Journal of Chromatography A, 1407 (2015) 100–105
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Influence of particle size and shell thickness of core–shell packing materials on optimum experimental conditions in preparative chromatography Krisztián Horváth a , Attila Felinger b,c,∗ a
Department of Analytical Chemistry, University of Pannonia, P.O. Box 158, H-8200 Veszprém, Hungary MTA–PTE Molecular Interactions in Separation Science Research Group, Ifjúság útja 6, H-7624 Pécs, Hungary c Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary b
a r t i c l e
i n f o
Article history: Received 31 March 2015 Received in revised form 9 June 2015 Accepted 14 June 2015 Available online 22 June 2015 Keywords: Core–shell particle Preparative chromatography Optimization of separation Simplex method Recovery yield Production rate
a b s t r a c t The applicability of core–shell phases in preparative separations was studied by a modeling approach. The preparative separations were optimized for two compounds having bi-Langmuir isotherms. The differential mass balance equation of chromatography was solved by the Rouchon algorithm. The results show that as the size of the core increases, larger particles can be used in separations, resulting in higher applicable flow rates, shorter cycle times. Due to the decreasing volume of porous layer, the loadability of the column dropped significantly. As a result, the productivity and economy of the separation decreases. It is shown that if it is possible to optimize the size of stationary phase particles for the given separation task, the use of core–shell phases are not beneficial. The use of core–shell phases proved to be advantageous when the goal is to build preparative column for general purposes (e.g. for purification of different products) in small scale separations. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The growing interest in the pharmaceutical industry for preparative chromatography that permits the purification of significant amounts of drug intermediates, peptides or proteins by eliminating closely related but unwanted compounds and impurities has made the optimization of the experimental conditions in preparative liquid chromatography a topic of serious current concern [1]. Therefore, a number of studies have focused recently on the determination of the optimum experimental conditions and column design parameters in preparative liquid chromatography. The nonlinear nature of preparative chromatography – due to column overload – complicates the separation process so much that the derivation of general conclusions regarding the determination of optimum conditions is a rather difficult – if not impossible – task. The optimization of preparative chromatography is further complicated by the choice of possible objective functions. In industrial
∗ Corresponding author at: Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifúság útja 6, H-7624 Pécs, Hungary. Tel.: +36 72 501 500x24582; fax: +36 72 501518. E-mail addresses: [email protected] (K. Horváth), [email protected] (A. Felinger). http://dx.doi.org/10.1016/j.chroma.2015.06.037 0021-9673/© 2015 Elsevier B.V. All rights reserved.
applications, the production cost is the major factor to consider. Many components of the production cost, however, are beyond the scope of the separation process itself. Accordingly, a more straightforward approach is chosen and usually simply the production rate is maximized [2–5]. Optimum experimental conditions were also determined considering economic consequences in situations where the cost of the solvent – a major cost factor in certain applications of preparative liquid chromatography – was also taken into account [6–8]. A hybrid objective function was introduced in order to weigh the importance of both the production rate (which should be as high as possible) and the solvent consumption (which should be as low as possible) [6]. Because all the modes of operation considered are usually applied as batch processes, the recovery yield during each run is lower than unity. Some optimization for maximum production rate were carried out with the constraint of a minimum yield [2,3,9]. The simple maximization of the production rate would often lead to scenarios where the yield is unacceptably low and some of the precious feed would remain unpurified. A rather attractive objective function was suggested: the product of the production rate and the recovery yield [10]. It was shown that the production rate only slightly decreased and the recovery yield significantly improved at the optimal experimental conditions found by that
K. Horváth, A. Felinger / J. Chromatogr. A 1407 (2015) 100–105
objective function. This trade-off of a slight decrease in the production rate for a considerable yield improvement would be most economical. The optimization of the different modes of preparative chromatography allowed the comparison of isocratic or gradient elution and displacement chromatography [9,11,12], revealing the relative advantages of either mode of separation. These studies suggested that elution can offer a larger production rate than displacement chromatography but delivers less concentrated fractions, which may significantly increase the cost of downstream processing. Core–shell particles have been extremely popular in analytical chromatography [13,14]. The optimization of the core radius fraction in preparative nonlinear liquid chromatography has been recently studied [15] but a holistic optimization has not yet been carried out. Recently, core–shell columns have been introduced to the market for preparative separations, which also calls for further studies in this area. The aim of this study is to investigate the particle size and core-to-shell ratio of core–shell packing materials for optimum separations in preparative chromatography. 2. Theory 2.1. Characterization of preparative separations
2.1.1. Loading factor (Lf ) Loading factor is a dimensionless unit that describes the sample size injected. Since adsorbents usually have a finite saturation capacity that corresponds to the formation of a monolayer of adsorbate, a convenient reference to express the sample size in dimensionless units is ratio of amount of sample injected to the amount of sample needed to cover the adsorbent with a monolayer (saturation capacity). The sample size is thus expressed as the loading factor Vinj Cinj (1 − εT ) S L qs
(1)
where S is the column cross-sectional area, L the column length, εT the total porosity of the bed, qs the column saturation capacity, Cinj the feed concentration, and Vinj the sample volume. 2.1.2. Cycle time (tc ) The cycle time is the time difference between two successive injections. tc can be defined in different ways. In this work, tc is defined as the sum of elution time and the time required for column regeneration and stabilization. 2.1.3. Recovery yield (Y) The recovery yield is the ratio between the amount of the desired component in the purified fraction, npur , and the amount injected in the column with the feed. Y=
npur Vinj Cinj
volume, the concentration of the corresponding component in the feed, and the recovery yield, divided by the cycle time. Pr =
Vinj Cinj Y tc
=
npur tc
(3)
2.1.5. Specific production (SP) The amount of solvent consumed per unit amount of purified product prepared is an important contribution to the total cost of production in many cases. The amount of solvent used during a cycle is the product of the cycle time and the flow rate. SP is the amount of purified component produced per unit volume of solvent used, and it can be calculated as SP =
npur Pr = F tc F
(4)
where F is the flow rate of the mobile phase. 2.1.6. Cut points The correct determination of the beginning and end of fraction collection is critical to the purity of products in preparative separations. Cut points represent the start and end of fraction collection. Cut points should be determined considering the purity requirement of the given products. 2.2. Equilibrium-dispersive model
As with any industrial process, preparative chromatography needs to be optimized. For the sake of clarity, it is important to state here the definitions of the main parameters employed to characterize the preparative separations simulated in this work [1].
Lf =
101
(2)
Y is a function of the purity at which the products must be prepared. 2.1.4. Production rate (Pr) The production rate is the amount of desired compound produced per unit time. It can be calculated as the product of the feed
Several mathematical models were developed to describe the chromatographic processes [1]. One of the most important models is the equilibrium-dispersive (ED) model which assumes that the mobile and the stationary phases are constantly in equilibrium. In this model, the contributions of different processes that cause band dispersion (e.g., mass transfer resistances, finite kinetics of adsorption-desorption). are lumped together in an apparent dispersion coefficient. Accordingly, the differential mass balance equation of the solute is given by 2
∂ c(z, t) ∂ q(z, t) ∂ c(z, t) ∂ c(z, t) +ϕ + u0 = Da ∂t ∂t ∂z ∂ z2
(5)
where q and c are the stationary and the mobile phase concentrations of the compound, respectively, t is the time, z the distance along the column, u0 the linear velocity, and ϕ = (1 − εT )/εT is the phase ratio with εT the total porosity of the column. u0 =
F L = SεT t0
(6)
where F is the flow rate of the mobile phase, t0 the column hold-up time, and S = dc2 /4 the cross-sectional area of the column with the dc column diameter. The total porosity can be calculated as εT = εe + (1 − εe )εp (1 − 3 )
(7)
where εe is external porosity of the column (fractional volume of the cavities in the bed that are around the particles), εp the porosity of particles (or internal porosity), the ratio of core radius to that of the particle ( = rcore /rp ). According to Eqs. (6) and (7), both the column hold-up time and phase ratio depends on the size of non-porous core. In Eq. (5), q is related to c through the isotherm equation, q = f(c). 3. Experimental All the calculations were carried out by a software written in house in C++ language using the GNU Scientific Library (GSL) [16]. The source code of the program was compiled by g++ shipped by GNU Compiler Collection ver. 4.5.3. O1 optimization level was set during the compilation since it turns on the most common forms
102
K. Horváth, A. Felinger / J. Chromatogr. A 1407 (2015) 100–105
of optimization that do not require any speed-space trade-offs. The calculations were performed on a computer equipped with Intel Celeron M CPU (1.6 GHz) running GNU Linux operating system (Debian Linux). In all the calculations, the column length (L) and diameter (dc ) were set to be 10 cm and 5 mm, respectively, while the particle diameters (dp ) were either optimized or changed from 3.0 to 14 m systematically. The external porosity of the column, εe , was 0.4 while the internal porosities, εp , were varied according to the diameter of solute molecules (see Section 4.1). The flow rate of the mobile phase, F, were always set to reach the maximum acceptable pressure of the system, Pmax , which was 100 bar during the calculations. dp2 Pmax 105
F=
L
εe S = 7.85 × 104 dp2
(cm3 /s)
qsS bS,i ci qsNS bNS ci + 1 + bNS (cA + cB ) 1 + bS,A cA + bS,B cB
(9)
where subscripts NS and S represents the non-selective and selective binding sites of the stationary phase, i represents compound A or B, qs is the saturation capacity of the stationary phase, and b the equilibrium constant. Eq. (9) typically describes the adsorption of racemic mixtures on a chiral stationary phase. In this work, the following parameters were used: qsNS = 55 mg/mL, qsS = 4 mg/mL, bNS = 0.0518 mL/mg, bS,A = 0.1765 mL/mg, and bS,B = 0.4175 mL/mg. Accordingly, the selectivity of the separation of the two compounds were ˛ = 1.27. The injected concentrations were set to be 30 mg/mL for both compounds, while the injection volumes were optimized. It is important to note that it was assumed that the shape and size distribution of the core–shell particles were the same, regardless of the shell thickness and that the quality of the column packing remained identical in all cases. 3.1. Solution of mass balance equation The solution of the mass balance equation [Eq. (5)] was carried out by Rouchon method [17,18]. The time and space increments (t, z) were chosen in order to accurately simulate the band dispersion for the two components by the numerical error of calculation [1]: z =
L N
(10)
t =
z (tR1 + tR2 ) L
(11)
tR,i
Lεe = u0 εT
3.2. Determination of cut-points The required purity of the products was set to 99%. The cut points were determined by the calculated band profiles in order to achieve the required purity. The following equations were solved for determining cut points tc,1 and tc,2
tc,1
(8)
The competitive retention behavior of compounds A and B was described by the following bi-Langmuir isotherm equation: qi =
plates, the difference between the calculated and measured band profiles and the proper cut-points were negligible. Even in case of extremely small efficiencies (N < 100), the difference in the cutpoints were ∼3%. Accordingly, the accuracy of the Rouchon method did not affect the results. In the algorithm used for the calculations, rectangular injection profiles were applied.
(1 − εe ) (1 − 3 ) 1+ (Ki (1 − εp ) + εp ) εe
Pur =
tc,1 t
Pur =
t
cA dt +
tc,1 t
(13) cB dt
2tR,B −t
2tR,B −t tc,2
cA dt
tc,2
cA dt +
cB dt
2tR,B −t tc,2
(14) cB dt
where Pur is the required purity (0.99), cA and cB the calculated band profiles for compounds A and B, respectively. Solution of Eqs. (13) and (14) were accomplished by the Brent method [22] provided by the GNU Scientific Library (Chap. 33 of Ref. [16]). The integrations were carried out by the QAG adaptive integration procedure of GSL (Chap. 17 of Ref. [16]) after fitting cubic splines on band profiles (Chap. 27 of Ref. [16]). 3.3. Simplex method for optimization of preparative separation The numerical method for the optimization of non-linear separations was carried out using the Simplex algorithm of Nelder and Mead [23] implemented in GSL (see Chap. 35 of Ref. [16]). With the simplex algorithm, an object function was minimized. During the calculations the negative specific production (−SP) and the negative product of recovery yield and production rate (−Y × Pr) were used as object functions. The parameters optimized (Vinj , and sometimes dp ) were changed by the simplex algorithm in order to minimize the objective function. The minimization was stopped when the overall size of the simplex decreased below 10−6 . The column regeneration and stabilization time was assumed to be 6t0 . Accordingly, the cycle time was set as tc = tR,2 + 6t0 . 4. Results and discussion 4.1. HETP curves of core–shell particles
(12)
where Ki is the Henry constant of the ith compound (Ki = qsNS bNS + qsS,i bS,i ). The number of theoretical plates, N, was calculated as it was described in Section 4.1 of Ref. [19]. Note that the Rouchon algorithm applied in this study tends to be unstable in case of convex isotherms (e.g. BET isotherm) and to give inaccurate results when the column efficiency is small [1,20]. Using, however, the isotherm equation defined by Eq. (9) and the parameters of numerical calculations defined by Eqs. (10)–(12), the algorithm was stable. Values of N were typically between 500 and 22000 depending on the diameters of particle and non-porous core. Only in few cases (19 out of 330), values of N were under 100. The accuracy of the Rouchon method was tested by comparing the calculated band profiles and cut-points to the results obtained by the Martin-Synge algorithm that is one of the most accurate methods in HPLC [21]. Above 1000
The height equivalent to a theoretical plate, HETP, can be calculated from the statistical moments of the peak eluted as H=L
2,col 21,col
(15)
The preparative separation power of superficially porous particles was investigated for three different cases: 1 M1H1: small molecules (molecular weight, M,
Contents lists available at ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Influence of particle size and shell thickness of core–shell packing materials on optimum experimental conditions in preparative chromatography Krisztián Horváth a , Attila Felinger b,c,∗ a
Department of Analytical Chemistry, University of Pannonia, P.O. Box 158, H-8200 Veszprém, Hungary MTA–PTE Molecular Interactions in Separation Science Research Group, Ifjúság útja 6, H-7624 Pécs, Hungary c Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary b
a r t i c l e
i n f o
Article history: Received 31 March 2015 Received in revised form 9 June 2015 Accepted 14 June 2015 Available online 22 June 2015 Keywords: Core–shell particle Preparative chromatography Optimization of separation Simplex method Recovery yield Production rate
a b s t r a c t The applicability of core–shell phases in preparative separations was studied by a modeling approach. The preparative separations were optimized for two compounds having bi-Langmuir isotherms. The differential mass balance equation of chromatography was solved by the Rouchon algorithm. The results show that as the size of the core increases, larger particles can be used in separations, resulting in higher applicable flow rates, shorter cycle times. Due to the decreasing volume of porous layer, the loadability of the column dropped significantly. As a result, the productivity and economy of the separation decreases. It is shown that if it is possible to optimize the size of stationary phase particles for the given separation task, the use of core–shell phases are not beneficial. The use of core–shell phases proved to be advantageous when the goal is to build preparative column for general purposes (e.g. for purification of different products) in small scale separations. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The growing interest in the pharmaceutical industry for preparative chromatography that permits the purification of significant amounts of drug intermediates, peptides or proteins by eliminating closely related but unwanted compounds and impurities has made the optimization of the experimental conditions in preparative liquid chromatography a topic of serious current concern [1]. Therefore, a number of studies have focused recently on the determination of the optimum experimental conditions and column design parameters in preparative liquid chromatography. The nonlinear nature of preparative chromatography – due to column overload – complicates the separation process so much that the derivation of general conclusions regarding the determination of optimum conditions is a rather difficult – if not impossible – task. The optimization of preparative chromatography is further complicated by the choice of possible objective functions. In industrial
∗ Corresponding author at: Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifúság útja 6, H-7624 Pécs, Hungary. Tel.: +36 72 501 500x24582; fax: +36 72 501518. E-mail addresses: [email protected] (K. Horváth), [email protected] (A. Felinger). http://dx.doi.org/10.1016/j.chroma.2015.06.037 0021-9673/© 2015 Elsevier B.V. All rights reserved.
applications, the production cost is the major factor to consider. Many components of the production cost, however, are beyond the scope of the separation process itself. Accordingly, a more straightforward approach is chosen and usually simply the production rate is maximized [2–5]. Optimum experimental conditions were also determined considering economic consequences in situations where the cost of the solvent – a major cost factor in certain applications of preparative liquid chromatography – was also taken into account [6–8]. A hybrid objective function was introduced in order to weigh the importance of both the production rate (which should be as high as possible) and the solvent consumption (which should be as low as possible) [6]. Because all the modes of operation considered are usually applied as batch processes, the recovery yield during each run is lower than unity. Some optimization for maximum production rate were carried out with the constraint of a minimum yield [2,3,9]. The simple maximization of the production rate would often lead to scenarios where the yield is unacceptably low and some of the precious feed would remain unpurified. A rather attractive objective function was suggested: the product of the production rate and the recovery yield [10]. It was shown that the production rate only slightly decreased and the recovery yield significantly improved at the optimal experimental conditions found by that
K. Horváth, A. Felinger / J. Chromatogr. A 1407 (2015) 100–105
objective function. This trade-off of a slight decrease in the production rate for a considerable yield improvement would be most economical. The optimization of the different modes of preparative chromatography allowed the comparison of isocratic or gradient elution and displacement chromatography [9,11,12], revealing the relative advantages of either mode of separation. These studies suggested that elution can offer a larger production rate than displacement chromatography but delivers less concentrated fractions, which may significantly increase the cost of downstream processing. Core–shell particles have been extremely popular in analytical chromatography [13,14]. The optimization of the core radius fraction in preparative nonlinear liquid chromatography has been recently studied [15] but a holistic optimization has not yet been carried out. Recently, core–shell columns have been introduced to the market for preparative separations, which also calls for further studies in this area. The aim of this study is to investigate the particle size and core-to-shell ratio of core–shell packing materials for optimum separations in preparative chromatography. 2. Theory 2.1. Characterization of preparative separations
2.1.1. Loading factor (Lf ) Loading factor is a dimensionless unit that describes the sample size injected. Since adsorbents usually have a finite saturation capacity that corresponds to the formation of a monolayer of adsorbate, a convenient reference to express the sample size in dimensionless units is ratio of amount of sample injected to the amount of sample needed to cover the adsorbent with a monolayer (saturation capacity). The sample size is thus expressed as the loading factor Vinj Cinj (1 − εT ) S L qs
(1)
where S is the column cross-sectional area, L the column length, εT the total porosity of the bed, qs the column saturation capacity, Cinj the feed concentration, and Vinj the sample volume. 2.1.2. Cycle time (tc ) The cycle time is the time difference between two successive injections. tc can be defined in different ways. In this work, tc is defined as the sum of elution time and the time required for column regeneration and stabilization. 2.1.3. Recovery yield (Y) The recovery yield is the ratio between the amount of the desired component in the purified fraction, npur , and the amount injected in the column with the feed. Y=
npur Vinj Cinj
volume, the concentration of the corresponding component in the feed, and the recovery yield, divided by the cycle time. Pr =
Vinj Cinj Y tc
=
npur tc
(3)
2.1.5. Specific production (SP) The amount of solvent consumed per unit amount of purified product prepared is an important contribution to the total cost of production in many cases. The amount of solvent used during a cycle is the product of the cycle time and the flow rate. SP is the amount of purified component produced per unit volume of solvent used, and it can be calculated as SP =
npur Pr = F tc F
(4)
where F is the flow rate of the mobile phase. 2.1.6. Cut points The correct determination of the beginning and end of fraction collection is critical to the purity of products in preparative separations. Cut points represent the start and end of fraction collection. Cut points should be determined considering the purity requirement of the given products. 2.2. Equilibrium-dispersive model
As with any industrial process, preparative chromatography needs to be optimized. For the sake of clarity, it is important to state here the definitions of the main parameters employed to characterize the preparative separations simulated in this work [1].
Lf =
101
(2)
Y is a function of the purity at which the products must be prepared. 2.1.4. Production rate (Pr) The production rate is the amount of desired compound produced per unit time. It can be calculated as the product of the feed
Several mathematical models were developed to describe the chromatographic processes [1]. One of the most important models is the equilibrium-dispersive (ED) model which assumes that the mobile and the stationary phases are constantly in equilibrium. In this model, the contributions of different processes that cause band dispersion (e.g., mass transfer resistances, finite kinetics of adsorption-desorption). are lumped together in an apparent dispersion coefficient. Accordingly, the differential mass balance equation of the solute is given by 2
∂ c(z, t) ∂ q(z, t) ∂ c(z, t) ∂ c(z, t) +ϕ + u0 = Da ∂t ∂t ∂z ∂ z2
(5)
where q and c are the stationary and the mobile phase concentrations of the compound, respectively, t is the time, z the distance along the column, u0 the linear velocity, and ϕ = (1 − εT )/εT is the phase ratio with εT the total porosity of the column. u0 =
F L = SεT t0
(6)
where F is the flow rate of the mobile phase, t0 the column hold-up time, and S = dc2 /4 the cross-sectional area of the column with the dc column diameter. The total porosity can be calculated as εT = εe + (1 − εe )εp (1 − 3 )
(7)
where εe is external porosity of the column (fractional volume of the cavities in the bed that are around the particles), εp the porosity of particles (or internal porosity), the ratio of core radius to that of the particle ( = rcore /rp ). According to Eqs. (6) and (7), both the column hold-up time and phase ratio depends on the size of non-porous core. In Eq. (5), q is related to c through the isotherm equation, q = f(c). 3. Experimental All the calculations were carried out by a software written in house in C++ language using the GNU Scientific Library (GSL) [16]. The source code of the program was compiled by g++ shipped by GNU Compiler Collection ver. 4.5.3. O1 optimization level was set during the compilation since it turns on the most common forms
102
K. Horváth, A. Felinger / J. Chromatogr. A 1407 (2015) 100–105
of optimization that do not require any speed-space trade-offs. The calculations were performed on a computer equipped with Intel Celeron M CPU (1.6 GHz) running GNU Linux operating system (Debian Linux). In all the calculations, the column length (L) and diameter (dc ) were set to be 10 cm and 5 mm, respectively, while the particle diameters (dp ) were either optimized or changed from 3.0 to 14 m systematically. The external porosity of the column, εe , was 0.4 while the internal porosities, εp , were varied according to the diameter of solute molecules (see Section 4.1). The flow rate of the mobile phase, F, were always set to reach the maximum acceptable pressure of the system, Pmax , which was 100 bar during the calculations. dp2 Pmax 105
F=
L
εe S = 7.85 × 104 dp2
(cm3 /s)
qsS bS,i ci qsNS bNS ci + 1 + bNS (cA + cB ) 1 + bS,A cA + bS,B cB
(9)
where subscripts NS and S represents the non-selective and selective binding sites of the stationary phase, i represents compound A or B, qs is the saturation capacity of the stationary phase, and b the equilibrium constant. Eq. (9) typically describes the adsorption of racemic mixtures on a chiral stationary phase. In this work, the following parameters were used: qsNS = 55 mg/mL, qsS = 4 mg/mL, bNS = 0.0518 mL/mg, bS,A = 0.1765 mL/mg, and bS,B = 0.4175 mL/mg. Accordingly, the selectivity of the separation of the two compounds were ˛ = 1.27. The injected concentrations were set to be 30 mg/mL for both compounds, while the injection volumes were optimized. It is important to note that it was assumed that the shape and size distribution of the core–shell particles were the same, regardless of the shell thickness and that the quality of the column packing remained identical in all cases. 3.1. Solution of mass balance equation The solution of the mass balance equation [Eq. (5)] was carried out by Rouchon method [17,18]. The time and space increments (t, z) were chosen in order to accurately simulate the band dispersion for the two components by the numerical error of calculation [1]: z =
L N
(10)
t =
z (tR1 + tR2 ) L
(11)
tR,i
Lεe = u0 εT
3.2. Determination of cut-points The required purity of the products was set to 99%. The cut points were determined by the calculated band profiles in order to achieve the required purity. The following equations were solved for determining cut points tc,1 and tc,2
tc,1
(8)
The competitive retention behavior of compounds A and B was described by the following bi-Langmuir isotherm equation: qi =
plates, the difference between the calculated and measured band profiles and the proper cut-points were negligible. Even in case of extremely small efficiencies (N < 100), the difference in the cutpoints were ∼3%. Accordingly, the accuracy of the Rouchon method did not affect the results. In the algorithm used for the calculations, rectangular injection profiles were applied.
(1 − εe ) (1 − 3 ) 1+ (Ki (1 − εp ) + εp ) εe
Pur =
tc,1 t
Pur =
t
cA dt +
tc,1 t
(13) cB dt
2tR,B −t
2tR,B −t tc,2
cA dt
tc,2
cA dt +
cB dt
2tR,B −t tc,2
(14) cB dt
where Pur is the required purity (0.99), cA and cB the calculated band profiles for compounds A and B, respectively. Solution of Eqs. (13) and (14) were accomplished by the Brent method [22] provided by the GNU Scientific Library (Chap. 33 of Ref. [16]). The integrations were carried out by the QAG adaptive integration procedure of GSL (Chap. 17 of Ref. [16]) after fitting cubic splines on band profiles (Chap. 27 of Ref. [16]). 3.3. Simplex method for optimization of preparative separation The numerical method for the optimization of non-linear separations was carried out using the Simplex algorithm of Nelder and Mead [23] implemented in GSL (see Chap. 35 of Ref. [16]). With the simplex algorithm, an object function was minimized. During the calculations the negative specific production (−SP) and the negative product of recovery yield and production rate (−Y × Pr) were used as object functions. The parameters optimized (Vinj , and sometimes dp ) were changed by the simplex algorithm in order to minimize the objective function. The minimization was stopped when the overall size of the simplex decreased below 10−6 . The column regeneration and stabilization time was assumed to be 6t0 . Accordingly, the cycle time was set as tc = tR,2 + 6t0 . 4. Results and discussion 4.1. HETP curves of core–shell particles
(12)
where Ki is the Henry constant of the ith compound (Ki = qsNS bNS + qsS,i bS,i ). The number of theoretical plates, N, was calculated as it was described in Section 4.1 of Ref. [19]. Note that the Rouchon algorithm applied in this study tends to be unstable in case of convex isotherms (e.g. BET isotherm) and to give inaccurate results when the column efficiency is small [1,20]. Using, however, the isotherm equation defined by Eq. (9) and the parameters of numerical calculations defined by Eqs. (10)–(12), the algorithm was stable. Values of N were typically between 500 and 22000 depending on the diameters of particle and non-porous core. Only in few cases (19 out of 330), values of N were under 100. The accuracy of the Rouchon method was tested by comparing the calculated band profiles and cut-points to the results obtained by the Martin-Synge algorithm that is one of the most accurate methods in HPLC [21]. Above 1000
The height equivalent to a theoretical plate, HETP, can be calculated from the statistical moments of the peak eluted as H=L
2,col 21,col
(15)
The preparative separation power of superficially porous particles was investigated for three different cases: 1 M1H1: small molecules (molecular weight, M,
