Fundamentals of Preparatiue and Nonlinear Chromatography Georges Guiochon University of Tennessee and Oak Ridge National Laboratory Distinguished Scientist Knoxville, Tennessee Sadroddin Golshan Shirazi Senior Scientist Applied Analytical Industries Wilmington, North Carolina Anita M. Katti Senior Development Engineer Mallinckrodt Chemical Inc. St. Louis, Missouri @ Academic Press Boston San Diego New York London Sydney Tokyo Toronto
TABLE OF CONTENTS Preface Acknowledgments xiii xv Chapter I. Introduction, Deflnitions, Goal 1 Introduction 1 I. History of Chromatography 3 1. Discovery by Tswett and Early Works 3 2. The Manhattan Project and the Purification of Rare Earth Elements 4 3. The API Project and the Extraction of Purified Hydrocarbons from Crude Oils 6 4- Preparative Chromatography as a Separation Process 6 IL Deflnitions 14 1. Linear and Nonlinear Chromatography 14 2. Ideal and Nonideal Chromatography 15 3. Separation, Extraction, and Purification 15 4- The Various Scales of Preparative Chromatography 16 III. Goal of the Book 17 References 18 Chapter II. The Mass Balance Equation of Chromatography and Its Properties..21 Introduction 21 I. The Mass and Heat Balance Equations of Chromatography 22 1. Derivation of the Differential Mass Balance of a Compound 22 2. Discussion of the Fundamental Assumptions 25 3. Relationship between the Concentrations in the Stationary and Mobile Phases 27 4- Near-Isothermal and Nonisothermal Systems 29 5. Initial and Boundary Conditions 30 II. Solution of the System of Mass Balance Equations 33 1. The Ideal Model 36 2. The Equilibrium-Dispersive Model 38 3. The Lumped Kinetic Models 41 4- Eguivalence between Equilibrium-Dispersive and Kinetic Models of Chromatography 43 References 47 Chapter III. Single-Component Equilibrium Isotherms 49 Introduction 49 I. Fundamentals of Adsorption Equilibria 51 1. Basic Thermodynamics of Adsorption and the Gibbs Isotherm 53 2. The Linear Isotherm 54 3. The Langmuir Isotherm in Gas-Solid Eguilibria 55 4- The Virial Isotherm 56 5. Statistical Thermodynamics of Adsorption 57 6. Liquid-Solid Equilibria 59 7. Surface Excess and Excess Isotherms 60 II. Models of Adsorption Isotherms 61 1. The Langmuir Isotherm in Liquid-Solid Equilibria 61 vii
viii Fundamentals of Preparative and Nonlinear Chromatography 2. The Bi-Langmuir Isotherm 68 3. The Fowler Isotherm 71 4- The Freundlich Isotherm 72 5. S-Shaped Isotherms and the Quadratic Isotherm Model 74 6. Other Useful Isotherm Equations 78 III. Determination of Single-Component Isotherms 80 1. Frontal Analysis (FA) 81 2. Frontal Analysis by Characteristic Point (FACP) 82 3. Elution by Characteristic Points (ECP) 83 4. Pulse Methods 84 5. The Retention Time Method 87 6. Computation of Elution Profiles (CEP) Method 88 7. Static Method 89 IV. Data Processing and Assessment 90 1. Processing Experimental Data into an Isotherm Equation 90 2. Accuracy and Precision 91 3. Comparison of the Main Chromatographie Methods 93 References 96 Chapter IV. Competitive Equilibrium Isotherms 99 Introduction 99 I. Multicomponent Adsorption and Competitive Isotherm Models 101 1. Competition for Adsorption 101 2. The Competitive Langmuir Isotherm Model and Its Properties 102 3. The Competitive Bi-Langmuir Isotherm 108 4- The Ideal Adsorbed Solution 108 5. The Statistical Isotherm 115 6. The Competitive Fowler Isotherm 116 7. The Competitive Freundlich-Langmuir Isotherm 117 8. Competitive Isotherm Models for Chromatography Modes Other than Adsorption.. 118 9. The Competitive Martire Isotherm 121 II. Determination of Competitive Isotherms 122 1. Competitive Frontal Analysis 123 2. Pulse Methods 129 3. The Simple Wave Method 132 References 134 Chapter V. Fundamentals of Transport Phenomena in Chromatography 137 Introduction 137 I. Diffusion 138 1. Diffusivity or Diffusion Coefficients 140 2. Influence of the Concentration on the Bulk Diffusion Coefficients 143 3. Influence of the Pressure on the Bulk Diffusion Coefficients 144 4- Influence of the Temperature on the Bulk Diffusion Coefficients 145 5. Measurement of the Diffusion Coefficients 146 II. Axial Dispersion and Mass Transfer Resistance in Packed Beds 146 1. Axial Dispersion in Packed Beds 147 2. Kinetics of Adsorption in Porous Adsorbents 148 III. The Viscosity of Liquids 152 1. The Viscosity of the Mobile Phase 154 2. Importance of the Mobile Phase Viscosity in Preparative Chromatography 158 3. Calculation of the Inlet Pressure in the Case of a Variable Viscosity 161 4- Feed Concentration, Mobile Phase Viscosity, and Inlet Pressure 162 5. Flow Instability and Viscous Fingering 163 References 167 Chapter VI. Linear Chromatography 169 Introduction 169 I. The Plate Models 171 1. Overview of the Approach 171
Table of Contents ix 2. The Martin and Synge Plate Model 3. The Craig Plate Model 172 174 4- Comparison of the Two Plate Models 176 II. The Solution of the Mass Balance Equation 178 1. The Equilibrium-Dispersive Model 179 2. Solution of the Lumped Kinetic Model 184 3. From the Lumped Kinetic Model back to the Equilibrium-Dispersive Model 189 III. The General Rate Model of Chromatography 189 1. Analytical Solution in a Particular Case 192 2. Inverse Laplace Transform of the Solution 195 3. Moment Analysis and Plate Height Equations 197 4- The Golay Plate Height Equation 203 5. Dispersion and Partitioning in Short Coated Tubes 204 IV. The Statistical Approach 205 1. Transport Equation in Chromatography with a Finite Speed of Signal Propagation. 206 V. Sources of Band Asymmetry and Tailing in Linear Chromatography VI. Extension of Linear Chromatography Models to Nonlinear Chromatography 207 213 References 215 Chapter VII. The Ideal Model of Chromatography I. Elution of Single-Component Bands 217 Introduction 217 I. Retrospective of the Solution of the Ideal Model 218 IL Migration and Evolution of the Band Profile 221 1. Continuous Part or Diffuse Boundary of the Profile 221 2. Origin of the Concentration Shock 224 3. Propagation of Concentration Shocks 226 III. Analytical Solution of the Ideal Model 229 1. General Closed-Form Solution 229 2. Case of the Langmuir Isotherm 230 3. Concentration Profile along the Column 233 4. Case of the Bi-Langmuir Isotherm 234 5. Case of the Freundlich Isotherm 237 6. Asymptotic Solution 240 IV. Practical Relevance of the Results of the Ideal Model 241 References 243 Chapter VIII. The Ideal Model of Chromatography II. Elution of Two-Component Bands 245 Introduction: Retrospective 245 I. General Principle of the Solution 248 1. Statement of the Problem and Its Constraints 248 2. Properties of the System of Mass Balance Equations 250 IL Analytical Solution for a Wide Band with Competitive Langmuir Isotherms 254 1. Position of the Two Concentration Shocks 254 2. Rear Diffuse Profiles of the Two Components 257 3. The Intermediate Plateau on the Rear Diffuse Profile of the Second Component... 258 III. Analytical Solution for a Narrow Band with Competitive Langmuir Isotherms 260 1. Retention Time of the Second Concentration Shock 262 2. Maximum Concentration of the Two Components in the Mixed Zone 263 3. Elution Profile of the First Component between the Two Shocks 264 4. Retention Time of the First Shock 265 IV. Method of Calculation of the Solution of the Ideal Model in a Specific Case 266 1. Case 1: Wide Injection 267 2. Case 2: Injection Plateau Eroded, Pure First Component Plateau Present 268 3. Case 3: Narrow Injection and Mixed Zone 269 4. Case 4'- Touching Bands, Second Component Plateau Present 270 5. Case 5: Resolved Bands 272 6. Influence of the Width of the Injection Pulse 273 V. Dimensionless Plot of a Two-Component Band System 276 VI. The Displacement Effect 276
x Fundamentals of Preparative and Nonlinear Chromatography 1. Origin of the Displacement Effect 278 2. Intensity of the Displacement Effect 278 VII. The Tag-Along Effect 279 1. Origin of the Tag-Along Effect 279 2. Intensity of the Tag-Along Effect 281 VIII. Practica! Relevance of the Results of the Ideal Model 281 References 284 Appendix 286 Chapter IX. The Ideal Model of Chromatography III. Displacement Chromatography 299 Introduction I. Steady State in the Displacement Mode. The Isotachic Train 299 301 1. The Operating Line 2. Influence of the Displacer Concentration 303 306 3. The Watershed Point 4- Case of a Trace Component 307 308 II. The Theory of Characteristics 308 1. Determination of the Characteristic Parameters 2. Application to Displacement Chromatography 309 312 3. Wave Interactions 313 4- Critical Value of the Displacer Concentration 316 5. Plateau Concentrations and Bandwidth 317 6. Critical Column Length for Isotachic Train Formation III. Practical Relevance of the Results of the Ideal Model 321 321 References 322 Chapter X. The Equilibrium Dispersive Model I. Elution of Single-Component Bands 325 Introduction 325 I. Fundamental Basis of the Model and Apparent Dispersion Coefficient 327 II. Approximate Analytical Solutions 331 1. The Houghton Solution 331 2. The Haarhoff- Van der Linde Solution 3. Range of Validity of the Haarhoff-Van der Linde and Houghton Equations 333 335 4- Influence of the Sample Size on the Bandwidth 337 5. Comparison of the Experimental Band Profiles and the Prediction of These Equations 341 III. Numerical Solutions of the Equilibrium-Dispersive Model 342 1. Principle of the Finite Difference Methods 2. Estimation of the Numerical Errors Made during the Calculation 344 346 3. First Method: Calculation of Numerical Solutions of the Mass Balance Equation 4- Second Method: Replacement of Axial Dispersion by Numerical Dispersion.. 347 348 5. Application of the Second Method 351 6. Finite Element Method 357 IV. Results Obtained with the Equilibrium-Dispersive Model 361 1. Comparison of Solutions of the Ideal and the Equilibrium-Dispersive Models 361 2. Comparison of the Results of Different Calculation Methods 364 3. Results of Computer Experiments 368 4- Comparison with Experimental Results 371 References 378 Chapter XI. The Equilibrium-Dispersive Model II. Isocratic Separations of Two-Component Bands and Gradient Elution 381 Introduction 381 I. Numerical Analysis of the Equilibrium-Dispersive Model 382 1. Finite Difference Methods. Principle 384 2. Finite Difference Methods. Errors in the Case of Two Components 384 3. Finite Element Method 390 IL Gradient Elution 393 1. Solution of the Ideal Model in Gradient Elution 395 2. Representation of the Isotherm in Gradient Elution 395
Table of Contents xi 3. Retention of the Modifier 397 4- Calculation of Elution Band Profiles 398 III. Applications of the Equilibrium-Dispersive Model 401 1. Comparison of Solutions of the Ideal and the Equilibrium-Dispersive Models 401 2. The Hodograph Transform and Its Application 403 3. Results of Computer Experiments 403 4- Calculation of Multicomponent Chromatograms 415 5. Comparison of Calculated Band Profiles and Experimental Results 6. Experimental and Calculated Band Profiles in Gradient Elution 417 429 References 432 Chapter XII. The Equilibrium-Dispersive Model III. Frontal Analysis and Displacement 435 Introduction 435 I. Displacement Chromatography with a Nonideal Column 436 1. Influence of the HETP 438 2. Influence of the Sample Size and the Displacer Concentration 438 3. Influence of the Column Length 442 4- Influence of the Separation Factor 444 5. Case of Trace Components 447 6. Influence of the Impurities in the Displacer Solution 451 7. Case of Selectivity Reversal 452 II. Applications of Displacement Chromatography 454 1. Separation of Rare Earths and Other Cations 456 2. Separation of Organic Compounds 457 3. Separation of Peptides and Proteins 459 4- Separation of Nucleic Acid Constituents 464 III. Comparison of Calculated and Experimental Results 467 References 471 Chapter XIII. The Equilibrium-Dispersive Model IV. System Peaks in Chromatography Introduction 473 473 I. System Peaks in Linear Chromatography 475 1. General Experimental Results on System Peaks 475 2. Theory of System Peaks 478 3. Indirect Detection Using System Peaks 487 4- Application of System Peaks to Analyte Peak Compression 491 5. Vacancy Chromatography 492 II. High-Concentration System Peaks 496 1. High-Concentration System Peaks for a Single-Component Sample 497 2. High-Concentration System Peaks for a Two-Component Sample 507 References 515 Chapter XIV. The Kinetic Models I. Frontal Analysis and Elution of Single-Component Bands 519 Introduction 519 I. Solution of the Breakthrough Curve under Constant Pattern Condition 521 1. Analytical Solution for the Constant Pattern Profile 522 2. Numerical Solution of the Breakthrough Curve under Constant Pattern Behavior..525 3. Effect of Axial Dispersion 526 4- The Shock Layer Theory 526 5. Shock Layer in the Case of the Langmuir Isotherm 530 6. Properties of the Shock Layer Thickness in Frontal Analysis 531 7. Range of Validity of the Equilibrium-Dispersive Model 537 II. Analytical Solution of the Reaction-Kinetic Model 539 1. Solution of the Reaction-Kinetic Model in the Case of a Step Injection 539 2. Numerical Solutions of the Kinetic Model for a Breakthrough Curve 540 3. Analytical Solution of the Reaction-Kinetic Model in the Case of a Pulse Injection 540 4- Numerical Solution of the Reaction-Dispersive Model for a Pulse Injection 543
xii Fundamentals of Preparative and Nonlinear Chromatography III. Comparison of the Various Kinetic Models of Nonlinear Chromatography 545 IV. Results of Computer Experiments 552 V. Comparison between Theoretical and Experimental Results 554 References 558 Chapter XV. The Kinetic Models II. Frontal Analysis, Flution, and Displacement of Multicomponent Bands 561 Introduction I. Analytical Solution under Constant Pattern Behavior 561 562 1. The Shock Layer Theory for a Binary Mixture 563 2. Shock Layer in the Case of Competitive Langmuir Isotherms 564 3. Shock Layer Thickness in Binary Frontal Analysis 567 4- Shock Layer Thickness in Displacement Chromatography 568 II. Linear Driving Force Model Approach 574 III. Numerical Solution of the General Rate Model of Chromatography 580 1. Formulation of the General Rate Model 581 2. Solutions of the General Rate Model 584 3. The VERSE Model IV. Comparison of Solutions of Rate Models and Experimental Results 584 587 V. Effect of the Particle Size Distribution References 591 592 Chapter XVI. Optimization of the Experimental Conditions in Preparative Chromatography 595 Introduction 595 I. Definitions 596 1. Throughput 596 2. Sample Size and Loading Factor 597 3. Cycle Time 597 4- Production Rate 598 5. Cut Points 599 6. Recovery Yield 602 7. Purity of a Component 8. Specific Production 602 603 II. The Economics of Chromatographie Separations 603 1. The Components of the Production Cost 604 2. The Different Objective Functions 3. Identification of the Experimental Parameters 607 608 III. Optimization Based on Theoretical Considerations 609 1. The Knox and Pyper Approach 610 2. Optimization for Touching Band Using the Ideal Model 612 3. Optimization for Overlapping Bands with No Yield Constraint 4- Optimization for Overlapping Bands with Yield Constraint 620 624 IV. Optimization Using Numerical Solutions 626 1. Maximum Production Rate in Elution 626 2. Minimum Solvent Consumption in Elution 632 3. Compromise between Maximum Production Rate and Minimum Solvent Consumption 636 4- Maximum Production Rate in Displacement Chromatography 637 5. Comparison between Elution and Displacement Chromatography 641 V. Recycling Procedures 648 VI. Comparison between Experimental and Calculated Results 649 VII. Practical Rules 651 References 656 Glossary of Symbols 659 Glossary of Terms 669 Subject Index 693