2564
Znd. Eng. Chem. Res. 1992,31, 2564-2572
Saturation and Regeneration of Ion Exchangers with Volume Changes Maria Inas G. Duriio School of Engineering, Polytechnic Institute of Porto, Porto, Portugal
Carlos A. V. Costa* and Alirio E. Rodrigues Department of Chemical Engineering, University of Porto, 4099 Porto Codex, Portugal
Models for fixed bed saturation and regeneration steps of monovalent ion exchange with shrinking/swelling of the resin are developed. The ion exchange Ag+/H+ over Duolite C20 is used as a test system: ion exchange equilibrium and swelling experimental data are measured as well as several fixed bed saturation and regeneration runs. An equilibrium model with variable interstitial velocity for the saturation step is proposed and solved analytically. This model shows good predictive capabilities for the time evolution of the resin volume, but due to the assumption of infinite mass-transfer rate it is only qualitative with respect to the prediction of breakthrough curves. A more complex model is presented for the regeneration step that accounts for bulk solution osmotic effecta and resin relaxation kinetics. This model was solved numerically and was able to quantitatively explain the experimentally observed time evolution of the resin volume but again is only qualitative relative to the predictions of the concentration histories.
Introduction Fixed ionogenic groups in polymeric ion exchangers are supported by a tridimensional structure that is insoluble in polar solvents. This structure has the consistency of a macromolecular gel. An increase in the content of the cross-linking agent decreases solubility, matrix elasticity, and ion mobility but increases the mechanical strength of the resin. Both the nature of the counterion (solvation) and matrix elasticity have an important influence on the force balance which determines the solvent penetration into the ion exchanger and thus its volume changes (Helfferich, 1962). The design of fEed beds containing ion exchangers with swelling/shrinking behavior has to take into account this phenomenon; otherwise some gross errors can be made. Modeling fixed beds with resin volume changes was the subject of a series of papers published several years ago (Mama and Cooney, 1973,1976,1978).In all these papers the possible compression of the resin (Janson and Hedman, 1982) is not considered. Their first paper (Marra and Cooney, 1973)describes an equilibrium model that allows for bed volume and interstitial velocity changes under the assumptions of constant fluid density and bed porosity and of the existence of a sharp front. Analytical expressions for the breakthrough time are given showing that the use of the constant-volume assumptions is not always valid. In the second paper (Marra and Cooney, 1976) a more complete model is developed considering porosity changes, a linear driving force law for intraparticle mass transfer, and a coordinate system based on the dry mass of the resin. Numerical solutions are shown and compared with experimental data for the H+/Na+exchange in AG DOWEX 50W-X8(Bio-Rad Labs). An analytical solution for the constant pattern is also given that agreea well with the full model solutions for sufficiently long columns. This model was further extended to the cases of multicomponent elution and ion-exclusion separation (Marra and Cooney, 1978). The objectives of this paper are (a) the development of simple models that can explain and predict the fundamental behavior of fixed bed monovalent ion exchange with moderate resin volume changes, especially the time evolution of the resin volume; (b) to show the experimental results obtained for the system Ag+/H+-Duolite C20,ion
*Towhom correspondence should be addressed.
exchange and swelling equilibrium data and fixed bed saturations and regenerations at various flow rates and feed concentrations; and (c) to evaluate the predictive performance of those models when tested against experimental results.
Modeling and Simulation Let us consider a fixed bed of an ion exchanger and assume plug flow, isothermal operation, constant interparticle porosity and fluid density, negligible mass-transfer resistances and pressure drop, and constant dry mass of the resin contained in the bed. Saturation and regeneration of the fixed bed are carried out with downward flow. The bed bottom is fixed, but the top is free to accommodate resin volume changes. Figure 1 shows the coordinate system where z is the axial coordinate, L(t) and 1 are the positions of the bed top and bottom, respectively, and Q is the cross-sectional area. The following model development assumes a monovalent ion exchange where the co-ion B is common to both counterions A and C that are used as feed in the saturation and regeneration, respectively. The maas balance for ion A in a volume element of the fixed bed, neglecting changes in velocity and in the dry mass of the resin, is
where cA and q A are the concentrations of A in the fluid and inside the resin, respectively, e is the bed porosity, ps is the dry density of the resin (mass of dry resinfresin volume), t is time, and u is the interstitial velocity. In the case of the saturation of a clean (or uniformly loaded) bed the boundary and initial conditions are z = L(t) CA = CAE (2) t=O CA = 0 (CAO) L(t) = Lo (3) In these conditions and with favorable equilibrium (A > C) the function psqA = f(cA) will normally show a negative second derivative and a shock wave will propagate along the bed. The concentration of A will then undergo a jump at a position z&), and the same will happen to the resin density. In this case the global material balance to the resin is Pso(l - Lo) = PsAl - L3 = Pso(l - z,(t)) + p,f(zll(t)- U t ) ) (4)
0888-5885f 92/2631-2564$03.O0/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 2565 saturation shock, i.e., u*+ and u*, respectively. The results are
r
1
Figure 1. Coordinate system for the fixed bed.
where ps and pd are the resin densities at cA0 and cm, respectively, and Lf is the final position of the fixed bed top. From eq 4 we obtain
Lf = 1 - (P,O/PSf)(l - Lo) (6) Equation 5 relates the position of the front with the position of the top, and eq 6 can be used to calculate the finalposition of the top once the solid densities at the two states (initial and feed) are known. The boundary and initial conditions for the regeneration of a saturated bed are 2=0 CA = 0 (CAO) (7) t=O CA CAE L(t) = Lf (8) The function p&A = f(cA) will now show a positive second derivative, and a dispersive wave will propagate along the bed. In this case we need a relationship between the density and the independent variables. The resin mass balance is now
In the above equations ACA,A(u*cA), A(PaqA),and A h are the difference between the values of the functions, upstream (+) and downstream (-) of the shock. These reaults can be simplified when ApR = Aps or when the velocity changes are negligible, Au* = 0. Let us consider the counterion C. A completely regenerated fixed bed is completely saturated with C. When ion A is fed, ion C is released by the resin and moves ahead of the saturation front. These ions contact the ion exchanger which is saturated with them and thus no net exchange occurs. In these conditions a mass balance for ion C in a volume element located downwards the saturation shock can be written
with
Solving eq 16 we obtain
USUdY Ody p, = f ( q A ) is known and eq 9 Cannot be Solved analytically. Saturation. Let us relax the assumption of constant interstitial velocity; the maw balance for ion A is
where z* = (2 - L(t))/(Z- L(t)),0 = t / T O , T~ = ( 1 - Lo)/uo, L*(O) = ( I - L(B))/(I-Lo), and u* = u/uo(uois the reference velocity). The mass balance for the solvent, neglecting solvent accumulation in the fluid phase and dry resin mass change in the volume element, is au* =--L*(@ 1 - E ~ P R (11) az* p t ae where p is the solvent density and pR is the resin wet density (massof wet reain/volume of resin). Equations 10 and 11 can be solved using the method of charactarbtics in order to obtain the position of the f i e d bed top, L*(e),the velocity of the shock, uSA*,the breakthrough time for ion A, &.A, and the relationship between the interstitial velocity upstream and downstream of the
A(u*cA)
'
E
A(u*cA)
'
The concentration history of ion C will show a plateau, between the feed (c,& and the initial (cco> plateaus, located between timea 0, and Osk The concentration of ion C (cc-) in this intermediate plateau can be calculated by
2566 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 Table I. Solution of the Equilibrium Model for Saturation Conditions and c A 0 = cCE = 0
Table 111. Solution of the Equilibrium Model for Saturation Conditions with cA0= cCE= 0 and u = Constant
1 eSc = Lf* [ 1 + 6 1 l n [1+--1 1-Lf* 1 1- Lf* Lf* 1 + Et
1.2
cc- =
Ff
CAE 1 - PR* i f +3 ; '
I
I
(20a)
Table 11. Solution of the Equilibrium Model for Saturation Conditions with CAO CCE 0 and ah, = b,
0.8 I
@SA =
-[ -1+ If [+ Lf* 1 Lr* - 1 u*+
u*- =
1
0.5
I 1 .o
P;i
1.5
Figure 2. Effect of resin volume changes on the interstitial velocity. In Lr* 1.2
-Iu*+ 1 Lf* Cf(1
+ If) 1 .o
5f
cc- =
1 - Lr*cAE
Ef+
(20b)
7
The results for co-ion B can be readily calculated by applying the electroneutrality principle. Tables 1-111 summarize the solutions for the following simplified cases where cA0= cCE= 0 (a) presented case; (b) APR = Aps;(c) constant interstitial velocity. In these ~ capacity parameter solutions & = (1- a ) p , r ~ ~ / eisc the calculated at the feed conditions, Cf = e p / ( l - e)psf is the ratio between the maea of solvent and the maas of dry resin inside the fued bed, lf* = c p / ( l - e ) p ~ and , PR* = PW/PW is the ratio between the wet densities of the regenerated and the saturated resin. The model solutions described in Table I were used to construct Figures 2 and 3, where u*-/u*+and cAE/cc-are plotted as a function of pR*. These figures show that as & increases the effect of resin volume changes on interstitial velocity and concentration of C in the intermediate plateau decreases. In swelling conditions pR* > 1, u*-/u*+< 1;i.e., the interstitial velocity decreases downstream the shock due to the net transfer of solvent to the resin and consequently a concentration effect appears on C,cAE/cC-< 1. The reverse is valid for shrinking.
0.8 L
0.5
1 .o
I p i
1.5
Figure 3. Effect of resin volume changes on the concentration of regenerating ion in the intermediate plateau.
The breakthrough time, &A, is also affected by volume changes. For instance,if we take the case of shrinking with constant feed flow rate (u*+= l),eq 14a shows that BSA is smaller than in the case of no volume changes. Regeneration. Even in the case of constant interstitial velocity the solution of eqs 1and 9 with conditions 7 and 8 is not analytical. The numerical solution of eq 1 with ps given by the integral eq 9 is quite difficult and requires special methods for hyperbolic equations. To circumvent this problem, eq 1 was transformed into a parabolic equation by adding an axial dispersion term:
Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 2667 1.o
1 .o
L'(9)
c,
0.8
0.5 0.6
Y L
0.4
0.0 0.0
9
1.5
0.0
3.0
Figure 4. Comparison of ions concentration histories when the resin = 0.5, -1 and when it remains constant (Lf*= volume changes (Lr*
I L', = 0.51 1
1 .o
2.0
8
3.0
Figure 5. Influence of Pe number on the predicted position of the fixed bed top.
1.0, 0 ) .
1 .o
where Pe = uo(l- Lo)/& and D, is the axial dispersion coefficient. The mass balance for component C is 0.5
z* = 1
-8%- - 0 az*
Two more equations are needed equilibrium relationship and resin density as a function of load: q A = f(cA,cC) (25) PS = f(qA) (26) The integral equation can be solved using a Gaussian quadrature (Villadsen and Michelsen, 1978) that gives
0.0 0.0
0.5 CIC, 1 .o Figure 6. Equilibrium data at 25 O C for the system Ag+/H+-Duolite
c20.
N+2
L*@) = P&/ C UiPsi
(27)
1
where wi are the quadrature weights, p j are the densities at the collocation pointa including the system boundary conditions, and N is the number of interior collocation pointa. &uations 21 and 23 were solved using the package PDECOL (Madsen and Sincovec, 1979). It is interesting to analyze the effect of volume changes and Peclet number on the breakthrough curves. Let us assume a fixed bed saturated with cA = 0.1 and washed that is going to be regenerated with a solution with concentration cc = cB = 1. We further assume a mass action type isotherm K=- Y(1 - x ) x(1 - Y) and a linear relationship between density and load
- -- 1 + - 1- Lf*Y PS
Lf* where y = q A / Q ,x = C A / C T , Q is the resin capacity, and cT is the total concentration of cations in bulk solution. Figure 4 compares the ions concentration histories when the resin volume changes (Lf*= 0.5)and when it does not. In this example the resin swells and we can observe that the ions breakthrough earlier than when there are no volume changes. Figure 5 shows that the influence of Peclet number on the predicted position of the fixed bed top is small. Pa0
Experimental Section The exchange H+/Ag+ in a strong cationic resin is an example of a system that shows volume changes. In this work we used H+/Ag+ exchange over Duolite C20 (Rohm and Haas). This resin is a copolymer of styrene and 8%
0.7 DUOLITE C20
0.5
0.0
3.0
qA (meq/g)
6.0
Figure 7. Variation of resin density with load.
divinylbenzenewith sulfonic groups. Prior to use the resin was regenerated with nitric acid and washed. Equilibrium Data. The equilibrium data points at 25 "C were measured by the batch equilibration procedure. After allowing sufficient time to reach equilibrium, solution was separated from the resin and analyzed. Then the solute concentrationin the resin was calculated by material balance. Figure 6 shows the experimental points and the fitting using eq 28. The value obtained for the equilibrium constant was K = 3.04 and that for the resin capacity was Q = 5.38 mequiv/g dry resin in RH form. Density Data. The resin densities in RH (pso; q A = 0) and RAg (psf; q A = Q)forms were obtained using a pycnometer to measure the volume and then oven drying and weighing the resin sample. In the case of the saturated form (RAg),the conversion to the equivalent mass in RH form was calculated using the capacity of the resin, Q. We obtained pa = 0.586 g of dry RH resin/cm3and pd = 0.723 g of dry RH resin/cm3. Densities at intermediate loads were determined by measuring initial and final resin volumes on a fixed bed fed with various solute concentrations; assuming constant porosity we have P s ( q A ) = PsO(Vinitial/ vtinnl) (30)
2568 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 Table IV. Exwrimental Omrating Conditions and Calculated Parameters for Fixed Bed Saturation Runs
U 1 2 3 4
0.100 0.080 0.100 0.010
5 30 100 30
23 20 23 23
21.5 21.3 21.3 21.7
17.5 17.6 17.4 17.5
13.4 3.8 122.7
Figure 7 shows pa as a function of qA. The data suggest that a linear relationship can be used (eq 29). The slope of the straight line is (ps0 - pSf)/Q and can be calculated directly giving 0.025 g2/(cm3/mequiv). The best fit of the data gives 0.022 g2/(cm3mequiv) close to the calculated value. The wet resin density, pR, can be calculated from the dry density, ps, by using PR
= Pa + (1 - ps/PE)P
0.575 0.496 0.510
0.41 0.11 0.37
35.0 1.240 38.5 0.902 363.9 0.954
32.7 38.5 331.6
0.995 0.996 1.001
1 1 1
1.2 0 V
A
0
0
0
0
0
0
,
0.9
0.6
0.3
(31)
where pE is the skeletal density (pE = 1.28 g of RNa/cm3 of resin matrix). A linear relationship equivalent to eq 29 is also valid in this case: PR = 1.074 + 0.0031q, (32)
Fixed Bed Experiments. Fixed bed runs were carried out in a standard experimental setup. A peristaltic pump feeds the top of a glass column (diameter = 1.1 cm; height = 28 cm), and the effluent is sampled with a fraction collector. &NO3 and HN03 solutions were used for saturation and regeneration, respectively. Before each saturation and regeneration the column was washed. The outlet concentrationsof H+,Ag+, and NO< were measured by potentiometry. Pairs of reference-selectiveelectrodes were used and potential differences were measured with a Crisson Model 517 voltmeter (sensitivity f O . l mV). The reference electrode was a double junction calomelanos electrode (Ingold Model 303-90-WTEKl) with saturated KC1 solution in the interior and 1M KNOBsolution in the exterior or 0.033 M K2S04when used for nitrate. H+ concentration was measured with a glass electrode (Ingold U202). A selective electrode S2-/Ag+with an electrically conductive resin (Lima and Machado, 1979) was used for the measurement of the silver cation concentration. Nitrate concentration was measured with a selective nitrate electrode with PVC membrane (Lima and Machado, 1985). Calibration of these electrodes was carried out using solutions of known ion concentration and with composition as close as possible to the composition of the solution to be measured. The height of the resin bed was monitored, too, by means of a ruler fixed to the column wall. Rseults and Discussion Saturation. Several saturation experiments were carried out at various feed concentrations and flow rates. Table IV shows the operating conditions as well as the parametera calculated using equilibrium,density, and f=ed bed experimentaldata. Figure 8 &play the concentration histories plottad as ion concentrations normalized by Ag+ feed concentration ( E i ) as a function of the dimensionless time, 8. In order to compare fmd bed experimental results with model predictions we need to calculate the experimental stoichiometric time, tSA: (33)
Using eqs 14a and 33 and remembering that in our experimental setup u*+ = 1,the bed porosity can be calculated and then T ~ Q, and 5;*.
0.0 0.0
20.0
40.0
e
60.0
20.0
40.0
e
60.0
1.2
0.9
0.6
0.3
0.0 0.0 1.2
h
0.9
0.6
0.3
0.0 0.0
20.0
e
40.0
60.0
1.5 1.2
0.9
0
125
250
375
e
500
Figure 8. Experimental saturation breakthrough curves and equilibrium model predictions (Ag+,A; NO3-, 0 ;H+, 0 ) : (e) run 1; (b) run 2; (c) run 3;(d) run 4.
Figure 8 shows the same fixed bed experiment carried out at various flow rates. Increasing the flow rate, the importance of dispersive and diffusional phenomena also increases and therefore the quality of equilibrium model predictions for Ag+ concentration histories decreases. The concentrations of H+ and NO3- in the plateau are always
Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 2669 Table V. Experimental Operating Conditions for Fixed Bed Regeneration Runs ~ u 1
1R 4R
CAO(M) 0.100 0.010
ccg(M) 1.0 1.0
U(&/min) 6.0 6.0
T(OC) 23 22
underpredicted by the modeL Also the influence of volume variation on break time is small. In fact, H+ concentration was measured with a glass electrode that is not completely stable for pH
