Fundamentals of Preparative Preparatiue and of Chromatography Nonlinear Chromatography Second Edition
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Fundamentals Preparatiue and of Preparative Chromatography Nonlinear Chromatography Second Edition
Georges Guiochon
Attila Felinger
Department of Chemistry University of Tennessee University Knoxville, TN 37996-1600 USA USA
Department of Analytical Chemistry University of Pécs Pecs University Pecs H-7624 Pécs Hungary
Dean G. Shirazi
Anita M. Katti
Director of Analytical Development Development Laboratories Laboratories AAIPharma Wilmington, NC 28405 USA
Department Chemistryand andPhysics Physics Department of Chemistry Purdue University Calumet Hammond, IN 46323-2094 USA
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Preface The first edition of this book was written to consolidate the fundamental concepts which form the basis for the development of preparative separations by chromatography on the laboratory, pilot and process scales. Our goal had been to present the many issues one may encounter while working in the field of preparative and process scale chromatography, and to bridge the gap between the basic mathematics associated with the modeling of chromatographic separations and the experimental phenomena one observes during experimental exploits. Emphasis was put on the illustration of equations and concepts to provide a physical image of the different chromatographic phenomena along with written narrative explanations. Considerable attention was also given to the application of these concepts to the optimization of the separation conditions. We hope that we succeeded in providing readers with an understanding of the fundamental concepts of chromatography and with the methods needed to apply them to the solution of practical problems. Since the first edition is now out of print and much progress was made in the last ten years that is relevant to our topic, it was decided to prepare an updated second edition. This new book includes most of the important results acquired in the fundamentals of chromatography since the first one was published. The study of reversed phase liquid chromatography (RPLC) showed that the surface of the adsorbents that are used in this method is heterogeneous, with serious consequences on the nature of the equilibrium isotherms. The development of practical methods of determination of adsorption energy distributions informs on this heterogeneity. The real adsorbed solution theory affords improved accuracy in the prediction of competitive adsorption isotherms from the single- component isotherms of the components of a mixture. The new, inverse method of isotherm determination is gaining acceptance. Surface diffusion, competitive pore diffusion, the concentration dependence of the coefficients of mass transfer kinetics have been investigated in depth and this new knowledge proven useful in the prediction of band profiles. The simulated moving bed process has become main stream in preparative chromatography. All these issues and more are covered in this revised version. The reader will find here a complete mathematical development of the models of chromatography and other physical laws which direct the chemical engineer in the design and scale-up of chromatographic processes1. For preparative chromatographic separations, our ultimate purpose is the optimization of the experimental conditions for maximum production rate, minimum solvent consumption, or minimum production cost, with or without constraints on the recovery yield. The considerable amount of work done on this critical topic is presented in the 1
We did not intend to write a book on the mathematics of chromatography; this has been done excellently by Rhee, Aris and Amundson.
v
vi
Preface
last chapter that was considerably expanded. In this sense, it is the culmination of this book. The availability of powerful computers enables the use of sophisticated models of chromatography which take into account most of its complexity. Users must keep in mind, however, that all the parameters used in a model must be determined with an accuracy compatible with the improvement in prediction accuracy of the new model. Thus, the use of the general rate model remains still rarely warranted. The direct determination of the numerous rate constants involved is highly time consuming and difficult at best, inaccurate at worst. Their determination by curve fitting procedures gives an illusory feeling of confidence, but the parameters obtained are empirical and do not deserve the physical sense that is often afforded to them. The use of these parameters should be reserved to those applications involving interpolation of previously obtained experimental results, such as in the optimization of the experimental conditions of a given separation that has been studied in detail (with accurate measurements of competitive isotherm data in a wide range of relative and absolute concentrations and precise determination of rate coefficients). These considerations explain the relative importance that we have given to the models of chromatography, in relation to their different degrees of sophistication. The ideal model which requires only the determination of the equilibrium isotherms, permits a rapid estimation of the individual band profiles for convex upwards isotherms. At high loading factors, with modern, high efficiency packing materials, its results are strikingly accurate. Nevertheless, actual columns have a finite efficiency, axial dispersion and mass transfer resistances disperse the profiles, and often these effects need to be taken into account. They reduce yields and production rates of purified products. In almost all cases of practical importance in preparative chromatography, the equilibrium-dispersive model is satisfactory. In addition to the determination of the equilibrium isotherms, it requires only the measurement of the column HETP under linear conditions as a function of the mobile phase velocity. When the mass transfer resistances are important, as in some protein separations, or when particles of unusually large size or complex pore structure are employed, a simple kinetic model based on the use of a lumped kinetic parameter or the POR model should give excellent results. Some issues of importance in preparative chromatography are not discussed in this volume, such as instrumentation, column technology, column packing procedures, safety considerations, hardware layout, or the selection of solvents and stationary phases. Among the fundamentals, the thermodynamics of phase equilibria, its concentration dependence and competitive nature were given the primary importance. Recognizing the existence and the fundamental importance of these same characteristics, concentration dependence and competitive nature, for the diffusion and the mass transfer coefficients, the authors have discussed them in considerably more detail in this second edition than in the first one. However, they warn that these phenomena have limited practical importance in the preparative chromatography of most chemicals and drugs and that there is still a paucity of reliable data on the transport properties of proteins in chromatographic systems. The extension to the separation and purification of proteins of the concepts and tools applied so effectively to the separation and purification of regular
Preface Preface
vii
chemicals will remain an important and fruitful area of research for years to come. Also, the chromatographic column has been considered, throughout all our work, as linear. The radial dependence of the stationary phase density, the porosity of the packed bed and its permeability have been neglected, as well as the thermal effects associated with band migration. Experimental evidence available so far suggests that the consequences of these simplifications are minor in almost all practical cases. Approaches useful to handle these issues when needed are presented in Chapter 2. They have not been pursued. This book stems from the work done in the group of Professor Georges Guiochon at the University of Tennessee and at Oak Ridge National Laboratory in the late 1980's, 1990's and early 2000's. It contains the many contributions of the students, post-doctoral fellows and visiting scientists who came from all over the world to East Tennessee to contribute to the advancement of this field. Their contributions add to those of the many scientists who have worked in this area over the last sixty years and have produced innumerable, valuable publications.
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Acknowledgments The advancement of science involves the input of a multitude of people. Similarly, the production of this book required the support of numerous individuals. We gratefully acknowledge those who have over the years, and to various degrees, contributed by their comments, questions, suggestions, results, and insights to the progress of our cooperative work. Some of these include Tarab Ahmad (Knoxville, TN), Klaus Albert (Tubingen, Germany), Leonid Asnin (Perm, Russia), Henning Boysen (Dormung, Germany), B. Scott Broyles (Richmond, VA), Alberto Cavazzini (Ferrara, Italy), Frederic Charton (Nancy, France), Sophie Charton (Nancy, France), Yibai Chen (Philadelphia, PA), Djamel E. Cherrak (Woburn, MA), Martin Czok (Paris, France), Moustapha Diack (Baton Rouge, LA), Francesco Dondi (Ferrara, Italy), Eric V. Dose (Chicago, IL), Eric C. Drumm (Knoxville, TN), Zoubair El Fallah (Newport Beach, CA), Tivadar Farkas (Torrance, CA), Torgny Fornstedt (Uppsala, Sweden), Joerg Fricke (Dortmund, Germany), Wilmer Galinada (Union, NJ), Samir Ghodbane (Nutley, NJ), Gustaf Gotmar (Uppsala, Sweden), Fabrice Gritti (Knoxville, TN), Jun-Xiong Huang (Beijing, China), Laurent Jacob (Paris, France), Stephen C. Jacobson (Bloomington, IN), Francois James (Orleans, France), Pavel Jandera (Pardubice, Czech Republic), Alain Jaulmes (Thiais, France), Krzysztof Kaczmarski (Rzeszow, Poland), Marianna Kele (Milford, MA), Saad Khattabi (Exton, PA), Hyunjung Kim (Knoxville, TN), Ervin sz. Kovats (Lausanne, Switzerland), BingChang Lin (Anshan, China), Xiaoda Liu (Beijing, China), Zidu Ma (Lafayette, IN), David V. McCalley (Bristol, UK), Nicola Marchetti (Ferrara, Italy), Michel Martin (Paris, France), Daniel E. Martire (Washington, D.C.), Kathleen Mihlbachler (Indianapolis, IN), Kanji Miyabe (Toyama, Japan), Uwe Neue (Milford, MA), Joan Newburger (Trenton, NJ), Wojciech Piatkowski (Rzeszow, Poland), Igor Quinones Garcia (Union, NJ), Roswitha Ramsey (Chapel Hill, NC), Jeffrey Roles (Orlando, FL), Pierre Rouchon (Paris, France), Hong and Peter Sajonz (Edison, NJ), Matilal Sarker (Bellefonte, PA), Andreas Seidel-Morgenstern (Magdeburg, Germany), Andrew Shalliker (Sydney, Australia), Brett J. Stanley (San Bernardino, CA), Pawel Szabelski (Lublin, Poland), Ulrich Tallarek (Magdeburg, Germany), Nobuo Tanaka (Kyoto, Japan), Patrick Valentin (Solaize, France), Jennifer Van Horn (Indianapolis, IN), Claire Vidal-Madjar (Thiais, France), Tong Yun (Knoxville, TN), Guoming Zhong (Tonawanda, NY), Dongmei Zhou (La Jolla, CA), and Jie Zhu (Kenilworth, NJ). It was difficult to name all those who have contributed; our deepest regrets go to those who are unmentioned. We are thankful for the assistance and support of our colleagues, particularly Professors C. E. Barnes, F. M. Schell, and M. J. Sepaniak, and the staff (especially Mr. Bill Gurley) of the Department of Chemistry at the University of Tennessee in the solution of the countless problems encountered in the conduct of our research activities. GG would like to thank Lois Ann Beaver, who urged him to embark upon this endeavor and provided constant and welcome support toward its completion. His secretary, Cathy Haggerty, was helpful in many opportunities. DS ix IX
x
Acknowledgments Acknowledgments
would like to recognize his sister, Shari Shirazi, for her encouragement during this project. AF would like to thank Gabriella Felinger for her continued patience and understanding. AMK acknowledges A. Rachel Prakash (Atlanta, GA) for her assistance with the figures, JBK and LZK for their love and patience. We pay tribute to a colleague, a friend, and a mentor. Csaba Horvath was a leader whose work contributed greatly to the advancement of separation methods and whose remarkable experimental and theoretical contributions have had profound effects on our understanding of analytical and preparative liquid chromatography. We miss him. We acknowledge the constant support of the National Science Foundation over the last twenty years, through grants CHE-8519789, CHE-8715211, CHE-8901382, CHE-9201663, CHE-9701680, CHE-00-70548, and CHE-02-44693. These grants enabled us to obtain the results which are our contribution to the Fundamentals of Preparative and Nonlinear Chromatography.
Contents Preface
v
Acknowledgements
ix
1 Introduction, Definitions, Goal 1.1 History of Chromatography 1.2 Definitions 1.3 Goal of the Book References
1 3 12 15 17
2 The Mass Balance Equation of Chromatography and Its General Properties 2.1 Mass and Heat Balance Equations in Chromatography 2.2 Solution of the System of Mass Balance Equations 2.3 Important Definitions References
19 21 42 57 63
3 Single-Component Equilibrium Isotherms 3.1 Fundamentals of Adsorption Equilibria 3.2 Models of Adsorption Isotherms in Liquid-Solid Equilibria . . . . 3.3 Adsorption and Affinity Energy Distribution 3.4 Influence of Experimental Conditions on Equilibrium Isotherms . . 3.5 Determination of Single-Component Isotherms 3.6 Data Processing and Assessment References
67 70 80 109 117 122 135 144
4 Competitive Equilibrium Isotherms 151 4.1 Models of Multicomponent Competitive Adsorption Isotherms . . 153 4.2 Determination of Competitive Isotherms 191 References 216 5 Transfer Phenomena in Chromatography 221 5.1 Diffusion 222 5.2 Axial Dispersion and Mass Transfer Resistance in Porous Media . . 240 5.3 The Viscosity of Liquids 257 References 275 6 Linear Chromatography 6.1 The Plate Models 6.2 The Solution of the Mass Balance Equation 6.3 The General Rate Model of Chromatography 6.4 Moment Analysis and Plate Height Equations 6.5 The Statistic Approach 6.6 Sources of Band Asymmetry and Tailing in Linear Chromatography 6.7 Extension of Linear to Nonlinear Chromatography Models References xi XI
281 283 290 301 310 328 335 341 342
xii
Contents Contents
7 Band Profiles of Single-Components with the Ideal Model 7.1 Retrospective of the Solution of the Ideal Model of Chromatography 7.2 Migration and Evolution of the Band Profile 7.3 Analytical Solutions of the Ideal Model 7.4 The Ideal Model in Gas Chromatography 7.5 Practical Relevance of Results of the Ideal Model
347 349 351 363 377 379
8 Band Profiles of Two Components with the Ideal Model 8.1 General Principle of the Solution 8.2 Elution of a Wide Band With Competitive Langmuir Isotherms . . 8.3 Elution of a Narrow Band with Competitive Langmuir Isotherms . 8.4 Method of Calculation of the Ideal Model Solution in a Specific Case 8.5 Dimensionless Plot of a Two-component Band System 8.6 The Displacement Effect 8.7 The Tag-Along Effect 8.8 The Ideal Model in Gas Chromatography 8.9 Practical Relevance of the Ideal Model References
387 390 395 401 407 414 416 419 421 423 436
9 Band Profiles in Displacement Chromatography with the Ideal Model 9.1 Steady State in the Displacement Mode. The Isotachic Train . . . . 9.2 The Theory of Characteristics 9.3 Coherence Theory 9.4 Practical Relevance of the Results of the Ideal Model References
437 439 450 461 467 468
10 Single-Component Profiles with the Equilibrium Dispersive Model 10.1 Fundamental Basis of the Equilibrium Dispersive Model 10.2 Approximate Analytical Solutions 10.3 Numerical Solutions of the Equilibrium-Dispersive Model 10.4 Results Obtained with the Equilibrium Dispersive Model References
471 473 476 492 509 527
11 Two-Component Band Profiles with the Equilibrium-Dispersive Model 531 11.1 Numerical Analysis of the Equilibrium-Dispersive Model 532 11.2 Applications of the Equilibrium-Dispersive Model 542 References 567 12 Frontal Analysis, Displacement and the Equilibrium-Dispersive Model 569 12.1 Displacement Chromatography with a Nonideal Column 570 12.2 Applications of Displacement Chromatography 587 12.3 Comparison of Calculated and Experimental Results 599 References 603
Contents Contents
xiii
13 System Peaks with the Equilibrium-Dispersive Model 13.1 System Peaks in Linear Chromatography 13.2 High-Concentration System Peaks References
605 606 626 647
14 Kinetic Models and Single-Component Problems 651 14.1 Solution of the Breakthrough Curve under Constant Pattern Condition 653 14.2 Analytical and Numerical Solutions of the Kinetic Models 669 14.3 Comparison Between the Various Kinetic Models 680 14.4 Results of Computer Experiments 687 14.5 Numerical Solution of the Lumped Pore Diffusion Model 689 14.6 The Monte Carlo Model of Nonlinear Chromatography 693 References 695 15 Gradient Elution Chromatography under Nonlinear Conditions 699 15.1 Retention Times and Band Profiles in Linear Chromatography . . . 701 15.2 Retention of the Organic Modifier or Modulator 705 15.3 Numerical Solutions of Nonlinear Gradient Elution 711 15.4 Gradient Elution in Ion-Exchange Chromatography 726 References 731 16 Kinetic Models and Multicomponent Problems 16.1 Analytical Solution for Binary Mixture; Constant Pattern Behavior . 16.2 Linear Driving Force Model Approach 16.3 Numerical Solution of The General Rate Model of Chromatography References
735 736 747 754 775
17 Simulated Moving Bed Chromatography 17.1 Introduction 17.2 Modeling of Simulated Moving Bed (SMB) Separations 17.3 Analytical Solution of the Linear, Ideal Model of SMB 17.4 Analytical Solution of the Linear, Nonideal Model of SMB 17.5 McCabe-Thiele Analysis 17.6 Optimization of the SMB Process 17.7 Nonlinear, Ideal Model of SMB 17.8 Recent Improvements in SMB Performance with New Operating Modes 17.9 Numerical Solutions for Nonlinear, Nonideal SMB References
779 780 783 785 806 808 809 816 826 836 845
18 Optimization of the Experimental Conditions 18.1 Definitions 18.2 The Economics of Chromatographic Separations 18.3 Optimization Based on Theoretical Considerations 18.4 Optimization Using Numerical Solutions 18.5 Recycling Procedures
849 851 857 867 883 915
xiv
Contents Contents 18.6 Practical Rules 18.7 Optimization of the SMB Process References
920 924 935
Glossary of Symbols
939
Glossary of Terms
949
Index
969
Chapter 1 Introduction, Definitions, Goal Contents 1.1 History of Chromatography 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5
Discovery by Tswett and Early Works The Rebirth of Chromatography The Manhattan Project and the Purification of Rare Earth Elements The API Project and the Extraction of Purified Hydrocarbons from Crude Oils . Preparative Chromatography as a Separation Process
1.2 Definitions 1.2.1 1.2.2 1.2.3 1.2.4
Linear and Nonlinear Chromatography Ideal and Nonideal Chromatography Separation, Extraction, and Purification The Various Scales of Preparative Chromatography
1.3 Goal of the Book References
3 3 4 5 6 8
12 13 13 14 15
15 17
Introduction Chromatography was born as a preparative technique [1] at the turn of the last century. This was a time when there were no physical methods of analysis, when the acquisition of physico-chemical data was slow and limited to a few parameters of low specificity (e.g., melting points, densities, refraction indices). Analytical methods were essentially based on chemical reactions, they were slow, and had a poor sensitivity. Originally designed to extract purified plant pigments from complex mixtures of vegetal origin, chromatography had to be used with sequential fraction collection, followed by the off-line analysis of these collected fractions. In the early 1950s, the development of sensors, transforming the variations of physical or physicochemical properties into a current or a voltage and now known as detectors, and that of recorders of the electric signals provided by these sensors transformed science. With the invention of these devices, modern instrumentation changed profoundly the way in which chemists use chromatography. In the early days of chromatography, the lack of sensors and the need to perform chemical reactions on the isolated fractions to identify and quantify their components imposed off-line detection. Then, the relative lack of sensitivity of most of the chemical methods of detection available at the time dictated the use of large-diameter columns, the injection of large samples, and the handling of only concentrated sample solutions. All these reasons combined made the chromatographic technique nonlinear, resulting in strongly unsymmetrical individual
2
Introduction, Definitions, Goal
band profiles. Moreover, under such conditions, the retention times and the band shapes depend not only on the amount of each component in the sample, but also on the composition of that sample. It was only in the mid-1950s that the development of gas chromatography [2] and the rapid progress made in the field of instrumentation permitted a considerable reduction in the column loading and made possible the operation of columns under linear conditions. Progressively, (1) the advancement of on-line detection, of on-line recording of the eluent composition, and later of the automatic integration of the peak area; (2) the development of new, extremely sensitive detectors; and (3) the progressive extension of on-line detection with very sensitive detectors to all the modes of chromatography led to the close association in the minds of analytical chemists between analytical chromatography and operation under linear conditions. Finally, a stage was reached where column overloading was considered to be an error, if not a sin, and was avoided at all costs. From there, preparative chromatography had to be rediscovered: a mental process that requires the painful revisitation of our knowledge and the reassessment of many rules and principles. For the last forty years, there has always been interest in the use of chromatography for the purification of valuable compounds. Several attempts at commercializing and popularizing preparative gas chromatography were made in the 1960s and 1970s. They met with little success [3-5]. The main reasons for these failures were economic. Few compounds are both valuable enough to justify the extraction/purification costs of this process and volatile enough to be purified at an affordable price by preparative gas chromatography. The rapid development of the fine chemical, pharmaceutical, and biotechnology industries during the last twenty years has combined with the pressure of the regulatory agencies. Their effort has led to the production of many high-purity chemicals to be used as pharmaceuticals or pharmaceutical intermediates, to the identification of the metabolites of these compounds, and to the completion of systematic studies of the toxicological properties of potential drugs and of their metabolites prior to their approval. This endeavor has generated the need to separate, extract, and purify many chemicals in the laboratory or at the industrial scale. Moreover, traditional techniques such as distillation, counter-current or centrifugal extraction, and crystallization used by the petroleum and the commodity chemical industries do not meet the needs of the pharmaceutical industry. No industrial separation technique is more versatile than chromatography, nor better suited for the rapid production of milligram to ton quantities of highly pure products. None has a comparable separation power. In recent years, preparative chromatography has been adopted as an industrial process in the pharmaceutical industry. Units with a production capacity ranging from a few pounds to more than a thousand tons per year have been built. In the meantime, the chemical industry has developed numerous processes based on the use of adsorption. The complexity of these processes has been increasing constantly. Some of the recent ones are based on the use of chromatographic principles. This is the case for separation processes based on the simulated moving bed concept [6-8]. Initially developed for the extraction of a few
1.1 History of Chromatography
3
specific compounds from complex mixtures, such as para-xylene from reforming streams or fructose from corn syrup, these processes are competing with the simpler chromatographic processes evolved from direct scaling-up of the laboratory procedures. Currently, equipments for overloaded elution may achieve production rates of up to 500-1000 ton/year or above and can handle most complex mixtures. Simulated moving bed units have been built with production rates between a few pound/year to more than a million ton/year. These equipments are unsurpassed for the separation of binary mixtures {e.g., enantiomers). A variety of recycling processes involving elution can fill in the gap.
1.1 History of Chromatography Tswett (1872-1919) was ahead of his time. A long induction period followed the tragic interruption of his work during the Russian civil war and his untimely death. Not until the early 1930s did the importance of chromatography became recognized among chemists involved in the study of natural products [9] and biochemists who continued to play a critical role at several stages of development {e.g., in the discoveries of paper chromatography [10], gas chromatography [2], size exclusion chromatography [11], and affinity chromatography [12], among others). With the progress made in the development of sensitive detection methods, analytical and preparative chromatography parted in the late 1940s. The first major preparative chromatography projects were the purification of rare earth elements by the group of Spedding [13] for the Manhattan project, and the isolation of pure hydrocarbons from crude oil by Mair et al. for the API project [14]. Later followed the development of the simulated moving bed technology by Broughton for UOP [6]. Finally, in the 1980s, the pharmaceutical industry began to show interest in high-performance preparative chromatography and this interest has been increasing steadily over the last twenty years [15]. Chromatography is now acknowledged as an industrial unit operation for the extraction and the purification of fine chemicals, particularly those used as pharmaceutical intermediates.
1.1.1 Discovery by Tswett and Early Works The story of the discovery of chromatography is classical [16,17]. A most lucid analysis of Tswett's work from the point of view of the preparative applications of chromatography, has been written by Verzele and Dewaele [18]. The Russian botanist Tswett discovered around 1902 that plant pigments could be separated by eluting a sample of plant extract with a proper solvent on a column packed with a suitable adsorbent [1]. Did he name the technique chromatography because it separates pigment mixtures into a rainbow of colored bands, or because "tswett" means color in Russian, or both? Nobody knows. What is remarkable, however, is the extreme care with which Tswett selected the adsorbents he used [19-21]. For the famous separation of a- and /3-carotenes, he tried 110 different adsorbents and selected inulin (a water-soluble polyfructose plant reserve material) as the
4
Introduction, Definitions, Goal
stationary phase, and ligroin (the 70-120° C distillation cut of crude oil) as the eluent [18-22]. Thus, the first publications on chromatography are masterpieces in method development. From its inception, the method suffered from a relatively poor production rate (poor relative to the size of the column used and to the amount of solvent that is needed per unit amount of products extracted). This limited production capacity was the main criticism that Willstater made to the work of Tswett [17]. To this criticism, Tswett rejoined that production rate could always be increased by using larger bore columns. After a century, the dilemma remains unchanged. The strength of the objection, the scientific stature of Willstater, and the late recognition of the extreme separation power of chromatography delayed its adoption as a laboratory technique for a third of a century. Tswett understood very well the importance of the nature of the adsorbent and the key role of differential adsorption in the separation [1,19-21]. His most popular result, the separation of the carotene and xanthophyll isomers on a calcium carbonate column, turned out to be most difficult to reproduce by later workers. Tswett had understood the importance of a number of characteristics of the packing material that had escaped the attention of the early followers, in part because the Russian text of his Ph.D. thesis was unavailable to most of them. The characteristics discussed by Tswett were (i) the purity of the adsorbents, (ii) the average particle size (which should be fine), and the size distribution (which should be narrow), (iii) the packing homogeneity, and (iv) the integrity of the top of the packed bed (which should not be disturbed when the sample is injected). He had also recognized the importance of avoiding chemical changes catalyzed by the packing, a serious risk in the analysis of carotenoids, and the particular inertness of polyol-based adsorbents. There are very few serious claims to a glimpse at chromatography predating Tswett's work or even to an independent later discovery, in spite of the 30-yearlong induction period that followed Tswett's earlier publications. The one who probably came closest was Day [22], who attempted to fractionate petroleum by filtration through columns of powdered limestone or fuller's earth.
1.1.2
The Rebirth of Chromatography
Unfortunately, the rare scientists who have used chromatography in the beginning of the 20th century [23-26] did as Tswett had done. They purified only very small amounts of a few natural pigments in solution, for further spectroscopic studies. For nearly thirty years, chemists remained reluctant to use chromatography. Its production rate was very small, at a time when organic chemistry reactions were carried out on a large scale and when new chemicals were produced in amounts that seem, now, for us, to be amazingly large. Accordingly, chromatography had to be run at a scale that would now be that of laboratory preparative chromatography. It was an expensive method requiring large volumes of solvents. Nonlinear effects made the results of the method difficult to understand and control. The lack of detectors made its application to colorless compounds impractical. Furthermore, serious doubts had been cast by Willstatter and Stoll [27] on the integrity of the collected products. This was due to their use of inadequate adsorbents that
1.1 History of Chromatography
5
catalyzed the isomerization of the labile carotenoids and the other plant pigments that they were studying. Tswett had warned against the use of silicagel in these separations but the reasons for this advice had not been understood. So, Tswett's findings were not reproduced and attracted little interest until Kuhn and Lederer [28,29] demonstrated the power of chromatography as a preparative method for the separation of many carotenoids, performing the separation of a- and /3carotene from carrots and separating egg yolk pigments [30] on a 7-cm i.d. column packed with calcium carbonate, using carbon disulfide as the mobile phase. This last work yielded 30 mg of carotene. Gram amounts were prepared soon afterward [31]. The interesting story of the rediscovery of chromatography in Heidelberg in 1931 is told in quite impressive detail by Lederer [9]. Among many contributions dating from that period, we must note the clear and systematic definitions by Tiselius and Claeson in 1943 [32] of the three modes of chromatography, already known to Tswett [19-21,33] — elution, frontal analysis and displacement. Displacement became very popular during the late 1930s and the 1940s, later to fade away and reemerge in the 1980s under the creative impulsion of Csaba Horvath [34]. The properties and performance of this method are discussed in Chapters 9 and 13.
1.1.3 The Manhattan Project and the Purification of Rare Earth Elements The progressive realization that adsorption chromatography could provide a high selectivity that few other separation techniques could offer paved the way for several large-scale applications, such as the isolation of oxides of rare earth elements for nuclear applications and the purification of petroleum products [35]. In the mid-1940s, the development of large-scale purification schemes for the separation of rare earth metals [13,35-41] stemmed from the demands of physicists who became interested in the unusual nuclear properties of the actinide elements in the early 1940s [38]. These investigations were carried out under the auspices of the Manhattan Project. Purification of the rare earth elements by ion-exchange chromatography was primarily conducted by Spedding et ah [13,35-41], who were able to separate various salts by displacement chromatography. Figure 1.1 presents a chromatogram whereby they separated three of the rare-earth ions, Samarium, Neodymium, and Praseodymium on Amberlite IR-100, using three 22 mm i.d., 30, 60, and 120 cm long columns, and a 0.1% buffer solution of citric acid and ammonium citrate at pH 5.30 as the displaces Other experiments show that the separation and the retention factors increase with decreasing pH, resulting in an optimum pH around 5. Numerous investigations on the effects of the various operating parameters were carried out (see further explanations in Chapter 12). A pilot plant, involving sixty-eight 4- and 6-inch-diameter columns, in a fourstep cascade, was developed to achieve the production of high-purity metals in large quantities [35,38-40]. The concentration of the citrate buffer was minimized to reduce costs. The advantage of using fine resin particles was noted, as well as the rather long column length needed to achieve formation of the isotachic
Introduction, Definitions, Goal
r
150 -
3 0 CM
100 -
50 ;
0 -.
/
.
.
i
10
¥
\
15
20
25
150 "
100 "
50 -
Figure 1.1 Separation of three rare-earth elements, Sm, Nd, and Pr, by displacement chromatography. Columns: 22mm i.d., 30, 60, and 120 cm long, packed with Amberlite IR-100 resin. Displacer solution: 0.01% citrate solution, pH 5.3. Reproduced with permission from F.H. Spedding, E.I. Fulmer, T.A. Butler and /.£. Powell, } . Am. Chem. Soc, 72 (1950) 2354 (Fig. 8). ©1950 American Chemical Society.
0
\r Y I, .V,
•
20
6 0 CM
1 . ,
30
25
39
150 120 CM
100 -
f\
50 -
i i i
0
. i , L 40
• V
'
•
•
V AL
1
45 50 55 Volume of eluate, liters.
1 .
1 .
1 1
60
train, and the improvement of the separation with increasing column length at constant loading factor.1 One major source of difficulties was the growth of molds in the column, efficiently controlled by adding 0.1% phenol to the mobile phase, an addition which had no effect on the separation [35].
1.1.4
The API Project and the Extraction of Purified Hydrocarbons from Crude Oils
In the late 1940s and early 1950s, the American Petroleum Institute (API) used displacement chromatography to fractionate samples of virgin crude oil and petroleum distillates to determine their content of paraffins, naphthenes, olefins, and aromatics and to isolate many pure compounds that were identified for the first time in crude oil or crude oil extracts [42]. In addition to working on the laboratory scale, 1
Ratio of the sample size to the amount of solute needed to saturate the resin.
1.1 History of Chromatography
THIRD FLOOR-
Lib SECOND FLOOR.
PENTHOUSE
Figure 1.2 Assembly of a 52-foot adsorption column. Aj, A2, connections to a source of pressurized nitrogen. B, Vent to roof. Cj, C2, Copper tubes, 1-inch o.d. for filling reservoirs. Dj-Dio, Packless diaphragm valves. Ej, E2, brass reservoirs to contain hydrocarbon and alcohol. F, steel plate for supporting reservoir. G, copper tubing. H, 1-liter receiver for recovering alcohol regenerant. I, prismatic sight glass. Ji, ~\i, top and bottom flanges. Kj, K2, top and bottom gaskets. L, petcock. M1-M4, transite collars to permit vertical movement. N1-N3, angle iron supports. O, aluminum foil. P, magnesia insulation. Q, nichrome heating wire. R, asbestos coated with resin. S, 1-inch o.d. stainless steel column. T, 200-325 mesh silicagel adsorbent. U, thermocouple well. V, steel plate supporting entire weight of column. W, glass wool. X, brass plug supporting adsorbent. Y, standard tapper. Z, receiver. Reproduced with permission from B.D. Mnir, A.L. Gaboriault, and F.D. Rossini, Ind. Eng. Chem., 39 (1947) 1072 (Fig. 1). ©1947 American Chemical Society.
a team led by Mair and Rossini [14,43-45] assembled and tested six 3/4 inch (19 mm) i.d., 52.4 foot (16 m) long stainless steel columns, a schematic of which is shown in Figure 1.2 [14]. These columns were placed in an elevator shaft. They were temperature controlled. Each column (volume, 5.52 L) was packed with 3.7 kg of 200-235 mesh silica gel, the total porosity being 77%. Columns were operated in the displacement mode, at first using isopropanol as the displacer. The typical sample size was 0.5 L. Sample introduction in the column took 12 hours; the first component broke through after 34 hours. Sample collection (1550 mL) took 21.5 hours and the average liquid flow rate was 72 mL/hour, with an inlet pressure of 15 psi at the
Introduction, Definitions, Goal
1 -methylnaphthalene
160
:
155 triethylbenzene
150
r
:
145 -
\
1 \
n-dodecane
140 -
isopropanol
135
:
1
.
.
1—-,
400
,
,
1
800
,
,
,
1
,
,
,
1200
1
1600 Volume, mL
Figure 1.3 API Project, separation of a synthetic mixture of n-dodecane, triethylbenzene, and 1-methylnaphthalene with isopropanol as the displacer. Column as in Figure 1.2, packed with silica gel. Reproduced with permission from B.D. Mair, A.L. Gaboriault, and F.D. Rossini, Ind. Eng. Chem., 39 (1947) 1072 (Fig. 5). ©1947 American Chemical Society.
beginning of the experiment and 90 psi at the end. An example of the composition profile of the column eluent is reproduced in Figure 1.3. Because the mixed zone between the last component and the displacer was too wide, isopropanol was replaced as the displacer by 3-methyl-l-butanol, which provided a much sharper separation [14]. The importance of using a displacer more viscous than the feed is noted but not justified (this was the first observation of the effects of viscous fingering in chromatography; see end of Chapter 5 for explanations).
1.1.5
Preparative Chromatography as a Separation Process
In the early 1970s, Union Oil developed and patented a chromatographic system based on the principle of a simulated moving bed (SMB) [6-8]. A schematic of a SMB unit is shown in Figure 1.4. Streams of the mobile phase (the 'desorbent') and of the feed to separate are continuously injected into the column while streams of the less retained (the 'raffinate') and the more retained components (the 'extract') are continuously withdrawn, all at constant flow rates. The rotary valves switch periodically the positions in the columns where these streams enter or exit. The operation of SMB units is discussed in detail in Chapter 17. Manufacturing facilities have been built and are operated for the fractionation of various petroleum distillates, for example, the selective separation of p-xylene, o-xylene and ethylbenzene from the C7-C8 aromatic fraction of light petroleum reformates, the separation of olefins from paraffins in feed mixtures of hydrocarbons having 10 to 14
1.1 History of Chromatography
LEGEND AC = ADSORBENT CHAMBER RV = ROTARV VALVE EC = EXTRACT COLUMN RC = RAFFIHATE COLUMN
Figure 1.4 Chromatographic system based on a simulated moving bed. ReprintedfromD.B, Broughton, Sep. Sci. Tech., 19 (1984) 723 (Fig. 5) by courtesy of Marcel Dekker Inc.
carbon atoms, and the separation of fructose and dextrose from corn syrup. It has been shown that chromatography, which is normally a batch process in both the elution and the displacement mode, could be turned into a continuous process if the stationary phase were forced to move along the column. Physically moving the stationary-phase bed is impractical (Chapter 17). However, the moving bed operation can be simulated, as accomplished in the Union Oil process by the use of a number of columns placed on-line and connected to a rotary valve [6]. The position of the desorbent, extract, feed, and raffinate can be switched in a practical way, permitting continuous unit operation (Chapter 17). In the late 1970s, Elf Inc. (France) developed a process to extract a large proportion of the n-alkanes (mainly n-pentane and n-hexane) contained in light petroleum distillates to prepare high-octane gasoline [46]. This process is based on the injection of large pulses of feed on a Molecular Sieve column. The branched pentanes and hexanes are not retained and elute as a large unretained pulse. In this process, the n-alkanes are strongly retained and elute as a very wide band. In practice, when feed pulses are injected sequentially at a sufficiently high frequency, the eluent contains a constant, low concentration of n-alkanes, which is given by the product of their concentrations in the feed and the ratio of the pulse duration to the pulse period. The elution bands of isoparaffins are collected as a highoctane gasoline. With this gas chromatographic process, it is easy to decrease the n-alkane concentration in the gasoline by a factor that slightly exceeds two. The plans to build a 100,000 tons per year plant were canceled after the first petroleum price shock, due to financial difficulties independent of the economics of the process [44]. Several designs were suggested in the 1970s for the continuous operation of the
Introduction, Definitions, Goal
10
FIXED FEED INLET
ELUENT
ELUENT INLET
STATIONARY ELUENT WASTE COLLECTION
- ANNULAR BED
Figure 1.5 Chromatographic system based on a rotating annular column. Reproduced with permission from A.}. Howard, G. Carta and CM. Byers, Ind. Eng. Chem., 27 (1988) 1873 (Figs. 1 and 2). ©1988 American Chemical Society.
chromatographic process, by rotating the column around its axis while the feed injector and the fraction collector remain fixed (Figure 1.5 [47]). Baxter and Deeble used a rectangular cross-section column, with an open side sliding along a plate pierced by several holes for the feed injection and the collection of a "retained" {i.e., moving backward, in the same direction as the column) and an "unretained" {i.e., moving forward, in the same direction as the carrier gas) fraction [48]. Scott et al. used a rotating, annular column [49]. The feed is injected at a fixed point. The solutes follow helicoidal paths in the annular column and exit at fixed places where the corresponding fractions are collected. Although this process is continuous, similar to elution chromatography, only a fraction of the column volume works at any given time. The implementations of these designs are complex. Perhaps because this complexity was considered excessive by chemical engineers, or because its performance was no better than that of competitive processes, or because the design was early for the times, or for some other reason, this process has not been accepted yet. New designs, including that of simulated rotating annular columns, are proposed occasionally in the literature or in communications made at meetings and a few implementations are made commercially available for some time. It seems that this solution is still looking for the problems that it can solve usefully. Over the last twenty years, the use of semipreparative and preparative chromatography has expanded considerably. A vast number of applications has been reported, mostly in the pharmaceutical industry, where preparative chromatography is an important general-purpose separation process. The amounts of purified
1.1 History of Chromatography Figure 1.6 Effects of a progressive consolidation of a column bed. (a) Photograph of a process-chromatography flanged-end column (no compression) after removal of the flange, (b) Chromatogram of a test sample in this column voids and after its repacking, (c) Chromatogram of a test sample in a chromatographic columns with and without compression. The exact amounts injected are different for all chromatograms.
11
Repack With Compression
Without Comprettiion
^ Voids
Time
products that are required for these applications are compatible with the use of columns ranging from a few inches to a few feet in diameter, which are technically quite feasible to build, pack, and operate. The purity requirements are often easier to meet in chromatography than with other separation methods. Most fine chemicals, particularly those to be used for the production of pharmaceuticals, can bear the cost of chromatographic separations. The purifications of enantiomers, peptides, and proteins are the most widely published applications, but many others have been reported. Several reviews [50-52], numerous books [53-58] and many special volumes of the Journal of Chromatography have discussed these applications [59]. One of the critical problems of large scale applications of preparative chromatography has been the achievement of large diameter columns having a separation power comparable to that of analytical columns. Long-term bed stability, high column efficiency, and low hydrodynamic resistance are important qualities required of industrial columns. Manufacturing groups have long complained that after a certain period of satisfactory operation runs lasting from a few days to several months, the performance, particularly the efficiency, of wide columns decreases sharply. When the column is opened, large voids are seen at the top of the bed. Filling them with fresh packing material can restore the initial column efficiency, provided suitably intense vibrations are applied. Figure 1.6 illustrates a large process-scale column with voids, as indicated by the arrow on the RHS of the photo. Below are chromatograms showing channeling and tailing as a result of void formation. To prevent this phenomenon and its often costly consequences, various implementations of bed compression have been proposed. Compression can be static or dynamic, the latter providing the large advantage of automatically eliminating the voids as they form, before their size can make them harmful for the column efficiency. With static methods, bed compression has to be applied by the operator when the column efficiency has dropped below a threshold and this reduces the performance of the unit compared to dynamic compression. Compression can be annular [60], axial [61] or radial [62]. Currently, implementations are commercially available for dynamic axial and radial compression and for static annular compression for semipreparative units but only for dynamic axial compression for large-scale industrial units.
12
Introduction, Definitions, Goal
The hardware employed in preparative liquid chromatography typically has the capability for feedback control of the pumps, automatic column switching, column backflushing, recycling, gradient mixing, online pressure, flow rate, UVabsorbance, and temperature monitoring, and automatic pneumatic actuation of fraction collection valves based on time, volume, or UV-absorbance thresholds. These capabilities are afforded by computer control. In conclusion, it is interesting to note that the early applications of preparative liquid chromatography in the 1940s and 1950s involved operating the column in the displacement mode [13,14,32,33,35-45], in which by definition chromatography is performed under nonlinear conditions. In the 1980s the approach to preparative scale chromatography was somewhat different, in fact nearly opposite. Twenty years of successful applications of liquid chromatography in the achievement of all kinds of analytical separations where column "overloading" was always scrupulously avoided had made chemists extremely reluctant to perform separations under conditions that deviate strongly from analytical practice. The early trend involved the nearly exclusive use of large-diameter columns operated in the elution mode, with the injection of rather large volumes of dilute solutions, in order to achieve "volume overloading" with no or only moderate "concentration overloading." In most cases, however, concentration overload is a far more economical approach. Realization of this situation led to thorough studies of the properties of nonlinear chromatography, and their mastery in the operation of columns under conditions of heavy volume and concentration overload. Still, elution is practically the only mode of chromatography used. In the last twenty years, proponents of displacement have relentlessly presented claims that pose serious advantages for certain separation problems [63,64]. However, demonstrations of practical applications of this technique in the regulated environment of the pharmaceutical industry have never been supplied. The error of the claim that displacement provides a 100% recovery yield because of the boxcar elution of its bands has been demonstrated (see Chapter 12). The progressive realization that isocratic elution, whenever possible, leads to larger production rates, higher recovery yields, and easier operation, albeit to the production of more dilute fractions than displacement [65-67] has ended the controversy. For proteins and moderate or large peptides, the balance of advantages and drawbacks of the displacement and elution methods has long remained uncertain as, in most cases, these biochemicals cannot be extracted or purified by isocratic elution. It seems now that the advantage goes to gradient or step gradient elution. The various recycling procedures available, as well as the simulated moving bed process, continue to undergo thorough investigations now that the fundamentals of nonlinear chromatography are better understood.
1.2
Definitions
In order to clarify matters, we present here a number of definitions of terms frequently used in this book.
1.2 Definitions
13
1.2.1 Linear and Nonlinear Chromatography In linear chromatography, the equilibrium concentrations of a component in the stationary and the mobile phases are proportional. In other words, the equilibrium isotherms are straight lines beginning at the origin. The individual band shapes and the retention times are independent of the sample composition and amount. The peak height is proportional to the amount of each component in the injected sample. Linear chromatography accounts well for most of the phenomena observed in the analytical applications of chromatography, as long as the injected amounts of the sample components are kept sufficiently low. Linear chromatography is discussed in Chapter 6. Since any isotherm can be expanded into a second degree polynomial, q(C) = aC + bC2, we consider any chromatographic experiment as carried out under linear conditions as long as bC tp
Thus, the maximum concentration, Co, can now be realistic. The injected amount is n = CotpFv, where Fv is the mobile phase flow rate. The loading factor along the column axis is Lj- = uotpCo/ (L(l — e)qs). The loading factor is the sample size referred to the monolayer capacity of the adsorbent in the column bed. It is often observed, however, that the actual injection profiles are far from the Dirac model, as illustrated in Figure 2.3b, which compares a rectangular pulse injection of 100 ]iL (solid line) and the injection profile recorded with a six-port Valco valve (Houston, TX) fitted with a 100-^L loop [42]. The Dirac injection is an acceptable model only if the width of the experimental injection is small compared to the standard deviation of the band profile under linear conditions. Usually, the experimental injection profile has a sharp front followed by a tailing decay (Figure 2.3b). This profile is also typical of those encountered in preparative chromatography, except that they include a concentration plateau lasting for a certain period of time (see Figure 2.3). Whenever numerical calculations are carried out to predict band profiles from equilibrium isotherms and kinetic data (see Chapter 10) or to derive the equilibrium isotherms from acquired band profiles (see Chapters 3 and 4), it is imperative accurately to model the actual boundary condition, i.e., to perform the calculations using the concentration profile of the feed as it enters into the column. The importance of the selection of the boundary condition, of its modeling in certain cases, has been demonstrated many times [42-45]. Figure 2.4 illustrates the importance of following this recommendation when comparing experimental and calculated band profiles. The extra-column band broadening—in the case of an infinitesimally narrow impulse injection—is usually accounted for by the exponentially modified Gaussian (EMG) function, which is the convolution of a Gaussian peak and an exponential decay function (see Chapter 6, Section 6.6.1). The former contribution describes the band broadening in the connecting tubes while the latter models the mixer-type extra volumes [46]. When the ideal inlet profile is a wide rectangular pulse, the true inlet concentration can be modeled by the convolution of the EMG function and a rectangular pulse of length tp. The resulting profile is _. . Cm
=
\ [ . m —t — < erfc—-= 2« I y/lff
tn + m — t ( o2 erfc——-= V exp —-= -\ V2CT * \2x2
m-t\
_ e r f,c / cr
V2T
m-t\ T +
x
)
m-
V2
where m is the residence time in the connecting tube, c is the Gaussian band width and T is the time constant of the mixer-type extra-column volume. The measured inlet concentration profile for a 0.9-minute injection of the content of a loop filled with a dilute solution of aniline is reported in Figure 2.3c (symbols). The fitted model described in Eq. 2.9 (solid lines) follows remarkably well the measured concentration profile.
32
The Mass Balance Equation of Chromatography and Its General Properties
2
3 4 Time (min)
6
7
3 4 Time (min)
6
7
Figure 2.4 Experimental and calculated band profiles of benzyl alcohol (a) and phenylethanol (b). Sample volume 0.5 mL, Co = 25 g/1. Column, Symmetry-Cis; mobile phase, methanol and water (1:1, v/v); flow rate, 1 ml/min. Experimental profiles (circles), profiles calculated with a rectangular injection (dashed line) and with the experimental injection profiles (solid line). Reproduced with permission from I. Quinones et al., Anal. Chem., 72 (2000) 1495 (Fig. 6). ©2000, American Chemical Society.
Different boundary conditions are needed for operation modes other than elution. Usually, there are no difficulties in translating the experimental procedure into a set of appropriate boundary conditions. If the injection is performed in a mobile phase containing an additive (initial condition as in Eq. 2.6), for example, the boundary condition for this additive after the injection is completed, is usually Ca{0,t)=C°a
for
t < 0 and tp 0
(2.12)
and then it models a staircase (Figure 2.2c). In displacement chromatography, the boundary condition for the feed components is given by Eq. 2.8 as in elution chromatography, while the boundary condition for the displacer is a step (Figure 2.2d) Cd(z,t)=0 Cd(O,t) = CJ
t < tp + 5t t > tp + 5t
where tp is the duration of the sample injection. There is usually a period of time, 6t, elapsed after the end of the injection and before the displacer is pumped into the column, in order to avoid mixing the sample and the displacer. For other experiments, more complicated boundary conditions may be necessary. This is particularly the case for simulated moving bed separations (see Chapter 17). 2.1.4.3 The Danckwerts Boundary Conditions There is a fundamental objection to the use of the boundary conditions described above. These conditions have vertical boundaries or shocks (Chapter 7) at both *Such phenomena take place only when the additive and some feed components are similarly adsorbed.
34
The Mass Balance Equation of Chromatography and Its General Properties
C
Figure 2.5 Illustration of the Danckwerts Boundary Condition. 1: Rectangular pulse injection. 2, 3, 4: Danckwerts injection conditions for the same sample amount; 2: D = 0.04 cm 2 /s; 3: D = 0.08 cm 2 /s; 4: D = 0.12 c m 2 / s . Reproduced with "permission from G. Guiochon, B. Lin, Modeling for Preparative Chromatography, Academic Press, San Diego, CA, USA, 2003 (Fig. 111-2).
ends. Axial diffusion proceeds at an infinite rate along vertical boundaries. Although chromatographic columns are usually highly efficient and shock layers are often observed on the front or rear side of bands (see Chapters 14 and 16), these boundary conditions are not realistic and their use raises serious fundamental difficulties. In practice, it is better to use the boundary conditions of Danckwerts [51], which are written: = uCn
(2.14)
=
(2.15)
x=0
dC
0
x=L
From a physical viewpoint, the Danckwerts condition states that the mass flux in the column at the column inlet where the injection is made, i.e., uC(0,t) — D^ , is equal to the mass flux that would be achieved in a pipe having the same diameter as the column, uCa(t). If the term D ^ is neglected in the equation above, we obtain the earlier coundary condition, C(0, t) = Co(t). This assumption constitutes the approximation of the ideal model. The Danckwerts boundary condition is illustrated in Figure 2.5 for three different values of the coefficient of apparent axial dispersion, D [2]. In actual columns, D is finite so the term D |^ is different from zero and dx
x=0
should not be neglected. In many numerical calculations, particularly those made for virtual chromatography, the boundary condition C(0, t) = Co(t) is often used instead of the Danckwerts condition. This is approximate but reasonable for efficient columns, e.g., those having more than a few thousand plates. By contrast, when the column efficiency is lower than a few hundred theoretical plates, significant errors may occur and the Danckwerts condition should be used. This situation is typically encountered in the numerical calculations carried out in Simulated Moving Bed separations (see Chapter 17). While the Danckwerts boundary
2.1 Mass and Heat Balance Equations in Chromatography
35
condition must always be considered for numerical calculations, it is rarely considered in fundamental studies. It is a mixed condition, involving values of both the function and its derivative in certain points (at the column inlet and exit). It is complex to handle in algebraic calculations and makes most integrations really difficult if not altogether impossible.
2.1.5 Near-isothermal and Nonisothermal Systems In linear chromatography, the solute concentrations are very low. The heat transfer resistances can be neglected and the chromatographic system considered as isothermal. This is no longer true in nonlinear chromatography, where the concentrations of the feed components are high. If the heat of adsorption is high enough, the system can deviate significantly from isothermal behavior. In such a case, it would be necessary to complete the system of partial differential equations of chromatography by two differential heat balance equations, for the mobile and the stationary phases. For a nonisothermal system, the differential heat balance for the mobile phase can be written as [52] d2Tm
dTm
dTm
dTs
(2.16) and the heat balance for the stationary phase as
where Cp/tn and CprS are the volumetric heat capacities of the mobile and stationary phases, respectively; Tm, Ts, and Tw are the absolute temperatures of the mobile phase, the stationary phase, and the column wall, respectively; and A^ is the axial thermal conductivity of the column, 4H, the heat of adsorption, hm the overall heat transfer coefficient at the column wall, h the overall heat transfer coefficient between stationary and mobile phases, dc the column inner diameter, and dp the average particle size. In an adiabatic system, the wall heat transfer coefficient, hw, is zero, so the last term on the RHS of Eq. 2.16 disappears. For nonisothermal systems, two main cases can be considered, when the system is near isothermal or adiabatic. In near-isothermal systems, it is assumed that the heat transfer between mobile and stationary phases is slow. This causes an additional band broadening contribution to appear [53]. Such a contribution can be especially important on the front of a sharp concentration profile. On the other hand, the heat transfer between the chromatographic column and the outside is fast enough to prevent the formation of a temperature front and of an associated secondary mass transfer zone. In adiabatic or near-adiabatic systems, since the coefficients of the adsorption isotherm depend on the temperature, there will be an additional mass transfer zone which propagates at the velocity of the thermal front. Thus, one may
36
The Mass Balance Equation of Chromatography and Its General Properties
consider that an n-component nonisothermal system is equivalent to an (n + 1)component isothermal system. However, Yun et al. have shown that this effect was negligible in almost all cases of practical importance in preparative HPLC [31].
2.1.6 Mass Balance in a Radially Heterogeneous Column Chromatographic columns are usually considered to be entirely homogeneous. Unfortunately, it is not possible to prepare such columns. By construction or because of the influence of the solute concentration, some physico-chemical properties of the column depend on the position along the column. Variations of some column parameters along the column length are easy to take into account by making small, straightforward changes in the mass balance equation (Eq. 2.2). For example, Piatkowski et al. accounted for a column porosity that depends on the sample concentration by keeping the phase ratio under the differential operator [54]. This work was done using the general rate model (see later) but the same approach could be used with the other models discussed in the next section. More difficult to handle is the well-known radial heterogeneity of chromatographic columns [55]. Because there is friction of the column bed along its wall during packing [56,57], the bed is not homogeneous [58]. The packing density, the external porosity, the phase ratio, the permeability, and the local velocity of the mobile phase are functions of the radial position in the column [55]. In other cases, the column is packed with two different lots of a given packing material [59]. The simplest model that deviates from an homogeneous column is an heterogeneous column having a cylindrical symmetry [59-61]. 2.1.6.1
Mass Balance Equation With Two Space Variables
In this case, the concentration distribution at a given time in a band migrating along the column depends on the position, z, and the distance, r, from the column axis. It does not depend on the azimuthal angle around the column axis. The differential element of the column is a ring of axial thickness dz, limited by the cylinders of radii r and r + dr [60,61]. In this element, the mass balance is written "^ _i_-pz5. i u ZJ! n " "" _i_ i l l v "* / (2 is) a 3t 9f 9z dz2 r dr where Dr is the radial dispersion coefficient. In the framework of the equilibriumdispersive model (see later, Section 2.2.2), it is natural to relate it to a radial plate height by Hr=2^L
(2.19)
The axial plate height is related to the apparent axial dispersion coefficient by Ha = 7^±
(2.20)
2.1 Mass and Heat Balance Equations in Chromatography
37
A relationship is expected between Hr and Ha. However, if the packing density depends on the radial position, the bed tortuosity and eddy diffusion may be different in the axial and radial directions. Furthermore, the mass transfer resistances do not affect Hr. Although, in the general case, Da and Dr could both be functions of the coordinates z and r and of the concentration, we assumed in writing Eq. 2.18 that they are constant. This would constitute the second order approximation of a model of physical columns, the other models discussed here being first order approximations since they all assume a homogeneous column. The presentation of numerical solutions of Eq. 2.18 and their discussion are greatly simplified if some reduced variables are introduced at this stage. These variables are the reduced axial position (x), the reduced radial position (p), the reduced time (T), the axial (Pea) and the radial (Per) column Peclet numbers (note that these two Peclet numbers are different from the conventional particle Peclet number or reduced velocity, v = udp / Dm), and the column aspect ratio (<J>), which are defined as follows
X = I
(2.21a)
p = Yc
(2 21b)
'
r = f Pea
=
(2.21c)
^ =
2
^
(2-21d)
Per = £=2±0
=
(2.21e)
A
(2.21f)
where L is the column length and Rc its radius. Combination of Eqs. 2.18 and 2.21a to 2.21f gives the mass balance equation in reduced coordinates:
ac
a, tc = 1_#c
dr
9T
dX
Pea dx2
^KP|) Per
dp
K
'
'
This equation has four parameters, F, Pea, Per, and <J>. The phase ratio, F, is a property of the packing material, related to its internal and external porosities. The internal porosity is a property of packing materials that is very difficult to adjust because it results from the preparation process of the material and any attempt at modifying it would be prone to result in some changes of the surface chemistry. The external porosity or fraction of the column volume available to the stream of mobile phase percolating through the column packing depends on the packing density but, in practice, this parameter is extremely difficult, if not impossible, to adjust. Recent results have shown that the packing density depends somewhat on the stress applied to the column packing during the preparation of the column [55-58]. Still, it does not seem possible to use an adjustable external stress to modify significantly, in a controlled fashion, the external porosity of the
38
The Mass Balance Equation of Chromatography and Its General Properties
column. By contrast, the other three parameters are easily adjustable. Note that both Dfl (Eq. 2.20) and Dr (Eq. 2.19) are functions of the flow velocity through the corresponding plate height equation. The ratio of the axial and radial Peclet numbers, Pea/Per, does not remain constant when this velocity is changed. 2.1.6.2 Initial and Boundary Conditions of Equation 2.22 The initial condition corresponds usually to a column empty of feed but containing mobile and stationary phases in equilibrium. A physical description of the column is the basis for writing the boundary condition. If the bed is radially heterogeneous, the general direction of the streamlines is no longer parallel to the column axis. These lines tend to avoid the low permeability regions and concentrate in the high permeability ones. At a time when sophisticated fluid dynamics programs were not yet available, Yun et ah [59-61] could not investigate the flow of mobile phase through a radially heterogeneous bed. So, they assumed that the column bed was mechanically homogeneous. With this simplification, there is no convective transport in the radial direction of the column and the radial profile of the mobile phase velocity is flat. This allowed the investigation of only columns in which there are no radial variations of the packing density, hence the column porosity and its permeability are constant throughout the bed. Two problems were investigated. In the first one, the injection profile is a cylindrical band with a diameter smaller than that of the column but coaxial to it [61]. The boundary condition is then: = O,t) = Co
= o,f) = o
o
Tp
(2.31b) v
'
42
The Mass Balance Equation of Chromatography and Its General Properties
2.1.7.3
Mass Balance in the Pores
The mass balance in the stagnant solution is written 3
m
3 [ 1 | (V^)] =0
(2.32)
In this equation, we have:
(233a)
evDv ,-L
* = "w~ csVii
=
-
- ^
(2.33b)
where ep is the particle or internal porosity and Dp!- is the effective diffusion coefficient in the pores. Equation 2.32 is the same in linear and nonlinear chromatography, the only difference being in the relationship between the concentrations in the stationary phase, cs •, and in the liquid phase, cpi[. hi the former case, this relationship is linear, in the latter it is given by the competitive isotherm model. The initial condition for the integration of Eq. 2.32 is T= 0
cP/i = cPii(Q,x)
(2.34)
and the boundary conditions are r= 0
- ^ = dr
0
r=l
- ^ =
Bii(cb/i -
(2.35a) cVii/r=i)
with B
'i =
)
T^r-
(2-35b)
This set of equations (Eqs. 2.25 to 2.35) constitutes the general rate model of chromatography.
2.2 Solution of the System of Mass Balance Equations The band profiles will be obtained as the solution of the relevant system of mass balance equations (Eq. 2.2), completed with a relationship between each stationary phase concentration, Gy, and the mobile phase concentrations, Q (Eq. 2.4 or 2.5), and with the proper set of initial (Eq. 2.6) and boundary conditions (Eqs. 2.8 to 2.15). There is an equation of each type for each component of the feed and of the mobile phase, except for the weak solvent in the mobile phase. However, the mass balance equations of the additives or strong solvents, whose retention factors
2.2 Solution of the System of Mass Balance Equations
43
Table 2.1 System of Equations of the Equilibrium-Dispersive Model. Binary Mixture and Pure (or weakly adsorbed) Mobile Phase Mass Balance Equations dqx
9Ci
_
n
Equilibrium Isotherms
