Distillation: Principles and Practice, Second Edition [2 ed.] 1119414660, 9781119414667

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DISTILLATION

DISTILLATION Principles and Practice Second Edition

Prof. Dr.-Ing. JOHANN STICHLMAIR Prof. Dr.-Ing. HARALD KLEIN Dr.-Ing. SEBASTIAN REHFELDT

Copyright © 2021 by American Institute of Chemical Engineers, Inc. All rights reserved. A Joint Publication of the American Institute of Chemical Engineers and John Wiley & Sons, Inc. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Edition History Wiley-VCH (1e, 1998) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Prof. Dr.-Ing. Johann Stichlmair, Prof. Dr.-Ing. Harald Klein, and Dr.-Ing. Sebastian Rehfeldt to be identified as the authors of this work has been asserted in accordance with law. Registered Office John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA Editorial Office 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Stichlmair, Johann, author. | Klein, Harald, author. | Rehfeldt, Sebastian, author. Title: Distillation : principles and practice / Johann Stichlmair, Harald Klein, Sebastian Rehfeldt. Description: Second edition. | Hoboken, New Jersey : Wiley-AIChE, [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020024694 (print) | LCCN 2020024695 (ebook) | ISBN 9781119414667 (cloth) | ISBN 9781119414698 (adobe pdf) | ISBN 9781119414681 (epub) Subjects: LCSH: Distillation. Classification: LCC TP156.D5 S85 2021 (print) | LCC TP156.D5 (ebook) | DDC 660/.28425–dc23 LC record available at https://lccn.loc.gov/2020024694 LC ebook record available at https://lccn.loc.gov/2020024695 Cover image: © Mina De La O/Getty Images Cover design by Wiley Set in 9.5/12.5pt TeXGyreTermes by SPi Global, Chennai, India 10 9 8 7 6 5 4 3 2 1

v

Contents xi

Preface Nomenclature

xiii

1

Introduction

1

1.1

Principle of Distillation Separation

1

1.2

Historical

3

2

Vapor–Liquid Equilibrium

7

2.1

Basic Thermodynamic Correlations

7

2.1.1

Measures of Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2

Equations of State (EOS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3

Molar Mixing and Partial Molar State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.4

Saturation Vapor Pressure and Boiling Temperature of Pure Components . . . . . . . . . . 13

2.1.5

Fundamental Equation and the Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.6

Gibbs–Duhem Equation and Gibbs–Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

18

2.2

Calculation of Vapor–Liquid Equilibrium in Mixtures

2.2.1

Basic Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2

Gibbs Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3

Correlations for the Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.4

Calculating Activity Coefficients with the Molar Excess Free Energy . . . . . . . . . . . . . . . . 23

2.2.5

Thermodynamic Consistency Check of Molar Excess Free Energy and Activity

2.2.6

Iso-fugacity Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.7

Fugacity of the Liquid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.8

Fugacity of the Vapor Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.9

Vapor–Liquid Equilibrium Using an Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.10

Fugacity of Pure Liquid as Standard Fugacity: Raoult’s Law . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.11

Fugacity of Infinitely Diluted Component as Standard Fugacity: Henry’s Law . . . . . . . . 48

2.2.12

Correlations Describing the Molar Excess Free Energy and Activity Coefficients . . . . 49

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

vi

CONTENTS 2.2.13

Using Experimental Data of Binary Mixtures for Correlations Describing the Molar

2.2.14

Vapor–Liquid Equilibrium Ratio of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2.15

Relative Volatility of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2.16

Boiling Condition of Liquid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.2.17

Condensation (Dew Point) Condition of Vapor Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Excess Free Energy and Activity Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

81

2.3

Binary Mixtures and Phase Diagrams

2.3.1

Boiling Curve Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.3.2

Condensation (Dew Point) Curve Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.3.3

(p, x , y )-Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.3.4

(T , x , y )-Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.3.5

McCabe–Thiele Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.3.6

Boiling and Condensation Behavior of Binary Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.3.7

General Aspects of Azeotropic Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

2.3.8

Limiting Cases of Binary Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

114

2.4

Ternary Mixtures

2.4.1

Boiling and Condensation Conditions of Ternary Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.4.2

Triangular Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.4.3

Boiling Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.4.4

Condensation Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

2.4.5

Derivation of Distillation Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.4.6

Examples for Distillation Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3

Single-Stage Distillation and Condensation

137 137

3.1

Continuous Closed Distillation and Condensation

3.1.1

Closed Distillation of Binary Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.1.2

Closed Distillation of Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

152

3.2

Batchwise Open Distillation and Open Condensation

3.2.1

Binary Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

3.2.2

Ternary Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3.2.3

Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

169

3.3

Semi-continuous Single-Stage Distillation

3.3.1

Semi-continuous Single-Stage Distillation of Binary Mixtures . . . . . . . . . . . . . . . . . . . . . . 169

4

Multistage Continuous Distillation (Rectification)

173 173

4.1

Principles

4.1.1

Equilibrium-Stage Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.1.2

Transfer-Unit Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

4.1.3

Comparison of Equilibrium-Stage and Transfer-Unit Concepts . . . . . . . . . . . . . . . . . . . . . 180

181

4.2

Multistage Distillation of Binary Mixtures

4.2.1

Calculations Based on Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

vii

CONTENTS 4.2.2

Calculation Based on Material and Enthalpy Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.2.3

Distillation of Binary Mixtures at Total Reflux and Reboil . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

4.2.4

Distillation of Binary Mixtures at Minimum Reflux and Reboil . . . . . . . . . . . . . . . . . . . . . . 198

4.2.5

Energy Requirement for Distillation of Binary Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

4.3

Multistage Distillation of Ternary Mixtures

4.3.1

Calculations Based on Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

206

4.3.2

Distillation of Ternary Mixtures at Total Reflux and Reboil . . . . . . . . . . . . . . . . . . . . . . . . . 215

4.3.3

Distillation of Ternary Mixtures at Minimum Reflux and Reboil . . . . . . . . . . . . . . . . . . . . . 224

4.3.4

Energy Requirement of Ternary Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

4.4

Multistage Distillation of Multicomponent Mixtures

4.4.1

Rigorous Column Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

5

Reactive Distillation, Catalytic Distillation

5.1

Fundamentals

5.1.1

Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

5.1.2

Stoichiometric Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

5.1.3

Non-reactive and Reactive Distillation Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

5.1.4

Reactive Azeotropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

5.2

Topology of Reactive Distillation Lines

5.2.1

Reactions in Ternary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

5.2.2

Reactions in Ternary Systems with Inert Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

5.2.3

Reactions with Side Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

5.2.4

Reactions in Quaternary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

5.3

Topology of Reactive Distillation Processes

5.3.1

Single Product Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

5.3.2

Decomposition Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

5.3.3

Side Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

5.4

Arrangement of Catalysts in Columns

5.4.1

Homogeneous Catalyst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

5.4.2

Heterogeneous Catalyst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

6

Multistage Batch Distillation

313

6.1

Batch Distillation of Binary Mixtures

314

6.1.1

Operation with Constant Reflux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

6.1.2

Operation with Constant Distillate Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

255

283 284

293

298

307

6.1.3

Operation with Minimum Energy Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

6.1.4

Comparison of Energy Requirement for Different Modes of Distillation . . . . . . . . . . . . . 327

6.2

Batch Distillation of Ternary Mixtures

6.2.1

Zeotropic Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

6.2.2

Azeotropic Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

327

viii

CONTENTS 6.3

Batch Distillation of Multicomponent Mixtures

336

6.4

Influence of Column Liquid Hold-up on Batch Distillation

337

6.5

Processes for Separating Zeotropic Mixtures by Batch Distillation

340

6.5.1

Total Slop Cut Recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

6.5.2

Binary Distillation of the Accumulated Slop Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

6.5.3

Recycling of the Slop Cuts at the Appropriate Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

6.5.4

Cyclic Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

341

6.6

Processes for Separating Azeotropic Mixtures by Batch Distillation

6.6.1

Processes in One Distillation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

6.6.2

Processes in Two Distillation Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

6.6.3

Process Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

6.6.4

Hybrid Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

7

Energy Economization in Distillation

357 358

7.1

Energy Requirement of Single Columns

7.1.1

Reduction of Energy Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

7.1.2

Reduction of Exergy Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

363

7.2

Optimal Separation Sequences of Ternary Distillation

7.2.1

Process and Energy Requirement of the a-Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

7.2.2

Process and Energy Requirement of the c-Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

7.2.3

Process and Energy Requirement of the Preferred a/c-Path . . . . . . . . . . . . . . . . . . . . . . 366

368

7.3

Modifications of the Basic Processes

7.3.1

Material (Direct) Coupling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

7.3.2

Processes with Side Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

7.3.3

Thermal (Indirect) Coupling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

391

7.4

Design of Heat Exchanger Networks

7.4.1

Optimum Heat Exchanger Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

7.4.2

Modifying the Optimum Heat Exchanger Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

7.4.3

Dual Flow Heat Exchanger Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

7.4.4

Process Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

8

Industrial Distillation Processes

407 407

8.1

Constraints for Industrial Distillation Processes

8.1.1

Feasible Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

8.1.2

Feasible Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

8.1.3

Feasible Dimensions of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

412

8.2

Fractionation of Binary Mixtures

8.2.1

Recycling of Diluted Sulfuric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

8.2.2

Ammonia Recovery from Wastewater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

8.2.3

Hydrogen Chloride Recovery from Inert Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

ix

CONTENTS 8.2.4

Linde Process for Air Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

8.2.5

Process Water Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

8.2.6

Steam Distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

8.3

Fractionation of Multicomponent Zeotropic Mixtures

8.3.1

Separation Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

8.3.2

Processes with Side Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

8.4

Fractionation of Heterogeneous Azeotropic Mixtures

435

8.5

Fractionation of Azeotropic Mixtures by Pressure Swing Processes

436

8.6

Fractionation of Azeotropic Mixtures by Addition of an Entrainer

439

8.6.1

Processes for Systems Without Distillation Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

8.6.2

Processes for Systems with Distillation Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

8.6.3

Hybrid Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

8.7

Industrial Processes of Reactive Distillation

8.7.1

Synthesis of MTBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

8.7.2

Synthesis of Mono-ethylene Glycol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

8.7.3

Synthesis of TAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

8.7.4

Synthesis of Methyl Acetate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

9

Design of Mass Transfer Equipment

481

9.1

Types of Design

482

9.1.1

Tray Columns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .482

9.1.2

Packed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

9.1.3

Criteria for Use of Tray or Packed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

9.2

Design of Tray Columns

9.2.1

Design Parameters of Tray Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

9.2.2

Operating Region of Tray Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

9.2.3

Two-Phase Flow on Trays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

9.2.4

Mass Transfer in the Two-Phase Layer on Column Trays . . . . . . . . . . . . . . . . . . . . . . . . . . 518

9.3

Design of Packed Columns

9.3.1

Design Parameters of Packed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

9.3.2

Operating Region of Packed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

9.3.3

Two-Phase Flow in Packed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

9.3.4

Mass Transfer in Packed Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

9.A

Appendix: Pressure Drop in Packed Beds

587

10

Control of Distillation Processes

601

10.1

Control Loops

602

10.1.1

Single Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

10.1.2

Ratio Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

10.1.3

Disturbance Feedforward Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

429

469

487

533

x

CONTENTS 10.1.4

Cascade Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

605

10.2

Single Control Tasks for Distillation Columns

10.2.1

Liquid Level Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

10.2.2

Split Stream Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

10.2.3

Pressure Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611

10.2.4

Product Concentration Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

613

10.3

Basic Control Configurations of Distillation Columns

10.3.1

Basic Control Systems Without Composition Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

10.3.2

One-Point Composition Control Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

10.3.3

Two-Point Composition Control Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626

629

10.4

Application Ranges of the Basic Control Configurations

10.4.1

Impact of Split Parameters According to Split Rule 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

10.4.2

Sharp Separations of Ideal Mixtures with Constant Relative Volatility at Minimum

10.4.3

Extended Application Ranges of the Basic Control Configurations . . . . . . . . . . . . . . . . . 643

Reflux and Boilup Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639

646

10.5

Examples for Control Configurations of Distillation Processes

10.5.1

Azeotropic Distillation Process by Pressure Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646

10.5.2

Distillation Process for Air Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

10.5.3

Distillation Process with a Main and a Side Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

10.5.4

Azeotropic Distillation Process by Using an Entrainer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

10.6

Control Configurations for Batch Distillation Processes

Index

651

655

xi

Preface Distillation is the most important and the most effective technology for the fractionation of multicomponent mixtures. Fields of application are all branches of the process industry, for instance, petroleum refineries, chemical industries, and food industries. The often very tall distillation towers dominate the view of many chemical sites. According to its great importance, distillation is a highly developed technology. The fundamental mechanism of distillation is the mass transfer between a gaseous and a liquid phase. The driving force for this interfacial mass transfer is the difference between the actual and the equilibrium concentration of the phases. The book consists of 10 chapters. Chapter 1 deals with the basic principle of distillation and with some historical aspects of the art. Chapter 2 concentrates on the thermodynamics of vapor–liquid equilibrium, since a good knowledge of vapor–liquid equilibrium is an indispensable prerequisite for the design of distillation processes. As compared to many other textbooks, the mixtures are not limited to two components, but ternary mixtures along with their boiling surfaces and triangular diagrams are considered. The Chapters 3 – 6 deal with the thermodynamics of single-stage distillation (Chapter 3) and multi-stage distillation (Chapter 4), which is often called rectification, reactive distillation (Chapter 5), and batch distillation (Chapter 6). Special attention is given as described above to ternary mixtures, since they represent a more general case than binary mixtures most textbooks on distillation focus on. In Chapter 7 the energy requirement of distillation processes is discussed. This chapter demonstrates how the large energy requirement of distillation processes can be drastically reduced by internal column coupling and intelligent process modifications. Important examples of industrial distillation processes are presented in Chapter 8. Here, special attention is given to processes for the fractionation of azeotropic mixtures. The design of distillation columns is treated in Chapter 9 with focus on tray columns and packed columns. Finally, the control of distillation columns is the objective of Chapter 10 where the concept of split stream control is applied. The prime intention of this textbook is to let the reader develop a deep understanding of the art of distillation. Many fully worked out examples demonstrate the easy applicability of the theoretical findings. These examples are arranged in boxes to

xii

PREFACE

facilitate the readability of the text. Of course, not only the authors are involved in the completion of such a comprehensive book. At this point we would like to thank everyone who contributed to the success of this project: Felicitas Engel, M.Sc., Philipp Fritsch, M.Sc., Patrick Haider, M.Sc., Florian Hanusch, M.Sc., Robert Kender, M.Sc., Thomas Kleiner, M.Sc., Maximilian Neumann, M.Sc., Dr.-Ing. Anna Reif, Marc Xia, M.Sc., Alexander Eder, B.Sc., Florian Kaufmann, M.Sc., and Jan Oettig, B.Sc., as well as the valuable expertise of Dr.-Ing. Volker Engel. Many thanks also to Stephan Korell for his helpful advice on the LATEX implementation.

Johann Stichlmair, Harald Klein, Sebastian Rehfeldt Munich, February 2020

xiii

Nomenclature Latin Symbols

a a aeff a/v 2 a aii aij ai A Ai Aij b b b b bi B B B˙ B

low boiler specific surface area specific effective interfacial area cohesion pressure van der Waals equation coefficient cubic equation of state of mixture coefficient cubic equation of state of pure component i cross coefficient cubic equation of state of component i and j activity of component i area Antoine or Wagner parameter of component i binary parameter Margules and van Laar equation of component i and j high boiler (binary mixture) or intermediate boiler (ternary mixture) constant co-volume van der Waals equation coefficient cubic equation of state of mixture coefficient cubic equation of state of pure component i mole amount of bottom product width of packing channel base bottom flow rate virial coefficient of mixture

– m2 /m3 m2 /m3 Pa J · m3 /kmol2 J · m3 /kmol2

J · m3 /kmol2 – m2 – – – – m3 3 m /kmol m3 /kmol

kmol m kmol/s m3 /kmol

xiv

NOMENCLATURE

Bi Bii Bij c c c, C C Ci CG Ch d d D D D D˙ DE Di e EOG EOGM E f fi fi0 F F F˙ Fi g g gi gi

Antoine or Wagner parameter of component i virial coefficient of pure component i cross virial coefficient of component i and j high boiler (ternary mixture) or intermediate boiler (quaternary mixture) specific or molar heat capacity constant second mixture virial coefficient of mixture Antoine or Wagner parameter of component i capacity factor empiric packing factor high boiler quaternary mixture diameter, distance diameter diffusion coefficient mole amount of overhead product (distillate) overhead (distillate) flow rate dispersion coefficient (eddy diffusion coefficient) Wagner parameter of component i entrainer overall gas-side point efficiency overall gas-side tray efficiency exergy friction factor fugacity of component i standard fugacity of component i F -factor (gas load) mole amount of feed feed flow rate surface area fraction/mole fraction UNIQUAC equation of component i gravitational acceleration g = 9.81 m/s2 molar Gibbs free energy gas flow rate of component i partial molar Gibbs free energy of component i

– m /kmol m3 /kmol – 3

J/(kg · K) J/(kmol · K) – m6 /kmol2 – m/s – – m m m2 /s kmol kmol/s m2 /s – – – – J – Pa Pa Pa0.5 kmol kmol/s –

m/s2 J/kmol kmol/s J/kmol

xv

NOMENCLATURE

∆g gE giE ∆gij G G˙ G GE Gij h hi ∆h h hdyn hdyn0 hf hL hL hp hstat hw H H˙ H Hij HL HETP HTU J k k kij

molar mixing Gibbs free energy molar excess free energy partial molar excess free energy of component i binary parameter NRTL equation of component i and j mole amount of vapor gas/vapor flow rate Gibbs free energy excess free energy binary parameter NRTL equation of component i and j specific or molar enthalpy partial molar enthalpy of component i molar mixing enthalpy height dynamic hold-up dynamic hold-up below loading point froth height clear liquid height liquid hold-up height of pressure drop static hold-up weir height enthalpy enthalpy flow rate tray spacing or packing height Henry coefficient of component i in component j molar hold-up of liquid height equivalent to one theoretical plate height of a transfer unit stripping factor numbers of components in mixture mass transfer coefficient binary parameter cubic equation of state of component i and j

J/kmol J/kmol J/kmol – kmol kmol/s J J – J/kg J/kmol J/kmol J/kmol m m3 /m3 m3 /m3 m m 3 m /m3 m m3 /m3 m J W m Pa kmol m m – – kmol/(m2 · s) –

xvi

NOMENCLATURE

Ki KR l li L L˙ Lp m m M M ˆ M M˙ n n N N˙ NTU p p pi p0i p+ Ph Poy i qi qF Q Q Q˙ ri r ˆ R R˙ RG

vapor–liquid equilibrium ratio of component i reaction equilibrium constant (path) length liquid flow rate of component i amount of liquid liquid flow rate wetted perimeter slope of equilibrium curve exponent mass mole amount of mixture in the middle vessel molecular weight mixture flow rate number of equilibrium stages exponent mole amount molar flow rate number of transfer units pitch pressure partial pressure of component i saturation vapor pressure of pure component i reference pressure number of phases Poynting correction of component i relative van der Waals surface UNIQUAC equation of component i caloric factor (thermal state) of the feed heat dimensionless concentration change heat flow relative van der Waals volume UNIQUAC equation of component i molar latent heat of vaporization ˆ = 8314 J/(kmol · K) ideal gas constant R reactor effluent flow rate external reboil (boilup) ratio

– – m kmol/s kmol kmol/s m – – kg kmol kg/kmol kmol/s – – kmol kmol/s – m Pa Pa Pa Pa – – – – J – W –

J/kmol J/(kmol · K) kmol/s –

xvii

NOMENCLATURE

RL s s S S˙ S S t T Ti0 u ∆uij U v vi ∆v V V˙ Vi wi wiG wiL W xi x ˜i X Xi yi y˜i z

external reflux ratio molar entropy plate thickness length of packing channel side molar flow rate after decanter, side product entropy correction factor (Example 2.2) time temperature boiling temperature of pure component i superficial velocity binary parameter UNIQUAC equation of component i and j internal energy molar volume partial molar volume of component i molar mixing volume volume volumetric flow rate volume fraction/mole fraction UNIQUAC equation of component i mass (weight) fraction of component i mass fraction of component i in gas phase mass fraction of component i in liquid phase work mole fraction liquid phase of component i transformed mole fraction complete chemical reaction of component i function transformed concentration of component i in reactive systems mole fraction vapor phase of component i estimated mole fraction vapor phase of component i (Example 2.3) number of interacting molecules UNIQUAC equation

– J/(kmol · K) m m kmol/s J/K – s K K m/s –

J m3 /kmol m3 /kmol m3 /kmol m3 m3 /s – – kg/kg kg/kg J – –

– – – –

xviii

NOMENCLATURE

z z zi Z Zf

number of particles or channels locus, dimensionless tray length mole fraction two-phase mixture of component i compressibility factor number of independent state variables (degrees of freedom)

– – – – –

Greek Symbols

α αi (T ) αij αij β γL ∆ ∆p ∆% ∆SgE ∆λij ε ζ ζE η ϑ π π Λij µi % σ

discharge coefficient temperature function cubic equation of component i non-randomness factor NRTL equation of component i and j relative volatility (separation factor) of component i and j mass transfer coefficient liquid-phase distribution difference pressure drop density difference sum of squares for g E binary parameter Wilson equation of component i and j porosity, voidage, relative content drag coefficient, orifice coefficient friction factor in Ergun equation dynamic viscosity contact angle pole on the enthalpy–concentration diagram circle constant π = 3.14159 binary parameter Wilson equation of component i and j chemical potential of component i density surface tension

– – – –

m/s – Pa kg/m3 2 J /kmol2 – – – – Pa · s ° J/mol – –

J/kmol kg/m3 kg/s2

xix

NOMENCLATURE

τ τL ϕ ϕi φj Φ Φfl γi γi∞ νi ωi τij

contact time liquid residence time in the two-phase layer relative free area of a tray fugacity coefficient of component i correction factor of component i Underwood parameter flooding factor activity coefficient of component i activity coefficient at infinite dilution of component i stoichiometric coefficient of component i acentric factor of component i binary parameter NRTL and UNIQUAC equation of component i and j

s s – – – – – – – – – –

Subscripts

a ac azeo b B c c c c cap cl cr C d d d d D e

low boiler active azeotrope high boiler (binary mixture) or intermediate boiler (ternary mixture) bottom product high boiler (ternary mixture) or intermediate boiler (quaternary mixture) critical state variable column continuous phase bubble cap clearance under downcomer critical condenser, cooling high boiler quaternary mixture dispersed phase downcomer dry overhead product (distillate) end

xx

NOMENCLATURE

e eq E Exp f fl F Fr G h h H irr j k lam L m m max min n n o o OG OL p P r r R s s S SP

entrainer equivalent entrainment experimental value froth flooding feed Froude number gas hole hydraulic heating irrigated stage number number of components laminar liquid intermediate mean maximum minimum number of plates or steps nominal openings overflow overall gas phase overall liquid phase particle pinch point residual reduced state variable (related to critical state variable) reboiler packing section particle swarm solid side pinch

NOMENCLATURE

t turb T vc v w α ω 0 ∞

plate/tray turbulent transition point vena contracta valve weir start end orifice, single particle infinity

Superscripts

0 azeo C E id id0 n new R 0 00

ˆ ∗ 0

pure component azeotrope combinatorial part UNIQUAC equation excess state variable ideal gas pure ideal gas iteration step or step on distillation line new estimate (Example 2.3) residual part UNIQUAC equation liquid phase vapor phase molar equilibrium state modified

Abbreviations

A-1 C-1 E-1 R-1 S-1

absorber column heat exchanger, extractor reactor decanter

xxi

xxii

NOMENCLATURE

Dimensionless Numbers

Bo =

%·g σ · a2

Bond number

Fr =

u2 g·d

Froude number

Pe =

l2 D·τ

Peclet number

Re =

%·u·d η

Reynolds number

Sc =

η %·D

Schmidt number

We =

u2 · % · d σ

Weber number

1

1 Introduction Distillation is a widely used method for separating liquid mixtures into their components. It is the workhorse for separation in the petroleum, petrochemical, chemical, and related industries. The consensus is that it will continue to dominate these industries in the future, too.

1.1

Principle of Distillation Separation

Distillation utilizes a very simple separation principle: an intimate contact is created between the starting mixture and a second phase in order to enhance an effective mass transfer between these two phases. The thermodynamic conditions are chosen so that primarily the component to be separated from the feed mixture enters the second phase. The phases are subsequently separated into two single phases with different compositions. Three steps are always involved in the implementation of this separation principle; see Figure 1.1:

Figure 1.1 General principle of fractionation in thermal separation technology. The essential mechanism is the mass transfer between two phases.

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

2

1 INTRODUCTION

• Generation of a two-phase system • Mass transfer across the interface • Separation of the phases Many separation techniques utilize this very effective separation principle. Absorption, desorption, evaporation, condensation, and distillation involve a gaseous and a liquid phase. Solvent extraction uses two liquid phases. Separation techniques that utilize a fluid phase and a solid phase include adsorption, crystallization, drying, and leaching. In most of these separation processes, the necessary two-phase system is generated by adding an auxiliary phase to the feed mixture. The substances to be separated collect in diluted form in the auxiliary agent. In distillation, however, the second phase is created by partial vaporization of the liquid feed. Hence, the use of an auxiliary substance (often called a mass separating agent), which requires costly recovery, is avoided, and the components to be separated are recovered as relatively pure substances. Indeed, distillation requires only energy in the form of heat, which can subsequently be easily removed from the system. This is an important advantage of distillation. In practice, distillation requires intimate contacting of vapor and liquid under such conditions that the desired components of the liquid enter the vapor phase. Governing these conditions is the vapor–liquid equilibrium. Many activities on the art of distillation are devoted to find out how closely the vapor–liquid equilibrium can be approached. In any case, it is necessary to separate the liquid and vapor phases afterward. The vapor and liquid are brought into intimate contact by countercurrent or crosscurrent flow, and mass exchange occurs because the two phases are not in thermodynamic equilibrium. The phases produced during distillation are formed by evaporation and condensation of the initial mixture. The separation process can be controlled only by the heat supply. The basis for planning distillation processes is the knowledge of the vapor–liquid equilibrium. As stated earlier, the separation depends primarily on the concentration of the individual substances in the vapor and liquid phases. In this book, principles of vapor–liquid equilibrium are discussed in Chapter 2, with special attention given to the equilibrium of ternary and multicomponent mixtures. Thermodynamic analysis of distillation and rectification is essential to establish the optimal conditions for mass transfer. The decisive factor is the driving force for mass transfer, i.e. the difference between the actual concentrations of the substances and their equilibrium concentrations. Operating conditions have to ensure that this difference is sufficiently large. Appropriate relationships and methods for determining mass transfer are described in Chapters 3 – 6. Examples of industrially important separation processes and energy requirement are discussed in Chapters 7 and 8, respectively. Since separation is achieved by bringing the two phases into intimate contact, in practice, the problems created by multiphase flow and mass transfer between phases must be confronted. The state of the art of multiphase flow is presently rather poor. Two-phase and multiphase flow is an underdeveloped field of fluid mechanics. As a result, just empirical approaches are presently available for practical equipment design, as described in

1.2 HISTORICAL

3

Chapter 9. Chapter 10 deals with the control of single distillation columns and of distillation processes.

1.2

Historical

Although several authors (e.g. Krell 1958) support the view that the art of distillation has been well known to ancient Greece, this opinion has never been proven by historians. Ancient philosophers have been, however, very close by the correct understanding of the principles of distillation, for instance, at the mere philosophical debate on the circulation of water in nature. Aristoteles (384-322 BCE) writes in his Meteorologia: ". . . several authors support a similar view on the origin of rivers. The water elevated by the sun and as rain condensed humidity collects . . . ". In the same book, he writes later on: "That evaporated sea water is drinkable and, after condensation, does not become sea water again, that can we state from experience." However, no practical applications of these theoretical considerations have been reported, and no device for performing the process of distillation is described in ancient literature. Ancient Egypt and ancient China as well had probably no knowledge of the art of distillation. Forbes 1970 agrees with several other authors (e.g. Underwood 1935) in the opinion that the art of distillation has been invented and pioneered in use in Alexandria, Egypt, in the first century CE.

Figure 1.2 Distillation and rectification equipment taken from The Alchemy of Andreas Libavius [Libavius 1964]: (a) boiler, (b) oven, (c) coolers, (d) receiver, (e) headpiece, and (f) receiver.

In the following centuries the knowledge of distillation spread widely and was used around the eleventh century for the first time in northern Italy to produce alcoholic beverages. The development of distillation equipment has been influenced tremendously by this special field of application. An interesting distillation equipment, de-

4

1 INTRODUCTION

scribed in the book The Alchemy of Andreas Libavius, published in 1597, is illustrated in Figure 1.2 [Libavius 1964]; it was used for the batch distillation of alcohol. Heat is supplied to the liquid contents of the boiler (a), built into the oven (b), and the vapor formed was allowed to condense in two coolers (c). Cooling water was changed periodically. The only visible process was the dripping of the condensate into the receiver (d). This separation technique was named after the Latin word destillare, which means "dripping or trickling down". Even in early times, it was well known that a higher alcohol content could be reached by using a second distillation step. In the apparatus shown in Figure 1.2, two distillations could be carried out simultaneously. Condensate from the first distillation is returned to the headpiece (e), the so-called rectificatorium, which is heated with vapor rising from the boiler. The vapor produced in the headpiece is condensed in the two coolers (c). A liquid with a higher alcohol content is then collected in a second receiver (f). The term rectification is derived from this process, which, especially in Europe, is used to describe multistage distillation. The Latin words recte facere mean "to rectify or improve". Indeed, up to this day, term rectification refers to a process by which a further concentration change is achieved after the first evaporation step. From such devices distillation columns have been finally developed during the following centuries. Many authors (e.g. Underwood 1935; Forbes 1970) agree in giving the credit of invention to the Frenchman Cellier-Blumenthal [CellierBlumenthal 1818]. Interesting are the circumstances that enhanced the development of improved distillation devices. In 1807 Napoleon organized a blockade against England, which answered by a blockade against the European continent. In consequence, goods from the oversea colonies no longer reached Europe, which resulted in shortages of sugarcane, among many other goods. It was well known that sugar can be produced from beets grown in Europe as well [Ullmann 1969]. However, the brown sugar from beets was much less attractive to the noblemen than the white sugar from canes. Napoleon opened a competition for producing white sugar from beets by setting a very high prize. A favorable process was extraction of sugar from the beets by alcohol instead of water – a process proposed again in recent years [Ullmann 1969]. The alcohol was recycled within this process. However, after longer periods of operation time, the alcohol needed purification since some water accumulated in the alcohol. CellierBlumenthal developed the first distillation column (a tray column) for this process. In the nineteenth and twentieth centuries, the art of distillation developed rapidly prompted by the oil and petrochemical industry [Deibele 1992] and by the chemical and pharmaceutical industry [Fair 1984]. The present importance of distillation is documented by the fact that approximately 40000 distillation columns are under operation in the United States [Humphrey and Seibert 1992]. These columns consume about 3 % of the total energy requirement of the United States [Gmehling et al. 1994].

REFERENCES

5

References Cellier-Blumenthal, J.B. (1818). Brevet d’invention et de perfectionnement de quinze ans, pour des appareils destinés à la distillation continue et à l’évaporation. French Patent No. 2266. Deibele, E. (1992). Die Entwicklung der Destillationstechnik im 19. Jahrhundert. PhD thesis. Technical University of Munich. Fair, J.R. (1984). Historical development of distillation equipment. AIChE Symposium Series 79 (235): 1–14. Forbes, R.J. (1970). A Short History of the Art of Distillation. Leiden: E. J. Brill. Gmehling, J., Menke, J., Krafczyk, J., and Fischer, K. (1994). Azeotropic Data, Part I and II. Weinheim: VCH Publishers. Humphrey, J.L. and Seibert, A.F. (1992). New

horizons in distillation. Chemical Engineering 99 (12): 86. Krell, E. (1958). Handbuch der Laboratoriumsdestillation. Berlin: VEB Deutscher Verlag der Wissenschaften. Libavius, A. (1964). Die Alchemie des Andreas Libavius. Ein Lehrbuch der Chemie aus dem Jahre 1597. Weinheim: Verlag Chemie. Ullmann, F. (1969). Ullmanns Enzyklopädie der Technischen Chemie, vol. 19, p. 201 and p. 217. München Berlin Wien: Urban & Schwarzenberg. Underwood, A.J.V. (1935). The historical development of distilling plant. Transactions – Institutions of Chemical Engineers 13: 34–63.

7

2 Vapor–Liquid Equilibrium The phase equilibrium between vapor and liquid is of significant importance in separation by distillation. In this chapter, a single prime 0 denotes the liquid phase, and double primes 00 denote the vapor phase, sometimes also labeled as gas phase. Vapor– liquid equilibrium is often abbreviated as VLE. In contrast to most thermodynamic textbooks, the presented examples of mixtures will not be limited to binary mixtures but also include ternary mixtures. Thorough knowledge of VLE and multicomponent mixtures is necessary to understand, develop, and improve industrial distillation processes.

2.1

Basic Thermodynamic Correlations

The required thermodynamic fundamentals to describe and characterize a mixture of chemical components, like measures of concentration, equations of state (EOS), saturation vapor pressure, fundamental equations, chemical potential, Gibbs–Duhem equation, and molar excess free energy are summarized in this section. 2.1.1

Measures of Concentration

A mixture that consists of the components i = 1, 2, . . . , k can be described with different measures of concentration. Here, the mass fraction wi , also labeled as weight fraction, and the mole fraction xi will be used. 2.1.1.1

Mass Fraction

The mass Mi of component i is related to the total mass M of the mixture to define the mass fraction wi as follows:

wi =

Mi M

with the closing condition

k X

wi = 1 .

(2.1)

i=1

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

8

2 VAPOR–LIQUID EQUILIBRIUM

2.1.1.2

Mole Fraction

The mole amount Ni of component i is related to the total mole amount N of the mixture to define the mole fraction xi with the following correlation:

xi =

Ni N

with the closing condition

k X

xi = 1 .

(2.2)

i=1

In a vapor–liquid mixture, xi will be used for the mole fraction of the liquid phase and yi for the mole fraction of the vapor phase. For a two-phase mixture, the mole fraction will be labeled as zi . 2.1.1.3

Conversions

The mole fraction xi can be converted into the mass fraction wi with

wi = xi ·

ˆi M ˆ M

with

ˆ = M

k X

ˆi . xi · M

(2.3)

i=1

ˆ of the mixture has to be calculated from For this conversion, the molecular weight M ˆ i of the components and the known mole fractions xi . The the molecular weights M mass fractions wi can be converted into the mole fraction xi as follows: xi = wi ·

ˆ M ˆi M

with

ˆ = M

1 k X wi ˆi M

k

or

X wi 1 = . ˆ ˆi M M

(2.4)

i=1

i=1

ˆ of the mixture has to be calculated from the molecular Now, the molecular weight M ˆ weights Mi of the components and the known mass fractions wi . In Example 2.3, the conversion from mass to mole fractions is shown for a ternary, liquid mixture. 2.1.2

Equations of State (EOS)

The state of a thermodynamic system can be described with state variables like mass M , mole amount N , pressure p, molar volume v , and temperature T . In this section, equations of state, often abbreviated as EOS, are presented as correlations between these state variables. In mixtures, the mole fractions xi and yi in the liquid and vapor phase have to be included. Equations of state that are able to describe both liquid and gas can be used to calculate the vapor–liquid equilibrium. 2.1.2.1

Ideal Gas Law

A vapor phase is characterized as an ideal gas if no attraction and repulsion forces between molecules exist and collisions of molecules happen without energy dissipation. For ideal gases, the correlation between the pressure p, the volume V , the mole amount N , and the absolute temperature T can be described with the ideal gas law:

ˆ·T p·V =N ·R

ˆ = 8314 with the ideal gas constant R

J . kmol · K

(2.5)

2.1 BASIC THERMODYNAMIC CORRELATIONS

9

With the molar volume v = V /N , it follows that

ˆ·T . p·v =R

(2.6)

If the pure components i = 1, 2, . . . , k behave like ideal gases, mixtures of these components also behave like an ideal gas. For ideal gases, the partial pressure pi is the pressure of component i when the mole amount Ni occupies the volume V at temperature T :

ˆ·T . p i · V = Ni · R

(2.7)

In addition, for mixtures of ideal gases, the component i with the mole fraction yi can be treated like being present alone in the mixture with the partial pressure

pi = yi · p .

(2.8)

For the sum of all partial pressures pi in Eq. (2.8), we obtain with the closing conPk dition i=1 yi = 1 the following correlation, which is also known as Dalton’s law:

p=

k X

pi .

(2.9)

i=1

The vapor phase can be treated as an ideal gas if the pressure p is well below the critical pressures pci and if the temperature T is well above the critical temperatures Tci of the components in the mixture, i.e. p  pci and T  Tci [Stephan et al. 2012]. 2.1.2.2

Virial Equations

For higher pressures, the vapor phase cannot be treated as an ideal gas anymore. The real gas behavior can be described by the compressibility factor Z , which is defined as follows:

Z=

p·v . ˆ R·T

(2.10)

For an ideal gas we have Z = 1. Deviations from this value are directly proportional to the degree of non-ideal behavior, which can be described with the virial equation:

Z =1+

B(T ) C(T ) + + ... . v v2

(2.11)

Intermolecular attraction forces are described by the virial coefficients B(T ), C(T ), etc., which, for pure components, are solely a function of the temperature T . With a virial equation, only gases can be described, and it is not applicable to liquids. As a rule of thumb, for pressures p < 0.5 · pc , using only the first virial coefficient B(T ) yields adequate accuracy [Stephan et al. 2012].

10

2 VAPOR–LIQUID EQUILIBRIUM

The following mixing rules can be applied to virial coefficients of mixtures [Stephan et al. 2017]:

B(T ) =

k X k X

yi · yj · Bij (T ) with Bij (T ) = Bji (T ) .

(2.12)

i=1 j=1

For a ternary mixture with the components a, b, and c and the mole fractions ya , yb , and yc , this yields

B(T ) = ya2 · Baa (T ) + yb2 · Bbb (T ) + yc2 · Bcc (T ) + 2 · ya · yb · Bab (T ) + 2 · ya · yc · Bac (T ) + 2 · yb · yc · Bbc (T )

(2.13)

with the pure component virial coefficients Baa (T ), Bbb (T ), and Bcc (T ). The cross virial coefficients Bij (T ) = Bji (T ) can be determined with a mixing rule using the virial coefficients of the pure components:

Bij (T ) =

1 · (Bii (T ) + Bjj (T )) . 2

(2.14)

In these correlations, no mixing parameters are required in order to describe gaseous mixtures at moderate pressures p < 0.5 · pc . 2.1.2.3

Cubic Equations of State

Various forms of cubic equations of state were developed in the past. All of them are based on the idea of the van der Waals equation, where the attraction and the volume of the molecules are considered by the cohesion pressure a/v 2 and the co-volume b.  a ˆ·T . (2.15) p + 2 · (v − b) = R v With the Soave–Redlich–Kwong (SRK) equation [Soave 1972], a high accuracy for pure components is obtained, which can be expressed in a pressure-explicit form as follows:

p=

ˆ·T R a − . v − b v · (v + b)

(2.16)

Cubic EOS can be used for both the vapor and liquid phase with the suitable molar volume v 0 of the liquid or v 00 of the vapor. It is, however, not possible to rewrite Eq. (2.16) to solve for the molar volume v such that iterative numerical solution schemes have to be applied [Asselineau et al. 1979, Prausnitz et al. 1980, Gmehling et al. 2012, Gmehling et al. 2019]. The SRK equation can also be expressed using the compressibility factor Z as defined above in Eq. (2.11):

Z=

v a − . ˆ · T · (v + b) v−b R

(2.17)

11

2.1 BASIC THERMODYNAMIC CORRELATIONS

The following pure component coefficients can be determined for the SRK equation from the critical pressure pci , temperature Tci , and the acentric factor ωi of the components i = 1, 2, . . . , k in the mixture:

aii (T ) = 0.42748 ·

ˆ2 · T 2 R ci · αi (T ) pci

(2.18)


0.5 2 αi (T ) = 1 + (0.48 + 1.574 · ωi − 0.176 · ωi2 ) · (1 − Tri ) ˆ · Tci R bi = 0.08664 · . pci

(2.19) (2.20)

Here, the reduced temperature Tri = T /Tci is used to calculate the temperature dependent function αi (T ). For cubic EOS, the pure component coefficients aii and bi can be used for the following mixing rules, depending on whether the parameters a and b for liquid phase (with the mole fraction xi ) or the vapor phase (with the mole fraction yi ) shall be calculated:

a0 =

k X k X

xi · xj · aij

a00 =

i=1 j=1

b0 =

k X

x i · bi

k X k X

yi · yj · aij

(2.21)

i=1 j=1

b00 =

i=1

k X

yi · bi .

(2.22)

i=1

The pure component coefficients aii and bi are, as described above in Eqs. (2.18) – (2.20), whereas the cross coefficients aij = aji can be determined using the binary parameters kij with the mixing rule

aij =

√

aii · ajj · (1 − kij ) .

(2.23)

The binary parameters kij are mixing properties and have to be adjusted to experimental data. The accuracy of a cubic EOS, describing the behavior of a mixture, strongly depends on the binary parameters kij . Furthermore, it is worthwhile to note that only parameters for binary mixtures are required in order to describe multicomponent mixtures. Example 2.3 demonstrates how pure component coefficients aii and bi , along with binary parameters kij , are applied to the mixing rules for a ternary mixture. Besides the Soave–Redlich–Kwong equation, the Peng–Robinson equation [Peng and Robinson 1976], using the same mixing rules and binary parameters, shows comparable accuracy:

p=

ˆ·T R a − . v − b v · (v + b) + b · (v − b)

(2.24)

For the Peng–Robinson equation, the pure component coefficients have to be calcu-

12

2 VAPOR–LIQUID EQUILIBRIUM

lated as follows:

aii (T ) = 0.45724 ·

ˆ2 · T 2 R ci · αi (T ) pci

0.5 αi (T ) = 1 + (0.37464 + 1.54226 · ωi − 0.26992 · ωi2 ) · (1 − Tri )

(2.25)
2 (2.26)

bi 2.1.2.4

ˆ · Tci R = 0.0778 · . pci

(2.27)

Enhanced and Predictive Equations of State

Especially for polar components and their mixtures, both the Soave–Redlich–Kwong and the Peng–Robinson equation, as described in the previous paragraph, do not yield the accuracy that is required for process calculations, and therefore many approaches to overcome this disadvantage can be found in literature [Gmehling et al. 2012]. One possibility is the PSRK (Predictive Soave–Redlich–Kwong) equation, developed by Holderbaum and Gmehling 1991 and Horstmann et al. 2000. Here, other mixing rules than the simple one in Eq. (2.23) are derived. These mixing rules are based on correlations for the molar excess free energy g E , as shown in Section 2.2.12, e.g. the mixing rules by Huron and Vidal 1979 based on the UNIQUAC equation. With the group contribution methods, as shown in the last paragraph of Section 2.2.12, the VTPR (Volume Translated Peng–Robinson) equation was developed and widens the scope of use of cubic EOS even for complex mixtures [Ahlers and Gmehling 2001, 2002]. A comprehensive overview of enhanced equations of state can be found in the textbook of de Hemptinne et al. 2012, which also includes a summary of the SAFT (Statistical Associating Fluid Theory) equation of state. Here, statistical mechanical methods to describe the interaction between the molecules in mixtures are used [Chapman et al. 1990]. Sadowsky and Gross 2001 developed the PC-SAFT (Pertubated Chain-Statistical Associating Fluid Theory) equation of state, which can be used for associating and polar molecules. The PC-SAFT equation of state has also been modified for use with polymers [Gross and Sadowski 2002]. CPA (Cubic–Plus–Association) is an EOS that is based on a combination of the Soave–Redlich–Kwong equation with an association term and has been successfully applied to a variety of complex VLE [Kontogeorgis and Folas 2010]. The ability of this equation of state to predict different types of VLE was tested, and satisfactory prediction, solely based on binary interaction parameters, was achieved for the multiphase equilibria of mixtures. 2.1.3

Molar Mixing and Partial Molar State Variables

Molar state variables in mixtures, such as the molar volume v and the molar enthalpy h, can be calculated using the pure component state variables vi0 and h0i of the components i = 1, 2, . . . , k and the molar mixing state variables, here the molar

2.1 BASIC THERMODYNAMIC CORRELATIONS

13

mixing volume ∆v and the molar mixing enthalpy ∆h, respectively:

v=

k X

xi · vi0 + ∆v

h=

k X

i=1

xi · h0i + ∆h .

(2.28)

i=1

Another method to calculate molar state variables is based on the partial molar state variable, here vi and hi , which are defined as follows:     ∂V ∂H vi = and hi = . (2.29) ∂ Ni p,T,Nj6=i ∂ Ni p,T,Nj6=i The index j 6= i at the partial derivatives denotes that the mole amounts Ni of all components other than i stay constant. With Euler’s theorem for homogeneous functions [Pfennig 2004] it follows that

V =

k X

Ni · v i

and

H=

i=1

k X

Ni · hi .

(2.30)

i=1

After dividing by the mole amount N , we obtain

v=

k X i=1

x i · vi

and

h=

k X

xi · hi .

(2.31)

i=1

The partial molar volume vk and the partial molar enthalpy hk of one particular component k can be calculated from the molar volume v and the molar enthalpy h of the mixture, respectively, as follows [Stephan et al. 2017]:   k−1 X ∂v vk = v − xi · (2.32) ∂ xi p,T,xj6=i i=1   k−1 X ∂h hk = h − xi · . (2.33) ∂ xi p,T,xj6=i i=1 During the course of this chapter, these correlations between a partial molar state variable and the corresponding molar state variable are also applied in Eqs. (2.41) and (2.92). If the molar mixing state variables are equal to zero, here ∆v = 0 and ∆h = 0, the mixture is called an ideal mixture. Mixtures of ideal gases are always ideal mixtures, but not all ideal mixtures are ideal gases, i.e. many liquids of chemically similar components form ideal mixtures. For ideal mixtures, we have vi = vi0 and hi = h0i . 2.1.4

Saturation Vapor Pressure and Boiling Temperature of Pure Components

The vapor–liquid equilibrium of a pure component i is determined by the saturation vapor pressure curve, which represents the dependence of the boiling temperature Ti0 (p) from the pressure or, vice versa, the dependence of the saturation vapor pressure p0i (T ) from the temperature.

14

2 VAPOR–LIQUID EQUILIBRIUM

The saturation vapor pressure p0i (T ) of component i can be determined analytically for a specified temperature T by the Antoine equation, for example, in the form

ln p0i (T ) = Ai −

Bi . T + Ci

(2.34)

The Antoine parameters Ai , Bi , and Ci of most components can be found in literature [Prausnitz et al. 1999; Perry et al. 1984]. Another form that can be found in literature is based on the common logarithm:

log10 p0i (T ) = Ai −

Bi . T + Ci

(2.35)

It has to be considered that Eqs. (2.34) and (2.35) as numerical value equations can be used only with variables in the same units as those of the tabulated Antoine parameters, i.e. in Table 2.1 the units K for temperature and bar for pressure, respectively. The Antoine equation in Eqs. (2.34) and (2.35) can be rewritten to solve for the boiling temperature Ti0 (p) for a specified pressure p:

Ti0 (p) =

Bi − Ci Ai − ln p

or

Ti0 (p) =

Bi − Ci . Ai − log10 p

(2.36)

The Wagner equation, which shows a higher accuracy than the Antoine equation, is also widely used in the following form [Prausnitz et al. 1999]:

ln

p0i 1  = · Ai · (1 − Tri ) + Bi · (1 − Tri )1.5 pci Tri  + Ci · (1 − Tri )2.5 + Di · (1 − Tri )5 .

(2.37)

The critical pressure pci and the critical temperature Tci of the component i are required. The reduced temperature is defined as Tri = T /Tci . Contrary to the Antoine equation, the Wagner equation cannot be rewritten to solve for the boiling temperature Ti0 (p) directly. The parameters of the Wagner equation can be found for an extensive list of components [Poling et al. 2001]. 2.1.5

Fundamental Equation and the Chemical Potential

The extensive state variable Gibbs free energy G with the definition

G=H −T ·S =U +p·V −T ·S

(2.38)

plays an important role in the thermodynamic of mixtures because the functional correlation G = G(p, T, N1 , N2 , . . . , Nk ) is a fundamental equation, which contains all thermodynamic information of a system. The fundamental equation in differential form  k  X ∂G dG = −S · dT + V · dp + · dNi (2.39) ∂ Ni p,T,Nj6=i i=1

15

2.1 BASIC THERMODYNAMIC CORRELATIONS

Table 2.1 Boiling temperatures at p = 1.013 bar and parameters Ai , Bi , and Ci for selected components for the Antoine equation in the form ln p0i (T ) = Ai − Bi /(T + Ci ) for p0i in bar and T in K. Boiling temperature at p = 1.013 bar in K

Formula

Name

Ar CCl4 CHCl3 CH2 Cl2 CH2 O CH2 O2 CH3 NO2 CH4 O CO CO2 C2 H3 N C2 H4 O C2 H4 O2 C2 H6 C2 H6 O C2 H7 N C3 H6 O C3 H8 C3 H8 O C3 H8 O C3 H8 O2 C4 H8 O C4 H8 O C4 H8 O2 C4 H8 O2 C4 H8 O2 C4 H9 NO C4 H10 C4 H10 C4 H10 O C4 H10 O2 C5 H5 N C5 H12 C6 H5 Cl C6 H5 NO2 C6 H6 C6 H6 O C6 H12 C6 H12 O C6 H12 O2 C6 H14 C7 H8 C7 H16 C8 H10 C8 H18 H2 O NH3 N2 O2

Argon Tetrachloromethane Chloroform Dichloromethane Formaldehyde Formic acid Nitromethane Methanol Carbon monoxide Carbon dioxide Acetonitrile Acetaldehyde Acetic acid Ethane Ethanol Ethylamine Acetone Propane n-Propanol Isopropanol 2-Methoxyethanol 2-Butanone Tetrahydrofuran 1,4-Dioxane Butyric acid Ethyl acetate Morpholine n-Butane Isobutane Diethyl ether 2-Ethoxyethanol Pyridine Pentane Chlorobenzene Nitrobenzene Benzene Phenol Cyclohexane Cyclohexanol Butyl acetate Hexane Toluene Heptane Ethylbenzene Octane Water Ammonia Nitrogen Oxygen

87.30 349.70 334.40 313.20 254.00 373.70 374.30 337.80 81.60 185.50 354.80 293.70 391.10 184.50 351.50 289.70 329.40 231.10 370.40 355.40 397.60 352.80 339.10 374.50 436.40 350.30 401.40 272.70 261.40 307.70 408.30 388.40 309.15 404.90 484.00 353.20 455.00 353.90 434.30 399.20 341.90 383.80 371.60 409.30 399.15 373.10 239.80 77.40 90.20

Ai 9.31039 9.22001 9.39360 10.44014 9.94883 9.37044 10.14657 11.98705 9.26679 10.77151 10.28058 9.97724 11.84896 9.27428 12.05896 10.38728 9.76775 9.10434 11.21152 13.82295 11.45476 9.64438 9.48686 10.49171 13.43588 9.73241 9.86713 9.05814 9.15169 9.91763 11.38407 9.52860 9.21312 9.77659 9.79835 9.22142 9.33802 9.15600 12.61197 9.79073 9.29213 9.38490 9.25363 9.41928 9.34011 11.96481 10.88865 8.74393 8.43907

Source: Gmehling et al. 1977 and Reid et al. 1987

Bi

Ci

832.778 2.361 2790.781 −46.741 2696.249 −46.918 3053.085 −20.530 2234.878 −29.026 2982.446 −55.150 3331.696 −45.550 3643.314 −33.424 769.935 1.637 1956.255 −2.112 3413.099 −22.627 2532.406 −39.205 4457.828 −14.699 1582.178 −13.762 3667.705 −46.966 2618.730 −37.300 2787.498 −43.486 1872.824 −25.101 3310.394 −74.687 4628.956 −20.514 4130.796 −36.273 2904.340 −51.181 2768.375 −46.896 3579.781 −32.813 5602.222 −17.961 2866.606 −55.269 3333.452 −63.150 2154.897 −34.420 2133.243 −28.162 2847.722 −20.110 4149.028 −43.150 3124.447 −60.495 2477.075 −39.945 3485.354 −48.327 4032.655 −71.814 2755.642 −53.989 3183.669 −113.657 2778.000 −50.014 5200.527 −21.526 3293.659 −62.405 2738.418 −46.870 3090.783 −53.963 2911.320 −56.510 3291.661 −59.383 3128.752 −63.295 3984.923 −39.724 2363.237 −22.621 648.592 −2.976 674.588 −10.093

Range of validity in K 73–133 287–350 263–333 233–313 164–373 299–381 329–409 288–357 52–121 146–285 246–355 191–293 291–391 145–284 293–366 215–316 260–328 192–331 333–378 247–356 290–447 316–361 296–373 293–378 293–423 289–349 317–443 196–292 165–345 308–432 336–407 254–388 223–331 260–335 317–484 281–353 336–455 280–354 317–434 333–399 243–443 246–384 270–400 263–409 259–399 274–373 200–379 65–115 60–125

16

2 VAPOR–LIQUID EQUILIBRIUM

exhibits the feature that the experimentally accessible state variables pressure p and temperature T appear as independent state variables. The partial molar Gibbs free energy gi with the definition   ∂G gi = (2.40) ∂ Ni p,T,Nj6=i describes how the Gibbs free energy G changes if, at constant pressure p and constant temperature T , the mole amount Ni of component i changes, while the mole amounts of all other components (Index j 6= i) are constant. The partial molar Gibbs free energy gk of one particular component k can be calculated with the molar Gibbs free energy g = G/N as follows [Stephan et al. 2017]1 :

gk = g −

k−1 X i=1



xi ·

∂g ∂ xi



.

(2.41)

p,T,xj6=i

Due to its important role, the partial molar Gibbs free energy gi is also titled as the chemical potential µi = gi . Hence, the fundamental equation in Eq. (2.39) can be written as

dG = −S · dT + V · dp +

k X

µi · dNi .

(2.42)

i=1

After division with the mole amount N of the mixture, we obtain the following molar form of the fundamental equation of mixtures:

dg = −s · dT + v · dp +

k X

µi · dxi .

(2.43)

i=1

For the Gibbs free energy G(p, T, N1 , N2 , . . . , Nk ), the following total differential can be derived:      k  X ∂G ∂G ∂G dG = · dT + · dp + · dNi . ∂ T p,Ni ∂ p T,Ni ∂ Ni p,T,Nj6=i i=1 (2.44) By comparison with Eq. (2.42), we obtain     ∂G ∂G S=− and V = . ∂ T p,Ni ∂ p T,Ni

(2.45)

After division with the mole amount N of the mixture, the following molar forms are obtained:     ∂g ∂g s=− and v= . (2.46) ∂ T p,xi ∂ p T,xi 1

This is analogous to the correlations in Eqs. (2.32) and (2.33).

17

2.1 BASIC THERMODYNAMIC CORRELATIONS

2.1.6

Gibbs–Duhem Equation and Gibbs–Helmholtz Equation

According to Euler’s theorem for homogeneous functions, the Gibbs free energy G can be calculated using the partial molar free energy gi = µi of the components i = 1, 2, . . . , k as follows:

G=

k X

Ni · gi =

i=1

k X

Ni · µi .

(2.47)

i=1

After division with the mole amount N of the mixture, we obtain the following molar form:

g=

k X

xi · gi =

i=1

k X

xi · µi .

(2.48)

i=1

The differential dg can be derived using the product rule of calculus:

dg =

k X

d (xi · µi ) =

i=1

k X

xi · dµi +

i=1

k X

µi · dxi .

(2.49)

i=1

Subtracting Eq. (2.43) yields the Gibbs–Duhem equation:

0 = s · dT − v · dp +

k X

xi · dµi .

(2.50)

i=1

The interpretation of the Gibbs–Duhem equation is that out of k + 2 state variables p, T , and µ1 , µ2 , . . . , µk , only k + 1 can be varied independently. For an isobaric–isothermal mixture with dT = dp = 0, it follows that

0=

k X

xi · dµi ,

(2.51)

i=1

i.e. only k − 1 chemical potentials µi can be varied independently in an isobaric– isothermal mixture. In order to describe the temperature dependence of the Gibbs free energy G, the Gibbs–Helmholtz equation can be applied [Gmehling et al. 2012]:   g! G H h ∂ ∂  T  = − 2 and =− 2. (2.52) T T T ∂ T p,xi ∂ T p,Ni

18

2.2

2 VAPOR–LIQUID EQUILIBRIUM

Calculation of Vapor–Liquid Equilibrium in Mixtures

If a vapor and a liquid are in intense contact for a long time, equilibrium is attained between the phases. This means that no net flow of heat, mass, and momentum occurs across the phase boundary. The complete formulation for a system at equilibrium is demonstrated below2 . 2.2.1

Basic Equilibrium Conditions

From the second law of thermodynamics, it follows that for a isobaric–isothermal system with the two phases 0 and 00 , as shown in Figure 2.1, the Gibbs free energy G reaches a minimum value at thermodynamic equilibrium conditions [Michelsen and Mollerup 2004, Gmehling et al. 2012]:

G = min at p = const and T = const .

(2.53)

This equilibrium state is reached when the mass transfer of the components i = 1, 2, . . . , k between the two phases 0 and 00 is finished.

Figure 2.1 Equilibrium in a mixture at isobaric–isothermal condition p, T = const with transfer of the components i = 1, 2, . . . , k between the two phases 0 and 00 .

By applying a differential approach, the mole amount dNi for each component i = 1, 2, . . . , k is assumed to be transferred between the two phases until equilibrium is reached. In Figure 2.1, the transfer dNi of component i from phase 00 to 0 is assumed as the positive direction. In order to keep isobaric–isothermal conditions, the differential heat dQ and the differential work dW have to be added or withdrawn 2

State variables in equilibrium state are not denoted with a particular index in this chapter. In subsequent chapters, where equilibrium and non-equilibrium states appear simultaneously, the index ∗ will be used to label the equilibrium state.

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

19

from the system in Figure 2.1. A necessary condition for thermodynamic equilibrium at constant temperature T = T 0 = T 00 and pressure p = p0 = p00 can be written as

dG = dG0 + dG00 = 0 .

(2.54)

The differential changes dG0 and dG00 of the Gibbs free energy in the two phases can be determined using the fundamental equation in Eq. (2.42). For an isobaric– isothermal system with dT = dp = 0, this results in

dG =

k X

µ0i · dNi0 +

i=1

k X

µ00i · dNi00 = 0 .

(2.55)

i=1

If no chemical reactions take place, the mole amounts Ni0 and Ni00 of the components i = 1, 2, . . . , k can only change via transfer between the two phases, i.e. dNi0 = +dNi and dNi00 = −dNi such that dNi00 = −dNi0 . Hence, Eq. (2.55) can be rewritten as

dG =

k X

(µ0i − µ00i ) · dNi0 = 0 at T 0 = T 00

and

p0 = p00 .

(2.56)

i=1

Finally, the thermodynamic equilibrium conditions between the two phases 0 and 00 are: Thermal equilibrium

T 0 = T 00

(2.57)

Mechanical equilibrium

p0 = p00

(2.58)

Phase equilibrium

µ0i

=

µ00i

for i = 1, 2, . . . , k .

(2.59)

Besides the pressure p and the temperature T , the chemical potentials µi of all components i = 1, 2, . . . , k in the mixture have to be equal in the two phases. In order to derive applicable correlations, the chemical potential has to be determined, as shown in Section 2.2.3. 2.2.2

Gibbs Phase Rule

From the Gibbs–Duhem equation in Eq. (2.50) and the thermodynamic equilibrium conditions in Eqs. (2.57) – (2.59), the following correlation can be derived [Stephan et al. 2017]:

Zf = 2 + k − P h .

(2.60)

This correlation is called Gibbs phase rule and describes the number Zf of independent state variables (degrees of freedom) in a mixture of k components with P h phases.

20

2.2.3

2 VAPOR–LIQUID EQUILIBRIUM

Correlations for the Chemical Potential

In order to evaluate the equilibrium conditions in Eq. (2.59), correlations for the chemical potentials µ0i and µ00i in the liquid and vapor phase are required. These can be based either on the ideal gas or on the pure components, as shown below, according to Stephan et al. 2017 and Prausnitz et al. 1999. 2.2.3.1

Ideal Gas as Pure Component

The fundamental equation for a pure component, i.e. dxi = 0, according to Eq. (2.43) for isothermal conditions with dT = 0 can be written as dg = v · dp. ˆ · T /p, this can be integrated between a For component i as an ideal gas with v = R + reference pressure p and the pressure p as follows: Z p dp id0 id0 + ˆ ˆ · T · ln p . gi (p, T ) − gi (p , T ) = R · T · =R (2.61) + p p+ p id0 Hence, the chemical potential µid0 i (p, T ) = gi (p, T ) of the pure component i as an ideal gas at temperature T and pressure p can be calculated as p 0 + ˆ µid0 (2.62) i (p, T ) = µi (p , T ) + R · T · ln + . p

The reference pressure p+ is usually selected sufficiently low such that ideal gas behavior can be assumed at p+ . 2.2.3.2

Real Gas as Pure Component

If the pure component i behaves like a real gas, the chemical potential can be calculated with the fugacity fi0 of the pure component i as follows: 0 ˆ · T · ln fi . µ0i (p, T ) = µ0i (p+ , T ) + R p+

(2.63)

The expression real gas, which can also include the liquid phase, is used here to describe the behavior differing from the ideal gas. Therefore, the expression real fluid could be used, too. The fugacity fi0 can be regarded as a “corrected pressure” and was introduced in order to make the simple correlation for ideal gases in Eq. (2.62) applicable for real gases as well [Gmehling et al. 2012]. With the fugacity coefficient ϕ0i of the pure component i

ϕ0i =

fi0 , p

(2.64)

and Eq. (2.62), we obtain from Eq. (2.63) 0 ˆ µ0i (p, T ) = µid0 i (p, T ) + R · T · ln ϕi .

(2.65)

For a real gas of the pure component i, the fugacity coefficient ϕ0i can be calculated with the following correlation:  Z p 0 Z p vi 1 Z −1 0 ln ϕi = − dp = dp . (2.66) ˆ p p R·T 0 0

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

21

ˆ · T ) and the integral contains the equation of Here, we have Z = p · vi0 /(R 0 ˆ · T is state vi (p, T ) of the pure component i. If ideal gas behavior with p · vi0 = R id0 assumed, it follows that ϕi = 1. For the fugacity fi0 = ϕ0i · p of the pure component i, we obtain  0 Z p vi0 ∂ fi vi0 ln fi0 = dp and = . (2.67) ˆ·T ˆ·T ∂p T R 0 R If the EOS is given in a pressure-explicit form p(vi0 , T ), e.g. for cubic equations of state, the following form can be used [Gmehling et al. 2012]: ! Z ∞ ˆ·T 1 R 0 ln ϕi = Z − 1 − ln Z + · p− 0 dvi0 . (2.68) ˆ·T vi R vi0 Here, the integration boundaries are the actual molar volume vi0 and the state of the ideal gas at vi0 → ∞. 2.2.3.3

Ideal Gas in a Mixture

If the component i is in a mixture of ideal gases, the mixture is also an ideal gas and the component behaves, according to Dalton’s law, like the sole component in the mixture at the pressure pi . Hence, the chemical potential of the component i can be calculated as pi 0 + ˆ µid (2.69) i (p, T, xi ) = µi (p , T ) + R · T · ln + , p where pi = xi · p is the partial pressure of component i in the mixture. With Eq. (2.62), we obtain id0 ˆ µid i (p, T, xi ) = µi (p, T ) + R · T · ln xi .

2.2.3.4

(2.70)

Real Gas in a Mixture with Fugacity Coefficients

If the component i is part of a mixture of real gases, the fugacity fi of component i in the mixture has to be used in Eq. (2.69) instead of the partial pressure3

ˆ · T · ln fi . µi (p, T, x1 , x2 , . . . , xk−1 ) = gi = µ0i (p+ , T ) + R p+

(2.71)

With the fugacity coefficient fi of component i in the mixture

ϕi =

fi pi

(2.72)

and Eqs. (2.69) and (2.70), we obtain

ˆ ˆ µi (p, T, x1 , x2 , . . . , xk−1 ) = µid0 i (p, T ) + R · T · ln xi + R · T · ln ϕi ˆ = µid (2.73) i (p, T, xi ) + R · T · ln ϕi . 3

The chemical potential µi formally depends on all mole fractions x1 , x2 , . . . , xk . Due to the closing condition in Eq. (2.2), this can be written as being dependent only on x1 , x2 , . . . , xk−1 .

22

2 VAPOR–LIQUID EQUILIBRIUM

The fugacity coefficient ϕi can be calculated with the correlation  Z p vi 1 ln ϕi = − dp , ˆ·T p R 0

(2.74)

which is equivalent to Eq. (2.66); however, the partial molar volume vi has to be used instead the molar volume v0i of the pure component. The partial molar volume vk of one particular component k can be calculated with Eq. (2.32). For an ideal gas, the partial molar volume is identical to the molar volume of the pure component such ˆ · T , it follows that ϕid = 1. that vi = vi0 . With p · vi0 = R i Usually, the EOS is given in the pressure-explicit form, which can be expressed as ˆ · T ), the following correlation for p(v, T, x1 , x2 , . . . , xk−1 ). With Z = p · v/(R the fugacity coefficients is obtained [Gmehling et al. 2012]:

ln ϕi (p, T, x1 , x2 , . . . , xk−1 ) = !  Z ∞  ˆ·T 1 ∂p R = · − dV − ln Z . ˆ·T ∂ Ni T,V,Nj6=i V R V

(2.75)

According to Stephan et al. 2017, we obtain the following correlation for one particular component k :

ln ϕk (p, T, x1 , x2 , . . . , xk−1 ) = = Z − 1 − ln Z + − 2.2.3.5

1 · ˆ R·T

Z v

1 ˆ·T R

∞ k−1 X i=1

·

Z v



xi ·

ˆ·T R p− v

∞

∂p ∂ xi

!

dv (2.76)



dv . T,v,xj6=i

Real Gas in a Mixture with Activity Coefficients

The difference between Eqs. (2.65) and (2.73) yields

µi (p, T, x1 , x2 , . . . , xk−1 ) − µ0i (p, T ) = ˆ · T · ln xi + R ˆ · T · ln ϕi − R ˆ · T · ln ϕ0i =R   ˆ · T · ln xi · ϕi . =R (2.77) ϕ0i With the activity coefficient γi of component i in the mixture

γi =

ϕi , ϕ0i

(2.78)

we obtain

ˆ · T · ln (xi · γi ) . µi (p, T, x1 , x2 , . . . , xk−1 ) = µ0i (p, T ) + R

(2.79)

With the activity ai = xi · γi , this can be written as

ˆ · T · ln ai . µi (p, T, x1 , x2 , . . . , xk−1 ) = µ0i (p, T ) + R

(2.80)

23

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

For pure components with xi = 1, we have γi = 1, and the activity ai is the same as the mole fraction xi . Hence, the activity can be regarded as a “corrected mole fraction”, which describes how “active” a component i is in a mixture. For γi > 1, the component i is “more active” in a mixture due to weaker attraction forces toward molecules of the other components. If the attraction forces between component i and the other components of a mixture are stronger, the molecules are “less active”, which results in γi < 1. If component i is infinitely diluted in the mixture, the activity coefficient at infinite dilution γi∞ is defined as

γi∞ = lim γi .

(2.81)

xi →0

2.2.3.6 Comparison of Fugacity and Activity Coefficients and their Applications

The fugacity coefficient ϕ0i can be considered as the thermodynamic departure function of the chemical potential µ0i of a pure real gas, which describes the difference to the behavior of a pure ideal gas. Equivalently, the fugacity coefficient ϕi describes the difference in the behavior of the component i as a real gas in a mixture to component i as an ideal gas in a mixture. Theoretically, both ϕ0i and ϕi can be used to describe the behavior of component i in the liquid phase, too. This is shown in Sections 2.2.7 and 2.2.9 with practical applications in Examples 2.2 and 2.3. However, in the majority of VLE calculations, fugacity coefficients are usually applied only to the vapor phase. The activity coefficients γi are always mixture properties, and for pure components, we have γi = 1 and ai = xi . This also valid for ideal mixtures: γi = 1 for

i = 1, 2, . . . , k .

(2.82)

Theoretically, activity coefficients can be used to describe the behavior of component i in a vapor mixture, too. However, in VLE calculations, activity coefficients are usually applied only to the liquid phase, as shown in Sections 2.2.4, 2.2.7, and 2.2.10 with a practical application in Example 2.6. 2.2.4

Calculating Activity Coefficients with the Molar Excess Free Energy

According to Euler’s theorem for homogeneous functions, the molar Gibbs free energy g of a mixture with the components i = 1, 2, . . . , k can be calculated using the partial molar free energies gi = µi , as shown in Eq. (2.48). Inserting Eq. (2.79) into Eq. (2.48) yields

g=

k X i=1

xi · µi =

k X i=1

ˆ·T · xi · µ0i + R

k X i=1

ˆ·T · xi · ln xi + R

k X

xi · ln γi .

i=1

(2.83) Another approach uses the molar values gi0 of the pure components and a molar mixing Gibbs free energy ∆g , as shown in Section 2.1.3 for the molar volume v and

24

2 VAPOR–LIQUID EQUILIBRIUM

the molar enthalpy h. Since the state variable Gibbs free energy G = H − T · S contains the entropy S , the molar mixing Gibbs free energy ∆g consists of a part resulting from mixing of ideal gases ∆g id and the molar excess free energy g E . Hence, with ∆g = ∆g id + g E , we obtain

g=

k X

xi · gi0 + ∆g =

i=1

k X

xi · gi0 + ∆g id + g E .

(2.84)

i=1

For ideal gases with giid = µid i , it follows that

g id =

k X

xi · giid =

i=1

k X

xi · µid i =

i=1

k X

xi · giid0 + ∆g id .

(2.85)

i=1

id With giid0 = µid0 i , and after inserting µi from Eq. (2.70), we obtain the molar id mixing Gibbs free energy ∆g of ideal gases as follows:

ˆ·T · ∆g id = R

k X

xi · ln xi .

(2.86)

i=1

With gi0 = µ0i , and after inserting Eq. (2.86) into Eq. (2.84), we obtain

g=

k X

ˆ·T · xi · µ0i + R

i=1

k X

xi · ln xi + g E .

(2.87)

i=1

By comparison of Eq. (2.87) with Eq. (2.83), we obtain for the molar excess free energy

ˆ·T · gE = R

k X

xi · ln γi .

(2.88)

i=1

This equation describes that all activity coefficients γ1 , γ2 , . . . , γk are correlated with one state variable, the molar excess free energy g E . The following dimensionless form is often used to describe analytical g E correlations: k

X gE = xi · ln γi . ˆ R·T

(2.89)

i=1

The next target is to find a correlation that enables the calculation of the activity coefficients γi using the molar excess free energy g E . Analogous to Eq. (2.48), the molar excess free energy g E can also be expressed using the partial molar free energy giE as follows:

gE =

k X i=1

xi · giE .

(2.90)

25

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

By comparison with Eq. (2.88), we obtain

ˆ · T · ln γi = g E , R i

(2.91)

i.e. the activity coefficient γi is directly correlated to the partial molar excess free energy giE , which can be determined for one particular component k , analogous to Eq. (2.41), as follows [Stephan et al. 2017]:

ˆ · T · ln γk = gkE = g E − R

k−1 X



xi ·

i=1

∂ gE ∂ xi



.

(2.92)

p,T,xj6=i

Hence, we obtain a correlation to calculate all activity coefficients γ1 , γ2 . . . , γk from one state variable, the molar excess free energy g E , in a thermodynamically consistent manner. With the excess free energy GE = N · g E and Ni = xi · N it follows from Eq. (2.88) that

ˆ·T · GE = R

k X

Ni · ln γi .

(2.93)

i=1

For the partial derivative of one particular component i, we obtain 

∂ GE ∂ Ni



ˆ·T · =R p,T,Nj6=i

 k  X ∂ (Nj · ln γj ) . ∂ Ni p,T,Nl6=j j=1

Using the product rule of calculus yields 

∂ GE ∂ Ni



ˆ·T · =R p,T,Nj6=i

k X



ln γj ·

j=1

∂ ln Nj ∂ Ni



+ Nj ·

∂ ln γj ∂ Ni

 p,T,Nl6=j

(2.94)

!



. p,T,Nl6=j

With

∂ Nj = 0 for j 6= i and ∂ Ni

∂ Nj = 1 for j = i , ∂ Ni

we obtain 

∂ GE ∂ Ni





ˆ · T · ln γi + =R p,T,Nj6=i

k X



Nj ·

j=1



ˆ · T · ln γi + N · =R

k X j=1



xj ·

∂ ln γj ∂ Ni



∂ ln γj ∂ Ni



 

p,T,Nl6=j

 .

p,T,Nl6=j

(2.95)

26

2 VAPOR–LIQUID EQUILIBRIUM

ˆ · T · d ln (xi · γi ) such that the Gibbs– From Eq. (2.79) it follows that dµi = R Duhem equation for isobaric–isothermal condition in Eq. (2.51) can be written as

0=

k X

xi · d ln (xi · γi ) =

i=1

k X

xi · d ln xi +

i=1

k X

xi · d ln γi .

With xi · d ln xi = dxi and the differential closing condition follows that

0=

k X

(2.96)

i=1

xi · d ln γi .

Pk

i=1

dxi = 0, it

(2.97)

i=1

This is called the Gibbs–Duhem equation for activity coefficients, which states that at isobaric–isothermal conditions, the k activity coefficients are not independent from each other. Hence, the sum at the right-hand side of Eq. (2.95) is zero, which yields the following correlation between the activity coefficient γi and the excess free energy GE :

ˆ · T · ln γi = R



∂ GE ∂ Ni



.

(2.98)

p,T,Nj6=i

This equation describes also how all activity coefficients γ1 , γ2 , . . . , γk can be calculated from one state variable, here the excess free energy GE = N · g E , in a thermodynamically consistent manner. Equation (2.98) is equivalent to Eq. (2.92), which is shown in Example 2.1 for a ternary mixture.

Example 2.1: Activity Coefficients for a Ternary Mixture For a ternary mixture with components a, b, and c, Poling et al. 2001 describe a two-suffix Margules equation for the molar excess free energy as follows:

g E = Aab · xa · xb + Aac · xa · xc + Abc · xb · xc .

(1)

Although this equation is not of practical importance any more, it can be used to show how activity coefficient correlations can be obtained from g E correlations. (1) Derive the correlations to determine the activity coefficients γa , γb , and γc with Eq. (2.98). (2) Using the activity coefficient γa , show that the correlations in Eqs. (2.92) and (2.98) are equivalent.

27

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

Solution:

(1) With GE = N · g E and xi = Ni /N , it follows from Eq. (1) that   Na · Nb Na · Nc Nb · Nc GE = N · Aab · + A · + A · ac bc N2 N2 N2 Na · Nb Na · Nc Nb · Nc = Aab · + Aac · + Abc · . N N N With N = Na + Nb + Nc and the quotient rule of calculus, we obtain 

∂ GE ∂ Na



Nb · N − Na · Nb Nc · N − Na · Nc + Aac · 2 N N2 −Nb · Nc + Abc · . N2

= Aab · p,T,Nb ,Nc

Replacing the mole amounts Ni by the mole fractions xi = Ni /N yields 

∂ GE ∂ Na



= Aab · (xb − xa · xb ) + Aac · (xc − xa · xc ) p,T,Nb ,Nc

− Abc · xb · xc . From Eq. (2.98), we obtain

ˆ · T · ln γa = R



∂ GE ∂ Na



, p,T,Nb ,Nc

such that

ˆ · T · ln γa = Aab · (xb − xa · xb ) + Aac · (xc − xa · xc ) − Abc · xb · xc . R (2) With

ˆ · T · ln γb = R



∂ GE ∂ Nb

 p,T,Na ,Nc

ˆ · T · ln γc = and R



∂ GE ∂ Nc



, p,T,Na ,Nb

we obtain similarly correlations for the activity coefficients of component b and c:

ˆ · T · ln γb = Aab · (xa − xa · xb ) + Abc · (xc − xb · xc ) − Aac · xa · xc R ˆ · T · ln γc = Aac · (xa − xa · xc ) + Abc · (xb − xb · xc ) − Aab · xa · xb . R (2) For a ternary mixture, it follows from Eq. (2.92) for the activity coefficient γa that  E  E ∂g ˆ · T · ln γa = g E − xb · ∂ g R − xc · . (3) ∂ xb p,T,xc ∂ xc p,T,xb

28

2 VAPOR–LIQUID EQUILIBRIUM

In order to get the correct dependencies in Eq. (1), the mole fraction xa has to be replaced by xa = 1 − xb − xc such that

g E = Aab · (1 − xb − xc ) · xb + Aac · (1 − xb − xc ) · xc + Abc · xb · xc = Aab · (xb − x2b − xb · xc ) + Aac · (xc − x2c − xb · xc ) + Abc · xb · xc . We obtain  E ∂g = Aab · (1 − 2 · xb − xc ) − Aac · xc + Abc · xc ∂ xb p,T,xc  E ∂g = Aac · (1 − 2 · xc − xb ) − Aab · xb + Abc · xb . ∂ xc p,T,xb

(4) (5)

Inserting Eqs. (1) – (5) into Eq. (3) yields

ˆ · T · ln γa = Aab · (xa · xb − xb + 2 · x2b + xb · xc + xb · xc ) R + Aac · (xa · xc − xc + 2 · x2c + xb · xc + xb · xc ) + Abc · (xb · xc − xb · xc − xb · xc ) = Aab · xb · (xa − 1 + 2 · (xb + xc )) + Aac · xc · (xa − 1 + 2 · (xc + xb )) − Abc · xb · xc . With xb + xc = 1 − xa , this simplifies to

ˆ · T · ln γa = Aab · (xb − xa · xb ) + Aac · (xc − xa · xc ) − Abc · xb · xc . R This is the same correlation as in Eq. (2), which shows the equivalence of Eqs. (2.92) and (2.98).

2.2.5

Thermodynamic Consistency Check of Molar Excess Free Energy and Activity Coefficients

The molar excess free energy g E depends on the mole fractions xi in the mixture, the temperature T , and the pressure p. Analogous to Eq. (2.46), the following partial derivatives of the molar Gibbs free energy g E can be obtained:  E ∂g sE = − ∂ T p,xi

and

vE =



∂ gE ∂p



. T,xi

(2.99)

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

29

With Eq. (2.88), the following correlations for the molar excess entropy sE and the molar excess volume v E can be established from Eq. (2.99): E

s

ˆ· = −R

k X

ˆ·T · xi · ln γi − R

i=1

ˆ·T · vE = R

k X i=1

k X



xi ·

i=1

∂ ln γi ∂p



xi ·

∂ ln γi ∂T

 (2.100) p,xi



.

(2.101)

T,xi

For the molar excess enthalpy hE = g E + T · sE , we obtain

ˆ · T2 · h =R E

k X i=1



xi ·

∂ ln γi ∂T



.

(2.102)

p,xi

Since the molar volume v and the molar enthalpy h do not have a molar mixing property for ideal gases, i.e. ∆v id = 0 and ∆hid = 0, it follows that ∆v = v E and ∆h = hE (see also Section 2.1.3). Therefore, from Eqs. (2.101) and (2.102), it can be concluded that the pressure and temperature dependence of the activity coefficients γ1 , γ2 , . . . , γk can be determined from volumetric measurements of the molar mixing volume ∆v and caloric measurements of the molar mixing enthalpy ∆h, respectively. With the dependence of g E and γi on the mole fractions x1 , x2 , . . . , xk being strong, the dependence on the pressure p can be neglected in most applications. The influence of the temperature T depends on the thermal mixing behavior of the components in mixture, i.e. the molar excess enthalpy hE . Correlations to calculate the molar excess free energy g E as a function of the mole fractions x1 , x2 , . . . , xk of a mixture are presented in Section 2.2.12. Usually, these correlations are based on isobaric or isothermal measurements of the vapor–liquid equilibrium (VLE) of the mixture. Experimental VLE data has to be checked for thermodynamic consistency before they are used to define g E correlations. From GE (p, T, N1 , N2 , . . . , Nk ), we obtain the following total differential:  E   E    G ∂ G ∂ GE d = · dT + · dp ˆ·T ˆ · T p,Ni ˆ · T T,Ni ∂T R ∂p R R  E  k X ∂ G + · dNi . ˆ ∂ N i R · T p,T,Nj6=i i=1

(2.103)

With the Gibbs–Helmholtz equation in Eq. (2.52), the correlations describing γi in Eq. (2.98), and v E in Eq. (2.99), this yields, after dividing by the mole amount N , a criterion that can be used to check the thermodynamic consistency of VLE data:

gE d ˆ·T R 



=−

k X hE vE · dT + · dp + ln γi · dxi . ˆ · T2 ˆ·T R R i=1

(2.104)

30

2.2.5.1

2 VAPOR–LIQUID EQUILIBRIUM

Binary Mixtures

Most important are correlations describing binary mixtures of the components a and b with dxb = −dxa . By integration from xa = 0 to xa = 1, the left side of Eq. (2.104) becomes zero because g E is zero for both pure components. Hence, we obtain the following correlation that can be used for consistency check [Kang et al. 2010]: Z xa =1 Z T (xa =1) Z p(xa =1) E γa hE v ln dxa = dT − dp . (2.105) 2 ˆ ˆ γ b xa =0 T (xa =0) R · T p(xa =0) R · T For isobaric or isothermal VLE measurements, i.e. dp = 0 or dT = 0, respectively, the corresponding term on the right-hand side can be canceled: • When considering isothermal data with dp 6= 0, the pressure dependence can usually be neglected due to small values of v E . • For isobaric data with dT 6= 0, the molar excess enthalpy hE has to be taken into account. If no data for hE is available, the influence can be estimated, e.g. using the method of Herington 1951 or Kang et al. 2010. With xb = 1 − xa , we obtain the following binary mixture correlations describing the activity coefficients γa and γb from Eq. (2.92):  E  E ∂g ∂g E E ˆ = g + (1 − xa ) · R · T · ln γa = g − xb · ∂ xb p,T ∂ xa p,T (2.106)  E ˆ · T · ln γb = g E − xa · ∂ g R . (2.107) ∂ xa p,T These correlations will be used in Section 2.2.12 to determine the activity coefficients for binary mixtures with the molar excess free energy g E . 2.2.6

Iso-fugacity Condition

With the correlation for the chemical potential µi of component i in a mixture in Eq. (2.71), the conditions for phase equilibrium in Eq. (2.59) lead to another equilibrium condition with equivalent form, known as the iso-fugacity condition:

fi0 = fi00

for i = 1, 2, . . . , k .

(2.108)

As shown below, evaluating Eq. (2.108) leads to different equilibrium conditions, depending on how the fugacities fi0 and fi00 in the liquid and vapor phase are determined. 2.2.7

Fugacity of the Liquid Phase

The fugacity fi0 of component i in the liquid phase can be calculated using the fugacity coefficient ϕ0i from Eq. (2.72) and the activity coefficient γi from Eq. (2.78)

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

31

along with pi = xi · p as follows:

fi0 = xi · p · ϕ0i = xi · p · γi · ϕ0i .

(2.109)

We use the definition of the fugacity in Eq. (2.64) to define a standard fugacity as fi0 = p · ϕ0i , i.e. the state of the pure liquid component i is used here as a suitable standard state. Hence, from Eq. (2.109), the following correlations of the fugacity of component i in the liquid are obtained4 :

fi0 = xi · p · ϕ0i fi0

= xi · γi ·

fi0

(2.110)

.

(2.111)

For practical calculation of the vapor–liquid equilibrium, both correlations can be used: (1) For the fugacity fi0 of component i in the liquid phase in Eq. (2.110), a suitable equation of state for the liquid phase has to be known in order to calculate the fugacity coefficient ϕ0i of component i in the liquid phase, e.g. by using a cubic EOS. This approach will be used in Section 2.2.9. The ideal gas law or a virial equation cannot be used to describe the fugacity of the liquid phase. (2) If the standard fugacity is known, the fugacity fi0 of component i in the liquid phase can be calculated. In the course of this section, two different approaches are shown: • The fugacity fi0 of the pure liquid component i is used as the standard fugacity, as described above. With Eq. (2.111), this will lead to the extended Raoult’s law and Raoult’s law, as shown in Section 2.2.10. • If, at a given temperature T , the pure component i cannot exist as a pure liquid, another fugacity has to be selected as standard fugacity. One feasible approach is to use the fugactiy fi∞ of component i, which is infinitely diluted in the liquid phase as another standard fugacity. This will lead to Henry’s law, as shown in Section 2.2.11. 2.2.8

Fugacity of the Vapor Phase

The fugacity fi00 of component i in the vapor phase can be calculated using the fugacity coefficient ϕ00i from Eq. (2.72) as follows5 :

fi00 = pi · ϕ00i = yi · p · ϕ00i .

(2.112)

In order to calculate the fugacity coefficient ϕ00i of component i in the vapor phase, using Eq. (2.112), an applicable equation of state is required. For moderate pressures, 4

5

Since the standard fugacity fi0 is used only for the liquid phase, the index 0 is omitted. This is valid also for the activity coefficients γi , which will be used only for the liquid phase. Since the partial pressure pi is used only for the vapor phase, the index 00 is omitted.

32

2 VAPOR–LIQUID EQUILIBRIUM

ideal gas behavior can be assumed, and with ϕ00i = 1, it follows that the fugacity of component i in the vapor phase is equal to the partial pressure pi of component i:

fi00 = pi = yi · p for ideal gases .

(2.113)

At higher pressures, either a virial equation or a cubic EOS from Section 2.1.2 can be used to calculate the fugacity coefficients ϕ00i 6= 1. 2.2.9

Vapor–Liquid Equilibrium Using an Equation of State

2.2.9.1

Mixtures

Starting from Eq. (2.108) and inserting Eqs. (2.110) and (2.112) yields an alternate form of the iso-fugacity condition:

xi · ϕ0i = yi · ϕ00i .

(2.114)

An equation of state that describes the liquid and the vapor phase has to be used to determine the fugacity coefficients in the two phases, e.g. a cubic EOS. With Eq. (2.76) and the Soave–Redlich–Kwong equation in Eq. (2.16), the following explicit correlation for the fugacity coefficient ϕ0j of component i in the liquid phase, dependent of the mole fractions x1 , x2 , . . . , xk , is obtained [Gmehling et al. 2012]:

ln ϕ0i

P 2 · kj=i xj · aij v0 bi v 0 + b0 + − = ln 0 · ln ˆ·T v − b0 v 0 − b0 v0 b0 · R   0 0 0 0 a · bi v +b b p · v0 + · ln − 0 − ln . 0 0 ˆ·T ˆ·T v v +b b02 · R R

(2.115)

For the fugacity coefficient ϕ00i of component i in the vapor phase, we obtain with the mole fractions y1 , y2 , . . . , yk the following correlation:

ln ϕ00i

P 2 · kj=1 yj · aij v 00 bi v 00 + b00 = ln 00 + − · ln ˆ·T v − b00 v 00 − b00 v 00 b00 · R   00 00 00 00 a · bi v +b b p · v 00 + · ln − 00 − ln . 00 00 ˆ·T ˆ·T v v +b b002 · R R

(2.116)

The parameters a0 and b0 in Eq. (2.115) have to be calculated with the mixing rules in Eqs. (2.21) – (2.22) using the mole fractions xj of the liquid phase. As shown in Section 2.1.2, the mole fractions yj of the vapor phase have to be used in Eqs. (2.21) – (2.22) to calculate the parameters a00 and b00 in Eq. (2.116)). The molar volumes v 0 and v 00 of the liquid and vapor phase have to be determined from Eq. (2.16) with an iterative, numerical solution scheme. Furthermore, the mole fractions xi and yi in the liquid and vapor phase have to be determined iteratively until the fugacity coefficients ϕ0 and ϕ00 with Eqs. (2.115) and (2.116), along with the mole fractions xi and yi , satisfy the iso-fugacity condition in Eq. (2.114). This iterative calculation is illustrated in Example 2.3 for a ternary mixture.

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

2.2.9.2

33

Pure Components

For a pure component i with xi = yi , we obtain from Eq. (2.114) the following correlation, which describes the vapor–liquid equilibrium6 :

ϕ0 = ϕ00 .

(2.117)

This is equivalent to

f 0 = f 00 ,

(2.118)

which follows also from Eq. (2.108) for the pure component i. Inserting the correlation for the pure component fugacity in Eq. (2.68) into Eq. (2.117) yields Z

v 00

p dv = p · (v 00 − v 0 ) .

v0

(2.119)

This is called Maxwell criterion [Gmehling et al. 2012], which also follows from Eq. (2.118) with the correlation in Eq. (2.67). The Maxwell criterion states that at VLE, the area below an isotherm of temperature T between the molar volume v 0 of the saturated liquid and the molar volume v 00 of the saturated vapor have to be identical to the rectangle framed by v 0 and v 00 and the pressure p in a (p, v )-diagram. If this is fulfilled, we have p = p0i (T ), i.e. the isotherm of a cubic EOS can be used to determine the saturation vapor pressure of a pure component, as shown for the Soave–Redlich–Kwong EOS in Example 2.2. The fugacity coefficients of the liquid and vapor phase for a pure component can be calculated using the Soave–Redlich–Kwong EOS as follows [Gmehling et al. 2012]:

p · (v 0 − b) a v0 + b − · ln ˆ·T ˆ·T v0 R b·R 00 p · (v − b) a v 00 + b ln ϕ00 = Z 00 − 1 − ln − · ln . ˆ·T ˆ·T v 00 R b·R

ln ϕ0 = Z 0 − 1 − ln

(2.120) (2.121)

The parameters a and b are the same for both phases and have to be calculated with Eqs. (2.18) – (2.20). The molar volumes v 0 and v 00 have to be determined from Eq. (2.16) with an iterative numerical solution scheme or graphically as illustrated ˆ · T ) and in Example 2.2. Afterward, the compressibility factors Z 0 = p · v 0 /(R 00 00 ˆ Z = p · v /(R · T ) can be calculated without any iteration.

6

For simplification, the component index i and the upper index 0 labeling a pure component state variable are omitted in this paragraph.

34

2 VAPOR–LIQUID EQUILIBRIUM

Example 2.2: Vapor–Liquid Equilibrium (VLE) of Pure Components: Antoine Equation and Soave–Redlich–Kwong (SRK) EOS The VLE of pure methane (CH4 ) shall be investigated using the SRK EOS. The following physical properties taken from the process simulator UniSim Design [Honeywell 2017] shall be used:

pc,CH4 = 46.41 bar ,

Tc,CH4 = 190.70 K ,

ωCH4 = 0.0115 .

Furthermore, the following Antoine equation for methane from the NIST Chemistry WebBook [National Institute of Standards and Technology 2018] shall be used:

BCH4 with p in bar and T in K T + CCH4 BCH4 = 443.028 CCH4 = −0.49 .

log10 p0CH4 = ACH4 − ACH4 = 3.9895

(1)

(1) Calculate the saturation vapor pressure of pure methane at the temperature T = −100 ◦C, i.e. p0CH4 (−100 ◦C) using the Antoine equation above. (2) Draw the isotherm for T = −100 ◦C into a (p, v )-diagram using the SRK EOS. Vary the molar volume v from 0.055 to 0.600 m3 /kmol and calculate the corresponding pressure p. (3) Determine graphically the molar volume v 0 of liquid methane at T = −100 ◦C and p = 30 bar as well as the molar volume v 00 for methane vapor at T = −100 ◦C and p = 20 bar with the (p, v )-diagram. Calculate the fugacity coefficients ϕ0 and ϕ00 of the liquid and the vapor phase. (4) Draw the isotherm for T = −100 ◦C and the isobaric line for the saturation vapor pressure p0CH4 (−100 ◦C), calculated in Task (1), into a new (p, v )-diagram. Check if the Maxwell criterion is fulfilled for this pressure and temperature. (5) Graphically determine the molar volume of the saturated liquid v 0 and vapor v 00 at T = −100 ◦C and the saturation vapor pressure p0CH4 (−100 ◦C) with the (p, v )-diagram, and calculate the fugacity coefficients ϕ0 and ϕ00 . (6) Using the SRK EOS, calculate the saturation vapor pressure for pure methane at the temperature T = −100 ◦C with an iterative solution scheme. Vary the pressure p starting at 20 bar, and, using the isotherm in the (p, v )-diagram, determine the molar volumes v 0 and v 00 of the saturated liquid and saturated vapor, respectively. In order to determine an improved estimation for the pressure p(n+1) , use the iteration formula

p(n+1) = p(n) ·

ϕ0(n) . ϕ00(n)

(2)

An iteration scheme with the numerical solution of the cubic EOS is shown in Excel sheet SRK_methane.xlsm, which can be downloaded from the homepage. In this sheet, direct numerical solutions for the saturation vapor pressure p0CH4 (T )

35

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

0 at a specified temperature T as well as the boiling temperature TCH (p) at a 4 specified pressure p are available.

Solution:

(1) The saturation vapor pressure can be obtained from Eq. (1) with the given Antoine parameters:

log10 p0CH4 (−100 ◦C) = 3.9895 −

443.028 = 1.4236 (273.15 − 100) − 0.49

p0CH4 (−100 ◦C) = 101.4236 bar = 26.52 bar . (2) The correlation between the pressure p and the molar volume v by SRK is given in Eq. (2.16):

p=

ˆ·T R a − . v − b v · (v + b)

(3)

The parameters a and b have to be determined in advance. In this example for a pure component, we have a = aii and b = bi , and the indices are omitted at all calculated values like Tr , α, a, and b for the sake of clarity. We obtain

Tr

=

T 173.15 K = = 0.91 Tc,CH4 190.7 K

2 α(T ) = 1 + (0.48 + 1.574 · ωCH4 − 0.176 · ωCH ) · (1 − Tr0.5 ) 4


2


2 = 1 + (0.48 + 1.574 · 0.0115 − 0.176 · 0.01152 ) · (1 − 0.910.5 )

= 1.047 ˆ2 · T 2 R c,CH4 · α(T ) pc,CH4
2 8314 kmolJ · K · (190.7 K)2 = 0.42748 · · 1.047 46.41 bar J · m3 = 2.43 × 105 kmol2

a

= 0.42748 ·

b

= 0.08664 · =

ˆ · Tc,CH R 4 pc,CH4

8314 kmolJ · K · 190.7 K m3 = 0.0296 . 46.41 bar kmol

36

2 VAPOR–LIQUID EQUILIBRIUM

Now the pressure can be calculated using Eq. (3). Exemplarily, this is done for v = 0.055 m3 /kmol:

p=

8314 kmolJ · K · 173.15 K 3

3

m m 0.055 kmol − 0.0296 kmol

3

−

3

m 0.055 kmol

J·m 2.43 × 105 kmol 2   3 m m3 · 0.055 kmol + 0.0296 kmol

= 45.46 bar . More results are summarized in the following table:

v in

m3 kmol

p in bar

0.055 0.060 0.080 0.100 0.200 0.300 0.400 0.500 0.600 45.46 22.36 9.00 17.33 31.66 28.71 24.75 21.44 18.82

This yields the isotherm at T = −100 ◦C in the following (p, v )-diagram:

(3) The desired molar volumes of the two states at T = −100 ◦C can be read off the (p, v )-diagram:

m3 kmol m3 00 ◦ Vapor: v (−100 C, 20 bar) = 0.552 . kmol

Liquid: v 0 (−100 ◦C, 30 bar) = 0.058

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

37

Next, the compressibility factors Z 0 and Z 00 can be calculated: 3

m 30 bar · 0.058 kmol p · v0 Z = = = 0.121 ˆ·T 8314 kmolJ · K · 173.15 K R 0

3

m 20 bar · 0.552 kmol p · v 00 Z = = = 0.767. ˆ·T 8314 kmolJ · K · 173.15 K R 00

These values are used to obtain the fugacity coefficients ϕ0 and ϕ00 with the correlations in Eqs. (2.120) and (2.121), respectively:

p · (v 0 − b) a v0 + b − · ln ˆ·T ˆ v0 R  b · R · T3  m m3 30 bar · 0.058 kmol − 0.0296 kmol = 0.121 − 1 − ln 8314 kmolJ · K · 173.15 K

ln ϕ0 = Z 0 − 1 − ln

− 0.0296 · ln ϕ0

J · m3 kmol2 m3 J kmol · 8314 kmol · K · 173.15 K 3 m m3 kmol + 0.0296 kmol = −0.399 m3 0.058 kmol

2.43 × 105

0.058

= e−0.399 = 0.671 . p · (v 00 − b) a v 00 + b − · ln ˆ·T ˆ v 00 R  b · R ·3T  m m3 20 bar · 0.552 kmol − 0.0296 kmol = 0.767 − 1 − ln 8314 kmolJ · K · 173.15 K

ln ϕ00 = Z 00 − 1 − ln

− 0.0296 · ln ϕ00

J · m3 kmol2 m3 J kmol · 8314 kmol · K · 173.15 K m3 m3 kmol + 0.0296 kmol = −0.210 m3 0.552 kmol

2.43 × 105

0.552

= e−0.210 = 0.811 .

The fugacity coefficients of the two states are different, which is not surprising: both states have the same temperature T = −100 ◦C, but the liquid is subcooled at p = 30 bar, and the vapor is superheated at p = 20 bar. The two states are not in vapor–liquid equilibrium and the iso-fugacity condition in Eq. (2.117) is not fulfilled.

38

2 VAPOR–LIQUID EQUILIBRIUM

(4) Adding the isobaric line for the saturation vapor pressure p0CH4 (−100 ◦C) to the (p, v )-diagram results in two areas enclosed by the isotherm and the isobaric line (see hatched areas in the following diagram). The Maxwell criterion is fulfilled if these two areas are the same size.

Graphically the following areas are determined:

A1 = 0.71

bar · m3 kmol

and

A2 = 0.71

bar · m3 . kmol

The two areas have the same size, and thus the Maxwell criterion is fulfilled for T = −100 ◦C and p = p0CH4 (T = −100 ◦C) = 26.52 bar. (5) The molar volumes can be read of from the (p, v )-diagram at the two intersection points of the isotherm and the isobaric line:

m3 kmol m3 00 ◦ Vapor: v (−100 C, 26.52 bar) = 0.354 . kmol

Liquid: v 0 (−100 ◦C, 26.52 bar) = 0.059

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

39

Using the same calculating method as in Task (3), the compressibility factors and the fugacity coefficients are obtained: 3

Z0

=

m 26.52 bar · 0.059 kmol p · v0 = = 0.109 J ˆ·T 8314 kmol · K · 173.15 K R

p · (v 0 − b) a v0 + b − · ln ˆ·T ˆ·T v0 R b·R   3 m m3 26.52 bar · 0.059 kmol − 0.0296 kmol = 0.109 − 1 − ln 8314 kmolJ · K · 173.15 K

ln ϕ0 = Z 0 − 1 − ln

− 0.0296 · ln ϕ0

J · m3 kmol2 m3 J kmol · 8314 kmol · K · 173.15 K m3 m3 kmol + 0.0296 kmol = −0.290 m3 0.059 kmol

2.43 × 105

0.059

= e−0.290 = 0.748 . 3

Z

00

m 26.52 bar · 0.354 kmol p · v 00 = = = 0.652 ˆ·T 8314 kmolJ · K · 173.15 K R

p · (v 00 − b) a v 00 + b − · ln ˆ·T ˆ v 00 R b · R · T 3  m m3 26.52 bar · 0.354 kmol − 0.0296 kmol = 0.652 − 1 − ln 8314 kmolJ · K · 173.15 K

ln ϕ00 = Z 00 − 1 − ln

− 0.0296 · ln ϕ00

J · m3 kmol2 m3 J kmol · 8314 kmol · K · 173.15 K m3 m3 kmol + 0.0296 kmol = −0.290 m3 0.354 kmol

2.43 × 105

0.354

= e−0.290 = 0.748 .

At these conditions, T = −100 ◦C and p = 26.52 bar, the fugacity coefficients ϕ0 and ϕ00 in the liquid phase and vapor phase, respectively, are equal, and the iso-fugacity condition in Eq. (2.117) is fulfilled. The two states are in vapor–liquid equilibrium. (6) Starting at the pressure p(n=0) = 20 bar, the saturation vapor pressure of methane, p0CH4 (−100 ◦C), can be determined by an iterative solution scheme. At each iteration step, the molar volumes v 0 and v 00 are read off the (p, v )-

40

2 VAPOR–LIQUID EQUILIBRIUM

diagram, and the fugacity coefficients ϕ0 and ϕ00 are calculated in the known manner. The pressure for the next iteration step n = 1 can be calculated with the ratio of the fugacity coefficients with Eq. (2):

p(n=1) = p(n=0) ·

ϕ0(n=0) 0.966 = 20 bar · = 23.83 bar . 0.811 ϕ00(n=0)

The results of the following iteration steps are tabulated below:

n

0 1 2 3 4 5 6 7 8

p in bar

v0 in m /kmol

v 00 in m /kmol

ϕ0

ϕ00

20.00 23.83 25.35 26.00 26.29 26.42 26.48 26.51 26.52

0.061 0.060 0.059 0.059 0.059 0.059 0.059 0.059 0.059

0.552 0.426 0.384 0.367 0.360 0.356 0.355 0.354 0.354

0.966 0.824 0.779 0.762 0.754 0.751 0.749 0.749 0.748

0.811 0.774 0.760 0.753 0.750 0.749 0.749 0.748 0.748

3

3

After n = 8 iteration steps, the relative change is considered to be small enough, and the saturation vapor pressure p0CH4 = 26.52 bar is obtained, which equals the value calculated in Task (1).

Example 2.3: Vapor–Liquid Equilibrium (VLE) of a Ternary Mixture with the Soave–Redlich–Kwong (SRK) EOS A ternary mixture of hydrogen (H2 , index a), methane (CH4 , index b), and ethane (C2 H6 , index c) shall be investigated. The mass fractions of the liquid mixture shall be wa = 0.00535 and wb = 0.59575. Calculate the saturation vapor pressure p(T ) of the mixture at the temperature T = 200 K using the SRK EOS with an iterative solution scheme. The following physical properties for the pure components taken from the process simulator UniSim Design [Honeywell 2017] shall be used: Hydrogen

pc,a = 13.16 bar

Tc,a = 33.45 K

ωa = −0.1201

Methane

pc,b = 46.41 bar

Tc,b = 190.70 K

ωb =

0.0115

Ethane

pc,c = 48.84 bar

Tc,c = 305.43 K

ωc =

0.0986

For the three combinations of binary mixtures, the following binary parameters can

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

41

be obtained from UniSim Design [Honeywell 2017] : Hydrogen–methane

kab = 0.00010

Hydrogen–ethane

kac = 0.00010

Methane–ethane

kbc = 0.00224

(1) Calculate the mass fraction wc of ethane in the liquid mixture and determine the mole fractions xa , xb , and xc in the liquid mixture. (2) Start the iterative numerical solution scheme at the pressure p = 60 bar with a reasonable estimate for the mole fractions ya , yb , and yc of the vapor, which is generated from the liquid at VLE. Hence, take into account that hydrogen is the low boiling component, whereas ethane is the high boiling component; thus ya > xa and yc < xc at boiling conditions. The iterative numerical solution scheme of the cubic SRK EOS for the molar volume v 0 of the liquid and for the molar volume v 00 of the vapor is shown in Excel sheet SRK_mixture.xlsm which can be downloaded from the homepage. (3) Calculate the fugacity coefficients ϕ0 and ϕ00 of the liquid and the vapor phase and check if the iso-fugacity condition in Eq. (2.114) is fulfilled. (4) Estimate new values for the mole fractions y˜a , y˜b , and y˜c of the vapor by rewriting the iso-fugacity conditions in Eq. (2.114) to solve for the mole fractions:

y˜i = xi ·

ϕ0i ϕ00i

for

i = a, b, c .

(1)

Check if these estimated mole fractions y˜i fulfill the closing condition of the vapor phase. Use the sum S = y˜a + y˜b + y˜c as a correction factor to calculate consistent new estimates for the mole fractions of the vapor. Use the correction factor S to calculate a new estimate for the pressure, too:

yinew =

y˜i S

for i = a, b, c

and

pnew = S · p .

(2)

(5) Use the values yanew , ybnew , and ycnew as well as pnew , and continue with the numerical solution of the cubic SRK EOS for the molar volumes v 0 and v 00 of the liquid and the vapor, respectively. Calculate the fugacity coefficients ϕ0 and ϕ00 of the liquid and the vapor and determine new estimates for yanew , ybnew , and ycnew as well as pnew . Stop the iteration if the iso-fugacity condition in Eq. (2.114) along with the closing condition of the vapor phase is fulfilled with a relative error of |1 − S| < 10−3 . Solution:

(1) Due to the closing condition, the sum of all mass fractions has to equal 1, and the mass fraction wc of ethane can be easily determined:

wc = 1 − wa − wb = 1 − 0.00535 − 0.59575 = 0.39890.

42

2 VAPOR–LIQUID EQUILIBRIUM

ˆ of the mixture can be calKnowing all mass fractions, the molecular weight M culated:  −1 ˆ = wa + wb + wc M ˆa ˆb ˆc M M M !−1 0.00535 0.59575 0.39890 kg = + + = 18.846 . kg kg kg kmol 2.016 kmol 16.04 kmol 30.07 kmol ˆ of the mixture, the mole fractions in the ternary With the molecular weight M mixture amount to xa = wa ·

ˆ ˆ ˆ M M M = 0.05 , xb = wb · = 0.70 , xc = wc · = 0.25 . ˆa ˆb ˆc M M M

(2) A reasonable estimate for the mole fractions yi in the vapor phase at p = 60 bar and T = 200 K due to the different boiling points would be

ya = 0.5 ,

yb = 0.4 ,

and

yc = 0.1.

With these estimates for the mole fractions of the vapor and the estimated pressure p = 60 bar, all parameters a and b of the SRK EOS can be calculated for the liquid and vapor phase, i.e. a0 , a00 , b0 , and b00 . With these values, the molar volumes v 0 and v 00 can be determined with an iterative numerical solution scheme of the cubic SRK EOS. First, the parameters αi (T ), aii , aij , and bi have to be calculated, which is shown below for i = a:

Tr,a

=

T 200 K = = 5.979 Tc,a 33.45 K

0.5 αa (T ) = 1 + (0.48 + 1.574 · ωa − 0.176 · ωa2 ) · 1 − Tr,a



2

= (1 + 0.48 + 1.574 · (−0.1201) − 0.176 · (−0.1201)2 √ · (1 − 5.979))2 = 0.3401 aaa

2 ˆ 2 · Tc,a R · αa (T ) pc,a
2 2 8314 kmolJ · K · (33.45 K) = 0.42748 · · 0.3401 13.16 bar 3 J·m = 8.54 × 103 kmol2

= 0.42748 ·




43

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

ˆ · Tc,a R pc,a

ba = 0.08664 ·

8314 kmolJ · K · 33.45 K m3 = 1.83 × 10−2 . 13.16 bar kmol

= 0.08664 ·

In order to calculate the cross coefficients aab = aba and aac = aca , the pure component coefficients abb and acc have to be calculated first:

Tr,b =

T 200 K = = 1.049 ; Tc,b 190.70 K

Tr,c =

T 200 K = = 0.655 Tc,c 305.43 K

2

= 0.9761



0.5 2

= 1.2563

0.5 αb (T ) = 1 + (0.48 + 1.574 · ωb − 0.176 · ωb2 ) · 1 − Tr,b

αc (T ) = 1 + (0.48 + 1.574 · ωc − 0.176 · ωc2 ) · 1 − Tr,c ˆ2 · T 2 R c,b · αb (T ) = 2.26 × 105 pc,b 2 ˆ 2 · Tc,c R = 0.42748 · · αc (T ) = 7.09 × 105 pc,c = 0.42748 ·

abb acc

J · m3 kmol2 J · m3 . kmol2

Now, the cross coefficients aab and aac can be calculated with the mixing rule in Eq. (2.23) and using the binary parameters kab and kac as follows:

aab = aba = s

√

aaa · abb · (1 − kab )

3 J · m3 5 J·m · (1 − 0.00010) 2 · 2.26 × 10 kmol kmol2 J · m3 = 4.39 × 104 kmol2

=

8.54 × 103

aac = aca = s

√

aaa · acc · (1 − kac )

3 J · m3 5 J·m · (1 − 0.00010) 2 · 7.09 × 10 kmol kmol2 J · m3 = 7.78 × 104 . kmol2

=

8.54 × 103

All pure component parameters and the cross coefficients are summarized in the following table:

αi

H2 0.3401 CH4 0.9761 C2 H6 1.2563

aij in J · m3 /kmol2 H2 8.54 × 103 4.39 × 104 7.78 × 104

CH4 4.39 × 104 2.26 × 105 3.99 × 105

bi in m3 /kmol C 2 H6 7.78 × 104 3.99 × 105 7.09 × 105

— 1.83 × 10−2 2.96 × 10−2 4.50 × 10−2

44

2 VAPOR–LIQUID EQUILIBRIUM

In the next step, the parameters a and b of the overall mixture in vapor and liquid phase can be obtained using the values from the table above. For the parameters a0 and b0 of the liquid phase, the mole fractions xi are used:

a0 = x2a · aaa + 2 · xa · xb · aab + 2 · xa · xc · aac + x2b · abb + 2 · xb · xc · abc + x2c · acc = 3.00 × 105 b0 = xa · ba + xb · bb + xc · bc = 3.29 × 10−2

J · m3 kmol2 m3 . kmol

For the parameters a00 and b00 of the vapor phase, the estimated values of the mole fractions yi are used, i.e. ya = 0.5, yb = 0.4, and yc = 0.1:

a00 = ya2 · aaa + 2 · ya · yb · aab + 2 · ya · yc · aac + yb2 · abb + 2 · yb · yc · abc + yc2 · acc = 1.03 × 105 b00 = ya · ba + yb · bb + yc · bc = 2.55 × 10−2

J · m3 kmol2 m3 . kmol

Now, the molar volume v 0 of the liquid phase and the molar volume v 00 of the vapor phase have to be determined. Therefore, the parameters of the liquid and the vapor phase have to be used in the following two equations:

ˆ·T R a0 − v 0 − b0 v 0 · (v 0 + b0 ) ˆ·T R a00 p = 00 − 00 . 00 v −b v · (v 00 + b00 ) p=

For the estimated pressure p = 60 bar, the molar volumes have to be determined with an iterative numerical solution since the equations above cannot be rewritten to solve for v 0 and v 00 . Alternatively, two isotherms for T = 200 K can be drawn for the liquid and vapor phase to obtain a graphical solution at the estimated pressure p = 60 bar for v 0 and v 00 , as shown in Example 2.2. From an numerical solution of the cubic equations above, the following values are obtained:

v 0 = 0.062

m3 kmol

and

v 00 = 0.246

m3 . kmol

45

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

(3) The fugacity coefficients in a ternary mixture can be calculated with Eqs. (2.115) and (2.116), which is shown exemplary for ln ϕ0a and ln ϕ00a :

ba 2 · (xa · aaa + xb · aab + xc · aac ) v0 + 0 − ˆ·T v 0 − b0 v − b0 b0 · R   0 0 0 0 0 v +b a · ba v +b b0 p · v0 · ln + · ln − − ln 0 0 0 0 ˆ·T ˆ·T v v v +b b02 · R R = 2.322

ln ϕ0a = ln

ϕ0a

= e2.322 = 10.195 . v 00 ba 2 · (ya · aaa + yb · aab + yc · aac ) + 00 − 00 00 ˆ·T −b v −b b00 · R   00 00 00 00 v +b a · ba v + b00 b00 p · v 00 · ln + · ln − − ln ˆ·T ˆ·T v 00 v 00 v 00 + b00 b002 · R R = 0.181

ln ϕ00a = ln

ϕ00a

v 00

= e0.181 = 1.199 .

The fugacity coefficients of the other components and of the vapor phase are calculated in the same way with the results summarized in the following table:

yi H2 CH4 C2 H6

ln ϕ0i

0.50 2.322 0.40 −0.397 0.10 −2.778

ϕ0i

ln ϕ00i

10.195 0.181 0.672 −0.333 0.062 −0.801

ϕ00i

xi · ϕ0i

yi · ϕ00i

1.199 0.717 0.449

0.510 0.471 0.016

0.599 0.287 0.045

It can be seen that the iso-fugacity condition in Eq. (2.114) is not fulfilled for any of the components, i.e. xi · ϕ0i 6= yi · ϕ00i for i = a, b, c. (4) First, new estimates for the mole fractions y˜i in the vapor phase are determined using the fugacity coefficients from Eq. (1):

y˜a = xa ·

ϕ0a 10.195 = 0.05 · = 0.425 ϕ00a 1.199

y˜b = xb ·

ϕ0b 0.672 = 0.70 · = 0.657 00 ϕb 0.717

y˜c = xc ·

ϕ0c 0.062 = 0.25 · = 0.035 . ϕ00c 0.449

It follows that

S = y˜a + y˜b + y˜c = 0.425 + 0.657 + 0.035 = 1.117 6= 1 ,

46

2 VAPOR–LIQUID EQUILIBRIUM

i.e. the closing condition of the vapor phase is not fulfilled with these calculated values. The correction factor S is used according to Eq. (2) to obtain an improved and consistent new estimate yinew for the mole fractions of the vapor phase:

yanew =

y˜a 0.425 = = 0.381 S 1.117

ybnew =

y˜b 0.657 = = 0.588 S 1.117

ycnew =

y˜c 0.035 = = 0.031 . S 1.117

With Eq. (2), a new estimated value for the pressure can be obtained as well:

pnew = p · S = 60 bar · 1.117 = 66.99 bar . (5) With the mole fractions yinew from the previous task, the SRK parameters a00 and b00 of the vapor phase have to be recalculated as illustrated in Task (2). Since the mole fractions xi of the liquid phase are given and stay constant, the values of the SRK parameters a0 and b0 of the liquid phase stay constant as well. With the new values of a00 and b00 and the pressure pnew = 66.99 bar, the values of the molar volumes v 0 and v 00 have to be determined with a numerical solution of the cubic SRK EOS as explained in Task (2). Then, as shown in Task (3), the fugacity coefficients ϕ0 and ϕ00 have to be recalculated, and the iso-fugacity condition has to be checked again. If these conditions are not fulfilled, new estimates for the mole fractions yinew and the pressure pnew can be calculated as shown in Task (4). The iteration scheme converges if the closing condition in the vapor phase is fulfilled with S ≈ 1. The results of this iterative calculation scheme are summarized in the following table: Iteration

p in bar

v0 in m3 /kmol

v 00 in m3 /kmol

ya

yb

yc

S

0

60.00

0.062

0.246

0.500

0.400

0.100

1.117

1

66.99

0.061

0.208

0.381

0.588

0.031

1.061

2

71.08

0.061

0.190

0.355

0.608

0.036

1.034

3

73.48

0.060

0.180

0.342

0.320

0.038

1.020

4

74.93

0.060

0.174

0.332

0.628

0.040

1.012

5

75.81

0.060

0.170

0.324

0.634

0.042

1.007

6

76.35

0.060

0.167

0.319

0.638

0.043

1.004

7

76.68

0.060

0.165

0.316

0.641

0.044

1.003

8

76.89

0.060

0.164

0.313

0.642

0.044

1.002

9

77.02

0.060

0.164

0.311

0.644

0.045

1.001

10

77.10

0.060

0.163

0.310

0.645

0.045

1.001

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

47

After 10 iteration steps, the required relative error of |1 − S| < 10−3 is achieved, and the iterative solution scheme is stopped. The resulting pressure p = 77.10 bar shows good agreement with values that were calculated by commercial software tools.

2.2.10

Fugacity of Pure Liquid as Standard Fugacity: Raoult’s Law

To calculate the fugacity fi0 of component i in the liquid phase with Eq. (2.111), the standard fugacity fi0 has to be determined. If, for a given temperature T , component i can exist as pure liquid, Eq. (2.67) can be integrated between the saturation vapor pressure p0i (T ) of the pure component i and the pressure p as follows [Stephan et al. 2017]:

ln fi0 (p, T ) − ln fi0 (p0i , T ) = | {z } ϕ0i ·p0i (T )

Z

p

p0i

|

vi0 dp . ˆ·T R {z }

(2.122)

ln Poy i

The integral on the right side is labeled as the natural logarithm of the Poynting correction Poy i , which has to be considered only at very high pressures. For moderate pressures, Poy i ≈ 1 is valid. At vapor–liquid equilibrium, the fugacity fi0 (p0i , T ) of the liquid at saturation vapor pressure p0i (T ) is, due to the iso-fugacity condition, the same as the fugacity in the vapor phase, which can be expressed with the fugacity coefficient ϕ0i of the pure saturated vapor and the saturation vapor pressure p0i (T ), which can be written as fi0 (p0i , T ) = ϕ0i · p0i (T ). Hence, with Poy i = 1, it follows from Eq. (2.122) that

fi0 (p, T ) = ϕ0i · p0i (T ) ,

(2.123)

which can be interpreted that the standard fugacity fi0 (p, T ) of component i is identical to its saturation vapor pressure p0i (T ) corrected with its fugacity coefficient ϕ0i at saturation conditions. Inserting Eq. (2.123) into Eq. (2.111) yields

fi0 = xi · γi · ϕ0i · p0i (T )

(2.124)

for the fugacity of component i in the liquid mixture. Finally, from the iso-fugacity condition in Eq. (2.108) and the fugacity fi00 of the vapor in Eq. (2.112), we obtain7

xi · γi · ϕ0i · p0i (T ) = yi · p · ϕi . 7

(2.125)

Here and in Eqs. (2.122) – (2.124), fugacity coefficients are used only for the vapor phase, and the index 00 is omitted at ϕi and ϕ0i .

48

2 VAPOR–LIQUID EQUILIBRIUM

For moderate pressures, it can be assumed that the vapor behaves like an ideal gas such that ϕ0i ≈ 1 and ϕi ≈ 1, which leads to the extended Raoult’s law8 :

xi · γi · p0i (T ) = yi · p = pi

for

i = 1, 2, . . . , k .

(2.126)

On the left-hand side, the fugacity fi0 of component i in the liquid phase is determined by the mole fraction xi , the saturation vapor pressure p0i (T ), and the activity coefficient γi [Stephan et al. 2017]. Hence, non-ideal liquid-phase behavior, i.e. different interactions between different molecules, is described by using the activity coefficient γi . The right-hand side shows that the fugacity fi00 of component i in the vapor phase at moderate pressures is equal to the partial pressure pi = yi · p. If the interactions between molecules of different components are of a similar kind as the interactions between molecules of the same component, the mixture is an ideal mixture, as described in Section 2.1.3, where ideal mixtures are defined on the basis of the molar mixing volume and enthalpy being zero, i.e. ∆v = v E = 0 and ∆h = hE = 0. This is equivalent to the condition that the activity coefficients are equal to one, i.e. γi = 1. Then, the extended Raoult’s law simplifies to Raoult’s law:

xi · p0i (T ) = yi · p = pi

for

i = 1, 2, . . . , k .

(2.127)

For xi → 1 it follows that γi → 1, i.e. if a mixture approaches the state of a pure component i, the activity coefficient γi is equal to one, and from Raoult’s law in Eq. (2.126), it follows xi · p0i (T ) = pi . Hence, the extended Raoult’s law in Eq. (2.126) approaches the form of Raoult’s law in Eq. (2.127) [Prausnitz et al. 1999]. This behavior is illustrated graphically in Figure 2.4. The saturation vapor pressure p0i (T ) of component i in Eqs. (2.126) and (2.127) can be calculated with the Antoine equation or Wagner equation, as described in Section 2.1.4, whereas the activity coefficients γi can be determined with the molar excess free energy g E , as described in Sections 2.2.4 and 2.2.12. Therefore, the determining parameters for the activity coefficients γi are the mole fractions x1 , x2 , . . . , xk in the liquid, while the pressure influence can be neglected. For the majority of mixtures, the dominating influence of the temperature T on VLE, according to Eqs. (2.126) and (2.127), is determined by the temperature dependent saturation vapor pressure p0i (T ). 2.2.11

Fugacity of Infinitely Diluted Component as Standard Fugacity: Henry’s Law

If the temperature T is higher than the critical temperature Tci of one of the components in the mixture, i.e. T > Tci , a saturation vapor pressure p0i of that component does not exist. Therefore, neither the extended Raoult’s law nor Raoult’s law, as described in Section 2.2.10, can be used to calculate the vapor–liquid equilibrium. 8

If not otherwise stated, it is assumed, in the course of this chapter, that the extended Raoult’s law can be applied, i.e. the Poynting correction is neglected, and the vapor behaves like an ideal gas: Poy i = 1 and ϕi = ϕ0i = 1.

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

49

Instead, another standard fugacity fi∞ = Hij (p, T ) for component i in the liquid phase has to be selected: component i being infinitely diluted in the liquid component j , i.e. xi → 0, serves as a new standard state. This leads to Henry’s law, which describes the solubility of the gaseous component i in the liquid component j as follows:

xi · Hij (T ) = yi · p = pi

for

i = 1, 2, . . . , k and i 6= j .

(2.128)

Here, Hij (T ) is the Henry coefficient that strongly depends on the temperature T . Furthermore, it has to be stated which liquid component j the gaseous component i is dissolved in. Henry’s law is strictly valid at infinite dilution xi → 0 and only for an ideal gas phase9 . Henry coefficients Hij (T ) can be found in literature for many combinations of i and j [Gmehling et al. 2012]. It follows from Eq. (2.128) that

Hij = lim

xi →0

pi , xi

which can be solved using the rule of L’Hospital to   dpi Hij = . dxi xi =0

(2.129)

(2.130)

Therefore, the Henry coefficient can be graphically determined from the slope of the tangent at the origin in a (p, x )-diagram [Prausnitz et al. 1999]. This is illustrated in Figure 2.4. From experimental investigations, it can be concluded that for many gases i and liquids j , Henry’s law is valid for a finite range of mole fractions, i.e. xi > 0. Therefore, Henry’s law can be applied to the typical process conditions of absorption and desorption, where gases are selectively dissolved in solvents. While being absolutely required for supercritical conditions with T > Tci , Henry’s law can also be used for subcritical conditions, i.e. T < Tci , to describe VLE in absorption and desorption processes. 2.2.12

Correlations Describing the Molar Excess Free Energy and Activity Coefficients

A vast variety of correlations for the molar excess free energy g E can be found in literature. Most of these correlations are empirical or semiempirical and for binary mixtures using two, in some rare cases three, parameters to fit experimental data from phase equilibrium measurements. The concept of local compositions in mixtures, 9

For real gas behavior, the right-hand side of Eq. (2.128) has to be multiplied with the fugacity coefficient ϕi . At states different from the state of infinite dilution, a modified activity coefficient γi∗ = ∞ ϕi /ϕ∞ i = γi /γi has to be included into the left-hand side of Eq. (2.128), causing the Henry coefficient Hij (p, T ) to not only depend on the temperature T but also on the pressure p [Prausnitz et al. 1999; Carroll 1991].

50

2 VAPOR–LIQUID EQUILIBRIUM

along with statistical thermodynamics, is increasingly employed for the modeling of the molar excess free energy g E . One important boundary condition for all correlations is g E = 0 at xi = 0, i.e. for pure components, the molar excess free energy g E has to disappear. A generic course ˆ · T ) as a function of the mole of the dimensionless molar excess free energy g E /(R fraction xa is shown in the diagram in Figure 2.2. It is shown how the logarithm of the activity coefficients γa and γb for one exemplary mole fraction xa can be determined ˆ · T ) and the graphically from the intersection of the tangent of the function g E /(R associated ordinate at xa = 1 and xb = 1 (xa = 0), respectively. This procedure follows directly from the correlations in Eqs. (2.106) – (2.107). As a consequence, the activity coefficients at infinite dilution γa∞ and γb∞ can be determined from the tangents at xa = 0 and xb = 0 (xa = 1), respectively. This graphical method to determine γa∞ and γb∞ is used in Example 2.7 and for sake of clarity not shown in Figure 2.2.

Figure 2.2 Graphical determination of the activity coefficients γa and γb from the ˆ · T ). dimensionless molar excess free energy g E /(R

2.2.12.1

Margules and van Laar Equation

A simple correlation for binary mixtures with two binary parameters Aab and Aba is the Margules equation, which can be written in the dimensionless form as follows:

gE = xa · xb · (Aba · xa + Aab · xb ) . ˆ·T R

(2.131)

For xa = 0 or xb = 0, it follows g E = 0. From Eqs. (2.106) and (2.107), alternatively from Eq. (2.98), we obtain the following correlations for the activity coefficients:

ln γa = x2b · (Aab + 2 · (Aba − Aab ) · xa ) ln γb =

x2a

· (Aba + 2 · (Aab − Aba ) · xb ) .

(2.132) (2.133)

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

51

For xa → 0 and xb → 0, it follows for the activity coefficients at infinite dilution that

ln γa∞ = Aab

(2.134)

ln γb∞

(2.135)

= Aba ,

i.e. the binary parameters Aab and Aba for the Margules equation can be directly determined from the activity coefficients at infinite dilution γa∞ and γb∞ . This procedure is illustrated in Example 2.7. The Margules equation and other correlations between the molar excess free energy g E and activity coefficients γi are summarized in Table 2.2. Similarly to the Margules equation, the van Laar equation can be used only for binary mixtures. For the binary parameters Aab and Aba of the van Laar equation, Eqs. (2.134) and (2.135) also apply. 2.2.12.2

Wilson, NRTL, and UNIQUAC Equation

The Margules and van Laar equations are solely empirical, whereas the correlations of this paragraph are based on the concept of the local compositions [Wilson 1964]. As compared with the Margules and van Laar equations, the Wilson equation in Table 2.2 can be used for ternary and multicomponent mixtures with k components, too. For a ternary mixture with the mole fractions xa , xb , and xc , we obtain

gE = − xa · ln (Λaa · xa + Λab · xb + Λac · xc ) ˆ·T R − xb · ln (Λba · xa + Λbb · xb + Λbc · xc )

(2.136)

− xc · ln (Λca · xa + Λcb · xb + Λcc · xc ) . From Eq. (2.92) or (2.98), we obtain for the activity coefficients ln γi with i = a, b, c

ln γi = 1 − ln (Λia · xa + Λib · xb + Λic · xc ) Λai · xa − Λaa · xa + Λab · xb + Λac · xc Λbi · xb − Λba · xa + Λbb · xb + Λbc · xc Λci · xc − . Λca · xa + Λcb · xb + Λcc · xc

(2.137)

The parameters Λij are calculated using the molar volumes vi0 and vj0 of the pure components10 and the binary parameters ∆λij as follows:   vj0 ∆λij Λij = 0 · exp − with Λii = 1 . ˆ·T vi R 10

(2.138)

In the context of the Wilson equation, the molar volumes of the pure components are labeled in literature mostly as vi and vj , which is used, however, in this chapter for the partial molar volume.

52

2 VAPOR–LIQUID EQUILIBRIUM

Hence, for multicomponent mixtures, the activity coefficients can be calculated by just using the pure component molar volumes vi0 and the interaction parameters ∆λij of the binary mixtures. For a ternary mixture, this reduces to the three binary mixtures a/b, a/c, and b/c with the pairs of binary parameters (∆λab , ∆λba ), (∆λac , ∆λca ), and (∆λbc , ∆λcb ) and the molar volumes va , vb , and vc . This is illustrated in Example 2.6. Besides the Wilson equation, the NRTL (Non Random Two Liquids) and the UNIQUAC (Universal Quasichemical) equations are listed in Table 2.2. Both correlations can be used for multicomponent mixtures, even though only binary parameters are required. The NRTL equation [Renon and Prausnitz 1968] requires the binary interaction parameters ∆gij as well as the non-randomness factor αij = αji as an additional third parameter for each binary mixture. Within the UNIQUAC equation [Abrams and Prausnitz 1975], the molar excess free energy g E and activity coefficients γi are split into a combinatorial (Index C ) and a residual (Index R) part. The latter depends on the binary interaction parameters ∆uij . Furthermore, the relative van der Waals volumes ri and surface areas qi of the components i = 1, 2, . . . , k are required as pure component parameters. With the NRTL and UNIQUAC equations, it is possible to predict the mixture behavior even if two immiscible liquid phases are formed. The Wilson equation, which will be used for most of the applied calculations in this chapter, is not capable of describing two immiscible liquid phases. Table 2.2 Correlations for molar excess free energy g E and activity coefficients γi .

Equation

Parameters

Margules

Aab Aba

van Laar

Aab Aba

gE = xa · xb · (Aba · xa + Aab · xb ) ˆ·T R ln γa = x2b · (Aab + 2 · (Aba − Aab ) · xa ) ln γb = x2a · (Aba + 2 · (Aab − Aba ) · xb ) gE Aab · xa · xb = ˆ xa · (Aab /Aba ) + xb R·T  2 Aba · xb ln γa = Aab ·  Aab · xa + Aba · xb 2 Aab · xa ln γb = Aba · Aab · xa + Aba · xb

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

53

Table 2.2 (continued) Equation Wilson

Parameters

∆λij

k k P P gE =− xi · ln Λij · xj ˆ·T R i=1 j=1

∆λji

ln γi = 1 − ln

k P

k P

Λij · xj −

j=1

l=1

Λli · xl k P Λlj · xj j=1

Λij

  vj0 ∆λij = 0 · exp − R·T with Λii = 1 ˆ vi

vi0 molar volume of pure component i E

NRTL

∆gij

g = ˆ R·T

k X

k P

xi ·

i=1

τji · Gji · xj

j=1 k P

Gji · xj

j=1 k P

∆gji

ln γi =

τji · Gji · xj

j=1 k P l=1

αij

·Gli · xl 

 τ · G · x k mj mj m  X Gij · xj   m=1 + · τ −   ij k k   P P j=1 Gli · xl Glj · xl k P

l=1

∆gij ˆ·T R

l=1

τij

=

Gij

= exp (−αij · τij )

αij

= αji

with

τii = 0 with

Gii = 1

non-randomness parameter

54

2 VAPOR–LIQUID EQUILIBRIUM

Table 2.2 (continued) Equation

Parameters

UNIQUAC11 ∆uij

∆uji

gE g E,C g E,R = + ˆ·T ˆ·T ˆ·T R R R k k g E,C X zX Fi = xi · ln Vi + qi · xi · ln ˆ 2 Vi R · T i=1 i=1 k k P P g E,R =− qi · xi · ln xj · Fj · τji ˆ·T R i=1 j=1

ln γi = ln γiC + ln γiR   z Vi Vi ln γiC = 1 − Vi + ln Vi − · qi · 1 − + ln 2 Fi Fi  k P qj · xj · τji  j=1  R ln γi = qi · 1 − ln k  P q j · xj j=1 

−

k X qj · xj · τij    k  P j=1 ql · xl · τlj l=1

Fi

=

qi k P j=1

Vi

=

k P

qj · xj ri

rj j=1 

τij

· xj

−∆uij = exp ˆ·T R

 with

τii = 1

ri relative van der Waals volume of component i qi relative van der Waals surface area of component i z number of interacting molecules for cubic z = 6, for z = 12 hexagonal mostly used in literature z = 10 Source: Adapted from Gmehling et al. 1977; Gmehling and Kolbe 1992; Gmehling et al. 2019. 11

In the DECHEMA Chemistry Data Series [Gmehling et al. 1977], the volume fraction xi · Vi and the surface area fraction xi · qi are used. Hence, Vi and Fi , as used here, are the volume and the surface area fraction divided by the mole fraction xi as in Gmehling and Kolbe 1992 and Gmehling et al. 2019.

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

2.2.12.3

55

Predictive Correlations Describing the Molar Excess Free Energy

For liquid mixtures where the molar mixing enthalpy is negligible, i.e. hE ≈ 0, it follows g E ≈ −T · sE . Hence, the molar excess free energy g E can be determined for these athermal mixtures from entropic considerations [Stephan et al. 2017]. In the theory of Flory and Huggins for solutions of large molecules (solute) in small molecules (solvent), values for g E and the activity coefficients γi can be determined from an perceptible volume fraction of the different molecules in the mixture [de Hemptinne et al. 2012]. Group contribution methods are obtained from known experimental data of welldefined pure components and mixtures. The first method was the ASOG (Analytical Solution of Groups), which was developed within Shell [Gmehling et al. 2012]. The UNIFAC (Universal Quasichemical Functional Group Activity Coefficients) equation was derived by Fredenslund et al. 1977 from the UNIQUAC equation (see previous paragraph) for multicomponent mixtures. Each molecule of the mixture is split into its composing functional groups. This enables the calculation of the combinatorial and residual part of the molar excess free energy g E . The required relative van der Waals volumes ri and surface areas qi of the most important functional groups can be found in Gmehling et al. 2012. Other common sources are thermophysical data banks like the Dortmund Data Bank [DDBST GmbH 2018] and the Design Institute for Physical Properties (DIPPR) from Design Institute for Physical Properties 2018. The Dortmund Data Bank contains also group data of the Modified UNIFAC (Dortmund) [Weidlich and Gmehling 1987]. A review of the most common group contribution methods is given in Kehiaian 1983. Nowadays, quantum-mechanical tools along with methods from statistical thermodynamics are available for describing the interactions between the molecules in a mixture. The method COSMO-RS (Conductor-like Screening Model for Real Solvents), developed by Klamt 1995, processes the screening charge density σ on the surface of the molecules to calculate the chemical potential µi of each component. The resulting chemical potentials are the basis for other thermodynamic equilibrium state variables such as activity coefficients. The method was developed to provide a general prediction method with no need for experiments. 2.2.13

Using Experimental Data of Binary Mixtures for Correlations Describing the Molar Excess Free Energy and Activity Coefficients

Based on experimental vapor–liquid equilibrium data of binary mixtures (VLE data), the binary interaction parameters for the Margules, van Laar, Wilson, NRTL, and UNIQUAC equation, as described in Section 2.2.12, can be determined. Depending on whether the VLE data is measured at isothermal or isobaric conditions, these parameters are independent from the mole fractions of the mixture but valid only for the corresponding constant temperature T or pressure p. However, the pressure dependence of the molar excess free energy g E and the activity coefficients γi can be neglected in most cases. For mixtures that exhibit a strong non-ideal behavior with high values of the molar excess enthalpy hE , the temperature dependence of

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2 VAPOR–LIQUID EQUILIBRIUM

the activity coefficients according to Eq. (2.102) cannot be neglected, as shown by Gmehling et al. 2012 for the binary acetone/water mixture. In these cases not only VLE data is required, but also experimental values, based on caloric measurements of the molar mixing enthalpies ∆h = hE . As an example for the binary mixture of low boiling acetone a and high boiling methanol b, experimental VLE data is shown in Figure 2.3. The data is taken from the DECHEMA Chemistry Data Series [Gmehling et al. 1977], which contains a collection of VLE data for numerous binary mixtures. The experimental VLE data consists of values for the mole fractions in the liquid phase (x1 in Figure 2.3) and the vapor phase (y1 in Figure 2.3)12 . In the example of Figure 2.3, which is given for isothermal conditions at T = 45 ◦C, the vapor and liquid phase are in equilibrium at the varying pressure p, also given in the data table. Before using experimental VLE data for determining the parameters for a correlation of the molar excess free energy g E , and thereby, the activity coefficients γa and γb , a thermodynamic consistency check of the data points with Eq. (2.105) shall be performed. For the isothermal example in Figure 2.3 with dT = 0 and v E ≈ 0, i.e. negligible values for the molar excess volume, we obtain from Eq. (2.105) the following correlation: Z

xa =1

ln xa =0

γa dxa = 0 . γb

(2.139)

If necessary, some of the data points have to be eliminated and dropped in order to fulfill Eq. (2.139) with the required accuracy. Two different consistency checks, indicated as Method 1 (point-by-point check) and Method 2 (integral check according to Eq. (2.139)), have been applied to the VLE data in Figure 2.3 [Gmehling et al. 1977]. To evaluate VLE data of binary mixtures at moderate pressures, and in order to obtain the interaction parameters for the aforementioned correlations for the molar excess free energy g E , the extended Raoult’s law in Eq. (2.126) is rewritten to solve for the activity coefficients:

γa =

ya · p xa · p0a (T )

and

γb =

yb · p (1 − ya ) · p = . xb · p0b (T ) (1 − xa ) · p0b (T )

(2.140)

Here, the saturation vapor pressures p0a (T ) and p0b (T ) of the pure components shall preferably be calculated with the given Antoine parameters also shown in tables like in Figure 2.3. From Eq. (2.140), for each data point experimental values for the activity coefficients γa and γb are obtained: • For isothermal data, the pressure p varies between the data points and the saturation vapor pressures p0a (T ) and p0b (T ) in Eq. (2.140) have to be calculated only for the given temperature T . 12

In Figure 2.3 the index 1 is used instead of a for the low boiling acetone and the index 2 instead of b for the high boiling methanol.

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57

Figure 2.3 VLE data for the binary acetone/methanol mixture. Antoine equation in the form log10 p0i (T ) = Ai − Bi /(T + Ci ) for p0i in mmHg and T in ◦C. Source: Gmehling et al. 1977.

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2 VAPOR–LIQUID EQUILIBRIUM

• For isobaric data, the pressure p in Eq. (2.140) stays constant. However, the temperature T varies such that the saturation vapor pressures p0a (T ) and p0b (T ) have to be calculated for each data point. The values of the activity coefficients γa and γb in Eq. (2.140) are the basis to obtain E experimental values for the molar excess free energy gExp . With Eq. (2.89) it follows 13 for the dimensionless form that E gExp = xa · γa + xb · γb = xa · γa + (1 − xa ) · γb . ˆ·T R

(2.141)

If one of the aforementioned theoretical correlations for the molar excess free energy g E of Table 2.2 is used, e.g. the Margules equation or the van Laar equation, the parameters Aab and Aba can be determined such that the theoretical values for g E E match with the experimental values gExp . Usually, a least square fit, with a sum considering all consistent data points, is the most accurate solution and yields the best fit. Here, the sum of squares∆SgE has to be minimized:

∆SgE =

X

E gExp gE − ˆ·T ˆ·T R R

!2

= min .

(2.142)

It is also possible to apply a least square fit to the experimental and theoretical values of the activity coefficients γa and γb or the mole fractions ya and yb as shown in Gmehling et al. 1977. Besides the dimensionless parameters Aab = b A12 and Aba = b A21 of the Margules and van Laar equations, Figure 2.3 also shows the binary interaction parameters fitted to the Wilson, NRTL, and UNIQUAC equations. Therefore, for the Wilson equation, we have14 ∆λab = b A12 and ∆λba = b A21 . For the NRTL equation the non-randomness parameter αab = b ALPHA12, along with ∆gab = b A12 and ∆gba = b A21 , is given. Finally, for the UNIQUAC equation, we have ∆uab = b A12 and ∆uba = b A21 . Instead of applying a least square fit to the dimensionless molar excess free energy g E , as shown in Eq. (2.142), it is also possible to use experimental data, in which one of the components is diluted in the other component. Thus, for the Margules and van Laar equations, the parameters can be calculated directly from the activity coefficients at infinite dilution γa∞ and γb∞ with Eqs. (2.134) and (2.135). This procedure is illustrated in Example 2.7. Although distillation is usually operated at close to isobaric conditions, fewer consistency problems arise with isothermal VLE data [Gmehling et al. 2012]. This can be explained with the fact that for many mixtures the molar mixing volume is negligible, i.e. v E ≈ 0 (see Section 2.2.5 with Eq. (2.105)). E In Eq. (2.141) only the molar excess free energy gExp is labeled as a property calculated from experimental values. The activity coefficients γa and γb on the right-hand side of Eq. (2.141) are also determined from experimental values with Eq. (2.140), but for sake of clarity not labeled explicitly. 14 All values in Figure 2.3 are in the unit cal/mol.

13

2.2 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM IN MIXTURES

2.2.14

59

Vapor–Liquid Equilibrium Ratio of Mixtures

The vapor–liquid equilibrium ratio (VLE ratio) Ki is defined as the ratio of the mole fraction yi of the vapor to the mole fraction xi of the liquid:

Ki =

yi xi

for i = 1, 2, . . . , k .

(2.143)

It is also labeled as the K -value of the mixture and can be calculated by using the extended Raoult’s law in Eq. (2.126) as follows:

Ki =

γi · p0i (T ) p

for

i = 1, 2, . . . , k .

(2.144)

It can be seen that Ki depends on the temperature T and the pressure p. Taking into account that the activity coefficients γi strongly depend on the mole fractions xi in the liquid, it follows that the VLE ratio also depends on the mole fractions, i.e. Ki (p, T, x1 , x2 , . . . , xk−1 ). For ideal mixtures with γi = 1, we obtain from Raoult’s law in Eq. (2.127)

Ki =

p0i (T ) p

for

i = 1, 2, . . . , k ,

(2.145)

in which the VLE ratio Ki (p, T ) depends only on the temperature T and the pressure p. If Henry’s law in Eq. (2.128) is used to describe vapor–liquid equilibrium in absorption and desorption processes, the VLE ratio can be calculated as

Ki = 2.2.15

Hij (T ) p

for

i = 1, 2, . . . , k and i 6= j .

(2.146)

Relative Volatility of Mixtures

In distillation the relative volatility αij can be used to describe how difficult a separation of the components i and j actually is:

αij =

Ki . Kj

(2.147)

The relative volatility αij is also labeled as separation factor and usually depends on the temperature T , the pressure p, and the mole fractions x1 , x2 , . . . , xk−1 . By using the extended Raoult’s law, it follows from Eq. (2.144) that

αij =

γi · p0i (T ) . γj · p0j (T )

(2.148)

Here, i is considered to be the low boiling component labeled here as low boiler as compared with the high boiling component j labeled here as high boiler j with

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p0i (T ) > p0j (T ) such that αij > 1. For azeotrope forming mixtures, this will change over the range of the mole fractions xi and yi , as described in Section 2.3. For ideal mixtures with γi = γj = 1, we obtain p0i (T ) . p0j (T )

αij =

(2.149)

Hence, the relative volatility of ideal mixtures depends only on the temperature T . It can be seen that the condition for constant relative volatility αij is, besides γi = γj = 1, a constant ratio of the temperature-dependent saturation vapor pressures p0i (T ) and p0j (T ) of the pure components i and j . The term ideal mixture will be used in subsequent chapters of this textbook for mixtures with constant relative volatilities, i.e. αij = const, too. For binary mixtures, with a as the low boiler and b as the high boiler, along with the closing conditions xb = 1 − xa and yb = 1 − ya , we obtain

αab =

Ka ya xb ya 1 − xa = · = · , Kb xa yb xa 1 − ya

(2.150)

which can be rewritten as

ya xa = αab · . 1 − ya 1 − xa

(2.151)

From these correlations we obtain the following forms to solve for ya and xa :

ya =

αab · xa 1 + (αab − 1) · xa

(2.152)

xa =

−1 αab · ya ya = . −1 α + (1 − αab ) · ya 1 + (αab − 1) · ya ab

(2.153)

For multicomponent mixtures with the components i = 1, 2, . . . , k , the following correlation can be derived from the definitions of αij and the closing conditions:

αij · xi

yi = 1+

k X

with αjj = 1 .

(2.154)

(αlj − 1) · xl

l=1

For ternary mixtures with a as the low boiler, b as the intermediate boiler, and c as the high boiler, we obtain with j = c and αcc = 1 the correlations

αac · xa 1 + (αac − 1) · xa + (αbc − 1) · xb αbc · xb yb = 1 + (αac − 1) · xa + (αbc − 1) · xb xc yc = . 1 + (αac − 1) · xa + (αbc − 1) · xb ya =

(2.155) (2.156) (2.157)

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61

It can be seen that the closing condition ya + yb + yc = 1 is fulfilled. Alternatively, similar correlations can be obtained for j = b and j = a. From the definition of the relative volatility αij in Eq. (2.147), we obtain for ternary mixtures

αab =

αac . αbc

(2.158)

The correlations above can be used to calculate the mole fraction yi of component i in the vapor phase as a function of the mole fractions x1 , x2 , . . . , xk in the liquid phase with the relative volatilities αij as the only determining thermodynamic parameters. It has to be considered, however, that the relative volatilities αij vary with the temperature, pressure, and the mole fractions. Nevertheless, these correlations are always valid and represent a simple method to describe the correlation of the mole fractions xi and yi in the liquid and vapor phase, which will be used subsequently in Section 2.4.5 for correlations that describe distillation lines of ternary mixtures. 2.2.16

Boiling Condition of Liquid Mixtures

For a liquid mixture with the mole fractions x1 , x2 , . . . , xk , the boiling condition can be determined by using the closing condition of the vapor phase: k X

yi = 1 .

(2.159)

i=1

With yi = Ki · xi from Eq. (2.143) and applying the extended Raoult’s law to determine the VLE ratio Ki in Eq. (2.144), we obtain the following boiling condition of the liquid mixture:

p=

k X

xi · γi · p0i (T ) .

(2.160)

i=1

If the temperature T is given, the corresponding boiling pressure p of the mixture can be calculated from Eq. (2.160) without iteration, assuming that the activity coefficients γi do not depend on the pressure (see Section 2.2.5). Thereafter, the mole fractions yi of the vapor phase can be calculated from the extended Raoult’s law in Eq. (2.126). In contrary, if the pressure p is given and the corresponding boiling temperature T of the liquid mixture has to be calculated, an iterative calculation is required, as shown in Example 2.4. The reason is the temperature dependence of the saturation vapor pressure p0i (T ), e.g. from an Antoine or Wagner equation in Eqs. (2.34), (2.35), and (2.37). In cases where the activity coefficients also vary with the temperature, the iterative calculation is even more complicated. Again, after the iteration, the mole fractions yi of the vapor phase can be calculated from the extended Raoult’s law in Eq. (2.126). For binary mixtures, the boiling pressure and temperature can be determined from phase diagrams using the boiling curve, as illustrated in Section 2.3.

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2.2.17

Condensation (Dew Point) Condition of Vapor Mixtures

For a vapor mixture with the mole fractions y1 , y2 , . . . , yk , the condensation condition, also labeled as dew point condition, can be calculated by using the closing condition of the liquid phase: k X

xi = 1 .

(2.161)

i=1

With xi = yi /Ki from Eq. (2.143) and applying the extended Raoult’s law to determine the VLE ratio Ki in Eq. (2.144), we obtain the following condensation condition of the vapor mixture: k

1 X yi = . p i=1 γi · p0i (T )

(2.162)

If the temperature T is given, the corresponding condensation pressure p of the mixture can be calculated from Eq. (2.160) without iteration only for ideal mixtures with γi = 1. Contrary, if the activity coefficients γi 6= 1 have to be considered for non-ideal mixtures, an iterative calculation is always required, since the activity coefficients strongly depend on the unknown mole fractions xi of the liquid phase. Hence, the correlations of the extended Raoult’s law in Eq. (2.126) for all components i = 1, 2, . . . , k have to be considered in the iterative calculation along with the condensation condition in Eq. (2.162). If the pressure p is given and the condensation temperature T of the vapor mixture has to be calculated, an iterative calculation, which is shown in Example 2.5, is required as well because of the temperature dependence of the saturation vapor pressure p0i (T ). For binary mixtures, the condensation pressure and temperature can be determined from phase diagrams using the condensation curve, as illustrated in Section 2.3.

Example 2.4: Boiling Temperature of an Ideal Ternary Mixture At the pressure p = 0.7 bar, a ternary non-ideal mixture of the components isobutane (index a), n-butane (index b), and n-pentane (index c) can be treated as an ideal mixture. The mole fractions of the liquid mixture are given as xa = 0.186, xb = 0.426, and xc = 0.388. For Task (1), a diagram with the saturation vapor pressure curves of the pure components is given. In this diagram with 1/T as abscissa and p as the logarithmic ordinate, the saturation vapor pressure curves are almost straight lines. In Task 3 the saturation vapor pressures p0i (T ) of the pure components shall be calculated with the following Antoine equation:

ln

p0i Bi = Ai − bar T /K + Ci

for i = a, b, c

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63

with the parameters Component

Ai

i-Butane a n-Butane b n-Pentane c

9.15169 9.05814 9.21312

Bi

Ci

2133.243 −28.162 2154.897 −34.420 2477.075 −39.945

(1) Using the given saturation vapor pressure curves of the pure components, determine the boiling temperature of the liquid mixture. As a first estimate of the boiling temperature T (n=0) , calculate an averaged boiling temperature using the mole fractions xi of the liquid. (2) Calculate the VLE ratios Ki of all components and determine the mole fractions yi of the first vapor bubble that arises from the liquid mixture. (3) Using the given Antoine equation, calculate the boiling temperature T of the mixture with an iterative method using T (n=0) = 1 ◦C as a first estimate. After each iteration step n = 0, 1, . . . , a correction factor φj shall be calculated for the ternary mixture as follows:

φj =

p(n) p0j (T (n) )

=

xa · p0a (T (n) ) + xb · p0b (T (n) ) + xc · p0c (T (n) ) . p0j (T (n) )

(1)

The component j is the component with the highest mole fraction yj in the vapor. This value, however, is not known yet but can be estimated by taking the highest value of the product xi · p0i (T (n) ). An “Antoine equation for the mixture” can be formulated with this correction factor φj as follows:   p0j p Bj = φj · = φj · exp Aj − . bar bar T /K + Cj

(2)

This correlation can be rewritten to solve for a better estimate T n+1 . If the correct component j is selected, the best convergence is obtained. Even if other components are selected, the correlation in Eq. (2) can still be used to improve the estimated temperature.

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Solution:

(1) For a first estimate of the boiling temperature T (n=0) for the iteration step n = 0, the boiling temperatures Ti0 (p) of the pure components at p = 0.7 bar shall be used. From the saturation vapor pressure curves within the diagram, we obtain • 1/Ta0 = 0.00396 1/K ⇒ Ta0 = 252.53 K • 1/Tb0 = 0.00380 1/K ⇒ Tb0 = 263.16 K • 1/Tc0 = 0.00335 1/K ⇒ Tc0 = 298.51 K.

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65

For a first estimate of the boiling temperature T (n=0) , we obtain

T (n=0) = xa · Ta0 + xb · Tb0 + xc · Tc0 = 0.186 · 252.53 K + 0.426 · 263.16 K + 0.388 · 298.51 K = 274.90 K . For an ideal ternary mixture with γi = 1, the boiling condition in Eq. (2.160) simplifies to

p = xa · p0a (T ) + xb · p0b (T ) + xc · p0c (T )

(3)

For T = T (n=0) = 274.90 K, we graphically obtain

p0a = 1.660 bar ,

p0b = 1.096 bar ,

p0c = 0.263 bar

from the diagram with 1/274.90 K = 0.00364 1/K. Hence, the boiling condition in Eq. (3) yields

p(n=0) = 0.186 · 1.660 bar + 0.426 · 1.096 bar + 0.388 · 0.263 bar = 0.309 bar + 0.467 bar + 0.102 bar = 0.878 bar 6= 0.7 bar . It can be seen that the boiling condition in Eq. (3) is not fulfilled. Therefore, a better estimate T (n=1) for the boiling temperature has to be determined, which can be achieved by plotting the point (1/T (n=0) , p(n=0) ) into the given diagram with the saturation vapor pressure curves, as shown in the diagram below. T (n=0) is not the boiling temperature at the pressure p = 0.7 bar; however, it is equal to the boiling temperature at the pressure p = 0.878 bar. Thus, a straight dashed line through the point (1/T (n=0) , p(n=0) ) roughly parallel to the saturation vapor pressure curves of the pure components is drawn. This dashed line represents an estimate for the “saturation vapor pressure curve of the mixture”, and a new estimate for the temperature T (n=1) can be read off at p = 0.7 bar. At p = 0.7 bar we obtain from the diagram 1/T (n=1) = 0.00372 1/K, and the new estimate is T (n=1) = 268.82 K. From the diagram, we obtain the saturation vapor pressures of the pure components at T (n=1) = 268.82 K as

p0a = 1.318 bar ,

p0b = 0.874 bar ,

and

p0c = 0.200 bar .

Hence, the boiling condition in Eq. (3) yields

p(n=1) = 0.186 · 1.318 bar + 0.426 · 0.874 bar + 0.388 · 0.200 bar = 0.245 bar + 0.372 bar + 0.078 bar = 0.695 bar ≈ 0.7 bar .

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Taking into account the limited precision that can be achieved from the readings in the diagram, this shall be considered accurately enough. Hence, we obtain T ≈ T (n=1) = 268.82 K as the boiling temperature of the mixture at p = 0.7 bar. (2) The VLE ratios can be calculated with Eq. (2.144), which, for an ideal mixture with γi = 1, simplifies to

Ki =

p0i (T ) p

for

i = a, b, c .

With the values from the previous task, we obtain

p0a 1.318 bar = = 1.896 p 0.695 bar p0 0.874 bar Kb = b = = 1.258 p 0.695 bar 0.200 bar p0 Kc = c = = 0.288 . p 0.695 bar

Ka =

From the definition of Ki in Eq. (2.143), it follows with yi = Ki · xi that

ya = Ka · xa = 1.896 · 0.186 = 0.353 yb = Kb · xb = 1.258 · 0.426 = 0.536 yc = Kc · xc = 0.288 · 0.388 = 0.112 . The closing condition is used for a final check:

ya + yb + yc = 0.353 + 0.536 + 0.112 = 1.001 ≈ 1 . It can be seen that ya > xa and Ka > 1 as well as yc < xc and Kc < 1, i.e. the low boiling component a (isobutane) is enriched in the vapor phase, whereas the high boiling component c (n-pentane) is enriched in the liquid phase.

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67

(3) With the Antoine equation, the saturation vapor pressures p0i (T ) can be calculated for the first estimated temperature T (n=0) = 1 ◦C as follows:   p0a Ba = exp Aa − (n=0) bar T /K + Ca   2133.243 = exp 9.15169 − = 1.6153 , p0a = 1.6153 bar 274.15 − 28.162

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  p0b Bb = exp Ab − (n=0) bar T /K + Cb   2154.897 = exp 9.05814 − = 1.0717 , p0b = 1.0717 bar 274.15 − 34.420   p0c Bc = exp Ac − (n=0) bar T /K + Cc   2477.075 = exp 9.21312 − = 0.2558 , p0c = 0.2558 bar . 274.15 − 39.945 Inserting into the boiling condition in Eq. (3) yields

p(n=0) = xa · p0a (T (n=0) ) + xb · p0b (T (n=0) ) + xc · p0c (T (n=0) ) = 0.186 · 1.6153 bar + 0.426 · 1.0717 bar + 0.388 · 0.2558 bar = 0.3004 bar + 0.4565 bar + 0.0993 bar = 0.8562 bar 6= 0.7 bar . It can be seen that xb · p0b (T (n=0) ) = 0.4565 bar represents the highest value such that component j = b will be chosen to calculate the correction factor of Eq. (1):

φb =

p(n=0) 0.8562 bar = = 0.7989 . 0 (n=0) 1.0717 bar pb (T )

The correlation in Eq. (2) can be rewritten to solve for a better estimate T (n=1) as follows:

T (n=1) = K

T (n=1)

Bb − Cb p/bar Ab − ln φb 2154.897 = + 34.420 = 268.90 0.7 9.05814 − ln 0.7989 = 268.90 K .

In the next step, the saturation vapor pressures p0i (T ) for T (n=1) = 268.90 K have to be calculated:   p0a Ba = exp Aa − (n=1) bar T /K + Ca   2133.243 = exp 9.15169 − = 1.3370 , p0a = 1.3370 bar 268.90 − 28.162

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69

  p0b Bb = exp Ab − (n=1) bar T /K + Cb   2154.897 = exp 9.05814 − = 0.8764 , p0b = 0.8764 bar 268.90 − 34.420   p0c Bc = exp Ac − (n=1) bar T /K + Cc   2477.075 = exp 9.21312 − = 0.2007 , p0c = 0.2007 bar . 268.90 − 39.945 Inserting into the boiling condition in Eq. (3) yields

p(n=0) = 0.186 · 1.3370 bar + 0.426 · 0.8764 bar + 0.388 · 0.2007 bar = 0.6999 bar ≈ 0.7 bar , i.e. the boiling temperature of the mixture at the pressure p = 0.7 bar is T ≈ T (n=1) = 268.90 K, which is very close to the value T ≈ 268.82 K obtained from the diagram with the saturation vapor pressure curves, as shown above.

Example 2.5: Condensation Temperature of an Ideal Ternary Mixture For the ternary mixture of the components i-butane (index a), n-butane (index b), and n-pentane (index c) from Example 2.4, the condensation temperature at the pressure p = 0.7 bar shall be calculated. Again, the mixture can be treated as an ideal mixture, and the mole fractions of the vapor mixture are given as ya = 0.186, yb = 0.426, and yc = 0.388. The saturation vapor pressures p0i (T ) of the pure components can be calculated with the Antoine equation given in Example 2.4. Contrary to the method used in Example 2.4, an iterative Newton–Raphson method with a numerical step size of ∆T = 1 K shall be applied. This method can also be applied to non-ideal mixtures where the mole fractions in the liquid have to be estimated as well in order to calculate the activity coefficients γi 6= 1. For ideal mixtures, the graphical solution and the iterative method with the correction factor φj used in Example 2.4 can also be applied to calculate the condensation temperature. (1) Calculate the condensation temperature of the given mixture. Use the temperature T (n=0) = 1 ◦C as an initial estimate. The iterative calculation can be terminated once the absolute value of the difference between the calculated pressure p(T (n) ) and the pressure p = 0.7 bar is less than 0.01 bar. (2) Calculate the VLE ratios Ki of all components and determine the mole fractions xi of the first liquid drop that is formed from the condensing vapor mixture.

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Solution:

(1) For an ideal ternary mixture with γi = 1, the condensation condition in Eq. (2.162) simplifies to

1 ya yb yc = 0 + + . p pa (T ) p0b (T ) p0c (T )

(1)

With the Antoine equation the saturation vapor pressures p0i (T ) can be calculated at the first estimated temperature T (n=0) = 1 ◦C as follows:   p0a Ba = exp Aa − (n=0) bar T /K + Ca   2133.243 = exp 9.15169 − = 1.6153 , p0a = 1.6153 bar 274.15 − 28.162   p0b Bb = exp Ab − (n=0) bar T /K + Cb   2154.897 = exp 9.05814 − = 1.0717 , p0b = 1.0717 bar 274.15 − 34.420   p0c Bc = exp Ac − (n=0) bar T /K + Cc   2477.075 = exp 9.21312 − = 0.2558 , p0c = 0.2558 bar. 274.15 − 39.945 The calculation of these values is accomplished analogous to Task (3) of Example 2.4. Inserting into the condensation condition in Eq. (1) yields

1 ya yb yc = 0 (n=0) + 0 (n=0) + 0 (n=0) p(T (n=0) ) pa (T ) pb (T ) pc (T ) 0.186 0.426 0.388 = + + 1.6153 bar 1.0717 bar 0.2558 bar 1 1 1 1 = 0.1151 + 0.3975 + 1.5168 = 2.0294 bar bar bar bar p(n=0) = 0.4928 bar 6= 0.7 bar . Since the condensation condition is not fulfilled, an improved estimate needs to be found using the Newton–Raphson method. This is an iterative approach to !

find the root of a given function, here f (T ) = 0. It is, therefore, necessary to formulate the problem at hand such that the solution can be obtained by finding the root of a function f (T ). At iteration step n the present problem requires !

f (T (n) ) = p(T (n) ) − p = 0 with p = 0.7 bar .

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71

For the initial estimate T (n=0) , this yields

f (T (n=0) ) = p(T (n=0) ) − p = 0.4928 bar − 0.7 bar = −0.2073 bar . Since |−0.2073 bar| = 0.2073 bar > 0.01 bar, the accuracy requirement is not satisfied. The Newton–Raphson method uses a numerical derivative at iteration step n

df (T (n) ) f (T (n) + ∆T ) − f (T (n) ) ≈ dT ∆T to find the root of a function. An improved estimate for the condensation temperature T (n=1) can be calculated using the following equation:

f (T (n=0) ) f (T (n=0) + ∆T ) − f (T (n=0) ) ∆T (n=0) f (T ) · ∆T = T (n=0) − . f (T (n=0) + ∆T ) − f (T (n=0) )

T (n=1) = T (n=0) −

To calculate the new estimate, a numerical step size ∆T is required. Here, a value of ∆T = 1 K is used. Before the improved estimate can be obtained, f (T (n) + ∆T ) needs to be calculated. This requires calculation of the saturation vapor pressure p0i of each component at T (n=0) + ∆T = 275.15 K:   p0a Ba = exp Aa − (n=0) bar (T + ∆T )/K + Ca   2133.243 = exp 9.15169 − = 1.6731 , p0a = 1.6731 bar 275.15 − 28.162   p0b Bb = exp Ab − (n=0) bar (T + ∆T )/K + Cb   2154.897 = exp 9.05814 − = 1.1125 , p0b = 1.1125 bar 275.15 − 34.420   p0c Bc = exp Ac − (n=0) bar (T + ∆T )/K + Cc   2477.075 = exp 9.21312 − = 0.2676 , p0c = 0.2676 bar. 275.15 − 39.945

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Inserting into the condensation condition in Eq. (1) yields

1 0.186 0.426 0.388 = + + 1.6731 bar 1.1125 bar 0.2676 bar p(T (n=0) + ∆T ) 1 1 1 = 0.1151 + 0.3975 + 1.5168 bar bar bar 1 = 1.9443 bar p(T (n=0) + ∆T ) = 0.5143 bar , and

f (T (n=0) + ∆T ) = p(T (n=0) + ∆T ) − p = 0.5143 bar − 0.7 bar = −0.1857 bar . Then the improved estimate T (n=1) can be calculated:

f (T (n=0) ) · ∆T f (T (n=0) + ∆T ) − f (T (n=0) ) −0.2073 bar · 1 K = 274.15 K − = 283.74 K . −0.1857 bar − (−0.2073 bar)

T (n=1) = T (n=0) −

Once again, the condensation condition is used to test the new estimated temperature T (n=1) . This requires calculation of the saturation vapor pressure p0i of each component using the Antoine equation at T (n=1) = 283.74 K. We obtain

p0a = 2.2369 bar ,

p0b = 1.5147 bar ,

and

p0c = 0.3878 bar .

Inserting into the condensation condition in Eq. (1) yields

1 p(T (n=1) )

=

0.186 0.426 0.388 1 + + = 1.3648 2.2369 bar 1.5147 bar 0.3878 bar bar

p(T (n=1) ) = 0.7327 bar 6= 0.7 bar , and for the root-finding algorithm

f (T (n=1) ) = p(T (n=1) ) − p = 0.7327 bar − 0.7 bar = 0.0327 bar > 0.01 bar . To find an improved estimate T (n=2) , the saturation vapor pressures p0i of all components need to be calculated at T (n=1) + ∆T = 284.74 K. We obtain

p0a = 2.3109 bar ,

p0b = 1.5679 bar ,

and p0c = 0.4043 bar .

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73

This yields

1 0.186 0.426 0.388 = + + 2.3109 bar 1.5679 bar 0.4043 bar p(T (n=1) + ∆T ) 1 = 1.3119 bar p(T (n=1) + ∆T ) = 0.7622 bar , and

f (T (n=1) + ∆T ) = p(T (n=1) + ∆T ) − p = 0.7622 bar − 0.7 bar = 0.0622 bar . Now the improved estimate T (n=2) can be calculated:

f (T (n=1) ) · ∆T f (T (n=1) + ∆T ) − f (T (n=1) ) 0.0327 bar · 1 K = 283.74 K − = 282.64 K . 0.0622 bar − 0.0327 bar

T (n=2) = T (n=1) −

Once again, the condensation condition is used to test the new estimate T (n=2) . The saturation vapor pressure p0i of the components at T (n=2) = 282.64 K are calculated using the Antoine equation. We obtain

p0a = 2.1571 bar ,

p0b = 1.4574 bar ,

and p0c = 0.3703 bar .

Inserting into the condensation condition in Eq. (1) yields

1 p(T (n=2) )

=

0.186 0.426 0.388 1 + + = 1.4264 2.1571 bar 1.4574 bar 0.3703 bar bar

p(T (n=2) ) = 0.7011 bar ≈ 0.7 bar , and the root-finding algorithm

f (T (n=2) ) = p(T (n=2) ) − p = 0.7011 bar − 0.7 bar = 0.0011 bar < 0.01 bar . According to the specified tolerance, the condensation condition is fulfilled. The calculated condensation temperature of the mixture is T ≈ T (n=2) = 282.64 K = 9.49 ◦C. (2) The VLE ratios Ki can be calculated with Eq. (2.144), which, for an ideal mixture with γi = 1, simplifies to

Ki =

p0i (T ) p

for

i = a, b, c .

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With the values from the previous task, we obtain

p0a 2.1571 bar = = 3.077 p 0.7011 bar 1.4574 bar p0 Kb = b = = 2.079 p 0.7011 bar p0 0.3703 bar Kc = c = = 0.528 . p 0.7011 bar

Ka =

From the definition of Ki in Eq. (2.143), it follows with xi = yi /Ki that

ya 0.186 = = 0.060 Ka 3.077 yb 0.426 xb = = = 0.205 Kb 2.079 yc 0.388 xc = = = 0.735 . Kc 0.528

xa =

The closing condition is used for a final check:

xa + xb + xc = 0.060 + 0.205 + 0.735 = 1.000 . It can be seen that xa < ya and Ka > 1 as well as xc > yc and Kc < 1, i.e. the light boiling component a (isobutane) is enriched in the vapor phase, whereas the high boiling component c (n-pentane) is enriched in the liquid phase.

Example 2.6: Calculation of Vapor–Liquid Equilibrium (VLE) Using the Wilson Equation for a Ternary Mixture For a liquid ternary mixture of the components acetone (index a), chloroform (index b), and methanol (index c), which exists at the temperature T = 45 ◦C, the VLE shall be calculated using the Wilson equation. The mole fractions of the liquid are given as xa = 0.25, xb = 0.35, and xc = 0.40, and the vapor phase can be treated as an ideal gas. The following binary interaction parameters ∆λij and pure component molar volumes vi0 can be obtained from the DECHEMA Chemistry Data Series [Gmehling et al. 1977]: Component Acetone a Chloroform b Methanol c

vi0 in ml/mol 74.04 80.67 40.73

∆λaj

∆λbj

– −437.9590 −33.8433 – −61.2123 −328.2534

∆λcj in cal/mol 495.6320 1822.9843 –

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75

We have the conversion 1 cal = 4.1868 J. Furthermore, the following Antoine equation and parameters are given in the DECHEMA Chemistry Data Series [Gmehling et al. 1977]:

log10

p0i Bi = Ai − mmHg T /◦C + Ci Component Acetone a Chloroform b Methanol c

for

i = a, b, c

(1)

Ai

Bi

Ci

7.11714 6.95465 8.08097

1210.595 1170.966 1582.271

229.664 226.232 239.726

We have the conversion 1.013 bar = 760 mmHg. (1) Calculate the activity coefficients γa , γb , and γc for the given mole fractions xi and the temperature T = 45 ◦C. (2) Calculate the boiling pressure p for the given mole fractions xi and temperature T = 45 ◦C. (3) Calculate the VLE ratios Ki at the calculated boiling pressure p from the previous task. (4) Calculate the mole fractions yi of the vapor formed from the liquid at the calculated boiling pressure p. (5) Calculate the relative volatilities αab , αac , and αbc and use them to recalculate the mole fractions ya , yb , and yc . Solution:

ˆ is required in the unit cal/(mol · K): (1) The ideal gas constant R ˆ = 8314 R

J cal kmol cal · · = 1.9858 . kmol · K 4.1868 J 1000 mol mol · K

Likewise the temperature has to be converted: T = 45 ◦C = 318.15 K. For acetone a in a ternary mixture, the activity coefficient γa can be calculated with the Wilson equation as follows:

ln γa = 1 − ln (Λaa · xa + Λab · xb + Λac · xc ) Λaa · xa − Λaa · xa + Λab · xb + Λac · xc Λba · xb − Λba · xa + Λbb · xb + Λbc · xc Λca · xc − . Λca · xa + Λcb · xb + Λcc · xc

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The parameters Λij are calculated, using the molar volumes vi0 of the pure components and the binary interaction parameters ∆λij , as follows:   vb0 ∆λab · exp − = ˆ·T va0 R   v0 ∆λac = c0 · exp − = ˆ·T va R   v0 ∆λba = a0 · exp − = ˆ·T vb R   vc0 ∆λbc = 0 · exp − = ˆ·T vb R   v0 ∆λca = a0 · exp − = ˆ·T vc R   v0 ∆λcb = b0 · exp − = ˆ·T vc R

Λab = Λac Λba Λbc Λca Λcb

  80.67 −33.8433 · exp − = 1.1495 74.05 1.9858 · 318.15   40.73 −61.2123 · exp − = 0.6061 74.05 1.9858 · 318.15   74.05 −437.9590 · exp − = 1.8358 80.67 1.9858 · 318.15   40.73 −328.2534 · exp − = 0.8489 80.67 1.9858 · 318.15   74.05 495.6320 · exp − = 0.8296 40.73 1.9858 · 318.15   80.67 1822.9843 · exp − = 0.1106 40.73 1.9858 · 318.15

With Λaa = Λbb = Λcc = 1, we obtain

ln γa = 1 − ln (0.25 + 1.1495 · 0.35 + 0.6061 · 0.40) 0.25 − 0.25 + 1.1495 · 0.35 + 0.6061 · 0.40 1.8358 · 0.35 − 1.8358 · 0.25 + 0.35 + 0.8489 · 0.40 0.8296 · 0.40 − = −0.2412 0.8296 · 0.25 + 0.1106 · 0.35 + 0.40 γa = exp (−0.2412) = 0.786 . For chloroform b and methanol c in a ternary mixture, the activity coefficients γb and γc can be calculated with the Wilson equation as follows:

ln γb = 1 − ln (Λba · xa + Λbb · xb + Λbc · xc ) Λab · xa − Λaa · xa + Λab · xb + Λac · xc Λbb · xb − Λba · xa + Λbb · xb + Λbc · xc Λcb · xc − Λca · xa + Λcb · xb + Λcc · xc

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77

ln γc = 1 − ln (Λca · xa + Λcb · xb + Λcc · xc ) Λac · xa − Λaa · xa + Λab · xb + Λac · xc Λbc · xb − Λba · xa + Λbb · xb + Λbc · xc Λcc · xc − . Λca · xa + Λcb · xb + Λcc · xc With the values from above, it follows that

ln γb = 1 − ln (1.8358 · 0.25 + 0.35 + 0.8489 · 0.40) 1.1495 · 0.25 − 0.25 + 1.1495 · 0.35 + 0.6061 · 0.40 0.35 − 1.8358 · 0.25 + 0.35 + 0.8489 · 0.40 0.1106 · 0.40 − = 0.1672 0.8296 · 0.25 + 0.1106 · 0.35 + 0.40 γb = exp (0.1672) = 1.182 ln γc = 1 − ln (0.8296 · 0.25 + 0.1106 · 0.35 + 0.40) 0.6061 · 0.25 − 0.25 + 1.1495 · 0.35 + 0.6061 · 0.40 0.8489 · 0.35 − 1.8358 · 0.25 + 0.35 + 0.8489 · 0.40 0.40 − = 0.3897 0.8296 · 0.25 + 0.1106 · 0.35 + 0.40 γc = exp (0.3897) = 1.476 . (2) According to the extended Raoult’s law in Eq. (2.126), the partial pressures pi of acetone a, chloroform b, and methanol c can be calculated as

pi = xi · γi · p0i (T ) for i = a, b, c . With Dalton’s law it follows with p = pa + pb + pc that the boiling pressure p can be calculated as

p = xa · γa · p0a (T ) + xb · γb · p0b (T ) + xc · γc · p0c (T ) , which also follows directly from the boiling condition in Eq. (2.160).

(2)

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The saturation vapor pressures p0a , p0b , and p0c are calculated with Eq. (1). For acetone a, it follows that

p0a 1210.595 = 7.11714 − = 2.7096 mmHg 45 + 229.664 p0a /mmHg = 102.7096 = 512.38 , p0a = 512.38 mmHg = 0.6829 bar . log10

Accordingly, from Eq. (1), we obtain

p0b = 433.95 mmHg = 0.5784 bar p0c = 334.04 mmHg = 0.4452 bar . Inserting all values for xi , γi , and p0i into Eq. (2) yields the required result for the boiling pressure at T = 45 ◦C:

p = 0.25 · 0.786 · 0.6829 bar + 0.35 · 1.182 · 0.5784 bar + 0.40 · 1.476 · 0.4452 bar = 0.6364 bar . As discussed in Section 2.2.16, the boiling pressure p of a mixture for a specified temperature T can be calculated without any iterations, even for a non-ideal mixture, as shown above. The reason is that the activity coefficients depend on the given temperature T and mole fractions xi but not on the unknown boiling pressure p. (3) The VLE ratios can be calculated with Eq. (2.144) as follows:

Ki =

γi · p0i (T ) p

for i = a, b, c .

With the values from the previous task, we obtain

γa · p0a 0.786 · 0.6829 bar = = 0.8432 p 0.6364 bar γb · p0b 1.182 · 0.5784 bar Kb = = = 1.0743 p 0.6364 bar γc · p0c 1.476 · 0.4452 bar Kc = = = 1.0330 . p 0.6364 bar

Ka =

(4) From the definition of Ki in Eq. (2.143), it follows with yi = Ki · xi that

ya = Ka · xa = 0.8432 · 0.25 = 0.2108 yb = Kb · xb = 1.0743 · 0.35 = 0.3760 yc = Kc · xc = 1.0330 · 0.40 = 0.4132 . The closing condition is used for a final check:

ya + yb + yc = 0.2108 + 0.3760 + 0.4132 = 1.000 .

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79

The results from a modern process simulator like UniSim Design [Honeywell 2017] are shown in the figures below. It can be seen that the Wilson equation is used to describe the liquid mixture.

Furthermore, the results from UniSim Design confirm the calculated values presented above.

The differences between the two calculation methods are in the low percent range and are a consequence of slightly different binary interaction parameters.

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2 VAPOR–LIQUID EQUILIBRIUM

(5) For the relative volatilities αij as defined in Eq. (2.147), we obtain

Ka 0.8432 = = 0.7849 Kb 1.0743 Ka 0.8432 = = = 0.8162 Kc 1.0330 Kb 1.0743 = = = 1.0399 . Kc 1.0330

αab = αac αbc

We obtain

αab · αbc = 0.7849 · 1.0399 = 0.8162 = αac , which is in accordance with Eq. (2.158). Applying these results for αac and αbc to the correlation in Eq. (2.155) yields

αac · xa 1 + (αac − 1) · xa + (αbc − 1) · xb 0.8162 · 0.25 = = 0.2108 . 1 + (0.8162 − 1) · 0.25 + (1.0399 − 1) · 0.40

ya =

Accordingly, from Eq. (2.156), it follows that

αbc · xb 1 + (αac − 1) · xa + (αbc − 1) · xb 1.0399 · 0.35 = = 0.3760 . 1 + (0.8162 − 1) · 0.25 + (1.0399 − 1) · 0.35

yb =

From Eq. (2.157) it follows that

xc 1 + (αac − 1) · xa + (αbc − 1) · xb 0.40 = = 0.4132 . 1 + (0.8162 − 1) · 0.25 + (1.0399 − 1) · 0.35

yc =

These values are identical to the ones calculated in the previos task, i.e. the correlations in Eqs. (2.155) and (2.156) are valid for all mixtures and are not limited to ideal mixtures with γi = 1 or mixtures with constant relative volatility αij , as mistakenly stated in literature sometimes.

2.3 BINARY MIXTURES AND PHASE DIAGRAMS

2.3

81

Binary Mixtures and Phase Diagrams

Binary mixtures with a low boiler a and a high boiler b are considered in this section. The vapor–liquid equilibrium is described with the extended Raoult’s law in Eq. (2.126), i.e. the vapor is treated as an ideal gas, and the behavior of the liquid is described with the activity coefficients γa and γb . Furthermore, the saturation vapor pressures p0a (T ) and p0b (T ) are selected as standard fugacities. 2.3.1

Boiling Curve Correlation

From the boiling condition in Eq. (2.160) and the closing condition xb = 1 − xa , we obtain the correlation15

p = γa · xa · p0a + γb · xb · p0b = γa · xa · p0a + γb · (1 − xa ) · p0b .

(2.163)

This is the boiling curve correlation in a (p, x )-diagram, which also follows from Dalton’s law p = pa + pb in Eq. (2.9), along with the correlations pa = xa · γa · p0a and pb = xb · γb · p0b from the extended Raoult’s law in Eq. (2.126). The boiling curve correlation in Eq. (2.163) can be written in the form

p = γb · p0b + xa · (γa · p0a − γb · p0b ) .

(2.164)

The boiling curves are shown in the (p, x )-diagrams in Figure 2.4 as solid lines with the label p for the two binary mixtures: methanol/water at 80 ◦C (left diagram) and dichloromethane/butanone at 60 ◦C (right diagram). In these diagrams, the mole fraction xa of the low boiler a is the abscissa, i.e. the left ordinate at xa = 0 (xb = 1) represents the pure high boiler b (here water or butanone), and the right ordinate at xa = 1 (xb = 0) represents the pure low boiler a (here methanol or dichloromethane). In the (p, x )-diagrams in Figure 2.4, the partial pressures pa and pb are also drawn as solid lines. It can be seen that pa = 0 at xa = 0 and pb = 0 at xb = 0. For all values along the abscissa, we have p = pa + pb , which is the graphical illustration of Dalton’s law, illustrated with the vertical double arrows of equal length. Furthermore, it can be seen that p = pa = p0a at xa = 1 and p = pb = p0b at xb = 1, i.e. the boiling pressure of the mixture p is equal to the partial pressures pa and pb , which are equal to the saturation vapor pressures p0a and p0b of the pure components. In the left (p, x )-diagram in Figure 2.4, it can be seen that the solid lines for the partial pressures pa and pb are always above the straight dash-dotted lines, which represent the linear correlations according to Raoult’s law in Eq. (2.127): pa = xa · p0a for the low boiler a and pb = xb · p0b = (1 − xa ) · p0b for the high boiler b. Hence, this behavior of a binary mixture where the partial pressures pa and pb are above the straight lines of Raoult’s law is known as mixtures with positive deviation from Raoult’s law. In these mixtures, molecules of the different components a and b 15

In this section, the temperature dependence of the saturation vapor pressures p0a (T ) and p0b (T ) is not always written explicitly.

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2 VAPOR–LIQUID EQUILIBRIUM

Figure 2.4 (p, x )-diagrams and activity coefficients of binary mixtures with low boiler a and high boiler b; left: methanol/water at 80 ◦C; right: dichloromethane/butanone at 60 ◦C.

attract each other in the liquid phase to a lesser extent than they attract the molecules of the same component: the molecules of component a and b are “more active” in the liquid mixture than in liquids of the pure components. This results in higher activity coefficients γa > 1 and γb > 1, which is shown in the left diagram below the (p, x )-diagram as ln γa > 0 and ln γb > 0. The interactions between molecules of different components are weaker compared with interactions between molecules of the same component. These effects result in an increase of the partial pressures pa and pb and, therefore, the boiling pressure p = pa + pb . In the right (p, x )-diagram in Figure 2.4, it can be seen that the solid lines for the partial pressures pa and pb are always below the straight dash-dotted line, given by linear correlations pa = xa · p0a and pb = xb · p0b = (1 − xa ) · p0b of Raoult’s law in Eq. (2.127). Hence, these binary mixtures with a behavior where the partial pressures pa and pb are below the straight lines of Raoult’s law are known as mixtures with negative deviation from Raoult’s law. In these mixtures, molecules of the different components a and b attract each other in the liquid phase to a higher extent than they attract the molecules of the same component: the molecules of component a and b are “less active” in the liquid mixture than in liquids of the pure components.

2.3 BINARY MIXTURES AND PHASE DIAGRAMS

83

This results in lower activity coefficients γa < 1 and γb < 1, which is shown in the right diagram below the (p, x )-diagram as ln γa < 0 and ln γb < 0. The interactions between molecules of different components are stronger than the interactions between molecules of the same component. These effects result in a decrease of the partial pressures pa and pb and, therefore, the boiling pressure p = pa + pb . In Section 2.2.10 it is shown that for xi → 1 with γi → 1 the extended Raoult’s law in Eq. (2.126) approaches the correlation in Eq. (2.127): if a mixture approaches the state of a pure component, Raoult’s law is always valid. This behavior can be seen for both mixtures in Figure 2.4, where the nonlinear curves of the partial pressures pa and pb approach the dash-dotted straight lines of Raoult’s law pa = xa · p0a for xa → 1 and pb = xb · p0b for xb → 0. The dotted lines in the (p, x )-diagrams in Figure 2.4 are the tangents of the partial pressure curves pa at xa = 0 and pb at xb = 0. The slope of these tangents equals the Henry coefficients of Henry’s law, which can be seen in Eq. (2.130). The value of this slope can be read off at the opposing ordinate at xa = 1 for Hab and xb = 1 for Hba . Hence, in the (p, x )-diagrams of Figure 2.4, the Henry coefficients Hba can be read off at xb = 1 (xa = 0) for the “solubility” of water b in methanol a (left diagram) and butanone b in dichloromethane a (right diagram)16 . The Henry coefficient Hab for the “solubility” of dichloromethane a in butanone b can be read off at xa = 1 in the right diagram, whereas in the left diagram, the Henry coefficient Hab for the “solubility” of methanol a in water b is out of scale at xa = 1. 2.3.2

Condensation (Dew Point) Curve Correlation

From the condensation condition in Eq. (2.162) and the closing condition for binary mixtures yb = 1 − ya , we obtain the correlation

p=

γa · p0a · γb · p0b . γa · p0a − ya · (γa · p0a − γb · p0b )

(2.165)

This is the correlation for the condensation curve in a (p, y)-diagram. The same correlation can be obtained by using the substitution xa = ya /Ka and Ka = γa · p0a /p from Eq. (2.145) into the boiling curve correlation in Eq. (2.164) and rewriting to solve for p. It is worth mentioning that using Eq. (2.165) requires the activity coefficients γa and γb , which can be calculated only if the mole fractions xa and xb = 1 − xa of the liquid phase are known. Hence, an iterative calculation scheme is required. For ideal mixtures with γa = 1 and γb = 1, the correlation for the condensation curve in Eq. (2.163) can be applied directly without iteration which is shown in Section 2.3.8; see Eq. (2.173). 16

For the binary mixtures shown in Figure 2.4, all components can exist as pure liquids such that the term “solubility” is usually not used as compared to mixtures where a component is dissolved in another, like in absorption and desorption processes.

84

2.3.3

2 VAPOR–LIQUID EQUILIBRIUM

(p, x , y )-Diagram

If the boiling curve and condensation curve are drawn into a common diagram, a (p, x , y)-diagram is obtained, which is valid for one specific temperature T . The (p, x , y)-diagrams for seven binary mixtures are shown in the bottom row of Figure 2.5 with a solid boiling curve and a dashed condensation curve. The boiling curve describes the state of the boiling saturated liquid (for a given value of xa ), whereas the condensation curve describes the state of the condensing saturated vapor (for a given value of ya ). The solid boiling curve always lies above the dashed condensation curve. For the binary benzene/toluene mixture, the boiling and condensation curves are given for three different temperatures. The boiling curves in the (p, x , y)-diagrams for the methanol/water and dichloromethane/butanone mixtures are identical to the ones in Figure 2.4. The region between the boiling and condensation curve represents the two-phase region with the mixture of saturated vapor, whose state is given by the condensation curve, and saturated liquid, whose state is given by the boiling curve. The horizontal connection between the boiling and condensation curve represents different ratios of saturated vapor and liquid, which always have the same state. On the horizontal connection, the ratio of saturated vapor and liquid can be determined via the opposite lever arms. At xa = 1 and xa = 0, the boiling and condensation curve intersect, i.e. for the pure components a and b, the boiling and condensation pressure are identical. From Gibbs phase rule in Eq. (2.60), it follows for binary mixtures with k = 2 that Zf = 2 + 2 − P h = 4 − P h. In the two-phase region with P h = 2, we obtain Zf = 4 − 2 = 2. Since a (p, x , y)-diagram as shown in Figure 2.5 is valid for one specific temperature T , for the number of independent state variables, we obtain Zf = 1. Hence, if the mole fraction xa of the liquid phase is specified, e.g. the pressure p and the mole fraction ya of the vapor phase cannot be selected as independent state variables any more but are fixed by the boiling and condensation curve. 2.3.4

(T , x , y )-Diagram

The top row in Figure 2.5 shows the boiling curves (solid) and condensation curves (dashed) for the same seven binary mixtures in (T , x , y)-diagrams, which are valid for a specific pressure p. Contrary to the correlations in Eqs. (2.164) and (2.165) for the boiling and condensation curves in a (p, x , y)-diagram, no explicit boiling and condensation curve correlations can be derived in a (T , x , y)-diagram. Hence, these curves have to be determined iteratively, similar to the iteration procedure illustrated in Examples 2.4 and 2.5. The boiling curve describes the state of the saturated liquid (for a given value of xa ), whereas the condensation curve describes the state of the saturated vapor (for a given value of ya ). The solid boiling curve always lies below the dashed condensation curve. It can be seen that T = Ta0 at xa = 1 and T = Tb0 at xa = 0, i.e. the boiling and condensation temperature of the pure components can be found at the right and left ordinate.

Figure 2.5 Phase diagrams of seven binary mixtures; top row: (T , x , y)-diagrams at p = const; middle row: McCabe–Thiele diagrams at p = const; bottom row: (p, x , y)-diagrams at T = const.

2.3 BINARY MIXTURES AND PHASE DIAGRAMS 85

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2 VAPOR–LIQUID EQUILIBRIUM

As in a (p, x , y)-diagram, the region between the boiling and condensation curve in a (T , x , y)-diagrams also represents the two-phase region with the mixture of saturated vapor and liquid. Again, the state of the saturated vapor is given by the condensation curve and the state of the saturated liquid by the boiling curve. On the horizontal connection between the boiling and condensation curve, the ratio of saturated vapor and liquid can be determined via the opposite lever arms. At xa = 1 and xa = 0, the boiling and condensation curve intersect, i.e. for the pure components a and b, the boiling and condensation temperature are identical. In the two-phase region with P h = 2, we obtain from Gibbs phase rule in Eq. (2.60) for the number of independent state variables Zf = 4 − 2 = 2. Since a (T , x , y)-diagram as shown in Figure 2.5 is valid for one specific pressure p, for the number of independent state variables, we obtain Zf = 1. Hence, if the mole fraction xa of the liquid phase is specified, e.g. the temperature T and the mole fraction ya of the vapor phase cannot be selected as independent state variables any more but are fixed by the boiling and condensation curve. 2.3.5

McCabe–Thiele Diagram

The middle row in Figure 2.5 shows (y, x )-diagrams, which can be derived, as indicated, from the boiling and condensation curves of the (T , x , y)-diagrams at a constant pressure. This diagram is also known as McCabe–Thiele diagram and is valid for a constant pressure17 . It would be possible to create McCabe–Thiele diagrams at constant temperature also. They are, however, not relevant for the application in distillation processes, commonly operated at approximately constant pressure. Hence, McCabe–Thiele diagrams are given at constant pressure like in the middle row of Figure 2.5. The difference ya − xa is of major importance to distillation, which can be clearly seen from the equilibrium curve in the McCabe–Thiele diagram. The difference increases if the relative volatility αab increases and the separation of the two components a and b with distillation is “easier” for a “wide boiling” mixture. This is the reason why αab is sometimes also labeled as separation factor. Along the equilibrium curve of a McCabe–Thiele diagram at constant pressure, the temperature T varies along the curve. However, the information of the different temperatures is “lost”, while the diagram is generated from the (T , x , y)-diagram. As described in Section 2.3.4, we have Zf = 1, which reflects that at the equilibrium curve only one of the mole fractions xa or ya can be specified independently. 2.3.6

Boiling and Condensation Behavior of Binary Mixtures

The phase behavior at moderate pressures of seven binary mixtures made up of components a and b is illustrated in Figure 2.5. The solid boiling and dashed condensation curves are based on calculations with the Margules equation. The diagrams 17

The constant pressure p, for which the given McCabe–Thiele diagrams are valid, is, for sake of clarity not specified here. It is identical to the pressure specified in the corresponding (T , x , y)-diagram in Figure 2.5.

2.3 BINARY MIXTURES AND PHASE DIAGRAMS

87

are similar to those shown by Mersmann 1980 and Mersmann et al. 2011. The parameters were adjusted to experimental VLE data. Significant differences in the behavior of the discussed binary mixtures at vapor–liquid equilibrium are due to the varying forces of interaction between the two components a and b. 2.3.6.1

Approximately Ideal Behavior

In the binary benzene/toluene mixture, shown in the center of Figure 2.5, the forces of interaction between the molecules a and b are the same as between molecules a and a or b and b, respectively. The molecule does not “realize” whether it collides with a molecule of the same or the other component. Such mixtures are close to ideal mixtures with γa ≈ 1 and γb ≈ 1 and can be described with Raoult’s law in Eq. (2.127). They have simple correlations for the boiling and condensation curve, as discussed in Section 2.3.8. 2.3.6.2

Positive Deviations from Raoult’s Law

If interactions between molecules of the low boiler a and the high boiler b are lower than between molecules of the same kind, a positive deviation from Raoult’s law is observed. Methanol/Water

Mixtures like methanol/water, in which the attraction between the molecules of component a and b in the liquid phase is weaker, sometimes turning into repulsion even, than the attraction between molecules of the same component, are shown on the left side of Figure 2.5. For both components, the partial pressures pa and pb increase, which leads to an increase of the boiling pressure p, as shown in the bottom row of Figure 2.5. This positive deviation from Raoult’s law with γa > 1 and γb > 1 results in a decrease of the boiling temperature T , as shown in the top row of Figure 2.5. Minimum Azeotrope of Diisopropyl Ether/Isopropanol

If the positive deviations from Raoult’s law are sufficiently large and the saturation vapor pressures p0a and p0b of the two components not too far apart, the curve of the boiling and condensation pressure at constant temperature may rise through a maximum at a certain mole fraction. This can be seen in Figure 2.5 for the diisopropyl ether/isopropanol mixture. The maximum of the boiling and condensation pressure causes a minimum of the boiling and condensation temperature, as shown in the top row in the (T , x , y)-diagram of this mixture, which is called a minimum azeotrope. At the minimum of the temperature, which is correlated to a maximum of the pressure, the boiling and condensation curve touch each other such that xa = ya : at the azeotrope, the mole fractions of the liquid and the vapor have the same value, and the equilibrium curve in the McCabe–Thiele diagram intersects with the diagonal. Left from the azeotrope, the boiling behavior is as expected with the mole fraction ya of the low boiler higher in the vapor than in the liquid: ya > xa . Right from the azeotrope, an inverse behavior with ya < xa can be observed.

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2 VAPOR–LIQUID EQUILIBRIUM

The diagram can be considered as split into two separate McCabe–Thiele diagrams left and right of the azeotrope. Here, the azeotropic mixture can be considered as the “low boiler” and the pure components a and b as high boiler, respectively. Miscibility Gap of Water/n-Butanol (Heteroazeotrope)

In the water/n-butanol mixture, the repulsive forces between the molecules a and b are so strong that a split into two immiscible liquid phases occurs for mole fractions approximately xa > 0.6. The dashed solubility lines in the (T , x , y)-diagram below the horizontal part of the boiling curve represent the two states of the two immiscible liquid phases. One liquid phase, represented by the right, over a wide range almost vertical dashed solubility line at xa ≈ 0.98 consists of almost pure water with a small amount of n-butanol miscible with water. The second liquid phase, represented by the left dashed solubility line, consists of water with a mole fraction of xa ≈ 0.6. For all mole fractions xa within the miscibility gap, the boiling curve is exactly horizontal, i.e. the mixture of the two immiscible liquids boils for all mole fractions at the same temperature, since only the ratios of the phases change and not the composition. The region where this phase splitting occurs, titled as miscibility gap, depends slightly on the temperature T , whereas the pressure p has almost no influence on the width of the miscibility gap, as shown with the vertical dashed solubility lines above the horizontal part of the boiling curve in the (p, x , y)-diagram in the bottom row. At the point where the condensation curve exhibits a minimum in the (T , x , y)diagram (and maximum in the (p, x , y)-diagram), it touches the horizontal boiling curve such that xa = ya , i.e. it behaves like an azeotrope. Therefore, a mixture like water/n-butanol with miscibility gap is also labeled as heteroazeotrope. The large positive deviation from Raoult’s law is reflected at very high values of the activity coefficients γa  1 and γb  1. In the (T , x , y)-diagram, the region above the solid boiling curve and below the dashed condensation curve represents a mixture that consists of saturated vapor and saturated liquid. The state of the saturated vapor is given by the left or right branch of the dashed condensation curve, depending on the mole fraction of the mixture. The state of the saturated liquid is given by the solid boiling curve, which is an almost vertical steep line for high values of xa at the right end of the (T , x , y)-diagram. This indicates that the saturated liquid consists of almost pure water18 . For intermediate values of approximately xa < 0.6, the boiling curve represents the state of a saturated liquid consisting of both water and n-butanol. In the (T , x , y)-diagram in Figure 2.6, the boiling and condensation curves are shown for three different pressures, along with the solubility lines describing the width of the temperature-dependent miscibility gap. It can be seen that the miscibility gap in the liquid disappears at higher temperatures, which is, however, possible only at higher pressures.

18

In Section 2.3.8 with the assumption of a complete miscibility gap, this part of the boiling curve will be vertical and identical to the right ordinate.

2.3 BINARY MIXTURES AND PHASE DIAGRAMS

89

Figure 2.6 (T , x , y)-diagram of the water/n-butanol mixture: boiling and condensation lines at different pressures; temperature dependent miscibility gap.

2.3.6.3

Negative Deviations from Raoult’s Law

If attractions between molecules of the low boiler a and the high boiler b are stronger than between molecules of the same kind, a negative deviation from Raoult’s law can be observed. Dichloromethane/Butanone

Binary mixtures like dichloromethane/butanone, shown on the right side of Figure 2.5, exhibit the opposite behavior as compared with the one described in Section 2.3.6.2: molecules of component a and b attract each other in the liquid phase to a higher extent than they attract the molecules of the same component. The attraction can be strong enough to cause the formation of complexes and initiation of chemical reactions. This leads to a decrease of the partial pressures pa and pb and thus to a decrease of the boiling pressure p, as shown in the bottom row. This negative deviation from Raoult’s law with γa < 1 and γb < 1 leads to an increase of the boiling temperature T , as shown in the top row of Figure 2.5. Maximum Azeotrope of Acetone/Chloroform

If the negative deviations from Raoult’s law are sufficiently large and the saturation vapor pressures p0a and p0b of the two components not too far apart, the curve of the boiling and condensation pressure at constant temperature may rise through a minimum at some mole fraction. This can be seen in Figure 2.5 for the acetone/chloroform mixture. The minimum of the boiling and condensation pressure causes a maximum of the boiling and condensation temperature, as shown in the top row in the (T , x , y)-diagram of this mixture, which is called a maximum azeotrope. At the maximum of the temperature, which is correlated to a minimum of the pressure, the boiling and condensation curve touch each other such that xa = ya : at the azeotrope, the mole fractions of the liquid and the vapor have the same value, and the equilibrium curve in the McCabe–Thiele diagram intersects with the diagonal. Right from

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2 VAPOR–LIQUID EQUILIBRIUM

the azeotrope, the boiling behavior is as expected with the mole fraction ya of the low boiler higher in the vapor than in the liquid: ya > xa . Left from the azeotrope, an inverse behavior with ya < xa can be observed. The diagram can be considered as split into two separate McCabe–Thiele diagrams left and right of the azeotrope with the azeotropic mixture as the “high boiler” and the pure components a and b as low boiler, respectively. Chemical Reaction of HNO3 /Water

In the mixture HNO3 /water, the attractive forces between the molecules a and b are so strong that the following chemical reaction occurs:

HNO3 + H2 O NO3 − + H3 O+ . The ionic reaction products NO3 – and H3 O+ are very high boiling components and can be treated as almost non-volatile. Therefore, the mixture shows a highly pronounced minimum of the boiling and condensation pressure, along with a highly pronounced maximum of the boiling temperature. Although other components, here NO3 – and H3 O+ , are present in the mixture due to the chemical reaction, it is still sufficient to consider only the two components HNO3 and H2 O. The strong interaction between the components HNO3 and H2 O results in very low values of the activity coefficients γa  1 and γb  1. 2.3.7

General Aspects of Azeotropic Mixtures

At an azeotrope, it follows from xa = ya and xb = yb such that ya ya Ka = and Kb = = 1. xa xb

(2.166)

The VLE ratios Ka and Kb can be calculated using the extended Raoult’s law according to Eq. (2.144) as follows:

Ka =

γa · p0a (T ) p

and

Kb =

γb · p0b (T ) . p

(2.167)

With Ka = Kb = 1, we obtain the following condition that has to be fulfilled at an azeotrope:

p0a (T ) γb = . p0b (T ) γa

(2.168)

If the boiling temperature of the low boiler a and the high boiler b are close together, i.e. p0a (T ) ≈ p0b (T ), and if the two components are chemically dissimilar, a mixture of a and b tends to form an azeotrope. Then, the activity coefficients are very large and have the same order of magnitude, i.e. γa ≈ γb . From Ka = Kb = 1 it follows for the relative volatility αab of the binary mixture that Ka αab = = 1. (2.169) Ka

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2.3 BINARY MIXTURES AND PHASE DIAGRAMS

Hence, at an azeotrope a separation of the components a and b with distillation is not possible, i.e. an azeotrope is a barrier for distillation.

Example 2.7: Estimation of Margules parameters from Experimental Vapor– Liquid Equilibrium Data For the azeotropic mixture of chloroform (low boiler, index a) and methanol (high boiler, index b), the following data from isothermal experimental measurements at the temperature T = 49.3 ◦C are available [Gmehling et al. 1994]:

p in mmHg 441.6 482.7 559.6 606.1 633.3 646.8 652.6 654.9 656.9 654.2 632.6 601.3

xa

ya

0.050 0.100 0.200 0.300 0.400 0.500 0.600 0.650 0.700 0.800 0.900 0.950

0.138 0.254 0.424 0.520 0.577 0.619 0.647 0.659 0.670 0.694 0.754 0.824

The Antoine equation and parameters as given in Example 2.6 shall be used to calculate the vapor saturation pressures p0a (T ) and p0b (T ) of the pure components chloroform a and methanol b. (1) Based on the given isothermal experimental measurements, draw a McCabe– Thiele diagram with the equilibrium curve at temperature T = 49.3 ◦C. Furthermore, draw a (p, x , y)-diagram at T = 49.3 ◦C with the boiling and condensation curve. Characterize the binary mixture of the low boiler chloroform and the high boiler methanol. (2) Check the given isothermal VLE data for thermodynamic consistency. (3) Show that the activity coefficients γa and γb are always thermodynamically consistent if they are calculated with the Margules equation according to Eqs. (2.132) and (2.133). E ˆ · T ) for all data (4) Calculate the dimensionless molar excess free energy gExp /(R points. (5) Draw a diagram that illustrates the values of the dimensionless molar excess free ˆ · T ) as a function of the mole fraction xa . With that diagram, energy g E /(R determine the coefficients Aab and Aba of the Margules equation.

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2 VAPOR–LIQUID EQUILIBRIUM

(6) Use the Margules equation with the parameters Aab and Aba from the previous ˆ · T ) as well task to calculate the dimensionless molar excess free energy g E /(R as the activity coefficients γa and γb for all data points. Calculate the sum of E ˆ ) and the theoretical squares ∆SgE between the experimental values gExp /(R·T values calculated with the Margules equation. (7) Derive two correlations that allow to calculate the parameters Aab and Aba of the Margules equation dependent on the activity coefficients γa and γb . (8) Use the correlations derived in the previous task to calculate the parameters Aab and Aba of the Margules equation. Apply these correlations to the data point which is closest to the azeotrope. (9) Use the Margules equation with the parameters Aab and Aba from the previˆ · T ) as ous task to calculate the dimensionless molar excess free energy g E /(R well as the activity coefficients γa and γb for all data points. Calculate sum of E ˆ · T ) and the theoretsquares ∆SgE between the experimental values gExp /(R ical values calculated with the Margules equation and compare with the results of Task (6). (10) Calculate the boiling pressure p at all data points using the Margules equation with the previously determined parameters Aab and Aba that yield the best fit between experimental and theoretical values. Furthermore, calculate the mole fraction ya of the vapor phase at all data points. Draw the equilibrium curve into the McCabe–Thiele diagram as well as the boiling and condensation curves into the (p, x , y)-diagram at T = 49.3 ◦C of Task (1). Compare these curves with the curves obtained from the experimental VLE data. Solution:

(1) The mixture of the low boiler chloroform a and the high boiler methanol b is an azeotropic mixture with a maximum of the boiling pressure and a minimum of the boiling temperature. Hence, it is labeled as a minimum azeotrope with the azeotrope close to the data point at xa = 0.65 and ya = 0.659 with the pressure p = 654.9 mmHg.

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93

(2) To check the given isothermal VLE data, the condition in Eq. (2.139) has to be fulfilled: Z xa =1 γa ln dxa = 0 . (1) γb xa =0 Hence, the activity coefficients γa and γb have to be determined from the given VLE data. From the extended Raoult’s law in Eq. (2.126), the correlations in Eq. (2.140) are obtained:

γa =

ya · p xa · p0a (T )

and

γb =

yb · p (1 − ya ) · p = . 0 xb · pb (T ) (1 − xa ) · p0b (T )

The isothermal VLE data is valid for the constant temperature T = 49.3 ◦C. Hence, the vapor saturation pressures p0a (T ) and p0b (T ) of the pure components a and b have to be calculated with the given Antoine parameters as follows:

p0a 1170.966 = 6.95465 − = 2.7048 mmHg 49.3 + 226.232 p0a /mmHg = 102.7048 = 506.76 , p0a = 506.76 mmHg log10

p0b 1582.271 = 8.08097 − = 2.6065 mmHg 49.3 + 239.726 p0b /mmHg = 102.6065 = 404.11 , p0b = 404.11 mmHg . log10

For the first experimental data point at the mole fractions xa = 0.05 and ya = 0.138 and the pressure p = 441.6 mmHg, we obtain the following activity coefficients:

ya · p 0.138 · 441.6 mmHg = = 2.405 xa · p0a (T ) 0.05 · 506.77 mmHg (1 − 0.138) · 441.6 mmHg (1 − ya ) · p γb = = = 0.992 . (1 − xa ) · p0b (T ) (1 − 0.05) · 404.11 mmHg γa =

For the integral in Eq. (1), the following value is required:

ln

γa 2.405 = ln = 0.886 . γb 0.992

Accordingly, we obtain the following table for all data points:

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2 VAPOR–LIQUID EQUILIBRIUM

p in mmHg 441.6 482.7 559.6 606.1 633.3 646.8 652.6 654.9 656.9 654.2 632.6 601.3

xa

ya

γa

γb

0.050 0.100 0.200 0.300 0.400 0.500 0.600 0.650 0.700 0.800 0.900 0.950

0.138 0.254 0.424 0.520 0.577 0.619 0.647 0.659 0.670 0.694 0.754 0.824

2.405 2.419 2.341 2.073 1.803 1.580 1.389 1.310 1.241 1.120 1.046 1.029

0.992 0.990 0.997 1.028 1.105 1.220 1.425 1.579 1.788 2.477 3.851 5.238

ln

γa γb

0.886 0.893 0.854 0.701 0.490 0.259 −0.026 −0.187 −0.365 −0.794 −1.303 −1.627

For the evaluation of Eq. (1), the following diagram can be used, which shows the value of ln(γa /γb ) as a function of the mole fraction xa .

At the left ordinate at xa = 0, which represents the state of pure component b, no data point is available, and the value of γa is unknown. Therefore, an extrapolation, indicated with the dashed line, is required. The same is valid for xa = 1 where the value of γb is unknown. It can be seen that the two indicated areas between the curve of ln(γa /γb ) and the abscissa are approximately the same, i.e. Area1 ≈ Area2 (with Area1 above and Area2 below the abscissa). This is equivalent to the fact that the integral in

2.3 BINARY MIXTURES AND PHASE DIAGRAMS

95

Eq. (1) is zero. According to Kang et al. 2010 the criteria

D = 100 ·

|Area1 − Area2 | y azeo a liquid phase with xa = 1 (pure low boiler a) is generated, and for mole fractions ya < y azeo , the generated liquid phase shows xa = 0 (pure high boiler b). The two vertical branches of the boiling curve can also be interpreted as follows: the boiling pressure increases already for a very small amount of component a in the almost pure component b from 1 to 3 bar. Vice versa, the boiling pressure increases already for a very small amount of component b in the almost pure component a from 2 to 3 bar. If the Antoine equation in Eq. (2.170) is inserted into the correlations of the right and left branch of the condensation curve in Eqs. (2.176) and (2.178), it follows that

T / ◦C =

Ba − Ca Aa − log10 (ya · p/mbar)

(2.180)

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109

and

T / ◦C =

Bb − Cb . Ab − log10 ((1 − ya ) · p/mbar)

(2.181)

These two explicit correlations for the two branches of the condensation curve are depicted in the left (T , x , y)-diagram of Figure 2.8. For ya = 1 and ya = 0, the two correlations in Eqs. (2.180) and (2.181) simplify to the Antoine equation in Eq. (2.171), and we obtain T = 70 ◦C and T = 100 ◦C, respectively. After inserting the Antoine equation in Eq. (2.170) into the boiling curve correlation p = p0a (T ) + p0b (T ) in Eq. (2.174), we obtain the following correlation:     Ba Bb Aa − A − b T /◦C + Ca + 10 T /◦C + Cb p/mbar = 10 (2.182) This equation cannot be rewritten to solve for an explicit correlation T = T (p). A numerical solution is required to determine the boiling temperature. For p = 1 bar a solution of Eq. (2.182) is T = 60.359 ◦C, which is drawn as the horizontal solid boiling curve in the left (T , x , y)-diagram of Figure 2.8 and labeled as T azeo . From the saturation vapor pressure curves in Figure 2.7, a graphical solution of T azeo can be read off at the point where the values of the two curves p0a (T ) and p0b (T ) add up to p = 1 bar. For the limiting case of a complete miscibility gap, a heteroazeotrope exhibits this characteristic minimum of the boiling temperature, which can be also observed for real mixtures with partial immiscibility of the two components like water/n-butanol in Figures 2.5 and 2.6. The correlation in Eq. (2.176) of the right branch of the condensation curve can be used to determine the mole fraction yaazeo at the azeotrope:

yaazeo =

p0a (T azeo ) . p

(2.183)

From the correlation in Eq. (2.178) of the left branch of the condensation curve, we obtain

ybazeo = 1 − yaazeo =

p0b (T azeo ) . p

(2.184)

Both correlations in Eqs. (2.183) and (2.184) yield the same result yaazeo = 0.776, which is shown as a dotted vertical line in the left (T , x , y)-diagram of Figure 2.8. This result can also be obtained from the intersection of the two branches of the condensation curve by equating Eqs. (2.180) and (2.181) and a numerical solution for yaazeo . In order to connect the condensation curve with the boiling curve at boiling temperatures of the pure components, the boiling curve is drawn as vertical lines at xa = 0 and xa = 1. This reflects the behavior that for mole fractions ya > yaazeo a liquid phase with xa = 1 (pure low boiler a) is generated, and for mole fractions ya < yaazeo , the generated liquid phase shows xa = 0 (pure high boiler b).

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2 VAPOR–LIQUID EQUILIBRIUM

The two vertical branches of the boiling curve can also be interpreted as follows: the boiling temperature decreases already for a very small amount of component a in almost pure component b from 100 to 60.359 ◦C. Vice versa, the boiling temperature decreases already for a very small amount of component b in almost pure component a from 70 to 60.359 ◦C. The left McCabe–Thiele diagram of Figure 2.8 at p = 1 bar exhibits a horizontal line at yaazeo = 0.776 and two vertical lines at xa = 0 and xa = 1. The horizontal line reflects that a vapor phase with the mole fraction yaazeo is generated for all mole fractions 0 < xa < 1 in the liquid phase. The vertical line at xa = 0 indicates that we obtain a liquid phase with xa = 0 (xb = 1) for all mole fractions ya < yaazeo in the vapor phase, i.e. a liquid phase of pure high boiler b. The vertical line at xa = 1 reflects that for all mole fractions ya > yaazeo , a liquid phase with xa = 1 is generated, i.e. a liquid phase of pure low boiler a. For the two components a and b, the extended Raoult’s law in Eq. (2.126) can be used to calculate the activity coefficients as follows: pa pb pb γa = and γb = = . (2.185) 0 0 xa · pa (T ) xb · pb (T ) (1 − xa ) · p0b (T ) With pa = p0a (T ) and pb = p0b (T ), this simplifies to

γa =

1 xa

and

γb =

1 1 = . xb 1 − xa

(2.186)

The activity coefficients γa and γb of the two components vary from γa = 1 and γb = 1, for pure components at xa = 1 and xb = 1, to γa = γa∞ → ∞ and γb = γb∞ → ∞, for xa = 0 and xb = 0. This is illustrated for γa and γb in the fourth row of the left diagrams of Figure 2.8. The limiting case of a complete miscibility gap represents the highest possible deviation from Raoult’s with the highest possible activity coefficients γa → ∞ and γb → ∞. All real binary mixtures with positive deviation from Raoult’s law have to be located somewhere between the two limiting cases ideal mixture and complete miscibility gap. In real mixtures there is always at least a small mutual solubility of the components a and b, and, therefore, the activity coefficients γa∞ and γa∞ at infinite dilution, i.e. at xa → 0 and xb → 0, are finite. 2.3.8.3

Complete Chemical Reaction to a Non-Volatile Component

The third limiting case for boiling and condensation behavior of binary mixtures is the complete chemical reaction of low boiler a and high boiler b in the liquid phase to a third component c as follows:

a + νb b → c . Here, νb is a generalized stoichiometric coefficient for component b reacting with one molecule of component a to one molecule of component c, which could also be labeled with molecular formula abνb . Furthermore, it is assumed that the liquid reaction product c is non-volatile with a very high boiling temperature:

p0c (T ) → 0 and Tc0 (p) → ∞ .

(2.187)

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111

The boiling behavior of such a mixture can be interpreted as a very strong negative deviation from Raoult’s law for components a and b. This is illustrated in the right diagrams of Figure 2.8. If the two components a and b are present in the liquid mixture at the stoichiometric ratio, they can react completely to the non-volatile component c, and the pressure above the liquid approaches zero. This minimum point of the boiling pressure is shown in the right (p, x , y)-diagram of Figure 2.8 at the mole fraction xazeo , which a can be determined as

xazeo = a

1 . 1 + νb

(2.188)

The labeling xazeo indicates that at this stoichiometric point the mole fractions in the a vapor and liquid phase are identical. For the example shown in Figure 2.8, the value of νb = 3 is selected such that xazeo = 0.25. If the mole fraction of component a a in the liquid is xa = 0.25, three times more molecules of component b are present, i.e. xb = 1 − xa = 0.75. Hence, with νb = 3 all molecules of component a react with all molecules of component b and only molecules of the reaction product c are present in the liquid phase. Since component c is assumed to be non-volatile, the pressure above the liquid phase equals to p = 0. For mole fractions xa > xazeo = 0.25, more of component a is present in the a mixture than required for the chemical reaction with component b. Therefore, besides the reaction product c, component a exists in the liquid phase. Hence, a binary mixture of the components a and c exists, which can be described with the transformed mole fraction

x ˜a =

xa − xazeo a 1 − xazeo a

for

xa > xazeo .

(2.189)

The boiling behavior of the liquid mixture of volatile component a and non-volatile component c can be described with p = pa and a modified form of extended the Raoult’s law:

x ˜a · γ˜a · p0a (T ) = pa = p .

(2.190)

A further assumption is required to determine the activity coefficient γ ˜a . Here, for sake of simplicity, it is assumed that γ ˜a = 1. Any other correlation, as described in Section 2.2.12, could also be used here without affecting the conclusions described below in this section. Thus, Eq. (2.190) simplifies to the linear correlation

x ˜a · p0a (T ) = pa = p .

(2.191)

This is illustrated in the right (p, x , y)-diagram of Figure 2.8 as the right branch of the solid linear boiling curve for mole fractions xazeo ≤ xa ≤ 1, which is equivalent a to 0 ≤ x ˜a ≤ 1. For xa = x ˜a = 1, it follows that p = pa = p0a (T ) = 2 bar, whereas for xa = xazeo , equivalent to x ˜a = 0, we obtain p = pa = 0. a For mole fractions xa < xazeo = 0.25 , more of component b is present in the a mixture than required for the chemical reaction with component a. Therefore, besides the reaction product c component b exists in the liquid phase such that we have

112

2 VAPOR–LIQUID EQUILIBRIUM

a binary mixture of the components b and c. This mixture can be described with the transformed mole fraction

x ˜b =

xazeo − xa a xazeo a

for

xa < xazeo .

(2.192)

The boiling behavior of the liquid mixture of volatile component b and non-volatile component c can be described with p = pb and a modified form of the extended Raoult’s law:

x ˜b · γ˜b · p0b (T ) = pb = p .

(2.193)

As described above, for component a in a liquid mixture with reaction product c, for component b, it is assumed that γ ˜b = 1 such that

x ˜b · p0b (T ) = pb = p .

(2.194)

This is illustrated in the right (p, x , y)-diagram of Figure 2.8 as the left branch of the solid linear boiling curve for mole fractions 0 ≤ xa ≤ xazeo , which is equivalent a to 1 ≥ x ˜b ≥ 0. For xa = 0, which is equivalent to x ˜b = 1, it follows that p = pb = p0b (T ) = 1 bar, whereas for xa = xazeo , equivalent to x ˜b = 0, we a obtain p = pb = 0. For the condensation curve it has to be considered that for all mole fractions in the range xa > xazeo , the vapor phase consists only of not reacted pure component a, a which results in the dashed vertical right branch of the condensation curve at xa = 1 in the right (p, x , y)-diagram of Figure 2.8. For mole fractions xa < xazeo , the a vapor phase consists only of not reacted pure component b, and the dashed condensation curve can be seen at xa = 1, equivalent to xb = 1. In order to connect the two branches of the condensation curve, a horizontal dashed line is drawn into the right (p, x , y)-diagram of Figure 2.8 at p = 0. This reflects the behavior that either pure component a or pure component b can be vaporized completely. If, however, a slight amount of the other component is present in the mixture, the reaction to the non-volatile component c occurs, a complete vaporization of the mixture for pressures p > 0 is not possible. If the Antoine equation in Eq. (2.170) is inserted into the correlations describing the right and left branch of the boiling curve in Eqs. (2.191) and (2.194), we obtain

T / ◦C =

Ba Aa − log10

p/mbar x ˜a

− Ca

(2.195)

− Cb .

(2.196)

and

T / ◦C =

Bb Ab − log10

p/mbar x ˜b

These two explicit correlations describing the two branches of the boiling curve are depicted in the right (T , x , y)-diagram of Figure 2.8. Both correlations in

2.3 BINARY MIXTURES AND PHASE DIAGRAMS

113

Eqs. (2.195) and (2.196) simplify for xa = x ˜a = 1 (pure component a with x ˜b = 1) and xa = 0 (pure component b) to the Antoine equations in Eq. (2.171), and we obtain T = 70 ◦C and T = 100 ◦C, respectively. For x ˜a = x ˜b = 0, which is equivalent to xa = xazeo , the correlations in a Eqs. (2.195) and (2.196) are undefined:

T (xa → xazeo ) = ∞. a

(2.197)

This is shown in the right (T , x , y)-diagram of Figure 2.8. For the limiting case of a complete chemical reaction to a non-volatile component c, the mixture exhibits a strong maximum of the boiling temperature, which can be observed also for real mixtures like HNO3 /water. Here, the attractive forces between the molecules HNO3 and water are so strong that a chemical reaction occurs and a maximum of the boiling temperature can be observed, as shown in Figure 2.5. For the condensation curve it has to be considered that for all mole fractions in the range xa > xazeo , the vapor phase consists only of not reacted pure component a, a which results in the dashed vertical right branch of the condensation curve at xa = 1 in the right (T , x , y)-diagram of Figure 2.8. For mole fractions xa < xazeo , the a vapor phase consists only of not reacted pure component b, and the dashed condensation curve can be seen at xa = 0, equivalent to xb = 1. The connection of the two branches of the condensation curve cannot be illustrated in the right (T , x , y)diagram of Figure 2.8 because this curve would be at T → ∞. This reflects the behavior that either pure component a or pure component b can be vaporized completely. If, however, a slight amount of the other component is present in the mixture, the reaction to the non-volatile component c occurs and the mixture cannot be vaporized completely. The right McCabe–Thiele diagram of Figure 2.8 at p = 1 bar for the limiting case of a complete chemical reaction of components a and b to a non-volatile component c exhibits a vertical line at xazeo = 0.25 and two horizontal lines at ya = 0 and a ya = 1. The vertical line reflects that for all mole fractions 0 < ya < 1 in the vapor phase, a liquid phase with the mole fraction xazeo is generated. The horizontal line a at ya = 0 indicates that for all mole fractions xa < xazeo in the liquid phase, a vapor a phase with ya = 0 (yb = 1), i.e. a vapor phase of pure component b, is obtained. The horizontal line at ya = 1 reflects that for all mole fractions xa > xazeo , a vapor a phase with ya = 1 is generated, i.e. a vapor phase of pure component a. For the two components a and b, extended Raoult’s law in Eq. (2.126) can be used to calculate the activity coefficients as follows:

γa =

ya · p xa · p0a (T )

and

γb =

yb · p . xb · p0b (T )

(2.198)

For xa > xazeo we have ya = 1 and yb = 0. From Eq. (2.198) it follows that a γb = 0. Furthermore, with p = x ˜a · p0a (T ) from Eq. (2.191), we obtain

γa =

x ˜a · p0a (T ) x ˜a = . xa · p0a (T ) xa

(2.199)

114

2 VAPOR–LIQUID EQUILIBRIUM

Replacing the transformed mole fraction x ˜a from Eq. (2.189) yields

γa =

1 xa − xazeo a · xa 1 − xazeo a

for

xa > xazeo . a

(2.200)

For xa = 1 we obtain γa = 1; for xa = xazeo it follows that γa = 0, which a is formally also valid for xa < xazeo , although no component a is present in the a mixture. For xa < xazeo we have ya = 0 and yb = 1. From Eq. (2.198) it follows that a γa = 0. Furthermore, with p = x ˜b · p0b (T ) from Eq. (2.194), we obtain

γb =

x ˜b · p0b (T ) x ˜b = . 0 xb · pb (T ) xb

(2.201)

Replacing the transformed mole fraction x ˜b from Eq. (2.192) yields

γb =

1 xazeo − xa 1 xazeo − xa · a azeo = · a azeo . xb xa 1 − xa xa

(2.202)

For xa = 0 we obtain γb = 1; for xa = xazeo it follows that γb = 0, which a is formally also valid for xa > xazeo , although no component b is present in the a mixture. The activity coefficients γa and γb of the two components vary from 1 (pure components at xa = 1 and xb = 1) to γa = γb = 0 at xa = xazeo . This is illustrated in a the fourth row of the right diagrams of Figure 2.8. The limiting case of a complete chemical reaction of component a and b to a non-volatile component c represents the highest negative deviation from Raoult’s for the lowest possible activity coefficients γa = γb = 0. All real binary mixtures with negative deviation from Raoult’s law have to be located somewhere between the two limiting cases: ideal mixture and complete chemical reaction.

2.4

Ternary Mixtures

For the understanding of the vapor–liquid equilibria of ternary mixtures, knowledge of the boiling and condensation behavior is required. In the following, this is mainly illustrated with boiling surfaces and distillation lines at constant pressure p. Additionally, the principle of condensation (dew point) surfaces is introduced. 2.4.1

Boiling and Condensation Conditions of Ternary Mixtures

The boiling condition in Eq. (2.160) can be written for ternary mixtures with low boiler a, intermediate boiler b, and high boiler c as follows:

p = xa · γa · p0a (T ) + xb · γb · p0b (T ) + xc · γc · p0c (T ) .

(2.203)

From the condensation condition Eq. (2.162), we obtain for a ternary mixture

1 ya yb yc = + + . p γa · p0a (T ) γb · p0b (T ) γc · p0c (T )

(2.204)

115

2.4 TERNARY MIXTURES

Furthermore, the extended Raoult’s law has to be fulfilled for all three components. With Eqs. (2.143) and (2.144), we can write

ya = xa · Ka yb = xb · Kb yc = xc · Kc

γa · p0a (T ) p γb · p0b (T ) with Kb = p γc · p0c (T ) with Kc = . p with

Ka =

(2.205) (2.206) (2.207)

In Table 2.3, the different cases (ideal mixtures and different dependence of the activity coefficients γi ) for calculating boiling and condensation pressure and temperature for ternary mixtures are summarized. Either the mole fractions xi of the liquid phase (boiling) or the mole fractions yi of the vapor phase (condensation) are assumed to be given. As shown in Sections 2.4.2 – 2.4.4, the boiling and condensation temperatures for ternary mixtures can be depicted as surfaces in three-dimensional triangular phase diagrams at a constant pressure p. Table 2.3 Calculation of boiling and condensation pressure p and temperature T .

Calculation of Boiling pressure p Eq. (2.203) Boiling temperature T Eq. (2.203) Condensation pressure p Eq. (2.204) Condensation temperature T Eq. (2.204)

Ideal mixtures

γi (xi )

γi (xi , T )

γi (xi , T, p)

No iteration (*)

No iteration (*)

No iteration (*)

Iteration (*)

Iteration (*)

Iteration (*)

Iteration (*)

Iteration (*)

No iteration (**)

Iteration (***)

Iteration (***)

Iteration (***)

Iteration (**)

Iteration (***)

Iteration (***)

Iteration (***)

(*) independent calculation of yi with Eqs. (2.205) – (2.207) (**) independent calculation of xi with Eqs. (2.205) – (2.207) (***) iteration of xi with Eqs. (2.205) – (2.207)

116

2.4.2

2 VAPOR–LIQUID EQUILIBRIUM

Triangular Diagrams

In a triangular diagram, as shown in Figure 2.9, the corners of the triangle represent the states of the pure components a (low boiler), b (intermediate boiler), and c (high boiler). The three sides of the triangle represent the three binary mixtures a/b, a/c, and b/c. Points inside the triangle represent mixtures that contain all three components a, b, and c. In literature, several ways to label the axis of triangular diagrams can be found. Here, two arrows are attached to the triangular diagram’s axes in order to illustrate the increasing mole fractions of the low boiler a and the intermediate boiler b. In two-dimensional triangular diagrams, as shown in Figure 2.11, the low boiler a is at the top corner of the triangle, the intermediate boiler b at the bottom right corner, and the high boiler c at the bottom left corner (clockwise sequence). 2.4.3

Boiling Surfaces

In Figure 2.9, three-dimensional plots of the boiling surfaces of eight ternary mixtures are shown. In these diagrams, the boiling temperatures T at constant pressures p are depicted as functions of the mole fractions xa , xb , and xc of the liquid phase. To determine the boiling temperature T , an iterative solution of the boiling condition in Eq. (2.203) is required. Most of the mixtures shown in Figure 2.9 are calculated using the Wilson equation with the parameters given by Gmehling et al. 1977 (see Section 2.2.12). From Gibbs phase rule in Eq. (2.60), it follows for ternary mixtures with k = 3 that Zf = 2 + 3 − P h = 5 − P h. At the boiling surface we have P h = 2 such that Zf = 5−2 = 3. Since the diagrams with the boiling surfaces in Figure 2.9 are valid for one specific pressure p, we obtain Zf = 2 for the number of independent state variables. Hence, if the mole fractions xa and xb of the liquid phase are specified, the temperature T cannot be selected as an independent state variable anymore but is fixed by the boiling surface. The boiling surface of the nitrogen/argon/oxygen mixture is shown in Figure 2.9A. At the three sides of the triangle, the well-known behavior of the three binary mixtures can be seen. The ternary mixture exhibits a boiling surface similar to the flank of a mountain. The highest point, i.e. the peak, lies at the pure high boiler c (oxygen at −183.0 ◦C) and the lowest point, i.e. the sink, at the low boiler a (nitrogen at −195.9 ◦C)19 . The intermediate boiler b (argon at −185.9 ◦C) forms a point somewhere in between. The boiling surface of the ternary benzene/cyclohexane/heptane mixture is shown in Figure 2.9B. The low boiler a (benzene at 80.2 ◦C) and the intermediate boiler b (cyclohexane at 80.8 ◦C) form a minimum azeotrope at 77.6 ◦C, and therefore, the boiling curve of that binary mixture exhibits a temperature minimum. Consequently, there exists a valley in the boiling surface of the ternary mixture that progresses 19

A peak is defined as a point where the temperature T is decreasing in every direction from the peak. Analogously, the temperature T is increasing in every direction from a sink.

2.4 TERNARY MIXTURES

117

Figure 2.9 Boiling surfaces of eight different ternary mixtures at p = 1 bar as a function of the mole fractions xa , xb , and xc of the liquid phase; o: azeotropes.

118

2 VAPOR–LIQUID EQUILIBRIUM

Figure 2.9 (Continued) Boiling surfaces of eight different ternary mixtures at p = 1 bar as a function of the mole fractions xa , xb , and xc of the liquid phase; o: azeotropes.

2.4 TERNARY MIXTURES

119

Figure 2.9 (Continued) Boiling surfaces of eight different ternary mixtures at p = 1 bar as a function of the mole fractions xa , xb , and xc of the liquid phase; o: azeotropes.

120

2 VAPOR–LIQUID EQUILIBRIUM

Figure 2.9 (Continued) Boiling surfaces of eight different ternary mixtures at p = 1 bar as a function of the mole fractions xa , xb , and xc of the liquid phase; o: azeotropes.

2.4 TERNARY MIXTURES

121

upward from the azeotrope to the high boiler c (heptane at 98.0 ◦C). The flanks on both sides of that valley have common highest and lowest point. This fact is of great importance, as will be shown in Section 4.3.2. A significantly different situation is encountered if a minimum azeotrope is formed by the intermediate boiler b and the high boiler c. This is shown for the tetrachloromethane/benzene/cyclohexane mixture in Figure 2.9C. Here, a valley progresses downward from the azeotrope at 77.6 ◦C to the low boiler a (tetrachloromethane at 76.8 ◦C). The flanks on both sides of the valley have different peaks, one at the intermediate boiler b (benzene at 80.2 ◦C) and one at the high boiler c (cyclohexane at 80.8 ◦C). This is an important difference to the mixture shown in Figure 2.9B. The boiling behavior of the ternary octane/2-ethoxyethanol/ethylbenzene mixture is illustrated in Figure 2.9D. Here, two binary mixtures form minimum azeotropes connected by a valley, which progresses upward from 116.1 to 127.1 ◦C. The flanks of this valley have different peaks such that the behavior is similar to the one shown in Figure 2.9C. The boiling surface of the ternary benzene/cyclohexane/isopropanol mixture is illustrated in Figure 2.9E. In this mixture, three binary minimum azeotropes are formed. Additionally, a ternary azeotrope, which has the lowest boiling temperature of the whole mixture at 69.1 ◦C, exists. Hence, the boiling surface of the mixture exhibits three valleys, each of them progressing from a binary azeotrope downward toward the ternary azeotrope. The three valleys divide the mixture into three regions with different peaks. In the n-propanol/water/n-butanol mixture, a partial miscibility gap is formed such that the mixture splits into two liquid phases. The boiling surface is depicted in Figure 2.9F with the miscibility gap as shaded area. In addition to the heteroazeotrope in the two-liquid phase region, a minimum azeotrope at 87.7 ◦C is formed in the binary mixture of the low boiler a (n-propanol) and the intermediate boiler b (water). A valley progresses between the two azeotropes, which is surrounded by two flanks with different peaks. The valley becomes wider and has a flat part within the miscibility gap at 92.7 ◦C. The boiling surface of the acetone/chloroform/acetonitrile mixture is shown in Figure 2.9G. Here, the binary mixture of a (acetone) and b (chloroform) form a maximum azeotrope at 64.4 ◦C. In the boiling surface a ridge exists that progresses from the high boiler c (acetonitrile at 81.6 ◦C) downward toward the maximum azeotrope. Hence, the ternary mixture is divided into two regions by this ridge. These regions have one common peak but different minima at the low boiler a (acetone at 56.2 ◦C) and the intermediate boiler b (chloroform at 61.2 ◦C). The highly complex acetone/chloroform/methanol mixture, also treated in Example 2.6, is shown in Figure 2.9H. Here, two binary minimum azeotropes (a/c at 55.7 ◦C and b/c at 53.4 ◦C) and a binary maximum azeotrope (a/b at 64.4 ◦C) are formed. On the boiling surface, a valley exists, which progresses from one binary minimum azeotrope to the other. Additionally, a ridge exists, which connects the high boiler c (methanol at 64.7 ◦C) with the maximum azeotrope. At the intersection point of valley and ridge, a ternary azeotrope is formed at 57.5 ◦C, which is referred

122

2 VAPOR–LIQUID EQUILIBRIUM

to as saddle azeotrope. From the three-dimensional plots in Figure 2.9, it can be seen clearly that the boiling surfaces of ternary mixtures can vary to a high extent. Binary azeotropes always result in the formation of valleys or ridges in the boiling surface. These valleys and ridges are of highest importance for distillation processes in regions surrounded by flanks with different boiling peaks or sinks (see Section 4.3.2). 2.4.4

Condensation Surfaces

A condensation surface can be plotted in a similar way as a boiling surface. To determine the condensation temperature T , an iterative solution of the condensation condition in Eq. (2.204), coupled with Eqs. (2.205) – (2.207), is required. The condensation temperatures T at a constant pressure p are depicted as functions of the mole fractions ya , yb , and yc of the vapor phase, which is shown exemplary for the acetone/chloroform/methanol mixture in Figure 2.10. Here, the condensation surface and the visible part of the related boiling surface from Figure 2.9H can be seen. The azeotropes of the three binary mixtures can be seen at the three sides of the triangle. The condensation surface touches the boiling surface at the pure components a (acetone), b (chloroform), and c (methanol) as well as at all azeotropes. At the valleys and ridges, the distance between condensation and boiling surfaces reaches a minimum but is not equal to zero.

Figure 2.10 Boiling and condensation surface of the acetone/chloroform/methanol mixture at p = 1 bar; boiling surface as a function of the mole fractions xa , xb , and xc of the liquid phase; condensation surface as a function the mole fractions ya , yb , and yc of the vapor phase; o: azeotropes.

The region between the boiling and condensation surface represents the two-phase

123

2.4 TERNARY MIXTURES

region, where saturated liquid is in equilibrium with saturated vapor. Here, we have P h = 2. From Gibbs phase rule in Eq. (2.60) with k = 3, it follows for the number of independent state variables that Zf = 5 − 2 = 3. Since the diagram with the boiling and condensation surfaces, as shown in Figure 2.10, is valid for one specific pressure p, we obtain Zf = 2 for the number of independent state variables. Hence, if the mole fractions xa and xb of the liquid phase are specified, e.g. the temperature T and the mole fractions of the vapor phase ya and yb cannot be treated as independent state variables anymore but are determined by the boiling and condensation surfaces. However, in the diagram in Figure 2.10, no information is included, which describes which point of the boiling surface with xa and xb is in VLE with which point of the condensation surface with ya and yb . Hence, other methods to describe the vapor–liquid equilibrium are required. This will be shown with the concept of distillation lines in Sections 2.4.5 and 2.4.6. 2.4.5

Derivation of Distillation Lines

In this section, the method for describing the VLE of ternary mixtures with distillation lines is presented. First, the derivation is generalized for multicomponent mixtures by applying Eq. (2.154), in which the relative volatilities αij are used to describe the correlation between the mole fractions xi and yi of the liquid and vapor phase. Distillation lines are based on the principle of consecutive multi-step partial evaporation of a first vapor bubble, followed by a complete condensation of said vapor bubble. 2.4.5.1

Multicomponent Mixtures (n=0)

A liquid multicomponent mixture with the mole fractions xi of the components i = 1, 2, . . . , k is partially evaporated in order to generate a first vapor bubble. (n=0) This vapor bubble with the mole fractions yi is in VLE with the liquid mixture such that Eq. (2.154) can be used to determine the mole fraction of the vapor20 : (n=0)

(n=0)

yi

αij · xi

= 1+

Pk

l=1 (αlj

(n=0)

− 1) · xl

.

(2.208)

(n=0)

If this vapor with the mole fractions yi is condensed completely, a liquid mixture (n=1) (n=0) with the mole fractions xi = yi is obtained: (n=0)

(n=1)

xi

αij · xi

= 1+

Pk

l=1 (αlj

(n=0)

− 1) · xl

.

(2.209)

This liquid mixture is once again evaporated in order to generate a first vapor bubble (n=1) with the mole fractions yi in VLE with the liquid. Again, Eq. (2.154) can be 20

For Eq. (2.208) and all similar successive equations, the correlation αjj = 1 is valid.

124

2 VAPOR–LIQUID EQUILIBRIUM

used to determine the mole fractions as (n=1)

(n=1)

yi

αij · xi

= 1+

Pk

l=1 (αlj

.

(n=1)

− 1) · xl

(2.210)

After inserting Eq. (2.209), we obtain the following correlation: (n=0)

αij · (n=1)

yi

=

αij · xi 1+

(n=0)

Pk

− 1) · xl

l=1 (αlj

(n=0)

1+

αlj · xl

Pk

l=1 (αlj − 1) ·

1+

Pk

m=1 (αmj

(n=0)

− 1) · xm

(n=0)

= 1+

αij · αij · xi P (n=0) (n=0) + kl=1 (αlj − 1) · αlj · xl l=1 (αlj − 1) · xl

Pk

(n=0)

2 αij · xi

= 1+

Pk

l=1 (αlj

(n=0)

2 −α )·x − 1 + αlj lj l (n=0)

2 αij · xi

= 1+

Pk

2 l=1 (αlj

(n=0)

− 1) · xl

.

(2.211)

This procedure described above is repeated: if the vapor with the mole frac(n=1) tions yi is condensed completely, a liquid mixture is obtained, which has (n=2) (n=1) the mole fractions xi = yi , and with Eq. (2.211), it follows that (n=0)

(n=2)

xi

2 αij · xi

= 1+

Pk

2 l=1 (αlj

(n=0)

− 1) · xl

.

(2.212)

This liquid mixture is once again evaporated in order to generate a first vapor bubble (n=2) with the mole fractions yi in VLE with the liquid. Again, Eq. (2.154) can be used to determine the mole fractions as (n=2)

(n=2)

yi

αij · xi

= 1+

Pk

l=1 (αlj

(n=2)

− 1) · xl

.

(2.213)

After inserting Eq. (2.212), we obtain, analogously to Eq. (2.211), the correlation (n=0)

(n=2) yi

3 αij · xi

= 1+

Pk

3 l=1 (αlj

(n=0)

− 1) · xl

.

(2.214) (n=3)

After complete condensation of this vapor mixture, it follows from xi that

(n=2)

= yi

(n=0)

(n=3) xi

3 αij · xi

= 1+

Pk

3 l=1 (αlj

(n=0)

− 1) · xl

.

(2.215)

125

2.4 TERNARY MIXTURES

Repeating this procedure of partial evaporation to get a first vapor bubble followed by a total condensation for n steps finally yields (n=0)

(n) xi

n αij · xi

= 1+

Pk

n l=1 (αlj

(n=0)

− 1) · xl

(n=0)

.

(2.216) (n=1)

(n=2)

(n=3)

Hence, a series of liquid states xi , xi , xi , xi , . . . is finally obtained; the mole fraction difference between two consecutive steps represents an equilibrium step. This procedure leads to a step-by-step enrichment of the more volatile component, which is accompanied by a decrease in boiling temperature. 2.4.5.2

Ternary Mixtures

For a ternary mixture with j = c and αcc = 1, we obtain the following correlations for the mole fractions xa , xb , and xc from Eq. (2.216): (n=0)

x(n) a =

n αac · xa (n=0)

n − 1) · x 1 + (αac a

(n=0)

(2.217)

(n=0)

(2.218)

n − 1) · x + (αbc b (n=0)

(n)

xb

=

n αbc · xb (n=0)

n − 1) · x 1 + (αac a

n − 1) · x + (αbc b

(n=0)

x(n) = c

xc

(n=0)

n − 1) · x 1 + (αac a

(n=0)

(n=0)

(n=0)

n − 1) · x + (αbc b (n=0)

.

(2.219)

= 1 − xa − xb , it can be seen that the closing condition (n) + + xc = 1 is also fulfilled after n steps. Hence, the closing condition can be used instead of Eq. (2.219). Alternatively, similar correlations can be obtained for j = b and j = a. (n=0) (n=1) (n=2) (n=3) The series of liquid states xi , xi , xi , xi , . . . for i = a, b, c can be plotted into a two-dimensional triangular diagram and joined together to form the (n=0) distillation line, which starts at xi . The distillation lines for mixtures with constant relative volatilities αab and αbc can easily be calculated by using Eqs. (2.217) – (2.219), which is illustrated in Example 2.8. For mixtures where the relative volatilities are not constant, the correlations in Eqs. (2.217) – (2.219) still can be used; however, at each step the changing values of αab and αbc have to be calculated with Eq. (2.148) for non-ideal mixtures or Eq. (2.149) for ideal mixtures with γi = 1. As shown in the triangular diagram of Example 2.8, an arrow that shows the direction of the decreasing boiling and condensation temperatures has to be added to the points on distillation lines. These points on distillations lines represent the different steps of the consecutive partial evaporation, generating a first vapor bubble that is completely condensed afterwards. Hence, the points on distillation lines describe the alternating equilibrium steps between states on the boiling and condensation surface of the ternary mixture, and the distances between the points show the

With xc (n) xa

(n) xb

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2 VAPOR–LIQUID EQUILIBRIUM

difference yi − xi of the mole fraction in the vapor and liquid phase. This difference increases if the relative volatility αij increases. Consequently, the separation of the three components a, b, and c with distillation is “easier” since it is a more “wide boiling” mixture. Hence, for ternary mixtures, distillation lines with points have the same information content about the VLE as equilibrium curves of a binary mixture in a McCabe–Thiele diagram as shown in Figure 2.5. In rough approximation, distillation lines follow the path of a rolling ball on the boiling or condensation surface21 . Distillation lines always start at a peak and end at a sink of the boiling surface. The direction of decreasing boiling and condensation temperatures is conveniently marked by an arrow. As shown in Example 2.8, the correlations in Eqs. (2.217) – (2.219) can be applied also for negative values of n to (n) (n=0) determine VLE values for mole fractions of xa < xa . Furthermore, in order to get a smooth curve for the distillation line, not only whole numbers for n are used but all real numbers. Then, however, it has to be considered that two adjacent points do not represent one equilibrium step anymore.

Example 2.8: Distillation Line for Constant Relative Volatilities For a ternary mixture with the components nitrogen a, argon b, and oxygen c, the relative volatilities can be regarded as constant and are given as follows:

αac = 4.032 and αbc = 1.339 .

(1)

Draw into a triangular diagram the distillation line through the point with the mole (n=0) (n=0) fractions xa = 0.2 and xb = 0.5 into a triangular diagram. Solution:

From the correlations in Eqs. (2.217) and (2.218), we obtain (n=0)

x(n) a =

n αac · xa (n=0)

n − 1) · x 1 + (αac a

(n=0)

n − 1) · x + (αbc b (n=0)

(n)

xb

=

n αbc · xb (n=0)

n − 1) · x 1 + (αac a (n)

21

(n)

(n)

= 1 − xa − xb = 1 − 0.2 − 0.5 = 0.3.

Furthermore, we have xc (n=0) xc

(n=0)

n − 1) · x + (αbc b

.

such that

In the course of this section, the illustration with the rolling ball will be utilized only with the boiling surface of a ternary mixture. However, as described above, this is a rough approximation since a distillation line does not follow exactly the steepest slope of the boiling surface.

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2.4 TERNARY MIXTURES

For n = 1, this yields

4.0321 · 0.2 = 0.4541 1 + (4.0321 − 1) · 0.2 + (1.3391 − 1) · 0.5 1.3391 · 0.5 = = 0.3770 1 + (4.0321 − 1) · 0.2 + (1.3391 − 1) · 0.5

x(n=1) = a (n=1)

xb

x(n=1) = 1 − 0.4541 − 0.3770 c

= 0.1689 .

For n = 2 we obtain

4.0322 · 0.2 = 0.7310 1+ − 1) · 0.2 + (1.3392 − 1) · 0.5 1.3392 · 0.5 = = 0.2015 2 1 + (4.032 − 1) · 0.2 + (1.3392 − 1) · 0.5

x(n=2) = a (n=2)

xb

(4.0322

x(n=2) = 1 − 0.7310 − 0.2015 c

= 0.0675 .

The following table summarizes the results for more values of n, which are graphically illustrated in the two-dimensional triangular diagram below:

n 3.5 3 2 1 0 −1 −2 −3 −4

xa

xb

xc

0.9397 0.8973 0.7310 0.4541 0.2000 0.0686 0.0208 0.0060 0.0017

0.0496 0.0822 0.2015 0.3770 0.5000 0.5165 0.4717 0.4973 0.3409

0.0107 0.0205 0.0675 0.1689 0.3000 0.4149 0.5075 0.5867 0.6574

128

2.4.6

2 VAPOR–LIQUID EQUILIBRIUM

Examples for Distillation Lines

For the eight different mixtures shown in Figure 2.9A–H before, the distillation lines are shown for the constant pressure p = 1 bar in two-dimensional triangular diagrams in Figure 2.11. For the nitrogen/argon/oxygen mixture, distillation lines are shown in Figure 2.11A. This mixture exhibits an almost ideal behavior such that all the distillation lines start at the corner of the heavy boiler c and end at the corner of the low boiler a. In Figure 2.11B distillations lines for the benzene/cyclohexane/heptane mixture are depicted. The low boiler a and the intermediate boiler b form a minimum azeotrope that has the lowest boiling temperature of all possible mixtures. Here, all distillation lines end at the a/b azeotrope at 77.6 ◦C. Similar to the mixture in Figure 2.11A, we have a single starting and a single ending point of all distillation lines. A different situation is encountered in the tetrachloromethane/benzene/cyclohexane mixture, which is shown in Figure 2.11C. Here, the intermediate boiler b and the high boiler c form a minimum azeotrope at 77.6 ◦C. The distillation lines start either at the corner of the heavy boiler c or at the corner of the intermediate boiler b and all end at the corner of the low boiler a. There exists a boundary distillation line that splits the ternary mixture into two regions with different starting points of the distillation lines. As a rough approximation, this boundary distillation line follows, as illustrated in Figure 2.9C, the course of the valley between the azeotrope at 77.6 ◦C and the corner of the low boiler a at 76.8 ◦C on the boiling surface. The distillation lines of the octane/2-ethoxyethanol/ethylbenzene mixture are

2.4 TERNARY MIXTURES

129

Figure 2.11 Distillation lines of the eight ternary mixtures from Figure 2.9A–B at p = 1 bar without boundary distillation lines; o: azeotropes.

130

2 VAPOR–LIQUID EQUILIBRIUM

Figure 2.11 (Continued) Distillation lines of the eight ternary mixtures from Figure 2.9C–D at p = 1 bar with boundary distillation lines; o: azeotropes.

2.4 TERNARY MIXTURES

131

Figure 2.11 (Continued) Distillation lines of the eight ternary mixtures from Figure 2.9E–F at p = 1 bar with boundary distillation lines; o: azeotropes.

132

2 VAPOR–LIQUID EQUILIBRIUM

Figure 2.11 (Continued) Distillation lines of the eight ternary mixtures from Figure 2.9G–H at p = 1 bar with boundary distillation lines; o: azeotropes.

2.4 TERNARY MIXTURES

133

shown in Figure 2.11D. Here, two binary minimum azeotropes are formed, and again a boundary distillation line proceeds from the b/c azeotrope at 127.1 ◦C to the a/b azeotrope at 116.1 ◦C, which divides the mixture into two regions with two different starting points of the distillation lines. Again, as a rough approximation, this boundary distillation line follows the course of the valley in the boiling surface, as illustrated in Figure 2.9D. The benzene/cyclohexane/isopropanol mixture with three binary azeotropes and one ternary minimum azeotrope is shown in Figure 2.11E. Each corner of the triangle is a starting point of the distillation lines and all end at the ternary azeotrope, which has the lowest boiling temperature at 69.1 ◦C. Boundary distillation lines proceed from each binary azeotrope toward the ternary azeotrope, dividing the mixture into three regions with different starting points of the distillation lines. In the n-propanol/water/n-butanol mixture, a partial miscibility gap is formed, which leads to a heteroazeotrope at 92.7 ◦C shown in Figure 2.11F with the miscibility gap as shaded area. Furthermore, a a/b minimum azeotrope is formed at 87.7 ◦C. Again, a boundary distillation line between the two azeotropes divides the mixture into two regions with two different starting points of the distillation lines. The distillation lines of the acetone/chloroform/acetonitrile mixture are shown in Figure 2.11G. In this mixture we have a a/b maximum azeotrope. All distillation lines start at the corner of the high boiler c and end either at the corner of intermediate boiler b or at the corner of low boiler a. A boundary distillation line proceeds from the high boiler c toward the maximum azeotrope. Approximately, this boundary distillation line follows the course of the ridge on the boiling surface shown in Figure 2.9G. In Figure 2.11H, the highly complex acetone/chloroform/methanol mixture is illustrated. Here, two binary minimum azeotropes, one binary maximum azeotrope, and one ternary saddle azeotrope are formed. There exist two starting points at 64.7 ◦C (high boiler c) and 64.4 ◦C (a/b maximum azeotrope) of the distillation lines along with two ending points at 55.7 ◦C (a/c minimum azeotrope) and 53.4 ◦C (b/c minimum azeotrope). The mixture is divided by boundary distillation lines into four regions with different starting and ending points. At 57.5 ◦C (ternary saddle azeotrope), all four boundary distillation lines converge and intersect. The formulation of the VLE of ternary mixtures by distillation lines is advantageous for process evaluation and design. In such diagrams, the existence of azeotropes can be seen very clearly. The starting or ending points of distillation lines are feasible top or bottom fractions of distillation columns, which will be demonstrated in Section 4.3.2.2. Boundary distillation lines are barriers that cannot be crossed by distillation in most cases. Boundary distillation lines are as important for ternary mixtures as azeotropes are for binary mixtures. It is essential to recognize that boundary distillation lines are, as a rough approximation, either valleys or ridges on the boiling surface. Thus, they run only between two local boiling temperature sinks (see Figure 2.11C–F and H) or two boiling temperature peaks (see Figure 2.11G); they never run between a local a local temperature peak and a local temperature sink. Usually, boundary distillation lines are not straight, but are more or less curved.

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2 VAPOR–LIQUID EQUILIBRIUM

References Abrams, D. and Prausnitz, J.M. (1975). Statistical thermodynamics of liquid mixtures. AIChE Journal 21: 116–128. Ahlers, J. and Gmehling, J. (2001). Development of a universal group contribution equation of state, 1. Prediction of liquid densities for pure components with a volume translated Peng–Robinson equation of state. Fluid Phase Equilibria 191: 177–188. Ahlers, J. and Gmehling, J. (2002). Development of a universal group contribution equation of state, 2. Prediction of vapor–liquid equilibria for asymmetric systems. Industrial and Engineering Chemistry Research 41: 3489–3498. Asselineau, L., Bogdanic, G., and Vidal, J. (1979). A versatile algorithm for calculating vapour–liquid equilibria. Fluid Phase Equilibria 3 (4): 273–290. Carroll, J.J. (1991). What is Henry’s law? Chemical Engineering Progress 87: 48–52. Chapman, W.G., Gubbins, K.E., Jackson, G., and Radosz, M. (1990). New reference equation of state for associating liquid. Industrial and Engineering Chemistry Research 29: 2284–2294. DDBST GmbH (2018). Dortmund Data Bank. http://www.ddbst.com/vle-databanks.html (accessed 3 April 2020). de Hemptinne, J.C., Ledanois, J.M., Mougin, P., and Barreau, A. (2012). Select Thermodynamic Models for Process Simulation. Editions TECHNIP. Design Institute for Physical Properties (2018). DIPPR data bank. https://www.aiche.org/dippr (accessed 3 April 2020). Fredenslund, A., Gmehling, J., and Rasmussen, P. (1977). Vapor–Liquid Equilibria Using UNIFAC. Amsterdam: Elsevier. Gmehling, J. and Kolbe, B. (1992). Thermodynamik. Weinheim, New York: VCH Publishers. Gmehling, J., Kolbe, B., Kleiber, M., and Rarey, J. (2012). Chemical Thermodynamics for Process Simulation. Weinheim, New York: VCH Publishers. Gmehling, J., Kolbe, B., Kleiber, M., and Rarey, J. (2019). Chemical Thermodynamics for Process Simulation. Weinheim, New

York: VCH Publishers. Gmehling, J., Menke, J., Krafczyk, J., and Fischer, K. (1994). Azeotropic Data, Part I and Part II. Weinheim: VCH Publishers. Gmehling, J., Onken, U., and Arlt, W. (1977). Vapor–Liquid Equilibrium Data Collection. In: Chemistry Data Series. Frankfurt/Main: DECHEMA. Gross, J. and Sadowski, G. (2002). Modeling polymer systems using the perturbed-chain statistical associating fluid theory equation of state. Industrial and Engineering Chemistry Research 41 (5): 1084–1093. Herington, E.F. (1951). Tests for the consistency of experimental isobaric vapor–liquid equilibrium data. Journal of the Institute of Petroleum 37: 457–470. Holderbaum, T. and Gmehling, J. (1991). A group-contribution equation of state based on UNIFAC. Fluid Phase Equilibria 70: 251–265. Honeywell (2017). UniSim Design Simulation Basis Reference Guide R451 Release, Honeywell. Horstmann, S., Fischer, K., and Gmehling, J. (2000). PSRK group contribution equation of state: revision and extension III. Fluid Phase Equilibria 167: 173–186. Huron, M.J. and Vidal, J. (1979). New mixing rules in simple equations of state for representing vapour–liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria 3: 255–271. Kang, J.W., Diky, V., Chirico, R.D., Magee, J.W., Muzny, C.D., Abdulagatov, I., Kazakov, A.F., and Frenkel, M. (2010). Quality assessment algorithm for vapor-liquid equilibrium data. Journal of Chemical & Engineering Data 55 (9): 3631–3640. Kehiaian, H.V. (1983). Group contribution methods for liquid mixtures: a critical review. Fluid Phase Equilibria 13: 243–252. Klamt, A. (1995). Conductor-like screening model for real solvents: a new approach to the quantitative calculation of solvation phenomena. Journal of Physical Chemistry 99: 2224–2235. Kontogeorgis, G.M. and Folas, G.K. (2010). Thermodynamic Models for Industrial Applications: From Classical and Advanced

REFERENCES Mixing Rules to Association Theories. Wiley. Mersmann, A. (1980). Thermische Verfahrenstechnik. Berlin: Springer-Verlag. Mersmann, A., Stichlmair, J., and Kind, M. (2011). Thermal Separation Technology. Berlin: Springer-Verlag. Michelsen, M.L. and Mollerup, J.M. (2004). Thermodynamic Models: Fundamental and Computational Aspects. Tie-Line Publications. National Institute of Standards and Technology (2018). NIST Chemistry WebBook. https://webbook.nist.gov/chemistry/ (accessed 3 April 2020). Peng, D.Y. and Robinson, D.B. (1976). A new two-constant equation of state. Industrial and Engineering Chemistry Fundamentals 15: 59–64. Perry, R.H., Green, D.W., and Maloney, J.O. (1984). Perry’s Chemical Engineer Handbook. New York: McGraw-Hill. Pfennig, A. (2004). Thermodynamik der Gemische. Berlin: Springer-Verlag. Poling, B.E., Prausnitz, J.M., and O’Connell, J. (2001). The Properties of Gases and Liquids. New York: McGraw-Hill. Prausnitz, J.M., Anderson, T.F., Grens, E.A., Eckert, C.A., R. Hsieh, R., and O’Connel, J.P. (1980). Computer Calculations for Multicomponent Vapor–Liquid and Liquid–Liquid Equilibria. Englewood Cliffs, NJ: Prentice-Hall. Prausnitz, J.M., Lichtenthaler, R.N., and de Azevedo, E.G. (1999). Molecular Thermodynamics of Fluid-Phase Equilibria.

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Prentice-Hall. Reid, R.C., Prausnitz, J.M., and Poling, B.E. (1987). The Properties of Gases and Liquids. New York: McGraw-Hill. Renon, H. and Prausnitz, J.M. (1968). Local composition in thermodynamic excess functions for liquid mixtures. AIChE Journal 14: 135–144. Sadowsky, G. and Gross, J. (2001). An equation of state based on a perturbation theory for chain molecules. Industrial and Engineering Chemistry Research 40 (4): 1244–1260. Soave, G. (1972). Equilibrium constants from a modified Redlich-Kwong equation of state. Chemical Engineering Science 27 (6): 1197–1203. Stephan, P., Schaber, K., Stephan, K., and Mayinger, F. (2012). Thermodynamik, Grundlagen und technische Anwendungen: Einstoffsysteme, Band 1. Berlin: Springer-Verlag. Stephan, P., Schaber, K., Stephan, K., and Mayinger, F. (2017). Thermodynamik, Grundlagen und technische Anwendungen: Mehrstoffsysteme und chemische Reaktionen, Band 2. Berlin: Springer-Verlag. Weidlich, U. and Gmehling, J. (1987). A modified UNIFAC model. 1. Prediction of VLE, hE, and gamma infinite. Industrial and Engineering Chemistry Research 26: 1372–1381. Wilson, G.M. (1964). Vapor–liquid equilibrium – a new expression for the excess free energy of mixing. Journal of the American Chemical Society 86: 127–130.

137

3 Single-Stage Distillation and Condensation As discussed in Chapter 2, the composition of the vapor usually differs from that of the liquid from which it originates. This concentration difference is the basis of distillation, which is a simple method of separating liquid mixtures. The method involves partial evaporation of the liquid followed by total condensation of the vapor. The resulting condensate is enriched in the more volatile components, in accordance with the vapor–liquid equilibrium of the system at hand. Thus, the starting mixture (feed) is separated into two fractions with different compositions. There exist three different modes of single-stage distillation; see Figure 3.1. The most important one is the cocurrent or closed distillation. Here, the leaving gas stream is in equilibrium with the exiting liquid stream. In the second mode, called open distillation, the gas is removed from the liquid immediately after its generation. This mode of operation is simplest performed by multistage batch distillation; see Chapter 6. The third mode is the countercurrent distillation. The leaving gas stream is supposed to be in equilibrium state with the liquid feed stream. Countercurrent distillation (in a reboiler) is very seldomly applied to industrial processes. Direct contact between vapor and liquid in a column is preferred here (see Chapter 4). All three modes of distillation have analogies in condensation; see Figure 3.2. The equations derived for distillation are, after small modifications, valid for partial condensation, too.

3.1

Continuous Closed Distillation and Condensation

Figure 3.3 presents the schematic of continuous closed distillation of a liquid mixture. The feed (molar flow rate F˙ ) is separated into an overhead product or distillate (molar ˙ ) and a bottom product (molar flow rate B˙ ). The composition difference flow rate D between these two products depends on the vapor–liquid equilibrium. 3.1.1

Closed Distillation of Binary Mixtures

A material balance around the reboiler (see Figure 3.3) yields

F˙ = G˙ + L˙ ,

(3.1)

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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Figure 3.1 Modes of single-stage distillation.

Figure 3.2 Modes of single-stage partial condensation.

3.1 CONTINUOUS CLOSED DISTILLATION AND CONDENSATION

139

where F˙ , G˙ , and L˙ are the molar flow rates of the feed, vapor and bottom liquid, respectively. Similarly, a material balance for the more volatile component a of a binary mixture a/b gives

F˙ · zFa = G˙ · ya + L˙ · xa ,

(3.2)

where zFa is the overall mole fraction of substance a in the feed, ya is the mole fraction of a in the vapor, and xa is the mole fraction of a in the bottom liquid. From both equations, it follows that   ˙ G˙ · xa + 1 + L/ ˙ G˙ · zFa . ya = −L/ (3.3)

˙ G˙ . When plotted on the This is the equation of a straight line with the slope −L/ (y, x )-diagram, it establishes the operating line, which intersects the diagonal at xa = zFa in Figure 3.3. With the closely met assumption that equilibrium exists between the vapor G˙ and the liquid L˙ in the reboiler, i.e. ya = ya∗ , the following equation holds for ideal mixtures (see Eq. (2.152)): ya∗ =

αab · xa , 1 + (αab − 1) · xa

(3.4)

where αab is the relative volatility. The concentrations xa and ya are determined by evaluating Eqs. (3.3) and (3.4). A quadratic equation is obtained, which can be

˙ Figure 3.3 Continuous closed distillation. The binary feed F˙ is split into two fractions D and B˙ with different compositions xDa and xBa , respectively.

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3 SINGLE-STAGE DISTILLATION AND CONDENSATION

˙ G˙ = F˙ /G˙ − 1, it follows that solved easily. With L/   p 1 xa = · −B − B 2 − 4 · A · C 2·A   ˙ F˙ − 1 · (αab − 1) with A = G/   ˙ F˙ − zFa ; C = zFa . −B = 1 + (αab − 1) · G/

(3.5)

The corresponding vapor or distillate concentration follows from Eq. (3.4). In most cases a graphical solution based on the (y, x )-diagram is preferred to Eq. (3.5) because this graphical method can also be applied to non-ideal mixtures. The desired concentrations of vapor ya and liquid xa are determined from the point of intersection of the operating line with the equilibrium line. Subsequent total condensation of the vapor does not effect a concentration change; i.e. the distillate concentration xDa is the same as the vapor concentration ya . This is represented graphically in the (y, x )-diagram by the intersection xDa of the horizontal line ya∗ = const with the diagonal. Figure 3.3 clearly shows the concentration difference between the distillate xDa and the bottoms xBa . The decisive factor is the course of the equilibrium curve, i.e. for ideal systems, the value of the relative volatility αab . Generally, only a limited difference between the concentrations of the two fractions can be obtained by single-stage distillation. 3.1.2

Closed Distillation of Multicomponent Mixtures

Boiling temperature and dew temperatures are identical for a single-component system. The vapor pressure line describes both states. In multicomponent mixtures, however, the vapor pressure curve splits into a boiling curve and a dew curve with an intermediate two-phase region; see Figure 3.4. The boiling line of a mixture is calculated by Eq. (2.160) derived in Section 2.2.16: X p0 = xi · γi · p0i with p0i = f (T ) . (3.6) The dew point line of a mixture follows from Eq. (2.162): X yi 1 = . 00 p γi · p0i

(3.7)

The intermediate two-phase region is characterized by the relative content of va˙ F˙ . Under the assumption that gas and liquid are in equilibrium state, the por G/ following set of equations holds: Material:

F˙ = G˙ + L˙ and F˙ · zFi = G˙ · yi + L˙ · xi

Equilibrium: yi = Ki · xi with Ki = γi ·

p0i /p .

(3.8) (3.9)

Combining Eqs. (3.8) and (3.9) yields

yi =

zFi · Ki . ˙ 1 + G/F˙ · (Ki − 1)

(3.10)

3.1 CONTINUOUS CLOSED DISTILLATION AND CONDENSATION

141

Figure 3.4 Dew point and boiling point lines of the mixture propane/isobutane/n-butane/ n-pentane. A two-phase vapor/liquid mixture exists in the region between boiling and dew line. Overall concentrations are za = 0.25, zb = 0.15, zc = 0.32, and zd = 0.28.

With the mole fraction constraint

P

yi = 1, it follows that

zFi · Ki = 1 or ˙ 1 + G/F˙ · (Ki − 1) X X zFi · p0i yi = = 1. ˙ F˙ · (p0 − p) p + G/ i X

yi =

X

(3.11)

Equation (3.11) is a very fundamental equation that formulates the state of multicomponent mixtures with coexisting vaporous and liquid phases. This situation is often encountered in distillation processes. The solution procedure for the flash Eqs. (3.10) and (3.11) depends on which parameters are known. Some cases are treated in the following sections. 3.1.2.1

Relative Amount of Vapor in a Vapor/Liquid Mixture

If the overall concentration zFi of the mixture, the total pressure p, and the temper˙ F˙ follows from ature T are set, the relative amount of vapor G/ X

yi =

X

zFi · Ki = 1. ˙ 1 + G/F˙ · (Ki − 1)

(3.12)

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3 SINGLE-STAGE DISTILLATION AND CONDENSATION

or with Ki = γi · p0i /p, i.e. ideal behavior in the vapor and non-ideal behavior in the liquid phase: X

yi =

X

zFi · γi · p0i = 1. ˙ F˙ ) + G/ ˙ F˙ · γi · p0 p · (1 − G/ i

(3.13)

The solution of the above equations is only possible by several iterations. First, a ˙ F˙ has to be estimated. The improvement of value of the relative amount of vapor G/ ˙ ˙ the estimated value of G/F is best performed by the Newton algorithm:   ˙ F˙ )(0) f (G/ ˙ F˙ )(1) = (G/ ˙ F˙ )(0) −   with (G/ ˙ F˙ )(0) f 0 (G/   X zFi · Ki ˙ F˙ )(0) = (3.14) f (G/ − 1 = 0 and ˙ 1 + (G/F˙ )(0) · (Ki − 1)   X zFi · Ki · (Ki − 1) ˙ F˙ )(0) = − f 0 (G/  2 . ˙ F˙ )(0) · (Ki − 1) 1 + (G/ Example 3.1 demonstrates the application of the calculation procedure to a quaternary mixture.

Example 3.1: State of a Multicomponent Mixture in a Vessel An ideal mixture of propane (za = 0.25), isobutane (zb = 0.15), n-butane (zc = 0.32), and n-pentane (zd = 0.28) is stored in a vessel at a pressure p = 2.5 bar and a temperature T = 20 ◦C. Find the state (vapor, liquid, two-phase) of the mixture in the vessel. Solution:

(1) State of the mixture: Boiling point pressure of the mixture, Eq. (3.6): P p0 = xi · p0i with p0i from Table 2.1.

p0a = 8.30 bar ; p0b = 3.00 bar ; p0c = 2.10 bar ; p0d = 0.57 bar p0 = 0.25 · 8.30 + 0.15 · 3.00 + 0.32 · 2.10 + 0.28 · 0.57 = = 3.356 bar ≥ 2.5 bar . Dew point pressure of the mixture, Eq. (3.7): 1/p00 =

P

yi /p0i

1/p00 = 0.25/8.30 + 0.15/3.00 + 0.32/2.10 + 0.28/0.57 = 0.7237 p00 = 1.38 bar ≤ 2.50 bar .

3.1 CONTINUOUS CLOSED DISTILLATION AND CONDENSATION

143

The state of the system is between boiling point and dew point. There exist two phases in the vessel; see Figure 3.4.

˙ F˙ , Eq. (3.12): (2) Relative amount of vapor G/ X X zFi · Ki yi = = 1 with ˙ F˙ · (Ki − 1) 1 + G/ 8.30 3.00 Ka = = 3.32 ; Kb = = 1.20 ; 2.50 2.50 2.10 0.57 Kc = = 0.84 ; Kd = = 0.23 . 2.50 2.50

Ki = p0i /p :

˙ F˙ )(0) = 0.20; evaluation of Eq. (3.12): Estimation of the amount of vapor: (G/ X 0.25 · 3.32 0.15 · 1.20 yi = + + 1 + 0.20 · (3.32 − 1) 1 + 0.20 · (1.20 − 1) 0.28 · 0.23 0.32 · 0.84 + + = 1.0938 6= 1.0 wrong. 1 + 0.20 · (0.84 − 1) 1 + 0.20 · (0.23 − 1) First iteration

˙ F˙ by Newton algorithm, Eq. (3.14): Better value for G/   ˙ F˙ )(0) f (G/ ˙ F˙ )(1) = (G/ ˙ F˙ )(0) −   (G/ ˙ F˙ )(0) f 0 (G/   X zFi · Ki ˙ F˙ )(0) = f (G/ − 1 = 0.0938 ˙ 1 + (G/F˙ )(0) · (Ki − 1)   X f 0 (G/F )(0) = −  "

=−

zFi · Ki · (Ki − 1) ˙ F˙ )(0) · (Ki − 1) 1 + (G/

0.25 · 3.32 · (3.32 − 1)

0.15 · 1.20 · (1.20 − 1)

2+ (1 + 0.20 · (3.32 − 1)) (1 + 0.20 · (1.2 − 1))  0.32 · 0.84 · (0.84 − 1) 0.28 · 0.23 · (0.23 − 1) + + = −0.8165 (1 + 0.3149 · (0.84 − 1))2 (1 + 0.3149 · (0.23 − 1))2

˙ F˙ )(1) (G/

2

+

2 =

  ˙ F˙ )(0) f (G/ 0.0938 ˙ F˙ )(0) −   = 0.20 − = (G/ = 0.3149 −0.8165 0 (0) ˙ ˙ f (G/F )

0.25 · 3.32 0.15 · 1.20 + + 1 + 0.3149 · (3.32 − 1) 1 + 0.3149 · (1.20 − 1) 0.32 · 0.84 0.28 · 0.23 + + = 1.0170 6= 1.0 wrong. 1 + 0.3149 · (0.84 − 1) 1 + 0.3149 · (0.23 − 1)

Eq. (3.12):

X

yi =

144

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Second iteration

˙ F˙ by the Newton algorithm, Eq. (3.14): Better value for G/   X zFi · Ki ˙ F˙ )(1) = f (G/ − 1 = 1.0170 − 1 = 0.0170 ˙ F˙ )(1) · (Ki − 1) 1 + (G/ f

0



˙ F˙ ) (G/

(1)



"

=−

0.25 · 3.32 · (3.32 − 1) 2

0.15 · 1.20 · (1.20 − 1) 2

(1 + 0.3149 · (1.20 − 1)) # 0.32 · 0.84 · (0.84 − 1) 0.28 · 0.23 · (0.23 − 1) + 2 + 2 = −0.5407 (1 + 0.3149 · (0.84 − 1)) (1 + 0.3149 · (0.23 − 1))   ˙ F˙ )(1) f (G/ 0.0170 ˙ F˙ )(2) = (G/ ˙ F˙ )(1) −   = 0.3149 − (G/ = 0.3463 −0.5407 0 (1) ˙ F˙ ) f (G/

Eq. (3.12):

X

(1 + 0.3149 · (3.32 − 1))

+

0.25 · 3.32 0.15 · 1.20 + + 1 + 0.3463 · (3.32 − 1) 1 + 0.3463 · (1.20 − 1) 0.32 · 0.84 0.28 · 0.23 + + = 1 + 0.3463 · (0.84 − 1) 1 + 0.3463 · (0.23 − 1) = 0.4602 + 0.1683 + 0.2846 + 0.0878 = 1.0009 ≈ 1.0 okay.

yi =

Result:

Relative amount of vapor in the vessel:

˙ F˙ = 0.3463 G/

Concentration of the vapor:

ya = 0.4602 ;

yb = 0.1683

yc = 0.2846 ;

yd = 0.0878

Concentration of the liquid with xi = yi /Ki : xa = 0.1386 ;

xb = 0.1403

xc = 0.3388 ;

xd = 0.3817

3.1.2.2

Partial Evaporation of a Multicomponent Liquid Mixture

The principal process of single-stage distillation is the partial evaporation of a liquid mixture; see Figure 3.7. For a multicomponent system the relevant equations are: Equation (3.11):

X

yi =

X

zFi · Ki =1 ˙ F˙ · (Ki − 1) 1 + G/

(3.15)

and

˙ F˙ = cL,m · (T − TF ) + G/ ˙ F˙ · rm (T ) . Enthalpy balance: Q/

(3.16)

The latent heat of vaporization ri (T ) can be taken from Figure 3.6. ˙ F˙ (= G/ ˙ F˙ ) is given. Again, an iterative Often, the relative amount of distillate D/ solution procedure must be applied. First, an approximation of the temperature in the

3.1 CONTINUOUS CLOSED DISTILLATION AND CONDENSATION

145

Figure 3.5 Schematic of closed distillation of a multicomponent liquid mixture.

Figure 3.6 Latent heat of vaporization of selected compounds. = normal boiling point; ◦ = melting point; • = critical point. Heats of vaporization at normal boiling point are an approximately linear function of the boiling point temperature TBP (Trouton’s rule). The temperature dependence of ri is described by ri = ri,BP · ((Tcr ,i − T )/(Tcr ,i − TBP ))0.38 .

146

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

˙ F˙ for this still is made. Second, Eq. (3.15) is solved iteratively, giving the value G/ ˙ ˙ temperature. If G/F is too far from the desired amount of distillate, the procedure has to be repeated with a better estimation of the temperature. The solution procedure is demonstrated in detail in Example 3.2. Example 3.2: Distillation of a Quaternary Ideal Mixture A liquid mixture of propane (a), i-butane (b), n-butane (c), and pentane (d) is continuously distilled at a pressure of 4 bar; see Figure 3.5. The relative amount of ˙ F˙ = 0.60. distillate is D/ Data:

xFa = 0.25 ; xFb = 0.15 ; xFc = 0.32 ; xFd = 0.28 ; TF = 20 ◦C

Find the temperature T in the evaporator, the concentrations of vapor and liquid, and the energy required. Solution:

(1) Estimation of the temperature in the evaporator The temperature in the two-phase region at a pressure of 4 bar is (see Figure 3.4) 26 ◦C ≤ T ≥ 53 ◦C ; estimation Test = 40 ◦C. (2) Vapor–liquid equilibrium X X Eq. (3.17): yi =

zFi · p0i = 1; ˙ F˙ · (p0 − p) p + G/ i

p0i from Antoine equation, Table 2.1: p0a = 13.45 bar ; p0b = 5.27 bar ; p0c = 3.75 bar ; p0d = 1.15 bar X

0.25 · 13.45 0.15 · 5.27 + + 4 + 0.6 · (13.45 − 4) 4 + 0.6 · (5.27 − 4) 0.32 · 3.75 0.28 · 1.15 + + = 4 + 0.6 · (3.75 − 4) 4 + 0.6 · (1.15 − 4) = 0.9660 6= 1.0 wrong, the estimation is too low.

yi =

Improved value of the temperature: Test = 43 ◦C. p0a = 14.43 bar ; p0b = 5.72 bar ; p0c = 4.09 bar ; p0d = 1.28 bar Eq. (3.15):

0.25 · 14.43 0.15 · 5.72 + + 4 + 0.6 · (14.43 − 4) 4 + 0.6 · (5.72 − 4) 0.32 · 4.09 0.28 · 1.28 + = + 4 + 0.6 · (4.09 − 4) 4 + 0.6 · (1.28 − 4) = 0.3507 + 0.1705 + 0.3228 + 0.1514 = 0.9954 ≈ 1.0 okay.

X

yi =

147

3.1 CONTINUOUS CLOSED DISTILLATION AND CONDENSATION

Concentration of the liquid xi = yi · p/p0i :

xa = 0.3507 · 4/14.43 = 0.0972 ;

xb = 0.1705 · 4/5.72 = 0.1192

xc = 0.3228 · 4/4.09 = 0.3157 ; X xi = 1.0052 .

xd = 0.1514 · 4/1.28 = 0.4731

˙ F˙ = cL,m · (T − TF ) + G/ ˙ F˙ · rm (T ) (3) Energy requirement, Eq. (3.16): Q/ Average heat capacity of liquid: cL,m ≈ 0.13 kJ/(mol · ◦C) P Latent heat of vaporization at 43 ◦C from Figure 3.6: rm = yi · ri rm = 0.3507 · 13.1 + 0.1705 · 17.7 + 0.3228 · 21.0 + 0.1514 · 26.1 = = 18.34 kJ/mol ˙ F˙ = 0.13 · (43 − 20) + 0.6 · 18.34 = 14.0 kJ/mol . Q/ Result:

˙ F˙ = 14.0 kJ/mol ; Temperature in the still: T = 43 ◦C ; energy requirement: Q/ concentrations of product streams: xDa = 0.3507 ; xDb = 0.1705 ; xDc = 0.3228 ; xDd = 0.1514 xBa = 0.0972 ;

3.1.2.3

xBb = 0.1192 ; xBc = 0.3157 ;

xBd = 0.4731

Adiabatic Flash

Pressure reduction in a nozzle causes partial evaporation of a liquid feed near boiling point; see Figure 3.7. Since the enthalpy remains constant by passing a nozzle, the relevant set of equations of an adiabatic flash is Eq. (3.12): X

yi =

X

zFi · Ki =1 ˙ 1 + G/F˙ · (Ki − 1)

(3.17)

and

G˙ cL,m = (TF − T ) · . ˙ r F m (T )

(3.18)

Since the vapor has a higher enthalpy than the liquid, the temperature of the system decreases. The solution procedure is demonstrated in detail in Example 3.3. First, the tem˙ F˙ is calculated by perature T after the nozzle is estimated. Second, the value of G/ ˙ F˙ is checked by Eq. (3.17) an approximate enthalpy balance. Third, the value G/ whether or not it meets the vapor–liquid equilibrium. The iteration procedure is rather complex since the enthalpy balance (step 2) requires knowledge of the vapor concentration yi , which is determined in step 3.

148

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Figure 3.7 Schematic of an adiabatic flash. Some vapor is generated by expanding a hot liquid in a nozzle and, hence, temperature decreases.

Example 3.3: Adiabatic Flash of a Saturated Liquid Part 1: Water flows through a nozzle.

• Feed state: temperature TF = 107 ◦C ; pressure pF = 1.30 bar. • State after the nozzle: pressure p = 0.50 bar. Find the temperature and the amount of vapor after the nozzle. Solution:

(1) Temperature T after the nozzle: Since the feed is at the boiling point, the state after the nozzle lies at the vapor pressure curve. For a pressure of 0.50 bar the temperature is determined by the rearranged Antoine equation (see Table 2.1): Bi 3984.923 T /K = −Ci + = 39.724 + = Ai − ln(pi /bar) 11.96481 − ln(0.50/bar)

= 354.54 K or T = 81.4 ◦C . (2) Enthalpy balance, Eq. (3.18): Heat capacity cL = 75.35 kJ/(kmol · K) Latent heat of vaporization r = 41 400 kJ/kmol from Figure 3.6

˙ F˙ = (TF − T ) · cL /r(T ) = (107 − 81.4) · 75.35/41400 = G/ = 0.0465 mol/mol . ˙ F˙ Result: Vapor content G/

= 0.0465 ; temperature T = 81.4 ◦C .

Part 2: A saturated quaternary liquid mixture flows through a nozzle.

Liquid mixture: Propane (xFa = 0.25) ; isobutane (xFb = 0.15) ; n-butane (xFc = 0.32) ; n-pentane (xFd = 0.28) .

3.1 CONTINUOUS CLOSED DISTILLATION AND CONDENSATION

149

• Feed state: Temperature TF = 20 ◦C ; pressure pF = 3.35 bar (= boiling pressure) . • State after the nozzle: pressure p = 1.20 bar . Find the temperature T , the amount of vapor, and the concentration of vapor and liquid after the nozzle. Solution:

(1) Temperature T after the nozzle: For a saturated liquid feed, the state after the nozzle is in the two-phase region between boiling and dew line in Figure 3.4. For the pressure of 1.20 bar, it follows that −12 ◦C ≤ T ≤ 16 ◦C. Estimation Test = 0 ◦C. ˙ F˙ = (TF − Test )·cL,m /rm (T ) (2) Approximate enthalpy balance, Eq. (3.18): G/ Heat capacity of the liquid cL,m = 0.13 kJ/(mol · K) P Heat of vaporization rm = yi · ri (T ) P Approximation rm ≈ xFi · ri (Test ) ; ri from Figure 3.6

rm = 0.25 · 16.50 + 0.15 · 20.50 + 0.32 · 22.30 + 0.28 · 27.80 = = 22.12 kJ/mol . ˙ F˙ = (20 − 0) · 0.13/22.12 = 0.1175 . Eq. (3.18): G/ (3) Vapor–liquid equilibrium, Eq. (3.11): X X zFi · p0i yi = =1 ˙ F˙ · (p0 − p) p + G/ i with p0i from Antoine equation, Table 2.1.

p0a = 4.73 bar ; p0b = 1.56 bar ; p0c = 1.032 bar ; p0d = 0.245 bar X

0.25 · 4.73 0.15 · 1.56 + + 1.2 + 0.1175 · (4.73 − 1.2) 1.2 + 0.1175 · (1.56 − 1.2) 0.32 · 1.032 0.28 · 0.245 + + = 1.2 + 0.1175 · (1.032 − 1.2) 1.2 + 0.1175 · (0.245 − 1.2) = 1.2635 6= 1.0 wrong

yi =

The estimated temperature is too high. First iteration

(1) Better estimation of the temperature after the nozzle Test = −10 ◦C (2) Approximate enthalpy balance: X X rm = yi · ri (−10 ◦C) with yi at 0 ◦C, corrected by yi = 1.2635

rm = (0.7323 · 17.0 + 0.1884 · 21.0 + 0.2798 · 22.8 + + 0.0631 · 28.3)/1.2635 = 19.45 kJ/mol ˙ F˙ = (20 − (−10)) · 0.13/19.45 = 0.2005 . Eq. (3.18): G/

150

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

(3) Vapor–liquid equilibrium, Eq. (3.11):

p0a = 3.445 bar ; p0b = 1.076 bar ; p0c = 0.696 bar ; p0d = 0.152 bar X 0.25 · 3.445 0.15 · 1.076 yi = + + 1.2 + 0.2005 · (3.445 − 1.2) 1.2 + 0.2005 · (1.076 − 1.2) 0.32 · 0.696 0.28 · 0.152 + + = 1.2 + 0.2005 · (0.696 − 1.2) 1.2 + 0.2005 · (0.152 − 1.2) = 0.9049 6= 1.0 wrong. The estimated temperature is too low. Second iteration

(1) Better estimation of the temperature after the nozzle: Test = −6.6 ◦C (2) Approximate enthalpy balance, Eq. (3.18): X X rm = yi · ri (−6.6 ◦C) with yi at −10 ◦C, corrected by yi = 0.9049

rm = (0.5219 · 16.9 + 0.1373 · 20.9 + 0.2027 · 22.7+ + 0.0430 · 28.2)/0.9049 = 19.34 kJ/mol ˙ F˙ = (20 − (−6.6)) · 0.13/19.34 = 0.1788 . Eq. (3.18): G/ (3) Vapor–liquid equilibrium, Eq. (3.11):

p0a = 3.84 bar ; p0b = 1.22 bar ; p0c = 0.797 bar ; p0d = 0.179 bar X 0.25 · 3.84 0.15 · 1.22 yi = + + 1.2 + 0.1788 · (3.84 − 1.2) 1.2 + 0.1788 · (1.22 − 1.2) 0.32 · 0.797 0.28 · 0.179 + + = 1.2 + 0.1788 · (0.797 − 1.2) 1.2 + 0.1788 · (0.179 − 1.2) = 0.5742 + 0.1520 + 0.2261 + 0.0493 = 1.0016 ≈ 1.0 okay. (4) Improved enthalpy balance: X rm = yi · ri (−6.6 ◦C) ; ri from Figure 3.6

rm = (0.5742 · 16.9 + 0.1520 · 20.9 + 0.2261 · 22.7+ + 0.0493 · 28.2)/1.0016 = 19.37 kJ/mol ˙ F˙ = (20 − (−6.6)) · 0.13/19.37 = 0.1785 . G/ Vapor–liquid equilibrium, Eq. (3.11): X 0.25 · 3.84 0.15 · 1.22 yi = + + 1.20 + 0.1785 · (3.84 − 1.20) 1.20 + 0.1785 · (1.22 − 1.20) 0.32 · 0.797 0.28 · 0.179 + + = 1.20 + 0.1785 · (0.797 − 1.20) 1.20 + 0.1785 · (0.179 − 1.20)

= 0.5742 + 0.1520 + 0.2261 + 0.0493 = 1.0016 ≈ 1.0 okay.

3.1 CONTINUOUS CLOSED DISTILLATION AND CONDENSATION

151

Result:

Relative amount of vapor:

T = −6.6 ◦C ˙ F˙ = 0.1785 G/

Vapor concentrations:

ya = 0.5742 ; yb = 0.1520

Temperature after the nozzle:

yc = 0.2261 ; yd = 0.0493 Liquid concentrations xi = yi ·

p/p0i :

xa = 0.1795 ; xb = 0.1495 xc = 0.3404 ; xd = 0.3298

Table 3.1 Carbon dioxide outburst at Lake Nyos, Cameroon, on 21 August 1986.

Lake Nyos, Cameroon, is a 200 m deep volcanic crater nowadays filled with water; see Figure 3.8. Carbon dioxide leaks from the volcanic rock into the water and accumulates in the lowest layer to a relatively high CO2 content (up to 2 mol%). On 21 August 1986, the instable layers in the lake suddenly turned over for any reason (e.g. falling rock, light earthquake). As the CO2 -rich water from the ground approached the surface, some CO2 degassed due to the pressure drop (= adiabatic flash). The bubbles enhanced the upward flow of the water, and, in turn, more and more CO2 degassed. A carbon dioxide cloud with a density higher than air emerged from the lake and floated down the hill in an approximately 20 m thick layer. At Nyos village and a neighbor village, 1700 people and all their livestock suffocated in this night (National Geographic Society No. 3, 1987).

Figure 3.8 Carbon dioxide outburst at Lake Nyos on 21 August 1986.

152

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Adiabatic flash caused by pressure drop is a very essential element of many industrial separation processes, e.g. desalination of seawater (Figure 7.6), thermal coupling of columns, integrated heat pump, etc. However, it is also encountered in nature. A very impressive example is the carbon dioxide outburst from Lake Nyos described in Table 3.1. Divers sickness caused by degassing air from blood is another important example of adiabatic flash.

3.2

Batchwise Open Distillation and Open Condensation

In continuous open distillation and in batchwise distillation, the vapor is removed from the system immediately after its generation. Both modes of operation are displayed in Figures 3.1 and 3.9, respectively. The time dependence of the variables of batch distillation is equal to the locus dependence of continuous open distillation. After elimination of time and locus, the same equations apply to both modes of operation. A schematic of batch distillation is shown in Figure 3.9. The vapor stream G˙ produced by heating L moles of liquid is continuously removed and condensed. The distillate D is collected in several receivers.

Figure 3.9 Schematic of batch distillation. Generally, the distillate is collected in several receivers.

3.2.1

Binary Mixtures

For the derivation of the essential equations, the case of batch distillation of binary mixtures is considered in this section. 3.2.1.1

Open Distillation (Batch Distillation)

The fundamental laws of batch distillation have been developed as early as at the turn of the nineteenth to the twentieth century [Ostwald 1900; Schreinemakers 1901; van der Waals 1902; Rayleigh 1902]. A material balance around the still

3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

153

(boiler) permits calculating the process of batch distillation of a binary mixture of substances a and b. For a differential element of time, dt, the following equation holds:

G˙ · dt + dL = 0 .

(3.19)

Here, G˙ denotes the molar stream of vapor generated by heating the system. A component balance gives

G˙ · dt · ya + d (L · xa ) = 0 ,

(3.20)

where the vapor and liquid mole fractions refer to the more volatile component a. With G˙ · dt = dG, it follows from Eq. (3.19) that

dG = −dL .

(3.21)

Combining Eqs. (3.19) and (3.20) yields

−dL · ya∗ + dL · xa + L · dxa = 0 .

(3.22)

After rearrangement

dL dxa = ∗ . L ya − xa

(3.23)

This is the well-known Rayleigh equation [Rayleigh 1902]. It formulates the relation between liquid level and liquid composition during open distillation. The vapor arising from the still is supposed to be in equilibrium state with the remaining liquid. Introducing the equilibrium ratio K = y ∗ /x yields

dxa dL = . L (Ka − 1) · xa

(3.24)

Since the equilibrium ratio Ka is a function of concentration and temperature, this equation cannot be integrated directly. For ideal mixtures, the equilibrium concentration ya∗ can be formulated with the relative volatility αab , which is often independent of the temperature:

ya∗ =

αab · xa . 1 + (αab − 1) · xa

(3.25)

From Eqs. (3.23) and (3.25), it follows that

dL dxa = . L αab · xa /(1 + (αab − 1) · xa ) − xa

(3.26)

The solution of the differential equation is

L = F



xa xFa

1/(αab −1)   1 − xFa αab /(αab −1) · , 1 − xa

(3.27)

154

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Figure 3.10 Batch distillation of an ideal binary mixture calculated with αab = 3. The system is depleted of the light component during operation.

where F represents the liquid in its initial state. Typical changes in the concentrations of the more volatile component a in the vapor and in the liquid are plotted versus the relative amount of distillate D/F = 1 − L/F in Figure 3.10. During operation both phases are increasingly depleted of substance a. The mean distillate concentration xDa,m is given by

xDa,m = xa +

xFa − xa . D/F

(3.28)

This equation can be graphically interpreted in Figure 3.10. Advantageously, the receivers of the distillate are changed periodically at constant values of D/F . Even for non-ideal mixtures, the mean distillate concentrations xDa,m are easily determined.

Example 3.4: Comparison of the Modes of Distillation An ideal binary mixture (xFa = 0.6 , αab = 2.5) is distilled in three modes: closed distillation, open distillation, and countercurrent distillation. Find the concentrations of gas and liquid and the yield of the low boiler a for ˙ F˙ = 0.5. G/

3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

155

(1) Closed distillation:

˙ F˙ = 0.5 , xFa = 0.6 , αab = 2.5 : Equation (3.5) with G/ p 1 xa = · (−B − B 2 − 4 · A · C) with 2·A ˙ F˙ − 1) · (αab − 1) = (0.5 − 1) · 1.5 = −0.75 A = (G/ ˙ F˙ − zFa ) = 1 + (2.5 − 1) · (0.5 − 0.6) = 0.85 −B = 1 + (αab − 1) · (G/ C = zFa = 0.6   q 1 · +0.85 − 0.852 − 4 · (−0.75) · 0.6 = 0.4922 2 · (−0.75) 2.5 · 0.4922 ya∗ = = 0.7079 1 + (2.5 − 1) · 0.4922

xa =

Yield: Ya =

G · ya 0.7079 = 0.5 · = 0.59 . F · xFa 0.6

(2) Open distillation:

Equation (3.27) with L/F = 0.5 , xFa = 0.6 :    xa 1/(αab −1) 1 − xFa αab /(αab −1) · xFa 1 − xa  x 1/(2.5−1)  1 − 0.6 2.5/(2.5−1) a 0.5 = · 0.6 1 − xa

L = F



Estimation: xa = 0.5:

L = F



0.5 0.6

0.6667   1 − 0.6 1.6667 · = 0.6105 6= 0.5 1 − 0.5

Estimation: xa = 0.46:

L = F



0.46 0.6

0.6667   1 − 0.6 1.6667 · = 0.5080 ≈ 0.50 okay. 1 − 0.46

Vapor concentration ya from a material balance:

F · xFa = G · ya + L · xa F L 1 ya = · xFa − · xa = · 0.6 − 1 · 0.46 = 0.740 G G 0.5 Yield: Ya =

G · ya 0.740 = 0.5 · = 0.6167 . F · xFa 0.6

156

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

(3) Countercurrent distillation:

2.5 · 0.6 = 0.7895 1 + (2.5 − 1) · 0.6 Liquid concentration xa from a material balance: Gas is in equilibrium with the feed: ya∗ =

F · xFa = G · ya + L · xa xa = F/L · xFa − G/L · ya = 1/0.5 · 0.6 − 1 · 0.7895 = 0.4105 Yield: Ya =

G · ya 0.7895 = 0.5 · = 0.6579 . F · xFa 0.6

Result: The sharpest separation is achieved by countercurrent distillation.

3.2.1.2

Open Condensation

In open partial condensation the condensate is removed from the system immediately after its formation; see Figure 3.2. The derivation of the fundamental equations is performed in strict analogy to Section 3.2.1.1 on open distillation. A material balance yields

dG + dL = 0 .

(3.29)

For the light component a,

d (G · ya ) + x∗a · dL = 0 .

(3.30)

Here, x∗a denotes the concentration of liquid in equilibrium state with the vapor concentration ya . From both equations, it follows that

dG · ya + G · dya − x∗a · dG = 0 .

(3.31)

After transformation,

dG dya = ∗ . G xa − ya

(3.32)

This equation fully corresponds with the Rayleigh equation (3.23) derived for open distillation. For ideal mixtures the vapor–liquid equilibrium is expressed by the relative volatility α (see Eq. (2.153)):

x∗a =

−1 αab · ya . −1 1 + (αab − 1) · ya

(3.33)

Combining Eqs. (3.32) and (3.33) yields

dG dya = −1 . −1 G αab · ya /(1 + (αab − 1) · ya ) − ya

(3.34)

3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

After integration, −1     −1 −1 G ya 1/(αab −1) 1 − yFa αab /(αab −1) = · . F yFa 1 − ya

157

(3.35)

This equation corresponds with Eq. (3.27) for open distillation; only L is replaced −1 by G, x by y , and αab by αab . During operation, the vapor G is depleted of high boiling components and, in turn, enriches in low boiling ones. Consequently, the vapor concentration ya is always higher than the feed concentration yFa . 3.2.2

Ternary Mixtures

Open distillation of ternary mixtures has found a wide interest in literature [Hausen 1935, 1952; Vogelpohl 1963]. 3.2.2.1

Open Distillation

The Rayleigh equation can be applied directly to ternary mixtures. For a ternary mixture, Eq. (3.23) becomes

dL dxa = ∗ L ya − xa

and

dL dxb = ∗ . L yb − xb

(3.36)

Elimination of dL/L yields

dxa y ∗ − xa = a∗ . dxb yb − xb

(3.37)

This very important differential equation describes the change of liquid concentrations in the still of an open distillation unit. It is called liquid residue curve or liquid residue [Doherty and Perkins 1979; Petlyuk 1986; Bernon et al. 1990]. Figure 3.11 displays the graphical interpretation of Eq. (3.37). The dashed dotted line is the liquid residue curve, i.e. the course of the change of liquid composition during open distillation. The dashed straight line between liquid concentration xi and the corresponding equilibrium vapor concentration yi∗ forms a tangent to the liquid residue curve. The liquid is depleted of low boiling components during operation. This is indicated by an arrow at the liquid residue curve. For an ideal ternary mixture with constant relative volatilities, the vapor concentrations follow from αac · xa ya∗ = and 1 + (αac − 1) · xa + (αbc − 1) · xb (3.38) αbc · xb yb∗ = . 1 + (αac − 1) · xa + (αbc − 1) · xb The solution of Eqs. (3.37) and (3.38) is [van der Waals 1902]  (αac −1)/(αac −αbc ) 1 − xa − xb xa · = xa xb   1 − xFa − xFb xFa (αac −1)/(αac −αbc ) = · . xFa xFb

(3.39)

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3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Figure 3.11 Batch distillation of an ideal ternary mixture (calculated with relative volatilities αac = 20, αbc = 2). The bold dashed pointed line describes the change of the liquid concentrations in the still. It is called liquid residue curve.

The evaluation of Eq. (3.39) is demonstrated in Example 3.5. The relative amount of liquid in the still, L/F , that corresponds with the actual liquid concentrations, xa and xb , is then determined by

L = F



xa xFa

αbc /(αac −αbc )   xFb αac /(αac −αbc ) · . xb

(3.40)

The profiles of concentration over the relative amount of distillate D/F = 1 − L/F is displayed in Figure 3.12. This plot is analogous to Figure 3.10 for a binary mixture. In ternary mixtures the presentation in a triangular diagram is preferred. Figure 3.13 presents such a plot for the non-ideal ternary mixture octane, 2-ethoxyethanol, and ethylbenzene. There exist two termini of residue curves and, consequently, a boundary residue curve. The course of liquid residue curves resembles the course of distillation lines; see Figure 2.11. However, the arrow that indicates the direction of change is reverse. Since liquid residue curves are only a function of vapor–liquid equilibrium, they are sometimes used for a graphical plot of vapor–liquid equilibrium. However, distillation lines are preferred for this purpose in this book because they are simpler to calculate. In addition, the length of each equilibrium step can be marked by points at distillation lines (see Figure 2.11).

3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

159

Figure 3.12 Batch distillation of an ideal ternary mixture (calculated with relative volatilities αac = 20, αbc = 2). The concentrations are plotted versus the relative amount of distillate. The intermediate boiling component b exhibits a concentration maximum. A) Liquid concentrations. B) Vapor concentrations.

160

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Figure 3.13 Liquid residue curves of the octane/2-ethoxyethanol/ethylbenzene mixture. The course of the liquid residue curves is very similar to that of distillation lines.

3.2.2.2

Open Condensation

In open condensation the liquid formed by down cooling the system is immediately removed from the vapor. In analogy to Eq. (3.36), the following holds:

dG dya = ∗ G xa − ya

and

dG dyb = ∗ . G xb − yb

(3.41)

Eliminating dG/G yields:

dya x∗ − ya = a∗ . dyb xb − yb

(3.42)

This equation describes the course of changes of vapor composition during open condensation. It is called vapor residue curve or vapor residue. It is represented by a bold dashed line in Figure 3.14. The straight line between actual vapor composition yi and the corresponding liquid composition x∗i forms a tangent to the vapor residue curve. For ideal systems the equilibrium is expressed by

x∗a =

−1 αac · ya −1 −1 1 + (αac − 1) · ya + (αbc − 1) · yb

x∗b =

−1 αbc · yb . −1 −1 1 + (αac − 1) · ya + (αbc − 1) · yb

and (3.43)

3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

161

Figure 3.14 Open condensation of a ternary mixture. The dotted bold line describes the change of vapor compositions during operation. It is called vapor residue curve.

Figure 3.15 Vapor residue curves of the octane/2-ethoxyethanol/ethylbenzene system.

162

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Figure 3.16 Comparison of the courses of liquid residue curves (− · −), vapor residue curve (− − −), and distillation lines (—–). Especially in close-boiling systems all three lines lie close together.

3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

Figure 3.16 (Continued)

163

164

3 SINGLE-STAGE DISTILLATION AND CONDENSATION

After integration of Eq. (3.42), −1 −1  (α−1 ac −1)/(αac −αbc ) 1 − ya − yb ya · = ya yb −1 −1   −1 1 − yFa − yFb yFa (αac −1)/(αac −αbc ) = · . yFa yFb The relative amount of remaining vapor is then determined by     G ya αac /(αbc −αac ) yFb αbc /(αbc −αac ) = · . F yFa yb

(3.44)

(3.45)

These equations fully correspond with Eqs. (3.39) and (3.40) for open distillation. Only the variables L, x, and α are substituted by G, y , and α−1 , respectively. Figure 3.15 displays the course of change of vapor composition during open condensation of the octane/2-ethoxyethanol/ethylbenzene system. The high boiling components condenses first, and, in turn, the remaining vapor enriches in low boiling components. This is indicated by an arrow on the vapor residue curve. Vapor residue curves are only a function of vapor–liquid equilibrium alike liquid residue curves and distillation lines. For close-boiling systems the course of these three lines is very similar as can be seen from Figures 3.16A–D. Vapor residue curves and liquid residue curves show the same characteristics as distillation lines. In azeotropic systems, boundary residue curves exist in analogy to boundary distillation lines. Generally, distillation lines lie between vapor and liquid residue curves.

Example 3.5: Liquid Residue Curve, Vapor Residue Curve, and Distillation Line Find the liquid residue curve, the vapor residue curve, and the distillation line through point zFa = 0.30 and zFb = 0.40 of an ideal ternary mixture with relative volatilities αac = 3.50 and αbc = 2.10. Solution: (1) Liquid residue curve:

 xa (αac −1)/(αac −αbc ) = xb   1 − xFa − xFb xFa (αac −1)/(αac −αbc ) = · xFa xFb

Equation (3.39):

1 − xa − xb · xa



Exponent (αac − 1)/(αac − αbc ) = (3.50 − 1)/(3.50 − 2.10) = 1.7857   1 − xFa − xFb xFa (αac −1)/(αac −αbc ) · = xFa xFb   1 − 0.30 − 0.40 0.30 1.7857 = · = 0.5983 . 0.30 0.40

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3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

Values for xa = 0.10: Estimation xb = 0.20:

1 − 0.10 − 0.30 · 0.10



0.10 0.30

1.7857

= 0.8436 6= 0.5983

 0.10 1.7857 = 0.5872 6= 0.5983 0.35   1 − 0.10 − 0.345 0.10 1.7857 Estimation xb = 0.345: · = 0.6080 6= 0.5983 0.10 0.345   1 − 0.10 − 0.343 0.10 1.7857 Estimation xb = 0.343: · = 0.5954 ≈ 0.5983 0.10 0.343

1 − 0.10 − 0.35 · 0.10



1 − 0.50 − 0.30 · 0.50



Estimation xb = 0.20:

1 − 0.50 − 0.40 · 0.50



Estimation xb = 0.40:

Estimation xb = 0.35:

Values for xa = 0.50:

0.50 0.30

1.7857

0.50 0.40

1.7857

= 0.9959 6= 0.5983 = 0.2979 6= 0.5983

 0.50 1.7857 = 0.6371 6= 0.5983 0.34  1.7857 1 − 0.50 − 0.345 0.50 Estimation xb = 0.345: · = 0.6014 ≈ 0.5983 0.50 0.345

1 − 0.50 − 0.34 Estimation xb = 0.34: · 0.50

Result:



xa = 0

xa = 0.10

xa = 0.30

xa = 0.50

xa = 1.0

xb = 0

xb = 0.343

xb = 0.40

xb = 0.345

xb = 0

(2) Vapor residue curve: −1 −1  −1 ya (αac −1)/(αac −αbc ) = yb −1 −1   −1 1 − yFa − yFb yFa (αac −1)/(αac −αbc ) = · yFa yFb

Equation (3.44):

1 − ya − yb · ya



−1 −1 −1 Exponent (αac − 1)/(αac − αbc ) = (3.50−1 − 1)/(3.50−1 − 2.10−1 ) =

= 3.750 1 − yFa − yFb · yFa =



yFa yFb

−1 −1 (α−1 ac −1)/(αac −αbc )

1 − 0.30 − 0.40 · 0.30

= 

0.30 0.40

3.750

= 0.3400 .

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3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Values for ya = 0.10:

1 − 0.10 − 0.20 · 0.10



Estimation yb = 0.20:

1 − 0.10 − 0.25 · 0.10



Estimation yb = 0.25:

0.10 0.20

3.750

0.10 0.25

3.750

= 0.5203 6= 0.3400 = 0.2092 6= 0.3400

 0.10 3.750 = 0.3533 6= 0.3400 0.22   1 − 0.10 − 0.222 0.10 3.750 Estimation yb = 0.222: · = 0.3407 ≈ 0.3400 0.10 0.222 Estimation yb = 0.22:

1 − 0.10 − 0.22 · 0.10



1 − 0.50 − 0.40 · 0.50



Values for ya = 0.50: Estimation yb = 0.40:

0.50 0.40

3.750

= 0.4618 6= 0.3400

 0.50 3.750 = 0.3077 6= 0.3400 0.42  3.750 1 − 0.50 − 0.415 0.50 Estimation yb = 0.415: · = 0.3413 ≈ 0.3400 0.50 0.415

1 − 0.50 − 0.42 Estimation yb = 0.42: · 0.50

Result:



ya = 0

ya = 0.10

ya = 0.30

ya = 0.50

ya = 1.0

yb = 0

yb = 0.222

yb = 0.40

yb = 0.415

yb = 0

(3) Distillation line: n αac · xFa αn · xFb ; xb,n = bc N N n n with N = 1 + (αac − 1) · xFa + (αbc − 1) · xFb

Equation (2.217): xa,n =

Values for n = −2:

N = 1 + (3.50−2 − 1) · 0.3 + (2.10−2 − 1) · 0.4 = 0.4152 xa =

3.50−2 · 0.3 2.10−2 · 0.4 = 0.0590 ; xb = = 0.2185 0.4152 0.4152

Values for n = 2:

N = 1 + (3.502 − 1) · 0.3 + (2.102 − 1) · 0.4 = 5.7390 xa =

3.502 · 0.3 2.102 · 0.4 = 0.6404 ; xb = = 0.3074 5.7390 5.7390

Result:

xa = 0

xa = 0.0590

xa = 0.30

xa = 0.6404

xa = 1.0

xb = 0

xb = 0.2185

xb = 0.40

xb = 0.3074

xb = 0

Please compare the values of all three lines with Figure 3.16A.

167

3.2 BATCHWISE OPEN DISTILLATION AND OPEN CONDENSATION

3.2.3

Multicomponent Mixtures

Open distillation and open condensation of multicomponent mixtures are of great industrial importance [Vogelpohl 1965]. 3.2.3.1

Open Distillation

The Rayleigh equation (3.23) can be applied to multicomponent systems also. With yi∗ = Ki · xi , it follows from Eq. (3.22) that

dL · Ki · xi = d (L · xi ) .

(3.46)

Here, the term d(L · xi ) causes problems because the amount of liquid L and the concentrations xi are changed during operation. More advantageous is the use of component amounts li that are defined by

li = L · xi .

(3.47)

The liquid concentrations xi are replaced by

xi = li /L .

(3.48)

With Eqs. (3.47) and (3.48), Eq. (3.46) becomes

dL · Ki · li /L = dli or

dL 1 dli = · . L Ki li

(3.49)

This equation is equivalent to Eq. (3.24). It also cannot be directly integrated. For ideal systems with constant relative volatilities αik , the changes of component i are referred to the highest-boiling component k :

dL 1 dli = · L K i li

and

dL 1 dlk = · . L K k lk

(3.50)

With Ki /Kk = αik

dli dlk = αik · . li lk

(3.51)

Integration yields  

li lFi

1/αik

la lFa

1/αak

= =

lk lFk 

or

lb

lFb

1/αbk



=

lc lFc

1/αck



= ... =

lk lFk

(3.52)

1

.

The amount of liquid L in the still is

L=

k X i=1

li .

(3.53)

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3 SINGLE-STAGE DISTILLATION AND CONDENSATION

The concentrations in the still follow from

xi = li /L .

(3.54)

Figure 3.17 displays the liquid concentrations in the still for a quaternary ideal mixture with the relative volatilities αad = 10, αbd = 3, and αcd = 2. Note that the intermediate boiling components b, c exhibit a concentration maximum. This is a very essential feature that will be encountered in multistage distillation, too.

Figure 3.17 Open distillation of an ideal quaternary mixture (calculated with relative volatilities αad = 10, αbd = 3, and αcd = 2). The intermediate boiling components b and c exhibit a concentration maximum over the relative amount of distillate.

3.2.3.2

Open Condensation

As has been outlined before, the relations of open distillation can be applied to open condensation by simply replacing L, x, α, and K by G, y , α−1 , and K −1 , respectively. Hence, Eq. (3.46) transfers into

dG · Ki−1 · yi = d (G · yi ) .

(3.55)

Here, the term d(G · yi ) causes problems because both the amount of vapor G and the concentrations yi change during operation. More advantageous is the use of component amounts gi :

gi = G · yi .

(3.56)

From both equations, it follows that

dG dgi = Ki · G gi

and

dG dgk = Kk · . G gk

(3.57)

3.3 SEMI-CONTINUOUS SINGLE-STAGE DISTILLATION

169

This equation is equivalent to Eq. (3.49). It also cannot be directly integrated. For ideal systems with constant relative volatilities αik , the changes of component i are referred to the highest-boiling component k :

dG dgi = Ki · G gi

and

dG dgk = Kk · . G gk

(3.58)

With Ki /Kk = αik ,

dgi −1 dgk = αik · . gi gk Integration yields   gi αik gk = or gFi gFk         ga αak gb αbk gc αck gk 1 = = = ... = . gFa gFb gFc gFk

(3.59)

(3.60)

The amount of remaining vapor is

G=

k X

gi .

(3.61)

i=1

The concentrations in the still follow from

yi = gi /G .

(3.62)

Figure 3.18 shows the liquid concentrations in the still for a quaternary ideal mixture with the relative volatilities αad = 10, αbd = 3, and αcd = 2. Alike Figure 3.16, the intermediate boiling components exhibit a concentration maximum during operation.

3.3

Semi-continuous Single-Stage Distillation

Semi-continuous distillation can often be advantageously applied to liquids containing small amounts of impurities with low volatility. As shown schematically in Fig˙ ure 3.19, the feed F˙ is fed continuously to the boiler, and the overhead product D is withdrawn continuously. During the course of distillation, high boiling impurities accumulate in the boiler and have to be drained off periodically. 3.3.1

Semi-continuous Single-Stage Distillation of Binary Mixtures

In a manner analogous to that formulated by Eq. (3.23), a material balance for a binary mixture yields

F dxa = , HL xFa − ya∗ (xa )

(3.63)

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3 SINGLE-STAGE DISTILLATION AND CONDENSATION

Figure 3.18 Open condensation of an ideal quaternary mixture (calculated with relative volatilities αad = 10, αbd = 3, and αcd = 2). The intermediate boiling components b and c exhibit a concentration maximum over the relative amount of condensate.

Figure 3.19 Schematic of a semi-continuous batch distillation. High boiling components accumulate in the boiler and have to be drawn off periodically.

REFERENCES

171

where mole fractions x and y refer to the more volatile component a. HL denotes the moles of liquid hold-up in the boiler. For ideal mixtures, integration gives:

F (αab − 1) · (xa − xFa ) = HL xFa · (αab − 1) − αab αab xa · [xFa · (αab − 1) − αab ] + xFa − . 2 · ln x Fa · [xFa · (αab − 1) − αab ] + xFa (xFa · (αab − 1) − αab ) (3.64) Hence, the concentration changes of impurities in the boiler and in the distillate can be determined. Variation of the concentration with time t is given in the relationship F = F˙ · t. In the special case of non-volatile impurities (i.e. αab → ∞), Eq. (3.64) is simplified to

F xa − xFa = . HL xFa − 1

(3.65)

Rearrangement gives

xa = xFa + F/HL · (xFa − 1) .

(3.66)

The concentration xa of the more volatile substance in the boiler decreases linearly with the amount of feed, while that of the less volatile substance (1 − xa ) increases linearly.

References Bernon, C., Doherty, M.F., and Malone, M.F. (1990). Patterns of composition change in multicomponent batch distillation. Chemical Engineering Science 45 (5): 1207–1221. Doherty, M.F. and Perkins, J.D. (1979). On the dynamics of distillation processes III. Chemical Engineering Science 34 (6): 1401–1414. Hausen, H. (1935). Rektifikation von Dreistoffgemischen. Forschung auf dem Gebiet des Ingenieurwesens 6: 9–22. Hausen, H. (1952). Rektifikation von Dreistoffgemischen. Zeitschrift für Angewandte Physik 4: 45–51. Ostwald, W. (1900). Dampfdrucke ternärer Gemische. Abhandlungen der Mathematisch-Physischen Classe der Königlich Sächsischen Gesellschaft der Wissenschaften 25: 413–453. Petlyuk, F.B. (1986). Rectification diagrams for ternary azeotropic mixtures. Theoretical

Foundations of Chemical Engineering 20 (3): 175–185. Rayleigh, L. (1902). On the distillation of binary mixtures. Philosophical Magazine (Sixth Series) 4 (23): 521–537. Schreinemakers, F.A.H. (1901). Dampfdrucke ternärer Gemische. Zeitschrift für Physikalische Chemie 36: 257–289 and 413–449. van der Waals, J.D. (1902). Ternary systems V. Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings 5, p. 228. Vogelpohl, A. (1963). Der Einfluß der Stoffaustauschwiderstände auf die Rektifikation von Dreistoffgemischen. Forschung auf dem Gebiete des Ingenieurwesen 29: 907–915 and 1033–1045. Vogelpohl, A. (1965). Offene Verdampfung idealer Mehrstoffgemische. Chemie Ingenieur Technik 37: 1144–1146.

173

4 Multistage Continuous Distillation (Rectification) As outlined in Chapter 3, just a limited separation of components in a mixture is achieved by single-stage distillation as the composition differences of the distillation products are often very small. Consequently, pure substances are not achieved in most cases. However, the most important task in the process industries is a total fractionation of liquid mixtures into pure substances. Very efficient techniques for total fractionation of liquid mixture are presented in Chapter 4 for continuous operation and in Chapter 6 for discontinuous operation.

4.1

Principles

The fundamental principle for total fractionation of liquid mixtures is the multiple distillation in countercurrently operated columns. Figure 4.1A represents a repeated single-stage distillation, called rectification. The overhead product of the first distillation is subjected to a second distillation, whose overhead product, in turn, is subjected to a third distillation, and so on. The same is done with the bottom product of the first distillation. As shown in Figure 4.1B, the vapor and liquid concentrations of each step can be easily determined at the (y, x )-diagram (Figure 4.1B) by methods developed in Section 3.1. This process yields very pure fractions, but it has two serious disadvantages: • Only a small part of the feed mixture is recovered in a highly concentrated state. • At each stage, by-products are obtained, which often cannot be processed further. Both of these disadvantages can be overcome if the by-products from each distillation step are returned to the previous step, as shown in Figure 4.2A. This process can be further improved, because the vapor needs not be condensed before each step and the liquid does not have to be subsequently reevaporized. Vapor and liquid are brought into direct contact (Figure 4.2B). In fact, intensive countercurrent contact between the vapor and liquid streams at each stage is important to provide high mass transfer rates between the two phases. The concentrations attainable at each step can be determined at the (y, x )-diagram by means of material balances around the top of the column, as shown in Figure 4.3. Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

174

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.1 Multiple distillation of a binary mixture a/b. A) Flow diagram. B) Determination of vapor and liquid mole fractions of component a at each stage on a (y, x )-diagram.

4.1 PRINCIPLES

175

Figure 4.2 Improvement of multiple distillation by returning liquid from each distillation step to the previous step with (A) and without (B) condensation and vaporization, respectively.

176

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

The smaller the distance between operating and equilibrium line, the more difficult is the separation. Two methods are usually employed to characterize the difficulty of a separation: the equilibrium-stage concept and the transfer-unit concept.

Figure 4.3 Equilibrium-stage concept for determining countercurrent separation efficiency. A) Schematic showing equilibrium stages and associated parameters: dotted-dashed line mark the envelopes of the material balance described in Eq. (4.1). B) McCabe–Thiele diagram: concentrations at each equilibrium stage and number of stages can be determined by stepping off the stages between the equilibrium line and the operating line.

4.1.1

Equilibrium-Stage Concept

The equilibrium-stage concept assumes that the vapor and liquid phases are mass transfer elements, where gas and liquid are brought into such intimate contact that they reach equilibrium with each other. The phases are subsequently separated; vapor flows upward to the stage above and liquid runs downward to the stage below (countercurrent phase flow), as shown in Figure 4.3. A material balance for the more volatile component a around the top of the column gives the following relationship ˙ L˙ between the indifor the vapor and liquid concentrations xa , ya and streams G, vidual stages n:

ya,n−1 =

L˙ n G˙ 0 L˙ 0 · xa,n + · ya,0 − · xa,0 . ˙ ˙ ˙ Gn−1 Gn−1 Gn−1

(4.1)

The subscripts n and n − 1 refer to the stages from which the stream originates, 0 being the overhead. When Eq. (4.1) is plotted on the (y, x )-diagram, it establishes

4.1 PRINCIPLES

177

the operating line (Figure 4.3B). The phase equilibrium is represented by a curve at the (y, x )-diagram. The liquid concentration xa,n is connected to the vapor concentration ya,n−1 by the material balance; the concentrations xa,n and ya,n−1 define a point on the operating line. Equilibrium exists between ya,n of the vapor above stage n, and the liquid concentration xa,n at that stage. Therefore, a vertical line at the liquid concentration xa,n intersects the equilibrium curve at point ya,n . Likewise, a horizontal line at the vapor concentration ya,n passes through the point xa,n+1 at the operating line. The point of intersection of xa,n+1 with the equilibrium curve gives the vapor concentration ya,n+1 at the stage above, and so on. Alternate use of the equilibrium curve and the operating line to step from one equilibrium stage to the other is illustrated on the (y, x )-diagram in Figure 4.3. The operating line gives the loci of states between the individual stages. The number n of equilibrium stages required to achieve a desired concentration change is a good index for the difficulty of a separation. The number of equilibrium stages, represented by steps, can be easily determined graphically on the (y, x )diagram; this is often called the McCabe–Thiele method, after the authors of the original graphical concept [McCabe and Thiele 1925]. Analytical solutions are available for a few special cases. A particularly important case occurs when the ˙ G˙ ) are straight lines; equilibrium curve (slope m) and the operating line (slope L/ the number of equilibrium stages n is given by (e.g. King 1980)    1 1 n= · ln 1 − ·Q+1 for J = 1 : n = Q. (4.2) ln J J The definitions of J and Q are given in Figure 4.4 for the stripping section and the rectifying section. When J = 1 (i.e. operating and equilibrium lines are parallel), then n = Q. Equation (4.2) can often be used to determine the number of stages for the separation of small impurities. Figure 4.4 shows a graphical plot of Eq. (4.2). 4.1.2

Transfer-Unit Concept

The concept of transfer units considers a differential mass transfer area dA of the column [Chilton and Colburn 1935]. It is illustrated schematically in Figure 4.5. Mass transfer is formulated as follows (see Chapter 9):

dN˙ = kOG · (ya∗ − ya ) · dA , where kOG is the overall mass transfer coefficient. stream dN˙ causes a change dya in vapor composition:

G˙ · dya = dN˙ .

(4.3) The molar mass transfer

(4.4)

Thus,

G˙ · dya = kOG · (ya∗ − ya ) · dA .

(4.5)

178

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.4 Graphical determination of the number of equilibrium stages n when both the operating line and the equilibrium line are straight.

Separating the variables gives

dya dA = kOG · . ya∗ − ya G˙

(4.6)

The term on the right-hand side of Eq. (4.6) contains quantities pertaining mainly to the apparatus. These factors are of interest primarily for column design and dimensioning (see Chapter 9). The difficulty of separation is characterized by the term on the left-hand side, which contains quantities pertaining mainly to the system itself. It is defined as the overall number of gas-phase transfer units NTUOG :

dNTUOG =

dya . ya∗ − ya

(4.7)

The number of transfer units can be interpreted as the ratio of the desired concentration change dya to the driving force (ya∗ − ya ) available for mass transfer. At a given

4.1 PRINCIPLES

179

liquid state xa , gas concentration ya lies at the operating line and ya∗ at the equilibrium curve. The hatched area in Figure 4.5 represents the driving force (ya∗ − ya ).

Figure 4.5 Transfer-unit concept for determining countercurrent separation efficiency. A) Flow diagram showing important parameters and material balance envelopes (I, II). B) A (y, x )-diagram: the number of transfer units NTUOG is determined by integrating between the equilibrium curve and the operating line (hatched area).

The difficulty of the required separation can be characterized in terms of transfer units NTUOG . If the value of NTUOG is very large, a very tall column is required. Transfer units are similar to equilibrium stages in that they are indicators for separation difficulty. The number of transfer units NTUOG is usually determined by integrating between the operating line and the equilibrium line, either graphically or numerically. Analytical solutions are available for a few special cases. Again, the solution for straight equilibrium and the operating lines on the (y, x )-diagram is important (e.g. King 1980):

NTUOG =

1 · ln 1 − 1/J

   1 1− · Q + 1 for J = 1 : NTUOG = Q . J (4.8)

The definitions of J and Q are given in Figure 4.6 for the stripping section and the rectifying section. When J = 1, then NTUOG = Q. Equation (4.8) is comparable to Eq. (4.2) for the number of equilibrium steps n. Figure 4.6 presents a graphical plot of Eq. (4.8).

180

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.6 Graphical determination of the number of transfer units NTUOG when both the operating line and the equilibrium line are straight.

4.1.3

Comparison of Equilibrium-Stage and Transfer-Unit Concepts

The equilibrium-stage concept for characterizing the difficulty of separation provides an adequate measure of separability for distinct mass transfer elements in countercurrent columns, e.g. in highly efficient tray columns. On the other hand, the transferunit concept is preferred when vapor and liquid are in continuous contact, e.g. in packed columns. However, the equilibrium-stage concept is used almost exclusively in practice because graphical determination of the number of equilibrium steps on the (y, x )-diagram is much easier than integration with respect to the driving force. The equilibrium curve and the operating line can almost always be linearized in narrow concentration ranges. The following relationship is then obtained from Eqs. (4.2) and (4.8):

NTUOG = n ·

ln J 1 − 1/J

with

NTUOG = n

for

m = 1 . (4.9) ˙ L/G˙

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

181

In the special case of parallel operating and equilibrium lines, the number of transfer units NTUOG is equal to the number of equilibrium stages n. This condition is often satisfied to a close approximation in difficult separations that require many equilibrium stages or transfer units.

4.2

Multistage Distillation of Binary Mixtures

A continuously operated distillation column is shown schematically in Figure 4.7. The binary feed (molar flow rate F˙ ) is introduced toward the middle of the column. ˙ ), which mainly contains the lower boiling compoThe distillate (molar flow rate D nent a, is withdrawn from the top of the column. The bottom product (molar flow rate B˙ ), which contains the higher boiling component b in higher concentration, is removed from the bottom of the column. Some overhead and bottom products are returned into the column to maintain countercurrent flow of vapor and liquid within the column. As discussed in Section 4.1, this is an indispensable prerequisite for the separation of feed into the substances a and b. The sections of the column above and below the feed are called the rectifying section and the stripping section, respectively.

Figure 4.7 Flow sheet of a continuously operated distillation column. Part of the overhead product and part of the bottom product are returned into the column to provide countercurrent flow within the column. Envelopes for material balances discussed in the text are indicated by dotted-dashed lines (I–IV).

182

4.2.1

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Calculations Based on Material Balances

Distillation processes can be modeled in the simplest way with the help of material balances by assuming that the vapor and liquid flow rates G˙ and L˙ are constant in the column (see Section 4.2.2). The overall column material balance (envelope I in Figure 4.7) yields

F˙ = D˙ + B˙

and

F˙ · zFa = D˙ · xDa + B˙ · xBa ,

(4.10)

where zFa , xDa , and xBa are the mole fractions of the more volatile substance a in the feed, distillate, and bottoms, respectively. Rearrangement gives

zFa − xBa D˙ = F˙ · xDa − xBa

and

xDa − zFa B˙ = F˙ · . xDa − xBa

(4.11)

˙ and bottoms B˙ These equations can be used to determine the quantities of distillate D for given feed and product specifications. A material balance for the upper section of the column (envelope II in Figure 4.7) gives a relationship for the states within the rectifying section of the column: G˙ = L˙ + D˙ and G˙ · ya = L˙ · xa + D˙ · xDa ,

(4.12)

where L˙ is the molar flow rate of the liquid returned to the column, called reflux. Thus: ! L˙ L˙ ya = · xa + 1 − · xDa . (4.13) G˙ G˙ The exact position in the column to which envelope II applies is not defined. Hence, Eq. (4.13) gives the relationship between the vapor concentration ya and the liquid concentration xa at any point within the column above the point of feed entry, i.e. ˙ G˙ is constant, Eq. (4.13) represents a straight line – in the rectifying section. If L/ ˙ G˙ on the (y, x )-diagram, which the rectifying operating line – with a slope of L/ intersects the ya = xa diagonal at xDa (Figure 4.8, McCabe–Thiele diagram). ˙ G˙ is known as the internal reflux ratio. The use of the external reflux The ratio L/ ratio RL is advantageous. The definition of RL is

˙ D˙ RL ≡ L/

with

˙ G˙ = L˙ + D;

L˙ RL = . ˙ R G L+1

(4.14)

Thus, Eq. (4.13) becomes

ya =

RL 1 · xa + · xDa . RL + 1 RL + 1

(4.15)

Analogously, material balance III in Figure 4.7 for the lower section of the column yields the following relationship for the stripping section: ! L˙ L˙ ya = · xa + 1 − · xBa . (4.16) G˙ G˙

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

183

Figure 4.8 McCabe–Thiele diagram for binary distillation. The area between the operating line and the equilibrium curve is decisive for operation. Equilibrium stages are also shown.

The definition of the external reboil ratio RG is

˙ B˙ RG ≡ G/

with

L˙ = G˙ + B˙ ;

L˙ RG + 1 = . ˙ RG G

(4.17)

The operating line Eq. (4.16) written with the external reboil ratio is

ya =

RG + 1 1 · xa − · xBa . RG RG

(4.18)

Equation (4.18) defines the relationship between the vapor concentration ya and the liquid concentration xa . If the flow rates G˙ and L˙ are assumed to be constant, it represents a straight line in the (y, x )-diagram (Figure 4.8), called stripping operating line. The feed is introduced into the column at the point of intersection of the rectifying and stripping operating lines defined by Eqs. (4.15) and (4.18) (see Figure 4.8). Figure 4.9 depicts the internal liquid concentration profiles xa and xb in the column. The low boiler a enriches in the upper section, and the high boiler b enriches in the lower section of the column. The feed F˙ enters the column at that point where the internal liquid concentration is equal to the feed concentration. There is also plotted a typical temperature profile in Figure 4.9. Material balance envelope IV in Figure 4.7 around the feed point gives the following equation for the point of intersection with the operating line (e.g. King 1980):

ya =

qF 1 · xa + · zFa . qF − 1 qF − 1

(4.19)

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.9 Internal concentration and temperature profiles of a binary distillation column.

The caloric factor qF characterizes the thermal state of the feed. It is defined by

qF ≡

enthalpy required for vaporization of 1 mol of feed . molar latent heat of vaporization of feed

If the feed is a vapor–liquid mixture, the caloric factor qF describes the fraction of liquid in the feed:

qF =

L˙ F F˙

and

1 − qF =

G˙ F F˙

for

0 ≤ qF ≤ 1 .

(4.20)

If the feed is introduced as a boiling liquid, the value of qF is 1 and Eq. (4.19) gives a vertical line in the (y, x )-diagram in Figure 4.10; i.e. the operating lines intersect at xa = zFa . However, if the feed is introduced as a saturated vapor, the value of qF is 0 and Eq. (4.19) yields a horizontal line. In turn, the two operating lines intersect at ya = zFa . If the feed is introduced into the column as a vapor–liquid mixture, the point of intersection of the two operating lines will be between the two extreme cases described above, as shown in Figure 4.8. The feed line can also represent a subcooled liquid (qF ≥ 1.0) and a superheated vapor feed (qF ≤ 0); see Figure 4.10. According to Eq. (4.19), the feed line defines the loci of all points of intersection of the two operating lines of the rectifying and stripping section of the column. Thus, if the feed conditions and the composition of overhead and bottom products have been specified, and one operating line is established, the position of the other operating line is also known. This means that a relationship exists between the external reflux ratio RL and the external reboil ratio RG . From the definitions of RL , RG , and qF ,

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

185

Figure 4.10 Feed lines for different values of the caloric state qF of the feed.

it follows (see Example 4.1) that:

RG =

˙ F˙ · (RL + 1) − (1 − qF ) D/ . ˙ F˙ 1 − D/

(4.21)

For the two important cases of boiling liquid feed (qF = 1) and saturated vapor feed (qF = 0), Eq. (4.21) simplifies

˙ F˙ · (RL + 1) D/ for qF = 1 , ˙ F˙ 1 − D/ ˙ F˙ · (RL + 1) − 1 D/ RG = for qF = 0 . ˙ F˙ 1 − D/ RG =

(4.22)

˙ F˙ is the relative quantity of overhead product, which can be determined The term D/ by using Eq. (4.11) after the product concentrations have been specified. The calculation of a rectification column with multiple feed entry and side stream withdrawal is conducted in an analogous manner (Figure 4.11). The operating line breaks at each feed or withdrawal point. Figure 4.11B shows the course of the operating lines when F˙ 1 is a vaporous feed stream (F˙ 1 > 0); Figure 4.11C shows the corresponding lines for a vaporous side stream (F˙ 1 < 0). Sometimes intermediate reboilers and/or intermediate condensers are used in distillation columns. The intermediate heat supply or heat removal causes changes of ˙ G˙ is internal gas and liquid flow rates. In consequence, the liquid-to-gas ratio L/ changed, and the operating line breaks. Depending on the details of heat supply or removal (e.g. total evaporation of a partial liquid stream or partial evaporation of the total liquid stream), the position of the operating line might also be shifted. As can be seen in Figure 4.12, the operating lines are shifted closer to the equilibrium line by the use of intermediate condensers or reboilers. The driving force for mass transfer becomes much smaller, and, in turn, the exergy losses decrease as will be shown in Chapter 7.

186

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.11 Distillation column and McCabe–Thiele diagram for multiple feed entries and side-stream withdrawal, respectively. A) Flow diagram. B) McCabe–Thiele diagram for two feeds. C) McCabe–Thiele diagram for one feed and one side stream.

Figure 4.12 McCabe–Thiele diagram for a distillation column with intermediate condenser in the rectifying section and intermediate reboiler in the stripping section.

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

187

If the positions of the operating lines and the equilibrium curve on the (y, x )diagram are known, the difficulty of separation is determined by using the number of equilibrium stages n (shown as steps in Figure 4.8) or the number of transfer units NTUOG (shaded area in Figure 4.8). The desired separation is possible only if the operating lines neither touch nor intersect the equilibrium curve. The smaller the space between equilibrium curve and operating lines, the more difficult is the separation.

Example 4.1: Relationship Between Reflux Ratio RL and Reboil Ratio RG Derive a relation between reflux ratio RL and reboil ratio RG for constant molar overflow.

˙ in each column section. Solution strategy: Consider the vapor flow rates G Vapor flow rate in the rectifying section from Eqs. (4.12) and (4.14):   G˙ 1 = L˙ + D˙ = (RL + 1) · D˙ Vapor flow rate in the stripping section from Eq. (4.17):   G˙ 2 = RG · B˙ = RG · F˙ − D˙ Vapor flow rate in the feed from Eq. (4.20):

G˙ 3 = (1 − qF ) · F˙ Overall balance with G˙ 1 = G˙ 2 + G˙ 3 yields   ˙ F˙ = RG · 1 − D/ ˙ F˙ + (1 − qF ) (RL + 1) · D/ The final result is

RG =

˙ F˙ · (RL + 1) − (1 − qF ) D/ ˙ F˙ 1 − D/

This relationship is very important for developing control configurations for distillation columns (see Chapter 10).

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.13 Feasible regions for operating lines of binary distillation. A) Zeotropic mixture. B) Azeotropic mixture.

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

189

The operating lines can only lie in the region between the equilibrium curve and the diagonal of the (y, x )-diagram (shaded area in Figure 4.13). The end points of the operating lines are fixed on the diagonal because the reflux L˙ and the reboil G˙ are generated by condensing or reboiling part of the top and bottom product, respectively. This mode of providing the second phase necessary for countercurrent flow also results in a limitation of the amount of reflux and reboil. Consequently, the slope of the equilibrium line in the rectifying section is always equal to or less than 1 and in the stripping section equal to or larger than 1. Mass transfer between the phases is possible only if the operating lines do not touch or intersect the equilibrium line. There has always to be a driving force |ya∗ − ya | = 6 0. Consequently, an azeotrope is a barrier for distillation (see Section 8.6.2). 4.2.2

Calculation Based on Material and Enthalpy Balances

The calculation of multistage distillation based on material balances alone is possible only if the vapor and liquid flow rates (G˙ and L˙ , respectively) are constant in the individual column sections (constant molar overflow). If this condition is not met, the enthalpy balance has also to be taken into account. An enthalpy–concentration diagram, shown in Figure 4.14 for a binary mixture a/b, is very convenient for this procedure [Bošnjaković and Knoche 1997]. The overall enthalpy balance I for the column shown in Figure 4.7 gives the following relationship:

F˙ · hF + Q˙ R = D˙ · hD + B˙ · hB + Q˙ C ,

(4.23)

where hF , hD , and hB are the specific enthalpies of the feed, distillate, and bottom, respectively. Q˙ R is the heat flow supplied by the reboiler, and Q˙ C is the heat removed by the condenser. Thus

F˙ · hF = D˙ · πD + B˙ · πB

with and

Q˙ C D˙ ˙ QR πB ≡ hB + . B˙

πD ≡ hD +

(4.24)

Equation (4.24) represents a straight line in Figure 4.15A that extends from point πD to point πB passing through F˙ . The amounts of heat Q˙ R and Q˙ C can be easily determined, as shown in the diagram, if the compositions of the feed and of the ˙ and the bottom desired overhead or bottom products are known. Both the distillate D ˙ product B are supposed to leave the column as a boiling liquid. Heat balance II in Figure 4.7 at the top of the column (rectifying section) gives

G˙ · hG = L˙ · hL + D˙ · πD .

(4.25)

Here, hG and hL are the specific enthalpies of the vapor and the liquid, respectively. This equation also represents a straight line through the point πD on the enthalpy– concentration diagram. If the vapor in the column is at the dew point and the liquid

190

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.14 Enthalpy–concentration diagram developed from the dew and boiling lines.

at the boiling point (an assumption that is met closely in most cases), the flow rates of the vapor G˙ and liquid L˙ streams as well as their concentrations y and x can be established as shown in Figure 4.15A. The concentrations ya and xa are points on the operating line on the (y, x )-diagram (Figure 4.15B). Analogously, if the balance lines in the stripping section are allowed to pass through πB , the vapor and liquid states are obtained from the points of intersection with the dew point and the boiling lines, respectively, as shown in Figure 4.15A. The operating lines in the rectifying and stripping sections are obtained by transferring the vapor and liquid concentrations to the (y, x )-diagram in Figure 4.15B. Generally, the operating lines are slightly curved. The quantities of vapor and liquid can be determined by using the lever rule:

G˙ · l2 = L˙ · (l1 + l2 )

or

L˙ l2 = , l1 + l2 G˙

(4.26)

where l1 is the distance between the boiling line and dew line and l2 is the distance between the pole πD and the dew line. For the special case of parallel dew and boiling lines in the enthalpy–concentration ˙ G˙ . Thus, the criteria for constant diagram, Eq. (4.26) yields a constant value for L/ vapor and liquid overflow are formulated as follows: • The heats of vaporization of the two substances a and b have to be equal.

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

191

Figure 4.15 Representation of distillation in an enthalpy–concentration diagram. A) The points of intersection of the straight lines through poles πD and πB with the dew line and the boiling line give the concentrations xa and ya , respectively, at the operating line. B) Operating lines in the McCabe–Thiele diagram developed from the enthalpy–concentration diagram.

192

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

• No heat of mixing occurs. • The boiling points of the substances should be equal. The first criterion is the most important one. Equal latent heats of vaporization can often be achieved by using molar enthalpies of vaporization, which, in turn, requires that the compositions are given in mole fractions. Therefore, distillation calculations generally employ these molar units. The enthalpy–concentration diagram of the binary ethanol/water system is shown in Figure 4.16. Weight fractions wGa and wLa are used in Figure 4.16A and, in turn, the enthalpies are referred to the mass of the components (see also Bošnjaković 1935; Perry et al. 1984). In this case, the dew curve and the boiling curve are far ˙ G˙ are not constant, and the operating from being parallel. Therefore, the values of L/ lines in the McCabe–Thiele diagram are curved. Figure 4.16B shows the enthalpy– concentration diagram for the same system with molar quantities for concentrations and enthalpies. In this case, the dew point and boiling lines are parallel in a first ˙ G˙ are constant in each column secapproximation. In consequence, the values of L/ tion, and the operating lines in the (y, x )-diagram are straight. The main reason for the differences of the two diagrams is the fact that the molar heats of vaporization of ethanol and water have nearly the same values. Generally, the molar heat of vaporization of most components are less different than the specific (mass related) heats of vaporization. This fact is one of the most decisive reasons for using mole fractions in the McCabe–Thiele diagram. ˙ G˙ are not met, the following If the criteria for a constant liquid-to-vapor ratio L/ procedure is recommended: • The poles πD and πB are established on the enthalpy–concentration diagram. • The vapor and liquid concentrations ya and xa are determined by extending straight lines that originate from these poles. The values obtained from the points of intersection are transferred into the McCabe–Thiele diagram. The curved operating line is obtained, point by point. • The number of equilibrium stages n or the number of transfer units NTUOG is determined in the usual manner on the (y, x )-diagram (Figure 4.15B). The number of equilibrium stages can also be determined on the (h, x )-diagram (e.g. Treybal 1968). However, this method is complex and inaccurate at low concentrations so that it has little practical importance. 4.2.3

Distillation of Binary Mixtures at Total Reflux and Reboil

If the internal streams G˙ and L˙ are much larger than those of the feed F˙ as well ˙ and B˙ , the internal reflux ratio L/ ˙ G˙ in all parts of the column as the products D approaches unity. Thus, according to Eqs. (4.14) and (4.17), the reflux and reboil ratio, respectively, are infinite:

RL → ∞

and

RG → ∞ .

(4.27)

In this special case, called total reflux, the energy input needed for separation is very large, but a minimum number of equilibrium stages n or transfer units NTUOG is

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

193

Figure 4.16 Enthalpy–concentration diagram of the ethanol/water system. A) Mass quantities for concentrations and enthalpies, i.e. mass fractions and mass related enthalpies. B) Molar quantities for concentrations and enthalpies, i.e. mole fractions and molar enthalpies.

194

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

required. As shown in Figure 4.17, the operating line coincides with the diagonal and is represented by a simple equation:

ya = xa .

(4.28)

Figure 4.17 McCabe–Thiele diagram of a binary mixture at total reflux and reboil.

The equilibrium state ya∗ is determined by starting with the liquid state xBa . The new liquid concentration x1 is now calculated from the vapor concentration according to Eq. (4.28). The vapor state y1∗ in equilibrium with this liquid concentration x1 is, in turn, determined, and so on. This procedure is identical with the determination of the distillation line described in Section 2.4.5. Equation (2.216) thus applies to ideal binary mixtures with constant relative volatility αab :

xDa =

n αab · xBa n − 1) · x 1 + (αab Ba

with

αab ≡ p0a /p0b .

(4.29)

Here, n is the number of equilibrium stages. After rearrangement, the minimum number of equilibrium stages nmin is obtained:   1 xDa 1 − xBa nmin = · ln · . (4.30) ln αab xBa 1 − xDa Analogously, the minimum number of transfer units N T UOG,min is given by     1 xDa 1 − xBa 1 − xBa NTUOG,min = · ln · + ln . (4.31) αab − 1 xBa 1 − xDa 1 − xDa

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

195

Equation (4.30) can be used for an analytical calculation of the number n of equilibrium stages even for column operation with non-total reflux and reboil; see Hausen 1952; Vogelpohl 2015. A new coordinate system is introduced, which makes the actual operating line the diagonal in the new (transformed) coordinate system. As the operating line is located closer to the equilibrium curve than the main diagonal, the relative volatility is smaller in the transformed concentration system. The following steps are required in the calculation procedure: • Evaluation of the points of intersection of the operating and the equilibrium lines (pinches). • Transformation of the mole fractions x, y into the new coordinates x ¯, y¯. • Calculation of the relative volatility α ¯ for the transformed coordinate system. • Evaluation of Eq. (4.30) for the transformed coordinate system. • Reverse transformation of the mole fractions from the transformed system x ¯, y¯ into the standard system x, y . The calculation procedure is explained in detail in Example 4.2. The results are plotted in Figure 4.18.

Figure 4.18 Plot of the results of Example 4.2. The dashed rectangles represent the transformed concentration space.

Example 4.2: Calculation of the Number of Equilibrium Stages of Binary Mixtures Find the number of equilibrium stages n and the concentration profile within the column by using Eq. (4.30).

196

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Data:

• Feed concentration: xFa = 0.35 , qF = 1.0 . • Product concentrations: xDa = 0.95 , xBa = 0.05 . • Relative volatility: αab = 2.50 , reflux ratio: RL = 2.3 reboil ratio: RG = 1.65 . Solution stripping section:

(1) Pinches:

αab · xa 1 + (αab − 1) · xa RG + 1 xBa Operating line: ya = · xa − ; Pinch ya = ya∗ RG RG Eliminating ya yields   RG + 1 RG + 1 xBa · (αab − 1) · x2a + − · (αab − 1) − αab · xa RG RG RG xBa − =0 RG Equilibrium line, Eq. (2.149): ya∗ =

RG + 1 1.65 + 1 · (αab − 1) = · (2.5 − 1) = 2.4091 RG 1.65 RG + 1 xBa B= − · (αab − 1) − αab = RG RG 1.65 + 1 0.05 = − · (2.5 − 1) − 2.5 = −0.9343 1.65 1.65 xBa 0.05 C=− =− = −0.0303 RG 1.65   p 1 xPa = · −B ± B 2 − 4 · A · C = 2·A   p 1 = · 0.9343 ± 0.93432 + 4 · 2.4091 · 0.0303 2 · 2.4091 xPa1 = 0.4199 ; xPa2 = −0.0299 αab · xa 2.5 · 0.4199 ∗ yPa1 = = = 0.6441 1 + (αab − 1) · xa 1 + 1.5 · 0.4199 2.5 · (−0.0299) ∗ yPa2 = = −0.0783 1 + 1.5 · (−0.0299)

A=

(2) Coordinate transformation:

xa − xPa2 ya − yPa2 ; y¯a ≡ xPa1 − xPa2 yPa1 − yPa2 0.35 + 0.0299 0.05 + 0.0299 = = 0.8464 ; x ¯Ba = = 0.1776 0.4199 + 0.0299 0.4199 + 0.0299

Definitions: x ¯a ≡

x ¯Fa

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4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

(3) Transformed relative volatility: Equilibrium at an arbitrary intermediate concentration xa,m :

xa,m ≡ (xPa1 + xPa2 )/2 = (0.4199 − 0.0299)/2 = 0.1950 2.5 · 0.1950 ∗ ya,m = = 0.3772 1 + 1.5 · 0.1950 0.3772 + 0.0783 ∗ x ¯a,m = 0.5 ; y¯a,m = = 0.6305 0.6441 + 0.0783 ∗ y¯a,m x ¯a,m 0.6305 Eq. (2.151): =α ¯ ab · ; α ¯ ab = = 1.7064 ∗ 1 − y¯a,m 1−x ¯a,m 1 − 0.6305 (4) Number of equilibrium stages:   1 x ¯Fa 1−x ¯Ba Eq. (4.30): n = · ln · = ln α ¯ ab 1−x ¯Fa x ¯Ba   1 0.8464 1 − 0.1776 = · ln · = 6.0617 ln(1.7064) 1 − 0.8464 0.1776 (5) Concentration profile: Equation (4.29): n

(¯ αab ) · x ¯Ba 1.7064n · 0.1776 = n 1 + (1.7064n − 1) · 0.1776 1 + ((α ¯ ab ) − 1) · x ¯Ba Reverse transformation: x ¯an =

xa = x ¯a · (xP a1 − xPa2 ) + xPa2 = x ¯a · (0.4199 + 0.0299) − 0.0299 n

1

2

3

4

5

6

x ¯a xa

0.2698 0.0915

0.3861 0.1438

0.5176 0.2030

0.6468 0.2611

0.7575 0.3109

0.8421 0.3489

Solution rectifying section:

(1) Pinches:

xPa1 = 1.0489 ; yPa1 = 1.0190 ; xPa2 = 0.2625 ; yPa2 = 0.4709 (2) Coordinate transformation:

x ¯Da = 0.8742 ; x ¯Fa = 0.1113 (3) Transformed relative volatility:

α ¯ ab = 1.8458

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

(4) Number of equilibrium stages:   1 x ¯Da 1−x ¯Fa n= · ln · = ln α ¯ ab 1−x ¯Da x ¯Fa   1 0.8742 1 − 0.1113 = · ln · = 6.55 ln (1.8458) 1 − 0.8742 0.1113 (5) Concentration profile:

n

1

2

3

4

5

6

6.55

x ¯a xa

0.1878 0.4102

0.2991 0.4977

0.4406 0.6090

0.5925 0.7284

0.7285 0.8354

0.8320 0.9168

0.8740 0.9498

Compare with Figure 4.18.

4.2.4

Distillation of Binary Mixtures at Minimum Reflux and Reboil

Column operation with minimum internal gas and liquid flow rates (i.e. minimum reflux or reboil) has considerable practical importance because it results in separation of a mixture by using the lowest energy input. Such operation is characterized on a McCabe–Thiele plot by the intersection of the rectifying and stripping operating lines at the same point on the equilibrium curve. This point is called pinch point, and an infinite number of equilibrium stages n or transfer units NTUOG is required to reach this state. Figure 4.19 shows the (y, x )-diagram for a binary system with minimum internal loads. If the vapor and liquid flow rates (G˙ and L˙ , respectively) in the individual sections of the column are constant, the operating lines are straight ones. The minimum slope of the rectifying operating line for the more volatile component a can be calculated as follows: ! ∗ L˙ xDa − yFa = for qF = 1 . (4.32) xDa − yFa G˙ min

˙ G˙ = RL /(RL + 1), the minimum reflux ratio becomes With the relationship L/ RL,min =

∗ xDa − yFa ∗ −x yFa Fa

for

qF = 1 .

(4.33)

A similar relationship is derived for the reboil ratio RG,min . The slope of the stripping line is ! L˙ y ∗ − xBa = Fa for qF = 1 . (4.34) ˙ xFa − xBa G max

199

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

Figure 4.19 McCabe–Thiele diagram of a binary mixture at minimum reflux ratio and minimum reboil ratio for a saturated liquid feed, qF = 1.0. The operating lines intersect the equilibrium line at the pinch point.

˙ G˙ = (RG + 1)/RG , the minimum reboil ratio becomes With the relation L/ RG,min =

xFa − xBa ∗ −x yFa Fa

for

qF = 1 .

(4.35)

∗ Analogously, the reflux ratio RL,min is the concentration change (xDa − yFa ) in the ∗ rectifying section of the column referred to the driving force (yFa − xFa ) at the feed point. Figure 4.19 allows a simple and clear interpretation of reflux and reboil ratios. The reboil ratio RG,min is the concentration change (xFa − xBa ) established in the ∗ stripping section of the column divided by the driving force (yFa − xFa ) at the feed point. It is important to be aware that above equations are valid for all components i in a mixture. Moreover, they also are valid for non-ideal mixtures as just one point of the equilibrium line is needed. Therefore, they are advantageously rewritten as follows:

RL,min =

∗ xDi − yFi ∗ yFi − xFi

and

RG,min =

xFi − xBi ∗ −x yFi Fi

for qF = 1 .

(4.36)

∗ Concentrations yFi and xFi are related via the phase equilibrium. This equation is applied to all processes as long as the pinch, i.e. the intersection of operating and equilibrium line, lies at the feed point. Especially in azeotropic systems a tangential pinch may occur (see Figure 4.20) whose concentration xPi is different from the feed concentration xFi .

200

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.20 Minimum reboil ratio with tangential pinch. In such systems the pinch concentration xPi is different from feed concentration xFi .

For ideal mixtures the following equation holds (see Eq. (2.217) in Section 2.4.5):

ya∗ = αab ·xa /N

and

yb∗ = 1·xb /N

N = 1+(αab − 1)·xa . (4.37)

with

The reflux ratio is formulated for the low and the high boiling component:

xDa − αab · xFa /N or αab · xFa /N − xFa xDb − xFb /N = for qF = 1 . xFb /N − xFb

RL,min = RL,min

(4.38)

For sharp separations (i.e. xDb = 0), it follows that

RL,min =

1 N −1

or RL,min =

1 (αab − 1) · xFa

qF = 1 .

(4.39)

with N = 1 + (αab − 1) · xFa .

(4.40)

for

The minimum reboil ratio is

RG,min =

xFa − xBa αab · xFa /N − xFa

For sharp separations (i.e. xBa = 0)

RG,min =

N αab − N

or

RG,min + 1 =

1 1−

−1 αab




· xFb

for

qF = 1 . (4.41)

201

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

Mind the feed concentration of component b in Eq. (4.41). The conditions for a vaporous feed (saturated vapor qF = 0) are shown in Figure 4.21. The basic equations for minimum reflux and minimum reboil are

RL,min =

xDi − yFi yFi − x∗Fi

and

RG,min =

x∗Fi − xBi yFi − x∗Fi

for

qF = 0 .

(4.42)

Figure 4.21 McCabe–Thiele diagram of a binary mixture at minimum reflux ratio and minimum reboil ratio for a saturated vapor feed, qF = 0. The operating lines intersect the equilibrium line at the pinch point.

These equations are valid for all components i in a mixture, too. They can be applied even to non-ideal mixture as only one point of the equilibrium curve is relevant. The following holds for ideal mixtures:

RL,min =

xDb − yFb yFb − yFb /N 0

with


−1 N 0 = 1 + αab − 1 · yFa

for

qF = 0 . (4.43)

For sharp separations with xDb = 0, it follows that

RL,min =

N0 1 − N0

or

RL,min + 1 =

1
−1 1 − αab · yFa

for qF = 0 . (4.44)

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

A similar relationship is derived for the minimum reboil ratio in ideal mixtures: −1 αab · yFa /N 0 − xBa −1 yFa − αab · yFa /N 0
−1 with N 0 = 1 + αab − 1 · yFa

RG,min =

(4.45)

qF = 0 .

for

In sharp separations with xBa = 0, the above equations become

RG,min =

−1 αab −1 N 0 − αab

RG,min =

or

(αab

1 − 1) · yFb

for qF = 0 . (4.46)

All these equations for sharp separations of ideal mixtures are compiled in Figure 4.22 in order to show the similarity of these equations. Noteworthy is the fact that the feed concentrations of substance a are decisive for the reflux ratio (at the top of the column) and that of substance b for the reboil ratio (at the bottom of the column). The general case of the caloric state of the feed is depicted in Figure 4.23. Here again, the basic equations apply:

RL,min =

∗ xDi − yFi ∗ ∗ yFi − xFi

and

RG,min =

x∗Fi − xBi ∗ − x∗ yFi Fi

for any values of qF . (4.47)

∗ yFi

x∗Fi

The relevant concentrations and are found by the intersection of the feed line Eq. (4.19) and the equilibrium curve Eq. (4.37). The following holds for ideal mixtures: 2

(αab − 1) · qF · (x∗Fa ) − (αab − (αab − 1) · (qF + zFa )) · x∗Fa − zFa = 0 . (4.48) The solution of this quadratic equation is:   p 1 x∗Fa = · −B + B 2 − 4 · A · C 2·A with A = (αab − 1) · qF

− B = (αab − (αab − 1) · (qFa + zFa )) The equilibrium concentration ∗ yFa = αab · x∗Fa /N

∗ yFa

with

(4.49)

;

C = −zFa .

comes from the vapor–liquid equilibrium:

N = 1 + (αab − 1) · x∗Fa .

(4.50)

The above equations are very helpful for calculating the energy requirement of binary distillations; see Section 4.2.5. Separation at minimum reflux requires an infinite number of equilibrium stages n or transfer units NTUOG . However, even a slight increase in the reflux of approximately 5 – 10 % results in economical operation. Thus, the special case and laws discussed in this section closely approximate actual distillations and have, in turn, great practical importance.

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

203

Figure 4.22 Minimum reflux and reboil ratios for sharp separations of ideal binary mixtures for saturated liquid feed and saturated vapor feed, respectively.

Figure 4.23 McCabe–Thiele diagram of a binary mixture at minimum reflux ratio and minimum reboil ratio for a vapor–liquid feed (0 ≤ qF ≤ 1.0). The operating lines intersect the equilibrium curve at the pinch point.

204

4.2.5

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Energy Requirement for Distillation of Binary Mixtures

In distillation columns, the heat Q˙ R is supplied in the reboiler at the bottom of the column and the heat Q˙ C is removed by the condenser at the top. The following equations hold:

Q˙ R = r · G˙ 0

and

Q˙ C = r · G˙ .

(4.51)

Here, G˙ 0 and G˙ denote the vapor stream at the bottom and the top of the column, respectively. Approximately equal molar latent heats r of vaporization of the substances involved are supposed in the following. Important relationships can be established for the quantities of heat Q˙ R and Q˙ C . The energy required by the reboiler Q˙ R is formulated by using the reboil ratio RG ≡ G˙ 0 /B˙ . The heat removed by the condenser Q˙ C is determined by using ˙ D˙ . Thus the reflux ratio RL ≡ L/

Q˙ R = r · RG · B˙

and

Q˙ C = r · (RL + 1) · D˙ .

(4.52)

The term (RL + 1) arises in the right-hand equation because both the reflux liquid and the overhead product have to be condensed at the top of the column. At the bottom, however, only the reboil has to be provided by the evaporator. ˙ can be expressed in terms of the feed stream F˙ The product flow rates B˙ and D and the composition of the products by using Eq. (4.11):

Q˙ R xDa − zFa = RG · ˙ x F ·r Da − xBa

and

Q˙ C zFa − xBa = (RL + 1) · . ˙ x F ·r Da − xBa

(4.53)

Consequently, Q˙ R and Q˙ C are directly proportional to the reboil ratio RG and the reflux ratio RL , respectively. These ratios cannot fall below a limiting value (for details, see Section 4.2.4). The minimal quantities of heat are calculated as follows:

Q˙ R,min xDa − zFa = RG,min · and ˙ x F ·r Da − xBa Q˙ C ,min zFa − xBa = (RL,min + 1) · . ˙ xDa − xBa F ·r

(4.54)

The above equations are compiled in Figure 4.24. Inserting Eqs. (4.39) and (4.41) for minimum reflux and reboil ratios in Eq. (4.54) enables the direct calculation of the energy requirement of a sharp separation of binary mixtures for a boiling liquid feed with qF = 1.0; see Figure 4.25. The result is:

Q˙ R,min 1 = xFa + ˙ α F ·r ab − 1 Q˙ C ,min 1 = xFa + ˙ αab − 1 F ·r

and (4.55) for qF = 1 .

4.2 MULTISTAGE DISTILLATION OF BINARY MIXTURES

205

˙ F˙ · r) of distillation Figure 4.24 Basic equations for calculating the energy requirement Q/( columns.

Figure 4.25 Compilation of the relevant equations for the energy requirement of binary mixtures with saturated liquid feed, qF = 1.0.

206

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Here, the minimum energy requirement is a linear function of the feed concentration xFa . The higher the concentration of the low boiling substance a, the higher is the energy requirement (see Figure 4.27A). The energy requirement of a binary mixture with vaporous feed is (see Figure 4.26)

Q˙ R,min 1 = ˙ α F ·r ab − 1

and

Q˙ C ,min 1 =1+ ˙ α F ·r ab − 1

for

qF = 0 . (4.56)

Here, the energy requirement does not depend on the concentration of the feed. A feed with 90 % low boiler a has the same energy requirement as a feed with 5 % low boiler a. This surprising fact is very important for process design. The reason is that the latent heat of the feed is utilized for the separation only in the rectifying section of the column (Figure 4.27B). Even better is the process depicted in Figure 4.27C. Here, the vaporous feed is first used to heat the reboiler. The condensate of the feed is fed as boiling liquid into the column to be separated into the pure components. In this process, the latent heat of the feed is utilized in the stripping section as well as in the rectifying section of the column. The result is

Q˙ R,min 1 = yFa + − 1 and αab − 1 F˙ · r Q˙ C ,min 1 = yFa + for qF = 0 . ˙ α F ·r ab − 1

(4.57)

In wide boiling systems (i.e. large values of αab ), a column can often be operated without external heat. Not recommended is a pre-evaporation of a liquid feed (see Figure 4.27D), since heat supplied to the pre-evaporator is utilized only in the rectifying section of the column. Here, the energy requirement is

Q˙ R,min 1 =1+ ˙ αab − 1 F ·r Q˙ C ,min 1 =1+ ˙ αab − 1 F ·r

and (4.58) for pre-evaporization.

A process with pre-evaporization of the feed has a higher energy requirement than a process with boiling liquid feed.

4.3

Multistage Distillation of Ternary Mixtures

The distillation of binary mixtures discussed in Section 4.2 represents a very special case seldom, if ever, encountered in process industry. In general, multicomponent mixtures have to be separated. Distillation of ternary mixtures with substances a, b, and c can be explained very clearly on a triangular diagram. Relationships derived for ternary mixtures are very helpful for a better understanding of distillation. Ternary distillation is more general than the very special case of binary distillation.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

207

Figure 4.26 Compilation of the relevant equations for the energy requirement of binary mixtures with saturated vapor feed, qF = 0.

Figure 4.27 Energy requirement of ideal binary mixtures as function of the feed conditions.

208

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

The results of a rigorous simulation of the distillation of a ternary mixture are shown in Figure 4.28. Depicted are the internal concentration profiles of the liquid, which are significantly different from that of binary mixtures. Noteworthy are the following features unknown in binary distillation: • There exist concentration maxima of the intermediate boiling substance b in each column section. • At the feed point, there is a gap in the internal concentration profile. The concentration of the feed is different from the internal concentrations. The end point of the internal concentration profile in the rectifying section is mixed with the feed to give the origin of the concentration profile in the stripping section. All three points have to be collinear in the concentration space. ˙ and B˙ underlie some strong restrictions. • The feasible products D The attainable products have to meet two conditions. First, all substances entering the column with the feed F˙ have to leave the column ˙ and B˙ . A material balance around the column yields in the product fractions D

F˙ = D˙ + B˙

F˙ · xDa = D˙ · xDa + B˙ · xDa ˙ · xDb + B˙ · xBb . and F˙ · xFb = D and

(4.59)

Equation (4.59) is represented in the concentrations space by a straight line where ˙ , and B˙ have to lie at; see Figure 4.28B. the points F˙ , D ˙ and B˙ have to be end points of the internal conSecond, the product fractions D centration profiles also shown in Figure 4.28B. The calculation of these internal concentration profiles is rather difficult, making the prediction of the feasible products very complex. 4.3.1

Calculations Based on Material Balances

Calculations of ternary mixtures follow the same basic principles as of binary mixtures described in Section 4.2. However, the graphical approach in the McCabe– Thiele diagram has to be replaced by a numerical approach. Distillation of ternary mixtures is conveniently calculated by first specifying the column and the operating conditions and then determining concentrations at the top and bottom of the column. If the required product specifications are not achieved, either the operating conditions or the column configuration has to be altered and the calculation repeated. The number of equilibrium stages n and the position of the feed entry point are required for column specification. The operating conditions are characterized suf˙ F˙ . ficiently by the reflux ratio RL and the relative amount of overhead product D/ The calculation is best carried out in a manner analogous to the stepwise graphical method used for binary mixtures described in Section 4.1.1 (McCabe–Thiele diagram). By starting with an estimated liquid composition (e.g. at the bottom), the

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

209

Figure 4.28 Internal liquid concentration profiles of a ternary mixture at finite reflux ratio. A) Liquid concentration profiles along column height. B) Presentation of the liquid concentration profiles in the triangular concentration diagram. A concentration jump occurs at the feed point due to admixing of the feed. The material balance is fulfilled if F˙ , ˙ , and B˙ lie at a straight line. D

210

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

equilibrium vapor phase state yi∗ is calculated. For ideal mixtures

ya∗ =

αac · xa N

and with

αbc · xb 1 · xc and yc∗ = N N N = 1 + (αac − 1) · xa + (αbc − 1) · xb .

yb∗ =

(4.60)

For non-ideal mixtures, the equilibrium state can be determined by more complex methods that have been described in Chapter 2. The next step involves determination of the new liquid state from the vapor state by using the operating line. A material balance around the bottom of the column gives the following equation for the operating line (see Eq. (4.18)):

RG 1 · ya∗ + · xBa RG + 1 RG + 1 RG 1 xb = · y∗ + · xb RG + 1 b RG + 1 xc = 1 − xa − xb .

xa =

(4.61)

The alternate use of Eqs. (4.60) and (4.61) produces a stepwise mathematical approach similar to the graphical construction for binary mixtures on the (y, x )diagram (see Section 4.2.1). In Figure 4.28, the concentration profile observed in the stripping section of the column is represented by dots. After the feed point is reached, admixing of the feed has to be considered in the calculation process. In most cases the feed concentrations differ from the internal concentrations. The following relationship is obtained from a material balance for the liquid composition (xi )0 directly above the feed point by assuming a boiling liquid feed and negligible temperature differences at the feed entry point:

˙ F˙ (RG + 1) · B/ · (xa − xFa ) + xFa ˙ F˙ − 1 (RG + 1) · B/ ˙ F˙ (RG + 1) · B/ 0 (xb ) = · (xb − xFb ) + xFb . ˙ (RG + 1) · B/F˙ − 1 0

(xa ) =

(4.62)

Equation (4.62) represents a straight line (mixing line) through F˙ in the triangular ˙ xi ) and above (L, ˙ xi )0 the concentration diagram. The states of the liquid below (L, feed point have to lie on the mixing line, as indicated in Figure 4.28. The equation of the operating line changes above the feed point because the quantity of liquid is altered. The following holds:

RG · ya∗ + xBa − xFa · F˙ /B˙ RG + 1 − F˙ /B˙ RG · yb∗ + xBb − xFb · F˙ /B˙ xb = . RG + 1 − F˙ /B˙

xa =

(4.63)

The calculation is continued by alternating Eqs. (4.60) and (4.63) working upward step by step to the top of the column. The details of these tray-to-tray calculations are shown in Example 4.3.

211

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

Example 4.3: Tray-to-Tray Calculation Find by tray-to-tray calculation the internal concentration profile and the products concentrations of the distillation of an ideal ternary mixture Data:

Feed concentrations:

xFa = 0.30 and xFb = 0.35

Relative volatilities:

αac = 2.1 and αbc = 1.4 ˙ ˙ D = F · 1/3 ; B˙ = F˙ · 2/3

Products:

Number of equilibrium stages: 5 in stripping section and 5 in rectifying section

RG = 2.5

Reboil ratio:

Solution strategy: Tray-to-tray calculation from bottom upward

Estimation of bottoms concentrations: xBa = 0.10 and

xBb = 0.40

(1) Equilibrium and operating lines in the stripping section Equilibrium line, Eq. (4.60):

ya∗ = 2.1 · xa /N with

and

yb∗ = 1.4 · xb /N

N = 1 + (2.1 − 1) · xa + (1.4 − 1) · xb

RG 1 · y∗ + · xBa = RG + 1 a RG + 1 2.5 1 = · y∗ + · 0.1 = 0.7143 · ya∗ + 0.0286 2.5 + 1 a 2.5 + 1 2.5 1 xb = · yb∗ + · 0.4 = 0.7143 · yb∗ + 0.1143 2.5 + 1 2.5 + 1

Operating line, Eq. (4.61): xa =

(2) Tray-to-tray calculation in the stripping section: Equilibrium line

Operating line Stage 1 N = 1 + 1.1 · 0.1 + 0.4 · 0.4 = 1.2700 ya∗ = 2.1 · 0.1/1.2700 = 0.1654 yb∗ = 1.4 · 0.4/1.2700 = 0.4409 xa = 0.7143·0.1654+0.0286 = 0.1467 xb = 0.7143 · 0.4409 + 0.1143 = 0.4292 Stage 2 N = 1 + 1.1 · 0.1467 + 0.4 · 0.4292 = 1.3331 ya∗ = 2.1 · 0.1467/1.3331 = 0.2311 yb∗ = 1.4 · 0.4292/1.3331 = 0.4507 xa = 0.7143·0.2311+0.0286 = 0.1937 xb = 0.7143 · 0.4507 + 0.1143 = 0.4362

212

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

ya∗ = 0.2932 ; yb∗ = 0.4401 ya∗ = 0.3487 ; yb∗ = 0.4187 ya∗ = 0.3965 ; yb∗ = 0.3935

Stage 3 xa = 0.2380 ; xb = 0.4287 Stage 4 xa = 0.2777 ; xb = 0.4134 Stage 5 xa = 0.3118 ; xb = 0.3954

(3) Admixing of the feed (Eq. (4.62)):

˙ F˙ (RG + 1) · B/ · (xa − xFa ) + xFa ˙ (RG + 1) · B/F˙ − 1 (2.5 + 1) · 2/3 = · (0.3118 − 0.30) + 0.30 = 0.3206 (2.5 + 1) · 2/3 − 1 ˙ F˙ (RG + 1) · B/ 0 (xb ) = · (xb − xFb ) + xFb ˙ F˙ − 1 (RG + 1) · B/ 0

(xa ) =

=

(2.5 + 1) · 2/3 · (0.3954 − 0.35) + 0.35 = 0.4295 (2.5 + 1) · 2/3 − 1

(4) Operating line in the rectifying section (Eq. (4.63)):

RG · ya∗ + xBa − xFa · F˙ /B˙ RG + 1 − F˙ /B˙ 2.5 · ya∗ + 0.1 − 0.3 · 3/2 = = 1.25 · ya∗ − 0.1750 2.5 + 1 − 3/2 RG · yb∗ + xBb − xFb · F˙ /B˙ xb = RG + 1 − F˙ /B˙

xa =

=

2.5 · yb∗ + 0.4 − 0.35 · 3/2 = 1.25 · yb∗ − 0.0625 2.5 + 1 − 3/2

(5) Tray-to-tray calculation in the rectifying section: Equilibrium line

Operating line Stage 6 N = 1 + 1.1 · 0.3206 + 0.4 · 0.4295 = 1.5244 ya∗ = 2.1 · 0.3206/1.5244 = 0.4416 yb∗ = 1.4 · 0.4295/1.5244 = 0.3945 xa = 1.25 · 0.4416 + 0.1750 = 0.3770 xb = 1.25 · 0.3945 + 0.0625 = 0.4306 Stage 7 N = 1 + 1.1 · 0.3770 + 0.4 · 0.4306 = 1.5869 ya∗ = 2.1 · 0.3770/1.5869 = 0.4989 yb∗ = 1.4 · 0.4306/1.5869 = 0.3799 xa = 1.25 · 0.4989 + 0.1750 = 0.4486 xb = 1.25 · 0.3799 + 0.0625 = 0.4124

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

213

Stage 8 xa = 0.5350 ; xb = 0.3726 Stage 9 ya∗ = 0.6466 ; yb∗ = 0.3002 xa = 0.6333 ; xb = 0.3128 Stage 10 ya∗ = 0.7300 ; yb∗ = 0.2399 xa = 0.7375 ; xb = 0.2374

ya∗ = 0.5680 ; yb∗ = 0.3481

The liquid concentration profile within the column is plotted in Figure 4.29. The end points of the concentration profile do not meet the external material balance. Therefore, a better value of the bottoms concentrations has to be chosen, and the calculation repeated. The exact values are xBa = 0.0975 ; xBb = 0.3985 From a material balance (Eq. (4.64)), it follows that xDa = 0.7050 ; xDb = 0.2530 .

Figure 4.29 Results of the tray-to-tray calculation of ternary distillation in Example 4.3. The concentration at stage 10 does not meet the specification of the distillate. Therefore, the calculation has to be repeated with a better estimation of bottom concentration.

The concentrations of components a, b, and c calculated in this way are plotted as a function of the number of equilibrium stages n, as shown in Figure 4.29. The intermediate boiling substance b can pass through a concentration maximum in both sections of the column. Furthermore, a concentration gap occurs near the feed point. The result of these stepwise calculations is the internal concentration profile. In

214

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

addition, the external material balance must be met:

F˙ = D˙ + B˙

F˙ · zFa = D˙ · xDa + B˙ · xBa ˙ · xb + B˙ · xBb . and F˙ · zFb = D and

(4.64)

Equation (4.64) is represented by a straight line in Figure 4.29 on which the feed F˙ , ˙ , and the bottoms B˙ have to lie at. Usually, this condition will not the overhead D be met in the first attempt, and it is necessary to repeat the whole procedure (see Example 4.3). A very important feature of ternary distillation is the fact that the intermediate boiling component b can exhibit a concentration maximum in the stripping as well as in the rectifying section of the column. Such a concentration maximum never occurs in binary distillation. At the concentration maximum, the effective equilibrium line intersects the operating line. 4.3.1.1

Problems Arising in Column Calculations

Calculation of a ternary multistage distillation, as discussed in Section 4.3.1, is basically very simple and can be easily done with a calculator or personal computer. However, a few problems often make the calculations very complex. Choice of Reflux Ratio

The values used in practice for the reflux ratio RL and the reboil ratio RG are about 10 – 30 % higher than the minimum values. Relationships for calculating RL,min and RG,min will be developed in Section 4.3.3; they apply to ideal mixtures and to weakly non-ideal mixtures. However, mixtures that deviate more strongly from ideal behavior can produce a pinch point above or below the feed point (tangential pinch). The reflux ratio can then be determined only by iterative column calculations. Choice of Bottom State

The bottom composition xBi is the iteration variable in the calculation process described in Section 4.3.1. A poorly estimated value of xBi often gives unreasonable concentrations in the first iteration, e.g. concentrations larger than 1 or smaller than 0. A useful approach requires knowledge of separation regions (Section 4.3.2.1). Even if the choice of bottom composition is satisfactory, unreasonable concentrations can still be obtained in the rectifying section of the column. This can be avoided by starting with estimated values of the distillate composition xDi for calculation of the rectifying section. The calculations begin at the top as well as the bottom of the column and continue step by step until the feed point is reached. Estimated values are checked with the mixing calculation at the feed point. For real mixtures, this procedure is effective; however, iterations are required at each step of the calculation in the rectifying section because equilibrium relationships cannot be directly solved for liquid concentrations (in non-ideal systems).

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

215

Improvement of Estimated Values

The material balance around the column is generally not fulfilled in the first calculation. A new bottom composition must be chosen, which complies better with the balance around the column. No physically based criteria are available for improving the estimated values; this is the most severe handicap for column calculations. Indeed, serious convergence problems often occur at this point, especially in the case of highly non-ideal systems. Choice of Feed Point

For binary mixtures, the feed is introduced at that point of the column where the internal liquid concentration is equal to the composition of the feed. This is often not possible in ternary distillation because the compositions in the column are usually different from that of the feed. As a rule, the concentration of only one component can be made equal to that of the feed. Other components undergo irreversible changes of state during mixing, which energetically impair separation. A series of criteria are described by Vogelpohl 1975 for optimal choice of feed point; most of the criteria aim at decreasing the irreversibilities. 4.3.2

Distillation of Ternary Mixtures at Total Reflux and Reboil

Operation at total reflux is of minor industrial importance, but in this mode of operation the feasible separations of ternary mixtures can be determined very easily [Stichlmair 1991; Stichlmair and Herguijuela 1992]. 4.3.2.1

Feasible Bottom and Top Fractions

As demonstrated in Section 4.3.1, the bottom and overhead fractions of a distillation column have to meet two conditions. The first condition requires the material balance around the column to be fulfilled. That is,

F˙ = D˙ + B˙

F˙ · zFa = D˙ · xDa + B˙ · xBa ˙ · xb + B˙ · xBb . and F˙ · zFb = D and

(4.65)

This system of linear equations is represented in Figure 4.28B by a straight line on ˙ , and the bottom product B˙ have to lie. which the feed F˙ , the top product D The second condition requires that the concentration xDi of the top fraction and xBi of the bottom fraction are end points of the concentration profile developed within the column. In general, the internal concentration profile is a very complex function:   ˙ F˙ , . . . . xi,n = f xFi , p0i , γi , T, p, ni , RL , D/ (4.66) The calculation of the internal concentration profile is very tedious, as shown in Example 4.3. If the reflux RL in a column is very high (RL → ∞), the internal concentration profile becomes identical with a section of a distillation line; see Example 4.4

216

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Example 4.4: Distillation Line of a Binary Mixture (1) Definition of distillation lines, Eq. (4.67):

x1 →

y ∗ (x1 )

=

x2 →

y ∗ (x2 )

=

x3 →

y ∗ (x3 )

...

(2) Graphical design of a binary distillation line:

(3) Result: Distillation lines describe the liquid concentration profile in a column at total reflux and reboil (compare Figure 4.17).

[Stichlmair 1988]. Distillation lines represent a sequence of equilibrium stages, which are easily calculated by the following formula:

xi,n

→

∗ yi,n

=

xi,n+1

→

∗ yi,n+1

=

xi,n+2

. . . . (4.67)

Thus, for ideal mixtures, the course of concentration profiles within the column is (Eq. (2.217)) n n αac · xBa αbc · xBb ; x = ; xc,n = 1 − xa,n − xb,n b,n 00 N N 00 n n with N 00 = 1 + (αac − 1) · xa + (αbc − 1) · xb ,

xa,n =

(4.68)

where the subscripts a, b, and c denote the three components of the mixture, n denotes the number of equilibrium stages, and B˙ denotes the bottom product. Evaluation of Eq. (4.68) with values n = 1, 2, 3, . . . (and −1, −2, −3, . . .) gives the course of the internal concentration profile at total reflux.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

Figure 4.30 Separation regions of a ternary zeotropic mixture at total reflux and reboil. A) Determination of feasible top and bottom fractions. B) Separation regions.

217

218

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

In Figure 4.30A four feasible overhead and bottom fractions of the distillation of ˙ and bottom B˙ fractions have to lie at a a zeotropic mixture are shown. The top D ˙ straight line through the point F and, due to the assumption of total reflux RL,min , at the same distillation line. The straight line representing Eq. (4.65) forms a chord to the distillation line. Obviously, there exist a large number of possible chords through the feed F˙ . The extreme positions go through the origin or the terminus of the distillation lines. Top and bottom fractions both can lie at another distillation line, also. It should be noted that the sides of the concentration triangle are themselves distillation lines. The multiplicity of feasible top and bottom fractions forms the separation regions shaded in Figure 4.30B. These regions are limited by the distillation line through the feed F˙ and by a section of the straight line through the feed F˙ and the origin and terminus, respectively, of the distillation line. This method of separation region determination is applied to a system with a b/c minimum azeotrope in Figure 4.31. This mixture exhibits two different origins of distillation lines (peaks of boiling point surface). Thus, there has to exist a valley between these two peaks. The position of the valley is marked by the course of the boundary distillation line, which runs from the minimum azeotrope to the low boiler a. Since the straight line through the feed F˙ is always a chord of a distillation ˙ and B˙ , respectively, can lie only at the same line, the top and bottom fractions D side of the boundary distillation line as does the feed F˙ . Hence, the boundary distillation line forms a barrier that cannot be crossed by distillation. As can be seen in Figure 4.31B the boundary distillation line divides the ternary mixtures into two regions (distillation fields). Separation by distillation is possible only within the same field. As a rule, only these components can be recovered in pure form from a ternary mixture where distillation lines either begin or end. In systems with more than one origin or terminus, there always exist boundary lines that cannot be crossed by distillation (at total reflux). There are a few exceptions of this rule. One of these exceptions is shown in Figure 4.32. Here, the origin and terminus of the distillation lines are both at the same side of the concentration triangle and, in turn, the outer distillation lines bypass two corners of the triangular diagram. In this case there exists a straight line through F˙ that intersects the distillation lines on the sides at a corner (here, the a-corner). Under these special conditions, pure substance a can be separated from a ternary mixture even though it is not an origin or terminus of distillation lines. Even pure intermediate boiler b can be separated if the feed contains only small amounts of high boiler c. The second exception is dealt with in Figure 4.33. If the feed lies in the concave region of the boundary distillation line, the straight line through F˙ can form a chord to distillation lines at the other side of the boundary distillation line. In these special conditions, the boundary distillation line does not form a barrier for distillation. The separation regions expand over both sides of the boundary distillation line. This exception is very important and is frequently used in distillation processes (see Section 8.6.2). Boundary distillation lines are valleys or ridges in the boiling surface (Figure 2.9). Both of them have the same consequences for distillation. They act as a barrier that

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

219

Figure 4.31 Separation regions of a ternary mixture with a b/c minimum azeotrope at total reflux and reboil. The boundary distillation line forms a barrier that cannot be crossed at total reflux. A) Determination of feasible top and bottom fractions. B) Separation regions.

220

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.32 Separation regions of a ternary mixture with an a/c minimum azeotrope. Pure component a can be recovered even though it is not an end point of distillation lines (at total reflux and reboil). A) Determination of possible top and bottom fractions. B) Separation regions.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

221

Figure 4.33 Separation regions of a ternary mixture with a b/c minimum azeotrope when the feed is lying in the concave section of the boundary distillation line (at total reflux and reboil). A) determination of possible top and bottom fractions. B) separation regions.

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

cannot be crossed by distillation at total reflux or reboil. Separation Regions at Non-total Reflux

˙ and B˙ do not necessarily lie exUnder finite reflux the top and bottom products D actly at the same distillation line. Thus, the boundary distillation line is not a strict barrier for distillation at finite reflux. Several authors report that at low reflux ratio, the barrier line of distillation is not identical with the boundary distillation line [Hausen 1935; Vogelpohl 1964a,b; Levy et al. 1985; Doherty and Caldarola 1985; Bernot et al. 1991; Wahnschafft et al. 1992]. The effective barrier for distillation depends on reflux ratio as well as on concentrations of feed and products. At low reflux ratios, the barrier line is (slightly) shifted to the convex side of the boundary distillation line; see Section 4.4.1. In turn, the space of feasible products becomes a little bit wider. However, the separation barrier does not vanish at all. Generally, the barrier line at finite reflux can be determined only by tiresome iterations of rigorous column calculations. It is a good strategy in process design to treat the boundary distillation line as the actual barrier for distillation even at finite reflux ratio. The effective separation barrier will be found later on by rigorous column simulation; see Figure 4.63C. 4.3.2.2

Feasible Pure Products

In most cases pure products have to be gained from a liquid mixture by distillation. The following provides a simple approach to figuring out which components of a mixture can be separated by a single distillation step either as top or as bottom product. Basis for this quick approach is the knowledge of the boiling points (low, intermediate, or high boiler) of the components and the concentration and boiling points of binary azeotropes existing in the system. With this information a triangular diagram can be sketched with the low boiler a at the top, the intermediate boiler b at bottom right, and the high boiler c at bottom left. The loci and types (maximum or minimum) of the azeotropes will be marked in this diagram (see Figure 4.34). Then, the directions of decreasing boiling temperatures are marked with arrows on the sides of the triangle (arrows pointing to decreasing boiling temperatures). If the arrows at both sides of a corner point away from this corner, then this corner is an origin of distillation lines (peak in boiling surface), and, in turn, this component can be recovered as a bottom product by distillation. If both arrows at the sides of a corner point toward that corner, then this corner is a terminus of distillation lines (hollow in boiling surface), and that substance can be recovered as a overhead product by distillation. If one arrow points toward and the other one away from the corner, then this corner is neither an origin nor a terminus of distillation lines. That component can in most cases not be recovered by a single distillation step from a mixture (but see last row of Figure 4.34). This approach is demonstrated in Figure 4.34A–L. In a zeotropic mixture (A), only the low boiler a and the high boiler c are end points of distillation lines. The approximate course of distillation lines can be sketched in the diagram. The distillation lines take a sharp turn near the b-corner. In system B with an a/b minimum azeotrope, only one substance can be separated in pure form (by a single distillation step). The

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

223

Figure 4.34 Determination of feasible pure products of a single distillation column (at total reflux and reboil). Arrows point in the direction of decreasing boiling temperatures. If both arrows at a corner point toward this corner, then this component can be separated in pure form as an overhead fraction. If both arrows at a corner point away from this corner, then this component can be recovered in pure form as a bottom fraction. If one arrow points toward the corner and the other one away from that corner, then this component cannot be recovered in pure form (but see Figure 4.32).

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

same holds for system B, E, and H. Two components can be separated from systems A, D, F, G, and I. All three components can be separated from a system with three binary azeotropes of the same type (not shown in Figure 4.34). Boundary distillation lines always exist, if there are more than one origin or terminus of distillation lines. They are always running between local boiling point extremata of the same kind (either peaks or hollows). Their approximate course can be sketched easily. In the systems G to I (third row of Figure 4.34), two boundary distillation lines occur (one ridge and one valley). At their intersection point a ternary saddle azeotrope is formed. A special situation is met in the systems J to L (last row of Figure 4.34). Here, origin and terminus of distillation lines are lying close together, and distillation lines make a sharp turn near two corners. As the straight line through the feed is always a chord of a distillation line (with top and bottom products at the intersection), substances b and a of system J, b and c of system K, and b and c of system L can be recovered. Under such conditions of operation, neither top nor bottom of the column is a boiling point extremum. Therefore, column control might be difficult in these cases. This easy evaluation of feasible pure products is very helpful in process design. The results are often quite astonishing. For example, in system H of Figure 4.34, only the low boiler a can be recovered in pure form by a single distillation step, and, surprisingly, it comes as the bottom product. With the technique described here, it is very easy to develop separation sequences for separating some components from a multicomponent mixture in pure form. 4.3.3

Distillation of Ternary Mixtures at Minimum Reflux and Reboil

Industrial distillation columns are preferentially operated at a low reflux ratio RL in order to minimize energy requirement. Hence, the special case of the minimum reflux ratio RL,min is of primary interest. A method described by Underwood 1948 permits determination of the minimum reflux ratio for ideal systems with constant relative volatilities αik . However, the equations can only be solved iteratively. This section shows that the minimum reflux ratio can be calculated directly (see Section 4.3.3.4). Operation of a distillation column with minimum reflux or reboil and very large (infinite) number of equilibrium stages is characterized by the existence of a pinch (i.e. point of intersection of operating and equilibrium line) within the column. At the pinch the vapor is in equilibrium with the liquid, and, in turn, mass transfer between the two phases comes to an end. Approaching the pinch requires an infinite number of equilibrium stages. In binary distillation there exists a double pinch immediately above and below the feed point (except in the cases where a tangential pinch occurs; see Section 4.2.4). As will be shown later on, the situation is more complex in ternary and multicomponent mixtures. Two restrictions are made in the following analysis:

˙ G˙ (slope of the operating line) is assumed to be constant • The liquid-to-gas ratio L/ in each column section.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

225

• Tangential pinches are not taken into account. Even though the space of feasible products is very limited in ternary distillation, several various separations can be realized by a single distillation column. In the following, three special cases of separations are considered; see Figure 4.35 [Stichlmair et al. 1993].

Figure 4.35 Preferred separation, low boiler separation, and high-boiler separation plotted into the region of feasible separations (at total reflux and reboil).

The first case of separations, here called preferred separation, is a sharp separation of substances a and c from a ternary mixture a/b/c. Overhead and bottom fractions are, in turn, binary mixtures a/b and b/c, respectively. The straight line, which represents the material balance around the column, passes through the points xFi ∗ and yFi in the triangular concentration diagram in Figure 4.35. The intersection of this straight line with the a/b and b/c sides of the concentration triangle determines the concentrations of the distillate xDa and the bottoms xBc of a sharp separation. This separation lies amidst the feasible separation region. The results of a rigorous column simulation are shown in Figure 4.36A. There exists a double pinch at the feed concentration xFi like in binary distillation (xPi = xFi ). The second case considered here is the removal of a pure low boiler a from a ternary mixture a/b/c, here called low boiler separation. This separation is a limiting case in the product space in Figure 4.35. The rigorous simulation reveals that only a single pinch exists located immediately beneath the feed point in the stripping section of the column; see Figure 4.36B. Mind that the concentration of the single pinch is different from that of the feed (xPi 6= xFi ). The third case dealt with is the removal of the high boiler c from the ternary mixture a/b/c, here called high boiler separation; see Figure 4.36C. This separation also

226

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.36 Flow sheet and internal liquid concentration profiles of a column for preferred (A), low boiler (B), and high boiler (C) separation of a ternary mixture at minimum reflux and reboil. The most important characteristics are the existence of internal concentration maxima and gaps in the concentration profiles.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

227

is a limiting case in the product space shown in Figure 4.35. Here too, there exists just a single pinch located immediately above the feed point. Its concentration is also different from the concentration of the feed (xPi 6= xFi ). In all three cases there exist concentration maxima of the intermediate boiling substance b in both column sections. Noteworthy is the fact that the internal concentration profiles of the liquid xi and the vapor yi∗ run on parallel straight lines. Types and loci of pinches in the column are marked in Figure 4.37 for all three processes. In the preferred separation, shown in Figure 4.37A, both sections of the column are operated with minimum reflux and minimum reboil, respectively. Hence, there exists a double pinch at the feed point like in binary distillation. In the process shown in Figure 4.37B, the rectifying section is operated with increased reflux in order to get a pure overhead fraction (xDa = 1). Just the stripping section is operated with minimum reboil. Therefore, there exists only a single pinch in the stripping section, which is located immediately below the feed point, with a concentration xPi 6= xFi . A similar situation exists in the process shown in Figure 4.37C. The stripping section is operated with increased reboil to achieve a pure high boiler c as bottoms. The rectifying section, however, is operated with minimum reflux. Again, the concentration of the pinch is different from the concentration of the feed (xPi 6= xFi ). The decisive feature for predicting minimum reflux and minimum reboil is the concentration xPi of the pinch, and not the concentration xFi of the feed. The pinch is that point where the operating line intersects the equilibrium line. Predicting the minimum reflux and the minimum reboil is easy whenever the pinch concentration

Figure 4.37 Types and loci of pinches at different modes of column operation with minimum reflux and reboil, respectively. The shaded sections are operated with minimum internal countercurrent flow.

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

is a priori known, i.e. when xPi = xFi . If the pinch concentration is different from the feed concentration xPi 6= xFi , the prediction of minimum reboil and minimum reflux is rather complex. 4.3.3.1

Minimum Reflux and Reboil of Preferred a/c-Separation

In the very special case of preferred separation of a ternary mixture (see Figure 4.38), there exists a double pinch, which has the same concentration as the feed, xPi = xFi (boiling feed, i.e. qF = 1.0). Therefore, preferred separation is very similar to binary distillation, where also a double pinch at the feed point exists. Consequently, the relations developed for binary mixtures in Section 4.2.4 can also be applied to preferred separation of ternary (and even multicomponent) mixtures. For a boiling liquid feed, the following holds (see Eq. (4.36)):

RL,min =

∗ xDi − yFi ∗ yFi − xFi

and

RG,min =

xFi − xBi ∗ −x yFi Fi

for

qF = 1 . (4.69)

∗ The concentration yFi is in equilibrium with xFi . For ideal mixtures the phase equilibrium is formulated as

ya∗ = αac · xa /N

;

yb∗ = αbc · xb /N

;

yc∗ = 1 · xc /N

with N = 1 + (αac − 1) · xa + (αbc − 1) · xb .

(4.70)

Figure 4.38 Preferred separation of a ternary mixture at minimum reflux. Both separation ˙ and B˙ , have to lie on the straight line through xFi and y ∗ (feed at boiling products, D Fi point). Hence, the separation region of preferred separation shrinks to a straight line (at total reflux and reboil).

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4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

Saturated Liquid Feed

For sharp preferred separations with xDc = 0, it follows from Eqs. (4.69) and (4.70), written for component c, that

−xFc /N 1 = or xFc /N − xFc N −1 1 = (αac − 1) · xFa + (αbc − 1) · xFb

RL,min = RL,min

(4.71) for

qF = 1 .

Equation (4.71) resembles the equations developed for binary mixtures. If the concentration of the intermediate boiling substance b approaches 0, then Eq. (4.71) becomes identical with Eq. (4.39). Equation (4.69) represents a system of three equations (one for each component in the mixture). Having determined the minimum reflux ratio by evaluating Eq. (4.69) for substance c, an evaluation for substance a delivers the concentration xDa of the distillate: ∗ ∗ xDa = RL,min · (yFa − xFa ) + yFa .

(4.72)

With Eq. (4.70), it follows that

αac − 1 · xFa or N −1  −1 (αbc − 1) · xFb = 1+ (αac − 1) · xFa

xDa = xDa

(4.73) for

qF = 1 .

The minimum reboil ratio comes from Eq. (4.69), written for substance a:

RG,min =

xFa − xBa ∗ −x yFa Fa

for

qF = 1 .

(4.74)

For ideal mixtures and sharp separations (xBa = 0), it follows that

xFa N = or αac · xFa /N − xFa αac − N 1

+1= −1 −1 1 − αab · xFb + 1 − αac · xFc

RG,min = RG,min

(4.75) for

qF = 1 .

For xFb → 0, the above equation transforms into the corresponding Eq. (4.41) for binary mixtures. The concentration of the binary bottom fraction is determined by writing Eq. (4.69) for substance c:

αac − 1 · xFc or αac − N  −1 (αac − αbc ) · xFb = 1+ (αac − 1) · xFc

xBc = xBc

(4.76) for

qF = 1 .

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Saturated Vapor Feed

Similar relationships for the minimum reflux ratio RL,min and the minimum reboil ratio RG,min can be derived for a vaporous feed with qF = 0. Here, the feed concentration yFi is given. The relevant basic equations (4.69) are rewritten as

RL,min =

xDi − yFi yFi − x∗Fi

and

RG,min =

x∗Fi − xBi yFi − x∗Fi

qF = 0 .

for

(4.77)

For ideal mixtures the equilibrium x∗i = f (yi ) is (see Chapter 2) −1 x∗b = αbc · yb /N 0 ; x∗c = 1 · yc /N 0

−1 −1 with N 0 = 1 + αac − 1 · xa + αbc − 1 · xb .

−1 x∗a = αac · ya /N 0

;

(4.78)

For sharp separations (i.e. xDc = 0), the basic equation is advantageously written for substance c. From Eqs. (4.77) and (4.78), it follows that

−yFc N0 = or yFc − yFc /N 0 1 − N0 1

+1= −1 −1 1 − αac · yFa + 1 − αbc · yFb

RL,min = RL,min

(4.79) for

qF = 0 .

Equation (4.77) forms a set of three equations. Evaluating it for substance a yields a relation for the concentration of the distillate:
−1 xDa = RL,min · yFa − αac · yFa /N 0 + yFa for qF = 0 . (4.80) Rearrangement gives
−1 1 − αac xDa = · yFa or 1 − N0 " #−1
−1 1 − αbc · yFb
xDa = 1 + −1 1 − αac · yFa

(4.81) for qF = 0 .

The relevant equations for the minimum reboil ratio are (for xBa = 0) −1 −1 αac · yFa /N 0 αac = or −1 −1 yFa − αac · yFa /N 0 N 0 − αac 1 = for qF = 0 . (αac − 1) · yFc + (αab − 1) · yFb

RG,min = RG,min

The concentration of the bottom fraction is
−1 1 − αac · yFc xBc = or −1 N 0 − αac " #−1
−1 −1 αbc − αac · yFb
xBc = 1 + for −1 1 − αac · yFc

(4.82)

(4.83)

qF = 0 .

All these equations are the basis for predicting the minimum energy requirement of preferred distillation of ternary mixtures; see Section 4.3.4.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

4.3.3.2

231

Minimum Reflux and Reboil of Low Boiler Separation

The result of a rigorous column simulation (see Section 4.4.1) for low boiler separation from an ideal ternary mixture is shown in Figure 4.39. The feed is a saturated liquid with a caloric factor qF = 1.0. Considered is a sharp separation with minimum reflux and reboil.

Figure 4.39 Separation of a pure low boiler as overhead fraction from a ternary mixture at minimum reflux and reboil. There exists only a single pinch point located directly below the feed point.

The internal concentration profiles are plotted in the triangular diagram. The concentration profiles show some characteristic features, which are very helpful for developing a deeper understanding of the process. There exists a single pinch xPi immediately below the feed point, whose mole fraction significantly differs from that of the feed, xPi 6= xFi . The concentration of the pinch is not a priori known, which makes the determination of the minimum reflux and reboil ratios rather complex as the basic equations (4.36) cannot be applied to this process. Near the column ends, in the upper part of the rectifying section and the lower part of the stripping section, there exist binary mixtures a/b and b/c, respectively. In these regions, the liquid concentration profiles run along the sides of the triangular diagram. The concentration profiles enter the ternary region of the concentration space at very distinctive points in the diagram. At these points, the operating lines intersect the equilibrium curve of the corresponding binary mixtures a/b and b/c, respectively. This situation is depicted in Figure 4.40 for the rectifying section of a column. Noteworthy is the parallelism of the xi -line and the yi∗ -line in the triangular diagram. At the transition points from binary to ternary mixtures (here called saddle pinches), the mole fraction of the intermediate boiling substance b reaches a maximal value, a feature unknown in binary distillation. The mole fractions of the saddle pin-

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.40 Parallelism of the y -line and the x-line of ideal ternary mixtures.

ches are easily found by rearranging the equations (4.39) developed for calculating the minimum reflux and reboil ratio of binary mixtures:

1 (αac − 1) · (RL,min )a 1 = (αab − 1) · (RL,min )a

(xSPa )ac = (xSPa )ab

and (4.84) for

qF = 1 .

In the ternary region of the concentration space, the internal liquid concentration profiles run along straight lines with distinctive orientation. This fact is explained by a very special feature of ideal ternary mixtures. Determination of vapor concentrations yi∗ in equilibrium with given liquid concentrations xi that lie at a straight line (for instance, xa = 0.2) delivers also a straight line for yi∗ . However, this line has a different slope in most cases; see Figure 4.41A. But for each point there exist two directions with parallel xi -lines and yi∗ -lines; see Figure 4.41B. The evidence for this fact is presented in Example 4.5. This very special feature of ideal mixtures meets ˙ G˙ = const, within the column. This the condition of constant molar overflow, L/ situation resembles the condition for constant molar overflow developed for binary mixtures in Section 4.2.2. In the enthalpy–concentration diagram in Figure 4.15, the ˙ G˙ = const. dew line and the boiling line have to be parallel to meet the condition L/ In sharp separations with minimum reflux and reboil, the internal profile of liquid concentrations xi runs in that direction where parallelism to vapor concentrations yi∗ exists. The feed xFi and the pinch xPi have to lie at this straight line. A very characteristic feature of the process is the fact that the liquid concentration profile in the rectifying section ends before the feed point F˙ is reached; see Figure 4.39. This fact is explained by a material balance around the rectifying section ∗ of the column. A vapor with yPi is fed into the rectifying section to be fractionated ˙ and the liquid L˙ exiting the rectifying section above the feed into the distillate D ˙ , the vapor with y ∗ , and the end point of the liquid point. Hence, the distillate D Pi concentration profile in the rectifying section have to be co-linear. They have to lie at the thin dashed line in Figure 4.39.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

233

Figure 4.41 Vapor–liquid equilibrium of an ideal ternary mixture. A) If the liquid state lies at a straight line, then the vapor state must also lie at a straight line. B) At any point xi of an ideal mixture, there exist two directions of vapor–liquid equilibrium parallelism (bold lines).

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

A liquid stream with mole fraction xPi is fed into the stripping section in the col∗ umn to be split into bottoms fraction B˙ and the vapor yPi exiting the stripping sec∗ ˙ tion. Therefore, the points B , yPi , and xPi also have to be co-linear and have to lie at the thin dashed line in Figure 4.39.

Example 4.5: Parallelism of Vapor and Liquid Concentrations Proof that, in ideal systems, the values of vapor concentrations yi∗ form a straight line if the corresponding liquid concentrations xi are lying at a straight line themselves. Find the direction of parallelism. Solution:

(1) Slope of the yi∗ -line: dya∗ /dyb∗ = f (xi ) In ideal systems,

ya∗ = αac · xa /N with

and

yb∗ = αbc · xb /N

N = 1 + (αac − 1) · xa + (αbc − 1) · xb

The total differential is

dya∗ =

∂ ya∗ ∂ y∗ ∂ yb∗ ∂ y∗ · dxa + a · dxb with dyb∗ = · dxa + b · dxb ∂ xa ∂ xb ∂ xa ∂ xb

The result of the differentiations is

∂ ya∗ ∂ xa ∂ ya∗ ∂ xb ∂ yb∗ ∂ xa ∂ yb∗ ∂ xb

C1 N2 C2 = 2 N C3 = 2 N C4 = 2 N =

with

C1 = (1 + (αbc − 1) · xb ) · αac

with

C2 = (1 − αbc ) · xa · αac

with

C3 = (1 − αac ) · xb · αbc

with

C4 = (1 + (αac − 1) · xa ) · αbc

The slope of the yi∗ -line is

dya∗ C1 /N 2 · dxa + C2 /N 2 · dxb C1 · dxa + C2 · dxb = = dyb∗ C3 /N 2 · dxa + C4 /N 2 · dxb C3 · dxa + C4 · dxb Case 1: The x-line is a horizontal line with xa = const and dxa = 0 :

dya∗ C2 · dxb (1 − αbc ) · xa · αac = = 6= f (xb ) dyb∗ C4 · dxb (1 + (αac − 1) · xa ) · αbc The slope of the yi∗ -line is constant for xa = const ⇒ straight line.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

235

Case 2: xb = const and dxb = 0 :

dya∗ C1 · dxa (1 + (αbc − 1) · xb ) · αac = = 6= f (xa ) dyb∗ C3 · dxa (1 − αac ) · xb · αbc Slope dya∗ /dyb∗ = const for xb = const ⇒ straight line in the concentration space. (2) Directions of parallel yi∗ -lines and x-lines: dxa dy ∗ Condition of parallelism: = a∗ with dxb dyb

dya∗ C1 · dxa + C2 · dxb C1 · (dxa /dxb ) + C2 = = dyb∗ C3 · dxa + C4 · dxb C3 · (dxa /dxb ) + C2 Rearrangement yields 

dxa dxb

2



· C3 + (C4 − C1 ) ·

dxa dxb



− C2 = 0

The solution of the quadratic equation is p dxa 1 = · (−B ± B 2 − 4 · A · C) dxb 2·A with A = C3 ; B = (C4 − C1 ) ; C = −C2

(1)

There are two directions of parallel yi∗ - and x-lines at each point of a ternary mixture. Evaluation with αac = 2.1 , αbc = 1.4 , xa = 0.29 , xb = 0.30 yields

C1 = (1 + (1.4 − 1) · 0.30) · 2.1 = 2.3520 C2 = (1 − 1.4) · 0.29 · 2.1 = −0.2436 C3 = (1 − 2.1) · 0.30 · 1.4 = −0.4620 C4 = (1 + (2.1 − 1) · 0.29) · 1.4 = 1.8466 From Eq. (1), it follows that

A = −0.4620 ; B = 1.8466 − 2.3520 = −0.5054 ; C = 0.2436   dxa 1 = · dxb 1/2 2 · (−0.4620) q · (+0.5054 ± 0.50542 − 4 · (−0.4620) · 0.2436)  Slope:

dxa dxb





= +0.362 1

and

dxa dxb



= −1.456 2

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Compare this result with Example 4.7. Slopes of x-lines at the pinch point:

0.40 − 0.1818 = 0.363 and 1 − 0.40 0.2727 − 1 (dxa /dxb )2 = = −1.455 0.50 (dxa /dxb )1 =

Saturated Liquid Feed

The most straightforward procedure for finding the minimum reflux ratio (RL,min )a for gaining pure low boiler a from a ternary mixture is as follows: • Estimation of a promising value for (RL,min )a . • Calculating the saddle pinches (xSPa )ac and (xSPa )ab with Eq. (4.84) and drawing a straight line between these two points in the concentration triangle. • Check whether or not the feed point xFi lies at that straight line. If not, a better estimation of (RL,min )a has to be made. This iterative procedure converges very fast in most cases. This graphical procedure can also be performed analytically. The result is a quadratic equation whose solution is   p 1 (RL,min )a = · −B + B 2 − 4 · A · C for qF = 1 2·A xFa + xFc xFa + xFb with A = xFa ; −B = + (4.85) αac − 1 αab − 1 1 C= . (αac − 1) · (αab − 1) This procedure for the determination of the minimum reflux ratio cannot be applied to non-ideal systems as in such systems the line between the binary saddle pinches is not a straight one. But in most cases this line is only slightly curved as seen in Figure 4.42. Hence, the graphical procedure yields a good first approximation of the minimum reflux ratio even for non-ideal mixtures. Saturated Vapor Feed

In the case of a saturated vapor feed (qF = 0), the feed concentrations yFi are given. Here the feed point F˙ has to lie on the dashed thin straight line in Figure 4.40. The mole fractions of the saddle pinches in the vapor phase are:

1
1− · (RL,min )a + 1 1

= −1 1 − αab · (RL,min )a + 1

(ySPa )ac = (ySPa )ab

−1 αac




and (4.86) for qF = 0 .

The easiest procedure for finding the minimum reflux ratio for gaining pure low boiler a from a vaporous ternary mixture is as follows:

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

237

• Estimation of a promising value for (RL,min )a . • Calculating the saddle pinch points (ySPa )ac and (ySPa )ab with Eq. (4.86) and drawing a straight line between these two points in the triangular concentration diagram. • Check whether or not the feed point (yFi ) lies at that straight line. If not, a better estimation of (RL,min )a has to be made. This graphical procedure can also be performed analytically. The result is a quadratic equation whose solution is   p 1 · −B + B 2 − 4 · A · C for qF = 0 2·A yFa + yFc yFa + yFb with A = yFa ; −B = −1 + −1 1 − αac 1 − αab 1 C= −1 −1 . (1 − αac ) · (1 − αab )

(RL,min )a + 1 =

(4.87)

Example 4.6: Minimum Reboil Ratio for Separating an Ideal Ternary Mixture Find the minimum reboil ratio RG,min for separating pure nitrogen from a nitrogen (a)/argon (b)/oxygen (c) mixture. The relative volatilities are αac = 4.032 , αbc = 1.339. Feed concentrations: xFa = 0.25 , xFb = 0.3. Consider a sharp separation with boiling liquid feed. Solution:

Eq. (4.85) for the minimum reflux ratio:   p 1 (RL,min )a = · −B + B 2 − 4 · A · C 2·A xFa + xFc xFa + xFb with A = xFa ; −B = + αac − 1 αab − 1 1 C= (αab − 1) · (αac − 1) With αab = αac /αbc = 4.032/1.339 = 3.0112 and

xFc = 1 − xFa − xFb = 1 − 0.25 − 0.30 = 0.45 0.25 + 0.45 0.25 + 0.30 A = 0.25 ; −B = + = 0.5043 4.032 − 1 3.0112 − 1 1 C= = 0.1640 (4.032 − 1) · (3.0112 − 1)   p 1 (RL,min )a = · 0.5043 + 0.50432 − 4 · 0.25 · 0.1640 = 1.6097 2 · 0.25

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

The minimum reboil ratio comes from Eq. (4.22):

(RG,min )a =

˙ F˙
D/ · (RL,min )a + 1 ˙ ˙ 1 − D/F

˙ F˙ = xFa = 0.25 : For a sharp separation D/ 0.25 The result is (RG,min )a = · (1.0697 + 1) = 0.6899 . 1 − 0.25

4.3.3.3

Minimum Reflux and Reboil Ratio of High Boiler Separation

The result of a rigorous column simulation (see Section 4.4.1) for high boiler separation from an ideal ternary mixture is shown in Figure 4.43. The feed is a boiling liquid with a caloric factor qF = 1.0. Considered is the operation with minimum reflux and reboil. The internal concentration profiles are plotted in the triangular diagram. In principle, the concentration profiles show some characteristic features similar to that ones in the process for low boiler separation. There exists a single pinch xPi , here located immediately above the feed point in the rectifying section of the column. The mole fraction of the pinch significantly differs from that of the feed, xPi 6= xFi . As the concentration of the pinch is not a priori known, the basic equations (4.69) cannot be applied in this case, too. Saturated Liquid Feed

Here too, the key for the determination of the minimum reboil ratio is the knowledge of the concentration of the binary saddle pinches. From Eq. (4.41), it follows that

1

and −1 1 − αac · (RG,min )c + 1 1

for qF = 1 . = −1 1 − αab · (RG,min )c + 1

(xSPc )ac = (xSPc )bc

(4.88)

Mind that the concentrations are given for the high boiler c in the mixture. The feed point xFi and the pinch xPi have to lie at the straight line between the two saddle pinches. The simplest procedure for finding the minimum reboil ratio (RG,min )c for gaining pure high boiler c from a ternary mixture is again as follows: • Estimation of a promising value for (RG,min )c . • Calculating the saddle pinches (xSPc )ac and (xSPc )bc with Eq. (4.88) and drawing a straight line between these two points in the concentration space. • Check whether or not the feed point xFi lies on that straight line. If not, a better estimation of (RG,min )c has to be made.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

239

Figure 4.42 In non-ideal mixtures the concentration profiles between the binary saddle pinches are curved. They have to be linearized near the feed point.

Figure 4.43 Separation of the high boiler as bottom fraction from a ternary mixture at minimum reflux and reboil. There exists only a single pinch located directly above the feed point.

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

This graphical procedure can also be performed analytically. The result is a quadratic equation whose solution is   p 1 (RG,min )c + 1 = · −B + B 2 − 4 · A · C for qF = 1 2·A xFa + xFc xFb + xFc with A = xFc ; −B = −1 + (4.89) 1−α 1 − α−1 ac

bc

1

C= −1 −1 . 1 − αac · 1 − αbc Saturated Vapor Feed

In the case of a saturated vapor feed (qF = 0), the feed concentrations yFi are the known quantities. Here the feed point F˙ has to lie at the dashed straight line between ∗ the saddle pinches ySPi in Figure 4.40. The mole fractions of the binary saddle pinches in the vapor phase are (see Eq. (4.46))

1 and (αac − 1) · (RG,min )c 1 = for qF = 0 . (αbc − 1) · (RG,min )c

(ySPc )ac = (ySPc )bc

(4.90)

The easiest procedure for finding the minimum reboil ratio for gaining pure high boiler c from a vaporous ternary mixture is as follows: • Estimation of a promising value for (RG,min )c . • Calculating of the saddle pinches (ySPc )ac and (ySPc )bc with Eq. (4.90) and drawing a straight line between these two points in the concentration space. • Check whether or not the feed point yFi lies at that straight line. If not, a better estimation of (RG,min )c has to be made. The analytical solution of this calculation procedure is   p 1 (RG,min )c = · −B + B 2 − 4 · A · C for qF = 0 2·A yFa + yFc yFb + yFc with A = yFc ; −B = + αac − 1 αbc − 1 1 C= . (αac − 1) · (αbc − 1)

(4.91)

This procedure for the determination of the minimum reflux ration can only be applied to ideal systems as only in such systems the line between the saddle pinch points is a straight one. In non-ideal systems this precondition is not met. But often the line between the binary saddle pinches is only slightly curved; see Figure 4.42. In consequence, the graphical procedure yields a good first approximation, even for non-ideal mixtures. An exact determination of minimum reflux and reboil ratios of non-ideal mixtures is possible by writing the phase equilibrium with activity coefficients and forming the total differential dyi∗ in a manner which is analoguous to Example 4.5 (e.g. Offers 1995).

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

241

Example 4.7: Determination of Pinch Points of Ideal Ternary Mixtures Find the ternary pinch point, the minimum reflux ratio, and the minimum reboil ratio of a sharp separation of a ternary mixture. Data:

• Relative volatilities: αac = 2.10 , αbc = 1.40 , αab = αac /αbc = 1.50 . • Boiling liquid feed: qF = 1.0 .

(RL,min )a and (RG,min )c and find the appropriate feed concentration. Use a graphical approach. Estimated values: (RL,min )a = 5.0 and (RG,min )c = 6.0 Binary saddle pinches from Eq. (4.84): Solution strategy: Start with estimated values for

1 1 = = 0.40 (αab − 1) · (RL,min )a (1.50 − 1) · 5.0 1 1 = = = 0.1818 (αac − 1) · (RL,min )a (2.10 − 1) · 5.0

(xSP a )ab = (xSPa )ac

Draw in Figure 4.44 a straight line between these two points in the triangular concentration diagram (line 1). Binary saddle pinches from Eq. (4.88):

1

−1 1 − αac · (RG,min )c + 1 1 = = 0.2727 (1 − 2.10−1 ) · (6 + 1) 1

= −1 1 − αbc · (RG,min )c + 1 1 = = 0.50 −1 (1 − 1.40 ) · (6 + 1)

(xSPc )ac =

(xSPc )bc

Draw a straight line between these two points in the triangular concentration diagram (line 2). The point of intersection of line 1 and line 2 is the ternary pinch: xPa = 0.29 and xPb = 0.30. The equilibrium concentration yP∗ i is

N = 1 + (αac − 1) · xPa + (αbc − 1) · xPb = 1 + (2.1 − 1) · 0.29 + (1.40 − 1) · 0.30 = 1.439 ∗ yPa = αac · xPa /N = 2.10 · 0.29/1.439 = 0.4232 and ∗ yPb = αbc · xPb /N = 1.40 · 0.30/1.439 = 0.2919

242

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Low boiler separation ∗ Draw in Figure 4.44 a straight line (line 3) between xPi and yPi . The intersection with the b/c side of the concentration triangle is concentration of the bottom product B˙ with xBb = 0.31. ˙ (a-corner). The point Draw a straight line from B˙ (xBb = 0.31) to the distillate D ˙ of intersection with line 1 is the concentration of the feed F . The result is xFa = 0.265 and xFb = 0.23 for (RL,min )a = 5.0. ˙ and y ∗ (line 5) is the material balance of the The straight line through points D Pi rectifying section. Therefore, the intersection of line 5 with line 1 is the end point of the concentration profile in the rectifying section of the column.

High boiler separation

The intersection of line 3 with the a/b side of the concentration triangle is the com˙ , xDa = 0.73. position of the distillate D ˙ and the c-corner. The interDraw in Figure 4.45 a straight line (line 4) between D section with line 2 is the composition of the feed. The result is xFa = 0.47 and xFb = 0.175 for (RG,min )c = 6.0. The straight line through points yP∗ i and B˙ (line 5) is the material balance of the stripping section. Therefore, the intersection of line 5 with line 2 is the end point of the concentration profile in the stripping section of the column.

Figure 4.44 Results of Example 4.7 for a sharp low boiler separation.

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

243

Figure 4.45 Results of Example 4.7 for a sharp high boiler separation.

4.3.3.4

Minimum Reflux and Reboil of All Other Separations

A classical method for calculating minimum reflux and minimum reboil of ideal mixtures has been developed by Underwood as early as in 1948 [Underwood 1948]. A revisit of the Underwood equations was made by Halvorsen and Skogestad 1999. The Underwood equations are written for multicomponent mixtures as X αij · xFi = (1 − qF ) , αij − Φ X αij · xDi X αij · xBi G˙ G˙ = or =− . ˙ ˙ αij − Φ αij − Φ D B

(4.92) (4.93)

Equation (4.92) is used to determine the Underwood parameter Φ, mostly by trial and error. Generally, there exist multiple roots for multicomponent mixtures. Equation (4.93) allows the determination of minimum reflux or minimum reboil. The Underwood equations can be applied to sharp as well as to non-sharp splits. The calculation procedure is demonstrated in detail in Example 4.8 for a ternary mixture. The Underwood equations look very simple, but their application is very tricky. The determination of the Underwood parameter Φ is very tiresome, since the roots have to be found by trial and error from Eq. (4.92). There is a high risk that not all roots are found. In ternary mixtures, an alternative procedure is solving a cubic equation. In the modified strategy presented in Example 4.8, the cubic equation is replaced by a quadratic equation, which significantly simplifies the calculation procedure. Furthermore, not each root can be used for calculating correct values of minimum

244

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

reflux and reboil of a given separation. In ternary mixtures, for instance, the larger value of Φ has to be used for low boiler separations and the smaller one for high boiler separations. Both values of Φ apply to preferred separations; see Example 4.8. Generally, large values of Φ give correct results for gaining overhead products richer in low boiler than the corresponding preferred separation. In turn, small values of Φ apply to bottom fractions richer in high boiler. This statement holds even for separating mixtures with more than three components. The Underwood equations require the complete specification of the product concentrations xDi or xBi . However, these concentrations underlie some severe restrictions: • The product concentrations have to lie within the feasible product space; see Figure 4.30B. • At near total reflux and reboil, the product concentrations have to lie on a distillation line; see Figure 4.30A. • In a preferred separation, the product concentrations have to lie on a straight line ∗ passing through xFi and yFi . • Generally, only one product concentration can be set at will. The other concentrations are a priori unknown in most cases. Because of these restrictions, there is a high risk that irrelevant values for minimum reflux and reboil will be determined. This problem is avoided by considering just sharp separations. The determination of minimum reflux and reboil of non-ideal mixtures is much more complex. An excellent review of the state of the art is given in Danilov et al. 2007. An exact determination of minimum reflux and reboil ratios of non-ideal mixtures is possible by writing the phase equilibrium with activity coefficients and forming the total differential dyi∗ in a manner which is analogous to Example 4.3 (e.g. Offers 1995).

Example 4.8: Underwood Equations Modification of Underwood equations for minimum reflux: (1) Underwood equations: For ternary mixtures the Underwood equations are written as Eq. (4.92):

αac · xFa αbc · xFb 1.0 · xFc + + = 1 − qF αac − Φ αbc − Φ 1.0 − Φ

(1)

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

˙ · (RL + 1) or G˙ = RG · B˙ : Eq. (4.93) with G˙ = D αac · xDa αbc · xDb 1.0 · xDc RL,min + 1 = + + or αac − Φ αbc − Φ 1.0 − Φ   αac · xBa αbc · xBb 1.0 · xBc RG,min = − + + αac − Φ αbc − Φ 1.0 − Φ

245

(2)

Determination of Φ from Eq. (1) by trial and error or by solving a cubic equation. (2) Improved strategy for ternary mixtures: Replacing Eq. (1) by Eq. (4.85), which is valid for sharp low boiler separations. Data: αac = 2.1 , αbc = 1.4 , αab = αac /αbc = 1.5 , xDa = 1.0 ,

xFa = 0.33 , xFb = 0.27 , xFc = 0.40 , qF = 1.0 . (2.1) Determination of Φ from Eq. (4.85):

A = xFa = 0.33 xFa + xFc xFa + xFb 0.33 + 0.40 0.33 + 0.27 −B = + = + = αac − 1 αab − 1 2.1 − 1 1.5 − 1 = 1.8636 1 1 C= = = 1.8182 (αac − 1) · (αab − 1) (2.1 − 1) · (1.5 − 1)   p 1 · −B ± B 2 − 4 · A · C = 2·A   p 1 = · 1.8636 ± 1.86362 − 4 · 0.33 · 1.8182 = 2 · 0.33 1 = · (1.8636 ± 1.0356) 0.66
(RL,min )a 1 = 4.3931 okay and
(RL,min )a 2 = 1.2542 irrelevant for low boiler separations

(RL,min )a =

Underwood parameter from Eq. (2) with xDa = 1.0 :

αac · 1.0 αac → Φ = αac − αac − Φ (RL,min )a + 1 2.1 Φ1 = 2.1 − = 1.7106 and 4.3931 + 1 2.1 Φ2 = 2.1 − = 1.1684 1.2542 + 1

(RL,min )a + 1 =

246

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

(2.2) Verification of the improved strategy: Check with Eq. (1) for Φ1 :

αac · xFa αbc · xFb 1 · xFc + + = αac − Φ1 αbc − Φ1 1 − Φ1 2.1 · 0.33 1.4 · 0.27 1.0 · 0.40 = + + = −0.0002 okay 2.1 − 1.7106 1.4 − 1.7106 1 − 1.7106 For Φ2 :

αac · xFa αbc · xFb 1 · xFc + + = αac − Φ2 αbc − Φ2 1 − Φ2 2.1 · 0.33 1.4 · 0.27 1.0 · 0.40 = + + = 0.0007 okay 2.1 − 1.1684 1.4 − 1.1684 1 − 1.1684 In this improved strategy, the cubic equation (1) is replaced by a quadratic equation. (3) Application of the Underwood equations: The Underwood parameters depend on feed concentration only. They are independent of product concentration. Therefore, they can be applied to all feasible separations of a given mixture xFi . But not each root is suited for all separations; see the resume at the end of this example. (3.1) Check on reboil ratio for sharp low boiler separation: Eq. (4.22): RG,min =

˙ F˙ D/ D˙ · (RL + 1) ; = xFa = 0.33 ˙ F˙ 1 − D/ F˙

0.33 · (4.3931 + 1) = 2.6563 1 − 0.33 Underwood equation (2): Φ = Φ1 = 1.7106   αbc · xBb αcc · xBc RG,min = − + αbc − Φ αcc − Φ 1 1 xBb = · xFb = · 0.27 = 0.4030 ˙ ˙ 1 − 0.33 1 − D/F xBc = 1 − xBb = 0.5970   1.4 · 0.4030 1.0 · 0.5970 RG,min = − + = 2.6566 okay 1.4 − 1.7106 1.0 − 1.7106 Φ = Φ2 = 1.1684 ; RG,min = 1.1090 wrong. RG,min =

(3.2) Check on sharp preferred separation: Eq. (4.70): N = 1 + (αac − 1) · xFa + (αbc − 1) · xFb =

= 1 + (2.1 − 1) · 0.33 + (1.4 − 1) · 0.27 = 1.4710

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

247

1 1 = = 2.1230 N −1 1.4710 − 1 αac − 1 2.1 − 1 = · xFa = · 0.33 = 0.7707 N −1 1.4710 − 1 = 1 − xDa = 0.2293

Eq. (4.71): RL,min = Eq. (4.73): xDa

xDb

Underwood equation (2) with xDc = 0 and Φ1 = 1.7106:

αac · xDa αbc · xDb + = αac − Φ1 αbc − Φ1 2.1 · 0.7707 1.4 · 0.2293 = + = 3.1228 2.1 − 1.7106 1.4 − 1.7106 = 2.1228 okay

RL,min + 1 =

RL,min

For Φ2 = 1.1684:

αac · xDa αbc · xDb + = αac − Φ2 αbc − Φ2 2.1 · 0.7707 1.4 · 0.2293 = + = 3.1234 2.1 − 1.1684 1.4 − 1.1684 = 2.1234 okay.

RL,min + 1 =

RL,min

(3.3) Check on sharp high boiler separation: Equation (4.89):

A = xFc = 0.40 xFa + xFc xFb + xFc 0.33 + 0.40 0.27 + 0.40 −B = + = −1 + −1 = 1 − 2.1−1 1 − 1.4−1 1 − αac 1 − αbc = 3.7386 C=

1 1 = 6.6818 −1 −1 = (1 − 2.1−1 ) · (1 − 1.4−1 ) (1 − αac ) · (1 − αbc ) p 1 · (3.7386 + 3.73862 − 4 · 0.40 · 6.6818) = 2 · 0.40 = 6.9393

(RG,min )c + 1 =

(RG,min )c = 5.9393 Underwood equation (2) with xBc = 1.0:

1.0 · xBc 1.0 · 1.0 =− = 1.4073 irrelevant 1.0 − Φ1 1.0 − 1.7106 1.0 · xBc 1.0 · 1.0 (RG,min )c = − =− = 5.9382 correct 1.0 − Φ2 1.0 − 1.1684 (RG,min )c = −

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Reflux ratio RL,min : Eq. (4.22): RL,min = RG,min ·

˙ F˙ 1 − D/ −1 ˙ D/F˙

D˙ = xFa + xFb = 0.33 + 0.27 = 0.60 F˙ 1 − 0.60 RL,min = 5.9393 · − 1 = 2.9595 0.60 Underwood Eq. (2): Φ = Φ2 = 1.1684 αac · xDa αbc · xDb RL,min = + αac − Φ2 αbc − Φ2 1 1 xDa = · xFa = · 0.33 = 0.55 ˙ F˙ 0.6 D/

xDb = 1 − xDa = 0.45 2.1 · 0.55 1.4 · 0.45 + − 1 = 2.9600 okay 2.1 − 1.1684 1.4 − 1.1684 Φ = Φ1 = 1.7106 ; RL,min = 0.9378 wrong. RL,min =

Φ1 and Φ2 are both correct for preferred separations. However, only Φ1 is correct for low boiler separations and, in turn, Φ2 for high boiler separations. Resume: The Underwood parameters

4.3.4

Energy Requirement of Ternary Distillation

The energy requirement is the most important process parameter of distillation columns, since the operating costs of distillation processes are dominated by the energy requirement of columns. The energy requirement, however, influences also the investment costs, since a high energy requirement results in high internal vapor flows and, in turn, in large column diameters. As will be shown in Chapter 9, the vapor flow is decisive for the column diameter. In most cases, the heat is supplied to the reboiler by steam and removed in the condenser by cooling water. As steam is more expensive than cooling water, the heat supplied to the reboiler is of primary interest in the following. In practical applications, only one of the two heats Q˙ R and Q˙ C is directly determined. The other one is found by an enthalpy balance around the column. In cases with liquid feed and liquid products, the two heats have approximately equal values. The basic equations for calculating the energy requirement have been developed in Section 4.2.5 for binary mixtures. These equations can also be applied to ternary

249

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

distillation (Eq. (4.53)):

Q˙ R xDi − zFi = RG · ˙ xDi − xBi F ·r

and

Q˙ C zFi − xBi = (RL + 1) · . ˙ xDi − xBi F ·r

(4.94)

These relationships enable the calculation of the energy requirement if the feed and product concentrations are known. The most difficult parameters are the reboil and the reflux ratios RG and RL , respectively. These quantities are the most important operation parameters of distillation processes. 4.3.4.1

Minimum Energy Requirement of Sharp Preferred a/c-Separation

The preferred a/c-separation of ternary mixtures resembles in many aspects binary distillation. A small modification of the corresponding equations derived for binary distillation enables the calculation of the energy requirement of ternary mixtures (and even of multicomponent mixtures). Considered is the special case of sharp separations with minimum reflux and reboil, and infinite number of equilibrium stages. Saturated Liquid Feed

The relevant quantities in Eq. (4.94) have been dealt with in Section 4.2.4. For saturated liquid feed, the concentrations zFi have to be replaced by xFi . The resulting equations are compiled in Figure 4.46. The final result is

Q˙ R,min 1 + (αac − 1) · xFa + (αbc − 1) · xFb = (αac − 1) F˙ · r ˙ QC ,min 1 + (αac − 1) · xFa + (αbc − 1) · xFb = ˙ (αac − 1) F ·r

and (4.95) for

qF = 1 .

With the assumptions made, the energy to be supplied to the reboiler is equal to the energy to be removed in the condenser. There exists a linear dependency of the energy requirement on feed concentration xFi , like in binary distillation. Equation (4.95) has been evaluated for an ideal mixture with αac = 1.887 and αbc = 1.329. This very close-boiling mixture is the reference system in this chapter (and in Chapter 7). The results are shown in Figure 4.47. Lines with constant values of Q˙ R,min /(F˙ · r) are plotted in a triangular diagram versus feed concentration xFi . The result is a group of equidistant and parallel straight lines. The energy requirement is lowest at very high concentrations of the high boiler c (Q˙ R,min /(F˙ · r) = 1.2) and highest at very high concentrations of the low boiler a (Q˙ R,min /(F˙ · r) = 2.1). Saturated Vapor Feed

For a saturated vapor feed, the feed concentrations zFi have to be replaced by yFi in Eq. (4.94). Again, the relevant equations are compiled in Figure 4.48. The final results for the energy requirement are

Q˙ R,min 1 = ˙ αac − 1 F ·r

and

Q˙ C ,min 1 = + 1 for qF = 0 . (4.96) ˙ αac − 1 F ·r

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.46 Compilation of the relevant equations for calculating the energy requirement of preferred separation with saturated liquid feed, qF = 1.0.

Figure 4.47 Energy requirement of preferred separation of the close-boiling ternary reference mixture with saturated liquid feed, qF = 1.0.

251

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

Figure 4.48 Compilation of the relevant equations for calculating the energy requirement of the preferred separation with saturated vapor feed, qF = 0.

For a saturated vapor feed, the energy to be removed in the condenser is higher than the energy supplied in the reboiler because the vaporous feed has to be condensed additionally. Surprising is the fact that the energy requirement of the column does not depend on feed concentration at all. The same result was found in binary distillation in Section 4.2.5. Even the equations are identical in both cases (but αab has to be replaced by αac ). Evaluation of Eq. (4.96) for the reference mixture with αac = 1.887 and αbc = 1.329 yields

Q˙ R,min 1 = = 1.274 and 1.887 − 1 F˙ · r Q˙ C ,min 1 = + 1 = 2.274 for qF = 0 . 1.329 − 1 F˙ · r

(4.97)

Hence, the energy requirement for a sharp preferred separation of a vaporous feed is much lower than that of a liquid feed; see Figure 4.47. Like in binary mixtures, this is due to the fact that the latent heat of the feed is automatically utilized in the rectifying section of the column. Even better would be a process with (partial) heating of the reboiler by the vaporous feed and subsequent separation of the condensate of the feed in the column; see Figure 4.27C. 4.3.4.2

Minimum Energy Requirement of Sharp Low Boiler Separation

A very important process is the recovery of the pure low boiler a from a ternary mixture as in most practical applications a pure product has to be gained. Equation (4.94)

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

can be applied to this process, also. Saturated Liquid Feed

For a saturated liquid feed with qF = 1.0, the feed concentrations zFi have to be replaced by xFi in Eq. (4.94). At sharp separations with xDa = 1.0, the following basic equations hold:
Q˙ C ,min = (RL,min )a + 1 · xFa and F˙ · r Q˙ R,min Q˙ C ,min = for qF = 1 . ˙ F ·r F˙ · r

(4.98)

Figure 4.49 Energy requirement of low boiler separation of the close-boiling reference mixture at saturated liquid feed, qF = 1.0.

The value for (RL,min )a comes from Eq. (4.85). The result of an evaluation of Eqs. (4.85) and (4.98) is shown in Figure 4.49 for the reference mixture with αac = 1.887 and αbc = 1.329. The energy requirement is plotted versus the feed concentration xFi . The parameter lines Q˙ C ,min /(F˙ · r) establish a group of curved lines. The values of Q˙ C ,min /(F˙ · r) vary from 1.2 in the c-corner to 3.2 in the a-corner of the diagram. In total, the values are significantly higher than those of preferred separation shown in Figure 4.47 for the same system. This is a big disadvantage of the process. However, the advantage of the process is the recovery of the pure substance a as overhead fraction.

253

4.3 MULTISTAGE DISTILLATION OF TERNARY MIXTURES

Saturated Vapor Feed

For a saturated vapor feed with qF = 0, the concentration zFi in Eq. (4.94) has to be replaced by yFi , which is a priori known:
Q˙ C ,min = (RL,min )a + 1 · yFa and F˙ · r Q˙ R,min Q˙ C ,min = + 1 for qF = 0 . F˙ · r F˙ · r

(4.99)

The minimum reboil ratio (RL,min )a comes from Eq. (4.87). The results of an evaluation of Eqs. (4.87) and (4.99) are shown in Figure 4.50 for the reference mixture. The energy requirement is plotted versus the feed concentration yFi . Interestingly, the parameter lines Q˙ R,min /(F˙ · r) = const establish a group of straight lines. All values of Q˙ R,min /(F˙ · r) are significantly smaller than those of the process with saturated liquid feed. At the binary edges of the concentration space, mixtures a/c and a/b, the values of Q˙ R,min /(F˙ · r) do no longer depend on feed concentration at all.

Figure 4.50 Energy requirement of low boiler separation of the close-boiling reference mixture at saturated vapor feed, qF = 0.

4.3.4.3

Minimum Energy Requirement of Sharp High Boiler Separation

The separation of a pure high boiler c from a ternary mixture is also of great practical importance. Equation (4.94) can be applied to this process, too.

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Saturated Liquid Feed

For a saturated liquid feed with qF = 1.0, the concentrations zFi have to be replaced by xFi . At sharp separations with xDc = 0, the following basic equations hold:

Q˙ R,min = (RG,min )c · xFc F˙ · r

and

Q˙ C ,min Q˙ R,min = ˙ F ·r F˙ · r

for

qF = 1 . (4.100)

The value of (RG,min )c comes from Eq. (4.89). The results of an evaluation of Eqs. (4.89) and (4.100) for the reference system with αac = 1.887 and αbc = 1.329 are presented in Figure 4.51. The parameter lines Q˙ R,min /(F˙ · r) establish a group of curved lines. The values of Q˙ R,min /(F˙ · r) vary from 1.8 at very low feed concentrations of component b up to 4.0 near the b-corner of the diagram. In total, the values are significantly higher than those of preferred separation. This is a big disadvantage of the process. However, the advantage of the process is the recovery of the pure substance c as bottoms fraction.

Figure 4.51 Energy requirement of high boiler separation of the close-boiling mixture with saturated liquid feed, qF = 1.0.

Saturated Vapor Feed

For a saturated vapor feed with qF = 0, the concentrations zFi have to be replaced by yFi in Eq. (4.94), as just yFi is a priori known:

Q˙ R,min = (RG,min )c · yFc F˙ · r

and

Q˙ C ,min Q˙ R,min = + 1 for qF = 0 . F˙ · r F˙ · r (4.101)

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

255

The minimum reboil ratio comes from Eq. (4.91). The results of an evaluation of Eqs. (4.91) and (4.101) are shown in Figure 4.52 for the close-boiling reference mixture. The energy requirement is plotted versus the feed concentration yFi . Surprisingly, the parameter lines Q˙ C ,min /(F˙ · r) = const establish a group of straight lines. All values of Q˙ C ,min /(F˙ · r) are significantly smaller than those of the process with liquid feed in Figure 4.51. The values of vary from 1.2 at very low feed concentrations of component b to 3.0 for feed concentrations yFa → 0. In total, the energy requirement is significantly higher than that of preferred separation. This is a big disadvantage of the process. However, the advantage of the process is the recovery of the pure substance c.

Figure 4.52 Energy requirement of high boiler separation of the close-boiling reference mixture with saturated vapor feed, qF = 0.

A comparison of the results of the energy requirement of preferred separation with that of low and high boiler separations reveals some interesting features with respect to the dependency on the composition of the feed; see Table 4.1. The recovery of pure products from liquid mixtures is always one degree more complex than from vaporous mixtures.

4.4

Multistage Distillation of Multicomponent Mixtures

A detailed simulation of the distillation of multicomponent mixtures has been possible only since the broad availability of computers. Highly sophisticated models for rigorous column simulation have been developed since then. Rigorous column simulation refers to the evaluation of the thermodynamic parameters of a column.

256

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Table 4.1 Dependency of the energy requirement of ternary distillation on feed concentrations.

Caloric state of the feed

Dependency of Q˙ min on feed concentration

Preferred separation

qF = 1 qF = 0

Q˙ min ∼ (xFi )1 Q˙ min ∼ (yFi )0

Low boiler separation

qF = 1 qF = 0

Q˙ min ∼ (xFi )2 Q˙ min ∼ (yFi )1

High boiler separation

qF = 1 qF = 0

Q˙ min ∼ (xFi )2 Q˙ min ∼ (yFi )1

Process

The results are the internal concentration profiles of all components, the temperature profile, and the profiles of vapor and liquid flow rates and the amount and concentration of the products. Typically, rigorous column simulation does not account for problems of two-phase flow in a column. 4.4.1

Rigorous Column Simulation

Rigorous column simulation requires specifying the column and the operating conditions. This includes setting the number n of equilibrium stages (which is different from the number of actual trays) or transfer units, the definition of the location of the feed, or in the case of multiple feed, the definition of the loci of the feed and, if necessary, of the withdrawal of side products. In the case of intermediate reboilers or condensers, their loci and duties have to be set also. The operating conditions are defined in most cases by setting the reflux or reboil ratio or, what is equivalent, the duties of condenser and reboiler. If the calculated product qualities do not meet the requirements, new column specifications have to be made followed by a new simulation. Normally, it is not advisable to specify the products and to search for the number of trays or transfer units in the simulation. Two different groups of models for column simulation are available: the group of equilibrium stage models and the group of the rate-based models. 4.4.1.1

Equilibrium Stage Model

Important contributions to the development of equilibrium stage models have been made by Wang and Henke 1966; Holland 1963; Goldstein and Stanfield 1970; Naphtali and Sandholm 1971; Block and Hegner 1976. The models developed until now differ primarily in the iteration procedure, i.e. in the iteration variables (e.g. flow rate profiles or concentration profiles) and the mathematical iteration routines. Some models allow for liquid-phase splitting [Block and Hegner 1976] and for liquid-phase chemical reaction [Block and Hegner 1977].

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

257

Steady State Operation

The concept of equilibrium stages leads to the well-known MESH equations for each stage j (e.g. Henley and Seader 1981). With the symbols from Figure 4.53, the relevant equations are: Material balances for i = 1 . . . k components:   L˙ j−1 · xi,j−1 + G˙ j+1 · yi,j+1 + F˙j · zi,j − L˙ j + S˙ L,j · xi,j   (4.102) − G˙ j + S˙ G,j · yi,j = 0 . Equilibria for i = 1 . . . k components:

yi,j − Ki,j · xi,j = 0 with Ki,j = f (Tj , pj , xi,j , yi,j ) .

(4.103)

Summation of mole fractions in liquid and in vapor phase for i = 1 . . . k components: k X

xi,j − 1 = 0 and

k X

yi,j − 1 = 0 .

(4.104)

i=1

i=1

Heat balances:   L˙ j−1 · hL,j−1 + G˙ j+1 · hG,j +1 + F˙j · hF ,j − L˙ j + S˙ L,j · hL,j   − G˙ j + S˙ G,j · hG,j + Q˙ j = 0 with

hL,j = f (Tj , pj , xi,j )

and

(4.105)

hG,j = f (Tj , pj , yi,j ) .

The subscript j runs from top, j = 1, to bottom, j = n. The MESH equations represent a set of n · (2 · k + 3) equations. Herein, k denotes the number of components and n the number of stages. The separation of a five-component mixture in a column with 50 equilibrium stages gives, for instance, a system of 650 equations. Furthermore, complex functions for vapor–liquid equilibrium ratios Ki and for vapor and liquid enthalpies, hL and hG , are required. Non-steady State Operation

Often, the non-steady state (dynamic) operation of a column has to be simulated, e.g. for batch distillation, for developing control schemes and for the start-up of a column. This requires a modification of some of the MESH equations. The material balance, Eq. (4.102), has to be expanded by the molar liquid holdup HLj on each equilibrium stage j . The result is   L˙ j−1 · xi,j −1 + G˙ j+1 · yi,j +1 + F˙j · zi,j − L˙ j + S˙ L,j · xi,j   (4.106) − G˙ j + S˙ G,j · yi,j = ∂ (HLj · xi,j ) /∂ t .

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.53 Schematic of a single equilibrium stage for the derivation of the MESH equations.

Analogously, the heat balance, Eq. (4.105), for dynamic operation gets   L˙ j −1 · hL,j −1 + G˙ j +1 · hG,j +1 + F˙j · hF ,j − L˙ j + S˙ L,j · hL,j   − G˙ j + S˙ G,j · hG,j + Q˙ j = ∂ (HLj · hL,j ) /∂ t with hL,j = f (Tj , pj , xi,j )

(4.107)

and hG,j = f (Tj , pj , yi,j ) .

Methods for calculating the liquid hold-up on a tray are described in Chapter 9. Stage Model with Tray Efficiencies

The modeling of a column can be made more realistic by accounting for tray efficiencies EOG (or EOL ) in the MESH equations. In particular, the equilibrium equations (4.103) have to be written for pseudo-equilibria that are actually reached on a tray (see Figure 9.27):

yi,j − yi,j +1 − (Ki,j · xi,j − yi,j +1 ) · EOG = 0 with

Ki,j = f (Tj , pj , xi,j , yi,j ) .

(4.108)

Generally, tray efficiencies will be different for each component i and each tray j . However, mean values are chosen in most cases. For close-boiling mixtures a mean value of EOG ≈ 0.6 to 0.7 is often recommended. Methods for evaluating tray efficiencies are developed in Chapter 9. The stage model with tray efficiencies is a first and a very robust form of a rate-based model; see Section 4.4.1.2. As written in Eqs. (4.102) – (4.105), the MESH equations look like an uncoupled linear system. However, even in ideal systems, the equilibrium ratio Ki is a nonlinear function of temperature. The higher the variation of temperature, the higher is the non-linearity of the MESH equations.

259

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

Furthermore, the equations are coupled since the temperature change caused by the mass transfer of one component influences the concentrations of the other components via the equilibrium ratio. In non-ideal systems the degree of non-linearity is even higher as the equilibrium ratio of component i depends on the concentration of all other components. If the degree of non-linearity of the MESH equations is very high, more than one solution may exist [Bekiaris et al. 1993], for instance, in any ordinary quadratic equation. 4.4.1.2

Non-equilibrium Models (Rate-based Models)

In recent years several rate-based models for rigorous column simulations have been developed [Krishnamurthy and Taylor 1985; Gorak 1990; Taylor et al. 1993; Seader 1989; Taylor and Krishna 1993; Kenig 2000]. These models use the concept of transfer units; see Section 4.1.2. In essence, rate-based models make a differential simulation of the mass transfer zone of a column; see Figure 4.54. The relevant equations are written in analogy to the equilibrium stage model in the following.

Figure 4.54 Schematic of a packed column for the derivation of a rate-based model.

Steady State Operation

Material balances for i = 1 . . . k components:

G˙ · dyi − L˙ · dxi = 0 .

(4.109)

Interfacial mass transfer for i = 1 . . . k components:

dyi = kOG · (yi∗ − yi ) ·

Ac · aeff · dl with yi∗ = Ki · xi . G˙

(4.110)

Summation of mole fractions in liquid and in vapor phase for i = 1 . . . k components: k X i=1

xi,j − 1 = 0 and

k X i=1

yi,j − 1 = 0 .

(4.111)

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Heat balances:

G˙ · dhG − L˙ · dhL = 0

with

hL,j = f (Tj , pj , xi,j )

and

hG,j = (Tj , pj , yi,j ) .

(4.112)

The material balance and the heat balance are simpler in the rate-based model than in the equilibrium stage model, since the admixing of feed and external heat and the withdrawal of side streams are established outside the mass transfer zone (e.g. between packing sections). The equilibrium equations of the stage model are replaced by the equations for the interfacial mass transfer, which involve the phase equilibrium as well as the mass transfer kinetics; see Chapter 9. These quantities depend on operational and geometrical parameters of the column, which are normally not known at thermodynamic column simulation. Non-steady State Operation

Like in the equilibrium stage model, the equations for material and heat balances have to be expanded by the molar liquid hold-up HL in the differential packing element for simulating the dynamic behavior of a column. Equation (4.109) gets

G˙ · dyi − L˙ · dxi = HL · ∂ xi /∂ t .

(4.113)

Analogously, the enthalpy balance equation (4.112) becomes

G˙ · dhG − L˙ · dhL = HL · ∂ hL /∂ t .

(4.114)

Methods for calculating the hold-up in a packing are developed in Chapter 9. 4.4.1.3

Comparison of the Stage Model and the Rate-based Model

In principle, the equilibrium stage model fits well with tray columns, and the ratebased model with packed columns (see Section 4.1.3). However, there are some severe differences between the two models. In the rate-based model, the change of the vapor concentration is directly proportional to the volumetric mass transfer coefficient (kOG · a) (see Chapter 9): 1

dyi ∼ (kOG · a) .

(4.115)

In the stage model, however, the change of the vapor concentration, ∆yi = yi,n − yi,n−1 , is proportional to tray efficiency EOG , which is less dependent on the volumetric mass transfer coefficient. From Figure 4.55, it follows that c

∆yi ∼ EOG ∼ (kOG · a)

with

0 ≤ c ≤ 1.

(4.116)

Realistic values of the exponent c are in the range from 0.3 – 0.4. The larger the volumetric mass transfer coefficient, the bigger are the differences between rate-based and stage models. Consequently, the influence of mass transfer kinetics is often overestimated in the rate-based model. This fact becomes very clear by considering the extreme case of (kOG · a) → ∞. In this case, the length of the mass transfer zone

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

261

(e.g. packing) is zero in the rate-based model. In the stage model, however, just the point efficiency of each tray reaches a value of 100 %. The obviously too strong influence of the volumetric mass transfer coefficient on mass transfer rate in the ratebased model is one of the reasons why the stage model is often applied to packed columns also. Another reason is the higher simplicity of the stage model. Rigorous column simulation is mathematically very complex. Therefore, it is generally not advisable to develop one’s own program. There are several companies that offer such programs either on sales or leasing basis (e.g. Aspen, Science Simulation, Process, Design 2009, Hysys, ChemCAD, Honeywell, ChemStation). Such commercial software contain not only very efficient mathematical solvers but also some subroutines for calculating the phase equilibria (even for non-ideal mixtures), the enthalpies of vapor and liquid, and other relevant system properties.

Figure 4.55 Dependency of point efficiency on mass transfer kinetics (Eq. (4.6)).

4.4.1.4

Procedures for Evaluating the MESH Equations

To demonstrate the complexity of rigorous column simulation, the evaluation of the simplest model, the equilibrium stage model with steady operation, is demonstrated here. First, the number of equations is reduced by the introduction of component flow rates for all streams:

l˙i,j ≡ L˙ j · xi,j

and

g˙ i,j ≡ G˙ j · yi,j

and f˙i,j ≡ F˙ j · zi,j .

(4.117)

The use of component rates l˙i and g˙ i is especially advantageous if the flow rates G˙ and L˙ are not constant within the column because concentration changes are caused by flow rate changes. Consequently, the material balance of component i depends on the concentration of all components. This is avoided by using component flow rates.

262

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

With Eq. (4.117), the MESH equations get the following form:

Mi,j = l˙i,j−1 + g˙ i,j+1 + f˙i,j − l˙i,j · (1 + s˙ L,j ) − g˙ i,j · (1 + s˙ G,j ) = 0 Ei,j = g˙ i,j ·

k X

l˙i,j − Ki,j · l˙i,j ·

i

k X

g˙ i,j = 0

i

with Ki,j = f (l˙i,j . . . l˙k,j , g˙ i,j . . . g˙ k,j , Tj )

Hj = hL,j−1 ·

k X

l˙i,j−1 + g˙ G,j+1 ·

i

− hL,j ·

k X

k X

g˙ i,j+1 + hF,j ·

i

l˙i,j · (1 + s˙ L,j ) − hG,j ·

i

k X

f˙i,j

i k X

g˙ i,j · (1 + s˙ G,j ) + Q˙ j = 0

i

with hL,j = f (l˙i,j . . . l˙k,j , Tj ) and hG,j = f (g˙ i,j . . . g˙ k,j , Tj ) . (4.118) The number of equations in a system with k components and n equilibrium stages is now n · (2 · k + 1). The functions Hj , Mj , and Ej are discrepancy functions, i.e. they are a quantitative measure of the failure of the values of l˙i,j , g˙ i,j , and Tj to satisfy the physical relationship. A solution to the problem has been obtained when one has values of the variables, which make Hj , Mj , and Ej zero [Naphtali and Sandholm 1971]. It is important to be aware of the dependencies of the functions on the variables involved.

Mi,j = f (l˙i,j−1 , l˙i,j , g˙ i,j , g˙ i,j+1 ) Ei,j = f (l˙i,j . . . l˙k,j , g˙ i,j . . . g˙ k,j ) Hj = f (l˙i,j−1 . . . l˙k,j−1 , l˙i,j . . . l˙k,j , g˙ i,j . . . g˙ k,j , g˙ i,j+1 . . . g˙ k,j+1 ,

(4.119)

Tj−1 , Tj , Tj+1 ) . The functions depend only on the variables of stage j and the neighboring stages j−1 and j +1, except the equilibrium functions that contain variables of stage j only. The material balance of component i depends on the flow rates of this component only, whereas the equilibrium and enthalpy functions are influenced by all components. Newton–Raphson Algorithm

A modified Newton–Raphson iteration procedure is used to solve the non-linear set of equations simultaneously [Naphtali and Sandholm 1971]. Its advantage is that this iteration procedure accelerates during approaching the solution. The Newton– Raphson algorithm is a very effective iteration procedure for solving systems of non-linear coupled equations. Its application to a system of two quadratic equations f1 (x1 , x2 ) = 0 and f2 (x1 , x2 ) = 0 is demonstrated in detail in Example 4.9. First, an estimation of the variables and has to be made:

x1 = x1,m

and x2 = x2,m .

(4.120)

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4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

Then, the non-linear equations are linearized in the vicinity of the estimated values x1,m and x2,m by Taylor series:     ∂ f1 ∂ f1 f1 + · ∆x1 + · ∆x2 = 0 and ∂ x1 ∂ x2     (4.121) ∂ f2 ∂ f2 f2 + · ∆x1 + · ∆x2 = 0 . ∂ x1 ∂ x2 After the partial derivatives of the functions with respect to all variables have been performed, Eqs. (4.121) form a system of linear equations for ∆x1 and ∆x2 that can be solved rigorously. The solution of the linear system is

∆x1 = ∆x1,m

and

∆x2 = ∆x2,m .

(4.122)

The solution of the non-linear system is

x1,m+1 = x1,m + ∆x1,m

and

x2,m+1 = x2,m + ∆x2,m .

(4.123)

Since the linear system is only an approximation of the non-linear system, the values x1,m+1 and x2,m+1 do not represent the exact solution. Thus, the procedure has to be repeated. Figure 4.56 displays the values of the functions f1 and f2 during the iteration. By the very effective Newton–Raphson method the values are driven to zero with increasing speed.

Example 4.9: Newton–Raphson Algorithm Solve a system of two quadratic equations by the Newton–Raphson algorithm. Data:

System of non-linear equation:

f1 = 3 · (x21 + x22 ) − 10 · x1 − 14 · x2 + 23 = 0 f2 =

x21

− 2 · x1 − x2 + 3 = 0

(1) (2)

Solution strategy:

Estimation of starting values: x1 = x1,0 and x2 = x2,0 Transformation into a linear system by Taylor series around the estimated values:     ∂ f1 ∂ f1 f1 + · ∆x1 + · ∆x2 = 0 ∂ x1 ∂ x2     ∂ f1 ∂ f1 with = 6 · x1 − 10 and = 6 · x2 − 14 ∂ x1 ∂ x2

⇒

f1 + (6 · x1 − 10) · ∆x1 + (6 · x2 − 14) · ∆x2 = 0

(3)

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)



f2 +

∂ f2 ∂ x1 

with

⇒





∂ f2 ∂ x2



· ∆x1 + · ∆x2 = 0    ∂ f2 ∂ f2 = 2 · x1,m − 2 and = −1 ∂ x1 ∂ x2

f2 + (2 · x1 − 2) · ∆x1 + (−1) · ∆x2 = 0

(4)

Solution of the linear systems (3) and (4): ⇒ ∆x1 and ∆x2 Improved values of the non-linear system: x1,m+1 = x1,m + ∆x1 and x2,m+1 = x2,m + ∆x2 Evaluation:

Estimated values:

x1,0 = 0 and x2,0 = 0

Check Eqs. (1) and (2): f1 (x1,0 , x2,0 ) = 23 6= 0 and f2 (x1,0 , x2,0 ) = 3 6= 0 The estimated values are wrong. 1. Iteration:

23 − 10 · ∆x1 − 14 · ∆x2 = 0 3 − 2 · ∆x1 − 1 · ∆x2 = 0 Solution of linear system (3) and (4): ∆x1 = 1.0556 and ∆x2 = 0.8889 Values of non-linear systems (1) and (2): x1 = 0 + 1.0556 = 1.0556 x2 = 0 + 0.8889 = 0.8889 Check Eqs. (1) and (2): f1 = 5.7127 6= 0 and f2 = 1.1142 6= 0 Linear system (3) and (4):

2. Iteration:

5.7127−3.6664·∆x1 −8.6666·∆x2 = 0 1.1142 + 0.1112 · ∆x1 − 1.0 · ∆x2 = 0 Solution of linear systems Eqs. (3) and (4): ∆x1 = −0.8517 and ∆x2 = 1.0195 Values of non-linear systems Eqs. (1) and (2): x1 = 1.0556 − 0.8517 = 0.2039 x2 = 0.8889 + 1.0195 = 1.9084 Check Eqs. (1) and (2): f1 = 5.2941 6= 0 and f2 = 0.7254 6= 0 Linear systems (3) and (4):

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

265

3. Iteration:

5.2941−8.7766·∆x1 −2.5496·∆x2 = 0 0.7254 − 1.5922 · ∆x1 − 1.0 · ∆x2 = 0 Solution of linear systems (3) and (4): ∆x1 = 0.7302 and ∆x2 = −0.4373 Values of non-linear systems (1) and (2): x1 = 0.2039 + 0.7302 = 0.9341 x2 = 1.9084 − 0.4373 = 1.4711 Check Eqs. (1) and (2): f1 = 2.1736 6= 0 and f2 = 0.5332 6= 0 Linear systems (3) and (4):

The values of the functions f1 and f2 of the first seven iterations are presented in Figure 4.54. The curves demonstrate the rapid convergence with increasing speed. The correct values of the variables are x1 = 1.0 and x2 = 2.0.

Figure 4.56 Iteration history for solving the two non-linear equations of Example 4.9 by Newton–Raphson algorithm.

Application of the Newton–Raphson Algorithm to the MESH Equations

Because of the large number of variables and functions in the modified MESH equations (4.118), the Newton–Raphson algorithm must be written with matrices. If the equations and variables are grouped according to stages, then the matrix of partial derivatives needed in the Newton–Raphson method takes a form that is easy to solve. The amount of storage and the number of calculations are drastically reduced.

266

4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

The vector x of variables is  h iT ˙1,1 . . . l˙k,1 , g˙ i,1 . . . g˙ k,1 , T1 x = l 1  h i     x = l˙ . . . l˙ , g˙ . . . g˙ , T T x1  2 1,2 k,2 i,2 k,2 2     x2   ..  .   .  .   h iT  .      xj = l˙1,j . . . l˙k,j , g˙ i,j . . . g˙ k,j , Tj    x =  xj  with   .   ..  ..   . h      x  n−1 = l˙1,n−1 . . . l˙k,n−1 , g˙ i,n−1 . . .  xn−1   T  xn . . . g˙ k,n−1 , Tn−1 ]   h iT xn = l˙1,n . . . l˙k,n , g˙ i,n . . . g˙ k,n , Tn

The vector F of functions is     F1      F2    .    .    .        F =  Fj  with    .    ..           Fn−1    Fn

                      . (4.124)

T



T

                

F1 = [M1,1 . . . Mk,1 , Ei,1 . . . Ek,1 , H1 ] F2 = [M1,2 . . . Mk,2 , Ei,2 . . . Ek,2 , H2 ] .. . Fj = [M1,j . . . Mk,j , Ei,j . . . Ek,j , Hj ] .. .

T

Fn−1 = [M1,n−1 . . . Mk,n−1 , Ei,n−1 . . . T . . . Ek,n−1 , Hn−1 ] Fn = [M1,n . . . Mk,n , Ei,n . . . Ek,n , Hn ]

T

. (4.125)

With this notation the Taylor series developed around the estimated values xm becomes

F (xm ) + J (xm ) · ∆x = 0 .

(4.126)

Herein is J the Jacobian matrix of the partial derivatives ∂ F /∂ x of all functions with respect to all variables. Figure 4.57 presents the Jacobian matrix in general form. It contains n2 terms ∂ F /∂ x (n is the number of stages). Each term ∂ F /∂ x represents a submatrix of the partial derivatives of all functions of a single stage with respect to all variables on any stage. These submatrices have (2 · k + 1)2 elements (k is the number of components). In consequence, the Jacobian matrix has n2 · (2 · k + 1)2 elements. For example, the separation of a five-component system in a column with 50 stages yields a Jacobian matrix with 302 500 elements. However, as will be outlined later, most elements are zero.

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Figure 4.57 Jacobian matrix in the general form. The matrix contains n2 submatrices (n is the number of stages).

Equation (4.126) represents a system of coupled equations that are linear with respect to ∆x. In order to solve this system, the inverse Jacobian matrix J −1 has to be formed. This is a very complex problem of matrix calculation. With the inverse matrix, the solution of the Taylor series is

∆x = −F (xm ) · J −1 (xm ) .

(4.127)

Improved values of the non-linear system of Eq. (4.125) are easily evaluated:

xm+1 = xm + ∆x .

(4.128)

Here, ∆x is the calculated correction. If the functions F were linear, this correction would make the value of each function zero. For non-linear equations the values xm+1 are just an approximation, and the procedure has to be repeated. Because of the large number of variables and functions, the procedure of solving the MESH equations is very involved and time consuming. Some details are presented in the following. Jacobian Matrix

The Jacobian matrix is very large in the general case; see Figure 4.57. However, in the modified MESH equations of a distillation column, the functions Fj depend only on the variables of three neighboring stages j − 1, j , and j + 1. Thus, only

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

the terms in the main diagonal and the diagonals below and above are relevant. The non-zero elements are shaded in the matrix of Figure 4.58. All unshaded elements in the Jacobian matrix are zero. Hence, the Jacobian matrix is of the special type of a tridiagonal matrix. Since the terms ∂ F /∂ x are matrices or blocks, the Jacobian matrix is a block tridiagonal matrix. The tridiagonal matrix contains n + 2 · (n − 1) non-zero submatrices each with (2 · k + 1)2 elements. However, many of these elements are zero. The submatrix for the term ∂ Fj /∂ xj−1 , i.e. the derivatives of the functions of stage j with respect to the variables of stage j − 1, is presented in Figure 4.59. In this submatrix most elements are zero because the functions of stage j are only influenced by the state of the liquid of stage j − 1, i.e. liquid flow rates l˙i,j−1 and temperature Tj−1 . As can be seen from Eq. (4.119), the material balance of component i depends only on the liquid rates of the same component of stage j − 1. The equilibrium function does not depend on any variable on stage j −1, but the enthalpy function depends on all liquid rates on stage j − 1 and on temperature Tj−1 . Thus, all but the shaded elements are zero in the submatrix of Figure 4.59. The submatrix for the term ∂ Fj /∂ xj , i.e. the derivatives of the functions of stage j with respect to the variables of the same stage j , is seen in Figure 4.60. According to Eq. (4.119), the material balance of component i depends on the liquid rates and the gas rates of the same component only. The equilibrium function Ei,j depends on all liquid and gas rates on stage j and on the temperature on stage j . The same holds for the enthalpy function. Here again, the non-zero elements are shaded in Figure 4.60. The submatrix for the term ∂ Fj /∂ xj+1 , i.e. the derivatives of the functions of stage j with respect to the variables of stage j + 1, is presented in Figure 4.61. The functions of stage j are only influenced by the state of the gas of stage j + 1, i.e. g˙ i,j+1 and Tj+1 . The Mj equations of components i depend only on the gas flow rates g˙ i,j+1 of the same component. However, the enthalpy function Hj is influenced by all gas flow rates of stage j + 1 and the temperature of stage j + 1. These relevant elements are shaded in the matrix in Figure 4.61. All the other elements are zero. Each submatrix in the side diagonals has 2·k +1 non-zero elements, whereas each submatrix in the main diagonal matrix contains k 2 + 5 · k + 1 non-zero elements. In total, the Jacobian matrix contains 2 · (n − 1) · (2 · k + 1) + n · (k 2 + 5 · k + 1) non-zero elements. In a system with 5 components and 50 stages, only 3628 out of 302 500 elements are non-zero. Hence, approximately 99 % of all elements of the Jacobian matrix are zero. Consequently, sparse techniques are advantageously applied to matrix transformations [Naphtali and Sandholm 1971]. Partial Derivatives

For the setup of the Jacobian matrix, the non-zero elements of the partial derivatives ∂ F /∂ x have to be generated for all functions. The results are listed in Table 4.2. The partial derivatives of the material balances Mi,j are easily performed. Their values are +1, (1 + s˙ L,j ) and (1 + s˙ G,j ), respectively, according to the rules of differentiation with a constant factor. The partial derivatives of the equilibrium functions are more complex because

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

269

Figure 4.58 Jacobian matrix of a distillation column. The matrix has a tridiagonal structure with n + 2 · (n − 1) non-zero submatrices (shaded elements).

Figure 4.59 Structure of the submatrices in the upper side diagonal of the Jacobian matrix. Only 2 · k + 1 elements are non-zero (k is the number of components).

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Table 4.2 Partial derivatives of material, equilibrium, and enthalpy functions [Herguijuela 1993].

Mi -functions:

∂ Mi,j = 1; ∂ l˙i,j−1

∂ Mi,j = −(1 + s˙ L,j ) ∂ l˙i,j

∂ Mi,j = −(1 + s˙ G,j ) ; ∂ g˙ i,j

∂ Mi,j =1 ∂ g˙ i,j+1

Ei -functions:

∂ Ei,j = g˙ i,j − ∂ l˙m,j

∂ Ki,j ∂ l˙m,j

!

k k P P · l˙i,j · g˙ z,j − Ki,j · δi,m · g˙ z,j z=1

z=1

with δi,m = 1 for i = m and δi,m = 0 for i 6= m   k k P P ∂ Ei,j ∂ Ki,j = δi,m · l˙z,j − · l˙i,j · g˙ z,j − Ki,j · l˙i,j ∂ g˙ m,j ∂ g˙ m,j z=1 z=1   k P ∂ Ei,j ∂ Ki,j = · l˙i,j · g˙ z,j ∂ tj ∂ tj z=1

H -functions: k ∂ Hj ∂ hL,j−1 P = · l˙z,j−1 + hL,j−1 ∂ l˙m,j−1 ∂ l˙m,j−1 z=1 k ∂ Hj ∂ hL,j−1 P = · l˙z,j−1 ∂ tj−1 ∂ tj−1 z=1

!

k ∂ hL,j P · l˙z,j + hL,j · (1 − s˙ L,j ) ∂ l˙m,j z=1   k ∂ Hj ∂ hG,j P =− · g˙ z,j + hG,j · (1 − s˙ G,j ) ∂ g˙ m,j ∂ g˙ m,j z=1 k P ∂ Hj ∂ hL,j =− · (1 + s˙ L,j ) · l˙z,j ∂ tj ∂ tj z=1 k P ∂ hG,j − · (1 + s˙ G,j ) · g˙ z,j ∂ tj z=1 k ∂ Hj ∂ hG,j+1 P = · g˙ z,j+1 + hG,j+1 ∂ g˙ m,j+1 ∂ g˙ m,j+1 z=1 k ∂ Hj ∂ hG,j+1 P =− · g˙ z,j+1 ∂ tj+1 ∂ tj+1 z=1

∂ Hj =− ∂ l˙m,j

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271

Figure 4.60 Structure of the submatrices in the main diagonal. The number of non-zero elements is k2 · 5 · k + 1.

differentiation of products has to be performed. Just a few of the derivatives can be performed analytically. The terms listed in Table 4.2 contain the partial derivatives of the equilibrium ratio Ki,j with respect to liquid concentration, gas concentration, and temperature. The expressions for these derivatives depend on the equations used to describe the gas-liquid equilibria either by excess enthalpies (e.g. Wilson, Uniquac; see Chapter 3) or by equations of state (e.g. Redlich–Kwong). In most programs for column simulation, these partial derivatives are evaluated numerically by

∂F f (x + ∆x) − f (x − ∆x) = . ∂x 2 · ∆x

(4.129)

The same is true for the partial derivatives of the enthalpies hL and hG , in particular with respect to concentrations. A good approximation for the partial derivative of the equilibrium ratio Ki = γi · p0i /pt with respect to temperature T can be made because the temperature dependency of the vapor pressure p0i is much stronger than that of the activity coefficient γi . With the Antoine equation p0i = exp(Ai − Bi /(T + Ci )), it follows that   ∂ Ki γi Bi Bi = · exp Ai − · . (4.130) ∂T pt T + Ci (T + Ci )2 An approximation for partial derivatives of enthalpies hL and hG with respect to

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.61 Structure of the submatrices in the lower side diagonal of the Jacobian matrix. Only 2 · k + 1 elements are non-zero.

temperature is

∂ hL = cL and ∂t

∂ hG = cp . ∂t

(4.131)

Here, cL and cp are the heat capacities of the liquid and gas mixture, respectively. They can often be considered as independent on temperature. Estimation of Starting Values of the variables

The most essential prerequisite for a good convergence of the Newton–Raphson algorithm is the setting of good starting values of all the variables. One possibility is the use of a distillation line through the feed. The higher the number of equilibrium stages is, the more difficult is convergence. A good strategy is to start with only a few equilibrium stages and to search the solution. Then, the number of equilibrium stages is increased, and the profiles of temperature and component flow rates are proportionally transferred to the higher number of stages. With these starting profiles, the calculation is performed again. This strategy can be applied several times till the correct number of equilibrium stages is reached. 4.4.1.5

Iteration History

A computer program has been developed by the author’s research group (Herguijuela 1993) for rigorous column simulation with simultaneous solution of the mod-

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

273

ified MESH equations as described in the previous sections. The iteration history for the simulations of the methanol/ethanol/n-propanol system is presented in Figure 4.62. Column specifications are number of trays, location ˙ F˙ , thermal state and concentration of of feed tray, relative amount of distillate D/ the feed, and reflux ratio. Starting values for the internal liquid and vapor states were generated by calculating the distillation line through the feed. Liquid and vapor concentrations, xi and yi , were transformed into component flow rates under the assumption of constant liquid and vapor flow rates within the column. The starting profile (a distillation line with 10 stages) is marked in the triangular diagram of Figure 4.62A by 0. The first iteration yields the dotted liquid concentration profile marked by 1. Number 2 indicates the result of the second iteration. This profile is nearly identical with the final one reached in four iterations. The iteration history for a column with 20 stages is presented in Figure 4.62B. Starting values were the previous results with 10 stages. In only three iterations convergence is reached. The transition from 20 to 40 stages also requires three iterations only as is seen in Figure 4.62C. The iteration history for a column with 40 stages with a distillation line through the feed as starting profile is presented in Figure 4.62D. Due to the great discrepancy between starting profile and end profile, eight iterations are required to reach convergence. Large changes occur from iteration to iteration, and it is really astonishing that convergence is reached finally. In Figure 4.63A–D, similar results for the acetone/chloroform/benzene system are presented. This system exhibits a binary maximum azeotrope and a boundary distillation line that runs from the high boiler to the maximum azeotrope. Here again, starting values were generated by calculation of the distillation line with 10 stages through the feed (marked by 0 in Figure 4.63A). This profile lies far left from the final result. The first iteration yields the dotted profile on the right-hand side of the final one. The profiles of the third, fourth, and fifth iteration lie very close together. Starting with this profile, the solution for a column with 20 stages is found in four iterations; see Figure 4.63B. The same number of iterations is required for the transition from 20 to 40 stages, which is presented in Figure 4.63C. However, iteration history is much more complex for a 40-stage column when the starting profile is the feed distillation line; see Figure 4.63D. Mind that the bottom product lies at the convex side of the boundary distillation (see Section 4.3.2.1). The methanol/ethanol/water system, depicted in Figure 4.64A–C, forms a binary minimum azeotrope with a boundary distillation line between azeotrope and low boiler. Starting with the profile of a distillation line through the feed, six iterations are required to reach convergence for a column with 10 stages (Figure 4.64A). Especially the liquid concentration profile after the first iteration is not a very good one. But in spite of that, convergence is reached. With the solution of the 10-stage column as starting profile, simulation of a 20-stage column converges in three iterations (Figure 4.64B). The transition from 20 to 40 stages is displayed in Figure 4.64C. No convergence was reached for direct simulation of the 40-stage column with the feed distillation line as starting profile.

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.62 Internal concentration profiles for the methanol/ethanol/n-propanol system. A) Iteration history for 10 stages with the feed distillation line as starting profile. B) Iteration history for 20 stages with the 10-stage solution as starting profile.

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

275

Figure 4.62 (Continued) C) Iteration history for 40 stages with the 20-stage solution as starting profile. D) Iteration history for 40 stages with the feed distillation line as starting profile.

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.63 Internal concentration profiles for the acetone/chloroform/benzene system. A) Iteration history for 10 stages with the feed distillation line as starting profile. B) Iteration history for 20 stages with the 10-stage solution as starting profile.

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

277

Figure 4.63 (Continued) C) Iteration history for 40 stages with the 20-stage solution as starting profile. D) Iteration history for 40 stages with the feed distillation line as starting profile.

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4 MULTISTAGE CONTINUOUS DISTILLATION (RECTIFICATION)

Figure 4.64 Internal concentration profiles for the methanol/ethanol/water system. A) Iteration history for 10 stages with the feed distillation line as starting profile. B) Iteration history for 20 stages with the 10-stage solution as starting profile.

4.4 MULTISTAGE DISTILLATION OF MULTICOMPONENT MIXTURES

279

Figure 4.64 (Continued) C) Iteration history for 40 stages with the 20-stage solution as starting profile.

Computer programs for column simulation are very valuable means for the design of distillation processes especially for non-ideal multicomponent mixtures. However, they cannot replace the distillation expert. One cannot hope that a complex problem can be solved without intensive work of an expert even when a good program is available. The program does not substitute the engineer; it is just an efficient tool for the engineer.

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References Bekiaris, N., Meski, G.A., Radu, C.M., and Morari, M. (1993). Multiple steady states in homogeneous azeotropic distillation. Industrial and Engineering Chemistry Research 32 (9): 2023–2038. Bernot, C., Doherty, M.F., and Malone, M.F. (1991). Feasibility and separation sequencing in multicomponent batch distillation. Chemical Engineering Science 46 (5-6): 1311–1326. Block, U. and Hegner, B. (1976). Development and application of a simulation model for three-phase distillation. AIChE Journal 22 (3): 582–589. Block, U. and Hegner, B. (1977). Simulationsmethode für homogene Flüssigkeitsphasenreaktionen mit überlagerter Destillation. VT Verfahrenstechnik 11 (2): 101–104. Bošnjaković, F. (1935). Technische Thermodynamik. Leipzig: Steinkopff Verlag. Bošnjaković, F. and Knoche, K.F. (1997). Technische Thermodynamik. Darmstadt: Steinkopff Verlag. Chilton, T.H. and Colburn, A.P. (1935). Distillation and absorption in packed columns. Industrial & Engineering Chemistry 27 (3): 255–260. Danilov, R.Y., Petlyuk, F.B., and Serafimov, L.A. (2007). Minimum reflux regime of simple distillation columns. Theoretical Foundations of Chemical Engineering 41 (4): 371–383. Doherty, M.F. and Caldarola, G.A. (1985). Design and synthesis of homogeneous azeotropic distillations, 3. The sequencing of columns for azeotropic and extractive distillations. Industrial and Engineering Chemistry Fundamentals 24 (4): 474–485. Goldstein, R.P. and Stanfield, R.B. (1970). Flexible method for the solution of distillation design problems using the Newton-Raphson technique. Industrial & Engineering Chemistry Process Design and Development 9 (1): 78–84. Gorak, A. (1990). Berechnungsmethoden der Mehrstoffrektifikation – Theorie und Anwendungen. Habilitation thesis. RWTH Aachen. Halvorsen, I.J. and Skogestad, S. (1999). Analytical expressions for minimum energy

consumption in multicomponent distillation. AIChE Annual Meeting, Dallas, TX. Hausen, H. (1935). Rektifikation von Dreistoffgemischen. Forschung auf dem Gebiet des Ingenieurwesens 6 (1): 9–22. Hausen, H. (1952). Rektifikation idealer Dreistoff-Gemische. Zeitschrift für Angewandte Physik 2: 41–51. Henley, E.J. and Seader, J.D. (1981). Equilibrium Stage Separation Operations in Chemical Engineering. New York: Wiley. Herguijuela, J.R. (1993). Konzepte zur Zerlegung azeotroper Gemische durch Rektifikation. PhD thesis. University of Essen. Düsseldorf: VDI Verlag, Fortschrittsberichte Reihe 3: Verfahrenstechnik, No. 375. Holland, C.D. (1963). Multicomponent Distillation. New York: Interscience. Kenig, E.Y. (2000). Modeling of multicomponent mass transfer in separation of fluid mixtures. Düsseldorf: VDI-Verlag . King, C.J. (1980). Separation Processes. New York: McGraw-Hill. Krishnamurthy, R. and Taylor, R. (1985). Simulation of packed distillation and absorption columns. Industrial & Engineering Chemistry Process Design and Development 24 (3): 513–524. Levy, S.G., van Dongen, D.B., and Doherty, M.F. (1985). Design and synthesis of homogeneous azeotropic distillations 2. Minimum reflux calculations for nonideal and azeotropic columns. Industrial and Engineering Chemistry Research 24 (4): 463–474. McCabe, W.L. and Thiele, E.W. (1925). Graphical design of fractionating columns. Industrial & Engineering Chemistry 17 (6): 605–611. Naphtali, L.M. and Sandholm, D.P. (1971). Multicomponent separation calculations by linearization. AIChE Journal 17 (1): 148–153. Offers, H. (1995). Berechnung des Mindestenergiebedarfs von Rektifikationskolonnen. PhD thesis. University of Essen. Düsseldorf: VDI Verlag, Fortschrittsberichte Reihe 3: Verfahrenstechnik, No. 414. Perry, R.H., Green, D.W., and Maloney, J.O.

REFERENCES (eds) (1984). Perry’s Chemical Engineers’ Handbook, New York: McGraw-Hill, chap. Physical & Chemical Data, pp. 3–189. Seader, J.D. (1989). The rate-based approach for modeling staged separations. Chemical Engineering Progress 85: 41–49. Stichlmair, J. (1988). Zerlegung von Dreistoffgemischen durch Rektifikation. Chemie Ingenieur Technik 60 (10): 747–754. Stichlmair, J. (1991). Separation of ternary mixtures by rectification. International Chemical Engineering 31 (3): 423–433. Stichlmair, J., Offers, H., and Potthoff, R.W. (1993). Minimum reflux and minimum reboil in ternary distillation. Industrial and Engineering Chemistry Research 32 (10): 2438–2445. Stichlmair, J.G. and Herguijuela, J.R. (1992). Separation regions and processes of zeotropic and azeotropic ternary distillation. AIChE Journal 38 (10): 1523–1535. Taylor, R. and Krishna, R. (1993). Multicomponent Mass Transfer. New York: Wiley. Taylor, R., Kooijman, H.A., and Woodman, M.R. (1993). Industrial applications of a nonequilibrium model of distillation and absorption operations. Institution of Chemical Engineers Symposium Series 128: A415–A427. Treybal, R.E. (1968). Mass-Transfer

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Operations. New York: McGraw-Hill. Underwood, A.J.V. (1948). Fractional distillation of multicomponent mixtures. Chemical Engineering Progress 44 (8): 603–614. Vogelpohl, A. (1964a). Rektifikation von Dreistoffgemischen. Teil 1: Rektifikation als Stoffaustauschvorgang und Rektifikationslinien idealer Gemische. Chemie Ingenieur Technik 36 (9): 907–915. Vogelpohl, A. (1964b). Rektifikation von Dreistoffgemischen. Teil 2: Rektifikationslinien realer Gemische und Berechnung der Dreistoffrektifikation. Chemie Ingenieur Technik 36 (10): 1033–1045. Vogelpohl, A. (1975). Die optimale Lage des Zulaufbodens bei der Vielstoffrektifikation. Chemie Ingenieur Technik 47 (21): 895–895. Vogelpohl, A. (2015). Distillation – The Theory. Berlin: Walter de Gruyter. Wahnschafft, O.M., Koehler, J.W., Blaß, E., and Westerberg, A.W. (1992). The product composition regions of single-feed azeotropic distillation columns. Industrial and Engineering Chemistry Research 31 (10): 2345–2362. Wang, J.C. and Henke, G.E. (1966). Tridiagonal matrix for distillation. Hydrocarbon Processing 45 (8): 155–163.

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5 Reactive Distillation, Catalytic Distillation A major task of chemical engineering is the downstream processing of the effluents of chemical reactors. Since most chemical reactions are reversible, i.e. the reactants (educts) are not completely converted into the desired products, the effluent of a reactor is a mixture of reactants (educts) and products. The non-converted reactants have to be separated from the products to enable an effective recycling into the reactor. The products have to be purified. In most cases, the fractionation of the reactor effluents is in most cases established by distillation. Often, downstream processing is more complex and more expensive than the reaction itself, especially in mixtures with azeotropes. A significant simplification of such processes is often possible by simultaneous realization of reaction and distillation in a countercurrently operated column [Sundmacher and Kienle 2003; Sakuth et al. 2008; Doherty and Malone 2001; Almeida-Rivera et al. 2004]. These processes are called reactive or catalytic distillation. The reaction takes place in a specific section of the column. The products, which hinder the progress of the reaction, are continuously removed from that section by the superimposed distillation. In consequence, a very high or even total conversion of the reactants can be reached even for reversible reactions [Frey and Stichlmair 1999b; Tuchlenski et al. 2001; Espinosa et al. 1999]. The objective of this chapter is to demonstrate principles and potentials of reactive distillation. Some essential simplifications are made in the following: • • • • •

The reaction is reversible, i.e. reactants as well as products exist in the effluent. The reaction takes place in the liquid phase only. The reaction is catalyzed either by a homogeneous or a heterogeneous catalyst. The catalyzed reaction is instantaneous. No reaction takes place without catalyst.

These assumptions simplify the real conditions. However, these simplifications allow an easy description and, in turn, a good understanding of the fundamental mechanism effective in reactive distillation.

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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5.1

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

Fundamentals

Knowledge of the equilibrium is the most fundamental prerequisite for a conceptual synthesis of non-reactive as well as reactive distillation processes. However, the equilibrium in reactive distillation systems is more complex since the chemical equilibrium is superimposed on the physical equilibrium. First, the laws of reactive and non-reactive equilibrium are presented. 5.1.1

Chemical Equilibrium

The following reaction mechanism is considered:

νa · a + νb · b + . . . ↔ νp · p + νq · q + . . . .

(5.1)

The letter νi denotes the stoichiometric factor of the substance i (= a, b, . . .). The reaction is reversible and instantaneous. In ideal mixtures the following holds for the chemical equilibrium:

KR =

ν ν k xpp · xq q . . . Y νi = x . νb νa xa · xb . . . i=1 i

(5.2)

By convention, the stoichiometric coefficients νi are negative for reactants and positive for products. The equilibrium constant of the reaction KR , is essentially a function of temperature. Hence, its value can be influenced by the pressure of the reactive distillation column since the mixture is at the boiling point. In non-ideal mixtures the concentrations need to be corrected by activity coefficients γi :

KR =

ν ν ν ν k xpp · xq q . . . γp p · γq q . . . Y ν · νa νb = (xi · γi ) i . νb νa xa · xb . . . γa · γb . . . i=1

(5.3)

Here, the value of the equilibrium constant KR is additionally influenced by the concentrations xi of the other compounds, especially at low concentrations. Figure 5.1 presents the chemical equilibrium for the following reaction mechanism:

a + b ↔ c with KR =

xc = 8. xa · xb

(5.4)

The letters a, b, and c denote the compounds in the order of increasing boiling points. Smaller values of the equilibrium constant KR lead to a smaller conversion rate of the reactants into the products. 5.1.2

Stoichiometric Lines

For the determination of the reaction products, the stoichiometry of the reaction has to be considered (see Example 5.1):

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5.1 FUNDAMENTALS

Figure 5.1 Chemical equilibrium of the reaction a + b ↔ c with KR = 8. The stoichiometric lines emerge from a pole π located outside the concentration triangle.

Example 5.1: Stoichiometric Lines

(1) Derive an equation for the change of mole fractions caused by a homogeneous reaction. (2) Find the coordinates of the pole π for the reaction a + b ↔ c. Mind that the stoichiometric coefficients of the reactants are negative and that of the products positive. Solution:

(1) Reaction: νa · a + νb · b + . . . ↔ νp · p + νq · q + . . . P Mole fraction xi ≡ Ni /N with N = Ni The nominator Ni is changed by the reaction of substance i, and the denominator N by the reaction of all components in the mixture. Changes of number N of moles: Ni = Ni,0 + νi · nRP (nRP is the number of the reaction progress)

N=

X

Ni =

X

= N0 · (1 + νtot

X

νi · nRP = N0 + νtot · nRP = X · nRP /n0 ) with νtot = νi

Ni,0 +

Mole fraction:

Ni,0 + νi · nRP Ni,0 /N0 + νi · nRP /N0 = = N0 · (1 + νtot · nRP /N0 ) 1 + νtot · nRP /N0 xi,0 + νi · nRP /N0 = 1 + νtot · nRP /N0

xi =

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5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

Extension with xi,0 :

xi,0 + νi · nRP /N0 1 + νtot · nRP /N0 − xi,0 · = 1 + νtot · nRP /N0 1 + νtot · nRP /N0 nRP /N0 = (νi − νtot · xi,0 ) · 1 + νtot · nRP /N0

xi − xi,0 =

Elimination of nRP /N0 by introducing a reference component r. The above equation is also valid for the reference component r.

xi − xi,0 νi /νtot − xi,0 = xr − xr,0 νr /νtot − xr,0 Group of straight lines through point xπ,i = νi /νtot (called pole π )

xi,0 · (νr /νtot − xr ) + νi /νtot · (xr − xr,0 ) νr /νtot − xr,0 P = νi = νa + νb + νc = −1 + (−1) + 1 = −1

Or xi = (2) νtot

The coordinates of the pole are

νa −1 νb −1 = = 1 ; xπ,b = = =1 νtot −1 νtot −1 νc +1 = = = −1 νtot −1

xπ,a = xπ,c

Check with Figure 5.1.

xi =

X xi,0 · (νr /νtot − xr ) + νi /νtot · (xr − xr,0 ) with νtot = νi . νr /νtot − xr,0 (5.5)

Here, r denotes a reference component. Equation (5.5) is represented by dotted straight lines in the triangular concentration diagram of Figure 5.1. These lines describe the direction of conversion of the reactants by the chemical reaction. All stoichiometric lines emerge from a pole π whose coordinates are (see Example 5.1)

xπ,i =

X νi with νtot = νi . νtot

(5.6)

The coordinates of the pole π are located outside the concentration triangle, i.e. the values of xπ,i can be larger than 1 and negative. In equimolar reactions (e.g. a+b ↔ 2 · c), the coordinates of the pole are infinite, xπ,i → ∞. In consequence, the stoichiometric lines are parallel ones for equimolar reactions. The reaction runs from

287

5.1 FUNDAMENTALS

Figure 5.2 Distillation lines of an ideal ternary mixture with αac = 3 and αbc = 2.

the feed components (reactants) to the equilibrium as indicated by the arrows at the stoichiometric lines. 5.1.3

Non-reactive and Reactive Distillation Lines

The adequate form for the presentation of the vapor–liquid equilibrium, in particular in ternary systems, are distillation lines. They constitute a sequence of equilibrium stages and run from a local boiling point maximum to a local boiling point minimum. Their general definition is (see Section 2.4.5)

xi,0

→

∗ yi,0

=

xi,1

→

∗ yi,1

=

xi,2

... . (5.7)

equilibrium condensation equilibrium condensation Starting at any liquid concentration xi,0 , the equilibrium concentration of the va∗ por yi,0 is determined; see Figure 5.2. Condensation of this vapor generates a new ∗ liquid with the concentration xi,1 = yi,0 . A sequence of such steps generates the distillation line that describes the maximal enrichment of low boilers in the mixture. The arrows at the distillation lines point to falling boiling temperatures. The length of an equilibrium step is marked by points. Distillation lines are very helpful for process synthesis since they represent the concentration profile within a distillation column at total (and high) reflux (see Section 4.3.2). The distillation lines of an ideal ternary mixtures follow from Eq. (4.68):

xa,0 xb,0 n and xb,n = αbc · 00 N 00 N n n with N 00 = 1 + (αac − 1) · xa,0 + (αbc − 1) · xb,0 .

n xa,n = αac ·

(5.8)

288

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

The letter n denotes the number of equilibrium stages by integer or real variables. Negative values of n are also permissible. Figure 5.2 presents the course of distillation lines for an ideal ternary mixture with the relative volatilities αac = 3 and αbc = 2. All distillation lines originate at the high boiler c and terminate at the low boiler a. These substances can easily be separated from a ternary mixture either as bottom fraction (c) or as overhead fraction (a) of a distillation column. The larger the distance between the points at the distillation lines, the easier is the separation. In reactive systems the chemical reaction is superimposed on the vapor–liquid equilibrium. The states in a column can be described by reactive distillation lines whose definition is:

xi,1

→

∗ yi,1

=

x∗i,1

→

xi,2

equilibrium condensation reaction

→

∗ yi,2

=

x∗i,3

... .

equilibrium condensation (5.9)

The procedure is graphically explained in Figure 5.3. The starting point 1 lies at the ∗ chemical equilibrium line. The concentration of the vapor yi,1 is in equilibrium with the liquid concentration xi,1 . It is marked in the diagram by point 1∗ . Condensation of the vapor generates a liquid with concentration x∗i,1 , which is off the chemical equilibrium line. Therefore, it is converted by the chemical reaction into the state xi,2 (point 2). The direction of conversion is given by the stoichiometric line through point 1∗ . Multiple application of this procedure leads to a sequence of liquid states at the chemical equilibrium line, which are marked by points 1, 2, 3, . . . in Figure 5.3. This sequence of liquid states constitutes a reactive distillation line running from the starting point 1 to the low boiler a. In ternary mixtures, the reactive distillation line coincides with the chemical equilibrium curve. Applying the same procedure to a different starting point, e.g. point 10 in Figure 5.3, leads to a reactive distillation line that runs from the starting point to the intermediate boiler b. The reason for the different direction of movement is obviously the different gradient of the vapor–liquid equilibrium. A very special situation exists if, at any point A at the chemical equilibrium line, the vapor–liquid equilibrium is co-linear with the stoichiometric line. In this case the liquid state remains unchanged since the vapor with different concentration (point A∗ ) is, after condensation, converted by the chemical reaction back to the original liquid state (point A). The concentration change caused by the physical equilibrium is compensated for by the chemical reaction. This special liquid state at the chemical equilibrium line is called reactive azeotrope (see Section 5.1.4) [Okasinski and Doherty 1997; Frey and Stichlmair 1999a]. Several authors [Ung and Doherty 1995; Espinosa et al. 1995] suggest a transformation of the liquid concentrations at the chemical equilibrium line. From Eq. (5.5), it follows for xr = 0 (r is the product) that:

Xi =

νr /νtot · xi,0 − νi /νtot · xr,0 νr /νtot − xr,0

with

νtot =

X

νi .

(5.10)

5.1 FUNDAMENTALS

289

Figure 5.3 Design of a reactive distillation line of an ideal ternary mixture: − − − vapor–liquid equilibrium; · · · stoichiometric lines.

P The transformed concentrations meet the condition Xi = 1. In ternary mixtures the values of Xi lie between 0 and 1; in quaternary mixtures one component has a value between −1 and +1. This transformation is, as indicated by dotted lines in Figure 5.3, a projection of the states at the reactive distillation line along the stoichiometric lines onto the right-hand side of the concentration triangle. The reference component r (here c) is not visible in the transformed coordinates. Such a transformation is advantageous because it effects a reduction of the dimensions of the concentration space. A triangle is transformed into a line, a tetrahedron into a plane, etc. In the transformed concentrations the lever rules are valid since the effect of the reaction is not visible. The disadvantage of this transformation is the loss of some important information. 5.1.4

Reactive Azeotropes

The condition for the existence of reactive azeotropes is that the vapor, which emerges from the liquid, is, after condensation, converted by the reaction into the original liquid concentration at the chemical equilibrium line. In mathematical terms, the vector of the vapor–liquid equilibrium and the vector of the stoichiometric line are co-linear. Whether or not this condition is met can be easily checked from a plot of liquid residue lines (see Section 3.2) and stoichiometric lines: the stoichiometric lines form tangents to the liquid residue lines; see Figure 3.11. Figure 5.4 shows the liquid residue lines of a ternary mixture. For the reaction a + b ↔ c all stoichiometric lines emerge from the pole π on the a/b side of

290

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

the concentration triangle in Figure 5.4A. Its coordinates are xπa = 1 , xπb = 1 , xπc = −1. The points of tangential contact of stoichiometric lines and residue lines form a curve of potential reactive azeotropes (dashed dotted line in Figure 5.4A). In most cases this line of co-linear vapor–liquid equilibrium and stoichiometric lines lies close by the baseline of the concentration triangle. Whether or not a reactive azeotrope is formed depends on the value of the chemical equilibrium constant KR . At large values of the equilibrium constant KR , a reactive maximum azeotrope is formed close to the c-corner of the diagram (see small triangle in Figure 5.4A). With decreasing values of the equilibrium constant KR , the reactive azeotrope moves to the b-corner of the triangle and vanishes finally. It is important to note that, in this case, reactive azeotropes exist only if the value of the chemical equilibrium constant KR is large. The equimolar reaction a+c ↔ 2·b is considered in Figure 5.4B. The coordinates of the pole are xπa = −∞; xπb = −∞; xπc = +∞. The stoichiometric lines are parallel lines oriented vertically to the a/c-side of the concentration triangle; see Figure 5.4B. The stoichiometric lines intersect the liquid residue lines with an angle between 30° and 120°. Consequently, no tangential points and, in turn, no reactive azeotropes exist in systems where the liquid residue lines and the chemical equilibrium lines have the same origin and terminus, here a-corner and c-corner. This feature is very important for the design of reactive distillation processes. In Figure 5.4C the reaction b + c ↔ 3 · a is considered. In this case the coordinates of the pole are xπa = 3 , xπb = −1 , xπc = −1. The points of the tangential contact of residue lines and stoichiometric lines constitute a line of potential reactive azeotropes, which lies in most cases close by the right-hand side of the concentration triangle. Again, the existence of reactive azeotropes and their location in the diagram depend on the value of the chemical equilibrium constant KR . Large values of KR always result in the formation of a reactive azeotrope; see small triangle of Figure 5.4C. The value of the equilibrium constant KR and, in turn, the location of the reactive azeotrope can be manipulated to a certain degree by the operating pressure of the reactive distillation column. The reaction a+b ↔ c is considered in a system with a non-reactive a/b-minimum azeotrope; see Figure 5.5. All residue lines start at the high boiler c and end at the minimum azeotrope (arrows point to falling boiling temperatures). Reactive azeotropes can be found by drawing stoichiometric tangents to the residue lines. Two curves of potential reactive azeotropes exist in this system. One lies close by the baseline of the triangle and the other one close by the upper part of the right-hand side of the concentration triangle. Reactions with a large value of the equilibrium constant KR exhibit a reactive azeotrope. The same holds for reactions with very small values of KR . Systems with intermediate values of the equilibrium constant KR may form no reactive azeotrope (see small concentration triangle in Figure 5.5). A system with a distillation boundary is considered in Figure 5.6A for the case of the non-equimolar reaction a + b ↔ c. The stoichiometric lines emerge from the pole π and form tangents to the residue lines only in the lower left section of the diagram. The curve of potential reactive azeotropes runs between the c-corner and the non-reactive minimum azeotrope. A reaction with a large equilibrium constant KR

5.1 FUNDAMENTALS

291

Figure 5.4 A) Determination of reactive azeotropes of the reaction a + b ↔ 2 · c. B) Determination of reactive azeotropes of the reaction a + c ↔ 2 · b. No reactive azeotropes exist since the stoichiometric pole π lies on the concave side of the residue lines.

292

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

Figure 5.4 (Continued) C) Determination of reactive azeotropes of the reaction b + c ↔ 2 · a.

Figure 5.5 Determination of reactive azeotropes of the reaction a + b ↔ c in a system with an a/b-minimum azeotrope.

5.2 TOPOLOGY OF REACTIVE DISTILLATION LINES

293

shows two reactive azeotropes, a maximum (left) and a minimum (right) azeotrope. If the value of the equilibrium constant KR is small, no reactive azeotrope exists. An interesting system is shown in Figure 5.6B with a distillation boundary running between the b/c- and the a/b-azeotrope. Two curves of potential azeotropes exist: one runs between the c-corner and the b/c-azeotrope and the other one between the a-corner and the a/b-azeotrope. In case of a large value of the equilibrium constant KR , two reactive azeotropes exist. In the case of a very low value of KR , just one reactive azeotrope is formed. Surprisingly, no reactive azeotrope exists at intermediate values of the equilibrium constant KR ; see small triangle in Figure 5.6B. In all cases studied here, the lines of potential reactive azeotropes run between singular points of the system, i.e. pure substances or azeotropes. One should be aware that the curves of potential reactive azeotropes may change drastically even by small variations of the residue line charts. Consequently, good knowledge of the vapor–liquid equilibrium is necessary for the localization of reactive azeotropes. Reactive azeotropes can be more precisely located in a McCabe–Thiele-type plot of the vapor and liquid concentrations in transformed coordinates Xi and Yi (see Eq. (5.12)) [Okasinski and Doherty 1997]. The reactive azeotropes always lie at the chemical equilibrium line. Since in reactive distillation the mixture is at its boiling point, the temperature and, in turn, the value of the reaction equilibrium constant KR are influenced by the operating pressure. As a consequence the location of reactive azeotropes can be influenced by changing the pressure of the system.

5.2

Topology of Reactive Distillation Lines

This section presents some typical examples for reactive distillation lines and reactive azeotropes. Again, it is assumed that a reversible and instantaneous reaction takes place in the liquid phase only. 5.2.1

Reactions in Ternary Systems

Figure 5.7 presents the graphical plot of the following reaction mechanism:

a + 2 · b ↔ c with KR =

xc = 4. xa · x2b

(5.11)

Assuming an approximately ideal ternary mixture with the relative volatilities αac = 3 and αbc = 2, a reactive maximum azeotrope exists at the reactive distillation line. Above that reactive azeotrope the reactive distillation line runs toward the low boiler a and below toward the intermediate boiler b. If the stoichiometric lines are considered as elements of the reactive distillation line, which is often done, the boundary is defined by the stoichiometric line through the reactive azeotrope. The reactive distillation lines describe the concentration profile within a column. The reactive azeotrope cannot be crossed by reactive distillation (but by non-reactive distillation).

294

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

Figure 5.6 A) Determination of reactive azeotropes of the reaction a + b ↔ c in a system with a b/c-minimum azeotrope and a distillation boundary. B) Determination of reactive azeotropes of the reaction a + b ↔ c in a system with an a/b- and a b/c-minimum azeotrope and a boundary residue line.

5.2 TOPOLOGY OF REACTIVE DISTILLATION LINES

295

Figure 5.7 Reactive distillation line of the reaction a + 2 · b ↔ c. There exists a reactive maximum azeotrope.

5.2.2

Reactions in Ternary Systems with Inert Components

Sometimes the feed mixture with the reactants a and b contains an inert species i. Such systems can be graphically presented by a tetrahedron as shown in Figure 5.8. The reaction mechanism is

a + b + i ↔ c + i with KR =

xc = 10 . xa · xb

(5.12)

If there is no inert species present, a similar situation as dealt with in the previous section exists. The concentration triangle a/b/c forms the base of a tetrahedron, and the chemical equilibrium line runs between substances a and b. In the presence of the inert substance i, the chemical equilibrium forms the envelope of a cone constituted by the inert-free chemical equilibrium and the peak of the tetrahedron. The reactive distillation lines lie at the cone surface. They run from the reactive maximum azeotrope to the peak of the tetrahedron provided the inert component is the lowest boiling species in the mixture. Hence, the chemical equilibrium line and the reactive distillation lines are no longer identical. The stoichiometric lines emerge from the pole π , which lies in the plane of the tetrahedron base, but off the concentration triangle. They allow an easy determination of the chemical equilibrium to any concentration of the reactants even when the inert substance i is present. Helpful is the projection of the states on the cone envelope into the plane i/a/b along the stoichiometric lines, as is shown in Figure 5.9. In these transformed co-

296

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

Figure 5.8 Reactive distillation lines of the reaction a + b ↔ c with a low boiling inert component i.

Figure 5.9 Reactive distillation lines of Figure 5.8 presented by transformed coordinates Xi .

5.2 TOPOLOGY OF REACTIVE DISTILLATION LINES

297

ordinates (according to Eq. (5.10)), the lever rule is valid since the influence of the reaction has been eliminated. 5.2.3

Reactions with Side Products

Chemical reactions are often accompanied by a (unwanted) side reaction between one of the reactants and the products. Considered are the following reactions:

xc = 8 and xa · xb xd a + c ↔ d with KR = = 4. xa · xc a + b ↔ c with KR =

(5.13)

The chemical equilibrium of the four species can be graphically represented by a tetrahedron. For the main reaction (Eq. (5.13), upper line), the substance d is an inert species. In analogy to the previous section, the chemical equilibrium lies on the envelope of a cone that is formed by the chemical equilibrium line of the system a/b/c and the d-corner of the tetrahedron; see Figure 5.10.

Figure 5.10 Reactive distillation line of the reactions 2 · a + b ↔ c and a + c ↔ d.

298

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

For the side reaction (Eq. (5.13), lower line), the species b is inert. Consequently, the chemical equilibrium of the side reaction lies on a cone envelope formed by the chemical equilibrium of the system a/c/d and the b-corner of the tetrahedron. The line of intersection of the two cone surfaces is the chemical equilibrium of the combined reactions. This line is identical with the reactive distillation line. Even here, a reactive azeotrope may exist. The stoichiometric lines that describe pairs of educts and products emerge from two poles πc and πd , which can be located by the procedure described before. 5.2.4

Reactions in Quaternary Systems

An important class of reactions is

a + b ↔ c + d with KR =

xc · xd = 1. xa · xb

(5.14)

The chemical equilibrium is graphically presented in Figure 5.11A. It forms a saddle area that extends between the sides a/c, a/d, b/c, and b/d of a tetrahedron. Since the number of moles is not changed by the reaction, the stoichiometric lines are all parallel to the main stoichiometric line running from the middle of the a/b side to the middle of the c/d side. The reactive distillation lines are qualitatively sketched in the saddle area. They originate at the high boiling products d and c and terminate at the low boiling educts a and b, respectively. A reactive saddle azeotrope exists as well as two reactive distillation boundary lines that connect the origins and the termini, respectively, of the distillation lines. An abstract but more exact presentation of the reactive distillation lines and the reactive azeotrope offers the use of transformed coordinates Xi (according to Eq. (5.10)). The states in the chemical equilibrium area are projected along the stoichiometric lines onto a plane oriented vertically to the stoichiometric lines. The tetrahedron transfers into a rhombus with component a and b at the top and at the bottom, respectively (see Figure 5.11B). The reactive saddle azeotrope lies at the intersection of the reactive boundary distillation lines. They cannot be crossed by reactive distillation. Hence, the mixture is divided into four reactive distillation regions.

5.3

Topology of Reactive Distillation Processes

Knowledge of the reactive distillation lines is the basis for a conceptual design of reactive distillation processes. The procedure is, in essence, the same as at the design of non-reactive distillation processes described in detail in Section 4.3. The most essential steps are: • Determination of the feasible products. • Formulation of the material balance around the column.

5.3 TOPOLOGY OF REACTIVE DISTILLATION PROCESSES

Figure 5.11 A) Reactive distillation lines of the reaction a + b ↔ c + d with KR = 1. B) Reactive distillation lines of the system of Figure 5.11A in transformed coordinates.

299

300

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

• Identification of separation barriers, e.g. azeotropes, boundary distillation lines, or boundary distillation areas. The feasible product fractions are determined from the knowledge of the distillation lines. Feasible pure products of distillation columns are either origins or termini of distillation lines. This principle holds also for reactive distillation. The material balance around the column is graphically represented by a straight line. The material balance is met in non-reactive distillation if the feed F˙ , the dis˙ , and the bottoms B˙ lie at the same straight line. In reactive distillation tillate D this principle does no longer hold since some of the species are converted into other ˙ , and the botspecies by the reaction. Hence, the states of the feed F˙ , the distillate D ˙ toms B are only co-linear after transformation by Eq. (5.10) along the stoichiometric lines. Barriers for the separation are azeotropes, boundary distillation lines, boundary distillation areas, etc. The effectiveness of reactive distillation is based on the fact that reactive distillation lines can cross non-reactive distillation boundaries. A disadvantage of reactive distillation is the risk of formation of reactive azeotropes and reactive distillation boundaries [Stichlmair and Frey 1999]. Reactive distillation is especially effective in catalytic reactions since non-reactive and reactive sections can be combined at will in a column. In the reactive sections non-reactive barriers can be crossed and vice versa. If possible, a heterogeneous catalyst should be favored against a homogeneous catalyst in order to avoid the separation of the catalyst from a homogeneous mixture. In the next sections the potentials of reactive distillation are presented in a general way by comparing processes of sequential and simultaneous reaction and distillation. 5.3.1

Single Product Reactions

The following reaction is considered:

a + b ↔ c.

(5.15)

Since not all of the feed (reactants, educt) is converted into the product, the effluent of the reactor is a ternary mixture that needs to be fractionated into a pure product fraction and an reactant fraction. Figure 5.12A presents the conventional process where the reaction and the separation are established sequentially. The stoichiometric ˙ consisting of reactant is fed into the reactor R-1 to be converted. The effluent R1 substance c and some unconverted substances a and b is subsequently separated in ˙ and a bottom fraction B1 ˙ . the distillation column C-1 into an overhead fraction D1 The overhead fraction is recycled into the reactor. If the chemical equilibrium lies very close by the reactant, a large stream has to be recycled, making the process very expensive. Figure 5.12B shows the process with simultaneous reaction and distillation in a countercurrently operated column. The unreacted reactants a and b being the low boilers move upward into the catalytic section of the column to be further converted. Since the high boiling reaction product c moves downward in the column, a total

5.3 TOPOLOGY OF REACTIVE DISTILLATION PROCESSES

301

Figure 5.12 A) Process with sequential realization of reaction and distillation. B) Process with simultaneous realization of reaction and distillation.

302

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

conversion of a stoichiometric reactant is finally reached. The product (low boiler c) is purified in the lower, non-catalytic section of the column. The concentration profile within the column is sketched in the triangular concentration diagram of Figure 5.12B by a bold line. In the reactive section (with catalyst), the liquid states always lie at the chemical equilibrium line. In the non-reactive section (without catalyst), they follow the concentration profile of a non-reactive distillation line, which originates from the high boiler c. The end points of the internal concentration profile do not lie at a straight line through the feed F˙ . However, a projection of the bottoms B˙ along the stoichiometric line to the a/b side of the triangle gives the feed F˙ , proving that the material balance is met. The advantage of reactive distillation is that no external recycle from the distillation column to the reactor is required since there exists a very effective internal recycle. Furthermore, in case of an exothermic reaction, the heat of reaction is automatically utilized for distillation. The merits of reactive distillation are even larger in systems with non-reactive azeotropes. It is supposed in Figure 5.13A that the chemical equilibrium lies very close by a distillation boundary. Hence, only a small part of the product c can be separated, and, in turn, the recycle stream of the non-reactive process becomes very large. In reactive distillation, the distillation boundary is no longer a barrier, and, consequently, the reactants are completely converted into the desired product c; see Figure 5.13B. 5.3.2

Decomposition Reactions

The following type of decomposition reactions is considered in Figure 5.14:

c ↔ a + b.

(5.16)

The products a and b have to be completely separated. The conventional process is shown in Figure 5.14A. The feed is partially converted in the reactor R-1. The ternary mixture is fed into column C-1 to be split into a bottom ˙ , which is separated fraction B˙ , which is recycled, and a binary overhead fraction D into pure a and b, respectively, in column C-2. Two distillation columns are required for downstream processing. In a reactive distillation process, shown in Figure 5.14B, the reaction and the separation are established in a single column with a catalyzed stripping section. The condition for such a simple process is that no reactive azeotrope exists. This condition is met when the chemical equilibrium lies very close by the right-hand side of the triangle in Figure 5.14B; see Section 8.7.1. The advantages of reactive distillation are especially significant if non-reactive azeotropes exist in the system that hinder the separation of the products. Such an example is presented in Figure 5.15A. The reaction is

c ↔ a + b.

(5.17)

The three species form two binary azeotropes that are connected by a boundary dis-

5.3 TOPOLOGY OF REACTIVE DISTILLATION PROCESSES

303

Figure 5.13 A) Process with sequential realization of reaction and distillation in a system with a distillation boundary. B) Realization of the process of Figure 5.13A by reactive distillation.

304

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

Figure 5.14 A) Conventional process for a decomposition reaction c ↔ a + b and separation of the products a and b. B) Realization of the process of Figure 5.14A by reactive distillation.

5.3 TOPOLOGY OF REACTIVE DISTILLATION PROCESSES

305

Figure 5.15 A) Conventional process for the decomposition reaction c ↔ a + b. The separation of the products a and b is very complex due to the existence of an a/b-minimum azeotrope. B) Realization of the process of Figure 5.15A by reactive distillation.

306

5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

tillation line. In a conventional process, the chemical equilibrium is reached in the reactor R-1. The ternary effluent is fed into column C-1 to be split into an overhead ˙ (mixture of a and b) and a bottom fraction B1 ˙ (pure high boiler c) that fraction D1 ˙ is recycled into the reactor. The fraction D1 is fed into column C-2 where the low ˙ has the concentration boiler a is separated as bottoms. The overhead fraction D2 of the a/b-azeotrope. Methods for separating azeotropic mixtures are dealt with in Section 8.5. Here the generalized process is used. The binary azeotropic mixture is ˙ ) to give a mixture M˙ that lies on mixed with a low boiling entrainer e (fraction D4 ˙ is split in column C-3 the left-hand side of the distillation boundary. The stream M ˙ ˙ that lies on the into the bottom fraction B3 (pure b) and an overhead fraction D3 ˙ concave side of the distillation boundary. Fraction D3 is finally separated in col˙ and B4 ˙ . The two fractions are recycled to column C-3 umn C-4 into fraction D4 and column C-2, respectively. The process effected in columns C-2, C-3, and C-4 is in essence identical with the general process for the separation of azeotropic mixtures developed in Section 8.6.2.2. Depending on the system properties, the separation of the azeotropic mixture a/b can also be effected by azeotropic or extractive distillation. In each version of the process, one reactor and four distillation columns are required. Figure 5.15B presents the process with reactive distillation. Since no reactive azeotrope exists, the internal concentration profile runs without any barrier from the intermediate boiler b to the low boiler a. No non-reactive section is necessary because the desired products are origin and a terminus, respectively, of the reactive distillation line. The educt c is completely converted and fractionated into the pure products a and b in a single column. The process shown in Figure 5.15B convincingly demonstrates the advantages offered by reactive distillation. 5.3.3

Side Reactions

In the simultaneous realization of reaction and distillation in a countercurrently operated column, the products of the reaction are steadily removed from the reaction zone. This is, in essence, the reason why the reactants are completely converted into the products even in reversible reactions. The steady removal of the reaction products can be further utilized for an effective suppression of unwanted side reactions. The following reaction mechanism is assumed:

a + b ↔ c and a + c ↔ d .

(5.18)

The substance c is the desired product and the substance d the unwanted side product. By instantaneous reaction performed in a totally mixed reactor, the chemical equilibrium of all four species will be inevitably reached, i.e. a significant amount of the unwanted species d will be present in the effluent; see Figure 5.10. The process with sequential realization of reaction and distillation is shown in Figure 5.16A. A reactor and two distillation columns are required, and the distillate and the bottoms of column C-2 have to be recycled to the reactor. Figure 5.16B presents the process with simultaneous reaction and distillation. The educts a and b are fed into the column. It is important to feed the high boiling re-

5.4 ARRANGEMENT OF CATALYSTS IN COLUMNS

307

Figure 5.16 A) Conventional process for the reaction a + b ↔ c and the side reaction a + c ↔ d. Two distillation columns are necessary to recover the main product c in pure form. B) Realization of the process of Figure 5.16A by reactive distillation. The side reaction is effectively suppressed by feeding the low boiling educt a below the high boiling educt b.

actants b above and the low boiling reactants a below the catalytic section of the column. This leads to the favorable situation that in the reaction zone the concentration of b in the liquid phase is very high. Most of the low boiler a concentrates in the vapor phase. The small percentage of low boiler a, which is dissolved in the liquid, is converted into the product c. This situation obviously favors the main reaction. As the high boiling product c is steadily removed from the reactive section, high concentrations of c exist only in the lower non-catalyzed zone. Hence, the main reaction (Eq. (5.18), left) with the desired product c is enhanced, and the side reaction (Eq. (5.18), right) with the unwanted side product d is effectively suppressed.

5.4

Arrangement of Catalysts in Columns

In principle, there are two different possibilities for catalyzing a liquid-phase chemical reaction: use of a homogeneous (liquid) catalyst or use of a heterogeneous (solid) catalyst. 5.4.1

Homogeneous Catalyst

Homogeneous catalysts are very effective for reactive distillation. The catalyst has to be a very high boiling liquid (for instance, sulfuric acid) as it should remain in the

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5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

liquid phase during the distillation process. The catalyst is fed into that section of the column in which the reaction should take place. Therefore, a homogeneous catalyst can only be used if the whole column or the lower section of the column is a reactive one; see Figs. 5.13B, 5.14B and 5.15B. A homogeneous catalyst cannot be used if the lowest section of a column is a non-reactive one; see Figure 5.12B. Homogeneous catalysts are suited for packed columns as well as for tray columns. Prioritized is the use in tray columns, since tray columns have a much higher liquid hold-up than packed columns. The longer liquid residence time in tray columns enhances the reaction. Of particular importance is the liquid hold-up in the downcomers. By a special design of the downcomer with respect to volume and liquid level, the hold-up can be increased nearly at will. Most of the reaction will take place in the downcomers in such a design. An increase of the weir height of the trays also effects an increase of liquid hold-up and, in turn, residence time, but at the expense of much higher pressure losses [Agreda et al. 1990]. The big disadvantage of homogeneous catalysts is that it leaves the column in the bottom fraction. It has to be removed from that fraction by an additional separation step for recycling. This drawback makes the use of homogeneous catalyst often uneconomical. 5.4.2

Heterogeneous Catalyst

Heterogeneous (solid) catalysts (for instance, acidic ion exchange resins) are arranged directly in the column and remain there for a long operation time. They can be used in packed columns as well as in tray columns, but use in packed columns is favored in most cases. The first idea is making the packing itself catalytic active by coating a conventional packing (random packing or structured packing) with catalysts [Flato and Hoffmann 1992; Gottlieb et al. 1993; Oudshoorn 1998]. However, the amount of catalyst that can be placed on the surface of the packing is rather small. It is too small for most reactions. Typically, the size of catalytic particles is in the range from 1 to 3 mm [Sundmacher and Kienle 2003]. Therefore, the catalytic particles have to be enveloped within wire gauze in form of balls, rings, tablets, doughnuts, etc. [Smith Jr. 1984, 1993; Buchholz et al. 1994]. An early version of a catalytic structured packing is described in Arganbright et al. 1986. Pockets are sewn into a fiberglass cloth that the catalytic particles are embedded in. The resulting belt is rolled to form a cylinder with the diameter of the column. The structure allows a good vapor–liquid contact. However, a general problem of cylindrical packing structures is a poor lateral mixing of vapor and liquid. Nevertheless, several industrial applications of these structures are known [Barchas et al. 1993; Shoemaker and Jones Jr. 1987; Johnson 1993; Crossland et al. 1995; Subawalla et al. 1997]. A modern design often used for reactive distillation is a sandwich type of corrugated sheet packing [Smith Jr. 1984; Gelbein and Buchholz 1991; Shelden and Stringaro 1995; Goetze and Bailer 1999; Goetze et al. 2001; Moritz and

5.4 ARRANGEMENT OF CATALYSTS IN COLUMNS

309

Hasse 1999]. The catalytic particles are layered between two corrugated sheets of metal wire gauze forming a sandwich element. The sandwiches are assembled together like the corrugated metal sheets of a conventional structured packing for mass transfer (e.g. Mellapak; see Chapter 9). The bed of particles in the sandwiches exerts a capillary attraction force to the liquid that enhances the contact of the liquid with the catalyst. The mass transfer between vapor and liquid can be improved by alternating arrangement of corrugated sandwich plates and corrugated metal plates. The performance parameters (two-phase flow and interfacial mass transfer) are similar to those of standard structured packings [Moritz and Hasse 1999; Goetze et al. 2001]. Well-known examples of such structured sandwich packings are marketed as Katamax and Katapak. Heterogeneous catalysts can, in principle, also be used in tray columns. The catalytic structures are positioned in the downcomers or in the froth on the trays [Haunschild 1971; Asselineau et al. 1994; Sanfilippo et al. 1996; Yeoman et al. 1995]. 5.4.2.1

Regeneration of Spent Catalyst

An essential problem of heterogeneous catalysts in distillation columns is the limited lifetime of the catalyst. The activity of each catalyst decreases during operation. Therefore, the catalytic elements have to be removed from the column to be reactivated or replaced by new catalytic particles. The column has to be shut down in most designs. There are several proposals in the patent literature to handle this problem. One idea is to suspend the catalytic particles in the froth on the trays that allows removing and replacing the catalyst without stopping column operation [Jones Jr. 1993]. The solid particulate catalyst is supported on the trays to approximately the depth of the liquid on the tray. Screens above the overflow weirs prevent the catalyst from overflow with the liquid into the downcomer. Each tray has a draw-off for removal of the slurry (mixture of liquid and catalyst particles) from that tray. A fresh slurry with regenerated catalyst is fed later on to that tray. The exchange of the catalyst is carried out tray after tray. This procedure allows, in principle, an exchange of all the spent catalyst without column shut down. However, such an operation is only feasible if the catalyst particles are mechanically very stable. Resume

A distillation column is a good separation device but not a good chemical reactor. Therefore reactive distillation should be applied preferably to simple reactions, i.e. moderate pressures and temperatures. Typical applications of reactive distillations are esterification, transesterifications or hydrolysis of esters, acetalyzations, and hydrolysis of acetates. Some systems are listed in Table 5.1. Some examples of industrial processes with reactive distillation are presented in detail in Section 8.7.

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5 REACTIVE DISTILLATION, CATALYTIC DISTILLATION

Table 5.1 Selected systems for reactive distillation after Doherty and Buzad 1992; Sundmacher et al. 1994; Ung and Doherty 1995.

Methyl acetate from methanol and acetic acid

Agreda et al. 1990

Ethyl acetate from ethanol and acetic acid

Komatsu 1977

Butyl acetate from butanol and acetic acid

Hartig and Regner 1971

MTBE from methanol and i-butene

Flato and Hoffmann 1992; Smith Jr. 1990

ETBE from ethanol and isobutene

Thiel et al. 1997

TAME from methanol and 2-methyl-2-butene

Bravo and Pyhälahti 1992; Thiel et al. 1997

TAME from methanol and 2-methyl-1-butene

Bravo and Pyhälahti 1992

Ethylene glycol from ethylene oxide and water

Gu and Ciric 1992

Isooctane from isobutane and 1-butene

Huss and Kennedy 1989

Ethylene benzene from benzene and ethylene

Smith Jr. et al. 1991

Cumene from benzene and propylene

Shoemaker and Jones Jr. 1987

tert-Butyl alcohol from isobutene and water

Velo et al. 1988

Nylon 6.6 pre-polymer from adipic acid and hexamethylenediamine

Jaswal and Pugi 1975

References Agreda, V.H., Partin, L.R., and Heise, W.H. (1990). High-purity methyl acetate via reactive distillation. Chemical Engineering Progress 86 (2): 40–46. Almeida-Rivera, C.P., Swinkels, P.L.J., and Grievink, J. (2004). Designing reactive distillation processes: present and future. Computers and Chemical Engineering 28 (10): 1997–2020. Arganbright, R.P., Hearn, D., Jones, E.M., and Smith, L.A. (1986). Novel process for methyl tertiary-buthyl ether. Patent DOE/CS/40454-T3. Asselineau, L., Mikitenko, P., Viltard, J.C., and Zuliani, M. (1994). Reactive distillation process and apparatus for carrying it out. US Patent 5,368,691 A. Barchas, R., Samarth, R., and Gildert, G. (1993). Zeolites as catalysts in industrial

processes. Fuel Reformation 3 (5): 44–49. Bravo, J.L. and Pyhälahti, A. (1992). Investigations in a catalytic distillation plant: VLE, kinetics and mass transfer issues. AIChE Annual Meeting, Miami. Buchholz, M., Pinaire, R., and Ulowetz, M.A. (1994). Structure and method for catalytically reacting fluid streams in mass transfer apparatus. US Patent 5,275,790. Crossland, C.S., Gildert, G.R., and Hearn, D. (1995). Catalytic distillation structure. US Patent 5,431,890. Doherty, M.F. and Buzad, G. (1992). Reactive distillation by design. Transactions of the Institution of Chemical Engineers 70 (Part A): 448–458. Doherty, M.F. and Malone, M.F. (2001). Conceptual Design of Distillation Systems. New York: McGraw-Hill.

REFERENCES

Espinosa, J., Aguirre, P., and Perez, G. (1995). The product composition regions of single-feed reactive distillation columns. Industrial and Engineering Chemistry Research 34: 853–861. Espinosa, J., Aguirre, P., Frey, T., and Stichlmair, J. (1999). Analysis of finishing reactive distillation columns. Industrial and Engineering Chemistry Research 38 (1): 187–196. Flato, J. and Hoffmann, U. (1992). Development and start-up of a fixed bed reaction column for manufacturing the antiknock enhancer MTBE. Chemical Engineering and Technology 15: 193. Frey, T. and Stichlmair, J. (1999a). Reactive azeotropes in kinetically controlled distillation. Transactions of the Institution of Chemical Engineers Part A 77 (7): 613–618. Frey, T. and Stichlmair, J. (1999b). Thermodynamic fundamentals of reactive distillation. Chemical Engineering and Technology 22 (1): 11–18. Gelbein, A.P. and Buchholz, M. (1991). Process and structure for effecting catalytic reactions in distillation structures. EP Patent 0,428,265. Goetze, L. and Bailer, O. (1999). Reactive Distillation with Katapak, Tech. Rep. 4, Sulzer Technical Review. Goetze, L., Bailer, O., Moritz, P., and von Scala, C. (2001). Reactive distillation with Katapak. Catalysis Today 69: 201–208. Gottlieb, K., Graf, W., Schaedlich, K., Hoffmann, U., Rehfinger, A., and Flato, J. (1993). Molded bodies comprised of macroporous ion exchange resins, and use of said bodies. US Patent 5,244,929. Gu, D. and Ciric, A.R. (1992). Optimization and dynamic operation of an ethylene-glycol reactive distillation column. AIChE Annual Meeting, Miami. Hartig, H. and Regner, H. (1971). Verfahrenstechnische Auslegung einer Veresterungskolonne. Chemie Ingenieur Technik 43 (18): 1001–1007. Haunschild, W.M. (1971). Separation of linear olefins from tertiary olefins. US Patent 3,629,478. Huss, A. and Kennedy, C.R. (1989). Hydrocarbon processes comprised of catalytic distillation using Lewis acid promoted inorganic oxide catalyst systems.

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US Patent 4,935,557. Jaswal, I. and Pugi, K. (1975). Preparation of polyamides by continuous polymerization. US Patent 3,900,450. Johnson, K.H. (1993). Catalytic distillation structure. US Patent 5,189,001. Jones Jr., E.M. (1993). Method for removing and replacing catalysts in a distillation column reactor. US Patent 5,198,196 A. Komatsu, H. (1977). Application of the relaxation method for solving reacting distillation problems. Journal of Chemical Engineering of Japan 10: 200–205. Moritz, D. and Hasse, A. (1999). Fluid dynamic in reactive distillation packing Katapak-S. Chemical Engineering Science 54 (10): 1367–1374. Okasinski, M.J. and Doherty, M.F. (1997). Thermodynamic behavior of reactive azeotropes. AIChE Journal 43 (9): 2227–2238. Oudshoorn, O.L. (1998). Zeolitic coatings applied in structured catalyst packings. PhD thesis. Delft University of Technology. Sakuth, M., Reusch, D., and Janowsky, R. (2008). Reactive distillation. In: Ullmann’s Encyclopedia of Industrial Engineering, pp. 263–276. Weinheim: Wiley-VCH. Sanfilippo, D., Lupieri, M., and Ancillotti, F. (1996). Process for prepairing tertiary alkyl ether and apparatus for reactive distillation. US Patent 5,493,059. Shelden, R. and Stringaro, J.P. (1995). Vorrichtung zur Durchführung katalysierter Reaktionen. EP Patent 0,396,650. Shoemaker, J.D. and Jones Jr., E.M. (1987). Cumene by catalytic distillation. Hydrocarbon Processing 66 (6): 55–58. Smith Jr., L.A. (1984). Catalytic distillation structure. US Patent 4,443,559. Smith Jr., L.A. (1990). Method for the preparation of methyl tertiary butyl ether. US Patent 4,978,807. Smith Jr., L.A. (1993). Method for operating a catalytic distillation process. US Patent 5,221,441. Smith Jr., L.A., Jones, E.M., and Hearn, D. (1991). Catalytic distillation – a new chapter in unit operations. AIChE Spring Meeting, Houston, TX, USA. Stichlmair, J. and Frey, T. (1999). Reactive distillation processes. Chemical Engineering and Technology 22 (2): 95–103.

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Subawalla, H., Gonzales, J., Seibert, A.F., and Fair, J.R. (1997). Capacity and efficiency of reactive distillation bale packing. Industrial and Engineering Chemistry Research 36: 3821–3832. Sundmacher, K. and Kienle, A. (2003). Reactive Distillation – Status and Further Directions. Weinheim: Wiley-VCH. Sundmacher, K., Rihko, L.K., and Hoffmann, U. (1994). Classification of reactive distillation processes by dimensionless numbers. Chemical Engineering Communications 127 (4): 151–167. Thiel, C., Sundmacher, K., and Hoffmann, U. (1997). Residue curve maps for heterogeneously catalyzed distillation of fuel ethers MTBE and TAME. Chemical Engineering Science 52 (6): 993–1005. Tuchlenski, A., Beckmann, A., Reusch, D., Düssel, R., Weidlich, U., and Janowsky, R.

(2001). Reactive distillation – industrial applications, process design & scale-up. Chemical Engineering Science 56 (2): 387–394. Ung, S. and Doherty, M.F. (1995). Synthesis of reactive distillation systems with multiple equilibrium chemical reactions. Industrial and Engineering Chemistry Research 34 (8): 2555–2565. Velo, E., Puigjaner, L., and Rescasens, F. (1988). Inhibition by-product in the liquid phase hydration of isobutene to tert-butyl alcohol: kinetics and equilibrium studies. Industrial and Engineering Chemistry Research 27: 2224–2231. Yeoman, N., Pinaire, R., Ulowetz, M.A., Nace, T.P., and Furse, D.A. (1995). Internals for distillation columns including those for use in catalytic reactions. US Patent 5,454,913.

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6 Multistage Batch Distillation Multistage batch distillation of liquid mixtures is of great importance; it is used extensively in laboratories and in small production units. The main advantage of batch distillation over continuous distillation is that a single apparatus can process many different liquid mixtures. A batch distillation column is in essence a multipurpose unit. Different product requirements can be taken into account by simply changing the reflux ratio with the same starting mixture. Even multicomponent mixtures can be separated by batch distillation in a single column if the components are collected separately in different receivers. A disadvantage of batch distillation is the long exposure of time the mixture to high temperatures. This increases the risk of thermal degradation or decomposition of the components. Furthermore, energy requirement is generally higher in batchwise than in continuous distillation. A setup for multistage batch distillation is shown in Figure 6.1A (e.g. Diwekar 1995). The still is filled with the liquid mixture F and heated. Since the still is at the bottom of the column, its molar amount of liquid is designated B . Vapor flows upward in the column and condenses at the top. Usually, the entire condensate is initially returned to the column as reflux (i.e. RL → ∞). This countercurrent contacting of vapor and liquid considerably improves the separation (cf. Section 4.1). After some time, one part of the overhead condensate is continuously withdrawn as distillate (the molar amount D), and the other part is recycled into the column as reflux. The liquid in the still is increasingly depleted of the more volatile components. As the amount of liquid in the still decreases, the concentration of the high boiler increases. In principle, a setup for batch distillation according to Figure 6.1B is also feasible (e.g. Sørensen and Skogestad 1996). Here, the vessel is located at the top of the column that is operated as a stripping column. The molar amount of liquid in the vessel is referred to here as D. During operation the high boiling components are primarily separated from the system as bottoms B . The liquid in the vessel is depleted of high boiling components and enriches with low boiling ones. This mode of batch distillation is seldom applied to industrial processes. However, as will be shown later on, it is necessary for separating mixtures with minimum azeotropes. A third feasible setup for batch distillation is shown in Figure 6.1C. The column consists of a rectifying and of a stripping section, and the vessel is located at the

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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middle of the column. The molar amount of liquid in the vessel is designated M . This and similar configurations (like multiple vessels between column sections) have aroused much interest in the literature in the last two decades (e.g. Davidyan et al. 1994; Diwekar 1995; Barolo et al. 1996; Skogestad et al. 1997; Warter and Stichlmair 1999, 2000; Warter et al. 2002; Skouras et al. 2005; Gruetzmann et al. 2006) but are still scarcely used in practice. In theoretical studies such setups prove to be more efficient with respect to energy utilization than a conventional setup (e.g. Meski and Morari 1995; Kim and Diwekar 2000; Low and Sørensen 2005).

Figure 6.1 Schematics of batch distillation devices. A) Bottom vessel with rectifying column. B) Top vessel with stripping column. C) Middle vessel with rectifying and stripping column.

6.1

Batch Distillation of Binary Mixtures

The calculation of multistage batch distillation of binary mixtures is based on the Rayleigh equation (3.23), derived in Section 3.2 for single-stage batch distillation. Application of this equation to multistage distillation requires the use of the mole fraction xD of the more volatile component in the distillate instead of the equilibrium vapor concentration y ∗ in Eq. (3.23) because the distillate is not in equilibrium with the composition xB of the liquid in the still:

dB dxB = . B xD − xB Here, B denotes the actual amount of liquid mixture in the still.

(6.1)

6.1 BATCH DISTILLATION OF BINARY MIXTURES

315

The concentration xD of the distillate depends on the concentration xB , the vapor– liquid equilibrium (i.e. the relative volatility α), the separation efficiency of the column (i.e. the number of equilibrium stages n or number of transfer units NTUOG ), and the reflux ratio RL :

xD = f (xB , α, n, RL ) .

(6.2)

For binary mixtures, the function f is usually graphically determined on the (y, x )diagram. Calculations depend on the mode of operation. 6.1.1

Operation with Constant Reflux

At operation with constant reflux, the value of the reflux ratio RL is kept constant during the process, while the distillate composition xD changes with time. For calculation, the separation efficiency of the column (e.g. number of equilibrium stages n), the reflux ratio RL , and the starting concentration xF are specified. Then, the amount of product D, the mean composition xDm of the product, and the energy requirement for distillation are determined. Integration of Eq. (6.1) gives Z xBe Be dxB ln = , (6.3) F x D − xB xF where F is the initial amount of liquid and Be the amount of liquid remaining in the still at the end. Integration of the right-hand side of this equation requires determination of the relationship xD = f (xB , α, n, RL ). In a binary system, this is usually done numerically or graphically on the (y, x )-diagram, as shown in Figure 6.2A. A distillate concentration xD1 is set, and an operating line is drawn, followed by the usual determination of the concentration xB1 in the still, e.g. by stepping off the stages. Subsequently, another distillate concentration xD2 is set, and the corresponding concentration xB2 is determined in an analogous manner. The operating lines are all parallel because the reflux ratio RL is supposed to be constant. In this way, the correlation between distillate concentration xD and bottoms concentration xB in the still is determined point by point. The integral in Eq. (6.3) is solved by using an auxiliary diagram in which the term 1/(xD − xB ) is plotted on the ordinate and the actual bottoms concentration xB on the abscissa; see Figure 6.2B. The area A under the curve, determined point by point from the starting concentration xF to the concentration xBe at the end, represents the solution to the integral. The following equation is obtained:

Be = F · eA .

(6.4)

With D = F − B , it follows that

De = F · (1 − Be /F ) = F · (1 − eA ) .

(6.5)

The mean distillate concentration xDm results from the material balance:

xDm =

xF − xBe · Be /F . 1 − Be /F

(6.6)

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6 MULTISTAGE BATCH DISTILLATION

Figure 6.2 Batch distillation of a binary mixture at constant reflux policy. A) Graphical determination of pairs of distillate and bottoms concentrations. B) Auxiliary diagram for graphical integration of Eq. (6.3).

6.1 BATCH DISTILLATION OF BINARY MIXTURES

317

The amount of vapor G generated in the reboiler is higher than the amount of overhead product D since some of the distillate is recycled into the column as reflux. With G = D · (RL + 1), the heat Q required for the separation is

Q = (1 − Be /F ) · (RL + 1) . F ·r Here, r is the latent heat of vaporization of the mixture. 6.1.1.1

(6.7)

Infinite Number of Stages

For ideal mixtures and infinite number of stages in the column, Eq. (6.3) can be analytically solved for a preset value of the reflux ratio RL . Two steps of the process have to be considered. In the first step, the concentration of the distillate is, under the assumption of an infinite number of stages, always xD = 1 if the reflux ratio is high. This holds as long as the concentration in the still is higher than x∗B , which is the liquid concentration at the intersection of the equilibrium curve and the operating line, which starts at xD = 1. The value of x∗B can be calculated with the relation (4.39) derived in Section 4.2.4 for the determination of the minimum reflux ratio. Here, the same equation applies, but the reflux ratio is preset and the feed concentration xF (i.e. bottoms concentration xB ) is the unknown quantity. After rearrangement we get

x∗B =

1 . (α − 1) · RL

(6.8)

For xF > x∗B , integrating Eq. (6.3) gives

B∗ xDm1 − xF = F xD − 1/ ((α − 1) · RL )

with

xDm1 = 1 .

(6.9)

Here, B ∗ denotes the amount of liquid in the still at the end of the first step of the process. If the concentration in the still falls below x∗B , pure low boiler is no longer separated from the mixture, i.e. xD < 1, and the second step of the process begins. During this step, both the bottoms and the overhead concentrations, xB and xD , decrease continuously. The relation between xD and xB is now formulated by Eq. (4.38). After transformation we get

xD =

α · xB · (RL + 1) − RL · xB . 1 + (α − 1) · xB

From Eqs. (6.3) and (6.10), it follows that   Z xBe Be 1 dxB ln = · . B∗ RL + 1 x∗B α · xB / (1 + (α − 1) · xB ) − xB

(6.10)

(6.11)

After integration, we obtain

Be = B∗



xBe xB ∗

!1/(RL +1) 1/(α−1)   1 − xB ∗ α/(α−1) · . 1 − xBe

(6.12)

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6 MULTISTAGE BATCH DISTILLATION

For a reflux ratio RL = 0, i.e. no reflux, this equation is simplified to Eq. (3.27) derived for single-stage distillation in Section 3.2. The mean distillation concentration xDm2 of the second step is calculated from

xDm2 =

x∗B − xBe · Be /B ∗ . 1 − Be /B ∗

(6.13)

The mean distillate concentration of the total process is

1 − B ∗/F + (B ∗ /F − Be /F ) · xDm2 1 − Be /F with Be /F = Be /B ∗ · B ∗/F .

xDm =

The energy requirement follows from   Qmin Be = (RL + 1) · 1 − . F ·r F

(6.14)

(6.15)

Figure 6.3 displays a plot of these equations for a mixture with a feed concentration xF = 0.5 and a relative volatility α = 2. Presented is the minimum energy requirement over the relative amount of distillate for several mean distillation concentrations xDm . The energy requirement increases sharply when nearly all low boiler has been separated from the mixture in the still. At the point xB = 0, i.e. D/F = xF /xDm , the energy requirement becomes infinite. Additionally, lines of constant reflux ratio RL are presented in the diagram. If, for instance, 60 % of the feed F has to be separated as overheads D with a mean concentration xDm = 0.8, the reflux ratio has to be as high as RL = 3.1 and the energy requirement Qmin /(F · r) is 2.5. Hence, the feed F has to be evaporated 2.5 times during the whole process. If only 40 % of the feed has to be separated from the mixture, the appropriate reflux ratio is RL = 1.3 (same mean distillate concentration). Here, the energy requirement is much lower with Qmin /(F · r) = 0.95. Operation at constant reflux ratio RL is relatively simple. However, the distillate concentration xD decreases with time, as illustrated in Figure 6.2. Thus, the desired product composition is achieved by having a higher starting purity than specified. However, if a high purity product is required, this is only possible to a severely limited extent. Hence, in order to prevent a steep drop in product purity, the reflux ratio RL is often increased several times during operation. 6.1.2

Operation with Constant Distillate Composition

A decrease of distillate concentration xD over time can be avoided by continuously increasing the reflux ratio RL throughout operation. The relation between still concentration xB and the corresponding reflux ratio RL is, in turn, graphically determined on the (y, x )-diagram. An initial reflux ratio RL and thus the position of the operating line are specified, and the corresponding still concentration xB is determined graphically. This is illustrated in Figure 6.4 for two different cases. The

6.1 BATCH DISTILLATION OF BINARY MIXTURES

319

Figure 6.3 Minimum energy requirement (at infinite number of stages) of batch distillation at constant reflux policy. When the still concentration of the low boiler approaches zero, the energy requirement increases infinitely.

correlation obtained depends on the separation efficiency of the column, i.e. number of equilibrium stages n and value of reflux ratio RL . Operation with constant distillate concentration xD allows direct integration of the Rayleigh equation in Eq. (6.1) to obtain

xD − xF . xD − xBe The amount of distillate is xF − xBe De = F · . xD − xBe Be = F ·

(6.16)

(6.17)

The energy requirement depends on the reflux ratio RL , which is variable. Hence, only a differential equation can be given:

dQ = dD · (RL + 1) · r .

(6.18)

By using the relation dD = −dB , the following relationship is obtained after differentiating Eq. (6.16) with respect to xBe and inserting it into Eq. (6.18):

dQ RL + 1 = (xD − xF ) · 2 dxB . F ·r (xD − xB )

(6.19)

Integration gives

Q = (xD − xF ) · F ·r

Z

xBe

xF

RL + 1 2

(xD − xB )

dxB .

(6.20)

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6 MULTISTAGE BATCH DISTILLATION

Figure 6.4 Batch distillation of a binary mixture at constant distillate concentration reflux policy. A) Graphical determination of pairs of bottoms concentrations and reflux ratios. B) Auxiliary diagram for graphical integration of Eq. (6.20).

6.1 BATCH DISTILLATION OF BINARY MIXTURES

321

The term on the right-hand side of Eq. (6.20) is graphically integrated by using an auxiliary diagram; see Figure 6.4B. The bottoms concentration xB in the still is plotted on the abscissa and the term (RL + 1)/(xD − xB )2 on the ordinate. The area A under the curve between xF and xBe gives the value of the integral. Hence, the energy requirement is

Q = (xD − xF ) · A . F ·r 6.1.2.1

(6.21)

Infinite Number of Stages

For infinite number of stages in the column, the minimum energy requirement can be analytically calculated for ideal mixtures with constant relative volatility α. The relation between still concentration xB and reflux ratio RL at operation with constant distillate concentration xD is given by   1 xD 1 − xD RL = · −α· . (6.22) α−1 xB 1 − xB Inserting into Eq. (6.20) yields Z xBe Qmin 1 = (xD − xF ) · 2· F ·r xF (xD − xB )     1 xD 1 − xD · · −α· + 1 dxB . α−1 xB 1 − xB (6.23) After solving the integral of the right-hand side, we obtain  Qmin xD − xF 1 − xF = · α · xD · ln F ·r (α − 1) · xD · (1 − xD ) 1 − xBe  xD − xF xF − ((α − 1) · xD + 1) · ln + (1 − xD ) · ln . xD − xBe xBe (6.24) A plot of this equation for a mixture with α = 2 and xF = 0.5 is presented in Figure 6.5. As with operation at constant reflux ratio, the energy requirement increases drastically when almost all low boiler has been removed from the mixture. Hence, separation with high yield is difficult in both modes of batchwise distillation. A comparison of Figure 6.3 with Figure 6.5 reveals that the energy requirement for distillation with constant distillate concentration is lower than for operation at constant reflux ratio. This difference is especially high for high purity distillations. 6.1.2.2

Mixtures with Partial Immiscibility in the Liquid Phase

In practice, operation with constant distillate concentration xD is more laborious because the reflux ratio RL changes (Figure 6.4) and has to be readjusted continuously. On the other hand, this situation occurs automatically in the fractionation of mixtures such as n-butanol/water, which exhibit a region of immiscibility and form

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Figure 6.5 Minimum energy requirement of batch distillation at constant distillate concentration reflux policy. The energy requirement becomes infinite when the concentration of low boiler a in the still approaches zero.

a heteroazeotrope (Figure 6.6). In this special case, one part of the separation effect is accomplished by the decanter. The reflux at the top of the column is n-butanol saturated with water. The concentration of the reflux is at the left edge of the shaded region of immiscibility in the McCabe–Thiele diagram of Figure 6.6. The state of the vapor is at the intersection of the operating line with the left edge of the shaded region of immiscibility. Hence, the vapor is enriched with water. After condensation the state of the liquid mixture is in the region of immiscibility. The two liquid phases are separated by a decanter into a butanol-rich fraction, which is recycled as reflux, and a water-rich fraction, which is withdrawn as distillate. Thus, the product has the same concentration throughout the entire process. The rates of product and distillate flow depend on the concentration of the vapor at the top of the column. At the beginning, the slope of the operating line is small, and, as shown in Figure 6.6, the concentration of the overhead vapor is high. This results in a high flow rate of the distillate D and, since D + L = const, in a low flow rate of reflux L. At the end of the process, the reflux is much higher, and the concentration of the overhead vapor is lower. After condensation, the overall liquid concentration is close to the left boundary of the region of immiscibility. According to the lever rule, the rate of the product D is much lower, and, consequently, the rate of the reflux L is much higher. Thus, the operation with constant distillate concentration and variable reflux is automatically established without any external action.

6.1 BATCH DISTILLATION OF BINARY MIXTURES

323

Figure 6.6 Batch distillation of the n-butanol/water mixture, which exhibits a region of immiscibility with an heteroazeotrope. Part of the separation is performed by the decanter. A) Column setup. B) McCabe–Thiele diagram. The shaded area denotes the region of immiscibility.

6.1.3

Operation with Minimum Energy Input

The two operating modes described in Sections 6.1.1 and 6.1.2 require different quantities of heat Q to obtain the same amount and composition of the product. Generally, less energy is required for the operation at constant distillate concentration. In the energetically optimal operation mode, however, the distillate concentration decreases, and the reflux ratio increases steadily. Pontryagin’s maximum principle [Pontryagin et al. 1962] can be used to calculate the relation between reflux ratio RL , distillate concentration xD , and still concentration xB [Coward 1967; Robinson 1969]. The basis of Pontryagin’s maximum principle is a system of differential equations that describes the time dependence of the state variables of the process. His principle is essentially a generalization of the Lagrange equations, developed for easy description of the dynamic behavior of kinematic systems (e.g. D’Souza and Garg 1984). Lagrange’s equations allow for the derivation of the differential equations of motion, dz/dt, by performing the partial derivatives of the total energy E with respect to the momentum m:

dz ∂E = . dt ∂m

(6.25)

The time dependence of the momentum m is in turn formulated by the partial derivatives of the energy E with respect to the locus z :

dm ∂E =− . dt ∂z

(6.26)

Equations (6.25) and (6.26) are essential characteristics of potential functions. The

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total energy E of a kinematic systems is only one example of such a potential function. In order to determine the potential function for non-kinematic systems, Pontryagin begins with the differential equations for the time dependence of the process variables and defines a new potential function, called the Hamiltonian function H . Function H represents the total driving force for any unsteady process. Alike total energy E in kinematic processes, the potential function H has to be kept constant during the process. The higher the value of H , the faster the process proceeds. Selection of the maximum value Hmax at the beginning gives the time optimum variation of the variables. Application of Pontryagin’s maximum principle to batch distillation starts with the physical description of the unsteady state behavior. The variation of the amount of liquid in the still dB with time dt follows from a material balance:

dB −G˙ = . dt RL + 1

(6.27)

Here, the vapor stream generated by the heating system is considered to be constant, i.e. G˙ 6= f (t). The component balance is described by the modified Rayleigh equation (6.1):

dB dxB = . B xD − xB

(6.28)

Substitution of dB with Eq. (6.27) and rearrangement gives

dxB −G˙ = · (xD − xB ) . dt (RL + 1) · B

(6.29)

From Eqs. (6.27) and (6.29), the Hamiltonian function follows:

H=

−G˙ −G˙ · π1 + · (xD − xB ) · π2 . RL + 1 (RL + 1) · B

(6.30)

The function H and the parameters π are analogous to the energy E and the momentum m, respectively, of a kinematic system. Hence, the derivatives ∂ H/∂ π1 and ∂ H/∂ π2 give the time dependence of the variables B and xB of batch distillation (see Eqs. (6.28) and (6.29)). A correlation for the time dependence of the parameters π1 and π2 is, according to Eq. (6.26), derived by the partial derivatives −∂ H/∂ B and −∂ H/∂ xB . Hence

dπ1 −G˙ = · (xD − xB ) · π2 dt (RL + 1) · B 2

(6.31)

and   dπ2 +G˙ ∂ xD = · − 1 · π2 . dt (RL + 1) · B ∂ xB

(6.32)

6.1 BATCH DISTILLATION OF BINARY MIXTURES

325

Generally, the integration of the system of differential Eqs. (6.27), (6.29), (6.31), and (6.32) has to be performed numerically, e.g. by Simpson’s rule or by a Runge– Kutta procedure. At each time step dt, the value of the Hamiltonian function H has to be constant. At varying variables B , xB , π1 , and π2 , this is only possible by additional time variation of the reflux ratio RL . If, for instance, in the beginning the reflux ratio has been chosen to maximize the Hamiltonian function, the time optimum variation of the process variables is found. Hence, the resulting variation of reflux ratio represents the time optimum reflux policy. At operation with constant energy input (i.e. Q˙ = const), the time optimum represents the energy minimum, too. For numerical integration the staring values of the parameters π1 and π2 have to be estimated. The starting value of π1 may always be set to 1 without any restrictions for the solution. Then, the value of π2 has to be negative. Since the term (xD − xB ) is also a function of the reflux ratio RL , the Hamiltonian function H exhibits a maximum when plotted versus the reflux ratio RL . Typically, the value of π1 increases monotonically with time, while the value of π2 possesses a maximum. The numerical integration starts with a value of RL that maximizes the Hamiltonian function. For each time step dt, the reflux ratio that maximizes the function H has to be calculated. The value of H is constant during the process. If the end values xBe and Be do not meet the material balance, the first estimate of π2 was wrong, and the procedure has to be repeated. After several iterations, the solution is found. A good check of the time and energy optimum is the constancy of the value of the Hamiltonian function during the last iteration. Figure 6.7 presents the course of the optimum reflux ratio RL,opt versus the relative amount of distillate D/F [Robinson 1969]. The reflux RL,opt increases steadily during operation. However, the increase is not as drastic as in the operation mode with constant product purity. In the beginning, the reflux is much lower than in the operation mode with constant reflux. The energy requirement is lowest in operation with RL,opt . Energy savings of up to approximately 15 % can be achieved [Coward 1967], which justifies this more complex method in special cases only. If a product with very high purity is demanded, only operation with constant distillate composition is feasible. Numerical integration of Eqs. (6.27), (6.29), (6.31), and (6.32) is tedious because it involves the numerical evaluation of the terms (xD − xB ) and ∂ xD /∂ xB , e.g. by tray-to-tray calculations. Furthermore, the search of a reflux ratio RL that maximizes the Hamiltonian function H is time consuming since the maximum flattens with increasing time or, even worse, may not exist at all (see Section 6.1.3.1). 6.1.3.1 Influence of Number of Equilibrium Stages on the Hamiltonian Function

The basis for the determination of the optimum reflux policy of batch distillation by Pontryagin’s maximum principle is the existence of a maximum in the Hamiltonian function H when plotted versus the reflux ratio RL . However, such a maximum only exists if the number of stages is small (approximately 2 to 8, depending on vapor–liquid equilibrium). This is outlined below, using the example of an ideal mixture with constant relative volatility. The limiting cases n → ∞ and n = 1 are

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Figure 6.7 Variation of the reflux ratio RL versus relative amount of distillate D/F for all three reflux policies. Typically, the optimum reflux policy lies between RL = const and xD = const.

considered. If the number of stages is high (approximately more than 8), the distillate concentration xD follows from Eq. (6.10):

xD =

α · xB · (RL + 1) − RL · xB . 1 + (α − 1) · xB

(6.33)

Here, the distillate concentration xD is a linear function of the reflux ratio RL . Combining Eqs. (6.30) and (6.33) yields the Hamiltonian function H for n → ∞:

H∞ =

−G˙ −G˙ · π1 + · RL + 1 B



α · xB − xB 1 + (α − 1) · xB



· π2 .

(6.34)

The structure of this function is

H∞ =

C1 + C2 . RL + 1

(6.35)

The value of function H∞ increases or decreases monotonically with increasing reflux ratio RL . No extremum exists. Hence, the optimum reflux policy is identical to operation with constant distillate concentration. If the number of equilibrium stages is 1, the distillate concentration xD is always in equilibrium state with the still concentration xB . Hence

xD =

α · xB . 1 + (α − 1) · xB

(6.36)

6.2 BATCH DISTILLATION OF TERNARY MIXTURES

327

The distillate concentration xD does not depend on reflux at all. Inserting Eq. (6.36) in Eq. (6.30) gives the Hamiltonian function for n = 1: !  ˙  1 − G α · x B H1 = · −G˙ · π1 + · − xB · π2 . (6.37) RL + 1 B 1 + (α − 1) · xB The structure of the function is

H1 =

C1 . RL + 1

(6.38)

The Hamiltonian function increases or decreases (depending on the sign of constant C1 ) monotonically with the reflux ratio RL , i.e. no maximum exists. Here, the optimum reflux policy is identical with the operation mode RL = const and xD = variable, which is the only feasible operation mode in this case. 6.1.4

Comparison of Energy Requirement for Different Modes of Distillation

A comparison of minimum energy requirement for different modes of distillation is presented in Figure 6.8. An ideal mixture with constant relative volatility α = 2, feed concentration xF = 0.5, and infinite number of stages is considered. Continuous distillation is also included in the comparison. The relevant equation is       Qmin 1 xD 1 − xD D = · −α· +1 · . (6.39) F · r cont α−1 xF 1 − xF F The diagram in Figure 6.8 shows that all modes of batch distillation require more energy than their counterparts in continuous distillation. The difference is especially high in case the concentration of the low boiler in the still approaches zero. At this point, given by D/F = xF /xDm , the term Qmin /(F · r) increases infinitely for batchwise distillation. In continuous distillation, however, the minimum energy requirement is a linear function of D/F , even at the limiting value D/F = xF /xDm , where the concentration of the low boiler in the still is zero. Hence, batchwise distillation is especially disadvantageous when all low boiler has to be separated, i.e. at high yield separations.

6.2

Batch Distillation of Ternary Mixtures

The application of the fundamental Eq. (6.1) to ternary mixtures yields

dB dxBa = B xDa − xBa

and

dB dxBb = . B xDb − xBb

(6.40)

These equations are similar to Eq. (3.36), but the distillate concentrations xDi replace the equilibrium concentrations yi∗ . In general, determination of the distillate

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Figure 6.8 Comparison of minimum energy requirement (at infinite number of stages) for continuous and batchwise distillation with either RL = const or xD = const. Especially in high yield separations, continuous distillation is superior to all modes of batch distillation.

concentrations xDi requires a rigorous simulation of the column, as described in Section 4.4.1. However, the results of such a tedious simulation are only reliable if the liquid hold-up in the column is much lower than the liquid hold-up in the vessel. If this condition is not met, the dynamic behavior of the column has to be simulated (see Section 6.4). Batch distillation of ternary (and multicomponent) mixtures is preferentially performed with constant reflux ratio. However, operation with periodically constant distillate concentration is also possible [Block et al. 1978]. The reflux ratio has to be carefully controlled over the entire operating time. In most cases, this mode of operation does not offer a big enough advantage to justify the cost of the complex control equipment required. 6.2.1

Zeotropic Mixtures

The results of a numerical evaluation of the equations in Eq. (6.40) for the methanol/ ethanol/isopropanol system is shown in Figure 6.9. The distillate concentrations xDi and bottoms concentrations xBi are plotted versus the relative amount of distillate D/F in Figure 6.9A,B, respectively. In case of a high number of equilibrium stages n and a high reflux ratio RL , pure low boiler a is recovered as overhead product in the first step of the process. Hence, the concentration of low boiler in the vessel xBa decreases. Simultaneously, the concentrations of the higher boiling components b and c increase. After the complete removal of the low boiler a, the

6.2 BATCH DISTILLATION OF TERNARY MIXTURES

329

intermediate boiler b is the more volatile component in the vessel and is therefore recovered as distillate in the second step of the process. At this point (D/F = xFa ), the distillate concentration switches sharply from xDa = 1 to 0 and xDb = 0 to 1. Now, the concentration of the intermediate boiler b decreases in the vessel. After complete removal of b (at D/F = xFa + xFb ), the concentration xDb decreases sharply to zero, and pure high boiler c is in the overhead product.

Figure 6.9 Concentration profiles for batch distillation of the zeotropic ternary methanol/ ethanol/isopropanol mixture, calculated with the reflux ratio RL = 20 and the number of equilibrium stages n = 30. A) Distillate concentrations. B) Bottoms concentrations.

All three components of the starting mixture are recovered in pure form by collecting the overhead fractions in different receivers. The width of the transition zone between different pure overhead products depends primarily on the reflux ratio RL and the number n of equilibrium stages. Figure 6.9 is well suited to display the result of a batch distillation process. However, a plot of distillate and bottoms concentrations in a triangular diagram provides a deeper understanding.

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Eliminating dB/B from the equations in Eq. (6.40) gives

dxBa xDa − xBa = . dxBb xDb − xBb

(6.41)

This linear correlation is represented by a straight line in the triangular diagram of Figure 6.10. According to Eq. (6.41), the distillate concentration xDi and the bottoms concentration xBi have to lie on a straight line that forms a tangent to the residue curve of the liquid in the still xBi . This behavior is analogous to the liquid residue curve of Figure 3.11. If the column provides a sharp separation (i.e. high reflux ratio and high number of equilibrium stages), the distillate concentration xDi takes an extreme position on the tangent, e.g. on a binary side of the triangular diagram.

Figure 6.10 Graphical interpretation of Eq. (6.41). The distillate concentration xDi has to lie on a straight line that forms a tangent to the curve of concentration change xBi of the liquid in the still (the residue curve).

Vice versa, if the pure component a, for instance, is removed from the system by batch distillation, the concentrations xBi of liquid in the still have to change along a straight line through the a-corner. The direction of concentration change points away from the a-corner. This is indicated in Figure 6.11 by an arrow. After the concentration xBi of the liquid in the still has reached the base of the diagram (i.e. xBa = 0), the intermediate boiler b is separated from the system, and the concentration changes along the straight line through the b-corner, moving toward the c-corner. Figures 6.9 and 6.11 represent the same batch distillation process of a zeotropic mixture. The intermediate fraction between two succeeding pure products is of particular interest for batch distillation processes. The influence of the reflux ratio on the course of the distillate concentration is presented in Figure 6.12. The number of equilibrium

6.2 BATCH DISTILLATION OF TERNARY MIXTURES

331

Figure 6.11 Presentation of the concentration profiles of Figure 6.9 in a triangular diagram. As long as a pure component is separated from the mixture, the change of the liquid concentration xBi has to follow a straight line through this pure component.

Figure 6.12 Influence of the reflux ratio RL on the distillate concentrations at n = 10 equilibrium stages. · · · · · RL = 5 ; − − − RL = 10 ; ——– RL = 20.

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stages is n = 10. It can be clearly seen that the distillate purity improves with an increasing reflux ratio. The higher the reflux ratio, the sharper the separation is. The intermediate fractions resulting from the transition from one to another pure product are smaller at high values of the reflux ratio. Figure 6.13 shows the influence of the number of equilibrium stages n on the distillate quality at a reflux ratio RL = 10. The separation is significantly improved by increasing the number of equilibrium stages from 5 to 10. A further increase from 10 to 20 has, however, a minor impact. An increase in the number of stages is always connected with an increase in the liquid hold-up in the column, which has been neglected here. However, a high liquid hold-up has, in most cases, a negative influence on the separation efficiency of a batch distillation column (see Section 6.4). Increasing the reflux ratio is thus the most effective means for improving the product quality.

Figure 6.13 Influence of the number of equilibrium stages n on the distillate quality at RL = 10. · · · · · n = 5 ; − − − n = 10 ; ——– n = 20.

6.2.2

Azeotropic Mixtures

A more complex situation arises in azeotropic mixtures due to the existence of distillation boundaries (see Section 4.3.2). Such a system is shown in Figure 6.14. Here, a distillation boundary curve runs from the high boiler benzene to the binary maximum azeotrope and divides the system into two distillation fields. If the starting mixture F is located in the upper field, pure acetone is recovered as overhead product during the first step of the batch distillation. The state of the bottoms B moves along a straight line away from the acetone corner. At the intersection of this line

6.2 BATCH DISTILLATION OF TERNARY MIXTURES

333

with the distillation boundary line, the concentration of acetone in the distillate decreases sharply to a value located at the tangent to the distillation boundary at the point of intersection (dashed line). During the second step of the process, the state of the bottoms changes along the distillation boundary, and due to its curvature, the distillate concentration xDa continues to decrease slowly (e.g. Ewell and Welch 1945; Reinders and De Minjer 1940). Figure 6.14B displays a plot of the distillate composition xDi versus the relative amount of distillate D/F . These concentration profiles can only be understood with the help of Figure 6.14A. Figure 6.15A displays the course of the residue of the batch distillation of the same system, but with a starting mixture located in the lower distillation field. Here, the intermediate boiler b is separated first as overhead product. Therefore, the composition of the liquid in the vessel follows a straight line through the b-corner. At the intersection of this line with the distillation boundary, pure component b can no longer be separated. Hence, its concentration decreases sharply to a value marked on the side of the triangular diagram by the tangent to the distillation boundary (dashed line). In the second step of operation, the composition of the bottoms changes along the distillation boundary. According to its curvature, the concentration of b in the distillate increases again. This behavior can be seen more clearly in Figure 6.15B. During the first step of the batch distillation, pure intermediate boiler b is recovered as distillate. In the second step, the concentration xDb decreases to approximately 0.7 and increases again later on. The concentration of the low boiler a shows a corresponding behavior. In the third step of the process, pure high boiler c is separated. Thus only high boiler c and part of the intermediate boiler b are separated in pure form. The low boiler a cannot be recovered as a pure product. Systems with a minimum azeotrope behave similarly. However, the vessel has to be located at the top of a stripping column (see Section 6.6.2). A special case is shown in Figure 6.16. This mixture exhibits a binary minimum azeotrope between intermediate boiler b and high boiler c. Hence, a distillation boundary curve runs from the minimum azeotrope to the low boiler a, dividing the mixture into two distillation fields. In the case shown, the starting mixture F is in the concave area of the distillation boundary curve in the left distillation field. When the batch distillation starts, the low boiler a can be recovered as distillate. Therefore, the composition in the still moves on a straight line away from the light boiler toward the distillation boundary curve. Normally, this boundary curve cannot be exceeded. In this case, however, the low boiler a forms the overhead product in both distillation fields. For the progress of the distillation, it does not matter in which distillation field the composition in the container is located. Therefore, the composition continues to change in the same direction and exceeds the boundary distillation curve until finally all of the low boiler a is separated. In the now binary mixture, the minimum azeotrope forms the distillate, and the composition in the still shifts toward pure middle boiler b. In the event that the product does not change when the distillation boundary curve is exceeded, it does not represent a barrier. This is a very important feature that can be very beneficial in designing processes.

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Figure 6.14 Concentration profiles for the batch distillation of the ternary system acetone/chloroform/benzene. This system exhibits a maximum azeotrope and a distillation boundary that divides the mixture into two distillation fields. A) Change of the state of the bottoms. B) Profiles of distillate concentrations xDi plotted versus the relative amount of distillate D/F .

6.2 BATCH DISTILLATION OF TERNARY MIXTURES

335

Figure 6.15 Concentration profiles for the batch distillation of the system acetone/chloroform/benzene. The feed is located in that distillation field where the intermediate boiler chloroform is separated first as the overhead fraction. When the state in the still reaches the distillation boundary, chloroform is no longer separated in pure form. A) Change of the state of the bottoms. B) Profiles of distillate concentrations xDi plotted versus the relative amount of distillate D/F .

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Figure 6.16 Concentration profiles for the batch distillation of a ternary system with a minimum azeotrope between heavy boiler c and intermediate boiler b. The feed is located in the concave area of the distillation boundary curve. The state in the still crosses the distillation boundary, because low boiler a forms the overhead product in both distillation fields.

6.3

Batch Distillation of Multicomponent Mixtures

Multicomponent batch distillation is also governed by the Rayleigh equation. For each component i in the mixture, we get

dB dxBi = . B xDi − xBi

(6.42)

This set of equations can only be solved numerically. The distillate concentrations xDi corresponding to the bottoms concentrations xBi are determined by rigorous column simulation (see Section 4.4.1). The result of an evaluation with constant RL = 5 and n = 20 for a quaternary system is shown in Figure 6.17. The changes in the distillate concentrations xDi clearly indicate that, as distillation progresses, the individual components of the feed mixture are successively recovered in the distillate with a high purity. The components are collected in order of decreasing volatilities by changing the receiver at the appropriate time. The still is gradually depleted of the low-boiling component a early in distillation; concentrations of the higher boiling components b, c, and d increase simultaneously. When most of component a has been separated, the still is depleted of component b, followed by component c. Finally, the highest boiling component d is recovered in pure form in both the still and the distillate. Batchwise single-stage distillation of the same mixture is shown in Figure 3.17. A comparison of Figures 3.17 and 6.17 clearly indicates that multistage distillation

6.4 INFLUENCE OF COLUMN LIQUID HOLD-UP ON BATCH DISTILLATION

337

Figure 6.17 Batch distillation of an ideal quaternary mixture with relative volatilities αad = 10, αbd = 3, and αcd = 2, number of stages n = 20, and reflux ratio RL = 5. The distillate concentrations xDi are plotted versus the relative amount of distillate D/F . Each component of the mixture is recovered with high purity in consecutive periods of time.

enables a much sharper separation than single-stage distillation. Furthermore, multistage distillation can be considerably improved by using a larger number of equilibrium stages n or a higher reflux ratio RL . For difficult separations, a special mode of operation can be effectively used. Here, the column is operated with total reflux for a certain period of time. Then, a small amount of top product is withdrawn. Both steps are repeated several times. This alternate operation yields a highly pure distillate at the expense of a high energy requirement [Heck et al. 1984]. The mode of operation for minimum energy requirement, described in Section 6.1.3, can thus, in principle, be extended to multicomponent mixtures. The calculation procedure is described by Robinson 1969. In addition, a chemical reaction in the liquid can be considered [Reuter et al. 1989]. The calculation methods contain a number of simplifications. For instance, a straight operating line is assumed in each case, but this is only valid if the vapor and liquid loads in the column are constant (see Section 4.2.2).

6.4

Influence of Column Liquid Hold-up on Batch Distillation

The liquid hold-up HL always present in a distillation column can greatly influence both the yield and the purity of the products of batch distillation. The yield is affect-

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ed by the column liquid hold-up since the sharp decrease and increase of distillate concentrations are hindered. Hence, the intermediate fractions that are off specifications are larger. The influence of column liquid hold-up on distillate purity is not so obvious. Information in literature is ambiguous [Mayur and Jackson 1971; Stewart et al. 1973; Luyben 1971]. Some authors find that a high liquid hold-up in the column may even enhance product purity (e.g. Zuiderweg 1953). In order to study the influence of liquid hold-up HL on batch distillation, the dynamic behavior of the column has to be considered. The fundamental MESH equations are: Material balances for i = 1 . . . k components:   L˙ j−1 · xi,j−1 + G˙ j+1 · yi,j+1 + F˙j · zi,j − L˙ j + S˙ L,j · xi,j (6.43)   d (HLj · xi,j ) − G˙ j + S˙ G,j · yi,j = . dt Equilibria for i = 1 . . . k components:

yi,j − Ki,j · xi,j = 0 with Ki,j = f (Tj , pj , xi,j , yi,j ) .

(6.44)

Summation of mole fractions in liquid and in vapor phase for i = 1 . . . k components: k X

xi,j − 1 = 0 and

i=1

k X

yi,j − 1 = 0 .

(6.45)

i=1

Heat balances:   L˙ j−1 · hL,j−1 + G˙ j+1 · hG,j+1 + F˙j · hF,j − L˙ j + S˙ L,j · hL,j   d (HLj · hL,j ) − G˙ j + S˙ G,j · hG,j + Q˙ j = dt with hL,j = f (Tj , pj , xi,j ) and hG,j = f (Tj , pj , yi,j ) .

(6.46)

These MESH equations differ from those for continuous column operation in Eqs. (4.102) – (4.105) only by the right-hand side terms of the material and heat balances that represent the possible accumulation of both material and heat in the column at non-steady-state operation. Equations (6.43) – (6.46) represent a non-linear system of mixed differential and algebraic equations. Mass and energy balances are given by differential equations. Further correlations between the variables, e.g. vapor–liquid equilibria and mole fraction constraints, are described by algebraic equations. Numerous calculation procedures for the solution of Eqs. (6.43) – (6.46) have been developed. They can be classified into two types. The first type of calculation procedures or computer programs uses a rigorous calculation algorithm for the integration of the differential equations. Numerical integration can be performed using the Euler method [Luyben 1971], the method of difference approximation [Meadows 1963], the second-order Runge–Kutta method

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[Robinson 1970], the fourth-order Runge–Kutta and fifth-order Milne methods [Goldman 1970], and the fifth-order Hamming method [Keith and Brunet 1971]. In all these methods, an accurate and stable performance of the numerical integration requires very short time intervals dt and, hence, a very long computation time [Sadotomo and Miyahara 1983]. The second type of computer programs simulates the non-steady-state behavior of a distillation column by a sequence of a finite number of steady states of short duration. For each time interval dt, the solving is carried out using a mathematical model corresponding to continuous distillation (see Section 4.4.1). Hold-up is taken into account [Rose 1979]. The robustness of this procedure is the same as for a continuous distillation model and is not influenced by the time integration procedure [Galindez and Fredenslund 1988]. The computation time required for the simulation is much shorter than with more rigorous simulations without any significant loss of accuracy [Nad and Spiegel 1987]. Figure 6.18 shows the result of evaluating the Eqs. (6.43) – (6.46) for the methanol/ ethanol/n-propanol system. Of primary interest is the influence of liquid hold-up in the column on the variation of distillate concentrations [Block et al. 1978]. A liquid hold-up that is high compared to the amount of liquid in the vessel is highly disadvantageous. The hold-up smooths the changes in distillate concentrations, thus impeding the recovery of highly concentrated top fractions in the distillate receivers. However, the recovery of the intermediate boiler ethanol is favored by a moderate relative liquid hold-up in the column.

Figure 6.18 Course of distillate concentration xDi versus D/F of the methanol/ethanol/ n-propanol mixture. The variations of distillate concentrations are significantly influenced by the liquid hold-up HL in the column based on the hold-up in the vessel. · · · · · HL = 20 % ; − − − HL = 10 % ; ——– HL = 0 %.

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Processes for Separating Zeotropic Mixtures by Batch Distillation

Batch distillation has the advantage of being able to completely separate multicomponent mixtures into their pure components in a single column. The pure components are drawn off as overhead fractions in increasing order of the boiling points. However, during the transition from one to another pure component, an intermediate fraction that is off specification is formed. This intermediate fraction is often called slop cut or off-cut. The amount and concentrations of the slop cuts depend on the separation efficiency and the hold-up of the column, as demonstrated in Sections 6.2.1 and 6.4 (see Figures 6.12, 6.13, and 6.18). In high efficiency columns, the slop cuts are small and approximately binary. This can be seen, for instance, in Figure 6.17. Thus, a slop cut is a prepurified fraction that does not meet the product specifications. However, its quality is much better than that of the feed. A complete distillation process for separating a zeotropic mixture consists of a sequence of operating cycles. Each of the cycles can be divided into several steps: • • • • • •

Start-up at total reflux. Withdrawal of the lightest component a. Removal of an intermediate a/b fraction (slop cut). Withdrawal of the second lightest component b. Removal of an intermediate b/c fraction (slop cut). Withdrawal of the third lightest component c.

The performance of a batch distillation process is significantly influenced by the way of slop cut handling. Several strategies of slop cut handling have been developed [Luyben 1988; Quintero-Marmol and Luyben 1990; Bonny 2006]. These strategies include total slop cut recycling, binary distillation of the accumulated slop cuts, and recycling of the slop cuts at the appropriate time. 6.5.1

Total Slop Cut Recycling

All slop cuts from the preceding batch are combined, recycled to the still, and mixed with the fresh feed of the following batch. Hence, both the amount and the concentrations of the subsequent batch are changed. This cyclic operation converges to a steady state after just a few cycles as Chiotti et al. 1993 have proved. Obviously, this mode of slop cut handling lacks efficiency since the pretreated slop cuts are remixed with the untreated feed. Recycling the binary slop cuts results in a multicomponent mixture. However, this is certainly the simplest and most widely used strategy in practice. 6.5.2

Binary Distillation of the Accumulated Slop Cuts

In this mode of slop cut handling, all slop cuts of the same mixture, e.g. a/b, b/c, c/d, etc., are collected in a separate tank each. After accumulation of sufficient material, each of the binary slop cuts is distilled in a batch distillation column. The resulting binary slop cuts are further processed with the next accumulated slop cut of the same

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mixture. 6.5.3

Recycling of the Slop Cuts at the Appropriate Time

Here, each of the individual slop cuts of the preceding cycle is fed into the column at an appropriate time and appropriate tray during the following batch distillation cycle. In optimal operation, the state of the liquid on the selected tray is identical with the state of the slop cut. Thus, any thermodynamic irreversibility due to mixing is effectively avoided. For a ternary mixture, a modification of this strategy is to fill the a/b slop cut into the reflux vessel and the b/c slop cut into the column for the start-up of the following cycle. The aim of all these modifications is to avoid spoiling of the prepurified slop cuts and to avoid mixing fractions of different compositions. However, these slop cut handling strategies require a highly sophisticated process control system. 6.5.4

Cyclic Operation

A zeotropic ternary mixture can also be separated by cyclic operation [Sørensen 1999; Warter 2001]. For this purpose, a middle vessel column, which is also equipped with vessels at the top and bottom, is required. The starting mixture (e.g. an equimolar mixture of methanol, ethanol, and n-propanol) is fed into the middle vessel, but also in the vessels at the top and bottom of the column; see Figure 6.19B. Then the column is operated at total reflux while no products are withdrawn. Over time, each component accumulates in the corresponding vessels. After sufficient operating time, methanol can be recovered in the vessel at the top, n-propanol at the bottom, and ethanol at the middle (see Figure 6.19A). Cyclic operation differs significantly from the other modes of operation. The big advantage is that the operation is simple and can be interrupted and resumed as desired. The energy requirement is not significantly higher than with conventional batch distillation. This procedure can also be applied to zeotropic mixtures with more components. The numbers of vessels and column sections depend on the number of desired fractions.

6.6

Processes for Separating Azeotropic Mixtures by Batch Distillation

The separation of azeotropic mixtures by distillation is difficult since the difference between the vapor and liquid concentrations is zero at the azeotropic point. Hence, there no longer exists a driving force for mass transfer in a distillation column. Separation can only be accomplished by more complex processes. The principles of such processes are outlined in detail in Section 8.6 for continuous distillation processes. Especially important and effective are those processes that involve a separating agent called entrainer e. As is explained in Section 8.6, two or even three distillation columns are required for separating a binary azeotropic mixture with the help of an entrainer. In batchwise operation, different separations are performed in a single distillation

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Figure 6.19 Separation of the ternary mixture methanol/ethanol/n-propanol with cyclic batch distillation with constant hold-up [Warter 2001]. A) Column setup. B) Triangular diagram with concentration profiles.

column at different periods of time. Thus, a sequence of columns of a continuous process is replaced by a sequence of operation cycles of a single column in a batch process. Hence, only a single column is required, operating at different operating conditions that vary over time. Three classes of processes can be distinguished between: • Processes in one distillation field. • Processes in two distillation fields. • Hybrid processes involving not only distillation but also other separation unit operations (e.g. decantation, absorption, desorption, extraction, adsorption, and membrane separations). 6.6.1

Processes in One Distillation Field

This class of processes has been extensively studied by Doherty and coworkers (e.g. Bernot et al. 1991). The starting mixture forms either a binary minimum or a binary maximum azeotrope. The entrainer e that enables separation has to be selected so that both products, a and b, are in the same distillation field. This condition is the most easily met if the boiling point of the entrainer lies between the boiling points of components a and b of a given binary mixture. Another possibility is the selection of an entrainer that forms either an intermediate boiling maximum azeotrope with the low boiler a or an intermediate boiling minimum azeotrope with the high boiler b (see Table 8.7).

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Figure 6.20 presents the process for the case of an a/b maximum azeotrope. The feed F consisting of a mixture of a and b is fed into the still together with the entrainer e. From the mixture B1α , the low boiler a is separated as a pure overhead fraction D1 of a rectifying column. The bottoms is depleted of low boiler during operation. Hence, the state of the liquid B1 in the still moves on a straight line from B1α to B1ω . When all low boiler has been separated, the intermediate boiling entrainer e forms the new overhead fraction D2 during the second period of a complete cycle. The state of the liquid B2 in the still moves on the base of the triangular diagram toward the b-corner. When all entrainer has been removed, pure high boiler b is left in the still. After removing the bottoms fraction B2ω from the still, a new process cycle is started with a fresh batch. An analogous process for separating a binary minimum azeotrope is presented in Figure 6.21. Here, the vessel is located at the top of a stripping column. By mixing the feed with the intermediate boiling entrainer e, the mixture D1α is obtained, from which the high boiler b is removed as bottoms B1 in a stripping column during the first period of a complete cycle. The second period begins when the entire high boiler b has been removed. The entrainer e, which is now the highest boiling component, forms the new bottoms fraction. At the end of this period, the low boiler a remains in the vessel. The processes illustrated in Figures 6.20 and 6.21 can advantageously be performed in a column with a middle vessel (see Figure 6.22). Here, the column consists of a rectifying and a stripping section, and the low boiler a and the high boiler b are simultaneously separated from the mixture. During operation, the residue in the middle vessel changes along a straight line in the direction of the entrainer e. Thus, the entrainer gets concentrated in the vessel. In the following batch, only the starting mixture F is fed into the vessel. The entrainer remains in the vessel and does not require external handling. This class of processes for separating azeotropic mixtures is simple and elegant. However, application to practical problems is rarely possible because the entrainer e has to be the intermediate boiler. Azeotropes are most likely formed by close-boiling mixtures. Therefore, most of the azeotropic mixtures to be separated by distillation are close-boiling ones. Hence, it might be difficult or even impossible to find an entrainer that boils in between. The schemes presented in Figures 6.20 – 6.22 can only be applied to mixtures a/b with significant boiling point differences. 6.6.2

Processes in Two Distillation Fields

In this class of processes, a distillation boundary curve always starts or ends at the a/b azeotrope, causing the components a and b to lie in different, but adjacent, distillation fields. The criteria for the selection of the entrainer are described in detail in Section 8.6.2. For a minimum azeotrope, the entrainer has to be the low boiler or has to form low boiling binary azeotropes. For a maximum azeotrope, a high boiler or a component that forms binary maximum azeotropes is suitable as an entrainer (see Table 8.8). A process that separates an a/b maximum azeotrope with a high boiling entrain-

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Figure 6.20 Batch process for separating a binary mixture a/b that exhibits a maximum azeotrope with the help of an intermediate boiling entrainer e. A full cycle consists of two periods. A) Operation sequence. B) Concentration change of the residue in the vessel.

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Figure 6.21 Batch process for separating a binary mixture a/b that exhibits a minimum azeotrope. The vessel has to be located at the top of the column. A full cycle consists of two periods. A) Operation sequence. B) Concentration changes of the residue in the vessel.

er e is shown in Figure 6.23 [Duessel and Stichlmair 1995]. Here, a distillation boundary curve runs from the entrainer e to the a/b azeotrope. The system is divided into two distillation fields with components a and b located in different fields. This process requires two vessels located at the bottom of the column and a rectifying column. The entire process cycle consists of three periods. At the beginning of the first period, the starting mixture F is fed into the vessel V-1 and completely mixed with the residues B2ω and B3ω of the preceding cycle’s second and third period, respectively. The mixture B1α is in the distillation field in which the component a has the lowest boiling temperature and is therefore separated in pure from as the overhead fraction. The state of the liquid in the vessel changes continuously from B1α to B1ω . All states lie on a straight line through B1α and D1. The first period of the cycle is finished when the state of B1 reaches the distillation boundary. In the second period of the cycle, the state of the liquid in the still changes along the distillation boundary from B2α to B2ω . The overhead fraction is located on the tangent to the distillation boundary at the current state of B2. If the separation

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Figure 6.22 Process for separating a binary azeotropic mixture by batch distillation with a middle vessel. A) Column setup. B) Triangular diagram with concentration change of the residue in the vessel.

performance is high, the state of the distillate is at the intersection of the tangent with the right side of the triangular diagram of Figure 6.23B. No pure component is separated during the second period of the cycle. Hence, the distillate D2 has to be collected in the vessel V-2. The material balance for the second period requires that the state of the distillate D2 lies on a straight line through B2α and B2ω (dotted line in Figure 6.23B). At the end of the second period, an entrainer-rich fraction B2ω remains in the vessel V-1. In the third period, the liquid in vessel V-2 is boiled, and component b is separated in pure form as the overhead fraction of the column. The composition of the liquid B3 in the vessel shifts from B3α to B3ω , which is the azeotropic point. At the end of period 3 of the cycle, the fraction B3ω is pumped from vessel V-2 into vessel V-1 and mixed with the residue B2ω of period 2. A new cycle begins by feeding the starting mixture F into the vessel V-1. An analogous process allows the separation of a binary mixture a/b that exhibits a minimum azeotrope. In this case the two vessels V-1 and V-2 are located at the top of the column, which is operated as a stripping column (see Figure 6.24). The starting liquid F is mixed with the residues of period 2 D2ω (= the entrainer) and period 3 D3ω (= the binary azeotrope), creating the fraction D1α . This fraction is in that distillation field where high boiler b can be removed as bottoms. The composition of the liquid D1 in the vessel V-1 shifts on a straight line from D1α to D1ω . During period 2 of the cycle, the composition of the liquid in the vessel V-1 moves along the distillation boundary curve from D2α to D2ω and a binary mixture B2, which lies in the other distillation field, is separated. This fraction is collected in vessel V-2. It is separated into pure low boiler a and the fraction D3ω in period 3 of

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Figure 6.23 Batch process for separating a binary mixture with a maximum azeotrope. The entrainer e is the high boiler in the system. A full cycle consists of three periods. Two vessels and one column are required. A) Operation sequence. B) Concentration changes of the residue in the vessel.

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the cycle. This process is similar to the continuous process presented in Figure 8.36. There are also three separation steps required. In the continuous process the separations are carried out in three different loci (i.e. columns) simultaneously. In the batch process all three separations are performed at the same locus (i.e. a single column) in three different periods of time. 6.6.3

Process Simplifications

The processes presented in Figures 6.23 and 6.24 require three separation steps, which are performed in the same column in different periods of time. In some special cases the process can be simplified. The methods of process simplification are analogous to those for continuous processes, dealt with in Section 8.6.2.3. The prerequisites for process simplifications are described in this section in detail. An example of great industrial importance is the separation of the HCl/H2 O mixture, which exhibits a maximum azeotrope. A suitable entrainer is sulfuric acid (H2 SO4 ). A H2 SO4 concentration of approximately 75 wt% is sufficient. The starting mixture F is fed into the vessel that contains the entrainer H2 SO4 as residue of the preceding cycle (Figure 6.25). The mixture B1α is in that distillation field where HCl can be separated as the overhead fraction. During operation the composition of the residue B1 shifts from B1α to B1ω . The bottoms fraction B1ω is essentially free of HCl since the distillation boundary closely joins the base of the triangular diagram. Water is separated as overheads during the second period of a cycle. The cycle is finished when the composition of the bottoms B1 in the vessel reaches a sufficiently high concentration to serve as entrainer for the following cycle. Here, the entire cycle consists of only two periods. This effective method of process simplification can often be applied. The only drawback is, in most cases, the large amount of entrainer used. However, the entrainer remains in the vessel and automatically performs the separation of the azeotropic mixture into the two pure components. The process is operated like a process for separating a zeotropic mixture. 6.6.4

Hybrid Processes

Process simplifications are often possible by replacing one of the distillation steps by a non-distillative separation operation. Some examples for continuous separation processes are given in Section 8.6.3. Here, the application to batch processes is demonstrated. For details, please refer to Section 8.6.3. 6.6.4.1

Azeotropic Batch Distillation

An interesting example of a hybrid process is the separation of the ethanol/water mixture with toluene as entrainer e. This type of process is called azeotropic distillation. All relevant characteristics of the system are discussed in Section 8.6.3.1. The continuous process presented in Figure 8.43 consists of two distillation columns and one decanter. In the batchwise process a cycle consists of two distillation and one

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Figure 6.24 Batch process for separating a binary mixture with a minimum azeotrope. The entrainer e is the low boiler in the system. A full cycle consists of three periods. Two vessels and one column are required. A) Operation sequence. B) Concentration changes of the residue in the vessel.

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Figure 6.25 Batch process for separating the HCl/H2 O mixture with the entrainer H2 SO4 . A full cycle requires only two periods since the distillation boundary curve approaches the base of the triangular diagram at high sulfuric acid concentrations. A) Operation sequence. B) Concentration changes of the residue in the vessel.

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decantation steps. The separations are performed in consecutive periods of time. In the first period of a cycle (see Figure 6.26), the ethanol/water mixture and the fraction S1 are fed into the vessel V-1 located at the top of the column. The mixture D1α is in that distillation field where pure water can be separated as bottoms of a stripping column. The composition of the liquid D1 in the vessel changes along a straight line from D1α to D1ω , which is close to the distillation boundary curve. At the start of the second period of a cycle, the residue D1ω is fed into the vessel V-2, located at the bottom of a rectifying column. The entrainer-rich fraction S2 is added, resulting in the composition B2α . From this mixture, the ternary minimum azeotrope, fraction D2, is removed. The composition of the residue in the vessel shifts from B2α to B2ω (pure ethanol) along a straight line; see Figure 6.26B. During this separation, the residue in the vessel crosses a distillation boundary. This is possible because in each distillation field the ternary azeotrope, fraction D2, is the low boiler. The overhead fraction D2 is cooled down and split into a water-rich fraction S1 and a toluene-rich fraction S2, which are filled into the vessels V-1 and V-2, respectively, for the following cycle. The process requires two vessels, two distillation columns, and one decanter. It is known to as azeotropic distillation. 6.6.4.2

Extractive Batch Distillation

Extractive distillation can also be applied to batchwise operation (e.g. Safrit and Westerberg 1997). The process is explained with the example ethanol/water that exhibits a minimum azeotrope (see Section 8.6.3.2 for the continuous process shown in Figure 8.44). In the first period of a cycle, the distillation step is replaced by an absorption step. The absorbent has to be a hygroscopic liquid. A good choice for this is ethylene glycol. The starting mixture F (ethanol/water) is fed into the vessel V-1; see Figure 6.27. The vapor rising from the boiling liquid in the still is enriched with ethanol but still contains some water. The water is absorbed by the ethylene glycol by countercurrent contacting in the column. Hence, pure (i.e. water-free) ethanol is the top product. The entrainer accumulates in the vessel. The composition of the bottoms shifts from the state of B1α to B1ω during the first period of a cycle. The second period of the cycle begins when all ethanol has been removed. The binary mixture water/ethylene glycol is separated by batch distillation. Water is the overhead fraction, and the residue in the vessel at the end of the cycle is pure ethylene glycol, which is used as entrainer in the following cycle. The processes presented here for the separation of azeotropic mixtures by batch distillation prove the efficiency of batch processes. They are as effective as continuous processes, but more complex in operation and less complex in equipment. The significant disadvantage of batch processes compared with continuous processes is the higher energy requirement (see Section 6.1.4).

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Figure 6.26 Batch process for separating the ethanol/water mixture by toluene. The shaded area represents the miscibility gap at 25 ◦C. A decantation step is involved in the second period of a full cycle. This process is known as azeotropic distillation. A) Operation sequence. B) Concentration changes of the residue in the vessel.

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Figure 6.27 Batch process for separating the ethanol/water mixture by ethylene glycol. In the first period of a cycle pure ethanol is separated by an absorption process. This process is known as extractive distillation. A) Operation sequence. B) Concentration changes of the residue in the vessel.

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References Barolo, M., Guarise, G.B., Rienzi, S., Trotta, A., and Macchietto, S. (1996). Running batch distillation in a column with a middle vessel. Industrial and Engineering Chemistry Research 35 (12): 4612–4618. Bernot, C., Doherty, M.F., and Malone, M.F. (1991). Feasibility and separation sequencing in multicomponent batch distillation. Chemical Engineering Science 46 (5-6): 1311–1326. Block, U., Schoenmakers, H., and Wolf, D. (1978). Entwurf und Fahrweise diskontinuierlicher Destillationen. VT Verfahrenstechnik 12 (10): 678–681. Bonny, L. (2006). Multicomponent batch distillations campaign: control variables and optimal recycling policy. Industrial and Engineering Chemistry Research 45 (26): 8998–9009. Chiotti, O.J., Salomone, H.E., and Iribarren, O.A. (1993). Selection of multicomponent batch distillation sequences. Chemical Engineering Communications 119 (1): 1–21. Coward, I. (1967). The time-optimal problem in binary batch distillation. Chemical Engineering Science 22 (4): 503–516. Davidyan, A.G., Kiva, V.N., and Meski, G. A .and Morari, M. (1994). Batch distillation in a dolumn with a middle vessel. Chemical Engineering Science 49 (18): 3033–3051. Diwekar, U.M. (1995). Batch Distillation: Simulation, Optimal Design and Control. Washington, DC: Taylor & Francis. D’Souza, A.F. and Garg, V.K. (1984). Advanced Dynamics: Modeling and Analysis. Prentice Hall. Duessel, R. and Stichlmair, J. (1995). Separation of azeotropic mixtures by batch distillation using an entrainer. Computers and Chemical Engineering 19: 113–118. Ewell, R.H. and Welch, L.M. (1945). Rectification in ternary systems containing binary azeotropes. Industrial & Engineering Chemistry 37 (12): 1224–1231. Galindez, H. and Fredenslund, A.A. (1988). Simulation of multicomponent batch distillation processes. Computers and Chemical Engineering 12 (4): 281–288. Goldman, M.R. (1970). Simulating multicomponent batch distillation. British

Chemical Engineering 15 (11): 1450. Gruetzmann, S., Fieg, S., and Kapala, T. (2006). Theoretical analysis and operating behaviour of a middle vessel batch distillation with cyclic operation. Chemical Engineering and Processing: Process Intensification 45 (1): 46–54. Heck, G., Arnold, D., and Piening, K. (1984). Entscheidungskriterien für die optimale Auslegung und Betriebsweise diskontinuierlicher Destillationsanlagen. Chemie Ingenieur Technik 56 (5): 414–415. Keith, F.M. and Brunet, J. (1971). Optimal operation of a batch packed distillation column. The Canadian Journal of Chemical Engineering 49 (2): 291–294. Kim, K.J. and Diwekar, U.M. (2000). Comparing batch column configurations: parametric study involving multiple objectives. AIChE Journal 46 (12): 2475–2488. Low, K.H. and Sørensen, E. (2005). Simultaneous optimal configuration, design and operation of batch distillation. AIChE Journal 51 (6): 1700–1713. Luyben, W.L. (1971). Some practical aspects of optimal batch distillation design. Industrial & Engineering Chemistry Process Design and Development 10 (1): 54–59. Luyben, W.L. (1988). Multicomponent batch distillation. 1. Ternary systems with slop recycle. Industrial and Engineering Chemistry Research 27 (4): 642–647. Mayur, D.N. and Jackson, R. (1971). Time-optimal problems in batch distillation for multicomponent mixtures and for columns with holdup. The Chemical Engineering Journal 2 (3): 150–163. Meadows, E.L. (1963). Multicomponent batch distillation calculations on a digital computer. Chemical Engineering Progress Symposium Series 59: 48–55. Meski, G.A. and Morari, M. (1995). Design and operation of a batch distillation column with a middle vessel. Computers and Chemical Engineering 19: 597–602. Nad, M. and Spiegel, L. (1987). Simulation of batch distillation by computer and comparison with experiment. The Use of Computers in Chemical Engineering, Sicily, Italy: 737–742.

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Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. (1962). The Mathematical Theory of Optimal Processes. New York: Interscience Publishers. Quintero-Marmol, E. and Luyben, W.L. (1990). Multicomponent batch distillation. 2. Comparison of alternative slop handling and operating strategies. Industrial and Engineering Chemistry Research 29 (9): 1915–1921. Reinders, W. and De Minjer, C.H. (1940). Vapour–liquid equilibria in ternary systems: III. The course of the distillation lines in the system acetone-chloroform-benzene. Recueil des Travaux Chimiques des Pays-Bas 59 (4): 392–406. Reuter, E., Wozny, G., and Jeromin, L. (1989). Modeling of multicomponent batch distillation processes with chemical reaction and their control systems. Computers and Chemical Engineering 13 (4-5): 499–510. Robinson, E.R. (1969). The optimisation of batch distillation operations. Chemical Engineering Science 24 (11): 1661–1668. Robinson, E.R. (1970). The optimal control of an industrial batch distillation column. Chemical Engineering Science 25 (6): 921–928. Rose, L.M. (1979). A simulation approach for the design of multicomponent batch distillation. Chimia 33 (2): 59–65. Sadotomo, H. and Miyahara, K. (1983). Calculation procedure of multicomponent batch distillation. International Chemical Engineering 23 (1): 56–64. Safrit, B.T. and Westerberg, A.W. (1997). Improved operational policies for batch extractive distillation columns. Industrial and Engineering Chemistry Research 36 (2): 436–443. Skogestad, S., Wittgens, B., Litto, R., and Sørensen, E. (1997). Multivessel batch distillation. AIChE Journal 43 (4): 971–978. Skouras, S., Skogestad, S., and Kiva, V. (2005). Analysis and control of heteroazeotropic

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batch distillation. AIChE Journal 51 (4): 1144–1157. Sørensen, E. (1999). A cyclic operating policy for batch distillation – theory and practice. Computers and Chemical Engineering 23 (4): 533 – 542. Sørensen, E. and Skogestad, S. (1996). Comparison of regular and inverted batch distillation. Chemical Engineering Science 51 (22): 4949–4962. Stewart, R.R., Weisman, E., Goodwin, B.M., and Speight, C.E. (1973). Effect of design parameters in multicomponent batch distillation. Industrial & Engineering Chemistry Process Design and Development 12 (2): 130–136. Warter, M. (2001). Batch-Rektifikation mit Mittelbehälter. PhD thesis. Technical University of Munich. Düsseldorf: VDI Verlag, Fortschrittsberichte Reihe 3: Verfahrenstechnik, No. 686. Warter, M. and Stichlmair, J. (1999). Batchwise extractive distillation in a column with a middle vessel. Computers and Chemical Engineering 23: 915–918. Warter, M. and Stichlmair, J. (2000). Batch distillation of azeotropic mixtures in a column with a middle vessel. In: European Symposium on Computer Aided Process Engineering-10, Computer Aided Chemical Engineering, vol. 8 (ed. S. Pierucci), pp. 691–696. Elsevier. Warter, M., Demicoli, D., and Stichlmair, J. (2002). Batch distillation of zeotropic mixtures in a column with a middle vessel. In: European Symposium on Computer Aided Process Engineering-12, Computer Aided Chemical Engineering, vol. 10 (eds J. Grievink and J. van Schijndel), pp. 385–390. Elsevier. Zuiderweg, F.J. (1953). Absatzweise Destillation. Einfluß der Bodenzahl, des Rücklaufverhältnisses und des Holdup auf die Trennschärfe. Chemie Ingenieur Technik 25 (6): 297–308.

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7 Energy Economization in Distillation The energy requirement W for fractionating a mixture into its pure components is calculated from [King 1980]: X ˆ·T · W = −R xFi · ln (γi · xFi ) . (7.1) This relationship is independent of the separation method and merely describes the reversal of the mixing process (see Eq. (2.86)). The energy required for distillation is supplied in the form of heat. The exergy E describes the ability of heat to be converted into work:

E =Q·

T1 − T0 , T1

(7.2)

where Q is the amount of heat, T1 the heat temperature, and T0 the ambient temperature. The work performed W is calculated from the exergy difference between the heat input temperature T1 and the heat output temperature T2 :   1 1 W = Q · T0 · − . (7.3) T2 T1 Thus, the energy requirement depends not only on the amount of heat Q but also on the temperature levels T1 and T2 . In a distillation column, a stream of heat Q˙ R is required at the bottom of the column at temperature TR , and a stream of heat Q˙ C is removed at the top at temperature TC . The following relationship holds:

Q˙ R + Q˙ C ≈ 0 with TR ≥ TC .

(7.4)

The temperature difference TR − TC is often in the range from 5 to 20 ◦C, so exergy losses are normally small. However, the heat required for distillation is usually supplied by steam at a temperature T1 > TR and is removed by cooling water at a temperature T2 < TC . Therefore, larger amounts of exergy are lost during separation than are actually necessary. Means of saving the energy requirement of distillation processes include reducing the heat requirement Q˙ R and minimizing the temperature difference (T1 − T2 ) between heat input and output. Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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7 ENERGY ECONOMIZATION IN DISTILLATION

Energy Requirement of Single Columns

This section deals with the energy requirement of processes for fractionating binary mixtures in a single column. 7.1.1

Reduction of Energy Requirement

The energy requirement Q˙ R of a single distillation column can be minimized best by effective heat exchange between the hot products and the cold feed and also by efficient insulation of the column. Furthermore, the operational reflux ratio RL should be kept as small as possible; optimum ratios are in the range of RL = (1.05 – 1.1) · RL,min [Lüdeker 1985]. The thermal state of the feed (caloric factor qF ) greatly influences the energy requirement of a column. The theoretical minimum values of the energy requirement Q˙ R,min /(F˙ · r) for ideal binary mixtures are seen in Figure 7.1. The energy requirement of a liquid feed (at boiling point) is larger by the additive term xFa than that of a vaporous feed (see Figure 7.1A,B). However, the latent heat of the vaporous feed is utilized only in the upper section of the column. Hence, the vaporous feed should be condensed in the reboiler before being introduced into the

Figure 7.1 Influence of thermal state qF of the feed on minimum energy requirement in sharp separations of an ideal binary mixture from Section 4.3. A) Saturated liquid feed qF = 1. B) Saturated vapor feed qF = 0. C) Condensation of vaporous feed in the reboiler. D) Pre-evaporization of liquid feed.

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359

column (Figure 7.1C). In this way, the latent heat is utilized in both sections of the column, which leads to a lower demand of external heat Q˙ R . The column can be operated without any external heat source if the feed concentration yFa is low and the relative volatility αab is high. This mode of operation is often employed in lowtemperature processes, but the pressure of the feed must be higher than the operating pressure of the column. Hence, a vapor feed is preferable to a liquid feed. However, a liquid should not be pre-evaporized because then the total energy requirement is higher than that with a liquid feed (Figure 7.1D). When the feed is composed of several streams with different composition, multiple feeding should be considered (see Figure 4.11). Optimal choice of feed points and careful column control are very important with respect to energy requirement. 7.1.2

Reduction of Exergy Losses

The use of intermediate reboilers and condensers (see Figure 4.12) has been proposed frequently. Although they do not lower the heat requirement of a column, part of the heat required can be supplied at a lower temperature and removed at a higher temperature, i.e. exergy losses are reduced. This method offers practical advantages only if different external heat sources (e.g. steam nets with different pressures) or several heat sinks are available or if thermal coupling is possible (Section 7.3.3). Exergy losses can be effectively reduced by utilizing waste heat obtained at the top of the column to produce low-pressure steam (Figure 7.2A) [Kleinhenz 1980]. No steam is “consumed” during distillation, just high-pressure steam is converted into low-pressure steam. Thus, exergy, and not energy, is consumed. If a low-pressure steam net is not available, steam is returned to the high-pressure net after compression (Figure 7.2B). Systematic development of this operation mode leads to incorporation of a heat pump with supplementary circulation (Figure 7.2C). Here, the question of finding an appropriate working medium for the heat pump arises. The substances separated in the distillation column can often be used to advantage as a working medium for the heat pump. Consequently, the heat pump has to be integrated into the distillation process. Utilization of the overhead product as a heat pump medium (Figure 7.3A) is usually more favorable than utilization of the bottom product (Figure 7.3B). Integrated heat pumps are used in many industrial low-temperature plants, e.g. ethylene plants [Kreuter 1973]. In high-temperature processes, however, a problem occurs because the high molecular substances involved have a very small value of the heat capacity ratio κ ≡ cp /cv . Figure 7.4 presents a plot of the heat capacity ratio versus the number of atoms in a molecule. The higher the number of atoms in a molecule, the lower is the heat capacity ratio. Consequently, the vapor temperature barely rises during compression of high molecular substances. Hence, the dew point may be crossed during compression causing machine damages due to condensation [Kleinhenz 1982]. In some cases this problem can be solved by using a compressor with a low efficiency. An example is the separation of 1, 2-dichlorethane from high boilers [Dummer and Schmidhammer 1991].

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Figure 7.2 Reduction of exergy losses of distillation columns. A) Generation of low-pressure steam from steam condensate. B) Compression and recycling of low-pressure steam. C) Heat pump with supplementary working fluid.

Figure 7.3 Schematic of integrated heat pumps that use the products of the distillation column as working fluid. A) The overhead product is the working fluid. B) The bottom product is the working fluid.

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361

Figure 7.4 Values of heat capacity ratios versus number of atoms in a molecule. Source: Data are taken from Perry et al. 1984.

A problem of columns with integrated heat pumps is the start-up. In most cases a special start-up reboiler, which is operated by steam, is installed. However, sometimes it is possible to start column operation with air or nitrogen. These gases have a very high heat capacity ratio, κ = 1.4. As a consequence, the temperature rises very high in the compressor, and the operating temperature increases gradually up to the boiling point of the mixture. A modification of heat pumps, which is often applied to columns with a pure water bottom fraction, is presented in Figure 7.5. The bottom liquid (water) is expanded by a live steam-driven ejector. At the lower pressure, the bottom liquid is in superheated state and, consequently, partly evaporated. The emerging vapor is mixed in the ejector with live steam and fed into the column. Such a process reduces the demand of live steam significantly. At the operating conditions given in Figure 7.5 the demand of live steam is reduced by approximately 30 %. A good example for the reduction of exergy losses is the multiple-stage-flash (MSF) process for the recovery of desalinated water from seawater (e.g. Greig and Wearmouth 1987). In principle, desalinated water can be produced by single-stage distillation as shown in Chapter 3 since the vapor is free of salt. However, energy consumption of such a process would be very high because the total heat of vaporization has to be provided by an external heat source. Figure 7.6 presents a simplified flow diagram of a typical MSF plant. The fresh seawater is first used as cooling agent in the evaporation stages and is hereby heated up to about 83 ◦C. It is then heated up to 90 ◦C in an external heat exchanger by low-pressure steam. The hot brine is flashed to about 0.07 bar in a multistage evaporator with typically 18 stages. The steam generated in each stage is condensed and withdrawn as product water. The liquid load of the evaporator is about 12.5 times as

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Figure 7.5 Modification of a heat pump applicable to columns that produce water as bottom product.

Figure 7.6 Multiple Stage Flash (MSF) process for the recovery of desalinated water from seawater.

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363

high as the product water. To reduce the expenses for seawater makeup, the greater part of the brine is recycled. Only 30 ◦C of the brine is blown down and replaced by fresh seawater. In the stages 1 – 15, the heat recovery section, an internal heat transfer from reboiler to condenser takes place with an average temperature difference of about 5 – 6 ◦C. The external heat supplied by low-pressure steam in the heat exchanger is finally removed in the condensers of stages 16 – 18 by additional seawater. This part is called the heat rejection section. After stage 16 the seawater is deaerated by live steam stripping in a packed column to avoid the accumulation of inert gases in the evaporator stages. Additives are necessary to prevent precipitation and foaming and to reduce the corrosiveness of the brine. The energetic effectiveness of the process is expressed by the performance ratio that represents the ratio of latent heat of vaporization to the external heat requirement of the process. Technically feasible are values of 7 to 8. Consequently, only 1/7 of the latent heat of vaporization of water (2330 kJ/kg) has to be provided by an external heat source. The energy consumption of the process is as low as 330 kJ (or 0.09 kWh) per kg product water. In addition, about 0.005 kWh/kg of electric energy is required. Such MSF plants for seawater desalination are built in huge units with a product water flow of more than 2000 m3 /d. The heat exchanger area of such a unit is as large as 80 000 m2 . The dimensions of a single-stage are typically 15 m in length, 4 m in width, and 4 m in height [El Saie et al. 1989].

7.2

Optimal Separation Sequences of Ternary Distillation

As described in Section 4.3.3, several different separating sequences exists for complete fractionation of ternary mixtures. Investment and operating costs depend on the separating path selected. Operating costs are determined primarily by the energy requirement of the processes. This is approximately also true for the investment cost since the amount of vapor determines the diameter of the columns. Heat requirements for the different separating paths in a ternary mixture are presented below. The minimum energy requirement Q˙ R,min /(F˙ · r) for separating an ideal ternary mixture a/b/c into pure substances by the a-path, c-path, and a/c-path can be formulated in general terms as described in Chapter 4. 7.2.1

Process and Energy Requirement of the a-Path

In the a-path (Figure 7.7), the low boiling substance a is removed as overhead fraction of column C-1. The remaining binary mixture b/c is fed into column C-2 to be separated into b and c. If the feed is a saturated liquid, then the following expression

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7 ENERGY ECONOMIZATION IN DISTILLATION

holds for the minimum energy requirement of column C-1 (see Eq. (4.98)):
Q˙ R1,min = (RL,min )a + 1 · xFa , F˙ · r

(7.5)

where r is the molar latent heat of vaporization of the feed and (RL,min )a the minimum reflux ratio, which can be calculated from Eq. (4.85). The minimum energy requirement of column C-2 is calculated with the equations in Figure 7.1 valid for separating binary mixtures. However, it has to be considered that the feed to column C-2 is smaller (F˙ 2 = B˙ 1 = F˙ · (1 − xFa )) and that component b is now the low boiling one. Furthermore, the relative volatility αab has to be replaced by αbc :   Q˙ R2,min 1 = xFb + · (1 − xFa ) . (7.6) αbc − 1 F˙ · r The energy requirement of the entire process is

Q˙ R,min Q˙ R1,min Q˙ R2,min = + . (7.7) ˙ ˙ F ·r F ·r F˙ · r In Figure 7.7, the energy requirement of the whole process is plotted in parametric form versus the feed composition of the mixture on a triangular concentration diagram. Values of Q˙ R,min /(F˙ · r) for an ideal mixture with relative volatilities αac =

Figure 7.7 Flow sheet and minimum energy requirement of the a-path, saturated liquid feed. The dimensionless energy requirement Q˙ R,min /(F˙ · r) is plotted versus the feed composition for the close-boiling reference system with relative volatilities αac = 1.887 and αbc = 1.329.

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365

1.887 and αbc = 1.329 are in the range of 3.4 – 6.2. This very close-boiling mixture is the reference system in this chapter (and in Chapter 4). The energy requirement increases drastically with higher concentrations of the intermediate boiling substance b. 7.2.2

Process and Energy Requirement of the c-Path

In the c-path, the high boiler c is removed as bottom fraction of column C-1 (Figure 7.8). The overhead product of column C-1 is fed – in vapor state – into column C-2, where pure products a and b are regained. For a saturated liquid feed, the energy required in column C-1 is (see Eq. (4.100)):

Q˙ R1,min = (RG,min )c · xFc , F˙ · r

(7.8)

where (RG,min )c is obtained from Eq. (4.89).

Figure 7.8 Flow sheet and minimum energy requirement of the c-path, saturated liquid feed. The dimensionless energy requirement Q˙ R,min /(F˙ · r) is plotted versus the feed composition for the close-boiling reference system with relative volatilities αac = 1.887 and αbc = 1.329.

The energy required by column C-2 for separating a vaporous binary mixture F˙ 2 = D˙ 1 = F˙ · (1 − xFc ) is taken from Figure 7.1:

Q˙ R2,min 1 − xFc = ˙ αab − 1 F ·r

for (qF )2 = 0 .

(7.9)

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7 ENERGY ECONOMIZATION IN DISTILLATION

For a liquid feed to column C-2,

Q˙ R2,min 1 − xFc = xFa + ˙ αab − 1 F ·r

for (qF )2 = 1.0 .

(7.10)

The energy requirement of the entire process, with vaporous feed (qF = 0) to column C-2, is expressed by

Q˙ R,min Q˙ R1,min Q˙ R2,min = + . F˙ · r F˙ · r F˙ · r

(7.11)

The values of Q˙ R,min /(F˙ · r) for the reference mixture are plotted in Figure 7.8. They are in the range from 3.0 – 6.2. Here again, the highest values are required at high concentrations of the intermediate boiler b. 7.2.3

Process and Energy Requirement of the Preferred a/c-Path

In the preferred a/c-path, three distillation columns are required (Figure 7.9). To obtain pure substances, the overhead product from column C-1 must be free of the high boiler c. Correspondingly, the bottom product of column C-1 must be free of the low boiler a. The overhead product is fed – in vapor state – into column C-2. The assumption is made in the following that a preferred separation is carried out in column C-1 (Section 4.3.3.1). Thus, the reflux ratio and the energy requirement of this column reach an absolute minimum. If the feed to column C-1 is a saturated liquid, the energy requirement of column C-1 follows from Eq. (4.95) or Figure 4.46:

Q˙ R1,min N = with N = 1+(αac − 1)·xFa +(αbc − 1)·xFb . (7.12) ˙ αac − 1 F ·r ˙1 = The feed to column C-2 is the overhead fraction of column C-1, i.e. F˙ 2 = D ˙ F · xFa /xDa . The relation for xDa is taken from Figure 4.46. The result for vapor feed (see Figure 7.1B) is Q˙ R2,min 1 xFa αac − 1 = · for (qF )2 = 0 with xDa = · xFa . (7.13) ˙ α − 1 x N −1 F ·r ab Da For liquid feed

Q˙ R2,min 1 xFa = xFa + · ˙ α − 1 x F ·r ab Da

for

(qF )2 = 1.0 .

(7.14)

The feed to column C-3 is the liquid bottom fraction of column C-1, i.e. F˙ 3 = B˙ 1 = F˙ · xFc /xBc with xBc from Figure 7.9. The energy requirement of column C-3 is   Q˙ R3,min 1 xFc = 1 − xBc + · ˙ αbc − 1 xBc F ·r

with xBc =

αac − 1 · xFc . αac − N (7.15)

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367

Figure 7.9 Flow sheet and minimum energy requirement of the a/c-path, saturated liquid feed. The dimensionless energy requirement Q˙ R,min /(F˙ · r) is plotted versus the feed composition for the close-boiling reference system with relative volatilities αac = 1.887 and αbc = 1.329.

Figure 7.10 Comparison of the minimum energy requirement Q˙ R,min /(F˙ · r) of all three separation paths of the ternary reference mixture. Each path can be the best choice. At high feed concentrations of the intermediate boiler b, the a/c-path has the lowest energy requirement.

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In Figure 7.9, the minimum energy requirement (with vaporous feed to column C-2) for the reference system is plotted versus the composition of the feed, giving a group of parallel lines. Values for Q˙ R,min /(F˙ · r) are in the range of 4.2 – 4.9. They increase only slightly with increasing concentration of the intermediate boiling substance b. A comparison of energy requirements of the three separating paths of a ternary mixture shows large variations depending on the composition of the feed. Each separation path can be the best choice (see Figure 7.10). If the feed contains a high concentration of the low boiler a, path a is the most favorable one. However, if the feed contains an excess of the high boiler c, the c-path is the better one. Path a/c is favored when the feed contains an excess of intermediate boiling substance b. Path a/c has three columns; i.e. it requires more equipment than the other two paths. However, this separation sequence is exergetically advantageous because heat is added or removed at several different temperature levels. Therefore, its energy-saving potential is high. If exploited properly, the a/c-path allows a more efficient separation than the other two paths (see Figure 7.26).

7.3

Modifications of the Basic Processes

The basic processes for separating ternary mixtures by distillation are often used in the process industry. However, they all have a high potential for process modification, especially with respect to minimizing the energy requirement. The most important modifications are material (or direct) coupling of columns [Schoenmakers 1986], use of side columns, and thermal coupling of columns. 7.3.1

Material (Direct) Coupling of Columns

A very important and efficient process modification is the material (or direct) coupling of columns in a process. This process modification is very effective as it allows a significant reduction of energy as well as investment costs. Therefore, this process modification should always be used whenever it is possible. In the a/c-path material coupling can be applied to columns C-2 and C-3 of the basic process as is shown in Figure 7.11. The reboiler of column C-2 has to provide the vapor flow in the stripping section of this column to maintain the countercurrent flow of vapor and liquid there. Analogously, the condenser in column C-3 has to produce the liquid flow in the rectifying section of column C-3. In sharp separations the concentrations of the streams at the bottom of column C-2 and at the top of column C-3 are equal (approximately pure substance b). In consequence, the vapor required in column C-2 can be directly taken from column C-3 discarding the evaporator. Analogously, the liquid for the rectifying section of column C-3 can be taken from the stripping section of column C-2 discarding the condenser there. The shaded elements in Figure 7.11 are no longer necessary. Material coupling is only possible if two conditions are met. First, the concentrations of the coupling streams have to be sufficiently equal. Second, the flow rates

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369

Figure 7.11 Modification of the a/c-path by material coupling of columns. The vapor from column C-3 is fed into column C-2. The liquid from column C-2 is fed into column C-3. The shaded elements of the basic a/c-path are discarded.

of the streams have to fit. The second condition is met in distillation columns when the energy requirements of the columns are nearly equal. Often, a column has to be operated with a surplus of energy to meet the condition for direct column coupling. The energy requirement of the a/c-path with material coupling is ! Q˙ R,min Q˙ R1,min Q˙ R2,min Q˙ R3,min = + Max or . (7.16) F˙ · r F˙ · r F˙ · r F˙ · r The individual amounts of energy are calculated by using Eqs. (7.12), (7.13), and (7.15). Direct coupling reduces the energy requirement, at best, to approximately two-thirds of the original amount. In Figure 7.12, Eq. (7.16) is evaluated for the reference mixture. The energy requirement is presented in parametric form versus feed composition. The values for Q˙ R,min /(F˙ · r) are in the range from 3.1 – 4.4, i.e. substantially lower than the values obtained without direct coupling; see Figure 7.9. The energy requirements are lowest in the intermediate concentration range where Q˙ R2,min /(F˙ · r) and Q˙ R3,min /(F˙ · r) are approximately equal. Hence, omission of the smaller value results in the largest energy savings. As a result of the direct coupling of columns C-2 and C-3, only two columns are necessary for the a/c-path. Thus, equipment requirements for all three paths for the separation of a ternary mixture are roughly similar. With regard to energy requirement, however, the a/c-path is more efficient over a wider concentration range than the other two paths. The shaded part of Figure 7.12 represents the optimal region. Comparison with Figure 7.10 clearly shows that direct coupling is particularly useful for increasing the optimal concentration range of this path. Material coupling is often possible in industrial separation processes. Some additional examples are presented in Chapter 8.

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Figure 7.12 Minimum energy requirement of the preferred a/c-path with material coupling of columns as a function of feed concentration for the reference system. In the shaded region of the triangular concentration diagram, the energy requirement of the a/c-path with material coupling is lower than that of the basic a-path and c-path, respectively.

7.3.2

Processes with Side Columns

The processes discussed so far are often used in the process industry. However, they have the drawback that – especially in sharp separations – nearly identical separations are performed in different columns. 7.3.2.1

a-Path with Side Column

In the a-path, for instance, the same mixture of components b and c is fractionated near the bottom of column C-1 and immediately above the feed point of column C-2; see concentration profiles in Figure 7.13. This energy-wasting twofold fractionation is avoided by using a side column as shown in Figure 7.14. In this modification of the basic process, the shaded part of the stripping section of column C-2 is arranged below the bottom of column C-1. The reboiler of column C-1 is discarded. The remainder of column C-2 is known as side column (rectifying side column). The internal concentration profiles of the process with side column are seen in Figure 7.14. The feed is fractionated in the modified column C-1 into pure low boiler a (overhead fraction) and pure high boiler c (bottom fraction). A vaporous side stream is withdrawn from the stripping section of column C-1 at the concentration maximum of component b and fed into the side column. The intermediate boiling component b is recovered as overhead fraction of the side column. No twofold fractionation exists, and, in turn, the energy requirement of the process with side column is lower than

7.3 MODIFICATIONS OF THE BASIC PROCESSES

371

Figure 7.13 Twofold separations in the sharp a-path.

Figure 7.14 Modification of the a-path by a side column (rectifying side column) to avoid twofold separations.

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that of the basic a-path. The energy requirement of the a-path with side column is calculated in several steps: • Evaluation of Eq. (4.85) to find the minimum reflux ratio (RL,min )a of column C-1 of the basic process in Figure 7.13. • Calculation of the minimum reboil ratio RG,min in the stripping section of this column from Eq. (4.22). • Determination of the concentration xSPb of the binary saddle pinch in the stripping section. Equation (4.40) has to be rearranged to xFa = f (RG,min , . . .). The result is a quadratic equation. • The minimum reboil ratio (RG,min )c in the lower stripping section of the main column in Figure 7.14 is found by exploiting Eq. (4.41). ∗ • Calculation of the vapor equilibrium concentration ySPb = f (xSPb , . . .) for the determination of the minimum reflux ratio (RL,min )b of the side column by Eq. (4.44). From the knowledge of (RL,min )a , (RG,min )c , and (RL,min )b , the energy requirement of the whole process is easily found. The result automatically meets the condition: Q˙ R1,min Q˙ C1,min Q˙ C2,min = + . (7.17) ˙ ˙ F ·r F ·r F˙ · r Example 7.1 demonstrates this calculation procedure in detail. Example 7.1: Minimum Energy Requirement of the a-Path with Side Column Find the minimum energy requirement for separating an ideal ternary mixture by the a-path with side column. Data:

• Relative volatilities: αac = 1.887 , αbc = 1.329 , αab = αac /αbc = 1.4199 . • Feed: xFa = 1/3 , xFb = 1/3 , xFc = 1/3 , saturated liquid feed. Solution:

(1) Main column C-1 (1.1) Minimum reflux ratio of a sharp low boiler separation in column C-1   √ 1 Eq. (4.85): (RL,min )a = · −B + B 2 − 4 · A · C with 2·A A = xFa = 1/3

xFa + xFc xFa + xFb 1/3 + 1/3 1/3 + 1/3 + = + = 2.3393 αac − 1 αab − 1 1.887 − 1 1.4199 − 1 1 1 C= = = 2.6849 (αac − 1) · (αab − 1) (1.887 − 1) · (1.4199 − 1)

−B =

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373

  q 1 · 2.3393 + 2.33932 − 4 · 1/3 · 2.6849 = 2 · 1/3 = 5.5724

(RL,min )a =

Figure 4.24:


Q˙ C,min = (RL,min )a + 1 · xFa = (5.5724 + 1) · 1/3 = ˙ F ·r = 2.1908 ; G˙ = 2.1908 · F˙

(1.2) Reboil ratio in column C-1 (basic form Eq. (4.22)) Eq. (4.22): RG,min =

˙ F˙ · (RL + 1) D/ 1/3 · (5.5724 + 1) = = 3.2862 ˙ ˙ 1 − 1/3 1 − D/F

Concentration of the bottom fraction:

xBb =

F˙ · xFb 1/3 = = 0.5 . ˙ 1/3 + 1/3 F · (xFb + xFc )

(1.3) Concentration maximum of component b in the stripping section of column C-1 Equation (4.40) resolved for xFa = f (RG , . . .):

A · x2Fa − B · xFa + C = 0 Consider that the mixture b/c is separated in the stripping section of column C-1.

A = (RG,min + 1) · (αbc − 1) = (3.2862 + 1) · (1.329 − 1) = 1.4102 −B = (αbc − 1) · xBb + RG,min · αbc − (RG,min + 1) = = (1.329 − 1) · 0.5 + 3.2862 · 1.329 − (3.2862 + 1) = 0.2457 = C = −xBb = −0.5   p 1 xSPb = · −B + B 2 − 4 · A · C = 2·A   q 1 2 = · 0.2457 + 0.2457 − 4 · 1/3 · (−0.5) = 0.6889 . 2 · 1.4102 (1.4) Minimum reboil ratio of the modified main column for sharp high boiler separation Eq. (4.41): N = 1 + (αbc − 1) · xSPb = 1 + (1.329 − 1) · 0.6889 =

= 1.2266 (RG,min )c =

N 1.2266 = = 11.9842 αbc − N 1.329 − 1.2266

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Figure 4.24:

Q˙ R,min = (RG,min )c · xFc = 11.9842 · 1/3 = 3.9947 F˙ · r

G˙ = 3.9947 · F˙ . (1.5) The side product of column C-1 is withdrawn at the concentration maximum of component b in the stripping section

G˙ = 3.9947 · F˙ − 2.1908 · F˙ = 1.8039 · F˙ Concentration of the vaporous side product

αbc · xSPb = 1 + (αbc − 1) · xSPb 1.329 · 0.6889 = = 0.7464 . 1 + (1.329 − 1) · 0.6889

∗ Equilibrium to xSPb : ySPb =

(2) Side column: C-2 The vaporous side product of column C-1 is the feed to the side column C-2. (2.1) Vapor and liquid flow in the side column

G˙ = 1.8039 · F˙ ˙ = G˙ − F˙ · xFb = 1.8039 · F˙ − F˙ · 1/3 = 1.4706 · F˙ . L˙ = G˙ − D2 (2.2) Minimum reflux ratio of the side column

1
∗ = −1 1 − αbc · ySPb 1 = = 5.412 (1 − 1.329−1 ) · 0.7464 (RL,min )b = 4.412 .

Eq. (4.44): (RL,min )b + 1 =

(2.3) Actual reflux ratio in the side column

˙ D˙ = 1.4706 · F˙ /(1/3 · F˙ ) = 4.4118 = (RL,min ) okay. (RL )actual = L/ b (3) Energy requirement of the whole process:

Q˙ R,min = 3.9947 F˙ · r Q˙ C,min Condenser of the main column (check with Figure 4.49): = 2.1908 F˙ · r Q˙ C,min G˙ Condenser of the side column: = = 1.8040 F˙ · r F˙ Reboiler of the main column (check with Figure 7.15):

7.3 MODIFICATIONS OF THE BASIC PROCESSES

Heat balance (check with Figure 7.15):

375

Q˙ R,min = 3.9947 and F˙ · r Q˙ C,min = 2.1908 + 1.8040 = 3.9948 . F˙ · r

Result: The energy supplied to the reboiler of the main column is sufficiently large

for the recovery of the low boiler a in the main column and the intermediate boiler b in the side column.

Figure 7.15 Energy requirement of the a-path and c-path with side column for the reference system. In the shaded region the paths with side column have a lower energy requirement than the preferred a/c-path with material coupling of columns and the basic a- and c-paths.

The energy requirement of the a-path with side column is seen in Figure 7.15 for the close-boiling reference system as a function of feed concentration xFi . The values of the dimensionless energy requirement Q˙ R,min /(F˙ · r) are in the range from 2.4 – 6.0. In the shaded region they are significantly lower than the values of the basic a-path in Figure 7.7. Especially advantageous is the process with side column when the concentration of the intermediate boiling component b is the feed is small. 7.3.2.2

c-Path with Side Column

The internal liquid concentration profiles of a sharp separation by the c-path are depicted in Figure 7.16. Here too, in the upper part of the rectifying section of column C-1 and in the upper part of the stripping section of column C-2, the same mixture of substances a and b is fractionated. By arranging the rectifying section of column C-2 atop column C-1, this energy-wasting twofold separation is avoided; see Figure 7.17. Furthermore, the condenser of column C-1 can be discarded. The

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7 ENERGY ECONOMIZATION IN DISTILLATION

remainder of column C-2 is a side column (stripping side column). The flow sheet of the process with side column and the internal liquid concentration profiles are seen in Figure 7.17. The liquid side stream of the main column is withdrawn at the concentration maximum of component b in the rectifying section. The low boiler a and the high boiler c are the overhead and bottom products of the main column, respectively. The intermediate boiling substance b is the bottom product of the side column. The energy requirement of the c-path with side column is also calculated in several steps: • Evaluation of Eq. (4.89) to find the minimum reboil ratio (RG,min )c of column C-1 of the basic process in Figure 7.16. • Calculation of the minimum reflux ratio RL,min in the stripping section of this column by Eq. (4.22). • Determination of the concentration xSPa of the binary saddle pinch in the rectifying section. Equation (4.38) has to be rearranged to xFa = f ((RL,min ) , . . .). The result is a quadratic equation. • Determination of the minimum reflux ratio (RL,min )a for separating pure low boiler a from the binary mixture xSPa by Eq. (4.39). • Calculation of the minimum reboil ratio (RG,min )b of the side column by Eq. (4.41). From the knowledge of (RG,min )c , (RL,min )a , and (RG,min )b , the energy requirement of the whole process is easily found. The result automatically meets the condition:

Q˙ R1,min Q˙ R2,min Q˙ C1,min + = . F˙ · r F˙ · r F˙ · r

Figure 7.16 Twofold separations in the sharp c-path.

(7.18)

7.3 MODIFICATIONS OF THE BASIC PROCESSES

377

Figure 7.17 Modification of the c-path by a side column (stripping side column) to avoid twofold separations.

The energy requirement of the c-path with side column for fractionating the reference system with the relative volatilities αac = 1.887 and αbc = 1.329 has been evaluated. Interestingly, the energy requirement of the c-path with side column is exactly equal to that of the a-path with side column. Hence, the values of Q˙ R,min /(F˙ · r) shown in Figure 7.15 are valid for the c-path with side column, too. In the shaded region of the concentration triangle in Figure 7.15, the processes with side columns have a lower energy requirement than the preferred a/c-path with material coupling seen in Figure 7.12. 7.3.2.3

Preferred a/c-Path with Side Column

In the basic a/c-path there exist two sections with twofold separations; see Figure 7.18. These twofold separations can be avoided by arranging the rectifying section of column C-2 atop column C-1 and by arranging the stripping section of column C-2 beneath column C-1. Two heat exchangers of the basic process in Figure 7.18 are discarded by this modification. The remainder of column C-2 is a side column (intermediate side column) without reboiler and condenser, but with side withdrawal of the intermediate boiler b. This configuration is often called Petlyuk column [Petlyuk et al. 1965]. The streams required for the internal countercurrent flow of vapor and liquid in the side column are taken from the main column. The flow sheet and the internal concentration profiles of the process with intermediate side column are depicted in Figure 7.20. No double fractionations exist in the process with side column. The feeds into the side column are taken at the concentration maxima of the intermediate boiling compound b from the rectifying and stripping section of the main column.

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Figure 7.18 Twofold separations in the sharp a/c-path.

Example 7.2: Minimum Energy Requirement of the a/c-Path with Side Column Find the minimum energy requirement for separating an ideal ternary mixture by the a/c-path with side column (dividing wall column). Data:

• Relative volatilities: αac = 1.887 , αbc = 1.329 , αab = αac /αbc = 1.4199 . • Feed: xFi = 1/3 , saturated liquid feed. Operation with minimum reflux and reboil. Solution:

(1) Vapor and liquid flow in the lower section of column C-1

G˙ Q˙ = F˙ F˙ · r Eq. (4.89): (RG,min )c + 1 =

  √ 1 · −B + B 2 − 4 · A · C with 2·A

A = xFc = 1/3 −B =

xFa + xFc xFb + xFc 1/3 + 1/3 1/3 + 1/3 + = 4.1113 −1 + −1 = 1 − 1.887−1 1 − 1.329−1 1 − αac 1 − αbc

7.3 MODIFICATIONS OF THE BASIC PROCESSES

C=

1−

−1 αac

379

1 1  = 8.5936
= −1 ) · (1 − 1.329−1 ) −1 (1 − 1.887 · 1 − αbc

  q 1 · 4.1113 + 4.11132 − 4 · 1/3 · 8.5936 = 9.6670 2/3 (RG,min )c = 8.6670

(RG,min )c + 1 =

Eq. (4.100):

Q˙ R,min = (RG,min )c · xFc = 8.6670 · 1/3 = 2.8890 F˙ · r

G˙ = 2.8890 · F˙ ;

L˙ = G˙ + B˙ = (2.8890 + 1/3) · F˙ = 3.2223 · F˙ .

(2) Saddle pinch in the stripping section

1 −1 (1 − αbc ) · ((RG,min )c + 1) 1 = = 0.4179 (1 − 1.329−1 ) · 9.6670 = 1 − 0.4179 = 0.5821 .

Eq. (4.88): (xSPc )bc =

(xSPc )bc (xSPb )bc

(3) Vapor and liquid flows in the upper section of column C-1

G˙ Q˙ = F˙ F˙ · r Minimum reflux ratio of a-separation (RL,min )a = ?   √ 1 Eq. (4.85): (RL,min )a = · −B + B 2 − 4 · A · C with 2·A

A = xFa = 1/3 xFa + xFc xFa + xFb 1/3 + 1/3 1/3 + 1/3 + = + = 2.3393 αac − 1 αab − 1 0.887 0.4199 1 1 C= = = 2.6849 (αac − 1) · (αab − 1) 0.887 · 0.4199   q 1 2 (RL,min )a = · 2.3393 + 2.3393 − 4 · 1/3 · 2.6849 = 5.5724 2/3 −B =

Eq. (4.98):


Q˙ C,min = (RL,min )a + 1 · xFa ˙ F ·r = (5.5724 + 1) · 1/3 = 2.1908 .

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7 ENERGY ECONOMIZATION IN DISTILLATION

The vapor flow in the stripping section is smaller than in the rectifying section. Both column sections have to be operated with the same vapor flow at vaporous feed. Therefore:

G˙ = 2.8890 · F˙ ;

L˙ = G˙ − D˙ = (2.8890 − 1/3) · F˙ = 2.5557 · F˙ .

(4) Saddle pinch in rectifying section

1 (αab − 1) · (RL,min )a 1 = = 0.4274 . (1.4199 − 1) · 5.5724

Eq. (4.84): (xSPa )ab =

(5) The middle section of the main column is the rest of the preferred separation of the basic process. The concentration profiles in that section are equal to the profiles of a preferred a/c-separation. Therefore, the internal vapor and liquid streams are identical with those of a preferred a/c-separation. Eq. (4.95):

G˙ Q˙ min N = = with ˙ ˙ αac − 1 F F ·r

N = 1 + (αac − 1) · xFa + (αbc − 1) · xFb = 1 + 0.887/3 + 0.329/3 = 1.4053 ˙ G/F˙ = 1.4053/(1.887 − 1) = 1.5844 ;

G˙ = 1.5844 · F˙ .

Rectifying section, Eq. (4.71):

RL,min =

1 1 = = 2.4673 N −1 1.4053 − 1

Eq. (4.14):

L˙ RL 2.4673 = = = 0.7116 ˙ R + 1 2.4673 +1 G L

L˙ = 0.7116 · 1.5844 · F˙ = 1.1274 · F˙ . Stripping section: Material balance, data from Figure 7.19:

L˙ = 1.1274 · F˙ + 1 · F˙ = 2.1274 · F˙ . (6) Vapor and liquid streams in the side column Vapor stream from material balance, data from Figure 7.19:

G˙ = (2.8890 − 1.5844) · F˙ = 1.3046 · F˙

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7.3 MODIFICATIONS OF THE BASIC PROCESSES

Liquid stream in upper section, data from Figure 7.19:

L˙ = (2.5557 − 1.1274) · F˙ = 1.4283 · F˙ Liquid stream in lower section, data from Figure 7.19:

L˙ = (1.4283 − 1/3) · F˙ = 1.0950 · F˙ . (7) Check of reflux ratio in the lower section of the side column, data from Figure 7.19 Figure 4.22: RL,min + 1 =

yb = yb∗ =

1 −1 (1 − αbc ) · yb

for vaporous feed qF = 0

αbc · (xSPb )bc 1.329 · 0.5821 = = 0.6493 1 + (αbc − 1) · (xSPb )bc 1 + 0.329 · 0.5821

1 = 6.2213 ; RL,min = 5.2213 (1 − 1.329−1 ) · 0.6493 L˙ RL 1 Actual reflux ratio Eq. (4.14): = = RL + 1 1 + 1/RL G˙ RL,min + 1 =

(RL )actual =

1 1 = = 5.2242 = RL,min okay. ˙ ˙ 1.3046/1.0950 −1 G/L − 1

(8) Check of reboil ratio in the upper section of the side column, data from Figure 7.19 Figure 4.22:

RG,min + 1 =

1 1 = = 5.9056 −1 −1 (1 − 1.4199 ) · (1 − 0.4274) (1 − αab ) · xb

RG,min = 4.9056 . Actual reboil ratio, data from Figure 7.19: L˙ RG + 1 1 Eq. (4.17): = =1+ ˙ R R G G G

(RG )actual =

1 1 = = 10.54 ≥ RG,min okay. ˙ ˙ 1.4283/1.3046 − 1 L/G − 1

Results: All sections of the main column and the side column are operated with

minimum reflux and reboil, respectively, except the upper rectifying section of the main column and the upper section of the side column. These are those sections where the binary mixture a/b is fractionated.

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Figure 7.19 Internal vapor and liquid flows of the preferred a/c-path with side column as calculated in Example 7.2. Approximately 45 % of the vapor in the main column is fed into the side column.

7.3 MODIFICATIONS OF THE BASIC PROCESSES

383

Figure 7.20 Modification of the a/c-path by a side column (intermediate side column) to avoid the twofold separations.

The energy requirement of the preferred a/c-path with side column can be directly calculated since the feed concentration is equal to the pinch concentration. The relevant equation is

Q˙ R,min = Max F˙ · r


(RL,min )a + 1 · xFa


or (RG,min )c · xFc .

(7.19)

The quantities (RL,min )a and (RG,min )c are calculated by Eqs. (4.85) and (4.89), respectively. The energy requirement of the preferred a/c-path with side column is shown in Figure 7.21 for the close-boiling reference mixture. The lines Q˙ R,min /(F˙ · r) = const are a strong function of the feed concentration xFi . At high contents of the low boiler in the feed, the energy requirement is equal to the energy requirement for separating a pure low boiler from a ternary mixture in a single column; see Figure 4.49. Therefore, no energy is required for fractionating the residual binary mixture b/c. The same holds for high concentrations of the high boiler c in the feed. Here, the energy requirement of the preferred a/c-path with side column is equal to the process for separating a pure high boiler from a ternary mixture in a single column; see Figure 4.51. The internal flow rates of vapor and liquid are shown in Figure 7.19 for a column operated with minimum reflux and reboil. The evaluation procedure is explained in detail in Example 7.2 for the reference system with a feed concentration xFi = 1/3. In this case, the minimum energy requirement of the reboiler is higher than that of the condenser. In consequence, the upper section of the main column and the upper section of the side column are operated with an excess of reflux and reboil, respectively. Therefore, the separation of the binary mixture a/b does not require any additional external energy. Moreover, the separation of the binary mixture a/b

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Figure 7.21 Minimum energy requirement Q˙ R,min /(F˙ · r) of the preferred a/c-path with intermediate side column as a function of feed composition. The values are valid for separating the reference system.

is performed with a reduced number of equilibrium stages. These features make the process of preferred a/c-path with side column the best process of all discussed so far. Just the preferred a/c-path with thermal coupling of columns has a lower energy requirement; see Section 7.3.3.3 and Table 7.1. 7.3.2.4

Dividing Wall Columns

With the exception of crude oil distillation and cryogenic argon recovery from air (see Chapter 8), side columns were very seldom used in industrial distillation processes. This situation has been changed by the concept of dividing wall columns. In processes with side columns, the gas load is reduced in that section of the main column where a side column is operated in parallel. For instance, in the process dealt with in Example 7.2, the vapor stream from the main column into the side column (and back) is as high as 45 % of the vapor stream in the main column. Since the diameter of distillation columns is primarily determined by the gas load (see Chapter 9), the main column should have a smaller diameter in the section with reduced gas load. Wright 1949 was the first to integrate an intermediate side column into the main column today known as dividing wall column. However, it was a paper of Kaibel 1987 that prompted the first industrial applications and, in turn, worldwide research activities on dividing wall columns. The integration of a side column into the main column is the principal idea behind the dividing wall columns shown in Figure 7.22. From a process point of view, dividing wall columns are identical with processes with side columns. Dividing wall columns have the same energy requirement as the corresponding processes depicted in Figures 7.15 and 7.21. They are just an investment cost-saving modification of the equipment. The biggest advantage offers the realization of the preferred a/c-path in

7.3 MODIFICATIONS OF THE BASIC PROCESSES

385

a dividing wall column. Here, the energy requirement is lowest in the full range of feed concentration, and, furthermore, only two heat exchangers (one reboiler and one condenser) are required.

Figure 7.22 Three examples of dividing wall columns for separating ternary mixtures.

In recent years, there is a flood of scientific papers on dividing wall columns [Yildirim et al. 2011]. Some papers concern the principles and fundamentals of these column configurations [Kaibel et al. 2003; Becker et al. 2001; Asprion and Kaibel 2010; Niggemann et al. 2010]. Others deal with the energy-saving potentials [Jansen et al. 2016; Jing et al. 2013; Vargas and Fieg 2013] and rigorous computer models of these processes [Niggemann et al. 2011]. There are also some papers regarding design and dimensioning the equipment [Olujić et al. 2012; Niggemann et al. 2011]. Dividing wall columns can be advantageously realized by packed columns. Special care has to be taken to provide the proper liquid and vapor flow rates in the sections at both sides of the dividing wall [Dwivedi et al. 2012b]. The heat transfer across the dividing wall has to be taken into account in detail engineering [Lestak et al. 1994; Jing et al. 2013; Ehlers et al. 2015]. Problems of optimal start-up, operation, and efficient control are considered in Niggemann et al. 2011; Wang and Wong 2007; Wolff and Skogestad 1995; Gupta and Kaistha 2015; Buck and Fieg 2012; Rewagad and Kiss 2012. Many authors report on industrial applications of this new type of columns [Dejanović et al. 2011; Halvorsen et al. 2016]. Among them are examples of azeotropic [Bude and Repke 2015; Wu et al. 2014], extractive [Bravo-Bravo et al. 2010] and even reactive distillation [Kaibel et al. 2005; Mueller and Kenig 2007; Dai et al. 2015; Ehlers et al. 2013; Geißler et al. 2006; Sun and Bi 2014; Hasse et al. 2007; Staak et al. 2014]. According to Olujić et al. 2016, more than 250 dividing wall columns are in industrial operation in 2016, one-third at BASF, Germany. Approximately 90 % of

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7 ENERGY ECONOMIZATION IN DISTILLATION

those are packed columns. The biggest dimensions are 6.5 m in diameter and 100 m in height. Recent developments are four-product columns with three dividing walls, two of them one above the other [Dwivedi et al. 2012a]. However, there are also some drawbacks of dividing wall columns. One of them is the fact that all the heat has to be supplied at the highest temperature level and removed at the lowest temperature level of the process. Therefore, the process offers no possibilities for thermal column coupling and, in turn, ranks second with respect to energy requirement (see Table 7.1), behind the thermally coupled a/c-path. Another drawback of dividing wall columns is that the separations at both sides of the dividing wall have to be established within the same column height. This fact often prevents their application to systems with significantly different relative volatilities. 7.3.3

Thermal (Indirect) Coupling of Columns

In thermal (or indirect) coupling of columns, waste heat from one column is used to heat another column. In most cases, the columns have to be operated with different pressures to provide the right temperature levels. Since the vapor pressure of liquids increases exponentially with temperature, comparatively large changes of pressure are needed to properly adjust the temperature levels of heat source and heat sink. Hence, thermal coupling of columns is of particular interest in close-boiling systems. A decisive criterion for thermal coupling is that the corresponding temperature levels do not differ too much. In the a-path, for instance, the heat from the condenser of column C-2 is used to heat the reboiler of column C-1; see Figure 7.23. In the c-path, however, the waste heat from the condenser of column C-1 is used to heat the reboiler of column C-2. In the a/c-path either the waste heat from column C-1 is used to heat the reboiler of column C-2, or the waste heat from column C-2 is used to heat column C-1, whichever has the lowest difference of temperature levels of heat source and heat sink.

Figure 7.23 Principle of thermal coupling of columns in the a-path, c-path, and the a/c-path.

7.3 MODIFICATIONS OF THE BASIC PROCESSES

7.3.3.1

387

Thermal Column Coupling of the a-Path

A process with thermal (indirect) coupling of the two columns of the a-path is shown in Figure 7.24. To demonstrate the principle of thermal coupling as clearly as possible, the two columns are – in the graph – combined in a single column with a heat exchanger in between. Column C-2 is operated with a higher pressure, so its waste heat can be used for heating column C-1. Thermal coupling reduces the energy requirement of the entire process to a value equivalent to the larger value of the two original columns. Hence: ! Q˙ R,min Q˙ R1,min Q˙ R2,min = Max or . (7.20) F˙ · r F˙ · r F˙ · r The quantities Q˙ R1,min /(F˙ · r) and Q˙ R2,min /(F˙ · r) can be calculated by using Eqs. (7.5) and (7.6). Thermal coupling reduces the energy requirement of the process by 50 % at best. The triangular concentration diagram in Figure 7.24 shows the values of the energy requirement Q˙ R,min /(F˙ · r) calculated from Eq. (7.20) for the reference system. The energy requirement, plotted in parametric form against the feed composition, is in the range from 1.8 – 3.9, i.e. considerably lower than the values obtained without thermal coupling; see Figure 7.7. In the intermediate concentration range, the energy required by each of the two columns is nearly equal. Therefore, omission of one quantity of energy results in the largest energy saving. Here, thermal coupling halves the energy requirement. In the region with high feed concentration of component a, the energy requirement of the process is equal to that of low boiler separation in a single column; see Figure 4.49. In the region of small low boiler concentrations in the feed, the energy requirement is equal to that of fractionating a binary mixture b/c, where a linear dependency on feed concentrations exists; see Figure 7.1. 7.3.3.2

Thermal Column Coupling of the c-Path

Analogously, a process with thermal coupling of the columns of the c-path is shown in Figure 7.25. The liquid overhead product of column C-1, which is operated at a higher pressure, is fed as saturated liquid into column C-2. The energy requirement of the entire process is ! Q˙ R,min Q˙ R1,min Q˙ R2,min = Max or . (7.21) F˙ · r F˙ · r F˙ · r The individual quantities of energy are calculated from Eqs. (7.8) and (7.10). The results of evaluating Eq. (7.21) for the reference system are seen at the triangular diagram in Figure 7.25. The parametric presentation of the dimensionless energy requirement clearly shows the transition between the two quantities of energy. Energy savings are highest in the transition range of the parameter lines. However, a 50 % reduction is not achieved because the feeding of a vapor into column C-2 represents a form of thermal coupling in the basic process (see Figure 7.8), which is, however, less effective than that one shown in Figure 7.25. In the region with low

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7 ENERGY ECONOMIZATION IN DISTILLATION

Figure 7.24 Flow sheet and energy requirement of the a-path with thermal coupling. The data of Q˙ R,min /(F˙ · r) are valid for the reference system.

Figure 7.25 Flow sheet and energy requirement of the c-path with thermal coupling. The data of Q˙ R,min /(F˙ · r) are valid for the reference system.

7.3 MODIFICATIONS OF THE BASIC PROCESSES

389

concentrations of component a, the energy requirement of the process is equal to that of high boiler separation from a ternary mixture in a single column; see Figure 4.51. At high concentrations of the low boiler a in the feed, the parameter lines are straight and parallel as the separation of the binary mixture a/b is decisive. 7.3.3.3

Thermal Column Coupling of the Preferred a/c-Path

A process with thermal coupling of the columns of the preferred a/c-path with material coupling is seen in Figure 7.26. Again, the columns are – in the graph – combined in a single column. In column C-1, which is operated at a higher pressure, a preferred a/c-separation of the ternary mixture is established. The heat exchanger at the top of column C-1 acts as condenser in column C-1 and as reboiler in column C-2. In principle, the three columns of the basic a/c-path (Figure 7.9) are arranged one above the other.

Figure 7.26 Flow sheet and energy requirement of the a/c-path with thermal coupling. The data of Q˙ R,min /(F˙ · r) are valid for the reference system.

The bottom fraction of C-1 is fed into the lower section of column C-2 to be split into pure substances b and c. The overhead fraction of column C-1 is fed – in liquid state – into the upper section of column C-2 to be fractionated into pure substances a and b. The flow sheet makes it clear that the heat supplied to the reboiler of column C-1 is used three times for fractionation before it is taken out by the condenser of column C-2 [Triantafyllou and Smith 1992]. The minimum energy requirement of the process is ! Q˙ R,min Q˙ R1,min Q˙ R2,min Q˙ R3,min = Max or or . (7.22) F˙ · r F˙ · r F˙ · r F˙ · r

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7 ENERGY ECONOMIZATION IN DISTILLATION

Table 7.1 Classification of processes for sharp separation of ideal ternary mixtures.

a-path

Figure 7.7

Advantageous at large values of xFa

c-path

Figure 7.8

Advantageous at large values of xFc

Preferred a/c-path

Figure 7.9

Advantageous at large values of xFb

Material coupling of columns

Preferred a/c-path

Figure 7.12

In the shaded concentration region better than the three basic processes

Side column

a-path and c-path

Figure 7.15

In the shaded concentration region better than all processes above

Side column with material coupling

Preferred a/c-path and dividing wall column

Figure 7.21

Energy requirement is equal to that of low boiler or high boiler separation in a single column. This process ranks second with respect to energy requirement.

Thermal coupling of columns (different operating pressures required)

a-path

Figure 7.24

At large values of xFa , the energy requirement is equal to that of low boiler separation in a single column (Figure 4.49). At small values of xFa , the energy requirement is equal to that of binary separation mixture b/c.

c-path

Figure 7.25

At small values of xFa , the energy requirement is equal to that of high boiler separation in a single column (Figure 4.51). At large values of xFa , the energy requirement is equal to that of binary separation of mixture a/b.

Preferred a/c-path

Figure 7.26

Best process of all with respect to energy requirement. Disadvantage: different operating pressures required.

Basic processes

Material and thermal coupling of columns

7.4 DESIGN OF HEAT EXCHANGER NETWORKS

391

The individual quantities of energy are calculated from Eqs. (7.12), (7.14), and (7.15). The results of evaluating Eq. (7.22) for the reference mixture are presented in parametric form on the triangular concentration diagram shown in Figure 7.26. A group of parallel straight lines is obtained in each region. Obviously, the value Q˙ R1,min /(F˙ · r) is always smaller than the values of Q˙ R2,min /(F˙ · r) and Q˙ R3,min /(F˙ · r) in the reference system. Hence, the binary separations a/b and b/c in column C-2 are decisive for the energy requirement of the whole process. As these binary distillations have a linear dependence on feed concentration, only two regions with different orientation of the straight parameter lines are obtained in the triangular concentration diagram. Values of the energy requirement Q˙ R,min /(F˙ · r) range from 1.7 – 3.2. They are lower than those obtained for the thermally coupled a-path and c-path; see Figures 7.24 and 7.25, respectively. The three processes with thermal coupling are equivalent only if the feed mixture has a very low content of intermediate boiler b. In all other cases, the preferred a/c-path with material and thermal coupling is the most favorable separation process. A classification with respect to energy requirement is listed in Table 7.1 for all processes for separating ternary mixtures dealt with in this chapter. Considered is the minimum energy requirement at minimum reflux and reboil and, in turn, infinite number of equilibrium stages. The listing makes it clear that the basic processes are not the best choice, even though they are often used in the process industry. A must is material coupling of the a/c-path. All separation paths can be further improved by side columns (dividing wall columns) and by thermal coupling of columns. Thermal column coupling, however, demands different operating pressures to properly adjust temperature levels of heat source and heat sink. Side columns and dividing wall columns do not have this drawback, but they have a higher energy requirement.

7.4

Design of Heat Exchanger Networks

Generally, separation processes consist of a large number of units that are interconnected by material streams. For proper function heat has to be supplied and removed at many places in different quantities as well as different temperature levels. From a thermodynamic point of view, any thermal separation process involves a heat exchanger network. The simplest design of such a network would apply hot utility to all streams that have to be heated (cold streams) and cold utility to all streams that have to be cooled (hot streams). However, a more sophisticated design would try to match cold streams and hot streams in order to reduce the quantity of external heat (or cold) required. The objective of any heat recovery network design is to select streams for thermal matching in such a way that the network is cheapest in terms of utility and capital cost required [Linnhoff 1983].

392

7.4.1

7 ENERGY ECONOMIZATION IN DISTILLATION

Optimum Heat Exchanger Networks

Development of systematic procedures for the design of heat exchanger networks has been an active area of interest in the chemical engineering literature for many years. A review is given by Nishida et al. 1981; Linnhoff 1993; Smith 1999; Agrawal and Shenoy 2006; Kemp 2007; Heck et al. 2014. Early applications of such methods are in cryogenic processes, e.g. air separation [Baldus et al. 1983]. All these methods are based on the graphical plot of temperature levels versus enthalpy streams similar to the design procedure for a single heat exchanger. These methods have been further developed and propagated by Linnhoff and coworkers [Linnhoff and Flower 1978; Linnhoff and Turner 1980; Linnhoff and Hindmarsh 1983; Linnhoff and Vredeveld 1984; Linnhoff and Sahde 1989]. The procedure will be explained in the following at the simple example of two hot and three cold streams without any phase change, e.g. neither evaporation nor condensation. Temperature levels and heat quantities of the streams are listed in Table 7.2. With the assumption of constant heat capacities, the streams are represented by straight lines in the temperature–enthalpy diagrams of Figure 7.27. The direction of temperature change is marked by arrows. First, the enthalpies of all hot streams are added at their individual temperature levels, resulting in a single line for all hot streams; see Figure 7.28A. According to Linnhoff and Flower 1978, the resulting line is called hot composite curve. The cold composite curve is constructed in the same way; see Figure 7.28B. In practical applications the number of streams involved is generally much larger and the individual lines might be curved. However, the design procedure of the composite curves remains essentially the same. In the next step, the cold and hot composite curves are combined in a single diagram. Since the abscissa represents enthalpy streams H˙ , the composite curves can be shifted in horizontal direction until a minimum temperature difference ∆TP (e.g. 10 ◦C) between hot and cold curve is reached; see Figure 7.29. The locus of the minimum distance between the composite curves is called pinch. The curves overlap along the abscissa. The quantity of heat in the overlapping section can be internally recovered by an optimum heat exchanger network. However, at each end, there exists an overhang so that the top of the cold composite curve needs external heat Q˙ H and the bottom of the hot composite curve needs external cold Q˙ C . These external heats are referred to as the hot and the cold utility targets [Linnhoff and Sahde 1989]. From the diagram in Figure 7.29, the quantities of external heat steams Q˙ H and Q˙ C are easily determined. In the example they are Q˙ H = 80 MW and Q˙ C = 60 MW. However, knowledge of heat quantities Q˙ H and Q˙ C is not sufficient for the design of the optimum heat exchanger network. As important as heat streams Q˙ H and Q˙ C are their temperature levels TH and TC . From a thermodynamic point of view, the heat stream Q˙ C should be withdrawn at a temperature level as high as possible, and, in turn, the heat stream Q˙ H should be supplied at a temperature level as low as possible in order to minimize the exergy losses of the heat exchanger network. It is obvious from Figure 7.29 that the cold utility Q˙ C needs not be withdrawn at the lowest temperature level (here 40 ◦C). The length of Q˙ C can be shifted upward

7.4 DESIGN OF HEAT EXCHANGER NETWORKS

Figure 7.27 Graphical plot of temperatures and enthalpies of the individual streams of Table 7.2. A) Hot streams. B) Cold streams.

Figure 7.28 Composite curves of the hot and cold streams. A) Hot streams. B) Cold streams.

393

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7 ENERGY ECONOMIZATION IN DISTILLATION

Table 7.2 Listing of temperature levels and enthalpies of five streams.

Stream

Type

Inlet temperature in ◦C

Outlet temperature in ◦C

Enthalpy stream in MW

1 2 3 4 5

Hot Hot Cold Cold Cold

150 200 40 110 110

60 100 130 130 180

200 100 120 140 60

Figure 7.29 Position of the composite curves at a pinch temperature difference of ∆TP = 10 ◦C. Heat can be internally recovered in the overlapping section (approximately 240 MW). Utility heats required are Q˙ H = 80 MW and Q˙ C = 60 MW.

7.4 DESIGN OF HEAT EXCHANGER NETWORKS

395

Figure 7.30 Determination of the optimal temperature levels of cold and hot utilities.

˙ )-diagram till the minimum temperature along the cold composite curve in the (T , H difference ∆TP between cold and hot composite curve is reached; see Figure 7.30. The value of the temperature level TC is determined easiest by shifting the lower part of the hot composite curve vertically downward by the distance ∆TP (here 10 ◦C). The lower part of the cold composite curve is shifted horizontally to the left by the distance of Q˙ C (here 60 MW). The point of intersection of these two auxiliary lines gives the value of TC (here 65 ◦C) in Figure 7.30. An analogous approach allows determination of the lowest value for temperature TH of the hot utility Q˙ H . The upper part of the cold composite curve is shifted vertically upward by the distance ∆TP (here 10 ◦C). The upper part of the hot composite curve is horizontally shifted to the right by the distance Q˙ H (here 80 MW). The intersection of both auxiliary lines gives the value of TH (here 134 ◦C). By implementation of the hot and cold utilities at their optimum temperature levels, the final courses of both composite curves result. They are seen in Figure 7.30. A different approach for determination of the optimum temperature levels of external duties has been proposed by Linnhoff and Flower 1978. He determines the grand composite curve in an auxiliary diagram; see Figure 7.31. First, the hot composite curve is shifted downward in vertical direction by the distance ∆TP /2. Analogously, the cold composite curve is shifted upward by the same distance till both curves touch. Then, the horizontal distances between the composite curves are ˙ )-diagram. The resulting line is referred to as the plotted in a separate (T ∗ , ∆H grand composite curve. The external heat stream Q˙ H has to be supplied above the ∗ pinch at the lowest temperature TH where the abscissa of the grand composite curve

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7 ENERGY ECONOMIZATION IN DISTILLATION

Figure 7.31 Construction scheme of the grand composite curve. The composite curves of Figure 7.29 are shifted vertically upward and downward, respectively, until they touch. The horizontal distances form the grand composite curve (right-hand side) from which the temperature levels TH and TC are determined.

reaches the value of Q˙ H . An analogous procedure yields the temperature TC∗ . Since the ordinate of the grand composite diagram is T ∗ , and not T , the temperature levels need to be corrected by ∗ TH = TH + 0.5 · ∆TP

and

TC = TC∗ − 0.5 · ∆TP .

(7.23)

The same values for the level of hot and cold utilities result from Figures 7.30 and 7.31. The approach of Figure 7.30 is, however, simpler and easier to understand. The optimum heat exchanger network is designed from a temperature–enthalpy diagram in which the utilities Q˙ H and Q˙ C are implemented in the hot and cold composite curve, respectively; see Figure 7.32. Here, the cold and the hot composite curves have the same length, and, generally, the minimum temperature difference ∆TP exists several times. This plot contains all information necessary for the design of the network presented in Figure 7.33. Each break of one of the composite curves is the end of a heat exchanger and the begin of the next one because at this point a stream is, in most cases, added or removed. An exception of this rule are processes with evaporation and condensation of pure substances. All temperatures and enthalpy differences necessary for heat exchanger dimensioning can be taken from the diagram of Figure 7.33. The characteristic values of the heat exchangers are listed in Table 7.3. For the very simple example with only 5 streams as many as 9 heat exchangers are required, most

7.4 DESIGN OF HEAT EXCHANGER NETWORKS

397

Figure 7.32 Temperature–enthalpy plot of streams and utilities. This diagram contains all information necessary for developing the optimum heat exchanger network. Nine heat exchangers are required. Their duties and mean temperature differences are listed in Table 7.3.

of them are multiple path heat exchangers. Such complex heat exchanger networks are difficult to realize in technical plants. However, there are examples for such networks, especially in cryogenic processes (see Figure 8.9). In most cases the optimum network cannot be realized and, in turn, needs to be simplified. For modifying the heat exchanger network, a simpler and more abstract graphical presentation of the network is convenient. All streams are represented by equally long, parallel horizontal lines with temperatures increasing from the left to the right. The cold streams are arranged below the hot streams. Heat exchangers are symbolized by circles that are connected by vertical lines between the streams involved. The external utilities are represented by circles also. The network of Figure 7.33 is shown in this more abstract form in Figure 7.34. This scheme of the network is very clear and allows easy modification. 7.4.2

Modifying the Optimum Heat Exchanger Network

For modifying the network the pinch principles developed by Linnhoff and Hindmarsh 1983 should be applied. The network can be considered as two separate systems, one system above the pinch temperature and the other one below the pinch temperature. The system above the pinch requires only external heat and is, therefore, a heat sink. The system below the pinch has heat to reject and is, consequently,

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7 ENERGY ECONOMIZATION IN DISTILLATION

Figure 7.33 Flow sheet of the optimal heat exchanger network with temperatures of the cold and hot streams of each heat exchanger. Most of the nine heat exchangers are multiple path exchangers.

Figure 7.34 Grid representation of the heat exchanger network of Figure 7.33. In this more abstract form, all streams are represented by horizontal lines of equal length. This grid presentation allows easy modification of the network.

399

7.4 DESIGN OF HEAT EXCHANGER NETWORKS

Table 7.3 Heat duties and mean logarithmic temperature differences of the optimized heat exchanger network seen in Figure 7.33.

Heat exchanger

Heat duty in MW

Mean logarithmic temperature difference in ◦C

E-1 E-2 E-3 E-4 E-5 E-6 E-7 E-8 E-9

34 56 4 61 43 80 52 7 43

14.4 19.9 36.0 20.6 13.6 13.6 14.8 24.3 23.7

a heat source. Hence, the following rules can be laid down [Linnhoff and Sahde 1989]: • Heat must not be transferred across the pinch. • There must be no external cooling above the pinch. • There must be no external heating below the pinch. One of the consequences of these rules is, for instance, that a heat pump has to be integrated into a process so that the heat is pumped across the pinch [Townsend and Linnhoff 1983] since it makes only sense to pump heat from a heat source to a heat sink. Many heat pumps installed in industrial plants violate this rule since only a single unit and not the entire process has been considered in the design procedure. Modifications of the optimum heat exchanger network are often prompted by the large number of heat exchangers required in such a network. The number can often be reduced by variation of the temperature levels of hot and cold utilities. From Figure 7.32, it follows, for instance, that the exchangers E-2 and E-3 can be united by simply decreasing the level of outside cooling from 65 ◦C to approximately 62 ◦C. The heat exchanger E-7 can be discarded by increasing the level of external heating from 134 to 150 ◦C. By these modifications the temperature–enthalpy diagram of Figure 7.35 results. The modified network is schematically depicted in Figure 7.36. Only 7 heat exchangers are required after changing the temperature levels of the hot and cold utilities. The feasibility of both measures depends on the temperature levels of external heat sink and heat source. The simplification of the heat exchanger network must, however, be paid by higher exergy losses of the process.

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7 ENERGY ECONOMIZATION IN DISTILLATION

Figure 7.35 Simplification of the heat exchanger network by decreasing the temperature level of cold utility and increasing the temperature level of hot utility to reduce the number of heat exchangers required.

Figure 7.36 Grid presentation of the simplified heat exchanger network of Figure 7.35. Only seven heat exchangers are required.

7.4 DESIGN OF HEAT EXCHANGER NETWORKS

7.4.3

401

Dual Flow Heat Exchanger Networks

One of the essential drawbacks of the optimum heat exchanger network of Figure 7.34 is the use of multiple path heat exchangers with up to five paths (see heat exchangers E-5 and E-7 in Figure 7.34). The standard design prefers dual path heat exchangers because they are cheaper and easier to install and to operate. For developing a heat exchanger network with dual flow heat exchangers, only the rules of Linnhoff and Hindmarsh 1983 are helpful. To avoid heat transfer across the pinch, the networks of exchangers below and above the pinch should be developed separately. First, the network below the pinch is developed because it is the simpler one in the example. In principle, the combined streams of the composite curves of Figure 7.32 must be replaced by an alternating sequence of small elements of the individual streams. The higher the temperature difference between hot and cold composite curve in Figure 7.32, the longer the elements of the individual streams can be. In the example the hot composite curve is a combination of the streams no. 1 ˙ )-diagram of the and 2 between 100 ◦C and the pinch (120 ◦C). Hence, in the (T , H individual streams in Figure 7.37 (left-hand side), only the hot streams no. 1 and 2 appear in this temperature range. The sequence of the streams is ordered so that the distance between hot and cold streams never falls below the value ∆TP (here 10 ◦C) at the pinch. The network of dual path heat exchangers above the pinch is developed from Figure 7.37 (right-hand side). The cold composite curve between pinch and 130 ◦C is a combination of streams no. 3, 4, and 5. Hence, the T /H˙ -curves of these streams are arranged in consecutive elements. The hot composite curve is combined from streams no. 1 and 2 in the temperature range from the pinch (120 ◦C) to 150 ◦C. Since the temperature differences between hot and cold composite curves in Figure 7.32 are very small, the T /H˙ -curves of the individual streams have to be alternatively arranged in small consecutive elements. The hot stream no. 1, for instance, is divided in two parts that are separated by the hot utility Q˙ H . The cold streams no. 4 and 5 are also separated in two elements that are arranged in mixed order with the stream no. 3 between them. ˙ )-diagram of the individual streams has the same minimum The resulting (T , H temperature difference ∆TP as the overall diagram of Figure 7.32. However, this minimum temperature difference exists more often. From the diagrams of Figure 7.37, the heat exchanger network that consists of dual flow heat exchangers only is easily developed. The network is seen in Figure 7.38. Now, the number of heat exchangers is 10 instead of 9, but only dual flow heat exchangers are involved. 7.4.4

Process Modifications

The energy requirement of separation processes can be reduced drastically by process modifications. The shape of the composite curves is, for instance, significantly changed by variation of the pressures of evaporators and condensers. Such modifications result in a different location of the pinch. They can manage to increase the

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7 ENERGY ECONOMIZATION IN DISTILLATION

Figure 7.37 Temperature–enthalpy diagram of a network with dual path heat exchangers only. The diagram is divided into sections below the pinch and above the pinch in order to safely avoid heat transfer across the pinch.

Figure 7.38 Grid presentation of the dual flow heat exchanger network of Figure 7.37.

REFERENCES

403

overlapping region in the enthalpy/temperature plot of Figure 7.29. In the optimum version the hot and cold composite curves do no longer form a sharp pinch. Both curves should have approximately the same distance everywhere. At a chemical site in Germany, 130 processes were examined in a large energysaving program [Körner 1988]. The heat exchanger networks of 70 out of these 130 processes were modified, resulting in a reduction of steam consumption of up to 700 × 103 kg/h. Thus, investment of a new 300 million dollar power station has been avoided. The study revealed that the potentials for energy savings lie primarily in process modifications, e.g. change of operation pressure, and not in mere stream matching. However, for a systematic approach to advantageous process modifications, the pinch technology as outlined in this section is the appropriate engineering tool.

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8 Industrial Distillation Processes In this chapter some typical examples of industrial distillation processes are presented. However, a few comments on additional non-thermodynamic constraints for industrial processes have to be made first.

8.1

Constraints for Industrial Distillation Processes

Industrial distillation processes underlie some restrictions due to operability of the units, to economic conditions, and to environmental constraints. Some aspects are dealt with in this chapter. 8.1.1

Feasible Temperatures

In industrial distillation processes there exists an upper limit for feasible operating temperatures. One reason is the thermal stability of the species in the mixtures to be separated. Many substances decompose gradually at higher temperatures. Some species are not even stable at their normal boiling points. Maximum allowable temperatures for selected substances are listed in Table 8.1. Thermal stability depends on time of exposure as well as on temperature level. The data in Table 8.1 are approximate values for typical distillation conditions. A second reason for a maximum temperature limitation is the means of heat supply in industrial processes. Except for a few very special cases, the heat required in distillation columns is supplied by condensing steam. The pressure of the steam available sets an upper limit on temperature levels that can be achieved on the process side of the reboilers and heaters. A practical temperature difference for heat transfer is in the range of about 10 – 30 ◦C. In special cases, e.g. in crude oil distillation or sulfuric acid concentration, higher temperatures are realized by heating with hot oil or with flue gases. Maximum feasible temperatures are as high as 300 and 400 ◦C, respectively. Electric heating of distillation columns is normally not practical. The maximum product side temperatures and the relative costs of some important heating media for distillation columns are listed in Table 8.2. The values of relative costs are very vague because they are not correctly evaluated in most companies [Cooper

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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Table 8.1 Maximum allowable operating temperatures of distillation columns to avoid thermal degradation or polymerization.

Maximum operation temperature in ◦C

Substance

170

Sulfuric acid H2 SO4 Hydrogen peroxide Crude oil Reduced crude oil Styrene Tall oil Nitric acid

60–80 350 420 100 260 Below normal

Triethylene glycol Acrylic acid

220 70–90

Operating pressure in bar

Comments

1

Formation of SO3

1 2.1 0.013

Explosive vapor Degradation Degradation Polymerization Degradation Formation of NOx that can be removed by air stripping

5 × 10−3

Polymerization

Table 8.2 Maximum product side temperatures and relative costs of some heating media for distillation columns.

Medium

Product side temperatures in ◦C

Steam - Exhaust - Low pressure (7 bar) - Medium pressure (15 bar) - High pressure (30 bar) - High pressure (80 bar)

90 140 170 200 260

Hot oil Hot oil (synthetic)

300 400

Flue gas

450

Relative costs in %

50 100 130 150

8.1 CONSTRAINTS FOR INDUSTRIAL DISTILLATION PROCESSES

409

1989]. In particular, there is a tendency to under-evaluate high-pressure steam. A lower limit of feasible temperatures in distillation columns is dictated by the temperature of the coolant for the overhead condenser. In most cases water is used as coolant, and practical temperature differences in the condenser lead to a minimum temperature in the column as low as 40 – 50 ◦C. With increasing tendency, air is used as coolant, which leads to minimum temperatures of about 60 – 70 ◦C. Seldom used coolants are chilled water or evaporating refrigerants such as ammonia, propane, or nitrogen. Even when available, such coolants are comparatively expensive, as may be noted from Table 8.3. Table 8.3 Minimum product side temperatures and relative costs of some cooling media for distillation columns.

Medium

Relative costs in %

Air

50–70

60

Cooling tower water Chilled water

40–50 20–30

100 150

10 −5 −30

300 400 500

0

300

Ammonia

Sole Propane/propylene Ethane/ethylene Methane 8.1.2

Product side temperatures in ◦C

30 75 150

Feasible Pressures

The temperature constraints set by the heating and cooling media can be met by selecting proper operating pressures or the addition of low boiling components. To decrease operating temperatures, the separation may require the use of a vacuum. It is technically feasible to operate distillation columns at a pressure as low as 0.06 – 0.08 bar with a single-stage rotating vacuum pump and as low as about 0.013 bar with a double-stage vacuum pump. Vacua higher than (2 – 5) × 10−3 bar are very seldom used in distillation columns [Strigle 1987] because of the high operating and capital costs of the vacuum-producing equipment. Instead of operating with vacuum, a low-boiling substance can be added to decrease the boiling temperature of the material to be distilled. Often used additives are inert gases (air, nitrogen) and, especially in organic systems, steam since it can be easily removed after distillate condensation (see Section 8.2.6).

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8 INDUSTRIAL DISTILLATION PROCESSES

To increase the boiling temperatures of low boiling mixtures, the column can be operated at higher pressures. However, the higher the pressure, the smaller is, in most cases, the concentration difference between gas and liquid. An upper limit for the operating pressure lies in the range of the critical pressures of the components. Very low boiling hydrocarbons such as methane are usually distilled at pressures of about 70 % of their critical ones. Ethylene and ethane are typically distilled at 40 – 55 % of their critical pressures, while for propylene and propane 35 – 50 % of the critical pressures are used [Strigle 1987]. The increase of operating pressure is a very fundamental problem in industrial processes since gas and liquid streams have to be compressed and pumped, respectively, to the higher pressure. Whenever possible, a pressure increase should be established in the liquid phase instead of in the vapor phase since power consumption is much lower then. A comparison of power consumption per mass flow is presented in Figure 8.1 for air and water. The diagram clearly shows that power consumption for compression of air is by 2 – 3 orders of magnitude higher than for pumping the same mass stream of water. This large difference results, in essence, from the density difference of air and water, and exists, therefore, in all systems.

Figure 8.1 Comparison of power consumption for compressing of air and pumping of water. Power consumption (kW/kg) for compressing a gas is much higher than for pumping the same mass stream of liquid due to the large density difference of gases and liquids.

8.1.3

Feasible Dimensions of Columns

The feasible number of equilibrium stages of industrial columns is limited, also. As can be seen in Table 8.4, more than 100 stages are very seldom installed in commer-

411

8.1 CONSTRAINTS FOR INDUSTRIAL DISTILLATION PROCESSES

Table 8.4 Key components for distillation processes of industrial importance and typical number of equilibrium stages.

Key components

Typical number of trays

Nitrogen/oxygen Argon/oxygen

100 150

Hydrocarbon systems: Crude oil (atmospheric tower inclusive side columns) Ethylene/ethane Propylene/propane Propyne/1,3-butadiene 1,3-Butadiene/vinyl acetylene Benzene/toluene Benzene/ethyl benzene Benzene/diethyl benzene Toluene/ethyl benzene Toluene/xylenes Ethyl benzene/styrene o-Xylene/m-xylene

51 73 138 40 130 34, 53 20 50 28 45 70 130

Organic systems: Methanol/formaldehyde Dichloroethane/trichloroethane Acetic acid/acetic anhydride Acetic anhydride/ethylene diacetate Vinyl acetate/ethyl acetate Ethylene glycol/diethylene glycol Cumene/phenol Phenol/acetophenone

23 30 50 32 90 16 38 39, 54

Aqueous systems: HCN/water Acetic acid/water Methanol/water Ethanol/water Isopropanol/water Vinyl acetate/water Ethylene oxide/water Ethylene glycol/water Source: Modified after Perry et al. 1984.

15 40 60 60 12 35 50 16

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8 INDUSTRIAL DISTILLATION PROCESSES

cial distillation columns. The largest diameters of industrial distillation columns are as high as 13 – 15 m. A very important problem in column design is a proper selection of the construction material. For very corrosive systems, e.g. acids, pyrex glass has to be used. In this cases the diameter of the column is restricted to 1 m or less, and the maximal operating pressure is 1 bar or less.

8.2

Fractionation of Binary Mixtures

A few examples of industrial binary distillation processes are given in the following sections. The examples have been selected to demonstrate the importance of the constraints discussed in Section 8.1. 8.2.1

Recycling of Diluted Sulfuric Acid

Sulfuric acid is one of the most important basic substances in the chemical industry. It is used in many processes to enhance a chemical reaction and, in most cases, leaves the process as a diluted impure acid. The boiling and dew temperatures of H2 O/H2 SO4 mixtures at a pressure of 1.0 bar are shown in Figure 8.2. A maximum azeotrope exists at a weight fraction of 98.4 % H2 SO4 . Up to a liquid concentration of about 75 wt% H2 SO4 , the vapor contains only water. The vapor pressures of H2 O/H2 SO4 mixtures at various temperatures are shown in Figure 8.3 [Perry et al. 1984]. This plot is very helpful for defining conditions of technical concentration processes. If cooling water is used for condensing the overhead product, the minimum overhead temperature is 40 – 50 ◦C (see Table 8.3). As nearly pure water is the overhead product, the minimum operating pressure is about 0.12 bar. The maximum bottom concentration of the acid at this pressure depends on the heating medium. With medium pressure steam (15 bar, 200 ◦C), the maximum process temperature is as high as 170 ◦C, which results in an acid concentration of about 86 wt%. Because of these temperature constraints, the feasible sulfuric acid concentrations in an industrial plant are by far lower than the thermodynamic limit set by the azeotrope. A two-pressure process for sulfuric acid concentration is seen in Figure 8.4. Initially, the diluted acid is concentrated up to 72 wt% at normal pressure by single-stage distillation. The vapor does not contain any H2 SO4 (or SO3 ) under these conditions. In a second step the acid is further concentrated up to 86 wt% at a vacuum of 0.12 bar. As the vapor contains some H2 SO4 (in the form of SO3 ), the separation is performed in a multistage distillation column. The SO3 in the vapor is absorbed by the sulfuric acid entering at the overhead of the column. The problem of recycling the H2 SO4 is not only the mere up-concentrating of H2 SO4 but also the removal of impurities. Organic substances are decomposed during up-concentrating and stripped off from the acid by the vapor. Inorganic substances often precipitate during the concentration step, which happens preferably in the concentration range of about 60 wt% H2 SO4 .

8.2 FRACTIONATION OF BINARY MIXTURES

Figure 8.2 Vapor–liquid equilibrium and dew and boiling temperatures of H2 O/H2 SO4 mixture at p = 1 bar [Perry et al. 1984]. There exists a maximum azeotrope at high H2 SO4 concentration.

Figure 8.3 Vapor pressure over a H2 O/H2 SO4 mixture [Perry et al. 1984]. The parameter is the H2 SO4 concentration in weight percent.

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.4 Process for concentrating diluted sulfuric acid.

8.2.2

Ammonia Recovery from Wastewater

Ammonia is a toxic chemical that is often present in wastewater. It has to be removed and recovered as a pure liquid. The content of NH3 in wastewater is typically as low as 1 wt%. As may be seen from the diagram in Figure 8.5 [Rizvi and Heidemann 1987], the dew point of ammonia is very low (−33.4 ◦C) at a pressure of 1 bar. To increase the dew point to 50 ◦C, the minimum temperature of a water-cooled condenser, the pressure has to be increased to about 20 bar. At this pressure the boiling point of water is as high as 212 ◦C, which is too high for a reboiler heated by steam. Therefore, the separation requires two steps as shown in Figure 8.6 [Wunder 1990]. In the first step most of the water is removed as bottoms from column C-1, which is operated at normal pressure. The distillate should have a boiling point not lower than 45 ◦C. From Figure 8.5 the maximum NH3 content of the distillate can be determined to be 20 wt%. The liquid distillate is pumped to 20 bar and fed into ˙ 2 is pure liquid ammonia. Considering the temperature column C-2. The overhead D ◦ limitation of 180 C in the reboiler, the minimum NH3 content of the bottoms is as low as 10 wt%. Therefore, the bottoms have to be recycled into column C-1. As water is the bottom fraction of column C-1, the reboiler can be discarded by providing heating by exhaust steam.

8.2 FRACTIONATION OF BINARY MIXTURES

415

Figure 8.5 Vapor pressure of ammonia/water solutions versus temperature [Rizvi and Heidemann 1987]. The parameter is the NH3 concentration in the liquid in weight percent.

Figure 8.6 Process for recovering pure ammonia from wastewater [Wunder 1990].

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Example 8.1: Ammonia Recovery

˙ and B2 ˙ of the process of Figure 8.6 for an Find the flow rates of the streams D1 ammonia concentration of 1 wt% in the feed. Solution:

˙ = F˙ · wLF = F˙ · 0.01 D2 ˙ = D2 ˙ + B2 ˙ The material balance over column C-2 yields D1 ˙ · 0.8 = B2 ˙ · 0.9 A water balance over column C-2 gives D1 A material balance of the unit gives

From Eqs. (1) – (3), it follows that

˙ = 0.08 · F˙ and B2 ˙ = 0.09 · F˙ D1

(1) (2) (3) (4)

˙ the feed to column C-1 is increased by 8 %. The feed to column C-2 By recycling B2 is as low as 9 % of the feed F˙ to the process.

8.2.3

Hydrogen Chloride Recovery from Inert Gases

Figure 8.7 presents a very interesting process for the recovery of hydrogen chloride, HCl, from inert gases (e.g. Stichlmair 1979). This process can be applied to HClrich gases with no or very low water content. The normal boiling point of HCl is −85 ◦C. To avoid a cryogenic process, the HCl is first absorbed in water and then recovered from the liquid solution by distillation. The solubility of HCl in water is very high, and a hydrochloric acid concentration up to 31 wt% can be achieved in the absorber C-1. A large heat of absorption is set free in the absorber, causing a rise of temperature. At higher temperature the water becomes volatile, and mass transfer of water from liquid to gas phase takes place, too. At the overhead of the absorber, the water is condensed and recycled into the column. The hydrochloric acid formed in the absorber is fed via a heat exchanger into distillation column C-2 to be separated into pure HCl gas as overhead and azeotropic acid (∼ 21 wt%) as bottom product, which is recycled into the lower section of the absorber C-1. Because water is continuously fed into the process, water or a water-rich fraction has to be withdrawn. In most cases the bottom product from the absorber is partly withdrawn as side products, because it meets the standards of commercial hydrochloric acid. The recovery of pure HCl gas is then as low as 45 %. The (wG , wL )-diagram of the process is shown in Figure 8.8. The only effect of the inert gas in the absorber is the reduction of the partial pressures of water and HCl. Consequently, the inert gas needs not be accounted for in the calculations. Only HCl and water have to be considered at a pressure (pt − pinert ). The boiling point of the liquid has to be determined for this reduced pressure, like in a vacuum column. As the heat of HCl absorption is nearly equal to the heat of water vaporization, the flow

8.2 FRACTIONATION OF BINARY MIXTURES

Figure 8.7 Process for recovery of HCl from waste inert gases [Stichlmair 1979].

Figure 8.8 McCabe–Thiele diagram for the process in Figure 8.7.

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rate of the gas (inert gas neglected) is nearly constant, and the equilibrium line for HCl water can be determined for constant pressure (pt −pinert ). As long as the partial pressure of HCl in the feed is higher than 0.1 bar, the equilibrium line is nearly the same in the absorption and in the distillation step. The process can be modified by withdrawal of the bottoms of the distillation col˙ has a lower content of HCl and a higher umn C-2 instead of the absorber. Since B2 ˙ content of water, the yield of pure HCl gas increases up to 70 %. But the bottoms B2 are less worthy than the 31 wt% side stream.

Example 8.2: Hydrogen Chloride Recovery Find the yield of pure HCl gas and the flow rates of the streams of the process in Figure 8.7. Solution:

The slope of the operating line in the upper section of the absorber can be determined ˙ G˙ HCl = 2 (inert gas neglected). In the lower section of the in Figure 8.8 to be L/ ˙ G˙ HCl = 4.7. The flow rate of water into the process is absorber, we find L/

L˙ H2 O = G˙ HCl with G˙ HCl = F˙ · wG,HCl .

(1)

Consequently, half of the liquid flow in the absorber is provided by the condenser. The rest is tap water. In the side stream S˙ of hydrochloric acid, all the water has to be withdrawn. A water balance over the process yields From (1), it follows that

S˙ · (1 − 0.31) = L˙ H2 O S˙ = 1.45 · G˙ HCl

˙ · 1 + S˙ · 0.31 A HCl balance over the process yields G˙ HCl = D2 ˙ = 0.55 · G˙ HCl . With Eqs. (3) and (4), it follows that D2

(2) (3) (4) (5)

The yield of pure HCl is as low as 55 wt%. The rest of the HCl in the feed gas is in the side stream S˙ of hydrochloric acid. The feed into column C-2 is F˙ 2 = 4.7 · G˙ HCl − 1.45 · G˙ HCl = 3.25 · G˙ HCl ˙ is: B2 ˙ = 3.25 · G˙ HCl − 0.45 · G˙ HCl = 2.8 · G˙ HCl . The bottoms B2

8.2.4

(6) (7)

Linde Process for Air Separation

The separation of air into nitrogen and oxygen (and often argon) is performed on a large scale in cryogenic processes. In 1980 the production of oxygen was as high

8.2 FRACTIONATION OF BINARY MIXTURES

419

as 112.2 × 106 t/a making air separation the most important separation process, second only to crude oil refination [Baldus et al. 1983]. Figure 8.9 shows a simplified flow sheet of the Linde process [Baldus et al. 1983]. First, air is compressed to about 6 bar and cooled down against the products nitrogen and oxygen. Water, carbon dioxide, and hydrocarbons are removed from the air by reversing molecular sieve adsorbers. The air is further cooled down till about 20 % of the feed is liquefied. The air enters the so-called Linde double column to be split into nitrogen and oxygen. Both products are warmed up in the heat exchangers E-1, E-2, and E-3 against the incoming air. The low temperatures required for the process are generated by expanding some nitrogen (approximately 12 % of the feed) in an expansion machine from 5.7 to 1.3 bar.

Figure 8.9 Linde process for air separation.

A very special feature of the process is the Linde double column. It is a distillation column without a reboiler at the bottom and without a condenser at the head. The function of the column is explained best in Figure 8.10. The separation of the binary mixture is performed in two steps. In the first step some pure nitrogen is separated from the air. As most of the feed is vaporous, only a rectifying section with an overhead condenser is required. The distillate of the column C-1 is used as reflux in column C-2. The bottoms of C-1 are fed into column C-2 to be split into nitrogen and oxygen. Since the reflux is provided by the first column C-1, no overhead condenser

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.10 Recovery of oxygen and nitrogen from air by distillation. A) Two separate columns C-1 and C-2. B) Linde double column.

Figure 8.11 Determination of the operating pressure for thermally coupling of the columns C-1 and C-2 of the process in Figure 8.10.

8.2 FRACTIONATION OF BINARY MIXTURES

421

is required. Column C-1 has only a condenser and column C-2 only a reboiler. Condenser and reboiler may be thermally coupled (see Section 7.3.3). The heat removed in the condenser of C-1 is used for heating the reboiler of column C-2. As the temperature at the head of column C-1 is lower than at the bottom of column C-1, the thermal coupling is only possible if column C-1 is operated at a higher pressure than column C-2. The minimum pressure of column C-2 is 1.3 bar to overcome the pressure losses in the heat exchangers E-1, E-2, and E-3 in the product lines. At a temperature difference of 2.5 K, which is required for heat transfer, the pressure of column C-1 has to be 5.7 bar (see Figure 8.11). In the Linde double column, the column C-2 is installed directly atop of column C-1 so that condenser and reboiler are combined in a single heat exchanger. Air separation by cryogenic distillation is a highly sophisticated process. The energy required for the separation is provided by the compression of the feed. The low temperatures are generated by the Joule–Thomson effect, which utilizes the fact that the enthalpy of low pressure gas is higher than that of high pressure gas. In modern air separation units, especially when parts of the products are withdrawn as cryogenic liquids, the expansion valve is replaced by an expansion machine. The specific energy requirement of modern cryogenic air separation is as low as 1.9 kWh/kmolair with expansion machine (but gaseous products) [Streich 1975]. By replacing the sieve trays in the column (typically 40 trays in the high pressure section and 60 trays in the low pressure section) by structured packings (see Section 8.3) the energy requirement of the process is reduced by about 8 % [Agrawal et al. 1992]. 8.2.5

Process Water Purification

Many chemical processes produce aqueous side streams polluted by organic substances with limited solubility in water. Such mixtures often form a heteroazeotrope, which is always a low boiling one. Consequently, these organics can be removed as overheads of a distillation column, even when they are higher boiling than water. The removal of toluene from water is considered as an example. It is assumed that water is saturated with toluene at a temperature of 80 ◦C. The toluene concentration in water is as high as 1.6 mol%. The boiling point of a two-liquid-phase toluene/water mixture is 84.1 ◦C. The vapor pressure of water at 84.1 ◦C is 0.55 bar; the vapor pressure of toluene is 0.45 bar (adding up to 1 bar; see Figure 8.17). In the region of immiscibility, the equilibrium is independent of liquid composition, as is seen in Figure 8.12. The azeotrope has a water content of 0.55 mol%. Important for the separation is that part of the equilibrium line that lies close by pure water, especially the region of toluene solubility in water. This region is depicted separately in Figure 8.12. The equilibrium line runs below the diagonal of the McCabe–Thiele diagram in this concentration range. Thus, the concentration of water (low boiler) is lower in the vapor than in the liquid phase. Toluene enriches up to 45 % in the vapor phase. After condensation two liquid phases are formed. The water-rich phase is recycled as reflux into the column; see Figure 8.13. The toluene-rich phase contains about 0.02 mol% water [Sorensen and Arlt 1979].

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Figure 8.12 McCabe–Thiele diagram for the separation of toluene from water.

Figure 8.13 Process for separating high boiling organics from water. In the region of immiscibility in the liquid phase, even high boiling organics go overhead.

8.2 FRACTIONATION OF BINARY MIXTURES

423

It is advantageous to heat the column by live steam because cheap exhaust steam can be utilized and the reboiler can be omitted. It must be recognized, however, that the end point of the operating line does not lie at the diagonal in this case, but rather at yH2 O = 1 and x = xB , as is shown in Figure 8.12. The process of Figure 8.13 is a standard process very often used for the separation of organic components from water. The process can be applied if the main organic component is at least partly immiscible with water. The amount of steam required is typically as low as 5 – 10 % of the liquid feed. If the organic phase contains too much water, it can be purified by a similar separation step. An example of such a process is dealt with in Section 8.4. In many cases the organic phase contains light components that are fully soluble in water (e.g. methanol or acetone). These components can be separated from the water as overhead fraction of a previous distillation step; see Figure 8.14. An example of such a system is the acetone/water/toluene mixture. It is assumed that the mixture is ˙ . saturated with toluene. Acetone is separated in column C-1 as overhead fraction D1 ˙ The bottom fraction B1 lies in the region of immiscibility. This fraction is split in

Figure 8.14 Process for separating low boiling and high boiling organics from water (see also Figure 8.28).

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Figure 8.15 Direct coupling of columns of the process in Figure 8.14 (see Section 7.3.1).

Figure 8.16 Process for separating low and high boiling organics from water with direct coupling of columns.

8.2 FRACTIONATION OF BINARY MIXTURES

425

the decanter S-1 into an organic phase and a water phase. The water is purified by steam stripping in analogy to the process seen in Figure 8.13. In the process of Figure 8.14, vapor is condensed at the overhead of column C-2, and vapor of the same composition is generated in the reboiler of column C-1. The process can be improved by material coupling of columns C-1 and C-2 (see Section 7.3.1). The vapor at the head of column C-2 is fed directly into column C-1 whereby the overhead condenser of C-2 and the reboiler of C-1 can be discarded (Figure 8.15). In the combined column the overhead product is acetone, and the bottom product is pure water. The heavy organics (e.g. toluene) concentrate in the middle of the column, forming two liquid phases there. The liquid is drawn off a few stages below the feed point and separated in a decanter. The water phase is recycled into the column. The organic phase is the side product. Depending on the solubility of water in the organic phase, the water content might be too high in many cases. The water can be easily removed by a second column (Figure 8.16). The overhead is recycled into the main column. The bottoms is pure organic product. 8.2.6

Steam Distillation

Steam distillation is a special case of distillation processes often applied to high boiling organic systems immiscible with water. As has been shown in Chapter 2, the existence of a region of immiscibility always results in a decrease of boiling temperature. The boiling point of such a mixture is always lower than the boiling temperatures of the components. Even very high boiling organic systems have, in a mixture with water, a boiling temperature lower than that of pure water. Thus, the problem of thermal degradation of organic compounds can often be circumvented by steam distillation. The boiling temperatures of substances immiscible with water can easily be determined by the diagram in Figure 8.17. The difference of system pressure pt and vapor pressure p0H2 O of water is plotted versus the temperature (here pt = 1 bar and pt = 0.1 bar). The intersection of the (pt − p0H2 O )-line with the vapor pressure curve of the organic component gives the boiling point of the mixture. At the point of intersection, the vapor pressure p0i of the organic substance allows the calculation of the vapor composition by yi∗ = p0i /pt . As can be seen from Figure 8.17, even very high boiling organic substances boil lower than the boiling point of water if liquid water is present. The schematic of a batch steam distillation process is shown in Figure 8.18 [Sattler 1988]. The steam is directly blown into the liquid in the still and partly condensed. As soon as the boiling point of the mixture is reached, the volatile organic components are stripped off. The vapor is condensed and two liquid phases are formed, which are split in a decanter. Water, the lower phase in the decanter, is drawn off. It contains at least small amounts of the organic components. This formation of polluted wastewater is the main disadvantage of steam distillation. The flow sheet of a continuous process of steam distillation is presented in Figure 8.19. In order to minimize the amount of live steam, which will be polluted, a reboiler with indirect heating is very often used. In addition, countercurrent flow of

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Figure 8.17 Vapor pressure curves for some high boiling organic compounds that are immiscible with water. The vapor pressure of water p0H2 O is presented as difference to the total pressure (pt − p0H2 O ). The intersection of the (pt − p0H2 O )-line with the vapor pressure curves of the organic compounds gives the boiling point of the mixture.

Figure 8.18 Schematic of batch steam distillation.

8.2 FRACTIONATION OF BINARY MIXTURES

427

Figure 8.19 Schematic of continuous multistage steam distillation.

Figure 8.20 Process for deodorizing of crude ester [Johannisbauer and Jeromin 1992].

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8 INDUSTRIAL DISTILLATION PROCESSES

liquid and vapor in a column increases mass transfer efficiency. Superheated steam is also very often used. Steam distillation is a very simple but effective means for purification of high boiling organic substances, e.g. essential oils for perfumery or long-chain fatty acids and for deodorization or decolorization of fats and oil [Ellerbe 1979]. Figure 8.20 shows a process for deodorization of esters by steam distillation. The volatile impurities in the crude ester have a concentration of typically less than 1 %. The amount of steam required for stripping is about 5 – 10 % of the crude feed [Johannisbauer and Jeromin 1992]. An important field of steam distillation is the raffination of palm oil [Drew and Propst 1981]. A very modern version of a process for deodorization and deacidification of palm oil is presented in Figure 8.21 [Stage 1988]. Typical operating conditions are a pressure of 2 – 3 Torr and a temperature of about 260 ◦C. The system is heated by hot oil. The reboiler is a bundle-type heat exchanger. The liquid is flowing inside the tubes as a thin film countercurrently to the steam. The residence time of the product is as low as about 10 minutes, thus avoiding thermal degradation. The countercurrent flow of liquid and steam reduces the amount of steam required for effective stripping. It is typical less than 1 wt% of the crude feed. The condensation of the vapor at the column head is performed by quenching with subcooled overhead product.

Figure 8.21 Process for refining palm oil [Stage 1988].

Steam stripping is also widely used in crude oil distillation (see Section 8.3.2.2). The advantage of steam distillation over conventional distillation is the low operation

8.3 FRACTIONATION OF MULTICOMPONENT ZEOTROPIC MIXTURES

429

temperature, which is essential for avoiding thermal degradation of the organic compounds, the use of low pressure steam (exhaust steam), and, in many cases, a simple device without a reboiler. The disadvantages of steam distillation are the high steam requirement, the generation of wastewater, and the risk of foam or emulsion formation.

8.3

Fractionation of Multicomponent Zeotropic Mixtures

Complete fractionation of a multicomponent liquid mixtures requires several distillation steps. As discussed in Chapter 4, a ternary mixture can be separated to only ˙ and a bottom product B˙ . Figure 4.30 a limited extent into an overhead product D shows the feasible range of compositions of the two fractions on a triangular concentration diagram. In special cases, pure overhead product a or pure bottom product c can be gained. The remaining binary fraction has to be separated into the pure substances in a second column. 8.3.1

Separation Paths

Figure 8.22 shows the low boiler path (a-path) and the high boiler path (c-path), respectively, for separating a zeotropic ternary mixture. In the a-path the light component a is separated first. The bottoms is fed into a second column, where the mixture b/c is split into pure substances. In the c-path the high boiler c is separated first, and the overhead fraction, mixture a/b, is vaporously fed into column C-2 for producing the substances a and b.

Figure 8.22 Paths for complete separation of a ternary mixture.

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Figure 8.23 Determination of number of separation paths of ternary, quaternary, and quinary mixtures. The brackets sign the two fractions of the first separation step. Several components can exist in both fractions.

In an alternative approach, the intermediate boiler path (a/c-path), the first column produces an overhead fraction without the high boiler c and a bottom fraction without the low boiler a. Two additional columns are required for separating the remaining binary fractions. The intermediate boiler b appears, in one case, as the bottom product and, in the other case, as the overhead product. As discussed in Section 7.2.3, this alternative process is energetically the most favorable one at higher feed concentrations of intermediate boiler b. For the complete separation of a four-component mixture, 22 different separation paths, shown in Figure 8.23, exist. Analogously, 719 different separation paths are feasible for the separation of a five-component mixture. Each of these separation paths has its own energy requirement that is significantly influenced by the concentration of the feed. Extensive studies are necessary to find the path with the lowest energy requirement. In practice, the separation sequence of multicomponent mixtures is often determined not by the energy requirement, but by other limiting conditions, e.g. stability or corrosiveness of individual substances, danger of explosion, toxicity, tendency to stick to column internals, etc. In all these cases, the most difficult substance should be separated first. Thus, the energy requirement is only one out of several aspects determining the choice of a separating sequence.

8.3 FRACTIONATION OF MULTICOMPONENT ZEOTROPIC MIXTURES

8.3.2

431

Processes with Side Columns

The processes shown in Figure 8.22, in particular the a-path and the c-path, are very often used in the process industry. However, these processes have some severe drawbacks as outlined in Chapter 7. Especially in sharp separations, the same fractionations are performed twice in different column sections. These twofold fractionations can be avoided by using side columns. Figure 8.25 depicts the principle of side columns for complete separation of a quaternary mixture by the high boiler path. In each column the actual heaviest component is separated as bottoms, first component d, then component c, and then component b. In the process modification with side columns, the upper section of column C-2 is combined with column C-10 ; the upper section of column C-3 is combined with the modified column C-10 . The gas and liquid load in column C-10 is highest in the upper section and – due to the side withdrawals of liquid and side feeds of vapor – lowest in the bottom section. 8.3.2.1

Argon Recovery from Air

An example for the use of a side column in an industrial distillation process is argon recovery in cryogenic air separation (see Section 8.2.4). Air contains about 1 mol% argon being the intermediate boiler in the nitrogen/argon/oxygen mixture. In the low pressure section of the Linde double column, shown in Figure 8.9, oxygen is the bottom, and nitrogen the overhead fraction. Argon concentrates in the mid-

Figure 8.24 Use of a rectifying side column for argon recovery in the Linde air separation process (see Figure 8.9).

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Figure 8.25 Use of side columns in the separation of a quaternary mixture. A) Conventional high boiler path. B) The shaded section of columns C-2 and C-3 are combined with column C-1. The reminders are stripping side columns.

8.3 FRACTIONATION OF MULTICOMPONENT ZEOTROPIC MIXTURES

433

dle section typically up to 8.5 mol% in the vapor phase. As shown in Figure 8.24 a vaporous side stream is withdrawn and fed into a rectifying side column. The overhead fraction is highly concentrated argon with a purity of about 96 mol%. The bottoms is recycled to the main column. The cooling medium of the side column’s condenser is the bottoms fraction of the high pressure column, after expansion to 1.3 bar. The liquid is rich in nitrogen and, in consequence, boils lower than argon. If high purity argon is required, the oxygen contained in the crude argon can be removed by catalytic combustion with hydrogen and condensation of the water [Baldus et al. 1983]. Application of structured packings with low pressure drop to the argon side column enables direct removal of high purity argon as overhead fraction, thus avoiding the subsequent purification of crude argon. Due to the unfavorable vapor–liquid equilibrium of argon and oxygen, up to 180 equilibrium stages are required in the side column. 8.3.2.2

Crude Oil Distillation

The first step in any petroleum refinery is the fractionation of crude oil into various fractions by distillation. These fractions may be products on their own or may be feedstocks for other refining or processing units [Watkins 1979; Meyers 1996]. Crude oil distillation is by far the most important distillation process worldwide. Crude oil consists of a large number of different hydrocarbons. The normal boiling points of the components range from −150 ◦C to more than 500 ◦C. This complex mixture has to be split into up to 10 fractions in a refinery. The fractions are not pure components but complex mixtures with about the same boiling point or boiling point range. The separation is performed in two distillation towers: the atmospheric tower and the vacuum tower. First, the oil is heated to the maximum temperature allowable to avoid thermal degradation and then fed to a fractionating tower, which operates slightly above atmospheric pressure. This tower is called the atmospheric tower. It yields several distillate products and a bottom product, which is the residual liquid material that could not be vaporized under the conditions of temperature and pressure existing in the atmospheric tower. This bottoms liquid is then reheated to its maximum allowable temperature (about 420 ◦C) and fed to a second fractionating tower, which operates at subatmospheric pressure (minimum overhead tower pressure about 0.013 bar). This tower is usually called the vacuum tower. Figure 8.26 shows the atmospheric tower with several stripping side columns of a petroleum distillation unit. The crude oil is considered as a mixture with 6 pseudo components to be separated into 6 fractions by the high boiler path. The actual highest boiling fraction is separated first (see Figure 8.25). The rectifying sections of all columns are combined in the main tower, and the stripping sections are formed by stripping side columns. Typically, the main tower has about 35 actual trays, and each side column has about 4 actual trays. The products are light naphtha, heavy naphtha, light distillate (kerosene), heavy distillate (diesel), atmospheric gas oil (AGO), and reduced crude, which is processed further in the vacuum tower. The amount of reduced crude is about 50 – 60 % of the crude oil feed. After being heated in a furnace, the crude oil enters the column as a vapor–liquid

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Figure 8.26 Use of stripping side columns in the atmospheric tower of crude oil distillation.

mixture. The liquid fraction of the feed containing primarily the high boilers but also some lower boiling components is stripped by superheated live steam (steam distillation; see Section 8.2.6). The products of the side columns are either reboiled or stripped off by live steam. Due to the compilation of several rectifying sections in the main column, the amount of vapor increases in the upper part of the column. This is disadvantageous because the diameter of the tower has to be increased and all the heat has to be withdrawn at the lowest temperature. To overcome these disadvantages, heat is removed from the main column by several pump-arounds. Some liquid is removed from the tower, cooled down against the cold crude oil, and fed three trays above the withdrawal point into the tower again. The subcooled liquid quenches part of the vapor in the column.

8.4 FRACTIONATION OF HETEROGENEOUS AZEOTROPIC MIXTURES

8.4

435

Fractionation of Heterogeneous Azeotropic Mixtures

Azeotropes constitute a barrier to separation by distillation as vapor and liquid have the same concentration at the azeotropic point. In consequence, no driving force for interfacial mass transfer exists at the azeotrope. Fractionation of azeotropic mixtures is only possible if either the concentration of the azeotrope is changed by any means, or alternative separation processes, which can overcome azeotropes, are combined with distillation. Mixtures with miscibility gaps in the liquid phase very often – but not always – form azeotropes within the miscibility gaps. These azeotropes, called heteroazeotropes, are always minimum (low boiling) azeotropes. A heteroazeotrope can be overcome by use of a decanter. For example, in Figure 8.27, the separation of the binary nitromethane/water mixture requires two columns, C-1 and C-2. Both pure substances are removed as bottom products. The vapor compositions at the overhead of each column are approximately equal to the azeotrope. Hence, both vapor streams can be combined and condensed in the same cooler. The resulting two-phase mixture goes through a decanter S-1.

Figure 8.27 Flow sheet and McCabe–Thiele diagram for separation of the binary water/nitromethane mixture. The water/nitromethane mixture forms an immiscibility region (shaded area) with a heteroazeotrope. Complete separation is possible by using two columns C-1 and C-2 and a decanter S-1. The vertically hatched region on the (y, x )-diagram represents the area between the operating line and the equilibrium line in which the number of equilibrium stages n or transfer units NTUOG is determined.

The nitromethane-rich phase with composition xF 2 is fed into column C-2, and the water-rich phase is returned to column C-1. Because water is the bottom product in column C-1, live steam used for heating can be fed directly into the column discarding the reboiler. This process can be applied to many systems. Some examples are listed

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8 INDUSTRIAL DISTILLATION PROCESSES

Table 8.5 Examples of binary systems processed by distillation and decantation.

Toluene/water Benzene/water Chloroform/water Dichloromethane/water Butyl alcohol/water

Ethyl propyl ether/water Butanol/water Ethyl acetate/water Nitromethane/water

in Table 8.5. According to Drew 1979, the organic component should be withdrawn a few plates above the column bottom as a vapor side stream when dirt or any high boilers are present in the feed. In ternary mixtures, several distillation fields can occur, which are divided by distillation boundaries (see Sections 4.3.2.1 and 8.6). Like in binary azeotropes, these distillation boundaries cannot be crossed by distillation. However, the distillation boundary can be crossed by separation in a decanter if phase splitting occurs. Separation of an acetone/water/butanol mixture illustrates this procedure (see Figure 8.28) Pucci et al. 1986. The water/butanol mixture forms a heteroazeotrope; the distillation boundary runs from the azeotrope to the low boiler acetone. Separation is ˙ from the carried out in two columns. Acetone is withdrawn as overhead fraction D1 ˙ first column C-1. The bottom fraction B1 lies in the shaded two-phase region and ˙ and a butanol-rich phase S2 ˙ is, consequently, separated into a water-rich phase S1 ˙ in the decanter S-1. Because S2 is located in another distillation field, butanol can ˙ . The overhead fraction from be withdrawn from column C-2 as bottom product B2 C-2 has approximately the concentration of the heteroazeotrope and can also be fed ˙ of the process in Figure 8.28 is, generinto the decanter. The water-rich fraction S1 ally, polluted with n-butanol. To meet quality standards of wastewater, separation of butanol from the mixture might be necessary. Some examples of these processes are listed in Table 8.5. The process in Figure 8.28 can be modified by material (or direct) coupling of column since the bottom fraction of column C-1 and the overhead fraction of column C-2 have approximately the same composition. Both fractions lie in the region of immiscibility in the liquid phase. Details of direct column coupling are described in Section 7.3.1.

8.5

Fractionation of Azeotropic Mixtures by Pressure Swing Processes

Many mixtures have azeotropes whose concentration can be significantly shifted by changing the system pressure. Especially in systems with low boiling azeotropes (minimum azeotropes), the azeotropic concentrations depend significantly on temperature, which can be changed by changing the operating pressure. An important example is the ethanol/water mixture [Gmehling and Kolbe 1988], whose azeotrope even vanishes at low temperatures, i.e. low operating pressures; see Figure 8.29.

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437

Figure 8.28 Flow sheet and phase diagram for separation of the ternary acetone/water/n-butanol mixture (p = 1.013 bar). The existence of an immiscibility region (shaded area) permits complete separation of the mixture in two columns C-1 and C-2 with an intermediate decanter S-1.

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Figure 8.29 Concentration of the azeotrope in the ethanol/water mixture as function of temperature [Gmehling and Kolbe 1988].

Figure 8.30 Flow sheet and McCabe–Thiele diagram for separation of the binary tetrahydrofuran/water mixture in two columns, C-1 and C-2, operating at different pressures. This mixture contains a minimum azeotrope, whose position can be shifted by changing system pressure. The hatched region represents the area between the equilibrium curves and the operating lines in which the number of equilibrium stages n or transfer units NTUOG is determined.

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439

Another example is the tetrahydrofuran (THF)/water mixture [Abu-Eishah and Luyben 1985]. As shown in Figure 8.30, this mixture forms a binary minimum azeotrope, which shifts toward the low-boiling substance (THF) at lower system pressure. The binary azeotrope can be crossed by firstly separating the high-boiling substance (water) at low pressure p1 = 1 bar in column C-1. The composition of the ˙ should be as close as possible by that of the azeotrope at this overhead product D1 pressure. After the pressure has been increased to p2 = 8 bar (0.8 MPa), the frac˙ is fed into a second column C-2. At the higher pressure in this column, the tion D1 azeotrope is formed at a lower concentration of low boiler (THF), which can then be ˙ . The overhead product D2 ˙ of column C-2 is, preferremoved as bottom product B2 ably in vapor state, returned to column C-1 after pressure reduction in a nozzle. Since ˙ , the operating line shows the column C-1 has a liquid feed F˙ and a vaporous feed D2 two breaks. The overhead temperature of column C-2 is higher than the boiling temperature of column C-1. Therefore, the requirements for indirect (thermal) column coupling are fulfilled (see Section 7.3). This process can be applied to many systems. Some examples are listed in Table 8.6. In addition, typical operation pressures of the two columns are given. Table 8.6 Examples of systems fractionated by pressure swing processes (Figure 8.30).

Systems Methyl ethyl ketone/water Methanol/acetone Methanol/methyl ethyl ketone Tetrahydrofuran/water Ethanol/water Water/2-butanone Water/isobutyl alcohol Ethanol/2-pentanone Methanol/acetone Methanol/2-butanone Hydrogen chlorid/water

8.6

Pressure p1 in bar

Pressure p2 in bar

1 1 1 1

7 0.25 7.6 7.6

Schweitzer 1979 Schweitzer 1979 Schweitzer 1979 Schweitzer 1979

6

0.1

Sulzer 2000

References

Fractionation of Azeotropic Mixtures by Addition of an Entrainer

Azeotropic mixtures can be separated into their pure components by the addition of an auxiliary substance called entrainer [Hoffman 1964; Tassios 1972]. The choice of the entrainer has to permit an easy separation of the mixture’s components and the recovery of the entrainer, which is recycled within the process. The principles of separating an azeotropic mixture by use of an entrainer are

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8 INDUSTRIAL DISTILLATION PROCESSES

demonstrated for the case of a binary azeotropic mixture in the following. By addition of the entrainer, a ternary system is formed. The process can be easily understood with the help of the rules of ternary distillation dealt with in Section 4.3. There are two different approaches to separate a ternary mixture with azeotropes. In the first approach the entrainer is selected so that no distillation boundary exists in the ternary system. In the second approach the entrainer is selected so that the components a and b of the feed mixture are both either origins or termini of distillation lines. In this cases the azeotrope is the origin or terminus of a boundary distillation line. 8.6.1

Processes for Systems Without Distillation Boundary

Fractionation of azeotropic mixtures is very simple, if no distillation boundary exists in the system. 8.6.1.1

Criteria for Entrainer Selection

Processes without distillation boundaries have been studied in detail by Doherty and Caldarola 1985. The entrainer has to be selected so that the components a and b are in the same distillation field. The a/b azeotrope must be either an origin or a terminus of distillation lines. Table 8.7 gives the criteria for entrainer selection resulting from these requirements. These criteria are minimum requirements. There may exist additional azeotropes in the system. A further possibility for processes without distillation boundary is the use of a high boiling entrainer for a minimum azeotrope to be separated (see Section 8.6.3.2). Table 8.7 Criteria for entrainer selection for processes without distillation boundaries (see process in Figure 8.31).

Entrainer for the separation of a mixture with a minimum azeotrope: • Intermediate boiler • Low boiler, which forms a medium boiling maximum azeotrope with component a Entrainer for the separation of a mixture with a maximum azeotrope: • Intermediate boiler • High boiler, which forms a medium boiling minimum azeotrope with component b

8.6.1.2

Processes

Figure 8.31 presents the flow sheet and the triangular composition diagram for the fractionation of a binary mixture with an a/b minimum azeotrope with an intermediate boiling entrainer e. All distillation lines originate at the high boiler b and terminate at the azeotrope because it has the lowest boiling temperature of the mixture.

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441

Figure 8.31 Separation of the azeotropic acetone/heptane mixture using benzene as entrainer. The entrainer boils intermediate, and, therefore, no distillation boundary exists in the ternary mixture. A) Flow sheet. B) Triangular concentration diagram.

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.32 Separation of the azeotropic acetone/heptane mixture using benzene as entrainer. Here, the column C-1 splits the feed in an overhead fraction without high boiler and a bottom fraction without low boiler. This modification of the process in Figure 8.31 is more economical. A) Flow sheet. B) Triangular concentration diagram. C) Column with internal wall (see Figure 7.22).

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443

Figure 8.33 Triangular concentration diagram for a process for separating azeotropic mixtures with a low boiling entrainer that forms an intermediate boiling maximum azeotrope. The flow sheet is the same as in Figure 8.31A.

˙ from which the high The feed F˙ is mixed with the entrainer e to form the mixture M ˙ ˙ boiler b is separated in column C-1 as bottom fraction B1. The overhead fraction D1 is (vaporously) fed into column C-2 to be split into the low boiler a as overhead and the entrainer e as bottom fraction. The entrainer is recycled to column C-1. This separation sequence is similar to the well-known separation of a zeotropic ternary mixture (see Section 8.3.1). According to Section 4.3.2.1, the low boiler a can also be separated first from the ˙ into overhead ternary mixture. In such a process the column C-2 splits the bottom B1 ˙ ˙ fraction D2 with the entrainer and bottom fraction B2, which consists of pure high boiler b. However, this variant of the process is in most cases less advantageous. A third modification of the process is presented in Figure 8.32. The feed F˙ and the entrainer e are fed into column C-1 and split in an overhead fraction free of b and a bottom fraction free of a. Both fractions are separately fed into column C-2 whose overhead fraction consists of pure substance a and bottom fraction of pure substance b. The entrainer e is removed from the middle of the column to be recycled to column C-1. According to Section 7.3.1, this process is in most cases more favorable due to a lower energy requirement. A very elegant modification of the process is to establish the separation in a column that is divided in the middle section by a vertical wall, called dividing wall column; see Figure 8.32C. In such a column the feed mixture a/b is fed into the left compartment, and the entrainer is regained in the right compartment of the column. From there the entrainer is internally recycled to the feed. The function of a column with a dividing wall is explained in detail in Section 7.3.2.4; see Figure 7.22. Such a modification allows complete separation of an azeotropic mixture in a single column. The

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separating agent, the entrainer, remains in the column and establishes the separation without any external action. The triangular concentration diagram for the separation of an a/b minimum azeotrope with a low boiling entrainer that forms an intermediate boiling maximum azeotrope with the component a is shown in Figure 8.33. The process is the same as in Figure 8.31, but the e/a azeotrope acts as the intermediate boiling entrainer. The processes without distillation boundary consist of two columns and one recycle. The problem of all these processes lies in the existence of an appropriate entrainer. It is well-known that the probability of azeotrope formation increases with decreasing boiling point differences of the components a and b. Consequently, most azeotropic mixtures to be fractionated by distillation are close-boiling ones. In many cases it will be difficult or even impossible to find a substance that boils in between and that does not form new azeotropes. But even if such a substance exists, the process might be uneconomical because of the small boiling point differences existing in the system. The entire process has to be performed in the boiling point range of the a/b mixture. The system presented in Figure 8.31 is one of the few examples this process works economically. 8.6.2

Processes for Systems with Distillation Boundary

Processes of systems with distillation boundaries are often used in the process industry because of their high flexibility. 8.6.2.1

Criteria for Entrainer Selection

This group of processes has been studied by Stichlmair 1988; Stichlmair and Herguijuela 1992. A good review has been published by Widago and Seider 1996. The entrainer has to be selected so that the components a and b of the azeotropic mixture to be separated are both either origins or termini of distillation lines. Such systems always have distillation boundaries. The entrainer has to allow the recovery of products a and b either at the head or at the bottom of distillation columns. This is only possible if the distillation lines in the triangular concentration diagram either start or end at points a and b (see Section 4.3.2.1). Figure 8.34 shows the minimum requirements to be met by the entrainer. If a minimum-boiling binary azeotrope exists (Figure 8.34A–C), substances a and b are high-boiling ones; i.e. the distillation lines start at a and b. This must also be the case in a ternary system. This condition is always met when the entrainer is a lowboiling substance (Figure 8.34A). If the entrainer has a boiling temperature between that of a and b, pure b can be separated; however, substance a can be recovered only if the mixture a/e forms a minimum azeotrope (Figure 8.34B). If the entrainer is a high-boiling substance (Figure 8.34C), a and b can be separated only if the entrainer forms minimum azeotropes with both substances. Substances a and b are obtained as bottom products, since they are origins of distillation lines. In the case of a maximum-boiling binary azeotrope (Figure 8.34D–F), components a and b in the feed mixture are low-boiling substances; i.e. the distillation lines terminate at points a and b. The entrainer used must ensure that the distillation lines

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Figure 8.34 Minimum requirements for the selection for the entrainer e for the processes with distillation boundaries. Separation of the azeotropic mixture into pure a and b is possible only when the distillation lines both either start or terminate at a and b. Table 8.8 Criteria for entrainer selection for processes with distillation boundaries (see process in Figure 8.35).

Entrainer for the separation of a mixture with a minimum azeotrope: • Low boiler (lower than the minimum azeotrope) • Intermediate boiler, which forms a new minimum azeotrope with the low boiling component of the given mixture • High boiler, which forms new minimum azeotropes with both components of the given mixture. At least one of them has to boil lower than the azeotrope of the given mixture. Entrainer for the separation of a mixture with a maximum azeotrope: • High boiler (higher than the maximum azeotrope) • Intermediate boiler, which forms a new maximum azeotrope with the high boiling component of the given mixture • Low boiler, which forms new maximum azeotropes with both components of the given mixture. At least one of them has to boil higher than the azeotrope of the given mixture.

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of both binary mixtures a/e and b/e also terminate at a and b, respectively. This condition is met only if the entrainer e is the high boiler of the system (Figure 8.34D). When the entrainer is an intermediate-boiling substance, component a can be separated, but component b can be recovered only if the mixture b/e forms a maximum azeotrope (Figure 8.34E). With a low-boiling entrainer (Figure 8.34F), mixtures b/e and a/e have to both form maximum azeotropes. Products a and b leave the rectification columns overhead, since they are termini of distillation lines, i.e. sinks in the boiling point surface (see Chapter 2). Thus, a binary mixture with a maximum azeotrope can generally be separated into its two components only if the entrainer is higher boiling than the azeotrope or if it forms high boiling azeotropes. On the other hand, a mixture with a minimum azeotrope requires a low-boiling entrainer or the formation of low-boiling azeotropes. In other words, only azeotropes of the same type occur in the process. The criteria listed in Table 8.8 are minimum requirements. As will be shown in the following examples, additional azeotropes and even ternary azeotropes may exist, but they have to be azeotropes of the same type. 8.6.2.2

Generalized Process

Figure 8.35 presents the generalized process for completely separating an a/b minimum azeotrope. According to Table 8.8, an entrainer has to be used that boils lower than the minimum azeotrope. Distillation lines start at pure substance a as well as at pure substance b and end at the entrainer e. Therefore, the components a and b of the given system can be recovered as bottom fractions of different distillation columns. First, it is assumed that the feed F˙ lies in that distillation field where the low boi˙ of column C-1. The overhead fraction D1 ˙ ler a can be separated as bottoms B1 is close by the azeotrope. By admixing the entrainer e, the distillation boundary is ˙ 2 lies on the other side of the boundary where substance b crossed, and the mixture M ˙ of column C-2. The overhead fraction D2 ˙ lies close can be separated as bottoms B2 by the distillation boundary. Assuming that both the distillation boundary is curved ˙ is lying at the concave side, the fraction D2 ˙ can be separated in column C-3 and D2 ˙ ˙ both lying at the other side of into an overhead fraction D3 and a bottom fraction B3 ˙ is recycled as entrainthe boundary line (see Figure 4.33). The overhead fraction D3 ˙ er. The bottom fraction B3 is recycled to column C-1. Depending on composition ˙ can be fed into column C-1 either separate or after being mixed with the fraction B3 ˙ the feed F . The generalized process of Figure 8.35 consists of three separation steps and two ˙ , recycles. A material balance around column C-2 and C-3 requires that fraction D1 ˙ pure substance b, and fraction B3 all lie at a straight line (thin dashed line in Figure 8.35). It is essential that the amount of entrainer to be recycled is as small as possible. ˙ , has to be as A more detailed study reveals that the feed to column C-3, fraction D2 small as possible since both fractions of C-3 are recycled. Decisive for the amount ˙ is the split of M˙ 2 in D2 ˙ and B2 ˙ . Since the bottom fraction B2 ˙ contains all of D2

8.6 FRACTIONATION OF AZEOTROPIC MIXTURES BY ADDITION OF AN ENTRAINER

447

Figure 8.35 Process for the separation of a binary mixture with a minimum azeotrope by using a low boiling entrainer. The feed is rich in a. A) Flow sheet: components a and b are obtained as pure bottom products from columns C-1 and C-2, respectively. B) Triangular concentration diagram: the ternary mixture is divided into two distillation fields by a distillation boundary running from the minimum azeotrope to the entrainer. The distillation boundary is crossed by the admixing of the entrainer e. ——– distillation boundary ; ——– external balance of distillation column ; − · − · − mixing of two streams ; − − − material balance around columns C-2 and C-3.

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8 INDUSTRIAL DISTILLATION PROCESSES

of substance b in the feed, the following holds:

˙ = F˙ · xFb . B2

(8.1)

The lever rule gives

˙ = F˙ · xFb · B2M 2/D2M 2 . D2

(8.2)

˙ and B3 ˙ are small when the length D2M 2 is large with respect to The recycles D3 the length B2M 2. This requires a heavily curved boundary distillation line and a ˙ 2 far away from the distillation boundary. position of M Figure 8.36 presents the generalized process when the feed lies in that distillation field, where the high boiler b can be separated. In this case just the sequence of the separations is different from that in Figure 8.35. Both modifications of the generalized process are characterized by two separations, one performed on the convex and the other one on the concave side of the boundary distillation line. The generalized process also works for the separation of a mixture with a maximum azeotrope if an upside down version of the flow sheet is used; see Figure 8.37. According to the criteria for entrainer selection listed in Table 8.8, a minimum azeotrope can be separated by an intermediate boiling or a high boiling entrainer instead of a low boiler, if additional azeotropes are formed. Figures 8.38 and 8.39 present the triangular concentration diagrams for these cases. A detailed study reveals that one of the azeotropes formed by the substance e acts as entrainer. Apart from that the process is the same as in Figure 8.35. The generalized process presented here allows the separation of two pure substances from a ternary mixture if the boundary line is curved. This condition is more or less met by real systems. However, the more the distillation boundary is curved, the more economical is the process. 8.6.2.3

Simplifications of the Generalized Process

The generalized process with three distillation columns and two recycles is expensive with respect to investment and operating costs. However, there exist several possibilities for process simplification. Some examples are presented in the following sections. Under special conditions one of the three separation steps of the generalized process can be discarded. This is possible when the distillation boundary is heavily ˙ 2 is lying close by the entrainer. In this case the separation curved and the stream M ˙ that contains performed in column C-2 (Figure 8.35) yields a bottom fraction B2 ˙ can be the substance a in high purity. Thus, the column C-3 and the recycle B3 discarded. This advantageous modification of the generalized process requires a very heavily curved distillation boundary that lies in part at a side of the triangular concentration diagram. Additionally, a large amount of entrainer has to be recycled within the process. This process modification is facilitated by the existence of one or two azeotropes formed by the entrainer with the components a and b. However, there should not exist any ternary azeotropes.

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Figure 8.36 Process for the separation of a binary mixture with a minimum azeotrope by using a low boiling entrainer. The feed is rich in b. The process is nearly identical with the process in Figure 8.35. Just the sequence of the columns is different. A) Flow sheet. B) Triangular concentration diagram.

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Figure 8.37 Process for the separation of a binary mixture with a maximum azeotrope by using a high boiling entrainer. The feed is rich in b. The process is an upside down version of the process in Figure 8.35A. A) Flow sheet. B) Triangular concentration diagram.

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451

Figure 8.38 Process for the separation of a binary mixture with a minimum azeotrope by using an intermediate boiling entrainer e that forms a minimum azeotrope with a. The flow sheet is the same as in Figure 8.35A.

Figure 8.39 Process for the separation of a binary mixture with a minimum azeotrope by using a high boiling entrainer e that forms binary minimum azeotropes with a and with b. The flow sheet is the same as in Figure 8.35A.

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Figure 8.40 Two-column process for the separation of hydrogen chloride (HCl) and water by using hydrogen sulfate (H2 SO4 ) as entrainer. In this system the distillation boundary lies in part on the base of the triangular concentration diagram. Thus, pure water is recovered in column C-2 and the third column of the generalized process can be discarded. A) Flow sheet. B) Triangular concentration diagram.

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453

Figure 8.41 Two-column process for the separation of a binary mixture (ethanol/water) with a minimum azeotrope by using a low boiling entrainer (tetrahydrofuran) that forms a minimum azeotrope with component b. The boundary line is so heavily curved that pure a is recovered as bottoms of column C-2. The third column of the generalized process can be discarded. A) Flow sheet. B) Triangular concentration diagram.

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Figure 8.40 presents the flow sheet and the triangular concentration diagram for concentrating diluted hydrochloric acid [Grewer 1971]. The system HCl/H2 O forms a maximum azeotrope (21 wt% HCl, 108.8 ◦C). According to Table 8.8 the entrainer has to be the high boiler in the system. Sulfuric acid is used in the process, which meets this condition. The distillation boundary line, a ridge in the boiling point surface, lies in part at the basic line of the triangular concentration diagram. The feed F˙ with a high water content is mixed with concentrated sulfuric ˙ 1 lies in that distillation field, where acid (approximately 75 wt%). The mixture M ˙ . If M˙ 1 HCl (or an HCl-rich mixture) can be separated as overhead fraction D1 ˙ is sufficiently close by the entrainer, then the bottom fraction B1 is nearly free of HCl. The subsequent separation step (single-stage distillation) yields an HCl-free water as overhead fraction. The process can be modified by feeding the entrainer directly on the overhead of column C-1 instead of mixing it with the feed (see Section 8.6.3.2). A process similar to this one is used for concentrating a diluted nitric acid (Figure 8.45). Figure 8.41 presents a two-column process for separating a mixture with a minimum azeotrope. The low boiling entrainer e forms a minimum azeotrope with substance b that is not required by the entrainer selection rules in Table 8.8. The e/b azeotrope, which boils lower than the azeotrope of the feed mixture, acts in the

Figure 8.42 Two-column process for the separation of a binary mixture with a minimum azeotrope by using a low boiling entrainer that forms minimum azeotropes with components a and b. The boundary lines are so heavily curved that pure a is recovered as bottoms of column C-2. The third column of the generalized process can be discarded. A) Flow sheet. B) Triangular concentration diagram.

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process as entrainer. If the distillation boundary between the two azeotropes is heav˙ lies on the concave side, then substance a ily curved and the overhead fraction D1 can be recovered in pure form as bottoms of column C-2. A similar process with a high boiling entrainer e is depicted in Figure 8.42. A ternary azeotrope not required by the entrainer selection rules is formed in this system. The distillation boundary running from the a/b azeotrope to the ternary azeotrope is so heavily curved that the separation in column C-2 yields the pure substance a as bottoms. While the generalized process with three separation steps can be performed even in systems with a slightly curved distillation boundary, the two-column process requires a heavily curved boundary. Thus, the two-column process is a special case of the generalized process. 8.6.3

Hybrid Processes

The generalized process can be simplified by performing one (or two) of the three separation steps not by distillation but by another separation technique, e.g. decantation, absorption, stripping, extraction, adsorption, or membrane permeation. Some examples will be shown in the following sections. 8.6.3.1

Distillation Combined with Decantation (Azeotropic Distillation)

Figure 8.43 presents a process for ethanol/water separation with toluene as entrainer. Ethanol and water form a low boiling azeotrope, and toluene is a suitable entrainer because, even though it is the high boiler in the system, it forms minimum azeotropes with water as well as with ethanol. Thus, the criteria for entrainer selection are met. The heterogeneous water/toluene azeotrope forms two liquid phases. In addition, there exists a ternary azeotrope, which has the lowest boiling temperature of the system. Each corner of the triangular concentration diagram is a starting point of a distillation line. All of them end at the ternary azeotrope. Distillation boundaries are running from each binary azeotrope to the ternary azeotrope dividing the mixture in three distillation fields that differ in the origins of the distillation lines. Generally, the feed F˙ is rich in water. It is situated in that distillation field where ˙ . This separation is performed in colwater can be separated as bottom product B1 ˙ umn C-1. The overhead fraction D1 lies very close by the distillation boundary. The ˙ 2 located in the concave region of the admixing of the entrainer produces a stream M ˙ in distillation boundary where pure ethanol can be obtained as a bottom product B2 ˙ column C-2. The overhead product D2 is very close by the ternary azeotrope, and, when condensed and subcooled, it forms two liquid phases that are separated in the ˙ serves as entrainer. The water-rich fracdecanter S-1. The toluene-rich fraction S2 ˙ tion S1 is recirculated to column C-1. This process is equivalent to the generalized process (Figure 8.35) with the exception that the third separation step is performed by decantation instead of distillation. A material balance around column C-2 and decanter S-1 (thin dashed line) leads to ˙ (ethanol), S1 ˙ , and D1 ˙ are collinear in the condition that the states of the streams B2 the triangular concentration diagram of Figure 8.43B. This condition fixes the state

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.43 Separation of the ethanol/water mixture using toluene as entrainer. This process is referred to as azeotropic distillation. A) Flow sheet showing distillation columns C-1 and C-2 and decanter S-1. B) Triangular concentration diagram: because the mixture contains three binary minimum azeotropes and one ternary azeotrope, it is divided into three distillation fields. ——– distillation boundary; ——– external balance of distillation column; − · − · − mixing of two streams; − − − material balance around columns C-2 and decanter S-1; - - - separation in the decanter. Shaded area represents the immiscibility range at 25 ◦C.

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˙ and, in turn, the state of the stream M˙ 1. The only free of the overhead fraction D1 ˙ . It parameter in the process is the circulation rate of the entrainer-rich fraction S2 ˙ has to be chosen so that the stream M 2 lies on a straight line between the ternary azeotrope and the pure ethanol. Use of a decanter instead of a distillation column results in a very significant simplification of the process. This trick for process simplification can be applied only to system with a minimum azeotrope since – as has been outlined in Section 8.6.2.1 – in a process only azeotropes of the same type can occur and azeotropes in a region of immiscibility (heteroazeotropes) are always minimum azeotropes. The process presented in Figure 8.43 is very often used. Some additional examples ˙ and S2 ˙ directly into the are listed in Table 8.9. It is possible to feed the recycles S1 overhead of columns C-1 and C-2, respectively, instead of first mixing them with F˙ ˙ . A critical point is sometimes the decanter. Some mixtures form a very stable and D1 emulsion, which cannot be easily separated into the two liquid phases by decantation. Experiments with the original feedstock should be made. Table 8.9 Examples of systems fractionated by combination of distillation and decantation (see process in Figure 8.43).

Mixture to be separated

Entrainer

References

Water / ethanol

Benzene Toluene Pentane Trichlorethylene Cyclohexane Ethyl acetate Ethyl ether

Keyes 1929 Pilhofer 1983 Black 1980 Fritzweiler 1938 Herfurth et al. 1987 Cairns and Furzer 1987 Othmer 1974

Acetic acid / formic acid

Chloroform

Hunsmann and Simmrock 1966

Water / pyridine

Benzene Toluene

Berg et al. 1945 Berg 1969

Water / acetic acid

Butyl acetate Propyl acetate

Hegner et al. 1973 Hegner et al. 1973

Water / propanol

Benzene

Bril et al. 1977

8.6.3.2

Distillation Combined with Absorption (Extractive Distillation)

The use of an absorption step instead of distillation is demonstrated in Figure 8.44. This process can be applied, for instance, to ethanol/water separation. In col˙ . The overhead fraction D1 ˙ umn C-1 most of the water is removed as bottoms B1 with azeotropic composition (approximately 90 mol% ethanol) is vaporously fed

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.44 Separation of a binary mixture a/b with a minimum azeotrope by using an absorption step. This process is generally known as extractive distillation although it actually involves absorption and not extraction. A) Flow sheet. B) Triangular concentration diagram.

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into the absorber A-1 where the water is removed by an appropriate absorbent (e.g. ethylene glycol, which is hygroscopic). The absorbent is regenerated in column C-3 by vacuum distillation. The vapor at the overhead of the absorption column A-1 contains small amounts of the absorbent e. For separation a small part of the overhead fraction is condensed and recycled as reflux some stages above the feed point of the absorbent. Analogously, some of substance a is dissolved in the liquid in the bottom, which can be separated by partial reboiling. Because of the existence of a condenser and a reboiler, the middle column of the process depicted in Figure 8.44 is often misinterpreted as a distillation ˙ is fed into the bottom and the column. However, since the low boiling fraction D1 ˙ high boiling liquid B2 is fed (in subcooled state) into the head of the column and, additionally, both flow rates are much higher than that of reflux and reboil, there is no doubt that the process is in fact an absorption. The section above the entrainer feed is a rectifying section to remove the entrainer e from the overhead product. The middle section is an absorption section; it is there where the azeotrope is crossed. The lower section is a stripping section where the lightest component is removed from the liquid. The three sections can be seen very clearly in the concentration profile in Figure 8.44B. The azeotrope is crossed by absorption in this process. Therefore, the criteria for entrainer selection (Table 8.8) cannot be applied. Here, the well-known rules for absorbent selection have to be used. They require a subcooled high boiling liquid with high capacity and selectivity for substance a or b. In addition, some other requirements have to be met [Blaß 1989]. The process depicted in Figure 8.44 is widely used not only for the separation of azeotropic mixtures but also for the separation of close-boiling mixtures. Table 8.10 lists some examples of industrial important systems. Depending on solubility either the light component a or the heavy component b is preferably absorbed. The process cannot be applied to systems with a maximum azeotrope. This process is well known as “extractive distillation”, even though it involves an absorption step and not an extraction step. 8.6.3.3

Distillation Combined with Desorption

An example for a hybrid process with the combination of distillation and desorption is the separation of the mixture of nitric acid and water. This system forms a maximum azeotrope with a nitric acid concentration of 37 mol% that boils at 121 ◦C. According to Table 8.7 a high boiling entrainer has to be selected. Sulfuric acid is used in the process presented in Figure 8.45 [Geriche 1973], which meets this condition. In this mixture a distillation boundary runs from the maximum azeotrope to the H2 SO4 -corner dividing the ternary mixture in two distillation fields. In one field pure water can be recovered as overhead fraction. In the other field HNO3 can be gained as overhead fraction. Normally, the feed is rich in water; it lies in the righthand-side distillation field in Figure 8.45. Most of the water is removed as distillate in ˙ , column C-1. By admixing concentrated sulfuric acid (46 mol%) to the fraction B1 ˙ the mixture M 1 lies in the left distillation field where pure HNO3 is recovered as

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.45 Concentration of nitric acid and water by using sulfuric acid as entrainer. A) Flow sheet. B) Triangular concentration diagram. ——– distillation boundary; ——– external material balance of distillation column; − · − · − mixing of two streams.

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Table 8.10 Examples of systems fractionated by extractive distillation (see process in Figure 8.44).

Mixture to be separated

Entrainer

References

Butane/butene/ butadiene

Furfural Acetonitrile Dimethylacetamide n-Methylpyrrolidone Dimethylformamide

Peters and Rogers 1968 Happe et al. 1946 Bittrich 1987 Coogler 1967 Volkamer et al. 1981 Hebel and Kaspareit 1971

Benzene/cyclohexane

Aniline n-Methylpyrrolidone n-Formylmorpholine

Blaß 1989 Mueller 1980 Lackner 1981

Butene/isoprene Acetone/methanol Tetrahydrofuran/water

Dimethylformamide Water Dimethylformamid

Ushio 1972 Drew 1979 Drew 1979

Ethanol/water

Ethylene glycol Salt (potassium acetate) Acrylonitrile

Black 1980 Schweitzer 1979

Propylene/propane

Hafslund 1969

overhead product of column C-2. The bottom fraction lies at the distillation boundary ˙ , at best. This boundary is crossed by stripping with live steam. The fraction B2 which is free of HNO3 , is fed to the single-stage distillation unit C-3 to be split into pure water as overheads and concentrated sulfuric acid as bottoms that is recycled to column C-2. The recovery of concentrated sulfuric acid has to be performed under vacuum (see Section 8.2.1). The process consists of three separation steps; two of them are performed by distillation and one by desorption. Nitric acid, when boiled, always forms some NOx that colors the liquid. To get clear liquid, the NOx is removed by stripping with air. 8.6.3.4

Distillation Combined with Extraction

The separation of an azeotropic mixture by using a combination of distillation and extraction is also often used in industrial processes. Figure 8.46 shows a process for the separation of tetrahydrofuran (THF) and water [Schoenmakers 1984]. This mixture exhibits a minimum azeotrope at about 81.7 mol% THF (see Section 8.5). From the water-rich feed F˙ water is removed as bottoms in column C-1. The distillate with azeotropic concentration is fed into the extraction column E-1. Concentrated sodium hydroxide (NaOH) is used as solvent because it is hygroscopic. The

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.46 Flow sheet of the process for the separation of the tetrahydrofuran/water mixture by distillation and extraction. The entrainer is concentrated natrium hydroxide (NaOH).

loaded solvent is regenerated by single-stage vacuum distillation. The concentrated sodium hydroxide is recycled to the extractor E-1; the distillate containing water and some THF is fed into column C-1 either together with the feed or separate from it. Solvent extraction has the disadvantage that no pure products can be obtained as the raffinate always contains some solvent. According to Schoenmakers tetrahydrofuran with 99 wt% can be gained at best. If a higher purity is required, an additional distillation step, performed in column C-2, has to be used. The bottoms of C-2 is pure tetrahydrofuran; the distillate is recycled to the extractor. Another example for the combination of distillation and extraction is the separation of the dichloromethane/methanol mixture [Drew 1979]. Here, the given mixture, which is usually rich in methanol, is fed into the extractor first; see Figure 8.47. The solvent used in the process is water that preferably extracts methanol. The raffinate is ˙ ) and an overhead fraction D1 ˙ split in column C-1 into dichloromethane (bottoms B1 with nearly azeotropic concentration. The extract, which is rich in methanol, is fed ˙ ) and methanol distillate D2 ˙ . into column C-2 to be separated into water (bottoms B2 This process produces highly pure dichloromethane, but the methanol fraction may still contain some dichloromethane. A very interesting hybrid process is presented in Figure 8.48 for the separation of a benzene/cyclohexane mixture [Seader 1984, pp. 13–58]. This mixture has a minimum azeotrope at about 53 mol% benzene. Acetone is added that forms a minimum azeotrope with cyclohexane. As can be seen in Figure 8.48B the boundary distillation line is running from the benzene/cyclohexane azeotrope to the acetone/cyclohexane azeotrope. Benzene is the origin of distillation lines and, consequently,

8.6 FRACTIONATION OF AZEOTROPIC MIXTURES BY ADDITION OF AN ENTRAINER

Figure 8.47 Process for the separation of the dichloromethane/methanol mixture by distillation and extraction. The entrainer is water. A) Flow sheet. B) Triangular concentration diagram.

463

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.48 Process for the separation of the benzene/cyclohexane mixture by distillation and extraction. Two entrainers, acetone and water, are used in this process. A) Flow sheet. B) Triangular concentration diagram for distillation.

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465

Figure 8.48 (Continued) C) Triangular concentration diagram for extraction.

˙ 1 as bottoms. The overhead fraction is can be removed from the ternary mixture M the acetone/cyclohexane azeotrope. This fraction is fed into the extractor E-1 where acetone is removed by extraction with water. As water is only slightly soluble in cyclohexane (xH2 O ≤ 0.1 mol%, Sorensen and Arlt 1979), nearly pure cyclohexane is gained. The acetone/water mixture is fed into the distillation column C-2 to be split into ˙ and water as bottoms B2 ˙ . Water is recycled to serve as acetone as overhead D2 solvent in the extractor. Acetone is recycled to column C-1 to serve as entrainer there. In this process two entrainers are used. By the first entrainer acetone, the benzene/cyclohexane azeotrope is replaced by the acetone/cyclohexane azeotrope. This azeotropic mixture can easily be separated by extraction with water, which is the second entrainer. 8.6.3.5

Distillation Combined with Adsorption

Adsorption is a very effective means for breaking azeotropes in a separation process. Several authors report on the separation of azeotropic alcohol/water mixtures. Adsorption in the liquid phase has been tested by Bindal and Misra 1986 using commercially available molecular sieves as adsorbers. The separation factors are very high (85 – 100). They appear to be better than results reported for pervaporation (see Section 8.6.3.6). Dehydration of ethanol in vapor phase using a 3A molecular sieve has been studied by Cartón et al. 1987. According to their results, vapor

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.49 Process for the separation of the ethanol/water mixture by distillation and adsorption. In order to ensure a pseudo-continuous product flow, three adsorbers have to be installed. Each adsorber is subsequently operated in the adsorption, regeneration, and pressure buildup mode.

phase adsorption is superior to liquid-phase adsorption in both kinetics (i.e. form of the breakthrough curve) and capacity. These findings have been confirmed by Sowerby and Crittenden 1991. Very low water concentrations (< 0.05 wt%) are achieved in the product provided that the temperature is not too high. Figure 8.49 depicts a process for the separation of an ethanol/water mixture by a hybrid process with distillation and adsorption Westphal 1987. The water-rich ˙ feed is separated in distillation column C-1 into pure water as bottoms fraction B1 ˙ and the ethanol/water azeotrope as overhead fraction D1. The vaporous overhead ˙ with approximately 10 mol% water is fed into the adsorber A-2 where fraction D1 the water is primarily adsorbed in a molecular sieve bed. The water-free ethanol vapor leaving the adsorber is used for heating the distillation column C-2, which is operated at a lower pressure. The liquid ethanol product is practically free of water. Adsorption in a fixed bed adsorber is a batch process. In order to provide a pseudocontinuous operation, three adsorbers are required: one for adsorption, one for regeneration, and one for pressure buildup. In a full cycle each adsorber is subsequently operated in all three modes. The regeneration of the loaded adsorber is best performed by desorption at low pressure, approximately 0.2 bar, with pure vaporous ethanol as stripping agent. The ethanol/water mixture formed during regeneration is fed into column C-2 to be sepa˙ and the ethanol/water azeotrope D2 ˙ . Column C-2 is operatrated into pure water B2

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467

ed at a vacuum pressure of approximately 0.2 bar. The overhead fraction is recycled as reflux of column C-1. The adsorbers have to be heated in order to avoid vapor condensation. An elegant possibility is to produce the vacuum in column C-2 by a steam-driven ejector and to utilize the waste steam for adsorber heating. The condensate is recycled into column C-2 because it contains some ethanol [Westphal 1987]. In this process the azeotrope is crossed by adsorption. Adsorption of water by a molecular sieve is so effective that water-free ethanol is achieved. The task of column C-2 is only the separation of the ethanol/water mixture formed during desorption. The energy requirement of column C-2 is covered by condensing the vaporous overhead fraction of column C-1. Hence, no additional heat is required for producing ˙ . the water-free ethanol from the azeotropic mixture D1 The advantage of the process is that the ethanol product will never be polluted by any liquid entrainer, even in case of operational disturbances. Thus, the process is well suited for the production of absolute ethanol for medical and pharmaceutical purposes. However, the feed must not contain any light organic components, like methanol, because they may pass the adsorber and collect in the ethanol product. 8.6.3.6

Distillation Combined with Membrane Permeation

The combination of distillation and membrane permeation is a promising hybrid process for separating azeotropic mixtures (e.g. Rautenbach and Albrecht 1989; Udriot et al. 1994; Rautenbach et al. 1996). Figure 8.50 shows a process in which the azeotrope is crossed by membrane permeation [Baker 1991]. This process can be applied, for instance, to the separation of organic compounds and water. The water-rich feed is split into water and azeotrope by distillation in column C-1. After compression to about 3 bar, the azeotropic mixture is fed into the membrane stack with a vacuum at the permeate side. A water-rich fraction permeates the membrane. Due to the vacuum the permeate flow is evaporated. It is first condensed and then pumped into column C-1. The organic-rich retentate is fed into column C-2 where the pure organic compound is recovered as bottoms. The overhead fraction is recycled to the membrane stack. Up to 5 – 7 membrane stacks are required. After each of them the retentate has to be heated to cover the heat of permeate vaporization. In this process the membrane is used only for crossing the azeotrope. The pure products are recovered by conventional distillation. Decisive for the effectiveness of the process is the availability of membranes with both high capacity and high selectivity. Membranes of cellophane, cellulose acetate, polytetra-fluoroethylene, acrylic acid, and acrylamide have been tested for separating alcohol/water mixtures [Ruckenstein and Park 1990]. Ionomeric materials of perfluorosulfonic acid proofed well for the separation of polar organic compounds from their azeotropic mixtures [Dutta and Sikdar 1991]. The combination of distillation and membrane separation can, for instance, be applied to processes for the production of bioethanol. In the process seen in Figure 8.51 [Weyd et al. 2010], the azeotropic overhead fraction of column C-1 is fed, in vapor state, to the a membrane stack. Favorable is a high pressure difference in the mem˙ is compressed to about 8 bar brane. Therefore, the vaporous overhead fraction D1

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.50 Process for the separation of mixtures of organics and water by distillation and membrane permeation. Since the permeate side of the membrane is operated with high vacuum, the permeate is vaporous.

Figure 8.51 Process for dewatering bioethanol by distillation and vapor permeation [Weyd et al. 2010]. Concentrations are given in wt% of ethanol.

8.7 INDUSTRIAL PROCESSES OF REACTIVE DISTILLATION

469

that increases the dew point of the mixture to 140 ◦C. The pressure at the permeate side should be as low as possible. A value of 0.1 bar is a good choice since it allows the condensation of the permeate (vaporous water) by cooling water. Under these operating conditions, the capacity of NaA-zeolite membranes is in the range of 10 – 12 kg/(m2 · h) [Weyd et al. 2010]. In this example, the membrane stack produces pure ethanol. ˙ offers the possibility of heating the The high pressure of the overhead fraction D1 column by condensation of most of the overhead fraction (integrated heat pump). Just the permeate has to be condensed with cooling water. The energy requirement of the process in Figure 8.51 is much lower than that of all alternative processes discussed so far (e.g. azeotropic distillation, extractive distillation, distillation combined with extraction, distillation combined with adsorption). However, since a compressor driven with electric energy is required, the cost advantages are rather small.

8.7

Industrial Processes of Reactive Distillation

The simultaneous realization of reaction and distillation in a countercurrently operated column is a very promising alternative to conventional sequential reaction and distillation. This so-called reactive distillation offers some important advantages [Sundmacher and Kienle 2003; Stichlmair and Frey 1999; Tuchlenski et al. 2001; Almeida-Rivera et al. 2004]: • • • •

Utilization of an exothermic heat of reaction for distillation. Total conversion of the reactants even in reversible reactions. Simple downstream processing of the reactor effluents. Overcoming of separation barriers like azeotropes or distillation boundary lines.

The use of reactive distillation lines or residue curves is the basis for the design of reactive distillation processes [Bessling et al. 1997]. Mixed integer programming allows the development of optimized processes [Frey and Stichlmair 1999, 2000; Poth et al. 2002; Stichlmair and Frey 2001; Doherty and Malone 2001].A large number of reactions favorably performed by reactive distillation are described in the literature. Table 8.11 lists some selected examples. A few industrially important processes are described in detail in the next sections. 8.7.1

Synthesis of MTBE

Methyl tertiary-butyl ether (MTBE, C5 H12 O) is a very effective antiknock additive to gasoline that is produced in amounts larger than 20 × 106 t/a [Rehfinger and Hoffmann 1990]. The feed mixture for MTBE synthesis is i-butene (C4 H8 ) and methanol (CH3 OH). The reaction is catalyzed either by a homogeneous (e.g. sulfuric acid) or a heterogeneous (strong acidic macroporous ion exchange resin) catalyst.

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8 INDUSTRIAL DISTILLATION PROCESSES

Table 8.11 Selected systems for reactive distillation after Doherty and Buzad 1992; Sundmacher et al. 1994; Ung and Doherty 1995.

Methyl acetate from methanol and acetic acid

Agreda et al. 1990

Ethyl acetate from ethanol and acetic acid

Komatsu 1977

Butyl acetate from butanol and acetic acid

Hartig and Regner 1971

MTBE from methanol and i-butene

Flato and Hoffmann 1992; Smith 1990 Thiel et al. 1997

ETBE from ethanol and isobutene TAME from methanol and 2-methyl-2-butene TAME from methanol and 2-methyl-1-butene

Bravo and Pyhalahti 1992; Thiel et al. 1997 Bravo and Pyhalahti 1992

Ethylene glycol from ethylene oxide and water

Gu and Ciric 1992

Isooctane from isobutane and 1-butene

Huss and Kennedy 1989

Ethylene benzene from benzene and ethylene

Smith et al. 1990

Cumene from benzene and propylene

Shoemaker and Jones 1987

tert-Butyl alcohol from isobutene and water

Velo et al. 1988

Nylon 6.6 prepolymer from adipic acid and hexamethylenediamine

Jaswal and Pugi 1975

Figure 8.52 System properties of the MTBE synthesis process.

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The reaction mechanism is

C4 H8 + CH3 OH ↔ C5 H12 O . a

b

(8.3)

c

In most cases there is some n-butane present in the feed mixture that is inert for this reaction. The system forms three binary minimum azeotropes; see Figure 8.52. The inert species, n-butane, is the peak of the concentration tetrahedron of Figure 8.52. The existence of the three binary azeotropes leads to a boundary distillation area that cannot be crossed by distillation. Consequently, sequential realization of reaction and distillation is very difficult. A reactive distillation process for MTBE synthesis is presented in Figure 8.53. The reaction parameters of the system are well known and, for instance, given by Espinosa et al. 1995; Thiel et al. 1997. The chemical equilibrium forms a section of a cone mantle as described in Section 8.7.2. All reactive distillation lines run on the cone surface. The origin of the reactive distillation lines is the high boiling component methanol, the terminus either the n-butane/methanol azeotrope or the ibutene. There exists a reactive boundary distillation line that divides the cone mantle in two sections with different termini of distillation lines. The process is effected at a pressure of 8 bar in a column with 14 trays. The upper 8 trays are equipped with a heterogeneous catalyst [Bessling et al. 1997]. The i-butene is totally converted into MTBE that is withdrawn as bottoms from the column. The overhead fraction is the minimum azeotrope formed by n-butane and small parts of methanol. The concentration profile in the column is qualitatively shown in Figure 8.53. It follows a reactive distillation line in the upper section and a non-reactive distillation line, which originates from pure MTBE in the lower section. As described earlier the material balance around the column forms a straight line only in the transformed coordinates. Therefore, the state of the product MTBE has to be projected along the stoichiometric line into the n-butane/i-butene/methanol plane. ˙ , and the bottom product B˙ are collinear. In this plane the feed F˙ , the overhead D The reactive distillation lines in transformed coordinates are shown in Figure 8.53B. This diagram is clearer, but the product MTBE is not visible there. 8.7.2

Synthesis of Mono-ethylene Glycol

The synthesis of mono-ethylene glycol (C2 H6 O2 ) from ethylene oxide (C2 H4 O) and water (H2 O) has been studied by Ciric and Gu 1994. The main reaction is

C 2 H 4 O + H 2 O ↔ C2 H 6 O 2 . a

b

(8.4)

c

A side reaction generates the unwanted by-product diethylene glycol (C4 H10 O3 ):

C2 H4 O + C2 H6 O2 ↔ C4 H10 O3 . a

c

d

(8.5)

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.53 A) Reactive distillation lines for MTBE synthesis. B) Reactive distillation lines for MTBE synthesis in transformed coordinates [Espinosa et al. 1995].

8.7 INDUSTRIAL PROCESSES OF REACTIVE DISTILLATION

473

The boiling temperatures of the four involved species increase from the low boiler ethylene oxide (a) to water (b), ethylene glycol (c), and diethylene glycol (d). Hence, the situation is the same as described in Section 8.7.3. The reaction is sufficiently fast even without a catalyst. Figure 8.54 shows the process established in a single column with 10 equilibrium stages. The low boiling reactant ethylene oxide is fed into the middle and the high boiling water into the head of the column. By rigorous simulation the authors proved that in the upper column section, in spite of stoichiometric amounts of reactants, the concentration of water in the liquid phase is about 200 times higher than the concentration of ethylene oxide. The ethylene oxide is present in the vapor phase mainly. These conditions favor the main reaction (Eq. (8.4)). Liquid concentrations of ethylene glycol are lower than 5 % in the reactive section of the column. In the lower column section, the ethylene glycol is concentrated up to 95 %. The rest is diethylene glycol with some unconverted water. The authors state that an excess of water as high as 60 % would be necessary to get the same selectivity in a plug flow reactor. However, this excess water would have to be separated by an additional distillation column.

Figure 8.54 Synthesis of ethylene glycol in a reactive distillation column.

8.7.3

Synthesis of TAME

Since there is a shortage of isobutene for MTBE synthesis, tertiary amyl methyl ether (TAME) is used as an antiknock additive to gasoline in increasing amounts. The advantage of TAME is its synthesis from C5 olefins. A modern process for TAME synthesis is presented in Figure 8.55 [Hoffmann et al. 1997]. The C5 olefin cut contains between 15 and 50 wt% isoamylene that reacts with methanol. In the first step of the process, impurities, which may poison the catalyst, are removed by a water scrubber. The major part (70 %) of the conver-

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8 INDUSTRIAL DISTILLATION PROCESSES

Figure 8.55 Process for synthesis of TAME.

sion takes place in the reactor R-1 with a sulfonic acid ion exchange resin bed. The final conversion is established in a reactive distillation column with a catalytic section above the feed. The high boiler TAME is the bottom fraction of the column. The ˙ is the minimum azeotrope formed by C5 olefins and methanol. overhead fraction D1 The methanol is separated from the olefin by extraction with water. The raffinate is split by distillation into an overhead fraction methanol and a bottom fraction water. Both fractions are recycled within the process to the reactor and the extractor, respectively. The physical and chemical properties of the systems are described in detail by Hoffmann et al. 1997. There exist a MeOH/C5 olefin and a methanol/TAME minimum azeotrope that are connected by a boundary distillation line. The reactive distillation column is, in the reactive section, equipped with a catalytic active random packing. An inert random packing is installed below and above the catalytic section. 8.7.4

Synthesis of Methyl Acetate

An important industrial application of reactive distillation is the production of methyl acetate from methanol and acetic acid. This system exhibits several azeotropes, which make the downstream processing of the products very complex [Siirola 1996]. The conventional process shown in Figure 8.56 consists of eight distillation columns, one extractor, and one decanter. In this case, it is possible to replace the conventional process by a reactive distillation column and a single-stage distillation; see Figure 8.57. Within the column there

8.7 INDUSTRIAL PROCESSES OF REACTIVE DISTILLATION

475

Figure 8.56 Conventional process for the synthesis of methyl acetate from methanol and acetic acid.

Figure 8.57 Process for the synthesis of methyl acetate from methanol and acetic acid via reactive distillation.

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8 INDUSTRIAL DISTILLATION PROCESSES

exist a rectifying section, a stripping section, an extraction section, and a reactive distillation section. Essential for the effectiveness of the process is that the reactants acetic acid and methanol are fed into the column at different locations. The high boiler acetic acid is fed above, and the low boiler methanol below the reactive section. The process in the reactive section is, in essence, a chemical absorption with superimposed distillation of the reaction products. The single-stage distillation effects the recovery of the homogeneous catalyst sulfuric acid. This process element can be discarded if a heterogeneous catalyst (e.g. acidic ion exchanger) is used. A comparison of the processes shown in Figures 8.56 and 8.57 demonstrates the high potential of reactive distillation for process simplification. This type of processes is generally applicable to systems with reversible chemical reactions, e.g. to esterification and etherification of alcohols, to alkylations, to dimerization of olefins, and to hydrogenation of aromatics [Sundmacher and Kienle 2003].

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Happe, J., Cornell, P.W., and Eastman, D. (1946). Extractive distillation of C4-hydrocarbons using furfural. AIChE Journal 4: 189–214. Hartig, H. and Regner, H. (1971). Verfahrenstechnische Auslegung einer Veresterungskolonne. Chemie Ingenieur Technik 43 (18): 1001–1007. Hebel, H. and Kaspareit, M. (1971). Die Gewinnung von Butadien nach dem Difex-Verfahren. Chemische Technik 23 (7): 419–423. Hegner, B., Hesse, D., and Wolf, D. (1973). Möglichkeiten der Berechnung bei heteroazeotroper Destillation. Chemie Ingenieur Technik 45 (14): 942–945. Herfurth, H., Meirelles, A., and Weiß, S. (1987). Azeotropdestillation von Ethanol/Wasser mit Cyclohexan als Schleppmittel. Chemische Technik 39 (8): 331–334. Hoffman, E.J. (1964). Azeotropic and Extractive Distillation. New York: Interscience Publishers. Hoffmann, U., Krummradt, H., Rapmund, P., and Sundmacher, K. (1997). Erzeugung des Kraftstoffethers TAME durch Reaktivdestillation. Chemie Ingenieur Technik 69 (4): 483–487. Hunsmann, W. and Simmrock, K.H. (1966). Trennung von Wasser, Ameisensäure und Essigsäure durch Azeotrop-Destillation. Chemie Ingenieur Technik 38 (10): 1053–1059. Huss, A. and Kennedy, C.R. (1989). Hydrocarbon processes comprised of catalytic distillation using Lewis acid promoted inorganic oxide catalyst systems. US Patent 4,935,577. Jaswal, I. and Pugi, K. (1975). Preparation of polyamides by continuous polymerization. US Patent 3,900,450. Johannisbauer, W. and Jeromin, L. (1992). Structured column packings in the oleochemical industry. Distillation Symposium, Birmingham: B77–B83. Keyes, D.B. (1929). The manufacture of anhydrous ethyl alcohol. Industrial & Engineering Chemistry 21 (11): 998–1001. Komatsu, H. (1977). Application of the relaxation method for solving reacting distillation problems. Journal of Chemical Engineering of Japan 10 (3): 200–205.

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Lackner, K. (1981). N-formyle-morpholine, the solvent for aromatics recovery and it ranges of applicability. Erdoel & Kohle Erdgas, Petrochemie 34 (1): 26–30. Meyers, R.A. (1996). Handbook of Petroleum Refining Processes. New York: McGraw-Hill. Mueller, E. (1980). Gewinnung von Aromaten durch Extraktivrektifikation. VT Verfahrenstechnik 14 (9): 551–555. Othmer, D.F. (1974). Azeotropic distillation, Part I: Background theory. Verfahrenstechnik 8 (3): 94–99. Perry, R.H., Green, D.W., and Maloney, J.O. (1984). Perry‘s Chemical Engineer Handbook. New York: McGraw-Hill. Peters, W.D. and Rogers, R.S. (1968). Improved furfural extraction process. Hydrocarbon Processing 47 (11): 131–134. Pilhofer, T. (1983). Energiesparende Alternativen zur Rektifikation bei der Rückgewinnung organischer Stoffe aus wässrigen Lösungen. VT Verfahrenstechnik 17 (9): 547–549. Poth, N., Frey, T., and Stichlmair, J. (2002). MINLP optimization of kinetically controlled reactive distillation processes. Computer-Aided Chemical Engineering (Proceedings of ESCAPE-11) vol. 9: 79–84. Pucci, A., Mikitenko, P., and Asselineau, L. (1986). Three-phase distillation. Simulation and application to the separation of fermentation products. Chemical Engineering Science 41 (3): 485–494. Rautenbach, R. and Albrecht, R. (1989). Membrane Processes. New York: Wiley. Rautenbach, R., Knauf, R., Struck, A., and Vier, J. (1996). Simulation and design of membrane plants with AspenPlus. Chemical Engineering & Technology 19 (5): 391–397. Rehfinger, A. and Hoffmann, U. (1990). Kinetics of methyl tertiary butyl ether liquid phase synthesis catalyzed by ion exchange resin. Chemical Engineering Science 45 (6): 1605–1617. Rizvi, S.S.H. and Heidemann, R.A. (1987). Vapor–liquid equilibria in the ammonia/water system. Journal of Chemical and Engineering Data 32 (2): 183–191. Ruckenstein, E. and Park, J.S. (1990). The separation of water/ethanol mixtures by pervaporation through hydrophilic–hydrophobic composite

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membranes. Journal of Applied Polymer Science 40: 213–220. Sattler, K. (1988). Thermische Trennverfahren. Weinheim: VCH Verlagsgesellschaft. Schoenmakers, H. (1984). Alternativen zur Aufarbeitung desorbierter Lösungsmittel. Chemie Ingenieur Technik 56 (3): 250–251. Schweitzer, P.A. (1979). Handbook of Separation Techniques for Chemical Engineers. New York: McGraw-Hill Book Company. Seader, J.D. (1984). Distillation. In: Perry’s Chemical Engineer Handbook (eds R.H. Perry, D.W. Green, and J.O. Maloney), pp. 13–1–13–97. New York: McGraw-Hill. Shoemaker, J.D. and Jones, E.M. (1987). Cumene by catalytic distillation. Hydrocarbon Processing 66 (6): 57–58. Siirola, J.J. (1996). Industrial applications of chemical process synthesis. Advances in Chemical Engineering 23: 1–62. Smith, L.A. (1990). Method for the preparation of methyl-tertiary butyl ether. US Patent 4,978,807. Smith, L.A., Jones, E.M., and Hearn, D. (1990). Catalytic distillation – a new chapter in unit operations. AIChE Spring Meeting Houston, TX, USA. Sorensen, J.M. and Arlt, W. (1979). Liquid–Liquid Equilibrium Data Collection. In: Chemistry Data Series. Frankfurt/Main: DECHEMA. Sowerby, B. and Crittenden, B.D. (1991). A vapor phase adsorption and desorption model for drying the ethanol/water azeotrope in small columns. Transactions of the Institution of Chemical Engineers 69 (Part A): 3–13. Stage, H. (1988). Richtlinien für Einsatz, Auslegung und Betrieb von Fallfilmrohrverdampfern zur Aufarbeitung thermisch sowie oxidationsempfindlicher hochsiedender Flüssigkeitsgemische. European Journal of Lipid Science and Technology 90 (10): 399–411. Stichlmair, J. (1979). Chlorwasserstoffabsorption. Chemie-Anlagen + Verfahren, No. 1 pp. 40–46. Stichlmair, J. (1988). Distillation and rectification. In: Ullmann‘s Encyclopedia of Industrial Chemistry, pp. 4–1–4–94. vol. B3, VCH Verlagsgesellschaft Weinheim.

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Stichlmair, J. and Frey, T. (1999). Reactive distillation processes. Chemical Engineering and Technology 22 (2): 95–103. Stichlmair, J. and Frey, T. (2001). Mixed integer non-linear programming: optimization of reactive distillation processes. Industrial and Engineering Chemistry Research 40 (25): 5978–5982. Stichlmair, J. and Herguijuela, J.R. (1992). Separation regions and processes of zeotropic and azeotropic ternary distillation. AIChE Journal 38 (10): 1523–1535. Streich, M. (1975). Exergieverluste bei thermischen Trennungen. VT Verfahrenstechnik 9 (5): 240–243. Strigle, R.F. (1987). Random Packings and Packed Towers, Design and Application. Houston, TX: Gulf Publishing Company, Book Division. Sulzer (2000). Separation Technology for the Hydrocarbon Process Industry. Winterthur: Sulzer Chemtech. Sundmacher, K. and Kienle, A. (2003). Reactive Distillation: Status and Future Directions. Weinheim: Wiley-VCH. Sundmacher, K., Rihko, L.K., and Hoffmann, U. (1994). Classification of reactive distillation processes by dimensionless numbers. Chemical Engineering Communications 127 (4): 151–167. Tassios, D.P. (1972). Extractive and Azeotropic Distillation. Advances in Chemistry Series 115, vol. 115. Washington, D.C.: American Chemical Society. Thiel, C., Sundmacher, K., and Hoffmann, U. (1997). Residue curve maps for heterogeneously catalysed reactive distillation of fuel ethers MTBE and TAME. Chemical Engineering Science 52 (6): 993–1005. Tuchlenski, A., Beckmann, A., Reusch, D., Düssel, R., Weidlich, U., and Janowsky, R. (2001). Reactive distillation – industrial applications, process design & scale-up. Chemical Engineering Science 56 (2): 387–394.

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Udriot, H., Araque, A., and von Stockar, U. (1994). Azeotropic mixtures may be broken by membrane distillation. The Chemical Engineering Journal and the Biochemical Engineering Journal (Lausanne, Switzerland) 54 (2): 87–93. Ung, S. and Doherty, M.F. (1995). Synthesis of reactive distillation systems with multiple equilibrium chemical reactions. Industrial and Engineering Chemistry Research 34 (8): 2555–2565. Ushio, S. (1972). Extract isoprene with DMF. Chemical Engineering 79 (5): 82–83. Velo, E., Puigjaner, L., and Recasens, F. (1988). Inhibition by product in the liquid-phase hydration of isobutene to tert-butyl alcohol: kinetics and equilibrium studies. Industrial and Engineering Chemistry Research 27 (12): 2224–2231. Volkamer, K., Schneider, K.J., and Lindner, A. (1981). Entwicklungsarbeiten am Butadienverfahren der BASF. Erdoel & Kohle Erdgas, Petrochemie 34 (8): 343–346. Watkins, R.N. (1979). Petroleum Refinery Distillation. Houston, TX: Gulf Publishing Company. Westphal, G. (1987). Kombiniertes Adsorptions-Rektifikationsverfahren zur Trennung eines Flüssigkeitsgemisches. Patent DE 3712291 A1. Weyd, M., Richter, H., Kuenert, J.T., Voigt, I., Tusel, E., and Brueschke, H. (2010). Effiziente Entwässerung von Ethanol durch Zeolith-Membranen in Vierkanalgeometrie. Chemie Ingenieur Technik 82 (8): 1257–1260. Widago, S. and Seider, W.D. (1996). Azeotropic distillation. AIChE Journal 42 (1): 96–130. Wunder, R. (1990). Abtrennung und Rückgewinnung von anorganischen Stoffen durch Absorption. In: Stofftrennverfahren in der Umwelttechnik. Düsseldorf: Preprints GVC-Gesellschaft für Verfahrenstechnik und Chemieingenieurwesen.

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9 Design of Mass Transfer Equipment In mass transfer equipment, the mass transfer always takes on the same principle, shown in Figure 9.1. Initially, a two-phase system must be created in the apparatus. In addition, the apparatus should be so constructed that it offers optimal conditions for a rapid mass transfer across the interface. Finally, both phases must be separated again.

Figure 9.1 Principle of mass transfer processes.

The basic requirements for a good mass transfer can best be explained by the following basic equation for mass transfer:

N˙ = kOG · Aeff · (y ∗ − y) .

(9.1)

The maximum mass transfer rate N˙ is achieved when all three terms on the righthand side of the above equation are as large as possible: • The mass transfer coefficient kOG depends on the flow conditions of the two phases and the physical properties; it increases with increasing velocities in both phases. The situation is improved if new fluid elements constantly reach the phase boundary since this results in a non-steady mass transfer with short transfer times. Columns that allow steady renewal of the interface area are particularly effective. • The effective exchange area (interfacial area) Aeff for mass transfer depends on the design of the apparatus as well as on the local velocities of both phases. The Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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internals of the apparatus affect the interfacial area significantly and thus offer the possibility to influence the mass transfer in the column through their design. • The driving force (y ∗ −y) for the mass transfer between the phases is greatest when the two phases are conducted in backmixing-free countercurrent to each other. The two phases should be distributed equally over the entire flow cross section.

9.1

Types of Design

Tray columns and packed columns are most often used for distillation [Kunesh et al. 1995] since they provide a very good counterflow conduction of the phases and allow large heights. As already described, a large interfacial area is of central importance. To generate this there are two possibilities. Either the apparatus provides a very large internal surface where the liquid phase can be distributed as a film, or one phase is highly dispersed, so that small droplets or bubbles are formed, which have a large surface. In packed columns the former is made. There, structured or random packings provide for an increase in the internal surface. In tray columns, the gas is introduced specifically into the liquid at the bottom of the tray and mixes with it intensely. The resulting two-phase layer (also called froth layer) has a large specific interfacial area, which is constantly renewed. The height of a column for a given separation task is obtained primarily from the required number of theoretical plates. These can be found, for example, in the McCabe–Thiele diagram by stage construction (see Section 4.2). The column diameter is calculated with the laws of fluid mechanics. Here, especially the gas volume flow is decisive. The fluid dynamics in the column also determine the load limits of the mass transfer apparatus. Their knowledge is essential to the design and operation of the apparatus. The use of mass transfer machines (based on the design proposed by Ramshaw and Mallinson 1983) has been proposed repeatedly for distillation. Although such devices have proved useful for extraction [Mersmann et al. 1986], they have not been fully accepted in distillation yet. However, there is ongoing research (e.g. Rao et al. 2004; Wang et al. 2011; Gładyszewski and Skiborowski 2018) on this topic as they can be very useful in special cases where high columns are not possible. 9.1.1

Tray Columns

Figure 9.2 shows the most important features of a tray column. The gas flows upward within the column through perforations in the horizontal trays, and the condensed liquid flows downward in countercurrent. However, on the individual trays, the two phases exhibit cross-flow relative to each other. The liquid is fed on the tray through a side outlet of the downcomer. The rising gas passes from below through the openings in the tray and forms a two-phase layer with a large interfacial area with the liquid flowing across the tray. The two-phase mixture then passes over the weir and flows into the downcomer.

9.1 TYPES OF DESIGN

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Figure 9.2 Cutaway section of a tray column: (a) downcomer, (b) integrated support, (c) sieve trays, (d) manhole, (e) outlet weir, (f) inlet weir, (g) long downcomer of bottom tray (to bypass the liquid at the reboiler inlet), and (h) seal pan.

The task of the downcomer is to degas the liquid and to convey it to the next tray. In the downcomer the gas is disengaged from the liquid, rises, and flows upward in the next upper tray. The liquid is passed downward to the inlet of the next lower tray. Backed-up liquid, which compensates for the pressure drop caused by the gas as it passes through the tray above, forms a seal and prevents the gas from this tray going through the downcomer. The liquid moves downward in the downcomer in countercurrent to the gas, but bypasses the gas. The implementation of the gas–liquid contact has produced a variety of designs in recent decades of industrial use. Over the decades, also the requirements changed.

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Initially, the pure functionality was in the focus; recently optimization of operating parameters and cost became more important. Bubble cap trays were used almost exclusively for many years. Usually they have chimneys with mounted caps fitted over the holes. The vapor rises upward through the chimney or riser, is deflected downward by the cap, and exits through slots on the side of the cap. The design of the bubble caps prevents liquid from leaking downward (weeping) through the tray. However, the complex bubble caps are relatively expensive and have a higher pressure drop than other designs. Afterward, sieve trays became increasingly important. They are inexpensive, provide a high separation efficiency, and have a low pressure drop across the tray. A disadvantage of this design is that the momentum of the gas passing through is directed vertically upward and thus can entrain liquid to the next tray. It is also very important to install sieve trays precisely leveled. Valve trays are again a more complex design. The cross section of the orifices in the tray is adjusted to prevent the liquid from leaking at low gas loads and to avoid excessive pressure drop at high gas loads. They are more expensive than sieve trays but offer a higher turndown ratio regarding the gas load. Fixed valve trays have become increasingly important in recent years. Since their openings are not cut out, but only bent upward, the gas enters horizontally into the liquid. This promotes thorough mixing. Some fixed valve types are so designed that the momentum of the gas also supports the liquid flow along the tray. In addition to these main types, there are numerous other variations (e.g. dual flow trays, tunnel trays, etc.) that are used particularly in special applications. 9.1.2

Packed Columns

Figure 9.3 shows the most important features of a packed column. The gas introduced at the bottom flows through the free cross section of the packing to the top, while the liquid flows downward in countercurrent to the gas. In packed columns, therefore, there is a perfect countercurrent, which is advantageous for mass transfer in comparison with tray columns (see last term of Eq. (9.1)). The liquid should wet the surface of the packing (random or structured) as completely as possible to maximize the interfacial contact area between the phases. The type of packing greatly influences the efficiency of the column. Often, random packings consisting of specially shaped particles (rings, saddles; see Figure 9.34) are used: their surface area should be as large as possible, they should have a low flow resistance, and they should be distributed homogeneously in the column (no large cavities). Structured packings (see Figure 9.35) comply more readily with these requirements and are widely used. Uniform distribution of the liquid over the entire cross-sectional area is very important. Initially, this is given by a properly designed liquid distributor (see Figure 9.3). With increasing run length of the liquid, usually the liquid flow becomes increasingly uneven distributed (which is called maldistribution), which can critically reduce the separation efficiency of the apparatus and presents an uncertainty in column design. Therefore, the height of the individual packed beds is limited to 6 – 8 m, also

9.1 TYPES OF DESIGN

485

depending on the type of packing and the separation task.

Figure 9.3 Cutaway section of a packed column with structured packing [Sulzer 1982]: (a) liquid distributor, (b) liquid collector, (c) structured packing, (d) support grid, and (e) manhole.

A partition of the packing is also required when liquid is to be fed or withdrawn. The liquid leaving the upper packing bed is collected and fed into an annular channel, where it is thoroughly mixed. It is then redistributed by a second distributor. The internals of the column should provide minimum resistance to the gas flow. Modern packings have a relative free cross-sectional area of more than 90 %. The liquid collector and distributor should also achieve high values here. The support grid of the packing is often a barrier to flow, especially when its free cross-sectional area is blocked by the lowest layer of random packing particles in random packed

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columns. 9.1.3

Criteria for Use of Tray or Packed Columns

The right choice of equipment is very important for an effective and economical separation. An unambiguous choice is not always possible because both tray and packed columns can be modified to meet different requirements. Tray columns are more likely to be used with larger column diameters. For a diameter less than 0.8 m, a cartridge design has to be employed [Kleiber 2016]. The gas load in tray columns has to be kept within a relatively narrow range; only valve trays allow greater operational flexibility. The liquid load, however, can be varied over a very wide range; see Figure 9.4. Tray columns can also be operated efficiently at very low liquid loads. A disadvantage, however, is their relatively high pressure drop, especially if the separation requires a large number of trays. Tray columns have a relatively high liquid hold-up in both the two-phase layer on the tray and the downcomer. This compensates for fluctuations in feed composition. However, the high hold-up results in a high liquid residence time in the column, which can lead to the decomposition of thermally unstable substances. The hold-up is also significant when a chemical reaction occurs during distillation. Special cap designs allow the liquid on the tray to be retained for short shutdown periods so that column operation can be resumed quickly. A further advantage of tray columns is that heat exchanger coils incorporated on the trays allow heat to be added or removed easily. Tray columns are also relatively insensitive to solid contaminants in the liquid and fouling. It is also easy to supply or remove gas or liquid side streams on any tray.

Figure 9.4 Sketch of the operating region of a tray column (A) and a packed column (B).

Random packed columns are used in towers with smaller diameter. However, structured packings can also be installed in columns with very large diameters. Packed columns are extremely flexible in terms of gas load, but they require a minimum liquid load (see Figure 9.4), which is a problem in vacuum operation. Newly developed structured packings, however, have brought substantial progress to this area,

9.2 DESIGN OF TRAY COLUMNS

487

too. The low pressure drop in packed columns, compared with tray columns, is a huge advantage. The risk of decomposition of thermally unstable substances is also lower in packed columns, since the liquid hold-up is very small. Liquids that contain particulate contaminants or tend to crystallize are not suitable for packed columns. However, packed columns are less sensitive to foaming than tray columns. Ceramic packings are resistant to corrosion.

9.2

Design of Tray Columns

In Figure 9.5, the schematic illustration of a tray column with standard values for the structural design is given. Shown are the three main types: bubble cap, valve, and sieve tray. In addition, there are a variety of special designs. On each tray the gas emerging from the openings in the plate penetrates the liquid and, thus, generates a two-phase mixture with a large interfacial area. The larger the interfacial area, the higher is the mass transfer rate. Very important for a high mass transfer rate is also a uniform gas flow through all openings in the tray (see Section 9.2.2.2). The column cross section is usually circular, while the downcomers are chordal (Ad ). The openings for the gas on the active surface Aac of the tray are distributed as evenly as possible. In most cases they have a square or equilateral triangular arrangement on the tray. The relative free area ϕ of the tray is defined as the ratio of the area Ao of all openings and the active area Aac :

ϕ=

Ao . Aac

(9.2)

The generation of the two-phase layer and thus the interfacial area is obtained by direct introduction of the gas into the liquid. This results in a pressure drop of the gas and is only possible by input of energy. A high pressure drop also means higher operating costs. A reduced pressure drop and at the same time a uniformly mixed two-phase layer with a large interfacial area are tried to obtain by improved structural design of the tray. This is usually only possible through higher investment costs. 9.2.1

Design Parameters of Tray Columns

Figure 9.6 shows some examples of fixed valves and floating valves. Typically, valves are smaller than bubble caps (dv < dcap ). Very important parameters are tray spacing H , weir height hw , and downcomer clearance hcl . Standard geometrical dimensions, developed by practical experience, are given in Table 9.1. These empirical findings will be theoretically consolidated in the following sections.

488

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.5 Schematic illustration of a tray column with nomenclature and typical dimensions.

489

9.2 DESIGN OF TRAY COLUMNS

Figure 9.6 Examples of fixed valves and floating valves. A) Round fixed valve (type VG0). B) Trapezoid fixed valve (type MVG). C) Mini fixed valve (type R-MV). D) Covered hole (type ProValve). E) Round floating valve (type V1). F) Caged valve (type A3). G) Rectangular floating valve (type BDH). H) Caged valve (type T). I) Special valve (type Vario Flex). Source: Data generated with TrayHeart [WelChem Process Technology 2020].

Table 9.1 Standard geometrical parameters of trays. Vacuum

Atmospheric pressure

Tray spacing H in m

0.4 – 0.6

0.4 – 0.6

0.3 – 0.4

Weir length lw

(0.5 – 0.6) · Dc

(0.6 – 0.7) · Dc

(0.6 – 0.8) · Dc

Weir height hw in mm

25 – 35

35 – 50

40 – 80

Downcomer clearance hcl

0.7 · hw

0.8 · hw

0.8 · hw

Bubble cap diam. dcap in mm

50 – 150

50 – 150

50 – 150

Bubble cap pitch pcap

1.25 · dcap

(1.25 – 1.4) · dcap

1.5 · dcap

Valve diameter dv in mm

25 – 50

25 – 50

25 – 50

Valve pitch pv

1.5 · dv

(1.7 – 2.2) · dv

(2 – 3) · dv

Hole diameter dh in mm

4 – 13

4 – 13

4 – 13

Hole pitch ph

(2.5 – 3) · dh

(3 – 4) · dh

(3.5 – 4.5) · dh

Relative free area ϕ

0.10 – 0.15

0.06 – 0.10

0.05 – 0.08

9.2.2

High pressure

Operating Region of Tray Columns

Careful column design requires the determination of the entire operating region and the proper selection of operating parameters in order to maintain a safety margin with respect to liquid and gas load limits. Hence, all the phenomena that limit column operation have to be studied first. From the proper understanding of the limiting factors, relations for the calculation of the boundary lines of the operating region can

490

9 DESIGN OF MASS TRANSFER EQUIPMENT

be developed. 9.2.2.1

Maximum Gas Load

The gas load in a tray column can be increased up to a point where the gas blows the liquid off the tray in form of fine droplets. Then, the liquid no longer flows in countercurrent to the gas, and the proper column operation ends. Souders and Brown 1934 were the first to develop a systematic approach to calculate the maximum gas load of a tray. They consider a single drop suspended above the tray and formulate the equilibrium of frictional force and gravitational force minus buoyant force:

d2 · π %G 2 d3 · π · ζ0 · · uG = · (%L − %G ) · g . 4 2 6

(9.3)

Here, uG is the superficial gas velocity, d is the diameter and ζ0 is the drag coefficient of the (spherical) drop, %L is the density of the liquid, and %G is the density of the gas. The difference between both densities is sometimes written as ∆% = %L − %G . The gas load is best expressed by the F -factor, which is a measure of the square root of the kinetic energy of the gas flow. It is defined by

F = uG ·

√

%G .

From Eqs. (9.3) and (9.4), it follows that s 4·d·g √ F = · %L − %G . 3·ζ

(9.4)

(9.5)

The first right-hand term contains the drop diameter d that is unknown in most cases. According to Souders and Brown 1934, this term is replaced by the gas capacity factor CG . Then Eq. (9.5) becomes √ F = CG · %L − %G . (9.6) The capacity factor CG has to be determined by experiments. Many correlations have been developed from experimental data. Some of them are listed in Table 9.2. The most commonly used correlation is that of Fair presented in Figure 9.7 [Fair 1961]. He found that CG is proportional to σ 0.2 (σ denotes the surface tension of the liquid) and depends on the relative free area ϕ. A theoretical approach based on first principles has been developed by Stichlmair 1974. He assumes that the gas on the tray forms the continuous phase while the liquid is dispersed into droplets (spray regime; see Section 9.2.3.1). The size of the droplets in the two-phase layer on a tray is determined by the gas jets emerging from the holes or slots of a sieve tray and a bubble cap tray, respectively. When the jets penetrate the two-phase mixture, they act alike a jet mill. The liquid drops are entrained into the jets, accelerated due to the frictional forces, and, simultaneously, deformed. After having lost their spherical shape, the drops are torn to pieces. Decisive for the drop stability in the gas jet is the ratio of frictional force to surface force, i.e. the Weber number. Drops are stable in a gas jet up to a critical value of the Weber number,

491

9.2 DESIGN OF TRAY COLUMNS

Figure 9.7 Correlation of the capacity factor CG for sieve trays according to Fair 1961. 0 has to be corrected with respect to surface tension σ and relative The capacity factor CG free area ϕ.

Table 9.2 Correlations for the gas capacity factor CG . References

Parameters

Range of validity

Souders and Brown 1934

CG = f (H, σ)

Bubble caps

Kirschbaum 1950

CG = f (H 1/2 , dcap )

Bubble caps

Kirschbaum 1960

CG = f (H 1/2 )

Sieve trays

Wenzel 1957

CG = f (H)

Cryogenic systems

Huang and Hodson 1958

CG = f (H, σ) CG = f (H, V˙ L /V˙ G · %L /%G )

Bubble caps Sieve trays

Smith et al. 1963

CG = f (H, V˙ L /V˙ G · %L /%G , σ 0.2 , ϕ) CG = f (H − hf , V˙ L /V˙ G · %L /%G )

Hoppe and Mittelstrass 1967

CG = f (H 1/2 , ϕ0.167 )

Sieve trays

Glitsch, Inc. 1974

CG = f (H, %G )

Valve trays

Koch 1968

CG = f (H)

Valve trays

Fair and Matthews 1958 Fair 1961

Bubble caps All tray design

492

9 DESIGN OF MASS TRANSFER EQUIPMENT

We cr , which is calculated with the gas velocity in the orifices uGh . From the general definition of the Weber number % · u2 · d , σ it follows that We =

d=

We cr · σ Fh2

or

(9.7)

d=

We cr · σ · ϕ2 , F2

(9.8)

where σ is the surface tension of the liquid, ϕ is the relative free area, F is the gas load factor defined in Eq. (9.4), and Fh is the gas load factor in the holes:

Fh =

F √ = uGh · %G . ϕ

Combining Eq. (9.8) with Eq. (9.5) yields  
1/4 4 · We cr 1/4 Fmax = · ϕ2 · σ · (%L − %G ) · g . 3 · ζ0

(9.9)

(9.10)

According to Wallis 1969, the value of the critical Weber number is approximately 12 for liquids with low viscosity. If the drag coefficient ζ0 is assumed to be 0.4 (a value valid for rigid spheres in turbulent flow), then Eq. (9.10) becomes
1/4 Fmax = 2.5 · ϕ2 · σ · (%L − %G ) · g . (9.11) This theoretically derived equation agrees very well with published experimental data and with the correlation of Fair 1961. In Eq. (9.11), there is a proportionality of σ 0.25 , while Fair 1961 empirically obtained a proportionality of σ 0.2 . The dependence of the correlation of Fair 1961 on the relative free area ϕ also corresponds well to the relation Fmax ∼ ϕ1/2 in Eq. (9.11). Equation (9.11) describes the process of blowing the liquid off the tray and is applicable only to columns with very large tray spacing (approximately H > 0.5 m). If the tray spacing is smaller, the height of the two-phase layer rather than the blowoff process is critical to the highest possible gas load. An equation for predicting the froth height (the height of the two-phase layer) is given in Section 9.2.3.3. For safe operation the froth height has always to be less than the tray spacing. 9.2.2.2

Minimum Gas Load

The gas load of the column tray can be reduced to a point where either the gas no longer flows evenly through all the tray openings or the liquid leaks through the tray (called weeping). Both modes of operation should be avoided as they reduce the tray efficiency. According to Mersmann 1963, a non-uniform gas flow through the holes of a sieve tray is avoided if the Weber number We = % · u2Gh · dh /σ calculated with the hole diameter dh exceeds a critical value of 2. From this condition it follows that s s 2·σ 2·σ Fh,min = or Fmin = ϕ · . (9.12) dh dh

493

9.2 DESIGN OF TRAY COLUMNS −1/2

Hence, the minimum gas load is Fh,min ∼ dh . According to Ruff et al. 1976, the liquid is prevented from weeping through a sieve tray if the Froude number Fr exceeds a critical value. From this condition it follows that s 1.25 (%L − %G ) Fh,min = 0.37 · dh · g · or %0.25 G (9.13) s 1.25 (%L − %G ) Fmin = ϕ · 0.37 · dh · g · . %0.25 G 1/2

Here, the minimum gas load is proportional to dh . Equations (9.12) and (9.13) are evaluated for three different systems in Figure 9.8. At small hole diameters up to approximately 2 or 3 mm, the uniform gas flow through all holes is the limiting mechanism. The minimum gas load Fh,min decreases with increasing hole diameter. At hole diameters greater than 2 – 3 mm, the usual case in most columns, weeping of liquid through the tray becomes the decisive factor; the minimum gas load then increases with increasing hole diameter.

Figure 9.8 Minimum gas load Fh,min of sieve trays for three different systems: (1) air/water, (2) benzene/toluene, and (3) isobutane/n-butane.

Equations (9.12) and (9.13) do not take into account the influence of the liquid flow on the tray, although, for instance, large liquid loads require a somewhat higher minimum gas load to prevent leaking of liquid through the tray. However, if some weeping occurs at high liquid loads, it does not affect separation efficiency significantly since there still exists a countercurrent flow of the two phases. A rule of thumb says that a weeping rate of up to 10 % of the liquid flow on the tray may be tolerated. For bubble cap trays, determining the minimum gas load is more complex because the bubble caps present obstacles to liquid flow across the tray. This flow resistance

494

9 DESIGN OF MASS TRANSFER EQUIPMENT

leads to a hydraulic gradient along the flow path of the liquid. Hence, the height of clear liquid on the tray varies, and the risk of non-uniform gas flow increases. A detailed study on the minimum gas load of bubble cap trays has been published by Kemp and Pyle 1949. They found that the minimum gas load depends significantly on liquid load per weir length and on the number of bubble cap rows to be crossed by the liquid flow across the tray. The values of Fmin are generally higher on bubble cap trays than on sieve trays. 9.2.2.3

Maximum Liquid Load

Liquid flows down the column through the downcomers due to gravity. This limited driving force allows only a limited liquid flow rate V˙ L . The following four empirical rules are often used to determine the maximum liquid flow rate (e.g. Hoppe and Mittelstrass 1967): • The weir load V˙ L /lw (lw = weir length) should be less than 60 m3 /(m · h). • The liquid velocity uLd in the downcomer should not exceed a value of 0.1 m/s. • The volume of the downcomer should allow a liquid residence time τLd = H/uLd of more than 5 s. • The height of the clear liquid in the downcomer should not exceed half of the tray spacing. These rules were developed by long-term practical experience. However, they do not allow any insight into the limiting phenomena. The flow of liquid through the downcomer is, in principle, comparable to the flow of liquid through the opening of a vessel. This classic flow problem was solved by Torricelli as early as in 1644 [Torricelli 1644]. In modern writing Torricelli’s equation is p uLd = α · 2 · g · H . (9.14) Here, α denotes the discharge coefficient, which is approximately 0.61 for a thin, sharp-edged outlet. Some additional phenomena have to be taken into account for applying Eq. (9.14) to the downcomer of a tray column: • The downcomer does not contain clear liquid but a liquid–gas mixture with a relative liquid content of εLd . • The velocity uLd of clear liquid in the downcomer outlet can be expressed by uLd = εLd · V˙ L /(lcl · hcl ). • The density %G of the gas must not be neglected against the density %L of the liquid. • The liquid flows onto a tray with a clear liquid height hL . • Between inlet and outlet of the downcomer, there is a pressure difference due to the pressure drop ∆p of a tray. It can be expressed in terms of clear liquid height that is defined by hp = ∆p/(%L · g).

9.2 DESIGN OF TRAY COLUMNS

495

With these considerations the Torricelli Equation (9.14) turns into [Stichlmair 1978] s !   V˙ L %L − %G hp + hL = 0.61·εLd ·hcl · 2 · g · H · · 1− , (9.15) lcl %L εLd · H max

where lcl is the downcomer clearance length and hcl the clearance height of the downcomer. The parameter εLd is unknown; an approximate value of εLd ≈ 0.4 can be employed for non-foaming systems. Further details are given by Volpert and Stichlmair 1986. 9.2.2.4

Minimum Liquid Load

In principle, a column tray can be operated even with very low liquid loads since the height of the two-phase layer on the tray is kept at a minimum value by the outlet weir. However, with extremely low liquid loads, the liquid flows unevenly across the tray (maldistribution), reducing the mass transfer efficiency (see Section 9.2.4). Picket fence weirs are utilized to counteract this. Accordingly, the minimum height of the weir overflow is usually set at how ≈ 5 mm; this corresponds to a minimum liquid weir load of (V˙ L /lw )min ∼ = 2 m3 /(m · h). In small diameter columns, however, this value may be considerably lower. Especially at high gas loads, entrainment of liquid by the gas may determine the minimum liquid load (see Section 9.2.2.5). 9.2.2.5

Operating Region of a Tray Column

Careful design of a tray column requires the complete determination of the entire operating region. Figure 9.9 presents a graphical plot of the operating region of a sieve tray column for the benzene/toluene system at a pressure of 1 bar. The gas load V˙ G /Aac is plotted versus the liquid load V˙ L /Aac according to Molzahn and Schmidt 1975; loads are referred to the active area of a tray Aac . The operating region has been calculated with the equations developed in the previous sections. The upper limits for the gas and liquid flow are absolute limits that can never be exceeded. The lower limits (dashed lines), however, may be exceeded to a certain extent without encountering any flow problems. However, the mass transfer efficiency may gradually decrease. Generally, the operating conditions should be selected so that there is a sufficient safety distance to the boundary lines. The selection of the geometric mean of the minimum and maximum gas load as the actual gas load fulfills this condition. It is very interesting to study the influence of some design parameters on the operating region of a column. An example is shown in Figure 9.10. Here, the influence of the relative free area ϕ on the upper and lower boundaries for gas load is shown. According to Eq. (9.11), the maximum gas load is proportional to ϕ1/2 , whereas the minimum gas load is proportional to ϕ1 (see Eq. (9.13)). Hence, the width of the operating region becomes smaller as the relative free area ϕ increases. For this reason, the relative free area should not be chosen higher than ϕ = 0.15 (see Table 9.1). The influence of surface tension σ on the upper and the lower gas load limit is shown in Figure 9.11. According to Eq. (9.11), the upper limit is proportional to σ 1/4 . The lower limit, however, shows a behavior that is different for sieve trays with

496

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.9 Operating region of a tray column. The upper limits for gas and liquid load cannot be exceeded. The lower limits may be crossed without encountering serious flow problems.

Figure 9.10 Influence of the relative free area ϕ on the maximum and minimum gas load.

9.2 DESIGN OF TRAY COLUMNS

497

small and with large holes, respectively. Equation (9.12), which is valid for small holes, contains a proportionality of σ 1/2 . Hence, the operating region is wide at low surface tension and small at high surface tension. For trays with large holes, Eq. (9.13) applies, which has no dependence on surface tension. In this case, the operating region is small at low surface tension and large at high surface tension. Hence, trays with small holes should be used in systems with low surface tensions. This agrees well with practical experience. For instance, columns for cryogenic air separation, a system with very low surface tension, are typically equipped with sieve trays with very small holes, e.g. 1 mm or less [Moll 2014].

Figure 9.11 Maximum and minimum gas load as a function of surface tension σ of the liquid.

9.2.3

Two-Phase Flow on Trays

After determining the operating region of the column, the phenomena of two-phase flow on the tray have to be investigated to ensure safe column operation. The behavior and properties of the two-phase layer on the trays are important for the effectiveness of tray columns. 9.2.3.1

Flow Regimes

Three different structures or regimes of the two-phase layer can be distinguished (see Figure 9.12): • Bubble regime: The liquid forms the continuous phase. The gas rises in the liquid

498

9 DESIGN OF MASS TRANSFER EQUIPMENT

in the form of discrete bubbles. • Spray regime: The gas forms the continuous phase, and the liquid is dispersed into fine droplets. • Froth regime: This regime represents the intermediate state between the bubble and the spray regime. The gas–liquid layer is agitated intensively, and there is no clearly dispersed phase.

Figure 9.12 Schematic illustration of bubble regime, froth regime, and spray regime.

Bubble regime can exist only at very high pressures without crossing the lower gas load limit. This statement is based on the following considerations. √ The gas load in the holes of a tray should not be lower than √ Fh ≈ p 5 Pa (see Figure 9.8); therefore, the gas velocity in the holes is uGh = 5 Pa/ %G . However, bubbles rise with a velocity of approximately 0.1 – 0.2 m/s. Only if the density %G of the gas is very high (i.e. at high pressure) the gas velocity in the holes does not significantly surpass the rising velocity of the bubbles, as is necessary with bubble regime. At low or moderate pressures, the superficial gas velocity is much higher than the rising velocity of the bubbles. Thus, bubbles accumulate above the orifices and form a region with very high gas content, which suddenly breaks through the two-phase layer and rises upward. This jerky mode of operation is called froth regime. The gas load at which the transition from froth to spray regime (phase inversion) occurs is of special interest. Stichlmair 1978 analyzed experimental data for phase inversion on sieve trays (Lockett et al. 1976; Loon et al. 1973; Pinczewski and Fell 1972; Porter and Wong 1969; Pinczewski et al. 1975). From the data it follows that if the liquid level on the tray hL is high, phase inversion occurs at F/Fmax ∼ = 0.65. However, if clear liquid height hL is very low, phase inversion may occur much earlier. Hence, column design requires a minimum gas penetration depth of approximately 20 mm. In most cases, tray columns are operated in the froth region. Spray regime may exist sometimes in vacuum distillation columns.

9.2 DESIGN OF TRAY COLUMNS

9.2.3.2

499

Relative Liquid Hold-up

The relative liquid hold-up εL is a very important characteristic parameter of the two-phase layer on a tray. It is defined as the ratio of clear liquid height hL and froth height hf :

εL =

hL . hf

(9.16)

With knowledge of the froth height hf and the relative liquid content εL , the height of clear liquid hL on a tray can be easily calculated. Clear liquid height hL is one of the most important parameters for the pressure drop of a tray and also determines, together with the liquid hold-up in the downcomer, the residence time of the liquid in the column. Table 9.3 lists some references for the relative liquid content εL together with the parameters involved and the range of validity. The correlations have been evaluated for air/water and a sieve tray with a relative free area ϕ = 0.1 and a weir height hw = 50 mm. The results are plotted versus the gas load F in Figure 9.13. Almost all correlations are within a narrow range. All correlations and most of the published data are summarized by the following equation: 

εL = 1 −

F

0.28

Fmax

.

(9.17)

The term F/Fmax accounts for gas load, several properties of the system, and tray design. The relative liquid content decreases significantly with increasing gas load. p This relationship is valid if the term ϕ · %L /%G is larger than 5. Determination of the relative liquid hold-up for smaller values of this parameter is described in Stichlmair 1978. 9.2.3.3

Froth Height

Knowing the froth height on a tray is very essential as it has to be considerably lower than tray spacing. If the froth extends up to the next tray, it may block the openings in the tray, causing a sharp increase in pressure drop and eventually flooding. Published correlations for the froth height are often neither dimensionally correct nor do they adequately represent the limiting conditions. Table 9.4 lists some references for froth height correlations together with their parameters and range of validity. These correlations have been evaluated for a sieve tray (hw = 50 mm, V˙ L /lw = 18 m3 /(m · h)). The results are plotted in Figure 9.14 versus the gas load factor F . The correlations differ significantly.

500

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.13 Comparison of published correlations for the relative liquid content εL in the two-phase layer on a tray (see Table 9.3). Equation (9.17) is presented by curve 2.

Table 9.3 Listing of some correlations for the relative liquid content εL in the froth.

1 2 3 4 5 6 7 8 9 10 11 12

References

Parameters

Range of validity

Kaštánek 1970 Stichlmair 1978 Williams et al. 1960 Mashelkar 1970 Crozier 1957 Solomakha et al. 1972 Gerster et al. 1958 Thomas and Campbell 1967 Takahashi et al. 1974 Andrew 1969 Rama Iyer and Murti 1965 Thomas and Haq 1976

uG F/Fmax F uG F F, hL , ηL , σ F, V˙ L /lw , hw F, V˙ L /lw , hw uG , hL uG F, V˙ L /lw , hw , %L F, V˙ L /lw , hw

Methanol/water Several Several Several — Several Air/water Air/water Air/water Air/water Air/water Air/water

501

9.2 DESIGN OF TRAY COLUMNS

Figure 9.14 Comparison of published correlations for the froth height hf (see Table 9.4). Equation (9.18) is presented by curve 5.

Table 9.4 Listing of some correlations for the froth height hf .

References

Parameters

1 2

Gerster et al. 1958 Gerster et al. 1958

F, V˙ L /lw , hw F 2 , hw

3 4 5 6 7 8 9 10 11

Bernard and Sargent 1966 Thomas and Campbell 1967 Stichlmair 1978 Kaštánek 1970 Andrew 1969 Weiss and John 1973 Mukhlenov 1958 Rama Iyer and Murti 1965 Thomas and Haq 1976

F, hw F, V˙ L /lw , hw F/Fmax , V˙ L /lw , hw uG , hw uG , V˙ L /lw , hw F 1.7 , hw uG , V˙ L /lw , σ, ηL /%L F, V˙ L /lw , hw , %L F, V˙ L /lw , hw

Validity Air/water Hexane/heptane Acetone/benzene Air/water Air/water Several Methanol/water Air/water Several Several Air/water Air/water

502

9 DESIGN OF MASS TRANSFER EQUIPMENT

Based on own and published experimental data, the following equation is recommended [Stichlmair 1978]:

1.45 hf = hw + 1/3 · g |

V˙ L lw · εL {z h0ow

!2/3

125 + · g · (%L − %G ) | }



F − 0.2 m/s · 1 − εL {z hFr

√

%G

2

. }

(9.18) Here, hw is weir height, lw weir length, and g the acceleration due to gravity. An illustrative explanation of Eq. (9.18) is given in Figure 9.15. The first influencing variable is, of course, the weir height hw . The second term h0ow of this relationship essentially takes into account the influence of the liquid load on the height of the two-phase layer. It represents the overflow height of the two-phase mixture over the outlet weir. The last term hFr , which is related to the Froude number Fr , takes into account the uptake of liquid through the gas, which occurs only in the bubble and the √ spray regime. Therefore, it may only be considered if F − 0.2 m/s · %G > 0.

Figure 9.15 Composition of the froth height hf based on the three terms of Eq. (9.18).

Equation (9.18) is valid only if the active surface extends to the outlet weir. If the outlet weir is preceded by a calming zone, gas–liquid separation can take place so that clear liquid flows over the outlet weir. The height of the two-phase layer is then expressed by

hw + hf =

2/3 1.45  ˙ · V /l L w g 1/3 . εL

(9.19)

Equation (9.19) usually gives a larger value of the froth height hf than Eq. (9.18). Figure 9.16 shows a graphical plot of Eq. (9.18). For a relative gas load F/Fmax → 1, the froth height hf rises drastically, and so does the tray spacing H required to avoid flooding since the condition H > hf has to be met. It follows from the diagram

9.2 DESIGN OF TRAY COLUMNS

503

that, depending on the liquid load, the most suitable tray spacing is between 0.3 and 0.6 m. A tray spacing greater than 0.6 m results in only a very small possible increase in gas load, while a tray spacing of less than 0.3 m requires a drastic reduction of the gas load. If the tray spacing is too low, this results in a large diameter of the column to handle a given gas flow.

Figure 9.16 Froth height hf versus the relative gas load F/Fmax . From this diagram it follows that the optimal tray spacing H is in the range of 0.3 – 0.6 m.

9.2.3.4

Liquid Entrainment

Gas flowing through the two-phase layer on a tray always entrains some liquid in the form of fine droplets. This liquid entrainment is unfavorable because it interferes with the countercurrent of gas and liquid in the column and, in turn, reduces the mass transfer efficiency of a tray. The key mechanism for liquid entrainment is the inevitable generation of liquid droplets in the two-phase layer. Even at very low gas load (e.g. bubble regime), fine droplets are formed by bursting bubbles. The jerky movement in the froth regime results in an increased drop formation rate. In spray regime all the liquid on the tray is dispersed into fine droplets. However, the drops do not have a uniform drop size as assumed in Section 9.2.2.1. Rather, there is an asymmetric drop size distribution, i.e. a swarm contains much more small than large drops. Liquid drops are entrained by the gas flow if their terminal velocity is lower than the gas velocity. However, due to the high-speed motion within the two-phase layer on a tray, even some large drops are thrown upward to form a layer of drops above the froth. After reaching a certain height, most droplets fall back again into the twophase layer on the tray. The smaller the drops, the higher they rise. Hence, at small tray spacing, even drops with a terminal velocity higher than the gas velocity may be entrained.

504

9 DESIGN OF MASS TRANSFER EQUIPMENT

Numerous investigations on liquid entrainment have been reported in literature (see Table 9.5). Frequently, the correlation of Fair 1961 is used. The liquid entrainment rate is a quantity that can vary over several orders of magnitude. All authors agree that the gas load factor F is the dominating factor. Increasing the gas load by a factor of 10 increases liquid entrainment by a factor of 103 or 104 . Another major factor is the surface tension σ , since it influences the drop size. Liquid viscosity is important only in highly viscous systems [Barker and Choudhury 1959], which are rarely encountered in distillation columns. Further important design parameters are the relative free area ϕ [Pinczewski et al. 1975] and tray spacing H , since they determine the clear vapor space between the top of the froth layer and the next tray, i.e. the distance that an entrained droplet has to cover. Figure 9.17 shows a plot of published experimental entrainment data of various systems (see Table 9.5). The relative entrainment V˙ E /V˙ G is plotted versus the relative gas load F/Fmax . According to Stichlmair and Hofer 1978 the quantity Fmax = 2.5 · (ϕ2 · σ · ∆% · g)1/4 accounts adequately for the influence of the surface tension σ and the relative free area ϕ of the tray. The data scatter is very large, since the important parameter tray spacing H is not considered. Dimensional 2 analysis yields the parameter (H − hf ) · ∆% · g/Fmax as dimensionless clear vapor space.

Figure 9.17 Plot of published entrainment data versus the relative gas load F/Fmax (see Table 9.5).

505

9.2 DESIGN OF TRAY COLUMNS Table 9.5 Liquid entrainment data plotted in Figure 9.17. References

Tray design and tray spacing H in m

System and surface tension σ in kg/s2

Liquid load V˙ L /lw in m3 /(m · h)

Peavy and Baker 1937

Caps, 0.15 – 0.3

Ethanol/water, 0.02

0.73 – 5.35

Ashraf et al. 1934

Caps, 0.38

Air/kerosene, 0.03

—

Holbrook and Baker 1934

Caps, 0.3 – 0.79

Steam/water, 0.062

0.2 – 1

Sherwood and Jenny 1935

Caps, 0.23

Air/water, 0.073

—

Hunt et al. 1955

Sieve, 0.51

Air/water, 0.073

—

Pinczewski et al. 1975

Sieve, 0.91

Air/water, 0.073

15

Calcaterra et al. 1968

Sieve, 0.33 – 0.66

Air/water, 0.073

10.9

Friend et al. 1960

Sieve, 0.15 – 0.3

Air/water, 0.073

7.18 – 29

Gerster et al. 1958

Caps, 0.61

Acetone/benzene, 0.02

8.7 – 18.6

Gerster et al. 1958

Caps, 0.61

Steam/water, 0.062

7.45 – 44.7

Gerster et al. 1958

Caps, 0.61

Cyclohexane/ n-heptane, 0.012

3.9 – 54.2

⊕

Piqueur and Verhoeye 1976

Valve, 0.3 – 0.5

Air/water, 0.073

0.45 – 4.5



Lockett et al. 1976

Sieve, 0.36

Air/water, 0.073

3.6 – 21.6





Figure 9.18 shows the correlation of the data of Figure 9.17 with the parameter dimensionless clear vapor space. The relative entrainment V˙ E /V˙ G increases significantly with increasing gas load F/Fmax (it is worth mentioning that the abscissa covers 1 decade and the ordinate 7 decades). The sharp increase of the entrainment at F/Fmax = 0.65 is apparently caused by the transition from froth to spray regime (phase inversion; see Section 9.2.3.1). For large values of the dimensionless 2 clear vapor space (H − hf ) · ∆% · g/Fmax (in air/water systems approximately H − hf > 0.5 m), the entrainment is no longer dependent on the clear vapor space, since only drops with a terminal velocity smaller than the gas velocity are entrained. However, with decreasing clear vapor space, the entrainment increases drastically since even large drops may be carried onto the next tray. The correlation of Figure 9.18 describes the major factors for entrainment quite well. More than 95 % of the data are correlated with an accuracy better than 50 %, which is very good keeping in mind the wide variation of the data from 10−7 – 10−2 . The accuracy of the correlation is limited by the strong influence of gas load on entrainment. For instance, a deviation of 10 % in the gas load causes a deviation

506

9 DESIGN OF MASS TRANSFER EQUIPMENT

of liquid entrainment of approximately 60 %. The database of the correlation (Table 9.5) contains data of bubble cap, sieve, and valve trays with different tray spacing and different liquid loads (accounting for the froth height hf ). Most of the data was collected using the air/water system but steam/water data and hot distillation data are also correlated very well.

Figure 9.18 Correlation of published entrainment data according to Stichlmair and Hofer 1978. Parameter is the dimensionless clear vapor space.

Liquid entrainment is inevitable in tray operation. A rule of thumb says that an entrainment rate of up to 10 % of the liquid flow on the tray may be tolerated. Even at slightly higher entrainment rates, the two-phase flow in the column is not severely disturbed; only the mass transfer efficiency is impaired to some extent. At excessive entrainment rates, however, the pressure drop increases very sharply, and the countercurrent flow in the column breaks down (entrainment flooding). 9.2.3.5

Liquid Mixing in the Two-Phase Layer

In tray columns, liquid flows across the tray to the outlet weir, i.e. there exists a cross-flow of liquid and gas. Mass transfer between the two phases generates a concentration profile along the liquid flow path, which greatly influences tray efficiency. The two-phase layer on the tray is intensely moved and agitated by the gas flow, particularly in the froth and spray regime (see Figure 9.19). This results in a partial backmixing of the liquid on the tray, which in turn reduces the differences in the liq-

9.2 DESIGN OF TRAY COLUMNS

507

uid concentration. According to the dispersion model, the degree of liquid mixing on the tray is represented by the dimensionless Péclet number, which is defined by

Pe =

2 lL , DE · τL

(9.20)

where lL is the flow path length of the liquid, DE is the dispersion coefficient, and τL is the residence time of the liquid in the two-phase layer on the tray. If the flow path length of the liquid and the length of the weir are approximately equal, lL ≈ lw , the Péclet number can be estimated with

Pe ≈

V˙ L . DE · hf · εL

(9.21)

A value of Pe = 0 means complete mixing of the liquid, i.e. the liquid concentration is the same everywhere on the tray. The opposite case (i.e. no mixing, plug flow) is characterized by Pe → ∞.

Figure 9.19 Liquid mixing in the two-phase layer on a sieve tray.

A result of liquid mixing experiments performed by Stichlmair and Weisshuhn 1973 is presented in Figure 9.20. The test tray, a rectangular sieve tray 60 mm wide and 255 mm long, was operated with warm water and air. On the tray the air becomes nearly saturated with water due to the high mass transfer rate in the two-phase layer. In this case, the mass transfer causes no change in the liquid concentration, but the liquid temperature, since the high heat of vaporization of water has to be covered by the liquid phase (similar to a cooling tower). The temperature profile in the froth has been measured by thermocouples. The data show that the liquid temperatures remain virtually unchanged in vertical direction. Hence, the liquid is well mixed vertically. In horizontal direction, however, there is a significant temperature gradient. In case of plug flow (i.e. Pe → ∞), the temperature profile on a tray can be easily calculated from the knowledge of the point efficiency EOG . The result is presented

508

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.20 Liquid mixing in the two-phase layer on a sieve tray. The dashed line is calculated for plug flow with an efficiency of EOG = 0.7. Data are measured 20 mm (•), 50 mm (N), and 70 mm () above the tray.

in Figure 9.20 by a dashed line (calculated with EOG = 0.7). The data points differ significantly from the calculated line indicating the degree of liquid mixing. A more detailed analysis of the data points (see Section 9.2.4.2) revealed a degree of liquid mixing of Pe ≈ 3. A series of such experiments with variation in tray length, gas load, and liquid load allowed Weisshuhn 1976 to develop a correlation for the dispersion coefficient DE (i.e. eddy diffusion coefficient): r εL DE = 1.06 · hf · F · . (9.22) %L − %G Equation (9.22) applies only to the froth and the spray regime. In the bubble regime, the degree of liquid mixing is typically so small that it can be neglected. 9.2.3.6

Maldistribution of Liquid

An undisturbed plug flow of liquid on the tray provides the highest possible mass transfer rate. For small diameter columns, plug flow is primarily disturbed by liquid mixing (see Section 9.2.3.5). For large diameter columns, however, liquid mixing is of minor importance. Here, maldistribution, i.e. nonuniform liquid flow across the tray, is the major mechanism adversely affecting plug flow. Liquid flow rates vary locally on the tray, and, in turn, there is a large variation in the liquid residence time in the froth. Maldistribution strongly affects the tray efficiency of large diameter columns. In sections with high liquid residence time, the liquid virtually reaches equilibrium with

9.2 DESIGN OF TRAY COLUMNS

509

the gas phase and does no longer contribute to the mass transfer due to the lack of driving force. The effect of maldistribution on tray efficiency can be much worse than that of liquid mixing because the tray efficiency of a large diameter tray can fall below the point efficiency. However, knowledge of the degree of maldistribution is very poor, and, even worse, there is no theory that quantitatively describes the relation between maldistribution and tray efficiency. There are several techniques for measuring maldistribution on trays. The simplest method is to suddenly inject a dye into the liquid at the inlet and to observe the spreading of the colored liquid across the tray by eye (e.g. [Lockett 1986]). Bell 1972 used a fluorescent dye and glass fiber detectors for more quantitative results. The movement of table tennis balls on the froth was observed by Porter et al. 1972. They found that the balls could stop and even circle in some areas of the tray. A good method to detect liquid maldistribution on a tray is to measure liquid temperature profiles in the froth [Stichlmair 1971] using either thermocouples or an infrared camera. Figure 9.21 shows some typical results of experiments conducted by Stichlmair and Ulbrich 1987 on a large diameter bubble cap tray operated with warm water and air (similar to the method used for Figure 9.20). The tray had a diameter of 2.3 m and 440 Bayer bubble caps. The liquid temperatures were measured by 182 thermocouples evenly distributed over the active area of the tray. The positions of the thermocouples are marked by crosses in Figure 9.21. From the data, lines of constant liquid temperatures (isotherms) were calculated via linear interpolation by a computer and plotted in the active area of the tray in the diagram. In a first approximation, these isotherms can be interpreted as lines of constant liquid residence time in the froth. In case of plug flow, the isotherms would be straight lines that are parallel to the inlet and the outlet weir. Any deviations from horizontal straight lines are caused by a non-uniform liquid flow, i.e. by maldistribution. Figure 9.21A shows two results with different gas loads. The course of the isotherms reveals a high degree of maldistribution on the tray. Near the wall and in the center, the liquid flow rate is increased, evidently by a higher bubble cap pitch in these areas. However, variations of the bubble cap pitch are inevitable in tray design, as larger trays consist of several segments. Experimental results with modifications of the inlet weir are depicted in Figure 9.21B. Blocking half of the inlet weir (left-hand side) completely changes the liquid flow across the tray. On the blocked side, the temperatures are significantly lower than on the open side, indicating a lower liquid flow rate. It should be noted that in these experiments, no maldistribution could be detected by eye. Central liquid feeding (right-hand side) also dominates the liquid flow across the entire tray. Modifications of the outlet weir have only little impact on liquid flow rate profiles across the tray as can be seen in Figure 9.21C. The same modifications were made as in Figure 9.21B but of the outlet weir instead of the inlet weir. Hence, the installation of notched outlet weirs, which are often applied to trays with very low liquid load, is a questionable method of improving the liquid flow pattern. More than 80 experiments of this kind were carried out with systematic variations of gas and liquid flow, of tray tilt angle, and of the form of the inlet and outlet weirs. In some experiments baffles were installed on the tray and especially near the column

510

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.21 Liquid isotherms in the two-phase layer on a bubble cap tray of 2.3 m diameter. Deviations from horizontal straight lines are caused by maldistribution. A) Modifications of the gas load. B) Modifications of the inlet weir. C) Modifications of the outlet weir.

9.2 DESIGN OF TRAY COLUMNS

511

wall. Bubble cap trays as well as sieve trays were tested with this technique. The results can be summarized as follows: • Even small variations of the bubble cap pitch cause severe liquid maldistribution. • Standard baffles often installed near the column wall are in most cases too small and ineffective. • Modifications of the form of the outlet weir, e.g. notched weir, have little or no impact. • Modifications of the form of the inlet weir are very effective. • Levelness of the plate and its horizontal installation are of major importance. Another way to influence the fluid flow profile on the tray is the use of push valves. A push valve can be shaped like a prompter’s box on a theater stage. Its orifice is oriented so that the momentum of the gas flow drives the liquid flow in the desired direction. Paschold et al. 2011 report that arranging push valves on the tray can improve the liquid flow profile on the tray at a certain operating point, but the same arrangement at a different gas load results in a significant increase in maldistribution. Thus, their use cannot be fully recommended if flexible operation is desired. In practice, a certain degree of liquid maldistribution is virtually unavoidable. Hence, mass transfer efficiency will never be as high as predicted for plug flow (see Section 9.2.4.2). However, the adverse effects of maldistribution are limited to each tray because in the subsequent downcomer the liquid is completely mixed and afterward evenly redistributed on the next tray. The effects of maldistribution do not accumulate over the height in tray columns as they do in packed columns (see Section 9.3.3.5). The permanent liquid mixing in the downcomers compensates for maldistribution to a certain degree and makes trays the preferred candidates for internals of columns with large diameter. 9.2.3.7

Interfacial Area

A high interfacial area in the two-phase layer on the tray is the most essential prerequisite for a good mass transfer. The interfacial area A is conveniently related to the volume Vf of the froth and expressed as specific area a:

a=

A . Vf

(9.23)

Knowledge of the interfacial area is very poor and no satisfactory correlation is available. Published data differ by a factor of up to 10, apparently due to different measuring techniques [Stichlmair 1978]. It is therefore attempted to estimate the magnitude of the interfacial area and the influencing factors. The basis is the following equation, which applies to a two-phase layer with spherically dispersed particles: εd a=6· . (9.24) d Here, εd is the volume fraction of the dispersed phase and d is the particle diameter. Equation (9.24) can be applied to the bubble and spray regime, but not to the froth regime.

512

9 DESIGN OF MASS TRANSFER EQUIPMENT

In bubble regime the diameter d of the bubbles can be calculated from the balance of frictional force and buoyant force: 

d=

6·σ (%L − %G ) · g

1/2

.

From Eq. (9.17) it follows that for the volume fraction εd of the bubbles:  0.28 F εd = 1 − εL = . Fmax

(9.25)

(9.26)

From Eqs. (9.24) – (9.26), it follows for the interfacial area in the bubble regime 

a=6·

(%L − %G ) · g 6·σ

1/2  0.28 F · . Fmax

(9.27)

In spray regime the liquid is the dispersed phase. The drop diameter d follows from Eq. (9.8), and the volume fraction εd from Eq. (9.17), respectively:  0.28  ϕ 2 F d = 12 · σ · and εd = εL = 1 − . (9.28) F Fmax Analogously, we get for the spray regime  0.28 ! F2 F a= · 1− , 2 · σ · ϕ2 Fmax

(9.29)

where ϕ is the relative free hole area, i.e. ratio of total hole area to the active area. Equation (9.27) for bubble regime and Eq. (9.29) for drop regime have been evaluated and plotted for three systems in Figure 9.22. The froth regime cannot be evaluated because there is no distinct dispersed phase. However, the interpolated values shown in Figure 9.22 should approximate the actual course. The interfacial area in aqueous systems is approximately a ≈ 500 m2 /m3 . In organic systems values of a ≈ 1000 m2 /m3 or more can be reached due to a lower surface tension. The surface tension σ has a significant influence on the interfacial area. The analysis of the above equations yields a proportionality of a ∼ σ −0.57 , which is in excellent agreement with the theory of Kolmogorov 1958 which predicts a proportionality of a ∼ σ −0.6 in dispersed systems with homogeneous turbulence. The above theoretical approach allows only a rough estimation of the interfacial area and the most important influencing factors. The values obtained are only partially applicable to the calculation of the mass transfer efficiency of a tray column. This is because it is not known whether all parts of the interface contribute equally to mass transfer between phases. Thus, the interfacial area is a weak point in the prediction of the tray efficiency. 9.2.3.8

Pressure Drop

The permanent pressure loss of a gas stream passing through a single tray is illustrated in Figure 9.23. The pressure drop of a tray is rather high, typically on the order of

9.2 DESIGN OF TRAY COLUMNS

513

Figure 9.22 Calculated values for the volumetric interfacial area a of the two-phase layer on column trays: - - - - - bubble regime; − · − · − spray regime; —— interpolation.

Figure 9.23 Pressure drop of a sieve tray versus gas load factor F . Parameter is the liquid flow per weir length V˙ L /lW .

514

9 DESIGN OF MASS TRANSFER EQUIPMENT

300 – 800 Pa (i.e. 30 – 80 mm clear liquid height). The main parameters that affect the pressure drop are tray design, gas load F , and liquid load per weir length, V˙ L /lw . The basic equation for calculating the pressure drop of a tray is ∆p = ∆pd + ∆pL + ∆pR .

(9.30)

The first term ∆pd accounts for the pressure drop of a dry tray. The second term ∆pL represents the pressure drop resulting from the clear liquid height hL and can be formulated by

∆pL = hL · %L · g = hf · εL · %L · g .

(9.31)

The froth height hf and the relative liquid content εL can be determined by Eqs. (9.17) and (9.18), respectively. The last term of Eq. (9.30), ∆pR , takes into account several residual factors, including bubble formation, liquid mixing, and vertical acceleration of the liquid. This term is much smaller than the first two terms and can be neglected in most cases. Dry Pressure Drop of Sieve Trays

The pressure drop ∆pd of the gas as it passes through a dry sieve tray is given by

∆pd = ζ ·

%G 2 · uGh 2

or

∆pd =

ζ · F2 . 2 h

(9.32)

The unknown quantity in Eq. (9.32) is the orifice coefficient ζ . For sieve trays, equations for the orifice coefficient can be derived from first principles. The mechanism of fluid flow through a sieve tray is easily understood by considering the flow through a single hole in a plate. In principle, such a flow consists of two essential parts, i.e. a converging part before the plate and a diverging part after the plate; see Figure 9.24.

Figure 9.24 Mechanism of fluid flow through a hole in a thin plate (A) and a thick plate (B).

515

9.2 DESIGN OF TRAY COLUMNS

In the converging part the fluid velocity increases, and, according to Bernoulli’s principle, the static pressure decreases. No significant pressure loss is generated in the converging part since fluid flow is directed toward a falling static pressure. Potential energy is transformed into kinetic energy without significant losses. At the end of the converging part, the fluid flow contracts to an area Avc (vena contracta) smaller than the hole area Ah . In the diverging part after the plate, the fluid velocity decreases, and consequently the static pressure increases. Flowing against increasing static pressure is accompanied by the generation of eddies that cause severe pressure losses. From the laws of conservation of momentum and energy, it follows the very fundamental Borda–Carnot equation for the permanent pressure loss during expansion (e.g. Perry et al. 1984):

∆pd =

%G 2 ·(uGvc − uG ) 2

or ∆pd =

  %G 2 uGvc uG 2 ·uGh · − . (9.33) 2 uGh uGh

A comparison with Eq. (9.32) gives for the orifice coefficient 

ζ=

uGvc uG − uGh uGh

2

.

(9.34)

The continuity equation allows calculating the velocity ratios uGvc /uGh and uG /uGh with knowledge of the flow cross sections. The ratio Avc /Ah is described by the discharge coefficient α:

α=

Avc . Ah

(9.35)

The continuity equation leads to

uGvc 1 = . uGh α

(9.36)

The ratio of the hole area Ah and the total area A is described by the relative free area ϕ:

ϕ=

Ah . A

(9.37)

With the continuity equation it follows that

uG = ϕ. uGh

(9.38)

For thin plates with large holes, the flow expands in a single step from Avc to A (Figure 9.24). Combining Eqs. (9.34), (9.36), and (9.38) yields 

ζ=

2 1 −ϕ α

for

s → 0. dh

(9.39)

516

9 DESIGN OF MASS TRANSFER EQUIPMENT

For thick plates with small holes, the flow expands in two steps, firstly from Avc to Ah and secondly from Ah to A (Figure 9.24). The pressure drops of the two expansions are added to give:  2 1 s 2 ζ= − 1 + (1 − ϕ) for  1. (9.40) α dh In the above equation the friction losses in the hole are neglected. For high Reynolds numbers and sharp-edged holes, the discharge coefficient has a value of α = π/(π + 2) = 0.611. For a single hole in a large diameter plate, the relative free area ϕ approaches zero, i.e. ϕ → 0. Evaluation of Eqs. (9.39) and (9.40) for ϕ = 0 yields  2 1 s ζ0 = − 0 = 2.67 for →0 (9.41) 0.611 dh and 

ζ0 =

2 1 2 − 1 + (1) = 1.41 for 0.611

s  1. dh

(9.42)

Hence, the orifice coefficient ζ0 of a thin plate is higher than that of a thick plate. This can easily be explained by the fact that the expansion of the flow in two small steps causes lower losses than in one big step. For example, the permanent pressure drop approaches zero with uniformly diverging cross section, since the expansion takes place in many small steps. In practice, values of the ratio of plate thickness and hole diameter s/dh are typically between the two special cases considered so far. Therefore, the orifice coefficient has to be determined by experiments. A correlation of the published data is presented in Figure 9.25. The orifice coefficient ζ0 , i.e. for ϕ → 0, is plotted versus the ratio of plate thickness and hole diameter s/dh . Parameter is the Reynolds number Re h = uGh · dh · %G /ηG in the hole. The increase of ζ0 with increasing Reynolds numbers at small values of s/dh results from the variation of the discharge coefficient α. The decrease in ζ0 with increasing Reynolds numbers at large values of s/dh is caused by the friction losses in the hole that were neglected in the theoretical considerations. The relationship between the orifice coefficient ζ and the relative free area ϕ differs for thin and thick plates. For thin plates it follows from Eq. (9.39) that p ζ = ζ0 + ϕ2 − 2 · ϕ · ζ0 for s/dh → 0 . (9.43) For thick plates, the following is obtained from Eq. (9.40):

ζ = ζ0 + ϕ2 − 2 · ϕ for s/dh  1 .

(9.44)

In practice, the differences between Eqs. (9.43) and (9.44) are very small because the value of ϕ usually does not exceed 0.15. Therefore, Eq. (9.43) can be used with sufficient accuracy for most tray thicknesses.

9.2 DESIGN OF TRAY COLUMNS

517

Figure 9.25 Dependence of the orifice coefficient ζ0 (ϕ → 0) of sieve trays with sharp-edged holes on the ratio of plate thickness to hole diameter s/dh . Parameter is the Reynolds number in the holes.

Dry Pressure Drop of Valve and Bubble Cap Trays

Bubble cap and valve trays have very complex geometrical structures that make it difficult or almost impossible to make a rigorous prediction of pressure drop. Since these elements are produced in large numbers, experimental data on the orifice coefficients are often provided by the suppliers. There are also attempts to determine the pressure loss by means of computational fluid dynamics. In the development of new valve types, these methods have proved to be useful (e.g. Glueer et al. 2018). A slightly different approach is recommended for calculating the dry pressure drop of bubble cap and valve trays. The orifice coefficients are not related to the velocity in the smallest geometric open area, but to the entire active area of the tray, since the quality of tray design depends not only on the pressure drop of a single element but also on the number of elements arranged on the tray. The following definition implies both quantities:

∆pd = ζt0 ·

F2 . 2

(9.45)

A compilation of the tray orifice coefficient ζt0 for different types of trays is shown in Figure 9.26 [Stichlmair 1978]. The symbols mark the closest installation of the elements as recommended by the suppliers. The lines allow estimation of the orifice coefficient when fewer elements per unit area were installed. The abscissa in Figure 9.26 is the smallest relative free area ϕ0 , which can be, e.g., either the slots or the riser of a bubble cap. For comparison purposes, the transformed orifice coefficient of a sieve tray with ζ0 = 1.7 is also shown. The orifice coefficients of

518

9 DESIGN OF MASS TRANSFER EQUIPMENT

valve trays and bubble cap trays are always greater than those of sieve trays. However, the relative free area ϕ of bubble cap and valve trays may be larger than that of sieve trays, due to the lower risk of liquid leaking through the tray. The quality of a tray design is determined by the distance of the corresponding parameter line from the line of a sieve tray with ζ0 = 1.0, which is the theoretical minimum of a sieve tray with rounded edges of the holes.

Figure 9.26 Dependence of the orifice coefficient ζt0 of bubble cap and valve trays on the relative free area ϕ0 . The symbols show the closest arrangement of the elements.

9.2.4

Mass Transfer in the Two-Phase Layer on Column Trays

The difficulty of a distillative separation is expressed either in the number of equilibrium stages or the number of transfer units (see Section 4.1). The higher the number of these quantities the more difficult is the separation to achieve. Both quantities are defined thermodynamically, i.e. the type and details of the equipment are not taken into account. However, it is obvious that a difficult separation requires a tall column. A major objective of column design is the conversion of equilibrium stages or transfer units into column height. For tray columns the concept of equilibrium stages is almost exclusively applied. However, it should be appreciated that the separation achieved by a tray is not identical with that of an equilibrium stage. In most cases the concentration changes achieved by an actual tray are significantly smaller. State of the art is the use of mass transfer efficiencies that are defined as the ratio of the concentration changes of an actual tray and an equilibrium stage. Two definitions of mass transfer efficiencies have to be distinguished: the point efficiency EOG and the tray efficiency EOGM (see Section 9.2.4.1). From the knowledge of the mass transfer efficiency of a tray, the number of trays

9.2 DESIGN OF TRAY COLUMNS

519

required to obtain the specified products is determined by plotting the effective gas concentration line in the McCabe–Thiele diagram; see Figure 9.27. The number of actual trays is determined by drawing steps between operating line and effective gas concentration line. The rule that the number of actual trays can be determined by dividing the number of equilibrium stages by the tray efficiency is only correct if operating line and equilibrium line are parallel. The more the slopes of the two lines differ, the more the number of actual trays is underestimated by this rule.

Figure 9.27 Determination of the number of actual trays with knowledge of tray efficiency in the McCabe–Thiele diagram.

9.2.4.1

Definitions of Mass Transfer Efficiencies

The different definitions of mass transfer efficiencies are best explained by Figure 9.28. In point efficiency EOG , the concentration change along an individual streamline of the gas moving vertically through the froth is considered. The definition is

EOG =

y − yn−1 . − yn−1

y ∗ (x)

(9.46)

The subscript OG indicates that the overall resistance is formally placed into the gas phase. The actual change in concentration of the gas y − yn−1 as it flows through the two-phase layer on tray n is related to the maximum concentration change attainable, i.e. y ∗ (x) − yn−1 . Here, y ∗ is the gas concentration in equilibrium with liquid concentration x at the point on the tray where the gas streamline passes through the two-phase layer. The point efficiency can reach a maximum value of 1 provided that the liquid is completely mixed throughout the height of the froth, an assumption

520

9 DESIGN OF MASS TRANSFER EQUIPMENT

that is well met (see Figure 9.20). Thus, point efficiency is a very useful theoretical quantity, but it has no immediate practical significance.

Figure 9.28 Sketch of a single sieve tray with notations for point efficiency and tray efficiency definitions.

In practice, the tray efficiency EOGM (Murphree efficiency) is used, which is defined by

EOGM =

yn − yn−1 . y ∗ (xn ) − yn−1

(9.47)

Here, the entire gas flow through the tray is considered. The actual concentration change of the whole gas flow yn − yn−1 is related to the concentration difference y ∗ (xn ) − yn−1 . Here, y ∗ is the gas concentration in equilibrium with the state of the liquid (concentration xn ) leaving tray n. This definition is arbitrary since the liquid concentration on the tray varies due to the cross-flow of the phases. The concentration xn of the liquid leaving tray n usually has the lowest value. Since this value is now used as reference for the whole tray, the tray efficiency can reach values above 100 %; this is not a theoretical limit in this definition. Although the theoretical definition of the tray efficiency EOGM is unsatisfactory, it is of practical interest because it describes the separation efficiency of a real tray; see Figure 9.27. 9.2.4.2

Relation Between Point Efficiency and Tray Efficiency

In the case of plug flow of the liquid, the following relationship between point and tray efficiency can be formulated [Lewis 1936]: ! ! ˙ G˙ EOGM L/ 1 m = · · exp · EOG − 1 . (9.48) ˙ G˙ EOG m EOG L/ Here, L˙ is the molar liquid flow rate, G˙ is the molar gas flow rate, and m is the slope of the equilibrium curve. The gas below each tray is assumed to be completely mixed. Plug flow prevails on very large trays only. On smaller trays, the liquid in the two-phase layer is at least partially mixed (see Section 9.2.3.5).

9.2 DESIGN OF TRAY COLUMNS

521

For complete liquid mixing, the tray efficiency and the point efficiency have the same value:

EOGM = 1. EOG

(9.49)

The normal state lies between the two extreme cases described by Eqs. (9.48) and (9.49). The degree of liquid mixing, which is expressed by the Péclet number Pe (see Section 9.2.3.5), then has to be considered. Liquid mixing on a tray is adequately described by the dispersion model. Here, the process of mixing is formulated in analogy to the process of molecular diffusion with the eddy diffusion coefficient DE instead of the molecular diffusion coefficient DL . An eddy diffusion stream is countercurrently superimposed on the liquid flow across the tray; see Figure 9.29. A mass balance at an element of the two-phase layer of differential length dz yields  1 d2 x dx m yn  · 2 − + · EOG · x − =0 ˙ G) ˙ Pe dz dz m (L/ dx with = 0 for z = 1 . dz

(9.50)

Here, z denotes the dimensionless tray length.

Figure 9.29 Application of the dispersion model to liquid mixing on a tray.

With knowledge of the concentration profile in the liquid phase, the mass transfer efficiency of the tray, EOGM , can be determined. The solution is graphically represented in Figure 9.30 [Gerster et al. 1958] with the Péclet number as parameter. In plug flow (no mixing), the Péclet number becomes infinite, Pe = ∞. The case of complete liquid mixing is expressed by Pe = 0. When both the parameter Pe and

522

9 DESIGN OF MASS TRANSFER EQUIPMENT

˙ G) ˙ are large, the ratio of the tray efficiency to the point efthe abscissa EOG ·m/(L/ ficiency EOGM /EOG becomes very large. However, in order for the abscissa to be very large, the liquid flow has to be small. The high residence time at low liquid flow rates causes considerable remixing of liquid on the tray so that the parameter Pe in Figure 9.30 becomes very small. Thus, the value of the tray efficiency is very rarely much higher than the point efficiency.

Figure 9.30 Relation between the ratio of overall tray efficiency EOGM to point efficiency EOG and the degree of liquid mixing represented by the Péclet number Pe .

In practice, tray efficiency is adversely affected by some other processes such as liquid leakage through the tray, entrainment of liquid to the next tray, and maldistribution. Careful column design should use a tray efficiency that is only slightly higher than the point efficiency. 9.2.4.3

Point Efficiency

The prediction of mass transfer efficiency is based on the following fundamental equation (see, for instance, Taylor and Krishna 1993):

EOG = 1 − exp(−NTUOG ) .

(9.51)

This equation formulates a relation between two definitions only, i.e. overall gas point efficiency EOG and the overall gas transfer units NTUOG . Figure 9.31 shows this relationship. The overall gas number of transfer units NTUOG can be calculated from the number of transfer units in the gas NTUG and in the liquid NTUL by using the following equation:

1 1 m 1 = + · . ˙ ˙ NTUOG NTUG L/G NTUL

(9.52)

523

9.2 DESIGN OF TRAY COLUMNS

Figure 9.31 Dependency of point efficiency on the number of transfer units.

˙ G) ˙ is known as the stripping factor. The number of transfer units The term m/(L/ in the gas phase is given by NTUG =

βG · a · hf · Aac . V˙ G

(9.53)

Here, βG is the mass transfer coefficient in the gas phase, a is the relative interfacial area (Figure 9.22), hf is the height of the two-phase layer (Eq. (9.18)), Aac is the active area of the tray, and V˙ G is the volumetric gas flow. Analogously, we obtain for the liquid phase

NTUL =

βL · a · hf · Aac . V˙ L

(9.54)

Inserting Eqs. (9.53) and (9.54) into Eq. (9.52) and replacing V˙ G /Aac by the superficial gas velocity uG gives

a · βG · NTUOG =

hf uG ˆL M

βG %G 1+m· · · ˆ βL M G % L

∼

hf . uG

(9.55)

ˆ L and M ˆ G are the molecular weights of gas and liquid phase, and %L and Here, M %G the densities of the liquid and gas, respectively. The numerator in this equation represents the mass transfer rate in the gas phase, while the denominator takes into account the additional resistance to mass transfer in the liquid phase. Equation (9.55) is generally applicable and contains four unknown quantities specific to the tray: the relative interfacial area a, the height of the two-phase layer hf , and the mass transfer coefficients in the gas βG and in the liquid βL .

524

9.2.4.4

9 DESIGN OF MASS TRANSFER EQUIPMENT

Mass Transfer Coefficients

The mass transfer coefficients in the gas, βG , and in the liquid, βL , can be estimated by assuming unsteady mass transfer: r r 4 DG 4 DL βG = · and βL = · . (9.56) π τ π τ Here, DG and DL are the diffusion coefficients in the gas phase and the liquid phase, respectively. Determination of the contact time τ is difficult. According to Stichlmair 1978, it is equivalent to the lifetime of the interface in the two-phase layer. Since the interface is generated by the penetration of gas in the liquid, the lifetime of the interface is identical with the residence time of the gas in the two-phase layer. Thus, the same value is obtained for the contact time τ in both the liquid and the gas phases. Hence, the mass transfer coefficient in the gas is s 4 DG uG βG = · · . (9.57) π hf εG Analogously, the mass transfer coefficient in the liquid is s 4 D L uG βL = · · . π hf εG

(9.58)

These relationships apply to the froth regime and, to a close approximation, to the spray regime, but not to the bubble regime since here the surface is renewed by the rising bubbles. Also in organic systems with low surface tension, Eqs. (9.57) and (9.58) sometimes give too low values because the interface is steadily renewed. Inserting Eqs. (9.57) and (9.58) into Eq. (9.55) gives s 4 DG hf a· · ·   π uG εG hf 1/2 NTUOG = ∼ . (9.59) r ˆ L %G uG DG M 1+m· · · ˆ G %L DL M The denominator in this equation takes into account the additional mass transfer resistance in the liquid phase. It contains thermodynamic properties only. While in the basic relationship given in Eq. (9.55) the overall gas mass transfer units NTUOG are directly proportional to the height hf of the two-phase layer and inversely proportional to the gas velocity uG , a proportionality to the square root of the ratio of these parameters is obtained in Eq. (9.59); this agrees well with experimental data. 9.2.4.5

Practical Determination of Tray Efficiencies

The theory of mass transfer on trays is well developed, as shown in the previous Sections 9.2.4.1 – 9.2.4.4. However, current knowledge of some of the most important parameters, e.g. interfacial area, mass transfer coefficients in the gas and the liquid

9.2 DESIGN OF TRAY COLUMNS

525

phase, liquid mixing, and liquid maldistribution, is quite poor. Hence, in the current state of the art, the mass transfer efficiency cannot be accurately predicted. In practice, empirical values of tray efficiencies are used, and, in most cases, quite large safety factors are included to reduce the risk of failure of poor column separation performance. Empirical tray efficiency data for distillation systems under normal conditions are on the order of 70 %. However, for wide boiling systems or high viscous systems, much lower values were found. In wide boiling systems often encountered in absorption and desorption, values of tray efficiencies are reported to be as low as 1 % (e.g. absorption of carbon dioxide in water) or even 0.1 % (e.g. desorption of oxygen from water). In order to include the empirical findings into the framework of the mass transfer theory, some experimental data were plotted in the diagram of Figure 9.32 [Stichlmair and Weisshuhn 1973]. The structure of the diagram results from Eqs. (9.51) and (9.52) with the overall gas point efficiency as ordinate, the stripping factor as abscissa, and the number of transfer units NTUG and NTUL as parameter. The regions containing empirical data are marked in the diagram. The parameter lines were determined by evaluating Eqs. (9.51) and (9.52) with different sets of gas-phase and liquid-phase transfer units, NTUG and NTUL , respectively. From the comparison of the parameter lines and the empirical data, it follows that the gas-side transfer units are in the range of NTUG ≈ 0.5 – 2 whereas the liquid-side transfer units vary from NTUL ≈ 0.5 – 20.

Figure 9.32 Typical experimental values of the gas-side point efficiency EOG versus the stripping factor.

526

9 DESIGN OF MASS TRANSFER EQUIPMENT

The most important parameter on gas-side tray efficiency is the stripping fac˙ G) ˙ . It represents the ratio of the slope of the equilibrium line, m, and tor m/(L/ ˙ G˙ , in the McCabe–Thiele diagram. The stripping the slope of the operating line, L/ factor varies over up to 4 decades. In distillation, however, the values of the stripping ˙ G) ˙ ≈ 1, i.e. the equilibrium line and factor seldom differ significantly from m/(L/ the operating line are nearly equidistant in the McCabe–Thiele diagram. Under these conditions, the tray efficiency is quite high. Figure 9.32 confirms the rule of thumb that the tray efficiency of close-boiling systems has a value of approximately 70 %. The very low efficiencies of absorption and desorption are primarily caused by the large value of the slope of the equilibrium line in such systems. It is important to note that Figure 9.32 is valid for the overall gas definitions of the mass transfer efficiencies only.

Example 9.1: Design of a Sieve Tray Column Determine the diameter Dc and the tray efficiency of a sieve tray column for the distillation of the benzene/toluene mixture at a pressure of p = 1 bar. The internal flow rates are G˙ = 0.05 kmol/s and L˙ = 0.034 kmol/s. Properties:

Properties Molecular weight Density Surface tension Viscosity Diffusion coefficient Slope of equilibrium curve

Liquid phase

Gas phase

ˆ L = 85.8 kg/kmol M %L = 800 kg/m3 σ = 0.02 N/m

ˆ G = 85.8 kg/kmol M %G = 2.6 kg/m3

−9

DL = 4.6 × 10

2

m /s

η = 8.9 × 10−6 Pa · s DG = 4.4 × 10−6 m2 /s

m = 0.7

Fluid dynamic design:

The column diameter is determined by the gas flow. Therefore, a gas load has to be found, which lies between the maximum and minimum gas load. (1) Maximum gas load: The limiting criterion for the maximum gas load for large tray spacing is given by Eq. (9.11):

Fmax = 2.5 · ϕ2 · σ · (%L − %G ) · g


1/4

.

527

9.2 DESIGN OF TRAY COLUMNS

A relative free area of ϕ = 0.1 is selected such that

1/4 Fmax = 2.5 · 0.12 · 0.02 N/m · 800 kg/m3 − 2.6 kg/m3 · 9.81 m/s2 √ = 2.8 Pa . (2) Minimum gas load: Two criteria have to be considered for the minimum gas load: the criterion for a non-uniform gas flow (Fmin,1 ) according to Eq. (9.12) and the criterion for weeping (Fmin,2 ) according to Eq. (9.13): r

Fmin,1 = ϕ ·

2·

σ . dh

A hole diameter of dh = 8 mm is selected for the sieve tray: s √ 0.02 N/m Fmin,1 = 0.1 · 2 · = 0.22 Pa 8 mm s

1.25

(%L − %G ) 0.37 · dh · g · %G 0.25 v
1.25 u u 800 kg/m3 − 2.6 kg/m3 2 t = 0.1 · 0.37 · 8 mm · 9.81 m/s ·
0.25 2.6 kg/m3 √ = 0.98 Pa .

Fmin,2 = ϕ ·

Since both criteria for the minimum gas load have to be fulfilled, the following minimum gas load is used: √ Fmin = max (Fmin,1 , Fmin,2 ) = 0.98 Pa . (3) Column diameter: The geometric mean of the minimum and √maximum gas load is used for the design value of the gas load F = 1.66 Pa. The column diameter will be calculated as follows: √ F 1.66 Pa m uG = √ =q = 1.03 %G s 2.6 kg/m3

ˆG M 85.8 m3 /kmol m3 V˙ G = G˙ · = 0.05 kmol/s · = 1.65 %G s 2.6 kg/m3 Aac =

V˙ G 1.65 m3 /s = = 1.6 m2 . uG 1.03 m/s

528

9 DESIGN OF MASS TRANSFER EQUIPMENT

The column cross section Ac = Aac + 2 · Ad . The area of the downcomer is a circular segment. From geometric relations, it follows that   s       Aac 2  lw lw 2 lw 4  = 1 − · arcsin − − Ac π Dc Dc Dc A relative weir length of lw /Dc = 0.7 is selected (see Figure 9.5):   q Aac 2 2 4 = 1 − · arcsin (0.7) − (0.7) − (0.7) = 0.825 Ac π

Ac

=

Aac = 1.94 m2 0.825

Afterward the column diameter Dc is calculated: r 4 · Ac Dc = = 1.57 m . π (4) Preliminary liquid flow check: Further on, the weir load has to be checked using the resulting weir dimensions. • First, the liquid flow per weir length is checked:

lw = 0.7 · Dc = 0.7 · 1.57 m = 1.1 m 3 ˆL M 85.8 kg/kmol −3 m V˙ L = L˙ · = 0.034 kmol/s · 3 = 3.65 × 10 %L s 800 kg/m

V˙ L 3.65 × 10−3 m3 /s m3 m3 = = 3.32 × 10−3 = 12.1 . lw 1.1 m m·s m·h The value of the liquid flow per weir length is higher than the minimum value of 2 m3 /(m · h) and lower than the maximum value of 60 m3 /(m · h). • Next, the liquid velocity uLd in the downcomer is checked:

Ad =

Ac − Aac 1.94 m2 − 1.60 m2 = = 0.17 m2 2 2

uLd =

V˙ L 3.65 × 10−3 m/s m = = 0.021 . 2 Ad s 0.17 m

The liquid velocity in the downcomer is lower than the maximum value of 0.1 m/s.

529

9.2 DESIGN OF TRAY COLUMNS

• Last, the liquid residence time τd in the downcomer is checked:

τd =

H . uLd

A tray spacing of H = 0.5 m is selected:

τd =

0.5 m = 23.8 s . 0.021 m/s

The residence time of the liquid in the downcomer is larger than the minimum residence time of 5 s. (5) Froth height: Before the froth height can be evaluated, the liquid hold-up in the froth is determined using Eq. (9.17). The froth height is calculated using Eq. (9.18). A weir height of hw = 0.05 m is selected: 

εL = 1 −

F Fmax

0.28

=1−

√ !0.28 1.66 Pa √ = 0.136 2.8 Pa

!2/3   1.45 V˙ L 125 hf = hw + 1/3 · + · lw · εL g · (%L − %G ) g  √ 2 F − 0.2 m/s · %G · 1 − εL  2/3 1.45 3.65 × 10−3 m3 /s = 0.05 m + ·
1/3 1.1 m · 0.136 9.81 m/s2 ! 125
+ 9.81 m/s2 · 800 kg/m3 − 2.6 kg/m3 q  2 √ 1.66 Pa − 0.2 m/s · 2.6 kg/m3  · 1 − 0.136

= 0.05 m + 0.057 m + 0.038 m = 0.145 m . The tray spacing is by far larger than the froth height. (6) Entrainment: The entrainment is determined from Figure 9.18: √ F 1.66 Pa √ = = 0.59 Fmax 2.8 Pa

530

9 DESIGN OF MASS TRANSFER EQUIPMENT

(H − hf ) · ∆% · g = 2 Fmax
(0.5 m − 0.14 m) · 800 kg/m3 − 2.6 kg/m3 · 9.81 m/s2 = = 354 .  √ 2 2.8 Pa Using Figure 9.18 yields

V˙ E = 1.3 × 10−5 V˙ G V˙ E V˙ E V˙ G 1.65 m3 /s = · = 1.3 × 10−5 · = 0.006 . 3.65 × 10−3 m3 /s V˙ L V˙ G V˙ L The value of the entrainment is far lower than the highest tolerable entrainment value of 0.1. (7) Pressure drop: The total pressure drop of a tray is determined using Eq. (9.30):

∆p = ∆pd + ∆pL + ∆pR • Dry pressure drop ∆pd

ζ ζ ∆pd = · Fh 2 = · 2 2



F ϕ

2

The orifice coefficient ζ is evaluated using Figure 9.25. A tray thickness of s = 2 mm is selected:

s 2 mm = = 0.25 dh 8 mm uGh · dh · %G uG · dh · %G 1.03 m/s · 8 × 10−3 m · 2.6 kg/m3 = = ηG ϕ · ηG 0.1 · 8.9 × 10−6 Pa · s 4 = 2.4 × 10

Reh =

ζ0

= 2.57 .

Since s/dh = 0.25 −→ 0, ζ is calculated using Eq. (9.43):

ζ = ζ0 + ϕ2 − 2 · ϕ ·

p

ζ0 = 2.57 + 0.12 − 2 · 0.1 ·

The pressure drop ∆pd equals

√ !2 2.26 1.66 Pa ∆pd = = 311 Pa . 2 0.1

√

2.57 = 2.26

531

9.2 DESIGN OF TRAY COLUMNS

• Pressure drop due to liquid on the tray ∆pL

hL

= hf · εL = 0.145 m · 0.136 = 0.02 m

∆pL = hL · %L · g = 0.02 m · 800 kg/m3 · 9.81 m/s2 = 157 Pa • Residual pressure drop ∆pR

∆pR ≈ 0 residual factors can be neglected. The total pressure drop per tray equals

∆p = 311 Pa + 157 Pa + 0 Pa = 468 Pa . (8) Downcomer capacity: The maximum downcomer capacity is evaluated using Eq. (9.15): s !   V˙ L %L − %G hp + hL = 0.61 · εLd · hcl · 2 · g · H · · 1− . lcl %L εLd · H max

For non-foaming systems, εLd = 0.4. The height hp equals

hp =

∆p 468 Pa = = 0.06 m . %L · g 800 kg/m3 · 9.81 m/s2

The following value is selected for the downcomer clearance:

hcl = 0.8 · hw = 0.8 · 0.05 m = 0.04 m . The maximum downcomer capacity equals ! V˙ L = 0.61 · 0.4 · 0.04 m · lcl max s

2 · 9.81 m/s2 · 0.5 m ·

·

800 kg/m3 − 2.6 kg/m3 · 800 kg/m3

r

0.06 m + 0.02 m 0.4 · 0.5 m = 0.0236 m3 /(m · s) = 85.11 m3 /(m · h) > 12.1 m3 /(m · h) . ·

1−

Since the weir length and the clearance length are equal, the weir load can be checked for maximum downcomer capacity. The actual liquid load is much lower than the maximum downcomer capacity.

532

9 DESIGN OF MASS TRANSFER EQUIPMENT

Mass transfer efficiency:

For the quantification of the mass transfer efficiency, two key quantities are presented: the point efficiency and the tray efficiency. (1) Gas-side point efficiency: For calculation of the relative interfacial area, we assume that spray regime is present on the tray. The relative interfacial area can then be calculated using Eq. (9.29):  0.28 ! F2 F a= · 1− 2 · σ · ϕ2 Fmax    2 √ √ !0.28  2 1.66 Pa 1.66 Pa  = 938 m . √ = · 1 − 2 2 · 0.02 N/m · 0.1 m3 2.8 Pa The overall gas point efficiency is calculated using Eq. (9.51):

EOG = 1 − exp (−NTUOG ) . The number of transfer units is calculated using Eq. (9.59): s 4 DG hf a· · · π uG εG NTUOG = = r ˆ L %G DG M 1+m· · · ˆ DL M %L s G 4 4.4 × 10−6 m2 /s 0.145 m 938 m2 /m3 · · · π 1.03 m/s 1 − 0.136 s = = 0.837 . 4.4 × 10−6 m2 /s 2.6 kg/m3 1 + 0.7 · ·1· 4.6 × 10−9 m2 /s 800 kg/m3 The overall gas point efficiency equals

EOG = 1 − exp (−0.837) = 0.57 . (2) Gas-side tray efficiency: The gas-side tray efficiency is evaluated depending on the degree of liquid mixing, described by the Péclet number given in Eq. (9.20). Since lL ≈ lw for a circular column with lw = 0.7 · Dc , Eq. (9.21) can be used to calculate Pe :

Pe =

V˙ L . DE · εL · hf

533

9.3 DESIGN OF PACKED COLUMNS

The dispersion coefficient is evaluated using Eq. (9.22): r εL DE = 1.06 · hf · F · %L − %G s √ 0.136 = 1.06 · 0.145 m · 1.66 Pa · 800 kg/m3 − 2.6 kg/m3

= 3.33 × 10−3

m2 . s

The Péclet number equals

Pe =

3.65 × 10−3 m3 /s = 55.5 . 3.33 × 10−3 m2 /s · 0.136 · 0.145 m

Comparing the Péclet number with Figure 9.30, plug flow can be assumed on the tray, and Eq. (9.48) is used for the evaluation of the overall gas tray efficiency: ! ! ˙ G˙ EOGM L/ 1 m = · · exp · EOG − 1 ˙ G˙ EOG m EOG L/

EOGM =

9.3

0.034 kmol/s 0.05 kmol/s

0.7









0.7 · 1 · exp  0.034 kmol/s · 0.57 − 1 = 0.78 . 0.05 kmol/s

Design of Packed Columns

In packed columns there is countercurrent flow of gas (upward flow) and liquid (downward flow) throughout the column. This feature makes a packed column, in principle, more effective for mass transfer than a tray column in which there is a cross-flow of gas and liquid on each tray. The countercurrent flow of the two phases utilizes the existing driving force for mass transfer more efficiently. However, the countercurrent flow of gas and liquid is not perfect in a packed column, since the liquid flow is usually not uniform over the cross section. There is a strong tendency for maldistribution of liquid, in particular due to a non-homogeneous structure of random packings. Well-known flow mechanisms of maldistribution are liquid channeling and wall flow [Hanusch et al. 2018b]. A good packing design is characterized by a high capacity as well as a high separation efficiency. Equally important is a great flexibility to gas and liquid throughput combined with almost constant separation efficiency. Figure 9.33 shows a rough comparison of capacity and separation efficiency of several metal packings. There are significant differences in the performance of different types of packings. Struc-

534

9 DESIGN OF MASS TRANSFER EQUIPMENT

tured packings are generally superior to random packings in both capacity and separation efficiency. Modern random packings perform better than the standard Pall ring packings. Also the standard corrugated sheet packing is surpassed by modern developments.

Figure 9.33 Qualitative comparison of capacity and separation efficiency of several metal packings based on Eiden 1997. Reference point is a corrugated metal sheet packing with a specific surface area of 250 m2 /m3 .

9.3.1

Design Parameters of Packed Columns

In order to provide a high interfacial area for mass transfer between gas and liquid, the column is filled with a bed of solid material with high porosity and large volumetric area. The liquid trickles downward in the bed in form of thin films, small rivulets, and even small droplets. Ideally, the entire surface of the bed is wetted by the liquid, and liquid flow through the bed is uniform. A low flow resistance of the bed against gas flow is important. Thus, the bed should have a high voidage and a shape that reduces pressure losses of the gas flow. If the flow resistance is too low, however, there is a risk of maldistribution of the gas flow, which can lead to a reduced mass transfer. 9.3.1.1

Types of Packings

The types of packings can be distinguished into random packings (dumped packings) and structured packings (ordered packings). Random Packings

Approximately one third of all packed columns use random packings consisting of a large number of individual particles. Some examples of packing elements are shown in Figure 9.34. The maximum flow rate and the separation efficiency of the column depend directly on the particle shape. Randomly dumped packings should be as

9.3 DESIGN OF PACKED COLUMNS

535

homogeneous as possible; bridging and large cavities between particles should be avoided. The particles should be sufficiently mechanical stable and should form a large packing surface with a low gas pressure drop. The feasible particle size depends on column diameter; depending on particle shape, it should not exceed 1/15 to 1/20 of the column diameter. Most particles are made of ceramics, metal, or plastics. Characteristic data of some random packings are compiled in Table 9.6. The data are average values with limited accuracy. If data published by the suppliers is available, it should be used instead of the values given in Table 9.6. One of the earliest technical packing elements is the Raschig ring patented as early as in 1914 [Raschig 1914]. The characteristic feature is that the length of the ring is equal to its diameter. This feature allows the particles to form a fairly homogeneous bed structure when filled into the column. Raschig rings (like all random packings of the first generation) have a rather high pressure drop because the walls of those rings in a horizontal position block the gas flow. The Pall ring (a second-generation representative) avoids this disadvantage as parts of the wall are punched out and bent into the interior of the ring. A Pall ring has approximately the same porosity and the same volumetric area as a Raschig ring, but a significantly lower pressure drop. For many years, Pall rings have been the standard random packing element. The third generation of random packing elements is characterized by the transition from planar structures to lattice structures [Billet 1993]. This reduces the pressure drop of the gas flow and in turn enhances surface renewal of the liquid phase by increased drop formation. About two decades ago, a fourth generation of random packings was introduced [Schultes 2003]. These packings have been further optimized in terms of efficiency, capacity, and pressure drop and are characterized by being significantly different from the original basic shapes (sphere, cylindrical ring, saddle) [Maćkowiak and Maćkowiak 2014]. Structured Packing

In any random packing, a certain degree of inhomogeneity is unavoidable. These inhomogeneities cause liquid maldistribution that adversely affects the mass transfer efficiency of the column. Thus, for years, attempts have been made to develop ordered packing structures [Fair 1988]. The first idea in developing an ordered structure is a concentric or rolled-up sheet structure that fits perfectly into the cylindrical shell of a column. However, all these structures failed in large diameter columns because the lateral mixing of gas and liquid was poor. Even slots or holes in the sheets to enhance radial mixing showed no improvement. A breakthrough was the development of the corrugated sheet structure by Sulzer company in the mid-1960s [Sulzer 1982]. Here, corrugated sheets are assembled parallel in vertical direction with alternating inclinations of the corrugations of adjacent sheets (see Figure 9.35). Lateral mixing of liquid is enhanced parallel to the sheets but suppressed vertical to the sheets. Therefore, the orientation of the sheets is changed in the subsequent layer (approximately 210 mm height) by 90° to enhance lateral mixing of the liquid in all directions. Since the column shell is not always perfectly cylindrical and there can be gaps between the packing and the column, additional wall wipers have to be installed.

536

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.34 Selected elements used in random packings: first row: ceramic, second row: plastic, third and fourth row: metal. Source: Data generated with TrayHeart [WelChem Process Technology 2020].

537

9.3 DESIGN OF PACKED COLUMNS Table 9.6 Typical values of specific surface area a and porosity ε depending on the nominal diameter dn for several random packings. Particle

Diameter dn

Ceramic

(mm)

Specific area a (m2 /m3 )

Balls

2 4 6 10 20 50

1800 900 600 360 180 72

0.4 0.4 0.4 0.4 0.4 0.4

Raschig rings

5 10 15 20 25 35 50 80

1000 440 330 240 195 140 98 60

0.56 0.65 0.70 0.72 0.73 0.76 0.77 0.77

Pall rings

10 15 25 35 50 80

220 165 120 75

Porosity ε

0.73 0.76 0.77 0.77

Bialecki rings

25 35 50

Torus saddles

25 35 50

255 166 120

0.74 0.76 0.79

Hiflow

20 25 35 50 75

285

0.76

140 97 61

0.83 0.81 0.85

Metal

Plastic

Specific area a (m2 /m3 )

Porosity ε

1000 500 350

0.87 0.89 0.92

220 150 110 65

0.92 0.93 0.95 0.96

515 360 215 145 105 78

0.92 0.93 0.94 0.94 0.95 0.96

225 155 110

0.95 0.95 0.96

185

0.86 0.87 0.87 0.90 0.91 0.91

350 220 160 110

0.88 0.91 0.93 0.93

0.96

218

0.92

93

0.98

114

0.93

15 25 40 50

282 225 150 100

0.96 0.97 0.98 0.98

Raschig Super-Ring

15 22 50 70

315 215 100 80

0.96 0.98 0.98 0.98

No.1 No.2 No.3

Cascade mini-rings

No.0 No.1 No.2 No.3 No.4

89 66 49

0.76 0.78

Porosity ε

350 290 220 150 110 65

IMTP

Tellerettes

Specific area a (m2 /m3 )

338 249 114 105 75

0.97 0.96 0.97 0.98 0.98

325 205 100 75

92 93 96 97

180 125 98

0.87 0.93 0.92

321 183 141 75

0.89 0.92 0.94 0.96

538

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.35 Structured packing assembled by corrugated sheets.

These packings provide a homogeneous bed structure and, due to the vertical orientation of the sheets, a low pressure drop of the gas flow. The first application of such structures with wire gauze elements was vacuum distillation. Meanwhile, such structures are widely used in low-pressure distillation with excellent results. Characteristic data of some important structured packings are given in Table 9.7. Most structured packings have a channel angle of 45° or 60° to the horizontal. The packings of different suppliers differ mainly in the microstructure of the sheet surface. A further development are packings in which the channel angle is steeper at the transition between the segments. Therefore, the capacity of the packing can be increased, since there is an unsteadiness at this point, which is counteracted with a reduced flow resistance [Brunazzi et al. 2002]. Additive manufacturing offers new possibilities for packing design but is still quite expensive. In the field of laboratory packings with very small diameters, such additively manufactured packings could soon be a good alternative [Neukäufer et al. 2019]. 9.3.1.2

Model Structures of Packings

The structure of any packing is too complex to model because of the large number of geometric dimensions. Thus, in the current state of the art, the complex real structure of the packing is substituted by a simple model structure having the same specific area a and the same porosity ε. Two simplified model structures are most common: the particle model structure and the channel model structure.

539

9.3 DESIGN OF PACKED COLUMNS

Table 9.7 Typical values of volumetric surface a and porosity ε for several structured packings.

Supplier

Designation

Specific area a (m2 /m3 )

Porosity ε

Sulzer (metal gauze)

BX CY

500 700

0.90 0.85

Sulzer (metal sheet)

Mellapak M250.X Mellapak M250.Y MellapakPLUS M252.Y Mellapak M350.Y MellapakPLUS M352.Y

250 250 250 350 300

0.988 0.988 0.988 0.975 0.985

Montz (metal sheet)

B1 100 B1 250 B1 250MN B1 350 B1 350MN

100 250 250 350 350

0.99 0.988 0.988 0.983 0.983

Raschig (metal sheet)

RPM 250X RPM 250Y RPM 250HC RSPM 250X

250 250 250 250

0.981 0.981 0.981 0.98

RVT (metal sheet)

RMP N250X RMP N250Y RMP S250 RMP N350Y

250 250 250 350

0.987 0.988 0.989 0.983

Koch–Glitsch (metal sheet)

Flexipak 3 Flexipak 2 Flexipak 1 Intalox 2T Intalox 1T

134 223 558 213 315

0.96 0.93 0.91 0.97 0.95

540

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.36 Model structures of packings.

Particle Model Structure

In the particle model structure, the real packing is represented by a collective of solid particles (i.e. balls). Thus, the solid material of the packing is considered as the dispersed phase; see Figure 9.36. At high porosities (ε > 0.4), the spheres do not touch each other, i.e. the model structure is very similar to a fluidized bed. For the porosity it follows that

ε=

VG z d3p · π =1− · , V V 6

(9.60)

with z the number of balls and V the volume of the bed. The surface area of the balls is

a=

z · d2p · π A = . V V

(9.61)

From both equations it follows that after elimination of z/V , we have

dp =

6 · (1 − ε) . a

(9.62)

The diameter dp characterizes the structure of the solid material of the packing. The particle model structure is quite realistic because the solid phase is considered as dispersed phase. Therefore, this model structure will be used in the following sections. Channel Model Structure

In the channel model structure, the real packing element is replaced by a system of parallel channels (see Figure 9.36), with the equivalent diameter deq . The equivalent diameter is chosen so that the porosity ε and the volumetric surface a of the model structure have the same values as the actual packing. For the porosity it follows that

ε=

z · H · d2eq · π VG = . V V ·4

(9.63)

9.3 DESIGN OF PACKED COLUMNS

541

Here, z denotes the number of channels and H the height. The specific area of the model structure is:

a=

A z · H · deq · π = . V V

(9.64)

Combining Eqs. (9.63) and (9.64) yields

deq =

4·ε . a

(9.65)

The diameter deq characterizes the open structure of the packing. Hence, the definitions and meanings of dp and deq are totally different. The two quantities must not be confused. The equivalent diameter deq of the channel can be expressed by the particle diameter dp . Eliminating the volumetric area a from Eqs. (9.62) and (9.65) yields

d∗eq =

2 ε · · dp . 3 1−ε

(9.66)

Equation (9.66) is in most cases used without the factor 2/3:

deq =

ε · dp . 1−ε

(9.67)

Equation (9.67) describes a relation between the particle diameter dp and the channel diameter deq . This is a roundabout way: first, the particle diameter is determined by Eq. (9.62) from knowledge of a and ε, and, second, the equivalent diameter is determined from the particle diameter. Equation (9.65) allows a more straightforward determination of the equivalent channel diameter deq . The channel model is very often used in modeling of packings. However, it is unrealistic since the solid material of the packing is treated as the continuous phase. In most real packings the solid material is the dispersed phase. Moreover, at high porosity values, i.e. ε > 0.907, the cross sections of the channels intersect, so the structure of the channel model cannot be represented geometrically. 9.3.1.3

Internals

Obviously, the most important internal component of a packed column is the packing itself. However, some supplementary elements are required for the proper operation of the column. These internals are schematically depicted in Figure 9.37. The most important elements are: • • • • • •

Liquid distributor and redistributor Liquid collector Wall wiper Support grid Hold-down plate Gas inlet

542

9 DESIGN OF MASS TRANSFER EQUIPMENT

All these elements require careful adjustment to the packing. Packing and internals form a complex and highly integrated system. Even a high efficiency packing can fail if, for instance, the liquid distributor is not properly designed. Therefore, most suppliers offer systems of packings and internals.

Figure 9.37 Cutaway section of a packed column with structured packing [Sulzer 1982]: (a) liquid distributor, (b) liquid collector, (c) structured packing, (d) support grid, and (e) manhole.

The basic requirement to all packing internals is that they must not hinder the countercurrent of gas and liquid through the column. Ideally, they should have a free area that is as high as the porosity of the bed. This requirement is difficult to meet, since the porosity of a modern structured packing can even exceed 98 %. Often the packing supplements give rise to an increased pressure drop and, even worse, to

9.3 DESIGN OF PACKED COLUMNS

543

flooding below the flooding point of the bed. Rix and Olujic 2008 give information on the pressure drop of some column internals. Liquid Distributor

A key element of any packed column design is the liquid distributor. Especially in vacuum distillation, which is characterized by a low liquid load, the liquid distributor is crucial for separation efficiency of the entire column. The requirements for a liquid distributor are: • • • • •

Uniform distribution of liquid across the entire cross section of the column. High flexibility to liquid flow rates. Low resistance against gas flow rate (low pressure drop). Low sensitivity to fouling. No entrainment of liquid at high gas flow rates.

High performance distributors provide a flow rate deviation per irrigation point of less than 5 – 6 % of the average flow [Strigle 1987]. Larger deviations between adjacent irrigation points can be tolerated since lateral liquid mixing flattens the flow rate profile. Particularly unfavorable are large-scale flow rate variations since they are not compensated by lateral mixing within the bed. Industrial liquid distributors use nozzles, weirs, or perforated plates and troughs as distribution elements. Nozzles require a high liquid pressure (∆p > 1 bar) for good operation and imply the risk of entrainment of small droplets [Ulrich et al. 1983]. Weirs are often equipped with dripping elements (noses) or zigzag edges. Plate distributors have simple holes or short pipes. The rule of thumb is 65 [Moore and Rukovena 1987] or 100 [Strigle 1987] irrigation points per square meter. Hanusch et al. 2019b show that the distributor design has to take into account the type of packing and the column diameter. Decisive for good operation is a precise leveling of weirs, plates, and troughs of the distributors. Up to a diameter of 0.8 m, the standard design is an orifice pan distributor consisting of a pan with small holes in the bottom for the downward flow of liquid. The gas flows upward through chimneys in the pan and the annular gap between pan and column shell. Columns larger than 0.8 m in diameter require a more complex distribution system with pipes or troughs. Distributors are made of metal or plastic, in very special cases of ceramics. Also important is the segmentation of the elements for installation in columns with small manhole diameters. Liquid Redistributor

After a certain height of the packing, the liquid needs redistribution in order to avoid a high degree of liquid maldistribution. Depending on the type of packing and the liquid load, the maximum feasible distance between redistribution is 6 – 8 m. In any case, the ratio of packing section height Hs and column diameter Dc should meet the condition H/Dc < 6. In small diameter columns maldistribution is mainly caused by wall flow. Here, simple wall wipers are used to improve the liquid flow profile. However, in large diameter columns, a system of liquid collector and liquid

544

9 DESIGN OF MASS TRANSFER EQUIPMENT

distributor has to be installed for redistribution as shown in Figure 9.37, for example. The height of such a system of liquid collector and redistributor can be up to 2 m, which significantly increases the total column height without contribution to mass transfer. Liquid Collector

Liquid collectors have to be installed for the withdrawal of side stream products and for pump arounds (see Section 8.3.2.2). In high efficiency packings, an effective liquid collector should be installed above each liquid redistributor. A liquid collector has to interrupt the liquid flow without any blocking of gas flow. A good design is shown in Figure 9.37. An essential additional task of a liquid collector is the effective mixing of the liquid before the redistribution since high differences of liquid concentrations may exist across the flow cross section (see Figure 9.47). Support Grid

A grid or a perforated plate is installed within the column to support the packing and, not to forget the liquid hold-up of the packing. The support grid has to allow both the downward-flowing liquid phase and the upward-flowing gas phase to pass over the whole operating range of the packing [Strigle 1987]. Therefore, the free area of the support grid should be as high as the porosity of the bed. However, the grid must not be so wide that packing elements can fall through. Especially in dumped packings, there is a high risk that the packing elements (e.g. Raschig rings) block some of the open area of the support grid. This increases the pressure drop of the column and may even cause flooding (see Section 9.2.3.1). Planar support grids are generally used for structured packings and also for random packings in small diameter columns (e.g. Dc < 0.6 m). Multi-beam packing support has a higher mechanical stability and, therefore, is used for random packings in columns with larger diameter. Hold-Down Plate

The main purpose of hold-down plates (or bed limiters) is to prevent expansion of the packed bed as well as to maintain a horizontal bed surface level [Strigle 1987]. Since the height of a dumped packing may shrink or settle during operation, the holddown plate is not fixed at the column wall but acts by its own weight. The hold-down plate should not restrict the gas and liquid flow through the column. Hence, the free area of these plates should be very high. The installation of hold-down plates is especially important above low weight packings as random metal and plastic packings. At structured packings, the liquid distributor is often installed directly atop the packing itself. In this case, no hold-down plate is required. Gas Inlet

A uniform gas flow across the column cross section is just as important as a uniform liquid flow. However, the risk of gas maldistribution is much lower than that of liquid maldistribution, since the gas flow rate profile is flattened by the pressure drop. Nevertheless, modern structured packings in large diameter columns require

9.3 DESIGN OF PACKED COLUMNS

545

careful initial gas distribution because of their low pressure drop. For column diameters up to Dc = 2.5 m, a simple inlet pipe is sufficient (see Figure 9.37). For a diameter larger than Dc = 2.5 m, a distributing system is recommended [Moore and Rukovena 1987]. 9.3.2

Operating Region of Packed Columns

The feasible operating region of packed columns is shown in Figure 9.38. There, the gas load V˙ G /Ac is plotted against the liquid load V˙ L /Ac (Ac is the cross-sectional area of the column). The operating region of packed columns differs considerably from that of tray columns (see Section 9.2.2.5). In packed columns, the gas load can be reduced to extremely low values, but the liquid load has to be maintained within an upper and a lower limit. This difference affects the choice of apparatus. Two principal mechanisms limit the gas and liquid flow rates: flooding and dewetting.

Figure 9.38 Operating region of a packed column.

9.3.2.1

Flooding

The upper capacity limit of a packed column is set by flooding. At the flooding point, the pressure drop of the gas flow through the bed increases so drastically that the liquid can no longer flow completely countercurrent to the gas in the packing. Thus, the countercurrent of gas and liquid collapses; liquid is intermittently discharged upward by the gas. Often the liquid forms a froth layer above the packing. The experimental determination of the flooding point is best carried out by increasing the gas load at constant liquid load. The maximum feasible gas load is easily determined from a plot of the pressure drop versus the gas load; see, for instance, Figure 9.46. This method becomes inaccurate only when operating with very

546

9 DESIGN OF MASS TRANSFER EQUIPMENT

low liquid load. However, the pressure drop at the flooding point can never be determined precisely as it increases infinitely approaching the flood point. Hence, very small variations in the gas flow drastically increase the pressure drop. Several attempts have been made to develop a general method for flooding point prediction. Sherwood et al. 1938 were the first to present an empirical flooding point correlation based on a variety of experimental data. They succeeded in obtaining a single flooding point line by plotting a modified gas load versus the flow parameter, i.e. the square root of the ratio of the kinetic energy of liquid and gas. Later on, Leva 1954 completed the Sherwood diagram by implementing the pressure drop in form of parameter lines. Several modifications of this diagram have been made by Eckert (e.g. Eckert 1970); see Figure 9.39. Kister 1992 adapted this diagram to modern packings. Surprisingly, the liquid viscosity ηL is combined with the gas load in the ordinate of all these diagrams.

Figure 9.39 Correlation of flooding point and irrigated pressure drop according to Eckert 1970.

A theoretically more profound flooding point correlation was developed by Mersmann 1965 and later on improved [Mersmann and Bornhütter 2002]. In this diagram (see Figure 9.40), the gas load, expressed as a dry pressure drop, is plotted versus a modified liquid load. The term for the liquid load was derived from the theory of film flow over a vertical plate. It includes the viscosity of the liquid. The diagram contains the flooding line, parameter lines for the pressure drop of the irrigated packing, and the loading line. The Mersmann diagram and the Sherwood diagram and its modifications claim, in principle, to be valid for all types of packings and to enable a genuine prediction

9.3 DESIGN OF PACKED COLUMNS

547

Figure 9.40 Correlation of flooding, loading, and irrigated pressure drop according to Mersmann and Bornhütter 2002.

of the fluid dynamic behavior of any packing. Hereby, the packing is characterized by the porosity, the volumetric surface, and the packing factor or the dry pressure drop. These parameters are, however, not sufficient to describe the complex structure of a packing precisely. Therefore, these generalized correlations imply rather high inaccuracies. Frequently, a plot of experimental flooding data versus the flow parameter as shown in Figure 9.41 is used. The flooding lines of different types of packings differ considerably. Such charts are often published by suppliers, mostly on the basis of air/water data. These diagrams, however, are not able to predict the fluid dynamic behavior of a new type of packing. They are just an adequate plot of experimental data. Even their applicability to other systems is questionable. A more rigorous approach to flooding point prediction should be based on a suitable model for the pressure drop of an irrigated packing. If such a model is also valid above the loading point (see Section 9.3.3.2), it will predict the sharp increase of pressure drop near the flooding point. Such models based on a channel model packing structure were developed by Maćkowiak 1991 and Billet 1995. A model developed by Stichlmair et al. 1989 and improved by Engel et al. 2001 that is based on the particle structure is outlined in Section 9.3.3.2.

548

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.41 Presentation of flooding point data [Billet 1995].

9.3.2.2

Minimum Liquid Load

In contrast to tray columns, packed columns require a minimum liquid load to ensure adequate mass transfer. Below this minimum value, only a small part of the packing surface is wetted, and so liquid and gas are no longer in intimate contact. This results in a considerable decrease in the separation efficiency. The lower capacity limit of liquid load depends on the type and the material of the packing, the quality of the liquid distributor, and the physical properties (e.g. surface tension) of the liquid. A rule of thumb gives the following values for random packings: • uL,min > 5 m3 /(m2 · h) in aqueous systems • uL,min > 2 m3 /(m2 · h) in organic systems Highly efficient structured packings with high quality liquid distributors allow an effective column operation with liquid loads as low as uL,min = 0.2 m3 /(m2 · h) in organic systems (e.g. Sulzer Chemtech 1994). This value is comparable to the flow rate of a heavy rainfall. A useful equation for estimating minimum liquid load has been published by Schmidt 1979:  2/9   %L · σ 3 g 1/2 −6 uL,min = 7.7 · 10 · · . (9.68) 4 ηL · g a where %L is the liquid density, σ the surface tension, g the acceleration due to gravity, ηL the liquid viscosity, and a the specific area of the packing. This equation accounts for the most important parameters and gives in most cases the right order of magnitude of the minimum liquid load.

9.3 DESIGN OF PACKED COLUMNS

9.3.3

549

Two-Phase Flow in Packed Columns

A packed column establishes a pure countercurrent of liquid and gas streams. The liquid trickles downward over the surface of the packing in the form of rivulets or films. Considerable drop formation can also occur within the bed if a large packing size is used. 9.3.3.1

Liquid Hold-up

The fluid dynamic behavior of a packed column is governed by the amount of liquid that accumulates within the packing during operation. The accumulated liquid is called liquid hold-up hL . The volume of the liquid VL is conveniently referred to the volume V of the packing:

hL =

VL . V

(9.69)

Hence, a packed column forms a three-phase system with a solid phase, a liquid phase, and a gas phase. During operation, the irrigated liquid flows through the bed in countercurrent to the gas flow. Whenever a liquid element touches the packing surface, it is slowed down due to the high viscosity of the liquid. Hence, most of the liquid trickles downward in the bed in the form of thin films or small rivulets. Only a small fraction of the liquid may move downward in the form of falling droplets. Since the liquid hold-up is not of prime interest for column operation as, for instance, pressure drop, flooding, or even interfacial area, the level of knowledge is also lower. Liquid hold-up is an intrinsic quantity that controls the fluid dynamics of the column. Thus, knowledge of the liquid hold-up is of great importance for a deeper understanding of the flow mechanism in the column. It facilitates the prediction of the fluid dynamic operation parameters. Many authors distinguish the liquid hold-up hL into a static hold-up hstat and a dynamic hold-up hdyn :

hL = hstat + hdyn .

(9.70)

The static hold-up results from the action of capillary forces that hold some liquid in narrow sections (pinches) of the packing. During operation, the static hold-up is in contact with the dynamic hold-up (see Figure 9.42), but when the irrigation is stopped, the static hold-up remains within the packing. The static hold-up is quite high in small packings with a plate structure, e.g. small Raschig rings with a high volumetric area a. Using a dimensionless Bond number

Bo =

%L · g , σ · a2

the static hold-up can be calculated [Engel et al. 1997]  %L · g  hstat = 0.033 · exp −0.22 · . σ · a2

(9.71)

(9.72)

550

9 DESIGN OF MASS TRANSFER EQUIPMENT

The dynamic hold-up strongly depends on liquid load. The higher the liquid load, the higher the dynamic hold-up. In modern packings with a lattice structure, the dynamic hold-up is much larger than the static hold-up.

Figure 9.42 Sketch for explanation of static and dynamic hold-up based on Brauer 1971.

Figure 9.43 shows a typical result of hold-up measurements [Billet 1995]. The hold-up hL is plotted versus the gas load uG with the liquid load uL as parameter. The hold-up hL is in the range of 1.5 – 20 % of the bed volume. Hence, it can be higher than the volume of the solid material of the packing (in the investigated packing approximately 6 % of the bed volume). At low gas loads the hold-up is a function of liquid load only. At the loading point the gas flow starts increasing the hold-up. As can be seen from the diagram, the loading point is reached at lower gas velocities with increasing liquid load.

Figure 9.43 Liquid hold-up data of a random packing of metal Bialecki rings [Billet 1995]. Figures 9.43 and 9.46 establish a full set of fluid dynamic data.

Hold-up Below the Loading Point

Several hold-up correlations have been published in literature. Most of them describe the hold-up below the loading point only. Here, the hold-up depends primarily on

551

9.3 DESIGN OF PACKED COLUMNS

the Froude number of the liquid. However, different definitions of the Froude number have been used by various authors. Since most of the liquid hold-up is attached to the surface of the packing, there should exist a close relationship between liquid hold-up hL and specific area a of the packing. Hence, the Froude number should be defined with the specific area a (and not with the particle diameter dp ). Further parameters that influence the liquid hold-up are the surface tension σ , the viscosity ηL , and the density %L of the liquid and the liquid load uL . Engel et al. 2001 express the dynamic liquid hold-up hdyn0 below the loading point by correlating relevant dimensionless parameters:

hdyn0 = 3.6 ·

uL · a1/2 g 1/2

!0.66

·

ηL · a3/2 %L · g 1/2

!0.25  0.1 σ · a2 · . %L · g

(9.73)

Most published data are correlated with a mean deviation about 15 % by this correlation. Similar correlations have been developed by several authors mostly in the form of potential equations. Table 9.8 lists the parameters accounted for in these equations and the values of the exponents. The large variation of the involved parameters and of their exponents demonstrates the poor and ambiguous knowledge in this field. Table 9.8 Correlations for the liquid hold-up below the loading point. Exponent of a

ε

σ

ηL

%L

uL

0.333

—

—

—

0.67

Billet and Bravo 1989

—

−0.67

— —

Blaß and Kurtz 1976

0.16 –

1

0.14

Bemer and Kalis 1978

0.22

Gelbe 1968

0.33 0.80

0.73 – −0.59

0.14

0.85

Mohunta and Laddha 1965

0.75

—

Bornhuetter 1991

0.57

−1.14

Maćkowiak 1991

0.5

−1

Billet and Schultes 1993

0.30 –

—

— — — —

0.57

−0.67

0.74 – −0.34 – −0.28

0.15 – −0.41 –

0.67 0.40 – 0.46 0.33 –

0.27

−0.29

0.45

0.25

−0.25

0.75

0.19

−0.19

0.57

0.17

−0.17

0.5

0.08 – −0.18 – 0.18

Stichlmair et al. 1989

0.33

1.55

—

—

Engel et al. 2001

0.91

—

0.10

0.25

−0.08

— −0.35

0.57 – 0.63 0.67 0.66

Hold-up Above the Loading Point

Above the loading point, the liquid hold-up is significantly influenced by the gas flow. The friction forces between gas and liquid retard the velocity of the surface of the liquid film and, in turn, increase both the thickness of the film and the pressure drop. Probably as important, however, may be the buildup of a pressure gradient

552

9 DESIGN OF MASS TRANSFER EQUIPMENT

along the column height. The liquid has to move downward in the bed against an increasing static pressure. The significance of the pressure gradient for the increase of liquid hold-up is illustrated in Figure 9.44. Here, the hold-up data of Figure 9.43 are plotted versus the dimensionless pressure drop ∆pirr /(H · %L · g) (and not versus the gas load uG ). The pressure drop data have been taken from Figure 9.46. Both diagrams represent a consistent set of fluid dynamic data because they have been collected during the same run of experiments. In the new plot the parameter lines are equidistant, and loading and flooding lines become independent of the parameter lines, i.e. of liquid load.

Figure 9.44 Modified plot of the hold-up data of Figure 9.43.

Based on the dependency of the increase in the total hold-up in Figure 9.44, the increase in the dynamic hold-up hdyn above the loading point can be correlated [Engel et al. 2001]:  2 ! ∆pirr hdyn = hdyn0 · 1 + 36 · . (9.74) H · %L · g The factor 36 and the exponent 2 have been adjusted to the experimental data. However, they should approximately be valid for all types of packings. It should be noted that the hold-up does not necessarily become infinite at the flooding point, as is often assumed. This is outlined in detail in Section 9.3.3.2. 9.3.3.2

Pressure Drop of a Dry Packing

The most important operating parameter of the two-phase flow in packed columns is the pressure drop of the gas. It is significantly influenced by the type of packing, the gas flow rate, and the liquid flow rate. First, the pressure drop of the gas in a dry

9.3 DESIGN OF PACKED COLUMNS

553

packing is treated. The pressure drop of the gas flow through a dry packed bed can be formulated either by the channel model or by the particle model. Channel Model Structure

In the channel model structure, the basis is the equation of the pressure drop in a pipe of length H :

∆pd ζ %G 2 = · ·u . H deq 2

(9.75)

Expressing the gas velocity in the channel by u = uG /ε (uG is the superficial gas velocity in the column) and the diameter d by 4 · ε/a (Eq. (9.65)) leads to

∆pd 1 %G 2 a = ·ζ · · uG · 3 . H 4 2 ε

(9.76)

Formulation of the equivalent diameter deq by Eq. (9.67) yields

∆pd 1−ε u2 = ζE · 3 · %G · G . H ε dp

(9.77)

This is the well-known Ergun equation [Ergun 1952] often used for calculating the pressure drop of a packed column. The friction factor ζE expresses the flow resistance of the bed. It has to be determined experimentally. Values for the friction factor ζE are, for instance, given in Kister 1992. It is important to be aware that the porosity term (1 − ε)/ε3 in the Ergun equation (9.77) is the result of a simplified model. This term has never been verified by comparison with experimental data since in a packed bed the porosity cannot be varied. The value of the porosity is only a function of the shape of the particles and the structure of the bed. Hence, the published values of the friction factor ζE are, in principle, only valid for the porosities of the tested beds. Particle Model Structure

Applying the particle model structure yields a different equation for the dry pressure drop. Basis for the development of an improved pressure drop equation is the pressure drop equation of a fluidized bed. The fundamental equation is derived in the appendix to this chapter in Section 9.A. The result is

∆pd 3 1−ε u2 = · ζ0 · 4.65 · %G · G . H 4 ε dp

(9.78)

This equation differs from the Ergun equation (9.77) in the friction factor ζ0 and in the porosity term. Here, ζ0 denotes the friction factor of a single particle and not, as ζE does, of the whole bed. Thus, the pressure drop of a bed of spheres, for instance, can be predicted with the knowledge of the friction factor of a single sphere. The porosity term (1 − ε)/ε4.65 was determined by experiments in fluidized beds with variation of the porosity from ε = 0.4 to 1.0. Hence, the porosity term in Eq. (9.78) is well proven, in contrast to the porosity term of the Ergun equation (9.77).

554

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.45 shows a correlation of the friction factor ζ0 versus the Reynolds number for some random packings and a structured packing [Stichlmair et al. 1989]. In a limited range of Reynolds numbers (which is sufficient because the Reynolds number in a packed column does not vary over a large range), the values of Figure 9.45 are correlated by

ζ0 = b · Re c0

with

Re 0 =

uG · dp · %G , ηG

(9.79)

where b and c are constants that depend on the type of packing; for values, see Table 9.9. The correlation of Figure 9.45 has been developed from published data of the dry pressure drop. Thus, in the technical important region, the dry pressure drop can be described with a two-parameter equation. Another possibility is to fit these parameters to the straight line of a logarithmic plot of the dry pressure drop the gas load (e.g. Figure 9.46):

∆pd = 10n · F m . H

(9.80)

The parameters n and m can easily be derived from experimental data. Engel 1999 has published an extensive list (pressure drop parameters, specific surface area, and porosity) of random and structured packings. Since even standard packings of different suppliers are slightly different, it is recommended to use dry pressure drop data published by the suppliers (as well as porosity and specific area data). Table 9.9 Values of the factor b and the exponent c of correlation Eq. (9.79).

Packing Spheres Raschig rings (ceramic) Raschig rings (metal) Pall rings (ceramic) Pall rings (metal) Pall rings (plastic) Saddles (ceramic) Saddles (plastic) 9.3.3.3

dn in mm

b

c

13 50 15 50 50 50 25 25

10.6623 4.2926 14.4952 15.3840 5.6802 8.5568 4.6338 2.5070

−0.8910 −0.0839 −0.1500 −0.2850 −0.1201 −0.1550 −0.1014 −0.0100

Pressure Drop of an Irrigated Packing

The pressure drop of an irrigated packing bed is always higher than that of a dry bed; see Figure 9.46. This increase of pressure drop is caused by the liquid that accumulates within the packing bed during irrigation. The liquid hold-up in the bed blocks a part of the free cross section and, in turn, increases the velocity of the gas flow. The pressure drop of an irrigated bed can be calculated either by the channel model or by the particle model.

9.3 DESIGN OF PACKED COLUMNS

555

Figure 9.45 Correlation of the friction factor of single particles of some packings [Stichlmair et al. 1989] according to the particle structure model.

Figure 9.46 Dry and irrigated pressure drop of a random packing of metal Bialecki rings [Billet and Maćkowiak 1984] (see also Figure 9.43).

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9 DESIGN OF MASS TRANSFER EQUIPMENT

Channel Model Structure

Most authors (e.g. Buchanan 1969; Bemer and Kalis 1978; Billet and Maćkowiak 1984; Bravo et al. 1986) use the channel model. They find

∆pirr ∆pd 1 = · . H H (1 − Ch · hL )5

(9.81)

The pressure drop of the dry bed is calculated by Eq. (9.76) or the equivalent Eq. (9.77). Hence

∆pirr 1 %G 2 a 1 = ·ζ · · uG · 3 · . H 4 2 ε (1 − Ch · hL )5

(9.82)

Here, Ch is an empirical factor that depends on the type of packing. Some values of Ch have been published by Bemer and Kalis 1978; Buchanan 1969; Billet and Maćkowiak 1984. Equation (9.82) has been used successfully by many authors for predicting the pressure drop of an irrigated packing. From a theoretical point of view, however, the equation appears inconsistent, since the influence of the volumetric fraction εs of the solid material (εs = 1 − ε) differs from the influence of the liquid hold-up hL , i.e. ∆p ∼ (1 − εs )−3 and ∆p ∼ (1 − Ch · hL )−5 , respectively. In a consistent model, the exponents in the two terms should not be different, as the hold-up of the liquid is considered as additional solid material in the model. Particle Model Structure

Applying the particle model structure to an irrigated packing leads to a more consistent formulation [Engel et al. 2001]. In analogy to the description of the real packing, the liquid hold-up should also be described by liquid particles in the model structure. For this derivation, volume, diameter, and friction factor of the liquid particles have to be determined. The volume of the liquid particles can be calculated from the knowledge of the liquid hold-up (Eq. (9.74)). The structure of the correlation of the diameter dL of the liquid particles can be derived from the balance of forces of a falling droplet (see Eq. (9.25)): s 6·σ dL = CL · . (9.83) (%L − %G ) · g The evaluation of the experimental data shows that random packings can be calculated with CL = 0.4 and structured packings with CL = 0.8. This difference can be explained by the different flow conditions in random packings (droplets and rivulets) and in structured packings (film flow). The friction factor of the liquid particles corresponds in a first approximation to the friction factor of the dry packing particles. This is based on the assumption that the liquid hold-up in the bed has no significant influence on the shape of the particles. Combining Eqs. (9.62) and (9.78) with these assumptions yields the following ba-

9.3 DESIGN OF PACKED COLUMNS

557

sic equations for the pressure drop ∆pirr of the irrigated packing:   ∆pirr 1 6 · hdyn %G · u2G = · ζ0 · a + · 4.65 . H 8 dL (ε − hdyn )

(9.84)

Conveniently, the pressure drop of an irrigated packing is related to the pressure drop of a dry packing (Eq. (9.78)):

∆pirr = ∆pd

6 · hdyn 4.65 +a  ε dL · . a ε − hdyn

(9.85)

The second term of Eq. (9.85) is very similar to Eq. (9.81), which has been derived by the channel model. As the liquid hold-up hdyn is a function of the irrigated pressure drop ∆pirr , Eq. (9.85) has to be solved by iteration. Equation (9.85) describes the pressure drop of an irrigated bed in the entire range of operation, i.e. below and above the loading point. It also correctly describes the steep increase of the pressure drop near the flood point. Thus, it should be possible to determine the flood point from the pressure drop equation. 9.3.3.4 Determination of the Flooding Point from the Pressure Drop Equation

The flooding point is reached when an infinitely steep increase in ∆pirr occurs with increasing gas load (e.g. Buchanan and Dixon 1987). Thus, if the dry pressure drop is used as a measure of the gas load, the flooding point condition is ∂ ∆pirr = ∞. (9.86) ∂ ∆pd fl The inverse form of the flooding point condition is expressed by ∂ ∆pd = 0. ∂ ∆pirr fl

(9.87)

In order to perform the derivative ∂ ∆pd /∂ ∆pirr , a description of the dry pressure drop as a function of the irrigated pressure drop is required. With Eq. (9.85), such a definition is available, although due to the implicit dependency of hold-up and irrigated pressure drop, the term is much more complex than in Eq. (9.85) assumes. The exact solution of Eq. (9.87) for the calculation of the irrigated pressure drop ∆pirr ,fl at the flooding point is given in the following two equations [Engel et al. 2001]:

∆pirr ,fl 1 = · %L · g · H 2988 · hdyn0

0.5 · 249 · hdyn0 · X 0.5 − 60 · ε − 558 · hdyn0 − 103 · dL · a (9.88)

558

9 DESIGN OF MASS TRANSFER EQUIPMENT

with

X = 3600 · ε2 + 186480 · hdyn0 · ε + 32280 · dL · a · ε + 191844 · h2dyn0 + + 95028 · dL · a · hdyn0 + 10609 · d2L · a2 . (9.89) It can be solved without any iteration. In general, values of (∆pirr ,fl /(H · %L · g)) are in the range of 0.1 – 0.3, which agrees well with experimental results. From the irrigated pressure drop ∆pirr ,fl at the flooding point, the dynamic holdup at the flooding point can be calculated analogously to Eq. (9.74):  ! ∆pirr ,fl 2 hdyn,fl = hdyn0 · 1 + 36 · . (9.90) H · %L · g The (fictive) dry pressure drop at the flooding point can be calculated by a transformation of Eq. (9.85):

∆pd,fl = ∆pirr ,fl ·

a 6 · hdyn,fl +a dL



hdyn,fl · 1− ε

4.65

.

(9.91)

For a turbulent flow regime, which is the normal state in random and structured packings, the flooding factor Φfl can be calculated by s

Φfl =

∆pd,fl . ∆pirr ,fl

(9.92)

The flooding factor Φfl describes the distance to the operating limit for flooding at a certain liquid load. The particle model developed by Engel et al. 2001 enables the prediction of the fluid dynamic behavior of packed columns with knowledge of the packing porosity ε, the specific area a, and the dry pressure drop ∆pd . No other empirical factors depending on the type of packing are required. The decisive factor in calculating the fluid dynamic behavior of an irrigated column is the liquid hold-up. Published data (usually for air/water) are not sufficient to provide a reliable equation for this; especially lacking are data at low surface tensions and viscosities, which are typical in distillation. 9.3.3.5

Maldistribution of Liquid

The aim of any packed column design is the realization of a countercurrent of gas and liquid in the form of a plug flow. The term plug flow implicates no axial backmixing and uniform flow profiles across the column cross section. However, this ideal flow regime is impossible to realize neither in random nor in structured packings. Deviations from plug flow are called maldistribution. In principle, maldistribution can exist in the gas phase as well as in the liquid phase. However, the pressure drop

9.3 DESIGN OF PACKED COLUMNS

559

of the gas flow flattens the flow rate profile of the gas, and thus the degree of maldistribution in the gas phase is practically quite low. In some cases, maldistribution of the gas can become a problem, e.g. in the case of dividing wall columns with large diameter, if the gas streams from the dividing wall sections mix insufficiently [Steffens 2017]. In the liquid phase, however, the degree of maldistribution can be very high. It often causes serious problems because it adversely affects the mass transfer efficiency of the column. The problem of maldistribution has attracted increased interest in recent years since packings are more and more installed in large diameter columns, which increases the risk of severe maldistribution. Present knowledge of maldistribution in packed columns is not yet sufficient. This is true for the degree of maldistribution present in an industrial-size column, but especially for the effect of maldistribution on the mass transfer efficiency of the entire column. Hurter 1893 was the first to perform maldistribution experiments on a large-scale column with a square cross section (1.2 × 1.2 m2 ). He used a single liquid feed at the top of a column with a coke bed and collected the liquid at 8 different locations below the bed. This technique has been used by most researchers since then (e.g. Kirschbaum 1931; Baker et al. 1935; Scott 1935; Staněk and Kolář 1973; Dutkai and Ruckenstein 1970; Onda et al. 1973; Hoek et al. 1986; Kouri and Sohlo 1987; Stoter et al. 1990; Yin et al. 2000; Hanusch et al. 2018b). Many attempts have also been made to study and model the influence of maldistribution on mass transfer efficiency of a column (e.g. Mullin 1957; Changez and Sawistowski 1963; Meier and Huber 1969; Schlünder 1979; Kunesh 1987; Porter and Jones 1987; Stoter et al. 1990; Zuiderweg et al. 1993; Schultes 2000; Pavlenko et al. 2010). A good overview of these activities is given by Hanusch et al. 2018b and Spiegel 2018. A simple experimental technique to answer both questions is to operate the test column with air and hot water (like a cooling tower) and to measure the liquid temperatures inside the packing as well as the states of gas and liquid at both ends of the column [Stichlmair and Stemmer 1987]. The mass transfer from liquid to gas in this system leads to a decrease in liquid temperatures instead of liquid concentrations like in conventional distillation systems (see also the experiment of Figure 9.20). Liquid temperatures can be easily measured in the packing with thermocouples. In the case of plug flow, the liquid temperatures should be uniform in each cross section of the column. Deviations of the isotherms from horizontal lines indicate the degree of maldistribution. Different packings were examined using his technique [Potthoff and Stichlmair 1991; Potthoff 1992; Schneider 2004; Kammermaier 2008]. Figure 9.47 shows an experimental result of experiments with a test column 0.63 m in diameter and 6.8 m in packing height [Mersmann et al. 2011]. 732 thermocouples have been installed among 12 cross sections of the column. Lines of constant liquid temperatures (i.e. isotherms) were determined by linear interpolation of the experimental data. The isotherms are presented in three longitudinal sections of the column. The crosses mark the positions of the thermocouples. Figure 9.47A shows results of a metal random packing of 35 mm Pall rings. With uniform liquid distribution at the top, the degree of maldistribution increases on the way of the liquid downward through the bed. However, the isotherms flatten in the

560

9 DESIGN OF MASS TRANSFER EQUIPMENT

lowest section of the bed. This surprising behavior is caused by the mass transfer between the liquid and the gas phase. The entering gas has a state (i.e. concentration and temperature) that is uniform across the bed. Due to the non-uniform liquid temperatures, the mass transfer rate is locally different, which leads to flattening of the isotherms. Thus, the highest deviations in the states of both liquid and gas exist in the middle section of the column [Zuiderweg and Hoek 1987]. An experimental result with the same packing, the same gas and liquid load, but point source initial liquid distribution at the top of the packing is presented in Figure 9.47B. The isotherms in the upper part of the column deviate strongly from horizontal lines; they are rather parallel to the column wall. Hence, most of the liquid flows downward in the center of the column. With increasing flow path length of the liquid, the degree of maldistribution decreases. Obviously, the packing has a certain ability for spreading the liquid, which improves the flow profile in the lower part. Thus, the degree of maldistribution depends on both the liquid distributor and the packed bed itself. A large variety of different random and structured packings has been tested with this technique. In all experiments a significant degree of maldistribution was detected, even with ideal liquid distribution atop the packing. Thus, the assumption of plug flow, which is always made in design of packed columns, is never correct. In these experiments the height of a transfer unit has been determined from the measurements of gas and liquid states at both ends of the column. Plug flow has been assumed in these calculations, but this assumption is false. Hence, not genuine but pseudo values of the height of a transfer unit have been evaluated. However, knowledge of these pseudo-transfer units is required for practical column design. The values for uniform and point source initial liquid distribution differ significantly. More results are presented in Section 9.3.4.3. These results demonstrate the critical impact of maldistribution on the height of a transfer unit, i.e. on the separation efficiency of a packed column. 9.3.3.6

Modeling of Liquid Maldistribution

Over the years, many authors have presented models for the description of liquid maldistribution in packed columns. However, all these models have in common that their parameters have to be adapted to experimental data. Thus, they are not applicable to not measured or newly developed packings. Wild and Engel 2007 proposed the WelChem Cell Model, which is implemented in WelChem GmbH’s hydraulic column design software TrayHeart. The distinctive feature of this model is that the liquid spread factors are determined by virtual 3D irrigation experiments of random packing elements in different orientations. Thus, with knowledge of the CAD data of the packing, they can be determined very quickly with little effort. In order to further develop and validate the model, extensive experimental measurements were carried out on a 1.2 m diameter column with a packing height of up to 3 m. The raw data of these experiments is publicly available and may be used for further modeling [Hanusch et al. 2018b]. Based on the experimental results, including different random packings and uniform and point source initial liquid distribution, the model was extended to reflect the effects of the other important influ-

9.3 DESIGN OF PACKED COLUMNS

561

Figure 9.47 Maldistribution in a random packing of 35 mm metal Pall rings [Mersmann et al. 2011]. Column diameter Dc = 0.63 m √, packing height H = 6.8 m, liquid load uL = 5.2 m/h, and gas load F = 1.1 Pa. Plot of liquid temperatures at three longitudinal sections of the column. Deviations of the isotherms from horizontal lines indicate the degree of maldistribution. A) Uniform liquid feed. B) Single-point liquid feed.

562

9 DESIGN OF MASS TRANSFER EQUIPMENT

encing factors. The TUM–WelChem cell model for the prediction of liquid distribution in random packed columns [Hanusch et al. 2019a] divides the column into layers of cells with a honeycomb base. The cell size is based on the bulk density of the packing and is chosen so that each cell corresponds to a single packing particle. Each cell is then assigned a random orientation of a packing particle, i.e. it is assigned the corresponding liquid spread factors and a free cross-sectional area that corresponds to the degree of opening of a projection of the packing particle in this orientation. In order to account for increased wall flow, random factors are assigned to the wall cells, which increase the respective free area and porosity and decrease the liquid spread factors to the adjacent cells. The phenomenon of increased lateral distribution at high liquid loading is taken into account by calculating a congestion level based on the free area and the local liquid load of each cell. If this level exceeds the cell height, the excess liquid is transferred to the adjacent cells in the next lower layer according to their free volume. The influence of the gas load is taken into account by carrying out a calculation of the fluid dynamics (see Section 9.3.3.4) for each cell and thus calculating a local flooding factor, which acts as a gas load-dependent dispersion coefficient and transfers liquid to the adjacent cells in the same layer. The initial liquid distribution is derived from the drip point coordinates of the liquid distributor, generating the liquid distribution in the top layer. The liquid distribution in the random packing follows then a top-down sequence for the total number of layers, generating a liquid distribution profile for the entire packing (see Figure 9.48). Results of the model calculations are in very good agreement with experimental data (see Figure 9.49) and allow insight into liquid distribution in random packed columns. Thus, the model provides a tool for further studies including design of liquid distributors [Hanusch et al. 2019b] and evaluation of mass transfer efficiencies using a two-column model [Hanusch et al. 2018a]. The most notable two aspects about the TUM–WelChem cell model are that the model does not resort to liquid spread factors from experiments, but uses a virtual 3D irrigation experiment to generate them itself, and that not a single fitting parameter is used in the whole model. Apart from the CAD data of the packing, only the bulk density, the porosity, the specific surface, and the dry pressure drop parameters according to Eq. (9.80) are required as input parameters. 9.3.3.7

Interfacial Area

The mass transfer efficiency of a column is determined by the interfacial area within the packing and the concentration gradients between gas and liquid. Knowledge of the value of the interfacial area is an essential prerequisite for predicting the separation achievable by the column. The interfacial area is first of all a quantity of two-phase flow within the column. It is defined as the area where gas and liquid are in intimate contact. It is important to know that the interfacial area between gas and liquid is not identical with the volumetric area a of the packing. However, there is a close relationship between these two quantities. Furthermore, not all of the interfacial area existing in the packing bed

9.3 DESIGN OF PACKED COLUMNS

563

Figure 9.48 Liquid distribution profile calculated with the TUM–WelChem cell model for uniform initial distribution [Hanusch et al. 2019a]. Packing: Raflux ring 50-5 metal, column diameter dc √ = 1.2 m, packing height H = 3 m, liquid load uL = 20 m/h, and gas load factor F = 1.3 Pa.

Figure 9.49 Comparison of experimental (top) and simulated (bottom) liquid distribution spectra [Hanusch et al. 2019a]. Packing: Raflux ring 50-5 metal, column diameter dc = 1.2 m, and packing height H = 3 √ m. A) Uniform initial distribution, liquid load uL = 20 m/h, gas load factor F = 1.3 √Pa. B) Point source initial distribution, liquid load uL = 15 m/h, gas load factor F = 1.3 Pa.

564

9 DESIGN OF MASS TRANSFER EQUIPMENT

is virtually effective for mass transfer. Of prime interest is the effective interfacial area aeff . Three groups of parameters that influence the effective interfacial area have to be distinguished [Schultes 1990]: • Design parameters of the packing, i.e. volumetric area, porosity and shape, and size and material of the packing. • Properties of the liquid, i.e. liquid density, liquid viscosity, and surface tension. • Operating parameters, i.e. liquid load, gas load, and acceleration due to gravity. Additional parameters, such as liquid distributors or wall wipers, only have influence if they are not properly designed. The significance of the packing material and the liquid properties for the effective interfacial area of a vertical plate is demonstrated in Figure 9.50. On a metal plate, water moves downward in the form of small rivulets that might even meander on the plate. However, an organic liquid such as heptane forms a broad thin film on such a plate; see Figure 9.50A. On a ceramic plate, the differences in the flow pattern of water and heptane are much smaller; see Figure 9.50B. Even water forms a fairly broad film here, which indicates a better wettability. A completely different behavior exists on a plastic plate (e.g. polypropylene); see Figure 9.50C. The water rivulet discretizes into a chain of droplets that move down the plate much faster than a rivulet or a film does. The wetting behavior of a packing material by a given liquid is often quantified by the angle ϑ formed by a drop sitting on a horizontal plate. The smaller this contact angle ϑ, the better the wettability. Figure 9.51 shows a plot of cos ϑ versus the surface tension σ of the liquid [Zech 1978]. The plate is completely wettable if the surface tension σ of the liquid is smaller than the critical surface tension σc of the plate, which is determined by extrapolating the parameter lines to cos ϑ = 1. In many empirical correlations for the effective interfacial area, the ratio σ/σc is included to account for the wettability of the packing material. Several researchers have developed correlations for the effective interfacial area. Some of them are presented in Section 9.3.4.2. The parameters considered in these correlations and their exponents are listed in Table 9.10. All authors agree that the liquid load uL is one of the major parameters with a dependence according to (aeff /a) ∼ u0.4 L . The values of the other exponents differ rather strongly, e.g. from −0.05 – +0.95 for liquid density %L and from −0.5 – +0.392 for liquid viscosity ηL . Figure 9.52 presents a plot of the relative interfacial area aeff /a versus the liquid load uL [Schultes 1990]. The correlations of different researchers have been evaluated for air/water in a packing of 35 mm metal Pall rings. The values of the correlations differ significantly, but the dependence on liquid load is fairly uniform. In principle, there should be a close relationship between the effective interfacial area and the liquid hold-up in the packing bed since it is the hold-up that forms the interfacial area. Wagner et al. 1997 developed the following relation for aeff /a

9.3 DESIGN OF PACKED COLUMNS

Figure 9.50 Flow pattern of water and heptane on a vertical plate. A) Metal plate. B) Ceramic plate. C) Polypropylene plate.

565

566

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.51 Wettability of several packing materials [Zech 1978].

Table 9.10 Exponents used in correlations for the effective interfacial area aeff by various investigators.

Bravo and Fair 1982 Kolev 1973 Puranik and Vogelpohl 1974 Shi and Mersmann 1985 Zech 1978 Billet and Schultes 1993

uL

a

0.392 0.392 0.307

0.608 0.696 0.826

0.4 0.5 0.4

1.2 — —

Exponent of %L ηL

σ

— 0.392 0.108 0.46 — −0.46 0.174 −0.041 −0.182

−0.05 0.2 0.95 −0.5 0.55 0.2

g — 0.264 —

−0.15 −0.15 −0.45 0.45 −0.75 +0.45

9.3 DESIGN OF PACKED COLUMNS

567

Figure 9.52 Interfacial area of a metal Pall ring packing (dn = 35 mm) predicted by different empirical correlations [Schultes 1990].

from the particle model structure:   aeff hL 2/3 = 1+ − 1. a 1−ε

(9.93)

This relation agrees quite well with the empirical correlations of Figure 9.52. The term hL /(1 − ε) describes the ratio of liquid and solid material within the packing bed. This ratio should be decisive for the interfacial area if the liquid hold-up in the packing bed has the same geometric structure as the solid material. Tsai et al. 2011 developed a correlation for the effective interfacial area aeff of structured packings:  !4/3 0.116 %  ˙L aeff V L 1/3  = 1.34 ·  ·g · , (9.94) a σ Lp using the wetted perimeter Lp :

Lp =

4·S · Ac , B·h

(9.95)

which depends on the packing geometry. Here, S is length of the packing channel side, B is the width of the packing channel base, h is packing crimp height, and Ac is the cross-sectional area of the column. It should be noted that the current understanding of the interfacial area, despite all the research activities described here, is rather inadequate. It is not enough to design industrial columns with sufficient accuracy (see Section 9.3.4.3).

568

9 DESIGN OF MASS TRANSFER EQUIPMENT

9.3.4

Mass Transfer in Packed Columns

The prediction of mass transfer in a packed column is based on the assumption that both gas and liquid move through the column bed in a kind of plug flow. This means that backmixing in the axial direction does not exist. Furthermore, both phases have to be distributed uniformly over the mass transfer area and throughout the packed height. However, these assumptions will never be fully met in practice. 9.3.4.1

Height of a Transfer Unit

From the transfer unit concept, the following relationship is obtained (cf. Section 4.1.2):

dNTUOG =

kOG · dA . G˙

(9.96)

With

dA = dH · Ac · aeff ,

(9.97)

where Ac is the cross-sectional area of the column and aeff is the effective interfacial area of the packing, the differential height dH of the packing is

dH = dNTUOG ·

kOG

G˙ . · Ac · aeff

(9.98)

The second term of this equation is defined as the height HTUOG of an overall gas transfer unit:

HTUOG =

kOG

G˙ . · Ac · aeff

(9.99)

˙ c is proportional to the superficial velocity uG of the gas. A simple The term G/A transformation gives G˙ %G = uG · . ˆG Ac M

(9.100)

The overall gas mass transfer coefficient kOG is calculated from the coefficients of the phases involved, i.e. gas phase kG and liquid phase kL :

1 kOG

=

1 m + , kG kL

(9.101)

where m is the slope of the equilibrium curve on the (y, x )-diagram. Furthermore

kG = βG ·

%G ˆG M

and

kL = βL ·

%L . ˆL M

(9.102)

9.3 DESIGN OF PACKED COLUMNS

569

By combining Eqs. (9.99) – (9.102), the height of a transfer unit can be expressed as ! ˆ L %G uG βG M HTUOG = · 1+m· · · . (9.103) ˆ G %L aeff · βG βL M In distillation, the slope m of the equilibrium line is very small. Hence, the second term in the parenthesis of Eq. (9.103) is very small, indicating that most of the mass transfer resistance is in the gas phase. Equation (9.103) contains three unknown quantities specific to the packed column: • The effective interfacial area aeff . • The mass transfer coefficient βG in the gas phase. • The mass transfer coefficient βL in the liquid phase. Relations for these three quantities are needed to predict the mass transfer efficiency of a packed column. The height H of the packing required to meet the product specifications follows from

H = HTUOG · NTUOG .

(9.104)

In this equation, NTUOG denotes the number of transfer units. It is a thermodynamic quantity describing the difficulty of the separation from knowledge of the system properties without any regard to the equipment; see Section 4.1.2. The height of a transfer unit HTUOG is the quantity that converts the thermodynamic quantity NTUOG into packing height H . Often the concept of equilibrium stages is used instead of the concept of transfer units. Then, the difficulty of the separation is expressed by the number n of equilibrium stages. The height of the packing required to perform a given separation is calculated as follows:

H = n · HETP .

(9.105)

The term HETP (= height equivalent to a theoretical plate) describes the mass transfer efficiency of the packing. The following relation exists (see Eq. (4.9)): ! m ln ˙ G˙ L/ HETP = HTUOG · m . (9.106) −1 ˙ G˙ L/ In the special case where operating line and equilibrium line are parallel, i.e. ˙ G) ˙ = 1, HETP and HTUOG have the same value. m/(L/ The key quantity for determining column height is the height of a transfer unit. It can be predicted by fundamental laws of mass transfer.

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9 DESIGN OF MASS TRANSFER EQUIPMENT

9.3.4.2

Mass Transfer Coefficients and Effective Interfacial Area

Several correlations are available in the literature for the mass transfer coefficients βG of the gas phase and βL of the liquid phase. Most of these correlations were developed from experimental mass transfer data in the form of the height of a transfer unit HTUOG . From these data, the volumetric mass transfer coefficient aeff · kOG can be determined rigorously assuming plug flow within the bed. There are no rigorous methods available for splitting the volumetric mass transfer coefficient into the effective interfacial area aeff and the mass transfer coefficient kOG . Hence, the researchers either start with a relation for the interfacial area and adjust the mass transfer coefficient, or they start with a model for the transfer coefficient and adjust the interfacial area. The correlations for mass transfer coefficients and interfacial area developed by a research group thus form a consistent set of relationships that should never be divided. Combining correlations of different authors to calculate the mass transfer in a packed column, there is a risk that the deviations of the individual correlations add up, while the deviations are compensated when using a consistent set. Most correlations for mass transfer coefficients and interfacial area take into account the dependence on different influencing factors with exponents (e.g. Eqs. (9.107) – (9.113)). Table 9.11 lists both the parameters involved in such correlations and the values of their exponents for the gas-phase mass transfer coefficient βG . 0.66 All authors agree on the exponent 0.66 of the diffusion coefficient, i.e. βG ∼ DG . The exponents of the gas load uG differ only in a small range from 0.65 – 0.8. A larger variation exists for the exponents of the viscosity ηG and density %G . Here, the values are in the range from −0.467 – −0.33 and from 0.33 – 0.65, respectively. Despite the good agreement of the values of the exponents, the absolute values of the gas-phase mass transfer coefficient βG differ significantly between the authors (see Example 9.2). Table 9.11 Exponents used in correlations for the gas-side mass transfer coefficient βG by various investigators.

Mersmann and Deixler 1986 Onda et al. 1968 Shulman and DeGouff 1952 van Krevelen and Hoftijzer 1948 Zech 1978 Billet and Schultes 1999

uG

Exponent of DG ηG

0.66 0.7 0.65 0.8 0.66 0.75

0.66 0.66 0.66 0.66 0.66 0.66

−0.33 −0.367 – −0.467 −0.33 −0.417

%G 0.33 0.367 0.65 0.467 0.33 0.417

A similar listing of the liquid-phase mass transfer coefficient βL is presented in Table 9.12. All authors agree on the exponent 0.5 of the diffusion coefficient, i.e. 0.5 βL ∼ DL . This results from the penetration model for mass transfer. The expo-

571

9.3 DESIGN OF PACKED COLUMNS

nents of the liquid load uL vary in a rather large range from 0.167 – 0.66. With regard to the liquid density %L and viscosity ηL , the authors do not even agree on the sign of the exponents. Some researchers find a dependence of the transfer coefficient on the surface tension σ , but others do not. Hence, the current knowledge about the liquid-phase mass transfer coefficient is rather ambiguous and inconsistent. However, since the mass transfer in distillation columns is dominated by the resistance in the gas phase and not in the liquid phase, this lack of knowledge has little effect in most cases. Table 9.12 Exponents used in correlations for the liquid-side mass transfer coefficient βL by various investigators.

uL 0.33 0.33 0.19 0.45 0.167 0.66

Billet 1985 Schultes 1990 Shi and Mersmann 1985 Shulman et al. 1955 Zech 1978 Onda et al. 1968

DL

Exponent of %L ηL

0.5 0.33 −0.33 0.5 0.167 −0.167 0.5 0.23 −0.23 0.5 −0.05 0.05 0.5 −0.15 — 0.5 0.167 −0.833

σ

g

— 0.33 — 0.167 0.05 0.22 — — 0.15 −0.017 — 0.33

A few important correlations for the effective interfacial area aeff and the mass transfer coefficients βG and βL are presented in detail in the following. Onda Correlation

Onda et al. 1968 published correlations for the effective interfacial area aeff (using a critical surface tension σc ; see Table 9.13) and the mass transfer coefficient in both phases:

aeff = 1 − exp a

− 1.45 ·

 σ 0.75 c

σ

·

  2 −0.05  2 0.2 ! uL · %L 0.1 uL · a uL · %L · · · a · ηL g σ·a | {z } | {z } | {z } Re L Fr L We L (9.107) 

Table 9.13 Values of critical surface tension σc of the Onda et al. 1968 correlation.

Material: σc in mN/m:

Polyethylene 33

Polyvinyl chloride 40

Ceramic 61

Glas 73

Metal 75

572

9 DESIGN OF MASS TRANSFER EQUIPMENT



βL = 0.0051 ·

%L g · ηL

−1/3    −1/2 %L · uL 2/3 ηL 2/5 · · · (a · dn ) aeff · ηL %L · D L | {z } | {z } Sc L Re L (9.108)

  1/3 %G · uG 0.7 ηG −2 · · (a · dn ) · (a · DG ) a · ηG %G · DG | {z } | {z } Re G Sc G with C = 5.23 except for dn < 15 mm then C = 2.0 . 

βG = C ·

(9.109)

Here, dn denotes the nominal diameter of the elements as defined by the suppliers. This correlation is valid only for random packings. It is considered to provide rather conservative results since it was developed at a time when only first-generation random packing particles (e.g. Raschig rings and Berl saddles) were used. Hence, the risk of underestimating the packing height is very low. Bravo/Fair Correlation

Bravo and Fair 1982 developed a new equation for the effective interfacial area and used the mass transfer coefficients of the correlation of Onda et al. 1968:   aeff σ 0.5 ηL · uL uG · %G 0.392 = 19.76 · · · . a σL a·η Hs 0.4 (9.110) | {z G } Re G The mass transfer coefficients are calculated with Eqs. (9.108) and (9.109). This correlation gives a dependence of the effective interfacial area on the surface tension σ and height Hs of a packing section to account for the influence of liquid maldistribution. These two quantities make the correlation dependent on the units. SI units have to be used, i.e. kg/s2 for the surface tension σ and m (meter) for the packing section height Hs . Billet/Schultes Correlation

Based on a very large database, Billet developed several models for interfacial area and mass transfer coefficients. The actual version is Billet and Schultes 1999:

aeff −0.5 = 1.5 · (a · dh ) · a    2 −0.45  2 0.75 uL · dh · %L −0.2 uL uL · %L · dh · · · ηL g·d σL | {z } | {z h } | {z } Re L Fr L We L (9.111)

573

9.3 DESIGN OF PACKED COLUMNS

βL = CL · 121/6 ·

uL · DL hL · dh



−1/2

βG = CG · (ε − hL )

with

dh = 4 ·

ε a



·

1/2 (9.112)

a dh

1/2

 and hL =

  1/3 uG · %G 3/4 ηG · a · ηG % ·D | {z } | G {z G } Re G Sc G 1/3 · uL · a2 . 

· DG ·

12 ·

ηL g · %L

(9.113) The factors CL and CG depend on the type of packings; see Table 9.14. Table 9.14 Selected values of the factors CL and CG of the Billet and Schultes 1999 correlation.

Type

Material

CL

CG

Raschig ring Pall ring

Bialecki ring Berl saddle Hiflow ring

Ceramic Metal Plastic Ceramic Metal Ceramic Ceramic

1.416 1.192 1.239 1.227 1.721 1.246 1.377

0.210 0.410 0.368 0.415 0.302 0.387 0.379

Ralu-pak Impuls pak Montz pak

Metal Metal Metal

1.334 0.983 1.165

0.385 0.270 0.422

Problems in Predicting Mass Transfer in Packed Columns

In the current state of the art, all models for mass transfer in a packed column assume ideal plug flow of gas and liquid within the packing. However, in a real column, the flow pattern of gas and, in particular, of liquid are much more complex. They deviate significantly from plug flow. Even in modern structured packings with a good initial liquid distribution, maldistribution cannot be completely avoided. The degree of maldistribution can be detected by the temperature measuring technique described in Section 9.3.3.5. All tests carried out so far revealed more or less drastic deviations from plug flow. Maldistribution may become the dominant factor especially in large diameter columns operated with low liquid loads, the conditions prevailing in vacuum distillation columns. It is generally agreed that maldistribution adversely affects the mass transfer efficiency of a packed column (e.g. Shariat and Kunesh 1995). This statement is

574

9 DESIGN OF MASS TRANSFER EQUIPMENT

based on many theoretical studies and, for instance, on the experiments described in Section 9.3.3.5. The measurement of temperature profiles within the bed enables a deep insight into the mass transfer zone, which is usually treated as a black box. This technique allows in the same experiments measurement of the maldistribution within the bed and determination of the mass transfer efficiency, expressed, for instance, as height of a transfer unit HTUOG . The values of HTUOG differ very much for uniform initial liquid distribution (i.e. low maldistribution) and single point source initial distribution (i.e. high maldistribution). Figure 9.53 shows typical results of tests with a random packing of 35 mm metal Pall rings; see also Figure 9.47. The ratio V˙ L /V˙ G was kept constant to hold the operation line parallel to the equilibrium line. Therefore, an increase in the gas load is accompanied by an increase in the liquid load. The values of HTUOG differ by a factor of up to five between uniform and point source initial distribution. At high gas and liquid loads, the column was operated above the loading point where the gas flow affects the liquid flow. The gas flow flattens the liquid flow rate profiles, i.e. it reduces the degree of liquid maldistribution, and, in turn, the differences of HTUOG between uniform and point source initial distribution vanish; see Figure 9.53.

Figure 9.53 Height of a transfer unit HTUOG of a random packing of metal Pll rings at uniform (white symbols) and point source (black symbols) initial distribution. Circles: H = 3.72 m, squares: H = 1.86 m [Potthoff 1992].

Similar results from experiments with a structured packing (Ralu-pak) are shown in Figure 9.54. Since this packing has a good ability for liquid distribution, the measured values of HTUOG are much less dependent on the mode of initial liquid distribution. However, not all structured packings have such a good ability for liquid distribution as further experiments prove [Potthoff 1992]. The influence of maldistribution on mass transfer efficiency of packed columns also depends on the distance between operating and equilibrium line in the McCabe– Thiele diagram. The smaller the distance, the greater the adverse effect on mass transfer efficiency. A special effect exists at the column ends. The quality of the bottom fraction is deteriorated by liquid maldistribution as well as the quality of

9.3 DESIGN OF PACKED COLUMNS

575

Figure 9.54 Height of a transfer unit HTUOG of a structured packing (Ralu-pak 250YC) at uniform (squares) and point source (circles) initial distribution. Black symbols: H = 3.72 m, white symbols: H = 1.86 m [Potthoff 1992].

the overhead fraction by gas maldistribution. This unfavorable effect is particularly strong when the required product purity is very high, since a locally too low purity is not compensated for by a too high purity. This explains the empirical knowledge of how difficult it is to obtain pure bottom products in packed columns. Tray columns are superior to packed columns in such separations. From these experiments it has to be concluded that most published experimental data on mass transfer efficiency (e.g. in form of HTUOG or aeff · KOG ) are not true values, but pseudo values, because they are falsified by maldistribution. Hence, the correlations for the effective interfacial area aeff and the mass transfer coefficients βG and βL are also not true values, as they include maldistribution effects. A sound mass transfer model should use true values of aeff , βG , and βL and separately account for the effect of maldistribution. However, such a model has not been developed yet because this subdivision is not measurable experimentally. One step in this direction is the direct numerical simulation of mass transport in a packed column. Hill et al. 2019 present the results of such a simulation for cryogenic air separation. From the huge amounts of data in the simulation, many insights into the mass transfer in the phase interface can be gained. Figure 9.55A shows the concentration profile in the gas phase and in the liquid phase and Figure 9.55B the phase distribution in a structured packing. It can be clearly seen in Figure 9.55B that the liquid (black) flows down as a film on the surface of the packing (white). The concentration profile shows that the lower part of Figure 9.55A is colored darker on average than the upper part. This shows the separation of the mixture above the column height, wherein the fraction of low boiler increases with increasing column height. The concentration jump at the phase boundary is indicated by a color contrast. It can be seen that the concentration of the low boiler is lower on the liquid-side than on the gas-side of the phase boundary, which corresponds to the local thermodynamic

576

9 DESIGN OF MASS TRANSFER EQUIPMENT

Figure 9.55 Result of direct numerical simulation of binary distillation in a structured packing [Hill et al. 2019]. A) Molar concentration of low boiler in gas and liquid phase. B) Phase distribution of gas (gray) and liquid (black).

equilibrium. It can also be seen that the liquid in the pinch between the packing sheets is colored very dark. Thus, the state of this liquid is already very close to equilibrium and hardly contributes to the mass transfer. This means that the phase interface does not count the same for mass transfer at every location. In the gas phase, turbulent structures are additionally recognizable, which decisively influence the mass transport. 9.3.4.3

Practical Determination of the Height of a Transfer Unit

The fundamental laws of mass transfer are sophisticated, and their application to packed columns is in principle clear. However, the most significant parameters, e.g. effective interfacial area and mass transfer coefficients, cannot be predicted from first principles at the current state of the art. Only empirical correlations are available, and most of these correlations have been developed by experiments in rather small-scale columns operated with air/water, a system that is not representative of distillation. Thus, it is not possible to predict mass transfer efficiency of a column with sufficient reliability. Presently, practical design of packed columns is always based on empirical knowledge either from industrial columns operated with similar systems or from the knowhow of the supplier of the packings. Even the methods applied to the scale-up are ambiguous. Hence, column design always involves some uncertainty, and, in turn, quite high safety factors have to be implemented. In random packings, the height of a transfer unit HTUOG depends strongly on

9.3 DESIGN OF PACKED COLUMNS

577

column size. A small-scale column is much more efficient than a large-scale column. This behavior is obviously caused by the fact that small-sized particles with a high volumetric area are used in small diameter columns. The larger the column diameter, the larger the packing elements and, in turn, the lower the volumetric area of the packing bed. In modern structured packings the same geometric dimensions are applied to small-scale (approximately Dc > 0.4 m) and large-scale columns. Hence, the dependency of mass transfer efficiency on column diameter is, if at all, quite small. The separation efficiency is only slightly affected by the gas and liquid load, provided that the column is operated within the operation region, i.e. with a sufficient distance to the limits that are set by flooding and dewetting, respectively. When approaching the flooding point, the height of a transfer unit increases drastically, which corresponds to a significant decrease in the separation efficiency. A large collection of published values of the height of a transfer unit HTUOG or the height equivalent to a theoretical plate HETP is included in the books of Kister 1992 and Billet 1995. These collections contain data of packing manufacturers and of research institutes. Empirical correlations for HTUOG and HETP , respectively, that are not based on the fundamentals of mass transfer have been developed by several authors (e.g. Porter and Jenkins 1979; Strigle 1987. The correlation of Strigle 1987, based on proprietary data of Norton, Fractionation Research Inc. (FRI), and published data from Billet, is

HETP = exp (n − 0.187 · ln σ + 0.213 · ln ηL ) .

(9.114)

This equation is the result of pure empiricism, and, moreover, it is not correct in the dimensions. Surface tension σ must be inserted in dyn/cm and liquid viscosity ηL in cP ( = 10−3 kg/(m · s)); the values of HETP have the unit ft (feet). The only parameter specific to the packing is n. Its values is in the range from 1.13 – 1.72 [Strigle 1987]. This correlation is valid for atmospheric distillation with surface tensions from 4 – 36 dyn/cm and liquid viscosities from 0.08 – 0.83 cP. The confidence of all methods for predicting the mass transfer efficiency of packed columns is rather poor. It is not sufficient for a save column design. In industry there is a huge demand for more reliable mass transfer data and correlations. This led to the foundation of mass transfer research institutes, e.g. FRI at Stillwater, Oklahoma, and Separations Research Program (SRP) at the University of Texas, Austin. Both institutions conduct mass transfer experiments with organic distillation systems in large diameter columns (0.4, 1.2, and 2.4 m, respectively). Most of the data are confidential, but some have been published (e.g. Kunesh 1987; Shariat and Kunesh 1995; Fair and Bravo 1990).

578

9 DESIGN OF MASS TRANSFER EQUIPMENT

Example 9.2: Design of a Packed Column Determine the diameter Dc and the height of a transfer unit HTUOG of a packed column for the distillation of the ethylbenzene/styrene mixture at a pressure of p = 133.3 mbar. The internal flows are G˙ = 1.017 × 10−2 kmol/s and L˙ = 1.162 × 10−2 kmol/s. The slope of the equilibrium curve is assumed to be m = 1. A thermodynamic calculation of the number of transfer units yields NTUOG = 56. Properties:

Properties

Liquid phase

Gas phase

Molecular weight Density Surface tension Viscosity Diffusion coefficient

ˆ L = 105 kg/kmol M %L = 835 kg/m3 σ = 0.021 N/m ηL = 3.76 × 10−4 Pa · s DL = 3.64 × 10−9 m2 /s

ˆ G = 105 kg/kmol M %G = 0.483 kg/m3 ηG = 7.68 × 10−6 Pa · s DG = 1.99 × 10−5 m2 /s

Fluid dynamic design:

The gas flow is in most cases the decisive factor for the column diameter. (1) Column diameter: √ Initially, the operational gas load is assumed to be F = 2 Pa. This value needs to be checked against flooding and if necessary be adapted. With the assumed gas load, the column diameter can be determined: s 4 · V˙ G Dc = π · uG

√ F 2 Pa m uG = √ =q = 2.88 %G s 0.483 kg/m3 ˆG M 105 kg/kmol m3 V˙ G = G˙ · = 1.017 × 10−2 kmol/s · 3 = 2.21 %G s 0.483 kg/m s

Dc =

4 · 2.21 m3 /s = 0.99 m . π · 2.88 m/s

579

9.3 DESIGN OF PACKED COLUMNS

A column diameter of Dc = 1 m is selected, and the superficial gas velocity and gas load are reevaluated using the continuity equation.

Ac =

π π 2 · Dc2 = · (1 m) = 0.785 m2 4 4

uG =

V˙ G 2.21 m3 /s m = 2 = 2.815 Ac s 0.785 m

F = 2.815 m/s ·

q

0.483 kg/m3 = 1.96

√ Pa .

(2) Minimum liquid load: First, a packing is selected. The selected packing consists of ceramic Pall rings with a standard diameter of dn = 50 mm. The volumetric area of the packing is a = 120 m2 /m3 ; the porosity of the packing is ε = 0.78: The minimum liquid load is determined using Eq. (9.68).

uL,min = 7.7 × 10−6 ·



%L · σ 3 ηL 4 · g

2/9   g 1/2 · a 3

835 kg/m3 · (0.021 N · m) = 7.7 × 10 ·
4 3.76 × 10−4 Pa · s · 9.81 m/s2  1/2 9.81 m/s2 · 120 m2 /m3 m3 = 4.982 × 10−4 m/s = 1.79 2 . m ·h

!2/9

−6

The liquid load is calculated using the continuity equation and needs to be checked:

uL =

ˆL V˙ L L˙ · M 1.162 × 10−2 m3 /s · 105 kg/kmol = = Ac Ac · % L 0.785 m2 · 835 kg/m3

= 1.86 × 10−3 m/s = 6.7 m3 /(m2 · h) > 1.79 m3 /(m2 · h) , i.e. the value of the actual liquid load uL is larger than the value of the minimum liquid load uL,min . (3) Pressure drop and hold-up: The pressure drop is calculated iteratively using the Eqs. (9.74) and (9.85):

∆pirr = ∆pd

6 · hdyn 4.65 +a  ε dL · a ε − hdyn

580

9 DESIGN OF MASS TRANSFER EQUIPMENT



hdyn = hdyn0 · 1 + 36 ·

∆pirr H · %L · g

2 !

The diameter of the liquid droplets of the particle model is calculated using Eq. (9.83): s 6·σ dL = CL · . (%L − %G ) · g For random packings, the constant CL is set to CL = 0.4: s 6 · 0.021 N/m
dL = 0.4 · = 1.57 × 10−3 m . 835 kg/m3 − 0.483 kg/m3 · 9.81 m/s2 The pressure drop of the dry packing ∆pd is calculated using Eq. (9.78):

∆pd 3 1−ε u2 = · ζ0 · 4.65 · %G · G . H 4 ε dp The equivalent spherical diameter of the packing elements is calculated using Eq. (9.62):

dp =

6 · (1 − ε) 6 · (1 − 0.78) = = 0.011 m . a 120 m2 /m3

The drag coefficient ζ0 is calculated using Eq. (9.79). The relevant packing coefficients can be looked up in Table 9.9 and are b = 15.3840 and c = −0.2850:

ζ0

= b · Re c0

Re 0 = ζ0

uG · dp · %G 2.815 m/s · 0.011 m · 0.483 kg/m3 = = 1947 ηG 7.68 × 10−6 Pa · s

= 15.3840 · 1947−0.2850 = 1.78 .

The dry pressure drop equals 2

∆pd 3 1 − 0.78 (2.815 m/s) Pa = · 1.78 · · 0.483 kg/m3 · = 324 . H 4 0.784.65 0.011 m m

581

9.3 DESIGN OF PACKED COLUMNS

The dynamic hold-up hdyn0 below the loading point is calculated using Eq. (9.73):

hdyn0

!0.25  0.1 σ · a2 = 3.6 · · · %L · g !
1/2 0.66 1.86 × 10−3 m/s · 120 m2 /m3 = 3.6 ·
1/2 9.81 m/s2
3/2 !0.25 3.76 × 10−4 Pa · s · 120 m2 /m3 ·
1/2 835 kg/m3 · 9.81 m/s2
2 !0.1 0.021 N/m · 120 m2 /m3 · 835 kg/m3 · 9.81 m/s2
0.66
0.25 0.1 = 3.6 · 6.505 × 10−3 · 1.890 × 10−4 · (0.0369)

uL · a1/2 g 1/2

!0.66

ηL · a3/2 %L · g 1/2

= 0.01094 . The pressure drop ∆pd of the dry packing is used as the initial value for the iteration: (0)

∆pirr ∆pd Pa = = 324 H H m The hold-up for the initial pressure drop equals (using Eq. (9.74))  !2  (0) ∆pirr /H  (0) hdyn = hdyn0 · 1 + 36 · %L · g  2 ! 324 Pa/m = 0.01094 · 1 + 36 · = 0.01156 . 835 kg/m3 · 9.81 m/s2 The pressure drop ∆pirr of the irrigated packing can now be evaluated iteratively (using Eqs. (9.74) and (9.85)):

582

9 DESIGN OF MASS TRANSFER EQUIPMENT

• 1. Iteration (0)

(1) ∆pirr

H

6 · hdyn  4.65 +a ∆pd ε dL  = · · (0) H a ε − hdyn 6 · 0.01156 m2 4.65 + 120 3  −3 0.78 m · = 324 Pa/s · 1.57 × 10 m 0.78 − 0.01156 120 m2 /m3 = 475 Pa/m 

(1)

hdyn

= hdyn0 · 1 + 36 ·

(1) ∆pirr /H

%L · g 

= 0.01094 · 1 + 36 ·

!2  

475 Pa/m 835 kg/m3 · 9.81 m/s2

2 !

= 0.01226

• 2. Iteration (1)

(2)

∆pirr H

6 · hdyn  4.65 +a ∆pd ε dL  = · · (1) H a ε − hdyn 6 · 0.01226 m2 4.65 + 120 3  −3 0.78 m · = 324 Pa/s · 1.57 × 10 m 0.78 − 0.01226 120 m2 /m3 = 485 Pa/m 

(2) hdyn

(2)

= hdyn0 · 1 + 36 ·

∆pirr /H %L · g 

= 0.01094 · 1 + 36 ·

!2  

485 Pa/m 835 kg/m3 · 9.81 m/s2

2 !

= 0.01232

The pressure drop of the irrigated packing and the liquid hold-up reach final values of

∆pirr Pa = 486 H m

and

hdyn = 0.01233 .

The total liquid hold-up in the column consists of the static hold-up hstat and the dynamic hold-up hdyn :

hL = hstat + hdyn

583

9.3 DESIGN OF PACKED COLUMNS

The static hold-up is evaluated using Eq. (9.72):  %L · g  hstat = 0.033 · exp −0.22 · σ · a2 ! 835 kg/m3 · 9.81 m/s2 = 0.033 · exp −0.22 ·
2 0.021 N/m · 120 m2 /m3

= 8.52 × 10−5 . The total liquid hold-up in the column reaches a final value of

hL = hstat + hdyn = 8.52 × 10−5 · 0.01233 = 0.01242 . (4) Check for flooding: The pressure drop ∆pirr ,fl of the irrigated packing at the flooding point is calculated by Eq. (9.88):

∆pirr ,fl %L · g = H 2988 · hdyn0 · 249 · hdyn0 · X 0.5 − 60 · ε − 558 · hdyn0 − 103 · dL · a



0.5

The value X from Eq. (9.89) equals

X = 3600 · ε2 + 186480 · hdyn0 · ε + 32280 · dL · a · ε + 191844 · h2dyn0 + 95028 · dL · a · hdyn0 + 10609 · dL 2 · a2 = 3600 · 0.782 + 186480 · 0.01094 · 0.78 + 32280 · 1.57 × 10−3 m · 120 m2 /m3 · 0.78 + 191844 · 0.010942 + 95028 · 1.57 × 10−3 m · 120 m2 /m3 · 0.01094
2
2 + 10609 · 1.57 × 10−3 m · 120 m2 /m3 = 9120.5 . The pressure drop of the irrigated packing at flooding conditions equals

∆pirr ,Fl 835 kg/m3 · 9.81 m/s2 = H 2988 · 0.01094 · 249 · 0.01094 · X 0.5 − 60 · 0.78 − 558 · 0.01094 −103 · 1.57 × 10−3 m · 120 m2 /m3 = 1992



0.5

Pa . m

The flooding factor Φfl , defined by Eq. (9.92), describes how far the operating conditions of the column are away from flooding: s ∆pd Φfl = ∆pd,fl

584

9 DESIGN OF MASS TRANSFER EQUIPMENT

The pressure drop ∆pd,fl /H of the dry packing at flooding conditions is calculated as follows (using Eqs. (9.90) and (9.91)):   ∆pd,fl ∆pirr ,fl a hdyn,fl 4.65 = · · 1− 6 · hdyn,fl H H ε +a dL  2 ! ∆pirr ,fl /H hdyn,fl = hdyn0 · 1 + 36 · %L · g  2 ! 1992 Pa/m = 0.01094 · 1 + 36 · = 0.03423 835 kg/m3 · 9.81 m/s2

∆pd,F l = 1992 Pa/m · H

= 774

Pa . m

120 m2 /m3 6 · 0.03423 + 120 m2 /m3 1.57 × 10−3 m   0.03423 4.65 · 1− 0.78

The flooding factor equals s 324 Pa/m Φfl = = 0.65 . 774 Pa/m The operating conditions are far away from flooding conditions. Mass transfer:

The aim of the mass transfer calculations is to find the required packing height H given by Eq. (9.104):

H = HTUOG · NTUOG . The height of a gas-side transfer unit is calculated using Eq. (9.103): ! ˆ L %G uG βG M HTUOG = · 1+m· · · . ˆ G %L aeff · βG βL M Different models are available to calculate the effective interfacial area and the mass transfer coefficients. It is important that one complete model is used for the calculation of HT UOG . Further on, the calculations using the Onda correlation are presented. (1) Effective interfacial area: The effective interfacial area according to the Onda correlation is calculated using

585

9.3 DESIGN OF PACKED COLUMNS

Eq. (9.107):   σ 0.75 aeff c = 1 − exp −1.45 · · a σ    2 −0.05  2 0.2 ! uL · %L 0.1 uL · a uL · %L · · · . a · ηL g σ·a The critical surface tension σc for ceramics equals σc = 61 mN/m (from Table 9.13).  0.75 aeff 0.061 N/m = 1 − exp −1.45 a 0.021 N/m  0.1 1.86 × 10−3 m/s · 835 kg/m3 · 120 m2 /m3 · 3.76 × 10−4 Pa · s !−0.05
2 1.86 × 10−3 m/s · 120 m2 /m3 · 9.81 m/s2 !0.2 
2 1.86 × 10−3 m/s · 835 kg/m3  · 0.021 N/m · 120 m2 /m3

= 0.860 . The effective interfacial area is equal to

aeff = a · 0.860 = 120 m2 /m3 · 0.860 = 103.2

m2 . m3

(2) Mass transfer coefficients: The mass transfer coefficients are calculated using the Eqs. (9.108) and (9.109): 

βL = 0.0051 · 

βG = C ·

%L g · ηL

%G · uG a · ηG

−1/3    −1/2 uL · %L 2/3 ηL 2/5 · · · (a · dn ) aeff · ηL %L · DL

0.7  ·

ηG %G · DG

1/3

−2

· (a · dn )

· (a · DG ) .

The mass transfer coefficient in the liquid phase equals −1/3 835 kg/m3 9.81 m/s2 · 3.76 × 10−4 Pa · s  2/3 1.86 × 10−3 m/s · 835 kg/m3 · 103.2 m2 /m3 · 3.76 × 10−4 Pa · s  −1/2
2/5 3.76 × 10−4 Pa · s · · 120 m2 /m3 · 0.05 m 835 kg/m3 · 3.64 × 10−9 m2 /s m = 1.80 × 10−4 . s 

βL = 0.0051 ·

586

9 DESIGN OF MASS TRANSFER EQUIPMENT

With the constant C = 5.23 the mass transfer coefficient in the gas phase equals 0.7 0.483 kg/m3 · 2.815 m/s 120 m2 /m3 · 7.68 × 10−6 Pa · s  1/3 7.68 × 10−6 Pa · s · 0.483 kg/m3 · 1.99 × 10−5 m2 /s
−2
· 120 m2 /m3 · 0.05 m · 120 m2 /m3 · 1.99 × 10−5 m2 /s m = 0.0532 . s 

βG = 5.23 ·

(3) Height of an overall gas transfer unit: The height of an overall gas transfer unit for the Onda correlation equals  2.88 m/s 0.0532 m/s HTUOG = · 1+1· 103.2 m2 /m3 · 0.0532 m/s 1.80 × 10−4 m/s

105 kg/kmol 0.483 kg/m3 · · 105 kg/kmol 835 kg/m3



= 0.60 m . Finally, the required packing height for the thermal separation process is calculated:

H = 0.60 m · 56 = 33.6 m . Applying different models to the calculations leads to the following results: Model Onda et al. 1968 Billet and Schultes 1999 Bravo and Fair 1982 (with Hs = 1 m) Bravo and Fair 1982 (with Hs = 6 m)

aeff

βG

βL

HTUOG

H

in m2 /m3

in m/s

in m/s

in m

in m

103.2

0.0532

1.80 × 10−4

0.600

33.6

0.1414

2.70 × 10−4

0.286

16.0

0.0532

1.77 × 10−4

0.556

31.1

0.0532

2.86 × 10−4

1.206

67.5

90.72 105.5 51.48

9.A APPENDIX: PRESSURE DROP IN PACKED BEDS

9.A

587

Appendix: Pressure Drop in Packed Beds

Fluid flow through a collective (i.e. swarm) of particles is an important case of twophase flow. However, this is a complex situation because the particles interact with each other, unlike a single particle in a flow, that can be described very well. Therefore, as a first step, a single particle in a tube is considered, which is kept in suspension by the superficial velocity u0 (see Figure 9.A.1A).

Figure 9.A.1 A) Single particle in suspension. B) Swarm of particles in suspension.

Since the particle is at rest, all forces acting on the particle must cancel each other out. The balance of forces for the particle is Frictional force = gravitational force − buoyant force . For a spherical particle with diameter d, this balance leads to Eq. (9.3), which can be converted to calculate the velocity u0 as follows: s 4 d · ∆% · g u0 = · . (9.A.1) 3 ζ0 · %c Here, ∆% is the density difference between the density %d of the disperse phase and the density %c of the continuous fluid phase, and ζ0 denotes the drag coefficient of a single particle. The absolute value of this velocity is identical to the terminal velocity of the particle if it were in a quiescent fluid. Thus, u0 is an important parameter of the flow characteristics of the particle. To be able to make statements about the behavior of many particles, a comparable approach is recommended. Figure 9.A.1B shows a homogeneous fluidized bed. For keeping the particle swarm in suspension, a superficial velocity us (the subscript stands for swarm) is required in the tube. Also us is an important parameter and contains information about the flow characteristics of the particle swarm. However, due to the complex interactions in the swarm, the calculation of us is sophisticated. Hence, experiments in fluidized beds can provide information for the calculation of us . Many researchers studied the fluid dynamics of fluidized beds (e.g. Wilhelm and Kwauck 1948; Richardson and Zaki 1954; Loeffler and Ruth 1959; Capes and McIlhinney 1968; Martin et al. 1981). A prime objective of their work was

588

9 DESIGN OF MASS TRANSFER EQUIPMENT

the development of a relation between superficial velocities us and u0 . Well accepted is the relation of Richardson and Zaki, developed as early as in 1954:

us = u0 · εn

with

n = f (Re) .

(9.A.2)

The superficial velocity us depends significantly on the porosity ε of the bed, which can vary over a wide range, i.e. from 0.4 – 1. The exponent n depends on the Reynolds number as is shown in Figure 9.A.2. This allows to make a statement about the behavior of the swarm by including the parameter u0 .

Figure 9.A.2 Dependence of the exponent n of Eq. (9.A.2) on the Reynolds number according to Richardson and Zaki 1954.

For a basic understanding of these relationships, a closer look at the interactions in the particle swarm is required. For one particle in the particle swarm, a balance can be written again in a first approach analogous to the basic Eq. (9.3):

d2 · π %c d3 · π · ζs · · us 2 = · (%d − %c ) · g . 4 2 6

(9.A.3)

Here, ζs is the drag coefficient of the particle in the swarm, which takes into account the influence of the swarm. However, this is considered separately below. The flow velocity at a particle in the swarm is increased because the many particles in the swarm lead to a narrowing of the flow cross section. This is taken into account by using the effective velocity us,eff of the fluid:

us,eff =

us . ε

(9.A.4)

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9.A APPENDIX: PRESSURE DROP IN PACKED BEDS

In addition, a pressure gradient occurs in the particle swarm according to Figure 9.A.1B, which enhances the buoyancy. This pressure gradient in the flow field results from the pressure drop that occurs when flowing through the particle swarm. Since the flow virtually carries the particle swarm, this pressure drop corresponds to the area-specific difference between weight and buoyancy of the particle swarm:

∆p =

(%d − %c ) · g · Vs . A

(9.A.5)

Here, Vs denotes the volume of all particles. For the pressure gradient it applies

∆p = (%d − %c ) · g · (1 − ε) . H

(9.A.6)

This covers the main effects that occur in the particle swarm. If these are applied in the basic Eq. (9.3), a more precise notation of the force balance for a particle in the particle swarm is

d2 · π %c  us 2 d3 · π · ζ0 · · = · (%d − %c ) · ε · g . 4 2 ε 6

(9.A.7)

A transformation provides

d2 · π ζ0 %c d3 · π · 3· · us 2 = · (%d − %c ) · g . 4 ε 2 6

(9.A.8)

To simplify, all swarm effects should now be captured in the drag coefficient ζs . A comparison of the coefficient with Eq. (9.A.3) yields

ζs =

ζ0 . εc 3

(9.A.9)

To confirm the correctness of these considerations, this result must be validated. For this, a special case is considered: in the case of laminar flow, according to Stokes 1851, for the drag coefficient, it follows that

ζ0 = 24/Re .

(9.A.10)

Inserting this in Eq. (9.A.1) leads to

u0 =

1 d2 · ∆% · g · . 18 η

(9.A.11)

Considering Eq. (9.A.9), for the swarm, it follows that

us =

1 d2 · ∆% · g 3 · ·ε . 18 η

(9.A.12)

If the ratio us /u0 of the terminal velocities of a single particle and particle swarm is calculated from this, the result is   us = ε3 . (9.A.13) u0 lam

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9 DESIGN OF MASS TRANSFER EQUIPMENT

This contradicts the result of Richardson and Zaki 1954, who give the exponent n = 4.65 in Eq. (9.A.2) for the laminar case. Thus, the exponent n = 3 is therefore wrong; it is disproved experimentally. The employed considerations do not seem to capture all effects occurring in the particle swarm. However, the fundamental dependence of the drag coefficient ζs on the porosity ε is correct. The exponent is adapted to the result of Richardson and Zaki 1954 for the region of laminar flow. Thus, the following relationship can be specified for the swarm law:

ζs =

ζ0 4.65 ε

.

(9.A.14)

The general validity of the swarm law is validated below. In the special case of fully developed turbulent flow, for the drag coefficient, it follows that

ζ0 = const .

(9.A.15)

Substituting the swarm law in Eq. (9.A.14) into Eq. (9.A.1), the following is obtained: s 4 d · ∆% · g 4.65 us = · · (1 − εd ) . (9.A.16) 3 ζ0 · %c Together with Eq. (9.A.1) the velocity ratio is given by   us = ε4.65/2 . u0 turb

(9.A.17)

This result is very close to the exponent n = 2.39 given by Richardson and Zaki 1954 for the range of fully developed turbulent flow. For these two special cases, the swarm law in Eq. (9.A.14) is in good agreement to the approach of Richardson and Zaki 1954 (Figure 9.A.2). A comparison of the swarm law in Eq. (9.A.14) with usage of the relation of Kaskas 1964 for the drag coefficient ζ0 of a sphere

ζ0 =

24 4 +√ + 0.4 . Re Re

(9.A.18)

to measured data in fluidized beds, which also cover the transition range between laminar and turbulent flow, is shown in Figure 9.A.3. The porosity of the beds has been varied from ε = 0.4 to 1. The accordance between the data of fluidized beds and the swarm law in Eq. (9.A.14) together with Eq. (9.A.18) is excellent. The empirical approach of Richardson and Zaki 1954 (Eq. (9.A.2)) as well as the physically based approach of the swarm law (Eq. (9.A.14)) can be used to describe the flow conditions of a particle swarm. For the calculation with the swarm law, however, only this single equation is required, whereas in the calculation according to Richardson and Zaki 1954 first the Reynolds number has to be calculated, and, dependent on it, the exponent n has to be determined. The physically based approach shows that the influence of the swarm is independent of the state of flow.

9.A APPENDIX: PRESSURE DROP IN PACKED BEDS

591

Figure 9.A.3 Comparison of experimental fluidization data with Eq. (9.A.14) together with Eq. (9.A.18).

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9 DESIGN OF MASS TRANSFER EQUIPMENT

For calculation of the pressure drop of a fluidized bed, the swarm law (Eq. (9.A.14)) can also be applied. Eliminating (%d − %c ) · g from Eqs. (9.A.3) and (9.A.5) yields

∆p 3 u2 = · ζs · (1 − ε) · %c · s . H 4 d

(9.A.19)

Replacing ζs by Eq. (9.A.14) leads to

∆p 3 1−ε u2 = · ζ0 · 4.65 · %c · s . H 4 ε d

(9.A.20)

This equation has been derived for fluidized beds. The porosity term (1 − ε)/ε4.65 is based on a large number of experimental data taken under a wide variation of the porosity. Hence, the porosity term is well proven.

Figure 9.A.4 Pressure drop data of fixed beds of sheers and granulates according to Coulson and Richardson 1976. The line represents Eq. (9.A.20).

However, the question whether or not Eq. (9.A.20) derived for fluidized beds can be applied to fixed beds has still to be answered. Figure 9.A.4 shows a large collection of pressure drop data of spheres and granulates in a fixed bed presented by Coulson and Richardson 1976. The line in the diagram represents Eq. (9.A.20), derived for fluidized beds. There is an excellent accordance. Rumpf and Gupte 1971 published data of the pressure drop in a fixed bed of spheres (Figure 9.A.5). They managed to vary the bed porosity from ε = 0.366 to 0.640 by shrinking the plastic particles of the bed. The dashed lines in Figure 9.A.5 represent Eq. (9.A.20). Here, the agreement between the experimental data and Eq. (9.82) is excellent, too. Hence, Eq. (9.A.20) is valid for both fluidized beds and fixed beds and can consequently be applied to column packings.

9.A APPENDIX: PRESSURE DROP IN PACKED BEDS

593

Figure 9.A.5 Pressure drop data of a fixed bed of spheres according to Rumpf and Gupte 1971. The porosity of the bed has been varied from ε = 0.366 to 0.640. The dashed lines represent Eq. (9.A.20).

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9 DESIGN OF MASS TRANSFER EQUIPMENT

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internals for packed columns. Chemical Engineering and Processing: Process Intensification 47 (9–10): 1520–1529. Ruff, K., Pilhofer, T., and Mersmann, A. (1976). Vollständige Durchströmung von Lochböden bei der Fluid-Dispergierung. Chemie Ingenieur Technik 48 (9): 759–764. Rumpf, H. and Gupte, A.R. (1971). Einflüsse der Porosität und Korngrößenverteilung im Widerstandsgesetz der Porenströmung. Chemie Ingenieur Technik 43 (6): 367–375. Schlünder, E.U. (1979). Über den Einfluß von Phasenverteilungen auf den Wirkungsgrad von Trennstufen. Berichte der Bunsengesellschaft für physikalische Chemie 83: 1184–1193. Schmidt, R. (1979). The lower capacity limits of packed columns. The Institution of Chemical Engineers, Symposium Series 56 2: 3.1/1–14. Schneider, O. (2004). Maldistribution in Packungskolonnen: Ausmaß, Auswirkungen und Gegenmaßnahmen. PhD thesis. Technical University of Munich. Düsseldorf: VDI Verlag, Fortschrittsberichte Reihe 3: Verfahrenstechnik, No. 823. Schultes, M. (1990). Einfluß der Phasengrenzfläche auf die Stoffübertragung in Füllkörperkolonnen. PhD thesis. University of Bochum. Düsseldorf: VDI Verlag, Fortschrittsberichte Reihe 3: Verfahrenstechnik, No. 230. Schultes, M. (2000). Influence of liquid redistributors on the mass-transfer efficiency of packed columns. Industrial and Engineering Chemistry Research 39 (5): 1381–1389. Schultes, M. (2003). Raschig super-ring: a new fourth generation packing offers new advantages. Chemical Engineering Research and Design 81 (1): 48–57. Scott, A.H. (1935). Liquid distribution in packed towers. Transactions of the Institution of Chemical Engineers 13: 211–217. Shariat, A. and Kunesh, J.G. (1995). Packing efficiency testing on a commercial scale with good (and not so good) reflux distribution. Industrial and Engineering Chemistry Research 34 (4): 1273–1279. Sherwood, T.K. and Jenny, F.J. (1935). Entrainment in plate columns. Industrial & Engineering Chemistry 27 (3): 265–272.

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10 Control of Distillation Processes The major objective of process control systems is to maintain the desired state of operation without any human action. This requires the ability to automatically prevent or suppress disturbances, caused by changes of the variables that the operation is subjected to. In general, a process control system consists of a large number of control loops that may interact with each other. A classical example of an interacting system is a distillation column with two-point composition control. The challenging control problems of distillation columns are caused by interactions of control loops and variables, nonlinearities of the process, process and measurement delays, and the large number of variables. In the past, a large variety of different control configurations for distillation columns has been developed [Luyben 2014]. This variety is a consequence of the different separation tasks realized in distillation processes and of the complex structure of distillation columns. Many publications about the advantages and the disadvantages of various configurations of process control systems for distillation columns are available; however, usually case studies predominate over theoretically established guidelines [Bezzo et al. 2004; Agaci et al. 2007]. The target of this chapter is assisting the process engineer in selecting the best process control configuration for a given distillation process. Utilizing the concept of split stream, design strategies for process control configurations are developed in a systematic manner, which are particularly helpful also in conflicting situations. The dynamic behavior of the control loops and the different possible controllers like P controllers, PI controllers, and PID controllers are not discussed here. Furthermore, the stability analysis of control loops is not subject of this chapter. Excellent reviews for this topic can be found in literature [Smith 2009; Lee et al. 2013]. In addition, dynamic process simulation tools can be used nowadays to develop feasible and robust control configurations [Luyben 2013]. Computer controls and advanced model predictive controls are outside the scope of this chapter. In the general literature about distillation process control, more details can be found [Shinskey 1977; Luyben 1992; Kister 1989; Skogestad 1992; Scattolini 2009; Christofides et al. 2013; Qin and Badgwell 2002; Camacho and Bordons 2007; Lee 2011]. For model predictive controls, dynamic simulation models are required, which reflect the dynamic behavior of distillation columns [Caspari

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

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et al. 2019]. In order to achieve feasible real-time performance, reduced dynamic simulation models for distillation columns have been developed [Schäfer et al. 2019; Ecker et al. 2019], whereas for dynamic simulation and control of startup and shutdown procedures of distillation processes, detailed models are available [Kender et al. 2019]. Standard symbols for control systems based on the standardization according to EN62424, DIN 19227-1, ISO 14617-6, and ANSI/ISA 5.1 are used in the following. The instrument is represented by a thin line circle. The purpose of the instruments is defined by a letter code that is placed inside the circle. This letter code is composed according to Table 10.1. The first letter denotes the measured process variable that is described in the second column of Table 10.1. The first letter can be modified with a second letter from the third column of Table 10.1 if required. Succeeding letters that characterize the display or output functions are listed in the fourth column of Table 10.1. If there are two or more succeeding letters, they are placed one after another in the sequence I, R, C, T, Q, S, Z, and A.

10.1

Control Loops

Different types of control loops are applied in distillation processes. The most important ones are described below. 10.1.1

Single Control Loop

In a single control loop, a fixed setpoint is specified. Figure 10.1A shows a simple example of a control loop where a constant liquid flow rate has to be maintained. This is labeled as flow controller FC, where, according to Table 10.1, the letter F stands for flow and C for control. If the actual and measured flow rate deviates from the specified setpoint, the flow controller adjusts position of the valve shaft to increase or decrease the flow rate. The target of the flow controller is to eliminate the deviation of the actual flow rate from the specified setpoint. The action of the controller on the valve is indicated by a thin, dashed line. In distillation processes, a decentralized control system with single loop controllers is preferred. This kind of control system is easier to understand and tune, more failure tolerant, and less sensitive to plant operation [Skogestad et al. 1990]. It is outside the scope of this chapter to examine the algorithm how the controller transforms the deviation of the actual measured value from the specified setpoint into a signal that causes the action on the valve. Usually, for control systems in distillation process, a PID controller shows satisfactory time response behavior. Here, from the deviation, a proportional, an integral, and a differential term is determined (P portion, I portion, and D portion). In most applications for distillation processes, a PI controller with a P portion and an I portion is sufficient, which stabilizes the control loop.

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Table 10.1 Letter code for identification of control instrument functions according to ISO 14617-6 (2002).

A

First letter

Second letter

Succeeding letter

Measured variable

Modifier

Display or output functions

Analysis

Alarm

C

Control

D

Density

Difference

E

Voltage

F

Flow rate

H

Hand operated

High

I

Current

Indicating

J

Power

K

Time

L

Level

M

Moisture, humidity

P

Pressure

Q

Quality

R

Radiation

Recording

S

Speed, frequency

Switching

T

Temperature

Transmitting

W

Weight, force

Sensor Ratio

Scan Low Test point connection Integrating, totalizing

Integrating, totalizing

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Figure 10.1 Examples of important control loops. A) Single loop control with flow controller FC. B) Ratio control loop with flow ratio controller FFC. C) Disturbance feedforward control loop FQC. D) Cascade control loop with flow controller FC and temperature controller TC.

10.1.2

Ratio Control Loop

In some distillation processes, the flow rates of the streams have to be kept in a certain ratio to each other. This flow ratio control is illustrated in Figure 10.1B. According to Table 10.1, the second F in the letter code stands for ratio that indicates that the controller keeps the flow rate of the second stream at a fixed specified ratio to the first one. The disadvantage of ratio control is that two flow rate measurements are required. This increases the risk of failure of the control system. Hence, flow rate control as shown in Figure 10.1A is preferred to the flow ratio control of Figure 10.1B. 10.1.3

Disturbance Feedforward Control Loop

In a disturbance feedforward control loop, the setpoint of the controller is adjusted to changes of an uncontrolled process variable. An example is depicted in Figure 10.1C.

10.2 SINGLE CONTROL TASKS FOR DISTILLATION COLUMNS

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The task of the control loop is a smooth flow of the lower stream. The setpoint of the flow controller depends on the temperature of the upper stream. The adjustment function of the controller is indicated by the letter Q. Disturbance feedforward control is rarely applied in distillation column control. 10.1.4

Cascade Control Loop

A cascade control loop consists of at least two loops, a master controller and a slave controller. The master controller adjusts the setpoint of the slave controller. For appropriate operation, the master controller has to be slow (low P portion, high I portion), whereas the slave controller has to be fast (high P portion, low I portion). An example of a cascade control loop is presented in Figure 10.1D. The setpoint of the flow controller FC is adjusted in a way that a temperature controller TC maintains the temperature of the upper stream constant. Contrary to disturbance feedforward control, two variables, here temperature and flow rate, are controlled by a cascade control loop. In some applications one master controller adjusts the setpoint of several slave controllers. Usually, cascade control loops show excellent process control behavior and are of major importance for distillation process control schemes.

10.2

Single Control Tasks for Distillation Columns

Typically, for distillation processes, the following control tasks are applied: • • • • •

Flow controller FC Level controller LC Pressure controller PC Temperature controller TC Product concentration controller AC (formerly also labeled as QC)

In this order, the complexity of the control task increases. For all flow controllers, split stream control schemes have to be applied in order to fulfill the material balance of the distillation column as shown in Section 10.2.2. 10.2.1

Liquid Level Control

Liquid level control is commonly used either as a single control loop as shown in Figure 10.2A or by a cascade control loop as illustrated in Figure 10.2B. The cascade control loop is preferred for large vessels since changes of the liquid level are very slow compared with the quick response of flow rate changes. Here, the slow master controller LC adjusts the setpoint of the fast slave controller FC.

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Figure 10.2 Level controller LC. A) Single control loop. B) Cascade control loop.

10.2.2

Split Stream Control

An essential control task is the split of a fluid stream into two or more smaller streams. Even if the flow rate of the feed stream before the split is absolutely constant, it is not possible to control the flow rate of each of the split streams independently. One reason for this results from the fact that the flow rate measurements are never absolutely precise. Hence, it is impossible that the material balance can be fulfilled completely with two independent flow control loops. Instead, only one of the split streams can be used for flow control, whereas the other split stream has to be left uncontrolled, which is shown in Figure 10.3A,B. Preferably, the stream with the smaller flow rate is directly manipulated and used for flow control. For example, if a 100 m3 /h stream is split into two streams of 90 and 10 m3 /h and the smaller stream is used for flow control, a control system with ±10 % accuracy keeps the streams in the range of 89 – 91 m3 /h and 9 – 11 m3 /h, respectively. Flow control of the larger stream results with the same ±10 % accuracy in flow rates of 81 – 99 m3 /h and 1 – 19 m3 /h. The more the flow rates of the split streams differ, the more important is that flow control is applied to the smaller stream. In many applications, the split of a liquid stream includes the use of a vessel, e.g. the reflux drum of a distillation column, which is illustrated in Figure 10.3B,C. Therefore, the flow rate of the directly manipulated stream that is used for flow control FC can be kept constant even if the feed stream to the reflux drum decreases for a short period of time. The resulting stream without flow controller is preferably used for level control LC as shown in Figure 10.3C. Flow control with split streams is one of the most essential control tasks in distillation processes. The above considerations can be summarized with the following two rules:

10.2 SINGLE CONTROL TASKS FOR DISTILLATION COLUMNS

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Figure 10.3 Flow controllers with stream split control; one of the two split streams is used for flow control FC, and the other stream is left uncontrolled or used for level control LC. A) Flow control of a gas stream and second stream left uncontrolled. B) Flow control of a liquid stream and overrun of the second stream. C) Flow control of a liquid stream and other stream used for level control LC.

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Rule 1: Only one stream can be directly used for flow control by a fixed setpoint. The other stream has to be left uncontrolled or used for level control. Rule 2: The stream that is used for flow control shall be the smaller one. In turn, the stream that is left uncontrolled or used for level shall be the larger one. In cases where one of the rules cannot be fulfilled, split Rule 1 takes priority since it results from the material balance, which can never be violated. Split Rule 2, however, results from a consideration of flow rate measurement accuracy and flow rate fluctuations and, hence, is not as strict as split Rule 1. If a well-tuned level controller is used, e.g. by applying a cascade control loop, the liquid level can be maintained even if the stream used for flow control is larger than the stream used for level control. As a rule of thumb, the stream used for flow control can be up to five times larger than the stream used for level control [Sieler 1970]. However, this violation of split Rule 2 should only be considered in conflicting situations where split Rule 1 would be violated otherwise, which is shown in Section 10.4.3. If one stream is much smaller than the other one, e.g. by a factor 10 or more, split Rule 2 has to be followed, and the stream with the smaller flow rate has to be used for flow control. 10.2.2.1

Split Stream Control at Top of a Distillation Column

Figures 10.4 and 10.5 show several control schemes at the top of a distillation column. All vapor from the top of the column is condensed, collected in the reflux drum, ˙ and reflux L˙ . If the column is operated with a low and then split into distillate D reflux ratio, i.e. RL < 1, the flow rate of the reflux L˙ is small and, thus, is directly manipulated and used for flow control FC, which is labeled in this chapter as reflux ˙ is used for level control LC of the controller. In this case, the distillate flow rate D reflux drum as shown in Figure 10.4A. ˙ is small At an operation with high reflux, i.e. RL > 1, the distillate flow rate D and, therefore, manipulated directly and used for flow control. Then, the reflux flow rate L˙ is used for level control of the reflux drum as illustrated in Figure 10.4B. By such a control system, an undisturbed operation is maintained even if the vapor stream arising in the column is smaller than the stream, which is used for flow control for a short time period. However, it is important to have enough hold-up in the reflux drum to provide effective damping of disturbances in the column heating system. A common rule of thumb requires a residence time of the liquid in the reflux drum of 5 – 10 minutes. In cases where the distillate has to be pumped, e.g. in vacuum distillation columns, the pump shall be located before the control valve to avoid suction problems as shown in Figure 10.5A. The principle of a reflux ratio controller is illustrated in Figure 10.5B. The disadvantages of such a ratio control loop are the requirement of two flow rate measure˙ . Flucments and the interaction with the level controller in the distillate stream D tuations of a poorly tuned level controller can affect the reflux flow rates. A cascade level control loop might be required to overcome these difficulties.

10.2 SINGLE CONTROL TASKS FOR DISTILLATION COLUMNS

609

Figure 10.4 Flow control with split of liquid condensate at the top of the column. A) Low reflux ratio RL < 1 with reflux L˙ used for flow control (reflux controller). B) High reflux ˙ used for flow control. ratio RL > 1 with distillate D

Figure 10.5 Flow control with split of liquid condensate at top of the column. A) Pump ˙ D ˙. installed before the control valve. B) Control of reflux ratio RL = L/

610

10.2.2.2

10 CONTROL OF DISTILLATION PROCESSES

Split Stream Control at Bottom of a Distillation Column

Figure 10.6 presents two possible control schemes for stream split at the bottom of a distillation column. The schematic in Figure 10.6A is applied if the flow rate of the bottoms B˙ is higher than the boilup rate G˙ , i.e. boilup ratio RG < 1, because the smaller flow G˙ should be used for flow control, which is labeled as boilup controller. The boilup rate G˙ cannot be controlled directly with a flow controller; instead, the flow rate of the heating steam to the reboiler is used for flow control. Therefore, the flow rate G˙ is indirectly manipulated and used for flow control FC. The bottoms B˙ is used for level control LC at the bottom of the distillation column.

Figure 10.6 Flow control with split of liquid at the bottom of the column. A) Low boilup ratio RG < 1 with boilup G˙ used for flow control (boilup controller). B) High boilup ratio RG > 1 with bottoms B˙ used for flow control.

If the boilup ratio is higher, i.e. RG > 1, the control system of Figure 10.6B is the better choice. Now, the flow rate of the heating stream is used for level control LC of the liquid level at the bottom of the column1 , whereas the bottoms B˙ is directly manipulated and used for flow control FC. 1

˙ . In a real distillation In Figure 10.6B, the level controller LC is drawn for clarity above the vapor inlet of G column, the level measurement is at the bottom of the column.

10.2 SINGLE CONTROL TASKS FOR DISTILLATION COLUMNS

10.2.3

611

Pressure Control

The simplest method of pressure control of an atmospheric distillation column is a vent of a vapor stream at the coldest point of the condenser. Another common solution is to manipulate the draw-off of non-condensable inert gases from the reflux drum as shown in Figure 10.7A. After the condenser there exists a two-phase flow that is indicated in the schematics of this section with parallel solid and dashed lines. An essential requirement is that there are enough non-condensable inert gases in the feed. This condition is always met with vapor feed or with a liquid feed that has not been exposed to temperatures higher than the bottom temperature of the column in a previous process.

Figure 10.7 Pressure control of a distillation column. A) Manipulating draw-off of inert gases. B) Vacuum distillation column.

The preferred control scheme for a vacuum column is illustrated in Figure 10.7B. Adding nitrogen avoids an extra nozzle in the suction pipe of the vacuum pump.

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10 CONTROL OF DISTILLATION PROCESSES

Thus, the vacuum pump is always in a safe range of operation. Sometimes, instead of a total condenser, a partial condenser is installed, and the ˙ overhead product is withdrawn as a vapor. In this case, the flow rate of the distillate D is directly manipulated and used for pressure control PC as illustrated Figure 10.8A. The cooling water flow is manipulated and used for level control LC, which maintains the liquid level in the reflux drum, whereas the reflux L˙ is used for flow control.

Figure 10.8 Pressure control of a distillation column. A) Manipulating flow rate of the vapor overhead product. B) Manipulating condensate temperature via cooling water flow.

If there are not enough inert gases in the feed, the column pressure can be manipulated via the temperature of the condensate. This is achieved by adjusting the cooling water flow as shown Figure 10.8B. However, such a control system has a long dead time. Sometimes, this results in a strong interference with temperature control loops with consequences for the product quality. In air-cooled condensers the control loops shown in Figure 10.9 are often applied. The withdrawal of the condensate is manipulated such that the liquid hold-up in the air-cooled condenser causes partial flooding of the heat exchanger tubes, which reduces the effective heat transfer surface area shown in Figure 10.9A. However, usu-

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

613

ally air-cooled condensers are arranged with horizontal tubes, with only two or four tube rows vertically above each other. Hence, no smooth control can be achieved by partial flooding of the tubes such that a bypass flow of part of the vapor as depicted in Figure 10.9B is the common practice for air-cooled condensers [Shinskey 1977].

Figure 10.9 Pressure control of a column with air-cooled condensers. A) Partial flooding of the condenser. B) Bypass flow of part of the vapor.

10.2.4

Product Concentration Control

A major problem of distillation processes is the control of the product concentrations that impacts directly the product quality. The first requirement for product concentration control is a suitable measurement. Product concentration measurement is often a difficult task; therefore, the concentration measurement and control is replaced by a temperature measurement and control. Usually, this concept is much easier, faster, and more reliable. In distillation columns, liquid and vapor are at their boiling and condensation point, respectively, such that under nearly isobaric conditions, the temperature depends only on the concentration. The required temperature sensors should be located in the vapor in order to get a quick response to temperature changes. A typical plot of a liquid temperature profile in a distillation column is shown in Figure 10.10. Toward the column ends, the temperature profile flattens. Here, the sensitivity of the temperature sensor may become the limiting factor. In this case, the effects of small pressure changes can superimpose the effects of concentration changes on the temperature. Hence, the sensor should be located in a sufficient distance from the column ends, where a significant temperature variation is caused when product concentration changes.

10.3

Basic Control Configurations of Distillation Columns

The process flow diagram of a distillation column is depicted in Figure 10.11A. From an abstract point of view, a distillation column is a system of three stream splits that are interconnected. Figure 10.11B illustrates an equivalent flow network with two vessels and five liquid streams where the distillation column has been discarded to

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Figure 10.10 Temperature profile in a distillation column with flattened temperature profiles toward at the top and bottom of the column.

simplify the flow structure and to show only the interconnections of the three stream splits. The pump shown in Figure 10.11B is included to reflect the flow from one vessel to the other. The three stream splits can be described as follows:

˙ and bottoms B˙ . This split is defined by the • Split of the feed F˙ into distillate D ˙ ˙ ˙ + B˙ , this split is also defined by product ratio D/B . Due to the balance F˙ = D ˙ ˙ ˙ F˙ . the ratio of distillate to feed D/F or bottoms to feed B/ ˙ ˙ • Split of the condensate into reflux L and distillate D. This stream split is defined ˙ D˙ . by the reflux ratio RL = L/ • Split of the bottoms liquid into boilup G˙ and bottoms B˙ . Here, the stream split is ˙ B˙ . defined by the boilup ratio RG = G/ The three stream splits in the flow network of a distillation column are interconnected. ˙ is part of the split of the feed as well as of the split of the overhead The distillate D condensate. The same holds for the split of the bottoms liquid since the bottoms B˙ is part of both the split of the feed and the split of the bottoms liquid. These interconnections of the three stream splits can be expressed by the following equation, which has been derived in Eq. (4.21):

RG =

˙ F˙ · (RL + 1) − (1 − qF ) D/ . ˙ F˙ 1 − D/

(10.1)

Here, qF is the caloric factor defined in Chapter 4. In the two-phase region, qF represents the fraction of saturated liquid in the feed. In the case of a saturated liquid feed, the caloric factor is qF = 1. In the case of a saturated vapor feed, the caloric factor is qF = 0. Equation (10.1) can be rewritten to solve for the reflux ratio RL as

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

615

Figure 10.11 A) Process flow diagram of a distillation column as interconnected system of three stream splits. B) Equivalent flow network showing the three stream splits.

follows:

RL =

˙ F˙ ) + (1 − qF ) RG · (1 − D/ − 1. ˙ F˙ D/

(10.2)

From the correlations in Eqs. (10.1) and (10.2), it can be concluded that only two ˙ F˙ can be chosen independently. In out of three split parameters RL , RG , and D/ other words, the flow network of a distillation column as shown in Figure 10.11B with three stream splits has a degree of freedom of only two. At a constant flow rate of the feed F˙ , as is assumed in this chapter, specification of the values of the split ˙ F˙ is equivalent to specifying the flow rates of the split parameters RL , RG , and D/ ˙ F˙ is equivalent to specifying streams. For example, the specification of the ratio D/ ˙ the flow rate of the distillate D. In principle, four streams can be adjusted and used for flow control in order to es˙ , reflux L˙ , boilup G˙ , and bottoms B˙ . tablish a column control system, i.e. distillate D However, only the flow rates of two of them can be independently manipulated and used for flow control. Depending on the streams that are used for flow control, different configurations for flow control result. They are labeled according to streams that are used for flow control. For example, in a D–G configuration, the flow rates ˙ and the boilup G˙ are used for flow control. It follows from the of the distillate D laws of permutations and combinations that six different pairs can be formed out of the set D, B , L, and G. These six different flow control configurations are listed in Table 10.2.

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Table 10.2 Check of flow control configurations for consistency with split Rule 1.

Configuration

Split Rule 1 for the stream split of

Judgment

Feed into D˙ and B˙

Condensate ˙ and L˙ into D

Bottoms liquid into G˙ and B˙

D–L

Fulfilled

Violated

Not fulfilled

Not feasible

D–G

Fulfilled

Fulfilled

Fulfilled

Favorable

D–B

Violated

Fulfilled

Fulfilled

Not feasible

B–L

Fulfilled

Fulfilled

Fulfilled

Favorable

B–G

Fulfilled

Not fulfilled

Violated

Not feasible

L–G

Not fulfilled

Fulfilled

Fulfilled

Feasible

However, not all of the six possible flow control configurations can be applied to column control since the split Rule 1 has to be additionally considered. This rule requires that in each stream split only one stream can be used for flow control and directly manipulated. The other stream has to be left uncontrolled or can be used for level control. In Table 10.2 the check whether this condition is fulfilled for each split is listed. In the D–L configuration, split Rule 1 is fulfilled for the stream split of the feed F˙ ˙ , and the since only one of the two streams is used for flow control, the distillate D ˙ second one is left uncontrolled, the bottoms B . For the condensate stream split, ˙ and L˙ ) are used for flow however, Rule 1 is violated since both split streams (D control and directly manipulated. Moreover, the stream split of the bottoms liquid does not fulfill split Rule 1 since none of the two split streams G˙ and B˙ is used for flow control. Thus, the D–L configuration is not feasible and has to be discarded. ˙ and the boilup G˙ are used for flow conIn the D–G configuration, the distillate D trol and are the directly manipulated streams. Thus, split Rule 1 is fulfilled for the ˙ and B˙ . The split of the condensate into distilsplit of the feed into the products D ˙ ˙ ˙ is used for flow control. The late D and reflux L fulfills split Rule 1 since only D same holds for the split of the bottoms liquid since only the boilup G˙ is used for flow control. Consequently, the D–G configuration fulfills all requirements and is a favorable control configuration. ˙ In the D–B configuration, split Rule 1 is violated since both product streams D ˙ and B of the feed split are used for flow control. Hence, this configuration is not feasible and has to be discarded. In the B–L configuration, the bottoms B˙ and the reflux L˙ are directly manipulated and used for flow control. Thus, split Rule 1 is fulfilled for the split of the feed into ˙ and B˙ . The split of the bottoms liquid into bottoms B˙ and boilup G˙ the products D fulfills split Rule 1 since only B˙ is used for flow control. The same holds for the split

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

617

of the condensate since only the reflux L˙ is used for flow control. Consequently, the B–L configuration fulfills all requirements and is a favorable control configuration. The B–L configuration is equivalent to the D–G configuration. The next configuration listed in Table 10.2 is the B–G configuration. Here, split Rule 1 violated for the split of the bottoms liquid into B˙ and G˙ . It is comparable with the D–L configuration and is, thus, not feasible. A special situation arises for the L–G configuration, which is the last configuration listed in Table 10.2. Split Rule 1 is fulfilled for the split of the condensate into D˙ and L˙ as well as in the split of the bottoms liquid into G˙ and B˙ . But none of ˙ and B˙ is used for flow control. Both prodthe two streams of the feed split into D ˙ ˙ uct streams D and B are left open such that this configuration is feasible but not as favorable as the D–G configuration and the B–L configuration [Wherry et al. 1984]. 10.3.1

Basic Control Systems Without Composition Control

The three feasible control configurations are illustrated in Figures 10.12 – 10.14. In order to minimize disturbances caused by the feed, its flow rate and temperature are kept constant by flow control FC and temperature control TC in all three control configurations. Furthermore, the column pressure is controlled in all three feasible control configurations by manipulating the draw-off of the inert gases from the condensate reflux drum with a pressure controller PC2 . For the D–G configuration and L–G configuration, it has to be considered that the boilup rate G˙ cannot be controlled directly with a flow controller; instead, the flow rate of the heating steam to the reboiler is used for flow control. Therefore, the flow rate G˙ is indirectly manipulated and used for flow control FC. Then, the bottoms B˙ is used for level control LC at the bottom of the column. For the B–L configuration, flow rate of the heating stream to the reboiler is used for level control LC at the bottom of the column, whereas the bottoms B˙ is directly manipulated and used for flow control FC. All three possible control configurations are evaluated if both the material and energy balance of the distillation column are manipulated independently with different controllers. The material balance can be manipulated with flow control of distil˙ or bottoms B˙ , whereas the energy balance of a distillation is determined by late D flow control of the reflux L˙ or the boilup G˙ . In this section, only binary mixtures with low boiler a and high boiler b are discussed. If not otherwise stated, the feed shall be a saturated liquid with qF = 1, which is achieved by a feasible setpoint of the temperature controller TC. Then, the following conditions are fulfilled:

G˙ = L˙ + D˙ and L˙ + F˙ = G˙ + B˙ . 2

(10.3)

In the schematics of this section, contrary to Figure 10.7A, the small vapor of non-condensable inert gases is not illustrated with a dashed line at the outlet of the condenser.

618

10.3.1.1

10 CONTROL OF DISTILLATION PROCESSES

D–G Configuration

˙ and Figure 10.12 shows the D–G configuration where the flow rate of the distillate D ˙ the boilup G are used for flow control. Hence, the material balance of the column is controlled by the withdrawal of one product, whereas the energy balance is controlled by the energy input to the reboiler to control the flow rate of the boilup G˙ . Both material balance and energy balance are manipulated independently, which results in a stable operation of the distillation column.

Figure 10.12 Basic D–G control configuration.

˙ has to be specified such that The setpoint of the flow controller of the distillate D the material balance of low boiler a is fulfilled. From the overall column material balance in Eqs. (4.10) and (4.11), we obtain zFa − xBa D˙ < F˙ · . xDa − xBa

(10.4)

Here, zFa denotes the mole fraction of low boiler a in the feed. For a sharp separation with pure distillate, i.e. xDa = 1, and no low boiler a in the bottoms, i.e. xBa = 0, we obtain the following condition:

D˙ < F˙ · zFa .

(10.5)

˙ has to be lower than This correlation expresses that the flow rate of the distillate D ˙ the flow rate F · xFa of low boiler a in the feed. If this condition is not fulfilled, no

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

619

˙ can be obtained, no matter how high the separation efficiency of the pure distillate D column is. Equation (10.5) defines a required condition for a pure distillate. However, this condition is required but not sufficient since the setpoint of the flow control of the boilup G˙ has to fulfill the condition G˙ > RG,min · B˙ .

(10.6)

The minimum boilup ratio RG,min is determined by the correlations as described in Sections 4.2.4 and 4.3.3. The setpoint of the flow control of the boilup G˙ is essential for the separation efficiency of the column. If this setpoint is too low, no pure products can be obtained even in a column with a very large number of equilibrium stages. The D–G configuration enables a stable column operation since the material balance and the energy balance are controlled by different control loops. For example, the variation of the setpoint of the flow controller of the boilup G˙ does not influence the material balance. Vice versa, the variation of the setpoint of the flow control of ˙ does not affect the energy balance. the distillate D According to split Rule 1, the D–G control configuration shall be applied if the ˙ is smaller than the flow rate of the bottoms B˙ . This will be distillate flow rate D discussed in detail in Section 10.4.1.1. 10.3.1.2

B–L Configuration

The B–L configuration is depicted in Figure 10.13. Here, the flow rates of the bottoms B˙ and the reflux L˙ are used for flow control. The setpoint of the flow controller of the bottoms B˙ has to be specified such that the material balance of the high boiler b is fulfilled. From the overall column material balance in Eqs. (4.10) and (4.11), we obtain

zFb − xDb B˙ < F˙ · . xBb − xDb

(10.7)

Here, zFb denotes the mole fraction of high boiler b in the feed. For a sharp separation with pure bottoms, i.e. xBb = 1, and no high boiler b in the distillate, i.e. xDb = 0, we obtain the following condition:

B˙ < F˙ · zFb .

(10.8)

This correlation expresses that the flow rate of the bottoms B˙ has to be lower than the flow rate F˙ · xFb of high boiler b in the feed. If this condition is not fulfilled, no pure bottoms B˙ with xBb = 1 can be obtained, no matter how high the separation efficiency of the column is. Furthermore, the setpoint of the flow control for the reflux L˙ has to be specified such that the following condition is fulfilled:

L˙ > RL,min · D˙ .

(10.9)

The minimum reflux ratio RL,min is determined by the correlations as derived in Section 4.2.4. The setpoint of the flow control of the reflux L˙ is essential for the

620

10 CONTROL OF DISTILLATION PROCESSES

Figure 10.13 Basic B–L control configuration.

separation efficiency of the column. If this setpoint is too low, no pure products can be obtained even in a column with a very large number of equilibrium stages. The B–L configuration is equivalent to the D–G configuration, i.e. the material balance and the energy balance are manipulated by different control loops. For example, the variation of the setpoint of the flow control of the reflux L˙ does not influence the material balance. Vice versa, the variation of the setpoint of the flow control of the bottoms B˙ does not affect the energy balance. According to split Rule 1, the B–L control configuration shall be applied if the ˙ . This will be discussed bottoms flow rate B˙ is smaller than the distillate flow rate D in detail in Section 10.4.1.2. 10.3.1.3

L–G Configuration

The basic L–G configuration is illustrated in Figure 10.14. Here, the flow rates of the reflux L˙ and the boilup G˙ are used for flow control. None of the two product ˙ and B˙ is used for flow control such that the material balance of the distilstreams D lation column is not directly controlled. The setpoint of the flow control of the reflux L˙ has to be specified such that the condition in Eq. (10.9) is fulfilled:

L˙ > RL,min · D˙ .

(10.10)

Analogously, the setpoint of the flow control of the boilup G˙ has to be specified such

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

621

Figure 10.14 Basic L–G control configuration.

that the condition in Eq. (10.6) is fulfilled:

G˙ > RG,min · B˙ .

(10.11)

It can be seen from Figure 10.14 that an increase of the setpoint of the flow control of the boilup G˙ increases both the energy input into the bottom of the column and ˙ . This can be seen directly from the correlation D˙ = G˙ − L˙ . the distillate flow rate D Hence, the energy balance and the material balance are not controlled separately. In the L–G configuration, the manipulation at one end of the column causes variations at the other end of the column, which is unfavorable. Vice versa, an increase of the setpoint of the flow control of the reflux L˙ increases both the separation efficiency of the column and the flow rate of the bottoms B˙ . This ˙ F˙ −G˙ . The increase of the reflux L˙ can be seen directly from the correlation B˙ = L+ improves the separation efficiency of the column, but the increase of the bottoms B˙ may violate the condition that for a sharp separation the flow rate of the bottoms B˙ has to be smaller than the flow rate F˙ ·xFb of high boiler b in the feed (see Eq. (10.8), for non-sharp separations Eq. (10.7) applies). Consequently, the L–G configuration is not as favorable as the D–G configuration and the B–L configuration. 10.3.1.4

B–L Configuration with Ratio Control

Until now, column operations with constant feed flow rate F˙ and constant feed compositions xFa and xFa have been considered. However, these requirements are rarely

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10 CONTROL OF DISTILLATION PROCESSES

fulfilled in industrial applications of distillation columns. Variations in feed flow rate F˙ can easily be compensated for when the setpoint of the flow rate is replaced ˙ F˙ for the B–L configuration, which is illusby a setpoint for flow ratios FFC, e.g. B/ ˙ D˙ , i.e. the reflux trated in Figure 10.15. Additionally, in Figure 10.15, the ratio L/ ratio RL , is used for flow ratio control. The configuration corresponds completely to the B–L configuration shown in Figure 10.13, however, with the exception that the two flow controllers FC are modified to a flow ratio controller FFC. This control configuration can be applied when the bottoms has to be recovered as pure high boiler b.

Figure 10.15 B–L configuration with ratio control FFC.

If the low boiler a has to be removed in pure form as distillate, an analogous ratio control structure, which is not shown here, can be developed from the D–G configuration shown in Figure 10.12. The main disadvantage of ratio control configuration is the requirement of more flow rate measurements, which makes them more difficult to implement and more sensitive to failure. Furthermore, the feed flow rate F˙ and the feed compositions xFa and xFa often vary in industrial applications of distillation columns. Hence, composition control with flow control of the feed as shown in Sections 10.3.2 – 10.3.3 is the better choice.

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

10.3.2

623

One-Point Composition Control Configurations

The basic control configurations developed in Section 10.3.1 are only applicable when the composition of the feed is constant. If, for instance, the amount of low boiler a in the feed varies, which is the normal situation in industrial applications, the setpoint of the flow control of the distillate has to be permanently adjusted to the changing feed concentration. This is particularly important for sharp separations with both a high purity in the distillate, i.e. xDa ≈ 1, and a high purity in the bot˙ is higher than the flow toms, i.e. xDb ≈ 1. For example, if the distillate flow rate D ˙ ˙ rate of low boiler a in the feed, i.e. D > F ·xFa , some high boiler b is forced into the distillate even when the separation efficiency of the column, i.e. number of stages n and the reflux ratio RL , is extremely high. Measuring the concentrations in the products may be time consuming and costly. Hence, temperature measurements are carried out at certain locations within the distillation column that is indicated in the related figures below with a small • showing the location of the temperature measurement. Regarding dead time it would be beneficial to place the temperature measurement point toward the upper end of the rectifying section or the lower end of the stripping section. However, it can be seen from Figure 10.10 that the temperature profiles flatten toward the ends of the column and the temperature changes that are to be expected due to compositions changes are rather small. Hence, the preferred location for the temperature measurements are in the region where the temperature profiles are steep. 10.3.2.1

D–G Configuration

Figure 10.16 shows the one-point composition control applied to the D–G configuration. As compared with the basic D–G control configuration shown in Figure 10.12, the setpoint of the flow control FC of the distillate is no longer specified to a fixed value. Instead, the setpoint is manipulated by a temperature control TC in the rectifying section of the distillation column. This arrangement represents a typical cascade ˙ is too high, control loop as described in Section 10.1.4. If the distillate flow rate D some high boiler b is forced into the distillate. Therefore, the temperature in the rectifying section increases, which is detected by the temperature measurement device, and the temperature control TC reduces the setpoint of the flow control FC of the distillate. By sensible location of the temperature measurement and feasible selection of the setpoint of the temperature control, for a sharp separation, the condition D˙ < F˙ · xFa can be fulfilled as long as the boilup G˙ is high enough to accomplish the separation task. Hence, the setpoint of the flow control of the boilup G˙ has to be specified externally to a feasible value. 10.3.2.2

B–L Configuration

In Figure 10.17 the one-point composition control is applied to the B–L control configuration, which is a variation of the basic B–L control configuration shown in Figure 10.13. Analogously to Figure 10.16, a temperature control TC in the stripping section of the column automatically adjusts the setpoint of the flow control FC of the bottoms B˙ for a sharp separation to the condition B˙ < F˙ · xFb . If the bottoms

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Figure 10.16 One-point composition control with D–G configuration.

Figure 10.17 One-point composition control with B–L configuration.

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

625

flow rate B˙ is too high, some low boiler a is forced into the bottoms. Therefore, the temperature in the stripping section decreases, which is detected by the temperature measurement device, and the temperature control TC reduces the setpoint of the flow control FC of the bottoms. In this control configuration the setpoint of the reflux L˙ has to be specified externally to a feasible value. 10.3.2.3

L–G Configuration

Application of one-point composition control to the L–G configuration shown in Figure 10.14 can be established in two different manners, either to the reflux L˙ or to the boilup G˙ . Figure 10.18 shows the configuration where the temperature control provides the setpoint of the reflux controller. This setpoint of the reflux controller FC is automatically adjusted by the temperature controller TC in order to compensate for perturbations of feed flow rate and feed composition. This configuration is preferably applied if a pure distillate is required, which of course requires that the setpoint of the boilup controller is high enough.

Figure 10.18 One-point composition control with L–G configuration and temperature control in cascade with the reflux controller.

An analogous control system is illustrated in Figure 10.19 where the temperature controller TC provides the setpoint of the boilup controller FC. It should be applied if a high purity of the bottoms is desired, which requires that the setpoint of the reflux controller is high enough. One-point composition control is a very favorable modification of the basic con-

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10 CONTROL OF DISTILLATION PROCESSES

Figure 10.19 One-point composition control with a L–G configuration and temperature control in cascade with the boilup controller.

trol configurations. Most industrial columns are controlled by such control configurations. 10.3.3

Two-Point Composition Control Configurations

If the feed flow rate and the feed composition are not constant and, furthermore, both pure distillate and pure bottoms have to be recovered, then two-point composition control might be considered. Here, the setpoints of two flow controllers FC are determined and controlled by two temperature controllers TC. 10.3.3.1

D–G Configuration, B–L Configuration, and L–G Configuration

Figures 10.20 and 10.21 show the application of two-point composition control to the D–G configuration and to the B–L configuration, respectively. Several authors report that such a control structure works well if the loops are carefully tuned and only PI controllers are used [Bogenstätter and Hengst 1959; Skogestad et al. 1990]. However, there is a high risk of severe control loop interactions. The two temperature controllers may act against each other. A common way to suppress interaction is to tune one loop very tight and the other loop loose. However, the performance of the slow loop is sacrificed to a certain extent. In Figure 10.22 the application of two-point composition control to the L–G con-

10.3 BASIC CONTROL CONFIGURATIONS OF DISTILLATION COLUMNS

Figure 10.20 Two-point composition control with a D–G configuration.

Figure 10.21 Two-point composition control with a B–L configuration.

627

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10 CONTROL OF DISTILLATION PROCESSES

figuration is shown. The principal problem of two-point composition control lies in the feasible selection of the setpoints of the two temperature controllers. Both setpoints have match with the temperature profile within the column during operation. Thus, the two setpoints cannot be chosen independently. However, a pair of setpoints that is consistent with the initial state of the feed might not be consistent when the state of the feed has changed. Here, application of advanced control systems that offer the possibility of calculating consistent pairs of setpoints might be required [Wozny et al. 1991; Scattolini 2009; Christofides et al. 2013; Qin and Badgwell 2002; Camacho and Bordons 2007; Lee 2011].

Figure 10.22 Two-point composition control with a L–G configuration.

10.3.3.2

D–B Configuration

The D–B control configuration has been classified in Table 10.2 as not feasible since it violates the material balance of the column. In rare cases the D–B configuration can be made workable in form of a two-point composition control shown in Figure 10.23. Both the bottoms and the distillate flow rates are manipulated by temperature controllers TC. If the bottoms flow rate is, for instance, higher than the amount of high boiler in the feed, some low boiler a is entrained into the bottoms, which decreases the temperature in the lower column section. If, vice versa, the flow rate of the distillate is too high, some high boiler b is entrained into the distillate, which makes the temperature arise in the upper column section. Consequently, violations of the material balance around the column are detected by the temperature sensors in the

10.4 APPLICATION RANGES OF THE BASIC CONTROL CONFIGURATIONS

629

Figure 10.23 Two-point composition control with a D–B configuration.

upper and in the lower column section. The D–B control configuration can be realized in this form. However, the D–B configuration with two-point composition control should be applied in very special cases only, e.g. very close-boiling systems with both pure distillate and bottoms. It is a very risky control configuration but better than no control at all.

10.4

Application Ranges of the Basic Control Configurations

The question which control configuration should be applied to which type of distillation has to be answered. The appropriate control configuration for a given distillation task depends on the nature and state of the feed xFa and on quality of the required products, i.e. xDa and xBb . Of great importance are, additionally, fluctuations of flow rate F˙ and composition zFa and zFb of the feed stream. 10.4.1

Impact of Split Parameters According to Split Rule 2

In this section it is discussed how the split Rule 2 of Section 10.2.2 is applied for the selection of a appropriate control configuration, i.e. that the stream with the smaller flow rate is directly manipulated and used for flow control. As described in Section 10.3 and illustrated in Figure 10.11, a distillation column represents, in essence,

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a system of three interconnected stream splits. Each stream split is characterized by ˙ F˙ , the value of the split parameters reflux ratio RL , boilup ratio RG , and the ratio D/ respectively. The parameters are related by the correlations in Eqs. (10.1) and (10.2). Hence, only two out of three split parameters can be chosen independently; the value of the third one results from Eqs. (10.1) or (10.2). The plot in Figure 10.24A illustrates graphically the correlation between the split parameters given by Eq. (10.2). Here, the reflux ratio RL is plotted in a logarithmic ˙ F˙ with the boilup ratio RG as parameter. scale versus the distillate to feed ratio D/ The values RL = 1 and RG = 1 are important and marked by bold lines. From the ˙ + B˙ , it follows that overall balance F˙ = D

˙ B˙ = D/

˙ F˙ D/ . ˙ F˙ 1 − D/

(10.12)

˙ B˙ can be illustrated as an addiHence, the ratio of the distillate to the bottoms D/ ˙ B˙ = 1, i.e. D/ ˙ F˙ = 0.5, tional abscissa as shown in Figure 10.24A. The value D/ is also important and marked by another bold line. In Figure 10.24A, the solid lines for a constant reflux ratio RL refer to a saturated liquid feed, i.e. qF = 1, the dashed lines refer to a saturated vapor feed, i.e. qF = 0. In Figure 10.24B, the boilup ratio RG is plotted versus the bottoms to feed ra˙ F˙ . The course of the lines in Figure 10.24B is identical to the ones in Figtio B/ ure 10.24A; however, the lines for a saturated liquid feed with q = 1 convert into the lines for vapor feed with qF = 0 and vice versa. This can be verified after inserting D˙ = F˙ − B˙ into Eq. (10.1), which yields ˙ F˙ ) · (RL + 1) − (1 − qF ) (1 − B/ ˙ F˙ ) 1 − (1 − B/ ˙ F˙ ) + 1 − B/ ˙ F˙ − 1 + qF RL · (1 − B/ = ˙ F˙ B/

RG =

=

˙ F˙ ) + qF RL · (1 − B/ − 1. ˙ F˙ B/

(10.13)

˙ F˙ and (1 − qF ) If this correlation is compared with Eq. (10.2), it can be seen that D/ ˙ ˙ in Eq. (10.2) are replaced by B/F and qF in Eq. (10.13). Again, the values RL = 1 and RG = 1 are important and marked by bold lines. From the overall balance F˙ = D˙ + B˙ , it follows that ˙ D˙ = B/

˙ F˙ B/ . ˙ F˙ 1 − B/

(10.14)

˙ D˙ can be illustrated as an addiHence, the ratio of the bottoms to the distillate B/ ˙ D˙ = 1, i.e. B/ ˙ F˙ = 0.5, is tional abscissa as shown in Figure 10.24B. The value B/ also important and marked by another bold line. For the selection of the appropriate control configuration, the split Rule 2 has to be considered. This rule requires that in each stream split the smaller of the two split

10.4 APPLICATION RANGES OF THE BASIC CONTROL CONFIGURATIONS

631

Figure 10.24 Graphical illustration of the correlations in Eqs. (10.1) and (10.2) between the split parameters for saturated liquid feed (qF = 1, solid lines) and saturated vapor feed ˙ F˙ and D/ ˙ B˙ with RG as parameter. (qF = 0, dashed lines). A) RL as a function of D/ ˙ F˙ and B/ ˙ D ˙ with RL as parameter. B) RG as a function of B/

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streams shall be directly manipulated and used for flow control. The values of the split parameters provide the information which of the two split streams is the smaller ˙ D˙ = 2, expresses, for instance, that one. A value of the reflux ratio RL = 2, i.e. L/ ˙ ˙. the flow rate of the distillate D is much smaller than that of the reflux L˙ = 2 · D ˙ Hence, the distillate flow rate D has to be used for flow control. At a value RL = 0.5, ˙ is the smaller stream and has to be used for flow however, the reflux L˙ = 0.5 · D control. The determining factor for the control configuration of any stream split is a value of less than one or larger than one of the relevant split parameter. 10.4.1.1

D–G Configuration

˙ and the boilup G˙ are the streams used for In the D–G configuration, the distillate D flow control. Split Rule 2 requires that the steams used for flow control are the small˙ has to be smaller than the bottoms B˙ to er ones in each split. Thus, the distillate D ˙ B˙ < 1, which is equivalent to fulfill split Rule 2 for the split of the feed F˙ , i.e. D/ ˙ ˙ D/F < 0.5. Applying split Rule 2 to the split of the condensate requires that the ˙ is smaller than the reflux flow L˙ , i.e. RL = L/ ˙ D˙ > 1. Analodistillate flow rate D ˙ gously, in the split at the bottom of the column, the boilup G is used for flow control ˙ B˙ < 1. and has to be smaller than the flow rate of the bottoms B˙ , i.e. RG = G/ Thus, the range of application of the D–G configuration is limited by the following conditions: • • •

˙ B˙ < 1 (equivalent to D/ ˙ F˙ < 0.5, B/ ˙ D˙ > 1, and B/ ˙ F˙ > 0.5) D/ RL > 1 RG < 1

These limitations define the upper left region labeled with D–G in the diagram of Figure 10.25, which shows the reflux ratio RL plotted versus the distillate to feed ˙ F˙ with the boilup ratio RG as parameter. The diagram in Figure 10.25A ratio D/ is valid for a saturated liquid feed with qF = 1 and the diagram in Figure 10.25B for a saturated vapor feed with qF = 0. The D–G configuration should be applied in the upper left region only. From the diagram in Figure 10.25A, it follows that for a saturated liquid feed, the condition RG < 1 is always more strict than the ˙ B˙ < 1. For a saturated vapor feed, it can be seen from Figure 10.25B condition D/ ˙ B˙ < 1 becomes the more strict conditions for 1 < RL < 2. that the condition D/ 10.4.1.2

B–L Configuration

In the B–L configuration, the bottoms B˙ and the reflux L˙ are the stream used for flow control. From split Rule 2 it follows that for the split of the overhead condensate, the ˙ . Hence, flow rate of the reflux L˙ has to be smaller than the flow rate of the distillate D ˙ ˙ ˙ we obtain for the reflux ratio RL = L/D < 1. In the split of the feed into distillate D ˙ and bottoms B , the flow rate of the bottoms has to be smaller than the distillate, i.e. ˙ B˙ > 1, which is equivalent to D/ ˙ F˙ > 0.5. Applying split Rule 2 to the split at D/ ˙ B˙ > 1. Consequently, the bottom of the column results in the condition RG = G/ the limitations for the application of the B–L configuration are as follows:

˙ B˙ > 1 (equivalent to D/ ˙ F˙ > 0.5, B/ ˙ D˙ < 1, and B/ ˙ F˙ < 0.5) • D/

10.4 APPLICATION RANGES OF THE BASIC CONTROL CONFIGURATIONS

633

Figure 10.25 Application ranges of control configurations using split Rule 2 dependent on ˙ F˙ , D/ ˙ B˙ , RL , and RG . A) Saturated liquid feed with qF = 1. split parameters D/ B) Saturated vapor feed with qF = 0.

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• RL < 1 • RG > 1 The region defined by these conditions is marked in the diagram of Figure 10.25 as the lower right region labeled with B–L. Here, for a saturated liquid feed shown in ˙ B˙ > 1 is more strict than the condition RG > 1. Figure 10.25A, the condition D/ For a saturated vapor feed shown in Figure 10.25B, the condition RG > 1 becomes ˙ F˙ > 0.667. the more strict condition for D/ 10.4.1.3

L–G Configuration

Limitations for the application of the L–G configuration can be derived in an analogous manner from split Rule 2. The reflux L˙ is used for flow control and has to ˙ , i.e. RL = L/ ˙ D˙ < 1. The second stream used for be smaller than the distillate D ˙ flow control is the boilup G. According to split Rule 2, it has to be smaller than the ˙ B˙ < 1. Consequently, the bottoms B˙ , which results in the condition RG = G/ range of application of the L–G configuration is defined as follows: • RL < 1 • RG < 1 These limitations define the lower left region labeled with L–G in the diagrams of Figure 10.25A,B. 10.4.1.4

Summary

The upper right regions in Figure 10.25A,B are described by the conditions RL > 1 ˙ and the bottoms B˙ and RG > 1. Hence, according to split Rule 2, the distillate D ˙ shall be used for flow control since both the reflux L and the boilup G˙ are larger ˙ and the bottoms B˙ , respectively. However, such a D–B control than the distillate D configuration violates split Rule 1 since both product streams of the feed split are used for flow control. Such a control configuration is forbidden as the material balance is not fulfilled as listed in Table 10.2. In Table 10.3 the limitations of the control configurations according to split Rule 2 are summarized. Table 10.3 Summary of application ranges of control configurations.

Control configuration D–G B–L L–G

˙ F˙ D/

˙ B˙ D/

˙ F˙ B/

˙ D˙ B/

RL

RG

< 0.5 > 0.5

1

> 0.5 < 0.5

>1 1 1, all and with D/ conditions for the B–L control configuration are fulfilled. (7) For zFa = 0.6 and n = 6, the reflux ratio has to be RL = 1.3, which results from an iterative stage construction in the McCabe–Thiele diagram. From the diagram in Figure 10.25A, we obtain that the forbidden D–B control configuration results. This can also be seen from the boilup ratio: RG =

0.4310 · (1.3 + 1) − (1 − 1) = 2.7237 . 1 − 0.4310

Hence, with RL = 1.3 > 1 and RG = 2.7237 > 1, it follows that the dis˙ and the bottoms B˙ have the smaller flow rates, as compared with the tillate D ˙ L = RL · D˙ and G˙ = RG · B˙ . Therefore, according to split Rule 2, these two product streams have to be used for flow control, which, however, violates split ˙ and B˙ . Rule 1 of the feed F˙ into D

10.4 APPLICATION RANGES OF THE BASIC CONTROL CONFIGURATIONS

639

˙ F˙ , RL , and RG are plotted into the diagram (8) If for all cases above the data for B/ in Figure 10.26A, which is valid for a boiling liquid feed with qF = 1, the same control configurations are obtained as described in the previous tasks.

10.4.2

Sharp Separations of Ideal Mixtures with Constant Relative Volatility at Minimum Reflux and Boilup Ratio

The selection of a suitable control configuration depends, as has been shown in the ˙ B˙ . Howprevious sections, on the values of the split parameters RL , RG , and D/ ever, these values cannot be chosen independently since they are determined by the thermodynamics of the distillation process. The values of the split parameters are the result of the thermodynamic calculation of the distillation process as shown in Chapter 4 and Example 10.1. ˙ B˙ The question arises which combinations of the split parameters RL , RG , and D/ typically exist in which mixtures to be separated by distillation. For ideal binary mixtures with a constant relative volatility αab and an infinite number of stages n, simple relations between the split parameters and the relative volatility αab , as shown in Section 4.2.4, can be used. For n → ∞, the reflux ratio RL approaches the minimum reflux ratio RL,min and the boilup ratio RG approaches the minimum boilup ratio RG,min . Usually, the operating point of distillation columns is only slightly above the minimum reflux and boilup ratio, respectively, such that the results presented in this section are also relevant to practical column operation. Here, only sharp separations into pure distillate and pure bottoms with xDa = 1 and xBb = 1 (equivalent to xBa = 0) are considered. With these assumptions, correlations for the distillate to ˙ F˙ and the bottoms to feed ratio B/ ˙ F˙ follow from the material balance feed ratio D/ around the whole column:

˙ F˙ = zFa D/

and

˙ F˙ = zFb . B/

(10.15)

If the feed is a saturated liquid with qF = 1, the minimum reflux and boilup ratio can be calculated from Section 4.2.4 according to Eqs. (4.39) and (4.41) as follows: For

1 (αab − 1) · zFa 1 = − 1. −1 (1 − αab ) · zFb

qF = 1: RL,min = RG,min

(10.16) (10.17)

According to Eq. (10.15), the concentration of the feed zFa is equivalent to the abscissa of Figure 10.25. This is also valid for zFb and the abscissa of Figure 10.26. Hence, the following correlations can be used to plot the minimum reflux ratio RL,min and

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the minimum boilup ratio RG,min into these diagrams: For qF = 1:

1 ˙ F˙ (αab − 1) · D/ 1 = − 1. ˙ F˙ (1 − α−1 ) · B/

RL,min = RG,min

(10.18) (10.19)

ab

If the feed is a saturated vapor with qF = 0, one obtains For qF = 0:

RL,min = RG,min =

1 (1 −

−1 αab )

˙ F˙ · D/

−1

1 . ˙ F˙ (αab − 1) · B/

(10.20) (10.21)

The resulting diagrams are shown in Figures 10.27 and 10.28, which reflect the following conditions: • Sharp separation with xDa = 1 and xBb = 1 (equivalent to xBa = 0) • Constant relative volatility αab • Minimum reflux ratio RL,min and minimum boilup ratio RG,min The lines with αab = const in Figures 10.27 and 10.28 characterize the nature of the mixture that has to be separated and show the required minimum reflux ratio RL,min and boilup ratio RG,min . It can be verified that a higher relative volatility αab leads to higher minimum reflux and boilup ratios. The regions of the control configurations D–G, B–L, and L–G, as developed in Section 10.4.1, are also marked in Figures 10.27 and 10.28. Here, compared with Figures 10.25 and 10.26, only the conditions that are more strict are shown as bold lines. 10.4.2.1

D–G Configuration

From Figure 10.27A it follows that the D–G configuration shall be applied to wide boiling mixtures with relative volatilities αab > 4, saturated liquid feed with qF = 1, and low concentration of the feed with zFa < 0.33. This can also be concluded from Figure 10.28A where the minimum boilup ratio RG,min is depicted as a function of the feed concentration zFb . Here, it can be seen that the concentration of the feed has to be zFb > 0.66 for the D–G configuration, which is equivalent to zFa < 0.33. For a saturated vapor feed with qF = 0 and feed concentrations as high as zFa = 0.5, the relative volatility has to be larger than αab = 3 for the D–G configuration. Generally, it can be seen from Figures 10.27B and 10.28B that for a saturated vapor feed with qF = 0, the region for the D–G configuration is larger than for a saturated liquid feed with qF = 1. 10.4.2.2

B–L Configuration

The B–L configuration can be applied to wide boiling systems with αab > 3 and high feed concentrations of the low boiler with zFa > 0.5. Here, from Figures 10.27B and 10.28B, it can be seen that the range of application of the B–L configuration is larger for a saturated liquid feed with qF = 1.

10.4 APPLICATION RANGES OF THE BASIC CONTROL CONFIGURATIONS

Figure 10.27 Application ranges of control configuration at minimum reflux and boilup ratio RL,min and RG,min for ideal mixtures with different constant volatilities αab ˙ F˙ = zFa and D/ ˙ B˙ . A) Saturated liquid feed with dependent on split parameters D/ qF = 1. B) Saturated vapor feed with qF = 0.

641

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Figure 10.28 Application ranges of control configuration at minimum reflux and boilup ratio RL,min and RG,min for ideal mixtures with different constant volatilities αab ˙ F˙ = zFb and B/ ˙ D ˙ . A) Saturated liquid feed with dependent on split parameters B/ qF = 1. B) Saturated vapor feed with qF = 0.

10.4 APPLICATION RANGES OF THE BASIC CONTROL CONFIGURATIONS

10.4.2.3

643

L–G Configuration

The application of the L–G configuration is restricted to intermediate feed concentrations zFa and zFb and to mixtures with very high values of relative volatilities αab > 4. Thus, this control configuration, although often considered in literature as a standard configuration, has only a very limited application range for practical distillation problems. The restriction to very wide boiling systems and the disadvantage of not decoupling the material balance and the energy balance lead to the recommendation not to apply L–G control configuration. As will be shown in Section 10.4.3, its application range is also covered by the expanded application ranges of the D–G and B–L control configuration. 10.4.3

Extended Application Ranges of the Basic Control Configurations

In this section again sharp separations of ideal mixtures with constant relative volatilities αab are considered at minimum reflux ratio RL,min and boilup ratio RG,min . As stated in Section 10.2.2, split Rule 2 requires that the smaller of the split streams has to be used for flow control. However, split Rule 2 is not a strict rule such that a control configuration also performs well if the stream used for flow control is somewhat larger than the uncontrolled stream. Hence, the application range of the different control configurations can be expanded to some extent. However, if the stream used for flow control is much larger than the uncontrolled one, the control configuration will fail at higher perturbances of flow rates. The expanded application range of all feasible control configurations is presented in this section only for the case of a saturated liquid feed with qF = 1. Analogous diagrams for a saturated vapor feed with qF = 0 can easily be developed. In these expanded application ranges, the values of the split parameters according to split Rule 1 (with a Factor 1.0) are changed by a Factor 2.0 and 4.0 and illustrated. As in Figures 10.27 and 10.28, only the conditions that are more strict are shown as bold lines as follows: Factor 1.0 Factor 2.0 Factor 4.0 10.4.3.1

–––– −−− ······

solid lines (spit Rule 2) dashed lines dotted lines

D–G Configuration

If the split parameters for the D–G configuration are changed by the Factors 2.0 and 4.0, we obtain the following limitations:

˙ B˙ < 1 (equivalent to D/ ˙ F˙ < 0.5) • Factor 1.0: RL > 1.0, RG < 1, and D/ ˙ ˙ ˙ F˙ < 0.667) • Factor 2.0: RL > 0.50, RG < 2, and D/B < 2 (equivalent to D/ ˙ ˙ ˙ F˙ < 0.8) • Factor 4.0: RL > 0.25, RG < 4, and D/B < 4 (equivalent to D/ Figure 10.29 shows the expanded application ranges of the D–G control configuration. This diagram shows that the D–G control configuration can be applied to systems with relative volatilities as low as αab = 1.4 with xFa < 0.3 and with feed concentrations as high as xFa = 0.75 with αab = 6.

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Figure 10.29 Expanded application range of the D–G configuration for a saturated liquid feed with qF = 1: Factor 1.0 –––, Factor 2.0 − − − , and Factor 4.0 · · · · · · .

10.4.3.2

B–L Configuration

After changing the split parameters for the B–L configuration by the Factors 2.0 and 4.0, we obtain the following limitations:

˙ B˙ > 1 (equivalent to D/ ˙ F˙ > 0.5) • Factor 1.0: RL < 1.0, RG > 1, and D/ ˙ ˙ ˙ F˙ > 0.333) • Factor 2.0: RL < 2, RG > 0.50, and D/B > 0.50 (equivalent to D/ ˙ ˙ ˙ F˙ > 0.2) • Factor 4.0: RL < 4, RG > 0.25, and D/B > 0.25 (equivalent to D/ The application range of the B–L control configuration is shown in Figure 10.30. It covers the region of high feed concentrations. Only for very close-boiling systems with relative volatilities αab < 1.3 this control configuration is not feasible. 10.4.3.3

L–G Configuration

If the split parameters for the L–G configuration are changed by the Factors 2.0 and 4.0, we obtain the following limitations: • Factor 1.0: RL < 1.0 and RG < 1 • Factor 2.0: RL < 2 and RG < 2 • Factor 4.0: RL < 4 and RG < 4 The expanded application range of the L–G configuration is shown in Figure 10.31. It mainly covers the range of small feed concentrations xFa . Most of its application range is also covered by the D–G and the B–L configurations. Whenever possible,

10.4 APPLICATION RANGES OF THE BASIC CONTROL CONFIGURATIONS

Figure 10.30 Expanded application range of the B–L configuration.

Figure 10.31 Expanded application range of the L–G configuration.

645

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10 CONTROL OF DISTILLATION PROCESSES

these configurations should be used instead of the L–G configuration since the material balance and the energy balance are coupled in the L–G configuration. 10.4.3.4

Summary

It can be stated that the D–G configuration is the best control configuration for low feed concentrations and, in turn, the B–L configuration is the best one for high feed concentrations. Both configurations cover all feed concentrations and nearly all relative volatilities αab . Only systems with very low volatilities αab < 1.3 are difficult to control by these configurations. Here, application of the D–B configuration with two-point composition control should be considered as described in Section 10.3.3.

10.5

Examples for Control Configurations of Distillation Processes

Distillation processes consist usually of a sequence of several distillations columns and many auxiliary devices, e.g. vessels, heat exchangers, pumps, and compressors. Typically, several streams are recycled within the process creating a very complex structure of the process. Hence, the control configuration of a distillation process is not only a compilation of the control schemes of single columns. Some examples for process control of distillation processes are presented in this section. 10.5.1

Azeotropic Distillation Process by Pressure Change

Binary mixtures with an azeotrope can be separated by changing the pressure from one distillation column to another as has been outlined in Section 8.5. Therefore, two columns are required for separating such a system. An important example is the tetrahydrofuran/water mixture. The basic process is depicted in Figure 10.32 with column C-1 that is operated at 1 bar and column C-1 that is operated at 8 bar. Since the mixture exhibits a minimum azeotrope, the pure products are drawn off as bottoms. In order to perform the separation, the two columns have to be operated with different pressures. In Chapter 7 it has been shown that thermal coupling of columns provides a very effective measure for energy saving. The most essential condition for this kind of coupled columns is that the temperature of the condenser of column C-2 is higher, here 136 ◦C, than the temperature of the reboiler of column C-1, here 100 ◦C. To fulfill this condition, the columns have to be operated at different pressures, here column C-1 at 1 bar and column C-2 at 8 bar. Thus, processes for separating azeotropic mixtures by changing system pressure are always candidates for thermal coupling of the columns. Details are presented in Figure 8.30 for the azeotropic tetrahydrofuran/water mixture. The control configuration of this process is shown in Figure 10.32 [Abu-Eishah and Luyben 1985]. In each column both the distillate and the bottoms are used for level controllers LC as shown in the basic L–G control configuration in Figure 10.14. Such an L–G control configuration is not an optimal way, but acceptable. Ratio control FFC is applied in both columns for the reflux controller. To avoid too many

10.5 EXAMPLES FOR CONTROL CONFIGURATIONS OF DISTILLATION PROCESSES

647

Figure 10.32 Control configuration of a distillation process for separating the azeotropic tetrahydrofuran/water mixture by pressure change and thermally coupled columns.

flow measurements, the flow rate of the distillate of the first column is used for the ratio control of the reflux controller of the second column, too. The setpoints of the boilup controllers are in cascade with temperature measurements in the stripping section of each column. For sake of clarity, the cascade control loops are not depicted in Figure 10.32. Thus, one-point composition control configurations, as described in Section 10.3.2, have been applied to the process. For complete illustration of the cascade control loops; see Figure 10.19. 10.5.2

Distillation Process for Air Separation

Air separation by distillation is performed in most cases by two thermally coupled columns as described in Section 8.2.4. In Figure 10.33, partially liquefied air is fed to the high-pressure column C-1, which consists of a rectifying section only. The overhead of column C-1 consists of almost liquid nitrogen that is condensed in a thermally coupled heat exchanger. This serves as condenser of column C-1 and reboiler of column C-2. A part of the condensed overhead of column C-1 is routed as reflux to column C-2. The bottoms B˙ 1 of column C-1 has an increased oxygen content of approximately 35 – 40 mol%. This fraction is fed as liquid into column C-2 to be separated into pure oxygen as bottoms B˙ 2 and pure nitrogen as vaporous distil˙ 2 . Besides this vaporous nitrogen product D˙ 2 , in the illustrated air separation late D process, liquid nitrogen is withdrawn as a product stream. Thermal coupling requires operation of column C-1 at a higher pressure to increase

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Figure 10.33 Control structure of an air separation double column with B–L configuration applied to the low-pressure column C-2.

the condensation temperature of nitrogen at the top of column C-1 above the boiling temperature of oxygen at the bottom of column C-2. This is achieved with the pressure controller PC shown in Figure 10.33 at the top of low-pressure column C-2. In most cases the high-pressure column C-1 is installed directly below the low-pressure column C-2. A well-approved control configuration is shown in Figure 10.33 [Zapp 1994]. The reflux of the high-pressure column C-1 is used for composition control AC and directly manipulated. The setpoint has to be selected such that the concentration in the middle of the column is kept constant. Untypically for distillation column control, a concentration measurement is used instead of a temperature measurement. A cascade loop should be used to provide a quick response. A slightly modified L–G configuration is applied to the low-pressure column C-2. This is in accordance with Figure 10.27 since the nitrogen/oxygen mixture is a wide boiling mixture with αN2 ,O2 = 4 and the concentration of the low boiler in the feed of column C-2 is high at xF N2 = 0.6 − 0.65. The reflux flow rate is directly manipulated to keep the concentration in the rectifying section of column C-2 constant shown with the AC above the feed point of column C-2. Additionally, the vapor flow at the bottom is manipulated by withdrawal of vapor oxygen product that keeps the concentrations in the stripping section constant, i.e. a two-point quality control with two AC is applied in the low-pressure column C-2. Essential for effective control is the appropriate location of the concentration mea-

10.5 EXAMPLES FOR CONTROL CONFIGURATIONS OF DISTILLATION PROCESSES

649

surement devices. The method of relative gain analysis has proven an efficient tool for that [Zapp 1994]. More details regarding advanced control configurations for air separation processes including model predictive control can be found in literature [Roffel et al. 2000; Zapp 2015; Zapp and Siebel 2017; Caspari et al. 2019]. 10.5.3

Distillation Process with a Main and a Side Column

Separation of a zeotropic ternary mixture into the pure substances a, b, and c requires two distillation columns. If the intermediate boiler b is present only in minor concentrations, the use of a side column is advantageous. The principles of such processes have been discussed in Section 8.3.2 (e.g. Figure 8.25). Control configurations for such processes have been studied by Glinos and Malone 1985; Alatiqi and Luyben 1986; Raisch et al. 1993. In ternary mixtures, the bubble point is not a definite function of concentration as it is at constant pressure in binary mixtures. Concentration control via temperature control is only feasible if the temperature sensors are located in that parts of the distillation column where mainly two components are present, i.e. toward the upper and lower ends of the column. An acceptable control configuration is presented in Figure 10.34 where one-point control is applied to both columns. In the main column, the reboiler duty is controlled by the setpoint of a temperature in the stripping section of the column. The level in the reboiler is manipulated by the flow rate of the bottoms. At the top of the column, the level in the reflux drum is controlled by the flow rate of the distillate. The reflux is controlled by ratio control. The rate of the vaporous side stream is controlled by a sensible temperature in the stripping section of the main column. In the side column

Figure 10.34 Control configuration of a process for separating a ternary mixture with a main and a side column.

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10 CONTROL OF DISTILLATION PROCESSES

the value of the reflux ratio is monitored by the setpoint of a temperature in the middle of the column. This and other control configurations have been studied experimentally with the methanol/ethanol/n-propanol mixture by Lang and Gilles 1989. The controller is able to stabilize the plant but becomes unstable if larger disturbances of feed rate and especially of feed concentrations are imposed. Depending on the type of disturbances, only two out of the three products meet the specifications at best. An improved control configuration suggested by Alatiqi and Luyben 1986 uses the temperature difference below and above the point of side stream withdrawal to control the flow rate of the side stream. For a better controller behavior, advanced control strategies have to be applied. Processes with side columns are prime candidates for multivariable robust control [Levien and Morari 1987]. 10.5.4

Azeotropic Distillation Process by Using an Entrainer

A very effective process for separating a binary mixture with a minimum azeotrope is azeotropic distillation using an entrainer; see Section 8.6.3.1. The separation of the feed into its components is accomplished by an entrainer that is not completely miscible with the feed. The process consists of two distillation columns that deliver the products as bottoms and one decanter that regains the entrainer. A classical example of an azeotropic distillation process is the separation of the azeotropic ethanol/water mixture using toluene as entrainer illustrated in Figure 10.35. In principle, the whole process is a split of the feed stream into the two

Figure 10.35 Control configuration of a distillation process for separating the azeotropic ethanol/water mixture with toluene as entrainer.

10.6 CONTROL CONFIGURATIONS FOR BATCH DISTILLATION PROCESSES

651

product streams. In most cases, the ethanol content of the feed is low, and, consequently, the ethanol product stream is smaller than the water product stream. Hence, the flow rate of the ethanol B˙ 2 has to be used for flow control FC, and, in turn, the flow rate of the water product B˙ 1 has be used as level controller LC. In order to account for a variable feed concentration, the ethanol flow B˙ 2 is manipulated by a temperature cascade control TC. Application of split Rule 1 to the column C-1 requires flow control FC of the small˙ 1 . Hence, a D–G control configuration is applied to column C-1. er flow of distillate D Analogously, in column C-2, the smaller flow of the bottoms B˙ 2 is used for flow con˙ 2 for level control LC. Thus, column C-2 is controlled by trol FC and the distillate D a B–L control configuration. Additional control loops are required to maintain constant levels in decanter S-1 and, especially important, to circulate the appropriate amount of the entrainer within the process. The proposed control scheme results from the split of the feed into the products. This split is superimposed on the splits of the columns and the decanter. It has to be checked whether or not the split rules are met for all these internal stream splits. In conflicting situations priority shall be given to the superimposed split of the feed into the products since it stabilizes the operation of the whole process.

10.6

Control Configurations for Batch Distillation Processes

In continuously operated distillation columns, there exists a steady non-linear temperature profile within the column. The temperature gradient varies significantly along the column. That tray that exhibits the highest temperature gradient is the optimal sensor point for quality control since disturbances that cause the temperature and concentration profiles move within the column are easiest detected there. In batch distillation, however, the temperature profile is not constant over time. The temperatures steadily increase, i.e. the temperature profile moves upward during operation. The point of the highest gradient of the temperature profile is no longer fixed within the column. According to Egly and Ruby 1980, the temperature sensors shall be located such that they characterize product quality over the entire time of operation. Furthermore, they shall exhibit the highest sensitivity to disturbances. The authors identified by rigorous simulation of the non-steady-state behavior of a 50 tray column 4 trays that meet these requirements. The resulting control configuration is depicted schematically in Figure 10.36 with 4 temperature sensors that are implemented in the column for identification of unwanted variations of the process. The actual temperatures at these 4 trays are compared with the theoretical values by a computer. The desired state of operation is maintained by manipulating the distillate flow rate and, thus, at constant heating, the reflux rate. The authors studied the effectiveness of this control configuration at the example of a binary mixture with constant product quality. The reflux ratio is automatically increased by the control system, and disturbances are compensated for very effectively.

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Figure 10.36 Control structure of a batch distillation process operated with constant product quality.

REFERENCES

653

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655

Index (T , x , y)-diagram, 84, 86–90, 106 (p, x )-diagram, 81–83 (p, x , y)-diagram, 83, 84, 86–92, 103, 106

Activity coefficient – correlations, 49, 52 – definition, 22 – experimental data, 55 – Margules equation, 26, 50–52, 55, 57, 58, 86 – NRTL equation, 51–53, 55, 57, 58 – UNIQUAC equation, 51, 52, 54, 55, 57, 58 – van Laar equation, 50–52, 55, 57, 58 – Wilson equation, 51–53, 55, 57, 58, 74 Adiabatic flash, 147, 148 Air separation, 418 Ammonia recovery, 414 Antoine equation, 14, 34, 57, 62, 67, 69, 74, 91, 104, 105, 108, 112 – parameters, 15, 57 Application ranges of basic control configurations, 629, 632, 634, 639, 640, 643 Area – active, 487 – interfacial, 511, 513, 562–567, 570–572 – relative free, 487 Argon recovery, 431 Arrangement of catalysts in columns, 307 Atmospheric tower, 433 Azeotrope – hetero, 88, 321, 435 – maximum, 89, 444 – minimum, 87, 444 – reactive, 289 – saddle, 122, 133 Azeotropic distillation – batchwise, 348 – continuous, 455 – control, 646 – list of systems, 457

Azeotropic mixture, 87–91, 117–120, 122, 129–132 Basic control configurations, 613, 617–621, 623, 625, 626 Batch distillation, 313–351 – control, 651 – singlestage, 152, 154 – steam distillation, 425, 426 Batch distillation, multistage, 313–351 – azeotropic mixtures, 332–333 – binary mixtures, 314–327 – control, 651 – energy requirement, 319, 322, 328 – modes of operation, 315–325 – multicomponent mixtures, 336 – processes, 340–351 – ternary mixtures, 327–333 Billet/Schultes correlation, 572 Boiling curve – correlation, 81 – diagram, 82, 84–90, 106 Boiling surface, 114, 116–122 Boundary distillation line, 128, 130–132 Bravo/Fair correlation, 572 Bubble cap trays, 484, 487, 493 Caloric factor, 184 Capacity factor, 490 Carbon dioxide outburst, 151 Cascade control, 605 Catalyst – arrangement, 307 – heterogeneous, 307, 308 – homogeneous, 307 – regeneration, 309 Channel model structure, 540 Chemical equilibrium, 284 Chemical potential

Distillation: Principles and Practice, Second Edition. Johann Stichlmair, Harald Klein, and Sebastian Rehfeldt. © 2021 American Institute of Chemical Engineers, Inc. Published 2021 by John Wiley & Sons, Inc.

656

– correlations, 20 – definition, 16 Chemical reaction, 90 – decomposition, 302 – in ternary systems, 293, 295 – single product, 300 – with side products, 297 Complete chemical reaction, 106, 110 Complete miscibility gap, 106, 107 Condensation – closed, 138 – open, 138, 152, 156, 160, 168 Condensation curve, 83 – correlation, 83 – diagram, 84–90, 106 Condensation surface, 122 Constant molar overflow, 189 Control – application ranges of basic control configurations, 629, 632, 634, 639, 640, 643 – basic control configurations, 613, 617–621, 623, 625, 626 – batch distillation, 651 – cascade control, 605 – control loop, 602 – example air separation, 647 – example azeotropic distillation, 646 – example azeotropic distillation entrainer, 650 – example side column, 649 – examples for control condigurations, 646 – extented application ranges of basic control configurations, 643, 644, 646 – feedforward control, 604 – letter code, 603 – liquid level control, 605 – pressure control, 611 – product concentration control, 613 – ratio control loop, 604 – rules split stream control, 606 – single control loop, 602 – split stream control, 606, 608, 610 Control loop, 602 Control of distillation processes, 601–653 COSMO-RS, 55 Criteria for entrainer selection, 445 Criteria for use of tray or packed columns, 486 Crude oil distillation, 433 Cubic equations of state, 10 Cubic–Plus–Association (CPA), 12 Dalton’s law, 9 DECHEMA Chemistry Data Series, 57

INDEX

Decomposition reactions, 302 Desalination, 361 Design – of mass transfer equipment, 481–586 – of packed columns, 533–586 – of tray columns, 487–533 Design parameters – of packed columns, 534 – of tray columns, 487 Distillation – batchwise, 152 – binary mixtures, 137, 181 – catalytic, 283 – closed, 137, 138, 140 – flash, 140 – multicomponent mixtures, 140 – multistage, 181, 206 – multistage continuous, 173 – multistage, batchwise, 313 – open, 138, 152, 157, 167 – reactive, 283 – semi-continuous, 169 – ternary mixtures, 206 Distillation line – derivation, 123–125 – examples, 126, 128–133 – non-reactive, 287 – reactive, 287 Dividing wall column, 384 Downcomer, 483, 494 Dumped packing, 534 Energy economization, 357 Energy requirement – a-path, 363 – c-path, 365 – batch distillation, 319, 322, 328 – binary mixtures, 204, 206 – of columns, 204 – preferred a/c-path, 366 – single column, 358 – ternary distillation, 248 Enthalpy balance, 189 Enthalpy–concentration diagram, 189, 191 Entrainer selection, 439 – binary maximum azeotrope, 440 – binary minimum azeotrope, 440 – systems with distillation boundary, 444 – systems without distillation boundary, 440 Entrainment, 503–506 Equation of state, 8 – cubic equations of state, 10 – Cubic–Plus–Association (CPA), 12

657

INDEX

– enhanced equations of state, 12 – ideal gas law, 8 – PC-SAFT, 12 – Peng–Robinson equation, 11 – predictive equations of state, 12 – Predictive Soave–Redlich–Kwong (PSRK), 12 – Soave–Redlich–Kwong equation, 10, 34, 40 – virial equation, 9 – Volume Translated Peng–Robinson (VTPR), 12 Equilibrium conditions, 19 Equilibrium curve, 86–90, 106 Equilibrium stage concept, 176, 180, 256 Ergun equation, 553 Excess free energy, 25 Extended application ranges of basic control configurations, 643, 644, 646 Extended Raoult’s law, 31, 48, 59 Extractive distillation – batchwise, 351 – continuous, 457 Feasible bottom and overhead factions, 215 Feedforward control, 604 Flooding – of packings, 545–547, 557–558 – of trays, 499, 506 Flooding factor, 558 Flow regimes on trays, 497 Froth height, 499–503 Fugacity – definition, 20 – liquid phase, 30 – mixtures, 21 – pure components, 20 – vapor phase, 31 Fugacity coefficient – correlations, 20, 22, 32 – definition, 20 – mixtures, 21 – pure components, 20 Gas capacity factor, 490 Gas inlet, 544 Gas load of trays – maximum, 490–492 – minimum, 492–494 – selection of, 495 Generalized process for azeotropic mixtures, 446 Gibbs phase rule, 19, 84, 86, 116, 122 Gibbs–Duhem equation, 17, 26 Gibbs–Helmholtz equation, 17, 29

Hamiltonian function, 324 Heat capacity ratio, 359 Heat exchanger – dual flow, 401 – network, 391, 397 Heat pump, 359 – bottom product is working fluid, 360 – overhead product is working fluid, 360 – with supplementary working fluid, 360 Height of a transfer unit, 568–569, 574–577 Henry coefficient, 49, 59, 83 Henry’s law, 31, 49, 59, 83 Heteroazeotrope, 88, 108, 321, 435 Heterogeneous catalyst, 308 HETP, 569, 577 History of distillation, 3 Hold-down plate, 544 Hold-up – influence on batch distillation, 337 – on trays, 499 – packed columns, 487, 549–552 – tray columns, 486 Homogeneous catalyst, 307 Hybrid processes, 455 – batchwise, 348–351 – distillation and absorption, 457 – distillation and adsorption, 465 – distillation and decantation, 455 – distillation and desorption, 459 – distillation and extraction, 461 – distillation and membrane permeation, 467 Hydrogen chloride recovery, 416 Ideal gas law, 8 Ideal Mixture, 62, 69 Ideal mixture, 48, 59, 60, 87, 104, 106, 107, 115 – boiling curve, 104 – condensation curve, 105 – definition, 13, 23 Industrial distillation processes, 407 – constraints, 407 – feasible dimensions of columns, 410 – feasible pressures, 409 – feasible temperatures, 407 Interfacial area – packings, 562–567, 570–572 – trays, 511, 513 Internals of packed columns, 541 Iso-fugacity condition, 30 Iteration history, 272 Jacobian matrix, 267

658

Latent heat of vaporization, 145 Limiting cases of binary mixtures, 104 Linde process, 418 Liquid collector, 544 Liquid distributor, 543 Liquid entrainment, 503–506 Liquid hold-up – influence on batch distillation, 337 – on trays, 499 – packed columns, 487, 549–552 – tray columns, 486 Liquid level control, 605 Liquid load of packings – minimum, 548 Liquid load of tray columns – maximum, 494 – minimum, 495 Liquid maldistribution – in packings, 558–562 – on trays, 508–511 Liquid mixing on trays, 506 Liquid redistributor, 543 Maldistribution of liquid – in packings, 558–562 – on trays, 508–511 Margules equation, 26, 50–52, 55, 57, 58, 86 Mass fraction – conversion, 8 – definition, 7 Mass transfer – efficiencies, 519–523, 526 – equipment, 481–586 – in Packed Columns, 568 – on trays, 518 Mass transfer coefficients – in packed columns, 570–573 – in tray columns, 524 Material balance, 182, 189, 208 Material coupling of columns, 368 Maximum gas load of trays, 490–492 Maximum liquid load – of tray columns, 494 McCabe–Thiele diagram, 85–92, 103, 106, 186 Membrane permeation, 467 MESH equations, 257, 261, 338 Middle vessel, 314, 341, 343, 346 Minimum energy requirement – saturated liquid feed, 249, 252 – saturated vapor feed, 249, 253, 254 – sharp high boiler separation, 253 – sharp low boiler separation, 251 – sharp preferred a/c-separation, 249

INDEX

Minimum gas load of trays, 492–494 Minimum liquid load of packings, 548 Minimum liquid load of tray columns, 495 Minimum reboil ratio, 237 Minimum reflux and reboil – binary mixtures, 198 – high boiler separation, 238 – low boiler separation, 231 – other separations, 243 – preferred a/c-separation, 228 – saturated liquid feed, 229, 236 – saturated vapor feed, 230, 236 – sharp separation of ideal binary mixtures, 203 – ternary mixtures, 224 Miscibility gap, 88 Mixing of liquid, 506 Model structures of packings, 538 – channel model structure, 540 – particle model structure, 540 Modifications of the general process, 368 Molar excess free energy – correlations, 49, 52 – definition, 24 – experimental data, 55 Molar mixing enthaply, 13 Molar mixing state variables, 12 Molar mixing volume, 13 Mole fraction – conversion, 8 – definition, 8 Multistage distillation – multicomponent mixtures, 255 Negative deviation from Raoult’s law, 89 Newton–Raphson algorithm, 262, 263, 265 Non-equilibrium models, 259 Non-steady state operation, 257, 260 NRTL equation, 51–53, 55, 57, 58 Number of equilibrium stages, 177 – binary mixtures, 195 Number of gas-phase transfer units, 178 Onda correlation, 571 Operating region – of packed columns, 486, 545–548 – of tray columns, 486, 489–497 Orifice coefficient, 514 Packed Columns, 484–486, 533–586 Packing – dumped, 534 – model structures, 538 – ordered, 535

659

INDEX

– random, 534 – structured, 535 Palm oil raffination, 428 Parallelism of vapor and liquid states, 234 Partial derivatives of MESH equations, 268 Partial molar excess free energy, 24 Partial molar state variables, 12 Particle model structure, 540 PC-SAFT, 12 Peng–Robinson equation, 11 Phase diagram, 81 Pinch point, 198, 224, 227, 231, 241, 242 Point efficiency, 519, 522 Pontryagin’s maximum principle, 323 Positive deviation from Raoult’s law, 87 Predictive Soave–Redlich–Kwong (PSRK), 12 Pressure control, 611 Pressure drop of packings – dry, 552–554 – irrigated, 554–557 Pressure drop of trays, 512 – bubble cap trays, 517 – dry, 514, 518 – sieve trays, 514–516 – valve trays, 517 Pressure swing processes, 436 Principle of distillation, 1 Process water purification, 421 Product concentration control, 613 Random packing, 534 Raoult’s law, 31, 48, 59 Rate-based models, 259 Ratio control loop, 604 Rayleigh equation, 153, 157, 167, 314 Reactions – decomposition, 302 – in ternary systems, 293, 295 – single product, 300 – with side products, 297 Reactive azeotrope, 289 Reactive distillation, 469 Reboil ratio, 184 – relationship with reflux ratio, 187 Rectification, 173 Reduction of energy requirement, 358 Reduction of exergy losses, 359 Reflux policy, 315, 318, 326 Reflux ratio, 184 – relationship with reboil ratio, 187 Relative volatility, 86, 107 – binary mixture, 60 – correlation, 59

– definition, 59 – ternary mixture, 61 Residue curve – liquid, 158, 160, 162 – vapor, 160, 162 Rigorous column simluation, 256 Rules split stream control, 606 Saturated liquid feed, 238, 254 Saturated vapor feed, 240 Saturation vapor pressure – Antoine equation, 14 – correlations, 13, 14 – parameters, 15 – Wagner equation, 14 Saturation vapor pressure curve, 105 Seawater desalination, 361 Separation factor (see relative volatility), 59 Separation paths, 429 Side column – a-path, 370 – c-path, 375 – preferred a/c-path, 377 Side columns, 370, 431 Side reactions, 306 Sieve trays, 487 Single control loop, 602 Single product reactions, 300 Slop cut, 340–341 Soave–Redlich–Kwong equation, 10, 34, 40 Split stream control, 606, 608, 610 Starting values, 272 Steady state operation, 257, 259 Steam distillation, 425 Stoichiometric lines, 284, 285 Structured packing, 535 Sulfuric acid, 412 Support grid, 544 Synthesis – of methyl-acetate, 474 – of mono-ethylene glycol, 471 – of MTBE, 469 – of TAME, 473 Ternary mixtures – feasible pure products, 222 Thermal column coupling, 386 – a-path, 387 – a/c-path, 389 – c-path, 387 Thermodynamic consistency check, 28–30, 56, 58 Topology of reactive distillation lines, 293

660

Topology of reactive distillation processes, 298 Total reflux and reboil – binary mixtures, 192 – ternary mixtures, 215 Transfer units, 522–526, 568, 576 – concept, 177–181 – total reflux, 194 Tray Columns, 482–484, 487–533 Tray efficiency, 258, 520 Tray-to-tray calculation, 208, 211 Two-phase flow – packed columns, 549–567 – trays, 487–514 Types of design – packed columns, 484

INDEX

– trays, 482 Underwood equation, 243, 244 UNIQUAC equation, 51, 52, 54, 55, 57, 58 Valve trays, 487 van Laar equation, 50–52, 55, 57, 58 Vapor–liquid equilibrium, 7–133 – basic correlations, 7 Virial equation, 9 Volume Translated Peng–Robinson (VTPR), 12 Wagner equation, 14 Wettability, 564 Wilson equation, 51–53, 55, 57, 58, 74