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N o n l i n ea r E f f ec t s i n LC and Chiral LC
The key to nonlinear effects lies in understanding the relationship between the solute concentrations at equilibrium in the two phases of the chromatographic system. o achieve high resolution and accurate peak integration, analytical chromatographers choose experimental conditions under which nonlinear effects are minimized. Careful use of these same effects allows them to design profitable separations. Nonlinear effects are ubiquitous. There are almost no experimental conditions under which they do not exist, although under many they are negligible. The key to understanding nonlinear effects lies in the equilibrium isotherms, the relationship between the solute concentrations at equilibrium in the two phases of the system. In most separations, particularly in chiral ones, the determination of these isotherms also allows more accurate investigation of the retention mechanism. We will show how band profiles can be modeled, why the model results are useful, and how systematic isotherm measurements allow a deeper understanding of certain retention mechanisms.
What is nonlinear chromatography? The goal of preparative chromatography is the rapid preparation of a significant amount of one component in a mixture to a sufficient degree of purity (1). Thus, large amounts of high-concentration samples are injected. Because of the finite surface area of the stationary phase in the column, the relative amount of a component adsorbed at equilibrium usually decreases as concentration increases, affecting retention. This phenomenon characterizes nonlinear chromatography. Peaks tail strongly in preparative chromatography, in contrast to the symmetrical peaks in analytical (i.e., linear) chromatography (2). Most often, nonlinear tailing peaks have sharp fronts and long, diffuse rear boundaries. The retention time of the high-concentration front decreases with increasing sample size, while the end of the peak tail has a con-
Torgny Fornstedt University of Uppsala (Sweden)
Georges Guiochon University of Tennessee and Oak Ridge National Laboratory
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FIGURE 1. Relationship between equilibrium isotherms and elution profiles. Component 1 is blue, component 2 is red. (a) Langmuir adsorption isotherms for two compounds on a homogenous surface; a1 = 10; b1 = 200 M–1; a2 = 20; b2 = 400 M–1; qs,1 = qs,2 = 0.05 M. The inset shows the isotherms in the low concentration range used in analytical applications. (b– d) Individual elution profiles of the two compounds with the isotherms in (a). Profiles calculated (b) assuming independence of the isotherms (i.e., no competition) and (c, d) with competitive Langmuir isotherms. Column efficiency is 4000 plates.
stant retention time, equal to that of the symmetrical analytical peak. Furthermore, solute bands compete with each other for access to the finite surface area, resulting in interactions between bands and strong profile changes during their migration along the column. Calculation programs are available that predict the response of a column to perturbations, including sample introduction at the column inlet. Writing these programs requires knowing the system’s thermodynamic and mass transfer kinetic parameters and the experimental conditions. These programs use a mass balance equation (1) for each solute, isotherm equations, and equations stating the kinetics of equilibration between the two phases in the column. The states of the column before sample introduction and at its boundaries (the initial and boundary conditions) are also required. The methods of solving these equation systems are discussed elsewhere (1, 3).
Thermodynamics At a given temperature, the equilibrium of a compound is characterized by the relationship between its stationary-phase concentration, q, at equilibrium with its mobile-phase concentration C, the equilibrium isotherm. Models are available to account for liquid–solid equilibrium data (1, 4–7). Single-component isotherms. Langmuir suggested the simplest isotherm model of physical relevance, assuming a ho-
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mogenous surface with only one type of adsorption site (4), which is q=
aC 1 + bC
(1)
in which a = bqs; b is the binding constant per surface area that characterizes the strength of the interaction energy of the adsorption sites, and qs is the monolayer saturation capacity. Figure 1a illustrates the Langmuir isotherm for two compounds. At very low concentrations (Figure 1a, inset), isotherms behave linearly. Their initial slopes are equal to their respective equilibrium constants at infinite dilutions (here, a1 = 10 and a2 = 20). At higher concentrations, the isotherms become nonlinear and, in most cases, convex upward, curving to the horizontal and tending toward saturation capacity (Figure 1a). Component 2 has a stronger curvature than component 1 because its b term is twice as large. Both compounds have the same monolayer capacity (qs = 0.05 M). The retention time (tR) of a compound is proportional to the slope of its isotherm (1) tR = t0(1 + F
dq ) dC
(2)
in which F is the phase ratio or the ratio of the volumes of the stationary and mobile phase in the column). F is easily
derived from t0, the flow rate, and the volume of the empty column. Under analytical conditions, the solute concentration is low, and Equation 2 reduces to tR = t0 (1 + Fa). At high concentrations, the retention time depends on this concentration and is given by Equations 1 and 2. Competitive isotherms. In separations, there are several components. Their isotherms are not independent, but are competitive. Thus, the amount of one component adsorbed at equilibrium depends on the concentrations of all the system components (1). For all separation problems, knowledge of the competitive isotherms is required for accurate modeling. Yet, reports on competitive isotherms remain few, mainly because they are more difficult to acquire than single-component isotherms. The Langmuir equation is easily extended to the multicomponent case q1 =
a1C1 (1 + b1C1 + ...bnCn)
(3)
The isotherm of the first component of a binary mixture in the presence of a constant concentration, C2, of the second one remains a Langmuir isotherm but with lower apparent values of the numerical coefficients [a´1 = a1/(1 + b2C2), b´1 = b1/(1 + b2C2)]. Influence of competition on band profiles. Figures 1b and 1c show the elution profiles of the two compounds whose isotherms are shown in Figure 1a, after injection of 135 µL of a solution (C1 = 20 mM, C2 = 60 mM). If we assume that there is no competition (Figure 1b), the two fronts coelute because of the particular combination of numerical values of the parameters. It would therefore be impossible to collect any pure fraction of component 1. Less than 50% of pure component 2 could be collected in a single run. Although this is not an actual situation, there is always competition between components for adsorption when their concentrations are high and their isotherms nonlinear. As mentioned earlier, the retention times of the rear endpoints are the retention times of the analytical peaks, tR1 = 7.02 and tR2 = 13.04 min (values derived from Equation 2 with t0 = 1.00 min and F = 0.602). Figure 1c shows the chromatogram calculated for the same experimental conditions, using the competitive Langmuir equation (Equation 3). The band of component 2 displaces that of component 1 whose maximum concentration is nearly doubled. The profile of component 2 changes less than that of component 1 because its amount is larger and it competes more strongly (b2 > b1). However, there is now a small hump on its rear boundary, at the end of the mixed region. This region is narrower in the competitive case than when we assumed no competition (Figure 1b), and larger amounts of pure fractions of both components can be collected.
The effect of band 2 on band 1 is due to the displacement effect (1) that arises because, in the presence of component 2, component 1 is less adsorbed at equilibrium and, therefore, moves faster along the column. The effect of band 1 on band 2 is due to the “tag-along effect” (1). In the presence of component 1, component 2 is less retained and moves faster. When the concentration of component 1 is tripled, and C1 = C2 = 60 mM, component 1 competes more strongly with component 2 (Figure 1d). The separation is less complete. The tag-along effect is stronger: The front of the second band moves in a region where the concentration of component 1 is higher, the competition is stronger, and the front elutes earlier. The second band tail remains unaffected and has the same retention time. The hump on the band of component 2 is stronger in Figure 1d than in Figure 1c. The hump in Figures 1b–d is a consequence of the tag-along effect. A more thorough explanation requires going through the math (1). The displacement and the tag-along effects are important features of separations under nonlinear conditions. They are used to optimize separation conditions. Single-component isotherms are represented by a curve in the (q, C) plane (Figure 1a). Binary competitive isotherms are represented by two surfaces in the three-dimensional space (qi, C1, C2), as illustrated in Figure 2 for the same two compounds that are assumed to follow competitive Langmuir isotherm behavior. The higher surface corresponds to component 2 (red), the lower one to component 1 (blue); the vertical axis corresponds to q1 for the lower surface, q2 for the higher one. The competitive interactions between the two components are illustrated by the shape of the surfaces (i.e., by the change in the curves obtained as sections of the isotherm surface by planes parallel to the vertical axis and to the axis of concentration of the competitor, the other component of the binary mixture). Because b2 > b1, the competition effect is more important for the first component than for the second. Experimental determination of isotherms. Several methods allow the determination of single-component adsorption isotherms (1). The traditional and most accurate method is frontal analysis (FA) (1, 5). Its modern implementation takes advantage of the high reliability of certain pumps (1). A highpressure gradient delivery system with two reservoirs is used—one reservoir contains the pure eluent and the other, a solution of a known concentration of the compound of interest in the eluent. Successive step-gradient changes of the solute concentration are performed at the column inlet (staircase mode), and the so-called breakthrough curves are recorded. The main drawbacks of the FA method are that it is tedious and costly in time and chemicals. Alternative methods can be found in the literature (1, 5). None yet has given
Competitive isotherms are difficult to acquire.
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results comparable in precision and accuracy. Acquiring competitive isotherms is much more difficult because it requires determining the amount of one component adsorbed as a function of its mobile-phase concentration in the presence of different concentrations of the other component. The FA method was adapted to measure competitive isotherms by Jacobson et al. (8). However, this approach is tedious and time-consuming, even for the binary case. Stepwise increases of the concentration of different binary mixtures of constant relative concentration are conducted, as in single-component FA. It is also necessary, however, to collect and analyze eluent fractions on the intermediate plateaus of the breakthrough curves (1, 8). Recent reports by Blümel and Fornstedt demonstrated the usefulness of the system peak method for the acquisition of competitive isotherms, but this also has some important pitfalls (9, 10), the most important being that the second system peak disappears under strongly nonlinear conditions, and the data cannot be used to draw an experimental isotherm. They merely allow the calculation of the best values of the parameters of a given isotherm model. Careful use of statistics becomes necessary to justify the choice of the best isotherm model.
Mass transfer kinetics The fastest computation program assumes instantaneous equilibrium and accounts for the influence of the mass transfer kinetics by pooling it with that of axial dispersion and using an apparent dispersion coefficient proportional to the height equivalent of a theoretical plate (HETP) (1). This is satisfac-
0.05
tory when the mass transfer kinetics is fast (1). However, in many cases involving the separation of enantiomers [e.g., on -blockers on the immobilized cellulase Cel7A (11), previously called CBH I (29)] or proteins (e.g., bovine serum albumin on anion exchanger) (12, 13), the mass transfer kinetics is slow. It is important to take the slow mass transfer kinetics into account, to measure its characteristics, and to implement it properly when modeling separations involving enantiomers, peptides, and proteins. The influence on the elution profiles of slow mass transfer kinetics was studied systematically using the simple kinetic rate model (1, 14, 15). This model assumes a rate of mass transfer proportional to the difference between the actual and equilibrium concentrations in the stationary phase. Results show that peak tailing increases with the decreasing rate coefficient, kf. The dimensionless Stanton number (St = kfL/u; L is the column length, and u is the mobile-phase linear velocity) characterizes the intensity of this tailing—it is the ratio of the column hold-up time, L/u, to the average residence time of the molecules in the stationary phase, 1/kf. Simulations show that peak tailing occurs only at very low Stanton numbers (St < 10), but the simple kinetic rate model is incorrect when St < 10 because it assumes that the concentration remains constant across the particles, the mass transfer resistance being localized at the particle/solution boundary. More complex models must be used to study very slow systems, although, admittedly, these systems are of little practical interest to separation scientists. When there are two retention mechanisms, each one has its own rate coefficient, the kinetics is heterogeneous, and there are two Stanton numbers. Peak tailing is intense whenever only one of the mechanisms is slow, and the typical profile resulting from active sites is obtained. In this case, simulations show that peak tailing takes place at St values as large as 100 for the slow sites (16, 17 ).
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Particular case of a chiral separation
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FIGURE 2. Three-dimensional plot of competitive adsorption isotherms. Langmuir parameters as in Figure 1 but implemented in the competitive Langmuir equation (Equation 3).
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As an example, we describe the separation of -blocker enantiomers on a protein (Cel7A) immobilized on silica gel. Cel7A is a cellulase that hydrolyzes the -1.4linkages at the reducing end of cellulose chains. Immobilized on silica, Cel7A separates many -blockers, often with large separation factors (18). The conventional eluent is an aqueous buffer with a low concentration of organic solvent. The retention times of both -blocker enantiomers decrease with increasing concentrations of organic modifier and with the increasing ionic strength of the buffer; however, the separation factor remains nearly constant. The retention times of the more retained
q (mM)
q (mM)
q (mM)
0.16 S enantiomers increase more rapidly with increasing mobile-phase pH than those of the cor0.12 responding R enantiomers, resulting in a higher 1.2 enantioselectivity. In the pH range 4–6, the 0.08 Cel7A molecule (pI 3.9) has a net negative 0.04 charge, whereas the amine group of -blockers, with a pKa of 9.4–9.7, is always positively 0 0.002 0.004 0.006 C (mM) charged. 0.8 We acquired the isotherms of R- and S-propranolol on immobilized Cel7A at pH = 4.7 (19). 8.0 The best fit of the data was to a bi-Langmuir 6.0 equation, consisting of the sum of two Langmuir terms (Equation 1). One of these terms had the 0.4 4.0 same parameters for both enantiomers; contributed only to their retention, not to their resolu2.0 tion; and corresponded to nonselective (type I) sites. The other Langmuir term corresponded to 0 0.4 0.8 1.2 1.6 C (mM) the enantioselective (type II) sites. Each type of 0.12 0.03 0.06 0.09 0 site has a corresponding binding interaction enC (mM) ergy and a monolayer capacity (b and qs in Equation 1). At pH 4.7, the S enantiomer has a 31- FIGURE 3. Single-component equilibrium isotherms for S-propranolol at fold larger interaction energy with the chiral increasing temperatures. than with the nonchiral sites, but the capacity of The figure shows the medium concentration range at which the highest mobile-phase conthe nonselective sites is 23-fold larger than that centration is 0.1 mM. The inset in the upper-left corner shows the low concentration range of the enantioselective sites. Thus, Cel7A offers below 5.0 µM and the inset at the lower-right corner shows the high concentration range a high density of type I sites with low interaction up to 1.7 mM. The data were calculated using the best bi-Langmuir isotherms. Stationary phase is immobilized Cel7A on silica; eluent is acetic acid buffered at pH 5.5. Black, 278 K; energies and a low density of type II sites with green, 288 K; red, 298 K; blue, 308 K; and yellow, 318 K. Adapted from Ref. 22. high interaction energies. When the pH increases from 4.7 to 6.0 at an ionic strength of 0.02 mM (1), the number of nonchiral type I of S-propranolol appears to be endothermic, while, at high sites increases (2), the monolayer capacity of the enantioselec- concentrations, it appears exothermic. The thermodynamic tive sites for R-propranolol increases significantly, and (3) the functions for each enantiomer were derived from the best bibinding constant of S-propranolol with the type II sites in- Langmuir estimates. The adsorption enthalpy on type I and creases considerably (19). Similar results were obtained at type IIR sites are negative, with ⌬Ho = –1.10 and –1.92, higher ionic strengths and with metoprolol and alprenolol kcal/mol, respectively. Adsorption on type IIS is endothermic, with ⌬Ho = +1.61 kcal/mole. The average adsorption (20). The separation of the propranolol enantiomers on Cel7A entropy ⌬So on the nonchiral type I sites, chiral type IIR sites, exhibits an unusual temperature effect (21). The retention and type IIS sites are +0.08, –2.55, and +11.55 cal/mol*K, time of the S enantiomer increases with increasing tempera- respectively. This last value is unusually high and explains why ture, while that of the less-retained R enantiomer decreases. an endothermic retention mechanism can take place. The term The retention of R-propranolol is exothermic at all pHs; that T⌬So outweighs ⌬Ho, and ⌬Go is negative. of S-propranolol is exothermic at pH < 5.5, but endothermic Relative importance of the hydrophobic and ionic conat pH > 5.5. Figure 3 shows the set of adsorption isotherms tributions. The adsorption isotherms of the enantiomers of acquired for S-propranolol at pH 5.5 between 5 °C and 45 three -blockers, metoprolol, alprenolol, and propranolol °C (22). The symbols show the experimental data and the (in order of increasing hydrophobicity), were measured on lines show the isotherms calculated from the best estimates Cel7A at pH = 5.0, 5.5, and 6.0 (20, 23). The following patof the bi-Langmuir parameters. The adsorption process is en- tern emerges from a comparison of the relationships between dothermic at low concentrations (upper-left inset, Figure 3). the different equilibrium constants of these compounds at the The top isotherm corresponds to the highest temperature different pHs and their hydrophobicities or distribution con(318 K (yellow), the bottom isotherm to the lowest tem- stants in the water/octanol system. The enantioselective inperature (278 K (black). The isotherms shown in the main teractions between type II sites and the S enantiomers are figure and in the lower-right corner inset (medium and high primarily ionic and secondarily hydrophobic, whereas those concentration ranges, respectively) are in the opposite order. of the R enantiomers are both strongly ionic and hydrophoA careful investigation of the main figure shows a transi- bic; the nonselective interactions with type I sites are mosttion range at ~0.025 mM where the order of the isotherms ly hydrophobic (20). reverses. Thus, at low concentrations, the global adsorption Finally, we found that the mass transfer kinetics was much
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FIGURE 4. Ball-and-stick representation of the active site of cellobiohydrolase I. The S-propranolol molecule (green C atoms); the protein residues (light brown C) whose atoms are within 4 Å of the ligand; and two water molecules (red). Probable hydrogen bonds are indicated with blue dots. The main interactions are the bidentate ion binding between the positively charged secondary amine of propranolol and the residues Glu212 and Glu217. Adapted with permission from Ref. 29.
slower on the enantioselective type II sites than on type I sites (11). This agrees with the qualitative observation of very broad and tailing peaks, especially for the most retained enantiomer. An important degree of tailing prevailed even at such low sample sizes that isotherm data demonstrated that type II sites operated in the linear range. Also, the calculation program (assuming fast mass transfer) gave band profiles in poor agreement with the experimental bands, in spite of the excellent agreement on the modeling of the experimental isotherms. Experimental band profiles were then compared with profiles calculated using a program implementing the simple kinetic model previously described to account for slow mass transfer kinetics, with two assumptions—a single rate constant and a rate constant for each type of site (11). The latter gave by far the best fits. Peak tailing caused by heterogeneous kinetics was theoretically predicted by Giddings (2, 24) but had remained unverified. X-ray crystallography revealed the detailed structure of Cel7A (25–29). The ␣-helices and -sheets of the molecule form an extended flat tunnel 40 Å long. The enantioselective site is inside this tunnel and overlaps with the catalytic site (28, 29). Because -blockers are cations in the pH range 614 A
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studied, ion binding requires anionic sites on either the protein or the silica matrix. The Cel7A molecule contains 43 carboxylic amino acid residues, 16 of which (11 aspartic and 5 glutamic acids) are located inside the tunnel. It has also 9 tryptophan (4 in the tunnel), 15 phenylalanine, and 24 tyrosine residues. However, like all nonpolar residues, phenylalanine residues tend to be hidden in the protein and unavailable for interactions with solutes. Although tryptophan is more hydrophobic, it is less hidden than phenylalanine because of its bulkiness. It was shown that the carboxylic groups of Glu217 and Glu212 are essential for both enzymatic and chiral activity (28, 29). Figure 4 shows a ball-and-stick representation of Cel7A and the binding of S-propranolol to its enantioselective site, suggesting that these two groups face each other on both sides of the protonated nitrogen atom of propranolol (29). The strongest hydrophobic interactions should be between the tryptophan residues on the Cel7A protein and the aromatic groups of the solutes. At least one tryptophan residue, Trp376, is sufficiently close to Glu212 and Glu217 to undergo hydrophobic stacking with the aromatic groups of -blockers. Our results agree with this picture. The chiral site is both ionic and hydrophobic. The number of nonselective interaction sites (either ionic or hydrophobic) between molecules of Cel7A and solute is much larger than that of enantioselective sites (one), consistent with type I and II monolayer capacities. With higher pH values, the number of anionic charges on the protein molecule grows rapidly, which explains the increase of the saturation capacity of type I sites with increasing pH. The binding energy on type IIS sites increases with higher pH because of the influence of the pH on the ionization degree of the two carboxylic groups at the selective site. The unusually large adsorption entropy of Spropranolol on type IIS sites suggests that this enantiomer more strongly perturbs the ordered structure of the water molecules surrounding the solute and the protein molecules, particularly those located in the Cel7A tunnel. This increased disorder of the water molecules explains the increased entropy and the origin of the hydrophobic nature of the interaction. It remains to be understood why the monolayer capacity of the enantioselective adsorption of R-propranolol is lower than that of S-propranolol but increases faster with increasing pH, up to that same level (at high pH, qsR tends to become equal to qsS). This might be due to a structural change of the microenvironment of the active site when the pH increases. Circular dichroism studies indicate a slight influence of the pH on the conformation of Cel7A (30). Too often, analytical chromatographers have ignored all nonlinear effects, although there are almost no experimental conditions under which these effects do not exist. Except for the perfunctory remark that nonselective interactions may contribute to retention on chiral stationary phases (CSPs) (31), few systematic investigations of this issue can be found in the literature. The works of Schurig (32, 33) and Boehm et al. (34) are most noteworthy. However, most ignore the possibility of determining and modeling isotherms, the only ac-
curate approach for separating the contributions of two different retention mechanisms. To pursue thermodynamic investigations of the important chiral retention mechanism, many chemists neglect the nonselective interactions, assuming without proof that they are negligible. Invariably, this leads to erroneous results (21, 35–37 ). For example, a method was recently suggested (35, 36) to “prove” whether both enantiomers interact with the same chiral site. The proposed method consists of equilibrating the chiral stationary phase with a solution of one of the enantiomers and injecting a sample of the other one. The operation is repeated with higher-concentration solutions of the first enantiomer. If a decrease in the retention times of the injected pulses is observed, this was claimed to be proof of competition for interactions with the same chiral site. This is incorrect: A decrease in retention would also originate from the unavoidable competition for the nonselective sites. Finally, interpretations and comparison between results obtained under different conditions must be made carefully. For example, we showed that the monolayer capacity of the enantioselective sites is lower for R than for S enantiomers at low pH (4.5), but it increases with higher pH and the two capacities becoming equal at high pH (6.0). At the same time, the chiral binding constant of S-propranolol increases considerably (19). Hedeland et al. reported that the association constant of S-alprenolol was 5.1 times larger than that of R-alprenolol (37 ). We also found the same saturation capacities and a ratio of 5.05 for the binding constants at pH 6.0 (20, 23). The agreement is excellent given the different methods used. The influence of the pH on these parameters is important, however, and was missed by Hedeland et al. (37 ). At pH 5.00, the ratio of the chiral capacities of S- and R-alprenolol is 4.85, but the chiral binding constants are similar. This illustrates the importance of pH on the different characteristics of some chiral retention mechanisms. At low pH, the chiral separation is mainly due to the difference in chiral monolayer capacities, not to that of the binding constants. The opposite is true at high pH.
described best the adsorption isotherms of these solutes. The monolayer capacity of the chiral type II sites was ~25-fold larger than those found for the -blockers on Cel7A, making Chiralcel OJ much more suitable for preparative applications. Later investigations using derivatized cellulose or other CSPs with a high density of chiral sites suggest a more complex picture. The complex isotherm models derived for studying heterogeneous surfaces are often better than the bi-Langmuir model to account for isotherm data (38, 39). The consequence is that it is impossible to separate, in the contributions of nonselective and enantioselective interactions in these cases. In the preparative separation of binary mixtures using traditional batch elution chromatography, the goal is generally the production of one component at a certain degree of purity, at either minimum cost (production) or maximum production rate (research and development). If the competitive isotherms and the mass transfer kinetic parameters are known, it is easy to calculate the profiles of the elution bands of the individual components in batch elution and the composition of the raffinate and extract streams during a cycle in simulated moving-bed (SMB) chromatography. These profiles and the purity criteria give the positions of the cut-points (i.e., the times when fraction collection should begin and end); the amounts produced during a cycle and the cycle time (the period between two successive injections), hence the production rate; the recovery yield (in batch elution); and the amount of solvent needed per unit of purified products. The production rate (in SMB) or a weighted function of production rate and yield (in batch elution) can be taken as the objective function in an optimization program. A recent study introduced the product of the production rate and the recovery yield as such a function (40).
There are almost no experimental conditions under which these effects do not exist.
High-capacity chiral phases Because the chiral monolayer capacities of immobilized proteins are very low, they are not the best CSPs for preparative applications. Rather, CSPs with large monolayer capacities are preferred. Charton et al. investigated the adsorption of methyl mandelate enantiomers on an immobilized cellulose phase (Chiralcel OJ) (38). The same bi-Langmuir equation
Batch elution systems Computer calculations of band profiles are an easy way to view displacement and tag-along effects (Figures 1b–d). Compared with “touching band operation”, which was prevalent in the early days, using these effects allows a large production rate increase for the first solute and a moderate one for the second. (Touching bands are defined as two peaks with a resolution of 1.) Ghodbane and Guiochon introduced this approach using the Knox HETP equation to relate mobilephase flow velocity, column efficiency, and a Simplex algorithm for optimization (41, 42). Felinger and Guiochon conducted systematic investigations of the simultaneous optimization of several parameters—those characterizing the column
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FIGURE 5. Schematics of (a) a traditional batch elution system, (b) a true continuous chromatographic system, and (c) a four-zone SMB unit with two columns in each section.
design (length, average particle size) and operating conditions (mobile-phase flow velocity, sample size)—and derived important guidelines (1, 40, 43). As an example, the optimum mobile-phase velocity for maximum production rate is always much higher than that for maximum column efficiency. The optimum retention factor of the first component is low, often
