Modeling for Preparative Chromatography

by Guiochon; Lin

• ISBN13: 9780120449835
• ISBN10: 0120449838
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-05-05
• Publisher: Elsevier Science

Summary

Nonlinear chromatography is a field that borders both chemical engineering and physical chemistry. in turn, the theory of nonlinear chromatography is the foundation of preparative chromatography, a separation process that has lately become of considerable interest in the pharmaceutical industry. Only chromatography is sufficiently flexible and powerful to satisfy the practical requirements encountered in most difficult separations of pharmaceuticals and pharmaceutical intermediates. Since "nonlinear" behaviour is strictly a mathematical concept, it is difficult to leave mathematics out of any fundamental study of nonlinear chromatography. Therefore, this book will describe the different mathematical models of chromatography, examine the assumptions on which they are based, consider their properties, and discuss their solutions. All this will be done from a mathematical analysis point of view, paying considerable attention to the influence of nonlinear behavior on the results. Clear and simple discussions of the basic physico-chemical phenomena involved in HPLC. Clear and complete presentation of the relevant properties of the mathematical tools involved. Detailed analysis of nonlinear effects.

Table of Contents

Preface	ix
Acknowledgments	xi
Introduction	1	(8)
The Principle of Chromatography	1	(2)
The Modeling of Chromatography	3	(1)
The Goals of this Book	4	(1)
Book Outline	5	(4)
Literature Cited	8	(1)
Physico--Chemical Basis of Chromatography	9	(39)
Equilibrium Isotherms	11	(15)
Single Component Isotherms	14	(1)
Single-Component Isotherm Models	15	(2)
Competitive Isotherms	17	(3)
Excess Isotherms	20	(1)
Empirical Isotherms	21	(1)
Equilibrium Isotherms and Statistical Mechanics	22	(3)
Statistical Thermodynamics Isotherms	25	(1)
Convection and Flow Velocity	26	(1)
Axial Dispersion	27	(4)
Molecular Diffusivity	28	(1)
Bed Tortuosity	29	(1)
Eddy Dispersion	30	(1)
Column Radial Heterogeneity	30	(1)
Mass Transfer Resistances	31	(9)
Radial Dispersion in the Mobile Phase Stream	32	(1)
Film Mass Transfer Resistance	33	(1)
Intraparticle Pore Diffusion	34	(2)
Surface Diffusion	36	(1)
Kinetics of Adsorption--Desorption	37	(3)
Physical Models of Chromatography	40	(8)
The Four Basic Combinations of Models	40	(2)
The Models Discussed in this Work	42	(2)
Exercises	44	(1)
Literature Cited	44	(4)
Mathematical Basis of Chromatography	48	(35)
Mathematical Models of Chromatography	49	(19)
The Fundamental Equations Used in Chromatography	49	(12)
The Quest for Solutions	61	(7)
General Properties of Partial Differential Equations	68	(15)
Definition	68	(2)
Solutions of a PDE	70	(4)
The Laplace transform and the Moments of Chromatographic Bands	74	(6)
Exercises	80	(1)
Literature Cited	81	(2)
The Profiles of Single Component Bands in Linear Nonideal Chromatography	83	(44)
Influence of Axial Diffusion	85	(12)
Frontal Analysis	87	(3)
Generalized Breakthrough Profile	90	(1)
Elution	91	(5)
Danckwerts condition	96	(1)
What is the Equation of the Elution Profile in Linear Chromatography?	96	(1)
Influence the Mass-transfer Resistances	97	(5)
Derivation of the Concentration Profiles	98	(3)
Derivation of the Moments	101	(1)
Linear Chromatography with axial Diffusion and Mass-transfer Resistances	102	(6)
Derivation of the Concentration Profile	103	(1)
Definition and Use of the Moments	104	(2)
The Van Deemter Concept of HETP	106	(2)
Influence of the Packing Irregularities on Chromatographic Performances	108	(19)
Appendix A. Derivation of the Simple Breakthrough Profile (Solution of Equation IV-2)	115	(1)
Transformation of Eqn. IV-2 into the Diffusion Equation	115	(1)
Duhamel principle	115	(1)
Solution of the Diffusion Equation	116	(2)
Transition from a Rectangular Pulse to a Dirac Pulse Injection	118	(1)
Solution for a Dirac Boundary Condition	119	(1)
Derivation of the Solution for the Dirac Pulse Injection Using the Laplace transform	120	(1)
Appendix B. Derivation of the Generalized Breakthrough Profile Equation (General Solution of Equation IV-2)	121	(1)
Calculation of f1(x, t)	121	(1)
Calculation of f2(x, t)	122	(1)
Solution of the Generalized Breakthrough Problem	123	(1)
Symbols Used	124	(1)
Exercises	124	(1)
Literature Cited	125	(2)
Single Component, Ideal, Nonlinear Chromatography	127	(46)
Analysis of the Mathematical Model of Nonlinear Chromatography	128	(26)
Characteristics Analysis, Shock and Shock Velocity	130	(7)
The Different Velocities Encountered in Chromatography	137	(1)
Time Needed to Form the Shock	138	(5)
Development of the Band Profile (Rectangular Pulse Injection)	143	(3)
Retention Time of the Shock on an Elution Band	146	(8)
Weak Solution	154	(5)
Definition of the Weak Solution	155	(1)
Proof of the First Property of Equation V-60	156	(1)
Proof of the Second Property of Equation V-60	157	(2)
Lax Solution of the Chromatography Equation	159	(3)
The Asymptotic Solution	162	(11)
Appendix A. Derivation of Lax Solution	166	(5)
Exercises	171	(1)
Literature Cited	172	(1)
Ideal Nonlinear Reaction Chromatography	173	(19)
Mathematical Model of Reaction Chromatography	174	(2)
Mathematical Analysis of the Model	176	(2)
The Breakthrough Time and the Elution Profile	178	(8)
Case I. Riemann Problem	178	(1)
Case II. Rectangular Pulse Injection Problem	179	(2)
Determination of the Integration Constant	181	(5)
Expression of the Parameters	186	(1)
Asymptotic Analysis	186	(6)
Appendix A. Demonstration of Equation VI-20	199
Exercises	191	(1)
Literature Cited	191	(1)
The Profiles of Single-Component Bands in Nonlinear Nonideal Chromatography	192	(42)
Influence of Axial Diffusion	194	(9)
Houghton Solution	195	(5)
Solution of Haarhoff and van der Linde	200	(1)
Comparison between the two solutions	201	(1)
Further Comments	202	(1)
Influence of the Mass-transfer Resistances in Nonlinear Chromatography	203	(5)
The Thomas Model and its Solution	204	(2)
Solutions of Other Lumped Kinetic Models	206	(2)
Nonlinear Chromatography with Both Axial Diffusion and Mass- Transfer Resistances (Perturbation Solution)	208	(6)
The Zero-order Moment and the Boundary Conditions of Chromatography	214	(4)
The Variation of the Zero-Order Moment of a Moving Band	214	(4)
The Zero-Order Moment and the Boundary Conditions	218	(1)
Shock Layer Analysis	218	(8)
Migration of a Breakthrough Profile in Nonideal, Nonlinear Chromatography	219	(2)
Shock Layer Analysis in Elution Chromatography	221	(5)
Transition Between Linear and Nonlinear Chromatography	226	(8)
Appendix A. Derivation of the Houghton Solution	229	(3)
Exercises	232	(1)
Literature Cited	232	(2)
Two-Component Ideal Nonlinear Chromatography	234	(32)
The Mathematical Foundations of the Theory of Ideal, Nonlinear Multicomponent Chromatography	235	(3)
Characteristic Lines	235	(2)
Characteristic Form	237	(1)
Two-component Nonlinear Chromatography	238	(5)
The Mathematical Model	238	(2)
Analysis of the Mathematical Model	240	(3)
Multiple-Component Problem	243	(2)
Solution with the Competitive Langmuir Isotherm	245	(3)
Simple Wave	248	(2)
Shock Wave	250	(4)
Development of the Bands of the Two Components of a Binary Mixture Following Langmuir Isotherm Behavior	254	(12)
Appendix A. The Hodograph transform	260	(4)
Exercises	264	(1)
Literature Cited	264	(2)
Numerical Analysis of Chromatography Problems	266	(68)
Discretization of the Chromatography Equation and Characteristic Scheme	270	(6)
Discretization of the Problem	270	(2)
Main Properties of the Calculation Schemes	272	(1)
Characteristic Scheme	273	(3)
Stability, Compatibility, and Convergence	276	(5)
Stability	276	(3)
Compatibility	279	(1)
Convergence	280	(1)
Artificial Dissipation and Diffusion Compensation	281	(3)
The Characteristic Scheme of the Ideal Two-component Chromatography Equation	284	(2)
The Conservation Type Difference Scheme of the Ideal Chromatography Equation	286	(8)
Single-Component Case	288	(1)
Two Component Case	289	(5)
The TVD Scheme of the Ideal Chromatography Equation	294	(5)
The TDV Scheme for Ideal Nonlinear Chromatography	295	(1)
The TDV Scheme for Ideal Nonlinear Two-component Chromatography	296	(3)
Numerical Analysis of the Equilibrium Dispersive Model of Chromatography	299	(7)
The Left Bias Explicit Scheme	299	(2)
The Lax-Wendroff Scheme	301	(1)
The Jump Point Scheme	302	(2)
The Predictor-Corrector Scheme	304	(1)
The Difference Scheme of the Convection Diffusion Problem When Convection Is Dominant	305	(1)
The Galerkin Method	306	(7)
The Orthogonal Collocation Method	313	(5)
Orthogonal Collocation Method	314	(1)
Application of Orthogonal Collocation to Chromatography	315	(3)
Comparison of the Accuracy of the Different Numerical Schemes	318	(9)
Comparison Between Experimental and Calculated Band Profiles	327	(7)
Exercises	332	(1)
Literature Cited	332	(2)
Glossary of Symbols	334	(3)
Subject Index	337