Unit Operations in Engineering - ICDSTdl.icdst.org/pdfs/files1/0d697ccbbfd1ef3ec6dbf714d56fa24c.pdfEditors: Pedro Fito, Amparo Chiralt, Jose M. Barat, Walter E. L. Spiess, and Diana

© 2003 by CRC Press LLC

UnitOperations inFoodEngineering


FOOD PRESERVATION TECHNOLOGY SERIES

Series Editor

Gustavo V. Barbosa-Cánovas

Innovations in Food ProcessingEditors: Gustavo V. Barbosa-Cánovas and Grahame W. Gould

Trends in Food EngineeringEditors: Jorge E. Lozano, Cristina Añón, Efrén Parada-Arias,and Gustavo V. Barbosa-Cánovas

Pulsed Electric Fields in Food Processing:Fundamental Aspects and ApplicationsEditors: Gustavo V. Barbosa-Cánovas and Q. Howard Zhang

Osmotic Dehydration and Vacuum Impregnation:Applications in Food IndustriesEditors: Pedro Fito, Amparo Chiralt, Jose M. Barat, Walter E. L. Spiess,and Diana Behsnilian

Engineering and Food for the 21st CenturyEditors: Jorge Welti-Chanes, Gustavo V. Barbosa-Cánovas,and José Miguel Aguilera

Unit Operations in Food EngineeringAlbert Ibarz and Gustavo V. Barbosa-Cánovas


CRC PR ESS

Boca Raton London New York Washington, D.C.

UnitOperations inFoodEngineeringAlbert Ibarz, Ph.D.University of LleidaLleida, Spain

Gustavo V. Barbosa-Cánovas, Ph.D.Washington State UniversityPullman, Washington


This book contains information obtained from authentic and highly regarded sources. Reprinted material

is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable

efforts have been made to publish reliable data and information, but the author and the publisher cannot

assume 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, electronic

or mechanical, including photocopying, microfilming, and recording, or by any information storage or

retrieval 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, for

creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC

for 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 are

used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com


No claim to original U.S. Government works

International Standard Book Number 1-56676-929-9

Library of Congress Card Number 2002017480

Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Ibarz, Albert.

[Operaciones unitarias en la engenierâia de alimentos. English]

Unit operations in food engineering / by Albert Ibarz, Gustavo V.

Barbosa-Cánovas.

p. cm. -- (Food preservation technology series)

Includes bibliographical references and index.

ISBN 1-56676-929-9

1. Food industry and trade. I. Barbosa-Cánovas, Gustavo V. II.

Title. III. Series.

TP370 .I2313 2002

664—dc21 2002017480

CIP

http://www.crcpress.com


To our families


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 efficient 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 engineeringfield, 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 specificto 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 first 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-Cánovas


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 Vélez-Ruiz, Universidad de las Américas-Puebla, Méxicofor his very important contributions in the preparation of Chapter 7.

María Luisa Calderón (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.

José Juan Rodríguez and Federico Harte (WSU) for their decisiveparticipation in the final review of the Spanish version. Both workedwith great care, dedication, enthusiasm, and professionalism.

The “translation team:” Lucy López (Universidad de las Américas-Puebla, México), Jeannie Anderson (WSU), Fernanda San Martín(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 finished work.

Albert Ibarz, Jr. for his careful collaboration in preparing many ofthe figures in the book and Raquel Ibarz for her invaluable helpand encouragement for making this book a pleasant reality.


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-Cánovas 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 films, food powdertechnology, and food dehydration.


CONTENTS

1 Introduction to Unit Operations: Fundamental Concepts ......... 11.1 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: Classification....................................................................6

1.7.1 Momentum Transfer Unit Operations ...........................................71.7.2 Mass Transfer Unit Operations.......................................................81.7.3 Heat Transfer Unit Operations .......................................................81.7.4 Simultaneous Mass–Heat Transfer Unit Operations...................81.7.5 Complementary Unit Operations...................................................9

1.8 Mathematical Setup of the Problems .........................................................9

2 Unit Systems: Dimensional Analysis and Similarity............... 112.1 Magnitude and Unit Systems.................................................................... 11

2.1.1 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 ..............................................................................15

2.2 Dimensional Analysis .................................................................................172.2.1 Buckingham’s π Theorem ..............................................................182.2.2 Dimensional Analysis Methods ....................................................20

2.2.2.1 Buckingham’s Method .....................................................202.2.2.2 Rayleigh’s Method............................................................222.2.2.3 Method of Differential Equations ..................................22

2.3 Similarity Theory .........................................................................................232.3.1 Geometric Similarity.......................................................................242.3.2 Mechanical Similarity .....................................................................25

2.3.2.1 Static Similarity .................................................................252.3.2.2 Kinematic Similarity.........................................................252.3.2.3 Dynamic Similarity...........................................................25

Problems.................................................................................................. 30


3 Introduction to Transport Phenomena ....................................... 433.1 Historic Introduction...................................................................................433.2 Transport Phenomena: Definition.............................................................443.3 Circulation Regimes: Reynolds’ Experiment ..........................................453.4 Mechanisms of Transport Phenomena.....................................................48

3.4.1 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........... 534.1 Introduction ..................................................................................................534.2 Momentum Transport: Newton’s Law of Viscosity...............................534.3 Energy Transmission: Fourier’s Law of Heat Conduction...................554.4 Mass Transfer: Fick’s Law of Diffusion ...................................................574.5 General Equation of Velocity.....................................................................61

5 Air–Water Mixtures....................................................................... 655.1 Introduction ..................................................................................................655.2 Properties of Humid Air ............................................................................655.3 Mollier’s Psychrometric Diagram for Humid Air .................................70

5.3.1 Psychrometric Chart ŝT – X...........................................................705.3.2 Psychrometric Chart X – T ............................................................74

5.4 Wet Bulb Temperature ................................................................................755.5 Adiabatic Saturation of Air........................................................................77Problems.................................................................................................. 80

6 Rheology of Food Products ......................................................... 896.1 Introduction ..................................................................................................896.2 Stress and Deformation ..............................................................................906.3 Elastic Solids and Newtonian Fluids .......................................................936.4 Viscometric Functions .................................................................................956.5 Rheological Classification of Fluid Foods ...............................................966.6 Newtonian Flow ..........................................................................................976.7 Non-Newtonian Flow .................................................................................99

6.7.1 Time Independent Flow.................................................................996.7.2 Time Dependent Flow ..................................................................103

6.8 Viscoelasticity .............................................................................................1076.9 Effect of Temperature................................................................................ 1136.10 Effect of Concentration on Viscosity ...................................................... 114

6.10.1 Structural Theories of Viscosity............................................... 1146.10.2 Viscosity of Solutions ................................................................ 1156.10.3 Combined Effect: Temperature–Concentration..................... 117

6.11 Mechanical Models .................................................................................. 118


6.11.1 Hooke’s Model ........................................................................... 1186.11.2 Newton’s Model......................................................................... 1186.11.3 Kelvin’s Model ........................................................................... 1186.11.4 Maxwell’s Model........................................................................1206.11.5 Saint–Venant’s Model................................................................1216.11.6 Mechanical Model of the Bingham’s Body ...........................121

6.12 Rheological Measures in Semiliquid Foods ........................................1216.12.1 Fundamental Methods ..............................................................123

6.12.1.1 Rotational Viscometers.............................................1236.12.1.2 Concentric Cylinders Viscometers .........................1236.12.1.3 Plate–Plate and Cone–Plate 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....................................135

6.12.2 Empirical Methods ....................................................................1366.12.2.1 Adams Consistometer..............................................1366.12.2.2 Bostwick Consistometer ..........................................1376.12.2.3 Tube Flow Viscometer..............................................137

6.12.3 Imitative Methods......................................................................137Problems................................................................................................ 138

7 Transport of Fluids through Pipes............................................ 1437.1 Introduction ................................................................................................1437.2 Circulation of Incompressible Fluids .....................................................144

7.2.1 Criteria for Laminar Flow ........................................................1447.2.2 Velocity Profiles..........................................................................147

7.2.2.1 Laminar Regime........................................................1497.2.2.2 Turbulent Regime .....................................................1537.2.2.3 Flow in Noncylindrical Piping ...............................155

7.2.3 Universal Velocity Profile .........................................................1577.3 Macroscopic Balances in Fluid Circulation ...........................................160

7.3.1 Mass Balance ..............................................................................1607.3.2 Momentum Balance...................................................................1617.3.3 Total Energy Balance .................................................................1627.3.4 Mechanical Energy Balance......................................................165

7.4 Mechanical Energy Losses .......................................................................1667.4.1 Friction Factors...........................................................................1667.4.2 Calculation of Friction Factors ................................................167

7.4.2.1 Flow under Laminar Regime..................................1687.4.2.2 Flow under Turbulent Regime ...............................170

7.4.3 Minor Mechanical Energy Losses ...........................................1737.4.3.1 Equivalent Length ....................................................1757.4.3.2 Friction Losses Factors.............................................175


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 ..............................................................................182

7.5.3.1 Parallel Piping Systems .................................................1827.5.3.2 Piping in Series ...............................................................1837.5.3.3 Branched Piping..............................................................184

7.6 Pumps..........................................................................................................1867.6.1 Characteristics of a Pump............................................................186

7.6.1.1 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 .......................189

7.6.2 Installation Point of a Pump .......................................................1907.6.3 Pump Power ..................................................................................1917.6.4 Pump Efficiency.............................................................................1917.6.5 Types of Pumps .............................................................................191

Problems................................................................................................ 193

8 Circulation of Fluid through Porous Beds: Fluidization ....... 2058.1 Introduction ................................................................................................2058.2 Darcy’s Law: Permeability .......................................................................2058.3 Previous Definitions ..................................................................................206

8.3.1 Specific Surface ..............................................................................2068.3.2 Porosity ...........................................................................................207

8.4 Equations for Flow through Porous Beds .............................................2108.4.1 Laminar Flow: Equation of Kozeny–Carman...........................2108.4.2 Turbulent Flow: Equation of Burke–Plummer.........................2128.4.3 Laminar-Turbulent Global Flow: Equations of Ergun and

Chilton–Colburn............................................................................2138.5 Fluidization.................................................................................................216

8.5.1 Minimal Velocity of Fluidization................................................2188.5.1.1 Laminar Flow ..................................................................2198.5.1.2 Turbulent Flow................................................................2198.5.1.3 Transition Flow................................................................220

8.5.2 Minimal Porosity of Fluizidation ...............................................2208.5.3 Bed Height......................................................................................221

Problems................................................................................................ 222

9 Filtration ...................................................................................... 2359.1 Introduction ................................................................................................2359.2 Fundamentals of Filtration.......................................................................235

9.2.1 Resistance of the Filtering Cake .................................................2369.2.2 Filtering Medium Resistance.......................................................239


9.2.3 Total Filtration Resistance ........................................................2409.2.4 Compressible Cakes ..................................................................241

9.3 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 ....................................... 26510.1 Introduction ..............................................................................................265

10.1.1 Stages of Mass Transfer ............................................................26710.1.2 Polarization by Concentration.................................................269

10.2 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.........................274

10.3.1 Hydraulic Model........................................................................27410.3.2 Osmotic Model ...........................................................................279

10.4 Reverse Osmosis ......................................................................................28010.4.1 Mathematical Model..................................................................28010.4.2 Polarization Layer by Concentration .....................................28310.4.3 Influence of Different Factors ..................................................284

10.4.3.1 Influence of Pressure ................................................28410.4.3.2 Effect of Temperature...............................................28510.4.3.3 Effect of Type of Solute............................................287

10.5 Ultrafiltration ............................................................................................28710.5.1 Mathematical Model..................................................................28810.5.2 Concentration Polarization Layer ...........................................28910.5.3 Influence of Different Factors ..................................................291

10.5.3.1 Influence of Pressure ................................................29110.5.3.2 Effect of Temperature...............................................29210.5.3.3 Effect of Type of Solute............................................293

10.6 Design of Reverse Osmosis and Ultrafiltration Systems ..................29310.6.1 First Design Method..................................................................29410.6.2 Second Design Method.............................................................297

10.7 Operative Layout of the Modules.........................................................29810.7.1 Single Stage.................................................................................29810.7.2 Simple Stages in Series .............................................................29910.7.3 Two Stages with Recirculation ................................................300

Problems................................................................................................ 301


11 Thermal Properties of Food....................................................... 30911.1 Thermal Conductivity .............................................................................30911.2 Specific Heat ............................................................................................. 31111.3 Density.......................................................................................................31311.4 Thermal Diffusivity .................................................................................316Problems................................................................................................ 319

12 Heat Transfer by Conduction .................................................... 32112.1 Fundamental Equations in Heat Conduction .....................................321

12.1.1 Rectangular Coordinates ..........................................................32112.1.2 Cylindrical Coordinates............................................................32412.1.3 Spherical Coordinates ...............................................................325

12.2 Heat Conduction under Steady Regime ..............................................32512.2.1 Monodimensional Heat Conduction ......................................326

12.2.1.1 Flat Wall......................................................................32712.2.1.2 Cylindrical Layer ......................................................32912.2.1.3 Spherical Layer..........................................................332

12.2.2 Bidimensional Heat Conduction .............................................33412.2.2.1 Liebman’s method ....................................................33612.2.2.2 Relaxation method....................................................337

12.2.3 Tridimensional Heat Conduction............................................33712.3 Heat Conduction under Unsteady State..............................................339

12.3.1 Monodimensional Heat Conduction ......................................33912.3.1.1 Analytical Methods ..................................................34012.3.1.2 Numerical and Graphical Methods .......................347

12.3.2 Bi- and Tridimensinal Heat Conduction: Newman’s Rule ..............................................................................................351

Problems................................................................................................ 352

13 Heat Transfer by Convection..................................................... 36713.1 Introduction ..............................................................................................36713.2 Heat Transfer Coefficients ......................................................................367

13.2.1 Individual Coefficients..............................................................36713.2.1.1 Natural Convection ..................................................37013.2.1.2 Forced Convection....................................................37113.2.1.3 Convection in Non-Newtonian Fluids..................373

13.2.2 Global Coefficients.....................................................................37413.3 Concentric Tube Heat Exchangers ........................................................378

13.3.1 Design Characteristics...............................................................37813.3.1.1 Operation in Parallel ................................................37813.3.1.2 Countercurrent Operation.......................................382

13.3.2 Calculation of Individual Coefficients ...................................38313.3.3 Calculation of Head Losses......................................................384


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 Coefficients ...................................389

13.4.3.1 Coefficients for the Inside of the Tubes ................39013.4.3.2 Coefficients on the Side of the Shell......................392

13.4.4 Calculation of Head Losses......................................................39513.4.4.1 Head Losses inside Tubes .......................................39513.4.4.2 Head Losses on the Shell Side................................395

13.5 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 Coefficients .......................40313.5.5 Calculation of Head Losses......................................................40613.5.6 Design Procedure.......................................................................407

13.6 Extended Surface Heat Exchangers ......................................................40913.6.1 Mathematical Model.................................................................. 41113.6.2 Efficiency of a Fin ......................................................................41213.6.3 Calculation of Extended Surface Heat Exchangers..............414

13.7 Scraped Surface Heat Exchangers.........................................................41513.8 Agitated Vessels with Jacket and Coils ................................................417

13.8.1 Individual Coefficient inside the Vessel.................................41713.8.2 Individual Coefficient inside the Coil ....................................41813.8.3 Individual Coefficient in the Jacket ........................................418

13.9 Heat Exchange Efficiency .......................................................................418Problems................................................................................................ 425

14 Heat Transfer by Radiation ....................................................... 46714.1 Introduction ..............................................................................................46714.2 Fundamental Laws ..................................................................................468

14.2.1 Planck’s Law...............................................................................46814.2.2 Wien’s Law..................................................................................46814.2.3 Stefan–Boltzmann Law .............................................................469

14.3 Properties of Radiation ...........................................................................46914.3.1 Total Properties ..........................................................................46914.3.2 Monochromatic Properties: Kirchhoff’s Law ........................47114.3.3 Directional Properties................................................................472

14.4 View Factors..............................................................................................47414.4.1 Definition and Calculation .......................................................47414.4.2 Properties of View Factors .......................................................475

14.5 Exchange of Radiant Energy between Surfaces Separated by Nonabsorbing Media...............................................................................478


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 ...........481

14.6 Coefficient of Heat Transfer by Radiation ...........................................48214.7 Simultaneous Heat Transfer by Convection and Radiation .............484Problems................................................................................................ 485

15 Thermal Processing of Foods .................................................... 49115.1 Introduction ..............................................................................................49115.2 Thermal Death Rate.................................................................................491

15.2.1 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..............................................................502

15.3.1 Heat Penetration Curve ............................................................50215.3.2 Methods to Determine Lethality .............................................505

15.3.2.1 Graphical Method.....................................................50515.3.2.2 Mathematical Method ..............................................506

15.4 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 ................................................... 53516.1 Freezing .....................................................................................................53516.2 Freezing Temperature..............................................................................537

16.2.1 Unfrozen Water ..........................................................................53816.2.2 Equivalent Molecular Weight of Solutes ...............................540

16.3 Thermal Properties of Frozen Foods ....................................................54116.3.1 Density.........................................................................................54116.3.2 Specific Heat ...............................................................................54116.3.3 Thermal Conductivity ...............................................................542


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 ............................................................................556

16.9.1 Systems with Two Compressors and One Evaporator........55916.9.2 Systems with Two Compressors and Two Evaporators......561

Problems................................................................................................ 563

17 Dehydration ................................................................................. 57317.1 Introduction ..............................................................................................57317.2 Mixing of Two Air Streams ....................................................................57417.3 Mass and Heat Balances in Ideal Dryers............................................575

17.3.1 Continuous Dryer without Recirculation ..............................57517.3.2 Continuous Dryer with Recirculation ....................................576

17.4 Dehydration Mechanisms.......................................................................57717.4.1 Drying Process ...........................................................................57717.4.2 Constant Rate Drying Period...................................................58017.4.3 Falling Rate Drying Period ......................................................582

17.4.3.1 Diffusion Theory.......................................................58217.5 Chamber and Bed Dryers.......................................................................584

17.5.1 Components of a Dryer ............................................................58517.5.2 Mass and Heat Balances ...........................................................587

17.5.2.1 Discontinuous Dryers ..............................................58717.5.2.2 Discontinuous Dryers with Air Circulation

through the Bed ........................................................58917.5.2.3 Continuous Dryers ...................................................592

17.6 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 ...........................................................602

17.7 Freeze Drying ...........................................................................................60417.7.1 Freezing Stage ............................................................................60717.7.2 Primary and Secondary Drying Stages..................................60717.7.3 Simultaneous Heat and Mass Transfer ..................................607

17.8 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 ................................................................617

Problems................................................................................................ 618


18 Evaporation .................................................................................. 62518.1 Introduction ..............................................................................................62518.2 Heat Transfer in Evaporators.................................................................626

18.2.1 Enthalpies of Vapors and Liquids...........................................62718.2.2 Boiling Point Rise.......................................................................62918.2.3 Heat Transfer Coefficients ........................................................631

18.3 Single Effect Evaporators........................................................................63218.4 Use of Released Vapor ............................................................................634

18.4.1 Recompression of Released Vapor..........................................63418.4.1.1 Mechanical Compression.........................................63418.4.1.2 Thermocompression .................................................636

18.4.2 Thermal Pump............................................................................63718.4.3 Multiple Effect ............................................................................638

18.5 Multiple-Effect Evaporators ...................................................................64018.5.1 Circulation Systems of Streams...............................................640

18.5.1.1 Parallel Feed ..............................................................64018.5.1.2 Forward Feed ............................................................64218.5.1.3 Backward Feed ..........................................................64218.5.1.4 Mixed Feed ................................................................642

18.5.2 Mathematical Model..................................................................64318.5.3 Resolution of the Mathematical Model..................................64518.5.4 Calculation Procedure...............................................................646

18.5.4.1 Iterative Method when there is Boiling Point Rise ..................................................................647

18.5.4.2 Iterative Method when there is No Boiling Point Rise ...................................................................648

18.6 Evaporation Equipment..........................................................................64918.6.1 Natural Circulation Evaporators.............................................649

18.6.1.1 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 .....................651

18.6.2 Forced Circulation Evaporators...............................................65118.6.3 Long Tube Evaporators.............................................................65218.6.4 Plate Evaporators.......................................................................654

Problems................................................................................................ 654

19 Distillation................................................................................... 67119.1 Introduction ..............................................................................................67119.2 Liquid–Vapor Equilibrium .....................................................................671

19.2.1 Partial Pressures: Laws of Dalton, Raoult, and Henry .......67419.2.2 Relative Volatility.......................................................................67619.2.3 Enthalpy Composition Diagram .............................................677

19.3 Distillation of Binary Mixtures ..............................................................678


19.3.1 Simple Distillation .....................................................................67819.3.2 Flash Distillation ........................................................................680

19.4 Continuous Rectification of Binary Mixtures......................................68219.4.1 Calculation of the Number of Plates ......................................684

19.4.1.1 Mathematical Model ................................................68419.4.1.2 Solution of the Mathematical Model: Method

of McCabe–Thiele .....................................................68719.4.2 Reflux Ratio ................................................................................691

19.4.2.1 Minimum Reflux Relationship ...............................69119.4.2.2 Number of Plates for Total Reflux.........................694

19.4.3 Multiple Feed Lines and Lateral Extraction..........................69419.4.4 Plate Efficiency ...........................................................................69719.4.5 Diameter of the Column...........................................................69819.4.6 Exhaust Columns.......................................................................701

19.5 Discontinuous Rectification....................................................................70219.5.1 Operation with Constant Distillate Composition ................70219.5.2 Operation under Constant Reflux Ratio ................................705

19.6 Steam Distillation.....................................................................................706Problems................................................................................................ 708

20 Absorption ................................................................................... 72320.1 Introduction ..............................................................................................72320.2 Liquid–Gas Equilibrium .........................................................................72420.3 Absorption Mechanisms .........................................................................726

20.3.1 Double Film Theory ..................................................................72720.3.2 Basic Mass Transfer Equations ................................................727

20.3.2.1 Diffusion in the Gas Phase......................................72820.3.2.2 Diffusion in the Liquid Phase.................................729

20.3.3 Absorption Velocity ...................................................................72920.4 Packed Columns ......................................................................................732

20.4.1 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..............................744

20.4.6 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


20.5 Plate Columns ..........................................................................................755Problems................................................................................................ 758

21 Solid–Liquid Extraction ............................................................. 77321.1 Introduction ..............................................................................................77321.2 Solid–Liquid Equilibrium.......................................................................774

21.2.1 Retention of Solution and Solvent ..........................................77621.2.2 Triangular and Rectangular Diagrams...................................777

21.2.2.1 Triangular Diagram ..................................................77721.2.2.2 Rectangular Diagram ...............................................781

21.3 Extraction Methods..................................................................................78221.3.1 Single Stage.................................................................................78221.3.2 Multistage Concurrent System ................................................78621.3.3 Continuous Countercurrent Multistage System...................792

21.4 Solid–Liquid 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.........................................................804

21.5 Applications to the Food Industry........................................................806Problems................................................................................................ 810

22 Adsorption and Ionic Exchange ................................................ 82322.1 Introduction ..............................................................................................823

22.1.1 Adsorption ..................................................................................82322.1.2 Ionic Exchange ...........................................................................823

22.2 Equilibrium Process.................................................................................82422.2.1 Adsorption Equilibrium ...........................................................82422.2.2 Ionic Exchange Equilibrium.....................................................827

22.3 Process Kinetics ........................................................................................82822.3.1 Adsorption Kinetics...................................................................82822.3.2 Ionic Exchange Kinetics ............................................................829

22.4 Operation by Stages ................................................................................82922.4.1 Single Simple Contact ...............................................................83022.4.2 Repeated Simple Contact .........................................................83122.4.3 Countercurrent Multiple Contact............................................832

22.5 Movable-Bed Columns............................................................................83422.6 Fixed-Bed Columns .................................................................................836

22.6.1 Fixed-Bed Columns with Phase Equilibrium .......................83722.6.2 Rosen’s Deductive Method ......................................................83722.6.3 The Exchange Zone Method....................................................838

22.6.3.1 Calculation of Height of Exchange Zone in an Adsorption Column .................................................842

22.6.3.2 Calculation of Height of Exchange Zone in an Ionic Exchange Column...........................................844

Problems................................................................................................ 846


References.............................................................................................................855

Appendix..............................................................................................................865


1

1Introduction 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 modifications 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 mankind’s social development, natural products wereused directly or with only small physical modifications. 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 definitions of foodprocess engineering can be given. Thus, according to one definition, “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.”


2 Unit Operations in Food Engineering

Food process engineering can also be defined 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 final products, or on conservation of materials and products.

1.3 Transformation and Commercialization

of Agricultural Products

For efficient 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 fields 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 flow charts. These arediagrams of all processes that indicate different manufacturing steps, as wellas the flow of materials and energy in the process.

There are different types of flow charts; the most common use “blocks” or“rectangles.” In these charts each stage of the process is represented by ablock or rectangle connected by arrows to indicate the way in which thematerials flow. The stage represented is written within the rectangle.


Introduction to Unit Operations: Fundamental Concepts 3

Other types of flow charts are “equipment” and “instrumentation.”Figures 1.1, 1.2, and 1.3 show some flow 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 system’spoint 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 1.1Extraction of olive oil.

Bagasse oil

CENTRIFUGATION

Oil frompress

Virgin oil

Exhaustedbagasse

DRYING

EXTRACTION

Bagasse

Olives

WASHING

PRESSING



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 equipment

The 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 1.2Production of fruit concentrated juices.

Fruit

Juice 12 ºBrix

Pulp

Water and

aromas

Juice 15 ºBrix

Water

Juice 70 ºBrix

CRUSHING

PRESSING

PRE-CONCENTRATION

ENZYMATICTREATMENT

CLARIFICATION

EVAPORATION

COOLING

STORAGE



In 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 difficult to reach an absolute steady state,since there may be some unavoidable fluctuations. An example of a contin-uous operation is the rectification of an alcohol–water mixture.

In some cases it is difficult 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, flour is loaded and thesolvent is fed in a continuous way; after some time, the flour runs out of oiland must be replaced.

FIGURE 1.3Elaboration of soluble coffee.

Roasted

coffee

Coffee exhaust

(diluted solution)

Coffee extract

(concentrated solution)

Soluble coffee

Hot

water

Solid

waste

Water

vapor

Water

EXTRACTION

GRINDING

EVAPORATION

DRYING



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 fluctuations.

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 fluctuations due to variations in the composition of the raw materials anddue to modifications 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 profitable to operate in a continuousway.

1.7 Unit Operations: Classification

When analyzing the flow 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 scientific 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:



• Physical stages: grinding, sieving, mixture, fluidization, sedimen-tation, flotation, filtration, rectification, absorption, extraction,adsorption, heat exchange, evaporation, drying, etc.

• Chemical stages: refining, chemical peeling• Biochemical stages: fermentation, sterilization, pasteurization,

enzymatic peeling

Hence, 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 classified into different groups depending on thetransferred property, since the possible changes that a body may undergoare defined by variations in either its mass, energy, or velocity. Thus, unitoperations are classified 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 classified 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 fluids: study of the movement of fluids throughthe interior of the tubing; also includes the study of equipmentused to impel the fluids (pumps, compressors, blowers, and fans)and the mechanisms used to measure the properties of fluids(diaphragms, venturi meters, rotameters, etc.).

External circulation of fluids: the fluid circulates through the externalpart of a solid. This circulation includes the flow of fluids throughporous fixed beds, fluidized beds (fluidization), and pneumatictransport.

Solids movement within fluids: the base for separation of solids with-in a fluid. This type of separation includes: sedimentation, filtration,and ultrafiltration, among others.



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 liquid–liquid or solid–liquid. Thelatter is also called washing, lixiviation, etc.

Adsorption: also called sorption, adsorption involves the eliminationof one or more components of a fluid (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 flows in the directionof temperature decrease and there is no macroscopic movement ofmass.

Convection: the enthalpy flow associated with a moving fluid is calledconvective flow 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 Mass–Heat Transfer Unit Operations

In these operations a concentration and a temperature gradient exist at thesame time:

Humidification and dehumidification: include the objectives of hu-midification and dehumidification of a gas and cooling of a liquid.

Crystallization: formation of solid glassy particles within a homoge-neous liquid phase.



Dehydration: 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 classification 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 is

found by optimizing the process.

Property entering the system Property exiting the system

Property that accumulates

( ) = ( )

+ ( )

E S A= +

E G S A+ = +



There are

design 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 finite 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= −


11

2Unit 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 defined beginning withthe three fundamental units. The force and energy units are detailed inTable 2.2.

When heat magnitudes are used, it is convenient to define 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 defined independently of work units. Later, it will be shownthat relating work and heat requires a factor called the mechanical equivalentof heat.



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

System

c.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

System

c.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

System

Metric 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 (θ) 1 degree Centigrade (°C) 1 degree Fahrenheit (°F)


Unit Systems: Dimensional Analysis and Similarity 13

When defining 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 (gc) should beinserted. The equation of this principle would be:

Observe that gc 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 Anglo–Saxoncountries 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 prefixes to indicate multiples and submultiples of thefundamental units. Generally, it is advisable to use these multiples and

TABLE 2.4

Engineering Unit Systems

Magnitude

System

Metric 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 (θ) 1 degree Centigrade (°C) 1 degree Fahrenheit (°F)

gc × ×Force = Mass Acceleration

Metric system: 9.81kgmass meter

kgforce second9.81

kg mkg s2 2

gC =( )( )( )( )

=⋅⋅

English system: 32.17lbmass foot

lbforce second32.17

lb ftlbf s2 2

gC =( )( )

( )( )=

⋅⋅



submultiples as powers of 103. 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, defined 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 ML2T–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 specific heat, whichdepends on the material and varies from one to another. The amount of heatis defined as a function of the material, with water taken as a reference andthe specific 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 MLT2

Energy Joule J ML2T–2

Power Watt W ML2T–3

Pressure Pascal Pa ML–1T–2

Frequency Hertz Hz T–1

Prefix Multiplication Factor IS Symbol

tera 1012 Tgiga 109 Gmega 106 Mkilo 103 khecto 102 hdeca 101 dadeci 10–1 dcenti 10–2 cmili 10–3 mmicro 10–6 µnano 10–9 npico 10–12 pfemto 10–15 fatto 10–18 a

Heat Mass Specific heat Temperature= × ×



The heat unit depends on the unit system adopted. Thus:

• Metric system:• Calorie: heat needed to raise the temperature of a gram of water

from 14.5 to 15.5°C• English systems:

• Btu (British thermal unit): quantity of heat needed to raise thetemperature of a pound of water one Fahrenheit degree (from60 to 61°F)

• 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. The

calorie can be defined as a function of the Joule: 1 calorie = 4.185Joules

Since heat and work are two forms of energy, it is necessary to define afactor that relates them. For this reason, the denominated mechanical equiv-alent of heat (Q) is defined 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 energy

Q = = =−

− −Mechanical energyHeat energy

MLT LM

L T2

2 2 1

θθ



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 mm2

1 square foot 0.09290 m2

Volume and Capacity

1 cubic foot 0.02832 m3

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

1°C = 1 K 1.8°F

Force

1 poundal (pdl) 0.138 N1 lbf 4.44 N

4.44 × 105 dina32.2 pdl

1 dyne 10–5 N

Pressure

1 technical atmosphere (at) 1 kgf/cm2

14.22 psi1 bar 100 kPa1 mm Hg (tor) 133 Pa

13.59 kgf/cm2

1 psi (lb/in2) 703 kgf/m2

Energy, Heat, and Power

1 kilocalorie (kcal) 4185 J426.7 kgfm



In 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 difficult to obtain equations

1 erg 10–7 J1 Btu 1055 J1 Chu 0.454 kcal

1.8 Btu1 horse vapor (CV) 0.736 kW

75 kgm/s1 horse power (HP) 0.746 kW

33,000 ft lb/min76.04 kgm/s

1 kilowatt (kW) 1000 J/s1.359 CV

1 kilowatt hour (kW.h) 3.6 × 106 J860 kcal

1 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 m2/s

Mass flow

1 lb/h 0.126 g/s1 ton/h 0.282 kg/s1 lb/(ft2.h) 1.356 g/s.m2

Thermal Magnitudes

1 Btu/(h.ft2) 3.155 W/m2

1 Btu/(h.ft2 °F) 5.678 W/(m2 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



of that type; therefore, in some cases it will be necessary to use equationsderived in an empirical form.

In the first 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 thefinal equation binding all the variables.

2.2.1 Buckingham’s � Theorem

Every term that has no dimensions is defined 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 influence thephenomenon, the dimensional constants that may correspond tothe selected unit system, and the universal constants related to thephenomenon treated.

This principle is denoted as Buckingham’s π 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 Buckingham’s π theorem, if the series q1, q2, …, qn is the set ofn independent variables that define a problem or a physical phenomenon,then there will always exist an explicit function of the type:



(2.1)

This way, a number of dimensionless factors p can be defined with all thevariables; hence:

TABLE 2.7

Dimensionless Modules

Modules Expression

Documents

Unit Operations in Engineering - ICDSTdl.icdst.org/pdfs/files1/0d697ccbbfd1ef3ec6dbf714d56fa24c.pdfEditors: Pedro Fito, Amparo Chiralt, Jose M. Barat, Walter E. L. Spiess, and Diana