Chemical Process Engineering Volume 1: Design, Analysis, Simulation, Integration, and Problem Solving with Microsoft Excel-UniSim Software for ... Fluid Flow, Equipment and Instrument Sizing [1, 1 ed.] 111951018X, 9781119510185

Citation preview

Volume 1 Chemical Process Engineering Design, Analysis, Simulation and Integration, and Problem Solving With Microsoft Excel – UniSim Design Software

Computation, Physical Property, Fluid Flow, Equipment & Instrument Sizing, Pumps & Compressors, Mass Transfer

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA 01915-6106

Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Volume 1 Chemical Process Engineering Design, Analysis, Simulation and Integration, and Problem Solving With Microsoft Excel – UniSim Design Software Computation, Physical Property, Fluid Flow, Equipment & Instrument Sizing, Pumps & Compressors, Mass Transfer

A. Kayode Coker and Rahmat Sotudeh-Gharebagh

This edition first published 2022 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2022 Scrivener Publishing LLC For more information about Scrivener publications please visit www.scrivenerpublishing.com. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http:// www.wiley.com/go/permissions. Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no rep­resentations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchant-­ability or fitness for a particular purpose. No warranty may be created or extended by sales representa­tives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further informa­tion does not mean that the publisher and authors endorse the information or services the organiza­tion, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 9781119510185 Cover image: Chemical Plant - Slidezero | Dreamstime.com Cover background: Flowing Image - Mikhail Sheleh | Dreamstime.com Cover design by Kris Hackerott Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines Printed in the USA 10 9 8 7 6 5 4 3 2 1

Companion Web Page This 2-volume set includes access to its companion web page, from which can be downloaded useful software, spreadsheets, and other value-added products related to the books. To access it, follow the instructions below: 1. Go to https://scrivenerpublishing.com/coker_volume_one/ 2. Enter your email in the username field 3. Enter “Refining” in the password field

Gratitude To the Almighty father for providing us with the abundance of nature, so that human beings can further develop and make use of the many things in nature for our well-being and for the world. To all process/chemical engineers worldwide utilizing this abundance of nature for the good of mankind. Keep the heart of your thoughts pure, by so doing you will bring peace and be happy To honour God in all things and to perform everything solely to the glory of God Abd-ru-shin (In the Light of Truth) A. Kayode Coker

Dedication “I am always obliged to a person who has taught me a single word.” In memory of my late father and to my respected family for their endless support To engineers and scientists for their commitment to inclusion and sustainability R. Sotudeh-Gharebagh

Contents Preface xvii Acknowledgments xix About the Authors

xxi

1 Computations with Excel Spreadsheet-UniSim Design Simulation SECTION I - NUMERICAL ANALYSIS INTRODUCTION Excel Spreadsheet Functions Trendline Coefficients Goal Seek SOLVER LINEAR REGRESSION Measuring Regression Quality MULTIPLE REGRESSION POLYNOMIAL REGRESSION SIMULTANEOUS LINEAR EQUATIONS NONLINEAR EQUATIONS INTERPOLATIONS INTEGRATIONS The Trapezoidal Rule Simpson’s 1/3 Rule Simpson’s 3/8 Rule DIFFERENTIAL EQUATIONS Nth Order Ordinary Differential Equations Solution of First-Order Ordinary Differential Equations Runge-Kutta Methods EXAMPLES AND SOLUTIONS SECTION II – PROCESS SIMULATION INTRODUCTION Thermodynamics for Process Simulators UNISIM Design Software EXAMPLES AND SOLUTIONS References

1 1 1 1 2 2 5 6 7 9 9 11 11 12 13 14 14 15 15 15 15 15 16 17 28 28 29 30 31 78

2 Physical Property of Pure Components and Mixtures PURE COMPONENTS Density of Liquid Viscosity of Liquid Heat Capacity of Liquid Thermal Conductivity of Liquid

81 81 82 83 85 87 ix

x  Contents Volumetric Expansion Rate Vapor Pressure Viscosity of Gas Thermal Conductivity of Gas Heat Capacity of Gases MIXTURES Surface Tensions Viscosity of Gas Mixture Enthalpy of Formation Enthalpy of Vaporization Gibbs Energy of Reaction Henry’s Law Constant for Gases in Water Coefficient of Thermal Expansion of Liquid DIFFUSION COEFFICIENTS Gas-Phase Diffusion Coefficients Liquid-Phase Diffusion Coefficients COMPRESSIBILITY Z-FACTOR SOLUBILITY AND ADSORPTION Solubility of Hydrocarbons in Water Solubility of Gases in Water Solubility of Sulfur and Nitrogen Compounds in Water Adsorption on Activated Carbon References 3 Fluid Flow INTRODUCTION Flow of Fluids in Pipes EQUIVALENT LENGTH OF VARIOUS FITTINGS AND VALVES Excess Head Loss Pipe Reduction and Enlargement PRESSURE DROP CALCULATIONS FOR SINGLE-PHASE INCOMPRESSIBLE FLUIDS Friction Factor Overall Pressure Drop Nomenclature COMPRESSIBLE FLUID FLOW IN PIPES Maximum Flow and Pressure Drop Critical or Sonic Flow and the Mach Number Mach Number Mathematical Model of Compressible Isothermal Flow Flow Rate Through Pipeline Pipeline Pressure Drop Nomenclature Subscripts TWO-PHASE FLOW IN PROCESS PIPING Flow Patterns Flow Regimes Pressure Drop Erosion-Corrosion Nomenclature VAPOR-LIQUID TWO-PHASE VERTICAL DOWNFLOW The Equations

90 91 93 94 95 97 98 99 101 103 105 107 108 109 109 110 111 116 116 117 118 119 119 121 121 121 123 123 124 124 127 128 130 130 131 131 132 134 136 138 139 139 139 140 142 142 145 145 146 147

Contents  xi The Algorithm Nomenclature LINE SIZES FOR FLASHING STEAM CONDENSATE The Equations Nomenclature FLOW THROUGH PACKED BEDS The Equations Nomenclature EXAMPLES AND SOLUTIONS References

147 147 148 148 149 150 151 152 152 162

4 Equipment Sizing INTRODUCTION SIZING OF VERTICAL AND HORIZONTAL SEPARATORS Vertical Separators Calculation Method for a Vertical Drum Calculation Method for a Horizontal Drum Liquid Holdup and Vapor Space Disengagement Wire Mesh Pad Standards for Horizontal Separators Piping Requirements Nomenclature SIZING OF PARTLY FILLED VESSELS AND TANKS The Equations Nomenclature PRELIMINARY VESSEL DESIGN Nomenclature CYCLONE DESIGN Introduction Cyclone Design Procedure The Equations Saltation Velocity Pressure Drop Troubleshooting Cyclone Maloperations Cyclone Collection Efficiency Cyclone Design Factor Cyclone Design Procedure Nomenclature GAS DRYER DESIGN The Equations Pressure Drop Desiccant Reactivation Nomenclature EXAMPLES AND SOLUTIONS References

165 165 166 166 168 170 171 171 172 172 172 173 173 175 176 177 178 178 178 179 180 181 182 182 182 183 183 184 186 187 188 188 189 194

5 Instrument Sizing INTRODUCTION Variable-Head Meters Macroscopic Mechanical Energy Balance Variable-Head Meters Orifice Sizing for Liquid and Gas Flows

195 195 195 196 196 200

xii  Contents Orifice Sizing for Liquid Flows Orifice Sizing for Gas Flows Orifice Sizing for Liquid Flow Orifice Sizing for Gas Flow Types of Restriction Orifice Plates Case Study 1 Nomenclature CONTROL VALVE SIZING Introduction Control Valve Characteristics Pressure Drop for Sizing Choked Flow Flashing and Cavitation Control Valve Sizing for Liquid, Gas, Steam and Two-Phase Flows Liquid Sizing Gas Sizing Critical Condition Steam Sizing Two-Phase Flow Installation Noise Control Valve Sizing Criteria Valve Sizing Criteria Self-Acting Regulators Types of Self-Acting Regulators Case Study 2 Rules of Thumb Nomenclature References

201 202 204 204 205 205 212 221 221 223 224 224 224 225 226 227 227 227 228 229 229 230 230 231 231 233 246 246 247

6 Pumps and Compressors Sizing 249 PUMPS 249 Introduction 249 Pumping of Liquids 249 Pump Design Standardization 252 Basic Parts of a Centrifugal Pump 253 Impellers 253 Casing 253 Shaft 254 CENTRIFUGAL PUMP SELECTION 255 Single-Stage (Single Impeller) Pumps 256 Hydraulic Characteristics for Centrifugal Pumps 260 Friction Losses Due to Flow 269 Velocity Head 269 Friction 271 NET POSITIVE SUCTION HEAD (NPSH) AND PUMP SUCTION 271 General Suction System 277 Reductions in NPSHR 279 279 Corrections to NPSHR for Hot Liquid Hydrocarbons and Water 280 Charting NPSHR Values of Pumps Net Positive Suction Head (NPSH) 280

Contents  xiii

Specific Speed “Type Specific Speed” Rotative Speed Pumping Systems and Performance System Head Using Two Different Pipe Sizes in Same Line POWER REQUIREMENTS FOR PUMPING THROUGH PROCESS LINES Hydraulic Power Relations Between Head, Horsepower, Capacity, Speed Brake Horsepower (BHP) Input at Pump AFFINITY LAWS Pump Parameters Specific Speed, Flowrate and Power Required by a Pump Pump Sizing of Gas-Oil Debutanizer Unit CENTRIFUGAL PUMP EFFICIENCY Centrifugal Pump Specifications Pump Specifications Steps in Pump Sizing Reciprocating Pumps Significant Features in Reciprocating Pump Arrangements Application Performance Discharge Flow Patterns HORSEPOWER Pump Selection Selection Rules-of-Thumb A CASE STUDY Pump Simulation on a PFD Variables Descriptions SIMULATION ALGORITHM Problem Discussion Pump Cavitation Factors in Pump Selection COMPRESSORS INTRODUCTION General Application Guide Specification Guides GENERAL CONSIDERATIONS FOR ANY TYPE OF COMPRESSOR FLOW CONDITIONS Fluid Properties Compressibility Corrosive Nature Moisture Special Conditions Specification Sheet PERFORMANCE CONSIDERATIONS Cooling Water to Cylinder Jackets Heat Rejected to Water Drivers Ideal Pressure – Volume Relationship Actual Compressor Diagram

282 285 286 286 288 291 292 293 293 296 298 299 301 303 306 311 311 312 313 314 316 316 317 318 318 318 321 321 322 322 323 324 332 333 334 334 334 337 337 338 338 338 339 339 339 339 339 339 340 341 343

xiv  Contents DEVIATIONS FROM IDEAL GAS LAWS: COMPRESSIBILITY 343 Adiabatic Calculations 346 Charles’ Law at Constant Pressure 346 Amonton’s Law at Constant Volume 346 Combined Boyle’s and Charles’ Laws 346 Entropy Balance Method 347 Isentropic Exponent Method 347 COMPRESSION RATIO 354 Horsepower 356 Single Stage 356 Theoretical Hp 356 Actual Brake Horsepower, Bhp 356 Actual Brake Horsepower, Bhp (Alternate Correction for Compressibility) 361 Temperature Rise – Adiabatic 363 Temperature Rise – Polytropic 365 A CASE STUDY USING UNISIM DESIGN R460.1 SOFTWARE FOR A TWO–STAGE COMPRESSION 365 CASE STUDY 2 365 Solution 365 1. Starting UniSim Design Software 366 2. Creating a New Simulation 366 Saving the Simulation 367 3. Adding Components to the Simulation 367 4. Selecting a Fluids Package 368 5. Select the Units for the Simulation 369 6. Enter Simulation Environment 369 Accidentally Closing the PFD 371 Object Palette 371 7. Adding Material Streams 371 8. Specifying Material Streams 372 9. Adding A Compressor 374 Specifications 381 COMPRESSION PROCESS 385 Adiabatic 385 Isothermal 385 Polytropic 385 Efficiency 388 Head 390 ADIABATIC HEAD DEVELOPED PER SINGLE-STAGE WHEEL 390 Polytropic Head 391 Polytropic 391 Brake Horsepower 393 Speed of Rotation 396 TEMPERATURE RISE DURING COMPRESSION 397 Sonic or Acoustic Velocity 399 MACH NUMBER 402 Specific Speed 402 COMPRESSOR EQUATIONS IN SI UNITS 403 Polytropic Compressor 405 Adiabatic Compressor 408 Efficiency 409

Contents  xv Mass Flow Rate, w Mechanical Losses Estimating Compressor Horsepower Multistage Compressors Multicomponent Gas Streams AFFINITY LAWS Speed Impeller Diameters (Similar) Impeller Diameter (Changed) Effect of Temperature AFFINITY LAW PERFORMANCE TROUBLESHOOTING OF CENTRIFUGAL AND RECIPROCATING COMPRESSORS NOMENCLATURE Greek Symbols Subscripts Nomenclature Subscripts Greek Symbols References Pumps Bibliography References Compressors Bibliography 7 Mass Transfer INTRODUCTION VAPOR LIQUID EQUILIBRIUM BUBBLE POINT CALCULATION DEW POINT CALCULATION EQUILIBRIUM FLASH COMPOSITION Fundamental The Equations The Algorithm Nomenclature TOWER SIZING FOR VALVE TRAYS Introduction The Equations Nomenclature Greek Letters PACKED TOWER DESIGN Introduction Pressure Drop Flooding Operating and Design Conditions Design Equations Packed Towers versus Trayed Towers Economic Trade-Offs Nomenclature Greek Letters

409 410 411 412 414 422 423 423 424 424 425 425 429 431 432 432 434 434 434 434 435 435 435 436 437 437 437 441 442 442 443 444 445 446 446 446 448 452 465 466 466 466 466 468 471 473 473 474 474

xvi  Contents

DETERMINATION OF PLATES IN FRACTIONATING COLUMNS BY THE SMOKER EQUATIONS Introduction The Equations Application to a Distillation Column Rectifying Section: Stripping Section: Nomenclature MULTICOMPONENT DISTRIBUTION AND MINIMUM TRAYS IN DISTILLATION COLUMNS Introduction Key Components Equations Surveyed Fractionating Tray Stability Diagrams Areas of Unacceptable Operation Foaming Flooding Entrainment Weeping/Dumping Fractionation Problem Solving Considerations Mathematical Modeling The Fenske’s Method for Total Reflux The Gilliland Method for Number of Equilibrium Stages The Underwood Method Equations for Describing Gilliland’s Graph Kirkbride’s Feed Plate Location Nomenclature Greek Letters EXAMPLES AND SOLUTIONS References

474 474 474 475 475 476 476 477 477 477 477 479 479 480 480 480 480 481 481 483 484 485 486 487 487 488 488 499

Index 501

Preface An increased use of computational resources by engineers has greatly expedited the design of equipment and process plants in refining, and the chemical process industries. In addition, the availability of commercial and open source process simulation packages, and spreadsheets has drastically reduced the requirement of programming with high-level languages, such as BASIC, C, C++, COBOL, Java, FORTRAN and Pascal. Situations may arise where a simulation package is not readily available, too expensive or of a limited scope, in this case, hands-on tools; e.g. spreadsheets could be used to review other alternatives or develop specific programs. Furthermore, the simulation packages can be integrated with spreadsheets to enlarge their scope to equipment and process design due to the infrastructure they provide on component databases, physical property estimations and unit operations. A text-book, including the theory with equations, tables, and figures and the use of advanced tools is not readily available to the process designer. Our aim in preparing this text is to present theory, along with Excel spreadsheet and UniSim Design software programs for solving a wide range of design problems. The book will, therefore, benefit chemical/process engineers, students, technologists and practitioners in the petroleum, petrochemical, pharmaceutical, biochemical and fine chemical industries. The structured approaches are provided that can be used to solve a wide range of process engineering problems and thus analyze and simulate process equipment regularly. The process design concepts, guidelines; codes and standards are clearly important, and several excellent books and references are available to address these issues and will be cited in the book where applicable. However, these two ­volume-sets are unique in that to date no textbook on chemical process design and simulation has been published, which provides adequate information (theory, equations, figures, tables and programs) to enable the reader to perform robust calculations using Excel spreadsheet - UniSim Design software program. The better use of these tools in the special format shapes the core of the book as: –– Microsoft Excel® is part of the Microsoft Office. It is widely recognized as the most versatile spreadsheet for problem solving, which enables a chemical engineer to make computation and visualizations in an easiest way possible. Process engineers can use Excel spreadsheet for equipment and process design, modeling, simulation and optimization. Data bases and pivot tables can be easily designed with Excel spreadsheet to ease making calculations, understanding technical reports, and preparing charts and figures. –– The new improved UniSim R480 Design of Honeywell, is a smart and intuitive software; it creates thermodynamics and unit operation steady-state and dynamic models. Process simulation is a tool used to design a new process, an existing process or debottleneck, monitor process conditions, troubleshoot current operations to compare theoretical results, optimize process conditions for enhanced throughput, and to reduce energy yields, and emissions. The contents of this book have been formed over many years through the concentrated research and industrial efforts of the authors. Furthermore, to assist the user and to demonstrate the validity of the methods, worked examples and case studies of practical relevance using the Excel spreadsheet and UniSim Design software programs are provided throughout the text, and the source files are provided at the publisher website. In this way, the students, and engineers can save time by using hands-on tools to ease the calculation and concentrate in understanding the fundamentals of the phenomena occurring in the process design sequences. These two volumes are fully extended version of the original title: Fortran Programs for Chemical Process Design, Analysis and Simulation by xvii

xviii  Preface A. Kayode Coker. In these volumes, we have provided examples in both Imperial and SI units, but we have intentionally kept most examples in Imperial units based on the following reasons: a) All problems solved in UniSim Design can be easily converted to any units of measurement and it makes no difference which unit is used. b) The use of both units is very useful from instructional viewpoint as students need to have some sort of practice with both units as the Imperial unit is still used in some countries. However, since all problems are carefully solved in Excel, the conversion is then easy, and we have also provided a conversion table to assist readers in their calculations. The book is primarily intended to serve senior students, early career engineers, university professors and practitioners, especially in the process, chemical, petrochemical, biochemical, mechanical, mining and metallurgical industries. However, other engineers, consultants, technicians and scientists concerned with various aspects of industrial design, and scale-up may also find it useful. It can be considered as a textbook to process design for senior and graduate students as well as a hands-on document for engineers at the entry level and practitioners. The content of this book can also be taught in intensive workshops in process industries.

Acknowledgments RSG wishes to express his profound gratitude to his students for reading the chapters and checking the programs. Further, he wishes to thank his current and former graduate students, Ms. Aghasi, Ms. Bakhshi and Messrs. Jabbari, Ahmadi, Moshiri, Khodabendehlou for checking the chapters and programs. Special credits are also extended to Professor Jamal Chaouki from Polytechnique de Montreal for hosting RSG for his sabbatical leave upon the completion of the book beside the main activity planned for sababtical. AKC expresses his gratitude to Ahmed Mutawa, formerly of SASREF for developing the conversion table software for the book. Thank you, Ahmed. Wherever it is required, permissions have been obtained to reproduce the works published by some organizations and companies. We acknowledge and thank the American Institute of Chemical Engineers, the Institution of Chemical Engineers (U.K.), Chemical Engineering (Mc-Graw Hill), Oil & Gas Journal, Tubular Exchanger Manufacturers’ Association, American Petroleum Institute, John Wiley & Sons, Nutter Engineering, and many other organizations that provided materials for this book. We express our gratitude to Honeywell Process Solutions for granting permission to incorporate the use of UniSim Design software simulation and many suites of software programs in the book. We wish to express our thanks to the Wiley-Scrivener team: Kris Hackerott- Graphics Designer, Bryan Aubrey – Copy editor, Myrna Ting – Typesetter and her colleagues. We are truly grateful for your professionalism, assistance and help in the production of this volume. Finally, very special thanks to Phil Carmical of Scrivener publishing company for his advice and helpful suggestions during the production of this volume. Finally, we should emphasize that process design is a creative, dynamic and challenging activity and for this reason, the design books like this one need continuous improvement with current digitalization outlook and abundant access to computation resources. We would appreciate and welcome any comments, suggestions, or feedback that you may have on this volume. A. Kayode Coker (www.akctechnology.com) A.K.C. TECHNOLOGY, U.K. Rahmat Sotudeh-Gharebagh ([email protected]) College of Engineering, University of Tehran, Iran

xix

About the Authors A. Kayode Coker PhD, is Engineering Consultant for AKC Technology, an Honorary Research Fellow at the University of Wolverhampton, U.K., a former Engineering Coordinator at Saudi Aramco Shell Refinery Company (SASREF) and Chairman of the Department of Chemical Engineering Technology at Jubail Industrial College, Saudi Arabia. He has been a chartered chemical engineer for more than 30 years. He is a Fellow of the Institution of Chemical Engineers, U.K., and a senior member of the American Institute of Chemical Engineers. He holds a B.Sc. honors degree in Chemical Engineering, a Master of Science degree in Process Analysis and Development and Ph.D. in Chemical Engineering, all from Aston University, Birmingham, U.K., and a Teacher’s Certificate in Education at the University of London, U.K., He has directed and conducted short courses throughout the world and has been a lecturer at the university level. His articles have been published in several international journals. He is an author of seven books in chemical engineering, a contributor to the Encylopedia of Chemical Processing and Design, Vol. 61, and a certified train – the mentor trainer. He is a Technical Report Assessor and Interviewer for chartered chemical engineers (IChemE) in the U.K. He is a member of the International Biographical Centre in Cambridge, U.K. (IBC) as Leading Engineers of the World for 2008. Also, he is a member of International Who’s Who for ProfessionalsTM and Marquis Who’s Who in the U.S. Rahmat Sotudeh-Gharebagh is currently a full Professor of Chemical Engineering at the University of Tehran (P.O. Box 11155–4563, Iran; email: [email protected]). He teaches process modeling and simulation, transport phenomena and fluidization, plant design and economics and soft skills. His research interests include computer-aided process design and simulation, fluidization, and engineering education. He holds a B.Eng. degree in chemical engineering from Iran’s Sharif University of Technology, plus a M.Sc. and a Ph.D. in Fluidization Engineering from Canada’s Polytechnique. He has been an invited Professor at Qatar University and Polytechnique de Montréal. Professor Sotudeh has more than 300 publications in major international journals and conferences, plus six books and four book chapters. He is the co-founder and Editor-in-Chief of Chemical Product and Process Modeling published by Walter de Gruyter GmbH, Germany, a Member of the Iranian Elite Foundation and an Expert Witness on the oil industry with the Iranian Expert Witness Organization and a winner of various awards and prizes.

xxi

1 Computations with Excel Spreadsheet-UniSim Design Simulation Engineers, technologists, and scientists have employed numerical techniques to solve a wide range of steady-state and transient problems, and energy and material balance equations. The fundamentals are essential in the basic operations of these techniques and the requirements are greater when new processes are designed. Engineers also need theoretical information and data from the literature to construct mathematical models with proper hypotheses. These mathematical models, which consist of a massive set of linear and non-linear equations, can be solved using established numerical analysis tools and process simulators. Furthermore, developing mathematical models sometimes involves experimental plans to obtain the required data for the models. Preparing an experimental plan is strongly dependent on the knowledge of the process with theory, where the whole modification can be produced by some form of mathematical models or regression analyses. This chapter deals with these important issues for chemical and process engineers in two sections: Section I: This section covers the modeling and numerical analysis with the Excel spreadsheet program which deals with curve fitting, approximation, interpolation, numerical solutions of simultaneous linear and nonlinear equations, and numerical differentiation and integration. Section II: This section covers process simulation using UniSim Design software which deals with the property estimation, equipment and process design, and case studies. These two tools (the Excel spreadsheet and UniSim Design software) are introduced in this chapter and are used throughout the book to ease calculations related to process design and analysis. In this way, students and engineers can save time by using hands-on tools to ease the calculations and concentrate on the fundamental understanding of the phenomena occurring in the process design sequences.

SECTION I - NUMERICAL ANALYSIS INTRODUCTION Excel Spreadsheet Microsoft Excel is part of the Microsoft Office. The Excel spreadsheet is today’s leading software for problem solving which enables a chemical engineer to make computations and visualizations in the easiest way. Most people often use the Excel spreadsheet program for business or financial purposes, but it is also a powerful and versatile tool that can be used for a wide variety of chemical engineering applications and numerical analysis. One of its key benefits is bringing techno-economic information together. Process engineers can use the Excel spreadsheet for equipment and process design, modeling, simulation and optimization. Databases and pivot tables can be easily designed with the Excel spreadsheet to perform calculations, understand technical reports, and prepare charts and figures. These make an engineer’s life easy in giving a presentation and trying to get a message across. The Excel spreadsheet program has the following unique characteristics [1, 2]: • • • •

1,048,576 rows by 16,384 columns Maximum number of cell characters (32,767 characters) 1,026 horizontal and vertical page breaks Undo levels: 100 and Iterations: 32,767

A. Kayode Coker and Rahmat Sotudeh-Gharebagh. Chemical Process Engineering: Design, Analysis, Simulation and Integration, and Problem-Solving With Microsoft Excel – UniSim Design Software, Volume 1, (1–80) © 2022 Scrivener Publishing LLC

1

2  Chemical Processing Engineering • Largest allowed number via formula: 1.8×10308 • Number of available worksheet functions: 341 Designing graphs and charts helps engineers with analyzing and predicting trends. The Excel spreadsheet applications make the engineers’ time usage more efficient and there is an enormous elimination of the time wasted on endless repetition of mathematical operations. The Excel spreadsheet program contains dozens of built-in functions, macros, add-ins and graphic modules that make it possible to develop powerful applications to solve process design problems. Students can also benefit from the Excel spreadsheet program to have a good adaption on its capabilities for their studies. It also makes mathematics very easy to understand in the rows and columns operations. Martín and Martín de Juan [3] provided Excellent coverage of the Excel spreadsheet program for chemical engineering. In their chapter, after introductory text, they discussed basics, built-in functions, operations with cells (columns and rows), fitting, plotting and solving, building functions in VBA (Visual Basic for Applications) and have also provided some comprehensive examples. Ferreira et al. [4] also reviewed the application of spreadsheets in chemical engineering education in process design and process integration. In their paper, the Excel SOLVER feature was used for the optimization of several problems on pollution prevention and mass-exchange networks. They examined three non-linear problems: the (a) recovery of benzene from a gaseous emission; (b) design of a chemical reactor network; and (c) solution of material balances in the production of vinyl chloride from ethylene. Dephenolization of aqueous wastes was also presented as a linear example. They concluded that the formulating and solving linear and non-linear problems within the Excel spreadsheet program is instrumental in teaching optimization and design concepts. In this chapter, we assumed that readers have basic knowledge of the Excel spreadsheet with hands-on experiences on cells (row and columns) and functions operations. This chapter covers selected basics and specific functions of Excel spreadsheet, calculation of trendline coefficients, Goal Seek and SOLVER add-in with some applied chemical engineering examples on numerical analysis. The key features of this chapter would be very useful for students and engineers at various levels and will be applied throughout the book whenever necessary.

Functions There is a large number of built-in functions in Excel spreadsheet, but there is no need to learn them all, especially at once. For the application in this book, selected functions are presented in Table 1.1 and Table 1.2 with their proper syntaxes. Worksheet functions are predefined formulas in Excel spreadsheet to make calculations in the order specified by its parameters. The functions have two basic parts: equal sign (=) and syntax which include a function name and arguments, and the functions mostly return numerical values. A detailed list and explanations of the Excel spreadsheet functions are given by Techonthenet [5].

Trendline Coefficients When one adds a trendline to a chart, the Excel spreadsheet program provides an option to display the trendline equation in the chart. The Excel spreadsheet functions can be also used to generate the trendline coefficients. This would allow calculating predicted y values for given values of x without a need to plot those data in the figure and read the coefficients from the Figures. Let’s assume two named ranges as x and y to show input arguments to Excel spreadsheet functions. Table 1.3 shows syntaxes for finding trendline coefficients. The combination of INDEX and LINEST functions can be also used to find the coefficients of the higher order polynomial. All functions shown in this table return the numeric values of the coefficients. For linear trendline, slope (b) and intercept (a) of the trend line is obtained by two ways [8]: • Using SLOPE and INTERCEPT functions • Using LINEST functions

Computations with Excel Spreadsheet-UniSim Design Simulation  3 Table 1.1  Selected basic functions of Excel spreadsheet [2, 5]. Function

Syntax

AVERAGE

AVERAGE (number1, [number2], ... [number_n]) number 1,… : numbers, named ranges, arrays, or references to numbers.

EXP

EXP (number) number: The power to raise e to.

IF

IF(condition, value_if_true, [value_if_false]) condition: value to test. value_if_true: It is the value returned if condition is TRUE. value_if_false: optional. It is the value returned if condition is FALSE.

INDEX

INDEX (table, row_number, column_number) row_number: Row position in the table for lookup. This is the relative row position in the table and not the actual row number in the worksheet. column_number: Column position in the table to lookup. This is the relative column position in the table and not the actual column number in the worksheet. Returns: returns any data type such as a string, numeric, date, etc.

LN or LOG

LN(number) or LOG(number) number: numeric value greater than 0.

MAX MIN

MAX or MIN(number1, [number2, ... number_n]) number1: It can be a number, named range, array, or reference to a number. number2, ... number_n: optional. These are numeric values (numbers, named ranges, arrays, or references to numbers) and can be up to 30 values entered.

PI

PI( ) with no arguments returns 3.14159265358979.

ROW COLUMN

ROW/COLUMN([reference]) Reference: optional. reference to a cell or range of cells. It returns row or column number of a cell reference.

SQRT

SQRT(number) number: positive number; If the number is negative, the function gives an error.

STDEV

STDEV(number1, [number2, ... number_n]) number1: It can be a number, named range, array, or reference to a number. number2, ... number_n: optional. These are numeric values (numbers, named ranges, arrays, or references to numbers) and can be up to 30 values entered.

SUM

SUM(number1, [number2, ... number_n]) or SUM (cell1:cell2, [cell3:cell4], ...) number: numeric value to sum; cell: range of cells to sum.

Let’s assume the range of x is A2:A13 and the range of y is B2:B13, the formulas would be: SLOPE(B2:B12,A2:A12) INTERCEPT(B2:B12,A2:A12) The same results can be obtained using the LINEST function. LINEST(C2:C13,B2:B13) However, with the use of the index function, we can find the values of the SLOPE (b) and INTERCEPT (a) by the following formulas in separate cells: INDEX(LINEST(C2:C13,B2:B13),1) INDEX(LINEST(C2:C13,B2:B13),2)

4  Chemical Processing Engineering Table 1.2  Selected applied functions of Excel spreadsheet for design and simulation [2, 5, 6]. Function

Syntax

CORREL

CORREL(array1, array2) array1 is a set of independent variables and array2 is a set of dependent variables. These arrays should be of equal length. returns the Pearson Product-Moment Correlation Coefficient(r)

FORECAST

FORECAST(x, known_y’s, known_x’s) x: value or data point whose value is interpolated/predicted by linear regression known_y’s: known range of y values. known_x’s: known range of x values.

INTERCEPT

INTERCEPT(known_y’s, known_x’s) known_y’s: y-values in data points (vector) known_x’s: x-values in data points (vector)

IRR

IRR(range, [estimated_irr]) range: range of cells that represent the series of cash flows. estimated_irr: optional. It is the guess of the internal rate of return. If omitted, it assumes irr = 0.1 or 10%.

LINEST

LINEST(y_values, [x_values], [constant], [additional_statistics]) y_values: Known set of “y values” x_values: optional. Known set of “x values”. If omitted, x_values is assumed to be [1,2,3,...] to have same length as y_values. For two or more independent variables, whole range of the known_x’s should be supplied. constant: optional. It is either TRUE or FALSE. If omitted, it is assumed TRUE. If it is TRUE, the intercept is calculated. If it is FALSE, the intercept becomes zero. additional_statistics: optional. It is either TRUE or FALSE. If omitted, the function will assume FALSE and it will return slopes and intercept. If it is TRUE, the function will return additional regression statistics.

MINVSERSE

MINVERSE(array) array: array of numbers with equal number of rows and columns. If any of the cells in this array are empty, contain non-numeric values or the array has unequal columns and rows, the function will return the #VALUE! error.

MMULT

MMULT(array1, array2) array1: array of numbers with same number of columns as number of rows in array2. array2: array of numbers with same number of columns as number of rows in array1.

NPV

NPV(discount_rate, value1, [value2, ... value_n]) discount_rate: The discount rate for the period. value1, value2, ... value_n: cash flows up to 29 values

PEARSON

PEARSON(y_values, x_values) y_values and x_values are two arrays of data with equal length. If any of these values text or logical values, or refer to empty cells, these values are excluded from the calculation. returns the Pearson Product-Moment Correlation Coefficient /(r)

SLOPE

SLOPE(known_y’s, known_x’s) known_y’s: y-values in data points (vector) known_x’s: x-values in data points (vector)

TREND

TREND( known_y’s, [known_x’s], [new_x’s]) x: value or data point whose value is interpolated/predicted by linear regression known_y’s: array known y-values. known_x’s: one or more arrays of known x-values news_x’s: optional value or data point whose value is predicted.

Computations with Excel Spreadsheet-UniSim Design Simulation  5 Table 1.3  Syntax for finding trendline coefficients [7]. Equation

Syntax for coefficients

Trendline

Y = a + bX

(1.1)

b: =SLOPE(y,x) a: =INTERCEPT(y,x)

Linear

Y = a + b LN(X)

(1.2)

b: =INDEX(LINEST(y,LN(x)),1) a: =INDEX(LINEST(y,LN(x)),2)

Logarithmic

Y = aXb(1.3)

b: =INDEX(LINEST(LN(y),LN(x)),1) a: = EXP(INDEX(LINEST(LN(y),LN(x)),2))

Power

Y = aebX(1.4)

b: =INDEX(LINEST(LN(y),x),1) a: =EXP(INDEX(LINEST(LN(y),x),2))

Exponential

Y = C0 + C1X + C2X2(1.5)

C2: =INDEX(LINEST(y,x^{1,2}),1) C1: =INDEX(LINEST(y,x^{1,2}),2) C0: =INDEX(LINEST(y,x^{1,2}),3) x^{1,2} means the range of two arrays where values of xi and xi2 are stored.

2nd Order Polynomial

All coefficients can be also obtained using: =LINEST(y,x^{1,2}) in one row and three columns. Y = C0 + C1X + C2X2 + C3X3(1.6)

C3: =INDEX(LINEST(y,x^{1,2,3}),1) C2: =INDEX(LINEST(y,x^{1,2,3}),2) C1: =INDEX(LINEST(y,x^{1,2,3}),3) C0: =INDEX(LINEST(y,x^{1,2,3}),4) x^{1,2,3} means the range of three arrays where values of xi , x i2 and x 3i are stored.

3rd Order Polynomial

All coefficients can be also obtained using: =LINEST(y,x^{1,2,3}) in one row and four columns.

Goal Seek The Excel GOAL SEEK function is part of Excel’s what-if analysis tool set. This allows the user to solve the nonlinear equations. To locate this function, one should navigate to the Data tab in the ribbon menu and at the far right, under the What-If Analysis group, the Goal Seek function is seen and it requires input for three parameters as seen in Figure 1.1:

Figure 1.1  The Excel Goal Seek Window.

6  Chemical Processing Engineering Where: Set cell: The address of the cell where the formula is entered. Let’s assume that is the cell $D$24 which holds the non-linear equation of (i.e. 1.35). To value: The real desired value of the equation to be solved by the GOAL SEEK function. Let’s set it to 0. This means that the non-linear equation is set to zero. By changing cell: This is the root of non-linear equation (let’s assume fD in Equation 1.35). The initial value is given in the cell, e.g. in cell $D$23 and then the Goal Seek iterates to find the final solution in the same cell.

SOLVER SOLVER is a Microsoft Excel add-in program used for what-if analysis. This is a widely used application to solve a set of algebraic equations, optimization, fitting experimental data to linear and non-linear equations and so on. It is generally used to find an optimal (maximum or minimum) value for a formula in one cell, called the objective cell, subject to constraints or limits on the values of other formula cells on a worksheet. It works with a group of cells, called decision variables or simply changing cells used in computing the formulas in the objective and constraint cells. SOLVER adjusts the values of the decision variable in cells to satisfy the constraint cells and produce the desired result for the objective cell. It comes pre-installed with Windows versions of the Excel spreadsheet program, but one must activate it manually before using. To activate the Excel SOLVER add-in, the following steps are needed: 1. C  lick the File tab, click Options, and then click the Add-ins item. 2. In the Manage box, click Excel Add-ins, and then click Go. 3. In the Add-Ins dialog box, select the SOLVER add-in check box and then click ok. Once activated, it can be visible on the Data sub menu on the far-right side of the ribbon bar. In Figure 1.2, an example of the SOLVER parameters window is shown. This window shows part of the solution of Example 1.4 where the system of 4 equations and unknowns are solved. The SOLVER parameters window should be properly configured with all parameters needed as explained below: 1. I n the Set Objective cell, the objective function is defined (cell $C$38). This is the address of the cell where equation is written which refers to four cells. The value of this cell can be set to Max, Min or a given value. 2. The decision variables or changing cells to be estimated are placed in By Changing Variable Cells section ($C$32:$C$35). 3. The constraints, which could be equality or non-equality equations, are introduced under the Subject to the Constraints dialogue box ($C$39 = $I$39, $C$40 = I$40, $C$41 = $I$41). 4. By clicking on the OK button, SOLVER will determine the values of the Changing Cells that maximize, minimize, or get closer to the exact value of the objective function. The Excel SOLVER uses several methods to find optimal solutions developed by Frontline Systems [9] as: a. G  RG Nonlinear Solving Method: This is the Generalized Reduced Gradient (GRG2) code used for nonlinear optimization. b. Simplex LP Solving Method: This is the Simplex method commonly used for linear programming with bounds on the variables and the dual Simplex methods, which is branch-and-bound type of algorithms, for mixed integer linear programming. c. Evolutionary Solving Method: This is a variety of genetic algorithms and local search methods.

Computations with Excel Spreadsheet-UniSim Design Simulation  7

Figure 1.2  The Excel Solver parameters Window.

The SOLVER can solve problems with up to 200 variables and 100 constraints. The user can control options and tolerances used by the optimization methods through the SOLVER Options button. Martín and Martín de Juan [3] explained these options in detail. By using SOLVER, one can perform all regressions analysis and measurement, data fitting and solve various linear and non-linear equations.

LINEAR REGRESSION Texts with computer programs and sometimes with supplied software are now available for scientists and engineers [10–13]. They have provided a function or functions to measure data that fluctuate, which result from random error of measurement. If the number of data points equals the order of the polynomial plus one, we can exactly fit a polynomial to the data points. Fitting a function to a set of data requires more data than the order of a polynomial. The accuracy of the fitted curve depends on many experimental data. In this chapter, we will use least-squares curve fitting programs. Also, we will use linear regression analyses to develop statistical relationships involving two or more variables. The most common type of relationship is a linear function. By transforming nonlinear functions, many functional relations are become linear. In this chapter, we will correlate X-Y data for the following equations:



Y = a + bX

(1.1)



Y = a + bX2

(1.7)



Y=a+

b X

(1.8)

8  Chemical Processing Engineering



Y = a + bX0.5



Y = aXb(1.3)



Y = aebX



Y = a + b log(X)

(1.10)



Y = a + beX

(1.11)

(1.9)

(1.4)

We can transform the nonlinear equations (1.4, 1.5, and 1.11) by linearizing them as follows:



Y = aXb → ln(Y) = ln(a) + b ln(X)

(1.12)



Y = aebX → ln(Y) = ln(a) + bX

(1.13)



Y = a + beX → ln(Y) = ln(a) + ln(b)X

(1.14)

Regression analysis uses statistical and mathematical methods to analyze experimental data and to fit mathematical models to these data. We can solve for the unknown parameters after fitting the model to the data. Suppose we want to find a linear function that involves paired observations on two variables, X the independent variable and Y the dependent variable. Where: n = the number of observations Xi = the ith observation of the independent variable Yi = the ith observation of the dependent variable We can develop a linear regression model that expresses Y as a function of X. We can further derive formula to determine the values of a and b that give the best fit of the equations. For each experimental point corresponding to an X-Y pair, there will be an element that represents the difference between the corresponding calculated value Yˆ and the original value of Y. This can be expressed as:

ri = Yˆi − Y



(1.15)

ri may be either negative or positive depending on the side of the fitted curve of X-Y points. Here, we can minimize the sum of the squares of the residuals (SSR) by the following expression: n

SSR =

∑r

i

2

= minimum

(1.16)

1

where n is the number of observations of X-Y points

Yˆi = a + bXi



(1.17)

ri = a + bXi – Yi and

n

SSR =

(1.18)

n

∑r = ∑(a + bX − Y ) i

1

2

i

1

i

2

(1.19)

Computations with Excel Spreadsheet-UniSim Design Simulation  9 The problem is reduced to finding the values of a and b so that the summation of Equation 1.19 is minimized. Coker [13] showed how the coefficients a and b can be determined from Cramer’s rules from Eq. 1.17-1.27 in his Fortran book. Here we use the Excel spreadsheet program which provides three methods using built-in functions to calculate the values of a and b. • With the SLOPE function we can calculate the value of b and with the INTERCEPT function, we can calculate the value of a using the syntaxes explained in Table 1.2. • The Excel LINEST function can be also used to calculate the unknown values. • The Excel SOLVER add-in can be also used to minimize equation 1.19 and find the values of a and b variables.

Measuring Regression Quality The quality of linear regression is measured by the Pearson product-moment correlation coefficient (r) which is a statistical measurement of the linear regression between two sets of data. For two sets of values (X and Y), the Pearson correlation coefficient (r) is given by the following formula [6]:

r=

∑( X − X )(Y − Y ) ∑( X − X )2 (Y − Y )2



(1.20)

where X and Y are the sample means of the two arrays of data. The Excel PEARSON function calculates the correlation coefficient for two supplied sets of values. This coefficient has the following properties: 1. I f the value of r is close to +1, this indicates a strong positive correlation. This means that there is a positive relationship between the variables; as one variable increases or decreases, the other tends to increase or decrease with it accordingly. 2. If the value of r equal to zero, there is no linear correlation. However, nonlinear correlations may exist, but the relationships cannot be measured using the Pearson product-moment correlation (r). 3. If the value of r is close to -1, this indicates a strong negative correlation. This means that when one of the variables increases, the other tends to decrease, and vice versa. Although the correlation coefficient gives a measure of the accuracy of fit, we should treat this method of analysis with great caution. Because the value of r is close to one does not always mean that the fit is necessarily good [13]. It is possible to obtain a high value of r when the underlying relationship between X and Y is not even linear. It is worth mentioning that the PEARSON function performs the same calculation as the CORREL Function. However, in earlier versions of the Excel spreadsheet program, the PEARSON function may exhibit some rounding errors. Therefore, the users of the earlier version of the Excel worksheet can use the CORREL rather than the PEARSON function. In more recent versions of the Excel worksheet; e.g., office 365, both functions give the same results.

MULTIPLE REGRESSION Inadequate results are sometimes obtained with a single independent variable. This shows that one independent variable does not provide enough information to predict the corresponding value of the dependent variable. We can approach this problem, if we use additional independent variables and develop a multiple regression analysis to achieve a meaningful relationship. Here, we can employ a linear regression model in cases where the dependent variable is affected by two or more independent variables.

10  Chemical Processing Engineering The linear multiple regression equation is expressed as:

Y = C0 + C1X1 + C2X2 + ⋯ + CkXk

(1.21)

where Y = the dependent variable X1, X2, …, Xk = the independent variables C1, C2, …, Ck = the unknown regression coefficients k = the number of independent variables The unknown coefficients are estimated based on n observation for the dependent variable Y, and for each of the independent variables Xj’s where j = 1, 2, 3,… , k. These observations are of the form:

Yj = C0 + C1X1j + C2X2j + ⋯ + CkXkj

(1.22)

For j = 1, 2, …, N where Yj = the jth observation of the dependent variable X1j ,…, Xkj = the jth observation of the X1, X2, …, Xk independent variables The LINEST function can be used to solve equation 1-21 and find the coefficients. This function returns the coefficients for two supplied sets of values with the syntax shown in Table 1.2. We can also use a least squares technique to calculate the coefficients C0, C1, …, CK by minimizing the following equation: N



S=

∑[Y − (C + C X j

0

1

1j

N

2

+ +C2 X 2 j +  + CK X Kj )] =

1

∑r

2 j



(1.23)

1

Equation 1.23 can be minimized by the SOLVER add-in to calculate the coefficients of C1, C2, …, Ck. With initial guesses for these coefficients, equation 1.23 is calculated in a cell and this cell is addressed as set objective in the SOLVER window and then it is solved by minimizing objective function by changing the cells related to initial values of the coefficients. It is also worth mentioning that the power equations have often been derived to calculate the parameters of experimental data. Such an equation can be expressed in the form:



Y = C0 . X1C1 . X 2C2 .... X kCk .

(1.24)

We can calculate the coefficients of the independent variables, if this equation is linearized by taking its natural logarithm to give:

ln(Y) = ln(C0) + C1ln(X1) + +C2ln(X2) + ⋯ + Ckln(Xk)

(1.25)

The same procedure explained before can be also used to find the coefficients of this equation with the LINEST function and SOLVER add-in.

Computations with Excel Spreadsheet-UniSim Design Simulation  11

POLYNOMIAL REGRESSION Some engineering data are often poorly represented by linear regression. If we know how Y depends on X, then we can develop some form of nonlinear regression [13], although total convergence of this iterative regression procedure cannot be guaranteed. However, if the form of dependence is unknown, then we can treat Y as a general function of X by trigonometric terms or polynomial function. The least-squares procedure can be readily extended to fit the data to an nth-degree polynomial:

Y = C0 + C1X + C2X2 + ⋯ + CnXn

(1.26)

where C0, C1, C2, …, Cn are constants. For this case, the sum of the squares of the residuals is minimized: N



S=

∑ n

Yj − (C0 + C1 X j + C2 X 2j +  + C j X nj )  = 2

N

∑r

2 j



(1.27)

1

linear equations generated by polynomial regression can be ill-conditioned when the coefficients have very small and very large numbers. This results in smooth curves that fit poorly [10]. The Excel LINEST function and SOLVER add-in can be used to find the coefficients of nth-degree polynomial as explained earlier.

SIMULTANEOUS LINEAR EQUATIONS Analyses of physiochemical systems often give us a set of linear algebraic equations. Also, methods of solution of differential equations and nonlinear equations use the technique of linearizing the models. This requires repetitive solutions of sets of linear algebraic equations. Linear equations can vary from a set of two to a set having 100 or more equations. In most cases, we can employ Cramer’s rule to solve a set of two or three linear algebraic equations. However, for systems of many linear equations, the algebraic computation becomes too complex and may require other methods of analysis. Coker [13] reported two methods of solving a set of linear algebraic equations, namely Gauss elimination and Gauss-Seidel iteration methods. Gauss elimination is most widely used to solve a set of linear algebraic equations. Other methods of solving linear equations are Gauss-Jordan and LU decomposition. The set of n algebraic equations of n unknowns is represented by:

a1,1X1 + a1,2X2 + a1,3X3 + ⋯ + a1,nXn = Y1 a2,1X1 + a2,2X2 + a2,3X3 + ⋯ + a2,nXn = Y1 ⋯ ⋯ ⋯ an,1 X1 + an,2 X2 + an,3 X3 + ⋯ + an,nXn = Yn

(1.28)

12  Chemical Processing Engineering The conventional Gauss elimination or Gauss-Seidel iteration methods can be used to solve this set of equation. Coker [13] provided a FORTRAN program using these methods to solve the set. The Excel worksheet can be also used to implement these two methods to solve the set of algebraic equations. Implementation of these methods in the Excel worksheet is a lengthy and tedious process. However, for the purpose of application in this book, the MINVERSE function as shown in Example 1.4 is used if we convert the set to the following form:

AX = Y  a1,1  A=    an,1 



  

 Y1  Y= .  Yn



(1.29)

a1,n     an,n  

(1.30)

   

(1.31)

Then the solution would be:

X = A−1Y

(1.32)

The Excel built-in MINVERSE function returns the inverse of the matrix A with the syntax shown in Table 1.2. The MINVERSE function is a worksheet function and can be entered as part of a formula in a cell of a worksheet. In order to use it, the empty Excel cells (same column and row equal to A matrix) are chosen and then the following syntax is used in the first cell to find the inverse of the matrix in the work sheet cell. In some versions of the Excel worksheet, the size of the inverse matrix is chosen automatically, otherwise, before entering the function, choose the right empty cells equal to the matrix size and then push Shift-Ctrl-Enter keys simultaneously, e.g.; if the matrix is located in cells $B$34:$E$37. =MINVERSE(B34:E37)

NONLINEAR EQUATIONS Solving problems in chemical and process engineering often requires finding the real root of a single nonlinear equation. Examples of such computations are in fluid flow, where pressure loss of an incompressible turbulent fluid is evaluated. The Colebrook [14] implicit equation of the Darcy friction factor, fD, for turbulent flow is expressed as:



1 2.51 ε D = −2log +  fD  3.7 N Re f D

 

(1.33)

where ε/D is the pipe roughness in feet, fD is the Darcy friction factor and NRe is the Reynolds number. Equation 1.33 is non-linear and involves trial and error solutions to achieve a solution for fD. The Equation can be further expressed as:

F( fD ) =

1 2.51 ε D + 2log +  fD  3.7 N Re f D

In the form of natural logarithm, the Equation becomes:

 

(1.34)

Computations with Excel Spreadsheet-UniSim Design Simulation  13

1 2.51  ε D + 0.86858ln +  fD  3.7 N Re f D 

F( fD ) =

(1.35)

In thermodynamics, the pressure-volume-temperature relationships of real gases are described by the equation of state. The compressibility factor from the reduced pressure and temperature can be rearranged from Redlich and Kwong [15] to two constant state equations in the form:



Z =1+ A −

where



A=

AB A2 B + A + Z AZ + Z 2

0.0867 Pr Tr

and B =

(1.36)

4.934 Tr1.5

Equation 1.36 is also nonlinear and can be expressed as:



 A2 B  AB + F(Z ) = Z −  1 + A −   A + Z AZ + Z 2 

(1.37)

For multicomponent separations, it is often necessary to estimate the minimum reflux ratio of a fractionating column. A method developed for this purpose by Underwood [16] requires the solution of the equation: N

f (θ ) =

∑ (αα −Xθ ) − 1 + q = 0 1

i

i

i



(1.38)

where N = the number of components in the feed xi = the mol fraction of component i. αi = the relative volatility of component i q = the thermal condition of the feed Equation 1.38 is highly nonlinear and must be solved for θ, the Underwood parameter, or its root. It has a singularity at each value of θ = α. If we can determine with some precision the degree of vaporization of the feed and the distillation composition, then we can also obtain a single value of θ by an iterative solution. The value of θ is generally bounded between the relative volatilities of the light and heavy keys (θLK < θ < θHK) and that is N

f (θ ) =

∑ (αα −Xθ ) − 1 + q = 0 1

i

i

i

(1.39)

Techniques to solve the non-linear equations have been published in various textbooks; e.g. [9, 11, 12]. The Goal seek function and SOLVER add-in of the Excel worksheet can be used to solve non-linear equations in this chapter.

INTERPOLATIONS Experimental and physical property data sometimes require values of their unknown functions that correspond to certain values of their independent variables. In certain cases, we may want to determine the behavior of the

14  Chemical Processing Engineering function. Alternatively, we may want to approximate other values of the function at values of the independent variables that are not tabulated. We can achieve these objectives either by interpolation or extrapolation of a polynomial fitted a selected data set of both variables (xi, f(xi)). Generally, the experimental data are approximated by a polynomial, the degree of which can often be calculated by constructing a difference table. The difference column that gives approximate constant value shows the degree of the polynomial that can be fitted to the data. When the polynomial is of the first degree, we have a linear interpolation. For polynomials of higher degrees, we can approximate functions if we construct a table with wider spacing. Such a table is known as a difference table. Coker [13] explained a fundamental approach as to how a difference table can be constructed. For simplicity, the Excel FORECAST, and TREND functions are used for the interpolations.

INTEGRATIONS We frequently use numerical techniques to integrate a function given in both analytical and tabular forms. For instance, we can use an integral method to determine the volumetric rate of a gas through a duct from the linear velocity distribution. In fluid mixing with residence time distribution theory, Danckwerts [17] showed that the fraction of material in the outlet stream that has been in the system for a period between t and t+dt is equal to Edt. E is a function of t, and E(t) is the residence time distribution function. We can express E(t) in integral form as:

∫



∞

0

E(t )dt = 1

(1.40)

The average time spent by material flowing at a rate, q, through a volume, V, equals V/q. The mean residence time, V t = , We can also express this equation in the form of dimensionless time where θ = tq/V, and this becomes: q

∫



∞

0

E(θ )dθ = 1

(1.41)

Numerical integral methods can be used to calculate areas under this or similar distribution functions. Various techniques can be implemented in the Excel worksheet for numerical integrations. Among the methods, the Trapezoidal Rule, Simpson’s 1/3 and 3/8 Rules are presented here.

The Trapezoidal Rule This is a numerical integration method derived by integrating the linear interpolation formula. It is expressed as:



I≅

∫

x i +1 xi

f ( x )dx ≅

f ( xi ) + f ( xi +1 ) h (∆x ) ≅ ( fi + fi +1 ) 2 2

(1.42)

For (a,b) subdivided into sub-intervals of size h, we can express the area as [13]:

I≅



∫

b

a

n

f ( x )dx ≅

∑ h2 ( f + f i

i +1

)

i =1

h ≅ ( f1 + 2 f 2 + 2 f3 +  + 2 fn + fn+1 ) 2



(1.43)

Computations with Excel Spreadsheet-UniSim Design Simulation  15

Simpson’s 1/3 Rule Simpson’s 1/3 rule is based on quadratic polynomial interpolation. For a quadratic integrated over two Δx intervals that are of uniform width or panels, the area is expressed as [13]:

I≅

∫

b

a

f ( x )dx

h ≅ [ f1 + 4( f 2 + f 4 +  fn ) + 2( f3 + f5 +  + 2 fn−1 ) + fn+1 ] 3





(1.44)

Simpson’s 3/8 Rule Simpson’s 3/8 rule is derived by integrating a third-order polynomial interpolation formula. For a domain (a, b) divided into three intervals, it is expressed as [13]:

I≅ ≅

∫

b

a

f ( x )dx

3h ( f1 + 3 f 2 + 3 f3 + 2 f 4 + 3 f5 + 3 f6 +  + 2 fn− 2 + 3 fn−1 + 3 fn + fn+1 ) 8 = width × average height

(1.45)

DIFFERENTIAL EQUATIONS In certain cases, we may need to find out the behavior of many dynamic and physical processes. This is often expressed mathematically by ordinary differential equations. The solutions of these equations are of great value to engineers and scientists and can be obtained by well-known analytical methods, although many physically differential equations are impossible to solve analytically. In this chapter, we consider the numerical techniques for solving differential equations.

Nth Order Ordinary Differential Equations We consider the solutions of the Nth order differential equations of the form:



 dy d 2 y d 3 y d n−1 y d n y  F  x , y, , 2 , 3 ,…, n−1 , n  = 0  dx dx dx dx dx 

(1.46)

Equation 1.46 has the highest derivative of the order n, and is ordinary because there is only one independent variable, x. We can obtain a unique solution when some additional information such as values of y(x) and its derivatives at some specific values of x are known. Therefore, for an Nth order equation, we require such N conditions to arrive at the unique solution y(x). If all N conditions are known at the same value of x, we can classify the problem as an initial value problem. However, when more than one value of x is involved, the problem is classified as a boundary value problem.

Solution of First-Order Ordinary Differential Equations We express a first-order equation as:



dy = f (x , y ) dx

(1.47)

and we require a solution y(x) that satisfies Equation 1.47 and one initial condition. Here, we can subdivide the interval in the independent variable in x into steps over which a solution is required (a,b). The value of the exact

16  Chemical Processing Engineering solution y(x) is then approximated at N+l evenly spaced values of x; i.e., (x0, x1, …, xn-1, xn). The step size, h, is expressed as:

h=



b−a n

(1.48)

and

xi = x0 + i × h, i=0,1,2,…,n

(1.49)

If we let the true solution y(x) be y(xi), and the computed approximations of y(x) at these same points be yi, so that

yi = y(xi)

(1.50)

Then, the exact derivative dy/dx can be approximated by f(xi,yi) and represented as fi such that

fi = f(xi, yi) = f(xi, y(xi))

(1.51)

The difference between the computed value and the true value is εi and can be expressed as:

εi = yi – y(xi)

(1.52)

εi is the local truncation error. The solution of a differential equation by direct Taylor’s expansion cannot be easily obtained as explained by Coker [13] if we retain the derivatives of a higher order. We can develop single-step procedures that involve only first-order derivative evaluations and produce results equivalent in accuracy to higher order Taylor formulas. These algorithms are named the Runge-Kutta methods after the German mathematicians Runge and Kutta.

Runge-Kutta Methods The second-order Runge-Kutta algorithm for the first-order differential equation can be expressed as:

dy = f (x , y ) dx k1 = hf ( xi , yi ) k2 = hf ( xi + h, yi + k1 )



(1.53)

1 yn+1 = yn + (k1 + k2 ) + O(h3 ) 2

The third-order Runge-Kutta method is:

dy = f (x , y ) dx k1 = hf ( xi , yi ) h k   k2 = hf xi + , yi + 1  2 2 k3 = hf ( xi + h, yi + 2k2 − k1 )

1 yn+1 = yn + (k1 + 4 k2 + k3 ) + O(h 4 ) 6



(1.54)

Computations with Excel Spreadsheet-UniSim Design Simulation  17 The fourth-order Runge-Kutta method is widely used in computer solutions of differential equations.

dy = f (x , y ) dx k1 = hf ( xi , yi ) h k   k2 = hf xi + , yi + 1  2 2



(1.55)

h k   k3 = hf xi + , yi + 2  2 2 k4 = hf ( xi + h, yi + k3 ) 1 yn+1 = yn + (k1 + 2k2 + 2k3 + k4 ) + O(h5 ) 6



k1, k2, k3, k4 are approximate derivative values computed on the interval of xi ≪ x ≪ xi+1 and h is the step size.

h = xi+1 – xi

(1.56)

There are, however, some other methods implementable in the Excel worksheet as: 1. R  unge-Kutta-Gill method which is the most widely used single-step method for solving ordinary differential equations. This provides an efficient algorithm for solving a system of first-order differential equations and makes use of much less computer memory when compared with other methods. 2. Runge-Kutta-Merson which outlines a process for deciding the step size for a better predetermined accuracy. For this method, five functions are evaluated at every step. The formulae for these methods were provided by Coker [13] and since these are somehow similar to the RungeKutta method, we did not explain them here. It is worth mentioning that the primary advantage of the single-step methods is that they are self-starting. We can also vary the step sizes. In contrast, the multistep methods [13], e.g. fourth-order Mine’s method, Adams-Moulton fourth-step method, require a single-step formula to start the calculations. Step size variation is difficult. However, the efficiency of both the Milne’s and Adams-Moulton methods is about twice that of the single-step methods. We need two function evaluations per step in the former while four or five are required with a single step. For the purpose of application in the book, the single-step method like, RungeKutta fourth-order method, can give sufficient accuracy and it is illustrated in example 1.8.

EXAMPLES AND SOLUTIONS Example 1.1 An experimental result in a liquid mixing experiment for the power correlation using a pitched-blade turbine shows that in the viscous regime, the power number is related to the Reynolds number by the following data (Table 1.4):

Solution The Excel Program, Example 1.1.xlsx, determines the slope, intercept, and correlation coefficient for the above experimental data using four Excel functions; SLOPE, INTERCEPT, LINEST and PEARSON. The syntax for using these functions is given in Table 1.2. For the data given in this Example, 8 equations are evaluated to find the perfect fit. We will use linear regression analyses to develop statistical relationships involving two variables. The

18  Chemical Processing Engineering Table 1.4  Input data for liquid mixing experiment. Reynolds number, NRe

Power number, P0

1

50

3

18

5

11

7

7.9

9

6.2

13

5

Figure 1.3  Snapshot of the Excel calculations.

Figure 1.4  A characteristic curve of power number as a function of Reynolds number.

Computations with Excel Spreadsheet-UniSim Design Simulation  19 most common type of relationship is a linear function. By transforming non-linear functions, many functional relations are made linear. The results show that equation 1.8 (Y = A + B *1 X ) gives the best fit with the slope b= 49.01, intercept a=1.12 and the correlation coefficient r = +0.9998. These are similar to values reported by Coker [13]. Figure 1.3 gives a plot of Reynolds number versus the power number with the regressed model and experiments. The figure shows that Equation 1.8 gives a perfect fit of the experimental data. Equation 1.8 also gives the better fit as compared with others. Figure 1.4 shows a characteristic curve of power number as a function of Reynolds number. The Excel Analysis Toolpak can be also used for the analysis of variance or ANOVA. This is a statistical method that separates observed variance data into different components to gain information about the relationship between the dependent and independent variables.

Example 1.2 In a fluid flow experiment, the volumetric rate of fluid through a pipe is dependent on the pipe diameter and slope by the following equation:

Q = C0 D C1 S C2



(1.57)

where Q = flow rate, ft3/sec D = pipe diameter, ft S = slope, ft/ft Determine the flow rate of fluid for a pipe with a diameter of 3.25 ft and a slope of 0.03 ft using the following data (Table 1.5):

Solution The Excel Program, Example 1.2.xlsx, determines the values of the coefficients C0, C1, and C2 and the correlation coefficient. For the data and function given in this Example, we will use linear regression analyses to develop statistical relationships involving variables. The nonlinear function is made linear with the following transformation:

ln(Q) = ln(C0) + C1 ln(D) + C2ln (S) or Table 1.5  Input data for fluid flow experiments. Diameter ft

Slope, ft/ft

Flowrate, ft3/s

1

0.001

1.5

2.5

0.005

9

3

0.01

25

4

0.01

5

1

0.05

30

3.5

0.05

100

(1.58)

20  Chemical Processing Engineering

y = C3 + C1x1 + C2x2

(1.59)

where

C0 = exp (C3)

(1.60)

By correlating data to equation 1.58 with INDEX and LINEST functions with the syntax given in Table 1.3, unknown values and correlation coefficient, r, are calculated as shown in Figure 1.5. The coefficients are: C0 = 563.25, C1 = 0.272, C2 = 0.87 and the correlation coefficient r = +0.9. The predicted flow for a pipe with a diameter of 3.25 ft and a slope of 0.03 ft/ft is 36.81 ft3/sec.

Example 1.3 The following data are obtained from y = x4 + 3x3 + 2x2 + x + 5. Show that a fourth-degree polynomial provides the best least squares approximation to the given data. Determine this polynomial with data shown in Table 1.6.

Solution The fourth-degree polynomial is written to the following form:

yˆ = C4 X 4 + C3 X 3 + C2 X 2 + C1 X1 + C0



(1.61)

Where: X4 = x4, X3 = x3, X2 = x2 and X1 = x

Figure 1.5  Snapshot of the Excel calculations for the fluid flow experiment.

Table 1.6  xy data for polynomial approximation. x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

y

5.123

5.306

5.569

5.938

6.437

7.098

7.949

9.025

10.363

Computations with Excel Spreadsheet-UniSim Design Simulation  21 Using the LINEST function, the coefficients of C4, C3, C2, C1 and C0 are found with data shown in Table 1.6 for equation 1.61 and the calculated polynomial from the Excel program is:

yˆ = x 4 + 3x 3 + 2 x 2 + x + 5



The Excel program, Example 1.3.xlsx, shows fitting a polynomial to a set of data N pairs of the independent and dependent variables. It calculates the coefficients of the polynomial and the correlation coefficient. The program shows that the fourth-degree polynomial gives the perfect fit with r = 1 as shown in Figure 1.6. The coefficients are the same as those of equation with which y was initially generated as the input data for the Example.

Example 1.4 The final product from a chemical factory is made by blending four liquids (α, β, γ and σ) together. Each of these liquids contains four components A, B, C, and D. The product leaving the factory must have a closely specified composition. Determine the relative quantities of α, β, γ and σ required to meet the blend specifications in the following data (Table 1.7).

Solution The Excel program, Example 1.4.xlsx, uses the matrix inversion capability of Excel and the SOLVER add-in to determine the quantities α, β, γ and σ required to meet the blend specifications. The syntax for MINVERSE function and SOLVER add-in are provided in the previous section of this chapter.

Figure 1.6  Snapshot of the Excel calculations for the LINEST function.

Table 1.7  Blend specifications. w/w composition of Component

α

β

γ

σ

(w/w) composition of specification

A

51.3

43.2

56.4

47.4

48.80

B

11.3

11.5

15.5

8.5

11.56

C

29.4

31.5

22.5

30.4

29.43

D

8.0

10.3

5.6

13.7

10.21

Source: B.Sc. Final year 1978, Aston University, Birmingham U.K.

22  Chemical Processing Engineering 1. By the inverse of Matrix A if the problem is formulated in the following form:

AX = B X = A−1B

(1.62)

2. U  sing SOLVER add-in with the initial values for α, β, and σ. The relevant cells to these values are iterated with this add-in to find the results. The following figure shows the results of using these two methods. The following quantities (α = 0.1172, β = 0.3789, γ = 0.2117 and σ = 0.3054) are required to meet the blend specifications. Similar results are calculated with two methods as shown in Figure 1.7. The SOLVER add-in is formulated with the procedure shown in the following snapshot. In previous sections, the application of this add-in was explained in more details. Figure 1.8 shows the snapshot of the SOLVER parameters. Coker [13] used the Gaussian elimination method with FORTRAN to determine the quantities of α, β, γ and σ required to meet the blending specification and the results are in agreement with the Excel spreadsheet calculations as shown. This method can be also programmed with Excel. Since Excel spreadsheet provides built-in functions to solve the system of linear equations, these functions are used for the sake of simplicity here.

Figure 1.7  Snapshot of the Excel spreadsheet calculations to determine the quantities α, β, γ and σ.

Figure 1.8  Snapshot of the SOLVER parameters.

Computations with Excel Spreadsheet-UniSim Design Simulation  23

Example 1.5 A first-order irreversible reaction takes place in a series of four continuous stirred tank reactors (CSTR) with recycle streams (Figure 1.9): The conditions of temperature in each reactor are such that the values of k and V are given in Table 1.8. Using the following assumptions, set up the material balance equations for the system and use the Excel spreadsheet built-in functions to determine the exit concentration from each reactor: a. b. c. d.

 e system is at steady-state conditions. Th The reactions are in the liquid phase. There is no change in the volume or density of the liquid. The rate of disappearance of component A in each reactor is given by –rA = kCA.

Solution The general unsteady state material balance for each reactor is:

Input = output + disappearance + accumulation by reaction

(1.63)

The mass balance for reactor 1 is:

qA0C A0 = qA0C A1 + (−rA )V1 + V1

CA0 qA0

k1

A

dC A1 dt

(1.64)

A3

B

V1 CA1 k1

CA1 qA0

qA4 V2 CA2 k2

CA2 qA0 + qA3

V3 CA3

CA3

k3

V4

qA0 + qA3 + qA4

CA4 k4

Figure 1.9  Chemical reaction with recycles in four continuous stirred tanks.

Table 1.8  Design and reaction data for continuous stirred tanks. Reactor

Volume, Vi, L

Rate Constant, kl. h-l

1

1000

0.1

2

1500

0.2

3

100

0.4

4

500

0.3

Source: Constantinides and Mostoufi [10].

CA4 qA0 + qA4

24  Chemical Processing Engineering Because the system is at steady state condition, the accumulation term is zero, therefore, the above equation becomes:

qA0CA0 = qA0CA1 + k1CA1V1

(1.65)

The mass balance for reactor 2 is:

qA0CA1 + qA3CA3 = (qA0 + qA3)CA2 + k2CA2V2

(1.66)

The mass balance for reactor 3 is:

(qA0 + qA3)CA2 + qA4CA4 = (qA0 + qA3 + qA4)CA3 + k3CA3V3

(1.67)

The mass balance for reactor 4 is:

(qA0 + qA4)CA3 = (qA0 + qA4)CA4 + k4CA4V4

(1.68)

Where: qA0=1000 liters/h, CA0=1 mol/liter, qA3=100 liters/h, qA4=100 liters/h. By replacing the values in above mentioned equations and rearranging, the following system can be obtained: 1100CA1 = 1000 1100CA1 – 1400CA2 + 100CA3 = 0 1100CA2 – 1240CA3 + 100CA4 = 0 1100CA3 – 1250CA4 = 0 The above is a set of four simultaneous linear algebraic equations which is similar to Example 1.4 and MINVERSE function, Example 1.5.xlsx, can be used to solve the set as shown in Figure 1.10. Since there is a diagonal system of equations in this Example, the conventional method of the Gauss-Seidel is used. Coker [13] developed a FORTRAN program based on this method for solving material balance equations. These are rearranged to solve for the unknown on the diagonal position of each equation. This algorithm can be programmed

Figure 1.10  Snapshot of the Excel spreadsheet calculations to determine exit concentrations in CSTRs.

Computations with Excel Spreadsheet-UniSim Design Simulation  25 in Excel spreadsheet, but for the sake of simplicity, the MINVERSE function is used to solve the equations. The results obtained by the Excel spreadsheet calculations agree with those by Coker [13].

Example 1.6 The Colebrook implicit equation for the Darcy friction factor, fD, for turbulent flow is:

F( fD ) =

1 2.51 ε D + 0.86858 Ln +  3.7 N Re f D fD

 

(1.69)

Determine the friction factor for NRe = 184,000, ε = 0.00015 ft and D = 0.17225ft (2.067 inch).

Solution The Excel program, Example 1.6.xlsx, solves the above explicit equation using Goal Seek as shown in the following figure. Coker [13] developed a FORTRAN program using Newton-Raphson method and found fD = 0.02063 which is identical to the value obtained by Excel spreadsheet in a very simple way as shown in Figure 1.11.

Example 1.7 In a fire tube boiler where gas flows inside the tubes and a steam-water mixture flows on the outside, the heat transfer coefficient inside hi (Btu/ ft2hr.0F) can be expressed as:



hi = 2.44

w 0.8F1 d 0.8

(1.70)

Where factors F1, F2 and F3 are:







 Cp  F1 =    µ 

0.4

 Cp  F2 =    µ 

0.3

k 0.6

(1.71)

k 0.7

(1.72)

F  F3 =  2  µ 0.15  Cp 

Figure 1.11  Snapshot of the Excel spreadsheet calculations to solve an implicit equation with Goal Seek.

(1.73)

26  Chemical Processing Engineering Table 1.9 shows Fl, F2, and F3 for flue gas at a temperature range 200 °F < T < 1200 °F. Determine the values of F1, F2, and F3 if the film temperature is 575 °F.

Solution The Excel program, Example 1.7.xlsx, uses the FORECAST and TREND functions, shown in Table 1.2, to determine the values of F1, F2, and F3 at T = 575 ° F. Figure 1.12 shows the snapshot of the Excel calculations. Coker [13] obtained the values of Fl, F2, and F3 from Stirling’s central difference formula as: Fl = 0.1932, F2 = 0.1160 and F3 = 0.6431 with FORTRAN programming. These values are in close agreement with those by the Excel spreadsheet calculations. Other techniques reported in the literature can be also used, but for the sake of simplicity, we employed the Excel built-in functions here.

Table 1.9  Flow gas data at various temperatures. Temperature (°F)

F1

F2

F3

200

0.1700

0.0954

0.5851

300

0.1770

0.1015

0.6059

400

0.1835

0.1071

0.6208

600

0.1943

0.1170

0.6457

800

0.2051

0.1264

0.6632

1000

0.2136

0.1340

0.6735

1200

0.2216

0.1413

0.6849

Source: Ganapathy [18].

Figure 1.12  Snapshot of the Excel calculations for the interpolation.

Computations with Excel Spreadsheet-UniSim Design Simulation  27

Example 1.8 Design of a batch reactor for the reaction scheme and determine the concentrations of A, B, C, and D over a period of ten minutes. A

k1

k2

B

C

k5 k3

k4

D

The rate constants and initial amounts for first-order reactions are shown in Table 1.10.

Solution The mass balances for the batch reactor involving components A, B, C and D are:

(−rA )net = k 1C A − k 5C B = −



dC A dt

(−rB )net = (k 2 + k 3 + k 5 )C B − k 1C A − k 4C D = − (−rC )net = k 2C B =



dC B dt

dC C dt

(−rD )net = k 4 C D − k 3C B = −



(1.74)

dC D dt

(1.75) (1.76) (1.77)

Rearranging the above equations



dC A = k 5C B − k 1C A dt

(1.78)



dC B = k 1C A + k 4 C D − (k 2 + k 3 + k 5 )C B dt

(1.79)



dC C = k 2C B dt

(1.80)

Table 1.10  Kinetic and concentration data for the batch reactor. Rate Constant [h-1]

Concentration at time t=0 [mol/m3]

k1 = 0.45

CA0 = 9.90

k2 = 0.16

CB0 = 0.0

k3 = 0.12

CC0 = 0.0

k4 = 0.08

CD0 = 0.5

k5 = 0.10

28  Chemical Processing Engineering



dC D = k 3C B − k 4 C D dt

(1.81)

The Excel spreadsheet program, Example 1.8.xlsx, uses the Runge-Kutta fourth-order methods to solve the differential equations. Figure 1.13 shows the profiles of concentrations from the start of the batch reaction to the final time of ten minutes. The intermediate results are shown in the Excel file.

Figure 1.13  Snapshot of the Excel spreadsheet for the solution of the set of differential equations.

SECTION II – PROCESS SIMULATION INTRODUCTION Simulation is the representation of a chemical process by a mathematical model and the solution of the resulting system of equations to obtain information on the performance of the proces. Applications for the simulation include heat and mass balance analysis, process design, sizing and costing, engineering studies, design audits, debottlenecking, control system check-out, process simulation, dynamic simulation, operator training simulator (OTS), pipeline management systems, production management systems and digital twins. Table 1.11 shows the benefits of using process simulators for engineers and scientists. Texts, literature and websites are now available for scientists and engineers regarding the basics and details of commercial process simulation tools; e.g., [19–23]. In these texts and literature, steady state and dynamic simulations, the differences and similarities among process simulators, library and hypothetical components, ideal and non-ideal units, convergence tips and tricks, and validation of simulation results are introduced and discussed in details. Complexity levels in process simulation are in the following order: material balances, energy balances, optimization, sizing and costing and profitability analysis, respectively. Wikipedia [24] has also provided a list of process simulators used to simulate the material and energy balances of chemical processes. Table 1.12 shows the name and website address of the commonly used process simulators. The concept of simulation in most leading simulators is based on Sequential Modular Simulation (SMS) approach [19]. In this approach, a sequential order of calculations is established which allows to simulate one unit at a time and in a forward manner. The chemical process with recycles is decomposed into one or several calculation sequences

Computations with Excel Spreadsheet-UniSim Design Simulation  29 Table 1.11  Benefits of using process simulators for engineers and scientists. Item

Application

1.

Estimate physical properties

2.

Obtain Information on intermediate streams

3.

Study operating parameters effect on process/plant

4.

Verify plant and process specifications

5.

Reduce down-time, start-up time, damage and injury

6.

Optimize material and energy usage and costs

7.

Decide on process control strategies

8.

Feed real-time plant data into the simulator to predict conditions

9.

Investigate real time optimization and parameters adjustment

10.

De-bottleneck each individual section of the process

Table 1.12  List of mostly used chemical process simulators. Simulator (Company)

Uniform Resource Locator (URL)

aspenONE (AspenTech)

www.aspentech.com

UniSim Design software (Honeywell)

www.honeywellprocess.com

Petro-SIM (KBC)

www.kbc.global

CHEMCAD (Chemstations)

www.chemstations.com

PRO/II (AVEVA SimSci)

www.sw.aveva.com

ProMax (Bryan)

www.bre.com

through tear streams and the solution is obtained by an iterative approach until the convergence criteria are satisfied for the whole process including units and streams. This approach is less flexible and more robust, the storage requirement is low, but the initialization is very important. It is worth mentioning that the model simplification is of prime importance in any simulation task. Complex models with many rigorous columns and recycles would be difficult to converge. The simplified model can be used to initialize tear streams in the complex model. The models can be simplified with fewer components, simpler unit operations (shortcut) (e.g. columns with separators and exchanger with heater) or replacing complex user models with simple models. Table 1.13 shows a good simulation practice for the successful simulation. It is important that after validation and verification of a simulation, we can use it confidently for process design. This can be performed through full-scale operational data, pilot plant, bench scale, laboratory and literature data depending on the availabilities, respectively. However, the choice of a proper thermodynamic model is critical in all stages of the simulation.

Thermodynamics for Process Simulators One of the basic tests of a thermodynamic model is to find the relevant experimental data and to choose a model that generates results, e.g., K-values or whatever is consistent with the experimental data [25]. The comparison helps

30  Chemical Processing Engineering Table 1.13  Good simulation practice. Step

Practice

1.

Start with a good physical property data

2.

Choose & validate a thermophysical model

3.

Build process one step at a time (Keep It SIMPLE!)

4.

Handle recycles with special care

5.

Use fewer pure components

6.

Exclude less important traces component

7.

Check overall and local heat and mass balance (manually)

8.

Converge the flowsheet first and then carry out a sensitivity analysis

decide whether the model is good, or another type would be more suitable. Process simulators have in-built fluid packages that provide accurate thermodynamic physical and transport properties for hydrocarbons, non-hydrocarbon, petrochemical and chemical fluids. Many of these simulators have databases of over 1,500 components and over 10,000 fitted binary coefficients. If a library component cannot be found within the database, the simulator provides methods for defining hypothetical components with a set of supplied data. Selecting the correct thermodynamic models from the property packages in the simulators (e.g., UniSim©, Hysys©, Aspen©, ChemCad©, Pro/II©, ProSim©, etc.) enables the designer to predict properties of mixtures ranging from well-defined light hydrocarbon systems to complex oil mixtures and highly non-ideal chemical systems. Proper use of thermodynamic property package parameters results in successfully simulating any chemical process as the effects of operating conditions can drastically alter the accuracy of a simulation yielding missing parameters or parameters fitted for different conditions. The thermodynamic models used to predict the chemical interaction properties, e.g., activity coefficients, volume and enthalpy of mixing, excess entropy of mixing, and K-values are the NRTL, Wilson, SRK, UNIQUAC, and UNIFAC models. These are incorporated in various simulation packages, e.g., Pro II, HYSYS, Aspen Plus, ProSim Plus, ChemCad and UniSim Design software (Honeywell). It is essential to select an appropriate thermodynamic model, otherwise erroneous results on the simulation of any process will occur. Various procedures were reported in the literature for the selection of proper thermodynamic models, and readers are strongly recommended to consult with literature for the reliable simulations. In main process simulators, guidelines were also proposed for choosing the selected thermodynamic model based on their own experiences. These are all very useful and one should always consult these sources as well [26–29].

UNISIM Design Software The UniSim Design software is among the widely used process simuators in the chemical process industry with an easy-to-use graphic user interface (GUI). It is a user-friendly, fast, smooth and interactive process simulator that enables engineers to create steady-state and dynamic models for equipment, plant design, performance monitoring, troubleshooting, optimizing and operational improvement. This section aims to provide a step-by-step guide for UniSim Design software simulator*. The UniSim Design software enjoys the same GUI as HYSYS of Hyprotech while AspenTech† markets its HYSYS simulator with different GUI from 2016 [19]. Herein, we assumed that the readers have basic knowledge of The UniSim Design software, and we only *www.honeywellprocess.com † www.aspentech.com

Computations with Excel Spreadsheet-UniSim Design Simulation  31

Figure 1.14  A new case in The UniSim Design software (Courtesy of Honeywell The UniSim Design software, Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

provide selected hands-on examples with detailed and step-by-step descriptions. These examples show how to estimate the physical properties and to perform the simulation of streams, equipment and chemical processes. Problems are also presented in other chapters of the book wherever relevant.

EXAMPLES AND SOLUTIONS Example 1.9 Calculate the properties of water at 25 ºC and 1 atm with a mass flow rate of 100 kg/hr.

Solution Follow the step-by-step instructions to solve the Example. 1. O  pen a new case (Figure 1.14). 2. Add a new component list (Figure 1.15). 3. Select water from the Components list and then close the active window by clicking on the cross button (Figure 1.16). If the active window is not seen on the figure, move it to see the cross button and close it. 4. 5. 6. 7.

S elect the Fluid Package (Make sure to select Component List -1 in the component list) (Figure 1.17). Add a new Fluid Package (Figure 1.18). Select the Peng-Robinson equation of state from fluid Package (Figure 1.19). Close the Fluid Package by clicking on the cross button. After this step, it is also possible to import/ export the Fluid Package. Enter to the Simulation Environment (Figure 1.20). 8. Drag a material stream to the PFD or click F11. (Choose the stream from the Object Palette‡ by pressing F4). Rename the stream if needed (Figure 1.21). ‡

All unit operations are classified in the Object Palette (Mode details in UniSim Reference Guides).

32  Chemical Processing Engineering

Figure 1.15  A new component list in UniSim Design software (Courtesy of Honeywell UniSim Design software, Honeywell are registered trademarks of Honeywell International Inc.).

Figure 1.16  List of components in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

and UniSim

9. Save the simulation work (e.g., Example 1.9) (Figure 1.22). 10. For the N components in the stream, N + 2 parameters are needed as input data in order the degree of freedom for the stream becomes zero. That allows calculating the stream properties. Click on stream 1, enter 2 out of 3 specifications (temperature, pressure and vapor fraction) and mass flowrate in the Worksheet/Conditions page (Figure 1.23). Enter the value of one for mol fraction in the Worksheet/Composition page (Figure 1.24). 11. The stream properties can now be calculated (seen in the Worksheet/Properties page). Note that in Worksheet/condition page, items in blue and black indicate user-defined and calculated properties, respectively (Figure 1.25). 12. By putting the cursor on the stream, the Fly-By window appears showing the main properties of the stream (Figure 1.26).

Computations with Excel Spreadsheet-UniSim Design Simulation  33

Figure 1.17  List of components in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Figure 1.18  A fluid package tab in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Figure 1.19  The choice of equation of state in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

34  Chemical Processing Engineering

Figure 1.20  Simulation Basis Manager in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.21  PFD window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell registered trademarks of Honeywell International Inc.).

and UniSim

and

are

Example 1.10 Plot the T-x-y diagram for the binary mixture of 1-butanol and water.

Solution Follow the step-by-step instructions to solve the Example. 1. O  pen a new case, add a new component list, select 1-butanol and water from the components list then close the active window by clicking on the cross button. 2. In the Fluid Package, select Component List -1 in the component list. 3. Add a new Fluid Package by selecting the UNIQUAC activity model from the Fluid Package. 4. Close the Fluid Package by clicking on the cross button. After this step, it is possible to import/export the Fluid Package. You may now enter the Simulation Environment as shown in Figure 1.20. 5. Drag a material stream to the PFD. (Choose the stream from the Object Palette (F4) or press F11.) Rename the stream to H2O (Figure 1.27). 6. Save the simulation work (e.g., Example 1.10.usc) (Figure 1.28).

Computations with Excel Spreadsheet-UniSim Design Simulation  35

Figure 1.22  Saving simulation in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Figure 1.23  Stream property window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

7. S elect another material stream for 1-butanol. Define both streams (100 °C and 1 atm) as explained in the previous Example. 8. Set the molar flow rate of water to 0.7 kmol/hr and its composition (mol fraction = 1) in order to define it. Enter the composition of 1-butanol (mol fraction = 1) (Figure 1.29). 9. Use the set function from the Object Palette to keep the total molar flow of these streams equal to 1. In this way, independent mol fraction variables could be defined for a mixture (Figure 1.30). 10. Double click on the set icon to define the target variable (Figure 1.31). 11. Choose H2O as a source stream (Figure 1.32).

36  Chemical Processing Engineering

Figure 1.24  Stream composition page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.25  Stream properties in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and

and UniSim

12. Click on the parameters section in the Set-1 to define the multiplier (-1) and offset (+1) in order to keep the total molar flow rate of these two streams at the outlet equal to 1. The molar flow rate of the second stream is adjusted so that the total molar flow rates of these two streams remain one where the molar flow rates represent the molar fractions (Figure 1.33). 13. There are different ways to keep the total molar flow rate of these streams equal to one instead of using Set function. For example, remove Set-1, use a mixer (from the object pallet) and then set the molar flow rate of the outlet stream to 1 kmol/hr. The flow rate of 1-butanol is adjusted accordingly (Figure 1.34).

Computations with Excel Spreadsheet-UniSim Design Simulation  37

Figure 1.26  Fly-By window for stream in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.27  Drag and define material stream in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

14. Connect the inlet and outlet streams by double clicking on the Mixer. (The stream properties can now be calculated) (Figure 1.35). 15. In order to plot the T-x-y diagram, the bubble and dew point should be calculated for the water stream flowrate varying from 0 to 1. A Heater and a Cooler are added to the flow sheet for this purpose. Add the unit operations and streams as shown in Figure 1.36. 16. Assume no pressure drop in the Heater and the Cooler. Click on E-100 and E-101 and put their pressure drops to zero (T-x-y at constant pressure) (Figure 1.37).

38  Chemical Processing Engineering

Figure 1.28  Adding both streams in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Figure 1.29  Definition of streams in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Figure 1.30  Addition of Set function in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Computations with Excel Spreadsheet-UniSim Design Simulation  39

Figure 1.31  Definition of Set function in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.32  Adding source stream to Set function in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

17. To calculate the bubble and dew point for a given mixture (mol fraction of water = 0.7 and mol fraction of 1-butanol = 0.3), set the vapor fraction at the exit streams of the Heater and Cooler to zero and one, respectively. With right click on streams, properties can be shown in the Table (Figure 1.38). 18. To plot the figure, press Ctrl + D to open Databook and click on the insert button in order to choose the required variables from the flowsheet (Figure 1.39). 19. Extract the relevant variables from PFD (temperatures for S-bubble and S-dew as dependent and flowrate of water stream as independent variables) (Figure 1.40). 20. Go to case studies and add a new case (Figure 1.41). 21. Choose the molar flow as independent (to represent x in T-x-y) and temperatures as dependent variables (to represent T in T-x-y) and then press view (Figure 1.42).

40  Chemical Processing Engineering

Figure 1.33  Entering Set function data in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.34  Adding Mixer to define 1-Butanol stream in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.35  Simulation of Mixer unit operation to determine 1-Butanol stream in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Computations with Excel Spreadsheet-UniSim Design Simulation  41

Figure 1.36  Adding Heater and Cooler to calculate bubble and dew points in UniSim Design software (Courtesy of Honeywell The UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.37  Pressure drop window for heater in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.38  Bubble and dew points calculation window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

42  Chemical Processing Engineering

Figure 1.39  Variable window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell registered trademarks of Honeywell International Inc.).

and UniSim

are

Figure 1.40  Selection of variables from flowsheet in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.41  Case studies window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Computations with Excel Spreadsheet-UniSim Design Simulation  43

Figure 1.42  Choice of dependent and independent variables for Case study in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

22. Specify the low, high bound and step size values of the independent variable (from 0-1 as the range of mol fractions and 0.02 as step size) and press start (Figure 1.43). 23. After the completion of the simulation, press the Results button to view the T-xy diagram (Figure 1.44). Different thermodynamic models may be selected to generate T-xy diagram and he simulation data can be also compared with the experimental data to figure out the proper physical property models employed in the simulation.

Example 1.11 Consider a stream of gas (T = 40 ºC and P = 30 kg/cm2) containing methane, ethane, propane, n-butane and n-pentane with molar flow rates of 40, 25, 15, 10 and 10 kmol/hr, respectively. Calculate: a. b. c. d.

 ressure of dew point at 40 ºC. P Pressure of bubble point at 40 ºC. Temperature of dew point at 30 kg/cm2. Temperature of bubble point at 30 kg/cm2.

Figure 1.43  Adding the data for independent variable in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim  are registered trademarks of Honeywell International Inc.).

44  Chemical Processing Engineering

Figure 1.44  A plot of T-x-y diagram in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

e. S tream enters a separator. Calculate properties of outlet streams. f. Plot molar flow rate of ethane in the vapor stream as a function of the operating temperature. g. Adjust the drum temperature to reach 50% liquid (in molar basis).

Solution Follow the step-by-step instructions to solve the Example. 1. 2. 3. 4. 5. 6. 7. 8.

 pen a new case. O Add a new component list Select components from the components list, then close the active window. If the cross button is not seen on the figure, move the active window to see and close it. Select the Fluid Package (Make sure to select Component List -1 in the component list). Add a new Fluid Package. Select the Peng-Robinson equation of state from Property Package. Close the Fluid Package by clicking on the cross button. After this step, it is possible to import/export the Fluid Package. You may now enter the Simulation Environment as shown in Figure 1.20. 9. Drag a material stream to the PFD and enter two out of three properties (temperature, pressure and vapor fraction) in the Worksheet/Conditions page. 10. Save the simulation work (e.g., Example 1.11.usc). 11. Enter the molar flow rate of components in Worksheet/Composition page (Figure 1.45). 12. By pressing the OK button, the properties of the stream will be calculated for a base case data (Figure 1.46). a. F  or calculating the dew point pressure at 40 ºC, first erase the pressure of the stream. Then, enter 1 in vapor/phase fraction of stream. (Pressure of dew point is 891.8 kPa) (Figure 1.47). b. For calculating the bubble point pressure at 40 ºC, enter 0 in vapor/phase fraction of stream. The pressure of bubble point is 9335 kPa (Figure 1.48). c. For calculating the temperature of dew point, erase the temperature, and then enter 30 kg/cm2 for pressure and 1 for vapor fraction. The dew point temperature is 76.29 ºC (Figure 1.49).

Computations with Excel Spreadsheet-UniSim Design Simulation  45

Figure 1.45  Input composition page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.46  Simulation of stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software Honeywell are registered trademarks of Honeywell International Inc.).

and

and UniSim

d. F  or calculating the temperature of bubble point at 30 kg/cm2, enter 0 for vapor/phase fraction of stream. The temperature of bubble point is calculated to be -63.42 ºC (Figure 1.50). e. Return to base case conditions (40 ºC and 30 kg/cm2). Then, put a separator on the PFD from the Object Palette (F4) (Figure 1.51). Double click on the separator to open it. Enter inlet and outlet vapor and liquid streams on the Design/Connections page. The calculation is now completed for an adiabatic separator immediately (Figure 1.52). The properties of streams are seen in the Worksheet/Conditions page (Figure 1.53). f. To complete a case study, go to Tools/Databook or press Ctrl + D.

46  Chemical Processing Engineering

Figure 1.47  Calculation of dew point pressure for stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.48  Calculation of bubble point pressure for stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

On the first page of Databook, variables appear, use insert to sample variables from flowsheet. Then, from the variable navigator, select the object and variable, e.g., choose 1 as object and its temperature as variable, press the add button to select another variable (Figure 1.54). Select the molar flow rate of ethane from the vapor outlet stream (stream vap) (Figure 1.55). Then, go to the Case Studies page and add a new case study Add button (Figure 1.56). Specify temperature as an independent and molar flow rate of ethane as a dependent variable and then click on the view button and enter low, high bonds and step size values for the independent variable (Figure 1.57). Press the Start button and then go to the Results page to see the plot or table. The results may be exported to any spreadsheet software for further processing (Figure 1.58). g. I n order to adjust the drum temperature to reach the 50% liquid, the duty should be specified for the drum to be able to run it isothermally as seen in Figure 1.59.

Computations with Excel Spreadsheet-UniSim Design Simulation  47

Figure 1.49  Calculation of dew point temperature for stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.50  Calculation of bubble point temperature for stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.51  Addition of Separator to PFD in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

48  Chemical Processing Engineering

Figure 1.52  Design window of the Separator in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.53  Worksheet/Conditions window for the Separator in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.54  Variable Navigator page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Computations with Excel Spreadsheet-UniSim Design Simulation  49

Figure 1.55  Choice of variable in the Variable Navigator page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.56  DataBook/Case Studies page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.57  Independent Variable Setup for Case Studies in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

50  Chemical Processing Engineering

Figure 1.58  A plot of ethane mol flow in vapor stream versus temperature in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.59  Setting the duty of the Separator in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

At this stage, the specifications for the drum are incomplete. The drum temperature (vap stream temperature) is now specified to run it (Figure 1.60). Close the active window. The drum temperature is now initiated. It could be changed by the Adjust function to control the bottom flow rate. Drag the adjust function from the object pallet to the PFD (Figure 1.61). Use the spreadsheet to define the new variable being the ratio of liquid stream to feed stream (Figure 1.62). Double click on the spreadsheet and import the flowrate variables from the flowsheet (Figure 1.63). Click on the spreadsheet button and create a new variable (liquid to feed ratio) and calculate its value (the formula used in cell B4 is the ratio) (Figure 1.64). Close the active window and double click on Adjust. Specify the drum temperature (vap stream) as the adjusted variable and the value calculated in the spreadsheet as the target value. Then click on start (Figure 1.65).

Computations with Excel Spreadsheet-UniSim Design Simulation  51

Figure 1.60  Adjusting drum temperature to reach 50 % liquid in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.61  Adding the Adjust function to PFD in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

The calculation is completed. Click on the Monitor button to view the results (Figure 1.66).

Example 1.12 Plot surface tension and vapor pressure of dimethylsulphide as a function of temperature in a desired range.

Solution Follow the step-by-step instructions to solve the Example. 1. O  pen a new case. 2. Add a new component list.

52  Chemical Processing Engineering

Figure 1.62  Addition of Spreadsheet to define new variable in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.63  Adding Imported Variables to Spreadsheet in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.64  Calculation of the liquid to feed ratio in the spreadsheet of UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Computations with Excel Spreadsheet-UniSim Design Simulation  53

Figure 1.65  Setting and calculating the Adjust function in the spreadsheet of UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.66  Monitoring the adjusted variable in the Adjust function of UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

3. 4. 5. 6.

S elect components from the components list (Figure 1.67). Select the Fluid Package (Make sure to select Component List -1 in the component list). Add a new Fluid Package. Select the Antoine equation from vapor pressure models in Fluid Package. Close the Fluid Package by clicking on the cross button. Enter the Simulation Environment as shown in Figure 1.20. 7. Drag a material stream to the PFD and rename it as “Feed”. Enter temperature, pressure and molar flow in the Worksheet/Conditions page. 8. On the Worksheet Compositions page, enter the mol fraction of the components. The stream properties are now calculated for a base case. Save the simulation work (e.g., Example 1.12.usc) (Figure 1.68). 9. To plot the surface tension of dimethylsulfide as a function of temperature, go to Tools/Databook or press Ctrl - D. 10. Click on the Insert button to choose and add the variables from the flowsheet. Herein, the temperature of feed stream is sampled from the variable navigator as shown in previous examples. 11. Also, insert the surface tension of the Feed stream (Figure 1.69).

54  Chemical Processing Engineering

Figure 1.67  Adding Dimethylsulphide from library to simulation in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.68  Definition of the mol fraction of Dimethylsulphide in Worksheet/Composition page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.69  Adding variables in DataBook in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Computations with Excel Spreadsheet-UniSim Design Simulation  55 12. Go to the case studies page and add a new case study. Select the temperature as an independent variable and the surface tension as the dependent variable. 13. Go to the view page and enter data for independent variables (Figure 1.70). 14. Click on the start button. Then go to the Results page to view the results. The results are shown both in the graph and table format (Figure 1.71). To plot the vapor pressure of dimethylsulphide as a function of temperature, drag the spreadsheet function to PFD. 15. Click on the spreadsheet, press the add import button to import the vapor pressure equation coefficients (10 coefficients from the navigator scope Basis) and Feed temperature from Flowsheet (Figure 1.72). 16. Use the Antoine vapor pressure expression where 6 coefficients from a-f are needed as extracted from the scope navigator (Basis) as shown in Figure 1.73. 17. The coefficients are now imported to the spreadsheet. Press the spreadsheet button in order to enter the equations (Figure 1.74). 18. Enter the equations in the spreadsheet to complete the calculation. In cell A9, the temperature is calculated in K (Figure 1.75).

Figure 1.70  Setting the range of the independent variable for Case Studies in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.71  A plot of surface tension versus temperature in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

56  Chemical Processing Engineering

Figure 1.72  Extracting Vapor Pressure Coefficients from the Basis object in DataBook in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.73  Antoine vapour pressure formula from the scope navigator in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.74  Antoine vapour pressure variables selection for the Spreadsheet in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Computations with Excel Spreadsheet-UniSim Design Simulation  57

Figure 1.75  Antoine vapour pressure coefficients exported to Spreadsheet in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.76  Antoine vapour pressure formula used in Spreadsheet of UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

19. By clicking on the Formula button, all equations used in the spreadsheet are shown in Figure 1.76. 20. Go to Tools/databook or press Ctrl-D. 21. Click on the Insert button to sample and add the variables. Herein, the vapor pressure of dimethylsulfide is sampled from the Spreadsheet object (Figure 1.77). 22. Press OK. The variable is now added to the Databook. Click on Case Studies (Figure 1.78). 23. Go to the Case Studies page and add a new case study. Select the temperature as an independent variable and the vapor pressure as the dependent variable (Figure 1.79). Go to the view page and enter data for independent variables (Figure 1.80). 24. Click on the Start button. After the completion of the simulation, go to the Results page to view the results. The results are shown both in the Graph and Table format (Figure 1.81). In this Example, the vapor pressure is calculated by extracting the coefficients of the Antoine equation; however, there is a simple way to plot vapor pressure versus temperature. If in the definition of a stream for a pure component for N+2 input data, we choose temperature, vapor fraction (zero, one or a value in between), mol flow and define

58  Chemical Processing Engineering

Figure 1.77  Sampling the vapor pressure of dimethylsulfide in the Spreadsheet object in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.78  List of variables in the DataBook of UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

composition as pure, the plot of pressure versus temperature in the stream will provide the vapor pressure. The reader can make this easy calculation and compare the results with the Antoine equation.

Example 1.13 Consider a stream of N2 and H2 at 300 ºF and 500 psia with a mol fraction of 0.3 and 0.7, respectively, with a molar flow rate of 100 lbmol/hr. This stream enters an isothermal reactor where the conversion of N2 is 30% for the following reaction:

N2 + 3H2 → 2NH3

Computations with Excel Spreadsheet-UniSim Design Simulation  59

Figure 1.79  Choice of the dependent and independent variables in DataBook/Case Studies tab in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.80  Setting the range of the independent variable for Case Studies in UniSim Design software (Courtesy of Honeywell UniSim software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Calculate the properties of outlet streams using the Peng-Robinson equation of state.

Solution Follow the step-by-step instructions to solve the Example. 1. O  pen a new case. 2. Add a new component list and choose all components from the Components list. 3. Select the Fluid Package (Make sure to select Component List -1 in the component list).

60  Chemical Processing Engineering

Figure 1.81  A plot of Vapor Pressure versus Temperature in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

4. 5. 6. 7. 8.

 dd a new Fluid Package and select the Peng-Robinson from Fluid Package. A Close the Fluid Package by clicking on the cross button. Go to the Reaction button, choose the add Rxn button to enter the reaction data (Figure 1.82). Select the Conversion Reaction and then click on the Add Reaction button (Figure 1.83). Select the components from the drop-down menu and enter the stoichiometry (negative for reactants and positive for products). The balance error should be zero (Figure 1.84). 9. Click on the Basis button, select Base Component from the drop down menu and enter the conversion for the base component (30%) (Figure 1.85). 10. Close the active window and return to the following window: The reaction Rxn-1 is added. Add a reaction set by clicking on the Add Set button (Figure 1.86).

Figure 1.82  Reactions Tab of the Simulation Basis Manager in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Computations with Excel Spreadsheet-UniSim Design Simulation  61

Figure 1.83  Adding Reaction to the Simulation Basis Manager in UniSim Design software (Courtesy of Honeywell UniSim software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.84  Entering the Stoichiometry Information of Conversion Reaction in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.85  Adding the Basis Information of Conversion Reaction in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

62  Chemical Processing Engineering

Figure 1.86  Adding the Reactions to the Reaction Set in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

11. In the Active List, select Rxn-1 from the drop down menu. Check mark the reaction set and close the active window (Figure 1.87). 12. Click on the Add FP button to have reaction available for the simulation (Figure 1.88). 13. Click on Add Set to Fluid Package Button (Figure 1.89). 14. The Basis-1 property model is now added to the Fluid Package. Then Enter the Simulation Environment (Figure 1.90). 15. Go to Tools/Preference to change the Unit Set to Field (Figure 1.91). 16. Go to the Variables button, select Field and close the window (The new unit set may also be cloned by clicking on the Clone button) (Figure 1.92). Drag a material stream to the PFD, click on stream and enter two out of three properties (temperature, pressure and vapor fraction) and molar flowrate in the Worksheet/Conditions Page. Save the simulation work (Example 1.13.usc) (Figure 1.93). 17. Enter the mol fractions of the components on the Worksheet Composition page. Click OK and close the active window to return to PFD (Figure 1.94).

Figure 1.87  Activating the Reaction Set in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Computations with Excel Spreadsheet-UniSim Design Simulation  63

Figure 1.88  Simulation Basis Manager/Reactions window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.89  Adding Set to the Fluid Package in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.90  Add Reaction to FP in Simulation Basis Manager of UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

64  Chemical Processing Engineering

Figure 1.91  Tools/Preference window to change the Unit Set to Field in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.92  Selecting the Unit Sets in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Figure 1.93  Definition of Stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and UniSim

Computations with Excel Spreadsheet-UniSim Design Simulation  65

Figure 1.94  Input Composition window for Stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

18. The stream properties can now be calculated (seen in the Worksheet/Properties page) (Figure 1.95). 19. Select a conversion reactor from the object palette (F4) and drag it to the PFD (Figure 1.96). 20. Double click on the reactor and enter Inlet, Vapor and Liquid Outlets and Energy stream in the Design/ Connections page (Figure 1.97). 21. Go to the Reactions page and select reaction set from drop down menu (Figure 1.98). 22. On the Worksheet page, either enter the temperature one outlet stream or define the duty of the heat stream. With this data, the reactor will be solved immediately (Figure 1.99). 23. One can also enter the duty of heat stream in the Design/Parameters section. (Make sure to delete the temperature entered in the previous section) (Figure 1.100). 24. The properties of all streams are viewed on the Worksheet Properties Page (Figure 1.101).

Figure 1.95  Worksheet/Properties page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

66  Chemical Processing Engineering

Figure 1.96  Adding Conversion reactor to PFD in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.97  Conversion Reactor Design window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.98  Reactions tab in the Conversion Reactor Design window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Computations with Excel Spreadsheet-UniSim Design Simulation  67

Figure 1.99  Worksheet/Conditions page in the Conversion Reactor Design window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.100  Design/Parameter page in the Conversion Reactor Design window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.101  Worksheet Properties page in the Conversion Reactor Design window in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

68  Chemical Processing Engineering

Example 1.14 In an ammonia plant, the feed gas (75% H2, 23.5% N2, 1% CH4, and 0.5% Ar in mol) at 275 °F and 500 psia enters to a process. The reaction N2 + 3H2 → 2NH3 is carried out in the reactor at T = 900 °F and P = 200 bar. The 60% of the N2 is converted in the reactor. The products of the reaction are refrigerated to separate 80% of NH3 product per pass. The remaining process stream is recycled back after being purged. Assume a total feed of 100 lbmol/hr and no pressure drop in the mixer. Simulate the process with the Peng-Robinson equation of state for the case when the purge fraction is 5%.

Solution Follow the step-by-step instructions to solve the Example. 1. O  pen a new case. 2. Add a new component list. 3. Select the components from the Components list and then close the active window by clicking on the cross button (all process components should be entered in this step). 4. If the cross button is not seen on the figure, you could move the active window to see the cross button in order to close it. 5. Select the Fluid Package (Make sure to select Component List -1 in the component list). 6. Add a new Fluid Package. Select the Peng-Robinson equation of state from the Fluid Package. 7. Close the Fluid Package by clicking on the cross button. Click on the Reaction button. 8. Go to the Add Rxn button to enter the conversion reaction data as the previous Example since the reaction is the same except there are two components (Methane and Argon) acting as inert. 9. In the Active List, select Rxn-1 from the drop down menu. Check mark the set and close the active window. 10. Enter the Simulation Environment as shown in Figure 1.20 11. Go to Tools/Preference to change the Unit Set to Field. 12. Go to the Variables button, select Field and close the window. 13. Drag a material stream to the PFD and change its name to feed. Enter its temperature, pressure and molar flowrate in the Worksheet/Conditions Page. 14. Enter the mol fractions of the components on the Worksheet/Composition Edit page. Click OK. The stream is now solved. Close the active window to return to PFD (Figure 1.102). 15. Save the simulation work (Example 1.14.usc).

Figure 1.102  Input Compositions page for Stream 1 in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Computations with Excel Spreadsheet-UniSim Design Simulation  69 16. Drag another stream named R* to the PFD. Define from the feed stream 1 (Figure 1.103). 17. Enter 0 for the molar flow rate of R* (Open-loop simulation). Close the active window (Figure 1.104). Select a Mixer from the object palette (F4) and drag it to the PFD (Figure 1.105). 18. Double click on the mixer and enter inlet and outlet streams. It is solved immediately. Close the active window and return to PFD (Figure 1.106). 19. Select a Compressor from the object palette and drag it in to the PFD (Figure 1.107). 20. Double click on Compressor and enter inlet, outlet and energy streams (Figure 1.108). 21. Go to the Worksheet and enter 200 bar as the pressure of outlet stream. The Compressor is solved. Close the active window (Figure 1.109). 22. Select a Heater from the object palette and drag it in to the PFD (Figure 1.110). 23. Double click on Heater and enter inlet, outlet and energy streams (Figure 1.111). 24. On the D esign/Parameters page enter 0 for Delta P (Figure 1.112).

Figure 1.103  Adding R* stream to PDF in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.104  Defining R* stream in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell are registered trademarks of Honeywell International Inc.).

and

and UniSim

70  Chemical Processing Engineering

Figure 1.105  Adding the Mixer to PDF in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.106  Simulation of the Mixer in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

and

Figure 1.107  Adding the Compressor to PFD in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Computations with Excel Spreadsheet-UniSim Design Simulation  71

Figure 1.108  Compressor Design/Connection page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.109  Simulation of the Compressor in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.110  Adding the Heater to PDF in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

and

72  Chemical Processing Engineering

Figure 1.111  The Heater Design/Connection page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.112  The Heater Design/Parameters page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

25. Go to the Worksheet page and enter 900 ºF for the outlet stream temperature. At the beginning, the Heater acts as a cooler. However, after the recycle stream completion, it will work as a Heater, because of the higher flow rate. Close the active window (Figure 1.113). 26. Select a Conversion reactor from the object palette and drag it to the PFD (Figure 1.114). 27. Double click on Reactor and enter the inlet, outlet and energy streams (Figure 1.115). 28. Go to the Reaction page and select the reaction set from the Reaction Set drop down (Figure 1.116). 29. Go to Worksheet and enter 900 ºF for the outlet stream temperature (Figure 1.117).

Computations with Excel Spreadsheet-UniSim Design Simulation  73

Figure 1.113  The Heater Design/Worksheet page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.114  Adding the Reactor to PFD in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.115  Reactor Design/Connection page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

74  Chemical Processing Engineering

Figure 1.116  Reactor Reactions/Details in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.117  Reactor Worksheet/Conditions page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

30. Select a Component splitter from the object palette and drag it in to the PFD (Figure 1.118). 31. Double click on the Splitter to enter the inlet, outlet and energy streams (Figure 1.119). 32. Go to the Design/Splits page and enter 0.20 for ammonia and 1 for the other components as the fraction to Overhead. Therefore, 80% of ammonia is separated as a pure product (Figure 1.120). 33. Go to the Worksheet tab in the X-100 unit operation and enter 900 ºF and 2901 psia for both outlet streams. The Splitter is now solved. Close the active window (Figure 1.121). 34. Select a Tee from the object palette and drag it to the PFD (Figure 1.122). 35. Double click on the Tee and enter the inlet and outlet streams (Figure 1.123). 36. On the Design Parameters page, enter 0.05 for the purge stream and close the active window (Figure 1.124). 37. Select a Recycle operation from the object palette and drag it to the PFD (Figure 1.125).

Computations with Excel Spreadsheet-UniSim Design Simulation  75

Figure 1.118  Adding Component Splitter to PDF in UniSim Design software (Courtesy of Honeywell UniSim Design software, Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.119  Splitter Design/Connections page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.120  Splitter Design/Splits page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

76  Chemical Processing Engineering

Figure 1.121  Simulation of the Splitter in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.122  Adding the Flow Splitter (TEE) to PFD in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.123  TEE Design/Connection page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Computations with Excel Spreadsheet-UniSim Design Simulation  77

Figure 1.124  TEE Design/Parameters page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

Figure 1.125  Adding Recycle Stream to PFD in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.126  Recycle Connections page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

78  Chemical Processing Engineering

Figure 1.127  Recycle simulation page in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

Figure 1.128  Simulation of the Process in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell UniSim are registered trademarks of Honeywell International Inc.).

and

and

38. Double click on Recycle and enter the inlet and outlet streams (Figure 1.126). 39. If the Recycle reached its iteration limit without converging, click Continue (set sensitivities of all recycle parameters to 0.01, this would be multiplied in internal tolerance to determine the accuracy) (Figure 1.127). 40. The process is now simulated (Rename stream 1 to Feed) (Figure 1.128).

References 1. AskVG, “[Did You Know] Microsoft Excel has a Limit of Maximum 1,048,576 Rows and 16,384 Columns,” February 2019. [Online]. Available: https://www.askvg.com/did-you-know-microsoft-excel-has-a-limit-of-maximum-1048576-rowsand-16384-columns/#:~:text=A%20worksheet%20in%20Microsoft%20Excel%20can%20contain%20maximum,limit%20 of%20maximum%2065%2C536%20rows%20and%20256%20columns. [Accessed October 2020]. 2. Microsoft, “Excel help & learning,” [Online]. Available: https://support.microsoft.com/en-us/excel. [Accessed October 2020]. 3. M. Martín and L. Martín de Juan, “EXCEL for Chemical Engineering,” in Introduction to Software for Chemical Engineers, M. Martín, Ed., Boca Raton, CRC Pressm Taylor & Francis Group, 2020, pp. 27-91.

Computations with Excel Spreadsheet-UniSim Design Simulation  79 4. E. C. Ferreira, R. Lima and R. L. Salcedo, “Spreadsheets in chemical engineering education - A tool in process design and process integration,” International Journal of Engineering Education, vol. 20, no. 6, pp. 928-938, January 2004. 5. Techonthenet, [Online]. Available: https://www.techonthenet.com/excel/tutorials.php. [Accessed October 2020]. 6. “Excel Functions,” 2020. [Online]. Available: https://www.excelfunctions.net/excel-correl-function.html. [Accessed October 2020]. 7. exceltoolset, “Getting Coefficients Of Chart Trendline,” [Online]. Available: http://www.exceltoolset.com/getting-coefficients-­ of-chart-trendline/. [Accessed october 2020]. 8. Ablebits, “Excel trendline types, equations and formulas,” 9 October 2020. [Online]. Available: https://www.ablebits.com/ office-addins-blog/2019/01/16/excel-trendline-types-equations-formulas/. [Accessed October 2020]. 9. Frontline Systems, “EXCEL SOLVER - ALGORITHMS AND METHODS USED,” Frontline Systems, 2008. [Online]. Available: https://www.solver.com/excel-solver-algorithms-and-methods-used. [Accessed October 2020]. 10. A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications, Prentice Hall, 1999. 11. Recktenwald and G. W., Numerical Methods with MATLAB: Implementation and Application, Prentice Hall, 2000. 12. Chapra and S. C., Applied Numerical Methods With MATLAB for Engineers and Scientists, McGraw-Hill College , 2017. 13. A. K. Coker, Fortran Programs for Chemical Process Design, Analysis, and Simulation, Gulf Professional Publishing, 1995. 14. E. C. Colebrook, “Turbulent Flow in Pipe with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws,” J. Inst. of Civi1 Eng., vol. 11, pp. 133-156, 1938-1939. 15. O. Redlich and J. N. S. Kwong, “On the Thermodynamics of Solutions: An Equation of State Fugacities of Gaseous Solutions,” Chem. Rev., vol. 44, p. 233, 1949. 16. A. J. V. Underwood, “Fractional Distillation of Multicomponent Mixtures Calculation of Minimum Reflux Ratio,” J. Inst. Petrol., vol. 32, p. 614, 1946. 17. P. V. Danckwerts, “Continuous Flow Systems: Distribution of Residence Times,” Chem. Eng. Sci., vol. 2, pp. 1–13, 1953. 18. V. Ganapathy, “Simplified Approach to Designing Heat Transfer Equipment,” Chem. Eng, pp. 81-87, April 13, 1987. 19. D. Foo, N. Chemmangattuvalappil, D. K. Ng, R. Elyas, C.-L. Chen, R. D. Elms, H.-Y. Lee, I.-L. Chien, S. Chong and C. H. Chong, Chemical Engineering Process Simulation, Elsevier, 2017. 20. SimulateLive, “Complete List of Process Simulators,” May 2017. [Online]. Available: http://www.simulatelive.com/product-­ reviews/simulation/complete-list-of-process-simulators-part-1-2. [Accessed October 2020]. 21. I. Lukec, “What is the Most Useful Software in Chemical Engineering?,” March 2019. [Online]. Available: http://www. simulatelive.com/product-reviews/simulation/what-is-the-most-useful-software-in-chemical-engineering. [Accessed October 2020]. 22. R. Ruiz-Femenía, C. R. Márquez, L. G. Hernández-Pérez, J. A. Caballero, M. Martín, J. M. Ponce Ortega and J. G. Segovia, “Modular Process Simulators,” in Introduction to Software for Chemical Engineers, Boca Raton, CRC Press, Taylor & Francis Group, 2020, pp. 498-583. 23. K. I. Al-Malah, Aspen Plus: Chemical Engineering Applications, Wiley, 2016. 24. Wikipedia, “List of chemical process simulators,” October 2020. [Online]. Available: https://en.wikipedia.org/wiki/List_ of_chemical_process_simulators. [Accessed October 2020]. 25. A. K. Coker, Petroleum Refining Design and Applications Handbook, vol. 1, John Wiley & Sons, 2018. 26. J. D. Seader and E. J. Henley, Separation Process Principles, 2nd ed., John Wiley & Sons Inc, 2006. 27. J. C. De Hemptinne and J. M. Ledanois, Select Thermodynamic Models for Process Simulation: A Practical Guide Using a Three Steps Methodology, Editions Technip, 2012. 28. E. Carlson, “Don’t Gamble with Physical Properties for Simulations,” Chem. Eng. Prog., pp. 34-56, October 1996. 29. S. I. Sandler, Using Aspen Plus in Thermodynamics Instruction: A Step-by-Step Guide, John Wiley & Sons, 2015.

2 Physical Property of Pure Components and Mixtures Physical property data are frequently required for laboratory, bench, and pilot plants or in design of industrial plants. These data are not easily accessible for process engineers in chemical plants or engineering firms and sometimes, it would be very tedious to obtain them using traditional approaches. Nowadays, simulation and design packages can be used as the prime sources of physical property information if used properly. However, the data obtained from these simulators should be used with caution since choosing the proper thermo-physical model is still a major issue for students, engineers and practitioners. It is true that these data are available in big libraries with some limitations and fast data mining is possible with the amazing speed of technology, which means that people may spend less time in libraries than was usual before. However, raw data mining cannot satisfy the need of engineers with the tight design one expects these days. The physical property data for pure components or mixtures can, of course, be obtained by experimentations, but this is often a time-consuming and expensive exercise. Therefore, engineers and practitioners may reply on correlations with enough accuracy or process simulators for design purposes. Physical property of this type is often required in most process design sizing and rating cases, such as vessels, pipelines, separation devices, gas processing, reactor design, etc. Many physical and thermodynamic property correlations have been published in the literature [1, 2]. Yaw [3], Poling et al. [4], Haynes [5], Matthias and Holger [6] have developed a number of executable programs to calculate the physical properties. This chapter reviews physical property data for liquids, gases and mixtures, and presents the Excel program and the UniSim Design software simulator for determining these properties for a range of temperature and correlation constants. Furthermore, compressibility and solubility are also calculated due to their importance in process design and safety. As shown in the chapter, the deviation between the results shown with these two approaches are in an acceptable range. It is important to mention that using correlations is tedious and time consuming since one must find the proper correlations for a given property considering various limitations applied to correlations. These days, user-friendly hands-on tools are extensively needed to ease the physical property calculations with proper flowsheets or procedures. These tools can be easily integrated into chemical engineering curricula in individual courses. In Chapter 1, the basics of two tools, the Excel Program and the UniSim Design software are explained in some detail. In this chapter, their snapshots are presented and the average deviation between the Excel Program and the UniSim Design software is calculated and explained. The Excel spreadsheet and UniSim Design software files can be downloaded from the companion website and the correlation constants used for the equations used in this chapter to calculate components/mixtures properties are obtained from Appendix C of the design book authored by Coker [2], pages 828-862 in a tabular format.

PURE COMPONENTS In this section, the selected physical properties for gas and liquid are introduced and two approaches are described in order to calculate these properties with the Excel spreadsheet and UniSim Design software simulator. These properties are density, viscosity, heat capacity, thermal conductivity, volumetric expansion rate, vapor pressure for liquid and viscosity, and thermal conductivity and heat capacity of gases. These are extensively required for laboratory, bench, and pilot plants or in design of industrial plants. A. Kayode Coker and Rahmat Sotudeh-Gharebagh. Chemical Process Engineering: Design, Analysis, Simulation and Integration, and Problem-Solving With Microsoft Excel – UniSim Design Software, Volume 1, (81–120) © 2022 Scrivener Publishing LLC

81

82  Chemical Process Engineering

Density of Liquid Saturated liquid densities at any temperature is calculated by [2]: n

ρL = AB−(1− Tr )



(2.1)

where: ρL = saturated liquid density, g/cm3 A, B, n = constants for a chemical compound T = temperature, K Tc = critical temperature, K  T Tr = reduced temperature,    TC  Many methods; e.g. [1, 7] have been reported in the literature to estimate densities of pure liquid. All are based on the law of corresponding states. They are rather complex algebraically and therefore, the aid of a computer is needed when many calculations are involved. The tabulated values of the correlation constants A and B for some chemical compounds are given in the literature [2, 3, 8].

Example 2.1 Estimate the density of saturated water in the range of 0 °C to 350 °C using the Excel spreadsheet and UniSim Design software simulator.

Solution Herein, the Excel program, Example 2.1.xlsx, employs Equation 2.1 for estimating the density of saturated water and the results are compared with those obtained from the UniSim Design software simulator. The results for water are illustrated in Figure 2.1 with the formulas written on the Excel program over 0 °C to 350 °C (below a critical value of 374 °C). In order to compare the results obtained by the Excel program with those of the UniSim Design software simulator, we use the following formula [9] for the average deviation:

∑iN=1 |x i − x| D= N



(2.2)

where D = average deviation percentage N = number of observations UniSim i − Exceli )  xi =  ∗100 for single prediction   UniSim i x = average of x vector in percentage The data shown in the UniSim Design software column in Figure 2.1 is obtained from the UniSim Design software simulator by defining the water stream as described in Chapter 1 and plotting the results by the proper choice of independent and dependent variables. Figure 2.2 shows the snapshot of the UniSim Design software simulator, Example 2.1.usc, for calculating the density of water (dependent variable) as a function of temperature (independent variable). Figure 2.3 compares the density of water calculated by the Excel spreadsheet and UniSim Design software from 0 to 350 °C. As seen in the Figure, the results are very identic, and one can easily use the Excel spreadsheet or UniSim Design software to obtain the density of liquid. There are some other correlations reported in the literature and

Physical Property of Pure Components and Mixtures  83

Figure 2.1  Snapshot of the Excel program for calculating the density of water as a function of temperature.

Figure 2.2  Snapshot of calculating the density of water as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

these can also be used to estimate the properties; i.e., DIPPR105 [10] which also provides similar results as shown in Figure 2.3. The correlation and the corresponding constants can be found in the snapshot of the Excel program in Figure 2.1. As seen in this Figure, the deviation calculated with formula (2.2) with the UniSim Design software is about 2% and for DIPPR105 is 1%. Although the results obtained by both correlations are in the acceptable range the DIPPR105 provides better results. This means that for design purposes, one should choose the correlation applicable in a broad range of operating conditions.

Viscosity of Liquid The viscosity of a saturated liquid as a function of temperature can be expressed by:



log µ L = A +

B + CT + DT 2 T

(2.3)

84  Chemical Process Engineering 1,200 EXCEL UniSim DESIGN SOFTWARE EXCEl-DIPPR105

Water density (kg/m3)

1,000

800

600

400

200

0

0

50

100

150

200

250

300

350

T (ᵒC)

Figure 2.3  Liquid density of water as a function of temperature.

where: μL = viscosity of saturated liquid, cP A, B, C and D = correlation constants T = temperature, K The viscosities of liquid decrease with temperature. Also, the variation is linear over a wide range of temperature from the freezing to the boiling point. This is expressed by the following correlation [1]:



ln µ L = lnA +

B T

(2.4)

This Equation is used to correlate the effect of temperature on liquid viscosity. However, it fails at low temperatures because liquids show a sharp increase in viscosity as the freezing point is approached. There are numerous methods in estimating liquid viscosities of compounds reported in the literature; e.g. [11, 12] and with the UniSim Design software simulator one can also estimate liquid viscosities of compounds.

Example 2.2 Estimate the viscosity of saturated water in the range of 0 °C to 350 °C using the Excel spreadsheet and UniSim Design software simulator and calculate the average deviation.

Solution Figure 2.4 shows the Snapshot of the Excel program, Example 2.2.xlsx, for the calculation of the viscosity of water as a function of temperature and the average deviation with the UniSim Design software simulator. The procedure for obtaining the physical properties using the UniSim Design software simulator, Example 2.2.usc, is explained in detail in Chapter 1. The average deviation between the Excel spreadsheet and UniSim Design software is about 7% which is still in the acceptable range. However, high error percentage shows that the correlation fails at low

Physical Property of Pure Components and Mixtures  85

Figure 2.4  Snapshot of the Excel spreadsheet for calculating the viscosity of water as a function of temperature.

temperature because liquids show a sharp decrease as the freezing point is approached. There are other methods in estimating liquid viscosities of compounds [4]. The reader can use other correlations to verify the viscosity of the water from 0-350 °C. The data shown in the UniSim Design software column in Figure 2.4 is obtained from the UniSim Design software simulator, by defining the water stream as described in Chapter 1 and plotting the results by the proper choice of independent and dependent variables. Figure 2.5 shows the snapshot of the UniSim Design software simulator for calculating the viscosity of water (dependent variable) as a function of temperature (independent variable). Figure 2.6 shows a plot of liquid viscosity of water versus temperature.

Heat Capacity of Liquid Heat capacity data for liquid are relevant in process design of heat exchanger equipments and chemical reactors. For example, heat capacity data are needed in the design of liquid-phase chemical reactors that involve energy input

Figure 2.5  Snapshot of calculating the viscosity of water as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

86  Chemical Process Engineering 2.0000 EXCEL

1.8000

UniSim DESIGN 1.6000

Water viscosity (cP)

1.4000 1.2000 1.0000 0.8000 0.6000 0.4000 0.2000 0.0000 0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

T (°C)

Figure 2.6  Liquid viscosity of water as a function of temperature.

for heating liquid reactants to the reaction temperature. Alternatively, these data are essential for sizing and rating reboilers and condensers and heat exchangers in general. The liquid heat capacity is expressed in a polynomial of the form:

CP = A + BT + CT2 + DT3

(2.5)

where: CP = liquid heat capacity, cal/g °C A, B, C and D = correlation constants for a chemical compound T = temperature, K T  Liquid heat capacities are not strongly dependent on temperatures, except at reduced temperatures,  Tr =  ,  TC  where Tr = 0.7 or 0.8. At high reduced temperatures, liquid heat capacities are large and strongly dependent on temperature. At the boiling point of most organic compounds, heat capacities are between 0.4-0.5 cal/g °C [1]. However, in this temperature range, there is no effect of pressure [12]. Herein, we also present the proposed by RowlinsonBondi [13] to calculate this property as:

C P = C 0P + 1.45R +



 1.742 25.2(1 − Tr )1/3  0.45R + 0.25ωR 17.11 + +  1 − Tr Tr 1 − Tr  

where CP = liquid heat capacity, J/mol.K C 0P = ideal Gas Specific heat capacity, J/mol.K R = universal Gas constants, J/mol.K Tr = reduced temperature Ω = acentric factor

(2.6)

Physical Property of Pure Components and Mixtures  87

Example 2.3 Estimate the heat capacity of saturated water in the range of 0 °C to 350 °C using the Excel spreadsheet and UniSim Design software simulator.

Solution The Excel program, Example 2.3.xlsx, uses Equations 2.5 and 2.6 for estimating the heat capacity of liquid. The UniSim Design software simulator, Example 2.3.usc, can also be used to calculate the heat capacity of liquid. Figure 2.7 shows the Snapshot of the Excel program where the heat capacity of water is calculated as a function of temperature. The average deviation between the Excel spreadsheet and UniSim Design software (Antoine as physical property model) is about 11%, which seems rather high. The difference can be attributed to the fact that the UniSim Design software uses updated correlations, which could be more accurate. The average deviation is significant at temperature well above 250 °C and it goes up to 16%. As seen in this Figure, the Rowlinson-Bondi Equation significantly improves the prediction and the average deviation goes down from 11-16% to 6.12%. Therefore, one should always use alternative correlations in process and equipment design. Figure 2.8 shows the snapshot of the UniSim Design software simulator for calculating the heat capacity of water (dependent variable) as a function of temperature (independent variable). Figure 2.9 shows the heat capacity of water from 0-350 °C using the Excel spreadsheet and UniSim Design software.

Thermal Conductivity of Liquid Liquid thermal conductivities, kL are required in many process design and engineering applications where heat transfer is prevalent. They are required to evaluate the Nusselt number, ( hd k ), and the Prandtl number, ( cµ k ), and in correlations to predict the idealized condensing film coefficient based upon laminar liquid flow over a cooled surface. The thermal conductivity of a saturated liquid is:

kL = A + BT + CT2 where: kL = thermal conductivity of saturated liquid, µcal/s.cm. °C A, B and C = correlation constants T = temperature, K

Figure 2.7  Snapshot of the Excel spreadsheet for calculating the heat capacity of water as a function of temperature.

(2.7)

88  Chemical Process Engineering

Figure 2.8  Snapshot of calculating the heat capacity of water as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

3.000 EXCEL-1 UniSim DESIGN EXCEL-2

Liquid heat capacity, (cal/g°C)

2.500

2.000

1.500

1.000

0.500

0.000

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

°C

Figure 2.9  Liquid heat capacity of water as a function of temperature.

Values of kL for most common organic liquids vary between 250-400 µcal/s.cm. °C at temperatures below the normal boiling point [1]. Water and other highly polar molecules have values that are two to three times larger. Except for water, aqueous solutions, and multi-hydroxyl molecules [14], the thermal conductivity of most liquids decreases with temperature. Below or close to the normal boiling point, the decrease is nearly linear. Various methods for calculating KL have been reviewed in the literature [12].

Physical Property of Pure Components and Mixtures  89

Example 2.4 Estimate the thermal conductivity of saturated water in the range of 0 °C to 350 °C using the Excel spreadsheet and UniSim Design software simulator.

Solution Figure 2.10 shows the Snapshot of the Excel program, Example 2.4.xlsx, for calculating kL of liquid as a function of temperature. The average deviation between the Excel spreadsheet and UniSim Design software is 1.6%, which is in acceptable range. Figure 2.11 shows the snapshot of the UniSim Design software simulator, Example 2.4.usc, for calculating the thermal conductivity of water (dependent variable) as a function of temperature (independent variable). Figure 2.12 provides a plot of thermal conductivity of water from 0 °C to 350 °C .

Figure 2.10  Snapshot of the Excel program for calculating the thermal conductivity of water.

Figure 2.11  Snapshot of calculating the thermal conductivity of water as a function of temperature by the UniSim Design software spreadsheet (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

90  Chemical Process Engineering

Thermal conductivity of water µcal/s.cm.°C

1800

EXCEL UniSim DESIGN

1600

1400

1200

1000

800

600 0.00

50.00

100.00

150.00 T (ᵒC)

200.00

250.00

300.00

350.00

Figure 2.12  Thermal conductivity of water as a function of temperature.

Volumetric Expansion Rate The volumetric expansion rate for a liquid in a process equipment that undergoes thermal expansion from heat input is [3]:



Qv =

Bliq UA(Text − T)m ρliq C P

(2.8)

where Qv = volumetric expansion rate Bliq = thermal expansion coefficients of liquid, 1/°C Cp = heat capacity of liquid U = overall heat transfer coefficient A = area of heat transfer Text = external temperature T = temperature of liquid, (K) This Equation is applicable for the design of relief systems and it describes the volumetric expansion rate at the beginning of the heat transfer. The relief systems should be sized to accommodate this volumetric flow.

Example 2.5 Calculate the volumetric expansion rate of benzene in the tubing in a reactor at 30 °C using the Excel spreadsheet.

Solution The Excel program, Example 2.5.xlsx, calculates the volumetric expansion rate of benzene in the tubing in a reactor at 30 °C (data from Coker [2]). Data and results are shown in Figure 2.13.

Physical Property of Pure Components and Mixtures  91

Figure 2.13  Snapshot of the Excel spreadsheet for calculating the volumetric expansion rate of benzene in the tubing in a reactor at 30 °C.

Vapor Pressure Vapor pressure data, PV, of pure components are important in vapor-liquid equilibria calculations, e.g., in the simplest case to predict the pressure in a closed vessel containing a specific liquid or a mixture of liquids. PV data are required for bubble and most dew point computations. These values are used in flash calculations involving mass transfer operations. Clearly, the design of pressure requirements for the storage tanks needs knowledge of the vapor pressure of the components as does the design of appropriate pressure relief systems. The vapor pressure of the saturated liquid as a function of temperature is:

log (PV ) = A +



B + C log(T) + DT + ET 2 T

(2.9)

where: PV = vapor pressure of the saturated liquid, mm Hg A, B, C, D and E = correlation constants for a chemical compound T = temperature, K In the table of vapor pressure, there are values for the acentric factor, ω, which is:



ω = −logPr– 1 at Tr = 0.7

(2.10)

where  P Pr = reduced pressure,    PC   T Tr = reduced temperature,    TC 

The acentric factor is used in thermodynamic correlations involving fugacity, compressibility factor, enthalpy, fugacity, and virial coefficients. The correlation constants for many compounds can be found in the literature [12].

92  Chemical Process Engineering A simplified Antoine Equation can often be used to estimate the vapor pressure [1]:



ln(PV ) = A −

B T+C

(2.11)

where PV = vapor pressure of liquid, mm Hg A, B and C = correlation constants for a chemical compound The constants A, B and C for many compounds are found in the data bank of some design simulation packages; e.g. Unism. Alternatively, these are found in the texts such as Reid et al. [12]. Equation 2.11 is only applicable for pressures with range of 0-1500 mm Hg [1]. Reid et al. [12] and Beaton and Hewitt [15] have discussed other methods in estimating vapor pressure of compounds, but none appears to offer any specific advantages.

Example 2.6 Estimate the vapor pressure of water versus temperature in the range of 0 °C to 350 °C using the Excel spreadsheet and UniSim Design software simulator.

Solution Figure 2.14 shows the Snapshot of the Excel program, Example 2.6.xlsx, for estimating the vapor pressure of water versus temperature. Figure 2.15 shows the snapshot of the UniSim Design softwares simulator, Example 2.6.usc, for calculating the vapor pressure of water (dependent variable) as a function of temperature (independent variable). Figure 2.16 demonstrates the vapor pressure of water versus temperature. The results obtained by correlation (2.11) in the Excel spreadsheet is very close to those estimated by the UniSim Design software with the average deviation of 1.3%.

Figure 2.14  Snapshot of the Excel spreadsheet for calculating the vapor pressure of water as a function of temperature.

Physical Property of Pure Components and Mixtures  93

Figure 2.15  Snapshot of calculating the vapor pressure of water as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.). 140,000 EXCEL

Vapor pressure Pv (mmHg)

120,000

UNISIM DESIGN

100,000

80,000

60,000

40,000

20,000

0

0

50

100

150

200

250

300

350

T (°C)

Figure 2.16  Vapor pressure of water as a function of temperature.

Viscosity of Gas Gas phase viscosity data, µG, are used in the design of compressible fluid flow and unit operations. For example, the viscosity of a gas is required to determine the maximum permissible flow through a given pipe size. Alternatively, the pressure loss of a given flowrate can be calculated. Viscosity data are needed for the design of process equipment involving heat, momentum and mass transfer operations. The gas viscosity of mixtures is obtained from data for the individual components in the mixture. The correlation for the viscosity of the gas at low pressure can be expressed as:

94  Chemical Process Engineering



μG = (A + BT + CT2) × 10−4

(2.12)

where: μG = viscosity of the gas at low pressure, cP A, B and C = constants for a given component T = temperature, K Reid et al. [12] have reviewed various methods for determining µG at low pressure.

Example 2.7 Estimate the vapor pressure of water versus temperature in the range of 0 °C to 350 °C using the Excel spreadsheet and UniSim Design software simulator.

Solution The Excel program, Example 2.7.xlsx, gives a routine for estimating gas viscosity of ethane. The average deviation between the Excel spreadsheet and the UniSim Design software simulator in this figure is about 1.7 %. This is acceptable, but one may also check with other correlations to ascertain about the results for design purposes. Figure 2.18 shows the snapshot of the UniSim Design software simulator, Example 2.7.usc, for calculating the viscosity of ethane (dependent variable) as a function of temperature (independent variable).

Thermal Conductivity of Gas The thermal conductivity, kG of low-pressure gases increases with temperature. In small temperature ranges, kG represents some form of linear relation. However, over wide temperature ranges, kG increases more rapidly with temperature than implied by a linear relation. Gas thermal conductivity data are used, e.g., in the design of process equipment and unit operations involving heat transfer, such as the rating of heat exchangers, fixed and fluidized bed operations handling gases. Thermal conductivity of a gas can be correlated as:

Figure 2.17  Snapshot of the Excel spreadsheet for calculating the gas viscosity.

Physical Property of Pure Components and Mixtures  95

Figure 2.18  Snapshot of calculating the viscosity of ethane as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

kG = A + BT + CT2 + DT3

(2.13)

where: kG = thermal conductivity of the gas at low pressure, µcal/cm.s.K A, B, C and D = correlation constants T = temperature, K Reid et al. [12] have reviewed other methods for estimating of kG.

Example 2.8 Calculate the thermal conductivity of propane as a function of temperature using the Excel spreadsheet and UniSim Design software simulator.

Solution The Excel program, Example 2.8.xlsx, employs Equation 2.13 in a routine for estimating the thermal conductivity of gases. Figure 2.20 shows the snapshot of the UniSim Design software simulator, Example 2.8.usc, for calculating the thermal conductivity of propane (dependent variable) as a function of temperature (independent variable). Figure 2.21 shows kG of propane as function of temperature with the Excel calculations and UniSim Design software Data. In this Figure, at temperatures well above 600 °C, the deviation seems significant, This means that the correlation does not predict the results obtained by the UniSim Design software satisfactorily and alternative correlations is needed for this zone.

Heat Capacity of Gases The heat capacity of gases is essential for some process engineering design, e.g., for air-cooled heat exchanger, and for gas-phase chemical reactions. Here, in the latter case, the heat capacities, CP° , is also used in the rating of heat exchangers and energy balance calculations. The heat capacity of a mixture of gases may be found from the hear capacities of the individual components contained in the mixture. The correlation for CP° of the ideal gas at low pressure is a third-degree polynomial, which is a function of temperature as:



CP° = A + BT + CT 2 + DT 3

(2.14)

96  Chemical Process Engineering

Figure 2.19  Snapshot of calculating the thermal conductivity of propane as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Thermal conductivity of propane (µcal/s.cm.K)

450.0 EXCEL

400.0

UNSIM DESIGN

350.0 300.0 250.0 200.0 150.0 100.0 50.0 0.0

0

100

200

300

400

500 T (°C)

600

700

800

900

1000

Figure 2.20  Thermal conductivity of propane as a function of temperature.

where CP° = heat capacity of ideal gas at low pressure, cal/gmol.K A, B, C and D = correlation constants for a chemical compound T = temperature, K The correlation constants A, B, C and D can be determined using the least square fit of the experimental data as explained in Chapter 1. Huang and Daubert [16] published constants for a fourth-order polynomial for heat capacities of many hydrocarbons. Yuan and Mok [17] used an exponential temperature function to correlate CP° with temperature and provided constants for many hydrocarbons and non-hydrocarbons. Tans [18] produced a

Physical Property of Pure Components and Mixtures  97 nomograph that allows a rapid approximation of CP° for paraffins. A compendium of CP° as a function of temperature for several hundreds of organic compounds is given by Stull et al. [19] and in NIST-JANAF Thermochemical Tables [20].

Example 2.9 Calculate the heat capacity of carbon dioxide as a function of temperature using the Excel spreadsheet and UniSim Design software simulator.

Solution Figure 2.21 shows the Snapshot of the Excel program, Example 2.9.xlsx, for estimating CP° of gases with carbon dioxide data with formula and data obtained from the UniSim. Figure 2.22 shows the snapshot of the UniSim Design software simulator, Example 2.9.usc for calculating the CP° of carbon dioxide (dependent variable) as a function of temperature (independent variable). Figure 2.23 is a plot of CP° with temperature for carbon dioxide. The average deviation between the Excel spreadsheet and UniSim Design software is 1.3%. This shows that the UniSim Design software simulator can predict the physical properties very close to the correlation and this is promising since it allows the fast calculation of physical properties.

MIXTURES In this section, the selected physical properties for mixtures are introduced and two approaches are described in order to calculate these properties with the Excel spreadsheet and UniSim Design software. These properties are surface tensions, viscosity of mixtures, enthalpy of formation, enthalpy of vaporization, Gibbs energy of reaction, Henry’s law constant for gases in water, viscosity of gas mixture and coefficient of thermal expansion of liquid. These are extensively important for laboratory, bench, and pilot plants operation or in design of industrial plants.

Figure 2.21  Snapshot of the Excel spreadsheet for estimating of C°p gases for carbon dioxide as a function of temperature.

98  Chemical Process Engineering

Figure 2.22  Snapshot of calculating the C°p of carbon dioxide as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

16.00

Heat capacity (cal/(g-mol.K))

14.00 12.00 10.00 EXCEL UniSim DESIGN

8.00 6.00 4.00 2.00 0.00 300.0

400.0

500.0

600.0

700.0 800.0 T (°C)

900.0

1,000.0

1,100.0 1,200.0

Figure 2.23  Vapor pressure of water as a function of temperature.

Surface Tensions Surface tension data of liquids are important in many process design calculations for situations where there is a twophase interface; e.g., two-phase flow, distillation, absorption and condensation. It can be expressed as: R

where: σ1 = surface tension at T1 in °C, dynes/cm R = correlation parameter

T −T  σ = σ 1  C  TC − T1 

(2.15)

Physical Property of Pure Components and Mixtures  99 Tc = critical temperature, K T = temperature, K For water, Table 2.1 shows the temperature range for which the surface tension is valid. The surface tension has been correlated with other physical parameters such as liquid compressibility, viscosity, mole fractions and the refractive index. Rao et al. developed a linear relationship between the surface tension at normal boiling point (log σb) and the reduced boiling point temperature (Tbr) [1]. Hadden presented a nomograph for the rapid calculation of σ for hydrocarbons that enables rapid calculation of σ [1]. Nowadays, these nomographs are less likely used due to the existence of simulators. For cryogenic liquids, Sprows and Prausnitz introduced the following Equation [1]:

σ = σ0(1 − Tr)p



(2.16)

where: σ = surface tension, (dynes/cm) σ0 and p were determined by the least-squares method from the experimental data.

Example 2.10 Calculate the surface tension of water as a function of temperature using the Excel spreadsheet and UniSim Design software simulator.

Solution Figure 2.24 shows the Snapshot of the Excel program, Example 2.10.xlsx, for calculating the surface tension of water as a function of temperature. Figure 2.25 shows the snapshot of the UniSim Design software simulator, Example 2.10.usc, for calculating the surface tension of water (dependent variable) as a function of temperature (independent variable). Figure 2.26 demonstrates the surface tension of the water from 0 to 350 °C. The average deviation between the Excel program and the UniSim Design software simulator is about 4 %. In some temperature ranges, the deviation is significant, and this can be improved using the alternative correlation in this range. In general, the surface tension of a liquid in equilibrium with its own vapor decreases with temperature and becomes zero at the critical point. In the reduced temperature of 0.45-0.65, σ for most organic liquids range from 20-40 dyne/cm and for water, σ = 71.97 dyne/cm at 25 °C [1].

Viscosity of Gas Mixture The viscosity of mixtures is obtained from data of the individual components present in the mixture. Many additional expressions can also be found in the literature for estimating the viscosity of gas mixtures [11]. Demirel [21] Table 2.1  The temperature range for the surface tension. Temperature range Parameter

0-100

100-374

σ0

71.97

58.91

T1(K)

298.16

373.16

R

0.8105

1.169

100  Chemical Process Engineering

Figure 2.24  Snapshot of the Excel spreadsheet for calculating the surface tension of water as a function of temperature.

Figure 2.25  Snapshot of calculating the surface tension of water as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

introduced the following semi-empirical formula proposed by Wilke in order to calculate the viscosity of gas mixtures, μmix, at low density. N

µmix =

∑ ∑xxµφ i =1

i

i

j ij



(2.17)

In which the dimensionless quantities ϕij are:



1  M  1+ i  φij =  8  Mj 

−0.5

2

  µi  0.5  M j  0.25   1 +  µ   M   i   j 

(2.18)

Physical Property of Pure Components and Mixtures  101

Surface tension dyne/cm

90.00 80.00

EXCEL

70.00

UniSim DESIGN

60.00 50.00 40.00 30.00 20.00 10.00 0.00

0

50

100

150

T (°C)

200

250

300

350

Figure 2.26  Surface tension of water as a function of temperature.

where N = number of chemical species xi = mole fraction of specie i μi = viscosity of species i Mi = molecular weight of species i

Example 2.11 Calculate the viscosity of mixture using the Excel spreadsheet and UniSim Design software simulator.

Solution Figure 2.27 shows the Snapshot of the Excel program, Example 2.11.xlsx, which employs Equations 2.17 and 2.18 for estimating viscosity of a mixture, μmix,. In the routine shown in Figure 2.27, mixture viscosity is calculated at 1 atm and 293 K based on pure components data and the results are compared with those obtained from the UniSim Design software. As seen in the Figure, the deviation is about 5% and this again shows the easiness of the UniSim Design software for calculating this value since the calculation by the Excel spreadsheet is rather time-consuming. Figure 2.28 shows the snapshot of the UniSim Design software simulator, Example 2.11.usc, for estimating μmix for the same mixture.

Enthalpy of Formation Enthalpy of formation (∆H °f ) for individual chemicals involved in chemical reactions are important to find the heat of reaction (∆H °f ) and associated heating and cooling requirements. If ∆H r° < 0 , then the chemical reaction is exothermic and will require cooling, while, ∆H r° > 0 , the reaction is endothermic, and heating is therefore required. The correlation of ∆H r° of the ideal gas at low temperature is expressed as:



∆H °f = A + BT + CT 2

(2.19)

102  Chemical Process Engineering

Figure 2.27  Snapshot of the Excel spreadsheet for estimating of μmix of a mixture and data from the UniSim Design software simulator.

Figure 2.28  Snapshot of estimating the viscosity of mixture by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell  and UniSim are registered trademarks of Honeywell International Inc.).

where ∆H °f = heat of formation of ideal gas at low temperature, kcal/mol A, B and C = correlation constants T = temperature, K Heats of formation are also found from the experimental data of the heat of combustion, ∆H c° or the enthalpy of reaction ∆H r° . Domalski reviewed values of ∆H °f ,298 K and ∆H c°,298 K for several organic compounds [1].

Example 2.12 Calculate the enthalpy of formation of methane as a function of temperature using the Excel spreadsheet and UniSim Design software simulator.

Physical Property of Pure Components and Mixtures  103

Solution Figure 2.29 shows the Snapshot the Excel program, Example 2.12.xlsx, which employs Equation 2.19 for estimating ∆H °f of methane as function of temperature Figure 2.30 shows the snapshot of the UniSim Design software simulator, Example 2.12.usc, for estimating the ∆H °f of methane (dependent variable) as a function of temperature (independent variable). Figure 2.31 is a plot of ∆H °f with temperature for methane as a function of temperature. The average deviation is 0.2% between the value calculated by the Excel spreadsheet and UniSim Design software, which is in an acceptable range. But, we should mention that using the UniSim Design software is very fast as compared with the correlation programed in the Excel program where one should look for formula, range of applicability and also finding the constants needed. This is a tedious and time-­consuming for designers and practionners. However, students need to know how to use correlations and compare the results with those obtained from process simulators.

Enthalpy of Vaporization The enthalpy of vaporization, ΔHV, also termed the latent heat of vaporization. ΔHV is the difference between the enthalpy of the saturated vapor and liquid at the same temperature. The enthalpy of vaporization data is used in process calculations such as the design of relief systems involving volatile compounds. In distillation columns, heat of vaporization values is needed to find the heat loads for the reboiler and condenser. Watson [22] has expressed a widely used correlation in calculating ΔHV: R



 T −T  ∆HV = ∆HV 1  C  TC − T1 

where ΔHV = heat of vaporization at a given temperature, cal/gr ΔHV1 = heat of vaporization at T1, cal/gr Tc = critical temperature, K T = temperature, K R = characteristic constant for the chemicals, R = 0.38

ΔHV decreases with temperature and it zero at the critical point.

Figure 2.29  Snapshot of the Excel spreadsheet for estimating ∆H°f of methane as a function of temperature.

(2.20)

104  Chemical Process Engineering

Figure 2.30  Snapshot of calculating the ∆H°f of methane as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Heat of formation of methane (kcal/mol)

–17.00

0

200

400

T (°C) 600

800

1000

1200

EXCEL UniSim DESIGN

–19.00

–21.00

–23.00

Figure 2.31  Heat of formation of methane as a function of temperature.

Example 2.13 Calculate the enthalpy of vaporization of water as a function of temperature using the Excel spreadsheet and UniSim Design software simulator.

Solution Figure 2.32 shows the Snapshot of the Excel program, Example 2.13.xlsx, for calculating ΔHV of water versus temperature. Figure 2.33 shows the snapshot of the UniSim Design software simulator, Example 2.13.usc, for calculating the enthalpy of vaporization of water (dependent variable) as a function of temperature (independent variable).

Physical Property of Pure Components and Mixtures  105

Figure 2.32  Snapshot of the Excel spreadsheet for calculating ΔHV of water as a function of temperature.

Figure 2.33  Snapshot of calculating the enthalpy of vaporization of water as a function of temperature by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 2.34 demonstrates the enthalpy of vaporization of water versus temperature. The shape of this Figure agrees with most other data vaporization data [1]. There is small average deviation of 1.3% between the Excel spreadsheet and UniSim Design software, which is acceptable and shows how reliable is the result obtained by the UniSim Design software simulator.

Gibbs Energy of Reaction Gibbs energy of formation is important in chemical reaction engineering. Values for all components including reactions and products are required to determine the changes in Gibbs energy for a given reaction. This change is significant due to its effect on chemical equilibrium for the reaction. If the change is negative, the thermodynamics for the chemical reaction are favorable and if it is highly positive, the thermodynamics for the reaction are not favorable. The chemical equilibrium for a reaction depends on the changes in Gibbs free energy for the reactions as:

106  Chemical Process Engineering 700.0 EXCEL

Heat of vaporization (cal/g)

600.0

UniSim DESIGN

500.0 400.0 300.0 200.0 100.0 0.0

0.0

50.0

100.0

150.0

T (°C)

200.0

250.0

300.0

350.0

Figure 2.34  Enthalpy of vaporization of water as a function of temperature.



∆Gr =

∑(n∆G ) f

products

−

∑(n∆G )

f reactants



(2.21)

The changes in Gibbs energy for a reaction may be used in preliminary work to determine whether a reaction is thermodynamically favorable at a given temperature. For thermodynamic equilibrium, one can use the following rough criteria for quick screening of chemical reactions:



∆Gr < 0

kJ mol

kJ mol kJ Gr > 50 mol

0 < Gr < 50

reaction is favorable reaction is possibly favorable

reaction is nor favorable

The correlation for Gibbs energy of formation, ΔGf, is expressed as:

ΔGf = A + BT + CT2

(2.22)

where ΔGf = Gibbs energy of formation of ideal gas at low temperature, kJ/mol A, B and C = correlation constants for chemical compounds, Appendix C, Table C-12 [2] T = temperature, K

Example 2.14 Calculate the Gibbs energy of formation for ethanol at 200-1000 K using the Excel spreadsheet and UniSim Design software simulator.

Physical Property of Pure Components and Mixtures  107

Solution The Excel program calculates Gibbs energy of formation for ethanol at 200-1000 K. The snapshot, the Excel program, Example 2.14.xlsx, is shown in Figures 2.35 and 2.36. The Excel program in Figure 2.36 calculates Gibbs energy for catalytic dehydrogenation of butane to butadiene based on Gibbs free energy of formation of products and reactants at two temperature. At lower temperature, the thermodynamics of the reaction are unfavorable while by increasing the temperature to 800 K, the thermodynamics becomes favorable.

Henry’s Law Constant for Gases in Water Henry’s law constant and the solubilities of gases in water are important and affect Health, Safety and Environment (HSE) in chemical industries, the threshold limit value (TLV) of gases for human exposure and the lower flammability limit (LFL) for flammable mixtures [23]. The correlation for Henry’s law constant for gases in water as a function of temperature is expressed as [24]:



log 10 H = A +

B + Clog10C + DT T

where H = Henry’s law constant at 1 atm, atm/mol-fraction A, B, C and D = correlation constants for gas T = temperature, K

Figure 2.35  Snapshot of the Excel spreadsheet for calculating and plotting Gibbs free energy of ethanol as a function of temperature.

Figure 2.36  Snapshot of the Excel spreadsheet for calculating Gibbs energy for catalytic dehydrogenation of butane.

(2.23)

108  Chemical Process Engineering

Example 2.15 Calculate the Henry’s law constant for various hydrocarbons using the Excel spreadsheet.

Solution The Excel program, Example 2.15.xlsx, shows the Henry’s law constant for various hydrocarbons (propane, ethylene, methane and propylene) using the correlations constants reported in Table C-15 as shown in Figure 2.37. The plots shown in this Figure demonstrate that Henry’s law constant increases with temperature. In this figure, the Henry’s law constant is used to calculate the partition coefficient, (Ki) as:

Ki =



H i yi = Pt xi

(2.24)

where Ki = partition coefficient Hi = Henry’s law constant at 1 atm, atm/mol-fraction Pt = total pressure, atm Application: Propylene (C3H6) is in contact with water at 25 °C (298 K) and 1 atm in an industrial process. If the concentration in the liquid at the surface of water (xi) is 15 ppmv, its concentration in the air at the surface of the water would be 12.1% (mol) as calculated in Figure 2.37.

Coefficient of Thermal Expansion of Liquid The thermal expansion coefficient of liquids is required in the design of relief systems. Liquids in process equipment may expand with an increase in the temperature. In order to accommodate such expansion, relief systems are normally designed to relieve thermally expanding liquid, and to prevent pressure build-up due to expansion. Such pressure buildup may lead to an eventual damage to the equipment if the pressure rise is excessive. The correlation of thermal expansion coefficient of liquid as a function of temperature is [3]: m



 T Bliq = a  1 −   Tc 

(2.25)

Figure 2.37  Snapshot of the Excel spreadsheet for calculating Henry’s law constant for various hydrocarbons (propane, ethylene, methane and propylene) as a function of temperature.

Physical Property of Pure Components and Mixtures  109

Figure 2.38  Snapshot of the Excel spreadsheet for calculating the thermal expansion liquids as a function of temperature.

where Bliq = thermal expansion coefficients of liquid, 1/ °C a & m = correlation constants for a chemical compound T = temperature, (K) Tc = critical temperature, (K)

Example 2.16 Calculate the Henry’s law constant for various hydrocarbons using the Excel spreadsheet.

Solution The Excel program, Example 2.16.xlsx, calculates the thermal expansion of compounds with the correlation constants, and respective minimum and maximum temperatures, as denoted by Tmin and Tmax in Table C-21 in Appendix C [2]. The tabular results and plot are shown in Figure 2.38.

DIFFUSION COEFFICIENTS Diffusion coefficients are important in the design of mass transfer operations equipment, such as gas absorption, distillation and liquid-liquid extraction. Experimental data for the common systems can be found in the literature, but for most design work, the values will have to be estimated through the correlations or process simulators.

Gas-Phase Diffusion Coefficients Diffusion coefficients for non-polar gases may be estimated from Chapman–Enskog, Gilliland, Fuller, Chen–Othmer, Wilke–Lee and Slattery–Bird correlations [1, 25–27]. Brid et al. [28] covers the effect of temperature and pressure on diffusivity, diffusion binary liquids and in gases at low density in sections 17.2-4. However, in most chemical engineering books, the correlations and expressions are also reported for estimating diffusions coefficients in different mixtures. Here, we choose the Slattery and Bird correlation which is valid for estimating DAB at low pressure and it can be expressed as:

110  Chemical Process Engineering



pDAB T  −4  1 5 1 = 2.745 × 10   TcATcB  ( pcA pcB ) 3 (TcATcB )12 (1 M A + 1 M B ) 2

1.823



(2.26)

where: p = pressure, atm DAB = diffusion coefficient of solute A in solvent B, cm2/s pcA, pcB = critical pressures for components A and B, atm TcA, TcB = critical temperatures for components A and B, K MA, MB = molecular Weight for components A and B, kg/kmol T = temperature, K

Example 2.17 Estimating the diffusion coefficient of CO-CO2 mixture as function of temperature using the Excel spreadsheet and the UniSim Design software simulator.

Solution Figure 2.39 shows the Snapshot of the Excel program, Example 2.17.xlsx, which employs Equations 2.26 for estimating the diffusion coefficient of CO-CO2 mixture as a function of temperature along with the input data obtained from the UniSim Design software simulator. The results are similar between the Excel spreadsheet and UniSim Design software, therefore, it can be noted that we can directly extract the value directly from the UniSim Design software. Figure 2.40 shows the snapshot of the UniSim Design software simulator, Example 2.17.usc, for estimating the diffusion coefficient of CO-CO2 mixture.

Liquid-Phase Diffusion Coefficients Diffusion coefficients in liquid phases depend on concentration and are valid for dilute solutions with solute concentrations smaller than 10% [1]. Also, the lower the solute concentration, the more accurate the calculated coefficients would be. For a binary mixture of solute A in solvent B, the diffusion coefficient can be represented as ° DAB for concentrations of A up to 5 or 10 mole percent [12]. Several correlations have been proposed for predict° ° in dilute liquid solutions [1, 29]. Here, the Wilke-Chang method [29] is employed for estimating DAB as: ing DAB



° DAB =

7.4 × 10−8 (ØMW ,B )0.5 T µ B × v A0.6

Figure 2.39  Snapshot of the Excel spreadsheet for estimating the diffusion coefficient of CO-CO2 mixture.

(2.27)

Physical Property of Pure Components and Mixtures  111

Figure 2.40  Snapshot of estimating the diffusion coefficient of CO-CO2 mixture by the UniSim Design software simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

where ° DAB = diffusion coefficient of solute A at very low concentrations in solvent B, cm2/s Mw,B = molecular weight of solvent B, kg/kmol T = temperature, K vA = molar volume of solute A in its normal boiling temperature, cm3/mol Ø = association factor of solvent B, (-) 2.6 for water (2.26 is also recommended), 1.9 for methanol, 1.5 for ethanol and 1.0 for unsaturated solvents The Le Bas additive volumes [4] can be used for the value of vA, if no experimental data are available.

Example 2.18 Calculate the diffusion coefficient of ethylbenzene in water as a function of molal volume of solute A in its normal boiling point using the Excel spreadsheet.

Solution The Excel program, Example 2.18.xlsx, calculates diffusion coefficient of ethylbenzene in water as a function of molal volume of solute A in its normal boiling point (T = 409.36 K). The result of calculation is compared with for one experimental point and it seems the correlation provides the results with an acceptable error around 5.4%.

COMPRESSIBILITY Z-FACTOR Many petroleum engineering and process design calculations dealing with natural gases require knowledge of deviation factors or compressibility Z factors. Experimental data from pressure-volume-temperature (P-V-T) measurements are seldom available. The Z factors are available in charts or tables as a function of pseudo-reduced

112  Chemical Process Engineering

Figure 2.41  Snapshot of the Excel spreadsheet for calculating diffusion coefficient of ethylbenzene in water as a function of molal volume of solute A.

temperatures, Tr and pressures Pr. However, use of these charts is often time consuming and involves complex calculations. Computer programs [30, 31] calculating the Z factors have been developed solely as a function of temperature and pressure of the gas. The compressibility factor Z of working fluid under different conditions, experimental measurement method of Z under high pressure and high temperature and data mining method were reported by Zhu et al. [32]. Furthermore, numerical methods and mathematical representations of the charts have been used to estimate the Z factors. Takacs [33] reviewed the various methods of estimating the Z factors. The compressibility factor Z of natural hydrocarbon gases can be estimated by using the revised Awoseyin method [1]. This method gives a compressibility factor within 5% for natural gases with specific gravities between 0.5 to 0.8 and for pressures up to 5000 psia [1]. The Z factor can be expressed as:

Z = F1(α−1 + F2 × F3) + F4 + F5

(2.28)

F1 = P(0.251 × Sg – 0.15) – 0.202 × Sg + 1.106

(2.29)



α =1+

1.785 S g

A6 P × 10 T 3.825

, A6 = 3.44 × 108

F2 = 1.4 × e−0.0054(T – 460)

(2.30) (2.31)

F3 = A1P5 + A2P4 + A3P3 + A4P2 + A5P A1 = 0.001946, A2 = −0.027635, A3 = 0.136315

(2.32)

A4 = −0.23849, A5 = 0.105168, F4 = (0.154 – 0.1525 × Sg)β – 0.02

(2.33)

β = P (3.18× Sg −1)e −0.5 P

(2.34)

F5 = 0.35(0.6 − Sg)eγ

(2.35)



Physical Property of Pure Components and Mixtures  113



γ = −1.039(P – 1.8)2

(2.36)



Sg = ( ρ g ρair )@60° F ,1atm or ( M w ,g MW ,air )

(2.37)

where

P = pressure, Kpsi Sg = specific gravity T = temperature, R MW = molecular weight, kg/kmol G = gas ρ = density, g/cm3

Example 2.19 Calculate the Z factor of natural gases for Sg = 0.72 as a function of pressure using the Excel spreadsheet and the UniSim Design software simulator.

Solution The Excel program, Example 2.19.xlsx, calculates the Z factor of natural gases for Sg = 0.72 as a function of pressure. Figure 2.42 shows the Snapshot of the Excel spreadsheet, which employs Equations 2.28-2.37, for estimating the Z factor of natural gases at 60 °F as a function of pressure.

Figure 2.42  Snapshot of the Excel spreadsheet for estimating the Z factor of natural gases.

The Excel program is written in such a way that the Z factor is calculated for any values of specific gravity. Figure 2.43 shows the Z factor for Sg = 0.55, 0.65, 0.72 obtained from the Excel program and the UniSim Design software simulator and the average error.

114  Chemical Process Engineering

Figure 2.43  Snapshot of the Excel spreadsheet and UniSim Design software Data of Z factor for three values of Sg = 0.55, 0.65, 0.72.

It is important to mention that the UniSim Design software simulator does not accept the specific gravity as the input parameter; instead, the feed compositions are provided. While for the Excel spreadsheet program, we need to calculate the specific gravity based on molecular weight and mole fraction of the component, and the molecular weight of the air as shown in the following Table. As it can be seen from the Figure 2.43, close agreement is obtained between Excel spreadsheet and UniSim Design software with the average deviations less than 2.6 %. The UniSim Design software simulator, Example 2.19.usc, employs the Peng Robinson Equation of State (EOS) for calculation; however, the reader can change the EOS to see the differences. Figure 2.44 shows the snapshot of the UniSim Design software simulator for calculating Sg for gas three gas compositions.

Figure 2.44  Snapshot of calculating Sg for gas three gas compositions by the UniSim Design software Simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 2.45 shows the snapshot of the UniSim Design software simulator for estimating Z factor for three values of Sg = 0.55, 0.65, 0.72.

Physical Property of Pure Components and Mixtures  115

Figure 2.45  Snapshot of estimating Z factor as a function of pressure by the UniSim Design software Simulator (Courtesy of Honeywell UniSim Design software. Honeywell and UniSim are registered trademarks of Honeywell International Inc.).

Figure 2.46 gives a plot of compressibility factor of natural gas as a function of pressure for three value Sg = 0.55, 0.65, 0.72. The plot can be generated for other specific gravities based on the procedures described above.

1.00 0.90

Comperessibility factor Z

0.80 0.70 0.60 0.50 0.40

Excel (SPG=0.55) Unisim (SPG=0.55) Excel (SPG=0.65) Unisim (SPG=0.65) Excel (SPG=0.72) Unisim (SPG=0.72)

0.30 0.20 0.10 0.00

0

500

1000

1500

2000

2500 P (psia)

Figure 2.46  Compressibility factor of natural gas as a function of pressure.

3000

3500

4000

4500

5000

116  Chemical Process Engineering Table 2.2 shows the specific gravity of natural gas calculated by Excel for three different compositions. Table 2.2  Calculation of specific gravity of natural gas. Composition Component

Molecular weight

Gas-1

Gas-2

Gas-3

Methane

16.04

1

0.83

0.8

Ethane

30.07

0

0.15

0.1

Propane

44.1

0

0.02

0.05

Butane

58.12

0

0

0.05

Air

28.95

0

0

0

0.55

0.65

0.72

Sg (Eq. 2.37)

SOLUBILITY AND ADSORPTION The solubility and adsorption are two critical properties used in process design and safety. The solubility of hydrocarbons in pure or salted water is important in initial engineering and environmental applications, such as the distribution of a hydrocarbon spill upon its contact with sea water [2]. Adsorption of volatile organic compounds on activated carbon is important in engineering and environmental studies and is highly useful in the design of carbon adsorption systems to remove trace pollutants, e.g., volatile organic compounds (VOC) from gases, especially in air pollution [3].

Solubility of Hydrocarbons in Water The following table summarizes the commonly used correlations to estimate the solubility of hydrocarbon in water at various conditions. Correlation log10S = A + BX + CX2 (2.38) S = solubility at 25 °C, ppm (wt.)1 X = concentration of salt (NaCl), ppm A, B and C = correlation constants

Description

Ref.

• solubility of hydrocarbons in salted water; e.g. sea water • hydrocarbons: alkanes, naphthenes, aromatics • other conditions to apply [2]

[2]

Figure 2.47  Snapshot of the Excel spreadsheet for calculating the solubility of benzene in pure and salt water at various conditions.

Physical Property of Pure Components and Mixtures  117

B C log10 S = A + + 2 (2.39) T T S = solubility in water, ppm(wt.) T = temperature, K A, B and C = correlation constants

• water solubility of organic compounds • air and steam stripping operations • temperature range of 25–121 °C

Parts per million by weight.

1

Example 2.20 Calculate the solubility of benzene in pure and salt water at various conditions using the Excel spreadsheet.

Solution The Excel program, Example 2.20.xlsx, show the solubility of benzene in pure and salt water at various conditions. The snapshot is shown in Figure 2.47.

Solubility of Gases in Water The correlation for the solubility of gases in water as a function of temperature is [24]:



log10 xwt = A +

B + C log10 C + DT T

(2.40)

where xwt = solubility of gases in water at 1 atm, weight fraction A, B, C & D = correlation constants for gas T = temperature, K

Example 2.21 Calculate the solubility for methane, propane, ethylene and propylene in water using.

Solution The Excel program, Example 2.21.xlsx, calculates the solubility for methane, propane, ethylene and propylene in water with correlation constants (Appendix C-16 [2]) as shown in Figure 2.48. These plots show that solubility decreases with temperature.

Figure 2.48  Snapshot of the Excel spreadsheet for the solubility for methane, propane, ethylene and propylene in water as a function of temperature.

118  Chemical Process Engineering

Solubility of Sulfur and Nitrogen Compounds in Water Sulfur and nitrogen emissions are considered as important safety, health, and environmental issues in the chemical process industries [2]. At low levels, sulfur can yield concentrations in air that exceed the TLV for human exposure. A correlation of water solubility of sulfur compounds based on the boiling point is [24]:



log10 S = A + BTB + CTB2 + DTB3

(2.41)

where S = solubility in water at 25 °C, ppm (wt.) TB = boiling point of compound, K.

Example 2.22 Calculate the solubility of sulfur compounds in water based on the boiling point using the Excel spreadsheet.

Solution The Excel program, Example 2.22.xlsx, calculates the solubility of sulfur compounds in water based on the boiling point and correlation constants of (Appendix C-17 [2]) as shown in Figure 2.49. These plots show that solubility decreases with temperature. More examples are provided by [2].

Figure 2.49  Snapshot of the Excel spreadsheet for calculating the solubility of sulfur compounds in water based on the boiling point.

Figure 2.50  Snapshot of the Excel program for calculating the adsorption capacity of VOC on activated carbon for the concentration of VOC ranged from 0-10,000 ppmv.

Physical Property of Pure Components and Mixtures  119 Coker [2] provided the correlation and examples for the solubility of nitrogen compounds in water which are also important as sulfur compounds in dealing with pollutions and emission issues in process design.

Adsorption on Activated Carbon The following correlation based on the concentration in the gas is used for the adsorption:

log10 Q = A + B[log10 y] + C[log10 y]2

(2.42)

where Q = adsorption capacity at equilibrium, 1 g compound/100g of carbon A, B, C and D = correlation constants y = concentration in gas at 25 °C, and at 1 atm, ppmv (parts per million, volume)

Example 2.23 Calculate the solubility of sulfur compounds in water based on the boiling point using the Excel spreadsheet.

Solution The Excel program, Example 2.23.xlsx, calculates the adsorption capacity of VOC on activated carbon with the constants provided in Table C-22 (Appendix C) [2].

References 1. A. K. Coker, Fortran Programs for Chemical Process Design, Analysis, and Simulation, Gulf Professional Publishing, 1995. 2. A. K. Coker, Ludwig’s Applied process design for chemical and petrochemical plants, 4th ed., vol. 1, Gulf Professional Publishing, 2007. 3. C. L. Yaws, Chemical Properties Handbook, McGraw-Hill, 1995. 4. B. E. Poling, J. M. Prausnitz and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, 2001. 5. Haynes W. M., CRC Handbook of Chemistry and Physics, 96th ed., CRC, 2015. 6. K. Matthias and H. Martin, VDI Heat Atlas, Series:VDI-Buch, Berlin Heidelberg: Springer-Verlag, 2010. 7. L. C. Yen and S. S. Woods, “A Generalized Equation for Computer Calculation of Liquid Densities,” AIChE J., vol. 12, pp. 95-99, 1966. 8. C. L. Yaws, Thermophysical Properties of Chemicals and Hydrocarbons, Gulf Professional Publishing, 2014. 9. G. Stephanie, “Probability and Statistics Topic Index: Elementary Statistics for the rest of us,” 12 December 2014. [Online]. Available: https://www.statisticshowto.com/probability-and-statistics/. [Accessed November 2020]. 10. DDBST GmbH, “Liquid Density Calculation by DIPPR105 Equation (Water),” [Online]. Available: http://ddbonline.ddbst. de/DIPPR105DensityCalculation/DIPPR105CalculationCGI.exe?compo​nent= Water. [Accessed 2020 November]. 11. D. S. Viswanath, T., K. Ghosh, D. H. L. Prasad, N. V. K. Dutt and K. Y. Rani, Viscosity of Liquids: Theory, Estimation, Experiment, and Data, Springer Netherlands, 2007. 12. R. C. Reid, J. M. Prausnitz and T. K. Sherwood, The Properties of Gases and Liquids, 3rd ed., New York: McGraw-Hill, 1977. 13. R. C. Reid, J. M. Prausnitz and B. E. Poling, The properties of gases and liquids, 4th ed., New York: McGraw-Hill, 1987. 14. J. A. Colapret, “Alcohols,” [Online]. Available: http://colapret.cm.utexas.edu/courses/Nomenclature_files/alcohols.htm. [Accessed November 2020]. 15. C. F. Beaton and G. F. Hewitt, Physical Property Data for the Chemical and Mechanical Engineer, New York: Hemisphere Publishing Corporation, 1989. 16. E. K. Huang and T. E. Daubert, “Extension of Passut-Danner Correlation of Ideal Gas Enthalpy, Heat Capacity, and Entropy,” Ind. Eng. Chem. Process Des. Dev., vol. 13, no. 2, p. 193, 1974. 17. S. C. Yuan and Y. I. Mok, “New Look at Heat Capacity Prediction,” Hydrocarbon Process, vol. 47, no. 3, 7, pp. 133-136, 153, 1968. 18. A. M. E. Tans, “Find Properties of n-Paraffins,” Hydrocarbon Process, vol. 47, no. 4, pp. 169-173, 1968. 19. D. R. Stull, E. E. Westrum and G. C. Sinke, The Chemical Thermodynamics of Organic Compounds, New York: Wiley, 1969.

120  Chemical Process Engineering 20. National Institute of Standards and Technology, “NIST-JANAF Thermochemical Tables; NIST Standard Reference Database 13,” 1998. [Online]. Available: https://janaf.nist.gov. [Accessed May 2020]. 21. Y. Demirel, Nonequilibrium Thermodynamics, Elsevier, 2014. 22. K. M. Watson, “Thermodynamics of the liquid State: Generalized Prediction of Properties,” Ind. Eng. Chem, vol. 35, no. 4, pp. 398-406, 1943. 23. C. Yaws, J. Hooper, X. Wang, A. Rathinsamy and R. Pike, “Calculating Solubility & Henry’s Law Constants for Gases in Water,” Chem. Eng, p. 102, June 1999. 24. C. L. Yaws, P. Bajaj, H. Singh and R. W. Pike, “Solubility & Henry’s Law Constants for Sulfur Compounds in Water,” Chem. Eng., p. 60, August 2003. 25. C. R. Wilke and C. Y. Lee, “Estimation of Diffusion Coefficients for Gases and Vapors,” Ind. Eng. Chem., vol. 47, no. 6, pp. 1253-1257, 1995. 26. S. K. Mitra and S. Chakraborty, Microfluidics and Nanofluidics Handbook, Taylor & Francis Group, 2012. 27. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, New York: Cambridge Univ. Press, 1961. 28. R. B. Bird, W. E. Stewart and E. N. and Lightfoot, Transport Phenomena, Revised 2nd ed., Wiley and Sons, 2002. 29. C. R. Wilke and P. Chang, “Correlation of Diffusion Coefficients in Dilute Solutions,” AIChE J., vol. 1, no. 2, p. 264–270, 1995. 30. M. M. EL-Gassier, “Fortran Program Computes Gas Compression,” Oil Gas J., p. 88, July 13, 1987. 31. V. N. Gopal, “Gas Z-factor Equations Developed for Computer,” Oil Gas J,. p. 59, August 8, 1977. 32. X. Zhu, B. Xu and Z. Han, “Measurement and Prediction Method of Compressibility Factor at High Temperature and High Pressure,” Mathematical Problems in Engineering, p. Article ID 4681918., 2016. 33. G. Takacs, “Comparing Methods for Calculating Z-factor,” Oil Gas J., pp. 43-46, May 15, 1989.

3 Fluid Flow INTRODUCTION Transportation of fluids is important in the design of chemical process industry (CPI). In CPI, piping and its accessories such as fittings, make up 20% to 30% of the total design costs and 10% to 20% of the total plant investment [1]. Maintenance requirements and energy usage in the shape of pressure drop (ΔP) in the fluids being pumped is also added to the cost. Also, these items escalate each year in line with inflation. As a result, sound pipe sizing practices can have a substantial influence on overall plant economics. It is the designer’s responsibility to optimize the pressure drops in piping and equipment and to assess the optimal conditions of operations. Figure 3.1 illustrates piping layouts in a chemical plant. The complexity of flow pattern is such that most flows are described by a set of empirical or semi-empirical equations. These relate the pressure drop in the flow system as a function of flow rate, pipe geometry, and physical properties of the fluids as detailed in Chapter 2. The aim in the design of fluid flow is to choose a line size and piping arrangement that achieve minimum capital and pumping costs. In addition, constraints on pressure drop and maximum allowable velocity in the process pipe should be maintained. These objectives require many computational efforts, which can be performed well by the Excel spreadsheet and process simulators like the UniSim Design software.

FLOW OF FLUIDS IN PIPES Pressure drop or head loss in a piping system is caused by fluid rising in elevation, friction, shaft-work (e.g., from a turbine) and turbulence due to sudden changes in direction or cross-sectional area. Figure 3.2 shows the distribution of energy between two points in a pipeline. Bernoulli’s equation expresses the conservation of the sum of pressure, kinetic and potential energies.



1 ρ

∫

2

1

1 dP + gc

2

2

∫ VdV + ∫ dZ = 0 1

1

(3.1)

Integrating Equation 3.1 gives:



1 V 2 − V12 (P2 − P1 ) + 2 + (Z 2 − Z1 ) = 0 2g c ρ

(3.2)

where P = Pressure of fluid, lbf/ft2   ρ = Density of fluid, lbm/ft3 V = Velocity of fluid, ft/s gc = dimensional constant, 32.174 (lbm/lbf )(ft/s2) Z = Elevation of fluid, ft subscript 1 - condition at initial point subscript 2 - condition at final point The first, second, and third terms in Equation 3.2 represent pressure head, velocity head, and static differences, respectively. Equation 3.2 is used for determining energy distributions or pressure differentials between any two A. Kayode Coker and Rahmat Sotudeh-Gharebagh. Chemical Process Engineering: Design, Analysis, Simulation and Integration, and Problem-Solving With Microsoft Excel – UniSim Design Software, Volume 1, (121–164) © 2022 Scrivener Publishing LLC

121

122  Chemical Process Engineering

Figure 3.1  Chemical plant piping layout. Source: I. Chem E. safer piping training package (Courtesy of the I. Chem. E., UK). Total head (energy grade line)

Head (ft) of fluid

v12 2g

hL

Hydraulic gr ade line

144 × P1 ρ1

v22 2g

144 × P2 ρ2

Pressure head, P

Velocity head 2

Flow

1 Z1

Z2

Elevation head

Arbitrary horizontal datum line Z1 +

v12

Pipe length (ft) v22

144 P2 144 P1 ρ1 + 2g = Z2 + ρ2 + 2g + hL

Figure 3.2  Distribution of fluid energy in a pipeline.

points in a pipeline. Incorporating the head loss due to friction, hL, with constant pipe diameter, i.e., V1= V2, Equation 3.2 becomes



1 (P2 − P1 ) + (Z 2 − Z1 ) = h L ρ



ΔP > hL + (Z2 – Z1)ρ

(3.3)

Equation 3.3 shows that the head loss, hL, is generated at the expense of pressure head or static head difference. The static head difference can be either negative or positive. However, for a negative static head difference, (3.4)

Fluid Flow  123 In general, pressure loss due to flow is the same whether the pipe is horizontal, vertical, or inclined. The change in pressure due to the difference in head must be considered in the pressure drop calculations.

EQUIVALENT LENGTH OF VARIOUS FITTINGS AND VALVES The effects of bends and fittings, such as elbow, valves, tees, and reduction or enlargement of pipes are determined empirically through a fictitious equivalent length of straight pipe having the same diameter which would develop the same pressure drop. The equivalent pipe length is the most convenient method for determining the overall ΔP in a pipe. The drawback to this approach is that, for a given fitting the equivalent length is not constant but depends on the Reynolds number, the pipe roughness, the pipe size, and the geometry of the fitting. The equivalent length is added to the length of actual straight pipe to give the total length of pipe

LTotal = Lst + Leq

(3.5)

where LTotal = total length of pipe, ft Lst  = length of straight pipe, ft Leq  = equivalent length of pipe, ft The pressure drop equation can be expressed as:

 L P = fD +  D



∑

2  V K ρ  2g C

(3.6)

2  V  2g C

(3.7)

and the head loss equation can be expressed as:

 L H = fD +  D



∑

K

where D = internal diameter of pipe, ft fD = darcy friction factor K = excess head loss L = pipe length, ft

Excess Head Loss K is a dimensionless factor defined as the excess head loss in a pipe fitting and expressed in velocity heads. The velocity head is the amount of kinetic energy of a stream or the amount of potential energy required to accelerate a fluid to its flowing velocity. Most published K values apply to fully developed turbulent flow because at high Reynolds number, K is found to be independent of Reynolds number. However, the two-K technique includes a correction factor for low Reynolds number. Hooper [2] gives a detailed analysis of his method compared to others [2, 3], and has shown that the two-K method is the most suitable for any pipe size. In general, the two-K method is independent of the roughness of the fittings, but it is a function of Reynolds number and of the geometry of the fitting. The method can be expressed as:



K=

K1  1 + K∞ 1 +  d NRe

(3.8)

124  Chemical Process Engineering where K1 = K for fitting at NRe = 1 K∞ = K for a large fitting at NRe = ∞ d   = internal diameter of attached pipe, in The conversion between equivalent pipe length and the resistance coefficient, K, can be expressed as:

K = fD



L eq D

(3.9)

Table 3.1 lists values of K1 and K∞ for the two-K method. There are several other methods for determining excess head loss in a pipe fitting. These include: the “constant Kf ” method which gives a rough estimate, the (L/D)eq method (which gives a better but limited estimate), the Crane method [4] which gives a good estimate for fully turbulent flow, 2-K method [5] and 3-K [6] method. The latter two provide good estimates over the greater range of fitting sizes and Reynolds numbers [1]. Coker [1] compares various loss coefficient methods for pipe fittings and valves. An improved 3-K method by Darby is discussed elsewhere [1].

Pipe Reduction and Enlargement The velocity head is V2/2gC. For any velocity profile, the true velocity represents the integral of the local velocity head across the pipe diameter. It is found by dividing the volumetric flow rate by the cross-sectional area of the pipe and multiplying by a correction factor. For laminar flow this factor is 2; for turbulent flow the factor depends on both the Reynolds number, the pipe roughness and (1 + 0.8fD) [5]. Therefore, when the pipe size changes, the velocity head also changes. Because the velocity is inversely proportional to the flow area and thus to the diameter squared, K is inversely proportional to the velocity squared [3]. 4

D  K 2 = K1  2   D1 



(3.10)

Although the potential energy provides the flowing fluid with kinetic energy at the pipe entrance, the kinetic energy is later recovered. This indicates that the measured pipe pressure will be lower than the calculated pressure by one velocity head. If the kinetic energy is not recovered at the pipe exit, the exit counts as a loss of one velocity head. Table 3.2 shows how K varies with changes in pipe size.

PRESSURE DROP CALCULATIONS FOR SINGLE-PHASE INCOMPRESSIBLE FLUIDS The pressure drop of flowing fluids can be calculated from the Darcy friction factor [3]

∆P f D ρV 2 = ⋅ L D 2g c

where: fD = Darcy friction factor D = internal pipe diameter, ft ρ  = density of fluid, lbm/ft3 V = velocity of fluid in the pipe, ft/s

(3.11)

Fluid Flow  125 Table 3.1  Values of K1 and K∞ for the two-K method for pipe fittings. Source: Hooper [2, 3]. Fitting

Type

Geometry

K1

K∞

Elbow 90°

Standard, Screwed

R/D = 1

800

0.40

Standard, Flanged/Welded

R/D = 1

800

0.25

Long Radius, all types

R/D = 1.5

800

0.20

1 Weld, 90° Angle

R/D = 1.5

1000

1.15

2 Welds, 45° Angle

R/D = 1.5

800

0.35

3 Welds, 30° Angle

R/D = 1.5

800

0.30

4 Welds, 22.5° Angle

R/D = 1.5

800

0.27

5 Welds, 18° Angle

R/D = 1.5

800

0.25

Standard, all types

R/D = 1

500

0.20

Long Radius, all types

R/D = 1.5

500

0.15

Mitered, 1 Weld, 45 Degree

R/D = 1.5

500

0.25

Mitered, 2 Welds, 22.5 Degree

R/D = 1.5

500

0.15

Standard, Screwed

R/D = 1

1000

0.60

Standard, Flanged/Welded

R/D = 1

1000

0.35

Long Radius, all types

R/D = 1.5

1000

0.30

Standard, Screwed

500

0.70

Long Radius, Screwed

800

0.40

Standard, Flanged/Welded

800

0.80

Stub-in Branch

1000

1.00

Elbow 90° Mitered

Elbow 45°

Elbows 180°

Tees

Through Branch (used as elbow)

Run through screwed

R/D = 1

200

0.10

Run through Flanged/Welded

R/D = 1

150

0.50

100

0.00

Run through stub-in type branch Valve

Gate

full line size, β = 1

300

0.10

Ball

reduced trim, β = 0.9

500

0.15

Plug

reduced trim, β = 0.9

1000

0.25

Globe

standard

1500

4.00

Globe

angle or Y type

1000

2.00

Diaphragm

dam-type

1000

2.00

800

0.25

Butterfly Check

lift

2000

10.00

Check

swing

1500

1.50

Check

tilting disk

1000

0.50

Use R/D = 1.5 values for R/D = 5 pipe bends, 45° to 180°, Use appropriate values for flow through crosses [3].

126  Chemical Process Engineering Table 3.2  Excess head loss “K” correlation due to changes in pipe size [1]. Type

Fitting

1 (A)

Inlet NRe NRe ≤ 2,500 NRe >2,500

D1 D2

Flow

NRe

All

D1 D2 NRe

 θ sin   for 45° < θ < 180°  2 NRe ≤ 2,500 NRe >2,500

D2

D1

Flow

NRe

NRe ≤ 4,000 NRe>4,000

D2 D1

Flow

NRe

All

D2 D1 NRe

θ

  D 4  K = 1 −  1     D2     D 2 K = [1 + 08f D ]1 −  1     D2  

Square expansion

5 (E)

2 2 4    D   120    D2    D1  − 1  − 1 1 −  K =  2.7 +  2        D1   NRe    D1    D2     2 2 4    D   4000    D2    D1  − 1  − K =  2.7 +  2   1         D1   NRe    D1    D2    

Thin, sharp orifice

4 (D)

Multiply K from Type 1 by   θ  1.6sin  2   for 0° < θ < 45° or  

Flow

θ

Tapered reduction

3 (C)

4   160  D1  − 1  K = 1.2 +   Re  D2    

2 2   D   D  K = [0.6 + 0.48f D ] 1   1  − 1  D2    D2  

Square reduction

2 (B)

K based on inlet velocity head

Flow

2

If θ> 45°, use K from Type 4, otherwise multiply K from Type 4 by   θ   2.6sin  2    

Tapered expansion

(Continued)

Fluid Flow  127 Table 3.2  Excess head loss “K” correlation due to changes in pipe size [1]. (Continued) Type

Fitting

6 (F)

Inlet NRe Rounded

D1

D2

All

Flow

Re1

K based on inlet velocity head  50 K = 0.1 + N Re 

 D1  4     − 1    D2   

Pipe reducer

7 (G)

D1

All

D2 L

Flow

If L/D2 > 5, use Case A and Case F; Otherwise multiply K from Case D by 0.0936    0.584 +  (L/D)1.5 + .225    

Thick orifice

8 (H)

All

D2

Use the K for Case F

D1 Flow

Pipe reducer

gC = dimensional constant, 32.174(lbm/lbf )(ft/s2) L = length of pipe, ft

Friction Factor The friction factor is related to the Reynolds number by a set of correlations and depends on whether the flow regime is laminar, transitional, or turbulent. The Reynolds number is:



NRe =

ρvd Qρ W = 50.6 = 6.31 µ dµ dµ

(3.12)

NRe =

ρvd Qρ W = 21.22 = 354 µ dµ dµ

(3.12a)

In SI units,

where

Q = volumetric flow rate, gal/min (l/min) d  = internal pipe diameter, in (mm) d  = internal pipe diameter, in (mm) v  = mean fluid velocity, ft/s (m/s) W = mass flow rate, lb/h (kg/h) μ  = fluid viscosity, cP (1 Pa.s=103 cP) ρ  = fluid density, lb/ft3 (kg/m3)

128  Chemical Process Engineering For laminar flow with NRe ≤ 2000



64 NRe

fD =

(3.13)

The Darcy friction factor is four times the Fanning friction factor, fP, i.e., fD = 4fF. For fully developed turbulent flow regime in smooth and rough pipes, the Colebrook or Chen equations can be used. The Colebrook equation is expressed as [3]:



1 2.51   ε = −0.8686ln  + fD  3.7D NRe f D 

(3.14)

Equation 3.14 is implicit in fD, as it cannot be rearranged to derive fD directly and thus requires an iterative solution and can be solved using the Excel spreadsheet built-in functions. Here, the Chen equation is used for calculating fD, (i.e, fD = 4fC). The Chen equation is explicit and easier than Equation 3.14. This can be expressed as:



0.9  ε 1 5.02  ε  6.7    log  = −4 Log  − +   fC  3.7D  NRe     3.7D NRe

(3.15)

where: ε = pipe roughness, ft A detailed review of other explicit equations is given by Gregory and Fogarasi [7]. Different piping materials are often used in the chemical process industries, and at a high Reynolds number, the friction factor is affected by the roughness of the surface. This is measured as the ratio ε/D of projections on the surface to the diameter of the pipe. Glass and plastic pipe essentially have ε = 0. Values of ε are shown in Table 3.3.

Overall Pressure Drop Designers involved in sizing of process piping often apply trial and error procedure. The designer first selects a pipe size and then calculates the Reynolds number, friction factor, and coefficient of resistance. The pressure drops per 100 feet of pipe is then computed. For a given volumetric rate and physical properties of a single-phase fluid, ΔP100 for laminar and turbulent flows is: laminar flow



∆P100 = 0.0273

µQ , psi/100 ft d4

(3.16)

∆P100 = 0.0679

µQ , bar/100m d4

(3.16a)

In SI units:

Turbulent flow



∆P100 = 0.0216f D

ρQ 2 , psi/100 ft d5

(3.17)

Alternatively, for a given mass flow rate and physical properties of a single-phase fluid, ΔP100 for laminar and turbulent flows respectively is:

Fluid Flow  129 Table 3.3  Equivalent roughness of various surfaces (Source [6]). Material

Condition

Roughness range

Recommended

Drawn brass, Copper, stainless Commercial steel

New

0.01-0.0015mm (0.0004-0.00006 in.)

0.002 mm (0.00008 in.)

New

0.1-0.02mm (0.004-0.0008 in.)

0.045 mm (0.0018 in.)

Light rust

1.0-0.15 mm (0.04-0.006 in.)

0.3 mm (0.015 in.)

General rust

3.0-1.0 mm

2.0 mm

Wrought, new

0.046 mm (0.002 in.)

0.046 mm (0.002 in.)

Cast, new

1.0-0.25 mm (0.04-0.01 in.)

0.30 mm (0.025 in.)

Galvanized

0.15-0.025 mm (0.006-0.001 in.)

0.15 mm (0.006 in.)

Asphalt-coated

1.0-0.1 mm (0.04-0.004 in.)

0.15 mm (0.006 in.)

Sheet metal

Ducts Smooth joints

0.1-0.02 mm (0.004-0.0008 in.)

0.03 mm (0.0012 in.)

Concrete

Very smooth

0.18-0.2 mm (0.007-0.001 in.)

0.04 mm (0.0016 in.)

Wood floated, Brushed

0.8-0.2 mm (0.03-0.007 in.)

0.3 mm (0.012 in.)

Rough, visible form marks

2.5-0.8 mm (0.1-0.03 in.)

2.0 mm (0.08 in.)

Wood

Stave, used

1.0-0.25 mm (0.035-0.01 in.)

0.5 mm (0.02 in.)

Glass or plastic

Drawn tubing

0.01-0.0015 mm (0.0004-0.00006 in.)

0.002 mm (0.00008 in.)

Rubber

Smooth tubing

0.07-0.006 mm (0.003-0.00025 in.)

0.01 mm (0.0004 in.)

Wire-reinforced

4.0-0.3 mm (0.15-0.01 in.)

1.0 mm (0.04 in.)

Iron

laminar flow



∆P100 = 0.0034

µW , psi/100 ft ρd 4

(3.18)

turbulent flow



W2 ∆P100 = 0.000336f D 5 , psi/100 ft ρd

(3.19)

130  Chemical Process Engineering Multiplying Equations 3.16, 3.17, 3.18, and 3.19 by the total length between two points and adding the pipe elevation yields the overall pressure drop.



∆P = ∆P ⋅

L Total ρ∆Z + , psi 100 144

(3.20)

Equation 3.20 is valid for compressible isothermal fluids of shorter lines where pressure drops are no more than 10% of the upstream pressure [1]. In general, pipe size for a given flow rate is often selected on the assumption that the overall pressure drop is close to or less than the available pressure difference between two points in the line.

Nomenclature A = pipe cross-sectional area, ft2 d = internal pipe diameter, inch D = internal pipe diameter, ft fc = Chen friction factor fD = Darcy friction factor gc = dimensional constant 32.174 (lbm/lb f * ft/s2) hL = head loss, ft K = excess head loss for a fitting, velocity heads K1 = K for fitting at NRe = 1, velocity heads K∞ = K for very large fitting at NRe = ∞, velocity heads Leq = equivalent length of pipe, ft Lst = actual length of pipe, ft LTotal = total length of pipe, ft n = number of fittings NRe = Reynolds number ΔP100= pressure drop per 100 ft of pipe, psi/100 ft ΔP = overall pressure drop of pipe, psi ρ = fluid density, lb/ft3 ε = absolute roughness of pipe wall, ft μ = fluid viscosity, cP

COMPRESSIBLE FLUID FLOW IN PIPES The flow of compressible fluids (e.g., gases and vapors) through pipelines and other restrictions are often affected by changing operating conditions and physical properties. The densities of gases and vapors vary with temperature and pressure as discussed in Chapter 2. During isothermal flow, i.e., constant temperature (PV = constant) density varies with pressure. Conversely, in adiabatic flow, i.e., no heat loss (PVk = constant), a decrease in temperature occurs when pressure decreases, resulting in a density increase. At high pressures and temperatures, the compressibility factor can be less than unity, which results in an increase in the fluid density. The condition of high ΔP in compressible flow frequently occurs in venting systems, vacuum distillation equipment, and long pipelines [8]. Some design situations involve vapor flows at very high velocities resulting in ΔP > 10% of the upstream pressure [3]. Such cases are vapor expanding through a valve, high speed vapor flows in narrow pipes, and vapors flowing in process lines under vacuum conditions. In many cases, ΔP is critical and requires accurate analysis and design. For instance, the inlet pipe ΔP of a safety relief valve should not exceed 3% of the relief valve set pressure (gauge) at its relieving capacity for stable operation [1]. This limit is to prevent the rapid opening and closing of the valve, a phenomenon known as chattering, resulting in lowered fluid capacity and subsequent damage of the value seating surfaces. Conversely, the tail pipe or vent line of a relief valve should be designed in such a way that ∆P < 10%

Fluid Flow  131

6" Tail pipe

Relief valve 3" K 4"

4"

Tube fluid in

3" P1

P2 d

Shell fluid in

Tube fluid out

Figure 3.3  Fluid flow through a heat exchanger, relief valve, and tail pipe.

of the relieving pressure, i.e., set pressure + over pressure in gauge [1]. Figure 3.3 shows a typical tail pipe and relief valve connected to a heat exchanger.

Maximum Flow and Pressure Drop Determining the maximum fluid flow rate or pressure drop for process design often has the dominant influence on density. As pressure decreases due to piping and other resistances, the gas expands and its velocity increases. A limit is reached when the gas velocity cannot exceed the sonic or critical velocity. Even if the downstream pressure is lower than the pressure required to reach sonic velocity, the flow rate will still not increase above that evaluated at the critical velocity. Therefore, for a given ΔP, the mass discharge rate through a pipeline is greater for an adiabatic condition (i.e., insulated pipes, where heat transfer is nil) than the rate for an isothermal condition by as much as 20% [1]. There is, however, no difference if the pipeline is more than 1,000 pipe diameters in long [9]. In practice, the actual flows are between adiabatic and isothermal conditions, and the inflow rates are well below 20% even for lines less than 1,000 pipe diameters [3].

Critical or Sonic Flow and the Mach Number The flow rate of a compressible fluid in a pipe at a given upstream pressure will approach a certain maximum rate even with reduced downstream pressure. The maximum velocity is limited by the velocity of propagation of a pressure wave that travels at the speed of sound in the fluid. Such velocity that a compressible fluid can attain in a pipe is known as the sonic velocity, Vs, and is expressed as:



 kT  Vs = 223   M W 

0.5

 kP  = 68.1 1   ρ1 

0.5

, ft/s

(3.21)

where T = temperature in Rankine With a high velocity vapor flow, the possibility of attaining critical or sonic flow conditions in a process pipe should be carefully investigated. These occur whenever the resulting pressure drop approaches the following values of ΔP as a percentage of the upstream pressure [10]:

132  Chemical Process Engineering 1. s aturated steam, ΔP = 42% 2. diatomic gases, (e.g., H2, N2, 02), ΔP = 47% 3. triatomic and higher molecular weight gases including hydrocarbon vapors and superheated steam, ΔP = 45%. Vapor flow at or near this maximum velocity should be avoided because a critical pressure, PC, is attained at the sonic velocity and any ΔP beyond PC will translate into shock waves and critical turbulence instead of being converted into useful kinetic energy. In the case of a high-pressure header, the flow may be sonic at the exit. Therefore, it is often necessary to check that the outlet pressure of each pipe segment is not critical. If PC is less than terminal P2, the flow is subcritical. If, however, PC is greater than P2, then the flow is critical. Although it may be impractical to keep the flow in high pressure sub headers below sonic, Mak [11] suggests that the main flare header should not be sized for critical flow at the outlet of the flare stack. This would obviate the undesirable noise and vibration resulting from sonic flow. Crocker’s equation for critical pressure can be expressed as [12]:

Pc =



R=



G  RT  2 ⋅ 11400d  k(k + 1) 

, psia

(3.22)

1544 , molar gas constant 29 ⋅ SpGr

SpGr =



0.5

molecular weight of gas moelcular weight of air

The upstream fluid velocity is:

V=



0.0509G , ft/s d 2ρ1

(3.23)

A recommended compressible fluid velocity for trouble-free operation is V 1/ k , the flow is supersonic Case 3 is produced under certain operating conditions in the throttling process (e.g., a reduction in the flow cross-sectional area). Kirkpatrick [13] indicates that there is a maximum length for which continuous flow is applied for an isothermal condition, and this corresponds to M = 1/ k . The limitation for isothermal flow, however, is the heat transfer required to maintain a constant temperature. Therefore, when

Fluid Flow  133 Table 3.4  Recommended fluid velocity and maximum ∆P for carbon steel vapor lines. Type of service 1.

Recommended velocity ft/sec.

Maximum ∆P psi/100ft.

General Recommendation Fluid pressure psig

2.

Sub-atmospheric

0.18

0 50 psia)

40-50

Atmospheric

60-100

Vacuum (P 300

1.5

High pressure Steam Lines Short (L < 600 ft)

1.0

Long (L > 600 ft)

0.5

Exhaust Steam lines

0.5

(P > atmosphere)

0.5

Leads to exhaust Header

1.5

Relief valve discharge

0.5Vs

Relief valve, Entry point at silencer

Vs

134  Chemical Process Engineering Table 3.5  Approximate k values for some common gases (68°F, 14.7 psia). Gas

Chemical formula or symbol

Approximate molecular weight

K(Cp/Cν)

Acetylene (Ethyne)

C₂H₂

26.0

1.30

Air

__

29.0

1.40

Ammonia

NH₃

17.0

1.32

Argon

Ar

39.9

1.67

Butane

C₄H₁₀

58.1

1.11

Carbon Dioxide

CO₂

44.0

1.30

Carbon Monoxide

CO

28.0

1.40

Chlorine

CI₂

70.9

1.33

Ethane

C₂H₆

30.0

1.22

Ethylene

C₂H₄

28.0

1.22

Helium

He

4.0

1.66

Hydrogen Chloride

HCI

36.5

1.41

Hydrogen

H₂

2.0

1.41

Methane

CH₄

16.0

1.32

Methyl Chloride

CH₃CI

50.5

1.20

Natural Gas

__

19.5

1.27

Nitric Oxide

NO

30.0

1.40

Nitrogen

N₂

28.0

1.41

Nitrous Oxide

N₂O

44.0

1.31

Oxygen

O₂

32.0

1.40

Propane

C₃H₈

44.1

1.15

Propylene (Propene)

C₃H₆

42.1

1.14

Sulfur Dioxide

SO₂

64.1

1.26

M < 1/ k , heat must be added to the stream to maintain constant temperature. For M > 1/ k , heat must be rejected from the stream. Depending on the ratio of specific heats, either condition could occur with subsonic flow. Therefore, to maintain isothermal flow during heat transfer, high temperatures require high Mach numbers and low temperatures require low Mach numbers. The ratio of the specific heat capacities of the gas, Cp/Cv can also be estimated by the UniSim Design software simulator.

Mathematical Model of Compressible Isothermal Flow The derivations of the maximum flow rate and pressure drop of compressible isothermal flow are based on the following assumptions: 1. i sothermal compressible fluid 2. no mechanical work done on or by the system

Fluid Flow  135 3. p  erfect gas law 4. a constant friction factor along the pipe 5. steady state flow Figure 3.4 illustrates the distribution of fluid energy with work done by the pump and heat added to the system. Table 3.6 gives friction factors for clean commercial steel pipes with flow in zones of complete turbulence.

Point 2

Point 1 +q (heat added to fluid)

P1 V1

Datum

Z2

Pump

Z1

P2 V2

–w (work done by pump on fluid)

Figure 3.4  Energy aspects of a single-stream piping system.

Table 3.6  Friction factors for total turbulence in new commercial steel pipes. Nominal size (inch)

Friction factor (f)

1

0.023

1.5

0.0205

2

0.0195

3

0.0178

4

0.0165

5

0.016

6

0.0152

8

0.0142

10

0.0136

12

0.0132

14

0.0125

16

0.0122

18

0.12

20

0.0118

24

0.0116

136  Chemical Process Engineering

Flow Rate Through Pipeline Bernoulli’s equation for the steady flow of a fluid is expressed as:



∫

2

1

dP ∆V 2 g + + ∆Z + h L + δ Ws = 0 ρ 2αg c g c

(3.25)

where α (the dimensionless velocity distribution)=1 for turbulent or plug flow. Assuming no shaft work is done (i.e., δW= 0), then Equation 3.25 becomes



P2 − P1 V22 − V12 g + (Z 2 − Z1 ) + h L = 0 + ρ 2αg c gc

(3.26)

For expanding gas flow, V2 ≠ Vl; with horizontal pipe, Z2 = Z1. Hence, the differential form of Bernoulli’s equation can be expressed as:



−

dP 1 = VdV + h L ρ gc

(3.27)

where ρ is constant, and the velocity head is

hL = K



V2 2g c

(3.28)

The mass flow rate through the pipe is G = ρVA or



G = ρV = constant A

(3.29)

Because both density and velocity change along the pipeline, Equation 3.29 can be expressed in differential form as:



ρdV + Vdρ = 0

(3.30)

In expanding gas flow, the pressure and density ratio are constant:



P dP = ρ dρ

(3.31)

P dP P1 = = ρ dρ ρ1

(3.32)

V dρ ρ

(3.33)

and, with respect to initial condition, P1 and ρ1,

From Equation 3.30



dV = −

Fluid Flow  137 and

dP dρ = P ρ

(3.34)

1 1  dP  1  P1  = = ρ P  dρ  P  ρ1 

(3.35)

Therefore



Substituting Equation 3.34 into Equation 3.33:

dV = −



V  dP  ρ ρ P 

(3.36)

Therefore

dV = − V



 dP   P

(3.37)

Substituting Equation 3.37 into Equation 3.27:





−

V2 dP  dP 1  = V −V +K 2g c P ρ gc 

(3.38)

−

V2 dP 1  dP  = − V2 +K  P 2g c ρ gc

(3.39)

dP V 2  K dP  = ρ − ρ gc  2 P 

(3.40)

Therefore

−

From the mass flow rate, G = ρVA

V=



G ρA

(3.41)

Substituting Equation 3.41 into Equation 3.40



−dP =

G2  K dP  − 2 2 ρ gc A ρ 2 P

(3.42)

In terms of the initial conditions of pressure and density, i.e., substituting Equation 3.35 into Equation 3.42



−dP =

2

1  G   1  P1    K dP  − g c  A   P  ρ1    2 P 

(3.43)

138  Chemical Process Engineering Integrating Equation 3.43 gives:



2

2

 G   P1  1  K−2  A   ρ1  g c 

2

dP   P

(3.44)

P  G   P1   1   K + 2 ln 1  P −P =  A   ρ1   g c   P2 

(3.45)

−2

∫

1

PdP =

∫

1

which is



2 1

2

2 2

Therefore, for maximum flow rate through the pipe, 0.5



   P12 − P22   g c A 2ρ1 G =  P1   P1    K TOTAL + 2 ln   P2   

(3.46)

where KTOTAL is the total velocity head due to friction, fittings, and valves:



L + D

K TOTAL = f D

∑K (pipefittings + valves) f

∑Kf (pipe fittings + valves) is the sum of the pressure loss coefficient of all fittings and valves in the line. Expressing the maximum fluid rate in pounds per hour, Equation 3.46 becomes 0.5

2 2  ρ1   P1 − P2   G = 1335.6d  P   P1     K TOTAL + 2 ln 1   P2    2



(3.47)

Pipeline Pressure Drop If ΔP is relatively small due to velocity acceleration compared with the frictional drop, then ln(P1/P2) may be neglected. Therefore Equation 3.47 becomes 0.5

 ρ1   P12 − P22   G = 1335.6d     K TOTAL   P1  

(3.48)

 P12 − P22  G2 ρ1 = ⋅ C 2 K TOTAL  P1 

(3.49)

2

Putting C = 1335.6d2

i.e.,

P12 − P22 =



P1G 2K TOTAL ρ1C 2

 P G 2K TOTAL  P2 = P12 − 1  ρ1C 2  

0.5



(3.50)

Fluid Flow  139 Therefore, the pressure drop

∆P = P1 − P2 0.5



 P G 2K TOTAL  , psi ∆P ≅ P1 − P12 ⋅ 1 2  ρ1C  

(3.51)

Nomenclature A = pipe cross-sectional area, ft2 d = internal pipe diameter, inch D = internal pipe diameter, ft = Chen friction factor fc fD = Darcy friction factor g = acceleration due to gravity (32ft/s2) gc = conversion factor (32.174(lbm/lbf ) (ft/s2)) G = fluid flow rate, lb/h k = ratio of specific heat capacities (Cp/Cv) K = resistance coefficient due to pipe fittings plus valves Kf = resistance coefficient due friction pipe fittings + valves KTotaI = total resistance coefficient Lst = length of straight pipe, ft MW = molecular weight of fluid (lb/lbmol) M = Mach number NRe = fluid Reynolds number (6.31G/dμ) P1 = inlet fluid pressure, psia P2 = outlet fluid pressure, psia PC = critical pressure, psia R = molar gas constant (1544/29 × SpGr) SpGr = molecular weight of gas / molecular weight of air T = fluid temperature, °F (R in Equation 3.21) V = fluid velocity, ft/s VS = fluid sonic or critical velocity, ft/s Z = compressibility factor α = correction factor (1 for turbulent or plug flow) ρ = fluid density, (lbm/ft3)(P.Mw/10.73ZRT) μ =fluid viscosity, cP ε = absolute pipe roughness, ft (0.00015 ft for carbon steel) Δ = difference

Subscripts 1 = upstream point in pipe 2 = downstream point in pipe

TWO-PHASE FLOW IN PROCESS PIPING Two-phase flow often presents design and operational issues not associated with liquid or gas flow. For example, several different flow patterns may exist along the pipeline. Frictional pressure losses are more tedious to estimate, and in the case of a cross-country pipeline, a full terrain profile is necessary to predict pressure drops due to elevation

140  Chemical Process Engineering changes. The downstream end of a pipeline often requires a device to separate the liquid and vapor phases, and a slug catcher may be required to remove liquid slugs. Static pressure losses in gas-liquid flow differ from those in single phase flow because an interface can be either smooth or rough, depending on the flow pattern. Two-phase pressure losses may be up to a factor of 10 higher than those in single-phase flow [1]. In the former, the two phases tend to separate and the liquid lags behind. Most published correlations for two-phase pressure drop are empirical, and therefore, limited by the range of data for which they were derived [14–17]. If two-phase situations are not properly detected, pressure drop problems may develop preventing the pipeline from operating.

Flow Patterns In determining the type of flow in a process pipeline, designers refer to a diagram similar to Figure 3.5, which is known as the Baker map. Figure 3.6 shows the types of flow regimes in a horizontal pipe, and Table 3.7 lists the characteristic linear velocities of the gas and liquid phases in each flow regime. Seven types of flow patterns are considered in evaluating two-phase flow, and only one type can exist in a pipeline at a time [1]. But as conditions change (e.g., velocity, roughness, and elevation), the type of flow pattern may also change. The pressure drop can also vary significantly between the flow regimes. The seven types of flow regimes in order of increasing gas rate at a constant liquid flow rate are: 1. Bubble or Froth Flow. Bubbles of gas are dispersed throughout the liquid moving along the upper part of the pipe at approximately the same velocity as the liquid. It occurs for liquid superficial velocities of about 5 to 15 ft/s (1.5-4.5 m/s) and gas superficial velocities of about 1 to 10 ft/s (0.3 to 3 m/s). 2. Plug Flow. Plugs of liquid and gas move along the upper part of the pipe while the liquid moves along the bottom of the pipe. This occurs for liquid velocities less than 2 ft/s (0.6 m/s) and gas velocities less than about 3 ft/s (0.9 m/s). 3. Stratified Flow. The liquid phase flows along the bottom of the pipe while the gas flows over a smooth liquid-gas interface. It occurs for liquid velocities less than 0.15 m/s and gas velocities of about 2-10 ft/s (0.6-3 m/s). 4. Wave Flow. Wave flow is like stratified flow except that the gas is moving at a higher velocity and the gas-liquid interface is distributed by waves moving in the direction of flow. It occurs for liquid velocities less than 1 ft/s (0.3 m/s) and gas velocities from about 1.5 ft/s (4.5 m/s).

100,000 DISPERSED C3 C2

WAVE

10,000

4" 6"

ANNULAR

BUBBLE OR FROTH

By

C4 C1 SLUG STRATIFIED

1,000

C5

C6

PLUG 100

.1

1

10

100

1,000

Bx

Figure 3.5  Baker parameters for two-phase flow regimes with modified boundaries. Source: Baker [14].

10,000

Fluid Flow  141 SEGREGATED

Stratified

Wavy

Annular INTERMITTENT

Plug

Slug DISTRIBUTED

Bubble

Mist

Figure 3.6  Flow regimes in horizontal flow.

Table 3.7  Characteristic linear Velocities of two-phase flow regimes. Regime

Liquid phase ft/s

Vapor phase ft/s

Bubble or Froth

5-15

0.5-2

Plug

2

200

142  Chemical Process Engineering 5. Slug Flow. This pattern occurs when waves are picked up periodically by the more rapidly moving gas. These form frothy slugs that move along the pipeline at a much higher velocity than the average liquid velocity. This type of flow causes severe and, in most cases, dangerous vibrations in equipment because of the high velocity slugs against fittings. 6. Annular Flow. In annular flow, liquid forms around the inside wall of the pipe and gas flows at a high velocity through the central core. It occurs for gas velocities greater than 20 ft/s (6 m/s). 7. Dispersed, Spray, or Mist Flow. Here, all of the liquid is entrained as fine droplets by the gas phase. Dispersed flow occurs for gas velocities greater than 200 ft/s (60 m/s).

Flow Regimes Establishing the two-phase flow regime involves determining the Baker parameters (Bx and By) from the system characteristics and physical properties. These parameters are:



0.5 1  WL   (ρg ρL )   3  B x = 531   µL  2  WG    ρL 3   σ L 

B y = 2.16

1  WG  ⋅  A  (ρg ρL )0.5

(3.52)

(3.53)

Bx depends on the weight ratio and the physical properties of the liquid and vapor phases. It is independent of pipe size; therefore, it remains constant once calculated from the characteristics of the liquid and vapor, and its position on the Baker map changes only if the liquid-vapor proportion changes. By depends on the vapor-phase flow rate, the vapor and liquid densities, and pipe sizes. The practical significance of the pipe size, however, is the effect on the frictional losses. The points Bx, By determine the flow regime for the calculated liquid-vapor ratio and the liquid’s and vapor’s physical properties. With increasing vapor content, Bx, By move up and to the left on the map. The boundaries of the various flow pattern regions depend on the mass velocity of the gas phase. These boundaries are represented by analytical equations developed by Coker [18]. These equations are used as the basis for determining the prevailing regime for any given flow rates and physical properties of the liquid and vapor. The mathematical models representing the boundaries of the flow regimes are:

 C1: ln By = 9.774459 – 0.6548 (ln Bx)

(3.54)



(3.55)

C2: ln By = 8.67694 – 0.1901 (ln Bx)

       C3: ln By = 11.3976 – 0.6084(ln Bx) + 0.0779 (ln Bx)2

(3.56)

        C4: ln By = 10.7448 – 1.6265 (ln Bx) + 0.2839 (ln Bx)2

(3.57)

 

(3.58)

C5: ln By = 14.569802 – 1.0173 (ln Bx)

C6: ln By = 7.8206 – 0.2189 (ln Bx)

(3.59)

Pressure Drop Various studies have been conducted in predicting the two-phase frictional pressure losses in pipes. The LockhartMartinelli correlations [19] shown in Figure 3.7 are employed. The basis of the correlations is that the two-phase pressure drop is equal to the single-phase pressure drop of either phase multiplied by a factor derived from the single-phase pressure drop of the two phases. The total pressure drop is based on the vapor-phase pressure drop based on the following assumptions:

Fluid Flow  143

10.000 6000 4000

YL

YG

Flow conditions Both liquid and gas turbulent Combination of turbulent and viscous Both liquid and gas viscous

2000 1000 600 400

Y

200 100 60 40 20 10 6 4 2 1 0.01 0.02

0.05

0.1

0.2

0.5

1 X

2

5

10

20

50

100

Figure 3.7  Lockhart-Martinelli pressure drop correlation. Source: Lockhart and Martinelli [19].

a. Th  e flow is isothermal and turbulent in both phases. b. The pressure loss is less than 10% of the absolute upstream pressure. The two-phase pressure drop can be expressed as:



∆PT PG = ⋅ YG 100ft 100ft

(3.60)

where YG is the two-phase flow modulus. YG is a function of the Lockhart-Martinelli two-phase flow modulus X, which is defined as:

 ∆PL  X=  ∆PL 

0.5

YG = f (x)



(3.61)

The value of YG for the different flow regimes is determined as follows For bubble or froth flow 2



 14.2X 0.75  YG =   (WL/A)0.1 



 27.315X 0.855  YG =   (WL/A)0.17 

(3.62)

For plug flow 2

(3.63)

144  Chemical Process Engineering For Stratified flow 2



 15 , 400X  YG =   (WL/A)0.8 

(3.64)

For wave flow, the Huntington friction factor [20] is used to determine the two-phase pressure loss:



HX =

 HL   µ L   WG   µ g 

(3.65)



ln(FH) = 0.2111ln(Hx) – 3.993

(3.66)



∆PT 0.000336(FH)(WG)2 = 100ft d 5ρG

(3.67)

For slug 2

 1190X 0.815  YG =   (WL/A)0.5 

(3.68)



YG = (aXb)2

(3.69)



a = 4.8 – 0.3125d



b = 0.343 – 0.021d



d = pipe inside diameter, inch



d = pipe inside diameter, inch



d = 10 for 12 inch and larger sizes

For annular flow

For dispersed or spray flow:



YG = [exp (C0 + C1(ln X) + C2(lnX)2 + C3(lnX)3)]2

(3.70)

C0 = 1.4695, C1 = 0.49138, C2 = 0.04887, C3 = −0.000349 Once the two-phase flow modulus (YG) for the particular flow regime has been calculated, Equation 3.60 can be used to calculate ΔPT/100ft, the two-phase, straight-pipe pressure drop. The pressure drop of liquid or gas flowing alone in a straight pipe can be expressed as:



∆P 0.000336f D Wx2 = , psi/100ft 100ft d 5ρG

(3.71)

Equation 3.71 can be modified for calculating the overall pressure drop of the two-phase flow for the total length of pipe plus fittings, L from Equation 3.5, based on the gas-phase pressure drop.

Fluid Flow  145



∆PToverall =

0.000336 ⋅ f D ⋅ Wx2 ⋅ YG ⋅ L 100d 5ρG

(3.72)

The velocity of the two fluid is



V=

0.0509  WG WL  + d 2  ρG ρL 

(3.73)

Erosion-Corrosion Depending on the flow regime, the liquid in a two-phase flow system can be accelerated to velocities approaching or exceeding the vapor velocity. In some cases, these velocities are higher than what would be desirable for piping. Such high velocities may lead to erosion-corrosion phenomena, where the corrosion rate of a material is accelerated by an erosive material or force (in this case, the force exerted by the high-velocity liquid). An index [20] based on velocity head can indicate whether erosion-corrosion may become significant at a given velocity. The index can be used to determine the range of mixture densities and velocities below which erosion-corrosion should not occur. The index is:



ρM U 2M ≤ 10 , 000

(3.74)

where the mixture density is:

ρM =

WL + WG  WL WG   ρ + ρ  L G

(3.75)

and the mixture velocity is:



WL   WG + U M = UG + UL =   3600ρG A 3600ρL A 

These properties can be calculated with the UniSim Design software simulator as explained in Chapter 2.

Nomenclature A = inside cross-sectional area of pipe, ft2 Bx, By = Baker parameters for determining for regime d = pipe inside diameter, inch D = pipe inside diameter, ft FC = Chen friction factor fD = Darcy friction factor = Fanning friction factor fF FH = Huntington friction factor Hx = Huntington flow factor ID = internal diameter of pipe, inch K = excess head loss for a fitting, velocity heads

(3.76)

146  Chemical Process Engineering K1 = K for fitting at NRe = 1 velocity heads = K for large fittings at NRe = ∞ velocity heads K∞ L = total fitting equivalent pipe length plus straight pipe length, ft = equivalent length of pipe due to fittings, Leq = straight pipe length, ft Lst = total length of pipe, ft LTotal MW = molecular weight = Reynolds number NRe = liquid Reynolds number NRe L = gas Reynolds number NRe G ΔPL/100ft = pressure drop of liquid if flowing alone in the pipe (psi/100 ft) ΔPG/100ft = pressure drop of gas if flowing alone in the pipe (psi/100 fi) ΔPT/100ft = pressure drop of the two-phase mixture in straight pipe (psi/100 ft) ΔPToverall = overall pressure drop of the two-phase mixture for the total length plus fittings, psi = liquid velocity, ft/s U L = gas velocity, ft/s U G = velocity of the two-phase mixture, ft/s UM V = velocity of fluid in a pipe, ft/s = liquid flow rate, lb/h W L = gas flow rate, lb/h WG = flow rate of either liquid or gas, lb/h Wx X = Lockhart-Martinelli two-phase flow modulus (APL/APG)0.5 YG = two-phase flow modulus ε = absolute roughness of pipe wall, ft (0.00015 ft for carbon steel) = liquid density, lb/ft3 ρL ρG = gas density, lb/ft3 ρM = density of liquid-gas mixture, lb/ft3 μx = viscosity of liquid or gas, cP = liquid viscosity, cP μL = gas viscosity, cP μ G = surface tension of liquid (dyne/cm) σL

VAPOR-LIQUID TWO-PHASE VERTICAL DOWNFLOW Understanding of two-phase flow is necessary for sound piping design since chemical process plants deals with two-phase flows. The two-phase vertical downflow presents its own problems in horizontal flow. In a vertical flow, large vapor bubbles or slugs are formed in the liquid stream. This flow regime is associated with pipe vibration and pressure pulsation. With bubbles greater than 1 inch in diameter and the liquid viscosity less than 100 cP, slug flow region can be represented by dimensionless numbers for liquid and vapor phases respectively (Froude numbers (NFr) L and (NFr)G). These numbers are:





(NFr )L =

VL  ρL  (gD)0.5  ρL − ρG 

0.5

(NFr )G =

VG  ρG  (gD)0.5  ρL − ρG 

0.5



(3.77)



(3.78)

The velocities VG and VL are superficial velocities based on the total pipe cross section. These Froude numbers exhibit several features in the range (NFr)L>0 and (NFr) G < 2.

Fluid Flow  147 Simpson [21] illustrates the values of (NFr)L and (NFr)G with water flowing at an increased rate from the top of an empty vertical pipe. As the flow rate further increases to the value (NFr) L= 2, the pipe floods and the total cross section is filled with water. If the pipe outlet is further submerged in water and the procedure is repeated, long bubbles will be trapped in the pipe below (NFr)L = 0.31. However, above (NFr)L = 0.31, the bubbles will be swept downward and out of the pipe. If large long bubbles are trapped in a pipe (d >1 inch) in vertically down flowing liquid having a viscosity less than 100 cP and the Froude number for liquid phase, (NFr)L < 0.3, the bubbles will rise. At higher Froude numbers, the bubbles will be swept downward and out of the pipe. A continuous supply of vapor causes the Froude number in the range 0.31 < (NFr)L < 1 to produce pressure pulsations and vibration. These anomalies are detrimental to the pipe and must be avoided. If the Froude number is greater than 1.0, the frictional force offsets the effect of gravity, and thus requires no pressure gradient in the vertical downflow liquid. This latter condition depends on the Reynolds number and pipe roughness [1].

The Equations The following equations will calculate Froude numbers for both the liquid and gas phases.

D=



Area =







d , ft 12

(3.79)

πD2 2 , ft 4

(3.80)

VL =

WL , ft/s 3600 × ρL × Area

(3.81)

VG =

WG , ft/s 3600 × ρG × Area

(3.82)

0.5

FRNL =

VL  ρL  (gD)0.5  ρL − ρG 

FRNG =

VG  ρG  (gD)0.5  ρL − ρG 

(3.83)

0.5

(3.84)

The Algorithm If FRNL < 0.31, vertical pipe is SELF-VENTING, else 0.3 < FRNL < 1.0, PULSE FLOW, and may result in pipe vibration, FRNL > 1.0, NO PRESSURE GRADIENT

Nomenclature Area = inside cross-sectional area of pipe, ft2 d = inside diameter of pipe, inch D = inside diameter of pipe, ft FRNL, (NFr)L = Froude number of liquid phase, dimensionless FRNG, (NFr)C = Froude number of vapor phase, dimensionless

148  Chemical Process Engineering g = gravitational constant, 32.2 ft/s2 VL = liquid velocity, ft/s VG = vapor velocity, ft/s WL = liquid flowrate, lb/h WG = vapor flowrate, lb/h ρL = liquid density, lb/ft3 ρG = vapor density, lb/ft3

LINE SIZES FOR FLASHING STEAM CONDENSATE When a liquid is flowing near its saturation point (i.e., the equilibrium or boiling point) in a pipe line, pressure decrease will cause vaporization. The higher the pressure difference, the greater the vaporization resulting in flashing of the liquid. Steam condensate lines cause a two-phase flow condition, with hot condensate flowing to a lower pressure through short and long lines. For small lengths with low pressure drops, and the outlet end being a few pounds per square inch of the inlet, the flash will be assumed as a small percentage. Consequently, the line can be sized as an all liquid line. However, caution must be exercised because 5% flashing can develop an important impact on the pressure drop of the system [22]. Sizing of flashing steam condensate return lines requires techniques that calculate pressure drop of two-phase flow correlations. Many correlations have been presented in the literature [14, 15, 19, 22]. Most flow patterns for steam condensate headers fall within the annular or dispersed region on the Baker map. Sometimes, they can fall within the slug flow region; however, the flashed steam in steam condensate lines is less than 30% by weight [1]. Ruskin [23] developed a method for calculating pressure drop of flashing condensate. The method employed here is based on a similar technique given by Ruskin. The pressure drop for flashing steam uses the average density of the resulting liquid-vapor mixture after flashing. In addition, the friction factor used is valid for complete turbulent flows in both commercial steel and wrought iron pipe. The pressure drop assumes that the vapor-­ liquid mixture throughout the condensate line is represented by mixture conditions near the end of the line. This assumption is valid because most condensate lines are sized for low pressure drop, with flashing occurring at the steam trap or valve close to the pipe entrance. If the condensate line is sized for a higher pressure drop, an iterative method must be used. For this case, the computations start at the end of the pipeline and proceed to the steam trap. Coker [3] provided a FORTRAN program to determine the followings: 1. t he amount of condensate flashed for any given condensate header from 15 to 140 psia. Initial steam pressure may vary from 40-165 psia. 2. the return condensate header temperature. 3. the pressure drop (psi/100 ft) of the steam condensate mixture in the return header. 4. the velocity of the steam condensate mixture and gives a warning message if the velocity is greater than 5000 ft/min, because this may present problems to the piping system.

The Equations The following equations are used to determine the pressure drop for flashed condensate mixture [24]



WFRFL = B(ln Pc)2 – A

(3.85)



A = 0.00671(ln Ph)2.27

(3.86)



B = ex · 10−4 + 0.0088

(3.87)

where

Fluid Flow  149 and



 16.919  X = 6.122 −   lnPh 

(3.88)



WG = WFRFL × W

(3.89)



WL = W – WG

(3.90)



TFL = 115.68(Ph)0.226

(3.91)



ρG = 0.0029Ph0.938

(3.92)



ρL = 60.827 − 0.078Ph + 0.00048Ph2 − 0.0000013Ph3

(3.93)

ρM =

WG + WL WG WL    ρ + ρ  G L

(3.94)

For turbulent flow

f=

0.25



(3.95)

0.000336f × W 2 d 5ρM

(3.96)

3.054  WG WL  + d 2  ρG ρL 

(3.97)

  0.000486    − log    d

2

where d=pipe diameter, inch Pressure drop



∆PT = V=

If V > 5000 ft/min, condensate may cause deterioration of the process pipeline.

Nomenclature d = internal pipe diameter, inch f = friction factor, dimensionless = steam condensate pressure before flashing, psia Pc = flashed condensate header pressure, psia Ph V = velocity of flashed condensate mixture, ft/min W = total flow of mixture in condensate header, lb/h WG = flashed steam flow rate, lb/h WL = flashed condensate liquid flow rate, lb/h WFRFL = weight fraction of condensate flashed to vapor TFL = temperature of flashed condensate, °F ΔPT = pressure drop of flashed condensate mixture, psi/100 ft

150  Chemical Process Engineering ρG = flashed steam density, lb/ft3 ρL = flashed condensate liquid density, lb/ft3 ρM = density of mixture (flashed condensate/steam), lb/ft3

FLOW THROUGH PACKED BEDS Flow of fluids through packed beds of granular particles occurs frequently in chemical processes. Examples are flow through a fixed-bed catalytic reactor, flow through a filter cake, and flow through an absorption or adsorption column. An understanding of flow through packed beds is also important in the study of sedimentation and fluidization. An essential factor that influences the design and operation of a catalytic or adsorption system is the energy loss (pressure drop). Factors determining the energy loss are many and investigators have made simplifying assumptions or analogies so that they could use some of the general equations. These equations represent the forces exerted by the fluids in motion (molecular, viscous, kinetic, static, etc.) to arrive at a useful expression correlating these factors. Ergun [25] developed a useful pressure drop equation caused by simultaneous kinetic and viscous energy losses and applicable to all types of flow. Ergun’s equation relates the pressure drop per unit of bed depth to dryer or reactor system characteristics, such as, velocity, fluid gravity, viscosity, particle size, shape, surface of the granular solids and void fraction. The original Ergun equation is:



 150(1 − ε)2   µV   1.75(1 − ε )   GV  ∆P gc =  ⋅  ⋅  2  +     D p  ε3 L ε3  Dp 

(3.98)

where ΔP = pressure drop   L = unit depth of packed bed  gc = dimensional constant 32.174(lbm/lbf )(ft/s2)   μ = viscosity of fluid, lb/ft.h   V = superficial fluid velocity, ft/s  Dp = effective particle diameter, ft   ε = void fraction of bed   ρ = fluid density, lb/ft3 Equation 3.98 gives the total energy loss in fixed beds as the sum of viscous energy loss (the first term on the right side of the equation) and the kinetic or turbulent energy loss (the second term on the right side of the equation). For gas systems, approximately 80% of the energy loss is dependent on turbulence and can be represented by the second term of Equation 3.98. In liquid systems, the viscous term is the major factor. Koekemoer and Luckos [26] modified the Ergun’s equation to include effect of material type and particle size distribution on pressure drop in packed beds of large particles as the following:



(1 − ε ) ρgU 2 (1 − ε)2 µU ∆P k ⋅ + = k1 2 L ε3 φd p ε 3 (φ d p )2

Where: ΔP = pressure drop across packed bed, Pa L = height of packed bed, m k1 and k2 = dimension less constant for various materials shown in Table 3.8 ε = voidage μ = dynamic viscosity, Pa.s U = superficial gas velocity, m/s ϕ = particle sphericity = particle diameter, m dp

(3.98a)

Fluid Flow  151 Table 3.8  Characteristic linear velocities of two-phase flow regimes [26]. Constant Bed materials

K1

K2

Coal

77.4

2.8

Char

160.4

2.8

Ash

229.7

2.3

dpSM = particle diameter, m ρg = gas density, kg/m3 In the case of beds with particle size distribution, dp in Equation 3.98a is replaced by the volume-surface mean diameter (Sauter mean diameter) [26]:

1

=

d pSM



∑ dx  i

pi

(3.98b)

The Equations The original Ergun equation of the total energy loss can be rearranged as follows:



BL (1 − ε ) G 2  150(2.419µ )(1 − ε )  ∆PT = × 3 × + 1.75   144 Dpρg c  D pG ε 

ρ c − ρb ρc

(3.100)

M WP 10.73ZT

(3.101)

6(1 − ε ) S

(3.102)

ε=



(3.99)

where ρc = density of catalyst, lb/ft3 ρb = density of the packed bed, lb/ft3

ρ=



DP =



S=



(1 − ε ) × Ap Vp

(3.103)

For cylindrical particles





1  2 π PD ⋅ ⋅ PL  1728  4 

(3.104)

1 π 2 PD + π ⋅ PD ⋅ PL   144  2 

(3.105)

Vp =

Ap =

152  Chemical Process Engineering Reynolds number



NRe =

GD p 2.419 × µ(1 − ε)

(3.106)

Friction factor For laminar flow with NRe10% of the upstream pressure or compressible fluids at high velocities. The program calculates the maximum fluid flow rate for a given pipe size, length of pipe, and the fluid’s physical properties. Figure 3.12 shows the input data and computer results for the maximum flow rate of natural gas in the 0.614-inch pipe.

Fluid Flow  157

Figure 3.12  Snapshot of the Excel spreadsheet calculations.

The computed maximum flow rate of natural gas through the 0.614-inch pipe is 8,397 lb/hr. The Mach number at the pipe inlet is 0.21. At critical condition, the Mach number is 0.89. The critical pressure of the compressible fluid is 242.3 psia, with a sonic velocity of 1,346.7 ft/sec. The compressible fluid flow pattern through the pipe is SUBSONIC. The results obtained here are identical to those reported by Coker [3] written using the FORTRAN language since the same equations are used in both cases. It is worth mentioning that with features provided in Excel spreadsheet, the programming in much easier than that prepared by the FORTRAN language that requires generating the programing codes.

Example 3.5 What is the overall pressure drop for natural gas flowing at 27,000 lb/hr through a 6-inch Schedule 40 tail pipe to a header during relieving condition as shown in Figure 3.3? Relief valve set pressure, P1 = 400 psig, during relieving condition is:

P1 = P1 + 10% accumulation + 14.7 = 454.7 psia The data are: • • • • • • • • • • •

pipe internal diameter, d, inch = 6.065 tail pipe length, ft = 10 gas flow rate, lb/hr = 27,000 gas viscosity, g, cP = 0.012 gas compressibility factor, Z= 0 .9 gas temperature, °F = 100 molecular weight of gas, Mw = 19.5 ratio of specific heat capacities k = 1.27 gas inlet pressure, psia = 454.7 pipe roughness, ε = 0.00015 ft resistance coefficient due to fittings and valves, Kf = 2.013

158  Chemical Process Engineering

Solution The Excel spreadsheet program, Example 3.5.xlsx, determines the overall pressure drop for the 6-inch tail pipe having a relieving rate of 27,000 lb/hr. Figure 3.13 illustrates both the input data and computer output. The Mach number at inlet condition is 0.017, and at the critical condition is 0.887. The critical pressure is 7.985 psia and overall pressure drop is 0.213 psi. The compressible fluid flow pattern through the pipe is SUBSONIC.

Example 3.6 Calculate the pressure drop of a 5-mile length (26,400 ft) in a 6-inch (Schedule 40, ID = 6.065 inch) pipe, for a twophase flow at a temperature of 110°F with the following physical properties: Parameter

Liquid

Gas

Flow rate, W, lb/hr

77,956

12,434

Density, ρ, lb/ft3

66.7

2.98

Viscosity, μ, cP

1

0.02

Surface tension, σ, dyne/com

70

Solution The Excel spreadsheet program, Example 3.6.xlsx, calculates the flow regime and the pressure drops for liquid and vapor phases. From the two-phase flow modulus, the overall pressure drop for the 6-inch pipe is calculated. Figure 3.14

Figure 3.13  Snapshot of the Excel spreadsheet calculations.

Fluid Flow  159

Figure 3.14  Snapshot of the Excel spreadsheet calculations.

gives the input data and computer results of the two-phase flow. The Lockhart-Martinelli two-phase flow modulus is 1.487. The Baker parameter in the liquid phase is 40.77, and the Baker parameter in the gas phase is 9,495. These results indicate that the flow regime is ANNULAR (and can be observed from the Baker’s plot, Figure 3.5). The two-phase modulus is 10.017 and the pressure drop of the two-phase mixture per 100 ft of pipe is 0.37 psi/100ft. The overall pressure drop of the two-phase mixture is 97.78 psi. In addition, the pipe is unlikely to encounter any corrosion-erosion because the computed index of 926 is less than 10,000. The results are identical to those calculated by Coker [3] using the FORTRAN language program.

Example 3.7 Calculate the Froude numbers and flow conditions for the 2, 4, and 6-inch (Schedule 40) vertical pipes having the following liquid and vapor flow rates and densities. Parameter

Liquid

Gas

Flow rate, W, lb/hr

6,930

1,444

Density, ρ, lb/ft3

61.8

0.135

Solution The Excel spreadsheet program, Example 3.7.xlsx, calculates the Froude numbers for the liquid and vapor phases. In addition, it determines whether the pipe is self-venting or whether pulsation flow is encountered. Figure 3.15 gives a typical input data and the Excel spreadsheet results for the three pipe sizes. The results are identical to those calculated by Coker [3] using the FORTRAN language.

Example 3.8 Determine the pressure drop for the 4, 6, and 8-inch (Schedule 40) condensate headers for the following conditions: Parameter

Value

Flow rate, W, lb/hr

10,000

Steam condensate pressure, psia

114.7

Header pressure, psia

14.7

160  Chemical Process Engineering

Figure 3.15  Snapshot of the Excel spreadsheet calculations.

Solution The Excel spreadsheet program, Example 3.8.xlsx, evaluates the pressure drop of any given condensate header. The program also determines whether the velocity of the flashed condensate mixture would cause deterioration in the header line. Figure 3.16 shows a typical input data and the Excel spreadsheet results for the 4-inch (Schedule 40) pipe. The computed results show that the 4-inch pipe gives the velocity of the flashed condensate mixture to be 7,055 ft/min. This indicates a possible deterioration in the pipe. For the 6- and 8-inch pipes, the velocities are 3,109 ft/min, and 1,795 ft/min respectively, indicating that the condensate pipe lines will not deteriorate.

Example 3.9 Calculate the pressure drop in a 60 ft length of 1.5-inch (Schedule 40, ID = 1.610 inch) pipe packed with catalyst pellets of 1/4 inch in diameter when 104.4 lb/h of gas is passing through the bed. The temperature is constant along the length of pipe at 260°C. The void fraction is 45% and the properties of the gas are like those of air at this temperature. The entering pressure is 10 atm [30]. How do the pressure drops vary with void fraction and pellets length?

Figure 3.16  Snapshot of the Excel spreadsheet calculations.

Fluid Flow  161

Solution-Excel The Excel spreadsheet program, Example 3.9.xlsx, calculates the pressure drop in a 1.610-inch ID packed bed for varying lengths of catalyst pellets. For air at 260°F and 10 atm, the density, ρ, = 0.413 lb/ft3 and viscosity, μ = 0.0278 cP. Figure 3.17 gives the Reynolds number, friction factor, and pressure drop of catalyst pellets of 0.25 inch. The simulation gives a pressure drop of 68.27 psi. By increasing the void fraction, the pressure drop decreases significantly as the amount of solid decreases in the bed.

Solution-UniSim Design Software The UniSim design simulation program, Example 3.9.usc, evaluates the total pressure drop in the packed bed with void fraction = 0.65 using the “plug flow” reactor as described in the user manuals [28]. This unit is used to simulate a wide variety of catalytic reactions with the Ergun’s correlations for the pressure drop calculations. The detailed steps on the simulation of plug flow reactors are provided in Chapter 11 and here only a brief description is presented. The following steps were taken to define the problem in the UniSim Design software simulator: Basis Environment: a. A  ir is defined as the main component. Since, the calculation of pressure drop in plug flow reactors needs a reaction. Three more components, namely, N2, O2 and NO are also added to the component list in order to define a reaction. b. The Peng-Robinson equation of state is used in the property package. c. The catalytic reaction is defined in a way that no reaction happens in the bed; through setting the values of K and E to zero. Simulation Environment In this environment, the inlet, outlet and energy streams, and “plug flow” reactor are defined as sketched below as described in Chapter 11. More details on process simulation using the UniSim Design software simulator were provided in Chapter 1 (Figure 3.18). The following sizing data, shown in Figure 3.19, on dimension, packing and catalyst are used to simulate the reactor: The total pressure drops calculated by the UniSim Design software simulator is:

Figure 3.17  Snapshot of the Excel spreadsheet calculations.

162  Chemical Process Engineering

PRF-100 In

Out

Figure 3.18  PFR Reactor in UniSim Design software (Courtesy of Honeywell UniSim Design software. Honeywell ® and UniSim ® are registered trademarks of Honeywell International Inc.).

Figure 3.19  Sizing properties of PFR Reactor in UniSim Design software (Courtesy of Honeywell UniSim Design software, Honeywell ® and UniSim ® are registered trademarks of Honeywell International Inc.).

Parameter

Excel spreadsheet

UniSim Design software

Total pressure drops, psi

14.31

15.11

The value obtained by the UniSim Design software simulator is very close to that calculated by the Excel spreadsheet. More pressure drop calculations are presented in Chapter 11.

References 1. A. K. Coker, Ludwig’s Applied process design for chemical and petrochemical plants, 4th ed., vol. 1, Gulf Professional Publishing, 2007. 2. W. B. Hooper, “The two-K Method Predicts Head Loss in Pipe Fittings,” Chem Eng., pp. 96-100, 1981. 3. A. K. Coker, Fortran Programs for Chemical Process Design, Analysis, and Simulation, Gulf Professional Publishing, 1995. 4. Crane Co., Engineering Div.,, “Flow of Fluids Through Valves, Fittings, and Pipe,” Crane Co., 1998. 5. W. B. Hooper, “Calculate Head Loss Caused by Change in Pipe Size,” Chem. Eng., vol. 7, no. 89-92, 1988. 6. R. Darby, Fluid Mechanics for Chemical Engineers, vol. 2, New York: Marcel Dekker, 2001. 7. G. A. Gregory and M. Fogarasi, “Alternate to Standard Friction Factor Equation,” Oil & Gas J., pp. 120-127, 1985. 8. M. B. Powley, “Flow of Compressible Fluids,” Can J. Chem. Eng., pp. 241-245, 1958.

Fluid Flow  163 9. API, “Recommended Practice for the Design and Installation of Pressure Relieving Systems in Refineries,” API, 1976. 10. A. K. Coker, “Determine Process Pipe Sizes,” CEP, vol. 3, 1991. 11. H. Y. Mak, “New Method Speeds Pressure Relief Manifold Design,” Oil & Gas J., vol. 11, pp. 166-172, 1978. 12. E. Janett, “How to Calculate Back pressure in Ventlines,” Chem. Eng., vol. 9, pp. 83-86, 1963. 13. D. M. Kirkpatrick, “Simpler Sizing of Gas Piping,” Hydrocarbon Process, vol. 12, pp. 135-138., 1969. 14. O. Baker, “Simulataneous Flow of Oil and Gas,” Oil & Gas J., vol. 7, pp. 185-189, 1954. 15. J. M. Chenoweth and M. W. Martin, “Turbulent Two-Phase Flow,” Petroleum Refiner, vol. 34, no. 10, pp. 151-155, 1955. 16. J. M. Mandhane, G. A. Gregory and K. Aziz, “A Flow Pattern Map for Gas-Liquid Flow in Horizontal Pipes,” Inst. J. Multiphase Flow, vol. 1, pp. 537-553, 1974. 17. L. S. Tong and Y. S. Tang, Boiling heat transfer and two-phase flow, Routledge: Taylor and Francis, 2018, pp. 119-245. 18. A. K. Coker, “Understand Two-Phase Flow in Process Piping,” CEP, pp. 60-65, 1990. 19. R. W. Lockhart and R. C. Martinelli, “Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes,” CEP, vol. 45, no. 1, pp. 39-48, 1949. 20. J. e. a. Coulson, Chem. Eng., 3 ed., New York: Pergamon Press, 1978, pp. 91-92. 21. L. L. Simpson, “Sizing Piping for Process Plants,” Chem. Eng., vol. 6, pp. 192-214, 1968. 22. A. E. Duckler, M. Wicks and R. C. Cleveland, “Frictional Pressure Drop in Two-Phase Flow” An Approach Through Similarity Analysis,” AIChE, vol. 10, pp. 44-51, 1964. 23. R. E. Ruskin, “Calculating Line Sizes for Flashing Steam Condensate,” Chem. Eng., vol. 8, pp. 101-103, 1975. 24. W. W. Blackwell, Chemical Process Design on a Programmable Calculator, New York: McGraw-Hill, p. 22, 1984. 25. S. Ergun, “Fluid Flow Through Packed Columns,” Chem. Eng. Prog., vol. 48, no. 2, pp. 89-92, 1952. 26. A. Koekemoer and A. Luckos, “Effect of material type and particle size distribution on pressure drop in packed beds of large particles: Extending the Ergun equation,” Fuel , vol. 158, pp. 232-238, 2015. 27. C. O. Bennett and J. E. Myers, Momentum, Heat and Mass Transfer, New York: McGraw-Hill, 1982. 28. Honeywell’s UniSim®, “UniSim – Software for Process Design and Simulation,” [Online]. Available: https://www.honey​ wellprocess.com/en-US/explore/products/advanced-applications/unisim/Pages/default.aspx. [Accessed September 2020]. 29. MAURER ENGINEERING INC., “MULTIPHASE FLOW PRODUCTION MODEL,” 1994. [Online]. Available: https:// www.bsee.gov/sites/bsee.gov/files/tap-technical-assessment-program//300ae.pdf. [Accessed September 2020]. 30. H. S. Fogler, Elements of Chemical Reaction Engineering,, Fifth ed., New Jersey: Prentice-Hall Inc., p. 175, 2016.

4 Equipment Sizing INTRODUCTION The process engineer is often required to design separators or knockout drums for removing liquids from gas streams. This chapter reviews the design of horizontal and vertical separators and the sizing of partly filled vessels. The chapter ends by reviewing the sizing of a cyclone and a solid-desiccant gas dryer. Vessels used for processing in the chemical process industries (CPI) are principally with or without internals. Empty separators are drums that provide intermediate storage or surge of a process stream for a limited or extended period. Alternatively, they provide phase separation by settling. The second category comprises equipment such as mixer, heat exchangers, reactors and separation columns. In some cases, it is important to separate liquid and gas flowing simultaneously through a pipe since the conditions of the flowing mixture and the efficiency of separation may vary widely. Therefore, a separator for such duty must be adequate. In addition, there are constraints due to space or weight that often affect the choice of separators, the need to handle solids or effects on three-phase separation, and the requirements for liquid hold-up. In practice, most separation problems are solved by the following types of separating equipment. Knockout or Surge Drums (Figures 4.1 and 4.2). A knockout drum is suitable for a bulk separation of gas and liquid, particularly when the liquid volume fraction is high with stratified or plug flow in the pipe. Also, it is useful when vessel internals are required to be kept to a minimum, e.g., in relief systems or in fouling service. It is unsuitable if a mist is being separated or if high separating efficiency is required. Cyclones (Figure 4.3). These are robust and not susceptible to fouling or wax. Multicyclones are compact and reasonably efficient for foam, but not for slugs. The efficiency of a cyclone decreases with increasing diameter, as Table 4.1 and Figure 4.3, which are some exprimental works, show, and cyclones are not applicable above 5 ft (1.5 m) in diameter [1]. They possess a good efficiency for smaller flow rates. However, cyclones are expensive to operate, especially under vacuum conditions, because they generate a higher pressure drop than either a knockout drum or a demister separator.

Note: T L = Tangent line

TL

TL

L (>2.5 D) d1

d2

D

Vortex breaker

d3

Figure 4.1  Horizontal knockout drum. A. Kayode Coker and Rahmat Sotudeh-Gharebagh. Chemical Process Engineering: Design, Analysis, Simulation and Integration, and Problem-Solving With Microsoft Excel – UniSim Design Software, Volume 1, (165–194) © 2022 Scrivener Publishing LLC

165

166  Chemical Process Engineering

A-A d2

Tangent line D 0.9 D (min 0.9 m)

d1 0.3 D (min 0.3 m)

LZA (HH)

Vortex breaker d3

Figure 4.2  Vertical knockout drum.

Demister Separators. A demister separator is fitted either with a vane demister package or with a wire mesh demister mat. The latter type is much preferred, although it is unsuitable for fouling service. The wire mesh demister is a widely applied type of separator, and is adequate for all gas-liquid flow regimes over a wide range of gas flow rates. A knockout drum or demister separator may be vertical or horizontal. A vertical vessel is generally preferred because its efficiency does not vary with the liquid level. Alternatively, a horizontal vessel is chosen when it offers a clear size advantage, if the headroom is restricted or if a three-phase separation is required. Knockout drums and cyclones are recommended for waxy and coking feeds. Demister mats are not suitable because of the danger of plugging. Vane demister packages are used as alternatives, but provision should be made for cleaning [2].

SIZING OF VERTICAL AND HORIZONTAL SEPARATORS Vertical Separators Vertical liquid-vapor separators are used to disengage a liquid from a vapor when the volume of liquid is small compared with that of vapor. The maximum allowable vapor velocity in a vertical separator to reduce the liquid carryover depends on the followings:

Equipment Sizing  167 Dc De

b

S a

h

H

B

Figure 4.3  A cyclone: design configurations.

Table 4.1  Design criteria for Reflux Distillate Accumulators. Minutes Instrument Factor F1

Labor Factor F2

Operation

w/Alarm

w/o Alarm

Good

Fair

Poor

FRC

1 2

1

1

1.5

2

LRC

1

1

1.5

2

1

1.5

2

TRC

1

1 1 2

2

1 2

• liquid and vapor densities. • a constant K based on surface tension, droplet size, and physical characteristics of the system. The proportionality constant, K, is 0.35 for oil and gas systems with at least 10 in. (254 mm) disengaging height between the mist eliminator bottom and gas liquid interface [1]. For vertical vessels, K can vary between 0.1-0.35, if mist eliminators (demisters) are used to enhance disentrainment. The value of K also depends on the operating pressure of the vessel. At pressures above 3 psig (0.21 barg), the K value decreases with pressure with an approximate value of 0.3 at 250 psig (17.2 barg) and 0.275 at 800 psig (55.2 barg). Watkins [3] has developed a correlation between the separation factor and K. Figure 4.4 illustrates Watkin’s vapor velocity factor chart, based on 5% of the liquid being entrained with the vapor. Backwell [4] has developed a polynomial equation using Watkin’s data to calculate the K value for a range of separation factors between 0.006 and 5.0. Watkins proposed a method for sizing reflux drums based upon several factors, as illustrated in the following Tables. Table 4.1 gives the recommended design surge times. Table 4.2 gives the multiplying factors for various operator efficiencies. The operating factor is based upon the external unit and its operation, its instrumentation and response to control, the efficiency of labor, and chronic mechanic problems, and the possibility of short- or long-term interruptions [3]. The multiplying factors F1 and F2 represent the instrument and the labor factors.

168  Chemical Process Engineering 0.6 0.4 0.2 KV

0.1 0.08 0.06 0.04 0.02 0.006 8 0.01

2

4

6 8 0.1 2 4 (WL/WV) (ρv/ρL)0.5

6 8 1.0

2

4

6

Figure 4.4  Design vapor velocity factor for vertical vapor-liquid separators at 85% of flooding.

Table 4.2  Operator factors for external units. Operating characteristics

Factor F3

Under good control

2

Under fair control

3

Under poor control

4

Feed to or from storage

1.25 Factor F4

Board mounted level recorder

1.0

Level indicator on board

1.5

Gauge glass at equipment only

2.0

A multiplying factor, F3, is applied to the net overhead product going downstream. F4 depends on the kind and location of level indicators. It is recommended that 36 inches plus one-half the feed nozzle, OD, (48 inches minimum) be left above the feed nozzle for vapor [1]. Below the feed nozzle, allowance of 12 inches plus one-half the feed nozzle, OD, is required for clearance between the maximum liquid level and the feed nozzle (minimum of 18 inches). At some value between L/D ratios of 3 and 5, a minimum vessel weight will occur resulting in minimum costs for the separator [1]. Figure 4.5 shows the dimensions of a vertical separator.

Calculation Method for a Vertical Drum The following steps are used to size a vertical drum [2]. Step 1. Calculate the vapor-liquid separation factor



0.5 S.Fac = ( WL WV )( ρv ρL )

(4.1)

Step 2. From Blackwell’s correlation, determine the design vapor velocity factor, Kv, and the maximum design vapor velocity.

Equipment Sizing  169

36" + 1/2 (FEED NOZZLE O.D.) HV

48" MIN. FEED NOZZLE

L

12" + 1/2 (FEED NOZZLE O.D.) 18" MIN.

MAX. LEVEL

HL D

Figure 4.5  Dimesion of a vertical separator.



X = Ln(S.Fac)

(4.2)

KV = exp (B + DX + EX2 + FX3 + GX4)

(4.3)

where B = -1.877478, D = -0.814580, E = -0.187074, F = -0.014523, G = -0.001015

 ρ −ρ  Vmax = k v  L V   ρV 



0.5



(4.4)

Step 3. Calculate the minimum vessel cross-sectional area.

 WV  [ft 3 /s] QV =    3600ρV 



AV =



QV [ft 2 ] VMax

(4.5)

(4.6)

Step 4. Set a vessel diameter based on 6-inch increments and calculate cross-sectional area.

Dmin =



 4A v   π 

0.5

[ft]

(4.7)

D= Dmin to next largest 6 inch



Area =

πD2min 4

Step 5. Estimate the vapor-liquid inlet nozzle based on the following velocity criteria.

(4.8)

170  Chemical Process Engineering



(U max )nozzle = (U min )nozzle =

100  ft  ρmix 0.5  s  60 ρmix

0.5

 ft   s



(4.9)

(4.10)

where



WL + WV   lb  ρmix =   WL WV    ft 3  +   ρ ρG    L 

(4.11)

Step 6. From Figure 4.5, make a preliminary vessel sizing for the height above the center line of a feed nozzle to top seam. Use 36 inches + 1/2 feed nozzle, OD, or 48 inches minimum. Use 12 inches + 1/2 feed nozzle, OD, or 18 inches minimum to determine the distance below the center line of the feed nozzle to the maximum liquid level. Step 7. From Table 4.1 or 4.2, select the appropriate full surge volume in seconds. Calculate the required vessel volume.



QL =

WL ft 3 3600ρL s

(4.12)

V = (QL)(design time to fill) = 60QLT

(4.13)

The liquid height is:



HL =

4V [ft] πD2

(4.14)

 HL + H V  must be between 3 and 5. For small volumes of liquid, it may be necessary to  D  provide more liquid surge than is necessary to satisfy the L/D > 3. If the required liquid surge volume is greater than that obtained in a vessel having L/D < 5, a horizontal drum must be provided.

Step 8. Check geometry.

Calculation Method for a Horizontal Drum Horizontal vessels are used for substantial vapor-liquid separation where the liquid holdup space must be large. Maximum vapor velocity and minimum vapor space are determined as in the vertical drum, except that KH for horizontal separators is generally set at 1.25KV. The following steps are carried out in sizing horizontal separators. Step 1. Calculate the vapor-liquid separation factor by Equation 4.l, and Kv by Equations 4.2 and 4.3. Step 2. For horizontal vessels

KH = 1.25 KV

(4.15)

Step 3. Calculate the maximum design vapor velocity.



 ρ −ρ  (U V )max = K H  L V   ρV 

0.5

[ft/s]

(4.16)

Equipment Sizing  171 Step 4. Calculate the required vaporflow area.



(A v )min =

QV ft 2 (U V )max

(4.17)

Step 5. Select appropriate design surge time from Table 4.1 or Table 4.2, and calculate full liquid volume by Equations 4.12 and 4.13. The remainder of the sizing procedure is carried out by trial and error as follows: Step 6. When the vessel is full, the separator vapor area can be assumed to occupy only 15 to 25% of the total cross-sectional area. Here, a value of 20% is used and the total cross-sectional area is expressed as:





(A total )min =

(A V )min 0.2

 4(A total )min  Dmin =   π

0.5

(4.18)



(4.19)

Step 7. Assume the length-to-diameter ratio of 3 (i.e, L/D= 3). Calculate the vessel length.



L = 3Dmin

(4.20)

Step 8. Because the vapor area is assumed to occupy 20% of the total cross-sectional area, the liquid area will occupy 80% of the total cross-sectional area.

AL = 0.8(Atotal)min, ft2

(4.21)

Step 9. Calculate the vessel volume.

VVES = (Atotal)min L, ft3

(4.22)

Step 10. Calculate liquid surge time.



T=

60A L LρL , min WL

(4.23)

Liquid Holdup and Vapor Space Disengagement The dimensions of both vertical and horizontal separators are based on rules designed to provide adequate liquid holdup and vapor disengaging space. For instance, the desired vapor space in a vertical separator is at least 1.5 times the diameter, with 6 inches as the minimum above the top of the inlet nozzle. In addition, a 6-inch minimum is required between the maximum liquid level and the bottom of the inlet nozzle. For a horizontal separator, the minimum vapor space is equal to 20% of the diameter or 12 inches.

Wire Mesh Pad Pads of fine wire mesh induce coalescence of impinging droplets into larger ones, which then separate freely from the gas phase. No standard equations have been developed for the pressure drop across wire mesh because there are no standardized mesh pads. However, as a rule of thumb, the pressure drop of a wire mesh is ΔP=1 in H2O. Every manufacturer makes a standard high efficiency, very high efficiency, or high-throughput mesh under various trade names, each for a specific requirement.

172  Chemical Process Engineering

Standards for Horizontal Separators The following specifications are generally accepted in the design of horizontal separators [5]: 1. Th  e maximum liquid level shall provide a minimum vapor space height of 15 in, but should not be below the center line of the separator. 2. The volume of dished heads is not considered in vessel sizing calculations. 3. The inlet and outlet nozzles shall be located as closely as practical to the vessel tangent lines. Liquid outlets shall have anti-vortex baffles.

Piping Requirements Pipes that are connected to and from the process vessels must not interfere with the good working of the vessels. Therefore, the following guidelines should be considered: • There should be no valves, pipe expansions or contractions within 10-pipe diameters of the inlet nozzle. • There should be no bends within 10-pipe diameters of the inlet nozzle except: • for knockout drums and demisters, a bend in the feed pipe is permitted if this is in a vertical plane through the axis of the feed nozzle. • for cyclones, a bend in the feed pipe is allowed if this is in a horizontal plane and the curvature is in the same direction as the cyclone vortex. • A pipe reducer may be used in the vapor line leading from the separator, but it should be no nearer to the top of the vessel than twice the outlet pipe diameter. • A gate or ball type valve that is fully opened in normal operation should be used, where a valve in the feed line near the separator cannot be avoided. • High pressure drops that cause flashing and atomization should be avoided in the feed pipe. • If a pressure reducing valve in the feed pipe cannot be avoided, it should be located as far upstream of the vessel as practicable. Coker [6] has provided detailed reviews and sizing of two-phase and three-phase flows in horizontal and vertical separtors.

Nomenclature A = cross-sectional area of vertical separator, ft2 AL = cross-sectional area of horizontal vessel occupied by liquid phase, ft2 AT = total cross-sectional area of horizontal vessel, ft2 AV = minimum cross-sectional area of vertical separator, ft2 D = vessel diameter, ft HL = liquid height, ft HV = vapor height, ft K = proportionality constant KH = vapor velocity factor for horizontal separator KV = vapor velocity factor for vertical separator L = horizontal vessel level, ft S.Fac = separation factor QL = liquid volumetric flow, ft3/s QV = vapor volumetric flow, ft3/s T = liquid surge time, min. Vmax = maximum vapor velocity, ft/s

Equipment Sizing  173 Vves WL WV ρL ρV

= horizontal vessel volume, ft3 = liquid flow rate, lb/h = vapor flow rate, lb/h = liquid density, lb/ft3 = vapor density, lb/ft3

SIZING OF PARTLY FILLED VESSELS AND TANKS Cylindrical vessels and horizontal tanks are used for the storage of fluids in the chemical process industries. Various level instruments are employed to determine the liquid level in these vessels. The exact liquid volume can be obtained either by calibration of the vessels or by tedious calculations. Partial volumes for horizontal, cylinders with flat, dished, elliptical, and hemispherical ends, and for vertical cylinders are employed for storing process fluids. Kowal [7] presented charts that allow designers to determine the volume of vessels for different dimensions. In addition, parameters for tray designs and tray layout in distillation columns, such as chord length and circle area, are computed in this chapter.

The Equations Calculate the area of the segment of a circle from Figure 4.6:

R −H R

(4.24)

 R − H  R 

(4.25)

Cos(α ) =



α = Cos −1

Area of the triangles:



Area of Δ = (R – H)(2HR – H2)0.5

(4.26)

Therefore, area of the segment of a circle from Figure 4.6 is:



A s = R 2Cos −1

 R − H − (R − H)(2HR − H 2 )0.5  R 

α

R–H

R

H

Figure 4.6  Area of the segment.

(4.27)

174  Chemical Process Engineering Figure 4.7 is a nomograph based upon the area of the segment of a circle [8]. It converts the height (rise) over diameter ratios directly to percent of area. The procedure in using the nomograph is: 1. S tart at the left-hand side of the nomograph with the trial diameter. 2. Align the internal diameter with the height of the vapor space and then with the rise of the minimum liquid level. Read the area occupied by these segments. 3. Subtract the sum of these segments from 100%. The difference is the area available for liquid holding time. Chord length of segmental area, CL:

CL = 2(DH – H2)0.5

(4.28)

Partial volume of a horizontal cylinder:

VHC = AS.L

(4.29)

1 0.04

12

9 8 7 6

5

3 4 5

0.2 20 0.3 30

90 SEGMENTAL AREA

10

10

70

4

9

0.1

7

40 50

100

3

8

0.08

0.4 0.6 0.8 1

20 RISE, INCHES

SEGMENTAL AREA, PERCENT OF DRUM AREA

10

7

0.06

RISE/DIAMETER RATION

11

D, DRUM INSIDE DIAMETER, FEET

6

2

30

40 50

+ 2

DIAMETER

RISE

60 70 80 90 100

1'–6"

Figure 4.7  Nomograph to find segmental area for liquid holding time.

150

Equipment Sizing  175 Partial volume of dished heads:

VDH = 0.215483H2(1.5D − H)

(4.30)

Partial volume of elliptical heads:

VELL = 0.5236H2(1.5D − H)

(4.31)

Partial volume of hemispherical heads:

VHS = 2VELL

(4.32)

The total volume VT:

VT = VHC + VH

(4.33)

where VH=head volume

Area =



πD2 4

(4.34)

Volume of a vertical tank:

VVT = 0.25πD2H Conversions: ft3 = in3/1728 gal = (in3)(0.004329) bbl = (in3)(0.00010307)

Nomenclature Area = circle area, in2 AS = segmental area of a circle, in2 CL = chord length of segmental area, inch bbl = volume in barrels d = vessel tank, or circle diameter, inch H = chord or liquid height, inch L = vessel length, inch R = radius of vessel or tank, inch VDH = volume of dished head, in3 VELL = volume of elliptical heads, in3 VH = head volumes, in3 VHC = volume of horizontal cylinder, in3 VHS = volume of hemispherical heads, in3 VT = total volume of horizontal vessel, in3 VVT = volume of vertical tank, in3

(4.35)

176  Chemical Process Engineering

PRELIMINARY VESSEL DESIGN Columns and towers are the most essential elements of the refineries in the chemical process industry (CPI). They have three separate functions: distillation, fractionation, and chemical operations. Stills are cylindrical chambers in which the application of heat to the charge stock changes from a liquid to a vapor. The vapor is then condensed in another vessel. Columns and towers are stills that increase the degree of separation obtained during the distillation of crude oil. Fractionation towers are used for light end products. Generally, towers are large cylindrical vessels that have plates (trays spaced inside the shell) to promote and enhance mixing of the downward flow of liquid with the upward flow of the vapors. These vessels require self-supporting, especially when they are exposed to high winds or to seismic risk. Process conditions in the CPI often change, and a tower that is initially designed for vacuum operation may at a later stage be employed at a pressure greater than atmospheric. Therefore, thin-walled vessels are often rated for both pressure and vacuum conditions. If the vessel is thick-walled and designed for high-pressure service, no vacuum rating will be required. Tall vessels have one of a number of closures such as hemispherical, ellipsoidal, torispherical, conical, or flat ends. The ellipsoidal shape is usual for closures of vessels that are 6ft (1.8 m) in diameter or greater. The hemispherical shape is preferred for vessels of a lesser diameter. Most pressure and vacuum vessels are closed by convex torispherical, ellipsoidal, or hemispherical heads. The geometry for torispherical head is dependent by two different meridional curvatures. Hence, such domes are called two-arc vessel heads. On the other hand, ellipsoidal and hemispherical domes are called one-arc heads. The basic relationships for thin cylindrical shells under internal pressure assume that circumferential stress is dependent on the pressure and vessel diameter, but independent of the shell thickness. That is:



f=

PD 2t

(4.36)

t=

PD 2f

(4.37)

or



Equation 4.36 is for membrane shells with a negligible thickness. As the pressure and the shell thickness increase, the stress distribution across the thickness is non-uniform. Therefore, some correction to the membrane theory is required. The modified equation as given by the ASME Unfired Pressure Code, Section VIII, Division 1 is:



t=

PR +c S.E − 0.6P

(4.38)

PR 2S.E − 0.2P

(4.39)

PD +c 2S.E − 0.2P

(4.40)

where c is the corrosion allowance. For a hemispherical head:



t HH =

For ellipsoidal dished head:



t ELL =

Equipment Sizing  177 For torispherical head:

0.885PR C +c S.E − 0.1P

(4.41)

π  t (OD + t) ⋅ SL ρ  12  s 12

(4.42)

OD ⋅ (H ⋅ Fac) + 2SF 12

(4.43)

 t ρ  12  s

(4.44)

t TOR =

The weight of the vessel shell is:

WTS =

The head blank diameter is:

BD =

Weight of head:

WTH = 0.5π(BD)2

The total weight:

WTOT = WTS + WTH

(4.45)

The total weight excludes the nozzles, attachments, and vessel internals. Table 4.3 shows the head blank diameter factor for different types of head.

Nomenclature BD = head blank diameter, inch c = corrosion allowance, inch D = inside diameter of shell or head, inch E = joint efficiency H.Fac = head blank diameter factor

Table 4.3  Head blank diameter factor (H.Fac). Head type

OD/t

H.Fac

Torispherical

>50

1.09

30–50

1.11

20–30

1.15

>20

1.24

10–20

1.30

>30

1.60

18–30

1.65

10–18

1.70

Ellipsoidal

Hemispherical

178  Chemical Process Engineering OD = outside diameter of head, inch P = internal design pressure, psig R = inside radius, inch Rc = crown radius of torispherical head, inch S = allowable stress, lb/in2 SF = straight flange length, inch SL = shell length, ft t = shell thickness, ft tELL = ellipsoidal head thickness, inch tHH = hemispherical head thickness, inch tTOR = torispherical head thickness, inch WTH = weight of head, lb WTS = weight of vessel shell, lb WTOT = total weight of vessel, lb = density of vessel material, lb/ft3 ρS

CYCLONE DESIGN Introduction Cyclones are widely used for the separation and recovery of industrial dusts from air or process gases. Pollution and emission regulations have compelled designers to study the efficiency of cyclones. Cyclones are the principal type of gas-solids separators using centrifugal force. They are simple to construct, of low cost, and are made from a wide range of materials with an ability to operate at high temperatures and pressures. These are also suitable for separating particles where agglomeration occurs. These types of equipment have been employed in the cement industry, and with coal gases for chemical feedstock, and gases from fluidized-bed reactors, as well as in the processing industry for the recovery of spray-dried products, and recently, in the oil industry for separating gas from liquid. Cyclone type reactors permit study of flow pattern and residence time distribution [9, 10]. See for example, the studies by Coker [11, 12] of synthetic detergent production with fast reaction. These devices are widely used to separate a cracking catalyst from vaporized reaction products. Reverse flow cyclones, in which the dust-laden gas stream enters the top section of the cylindrical body either tangentially or via an involute entry, are the most common design. The cylindrical body induces a spinning, vortexed flow pattern to the gas-dust mixture. Centrifugal force separates the dust from the gas stream; the dust travels to the walls of the cylinder and down the conical section to the dust outlet. The spinning gas also travels down the wall toward the apex of the cone, then reverses direction in an air-core and leaves the cyclone through the gás outlet tube at the top. This tube consists of a cylindrical sleeve and the vortex finder, whose lower end extends below the level of the feed port. Separation depends on particle settling velocities, which are governed by size distribution, density, and shape. Stairmand [13], Strauss [14] and Koch and Licht [15] have given guidelines for designing cyclones. The effects of feed and cyclone parameters on the efficiency are rather complex, because many parameters are interdependent. Figure 4.3 shows the design dimensions of a cyclone. Table 4.4 gives the effects on cyclone performance in the important operating and design parameters.

Cyclone Design Procedure The computation of a cyclone fractional or grade efficiency depends on design parameters and flow characteristics of particle-laden gases. The procedure involves a series of equations containing exponential and logarithmic functions. Koch and Licht [15] described a cyclone using seven geometric ratios in terms of its diameter as:



a b DC S h H B . . . . . . DC DC DC DC DC DC DC

Equipment Sizing  179 Table 4.4  Effect of variables on Cyclone Performance. Variable

Effect

Pressure drop increase

Cut size (diameter of particles of which 50% are collected) decreases, flow rate increases; sharpness increases

Solids content of feed increases

Cut size increases (large effect above 15 – 20% v/v)

(ρp – ρf ) increases

Cut size decreases

Viscosity of liquid phase increases

Little effect below 10 mPas.

Cyclone diameter (De) increases

Cut size increases; pressure drop usually decreases.

Cyclone inlet (a) diameter increases

Gravitational force in cyclone decreases; cut size increases; capacity falls; pressure drop decreases.

Overflow diameter increases

Cut size increases; risk of coarse size appearing.

Underflow diameter increases

Brings excess fines from liquid phase into underflow.

Cyclone shape becomes longer

Decrease cut size; sharpens separation.

They further stated that certain constraints are observed in achieving a sound design and these are: a < S

1 b < (DC − De ) 2

S + 1 ≤ H S < h < H ΔP < 10 in H2O vi ≤ 1.35 vs

vi ≅ 1.25 vs

To prevent short circuiting To avoid sudden contraction To keep the vortex inside of the cyclone

To prevent reentrainment For optimum efficiency

The Equations Natural length, l: the distance below the gas outlet where the vortex turns. 1

 D2  3 l = 2.3 De  C   ab 



(4.46)

For 1 < (H − S), the cyclone volume at the natural length (excluding the core) is Vnl:



Vnl =

πDC2 πDC2  1 + S − h   d d 2  πDe2 l 1 + + (h − S) + − 4 4  3   DC DC2  4

(4.47)

The diameter of a central core at a point of vortex turns, d:



d = Dc − (Dc − B)

 S− l− h  H−h 

(4.48)

180  Chemical Process Engineering For 1 > (H – S), the cyclone volume below the exit duct (excluding the core) VH:



VH =

πDC2 πDC2  H − h   B B2  πDe2 (h − S) + 1 + + (H − S) − 4 4  3   DC DC 2  4

(4.49)

For a vortex exponent, n:



 (12DC )0⋅14   T + 460  0.3 n = 1 − 1 −   2.5   530 

(4.50)

The cyclone volume constant, Kc, using Vnl or VH:

where



KC =

2VS + Vnl.H 2D3C

a  π S − ( Dc2 − De2 )  2 VS = 4

(4.51)

(4.52)

The relaxation time (τi) for particle species i of diameter dpi is

τi =



ρpd 2pi 18µ

(4.53)

Cyclone configuration factor G: This is specified by the geometric ratios that describe the cyclone’s shape. The cyclone configuration factor G is only a function of the configuration and is specified by the seven geometrical ratios that describe its shape. G is expressed as:

8K C K a2K 2b

(4.54)

a b , Kb = DC Dc

(4.55)

G=

where



Ka =

Substituting the values of Ka, Kb, and KC in Equation 4.54 gives:



a    D G =  2π S − ( DC2 − De2 ) + 4VH.nl  2 c 2   2  a b

(4.56)

The fractional or grade efficiency, ηi, can be expressed as:



0⋅5   (n +1) G τ   iQ  ηi = 1 − exp  −2  3 (n + 1)   DC  

Saltation Velocity Koch and Licht [15] expressed the saltation velocity as:

(4.57)

Equipment Sizing  181 • The minimum fluid velocity necessary to prevent the settling out of solid particles carried in the stream. • The necessary velocity that picks up deposited particles and transports them without settling. Zenz [16] has shown that the velocity given by the latter differs from the former by a factor of 2 to 2.5. Kalen and Zenz [17] have applied the saltation concept to cyclone design by assuming: • There is no slippage between fluid and particles. The cyclone inlet width is the effective pipe diameter for calculating saltation effects. • Grain loading (dust concentration) is less than 10 grains/ft3. • The diameter effect on the saltation velocity is proportional to the 0.4 power of the inlet width. The saltation velocity, Vs, is dependent on cyclone dimensions, particle and fluid properties and it is expressed as: 2

b/DC  0⋅067 3  Vs = 2.055ω  DC Vi , ft/s 1  (1 − b/DC ) 3 



(4.58)

where 1

 4gµ(ρp − ρf )  3 ω=  3ρf 2 



(4.59)

Inlet velocity, Vi: ft/s

Vi =



Q ab

(4.60)

Kalen and Zenz [17] have shown that maximum cyclone collection efficiency occurs at vi/vs=1.25, and Zenz [16] has found experimentally that fluid reentrainment occurs at vi/vs =1.36.

Pressure Drop Several approaches have been proposed to calculate the frictional loss or pressure drop (ΔP) of a cyclone, although none has been very satisfactory. These are because assumptions made have not considered entrance compression, wall friction, and exit contraction, all of which have a major effect. Consequently, no general correlation of cyclone ΔP has been adopted. Pressure drop in a cyclone with collection efficiency is important in evaluating its cost. Correlations for the pressure drop have been empirical and are acceptable up to ΔP=10 in H2O. The pressure drop (ΔP) or the frictional loss is expressed in terms of the velocity head based on the cyclone inlet area. The frictional loss through cyclones is from 1 to 20 inlet velocity heads and depends on the geometric ratios. ΔP through a cyclone is given by:



∆P = 0.003ρf Vi2 NH



 ab  NH = K  2   De 

(4.61)

where K = 16 for no inlet vane K = 7.5 with a neutral inlet vane

182  Chemical Process Engineering ΔP depends strongly on the inlet velocity and high velocities can cause both reentrainment and high pressure drop. However, entrainment can be reduced to a minimum, if the cyclone has a small base angle.

Troubleshooting Cyclone Maloperations In general, cyclones are used to separate particles from the gas stream, but recent developments have enabled cyclones to function as reactors. Some cyclones can separate cracking catalyst from vaporized reaction products in the range of 950°F (510°C) and 1000°F (538°C) or can function as regenerators for flue gases between 1250°F (677°C) and 1500°F (816°C). In both cases, the high particle velocities can cause rapid erosion of the cyclone material. This often results in poor performance of the cyclone. Other causes of poor cyclone performance are: • • • • •

Hole in cyclone body Cyclone volute plugged Dipleg unsealed Dipleg plugged Dipleg failure

Lieberman [18] has reviewed the causes of these maloperations, which often result in catalyst lost and reduced efficiency. A deficient cyclone is identified by bottom sediment and water levels in the slurry oil product. For a regenerator cyclone, problems are visibly identified by the increased opacity of the regenerator flue gas or by reduced rates of spent catalyst withdrawal.

Cyclone Collection Efficiency Many theories have been proposed to predict the performance of a cyclone, although no fundamental relationship has been accepted. Attempts have been made to predict the critical particle diameter, (Dp)crit. This is the size of the smallest particle theoretically separated from the gas stream with 50% efficiency. The critical particle diameter is defined by [19]: 0.5



9µDc  (DP )crit =    4 πN t Vi (ρp − ρf ) 

(4.62)

where Nt = effective number of turns made by the gas stream in the cyclone, and is defined by:



N t = v i (0.1079 − 0.00077v i + 1.924 × 10−6 v i2 )

(4.63)

and vi is the inlet linear velocity. Figure 4.8 shows the percentage removal of particles in a cyclone as a function of the ratio of the particle to the critical diameter.

Cyclone Design Factor Cyclones are designed to meet specified ΔP limitations. The factor that controls the collection efficiency is the cyclone diameter, and a smaller diameter cyclone at a fixed ΔP will have a higher efficiency. Therefore, small diameter cyclones require a multiple of units in parallel for a given capacity. Reducing the gas outlet diameter results in an increased collection efficiency and ΔP, High-efficiency cyclones have cone lengths in the range of 1.6 to 3.0 times the cyclone diameters. Collection efficiency increases as the gas throughput increases. Kalen and Zenz [17] reported that the collection efficiency increases with increasing gas inlet velocity to a minimum tangential velocity. This reaches the point where the dust is re-entrained or not deposited because of saltation. Koch and Licht [15] showed that saltation velocity is consistent with cyclone inlet velocities in the range of 50 - 90 ft/s (15.2 – 27.4 m/s).

Equipment Sizing  183 10 8 6 4

Dp/(Dp)critical

2

1 0.8 0.6 0.4

0.2

5

10

20

40

60

80

90

95

98

99.8

2

99.0 99.5

0.1

Percent removed

Figure 4.8  Percentage removal of particles as a function of particle diameter relative to the critical diameter.

Cyclones offer the least expensive means of dust collection. They give low efficiency for collection of particles smaller than 5µm. A high efficiency of 98% can be achieved on dusts with particle sizes of 0.1 to 0.2µm that are highly flocculated.

Cyclone Design Procedure The following design procedures can be used to size a cyclone with a specified fluid flow rate and physical property data: 1. 2. 3. 4. 5. 6.

S elect either the high-efficiency or high-throughput design depending on the performance required. Obtain an estimate of the particle size distribution of the solids in the stream to be treated. Estimate the number of cyclones needed in parallel. Calculate the cyclone diameter for an inlet velocity of 50 ft/s (15.24 m/sec). Calculate the scale-up factor for the transposition of grade efficiency and particle size. Calculate the cyclone performance and overall efficiency (recovery of solids). If unsatisfactory, try a smaller diameter. 7. Calculate the cyclone pressure drop and, if required, select a suitable blower. 8. Cost the system and optimize to make the best use of the pressure drop available or, if a blower is required, to give the lowest operating cost.

Nomenclature a b B

= inlet height, ft = inlet width, ft = cyclone dust-outlet diameter, ft

184  Chemical Process Engineering d dpi Dc De g G h i Kc l n NH ΔP Q S T v i vs VH Vnl Vs ηi µ ρf ρp τ

= diameter of central core at a point where vortex turns, ft = diameter of particle in size range i, ft = cyclone diameter, ft = cyclone gas-outlet diameter, ft = acceleration due to gravity, 32.2 ft/s2 = cyclone configuration factor = cylindrical height of cyclone, ft = subscript denotes interval in particle size range = cyclone volume constant = natural length (distance below gas outlet where vortex turns), ft = vortex exponent = number of inlet velocity heads = pressure drop, in H2O = total gas flowrate, actual ft3/s = gas outlet length, ft = temperature, °F = inlet velocity, ft/s = saltation velocity, ft/s = volume below exit duct (excluding core), ft3 = volume at natural length (excluding core), ft3 = annular volume above exit duct to middle of entrance duct, ft3 = grade efficiency for particle size at midpoint of interval i, % = fluid viscosity, lbm/ft.s = fluid density, lbm/ft3 = particle density, lbm/ft3 = relaxation time, s

GAS DRYER DESIGN Liquid water and sometimes water vapor are removed from natural gas to prevent corrosion and formation of hydrates in transmission lines and to attain a water dew point requirement of the sales of gas. Many sweetening agents employ an aqueous solution for treating the gas. Therefore dehydrating the natural gas that normally follows the sweetening process involves: • dehydration by refrigeration • absorption by liquid desiccants • adsorption by solid desiccants Adsorption is a process that involves the transfer of a material from one phase to a surface where it is bound by intermolecular forces. The process involves the transfer from a gas or liquid to a solid surface. It could also involve the transfer from a gas to a liquid surface. The adsorbate is the material being concentrated on the surface, and the material that it accumulates is defined as the adsorbent. The oil and chemical industries use the adsorption process in the cleanup and purification of wastewater streams and for the dehydration of gases. The process is also used in gas purification involving the removal of sulfur dioxide (SO2) from a stack gas. In addition, adsorption is employed to fractionate fluids that are difficult to separate by other separating methods. The amount of adsorbate that is collected on a unit of surface area is negligible. Therefore, porous desiccants (adsorbent) having a large internal surface area are used for industrial applications. There are many solid desiccants that can adsorb water from natural gas. The common commercial desiccants are alumina, silica gel, and molecular sieves. The molecular sieves possess the highest water capacity, will give the lowest water dew points, and can be applied to sweeten dry gases and liquids. Figure 4.9 shows how desiccants can be used

Equipment Sizing  185

COMPRESSOR 450 to 600°F

COOLER WET FEED

SEPARATOR

REGENERATING & COOLING

ABSORBING

VALVE OPEN VALVE CLOSED

600°F HEATER

DRY GAS

Figure 4.9  Solid desiccant dehydrator twin tower system. Courtesy of Gas Processors Suppliers Association [20].

in dehydrators containing two or more towers. One of the towers operates using steam to adsorb water from the gas, while the other tower is regenerated and cooled. The regenerated gas is heated to a range of 450°F (232°C) and 600°F (316°C) depending on the type of solid desiccant and the nature of service [20]. The use of solid desiccant is limited to applications such as those with very low water dewpoint requirements, simultaneous control of water and hydrocarbon dewpoints, and in very sour gases. In cryogenic plant, solid-­desiccant dehydration is much preferred over methanol injection to prevent hydrate and ice formation. The design procedures for dehydrating saturated natural gas at a specified dew point are: 1. 2. 3. 4. 5.

 etermine the process conditions and the dryer process flow diagrams. D Select a drying cycle and calculate the water load. Select the type of desiccant and compute the capacity and volume required. Size the dryer and check for the pressure drop. Calculate the desiccant reactivation heating and cooling requirements.

Here, design procedures are presented for sizing a solid desiccant dryer in removing moisture from gas streams. The process involves the following calculations. 1. 2. 3. 4. 5. 6.

t he water pickup desiccant volume desiccant bed gas velocity through the bed vessel weight dryer regeneration requirements

The design is based upon the following assumptions [4]: 1. Th  e temperature difference between the heater outlet temperature and the peak vessel outlet temperature is 50oF. 2. The average bed temperature is based on 75% of the bed at heater outlet temperature and 25% at the peak vessel outlet temperature Equation 4.74.

186  Chemical Process Engineering 3. 4. 5. 6. 7.

S pecific heats: steel, 0.12 Btu/lb°F and desiccant 0.25 Btu/lb°F. Heat of water desorption 1400 Btu/lb H2O adsorbed. Flat heads used on vessel ends; steel density is 480 lb/ft3. Total vessel weight increased by 10% for supports. Heat losses to the dryer during heating period calculated at 5%.

The Equations The equations used for calculating these requirements are: The total water adsorbed, lb



lb MMstdft 3 ×h× MMstdft 3 day Flow × Cycle × H 2O Cont H 2O load = = h/day 24

(4.64)

Desiccant volume

VDES =



H 2O load × 100 pickup × ρD

(4.65)

VDES L

(4.66)

The calculated area based on desiccant volume

A DES =

The bed diameter

DBED =



0.5

 4A DES   π 

(4.67)

actual desiccant bed área 2 πDBED 4

(4.68)

Flow × 106 × 14.7ZT 19.6314 × Flow × ZT = 520 × 1440(P + 14.7) (P + 14.7)

(4.69)

A=



The actual gas flowrate at flowing conditions, ft3/min



Q=

T in above formula is Rankine (R) (i.e., R = °F + 460). Superficial gas velocity, ft/min



VGAS =

Q A

(4.70)

Superficial gas velocities for adequate contact time should range between 30 and 60 ft/min. Dryer shell thickness, inch



t=

PR SE − 0 ⋅ 6P

(4.71)

Equipment Sizing  187 Vessel weight



WVES = 1.1 × 480πD

 t (L + h + R)  12 

(4.72)

Add an extra 2 ft (h=2) of straight side (i.e., for 6 in. of 3/8-in. inert balls above and below the desiccant bed and 1 ft for an inlet-gas distributor). Temperature of desiccant bed, °F

TBED = 0.75(HT) + 0.25(HT − 50)

(4.73)

Total heat required to regenerate the dryer (ideal), Btu

HTOT = 1.05[(0.12WVES + 0.25WD)(TBED – T) + 1400 × H2O load]

(4.74)

WD = ρDVDES loss factor for non-steady state heating, F

F = ln



 HT − T   50 

(4.75)

Heat added to regeneration gas to regenerate the desiccant bed

HREG = HTOTF

(4.76)

The total regeneration gas requirement, lb



HGAS =

HREG H1 − H 2

(4.77)

SCF =

380HGas MW

(4.78)

Standard cubic feet of regeneration gas:

Pressure Drop

The pressure drop (ΔP) is estimated from the desiccant manufacturer’s data and correlations. In addition, ΔP can be calculated by Wunder’s [21] graphical correlation of gas superficial velocity against ΔP in ft water per ft bed. Wunder used a factor of 1.6 for a fouled bed. ΔP through a new desiccant bed is initially low. After a short time in service, ΔP rises because of bed settling and then slowly increases over the active life of the desiccant because of more settling and some attrition. Figure 4.10 shows ΔP for an 8-inch silica gel desiccant. These data are based on air. For other 0.9 MWgas  . The pressure drop for a clean bed is given by: gases, the given values should be multiplied by   MWair 



 ft H 2O   0.4335psi   MWgas  ∆P =     ft bed   ft H 2O   MWair 

0.9



(4.79)

Fouled bed:



ΔP = 1.6ΔPclean bed

(4.80)

188  Chemical Process Engineering

Superficial velocity fpm

1,000

sig

6p

sig

0p

15

100

ig

ps

50

ig

ps

sig 0 p ig s 0p g 20 si 0 p ig 40 s 0 p ig 60 0 ps sig 80 00 p 1,0

10

0 0.1

1.0 Pressure drop ft. water per ft. bed

Figure 4.10  Pressure drop for an 8 mesh silica gel desiccant (by permission Oil & Gas Journal) [21].

Desiccant attrition due to high gas velocities is controlled by proper bed design. The superficial gas velocity is used to estimate Alcoa’s momemtum downflow velocities [21]. However, desiccant attrition should not be a problem when the momentum number is equal to or less than 30,000. Although, this number is based on granular alumina, it can be used for silica gel, which has crushing characteristics similar to alumina. The momentum is expressed as:

Momentum = (VGAS )(MW)



 P + 14.7  ≤ 30.000  14.7 

(4.81)

where VGAS = superficial gas velocity, ft/min MW = molecular weight of gas P = system pressure in psig

Desiccant Reactivation The desiccant bed is reactivated at as high a temperature as permissible to start optimum capacity and minimum dew points. In natural gas drying service, the normal reactivation temperature for silica gel is between 350°F and 400°F (177 and 204°C). Because the feed gas contains high pentanes-plus materials, Wunder [21] suggested that reactivation can be carried out at 450°F (232oC) to desorb the heavy hydrocarbons. At a temperature above 450°F (232oC) the attrition rate for silica gel is high. However, occasionally at 600°F (316°C), reactivation will restore some desiccant’s initial activity thus extending the life of the silica gel. Therefore, the reactivation procedure is designed to periodically heat the desiccant to 600°F (316°C) [21].

Nomenclature A ADES

= actual desiccant bed area, ft2 = calculated area based on desiccant volume, ft2

Equipment Sizing  189 Cycle = drying cycle, h = calculated bed diameter, ft DBED E = joint efficiency F = loss factor for non-steady state heating Flow = gas flow rate, 106std ft3/day h = additional shell height for desiccant supports and distributor, ft = enthalpy of regeneration gas before and after regeneration heater, Btu/lb H1, H2 = total regeneration gas requirement, lb HGAs = heat added to regeneration gas to regenerate desiccant bed HREG = heat required to regenerate dryer (ideal), Btu HTOT HT = heater temperature, oF H2O load = total water adsorbed, lb H2O Cont = water content of gas, lb/106std ft3 L = desiccant bed length, ft MW = molecular weight of regeneration gas P = dryer operating pressure, psig PICKUP = useful design capacity of desiccant, % Q = actual gas flow rate at flowing conditions, ft3/min R = radius of desiccant bed, ft S = allowable stress, lb/in2 SCF = standard cubic feet of regeneration gás t = dryer shell thickness, inch T = gas flowing temperature, oR TB = temperature of desiccant bed, oF = desiccant volume, ft3 VDES VGAS = superficial gas velocity, ft/min = desiccant weight, lb WD WS = steel weight of shell and heads, lb = vessel weight, lb WVES Z = compressibility factor = desiccant density, lb/ft3 ρD

EXAMPLES AND SOLUTIONS Example 4.1 Size a vertical and horizontal separators under the following conditions: Parameter

Symbol

Vertical separator

Horizontal separator

Liquid flow rate, lb/h

WL

5,000

56,150

Liquid density, lb/ft3

ρL

61.87

60

Vapor flow rate, lb/h

WV

37,000

40,000

Vapor density, lb/ft3

ρV

0.374

1.47

Length to diameter ratio

L/D

3

3

Liquid surge time

τ

5

6

190  Chemical Process Engineering

Solution The Excel program, Example 4.1.xlsx, sizes both the vertical and horizontal separators. Figure 4.11 shows the snapshot of Excel spreadsheet for both separators. From the computed results, the diameter of the vertical separator with the given flow data and physical properties is 3 ft (D=Dmin (2.88) set to next largest 6 inch, the Excel spreadsheet roundup function is used for this end), the liquid height is 0.95 ft and the vessel volume is 6.73 ft3. The computer output for the horizontal separator gives its diameter as 4.13 ft, the vessel length is 12.39 ft, and volume is 165.82 ft3. Figure 4.7 gives a nomograph of the vapor rise in horizontal vessels. A check of the vapor area with 20% of the total cross-sectional area and the diameter of 4.13ft shows that the vapor rise is approximately 12.2 inches as illustrated in Figure 4.7. By rearranging and solving the equation 4.27 with Goal Seek function in Excel spreadsheet, the vapor rise value, H, can be found if it is arranged to the following form:



f (H) = A s − R 2Cos −1

 R − H + (R − H)(2HR − H 2 )0.5  R 

(4.82)

The Goal Seek function uses a single variable input value. This means that there’s only one Changing Cell. In the Goal Seek Dialog Box, guess the value for H (changing cell, $N#38) and set the value of f(H) in cell $N#39 (the dependent cell containing the formula) to 0 by Goal Seek, this gives the rise of H=12.6 inch for the vapor (Figure 4.12) which is almost the same as the value obtained by nomograph.

Example 4.2 Estimate the liquid volume in a horizontal storage tank that has a diameter of 120 inches, a length of 200 inches, and contains 40 inches of liquid. Calculate the liquid volume for a vertical tank and for tanks with either flat heads, dished heads, elliptical and hemispherical.

Solution The Excel spreadsheet program, Example 4.2.xlsx, calculates the liquid volume in a horizontal tank and for tanks with flat, dished, elliptical and hemispherical heads. Figure 4.13 shows the results of the input data and computer output for a vertical tank and tanks with flat, dished, elliptical and hemispherical heads.

Figure 4.11  Snapshot of the Excel spreadsheet calculations for both separators.

Equipment Sizing  191

Figure 4.12  Snapshot of the Excel spreadsheet calculations for vapor rise.

Figure 4.13  Snapshot of the Excel spreadsheet calculations for tank volumes.

Example 4.3 Estimate the weight of a pressure vessel with hemispherical and ellipsoidal heads. The vessel is 2.5 ft in ID and 10 ft in length and has a internal design pressure of 1000 psig. SF=2 in

Eshell=0.85

C=0.125

Ehead=1

ρS=489.024 lb/ft3

S=17,500 lb/in2

Solution The Excel spreadsheet program, Example 4.3.xlsx, determines the vessel weight of the pressure vessel with hemispherical, elliptical, and torispherical heads. Figure 4.14 gives input data and the Excel spreadsheet result for 30 inch-diameter

192  Chemical Process Engineering

Figure 4.14  Snapshot of the Excel spreadsheet calculations for total weight of vessels with two heads.

vessel. The result gives the shell and head thicknesses, the weights of the shell and head, and the total weight of the vessel with hemispherical and elliptical heads. The results are identical to those reported in the literature [1]. Note that the following syntaxes are used for Excel spreadsheet if function in cell $H$20 and $I$20 to choose value of H.fac from Table 4.3: =IF(10 1 i

(7.36)

i

i

and

n

f2 =

∑ Kn > 1 i

i

(7.37)

i

Table 7.2  Equilibrium flash criteria. f1 (Equation 7.36)

f2 (Equation 7.37)

Subcooled liquid

1

Bubble point

=1

>1

Two-phase condition

>1

>1

Dew point

>1

=1

Superheated vapor

>1