
51.6 44.6
44
1,lO
0.0540 0.0539
8
91.5
1.10
0.0542 0.0538 9
188
1.06
0.0556 0.0500 10
11
15
I
i'
.
.
0.0486 0.0499 ... 0.0492 0.0327 0,0337 0.0336
Reservoir volume before downflow, ml TOP BOT
53.5 43.7 53.5 43.7 53.5 43.7
Final reservoir concn, mequiv/ml TOP Temp range, OC
No. of cycles
4-80
30
3-55
30
3-55
3-55 3-55
3-55
140
43
1.62
53.3 19.2
3-55
140
was placed inside each syringe and a n electrically driven magnetic stirrer was supported above each syringe. The ion-exchange resin, Dowex 50x8, 50-100 mesh, was washed, then alternately cycled between KC1 and HC1 solutions a number of times to remove organic matter present from the resin's manufacture. The column was packed with K + form resin, arid then a solution of equimolar K + and H + chlorides (0.05 -11 each) was flowed through the bed a t room temperature until the resin was in equilibrium with this solution. The column volume, measured by experiment, was 60.16 cm3, which together with the void fraction from Table 11, gave a void (Le., interstitial) volume of 19.5 cm3. The top reservoir (syringe plus connecting tubing) was filled with the equimolar solution to a total volume of 53.3 cm3 and the bottom reservoir to a volume of 19.2 em3. (These last data apply to one particular experiment, run 11. See Table I11 for other runs.) Aiseparation experiment begins when the column is cooled by cold water (3'C) flowed through the jacket. At the same time, a pump displaces the reservoir syringes, forcing liquid downward t'hrough the bed. This downflow continues until a preselected volume of fluid is displaced. At this point the bed is heated (55°C) aiid upflow begins uiitil the same amount of fluid is displaced in this direction as before. N o solution is added to nor removed from the system in this batch process. (Occasionally, small samples are removed from the reservoirs for analysis, I
