Chemical Process Design and Integration, Notas de estudo de Engenharia Química

Baixe Chemical Process Design and Integration e outras Notas de estudo em PDF para Engenharia Química, somente na Docsity! Chemical Process Design and Integration Robin Smith Centre for Process Integration, School of Chemical Engineering and Analytical Science, University of Manchester. Chemical Process Design and Integration Robin Smith Centre for Process Integration, School of Chemical Engineering and Analytical Science, University of Manchester. Previous edition published by McGraw Hill Copyright  2005 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. 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Library of Congress Cataloging-in-Publication Data Smith, R. (Robin) Chemical process design and integration / Robin Smith. p. cm. Includes bibliographical references and index. ISBN 0-471-48680-9 (HB) (acid-free paper) – ISBN 0-471-48681-7 (PB) (pbk. : acid-free paper) 1. Chemical processes. I. Title. TP155.7.S573 2005 660′.2812 – dc22 2004014695 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-471-48680-9 (cloth) 0-471-48681-7 (paper) Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Spain by Grafos, Barcelona This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. To my family viii Contents 5.8 Choice of Reactor Performance – Summary 94 5.9 Exercises 95 References 96 Chapter 6 Choice of Reactor II - Reactor Conditions 97 6.1 Reaction Equilibrium 97 6.2 Reactor Temperature 100 6.3 Reactor Pressure 107 6.4 Reactor Phase 108 6.5 Reactor Concentration 109 6.6 Biochemical Reactions 114 6.7 Catalysts 114 6.8 Choice of Reactor Conditions – Summary 117 6.9 Exercises 118 References 120 Chapter 7 Choice of Reactor III – Reactor Configuration 121 7.1 Temperature Control 121 7.2 Catalyst Degradation 123 7.3 Gas–Liquid and Liquid–Liquid Reactors 124 7.4 Reactor Configuration 127 7.5 Reactor Configuration for Heterogeneous Solid-Catalyzed Reactions 133 7.6 Reactor Configuration from Optimization of a Superstructure 133 7.7 Choice of Reactor Configuration – Summary 139 7.8 Exercises 139 References 140 Chapter 8 Choice of Separator for Heterogeneous Mixtures 143 8.1 Homogeneous and Heterogeneous Separation 143 8.2 Settling and Sedimentation 143 8.3 Inertial and Centrifugal Separation 147 8.4 Electrostatic Precipitation 149 8.5 Filtration 150 8.6 Scrubbing 151 8.7 Flotation 152 8.8 Drying 153 8.9 Separation of Heterogeneous Mixtures – Summary 154 8.10 Exercises 154 References 155 Chapter 9 Choice of Separator for Homogeneous Fluid Mixtures I – Distillation 157 9.1 Single-Stage Separation 157 9.2 Distillation 157 9.3 Binary Distillation 160 9.4 Total and Minimum Reflux Conditions for Multicomponent Mixtures 163 9.5 Finite Reflux Conditions for Multicomponent Mixtures 170 9.6 Choice of Operating Conditions 175 9.7 Limitations of Distillation 176 9.8 Separation of Homogeneous Fluid Mixtures by Distillation – Summary 177 9.9 Exercises 178 References 179 Chapter 10 Choice of Separator for Homogeneous Fluid Mixtures II – Other Methods 181 10.1 Absorption and Stripping 181 10.2 Liquid–Liquid Extraction 184 10.3 Adsorption 189 10.4 Membranes 193 10.5 Crystallization 203 10.6 Evaporation 206 10.7 Separation of Homogeneous Fluid Mixtures by Other Methods – Summary 208 10.8 Exercises 209 References 209 Chapter 11 Distillation Sequencing 211 11.1 Distillation Sequencing Using Simple Columns 211 11.2 Practical Constraints Restricting Options 211 11.3 Choice of Sequence for Simple Nonintegrated Distillation Columns 212 11.4 Distillation Sequencing Using Columns With More Than Two Products 217 11.5 Distillation Sequencing Using Thermal Coupling 220 11.6 Retrofit of Distillation Sequences 224 11.7 Crude Oil Distillation 225 11.8 Distillation Sequencing Using Optimization of a Superstructure 228 11.9 Distillation Sequencing – Summary 230 11.10 Exercises 231 References 232 Contents ix Chapter 12 Distillation Sequencing for Azeotropic Distillation 235 12.1 Azeotropic Systems 235 12.2 Change in Pressure 235 12.3 Representation of Azeotropic Distillation 236 12.4 Distillation at Total Reflux Conditions 238 12.5 Distillation at Minimum Reflux Conditions 242 12.6 Distillation at Finite Reflux Conditions 243 12.7 Distillation Sequencing Using an Entrainer 246 12.8 Heterogeneous Azeotropic Distillation 251 12.9 Entrainer Selection 253 12.10 Trade-offs in Azeotropic Distillation 255 12.11 Multicomponent Systems 255 12.12 Membrane Separation 255 12.13 Distillation Sequencing for Azeotropic Distillation – Summary 256 12.14 Exercises 257 References 258 Chapter 13 Reaction, Separation and Recycle Systems for Continuous Processes 259 13.1 The Function of Process Recycles 259 13.2 Recycles with Purges 264 13.3 Pumping and Compression 267 13.4 Simulation of Recycles 276 13.5 The Process Yield 280 13.6 Optimization of Reactor Conversion 281 13.7 Optimization of Processes Involving a Purge 283 13.8 Hybrid Reaction and Separation 284 13.9 Feed, Product and Intermediate Storage 286 13.10 Reaction, Separation and Recycle Systems for Continuous Processes – Summary 288 13.11 Exercises 289 References 290 Chapter 14 Reaction, Separation and Recycle Systems for Batch Processes 291 14.1 Batch Processes 291 14.2 Batch Reactors 291 14.3 Batch Separation Processes 297 14.4 Gantt Charts 303 14.5 Production Schedules for Single Products 304 14.6 Production Schedules for Multiple Products 305 14.7 Equipment Cleaning and Material Transfer 306 14.8 Synthesis of Reaction and Separation Systems for Batch Processes 307 14.9 Optimization of Batch Processes 311 14.10 Storage in Batch Processes 312 14.11 Reaction and Separation Systems for Batch Processes – Summary 313 14.12 Exercises 313 References 315 Chapter 15 Heat Exchanger Networks I – Heat Transfer Equipment 317 15.1 Overall Heat Transfer Coefficients 317 15.2 Heat Transfer Coefficients and Pressure Drops for Shell-and-Tube Heat Exchangers 319 15.3 Temperature Differences in Shell-and-Tube Heat Exchangers 324 15.4 Allocation of Fluids in Shell-and-Tube Heat Exchangers 329 15.5 Extended Surface Tubes 332 15.6 Retrofit of Heat Exchangers 333 15.7 Condensers 337 15.8 Reboilers and Vaporizers 342 15.9 Other Types of Heat Exchange Equipment 346 15.10 Fired Heaters 348 15.11 Heat Transfer Equipment – Summary 354 15.12 Exercises 354 References 356 Chapter 16 Heat Exchanger Networks II – Energy Targets 357 16.1 Composite Curves 357 16.2 The Heat Recovery Pinch 361 16.3 Threshold Problems 364 16.4 The Problem Table Algorithm 365 16.5 Nonglobal Minimum Temperature Differences 370 16.6 Process Constraints 370 16.7 Utility Selection 372 16.8 Furnaces 374 16.9 Cogeneration (Combined Heat and Power Generation) 376 16.10 Integration Of Heat Pumps 381 16.11 Heat Exchanger Network Energy Targets – Summary 383 x Contents 16.12 Exercises 383 References 385 Chapter 17 Heat Exchanger Networks III – Capital and Total Cost Targets 387 17.1 Number of Heat Exchange Units 387 17.2 Heat Exchange Area Targets 388 17.3 Number-of-shells Target 392 17.4 Capital Cost Targets 393 17.5 Total Cost Targets 395 17.6 Heat Exchanger Network and Utilities Capital and Total Costs – Summary 395 17.7 Exercises 396 References 397 Chapter 18 Heat Exchanger Networks IV – Network Design 399 18.1 The Pinch Design Method 399 18.2 Design for Threshold Problems 404 18.3 Stream Splitting 405 18.4 Design for Multiple Pinches 408 18.5 Remaining Problem Analysis 411 18.6 Network Optimization 413 18.7 The Superstructure Approach to Heat Exchanger Network Design 416 18.8 Retrofit of Heat Exchanger Networks 419 18.9 Addition of New Heat Transfer Area in Retrofit 424 18.10 Heat Exchanger Network Design – Summary 425 18.11 Exercises 425 References 428 Chapter 19 Heat Exchanger Networks V – Stream Data 429 19.1 Process Changes for Heat Integration 429 19.2 The Trade-Offs Between Process Changes, Utility Selection, Energy Cost and Capital Cost 429 19.3 Data Extraction 430 19.4 Heat Exchanger Network Stream Data – Summary 437 19.5 Exercises 437 References 438 Chapter 20 Heat Integration of Reactors 439 20.1 The Heat Integration Characteristics of Reactors 439 20.2 Appropriate Placement of Reactors 441 20.3 Use of the Grand Composite Curve for Heat Integration of Reactors 442 20.4 Evolving Reactor Design to Improve Heat Integration 443 20.5 Heat Integration of Reactors – Summary 444 Reference 444 Chapter 21 Heat Integration of Distillation Columns 445 21.1 The Heat Integration Characteristics of Distillation 445 21.2 The Appropriate Placement of Distillation 445 21.3 Use of the Grand Composite Curve for Heat Integration of Distillation 446 21.4 Evolving the Design of Simple Distillation Columns to Improve Heat Integration 447 21.5 Heat Pumping in Distillation 449 21.6 Capital Cost Considerations 449 21.7 Heat Integration Characteristics of Distillation Sequences 450 21.8 Heat-integrated Distillation Sequences Based on the Optimization of a Superstructure 454 21.9 Heat Integration of Distillation Columns – Summary 455 21.10 Exercises 456 References 457 Chapter 22 Heat Integration of Evaporators and Dryers 459 22.1 The Heat Integration Characteristics of Evaporators 459 22.2 Appropriate Placement of Evaporators 459 22.3 Evolving Evaporator Design to Improve Heat Integration 459 22.4 The Heat Integration Characteristics of Dryers 459 22.5 Evolving Dryer Design to Improve Heat Integration 460 22.6 Heat Integration of Evaporators and Dryers – Summary 461 Preface This book deals with the design and integration of chemical processes, emphasizing the conceptual issues that are fundamental to the creation of the process. Chemical process design requires the selection of a series of processing steps and their integration to form a complete manufacturing system. The text emphasizes both the design and selection of the steps as individual operations and their integration to form an efficient process. Also, the process will normally operate as part of an integrated manufacturing site consisting of a number of processes serviced by a common utility system. The design of utility systems has been dealt with so that the interactions between processes and the utility system and the interactions between different processes through the utility system can be exploited to maximize the performance of the site as a whole. Thus, the text integrates equipment, process and utility system design. Chemical processing should form part of a sustainable industrial activity. For chemical processing, this means that processes should use raw materials as efficiently as is economic and practicable, both to prevent the production of waste that can be environmentally harmful and to preserve the reserves of raw materials as much as possible. Processes should use as little energy as is economic and practicable, both to prevent the buildup of carbon dioxide in the atmosphere from burning fossil fuels and to preserve reserves of fossil fuels. Water must also be consumed in sustainable quantities that do not cause deterioration in the quality of the water source and the long-term quantity of the reserves. Aqueous and atmospheric emissions must not be environmentally harmful, and solid waste to landfill must be avoided. Finally, all aspects of chemical processing must feature good health and safety practice. It is important for the designer to understand the limitations of the methods used in chemical process design. The best way to understand the limitations is to understand the derivations of the equations used and the assumptions on which the equations are based. Where practical, the derivation of the design equations has been included in the text. The book is intended to provide a practical guide to chemical process design and integration for undergraduate and postgraduate students of chemical engineering, practic- ing process designers and chemical engineers and applied chemists working in process development. For undergrad- uate studies, the text assumes basic knowledge of mate- rial and energy balances, fluid mechanics, heat and mass transfer phenomena and thermodynamics, together with basic spreadsheeting skills. Examples have been included throughout the text. Most of these examples do not require specialist software and can be solved using spreadsheet soft- ware. Finally, a number of exercises have been added at the end of each chapter to allow the reader to practice the calculation procedures. Robin Smith Acknowledgements The author would like to express gratitude to a number of people who have helped in the preparation and have reviewed parts of the text. From The University of Manchester: Prof Peter Heggs, Prof Ferda Mavituna, Megan Jobson, Nan Zhang, Constanti- nos Theodoropoulos, Jin-Kuk Kim, Kah Loong Choong, Dhaval Dave, Frank Del Nogal, Ramona Dragomir, Sungwon Hwang, Santosh Jain, Boondarik Leewongtanawit, Guil- ian Liu, Vikas Rastogi, Clemente Rodriguez, Ramagopal Uppaluri, Priti Vanage, Pertar Verbanov, Jiaona Wang, Wen- ling Zhang. From Alias, UK: David Lott. From AspenTech: Ian Moore, Eric Petela, Ian Sinclair, Oliver Wahnschafft. From CANMET, Canada: Alberto Alva-Argaez, Abde- laziz Hammache, Luciana Savulescu, Mikhail Sorin. From DuPont Taiwan: Janice Kuo. From Monash University, Australia: David Brennan, Andrew Hoadley. From UOP, Des Plaines, USA: David Hamm, Greg Maher. Gratitude is also expressed to Simon Perry, Gareth Maguire, Victoria Woods and Mathew Smith for help in the preparation of the figures. Finally, gratitude is expressed to all of the member companies of the Process Integration Research Consortium, both past and present. Their support has made a considerable contribution to research in the area, and hence to this text. xviii Nomenclature CP Capacity parameter in distillation (m·s−1) or heat capacity flowrate (kW·K−1, MW·K−1) CPEX Heat capacity flowrate of heat engine exhaust (kW·K−1, MW·K−1) CW Cooling water d Diameter (µm, m) di Distillate flowrate of Component i (kmol·s−1, kmol·h−1) dI Inside diameter of pipe or tube (m) D Distillate flowrate (kmol·s−1, kmol·h−1) DB Tube bundle diameter for shell-and-tube heat exchangers (m) DS Shell diameter for shell-and-tube heat exchangers (m) DCFRR Discounted cash flowrate of return (%) E Activation energy of reaction (kJ·kmol−1), or entrainer flowrate in azeotropic and extractive distillation (kg·s−1, kmol·s−1), or extract flowrate in liquid–liquid extraction (kg·s−1, kmol·s−1), or stage efficiency in separation (–) EO Overall stage efficiency in distillation and absorption (–) EP Economic potential ($·y−1) f Fuel-to-air ratio for gas turbine (–) fi Capital cost installation factor for Equipment i (–), or feed flowrate of Component i (kmol·s−1, kmol·h−1), or fugacity of Component i (N·m−2, bar) fM Capital cost factor to allow for material of construction (–) fP Capital cost factor to allow for design pressure (–) fT Capital cost factor to allow for design temperature (–) F Feed flowrate (kg·s−1, kg·h−1, kmol·s−1, kmol·h−1), or future worth a sum of money allowing for interest rates ($), or volumetric flowrate (m3·s−1, m3·h−1) FLV Liquid–vapor flow parameter in distillation (–) FT Correction factor for noncountercurrent flow in shell-and-tube heat exchangers (–) FTmin Minimum acceptable FT for noncountercurrent heat exchangers (–) g Acceleration due to gravity (9.81 m·s−2) gij Energy of interaction between Molecules i and j in the NRTL equation (kJ·kmol−1) G Free energy (kJ), or gas flowrate (kg·s−1, kmol·s−1) Gi Partial molar free energy of Component i (kJ·kmol−1) G O i Standard partial molar free energy of Component i (kJ·kmol−1) h Settling distance of particles (m) hC Condensing film heat transfer coefficient (W·m−2·K−1, kW·m−2·K−1) hI Film heat transfer coefficient for the inside (W·m−2·K−1, kW·m−2·K−1) hIF Fouling heat transfer coefficient for the inside (W·m−2·K−1, kW·m−2·K−1) hL Head loss in a pipe or pipe fitting (m) hNB Nucleate boiling heat transfer coefficient (W·m−2·K−1, kW·m−2·K−1) hO Film heat transfer coefficient for the outside (W·m−2·K−1, kW·m−2·K−1) hOF Fouling heat transfer coefficient for the outside (W·m−2·K−1, kW·m−2·K−1) hW Heat transfer coefficient for the tube wall (W·m−2·K−1, kW·m−2·K−1) H Enthalpy (kJ, kJ·kg−1, kJ·kmol−1), or height (m), or Henry’s Law Constant (N·m−2, bar, atm), or stream enthalpy (kJ·s−1, MJ·s−1) HT Tray spacing (m) Nomenclature xix H O i Standard heat of formation of Component i (kJ·kmol−1)
HO Standard heat of reaction (J, kJ)
HCOMB Heat of combustion (J·kmol−1, kJ·kmol−1)
HOCOMB Standard heat of combustion at 298 K (J·kmol−1, kJ·kmol−1)
HP Heat to bring products from standard temperature to the final temperature (J·kmol−1, kJ·kg−1)
HR Heat to bring reactants from their initial temperature to standard temperature (J·kmol−1, kJ·kmol−1)
HSTEAM Enthalpy difference between generated steam and boiler feedwater (kW, MW)
HVAP Latent heat of vaporization (kJ·kg−1, kJ·kmol−1) HETP Height equivalent of a theoretical plate (m) HP High pressure HR Heat rate for gas turbine (kJ·kWh−1) i Fractional rate of interest on money (–), or number of ions (–) I Total number of hot streams (–) J Total number of cold streams (–) k Reaction rate constant (units depend on order of reaction), or thermal conductivity (W·m−1·K−1, kW·m−1·K−1) kG,i Mass transfer coefficient in the gas phase (kmol·m−2·Pa−1·s−1) kij Interaction parameter between Components i and j in an equation of state (–) kL,i Mass transfer coefficient in the liquid phase (m·s−1) k0 Frequency factor for heat of reaction (units depend on order of reaction) K Overall mass transfer coefficient (kmol·Pa−1·m−2·s−1) or total number of enthalpy intervals in heat exchanger networks (–) Ka Equilibrium constant of reaction based on activity (–) Ki Ratio of vapor to liquid composition at equilibrium for Component i (–) KM,i Equilibrium partition coefficient of membrane for Component i (–) Kp Equilibrium constant of reaction based on partial pressure in the vapor phase (–) KT Parameter for terminal settling velocity (m·s−1) Kx Equilibrium constant of reaction based on mole fraction in the liquid phase (–) Ky Equilibrium constant of reaction based on mole fraction in vapor phase (–) L Intercept ratio for turbines (–), or length (m), or liquid flowrate (kg·s−1, kmol·s−1), or number of independent loops in a network (–) LB Distance between baffles in shell-and-tube heat exchangers (m) LP Low pressure m Mass flowrate (kg·s−1), or molar flowrate (kmol·s−1), or number of items (–) M Constant in capital cost correlations (–), or molar mass (kg·kmol−1) MP Medium pressure MC STEAM Marginal cost of steam ($·t−1) n Number of items (–), or number of years (–), or polytropic coefficient (–), or slope of Willans’ Line (kJ·kg−1, MJ·kg−1) N Number of compression stages (–), or number of moles (kmol), or number of theoretical stages (–), or rate of transfer of a component (kmol·s−1·m−3) Ni Number of moles of Component i (kmol) Ni0 Initial number of moles of Component i (kmol) NPT Number of tube passes (–) NR Number of tube rows (–) xx Nomenclature NSHELLS Number of number of 1–2 shells in shell-and-tube heat exchangers (–) NT Number of tubes (–) NUNITS Number of units in a heat exchanger network (–) NC Number of components in a multicomponent mixture (–) NPV Net present value ($) p Partial pressure (N·m−2, bar) pC Pitch configuration factor for tube layout (–) pT Tube pitch (m) P Present worth of a future sum of money ($), or pressure (N·m−2, bar), or probability (–), or thermal effectiveness of 1–2 shell-and-tube heat exchanger (–) PC Critical pressure (N·m−2, bar) Pmax Maximum thermal effectiveness of 1–2 shell-and-tube heat exchangers (–) PM,i Permeability of Component i for a membrane (kmol·m·s−1·m−2·bar−1, kg solvent ·m−1·s−1·bar−1) P M,i Permeance of Component i for a membrane (m3·m−2·s−1·bar−1) PN−2N Thermal effectiveness over NSHELLS number of 1–2 shell-and-tube heat exchangers in series (–) P1−2 Thermal effectiveness over each 1–2 shell-and-tube heat exchanger in series (–) P SAT Saturated liquid–vapor pressure (N·m−2, bar) Pr Prandtl number (–) q Heat flux (W·m−2, kW·m−2), or thermal condition of the feed in distillation (–), or Wegstein acceleration parameter for the convergence of recycle calculations (–) qC Critical heat flux (W·m−2, kW·m−2) qC1 Critical heat flux for a single tube (W·m−2, kW·m−2) qi Individual stream heat duty for Stream i (kJ·s−1), or pure component property measuring the molecular van der Waals surface area for Molecule i in the UNIQUAC Equation (–) Q Heat duty (kW, MW) Qc Cooling duty (kW, MW) Qcmin Target for cold utility (kW, MW) QCOND Condenser heat duty (kW, MW) QEVAP Evaporator heat duty (kW, MW) QEX Heat duty for heat engine exhaust (kW, MW) QFEED Heat duty to the feed (kW, MW) QFUEL Heat from fuel in a furnace, boiler, or gas turbine (kW, MW) QH Heating duty (kW, MW) QHmin Target for hot utility (kW, MW) QHE Heat engine heat duty (kW, MW) QHEN Heat exchanger network heat duty (kW, MW) QHP Heat pump heat duty (kW, MW) QLOSS Stack loss from furnace, boiler, or gas turbine (kW, MW) QREACT Reactor heating or cooling duty (kW, MW) QREB Reboiler heat duty (kW, MW) QREC Heat recovery (kW, MW) QSITE Site heating demand (kW, MW) QSTEAM Heat input for steam generation (kW, MW) r Molar ratio (–), or pressure ratio (–), or radius (m) ri Pure component property measuring the molecular van der Waals volume for Molecule i in the UNIQUAC Equation (–), or rate of reaction of Component i (kmol−1·s−1), or recovery of Component i in separation (–) R Fractional recovery of a component in separation (–), or Nomenclature xxiii cont Contribution C Cold stream, or contaminant CN Condensing COND Condensing conditions CP Continuous phase CW Cooling water D Distillate in distillation DS De-superheating e Enhanced, or end zone on the shell-side of a heat exchanger, or environment E Extract in liquid–liquid extraction EVAP Evaporator conditions EX Exhaust final Final conditions in a batch F Feed, or fluid G Gas phase H Hot stream HP Heat pump, or high pressure i Component number, or stream number I Inside IS Isentropic in Inlet j Component number, or stream number k Enthalpy interval number in heat exchanger networks L Liquid phase LP Low pressure m Stage number in distillation and absorption max Maximum min Minimum M Makeup MIX Mixture n Stage number in distillation and absorption out Outlet O Outside, or standard conditions p Stage number in distillation and absorption prod Products of reaction P Particle, or permeate react Reactants R Raffinate in liquid–liquid extraction REACT Reaction S Solvent in liquid–liquid extraction SAT Saturated conditions SF Supplementary firing SUP Superheated conditions T Treatment TW Treated water V Vapor w Window section on the shell-side of a heat exchanger W Conditions at the tube wall, or water ∞ Conditions at distillate pinch point SUPERSCRIPTS I Phase I II Phase II III Phase III L Liquid O Standard conditions V Vapor * Adjusted parameter 1 The Nature of Chemical Process Design and Integration 1.1 CHEMICAL PRODUCTS Chemical products are essential to modern living standards. Almost all aspects of everyday life are supported by chemical products in one way or another. Yet, society tends to take these products for granted, even though a high quality of life fundamentally depends on them. When considering the design of processes for the manufacture of chemical products, the market into which they are being sold fundamentally influences the objectives and priorities in the design. Chemical products can be divided into three broad classes: 1. Commodity or bulk chemicals: These are produced in large volumes and purchased on the basis of chemical composition, purity and price. Examples are sulfuric acid, nitrogen, oxygen, ethylene and chlorine. 2. Fine chemicals: These are produced in small volumes and purchased on the basis of chemical composition, purity and price. Examples are chloropropylene oxide (used for the manufacture of epoxy resins, ion-exchange resins and other products), dimethyl formamide (used, for example, as a solvent, reaction medium and interme- diate in the manufacture of pharmaceuticals), n-butyric acid (used in beverages, flavorings, fragrances and other products) and barium titanate powder (used for the man- ufacture of electronic capacitors). 3. Specialty or effect or functional chemicals: These are purchased because of their effect (or function), rather than their chemical composition. Examples are pharma- ceuticals, pesticides, dyestuffs, perfumes and flavorings. Because commodity and fine chemicals tend to be pur- chased on the basis of their chemical composition alone, they are undifferentiated. For example, there is nothing to choose between 99.9% benzene made by one manufacturer and that made by another manufacturer, other than price and delivery issues. On the other hand, specialty chemicals tend to be purchased on the basis of their effect or function and are therefore differentiated. For example, competitive pharmaceutical products are differentiated according to the efficacy of the product, rather than chemical composition. An adhesive is purchased on the basis of its ability to stick things together, rather than its chemical composition and so on. Chemical Process Design and Integration R. Smith  2005 John Wiley & Sons, Ltd ISBNs: 0-471-48680-9 (HB); 0-471-48681-7 (PB) However, undifferentiated and differentiated should be thought of as relative terms rather than absolute terms for chemical products. In practice, chemicals do not tend to be completely undifferentiated or completely differentiated. Commodity and fine chemical products might have impurity specifications as well as purity specifications. Traces of impurities can, in some cases, give some differentiation between different manufacturers of commodity and fine chemicals. For example, 99.9% acrylic acid might be considered to be an undifferentiated product. However, traces of impurities, at concentrations of a few parts per million, can interfere with some of the reactions in which it is used and can have important implications for some of its uses. Such impurities might differ between different manufacturing processes. Not all specialty products are differentiated. For example, pharmaceutical products like aspirin (acetylsalicylic acid) are undifferentiated. Different manufacturers can produce aspirin and there is nothing to choose between these products, other than the price and differentiation created through marketing of the product. Scale of production also differs between the three classes of chemical products. Fine and specialty chemicals tend to be produced in volumes less than 1000 t·y−1. On the other hand, commodity chemicals tend to be produced in much larger volumes than this. However, the distinction is again not so clear. Polymers are differentiated products because they are purchased on the basis of their mechanical properties, but can be produced in quantities significantly higher than 1000 t·y−1. When a new chemical product is first developed, it can often be protected by a patent in the early years of commercial exploitation. For a product to be eligible to be patented, it must be novel, useful and unobvious. If patent protection can be obtained, this effectively gives the producer a monopoly for commercial exploitation of the product until the patent expires. Patent protection lasts for 20 years from the filing date of the patent. Once the patent expires, competitors can join in and manufacture the product. If competitors cannot wait until the patent expires, then alternative competing products must be developed. Another way to protect a competitive edge for a new product is to protect it by secrecy. The formula for Coca- Cola has been kept a secret for over 100 years. Potentially, there is no time limit on such protection. However, for the protection through secrecy to be viable, competitors must not be able to reproduce the product from chemical analysis. This is likely to be the case only for certain classes of specialty and food products for which the properties of 4 The Nature of Chemical Process Design and Integration ill-defined problem, the design team must create a series of very specific options and these should then be compared on the basis of a common set of assumptions regarding, for example, raw materials prices and product prices. Having specified an option, this gives the design team a well- defined problem to which the methods of engineering and economic analysis can be applied. In examining a design option, the design team should start out by examining the problem at the highest level, in terms of its feasibility with the minimum of detail to ensure the design option is worth progressing4. Is there a large difference between the value of the product and the cost of the raw materials? If the overall feasibility looks attractive, then more detail can be added, the option re- evaluated, further detail added, and so on. Byproducts might play a particularly important role in the economics. It might be that the current process produces some byproducts that can be sold in small quantities to the market. But, as the process is expanded, there might be market constraints for the new scale of production. If the byproducts cannot be sold, how does this affect the economics? If the design option appears to be technically and eco- nomically feasible, then additional detail can be considered. Material and energy balances can be formulated to give a better definition to the inner workings of the process and a more detailed process design can be developed. The design calculations for this will normally be solved to a high level of precision. However, a high level of preci- sion cannot usually be justified in terms of the operation of the plant after it has been built. The plant will almost never work precisely at its original design flowrates, temperatures, pressures and compositions. This might be because the raw materials are slightly different than what is assumed in the design. The physical properties assumed in the calculations might have been erroneous in some way, or operation at the original design conditions might create corrosion or foul- ing problems, or perhaps the plant cannot be controlled adequately at the original conditions, and so on, for a mul- titude of other possible reasons. The instrumentation on the plant will not be able to measure the flowrates, tem- peratures, pressures and compositions as accurately as the calculations performed. High precision might be required for certain specific parts of the design. For example, the polymer precursor might need certain impurities to be very tightly controlled, perhaps down to the level of parts per million. It might be that some contaminant in a waste stream might be exceptionally environmentally harmful and must be extremely well defined in the design calculations. Even though a high level of precision cannot be justified in many cases in terms of the plant operation, the design calculations will normally be carried out to a reasonably high level of precision. The value of precision in design calculations is that the consistency of the calculations can be checked to allow errors or poor assumptions to be identified. It also allows the design options to be compared on a valid like-for-like basis. Because of all the uncertainties in carrying out a design, the specifications are often increased beyond those indicated by the design calculations and the plant is overdesigned, or contingency is added, through the application of safety factors to the design. For example, the designer might calculate the number of distillation plates required for a distillation separation using elaborate calculations to a high degree of precision, only to add an arbitrary extra 10% to the number of plates for contingency. This allows for the feed to the unit not being exactly as specified, errors in the physical properties, upset conditions in the plant, control requirements, and so on. If too little contingency is added, the plant might not work. If too much contingency is added, the plant will not only be unnecessarily expensive, but too much overdesign might make the plant difficult to operate and might lead to a less efficient plant. For example, the designer might calculate the size of a heat exchanger and then add in a large contingency and significantly oversize the heat exchanger. The lower fluid velocities encountered by the oversized heat exchanger can cause it to have a poorer performance and to foul up more readily than a smaller heat exchanger. Thus, a balance must be made between different risks. In summary, the original problem posed to process design teams is often ill-defined, even though it might appear to be well defined in the original design specification. The design team must then formulate a series of plausible design options to be screened by the methods of engineering and economic analysis. These design options are formulated into very specific design problems. Some design options might be eliminated early by high-level arguments or simple calculations. Others will require more detailed examination. In this way, the design team turns the ill-defined problem into a well-defined one for analysis. To allow for the many unquantifiable uncertainties, overdesign is used. Too little overdesign might lead to the plant not working. Too much overdesign will lead to the plant becoming unnecessarily expensive, and perhaps difficult to operate and less efficient. A balance must be made between different risks. Consider the basic features of the design of chemical processes now. 1.3 CHEMICAL PROCESS DESIGN AND INTEGRATION In a chemical process, the transformation of raw materials into desired chemical products usually cannot be achieved in a single step. Instead, the overall transformation is bro- ken down into a number of steps that provide intermediate transformations. These are carried out through reaction, sep- aration, mixing, heating, cooling, pressure change, particle size reduction or enlargement. Once individual steps have been selected, they must be interconnected to carry out the The Hierarchy of Chemical Process Design and Integration 5 Feed Streams Feed Streams Product Streams Product Streams (a) Process design starts with the synthesis of a process to convert raw materials into desired products. (b) Simulation predicts how a process would behave if it was constructed. d. ? Figure 1.2 Synthesis is the creation of a process to transform feed streams into product streams. Simulation predicts how it would behave if it was constructed. overall transformation (Figure 1.2a). Thus, the synthesis of a chemical process involves two broad activities. First, indi- vidual transformation steps are selected. Second, these indi- vidual transformations are interconnected to form a complete process that achieves the required overall transformation. A flowsheet is a diagrammatic representation of the process steps with their interconnections. Once the flowsheet structure has been defined, a simulation of the process can be carried out. A simulation is a mathematical model of the process that attempts to predict how the process would behave if it were constructed (Figure 1.2b). Having created a model of the process, the flowrates, compositions, temperatures and pressures of the feeds can be assumed. The simulation model then predicts the flowrates, compositions, temperatures, and pressures of the products. It also allows the individual items of equipment in the process to be sized and predicts, for example, how much raw material is being used or how much energy is being consumed. The performance of the design can then be evaluated. There are many facets to the evaluation of performance. Good economic performance is an obvious first criterion, but it is certainly not the only one. Chemical processes should be designed as part of a sustainable industrial activity that retains the capacity of ecosystems to support both life and industrial activity into the future. Sustainable industrial activity must meet the needs of the present, without compromising the needs of future generations. For chemical process design, this means that processes should use raw materials as efficiently as is economic and practicable, both to prevent the production of waste that can be environmentally harmful and to preserve the reserves of raw materials as much as possible. Processes should use as little energy as is economic and practicable, both to prevent the build-up of carbon dioxide in the atmosphere from burning fossil fuels and to preserve the reserves of fossil fuels. Water must also be consumed in sustainable quantities that do not cause deterioration in the quality of the water source and the long-term quantity of the reserves. Aqueous and atmospheric emissions must not be environmentally harmful, and solid waste to landfill must be avoided. The process must also meet required health and safety criteria. Start-up, emergency shutdown and ease of control are other important factors. Flexibility, that is, the ability to operate under different conditions, such as differences in feedstock and product specification, may be important. Availability, that is, the number of operating hours per year, may also be critically important. Uncertainty in the design, for example, resulting from poor design data, or uncertainty in the economic data, might guide the design away from certain options. Some of these factors, such as economic performance, can be readily quantified; others, such as safety, often cannot. Evaluation of the factors that are not readily quantifiable, the intangibles, requires the judgment of the design team. Once the basic performance of the design has been eval- uated, changes can be made to improve the performance; the process is optimized. These changes might involve the synthesis of alternative structures, that is, structural opti- mization. Thus, the process is simulated and evaluated again, and so on, optimizing the structure. Alternatively, each structure can be subjected to parameter optimization by changing operating conditions within that structure. 1.4 THE HIERARCHY OF CHEMICAL PROCESS DESIGN AND INTEGRATION Consider the process illustrated in Figure 1.35. The process requires a reactor to transform the FEED into PRODUCT (Figure 1.3a). Unfortunately, not all the FEED reacts. Also, part of the FEED reacts to form BYPRODUCT instead of the desired PRODUCT. A separation system is needed to isolate the PRODUCT at the required purity. Figure 1.3b shows one possible separation system consisting of two distillation columns. The unreacted FEED in Figure 1.3b is recycled, and the PRODUCT and BYPRODUCT are removed from the process. Figure 1.3b shows a flowsheet where all heating and cooling is provided by external utilities (steam and cooling water in this case). This flowsheet is probably too inefficient in its use of energy, and heat would be recovered. Thus, heat integration is carried out to exchange heat between those streams that need to be cooled and those that need to be heated. Figure 1.45 shows two possible designs for the heat exchanger network, but many other heat integration arrangements are possible. The flowsheets shown in Figure 1.4 feature the same reactor design. It could be useful to explore the changes in reactor design. For example, the size of the reactor could be increased to increase the amount of FEED that reacts5. 6 The Nature of Chemical Process Design and Integration BYPRODUCT Steam FEED FEED PRODUCT BYPRODUCT CW Steam Reactor CW CW Steam PRODUCT BYPRODUCT (b) To isolate the PRODUCT and recycle unreacted FEED a separation system is needed. (a) A reactor transforms FEED into PRODUCT and BYPRODUCT. Unreacted FEED PRODUCT Steam FEED FEED PRODUCT BYPRODUCT Reactor Figure 1.3 Process design starts with the reactor. The reactor design dictates the separation and recycle problem. (From Smith R and Linnhoff B, 1998, Trans IChemE ChERD, 66:195 by permission of the Institution of Chemical Engineers). Now, there is not only much less FEED in the reactor effluent but also more PRODUCT and BYPRODUCT. However, the increase in BYPRODUCT is larger than the increase in PRODUCT. Thus, although the reactor has the same three components in its effluent as the reactor in Figure 1.3a, there is less FEED, more PRODUCT and significantly more BYPRODUCT. This change in reactor design generates a different task for the separation system, and it is possible that a separation system different from that shown in Figures 1.3 and 1.4 is now appropriate. Figure 1.5 shows a possible alternative. This also uses two distillation columns, but the separations are carried out in a different order. Figure 1.5 shows a flowsheet without any heat integra- tion for the different reactor and separation system. As before, this is probably too inefficient in the use of energy, and heat integration schemes can be explored. Figure 1.65 shows two of the many possible flowsheets. Different complete flowsheets can be evaluated by simulation and costing. On this basis, the flowsheet in Figure 1.4b might be more promising than the flowsheets in Figures 1.4a, 1.6a and b. However, the best flowsheet can- not be identified without first optimizing the operating con- ditions for each. The flowsheet in Figure 1.6b might have greater scope for improvement than that in Figure 1.4b, and so on. Thus, the complexity of chemical process synthesis is twofold. First, can all possible structures be identified? It might be considered that all the structural options can be found by inspection, at least all of the significant ones. The fact that even long-established processes are still being improved bears evidence to just how difficult this is. Second, can each structure be optimized for a valid comparison? When optimizing the structure, there may be many ways in which each individual task can be performed and many ways in which the individual tasks can be interconnected. This means that the operating conditions for a multitude of structural options must be simulated and optimized. At first sight, this appears to be an overwhelmingly complex problem. It is helpful when developing a methodology if there is a clearer picture of the nature of the problem. If the process requires a reactor, this is where the design starts. This is likely to be the only place in the process where raw materials are converted into products. The chosen reactor design produces a mixture of unreacted feed materials, products and byproducts that need separating. Unreacted feed material is recycled. The reactor design dictates the separation and recycle problem. Thus, design of the separation and recycle system follows the reactor design. The reactor and separation and recycle system designs together define the process for heating and cooling duties. Continuous and Batch Processes 9 Separation and. Recycle System Heat Recovery System Reactor Heating and Cooling Utilities Water and Effluent Treatment Figure 1.7 The onion model of process design. A reactor is needed before the separation and recycle system can be designed and so on. process design. Usually, there are many options, and it is impossible to fully evaluate them unless a complete design is furnished for the “outer layers” of the onion. For example, it is not possible to assess which is better, the basic scheme from Figure 1.3b or that from Figure 1.5, without fully evaluating all possible designs, such as those shown in Figures 1.4a and b and Figures 1.6a and b, all completed, including utilities. Such a complete search is normally too time consuming to be practical. Later, an approach will be presented in which some early decisions (i.e. decisions regarding reactor and separator options) can be evaluated without a complete design for the “outer layers”. 1.5 CONTINUOUS AND BATCH PROCESSES When considering the processes in Figures 1.3 to 1.5, an implicit assumption was made that the processes operated continuously. However, not all processes operate continuously. In a batch process, the main steps operate discontinuously. In contrast with a continuous process, a batch process does not deliver its product continuously but in discrete amounts. This means that heat, mass, temper- ature, concentration and other properties vary with time. In practice, most batch processes are made up of a series of batch and semicontinuous steps. A semicontinuous step runs continuously with periodic start-ups and shutdowns. Consider the simple process shown in Figure 1.8. Feed material is withdrawn from storage using a pump. The feed material is preheated in a heat exchanger before being fed to a batch reactor. Once the reactor is full, further heating takes place inside the reactor by passing steam into the reactor jacket, before the reaction proceeds. During the later stages of the reaction, cooling water is applied to the reactor jacket. Once the reaction is complete, the reactor product is withdrawn using a pump. The reactor product is cooled in a heat exchanger before going to storage. The first two steps, pumping for reactor filling and feed preheating are both semicontinuous. The heating inside the reactor, the reaction itself and the cooling using the reactor jacket are all batch. The pumping to empty the reactor and the product-cooling step are again semicontinuous. The hierarchy in batch process design is no different from that in continuous processes and the hierarchy illustrated in Figure 1.7 prevails for batch processes also. However, the time dimension brings constraints that do not present a problem in the design of continuous processes. For example, heat recovery might be considered for the process in Figure 1.8. The reactor effluent (that requires cooling) could be used to preheat the incoming feed to the reactor (that requires heating). Unfortunately, even if the reactor effluent is at a high enough temperature to allow this, the reactor feeding and emptying take place at different times, meaning that this will not be possible without some way to store the heat. Such heat storage is possible but usually uneconomic, especially for small-scale processes. If a batch process manufactures only a single product, then the equipment can be designed and optimized for Feed Storage Steam Steam Cooling Water REACTOR Cooling Water Product Storage Figure 1.8 A simple batch process. 10 The Nature of Chemical Process Design and Integration that product. The dynamic nature of the process creates additional challenges for design and optimization. It might be that the optimization calls for variations in the conditions during the batch through time, according to some profile. For example, the temperature in a batch reactor might need to be increased or decreased as the batch progresses. Multiproduct batch processes, with a number of different products manufactured in the same equipment, present even bigger challenges for design and optimization7. Different products will demand different designs, different operating conditions and, perhaps, different trajectories for the operating conditions through time. The design of equipment for multiproduct plants will thus require a compromise to be made across the requirements of a number of different products. The more flexible the equipment and the configuration of the equipment, the more it will be able to adapt to the optimum requirements of each product. Batch processes • are economical for small volumes; • are flexible in accommodating changes in product formulation; • are flexible in changing production rate by changing the number of batches made in any period of time; • allow the use of standardized multipurpose equipment for the production of a variety of products from the same plant; • are best if equipment needs regular cleaning because of fouling or needs regular sterilization; • are amenable to direct scale-up from the laboratory and • allow product identification. Each batch of product can be clearly identified in terms of when it was manufactured, the feeds involved and conditions of processing. This is particularly important in industries such as pharmaceuticals and foodstuffs. If a problem arises with a particular batch, then all the products from that batch can be identified and withdrawn from the market. Otherwise, all the products available in the market would have to be withdrawn. One of the major problems with batch processing is batch- to-batch conformity. Minor changes to the operation can mean slight changes in the product from batch to batch. Fine and specialty chemicals are usually manufactured in batch processes. Yet, these products often have very tight tolerances for impurities in the final product and demand batch-to-batch variation being minimized. Batch processes will be considered in more detail in Chapter 14. 1.6 NEW DESIGN AND RETROFIT There are two situations that can be encountered in process design. The first is in the design of new plant or grassroot design. In the second, the design is carried out to modify an existing plant in retrofit or revamp. The motivation to retrofit an existing plant could be, for example, to increase capacity, allow for different feed or product specifications, reduce operating costs, improve safety or reduce environmental emissions. One of the most common motivations is to increase capacity. When carrying out a retrofit, whatever the motivation, it is desirable to try and make as effective use as possible of the existing equipment. The basic problem with this is that the design of the existing equipment might not be ideally suited to the new role that it will be put to. On the other hand, if equipment is reused, it will avoid unnecessary investment in new equipment, even if it is not ideally suited to the new duty. When carrying out a retrofit, the connections between the items of equipment can be reconfigured, perhaps adding new equipment where necessary. Alternatively, if the existing equipment differs significantly from what is required in the retrofit, then in addition to reconfiguring the connections between the equipment, the equipment itself can be modified. Generally, the fewer the modifications to both the connections and the equipment, the better. The most straightforward design situations are those of grassroot design as it has the most freedom to choose the design options and the size of equipment. In retrofit, the design must try to work within the constraints of existing equipment. Because of this, the ultimate goal of the retrofit design is often not clear. For example, a design objective might be given to increase the capacity of a plant by 50%. At the existing capacity limit of the plant, at least one item of equipment must be at its maximum capacity. Most items of equipment are likely to be below their maximum capacity. The differences in the spare capacity of different items of equipment in the existing design arises from errors in the original design data, different design allowances (or contingency) in the original design, changes to the operation of the plant relative to the original design, and so on. An item of equipment at its maximum capacity is the bottleneck to prevent increased capacity. Thus, to overcome the bottleneck or debottleneck, the item of equipment is modified, or replaced with new equipment with increased capacity, or a new item is placed in parallel or series with the existing item, or the connections between existing equipment are reconfigured, or a combination of all these actions is taken. As the capacity of the plant is increased, different items of equipment will reach their maximum capacity. Thus, there will be thresholds in the plant capacity, created by the limits in different items of equipment. All equipment with capacity less than the threshold must be modified in some way, or the plant reconfigured, to overcome the threshold. To overcome each threshold requires capital investment. As capacity is increased from the existing limit, ultimately, it is likely that it will be prohibitive for the investment to overcome one of the design thresholds. This is likely to become the design Approaches to Chemical Process Design and Integration 11 limit, as opposed to the original remit of a 50% increase in capacity in the example. 1.7 APPROACHES TO CHEMICAL PROCESS DESIGN AND INTEGRATION In broad terms, there are two approaches to chemical process design and integration: 1. Building an irreducible structure: The first approach follows the “onion logic”, starting the design by choosing a reactor and then moving outward by adding a separation and recycle system, and so on. At each layer, decisions must be made on the basis of the information available at that stage. The ability to look ahead to the completed design might lead to different decisions. Unfortunately, this is not possible, and, instead, decisions must be based on an incomplete picture. This approach to creation of the design involves making a series of best local decisions. This might be based on the use of heuristics or rules of thumb developed from experience4 on a more systematic approach. Equipment is added only if it can be justified economically on the basis of the information available, albeit an incomplete picture. This keeps the structure irreducible, and features that are technically or economically redundant are not included. There are two drawbacks to this approach: (a) Different decisions are possible at each stage of the design. To be sure that the best decisions have been made, the other options must be evaluated. However, each option cannot be evaluated properly without com- pleting the design for that option and optimizing the operating conditions. This means that many designs must be completed and optimized in order to find the best. (b) Completing and evaluating many options gives no guarantee of ultimately finding the best possible design, as the search is not exhaustive. Also, complex interactions can occur between different parts of a flowsheet. The effort to keep the system simple and not add features in the early stages of design may result in missing the benefit of interactions between different parts of the flowsheet in a more complex system. The main advantage of this approach is that the design team can keep control of the basic decisions and interact as the design develops. By staying in control of the basic decisions, the intangibles of the design can be included in the decision making. 2. Creating and optimizing a superstructure. In this approach, a reducible structure, known as a superstructure, is first created that has embedded within it all feasible pro- cess options and all feasible interconnections that are can- didates for an optimal design structure. Initially, redundant features are built into the superstructure. As an example, consider Figure 1.98. This shows one possible structure of a process for the manufacture of benzene from the reaction between toluene and hydrogen. In Figure 1.9, the hydro- gen enters the process with a small amount of methane as an impurity. Thus, in Figure 1.9, the option of either puri- fying the hydrogen feed with a membrane or of passing it directly to the process is embedded. The hydrogen and toluene are mixed and preheated to reaction temperature. Only a furnace has been considered feasible in this case because of the high temperature required. Then, the two alternative reactor options, isothermal and adiabatic reac- tors, are embedded, and so on. Redundant features have been included in an effort to ensure that all features that could be part of an optimal solution have been included. The design problem is next formulated as a mathematical model. Some of the design features are continuous, describing the operation of each unit (e.g. flowrate, composition, temperature and pressure), its size (e.g. volume, heat transfer area, etc.) as well as the costs or profits associated with the units. Other features are discrete (e.g. a connection in the flowsheet is included or not, a membrane separator is included or not). Once the problem is formulated mathematically, its solution is carried out through the implementation of an optimization algorithm. An objective function is maximized or minimized (e.g. profit is maximized or cost is minimized) in a structural and parameter optimization. The optimization justifies the existence of structural features and deletes those features from the structure that cannot be justified economically. In this way, the structure is reduced in complexity. At the same time, the operating conditions and equipment sizes are also optimized. In effect, the discrete decision-making aspects of process design are replaced by a discrete/continuous optimization. Thus, the initial structure in Figure 1.9 is optimized to reduce the structure to the final design shown in Figure 1.108. In Figure 1.10, the membrane separator on the hydrogen feed has been removed by optimization, as has the isothermal reactor and many other features of the initial structure shown in Figure 1.9. There are a number of difficulties associated with this approach: (a) The approach will fail to find the optimal structure if the initial structure does not have the optimal structure embedded somewhere within it. The more options included, the more likely it will be that the optimal structure has been included. (b) If the individual unit operations are represented accu- rately, the resulting mathematical model will be extremely large and the objective function that must be optimized will be extremely irregular. The profile of the objective function can be like the terrain in a 14 The Nature of Chemical Process Design and Integration If the transitory states were to have a significant effect on the overall process performance in terms of the objec- tive function being optimized, then the process design and control system design would have to be carried out simul- taneously. Simultaneous design of the process and the con- trol system presents an extremely complex problem. It is interesting to note that where steady-state optimization for continuous processes has been compared with simultaneous optimization of the process and control system, the two pro- cess designs have been found to be almost identical10 – 12. Industrial practice is to first design and optimize the process configuration (taking into account multiple states, if necessary) and then to add the control system. However, there is no guarantee that design decisions made on the basis of steady-state conditions will not lead to control problems once process dynamics are considered. For example, an item of equipment might be oversized for contingency, because of uncertainty in design data or future debottlenecking prospects, based on steady-state considerations. Once the process dynamics are considered, this oversized equipment might make the process difficult to control, because of the large inventory of process materials in the oversized equipment. The approach to process control should adopt an approach that considers the control of the whole process, rather than just the control of the individual process steps in isolation13. This text will concentrate on the design and optimization of the process configuration and will not deal with process control. Process control demands expertise in different techniques and will be left to other sources of information13. Thus, the text will describe how to develop a flowsheet or process flow diagram, but will not take the final step of adding the instrumentation, control and auxiliary pipes and valves required for the final engineering design in the piping and instrumentation diagram (P & I D). Batch processes are, by their nature, always in a transitory state. This requires the dynamics of the process to be optimized, and will be considered in Chapter 14. However, the control systems required to put this into practice will not be considered. 1.9 THE NATURE OF CHEMICAL PROCESS DESIGN AND INTEGRATION – SUMMARY Chemical products can be divided into three broad classes: commodity, fine and specialty chemicals. Commodity chemicals are manufactured in large volumes with low added value. Fine and specialty chemicals tend to be manufactured in low volumes with high added value. The priorities in the design of processes for the manufacture of the three classes of chemical products will differ. The original design problem posed to the design team is often ill-defined, even if it appears on the surface to be well-defined. The design team must formulate well-defined design options from the original ill-defined problem, and these must be compared on the basis of consistent criteria. The design might be new or a retrofit of an existing process. If the design is a retrofit, then one of the objectives should be to maximize the use of existing equipment, even if it is not ideally suited to its new purpose. Both continuous and batch process operations can be used. Batch processes are generally preferred for small- scale and specialty chemicals production. When developing a chemical process design, there are two basic problems: • Can all possible structures be identified? • Can each structure be optimized such that all structures can be compared on a valid basis? Design starts at the reactor, because it is likely to be the only place in the process where raw materials are converted into the desired chemical products. The reactor design dictates the separation and recycle problem. Together, the reactor design and separation and recycle dictate the heating and cooling duties for the heat exchanger network. Those duties that cannot be satisfied by heat recovery dictate the need for external heating and cooling utilities. The process and the utility system both have a demand for water and create aqueous effluents, giving rise to the water system. This hierarchy is represented by the layers in the “onion diagram”, Figure 1.7. Both continuous and batch process design follow this hierarchy, even though the time dimension in batch processes brings additional constraints in process design. There are two general approaches to chemical pro- cess design: • building an irreducible structure; • creating and optimizing a super structure. Both of these approaches have advantages and disadvan- tages. REFERENCES 1. Sharratt PN (1997) Handbook of Batch Process Design, Blackie Academic and Professional. 2. Brennan D (1998) Process Industry Economics, IChemE, UK. 3. Cussler EL and Moggridge GD (2001) Chemical Product Design, Cambridge University Press. 4. Douglas JM (1985) A Hierarchical Decision Procedure for Process Synthesis, AIChE J, 31: 353. 5. Smith R and Linnhoff B, (1988) The Design of Separators in the Context of Overall Processes, Trans IChemE ChERD, 66: 195. References 15 6. Linnhoff B, Townsend DW, Boland D, Hewitt GF, Thomas BEA, Guy AR and Marsland RH (1982) A User Guide on Process Integration for the Efficient Use of Energy, IChemE, Rugby, UK. 7. Biegler LT, Grossmann IE and Westerberg AW (1997), Sys- tematic Methods of Chemical Process Design, Prentice Hall. 8. Kocis GR and Grossmann IE (1988) A Modelling and Decomposition Strategy for the MINLP Optimization of Process Flowsheets, Comput Chem Eng, 13: 797. 9. Floudas CA (2000) Deterministic Global Optimization: The- ory, Methods and Applications, Kluwer Academic Publishers. 10. Bansal V, Perkins JD, Pistikopoulos EN, Ross R and van Shijnedel JMG (2000) Simultaneous Design and Control Optimisation Under Uncertainty, Comput Chem Eng, 24: 261. 11. Bansal V, Ross R, Perkins JD, Pistikopoulos EN (2000) The Interactions of Design and Control: Double Effect Distillation, J Process Control, 10: 219. 12. Kookos IK and Perkins JD (2001) An Algorithm for Simul- taneous Process Design and Control, Ind Eng Chem Res, 40: 4079. 13. Luyben WL, Tyreus BD and Luyben ML (1999) Plant-wide Process Control, McGraw Hill. Capital Cost for New Design 19 magazine and the Nelson–Farrar Cost Indexes for refinery construction (1946 index = 100) published in the Oil and Gas Journal. The Chemical Engineering (CE) Indexes are particularly useful. CE Indexes are available for equipment covering: • Heat Exchangers and Tanks • Pipes, Valves and Fittings • Process Instruments • Pumps and Compressors • Electrical Equipment • Structural Supports and Miscellaneous. A combined CE Index of Equipment is available. CE Indexes are also available for: • Construction and Labor Index • Buildings Index • Engineering and Supervision Index. All of the above indexes are combined to produce a CE Composite Index. Table 2.1 presents data for a number of equipment items on the basis of January 2000 costs7 (CE Composite Index = 391.1, CE Index of Equipment = 435.8). Cost correlations for vessels are normally expressed in terms of the mass of the vessel. This means that not only a preliminary sizing of the vessel is required but also a preliminary assessment of the mechanical design9,10. Materials of construction have a significant influence on the capital cost of equipment. Table 2.2 gives some approximate average factors to relate the different materials of construction for equipment capital cost. It should be emphasized that the factors in Table 2.2 are average and only approximate and will vary, amongst other things, according to the type of equipment. As an example, consider the effect of materials of construction on the cap- ital cost of distillation columns. Table 2.3 gives materials of construction cost factors for distillation columns. The cost factors for shell-and-tube heat exchangers are made more complex by the ability to construct the differ- ent components from different materials of construction. Table 2.2 Typical average equipment materials of construction capital cost factors. Material Correction factor fM Carbon steel 1.0 Aluminum 1.3 Stainless steel (low grades) 2.4 Stainless steel (high grades) 3.4 Hastelloy C 3.6 Monel 4.1 Nickel and inconel 4.4 Titanium 5.8 Table 2.3 Typical materials of construction capital cost factors for pressure vessels and distillation columns9,10. Material Correction factor fM Carbon steel 1.0 Stainless steel (low grades) 2.1 Stainless steel (high grades) 3.2 Monel 3.6 Inconel 3.9 Nickel 5.4 Titanium 7.7 Table 2.4 Typical materials of construction capital cost factors for shell-and-tube heat exchangers2. Material Correction factor fM CS shell and tubes 1.0 CS shell, aluminum tubes 1.3 CS shell, monel tubes 2.1 CS shell, SS (low grade) tubes 1.7 SS (low grade) shell and tubes 2.9 Table 2.4 gives typical materials of construction factors for shell-and-tube heat exchangers. Its operating pressure also influences equipment capital cost as a result of thicker vessel walls to withstand increased pressure. Table 2.5 presents typical factors to account for the pressure rating. As with materials of construction correction factors, the pressure correction factors in Table 2.5 are average and only approximate and will vary, amongst other things, according to the type of equipment. Finally, its operating temperature also influences equipment capital cost. This is caused by, amongst other factors, a decrease in the allow- able stress for materials of construction as the temperature increases. Table 2.6 presents typical factors to account for the operating temperature. Thus, for a base cost for carbon steel equipment at moderate pressure and temperature, the actual cost can be Table 2.5 Typical equipment pressure capital cost factors. Design pressure (bar absolute) Correction factor fP 0.01 2.0 0.1 1.3 0.5 to 7 1.0 50 1.5 100 1.9 20 Process Economics Table 2.6 Typical equipment temperature capital cost factors. Design temperature (◦C) Correction factor fT 0–100 1.0 300 1.6 500 2.1 estimated from: CE = CB ( Q QB )M fMfP fT (2.3) where CE = equipment cost for carbon steel at moderate pressure and temperature with capacity Q CB = known base cost for equipment with capacity QB M = constant depending on equipment type fM = correction factor for materials of construction fP = correction factor for design pressure fT = correction factor for design temperature In addition to the purchased cost of the equipment, investment is required to install the equipment. Installation costs include: • cost of installation • piping and valves • control systems • foundations • structures • insulation • fire proofing • electrical • painting • engineering fees • contingency. The total capital cost of the installed battery limits equipment will normally be two to four times the purchased cost of the equipment11,12. In addition to the investment within the battery limits, investment is also required for the structures, equipment and services outside of the battery limits that are required to make the process function. 2. Utility investment: Capital investment in utility plant could include equipment for: • electricity generation • electricity distribution • steam generation • steam distribution • process water • cooling water • firewater • effluent treatment • refrigeration • compressed air • inert gas (nitrogen). The cost of utilities is considered from their sources within the site to the battery limits of the chemical process served. 3. Off-site investment: Off-site investment includes • auxiliary buildings such as offices, medical, person- nel, locker rooms, guardhouses, warehouses and main- tenance shops • roads and paths • railroads • fire protection systems • communication systems • waste disposal systems • storage facilities for end product, water and fuel not directly connected with the process • plant service vehicles, loading and weighing devices. The cost of the utilities and off-sites (together sometimes referred to as services) ranges typically from 20 to 40% of the total installed cost of the battery limits plant13. In general terms, the larger the plant, the larger will tend to be the fraction of the total project cost that goes to utilities and off-sites. In other words, a small project will require typically 20% of the total installed cost for utilities and off-sites. For a large project, the figure can be typically up to 40%. 4. Working capital: Working capital is what must be invested to get the plant into productive operation. This is money invested before there is a product to sell and includes: • raw materials for plant start-up (including wasted raw materials) • raw materials, intermediate and product inventories • cost of transportation of materials for start-up • money to carry accounts receivable (i.e. credit extended to customers) less accounts payable (i.e. credit extended by suppliers) • money to meet payroll when starting up. Theoretically, in contrast to fixed investment, this money is not lost but can be recovered when the plant is closed down. Stocks of raw materials, intermediate and product inventories often have a key influence on the working capital and are under the influence of the designer. Issues relating to storage will be discussed in more detail in Capital Cost for New Design 21 Chapters 13 and 14. For an estimate of the working capital requirements, take either14: (a) 30% of annual sales, or (b) 15% of total capital investment. 5. Total capital cost: The total capital cost of the process, services and working capital can be obtained by applying multiplying factors or installation factors to the purchase cost of individual items of equipment11,12: CF = ∑ i fiCE,i (2.4) where CF = fixed capital cost for the complete system CE,i = cost of Equipment i fi = installation factor for Equipment i If an average installation factor for all types of equipment is to be used11, CF = fI ∑ i CE,i (2.5) where fI = overall installation factor for the complete system. The overall installation factor for new design is broken down in Table 2.7 into component parts according to the dominant phase being processed. The cost of the installation will depend on the balance of gas and liquid processing versus solids processing. If the plant handles only gases and liquids, it can be characterized as fluid processing. A plant can be characterized as solids processing if the bulk of the material handling is solid phase. For example, a solid processing plant could be a coal or an ore preparation plant. Between the two extremes of fluid processing and solids processing are processes that handle a significant amount of both solids and fluids. For example, a shale oil plant involves preparation of the shale oil followed by extraction of fluids from the shale oil and then separation and processing of the fluids. For these types of plant, the contributions to the capital cost can be estimated from the two extreme values in Table 2.7 by interpolation in proportion of the ratio of major processing steps that can be characterized as fluid processing and solid processing. A number of points should be noted about the various contributions to the capital cost in Table 2.7. The val- ues are: • based on carbon steel, moderate operating pressure and temperature • average values for all types of equipment, whereas in practice the values will vary according to the type of equipment • only guidelines and the individual components will vary from project to project • applicable to new design only. Table 2.7 Typical factors for capital cost based on delivered equipment costs. Item Type of process Fluid processing Solid processing Direct costs Equipment delivered cost 1 1 Equipment erection, fER 0.4 0.5 Piping (installed), fPIP 0.7 0.2 Instrumentation & controls (installed), fINST 0.2 0.1 Electrical (installed), fELEC 0.1 0.1 Utilities, fUTIL 0.5 0.2 Off-sites, fOS 0.2 0.2 Buildings (including services), fBUILD 0.2 0.3 Site preparation, fSP 0.1 0.1 Total capital cost of installed 3.4 2.7 equipment Indirect costs Design, engineering and construction, fDEC 1.0 0.8 Contingency (about 10% of fixed capital costs), fCONT 0.4 0.3 Total fixed capital cost 4.8 3.8 Working capital Working capital (15% of total capital cost), fWC 0.7 0.6 Total capital cost, f I 5.8 4.4 When equipment uses materials of construction other than carbon steel, or operating temperatures are extreme, the capital cost needs to be adjusted accordingly. Whilst the equipment cost and its associated pipework will be changed, the other installation costs will be largely unchanged, whether the equipment is manufactured from carbon steel or exotic materials of construction. Thus, the application of the factors from Tables 2.2 to 2.6 should only be applied to the equipment and pipework: CF = ∑ i [fMfP fT (1 + fPIP )]iCE,i + (fER + fINST + fELEC + fUTIL + fOS + fBUILD + fSP + fDEC + fCONT + fWS ) ∑ i CE,i (2.6) Thus, to estimate the fixed capital cost: 1. list the main plant items and estimate their size; 2. estimate the equipment cost of the main plant items; 24 Process Economics detailed examination of the individual features of retrofit projects is necessary. Example 2.2 An existing heat exchanger is to be repiped to a new duty in a retrofit project without moving its location. The only significant investment is piping modifications. The heat transfer area of the existing heat exchanger is 500 m2. The material of construction is low-grade stainless steel, and its design pressure is 5 bar. Estimate the cost of the project (CE Index of Equipment = 441.9). Solution All retrofit projects have individual characteristics, and it is impossible to generalize the costs. The only way to estimate costs with any certainty is to analyze the costs of all of the modifications in detail. However, in the absence of such detail, a very preliminary estimate can be obtained by estimating the retrofit costs from the appropriate installation costs for a new design. In this case, piping costs can be estimated from those for a new heat exchanger of the same specification, but excluding the equipment cost. For Example 2.1, the cost of a new stainless steel heat exchanger with an area of 500 m2 was estimated to be $11.6 × 104. The piping costs (stainless steel) can therefore be estimated to be: Piping cost = fMfPIPCE = 2.9 × 0.7 × 11.6 × 104 = 2.03 × 11.6 × 104 = $2.35 × 105 This estimate should not be treated with any confidence. It will give an idea of the costs and might be used to compare retrofit options on a like-for-like basis, but could be very misleading. Example 2.3 An existing distillation column is to be revamped to increase its capacity by replacing the existing sieve trays with stainless steel structured packing. The column shell is 46 m tall and 1.5 m diameter and currently fitted with 70 sieve trays with a spacing of 0.61 m. The existing trays are to be replaced with stainless steel structured packing with a total height of 30 m. Estimate the cost of the project (CE Index of Equipment = 441.9). Solution First, estimate the purchase cost of the new structured packing from Equation 2.1 and Table 2.1, which gives costs for a 5-m height of packing: CE = CB ( Q QB )M = 1.8 × 104 × 30 5 ( 1.5 0.5 )1·7 = $6.99 × 105 Adjusting the cost to bring it up-to-date using the ratio of cost indexes: CE = 6.99 × 105 ( 441.9 435·8 ) = $7.09 × 105 From Table 2.8, the factor for removing the existing trays is 0.1 and that for installing the new packing is 0.5 to 0.8 (say 0.7). Estimated total cost of the project: = (1 + 0.1 + 0.7)7.09 × 105 = $1.28 × 106 2.4 ANNUALIZED CAPITAL COST Capital for new installations may be obtained from: a. Loans from banks b. Issue by the company of common (ordinary) stock, preferred stock or bonds (debenture stock) c. Accumulated net cash flow arising from company profit over time. Interest on loans from banks, preferred stock and bonds is paid at a fixed rate of interest. A share of the profit of the company is paid as a dividend on common stock and preferred stock (in addition to the interest paid on preferred stock). The cost of the capital for a project thus depends on its source. The source of the capital often will not be known during the early stages of a project, and yet there is a need to select between process options and carry out optimization on the basis of both capital and operating costs. This is difficult to do unless both capital and operating costs can be expressed on a common basis. Capital costs can be expressed on an annual basis if it is assumed that the capital has been borrowed over a fixed period (usually 5 to 10 years) at a fixed rate of interest, in which case the capital cost can be annualized according to Annualized capital cost = capital cost × i(1 + i) n (1 + i)n − 1 (2.7) where i = fractional interest rate per year n = number of years The derivation of Equation 2.7 is given in Appendix A. As stated previously, the source of capital is often not known, and hence it is not known whether Equation 2.7 is appropriate to represent the cost of capital. Equation 2.7 is, strictly speaking, only appropriate if the money for capital expenditure is to be borrowed over a fixed period at a fixed rate of interest. Moreover, if Equation 2.7 is accepted, then the number of years over which the capital is to be annualized is known, as is the rate of interest. However, the most important thing is that, even if the source of capital is not known, and uncertain assumptions are necessary, Equation 2.7 provides a common basis for the comparison of competing projects and design alternatives within a project. Example 2.4 The purchased cost of a new distillation column installation is $1 million. Calculate the annual cost of installed capital if the capital is to be annualized over a five-year period at a fixed rate of interest of 5%. Operating Cost 25 Solution First, calculate the installed capital cost: CF = fiCE = 5.8 × (1,000,000) = $5,800,000 Annualization factor = i(1 + i) n (1 + i)n − 1 = 0.05(1 + 0.05) 5 (1 + 0.05)5 − 1 = 0.2310 Annualized capital cost = 5, 800, 000 × 0.2310 = $1,340,000 y−1 When using annualized capital cost to carry out optimiza- tion, the designer should not lose sight of the uncertainties involved in the capital annualization. In particular, changing the annualization period can lead to very different results when, for example, carrying out a trade-off between energy and capital costs. When carrying out optimization, the sen- sitivity of the result to changes in the assumptions should be tested. 2.5 OPERATING COST 1. Raw materials cost: In most processes, the largest indi- vidual operating cost is raw materials. The cost of raw mate- rials and the product selling prices tend to have the largest influence on the economic performance of the process. The cost of raw materials and price of products depends on whether the materials in question are being bought and sold under a contractual arrangement (either within or outside the company) or on the open market. Open market prices for some chemical products can fluctuate considerably with time. Raw materials might be purchased and products sold below or above the open market price when under a con- tractual arrangement, depending on the state of the market. Buying and selling on the open market may give the best purchase and selling prices but give rise to an uncertain economic environment. A long-term contractual agreement may reduce profit per unit of production but gives a degree of certainty over the project life. The values of raw materials and products can be found in trade journals such as Chemical Marketing Reporter (pub- lished by Schnell Publishing Company), European Chemi- cal News and Asian Chemical News (published by Reed Business Information). However, the values reported in such sources will be subject to short-term fluctuations, and long-term forecasts will be required for investment analysis. 2. Catalysts and chemicals consumed in manufacturing other than raw materials: Catalysts will need to be replaced or regenerated though the life of a process (see Chapter 7). The replacement of catalysts might be on a continuous basis if homogeneous catalysts are used (see Chapters 5 and 7). Heterogeneous catalysts might also be replaced continuously if they deteriorate rapidly, and regeneration cannot fully reinstate the catalyst activity. More often for heterogeneous catalysts, regeneration or replacement will be carried out on an intermittent basis, depending on the characteristics of the catalyst deactivation. In addition to the cost of catalysts, there might be significant costs associated with chemicals consumed in manufacturing that do not form part of the final product. For example, acids and alkalis might be consumed to adjust the pH of streams. Such costs might be significant. 3. Utility operating cost: Utility operating cost is usually the most significant variable operating cost after the cost of raw materials. This is especially the case for the production of commodity chemicals. Utility operating cost includes: • fuel • electricity • steam • cooling water • refrigeration • compressed air • inert gas. Utility costs can vary enormously between different pro- cessing sites. This is especially true of fuel and power costs. Not only do fuel costs vary considerably between different fuels (coal, oil, natural gas) but costs also tend to be sen- sitive to market fluctuations. Contractual relationships also have a significant effect on fuel costs. The price paid for fuel depends very much on how much is purchased and the pattern of usage. When electricity is bought from centralized power- generation companies under long-term contract, the price tends to be more stable than fuel costs, since power- generation companies tend to negotiate long-term contracts for fuel supply. However, purchased electricity prices (and sales price if excess electricity is generated and exported) are normally subject to tariff variations. Electricity tariffs can depend on the season of the year (winter versus summer), the time of day (night versus day) and the time of the week (weekend versus weekday). In hot countries, electricity is usually more expensive in the summer than in the winter because of the demand from air conditioning systems. In cold countries, electricity is usually more expensive in the winter than in the summer because of the demand from space heating. The price structure for electricity can be complex, but should be predictable if based on contractual arrangements. If electricity is purchased from a spot market in those countries that have such arrangements, then prices can vary wildly. Steam costs vary with the price of fuel and electricity. If steam is only generated at low pressure and not used for power generation in steam turbines, then the cost can be estimated from fuel costs assuming an efficiency of generation and distribution losses. The efficiency of 26 Process Economics generation depends on the boiler efficiency and the steam consumed in boiler feedwater production (see Chapter 23). Losses from the steam distribution system include heat losses from steam distribution and condensate return pipework to the environment, steam condensate lost to drain and not returned to the boilers and steam leaks. The efficiency of steam generation (including auxiliary boiler- house requirements, see Chapter 23) is typically around 85 to 90% and distribution losses of perhaps another 10%, giving an overall efficiency for steam generation and distribution of typically around 75 to 80% (based on the net calorific value of the fuel). Care should be exercised when considering boiler efficiency and the efficiency of steam generation. These figures are most often quoted on the basis of gross calorific value of the fuel, which includes the latent heat of the water vapor from combustion. This latent heat is rarely recovered through condensation of the water vapor in the combustion gases. The net calorific value of the fuel assumes that the latent heat of the water vapor is not recovered and is therefore the most relevant value. Yet, figures are most often quoted on the basis of gross calorific value. If high-pressure steam mains are used, then the cost of steam should be related in some way to its capacity to generate power in a steam turbine rather than simply to its additional heating value. The high-pressure steam is generated in the utility boilers, and the low-pressure steam is generated by reducing pressure through steam turbines to produce power. This will be discussed in more detail in Chapter 23. One simple way to cost steam is to calculate the cost of the fuel required to generate the high-pressure steam (including any losses), and this fuel cost is then the cost of the high-pressure steam. Low-pressure mains have a value equal to that of the high-pressure mains minus the value of power generated by letting the steam down to the low pressure in a steam turbine. To calculate the cost of steam that has been expanded though a steam turbine, the power generated in such an expansion must be calculated. The simplest way to do this is on the basis of a comparison between an ideal and a real expansion though a steam turbine. Figure 2.1 shows a steam turbine expansion on an enthalpy-entropy plot. In an ideal turbine, steam with an initial pressure P1 and enthalpy H1 expands isentropically to pressure P2 and enthalpy H2. In such circumstances, the ideal work output is (H1 – H2). Because of the frictional effects in the turbine nozzles and blade passages, the exit enthalpy is greater than it would be in an ideal turbine, and the work output is consequently less, given by H ′2 in Figure 2.1. The actual work output is given by (H1 – H ′2). The turbine isentropic efficiency ηIS measures the ratio of the actual to ideal work obtained: ηIS = H1 − H ′ 2 H1 − H2 (2.8) Real Expansion Isentropic Expansion X = 1.0 X = 0.9 X = 0.85 P1 P2 H S H1 H2 P1 P2 W H2′ Figure 2.1 Steam turbine expansion. The output from the turbine might be superheated or partially condensed, as is the case in Figure 2.1. The following example illustrates the approach. Example 2.5 The pressures of three steam mains have been set to the conditions given in Table 2.9. High-pressure (HP) steam is generated in boilers at 41 barg and superheated to 400◦C. Medium-pressure (MP) and low-pressure (LP) steam are generated by expanding high-pressure steam through a steam turbine with an isentropic efficiency of 80%. The cost of fuel is $4.00 GJ−1, and the cost of electricity is $0.07 kW−1·h−1. Boiler feedwater is available at 100◦C with a heat capacity of 4.2 kJ·kg−1·K−1. Assuming an efficiency of steam generation of 85% and distribution losses of 10%, estimate the cost of steam for the three levels. Table 2.9 Steam mains pressure settings. Mains Pressure (barg) HP 41 MP 10 LP 3 Solution Cost of 41 barg steam. From steam tables, for 41 barg steam at 400◦C: Enthalpy = 3212 kJ·kg−1 For boiler feedwater: Enthalpy = 4.2(100 − 0)(relative to water at 0◦C) = 420 kJ·kg−1 To generate 41 barg steam at 400◦C: Heat duty = 3212 − 420 = 2792 kJ·kg−1 Project Cash Flow and Economic Evaluation 29 Two simple economic criteria are useful in pro- cess design: 1. Economic potential (EP): EP = value of products − fixed costs − variable costs − taxes (2.11) 2. Total annual cost (TAC): T AC = fixed costs + variable costs + taxes (2.12) When synthesizing a flowsheet, these criteria are applied at various stages when the picture is still incomplete. Hence, it is usually not possible to account for all the fixed and variable costs listed above during the early stages of a project. Also, there is little point in calculating taxes until a complete picture of operating costs and cash flows has been established. The preceding definitions of economic potential and total annual cost can be simplified if it is accepted that they will be used to compare the relative merits of different structural options in the flowsheet and different settings of the operating parameters. Thus, items that will be common to the options being compared can be neglected. 2.7 PROJECT CASH FLOW AND ECONOMIC EVALUATION As the design progresses, more information is accumulated. The best methods of assessing the profitability of alterna- tives are based on projections of the cash flows during the project life18. Figure 2.2 shows the cash flow pattern for a typical project. The cash flow is a cumulative cash flow. Consider Curve 1 in Figure 2.2. From the start of the project at Point A, cash is spent without any immediate return. The early stages of the project consist of development, design and other preliminary work, which causes the cumulative curve to dip to Point B. This is followed by the main phase of capital investment in buildings, plant and equipment, and the curve drops more steeply to Point C. Working capital is spent to commission the plant between Points C and D. Production starts at D, where revenue from sales begins. Initially, the rate of production is likely to be below design conditions until full production is achieved at E. At F , the cumulative cash flow is again zero. This is the project breakeven point. Toward the end of the projects life at G, the net rate of cash flow may decrease owing to, for example, increasing maintenance costs, a fall in the market price for the product, and so on. Ultimately, the plant might be permanently shut down or given a major revamp. This marks the end of the project, H . If the plant is shut down, working capital is recovered, L K F E D C B A 0 G J 1 2 3 Time (y) Cumulative Cash Flow Figure 2.2 Cash flow pattern for a typical project. (From Allen DH, 1980, A Guide to the Economic Evaluation of Projects, IChemE, reproduced by permission of the Institution of Chemical Engineers.) and there may be salvage value, which would create a final cash inflow at the end of the project. The predicted cumulative cash flow curve for a project throughout its life forms the basis for more detailed eval- uation. Many quantitative measures or indices have been proposed. In each case, important features of the cumula- tive cash flow curve are identified and transformed into a single numerical measure as an index. 1. Payback time: Payback time is the time that elapses from the start of the project (A in Figure 2.2) to the breakeven point (F in Figure 2.2). The shorter the payback time, the more attractive is the project. Payback time is often calculated as the time to recoup the capital investment based on the mean annual cash flow. In retrofit, payback time is usually calculated as the time to recoup the retrofit capital investment from the mean annual improvement in operating costs. 2. Return on Investment (ROI): Return on investment (ROI) is usually defined as the ratio of average yearly income over the productive life of the project to the total initial investment, expressed as a percentage. Thus, from Figure 2.2 ROI = KH KD × 100 LD %per year (2.13) Payback and ROI select particular features of the project cumulative cash flow and ignore others. They take no account of the pattern of cash flow during a project. 30 Process Economics The other indices to be described, net present value and discounted cash flow return, are more comprehensive because they take account of the changing pattern of project net cash flow with time. They also take account of the time value of money. 3. Net present value (NPV): Since money can be invested to earn interest, money received now has a greater value than money if received at some time in the future. The net present value of a project is the sum of the present values of each individual cash flow. In this case, the present is taken to be the start of a project. Time is taken into account by discounting the annual cash flow ACF with the rate of interest to obtain the annual discounted cash flow ADCF . Thus at the end of year 1 ADCF1 = ACF1 (1 + i) at the end of year 2, ADCF2 = ACF2 (1 + i)2 and at the end of year n, ADCFn = ACFn (1 + i)n (2.14) The sum of the annual discounted cash flows over n years ADCF is known as the net present value (NPV) of the project. NPV = ∑ ADCF (2.15) The value of NPV is directly dependent on the choice of the fractional interest rate i and project lifetime n. Returning to the cumulative cash flow curve for a project, the effect of discounting is shown in Figure 2.2. Curve 1 is the original curve with no discounting, that is, i = 0, and the project NPV is equal to the final net cash position given by H . Curve 2 shows the effect of discounting at a fixed rate of interest, and the corresponding project NPV is given by J . Curve 3 in Figure 2.2 shows a larger rate of interest, but it is chosen such that the NPV is zero at the end of the project. The greater the positive NPV for a project, the more economically attractive it is. A project with a negative NPV is not a profitable proposition. 4. Discounted cash flow rate of return: Discounted cash flow rate of return is defined as the discount rate i, which makes the NPV of a project to zero (Curve 3 in Figure 2.2): NPV = ∑ ADCF = 0 (2.16) The value of i given by this equation is known as the discounted cash flow rate of return (DCFRR). It may be found graphically or by trial and error. Example 2.7 A company has the alternative of investing in one of two projects, A or B. The capital cost of both projects is $10 million. The predicted annual cash flows for both projects are shown in Table 2.10. Capital is restricted, and a choice is to be made on the basis of discounted cash flow rate of return, based on a five-year lifetime. Table 2.10 Predicted annual cash flows. Cash flows ($106) Year Project A Project B 0 −10 −10 1 1.6 6.5 2 2.8 5.2 3 4.0 4.0 4 5.2 2.8 5 6.4 1.6 Project A Start with an initial guess for DCFRR of 20% and increase as detailed in Table 2.11. Table 2.11 Calculation of DCFRR for Project A. DCF 20% DCF 30% DCF 25% Year ACF ADCF ADCF ADCF ADCF ADCF ADCF 0 −10 −10 −10 −10 −10 −10 −10 1 1.6 1.33 −8.67 1.23 −8.77 1.28 −8.72 2 2.8 1.94 −6.73 1.66 −7.11 1.79 −6.93 3 4.0 2.31 −4.42 1.82 −5.29 2.05 −4.88 4 5.2 2.51 −1.91 1.82 −3.47 2.13 −2.75 5 6.4 2.57 0.66 1.72 −1.75 2.10 −0.65 Twenty percent is too low since ADCF is positive at the end of year 5. Thirty percent is too large since ADCF is negative at the end of year 5, as is the case with 25%. The answer must be between 20 and 25%. Interpolating on the basis of ADCF the DCFRR ≈ 23%. Project B Again, start with an initial guess for DCFRR of 20% and increase as detailed in Table 2.12. From ADCF at the end of year 5, 20% is to low, 40% too high and 35% also too low. Interpolating on the basis of ADCF , the DCFRR ≈ 38%. Project B should therefore be chosen. 2.8 INVESTMENT CRITERIA Economic analysis should be performed at all stages of an emerging project as more information and detail become available. The decision as to whether to proceed with a project will depend on many factors. There is most often Process Economics – Summary 31 Table 2.12 Calculation of DCFRR for Project B. DCF 20% DCF 40% DCF 35% Year ACF ADCF ADCF ADCF ADCF ADCF ADCF 0 −10 −10 −10 −10 −10 −10 −10 1 6.4 5.42 −4.58 4.64 −5.36 4.81 −5.19 2 5.2 3.61 −0.97 2.65 −2.71 2.85 −2.34 3 4.0 2.31 1.34 1.46 −1.25 1.63 −0.71 4 2.8 1.35 2.69 0.729 −0.521 0.843 0.133 5 1.6 0.643 3.33 0.297 −0.224 0.357 0.490 stiff competition within companies for any capital available for investment in projects. The decision as to where to spend the available capital on a particular project will, in the first instance but not exclusively, depend on the economic criteria discussed in the previous section. Criteria that account for the timing of the cash flows (the NPV and DCFRR) should be the basis of the decision-making. The higher the value of the NPV and DCFRR for a project, the more attractive it is. The absolute minimum acceptable value of the DCFRR is the market interest rate. If the DCFRR is lower than market interest rate, it would be better to put the money in the bank. For a DCFRR value greater than this, the project will show a profit, for a lesser value it will show a loss. The essential distinction between NPV and DCFRR is: • Net Present Value measures profit but does not indicate how efficiently capital is being used. • DCFRR measures how efficiently capital is being used but gives no indication of how large the profits will be. If the goal is to maximize profit, NPV is the more important measure. If the supply of capital is restricted, which is usual, DCFRR can be used to decide which projects will use the capital most efficiently. Both measures are therefore important to characterize the economic value of a project. Predicting future cash flows for a project is extremely difficult. There are many uncertainties, including the project life. Also, the appropriate interest rate will not be known with certainty. The acceptability of the rate of return will depend on the risks associated with the project and the company investment policy. For example, a DCFRR of 20% might be acceptable for a low risk project. A higher return of say 30% might be demanded of a project with some risk, whereas a high-risk project with significant uncertainty might demand a 50% DCFRR. The sensitivity of the economic analysis to the underlying assumptions should always be tested. A sensitivity analysis should be carried out to test the sensitivity of the economic analysis to: • errors in the capital cost estimate • delays in the start-up of the project after the capital has been invested (particularly important for a high capital cost project) • changes in the cost of raw materials • changes in the sales price of the product • reduction in the market demand for the product, and so on. When carrying out an economic evaluation, the magnitude and timing of the cash flows, the project life and interest rate are not known with any certainty. However, providing that consistent assumptions are made for projections of cash flows and the assumed rate of interest, the economic analysis can be used to choose between competing projects. It is important to compare different projects and options within projects, on the basis of consistent assumptions. Thus, even though the evaluation will be uncertain in an absolute sense, it can still be meaningful in a relative sense for choosing between options. Because of this, it is important to have a reference against which to judge any project or option within a project. However, the final decision to proceed with a project will be influenced as much by business strategy as by the economic measures described above. The business strategy might be to gradually withdraw from a particular market, perhaps because of adverse long-term projections of excessive competition, even though there might be short- term attractive investment opportunities. The long-term business strategy might be to move into different business areas, thereby creating investment priorities. Priority might be given to increasing market share in a particular product to establish business dominance in the area and achieve long-term global economies of scale in the business. 2.9 PROCESS ECONOMICS – SUMMARY Process economics is required to evaluate design options, carry out process optimization and evaluate overall project profitability. Two simple criteria can be used: • economic potential • total annual cost. These criteria can be used at various stages in the design without a complete picture of the process. The dominant operating cost is usually raw materials. However, other significant operating costs involve catalysts and chemicals consumed other than raw materials, utility costs, labor costs and maintenance. Capital costs can be estimated by applying installation factors to the purchase costs of individual items of equipment. However, there is considerable uncertainty associated with cost estimates obtained in this way, as equipment costs are typically only 20 to 40% of the total installed costs, with the remainder based on factors. Utility investment, off-site investment and working capital are also needed to complete the capital investment. The capital cost can be annualized by considering it as a loan over a fixed period at a fixed rate of interest. 3 Optimization 3.1 OBJECTIVE FUNCTIONS Optimization will almost always be required at some stage in a process design. It is usually not necessary for a designer to construct an optimization algorithm in order to carry out an optimization, as general-purpose software is usually available for this. However, it is necessary for the designer to have some understanding of how optimization works in order to avoid the pitfalls that can occur. More detailed accounts of optimization can be found elsewhere1 – 3. Optimization problems in process design are usually con- cerned with maximizing or minimizing an objective func- tion. The objective function might typically be to maximize economic potential or minimize cost. For example, con- sider the recovery of heat from a hot waste stream. A heat exchanger could be installed to recover the waste heat. The heat recovery is illustrated in Figure 3.1a as a plot of tem- perature versus enthalpy. There is heat available in the hot stream to be recovered to preheat the cold stream. But how much heat should be recovered? Expressions can be written for the recovered heat as: QREC = mH CP ,H (TH ,in −TH ,out) (3.1) QREC = mCCP ,C (TC ,out − TC ,in) (3.2) where QREC = recovered heat mH, mC = mass flowrates of the hot and cold streams CP,H , CP,C = specific heat capacity of the hot and cold streams TH ,in , TH ,out = hot stream inlet and outlet temperatures TC ,in , TC ,out = cold stream inlet and outlet temperatures The effect of the heat recovery is to decrease the energy requirements. Hence, the energy cost of the process: Energy cost = (QH − QREC )CE (3.3) where QH = process hot utility requirement prior to heat recovery from the waste stream CE = unit cost of energy There is no change in cost associated with cooling as the hot stream is a waste stream being sent to the environment. Chemical Process Design and Integration R. Smith  2005 John Wiley & Sons, Ltd ISBNs: 0-471-48680-9 (HB); 0-471-48681-7 (PB) An expression can also be written for the heat transfer area of the recovery exchanger: A = QREC U
TLM (3.4) where A = heat transfer area U = overall heat transfer coefficient
TLM = logarithmic mean temperature difference = (TH ,in − TC ,out ) − (TH ,out − TC ,in) ln [ TH ,in − TC ,out TH ,out − TC ,in ] In turn, the area of the heat exchanger can be used to estimate the annualized capital cost: Annualized capital cost = (a + bAc)AF (3.5) where a, b, c = cost coefficients AF = annualization factor (see Chapter 2) Suppose that the mass flowrates, heat capacities and inlet temperatures of both streams are fixed and the current hot utility requirement, unit cost of energy, overall heat transfer coefficient, cost coefficients and annualization factor are known. Equations 3.1 to 3.5, together with the specifications for the 13 variables mH , mC , CP,H , CP ,C , TH ,in , TC ,in , U , a, b, c, AF, QH and CE , constitute 18 equality constraints. In addition to these 13 variables, there are a further six unknown variables QREC , TH ,out , TC ,out , Energy cost, Annualized capital cost and A. Thus, there are 18 equality constraints and 19 variables, and the problem cannot be solved. For the system of equations (equality constraints) to be solved, the number of variables must be equal to the number of equations (equality constraints). It is underspecified. Another specification (equality constraint) is required to solve the problem; there is one degree of freedom. This degree of freedom can be optimized; in this case, it is the sum of the annualized energy and capital costs (i.e. the total cost), as shown in Figure 3.1b. If the mass flowrate of the cold stream through the exchanger had not been fixed, there would have been one fewer equality constraint, and this would have provided an additional degree of freedom and the optimization would have been a two-dimensional optimization. Each degree of freedom provides an opportunity for optimization. Figure 3.1b illustrates how the investment in the heat exchanger is optimized. As the amount of recovered heat increases, the cost of energy for the system decreases. On the other hand, the size and capital cost of the heat exchange equipment increase. The increase in size of heat 36 Optimization Total Cost T Energy Cost Optimum Heat Recovered Capital Cost H Cost QREC TC,in TH,out TC,out (a) Conditions in heat recovery exchanger. (b) Cost trade-offs. Increase Heat Recovery TH,in QH Figure 3.1 Recovery of heat from a waste steam involves a trade-off between reduced energy cost and increased capital cost of heat exchanger. x1 x2 x3 x x1 x2 x3 x4 x f(x) f (x) (a) Discontinuous function. (b) Multimodal function. Figure 3.2 Objective functions can exhibit complex behavior. exchanger results from the greater amount of heat to be transferred. Also, because the conditions of the waste stream are fixed, as more heat is recovered from the waste stream, the temperature differences in the heat exchanger become lower, causing a sharp rise in the capital cost. In theory, if the recovery was taken to its limit of zero temperature difference, an infinitely large heat exchanger with infinitely large capital cost would be needed. The costs of energy and capital cost can be expressed on an annual basis, as explained in Chapter 2, and combined as shown in Figure 3.1b to obtain the total cost. The total cost shows a minimum, indicating the optimum size of the heat exchanger. Given a mathematical model for the objective function in Figure 3.1b, finding the minimum point should be straightforward and various strategies could be adopted for this purpose. The objective function is continuous. Also, there is only one extreme point. Functions with only one extreme point (maximum or minimum) are termed unimodal. By contrast, consider the objective functions in Figure 3.2. Figure 3.2a shows an objective function to be minimized that is discontinuous. If the search for the minimum is started at point x1, it could be easily concluded that the optimum point is at x2, whereas the true optimum is at x3. Discontinuities can present problems to optimization algorithms searching for the optimum. Figure 3.2b shows an objective function that has a number of points where the gradient is zero. These points where the gradient is zero are known as stationary points. Functions that exhibit a number of stationary points are known as multimodal. If Multivariable Optimization 39 1. Deterministic methods. Deterministic methods follow a predetermined search pattern and do not involve any guessed or random steps. Deterministic methods can be further classified into direct and indirect search methods. Direct search methods do not require derivatives (gradients) of the function. Indirect methods use derivatives, even though the derivatives might be obtained numerically rather than analytically. (a) Direct search methods. An example of a direct search method is a univariate search, as illustrated in Figure 3.7. All of the variables except one are fixed and the remaining variable is optimized. Once a minimum or maximum point has been reached, this variable is fixed and another variable optimized, with the remaining variables being fixed. This is repeated until there is no further improvement in the objective function. Figure 3.7 illustrates a two-dimensional search in which x1 is first fixed and x2 optimized. Then 200 100 50 20 Local Optimum x1 x2 Global Optimum Starting Point Figure 3.7 A univariate search. x2 is fixed and x1 optimized, and so on until no further improvement in the objective function is obtained. In Figure 3.7, the univariate search is able to locate the global optimum. It is easy to see that if the starting point for the search in Figure 3.7 had been at a lower value of x1, then the search would have located the local optimum, rather than the global optimum. For searching multivariable optimization problems, often the only way to ensure that the global optimum has been reached is to start the optimization from different initial points. Another example of a direct search is a sequential simplex search. The method uses a regular geometric shape (a simplex) to generate search directions. In two dimensions, the simplest shape is an equilateral triangle. In three dimensions, it is a regular tetrahedron. The objective function is evaluated at the vertices of the simplex, as illustrated in Figure 3.8. The objective function must first be evaluated at the Vertices A, B and C. The general direction of search is projected away from the worst vertex (in this case Vertex A) through the centroid of the remaining vertices (B and C), Figure 3.8a. A new simplex is formed by replacing the worst vertex by a new point that is the mirror image of the simplex (Vertex D), as shown in Figure 3.8a. Then Vertex D replaces Vertex A, as Vertex A is an inferior point. The simplex vertices for the next step are B, C and D. This process is repeated for successive moves in a zigzag fashion, as shown in Figure 3.8b. The direction of search can change as illustrated in Figure 3.8c. When the simplex is close to the optimum, there may be some repetition of simplexes, with the search going around in circles. If this is the case, then the size of the simplex should be reduced. (b) Indirect search methods. Indirect search methods use derivatives (gradients) of the objective function. The D C A B D C A B (a) Reflection of a simplex to a new point. (b) Search proceeds in a zig–zag pattern. (c) Change of direction. . x1x1x1 x2 x2 x2 Figure 3.8 The simplex search. 40 Optimization 200 100 50 20 Local Optimum x1 x2 Global Optimum Starting Point Figure 3.9 Method of steepest descent. derivatives may be obtained analytically or numerically. Many methods are available for indirect search. An example is the method of steepest descent in a minimization problem. The direction of steepest descent is the search direction that gives the maximum rate of change for the objective function from the current point. The method is illustrated in Figure 3.9. One problem with this search method is that the appropriate step size is not known, and this is under circumstances when the gradient might change significantly during the search. Another problem is that the search can slow down significantly as it reaches the optimum point. If a search is made for the maximum in an objective function, then the analogous search is one of steepest ascent. The method of steepest descent uses only first-order derivatives to determine the search direction. Alternatively, Newton’s method for single-variable optimization can be adapted to carry out multivariable optimization, taking advantage of both first- and second-order derivatives to obtain better search directions1. However, second- order derivatives must be evaluated, either analytically or numerically, and multimodal functions can make the method unstable. Therefore, while this method is potentially very powerful, it also has some practical difficulties. One fundamental practical difficulty with both the direct and indirect search methods is that, depending on the shape of the solution space, the search can locate local optima, rather than the global optimum. Often, the only way to ensure that the global optimum has been reached is to start the optimization from different initial points and repeat the process. 2. Stochastic search methods. In all of the optimization methods discussed so far, the algorithm searches the objec- tive function seeking to improve the objective function at each step, using information such as gradients. Unfortu- nately, as already noted, this process can mean that the search is attracted towards a local optimum. On the other hand, stochastic search methods use random choice to guide the search and can allow deterioration of the objective func- tion during the search. It is important to recognize that a randomized search does not mean a directionless search. Stochastic search methods generate a randomized path to the solution on the basis of probabilities. Improvement in the objective function becomes the ultimate rather than the immediate goal, and some deterioration in the objec- tive function is tolerated, especially during the early stages of the search. In searching for a minimum in the objec- tive function, rather than the search always attempting to go downhill, stochastic methods allow the search to also sometimes go uphill. Similarly, if the objective function is to be maximized, stochastic methods allow the search to sometimes go downhill. As the optimization progresses, the ability of the algorithm to accept deterioration in the objec- tive function is gradually removed. This helps to reduce the problem of being trapped in a local optimum. Stochastic methods do not need auxiliary information, such as derivatives, in order to progress. They only require an objective function for the search. This means that stochastic methods can handle problems in which the calculation of the derivatives would be complex and cause deterministic methods to fail. Two of the most popular stochastic methods are simu- lated annealing and genetic algorithms. (a) Simulated annealing. Simulated annealing emulates the physical process of annealing of metals4,5. In the physical process, at high temperatures, the molecules of the liquid move freely with respect to one another. If the liquid is cooled slowly, thermal mobility is lost. The atoms are able to line themselves up and form perfect crystals. This crystal state is one of minimum energy for the system. If the liquid metal is cooled quickly, it does not reach this state but rather ends up in a polycrystalline or amorphous state having higher energy. So the essence of the process is slow cooling, allowing ample time for redistribution of the atoms as they lose mobility to reach a state of minimum energy. With this physical process in mind, an algorithm can be suggested in which a system moves from one point to another, with the resulting change in the objective function from E1 to E2. The probability of this change can be assumed to follow a relationship similar to the Boltzmann probability distribution (maintaining the analogy with physical annealing)4: P = exp[−(E2 − E1)/T ] (3.11) where P is the probability, E is the objective function (the analogy of energy) and T is a control parameter (the analogy of annealing temperature). Relationships other than Equation 3.11 can be used. If the objective function is being minimized and E2 is less than E1 (i.e. the objective function improves as a result of the move), then the probability from Equation 3.11 is greater than unity. In Multivariable Optimization 41 such cases, the move is arbitrarily assigned to a probability of unity, and the system should always accept such a move. If the probability in Equation 3.11 is 0, the move is rejected. If the probability is between zero and unity (i.e. the objective function gets worse as a result of the move), then a criterion is needed to dictate whether the move is accepted or rejected. An arbitrary cutoff point (e.g. reject any move with a probability of less than 0.5) could be made, but the most appropriate value is likely to change from problem to problem. Instead, the analogy with physical annealing is maintained, and the probability from Equation 3.11 is compared with the output of a random number generator that creates random numbers between zero and unity. If Equation 3.11 predicts a probability greater than the random number generator, then the move is accepted. If it is less, then the move is rejected and another move is attempted instead. Thus, Equation 3.11 dictates whether a move is accepted or rejected. In this way, the method will always accept a downhill step when minimizing an objective function, while sometimes it will take an uphill step. However, as the optimization progresses, the possibility of accepting moves that result in a deterioration of the objective function needs to be removed gradually. This is the role of the control parameter (the analogy of temperature) in Equation 3.11. The control parameter is gradually decreased, which results in the probability in Equation 3.11 gradually decreasing for moves in which the objective function deteriorates. When this is compared with the random number, it gradually becomes more likely that the move will be rejected. An annealing schedule is required that controls how the control parameter is lowered from high to low values. Thus, the way the algorithm works is to set an initial value for the control parameter. At this setting of the control parameter, a series of random moves are made. Equation 3.11 dictates whether an individual move is accepted or rejected. The control parameter (annealing temperature) is lowered and a new series of random moves is made, and so on. As the control parameter (annealing temperature) is lowered, the probability of accepting deterioration in the objective function, as dictated by Equation 3.11, decreases. In this way, the acceptability for the search to move uphill in a minimization or downhill during maximization is gradually withdrawn. Whilst simulated annealing can be extremely powerful in solving difficult optimization problems with many local optima, it has a number of disadvantages. Initial and final values of the control parameter, an annealing schedule and the number of random moves for each setting of the control parameter must be specified. Also, because each search is random, the process can be repeated a number of times to ensure that an adequate search of the solution space has been made. This can be very useful in providing a number of solutions in the region of the global optimum, rather than a single solution. Such multiple solutions can then be screened not just on the basis of cost but also on many other issues, such as design complexity, control, safety, and so on, that are difficult to include in the optimization. (b) Genetic algorithms. Genetic algorithms draw their inspiration from biological evolution6. Unlike all of the optimization methods discussed so far, which move from one point to another, a genetic algorithm moves from one set of points (termed a population) to another set of points. Populations of strings (the analogy of chro- mosomes) are created to represent an underlying set of parameters (e.g. temperatures, pressures or concentrations). A simple genetic algorithm exploits three basic operators: reproduction, crossover and mutation. Reproduction is a process in which individual members of a population are copied according to the objective function in order to generate new population sets. The operator is inspired by “natural selection” and the “survival of the fittest”. The easiest way to understand reproduction is to make an analogy with a roulette wheel. In a roulette wheel, the probability of selection is proportional to the area of the slots in the wheel. In a genetic algorithm, the probability of reproduction is proportional to the fitness (the objective function). Although the selection procedure is stochastic, fitter strings are given a better chance of selection (survival). The reproduction operator can be implemented in a genetic algorithm in many ways6. Crossover involves the combination of genetic mate- rial from two successful parents to form two offspring (children). Crossover involves cutting two parent strings at random points and combining differently to form new offspring. The crossover point is generated randomly. Crossover works as a local search operator and spreads good properties amongst the population. The fraction of new population generated by crossover is generally large (as observed in nature) and is controlled stochastically. Mutation creates new strings by randomly changing (mutating) parts of strings, but (as with nature) with a low probability of occurring. A random change is made in one of the genes in order to preserve diversity. Mutation creates a new solution in the neighborhood of a point undergoing mutation. A genetic algorithm works by first generating an initial population randomly. The population is evaluated according to its fitness (value of the objective function). A repro- duction operator then provides an intermediate population using stochastic selection but biased towards survival of the fittest. Crossover and mutation operators are then applied to the intermediate population to create a new generation of population. The new population is evaluated according to its fitness and the search is continued with further repro- duction, crossover and mutation until the population meets the required convergence criterion (generally the difference between the average and maximum fitness value or number of generations). 44 Optimization 600 500 400 300 200 1000 200 300 400 500 600 n1 10n1 + 20n2 = 6000 Increasing Annual Revenue Optimum 25n1 + 10n2 = 5000 A B C D n2 100 Figure 3.12 Graphical representation of the linear optimization problem from Example 3.1. On a plot of n1 versus n2 as shown in Figure 3.12, lines of constant annual revenue will follow a straight line given by: n2 = −3 2 n1 + A 2000 Lines of constant annual revenue are shown as dotted lines in Figure 3.12, with revenue increasing with increasing distance from the origin. It is clear from Figure 3.12 that the optimum point corresponds with the extreme point at the intersection of the two equality constraints at Point C. At the intersection of the two constraints: n1 = 100 n2 = 250 As the problem involves discrete batches, it is apparently fortunate that the answer turns out to be two whole numbers. However, had the answer turned out not to be a whole number, then the solution would still have been valid because, even though part batches might not be able to be processed, the remaining part of a batch can be processed the following year. Thus the maximum annual revenue is given by: A = 3000 × 100 + 2000 × 250 = $800,000 $ y−1 Whilst Example 3.1 is an extremely simple example, it illustrates a number of important points. If the optimization problem is completely linear, the solution space is convex and a global optimum solution can be generated. The optimum always occurs at an extreme point, as is illustrated in Figure 3.12. The optimum cannot occur inside the feasible region, it must always be at the boundary. For linear functions, running up the gradient can always increase the objective function until a boundary wall is hit. Whilst simple two variable problems like the one in Example 3.1 can be solved graphically, more complex problems require a more formal nongraphical approach. This is illustrated by returning to Example 3.1 to solve it in a nongraphical way. Example 3.2 Solve the problem in Example 3.1 using an analytical approach. Solution The problem in Example 3.1 was expressed as: A = 3000n1 + 2000n2 25n1 + 10n2 ≤ 5000 10n1 + 20n2 ≤ 6000 To solve these equations algebraically, the inequality signs must first be removed by introducing slack variables S1 and S2 such that: 25n1 + 10n2 + S1 = 5000 10n1 + 20n2 + S2 = 6000 In other words, these equations show that if the production of both products does not absorb the full capacities of both steps, then the slack capacities of these two processes can be represented by the variables S1 and S2. Since slack capacity means that a certain amount of process capacity remains unused, it follows that the economic value of slack capacity is zero. Realizing that negative production rates and negative slack variables are infeasible, the problem can be formulated as: 3000n1 + 2000n2 + 0S1 + 0S2 = A (3.14) 25n1 + 10n2 + 1S1 + 0S2 = 5000 (3.15) 10n1 + 20n2 + 0S1 + 1S2 = 6000 (3.16) where n1, n2, S1, S2 ≥ 0 Equations 3.15 and 3.16 involve four variables and can there- fore not be solved simultaneously. At this stage, the solution can lie anywhere within the feasible area marked ABCD in Figure 3.12. However, providing the values of these variables are not restricted to integer values; two of the four variables will assume zero values at the optimum. In this example, n1, n2, S1 and S2 are treated as real and not integer variables. The problem is started with an initial feasible solution that is then improved by a stepwise procedure. The search will be started at the worst possible solution when n1 and n2 are both zero. From Equations 3.15 and 3.16: S1 = 5000 − 25n1 − 10n2 (3.17) S2 = 6000 − 10n1 − 20n2 (3.18) When n1 and n2 are zero: S1 = 5000 S2 = 6000 Substituting in Equation 3.14: A = 3000 × 0 + 2000 × 0 + 0 × 5000 + 0 × 6000 = 0 (3.19) This is Point A in Figure 3.12. To improve this initial solution, the value of n1 and/or the value of n2 must be increased, because Nonlinear Programming 45 x2 x2 x2 x1 x1 x1 Increasing Value of Objective Function Increasing Value of Objective Function (a) Objective function and constraints are parallel. (b) Feasible region only partially bounded. (c) No feasible region. Figure 3.13 Degenerate linear programming problems. these are the only variables that possess positive coefficients to increase the annual revenue in Equation 3.14. But which variable, n1 or n2, should be increased first? The obvious strategy is to increase the variable that makes the greatest increase in the annual revenue, which is n1. According to Equation 3.18, n1 can be increased by 6000/10 = 600 before S2 becomes negative. If n1 is assumed to be 600 in Equation 3.17, then S1 would be negative. Since negative slack variables are infeasible, Equation 3.17 is the dominant constraint on n1 and it follows that its maximum value is 5000/25 = 200. Rearranging Equation 3.17: n1 = 200 − 0.4n2 − 0.04S1 (3.20) which would give a maximum when n1 and S1 are zero. Substituting the expression for n1 in the objective function, Equation 3.14, gives: A = 600,000 + 800n2 − 120S1 (3.21) Since n2 is initially zero, the greatest improvement in the objective function results from making S1 zero. This is equivalent to making n1 equal to 200 from Equation 3.20, given n2 is initially zero. For n1 = 200 and n2 = 0, the annual revenue A = 600,000. This corresponds with Point B in Figure 3.12. However, Equation 3.21 also shows that the profit can be improved further by increasing the value of n2. Substituting n1 from Equation 3.20 in Equation 3.18 gives: n2 = 250 + 0.025S1 − 0.0625S2 (3.22) This means n2 takes a value of 250 if both S1 and S2 are zero. Substituting n2 from Equation 3.22 in Equation 3.20 gives: n1 = 100 − 0.05S1 + 0.025S2 (3.23) This means n1 takes a value of 100 if both S1 and S2 are zero. Finally, substituting the expression for n1 and n2 in the objective function, Equation 3.14 gives: A = 800,000 − 100S1 − 50S2 (3.24) Equations 3.22 to 3.24 show that the maximum annual revenue is $800,000 y−1 when n1 = 100 and n2 = 250. This corresponds with Point C in Figure 3.12. It is also interesting to note the Equation 3.24 provides some insight into the sensitivity of the solution. The annual revenue would decrease by $100 for each hour of production lost through poor utilization of Step I. The corresponding effect for Step II would be a reduction of $50 for each hour of production lost. These values are known as shadow prices. If S1 and S2 are set to their availabilities of 5000 and 6000 hours respectively, then the revenue from Equation 3.24 becomes zero. While the method used for the solution of Example 3.1 is not suitable for automation, it gives some insights into the way linear programming problems can be automated. The solution is started by turning the inequality constraints into equality constraints by the use of slack variables. Then the equations are solved to obtain an initial feasible solution. This is improved in steps by searching the extreme points of the solution space. It is not necessary to explore all the extreme points in order to identify the optimum. The method usually used to automate the solution of such linear programming problems is the simplex algorithm1,2. Note, however, that the simplex algorithm for linear programming should not be confused with the simplex search described previously, which is quite different. Here the term simplex is used to describe the shape of the solution space, which is a convex polyhedron, or simplex. If the linear programming problem is not formulated properly, it might not have a unique solution, or even any solution at all. Such linear programming problems are termed degenerate1. Figure 3.13 illustrates some degen- erate linear programming problems1. In Figure 3.13a, the objective function contours are parallel with one of the boundary constraints. Here there is no unique solution that maximizes the objective function within the feasible region. Figure 3.13b shows a problem in which the feasible region is unbounded. Hence the objective function can increase without bound. A third example is shown in Figure 13.3c, in which there is no feasible region according to the spec- ified constraints. 3.6 NONLINEAR PROGRAMMING When the objective function, equality or inequality con- straints of Equation 3.7 are nonlinear, the optimization 46 Optimization becomes a nonlinear programming (NLP) problem. The worst case is when all three are nonlinear. Direct and indi- rect methods that can be used for nonlinear optimization have previously been discussed. Whilst it is possible to include some types of constraints, the methods discussed are not well suited to the inclusion of complex sets of con- straints. The stochastic methods discussed previously can readily handle constraints by restricting moves to infeasi- ble solutions, for example in simulated annealing by setting their probability to 0. The other methods discussed are not well suited to the inclusion of complex sets of constraints. It has already been observed in Figure 3.11 that, unlike the linear optimization problem, for the nonlinear optimization problem the optimum may or may not lie on the edge of the feasible region and can, in principle, be anywhere within the feasible region. One approach that has been adopted for solving the general nonlinear programming problem is successive linear programming. These methods linearize the problem and successively apply the linear programming techniques described in the previous section. The procedures involve initializing the problem and linearizing the objective function and all of the constraints about the initial point, so as to fit the linear programming format. Linear programming is then applied to solve the problem. An improved solution is obtained and the procedure repeated. At each successive improved feasible solution, the objective function and constraints are linearized and the linear programming solution repeated, until the objective function does not show any significant improvement. If the solution to the linear programming problem moves to an infeasible point, then the nearest feasible point is located and the procedure applied at this new point. Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x1 and x2 would be of the general form: f (x1, x2) = a0 + a1x1 + a2x2 + a11x21 + a22x22 + a12x1x2 (3.25) where aij are constants. Quadratic programming problems are the simplest form of nonlinear programming with inequality constraints. The techniques used for the solution of quadratic programming problems have many similarities with those used for solving linear programming problems1. Each inequality constraint must either be satisfied as an equality or it is not involved in the solution of the problem. The quadratic programming technique can thus be reduced to a vertex searching procedure, similar to linear programming1. In order to solve the general nonlinear programming problem, quadratic programming can be applied successively, in a similar way to that for successive linear programming, in successive (or sequential) quadratic programming (SQP). In this case, the objective function is approximated locally as a quadratic function. For a function of two variables, the function would be approximated by Equation 3.25. By approximating the function as a quadratic and linearizing the constraints, this takes the form of a quadratic programming problem that is solved in each iteration1. In general, successive quadratic programming tends to perform better than successive linear programming, because a quadratic rather than a linear approximation is used for the objective function. It is important to note that neither successive linear nor successive quadratic programming are guaranteed to find the global optimum in a general nonlinear programming problem. The fact that the problem is being turned into a linear or quadratic problem, for which global optimality can be guaranteed, does not change the underlying problem that is being optimized. All of the problems associated with local optima are still a feature of the background problem. When using these methods for the general nonlinear programming problem, it is important to recognize this and to test the optimality of the solution by starting the optimization from different initial conditions. Stochastic optimization methods described previously, such as simulated annealing, can also be used to solve the general nonlinear programming problem. These have the advantage that the search is sometimes allowed to move uphill in a minimization problem, rather than always search- ing for a downhill move. Or, in a maximization problem, the search is sometimes allowed to move downhill, rather than always searching for an uphill move. In this way, the technique is less vulnerable to the problems associated with local optima. 3.7 PROFILE OPTIMIZATION There are many situations in process design when it is nec- essary to optimize a profile. For example, a reactor design might involve solid catalyst packed into tubes with heat removal via a coolant on the outside of the tubes. Prior to designing the heat transfer arrangement in detail, the designer would like to know the temperature profile along the tube that optimizes the overall reaction conditions7. Should it be a constant temperature along the tube? Should it increase or decrease along the tube? Should any increase or decrease be linear, exponential, and so on? Should the profile go through a maximum or minimum? Once the opti- mum profile has been determined, the catalyst loading and heat transfer arrangements can then be designed to come as close as possible to the optimum temperature profile. Rather than a parameter varying through space, as in the example of the temperature profile along the reactor, the profile could vary through time. For example, in a batch reactor, the reactants might be loaded into the reactor at the beginning of the batch, the reaction initiated by heating the contents, adding a catalyst, and so on, and Structural Optimization 49 a feature exists or not. If a feature exists, its binary variable takes the value 1. If the feature does not exist, then it is set to 0. Consider how different kinds of decisions can be formulated using binary variables2. a. Multiple choice constraints. It might be required to select only one item from a number of options. This can be represented mathematically by a constraint: J∑ j=1 yj = 1 (3.28) where yj is the binary variable to be set to 0 or 1 and the number of options is J . More generally, it might be required to select only m items from a number of options. This can be represented by: J∑ j=1 yj = m (3.29) Alternatively, it might be required to select at most m items from a number of options, in which case the constraint can be represented by: J∑ j=1 yj ≤ m (3.30) On the other hand, it might be required to select at least m items from a number of options. The constraint can be represented by: J∑ j=1 yj ≥ m (3.31) b. Implication constraints. Another type of logical con- straint might be that if Item k is selected, Item j must be selected, but not vice versa, then this is represented by the constraint: yk − yj ≤ 0 (3.32) A binary variable can be used to set a continuous variable to 0. If a binary variable y is 0, the associated continuous variable x must also be 0 if a constraint is applied such that: x − Uy ≤ 0, x ≥ 0 (3.33) where U is an upper limit to x. c. Either–or constraints. Binary variables can also be applied to either–or constraints, known as disjunctive constraints. For example, either constraint g1(x) ≤ 0 or constraint g2(x) ≤ 0 must hold: g1(x) − My ≤ 0 (3.34) g2(x) − M(1 − y) ≤ 0 (3.35) where M is a large (arbitrary) value that represents an upper limit g1(x) and g2(x). If y = 0, then g1(x) ≤ 0 must be imposed from Equation 3.34. However, if y = 0, the left-hand side of Equation 3.35 becomes a large negative number whatever the value of g2(x), and as a result, Equation 3.35 is always satisfied. If y = 1, then the left-hand side of Equation 3.34 is a large negative number whatever the value of g1(x) and Equation 3.34 is always satisfied. But now g2(x) ≤ 0 must be imposed from Equation 3.35. Some simple examples can be used to illustrate the application of these principles. Example 3.3 A gaseous waste stream from a process contains valuable hydrogen that can be recovered by separating the hydrogen from the impurities using pressure swing adsorption (PSA), a membrane separator (MS) or a cryogenic condensation (CC). The pressure swing adsorption and membrane separator can in principle be used either individually, or in combination. Write a set of integer equations that would allow one from the three options of pressure swing adsorption, membrane separator or cryogenic condensation to be chosen, but also allow the pressure swing adsorption and membrane separator to be chosen in combination. Solution Let yPSA represent the selection of pressure swing adsorption, yMS the selection of the membrane separator and yCC the selection of cryogenic condensation. First restrict the choice between pressure swing adsorption and cryogenic condensation. yPSA + yCC ≤ 1 Now restrict the choice between membrane separator and cryogenic condensation. yMS + yCC ≤ 1 These two equations restrict the choices, but still allow the pressure swing adsorption and membrane separator to be cho- sen together. Example 3.4 The temperature difference in a heat exchanger between the inlet temperature of the hot stream TH ,in and the outlet of the cold stream TC ,out is to be restricted to be greater than a practical minimum value of
Tmin , but only if the option of having the heat exchanger is chosen. Write a disjunctive constraint in the form of an integer equation to represent this constraint. Solution The temperature approach constraint can be written as TH ,in − TC ,out ≥
Tmin But this should apply only if the heat exchanger is selected. Let yHX represent the option of choosing the heat exchanger. TH ,in − TC ,out + M(1 − yHX) ≥
Tmin where M is an arbitrary large number. If yHX = 0 (i.e. the heat exchanger is not chosen), then the left-hand side of this equation is bound to be greater than
Tmin no matter what the values of TH ,in and TC ,out are. If yHX = 1 (i.e. the heat exchanger is chosen), then the equation becomes (TH ,in − TC ,out) ≥
Tmin and the constraint must apply. When a linear programming problem is extended to include integer (binary) variables, it becomes a mixed inte- ger linear programming problem (MILP). Correspondingly, 50 Optimization when a nonlinear programming problem is extended to include integer (binary) variables, it becomes a mixed inte- ger nonlinear programming problem (MINLP). First consider the general strategy for solving an MILP problem. Initially, the binary variables can be treated as continuous variables, such that 0 ≤ yi ≤ 1. The problem can then be solved as an LP. The solution is known as a relaxed solution. The most likely outcome is that some of the binary variables will exhibit noninteger values at the optimum LP solution. Because the relaxed solution is less constrained than the true mixed integer solution in which all of the binary variables have integer values, it will in general give a better value for the objective function than the true mixed integer solution. In general, the noninteger values of the binary variables cannot simply be rounded to the nearest integer value, either because the rounding may lead to an infeasible solution (outside the feasible region) or because the rounding may render the solution nonoptimal (not at the edge of the feasible region). However, this relaxed LP solution is useful in providing a lower bound to the true mixed integer solution to a minimization problem. For maximization problems, the relaxed LP solutions form the upper bound to the solution. The noninteger values can then be set to either 0 or 1 and the LP solution repeated. The setting of the binary variables to be either 0 or 1 creates a solution space in the form of a tree, Figure 3.163. As the solution is stepped through, the number of possibilities increases by virtue of the fact that each binary variable can take a value of 0 or 1, Figure 3.16. At each point in the search, the best relaxed LP solution provides a lower bound to the optimum of a minimization problem. Correspondingly, the best true mixed integer solution provides an upper bound. For maximization problems the best relaxed LP solution forms an upper bound to the optimum and the best true mixed integer solution provides the lower bound. A popular method of solving MILP problems is to use a branch and bound search7. This will be illustrated by a simple example from Edgar, Himmelblau and Lasdon1. Example 3.5 A problem involving three binary variables y1, y2 and y3 has an objective function to be maximized1. maximize: f = 86y1 + 4y2 + 40y3 subject to : 774y1 + 76y2 + 42y3 ≤ 875 67y1 + 27y2 + 53y3 ≤ 875 y1, y2, y3 = 0, 1 (3.36) Solution The solution strategy is illustrated in Figure 3.17a. First the LP problem is solved to obtain the relaxed solution, allowing y1, y2 and y3 to vary continuously between 0 and 1. Both y1 = 1 and y3 = 1 at the optimum of the relaxed solution but the value of y2 is 0.776, Node 1 in Figure 3.17a. The objective function for this relaxed solution at Node 1 is f = 129.1. From this point, y2 can be set to be either 0 or 1. Various strategies can be adopted to decide which one to choose. A very simple strategy will be adopted here of picking the closest integer to the real number. Given that y2 = 0.776 is closer to 1 than 0, set y2 = 1 and solve the LP at Node 2, Figure 3.17a. Now y2 = 1 and y3 = 1 at the optimum of the relaxed solution but the value of y1 is 0.978, Node 2 in Figure 3.17a. Given that y1 = 0.978 is closer to 1 than 0, set y1 = 1 and solve the LP at Node 3. This time y1 and y2 are integers, but y3 = 0.595 is a noninteger. Setting y3 = 1 yields an infeasible solution at Node 4 in Figure 3.17a as it violates the first inequality constraint in Equation 3.36. Backtracking to Node 3 and setting y3 = 0 yields the first feasible integer solution at Node 5 for which y1 = 1, y2 = 1, y3 = 0 and f = 90.0. There is no point in searching further from Node 4 as it is an infeasible solution, or from Node 5 as it is a valid integer solution. When the search is terminated at a node for either reason, it is deemed to be fathomed. The search now backtracks to Node 2 and sets y1 = 0. This yields the second feasible integer solution at Node 6 for which y1 = 0, y2 = 1, y3 = 1 and f = 44.0. Finally, backtrack to Node 1 and set y2 = 0. This yields the third feasible y2 = 1y2 = 1 y2 = 0y2 = 0 y3 = 0 y3 = 0 y3 = 0 y3 = 0y3 = 1 y3 = 1 y3 = 1 y3 = 1 y1 = 0 y1 = 1 Figure 3.16 Setting the binary variables to 0 or 1 creates a tree structure. (Reproduced from Floudas CA, 1995, Nonlinear and Mixed-Integer Optimization, by permission of Oxford University Press). Structural Optimization 51 f = 129.1 f = 126.0 f = 128.1 f = 113.8f = 44.0 f = 90.0 [1,0.776,1] y2 = 1 y1 = 0 y1 = 1 y3 = 0 y3 = 1 y2 = 0 [0.978,1,1] [1,1,0.595] [1,1,0] [0,1,1] [1,0,1] 6 1 2 3 45 7 Infeasible f = 129.1 f = 126.0 f = 128.1 f = 113.8f = 44.0 y2 = 1 y1 = 0 y1 = 1 y2 = 0 [1,0.776,1] [0.978,1,1] [1,1,0.595][0,1,1] [1,0,1] 4 1 3 5 2 (a) Depth first search. (b) Breadth first search. Figure 3.17 Branch and bound search. integer solution at Node 7 for which y1 = 1, y2 = 0, y3 = 1 and f = 126.0. Since the objective function is being maximized, Node 7 is the optimum for the problem. Searching the tree in the way done in Figure 3.17a is known as a depth first or backtracking approach. At each node, the branch was followed that appeared to be more promising to solve. Rather than using a depth first approach, a breadth first or jumptracking approach can be used, as illustrated in Figure 3.17b. Again start at Node 1 and solve the relaxed problem in Figure 3.17b. This gives an upper bound for the maximization problem of f = 129.1. However, this time the search goes across the tree with the initial setting of y2 = 0. This yields a valid integer solution at Node 2 with y1 = 1, y2 = 0, y3 = 1 and f = 126.0. Node 2 now forms a lower bound and is fathomed because it is an integer solution. In this approach, the search now backtracks to Node 1 whether Node 2 is fathomed or not. From Node 1 now set y2 = 1. The solution at Node 3 in Figure 3.17b gives y1 = 0.978, y2 = 1, y3 = 1 and f = 128.1, which is the new upper bound. Setting y1 = 0 and branching to Node 4 gives the second valid integer solution. Now backtrack to Node 3 and set y1 = 1. The solution at Node 5 has y1 = 1, y2 = 1, y3 = 0.595 and f = 113.8. At this point, Node 5 is fathomed, even though it is neither infeasible nor a valid integer solution. The upper bound of this branch at Node 5 has a value of the objective function lower than that of the integer solution at Node 2. Setting the values to be integers from Node 5 can only result in an inferior solution. In this way, the search is bounded. In this case, the breadth first search yields the optimum with a fewer number of nodes to be searched. Different search strategies than the ones used here can readily be used10. It is likely that different problems would be suited to different search strategies. Thus, the solution of the MILP problem is started by solving the first relaxed LP problem. If integer values are obtained for the binary variables, the problem has been solved. However, if integer values are not obtained, the use of bounds is examined to avoid parts of the tree that are known to be suboptimal. The node with the best noninteger solution provides a lower bound for minimization problems and the node with the best feasible mixed integer solution provides an upper bound. In the case of maximization problems, the node with the best noninteger solution provides an upper bound and the node with the best feasible mixed integer solution provides a lower bound. Nodes with noninteger solutions are fathomed when the value of the objective function is inferior to the best integer solution (the lower bound). The tree can be searched by following a depth first approach or a breadth first approach, or a combination of the two. Given a more complex problem than Example 3.5, the search could for example set the values of the noninteger variables to be 0 and 1 in turn and carry out an evaluation of the objective function (rather than an optimization). This would then indicate the best direction in which to go for the next optimization. Many strategies are possible10. The series of LP solutions required for MILP problems can be solved efficiently by using one LP to initialize the next. An important point to note is that, in principle, a global optimum solution can be guaranteed in the same way as with LP problems. The general strategy for solving mixed integer nonlinear programming problems is very similar to that for linear problems3. The major difference is that each node requires the solution of a nonlinear program, rather than the solution of a linear program. Unfortunately, searching the tree with a succession of nonlinear optimizations can be extremely expensive in terms of the computation time required, as information cannot be readily carried from one NLP to the next as can be done for LP. Another major problem is that, because a series of nonlinear optimizations is being carried out, there is no guarantee that the optimum will even be close to the global optimum, unless the NLP problem being solved at each node is convex. Of course, different initial points can be tried to overcome this problem, but there can still be no guarantee of global optimality for the general problem. 54 Optimization mountains. If the objective function is to be maximized, each peak in the mountain range represents a local optimum in the objective function. The highest peak represents the global optimum. Optimization requires searching around the mountains in a thick fog to find the highest peak, without the benefit of a map and only a compass to tell direction and an altimeter to show height. On reaching the top of any peak, there is no way of knowing whether it is the highest peak because of the fog. When solving such nonlinear optimization problems, it is not desirable to terminate the search at a peak that is grossly inferior to the highest peak. The solution can be checked by repeating the search but starting from a different initial point. However, the shape of the optimum for most optimiza- tion problems bears a greater resemblance to Table Moun- tain in South Africa, rather than to Mount Everest. In other words, for most optimization problems, the region around the optimum is fairly flat. Although on one hand a grossly inferior solution should be avoided, on the other hand the designer should not be preoccupied with improving the solution by tiny amounts in an attempt to locate exactly the global optimum. There will be uncertainty in the design data, especially economic data. Also, there are many issues to be considered other than simply maximizing economic potential or minimizing cost. There could be many reasons why the solution at the exact location of the global optimum might not be preferred, but a slightly suboptimal solution preferred for other reasons, such as safety, ease of control, and so on. Different solutions in the region of the optimum should be examined, rather than considerable effort being expended on finding the solution at the exact location of the global optimum and considering only that solution. In this respect, stochastic optimization has advantages, as it can provide a range of solutions in the region of the optimum. 3.11 SUMMARY – OPTIMIZATION Most design problems will require optimization to be carried out at some stage. The quality of the design is characterized by an objective function to be maximized (e.g. if economic potential is being maximized) or minimized (e.g. if cost is being minimized). The shape of the objective function is critical in determining the optimization strategy. If the objective function is convex in a minimization problem or concave in a maximization problem, then there is a single optimum point. If this is not the case, there can be local optima as well as the global optimum. Various search strategies can be used to locate the optimum. Indirect search strategies do not use information on gradients, whereas direct search strategies require this information. These methods always seek to improve the objective function in each step in a search. On the other hand, stochastic search methods, such as simulated annealing and genetic algorithms, allow some deterioration in the objective function, especially during the early stages of the search, in order to reduce the danger of being attracted to a local optimum rather than the global optimum. However, stochastic optimization can be very slow in converging, and usually needs to be adapted to solve particular problems. Tailoring the methods to suit specific applications makes them much more efficient. The addition of inequality constraints complicates the optimization. These inequality constraints can form convex or nonconvex regions. If the region is nonconvex, then this means that the search can be attracted to a local optimum, even if the objective function is convex in the case of a minimization problem or concave in the case of a maximization problem. In the case that a set of inequality constraints is linear, the resulting region is always convex. The general case of optimization in which the objective function, the equality and inequality constraints are all linear can be solved as a linear programming problem. This can be solved efficiently with, in principle, a guarantee of global optimality. However, the corresponding nonlinear programming problem cannot, in general, be solved efficiently and with a guarantee of global optimality. Such problems are solved by successive linear or successive quadratic programming. Stochastic optimization methods can be very effective in solving nonlinear optimization, because they are less prone to be stuck in a local optimum than deterministic methods. One of the approaches that can be used in design is to carry out structural and parameter optimization of a superstructure. The structural optimization required can be carried out using mixed integer linear programming in the case of a linear problem or mixed integer nonlinear programming in the case of a nonlinear problem. Stochastic optimization can also be very effective for structural optimization problems. 3.12 EXERCISES 1. The overhead of vapor of a distillation column is to be condensed in the heat exchanger using cooling water. There is a trade-off involving the flowrate of cooling water and the size of the condenser. As the flowrate of cooling water increases, its cost increases. However, as the flowrate increases, the return temperature of the cooling water to the cooling tower decreases. This decreases the temperature differences in the condenser and increases its heat transfer area, and hence its capital cost. The condenser has a duty of 4.1 MW and the vapor condenses at a constant temperature of 80◦C. Cooling water is available at 20◦C with a cost of $ 0.02 t−1. The overall heat transfer coefficient can be assumed to be 500 W·m−1·K−1. The cost of the condenser can be assumed to be $2500 m−2 with an installation factor of 3.5. Annual capital charges can be assumed to be 20% of the capital costs. The heat capacity of the cooling water can be assumed constant at 4.2 kJ·kg−1·K−1. The distillation column operates for 8000 h·y−1. Set up an equation for the heat transfer area of the condenser, and hence the annual capital cost of the condenser, in terms of the cooling Exercises 55 water return temperature. Using this equation, carry out a trade- off between the cost of the cooling water and the cost of the condenser to determine approximately the optimum cooling water return temperature. The maximum return temperature should be 50◦C. The heat exchange area required by the condenser is given by: A = Q U
TLM where A = heat transfer area (m2) Q = heat duty (W) U = overall heat transfer coefficient (W·m−1·K−1)
TLM = logarithmic mean temperature difference = (TCOND − TCW2 ) − (TCOND − TCW1 ) ln ( TCOND − TCW2 TCOND − TCW1 ) TCOND = condenser temperature (◦C) TCW1 = inlet cooling water temperature (◦C) TCW2 = outlet cooling water temperature (◦C) 2. A vapor stream leaving a styrene production process contains hydrogen, methane, ethylene, benzene, water, toluene, styrene and ethylbenzene and is to be burnt in a furnace. It is proposed to recover as much of the benzene, toluene, styrene and ethylbenzene as possible from the vapor using low- temperature condensation. The low-temperature condensation requires refrigeration. However, the optimum temperature for the condensation needs to be determined. This involves a trade- off in which the amount and value of material recovered increases as the temperature decreases, but the cost of the refrigeration increases as the temperature decreases. The material flows in the vapor leaving the flash drum are given in Table 3.2, together with their values. Table 3.2 Stream flowrates and component values. Component Flowrate (kmol·s−1) Value ($·kmol−1) Hydrogen 146.0 0 Methane 3.7 0 Ethylene 3.7 0 Benzene 0.67 21.4 Water 9.4 0 Toluene 0.16 12.2 Ethylbenzene 1.6 40.6 Styrene 2.4 60.1 Total 167.63 The low-temperature condensation requires refrigeration, for which the cost is given by: Refrigeration cost = 0.033QCOND ( 40 − TCOND TCOND + 268 ) where QCOND = condenser duty (MW) TCOND = condenser temperature (◦C) The fraction of benzene, toluene, styrene and ethylben- zene condensed can be determined from phase equilibrium calculations. The percent of the various components entering Table 3.3 Condenser performance. Percent of Component Entering Condenser that Leaves with the Vapor Condensation temperature (◦C) 40 30 20 10 0 −10 −20 −30 −40 −50 −60 Benzene 100 93 84 72 58 42 27 15 8 3 1 Toluene 100 80 60 41 25 14 7 3 1 1 0 Ethyl- 100 59 33 18 9 4 2 1 0 0 0 benzene Styrene 100 54 29 15 8 4 2 1 0 0 0 Table 3.4 Stream enthalpy data. Temperature (◦C) Stream enthalpy (MJ·kmol−1) 40 0.45 30 −1.44 20 −2.69 10 −3.56 0 −4.21 −10 −4.72 −20 −5.16 −30 −5.56 −40 −5.93 −50 −6.29 −60 −6.63 the condenser that leave with the vapor are given in Table 3.3 as a function of temperature. The total enthalpy of the flash drum vapor stream as a function of temperature is given in Table 3.4. Calculate the optimum condenser temperature. What practical difficulties do you foresee in using very low temper- atures? 3. A tank containing 1500 m3 of naphtha is to be blended with two other hydrocarbon streams to meet the specifications for gasoline. The final product must have a minimum research octane number (RON) of 95, a maximum Reid Vapor Pressure (RVP) of 0.6 bar, a maximum benzene content of 2% vol and maximum total aromatics of 25% vol. The properties and costs of the three streams are given in the Table 3.5. Table 3.5 Blending streams. RON RVP (bar) Benzene (% vol) Total aromatics (% vol) Cost $·m−3 Naphtha 92 0.80 1.5 15 275 Reformate 98 0.15 15 50 270 Alkylate 97.5 0.30 0 0 350 Assuming that the properties of the mixture blend are in proportion to the volume of stream used, how much reformate and alkylate should be blended to minimize cost? 56 Optimization 4. A petroleum refinery has two crude oil feeds available. The first crude (Crude 1) is high-quality feed and costs $30 per barrel (1 barrel = 42 US gallons). The second crude (Crude 2) is a low-quality feed and costs $20 per barrel. The crude oil is separated into gasoline, diesel, jet fuel and fuel oil. The percent yield of each of these products that can be obtained from Crude 1 and Crude 2 are listed in Table 3.6, together with maximum allowable production flowrates of the products in barrels per day and processing costs. Table 3.6 Refinery data. Yield (% volume) Value of product Maximum production Crude 1 Crude 2 ($·bbl−1) (bbl·day−1) Gasoline 80 47 75 120,000 Jet fuel 4 8 55 8,000 Diesel 10 30 40 30,000 Fuel oil 6 15 30 – Processing cost ($ bb1−1) 1.5 3.0 The economic potential can be taken to be the difference between the selling price of the products and the cost of the crude oil feedstocks. Determine the optimum feed flowrate of the two crude oils from a linear optimization solved graphically. 5. Add a constraint to the specifications for Exercise 4 above such that the production of fuel oil must be greater than 15,000 bbl·day−1. What happens to the problem? How would you describe the characteristics of the modified linear program- ming problem? 6. Devise a superstructure for a distillation design involving a single feed, two products, a reboiler and a condenser that will allow the number of plates in the column itself to be varied between 3 and 10 and at the same time vary the location of the feed tray. 7. A reaction is required to be carried out between a gas and a liquid. Two different types of reactor are to be considered: an agitated vessel (AV) and a packed column (PC). Devise a superstructure that will allow one of the two options to be chosen. Then describe this as integer constraints for the gas and liquid feeds and products. REFERENCES 1. Edgar TF, Himmelblau DM and Lasdon LS (2001) Optimiza- tion of Chemical Processes, 2nd Edition, McGraw-Hill. 2. Biegler LT, Grossmann IE and Westerberg AW (1997) Sys- tematic Methods of Chemical Process Design, Prentice Hall. 3. Floudas CA (1995) Nonlinear and Mixed-Integer Optimiza- tion, Oxford University Press. 4. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH and Teller E (1953) Equation of State Calculations by Fast Computing Machines, J Chem Phys, 21: 1087. 5. Kirkpatrick S, Gelatt CD and Vecchi MP (1983) Optimiza- tion by Simulated Annealing, Science, 220: 671. 6. Goldberg DE (1989) Genetic Algorithms in Search Optimiza- tion and Machine Learning, Addison-Wesley. 7. Mehta VL and Kokossis AC (1988) New Generation Tools for Multiphase Reaction Systems: A Validated Systematic Methodology for Novelty and Design Automation, Comput Chem Eng, 22S: 5119. 8. Choong KL and Smith R (2004) Optimization of Batch Cooling Crystallization, Chem Eng Sci, 59: 313. 9. Choong KL and Smith R (2004) Novel Strategies for Opti- mization of Batch, Semi-batch and Heating/Cooling Evapo- rative Crystallization, Chem Eng Sci, 59: 329. 10. Taha HA (1975) Integer Programming Theory and Applica- tions, Academic Press. 11. Floudas CA (2000) Deterministic Global Optimization: The- ory, Methods and Applications, Kluwer Academic Publishers. 12. Williams HP (1997) Model Building in Mathematical Pro- gramming, 3rd Edition, John Wiley. Phase Equilibrium for Single Components 59 Table 4.1 Solution of the Peng–Robinson equation of state for nitrogen. Pressure 1.013 bar Pressure 5 bar Pressure 50 bar κ 0.43133 0.43133 0.43133 α 0.63482 0.63482 0.63482 a 0.94040 0.94040 0.94040 b 2.4023 × 10−2 2.4023 × 10−2 2.4023 × 10−2 A 1.8471 × 10−3 9.1171 × 10−3 9.1171 × 10−2 B 1.0716 × 10−3 5.2891 × 10−3 5.2891 × 10−2 β −0.99893 0.99471 −0.94711 γ −2.9947 × 10−4 −1.5451 × 10−3 −2.3004 × 10−2 δ −8.2984 × 10−7 −2.0099 × 10−5 −1.8767 × 10−3 q 0.11097 0.11045 0.10734 r −3.6968 × 10−2 −3.6719 × 10−2 −3.6035 × 10−2 q3 − r2 −2.9848 × 10−8 −7.1589 × 10−7 −6.1901 × 10−5 Z1 0.9993 0.9963 0.9727 RT/P (m3·kmol−1) 22.42 4.542 0.4542 Z1RT/P (m3·kmol−1) 22.40 4.525 0.4418 this case, q3 − r2 > 0 and there are three roots given by Equations 4.12 to 4.14. The parameters for the Peng–Robinson equation are given in Table 4.2. Table 4.2 Solution of the Peng–Robinson equation of state for benzene. Pressure 1.013 bar κ 0.68661 α 1.41786 a 28.920 b 7.4270 × 10−2 A 4.9318 × 10−2 B 3.0869 × 10−3 β −0.99691 γ 4.3116 × 10−2 δ −1.4268 × 10−4 q 9.6054 × 10−2 r −2.9603 × 10−2 q3 − r2 9.9193 × 10−6 Z1 0.0036094 Z2 0.95177 Z3 0.04153 From the three roots, only Z1 and Z2 are significant. The smallest root (Z1) relates to the liquid and the largest root (Z2) to the vapor. Thus, the density of liquid benzene is given by: ρL = 78.11 Z1RT/P = 78.11 0.0036094 × 24.0597 = 899.5 kg m−3 This compares with an experimental value of ρL = 876.5 kg·m−3 (an error of 3%). 4.2 PHASE EQUILIBRIUM FOR SINGLE COMPONENTS The phase equilibrium for pure components is illustrated in Figure 4.1. At low temperatures, the component forms a solid phase. At high temperatures and low pressures, the component forms a vapor phase. At high pressures and high temperatures, the component forms a liquid phase. The phase equilibrium boundaries between each of the phases are illustrated in Figure 4.1. The point where the three phase equilibrium boundaries meet is the triple point, where solid, liquid and vapor coexist. The phase equilibrium boundary between liquid and vapor terminates at the critical point. Above the critical temperature, no liquid forms, no matter how high the pressure. The phase equilibrium boundary between liquid and vapor connects the triple point and the Liquid Solid Triple Point Vapor Critical Point Pressure PSAT T Temperature Figure 4.1 Phase equilibrium for a pure component. 60 Thermodynamic Properties and Phase Equilibrium critical point, and marks the boundary where vapor and liquid coexist. For a given temperature on this boundary, the pressure is the vapor pressure. When the vapor pressure is 1 atm, the corresponding temperature is the normal boiling point. If, at any given vapor pressure, the component is at a temperature less than the phase equilibrium, it is subcooled. If it is at a temperature above the phase equilibrium, it is superheated. Various expressions can be used to represent the vapor pressure curve. The simplest expression is the Clausius–Clapeyron equation1,2: ln P SAT = A − B T (4.18) where A and B are constants and T is the absolute temperature. This indicates that a plot of ln P SAT versus 1/T should be a straight line. Equation 4.18 gives a good correlation only over small temperature ranges. Various modifications have been suggested to extend the range of application, for example, the Antoine equation1 – 3: ln P SAT = A − B C + T (4.19) where A, B and C are constants determined by correlating experimental data3. Extended forms of the Antoine equation have also been proposed3. Great care must be taken when using correlated vapor pressure data not to use the correlation coefficients outside the temperature range over which the data has been correlated; otherwise, serious errors can occur. 4.3 FUGACITY AND PHASE EQUILIBRIUM Having considered single component systems, multicompo- nent systems need to be addressed now. If a closed system contains more than one phase, the equilibrium condition can be written as: f Ii = f IIi = f IIIi i = 1, 2, . . . , NC (4.20) where fi is the fugacity of Component i in Phases I , II and III and NC is the number of components. Fugacity is a thermodynamic pressure, but has no strict physical significance. It can be thought of as an “escaping tendency”. Thus, Equation 4.20 states that if a system of different phases is in equilibrium, then the “escaping tendency” of Component i from the different phases is equal. 4.4 VAPOR–LIQUID EQUILIBRIUM Thermodynamic equilibrium in a vapor–liquid mixture is given by the condition that the vapor and liquid fugacities for each component are equal2: f Vi = f Li (4.21) where f Vi = fugacity of Component i in the vapor phase f Li = fugacity of Component i in the liquid phase Thus, equilibrium is achieved when the “escaping ten- dency” from the vapor and liquid phases for Component i are equal. The vapor-phase fugacity coefficient, φVi , can be defined by the expression: f Vi = yiφVi P (4.22) where yi = mole fraction of Component i in the vapor phase φVi = vapor-phase fugacity coefficient P = system pressure The liquid-phase fugacity coefficient φLi can be defined by the expression: f Li = xiφLi P (4.23) The liquid-phase activity coefficient γi can be defined by the expression: f Li = xiγif Oi (4.24) where xi = mole fraction of Component i in the liquid phase φLi = liquid-phase fugacity coefficient γi = liquid-phase activity coefficient f Oi = fugacity of Component i at standard state For moderate pressures, f Oi can be approximated by the saturated vapor pressure, P SATi ; thus, Equation 4.24 becomes3: f Li = xiγiP SATi (4.25) Equations 4.21, 4.22 and 4.23 can be combined to give an expression for the K-value, Ki , that relates the vapor and liquid mole fractions: Ki = yi xi = φ L i φVi (4.26) Equation 4.26 defines the relationship between the vapor and liquid mole fractions and provides the basis for vapor–liquid equilibrium calculations on the basis of equations of state. Thermodynamic models are required for φVi and φ L i from an equation of state. Alternatively, Equations 4.21, 4.22 and 4.25 can be combined to give Ki = yi xi = γiP SAT i φVi P (4.27) This expression provides the basis for vapor–liquid equilib- rium calculations on the basis of liquid-phase activity coeffi- cient models. In Equation 4.27, thermodynamic models are required for φVi (from an equation of state) and γi from a liquid-phase activity coefficient model. Some examples will be given later. At moderate pressures, the vapor phase becomes ideal, as discussed previously, and φVi = 1. For Vapor–Liquid Equilibrium 61 an ideal vapor phase, Equation 4.27 simplifies to: Ki = yi xi = γiP SAT i P (4.28) When the liquid phase behaves as an ideal solution, • all molecules have the same size; • all intermolecular forces are equal; • the properties of the mixture depend only on the proper- ties of the pure components comprising the mixture. Mixtures of isomers, such as o-, m- and p-xylene mixtures, and adjacent members of homologous series, such as n-hexane–n-heptane and benzene–toluene mixtures, give close to ideal liquid-phase behavior. For this case, γi = 1, and Equation 4.28 simplifies to: Ki = yi xi = P SAT i P (4.29) which is Raoult’s law and represents both ideal vapor- and liquid-phase behavior. Correlations are available to relate component vapor pressure to temperature, as dis- cussed above. Comparing Equations 4.28 and 4.29, the liquid-phase nonideality is characterized by the activity coefficient γi . When γi = 1, the behavior is ideal. If γi = 1, then the value of γi can be used to characterize the nonideality: • γi < 1 represents negative deviations from Raoult’s Law; • γi > 1 represents positive deviations from Raoult’s Law. The vapor–liquid equilibrium for noncondensable gases in equilibrium with liquids can often be approximated by Henry’s Law1 – 3: pi = Hixi (4.30) where pi = partial pressure of Component i Hi = Henry’s Law constant (determined experimentally) xi = mole fraction of Component i in the liquid phase Assuming ideal gas behavior (pi = yiP ): yi = Hixi P (4.31) Thus, the K-value is given by: Ki = yi xi = Hi P (4.32) A straight line would be expected from a plot of yi against xi . The ratio of equilibrium K-values for two components measures their relative volatility: αij = Ki Kj (4.33) where αij = volatility of Component i relative to Compo- nent j These expressions form the basis for two alternative approaches to vapor–liquid equilibrium calculations: a. Ki = φLi /φVi forms the basis for calculations based entirely on equations of state. Using an equation of state for both the liquid and vapor phase has a number of advantages. Firstly, f Oi need not be specified. Also, in principle, continuity at the critical point can be guaranteed with all thermodynamic properties derived from the same model. The presence of noncondensable gases, in principle, causes no additional complications. However, the application of equations of state is largely restricted to nonpolar components. b. Ki = γiP SATi /φVi P forms the basis for calculations based on liquid-phase activity coefficient models. It is used when polar molecules are present. For most systems at low pressures, φVι can be assumed to be unity. If high pressures are involved, then φVι must be calculated from an equation of state. However, care should be taken when mixing and matching different models for γ i and φVι for high-pressure systems to ensure that appropriate combinations are taken. Example 4.2 A gas from a combustion process has a flowrate of 10 m3·s−1 and contains 200 ppmv of oxides of nitrogen, expressed as nitric oxide (NO) at standard conditions of 0◦C and 1 atm. This concentration needs to be reduced to 50 ppmv (expressed at standard conditions) before being discharged to the environment. It can be assumed that all of the oxides of nitrogen are present in the form of NO. One option being considered to remove the NO is by absorption in water at 20◦C and 1 atm. The solubility of the NO in water follows Henry’s Law, with HNO = 2.6 × 104 atm at 20◦C. The gas is to be contacted countercurrently with water such that the inlet gas contacts the outlet water. The concentration of the outlet water can be assumed to reach 90% of equilibrium. Estimate the flowrate of water required, assuming the molar mass of gas in kilograms occupies 22.4 m3 at standard conditions of 0◦C and 1 atm. Solution Molar flowrate of gas = 10 × 1 22.4 = 0.446 kmol·s−1 Assuming the molar flowrate of gas remains constant, the amount of NO to be removed = 0.446(200 − 50) × 10−6 = 6.69 × 10−5 kmol·s−1 Assuming the water achieves 90% of equilibrium and contacts countercurrently with the gas, from Henry’s Law (Equation 4.38): xNO = 0.9y ∗ NOP HNO 64 Thermodynamic Properties and Phase Equilibrium 4.6 VAPOR–LIQUID EQUILIBRIUM BASED ON EQUATIONS OF STATE Before an equation of state can be applied to calcu- late vapor–liquid equilibrium, the fugacity coefficient φi for each phase needs to be determined. The relationship between the fugacity coefficient and the volumetric proper- ties can be written as: ln φi = 1 RT ∫ ∞ V [( ∂P ∂Ni ) T,V ,Nj − RT V ] dV − RT ln Z (4.46) For example, the Peng–Robinson equation of state for this integral yields2: ln φi = bi b (Z − 1) − ln(Z − B) − A 2 √ 2B ( 2 ∑ xiai a + bi b ) ln ( Z + (1 + √2)B Z + (1 − √2)B ) (4.47) where A = aP R2T 2 B = bP RT Thus, given critical temperatures, critical pressures and acentric factors for each component, as well as a phase composition, temperature and pressure, the compressibility factor can be determined and hence component fugacity coefficients for each phase can be calculated. Taking the ratio of liquid to vapor fugacity coefficients for each com- ponent gives the vapor–liquid equilibrium K-value for that component. This approach has the advantage of consistency between the vapor- and liquid-phase thermodynamic mod- els. Such models are widely used to predict vapor–liquid equilibrium for hydrocarbon mixtures and mixtures involv- ing light gases. A vapor–liquid system should provide three roots from the cubic equation of state, with only the largest and smallest being significant. The largest root corresponds to the vapor compressibility factor and the smallest is the liquid compressibility factor. However, some vapor–liquid mixtures can present problems. This is particularly so for mixtures involving light hydrocarbons with significant amounts of hydrogen, which are common in petroleum and petrochemical processes. Under some conditions, such mixtures can provide only one root for vapor–liquid systems, when there should be three. This means that both the vapor and liquid fugacity coefficients cannot be calculated and is a limitation of such cubic equations of state. If an activity coefficient model is to be used at high pressure (Equation 4.27), then the vapor-phase fugacity coefficient can be predicted from Equation 4.47. However, this approach has the disadvantage that the thermodynamic models for the vapor and liquid phases are inconsistent. Despite this inconsistency, it might be necessary to use an activity coefficient model if there is reasonable liquid-phase nonideality, particularly with polar mixtures. 4.7 CALCULATION OF VAPOR–LIQUID EQUILIBRIUM In the case of vapor–liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found? In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. Consider a simple process in which a multicomponent feed is allowed to separate into a vapor and a liquid phase with the phases coming to equilibrium, as shown in Figure 4.2. An overall material balance and component material balances can be written as: F = V + L (4.48) Fzi = Vyi + Lxi (4.49) where F = feed flowrate (kmol·s−1) V = vapor flowrate from the separator (kmol·s−1) L = liquid flowrate from the separator (kmol·s−1) zi = mole fraction of Component i in the feed (–) yi = mole fraction of Component i in vapor (–) xi = mole fraction of Component i in liquid (–) The vapor–liquid equilibrium relationship can be defined in terms of K-values by: yi = Kixi (4.50) VaporVyi Feed Liquid Lxi F zi Figure 4.2 Vapor–liquid equilibrium. Calculation of Vapor–Liquid Equilibrium 65 Equations 4.48 to 4.50 can now be solved to give expressions for the vapor- and liquid-phase compositions leaving the separator: yi = zi V F + ( 1 − V F ) 1 Ki (4.51) xi = zi (Ki − 1)V F + 1 (4.52) The vapor fraction (V/F) in Equations 4.51 and 4.52 lies in the range 0 ≤ V/F ≤ 1. For a specified temperature and pressure, Equations 4.51 and 4.52 need to be solved by trial and error. Given that: NC∑ i yi = NC∑ i xi = 1 (4.53) where NC is the number of components, then: NC∑ i yi − NC∑ i xi = 0 (4.54) Substituting Equations 4.51 and 4.52 into Equation 4.54, after rearrangement gives7: NC∑ i zi(Ki − 1) V F (Ki − 1) + 1 = 0 = f (V/F ) (4.55) To solve Equation 4.55, start by assuming a value of V/F and calculate f (V/F ) and search for a value of V/F until the function equals zero. Many variations are possible around the basic flash calculation. Pressure and V/F can be specified and T calculated, and so on. Details can be found in King7. However, two special cases are of particular interest. If it is necessary to calculate the bubble point, then V/F = 0 in Equation 4.55, which simplifies to: NC∑ i zi(Ki − 1) = 0 (4.56) and given: NC∑ i zi = 1 (4.57) This simplifies to the expression for the bubble point: NC∑ i ziKi = 1 (4.58) Thus, to calculate the bubble point for a given mixture and at a specified pressure, a search is made for a temperature to satisfy Equation 4.58. Alternatively, temperature can be specified and a search made for a pressure, the bubble pressure, to satisfy Equation 4.58. Another special case is when it is necessary to calculate the dew point. In this case, V/F = 1 in Equation 4.55, which simplifies to: NC∑ i zi Ki = 1 (4.59) Again, for a given mixture and pressure, temperature is searched to satisfy Equation 4.59. Alternatively, tempera- ture is specified and pressure searched for the dew pressure. If the K-value requires the composition of both phases to be known, then this introduces additional complications into the calculations. For example, suppose a bubble- point calculation is to be performed on a liquid of known composition using an equation of state for the vapor–liquid equilibrium. To start the calculation, a temperature is assumed. Then, calculation of K-values requires knowledge of the vapor composition to calculate the vapor-phase fugacity coefficient, and that of the liquid composition to calculate the liquid-phase fugacity coefficient. While the liquid composition is known, the vapor composition is unknown and an initial estimate is required for the calculation to proceed. Once the K-value has been estimated from an initial estimate of the vapor composition, the composition of the vapor can be reestimated, and so on. Figure 4.3 shows, as an example, the vapor–liquid equilibrium behavior for a binary mixture of benzene and toluene8. Figure 4.3a shows the behavior of temperature of the saturated liquid and saturated vapor (i.e. equilibrium pairs) as the mole fraction of benzene is varied (the balance being toluene). This can be constructed by calculating the bubble and dew points for different concentrations. Figure 4.3b shows an alternative way of representing the vapor–liquid equilibrium in a composition or x–y diagram. The x–y diagram can be constructed from the relative volatility. From the definition of relative volatility for a binary mixture of Components A and B: αAB = yA/xA yB/xB = yA/xA (1 − yA)/(1 − xA) (4.60) Rearranging gives: yA = xAαAB 1 + xA(αAB − 1) (4.61) Thus, by knowing αAB from vapor–liquid equilibrium and by specifying xA, yA can be calculated. Figure 4.3a also shows a typical vapor–liquid equilibrium pair, where the mole fraction of benzene in the liquid phase is 0.4 and that in the vapor phase is 0.62. A diagonal line across the x–y diagram represents equal vapor and liquid compositions. The phase equilibrium behavior shows a curve above the diagonal line. This indicates that benzene has a higher concentration in the vapor phase than toluene, that is, 66 Thermodynamic Properties and Phase Equilibrium 80 90 100 110 120 0 0.2 0.4 0.8 Mole Fraction of Benzene T em pe ra tu re ( °C ) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.8 Mole Fraction of Benzene (Liquid Phase) M ol e Fr ac tio n of B en ze ne (V ap or P ha se ) Saturated Vapor Saturated Liquid 0.5 0.6 95 Equilibrium Pair P (a) Temperature−composition behaviour. (b) x−y representation. Q 1 R 10.6 Figure 4.3 Vapor–liquid equilibrium for a binary mixture of benzene and toluene at a pressure of 1 atm. (From Smith R and Jobson M, 2000, Distillation, Encyclopedia of Separation Science, Academic Press; reproduced by permission). benzene is the more volatile component. Figure 4.3b shows the same vapor–liquid equilibrium pair as that shown in Figure 4.3a8. Figure 4.3a can be used to predict the separation in a single equilibrium stage, given a specified feed to the stage and a stage temperature. For example, suppose the feed is a mixture with equal mole fractions of benzene and toluene of 0.5 and this is brought to equilibrium at 95◦C (Point Q in Figure 4.3a). Then, the resulting liquid will have a mole fraction of benzene of 0.4 and the vapor, a mole fraction of 0.62. In addition, the quantity of each phase formed can be determined from the lengths of the lines PQ and QR in Figure 4.3a. An overall material balance across the separator gives: mQ = mP + mR (4.62) A material balance for Component i gives: mQxi,Q = mpxi,P + mRxi,R (4.63) Substituting Equation 4.62 into Equation 4.63 and rearrang- ing gives: mp mR = xi,R − xi,Q xi,Q − xi,P (4.64) Thus, in Figure 4.3: mp mR = PQ QR (4.65) The ratio of molar flowrates of the vapor and liquid phases is thus given by the ratio of the opposite line segments. This is known as the Lever Rule, after the analogy with a lever and fulcrum7. Consider first, a binary mixture of two Components A and B; the vapor–liquid equilibrium exhibits only a moderate deviation from ideality, as represented in Figure 4.4a. In this case, as pure A boils at a lower temperature than pure B in the temperature–composition diagram in Figure 4.4a, Component A is more volatile than Component B. This is also evident from the vapor–liquid composition diagram (x–y diagram), as it is above the line of yA = xA. In addition, it is also clear from Figure 4.4a that the order of volatility does not change as the composition changes. By contrast, Figure 4.4b shows a more highly nonideal behavior in which γi > 1 (positive deviation from Raoult’s Law) forms a minimum-boiling azeotrope. At the azeotropic composition, the vapor and liquid are both at the same composition for the mixture. The lowest boiling temperature is below that of either of the pure components and is at the minimum-boiling azeotrope. It is clear from Figure 4.4b that the order of volatility of Components A and B changes, depending on the composition. Figure 4.4c also shows azeotropic behavior. This time, the mixture shows a behavior in which γi < 1 (negative deviation from Raoult’s Law) forms a maximum- boiling azeotrope. This maximum-boiling azeotrope boils at a higher temperature than either of the pure components and would be the last fraction to be distilled, rather than the least volatile component, which would be the case with nonazeotropic behavior. Again, from Figure 4.4c, it can be observed that the order of volatility of Components A and B changes depending on the composition. Minimum- boiling azeotropes are much more common than maximum- boiling azeotropes. Some general guidelines for vapor–liquid mixtures in terms of their nonideality are: a. Mixtures of isomers usually form ideal solutions. b. Mixtures of close-boiling aliphatic hydrocarbons are nearly ideal below 10 bar. c. Mixtures of compounds close in molar mass and struc- ture frequently do not deviate greatly from ideality (e.g. ring compounds, unsaturated compounds, naph- thenes etc.). d. Mixtures of simple aliphatics with aromatic compounds deviate modestly from ideality Calculation of Vapor–Liquid Equilibrium 69 Table 4.10 Bubble-point calculation for a methanol–water mixture using the Wilson equation. zi T = 340 K T = 350 K T = 346.13 K γi Ki ziKi γi Ki ziKi γi Ki ziKi 0.5 1.1429 1.2501 0.6251 1.1363 1.8092 0.9046 1.1388 1.5727 0.7863 0.5 1.2307 0.3289 0.1645 1.2227 0.5012 0.2506 1.2258 0.4273 0.2136 1.00 0.7896 1.1552 0.9999 Solution a. K-values assuming ideal vapor and liquid behavior are given by Raoult’s Law (Equation 4.66). The composition of the liquid is specified to be x1 = 0.5, x2 = 0.5 and the pressure 1 atm, but the temperature is unknown. Therefore, a bubble-point calculation is required to determine the vapor composition. The procedure is the same as that in Example 4.2 and can be carried out in spreadsheet software. Table 4.9 shows the results for Raoult’s Law. Table 4.9 Bubble-point calculation for an ideal methanol–water mixture. zi T = 340 K T = 360 K T = 350 K Ki ziKi Ki ziKi Ki ziKi 0.5 1.0938 0.5469 2.2649 1.1325 1.5906 0.7953 0.5 0.2673 0.1336 0.6122 0.3061 0.4094 0.2047 1.00 0.6805 1.4386 1.0000 Thus, the composition of the vapor at 1 atm is y1 = 0.7953, y2 = 0.2047, assuming an ideal mixture. b. The activity coefficients for the methanol and water can be calculated using the Wilson equation (Equations 4.37 to 4.38). The results are summarized in Table 4.10. Thus, the composition of the vapor phase at 1 atm is y1 = 0.7863, y2 = 0.2136 from the Wilson Equation. For this mixture, at these conditions, there is not much difference between the predictions of Raoult’s Law and the Wilson equation, indicating only moderate deviations from ideality at the chosen conditions. Example 4.5 2-Propanol (isopropanol) and water form an azeotropic mixture at a particular liquid composition that results in the vapor and liquid compositions being equal. Vapor–liquid equi- librium for 2-propanol–water mixtures can be predicted by the Wilson equation. Vapor pressure coefficients in bar with tempera- ture in Kelvin for the Antoine equation are given in Table 4.113. Data for the Wilson equation are given in Table 4.126. Assume the gas constant R = 8.3145 kJ·kmol−1·K−1. Determine the azeotropic composition at 1 atm. Table 4.11 Antoine equation coefficients for 2-propanol and water6. Ai Bi Ci 2-propanol 13.8228 4628.96 −20.524 Water 11.9647 3984.93 −39.734 Table 4.12 Data for 2-propanol (1) and water (2) for the Wilson equation at 1 atm6. V1 (m3·kmol−1) V2 (m3·kmol−1) (λ12 – λ11) (kJ·kmol−1) (λ21 – λ22) (kJ·kmol−1) 0.07692 0.01807 3716.4038 5163.0311 Solution To determine the location of the azeotrope for a spec- ified pressure, the liquid composition has to be varied and a bubble-point calculation performed at each liquid composition until a composition is identified, whereby xi = yi . Alternatively, the vapor composition could be varied and a dew-point calcu- lation performed at each vapor composition. Either way, this requires iteration. Figure 4.5 shows the x–y diagram for the 2- propanol–water system. This was obtained by carrying out a bubble-point calculation at different values of the liquid com- position. The point where the x–y plot crosses the diagonal line gives the azeotropic composition. A more direct search for the azeotropic composition can be carried out for such a binary sys- tem in a spreadsheet by varying T and x1 simultaneously and by solving the objective function (see Section 3.9): (x1K1 + x2K2 − 1)2 + (x1 − x1K1)2 = 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Azeotrope x1 y1 Figure 4.5 x–y plot for the system 2-propanol (1) and water (2) from the Wilson equation at 1 atm. 70 Thermodynamic Properties and Phase Equilibrium The first bracket in this equation ensures that the bubble-point criterion is satisfied. The second bracket ensures that the vapor and liquid compositions are equal. The solution of this is given when x1 = y1 = 0.69 and x2 = y2 = 0.31 for the system of 2-propanol-water at 1 atm. 4.8 LIQUID–LIQUID EQUILIBRIUM As the components in a liquid mixture become more chemically dissimilar, their mutual solubility decreases. This is characterized by an increase in their activity coefficients (for positive deviation from Raoult’s Law). If the chemical dissimilarity, and the corresponding increase in activity coefficients, become large enough, the solution can separate into two-liquid phases. Figure 4.6 shows the vapor–liquid equilibrium behav- ior of a system exhibiting two-liquid phase behav- ior. Two-liquid phases exist in the areas abcd in Figures 4.6a and 4.6b. Liquid mixtures outside of this two- phase region are homogeneous. In the two-liquid phase region, below ab, the two-liquid phases are subcooled. Along ab, the two-liquid phases are saturated. The area of the two-liquid phase region becomes narrower as the temperature increases. This is because the mutual sol- ubility normally increases with increasing temperature. For a mixture within the two-phase region, say Point Q in Figures 4.6a and 4.6b, at equilibrium, two-liquid phases are formed at Points P and R. The line PR is the tie line. The analysis for vapor–liquid separation in Equations 4.56 to 4.59 also applies to a liquid–liquid sepa- ration. Thus, in Figures 4.6a and 4.6b, the relative amounts of the two-liquid phases formed from Point Q at P and R follows the Lever Rule given by Equation 4.65. In Figure 4.6a, the azeotropic composition at Point e lies outside the region of two-liquid phases. In Figure 4.6b, the azeotropic composition lies inside the region of two- liquid phases. Any two-phase liquid mixture vaporizing along ab, in Figure 4.6b, will vaporize at the same tem- perature and have a vapor composition corresponding with Point e. This results from the lines of vapor–liquid equi- librium being horizontal in the vapor–liquid region, as shown in Figure 4.3. A liquid mixture of composition e, in Figure 4.6b, produces a vapor of the same composi- tion and is known as a heteroazeotrope. The x –y diagrams in Figure 4.6c and 4.6d exhibit a horizontal section, corre- sponding with the two-phase region. For liquid–liquid equilibrium, the fugacity of each component in each phase must be equal: (xiγi) I = (xiγi)II (4.67) where I and II represent the two-liquid phases in equi- librium. The equilibrium K-value or distribution coefficient Bubble Point Dew Point xA xA yA yA yA xA xA yA T b b P a e e Prediction of Vapor−liquid Equilibrium Model Two Liquids Liquid Vapor Liquid Bubble Point Dew Point Minimum Boiling Azeotrope T b b a a e e Prediction of Vapor−liquid Equilibrium Model Two Liquids Liquid Vapor Liquid (a) (b) cd d c Heterozeotrope a Q R P Q R (c) (d) Figure 4.6 Phase equilibrium featuring two-liquid phases. Calculation of Liquid–Liquid Equilibrium 71 for Component i can be defined by Ki = x I i xIIi = γ II i γ Ii (4.68) 4.9 LIQUID–LIQUID EQUILIBRIUM ACTIVITY COEFFICIENT MODELS A model is needed to calculate liquid–liquid equilibrium for the activity coefficient from Equation 4.67. Both the NRTL and UNIQUAC equations can be used to predict liquid–liquid equilibrium. Note that the Wilson equation is not applicable to liquid–liquid equilibrium and, therefore, also not applicable to vapor–liquid–liquid equilibrium. Parameters from the NRTL and UNIQUAC equations can be correlated from vapor–liquid equilibrium data6 or liquid–liquid equilibrium data9,10. The UNIFAC method can be used to predict liquid–liquid equilibrium from the molecular structures of the components in the mixture3. 4.10 CALCULATION OF LIQUID– LIQUID EQUILIBRIUM The vapor–liquid x–y diagram in Figures 4.6c and d can be calculated by setting a liquid composition and calculating the corresponding vapor composition in a bubble point calculation. Alternatively, vapor composition can be set and the liquid composition determined by a dew point calculation. If the mixture forms two-liquid phases, the vapor–liquid equilibrium calculation predicts a maximum in the x–y diagram, as shown in Figures 4.6c and d. Note that such a maximum cannot appear with the Wilson equation. To calculate the compositions of the two coexisting liquid phases for a binary system, the two equations for phase equilibrium need to be solved: (x1γ1) I = (x1γ1)II , (x2γ2)I = (x2γ2)II (4.69) where xI1 + xI2 = 1, xII1 + xII2 = 1 (4.70) Given a prediction of the liquid-phase activity coeffi- cients, from say the NRTL or UNIQUAC equations, then Equations 4.69 and 4.70 can be solved simultaneously for xI1 and x II 1 . There are a number of solutions to these equations, including a trivial solution corresponding with xI1 = xII1 . For a solution to be meaningful: 0 < xI1 <1, 0 < x II 1 <1, x I 1 = xII1 (4.71) For a ternary system, the corresponding equations to be solved are: (x1γ1) I = (x1γ1)II , (x2γ2) I = (x2γ2)II , (x3γ3)I = (x3γ3)II (4.72) FeedF zi Liquid I LI xi I Liquid II LII xi II Figure 4.7 Liquid–liquid equilibrium. These equations can be solved simultaneously with the material balance equations to obtain xI1 , x I 2 , x II 1 and x II 2 . For a multicomponent system, the liquid–liquid equilibrium is illustrated in Figure 4.7. The mass balance is basically the same as that for vapor–liquid equilibrium, but is written for two-liquid phases. Liquid I in the liquid–liquid equilibrium corresponds with the vapor in vapor–liquid equilibrium and Liquid II corresponds with the liquid in vapor–liquid equilibrium. The corresponding mass balance is given by the equivalent to Equation 4.55: NC∑ i zi(Ki − 1) LI F (Ki − 1) + 1 = 0 = f (LI /F ) (4.73) where F = feed flowrate (kmol·s−1) LI = flowrate of Liquid I from the separator (kmol·s−1) LII = flowrate of Liquid II from the separator (kmol·s−1) zi = mole fraction of Component i in the feed (–) xIi = mole fraction of Component i in Liquid I (–) xIIi = mole fraction of Component i in Liquid II (–) Ki = K-value, or distribution coefficient, for Component i (–) Also, the liquid–liquid equilibrium K-value needs to be defined for equilibrium to be Ki = x I i xIIi = γ II i γ Ii (4.74) xI1 , x I 2 , . . . , x I NC −1; x II 1 , x II 2 , . . . , x II NC −1 and L I/F need to be varied simultaneously to solve Equations 4.73 and 4.74. Example 4.6 Mixtures of water and 1-butanol (n-butanol) form two-liquid phases. Vapor–liquid equilibrium and liquid–liquid equilibrium for the water–1-butanol system can be predicted by the NRTL equation. Vapor pressure coefficients in bar with temperature in Kelvin for the Antoine equation are given in Table 4.136. Data for the NRTL equation are given in Table 4.14, for a pressure of 1 atm6. Assume the gas constant R = 8.3145 kJ·kmol−1·K−1.