Ind. Eng. Chem. Process Des. Dev. 1982, 21, 197-203
197
Simplified Method for Calculating Cyclic Exhaustion-Regeneration Operations in Fixed-Bed Adsorbers Lorenro Llbertl’ and Roberto Passlno Istituto di Rlcerca Sulle Acque -Consigllo Nazlonale delle Ricerche, 5, Via De Blasio, 70 123 Bari, Italy
An idealized model for cyclic operation of fixed-bed ionexchange processes with favorable and unfavorable equilibrium conditions is suggested. The model views the fixed bed as a continuous system in which the solid and fluid phases flow countercurrently to one another at their respective average concentrations. With this approach, process feasibility and practical performance are easily determined. The model reproduces experimental results with good accuracy, as illustrated by the example of sulfate removal from seawater by an anionexchange resin.
Introduction Despite the profusion of fixed-bed operations (adsorption, ion exchange, etc.) operated in the exhaustion-regeneration cycle, their design still largely relies on semiempirical methods. Great difficulties are encountered in treating mass transfer data, even for once-through operations, since the concentration of the transferred species changes with time and space as they travel along the bed. A step forward was taken with the introduction of the exchange zone (EZ)concept (Michaels, 1952) for fixed bed ion-exchange processes, widely accepted for theoretical and practical applications (Treybal, 1955; Lukchis, 1973). It essentially restricts the exchange to a limited portion of the bed, maintaining the same concentration profile while traveling through the column (see Figure 1A). Mass balances refer to this zone. A major limitation of Michaels’ method is that it is not applicable to processes with “unfavorable” equilibrium (for instance, desorption when adsorption is favorable): here the difference parts of EZ travel at different velocities along the bed, so that the concentration profile changes its shape and ion concentrations depend both on time and space (Helfferich, 1962) (see Figure 1B). An attempt to predict fixed-bed ion-exchangeperformances under equilibrium conditions using unit operation procedures was made by Hiester et al. (1973). In this paper a simplified method is presented which view the behavior of a fixed-bed process as analogous to a steady-state countercurrent continuous process, independent of the nature of the equilibrium involved. Description of t h e Method Let us consider a simple fixed-bed liquid-solid mass transfer process such as binary ion exchange, i.e., the exchange of ion B in solution by a resin (R) in A form, represented by the reaction RA + B s RB + A (1) Neglecting minor operations (backwash, rinse, etc.), essentially two operations alternate discontinuously in this process: resin exhaustion and regeneration. During the exhaustion, a solution containing the species B to be exchanged is fed at a certain flow rate to a column containing a given volume of exchanger in A form. Then the operation is discontinued when the capacity of the bed has been exhausted. [Usually this point, called the breakthrough point, corresponds to reaching a prefixed leakage (say 5 to 10%) of the exchanged species in the effluent.] Regeneration is then performed: a concentrated solution of A is fed at a certain flow rate to the column to restore it to its original condition, in preparation for the next cycle. 0196-4305/82/1121-0 197$01.25/0
Column performance in these processes is commonly described in terms of effluent concentration histories (or breakthrough curves; see Figure l), displaying the concentration in the effluent vs. the volume V of fluid that has passed (or time, if the flow rate is constant). The average composition of the effluent can be easily calculated by graphical integration of these curves. By reference to Figure lA, integration between the extremes 0 and V of the breakthrough curve leads to the area A2,which is equal to the area Al that would have been obtained if an effluent with a constant, average concentration X2C had been exited from the bed throughout the operation. The average is given by
xz =
f v X dV
JO
XI v
Let us now make the following assumptions: (1) In each operation the fixed phase moves countercurrently to the fluid phase at a relative flow rate given by the ratio between the bed volume and the operation time. (2) The effluent throughout the operation has a constant composition X2C, corresponding to the average calculated by graphic integration of the breakthrough curve. These assumptions correspond to an idealized process in which the solid phase, as soon as it becomes exhausted, is withdrawn continuously from one end of the contact apparatus and is replaced by regenerated material at the opposite end. The second assumption corresponds to the use of a suitable holding tank, in which all the effluent is collected and mixed before being discharged. With these assumptions, the fixed bed process can be treated as a continuous countercurrent one, whatever the nature of the equilibrium involved, the concentrations at the extreme sections of the contactor being constant with time. Assuming that composition changes in the intraparticle liquid are irrelevant, the following mass balance can be written for any finite portion of the contact bed (see Figure 2) VCAX = uCAY (3) Introducing the (apparent) solid and liquid phase flow rates Fs = U / t o and FL = V / t o (4) eq 3 may be written as
0 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982
198
A
\
\'Y
'\
i
0
B
Figure 1. Schematic representation of concentration profiles inside bed during a fied bed ion exchange process (left) and corresponding breakthrough curves of the effluent (right): A, fixed-bed process with favorable exchange equilibrium; B, fixed-bed process with unfavorable exchange equilibrium; EZ, exchange zone. i
_..
,
-
I
4
I S 1
relative to the equilibrium line (EL), as driving forces are positive if the OL lies below the corresponding EL during exhaustion and above during regeneration. The optimization of operating variables is also simplified. For a better illustration of this method, a binary ion-exchangeprocess such as Ca2+removal from a hard water by a fixed bed of a cation resin, regenerated by NaCl solution, depicted in Figure 3, may serve as an example. The upper part of the diagram shows the fixed bed contactor containing a volume u of resin to which during exhaustion (left side) a volume Vexof a hard water (total concentration Co) is fed at a flow rate (FL)ex for an exhaustion time t,,, so that its average Ca2+ equivalent fraction is reduced from Xexlto Xex2. Correspondingly, the average Ca2+equjvalent fraction in the resin (with total exchange capacity C) increases from Yex2to Yexl.In the model, fresh resin is continuously supplied throughout the operation to the contactor from the left (opposite to the solution entrance) at a flow rate (Fs)ex= u / t 0 . Similar considerations apply during the regeneration operation (right side of Figure 3), performed with a volume V , of concentrated NaCl solution (totalconcentration C >> Co), fed at a flow rate (FL), for a time t,. At the en2 of that operation, the average ?!a2+ equivalent fraction has increased from X,,, to XW2 in the solution and decreased from Yregz to Yreglin the resin phase. The lower part of the diagram shows the OLs for these operations and the ELs for the Ca2+/Na+exchange on the cationic resin. EL, usually described by Henry's, Langmuir, or other equations of the type Y* = f ( X ) ,presents a peculiarity: typically, in heterovalent exchanges, such as Ca2+/Na+exchange, resin selectivity toward the polyvalent ion decreases with the ionic strength of solution (Helfferich, 1962). Accordingly, the EL at C,, is lower than that at Co, lying above the corresponding b L for exhaustion, and below for regeneration. The coordinates of the OLs for these operations are easily calculated. (a) Exhaustion.
Xexl= X o (Ca in the feed) J-&b
T
Yes1 =
(64
cov e x
Yexz+ (Xexl- X e X 2 ) 7 y (Ca in the exhausted resin) (6b) Xexz
=
L " ' X du X e x l Vex 2
Yexz (Ca in the regenerated resin)
f
-~~ x,:
I
(average Ca leakage in the effluent) (6c) (64
i :
2 LFigure 2. Schematic representation of continuous countercurrent unit.
This represents the operating line (OL) of the process considered, with slope proportional to the specific output V / u (i.e., volume of solution treated per volume of bed), or to the ratio of flow rates, FL/Fs,for a given ratio of total solution concentration, C, to total resin exchange capacity, C. These quantities are usually constant during each ion-exchange operation, and so is the slope of the OL representing that operation. The graphical description of the process greatly simplifies the solution of the mass transfer problems. First of all, process feasibility, for any given set of conditions, can be ascertained by considering the position of the OL
slope
The values of Co, C, and u are given, so that for any Vex value the OL slope is determined by eq 7. As for the coordinates, Xo is also given, Xexzcan be obtained by graphical integration from the breakthrough curve, and YeX2 is usually specified: the OL for exhaustion is thus completely defined. [Regeneration procedures are normally set up to give a residual amount of Ca in the regeneration resin, Yex2,below 2-570, a value considered acceptable for most practical applications.] (b) Regeneration. Assuming that no chemical changes occur in the interval between exhaustion (or regeneration)
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982
A
199
B
E t h a d s tic n
qegereratlcr
Figure 3. Representation of softening of a hard water on ideally continuousbed of strong cation resin in Na cycle (DILand CONC are contactor diluted and concentrated ends).
and regeneration (or exhaustion), the coordinates of the end point of the OL for regeneration are
Xrql (Ca in fresh regenerant solution, assigned) (8a) Yregl= Ye12 (Ca in the regenerated resin) XrenP
(8b)
volume of resin seawater total concentration, C, equivalent fraction of sulfate in seawater, Xe,,
=
solution flow rate during exhaustion
(Yree -
Yregl)
Cu
cre,vre,
+
(FL)eF
(Ca in the spent regenerant) (84
Yreg2
=
Ye11
(Ca in the exhausted resin)
(8d)
while the slope is given by Y r e g 2 - Yregl (%)reg
Table I. Operating Conditions for Seawater Desulfation on Fixed Bed of Kastel A 102 Anion Resin (Laboratory Pilot Plant)
= Xreg2- Xregl
exhaustion time, t,, exhaustion throughput ( V/U),, average equivalent fraction of sulfate in desulfated seawater, X,,, evaporation temperature concentration factor, n solution ,. flow rate during regeneration (b'L)reg
CregVreg
=-=-
cu
Creg
C (FdFdreg (9)
The OL for regeneration can accordingly be easily drawn. Once each operation has been characterized graphically by the proper OL and the corresponding EL (the latter taken from the literature or determined experimentally), several familiar graphical procedures for design may be applied. More generally, the many established theories, such as those concerned with linear driving force kinetics, number of transfer units (NTU), height of transfer units (HTU), and bed height (&), as described in chemical engineering textbooks (Perry, 1973), may become more readily applicable to cyclic fixed bed operations, like those described, when used with the present method. Of course, allowance must be made for the simplifications involved.
Application of the OL Method to Seawater Desulfation To demonstrate the validity of the proposed method, the application to seawater desulfation is shown as an example. Desulfation is practiced to prevent formation of CaS04 scale in evaporation plants (Boari et al., 1974a; Aveni et al., 1975; Boari et al., 1977). In this process a fEed bed of an anion exchange resin in C1 form is used to remove
regeneration time, t,, regenerant solution volume ( V / u ) ,
0.4 L, 0.59 N 0.089 2-14 L/h (1 ) 4-0.5 h 8L 0.022-0.010 120-150 "C 2
1-7 L/h ( t ) 4-0.5 h 4L
SO-: from sea water intended as feed for evaporation plants. The resin is regeneration by the concentrated brine discharged from the evaporator. The exchange reaction 2RC1 + S042-s R2S04+ 2Cl(10) occurs alternatingly left to right (exhaustion) and right to left (regeneration). As previously stated, the selectivity of the resin for S042-over C1- (heterovalent exchange) decreases with increasing ionic strength of the solution. This can be seen from the lower part of Figure 4, where SO:-/Cl- ELs corresponding to seawater (0.59 N) and to a concentrated by a factor of 2 brine (1.18 N), experimentally determined at 25 f 0.1 "C for the anion resin Kastel A 102 (Boari et al., 1974b), are shown. This fundamental characteristic of the system makes the whole process feasible. Under the chosen experimental conditions, equilibrium is favorable for S042uptake from the solution during exhaustion and is unfavorable for regeneration. In a typical laboratory experiment (operating conditions summarized in Table I), a single plexiglass column ( h = 80 cm, S = 2.47 cm2, u = 400 cm3 of resin) containing a fixed bed of Kastel A 102 resin (C = 2.7 equiv/L) was used, to which 8 L of natural seawater (C, = 0.59 N) were fed
200
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 S E A WATER
P E J I C i E 3 BRINE
t
T
32,
r so
..I o n
53;
n s d u t an
Figure 4. Representation of combined ion-exchange-evaporationprocess for sea water desulfation on ideally continuous resin bed (E, exhaustion; R, regeneration; EV, evaporator; CONC, concentrated end; DIL diluted end).
downflow at a flow rate (FL)ex= 16 L/h for an exhaustion time t,, = 'Iz h. The average equivalent fraction of sulfate was decreased from 0.089 in natural seawater to 0.022 (75% of sulfates replaced in solution by an equivalent amount of chlorides). This corresponds to a sulfate uptake of 1.29 equiv/L of bed and thus an equivalent fraction of sulfate in the resin equal to 1.29/2.7 = 0.475. [Unless a complete equilibration of the resin with the solution occurs (Le., the complete breakthrough of EZ is obtained), sulfate distribution throughout the bed is not uniform, for there is less sulfate in the very bottom of the bed. However, a situation of uniform loading of the bed can be assumed in this case.] The ion-exchange column was then withdrawn from service and the desulfated seawater fed to a pressurized evaporator (evaporation temperature about 150 "C) where its total concentration was increased twofold ( n = 2) and its volume decreased to 4 L. No precipitation occurred (enough sulfates had been exchanged to prevent reaching the CaS04 solubility product) so that the equivalent fraction of sulfates in solution remained unchanged. The concentrated brine was then employed to regenerate the exhausted resin, to which it was fed at a flow rate of 8 L/ h, so that time spans required for exhaustion and regeneration were equal (llzh). The operation is illustrated in Figure 4. Seawater (total solution concentration C,) enters the fixed bed of fresh resin (E) at a flow rate and its equivalent fraction of sulfate is reduced from Xexlto XerP.The desulfated water then reaches the evaporation plant (EV), where its total concentration is increased to nCo, its equivalent fraction of sulfate remaining constant (Xex2= Xregl). Before being discharged, the concentrated brine is used to regenerate the exhausted resin (R): the sulfates previously exchanged are now released and the resin is ready for a new cycle. The OL method models the process as a countercurrent flow of resin and solutions: fresh resin (YWl = Yea) enters the exhaustion contactor (E) and, after exhaustion ( Yexl = Y ), reaches the regeneration contactor (R) where its initi2composition is restored.
In such a "closed" system, X,, = X e d , YWl = Y d , and YeX1= Yregzwhile, according to (7) and (91, the slopes of the two OL are in the ratio (AY/AX)reg = -nvreg (FL/Fdreg - n (11) (AY/Ax)ex Vex (FL/Fdex Since desulfated seawater which has been concentrated n-fold in the evaporator is used as regenerant, V , = Vex/n, or (FL/Fs)reg = (FL/Fs)ex/n, SO that the exhaustion and the regeneration operations are described by the same operating line (12) Application of the OL method to evaluation of process feasibility, sizing of the bed, calculation of breakthrough curves, and selection of the most suitable resin is now described. (a) Process Feasibility. Process variables for exhaustion and regeneration, i.e., desulfation percentage, specific outputs, duration of operations, etc., should always be set to such values that the OL lies between the two ELs. From the experimental figures referred to the experiment described, OL can be easily found from the exhaustion operation data: slope VCIvC = 8 X 0.5910.4 X 2.7 = 4.37; upper coordinated, Xexl= 0.089, Yell = 0._475;lower coordinates, X e d = 0.022, Yed = Yexl- VC/vC (Xexl- Xexz) = 0.178. The resulting OL is shown in the lower part of Figure 4A. According to statement (12), the above are also the coordinates of the end points on the OL for regeneration, provided that only the blow-down from the evaporator is used as regenerant (see Figure 4B). The described graphical representation thus shows whether the process is feasible: to be so, the single OL must lie between the two ELs. In practice, a concentration increase of desulfated water in the evaporator by only a factor of 2 allows the S042-/Cl-EL with this resin to be shifted down enough to fall below the OL. Hence, a 75% degree of desulfation, sufficient to prevent CaS04 scale formation under the conditions investigated, can be obtained by this process without consumption of chemicals. If higher degrees of desulfation are required, say, in connection with higher concentration ratios or evaporation
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982 201 A!
Exi-aust i o n
8 ) Rege-e-Ph3: I
X ex
X reg
Figure 5. Graphical evaluation of NTU for desulfation process, from Figure 4. Table 11. Driving Forces for NTU Computation Evaluated Graphicallv from Figure 4 exhaustion xex
0.089 0.080 0.070 0.060 0.050 0.040 0.030 0.022
(Xexl)
( X e x ,1
regeneration
X*ex
1/(Xex- X*ex)
Xreg
0.078 0.061 0.048 0.038 0.030 0.022 0.015 0.012
90.9 52.6 45.5 45.5 50.0 55.6 66.7 100.0
0.089 0.080 0.070 0.060 0.050 0.040 0.030 0.022
temperatures, the corresponding performance can be immediately determined by the same graphical procedure. The above figures were experimentally confirmed at the pilot plant level (Aveni et al., 1975), where flow rates of 25 and 12.5 m3/m: h at n = 2, 120 O C , and an equal duration of 1 h for exhaustion and regeneration of the resin were adopted. The same procedure was also used for the design of a 1700 m3/d desulfation plant now in operation (Boari et al., 1977). (b) Contact Bed Design. By the well-known theories for liquid-solid mass transfer calculation in packed columns (Perry, 1973), the height of the contact bed can bc: calculated as h, = NTU X HTU (13) The number of transfer units, NTU, may be obtained by a graphical integration procedure from the distance (linear driving force) between the OL and the corresponding EL. As shown elsewhere (Boari et al., 1980), in this process the mass transfer rate is controlled by the resistance in the liquid film (due mainly to low mass velocity of fluid, less than 1 cm/s). Driving forces are accordingly the orizontal distances [(X - X*) and (X*- X) in exhaustion and regeneration, respectively] between the operating and equilibrium lines. On the other hand, the height of the transfer unit, HTU, is experimentally obtained or calculated as a function of the mass velocity and of the mass transfer rate coefficient. (i) Exhaustion. With reference to the experimental conditions described in Figure 4, the NTU can be obtained as
The distances between the abscissas X,, (on the OL) and X,* (on the EL) corresponding to various Ye, values may be graphically evaluated between the extremes 1 and 2 in
(XlW 1
1
(Xrq,
x*,g
U ( X * r e g - Xreg)
0.172 0.141 0.113 0.091 0.073 0.057 0.043 0.033
12.1 16.4 23.3 32.3 43.5 58.8 76.9 90.9
diagram 4A. The values obtained are summarized in Table 11, while Figure 5A describes the graphical evaluation of this integral. The result is NTU,, N 4. As for the HTU, it can be calculated by the well known relation (Perry, 1973) HTU = (FL/S)/KLu (cm) (15) In the investigated conditions, mass velocity was (FLIS),, = (16000)/(3600)(2.47) = 1.8 cm/s. Using for the overall mass transfer coefficient the expression KLu = 0.15(FL/S)0'5 (s-') (16) previously found for this system (Boari et al., 1980), one obtains HTU,, = 9 cm; hence (hZ),,= 36 cm. (ii) Regeneration. With eq 14 in which the driving force now is (X*- X)referred to the regeneration EL, the use of an analogous procedure (see Table I1 and Figure 5B) leads to NTU,, r 3. For (FL/S)reg= (8000)/(3600)(2.47) = 0.9 cm/s, eq 15 and 16 give HTU,,, = 6.3 cm, so that (hz)regr 19 cm. (c) Calculation of Breakthrough Curves. The graphical description outlined by the present method and the knowledge of the fundamental kinetic parameters allow for Sod2breakthrough curves under various experimental conditions to be easily predicted. By use of linear driving force theory for liquid-solid mass transfer kinetics (Perry, 1973), the exchange rate of this film-diffusion controlled process can be expressed as
while the continuity condition in this mixed feed system should be written as Yo d X = Xo dY (18) Solving simultaneous eq 16-18 and evaluating the corresponding linear driving forces graphically from proper OL and EL, the breakthrough curves for exhaustion and regeneration may be calculated. Figure 6 shows the calcu-
202
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 2, 1982
05
04
33 3i
4c
85 tx
2
*
Ti
1 5
Figure 6. Calculated SO4*- breakthrough curve and experimental points (0) during seawater desulfation on fixed bed of resin Kastel A 102 for (FL/& = 0.222 cm/s.
lated curve and the experimental points for an exhaustion experiment at (FL/S)= = 0.222 cm/s. The close agreement with experimental data confirms the validity of the proposed method. (a) Selection of Resin. Finally, the OL method has been applied to select the resin which best suits the desulfation process according to a limiting criterion. Potential performances of different resins are compared under equilibrium conditions. In the case of seawater desulfation, the selectivity of the C1-resin toward sulfates decreases with total solution concentration. Accordingly, the minimum value for n (i.e., the minimum concentration to be reached in the evaporator for the desulfated seawater to be used as regenerant) corresponds to the regeneration EL tangent to OL at its lower end ( X r e 1, Yregl). If the performances of various candidate resins are compared under equilibrium conditions, as may be approximated by a desulfation cycle run at very low flow rate on a fully regenerated resin, complete exhaustion will occur with virtual zero leakage of sulfates in the effluent. Under such “limiting” conditions, the OL coordinates are Xexl = X,,Yexl = Yo, and Xed= Yex2= 0 (see OLh in Figure 7). A specific output of completely desulfated seawater, equal to the “equivalent volume” of the resin -vex =u
YoC
xoco
(19)
is produced. This “equivalent volume” corresponds to the ratio of the maximum amount of exchangeable sulfates per liter of resin ( YoC) to the sulfate concentration per liter of seawater (XoCo). A complete regeneration of the resin is performed with a VJnu brine volume. In this case, the limiting condition is reached when the OL is tangent to the regeneration EL at the origin, i.e., when the slopes of OL and EL,,, at the origin are equal. According to eq 7 and 19, the slope of the OL is now Yo/Xoso that the condition is pn -dY = - yo +OX Xo C1-/S0,2- exchange ELs have been found to be satisfactorily described by the equation (Boari et al., 1974b)
whose derivative at the origin is
32
GI
0 0
0 025
0 050
0075
xo
3’
X
Figure 7. Experimental equilibrium lines (EL) for the C1-/S042exchange on anion resin Kastel A 102 and limiting operation line (04,). (Totalsolution concentration C = 0.59 N for n = 1, 1.18 N for n = 2, etc; 25 f 0.1 “C).
( Ris the mean value of the SO,2-/C1- selectivity coefficient for each salinity). Combining (20) and (22) one finds for the minimum value of n
Equation 23 relates the equilibrium properties of the resin (Yo, K,and 6 ) to their regenerability with the limited amount of the brine discharged from evaporative plants and thereby provides a limiting criterion for the comparison of different anion resins. With this criterion, the Kastel A 102 resin was successfully selected as the best suited for the desulfation process. It is highly selective toward sulfates (R= 4.20 L,/L at seawater concentration), yet regenerable with moderately concentrated brines (nk = 2.7)(Boari et al., 1974~).A similar criterion was independently derived by Klein and Vermeulen, who used the K C parameter for scale prevention in evaporation and reverse osmosis plants by either anionic (Sassi, et al., 1971) or cationic (Klein et al., 1964; 1972) ion exchange. Further practical applications of the present OL method have been described elsewhere (Liberti et al., 1978). Conclusions With the proposed method, cyclic exhaustion-regeneration operation of fixed bed ion-exchange processes under conditions of “favorable” or “unfavorable”equilibrium can be calculated. The method uses a model in which solid and fluid are viewed as contacting one another countercurrently and continuously at their respective average concentrations. Once the operating and equilibrium lines for a given ion-exchange operation have been determined, the de-
Ind. Eng. Chem. Process Des, Dev., Vol. 21, No. 2, 1982
203
scribed method allows fixed bed performances over a wide range of operating conditions to be predicted and the basic kinetic parameters for bed design can be calculated with acceptable precision. This avoids tedious calculations and extensive laboratory experimentation usually required for unfavorable equilibrium processes and provides the basic data for plant design with reasonable approximation. The method has been applied to seawater desulfation by fixed bed ion exchange. In addition to two major idealizations inherent in the model (use of average concentrations and continuous countercurrent movement of phases), the following simplifications were introduced constancy of liquid phase flow rate and of total concentration of both phases, uniform saturation of the fixed bed, uniform fluid distribution (no channelling) in the column, and negligible composition gradients in the intraparticle liquid. The method is expected to be of great utility for other fiied bed mass transfer processes, such as multicomponent ion exchange or adsorption, provided the effect of the approximations is carefully analyzed.
nlh = minimum n value for resin regeneration under quasiequilibrium conditions OL = operating line describing specified process step OLh = operatingline describing the desulfation process under quasiequilibrium conditions NTU = number of transfer units S = cross sectional area of column, cm2 t = time, h to = operation time, h = volume of treated solution, L V e x / u= equivalent volume of resin (i.e., maximum solution volume that can be treated by unit volume of resin without leakage), L/L, v = resin volume L, X = equivalent fraction of exchanged ion in solution Y = equivalent fraction of exchanged ion in resin * = refers to equilibrium conditions ex = refers to exhaustion reg = refers to regeneration 0 = refers to seawater conditions L = refers to liquid phase S = refers to solid phase Literature Cited
Acknowledgment
Aveni, A,; Boari, G.; Liberti, L.; Monopoli, B.; Santori, M. Desalination 1975, 76, 135-49. Boari. G.; Liberti, L.; Passino, R. J . Chromfogr. 1974a, 702, 393-401. Boari, G.; Liberti, L.; Merli, C.; Passino, R. Desalinafion 1974b, 15, 145-66. Boari, 0.; Liberti, L.; Merli, C.; Passino, R. I o n Ex&. M m b r . 1974c, 2, 59-66. Boari, G.; Liberti, L.; Santori, M.; Splnosa, L. Desalination 1977, 79, 283-98. Boari, G.; Llberti, L.; Merli, C.; Passino, R. Envlron R o t . Eng. 1980, 6 , 251-256. Helfferich, F. “Ion Exchange”, McGraw-Hili: New York, 1962. (University Microfilms International: Ann Arbor, Mich No. 2003414). Hiester, N. K.; Vermeulen, T.; Klein, G. “Adsorption and Ion Exchange”, Sec. 16 in “Chemical Engineers’ Handbook”, 5th ed.; Perry, J. H., Ed.; McGrawHill: New York, 1973. Klein, G.; Villena-Blanco, M.; Vermeulen, T. Ind. Eng. Chem. Process D e s . D e v . 1964,3,280. Klein, G.; Vermeulen, T. ”The Role of Ion Exchange in Saline Water Pretreatment”, Research Conference on Pretreatment, Scale Control and Prevention of Fouling, UCLA, Los Angeles, Dec 11-12, 1972. Liberti, L.; Boari, G.; Passino, R. “An Approach to the Design of Mass Transfer Processes”, 6th International Congress, CHISA 78, F’raha, Czechosb vakia, Aug 21-25. 1978, J3.10. Lukchis, G. M. Chem. Eng. June 1973, 111-16. Michaels, A. S. Ind. Eng. Chem. 1952,4 4 , 1922. Perry, J. H. “Chemical Engineers’ Handbook”, Int. Student Ed., 5th ed.; McGraw-Hili: New York, 1973. Sassi, A. G.; Vermeulen, T.; Klein G. AEC Contract noW-7405+ng48, Lawrence Berkeley Laboratory 299, University of California, Berkeley, 1971. Treybal, R. E. “Mass Transfer Operations”; McGraw-Hill: New York, 1955.
Thanks are due to Professor Friedrich G. Helfferich, Pennsylvania State University, for valuable comments about the method and for carefully assisting in the editing of this paper. Nomenclature particle surface per unit volume, cm2/cm3 C = total solution concentration, equiv/l C = total exchange capacity of resin, equiv/L, EL = equilibrium line (or equilibrium isotherm) describing a mass transfer process in equilibrium conditionsat a given temperature EZ = exchange zone F = volumetric flow rate, L/h h = distance from inlet of exchanger, cm ho = total height of exchanger, cm h, = height of the contact bed, cm HTU = height of transfer unit, cm K = mean value of selectivity coefficient for C1-/S04*- exchange at given salinity (L,/L) defined as K = [Yso; (xc,)2/xso(YcJ21 IC/CI K L = liquid phase mass transfer coefficient, cm/s n = concentrationratio of regenerant and exhaustion solutions a =
Received for review August 19, 1980 Accepted October 6, 1981
