Introduction to Food Engineering Fourth Edition PDF

Introduction to Food Engineering Fourth Edition Food Science and Technology International Series Series Editor Steve L. Taylor University of Nebraska—Lincoln, USA Advisory Board Ken Buckle The University of New South Wales, Australia Mary Ellen Camire University of Maine, USA Roger Clemens University of Southern California, USA Hildegarde Heymann University of California—Davis, USA Robert Hutkins University of Nebraska—Lincoln, USA Ron S. Jackson Quebec, Canada Huub Lelieveld Bilthoven, The Netherlands Daryl B. Lund University of Wisconsin, USA Connie Weaver Purdue University, USA Ron Wrolstad Oregon State University, USA A complete list of books in this series appears at the end of this volume. Introduction to Food Engineering Fourth Edition R. Paul Singh Department of Biological and Agricultural Engineering and Department of Food Science and Technology University of California Davis, California Dennis R. Heldman Heldman Associates Mason, Ohio AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK Copyright © 2009, 2001, 1993, 1984 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (44) 1865 843830, fax: (44) 1865 853333, E-mail: [email protected]. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data APPLICATION SUBMITTED British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-0-12-370900-4 For information on all Academic Press publications visit our Web site at www.elsevierdirect.com Printed in China 08 09 10 9 8 7 6 5 4 3 2 1 About the Authors R. Paul Singh and Dennis R. Heldman have teamed up here once again, to produce the fourth edition of Introduction to Food Engineering; a book that has had continu- ing success since its ﬁrst publication in 1984. Together, Drs. Singh and Heldman have many years of experience in teaching food engineering courses to students, both under- graduates and graduates; along with Dr. Heldman’s experience in the food processing industry, is once again apparent in their approach within this book. The authors’ crite- ria for the careful selection of topics, and the way in which this material is presented, will enable students and faculty to reap the full beneﬁts of this combined wealth of knowledge. Singh is a distinguished professor of food engineering at the University of California, Davis, where he has been teaching courses on topics in food engineering since 1975. The American Society of Agricultural Engineers (ASAE) awarded him the Young Educator Award in 1986. The Institute of Food Technologists (IFT) awarded him the Samuel Cate Prescott Award for Research in 1982. In 1988, he received the International Award from the IFT, reserved for a member of the Institute who “has made outstand- ing efforts to promote the international exchange of ideas in the ﬁeld of food technol- ogy.” In 1997, he received the Distinguished Food Engineering Award from the Dairy and Food Industry Suppliers Association and ASAE, with a citation recognizing him as a “world class scientist and educator with outstanding scholarly distinction and international service in food engineering.” In 2007, ASAE awarded him the Kishida International Award for his worldwide contributions in food engineering education. He was elected a fellow of both IFT and ASAE in 2000 and the International Academy of Food Science and Technology in 2001. He has helped establish food engineering pro- grams in Portugal, Indonesia, Argentina, and India and has lectured extensively on food engineering topics in 40 different nations around the world. Singh has authored, or co-authored, fourteen books and published more than two hundred technical papers. His research program at Davis addresses study of heat and mass transfer in foods dur- ing processing using mathematical simulations and seeking sustainability in the food supply chain. In 2008, Singh was elected to the National Academy of Engineers “for innovation and leadership in food engineering research and education.” The honor is one of the highest professional distinctions for engineers in the United States. Currently, Heldman is the Principal of Heldman Associates, a consulting business dedicated to applications of engineering concepts to food processing for educational institutions, industry and government. He is an Adjunct Professor at the University of California-Davis and Professor Emeritus at the University of Missouri. His research interests focus on use of models to predict thermophysical properties of foods and the development of simulation models for processes used in food manufacturing. He has been author or co-author of over 150 research papers and is Co-Editor of the v vi About the Authors Handbook of Food Engineering, and Editor of the Encyclopedia of Agricultural, Food and Biological Engineering and an Encyclopedia of Biotechnology in Agriculture and Food to be published in 2009. Heldman has taught undergraduate and graduate food engineering courses at Michigan State University, University of Missouri and Rutgers, The State University of New Jersey. He has held technical administration positions at the Campbell Soup Company, the National Food Processors Association, and the Weinberg Consulting Group, Inc. He has been recognized for contributions, such as the DFISA- ASAE Food Engineering Award in 1981, the Distinguished Alumni Award from The Ohio State University in 1978, the Young Researcher Award from ASAE in 1974, and served as President of the Institute of food Technologists (IFT) in 2006–07. In addition, Heldman is Fellow in the IFT (1981), the American Society of Agricultural Engineers (1984), and the International Academy of Food Science & Technology (2006). Foreword Nine out of ten Food Science students would probably claim the Food Engineering course as the most difﬁcult one in their undergraduate curriculum. Although part of the difﬁculty may be related to how food engineering is taught, much of the difﬁculty with food engineering stems from the nature of the material. It’s not necessarily that food engineering concepts are more difﬁcult than other food science concepts, but food engi- neering is based on derivations of equations, and the quantitative manipulation of those equations to solve problems. From word problems to integral calculus, the skills required to master food engineer- ing concepts are difﬁcult for many Food Science students. However, these concepts are integral to the required competencies for an IFT-approved Food Science program, and are the cornerstone for all of food processing and manufacturing. It is critical that Food Science graduates have a good understanding of engineering principles, both because they are likely to need the concepts during the course of their career but also because they will most certainly need to interact with engineers in an educated manner. Food Science graduates who can use quantitative engineering approaches will stand out from their co-workers in the ﬁeld. Fortunately, two of the leading food engineers, Paul Singh and Dennis Heldman, have teamed up to write a textbook that clearly and simply presents the complex engineering material that Food Scientists need to know to be successful. In this fourth edition of a classic Food Engineering textbook, Singh and Heldman have once again improved the book even further. New chapters on process control, food packaging, and process opera- tions like ﬁltration, centrifugation and mixing now supplement the classic chapters on mass and energy balances, thermodynamics, heat transfer and ﬂuid ﬂow. Furthermore, numerous problems have now been solved with MATLAB, an engineering mathematical problem solver, to enhance student’s math skills. A good textbook should clearly and concisely present material needed by the students and at a level appropriate to their backgrounds. With chapters that are broken down into short, manageable sections that promote learning, the easy-to-follow explanations in the 4th Edition of Singh and Heldman are aimed at the perfect level for Food Scientists. Numerous example problems, followed by practice problems, help students test their understanding of the concepts. With ﬁfteen chapters that cover the fundamental aspects of engineering and their practical application to foods, this book is an ideal text for courses in both food engineering and food processing. It will also serve as a useful refer- ence for Food Science graduates throughout their career. Richard W. Hartel Professor of Food Engineering University of Wisconsin-Madison vii This page intentionally left blank Preface The typical curriculum for an undergraduate food science major in the United States and Canada requires an understanding of food engineering concepts. The stated con- tent of this portion of the curriculum is “Engineering principles including mass and energy balances, thermodynamics, ﬂuid ﬂow, and heat and mass transfer”. The expec- tations include an application of these principles to several areas of food processing. Presenting these concepts to students with limited background in mathematics and engineering science presents a signiﬁcant challenge. Our goal, in this text book, is to provide students, planning to become food science professionals, with sufﬁcient back- ground in engineering concepts to be comfortable when communicating with engineer- ing professionals. This text book has been developed speciﬁcally for use in undergraduate food engineer- ing courses taken by students pursuing a four-year degree program in food science. The topics presented have been selected to illustrate applications of engineering during the handling, processing, storage, packaging and distribution of food products. Most of the topics include some descriptive background about a process, fundamental engineering concepts and example problems. The approach is intended to assist the student in appre- ciating the applications of the concepts, while gaining an understanding of problem- solving approaches as well as gaining conﬁdence with the concepts. The scope of the book ranges from basic engineering principles, based on fundamental physics, to several applications in food processing. Within the ﬁrst four chapters, the concepts of mass and energy balance, thermodynamics, ﬂuid ﬂow and heat transfer are introduced. A signiﬁcant addition to this section of the fourth edition is an introduc- tion to the concepts of process control. The next four chapters include applications of thermodynamics and heat transfer to preservation processes, refrigeration, freezing pro- cesses and evaporation processes used in concentration of liquid foods. Following the chapters devoted to the concepts of psychrometrics and mass transfer, several chapters are used to present applications of these concepts to membrane separation processes, dehydration processes, extrusion processes and packaging. Finally, a new chapter in this edition is devoted to supplemental processes, including ﬁltration, centrifugation and mixing. Most features of the ﬁrst three editions of this book are included in this fourth edition. Chapters include modest amounts of descriptive material to assist the student in appre- ciating the process applications. Although equations are developed from fundamental concepts, the equations are used to illustrate the solution to practical problems. Most chapters contain many example problems to illustrate various concepts and applica- tions, and several examples are presented in spreadsheet program format. At the end of most chapters, lists of problems are provided for the student to use in gaining conﬁ- dence with problem-solving skills, and the more difﬁcult problems are identiﬁed. ix x Preface The focus of additions to the fourth edition has been on evolving processes and related information. Chapter 2 has been expanded to include information on properties of dry food powders and applications during handling of these products. The new material on process controls in Chapter 3 will assist students in understanding the systems used to operate and control food manufacturing operations. Numerous revisions and additions in the preservation process chapter provide information on applications of evolving technol- ogies for food preservation. Completely new chapters have been included on the subjects of supplemental processes (ﬁltration, centrifugation, mixing) and extrusion processes. Finally, a separate chapter has been devoted to food packaging, to emphasize applications of engi- neering concepts in selection of packaging materials and prediction of product shelf-life. The primary users of this book are the faculty involved in teaching students pursuing an undergraduate degree in Food Science. The approaches used to present the concepts and applications are based on our own combined teaching experiences. Faculty members are encouraged to select chapters and associated materials to meet the speciﬁc objectives of the course being taught. The descriptive information, concepts and problems have been organized to provide maximum ﬂexibility in teaching. The organization of the informa- tion in the book does serve as a study guide for students. Some students may be able to solve the problems at the end of chapters after independent study of the concepts pre- sented within a given chapter. For the purposes to enhance learning, many illustrations in the book are available in animated form at www.rpaulsingh.com. This website also contains most of the solved examples in an electronic form that allow “what-if” analysis. The topics presented in this book can be easily organized into a two-course sequence. The focus of the ﬁrst course would be on engineering concepts and include information from Chapters 1 through 4, and the second course would focus on applications using Chapters 5 to 8. Alternatively, Chapters 9 and 10 could be added to the course on fundamentals, and the applications from Chapters 11 through 15 would be used in the second course. The chapters on applications provide an ideal basis for a process-based capstone course. A new feature in this edition is the inclusion of several problems that require the use of MATLAB®. We are indebted to Professor Thomas R. Rumsey for generously sharing several of these problems that he has used in his own teaching. We thank Ms. Barbara Meierhenry for her valuable assistance in editing the original manuscript. We appreciate the many recommendations from colleagues, and the encouragement from students, as received over a period of nearly 25 years. All of these comments and suggestions have been valuable, and have made the continuous development of this book a rewarding experience. We will continue to respond to communications from faculty members and students as the concepts and applications of food engineering continue to evolve. R. Paul Singh Dennis R. Heldman Contents About the Authors................................................................................................................. v Foreword.............................................................................................................................vii Preface..................................................................................................................................ix CHAPTER 1 Introduction....................................................................................................... 1 1.1 Dimensions.............................................................................................1 1.2 Engineering Units...................................................................................2 1.2.1 Base Units....................................................................................2 1.2.2 Derived Units..............................................................................3 1.2.3 Supplementary Units..................................................................4 1.3 System...................................................................................................10 1.4 State of a System.................................................................................... 11 1.4.1 Extensive Properties..................................................................12 1.4.2 Intensive Properties..................................................................13 1.5 Density...................................................................................................13 1.6 Concentration.......................................................................................15 1.7 Moisture Content..................................................................................17 1.8 Temperature...........................................................................................20 1.9 Pressure..................................................................................................22 1.10 Enthalpy.................................................................................................26 1.11 Equation of State and Perfect Gas Law................................................26 1.12 Phase Diagram of Water.......................................................................27 1.13 Conservation of Mass...........................................................................29 1.13.1 Conservation of Mass for an Open System.............................30 1.13.2 Conservation of Mass for a Closed System.............................32 1.14 Material Balances..................................................................................32 1.15 Thermodynamics..................................................................................41 1.16 Laws of Thermodynamics.....................................................................42 1.16.1 First Law of Thermodynamics..................................................42 1.16.2 Second Law of Thermodynamics.............................................42 1.17 Energy...................................................................................................43 1.18 Energy Balance......................................................................................45 1.19 Energy Balance for a Closed System....................................................45 1.19.1 Heat...........................................................................................45 1.19.2 Work..........................................................................................46 1.20 Energy Balance for an Open System....................................................55 1.20.1 Energy Balance for Steady Flow Systems.................................56 1.21 A Total Energy Balance..........................................................................56 1.22 Power.....................................................................................................59 xi xii Contents 1.23 Area........................................................................................................59 Problems.........................................................................................................60 List of Symbols...............................................................................................62 Bibliography...................................................................................................63 CHAPTER 2 Fluid Flow in Food Processing........................................................................... 65 2.1 Liquid Transport Systems........................................................................66 2.1.1 Pipes for Processing Plants...........................................................67 2.1.2 Types of Pumps.............................................................................68 2.2 Properties of Liquids............................................................................... 71 2.2.1 Terminology Used in Material Response to Stress......................72 2.2.2 Density...........................................................................................72 2.2.3 Viscosity.........................................................................................73 2.3 Handling Systems for Newtonian Liquids.............................................81 2.3.1 The Continuity Equation..............................................................81 2.3.2 Reynolds Number.........................................................................84 2.3.3 Entrance Region and Fully Developed Flow................................88 2.3.4 Velocity Proﬁle in a Liquid Flowing Under Fully Developed Flow Conditions........................................................90 2.3.5 Forces Due to Friction...................................................................96 2.4 Force Balance on a Fluid Element Flowing in a Pipe—Derivation of Bernoulli Equation............................................................................100 2.5 Energy Equation for Steady Flow of Fluids.......................................... 107 2.5.1 Pressure Energy........................................................................... 110 2.5.2 Kinetic Energy............................................................................. 110 2.5.3 Potential Energy.......................................................................... 112 2.5.4 Frictional Energy Loss................................................................. 112 2.5.5 Power Requirements of a Pump................................................. 115 2.6 Pump Selection and Performance Evaluation..................................... 119 2.6.1 Centrifugal Pumps...................................................................... 119 2.6.2 Head............................................................................................121 2.6.3 Pump Performance Characteristics............................................121 2.6.4 Pump Characteristic Diagram................................................... 125 2.6.5 Net Positive Suction Head......................................................... 126 2.6.6 Selecting a Pump for a Liquid Transport System..................... 129 2.6.7 Afﬁnity Laws............................................................................... 135 2.7 Flow Measurement............................................................................... 136 2.7.1 The Pitot Tube............................................................................ 140 2.7.2 The Oriﬁce Meter....................................................................... 142 2.7.3 The Venturi Meter....................................................................... 146 2.7.4 Variable-Area Meters.................................................................. 146 2.7.5 Other Measurement Methods................................................... 147 Contents xiii 2.8 Measurement of Viscosity.................................................................. 148 2.8.1 Capillary Tube Viscometer.................................................... 148 2.8.2 Rotational Viscometer........................................................... 150 2.8.3 Inﬂuence of Temperature on Viscosity................................. 153 2.9 Flow Characteristics of Non-Newtonian Fluids............................... 155 2.9.1 Properties of Non-Newtonian Fluids................................... 155 2.9.2 Velocity Proﬁle of a Power Law Fluid....................................161 2.9.3 Volumetric Flow Rate of a Power Law Fluid......................... 162 2.9.4 Average Velocity in a Power Law Fluid.................................. 163 2.9.5 Friction Factor and Generalized Reynolds Number for Power Law Fluids............................................................. 163 2.9.6 Computation of Pumping Requirement of Non-newtonian Liquids........................................................ 166 2.10 Transport of solid foods.................................................................... 169 2.10.1 Properties of Granular Materials and Powders.....................170 2.10.2 Flow of Granular Foods......................................................... 175 Problems...................................................................................................... 178 List of Symbols............................................................................................ 183 Bibliography................................................................................................ 185 CHAPTER 3 Energy and Controls in Food Processes........................................................... 187 3.1 Generation of Steam.......................................................................... 187 3.1.1 Steam Generation Systems.................................................... 188 3.1.2 Thermodynamics of Phase Change...................................... 190 3.1.3 Steam Tables........................................................................... 194 3.1.4 Steam Utilization................................................................... 200 3.2 Fuel Utilization.................................................................................. 204 3.2.1 Systems................................................................................... 206 3.2.2 Mass and Energy Balance Analysis.........................................207 3.2.3 Burner Efﬁciencies..................................................................209 3.3 Electric Power Utilization................................................................... 210 3.3.1 Electrical Terms and Units......................................................212 3.3.2 Ohm’s Law..............................................................................213 3.3.3 Electric Circuits.......................................................................214 3.3.4 Electric Motors........................................................................216 3.3.5 Electrical Controls...................................................................217 3.3.6 Electric Lighting......................................................................218 3.4 Process Controls in Food Processing................................................ 220 3.4.1 Processing Variables and Performance Indicators................ 222 3.4.2 Input and Output Signals to Control Processes................... 224 3.4.3 Design of a Control System................................................... 224 3.5 Sensors................................................................................................ 232 xiv Contents 3.5.1 Temperature............................................................................. 232 3.5.2 Liquid Level in a Tank.............................................................. 234 3.5.3 Pressure Sensors....................................................................... 235 3.5.4 Flow Sensors............................................................................. 236 3.5.5 Glossary of Terms Important in Data Acquisition................. 237 3.6 Dynamic Response Characteristics of Sensors.................................... 237 Problems.......................................................................................................241 List of Symbols............................................................................................ 244 Bibliography................................................................................................ 245 CHAPTER 4 Heat Transfer in Food Processing.................................................................... 247 4.1 Systems for Heating and Cooling Food Products............................... 248 4.1.1 Plate Heat Exchanger............................................................... 248 4.1.2 Tubular Heat Exchanger.......................................................... 252 4.1.3 Scraped-surface Heat Exchanger.............................................. 253 4.1.4 Steam-infusion Heat Exchanger.............................................. 255 4.1.5 Epilogue.................................................................................... 256 4.2 Thermal Properties of Foods................................................................ 257 4.2.1 Speciﬁc Heat............................................................................. 257 4.2.2 Thermal Conductivity.............................................................. 260 4.2.3 Thermal Diffusivity.................................................................. 262 4.3 Modes of Heat Transfer........................................................................ 264 4.3.1 Conductive Heat Transfer........................................................ 264 4.3.2 Convective Heat Transfer......................................................... 267 4.3.3 Radiation Heat Transfer........................................................... 269 4.4 Steady-State Heat Transfer.....................................................................270 4.4.1 Conductive Heat Transfer in a Rectangular Slab.....................271 4.4.2 Conductive Heat Transfer through a Tubular Pipe................ 274 4.4.3 Heat Conduction in Multilayered Systems............................. 277 4.4.4 Estimation of Convective Heat-Transfer Coefﬁcient.............. 285 4.4.5 Estimation of Overall Heat-Transfer Coefﬁcient.....................302 4.4.6 Fouling of Heat Transfer Surfaces........................................... 306 4.4.7 Design of a Tubular Heat Exchanger.......................................312 4.4.8 The Effectiveness-NTU Method for Designing Heat Exchangers................................................................................ 320 4.4.9 Design of a Plate Heat Exchanger........................................... 325 4.4.10 Importance of Surface Characteristics in Radiative Heat Transfer............................................................................ 332 4.4.11 Radiative Heat Transfer between Two Objects....................... 334 4.5 Unsteady-State Heat Transfer............................................................... 337 4.5.1 Importance of External versus Internal Resistance to Heat Transfer..................................................................................... 339 Contents xv 4.5.2 Negligible Internal Resistance to Heat Transfer (NBi 0.1)—A Lumped System Analysis................................ 340 4.5.3 Finite Internal and Surface Resistance to Heat Transfer (0.1 NBi 40)....................................................................... 345 4.5.4 Negligible Surface Resistance to Heat Transfer (NBi 40)..... 348 4.5.5 Finite Objects............................................................................. 348 4.5.6 Procedures to Use Temperature–Time Charts.......................... 350 4.5.7 Use of fh and j Factors in Predicting Temperature in Transient Heat Transfer............................................................. 358 4.6 Electrical Conductivity of Foods.......................................................... 366 4.7 Ohmic Heating..................................................................................... 369 4.8 Microwave Heating................................................................................371 4.8.1 Mechanisms of Microwave Heating.......................................... 372 4.8.2 Dielectric Properties.................................................................. 373 4.8.3 Conversion of Microwave Energy into Heat............................ 374 4.8.4 Penetration Depth of Microwaves............................................ 375 4.8.5 Microwave Oven........................................................................ 377 4.8.6 Microwave Heating of Foods.................................................... 378 Problems...................................................................................................... 380 List of Symbols............................................................................................ 397 Bibliography................................................................................................ 399 CHAPTER 5 Preservation Processes................................................................................... 403 5.1 Processing Systems............................................................................... 403 5.1.1 Pasteurization and Blanching Systems..................................... 404 5.1.2 Commercial Sterilization Systems............................................ 406 5.1.3 Ultra-High Pressure Systems...................................................... 410 5.1.4 Pulsed Electric Field Systems.....................................................412 5.1.5 Alternative Preservation Systems...............................................413 5.2 Microbial Survivor Curves.....................................................................413 5.3 Inﬂuence of External Agents.................................................................418 5.4 Thermal Death Time F......................................................................... 422 5.5 Spoilage Probability............................................................................. 423 5.6 General Method for Process Calculation............................................ 424 5.6.1 Applications to Pasteurization.................................................. 426 5.6.2 Commercial Sterilization.......................................................... 429 5.6.3 Aseptic Processing and Packaging............................................ 432 5.7 Mathematical Methods........................................................................ 440 5.7.1 Pouch Processing....................................................................... 444 Problems...................................................................................................... 447 List of Symbols............................................................................................ 450 Bibliography.................................................................................................451 xvi Contents CHAPTER 6 Refrigeration.................................................................................................. 455 6.1 Selection of a Refrigerant..................................................................... 456 6.2 Components of a Refrigeration System............................................... 460 6.2.1 Evaporator...................................................................................461 6.2.2 Compressor................................................................................ 463 6.2.3 Condenser.................................................................................. 466 6.2.4 Expansion Valve......................................................................... 468 6.3 Pressure–Enthalpy Charts.....................................................................470 6.3.1 Pressure–Enthalpy Tables.......................................................... 474 6.3.2 Use of Computer-Aided Procedures to Determine Thermodynamic Properties of Refrigerants.............................. 475 6.4 Mathematical Expressions Useful in Analysis of Vapor-Compression Refrigeration....................................................... 478 6.4.1 Cooling Load.............................................................................. 478 6.4.2 Compressor................................................................................ 480 6.4.3 Condenser.................................................................................. 480 6.4.4 Evaporator...................................................................................481 6.4.5 Coefﬁcient of Performance.........................................................481 6.4.6 Refrigerant Flow Rate..................................................................481 6.5 Use of Multistage Systems.................................................................... 490 6.5.1 Flash Gas Removal System.........................................................491 Problems.................................................................................................... 495 List of Symbols............................................................................................ 498 Bibliography................................................................................................ 498 CHAPTER 7 Food Freezing................................................................................................. 501 7.1 Freezing Systems....................................................................................502 7.1.1 Indirect Contact Systems............................................................502 7.1.2 Direct-Contact Systems...............................................................507 7.2 Frozen-Food Properties......................................................................... 510 7.2.1 Density......................................................................................... 510 7.2.2 Thermal Conductivity................................................................. 511 7.2.3 Enthalpy...................................................................................... 511 7.2.4 Apparent Speciﬁc Heat...............................................................513 7.2.5 Apparent Thermal Diffusivity.....................................................513 7.3 Freezing Time.........................................................................................514 7.3.1 Plank’s Equation.........................................................................516 7.3.2 Other Freezing-Time Prediction Methods................................ 520 7.3.3 Pham’s Method to Predict Freezing Time................................. 520 7.3.4 Prediction of Freezing Time of Finite-Shaped Objects............ 524 7.3.5 Experimental Measurement of Freezing Time.......................... 528 Contents xvii 7.3.6 Factors Inﬂuencing Freezing Time............................................ 528 7.3.7 Freezing Rate.............................................................................. 529 7.3.8 Thawing Time............................................................................. 529 7.4 Frozen-Food Storage............................................................................. 530 7.4.1 Quality Changes in Foods during Frozen Storage................... 530 Problems............................................................................................... 534 List of Symbols............................................................................................ 538 Bibliography............................................................................................... 539 CHAPTER 8 Evaporation.................................................................................................... 543 8.1 Boiling-Point Elevation........................................................................ 545 8.2 Types of Evaporators............................................................................ 547 8.2.1 Batch-Type Pan Evaporator....................................................... 547 8.2.2 Natural Circulation Evaporators............................................... 548 8.2.3 Rising-Film Evaporator.............................................................. 548 8.2.4 Falling-Film Evaporator............................................................. 549 8.2.5 Rising/Falling-Film Evaporator................................................. 550 8.2.6 Forced-Circulation Evaporator...................................................551 8.2.7 Agitated Thin-Film Evaporator...................................................551 8.3 Design of a Single-Effect Evaporator................................................... 554 8.4 Design of a Multiple-Effect Evaporator............................................... 559 8.5 Vapor Recompression Systems............................................................. 565 8.5.1 Thermal Recompression............................................................ 565 8.5.2 Mechanical Vapor Recompression............................................ 566 Problems............................................................................................... 566 List of Symbols............................................................................................ 569 Bibliography............................................................................................... 569 CHAPTER 9 Psychrometrics................................................................................................ 571 9.1 Properties of Dry Air..............................................................................571 9.1.1 Composition of Air.....................................................................571 9.1.2 Speciﬁc Volume of Dry Air........................................................ 572 9.1.3 Speciﬁc Heat of Dry Air............................................................. 572 9.1.4 Enthalpy of Dry Air.................................................................... 572 9.1.5 Dry Bulb Temperature............................................................... 573 9.2 Properties of Water Vapor.................................................................... 573 9.2.1 Speciﬁc Volume of Water Vapor................................................ 573 9.2.2 Speciﬁc Heat of Water Vapor..................................................... 573 9.2.3 Enthalpy of Water Vapor............................................................ 574 9.3 Properties of Air–Vapor Mixtures........................................................ 574 9.3.1 Gibbs–Dalton Law..................................................................... 574 9.3.2 Dew-Point Temperature............................................................. 574 xviii Contents 9.3.3 Humidity Ratio (or Moisture Content).............................. 575 9.3.4 Relative Humidity................................................................ 576 9.3.5 Humid Heat of an Air–Water Vapor Mixture...................... 576 9.3.6 Speciﬁc Volume.................................................................... 577 9.3.7 Adiabatic Saturation of Air.................................................. 577 9.3.8 Wet Bulb Temperature.......................................................... 579 9.4 The Psychrometric Chart.................................................................. 582 9.4.1. Construction of the Chart.................................................... 582 9.4.2 Use of Psychrometric Chart to Evaluate Complex Air-Conditioning Processes................................................. 584 Problems...................................................................................................... 589 List of Symbols............................................................................................ 592 Bibliography............................................................................................... 593 CHAPTER 10 Mass Transfer................................................................................................ 595 10.1 The Diffusion Process....................................................................... 596 10.1.1 Steady-State Diffusion of Gases (and Liquids) through Solids...................................................................... 599 10.1.2 Convective Mass Transfer..................................................... 600 10.1.3 Laminar Flow over a Flat Plate............................................ 604 10.1.4 Turbulent Flow Past a Flat Plate.......................................... 608 10.1.5 Laminar Flow in a Pipe........................................................ 608 10.1.6 Turbulent Flow in a Pipe..................................................... 609 10.1.7 Mass Transfer for Flow over Spherical Objects................... 609 10.2 Unsteady-State Mass Transfer........................................................... 610 10.2.1 Transient-State Diffusion...................................................... 611 10.2.2 Diffusion of Gases.................................................................616 Problems....................................................................................................619 List of Symbols..........................................................................................621 Bibliography............................................................................................. 622 CHAPTER 11 Membrane Separation................................................................................. 623 11.1 Electrodialysis Systems..................................................................... 625 11.2 Reverse Osmosis Membrane Systems.............................................. 629 11.3 Membrane Performance................................................................... 636 11.4 Ultraﬁltration Membrane Systems.................................................. 637 11.5 Concentration Polarization............................................................. 639 11.6 Types of Reverse-Osmosis and Ultraﬁltration Systems.................. 645 11.6.1 Plate and Frame.................................................................... 646 11.6.2 Tubular.................................................................................. 646 11.6.3 Spiral-Wound....................................................................... 646 11.6.4 Hollow-Fiber........................................................................ 649 Contents xix Problems................................................................................................... 649 List of Symbols......................................................................................... 650 Bibliography..............................................................................................651 CHAPTER 12 Dehydration................................................................................................. 653 12.1 Basic Drying Processes..................................................................... 653 12.1.1 Water Activity....................................................................... 654 12.1.2 Moisture Diffusion.............................................................. 657 12.1.3 Drying-Rate Curves.............................................................. 658 12.1.4 Heat and Mass Transfer....................................................... 658 12.2 Dehydration systems........................................................................ 660 12.2.1 Tray or Cabinet Dryers........................................................ 660 12.2.2 Tunnel Dryers.......................................................................661 12.2.3 Puff-Drying.......................................................................... 662 12.2.4 Fluidized-Bed Drying.......................................................... 663 12.2.5 Spray Drying........................................................................ 663 12.2.6 Freeze-Drying....................................................................... 664 12.3 Dehydration System Design............................................................. 665 12.3.1 Mass and Energy Balance.................................................... 665 12.3.2 Drying-Time Prediction.......................................................670 Problems................................................................................................... 680 List of Symbols......................................................................................... 685 Bibliography.............................................................................................. 686 CHAPTER 13 Supplemental Processes............................................................................... 689 13.1 Filtration........................................................................................... 689 13.1.1 Operating Equations........................................................... 689 13.1.2 Mechanisms of Filtration.................................................... 695 13.1.3 Design of a Filtration System.............................................. 696 13.2 Sedimentation.................................................................................. 699 13.2.1 Sedimentation Velocities for Low-Concentration Suspensions......................................................................... 699 13.2.2 Sedimentation in High-Concentration Suspensions..........702 13.3 Centrifugation...................................................................................705 13.3.1 Basic Equations.....................................................................705 13.3.2 Rate of Separation................................................................705 13.3.3. Liquid-Liquid Separation.....................................................707 13.3.4 Particle-Gas Separation........................................................709 13.4 Mixing...............................................................................................709 13.4.1 Agitation Equipment............................................................ 711 13.4.2 Power Requirements of Impellers........................................714 xx Contents Problems....................................................................................................718 List of Symbols..........................................................................................719 Bibliography............................................................................................. 720 CHAPTER 14 Extrusion Processes for Foods....................................................................... 721 14.1 Introduction and Background..........................................................721 14.2 Basic Principles of Extrusion............................................................ 722 14.3 Extrusion Systems............................................................................. 729 14.3.1 Cold Extrusion...................................................................... 730 14.3.2 Extrusion Cooking................................................................731 14.3.3 Single Screw Extruders......................................................... 732 14.3.4 Twin-Screw Extruders........................................................... 734 14.4 Extrusion System Design.................................................................. 735 14.5 Design of More Complex Systems................................................... 740 Problems....................................................................................................741 List of Symbols......................................................................................... 742 Bibliography.............................................................................................. 742 CHAPTER 15 Packaging Concepts..................................................................................... 745 15.1 Introduction...................................................................................... 745 15.2 Food Protection................................................................................ 746 15.3 Product Containment...................................................................... 747 15.4 Product Communication................................................................. 748 15.5 Product Convenience....................................................................... 748 15.6 Mass Transfer in Packaging Materials.............................................. 748 15.6.1 Permeability of Packaging Material to “Fixed” Gases.........751 15.7 Innovations in Food Packaging....................................................... 754 15.7.1 Passive Packaging................................................................. 755 15.7.2 Active Packaging................................................................... 755 15.7.3 Intelligent Packaging............................................................ 756 15.8 Food Packaging and Product Shelf-life........................................... 758 15.8.1 Scientiﬁc Basis for Evaluating Shelf Life.............................. 758 15.9 Summary........................................................................................... 766 Problems................................................................................................... 766 List of Symbols......................................................................................... 767 Bibliography............................................................................................. 768 Appendices........................................................................................................................771 A.1 SI System of Units and Conversion Factors.......................................771 A.1.1 Rules for Using SI Units...........................................................771 Table A.1.1: SI Preﬁxes........................................................................771 Table A.1.2: Useful Conversion Factors............................................ 774 Table A.1.3: Conversion Factors for Pressure.................................... 776 Contents xxi A.2 Physical Properties of Foods.............................................................. 777 Table A.2.1: Speciﬁc Heat of Foods................................................... 777 Table A.2.2: Thermal Conductivity of Selected Food Products......................................................................... 778 Table A.2.3: Thermal Diffusivity of Some Foodstuffs...................... 780 Table A.2.4: Viscosity of Liquid Foods...............................................781 Table A.2.5: Properties of Ice as a Function of Temperature................................................................... 782 Table A.2.6: Approximate Heat Evolution Rates of Fresh Fruits and Vegetables When Stored at Temperatures Shown................................................. 782 Table A.2.7: Enthalpy of Frozen Foods............................................. 784 Table A.2.8: Composition Values of Selected Foods........................ 785 Table A.2.9: Coefﬁcients to Estimate Food Properties..................... 786 A.3 Physical Properties of Nonfood Materials........................................ 787 Table A.3.1: Physical Properties of Metals........................................ 787 Table A.3.2: Physical Properties of Nonmetals................................. 788 Table A.3.3: Emissivity of Various Surfaces....................................... 790 A.4 Physical Properties of Water and Air................................................. 792 Table A.4.1: Physical Properties of Water at the Saturation Pressure........................................................ 792 Table A.4.2: Properties of Saturated Steam....................................... 793 Table A.4.3: Properties of Superheated Steam.................................. 795 Table A.4.4: Physical Properties of Dry Air at Atmospheric Pressure.......................................................................... 796 A.5 Psychrometric Charts......................................................................... 797 Figure A.5.1: Psychrometric chart for high temperatures................. 797 Figure A.5.2: Psychrometric chart for low temperatures.................. 798 A.6 Pressure-Enthalpy Data...................................................................... 799 Figure A.6.1: Pressure–enthalpy diagram for Reﬁgerant 12............. 799 Table A.6.1: Properties of Saturated Liquid andVapor R-12............ 800 Figure A.6.2: Pressure–enthalpy diagram of superheated R-12 vapor.................................................................... 803 Table A.6.2: Properties of Saturated Liquid and Vapor R-717 (Ammonia).......................................................... 804 Figure A.6.3: Pressure-enthalpy diagram of superheated R-717 (ammonia) vapor..........................807 Table A.6.3: Properties of Saturated Liquid and Vapor R-134a...................808 Figure A.6.4: Pressure–enthalpy diagram of R-134a......................... 811 Figure A.6.5: Pressure–enthalpy diagram of R-134a (expanded scale)...........................................................812 A.7 Symbols for Use in Drawing Food Engineering Process Equipment...........................................................................................813 xxii Contents A.8 Miscellaneous......................................................................................818 Table A.8.1: Numerical Data, and Area/Volume of Objects..................................................................818 Figure A.8.1: Temperature at geometric center of a sphere (expanded scale).............................819 Figure A.8.2: Temperature at the axis of an inﬁnitely long cylinder (expanded scale)........................ 820 Figure A.8.3: Temperature at the midplane of an inﬁnite slab (expanded scale)............................821 A.9 Dimensional Analysis........................................................................ 822 Table A.9.1: Dimensions of selected experimental variables.............................................................. 823 Bibliography.............................................................................................. 826 Index.......................................................................................................... 829 Food Science and Technology: International Series............................. 839 Chapter Introduction 1 Physics, chemistry, and mathematics are essential in gaining an understanding of the principles that govern most of the unit opera- tions commonly found in the food industry. For example, if a food engineer is asked to design a food process that involves heating and cooling, then he or she must be well aware of the physical principles that govern heat transfer. The engineer’s work is often expected to be quantitative, and therefore the ability to use mathematics is essential. Foods undergo changes as a result of processing; such changes may be physical, chemical, enzymatic, or microbiological. It is often neces- sary to know the kinetics of chemical changes that occur during pro- cessing. Such quantitative knowledge is a prerequisite to the design and analysis of food processes. It is expected that prior to studying food engineering principles, the student will have taken basic courses in mathematics, chemistry, and physics. In this chapter, we review some selected physical and chemical concepts that are important in food engineering. All icons in this chapter refer to the author’s web site, which 1.1 DIMENSIONS is independently owned and A physical entity, which can be observed and/or measured, is deﬁned operated. Academic Press is not responsible for the content or qualitatively by a dimension. For example, time, length, area, volume, operation of the author’s web mass, force, temperature, and energy are all considered dimensions. site. Please direct your web The quantitative magnitude of a dimension is expressed by a unit; a site comments and questions unit of length may be measured as a meter, centimeter, or millimeter. to the author: Professor R. Paul Singh, Department of Biological Primary dimensions, such as length, time, temperature, and mass, and Agricultural Engineering, express a physical entity. Secondary dimensions involve a combina- University of California, Davis, tion of primary dimensions (e.g., volume is length cubed; velocity is CA 95616, USA. distance divided by time). Email: [email protected]. 1 2 CHAPTER 1 Introduction Equations must be dimensionally consistent. Thus, if the dimension of the left-hand side of an equation is “length,” the dimension of the right-hand side must also be “length”; otherwise, the equation is incorrect. This is a good method to check the accuracy of equations. In solving numerical problems, it is also useful to write the units of each dimensional quantity within the equations. This practice is help- ful to avoid mistakes in calculations. 1.2 ENGINEERING UNITS Physical quantities are measured using a wide variety of unit systems. The most common systems include the Imperial (English) system; the centimeter, gram, second (cgs) system; and the meter, kilogram, second (mks) system. However, use of these systems, entailing myr- iad symbols to designate units, has often caused considerable confu- sion. International organizations have attempted to standardize unit systems, symbols, and their quantities. As a result of international agreements, the Système International d’Unités, or the SI units, have emerged. The SI units consist of seven base units, two supplementary units, and a series of derived units, as described next. 1.2.1 Base Units The SI system is based on a choice of seven well-deﬁned units, which by convention are regarded as dimensionally independent. The deﬁ- nitions of these seven base units are as follows: 1. Unit of length (meter): The meter (m) is the length equal to 1,650,763.73 wavelengths in vacuum of the radiation corre- sponding to the transition between the levels 2p10 and 5d5 of the krypton-86 atom. 2. Unit of mass (kilogram): The kilogram (kg) is equal to the mass of the international prototype of the kilogram. (The inter- national prototype of the kilogram is a particular cylinder of platinum-iridium alloy, which is preserved in a vault at Sèvres, France, by the International Bureau of Weights and Measures.) 3. Unit of time (second): The second (s) is the duration of 9,192,631,770 periods of radiation corresponding to the transi- tion between the two hyperﬁne levels of the ground state of the cesium-133 atom. 4. Unit of electric current (ampere): The ampere (A) is the constant current that, if maintained in two straight parallel conductors 1.2 Engineering Units 3 Table 1.1 SI Base Units Measurable attribute of phenomena or matter Name Symbol Length meter m Mass kilogram kg Time second s Electric current ampere A Thermodynamic temperature kelvin K Amount of substance mole mol Luminous intensity candela cd of inﬁnite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between those conductors a force equal to 2 107 newton per meter length. 5. Unit of thermodynamic temperature (Kelvin): The Kelvin (K) is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. 6. Unit of amount of substance (mole): The mole (mol) is the amount of substance of a system that contains as many ele- mentary entities as there are atoms in 0.012 kg of carbon 12. 7. Unit of luminous intensity (candela): The candela (cd) is the luminous intensity, in the perpendicular direction, of a surface of 1/600,000 m2 of a blackbody at the temperature of freezing platinum under a pressure of 101,325 newton/m2. These base units, along with their symbols, are summarized in Table 1.1. 1.2.2 Derived Units Derived units are algebraic combinations of base units expressed by means of multiplication and division. For simplicity, derived units often carry special names and symbols that may be used to obtain other derived units. Deﬁnitions of some commonly used derived units are as follows: 1. Newton (N): The newton is the force that gives to a mass of 1 kg an acceleration of 1 m/s2. 4 CHAPTER 1 Introduction 2. Joule (J): The joule is the work done when due to force of 1 N the point of application is displaced by a distance of 1 m in the direction of the force. 3. Watt (W): The watt is the power that gives rise to the produc- tion of energy at the rate of 1 J/s. 4. Volt (V): The volt is the difference of electric potential between two points of a conducting wire carrying a constant current of 1 A, when the power dissipated between these points is equal to 1 W. 5. Ohm (Ω): The ohm is the electric resistance between two points of a conductor when a constant difference of potential of 1 V, applied between theses two points, produces in this conductor a current of 1 A, when this conductor is not being the source of any electromotive force. 6. Coulomb (C): The coulomb is the quantity of electricity trans- ported in 1 s by a current of 1 A. 7. Farad (F): The farad is the capacitance of a capacitor, between the plates of which there appears a difference of potential of 1 V when it is charged by a quantity of electricity equal to 1 C. 8. Henry (H): The henry is the inductance of a closed circuit in which an electromotive force of 1 V is produced when the electric current in the circuit varies uniformly at a rate of 1 A/s. 9. Weber (Wb): The weber is the magnetic ﬂux that, linking a cir- cuit of one turn, produces in it an electromotive force of 1 V as it is reduced to zero at a uniform rate in 1 s. 10. Lumen (lm): The lumen is the luminous ﬂux emitted in a point solid angle of 1 steradian by a uniform point source having an intensity of 1 cd. Examples of SI-derived units expressed in terms of base units, SI-derived units with special names, and SI-derived units expressed by means of special names are given in Tables 1.2, 1.3, and 1.4, respectively. 1.2.3 Supplementary Units This class of units contains two purely geometric units, which may be regarded either as base units or as derived units. 1. Unit of plane angle (radian): The radian (rad) is the plane angle between two radii of a circle that cut off on the circum- ference an arc equal in length to the radius. 1.2 Engineering Units 5 Table 1.2 Examples of SI-Derived Units Expressed in Terms of Base Units SI Unit Quantity Name Symbol Area square meter m2 Volume cubic meter m3 Speed, velocity meter per second m/s Acceleration meter per second squared m/s2 Density, mass density kilogram per cubic meter kg/m3 Current density ampere per square meter A/m2 Magnetic field strength ampere per meter A/m Concentration (of amount of substance) mole per cubic meter mol/m3 Specific volume cubic meter per kilogram m3/kg Luminance candela per square meter cd/m2 Table 1.3 Examples of SI-Derived Units with Special Names SI Unit Expression in terms Expression in terms Quantity Name Symbol of other units of SI base units Frequency hertz Hz s1 Force newton N m kg s2 Pressure, stress pascal Pa N/m2 m1 kg s2 Energy, work, quantity of heat joule J Nm m2 kg s2 Power, radiant flux watt W J/s m2 kg s3 Quantity of electricity, electric charge coulomb C sA Electric potential, potential difference, volt V W/A m2 kg s3 A1 electromotive force Capacitance farad F C/V m2 kg1 s4 A2 Electric resistance ohm Ω V/A m2 kg s3 A2 Conductance siemens S A/V m2 kg1 s3 A2 Celsius temperature degree Celsius C K Luminous flux lumen lm cd sr Illuminance lux lx lm/m2 m2 cd sr 6 CHAPTER 1 Introduction Table 1.4 Examples of SI-Derived Units Expressed by Means of Special Names SI Unit Expression in terms Quantity Name Symbol of SI base units Dynamic viscosity pascal second Pa s m1 kg s1 Moment of force newton meter Nm m2 kg s2 Surface tension newton per meter N/m kg s2 Power density, heat flux density, watt per square meter W/m2 kg s3 irradiance Heat capacity, entropy joule per kelvin J/K m2 kg s2 K1 Specific heat capacity joule per kilogram kelvin J/(kg K) m2 s2 K1 Specific energy joule per kilogram J/kg m2 s2 Thermal conductivity watt per meter kelvin W/(m K) m kg s3 K1 Energy density joule per cubic meter J/m3 m1 kg s2 Electric field strength volt per meter V/m m kg s3 A1 Electric charge density coulomb per cubic meter C/m3 m3 s A Electric flux density coulomb per square meter C/m2 m2 s A Table 1.5 SI Supplementary Units SI Unit Quantity Name Symbol Plane angle radian rad Solid angle steradian sr 2. Unit of solid angle (steradian): The steradian (sr) is the solid angle that, having its vertex in the center of a sphere, cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere. The supplementary units are summarized in Table 1.5. 1.2 Engineering Units 7 Determine the following unit conversions to SI units: Example 1.1 a. a density value of 60 lbm/ft3 to kg/m3 b. an energy value of 1.7 103 Btu to kJ c. an enthalpy value of 2475 Btu/lbm to kJ/kg d. a pressure value of 14.69 psig to kPa e. a viscosity value of 20 cp to Pa s Solution We will use conversion factors for each unit separately from Table A.1.2. a. Although a composite conversion factor for density, 1 lbm/ft3 16.0185 kg/m3, is available in Table A.1.2, we will first convert units of each dimension sepa- rately. Since 1 lbm 0.45359 kg 1 ft 0.3048 m Thus, ⎛ 1 ⎞3 ( 60 lbm /ft 3 )( 0.45359 kg/lbm ) ⎜⎜ m/ft ⎟⎟⎟ ⎜⎝ 0.3048 ⎠ 961.1 kg/m3 An alternative solution involves the direct use of the conversion factor for density, ( 60 lbm /ft 3 )(16.0185 kg/m3 ) 961.1 kg/m3 (1 lbm /ft 3 ) b. For energy 1 Btu 1.055 kJ Thus, (1.7 10 3 Btu)(1.055 kJ) 1.8 10 3 kJ (1 Btu) c. For enthalpy, the conversion units for each dimension are 1 Btu 1.055 kJ 1 lbm 0.45359 kg Thus, ⎛ 1 ⎞⎟ ( 2475 Btu/lbm )(1.055 kJ/Btu) ⎜⎜⎜ ⎟ ⎜⎝ 0.45359 kg/lbm ⎟⎟⎠ 5757 kJ/kg 8 CHAPTER 1 Introduction Alternately, using the composite conversion factor for enthalpy of 1 Btu/lbm 2.3258 kJ/kg ( 2475 Btu/lbm )( 2.3258 kJ/kg) 5756 kJ/kg (1 Btu/lbm ) d. For pressure psia psig 14.69 The gauge pressure, 14.69 psig, is first converted to the absolute pressure, psia (see Section 1.9 for more discussion on gauge and absolute pressures). 14.69 psig 14.69 29.38 psia The unit conversions for each dimension are 1 lb 4.4483 N 1 in 2.54 102 m 1 Pa 1 N/m2 Thus, ⎛ 1 ⎞⎟2 ⎛ 1Pa ⎞⎟ (29.28 lb/in2 )(4.4482 N/lb) ⎜⎜ ⎜⎜⎝ 2.54 102 m/in ⎟⎟⎠ ⎟ ⎜⎜ ⎟ ⎜⎜⎝1 N/m2 ⎟⎟⎠ 201877 Pa 201.88 kPa Alternatively, since 1 psia 6.895 kPa (29.28 psia)(6.895 kPa) 201.88 kPa (1Psia) e. For viscosity 1 cp 103 Pa s Thus, (20 cp)(103 Pa s ) 2 102 Pa s (1 cp) 1.2 Engineering Units 9 Starting with Newton’s second law of motion, determine units of force and Example 1.2 weight in SI and English units. Solution a. Force Newton’s second law of motion states that force is directly proportional to mass and acceleration. Thus, F ma Using a constant of proportionality k, F kma where in SI units, N k 1 kg m/s 2 Thus, ⎛ N ⎞⎟ F 1⎜⎜⎜ ⎟ (kg)(m/s 2 ) ⎜⎝ kg m/s 2 ⎟⎟⎠ F 1N In English units the constant k is defined as 1 lbf k 32.17 lbm ft/s 2 More commonly, another constant gc is used where ⎛ lb ⎞ ⎛ ft ⎞⎟ gc 1/k 32.17 ⎜⎜⎜ m ⎟⎟⎟ ⎜⎜ ⎟ ⎜⎝ lbf ⎟⎠ ⎜⎝ s 2 ⎟⎠ Thus ma F gc or 1 ⎛⎜ lbf ⎞⎟ F ⎜ ⎟⎟ (lbm )( ft/s 2) 32.17 ⎜⎜⎝ lbm ft/s 2 ⎟⎠ 1 F lbf 32.17 10 CHAPTER 1 Introduction b. Weight Weight W is the force exerted by the earth’s gravitational force on an object. Weight of 1 kg mass can be calculated as In SI units, W kmg ⎛ N ⎞⎟ ⎛ m⎞ ⎜⎜⎜1 ⎟⎟ (1 kg) ⎜⎜9.81 ⎟⎟⎟ ⎜⎝ kg m/s ⎟⎠ 2 ⎜ ⎝ s2 ⎠ 9.81 N In English units, W kmg 1 ⎛ lb ⎞⎟ ⎜⎜ f ⎟ 2 32.17 ⎜⎜⎝ lb ft/s 2 ⎟⎟⎠ (1 lbm )( 32.17 ft/s ) m 1lbf 1.3 SYSTEM A system is any region prescribed in space or a ﬁnite quantity of mat- ter enclosed by a boundary, real or imaginary. The boundary of a sys- tem can be real, such as the walls of a tank, or it can be an imaginary surface that encloses the system. Furthermore, the boundary may be stationary or moveable. For example, in Figure 1.1, the system bound- ary encloses a tank, piping, and a valve. If our analysis had concerned

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