Application of the general rate model with the Maxwell–Stefan equations for the prediction of the band profiles of the 1-indanol enantiomers

Chemical Engineering Science 58 (2003) 2325 – 2338

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Application of the general rate model with the Maxwell–Stefan equations for the prediction of the band pro&les of the 1-indanol enantiomers Krzysztof Kaczmarskia , Ma lgorzata Gubernaka , Dongmei Zhoub; c , Georges Guiochonb; c;∗ a Faculty

of Chemistry, Rzeszow University of Technology, 35-959 Rzeszow, Poland of Chemistry, The University of Tennessee, 552 Buehler Hall, Knoxville, TN 37996-1600, USA c Division of Chemical and Analytical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, USA

b Department

Received 30 April 2002; received in revised form 27 December 2002; accepted 8 January 2003

Abstract The adsorption isotherm data of R- and S-1-indanol and of their racemic mixture on cellulose tribenzoate were measured by frontal analysis. These experimental data were &tted to the single-component and the modi&ed competitive Bilangmuir isotherms. The overloaded elution pro&les of bands of the pure enantiomers and of the racemic mixture were calculated for di8erent sample sizes, using the best competitive isotherm model and the General Rate Model of chromatography coupled with the generalized Maxwell–Stefan equation that describes the surface di8usion 9ux. The calculated and the experimental pro&les were found to be in excellent agreement in all cases. The parameters of the model of the mass transfer kinetics were derived from the band pro&les obtained for the pure enantiomers. The same values of these parameters give an excellent prediction of the pro&les of multicomponent bands. The new model described here allows a satisfactory interpretation of the competitive mass transfer kinetics. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: Enantiomeric separations; Equilibrium isotherm; Kinetic studies; General rate model; Maxwell–Stefan equation; 1-indanol

1. Introduction Preparative liquid chromatography, particularly as implemented in the simulated moving bed process (SMB), has become the preferred method to carry out preparative enantiomeric separations or puri&cations. This method has been the topic of numerous research papers in the recent years (Sellergren, 1989; Charton, Jacobson, & Guiochon, 1993; Seidel-Morgenstern & Guiochon, 1993; Charton, Bailly, & Guiochon, 1994; Edge, Heaton, & Bartle, 1995; Maris, Vervootr, & Hindriks, 1991; Krause & Galensa, 1988; Xu & Tran, 1991; Kunath, HBoft, & Hamann, 1991; Hampe et al., 1993; Kunath, Theil, & Wagner, 1994; Wang, Lu, Chen, & Li, 1997; Armstrong, Gahm, & Chang, 1997; Jacobson, Shirazi, & Guiochon, 1990) Enantiomeric separations are diDcult because they require a dedicated chiral ∗

Corresponding author. Current address: Department of Chemistry, The University of Tennessee, 552 Buehler Hall, Knoxville, TN 37996-1600, USA. Tel.: + 1-865-9740-733; fax: +1-865-9742-667. E-mail address: [email protected] (G. Guiochon).

stationary phase (CSP), with a chiral selector suitable for the speci&c separation required, and because the enantioselectivity of the CSP’s used for industrial applications is often rather small. Conventional trial-and-error methods of optimization of such separations and particularly of the SMB separations are very wasteful of time and chemicals. Accordingly, computer-assisted optimization is becoming important for the economics of these separations. This approach, however, requires the prior determination of the column characteristics, including the competitive equilibrium isotherms of the feed components in rather wide concentration ranges, extending up to the saturation of the mobile phase or to a value exceeding signi&cantly the largest concentrations contemplated in practical applications. In a separate report (Zhou, Cherrak, Kaczmarski, Cavazzini, & Guiochon, 2003), we investigated the thermodynamics and the mass transfer kinetics of the chiral separation of R- and S-1-Indanol on cellulose tribenzoate. We showed that the competitive Bilangmuir isotherm model is the one that most accurately describes the adsorption behavior of the single enantiomers and of the racemic

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0009-2509(03)00096-4

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K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

mixture. However, to obtain an excellent agreement between the experimental band pro&les of binary mixtures and the pro&les calculated using this isotherm model, we had to use model parameters estimated simultaneously from single and competitive isotherm data. Such a method of estimation of the model parameters is common in chromatography (see e.g. Cherrak, Khattabi, & Guiochon, 2000; Cavazzini et al., 2001; Mihlbachler, Kaczmarski, Seidel-Morgenstern, & Guiochon, 2002). In many cases it gives calculated band pro&les that agree very well with experimental ones. In this earlier report (Zhou et al., 2003), we used a Lumped Pore Di8usion model (POR) to calculate the band pro&les. However, the excellent agreement obtained in this case between the experimental and the calculated band pro&les could be obtained only because we adjusted the model parameters for each sample size. This suggested an inadequacy of the POR model to account for the mass transfer kinetics in the case studied. In this new report, we present a more detailed analysis of the thermodynamics of adsorption and of the kinetics of the mass transfer of the two enantiomers of 1-indanol, using the same experimental data as presented earlier (Zhou et al., 2003). Our aim is to obtain the same good agreement between the calculated and the experimental band pro&les for samples of binary mixtures in a wide size range. However, we wanted to achieve this goal using a single set of parameters for each of the models of the isotherm and of the mass transfer kinetics. These parameters could be derived from experimental data acquired for single component and binary mixtures but they are not adjusted for each new case. A kinetic model more sophisticated than the POR model must be used. 2. Theory In a previous paper (Zhou et al., 2003), the POR model was used for the modeling of the band pro&les of the Indanol enantiomers. In spite of the fact that the conditions of validity of the POR model were ful&lled (Kaczmarski & Antos, 1996; Zhou et al., 2003), a good agreement between calculated and experimental pro&les was achieved only if a di8erent value of the molecular di8usivity was used for each value of the loading factor, Lf . This required that the molecular di8usivity of each enantiomer be adjusted for each new experiment. This need suggested that the POR model was not adequate and, more speci&cally, that the molecular 9ux inside the adsorbent particles was a function of the solute concentration. A similar result was obtained when the general rate model (GR) (Kaczmarski, Antos, Sajonz, Sajonz, & Guiochon, 2001) was used instead of the POR model, assuming a Fickian molecular di8usion 9ux inside the adsorbent particles. Recently, we applied a new variant of the GR model to the modeling of the separation process of the enantiomers

of 1-phenol-1-propanol (Kaczmarski et al., 2002). Instead of a Fickian molecular di8usion 9ux, this model used the generalized Maxwell–Stefan (GMS) model to describe the surface di8usional 9ux of the compound studied inside the particles. In this paper, we apply a very similar GR-GMS model to the calculation of the band pro&les of the Indanol enantiomers. 2.1. General rate model of chromatography (GR) The GR model used in this paper was discussed in detail previously (Kaczmarski et al., 2002). We present below a short description of this model. In the following discussion of this GR model, we ignore the external mass transfer resistances because, as showed previously (Zhou et al., 2003), the internal mass transfer resistances dominate the mass transfer kinetics in the chromatographic system under consideration. In writing the equations of this model, we made the following assumptions: 1. The multicomponent &xed-bed process is isothermal. 2. The velocity of the mobile phase is constant. Its compressibility is negligible. 3. The packing material has porous, spherical, particles of uniform size. 4. The concentration gradient in the radial direction of the bed is negligible. 5. Local equilibrium exists for each component between the pore surface and the stagnant 9uid phase in the macropores. 6. The dispersion coeDcient is constant. 7. The external mass transfer resistances are negligible. With these assumptions, the di8erential mass balance of each feed component in the mobile and the solid phase can be formulated as follows: e

@CR p; i @Ci @Ci @ 2 Ci +u = e DL ; − (1 − e ) 2 @t @z @z @t

(1)

p

@Cp; i @qi 1 @ 2 + (1 − p ) + 2 (r JT; i ) = 0; @t r @r @t

(2)

where 3 CR p; i = 3 Rp


 0

Rp

2

p Cp; i r dr +

 0

Rp

2



(1 − p )qi r dr ; (3)

where Ci and Ci; p are the concentrations of component i in the mobile and the stagnant liquid phases, respectively, u is the super&cial velocity, DL is the axial dispersion coeDcient, z is the distance along the column, r is the distance from the particle center, Rp is the particle diameter, and p , e are the particle and bulk porosity, respectively. The molecular 9ux, JT; i , is given by JT; i = p Jm; i + (1 − p )Ns; i ;

(4)

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

where Jm; i is the pore molecular di8usion 9ux and Ns; i is the surface molecular 9ux. As initial conditions for Eqs. (2) and (3), we assumed that the concentrations of the feed components in the liquid phase and on the surface of the solid phase are equal to zero. As boundary conditions for the &rst mass balance equation (Eq. (1)) we used the well know Danckwerts condition, so for t ¿ 0 we have at z = 0: @Ci
uf Cfi − u(0) C(0) = −e DL @z with
Cfi = Cfi for 0 ¡ t ¡ tp ; (5)
Cfi = 0 for tp ¡ t and for t ¿ 0, we have at z = L: @Ci = 0: (6) @z For the second mass balance equation (Eq. (2)), we used typical boundary conditions. For t ¿ 0, at r = Rp , we have Cp; i (t; r) = Ci

(7)

and for t ¿ 0, at r = 0, we have @Cp; i (t; r) = 0: (8) @z Eqs. (1)–(8), together with the suitable isotherm model constitute the GR model. This model must be completed with the relationships required to express the di8usion 9ux inside the particles. 2.2. The di
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mass transfer on the sample concentration can be explained by assuming that surface di8usion plays a dominant role in the overall mass transfer kinetics. Assuming that the dusty gas-like model (Mason & Malinauskas, 1983; Krishna, 1993a, b; Kapteijn, Moulijn, & Krishna, 2000) gives a good description of the surface di8usion of adsorbed molecules, we may use the following equation for the 9ux n

−

 j Ns; i − i Ns; j Ns; i i + ; ∇i = RT q D q s ij s Di j=1

(10)

j=i

where i = qi =qs is the fractional surface occupancy of species i, ∇i is the surface chemical potential gradient of species i at constant temperature and spreading pressure, R is the universal gas constant, T is the temperature, qs is the saturation capacity of the adsorbent (common to both enantiomers), Dij are the GMS counter-sorption di8usivity coeDcients, Di is the corrected di8usivity (Krishna, 1990) or Maxwell–Stefan di8usivity, describing the interaction between component i and the adsorbent (Krishna, 1993a, b; Kapteijn et al., 2000). Assuming a state of equilibrium between the adsorbed species and the bulk 9uid, the surface chemical potential, is , of species i can be expressed as is = il = i0; l + RT ln fi ;

(11)

where fi is the fugacity of component i in the bulk 9uid phase in equilibrium with the sorbent. After some simple mathematical manipulations, assuming the mobile phase to be ideal and the fugacity, fi , of the mobile phase to be equal to its concentration, Cp; i , we obtain the following equations for the surface molecular 9ux (Kapteijn et al., 2000; Kaczmarski et al., 2002):   @Cp; 1 @Cp; 2 ; (12) Ns; 1 = − Ds; 11 + Ds; 12 @r @r   @Cp; 1 @Cp; 2 Ns; 2 = − Ds; 21 + Ds; 22 ; (13) @r @r where

q1 D1 [ 1 D2 + D1; 2 ]=[ 1 D2 + 2 D1 + D1; 2 ]; Cp; 1 q2 Ds; 12 = 1 D2 D1 =[ 1 D2 + 2 D1 + D1; 2 ]; Cp; 2 q1 Ds; 21 = 2 D1 D2 =[ 1 D2 + 2 D1 + D1; 2 ]; Cp; 1 q2 Ds; 22 = D2 [ 2 D1 + D1; 2 ]=[ 1 D2 + 2 D1 + D1; 2 ]: Cp; 2

Ds; 11 =

(14) (15) (16) (17)

The counter-exchange di8usion coeDcient, D1; 2 , was derived from the equation suggested by Krishna (1990), based on the generalization of Vignes (1966) relationship for diffusion in bulk liquid mixtures. D1; 2 = (D1 )

1 =( 1 + 2 )

(D2 )

2 =( 1 + 2 )

:

(18)

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Table 1 Values of the parameters used in the GR-GSM model Parameter Dispersion coeDcient, DL Total porosity, t External porosity, e Internal porosity, p Tortuosity, 

Numerical value (cm2 =min)

0.00437 0.705 0.35 0.5461 3.87

is written as qs2 K2 C qs1 K1 C + ; (20) q= 1 + K1 C 1 + K2 C where qs; i is the saturation capacity of the type of sites I and Ki is the equilibrium constant on these sites. Since this model accounts accurately for the experimental data, we do not consider any other model in this report. 2.6. Competitive equilibrium isotherm model

The model that was presented in this section is called the GR-GMS model. 2.3. Parameters of the GR-GMS model The parameters needed to calculate numerical solutions of the GR-GMS model, like the axial dispersion coeDcient and the tortuosity parameter, were derived from correlations given in previous papers (Zhou et al., 2003). The values obtained are listed in Table 1. 2.4. Numerical solution of GR-GMS model The GR-GMS model has no closed-form solutions. Numerical solutions were calculated using a computer program based on an implementation of the method of orthogonal collocation on &nite elements (Kaczmarski et al., 2002; Guiochon, Shirazi, & Katti, 1994; Kaczmarski, Storti, Mazzotti, & Morbidelli, 1997; Berninger, Whitley, Zhang, & Wang, 1991). The set of discretized ordinary di8erential equations was solved with the Adams–Moulton method, implemented in the VODE procedure (Brown, Hindmarsh, & Byrne, 1989). The relative and absolute errors of the numerical calculations were 1 × 10−6 and 1 × 10−8 , respectively. 2.5. Single component equilibrium isotherm model To acquire the equilibrium concentration data, singleand two-component frontal analysis measurements was performed (Zhou et al., 2003). The calculation of the equilibrium concentration in the solid phase were made using the well know integral mass balance equation (Guiochon et al., 1994). For single-component frontal analysis, this mass balance reads VR − V 0 0 C ; (19) q= Va where q is the concentration in the solid phase at equilibrium with the feed concentration C o in the 9uid phase, Vo is a the hold-up volume, Va is the volume of adsorbent in the column, and VR is the retention volume of the breakthrough curve. The Bilangmuir single-component isotherm model proved to account best for the sets of single-component experimental data acquired (Zhou et al., 2003). This isotherm model

In this work as previously (Zhou et al., 2003), we assumed that there are only two di8erent types of sites. We also assumed that the saturation capacity on each type of sites is the same for each of the two enantiomers and that their equilibrium constants are di8erent on both the high and low energy sites—see Section 4 for more details. Therefore, the low energy sites are not necessarily achiral. So, the Bilangmuir competitive model reads as qs; 1 K1; i Ci qs; 2 K2; i Ci qi = + ; (21) 1 + K1; 1 C1 + K1; 2 C2 1 + K2; 1 C1 + K2; 2 C2 where qs; i are the two saturation capacities and Ki; j the equilibrium constants. The equilibrium concentration of both components in two-component frontal analysis were calculated using the following equations (Guiochon et al., 1994): (V2 − V0 ) ∗ C10 − (V2 − V1 ) ∗ C1; m q1 = ; (22) Va (V2 − V0 ) ∗ C20 q2 = ; (23) Va where q1 and q2 are the solid phase concentrations at equilibrium with the feed concentrations Cio , V1 and V2 are the retention volumes of the two fronts. The concentration C1; m is the concentration of the intermediate plateau—see later, Fig. 2. As we show later, this model accounts reasonably well for the experimental data although small but signi&cant deviations take place at high concentrations. More sophisticated binary isotherm models must be considered. These can be based on the ideal adsorbed solution theory or on the consideration of molecular interactions between the two enantiomers in the adsorbed phase. Models implementing these two possibilities are discussed later, as part of the interpretation of the experimental data. 3. Experimental Complete details regarding the experimental work were previously published (Zhou et al., 2003). We report here only the most important experimental conditions. 3.1. Equipment A HP 1090 liquid chromatography system was used (Agilent Technologies, Palo Alto, CA, USA). This system is equipped with a multi-solvent delivery system, an

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

automatic injector with a 25 l sample loop, a column oven, a diode-array detector, and a data acquisition system. It is controlled and operated with a microcomputer. 3.2. Materials The mobile phase was a solution of n-hexane and 2-propanol (92.5:7.5, v/v), both HPLC grade solvents from Fisher Scienti&c, Fair Lawn, NJ, USA. 1; 3; 5-tri-tert-butylbenzene and 1-indanol were from Aldrich (Milwaukee, WI, USA). Samples of pure R-1- and S-1-indanol were also purchased from Aldrich and were puri&ed in our laboratory. 3.3. Column The column used for the experiment was a 20 × 1:0 cm column, packed in house with Chiracel OB (cellulose tribenzoate coated on a silica support; Daicel, Tokyo, Japan). The average particle diameter of the packing material was 20 m; The total column porosity, t , was 0.705. The external porosity, e , was assumed to be 0.35. 3.4. Measurements of the isotherm data All experimental data were measured at room temperature (ca 25◦ C) with a 2:5 ml=min mobile phase 9ow-rate. Single and binary frontal analyses were performed in the conventional staircase mode (Guiochon et al., 1994), using the multi-channel solvent delivery system, one channel of this system delivering the sample solution, the other the pure mobile phase. A program adjusted periodically the ratio of the 9ow rates of the two streams to increase the concentration of sample solution by 10% increments from 0% to 100%. The minimum sample volumes necessary to reach the plateau concentration were approximately 12:5 ml in single-component and 17:5 ml in binary frontal analysis. Thus, the period of the step gradients were 5 and 7 min, respectively. The concentration range investigated was approximately 0.2–20 g=l. In this range, 19 data points were acquired, 10 between 0.2 and 2:5 g=l, nine between 2.5 and 20 g=l. 3.5. Modeling of the experimental isotherm data The best numerical values of the Bilangmuir and modi&ed Bilangmuir (see Section 4) isotherm models were estimated by &tting the experimental adsorption data to the model equation, using the least-squares Marquardt method modi&ed by Fletcher (1971). 4. Results and discussion 4.1. Equilibrium isotherms 4.1.1. Single component equilibrium isotherm model The adsorption equilibrium data used previously (Zhou et al., 2003; Kaczmarski et al., 2002) were derived by

2329

determining the retention volume VR from the in9ection point of the breakthrough curves recorded. Although this widespread method is accurate when the mass transfer kinetics is fast, it becomes inaccurate for relatively slow kinetics, like in the case of the chiral separation of the 1-indanol enantiomers. In this work, we recalculated the equilibrium data using as value of the retention volume the one calculated using the area method (Sajonz, Zhong, & Guiochon, 1996). The data obtained for the equilibrium concentrations of the single components were unchanged. By contrast, however, the adsorption data calculated for binary mixtures, particularly for R-Indanol, using the area method were systematically di8erent from those obtained using the in9ection point method—see Section 4.1.2. The best estimates of the isotherm parameters are reported in Table 2a. The errors made on these estimates were calculated using the t-Student test, for a 95% con&dence level. We observed that the saturation capacities obtained for S-1-indanol on both types of sites are included in the con&dence level of the saturation capacities obtained for R-1-indanol. Taking this fact into account, we assumed that the saturation capacities of each of the two types of sites is the same for both enantiomers. The best estimates of the parameters made with this assumption are listed in Table 2b. The comparison of the data presented in Table 2a and b shows that they are consistent and that the values given in Table 2b are included within the con&dence interval of the same parameters given in Table 2a. The values of the Fisher coeDcient are also consistent. It is thus reasonable to assume that the saturation capacities of each type of sites are the same for both enantiomers. This allows the use of the simple competitive bilangmuir model as a &rst approximation of the competitive adsorption behavior of the two enantiomers in the system used. The comparison of the equilibrium constants (Table 2b) shows that the major role in the separation of this racemic mixture is plaid by the high-energy sites. This con&rms the generally accepted model of enantiomeric separation. Many earlier studies have found that the low-energy sites are not selective. This is often postulated, although this assumption should always be checked carefully. In the present case, this assumption is not valid. It leads to excessively high values of the saturation capacity of the low-energy sites (larger than 5000) and to relatively small value of the Fisher coeDcient. So, we rejected this hypothesis. In the following we use the parameters given in Table 2b to interpret the competitive isotherm data. 4.1.2. Competitive equilibrium isotherm models In our previous paper (Zhou et al., 2003), we showed that the experimental band pro&les of large-size samples of pure R-1- or S-1-Indanol are in excellent agreement with the corresponding pro&les calculated as numerical solution of the POR model, using the values of the Bilangmuir isotherm

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K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

Table 2 Best estimates of the parameters of the Bilangmuir isotherm model for the two enantiomers and Fisher’s test values: (a) Independent parameters; (b) Same saturation capacity for both isomers

(b) Bilangmuir

Parameters

Fisher coeDcient

qs1

K1

qs2

K2

R-1-indanol S-1-indanol

106:4 ± 26:9 89:34 ± 2:10

0:0118 ± 0:0064 0:0201 ± 0:0010

13:55 ± 6:71 8:65 ± 0:50

0:108 ± 0:063 0:360 ± 0:012

2:94 × 105 6:69 × 105

R-1-indanol S-1-indanol

91:05 ± 2:33

0:01693 ± 0:00090 0:01929 ± 0:00109

8:988 ± 0:54

0:1315 ± 0:0035 0:3486 ± 0:0192

3:97 × 105

parameters given in Table 2a. The same is true (not shown) when the values of the Bilangmuir model parameters are those in Table 2b. However, in spite of the equality of the saturation capacities of each Langmuir term for the two compounds, the use of the coeDcients of the single component Bilangmuir model to describe the competitive Bilangmuir isotherm failed. Important di8erences were observed between the experimental band pro&les of samples of the binary mixture and those calculated with these parameters. Conversely, single component pro&les calculated using the best estimates of the parameters of the competitive Bilangmuir isotherm model failed properly to describe the experimental band pro&les of the pure enantiomers. To increase the accuracy of the isotherm model description, the parameters are estimated from both single and competitive experimental isotherm data (Cherrak et al., 2000; Cavazzini et al., 2001; Mihlbachler et al., 2002). This method of approximation of real isotherms turns out to be useful in engineering practice as was showed in our recent paper (Zhou et al., 2003). However such a method does not disclose any useful detail of the mechanism of adsorption. Later, we will try to analyze some other possible mechanism of the competitive adsorption of 1-indanol but, &rst, we discuss the accuracy of the competitive equilibrium data and their recalculation from the experimental frontal analysis data. As was stated earlier, we recalculated the adsorption equilibrium data using the value of the retention volume of the breakthrough curve obtained with the area method. In Fig. 1, we compare the experimental competitive isotherms of both enantiomers calculated with the in9ection point method (squares) and the area method (triangles), using Eqs. (22) and (23). The &gure shows important di8erences between the results of the two methods, especially for the less retained component. To explain the origin of such large di8erences, consider Eqs. (22) and (23) and the pro&le of a typical adsorption step for a binary sample of R- and S-1-Indanol (see Fig. 2). An error made on the second term of the nominator of Eq. (22) has a strong in9uence on the accuracy of the calculation of the surface concentration q1 , especially

16 14 12 10

q[g/l]

(a) Bilangmuir

Enantiomers

8 6 4 2 0 0

1

2

3

4

5

6

7

C[g/l]

Fig. 1. Comparison between the experimental competitive isotherms calculated with the in9ection point method (squares) and the area method (triangles). The upper two lines are for S-indanol, the lower two lines for R-indanol. 16 14

V2 =t 2*u* Π*R

C1,m

12

2

10

C [g/l]

Isotherm type

8 6 4 2

tC

0 6

7

8

9

t [min]

Fig. 2. Typical breakthrough curve obtained for a mixture of the 1-indanol enantiomers at high concentrations.

when the concentration C1; m of the intermediate plateau is high compared to the feed concentration of the &rst component, C1 (see Fig. 2). A relatively small error in the determination of the concentration C1; m and the retention

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

2331

18

16

16

14 14

12 12

q[g/l]

q[g/l]

10 8 6

10 8 6

4

4

2

2

0

0

0

1

2

3

4

5

6

7

0

1

2

3

C[g/l]

4

5

6

7

8

C[g/l]

Fig. 3. Comparison between the experimental competitive isotherm data (symbols) and the competitive Bilangmuir isotherm derived from Eq. (21) with the values of the parameters in Table 2b (solid lines). Circles, S-1-indanol, squares, R-1-indanol.

Fig. 4. Comparison between the experimental competitive isotherm data (symbols) and the IAS isotherm model derived from Eq. (21), using the values of the parameters in Table 2a (solid lines). Circles, S-1-indanol, squares, R-1-indanol.

volume V2 can produce a large error on q1 . To derive the value of C1; m , we calculated numerically the derivative of the concentration pro&le by respect to time and adopted as value of the plateau concentration C1; m the value for which the derivative was minimum (close to the center of the plateau). To determine the retention volume, V2 , the second step area was calculated from the time tC at which C1; m is eluted to the time when the concentration becomes equal to the feed concentration—see Fig. 2. Time t2 was chosen such that the areas between the vertical straight lines and the breakthrough curve are equal. The retention volume, V1 of the &rst breakthrough curve was similarly calculated. All the adsorption data used in this work were derived from the frontal analysis results using the area method. Fig. 3 compares the experimental data for the competitive adsorption isotherm derived as explained above (symbols) and the line calculated using the competitive Bilangmuir isotherm in Eq. (21) and the parameters in Table 2b. A slight di8erence between the two series subsists. The solid phase concentration at equilibrium is systematically lower than the experimental one for both isomers. This discrepancy is a consequence of using too simple a model of competitive isotherm behavior. A simple competitive Langmuir isotherm built from two single-component Langmuir isotherms is thermodynamically consistent when the saturation capacities for each compound are the same, as they are in the present case (see Table 2b) (Schwab, 1928). This does not guarantee that this model will account for the competitive isotherm data. A more sophisticated model may be designed, using the ideal adsorbed solution (IAS) theory (Radke & Prausnitz, 1972). Alternately, we may assume that some molecular interactions can take place between the two enantiomers in the adsorbed phase. Obviously, there can be no clues regarding the possibility, let alone the strength, of such a type of interaction taking place between the two enantiomers in the single component isotherm behavior.

The &rst approach was not successful, as illustrated in Fig. 4. Accordingly work was not pursued along this line. The second one is more promising as we show below by assuming that a molecule of R-indanol can interact with one of S-indanol that is already adsorbed on a high energy site. If the adsorption–desorption process is in&nitely fast, the equilibrium isotherm for the high energy sites can be derived from the following equations: K2; 1 ∗ C1 ∗ (1 − 2; 1 − 2; 2 − 2; 21 ) − 2; 1 = 0;

(24)

K2; 2 ∗ C2 ∗ (1 − 2; 1 − 2; 2 − 2; 21 ) − 2; 2 = 0;

(25)

K2; d21 ∗ C1 ∗ 2; 2 − 2; 21 = 0:

(26)

The &rst two equations describe the adsorption equilibrium of the &rst and second component on the high energy sites. The &rst subscript is the site index, the second and third subscripts denote the components involved. The third equation describes the interaction of the &rst and second component, with an equilibrium constant K2; d21 . The total amounts of each enantiomer adsorbed on the high energy sites is given by the following equations: q2; 1 = qs2 ∗ (2; 1 + 2; 21 );

(27)

q2; 2 = qs2 ∗ (2; 2 + 2; 21 ):

(28)

After introducing Eqs. (24)–(26) into Eqs. (27) and (28), simple arithmetic manipulations and incorporating the term describing the adsorption on the low energy sites, one can obtain the adsorption isotherms of the two compounds: K1; 1 ∗C1 q1 =qs1 ∗ +qs2 ∗ 1+K1; 1 ∗ C1 +K1; 2 ∗C2 ×

K2; 1 ∗C1 +K2; 2 ∗ C2 ∗ K2; d21 ∗ C1 ; (29) 1+K2; 1 ∗C1 +K2; 2 ∗ C2 +K2; 2 ∗ K2; d21 ∗ C1 ∗ C2

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K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338 16

0.00013

14

0.00012

12

0.00011 0.00010

Dm

q[g/l]

10 8

0.00009 0.00008

6 0.00007

4 0.00006

2 0.00005

0 0

1

2

3

4

5

6

0.00

7

Fig. 5. A comparison between competitive isotherm data and theoretical calculated from Eqs. (29) and (30). Upper line refers to S-1-indanol, lower line to R-1-indanol. Symbols represent theoretical data, solid line theory.

q2 =qs1 ∗ ×

K1; 2 ∗ C2 + qs2 ∗ 1+K1; 1 ∗ C1 +K1; 2 ∗ C2

K2; 2 ∗C2 ∗(1+K2; d21 ∗C1 ) : 1+K2; 1 ∗C1 +K2; 2 ∗C2 +K2; 2 ∗K2; d21 ∗C1 ∗C2

0.02

0.04

0.06

0.08

0.10

0.12

Lf

C[g/l]

(30)

The value of K2; d21 estimated from the competitive data is 0:028 ± 0:0044 and the Fisher coeDcient is equal to 2007. This result suggests that the interpretation suggested above is plausible. Fig. 5 compares the competitive isotherm data and the best isotherms in Eqs. (29) and (30). Very similar agreement was obtained assuming any other of several possible associations of R-indanol on S-indanol or S-indanol on R-indanol on both active sites. The isotherm model expressed by Eqs. (29) and (30) was used to solve the GR-GSM model. To calculate solutions of this model, i.e., high concentration elution band pro&les, the surface di8usion coeDcients de&ned by Eqs. (14)–(18) must be calculated. These coeDcients are functions of the concentrations of the two solutes in the mobile phase that is inside the pores of the particles, Cp , and on their concentrations in the adsorbed phase, q. The concentration q in Eqs. (14)–(17) is the concentration in the adsorbed phase when in equilibrium with the concentration Cp in the liquid &lling the pores; it is given by Eqs. (29) and (30) when Cp is known. The calculation of q is done using the isotherm parameters in Table 2b. The fractional surface occupancy of species i is the quotient of its adsorbed phase concentration and the total saturation capacity, qs; 1 + qs; 2 , also given in Table 2b. 4.2. Validation of the isotherm model 4.2.1. Single component peaks proBles Previously (Zhou et al., 2003), the POR model was used to predict the separation of the 1-indanol enantiomers. There was an excellent agreement between experimental and

Fig. 6. The dependence of Dm on the loading factor, Lf , for S-1-indanol.

calculated band pro&les. However, this good agreement was achieved only when a di8erent molecular di8usion coeDcient was selected for each component and for each sample concentration, a result that proves that the mass transfer resistances in the particles are a function of the solute concentration and suggests a model error in accounting for the mass transfer kinetics in this earlier work, due to the use of too simple a model. We now make a more detailed analysis of the mass transfer kinetics of the 1-indanol enantiomers in the adsorbent particles. First, we optimized the molecular di8usion coeDcient, Dm , inside the pores in order to obtain the best possible agreement between the calculated and experimental band pro&les. Obviously, the values obtained are actually apparent di8usion coeDcients because in the process of their derivation the contributions of other mass transfer processes, e.g., surface di8usion, were neglected. For the pro&le calculations, we used the POR model and neglected the external mass transfer resistance because, as was demonstrated previously, these resistances have a negligible in9uence on the band pro&les unless the mass transfer kinetics is very fast. Figs. 6 and 7 illustrate the dependency of Dm on the loading factor, Lf , for R-1- and S-1-Indanol. The loading factor Lf is given by the relation: n Lf = ; (31) (1 − T )SLqs where n is the sample size, T the total column porosity, S the column cross-section area, L the column length and qs the total saturation capacity of the stationary phase. There is an important scatter of the data points in these two &gures. We found that a signi&cant reason for this scatter originates from errors made in the estimate of the amounts of sample injected. Comparisons of the peak areas and the estimated injected amounts suggested an average relative error of about 5%. This error can arise from any combination of an imperfect mixing of the two liquid streams in the mixing chamber, 9uctuations in the mobile phase velocity and errors of measurements made in the calibration curves.

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338 0.00012

0.00013

0.00011

0.00012

0.00010

0.00011 0.00010

Dm

Dm

0.00009 0.00008

0.00009 0.00008

0.00007

0.00007

0.00006

0.00006

0.00005

0.00005

0.00

0.02

0.04

0.06

0.08

0.00

0.10

0.02

0.04

Fig. 7. The dependence of Dm on the loading factor, Lf , for R-1-indanol.

0.08

0.10

Fig. 9. Dependence of Dm on the corrected value of the loading factor, Lf , for R-1-indanol. Table 3 Feed concentration, calculated injection time, and molecular di8usivity for R-1-indanol

0.00011 0.00010 0.00009 0.00008 0.00007 0.00006 0.00005 0.00004 0.00

0.06

Lf

Lf

Dm

2333

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Lf Fig. 8. Dependence of Dm on the corrected value of the loading factor, Lf , for S-1-indanol.

So, we recalculated the loading factors by estimating the sample size n from the following equation:  ∞ Cu dt; (32) n=S 0

where C is the concentration of the enantiomers in the eluent at the column outlet. Figs. 8 and 9 show the dependence of Dm on the corrected value of the loading factor, Lf , for both enantiomers. The relationship obtained between Dm and the corrected value of the loading factor (Figs. 8 and 9) seems more realistic than the one without correction (Figs. 6 and 7). In the following, all further calculations are based on corrected loading factor data. In practice, this means that the band pro&le are calculated with the same values of the column characteristics, mobile phase 9ow velocity, and feed concentration as noted but that the injection time is adjusted in proportion to the correction of the loading factor. We report in Table 3 the feed concentration, calculated (corrected) injection time and molecular di8usivity Dm for R-1-Indanol. From the data presented in Figs. 8 and 9 and in Table 3, we conclude that

C0 (g/l)

Lf (dimensionless)

tinj (min)

Dm (cm2 =min)

1.80502 1.80502 1.80502 1.80502 1.80502 3.61004 3.61004 3.61004 3.61004 3.61004 6.31757 6.31757 6.31757 6.31757 6.31757 9.02510 9.02510 9.02510 9.02510 9.02510 18.05020 18.05020 18.05020 18.05020 18.05020

0.00183 0.00372 0.00555 0.00739 0.00992 0.00385 0.00747 0.0111 0.0148 0.0185 0.00637 0.0127 0.0196 0.0263 0.0332 0.0099 0.0192 0.0287 0.0383 0.0477 0.0196 0.039 0.0583 0.0776 0.097

0.188 0.383 0.57 0.76 1.02 0.198 0.384 0.573 0.759 0.95 0.187 0.374 0.575 0.771 0.977 0.203 0.396 0.59 0.788 0.982 0.202 0.401 0.6 0.798 0.997

1:25 × 10−4 1:16 × 10−4 1:09 × 10−4 9:92 × 10−5 9:78 × 10−5 1:15 × 10−5 1:06 × 10−5 9:78 × 10−5 8:79 × 10−5 7:8 × 10−5 9:78 × 10−5 8:22 × 10−5 7:51 × 10−5 6:95 × 10−5 6:38 × 10−5 9:21 × 10−5 7:51 × 10−5 6:80 × 10−5 6:38 × 10−5 5:95 × 10−5 7:23 × 10−5 6:1 × 10−5 5:81 × 10−5 5:53 × 10−5 5:1 × 10−5

the molecular di8usivity Dm decreases strongly with increasing sample concentration. However, there are no fundamental reasons to explain a dependence of the bulk molecular di8usivity on the concentration. This is all the more surprising because the concentrations of 1-indanol enantiomers in the mobile phase in all our experiments are low (the largest was 2%, w/w). It seems that the trend observed, an increase of the mass transfer resistances with increasing concentration, can be explained by assuming that surface di8usion plays a dominant role in the mass transfer inside the adsorbent particles.

2334

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338 Table 4 Estimated values of Dso , n, and *—see Eqs. (35) and (37)

14

12

R-1-indanol

C[g/l]

10

8

6

4

Parameters of Eq. (35) Dso (cm2 =min) 1:47 × 10−5

n 3

Parameters of Eq. (37) Dso (cm2 =min) 1:65 × 10−5

* 0.029

2

0 4

6

8

10

12

14

t [min]

Fig. 10. Comparison of experimental (symbol) and calculated (solid lines) band pro&les for S-1-indanol, for values of the loading factor equal to 12.9%, 5.0%, and 0.20%, respectively. To make possible the comparison of bands with so di8erent loading factors, the concentrations of the far right peak were multiplied by 50 and those of the central peak by 6.

4.2.1.1. Single component kinetic mass transfer In the following we assumed that the molecular 9ux by di8usion through the bulk solution inside the particles can be neglected in comparison with the surface molecular 9ux. So, for single component chromatography, the surface 9ux reads: Ns = −

qDs @Cp : Cp @r

(33)

Eq. (33) follows directly from Eqs. (12) and (14). The surface di8usion coeDcient was adjusted in order to obtain the best possible agreement between the experimental and the calculated band pro&les. We have obtained for S-1-indanol a value of Ds equal to 7:3 × 10−5 cm2 =min. Fig. 10 illustrates typical results obtained in the comparison of experimental and calculated pro&les for very di8erent sample sizes. However, in the case of R-1-indanol, the surface di8usion coeDcient, Ds , turned out to decrease with increasing concentration (data not shown). The variations of the surface di8usion coeDcient with the concentration can be explained in two di8erent ways. (1) Mechanistically, the Maxwell–Stefan di8usivity is related to the average displacement of the adsorbed molecular species when it jumps from one site to another, (, and to the jump frequency, ), with Ds =

1 2 ( ); z

(34)

where z is the number of nearest neighbor sites. The two parameters in Eq. (34) can be expected to depend on the total surface coverage (Reed & Ehrilch, 1981a, b; Riekert, 1971; Zhdanov, 1985). The jump frequency can be expected to decrease with increasing occupancy of the surface because it is usually assumed that a molecule can migrate from one site to another one only if this other site is vacant. The

probability that this occurs decreases with increasing fraction of occupied sites. In this work we assumed that the surface di8usion coef&cient is given by the following equation: Ds = Dso (1 − )n ;

(35)

where n is the number of nearest neighbor sites. This model was developed for gas–solid adsorption. However, if the adsorption of the mobile phase components on the stationary phase is much weeker than that of the compound studied, the model accounted for by Eq. (35) can be used also for liquid–solid adsorption. (2) Miyabe and Guiochon (Miyabe & Guiochon, 1999, 2000) assumed that the surface di8usion coeDcient is proportional to the isosteric heat of adsorption, st , according to the following equation:   +| st | ; (36) D = Dso exp − RT where + is an empirical parameter. Assuming that the isosteric heat of adsorption is a function of the surface concentration of the compound studied, the surface di8usion coeDcient is given by the following relationship  * ∗ q : (37) D = Dso exp − RT Miyabe and Guiochon (1999, 2000) suggested also that the parameter * should be less than zero because the isosteric heat of adsorption is expected to decrease with increasing surface coverage. However, an increase of the isosteric heat of adsorption with increasing surface coverage has been reported in some cases for the adsorption of gases (Kelcev, 1976). We tested both Eqs. (35) and (37). In each case, it was possible to &nd values of the parameters Dso , n, and * for which the agreement between the calculated and the experimental band pro&les for R-1-indanol was similar to the one shown in Fig. 10 for S-1-indanol. The best values determined for Dso , n, and * are reported in Table 4. Note that the two estimates of Dso di8er by only approximately 10%. The excellent agreement between the calculated and the experimental pro&les obtained in the single-component case con&rms the validity of the proposed Bilangmuir adsorption mechanism.

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

2335

8 4.0

7 3.5

6 3.0

5

C[g/l]

C[g/l]

2.5 2.0

4 3

1.5

2

1.0

1

0.5

0

0.0 0

2

4

6

8

(A)

10

12

14

16

0

18

2

4

6

8

10

(B)

t [ min ]

12

14

16

18

20

t [ min ]

12

5

10

4

C[g/l]

C[ g / l ]

8 3

2

6

4 1

2

0

0 0

2

4

6

8

10

14

16

18

20

0

4

6

8

5

8

4

6

3

4

10

12

14

16

18

20

t [ min ]

10

2

2

1

0

0 0

(E)

2

(D)

t [ min ]

C[g/l]

C[g/l]

(C)

12

2

4

6

8

10

12

14

16

18

0

20

2

4

6

8

(F)

t [ min ]

10

12

14

16

18

20

t [ min ]

8 7 6

C[g/l]

5 4 3 2 1 0 0

(G)

2

4

6

8

10

12

14

16

18

20

t [ min ]

Fig. 11. Comparison of the calculated (solid lines) and experimental (symbols) pro&les for the overloaded band pro&les of di8erent R- and S-1-indanol mixtures and isotherm calculated from Eqs. (29) and (30). The numerical parameters used in the calculations and the experimental conditions are given in Table 5.

2336

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

Table 5 Corrected loading factor, Lf , injection time, tinj , and experimental conditions for the elution of large samples of di8erent 1-indanol mixtures

A B C D E F G

5

Experimental conditions C(g=l)(R + S)

tinj (min)

Lf (%)

7.131 20.37 20.37 20.37 20.37 20.04 20.37

1 0.41 0.61 0.8 1 0.54 0.54

3.8 4.5 6.7 8.8 10.9 5.8 5.9

R R R R R R R

: S = 1:1 : S = 1:1 : S = 1:1 : S = 1:1 : S = 1:1 : S = 3:1 : S = 1:3

4

C[g/l]

Fig. 11

6

3

2

1

0 0

2

4

6

8

10

12

14

16

18

20

t [ min ]

4.2.2. Competitive kinetic mass transfer For a &nal test of the validity of the mechanisms of adsorption and of the mass transfer kinetics proposed in this work, we compared the band pro&les of the binary mixture and those calculated using the GR-GMS model—see Eqs. (1)–(8), (12)–(18) and (37). This model assumes that R-1-indanol can adsorb on S-1-indanol when the latter is adsorbed on the high energy sites, as described by Eqs. (29) and (30). The converse is not true, S-1-indanol does not adsorb on R-1-indanol when the latter is adsorbed on the high energy sites. Figs. 11a–g compare the experimental and the calculated band pro&les of the racemic mixture of 1-indanol. The experimental parameters corresponding to these di8erent &gures are listed in Table 5. These &gures validate the isotherm model because of the excellent agreement between the calculated and experimental values of retention times of the front shock layers—which should be the same as those of the shocks in the ideal model (Guiochon et al., 1994)—and of the rear di8use boundaries. Note that this agreement was expected in the case of the racemic mixture since the model of competitive isotherm used was established by analysing the frontal analysis data for this mixture (Figs. 3 and 5). For further veri&cation of the model, the experimental and calculated band pro&les of large samples of 1:3 and 3:1 1-indanol enantiomer mixtures (Figs. 11f, g) were compared. In all cases the agreement between the two sets of pro&les is excellent. It must be emphasized that such an agreement was obtained using isotherm and kinetic parameters estimated from single component data, except for the equilibrium constant characterizing the adsorbate— adsorbate interactions between R- and S-1-indanol, constant that was estimated from the competitive adsorption isotherm data. We also checked whether some other possible isotherm models could account for the experimental pro&les as well as or possibly better than the one described earlier and used in the calculations leading to the results shown in Figs. 11a– g. For example, we designed competitive models assuming the adsorption of R-1-indanol on S-1-indanol previously adsorbed on the low energy sites or on both the low and

Fig. 12. As Fig. 10f but assumed of S-1-indanol absorption on previously adsorbed R-1-Indalon on high energy active sites.

the high energy sites. In either case, the accuracy of the approximation of the experimental peak pro&les of the binary mixtures was generally very similar to the one seen in Figs. 11a–g, albeit slightly less good. On other hand, assuming a possible adsorption of S-1-indanol on previously adsorbed R-1-indanol gives in general calculated pro&les in a rather poor agreement with the experimental pro&les, as exempli&ed in Fig. 12. All the results obtained in Figs. 11a–12 were calculated using Eq. (37) to account for the contribution of surface di8usion to the mass transfer kinetics. By contrast and in general, we did not obtain a good agreement between the experimental band pro&les of binary mixtures and the calculated pro&les when estimating the surface di8usion coef&cient using Eq. (35).

5. Conclusions This work has two important consequences. First it shows that the adsorption mechanism of enantiomers at high concentrations may be more complex than we have tended to believe so far. In the case studied here, the separation of the enantiomers of 1-indanol by preparative liquid chromatography, the Bilangmuir isotherm model seemed at &rst to &t the experimental data very well (Zhou et al., 2003), as it did in many other enantiomeric cases (Fornstedt, Sajonz, & Guiochon, 1998). In this case, however, it was not possible to associate the high energy sites only to the enantiomeric separation, the low energy sites being achiral. Furthermore, there was a small but signi&cant di8erence between the best values of the numerical coef&cients of the bilangmuir isotherm derived from the single-component and from the binary or competitive set of data (Zhou et al., 2003). The reexamination of the raw data showed that cooperative adsorption takes place to a modest but signi&cant degree in the case of the 1-indanol

K. Kaczmarski et al. / Chemical Engineering Science 58 (2003) 2325 – 2338

enantiomers. Albeit markedly less important, this e8ect is similar to the one already reported in the case of the competitive adsorption of the enantiomers of TrBoger’s base on amylose tri-(dimethyl carbamate) (Mihlbachler et al., 2002). This stationary phase has a structure and properties that are not unlike those of the cellulose tribenzoate used in the present work. In both cases, the less retained enantiomer can form weak associations with the more retained one when the latter is already adsorbed. This illustrates the complexity of chiral interactions and opens new avenues of investigation. Second, considerable progress have been made recently in the understanding of mass transfer kinetics in reversed phase liquid chromatography. The agreement observed between the experimental overloaded elution band pro&les of samples of pure R-1-indanol and S-1-indanol of their mixtures and the pro&les calculated (Figs. 11a–g) is excellent. This agreement was achieved by using for the calculations the general rate model of chromatography coupled with the general Maxwell–Stefan surface di8usion model. The Bilangmuir isotherm model discussed above was used. However, the column eDciency was moderate and the agreement observed was due to the use of the competitive GR-GMS model of mass transfer kinetics (Kaczmarski et al., 2002) and of the restricted di8usion model (Miyabe & Guiochon, 1999, 2000) to calculate the coeDcient of surface di8usion. This demonstrate the importance of surface di8usion in RPLC (see Eq. (37)). Acknowledgements This work was supported in part by Grant CHE-00-70548 of the National Science Foundation, and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. The authors are grateful to Chiral Technologies (Exton, PA, USA) for the generous gift of Chiracel OB stationary phase. References Armstrong, D. W., Gahm, K. H., & Chang, L. W. (1997). Synthesis, enantioselective separation, and identi&cation of racemic tetraline, indan, and benzosuberan derivatives. Microchemical Journal, 57, 149–165. Berninger, J. A., Whitley, R. D., Zhang, X., & Wang, N. H. L. (1991). A versatile model for simulation of reaction and nonequilibrium dynamics in multicomponent &xed-bed adsorption process. Computers and Chemical Engineering, 15, 749–768. Brown, P. N., Hindmarsh, A. C., & Byrne, G. D. (1989). Variable coeDcient ordinary di
2337

Charton, F., Jacobson, S. C., & Guiochon, G. (1993). Modeling of the adsorption behavior and the chromatographic band pro&les of enantiomers. Behavior of methyl mandelate on immobilized cellulose. Journal of Chromatography, 630, 21–35. Cherrak, D. E., Khattabi, S., & Guiochon, G. (2000). Adsorption behavior and prediction of the band pro&les of the enantiomers of 3-chloro-1-phenyl-1-propanol. In9uence of the mass transfer kinetics. Journal of Chromatography A, 877, 109. Edge, A. M., Heaton, D. M., & Bartle, K. D. (1995). Computational study of the chromatographic separation of racemic mixtures. Chromatographia, 41, 161–166. Fletcher, R. (1971). A modiBed Marquardt sub-routine for nonlinear least squares. AERE-R6799-Harwell. Fornstedt, T., Sajonz, P., & Guiochon, G. (1998). A closer study of chiral retention mechanisms. Chirality, 10, 375. Guiochon, G., Shirazi, S. G., & Katti, A. M. (1994). Fundamentals of preparative and nonlinear chromatography. Boston: Academic Press. Hampe, Th. R. E., SchlButer, J., Brandt, K. H., Nagel, J., Lamparter, E., & Blaschke, G. (1993). Rapid and complete chiral chromatographic separation of racemic quaternary tropane alcaloids. Complementary use of a cellulose-based chiral stationary phase in reversed-phase and normal-phase modes. A mechanistic study. Journal of Chromatography, 634, 205–214. Jacobson, S., Shirazi, S. G., & Guiochon, G. (1990). Chromatographic band pro&les and band separation of optical isomers at high concentration. Journal of the American Chemical Society, 112, 6492. Kaczmarski, K., & Antos, D. (1996). Modi&ed Rouchon and Rouchon-like algorithms for solving di8erent models of multicomponent preparative chromatography. Journal of Chromatography A, 756, 73. Kaczmarski, K., Antos, D., Sajonz, H., Sajonz, P., & Guiochon, G. (2001). Comparative modeling of breakthrough curves of bovine serum albumin in anion-exchange chromatography. Journal of Chromatography A, 925, 1. Kaczmarski, K., Cavazzini, A., Szabelski, P., Zhou, D., Liu, X., & Guiochon, G. (2002). Application of the general rate model and the generalized Maxwell–Stefan equation to the study of the mass transfer kinetics of a pair of enantiomers. Journal of Chromatography A, in press. Kaczmarski, K., Storti, G., Mazzotti, M., & Morbidelli, M. (1997). Modeling &xed-bed adsorption columns through orthogonal collocations on moving &nite elements. Computers & Chemical Engineering, 21, 641–660. Kapteijn, F., Moulijn, J. A., & Krishna, R. (2000). The generalized Maxwell–Stefan model for di8usion in zeolites: Sorbate molecules with di8erent saturation loadings. Chemical Engineering Science, 55, 2923. Kelcev, N. V. (1976). Osnovy adsorbcionnoj tekhniki. Moscow: Khimija. Krause, M., & Galensa, R. (1988). Direct enantiomeric separation of racemic 9avanones by high performance liquid chromatography using cellulose triacetate as a chiral stationary phase. Journal of Chromatography, 441, 417–422. Krishna, R. (1990). Multicomponent surface di8usion of adsorbed species: A description based on the generalized Maxwell–Stefan equations. Chemical Engineering Science, 45, 1779. Krishna, R. (1993a). Problems and pitfalls in the use of the &ck formulation for intraparticle di8usion. Chemical Engineering Science, 48, 845. Krishna, R. (1993b). A uni&ed approach to the modelling of intraparticle di8usion in adsorption process. Gas Separation and PuriBcation, 7, 91. Kunath, A., HBoft, E., & Hamann, H. (1991). Enantiomeric separation of racemic hydroperoxides and related alcohols. Journal of Chromatography, 588, 352–355. Kunath, A., Theil, F., & Wagner, J. (1994). Diastereomeric and enantiomeric separation of monoesters prepared from meso-cyclopentadienols and racemic carboxylic acids on a silica phase

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