Preparative Chromatography

Transcription

1 Edited by H. Schmidt-Traub, M. Schulte and A. Seidel-Morgenstern Preparative Chromatography Second, Completely Revised and Enlarged Edition

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3 Edited by Henner Schmidt-Traub, Michael Schulte, and Andreas Seidel-Morgenstern Preparative Chromatography

4 Related Titles Seidel-Morgenstern, A. (ed.) Membrane Reactors Distributing Reactants to Improve Selectivity and Yield 2010 ISBN: Miller, J. M. Chromatography Concepts and Contrasts 2008 E-Book ISBN: Afonso, C. A. M., Crespo, J. P. G. (eds.) Green Separation Processes Fundamentals and Applications 2005 ISBN: Sundmacher, K., Kienle, A., Seidel- Morgenstern, A. (eds.) Integrated Chemical Processes Synthesis, Operation, Analysis, and Control 2005 ISBN: Sinaiski, E.G., Lapiga, E. J. Separation of Multiphase, Multicomponent Systems 2007 ISBN:

5 Edited by Henner Schmidt-Traub, Michael Schulte, and Andreas Seidel-Morgenstern Preparative Chromatography Second, Completely Revised and Updated Edition

6 The Editors Prof. Dr.-Ing. Henner Schmidt-Traub TU Dortmund Fakultät für Bio- und Chemieingenieurwesen Lehrstuhl für Anlagen- und Prozesstechnik Emil-Figge-Str Dortmund Germany Dr. Michael Schulte Merck KGaA R&D Performance & Life Science Chemicals Frankfurter Str Darmstadt Germany Prof. Dr.-Ing. Andreas Seidel-Morgenstern Otto-von-Guericke-Universität Institut für Verfahrenstechnik Lehrstuhl für Chemische Verfahrenstechnik and Max-Planck-Institut für Dynamik komplexer technischer Systeme Sandtorstraße 1 Universitätsplatz Magdeburg Germany All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at dnb.d-nb.d e. #2012 Wiley-VCH Verlag & Co. KGaA, Boschstr. 12, Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form by photoprinting, microfilm, or any other means nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Composition Thomson Digital, Noida, India Printing and Binding Markono Print Media Pte Ltd, Singapore Cover Design Formgeber, Eppelheim Cover The cover figure has been kindly provided by Novasep, France. Print ISBN: epdf ISBN: epub ISBN: mobi ISBN: obook ISBN: Printed in Singapore Printed on acid-free paper

7 jv Contents Preface XV About the Editors XVII List of Contributors XIX List of Abbreviations XXI Notations XXV 1 Introduction 1 Henner Schmidt-Traub and Reinhard Ditz 1.1 Development of Chromatography Focus of the Book Recommendation to Read this Book 4 References 6 2 Fundamentals and General Terminology 7 Andreas Seidel-Morgenstern, Michael Schulte, and Achim Epping 2.1 Principles of Adsorption Chromatography Adsorption Process Chromatographic Process Basic Effects and Chromatographic Definitions Chromatograms and Parameters Voidage and Porosity Influence of Adsorption Isotherms on Chromatogram Shapes Fluid Dynamics Extra Column Effects Column Fluid Distribution Packing Nonidealities Sources for Nonideal Fluid Distribution Column Pressure Drop Mass Transfer Phenomena Principles of Mass Transfer Efficiency of Chromatographic Separations Resolution Equilibrium Thermodynamics 30

8 VIj Contents Definition of Isotherms Models of Isotherms Single-Component Isotherms Multicomponent Isotherms Based on the Langmuir Model Competitive Isotherms Based on the Ideal Adsorbed Solution Theory Steric Mass Action Isotherms for Ion Exchange Equilibria Thermodynamic Effects on Mass Separation Mass Load Linear and Nonlinear Isotherms Elution Modes 43 References 45 3 Stationary Phases and Chromatographic Systems 47 Michael Schulte, Matthias J ohnck, Romas Skudas, Klaus K. Unger, Cedric du Fresne von Hohenesche, Wolfgang Wewers, Jules Dingenen, and Joachim Kinkel 3.1 Column Packings Survey of Packings and Stationary Phases Generic, Designed, and Customized Adsorbents Generic Adsorbents Designed Adsorbents Customized Adsorbents Reversed Phase Silicas Silanisation of the Silica Surface Chromatographic Characterization of Reversed Phase Silicas Cross-Linked Organic Polymers General Aspects Hydrophobic Polymer Stationary Phases Hydrophilic Polymer Stationary Phases Ion Exchange (IEX) Mixed Mode Chiral Stationary Phases Antibiotic CSP Synthetic Polymers Targeted Selector Design Further Developments Properties of Packings and their Relevance to Chromatographic Performance Chemical and Physical Bulk Properties Mass Loadability Comparative Rating of Columns Sorbent Maintenance and Regeneration Cleaning in Place (CIP) Conditioning of Silica Surfaces 106

9 Sanitization in Place (SIP) Column and Adsorbent Storage Selection of Chromatographic Systems Definition of the Task Mobile Phases for Liquid Chromatography Stability Safety Concerns Operating Conditions Aqueous Buffer Systems Adsorbent and Phase Systems Choice of Phase System Dependent on Solubility Improving Loadability for Poor Solubilities Dependency of Solubility on Sample Purity Generic Gradients for Fast Separations Criteria for Choosing NP Systems Pilot Technique Thin-layer Chromatography Retention in NP Systems Solvent Strength in Liquid Solid Chromatography Selectivity in NP Systems Mobile-Phase Optimization by TLC Following the PRISMA Model Strategy for an Industrial Preparative Chromatography Laboratory Criteria for Choosing RP Systems Retention and Selectivity in RP Systems Gradient Elution for Small amounts of Product on RP Columns Rigorous Optimization for Isocratic Runs Rigorous Optimization for Gradient Runs Practical Recommendations Criteria for Choosing CSP Systems Suitability of Preparative CSP Development of Enantioselectivity Optimization of Separation Conditions Practical Recommendations Downstream Processing of Mabs using Protein A and IEX Size Exclusion (SEC) Overall Chromatographic System Optimization Conflicts During Optimization of Chromatographic Systems Stationary Phase Gradients 184 References Chromatography Equipment: Engineering and Operation 199 Abdelaziz Toumi, Jules Dingenen, Joel Genolet, Olivier Ludemann-Hombourger, Andre Kiesewetter, Martin Krahe, Michele Morelli, Henner Schmidt-Traub, Andreas Stein, and Eric Valery 4.1 Introduction Engineering and Operational Challenges 201 Contents jvii

10 VIIIj Contents 4.3 Chromatography Columns Market Generalities The Suppliers General Design High- and Low-Pressure Columns Chemical Compatibility Frits Design Special Aspects of Bioseparation Chromatography Systems Market Generalities The Suppliers General Design Aspects High Performance and Low-Pressure Systems Material Batch Low-Pressure Liquid Chromatography (LPLC) Systems Inlets Valves to Control Flow Direction Pumps Pump(s) Valves and Gradient Formation Batch High-Pressure Liquid Chromatography (HPLC) Systems General Layout Inlets and Outlets Pumps Valves and Pipes Batch SFC Systems General Layout Inlets Pumps, Valves, and Pipes Continuous Systems Simulated Moving Bed General Layout A Key Choice: The Recycling Strategy Pumps, Inlets, and Outlets Valves and Piping Auxiliary Systems Slurry Preparation Tank Slurry Pumps and Packing Stations Cranes and Transport Units Filter Integrity Test Process Control Standard Process Control Advanced Process Control Detectors Packing Methods Column and Packing Methodology Selection Slurry Preparation Column Preparation Flow Packing 246

11 Contents jix Dynamic Axial Compression (DAC) Packing Stall Packing Combined Method (Stall þ DAC) Vacuum Packing Vibration Packing Column Equilibration Column Testing and Storage Test Systems Hydrodynamic Properties and Column Efficiency Column and Adsorbent Storage Process Troubleshooting Technical Failures Loss of Performance Pressure Increase Loss of Column Efficiency Variation of Elution Profile Loss of Purity/Yield Column Stability Disposable Technology for Bioseparations Market Trend Prepacked Columns Membrane Chromatography Membrane Technology 269 References Process Concepts 273 Malte Kaspereit, Michael Schulte, Klaus Wekenborg, and Wolfgang Wewers 5.1 Discontinuous Processes Isocratic Operation Flip-Flop Chromatography Closed-Loop Recycling Chromatography Steady-State Recycling Chromatography Gradient Chromatography Chromatographic Batch Reactors Continuous Processes Column Switching Chromatography Annular Chromatography Multiport Switching Valve Chromatography (ISEP/CSEP) Isocratic Simulated Moving Bed (SMB) Chromatography SMB Chromatography with Variable Process Conditions VariCol PowerFeed Partial-Feed, Partial-Discard, and Fractionation-Feedback Concepts Improved/Intermittent SMB (ismb) 293

12 Xj Contents ModiCon FF-SMB SMB Chromatography with Variable Solvent Conditions Gradient SMB Chromatography Supercritical Fluid SMB Chromatography Multicomponent Separations Multicolumn Systems for Bioseparations Sequential Multicolumn Chromatography (SMCC) Multicolumn Countercurrent Solvent Gradient Purification (MCSGP) Countercurrent Chromatographic Reactors SMB Reactor Processes with Distributed Functionalities Choice of Process Concepts Scale Range of k Number of Fractions Example 1: Lab Scale; Two Fractions Example 2: Lab Scale; Three or More Fractions Example 3: Production Scale Wide Range of k Example 4: Production Scale; Two Main Fractions Example 5: Production Scale; Three Fractions Example 6: Production Scale; Multi-Stage Process 312 References Modeling and Model Parameters 321 Andreas Seidel-Morgenstern, Henner Schmidt-Traub, Mirko Michel, Achim Epping, and Andreas Jupke 6.1 Introduction Models for Single Chromatographic Columns Classes of Chromatographic Models Derivation of the Mass Balance Equations Mass Balance Equations Convective Transport Axial Dispersion Intraparticle Diffusion Mass Transfer Adsorption Kinetics Adsorption Equilibrium Equilibrium ( Ideal ) Model Models with One Band Broadening Effect Dispersive Model Transport Model Reaction Model Lumped Rate Models 338

13 Transport-Dispersive Model Reaction-Dispersive Model General Rate Models Initial and Boundary Conditions of the Column Models of Chromatographic Reactors Stage Models Assessment of Different Model Approaches Dimensionless Model Equations Modeling HPLC Plants Experimental Setup and Simulation Flow Sheet Modeling Extra Column Equipment Injection System Piping Detector Calculation Methods Analytical Solutions Numerical Solution Methods General Solution Procedure Discretization Parameter Determination Parameter Classes for Chromatographic Separations Design Parameters Operating Parameters Model Parameters Determination of Model Parameters Evaluation of Chromatograms Moment Analysis and HETP Plots Parameter Estimation Peak Fitting Functions Detector Calibration Plant Parameters Determination of Packing Parameters Void Fraction and Porosity of the Packing Axial Dispersion Pressure Drop Isotherms Determination of Adsorption Isotherms Determination of the Henry Coefficient Static Isotherm Determination Methods Dynamic Methods Frontal Analysis Analysis of Disperse Fronts (ECP/FACP) Peak Maximum Method Minor Disturbance/Perturbation Method Curve Fitting of the Chromatogram 394 Contents jxi

14 XIIj Contents Prediction of Mixture Behavior from Single-Component Data Data Analysis and Accuracy Mass Transfer Identification of Isotherms and Mass Transfer Resistance by Neural Networks Experimental Validation of Column Models Batch Chromatography SMB Chromatography Model Formulation and Parameters Experimental Validation of SMB Models 410 References Model-Based Design, Optimization, and Control 425 Henner Schmidt-Traub, Malte Kaspereit, Sebastian Engell, Arthur Susanto, Achim Epping, and Andreas Jupke 7.1 Basic Principles and Definitions Performance, Costs, and Optimization Performance Criteria Economic Criteria Objective Functions Degrees of Freedom Optimization Parameters Dimensionless Operating and Design Parameters Scaling by Dimensionless Parameters Influence of Different HETP Coefficients for Every Component Influence of Feed Concentration Examples for a Single Batch Chromatographic Column Examples for SMB Processes Batch Chromatography Fractionation Mode (Cut Strategy) Design and Optimization of Batch Chromatographic Columns Design and Optimization Strategy Process Performance Depending on Number of Stages and Loading Factor Other Strategies Recycling Chromatography Design of Steady-State Recycling Chromatography Scale-Up of Closed Loop Recycling Chromatography Conventional Isocratic SMB Chromatography Optimization of Operating Parameters Process Design Based on TMB Models (Shortcut Methods) Process Design Based on Rigorous SMB Models Optimization of Design Parameters 476

15 7.5 Isocratic SMB Chromatography under Variable Operating Conditions Gradient SMB Chromatography Multicolumn Systems for Bioseparations Advanced Process Control Online Optimization of Batch Chromatography Advanced Control of SMB Chromatography Purity Control for SMB Processes Direct Optimizing Control of SMB Processes Advanced Parameter and State Estimation for SMB Processes 509 References 510 Appendix A: Data of Test Systems 519 Index 527 Contents jxiii

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17 jxv Preface Over 7 years have passed since the 1st edition of this book was published, and practical application as well as theoretical research on preparative chromatography has since then progressed rapidly. This motivated us to revise the content of the 1st edition. We decided to rearrange the structure in this 2nd edition. Our intention was to present the aspects of practical equipment design and operation together in a separate chapter, to merge the discussion on stationary phases and the selection of chromatographic systems in one chapter, and to reduce the content concerning chromatographic reactors because of their specific features and the still limited practical relevance. These changes provided room for important new sections on ion exchange, bioseparation, and new process concepts and calculation methods. What else is new in this revised second edition? First of all, the team did change significantly. Besides the additional editors, there are several new authors from industry and academia. The former crew from Dortmund University went to industries and is now active in other fields of chemical engineering. Their names as well as the names of other authors of the first edition are marked by asterisk in the byline of the corresponding chapters. We are grateful to Klaus Unger, Jules Dingenen, and Reinhard Ditz that they agreed to join us as senior authors. The most challenging task to tackle is presented in Chapter 4 that has been efficiently handled by Abdelaziz Toumi, Joel Genolet, Andre Kiesewetter, Martin Krahe, Michele Morelli, Olivier Ludemann- Hombourger, Andreas Stein, and Eric Valery. It is in the nature of practical design and plant operation that the experience and interests are sometimes different. Additionally, the limited volume further constrains the content. But we hope to meet most of the practical aspects related to design and operation of chromatographic plants. In Chapter 3, Matthias J ohnck and Romas Skudas with the team of Michael Schulte combined the formerly separated topics on stationary phases and chromatographic systems to a unique and completely revised chapter and also extended it to ion exchange. We are especially indebted to Malte Kaspereit for his valuable contributions to Chapters 5 and 7. Sebastian Engell provided in Chapter 7 an overview of the latest research results on advanced process control. We hope that this will motivate practitioners to have a closer look at these promising methods.

18 XVIj Preface Finally, we want to acknowledge the assistance of Fabian Thygs, who produced the new drawings and was patient enough to handle all our revisions. As in the 1st edition, we have summarized the recently published results. In addition, we have made efforts to address preparative and process chromatographic issues from both the chemist and the process engineer viewpoints in order to improve the mutual understanding and to transfer knowledge between both disciplines. With this book we want to reach colleagues from industries as well as universities interested in chromatographic separation with preparative purpose. Students and other newcomers looking for detailed information about design and operation of preparative chromatography are hopefully other users. Our message to all of them is that chromatography is nowadays rather well understood and not that difficult and expensive as it is often said and perceived. On the other hand, it is of course not the solution for all separation problems. We would like to thank all authors for their contributions. We apologize for sometimes getting on their nerves pressing them to meet time limits. Last but not least, we thank our families and friends for their patience and cooperation in bringing out this book. August 2012 Henner Schmidt-Traub Michael Schulte Andreas Seidel-Morgenstern

19 jxvii About the Editors Henner Schmidt-Traub was Professor of Plant and Process Design at the Department of Biochemical and Chemical Engineering, TU Dortmund University, Germany, until his retirement in He is still active in the research community and his main areas of research focus on preparative chromatography, downstream processing, integrated processes, and plant design. Prior to his academic appointment, Prof. Schmidt-Traub gained 15 years of industrial experience in plant engineering. Michael Schulte is Senior Director, Emerging Businesses Energy, at Merck KGaA Performance Materials, Darmstadt, Germany. In his PhD thesis at the University of M unster, Germany, he developed new chiral stationary phases for chromatographic enantioseparations. In 1995 he joined Merck and since then he has been responsible for research and development in the area of preparative chromatography, including the development of new stationary phases, new separation processes, and the implementation of Simulated Moving Bed technology at Merck. In his current position, one of the areas of his research is the use of ionic liquids for separation processes. Andreas Seidel-Morgenstern is Director at the Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany, and holds the Chair in Chemical Process Engineering at the Otto-von-Guericke-University, Magdeburg, Germany. He received his PhD in 1987 at the Institute of Physical Chemistry of the Academy of Sciences in Berlin. From there he went on to work as postdoctoral fellow at the University of Tennessee, Knoxville, TN. In 1994 he finished his habilitation at the Technical University in Berlin. His research is focused on new reactor concepts, chromatographic reactors, membrane reactors, selective crystallization, adsorption and preparative chromatography, and separation of enantiomers among others.

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21 jxix List of Contributors Jules Dingenen Horststraat Arendonk Belgium Reinhard Ditz Merck KGaA Technology Office Chemicals (TO-I) Frankfurter Str Darmstadt Germany Sebastian Engell TU Dortmund Fakult at Bio- und Chemieingenieurwesen Lehrstuhl für Systemdynamik und Prozessführung Emil-Figge-Str Dortmund Germany Joel Genolet Merck Serono S.A. Corsier sur Vevey Zone Industrielle B 1809 Fenil sur Corsier Switzerland Matthias J ohnck Merck KGaA R&D Performance & Life Science Chemicals Frankfurter Str Darmstadt Germany Malte Kaspereit Friedrich-Alexander-Universit at Erlangen-N urnberg Lehrstuhl f ur Thermische Verfahrenstechnik Egerlandstr Erlangen Germany Andre Kiesewetter Merck KGaA PC-SRG-Bioprocess Chromatography Frankfurter Str Darmstadt Germany Martin Krahe Bideco AG Bankstr Uster Switzerland

22 XXj List of Contributors Olivier Ludemann-Hombourger Polypeptide laboratories France 7 rue de Boulogne Strasbourg France Michele Morelli Merck-Millipore SAS 39 Route Industrielle de la Hardt Bldg E Molsheim France Henner Schmidt-Traub TU Dortmund Fakultät für Bio- und Chemieingenieurwesen Lehrstuhl f ur Anlagen- und Prozesstechnik Emil-Figge-Str Dortmund Germany Michael Schulte Merck KGaA R&D Performance & Life Science Chemicals Frankfurter Str Darmstadt Germany Andreas Seidel-Morgenstern Otto-von-Guericke-Universit at Lehrstuhl f ur Chemische Verfahrenstechnik Universit atsplatz 2 and Max-Planck-Institut für Dynamik komplexer technischer Systeme Sandtorstraße Magdeburg Germany Romas Skudas Merck KGaA R&D Performance & Life Science Chemicals Frankfurter Str Darmstadt Germany Andreas Stein Merck KGaA Chromatography Global Applied Technology Frankfurter Str Darmstadt Germany Abdelaziz Toumi Merck Serono S.A. Corsier sur Vevey Zone Industrielle B 1809 Fenil sur Corsier Switzerland Klaus K. Unger Am alten Berg Seeheim Germany Eric Valery Novasep Process Boulevard de la Moselle BP Pompey France

23 jxxi List of Abbreviations ACD: AIEX: ARX: ATEX: BET: BJH: BR: BV: CACR: CD: CEC: CFD: cgmp: CIEX: CIP: CLP: CLRC: COGS: CPG: CSEP 1 : CSF: CSP: CTA: CTB: DAC: DAD: DMF: DMSO: DTA: DVB: EC: ECP: EDM: At-column dilution Anion exchanger Autoregressive exogenous Explosion proof (French: ATmospheres EXplosibles) Brunauer Emmet Teller Barrett Joyner Halenda Chromatographic batch reactor Bed volume Continuous annular chromatographic reactor Circular dichroism (detectors) Capillary electrochromatography Computational fluid dynamics Current good manufacturing practice Cation exchanger Cleaning in place Column liquid chromatography Closed-loop recycling chromatography Cost of goods sold Controlled pore glass Chromatographic separation Circle suspension flow Chiral stationary phase Cellulose triacetate Cellulose tribenzoate Dynamic axial compression Diode array detector Dimethyl formamide Dimethyl sulfoxide Differential thermal analysis Divinylbenzene Elution consumption Elution by characteristic points Equilibrium dispersive model

24 XXIIj List of Abbreviations EMG: FACP: FDM: FFT: FT: GC: GMP: GRM: HCP: HETP: HFCS: HIC: H-NMR: HPLC: HPW: IAST: ICH: IEX: IMAC: IR: ISEC: ISEP 1 : ISMB: LC: LGE: LHS: LOD: LOQ: LPLC: LSB: MaB: mabs: MD: MPC: MS: MW: NMPC: NMR: NN: NP: NPLC: NSGA: OC: OCFE: ODE: Exponential modified Gauss (function) Frontal analysis by characteristic points Finite difference methods Forward flow test Flow through Gas chromatography Good manufacturing practice General rate model Health care provider Height of an equivalent theoretical plate High fructose corn syrup Hydrophobic interaction chromatography Hydrogen nuclear magnetic resonance (spectroscopy) High-performance liquid chromatography Highly purified water Ideal adsorbed solution theory International Guidelines for Harmonization Ion exchange Immobilized metal affinity chromatography Infrared Inverse size exclusion chromatography Ion exchange separation Improved/intermittent simulated moving bed Liquid chromatography Linear gradient elution Liquid-handling station Limit of detection Limit of quantification Low-pressure liquid chromatography Large Scale Biotech project Monoclonal antibody monoclonal antibodies Molecular dynamics Model predictive control Mass spectroscopy Molecular weight Nonlinear model predictive control Nuclear magnetic resonance (spectroscopy) Neural network Normal phase Normal-phase liquid chromatography Non-dominating sorting generic algorithm Orthogonal collocation Orthogonal collocation on finite elements Ordinary differential equation

25 List of Abbreviations jxxiii PAT: PDE: PDT: PEEK: PES: PLC: PMP: PSD: QC: R&D: RI: RMPC: RP: S/N: SEC: SEM: SFC: SIP: SIP: SMB: SMBR: SOP: SQP: SSRC: St-DVB: TDM: TEM: TEOS: TFA: TG/DTA: THF: TLC: TMB: TMBR: TPX TM : UPLC: USP: UV: VSP: WFI: WIT: Process analytical technology Partial differential equation Pressure decay test Poly(ether ether ketone) Poly(ethoxy)siloxane Programmable logic controller Polymethylpentene Particle size distribution Quality control Research and Development Refractive index Repetitive model predictive control Reversed phase Signal-to-noise ratio Size exclusion chromatography Scanning electron microscopy Supercritical fluid chromatography Sanitization in place Steaming in place Simulated moving bed Simulated moving bed reactor Standard operation procedure Sequential quadratic programming Steady-state recycling chromatography Styrene-divinylbenzene Transport dispersive model Transmission electron microscopy Tetraethoxysilane Trifluoroacetic acid Thermogravimetric/differential thermal analysis Tetrahydrofuran Thin-layer chromatography True moving bed process True moving bed reactor Transparent polymethylpentene Ultrahigh-performance liquid chromatography United States pharmacopoeia Ultraviolet Volume-specific productivity Water for injection Water intrusion test

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27 jxxv Notation Symbols Symbol Description Units a i Coefficient of the Langmuir isotherm cm 3 g 1 a s Specific surface area cm 2 g 1 A Area cm 2 A c Cross section of the column cm 2 A i Coefficient in the Van Deemter equation cm A s Surface area of the adsorbent cm 2 ASP Cross section-specific productivity g cm 2 s 1 b i Coefficient of the Langmuir isotherm cm 3 g 1 B Column permeability m 2 B i Coefficient in the Van Deemter equation cm 2 s 1 c i Concentration in the mobile phase g cm 3 c p,i Concentration of the solute inside the particle gcm 3 pores C Annual costs D C i Coefficient in the Van Deemter equation s C DL,i Dimensionless concentration in the liquid phase C p,dl,i Dimensionless concentration of the solute inside the particle pores C spec Specific costs D g 1 d c Diameter of the column cm d p Average diameter of the particle cm d pore Average diameter of the pores cm D an Angular dispersion coefficient cm 2 s 1 D app,i Apparent dispersion coefficient cm 2 s 1 D app,pore Apparent dispersion coefficient inside the pores cm 2 s 1

28 XXVIj Notation Symbol Description Units D ax Axial dispersion coefficient cm 2 s 1 D m Molecular diffusion coefficient cm 2 s 1 D pore,i Diffusion coefficient inside the pores cm 2 s 1 D solid,i Diffusion coefficient on the particle surface cm 2 s 1 Da Damkoehler number EC Eluent consumption cm 3 g 1 F Prices D l 1, D g 1 f i Fugacity h Reduced plate height hr f Retardation factor Dh vap Heat of vaporization kj mol 1 H i Henry coefficient H p Prediction horizon H r Control horizon HETP Height of an equivalent theoretical plate cm k ads,i Adsorption rate constant cm 3 g 1 s 1 k des,i Desorption rate constant cm 3 g 1 s 1 k eff,i Effective mass transfer coefficient cm 2 s 1 K eq Equilibrium constant Miscellaneous K EQ Dimensionless equilibrium coefficient k film,i Boundary or film mass transfer coefficient cm s 1 k 0 i Retention factor 0 ~ k i Modified retention factor k 0 Pressure drop coefficient k reac Rate constant Miscellaneous LF Loading factor L c Length of the column cm _m i Mass flow g s 1 m i Mass g m j Dimensionless mass flow rate in section j m s Total mass g n i Molar cross section of component i n T Pore connectivity N Column efficiency, number of plates N col Number of columns N comp Number of components N p Number of particles per volume element Dp Pressure drop Pa Pe Peclet number Pr i Productivity g cm 3 h 1 P s Selectivity point Pu i Purity %

29 Symbols jxxvii Symbol Description Units q i Solid load g cm 3 q i Total load g cm 3 q i Averaged particle load g cm 3 q sat,i Saturation capacity of the stationary phase g cm 3 Q DL,i Dimensionless concentration in the stationary phase r Radial coordinate cm r i Reaction rate Miscellaneous r p Particle radius cm R f Retardation factor R i Regulation term R s Resolution Re Reynolds number S BET Specific surface area m 2 g 1 Sc Schmidt number Sh Sherwood number St Stanton number t Time s t 0 Dead time of the column (for total liquid holdup) s t 0,int Dead time of the column (for interstitial liquid s holdup) t cycle Cycle time s t g Gradient time s t inj Injection time s t life Lifetime of adsorbent h t plant Dead time of the plant without column s t R,i Retention time s t R,i,net Net retention time s t shift Switching time of the SMB plant s t total Total dead time s T Temperature K T Degree of peak asymmetry u 0 Velocity in the empty column cm s 1 u int Interstitial velocity in the packed column cm s 1 u m Effective velocity (total mobile phase) cm s 1 v sp Specific pore volume cm 3 g 1 V Volume cm 3 _V Volume flow cm 3 s 1 V ads Volume of the stationary phase within a column cm 3 V c Total volume of a packed column cm 3 V i Molar volume cm 3 mol 1 V int Interstitial volume cm 3 V m Overall fluid volume cm 3

30 XXVIIIj Notation Symbol Description Units V pore Volume of the pore system cm 3 V solid Volume of the solid material cm 3 VSP Volume-specific productivity g cm 3 s 1 w i Velocity of propagation cm s 1 x Coordinate cm x i State of the plant X i Mole fraction X Conversion % X cat Fraction of catalyst of the fixed bed Y i Yield % Z Dimensionless distance Greek Symbols Symbol Description Units a Selectivity a exp Ligand density mmol m 2 b Modified dimensionless mass flow rate c Obstruction factor for diffusion or external tortuosity C Objective function e Void fraction e 0 Solvent strength parameter e p Porosity of the solid phase e t Total column porosity g Dynamic viscosity mpa s H Angle of rotation L Total ion exchange capacity mm l Irregularity in the packing m i Chemical potential J mol 1 m t First absolute moment n Kinematic viscosity cm 2 s n i Stoichiometric coefficient p Spreading pressure Pa r Density g cm 3 s t Standard deviation s i Steric shielding parameter t Dimensionless time