Mass Transfer Performance of Centrifugal Adsorption Technology

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Ind. Eng. Chem. Res. 2000, 39, 4376-4382

Mass Transfer Performance of Centrifugal Adsorption Technology Marc A. T. Bisschops,† Stef H. van Hateren, Karel Ch. A. M. Luyben, and Luuk A. M. van der Wielen* Kluyver Laboratory for Biotechnology, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands

Centrifugal adsorption technology (CAT) provides a new way of countercurrently contacting micrometer range adsorbent particles and liquid feed. As a result of the fast mass transfer kinetics and countercurrent contact, CAT can lead to very compact adsorption equipment with high throughputs and very good separation efficiencies. A model for predicting the performance of CAT is described. The model calculations are based on the steady-state continuity equations and include mass transfer resistance in both phases as well as longitudinal dispersion effects in both phases. Data for the degree of backmixing have been taken from earlier work on CAT. The model predictions are compared with experimental data for Ca2+/Na+ ion exchange on Dowex 50 WX8 cation-exchange resin in a pilot scale CAT rotor. The experimentally observed separation efficiencies range from 90% for a separation factor equal to one, to 99% and beyond for higher separation factors. The model is shown to give accurate predictions of the performance of CAT. Model calculations demonstrate that the performance of CAT is limited by the degree of backmixing in the adsorbent phase mainly. Introduction Centrifugal adsorption technology (CAT) is a new method for carrying out adsorption and ion exchange processes. It involves countercurrent contact between micrometer range adsorbent particles and process liquid under the influence of a centrifugal force. The small adsorbent particles lead to a very large interfacial area and short distances for diffusion. As a result, mass transfer kinetics in CAT can be extremely fast, which leads to very high mass transfer rates. The countercurrent contact leads to an optimal driving force over the entire contact length. These combined effects allow the development of extremely compact separation equipment. On the basis of an analysis of the mass transfer kinetics, a decrease in contactor volume of 2-3 orders of magnitude can be expected with reference to ion exchange equipment presently in use in the industry.1 A schematic view of the concept of CAT is shown in Figure 1. Fresh adsorbent particles enter the column close to the axis of rotation and are dragged outward toward the rim by the centrifugal force. The liquid feed enters the column at the outside and is pumped toward the axis of rotation. The liquid flow is chosen such that the adsorbent material is fluidized in the contact zone. The saturated adsorbent is collected at the rim and is transported back by means of a separate transport channel toward the axis of rotation where it is discharged. Since liquid and adsorbent enter and leave the rotor via the axis of rotation, the pressure drop over the contactor is kept very low. The advantages of centrifugal adsorption technology include the following: •Compact separation equipment to be applied on sites with lack of space or in mobile installations. * To whom correspondence should be addressed. Fax: +31 15 278 2355. E-mail: [email protected]. † Current address: BIRD Engineering B. V.; P.O. Box 149, 3100 AC Schiedam, The Netherlands. Fax: +31 10 437 9648. E-mail: [email protected].

Figure 1. Schematic concept of countercurrent flow in a centrifugal field.

•Short residence times allowing for the recovery of sensitive products from a relatively harsh environment without suffering excessive product degradation. •Low adsorbent inventory with an economical use of more expensive and more selective materials, thereby facilitating further downstream processing. •Rapidly achieved steady-state operation, which is beneficial for the application in semibatch processes. •Operation in expanded bed mode resulting in a lowpressure drop and the ability of processing turbid liquids without blocking the contactor. •Increased flexibility with respect to capacity and the type of adsorbent. The variable speed of rotation can be adjusted to accommodate for the desired process specifications. These advantages are beneficial in the recovery of valuable products from diluted process streams, such as fermentation broths, as well as in removal of low concentrated contaminants from wastewater. The performance of CAT is determined by the phase equilibrium, the hydrodynamics of the two-phase flow in the centrifugal field, the degree of backmixing in both phases and by mass transfer kinetics. The hydrodynamic capacity of CAT has been described earlier2,3 as well as the degree of backmixing in the adsorbent phase4 and liquid phase.5

10.1021/ie990927b CCC: $19.00 © 2000 American Chemical Society Published on Web 09/23/2000

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4377

In this paper, we investigate the mass transfer performance of centrifugal adsorption technology. The analysis includes a comparison of model calculations, based on steady-state continuity equations, with experimental data. This study involves the exchange of Ca2+ vs Na+ cations in a 0.10 N solution. The ion exchange equilibrium of these ions is well described, and the system does not involve chemical reaction or pH effects. It is therefore particularly appropriate for focusing on the mass transfer characteristics purely in CAT. Design Considerations. In general, the design of any separation process starts with the definition of the feed and the required separation.6 The second step is the choice for the separation process and the separating agent. In this case, we focus on continuous ion exchange. The choice of an appropriate ion-exchange resin depends on criteria such as loading capacity, selectivity, and chemical and mechanical stability under the relevant process conditions. In addition, each application may pose its specific requirements for the resin, in which regenerability and economic factors may play a role. Centrifugal adsorption technology involves differential countercurrent contact between a dispersed resin and a continuous process liquid. The design sequence for CAT processes thus involves the following major steps: 1. Evaluation of the equilibrium flows. 2. Evaluation of the hydrodynamic capacity of the contactor in relation to the physical properties of the system and the process conditions, such as the speed of rotation and the flow ratio. 3. The influence of the degree of backmixing in both the continuous (liquid) and dispersed phase (resin) on the efficiency of the countercurrent contact. 4. Evaluation of the mass transfer kinetics. The key advantage of CAT is the application of very small adsorbent particles, which is claimed to result in very fast mass transfer kinetics. These phenomena provide the information required to determine the main dimensions of the separation equipment in relation to its performance. These topics will be discussed separately below. A more detailed design would involve additional steps, such as the choice of the fluid and resin transport systems, control system etcetera. These topics are not addressed in this work. Equilibrium Flows. Efficient utilization of the ionexchange resin requires an optimal flow ratio between the resin and liquid feed. This is commonly expressed in terms of a separation factor or capacity ratio

S)

mφS φL

(1)

in which φS and φL represent the volumetric flows of adsorbent and liquid feed respectively and m represents the volumetric distribution coefficient. When the separation factor equals unity (S ) 1), the adsorbent flow is exactly high enough to adsorb all adsorbate present in the liquid feed. Generally, the separation factor is chosen somewhat above unity in order to compensate for the fact that the efficiency of the process is reduced due to backmixing and mass transfer resistance. King suggests S ) 1.4-2.0.7 A process in which the separation factor is chosen significantly higher than unity operates with a too high adsorbent flow, which will lead to poor loading efficiency of the resin. As a consequence, the regeneration may be more difficult and expensive.

In ion exchange processes, the isotherm usually is not linear. As a consequence, the volumetric distribution coefficient depends on the concentration. Since the separation factor is defined as the capacity ratio of incoming adsorbent and liquid feed, the separation factor can be calculated on base of the incoming concentrations of adsorbate

S)

(

)

φS (1 - yin)Qwet φL xinctot

(2)

in which Qwet represents the total ionic capacity of the wet resin expressed in equivalents per unit volume of wet resin, ctot represents the total salt concentration of the feed solution expressed in equivalents per unit volume. The dimensionless concentrations yin and xin represent the equivalent fraction in the adsorbent feed and liquid feed, respectively:

yi )

qiZi ciZi and xi ) Qwet ctot

(3)

In eq 3, Zi represents the valence of ion i involved. The ion exchange equilibrium for simple systems, such as the exchange of monovalent-divalent metal ions, is described using the mass action law8

KCa,Na )

a j CaaNa2 aCaa j Na2

(4)

in which ai and a j i indicate the activity of the ion i in the solution and the resin phase, respectively. For reasons of simplicity, the activity coefficients are assumed to be unity, which allows the isotherm to be expressed in terms of equivalent fractions as follows:

KCa,Na )

( )( ) yCaxNa2 Qwet x y 2 ctot

(5)

Ca Na

Vermeer et al.9 reported equilibrium data for Ca2+/Na+ exchange in a 0.10 N solution on a sulfonated polystyrene resin (Dowex 50 WX12). This resin is identical to the adsorbent used in this work, except for having 12% cross-linking agent (divinylbenzene, DVB), whereas Dowex 50 WX8 has 8% cross-linking agent. They found an ion exchange equilibrium constant of KCa,Na ) 2.90. Bonner and Smith published ion exchange equilibrium constants for various mono- and divalent cations on Dowex 50 WX8 with lithium (Li+) as reference ion.10 They found KCa,Li ) 5.16 and KNa,Li ) 1.98, from which can be derived that KCa,Na ) 2.61. The data from Vermeer et al.9 and both equilibrium curves are drawn in Figure 2. It can be seen that the agreement is satisfactory. In this work, the equilibrium constant reported by Bonner and Smith10 will be used. Hydrodynamic Capacity. The countercurrent flow of micrometer range adsorbent particles and water is controlled by the centrifugal force, induced by the speed of rotation. This topic has been subject to investigations, which are reported elsewhere.2,3 It was found that application of the correlations commonly used for describing homogeneous two-phase flow leads to a systematic underestimation of the capacity of the rotor. Furthermore, it was demonstrated that countercurrent flow can be established in two hydrodynamic states, depending on the configuration of the solids distribu-

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purposes, the assumption of the two-film model11 gives satisfactory results.6,12,13 According to this simplification, the mass transfer resistances in the liquid and resin particles can be expressed in terms of partial mass transfer coefficients (kL at the liquid phase and kS in the dispersed phase), which are defined as

kL )

Figure 2. Ion exchange isotherm for Ca2+/Na+ exchange on Dowex 50 cation-exchange resin from a 0.10 N solution. Concentrations are expressed in terms of equivalent fractions.

tor: (1) the “free settling state”, which is characterized by a low adsorbent hold-up in the rotor, and (2) the “fluidized state”, which is characterized by a high adsorbent hold-up in the rotor. The latter is preferred since it gives a larger interfacial area and hence better mass transfer performance. The fluidized state can be obtained if the flow of adsorbent phase is limited at the exit rather than at the entrance. Backmixing. Typically, the length of the contact zone in the CAT rotor is 10 cm, which is 1 or 2 orders of magnitude smaller than in conventional separation equipment, such as moving packed beds or multistage fluidized bed columns. As a consequence, the concentration gradient is much steeper. Therefore, even small disturbances in the plug flow behavior of any of the two phases may seriously influence the performance of the separation process. CAT involves differential countercurrent contact between the two phases. The degree of backmixing in both phases can be described using the axial dispersion model. The degree of backmixing is usually expressed in terms of Pe´clet numbers for the separate phases:

LuL LuS PeL ) and PeS ) EL ES

(6)

In eq 6, E represents the longitudinal dispersion coefficient, L is the length of the contact zone, and u is the superficial velocity of the particular phase. The degree of backmixing of both phases in a centrifugal field has been measured by means of pulseresponse measurements. This work is described in more detail elsewhere.4,5 For the liquid phase, Pe´clet numbers were found in the range PeL ) 5-20, and for the solid phase PeS ) 1-5. The variations in the tangential velocity in the contactor were held responsible for the high degree of backmixing. These variations lead to the formation of eddies, which contribute significantly to the degree of backmixing. Moreover, mixing effects at the liquid and solid phase distributors and the relatively low L/D ratio may. Estimation of the Mass Transfer Kinetics. A rigorously correct approach to the mass transfer phenomena should take into account the effect of the two cations moving in opposite direction in- and outside the resin matrix. This would also involve evaluation of the intraparticle concentration gradients. Yet, for design

ShLDL ShSDS and kS ) d d

(7)

in which DL and DS represent the diffusion coefficients in the liquid phase and resin particle respectively and d represents the particle diameter. For countercurrent separation processes, the dispersed phase Sherwood number equals ShS ) 10. The liquid-phase Sherwood number (ShL) depends on process conditions and is usually correlated to the particle Reynolds number (Re) and the Schmidt number (Sc). Various relations have been proposed.14 We will use the Snowdon-Turner relation for estimating the liquid-phase Sherwood number:15

ShL )

0.86 1/2 1/3 Re Sc L

(8)

The mass transfer kinetics is expressed in a dimensionless number, which represents the ratio between mass transfer between the two phases and convective flow of either phase. This ratio is given as the number of mass transfer units (NTU). The NTU can be defined for either of the two phases

NTUL )

kLaV kSaV and NTUS ) φL φS

(9)

where V represents the volume of the contactor and a represents the interfacial area per unit volume of the contactor. These numbers are related according to

NTUL S × Bi ) NTUS ShS

(10)

in which Bi represents the Biot number. Calculation of the Separation Performance. The separation performance can be calculated by using the steady-state mass balance. In dimensionless form, the mass balances for the liquid and solid phase in the CAT system read as follows:

1 d2x dx - NTUL(x - xi) ) 0 PeL dz2 dz

(11)

1 d2y dy + NTUS(yi - y) ) 0 + PeS dz2 dz In eq 11, xi and yi are the local equivalent fractions at the phase boundary and z represents the dimensionless coordinate in the contact zone, ranging from z ) 0 at the liquid-phase entrance to z ) 1 at the adsorbent inlet position. At the interface of the particles, equilibrium is assumed. As a consequence, the flux from the liquid phase toward the phase boundary equals the flux from the phase boundary into the resin particle. This is expressed in dimensionless numbers as follows:

Bi(x - xi) ) ShS(yi - y)

(12)

Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 4379 Table 1. Physical Properties of the System Ca2+/Na+ in Water and Dowex 50-WX8 liquid (kg/m3)

density viscosity (Pa‚s) particle diameter (µm) diffusion coeff (m2/s) capacity (mequiv/g) (dry Na+ form) (wet Na+ form)

F µ dp DNa DCa Qdry Qwet

1000 1 × 10-3 13.3 × 10-10 7.9 × 10-10

resin

ref

1330

exptl

91 16.1 × 10-11 6 × 10-11 4.53 2.24

exptl Atkins17 Takaota18 Vermeer9 exptl

The boundary conditions for the set of continuity equations now read as follows:

dx dy | )0 | ) PeS(yin - y|z)1) dz z)1 dz z)1

(13)

dy dx | ) PeL(x|z)0 - xin) | )0 dz z)0 dz z)0 Miyauchi and Vermeulen16 derived an analytical expression for the set of equations given above under the condition a linear isotherm. For curved isotherms, however, numerical methods need to be employed. In this work, the set of differential equations was solved using MatLab with a routine from the Numerical Algorithm Group (NAG) Foundation toolbox. This routine solves the two-point boundary value problem, using a finite difference technique and a Newton iteration (NAG). Physical Properties of the System. The case study presented in this work focuses on the exchange of Ca2+/ Na+ ions in a 0.10 N solution on a strong cationexchange resin (Dowex 50 WX8). The capacity of the resin was determined by a titration experiment, yielding Qdry ) 4.53 mequiv/g dry resin in Na+ form. The water content of the sample was determined at 50 wt % by drying, from which the capacity of the wet settled resin was calculated to be Qwet ) 2.24 mequiv/g wet settled resin in Na+ form. The relevant physical properties of the system are summarized in Table 1. The values are either taken from the literature or determined experimentally. The wet density of the resin hardly changes during the exchange Na+ for Ca2+. The moisture content of the resin in both ionic forms is almost equal and the atomic mass of the two ions expressed in terms of mass per equivalent, corresponds closely (MCa/ZCa ) 20.04 g/eq and MNa/ZNa ) 22.99 g/equiv). Experimental Section An experimental CAT rotor was constructed by the Mechanical Workshop of the Kluyver Laboratory for Biotechnology. It consists of two straight contacting columns that can rotate around a horizontal axis, with an inner diameter of 25 mm and a length of 125 mm. The rotor is driven by an electromotor via a belt and is limited to a maximum speed of rotation of ω ) 2500 rpm, which corresponds to a maximum centrifugal acceleration of 1000g in the rotor. A schematic layout of the experimental setup is shown in Figure 3. Liquid is fed to the rotor via a rotary seal and is directed toward the distributor at the outer end of the contactor (1), where it enters the contact zone. The adsorbent is fed to the rotor via a rotary seal and enters the contact zone via a pipe (2). The adsorbent is dragged toward the rim, in opposite direction to the liquid flow, and is discharged via a pipe at the heart line of the contact zone (3). The

Figure 3. Schematic layout of the experimental CAT rotor.

Figure 4. Schematic view of the internals in the columns.

treated liquid is discharged from the contact zone close to the axis of rotation (4). The rotor is equipped with two columns, each having an internal diameter (Dc) of 25 mm and a length (L) of 125 mm (measured from the solids inlet position (2) to the solids effluent position (3)). To decrease the degree of backmixing, the column was divided by plates in four sections in the longitudinal direction. This is schematically shown in Figure 4. The adsorbent is fed to the rotor in a concentrated suspension, which is contained in a fluidized bed. The liquid feed flow rate is monitored by a magnetoinductive flow meter (FlowTec Picomag DMI 6530; range 0-100 L/h). The liquid and solids effluent are guided through a 2.5 L flask, in which the solids are allowed to settle. By monitoring the mass of the settler, the adsorbent flow could be determined. The liquid that is used to drag the adsorbent out of the rotor is recycled after the adsorbent has been separated. The recycle flow is measured by a magneto-inductive flow meter as well (FlowTec Picomag DMI 6530; range 0-50 L/h). A simplified scheme of the entire setup is shown in Figure 5. It was found that an air bubble in the CAT rotor might form an air lock, thereby causing maloperation. As a consequence, all flows to the rotor were equipped with air traps. These traps also damp fluctuations in the flows that are caused by the peristaltic pumps.

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Ind. Eng. Chem. Res., Vol. 39, No. 11, 2000 Table 2. Process Conditions for the Mass Transfer Experiments expt ω (rpm) φL (L/h) φRec (L/h) φS (L/h) ∆P (mbar) A1 A2 A3 A4 B1 B2 C1 C2

700 700 700 700 700 700 700 700

23.998 23.896 23.850 23.743 20.985 20.863 14.597 14.529

1.709 1.498 3.027 3.832 3.009 4.533 2.983 3.791

0.850 0.739 1.380 1.520 1.457 2.042 1.653 1.866

113 113 111 101 85 90 80 77

L 0.735 0.735 0.737 0.749 0.766 0.757 0.756 0.762

Table 3. Estimated Mass Transfer Parameters, Required for the Evaluation of the Separation Performance expt

Figure 5. Simplified schematic outline of the experimental setup.

The following data are monitored on-line by a computer: the liquid feed flow rate; the solids withdrawal flow rate; balance measuring the mass of the settler at solids discharge; balance measuring the mass of the settler at liquid discharge (only in the second series of experiments); pressure difference between liquid feed and discharge. The pressure drop in the column is a result of the hydrostatic contribution due to the slightly higher average density of the slurry in the column and the friction between liquid and adsorbent. Assuming that the friction can be described by the Ergun relation, the mean void fraction in the contactor is estimated by solving the pressure drop equation:

To ensure operation in the fluidized state regime, the flow of the adsorbent was controlled at the discharge rather than at the feed. Therefore, the recycle flow pump was used to control the solids flow rate. The adsorbent feed pump was controlled such that the level of the fluidized zone inside the column was maintained at the same position. This was monitored by the pressure drop. During the start-up procedure, the adsorbent feed flow rate was significantly higher than the withdrawal rate, allowing a fluidized zone to build up in the contactor. At the moment that the level of the fluidized zone reached the liquid outlet position, adsorbent started to come out of the rotor via the liquid effluent and the pressure drop over the contactor did no longer increase. The pressure drop set-point is set at 90-95% of the maximum pressure drop. During the operation of the CAT rotor, it appeared to be very difficult to operate two columns in parallel. A small disturbance in either of the flows could cause a collapse of the fluidized zone in either of the columns, which resulted in a packed bed in that column. As a consequence, all liquid would pass through the other column, forcing all solids to flow directly toward the liquid outlet and leave the rotor via the liquid effluent. To avoid these operational stability problems, one

A1 A2 A3 A4 B1 B2 C1 C2

S 1.03 0.90 1.68 1.86 2.02 2.84 3.29 3.73

Bi 5.4 5.4 5.4 5.3 4.9 4.9 4.1 4.1

ShL

kL (m/s)

13.9 13.9 13.9 13.7 12.6 12.7 10.6 10.7

10-4

1.2 1.2 10-4 1.2 10-4 1.2 10-4 1.1 10-4 1.1 10-4 9.3 10-5 9.2 10-5

a (m2/m3)

NTUL

NTUS

17653 17674 17518 16732 15623 16172 16238 15849

17.6 17.7 17.5 16.5 16.0 16.8 20.2 19.6

31.7 36.4 19.3 16.8 16.3 12.1 15.0 12.9

column was taken out of operation. After all connections to this column were sealed, it was filled with water to ensure mechanical stability of the rotor. After steady state operation was obtained, samples for analysis of the liquid effluent were taken directly at the liquid outlet. These samples were analyzed for Ca2+ and Na+ by means of inductive coupled plasma (ICP) emission spectrometry. Furthermore, the particle size distribution of the ion-exchange resin was analyzed during the experiments. All samples were taken in duplo with intervals of approximately 2 min. Calculated liquid phase residence times are in the range of approximately 6-10 s and adsorbent phase residence times are in the range of 30-75 s. The experimental conditions are listed in Table 2. Between series A and B, the entire batch of ionexchange resin was regenerated by repeated extensive washing with a concentrated 1.0 N NaCl solution. After regeneration, the resin was washed thoroughly in the fluidized bed with tap water in order to remove all remaining salt. Results and Observations The relevant parameters for estimating the separation performance of the CAT process under the given set of process conditions have been calculated and are listed in Table 3. The mass transfer numbers (NTUL and NTUS) are evaluated on base of the diffusion coefficients of calcium. Since the diffusivities of Ca2+ in water and resin are lower than those of Na+, this leads to a slight underestimation of these characteristics. The experimentally determined separation efficiencies are listed in Table 4. The efficiency is reported as the fraction of Ca2+ ions that has been removed and exchanged by Na+ ions:

ξ)

cCa,in - cCa,out ) 1 - xCa,out cCa,in

(15)

All separation efficiencies are above 90%, and the reproducibility of the experiments is very good. All duplicate experiments agree very closely. Experiment

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Figure 6. Efficiency of Ca2+ removal vs separation factor. Table 4. Experimental Separation Efficiencies, Expressed as the Fraction of Ca2+ That Has Been Exchanged for Na+ efficiencies

expt A1 A2 A3 A4 B1 B2 C1 C2

Figure 7. Separation efficiency as a function of the mass transfer kinetics (expressed in NTUL) for varying degree of backmixing in the liquid phase.

92.2% 91.1% 90.6% 98.8% 99.4% 99.0% 99.3% 99.5% 99.7%

91.4% 91.9% 90.5% 99.4% 99.5% 98.6% 99.3% 99.6% 99.7%

A1 was extended over a longer period of time in order to verify that steady-state operation was obtained. The experimentally determined separation efficiency is plotted vs the separation factor (S) in Figure 6. The graph also shows some model curves. All model calculations have been done with NTUL ) 20, Bi ) 5, and ShS ) 10. The lower curve corresponds to the efficiency of a single ideally mixed nonequilibrium stage. The two higher curves are drawn for PeL ) 5 and PeS ) 2 and for PeL ) 10 and PeS ) 2. Higher values for the solidphase Pe´clet numbers yield higher efficiencies and less accurate correspondence with the experimental data. The Pe´clet numbers used in the simulations fall within the range observed by means of pulse-response measurements.4,5 It is clearly seen that, despite the low Pe´clet numbers, the contactor performs significantly better than a single-stage contactor. The model can be used to identify the phenomenon that limits the performance of CAT as well. In Figure 7 the influence of the mass transfer kinetics on the overall separation performance is shown. The model calculations are done for separation factor equal to unity (S ) 1), because the sensitivity of the performance to changes in the process conditions is maximal at this point. The calculations are based on the same process conditions as used above (Bi ) 5, PeS ) 2; PeL ) 5 and 10). The graph indicates that an enhancement of the mass transfer kinetics beyond NTUL ) 50 does not lead to a significant increase in the separation efficiency. The influence of the degree of backmixing in the adsorbent phase is shown in Figure 8. This graph clearly demonstrates that the separation performance is very sensitive to changes in the value of the Pe´clet number in the current range (PeS ) 1-5).

Figure 8. Separation efficiency as a function of the degree of backmixing in the solid phase (expressed in PeS) for varying degrees of backmixing in the liquid phase.

Conclusions The concept of centrifugal adsorption technology has been demonstrated to be successful for exchanging Ca2+ for Na+ in a 0.10 N solution using a strong cationexchange resin. Separation efficiencies range from 90% at a separation factor of unity, up to 99% for higher separation factors. The model, in which convective flow, dispersive transport, and mass transfer kinetics are combined in the continuity equations, allows accurate prediction of the separation performance of CAT. The model requires knowledge of the ion exchange isotherm and all separate transport phenomena that take place in the CAT rotor, expressed in terms of dimensionless numbers (NTUL, Bi, PeL, and PeS). These dimensionless numbers can be used for the design of CAT processes. Model calculations indicate that the performance of CAT at present is limited by the degree of backmixing in the adsorbent phase. It has been discussed elsewhere, that the current rotor can be improved significantly with this respect. Both in- and outlet effects may contribute severely to the degree of backmixing. Moreover, the effective L/D ratio in the contact zone can be enlarged by installing internals in the rotor. Although the particle diameter employed in this study does not give the full benefits of CAT, the mass transfer

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rates are already satisfactory to allow reasonable separation efficiencies in a very compact installation. The model system investigated in this work is characterized by fairly high diffusion coefficients. With particle diameters of 100 µm this already leads to reasonable mass transfer numbers (NTUL and NTUS). For larger molecules, such as proteins, the lower diffusion coefficients can be compensated by applying smaller ion-exchange resin beads and a higher speed of rotation. Nomenclature a ) interfacial area (m-1) c ) concentration in the liquid phase (mol/m3) c0 ) total salt concentration in the liquid phase (equiv/m3) D ) diffusion coefficient (m2/s) d ) particle diameter (m) E ) longitudinal dispersion coefficient (m2/s) K ) ion exchange equilibrium constant k ) mass transfer coefficient (m/s) m ) overall volumetric distribution coefficient Qwet ) volumetric capacity of the wet resin (equiv/m3) Qdry ) capacity of the dry resin (equiv/kg) q ) concentration in the adsorbent (mol/m3) u ) superficial velocity (m/s) V ) l of the contact zone (m3) x ) equivalent fraction in the liquid phase y ) equivalent fraction in the resin phase Z ) valence of the ion z ) dimensionless length in the contactor L ) void fraction in the column φ ) volumetric flow (m3/h) Dimensionless Groups Bi ) Biot number ()kLd/DS/m) NTU ) number of transfer units ()kaV/φ) Pe ) Pe´clet number ()EL/u) Sh ) Sherwood number ()kd/D) S ) separation factor or capacity ratio ()mφS/φL) Subscripts Ca ) for the calcium ion (Ca2+) i ) at the phase boundary L ) liquid phase Na ) for the sodium ion (Na+) S ) adsorbent phase

Acknowledgment The authors wish to express their thanks to the Mechanical Workshop of the Kluyver Laboratory for Biotechnology for the construction and maintenance of the CAT rotor. The project is supported financially by the Ministry of Economic Affairs (IOP-MP) through Senter.

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Received for review December 30, 1999 Accepted July 22, 2000 IE990927B