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Ind. Eng. Chem. Res. 2005, 44, 1906-1913

New Design of Simulated Moving Bed (SMB) for Ternary Separations Jin Seok Hur and Phillip C. Wankat* Purdue University, School of Chemical Engineering, Forney Hall of Chemical Engineering (FRNY), 480 Stadium Mall Drive, West Lafayette, Indiana 47907-2100

A semicontinuous two-zone SMB/chromatography hybrid system was designed for ternary separations, and compared to cascades with two two-zone SMBs and two four-zone SMBs. Operating conditions, productivity, and minimum Dtotal/F values were determined with the local equilibrium model, and purities were simulated using Aspen Chromatography version 11.1. Phenolic mixtures and sugar mixtures were used as models of moderately difficult and difficult linear separations, respectively. At the same productivity, the two-zone SMB/chromatography system gives at least 95% purities with reasonable Dtotal/F values and shows comparable performance with the two four-zone SMB cascade for the moderately difficult separation. For the difficult separation, the two-zone SMB/chromatography system has low purities with large Dtotal/F values, while the two four-zone SMB cascade gives higher purities at lower Dtotal/F values. Introduction Chromatography has been widely used not only for chemical analysis but also on a large industrial scale for separation. Elution chromatography was originally developed as an analytical separation. On a large scale, countercurrent chromatography is usually more economical than elution chromatography. True moving bed (TMB, Figure 1a) and simulated moving bed (SMB, Figure 1b) systems have been proposed. In the SMB system, the location of feed and product ports is switched instead of moving solids to minimize axial mixing and attrition of packing materials, which are common problems in TMB systems. Modern industrial applications of the SMB were developed by Universal Oil Products (UOP) in the late 1950s and early 1960s.1,2 The standard four-zone SMB (Figure 1b) has been commercially used in the petroleum industry, the corn wet-milling industry, the beet sugar industry, and pharmaceuticals (particularly chiral separations) because of its high efficiency and minimum desorbent usage.2-4 For binary separations, the standard four-zone SMB (Figure 1b) has been thoroughly studied and commercialized in various areas.1-6 However, applications of the SMB for ternary separations have not been extensively studied yet. In general, two coupled SMB cascades (Figure 2) can be used for ternary separations.7-11 Wankat12 developed alternate SMB cascades for ternary separations and determined the minimum D/F for 10 different separation cases. Masuda et al.13 patented a method for multicomponent separation that has been commercialized by Organo Corp., Tokyo, Japan. Wooley et al.14 proposed a nine-zone system that consists of two coupled binary SMB rings to recover glucose and xylose from biomass hydrolyzate. Applications of five-zone systems with side streams for ternary separations have been extensively studied.4,15-18 In the SMB separation process, the key items that determine cost are the amount of adsorbent, the number * To whom correspondence should be addressed. Tel: 765494-7422. E-mail: [email protected].

Figure 1. (a) True moving bed (TMB) and (b) simulated moving bed (SMB) for binary separations. Switching of ports in the SMB is not shown. Multiple columns can be employed in each zone.

of columns, and the desorbent use or desorbent-to-feed ratio, D/F. Although desorbent recovery costs usually dominate, the adsorbent for optical isomer separations and columns for high-pressure separation are expensive. In this study, we designed a high productivity, two-zone SMB/chromatography system for ternary separations and compared the results to a cascade of two two-zone SMBs and to a cascade of two four-zone SMB systems. The local equilibrium model with linear isotherms was used to determine separation conditions for the systems. At the same productivity, Dtotal/F and purity of the systems were compared, and the change in purities was studied with increasing Dtotal/F by simulation of the processes with Aspen Chromatography version 11.1.

10.1021/ie040164e CCC: $30.25 © 2005 American Chemical Society Published on Web 02/10/2005

Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1907

Figure 2. “Standard” coupled cascade with two four-zone SMBs for ternary separations. Switching of ports in the two SMBs is not shown.

SMB Designs for Ternary Separations In a typical cascade with two four-zone SMBs (Figure 2) for ternary separations, all zones have the same functions as those in the binary SMB shown in Figure 1b. Components A-C are the least, intermediate, and most sorbed components, respectively. In the first train, the ternary feed is separated to form the AB [or A] and C [or BC] product streams as the raffinate and extract streams. Train 2 receives the AB [or BC] stream as a feed and separates it into A and B [or B and C] streams. Desorbent is purified for recycle in zones 1 and 1′ (Figure 2).

Figure 3 shows a semicontinuous four-zone system consisting of two two-zone SMBs. In Figures 3 and 4 the principal components in the intermediate streams are shown in parentheses. The binary two-zone SMB was developed by Jin and Wankat.19 The separation procedure of the system in Figure 3 is almost the same as that of the cascade with two four-zone SMBs shown in Figure 2. AB [or A] and C [or BC] products are removed by train 1, and then the AB [or BC] stream is separated in train 2. The tank is used for storage (step a) and reuse (step b) of desorbent. Under ideal conditions this desorbent will be pure; however, when mass transfer is not very rapid, some impurities will appear in the tank. We have also developed a number of different twozone systems that combine SMB/chromatography characteristics for ternary separations. One such system is shown in Figure 4. In this integrated system, A and B are separated by the SMB approach (the A-B mass transfer zone is remixed with the feed) while the B-C separation is chromatographic (the B-C mass transfer zone leaves the system). This system may be advantageous when the B-C separation is easy. Capital investment is probably lower, because only two columns are needed. Detailed operating conditions for the above SMB systems are developed in the next section. Simulation Model The solute mass balance for the single porosity model is20

∂Ci ∂2Ci ∂qi ∂(vCi) + (1 - ) + - Dax,i 2 ) 0 ∂t ∂t ∂z ∂z



(1)

The subscript i indicates the different solutes, C and q are the solute concentrations in liquid phase and solid phase, and Dax is the axial dispersion coefficient. The expression for the mass transfer between liquid phase

Figure 3. Coupled cascade analogous to Figure 2 but with two two-zone SMBs for ternary separations. Switching of ports in the cascade with two two-zone SMBs is not shown.

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Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005

Figure 4. Complete ternary separation with integrated two-zone SMB/chromatography system.

Figure 5. Local equilibrium solution for the first train of the cascade with two two-zone SMBs (Figure 3): (s) A; (‚‚‚) B; (- - -) C.

and solid surface is21

∂qi ) km,iap(Ci - Ci*) ∂t

(2)

where km is the mass-transfer coefficient, ap ) 3/Rp for spherical particles, and qi ) KiCi for linear isotherms. The numerical solution of these differential equations was obtained by a second-order central differencing scheme (CDS2) using Aspen Chromatography version 11.1.22 Aspen Chromatography also allows one to solve these equations for linear systems with the number of theoretical plates, Np. The combined mass transfer and

dispersion effects can be described as the equivalent number of theoretical plates.24 The initial values for velocities were determined by the local equilibrium model.1,20 For linear systems this model is equivalent to the triangle theory.23 Since the model assumes rapid mass transfer and negligible dispersion, the solute velocity for a single porosity model of a linear system is1

usolute i,zone j,step k )

vjk ) Givjk 1- 1+ Ki 

[

]

(3)

Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1909

where vjk is the superficial velocity in column j for step k,  is the total bed void fraction, K is the linear equilibrium constant, and G is the constant to determine the velocity of solute. Since in this study separations of components B and C in the ternary mixtures are easier than those of A and B, the first train of both SMB cascades was designed to separate ternary feeds into AB and C product. (1) Cascade with Two Four-Zone SMBs. In the cascade with two four-zone SMBs (Figure 2) the constraints for the first train are

uA1 e uport (A not breakthrough)

(4a)

uA2 g uport uA3 g uport (trailing edge of A exits column) (4b,c) uB1 e uport (B not breakthrough)

(4d)

uB2 g uport uB3 g uport (trailing edge of B exits column) (4e,f) uC2 e uport uC3 e uport (C not breakthrough) (4g,h) uC4 g uport (trailing edge of C exits column) (4i) where the average port velocity uport ) L/tsw and uij is the velocity of solute i in column j. Since uA1 > uB1, uB2 < uA2, uB3 < uA3, uB3 < uB2, and uC2 > uC3, constraints (4b), (4c), (4d), (4e), and (4h) are automatically satisfied if constraints (4a), (4f), (4g), and (4i) are satisfied. The remaining constraints can be written as equations:

uA1 ) MAFuport (MAF e 1)

(5a)

uB3 ) MBTuport (MBT g 1)

(5b)

uC2 ) MCFuport (MCF e 1)

(5c)

uC4 ) MCTuport (MCT g 1)

(5d)

Mix are the multipliers for the front (F) and trailing edges (T) of component i. Similar linear multipliers were used by Ruthven and Ching25 (called γ) and Zhong and Guiochon26 (called β). The constant density mass balances are

Figure 6. Local equilibrium solution for the integrated two-zone SMB/chromatography system (Figure 4): (s) A; (‚‚‚) B; (- - -) C.

in detailed by Jin and Wankat.19 Briefly, for the first train of the cascade (Figure 3) all constraints are satisfied if

v1aGAta ) MAFL (MAF e 1, A not breakthrough) (7a) v1bGBtb + v2aGBta ) MBTL (MBT g 1, trailing edge of B exits column) (7b) v1aGCta + v1bGCtb ) MCFL (MCF e 1, C not breakthrough) (7c) v2aGCta + v2bGCtb ) MCTL (MCT g 1, trailing edge of C exits column) (7d) where ta and tb are switching times for steps a and b, respectively, and L is the column length. The mass balances for this cascade (with constant density) are

v1a ) v2a + vfeed

(8a)

v2 ) v3 + vfeed

(6a)

v1b ) vAB product

(8b)

v1 ) v2 - vAB product

(6b)

v1b ) v2b - vC product

(8c)

v3 ) v4 - vC product

(6c)

In the same manner as for Figure 2, when the column diameter and length and the flow rate of the feed stream are specified, four unknown interstitial velocities (v1a, v2a, v1b, v2b) and ta ()tb) can be calculated with eqs 7a-d and 8a by choosing proper Mix. Flow rates of products vAB product and vC product can be determined from eqs 8b,c. The solute movement in the first train of the system is shown in Figure 5. During step a, the mass transfer zone is remixed with the feed by recycling and part of the desorbent for recycle is stored in tank 1. While AB is separated from zone 1, C is removed from zone 2, and the desorbent in the tank is recycled during step b. The constraints for the second train of the cascade are the same as those for binary separations.19 (3) Integrated Two-Zone SMB/Chromatographic System. Figure 6 shows the solute movement in the semicontinuous two-zone SMB/chromatography system

where vfeed, vAB product, and vC product are the interstitial velocities that feed, AB product, and C product streams, respectively, would have in the column. If the column diameter and length and the flow rate of the feed stream are specified, we have five unknown variables (v1, v2, v3, v4, tsw) and equations (eqs 5a-d and 6a); thus, the unknown variables can be decided by choosing proper Mix. Flow rates of product streams can be calculated from eqs 6b,c. The second four-zone cascade separates the AB mixture into A and B products, and constraints for the separations are exactly the same as those for binary separations. (2) Cascade with Two Two-Zone SMBs. The separation conditions for two-zone SMBs are explored

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Table 1. System and Operating Parametersa phenolic system (phenol (A), 2-phenylethanol (B), and 3-phenyl-1-propanol (C))

sugar system (dextran T9 (A), dextran T6 (B), and fructose (C))

25 0.46 0.54 5 1.0 1.9 NpA ) 3500 NpB ) 4000 NpC ) 6000 KA ) 2.15, GA ) 0.654 KB ) 3.61, GB ) 0.454 KC ) 6.85, GC ) 0.271 5

47.5 1.4 0.45 11 1.0 1.0 kmAap ) 0.60 kmBap ) 2.84 kmCap ) 5.52 KA ) 0.13, GA ) 1.918 KB ) 0.23, GB ) 1.735 KC ) 0.69, GC ) 1.206 5

length of column (L), cm column diameter (d), cm voidage () particle radius (Rp), µm density (F), g/cm3 viscosity (µ), cP dispersion (Np)/mass transfer (kmap, 1/min) linear isotherm at T ) 21 °C (phenol), 25 °C (sugar) feed concn, g/L a

Data for equilibrium, , and mass transfer are from ref 21 for the phenolic system and from ref 22 for sugars.

(Figure 4). During step b, A and B are produced at the exits of zones 1 and 2, respectively. During step a, C from the previous feed step exits from zone 1 and AB is recycled by remixing with feed. This remixing with feed eliminates the A-B mass transfer zone. The local equilibrium constraints for complete separation are

v1aGAta ) MAFL (MAF e 1, A not breakthrough) (9a) v1bGAtb + v2aGAta ) MATL (MAT g 1, trailing edge of A exits column) (9b) v1aGBta + v1bGBtb ) MBFL (MBF e 1, B breakthrough) (9c) v1bGBtb + v2aGBta + v2bGBtb ) MBTL (MBT g 1, trailing edge of B exits column) (9d) v1aGCta + v1bGCtb + v2aGCta + v2bGCtb ) MCFL (MCF e 1, C breakthrough) (9e) v1bGCtb + v2aGCta + v2bGCtb + v1aGCta ) MCTL (MCT g 1, trailing edge of C exits column) (9f) The left-hand sides of eqs 9e,f are identical, so that MCF and MCT should both be 1 for complete separation. In addition, the superficial velocities must satisfy the constant density mass balances:

v1a ) v2a + vfeed

(10a)

v1b ) vA product

(10b)

v2b ) vB product

(10c)

v1a ) vC product

(10d)

If we specify feed flow rate and column length and diameter, we have five unknown variables (v1a, v2a, v1b, v2b, ta ) tb) and six equations (eqs 9a-d, 9e ) 9f, 10a) for certain Mix values. Thus, one of them is not independent. We have chosen eqs 9b-d, 9e ()9f), and 10a to find the operating conditions and then check whether eq 9a is satisfied. By using Mix ) 1 on the appropriate set of equations (e.g. eqs 3, 9, and 10 for Figure 4), the superficial velocities vjk for each system can be calculated to find

Table 2. Minimum Dtotal/F of Each System with Productivity ) 3.01 × 10-2 cm3/(cm3 min) for the Phenolic Mixture and 1.71 × 10-4 cm3/(cm3 min) for the Sugar Mixture min Dtotal/F

phenol mixture sugar mixture

two-zone SMB/chr

two two-zone SMBs

two four-zone SMBs

3.28 10.48

2.90 2.43

2.90 2.43

the minimum desorbent usage. These calculated feed flow rates and switching times were used for the initial simulation of the systems with Aspen Chromatography version 11.1. Higher purities were obtained by increasing Dtotal/F, adjusting the flow rates, and repeating the simulations. Results The system and operating parameters are given in Table 1. The number of theoretical plates Np based on the apparent dispersion coefficients Dax was used to evaluate the mass transfer and dispersion effects for the phenolic mixture (phenol, 2-phenylethanol, and 3-phenyl-1-propanol),24 and the effective mass transfer coefficient kmap was used for the sugar mixture (dextran T9, dextran T6, and fructose).27 Although the number Np can be converted into the coefficient kmap,28 we used Np because Aspen Chromatography has an option for this input. Although the difficulty of separation is generally classified by the selectivity, RBA ) KB/KA, this measure does not properly represent the difficulty for small Ki. Thus, we have used R′AB defined by12

R′AB )

uA GA ) g 1.0 u B GB

(11)

The difficulties of separation of the two binary pairs of the phenolic mixture (R′AB ) 1.44 and R′BC ) 1.68) were both considered moderate, while those of the sugar mixture (R′AB ) 1.11 and R′BC ) 1.44) were considered difficult and moderately difficult, respectively. The performance of each system is represented by Dtotal/F, the purity index PI, and the productivity. The purity index PI is the average purity of the desired components

Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1911 Table 3. Purities of Each System with Dtotal/F ) 3.27 and Productivity ) 3.01 × 10-2 cm3/(cm3 min) for the Phenol Mixture and with Dtotal/F ) 10.48 and Productivity ) 1.71 × 10-4 cm3/(cm3 min) for the Sugar Mixture phenolic system (phenol (A), 2-phenylethanol (B), and 3-phenyl-1-propanol (C))

purity of A (%) in A product purity of B (%) in B product purity of C (%) in C product PI

two-zone SMB/chr

two two-zone SMBs

two four-zone SMBs

two-zone SMB/chr

two two-zone SMBs

two four-zone SMBs

95.1 95.6 95.4 95.4

96.7 96.1 98.7 97.2

98.2 98.1 99.2 98.5

90.8 89.5 89.2 89.8

95.8 95.9 99.5 97.1

98.9 97.6 99.8 98.8

Table 4. Optimized Flow Rates (cm3/min) for the Two-Zone SMB/Chromatography System

in the three products:

PI ) purity of A (%) purity of B (%) purity of C (%) in A product + in B product + in C product 3 (12) and the productivity is defined as

productivity )

sugar system (dextran T9 (A), dextran T6 (B), and fructose (C))

volumetric flow rate of feed total adsorbent volume

(a) Phenolic Mixture (Qfeed ) 0.5 cm3/min; CA ) CB ) CC ) 5.0 g/L) Dtotal/ F

Tsw (min)

QD1

QD2

QD3

QA

QB

QC

3.28 4 5 6 7

11.16 9.44 7.55 6.26 5.41

0.610 0.748 1.061 1.396 1.673

0.529 0.611 0.642 0.673 0.736

0.500 0.641 0.797 0.931 1.091

0.529 0.611 0.642 0.673 0.736

0.500 0.6413 0.797 0.931 1.091

1.110 1.248 1.561 1.896 2.173

(b) Sugar Mixture (Qfeed ) 0.05 cm3/min; CA ) CB ) CC ) 5.0 g/L)

(13)

The systems were compared at the same productivities: 3.01 × 10-2 cm3/(cm3 min) for the phenolic mixture and 1.71 × 10-4 cm3/(cm3 min) for the sugar mixture. Table 2 shows the minimum Dtotal/F of each system determined from the local equilibrium analysis by using Mix ) 1 at the same productivity. At minimum Dtotal/F the separation will be perfect if mass transfer rates are infinite and there is no axial dispersion. For the phenolic mixture, the minimum Dtotal/F of the integrated twozone SMB/chromatography system is slightly larger than those of the other systems, while it is quite a bit larger for the sugar mixture. To compare the purities at the same Dtotal/F and productivity, the cascade with two four-zone SMBs and the cascade with two two-zone SMBs were simulated with the minimum Dtotal/F of the integrated two-zone SMB/chromatography system. In the simulations, train 1 was simulated with various D/F values, and then four D/F values were chosen between the minimum D/F and D/F where the PI for train 1 is almost constant. Train 2 was simulated with the results of train 1 to find maximum purity indexes. Flow rates were varied to optimize the purity indexes at each Dtotal/F value by changing Mix one after the other. Switching time and flow rates of the integrated two-zone SMB/chromatography system are shown in Table 4, and data for other cases are available from the authors. The results of the simulations are shown in Table 3. The purity indexes of the integrated two-zone SMB/chromatography is slightly smaller than those of the cascade with two fourzone SMBs and two two-zone SMBs for the phenolic mixture. However, for the sugar mixture, the purity index of the cascade with two four-zone SMBs is distinctively larger than for the integrated two-zone SMB/chromatography system. For the cascade with two two-zone SMBs the slow mass transfer rates of the sugar mixture causes impurities to occur in the tank, and they contaminate products when they are recycled in step b. The low purities obtained with the integrated two-zone SMB/chromatography are due to not only the slow mass transfer rates of the sugar mixture but also the constraints for C product (eqs 9e,f). In these equa-

Dtotal/ F

Tsw (min)

10.48 160.87 12.5 139.03 18 97.11

Qdesorbent 1 Qdesorbent 2 Qdesorbent 3 0.180 0.198 0.300

0.294 0.351 0.488

0.050 0.076 0.113

QA

QB

QC

0.294 0.050 0.230 0.351 0.076 0.248 0.488 0.113 0.350

tions Mix values are both 1, so that dispersion of C makes the A and B products impure (see Figure 6). Figure 7 shows the concentration profiles exiting train 1 of the cascade with two two-zone SMBs for the phenolic mixture separation. In this figure A and B overlap and move together from zone 1 at step b. If the isotherms of A and B are similar (R′AB close to 1), they will move like one component. This makes the separation of AB product from C in train 1 easier and the separation will require less desorbent. This is the reason that the minimum Dtotal/F values of the cascades for the sugar mixture are smaller than those for the phenolic mixture in Table 2. Figure 8 shows the concentration profiles exiting the integrated two-zone SMB/chromatographic system for the phenolic mixture. The profiles in these figures directly correspond to the predicted flow trends in Figures 5 and 6. The effect of Dtotal/F on purity index is presented in Figures 9 and 10. For the phenolic mixture (Figure 9), the purity indexes of the cascades with two two-zone SMBs and two four-zone SMBs are almost constant for Dtotal/F g 5.0, while the purity index of the integrated two-zone SMB/chromatography increases until Dtotal/F ) 4.0. For the sugar mixture (Figure 10) the purity index of the two-zone SMB/chromatography system is lower than for the other systems. Differences among the purities of the different cascades are larger for the sugar mixture than for the phenolic mixture. As an another way to compare the systems, Table 5 shows Dtotal/F values for purity indexes of 96% for the phenolic mixture and 92% for the sugar mixture. Although more desorbent is required for the integrated two-zone SMB/ chromatography system than for the other systems, with only two zones it is considerably simpler than the other systems. The differences among the Dtotal/F values for the phenolic mixture are small. In the same way, the cascade with two two-zone SMBs has reasonable Dtotal/F

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Figure 7. Concentration profiles exiting the first train of the cascade with two two-zone SMBs (Figure 3) for the phenolic mixture: (O) A; (0) B; (3) C.

Figure 9. Purity index (PI) vs Dtotal/F for the phenolic mixture. Flow rates are optimized for each Dtotal/F value (Table 4).

Figure 10. Purity index (PI) vs Dtotal/F for the sugar mixture. Flow rates are optimized for each Dtotal/F value (Table 4). Table 5. Dtotal/F of Each System with PI ) 96% for the Phenolic Mixture and 92% for the Sugar Mixture Figure 8. Concentration profiles exiting the integrated two-zone SMB/chromatographic system (Figure 4) for the phenolic mixture: (O) A; (0) B; (3) C.

for the sugar and is simpler than the cascade with two four-zone SMBs. Conclusion The results of our simulations show that the integrated two-zone SMB/chromatography system can be used for moderately difficult ternary separations with reasonable purities and Dtotal/F values. The separation using this system is carried out with only two zones, so that the system is simple and should be relatively easy to control. In addition, the cost of the columns will be smaller than for the cascades with two SMBs. Unfortunately, the equilibrium constraints cannot be always satisfied because eq 9a is not independent. In addition, since eq 9e should be equal to eq 9f, purities are limited. For these reasons, the system will require large Dtotal/F values for separations of ternary mixtures which have small selectivities such as the sugar mixture. The cascade with two two-zone SMBs shows reasonable

Dtotal/F

phenol mixture sugar mixture

two-zone SMB/chr

two two-zone SMBs

two four-zone SMBs

3.46 11.53

3.11 3.67

2.90 2.43

performance for both separation cases. This cascade requires more columns but has a lower minimum desorbent ratio than the integrated two-zone SMB/ chromatography system for the sugar mixture. The cascade with two two-zone SMBs has fewer columns and the same minimum desorbent usage as the cascade with two four-zone SMBs for the sugar mixture, although product streams were less pure. The cascade with two two-zone SMBs may be preferred for difficult separations that do not require very high purities but need to minimize the number of columns. Although the simulations in this study were done for laboratory-scale systems, they can easily be scaled up.29 Since it is well-known that four-zone SMB systems can be improved by using two or three columns per zone, the use of multiple columns will be explored in future work. In addition, techniques such as partial feed,

Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1913

selective withdrawal, and variable flow rates will also be studied.21 Acknowledgment This research was supported by NSF Grant No. CTS0211208. We thank our colleagues Nadia Abunasser, Weihua Jin, and Jeung Kun Kim, who offered useful suggestions in this research. Notation ap ) external surface area/volume, cm2/cm3 C ) solute concentration in the liquid phase, g/cm3 C* ) solute equilibrium liquid concentration corresponding to the solid concentration, g/cm3 Dax ) axial dispersion coefficient, cm2/s Dtotal/F ) ratio of total flow rate of desorbent to feed flow rate d ) column diameter, cm G ) constant for determining the velocity of solute K ) linear equilibrium parameter km ) mass transfer coefficient, cm/s L ) column length, cm M ) multiplier Np ) number of theoretical plates Q ) volumetric flow rate q ) solute concentration on the solid phase, g/(cm3 of particles) Rp ) radius of particle, cm t ) time, s u ) solute velocity, cm/s v ) interstitial velocity, cm/s z ) axial coordinate, cm Greek Symbols RAB ) selectivity based on isotherms R′AB ) selectivity based on solute velocities  ) total bed void fraction F ) fluid density, g/cm3 µ ) fluid viscosity, g/(cm s) Subscripts i (A, B, C) ) solute j (1, 1′, 2, 2′, 3, 3′, 4, 4′) ) zones in SMB k (a, b) ) step x (F, T) ) front and trailing edge

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Received for review May 21, 2004 Revised manuscript received December 15, 2004 Accepted January 3, 2005 IE040164E