Encyclopedia of Chemical Processing and Design: Volume 16 - Dimensional Analysis to Drying of Fluids with Adsorbants 0824724666, 9780824724665

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Encyclopedia of Chemical Processing and Design

16

EXECUTIVE EDITOR

JOHN J. McKETTA The University of Texas Austin, Texas

ASSOC IATE EDITOR

WILLIAM A. CUNNINGHAM The University of Texas Austin, Texas EDITORIAL ADV ISORY BOARD

LYLE F. ALBRIGHT Purdue University Lafayette, Indiana

JAMES R. FAIR Professor of Chemical Engineering The University of Texas Austin, Texas

JOHN HAPPEL Columbia University New York, New York

ERNEST E. LUDWIG Ludwig Consulting Engineers, Inc. Baton Rouge, Louisiana

Encyclopedia of Chemical Processing and Design EXECUTIVE EDITOR AssociATE EDITOR

16

John J. McKett a William A. Cunnin gham

Dimens ional Analys is to Drying of Fluids with Adsorb ants

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

First published 1982 by Marcel Dekker, Inc. Published 2021 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1982 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an lnforma business No claim to original U.S. Government works ISBN 13: 978-0-8247-2466-5 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Library of Congress Cataloging in Publication Data (Revised) Main entry under title: Encyclopedia of chemical processing and design. Includes bibliographical references. I. Chemical engineering-Dictionaries. 2. Technical-Dictionaries. I. McKetta, John J. II. Cunningham, William Aaron. 660.2'8'003 TP9.E66 ISBN 0-8247-2451-8 (v. I)

Chemistry,

LIBRARY OF CONGRESS CATALOG CARD NUMBER: DOI: 10.1201/9781003209799

75 40646

75 40646

International Advisory Board

RAY C. ADAM Chairman of the Board N L Industries, Inc. New York, New York

LUCIANO BENINCAMPl Manager of Public Relations CTIP--Compagnia Tecnica Industrie Rome, Italy

M. A. ALLAWALA Managing Director National Refinery Ltd. Karachi, Pakistan

LLOYD BERG Professor Department of Chemical Engineering Montana State University Bozeman, Montana

HAMED H. AMER Chairman Agiba Petroleum Co. Cairo, Egypt R. G. ANTHONY Professor, Department of Chemical Engineering Texas A & M University College Station, Texas H. J. AROYAN Vice President Chevron Research Company Richmond, California F. SlD ASKARI President Technolog, Inc. Engineering and sultants Tehran, Iran

Industrial

NEIL S. BERMAN Professor of Engineering Engineering Center Arizona State University Tempe. Arizona D. J. BLICKWEDE Vice President and Director of Research Bethlehem Steel Corp. Bethlehem. Pennsylvania M. J. P. BOGART Fluor Engineers and Constructors, Inc. Santa Ana, California

Con-

AZARPOOR AZARNOOSH Bechtel National, Inc. San Francisco, California DONALD L. BAEDER Executive Vice President-Science and Technology Occidental Petroleum Corporation Los Angeles, California Wm. A. BAILEY. Jr. Former Director, MTM Process Research and Development Lab Shell Development Company Houston, Texas TRAVIS W. BAIN Vice President National Sales, Inc. Jackson, Mississippi GAREN BALEKJIAN C. F. Braun

Arcadia, California

CESAR BAPTISTE Vice President Petroleos Mexicanos Mexico City, Mexico

Z. D. BONNER Vice Chairman of the Board Tesoro Petroleum Corp. San Antonio, Texas JOSEPH F. BOSICH Consultant Humble, Texas ARCHIE BROODO President AID, Inc. Dallas, Texas WARREN B. BROOKS Manager, Computer and Telecommunications Department Vice President Mobil Telecom Inc. MobilOil New York, New York ARTHUR W. BUSCH Environmental Engineer Consultant Dallas, Texas ROBERT C. BUTLER Administrative Assistant and Planning Manager, Petroleum Chemicals Division E. I. du Pont de Nemours and Co. Wilmington, Delaware

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J. MORSE CAVENDER President The Mactan Company Dusseldorf, Federal Republic of Germany

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PRAMOTE CHAIYAVICH Chief Technologist The Tahi Oil Refinery Co., Ltd. Bangkok, Thailand

FRANK CHRENCIK Vulcan Materials Co. Birmingham, Alabama C. W. COOK Chairman, Executive Committee General Foods Corp. White Plains, New York EARL J. COUCH Research Associate Mobil Research and Development Corp. Dallas, Texas JAMES R. COUPER Professor Department of Chemical Engineering University of Arkansas Fayetteville, Arkansas HORACE R. CRAWFORD Senior Staff Scientist CONOCO Corp. Houston, Texas CALVIN S. CRONAN Editorial Director Chemical Engineering, A McGraw Hill Publication New York, New York ORAN L. CULBERSON Chemical Engineer Oak Ridge National Lab Chemical Technology Division Oak Ridge, Tennessee DONALD A. DAHLSTROM Vice President, Research and Development Process Equipment Group Envirotech Corp. Salt Lake City, Utah PERRY P. DAWSON Production Engineer Dow Chemical Co. Freeport, Texas ELBERT M. DeFOREST Director of Technology. Chemicals and Metals Vulcan Materials Co. Wichita, Kansas ROBERT G. DENKEWALTER Corporate Vice PresidentTechnology Allied Corp. Morristown. New Jersey J. P. de SOUSA Publisher Chemical Age of India Techmcal Press Publication Bombay. India

iii

iv

JAMES D. D'IANNI Former D~rector of Research The Goodyear Tire and Rubber Co. Akron, Oh10 JUAN M. DIAZ Producuon General Manager Rohm and Haas Mexico. S. A C. V Mexico CI!y, Mex1co WERNER DIMMLING Dipi-Chem1st Friedrich Uhde GmbH Dortmund, Federal Republic of Germany BARRETT S. DUFF Barrel! S. Duff and Associates South Pasadena, Califorma P. K. DUTTA Project Manager Chemical and Metallurgical Design Company, Pnvate Ltd. New Delhi, India WALTER EMRICH Consultant Teterboro, New Jersey E. FREDERICO ENGEL Member of the Board of Management Chemische Werke Hiils AG Marl, Federal Republic of Germany P. E. G. M. EVERS Operations Manager Anzo Salt Chemical Delfzyl Delfzyl, The Netherlands ALEXANDRE EVSTAFIEV Director, Div1sion of Technological Research and Higher Education UNESCO- Paris Paris, France GERALD l. FARRAR Editorial Director of Engineering The Petroleum Publishing Co. Tulsa, Oklahoma LOUIS FEUVRAIS Directeur General Ecole Nationale Superieure D' Arts et Metiers Paris, France R. A. FINDLAY Former Director, Fuels and Lubricants, Research and Development Phillips Petroleum Company Bartlesville, Oklahoma

International Advisory Board

ION GHEJAN Department of Chemical Engineering Institute of Petroleum, Gas. and Geology Bucharest, Romama B. GENE GOAR Goar, Arrington, and Associates, !ne Tyler, Texas MARCEL GOLDENBERG SAMIN Corp., Inc. New York, New York WILHELM GRAULICH Directbr, Manager, Rubber Divis1on Bayer AG Leverkusen, Federal Republic of Germany E. HENRY GROPPE Groppe, Long, & Littell Houston, Texas GIANFRANCO GUERRERI INGECO Altech Group Societa per Az10ni Con Sede m Milano Milan, Italy KENNETH M. GUTHRIE Cost Consultant Manna Del Rey, California NORMAN HACKERMAN PreSident Rice University Houston, Texas VLADIMIR HAENSEL Vice President, Science and Technology Universal Oil Products Co. Des Pla'ines, Illinois HENRY E. HALEY Vice President Arthur D. Little, Inc. Cambridge, Massachusetts

A. B. HORN, Jr. Director- Vice President Ethyl Corporation Baton Rouge, Louisiana NORBERT IBL Professor Eidg. Techn. Hochschule Ziinch Techn.-Chemie Ziinch, Switzerland RUBEN F. INGA Jefe Division Technica-Technical Director de Refineria La Pampilla Petroleos del Peru Lima, Peru JAMES R. JOHNSON Former Executve Scienctist and Director, Advanced Research Programs Laboratory 3M Company, Central Research Labs Saint Paul, Minnesota NAJI A. KADIR President Scientific Research Council Baghdad, Iraq JOHN E. KASCH Vice Pres1dent Standard Oil Indiana Escondido, California RAPHAEL KATZEN Managing Partner Ralph Katzen Associates Cincinnau, Ohio JOHN J. KELLY Department of Chemical Engineering University College, Dublin Dublin, Ireland 0. P. KHARBANDA 0. P. Kharbanda & Associates Cost and Management Consultants Bombay, India

R. l. HARVEL Project Manager Dow Chemical International Ltd. Tokyo, Japan

WLODZIMIERZ KISIELOW Professor of Petroleum Technology. Director of Research Department of Petroleum and Coal Centre of Polish Academy of Sciences Krzywoustego, Poland

J. W. HAUN Vice President and Director of Engineering General Mills, Inc. Minneapolis, Minnesota

MOHAN SINGH KOTHARI Chief Consultant Punjab Industrial Consultancy Organisation Chandigarh, India

RUSSEll F. HECKMAN Consulting Engineer Littleton, Colorado

A. P. KUDCHADKER Professor of Chemical Engineenng and Dean of Student Affairs lnd1an Institue of Technology, Kanpur Kanpur. India

WILLIAM B. FRANKLIN Former Technical Manager of Baytown Refinery Exxon Company USA Baytown, Texas

TERUAKI HIGUCHI President Japan Fody Corp. Osaka, Japan

DONALD E. GARRETT President Saline Processors Ojai, California

JOHN R. HILL Jr. President and Chief Executive Officer Gifford-Hill & Co., Inc. Dallas, Texas

L W. GARRETT. Jr. President Garrett Associates, Inc. San Mateo, California

HAROLD l. HOFFMAN Refining Editor Hydrocarbon Porcessing Houston, Texas

RALPH LANDAU Cha~rman

Halcon lnternauonal. Inc. New York. New York W. S.LANIER Project Manager Seadnft ExpansiOn Project' Umon Carb1de Corp Houston. Texa~

International Advisory Board

V

CLARK P. LATTIN, Jr. President The M. W. Kellogg Company Houston, Texas

GUY McBRIDE President Colorado School of Mines Golden, Colorado

MARCELLO PICCIOTTI Technical Promotion Manager TechniPetrol-Rome Rome, Italy

ISIDORO LAZARRAGA Chief of Engineering and Control Empresa Nacional del Petroleo Viiia del Mar, Chile

R. A. McKETTA Chemical Engineer Purvin & Gertz, !ne. Houston, Texas

THOMAS C. PONDER Petrochemicals Editor Hydrocarbon Processing Houston, Texas

JEAN Le BRETON Coordinator Research Development Society Nationales des Petroles d'Aquitaine (SNPA) Pau, France

CLYDE McKINELY Director, Allentown Labs Air Products and Chemicals, Inc. Allentown, Pennsylvania

R. G. H. PRINCE Professor, Head of Department Chemical Engineering University of Sydney Sydney, Australia

IRV LEIBSON Bechtel Corp. San Francisco, California PIERRE Le PRINCE Director of Refining and Engineering Center lnstitut Francaise du Petrole Malmaison, France C. E. LETSCHER Caltex Petroleum Company New York, New York C. J. LIDDLE White Young & Partners Ltd. Herts, England DAVID C. K. LIN Senior Engineer Owens Coming Fiberglass Corp. Newark, Ohio CHARLES E. LOEFFLER Technical Manager Celanese Chemical Company Pampa, Texas T. N. LOLADZE Vice-Rector, Professor of the Georgian Polytechnic lnstitue Tbilisi, USSR STANLEY L. LOPATA Chairman of the Board Carboline Company Saint Louis, Missouri PHILIPS S. LOWELL Consultant Austin, Texas W. D. LUEDEKE Former Planning Manager E. I. du Pont de Nemours Wilmington. Delaware BRYCE I. MacDONALD Manager, Environmental Engineering General Electric Company Fairfield, Connecticut R. N. MADDOX Sheerar Professor School of Chemical Engineering Oklahoma State University Stillwater, Oklahoma F. DREW MAYFIELD Drew Mayfield & Associates Baton Rouge, Louisiana

RICARDO MILLARES President Papel Satinado, S. A. Mexico City. Mexico ROBERT L. MITCHELL Vice Chairman of the Board Celanese Corp. New York. New York RICHARD MOLLISON General Manager Colpapel, S.A. Pereira, Columbia

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TAKAYUKI NATE Plastics Sales Department Tonen Petrochemical Co. Ltd. Tokyo, Japan JAMES K. NICKERSON Research Associate Esso Research and Engineering Company' Summit, New Jersey ALEX G. OB LAD Distinguished Professor of Chemistry Mining, and Fuels Engineering University of Utah Salt Lake City, Utah

Saint Louis, Missouri

LOUIS R. ROBERTS Director, Planning and Source Evaluation Texas Air Control Board Austin, Texas RICCARDO ROBITSCHEK Direttore Divisione Resine

Societa ltaliana Resine Milano. Italy

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GEORGE E. SCHAAL Manager, Research and Development Produits Chimiques Ugine Kuhlmann Pierre-Benite, France

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GERT G. SCHOLTEN Managing Director Edeleanu Gesellschaft mbH Frankfurt/Main, Federal Republic of Germany

H. E. O'CONNEL Former President Tenneco Chemicals Inc. Houston, Texas

vi MICHAEL W. SCHWARTZLANDER Engineer Union Carbide Corp. South Charleston, West Virginia M. L. SHARRAH Senior Vice President Continental Oil Company Stamford, Connecticut JOHN W. SHEEHAN Vice President, Manufacturing and Marketing Champlin Petroleum Company Kerrville, Texas PIERRE SIBRA Designer Esso Engineering Services Ltd. Surrey, England PHILLIP M. SIGMUND R. M. Hardy & Associates Ltd. Alberta, Canada ARTHUR L. SMALLEY, Jr. President Matthew Hall Inc. Houston, Texas CARL I. SOPCISAK Technical Consultant Synthetic Fuels Wheat Ridge, Colorado PETER H. SPITZ Chemicals Systems Inc. New York, New York JOSEPH E. STEINWINTER Personnel Senior Coordinator C. F. Braun & Company Alhambra, California SAM STRELZOFF Consultant Marlboro, Vermont V. S. SURY CIBA-Geigy Chemical Corp. Saint Gabriel, Louisiana T. SZENTMARTONY Associate Professor Technical University Budapest Budapest, Hungary M. TAKENOUCHI General Manager of Manufacturing Department Maruzen Oil Co., Ltd. Tokyo, Japan

International Advisory Board SOONTHORN THAVIPHOKE Managing Director S. Engineering Services Co., Ltd. Bangkok, Thailand ROBERT S. TIMMINS Core Laboratory Aurora, Colorado T. W. TOMKOWIT Manager-Logistics Section Freon Products Division E. I. du Pont de Nemours Wilmington, Delaware A. A. TOPRAC Interchem-Hellas Athens, Greece YORGIA.TOPRAKCLOGLU Chairman of the Board of Directors Marshall Boya ve Vernik Sanayii A. S. Istanbul, Turkey GOPAL TRIPAHTI Vice Chancellor Lucknow University Lucknow, India

JAMES D. WALL Gas Processing Editor Hydrocarbon Processing Gulf Publishing Company Houston, Texas J. C. WALTER, Jr. J. C. Waiter Interests Houston, Texas THEODORE WEAVER Director of Licensing Fluor Corporation Los Angeles, California ALBERT H. WEHE Chief, Cost and Energy U. S. Government Raleigh, North California GUY. E. WEISMANTEL Manager, PNA-Oiefins Dewitt & Company Houston, Texas

HERNANCO VASQUEZ-SILVA President Hernando Vasqez & Associates. Ltd. Bogota. Colombia

PAUL B. WEISZ Manger, Central Research Division Mobil Research and Development Corp. Princeton, New Jersey

M. A. VELA President VELCO Engineering, Inc. Houston, Texas

D. L. WILEY Senior Vice President Union Carbide Corp. Danbury, Connecticut

JUAN JOSE URRUELA VILLACORTA Ingeniero Fabrica de Jabon "La Luz, S.A." Guatemala

JACK C. WILLIAMS Former Vice President Texaco, Inc. Houston, Texas

S. P. VOHRA Managing Director Bakelite Hylam, Ltd. Bombay, India

MASAMI YABUNE Section Head, Technical Section Tonen Pertochemical Co., Ltd. Tokyo, Japan

A. L. WADDAMS Manager, Marketing Services Division BP Chemicals International Ltd. London, England

LEWIS C. YEN Manager, Technical Data M. W. Kellogg Company Houston, Texas

T. J. WALKER Production Manager Dow Chemical Europe S. A Ziirich, Switzerland

STANLEY B. ZDONIK Vice President and Manager Process Department Stone and Webster Engineering Corp. Boston, Massachusetts

Contributors to Volume 16

R. H. Baldwin Manager, Process Engineering, Amoco Chemicals Corporation, Chicago, Illinois: Dimethyl Terephthalate B. L. Bauman Mobil Research and Development Corporation, Princeton, New Jersey: Distillation Simulation W. L. Bolles Senior Engineering FeJlow, Monsanto Company, Saint Louis, Missouri: Distillation Adam T. Bourgoyne, Jr. Chairman, Petroleum Engineering Department, Louisiana State University, Baton Rouge, Louisiana: Drilling Practices D. B. Broughton UOP Process Division, Des Plaines, Illinois: Distillation, Estimates for Naphtha Cuts Kenneth M. Brown Director, Treating Services (Retired), UOP Process Division, Des Plaines, Illinois: Doctor Sweetening A. Chou Mobil Research and Development Corporation, Princeton, New Jersey: Distillation Simulation Ernest F. Cooke Professor, Department of Marketing, CoJlege of Business Administration, Memphis States University, Memphis, Tennessee: Drives, Adjustable Speed R. W. Ellerbe Project Manager, Rust International Corporation, Birmingham, Alabama: Distillation, Steam J. H. Erbar Professor of Chemical Engineering, Department of Chemical Engineering, Oklahoma State University, StiJlwater, Oklahoma: DistiJlation, Crude Stabilization James R. Fair Professor, Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas: Distillation Mobil Chemical Company, Houston, Texas: Distillation Simulation S. E. Gallun Exxon Chemicals Americas, Baytown, Texas: DistiJlation, Azeotropic and Extractive G. E. Hales Region Manager, LINDE Molecular Sieves Department, Union Carbide Corporation, Houston, Texas: Drying of Fluids with Adsorbants C. D. Holland Professor and Head, Department of Chemical Engineering, Texas A & M University CoJlege of Engineering. College Station, Texas: Distillation, Azeotropic and Extractive W. G. Kingsley (Deceased) American Hoechst Corporation, Leominster, Massachusetts: Divinylbenzene H. A. Leipold Group Project Manager, Amoco Chemicals Corporation, Chicago. Illinois: Dimethyl Terephthalate C. J. liddle Managing Director, White-Young Project Development, Herts. England: Distillation, Flash M. J. Lockett University of Manchester, Institute of Science & Technology, Manchester, England: Distillation, Azeotropic and Extractive

A. M. Fayon

vii

viii

Contributors to Volume 16

Professor, Department of Chemical Engineering, University of Southern California, Los Angeles, California: Distillation, Therma)ly Coupled

F. J. Lockhart

Leonard F. Sheerar Professor, Physical Properties Laboratory, Oklahoma State University, Stillwater, Oklahoma: Distillation, Crude Stabilization

R. N. Maddox

Professor, Chemical Engineering Department, The University of Texas at Austin, Austin, Texas: Dimensional Analysis

John J. McKetta W. T. Mitchell

Optimization

Celanese Chemical Company, Corpus Christi, Texas: Distillation,

Director (Research and Development) The Fertiliser Association of India, New Delhi, India: Distillery Wastes-Disposal and By-Product Recovery

V. Pachaiyapan

Research Manager, Petrochemicals Division, American Hoechst Corporation, Leominster, Massachusetts: Divinylbenzene

R. Partos

I. H. Silberberg Senior Lecturer, Petroleum Engineering Department, The University of Texas at Austin, Austin, Texas: Dimensional Analysis

Associate Professor, School of Engineering, Shiraz University, Shiraz, Iran: Distillation, Crude Stabilization

A. Shariat

Systems Development Consultant, The Foxboro Company, Foxboro, Massachusetts: Distillation Control Wm. P. Southard President and Executive Consultant, Wm. P. Southard & Associates, Broaddus, Texas: Drawings, Engineering F. G. Shinskey

Petrochemicals Division, American Hoechst Corporation, Leominster, Massachusetts: Divinylbenzene Waiter J. Stupin Manager, Process Engineering, C. E Braun Co, Alhambra, California: Distillation, Thermally Coupled L. Steib

Manager, Process Engineering, Allied Chemical, Marcus Hook, Pennsylvania: Distillation, Batch N. K. Tschirley San Antonio, Texas: Drilling Fluids

G. A. R. Trollope

UOP Process Division, Des Plaines, Illinois: Distillation, Estimates for Naphtha Cuts

K. D. Uitti

Contents of Volume 16

Contributors to Volume 16

vii

Conversion to SI Units

xi

Dimensional Analysis I. H. Silberberg and John J. McKetta

1

Dimethyl Terephthalate R. H. Baldwin and H. A. Leipold

38

Distillation W. L. Bolles and James R. Fair

42

Distillation, Azeotropic and Extractive C. D. Holland, S. E. Gallun, and M. J. Lockett

96

Distillation, Batch G. A. R. Trollope

134

Distillation Control F. G. Shinskey

141

Distillation, Crude Stabilization R. N. Maddox, J. H. Erbar, and A. Shariat

174

Distillation, Estimates for Naphtha Cuts D. B. Broughton and K. D. Uitti

186

Distillation, Flash C. J. Liddle

198

Distillation, Optimization W. T. Mitchell

224

Distillation Simulation A. Chou, B. L. Bauman, and A. M. Fayon

233

Distillation. Steam R. W. Ellerbe

258

Distillation, Thermally Coupled Waiter J. Stupin.and F. J. Lockhart

279

Distillery Wastes-Disposal and By-Product Recovery V. Pachaiyapan

299

Divinylbenzene W. G. Kingsley, L. Steib, and R. Partos

306

Doctor Sweetening Kenneth M. Brown

315

Drawings, Engineering Wm. P. Southard

338

Drilling Fluids N. K. Tschirley

345

Drilling Practices Adam T. Bourgoyne, Jr.

359

ix

Contents of Volume 16

X

Drives, Adjustable Speed

394

Drying of Fluids with Adsorbants

416

Ernest F. Cooke

G. E. Hales

Conversion to SI Units To convert from

To

Multiply by

acre angstrom are atmosphere bar barrel (42 gallon) Btu (International Steam Table) Btu (mean) Btu (thermochemical) bushel calorie (International Steam Table) calorie (mean) calorie (thermochemical) centimeter of mercury centimeter of water cubit degree (angle) denier (international) dram (avoirdupois) dram (troy) dram (U.S. fluid) dyne electron volt erg fluid ounce (U.S.) foot furlong gallon (U.S. dry) gallon (U.S. liquid) gill (U.S.) grain gram horsepower horsepower (boiler) horsepower (electric) hundred weight (long) hundred weight (short) inch inch mercury inch water kilogram force

square meter (m 2) meter (m) square meter (m 2) newton/square meter (N/m 2) newton/square meter (N/m 2) cubic meter (m 3 ) joule (J) joule (J) joule (J) cubic meter (m 3 ) joule (J) joule (J) joule (J) newtonjsquare meter (Njm 2) newton/square meter (N/m 2) meter (m) radian (rad) kilogram/meter (kg/m) kilogram (kg) kilogram (kg) cubic meter (m 3 ) newton (N) joule (J) joule (J) cubic meter (m 3 ) meter (m) meter (m) cubic meter (m 3 ) cubic meter (m 3 ) cubic meter (m 3) kilogram (kg) kilogram (kg) watt (W) watt (W) watt (W) kilogram (kg) kilogram (kg) meter (m) newtonjsquare meter (N/m 2 ) newton/square meter (N/m 2) newton (N)

4.046 X 10 3 1.0 x 10~ ' 0 1.0 X 10 2 1.013 X 10 5 1.0 X 10 5 0.159 1.055 X 10' 1.056 X 103 1.054 X 10 3 3.52 X 10~2 4.187 4.190 4.184 1.333 X 10 3 98.06 0.457 1.745 X 10~ 2 1.0 X 10~ 7 1.772 X 10~' 3.888 X 10~ 3 3."697 X 10~ 6 1.0 X 10~ 5 1.60 X IO~lo 1.0 X 10~ 7 2.96 X 10~ 5 0.305 2.01 X 102 4.404 X 10~' 3.785 X 10~ 3 1.183 X 10~ 4 6.48 x 10~ 5 1.0 X 10~ 3 7.457 X 10 2 9.81 X 103 7.46 X 102 50.80 45.36 2.54 X 10~ 2 3.386 X J0 3 2.49 X J02 9.806

xi

Conversion to SI Units

xii To convert from

To

Multiply by

kip knot (international) league (British nautical) league (statute) light year !iter micron mil mile (U.S. nautical) mile (U.S. statute) millibar millimeter mercury oersted ounce force (avoirdupois) ounce mass (avoirdupois) ounce mass (troy) ounce (U.S. fluid) pascal peck (U.S.) pennyweight pint (U.S. dry) pint (U.S. liquid) poise pound force (avoirdupois) pound mass (avoirdupois) pound mass (troy) poundal quart (U.S. dry) quart (U.S. liquid) rod roentgen second (angle) section slug span stoke ton (long) ton (metric) ton (short, 2000 pounds) torr yard

newton (N) meter/second (m/s) meter (m) meter (m) meter (m) cubic meter (m 3 ) meter (m) meter (m) meter (m) meter (m) newton/square meter (Njm 2) newtonjsquare meter (Njm 2 ) amperejmeter (A/m) newton (N) kilogram (kg) kilogram (kg) cubic meter (m 3 ) newton/square meter (Njm 2) cubic meter (m 3 ) kilogram (kg) cubic meter (m 3 ) cubic meter (m 3 ) newton second/square meter (N · sjm 2) newton (N) kilogram (kg) kilogram (kg) newton (N) cubic meter (m 3 ) cubic meter (m 3 ) meter (m) coulomb/kilogram (cjkg) radian (rad) square meter (m2) kilogram (kg) meter (m) square meter/second (m 2 js) kilogram (kg) kilogram (kg) kilogram (kg) newtonjsquare meter (Njm 2 ) meter (m)

4.45 X 10 3 0.5144 5.559 X 10 3 4.83 X 10 3 9.46 X 10 1 S 0.001 1.0 x 10- 6 2.54 x 10- 6 1.852 X !0 3 1.609 X 10 3 100.0 1.333 X !0 2 79.58 0.278 2.835 x 10- 2 3.11 x 10- 2 2.96 x 10- s 1.0 8.81 X 10- 3 1.555 X 10-' 5.506 X 10-4 4.732 X 10-4 0.10 4.448 0.4536 0.373 0.138 l.IOx 10- 3 9.46 X 10·-4 5.03 2.579 x 10- 4 4.85 X 10- 6 2.59 X 106 14.59 0.229 1.0 x 10- 4 1.016 X 10 3 1.0 X 10 3 9.072 X 10 2 1.333 X 10 2 0.914

Dimensional Analysis Dimensional analysis, or the principle of similitude or the method of dimensions, as it has variously been called, is by no means new. According to Murphy [18], the principle was first published by Dupre in 1869 in Theorie mecanique de la chaleur. In 1915, Rayleigh published an article dealing solely with dimensional analysis [25], but as early as 1892 he was applying the method to certain physical problems [24]. Buckingham's "Pi Theorem" [7] of 1914 was a very important contribution. Since then, Porter [22], Eshbach [10], Bridgman [5], Van Driest [26], Murphy [19], Langhaar [15], and others have written on the subject. All of these authors have done very creditable jobs in presenting the mathematical basis of dimensional analysis and examples of its use. It is not intended here to develop the mathematical theory but rather to present the basic concepts of dimensional analysis and some of their ramifications to illustrate the use of the Rayleigh and Buckingham methods and to show how dimensional analysis is used in planning experiments and developing engineering correlations. When only two variables are to be correlated, a simple plot of one against the other suffices as a correlation, a single curve resulting. When three variables are involved, their relationship constitutes a surface, and reduction to two dimensions requires one of the variables to be held at constant values. Consequently, the correlation would normally take the form of a set of curves, or a chart; each curve would be characterized by a constant value of one of the variables. To this point, correlation is a fairly simple matter. The addition of a fourth variable further complicates the experimental work and correlation, requiring a chart (set of curves) for each value of the fourth variable. Beyond this level of complexity, simple graphical correlation of the variables becomes virtually impossible and dimensional analysis a necessity. As a simple example, the hypothetical problem illustrated in Fig. 1 involves two forces F and P and two lengths X and Y. In an experimental investigation of the interdependence among these four variables, F would have to be held constant at F 1 while the variation of Y with X for various constant values of P(P = P 1 , P 2 , P 3 , etc.) was determined. ThenFwould have to be held constant at F2 while the variation of Y with X was again determined for constant values of P. The procedure would have to be repeated again and again until the range of values of Fhad been covered. Two such charts as would result are shown in

p

F

FIG. 1.

DOI: 10.1201/9781003209799-1

A simple problem.

1

2

Dimensional Analysis

X FIG. 2.

X

Part of the long solution.

Fig. 2. Application of the basic principle of dimensional analysis, however, would show that, since F and P are both forces and X and Y both lengths, correlation ofthe four variables as a single curve of FjP versus X/Y, such as Fig. 3, is sufficient. For such a correlation, no variable would have to be held at a constant value during the experimental work, and fewer determinations would be necessary. Such simple problems in engineering work are rare. A more typical engineering problem is the one involving pressure drop accompanying the flow of a fluid through a closed conduit. In this problem, seven variables are involved: pressure drop, fluid velocity, fluid density, fluid absolute viscosity, conduit length, conduit diameter, and conduit wall roughness. A general correlation of these seven variables in the form of charts such as those of Fig. 2 would be a tremendous task. Application of dimensional analysis to the problem reduces the number of quantities to be correlated from seven variables to four dimensionless groups of variables, and a consideration of the mechanics of fluid friction further reduces the number to only three dimensionless groups. A single chart then becomes sufficient for a general -correlation of the seven variables. The resulting saving of time in the experimentation and correlation and in the use of the correlation is considerable. In the simple problem of Fig. 1, the four variables might conceivably be arranged intuitively into the two dimensionless groups and correlated as such. The success of the correlation might then be considered a justification of the arrangement into dimension less groups. If the problem were being confronted for the first time, a rather remarkable intuition would be required to arrange the seven variables of the fluid friction problem into the necessary number of

X;.y FIG. 3.

Complete solution using dimensional analysis.

Dimensional Analysis

3

dimensionless groups. Not only will the method of dimensional analysis arrange the variables in any problem into a complete set of dimensionless groups, but the theory of dimensional analysis proves rigorously that correlation of such a complete set is entirely equivalent to correlation of the individual variables.

Dimensions and Dimensional Systems Dimensions and Units

An understanding of the nature of dimensions and units is essential to the understanding and use of the principles of dimensional analysis. The word "dimensions" as used here must be distinguished from the word "units." For example, time and length are dimensions. Seconds, days, and years are all units of time that bear fixed quantitative relations among themselves. Similarly, inches, centimeters, and miles are all units of length and of course also bear quantitative relations among themselves. Velocity is defined as the derivative of position with respect to time; consequently, it has the dimensions of length divided by time. However, velocity may be expressed in the units of feet per second, miles per hour, kilometers per minute, millimeters per year, etc. The units of measurement of mass, length, time, temperature, and electrical charge are prescribed by international agreement. Other units may be defined in terms of these standardized units of measurement. The definition may be arbitrary; for example, the international gram is defined as the thousandth part of the international kilogram. On the other hand, the definition may be based on a fundamental physical equation which governs the phenomenon in which the particular unit of measurement is involved. The internationally standardized unit of electrical charge, the coulomb, is defined in this manner. In general, units may be classified into three systems on the basis ofthe types of units of mass, length, and temperature used. In the English "fps" system, mass is expressed in pounds, length in feet, and temperature in degrees Fahrenheit. In the metric "cgs" system of units, mass is expressed in grams, length in centimeters, and temperature in degrees Centigrade. In the newer International System of Units ("SI"), the units are mass in kilograms, length in meters, and temperature in degrees Kelvin. Time is expressed in seconds in all three systems. In addition, special systems of units exist for use with special dimensional systems. Dimensional systems consist of primary (independent) dimensions and secondary (dependent) dimensions. In general, the primary dimensions are of such a simple nature that other dimensions (secondary) may be expressed in terms of them. As in the case of some units of measurement, the relation between the primary and secondary dimensions is generally established through the fundamental physical equation governing the phenomenon which the dimensions are to describe or through definitions. One absolute requirement for such a dimensional system, however, is that it be of such a nature that the dimensions selected as primary may be expressed in units that in some way

Dimensional Analysis

4

may be related to the internationally prescribed standards of mass, length, time, temperature, and electrical charge. Otherwise, the system would be useless in practice. Obviously then, time, length, and temperature are ideally suited as primary dimensions. Time and length are primary dimensions in most of the dimensional systems employed with descriptions of physical phenomena, and the dimension of temperature is also primary in most phenomena in which thermal effects are involved. The dimension of time is given the symbol e, the dimension of length the symbol L, and the dimension of temperature the symbol T. Newton's Law of Motion

The engineer is primarily interested in force, mass, length, time, and temperature, for these dimensions are capable of describing all of the quantities with which he must deal, with the exception of those encountered in work with electricity and magnetism. The fundamental physical law relating force, mass, length, and time dimensionally is Newton's Law of Motion, which may be written as Force= }(mass)(acceleration) where

f3 =Newton's proportionality constant

If length and time are chosen as two of the primary dimensions, the dimensions of acceleration are fixed; that is, acceleration will have the dimensions of (L/8)/(8), or Lje 2 . Three choices then remain on the basis of Newton's equation. Mass may be selected as a primary dimension and force as a secondary dimension. Conversely, force may be selected as a primary dimension and mass as a secondary dimension. In addition, it is possible to select both force and mass as primary dimensions. Consequently, there result the three systems of dimensions most often encountered by engineers. The Absolute System

In the absolute (dynamical, physical) system, mass is a primary dimension, with the symbol M, and force is a secondary dimension. In the English system of units, the primary unit of mass is the standard avoirdupois pound, the masspound. The unit of force is called the poundal and is defined as the force required to give one avoirdupois pound an acceleration of 1 ftjs 2 • In the metric "cgs" system, the primary unit of mass is the international gram, and the unit of force is called the dyne and is defined as the force required to give one gram an acceleration of 1 cmjs 2 • In the International System, the primary unit of mass is the international kilogram, and the unit of force is called the newton and is defined as the force required to give one kilogram an acceleration of 1 mjs 2 . For all three of these definitions, Newton's equation gives the same result.

5

Dimensional Analysis 1 . F orce = -(mass)(acceleratwn)

f3

1 1 =p(1)(1)

Therefore, f3 = 1. Consequently, for the absolute dimensional system employing these primary units of mass, length, and time, and the corresponding secondary units of force, Newton's equation becomes simply Force= (mass)(acceleration) The secondary dimension of force F may then be determined in terms of the primary dimensions.

Force therefore has the dimensions of M L/0 2 in the absolute dimensional system. In terms of the units of measurement, 1 poundal = 1 (pound)(foot)/(second) 2 1 dyne

= 1 (gram)(centimeter)/(secondf

1 newton

= 1 (kilogram)(meter)/(second) 2

The Gravitational System

In the gravitational or technical system, force is a primary dimension and mass a secondary dimension. Since there are no internationally prescribed standard units of force, however, recourse must be made to a standard mass only for the purpose of defining the primary units offorce. In the English system of units, the primary unit of force is the standard force-pound, which is defined as the force which will give the avoirdupois pound an acceleration equal to the standard acceleration of gravity, 32.1740 ft/s 2 . The secondary unit of mass is called the slug and is defined as the amount of mass which is accelerated 1 ft/s 2 when acted upon by one standard force-pound. In the metric system of units, the primary unit of force is the force-kilogram, which is defined as the force which will give the standard kilogram an acceleration equal to the standard acceleration of gravity 980.665 cm/s 2 or 9.80665 m/s 2 . The secondary unit of mass in the metric system has not been given a special name but is defined as the amount of mass which is accelerated 1 m/s 2 when acted upon by a force of one force-kilogram. For both the English and metric systems of units, Newton's law gives the same result. Force= .!._(mass)(acceleration)

f3

1 = .!._(1)(1)

f3

6

Dimensional Analysis

Therefore, f3 = 1. Consequently, for the gravitational dimensional system employing these primary units of force, length, and time, and the corresponding secondary units of mass, Newton's equation becomes Force = (mass)(acceleration) The secondary dimensions of mass may then be determined in terms of the primary dimensions. or Mass therefore has the dimensions of F8 2 /L in the gravitational dimensional system. In the English system of units, the slug has the units of measurement of (force-pounds)(secondsfj(foot), and the units of mass in the metric system are (force-kilograms)( second) 2 j (meter).

The Engineering System

In the engineering system of dimensions, both force and mass are primary dimensions. The obvious consequence is that Newton's proportionality constant becomes a dimensional constant, as is shown by the following: Force= .!_(mass)(acceleration)

f3

F

= ~(M)(L/8 2 )

Therefore, f3 has the dimensions of M LjF8 2 in the engineering system of dimensions. The numerical value of f3 follows from the definition of the sizes of the primary units of force and mass. As in the gravitational system, the primary unit of force must be defined in terms of a standard mass. In the English system of units, the primary unit of force is called the force-pound and is defined as the force which will give the avoirdupois pound an acceleration of 9o· where 9o = 32.1740 ftjs 2 ' the standard acceleration of gravity. The primary unit of mass is the avoirdupois pound, or the mass-pound. Equivalent definitions might be made for the metric system of units, but actually the metric system is not conventionally used with the engineering system of dimensions. According to Newton's equation, 1 . F orce = -(mass)(accelerat10n)

f3

1 1 force-pound= -(1 mass-pound)(32.1740 ft/s 2 )

f3

Therefore,

7

Dimensional Analysis

{3 = 32 _1740 (mass-pounds) (ft) (force-pound) (s) 2 Since for the engineering system Newton's proportionality constant {3 is numerically equal to the standard acceleration of gravity in feet/(second) 2 , the constant is usually denoted by the symbol g0 thereby distinguishing it from a local acceleration of gravity g or the standard acceleration of gravity g 0 . The importance of the difference denoted by the subscript "c" can hardly be overstressed. Whereas g and g 0 are true accelerations (the latter being a constant by international standards) with dimensions of Lj() 2 , the constant gc must be considered a true dimensional constant expressing the fundamental relation, as set forth by Newton's law of motion, between force, mass, length, and time expressed in the engineering system of dimensions. Newton's law for the engineering system may then be written as

. Force = -1 (mass)( acce 1era twn) gc where gc

= 32 _1740 (mass-pounds)(ft) (force-pound)(s) 2

The units in which force, mass, length, and time must be expressed in this equation are obviously those units which characterize this numerical value of gcOther Dimensional Systems

In order better to understand what really constitutes a dimensional system, a closer look at dimensions themselves is helpful. All physical quantities have dimensions. Because of the fundamental nature of some of these quantities, namely mass, force, length, time, temperature, and electrical charge, there is a tendency to think in terms of their dimensions. Furthermore, the sizes of their units are fixed by international agreement, thereby creating international standards of reference and comparison. Other physical quantities are defined in terms of these quantities of fundamental nature and consequently have dimensions and units which may be related to those of the fundamental quantities. In addition, such governing physical equations as may apply to a particular phenomenon also establish relations between these fundamental quantities. Such an equation is Newton's equation of motion, from which the units of force have been defined and the three common dimensional systems derived. However, when it is desired to create a dimensional system to be used in describing a certain phenomenon, the choice of primary dimensions need not be restricted to these fundamental dimensions. Convenience should be the determining factor in this choice. Certain dimensions may be arbitrarily designated as primary, with the obvious restriction that it must be possible to

8

Dimensional Analysis

derive from those dimensions selected as primary the dimensions of all the other quantities involved. The derivation may be made from the fundamental physical law (or laws) or from definitions of these quantities. When mass, length, and time were selected as primary dimensions in the absolute system, Newton's equation of motion established force as a secondary dimension and fixed the dimension of force in terms of those of the primary dimensions. When force was selected as primary rather than mass, mass became a secondary dimension in the gravitational system. Similarly, by definition the dimension of velocity is length divided by time. If length and time were both selected as primary, the dimension of velocity might logically be considered as secondary and described in terms of the primary dimensions oflength and time. However, velocity and time might be selected as the primary dimensions, and then the secondary dimension of length could be described in terms of the primary dimensions of velocity and time. In the same manner that the secondary dimensions are described in terms of the primary dimensions, the units of the secondary dimensions must be described in terms of the units of the primary dimensions. In the absolute system with English units, the unit of the secondary dimension of force has been given the name of poundal, but a poundal is in reality 1 (pound)( foot )/(second )2 . If length in feet and time in seconds are primary, the secondary dimension of velocity must have the units of feet/second. On the other hand, if velocity in yards/second and time in seconds were chosen as primary, the secondary dimension of length must have the units of yards. Obviously, a special name given to a unit of a secondary dimension is without significance; only the definition of the unit in terms of primary units is important. A number of dimensional systems based on Newton's equation of motion but differing from the three common systems are described in the literature. Brown [6] modified the gravitational system of dimensions to produce a system in which the primary dimensions are those of force, energy, time, and temperature. The usefulness of this system stems from the fact that the concept of energy as a product of a force and the distance through which that force operates is a fundamental one. The dimension of length is secondary and is defined as the dimension of energy E divided by the dimension of force F, or L = E/F. Since length is secondary, the unit oflength must be equal to one unit of energy divided by one unit of force. Mass is also a secondary dimension and is expressed in terms of the primary dimensions as M = F2 82 / E. One unit of mass must be equal to the product of one force unit squared and one time unit squared divided by one energy unit. If force were expressed in pounds (defined as for the gravitational system), energy in Btu's, and time in seconds, the unit oflength would be the Btu/pound, and the unit of mass would be the (pound) 2 (second) 2 /(Btu). If force were again expressed in pounds, energy in foot-pounds, and time in seconds, the unit of length would be the foot-pound/pound, or simply the foot, and the unit of mass would be the (pound) 2 (second) 2 /(foot-pound), or more simply the (pound)(second) 2 /(foot), which is the slug. In this latter case the system becomes identical in effect with the gravitational system. McAdams [16] proposed a dimensional system in which force, mass, time,

Dimensional Analysis

9

length, temperature, and energy are all primary dimensions. This system is basically the engineering system modified by the inclusion of the additional primary dimension of energy E. Since only four primary dimensions, including temperature, are required, two dimensional constants must be associated with this system. If the units of force and mass were defined as for the engineering system, one such constant would obviously be gc. The other is the proportionality constant in the equation defining energy as a product of force and distance, the dimensions of this constant being FLjE. This proportionality constant may not have the value of unity; in other words, one unit of energy must not be defined as the product of one unit of force and one unit of length. If force were expressed in force-pounds, mass in masspounds, length in feet, and time in seconds, a satisfactory unit of energy would be the Btu, and the corresponding dimensional constant would be the so-called mechanical equivalent of heat, J = 778 (feet)(force-pounds)/(Btu). Langhaar [15] discussed the astronomical system of dimensions, a system employed to describe phenomena in which two fundamental equations are applicable, Newton's laws of motion and gravitation. This system illustrates not only the importance of units but also the fact that dimensions need not have physical significance. The fact that two physical equations apply permits the selection of only two primary dimensions, length and time, to describe the four fundamental quantities of force, mass, length, and time. The secondary dimension of force has the dimensions of L 4 j8 4 , and the secondary dimension of mass has the dimensions of L 3 /8 2 • The astronomical unit for force is called the "asf," and the astronomical unit of mass is called the "asm." These two units are so defined that, in the English system of units, 1 asf = 1 mi 4 /s 4 1 asm = 1 mi 3 /s 2 Obviously, then, the primary units of length and time are miles and seconds, respectively. Similar relations exist in the metric system of units, in which length is expressed in kilometers and time in seconds, but of course the metric asf and asm are units that differ in size from the English asf and asm.

The Correct Use of Dimensional Systems The correct use of a dimensional system requires the proper expression of the dimensions of all the quantities involved in terms of the primary dimensions of that system. In addition and of equal importance is the requirement that, if the system involves specially defined units of measurement, these units and no others must be used. If certain dimensional constants, such as gc or J, are necessitated by the dimensional system used, these must also be taken into proper consideration. The last and most obvious condition is that the dimensional system to be used must be capable of describing dimensionally all of the quantities involved in the problem.

10

Dimensional Analysis TABLE 1

Characteristics of Absolute, Gravitational, and Engineering Systems

System

Primary Dimensions

Units of Force

Units of Mass pound, gram, kilogram kilogram-second 2 slug, meter pound

Absolute

MLOT

poundal, dyne, newton

Gravitational

FLOT

pound, kilogram

Engineering

FM LOT

pound

The three dimensional systems with which the engineer is most frequently in contact are the absolute, gravitational, and engineering systems. The absolute system is most commonly used in physics. The gravitational system is used in various fields of engineering. The engineering system is commonly employed in chemical and petroleum engineering and in engineering thermodynamics. In addition to the primary dimensions determined on the basis of Newton's law of motion, all three of these systems also employ temperature (symbol, T) as a primary dimension. The main features of these systems are summarized in Table 1. The dimensions of some common engineering quantities in the three systems are given in Table 2. The following example illustrates how secondary dimensions may be expressed in terms of the primary dimensions of each system. Example 1: Expression in Terms of Primary Dimensions. (a) Energy E is defined as the product of a force and a distance. E=FL

(1)

Absolute system: F = M LjfP

:. E=(MLjB 2 )(L)=ML 2W

Gravitational system: E = F L (3) Engineering system : E = F L

(2)

(b) Absolute viscosity, J-1, is defined as the proportionality constant between the shear stress and the velocity gradient, or mathematically: Shear stress

= J-l(velocity gradient)

or force dx J-1=---­ area dV

Dimensionally,

11

Dimensional Analysis (1)

Absolute system: F = M L/8 2 (ML/8 2 )(8) :. J1. = (L2) = MjL8

(2) (3)

Gravitational system: 11. = F8jL 2 Engineering system: 11. = F8jL 2 or, since M is also primary, 11. = MjL()

(c)

Specific heat c is defined most simply by Heat absorbed

= c(mass)(temperature increase)

or (heat absorbed)

c = -----'----------'----(mass)(temperature increase)

Dimensionally,

c = EjMT

TABLE 2

Some Quantities and Their Dimensions in Three Systems Name of System

Primary dimensions Length Time Mass Force Newton's constant, fJ Pressure, p Mass velocity, G Linear velocity, V Linear acceleration, a Linear momentum Angular velocity, w Angular acceleration, a Angular momentum Energy, work, E Energy junit mass Power, P Density, p Surface tension, a Absolute viscosity, 11. Absolute viscosity, 11. Kinematic viscosity, v = 11./p Specific heat, c Thermal conductivity, k Heat transfer film coefficient, h Diffusivity, Dv

Absolute

Gravitational

Engineering

M,L,e, T L e M MLj8 2 None MjL8 2 MjL 28 L/8 Lj()2 MLj()

F,L, 8, T L e F8 2 /L F None FjL 2 F8jL 3 LJe Lj()z Fe

F, M, L, e, T L e M F MLjF8 2 F/L 2 MjL 2 8 Lj() L/8 2 ML/8

lj82

lj82

FL() FL Lz;e2 FL/8 F8 2/L 4 FjL F8jL 2

ML 2/8 FL FLjM FL/8 MjL 3 FjL Fe;u MjL() L 2 /8 FLjMT Fj8T FjL8T L 2 /8

1/8 1/82

ML 2/8 ML 2j8 2 L2/82 ML 2 j8 3 MjL 3 M/82 MjL8 L 2 /8 L 2 j8 2T MLj8 3 T Mj8 3 T L 2 /8

1/8

u;e L 2 j8 2 T Fj8T FjL8T L 2 /8

1/8

12

Dimensional Analysis (1)

(2)

Absolute system: E = FL =M L 2 j8 2 (ML 2 j8 2 ) · c= = L 1 j8 2 T (MT) Gravitational system: E = F L and M

= F8 2 j L

· c= (FL) = L 2WT .. (F8 2 jL)(T) (3)

Engineering system: E :. c = FLjMT

= FL

For the proper use of dimensional systems, care should be taken to adhere to a complete system and not to mix systems. One of the greatest sources of error in this respect lies in the tendency to mix units of different systems. For example, the English system unit of mass for the gravitational system is the slug. The slug is defined as the amount of mass accelerated 1 ft/s 2 by 1 force-pound, which is in turn defined as the force which will give the standard avoirdupois pound of mass an acceleration of 32.1740 ft/s 2 • Clearly the slug is 32.1740 times as large as the mass-pound. Knowledge of this relationship is useful and necessary, for example, to convert a density in pounds/cubic foot to a density in slugs/cubic foot. However, the mass-pound is otherwise alien to the gravitational system. To speak of a mass-pound and a force-pound simultaneously implies the engineering system of dimensions, which requires the use of the dimensional constant gC' Equations derived for a one dimensional system are not always correct for a different system. Example 2 illustrates this point.

The Bernoulli Equation in Three Systems of Dimensions. Figure 4 illustrates the situation in which a fluid is flowing under reversible conditions in a conduit between points "1" and "2" such that no mechanical work is done. If the fluid is incompressible and the flow is isothermal, the flow equation takes on a form frequently called the Bernoulli equation. In its general form, the Bernoulli equation may be written as

Example 2:

~(kinetic

energy)+

~(potential

energy)+ vmbP = 0

The basis of each term is m units of mass; vm is the volume of m units of mass. The symbol ~indicates the value of the property at condition "2" minus that at condition "1."

Kinetic Energy: When a force F is applied to a mass m through a small increment of distance dsin such a manner that only the kinetic energy of the mass is changed, the work done is equal to the change in kinetic energy, d(K.E.), of the mass: d(K.E.)

= F ds

But F = (1//3) ma, and the acceleration a may be expressed in terms of the velocity Vas dV dsdV dV a=-=--=Vdt dtds ds

13

Dimensional Analysis

FIG. 4.

A nonwork flow system (Example 2).

Therefore, 1 d(K.E.) =pm V d V

Integrating between the limits of velocity = V 1 and velocity= V 2 , the change in kinetic energy of the mass m is expressed by ~{K.E . ) =m~( VZj 2{J)

Potential Energy : When a constant force F is applied to a mass m through a small increment of distance dZ in such a manner that only the potential energy of the mass relative to a reference plane is changed, the work done is equal to the change in potential energy, d{P.E.), of the mass: d(P.E.) = F dZ

But F = (1/ fl)ma, and the acceleration in this case is that of gravity, g. Therefore, 1

d(P.E.) = pmgdZ Integrating from Z 1 to Z 2 and assuming g to be constant, the change in potential energy of the mass m is given by ~{P.E.)

=

g

m - ~Z

fJ

Bernoulli's equation may then be expressed in the form

This equation may be written as an energy balance per unit of mass, rather than for m units of mass, by division of each term by m . In this case, vmfm = v, a specific volume.

14

Dimensional Analysis

This form of Bernoulli's equation is general insofar as dimensional systems are concerned; any system may be used with it so long as f3 is correctly evaluated. For the three dimensional systems being considered here, this equation takes the forms shown below. Absolute and gravitational systems: ~(V2 /2)

+ g~Z + v~p =

0

Engineering system :

When English units are employed, g,

(mass-pounds)(ft)

= 32.1 740 -'----.:__---=(force-pound)(s)2

for the engineering system equation, and the other quantities have the units presented in Table 3.

The resulting equations of Example 2 are quite similar, a fact that no doubt accounts for most of the misuse of the Bernoulli equation. The forms of the equation for the absolute and gravitational systems are identical, but the units of force and mass are different. The equation for the engineering system differs by the intrusion of g, and also by the units of force and mass required. Through the use of the dimensions given in Table 2, it can easily be shown that the dimensions of each energy term, representing energy per unit mass, are L 2 /8 2 in both the absolute and gravitational systems and FL/M in the engineering system.

TABLE 3

Units Commonly Used with the Bernoulli Equation (Example 2) Quantity

Absolute System

Velocity, V Acceleration of gravity, g Elevation above reference plane, Z Pressure, p Specific volume, v

ft/s ft/s 2 ft poundalsjftl ft3jpound

Gravitational System ftjs

ft/s 2 ft poundsjftl ft 3 /slug

Engineering System ft/s ft/s 2 ft force-poundsjft 2 re /mass-pound

15

Dimensional Analysis

In English units, energy per unit mass in the absolute system has the units of (feet)(poundals)j(pound), and in the gravitational system the units of (feet)(pounds)j(slug). In terms of primary units only, energy per unit mass has the units of (feet) 2 /(second) 2 in both these systems. In the engineering system, the English units are (force-pounds)(feet)/(mass-pound). The Bernoulli equation is often seen written as

and it is a common practice for engineers to refer to the terms in the equation as having the units of "feet of head." This peculiar appearance ofthe acceleration of gravity in the kinetic energy term and absence in the potential energy term led Lord Rayleigh more than 60 years ago to observe: "When the question under consideration depends essentially upon gravity, the symbol of gravity (g) makes no appearance, but when gravity does not enter into the question at all, g obtrudes itself conspicuously" [25]. Although the poundal is so defined that one force-pound is equal to g 0 poundals, and the slug is so defined that one slug equals g0 mass-pounds, this form of the equation may not be justified as a conversion either of poundals to force-pounds in the absolute system or of slugs to mass-pounds in the gravitational system. Force-pounds and mass-pounds are alien, respectively, to the absolute and gravitational systems. Furthermore, g 0 and g are not necessarily numerically equal, although their difference is usually quite small. Neither may this form of the equation be justified for the engineering system, for even though g and 9c might happen to be numerically equal, they are not dimensionally equivalent, and one may not be substituted for the other in the Bernoulli equation. However, one mass-pound does weigh (exert a force of) g poundals in the absolute system, and one slug weighs g force-pounds in the gravitational system. Furthermore, one mass-pound weighs gfgc force-pounds in the engineering system. Consequently, division of each term in the equations for the absolute and gravitational systems by g and for the engineering system by gfgc represents a conversion of the terms from a mass-basis to a weight-basis. The Bernoulli equation may then be correctly written for all three systems as A( V 2 j2g)

1 w

+ AZ + -Ap = 0

where w is the specific weight, the weight of a unit volume. The dimensions of each term in this equation are energy per unit weight, or length. Fortunately, one pound of weight and one pound of mass represent very nearly the same quantity of matter in fluid flow calculations, so the numerical results from the erroneous use of the Bernoulli equation do not as a rule differ appreciably from the results of the correct use. This example does, however, serve to illustrate the importance of the correct use of dimensional systems and the possible confusion and errors that may result from the careless use of these systems and their units.

16

Dimensional Analysis

Dimensional Homogeneity

An equation is said to be dimensionally homogeneous if each term has the same resultant dimensions. Such an equation is applicable regardless of the units used, provided that the units are compatible with the dimensional system employed with the equation. For example, for a body falling from rest in a vacuum and under the acceleration of gravity, the velocity V at elapsed time eis given by V=g8

This equation is dimensionally homogeneous and requires two dimensions, conventionally those of length and time. The acceleration of gravity g may be expressed in feetfsecond 2 , time in seconds, and velocity in feet/second. Similarly, g may be expressed in centimetersfyear 2 , time in years, and velocity in centimetersfyear. On the other hand, V = 32.28 is not dimensionally homogeneous and is applicable within the vicinity of the earth's surface only when time is expressed in seconds and velocity in feet/second. The fundamental equations of physics, and all equations derived therefrom, are dimensionally homogeneous. It is upon this fact that dimensional analysis is based, for the fundamental hypothesis of dimensional analysis is that the solution to a certain problem is expressible in the form of a dimensionally homogeneous equation involving the variables concerned in the problem. As a consequence, any sort of relation obtained through dimensional analysis will be dimensionally homogeneous. In fact, it is the purpose of dimensional analysis to arrange the variables of a problem into sets of products which are themselves dimensionless. These dimension less products or groups, as they are often called, may then become the arguments of any type of function, polynomial, trigonometric, or expontial, without violating the fundamental hypothesis of a dimensionally homogeneous solution to the problem. In general, any equation which is not dimensionally homogeneous should be used with caution, for the equation is restricted somehow in its application. Perhaps only certain sizes of units will give the correct numerical results, or perhaps it applies only under certain limited conditions. It is always wise to know the background of a dimensionally nonhomogeneous equation before using it. Example 3 illustrates how the dimensional homogeneity of an equation may be checked.

Example 3:

Checking for Dimensional Homogeneity. Check the following equation for dimensional homogeneity in the absolute, gravitational, and engineering system of dimensions:

17

Dimensional Analysis where

a = dimensionless constant

L =length

p =density g =acceleration of gravity

A. k J1 .1. t

= latent heat of vaporization per unit mass =thermal conductivity = absolute viscosity = temperature difference

From the dimensions given in Table 2, for the absolute system: [L3pzgA.] = (L)3(M/L3)2(L/Bz)(Lz;ez) = Mz;e4 [rxkJ1M] = (l)(MLj8 3T)(MjL8)(T)

= M 2j8 4

for the gravitational system: [L3pzgA.] = (L)3(F8z/L4)z(Lj8z)(Lzjez)= pz;Lz [rxkJ1.1.l] = (l)(Fj8T)(F8jL 2 )(T)

= F2 jLz

for the engineering system: [L 3p 2 gA.] = (L) 3(M/L 3)2 (Lj8 2 )(FL/M) = MF/L8 2 [~kJ1.1.t] = (l)(F/8T)(F8/L 2 )(T)

= PjL 2

It can be seen that the equation as given is dimensionally homogeneous in the absolute

and gravitational systems of dimensions but not in the engineering system. The reason for this inhomogeneity lies in the dimensions of viscosity used, FB/U. For viscosity expressed in such units, the equation must be written as

for which the dimensions of each side are PjL 2 (same as gravitational) or as

for which the dimensions of each side are M F/L8 2 . It should be apparent that gcJl is nothing other than viscosity expressed in the dimensions M/LB, shown in Table 2 as an alternative to FBjL 2 in the engineering system. In fact, many relations intended for use with the engineering system of dimensions require viscosity to have the dimensions of MjL().

Methods of Dimensional Analysis The Rayleigh Method

This method of dimensional analysis was described in the literature as early as 1892, when Lord Rayleigh applied the technique to the problem of pressure drop due to friction in fluid flow [24], but it was not until 1915 that Rayleigh

18

Dimensional Analysis

published an article concerned primarily with the principles of his method [25]. Rayleigh's method is based upon the premise that, ifn quantities Q1 , Q2 , Q3 , ... , Qn are involved in a certain physical phenomenon, for the purpose of the dimensional analysis their mutual dependence may be expressed as a power product of the following type: (1)

where K is a dimensionless constant. In this equation, Q1 might be construed to be the quantity which is of principal interest, although such an interpretation is entirely unessential to the method. The set of Q's would include all ofthe variables known to enter into the particular phenomenon and, in addition, all dimensional constants either demanded by the dimensional system employed or otherwise known to be involved. The requirement of dimensional homogeneity places some restrictions upon the values that then - 1 exponents a 2 , a 3 , • •• , an may have. If the dimensional system required to describe completely the n variables and dimensional constants consists of r primary dimensions, then there exists a maximum of r conditions which the constant exponents of Eq. (1) must satisfy. The word "maximum" is used advisedly, for in some cases, depending upon the dimensional system used or upon the dimensional nature of the quantities involved, two or more of the r conditions may in effect be identical, thereby reducing the number of actual conditions to a value less than r. Consequently, of the n - 1 exponents, a minimum of n - 1 - r are not restricted by the requirement of dimensional homogeneity. The final result of dimensional analysis by the Rayleigh method is an arrangement of the n quantities into such a form that a dimensionless product or group containing Q1 is equated to the product of a minimum of n- 1 - r other dimensionless groups, each raised to the power represented by one ofthe n - 1 - r unrestricted exponents. As a consequence, there results an arrangement of the n quantities into a minimum of n - 1 - r + 1 = n - r dimensionless groups. Example 4: The Simple Pendulum. It is desired to obtain a relation for the period (time of swirlg) of a simple pendulum operating in a vacuum. The length of the pendulum and the acceleration of gravity are known to be variables. The opinion has also been advanced that the mass of the pendulum may affect its period. Using the engineering system of dimensions and including the dimensional constant Uc• the quantities and their dimensions are given by the following:

Period = t (8) Length= l(L) Acceleration of gravity= g(L/8 2 ) Mass= m( M) Dimensional constant= Uc(M L/F0 2 )

Dimensional Analysis

19

Combining these quantities in the manner indicated by Eq. (1), using a, b, c, and d as the constant exponents, there results the following equation: t

= Kl"lm'g/

(2)

The substitution of the engineering system dimensions ofthe variables into Eq. (2) gives (3)

The requirement of dimensional homogeneity indicates that the algebraic sum of the exponents of each ofthe primary dimensions F, M, L, and 0, respectively, on the left side of Eq. (3) must equal that on the right side. This requirement may be expressed symbolically as IF= 0, IM = 0, IL = 0, and IO = 0.

IF

=0:

IM=O:

IL IO

0= -d

O=c+d

=0:

O=a+b+d

=0:

1 = -2b- 2d

The solution to this system of equations is a = related by the equation

t and b = -t. The variables are then (4)

or t(g/1)112 = K

(4a)

This arrangement of the variables of this problem in Eq. (4a) is called a dimensionless group or dimensionless number. In this simple example, neither the mass of the pendulum nor the dimensional constant g, appears in the final solution. An inspection of the other variables offers an explanation. The dimension F appeared nowhere in the quantities excepting,; consequently, g, was dimensionally noncombinable with the remaining variables. Similarly, the dimension M appeared only in the variable m and in g,, which was itself excluded from the final result by virtue of the dimension F. Consequently, neither the mass of the pendulum nor the engineering system dimensional constant are involved in the equation for the period of a simple pendulum in a vacuum. Only when a quantity is dimensionally noncombinable with the other quantities will it not appear in the final equation obtained by dimensional analysis. The question of when the dimensional constant g, should be included among the quantities being related is not an easy one to answer. Obviously it is required only when the engineering system of dimensions is being used. If the dimension F and the dimension M do not both appear in the dimensions of the quantities, then g, is not required, for its purpose in reality is merely to establish the dimensional relationship between Fand M. Conversely, if the dimensions of

20

Dimensional Analysis

one variable contain the dimension F and those of a different variable contain the dimension M, and the dimensions of none of the other variables contain either M or F, then gc is required for a relation between these two variables to be possible. Beyond these simple cases, it is impossible to generalize. However, in all problems in which the engineering system of dimensions is being used, the dimensional constant gc may be included as a precaution without prejudicing the results of the analysis. If gc is required in the problem, it will appear in the results; if it is not required, as in Example 4, it will not appear in the results. Another important point illustrated by Example 4 is the fact that it is not necessary to calculate how many dimensionless groups are to be expected from the analysis. Obviously, had this been done before the analysis, since there were five variables assumed (n = 5) and four dimensions were necessary to describe them (r = 4), one dimensionless group would have been predicted (n- r = 1). The fact that only one dimensionless group t(g/1) 112 was actually obtained was coincidental; the inclusion of two quantities that were not involved in this case required the use of two more primary dimensions than otherwise would have been needed ( F and M). A two-dimensional system involving only L and (} is adequate for this problem to describe the three variables t, l, and g. Consequently, n- r = 3- 2 = 1, and the dimensionless group t(g/1) 112 was the only one obtained.

Pressure Drop by Friction in Fluid Flow. When a fluid is flowing in a straight length L of a pipe with internal diameter D, a pressure drop - ~ p 1 occurs as a result of friction. The variables believed to be involved and their dimensions in the engineering system are given below.

Example 5:

Pressure drop= -~p 1 (F/L 2 ) Pipe internal diameter= D(L) Pipe length = L ( L) Pipe roughness = e( L) Fluid linear velocity= V(L/8) Fluid absolute viscosity= J1(MjL8) Fluid density= p(MjL 3 ) Dimensional constant = gc( M LjF8 2 ) An inspection of the dimensions of these quantities indicates that all are dimensionally combinable and should appear in the result of the dimensional analysis. Therefore, since there a're 8 quantities and 4 primary dimensions, a minimum of 3 (n - r - 1 = 3) unrestricted exponents and a minimum of 4 (n- r = 4) dimensionless groups may be expected. The basic equation relating the variables is (5)

The corresponding dimensional equation is

21

Dimensional Analysis

(6)

Applying the condition of dimensional homogeneity, there results the following set of equations:

"i.F = 0:

1 = -j

IM=O:

O=e+h+j

l:L

=0:

- 2

L:e

=0:

0 = -d- h- 2j

=a+b+c +d-

3e - h + j

(6a)

No two of these four equations are identical, nor may any be obtained from a linear combination oftwo others. Therefore, the number of conditions is the maximum (r = 4 ), and the minimum number of dimensionless groups may be expected (n - r = 4). Since there are four independent equations relating the seven constants, the values of four of the constants may be determined in terms of the remaining three. Letting these three unrestricted constants be b, c, and h, the solutions to Eq. (6a) then become

a= -b-c-h d=2-h

(6b)

e=1-h j= -1 The insertion of the solutions represented by Eq. (6b) into Eq. (5) results in the expression (7)

or (7a) The seven variables plus g, are now correlated in the form of four dimensionless groups. From theoretical considerations, it can be shown that the pressure drop-~p1 must be directly proportional to the pipe length L ; consequently, the exponent b must have the value of unity. The dimensionless group g,( -~p1 )jp V 2 is called the Euler number, N Eu· The dimensionless group D V pjJ1 is called the Reynolds number, N Re· Both of these numbers have general application in fluid flow. The dimensionless group ejD is called the roughness factor of the pipe. For reasons which will be discussed briefly later, in place ofEq. (7a) with b = 1 there may be written the more general functional equation

(8) If the function (NRe• ejD) is given the symbol f/2, Eq. (8) may be written as (8a)

22

Dimensional Analysis The function of the Reynolds number and the roughness factor denoted by the symbol! is called the friction factor. Moody [17] presented a chart that permits evaluation off when the Reynolds number and the roughness factor are known. (See Fig. 5.)

In Example 5, the exponents b, c, and h were purposely selected as the unrestricted exponents in order to obtain the results in the conventional form. Other three exponents might have been so selected, with the obvious exception of j, which must have the value of- 1 for this example. Inspection of the equations for l:M and 2:8 also shows that, of the exponents d, e, and h, no more than one may be selected as an unrestricted exponent. For example, the selection of a, b, and d as the unrestricted exponents would have resulted in the following four dimensionless groups :

e D'

L D'

eVp 11

The appearance of e, a quantity which is difficult to determine with exactness, in three of these four dimensionless groups is undesirable. If eV pjp. is divided by ejD, the Reynolds number DV pjp. results. If the group g,( -tJ.p1 )e 2 pjp. 2 is divided by (eV p/p.) 2 , the Euler number g,( -!J.pf)/p V 2 results. In this manner the results of dimensional analysis may be converted from dimensionless groups that are not desired to groups that are more convenient and desirable. The usefulness of such conversions will be discussed later. The transformation of (DV pjp.)-h(ejD)' in Eq. (7a) to an unspecified function ofthese two dimensionless groups tl>(NRe• ejD) in Eq. (8) is a definite feature of the Rayleigh method. Rayleigh [25] pointed out that solutions of the

a:: 0 t-

u

N

CD

~ ..,

0 "0

(;'

I»

:;,

c.

..

m

.:c· )(

iil(')

CD

Distillation, Azeotropic and Extractive TAB LE 7

127

Antoine Constants [20] log 10 P; =A;-

TABLE 8

( P; in mmHg, Tin

T)

oq

Component

A;

B; x 10- 3

C; x 10- 2

Methanol Acetone Ethanol Water

7.87863 7.02447 8.04494 7.96681

1.47311 1.16000 1.55430 1.66821

2.3000 2.24000 2.22640 2.28000

Wilson Parameters [20] A;j- ).il

Component

(methanol)

(methanol) 2 (acetone) 3 (ethanol) 4 (water)

0.0

TABLE 9

(C;:

(caljg-mol)

2 (acetone) 0.66408

10 3

X

-0.21495

X

10 3

0.0

0.51139

X

!0 3

0.41896

0.48216

X

!0 3

0.140549

3 (ethanol)

10 3

X

X

4 (water)

0.59844

X

10 3

0.20530

X

10 3

0.38170

X

10 2

0.43964

X

10 3

0.38230

X

10 3

0.0

10 3

0.95549

X

10 3

0.0

Molar Volume Constants• ex; =a;

Component Methyl alcohol Acetone Ethanol Water

(ex; in cc/g-mol, Tin oR)

b;

a;

0.6451094 0.5686523 0.5370027 0.2288687

+ b; T + c; T 2

X

X X

X

10 2 10 2 10 2 10 2

-0.1095359 0.468039 x 10- 2 -0.1728176 x w-' -0.2023121 x w-'

C;

0.1195526 0.5094978 0.4938200 0.2115899

"These curve fits are by Gallun [8] who based them on data taken from Holmes and Van Winkle [20].

x w- 3 x w- 4 x 10- 4 x

w- 4

128

Distillation, Azeotropic and Extractive

TABLE 10

Solution Values of the Product Flow Rates for Examples I and 2

Component

di,l

Methanol Acetone Ethanol Water

8.9252 8.4422 1.1318 3.6008

Methanol Acetone Ethanol Water

5.0092 24.9999 0.3614 1.7297

Example 1 : Product Flow Rates (moljh) 98.9252 68.3395 16.5579 16.5577 12.0196 3.34823 157.4972 1.75426 Example 2: Product Flow Rates (moljh) 64.9705 74.6729 0.1981 X J0- 4 0.1981 X J0- 4 4.8876 4.6227 0.7042 205.1428

1

Example No. 1

2

45.5857 0.0001267 8.6713 160.7429 5.2976 0.0 0.2648 209.4387

Temperature (°F) of Steam Indicated

Example No. 2

b;, 2

d,, 2

bi,l

138.38 132.07

159.14 176.37

Column Modular Method

141.52 151.13

177.04 203.43

Almost Band Algorithm

No of Trials

Time (s)

No of Trials

Time (s)

14 (at B 1 = 277) 70 (at B 1 = 275)

4 25.3

9 (at B 1 = 277) 65 (at B 1 = 275)

3.7 27.3

specified, this problem becomes very sensitive to the value specified for B 1 • For B 1 = 277 instead of the specified value B 1 = 275, Example 2 is relatively easy to solve by use of the combination of the capital ()method and the almost band algorithm. When B1 was taken to equal 275, Example 2 could not be solved by either of the two proposed methods. However, Example 2 could be solved by the column modular method by using the following procedure. First, a modified version of Example 2 in which B 1 was taken equal to 277 was solved. The solution so obtained was then used as the starting values for Example 2. Selected values of the solution so obtained are presented in Tables 10 through 12. However, when the solution at B1 = 277 was used as the starting values for Example 2, a solution could not be obtained by use of the almost band algorithm for systems. In order to solve Example 2 by the almost band algorithm for systems, the following procedure was used. First, a solution to the following modified form of Example 2 was obtained. The recycle stream B 2 was replaced by an additional independent feed. This feed was taken to be liquid at 170°F and was assigned the following composition. The flow rate of B 1 was again set equal to 277.

Distillation, Azeotropic and Extractive

TAB LE 11

129

Final Temperature and Liquid Rate Profiles and Other Variables for Column 1 of Example 2

Plate No.

Temperature ("F)

Liquid Flow Rate (lb-mol/h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

132.07 133.11 135.00 139.39 148.86 171.36 172.43 172.89 173.09 173.18 173.21 173.21 173.20 173.15 173.05 172.85 172.45 171.66 170.00 166.12 154.94 155.66 156.73 158.23 160.17

80.247 78.563 75.795 76.965 69.805 265.91 266.02 266.07 266.08 266.10 266.10 266.10 266.11 266.12 266:14 266.18 266.26 266.44 266.90 268.54 372.96 372.65 372.25 371.80 371.38

I. . ProfileS'

Plate No.

Temperature (oF)

Liquid Flow Rate (lb-mol/h)

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

162.38 164.56 166.45 167.89 168.90 169.57 169.98 170.23 170.38 170.48 170.53 170.56 170.58 170.59 170.60 170.560 170.60 170.61 170.61 170.61 170.62 170.65 170.81 171.63 176.37

371.07 370.92 370.88 370.90 370.94 370.97 371.00 371.02 371.03 371.03 371.04 371.04 371.04 371.04 371.04 371.04 371.04 371.04 371.04 371.04 371.04 371.02 370.89 370.13 275.00

ll. Final Values of Other Variablesh Component Ytic

Methanol Acetone Ethanol Water "Q,. bQ, c

= 1.55154 x 10" Btujh; QR = 1.602273 x 10" Btu/h. = 1.5515 x 10" Btu/h; QR = 1.6023 x 10" Btujh.

0.150667 0.809183 0.0062675 0.033923

Mole fraction of each component in the vapor above the liquid in the accumulator.

130 TABLE 12

Distillation, Azeotropic and Extractive Final Temperature and Liquid Rate Profiles and Other Variables for Column 2 of Example 2

'Lite '-;o.

Temperature (oF)

Liquid Flow Rate (lb-mol/h)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

151.13 152.05 152.93 153.73 154.44 155.06 155.60 156.07 156.65 156.91 157.31 157.75 158.25 158.85 159.66 160.74 162.20 164.24 167.02 170.58

120.00 119.85 119.69 119.54 119.37 119.20 119.02 118.82 118.60 118.35 118.07 117.73 117.33 116.84 116.22 115.44 114.46 113.25 111.85 110.45

I.

Profile

Plate No.

Temperature (oF)

Liquid Flow Rate (lb-mol/h)

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

174.48 174.47 174.45 174.43 174.41 174.40 174.38 174.36 174.34 174.32 174.29 174.22 174.08 173.73 172.79 173.47 175.31 180.22 190.81 203.41

385.37 385.37 385.37 385.37 385.37 385.37 385.37 385.37 385.37 385.37 385.38 385.42 385.53 385.82 406.87 406.33 404.86 402.04 400.25 215.00

ll. Final Values of Other VariableS' Component Yub

Methanol Acetone Ethanol Water

0.960544 0.625244 x 0.0351114 0.0043440

w-

6

"Qc = 3.132158 x 10 7 Btu/h; Q, = 3.272744 x 10 7 Btu/h. bMole fraction of each component in the vapor above the liquid in the accumulator.

Component

moljh

Methanol Acetone Ethanol Water

0.25 0.50 5.00 189.50

After Column 1 had been solved by use of the almost band algorithm, the bi,l 's so obtained were used to solve Column 2 by use of the almost band algorithm. The solutions so obtained for the respective columns were used as the starting

Distillation. Azeotropic and Extractive

131

values for Example 2. The final solution values of selected variables for this example are presented in Tables 10 through 12. The sensitivity of the system of columns of Example 2 to the specified value of B 1 is reflected by the fact that when the value of B 1 was changed from B 1 = 277 to B 1 = 275, the temperature profile and the mole fraction of acetone in B 1 changed markedly. For example, for B 1 = 277 and B 1 = 275, the corresponding temperatures of Stage 30 were 168.90 and 154.42°F, respectively, and the corresponding molar flow rates of acetone in B 1 were 1.0348 and 0.0000198, respectively. The difficulty experienced in solving this example is attributed to the behavior of this system in the near neighborhood of B 1 = 275. Example 2 was included in order to demonstrate that the selection of an initial set of values of the variables js difficult for some problems. Experience thus far suggests that it is generally e;.1sier to pick a suitable set of starting values of the variables for the column modular method (the combination of the capital e method and the almost band algorithm) than it is for the almost band algorithm for systems. Although this problem was originally devised by Gallun et al. [14] to test the proposed calculational procedures, it does serve to illustrate extractive distillation. Water is the extractive distillation solvent. The acetone is recovered in the first column and the alcohols are recovered by the second column. The purity of acetone in the distillate of the first column could be improved, perhaps, by changing the specifications on the total flow rates. A program of the type used to solve this problem may be utilized to find the optimum set of operating conditions needed to effect a given separation. In conclusion, the techniques of azeotropic and extractive distillation have come of age. The advent of high-speed computers and the development of the mathematical techniques to solve problems involving these types of separations can only serve to increase and to quantify the application of azeotropic and extractive distillation.

Symbols b,

B c d, D

jj; f

J."

molar flow rate at which component i leaves the reboiler total molar flow rate at which the bottom product is withdrawn from a column total number of components molar flow rate at which component i leaves the column total molar flow rate of the distillate equilibrium function for component i and plate j vector of functions; see Eq. (26) fugacity of component i in a vapor mixture; evaluated at the temperature, pressure, and composition of the mixture evaluated at the temperature, pressure, and composition of the liquid fugacity of pure component i in the vapor phase at the temperature and pressure of the mixture fugacity of pure component i in the liquid phase at the temperature and pressure of the mixture

132

Distillation, Azeotropic and Extractive F g!, 92

Gj

Hf Hi hi

ri,l' ri, 2

SI

SN

Yi xi

xi

X! X2

X

vector of functions; see Eq. (12) functions; defined by Eqs. (44) and (45), respectively enthalpy balance function; see Eq. (10) enthalpy of component i in the perfect gas state at temperature T virtual enthalpy of component i in the vapor state at the temperature, pressure, and composition of the liquid phase virtual enthalpy of component i in the liquid state at the temperature, pressure, and composition of the liquid phase equilibrium ratio; defined below Eq. (4) molar flow rate at which component i leaves platej in the liquid phase total molar flow rate at which the liquid leaves plate j material balance for component i and plate j vector of material balances; defined below Eq. (13) total number of stages ratios; defined by Eqs. (39) and (40), respectively condenser duty reboiler duty functions of 8 1 and 8 2 , respectively; see Eqs. (39) and (40) specification function; see Eq. ( 17) specification function; see Eq. ( 18) mole fraction of component i in the vapor phase mole fraction of component i in the liquid phase total mole fraction of component i in the feed to the column vector of variables; see Eqs. (12) and (13) vector of variables; see Eqs. (12) and (13) vector of variables; see Eq. (25)

Greek Letters IX,

yt

f3

'liL

81, 82

fractions of streams L 1 1 and L 1 2 withdrawn; see Fig. 16 activity coefficient of component i in the vapor phase; see Eq. (2) activity coefficient of component i in the liquid phase; see Eq. (3) multipliers; defined by Eqs. (33) and (34)

References Atkins, G. T., and Boyer, C. M., "Application of the McCabe-Thiele Method to Extractive Distillation Calculations," Chem. Eng. Prog., 45, 553 (1949). 2. Benedict, M., and Rubin, L. C., "Extractive and Azeotropic Distillation, Theoretical Aspects 1," Trans. AIChE, 45, 353 (1945). 3. Berg, L., "Azeotropic and Extractive Distillation-Selecting the Agent for Distillation," Chem. Eng. Prog., 65(9), 52 (1969). 4. Carlson, C. S., Smith, P. V., Jr., and Morrell, C. E., "Separation of Oxygenated Organic Compounds by Water Extractive Distillation," Ind. Eng. Chem., 46, 350 1.

5.

6.

(1954).

Carnahan, B., Luther, H., and Wilkes, J. 0., Applied Numerical Methods, Wiley, New York, 1969. Dunn, C. L., Millar, R. W., Pierotti, G. J., Shiras, R. N., and Souders, M., Jr., "Toluene Recovery by Extractive Distillation," Trans. A!ChE, 41, 631 (1945).

Distillation, Azeotropic and Extractive 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

133

Gallun, S. E., and Holland, C. D., "Solve More Distillation Problems, Part 5," Hydrocarbon Process., 55(1), 137 (1976). Gallun, S. E., M.S. Thesis, Texas A&M University, 1975. Gallun, S. E., Ph.D. Dissertation, Texas A&M University, 1977. Guillaume, E., U.S. Patent 887,793 (May 19, 1908). Guinot, H., and Clark, F. W., "Azeotropic Distillation in Industry," Trans. Inst. Chem. Eng. (London), 16, 187 (1938). Happel, J., Cornell, P. W., Eastman, D. B., Fowle, M. J., Porter, C. A., and Schutte, A. H., Trans. AIChE., 42, 189 (1946). Hess, H. V., Naragon, E. A., and Coghlan, C. A., "Extractive Distillation and Separation of n-Butane from Butenes-2," Chem. Eng. Prog. Symp. Ser., 2, 72 (1952). Hess, F. E., Gallun, S. E., Bentzen, G. W., and Holland, C. D., "Solve More Distillation Problems, Part 8-Which Method to Use," Hydrocarbon Process., 56(6), 181 (1977). Hess, F. E., Gallun, S. E., and Bentzen, G. W., Calculational Procedures for Systems of Distillation Columns, Presented at the 83rd National Meeting of the American Institute of Chemical Engineers, Houston, Texas, March 20-24, 1977. Holland, C. D., Fundamentals ofMulticomponent Distillation, McGraw-Hill, New York, 1981. Holland, C. D., and Eubank, P. T., "Solve More Distillation Problems, Part 2," Hydrocarbon Process. 53(11), 174 (1974). Hopkins, W. C., and Fritsch, J. J ., "How Celanese Separates Complex Petrochemical Mixtures," Chem. Eng., 51(8), 361 (1955). Horsley, L. H., Azeotropic Data, American Chemical Society, Washington, D.C., 1952. Holmes, M. J., and Van Winkle, M., "Prediction of Ternary Vapor-Liquid Equilibria from Binary Data," Ind. Eng. Chem., 62(1), 20 (1970). Hutchison, H. P., and Shewchuk, C. F., "A Computational Method for Multiple Distillation Towers," Trans. Inst. Chem. Eng. (London), 52, 325 (1974). Kubicek, M., Hlavacek, V., and Prochaska, F., "Global Modular NewtonRaphson Technique for Simulation of an Interconnected Plant Applied to Complex Rectifying Columns," Chem. Eng. Sci., 31, 277 (1976). Null, H. R., and Palmer, D. A., "Azeotropic and Extractive DistillationPredicting Phase Equilibria," Chem. Eng. Prog., 66(9), 47 (1969). Poffenberger, N., Horsley, L. H., Nutting, H. S., and Britton, C. E., "Separation of Butadiene by Azeotropic Distillation with Ammonia," Trans. AIChE, 42, 815 (1946). Prausnitz, J. M., Echert, C. A., Orye, R. V., and O'Connell, J. P., Computer Calculations for Multi-Component Vapor-Liquid Equilibria, Prentice-Hall, Englewood Cliffs, New Jersey, 1967. Robertson, N. C., U.S. Patent 2,477,087 (July 16, 1949), to Celanese Corporation of America. Tassios, D., "Choosing Solvents for Extractive Distillation," Chem. Eng., 118 (February 10, 1969). Tewarson, R. P., Sparse Matrices, Academic, New York, 1973. Van Winkle, M., Distillation, McGraw-Hill, New York, 1967. C. D. HOLLAND S. E. GALLUN M. J. LOCKETT

Distillation, Batch

134

Distillation, Batch

Introduction In attempting to grasp how a batch still system operates functionally, it is helpful to consider it as three distinct subsystems: 1. 2.

3.

A pot of boiling liquid that produces a vapor higher in the composition of the more volatile components than the liquid. A vapor-liquid contacting device that accepts a vapor and divides it into two liquid streams, one usually considerably enriched in the more volatile components, the other depleted. This subsystem actually consists of three major components: a rectification column, a condenser, and a condensate splitter. The latter device adjusts the relative proportions of the two liquid streams, but this proportion is not usually related in a simple fashion to the relative proportions of the streams passing out of the splitter itself. A vessel to receive the enriched liquid, hereinafter the "distillate."

The whole art of batch distillation is summarized by the art of applying the correct strategy of adjusting the condensate splitter control during the batch cycle to effect the required separation most expeditiously. As the calculations must always proceed by testing a proposed strategy, this is true even for the design of new installations.

The Material Balance Limitation

By way of introduction, let us consider the separation of two otherwise pure components where the more volatile component is present to the extent of, say, 2% of the batch. Further, let the vapor-liquid equilibrium be such that the composition of the vapor issuing from the pot be 5%, and let us assume there is negligible holdup in the still subsystem. Some concepts can now begin to be crystallized. It should be clear that for essentially pure distillate, the distillate flow cannot possibly exceed 5% of the pot vapor flow-the still would be performing a perfect separation of the vapor stream. In reality, the still system is incapable of separating a reflux stream* purer than that of pot liquid, in this case 2%. A moment's reflection will show that the distillate flow thereby cannot exceed about 3% of the vapor stream for essentially pure distillate. The *The liquid stream returning to the pot.

DOI: 10.1201/9781003209799-5

Distillation, Batch

135

calculations may be formalized by performing a material balance over the column subsystem: Vy.,. = Dxd +(V- D)xw

or where

D = distillate rate V = vaporization rate x, y = liquid, vapor composition w, d = re boiler, distillate streams

This equation, then, relates the maximum distillate rate beyond which it will be impossible to recover distillate purer than xd to the pot liquid and vapor compositions, or by substituting a suitable relationship between these compositions, to the pot composition alone. The writer believes this to be the single most important equation of batch distillation. What if one exceeds the critical rate? Some juggling of the equations will show that the distillate can be considered to consist of a blend of two separate streams, the first essentially pure at the critical rate, the second the balance with a composition equal to the pot liquid composition. Below the critical rate the situation may be shown to be the equivalent of returning the excess distillate to the pot, raising the composition of the pot reflux in the direction of the vapor. It may be useful to note that the composition of the reflux is strictly bounded by the pot liquid and vapor compositions for a binary system. We have been assuming that the distillation column has the ability to separate the vapor into the two liquid streams; one essentially pure, the other of the pot composition. This situation is not too uncommon for a multipurpose batch still. A rule of thumb that seems to work quite well is that the number of theoretical trays in the still should exceed ajlog 10 ()(, where ()( is the relative volatility calculated at the pot conditions and a is a constant from 3 to 6 which depends on whether one or both of the liquid streams approaches the theoretical limit, and the actual purity requirements. The example would thus require about 20 trays. As the distillation proceeds and the pot becomes depleted of the more volatile material, so the critical distillate rate will decrease. Any strategy of distillate control which departs significantly from the critical distillate rate as it changes during the cycle is doomed to inefficiency. Indeed, the optimum strategy consists of continuously adjusting the distillate rate to the critical value. Tsziui-Fu [1] has shown how to integrate the expressions analytically for the case of a constant relative volatility and the more general case of any specified constant distillate composition. Implementation is conceptually simple-a composition controller, derived usually from a temperature high in the column, is used to control the splitter, usually in an onjoffmode. However, a feed forward control, based on the pot temperature, has been used with some success. The problem with most implementations is that it is difficult to find a condensate splitter system that does not leak, etc., into the distillate takeoff, making small drawoff rates impractical.

Distillation, Batch

136 Constant Drawoff Rate

The simplest condensate splitter control implementation utilizes a constant split for the entire batch. This works well typically for separating a pure material from a pot containing only a small quantity of high boiler. The reflux ratio chosen will be close to critical at the end of the cycle, which may occur because the pot heat transfer system becomes inefficient as the pot level falls-it is rarely desirable to distill off more than about 80% of a batch. Tsziui-Fu gives an analytical integration of the equation relating to the distillate composition to pot composition under these circumstances as well, but only for the case where the distillate rate exceeds the critical value throughout the distillation. Fortunately, the work can be extended readily to the more general case-it is a simple material balance calculation. If the pot undergoes a significant change in composition during the distillation, it will usually be desirable to change the draw off rate at least once during the distillation. The selection of a pair, etc., of draw off rates and suitable times for each period is an interesting optimization problem-note one would not be able to improve upon the continuously adjusted strategy, but one would try only if the latter were appreciably better than the single constant drawoff rate strategy. Top Charging

If the high boiler constitutes significantly less than 20% of the charge, the heel from the distillation will typically still be rich in the more volatile material. The desirability of "top charging" this heel must depend upon a balance ofthe cost of withdrawing and saving the heels for a full rework batch compared to the cost of the reduced distillate rate required for the top charged batch. Continuously top charged batch stills systems have been successfully implemented: when the distillate specification is no longer met, the feed is turned off, and the batch is distilled in conventional fashion (at a lower distillate rate, naturally). Limited Number of Theoretical Trays

With a limited number of theoretical trays, the column will be unable to separate the mixture to the material balance limitation, so that, in general, the blended distillate and final heel compositions will not be as well resolved as predicted by Tsziui-Fu. Often one may compensate by operating at a somewhat lower distillate rate than predicted to achieve the same resolution, which technique might offer adequate guidance for the operating of an unusual plant batch, for example. These fundamental concepts of the material balance limitation have been stressed, as no matter how esoteric the simulation of the still, it will always be bounded by these rules, whether there be only a few trays, there will be significant holdup in the still, or more than two components be treated. The

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137

concepts can often be used to provide an initial value to an iterative procedure, for example, or as we have seen, to guide the selection of a reflux ratio to be used for plant batches.

Plate-to- Plate Calculations As soon as one enters the realm of plate-to-plate calculations to try to obtain a more precise picture of how the separation progresses, one must, these days, embrace the computer. Consequently, the techniques described in the remainder of this article are intended for computerized calculation. One must, however, resist the temptation to develop only a single sophisticated model of a distillation column which incorporates every possible transient occurring in the still, for in general this will be too expensive to run more than a few times-one will not obtain a good grasp of even the material balance limitation.lt is better, therefore, to consider a series of models, each incorporating a feature more sophisticated than the previous. By comparison of similar runs between adjacent pairs of models, one obtains a feel for the importance of any specific transient. The process is analogous to that of converting a part of a tree into a table leg: The carpenter will proceed with a series of ever more refined tools. It would be remiss, therefore, to possess a set of batch distillation programs that did not evaluate the Tsziui-Fu equations. Constant Relative Volatility Equimolecular Overflow

The next program should be one that uses a relative volatility to describe the vapor-liquid equilibrium, and obtains the material balance in the column assuming a mole of less volatile material in condensing will evaporate exactly 1 mol ofthe more volatile. The success of this approximation depends upon there usually being slightly more than 1 mol of more volatile material evaporated, the excess being condensed by heat losses from the column. Column, etc., holdup is ignored. Typically, the calculation will proceed by first finding the distillate composition corresponding to the initial charge. Since one can only calculate from the top down, the procedure involves a search. The highest composition to try is a sequence of 9's (0.999 .... ) no longer than about one-half the significant digits carried by the computer, to allow the remaining accuracy in the term (1 - xd).lfthis is too high, a number of other lower values should be tried until one can interpolate for the initial value. Searches in the concentration domain can be quite difficult owing to its convexity-one might wish to try .9999, .999, .99, .9, ... (then what?). It is preferable to perform a linear search in a transformed domain given by x' = lg (x/(1 - x)); the above sequence is roughly 9, 7, 5, 3, .... The interpolation procedure should of course use the transform, converting back to the concentration domain by x = exp x' j ( 1 + exp x'). Similarly, the integration proper may proceed stepwise as follows:

138

Distillation, Batch 1.

2. 3. 4. 5.

For a somewhat lower value of distillate composition transform, perform a set of plate-to-plate calculations to give the pot composition at the end of the step. Calculate an average distillate composition during the step as the mean of the transforms at the beginning and end of the step. Perform a material balance to calculate how much of this material was removed from the mean composition, the initial and final pot compositions, and the quantity of material initially in the pot. Perform inventory calculations for the pot and the receiver. Additional steps are taken until the receiver, the pot, or the distillate composition passes a preselected value, when the final conditions can be obtained by an interpolation using the last few values.

The results of these calculations may be presented in two forms. The first of the forms is as a distillation curve, preferably plotted, that enables one to see how sharply the distillate composition curve breaks as the distillation passes through the critical condition. In the second, the results are presented in tabular form as a parameter study of the various variables. This enables one to get a better grasp of the number of trays required for various possible values of relative volatility, initial pot composition, distillate draw off rate, and distillation termination specification. If one routinely deals with systems which are highly nonideal, it would be worthwhile to prepare a similar pair of programs utilizing a more rigorous correlation of the vapor-liquid equilibrium. Binary Distillation Using a Column with Significant Holdup

Consider the problem of charging 5000 gal of a material containing 100 gal of a low boiling impurity to a still with 50 actual trays, each having a hold up of 5 gal, and a reflux system with a hold up of 50 gal. Given the relative volatility of the system, 2, and the tray efficiency, 80%, predict the compositions within the system when they become steady at total reflux. Stop now and contemplate the problem: the hold up of a highly efficient still system is three times that of the boiler in the charge. Rest assured that most of the impurity will be in the reflux system and the top of the column, and the pot and the lower part of the column will contain only a few parts per million. For all practical purposes, then, the separation will be complete at the end of the total reflux period, if only one could drain the reflux tank and column. Calculations show that this can be done by operating at a high drawoff rate* and withdrawing a volume from the system equal to somewhat more than the volume of the column, or 5 to 7 times the volume of the reflux tank, whichever is larger. Under these conditions the system is effectively eluted of the low boiler. *Provided there is no undue hydraulic upset of the column, there is no way of manipulating the condensate splitter to cause the composition of the low boiler in the pot to increase with time. Nor is it possible to cause a composition inversion in the column.

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The product may then be taken "up and over" to a second receiver, for example. What could be a simpler way of running the distillation? The product composition will ultimately contain only a few ppm of low boiler, probably much purer than necessary-0.1% might be a more reasonable requirement. One would like to interrupt the total reflux period when the pot was just a shade purer than the requirement. The question is, of course, to estimate the length of the total reflux period. In this kind of situation the column subsystem will accept vapor from the pot and return a liquid equal in composition to that in the pot by substituting materials from its own inventory. The time period can thus be estimated by a simple integration of the net flux of low boiler from the pot. An analytical integration should be possible for a system described by a constant relative volatility, using techniques similar to those of Tsziui- Fu. For this approximation to be useful, the cutoff pot composition should be significantly different from the steady state value, say 3 or 4 transform units higher. The converse situation is that of separating the main cut from a small amount of high boiler. Here, during the total reflux period, the pot composition will change little. After boiling up a volume ofliquid equal to two or three times the holdup in the column and reflux system, the compositions will become essentially steady. After this time, the effect of holdup can be ignored. Two generalizations : 1.

2.

Holdup is unimportant if the main cut is to be separated from a small quantity of high boiler. The best strategy is to allow a short time at total reflux and follow up with a strategy devised ignoring the effects of holdup. Hold up transients are all-important if a small quantity oflow boiler is to be removed from a main cut. There will be a protracted period at total reflux, followed by a short period at a high takeoff rate.

An important transition between these two extremes occurs when the quantity of more volatile material charged is equal to the holdup in the column and reflux system. The distillation time will be roughly evenly divided between a total reflux period and a period of distillate takeoff at low rate. It is an interesting optimization problem to divide the time between the two periods, assuming an equal amount will be removed regardless of the time split, to obtain the best resolution of the composition of the distillate and the overall composition of the material remaining in the pot and still system-the inventory composition. Regardless, the drawoff rate during the second period will be higher than that calculated ignoring hold up for the same resolution, but the average drawoff rate for the entire distillation will be less than this figure. How then can we model the transients of the hold up of a batch still? The integration can be performed stepwise by breaking the distillation time up into a number of short increments of time whose length is given by the time necessary to boil up from the pot a quantity of material equal to about 20 to 30% that held on each tray. Inventories are maintained for the pot, each tray, the condensate system, and the distillate receiver. These are initialized to the charge composition. For each unit of time, calculate:

140

Distillation, Batch

1.

2.

3.

For the pot, the new pot composition by material balance of its inventory, the loss of a quantity of vapor in equilibrium with the initial pot composition, and the receipt of a quantity of liquid from the bottom tray inventory. For each tray, a new tray composition by material balance of its inventory, the loss of a quantity of vapor in equilibrium with its initial composition, the loss of a quantity of liquid of composition equal to its initial composition, the receipt of a quantity ofvapor from below, and the receipt of a quantity of liquid from the tray above. Similar calculations can be performed for the pump tank and receiver; the final adjustments to these inventories and that of the pot are best carried out in double precision arithmetic.

Again, the reader is cautioned against including such transients as change in hold up due to density changes or additional reflux due to thermal transients, as these make the calculations too cumbersome. Multicomponent Batch Distillation

One ofthe toughest assignments is the design of a batch still that will be used to recover a small quantity of low boiler, followed by an isomer separation to recover the more volatile isomer from a mixture initially containing about equal amounts of the two. Of course, if one can afford to be conservative, one might assume that the low boiler cut would contain principally only the more volatile isomer. Otherwise, there is little alternative but to use a multicomponent batch distillation routine that acknowledges holdup. An algorithm for a suitable program is very similar to that described above for the binary case. To summarize: Look at the still from the point of view of the pot, and resist the urge to make the calculations too complicated at the risk of causing changes in the time scheduling algorithm of the Corporate Timesharing Computer system.

Reference

1.

Tsziui-Fu, J. App/. Chem. USSR, 31, 518-524 (1958).

G.A.R.TROLLOPE

Distillation Control

Distillation Control

141

(see also Process Control)

Objectives The primary objective for which any distillation unit is designed is to separate a given feedstock into fractions which meet certain specifications placed on them by their user or purchaser. But this cannot be the sole objective, for if those specifications can only be satisfied while operating the unit at a financial loss, the project is a failure. Two other objectives then emerge-the operation must be profitable and it must be capable of meeting certain production requirements as well. While the capability of achieving these goals may be designed into a separation unit, operating that unit to fulfil! that potential is another matter. Columns are designed to satisfy a specific set of steady-state conditions. In operation, however, a steady state is seldom reached, and the original design conditions may, in fact, never be experienced. As feed rate and composition vary, and as ambient conditions change, product qualities will most certainly follow suit, unless control is effectively applied. A train of columns can respond erratically even when the feedstock is uniform in rate and composition. Inability to control the first columns in the train may in fact generate disturbances which are propagated and even amplified through the remainder of the unit. Consequently, control is an essential ingredient in the success of every distillation unit. Product Quality Specifications

Specifications on product quality may be set in terms of purity, component impurities, or both. These purity or impurity specifications may be expressed in mole (gas-volume) percent, weight percent, or liquid-volume percent, depending on the tradition within the industry using the product. If the product is soid in the gaseous state, such as natural gas and nitrogen, it is accounted in volumetric units of both quantity and purity. Most chemical products are sold in mass units, and therefore the concentrations of components they contain are given in weight percent. Liquid petroleum products are typically sold by the barrel or gallon, and therefore their purities are reported in liquid-volume percent. It is not unusual for a product to meet multiple specifications. For example, a propane product might have to be at least 95% pure while containing less that 2% isobutane. A physical-property specification may also be applied: that same propane product may have a vapor-pressure limit of 220 lbjin. 2 abs at 100°F. This places a limit on the concentration of lower-boiling components such as ethane and methane in the product. Recognize that control can only be enforced over one specification at the point where the product emerges from the distillation unit. Figure 1 shows the DOI: 10.1201/9781003209799-6

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142

Propane

Ethane and lighter

Feed

lsobutone and heavier Oeethonizer

FIG. 1.

Deproponizer

Control over the isobutane in the propane product may be accomplished in the depropanizer, but its ethane content must be controlled at the deethanizer.

propane product leaving the top ofthe depropanizer where butanes and heavier components are separated from it. The concentrations of the heavy key (isobutane) and heavier components in the distillate (propane) are controllable in that column. However, components lighter than the light key (propane) are inseparable from it in the single-product column, and their concentrations must be controlled elsewhere. Similarly, the concentrations of the light key and lighter components in the bottom product may be controlled at that point, but components heavier than the heavy key must be controlled elsewhere. To control the ethane content in the propane product, the ethane-propane ratio in the bottom of the deethanizer must be held constant. If neither of these components were to appear in the depropanizer bottoms, then their ratio in the deethanizer bottoms and propane product would be identical. In practice, however, some propane is lost in the depropanizer bottoms and therefore the ethane-propane ratio in the propane product will tend to be slightly higher than in the deethanizer bottoms. To control propane purity, the sum of the impurities must be controlled. The accuracy of analyzers is typically stated on a percent-of-range basis. Therefore, control over the combination of ethane and isobutane, both reported over a range of 0 to 5%, will give superior accuracy to controlling propane content reported over the full range of 0 to 100%. To summarize, a specific component impurity or a physical property of a final product can be controlled using a single quality measurement. Product purity more often requires two or more measurements which are added to determine total impurity. And intermediate products whose compositions have a bearing on the quality of a final product require control of the ratio of the component impurity to the main component in the final product.

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Operating Costs

There are operating costs associated with meeting specifications. If control is poor, variation in weather conditions, feed rate, etc., will upset product quality, possibly beyond specifications. To minimize the possibility of exceeding specifications, operators tend to allow ample margins of safety, typically manufacturing products which are excessively pure. These excessive purities may add to operating costs in several ways, one of which is an increase in energy consumption. Reference 1 derives a mathematical relationship between impurity levels and energy that is logarithmic in nature: (1)

where V/Fis the column's boilup-to-feed ratio, y 1 and yh are the concentration of light and heavy keys in the distillate product, and x 1 and xh are their concentrations in the bottom product; coefficient f3 depends on relative volatility, the number of trays, and their efficiency. The component impurities in the above expression are Yh and x 1, whereas y 1 and xh would be principal constituents in the products. Then a decade reduction in either Yh or x 1 would require a doubling in VjF, or a three-fold reduction in both yh and x 1 would require a doubling in VjF. Allowing safe operating margins below specifications is then costly in terms of energy consumption, which constitutes a large portion of the operating budget, in addition to being an important national concern. When only one product from a distillation column needs to meet a specification, its composition may be altered at the expense of the other product. This can be illustrated again by using Eq. ( 1). Energy consumption V/ F can be held constant and yh can be adjusted by a proportional change in x 1 or vice versa. While this does not use additional energy, it results in lost product. If Yh is a specification which must be met on a valuable overhead product, then a reduction in Yh necessitates an increase in x 1, indicating a diversion of distillate product to the bottom ofthe column. Assuming that the specification product is the more-valuable, revenue is lost proportional to the diversion ofthat product into the less-valuable stream. An associated debit is the reduction in productivity of equipment, energy, and the labor dedicated to that separation. Certain variations in product quality are allowed if they can be attenuated through blending in downstream tankage. In some cases, tankage of sufficient capacity may not be available. The time duration during which off-specification product is made must be minimized if the blend is to be acceptable. Additionally, alternately making underpure and overpure product consumes more energy than continuously yielding product of their average purity, owing to the logarithmic nature of Eq. ( 1). For example, stepwise cycling between 2 and 8% impurity (yh) to meet a specification of 5% uses 7% more energy than that required to made a product having a uniform 5% impurity. This same argument rules against blending overpure and underpure products from parallel processes to meet an average specification.

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The penalties for failing to meet specifications can be quite severe. A lowgrade product may be sold for less, but only if a market exists for it. Alternately, it may be reprocessed or used elsewhere in the plant, viz., as fuel. Reprocessing not only increases manufacturing cost by using additional energy, but it cuts into production capacity as well. Furthermore, returning a finished product to a feed stream can impose a substantial composition disturbance on the unit. To avoid detrimental upset to the plant, off-grade products can only be processed at a fraction of the rate they were produced. Hence tankage must be provided to store them. Reprocessing is discouraged for justifiable reasons. Production Capacity

A most important goal of any distillation umt 1s to maximize production capacity. Although the demand for products may change with time or seasonally, it is essential that the maximum capability of the plant equipment be available for projected increases in flow rates. Naturally, there are limitations in all plant equipment. Each column has a maximum vapor and liquid carrying capacity, and each heat exchanger a maximum heat-transfer rate for any given set of conditions. When a limit is encountered, control over a variable is lost. For example, when the ambient temperature rises sufficiently high that a given vapor load cannot be condensed, control over column pressure may be lost. Or when a certain feed rate demands more heat than a reboiler is capable of transferring, control of temperature or composition may be lost. Variables like temperature and pressure have a certain degree of self-regulation, however. The pressure, in rising slightly above the control point, increases the dew point of the overhead vapor which increases the rate of heat transfer. Consequently the pressure eventually reaches a new steady state. The column cannot regulate itself in a limited condition, however. A column may "flood" in different ways. If the vapor velocity is too high, liquid may be entrained. Physically lifting liquid from a lower to a higher tray reduces the efficiency of the trays, causing product quality to deteriorate. Since the entrained liquid must return, liquid rates are increased, raising the level on trays and aggravating the problem. In addition, temperature and composition controllers will respond to degrading compositions by increasing boilup and reflux, driving them further off-specification. Downcomer flooding occurs when downcomers cannot carry the necessary flow of liquid, either because reflux is too high or because the liquid density is low due to entrained vapor. Then liquid accumulates on the trays, with the vapor carrying froth upward, which further increases the liquid loading. At this point the differential pressure measured across the column starts to rise sharply, indicating it is filling with liquid; base level also falls. The normal functioning of the quality controls must be interrupted to correct a flooding condition. Boilup and/or reflux must be reduced until differential pressure returns to normal, or limited when differential pressure begins to rise. In either case, quality control must be sacrificed, signaling that a production limit has been reached.

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In some columns, flooding always begins at the same point, e.g., at the top tray. However, the point most likely to flood can shift between the top and bottom sections depending on feed composition and enthalpy. If the operator can identify which section is flooding, he may be able to correct it by more effectively balancing the vapor rates between the two sections using feed preheat. Seldom is this ;nformation available along with the ability to adjust preheat. These constraints vary with the condition of the equipment. compositions, and ambient conditions. Condensers which reject heat to the atmosphere are likely' o limit production in summer, and are capable of condensing more vapor at 111ght than during the day. When light components accumulate, condensation is impeded, and they may have to be vented. Similarly the accumulation of heavy components in the reboiler can limit its heat-transfer rate. Column capacity increases with pressure for components heavier than propane and decreases for lighter components. However, relative volatility decreases with increasing pressure for most mixtures, so that maximum production can be attained at minimum pressure for the separation of pentane-isopentane and most lighter mixtures [2].

Control of Flow and Inventory Variables In order to meet the objectives outlined above, certain measurable variables must be brought under control. While its heat exchangers may be selfregulating, the column itself is not, such that it cannot satisfactorily function without at least some automatic control. There are basically three groups of variables to be controlled in a distillation unit: flow rates, materials inventory, and compositions. External Flow Rates

The quality of the final products depends on matching flow rates of feed, products, reflux, and energy. It is possible to control quality without measuring these flow rates, but only if the quality analyzer is accurate and responsive, and the valves affect the flow rates linearly. But it is also possible to control quality with an intermittent or off-line analyzer when flow controls are applied judiciously. The combination of flow control with feedback from product quality gives results superior to either acting alone. Flow measurements are also necessary to maximize production. Limits of vapor and liquid flow rates need to be established by predicting or observing conditions which promote flooding. Then the intelligent operator can locate the plant's current load relative to these limits and estimate its remaining available capacity. Heat input and reflux flow rates should be measured and controlled to establish the column's internal material balance. Controlling one of the product

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146

flow rates allows manipulation of the external material balance, which greatly facilitates product-quality control for columns having reflux ratios greater than unity. Feedforward control requires a feed rate measurement as well. The orifice plate and differential pressure transmitter are used almost universally to measure flow rates in distillation units. Their limited rangeability (4: 1) is not considered a detriment, in that the efficiency of most tray or packing designs does not generally extend as far. Internal Flow Rates

Internal flow rates are much more difficult to measure. The return flow from re boilers is usually in two phases and so cannot be conveniently measured. The overhead vapor leaving the column is sometimes measured, although with difficulty-a very large orifice is needed to minimize pressure drop because of its effect on the capacity of the condenser. The differential pressure measured across the column varies with vapor and liquid flow rates, acting essentially like an orifice. It is more sensitive to vapor flow due to its higher velocity. However, its sensitivity varies greatly from one type of tray to another. Perforated trays behave more like orifices than others. The openings in valve trays change with vapor flow, tending to regulate column differential. Despite the reduced sensitivity provided by valve trays in contrast to perforated trays, differential pressure can still be controlled by manipulating heat input. Bubble-cap trays exhibit a static head even with no vapor flow, which other trays do not. However, under ordinary operating conditions the response of differential pressure to changes in boil up is similar to that obtained with perforated trays. Entrainment is not detectable as a change in differential pressure, although higher values indicate increased velocity which promotes entrainment. Downcorner flooding is indicated by and directly related to differential pressure with any type of tray. When differential pressure rises to equal the head of liquid in the downcomers, liquid downflow stops. At this point, differential pressure rises sharply as liquid accumulates on the trays. The normal pressure drop per tray averages perhaps 2 in. of water. Then a 50-tray column approaching 100ft in height would develop roughly 100 in. of water head. To obtain an accurate differential-pressure measurement across the full height of a column requires careful installation. Liquid must not be allowed to accumulate in either pressure connection; yet gas purges must be avoided. The upper (low-pressure) connection of the transmitter should be close-coupled to the overhead vapor line, preferably at its lowest point; this could be where the vapor line enters the condenser as shown in Fig. 2. The lower (highpressure) connection should drain freely back to the column. Vapor from the base of the column can condense at ambient temperature and its condensate must be allowed to flow back into the column. The connecting line should be at least %-in. internal diameter, without horizontal sections, and insulated to minimize condensation. Valves should be full opening such as a gate, ball, or plug style. Internal liquid flow rates cannot be measured directly but can be computed

Distillation Control

FIG. 2.

147

Connections to the differential-pressure transmitter should be as short as possible, should drain freely, and should be insulated to minimize refluxing.

from external measurements. If reflux enters the column at its bubble point, the flow of liquid leaving the top tray will be the same. If the reflux is subcooled however, in rising to its bubble point it will condense some vapor, thereby increasing the liquid flow leaving the top tray. The percentage increase in liquid flow is the product of the temperature depression of the reflux below its bubble point and the ratio of its heat capacity to its latent heat. For light hydrocarbons this correction is in the order of 0.4% per oy depression. Reference 3 describes an analog computer which calculates internal reflux based on measured external reflux flow and its temperature difference below that of the overhead vapor. While intended to correct for variations in subcooling, it creates a positive feedback loop through overhead vapor temperature. An increase in vapor temperature caused by a rising concentration of higher-boiling components creates a larger temperature difference which indicates an increasing internal reflux. If the internal-reflux controller reacts by decreasing external reflux, the composition disturbance will be augmented. However, if a temperature controller is used to set the internal reflux flow controller, it can apply enough negative feedback to stabilize the system. Column Pressure

Inventories of both liquid and vapor phases must be measured and controlled. Because the volume of the column and its auxiliaries is fixed, variation in vaporphase inventory is reflected in measured pressure. If the rate of condensing and venting exceeds the rate of boiling and vapor feeding, column pressure will fall. Some columns float on atmospheric pressure-then instead of the pressure falling, air is inhaled. Eventually conditions will change where vapor generation exceeds the rate of condensing, in which case the air must be exhaled. Some volatile vapors are lost in each of these cycles.

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148

Column pressure is more often controlled by manipulating the rate of condensing, either bypassing or flooding the condenser. In general the use of a bypass valve as shown in Fig. 3 (left) provides responsive pressure control, although its range is limited. When cooling is plentiful, pressure may not be held up to the control point even with the bypass valve fully open. If cooling is provided by fans, some may be turned off or slowed until control is restored. A valve located in the line leaving the condenser, as in Fig. 3 (right), can be used to accumulate liquid, thereby covering heat-transfer surface. The rangeability of this system is greater, although pressure cannot respond as fast to changes in valve opening because liquid level must change first. With both of the systems in Fig. 3, the pressure controller can be adjusted to an extremely narrow proportional band, e.g., < 10%, giving very tight control of pressure. Another possibility is the use of a valve located between column and condenser. While this is occasionally seen, and provides excellent pressure control, it must be quite large to avoid excessive pressure drop when fully open. Pressure control can also be achieved by manipulating boilup rate. However, Rademaker et al. [4] describe a column where an increase in heat input actually brought about a steady-state reduction in pressure owing to a high degree of self-regulation in both re boiler and condenser. Consequently this control loop may function differently from one column to another. When the column contains noncondensible gases, pressure may be controlled by venting them. If their presence is variable, one of the above pressurecontrol schemes may be applied, with venting exercised only when the normal pressure-controlling valve has reached its limit. The vent valve may be operated sequentially with the normal control valve if desired. Liquid-Level Controls

Liquid levels must be controlled in the reflux accumulator and in the base ofthe column. In most cases they are controlled by manipulating the flows of products leaving them, as shown in Fig. 4. While the liquid levels are relatively easy to control with these flows, this is not the only, nor necessarily the best, arrangment of loops. In the discussion of quality control which follows, it is

FIG. 3.

Column pressure can most easily be controlled by bypassing (left) or flooding (right) the condenser.

Distillation Control

149 Heot remove I Q

Feed F

TC

Distillate D

Bottoms B

FIG. 4.

In most columns the liquid levels are controlled by manipulating the flows of the products.

pointed out that in columns having a high reflux ratio, one of these levels should be controlled by either reflux or boilup. Because of its nonself-regulating nature, liquid level is most-effectively controlled by proportional action. Integral action is usually not required, and is known to cause limit cycling in conjunction with valve hysteresis [5]. This possibility is eliminated with a valve position er, or better, by using the levelcontroller to set flow in cascade. Base level is more difficult to control than accumulator level due to its smaller capacity and the continuous disturbances induced by the re boiler. The proportional band of the level controller must be wider than used for the accumulator, e.g., 50% compared to 10%, and integral action is usually required to keep the reboiler tubes properly immersed. The response of base level to control action differs from one style ofreboiler to another. Kettle reboilers (Fig. 5) are partitioned into a boiling zone and an inactive zone by a weir. The weir keeps the tubes immersed, while the liquid level is controlled in the smaller volume downstream. The small size of this volume and the randomness with which boiling liquid overflows the weir makes the level record appear quite ragged. A nonlinear controller having an adjustable region oflow gain centered around zero deviation can minimize the amount of this noise passed on to the valve without sacrificing control action to large disturbances. In columns having thermosiphon reboilers of either the vertical or horizontal variety (Fig. 6), bottom product is accumulated in the column base. The larger volume of the column base and the absence of a weir gives a much more uniform level record than possible with a kettle, resulting in reduced variations in bottom product flow. A train of columns fitted with thermosiphon

Distillation Control

150

Heating medium

Bottom product

FIG. 5.

In kettle reboilers the level is controlled in the small terminal volume by manipulating the bottom product valve.

reboilers will therefore tend to be much more stable than a similar train having kettle reboilers. Buckley [6] reports that heat transfer in a vertical thermosiphon reboiler is maximized when the level in the base ofthe column is controlled at a point onethird ofthe distance below the top tube-sheet of the re boiler. Perhaps only 5% of the liquid flowing through a vertical thermosiphon reboiler vaporizes in a single pass. At increasing heat transfer rates, the percentage vaporization increases. Then an increase in heat flux, by increasing the vapor volume in the reboiler, forces more liquid from the reboiler into the column base, raising its level. Eventually the higher boilup rate reduces liquid inventory in the column base. However, the momentary increase in liquid level, followed by a falling level, can present a severe control problem should one want to control level by heat input. Buckley et al. [7] have also observed this "inverse response" in column trays, attributing it to the variation ofliquid holdup with vapor rate. In the column they observed, the time required for the liquid level to cross its original value following a step increase in heat input was 11 minutes. With this delay it was not possible to control base level with heat input.

Hea;;;n

me~ Bottom product

FIG. 6.

The level maintained in the base of the column affects heat transfer through the vertical thermosiphon reboiler (left), but horizontal units (right) usually receive liquid directly from the bottom tray.

151

Distillation Control

The percentage vaporization in a horizontal thermosiphon reboiler is greater than in a vertical unit. To sustain circulation, liquid is trapped from the bottom tray rather than taken from the base of the column. This gives a greater head on the inlet to the reboiler, and also a higher percentage of volatile components. However, the column base is then dead-ended-if the bottom product is contaminated with volatiles, they cannot be removed by the re boiler. The same was true ofthe kettle re boiler. Hence the response of bottom-product composition to changes in heat input will be slower with these reboilers than with the vertical-thermosiphon unit. Forced circulation is used primarily with vacuum columns and with directfired reboilers. For safety reasons the latter are located at least lOO ft from the column. The return line to the column carries a two-phase mixture whose percent vapor varies directly with boilup, since the liquid flow to the heater is constant. As with the vertical-thermosiphon reboiler, an increase in heat input forces liquid from the re boiler and return line into the column base, temporarily raising base level. In some cases the liquid contained in the return line may actually exceed the capacity of the column base. Base level is then severly upset by changes in heat input; furthermore the inverse response experienced prohibits base level from being controlled by heat input.

Control System Structure A typical two-product column has five variables which may be manipulated for control: 1.

2. 3. 4. 5.

Top product flow Bottom product flow Reflux flow Heat input Heat removal

(Feed rate is not included in this list, because more often than not, it is the top or bottom product from an upstream column or is otherwise subject to independent manipulation.) The principal challenge in designing a control system for a column is the assignment of these manipulated variables to the controlled variables having the greatest response to them. The five principal controlled variables for the same column are: a. b. c.

d.

e.

Top-product composition Bottom-product composition Column pressure Accumulator level Base level

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152

These variables may be paired in 5! or 120different ways, many of which are likely to be totally ineffective. By a process of elimination, the viable alternatives can be reduced to a handful. Attempts to narrow the selection further are thwarted by the realization of the variety of two-product separations encountered with regard to specifications, constraints, equipment characteristics, etc. For example, a column with a reflux-to-distillate ratio of 20 has different requirements and responses than one with a ratio of 0.5. The relative portions of light and heavy components in the feed also has a bearing on the response of controlled variables to manipulated flows. And last but equally important are the practical limitations of the equipment being used; for example, base level cannot be controlled by heat input to a fired heater. Because of this variability, loop selections cannot be made a priori. Instead, the principles governing composition control are here presented to illustrate how the engineer can make an intelligent selection of system structure to suit his individual applications. Factors Affecting Product Quality

Examining a column as a whole (Fig. 7), it is easy to visualize that the quality of a product varies inversely with its rate of flow. This can be demonstrated using an overall material balance:

(2) and a component balance: Fz,

= Dy, + Bx,

(3)

Energy out

Distillate

Feed

Energy in

FIG. 7.

An overview of a column reveals that product composition is related to its flow and energy in proportion to the feed rate.

153

Distillation Control

where F, D, and Bare the flow rates of feed, distillate, and bottoms, and z;, Yi> and X; are the concentrations of any component i in their respective streams. Combining these two equations reveals the relationship between product compositions and their flow rates, relative to the feed: D

Z; -X;

F

Y i - X;

B

Y i - Z;

F

Y; -X;

(4) (5)

Recognize that these two equations are not ind('pendent-if D/F is manipulated for composition control, B/Fcannot be. In essence, Eqs. (4) and (5) are but a single equation with two unknowns, X; and Y;· Typically, F and z; are known or at least independent variables, and D or B is manipulated to control X; or Y;A second relationship between X; and Y; was given in Eq. ( 1); however, it introduces a second set of compositions. For the binary case, Yh = 1 - y 1 and xh = 1 - x 1. Making this substitution in Eq. ( 1), and identifying component i as the light key I in Eq. (4) or (5), gives two equations in two unknowns, x 1 and y 1• Then x 1 or xh and y 1 or Yh are controllable by setting DfF or B/F and V/F. In a multicomponent system the above substitution cannot be made, in which case Eq. (1) remains with four unknowns. However, Eq. (4) or (5) can be stated for each component, although only three are needed. The first may be written in terms of the heavy key, the second for the light key, and the third for all others lumped together. In the third, X; would be 1 - x 1 - xh, etc. Then four independent equations containing the four unknowns can be solved. They do not need to be solved here-the point to be made is that regardless of the number of components in the system, x 1 or xh and y 1 or Yh can be controlled by manipulating D/F or B/F and V/F. While D and B are externally measurable flow rates as is F, V is an internal vapor rate, not easily measured with accuracy. Boilup is manipulable, however, because it is proportional to the heat input to the reboiler through the latent heat of vaporization of the bottom product. Even then, heat input itself is not directly measurable and must be inferred, e.g., from steam flow, or calculated, e.g., from hot oil flow and its change in temperature. In actual practice, D and B have no direct influence on their respective compositions, notwithstanding the external material balance. Figure 4 shows that the immediate effect of changing D and B is only to alter the liquid level in the vessels from which they are drawn. Compositions can only be affected by adjusting the relative flow rates of vapor and liquid within the column itself. This internal material balance may be adjusted directly by manipulating boilup or reflux, or indirectly by operating on the external material balance. In the latter case, changes in the external balance are imposed on the internal balance by one of the level controllers. These two basic methods of distillation control-internal and external material balance-are fundamentally different. The control system designer must decide which one to choose at the outset, because the formation of the balance of his system structure depends entirely on that choice.

Distillation Control

154 Internal Material- Balance Control

The steady-state relationships of Eqs. (1) through (5) must be satisfied regardless of which control arrangement is used. However, the two methods differ in the means used to attain that end. The choice depends on many factors, principally the reflux ratio (L/D) and the type of disturbances to which the column is subject. It should be understood that in the absence of disturbances, no control whatever is needed-therefore the control system providing the most-effective control in the face of disturbances would be the preferred choice. The system structure shown in Fig. 4 is of the internal material-balance type. Neither product flow is controlled, both being dependent on the flows of vapor and liquid entering and leaving their respective ends of the column. Internal liquid flow is the controlled external reflux, augmented by internal condensation relative to its subcooling, and by liquid in the feed. The internal balance is adjusted by the heat input. To satisfy the external balance required by Eqs. (2) to (4), adjustments Ll V are made in boilup, which produce identical changes L1D in net overhead product as long as reflux L is constant. It is very difficult to establish exact values of heat input and reflux needed to produce desired distillate and bottom-product compositions. Although DJF and V/F may be calculated accurately from Eqs. (1) and (4), even for the multicomponent situation, the inability to measure internal liquid and vapor rates discourages positioning reflux and boilup on a calculated basis. The success of the internal material-balance control system hinges on feedback control from a measurement of tray temperature providing a rapid correction for shifts in the internal balance. Should any disturbance, in feed rate or composition, reflux flow or temperature, or heat input itself, cause a change in the vapor-liquid difference, the temperature profile in the column will start to shift. In adjusting boilup rate, the temperature controller can readjust the internal balance before a substantial change in product quality appears. The alternate arrangement of the internal material-balance system would have boilup held constant, with reflux flow manipulated by a temperature controller. While in the steady-state the two arrangements produce the same result, response of temperature to reflux flow is much slower. A change in vapor flow passes entirely through the tower in a few seconds. However, a change in liquid flow must raise the level on a tray before being passed down to the next tray. While this delay may be only the order of 5 sjtray, it is multiplied by the number of trays between the points of reflux entry and temperature measurement. Heat input can therefore be used effectively to control temperatures at either end of the column, while reflux is only effective for control near the top. Internal material-balance control is more commonly applied than external material-balance control, owing to its simplicity, speed of response, and ease of operation. However, it also has substantial limitations. The temperature measurement must be located carefully in a region of sensitivity. In the terminal trays of a column the products approach purity, and therefore temperature changes tend to be small. To attain a reasonable sensitivity, then, the measurement must be located a few trays from the end of the column, in which case it is less representative of the true condition of the product. Variations in feed rate and composition are known to produce some variations in product

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155

composition even when the temperature is constant. Second, in a multicomponent mixture a change in the concentration of components heavier than the heavy key will be offset by a change in the light key in the bottom product, to keep temperature constant. These limitations can be overcome if the temperature set point is adjusted in cascade by a controller acting on an analysis of final product quality. In this arrangement, shown in Fig. 8, the speed of response of the temperature-control loop is used to correct upsets in feed rate and energy flows. The composition control loop is limited in response by the time required to sample and analyze, and by the holdup in the column base. While it cannot react quickly enough to counter disturbances in the internal material balance, it should be able to cope with feed composition changes which would alter the relationship between temperature and composition only slowly. It is virtually impossible to control temperatures at both ends of a column by manipulating boilup and reflux. The interaction between these two control loops is known to be severe, a change in boilup producing very nearly the same effect on both top temperature and bottom temperature. In attempting to correct for a top-temperature upset, reflux tends to cause a similar disturbance in bottom temperature, completing the cycle. Higher reflux ratios augment the severity of this interaction. Separations requiring high reflux-to-distillate ratios require the subtraction of two relatively large numbers ( V and L) to satisfy the material balance. Then variations in V and L that are small percentagewise can produce variations in V-L which are large, percentagewise. Variations in reflux temperature, temperature and pressure of the heating medium, feed rate, and enthalpy all affect the internal balance between vapor and liquid. At a 1-to-1 reflux ratio, these disturbances produce the same effect in the external balance as on the internal balance. However, when the reflux ratio is 10-to-1, each disturbance is amplified by 10 on its influence over the external balance. Engineers have found it necessary to change to external material-balance controls to provide regulation in the face of even minor disturbances when amplified by high reflux ratios [8, 9].

FIG. 8.

In cases where the temperature is not completely representative of product composition, a composition controller may be used to set temperature in cascade.

156

Distillation Control External Material-Balance Control

An external material-balance control system has one of the product flow rates manipulated to control composition. The choice is usually, but not always, given to the smaller product flow. The reasoning again is based on accuracy. If the bottom product flow is only half as great as the distillate, for example, percentage errors in its measurement and control would have only half the effect of the same percentage errors in distillate flow. Assigning a product flow to composition control requires another variable to be used to control level. If distillate flow is manipulated for composition control, the level in the reflux drum must be controlled by reflux or heat input. If bottom-product flow is manipulated to control composition, base level must be controlled by heat input or reflux. These arrangements are not without their problems. As mentioned earlier, in cases where inverse response appears between base level and heat input, that loop cannot be closed with confidence. And because of the long dead time between reflux and base level, that loop is not recommended, either. These considerations may force a decision in favor of manipulating distillate flow, or even to an internal material-balance scheme. Dynamic response with an external material-balance system is inherently slower than internal-balance manipulation. Changing distillate or bottomproduct flow has no direct effect on the internal balance which must change before compositions can be altered. Without providing additional loops, the internal balance can only respond to external-balance manipulation through the action of level controllers. A change imposed in distillate flow must first cause the liquid level in the accumulator to start changing before the level controller can react by adjusting reflux or boilup. If the level control loop had an infinite gain and no dead band, changes in the external balance would be converted by the controller immediately into corrections in the internal balance, but this is not practical. Noise in the level measurement, interaction with column pressure, and the dynamic stability of the level loop itself limit the controller's proportional band to perhaps 10%. As a result, reflux (or boil up) can respond to changes in distillate flow at best with a time Jag of 10% of the hold up in the reflux accumulator. Integral action cannot improve on this response, and derivative action cannot be used due to the noise on the level measurement. When bottom-product flow is manipulated for composition control, heat input must be used to maintain base level. The natural turbulence in the reboiler, along with a tendency toward inverse response, requires a wider proportional band in the level controller. However, the base hold up is typically less than that of the accumulator, resulting in a similar time Jag. Because tray temperatures respond faster to boil up than to reflux, except at the top tray, a temperature almost anywhere in the column can be controlled with bottom flow if base level controls heat input. On the other hand, temperature control by distillate manipulation is only effective near the top of the column when reflux is under level control. In cases where heat input is under control of accumulator level, temperatures everywhere should respond equally to distillate flow. Figure 9 illustrates what is probably the most common embodiment of

Distillation Control

FIG. 9.

157

This external material-balance system depends on the action of the accumulator level controller for its success.

external material-balance control. Heat input is under flow control, set at some desirable or maximum value that the column and its auxiliaries can accommodate. With the internal-balance scheme of Fig. 4, changes in heat input were converted directly into changes in the external balance. But the system shown in Fig. 9 holds the external balance constant except as adjusted by the temperature controller. The heat input has very little effect on composition. From the material-balance Eqs. (4) and (5) it can be seen that a 1% shift in D/Fand B/F will produce approximately the same change in product compositions. This requires a high level of accuracy in the control of the balance, but it also provides an immunity to disturbances which would shift that balance. By contrast, the sensitivity of product compositions to the boilup-to-feed ratio as related through Eq. (1) is seen to be much less. For example, a 1% change in Yh from 5 to 4%, all other things being equal, would require about 4% more energy. If the composition were changed from 2 to 1%,9% more energy would be required. High-purity products are apparently more sensitive to changes in the material balance than in the energy used to make the separation. It is important, then, to protect them from material-balance upsets caused by variations in heat input, reflux, heat losses, etc. The system shown in Fig. 9 does just this. Should the heat input change, it will affect the internal vapor-liquid balance only temporarily. An increase in vapor flow to the condenser will raise accumulator level and increase reflux flow to reach a new balance. Compositions that started to change as a result of the increase in vapor flow will return nearly to their former values when the subsequent increase in liquid flow has reached all traps. A chance in reflux temperature or feed enthalpy will be countered in much the same manner. There is then no need for such functions as internal-reflux calculations in an external material-balance system.

Distillation Control

158

The very factors which protect the externally balanced column against upset make it sluggish to respond to direction. Changing the heat input to the system in Fig. 9 has very little permanent effect on product compositions, unless the distillate flow is also changed. So the selected product flow must be manipulated for composition control. Feedforward Control

The inferior dynamic response of external material-balance systems can be overcome by adding a feedforward control loop to manipulate the internal balance. This concept first appeared in the literature in a paper by Van Kampen [8], where he reported having achieved a reduction in the period of oscillation of his product-quality loop from 5 h to 30 min. His scheme is shown in Fig. 10. In effect, it combines the dynamic responsiveness of the internal material-balance system with the accuracy of the external material-balance system. The summing device calculates the reflux set point L* as a function of the output of the level controller mL and measured distillate flow D: (6)

Thus reflux is ultimately under the jurisdiction of the level controller, yet it is able to respond directly to the quality or temperature controller. Variations in heat input, feed enthalpy, reflux temperature, etc. are compensated by the action of the level controller, yet demands for changing composition are satisfied, too. The selection of coefficient kD is a prime importance. If it is set at zero, reflux manipulation is left entirely up to the level controller, and reflux responds to changes in distillate flow as a dominant lag. If kD is set at unity, then each

Distillate

FIG. 10.

Feeding forward distillate flow to adjust reflux flow combines the best features of internal and external material-balance methods.

159

Distillation Control

change in distillate flow will evoke an exactly equal and opposite change in reflux flow. In this case, level should remain unaffected, and the level controller need only adjust for errors in the measurements and calculations. (It still needs to adjust reflux for variations in boil up, enthalpy, etc.) The direct response of reflux to distillate thus achieved essentially eliminates the lag of the reflux accumulator. Further improvement is possible by increasing kv beyond unity. The , adjustments to distillate cause overcorrections in reflux, which results in le~ ,i­ lag action, adding a derivative effect to the response. Step responses for thtse three settings are shown in Fig. 11. In essence, the lag of the accumulator remains, but it is complemented by a lead. The lag is 100 YL! P, where Y L is the volume of the accumulator divided by the span of the reflux flowmeter, and Pis the percent proportional band ofthe level controller. The lead time constant is 100kv YLfP. Feedforward control may also be applied to protect product quality from disturbances in feed rate and composition. This type of calculation lends itself to external material-balance manipulation owing to the simplicity of the relationships. Distillate flow may be calculated directly from Eq. (4), using set point or nominal values for Y; and X;, or the equation may be reduced to the form D*

= RF'Lz;

(7)

1

where D* is the calculated distillate set point, R is designated as the recovery factor, and k indicates the summation of the light key and all lighter components in the feed. While the recovery factor may not be known beforehand and is subject to change with feed composition, boilup-to-feed ratio, and pressure, it may be adjusted by a composition controller as shown in Fig. 12. Although Fig. 12 shows a dynamic compensator g(t), it would not normally be required if the second forward loop to reflux described in Fig. 10 is

Distillate

-- 1------k0 =

Reflux

I

o

2

1~,..-- - - -

--

?_.----- - - - - -

Level

-- ---------

-- __. --I

--o

Time

FIG. 11.

The Jag of the accumulator can be compensated for with a lead proportional to adjustable coefficient k 0 .

160

Distillation Control

Feed conditions

FIG. 12.

Distillate set point

Feedforward control can avoid upsets to product quality caused by changes in feed conditions.

used. The latter provides adjustment to lead and Jag functions required to achieve a dynamic balance in the column. If heat input is held constant while feed rate is changed, the separation between the two keys will vary per Eq. (1). This has an effect on the recovery factor, causing R to decrease as F increases. While the exact relationship is rather complex, a relatively simple model may be derived to effect the required compensation. Equation (7) is modified as follows: D* = R 0 F

Iz; -

bF 2

(8)

l

where R 0 is the recovery factor at zero feed rate and b is a coefucient governing the nonlinearity of the expression, adjusted to match the model to the column under control. This model has the advantage of requiring no additional inputs and being adjustable through a single coefficient (assuming R 0 is adjusted automatically by the quality controller). A block diagram illustrating its implementation appears in Fig. 13. Only a summing device containing the adjustable coefficient b is required to incorporate the nonlinear function-F2 is already available as the differential pressure across the feed orifice. If boil up is varied in ratio to the feed, then the linear Eq. (7) suffices; however, an additional feedforward path must be added between feed rate and boilup. The system shown in Fig. 14 sets boilup in ratio K to total feed (regardless of its composition). Boilup causes such a rapid change in the internal balance that a predominant Jag must be used for dynamic compensation in the forward path. Without this Jag, an increase in feed will cause a simultaneous increase in boilup, driving heavy components up the column and reducing base level. The Jag should be adjusted to minimize the effect of a feed rate change on base level. When heat input becomes variable, its effect on the internal balance should be included in the reflux flow calculation: (9)

Distillation Control

161

Feed

FIG. 13.

The nonlinearity in a constant-boilup system is provided by orifice differential pressure.

Here Q is the measured heat input to the reboiler and kQ converts its signal into an equivalent flow of reflux. The bias of 0.5 is necessary to keep mL near midscale under a balanced condition, i.e., when L = kQQ- D. If feed composition is not readily measurable, or if its variation is slow enough to be corrected by feedback, it may be left out of the feedforward calculations with little loss in performance. Applying Constraints

In actual operation, constraints are encountered on all sides. Quite often the control valve on heat input or reflux will be driven fully open without being able

Distillate set point

Fit)

Q*

K~--­

Heat input set point

FIG. 14.

Use of the linear model requires concurrent manipulation of boilup with lag compensation.

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162

to satisfy its flow controller. Similarly it is not uncommon to find the coolant control valve fully open, or the condenser bypass valve closed, with the column pressure floating above its set point. These flows and column pressure are all self-regulating, so that they will reach a steady state while not being controlled. The difficulty that arises in not being able to control pressure is that the relief valves may be lifted. This not only loses product but it upsets the entire plant. Therefore, when the normal avenues of control are unable to maintain pressure near its desired value, another mechanism must be applied. For example, loss of control by cooling may be caused by an accumulation of noncondensibles. They may be removed by venting, with the vent valve operating in sequence with the coolant control valve, so that it only opens when the coolant valve has reached its limit. If noncondensibles are known to be absent, the only alternative may be to reduce boilup. Then a second pressure controller having a set point above that of the normal pressure controller may be connected to heat input through a low selector as shown in Fig. 15. Should the pressure exceed its set point, the controller output will fall below that of the other controllers and will therefore be selected to set boilup. Under normal conditions, pressure would be controlled below the high set point, in which case the pressure controller in Fig. 15 will raise its output, thus failing to be selected for control. Because the pressure controller sees a sustained deviation from set point under normal conditions, it will tend to integrate its output upward to saturation. Then the output of the controller would remain in saturation until the pressure actually rose above the set point-this condition is called integral "wind up." It can be avoided by feedback of the heat flow measurement into the integral modes of all primary controllers in the network. The selected controller sees its own output and functions normally. Those controllers not selected receive a signal different from their own outputs-this breaks the feedback loop, interrupting integral action.

Alarm an-_ override

Differential pressure

FIG. 15.

Constraints to heat input may be implemented through a low selector; external feedback from the flow measurement should be applied to all primary controllers to avoid windup.

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163

This arrangement also protects the selected controller against wind up when the flow controller is unable to deliver the required flow. In this case the flow measurement feedback is lower even than the output of the selected controller, stopping its integral action. As soon as flow control is restored, integral action in the primary controller resumes. Primary windup is then avoided when the flow controller is set locally or operated in manual. A column differential pressure controller is also shown having a limiting effect on heat input. The purpose of this controller is to keep the column from flooding. If due to an unusually high feed rate more boil up is needed to make the separation than the column can carry, the differential pressure controller will take control away from temperature. Loss of temperature control means loss of composition control. Because composition tends to be self-regulating, a new steady state may be reached. However, an alarm should be provided on the deviation between the output of the temperature controller and the flow measurement to signal the operator when temperature control is lost. In the situation where heat input is used to control base level, overrides cannot be applied because base level control must be maintained. Then a differential-pressure override may be imposed on reflux. Reducing reflux flow will lower both differential pressure and base level, which in turn will cause heat input to be reduced. The effect of reflux on pressure is not so clear. Reducing reflux flow will tend to raise pressure, particularly if it is subcooled. Eventually, the reduction in base level which follows can cause heat input and pressure to fall, but this reaction is too late to provide effective control.

Composition Feedback The most comprehensive feed forward control system can be outperformed by a simple feedback system if the controlled variable can respond quickly to corrective action, and if it is representative of true product quality. The principal limitation of a feed forward system is that its accuracy cannot be as high as needed to satisfy specifications on pure products. Typically, a l% change in the material balance will cause a l% shift in product compositions. The combination of feed and product flowmeter errors, together with variations in feed composition (if it is not analyzed), or analyzer errors (if it is), will be in the neighborhood of 2%. While their absolute accuracy could be worse (material balances on columns seldom close within 5%), their repeatability could be somewhat better. Nonetheless, feed forward by itself can probably not control composition closer than 2%. There are exceptions, however. In some columns, one of the product streams is very small-a trace of a heavy or light impurity being removed from a final product. Its small rate relative to the feed allows much closer control of the material balance. Also, many columns are used to make rough cuts between fractions. For example, a naphtha stream may be split into light and heavy fractions to be fed to different processes. One could almost dictate beforehand

164

Distillation Control

the percentage of each in the feed; in this case errors in the order of 2% will tend to have little effect on product quality because it does not have to be cut that closely. But when close-boiling products must be controlled precisely to specifications, the importance of a responsive feedback loop cannot be overstated. External material-balance controls can help protect against disturbances in the heat balance, and feedforward can help protect against feed rate and composition upsets. But to reach set point initially, and to keep product composition there stably while correcting for errors in the feedforward system, can only be accomplished by a responsive feedback loop. Boiling-Point Control

Complex mixtures of hydrocarbons, such as the naphthas described above and crude-oil fractions, are ordinarily specified as to boiling point. The composition measurement then usually consists of temperature in association with pressure. With wide boiling-range fractions such as those separated from crude oil, it may be necessary to specify the initial or final boiling point rather than the average boiling point which a temperature measurement would give. Distillation analyzers then must be provided for these products. The very substantial advantages of a temperature measurement are its simplicity and speed of response. It does have limitations, however. Boiling point varies with pressure, requiring that pressure be controlled at the point of measurement, or compensation be applied if it cannot be. Boiling point is also influenced by the presence of off-key components. When the difference between the boiling points of the key components is small, minor concentrations of offkey components can contribute substantially to temperature. A variation of 1 mol% isobutane inn-butane will cause its boiling point to change only 0.25°F at 60 lb/in. 2 gauge, yet that same 0.25°F change could also be caused by a pressure variation of 0.25 lb/in. 2 or a change of 0.6 mol% isopentane. Compensation for variable pressure has been applied in a number of ways. If a vapor-pressure curve for the product is available, its boiling point at any given pressure is predictable. While the relationship between temperature and pressure is logarithmic, it can be approximated satisfactorily by a linear equation over the pressure ranges encountered in most separations: Tb

=

T- a(p - Pb)

(10)

Here Tb is the boiling point referred to base pressure pb, and T and p are measured temperature and pressure. Coefficient a is the slope of the vaporpressure curve at Pb· In cases where the mixture is complex and a vapor-pressure curve is unavailable, a can be found by measuring the vapor pressure of a sample in a laboratory at two controlled temperatures in the operating range. An instrument is also available for measuring the differential pressure between the column and a sample of product exposed to the same temperature. One side of this differential-vapor-pressure device [10] is fitted with a

Distillation Control

165

temperature bulb containing a sample of the product desired from the column. When exposed to the temperature inside the column, the bulb develops the vapor pressure of the sample at that temperature. The difference between sample vapor pressure and column pressure is linear with boiling-point difference. This instrument therefore compensates for changes in column pressure, and features very high sensitivity in addition. For example, the boiling point of acetic acid changes 1oF when contaminated with 0.08 wt.% water. But this small change produces a 7.3 in. of water vapor-pressure difference-a substantial portion of its minimum 20-in. span. Another compensating method is the use of two temperature measurements-one near the end of the tower and another close to the center. Column pressure tends to affect both equally, so control of temperature difference is largely immune to pressure variations. In effect, it is the composition gradient which is being controlled in the section of the column where the temperature bulbs are located. However, the composition gradient and therefore the temperature gradient tends to be quite nonlinear, flattening near the ends of the column and usually near the feed tray. The possibility then exists that the same difference between two temperatures can be attained with two entirely different composition profiles, one of which is undesirable. While temperature-difference control may ordinarily be satisfactory, it could, during a severe upset, result in complete loss of control. The art of temperature-difference control has been refined by Boyd [11] as applied to aromatic hydrocarbon separations. He subtracts the temperature difference measured across the top section of the column from that measured across the bottom section to manipulate distillate flow. He reports control of product qualities in the low parts-per-million level, with a proportional band of only 20% in his double-differential-temperature controller. Feedforward is apparently not required with this type of system, presumably because the temperature measurements located near the middle of the column correct for feed disturbances rapidly enough. As with other external material-balance systems, performance depends heavily on the responsiveness of the accumulator level controller manipulating reflux flow. Several other engineers have attempted to use multiple temperature measurements for control [12] with mixed results. The relationship between temperature profile and product compositions must be characterized for any given separation, using either a computer model or plant data. Additionally, feed-composition changes may alter the relationship, particularly in columns having low boilup-to-feed ratios. Using Analyzers for Feedback

While a temperature measurement may not accurately report the quality of the product, its dynamic responsiveness is important in achieving stable control. By contrast, an analyzer can report the composition of the final product accurately, although too late to control it effectively. It is not unusual for a chromatograph to report a change in product composition half an hour after a change in temperature is observed. While some delay in the analysis is

166

Distillation Control

unavoidable, much is due to remote location of analyzers. sampling liquids at low flow rates, etc. The deterioration in performance due to these delays is considerable. The natural period of oscillation of a temperature-control loop manipulating heat input is typically 20 min whereas that of a composition loop with a chromatographic analyzer on the same process may be 60 to 90 min. Reference 13 relates the performance of a feedback loop (as indicated by the time integral of the deviation from set point) to the square of the natural period of oscillation. Then feedback control by the chromatograph may be only 5 to 10% as effective as the temperature-control loop in correcting for upsets of all kinds. To minimize delays in obtaining a measurement, the analyzer should ideally be located at the column, connected directly to an equilibrium stage. This is not ordinarily possible, most analyzers requiring a safe area for installation. Then vapor samples should be taken to maximize sample-line velocity. Reducing its pressure as soon as a sample leaves the column will further increase its velocity and minimize the possibility of fractionation. An overhead sample should be taken from the point where vapor enters the condenser. A bottom-product sample should be withdrawn from the lowest tray. These points are selected to avoid the time lags of the accumulator and column base respectively. Additional delays may exist within the analyzer, particularly in chromatographs. In order to separate a sample into its components, it must travel through a packed column at a controlled velocity. The results of an analysis may be reported from a few seconds to a few minutes after the sample is injected, depending on the volatility of the components. This delay is pure dead time, which will add four times its value to the natural period of oscillation of the control loop. Additionally, an effective delay is introduced by the act of sampling on a periodic basis. Analyses are only reported at specific intervals; then a given analysis becomes increasingly old until a later one is reported. The average age of the reported information is then one-half sample interval; consequently sampling introduces an additional dead time equivalent to half the sample interval. Time-sharing of analyzers among streams does not affect the dead time, but does increase the sample interval, thereby reducing controlloop performance. The intermittent nature of the chromatographic output creates other control problems as well. Derivative action cannot be used because the measurement to the controller changes stepwise. Each step would produce a derivative spike of large amplitude and short duration, which has a disturbing rather than stabilizing influence on the column. However, if th:> action of the controller is synchronized with the output of the analyzer, the effects of sampling can be largely overcome. In practice, the controller should be left in manual throughout most of the sample interval, switching to automatic only for a short time following the arrival of a new analysis. This allows the integral time constant to be reduced proportionate to the percentage of time that the controller is in automatic. Furthermore, derivative is able to produce a sustained contribution if the time during which the controller is in automatic is less than the decay time of the derivative spike. Chromatographs, of course, can report on several components in a mixture. They therefore allow controlling the ratio or the sum of two components, as required for the multiproduct separation described in connection with Fig. 1.

167

Distillation Control

Nonlinear Compensation

Boiling point and composition are generally linear with respect to each other, but both vary nonlinearly with tray location. Composition changes are small near the ends of the column where purities are high, and greater elsewhere. Should the composition of the feed not match that on the feed tray, the gradient is again reduced around the feed tray. These non linear characteristics appear when tray temperature is controlled. Under normal conditions a certain sensitivity exists in temperature responding to heat input. But during a severe upset the temperature profile may shift in one direction or the other, reducing the gradient near the control point. This is observed as a sharp rise or fall in temperature to a level where sensitivity is markedly reduced. Integral control action then continues to drive the heat input to correct the imbalance, but the low sensitivity of the measurement under these conditions indicates control action to be ineffective. Eventually, enough corrective action is applied to drive the temperature rapidly back through its sensitive region, possibly into the insensitive region on the other side of set point. The result is a large-amplitude, slow cycle, quite different from the response attained near the set point. The controller cannot be adjusted satisfactorily for both the small- and large-amplitude conditions. The large-amplitude cycle is sustained by the integral action of the controller whose time constant would normally be selected for the smallamplitude, faster cycle. The large-amplitude cycle can be eliminated by using a very long integral time constant or eliminating integral action altogether. The latter choice will result in a permanent offset between temperature measurement and set point-normally an unacceptable alternative. However, if the temperature controller is being set in cascade from a product composition controller as in Fig. 8, it does not need integral action. Another solution to the problem is to use a nonlinear controller whose characteristic is shown in Fig. 16. The function, consisting essentially of three segments, is applied to the deviation from set point. The deviation thus

Characterized deviation 0 t-------:::::::>"-i"""=:.__ _ _ __

tiel

0 Deviation from setpoint e

FIG. 16.

A nonlinear characteristic of this type can help overcome the natural nonlinear response of tray temperature.

Distillation Control

168

characterized is then acted on by the conventional control modes. The width of the low-gain region and its gain are adjustable to match the characteristic of the process. Where terminal composition is controlled, the nonlinear response is seen only in the direction of increasing purity. The sensitivity of composition to a change in the material balance varies directly with the concentration of the key impurity. As the key impurity approaches zero, so does the sensitivity. The effect ofthis variable sensitivity appears in the record of controlled composition as a distorted sine wave, fiat near zero impurity and sharply peaked at higher levels. The ideal compensating function for this characteristic is a logarithm. In effect, the logarithm of the impurity would tend to respond linearly to control action. (This relationship is also verified by Eq. 1.) However, an adequate compensating function can be achieved using a combination of summation and division: f(e)

=

Yh

Yh

+ Yi:'

1

( 11)

2

where f(e) is the characterized deviation of impurity Yh from its set point yi:'. Observe that when Yh = yi:', f(e) = 0. The calculation may be made outside of the controller, with a set point of 1/2 applied to the controller as shown in Fig. 17. This function provides the same change in gain as the logarithmic function for geometric variations in composition about the desired value, yet is much easier to implement, both with analog or digital computation.

Controlling Two or More Compositions

The existence of multiple control loops on a process unit presents the possibility of their mutual interaction. The principal objective in identifying the best control system structure is in fact to minimize counterproductive interaction

Measured composition

Set 1/2

Desired composition

FIG. 17.

This calculation applies effective nonlinear compensation to the product composition controller.

169

Distillation Control

among the four or five loops used. At this point, special consideration must be given to potential interaction among multiple quality-control loops. Considerable attention is given to this subject in Ref. 1. The results of that work are summarized as follows: 1. 2.

3.

Interaction between two composition controllers manipulating boilup and reflux tends to be quite severe, particularly with increasing reflux ratio. Interaction between two composition loops can be minimized if the composition of the less-pure stream is controlled by manipulating distillate or bottom-product flow, with that of the more-pure stream controlled by manipulating heat input or reflux. Interaction between the compositions of multiple sidestreams also tends to be severe.

Severe interaction is marked by the onset of instability when both controllers are placed in automatic. Boilup and reflux each tend to drive both compositions in the same direction. Closure of both these loops is accompanied by a progressive increase or decrease of both manipulated flows. However, an external material-balance variable such as distillate tends to increase the purity of one product while decreasing that of the other. Interaction with another composition under reflux or boilup control then appears as an oscillation around the normal operating point. Interaction between pairs of sidestreams is similar to that observed with reflux and boilup. Severe interaction can be expected when the purities of the products are quite similar. For example, a distillate product containing 5% of the heavy key and a bottom product with 0.5% of the light key can both be controlled by manipulating distillate flow and heat input respectively. But if their specifications were identical, e.g., both at 2%, interaction would be severe. If the distillate had to be purer than the bottom product, reflux and bottom flow should be manipulated-otherwise both loops could not be closed satisfactorily. In cases where purities are similar, decoupling may be added. A decoupling system is like a feedforward system in that it attempts to compensate one loop for upsets induced by another loop. Partial (i.e., unilateral) decoupling is sufficient to break the interaction, and is much easier to implement and adjust than bilateral decoupling. A partial decoupler specifically developed for distillation control is shown in Fig. 18. In this system, measured heat input and feed rate are used to calculate distillate set point: D*

= F(t)[m

- bF(t)/Q]

(12)

Note the similarity between Eq. (12) and the nonlinear feedforward model Eq. (8). The decoupling is applied nonlinearly to match the actual process as well as possible within a simple structure. Additionally, only coefficient b needs to be adjusted to match the model to the process. But the most important feature of this decoupling scheme is that its inputs

170

Distillation Control

Distillate set point

Feed flow

FIG. 18.

Variations in heat input caused by the bottom-composition controller, by an override or by operator intervention, automatically adjusts distillate flow to avoid upsetting its composition.

are taken from process measurement instead of the bottom-composition controller. The distillate loop is protected from heat-input upsets whether th-::y are induced by the bottom-composition controller, an override controller, a heat-input limit, or operator intervention. Similar schemes are described in Ref. 1 for decoupling multiple-sidestreams.

Optimization Only after all product compositions in the distillation unit have been controlled satisfactorily at specifications should optimization be considered. Optimization entails adjustment to certain plant parameters in an effort to increase profits while still meeting product specifications. Since adjustment to these selected parameters can be expected to upset quality control, optimization can only be brought about slowly. There are two phases of optimization described below. The first considers adjusting column pressure to maximize relative volatility. This can save energy and even increase capacity, but it requires control of both product compositions to be effective. The second minimizes operating cost for a particular separation by balancing product losses against energy costs. Floating Pressure Control

Most mixtures are easier to separate as their pressure is reduced. The enhanced relative volatility thus achieved more than offsets the increased heat of vaporization, such that less energy is required to meet specifications at lower pressure. Column pressure is determined by the temperature of the coolant, the condition of the condenser, and the heat load placed on it. In order to control column pressure, the condenser is bypassed, flooded, blanketed with a noncondensible gas, or otherwise restricted. If pressure is to be minimized, the restrictions placed on it to achieve control must be largely removed.

Distillation Control

171

There are both short-term and long-term consequences of removing pressure control. In the short-term, a pressure cycle can develop sympathetic with accumulator level control by reflux manipulation. In addition, the sudden increase in condenser cooling brought about by rain pelting a fan condenser can drop pressure quickly enough to flood a column. In the long-term, low ambient temperatures experienced at night, during rainy weather, and in the winter can be used to reduce the energy required to make a separation. Tight control of pressure can be maintained in the short-term, while minimizing its average value in the long-term, using the system in Fig. 19. The pressurt' wntroller functions normally; however, its set point is slowly adjusted by a valve-position controller to keep the pressure control valve in an average position of 10%. When excessive cooling is available, then the pressure controller will tend to increase the valve position, which causes the valveposition controller to slowly reduce the pressure set point. In order to avoid interference with short-term pressure control, the valve-position controller requires only integral action, with a time constant of an hour or more. It must be protected against wind up when the pressure controller is in manual, local set, or when the control valve is limited in an extreme position. This is achieved by taking its integral feedback from the actual pressure measurement. While this scheme can provide minimum pressure operation in the long term, it cannot of itself save energy. If no other control action is taken, the result of reduced pressure will simply be improved purity for one or both products. If energy is to be saved, the purity of both products must be controlled. If this is left up to product composition controllers to accomplish, changes in column pressure will tend to upset composition, requiring an hour or more for the process to recover. In this case the integral time of the valve-position controller must exceed an hour. But, since the relationship between the energy required for separation and column pressure is predictable, feedforward compensation may be applied. Over a typical operating range, the relationship is found to be linear: Q

F

= m(p- Po)

( 13)

where m is the slope of the line and p 0 is the intercept. In practice,p 0 is a function

Feedback

To control valve

FIG. 19.

The pressure setpoint is slowly adjusted to keep the average position ofthe control valve at 10%.

Distillation Control

172

of the components being separated: their relative volatility, heat of vaporization, and vapor pressures. For a given separation, p 0 will be constant regardless of the characteristics of the column. On the other hand, m is a function of the number of trays and their efficiency, which change from one column to another and are less predictable. Figure 20 shows a feedforward system manipulating heat input, with coefficient m adjusted by a composition controller. While reducing column pressure tends to decrease vapor density, it also increases the heat of vaporization, liquid density, and surface tension. The combination of all of these factors tends to promote flooding as pressure is reduced in columns separating butanes and heavier materials, and as pressure is increased in columns separating propane, propylene, and lighter products. Nonetheless, Ref. 2 indicates that the reduced energy requirements brought about by enhanced relative volatility at lower pressures can actually move the column farther away from the flood point, even for pentane separations. The energy savings possible by reducing the pressure of vacuum columns or those using refrigeration are considered not to be worthwhile pursuing. Balancing Product Losses Against Energy

The specifications on a single product can be met by manipulating either the material balance per Eq. (4) or (5), or the heat input per Eq. (1), or some combination of the two. When there are more manipulated than controlled variables as in this example, they may be adjusted together in such a way as to minimize operating costs. If very little energy is used, the separation between the two products will be low, resulting in considerable loss ofthe more-valuable product into the less-valuable stream. While raising the heat input can reduce this loss, increased costs for heating and cooling are incurred. The dollar loss of distillate product out the bottom of the column can be calculated as (14)

Set point to heat input

Column

Feed flow

~--P_-_Po----

·'\.

70

(~zJ1 (=

120)

60

'\.

so 40 30 20 10 ·0

0

10

20

30

40

so

...,_____________ ----x-

Fz L

(:120) FIG. 3.

Graphical solution of Illustration I.

203

Distillation, Flash

V = 0·28, y = 0·784

V= 1, y = 0·91

I F:2 z = 0·6

I T= 80°C

T: 67·8 °C

I

I

L:0·72,

L=1,x=0·29

X=

0·10

2nd Stage

1st Stage FIG. 4.

Results for Illustration I.

Physical Properties of Multicomponent Mixtures k-Values

As is well understood, the equilibrium composition of a multicomponent mixture can be characterized by a constant known as the k-value. Thus, if the mole fraction of component i in the vapor phase is Y; and that in the liquid is X;, then the equilibrium constant for i is k; such that or

Yi =k;X;

(6)

Now setting the total pressure of the system at P, and denoting the vapor pressure of the pure component i as pt, a combination of Raoult's and Dalton's laws gives (7)

where p; is the partial pressure of i in the vapor phase. Equation (7) applies to an ideal solution with a perfect gas phase. For nonideal systems, an activity coefficient, y;, is added and so the definition of k; is either

= ptJP (ideal) = y;jx;

(8)

= y;pt/P (nonideal) = y;Jx;

(9)

k;

or k;

For certain ideal systems, e.g., hydrocarbons, nomographs exist which enable us to obtain the k-values for specified components at given temperatures

Distillation, Flash

204

and pressures. An instance of this is the so-called DePriester chart [3]. An example of this in the high temperature range is shown in Fig. 5. pt may often be accurately and conveniently obtained from the Antoine equation [2], viz.:

In pt =A;

+ B;((C; + t)

(10)

where t is usually in oc and p* is in mmHg. Clearly, for any given temperature and pressure, and knowing the constants A, B, and C for component i, k; can be calculated from Eqs. (8) and (10). If sufficient thermodynamic data are available,)'; may be included (for nonideal systems) via the equations of van Laar, Margules, Wilson, or the NRTL equation. Henley and Rosen [4] present problems of this nature. Table 1 gives a list of Antoine constants for a few selected substances. The source material for this has been taken from several major works [5-9] wherein many further values are available. In some cases the constants themselves are given whereas in others they may have to be computed from vapor pressure data (see again Ref. 2). Bubble (or Boiling) Points

Unlike the composition of the liquid, X;, that of the vapor, Y;, at the bubble point of a liquid mixture will not be known except that the condition

LrnYi

=

( 11)

1

must apply-the mixture has been taken as containing m constituents. Thus, from Eq. (6) it follows that (12) must be satisfied. We therefore choose the temperature (by trial-and-error) so that Eq. (12) is satisfied. Equation (6) then yields the composition of the vapor phase at the bubble point. It is clear that the feed to the system must be above its bubble point for flash distillation to be possible. To aid the type of iterative calculation suggested by Eq. (12), the NewtonRaphson method may be used in conjunction with the Antoine equation. Thus the n-th iteration on temperature becomes sbn

with the derivative

whence

= 1-

(1/P) LmXi exp [A;+ BJ(C;

+ tn)J

(13)

205

Distillation, Flash

~

.E ...

"'

~

147 15

:?! >.. "' .c UJ

"'c:

"'

c:

... 0

.c

icr SI ,

:::1

.Q

'

0

c:

o.S

0

0

"'>< ..."'c: "'c: I"' I O

~ ~

"'c: ... ...::IlD

0

"'

~

c:

0

V

0

0

c:

z ~ z z z z z V

"'

-Z

~z

0 0 0. ...

400

~18 no

20

u.o

'~ ,40

,!10

~

'0

,10

40 ?0

?0

..Q

'

8

20

7

" s

:::1

"'"'

l.

;

/5

10 ~

s

"

4

'

I·

;

z

·8

·9

"'

... 0

~

·2 .4

., -4 ., ·'

.Of

"'

LIJ

210

.02 01'1

200 I~

ISO

1'70 160 ISO

I

,~

.07 .o6

.IS

IL.

140

.0'1 .01

120

-I~

.001

.11

110 100

}

~0

z

eo 70

,,_

60

.01

·8

.l#'

c: 0

~ ~-oz

.s..S :::1:::1

"'c:

.o~

.007 .00'

-z'

~

~

so

...

40

-~~

"§ID .So"'c:

...~ "'c: I"' ..."'c: ' >
- .c .c

.c

I.

2~

240 230 220

.06 ·l'i .01 .0, .o1 .Oio .Cl? .02 .006

I. . I ~

3

2

.2

·4

·4

1.'

·"".04

1.5

I')

.of.

' I.S

.08

,0';

.7

l.o;7

/.')

2

;

2

4

7

.a

2 I.

-~

'

(,

.,

-~

.;

I.

280 270 2fi0

.08

.7

·' ,, ., ., ., ., .. .,.a .,.,

4

4

8 7

10

.E

~-

'

~

0

N

...... :.9

10

15

.,;

',,

I

.0';

.s

/.;

\~

"; 's '

40

80

150

IS

2.0

70

100

7 4

2-,o

15

-~

I.

2

4z

'8 ?

~

60

~0

b

10

JOO

·l

0

z

~

V

0

z'

DePreister-type nomograph of k-values for light hydrocarbons.

20

206

Distillation, Flash TABLE 1

Antoine Constants for Some Miscellaneous Compounds

Compound Acetaldehyde Acetic acid Benzene i-Butylene Methyl acetate Methyl alcohol Methyl formate Propylene Propylene oxide Toluene o-Xylene m-Xylene p-Xylene Water

Boiling Points• CC)

Temperature Range ( C)

A

B

-24 to 28 30 to 170 0 to 160 -68 to 39 57 to 220 65 to 240 40 to 200 -lOO to 0.6 12 to 49 -26 to 136 50 to 200 45 to 195 45 to 190 60 to 150

18.008 16.5512 15.904 15.7556 16.726 18.5751 16.9289 15.7028 17.685 16.586 16.119 16.142 16.099 18.3443

-3332.185 -3262.0934 -2788.940 -2126.1296 -2921.778 -3632.649 -2878.6253 -1807.5410 -3395.184 -3448.107 -3396.186 -3367.599 -3347:249 -3841.2203

0

c

( l)

(2)

273.0 20.0 21.0 211.0 117.9 118.1 220.790 79.8 80.1 240.0 -6.3 -6.9 57.6 57.7 231.9 64.7 239.2 65.0 248.2 31.6 31.5 -47.7 -47.7 247.0 34.5 273.0 34.2 235.880 ll0.6 110.6 213.686 144.3 144.4 215.105 139.0 139.1 215.307 138.4 138.4 228.0 100.0 100.0

•columns (I) and (2) give the calculated and actual boiling points, respectively, for the various compounds at atmospheric pressure. The calculated values are obtained from the Antoine equation using the relevant figures for A, B, and C.

( 15)

and the iteration con verges rapidly. Dew Points The dewpoint of a vapor (i.e., the temperature at which the first drops of liquid condense out of the mixture) may be similarly evaluated since Eqs. (6) and (7) hold subject to the constraint (16)

We therefore have to solve ( 17)

to find the temperature and relevant liquid composition. The feed to the system must evidently be below its dew point for flash distillation to be possible. Using the approach given for bubble point, we have (18)

207

Distillation, Flash

and

Equations (15) [with the subscript d replacing b], (18), and (19) then provide the iterative solution for the dew point. Illustration 11 Problem. A bottled gas has the following composition (mole%): ethane 2%, propane 54%, i-butane 21%, n-butane 16%, i-pentane 3%, and n-pentane 4%. Confirm that its boiling point is about 50°F if the respective k-values are 5.08, 1.34, 0.52, 0.373, 0.133, and 0.101 at that temperature and the prevailing pressure.

Solution. (0.02

X

Evaluating ~);x; is as follows:

5.08)

+ (0.54

X

1.34)

+ (0.21 X 0.52) + (0.16 X 0.373) + (0.03 + (0.04 X 0.101) = l.OOi

X

0.133)

Note: Each of the parenthetical terms gives the y;'s. This is near enough to unity and therefore the bubble point is 50°F as suggested.

Illustration Ill Problem.

A natural gas has the following (volume %) composition:

The following k-values were found at 10 atm absolute:

Assuming ideal solution laws and a perfect gas phase, obtain an estimate of the maximum pressure at which the gas can be pumped in order to avoid condensation in the pipeline. The ambient temperature is the same as that at which the k-values were determined. Solution.

Raoult's law applies so that k; =pt/P

or

k; 1. Also 'f.z;/k; = 1.103

therefore the mixture is below its dew point since 'f.z;/k; > 1. Remembering Eqs. (24)-(27), we can write S=

0.15

X

0.87

R+l.87

0.40

X

0.42

+---R+l.42

0.35

X

0.29

R+0.71

Guessing at a liquid to vapor split of I: 3, i.e., R similar calculation, s; = 0.12088, whence

0.06

X

0.40

0.04

R+0.60

X

0. 73

R+0.27

= 0.33, then S1 = -

0.01675 and, by a

Likewise

s2 = s3 =

-0.003355;

S2

-o.oo02142;

s; =

= 0.07665;

R~

o.06704;

= 0.5124

R4 = 0.5156

We would normally terminate the computation at this juncture because of the closeness of S 3 to zero. However, one more cycle has been calculated from which we again observe that R 4 = 0.5156 and we have achieved convergence to four decimal places. As R = 0.5156, it follows that V= 65.98 mol/100 mol feed and L = 34.02 mol/ I 00 mol feed. The composition of the liquid and vapor products also follow as

X;

(mol%)

Y; (mol%)

9.53 17.82

2 31.32 44.47

3 43.28 30.73

4 8.15 4.89

5 7.72 2.08

Totals 100.00 99.99

Illustration VI Employing the fluid described in Illustration IV, it is desired to find the temperature at which a split according to R = 0.5 is achieved in a flash distillation process if it is operated at atmospheric pressure.

Problem.

Solution. Using the method described in Eqs. (28)-(32), aided by a simple computer program, the computation yields t as 110.44°C within a 0.5% limit of error imposed on the summation. A maximum of four iterations were required to attain this. A precis of the results is given in Table 3 and Fig. 8.

Distillation. Flash

213

TABLE 3

Solution of Illustration VI [feed composition (mol%): (I) 41.0 benzene, (2) 29.0 toluene, (3) 11.0 a-xylene, (4) 10.0 m-xylene, (5) 9.0 p-xylene. R = 0.50, F = 1.00, V= 0.67, L = 0.33. Operating pressure, P = 760.00 mmHg. Flash temperature, t = !I 0.44°C] Vapor Composition (mol%)

Liquid Composition (mol%)

(I)

(2) (3) (4) (5)

50.67 28.96 7.02 6.96 6.35

21.64 29.07 18.94 16.06 14.28

Total

99.96

99.99

Component

First Trial

Second Trial

Third Trial CC)

50.00 112.52 110.43

100.00 109.91 110.45

150.00 93.34 109.20 110.44

CC)

ln

ft 12 (3

CC)

(4

150

r-------------------------------~

- - - Flash calculations ------ Limiting value

t

u 0

~

~ 100 ~

~

E

~

so

0

2

3

4

Number of iterations FIG. 8.

Convergence for typical flash distillation calculation (Illustration VI).

Distillation, Flash

214

The Treatment of Complex Mixtures The computational procedures that we have already discussed are relevant only to well-defined binary or multicomponent mixtures, the properties of whose components are known or can easily be evaluated in the laboratory. However, one of the most important fluids in the world, crude oil, together with its derivatives, gasoline and other fractions, are not amenable to this type of treatment. Most crudes contain many thousands of different compounds, and a very different approach is normally required for dealing with all but the lightest cuts. It is possible, of course, to determine an experimental flash-vaporization curve for a specific pressure but the results of such an exercise are clearly of limited value. A better approach for the designer is to use the several empirical relationships which have been devised to attack this difficult problem. In order to consider these. we will first need to understand something of the characterization of refinery products. Essentially, our aim throughout will be to use a single. common measure of an oil fraction, its distillation curve, as the principal means of calculating all possible splits obtainable by flash distillation at various chosen temperatures and pressures. The reader should note the practical applications of this type of calculation are among the most important in petroleum refinery engineering. These include the performance and assessment of pipestills and fractionator feed preheaters. tower top, bottom and plate temperatures, heat transfer rates, and pressure drop in relation to two-phase flow.

Some Aspects of the Characterization of Oil Stocks Gravity

Specific gravity (SG) and API (American Petroleum Institute) gravity are expressions of the density of an oil stock. Normally evaluated at 60°F, the relationship between the two is 0

API SG

= (141.5/SG)- 131.5 = 141.5W API + 131.5)

(33)

As will be generally known, specific gravity is the ratio of the weight of a unit volume of oil to the weight of the same volume of water at the standard temperature. ASTM Distillations

These are standard tests applied to gasolines, naphthas, kerosines, and gas oils wherein the volumetric percentage distilled is recorded as the vapor tempera-

215

Distillation, Flash

ture changes. Virtually no fractionation occurs in this distillation and the hydrocarbons in the oil do not distill one by one in the order of their boiling points but as successively higher boiling mixtures. TBP Distillations

The true boiling point (TBP) curves of oil stocks may be obtained in any equipment that accomplishes a good degree of fractionation. Thus the separate hydrocarbons (or groups of them) are taken off overhead, one by one, in order of their boiling points and, in the simplest of cases, the curve of the results (temperature against percentage distilled) may show plateaus corresponding to the individual components. Figure 9 shows the relationship between TBP curves and API gravity for a representative sample of normal crudes. Furthermore, Fig. 10 gives a reasonable correlation between the slopes of the TBP, EFV and ASTM curves; Table 4 correlates their 50% boiling points.

EFV Curves

Equilibrium flash vaporization curves (EFV) can, as we have said, be established experimentally. It is much more convenient, however, to use the ASTM or TBP distillation data to generate these, and this is done via the 50% temperature, as represented in Fig. 11. Even so, the result only applies at the pressure-usually atmospheric-at which the distillation was perf0rmed.

1200~----------------------------~

1000

1

IJ..

0

API gravity of crude oil

I

I

I

I

I

I

I

I

lf/1/

////tit 1111/ltl

I 1111/1/ /,/////,,

800

~-

1600 ~

400

~

0~----L-----~----~----~----~

0

20

40

60

Pt:rcentage distillt:d

FIG. 9.

80

100

-

Average true boiling point distillation curves of crude oils.

Distillation, Flash

216

...c: ""

u

12 11

'-

~10 LL

0

v!" 9 > '-

"":::>

u c: .Q

...a "'

·;: 0

/

8

--------'

,.~Equilibrium " flash vaporisation

7

6 5

a. ~ 4

..."" '-

.c 3 0 I

~

2

0

l7i 3

4

5

6

7

8

9

10

11

12

13

14 15

Slope-true-boiling-point curve. °F/per centFIG. 10.

Relationships between the slope of the TBP and the slopes of the ASTM and EFV distillation curves.

Critical Properties

Although pseudocritical temperatures and pressures are thought to be best for the computation of the compressibility behavior of gases and vapors, they do not represent the true critical point which is required for studying phase

TABLE 4

TBP 50% Boiling Point (°F) lOO 200 300 400 500 600 700 800 900

Relation of 50% Boiling Points ("F) for TBP and ASTM Distillation Curves TBP 50% Boiling Point Minus ASTM 50% Boiling Point TBP Slope of 1

TBP Slope of 3

-5 -2

-17

-I

+1 +2 +4

-6

-3 +2 +4 +9 +14

TBP Slope of 5

TBP Slope of 7

TBP Slope of 9

TBP Slope of 11

-12 -6 -2 +2 +6 + 15 +21

-39 -24 -16 -7 -1 + 10 +20 +37

-40 -30 -18 -10 +1 + 16 +44

-42 -30 -21 -10 + 10 +50

217

Distillation, Flash

4

c: 8.

....c:

...u

70

'\ \

--TBP ----- ASTM

~ 60

I

\

\

0

Ll1

.g"' "'c:

::>

E

...0..

''

20

'

'''

0

Ll1

c:

.Q

....

,g "' Ci ·~

10

\

\

I

\

\ Slopes \above

\

\

I 8 \

\

\

\

\

\

\

\

\7

\ \

\

\

\

\

\

I

''

'''

'''

0

100

200

300

400

50 per cent boiling point FIG. 1 1.

\

40

....c: 8_ 30 ....c: '--

\

50

.t:::

...u

\

500

600

700

BOO

900

( T B.P or A.S.TM ) -

Relationship between distillation temperatures at 50% vaporized and the flash (EFV) temperature at 50%.

be ha vi or. The true critical temperature and true critica' pressure can be estimated from Figs. 12 and 13 as a function of ASTM boiling range and API gravity. With the aid of a Cox-type chart, i.e., a logarithmic pressure scale plotted against a reciprocal absolute temperature in the form of 1/(382 + t°F), and Figs. 14 and 15, the focal pressure and temperature may be entered on a phase diagram. Because of the form of the chart, all phase relationships are linear and the atmospheric EFV data then yield a complete spectrum of results.

The Flash Distillation of Petroleum Fractions Summary of Method Given the API gravity and a distillation test (usually TBP but preferably ASTM) on a sample of an oil fraction, it is possible to calculate the percentage of the mixture that could be flashed-off at any specified temperature and pressure. The method is summarized:

Distillation, Flash

218

Critical pressures of petroleum fractions, psia

----

100

200

300

400

500

600

700

100

200

300

400

500

600

700

ASTM volumetric average boiling point, °F FIG. 12.

BOO

900

800

---

True critical pressures of petroleum fractions.

4BO r---------------------------------------~ lL

0

Lines of constant 0API gravity

460

CL 440

a:i I

420 400

;:J..l380

~

360

[

340t=====

~

E 320

!!I

~

8

~----------

r--:=-::;,.....::.:.

300

·.;; 280 ·;::

u

260 240 .___

100

_.__ __._ _....__ __.__ _..__ _.__ _.____...J 200 300 400 500 600 700 BOO 900

AS.T.M. volumetric average boiling point FIG. 13.

1.

2.

°F

--

True critical temperatures of petroleum fractions.

Calculate the average volumetric true boiling point (graphically). Find the corresponding ASTM curve (i.e., slope, average volumetric boiling point): Fig. 10 and Table 4.

Distillation, Flash

219

3.

Establish the atmospheric equilibrium flash vaporization curve (i.e., slope, 50% flash temperature): Figs. 10 and 11. Estimate the critical properties of the mixture from which the focal point may be determined: Figs. 12 to 15. Draw the Cox-type chart.

4.

5.

600 r---r--.:--.:-.-~...--.----------------,

-~ Q.

O::u 500

~

iil

..."' L.

400

Q.

0u

:;::;

5

300

Lines of constant A.STM. 10% to 90% slope .°F/%

~

a_U..

FIG. 14.

Phase diagram focal pressure of petroleum fractions.

Lines of constant ASTM 10% to 90% slope.°F/%

FIG. 15.

Phase diagram focal temperature of petroleum fractions.

Distillation, Flash

220

An illustration of this procedure is given below and is abstracted from Nelson [1] as are most of the related graphs. This book also contains enthalpy charts which permit an energy balance to be carried out on the flashing operation. While it is possible to calculate the split that may be expected from the flash distillation of an oil, as indicated above, it is not so convenient to find the distillation curves of the flash products. Probably the best method is to apply the approach, described in the foregoing, for multicomponent distillation. This is done by considering the feed as a series of narrow boiling fractions. Percentages are converted to mole fractions and then equilibrium ratios are selected from the best available data (vapor pressures or k-values). An iterative procedure then follows in order to match the calculated with the empirical EFV curve. Illustration VII Problem. A 28° API oil has the following TBP distillation curve. Construct the flash distillation phase diagram.

Volume distilled, % Temperature, oF

5 428

10 447

30 494

50 549

70 590

90 672

Solution. The (10-90) TBP slope is (672 - 447)/80 = 2.82°F /%. According to Fig. 10, the ASTM slope should be about 1.97. Now the average volumetric true boiling point is the mean height of the TBP curve. Calculating the area of rectangles lying beneath this curve gives 54,990, and dividing this by the length of the base line (i.e., 100), we find a temperature of 550°F (see Fig. 16). It should be noted that the rectangles are arranged so that the triangular portions above and below the curve are of equal area as near as can be judged by eye. Table 4 shows that the ASTM average volumetric boiling point will usually be about S'F lower, i.e., 545°F, for these conditions. We now need to establish the atmospheric equilibrium flash vaporization curve. To do this, Fig. 10 gives a slope of 1.03 and Fig. 11 a 50% temperature of about 5°F lower than the TBP 50°io temperature, i.e., 545°F. Using this information, the flashing at 14.7lb/in. 2 abs is given by Vaporization, % Temperature, oF

0 494

20 514

50 545

80 576

100 596

Since we need to find the EFV properties of this oil under all conditions, we must next consider the critical properties of the mixture. The critical pressure is read from Fig. 12 by starting at 545°F, reading up to 28° API, then to the left to a slope of 1.97 (the ASTM slope found above) and finally at the top, we locate 315lb/in. 2 abs. Next, the critical temperatue from Fig. 13 appears as 365°F above the ASTM volumetric average boiling point (again see above), i.e., 910°F. Finally, Figs. 14 and 15 give the focal point as: For pressure, lb/in. 2 abs = 315 For temperature, oF= 910

+ 80 =

+ 48 =

958

395

Distillation, Flash

221 800 ~--------------------------------------~

Area=

15 x 428 + 20

X

+

25 x 494 +

590 + 672

Hence average volumetric

u.

20 x 549

20

= 54990

700

)

X

boiling

point

true

54990 I 100

= 550 °F 600

0

~::>

+'

...c

~

E ~

500

400

0

10

20

30

40

50

60

70

80

90

100

Volume Ofo distilled

FIG. 16.

Illustrative TBP distillation and calculation of average true boiling point.

The focal point is plotted on the phase diagram (pressure vs temperature) and connected by straight lines to the data for atmospheric flash vaporization. Figure 17 represents the complete solution to this problem. From such a diagram it is possible to predict the percentage vaporization under any conditions of temperature and pressure up to the focal point.

Equipment The general configuration of equipment for flash distillation is shown schematically in Fig. 1 and is seen to be uncomplicated. The type of flash vessel to be used and the materials of construction are usually selected on the basis of experience with the processed medium together with the envisaged operating conditions. The criteria for the successful design of such a drum are:

222

Distillation, Flash

6oor----------------------------,

500 400 300

Focal point-.. Critical poi1t' /

200 150

-~100 a. ~- 70 ::>

::l so

~ 40

30 20 15 10 L-----'----L----'----'--'---'---'-----'----' 450 500 550 600 650 700 750 800 900 1000

Temperature, °F FIG. 17.

1.

2. 3.

-

Phase diagram derived from Illustration VII.

That the residence time should be adequate for equilibrium to be attained. That the geometry permits satisfactory process control. That the vapor and liquid phases are completely disengaged.

A residence time of around 15 min (based on an ullage of 50/~) will normally be found to be sufficient to satisfy the above requirements. In addition, a demister or entrainment separator may be required. Details of this type of approach have been given recently for reflux drums by Sigales [10]. A further article by the same author [11] gives a method of tackling three-phase systems. Although not described in the foregoing, this type of system is occasionally encountered in flashing operations; see also Ref. 4. Finally, we might observe that several of the comments on the design of equipment for desorption processes are also appropriate here (see Desorption). In particular, the remarks concerning the hardened trim of the control valve and the design of a vessel for degassing by pressure reduction are relevant.

Alternative Method for the Flash Distillation of Petroleum Fractions The method outlined above gives reasonable results at pressures close to atmospheric but sometimes yields doubtful values under vacuum or pressure conditions.

223

Distillation. Flash

An alternative approach was proposed as long ago as 1933 by Katz and Brown [12]. In this method the TBP curve is divided into a large number of sections in such a way that the average boiling point of each section corresponds to the boiling point of an identifiable hydrocarbon. The total range is thus represented by a large number of pseudo-components and can be treated as if it was a mixture of these. The method already described for multicomponent mixtures of known compounds may then be applied. The disadvantage of this method is that a large number of pseudocomponents must be used to represent adequately the TBP curve of a crude oil. This results in an extremely tedious trial-and-error calculation. The method has therefore only become practical as computers have been made available. Actual k-values for the lightest hydrocarbons can be used directly as functions of both temperature and pressure, while for the heavier pseudocomponents they can be estimated [13]. This approach is claimed to be more rapid and accurate than the empirical methods [14]; it also gives the properties of the liquid and vapor phases.

Symbols 0

API

A,B,C F

k L p p q

R

s

1

V X

y

z y

degrees API (defined in Eq. 33) Antoine constant (defined in Eq. 10) molar (usually) feed flow rate (mole/time) k-value (defined in Eqs. 6, 8, and 9) molar (usually) liquid product flow rate (molejtime) vapor pressure or partial pressure (atm, lb/in. 2 abs, mmHg as appropriate) total system pressure (atm, lbjin. 2 abs, mmHg as appropriate) fraction of feed which remains liquid after flash distillation ratio ( =L/V) sum (see subscripts b, d, f, F, and n) temperature (°C, op as appropriate) molar (usually) vapor product flow rate (mole/time) mole (usually) fraction of more volatile component in the liquid product mole (usually) fraction of more volatile component in the vapor product mole (usually) fraction of more volatile component in the feed activity coefficient

Subscripts

b c d

f

F F

m n

bubble-point iteration critical property dew-point iteration flash iteration when R is unknown flash iteration when t is unknown focal property refers to component i total components iteration number

Distillation, Flash

224 Superscripts

*

refers to pure component indicates first derivative

References 1.

2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12.

13. 14.

W. L. Nelson, Petroleum Refinery Engineering, 4th ed., McGraw-Hill, New York, 1969. C. J. Liddle, Br. Chem. Eng., 16(2/3), 193(1971). C. L. DePriester, Chem. Eng. Prog., Symp. Ser. 7, 49, 1(1953). E. J. Henley and E. M. Rosen, Material and Energy Balance Computations, Wiley, New York, 1969. R. R. Driesbach, Physical Properties of Chemical Compounds, Vol. 1, American Chemical Society, Washington, D.C., 1955. Ref. 5, Vol. 2, 1959. Ref. 5, Vol. 3, 1961. T. E. Jordan, Vapor Pressure of Organic Compounds, Interscience, New York, 1954. J. H. Perry (ed.), Chemical Engineers' Handbook, 5th ed., McGraw-Hill, New York, 1973. B. Sigales, Chem. Eng., 82(5), 157 (March 3, 1975). B. Sigales, Chem. Eng., 82(20), 87 (September 29, 1975). D. L. Katz and G. G. Brown, Ind. Eng. Chem., 25, 1373 (1933). S. T. Hadden and H. G. Grayson, Pet. Refiner, 40(9), 207 (1961). 0. H. Hariu and R. C. Sage, Hydrocarbon Process., 48(4), 143 (1969). C. J. LIDDLE

Distillation, Optimization

Introduction Development of a vapor-liquid equilibrium model for use in distillation optimization is demonstrated with a heterogeneous azeotropic separation. Basic data are taken from a calibrated laboratory Oldershaw column operated DOI: 10.1201/9781003209799-10

225

Distillation, Optimization

at total reflux over the concentration range to be used in the design. A computer program utilizes data to optimize trays vs reflux design. The optimum design is verified on a laboratory Oldershaw column. Attempts to use computer methods for optimum distillation tower design in petrochemical purification are plagued with insufficient vapor-liquid equilibrium data. Typical separations involve highly polar compounds, water, alcohols, aldehydes, ketones, acids, and esters along with hydrocarbon compounds. Azeotropes, especially those involving water, are common. Therefore, ideal vapor pressure behavior rarely exists in such systems. Design of distillation separation for such systems has traditionally been made by trial-and-error laboratory development. Limitations in development time rarely permit full economic optimization of such designs. Considerable effort has been made in recent years to develop methods for converting binary pair vapor-liquid equilibrium data into full multicomponent data. Although use ofthese techniques has increased, the extent of binary pair data available from the literature and proprietary sources is insufficient to permit computer design of a large percentage of distillation separations. A design method has evolved which is based on determining vapor-liquid equilibrium data from total reflux column operation using the Fenske equation. The key to the method is the conversion of average relative volatility data to temperature-dependent, composition-implicit, vapor pressure correction factors. Utility of the method is illustrated with the design and verification of the azeotropic separation of n-butyl acetate from acetic acid using water as the azeotroping agent. The system possesses a heterogeneous liquid phase over a portion of the concentration range and was chosen because of its highly nonideal phase equilibria. Target specifications for the separation were 0.01 wt.% acetic acid overhead and 0.5 wt.% n-butyl acetate in the base. Water to satisfy the n-butyl acetate-water azeotrope was added to the reflux. Up to about 5% excess water could be tolerated in the base stream.

Vapor-Liquid Equilibrium Method Fenske Equation

The average relative volatility (riii) for the column section tray m to tray n can be calculated from the Fenske equation: _ log rx;j

=

log[( Y;/Yj)n/( Y;/YJm-d E[n- (m- 1)]

(1)

where E is average tray efficiency for the tray zone. Note that Eq. (1) is expressed in terms of vapor mole fraction and avoids the need to sample the heterogeneous liquid phase sometimes present. Average relative volatilities are determined from laboratory total reflux

226

Distillation, Optimization

distillation data using Eq. ( 1). To insure applicability of the data, the total reflux distillation is carried out by using actual feed and by establishing overhead and base compositions approximately equal to those of the projected distillation design. Vapor Pressure Correction

The computer method used to make tray-to-tray calculations accepts vapor pressure corrections as a method of inputing vapor-liquid equilibrium data. The data are inputed at a series of temperatures covering a range which exceeds the column temperature range. The correction factors are curve fitted by the computer program using a conventional polynomial method. Average relative volatility data from Eq. ( 1) are converted to vapor pressure corrections by the following method. A vapor pressure correction f3 is defined as follows: (2)

where Pt ( T) is actual partial pressure of component i in the distillation mixture corresponding to temperature T. Pi( T) is pure component i vapor pressure. The bubble-point expression in terms of Eq. (2) is:

I

i= 1

Pt(T)

=

I

i= 1

{3;(T)X;(T)Pi(T)

= n(T)

(3)

where n is the total pressure in the column at the tray corresponding to temperature T. Also, from Eq. (2): {3;(T)Pi(T) _ rx· T f3j(T)Pj(T)- ,,( )

(4)

Experimental composition data used to define &"in Eq. ( 1) can also be used in the calculation of sets of {3. For example, vapor composition at trays m- 1 and n establishes average relative volatilities for the tray zone mn. Since, at total reflux, X; m = Y;m- 1. liquid-phase molar concentrations at trays m and n + 1 are available from the vapor samples. Thus, using the &"ij for the tray zone mn and the vapor compositions at trays m - 1 and n, vapor pressure correction for each component can be calculated for the temperatures of trays m and n + 1. Note that composition dependence is implicit in the vapor-pressure corrections since actual composition data are used in the calculations. Comparing Eq. (2) to Raoult's law shows that f3 is a correction for nonideal vapor pressure. At low pressure (e.g., 1 atm) f3 is approximately equal to the liquid-phase activity coefficient [2].

227

Distillation, Optimization

Experimental Data Relative volatility data were taken using a 2-in. vacuum-jacketed Oldershaw column. The experimental setup is shown in Fig. I. Surge volume in the re boiler and surge vessels for both overhead phases permitted adjustment of overheadto-base split and column water inventory. The number of trays used in the total reflux runs is ideally the number which exactly gives the target overhead and base specifications with proper adjustment of overhead-base split. In this way the relative volatility data apply exactly to the composition range of interest. However, to make the trial-and-error selection of trays practical, it is assumed that the data are applicable if the data approximately cover the desired range. Using a 15-tray column and with overhead-to-base inventory adjusted to given-butyl acetate and water-base vapor concentrations of 0.26 and o.l wt.%, respectively, the composite acetic acid overhead was 0.5 wt.%. The base

SYPHON BREAKER

REFLUX HEATER

VAPOR

AQUEOUS PHASE RECEIVER

+

THERMO SYPHON REBOILER

FIG. 1.

Laboratory total reflux column.

Distiilation, Optimization

228

compos1t10n was close to the target; however, overhead acetic acid concentration was somewhat higher than the target value of 0.01 wt.%. It was assumed that the range covered by these data was sufficiently close to the entire range for the data to be applicable. Tray efficiency measurements were made using the methanol-water binary system at atmospheric pressure. Relative volatility data for the methanolwater system were taken from Ref. 1. The system was chosen because of the similarity of liquid viscosity.

Results and Discussion Relative Volatility

The composition ofvapor samples from the total reflux run are shown in Table 1. Consistency of analyses at 6 and 7~ h after start-up showed that the column was at steady-state. Average relative volatility data calculated using Eq. ( 1) are listed in Table 2. An overall tray efficiency of 63% was used in the calculation of relative volatility. All data needed to calculate f3 are listed in Table 3. Calculated vapor pressure corrections are shown in Fig. 2. The temperature dependence of relative volatility was included in the calculation of the correction factors as shown in Table 3. The magnitude of f3 for n-butyl acetate (i.e., from 3 to 5 in Table 3) shows the degree of nonideality exhibited by this system. Computer Design

The program was set up for liquid feed at the bubble point with concentrations of 60 wt.% acetic acid and 40 wt.% n-butyl acetate. Reflux was returned to the top tray at the bubble point. The water phase, corresponding to the organic

TABLE 1

Total Reflux Data: Vapor Composition (mole fraction)

Sample 1 Acetic acid n-Butyl acetate Water Sample 2 Acetic acid n-Butyl acetate Water

Base

Tray 5

Composite Overhead

0.82 0.0012 0.18

0.41 0.0054 0.58

0.004 0.26 0.74

0.83 0.0010 0.17

0.41 0.0056 0.58

0.004 0.26 0.74

229

Distillation. Optimization TABLE 2

Calculated Relative Volatility: Effective Relativity (Eq. 1)

Sample 1 Acetic acid n-Butyl acetate Water Sample 2 Acetic acid n-Butyl acetate Water

Base, Tray 5

Overall

Tray 5, Overhead

1.0 2.06 1.81

1.0 3.15 2.06

1.0 3.91 2.20

1.0 2.16 1.85

1.0 3.16 2.05

1.0 3.38 2.16

phase product, and the excess water, corresponding to approximately 5 wt.% of the base stream, were fed to the top tray at the bubble point. Composition of the auxiliary feed was set at 0.003 wt.% acetic acid and 0.5 wt.% n-butyl acetate. This composition closely matched the actual water-phase composition. Vaporpressure corrections were included in the program as previously discussed. The program was checked against the original total reflux data as a preliminary test. Using near total reflux conditions (9 .0 theoretical trays-14.3 actual trays compared to 15 experimental and 1000 reflux ratio), the calculated acetic acid overhead and n-butyl acetate in the base compared closely with the original experimental data. Calculated acetic acid overhead was 0.42 wt.%

TABLE 3

Calculated Vapor Pressure Corrections

Temperature ("F)

Vapor Pressure (mmHg)

Total Column Pressure (mmHg)

Effective Liquid Phase Mole Fraction (Table 1)

Effective Relative Volatility (Table 2)

Vapor Pressure Correction (Eqs. 3 and 4)

1.0 1.0 1.0

1.027 1.063 1.135

2.91 3.15 2.06

5.11 4.22 2.93

2.20 2.06 1.81

1.25 1.19 1.11

Acetic Acid

192 219 237

281 476 662

760 825 860

0.0036 0.41 0.82 n-Butyl Acetate

192 219 237

220 378 528

760 825 860

192 219 237

506 875 1227

760 825 860

0.26 0.0054 0.0012 Water

0.74 0.58 0.18

Distillation, Optimization

230

- -r--.. - n-BuOAc

4

fJ

3

2

...

WATER

""""'

-

HOAc 0

190

FIG. 2.

200

--

~

210

220

230

240

Experimental vapor pressure corrections.

compared to an actual value of0.50 wt.%; calculated n-butyl acetate in the base was 0.18 wt.% compared to 0.26 wt.%. The close check indicated that the program is consistent with the data on which it is based. Experimental Verification

The final test was to check a finite reflux rate design with a laboratory continuous run. A 60/40 wt.% acetic acid/n-butyl acetate feed was separated continuously in a 40-tray, 2-in. Oldershaw column. Hourly samples showed column lineout about 4 h into the run. The reflux ratio was set at 1.0; the liquid flow including internal reflux (resulting from heat losses) at the column base, however, corresponded to 1.5 reflux ratio. Assuming uniform heat losses along the column, since feeds were introduced at their bubble points, the effective reflux ratio was 1.25. Table 4 summarizes the test and compares the experimental results to the computer design for the same conditions. The experimental overhead organic phase contained 50 pp m acetic acid; the base contained 0.4 wt.% n-butyl acetate. The calculated separation for the 40 (actual) tray, 1.25 reflux ratio case was 30 ppm acetic acid overhead (composite of both phases) and 0.25 wt.% n­ butyl acetate in the base. The molar partition coefficient for acetic acid between the overhead phases was measured at 1.66 organic/aqueous. The calculated acetic acid in the organic phase based on the two-phase composite was 20 ppm. The calculated temperature profile is shown in Fig. 3 and compared with measured temperature points. The column pressure drop used in Eq. (3) in the evaluation of correction factors was 1.5lb/in 2 • A similar computer design was made using corrections based on overall column average r:x values. As would be expected, the second set varied less over the temperature range. The set for n-butyl acetate, for instance, varied from 3.79 at 248°F to 4.96 at 185°F (ti = 3.15) compared to 2.96 and 5.94 in the case of variable r:x. The calculated separations using the two sets of vapor-pressure corrections were essentially identical. However, intermediate compositions deviate from actual when average r:x values are used in the calculation.

231

Distillation, Optimization TABLE 4

Comparison of Calculated and Experimental Separation Computer Design

Lab Run

Trays, 40 actual (25 theoretical) Reflux ratio, 1.25 Auxiliary feed, %of feed, 19 Overhead/base, wt. 0.61 Feed tray, 20 actual (13 theoretical) Products, wt.% Overhead Composite of phases 0.003 Acetic acid 71.0 n-Butyl acetate 28.9 Water Acetic acid : Organic, 0.002 (calc) Aqueous, 0.005 (calc) Base 95.20 Acetic acid n-Butyl acetate 0.25 4.55 Water

40 1.25 Water phase + excess water (equivalent to 19) 0.60 20

0.0055 (calc) 73.7 26.3 0.005 0.007 93.5 0.44 6.0

The heterogeneous azeotrope was handled like ordinary components. The column material balance fixed overhead water and n-butyl acetate at the azeotropic composition. The program with corrected vapor pressure would not

CALCULATED •

EXPERIMENTAL

250

~ 200

150

0

~~

10

20

r-., 30

TRAY

FIG. 3.

Calculated temperature profile.

40

232

Distillation, Optimization

be applicable to the case of higher n-butyl acetate in the base. To calculate higher re boiler concentrations of n-butyl acetate forces the overhead water jn­ butyl ratio away from the actual azeotropic composition. The case of higher reboiler concentration of n-butyl acetate could be designed using vaporpressure corrections taken from total reflux data with the appropriate n-butyl acetate profile. This points to the importance of establishing product specifications before the total reflux step. Column optimization using the tray-to-tray computer program with corrected vapor pressure is a straightforward procedure and is not included. This article is adapted from Chemical Engineering Progress, 68(10), 62-65

(1972).

Symbols average tray efficiency of column zone where average relative volatility is being evaluated Y;n = actual partial pressure pure component vapor pressure mole fraction in vapor phase effective liquid mole fraction

E

Pt

Pi Y;

X;

Greek Letters

average relative volatility vapor-pressure correction column total pressure

Ci

/3

n

Subscripts i, j

m, n

refers to component refers to tray number

References 1.

2.

P. Dalager, J. Chem. Eng. Data, 14, 298 (1969). H. R. Null, Phase Equilibrium in Process Design, Wiley-Interscience, New York, 1970, pp. 93-97, 126-132. W. T. MITCHELL

Distillation Simulation

233

Distillation Simulation Simulation is a powerful method for qualitative and quantitative analysis of distillation process operating performance as well as for the initial design and revamp of distillation processes [14, 24, 39, 42, 60, 66, 75, 90, 92, 104, 105, 107]. The continuing increase in the use of distillation simulation is a result of the improved cost/capability of the computers used for simulation as well as the ever-increasing value ofthe results. With the rapid rise in fuel costs, distillation studies with special emphasis to energy conservation have become increasingly important [83, 101]. Other benefits of distillation simulation studies include increased capacity, improved product recovery, reduced pollutant discharge, more effective designs and improved operability, and in the case of dynamic simulation, the design of improved control systems.

Types of Distillation Columns Distillation columns can be classified as simple or complex. Simple columns separate a single feed into an overhead and a bottom product whereas complex columns handle multifeed streams and/or produce sidestream products. Another classification is based on the number of components in the feed, i.e., binary or multicomponent. Complex columns, such as crude units, contain several hundred components which are separated into four to eight product streams. In terms of simulation, these systems are simiiar except that the higher dimensionality introduces mathematical complications. Other variations on distillation columns are batch vs continuous, packed vs tray, absorption, stripping, azeotropic, extracting, or simultaneous reaction in distillation columns [35, 43, 102]. To illustrate the fundamentals of distillation simulation, we shall limit our considerations to continuous tray distillation .;olumns.

Types of Simulation Distillation simulation can be either steady state or dynamic. Steady-state simulation, not time dependent, is generally used in design and process studies. Steady-state simulation may be used to study different schemes for energy conservation, such as transporting heat from one unit to another by heat exchange or pollutant control by improved fractionation. Dynamic simulation, not as commonly used as steady-state simulation, plays a vital part in analyzing distillation control problems. Since dynamic simulation includes time as a variable, transient behavior of distillation column can be assessed [52, 103]. While the industrial use of steady-state distillation DOI: 10.1201/9781003209799-11

Distillation Simulation

234

simulation has become routine, dynamic simulation remains a tool used only for special studies.

Simulation Tools Digital computers are the predominant tool used in both steady-state and dynamic simulation studies [7, 9, 13, 25, 36, 44, 47, 50, 56, 58, 63, 64, 67, 69, 95]. Software packages for steady-state and dynamic simulation are available [29, 37, 45, 66]. Simulation routines are also used for composing different distillation models [2, 3, 6, 11, 20, 28, 31, 48, 49, 70, 72, 80, 91]. Analog and hybrid computers are other choices for simulation studies [26, 30, 32, 57, 76, 97]. Analog computers, as the name implies, are composed of electronic components. Hybrid computers include both analog and digital portions. Both types of computers are limited by the accuracy of the analog components. However, they are useful where fast solutions are desired. An operator-training simulator [8] is one example.

Simulation Techniques Detailed description of the techniques used in distillation column simulation is beyond the scope here. The intent is to provide an understanding ofthe value of distillation simulation, the nature of the task, the approaches that have been used successfully, and references to sources for details. Steady-State Simulation

Shortcut methods can be used for approximate analysis of distillation columns. Shinskey [93, 94] describes the effect of material and energy balance control using Fenske's equation. Generally, steady-state simulation of distillation columns involves tray-to-tray calculation to obtain heat and material balances at each tray and for the overall column. The heat and material balances for a simple column (Fig. 1) are represented by the following equations: F - D - B= 0

Material balance : Heat balance: where

F =feed rate

Fhf- Dhd- Bhb- Qc

D = distillate product rate B =bottom product rate

+ Qr = 0

235

Distillation Simulation

COOLING WATER

REFLUX

FEED

DISTILLATE (]))

COLUI"'i

BOTTOMS

FIG. 1.

Simple distillation column.

hf, hd, hb = enthalpies of feed, distillate, and bottom, respectively Qc = heat removed by condenser Qr = heat added to reboiler

A rigorous method for distillation simulation is tray-to-tray calculation. The simulation model is related to the actual column by tray efficiency. Murphree tray efficiency [73] is a localized representation for individual trays. Generally, overall tray efficiency is used; i.e., the ratio of simulation (or theoretical) tray number to actual column tray number. Drickamer-Bradford [22] relate tray efficiency to feed stock viscosity. O'Connell [74] uses the product of relative volatility and viscosity. AIChE efficiency correlations [1, 18] include factors such as mass-transfer, mixing, and entrainment. For operating units, overall tray efficiency can be determined from plant data. The heat and material balances for tray-to-tray simulation can either be represented by individual tray or by envelope techniques. An individual tray balance is developed below (Fig. 2) where tray n is the reference tray. Material balance: Heat balance:

L. = 0 + Ln-lhn-1- L.h. = 0

v. + 1 - v. + L. - 1 Vn+IHn+l- V.H.

236

Distillation Simulation

vn-1

Ln-2

n-11

vn

TRAY

ln-1

TRAY Vn+t

Ln

n+1

t vn+2

FIG. 2.

where

TRAY

l

Ln+1

Material balance for tray-to-tray.

V = vapor rate L = liquid rate H = enthalpy of vapor h = enthalpy of liquid n- 1, n, n + 1 =denote tray number above and below reference tray, respectively

Similar balances can be made for the top, feed, and bottom tray with the condenser, feed, and reboiler included in these equations. The envelope technique is shown in Fig. 3 where equations are developed for a section of the column. For the top section, the following heat and material balance equations are used for tray n: Material balance: Heat balance :

V"+ 1 - Ln - D Vn+1Hn+1- Lnhn- Dhd- Qc

=0 =0

Similar equations are used for feed, bottom, or other trays by simply drawing an envelope around the column section in question. In addition to the above equations, a material balance of each component can be made: Overall: Tray balance: Envelope:

Vn+1Yi,n+1-

F!; - Ddi - B bi = 0 VnYi,n + Ln-1Xi,n-1- LnXi,n = 0 Vn + 1 Yi, n+ 1 - LnXi, n - Ddi = 0

237

Distillation Simulation CONDENSER (Qc)

REFLUX

DISTILLATE CD)

COLUMN

n Tray

FIG. 3.

where

Envelope technique for column section.

!;, d;, b; =mole fractions of i-th component in feed, distillate, and bottom products Y;.n~ 1 , X;,n-l =mole fractions of i-th component in vapor and liquid leaving tray n - 1

Besides heat and material balance equations, selection of thermodynamic properties for the fluid mixtures is an important consideration. For most fluid mixtures, departure from the ideal state is a rule rather than an exception. Textbooks [23, 46, 82, 85] provide detailed discussions. The API Technical Data Book [4] contains this information for hydrocarbons. Vapor-liquid equilibrium and enthalpy data are the most important physical properties needed for distillation simulation. The vapor and liquid streams leaving each tray are usually considered to be in equilibrium and related by the equilibrium constant (K) with the following equation:

where K;,n is the equilibrium constant for i-th component at tray n conditions. Instead of K values, relative volatilities (a) are frequently used and are as defined below:

=K K; n

ai-r,n

r,n

Y;,nXr,n X y i,n r,n

where a;~,,n is the relative volatility of i-th component relating tor component at tray n conditions.

238

Distillation Simulation

No general quantitative equilibrium relationship for all mixtures encountered in the chemical industry is available. Thermodynamics provides a coarse but reliable framework. Relationships for any specific mixture are generally obtained experimentally. A paper [55] shows relationships developed for hydrocarbons. Similarly, enthalpy data need careful screening. Some mixtures behave ideally so that properties can be computed from the properties of the pure components. For most mixtures, nonideal behavior should be considered in computing the enthalpy of the mixture. Reference 54 shows some developments in correlating hydrocarbon enthalpy data. Most frequently, the solution of distillation simulation equations reduces to a two-point boundary problem where some variables are known at each end of the column. These end conditions are linked by the heat and material balance of each tray as described earlier. Iteration techniques are necessary to obtain solution to the problem. With these inherent characteristics of distillation columns, the major difficulty in simulation besides the dimensionality is the convergence of the solution. LewisjMatheson [59] and Thiele/Geddes [99] are two common approaches used for tray-to-tray simulation. The former method assumes end conditions of the column and computes the heat and material balance for each tray from both ends of the column toward the feed tray. If matching is not achieved at the feed tray, computation would be reiterated with new end condition values. Convergence methods [34, 65] are used for selecting these new conditions. The ThielejGeddes method assumes a temperature profile for the column and computes the conditions for each plate. If agreement is not achieved at all trays, new temperature profiles will be selected based on convergence methods. There are numerous convergence methods of which the cjJ-method [100] and mismatch [12] are two examples. Other authors [10, 77, 78, 81] extend or modify these methods. Distillation columns can be represented by a set of simultaneous equations that can be solved using matrix algebra [5]. The Newton-Raphson method [33] is a means for the successive approximation of the constants in the matrix. Relaxation methods [86] are uniquely different from above. They are basically simplified dynamic simulations of distillation columns following the general form Input - output

= accumulation

The accumulation term compensates for the mismatch in heat and material balance from tray-to-tray. As in actual operation, the steady-state solution is obtained when the column has reached stable conditions. Dynamic Simulation

Dynamic simulation is a powerful tool for studying the transient behavior of distillation columns. Dynamic simulation has been used effectively not only for control system design and evaluation, but also for operator training, determin-

Distillation Simulation

239

ing the consequences of equipment failure (e.g., failure of a pump supplying cooling water to condensers), investigating controlled-cycle operations, and determining the causes of operating problems. As a natural outgrowth of the increased capability and lower cost of digital computers, more definitive mathematical models can be used to provide more accurate solutions to problems economically. However, that is not to suggest that a rigorous solution of a comprehensive model is the proper approach to all problems. In contrast to the relative ease of conducting a steady-state study, a considerable investment in manpower must be made for a detailed dynamic simulation, even for such a mundane task as collecting the required physical property and equipment mechanical data. The approach used should be tailored to the problem. Judgment should be exercised so that an engineering solution is obtained without expending effort out of proportion to the value of the results. For example, for operator training, the model should represent a broad range of operations, i.e. start-up, shutdown, normal operations, and emergency conditions. However, the model need not be very accurate as long as key variables respond in the correct direction and time sequence with appropriate interactions. On the other hand, for studying the ability to closely control product quality using (1) a temperature controller alone, or (2) a discontinuous analyzer alone, or (3) an analyzer-temperature cascade, the range of operations to be represented is narrower. However, the model must be accurate for the comparison to be meaningful. Approaches that are used for dynamic simulation of distillation columns are described below. The literature references cited include mathematical models of varying degrees of complexity. Caution should be exercised when applying these models to specific problems. For example, some authors are quite rigorous in developing the models for the column itself but make gross simplifications for the condenser and reboiler. Such models would be totally inadequate for determining whether the column bottom level can be controlled satisfactorily by manipulating fuel rate to a fired reboiler or whether a flooded condenser provides better pressure control than a hot gas bypass system. A mathematical model representing the trays of a distillation column must consider dynamic mass balances for the components, dynamic energy balances, tray fluid dynamics, and physical properties. The complexity of the model developed will depend on the set of assumptions used. References are cited below that provide perspective to the range of approaches that have been used to effectively simulate distillation column dynamics. More thorough reviews are provided by such sources as Holland [45] and Rademaker, Rijnsdorp, and Maarleveld [84]. The transfer function approach [38, 41, 84] greatly simplifies the dynamics by lumping the sections ofthe column rather than representing individual trays. This approach is well illustrated in several papers. Franks [27] describes hybrid simulation for a four component, 40 theoretical tray column. The steady-state tray-to-tray equations are solved iteratively at high speed. The dynamics ofthe trays are represented by eight lumped transfer functions with the reboiler, condenser, and feed tray programmed on an analog computer based on the relevant equilibrium, heat, and mass balance equations. For a conventional analog simulation, Johnson and Lupfer [51] developed the necessary steady-

240

Distillation Simulation

state relationships by simple mass balances and regression equations generated from tray-to-tray runs made on a digital computer and used transfer functions for the dynamics. Portable instruments based on microprocessors are available for determining transfer functions experimentally [40]. Wahl and Harriott [106] made a comprehensive study of transfer functions for binary distillation columns and present correlations for estimating gains and time constants. As an example, the optimum control tray is determined for feed composition changes. The use of on-line identification to estimate transfer function parameters is considered by Smith and Corripio [96]. Lupfer and Parsons [62] describe use of a transfer function model of a distillation column to design a feedforward control system. Another study illustrating the use of simplified dynamics to evaluate control system performance is that of Conover, Nisenfeld, and Miyasaki [19]. Experimental data were used to determine the configuration of transfer functions and values for the parameters. An analog computer simulation was used to compare the initial response of distillate rate to load changes for several control systems. A noteworthy earlier paper is that of Lamb, Pigford, and Rippin [57] wherein the linearized perturbation equations for a binary distillation column are derived. An analog computer was used to estimate the frequency response of tray compositions to variations in feed composition. The transfer function approach can be used effectively for dynamic simulation of distillation columns for the set of problems where simplified and specific dynamics will suffice. This approach is especially useful in representing peripheral equipment such as instrumentation, pumps, and hold-up tanks. The development of more rigorous models suitable for numerical solutions using digital computers is well described in several references [28, 45, 68, 98]. In general, such models do not contain as many simplifying assumptions as do the transfer function models. Such factors as individual tray dynamics, multicomponent systems, and nonlinear effects can be treated readily. However, the tradeoff between modeling rigor and computational effort still must be made. Noteworthy references providing a cross section of methods are cited below. In principle, dynamic tray simulation is represented by the same set of basic equations as given for steady state with an added term for accumulation. This time-dependent accumulation term replaces the zero as given on the right-hand side of the steady-state equations. Peiser and Grover [79] developed a comprehensive, nonlinear model that included tray hydraulics, variable liquid holdup, and condenser and reboiler dynamics as well as the mass and energy balances and equilibrium relationships for a multicomponent system. The model was developed to help solve operating problems being encountered. These problems included flooding, unstable product composition, and erratic bottom liquid level. The nature of these problems dictated the need for the comprehensive model employed. The simulation study led to field changes which overcame the problems. Distefano et al. [21] simulated separation of a binary mixture using a nonlinear model. Using a predictor-corrector numerical integration method programmed in FORTRAN, response to a sequence of upsets was simulated. The simulation was verified by comparison with experimental data. Another binary system study based on a nonlinear model and including experimental

Distillation Simulation

241

verification is that of Sastry et al. [89]. They used dynamic simulation to demonstrate the effectiveness of a self-tuning regulator for control of product composition. Rosenbrock et al. [88] and Howard [47] present experimental results for multicomponent systems. Earlier work by Rosenbrock [87] presented the model and numerical techniques as applied to a binary system. Holland and eo-workers [45, 98] went beyond previously cited studies in representing tray phenomena. Perfect mixing is not assumed. Rather, the effects of transfer Jag, channeling, mass transfer, and mixing, are all considered. The numerical technique used for solution is an adaption of the e convergence method developed for steady-state simulation. The book [45] represents one of the most comprehensive sources of information on distillation dynamics. The book by Rademaker et al. [84] is another very comprehensive source. The approach taken by these authors is to be sparing in making simplifying assumptions to avoid neglecting potentially significant phenomena and then linearizing the resulting set of nonlinear differential equations to ease computation. The book includes extensive treatment of model development and control system evaluation with considerable experimental data supporting conclusions. A mammoth literature review is incorporated. For those seeking readily available models with which to conduct simulation studies, Franks [28] and Ju and Finkelstein [53] should be consulted. Both contain program listings, the former being FORTRAN based and the latter CSMP based. Franks develops models for routine stages as well as for feed stages, sidestream stages, reboilers, and condensers. These subroutines can be assembled to represent a specific distillation system. Simulation examples are presented of a batch distillation, start-up transient to steady state, and control system evaluation. The Ju and Finkelstein work was not directed to providing a model for general application. Rather the thrust was to describe a method for treating nonlinear, vapor~liquid equilibria without linearization by an iterative solution technique. However, a CSMP listing is presented of a distillation column model for a ternary mixture being separated in a system with 11 trays, a condenser, and a re boiler.

Examples of Application Four examples are presented to illustrate the application of simulation to solving industrial problems. The first two are examples of steady-state and last two of dynamic simulations. Steady-state simulation is used here to analyze the performance of a refrigerated multicolumn system called an ethylene rectifier as shown in Fig. 4. This configuration of columns resulted from expansion of an ethylene plant. The goal of the study is to minimize refrigeration consumption in the separation of ethane and ethylene [17]. Since the distillation system is composed of columns in parallel and in series, it is important to find the interactions for this complex configuration of columns by simulation. Example 1.

Distillation Simulation

242

FEED

FIG. 4.

Rectifier-section control.

The simulation model of the rectifier section, as shown in Fig. 5, was constructed with a packaged computer program for steady-state simulation of process equipment. Ratios of feed, intermediate reflux (reflux from bottom of Column 3), and re boiler duties to the parallel rectifying columns are designated as Rf, RI, and Rq, respectively. In addition to the normal operating variables of feed rate, product rates, and heat duties, operators need to specify these three ratios for optimal operation. Figure 6 shows the response of ethane content in the overhead product to the heatduty variations. The design value is very close to the knee of the curve. In other words, increasing the separation by increasing condenser duty has little influence upon overhead purity. However, decreasing the QcfD (condenser-duty/overhead-product) ratio to below 16 will drastically increase ethane content. This type of response is typical of distillation columns, except that different columns are designed to operate at different portions of this curve. With Rf, RI, and Rq properly selected and fixed, the rectifier section can be manipulated as a single column according to accepted distillation technology. The effects of Rf, RI and Rq were studied as described below: Rf and Rq were SE' constant at 0.79, based on the column diameters, and RI was varied while maintair..,tg overhead ethane content at 0.07 mol% and re boiler duties at a fixed value. The effects of RI changes on overhead product rate and Qc/D are shown in Fig. 7. An optimal RI value of 0.8 was obtained such that a maximum overhead product was produced and

Distillation Simulation

243

RF • 1121113 RA

a

D

QR/QT

RLe #10/#11

Tr= NUMBER OF THEORETICAL TRAYS

No. 3 3

FIG. 5.

Rectifier-section model.

:

mm1mum condenser duty or refrigerant was used. The shape of the curves was anticipated, but the sharpness in curvature was surprising. The data show that changing R 1 from 0.8 to 0. 75 will result in more than an 8% increase in Qc/D. This phenomenon can be attributed to two causes. As shown in Fig. 6, the reflux effect is highly nonlinear. For a given amount of reflux from the second column, there is an optimum distribution of the intermediate reflux between the two parallel columns in terms of separation. Wasting reflux on one column, with little improvement in purity, will result in a drastic reduction in purity for the other column. The combined overhead product from the parallel columns will show higher ethane content than it would if R1 were properly selected. In addition, the parallel columns, as shown in Fig. 4, are not independent-i.e., the overheads are fed to a common column (No. 3). The intermediate reflux to the parallel columns is the bottom liquid from the No. 3 column. For given reboiler duties and overhead-product specification, ethane content in the intermediate reflux decreases and then increases with indreasing R1 values, as shown in Fig. 8. This is caused in part by the nonlinear characteristics of distillation columns, as was explained earlier. Example 2. A distillation column containing a heat pump [17] illustrates steady-state simulation used to derive operational philosophy. This distillation system is a C 2 splitter which separates an ethane/ethylene mixture to produce 99.9% ethylene. This distillation

Distillation Simulation

244 3.0

x REBOILER VARIATIONS

o REFLUX VARIATIONS

2.0

1.0

0

14

19 16 17 18 15 CONDENSER DUTY/OVERHEAD PRODUCT, THOUSAND BTU/MOLE FIG. 6.

,_: u

§ ""

20

Rectifier-section response.

19

CL

;~

!S!i= 0.., ...... ~"" ~~

..:iS -..::z:: " " (I)

18 17

.., >-

15 ""i!5u

16 2,400

~

::z:: ::z::

""!$!~ ...... 0~

.... - >-

a::

2,300

u.. u

-~

>-""

~~

2,200 2,100

0.8 INTERMEDIATE REFLUX RATIO, RL

Ethane content in intermediate reflux.

column is heat integrated by using the overhead vapor, after recompression, as the heat medium for the bottom reboiler. Simulation was used to derive operator guidelines. The C 2 splitter contains two columns, as shown in Fig. 9. The overhead vapor, compressed and chilled with propylene refrigerant, exchanges heat with column bottoms to supply reboiler heat. This results in considerable ene1gy savings. At the same time, integration of the overhead condenser with the column re boiler by heat exchanging of these two streams restricts the manipulation of two most-important variables in distillation column control-reflux and boilup rates. To provide an additional degree of freedom, a chiller is added to the reflux cooling train. This unit uses zero-degree refrigerant to precool the reflux vapor before it enters the refluxjreboiler (R/R) heat exchanger. By doing so, the vapor and liquid traffic within the column can be manipulated. This series arrangement eliminates the need for a lowtemperature refrigerant as used in a parallel trimmer [71]. In this setup the two variables controlling the C 2 splitters are the chiller duty and the reflux vapor rate. The former is controlled by the refrigerant level in the accumulator, and the latter is on flow control. A good start for an analysis of Cz-splitter operation is the R/R exchanger. Figure 10 shows a typical R/R exchanger profile. Since the reflux vapor is nearly pure ethylene (99.9%), during cooling it exhibits a nonlinear temperature profile comprised of desuperheating, condensing, and subcooling regions. On the reboiler side, the liquid contains mainly ethane; a nearly constant boiling point is obtained, as shown by the vaporizing curve in Fig. 10. This nonlinear temperature profile on the reflux-vapor side leads to very interesting behavior of the R/R exchanger. To illustrate the effect of the R/R exchanger upon a distillation column with a heat pump, it is best first to explain the heat exchanger performance in the partial condensation and subcooling regions in terms of reflux vapor inlet (chiller outlet) temperature.

246

Distillation Simulation At partial-condensing reflux-outlet conditions, a reduction in reflux-vapor inlet temperature will result in an increase in the heat transferred. This is best explained with a schematic diagram of a heat-exchanger profile, as given in Fig. 11. Solid lines show the temperature profile for a given operation. If the reflux vapor is chilled, as shown by the dashed line, less desuperheating is required. The additional heat-exchanger surface can now be applied for condensation. Since the heat-transfer coefficient for condensation is considerably larger than for desuperheating, the overall heat transferred in the exchanger increases, despite the reduction in temperature driving force. In the case of subcooling, the converse is true, as shown in Fig. 12. A lower refluxvapor temperature will result in less desuperheating as before. Since the condensation area is fixed as a result of a fixed heat transfer coefficient and a nearly constant temperature difference between condensing vapor and boiling liquid, the additional exchanger area will be used for subcooling. It is easy to visualize that for a high degree of subcooling, the outlet temperature of the reflux will approach the inlet temperature of the boiling liquid. The overall effect is less heat transferred as the reflux-vapor temperature is reduced.

C_f'(l•F

REFRIGERANT c2 SPUTTER

CHILLER CzSPLITTER

RECYCLE ETHANE

FIG. 9.

C 2 spiitter.

247

Distillation Simulation

REFLUX

REBQ!LER "----_/

/

/

/

/ /

/

/

/

/

/

EXCHANGER LENGTH FIG. 12.

Temperature profile for subcooling.

To explain this heat-exchanger behavior, the chiller temperature is plotted against re boiler duty, condenser duty, and total heat removal as shown in Fig. 13. As expected, the reboiler duty increases and then decreases as the chiller temperature is reduced. Maximum re boiler duty is the total condensation point of the reflux vapor. Total heat removal and condenser duty increase as the chiller temperature is reduced. The trend indicates a leveling of these values at low chiller temperature. Therefore, manipulating the chiller outlet temperature alone is not sufficient for the column control to produce the desired product rate and quality. A simulation such as this is valuable for developing guidelines to aid operators. Example 3. The following example illustrates the use of digital dynamic simulation for analyzing the effect of noise upon temperature control of a depropanizer [15]. In summary, a seven component system of C 2 , C 3 , C/- n-C4 , i-C 4 , i-C4 2 -, i-C 5 , is simplified with 40 trays lumped into 10 simulation trays. Figure 14 shows a schematic diagram of the model with associated controls. The overhead product is on accumulator level control while the reflux is fixed on flow control. The re boiler heat input is on column bottom level control. The bottom product rate is manipulated to control the bottom temperature which is used to infer combined C 3 's in the bottoms. The disturbances were introduced to the model as step functions. The feed composition and feed temperature were perturbed. Noise in a Gaussian distribution was introduced to the bottom temperature for test purposes. For feedback control, a PID (proportional, integral, derivative) algorithm in digital form is used:

249

Distillation Simulation

10.0

""0 ::::> ,_ ..,

::::>

9.0

:......

~

::::;

8.0

-' ;:

~

::::>

""

s

7.0

:

6.0

0

20 40 60 CHILLER OUTLET TEMPERATE "F. FIG. 13.

~

where

80

Effect of chiller on heat duties.

= K [ e - e'

t'lt + -(e Td + -e T,

t'lt

2e'

+ e")

J

~ = incremental change of manipulated variable K = proportional gain t'lt = sampling time interval T, = integral control constant Td = derivative control constant e =error = (setpoint - feedback) e, s', e" =present, old, and old-old errors

This is a velocity type of PID where the output of the algorithm is added to the present controller position. The individual control modes are used alone or in combination for best performance. The performance criterion selected is integral error of C 3 's content of the bottoms as given below:

Q= where

Q = performance criterion

t =time X= C 3 's content in bottoms s = C 3 's setpoint value

f,, ISt2

Xldt

Distillation Simulation

250

TOPS DIST.

COLUMN

FEED

BOTTOI'IS FIG. 14.

Schematic diagram of depropanizer.

In digital form, it is reduced to Q=

I

IS-X;!

i=O

where n is the total samples, fixed for these test runs. The tuning of controllers is based on the Ziegler-Nichols ultimate gain method [61]. Though the obtained control constants may not be optimum, it provides a consistent means for comparing performance. Figure 15 shows a temperature feedback control loop. The column bottom temperature is the feedback signal to be controlled to the setpoint value and the manipulated variable is the bottom flow controller setpoint. The control frequency for the temperature loop is once per minute and the flow controller, an analog controller, is continuous. Figure 16 demonstrates the shortcomings of temperature control inferring composition. Both P and PI controllers stabilize near a column temperature setpoint of 211.5°F, but with feed composition disturbance of this column the temperature no longer corresponds to 0.09 mole fraction of C 3 's content. These effects in the final C 3 's composition obtained are the result of the changing temperature-composition relationship.

251

Distillation Simulation

Since signal noise is inevitable in plant operation, the temperature control loop with noise having a mean value of ±0.035cF added to the bottom temperature signals is simulated. Figure 17 shows the responses in C 3 's content variations from a temperature controller using P or PI with disturbances. Comparing Fig. 17 with Fig. 16 confirms the common belief that noise affects control. Considerably more oscillations are observed with noise. Filtering can be used to reduce these oscillations. Example 4. Analog simulation using a transfer function approach is applied here to study the thermal coupling effect for the bottom control of a distillation column [16]. A refinery distillation train contains a stabilizer and a splitter, as shown in Fig. 18. Analysis of plant data indicated oscillatory feed rate (stabilizer bottoms) to the splitter. This can be caused by improper tuning of the level controller and further aggravated by the thermal coupling of the stabilizer bottom product to preheat the stabilizer feed in the following manner:

Any disturbance upon bottom level, such as a feed rate change, will affect the bottom product flow rate via the bottom level controller (Fig. 19). This changes the degree of preheating of the feed which in turn results in more or less liquid flow to the column bottom. The disturbance can be recycled through the column, producing oscillation in the bottom level and bottom product flow. Under some conditions, this oscillation can become unstable.

COI'li'UTER SET POINT STATION

BOTTOM PRODUCT

SET POINT TEMPERATURE

FIG. 15.

FLOW

Schematic diagram of temperature feedback control.

N

U'l N

§

8......

"' ~

g:

U"l

P ONLY

g: 1---~

....;~

~u. -

~~~

u~

"'

~

z: u..

0

20

40

60

80

100

_J

~ 0 00 0

120

Tll'lE

8......

8......

g; "'

"' ~

~

i!

20 TIME

250

Temperature (°C) FIG. 2.

Vapor pressure vs temperature plot for a system made up of water and bromobenzene. (Redrawn from Ref. I.)

liquids distill together. When either one of the liquids distill away, the vapor pressure drops to that of the remaining liquid. Ordinarily bromobenzene boils at 156.2aC and atmospheric pressure, but the presence of a little water (or steam) makes bromobenzene boil at 95°C, a temperature that is easy to achieve even with low-pressure steam. In 1918, Hausbrand published a vapor pressure diagram that proved to be very useful in steam distillation calculations. Figure 3 shows that diagram. It plots n-p. at three system pressures (760, 300, and 70 mmHg) versus temperature where Psis the vapor pressure of water. These curves cut across the ordinary vapor pressure curves of the materials to be distilled. The intersection of the water curve with the curves of other materials gives the temperature at which steam distillations can take place. Suppose an atmospheric still contains some impure toluene, and steam is blown into the still. Suppose also that the impurities are very high boiling compounds with negligible vapor pressure. Further, consider the case where the liquid in the still is heated solely by the condensation ofthe open steam. A water layer will therefore accumulate. As the temperature rises, the vapor pressures of the toluene and the water layers rise. When the sum of the two vapor pressures equals 760 mmHg, the mixture begins to distill. At this point the pure component vapor pressure of the toluene is p* mmHg and the vapor pressure of water is n-p* mmHg; Fig 3 shows that this occurs at a temperature of about 84°C. In this case the vapor passing over consists of toluene with a partial pressure of 350 mmHg and water vapor with a partial pressure of 410 mmHg. The molar ratio of toluene to water would therefore be 350/410, or 85.4 parts toluene to 100 parts water.

N

0)

N

700

600

c,

500

:r:

E

.§ Q)

~ 400

a. Q)

0a.

>"' 300

200

c C/1 z.

100

Ill

z.

0

50

0

FIG. 3.

100 Temperature(°C)

150

Hausbrand diagram for various liquids at three system pressures. (Redrawn from Ref. I.)

-200

0

::l C/)

r+ CD Ill

3

263

Distillation. Steam

The Hausbrand diagram permits a quick determination of the temperature at which steam d_istillation occurs. It also graphically shows the molar concentrations of the vapor. Since most substances on the diagram have a molecular weight considerably greater than that of water, the composition by weight is much richer than would appear from the diagram. Batch Steam Distillation/Stripping

Carey has outlined an approach to developing a general equation for batch stripping, For a small time interval, d8, a constant sparging rate for the steam, and enough additional heat added to the charge so that only the steam and volatile component enter the vapor phase, the following equation describes a differential material balance for the volatile component: (1)

where

Ni = mole of inert component in the still charge X= instantaneous composition of the liquid in the still in moles of volatile component per mole of inert component si =steam rate in moles per unit time Y =instantaneous composition of the vapor leaving the still in moles of volatile component per mole of steam,

Equation (1) may be reduced to two variables by solving for Yin terms of other variables. Carey notes that the actual partial pressure of the volatile component in the vapor is usually less than the theoretical because of massand heat-transfer resistance in the process, He therefore defines a vaporization efficiency as

E where

=

pjp*

(2)

p = partial pressure of the volatile component in the vapor p* =equilibrium partial pressure of the volatile component

Applying Dalton's law to the components in the vapor, Y is the ratio ofthe partial pressure of the volatile component p, and the partial pressure of the steam is n - p, where n is the total pressure maintained in the vapor space of the stilL Then, by appropriate substitutions in Eq. (2):

Y=-p-= Ep* n- p n- Ep*

(3)

Gibbs' Phase Rule, when applied to a three-component system with two phases present and with pressure specified, says that p* is a function of two variables; that is, temperature ( T) and X, so

p*

= p*(T, X)

(4)

264

Distillation, Steam

Using this implicit equation for p*, Eqs. (3) and (1) lead to the general expression for batch stripping:

NIX/ n-E Ep*(T,X) dX = Io S-de = s.e ' *(T X) ' ' x.

where

p

'

(5)

0

Xs =composition of residue in still, moles volatile component per mole of inert component XJ = composition of feed, moles volatile component per mole of inert component e = stripping period, units of time consistent with steam sparging rate

Implicit Eq. (5) involves assumptions that are usually valid. If data for the influence ofT and X on p* and E for the process, normally assumed constant, are substituted into Eq. (5), an implicit expression for the total steam usage derives. When two liquid layers are present in the still, the variables reduce to two, so that when pressure is fixed, only one degree of freedom remains. If X is the remaining independent variable, Tis no longer subject to choice. Therefore, when two liquid layers are present,p* is a function of X only, once n has been set. Returning to the bromobenzene-water example used earlier, if water is present as a liquid phase, there are three distinct phases to the system. Therefore, from the phase rule, when two components are present there is only one degree of freedom. So, in this case, fixing the temperature or the pressure fixes the state of the system. If we choose to distill at atmospheric pressure, and since the sum of the partial pressures necessarily equals the total pressure (760 mmHg), the system must boil at 95°C. However, if no liquid water is present, there are two components and two phases and therefore, two degrees of freedom. Both temperature and pressure can be independently varied. (A little later it will be apparent that a liquid water phase has important economic and operational implications.) If all or almost all of the volatile component distills away, the partial pressure of that component becomes very small and the total pressure becomes essentially equal to the partial pressure of the steam, n ~ Ps = P•. Therefore, in order to prevent condensation of the steam, its temperature should be selected to be higher than the saturation temperature at the pressure of the distillation. Since steam distillations are conducted so that low temperatures can be maintained, it is not economical to distill at low pressures when the ideal gas law applies. Thus the solving of steam-distillation problems may be based on partial pressures. This is true provided the total pressure is not high enough to cause appreciable deviations from the ideal gas Jaw. This low is applicable for pressures below about 3 atm. Estimates of the steam usage for the batch process may be obtained by using some simple or approximate function for p* in Eq. (5). Three such cases would include Raoult's, Henry's, or Lewis and Luke's laws. All three cases can be dealt with at the same time since the three Jaws have the same analytic form:

265

Distillation, Steam

Raoult's law

p*=P(~) 1 +X

Henry's law

p*=H(~) 1+X

Lewis and Luke's law

p*=K(~) 1+X

This leads to the general expression for Y in terms of X:

Y=

X

(6)

(k- l)X + k

when k = n/EP, n/EH, or n/EK when Raoult's, Henry's or Lewis and Luke's laws, respectively, apply. Substituting this Y into Eq. (5) and integrating over the composition range specified gives the steam usage: (7)

The moles of volatile component vaporized is N;(XJ -X.), so the ratio of moles of steam used to moles of volatile component recovered is

s.e I

N;(XJ -X.)

k = [ -+ (K -1) X1n

J

(8)

where X1n is the log mean value of Xf and X •. Equations (7) and (8) are often used for steam distillation approximations. For more exact calculations, the true relationship for p* required by Eq. (4) must be known (usually from experiments). Equation (7) may be expressed in mass rather than mole quantities. Such an expression suggests that steam usage will be low when the ratio ofthe molecular weights of the volatile component to the inert component is low, and when the volatility of the volatile component is high, i.e., when k is small. It also indicates the quantitative effect of operating at low pressures and high temperatures to reduce steam usage. Multicomponent Batch Steam Distillation

The most common situation encountered in multicomponent batch steam distillation is that in which n components are present. Some are volatile and some nonvolatile. All components of the mixture are miscible in the liquid phase. Holland and Welch [5] derived the following equation to describe the process:

266

Distillation, Steam

When the mole fractions of the volatile components are not appreciably affected by the nonvolatile components, Eq. (9) becomes ( 10)

where

E; = p;jp* = vaporization efficiency of i L = moles of volatile components in still at any time L = L 0 at start of distillation L~ = moles of nonvolatile material in still P; = vapor pressure of i at still temperature Pi =partial pressure of i in the vapor B; = (E;/Eb)A;b A;b =relative volatility of i referred to b = P;/Pb Li*s.r =summation of all components except steam and nonvolatiles b = reference component r = nonvolatile component S =steam required, moles

Continuous Steam Distillation

Steam distillation or steam stripping can be conducted as a continuous, steadystate operation. The flow of steam may be countercurrent to the flow of feed or cocurrent with it. In the countercurrent case the vapor contacts the richest liquid in the system last and tends to be richer in the volatile component than in the cocurrent case. Definitions of variables already used will be modified to enable an easy comparison of the different processes. Ne is the moles of steam blown into the still per unit time in the countercurrent case. If steady-state conditions prevail, Ne is also the moles of inert component removed from the still per unit time. The volatile component feed rate is NeXf and the moles of this component leaving per unit time in the still residue is NeX,. The difference between these two quantities, Ne(X1 -X,), is the mole rate at which the volatile component passes into the vapor phase. In the countercurrent process the vapor last engages the feed liquid having a composition of X 1 . For equilibrium operations the vapor composition is predicted by the equilibrium properties of the system, e.g., p* = p*(T, X 1 ). However, the partial pressure of the volatile component seldom reaches the full equilibrium value, only a fraction of it. If this fraction is E, as in Eq. (2), the vapor leaving the still has the composition

y _ _ E_:_p_*-'--(T_,_X_::_f:_)_ 1 - n- Ep*(T,X1 )

( 11)

267

Distillation. Steam

For the cocurrent process, the vapor leaving the stripper last encounters liquid of composition X, and its composition is expressed by yf

Ep*(T,X,) - E p* ( T, X,)

= -----'-11:

( 12)

By material balance on the volatile component, the steam usage equation follows: (13) (14) The subscripts c and p refer to the countercurrent and the parallel flow (cocurrent) cases, respectively. The ratio of steam usage to the volatile component removed is Se Ne(XJ- X,)

n- Ep*(T,X1 ) Ep*(T,X1 )

(15)

se Np(X1 - X,)

n- Ep*(T,X,) Ep*(T,X,)

(16)

Equations (13) and (14) are implicit exact expressions and are comparable to Eq. (5) for batch stripping. They and Eqs. (15) and (16) may be rearranged into an explicit approximate form for the three cases previously considered for batch stripping. By using Eq. (6) we get (17) (18) (19) (20) Depending on which solution law most closely fits the situation, the approximate value of k may be used in Eqs. (17), (18), (19), and (20). Equation (8) for the batch process is very similar in form to Eqs. (19) and (20) for the continuous processes. The difference occurs only in the first term of the bracketed quantity, 1/X1n being replaced by 1jX1 and 1/X, for the countercurrent and cocurrent flow processes. The similarities naturally lead to a

268

Distillation, Steam

comparison of steam usage for equal amounts of stripping for the three processes. Since X 1 > X 1" >X, then (1/X1 ) < (1/X1n) < (1/X,). When applied to Eqs. (8), (19), and (20), these inequalities show that for the same operating conditions, but assuming different contacting processes, and by assuming the same value of E for each case, the steam usage is largest for the parallel flow and least for the countercurrent process.

Comparison of Steam Usage It is interesting to compare the difference in steam usage for the batch and

countercurrent processes for the same operating conditions, since these are more widely applied. Calling this difference /lS, /lS =

s.esc '

N;(XJ- X,)

=

1

(XJ- X,)

IXJ X,

n- Ep*(T X) ' Ep*(T,X)

dX-

n- Ep*(T XJ) ' Ep*(T,XJ)

(21)

Equation (21) may be used in its approximate form for the three cases already considered. If a is the ratio of final to initial concentration of the volatile component in the liquid, (22)

Then t.S =

~[ln(lja) _ 1] X 1 (1 -a)

(23)

This expression for t. S suggests how the ratio of the final to initial concentration of the volatile component influences steam usage. Equation (23) also implies that /). S increases directly as n increases, and varies inversely with X 1 . Temperature and escaping tendency also inversely affect /). S. A quantitative picture of the effect of a and X1 above on/). S is given in Fig. 4; the other variables are held constant. This graph shows that the countercurrent process is sharply more efficient than the batch process. This is particularly true where the concentration of the volatile component in the residual liquid is reduced to a very low value, as in the deodorization of edible oils.

Theoretical Stage Requirements The differential pressure across plate stripping columns operating at high pressure may be neglected in any calculations. This is not true for high vacuum

Distillation, Steam

269 1000

100

)(' "'1o 1 ~....:.07

-

1-- r---...

~

r--r--.....

)(f "'0

--:_Os

-~

)(' "'1o 1 ~....;,1

to....

j"-... 1--

r- ....

r-

- --0. 1

~

r-.....

r----..

" ,..... r--r--.....

)(f "'0

"""'

-..:,5 1

)(f "' 7 -.:......_

to....

~

----

r-.....

~

..............

I'"""-.....

....

1'-o..

j"-...

r-......

~

~""-.... ~

f'\ "'

\

, ''l'"

r--~

1',

)(f "'5

)(f "' 17o

,.....

"'

'

_).

'

~"\ ·'' ' \'1\ '".... ~"\ I'.)

,

1'\.

'

"' f\1\ '

'·', 1\

0.0 1 0.0001 FIG. 4.

0.1 0.01 0.001 a, ratio of final to initial concentration of the volatile component

Relative stripping superiority for batch vs continuous countercurrent systems. (Redrawn

from Ref. 4.)

operations; pressure drops cannot be neglected in the design calculations without introducing large errors. The pressure differential across the column may amount to several hundred percent more than the pressure maintained at the still head for industrial units having pressure drops of 1 to 4 mm of mercury per plate. Under these circumstances the bottom plates are operating at higher pressures than the plates near the still head. These plates are, therefore, less effective and the enriching ofthe vapor per plate drops off quickly as the column bottom is approached. This must be taken into account in the design of stripping columns or too few plates will be specified. Figure 5 shows the construction of vapor-liquid equilibrium curves for each plate. This construction requires the knowledge of the temperature on each plate in the column. For most engineering purposes involving preheated

Distillation, Steam

270

feed and superheated steam, the temperature differential across the column is small. So it is usually accurate enough to assume that the temperature of the liquid on the plate is about the same throughout the column. The usual McCabe-Thiele procedure for binary mixtures is used to solve for the number of theoretical stages. Figure 5 is constructed with a separate vapor-liquid composition for each plate, since the pressure above each plate is different. One curve applies to one plate only, as specified by the pressure at the still head plus the cumulative pressure drops down to the plate under consideration.

E :l 'i5 Ql E

"' ·c. c

·E "'

Ql

0

_g ..!!!

·;:;

0"'

.,> Ql

0

.§.

:.."" c' 0

·;:;

'§

a.

E 0

"0 a. >"'

Liquid composition, X, (moles volatile/mole inert)

FIG. 5.

Graphical determination of actual plates required for a stripping column at high vacuums. {Redrawn from Ref. 4.)

Distillation, Steam

271

Sample Problems Problem 1. An amount of 100,000 lb of food extract-approximate molecular weight is 450-containing 1.3 wt.% hexane is to be subjected to steam distillation to reduce the hexane content to not more than 0.01 wt.%. Assume E = 0.7, total pressure 1000 mmHg, t = 100°C. Determine the steam consumption for the continuous countercurrent process.

Solution: Se= Ne [

n

(X1 - X,)

Ep*

X1

n +(-

Ep*

)

1 (X,- X,)

J

Assume feed rate equals 1000 lb/h:

Ne= X1 =

Xz

=

1000- 1000(0.013) 450

450

(1000)(0.013)/(86) 2.19 0.0986/86 2.19

987

= - = 2.19 moljh

= 0.069 mol/mol

= 0.00053 moljmol 1000

1t

1t

Ep*

E(~)P. I+ X

Se= 2.19(11.9)(

----,--------,--- = 11.9

0.7( 0 ' 069 )(1860) 1 + 0.069

0.069 - 0.00053) 0.069

+ (11.9-

1)(0.069- 0.00053)

= 25.1(0.99) + 10.9(0.0685) = 25.5 mol steam/h

We= 25.5(18) = 459lb steam/1000 lb feed

Problem 2. It is desired to separate three high boiling materials from a nonvolatile material by batch steam distillation. At least 95% of the volatile material is to be removed. Determine the moles of steam required when the distillation is carried out at 100°C and 200 mmHg pressure. The data to be used are:

Initial mixture

Moles

Component 1 Component 2 Component 3 Nonvolatile

30 25 25 20

P at 200 mmHg and 100°C

E

20 14 8

0.9 0.9 0.9

Distillation. Steam

272 Using Component 3 as the base component E 1 P 1 0.9 B3 = Bb = l.OB1 = - - = EbPb 0.9

20

X -

8

= 2.5

E 2 P 2 0.9 14 B2 = - - = - x - = 1.75; EbPb 0.9 8

L, =Total moles in still at end of distillation L L,=L? ( L~

)B' +Lg (LL~ )B' +L~ (LL~ )B3 +L, (LL~ )B,

0 05(25)] = 3{ . 25

2 •5

+ 25(0.05)1.7 5 + 25(0.05)1. 0 + 20

= 21.4

L = L, - L~ = 21.4 - 20 = 1.4

I -L ?[ 1 -

i*'·' B1

8 (Lb) ' 0 Lb

=30 -[

2.5

1 - (o.o5) 2 · 5

J+

-25l 1 - (o.o5)1. 75 J 1.75

25

+ -[1- (0.05)1. 2 ] = 12 + 14.3 + 23.75 = 50.05 1

Lg 25 L~ In-= 20 In = 60.1 Lb (0.05)(25) 50.05 + 60.1 = EbPb (L 0

n

-

L + S) = 0. 9(8 ) (80- 1.4 + S) 200

110.15- 0.036(78.602) = 0.036S 110.15 - 2.83

----- =

0.036

S = 2980 mol steam

Example from Industry What appears simple today was a difficult problem 40 years ago. Back in the early 1930s, continuous processes in the fat and oil industry were not as generally available as they are now. The separation of many crude fatty acid mixtures into relatively pure chemical components had been considered. However, these separations had not been successfully commercialized because batch operation was a major limiting factor for most feedstocks. Application of heat to a batch of crude fatty acids at high temperatures over long periods of time (measured in hours) caused coking, polymerization, and breakdown of the fatty acids. Because of these difficulties, some fatty acids were separated using crystallization techniques followed by hydraulic pressing. By applying heat to a continuous feedstock and by cooling the products as

273

Distillation, Steam

soon as possible after separation, the heating and cooling times were greatly reduced compared to batch distillation. In fact, the retention time at elevated temperatures after heating and before cooling in a commercial continuous fractional distillation system was much less than in a batch still of comparable size. This background briefly highlights the chief problems relating to fatty acid distillation. Other factors, with only indirect or even no effect on operating temperatures, are important to assure product quality. The first continuous fractional distillation unit for the commercial separation of fatty acid mixtures was installed in 1933. This still consisted of a main tower, two smaller side-stripping towers, air ejectors and boosters, condensers, coolers, and a direct-fired fatty acid heater. The flow sheet for this unit is shown in Fig. 6. Though commercially successful, operating difficulties were experienced, particularly in coking of the heater and by corrosive attack on the equipment by the fatty acids themselves. The next development was the replacement of the direct-fired fatty acid heater with an indirect heating system, using condensing Dowtherm vapor.

F racti onati ng tower

Cottonseed 'oi I crude fatty acid Charge

+

0 Direct-fired heater

Hotwell

Pitch bottoms FIG. 6.

No. 2 No. 1 cut cut

Flow sheet for first continuous fatty acid distillation unit. (Redrawn from Ref. 3.)

274

Distillation, Steam

Dowtherm systems were not available when the distillation system was originally designed. Its use made possible the installation of spare shell-andtube heaters for plant shutdown. In addition, the relatively gentle indirect Dowtherm heat lowered fatty acid film temperatures and reduced corrosion and the amount of coke formed. The general arrangement of this modified system is shown in Fig. 7. When a completely new, second fatty acid fractional distillation system was installed in 1941, it included many design features not found in the original unit. It consisted of a three-tower system, similar in principle but quite different in detail from the first unit in its modified form. These same comments apply to subsequent more modern units built since 1941 when comparing them to the second unit. In 1948 a complete fat and oil processing plant began operation in Kankakee, Illinois. It used low-grade fats, oils, acid oils, and tall oil to produce finished fractionated fatty acids, fatty acid esters, and their derivatives. The fractional distillation section is representative of modern practice and is shown schematically in Fig. 8. The distillation system appears to have only two towers; however, three towers are actually present, with the second superimposed on the third to conserve space and construction costs.

3-stage vacuum unit

Charge stock

Steam

i

Super-~ heater 1

Run-down tank

("\I

Dowtherm vaporizer

0

II

I

1!

I

1 I i' 11 1

~- ....J

1

~

I

Hotwell

\._1_.)

Pitch No. 2 bottoms cut FIG. 7.

cut

Flow sheet for next evolution of continuous fatty acid distillation unit. (Redrawn from Ref. 3.)

Distillation, Steam

Light ends condenser

t

275

-

Barometric condenser with an ejector

Booster ejector Medium fatty acid condenser Light 11'/ ends

Main

+ fractionating

Fatty acidfeed

+

tow"l

Booster ejector

1---:: ..:---1-~-.. -.::--+~-;.. :....Reboiler I ..... --, ~-~-:f!....!:""' __ ,

Super heater

-

t ~ ~- ._J I I

I

--1---

Downtherm vaporizer FIG. 8.

--

f

-- -- Coolers

Flow sheet for modern continuous fatty acid distribution unit. (Redrawn from Ref. 3.)

In this plant, as in all subsequent ones, the fatty acid preheater was eliminated. This piece of equipment proved to be a high maintenance item because of tube plugging inside its heater tubes. Inorganic material in water solution, carried along by the fatty acid feedstock, deposited inside the tubes as soon as the water vaporized. Complete dehydration of the feedstock before distillation would eliminate this difficulty but was not adopted due to its added complexity. The problem was solved by introducing fatty acid feed directly into the first distilling tower. Sensible heat to preheat the feedstock, as well as latent heat, was provided by the fatty acid vapors rising from the reboiler in the base of the tower. Thus dissolved solids were dispersed harmlessly in the tower and were easily carried along with the bottoms stream. This made it possible to operate satisfactorily with a wet feed stock. The sensible preheat represents a significant part of the load, so the base of the first tower must be larger than the section above the feed plate.

Distillation. Steam

276

Fatty acids containing a small amount of moisture are fed directly to the light ends tower in which a low-boiling distillate is removed. This distillate is a concentrated odor and dark-color bearing fraction, whose removal upgrades the remaining fatty acids. Bottoms from this tower are transferred to the main fractioning tower by gravity and vacuum. In the main tower, if feedstock is similar to tallow or cottonseed oil fatty acids, a fraction high in palmitic acid is removed from the overhead condenser. The bottoms stream flows by gravity from the base of the main tower to the third tower directly below. In this tower, higher boiling fatty acids, such as oleic acid, are recovered from the condenser. Residue from the base of the third tower may be converted into pitch. Similar installations have been built in several other locations. Materials of Construction

Materials of construction for these plants originally posed a major obstacle to successful operation. In fact, this problem has not been completely solved since reboiler tubes require periodic replacement. At one time ordinary 304 stainless steel, containing about 18 to 20% chromium, 8 to 11% nickel, a small percentage of carbon, and the balance mainly iron, was thought to be adequate. Commercial operation quickly changed this belief. Type 316 stainless steel, containing about the same constituents but with the addition of molybdenum, was substituted. Experience indicates a high molybdenum content is desirable and that a minimum value of 2.5% looks right. Low carbon content is desirable to minimize chromium carbide precipitation in the metal. These points of carbide precipitation are more susceptible to fatty acid corrosion than other portions of the metal. Important Design Factors

As stated earlier, the need for minimum temperatures and minimum holding periods while the fatty acid stock is hot govern the design of fatty acid distillations systems. It is necessary to: 1. Provide good vacuum equipment to permit operation at low absolute pressures 2. Minimize the pressure drop in the tower and tower reboiler by careful design and generous proportioning of vapor passages 3. Reduce liquid hold-up to a minimum 4. Use stripping steam as an inert medium to reduce the fatty acid partial pressure needed for boiling The last factor is one which relates to the designer's attempt to cause boiling at the lowest possible temperature consistent with equipment size and vapor losses. The use of stripping steam to promote boiling at low temperatures has a

Distillation, Steam

277

good analogy in the deodorization of vegetable oil. Stripping steam is used in this process to help remove volatile undesirables in the oil, such as fatty acids. There is, of course, a proper balance between the degree of vacuum maintained and the quantity of stripping steam used.

Symbols A a

B E

H k K N;

Ne

NP

n P p p*

S; Se SP ~S

T X X1 X.

relative volatility of volatile component referred to reference component = PdPb ratio of final to initial concentration of the volatile component in the liquid constant vaporization efficiency; the partial pressure of the volatile component in the vapor leaving the still divided by the equilibrium partial pressure of the volatile component as specified by the temperature and composition of the liquid last encountered by the vapor; used in the same manner for each plate in discussions concerning plate columns Henry's coefficient in the form p* = Hx constant in the liquid vapor expression when one of the three simple solution laws applies to the system equilibrium constant in Lewis and Luke's law in the form y = Kx number of moles of nonvolatile or inert component present in the still at any time in the batch process number of moles of nonvolatile or inert component introduced into the still per unit time in the continuous countercurrent process number of moles of nonvolatile or inert component introduced into the still per unit time in the continuous parallel-flow process plate number, counting up from bottom of column vapor pressure of the volatile component partial pressure of the volatile component in the vapor leaving the still or a particular plate equilibrium partial pressure of the volatile component that results from a liquid whose temperature and composition is that of the liquid last contacted by the vapor leaving the still; also used in the same manner for each plate of a plate column moles of stripping medium introduced per unit time in the batch process moles of stripping medium introduced per unit time in the continuous countercurrent process moles of stripping medium introduced per unit time in the continuous parallelflow process difference in moles of stripping medium required per mole of volatile component stripped by the batch and continuous countercurrent process temperature variable concentration of the volatile component in the liquid, moles per mole of inert component concentration of the volatile component in the feed liquid moles per mole of inert component concentration of the volatile component in the liquid on the n-th plate, moles per mole of inert component

Distillation, Steam

278 X, X

Y, Yn Y, y

e

concentration ofthe volatile component in the residue or stripped liquid, moles per mole of inert component mole fraction of the volatile component in the liquid concentration of the volatile component in the vapor last encountering liquid of composition X 1 , moles per mole of stripping medium concentration of the volatile component in the vapor over the n-th plate, moles per mole of stripping medium concentration of the volatile component in the vapor last encountering liquid of composition X,, moles per mole of stripping medium mole fraction of the volatile component in the vapor time

Subscripts

c

I

n p

In

batch process countercurrent flow process feed n-th plate parallel flow process residue logarithmic average

References 1.

2. 3. 4. 5.

R. W. Ellerbe, "Steam Distillation Basics." Chem. Eng., pp. 105-112 (March 4. 1974). K. E. Coulter. "Applied Distillation. Part III," Pet. Refiner, 31(11), 156-158 (November 11, 1952). R. H. Potts and F. B. White, "Fractional Distillation of Fatty Acids," J. Am. Oil Chem. Soc., 30(2). 49-53 (1953). H. J. Garber and F. Lerman, "Principles of Stripping Operations." Trans. Am. Inst. Chem. Eng., 39, 113-131 (1943). C. D. Holland and N. E. Welch, Pet. Refiner. p. 251 (May 1957). R. W. ELLERBE

Distillation. Thermally Coupled

279

Distillation, Thermally Coupled

A distillation system contains a thermal coupling when a heat flux is utilized for more than one fractionation and the heat transfer between fractionation sections occurs by a direct contact of vapo: and liquid. Compared with a conventional system separating a multicomponent mixture, the thermally coupled distillation system can separate close boiling components with considerable savings in heating and cooling costs. The limitation is that all fractionation sections are at approximately the same operating pressure, whereas in the conventional system the columns can be at different pressures. In 1937 Burgm a [ 1, 2] patented such a process. Recently the concept has been rediscovered by several authors [3, 6-8]. Cahn and Di Miceli [3] describe the separation of mixed pentanes and hexanes by such a system. Petlyuk et al. [6] describe the use of such systems for separating ethylbenzene-xylene mixtures and numerous others. Stupin and Lockhart have developed a rough design method and presented a case study [11-13]. The separation of a multicomponent mixture is conventionally accomplished·in a series of columns, numbering one less than the number of products, each having a condenser and a reboiler. In contrast, the thermally coupled system requires only one condenser and one reboiler and a number of column sections which may or may not be in the same column. Consider the separation of a given feed into three products, A, B, and C, arranged in order of decreasing volatility. There are two schemes using conventional columns that will produce the desired fractionation as shown in Fig. 1. In the first scheme, Component A is fractionated from the mixture ofB and C, then Components B and C are separated in a second column. In the thermally coupled system the initial separation is made between Components A and C which are the extremes in volatility and are therefore easily separated. Then Components A and B are separated in the upper part of the second column and Components B and C are separated in the lower section of this column. The separations of the three components in the second column are essentially binary and are carried out without interference from the third component.

The Design of Thermally Coupled Distillation In a conventional distillation column equipped with a partial reboiler for separating a single feed into two products at their bubble points, there are 2N + I+ 4 independent variables [10, 11]. Early in the design, pressures are chosen for each stage and heat duties (losses are zero) are fixed for each stage except the re boiler and condenser. This specifies 2N - 2 variables and therefore DOI: 10.1201/9781003209799-13

280

Distillation, Thermally Coupled

A, B,C

A

CONVENTIONAL SCHEME

B

CONVENTIONAL SCHEME (ALTERNATIVE SEPARAT;ON SEQUENCE!

FEED

A, B, C

NOTE CW - COOLING FLUID ST - HEATING FLUID C

FIG- 1.

THERMALLY COUPLED SCHEME

Flow diagrams for the separation of ternary mixtures by conventional and thermally coupled distillation schemes.

leaves I + 6 independent variables to define the fractionation. Of these, I + 2 are used to describe the feed. Four independent variables remain to specify the separation and the details of the process. For this conventional column the separation is specified by the distributions of two key components. The final two independent variables are taken to be the reflux and feed location. The design process then determines the reflux and feed location for the specified separation of the given feed. For the conventional column, the total cost, including capital and operating, is relatively insensitive to reflux over the range of 1.05 to 1.5 times the minimum reflux. Anywhere in

Distillation, Thermally Coupled

281

this range the costs are close to the minimum. Because of this, the reflux ratio is commonly specified as some percentage over the minimum, and the design is completed at this "design reflux." This eliminates the problems associated with developing, maintaining, and using accurate cost data, and reduces the problem to one of finding the fewest stages at the specified design reflux. The equilibrium stage model of Fig. 2 represents the three-product thermally coupled distillation system equipped with a partial reboiler and producing all three products as bubble point liquids. Applying the methods of Smith [10] or Stupin [11] to this system, we find a total of 2N + I+ 11 independent variables. Then pressures are chosen for each stage, and the heat duties are fixed for each stage except the reboiler and the condenser. This reduces the set of independent variables to be specified by 2N- 2 to I+ 13. The feed fixes I + 2 variables and leaves 11 independent design variables. If, in the thermally coupled system, we specifiy the separation by four quantities for the three key components, one each for the lightest and heaviest products, and two for the intermediate product, then there are 11 - 4 = 7 independent variables that determine the configuration and reflux. Under these circumstances the determination of the optimum design becomes one of a ?­ dimensional optimization. Of these, one relates to the overall energy input to the system and can be considered the external reflux rate, four relate to the allocation of stages to the various column sections of the system, and two relate to the distribution of reflux and vapor to Column 1. As with the conventional column, the thermally coupled system can be designed at some given percentage over the minimum reflux. However, this problem involves finding values for the six degrees of freedom that relate to the internal distribution of stages and reflux. In contrast, the conventional column under the corresponding conditions had only one degree of freedom, the feed location. In any case, determining the optimum design is very difficult. For this reason we have chosen the approach of trying to find a feasible design.

Simplified Model for Design Comparison of the conventional and thermally coupled distillation schemes shows there are two major differences, the processing sequence and the heat exchange. However, instead of the conventional two-column scheme for the three-product separation, a three-column scheme using the sequence of the thermally coupled system can be constructed. Then, if the thermal couplings are replaced by heat exchangers, a system of three conventional columns could simulate, at least approximately, the thermally coupled system. Such a simplified model is illustrated in Fig. 3. In this model the columns have been uncoupled as far as the recycle of material between the columns. Each of the three columns can be treated separately with the following constraints. The heat removal needed to provide reflux for Column 1 is accounted for by adding it to the feed to Column 2. Therefore Column 2 acts as

282

Distillation, Thermally Coupled

CONDENSER

VAPOR COLUMN 2 LIQUID

RECTIFYING

FEED

STRIPPING

1----L-I_Q_u_lD----·~ INTER MED lATE PRODUCT

FEED

s

COLUMN

STRIPPING

I

RECTIFYING

VAPOR

LIQU.I D

COLUMN 3 FEED

STRIPPING

QR _ _ _ __.

tVAPOR FLOW

~LIQUID

REBOILER IS THE BOTTOM STAGE OF COLUMN 3

LIQUID

~----------i-.

BOTTOM PRODUCT

FLOW

FIG. 2.

Equilibrium stage model of the three-product thermally coupled distillation system.

283

Distillation, Thermally Coupled

/

/

-

/

I

"'1--.....,'---~\-----i~ 0 VER HEAD

I

,..

I

I

PRODUCT

I

I

I

I

I

I

I

____

14----//

I

//

/

FEED

///

//

I

I

II

'

L.....-.,...---1

I

/

I

I

t-------~ INTERMEDIATE

I \

COLUMN 1

COLUMN 2

1-------~

'\

PRODUCT

\

\

'

\ \

I

COLUMN 3

I

I

I

I

--c:;),...

-

HYPOTHETICAL INTERMEDIATE HEAT EXCHANGERS

FIG. 3.

-

I /

BOTTOM

~-/-~-----""""l'~ PRODUCT

Uncoupled model of the thermally coupled distillation system.

Distillation, Thermally Coupled

284

if it were supplied with a very superheated feed. The heat needed to provide stripping vapor in Column 1 is accounted for by removing it from the feed to Column 3. Therefore Column 3 acts as if it were fed with a very subcooled feed.

Sample Problem The design of a thermally coupled system is illustrated using the separation of the hypothetical mixture of Components A, B, and C, as shown in Table 1. Assumptions of constant molal overflow and constant relative volatility are used throughout. To simplify the following presentation, the components in the feed have been classified as light components, mid components, or heavy components, depending on their volatilities. The light components are most volatile and concentrate in the overhead product. The mid components are intermediate in volatility and concentrate in the intermediate product. And the heavy components are least volatile and concentrate in the bottoms product.

Preliminary Material Balance In a system producing three products there are 3I unknown component rates that describe these products. For a system with a feed containing I components, I independent material balances may be written. In addition, the specification of the separation can be fixed by four quantities. This leaves a set of I equations with 3I- 4 unknowns. Therefore, a unique solution cannot be found unless additional relationships are included. The following rules are useful to simplify the completion of a preliminary material balance.

TABLE 1

Feed-Bubble Point Liquid

Component Mole fraction in feed Relative volatility Mole fraction in liquid Mole fraction in vapor Objectives: 1. Overhead product 2. Intermediate product 3. Bottoms product

A

B

c

Total

0.3333 9.0 0.3333 0.6924

0.3334 3.0 0.3334 0.2308

0.3333 1.0 0.3333 0.0768

1.000 1.000 1.000

90% A 90% B; about equal amounts of A and C 90% c

285

Distillation, Thermally Coupled 1.

2. 3. 4.

Components lighter than the light key appear only in very small quantities in the intermediate and bottoms products. Their mole fractions in these streams are set to zero. Components lighter than the mid key appear in only very small quantities in the bottoms product. Their mole fractions in this stream are set to zero. Components heavier than the mid key appear in only very small quantities in the overhead product. Their mole fractions in the overhead are set to zero. Components heavier than the heavy key appear in very small amounts in the overhead and intermediate products. Their mole fractions in these streams are set to zero.

By using these rules the preliminary material balance for the example shown in Table 2 is completed.

Minimum Stages As the reflux to the system is increased, the required number of stages decreases. When the ratio of reflux to products approaches infinity, a minimum number of stages is reached. At this limiting condition of total reflux, any product withdrawn from the column or any feed added to the column is insignificant compared to the internal vapor and liquid traffic. Therefore, at infinite reflux, the thermally coupled distillation is reduced to two total reflux sections stacked

TABLE 2

Preliminary Material Balance Component

Feed: Moles/mole of feed Mole fraction Overhead product: Moles/mole of feed Mole fraction Intermediate product: Moles/mole of feed Mole fraction Bottoms product: Moles/mole of feed Mole fraction

A

B

c

Total

0.3333 0.3333

0.3334 0.3334

0.3333 0.3333

1.000 1.000

0.3187 0.900

0.0354 0.100

0 0

0.3541 1.000

0.0146 0.050

0.2626 0.900

0.0146 0.050

0.2918 1.000

0 0

0.0354 0.100

0.3187 0.900

0.3541 1.000

Distillation, Thermally Coupled

286

one above the other. At this very high reflux condition, Column 1 is not needed, and the actual feed entry has no effect on the separation. The minimum number of stages can be calculated by one of the total reflux methods for conventional columns. Since we are restricting the example by assuming constant relative vola .J ity, the Fenske-Underwood [5, 14, 15] equation applies.

and

(1)

By using the values from Tables 1 and 2 we find N 2 (min) = 4.63 and N 3 (min) = 4.63, and since there are no stages in Column 1, N (min) = 9.26 stages. At this point we should note that both the conventional and thermally coupled systems have about the same number of stages at total reflux. This will generally be the case except when the pressures in the two columns are very different, or when the relative volatilities are far from being constant. When the pressures of the two columns of the conventional system are considerably different, the chances are that the thermally coupled system would not be applicable.

Minimum Reflux

In general, the reflux requirement of a distillation system can be reduced by increasing the number of stages in the system. In the limit, a minimum reflux condition is attained. If infinite numbers of stages are specified in all sections of the thermally coupled system and if constant molal overflow and constant relative volatility are assumed, then Underwood's [16-18] analytical solution for minimum reflux can be extended to cover the thermally coupled system. The details of this development are given elsewhere [11]. The following equations describe column K at minimum reflux for conditions of constant molal overflow and constant relative volatility: (2) (3)

(4) Where fh k are referred to as the common roots in column K. These common roots lie between the volatilities ofthe components that are distributed between the overhead and bottoms of column K at minimum reflux.

287

Distillation, Thermally Coupled

In Column 1 of the thermally coupled distillation system, the common roots may be determined by solving Eq. (2), using the thermal condition and composition of the feed to the system. If a pair of components is distributed between both the overhead and bottoms of Column 1, then a common root, 81.1, will lie between their volatilities. This root will also be a root to the equation for the feed to Column 2. Then, if these two components are distributed in Column 2 between the overhead product ofthe system and the intermediate product, this same root is a common root in Column 2 and can be used along with the overall material balance and Eq. (3) to determine the minimum reflux. Similarly, if a pair of components is distributed between the overhead and bottoms of Column 1, and also between the intermediate product and bottoms from Column 3, the common root calculated from the feed to the system can be used along with the overall material balance and Eq. (4) to determine the minimum boilup. Generally, a number of components are assumed to be distributed in Column 1. The more volatile of these components will be distributed in Column 2, and the less volatile will be distributed in Column 3. However, the minimum reflux calculated by the common roots lying between the volatilities of the more volatile components applied to Column 2, and that determined from the common root lying between the volatilities of the less volatile components applied to Column 3, will generally be different. Under these conditions, the larger minimum reflux applies. The lower value has arisen due to an incorrect assumption about how the components are distributed in Column 1. The following equation, which relates the stripping vapor of Column 3 to the rectifying vapor of Column 2, is useful in establishing the proper minimum flows.

(5) This method is illustrated by applying it to the sample problem of Table 1. For this sample problem, the feed is specified a bubble-point liquid. Then the change in liquid rate going from the rectifying to the stripping section in Column 1 is equal to the feed rate and q1 = 1. Applying Eq. (2), we get

Substituting in the values and solving for the common roots, 81,1

= 4.6641

82,1

=

1.3359

If all three components, A, B, and C, are distributed in Column 1, then both of the above roots are common roots in Column 1. Since both A and B distributed in Column 2, and if we assume they are also distributed in Column 1, then

Distillation, Thermally Coupled

288

e1.2 = e1,1 = 4.6641 Applying this into Eq. (3), we find

and substituting in the values, ( V 2 )m = 0.5977 moljmol of feed. If we assume Components Band Care distributed in both Columns 1 and 3, then the common root 8 2 , 1 applies in Columns 1 and 3.

e1,3 = e2.1 =

1.3359

Combining this with the overall material balance in Eq. (4), we find

Then, substituting in the values, ( V3 )m = 0.8850 mol/mol of feed. In our system the feed is a bubble-point liquid, and it follows from Eq. (5) for constant molal overflow that

On this basis the minimum vapor rate in the rectifying section of Column 2 is V 2 = 0.8850 mol/mol of feed and the minimum reflux is L 2 = V 2 - D 2 = 0.8850 - 0.3541 = 0.5309 mol/mol of feed. The liquid and vapor in the stripping section of Column 3 are V3 = 0.8850 mol/mol of feed and £ 3 = 1.2391 moljmol of feed. Since our calculations of minimum reflux based on Column 3 show a higher required vapor V 2 than that based on Column 2, our assumption that A and Bare both distributed in Column 1 is incorrect. The root 81 , 1 is not a common root, and Component A is not distributed at minimum reflux between the overhead and bottoms of Column 1.

Design Method The design is initiated by assuming an operating reflux rate at some percentage over the minimum. Then splits in liquid between the rectifying section of Column 1 and the stripping section of Column 2 are chosen so that a reasonable loading occurs in both sections of Column 1, the rectifying section of Column 3, and the stripping section of Column 2. Similarly, the split in vapor between the stripping section of Column 1 and the rectifying section of Column 3 is chosen so that vapor loadings are relatively well balanced throughout the system.

289

Distillation, Thermally Coupled

The key to the design procedure is setting up the material balance around Column I. The function of this column is to split the feed into a fraction containing the light and mid components and a fraction containing the mid and heavy components. A reasonable balance around the column is obtained by assuming that all light materials appear in the overhead vapor and none in the bottoms liquid, and that all heavy materials appear in the bottoms liquid and none in the overhead vapor. The net flows of mid components are found by difference. If there is more than one mid component, they are distributed so that the more volatile ones appear predominantly in the feed to Column 2 and the heavie1 ones appear in the feed to Column 3. Columns 2 and 3 are designed as conventional columns, based on the simplified model given in Fig. 3, using any of the many published methods. If the system is separating a ternary mixture into three relatively pure products, then the separation in Columns 2 and 3 are binaries and may be treated by the McCabe-Thiele graphical or by other simple methods. Since the function of Column 1 is to remove the light components from the feed to Column 3 and the heavy components from the feed to Column 2, it is necessary to estimate the amounts oflight components to allow in the bottoms of Column 1 and of heavy components to allow in the overhead of Column 1. The degree of separation needed in Column 1 is a function of recycle to the system. At one limit, infinite recycle or total reflux, no separation is necessary; at the other limit, minimum reflux, a very good or complete separation is needed. The following assumptions give estimates that result in reasonable designs. For this purpose a light key component in Column 1 is defined as the least volatile of the light components that are distributed to the intermediate product, and a heavy key is defined as the most volatile of the heavy components that are distributed to the intermediate product. 1.

2.

3.

For the light key in the bottoms liquid from Column 1: (a) At total reflux its mole fraction is the mole fraction in the intermediate product. (b) At minimum reflux its mole fraction is zero. For the heavy key in the overhead vapor from Column 1: (a) At total reflux its mole fraction is the mole fraction in the vapor in equilibrium with the intermediate product. (b) At minimum reflux its mole fraction is zero. These mole fractions are given at intermediate reflux ratios by a linear interpolation on the reciprocal of the system reflux. Converting this into an algebraic equation: (6)

and

_ Rs - ( Rs )m K X R H.S H,S

YH.l,l-

s

(7)

290

Distillation, Thermally Coupled

Since the separation in Column 1 is being made between components of extreme volatilities, the degree of separation is not high. Therefore Column 1 is approximated by a sloppy separation in which generally all components are distributed between the overhead and bottoms at minimum reflux. This corresponds to the Class 1 separations as described by Shiras et al. [9], in which the composition of the liquid part ofthe feed is the same as the composition on the feed stage. With this basis the number of stages in Column 1 is estimated using the following arrangements and simplification of the Kremser equations [12]. The equation for a light component in the stripping section is (8)

where SL is the average stripping factor. The equation for a heavy component in the rectifying section is

(9) where A. is the average absorption factor. The design of a system for the sample problem is initiated by choosing an operating reflux, say 30% over the minimum. Then L 2 = 1.3(L 2 )m = 1.3(0.5309) = 0.693 moljmol of feed and the vapor rate is V 2 = L 2 +D. = 0.693 + 0.354 = 1.047 moljmol of feed. The internal flow throughout the system is determined by choosing reasonable recycles to Column 1, such as L 1 = 0.221 moljmol of feed and V1 = 0.635 moljmol of feed. The flows throughout the system are then determined by material balances and are summarized in Table 3. These flows should always be such that the net flow of material is in the direction of the products. For instance, if L 1 = 0.221 and V1 = 0.4, then the net flow in the stripping section of Column 1 would be upward. It might be very difficult, if not impossible, to make the system work under these conditions. The net overhead of Column 1 is D 1 = V1 - L 1 or D 1 = 0.635 0.221 = 0.414 moljmol of feed. Assuming none of Component C gets into this net overhead and none of A gets into the net bottoms, the material balance is completed as shown in Table 4. The number of stages in Columns 2 and 3 are estimated as if they were conventional columns. However, there are differences around the intermediate product withdrawal point. In general, the composition at the bottom of Column 2 will not correspond to the net bottoms, and the composition of the net overhead of Column 3 will not correspond to the overhead vapor at the top of this column. The stagewise calculations are illustrated in Fig. 4 by the McCabe-Thiele construction for Column 2. The figure indicates six stages are required in Column 2 with stage number two as the feed stage. A similar construction for Column 3 shows that nine stages are required, with the feed introducted five stages from the top.

291

Distillation, Thermally Coupled TABLE 3

Vapor and Liquid Flows: Example at 1.3 Times Minimum Reflux Rates (moles/mole of feed)

Column 1 Column 2 Column 3

Rectifying section Stripping section Rectifying section Stripping section Rectifying section Stripping section

Vapor

Liquid

0.635 0.635 1.047 0.412 0.412 1.047

0.221 1.221 0.693 0.472 0.180 1.401

0.8 < >-

MOLE FRACTION IN OVERHEAD PRODUCT

a:: 0

"-

< >

z

0. 6