Ind. Eng. Chem. Res. 2003, 42, 6171-6180
6171
Determination of the Competitive Adsorption Isotherms of Nadolol Enantiomers by an Improved h-Root Method Xin Wang* and Chi Bun Ching Chemical and Process Engineering Centre (CPEC), Block E5 Basement 08, National University of Singapore, 4 Engineering Drive 4, Singapore 117576
Nadolol, a β-blocker used in the management of hypertension and angina pectoris, has three chiral centers and is currently marketed as an equal mixture of its four stereoisomers. Resolution of three of the four stereoisomers of nadolol was obtained by HPLC, with a complete separation of the most active enantiomer, (RSR)-nadolol, on a column packed with perphenyl carbamoylated β-cyclodextrin (β-CD) immobilized onto silica gel. The h-root method without the introduction of dummy species was presented and applied to determine the nonlinear competitive Langmuir isotherms of the three components of nadolol. In this method, h-root transformation was applied directly to the n-component nonstoichiometric system, without introduction of dummy species. The experiments consist of linear elution and nonlinear frontal chromatography. The solid film linear driving force model was used to simulate the response of a column to a pulse injection in the nonlinear region. The experimental and simulated profiles matched well, which confirmed the validity of the obtained Langmuir isotherm coefficients. On the basis of the isotherm obtained, complete (1, 2) and (2, 3) simulated moving bed (SMB) separation regions were determined to separate the ternary mixture of nadolol into different fractions. 1. Introduction Nowadays more research efforts have been concentrated on the production of optically pure products due to increasing demand that such drugs are administered in optically pure form.1 Compared to the conventional approach of designing a stereoselective organic synthesis, preparative-scale chromatography is becoming widely used for the purification of enantiomers because of the improved availability of highly selective stationary phases and the low cost of this method.2-5 Nadolol, 5-{3[(1,1-dimethylethyl)amino]-2-hydroxypropoxy}-1,2,3,4tetrahydro-cis-2,3-naphthalenediol is a β-blocker drug widely used in the management of hypertension and angina pectoris. Its chemical structure has three stereogenic centers, which allows for eight possible stereoisomers. However, the two hydroxyl substituents on the cyclohexane ring are fixed in the cis configuration, which precludes four stereoisomers. Nadolol is currently marketed as an equal mixture of four stereoisomers, designated as diastereomers of “racemate A” and “racemate B”, as shown in Figure 1. Racemate A is a mixture of the most active stereoisomer I ((RSR)-nadolol) and its enantiomer II ((SRS)-nadolol) in a 1:1 molar ratio, whereas racemate B is a mixture of stereoisomer III ((RRS)-nadolol) and its enantiomer IV ((SSR)-nadolol) also in a 1:1 molar ratio. For safer and more effective use, it is better to separate the enantiomer (RSR)nadolol before use. The simulated moving bed (SMB) technology has demonstrated its considerable advantages with respect to traditional batchwise preparative chromatography methods.6-8 It is well-known that SMB separation performance depends primarily on the adsorption isotherms, which are the most important parameters for * To whom correspondence should be addressed. Telephone: 65-6874.2196. Fax: 65-6873.1994. E-mail: cpewx@ nus.edu.sg.
Figure 1. Chemical structures of stereoisomers of nadolol.
the proper design, operation, and modeling of SMB processes. Especially, for Langmuir type isotherms, constraints on the flow rate ratios (i.e., m1, m2, m3, and m4) in a SMB unit can be determined explicitly only by the adsorption isotherm and feed concentrations (in the frame of equilibrium theory).9 The determination of single-component isotherms by chromatographic methods has been extensively investigated.2,10-13 The methods of frontal analysis, frontal analysis by characteristic points, elution by characteristic points, elution of a pulse on a plateau, as well as elution of an isotopic pulse on a plateau have been used successfully in liquid chromatography. Since the SMB method is preferably conducted in a nonlinear region to achieve higher productivity, it is more important to determine the competitive adsorption behavior among the consisting species. When applied to determine multicomponent adsorption isotherms, only the singlesolute isotherms are determined experimentally and the competitive coefficients have to be predicted using suitable isotherm models (i.e., Langmuir, modified Langmuir, or bi-Langmuir isotherms) with limited accuracy.14 In the case of enantiomeric separations, the
10.1021/ie0303698 CCC: $25.00 © 2003 American Chemical Society Published on Web 10/23/2003
6172 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
pure individual isomers are expensive and, in most cases, not commercially available (as in the case of this work), so the applicability of single-component methods for the determination of the competitive adsorption coefficients is restricted. For a multicomponent system, two chromatographic procedures were commonly used for obtaining equilibrium data. They were the tracer pulse (TP) method proposed by Helfferich and Peterson15 and the elution on a plateau (EP) method (also named perturbation chromatography) introduced by Reilley et al.16 These two methods have been evaluated recently by Ma et al., who proposed a method for determination of binary competitive isotherms based on the so-called hodograph transform of the elution profiles obtained in response to a wide rectangular injection pulse.17 Jacobson et al. also contributed to the measurement of competitive adsorption isotherms by presenting two methods both based on frontal chromatography which are the method of mass balance (MMB) and the method of composition velocities (MCV).18 In other methods,19 experimental single-component and competitive adsorption data were measured by frontal analysis and fitted to different isotherm models by a nonlinear regression technique. Essentially the competitive adsorption data measured by racemic mixtures suffice to obtain the competitive bi-Langmuir or Langmuir model parameters. This feature makes it a good alternative to determine the competitive adsorption isotherm for enantiomeric separation regardless of the complication of the regression procedure. On the basis of the coherence theory of chromatography, Jen and Pinto modified the h-root method20,21 for measuring the competitive Langmuir coefficients in an attempt to minimize experimental effort. Advantages of the h-root method for determination of nonlinear competitive isotherms of racemates include the following: pure enantiomers are not required and the needed amount of racemate sample is relatively low because, in principle, only one frontal chromatography experiment is sufficient to determine the competitive bcoefficients. Furthermore, the complete effluent composition history is superfluous because only the breakthrough time for the frontal boundaries is needed. However, the introduction of a fictitious species (dummy species) was needed in their method development. It is worth noting that the introduction of the dummy species was originally proposed by Helfferich with the purpose of establishing the equivalence of an n-component nonstoichiometric adsorption system (i.e. Langmuir competitive isotherm) with a (n + 1)-component system characterized by stoichiometric ion exchange. Furthermore, on the basis of this established equivalence, h-transformation of concentration variables has been applied to Langmuir isotherm systems directly (without the dummy species) and was used to deduce chromatographic behavior knowing the equilibrium isotherms.22 Thus, the necessity of introducing the dummy species in the method proposed by Jen and Pinto remains to be investigated. On the other hand, for the three chiral center drug nadolol, no work has been done to determine the competitive adsorption isotherms on a perphenyl carbamoylated β-cyclodextrin bonded chiral stationary phase. This is probably due to the difficulty of the enantiomeric separation of a multichiral center drug like nadolol as well as the complicated nature of determination of the multicomponent isotherm.
In this paper, the h-root transformation is applied directly to the n-component chromatographic system, without introduction of the dummy species. Detailed derivation of the transformation without the aid of dummy species is presented, and it will be used to determine the nonlinear competitive Langmuir coefficients of the three stereoisomers of nadolol. The validity of the isotherms obtained is testified by comparing the experimental and simulated elution profiles in the nonlinear region. Last, different complete separation regions were determined for the four-zone simulated moving bed (SMB) process to separate the ternary mixture of nadolol into different fractions, for example, pure component 1 in the raffinate and a binary mixture of components 2 and 3 in the extract for the (1, 2) complete separation regime and pure component 3 in the extract and a binary mixture of components 1 and 2 in the raffinate for the (2, 3) complete separation, respectively. 2. Theoretical Section 2.1. Adsorption Isotherms. The nonstoichiometric Langmuir isotherms can be expressed as
q/j )
ajcj n
1+
(1)
bici ∑ i)1
The ai are measures of the intrinsic affinities of the respective species for the sorbent, and the bi are characteristic of the nature and strength of the interference produced by the species. It is worth noticing that the linear isotherm and stoichiometric Langmuir isotherm can be regarded as particular cases of the nonstoichiometric Langmuir model. 2.2. Methodology of the Improved h-Root Method. The h-root method is based on nonlinear wave theory and the coherence principle (Helfferich has conducted pioneering studies in this area,23 and no attempts were made to outline the theory in detail here). The assumptions of ideal chromatography (e.g. local equilibrium between the stationary and mobile phases, ideal plug flow, no dispersion, isothermal behavior of the column) are taken for granted. According to the coherence theory, a chromatographic column exposed to a disturbance will, after a certain time, come back into a stable state which is called coherence. In the coherence state, concentration velocities of all species have to be equal:
vCi ) vCj
(2)
The wave velocity refers to the propagation of dependent variables (i.e., concentrations) with distance and time, as distinct from the species velocity, which refers to travel of molecules. Wave velocities can be derived from a differential mass balance of component i:
v Ci )
v dCS,i 1+F dCi
(3)
The coherence condition defines composition path grids to which the system is restricted. For adsorption
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6173
Figure 2. Schematic representation of column behavior in h-space for frontal chromatography for an n-component mixture.
isotherms described by the Langmuir approach, the concentration path grid consists of straight lines and can be orthogonalized by a nonlinear transformation.23 In an n-component system, the n transformed variables, hi, are obtained as the n-roots of a polynomial called the H function: n
bici
∑ i)1ha
i
-1
)1
(4)
Equation 4 replaces the concentrations of a multicomponent system by the roots hi of a polynomial. The roots of the polynomial are restricted to the intervals as
1 1 1 1 < h1 < < h2 < < ... < < hn a1 a2 a3 an If the concentration of a solute j is zero, the Cj term of the solute in the H function is zero, so that eq 4 has one root less. The corresponding missing h-root is equal to
hj )
1 if cj ) 0 (1 e j e n) aj
other individually, since a variation in one h-root can change several concentrations and vice versa. Both of them describe the stream compositions. The h-root method can be applied to nonlinear frontal analysis, where the equilibrated and solute free column is disturbed by a step change in the concentration of the solutes in the mobile phase. Figure 2 shows a schematic representation of the column profile for frontal development. With n solutes, there may be at most n waves partitioned by n - 1 composition plateaus. The horizontal lines represent sharp waves, and the compositions of the plateau regions are expressed in terms of h-roots. Primes and double primes indicate the upstream and downstream compositions of the waves, respectively. It is worth noticing that across each wave only one h-root varies, which is an important advantage of h transformation. In Figure 2, the plateau compositions in terms of hi (noting both hi and Ci describe the plateau compositions) can be given entirely by the h-roots of the entering and initial fluid compositions, h′i and h′′i, respectively. The column contains no adsorbates at the beginning, and h′′ of the initial fluid can be readily calculated from eq 5:
(5)
The reverse transformation from h-roots to concentrations was also derived.23 It should be noted that h-roots (hi) and concentrations (Ci) cannot be compared to each
h′′j )
1 j ) 1, 2, ..., n aj
(6)
In ideal chromatography, shocks (defined as selfsharpening waves) are ideally sharp concentration steps
6174 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
(i.e., concentration discontinuities). Equations for adjusted wave velocities of the shocks in terms of h-roots were originally derived by Helfferich:22,23 n
n
∏ ∏
Uj ) v h′jh′′j hi ai 1- i)1 i)1
(7)
where the adjusted wave velocities Uj of the shocks can be calculated from the adjusted emergence time:
L τj
(8)
The adjusted breakthrough time τj can be calculated from the real emergence time Tj of the shocks:
τ j ) Tj -
L v
(9)
Also, the relation between true velocities, uj, and adjusted velocities, Uj, can be derived from eq 9:
L ) uj ) Tj
v 1+
v Uj
(10)
For the h′i of the entering fluid, it can be derived from eq 7 that
1- Un v
(11)
1 Uj j ) 1, 2, ..., n - 1 aj+1 Uj+1
(12)
h′n ) h′j )
Now we define the frontal capacity factor K′i as
K′i )
Ti - T0 T0
(13)
Substituting eq 13 into eq 10 gives
Uj )
v K′j
(14)
Combining eq 14 with eqs 11 and 12, respectively, we obtain the following equations:
1 1- j)n K′n
(15)
1 K′j+1 j ) 1, ..., n - 1 a′j+1 K′j
(16)
h′n ) h′j )
Up to now, we obtain the expressions of the h space for the feed compositions in eqs 15 and 16. Substituting these equations into eq 4 gives n
∑ i)1
( ) cfi
k′i
K′n
-1
bi ) 1
( )
∑ i)1 k′K′ i
cfi
j+1
k′j+1K′j
bi ) 1 j ) 1, 2, ..., n - 1
(18)
-1
The elution capacity factor k′i in eqs 17 and 18 can be calculated from the linear elution experiments:
i*j
Uj )
n
(17)
k′i )
1- ai
(19)
Now all the terms in eqs 17 and 18 are known or can be experimentally determined, except that of the Langmuir competitive adsorption coefficients bi. Thus, n equations can be used to determine the unknown bi (i ) 1, 2, ..., n). It is worth noticing that, compared with the case of the modified h-root method proposed by Jen and Pinto,20 the (n + 1) component (also known as the dummy species) was not introduced in this method. Thus, the expressions for wave velocity and the h-root of the entering fluid (h′i, i ) 1, 2, ..., n) do not contain this species. The dummy species, first introduced by Helfferich,23 was aimed to make an n-component nonstoichiometric adsorption system equivalent with a (n + 1) component system with stoichiometric exchange and constant total concentrations in both phases. For example, a column receiving influents of different total concentrations of the physically existing species receives, in the equivalent stoichiometric system, influents of the same total concentrations (an arbitrary mathematical constant) but with varying proportions of the dummy species. It is not surprising that the same final expressions were obtained in this study as those of Jen and Pinto’s method; however, the application of the latter one needed the introduction of a dummy species and therefore rendered unnecessary complications, since the equivalence of an n-component nonstoichiometric Langmuir adsorption system with an (n + 1)-component system with stoichiometric exchange has been established. Obviously, the method presented in this paper is simpler and clearer because the introduction of the confusing (the dummy species is assumed to be the least retained one and is sometimes confused with the displacer in displacement chromatography) and unnecessary dummy species is avoided. The method is termed as the improved h-root method for distinction from the modified h-root method proposed by Pinto et al. 2.3. Lumped Kinetic Model. In this study, the solid film linear driving force model was used to account for the mass transfer kinetics. In this model, the mass transfer resistance is assumed to be located in the thin solid film in which the stationary phase concentration varies from the equilibrium concentration q/i at the contacting surface to the stationary phase average concentration CS,i. Since the mass transfer rate is believed to be proportional to the concentration difference, the rate equation is written as
∂CS,i ) ki(q/i - CS,i) ∂t
(20)
The differential mass balance equation is as follows:
∂CS,i ∂Ci ∂Ci ∂2Ci +F +v ) DL 2 ∂t ∂t ∂z ∂z
(21)
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6175
With suitable initial and boundary conditions, eqs 20 and 21 constitute the basis to calculate the eluted band profiles. For the solid film linear driving force model, the corresponding expressions for the first and second moments are24
µ1 ) µ2 )
{
1- L 1+ K v
[ (
)]
(22)
}
2L DL 1- 2 1-K 1+ K + 2 v v k
[ (
)] (
)
( ) (
1 L 2DL ) + 2v 1+ N v 1 - kK (1 - )K
)
-2
Straight line wr,err,e: m2 )
Curve rr,ede: m2 )
(ωFe < ω < ae)
E′(ar,ω)
(25)
E′′(ω,ωFe ) E′(ω,ωFe )
(an < ω < ωFe ) (26)
E′′(ω,ω) (ωFe < ω < ae) (27) E′(ω,ω)
Point cr: m2 ) m3 ) ar
(28)
Point de: m2 ) m3 ) ae
(29)
Optimal point wr,e: m2 )
E′′(ωFe ,ωFe ) , E′(ωFe ,ωFe ) m3 )
Point rr,e: m2 )
1 + E′′(ωFe ,ωFe ) E′(ωFe ,ωFe )
∑ i)e a
-ω
i
aibiCFi
n
E′(ω1,ω2) )
∑ i)e (a
- ω1)(ai - ω2) biCFi
n
The equilibrium constants as well as the axial dispersion coefficient and the overall mass transfer coefficients for the chromatographic enantiomeric separation can be evaluated by the well-proved method of moment analysis (from eqs 22 and 24, respectively) on the basis of solid film linear driving force model. 2.4. SMB Process Conditions for Ternary Separation. Unlike binary separation, the complete separation of n (n > 2) component mixtures by the four-zone SMB method implies that the solutes are separated into two fractions; that is, components 1 to r are collected in the raffinate whereas components e (e ) r + 1) to n are collected in the extract. For the complete separation of a ternary mixture, two regimes exist which are r ) 1, e ) 2 and r ) 2, e ) 3. The parametric equations of the boundaries of the complete (e, r) separation region and the coordinates of the intersection points have been obtained on the basis of equilibrium theory25 (Refer to Figure 5 for the definition of boundaries and intersection points):
E′′(ar,ω)
E(ω) )
(32)
(33)
(23)
(24)
Straight line crwr,e: m2 )
biCFi
n
i
The expression for HETP was derived from the moment analysis of the solution of the general rate model in the Laplace domain:24
HETP )
where the function E(ω) and its first and second derivatives E′(ω) and E′′(ω) are defined as
(30)
E′′(an,ωFe ) 1 + E′′(ar,ωFe ) , m ) 3 E′(an,ωFe ) E′(ar,ωFe ) (31)
E′′(ω1,ω2) ) ω1ω2 n
ω
∑ i)e (a
biCFi
∑ i)1a
i
-ω
i
- ω1)(ai - ω2)
)1
(34)
(35)
In these equations, ωi (i ) 1, ..., n) are the roots of eq 35, which define a one-to-one mapping between the space of the fluid phase concentrations and the vectors ωi (i ) 1, ..., n) in the n-dimensional spaces. It is worth noting that the ω-transformation introduced by Rhee26 rather than the h-transformation discussed early in the paper is adopted by Migliorini et al. In fact, despite differences in terminology and scope of these two approaches, they are mathematically equivalent and the procedures of their application are identical.27 It is noted that, knowing the Langmuir isotherm and feed concentration, a complete separation region for any ternary mixture can be constructed in the (m2, m3) plane for both (1, 2) and (2, 3) separation regimes. 3. Materials, Instrumentation, and Experimental Procedure 3.1. Materials. HPLC-grade methanol was obtained from Fisher Scientific (Leics, U.K.). Glacial acetic acid and triethylamine were obtained from Merck (Schuchardt, Germany). HPLC water was made in the laboratory using a Millipore ultrapure water system. 1,3,5-Tri-tert-butylbenzene (TTBB) was purchased from Aldrich (USA). The racemate mixture of nadolol was purchased from Sigma (St. Louis, MO). All purchased products are used without further purification. 3.2. Apparatus. The experiments were carried out with a Shimadzu SCL-10AVP chromatographic system (Kyoto, Japan). The system consists of two LC-10ATvp pumps (A and B), an online degasser DGU-14A, a SILI0ADvp autoinjector, and a SPD-10Avp UV-vis detector. The software CLASS-VP 5.032 was used to control the system and record the detector signal. The column (250 mm × 4.6 mm i.d.) was packed with 15 µm spherical silica gel on which perphenyl carbamoylated β-CD was covalently bonded in our lab. The experiment was conducted at room temperature (about 24 °C). 3.3. Experimental Procedure. All linear elution and nonlinear frontal experiments were carried out in the reversed phase mode. The mobile phase was a binary mixture of methanol and acetate buffer (triethylamine aqueous solution adjusted by acetic acid to the desired pH value, briefly TEAA). The optimum mobile phase composition was found previously,28 which contains 80% buffer solution (1% TEAA, pH ) 5.5) and 20% methanol. The mixed solvents (mobile phase) were delivered by pump A until the column was equilibrated
6176 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
Figure 3. Experimental nonlinear frontal chromatogram (concentration ) 0.18 mg/mL; mobile phase flow rate ) 0.5 mL/min).
with the mobile phase (as monitored in the UV detector). After that, a step change in the concentration of nadolol over the mobile phase was introduced into the column by pump B (pump A stopped and pump B started simultaneously, realized by the time program). TTBB was dissolved in methanol at a concentration of 0.13 mg/ mL, which was used to determine the column hold up time. 33.75 and 44.67 mg of racemic nadolol (weighed with a Mettler Toledo AG 135 balance) were dissolved in 250 mL of mobile phase to make sample solutions at the concentrations 0.14 and 0.18 mg/mL, respectively, both in the nonlinear region of the adsorption isotherm. TTBB and nadolol solutions were filtered with a vacuum filtration device using a 0.45 µm size membrane and degassed in a model LC 60H ultrasonic bath before the experiments. Signals were detected with the UV detector with the wavelength set at 220 and 280 nm for TTBB and nadolol, respectively. 4. Results and Discussions 4.1. Determination of Competitive Langmuir Coefficients. The bed voidage, axial dispersion coefficient, overall mass transfer coefficients, and equilibrium constants (the Langmuir coefficient a) for the chromatographic enantiomeric separation were evaluated by moment analysis on the basis of a solid film linear driving force model in a previous study.28 The equilibrium constants were found to be 2.94, 3.46, and 5.45 for (SRS)- and (SSR)-nadolol, (RRS)-nadolol, and (RSR)-nadolol, respectively, which correspond to the first, second, and third eluted peaks. Their overall mass transfer coefficients were found to be 669.3, 846.3, and 106.4 min-1, respectively. The bed voidage was 0.58. Among these parameters, the mass transfer coefficients deserve more discussion. It has been demonstrated that the contributions of the axial dispersion and of the other mass transfer processes are additive in both linear and nonlinear chromatography when assuming the solid film linear driving force model. The contributions of each mass
transfer process, that is, (1) the fluid-to-particle mass transfer, (2) the intraparticle diffusion, and (3) the adsorption/desorption, are lumped into only one kinetic parameter, k, which is a considerable simplification. Although the mass transfer coefficients k have been reported to be not slightly concentration dependent in some system,29 it was found in other cases that k increases only moderately with increasing concentration, for example, of S-Troger’s base in a chromatographic system of ethanol and microcrystalline cellulose triacetate used as mobile and stationary phases, respectively.30 In that work, it was found that when the solute concentration increases 20-fold, the mass transfer coefficient k increases only by a factor of about 1.6. It is obvious that the concentration dependence of the mass transfer coefficient depends on the solute-mobile phasestationary phase system. In this study, the lumped mass transfer coefficients obtained in the linear range were used for the modeling and simulation of the nonlinear elution chromatography, since mass transfer resistance contributes mainly to the diffusion of peaks rather than their position. In frontal chromatography experiments of nadolol, the step change in the mobile phase composition (the column was initially equilibrated with solute free mobile phase) involves three solutes, and thus, the effluent history consists of three concentration shocks that are partitioned by intermediate plateaus. Figure 3 illustrates typical experiment results of the frontal analysis. The experiment was conducted at concentrations of nadolol at 0.18 mg/mL, and the flow rate of the mobile phase is 0.5 mL/min. It was also found from Figure 3 that all three fronts were diffused waves rather than being shocks, as predicted by ideal chromatography theory. This is because in reality the mobile phase and the stationary phase are not always in equilibrium and the effects of dispersion and mass transfer resistance tend to flatten the shock profiles to diffused layers which travel with the same velocity. For this reason, the inflection points
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6177 Table 1. Experimental Results of the Nonlinear Frontal Analysis (Mobile Phase Flow Rate at 0.5 mL/min) for Different Concentrations of the Step Changesa 0.14 mg/mLb 0.18 mg/mLb
TD
T′0
T′1
T′2
T′3
T0
T1
T2
T3
K′1
K′2
K′3
6.12 6.12
11.80 11.80
18.84 18.80
19.88 19.86
23.10 22.85
5.68 5.68
12.72 12.68
13.76 13.74
16.98 16.73
1.24 1.23
1.42 1.42
1.99 1.95
a Where T is the dead time of the system without a column, T′ is the breakthrough time of unretained TTBB measured in the D 0 experiment, T′i is the breakthrough time of the boundary of component i measured in the frontal experiments, Ti is the breakthrough time of component i of nadolol generated by the column, and T0 is the column hold up time. b It should be noted that, in the calculation of final results of bi by eqs 36-38, more significant figures of concentration (i.e., 0.1350 and 0.1787 mg/mL) were used to give reliable results.
Table 2. Experimental Results of the Linear Elution Chromatography
Table 3. Competitive Langmuir Coefficients of the Stereoisomers of Nadolol
component
ai
k′i
1 ((SRS)-nadolol and (SSR)-nadolol) 2 ((RRS)-nadolol) 3 ((RSR)-nadolol)
2.94 3.46 5.45
2.13 2.51 3.95
of the breakthrough curves were used to determine the breakthrough time of the frontal boundaries. It is also worth noticing that the detector’s response is sufficient to provide the required information and no calibration work is needed. Although ideally only one frontal experiment is necessary to determine the competitive Langmuir coefficients bi, the possibility of experimental error and the difficulty to determine Ti accurately necessitate other confirming frontal experiments, which may be conducted at different concentrations of the step changes of the solutes and at different flow rates of the mobile phase. The experiments were conducted at concentrations of nadolol at 0.14 and 0.18 mg/mL, respectively, and the flow rate of the mobile phase was 0.5 mL/min. The experimental results of the nonlinear frontal and linear elution chromatography are shown in Tables 1 and 2, respectively. The three equations used to determine the competitive Langmuir coefficients of nadolol are given as follows:
( ) cfi
n)3
∑ i)1
k′i
K′3
cfi
k′iK′2
k′2K′1
n)3
∑ i)1
bi ) 1
(36)
-1
cfi
k′iK′3
k′3K′2
(37)
bi ) 1
(38)
-1
In eqs 36-38, i ) 1, 2, and 3 refer to the first, second, and third breakthrough components of nadolol from the column. They correspond to (SRS)- and (SSR)-nadolol, (RRS)-nadolol, and (RSR)-nadolol, respectively. In the above equations, the elution capacity factor k′i, the frontal capacity factor K′i, and the feed concentrations Cfi are all known, so they can be easily solved (e.g., by MATLAB 5.3.1 in this work) to give the results of the competitive Langmuir coefficients b1, b2, and b3. The
conc ) 0.18 mg/mL
average
0.0913 1.972 19.111
0.0859 1.959 14.154
0.0886 1.965 16.632
results of bi are shown in Table 3. It was found that there was a noticeable difference between the obtained b3 values from the two experiments. This could be explained by the fact that separation of nadolol is essentially a difficult separation and the fronts of components 1 and 2 are not completely partitioned by an intermediate concentration plateau. In addition, they are diffused waves rather than shocks due to axial dispersion and mass transfer resistance. All these reasons made it very difficult to determine Tj (i ) 1, 2, 3) accurately, whose fluctuations were found to be sensitive to the final results of bi (i ) 1, 2, 3). The competitive Langmuir coefficients of the three components of nadolol were evaluated at the average of b1, b2, and b3, and the final isotherms were given as
-1
( ) ( )
n)3
∑ i)1
bi ) 1
conc ) 0.14 mg/mL b1 b2 b3
q/1 )
2.94c1 (39) 1 + 0.0886C1 + 1.965C2 + 16.632C3
q/2 )
3.46c2 (40) 1 + 0.0886C1 + 1.965C2 + 16.632C3
q/3 )
5.45c3 (41) 1 + 0.0886C1 + 1.965C2 + 16.632C3
It is found from the value of bi that the competitive interference among the three stereoisomers of nadolol is high, which demonstrates strong competitiveness among the three components for the adsorbents. 4.2. Validation of the Adsorption Isotherms of Nadolol. The adsorption isotherms determined in the previous section needed to be tested for their validity. In the modeling of chromatographic processes, a model is only as good as its ability to predict band profiles, not only for the single components but, most importantly, also for the separations of mixtures. For this, the partial differential equation (PDE) systems with the initial and boundary conditions were solved by a code FEMLAB 2.0, which is based on MATLAB and solves PDEs using the finite element method, to simulate the response to a pulse injection of nadolol. Applying Danckwerts boundary conditions,
DL
∂Ci (z ) 0, t) ) -v[Ci(z ) 0- ,t) - Ci(z ) 0 + ,t)] ∂z (42) ∂Ci (z ) L, t) ) 0 ∂z
(43)
6178 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
Figure 4. Comparison of the simulated and experimental band profiles of the racemate nadolol. Conditions: flow rate ) 0.3 mL/min; C0 ) 0.14 mg/mL (nonlinear region).
Initial conditions are
Ci(z, t ) 0) ) CS,i(z, t ) 0) ) 0 The PDE equations, together with the adsorption isotherms of eqs 39-41, were solved by the FEMLAB program. The simulations were performed in the nonlinear region of nadolol at concentrations of 0.14 and 0.18 mg/mL. They were compared with the experimental results. Figure 4 gives the comparison between experimental and simulated profiles for nonlinear elution chromatography at the concentration 0.14 mg/mL and the mobile phase flow rate 0.3 mL/min. It was found that the experimental and simulated results matched reasonably well. This confirms the validity of the obtained adsorption isotherms. 4.3. SMB Complete Separation Regions for Ternary Separation of Nadolol. Assuming adsorbent and fluid to be regenerated completely in sections 1 and 4, respectively (m1 > m1,cr ) a3 and m4 < m4,cr), we are mostly concerned with the projection of the separation regions onto the (m2, m3) plane, that is, the plane in the operating parameter space spanned by the flow rate ratios of the two key sections of the SMB unit. From the adsorption isotherms obtained, the complete separation region for a ternary mixture of nadolol was constructed for both (1, 2) and (2, 3) separation regimes, as shown in Figure 5. If one decreases m2 and m3 along an operation line parallel to the diagonal (feed flow rate remains unchanged), one would undergo different regions of pure extract of the (2, 3) separation regime, complete (2, 3) separation, pure raffinate of the (2, 3) separation regime, complete (1, 2) separation, and pure raffinate of the (1, 2) separation regime, as summarized in Table 4. It is worth noting that in Table 4 the pure raffinate region of the (2, 3) separation regime is identical to the pure extract region of the (1, 2) separation regime. Figure 5 shows that both linear and nonlinear complete separation regions for the (2, 3) separation regime
Figure 5. Complete SMB separation regions of nadolol and effect of feed concentrations. Table 4. Component (1, 2, 3) Distributions in Product Streams for Different Separation Regions of the Four-Zone SMBa (n, p) regime
separation regions
extract
raffinate
n ) 1, p ) 2
complete separation pure raffinate pure extract complete separation pure raffinate pure extract
2, 3 1, 2, 3 2, 3 3 2, 3 3
1 1 1, 2 1, 2 1, 2 1, 2, 3
n ) 2, p ) 3
a Note that the pure raffinate region of the (2, 3) separation regime is identical to the pure extract region of the (1, 2) separation regime.
are larger than that for the (1, 2) separation regime, because a3 - a2 is greater than a2 - a1. This explained the reason it is easier to split a ternary mixture between components with a larger equilibrium constant difference (i.e., between components 2 and 3 of nadolol) than that with a smaller difference (i.e., between components
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6179
1 and 2) in SMB separation. It was also noted that, with the increase of feed concentrations, the optimal points of wr,e move toward the lower left corner of the plane and both complete (1, 2) and (2, 3) separation regions become smaller and sharper with a long tail (curve rn,pdp). In other words, the complete separation regions gain increasing nonlinear character, which requires that the flow rate ratios of m2 and m3 must be decreased. It should be pointed out that the complete separation regions obtained in this study can only provide a theoretical guideline for selecting operation conditions in real applications. Model assumptions of equilibrium theory (i.e., neglecting axial dispersion and mass transfer resistances), inaccuracies of the isotherm parameters obtained, and perturbation of operation conditions (e.g., pump flow rate accuracy and stability) could make the complete separation region slightly different, which must be considered in practical operation of an SMB. 5. Conclusions In this paper, the h-root method was used to determine the competitive Langmuir isotherm for stereoisomers of nadolol on a perphenyl carbamoylated β-cyclodextrin bonded chiral stationary phase. Although, in the range of 0 and the intrinsic affinity coefficient of the least retained component, the arbitrary choice of dummy species has no effect on the final results, the dummy species can give rise to confusion and this disadvantage is avoided by applying the h-root transformation directly to the multicomponent chromatographic system without introduction of the dummy species. Furthermore, without the introduction of dummy species, wave velocities of shocks in frontal development do not depend on the adjustable parameter and can be calculated directly, and thus, one can have insight into the process development. In addition, the individual isomers of nadolol, which are not commercially available, are not required, and only a very small amount of a racemic mixture of nadolol is needed. This facilitates the determination of isotherms for racemic drugs. This method divides the determination of Langmuir parameters into two parts. The intrinsic affinity coefficients ai were obtained from linear elution chromatography, and competitive interference coefficients bi were obtained from nonlinear frontal chromatography. The nonlinear frontal experiments were performed at different step changes of the solute concentrations in the mobile phase, which was aimed to eliminate the possible experimental errors and to obtain average b coefficients of Langmuir isotherms. Through the simulation, it was found that the experimental and simulated results match well; this confirms that the adsorption of nadolol on a perphenyl carbamoylated β-cyclodextrin bonded chiral stationary phase conforms to Langmuir isotherm behavior as well as the validity of the isotherm coefficients. Last, on the basis of the obtained isotherms, complete separation regions were determined for the four-zone SMB process to separate the ternary mixture of nadolol into different fractions. Nomenclature ai ) intrinsic affinity coefficients (dimensionless) bi ) Langmuir competitive interference coefficient (mL/mg) ci ) mobile phase concentration based on fluid volume (mg/mL) CS,i ) concentration in stationary phase (mg/mL)
cfi ) feed concentration based on fluid volume (mg/mL) F ) phase ratio, equal to (1 - )/ hi ) h-root (dimensionless) k′i ) elution capacity factor at infinite dilution (dimensionless) K ) Henry’s law adsorption equilibrium constant K′i ) frontal capacity factor (dimensionless) L ) column length (cm) mi ) flow rate ratio in section i of the SMB m1,cr ) critical flow rate ratio in section 1 of the SMB, m1,cr ) a3 m4,cr ) critical flow rate ratio in section 4 of the SMB, 0 < m4,cr < a1; m4,cr is a function of feed concentration and operating point in the (m2, m3) plane.25 / qi ) equilibrium concentration in the stationary phase expressed by the Langmuir isotherm of eq 1 (mg/mL) Ti ) breakthrough time of the waves in frontal experiments (min) T0 ) column hold up time (min) uj ) real velocity of waves (cm/s) Uj ) adjusted wave velocities (cm/s) v ) interstitial fluid velocity of the mobile phase (cm/s) Greek Symbols ) bed voidage of the column (dimensionless) τj ) adjusted emergence times of the waves (min) ω ) equilibrium theory parameter, defined by eq 35 µ1 ) the first moment µ2 ) the second central moment Superscripts and Subscripts 1 ) least retained component in ternary mixtures 2 ) intermediary retained component in ternary mixtures 3 ) strongest retained component in ternary mixtures f, F ) feed i, j ) species r ) strongest adsorbed component in raffinate stream e ) least adsorbed component in extract stream
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Received for review April 28, 2003 Revised manuscript received September 9, 2003 Accepted September 17, 2003 IE0303698
