2003 by CRC Press LLCUnitOperations inFoodEngineering 2003 by CRC Press LLCFOOD PRESERVATION TECHNOLOGY SERIESSeries EditorGustavo V. Barbosa-CnovasInnovations in Food ProcessingEditors: Gustavo V. Barbosa-Cnovas and Grahame W. GouldTrends in Food EngineeringEditors: Jorge E. Lozano, Cristina An, Efrn Parada-Arias,and Gustavo V. Barbosa-CnovasPulsed Electric Fields in Food Processing:Fundamental Aspects and ApplicationsEditors: Gustavo V. Barbosa-Cnovas and Q. Howard ZhangOsmotic Dehydration and Vacuum Impregnation:Applications in Food IndustriesEditors: Pedro Fito, Amparo Chiralt, Jose M. Barat, Walter E. L. Spiess,and Diana BehsnilianEngineering and Food for the 21st CenturyEditors: Jorge Welti-Chanes, Gustavo V. Barbosa-Cnovas,and Jos Miguel AguileraUnit Operations in Food EngineeringAlbert Ibarz and Gustavo V. Barbosa-Cnovas 2003 by CRC Press LLCCRC PRESSBoca Raton London New York Washington, D.C.UnitOperations inFoodEngineeringAlbert Ibarz, Ph.D.University of LleidaLleida, SpainGustavo V. Barbosa-Cnovas, Ph.D.Washington State UniversityPullman, Washington 2003 by CRC Press LLC This book contains information obtained from authentic and highly regarded sources. Reprinted materialis quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonableefforts have been made to publish reliable data and information, but the author and the publisher cannotassume responsibility for the validity of all materials or for the consequences of their use.Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronicor mechanical, including photocopying, microlming, and recording, or by any information storage orretrieval system, without prior permission in writing from the publisher.The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, forcreating new works, or for resale. Specic permission must be obtained in writing from CRC Press LLCfor such copying.Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and areused only for identication and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com 2003 by CRC Press LLC No claim to original U.S. Government worksInternational Standard Book Number 1-56676-929-9Library of Congress Card Number 2002017480Printed in the United States of America 1 2 3 4 5 6 7 8 9 0Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Ibarz, Albert.[Operaciones unitarias en la engenieria de alimentos. English]Unit operations in food engineering / by Albert Ibarz, Gustavo V. Barbosa-Cnovas.p. cm. -- (Food preservation technology series)Includes bibliographical references and index.ISBN 1-56676-929-91. Food industry and trade. I. Barbosa-Cnovas, Gustavo V. II. Title. III. Series.TP370 .I2313 2002664dc21 2002017480 CIP TX69299 fm frame Page 4 Friday, September 20, 2002 8:01 AM 2003 by CRC Press LLC To our families TX69299 fm frame Page 5 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC Preface One of the primary objectives of the food industry is to transform, by a seriesof operations, raw agricultural materials into foods suitable for consumption.Many different types of equipment and several stages are used to perform thesetransformations. The efcient calculation and design of each stage calledunit or basic operation is one of the main purposes of food engineering.The systematic study of unit operations began in the chemical engineeringeld, where calculation tools were developed to describe, based on engineer-ing principles, the changes taking place in each processing step. This knowl-edge has been applied to food engineering and, at the same time, has beenadapted to the particular and distinctive nature of the raw materials used.The goal of any series of operations is not just to obtain optimum production,but also a food product suitable for consumption and of the highest quality.Thus, in the application of unit operations to a food process, exhaustive andcareful calculation is essential to obtaining process stages that cause mini-mum damage to the food that is being processed.The main objective of this book is to present, in progressive and systematicform, the basic information required to design food processes, including thenecessary equipment. The number of food engineering unit operations isquite extensive, but some are rarely applied because they are quite specicto a given commodity or process. This book covers those unit operationsthat, in the opinion of the authors, are most relevant to the food industry ingeneral. The rst chapters contain basic information on transport phenomenagoverning key unit operations, followed by chapters offering a detaileddescription of those selected unit operations. To facilitate the understandingof all the studied unit operations, each chapter concludes with a set of solvedproblems.We hope this book will be useful as a reference for food engineers and asa text for advanced undergraduate and graduate students in food engineer-ing. We also hope this book will be a meaningful addition to the literaturedealing with food processing operations. Albert IbarzGustavo V. Barbosa-Cnovas TX69299 fm frame Page 7 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC Acknowledgments The authors wish to express their gratitude to the following institutions andindividuals who contributed to making this book possible:Interministerial Commission of Science and Technology (CICYT)of Spain for supporting the preparation of this book through projectTXT96-2223.The University of Lleida and the Washington State University(WSU) for supplying the facilities and conducive framework forthe preparation of this book.Dr. Jorge Vlez-Ruiz, Universidad de las Amricas-Puebla, Mxicofor his very important contributions in the preparation of Chapter 7.Mara Luisa Caldern (WSU) for her professionalism and dedica-tion in revising the Spanish version of the book from beginning toend. Her commentaries and suggestions were very valuable.Jos Juan Rodrguez and Federico Harte (WSU) for their decisiveparticipation in the nal review of the Spanish version. Both workedwith great care, dedication, enthusiasm, and professionalism.The translation team: Lucy Lpez (Universidad de las Amricas-Puebla, Mxico), Jeannie Anderson (WSU), Fernanda San Martn(WSU), and Gipsy Tabilo (WSU) for their incredible dedication totransforming this book into the English version.All the students who attended our unit operations in food engi-neering courses; they provided a constant stimulus for conceivingand developing the nished work.Albert Ibarz, Jr. for his careful collaboration in preparing many ofthe gures in the book and Raquel Ibarz for her invaluable helpand encouragement for making this book a pleasant reality. TX69299 fm frame Page 9 Friday, September 20, 2002 7:41 AM 2003 by CRC Press LLC Authors Albert Ibarz earned his B.S. and Ph.D. in chemical engineering from theUniversity of Barcelona, Spain. He is a Professor of Food Engineering at theUniversity of Lleida, Spain and the Vice-Chancellor for Faculty Affairs. Hiscurrent research areas are: transport phenomena in food processing, reactionkinetics in food systems, physical properties of foods, and ultra high pressurefor food processing. Gustavo V. Barbosa-Cnovas earned his B.S. in mechanical engineering fromthe University of Uruguay and his M.S. and Ph.D. in food engineering fromthe University of Massachusetts at Amherst. He is a Professor of Food Engi-neering at Washington State University and Director of the Center for Non-thermal Processing of Food. His current research areas are: nonthermalprocessing of foods, physical properties of foods, edible lms, food powdertechnology, and food dehydration. TX69299 fm frame Page 11 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC CONTENTS 1 Introduction to Unit Operations: Fundamental Concepts ......... 1 1.1 Process ............................................................................................................. 11.2 Food Process Engineering............................................................................ 11.3 Transformation and Commercialization of Agricultural Products ....... 21.4 Flow Charts and Description of Some Food Processes........................... 21.5 Steady and Unsteady States......................................................................... 31.6 Discontinuous, Continuous, and Semicontinuous Operations.............. 31.7 Unit Operations: Classication.................................................................... 61.7.1 Momentum Transfer Unit Operations ........................................... 71.7.2 Mass Transfer Unit Operations....................................................... 81.7.3 Heat Transfer Unit Operations ....................................................... 81.7.4 Simultaneous MassHeat Transfer Unit Operations................... 81.7.5 Complementary Unit Operations................................................... 91.8 Mathematical Setup of the Problems ......................................................... 9 2 Unit Systems: Dimensional Analysis and Similarity............... 11 2.1 Magnitude and Unit Systems.................................................................... 112.1.1 Absolute Unit Systems ................................................................... 112.1.2 Technical Unit Systems................................................................... 122.1.3 Engineering Unit Systems ............................................................. 122.1.4 International Unit System (IS) ...................................................... 132.1.5 Thermal Units .................................................................................. 142.1.6 Unit Conversion .............................................................................. 152.2 Dimensional Analysis ................................................................................. 172.2.1 Buckinghams Theorem.............................................................. 182.2.2 Dimensional Analysis Methods.................................................... 202.2.2.1 Buckinghams Method..................................................... 202.2.2.2 Rayleighs Method............................................................ 222.2.2.3 Method of Differential Equations .................................. 222.3 Similarity Theory......................................................................................... 232.3.1 Geometric Similarity....................................................................... 242.3.2 Mechanical Similarity..................................................................... 252.3.2.1 Static Similarity ................................................................. 252.3.2.2 Kinematic Similarity......................................................... 252.3.2.3 Dynamic Similarity........................................................... 25Problems.................................................................................................. 30 TX69299 fm frame Page 13 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 3 Introduction to Transport Phenomena ....................................... 43 3.1 Historic Introduction................................................................................... 433.2 Transport Phenomena: Denition............................................................. 443.3 Circulation Regimes: Reynolds Experiment .......................................... 453.4 Mechanisms of Transport Phenomena..................................................... 483.4.1 Mass Transfer................................................................................... 493.4.2 Energy Transfer ............................................................................... 503.4.3 Momentum Transport..................................................................... 503.4.4 Velocity Laws................................................................................... 503.4.5 Coupled Phenomena ...................................................................... 51 4 Molecular Transport of Momentum, Energy, and Mass........... 53 4.1 Introduction.................................................................................................. 534.2 Momentum Transport: Newtons Law of Viscosity............................... 534.3 Energy Transmission: Fouriers Law of Heat Conduction................... 554.4 Mass Transfer: Ficks Law of Diffusion ................................................... 574.5 General Equation of Velocity..................................................................... 61 5 AirWater Mixtures....................................................................... 65 5.1 Introduction.................................................................................................. 655.2 Properties of Humid Air ............................................................................ 655.3 Molliers Psychrometric Diagram for Humid Air ................................. 705.3.1 Psychrometric Chart sT X........................................................... 705.3.2 Psychrometric Chart X T ............................................................ 745.4 Wet Bulb Temperature ................................................................................ 755.5 Adiabatic Saturation of Air........................................................................ 77Problems.................................................................................................. 80 6 Rheology of Food Products ......................................................... 89 6.1 Introduction.................................................................................................. 896.2 Stress and Deformation .............................................................................. 906.3 Elastic Solids and Newtonian Fluids ....................................................... 936.4 Viscometric Functions................................................................................. 956.5 Rheological Classication of Fluid Foods ............................................... 966.6 Newtonian Flow.......................................................................................... 976.7 Non-Newtonian Flow................................................................................. 996.7.1 Time Independent Flow................................................................. 996.7.2 Time Dependent Flow.................................................................. 1036.8 Viscoelasticity ............................................................................................. 1076.9 Effect of Temperature................................................................................ 1136.10 Effect of Concentration on Viscosity ...................................................... 1146.10.1 Structural Theories of Viscosity............................................... 1146.10.2 Viscosity of Solutions ................................................................ 1156.10.3 Combined Effect: TemperatureConcentration..................... 1176.11 Mechanical Models.................................................................................. 118 TX69299 fm frame Page 14 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 6.11.1 Hookes Model ........................................................................... 1186.11.2 Newtons Model......................................................................... 1186.11.3 Kelvins Model ........................................................................... 1186.11.4 Maxwells Model........................................................................ 1206.11.5 SaintVenants Model................................................................ 1216.11.6 Mechanical Model of the Binghams Body ........................... 1216.12 Rheological Measures in Semiliquid Foods ........................................ 1216.12.1 Fundamental Methods.............................................................. 1236.12.1.1 Rotational Viscometers............................................. 1236.12.1.2 Concentric Cylinders Viscometers ......................... 1236.12.1.3 PlatePlate and ConePlate Viscometers.............. 1266.12.1.4 Error Sources ............................................................. 1286.12.1.5 Oscillating Flow........................................................ 1306.12.1.3 Capillary Flow........................................................... 1326.12.1.7 Back Extrusion Viscometry ..................................... 1326.12.1.8 Squeezing Flow Viscometry.................................... 1356.12.2 Empirical Methods .................................................................... 1366.12.2.1 Adams Consistometer.............................................. 1366.12.2.2 Bostwick Consistometer .......................................... 1376.12.2.3 Tube Flow Viscometer.............................................. 1376.12.3 Imitative Methods...................................................................... 137Problems................................................................................................ 138 7 Transport of Fluids through Pipes............................................ 143 7.1 Introduction................................................................................................ 1437.2 Circulation of Incompressible Fluids ..................................................... 1447.2.1 Criteria for Laminar Flow........................................................ 1447.2.2 Velocity Proles.......................................................................... 1477.2.2.1 Laminar Regime........................................................ 1497.2.2.2 Turbulent Regime ..................................................... 1537.2.2.3 Flow in Noncylindrical Piping............................... 1557.2.3 Universal Velocity Prole......................................................... 1577.3 Macroscopic Balances in Fluid Circulation........................................... 1607.3.1 Mass Balance .............................................................................. 1607.3.2 Momentum Balance................................................................... 1617.3.3 Total Energy Balance................................................................. 1627.3.4 Mechanical Energy Balance...................................................... 1657.4 Mechanical Energy Losses ....................................................................... 1667.4.1 Friction Factors........................................................................... 1667.4.2 Calculation of Friction Factors ................................................ 1677.4.2.1 Flow under Laminar Regime.................................. 1687.4.2.2 Flow under Turbulent Regime ............................... 1707.4.3 Minor Mechanical Energy Losses ........................................... 1737.4.3.1 Equivalent Length .................................................... 1757.4.3.2 Friction Losses Factors............................................. 175 TX69299 fm frame Page 15 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 7.5 Design of Piping Systems......................................................................... 1797.5.1 Calculation of Velocity and Circulation Flow Rate................. 1797.5.2 Calculation of Minimum Diameter of Piping .......................... 1817.5.3 Piping Systems .............................................................................. 1827.5.3.1 Parallel Piping Systems ................................................. 1827.5.3.2 Piping in Series ............................................................... 1837.5.3.3 Branched Piping.............................................................. 1847.6 Pumps.......................................................................................................... 1867.6.1 Characteristics of a Pump............................................................ 1867.6.1.1 Suction Head ................................................................... 1877.6.1.2 Impelling Head ............................................................... 1887.6.1.3 Total Head of a Pump.................................................... 1887.6.1.4 Net Positive Suction Head: Cavitation ....................... 1897.6.2 Installation Point of a Pump ....................................................... 1907.6.3 Pump Power .................................................................................. 1917.6.4 Pump Efciency............................................................................. 1917.6.5 Types of Pumps ............................................................................. 191Problems................................................................................................ 193 8 Circulation of Fluid through Porous Beds: Fluidization....... 205 8.1 Introduction................................................................................................ 2058.2 Darcys Law: Permeability....................................................................... 2058.3 Previous Denitions.................................................................................. 2068.3.1 Specic Surface.............................................................................. 2068.3.2 Porosity ........................................................................................... 2078.4 Equations for Flow through Porous Beds ............................................. 2108.4.1 Laminar Flow: Equation of KozenyCarman........................... 2108.4.2 Turbulent Flow: Equation of BurkePlummer......................... 2128.4.3 Laminar-Turbulent Global Flow: Equations of Ergun and ChiltonColburn............................................................................ 2138.5 Fluidization................................................................................................. 2168.5.1 Minimal Velocity of Fluidization................................................ 2188.5.1.1 Laminar Flow.................................................................. 2198.5.1.2 Turbulent Flow................................................................ 2198.5.1.3 Transition Flow................................................................ 2208.5.2 Minimal Porosity of Fluizidation ............................................... 2208.5.3 Bed Height...................................................................................... 221Problems................................................................................................ 222 9 Filtration ...................................................................................... 235 9.1 Introduction................................................................................................ 2359.2 Fundamentals of Filtration....................................................................... 2359.2.1 Resistance of the Filtering Cake ................................................. 2369.2.2 Filtering Medium Resistance....................................................... 239 TX69299 fm frame Page 16 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 9.2.3 Total Filtration Resistance ........................................................ 2409.2.4 Compressible Cakes .................................................................. 2419.3 Filtration at Constant Pressure Drop.................................................... 2419.4 Filtration at Constant Volumetric Flow................................................ 2449.5 Cake Washing........................................................................................... 2459.6 Filtration Capacity ................................................................................... 2489.7 Optimal Filtration Conditions at Constant Pressure ......................... 2489.8 Rotary Vacuum Disk Filter..................................................................... 250Problems................................................................................................ 253 10 Separation Processes by Membranes ....................................... 265 10.1 Introduction .............................................................................................. 26510.1.1 Stages of Mass Transfer ............................................................ 26710.1.2 Polarization by Concentration................................................. 26910.2 Mass Transfer in Membranes................................................................. 27010.2.1 Solution Diffusion Model ......................................................... 27010.2.2 Simultaneous Diffusion and Capillary Flow Model............ 27010.2.3 Simultaneous Viscous and Friction Flow Model.................. 27110.2.4 Preferential Adsorption and Capillary Flow Model ............ 27210.2.5 Model Based on the Thermodynamics of Irreversible Processes...................................................................................... 27310.3 Models for Transfer through the Polarization Layer......................... 27410.3.1 Hydraulic Model........................................................................ 27410.3.2 Osmotic Model ........................................................................... 27910.4 Reverse Osmosis ...................................................................................... 28010.4.1 Mathematical Model.................................................................. 28010.4.2 Polarization Layer by Concentration ..................................... 28310.4.3 Inuence of Different Factors .................................................. 28410.4.3.1 Inuence of Pressure................................................ 28410.4.3.2 Effect of Temperature............................................... 28510.4.3.3 Effect of Type of Solute............................................ 28710.5 Ultraltration............................................................................................ 28710.5.1 Mathematical Model.................................................................. 28810.5.2 Concentration Polarization Layer ........................................... 28910.5.3 Inuence of Different Factors .................................................. 29110.5.3.1 Inuence of Pressure................................................ 29110.5.3.2 Effect of Temperature............................................... 29210.5.3.3 Effect of Type of Solute............................................ 29310.6 Design of Reverse Osmosis and Ultraltration Systems .................. 29310.6.1 First Design Method.................................................................. 29410.6.2 Second Design Method............................................................. 29710.7 Operative Layout of the Modules......................................................... 29810.7.1 Single Stage................................................................................. 29810.7.2 Simple Stages in Series ............................................................. 29910.7.3 Two Stages with Recirculation ................................................ 300Problems................................................................................................ 301 TX69299 fm frame Page 17 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 11 Thermal Properties of Food....................................................... 309 11.1 Thermal Conductivity............................................................................. 30911.2 Specic Heat ............................................................................................. 31111.3 Density....................................................................................................... 31311.4 Thermal Diffusivity ................................................................................. 316Problems................................................................................................ 319 12 Heat Transfer by Conduction .................................................... 321 12.1 Fundamental Equations in Heat Conduction ..................................... 32112.1.1 Rectangular Coordinates .......................................................... 32112.1.2 Cylindrical Coordinates............................................................ 32412.1.3 Spherical Coordinates ............................................................... 32512.2 Heat Conduction under Steady Regime.............................................. 32512.2.1 Monodimensional Heat Conduction...................................... 32612.2.1.1 Flat Wall...................................................................... 32712.2.1.2 Cylindrical Layer ...................................................... 32912.2.1.3 Spherical Layer.......................................................... 33212.2.2 Bidimensional Heat Conduction............................................. 33412.2.2.1 Liebmans method.................................................... 33612.2.2.2 Relaxation method.................................................... 33712.2.3 Tridimensional Heat Conduction............................................ 33712.3 Heat Conduction under Unsteady State.............................................. 33912.3.1 Monodimensional Heat Conduction...................................... 33912.3.1.1 Analytical Methods .................................................. 34012.3.1.2 Numerical and Graphical Methods....................... 34712.3.2 Bi- and Tridimensinal Heat Conduction: Newmans Rule .............................................................................................. 351Problems................................................................................................ 352 13 Heat Transfer by Convection..................................................... 367 13.1 Introduction .............................................................................................. 36713.2 Heat Transfer Coefcients ...................................................................... 36713.2.1 Individual Coefcients.............................................................. 36713.2.1.1 Natural Convection.................................................. 37013.2.1.2 Forced Convection.................................................... 37113.2.1.3 Convection in Non-Newtonian Fluids.................. 37313.2.2 Global Coefcients..................................................................... 37413.3 Concentric Tube Heat Exchangers ........................................................ 37813.3.1 Design Characteristics............................................................... 37813.3.1.1 Operation in Parallel ................................................ 37813.3.1.2 Countercurrent Operation....................................... 38213.3.2 Calculation of Individual Coefcients ................................... 38313.3.3 Calculation of Head Losses...................................................... 384 TX69299 fm frame Page 18 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 13.4 Shell and Tube Heat Exchangers........................................................... 38413.4.1 Design Characteristics............................................................... 38513.4.2 Calculation of the True Logarithmic Mean Temperature Difference .................................................................................... 38813.4.3 Calculation of Individual Coefcients ................................... 38913.4.3.1 Coefcients for the Inside of the Tubes ................ 39013.4.3.2 Coefcients on the Side of the Shell...................... 39213.4.4 Calculation of Head Losses...................................................... 39513.4.4.1 Head Losses inside Tubes ....................................... 39513.4.4.2 Head Losses on the Shell Side................................ 39513.5 Plate-Type Heat Exchangers .................................................................. 39613.5.1 Design Characteristics............................................................... 39913.5.2 Number of Transfer Units ........................................................ 40113.5.3 Calculation of the True Logarithmic Mean Temperature Difference .................................................................................... 40213.5.4 Calculation of the Heat Transfer Coefcients ....................... 40313.5.5 Calculation of Head Losses...................................................... 40613.5.6 Design Procedure....................................................................... 40713.6 Extended Surface Heat Exchangers...................................................... 40913.6.1 Mathematical Model.................................................................. 41113.6.2 Efciency of a Fin ...................................................................... 41213.6.3 Calculation of Extended Surface Heat Exchangers.............. 41413.7 Scraped Surface Heat Exchangers......................................................... 41513.8 Agitated Vessels with Jacket and Coils................................................ 41713.8.1 Individual Coefcient inside the Vessel................................. 41713.8.2 Individual Coefcient inside the Coil .................................... 41813.8.3 Individual Coefcient in the Jacket ........................................ 41813.9 Heat Exchange Efciency ....................................................................... 418Problems................................................................................................ 425 14 Heat Transfer by Radiation ....................................................... 467 14.1 Introduction .............................................................................................. 46714.2 Fundamental Laws .................................................................................. 46814.2.1 Plancks Law............................................................................... 46814.2.2 Wiens Law.................................................................................. 46814.2.3 StefanBoltzmann Law............................................................. 46914.3 Properties of Radiation ........................................................................... 46914.3.1 Total Properties .......................................................................... 46914.3.2 Monochromatic Properties: Kirchhoffs Law........................ 47114.3.3 Directional Properties................................................................ 47214.4 View Factors.............................................................................................. 47414.4.1 Denition and Calculation....................................................... 47414.4.2 Properties of View Factors ....................................................... 47514.5 Exchange of Radiant Energy between Surfaces Separated by Nonabsorbing Media............................................................................... 478 TX69299 fm frame Page 19 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 14.5.1 Radiation between Black Surfaces .......................................... 47914.5.2 Radiation between a Surface and a Black Surface Completely Surrounding It ...................................................... 47914.5.3 Radiation between Black Surfaces in the Presence of Refractory Surfaces: Refractory Factor................................... 48014.5.4 Radiation between Nonblack Surfaces: Gray Factor ........... 48114.6 Coefcient of Heat Transfer by Radiation........................................... 48214.7 Simultaneous Heat Transfer by Convection and Radiation............. 484Problems................................................................................................ 485 15 Thermal Processing of Foods .................................................... 491 15.1 Introduction .............................................................................................. 49115.2 Thermal Death Rate................................................................................. 49115.2.1 Decimal Reduction Time D ...................................................... 49215.2.2 Thermal Death Curves.............................................................. 49315.2.3 Thermal Death Time Constant z ............................................. 49315.2.4 Reduction Degree n ................................................................... 49715.2.5 Thermal Death Time F .............................................................. 49815.2.6 Cooking Value C ........................................................................ 50115.2.7 Effect of Temperature on Rate and Thermal Treatment Parameters................................................................................... 50115.3 Treatment of Canned Products.............................................................. 50215.3.1 Heat Penetration Curve............................................................ 50215.3.2 Methods to Determine Lethality............................................. 50515.3.2.1 Graphical Method..................................................... 50515.3.2.2 Mathematical Method.............................................. 50615.4 Thermal Treatment in Aseptic Processing........................................... 50815.4.1 Residence Times......................................................................... 51015.4.2 Dispersion of Residence Times................................................ 51115.4.3 Distribution Function E under Ideal Behavior ..................... 51315.4.4 Distribution Function E under Nonideal Behavior ............. 51615.4.5 Application of the Distribution Models to Continuous Thermal Treatment .................................................................... 519Problems................................................................................................ 521 16 Food Preservation by Cooling................................................... 535 16.1 Freezing ..................................................................................................... 53516.2 Freezing Temperature.............................................................................. 53716.2.1 Unfrozen Water .......................................................................... 53816.2.2 Equivalent Molecular Weight of Solutes ............................... 54016.3 Thermal Properties of Frozen Foods .................................................... 54116.3.1 Density......................................................................................... 54116.3.2 Specic Heat ............................................................................... 54116.3.3 Thermal Conductivity............................................................... 542 TX69299 fm frame Page 20 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 16.4 Freezing Time ........................................................................................... 54316.5 Design of Freezing Systems ................................................................... 54916.6 Refrigeration ............................................................................................. 55016.7 Refrigeration Mechanical Systems ........................................................ 55116.8 Refrigerants............................................................................................... 55516.9 Multipressure Systems ............................................................................ 55616.9.1 Systems with Two Compressors and One Evaporator........ 55916.9.2 Systems with Two Compressors and Two Evaporators...... 561Problems................................................................................................ 563 17 Dehydration................................................................................. 573 17.1 Introduction .............................................................................................. 57317.2 Mixing of Two Air Streams .................................................................... 57417.3 Mass and Heat Balances in Ideal Dryers............................................ 57517.3.1 Continuous Dryer without Recirculation.............................. 57517.3.2 Continuous Dryer with Recirculation.................................... 57617.4 Dehydration Mechanisms....................................................................... 57717.4.1 Drying Process ........................................................................... 57717.4.2 Constant Rate Drying Period................................................... 58017.4.3 Falling Rate Drying Period ...................................................... 58217.4.3.1 Diffusion Theory....................................................... 58217.5 Chamber and Bed Dryers....................................................................... 58417.5.1 Components of a Dryer ............................................................ 58517.5.2 Mass and Heat Balances........................................................... 58717.5.2.1 Discontinuous Dryers .............................................. 58717.5.2.2 Discontinuous Dryers with Air Circulation through the Bed ........................................................ 58917.5.2.3 Continuous Dryers ................................................... 59217.6 Spray Drying ............................................................................................ 59417.6.1 Pressure Nozzles........................................................................ 59517.6.2 Rotary Atomizers....................................................................... 59817.6.3 Two-Fluid Pneumatic Atomizers............................................. 60017.6.4 Interaction between Droplets and Drying Air...................... 60217.6.5 Heat and Mass Balances........................................................... 60217.7 Freeze Drying ........................................................................................... 60417.7.1 Freezing Stage ............................................................................ 60717.7.2 Primary and Secondary Drying Stages.................................. 60717.7.3 Simultaneous Heat and Mass Transfer .................................. 60717.8 Other Types of Drying............................................................................ 61417.8.1 Osmotic Dehydration................................................................ 61417.8.2 Solar Drying................................................................................ 61517.8.3 Drum Dryers............................................................................... 61617.8.4 Microwave Drying..................................................................... 61617.8.5 Fluidized Bed Dryers ................................................................ 617Problems................................................................................................ 618 TX69299 fm frame Page 21 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 18 Evaporation.................................................................................. 625 18.1 Introduction .............................................................................................. 62518.2 Heat Transfer in Evaporators................................................................. 62618.2.1 Enthalpies of Vapors and Liquids........................................... 62718.2.2 Boiling Point Rise....................................................................... 62918.2.3 Heat Transfer Coefcients ........................................................ 63118.3 Single Effect Evaporators........................................................................ 63218.4 Use of Released Vapor ............................................................................ 63418.4.1 Recompression of Released Vapor.......................................... 63418.4.1.1 Mechanical Compression......................................... 63418.4.1.2 Thermocompression................................................. 63618.4.2 Thermal Pump............................................................................ 63718.4.3 Multiple Effect ............................................................................ 63818.5 Multiple-Effect Evaporators ................................................................... 64018.5.1 Circulation Systems of Streams............................................... 64018.5.1.1 Parallel Feed .............................................................. 64018.5.1.2 Forward Feed ............................................................ 64218.5.1.3 Backward Feed.......................................................... 64218.5.1.4 Mixed Feed ................................................................ 64218.5.2 Mathematical Model.................................................................. 64318.5.3 Resolution of the Mathematical Model.................................. 64518.5.4 Calculation Procedure............................................................... 64618.5.4.1 Iterative Method when there is Boiling Point Rise .................................................................. 64718.5.4.2 Iterative Method when there is No Boiling Point Rise ................................................................... 64818.6 Evaporation Equipment.......................................................................... 64918.6.1 Natural Circulation Evaporators............................................. 64918.6.1.1 Open Evaporator....................................................... 64918.6.1.2 Short Tube Horizontal Evaporator ........................ 64918.6.1.3 Short Tube Vertical Evaporator .............................. 65018.6.1.4 Evaporator with External Calandria ..................... 65118.6.2 Forced Circulation Evaporators............................................... 65118.6.3 Long Tube Evaporators............................................................. 65218.6.4 Plate Evaporators....................................................................... 654Problems................................................................................................ 654 19 Distillation................................................................................... 671 19.1 Introduction .............................................................................................. 67119.2 LiquidVapor Equilibrium..................................................................... 67119.2.1 Partial Pressures: Laws of Dalton, Raoult, and Henry ....... 67419.2.2 Relative Volatility....................................................................... 67619.2.3 Enthalpy Composition Diagram............................................. 67719.3 Distillation of Binary Mixtures .............................................................. 678 TX69299 fm frame Page 22 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 19.3.1 Simple Distillation ..................................................................... 67819.3.2 Flash Distillation........................................................................ 68019.4 Continuous Rectication of Binary Mixtures...................................... 68219.4.1 Calculation of the Number of Plates...................................... 68419.4.1.1 Mathematical Model ................................................ 68419.4.1.2 Solution of the Mathematical Model: Method of McCabeThiele ..................................................... 68719.4.2 Reux Ratio ................................................................................ 69119.4.2.1 Minimum Reux Relationship ............................... 69119.4.2.2 Number of Plates for Total Reux......................... 69419.4.3 Multiple Feed Lines and Lateral Extraction.......................... 69419.4.4 Plate Efciency ........................................................................... 69719.4.5 Diameter of the Column........................................................... 69819.4.6 Exhaust Columns....................................................................... 70119.5 Discontinuous Rectication.................................................................... 70219.5.1 Operation with Constant Distillate Composition ................ 70219.5.2 Operation under Constant Reux Ratio................................ 70519.6 Steam Distillation..................................................................................... 706Problems................................................................................................ 708 20 Absorption ................................................................................... 723 20.1 Introduction .............................................................................................. 72320.2 LiquidGas Equilibrium......................................................................... 72420.3 Absorption Mechanisms......................................................................... 72620.3.1 Double Film Theory .................................................................. 72720.3.2 Basic Mass Transfer Equations ................................................ 72720.3.2.1 Diffusion in the Gas Phase...................................... 72820.3.2.2 Diffusion in the Liquid Phase................................. 72920.3.3 Absorption Velocity................................................................... 72920.4 Packed Columns ...................................................................................... 73220.4.1 Selection of the Solvent............................................................. 73220.4.2 Equilibrium Data ....................................................................... 73320.4.3 Mass Balance .............................................................................. 73320.4.4 Enthalpy Balance ....................................................................... 73620.4.5 Selection of Packing Type: Calculation of the Column Diameter ...................................................................................... 73820.4.5.1 Packing Static Characteristics ................................. 74020.4.5.2 Packing Dynamic Characteristics........................... 74120.4.5.3 Determination of Flooding Rate............................. 74220.4.5.4 Determination of Packing Type.............................. 74420.4.6 Calculation of the Column Height ......................................... 74520.4.6.1 Concentrated Mixtures ............................................ 74620.4.6.2 Diluted Mixtures....................................................... 74920.4.6.3 Calculation of the Number of Transfer Units...... 75120.4.6.4 Calculation of the Height of the Transfer Unit.... 754 TX69299 fm frame Page 23 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 20.5 Plate Columns .......................................................................................... 755Problems................................................................................................ 758 21 SolidLiquid Extraction ............................................................. 773 21.1 Introduction .............................................................................................. 77321.2 SolidLiquid Equilibrium....................................................................... 77421.2.1 Retention of Solution and Solvent .......................................... 77621.2.2 Triangular and Rectangular Diagrams................................... 77721.2.2.1 Triangular Diagram.................................................. 77721.2.2.2 Rectangular Diagram............................................... 78121.3 Extraction Methods.................................................................................. 78221.3.1 Single Stage................................................................................. 78221.3.2 Multistage Concurrent System................................................ 78621.3.3 Continuous Countercurrent Multistage System................... 79221.4 SolidLiquid Extraction Equipment ..................................................... 79921.4.1 Batch Percolators........................................................................ 80021.4.2 Fixed-Bed Multistage Systems................................................. 80121.4.3 Continuous Percolators............................................................. 80121.4.4 Other Types of Extractors......................................................... 80421.5 Applications to the Food Industry........................................................ 806Problems................................................................................................ 810 22 Adsorption and Ionic Exchange ................................................ 823 22.1 Introduction .............................................................................................. 82322.1.1 Adsorption.................................................................................. 82322.1.2 Ionic Exchange ........................................................................... 82322.2 Equilibrium Process................................................................................. 82422.2.1 Adsorption Equilibrium ........................................................... 82422.2.2 Ionic Exchange Equilibrium..................................................... 82722.3 Process Kinetics........................................................................................ 82822.3.1 Adsorption Kinetics................................................................... 82822.3.2 Ionic Exchange Kinetics............................................................ 82922.4 Operation by Stages ................................................................................ 82922.4.1 Single Simple Contact ............................................................... 83022.4.2 Repeated Simple Contact ......................................................... 83122.4.3 Countercurrent Multiple Contact............................................ 83222.5 Movable-Bed Columns............................................................................ 83422.6 Fixed-Bed Columns ................................................................................. 83622.6.1 Fixed-Bed Columns with Phase Equilibrium ....................... 83722.6.2 Rosens Deductive Method...................................................... 83722.6.3 The Exchange Zone Method.................................................... 83822.6.3.1 Calculation of Height of Exchange Zone in an Adsorption Column ................................................. 84222.6.3.2 Calculation of Height of Exchange Zone in an Ionic Exchange Column........................................... 844Problems................................................................................................ 846 TX69299 fm frame Page 24 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC References............................................................................................................. 855Appendix.............................................................................................................. 865 TX69299 fm frame Page 25 Tuesday, September 10, 2002 1:42 PM 2003 by CRC Press LLC 1 1 Introduction to Unit Operations: Fundamental Concepts 1.1 Process Process is the set of activities or industrial operations that modify the prop-erties of raw materials with the purpose of obtaining products to satisfy theneeds of a society. Such modications of natural raw materials are directedto obtain products with greater acceptance in the market, or with betterpossibilities of storage and transport.The primary needs of every human being, individually or as a society,have not varied excessively throughout history; food, clothing, and housingwere needed for survival by prehistoric man as well as by modern man. Thesatisfaction of these necessities is carried out by employing, transforming,and consuming resources available in natural surroundings.In the early stages of mankinds social development, natural products wereused directly or with only small physical modications. This simple produc-tive scheme changed as society developed, so that, at the present time, rawmaterials are not used directly to satisfy necessities, but rather are subjectedto physical and chemical transformations that convert them into productswith different properties.This way, not only do raw materials satisfy the necessities of consumers,but also those products derived from the manipulation of such raw materials. 1.2 Food Process Engineering By analogy with other engineering branches, different denitions of foodprocess engineering can be given. Thus, according to one denition, foodprocess engineering includes the part of human activity in which the knowl-edge of physical, natural, and economic sciences is applied to agriculturalproducts as related to their composition, energetic content, or physical state. TX69299 ch01 frame.book Page 1 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 2 Unit Operations in Food Engineering Food process engineering can also be dened as the science of conceiving,calculating, designing, building, and running the facilities where the trans-formation processes of agricultural products, at the industrial level and aseconomically as possible, are carried out.Thus, an engineer in the food industry should know the basic principlesof process engineering and be able to develop new production techniquesfor agricultural products. He should also be capable of designing the equip-ment to be used in a given process. The main objective of food processengineering is to study the principles and laws governing the physical,chemical, or biochemical stages of different processes, and the apparatus orequipment by which such stages are industrially carried out. The studiesshould be focused on the transformation processes of agricultural raw mate-rials into nal products, or on conservation of materials and products. 1.3 Transformation and Commercialization of Agricultural Products For efcient commercialization, agricultural products should be easy to han-dle and to place in the market. As a general rule, products obtained directlyfrom the harvest cannot be commercialized as they are, but must undergocertain transformations. Products that can be directly used should be ade-quately packaged according to requirements of the market. These productsare generally used as food and should be conveniently prepared for use.One problem during handling of agricultural products is their transportfrom the elds to the consumer. Since many agricultural products have ashort shelf life, treatment and preservation methods that allow their lateruse should be developed. As mentioned earlier, many of these productscannot be directly used as food but can serve as raw material to obtain otherproducts. Developed countries tend to elaborate such products in the harvestzone, avoiding perishable products that deteriorate during transport fromthe production zone to the processing plant. 1.4 Flow Charts and Description of Some Food Processes Food processes are usually schematized by means of ow charts. These arediagrams of all processes that indicate different manufacturing steps, as wellas the ow of materials and energy in the process.There are different types of ow charts; the most common use blocks orrectangles. In these charts each stage of the process is represented by ablock or rectangle connected by arrows to indicate the way in which thematerials ow. The stage represented is written within the rectangle. TX69299 ch01 frame.book Page 2 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Introduction to Unit Operations: Fundamental Concepts 3Other types of ow charts are equipment and instrumentation.Figures 1.1, 1.2, and 1.3 show some ow charts of food processes. 1.5 Steady and Unsteady States A system is said to be under steady state when all the physical variablesremain constant and invariable along time, at any point of the system; how-ever, they may be different from one point to another. On the other hand,when the characteristic intensive variables of the operation vary through thesystem at a given moment and the variables corresponding to each systemspoint vary along time, the state is called unsteady. The physical variables toconsider may be mechanical or thermodynamic. Among the former are vol-ume, velocity, etc., while the thermodynamic variables are viscosity, concen-tration, temperature, pressure, etc. 1.6 Discontinuous, Continuous, and Semicontinuous Operations The operations carried out in the industrial processes may be performed inthree different ways. In a discontinuous operation the raw material is loaded FIGURE 1.1 Extraction of olive oil.Bagasse oilCENTRIFUGATIONOil frompressVirgin oilExhaustedbagasseDRYINGEXTRACTIONBagasseOlivesWASHINGPRESSING TX69299 ch01 frame.book Page 3 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 4 Unit Operations in Food Engineering in the equipment; after performing the required transformation, the obtainedproducts are unloaded. These operations, also called batch or intermit-tent, are carried out in steps:1. Loading of equipment with raw materials2. Preparation of conditions for transformation3. Required transformation4. Unloading products5. Cleaning equipmentThe batch operation takes place under an unsteady state, since its intensiveproperties vary along time. An example of this batch process is the crushingof oily seeds to obtain oil. FIGURE 1.2 Production of fruit concentrated juices.FruitJuice 12 BrixPulpWater andaromasJuice 15 BrixWaterJuice 70 BrixCRUSHINGPRESSINGPRE-CONCENTRATIONENZYMATICTREATMENTCLARIFICATIONEVAPORATIONCOOLINGSTORAGE TX69299 ch01 frame.book Page 4 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Introduction to Unit Operations: Fundamental Concepts 5In continuous operations the loading, transformation, and unloadingstages are performed simultaneously. Equipment cleaning is carried outevery given time, depending on the nature of the process and the materialsused. To carry out the cleaning, production must be stopped. Continuousoperations take place under steady state, in such a way that the characteristicintensive variables of the operation may vary at each point of the systembut do not vary along time. It is difcult to reach an absolute steady state,since there may be some unavoidable uctuations. An example of a contin-uous operation is the rectication of an alcoholwater mixture.In some cases it is difcult to have a continuous operation; this type ofoperation is called semicontinuous. A semicontinuous operation may occurby loading some materials in the equipment that will remain there for agiven time in a discontinuous way, while other materials enter or exit con-tinuously. Sometimes it is necessary to unload those accumulated materials.For example, in the extraction of oil by solvents, our is loaded and thesolvent is fed in a continuous way; after some time, the our runs out of oiland must be replaced. FIGURE 1.3 Elaboration of soluble coffee.RoastedcoffeeCoffee exhaust(diluted solution)Coffee extract(concentrated solution)Soluble coffeeHotwaterSolidwasteWatervaporWaterEXTRACTIONGRINDINGEVAPORATIONDRYING TX69299 ch01 frame.book Page 5 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 6 Unit Operations in Food Engineering These different ways of operation present advantages and disadvantages.Advantages of continuous operation include:1. Loading and unloading stages are eliminated.2. It allows automation of the operation, thus reducing the work force.3. Composition of products is more uniform.4. There is better use of thermal energy.Disadvantages of continuous operation are:1. Raw materials should have a uniform composition to avoid oper-ation uctuations.2. Is usually expensive to start the operation, so stops should beavoided.3. Fluctuations in product demand require availability of consider-able quantities of raw materials and products in stock.4. Due to automation of operation, equipment is more expensive anddelicate.Continuous operation is performed under an unsteady state during startsand stops but, once adequately running, may be considered to be workingunder steady state. This is not completely true, however, since there couldbe uctuations due to variations in the composition of the raw materials anddue to modications of external agents.When selecting a form of operation, the advantages and disadvantages ofeach type should be considered. However, when low productions arerequired, it is recommended to work under discontinuous conditions. Whenhigh productions are required, it is more protable to operate in a continuousway. 1.7 Unit Operations: Classication When analyzing the ow charts of different processes described in othersections, it can be observed that some of the stages are found in all of them.Each of these stages is called basic or unit operation, in common with manyindustrial processes. The individual operations have common techniquesand are based on the same scientic principles, simplifying the study of theseoperations and the treatment of these processes.There are different types of unit operations depending on the nature ofthe transformation performed; thus, physical, chemical, and biochemicalstages can be distinguished: TX69299 ch01 frame.book Page 6 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Introduction to Unit Operations: Fundamental Concepts 7 Physical stages: grinding, sieving, mixture, uidization, sedimen-tation, otation, ltration, rectication, absorption, extraction,adsorption, heat exchange, evaporation, drying, etc. Chemical stages: rening, chemical peeling Biochemical stages: fermentation, sterilization, pasteurization,enzymatic peelingHence, the group of physical, chemical, and biochemical stages that takeplace in the transformation processes of agricultural products constitute theso-called unit operations of the food industry, the purpose of which is theseparation of two or more substances present in a mixture, or the exchangeof a property due to a gradient. Separation is achieved by means of a sepa-rating agent that is different, depending on the transferred property.Unit operations can be classied into different groups depending on thetransferred property, since the possible changes that a body may undergoare dened by variations in either its mass, energy, or velocity. Thus, unitoperations are classied under mass transfer, heat transfer, or momentumtransfer.Besides the unit operations considered in each mentioned group, thereexist those of simultaneous heat and mass transfer, as well as other opera-tions that cannot be classied in any of these groups and are called comple-mentary unit operations.All the unit operations grouped in these sections are found in physicalprocesses; however, certain operations that include chemical reactions canbe included. 1.7.1 Momentum Transfer Unit Operations These operations study the processes in which two phases at different veloc-ities are in contact. The operations included in this section are generallydivided into three groups:Internal circulation of uids: study of the movement of uids throughthe interior of the tubing; also includes the study of equipmentused to impel the uids (pumps, compressors, blowers, and fans)and the mechanisms used to measure the properties of uids(diaphragms, venturi meters, rotameters, etc.).External circulation of uids: the uid circulates through the externalpart of a solid. This circulation includes the ow of uids throughporous xed beds, uidized beds (uidization), and pneumatictransport.Solids movement within uids: the base for separation of solids with-in a uid. This type of separation includes: sedimentation, ltration,and ultraltration, among others. TX69299 ch01 frame.book Page 7 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 8 Unit Operations in Food Engineering 1.7.2 Mass Transfer Unit Operations These operations are controlled by the diffusion of a component within amixture. Some of the operations included in this group are:Distillation: separation of one or more components by taking advan-tage of vapor pressure differences.Absorption: a component of a gas mixture is absorbed by a liquidaccording to the solubility of the gas in the liquid. Absorption mayoccur with or without chemical reaction. The opposite process iscalled desorption.Extraction: based on the dissolution of a mixture (liquid or solid) ina selective solvent, which can be liquidliquid or solidliquid. Thelatter is also called washing, lixiviation, etc.Adsorption: also called sorption, adsorption involves the eliminationof one or more components of a uid (liquid or gas) by retentionon the surface of a solid.Ionic exchange: substitution of one or more ions of a solution withanother exchange agent. 1.7.3 Heat Transfer Unit Operations These operations are controlled by temperature gradients. They depend onthe mechanism by which heat is transferred:Conduction: in continuous material media, heat ows in the directionof temperature decrease and there is no macroscopic movement ofmass.Convection: the enthalpy ow associated with a moving uid is calledconvective ow of heat. Convection can be natural or forced.Radiation: energy transmission by electromagnetic waves. No mate-rial media are needed for its transmission.Thermal treatments (sterilization and pasteurization), evaporation, heatexchangers, ovens, solar plates, etc. are studied based on these heat transfermechanisms. 1.7.4 Simultaneous MassHeat Transfer Unit Operations In these operations a concentration and a temperature gradient exist at thesame time:Humidication and dehumidication: include the objectives of hu-midication and dehumidication of a gas and cooling of a liquid.Crystallization: formation of solid glassy particles within a homoge-neous liquid phase. TX69299 ch01 frame.book Page 8 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Introduction to Unit Operations: Fundamental Concepts 9Dehydration: elimination of a liquid contained within a solid. Theapplication of heat changes the liquid, contained in a solid, into avapor phase. In freeze-drying, the liquid in solid phase is removedby sublimation, i.e., by changing it into a vapor phase. 1.7.5 Complementary Unit Operations One series of operations is not included in this classication because theseare not based on any of the transport phenomena cited previously. Theseoperations include grinding, milling, sieving, mixing of solids and pastes, etc. 1.8 Mathematical Setup of the Problems The problems set up in the study of unit operations are very diverse,although in all of them the conservation laws (mass, energy, momentum,and stochiometric) of chemical reactions apply. Applying these laws to agiven problem is done to perform a balance of the property studied insuch a problem. In a general way, the expression of the mass, energy, andmomentum balances related to the unit time can be expressed as:This is, that which enters into the system of the considered property isequal to that which leaves what is accumulated. In a schematic way:In cases where a chemical reaction exists, when carrying out a balance fora component, an additional generation term may appear. In these cases thebalance expression will be:When solving a given problem, a certain number of unknown quantitiesor variables ( V ) are present, and a set of relationships or equations ( R ) isobtained from the balances. According to values of V and R , the followingcases can arise: If V < R , the problem is established incorrectly, or one equation isrepeated. If V = R, the problem has only one solution. If V > R , different solutions can be obtained; the best solution isfound by optimizing the process.Property entering the system Property exiting the systemProperty that accumulates( )=( )+( ) E S A = +E G S A + = + TX69299 ch01 frame.book Page 9 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 10 Unit Operations in Food Engineering There aredesign variables. The different types of problems presented depend on thetype of equation obtained when performing the corresponding balances.Thus, Algebraic equations have an easy mathematical solution obtainedby analytical methods. Differential equations are usually obtained for unsteady continu-ous processes. The solution of the mathematical model establishedwith the balances can be carried out through analytical or approx-imate methods. In some cases, differential equations may have ananalytical solution; however, when it is not possible to analyticallysolve the mathematical model, it is necessary to appeal to approx-imate methods of numerical integration (digital calculus) orgraphic (analogic calculus). Equations in nite differences are solved by means of analogiccomputers which give the result in a graphic form. In some casesthe exact solution can be obtained by numerical methods. F V R = TX69299 ch01 frame.book Page 10 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 11 2 Unit Systems: Dimensional Analysis and Similarity 2.1 Magnitude and Unit Systems The value of any physical magnitude is expressed as the product of twofactors: the value of the unit and the number of units. The physical propertiesof a system are related by a series of physical and mechanical laws. Somemagnitudes may be considered fundamental and others derived. Fundamen-tal magnitudes vary from one system to another.Generally, time and length are taken as fundamental. The unit systemsneed a third fundamental magnitude, which may be mass or force. Thoseunit systems that have mass as the third fundamental magnitude are knownas absolute unit systems, while those that have force as a fundamental unitare called technical unit systems. There are also engineering unit systemsthat consider length, time, mass, and force as fundamental magnitudes. 2.1.1 Absolute Unit Systems There are three absolute unit systems: the c.g.s. (CGS), the Giorgi (MKS),and the English (FPS). In all of these, the fundamental magnitudes are length,mass, and time. The different units for these three systems are shown inTable 2.1. In these systems, force is a derived unit dened beginning withthe three fundamental units. The force and energy units are detailed inTable 2.2.When heat magnitudes are used, it is convenient to dene the temperatureunit. For the CGS and MKS systems, the unit of temperature is degreesCentigrade (C), while for the English system it is degrees Fahrenheit (F).Heat units are dened independently of work units. Later, it will be shownthat relating work and heat requires a factor called the mechanical equivalentof heat. TX69299 ch01 frame.book Page 11 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 12 Unit Operations in Food Engineering 2.1.2 Technical Unit Systems Among the most used technical systems are the metric and the Englishsystems. In both, the fundamental magnitudes are length, force, and time.In regard to temperature, the unit of the metric system is the Centigradedegree, and that of the English system is the Fahrenheit. Table 2.3 shows thefundamental units of the metric and English systems.In engineering systems, mass is a derived magnitude, which in the metricsystem is 1 TMU (technical mass unit) and in the English system is 1 slug. 2.1.3 Engineering Unit Systems Until now, only unit systems that consider three magnitudes as fundamentalhave been described. However, in engineering systems, four magnitudes areconsidered basic: length, time, mass, and force. Table 2.4 presents the differ-ent units for the metric and English engineering systems. TABLE 2.1 Absolute Unit Systems Magnitude Systemc.g.s. Giorgi English(CGS) (MKS) (FPS) Length (L) 1 centimeter (cm) 1 meter (m) 1 foot (ft)Mass (M) 1 gram (g) 1 kilogram (kg) 1 pound-mass (lb)Time (T) 1 second (s) 1 second (s) 1 second(s) TABLE 2.2 Units Derived from Absolute Systems Magnitude Systemc.g.s. Giorgi English(CGS) (MKS) (FPS) Force 1 dyne 1 Newton (N) 1 poundalEnergy 1 erg 1 Joule (J) 1 (pound)(foot) TABLE 2.3 Technical Unit Systems Magnitude SystemMetric English Length (L) 1 meter (m) 1 foot (ft)Force (F) 1 kilogram force (kp or kgf) 1 pound force (lbf)Time (T) 1 second (s) 1 second (s)Temperature ( ) 1 degree Centigrade (C) 1 degree Fahrenheit (F) TX69299 ch01 frame.book Page 12 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Unit Systems: Dimensional Analysis and Similarity 13When dening mass and force as fundamental, an incongruity may arise,since these magnitudes are related by the dynamics basic principle. To avoidthis incompatibility, a correction or proportionality factor ( g c ) should beinserted. The equation of this principle would be:Observe that g c has mass units (acceleration/force). The value of thiscorrection factor in the engineering systems is: 2.1.4 International Unit System (IS) It was convenient to unify the use of the unit systems when the AngloSaxoncountries incorporated the metric decimal system. With that purpose, theMKS was adopted as the international system and denoted as IS. Althoughthe obligatory nature of the system is recognized, other systems are still used;however, at present many engineering journals and books are edited onlyin IS, making it more and more acceptable than other unit systems. Table 2.5presents the fundamental units of this system along with some supplemen-tary and derived units.Sometimes the magnitude of a selected unit is too big or too small, makingit necessary to adopt prexes to indicate multiples and submultiples of thefundamental units. Generally, it is advisable to use these multiples and TABLE 2.4 Engineering Unit Systems Magnitude SystemMetric English Length (L) 1 meter (m) 1 foot (ft)Mass (M) 1 kilogram (kg) 1 pound-mass (lb)Force (F) 1 kilogram force (kp or kgf) 1 pound force (lbf)Time (T) 1 second (s) 1 second (s)Temperature ( ) 1 degree Centigrade (C) 1 degree Fahrenheit (F)gc Force = Mass Acceleration Metric system: 9.81kgmass meterkgforce second9.81kg mkg s2 2gC = ( )( )( )( ) = English system: 32.17lbmass footlbforce second32.17lb ftlbf s2 2gC = ( )( )( )( ) = TX69299 ch01 frame.book Page 13 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 14 Unit Operations in Food Engineering submultiples as powers of 10 3 . Following is a list of the multiples and sub-multiples most often used, as well as the name and symbol of each.It is interesting that, in many problems, concentration is expressed by usingmolar units. The molar unit most frequently used is the mole, dened as thequantity of substance whose mass in grams is numerically equal to its molec-ular weight. 2.1.5 Thermal Units Heat is a form of energy; in this way, the dimension of both is ML 2 T 2 .However, in some systems temperature is taken as dimension. In such cases,heat energy can be expressed as proportional to the product mass timestemperature. The proportionality constant is the specic heat, whichdepends on the material and varies from one to another. The amount of heatis dened as a function of the material, with water taken as a reference andthe specic heat being the unit, so: TABLE 2.5 International Unit System Magnitude Unit Abbreviation Dimension Length meter m LMass kilogram kg MTime second s TForce Newton N MLT 2 Energy Joule J ML 2 T 2 Power Watt W ML 2 T 3 Pressure Pascal Pa ML 1 T 2 Frequency Hertz Hz T 1 Prex Multiplication Factor IS Symbol tera 10 12 Tgiga 10 9 Gmega 10 6 Mkilo 10 3 khecto 10 2 hdeca 10 1 dadeci 10 1 dcenti 10 2 cmili 10 3 mmicro 10 6 nano 10 9 npico 10 12 pfemto 10 15 fatto 10 18 a Heat Mass Specific heat Temperature = TX69299 ch01 frame.book Page 14 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Unit Systems: Dimensional Analysis and Similarity 15The heat unit depends on the unit system adopted. Thus: Metric system: Calorie: heat needed to raise the temperature of a gram of waterfrom 14.5 to 15.5C English systems: Btu (British thermal unit): quantity of heat needed to raise thetemperature of a pound of water one Fahrenheit degree (from60 to 61F) Chu (Centigrade heat unit or pound calorie): quantity of heatneeded to raise the temperature of one pound of water onedegree Centigrade International system: Calorie: since heat is a form of energy, its unit is the Joule. Thecalorie can be dened as a function of the Joule: 1 calorie = 4.185JoulesSince heat and work are two forms of energy, it is necessary to dene afactor that relates them. For this reason, the denominated mechanical equiv-alent of heat ( Q ) is dened so that:so: 2.1.6 Unit Conversion The conversion of units from one system to another is easily carried out ifthe quantities are expressed as a function of the fundamental units mass,length, time, and temperature. The so-called conversion factors are used toconvert the different units. The conversion factor is the number of units ofa certain system contained in one unit of the corresponding magnitude ofanother system. The most common conversion factors for the different mag-nitudes are given in Table 2.6.When converting units, it is necessary to distinguish the cases in whichonly numerical values are converted from those in which a formula shouldbe converted. When it is necessary to convert numerical values from oneunit to another, the equivalencies between them, given by the conversionfactors, are used directly.Q = Heat energy Mechanical energyQ = = = Mechanical energyHeat energyMLT LML T22 2 1 TX69299 ch01 frame.book Page 15 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 16 Unit Operations in Food Engineering Table 2.6 Conversion Factors Mass 1 lb 0.4536 kg(1/32.2) slug Length 1 inch 2.54 cm1 foot 0.3048 m1 mile 1609 m Surface 1 square inch 645.2 mm 2 1 square foot 0.09290 m 2 Volume and Capacity 1 cubic foot 0.02832 m 3 1 gallon (imperial) 4.546 l1 gallon (USA) 3.786 l1 barrel 159.241 l Time 1 min 60 s1 h 3600 s1 day 86,400 s Temperature difference 1C = 1 K 1.8F Force 1 poundal (pdl) 0.138 N1 lb f 4.44 N4.44 10 5 dina32.2 pdl1 dyne 10 5 N Pressure 1 technical atmosphere (at) 1 kgf/cm 2 14.22 psi1 bar 100 kPa1 mm Hg (tor) 133 Pa13.59 kgf/cm 2 1 psi (lb/in 2 ) 703 kgf/m 2 Energy, Heat, and Power 1 kilocalorie (kcal) 4185 J426.7 kgfm TX69299 ch01 frame.book Page 16 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Unit Systems: Dimensional Analysis and Similarity 17In cases of conversion of units of a formula, the constants that appear inthe formula usually have dimensions. To apply the formula in units differentfrom those given, only the constant of the formula should be converted. Incases in which the constant is dimensionless, the formula can be directlyapplied using any unit system. 2.2 Dimensional Analysis The application of equations deducted from physical laws is one method ofsolving a determined problem. However, it may be difcult to obtain equations 1 erg 10 7 J1 Btu 1055 J1 Chu 0.454 kcal1.8 Btu1 horse vapor (CV) 0.736 kW75 kgm/s1 horse power (HP) 0.746 kW33,000 ft lb/min76.04 kgm/s1 kilowatt (kW) 1000 J/s1.359 CV1 kilowatt hour (kW.h) 3.6 10 6 J860 kcal1 atm.liter 0.0242 kcal10.333 kgm Viscosity 1 poise (P) 0.1 Pa s1 pound/(ft.h) 0.414 mPa s1 stoke (St) 10 4 m 2 /s Mass ow 1 lb/h 0.126 g/s1 ton/h 0.282 kg/s1 lb/(ft 2 .h) 1.356 g/s.m 2 Thermal Magnitudes 1 Btu/(h.ft 2 ) 3.155 W/m 2 1 Btu/(h.ft 2 F) 5.678 W/(m 2 K)1 Btu/lb 2.326 kJ/kg1 Btu/(lb.F) 4.187 kJ/(kg.K)1 Btu/(h.ft. F) 1.731 W/(m.K) Table 2.6 (continued) Conversion Factors TX69299 ch01 frame.book Page 17 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC 18 Unit Operations in Food Engineering of that type; therefore, in some cases it will be necessary to use equationsderived in an empirical form.In the rst case, the equations are homogeneous from a dimensional pointof view. That is, their terms have the same dimensions and the possibleconstants that may appear will be dimensionless. This type of equation canbe applied in any unit system when using coherent units for the samemagnitudes. On the other hand, equations experimentally obtained may notbe homogeneous regarding the dimensions, since it is normal to employdifferent units for the same magnitude.The objective of dimensional analysis is to relate the different variablesinvolved in the physical processes. For this reason, the variables are groupedin dimensionless groups or rates, allowing discovery of a relationship amongthe different variables. Table 2.7 presents the dimensionless modules usuallyfound in engineering problems. Dimensional analysis is an analyticalmethod in which, once the variables that intervene in a physical phenome-non are known, an equation to bind them can be established. That is, dimen-sional analysis provides a general relationship among the variables thatshould be completed with the assistance of experimentation to obtain thenal equation binding all the variables. 2.2.1 Buckinghams Theorem Every term that has no dimensions is dened as factor . According toBridgman, there are three fundamental principles of the dimensional analysis:1. All the physical magnitudes may be expressed as power functionsof a reduced number of fundamental magnitudes.2. The equations that relate physical magnitudes are dimensionallyhomogeneous; this means that the dimensions of all their termsmust be equal.3. If an equation is dimensionally homogeneous, it may be reducedto a relation among a complete series of dimensionless rates orgroups. These induce all the physical variables that inuence thephenomenon, the dimensional constants that may correspond tothe selected unit system, and the universal constants related to thephenomenon treated.This principle is denoted as Buckinghams theorem. A series of dimen-sionless groups is complete if all the groups among them are independent;any other dimensionless group that can be formed will be a combination oftwo or more groups from the complete series.Because of Buckinghams theorem, if the series q 1 , q 2 , , q n is the set of n independent variables that dene a problem or a physical phenomenon,then there will always exist an explicit function of the type: TX69299 ch01 frame.book Page 18 Wednesday, September 4, 2002 2:13 PM 2003 by CRC Press LLC Uni