Heidelberg,Tingyue Gu,Mathematical Modeling,and Scale up. of Liquid Chromatography,With 75 Figures,Prof Dr Tingyue Gu. Ohio University,Dept of Chemical Engineering,Stocker Center. Athens Ohio 45701 2979,ISBN 13 978 3 642 79543 5,DOl 10 1007 978 3 642 79541 1. CIP data applied for, This work is subject to copyright All rights are reserved whether the whole or part of the materials is concerned. specifically the rights of translation reprinting reuse of illustrations recitation braodcasting reproduction on microfIlm. or in other ways and storage in data banks Duplication of this publication or parts thereof is permitted only under the. provision of the German Copyright Law of September 9 1965 in its current version and permission for use must. always be obtained from Springer Verlag Violations are liable for prosecution act under German Copyright Law. Springer Verlag Berlin Heidelberg 1995, Softcover reprin to the hardcover 1St edition 1995. The use of general descriptive names registered names trademarks etc in this publication does not imply even in the. absence of a specific statement that such names are exempt from the relevant protective laws and regulations and. therefore free for general use, Production PRODUserv Springer Produktions Gesellschaft Berlin. Cover layout Design Production Heidelberg, Typesetting Dataconversion by Satztechnik Neuruppin GmbH Neuruppin. SPIN 10100713 02 3020 5 4 3 2 1 0 Printed on acid free paper. Liquid chromatography is no longer limited to chemical analysis It has. become an indispensable tool for the preparative and large scale purifica. tions of proteins and other fine chemicals So far the scale up of liquid. chromatography relies mostly on trial and error and a few scaling rules that. are more of a rule of thumb nature, This book provides numerical solutions to a series of general multi. component rate models for liquid chromatography The models consider dis. persion interfacial film mass transfer intra particle diffusion and nonlinear. multicomponent isotherm or the second order kinetics The models can be. used to simulate various chromatographic operations They provide more. realistic descriptions of preparative and large scale liquid chromatography. than the equilibrium theory and plate models because various mass transfer. mechanisms are included, The applications of the Fortran 77 codes for the models are explained. Parameter estimation for the models is discussed The codes should be helpful. in both the understanding of the dynamics of liquid chromatography and its. scale up The codes are available to readers upon request by a letter or pre. ferably an electronic mail to guting ent ohiou edu. Most of this book is based on the theoretical part of the author s Ph D thesis. work at Purdue University West Lafayette Indiana U S A I am deeply indebt. ed to my former advisor Prof George T Tsao of the Laboratory of Renewable. Resources Engineering at Purdue,Spring 1995 Tingyue Gu. Table of Contents,List of Symbols and Abbreviations xi. Introduction 1,2 Literature Review 4, 2 1 Theories for Nonlinear Multicomponent Liquid Chromato. 2 1 1 Equilibrium Theory 4,2 1 2 Plate Models 4,2 1 3 Rate Models 5. 2 1 3 1 Rate Expressions 5, 2 1 3 2 Governing Equation for the Bulk Fluid Phase 6. 2 1 3 3 General Multicomponent Rate Models 6,2 1 3 4 Numerical Solutions 7. 2 1 3 5 Solution to the ODE System 7,2 2 Scale Up of Liquid Chromatography 8. 3 A General Multicomponent Rate Model for Column Liquid. Chromatography 9,3 l Model Assumptions 9,3 2 Model Formulation 10. 3 3 Finite Element Formulation for the Bulk Fluid Phase. Governing Equation 14, 3 4 Orthogonal Collocation Formulation of the Particle Phase. Governing Equation 15,3 5 Solution to the ODE System 15. 3 6 Fortran 77 Code for the General Multicomponent Rate. 3 7 CPU Time for the Simulation 20, 3 8 Extension of the General Multicomponent Rate Model 22. 3 8 1 Second Order Kinetics 23, 3 8 2 Addition of Size Exclusion Effect to the Rate Model 24. 3 9 The Question of Choosing Column Boundary Conditions 27. viii Table of Contents,4 Mass Transfer Effects 32,4 1 Effects of Parameters PeL Bi and 1 33. 4 2 Effect of Flow Rate 32, 4 3 Effect of Mass Transfer in a Case with Unfavorable Isotherm 37. 5 Interference Effects in Multicomponent Chromatography 39. 5 1 Introduction 39,5 2 Computer Simulation and Discussion 40. 5 2 1 Displacement Mode 40,5 2 2 Frontal Adsorption Mode 42. 5 2 3 Elution Mode 44,5 2 3 1 Adsorption Equilibrium Constants 47. 5 2 3 2 Low Adsorption Saturation Capacity 47, 5 2 3 3 High Sample Feed Concentration Concentration Overload 48. 5 2 3 4 Large Sample Size Volume Overload 48,5 2 3 5 More Component s 50. 5 3 Summary 51,6 System Peaks In Multicomponent Elution 52. 6 1 Introduction 52, 6 2 Boundary Conditions for the General Rate Model 53. 6 3 Results and Discussion 53,6 3 1 Modifier is Weaker than Sample Solutes 53. 6 3 2 Modifier Affinity is Between those of Sample Solutes 56. 6 3 3 Modifier is Stronger than Sample Solutes 58, 6 3 4 Effect of Modifier Concentration on System Peak Patterns 60. 6 3 5 Effect of Modifier on Sample Solutes 62,6 3 6 Effect of Sample Type 64. 6 3 7 Effect of Sample Solutes on the Modifier 65,6 3 8 Summary of System Peak Patterns 66. 6 3 9 Binary Elution with Two Different Modifiers 68. 6 4 Concluding Remarks 68, 7 Multicomponent Adsorption with Uneven Saturation. Capacities 71,7 1 Introduction 71,7 2 Kinetic and Isotherm Models 72. 7 3 Isotherm Crossover 74,7 4 Summary 80,8 Modeling of Affinity Chromatography 81. 8 1 Introduction 81,8 2 Effect of Reaction Kinetics 82. 8 3 Effect of Size Exclusion 87, 8 4 Interaction Between Soluble Ligand and Macromolecule 89. Table of Contents ix,8 4 1 Modeling of Reaction in the Fluid 89. 8 4 2 Solution Strategy 91, 8 5 Modeling of the Three Stages in Affinity Chromatography 91. 8 6 How to Use the Fortran 77 Code AFFINITY F 93,8 7 Summary 94. 9 Modeling of Multicomponent Gradient Elution 95,9 1 Introduction 95. 9 2 General Rate Model for Multicomponent Gradient Elution 96. 9 3 Numerical Solution 97,9 4 How to Use the Fortran 77 Code GRADIENT F 97. 9 5 Summary 101,10 Multicomponent Radial Flow Chromatography 102. 10 1 Introduction 102, 10 2 General Multicomponent Rate Model for RFC 104. 10 3 Numerical Solution 106,lOA How to Use the Fortran 77 Code RATERFC F 106. 10 5 Extensions of the General Multicomponent Rate Model. for RFC 108,10 6 Summary 110, 11 Scale Up of Liquid Chromatography using General Rate. Models 111,11 1 Isotherms 111,11 1 1 Batch Adsorption Equilibrium Method 111. 11 1 2 Column Method 112,11 1 3 Langmuir Isotherm 112. 11 1 4 Other Isotherm Models 115,11 2 Mass Transfer Parameters 116. 11 3 Evaluation of Pe j 11 j and Bi j 117,11 4 General Procedure for Scale Up 118. 12 References 121,13 Subjectlndex 125,List of Symbols and Abbreviations. constant in Langmuir isotherm for component i hi Ci. adsorption equilibrium constant for component i kai I kru. Biot number of mass transfer for component i k Rp I EpDpi. concentration used for nondimensionalization max Cfi t. bulk fluid phase concentration of component i, feed concentration profile of component i a time dependent. mobile phase concentration of component i in ion exchange. concentration of component i in the stagnant fluid phase inside. particle macropores, critical concentration for concentration crossover in a binary. concentration of component i in the solid phase of particle based. on unit volume of particle skeleton, adsorption saturation capacity for component i based on unit. volume of particle skeleton,saturation capacity in ion exchange. stationary phase concentration of component i in ion exchange. axial or radial dispersion coefficient of component i. molecular diffusivity, effective diffusivity of component i porosity not included. Damkolher number for adsorption L kaiCod I v,Damkolher number for desorption L kdi I v. inner diameter of a column, size exclusion factor for component i J ex O means complete. exclusion lOpi I lOp,film mass transfer coefficient of component i. adsorption rate constant for component i,desorption rate constant for component i. retention factor capacity factor for component i,column length. xii List of Symbols and Abbreviations,N number of interior collocation points. Ne number of quadratic elements,Ns number of components. PeLi Peclet number of axial dispersion for component i vLlDbi. Q mobile phase volumetric flow rate,R radial coordinate for particle. Rp particle radius,Re Reynolds number VPf 2Rp I 1,Sc Schmidt number J11 pfDm. Sh Sherwood number k 2Rp I Dm, dimensional time t O is the moment a sample enters a column. to dead volume time of unretained small molecules such as salts and. td dead volume time of unretained large molecules such as blue. tR dimensional retention time,v interstitial velocity 4Q nd2E b. W weight of adsorbent,Z axial coordinate,z dimensionless axial coordinate ZI L. Greek Letters, a 2 V Vo I l Vo Fa for radial flow chromatography RFC. a ij separation factor of components i and j in ion exchange. ai f3i X parameters for the eluite modulator correlation. Eb bed void volume fraction,Ep particle porosity,1Ji dimensionless constant EpDpiLl R v. Oij discount factors for extended multicomponent Langmuir isotherm. 1 mobile phase viscosity, i dimensionless constant for component i 3Bi i1Ji 1 Eb I Eb. Pf mobile phase density, Pp particle density based on unit volume of particle skeleton. f dimensionless time vt L,fR dimensionless retention time. fimp dimensionless time duration for a rectangular pulse of the sample. phase ratio stationary phase to mobile phase,l Eb l Ep Eb l Eb Epl. Subscripts,a adsorption reaction,b bulk fluid phase. d desorption reaction,List of Symbols and Abbreviations xiii. i th component,imp sample impulse,L bulk fluid phase. p particle phase,Superscripts,a adsorption,c isotherm crossover concentration point. d desorption,particle phase concentration,00 saturation capacity.
Algorithmic Information Theory and Kolmogorov Complexity Alexander Shen, ? Uppsala University, Computing Science Department, Independent University of Moscow, [email protected] November 19, 2007 Abstract This document contains lecture notes of an introductory course on Kolmogorov complexity.
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